STOCHASTIC AND DETERMINISTIC FINITE-TIME SYSTEM IDENTIFICATION By Farzaneh Tatari A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering – Doctor of Philosophy 2023 ABSTRACT Identifying a high-fidelity model of nonlinear dynamic systems is a prerequisite for achieving desired specifications in any model-based control design technique. This is because, most control design methods rely on the availability of an accurate model of the system dynamics and coarse dynamics models without generalization guarantees typically induce controllers that are either overly conservative with poor performance or violate spatiotemporal constraints imposed on the system when applied to the true system. This dissertation investigates the finite-time identification of deterministic and stochastic sys- tems. First in Chapter 2, a novel finite-time distributed identification method is introduced for nonlinear interconnected systems. A distributed concurrent learning (CL) based discontinuous gradient descent (GD) update law is presented to learn uncertain interconnected subsystems’ dy- namics by minimizing the identification error for a batch of previously recorded data collected from each subsystem as well as its neighboring subsystems. The state information of neighboring interconnected subsystems is acquired through direct communication. Finite-time Lyapunov sta- bility analysis is performed and easy-to-check rank conditions on the distributed memories data of subsystems are obtained, under which finite-time stability of the distributed identifier is guaranteed. These rank conditions replace the restrictive persistence of excitation (PE) conditions which are hard and even impossible to achieve and verify. Next, Chapter 3 presents a fixed-time system identifier for continuous-time nonlinear systems. A novel adaptive update law with discontinuous gradient flows of the identification errors is presented that leverages CL to guarantee the learning of uncertain dynamics in a fixed time. The CL approach retrieves a batch of samples stored in a memory and the update law simultaneously minimizes the identification error for current stream of samples as well as past memory samples. Fixed-time Lyapunov stability analysis certifies fixed-time convergence to the stable equilibria of the GD flow of the system identification error under easy-to-verify rank conditions. In Chapter 4, an online data-regularized CL-based stochastic GD is presented for function approximation with noisy data. A fixed-size memory of past experiences is repeatedly used in the update law along with the current streaming data to provide probabilistic convergence guarantees with much-improved convergence rates (i.e, linear instead of sublinear) and less restrictive data- richness requirements. This approach allows us to leverage the Lyapunov theory to provide probabilistic guarantees that assure convergence of the parameters to a probabilistic ultimate bound exponentially fast, provided that a rank condition on the stored data is satisfied. This analysis shows how the quality of the memory data affects the ultimate bound and can reduce the effects of the noise variance on the error bounds. In Chapter 5, deterministic and stochastic fixed-time stability of autonomous nonlinear discrete- time (DT) systems are studied. Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT systems is certified. Extensions to systems under deterministic perturbations as well as stochastic noise are then considered. For the former, the sensitivity to perturbations for fixed-time stable DT systems is analyzed, and it is shown that fixed-time attractiveness is resulted from the presented Lyapunov conditions. For the latter, sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented. The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed- time attractive systems, and the stochastic settling-time function fixed upper bound is derived for stochastic DT systems. Finally, using the results of Chapter 5, in Chapter 6, a fixed-time identifier for modeling unknown DT nonlinear systems without requiring the PE condition is developed. A data-driven update law based on a modified GD update law is presented to learn the system parameters, which relies on CL. Fixed-time convergence guarantees are provided for the modified GD update law under a rank condition on the recorded data. To guarantee fixed-time convergence, fixed-time Lyapunov analysis is leveraged. To my deceased father who was the best inspiration in my whole life, to my kind mother for all her life support, and to my lovely husband, Majid, who is always the best friend of mine! iv ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. Hamidreza Modares, for his continuous support, time, and superb guidance throughout my Ph.D. studies. I would like to express my appreciation to the members of my Ph.D. committee, Prof. Guoming G Zhu, Prof. Ranjan Mukherjee, and Dr. Bahareh Kiumarsi, for their time and support. I would also like to thank all faculty and staff at Michigan State University. A special appreciation to my family for their love and support throughout my life, to my beloved mother and father for empowering me, and for their consistent support, to my dear siblings, Farin, Farzad, Farshad, and Farnoosh. A special thanks to my beloved husband, Majid for his support, encouragement, and standing by my side. v TABLE OF CONTENTS CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW . . . . . . . . . . . . . . 1 1.1 Organization of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 CHAPTER 2 FINITE-TIME DISTRIBUTED IDENTIFICATION FOR NONLINEAR INTERCONNECTED SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Preliminaries and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Finite-time Distributed Concurrent Learning . . . . . . . . . . . . . . . . . . . . . 23 2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 CHAPTER 3 FIXED-TIME SYSTEM IDENTIFICATION USING CONCURRENT LEARNING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Preliminaries and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Fixed-time Concurrent Learning Identifier . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 CHAPTER 4 ONLINE IDENTIFICATION OF NOISY FUNCTIONS VIA A DATA-REGULARIZED LEARNING APPROACH . . . . . . . . . . . . . 62 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Problem Formulation and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Data-regularized Concurrent Learning-based SGD for Function Identifier with noisy measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 CHAPTER 5 DETERMINISTIC AND STOCHASTIC FIXED-TIME STABILITY OF DISCRETE-TIME AUTONOMOUS SYSTEMS . . . . . . . . . . . . . . . 88 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Fixed-time Stability for Deterministic Discrete-time Systems . . . . . . . . . . . . 89 5.3 Sensitivity to Deterministic Perturbation for Fixed-time Stable Discrete-time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Fixed-time Stability in Probability for Stochastic Discrete-time Systems . . . . . . 99 5.5 Example Illustration and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.6 Conclusion and Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 CHAPTER 6 DISCRETE-TIME NONLINEAR SYSTEM IDENTIFICATION: A FIXED-TIME CONCURRENT LEARNING APPROACH . . . . . . . . . . . . . . . . . . 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 vi 6.4 Fixed-time Concurrent Learning of the Unknown Discrete-time Dynamics . . . . . 118 6.5 Fixed-time Convergent Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.6 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 CHAPTER 7 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . 130 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 vii CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW System identification approaches are typically categorized into batch (offline) or incremental (on- line) identification methods. Batch identification relies on the availability of a rich set of samples that are collected offline. In offline learning settings, rich data are assumed to be available and collected a priori for learning. This includes system identification using the subspace method and its variants [1–5] and the RL value function learning using the least-square temporal difference (LSTD) leaning [6–9]. In both cases, the least-squares (LS) method [10,11] is most commonly used to estimate unknown parameters by minimizing the estimation error. In offline learning settings, rich data are assumed to be available and collected a priori for learning. This includes system identification using the subspace method and its variants [1–5] and the RL value function learning using the least-square temporal difference (LSTD) leaning [6–9]. In both cases, the least-squares (LS) method [10, 11] is most commonly used to estimate unknown parameters by minimizing the estimation error. While classical offline learning methods provide asymptotic-sample guarantees (i.e, conver- gence to the actual parameters under infinite number of samples), finite-sample guarantees have been widely considered recently [1–3,12–26]. These methods provide error bounds for every finite number of samples, only after a sufficiently large number of samples that satisfy the PE condition. Existing finite-sample results for offline system identification are typically limited to linear sys- tems, as they are inspired by the subspace method, in which the linear dynamics structure is used to constructs a Hankel matrix from the input–output pairs. Finally, the offline setting does not comply with adaptive control settings for which the data samples are streaming and rich PE data are not available a priori. To adapt to a new situation in adaptive control settings, offline or least-square methods are not satisfactory, since one has to compute the new estimate from the scratch after a new sample becomes available. To circumvent this issue, recursive methods leverage the new data and modify online the immediately past estimate accordingly. The LS estimate for linearly parameterized approximators (e.g. linear in parameter system identification [27] and linearized 1 parametrization of value functions in RL using a single-layer network [28]) can be derived in a recursive manner. However, in general, the recursive implementation of LS estimate for nonlinear systems is a daunting challenge. However, samples must satisfy restrictive independent and identically distributed (i.i.d) con- ditions which are hard or even impossible to verify and obtain in closed-loop control systems. Moreover, offline learning cannot account for changes in system dynamics. On the other hand, online learning provides a framework to learn a system model on the fly and using the stream of data collected from the system dynamics in real-time. Nevertheless, restrictive persistence of excitation (PE) conditions [10, 29] must be satisfied to guarantee parameter convergence (which turns to assure generalization guarantee). Satisfying and verifying PE conditions in real-time pose limitations on certifying parameter convergence of online system identifiers. Moreover, parameter convergence guarantees are mainly achieved asymptotically or exponentially. Concurrent learning (CL) technique has been leveraged to relax the PE condition [30–43]. Chowdhary et al. [35, 36] presented a CL update law for adaptive control systems in which the identification error is minimized for not only current samples but also a batch of recorded samples. An easy-to-verify condition on the richness of data is then derived to guarantee the exponential parameter error convergence, which replaces the restrictive PE condition with a rank condition on the recorded samples. In CL methods, past recorded data are replayed along with the current stream of data in the update law to not only minimize the identification error for the current data but also for the batch of recorded data. CL has been recently extended to adaptive control [33, 44, 45], optimal and robust control [39, 46] networked control [32, 38, 47], continuous and discrete-time system identification [37, 40–42, 48]. In the most of previously mentioned studies, the asymptotic convergence of the estimated parameters is guaranteed under an easy-to-verify rank condition rather than PE condition. Recently, a few CL-based methods [40–43] provided the finite-time convergence for the estimated parameters. However, all the aforementioned identification approaches are dealing with identification of a single dynamic system. A class of nonlinear multi-agent systems is interconnected systems which are composed of sev- 2 eral (possibly heterogeneous) physically connected subsystems influencing each other’s behavior. Numerous engineering systems with practical relevance belong to this class of systems, including intelligent buildings, power systems, transportation infrastructure and urban traffic systems. Typi- cally, distributed control and monitoring methods for interconnected systems rely on high-fidelity models of the subsystems. Designing controllers based on coarse dynamic models and without generalization guarantees may induce closed-loop systems with poor performance or may even result in instability. Moreover, failure in accurate and timely identification of the dynamics of a sin- gle subsystem may snowball into an entire network instability due to the physical interconnections among subsystems. However, identifying the dynamics of interconnected systems is challenging due to the physical interconnections among the subsystems. This makes the existing system identification methods for single-agent systems not directly applicable to interconnected systems. Developing system iden- tifiers with finite-time guarantees for interconnected systems is of utmost importance in practice, since it allows the designer to preview and quantify the identification errors. The preview and qualification of the error bounds can in turn be leveraged by the control and/or monitoring systems to avoid conservationism. Otherwise, the conservationism introduced due to slow or asymptotic convergence can degrade the interconnected system performance. Different types of multi-agent systems’ learning approaches, classified as centralized, de- centralized, and distributed identification methods, typically employed in control of multi-agent systems [49–53], can be adopted to identify interconnected system dynamics. Centralized identifi- cation methods rely on the existence of a learning center that receives data from all subsystems and identifies the dynamics of the entire network. The centralized approach, however, comes at a high computation and communication cost and requires access to the global knowledge of the subsys- tems’ interconnection network. By contrast, in the decentralized learning, an independent identifier is allocated for every subsystem which only relies on the subsystem’s own information to identify its dynamics. Since there is no exchange of state information among the subsystems, decentralized identifiers are unable to identify the interconnection terms in the dynamics of subsystems. On the 3 other hand, distributed learning methods can accurately identify the interconnected system dynam- ics by employing a local identifier for every subsystem while allowing it to communicate its state information with its neighboring interconnected identifiers. In contrast to the centralized method, in the distributed identification approach, no access to the global knowledge of the interconnection network is required. The distributed identification of interconnected systems can be performed either online or offline. Generally, in offline (batch) distributed identification [54], a rich batch of data must be collected from each subsystem and its neighboring subsystems to provide high confidence generalization guarantees across the entire operating regimes of subsystems [12–16]. In the batch learning, where finite-time or non-asymptotic convergence refers to generalized guarantees provided by finite number of samples, satisfying the condition of independent and identically distributed (i.i.d) is difficult to obtain and hard to verify in closed-loop interconnected systems. On the other hand, online distributed identification, which is the problem of interest in this dissertation, uses online data from each subsystem and its neighboring subsystems to learn the dynamics of interconnected system in real time. Nevertheless, standard approaches for online identification require the restrictive persistence of excitation (PE) condition [10, 29] to ensure generalization and exact parameter convergence. This includes online identification of interconnected systems using both decentralized [55, 56] and distributed [57, 58] learning approaches. The PE condition, however, is hard to achieve and to verify online and its satisfaction is much more challenging for interconnected subsystems compared to single-agent systems. This is because the regressor’s PE condition for a subsystem not only depends on the richness of its own data but also the interactive data collected from its interaction with its neighboring subsystems. To satisfy the regressor’s PE condition in interconnected systems, all subsystems must syn- chronously inject probing noises into their control systems to excite their dynamics and conse- quently to produce rich data for the entire network of subsystems. Designing such a probing noise for every subsystem to collectively satisfy the regressors’ PE conditions for all subsystems while not jeopardizing the overall system stability is a daunting challenge due to the subsystems’ 4 interconnectivity: the probing noise can snowball in the entire network and lead to the system instability. Therefore, designing an identification method for interconnected systems without re- quiring restrictive PE conditions, for which their satisfaction can deteriorate the system’s stability and performance, is of vital importance. For interconnected systems, the identification error dynamics are only guaranteed to be locally uniformly ultimately bounded [55–58]. Based on the concept of finite-time stability [59], several finite-time control methods have been developed for output feedback control [60] and multi-agent system consensus [61, 62]. In finite-time control design methods, the controllers are designed to guarantee finite-time stability of the system dynamics or tracking error dynamics where either no learning is accomplished or some observers are used along with identifiers whose identification precision is not taken into account; therefore, there is no requirement on the data richness. Moreover, several distributed asymptotic-convergent estimators have been designed in [63, 64] to estimate the system states or a specific parameter for multi-agent systems with known dynamics for which the learning objectives and therefore the rich data recording do not exist. In contrast, in the multi-agent system identification, a precise model of the system is not available and the richness of the employed data affects the identification results. For the multi-agent system identification, specifically interconnected systems, finite-time approaches are essential to assure collecting rich data to identify the system dynamics in finite time. However, finite-time identification of interconnected systems is unsettled. Therefore, in the second chapter of this dissertation, we aim to identify the interconnected system dynamics in finite time by proposing a novel distributed discontinuous CL-based estimation law without requiring the standard regressors’ PE condition. In finite-time CL-based system identifiers [40,41], the convergence settling-time is a function of initial parameters’ estimation error and varies with initial parameter estimation variations. In finite- time convergence, the amplitude of the initial parameter estimation error is of great importance because if the initial error is not bounded, then it is hard to guarantee convergence of the parameters 5 to their true values in a limited time. Moreover, the settling time of their convergence depends on the initial parameters’ estimation error and thus cannot be computed a priori since the true values of the system parameters are unknown. In practice, having an accurate fixed-time identification method, for which the convergence time is independent of the initial errors is of utmost importance and allows to preview and quantify the identification errors, which can be leveraged by the control system to avoid conservatism. Based on the notion of fixed-time stability [65], various fixed-time control methods are ex- tensively developed in neural control [66], event/self-triggered consensus [67], team-triggered consensus [68], and prescribed performance control [69, 70]. It is worth noting that in fixed-time control design methods, the controller is designed to assure fixed-time tracking error or stability for known system dynamics: No learning is taken placed and conditions on the richness of data are therefore not required. In sharp contrast, system identification requires learning unknown dynam- ics while the controller is usually not designed for the sake of identifying dynamics. Therefore, designing data-efficient system identifier that requires limited access to samples collected from system dynamics is of vital importance. Fixed-time observer-based controllers [66] and observers [71–76] are investigated to estimate the system states [66, 71–73], disturbance [74, 75] and uncertainty [76] where the settling time usually depends on the observer gains satisfying a Hurwitz condition on the observer gains matrix. Moreover, a high-fidelity model of the system is assumed to be known in existing fixed-time observers and/or the controller is designed to achieve fixed-time convergence. Therefore, the problem of learning and rich data collection does not appear in these approaches. However, in sharp contrast, in system identification, neither a high-fidelity model of the system is available, nor the controller is usually designed for the sake of learning the dynamics. Therefore, novel approaches are required to assure collecting rich data for identifying system dynamics in fixed time, which is surprisingly unsettled. Although [77] and [78] presented fixed-time identification methods, they rely on the PE condi- tion which is hard to verify and certify in real-time. The authors in [79] and [80] introduced two 6 short fixed-time stable parameter estimation algorithms by relaxing the PE condition to an interval excitation condition; however, short fixed-time stability which ensures stability in a finite interval of time, is a weak form of fixed-time stability. A fixed-time convergent method for time-varying parameter identification is given in [81] that requires an analogous condition to the PE condition called injectivity which requires the minimum singular value of the regressor to be always strictly positive. Furthermore, in [81], the learning rate must satisfy some constraints and to check these constraints the minimum and maximum singular values of the regressor and the upper bound of the unknown parameters are needed to be known which are hard to compute and check online. Despite, the emergent need for an accurate fixed-time identification method in practical cases, to the best of our knowledge, no fixed-time identification algorithm without requiring the PE condition is presented in the literature. Therefore, in chapter 3 of this dissertation, we present a fixed-time identification algorithm independent of the initial estimation errors that eliminates the PE condition by integrating the capabilities of CL and fixed-time learning. The novel idea of employing the recorded data in a discontinuous gradient flow along with the current data in the update law has overcome the challenge of proposing a fixed-time CL method without requiring a PE condition. In control theory, function mapping in the presence of noisy data has been a long-standing challenge and amounts to a joint optimization over data-richness satisfaction and function error reduction. This is because, learning a set of parameters by minimizing a loss function does not necessarily minimizes the expected parameter estimation error, unless a set of rich data is used for learning. For instance, in system identification, for which the aim is to learn the unknown dynamics of a system from collected data, to ensure the system parameters’ convergence to their actual values, data samples must be persistently exciting (PE) [10]. Otherwise, a set of system parameters is learned without any convergence guarantee, even though the estimation error for the set of collected data is minimized. The parameters convergence cannot be guaranteed because the set of collected data used for leaning the system parameters is not a good representative of the entire state space. As another example, the PE condition over collected data must be satisfied in reinforcement learning (RL) to assure convergence of the RL agent to an optimal policy that 7 minimizes the cumulative cost of control actions [82]. In online settings, simultaneous satisfaction of the PE condition and learning of the function parameters requires solving a joint optimization over the data and the function error to assure rich data collection and learning an optimal set of parameters, respectively. A popular approach for online learning for streaming settings is the stochastic gradient descent (SGD) method [83–88]. However, when the data have temporal dependencies, as shown in [84], naive implementation of SGD does not show a satisfactory performance. The data-drop technique drops a large number of samples from the stream to obtain nearly independent samples [84]. However, there is no systematic approach to check whether the stream data are independent and can provide convergence guarantees and which samples to drop or collect to satisfy the data richness conditions, which can result in wasting a lot of samples. Recently, several studies have focused on accelerating SGD methods for linear regressions corrupted by noise. In [89], a projected SGD based algorithm with weighted iterate-averaging is presented. The convergence rate, however, is sublinear and the function under optimization is assumed to be strongly convex. A high-order tuner is presented in [90] for time-varying regressors that guarantees exponential convergence of parameter estimates to a bound depending on the noise statistics. Nevertheless, a regularization term is added to penalize the deviation of the parameters from their initial values, which can lead to a bias from the optimal value. To overcome the problem of highly correlated steaming data, a SGD with reverse experience replay is developed in [84] that divides data into small buffers and runs SGD backwards on the data stored in the individual buffers. This method guarantees a sublinear convergence rate for linear regressors. A non-asymptotic convergence analysis of a variant of SGD is presented in [91], in which the learning rate is selected according to the expected data streams to improve the convergence rate. In [92] a stochastic average gradient method is presented for optimizing strongly convex functions to achieve a linear convergence rate. The work of [93] leverages the importance sampling approach to improve the convergence rate of the SGD. Most of these existing results are presented for finite training sets for which the loss function is sum of a finite set of strongly convex functions. However, as shown 8 later, for online time-varying regression, under which the data samples are streaming, the strong convexity is satisfied under the PE condition on the streaming data. Besides, in existing mentioned results, the bounds on the smoothness of the function and Lipschitzness of its gradient are assumed fixed. In sharp contrast, we aim to pave the way to change these bounds and thus to improve the convergence rate and reduce the ultimate bound of the parameters’ estimation error while reducing the PE condition to a rank condition on the stored data. Therefore, in chapter 4 of this dissertation, an online data-regularized concurrent learning-based stochastic gradient descent (CL-based SGD) update law is presented for function approximation with noisy measurements. The Lyapunov stability theory has a longstanding history as a powerful tool in control theory to obtain many important results in the design of a variety of controllers and adaptation laws. The basic framework of the Lyapunov stability theory provides conditions under which their satisfaction guarantees the stability of the system in some sense. While finding a function satisfying these conditions, called Lyapunov function, is generally challenging, controllers and update laws can be developed to make a candidate Lyapunov function enforce the stability conditions. The Lyapunov theory generally provides conditions to assure the states of a system convergence to an equilibrium state. The qualitative guarantees that are provided for the convergence time determine the stability type, ranging from asymptotic stability, exponential stability, finite-time stability to fixed-time stability. While asymptotic stability and exponential stability provide as- surance that the system’s states eventually converge to an equilibrium, many real-world practical systems demand intense time response constraints, which makes these types of stabilities insuffi- cient. Therefore, a surge of interest has emerged in the control community in studying finite-time stability to design control systems and adaptation laws that exhibit finite-time convergence to an equilibrium point. Finite-time stability [59] has been studied for continuous-time (CT) and discrete-time (DT) deterministic and stochastic systems [94–97]. Moreover, finite-time stability concept has been extensively applied for the finite-time control of DT [98–100] and CT [101–103] systems, as well as finite-time identification [40–42, 104–110]. In the finite-time stability, however, the settling 9 (i.e., convergence) time, depends on the system’s initial condition, and, thus, cannot be specified a priori. Moreover, when the magnitude of the initial condition is large, it can lead to an unacceptable convergence time guarantee. Fixed-time stability, on the other hand, imposes a stronger requirement on the settling time, because it requires convergence guarantees with a pre-specified bound on the settling-time function, independent of the initial condition. Fixed-time stability of deterministic and stochastic CT systems, respectively, studied in [65] and [111], have been widely studied within the frameworks of fixed-time control design [66–70, 112–116], fixed-time observer design [71–76] and fixed-time identification [43, 77–81]. While most real-world systems are CT in nature, DT systems are of great importance since systems are typically discretized and controlled with digital computers and micro-controllers in real-world applications. DT Lyapunov analysis is different from its CT counterpart and the analysis applied for CT systems’ fixed-time stability can not be employed for DT systems. Moreover, development of Lyapunov conditions that guarantee fixed-time stability of DT deterministic and stochastic systems is challenging due to the requirement of having a fixed upper bound for the convergence time. This fixed-time bound represents a priori computable time of convergence independent of the initial conditions. Even though finite-time stability of DT deterministic [42, 97, 117] and stochastic [118, 119] systems are recently studied, fixed-time stability of DT deterministic and stochastic systems is surprisingly unsettled, despite its practical importance. This gap motivates us to present fixed-time Lyapunov stability conditions that pave the way for the realization of fixed-time control and identification of DT systems through designing appropriate controllers and adaptation laws, respectively. Lyapunov theory can also be leveraged to study the behavior of uncertain systems. There are typically two types of uncertainties in control systems: randomness which is caused by a noise in a stochastic system, and deterministic unknown perturbations with known bounds (here, we call the deterministic systems affected by deterministic perturbations as perturbed deterministic systems). The stability results are typically presented in terms of stability in probability for stochastic systems’ stability [94,95,111,119,120], which guarantees convergence in probability to an equilibrium point, 10 and in terms of attractiveness to a bounded set for perturbed systems. Thus, in Chapter 5 of this dissertation, we develop fixed-time stability conditions for both deterministic and stochastic DT autonomous nonlinear systems. To relax the PE condition, concurrent learning (CL) has been widely leveraged [36] [32,35,37– 39]. In this approach, the identification error is minimized for not only current samples but also a set of recorded samples. using recorded past data during learning allows us to replace a verifiable rank condition on the memory data with the strong PE condition. The convergence guarantees, however, are limited to the exponential or asymptotic convergence of parameters’ errors. Besides, most of these results are presented for the identification of continuous-time systems. Nevertheless, in practice, due to the employment of digital computers for controlling the systems, a discrete-time model is typically needed. Despite its importance, few results are available on the identification of discrete-time systems [42, 48, 108, 109, 121, 122]. To improve the convergence, the work of [108] presented a finite- time identifier for discrete-time systems. However, it requires online invertibility of a regressor matrix and its inverse computation, which makes it inapplicable for online learning of a large number of unknown parameters’ identification. The work of [121] presented an estimation method using dynamic regressor extension and mixing for both continuous-time and discrete-time systems; however, their results on finite-time convergence are limited to continuous-time systems. The work in [48] presented a concurrent learning-based function approximator for discrete-time systems without the PE condition requirement and ensured the asymptotic convergence of the estimated parameters. The authors in [122] presented a framework for processing gradient algorithms where finite-time algorithms are given using nabla fractional-order calculus. In [122], time-varying learning rates that reach zero along with converging to the optimal solution are employed to converge to the optimal solution regardless of the initial conditions. Although, no fixed time of convergence and fixed-time Lyapunov analysis are given in [122]. The works of [42] and [109] proposed finite- time CL identifiers for discrete-time systems’ dynamics identification where rigorous finite-time Lyapunov analysis guaranteed finite-time convergence. 11 While finite-time identifiers have significantly improved the convergence time of classical system identifiers that rely on standard gradient descent, the settling-time upper bound in these methods is a function of the initial parameters’ estimation error. Therefore, the settling time of convergence becomes unbounded as the initial condition’s norm approaches infinity. Moreover, in finite-time convergence, a bound for the settling time cannot be computed because it would depend on the unknown true values of the system parameters. Therefore, it is of vital importance to develop a fixed-time identification method in which the settling-time function upper bound is independent of the initial errors. This will allow us to quantify the identification errors over time, which leads to less conservative control design methods that rely on the fixed-time identified system model. This motivates us to propose an online identifier for discrete-time systems with guaranteed fixed-time convergence properties using CL with a rank condition on the memory data which eliminates the requirement of restrictive PE condition. Although fixed-time controls, identifiers, and observers have been extensively employed for continuous-time systems [43,66,68–70,72,73,123–127], fixed-time methods for discrete-time sys- tems are generally unsettled due to the lack of fixed-time stability analysis of discrete-time systems. The extension of fixed-time stability analysis from continuous-time systems to discrete-time sys- tems is far from trivial. Recently, we presented fixed-time stability for stochastic and deterministic discrete-time systems in [128], which opens the door to developing fixed-time learning algorithms. Therefore, it is desirable to present a fixed-time learning method that can eliminate the restrictive PE condition for discrete-time systems’ identification due to the need for an accurate fixed-time identification method in real-world applications. Therefore, in Chapter 6 of this dissertation, a fixed-time concurrent learning (FxTCL) algorithm for discrete-time systems is presented to 1) ensure fixed-time parameter convergence independent of the initial estimation errors and 2) relax the PE condition to a rank condition on the recorded data using CL. 1.1 Organization of the dissertation Based on the above-elaborated problems, the brief contribution and organization of this disser- tation are as follows. 12 Chapter 2 presents a novel distributed discontinuous CL-based estimation law without requiring the standard regressors’ PE condition, to identify the interconnected system dynamics in finite time. To this end, a distributed finite-time identifier is allocated to every subsystem that leverages local communication to not only learn the subsystem’s own dynamics but also the interconnected dynamics based on its own state and input data, and its neighbors’ state information. Moreover, in order to relax the regressors’ PE condition and guarantee finite-time convergence, a discontinuous distributed CL-based gradient descent update law is presented. Using the presented update law, every local identifier minimizes the identification error at the current time based on the current stream of data from its own state and that of its neighbors as well as the identification error for data collected in a rich distributed memory. The dynamics of the gradient flows are analyzed using finite-time stability and it is shown that for every subsystem an easy-to-verify rank condition on the matrix containing the recorded filtered regressor data (that is used to avoid state derivative measurements) is sufficient to ensure finite-time convergence. Two different cases are considered in this chapter: 1) Realizable system identification for which there is a set of model parameters that can make the identification error zero. That is, the minimum functional approximation error (MFAE) is zero and is realized by an optimal set of unknown system parameters; and 2) non-realizable system identification for which there are no model parameters that result in zero identification error. For case 2, the subsystems have mismatch identification errors and their MFAEs are nonzero. In both cases, linearly parameterized universal approximators such as radial basis function neural networks are used to model the uncertain system functions. It is shown that under a verifiable rank condition, the proposed approach results in finite-time zero identification error for case 1 (which is a special form of case 2) and finite-time attractiveness to a bound near zero for case 2. Chapter 3 presents a a fixed-time identification algorithm independent of the initial estimation errors that eliminates the PE condition by integrating the capabilities of CL and fixed-time learning. The novel idea of employing the recorded data in a discontinuous gradient flow along with the current data in the update law has overcome the challenge of proposing a fixed-time CL method without requiring a PE condition. Here, by leveraging the CL technique, unlike [81], no persistence 13 of excitation or injectivity condition on the regressor and no upper bound knowledge of the unknown parameters is required for fixed time convergence. In the proposed fixed-time concurrent learning (FxTCL), the settling time is independent of the initial parameter estimation error. Therefore, given the recorded data, the settling time of convergence can be computed a priori regardless of the initial parameter estimation error. Consequently, the presented FxTCL update law guarantees learning a high-fidelity model of the system with a priori computable and fixed time of convergence. A fixed settling time of convergence for the identifier provides a priori computable convergence bound. This, in turn, allows quantifying system uncertainty for the control design and provides mechanisms to avoid designing overly conservative controllers caused by long-lasting large model estimation errors. Chapter 4 presents an online data-regularized concurrent learning-based stochastic gradient descent (CL-based SGD) update law is presented for function approximation with noisy measure- ments. Inspired by the concurrent learning for deterministic settings, a novel parameter estimation update law is presented that replaces the typical gradient estimation methods with a memory- augmented gradient update law. That is, the gradient update law not only minimizes the current estimated estimation error but also the estimation error for past historic data stored in a fixed-size memory. This is in sharp contrast with mini-batch SGD in which a mini-batch of data are randomly selected to estimate the noisy gradient. Using the Lyapunov theory, probabilistic guarantees are provided for the parameters estimation errors, provided that a rank condition on the stored data is satisfied. It is also shown that the parameters estimation errors converge exponentially to a probabilistic ultimate bound. The ultimate bound depends on the noise variance of the function approximation as well as approximation error and richness of the recorded memory data. Chapter 5 presents fixed-time stability conditions for both deterministic and stochastic DT autonomous nonlinear systems. First, fixed-time stability for equilibria of deterministic DT au- tonomous systems is defined. That is, a settling-time function is defined with a fixed upper bound independent of the initial condition. We then present Lyapunov theorems for fixed-time stability of both unperturbed and perturbed deterministic DT systems. Moreover, the sensitivity of fixed-time 14 stability properties to perturbations of systems is investigated under the assumption of the existence of a locally Lipschitz discrete Lyapunov function. It is ensured that fixed-time stability is preserved under perturbations in the form of fixed-time attractiveness. Furthermore, sufficient Lyapunov conditions for fixed-time stability in probability of stochastic DT systems and their stochastic settling-time function are presented. Chapter 6 of this dissertation presents a fixed-time concurrent learning (FxTCL) algorithm for discrete-time systems to 1) ensure fixed-time parameter convergence independent of the initial estimation errors and 2) relax the PE condition to a rank condition on the recorded data using CL. In the presented FxTCL, the settling-time upper bound is independent of the initial parameter estimation error. To achieve this goal, a modified gradient-descent update law is presented for learning the unknown system parameters. This update law reuses past collected data at every time instance and leverages discontinuous and non-integer powers of the identification errors. The Lyapunov analysis presented in Chapter 5 is then leveraged to guarantee fixed-time convergence of the system parameters to their true values. Chapter 7 summarizes and concludes this dissertation and provides future research directions. The contributions of this dissertation are published or submitted in the following journal papers. Farzaneh Tatari, Hamidreza Modares, Christos Panayiotou, Marios Polycarpou, “Finite-time Distributed Identification for Nonlinear Interconnected Systems”, IEEE/CAA Journal of Automat- ica Sinica, vol. 9, no. 7, pp. 1–12, Jul. 2022. Farzaneh Tatari, Majid Mazouchi, Hamidreza Modares, “Fixed-time System Identification Using Concurrent Learning”, IEEE Transactions on Neural Networks and Learning Systems, 2021, doi: 10.1109/TNNLS.2021.3125145. Farzaneh Tatari, and Hamidreza Modares, " Online Function Identification with Noisy Data via Data-regularized Stochastic Concurrent Learning," Under review in IEEE Transactions on Neural Networks and Learning Systems. Farzaneh Tatari, and Hamidreza Modares, "Deterministic and Stochastic Fixed-Time Stability 15 of Discrete-time Autonomous Systems," in IEEE/CAA Journal of Automatica Sinica, vol. 10, no. 4, pp. 945-956, April 2023, doi: 10.1109/JAS.2023.123405. Farzaneh Tatari, and Hamidreza Modares, "Discrete-time Nonlinear System Identification: A Fixed-time Concurrent Learning Approach," Under review in IEEE Transactions on Systems, Man and Cybernetics: Systems. 16 CHAPTER 2 FINITE-TIME DISTRIBUTED IDENTIFICATION FOR NONLINEAR INTERCONNECTED SYSTEMS 2.1 Introduction In this chapter, first, a novel finite-time distributed CL identification method is presented for nonlinear interconnected systems. The proposed discontinuous distributed CL estimation law en- sures the finite-time convergence of the approximated parameters without requiring the regressors’ PE condition. In the proposed distributed CL, every distributed identifier leverages a local state communication with its neighboring subsystems to collect and employ a rich distributed memory to relax the regressor’s PE condition and identify its own interconnected subsystem dynamics in finite time. Then, based on finite-time Lyapunov analysis, when there is zero MFAE, the finite- time convergence of interconnected system parameters is ensured through rigorous proofs. For the case with non-zero MFAE, finite-time attractiveness of the interconnected system parameters’ estimation error is guaranteed. Finally, the upper bounds of the settling-time functions for the finite convergence time are provided as a function of distributed memory data richness. Notation The network of subsystems in an interconnected system is shown by a bidirectional graph 𝐺(V, Σ), where V = {1, 2, . . . , 𝑁 } is the set of vertices representing 𝑁 subsystems and Σ ⊂ V × V is the set of graph edges. (𝑖, 𝑗) ∈ Σ indicates that there exists an edge from node 𝑖 to node 𝑗 which indicates the interconnection between subsystems 𝑖 and 𝑗. The set of  neighbors of node 𝑖 is shown by 𝑁𝑖 = 𝑗 : ( 𝑗, 𝑖) ∈ Σ and |𝑁𝑖 | is the cardinality measure of the set 𝑁𝑖 , 𝑖 = 1, ..., 𝑁. Throughout this chapter, 𝐼 is the identity matrix of appropriate dimension. 𝑠𝑡𝑎𝑐𝑘(𝑥, 𝑦) is an operator which stacks the columns of 𝑥 and 𝑦 vectors on top of one another. k𝑥k denotes the vector norm for 𝑥 ∈ R𝑛 , k 𝐴k shows the induced 2-norm of the matrix 𝐴. 𝜆 𝑚𝑖𝑛 (𝐴) and 𝜆 𝑚𝑎𝑥 (𝐴) denote the minimum and maximum eigenvalues of the matrix 𝐴, respectively. 17 2.2 Preliminaries and Problem Formulation Preliminaries Consider the following nonlinear system with the equilibrium point in the origin, 𝑦¤ (𝑡) = 𝐹(𝑡, 𝑦), 𝑦(0) = 𝑦 0 , (2.1) where 𝑦 ∈ D 𝑦 , 𝐹 : R+ × D 𝑦 ↦→ D 𝑦 and D 𝑦 ⊂ R𝑛 is an open neighborhood of the origin. Definition 1 (Persistence of excitation [29]) A signal 𝑦(𝑡) is persistently exciting if there are positive scalars 𝜂1 , 𝜂2 and T ∈ R+ , such that the following condition on 𝑦(𝑡) (PE condition) is satisfied for ∀𝑡 ∈ R+ , Z 𝑡+T 𝜂1 𝐼 ≤ 𝑦(𝜏)𝑦𝑇 (𝜏)𝑑𝜏 ≤ 𝜂2 𝐼. 𝑡 Definition 2 (Finite-time stability [59]) The system (2.1) is said to be 1) finite-time stable, if it is asymptotically stable and any solution 𝑦(𝑡, 𝑦 0 ) of (2.1) reaches the equilibrium point in finite time, i.e., 𝑦(𝑡, 𝑦 0 ) = 0, ∀𝑡 ≥ 𝑇(𝑦 0 ), where 𝑇 : D 𝑦 ↦→ R+ ∪ {0} is the settling-time function. 2) finite-time attractive to an ultimate bounded set 𝑌 around origin, if any solution 𝑦(𝑡, 𝑦 0 ) of (2.1) reaches 𝑌 in finite-time and stays there ∀𝑡 ≥ 𝑇(𝑦 0 ) where 𝑇 : D 𝑦 ↦→ R+ ∪ {0} is the settling-time function. Lemma 1 [59] Suppose that there exists a positive definite continuous function 𝑉 : D 𝑦 ↦→ R+ ∪ {0} in an open neighborhood of the origin and there exist real numbers 𝛼 > 0 and 0 < 𝑟 1 < 1 such that 𝑉(𝑦) is positive definite and ¤ 𝑉(𝑦) ≤ −𝛼𝑉 𝑟 1 (𝑦). Then, the system (2.1) is finite-time stable with a finite settling-time 1 𝑇(𝑦 0 ) ≤ 𝑉 1−𝑟 1 (𝑦 0 ), 𝛼(1 − 𝑟 1 ) for all 𝑦 0 ∈ D 𝑦 . Fact 1: In general, for a vector 𝑥 = [𝑥 1 , 𝑥 2 , ..., 𝑥 𝑛 ]𝑇 ∈ R𝑛 , the 𝑝-norm is defined as k𝑥k 𝑝 = P𝑛 1 ( 𝑖=1 |𝑥𝑖 | 𝑝 ) 𝑝 . Moreover, for positive constants 𝑟 and 𝑠, if 0 < 𝑟 < 𝑠, then based on Hölder 18 inequality [129], one obtains 1 1 k𝑥k 𝑠 ≤ k𝑥k 𝑟 ≤ 𝑛 𝑟 − 𝑠 k𝑥k 𝑠 . Problem Formulation Consider the following nonlinear interconnected system composed of 𝑁 uncertain subsystems described by 𝑥¤𝑖 (𝑡) = 𝑓𝑖 (𝑥𝑖 (𝑡)) + 𝑔𝑖 (𝑥𝑖 (𝑡))𝑢𝑖 (𝑡) + ∆𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ), 𝑖 = 1, ..., 𝑁, (2.2) where 𝑥𝑖 = [𝑥𝑖1 , 𝑥𝑖2 , ..., 𝑥𝑖𝑛 ] ∈ D𝑖 ⊂ R𝑛 is the state and 𝑢𝑖 ∈ D𝑢 ⊂ R𝑚 is the control input of subsystem 𝑖, 𝑖 = 1, ..., 𝑁; D𝑖 and D𝑢 are compact sets. 𝑓𝑖 : D𝑖 ↦→ R𝑛 , 𝑔𝑖 : D𝑖 ↦→ R𝑛×𝑚 and ∆𝑖 : D 𝑁𝑖 ↦→ R𝑛 are the unknown nonlinear drift, input and interconnection terms with D ⊂ R𝑛 ( |𝑁𝑖 | +1) , respectively. 𝑁𝑖 This chapter aims to present an identification method to learn the unknown dynamics of the nonlinear interconnected system (2.2) in finite time and in a distributed fashion. Assumption 1 𝑓𝑖 (𝑥𝑖 (𝑡)) and 𝑔𝑖 (𝑥𝑖 (𝑡)) are both locally Lipschitz in 𝑥𝑖 (𝑡) and ∆𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ) is locally Lipschitz in 𝑥 𝑁𝑖 (𝑡) where 𝑥 𝑁𝑖 (𝑡) = 𝑠𝑡𝑎𝑐𝑘 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)) 𝑗 ∈𝑁 . 𝑖 In order to learn the subsystems’ uncertain dynamics in a distributed fashion, first, every subsystem dynamics (2.2) is formulated into a distributed filtered regressor form. The distributed filtered-regressor form presents the subsystems’ states with a time-varying regressor for which the dynamic flow of regressors are known and depend on the subsystem’s states and inputs, as well as its neighbors’ states. This will allow to present update laws without requiring to measure the state derivatives of the subsystems and their neighbors [10]. To develop filtered regressors, linearly parameterized adaptive approximation models are first used to respectively represent 𝑓𝑖 (𝑥𝑖 ), 𝑔𝑖 (𝑥𝑖 ) and ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ) for every subsystem 𝑖, 𝑖 = 1, ..., 𝑁 as follows, 𝑓𝑖 (𝑥𝑖 (𝑡)) = ˆ𝑓𝑖 (𝑥𝑖 (𝑡), Θ𝑖∗ ) + 𝑒 𝑓𝑖 (𝑥𝑖 (𝑡)), (2.3) 𝑔𝑖 (𝑥𝑖 (𝑡)) = 𝑔ˆ𝑖 (𝑥𝑖 (𝑡), Φ𝑖∗ ) + 𝑒 𝑔𝑖 (𝑥𝑖 (𝑡)), (2.4) 19 ∆𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ) = ∆ ˆ 𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁 , Ψ∗ ) + 𝑒 ∆ (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁 ), (2.5) 𝑖 𝑖 𝑖 𝑖 where ˆ𝑓𝑖 (𝑥𝑖 (𝑡), Θ∗ ) = Θ∗𝑇 𝜑𝑖 (𝑥𝑖 (𝑡)), (2.6) 𝑖 𝑖 𝑔ˆ𝑖 (𝑥𝑖 (𝑡), Φ𝑖∗ ) = Φ𝑖∗𝑇 𝜒𝑖 (𝑥𝑖 (𝑡)), (2.7) ∆ˆ 𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁 , Ψ∗ ) = Ψ∗𝑇 𝜐𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁 ), (2.8) 𝑖 𝑖 𝑖 𝑖 𝑒 𝑓𝑖 (𝑥𝑖 ) = 𝑠𝑢 𝑝 𝑓𝑖 (𝑥𝑖 ) − 𝑓ˆ𝑖 (𝑥𝑖 , Θ𝑖 ) , (2.9) 𝑥𝑖 ∈D𝑖 𝑒 𝑔𝑖 (𝑥𝑖 ) = 𝑠𝑢 𝑝 𝑔𝑖 (𝑥𝑖 ) − 𝑔ˆ𝑖 (𝑥𝑖 , Φ𝑖 ) , (2.10) 𝑥𝑖 ∈D𝑖 𝑒 ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) = 𝑠𝑢 𝑝 ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) − ∆ˆ 𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁 , Ψ𝑖 ) . (2.11) 𝑖 𝑥𝑖 ∈D𝑖 ,𝑥 𝑗 ∈D 𝑗 The matrices Θ𝑖∗ ∈ D 𝑓 ⊂ R 𝑝𝑖 ×𝑛 , Φ𝑖∗ ∈ D𝑔 ⊂ R𝑞𝑖 ×𝑛 , Ψ𝑖∗ ∈ D∆ ⊂ R𝑟𝑖 ×𝑛 represent the unknown optimal adaptive parameters for the approximators given as follows: Θ𝑖∗ = arg min {𝑒 𝑓𝑖 (𝑥𝑖 )}, (2.12) Θ𝑖 ∈D 𝑓 Φ𝑖∗ = arg min {𝑒 𝑔𝑖 (𝑥𝑖 )}, (2.13) Φ ∈D𝑔 𝑖 Ψ𝑖∗ = arg min {𝑒 ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 )}, (2.14) Ψ𝑖 ∈D∆ and 𝜑𝑖 : D𝑖 ↦→ R 𝑝𝑖 , 𝜒𝑖 : D𝑖 ↦→ R𝑞𝑖 , 𝜐𝑖 : D 𝑁𝑖 ↦→ R𝑟𝑖 are the basis functions, such that 𝑝𝑖 , 𝑞𝑖 and 𝑟𝑖 are the number of linearly independent basis functions to approximate 𝑓𝑖 (𝑥𝑖 ), 𝑔𝑖 (𝑥𝑖 ) and ∆(𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ), respectively. The quantities 𝑒 𝑓𝑖 (𝑥𝑖 ), 𝑒 𝑔𝑖 (𝑥𝑖 ) and 𝑒 ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ), defined in (2.9)-(2.11) are, respectively, the MFAEs for 𝑓𝑖 (𝑥𝑖 ), 𝑔𝑖 (𝑥𝑖 ) and ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ), denoting the 20 residual approximation errors for the case of optimal parameters. As a special case, if the adaptive approximation models ˆ𝑓𝑖 (𝑥𝑖 , Θ𝑖 ), 𝑔ˆ𝑖 (𝑥𝑖 , Φ𝑖 ) and ∆ ˆ 𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁 , Ψ𝑖 ) can exactly approximate the 𝑖 unknown functions 𝑓𝑖 (𝑥𝑖 ), 𝑔𝑖 (𝑥𝑖 ) and ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ), respectively, then, 𝑒 𝑓𝑖 = 𝑒 𝑔𝑖 = 𝑒 ∆𝑖 = 0. Remark 1 Generally, adaptive approximators can be classified into linearly parameterized and nonlinearly parameterized [10]. Linearly parameterized approximators are more common in the literature of adaptive control because they provide mechanism to derive stronger analytical results for stability and convergence. Linearly parameterized approximators are different and more general than linear models. In linear models, the entire structure of the system is assumed to be linear. In linearly parameterized approximators, the unknown nonlinearities are estimated by nonlinear approximators, where the weights (parameter estimates) appear linearly with respect to nonlinear basis functions. Remark 2 The linearly parameterized approximation models as given in (2.6), (2.7) and (2.8), are linear in parameters Θ𝑖∗ , Φ𝑖∗ , and Ψ𝑖∗ , respectively, and their corresponding basis functions 𝜑𝑖 (𝑥𝑖 (𝑡)), 𝜒𝑖 (𝑥𝑖 (𝑡)) and 𝜐𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ), respectively, contain some nonlinear functions. We consider two different cases [130]: 1) in the first case, the hypothesis class is assumed to be realizable. That is, the identification is realizable as there is a perfect hypothesis within the hypothesis class (i.e, basis functions and their corresponding optimum weights) that generates no error. 2) in the second case, the hypothesis class is assumed to be not realizable (all system parameters make some identification error). For the first case, the nonlinear basis functions completely capture the subsystem dynamics (i.e., 𝑒 𝑓𝑖 (𝑥𝑖 (𝑡)), 𝑒 𝑔𝑖 (𝑥𝑖 (𝑡)) 𝑒 ∆𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 , given in (2.9)-(2.11), are zero) and only parametric uncertainty exists and therefore, the MFAE is zero. For the second case, the basis functions cannot fully capture the dynamics of the subsystems and mismatch error exists and therefore, the MFAE is nonzero for all hypotheses. By using (2.3)-(2.8), each subsystem dynamics (2.2) can be rewritten as 𝑥¤𝑖 (𝑡) =𝑊𝑖∗𝑇 𝑧𝑖 (𝑥𝑖 (𝑡), 𝑢𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ) + 𝜀𝑖 (𝑥𝑖 (𝑡), 𝑢𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ), (2.15) 21 where 𝑊𝑖∗ ∈ R(𝑝𝑖 +𝑞𝑖 +𝑟𝑖 )×𝑛 , 𝑧𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) ∈ R(𝑝𝑖 +𝑞𝑖 +𝑟𝑖 ) , 𝑊𝑖∗ = [Θ𝑖∗𝑇 , Φ𝑖∗𝑇 , Ψ𝑖∗𝑇 ]𝑇 , 𝑧𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) =[𝜑𝑇𝑖 (𝑥𝑖 ), 𝑢𝑇𝑖 𝜒𝑖𝑇 (𝑥𝑖 ), 𝜐𝑇𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 )]𝑇 , 𝜀𝑖 (𝑥𝑖 (𝑡), 𝑢𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ) =𝑒 𝑓𝑖 (𝑥𝑖 (𝑡)) + 𝑒 𝑔𝑖 (𝑥𝑖 (𝑡))𝑢𝑖 + 𝑒 ∆𝑖 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ). Assumption 2 The approximation error 𝜀𝑖 are bounded inside compact sets D𝑖 , D𝑢 and D 𝑁𝑖 . That is, sup k𝜀𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 )k≤ 𝑏 𝜀 with 𝑏 𝜀 ≥ 0. The approximators’ basis 𝑥𝑖 ∈D𝑖 ,𝑥 𝑗 ∈D 𝑗 ,𝑢𝑖 ∈D𝑢 functions are also bounded in the mentioned compact sets. A distributed filtered regressor is now formulated to circumvent the requirement of measuring 𝑥¤𝑖 (𝑡), and is leveraged by the update law later. For regressor filtering, the dynamics (2.15) is rewritten as 𝑥¤𝑖 = −𝐶𝑥𝑖 + 𝑊𝑖∗𝑇 𝑧𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) + 𝐶𝑥𝑖 + 𝜀𝑖 , (2.16) where 𝐶 = 𝑐𝐼, 𝑐 > 0 and 𝑖 = 1, ..., 𝑁. The state space solution to state model (2.16) can be expressed as Z𝑡 𝑥𝑖 =𝑒 −𝐶𝑡 𝑥𝑖 (0) + 𝑒 −𝐶(𝑡−𝜏) [𝑊𝑖∗𝑇 𝑧𝑖 (𝑥𝑖 (𝜏), 𝑢𝑖 (𝜏), 𝑥 𝑗 (𝜏)| 𝑗 ∈𝑁𝑖 ) + 𝐶𝑥𝑖 (𝜏) + 𝜀𝑖 (𝜏)]𝑑𝜏. (2.17) 0 Now, one can rewrite (2.17) as follows, 𝑥𝑖 =𝑊𝑖∗𝑇 𝑑𝑖 (𝑡) + 𝑐𝑙𝑖 (𝑥𝑖 ) + 𝑒 −𝐶𝑡 𝑥𝑖 (0) + 𝜀 𝑥𝑖 (𝑡), (2.18) 𝑑¤𝑖 (𝑡) = −𝑐𝑑𝑖 (𝑡) + 𝑧𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ), 𝑑𝑖 (0) = 0, 𝑙¤𝑖 (𝑡) = −𝐶𝑙𝑖 (𝑥𝑖 ) + 𝑥𝑖 (𝑡), 𝑙𝑖 (0) = 0, 𝑖 = 1, ..., 𝑁, (2.19) R𝑡 R𝑡 where 𝑙𝑖 (𝑡) = 0 𝑒 −𝐶(𝑡−𝜏) 𝑥𝑖 (𝜏)𝑑𝜏 is the filtered regressor version of 𝑥𝑖 (𝑡), 𝜀 𝑥𝑖 = 0 𝑒 −𝐶(𝑡−𝜏) 𝜀𝑖 (𝜏)𝑑𝜏 R𝑡 and 𝑥𝑖 (0) is the initial state of (2.16), 𝑑𝑖 (𝑡) = 0 𝑒 −𝑐(𝑡−𝜏) 𝑧𝑖 (𝑥𝑖 (𝜏), 𝑢𝑖 (𝜏), 𝑥 𝑗 (𝜏)| 𝑗 ∈𝑁𝑖 )𝑑𝜏 is the dis- tributed filtered regressor of 𝑧𝑖 (𝑥𝑖 (𝑡), 𝑢𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ). 22 Dividing (2.18) by 𝑛𝑖 = 1 + 𝑑𝑖𝑇 (𝑡)𝑑𝑖 (𝑡) + 𝑙𝑖𝑇 (𝑡)𝑙𝑖 (𝑡) as a normalizing signal, one has, 𝑥¯𝑖 (𝑡) =𝑊𝑖∗𝑇 𝑑¯𝑖 (𝑡) + 𝑐 𝑙¯𝑖 (𝑡) + 𝑒 −𝐶𝑡 𝑥¯𝑖 (0) + 𝜀¯𝑖 (𝑡), (2.20) 𝑑 𝑙 𝑥 𝜀𝑥 where 𝑑¯𝑖 = 𝑛𝑖 , 𝑙¯𝑖 = 𝑛𝑖 , 𝑥¯𝑖 = 𝑛𝑖 and 𝜀¯𝑖 = 𝑛 𝑖 . It is implied by Assumption 2 that 𝜀¯𝑖 (𝑡) is also 𝑖 𝑖 𝑖 𝑖 bounded, i.e., sup k 𝜀¯𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 )k≤ 𝑏 𝜀¯ 𝑥𝑖 ∈D𝑖 ,𝑥 𝑗 ∈D 𝑗 ,𝑢𝑖 ∈D𝑢 for some 𝑏 𝜀¯ ≥ 0 and k 𝑑¯𝑖 (𝑡)k< 1. To approximate the uncertainties 𝑓𝑖 (𝑥𝑖 ), 𝑔𝑖 (𝑥𝑖 ) and ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) in a distributed finite-time fashion without the need for satisfaction of the PE condition on the regressor, the chapter objective is to propose a finite-time distributed CL approach that guarantees every interconnected subsystem 𝑖 parameter estimation error, 𝑊 ˜ 𝑖 (𝑡) := 𝑊 ˆ 𝑖 (𝑡) − 𝑊 ∗ , is: 𝑖 1) finite-time stable for distributed adaptive approximators with zero MFAE; 2) finite-time attractive to a bounded set around zero for distributed adaptive approximators with non-zero MFAE; where 𝑊 ˆ 𝑖 (𝑡) = [Θ̂𝑇 (𝑡), Φ̂𝑇 (𝑡), Ψ̂𝑇 (𝑡)]𝑇 ∈ R(𝑝𝑖 +𝑞𝑖 +𝑟𝑖 )×𝑛 , Θ̂𝑖 (𝑡), Φ̂𝑖 (𝑡) and Ψ̂𝑖 (𝑡) 𝑖 𝑖 𝑖 are, respectively, the estimated parameter matrices of 𝑊𝑖∗ , Θ𝑖∗ , Φ𝑖∗ and Ψ𝑖∗ at time 𝑡 for the subsystem 𝑖 and 𝑊 ˜ 𝑖 (𝑡) := 𝑊 ˆ 𝑖 (𝑡) − 𝑊 ∗ := [Θ̃𝑇 (𝑡), Φ̃𝑇 (𝑡), Ψ̃𝑇 (𝑡)]𝑇 such that Θ̃𝑖 (𝑡) := Θ̂𝑖 (𝑡) − Θ∗ , 𝑖 𝑖 𝑖 𝑖 𝑖 Φ̃𝑖 (𝑡) := Φ̂𝑖 (𝑡) − Φ𝑖∗ , Ψ̃𝑖 (𝑡) := Ψ̂𝑖 (𝑡) − Ψ𝑖∗ , 𝑖 = 1, ..., 𝑁. 2.3 Finite-time Distributed Concurrent Learning In this section, a finite-time distributed parameter estimation law for approximating the uncer- tainties of the nonlinear interconnected system (2.2) is presented. The convergence analysis of the proposed method is presented based on the Lyapunov approach. Consider the distributed approximator for subsystem 𝑖 to be of the form 𝑥ˆ¯𝑖 (𝑡) =𝑊 ˆ 𝑇 (𝑡)𝑑¯𝑖 (𝑡) + 𝑐 𝑙¯𝑖 (𝑡) + 𝑒 −𝐶𝑡 𝑥¯𝑖 (0). (2.21) 𝑖 The state estimation error of the subsystem 𝑖 is obtained as 𝑒𝑖 (𝑡) = 𝑥ˆ¯𝑖 (𝑡) − 𝑥¯𝑖 (𝑡). (2.22) 23 The state estimation error 𝑒𝑖 (𝑡), which is later employed in the proposed parameter update law, is accessible online, because 𝑥ˆ¯𝑖 (𝑡) is computed online by the approximator (2.21) and 𝑥¯𝑖 (𝑡) is the normalized measurable state of the system. However, for the sake of parameter convergence analysis, using (2.20) and (2.21), 𝑒𝑖 (𝑡) in (2.22) is rewritten as 𝑒𝑖 (𝑡) = 𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡) − 𝜀¯𝑖 (𝑡). (2.23) 𝑖 To use CL, that employs experienced data along with current data in the update law of the distributed identifier parameters, the memory data is recorded in the memory stacks 𝑀𝑖 ∈ R(𝑝𝑖 +𝑞𝑖 +𝑟𝑖 )×𝑃𝑖 , 𝐿 𝑖 ∈ R𝑛×𝑃𝑖 and 𝑋𝑖 ∈ R𝑛×𝑃𝑖 for each interconnected subsystem 𝑖, 𝑖 = 1, ..., 𝑁 at times 𝜏1 , ..., 𝜏𝑃𝑖 as 𝑀𝑖 = [𝑑¯𝑖 (𝜏1 ), 𝑑¯𝑖 (𝜏2 ), ..., 𝑑¯𝑖 (𝜏𝑃𝑖 )], 𝐿 𝑖 = [𝑙¯𝑖 (𝜏1 ), 𝑙¯𝑖 (𝜏2 ), ..., 𝑙¯𝑖 (𝜏𝑃𝑖 )], 𝑥𝑖 (𝜏1 ), 𝑥¯𝑖 (𝜏2 ), ..., 𝑥¯𝑖 (𝜏𝑃𝑖 )], 𝑋𝑖 = [¯ (2.24) where 𝑃𝑖 denotes the number of data points recorded in each stack of subsystem 𝑖. The memory stack 𝑀𝑖 captures the interactive data samples for which their richness depends on collective richness of the subsystem’s state itself as well as its neighbors. The number of data points 𝑃𝑖 , for 𝑖 = 1, ..., 𝑁, is chosen such that 𝑀𝑖 is full-row rank and contains as many linearly independent elements as the dimension of the distributed filtered regressor 𝑑𝑖 (𝑡) (i.e., the total number of linearly independent basis functions for 𝑓𝑖 (𝑥𝑖 ), 𝑔𝑖 (𝑥𝑖 ) and ∆𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 )), given in (2.18), that is called as rank condition on 𝑀𝑖 and requires 𝑃𝑖 ≥ 𝑝𝑖 + 𝑞𝑖 + 𝑟𝑖 , for 𝑖 = 1, ..., 𝑁. In order for the matrix 𝑀𝑖 to be full-row rank, one needs to collect at least 𝑝𝑖 + 𝑞𝑖 + 𝑟𝑖 number of data samples. Therefore, one can check the full-row rank condition on the data matrix 𝑀𝑖 online after recording 𝑝𝑖 + 𝑞𝑖 + 𝑟𝑖 number of data points in the memory stacks of the subsystem 𝑖. Whenever the full-row rank condition on 𝑀𝑖 is satisfied, i.e., 𝑟𝑎𝑛𝑘(𝑀𝑖 ) = 𝑝𝑖 + 𝑞𝑖 + 𝑟𝑖 , one can stop recording data samples in the corresponding subsystem’s memory stacks. 24 The error 𝑒𝑖ℎ (𝑡) for the ℎ𝑡ℎ recorded data is defined as follows 𝑒𝑖ℎ (𝑡) = 𝑥ˆ¯𝑖ℎ (𝑡) − 𝑥¯𝑖 (𝜏ℎ ), (2.25) where 𝑥ˆ¯𝑖ℎ (𝑡) =𝑊 ˆ 𝑇 (𝑡)𝑑¯𝑖 (𝜏ℎ ) + 𝑐 𝑙¯𝑖 (𝜏ℎ ) + 𝑒 −𝐶𝑡 𝑥¯𝑖 (0), (2.26) 𝑖 is the state estimation at time 0 ≤ 𝜏ℎ < 𝑡, ℎ = 1, ..., 𝑃𝑖 employing the current estimated parameters in 𝑊ˆ 𝑖 (𝑡) and the recorded 𝑑¯𝑖 (𝜏ℎ ) and 𝑙¯𝑖 (𝜏ℎ ). The error 𝑒𝑖ℎ (𝑡), which is later employed in the proposed parameter update law, is accessible online, since, 𝑥¯ˆℎ (𝑡) is computed online by (2.26) using the online estimated 𝑊 ˆ 𝑖 (𝑡) and the memory 𝑖 stacks’ elements 𝑀𝑖 and 𝐿 𝑖 , and 𝑥¯𝑖 (𝜏ℎ ) is accessible from the memory stack 𝑋𝑖 of the corresponding subsystem 𝑖. For analysis purposes, using (2.20) and (2.26), one can rewrite (2.25) as follows 𝑒𝑖ℎ (𝑡) = 𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝜏ℎ ) − 𝜀¯𝑖 (𝜏ℎ ). (2.27) 𝑖 Remark 3 In the distributed approximator (2.21), the received neighboring states appear in the distributed filtered regressor 𝑑¯𝑖 (𝑡), as given in (2.19). Therefore, the richness of the local neighboring data affects the richness and rank condition satisfaction of the distributed data stored in memory 𝑀𝑖 . Finite-time Distributed Concurrent Learning Estimation Law Now, the finite-time distributed estimation law for the unknown parameters in the interconnected subsystem 𝑖 approximator (2.21) is proposed as X𝑃𝑖 𝑊¤̂ 𝑖 (𝑡) = −Γ𝑖 (Ξ𝐺 𝑑¯𝑖 (𝑡)⌊𝑒𝑇𝑖 (𝑡)⌉ 𝛾𝑖 + Ξ𝐶 𝑑¯𝑖 (𝜏ℎ )⌊𝑒𝑖ℎ𝑇 (𝑡)⌉ 𝛾𝑖 ), (2.28) ℎ=1 where ⌊.⌉ 𝛾𝑖 := |.| 𝛾𝑖 𝑠𝑖𝑔𝑛(.) with |.| and 𝑠𝑖𝑔𝑛(.) understood in component-wise sense and 0 ≤ 𝛾𝑖 < 1. The matrices Γ𝑖 , Ξ𝐺 , Ξ𝐶 ∈ R(𝑝𝑖 +𝑞𝑖 +𝑟𝑖 )×(𝑝𝑖 +𝑞𝑖 +𝑟𝑖 ) are positive definite, Γ𝑖 > 0 is the learning rate matrix, Ξ𝐶 = 𝜉𝐶 𝐼 and Ξ𝐺 = 𝜉𝐺 𝐼 with scalars 𝜉𝐶 > 0 and 𝜉𝐺 > 0. The proposed estimation law is distributed since the current 𝑑¯𝑖 (𝑡) and recorded 𝑑¯𝑖 (𝜏ℎ ) are distributed filtered regressors depending not only on the subsystem 𝑖 states but also on its neighboring states. In (2.28), the first 25 term Ξ𝐺 𝑑¯𝑖 (𝑡)⌊𝑒𝑇𝑖 (𝑡)⌉ 𝛾𝑖 is a gradient descent term, containing the current state approximation error P𝑃𝑖 ¯ for the subsystem 𝑖, and the second term Ξ𝐶 ℎ=1 𝑑𝑖 (𝜏ℎ )⌊𝑒𝑖ℎ𝑇 (𝑡)⌉ 𝛾𝑖 , containing the experienced data of subsystem 𝑖, is the distributed CL term. Remark 4 In (2.28), the weights Ξ𝐶 and Ξ𝐺 are not necessarily equal and one of the two estimation terms can be prioritized over the other by choosing appropriate 𝜉𝐶 and 𝜉𝐺 , respectively. Generally, in (2.28) choosing high learning rates Γ𝑖 or weights 𝜉𝐶 can increase the convergence rate. However, it may also lead to chattering in the estimated parameters. Once combined with the control design, this chattering can result in poor control performance or even instability. Remark 5 For every distributed identifier that uses (2.28), the shared neighboring states on the learning time length, not only affect the current value of the distributed regressor, 𝑑¯𝑖 (𝑡), but are also influential on the richness of the distributed memory employed in the second term of (2.28). This entirely discriminates the current work from single system’s finite-time CL-based identification methods [40–43]. In the following, the convergence properties for distributed adaptive approximators with zero and nonzero MFAEs are investigated. Finite-time Convergence Properties for Distributed Adaptive Approximators with Zero MFAEs (¯ 𝜀𝑖 (𝑡) = 0) The theorem below shows that using the proposed finite-time distributed concurrent learning method (2.28), for distributed adaptive approximators with zero MFAEs, i.e. 𝜀¯𝑖 (𝑡) = 0, the estimated parameters 𝑊 ˆ 𝑖 (𝑡) converge to their optimal values in finite time. Theorem 1 Let the distributed approximator for every nonlinear interconnected subsystem 𝑖 in (2.2), be given by (2.21), whose parameters are adjusted by the update law of (2.28) with 0 ≤ 𝛾𝑖 < 1 and a distributed filtered regressor given by (2.19), for 𝑖 = 1, ..., 𝑁. Let Assumptions 1-2 hold. Once the full-row rank condition on 𝑀𝑖 , 𝑖 = 1, .., 𝑁 is satisfied, then for every 𝑖 𝑡ℎ adaptive distributed approximator with zero MFAE, i.e., 𝜀¯𝑖 (𝑡) = 0, the distributed parameter estimation law (2.28) ensures finite-time convergence of 𝑊 ˜ 𝑖 (𝑡) to zero for all interconnected subsystems within the 26 following settling-time function 2k𝑊 ˜ 𝑖 (0)k 1−𝛾𝑖 𝑇 ≤ max , (2.29) 𝑖=1,...,𝑁 𝜁𝑖 𝛽𝑖 (1 − 𝛾𝑖 ) 𝛾𝑖 +1 P𝑃𝑖 where 𝜁𝑖 = 𝜉𝐶 𝜆 𝑚𝑖𝑛 2 (𝑆 ), 𝛽 = 2𝜆 𝑖 𝑖 𝑚𝑖𝑛 (Γ𝑖 ) and 𝑆𝑖 = 𝑑¯ (𝜏 )𝑑¯𝑇 (𝜏 ). ℎ=1 𝑖 ℎ 𝑖 ℎ Proof 1 Choosing the following Lyapunov function candidate X 𝑁 1X 𝑁 𝑉(𝑡) = 𝑉𝑖 (𝑡) = 𝑡𝑟{𝑊 ˜ 𝑇 (𝑡)Γ−1𝑊 ˜ 𝑖 (𝑡)}, (2.30) 2 𝑖=1 𝑖 𝑖 𝑖=1 one has 𝛼𝑖−1 k𝑊 ˜ 𝑖 (𝑡)k 2 ≤𝑉𝑖 (𝑡) ≤ 𝛽−1 k𝑊 ˜ 𝑖 (𝑡)k 2 , (2.31) 𝑖 where 𝛼𝑖 = 2𝜆 𝑚𝑎𝑥 (Γ𝑖 ), 𝛽𝑖 = 2𝜆 𝑚𝑖𝑛 (Γ𝑖 ). The time derivative 𝑉¤𝑖 for 𝑖 = 1, ..., 𝑁, using (2.23), (2.27) and (2.28), yields, 𝑉¤𝑖 (𝑡) = 𝑡𝑟{𝑊˜ 𝑇 (𝑡)Γ−1𝑊¤̂ 𝑖 (𝑡)} 𝑖 𝑖 𝑃𝑖 X = 𝑡𝑟{ − Ξ𝐺 𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)⌊ 𝑑¯𝑇 (𝑡)𝑊 𝑖 𝑖 ˜ 𝑖 (𝑡)⌉ 𝛾𝑖 − Ξ𝐶 𝑊 ˜ 𝑇 (𝑡) 𝑖 𝑑¯𝑖 (𝜏ℎ )⌊ 𝑑¯𝑖𝑇 (𝜏ℎ )𝑊 ˜ 𝑖 (𝑡)⌉ 𝛾𝑖 }. (2.32) ℎ=1 One knows that X 𝑛 𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)⌊ 𝑑¯𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)⌉ 𝛾𝑖 = |(𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡))𝑖 | 𝛾𝑖 +1 𝑖 𝑖 𝑖 𝑖=1 = k𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)k 𝛾𝑖 +1 , (2.33) 𝑖 𝛾𝑖 +1 and based on Fact 1 k𝑊˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)k≤ k𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)k 𝛾 +1 , (2.34) 𝑖 𝑖 𝑖 holds for 0 < 𝛾𝑖 + 1 < 2. Therefore, using (2.32)-(2.34), one obtains, X 𝑃𝑖 𝑉¤𝑖 (𝑡) ≤ −𝜉𝐺 k𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)k 𝛾𝑖 +1 −𝜉𝐶 k𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝜏ℎ )k 𝛾𝑖 +1 𝑖 𝑖 ℎ=1 X𝑃𝑖 𝛾𝑖 +1 ≤ −𝜉𝐶 (𝑊˜ 𝑇 (𝑡)𝑑¯𝑖 (𝜏ℎ )𝑑¯𝑇 (𝜏ℎ )𝑊 ˜ 𝑖 (𝑡)) 2 . (2.35) 𝑖 𝑖 ℎ=1 27 Therefore, 𝛾𝑖 +1 2 (𝑆 )k𝑊 𝛾 +1 𝑉¤𝑖 (𝑡) ≤ −𝜉𝐶 𝜆 𝑚𝑖𝑛 𝑖 ˜ 𝑖 (𝑡)k 𝑖 , (2.36) P𝑃𝑖 where 𝑆𝑖 = 𝑑¯ (𝜏 )𝑑¯𝑇 (𝜏 ). Using (2.31), (2.36) gives ℎ=1 𝑖 ℎ 𝑖 ℎ 𝛾𝑖 +1 𝛾𝑖 +1 𝑉¤𝑖 (𝑡) ≤ −𝜁𝑖 𝛽𝑖 2 𝑉𝑖 2 (𝑡), 𝛾𝑖 +1 where 𝜁𝑖 = 𝜉𝐶 𝜆 𝑚𝑖𝑛 2 (𝑆 ) and based on Lemma 1, it is proved that for every subsystem 𝑖, 𝑖 = 1, ..., 𝑁, 𝑖 𝑊˜ 𝑖 (𝑡) is finite-time stable with the following settling-time function ˜ 1−𝛾𝑖 𝑇𝑖 (𝑊 ˜ 𝑖 (0)) ≤ 2k𝑊𝑖 (0)k . 𝜁𝑖 𝛽𝑖 (1 − 𝛾𝑖 ) Therefore, the whole interconnected system dynamics can be identified in finite time within the following settling time, 2k𝑊 ˜ 𝑖 (0)k 1−𝛾𝑖 𝑇 ≤ max 𝑇𝑖 (𝑊 ˜ 𝑖 (0)) = max . 𝑖=1,...,𝑁 𝑖=1,...,𝑁 𝜁𝑖 𝛽𝑖 (1 − 𝛾𝑖 ) This completes the proof. Corollary 1 Let the assumptions and statements of Theorem 1 hold. Then, for adaptive distributed approximators with zero MFAEs, i.e., 𝜀¯𝑖 (𝑡) = 0, the state estimation error 𝑒𝑖 (𝑡) for every subsystem 𝑖, 𝑖 = 1, ..., 𝑁, is finite-time stable. Proof 2 The proof is a direct consequence of Theorem 1. Remark 6 As shown in (2.29), the settling time function of the identifier depends on the minimum eigenvalue of the distributed memory matrix, 𝜆 𝑚𝑖𝑛 (𝑆𝑖 ). Therefore, to improve the convergence speed, an optimization over recorded data can be performed to replace old data with new ones as more data becomes available to maximize the minimum eigenvalue of the distributed memory matrix to reduce the convergence time. Finite-time Convergence Properties for Distributed Adaptive Approximators with Non- zero MFAEs (¯ 𝜀𝑖 (𝑡) 6= 0) The following theorem gives the finite-time convergence properties for the distributed parameter estimation law (2.28) of distributed adaptive approximators with non-zero MFAEs, 𝜀¯𝑖 (𝑡) 6= 0, in interconnected systems’ identification. 28 Theorem 2 Let the distributed approximator for nonlinear interconnected subsystem (2.2) given by (2.21), whose parameters are adjusted by the update law of (2.28) with 0 < 𝛾𝑖 < 1 and a regressor given in (2.19). Consider that Assumptions 1-2 hold. Once the full-row rank condition on 𝑀𝑖 for 𝑖 = 1, ..., 𝑁 is met, then for adaptive distributed approximators with non-zero MFAEs, i.e., 𝜀¯𝑖 (𝑡) 6= 0, the proposed parameter estimation law (2.28) guarantees that for every subsystem 𝑖, 𝑖 = 1, ..., 𝑁, the 𝑊 ˜ 𝑖 (𝑡) is finite-time attractive by the following bounded set, s 𝑆𝑖˜ = {𝑊 ˜ 𝑖 (𝑡) : k𝑊 ˜ 𝑖 (𝑡)k≤ 𝜆 𝑚𝑎𝑥 (Γ𝑖 ) 𝜇¯𝑖 }, ∀𝑡 ≥ 𝑇, (2.37) 𝑊 𝜆 𝑚𝑖𝑛 (Γ𝑖 ) where   𝜔 1 𝑏 𝜀¯    𝑚𝑎𝑥{ , ( 𝜁 𝑖𝛿 ) 𝛾𝑖 }, 𝑑¯𝑖 (𝑡) 6= 0,    1 𝑖 𝜇¯𝑖 = min 𝜆 2 (𝐷 𝑖 (𝑡)),𝜆¯𝑖 (2.38)  𝑚𝑖𝑛   1 𝑏 𝜀¯ 𝜔𝑖 𝛾𝑖   𝑚𝑎𝑥{ 𝜆¯ , ( 𝜁 𝛿 ) }, 𝑑¯𝑖 (𝑡) = 0,  𝑖 𝑖 2k𝑊 ˜ 𝑖 (0)k 1−𝛾𝑖 𝑇 ≤ max , (2.39) 𝑖=1,...,𝑁 𝜁𝑖 𝛽𝑖 (1 − 𝛿)(1 − 𝛾𝑖 ) 1 1−𝛾𝑖 2 ¯ ¯ 𝑇 𝛾 𝜆¯𝑖 = min (𝜆 𝑚𝑖𝑛 (𝐷 𝑖 (𝜏ℎ ))), 𝐷 𝑖 (𝑡) = 𝑑𝑖 (𝑡)𝑑𝑖 (𝑡), 𝜔𝑖 = 𝑛 2 𝑏 𝜀¯𝑖 (𝜉𝐺 + 𝑃𝑖 𝜉𝐶 ), ℎ=1,...,𝑃𝑖 and 0 < 𝛿 < 1. Proof 3 Choose the same Lyapunov function (2.30) that satisfies (2.31). The time derivation of 𝑉𝑖 employing (2.23), (2.27) and (2.28) gives, 𝑉¤𝑖 (𝑡) = 𝑡𝑟{ − Ξ𝐺 𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)⌊ 𝑑¯𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡) − 𝜀¯𝑇 (𝑡)⌉ 𝛾𝑖 𝑖 𝑖 𝑖 X𝑃𝑖 − Ξ𝐶 𝑊 ˜ 𝑇 (𝑡) 𝑖 𝑑¯𝑖 (𝜏ℎ )⌊ 𝑑¯𝑖𝑇 (𝜏ℎ )𝑊 ˜ 𝑖 (𝑡) − 𝜀¯𝑇 (𝜏ℎ )⌉ 𝛾𝑖 }. 𝑖 (2.40) ℎ=1 Consider in the component-wise sense that |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) 𝑘 |≥ |(¯ 𝜀𝑖 (𝑡)) 𝑘 |, for 𝑘 = 1, ..., 𝑛. Note that the previous inequality is required for 𝑑¯𝑖 (𝑡) 6= 0. If 𝑑¯𝑖 (𝑡) = 0 then the first term in (2.28) is zero and in the second term of (2.28), the data collection assures that 𝑑¯𝑖 (𝜏ℎ ) 6= 0, ℎ = 1, ..., 𝑃𝑖 . Therefore, 𝑠𝑖𝑔𝑛(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡) − 𝜀¯𝑇 (𝑡)) = 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝑡)𝑊 𝑖 𝑖 ˜ 𝑖 (𝑡)) is obtained. Then, for any 𝑦, 𝑦¯ ∈ R and 0 < 𝛾𝑖 < 1, one has [129], |𝑦 + 𝑦¯| 𝛾𝑖 < |𝑦| 𝛾𝑖 +|¯ 𝑦 | 𝛾𝑖 . 29 Therefore, defining 𝑦 = (𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) 𝑘 − (¯ 𝜀𝑖 (𝑡)) 𝑘 and 𝑦¯ = (¯ 𝜀𝑖 (𝑡)) 𝑘 , one obtains that for all 𝑘 = 1, ..., 𝑛, |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) 𝑘 | 𝛾𝑖 = |(𝑑¯𝑇 (𝑡)𝑊 𝑖 ˜ 𝑖 (𝑡)) 𝑘 − (¯ 𝜀𝑖 (𝑡)) 𝑘 + (¯ 𝜀𝑖 (𝑡)) 𝑘 | 𝛾𝑖 ≤ |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) 𝑘 − (¯ 𝜀𝑖 (𝑡)) 𝑘 | 𝛾𝑖 +|(¯ 𝜀𝑖 (𝑡)) 𝑘 | 𝛾𝑖 ⇒ |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) 𝑘 | 𝛾𝑖 −|(¯ 𝜀𝑖 (𝑡)) 𝑘 | 𝛾𝑖 ≤ |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) 𝑘 − (¯ 𝜀𝑖 (𝑡)) 𝑘 | 𝛾𝑖 , and then in the component-wise sense, −| 𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡) − 𝜀¯𝑖 (𝑡)| 𝛾𝑖 ≤ −| 𝑑¯𝑇 (𝑡)𝑊 𝑖 ˜ 𝑖 (𝑡)| 𝛾𝑖 +| 𝜀¯𝑖 (𝑡)| 𝛾𝑖 . (2.41) Now, using (2.41), (2.40) is upper bounded by 𝑉¤𝑖 (𝑡) ≤ 𝑡𝑟{ − Ξ𝐺 𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)(⌊ 𝑑¯𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)⌉ 𝛾𝑖 − | 𝜀¯𝑖 (𝑡)| 𝛾𝑖 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡))) 𝑖 𝑖 𝑖 X𝑃𝑖 − Ξ𝐶 𝑊 ˜ 𝑇 (𝑡) 𝑖 𝑑¯𝑖 (𝜏ℎ )(⌊ 𝑑¯𝑖𝑇 (𝜏ℎ )𝑊 ˜ 𝑖 (𝑡)⌉ 𝛾𝑖 − | 𝜀¯𝑖 (𝜏ℎ )| 𝛾𝑖 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝜏ℎ )𝑊 𝑖 ˜ 𝑖 (𝑡)))}. (2.42) ℎ=1 Recall that in ⌊.⌉ 𝛾𝑖 , |.| and 𝑠𝑖𝑔𝑛(.) are employed in the component-wise sense, i.e. ⌊ 𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)⌉ 𝛾𝑖 =[|(𝑑¯𝑇 (𝑡)𝑊 𝑖 ˜ 𝑖 (𝑡))1 | 𝛾𝑖 𝑠𝑖𝑔𝑛((𝑑¯𝑇 (𝑡)𝑊 𝑖 ˜ 𝑖 (𝑡))1 ), |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡))2 | 𝛾𝑖 𝑠𝑖𝑔𝑛((𝑑¯𝑇 (𝑡)𝑊 𝑖 ˜ 𝑖 (𝑡))2 ), .. . |(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡))𝑛 | 𝛾𝑖 𝑠𝑖𝑔𝑛((𝑑¯𝑇 (𝑡)𝑊 𝑖 ˜ 𝑖 (𝑡))𝑛 )], | 𝜀¯𝑖 (𝑡)| 𝛾𝑖 𝑠𝑖𝑔𝑛(𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡)) =[|(¯ 𝜀𝑖 (𝑡))1 | 𝛾𝑖 𝑠𝑖𝑔𝑛((𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡))1 ), |(¯ 𝜀𝑖 (𝑡))2 | 𝛾𝑖 𝑠𝑖𝑔𝑛((𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡))2 ), . . . , |(¯𝜀𝑖 (𝑡))𝑛 | 𝛾𝑖 𝑠𝑖𝑔𝑛((𝑑¯𝑖𝑇 (𝑡)𝑊 ˜ 𝑖 (𝑡))𝑛 )]. Therefore, using (2.33), (2.34), k 𝑑¯𝑖 (𝑡)k≤ 1 and (2.42), one obtains, 𝑉¤𝑖 (𝑡) ≤ −𝜉𝐺 k𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝑡)k 𝛾𝑖 +1 +𝜉𝐺 k𝑊 ˜ 𝑖 (𝑡)kk| 𝜀¯𝑖 (𝑡)| 𝛾𝑖 k 𝑖 X𝑃𝑖 − 𝜉𝐶 k𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝜏ℎ )k 𝛾𝑖 +1 +𝜉𝐶 𝑃𝑖 k| 𝜀¯𝑖 (𝜏ℎ )| 𝛾𝑖 kk𝑊 ˜ 𝑖 (𝑡)k. (2.43) 𝑖 ℎ=1 30 qP Since k| 𝜀¯𝑖 (𝑡)| 𝛾𝑖 k= 𝑛 |(¯ 𝜀 (𝑡)) | 2𝛾𝑖 = k 𝜀¯ (𝑡)k 𝛾𝑖 and by Hölder’s inequality 𝑘=1 𝑖 𝑘 𝑖 2𝛾𝑖 1−𝛾𝑖 k 𝜀¯𝑖 (𝑡)k 2𝛾𝑖 ≤ 𝑛 2𝛾𝑖 k 𝜀¯𝑖 (𝑡)k, (2.44) holds for all 0 < 2𝛾𝑖 < 2, it is given that X𝑃𝑖 𝛾𝑖 +1 1−𝛾𝑖 𝑉¤𝑖 (𝑡) ≤ − 𝜉𝐶 (𝑊 ˜ 𝑇 (𝑡)𝑑¯𝑖 (𝜏ℎ )𝑑¯𝑇 (𝜏ℎ )𝑊 ˜ 𝑖 (𝑡)) 2 + 𝜉𝐺 𝑛 2 k 𝜀¯𝑖 (𝑡)k 𝛾𝑖 k𝑊 ˜ 𝑖 (𝑡)k 𝑖 𝑖 ℎ=1 1−𝛾𝑖 + 𝜉𝐶 𝑃𝑖 𝑛 2 k 𝜀¯𝑖 (𝜏ℎ )k 𝛾𝑖 k𝑊 ˜ 𝑖 (𝑡)k. Therefore, 𝑉¤𝑖 (𝑡) ≤ −𝜁𝑖 k𝑊 ˜ 𝑖 (𝑡)k 𝛾𝑖 +1 +𝜔𝑖 k𝑊 ˜ 𝑖 (𝑡)k, (2.45) where 1−𝛾𝑖 𝛾 𝜔𝑖 = 𝑛 2 𝑏 𝜀¯𝑖 (𝜉𝐺 + 𝑃𝑖 𝜉𝐶 ) . In the following, (2.45) is rewritten as 𝑉¤𝑖 (𝑡) ≤ − 𝜁𝑖 k𝑊 ˜ 𝑖 (𝑡)k 𝛾𝑖 +1 +𝜔𝑖 k𝑊 ˜ 𝑖 (𝑡)k ≤ − 𝜁𝑖 (1 − 𝛿)k𝑊 ˜ 𝑖 (𝑡)k 𝛾𝑖 +1 −𝜁𝑖 𝛿k𝑊 ˜ 𝑖 (𝑡)k 𝛾𝑖 +1 +𝜔𝑖 k𝑊 ˜ 𝑖 (𝑡)k, where 0 < 𝛿 < 1. Hence, 𝑉¤𝑖 (𝑡) ≤ −𝜁𝑖 (1 − 𝛿)k𝑊 ˜ 𝑖 (𝑡)k 𝛾𝑖 +1 , 𝜇¯𝑖 ≤ k𝑊 ˜ 𝑖 (𝑡)k, (2.46) where   𝜔 1 𝑏 𝜀¯    𝑚𝑎𝑥{ , ( 𝜁 𝑖𝛿 ) 𝛾𝑖 }, 𝑑¯𝑖 (𝑡) 6= 0,    1 𝑖 𝜇¯𝑖 = min 𝜆 2 (𝐷 𝑖 (𝑡)),𝜆¯𝑖  𝑚𝑖𝑛   1 𝑏 𝜀¯ 𝜔𝑖 𝛾𝑖   𝑚𝑎𝑥{ 𝜆¯ , ( 𝜁 𝛿 ) }, 𝑑¯𝑖 (𝑡) = 0.  𝑖 𝑖 From (2.31) and (2.46), it follows that 𝛾𝑖 +1 𝛾𝑖 +1 𝑉¤𝑖 (𝑡) ≤ −𝜁𝑖 (1 − 𝛿)𝛽𝑖 2 𝑉𝑖 2 (𝑡), (2.47) 31 and by comparison principle one obtains 𝛾𝑖 +1 1−𝛾𝑖 𝜁𝑖 (1 − 𝛿)(1 − 𝛾𝑖 )𝛽𝑖 2 2 𝑉𝑖 (𝑡) ≤ (𝑉𝑖 2 (0) − 𝑡) 1−𝛾𝑖 , 2 then using (2.31), the above inequality ensures that 𝑊 ˜ 𝑖 (𝑡) satisfies 𝛾𝑖 +1 𝛾𝑖 −1 𝜁 (1 − 𝛿)(1 − 𝛾𝑖 )𝛽𝑖 2 1 k𝑊 ˜ 𝑖 (𝑡)k≤ √𝛼𝑖 (𝛽 2 k𝑊 ˜ 𝑖 (0)k 1−𝛾𝑖 − 𝑖 𝑡) 1−𝛾𝑖 , 𝑖 2 ˜ 𝑖 (0)). Then, for all 𝑡 > 𝑇𝑖 (𝑊 for all 𝑡 < 𝑇𝑖 (𝑊 ˜ 𝑖 (0)), from (2.31), one obtains that 𝑊 ˜ 𝑖 (𝑡) is bounded as s k𝑊˜ 𝑖 (𝑡)k≤ 𝜆 𝑚𝑎𝑥 (Γ𝑖 ) 𝜇¯𝑖 , ∀𝑡 ≥ 𝑇𝑖 (𝑊 ˜ 𝑖 (0)). (2.48) 𝜆 𝑚𝑖𝑛 (Γ𝑖 ) Therefore, for every subsystem 𝑖, 𝑖 = 1, ..., 𝑁, the solutions of 𝑊 ˜ 𝑖 (𝑡) are finite-time attractive to the bound in (2.48) where 2k𝑊 ˜ 𝑖 (0)k 1−𝛾𝑖 𝑇𝑖 (𝑊˜ 𝑖 (0)) ≤ . 𝜁𝑖 𝛽𝑖 (1 − 𝛿)(1 − 𝛾𝑖 ) Therefore, all the solutions of 𝑊 ˜𝑖 , 𝑖 = 1, ..., 𝑁 for the interconnected system are finite-time attractive to the bound given in (2.37) in the following settling time, 2k𝑊 ˜ 𝑖 (0)k 1−𝛾𝑖 𝑇 ≤ max 𝑇𝑖 (𝑊 ˜ 𝑖 (0)) = max . 𝑖=1,...,𝑁 𝑖=1,...,𝑁 𝜁𝑖 𝛽𝑖 (1 − 𝛿)(1 − 𝛾𝑖 ) This completes the proof. Corollary 2 Let the assumptions and statements of Theorem 2 hold. Then, for adaptive distributed approximators with non-zero MFAEs, i.e., 𝜀¯𝑖 (𝑡) 6= 0, the state estimation error 𝑒𝑖 (𝑡) for every subsystem 𝑖, 𝑖 = 1, ..., 𝑁, is finite-time attractive. Proof 4 The proof is a direct consequence of Theorem 2. Remark 7 In Theorem 2, for 𝛾𝑖 = 0, using (2.31) and (2.45) it can be shown that for every interconnected subsystem 𝑖, 𝑖 = 1, ..., 𝑁, if 𝜁𝑖 > 𝜔𝑖 , 𝑊 ˜𝑖 (𝑡) is finite-time stable and the interconnected system can be exactly identified with zero MFAE. Remark 8 As discussed in [79], the concurrent learning approach is based on the combination of a gradient descent algorithm with an auxiliary static feedback update law, which can be viewed 32 as a type of 𝜎-modification [10] and allows the requirement on persistence of excitation to be relaxed by keeping enough measurements in memory. Here, the same extension is applied to the proposed distributed finite-time concurrent learning in (2.28). Theoretical support of this claim is provided in Theorem 2 to show the finite-time attractiveness of the proposed parameter update law (2.28) in case of nonzero MFAEs. Remark 9 For distributed adaptive approximators with non-zero MFAEs, the richness of the distributed data stored in 𝑀𝑖 influences the finite settling time as well as the error bound. Accordingly, 𝑊 ˜ 𝑖 (𝑡) converges to a narrower bound in faster time by maximizing 𝜆 𝑚𝑖𝑛 (𝑆𝑖 ) that minimizes the error bound and the settling time respectively given in (2.37) and (2.39). Therefore, after the rank condition satisfaction, optimization over recorded data can improve the convergence results for every subsystem where one can replace new distributed samples with old ones in 𝑀𝑖 , 𝑖 = 1, ..., 𝑁, if 𝜆 𝑚𝑖𝑛 (𝑆𝑖 ) increases to result in a faster convergence to a lower error bound. Remark 10 Similar to the concurrent learning literature [30, 33–37, 39, 41] and most sys- tem identifiers for nonlinear systems, in this chapter it is assumed that the subsystem states are measurable. Even though the subsystems’ states are measurable, the finite-time identification of interconnected systems without the persistency of excitation requirement is challenging. Online finite-time identification of the interconnected system dynamics under output measurements as- sumption is a direction for future research. This requires coupled distributed identifier and observer design for every subsystem to be able to identify the subsystem dynamics and observe its states interactively in finite-time. Remark 11 In the proposed finite-time distributed concurrent learning estimation law (2.28), if the concurrent learning term regarding the past historical data is eliminated, the following finite- time distributed gradient descent estimation law that only depends on the current distributed data is obtained as 𝑊¤̂ 𝑖 (𝑡) = −𝐾𝑖 𝑑¯𝑖 (𝑡)⌊𝑒𝑇𝑖 (𝑡)⌉ 𝛾𝑖 , (2.49) with 𝐾𝑖 > 0. According to the analysis provided in the previous theorems, similar results are obtained for the estimation law (2.49) provided that 𝑑¯𝑖 (𝑡) is persistently excited for every subsystem 33 𝑖. The finite-time distributed gradient descent law (2.49) is similar to the estimation law for a single system in Algorithm 1 of [106] where the short finite-time input to state stability of the mentioned learning law (ensuring stability in a finite and limited time interval) has been proven, provided that the regressor 𝑑¯𝑖 (𝑡) is nullifying in finite time. 2.4 Simulation Results Now, the proposed finite-time distributed CL method performance for a nonlinear intercon- nected system identification is examined in comparison with the finite-time distributed gradient descent estimation method given in (2.49). The considered nonlinear interconnected system con- tains 3 inverted interconnected pendulums as depicted in Fig. 2.1. Every inverted pendulum 𝑖 [131], subject to control input 𝑢𝑖 is described by     𝑥¤𝑖1 = 𝑓𝑖1 (𝑥𝑖 ) + 𝑔𝑖1 (𝑥𝑖 ) + ∆𝑖1 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 ),       = 𝑥𝑖2    (2.50)   𝑥¤𝑖2 = 𝑓𝑖2 (𝑥𝑖 ) + 𝑔𝑖2 (𝑥𝑖 ) + ∆𝑖2 (𝑥𝑖 (𝑡), 𝑥 𝑗 (𝑡)| 𝑗 ∈𝑁𝑖 )      𝑔 𝑢𝑖 P 𝑘 𝑖, 𝑗 𝑎 2   = 𝑠𝑖𝑛𝑥 𝑖1 + + 2 (𝑠𝑖𝑛𝑥 𝑗1 𝑐𝑜𝑠𝑥 𝑗1 − 𝑠𝑖𝑛𝑥𝑖1 𝑐𝑜𝑠𝑥𝑖1 ),   𝑙 𝑚𝑖 𝑙 2 𝑗 ∈𝑁 𝑗 𝑚𝑖 𝑙  where 𝑥𝑖1 = 𝜃𝑖 (𝑟𝑎𝑑) is the angular position and 𝑥𝑖2 = 𝜃¤𝑖 (𝑟𝑎𝑑/𝑠) is the angular velocity, for the inverted pendulum 𝑖, 𝑖 = 1, 2, 3. The gravity acceleration 𝑔 is 𝑔 ≈ 10 𝑚2 , 𝑚𝑖 is the mass of 𝑠 the 𝑖 𝑡ℎ rod (𝑚𝑖 = 0.25 𝑘𝑔, 𝑖 = 1, 2, 3), 𝑙 is the length of each rod (𝑙 = 2 𝑚), 𝑎 is the distance 𝑘𝑔 from the pivot to the center of gravity of the rod (𝑎 = 1 𝑚), 𝑘 𝑖, 𝑗 ( 2 ) is the spring constant which 𝑠 interconnects subsystem 𝑖 to subsystem 𝑗, 𝑗 ∈ 𝑁𝑖 , with 𝑘 𝑖, 𝑗 = 𝑘 𝑗,𝑖 and 𝑘 1,2 = 𝑘 1,3 = 1.5, 𝑘 2,3 = 2. In this system, due to the physical limitations, 𝑥𝑖 domain is defined by 𝐷 𝑖 = [𝐷 𝑖1 , 𝐷 𝑖2 ]𝑇 where 𝐷 𝑖1 = [−6, 6] and 𝐷 𝑖2 = [−4, 4] for 𝑖 = 1, 2, 3. The initial states and parameters are chosen from the interval [−2, 2] and stabilizing controllers 𝑢𝑖 = −0.06𝑥𝑖 for 𝑖 = 1, 2, 3 are employed. Every interconnected subsystem 𝑖 dynamics in (2.50) is unknown. For every subsystem 𝑖, 𝑖 = 1, 2, 3, the proposed finite-time distributed concurrent learning identifier employs the following basis functions, 𝑧𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) = [𝑥𝑖2 , sin 𝑥𝑖1 , 𝑢𝑖 , sin 𝑥𝑖1 cos 𝑥𝑖1 , sin 𝑥 𝑗1 cos 𝑥 𝑗1 ]𝑇𝑗∈𝑁 . (2.51) 𝑖 34 Figure 2.1: Interconnection network of the physically interconnected inverted pendulums. While the regressor vector 𝑧𝑖 (𝑥𝑖 , 𝑢𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁𝑖 ) is exciting over some time period, it is not persistently exciting. The relaxed excitation condition without the PE requirement is achieved without injecting any exciting probing noise to the subsystems’ controllers. Therefore, the approximation of (2.50) for every subsystem 𝑖 is as follows,   𝑖   𝑥¤𝑖1 = 𝑝 1 𝑥𝑖2 ,   P (2.52)   𝑖 𝑖 𝑖  𝑥¤𝑖2 = 𝑝 2 𝑠𝑖𝑛𝑥𝑖1 + 𝑝 3 𝑢𝑖 + 𝑝 4 𝑠𝑖𝑛𝑥𝑖1 𝑐𝑜𝑠𝑥𝑖1 + 𝑝𝑖𝑟 𝑠𝑖𝑛𝑥 𝑗1 𝑐𝑜𝑠𝑥 𝑗1 ,  𝑗 ∈𝑁 𝑗 ,𝑟=5,...,4+|𝑁 𝑗 |  where based on the system descriptions, the true parameters for the three interconnected subsystems are as follows, [𝑝 11 , 𝑝 12 , 𝑝 13 , 𝑝 14 , 𝑝 15 , 𝑝 16 ] = [1, 5, 1, −3, 1.5, 1.5], [𝑝 21 , 𝑝 22 , 𝑝 23 , 𝑝 24 , 𝑝 25 , 𝑝 26 ] = [1, 5, 1, −3.5, 1.5, 2], [𝑝 31 , 𝑝 32 , 𝑝 33 , 𝑝 34 , 𝑝 35 , 𝑝 36 ] = [1, 5, 1, −3.5, 2, 1.5]. We set Γ𝑖 = 3𝐼, 𝜉𝐺 = 1, 𝜉𝐶 = 0.1, 𝛾𝑖 = 0.5 for 𝑖 = 1, 2, 3. We chose 𝜉𝐺 > 𝜉𝐶 , to prioritize current data to the recorded data in the proposed learning method (2.28) and 𝑃𝑖 = 10, 𝑖 = 1, 2, 3, which are set to be greater than 6, the number of independent basis functions for every subsystem. To have a fair speed and the precision comparison of the mentioned methods for approximating 𝑓ˆ𝑖 (𝑥𝑖 ) and 𝑔ˆ𝑖 (𝑥𝑖 ) on the domain 𝐷 𝑖 , and ∆ ˆ 𝑖 (𝑥𝑖 , 𝑥 𝑗 | 𝑗 ∈𝑁 ) on the domain of 𝐷 𝑁 , the following online 𝑖 𝑖 35 learning errors are computed for every subsystem 𝑖, 𝑖 = 1, 2, 3, Z Z 𝐸 𝑓𝑖 (𝑡) = k𝑒 𝑓𝑖 (𝑥𝑖 (𝑡))k𝑑 𝑛 𝑥𝑖 , 𝐸 𝑔𝑖 (𝑡) = k𝑒 𝑔𝑖 (𝑥𝑖 (𝑡))k𝑑 𝑛 𝑥𝑖 , D D𝑖 Z 𝑖 𝐸 ∆𝑖 (𝑡) = k𝑒 ∆𝑖 (𝑥 𝑁𝑖 (𝑡))k𝑑 𝑛(𝑁𝑖 +1) 𝑥 𝑁𝑖 , (2.53) D𝑁 𝑖 R where the notation D k𝑒 𝑓𝑖 (𝑥𝑖 (𝑡))k𝑑 𝑛 𝑥𝑖 indicates that the integral of k𝑒 𝑓𝑖 (𝑥𝑖 (𝑡))k is calculated over 𝑖 an 𝑛-dimensional region D𝑖 . The simulations are done in MATLAB with Euler integration and the sample time is equal to 0.05 seconds. In the simulation results, the proposed finite-time distributed concurrent learning method and finite-time distributed gradient descent approach, given in (2.49), are respectively labeled by FTDCL and FTDGD. Fig. 2.2 shows the approximated parameters using the proposed finite-time distributed concurrent learning approach and finite-time distributed gradient descent method (given in (2.49)) for three interconnected subsystems. Fig. 2.2 clearly shows that the approximated parameters using the proposed finite-time distributed concurrent learning method have converged to the true parameters, while because of the lack of persistence of excitation, the estimated parameters for the finite-time distributed gradient descent failed to converge to the true parameters. Figs. 2.3 to 2.5 depict the online learning errors 𝐸 𝑓𝑖 (𝑡), 𝐸 𝑔𝑖 (𝑡), and 𝐸 ∆𝑖 (𝑡) for, respectively, the three interconnected subsystems, 𝑖 = 1, 2, 3, where the results of the proposed finite-time distributed concurrent learning show the finite-time convergence of all errors to zero while the learning errors for finite-time distributed gradient descent method did not converge to the origin due to the lack of regressor’s PE condition. 36 Figure 2.2: (a). Parameters of finite-time distributed concurrent learning (FTDCL) identifiers for subsystems 1, 2 and 3. (b). Parameters of finite-time distributed gradient descent (FTDGD) identifiers for subsystems 1, 2 and 3. 400 FTDCL E f (t) 1 200 FTDGD 0 0 50 100 150 200 250 300 350 400 450 500 200 FTDCL E g (t) 1 100 FTDGD 0 0 50 100 150 200 250 300 350 400 450 500 100 FTDCL (t) 1 FTDGD 50 E 0 0 50 100 150 200 250 300 350 400 450 500 Time (s) Figure 2.3: Online learning errors 𝐸 𝑓1 (𝑡), 𝐸 𝑔1 (𝑡), and 𝐸 ∆1 (𝑡) for interconnected subsystem 1. 37 400 FTDCL E f (t) 2 200 FTDGD 0 0 50 100 150 200 250 300 350 400 450 500 200 FTDCL E g (t) 2 100 FTDGD 0 0 50 100 150 200 250 300 350 400 450 500 100 FTDCL (t) 2 FTDGD 50 E 0 0 50 100 150 200 250 300 350 400 450 500 Time (s) Figure 2.4: Online learning errors 𝐸 𝑓2 (𝑡), 𝐸 𝑔2 (𝑡), and 𝐸 ∆2 (𝑡) for interconnected subsystem 2. 400 FTDCL E f (t) 3 200 FTDGD 0 0 50 100 150 200 250 300 350 400 450 500 200 FTDCL E g (t) 3 100 FTDGD 0 0 50 100 150 200 250 300 350 400 450 500 200 FTDCL (t) 3 FTDGD 100 E 0 0 50 100 150 200 250 300 350 400 450 500 Time (s) Figure 2.5: Online learning errors 𝐸 𝑓3 (𝑡), 𝐸 𝑔3 (𝑡), and 𝐸 ∆3 (𝑡) for interconnected subsystem 3. 2.5 Conclusion In this chapter, a finite-time distributed concurrent learning method for interconnected sys- tems’ identification in finite time is introduced. Leveraging local state communication among interconnected subsystems’ identifiers enabled them to identify every subsystem’s own dynamics as well as its interconnections’ dynamics. In this method, distributed concurrent learning relaxed the regressors’ persistence of excitation (PE) conditions to rank conditions on the recorded dis- tributed data in the memory stack of the subsystems. It is shown that the precision and convergence speed of the proposed finite-time distributed learning method depends on the spectral properties of the distributed recorded data. Simulation results show that the proposed finite-time distributed concurrent learning has outperformed the finite-time distributed gradient descent in both terms of 38 precision and convergence speed. 39 CHAPTER 3 FIXED-TIME SYSTEM IDENTIFICATION USING CONCURRENT LEARNING 3.1 Introduction In this chapter, first, a novel discontinuous update law is presented that employs CL to identify system uncertainties in a fixed time that can be computed a priori. Fixed-time convergence guar- antee is certified under a rank condition on recorded experienced data rather than PE condition. Second, the rigorous analysis based on fixed-time Lyapunov stability certifies the convergence of the discontinuous gradient flow equipped with CL to zero for the case where minimum functional approximation error (MFAE) is zero and under a rank condition on stored data. Moreover, for adaptive approximators with non-zero MFAE, it is ensured that by employing the proposed al- gorithm, the parameters estimation errors are fixed time attractive to an ultimate bound. Third, the fixed-time upper bounds of the estimated parameters’ settling-times, independent of the initial parameter estimation error, are derived for adaptive approximators with zero and non-zero MFAEs. Notation Throughout this chapter, the following notation is adopted. R and R+ denote the set of real and positive real numbers, respectively. k.k is used to denote the Euclidean norm for a vector and induced 2-norm for a matrix. 𝑡𝑟(.) indicates trace of a matrix. 𝜆 𝑚𝑖𝑛 (𝐴) and 𝜆 𝑚𝑎𝑥 (𝐴) denote the minimum and maximum eigenvalues of matrix 𝐴 respectively. 𝐼 is the identity matrix of appropriate dimension. 3.2 Preliminaries and Problem Formulation Preliminaries Consider 𝑦¤ (𝑡) = 𝐹(𝑡, 𝑦), 𝑦(0) = 𝑦 0 , (3.1) where 𝑦 ∈ D 𝑦 , 𝐹 : R+ × D 𝑦 ↦→ D 𝑦 , is a nonlinear function on the open neighborhood D 𝑦 of the origin. Assume the origin is an equilibrium point of (3.1). Definition 3 [29] The bounded signal 𝑦(𝑡) is said to be persistently exciting if there exist positive R 𝑡+T scalars 𝜇1 , 𝜇2 and T ∈ R+ such that ∀𝑡 ∈ R+ , 𝜇1 𝐼 ≤ 𝑡 𝑦(𝜏)𝑦𝑇 (𝜏)𝑑𝜏 ≤ 𝜇2 𝐼. 40 Definition 4 [65] The system (3.1) is said to be 1) fixed-time stable, if it is asymptotically stable and ∀𝑦 0 ∈ D 𝑦 any solution 𝑦(𝑡) of (3.1) reaches the equilibrium point at some finite-time moment, i.e., 𝑦(𝑡) = 0, ∀𝑡 ≥ 𝑇(𝑦 0 ), where 𝑇 : D 𝑦 ↦→ R+ ∪{0} is the settling-time function and the settling-time function 𝑇(𝑦 0 ) is bounded, i.e., ∃𝑇𝑚𝑎𝑥 > 0 : 𝑇(𝑦 0 ) ≤ 𝑇𝑚𝑎𝑥 , ∀𝑦 0 ∈ D 𝑦 . 2) fixed-time attractive by a bounded set Y around zero, if ∀𝑦 0 ∈ D 𝑦 any solution 𝑦(𝑡) of (1) reaches Y in some finite-time moment 𝑡 = 𝑇(𝑦 0 ) and remains there, ∀𝑡 ≥ 𝑇(𝑦 0 ), 𝑇 : D 𝑦 ↦→ R+ ∪ {0} is the settling-time function and the settling-time function 𝑇(𝑦 0 ) is bounded by some 𝑇𝑚𝑎𝑥 > 0. Lemma 2 [65] Let there exist a continuous positive definite function 𝑉 : D 𝑦 ↦→ R+ ∪ {0} in an open neighborhood of the origin and real positive numbers 𝛼, 𝛽, 𝑟 1 , 𝑟 2 > 0 such that 0 < 𝑟 1 < 1 and 1 < 𝑟 2 . Let also any solution 𝑦(𝑡) of (3.1) satisfy the inequality, ¤ 𝑉(𝑦(𝑡)) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝑡)) − 𝛽𝑉 𝑟 2 (𝑦(𝑡)). (3.2) Then, the system (3.1) is fixed-time stable and 1 1 𝑇(𝑦 0 ) ≤ + . (3.3) 𝛼(1 − 𝑟 1 ) 𝛽(𝑟 2 − 1) Fact 1. In general, ∀𝑥 ∈ R𝑛 with 0 < 𝑟 < 𝑠, one has 1 1 k𝑥k 𝑠 ≤ k𝑥k 𝑟 ≤ 𝑛 𝑟 − 𝑠 k𝑥k 𝑠 . This is a consequence of Hölder inequality [129]. Problem Formulation Consider the following nonlinear system, ¤ = 𝑓 (𝑥(𝑡)) + 𝑔(𝑥(𝑡))𝑢(𝑡), 𝑥(𝑡) (3.4) where 𝑥 ∈ D𝑥 ⊂ R𝑛 and 𝑢 ∈ D𝑢 ⊂ R𝑚 are the state and input vectors, respectively; D𝑥 and D𝑢 are compact sets. Let 𝑓 : D𝑥 ↦→ R𝑛 and 𝑔 : D𝑥 ↦→ R𝑛×𝑚 be the unknown nonlinear system and input dynamics, respectively. The overarching objective of this chapter is to present novel model learning approaches to learn these uncertain dynamics in a fixed time. Assumption 3 𝑥(𝑡) is a measurable state vector, and 𝑓 (𝑥(𝑡)) and 𝑔(𝑥(𝑡)) are both locally Lipschitz in 𝑥(𝑡). 41 Here, linearly parameterized adaptive approximation models [10] are used to, respectively, represent 𝑓 (𝑥(𝑡)) and 𝑔(𝑥(𝑡)) as follows, 𝑓 (𝑥(𝑡)) = 𝑓ˆ(𝑥(𝑡), Θ∗𝑓 ) + 𝑒 𝑓 (𝑥(𝑡)), (3.5) 𝑔(𝑥(𝑡)) = 𝑔ˆ(𝑥(𝑡), Θ∗𝑔 ) + 𝑒 𝑔 (𝑥(𝑡)), (3.6) where 𝑓ˆ(𝑥(𝑡), Θ∗𝑓 ) = Θ∗𝑇𝑓 𝜑(𝑥(𝑡)), (3.7) 𝑔ˆ(𝑥(𝑡), Θ∗𝑔 ) = Θ∗𝑇𝑔 𝜒(𝑥(𝑡)). (3.8) The matrices Θ 𝑓 ∗ ∈ D 𝑓 ⊂ R 𝑝×𝑛 and Θ𝑔 ∗ ∈ D𝑔 ⊂ R𝑞×𝑛 denote the unknown optimal parameters of the adaptive approximation models, defined as follows Θ∗𝑓 = arg min { 𝑠𝑢 𝑝 𝑓 (𝑥(𝑡)) − 𝑓ˆ(𝑥(𝑡), Θ 𝑓 ) }, (3.9) Θ 𝑓 ∈D 𝑓 𝑥(𝑡)∈D𝑥 Θ∗𝑔 = arg min { 𝑠𝑢 𝑝 𝑔(𝑥(𝑡)) − 𝑔ˆ(𝑥(𝑡), Θ𝑔 ) }, (3.10) Θ𝑔 ∈D𝑔 𝑥(𝑡)∈D𝑥 where D 𝑓 and D𝑔 are compact sets. The vectors 𝜑 : D𝑥 ↦→ R 𝑝 and 𝜒 : D𝑥 ↦→ R𝑞 , denote the basis functions, while 𝑝 and 𝑞 are the number of linearly independent basis functions for approximating 𝑓 (𝑥(𝑡)) and 𝑔(𝑥(𝑡)), respectively. The quantities 𝑒 𝑓 (𝑥(𝑡)) ∈ R𝑛 and 𝑒 𝑔 (𝑥(𝑡)) ∈ R𝑛×𝑚 are the MFAEs for 𝑓 (𝑥(𝑡)) and 𝑔(𝑥(𝑡)), respectively, representing the residual approximation error in the case of optimal parameters. In the special case that the unknown functions 𝑓 (𝑥(𝑡)) and 𝑔(𝑥(𝑡)) can be approximated exactly by the adaptive approximation models 𝑓ˆ(𝑥(𝑡), Θ 𝑓 ) and 𝑔ˆ(𝑥(𝑡), Θ𝑔 ), respectively, then 𝑒 𝑓 (𝑥(𝑡)) = 𝑒 𝑔 (𝑥(𝑡)) = 0. Using (3.5)-(3.8), the system dynamics (3.4) can be written as ¤ = Θ∗𝑇 𝑧(𝑥(𝑡), 𝑢(𝑡)) + 𝜀(𝑥(𝑡), 𝑢(𝑡)), 𝑥(𝑡) (3.11) 42 where Θ∗ = [Θ∗𝑇 , Θ∗𝑇 𝑇 (𝑝+𝑞)×𝑛 , 𝑧(𝑥(𝑡), 𝑢(𝑡)) = [𝜑𝑇 (𝑥(𝑡)), 𝑢𝑇 (𝑡)𝜒𝑇 (𝑥(𝑡))]𝑇 ∈ R(𝑝+𝑞) , and 𝑓 𝑔 ] ∈R 𝜀(𝑥(𝑡), 𝑢(𝑡)) = 𝑒 𝑓 (𝑥(𝑡)) + 𝑒 𝑔 (𝑥(𝑡))𝑢(𝑡). Assumption 4 For the given compact sets D𝑥 and D𝑢 , the approximators’ basis functions are bounded. Moreover, the approximation error 𝜀(𝑥(𝑡), 𝑢(𝑡)) is bounded by an upper bound 𝑏 𝜀 ≥ 0, i.e., sup k𝜀(𝑥, 𝑢)k≤ 𝑏 𝜀 𝑥∈D𝑥 ,𝑢∈D𝑢 . Remark 12 Assumption 3 ensures the existence and uniqueness of the solution of system (3.4) and Assumption 4 is standard in the literature based on universal approximator characteristics [29]. ¤ is not available for measurement, a regressor filtering method [39] is In this chapter, since, 𝑥(𝑡) used to obviate its requirement in the presented update law. Therefore, to proceed with regressor filtering, dynamics (3.11) is written as ¤ = −𝐶𝑥(𝑡) + Θ∗𝑇 𝑧(𝑥(𝑡), 𝑢(𝑡)) + 𝐶𝑥(𝑡) + 𝜀(𝑡), 𝑥(𝑡) (3.12) where 𝐶 = 𝑐𝐼, 𝑐 > 0. The solution of (3.11) can be expressed as 𝑥(𝑡) =Θ∗𝑇 𝑑(𝑡) + 𝐶𝑙(𝑡) + 𝑒 −𝐶𝑡 𝑥(0) + 𝜀 𝑓 (𝑡), (3.13) ¤ = −𝑐𝑑(𝑡) + 𝑧(𝑥(𝑡), 𝑢(𝑡)), 𝑑(0) = 0, 𝑑(𝑡) ¤ = −𝐶𝑙(𝑡) + 𝑥(𝑡), 𝑙(0) = 0, 𝑙(𝑡) (3.14) R𝑡 R𝑡 where 𝑙(𝑡) = 0 𝑒 −𝐶(𝑡−𝜏) 𝑥(𝜏)𝑑𝜏 is the filtered regressor of 𝑥(𝑡), 𝑑(𝑡) = 0 𝑒 −𝑐(𝑡−𝜏) 𝑧(𝑥(𝜏), 𝑢(𝜏))𝑑𝜏 R𝑡 is the filtered regressor of 𝑧(𝑥(𝑡), 𝑢(𝑡)), 𝜀 𝑓 (𝑡) = 0 𝑒 −𝐶(𝑡−𝜏) 𝜀(𝜏)𝑑𝜏, and 𝑥(0) is the initial state of (3.12). Dividing (3.13) by the normalizing signal 𝑛 𝑠 = 1 + 𝑑𝑇 (𝑡)𝑑(𝑡) + 𝑙 𝑇 (𝑡)𝑙(𝑡), one has, 𝑥¯(𝑡) =Θ∗𝑇 𝑑(𝑡) ¯ + 𝑒 −𝐶𝑡 𝑥¯(0) + 𝜀¯(𝑡), ¯ + 𝑐 𝑙(𝑡) (3.15) 𝜀 𝑓 where 𝑑¯ = 𝑛𝑑𝑠 , 𝑙¯ = 𝑛𝑙𝑠 , 𝑥¯ = 𝑛𝑥𝑠 and 𝜀¯ = 𝑛 𝑠 . Note that Assumption 4 implies that 𝜀¯(𝑡) is also bounded, i.e., sup k 𝜀¯(𝑥, 𝑢)k≤ 𝑏 𝜀¯ 𝑥∈D𝑥 ,𝑢∈D𝑢 43 ¯ and k 𝑑(𝑡)k< 1. Now, let the approximator of (3.15) for the system (3.4) be of the form ¯ + 𝑐 𝑙(𝑡) 𝑥ˆ¯(𝑡) =Θ̂𝑇 (𝑡)𝑑(𝑡) ¯ + 𝑒 −𝐶𝑡 𝑥¯(0), (3.16) where Θ̂(𝑡) = [Θ̂𝑇𝑓 (𝑡), Θ̂𝑇𝑔 (𝑡)]𝑇 ∈ R(𝑝+𝑞)×𝑛 , Θ̂ 𝑓 (𝑡) and Θ̂𝑔 (𝑡) are, respectively, the estimation of parameters matrices Θ∗ , Θ∗𝑓 and Θ∗𝑔 at time 𝑡. The state estimation error, 𝑒(𝑡), for system (3.4) is defined as ¯ − 𝜀¯(𝑡), 𝑒(𝑡) = 𝑥ˆ¯(𝑡) − 𝑥¯(𝑡) = Θ̃𝑇 (𝑡)𝑑(𝑡) (3.17) where Θ̃(𝑡) := Θ̂(𝑡) − Θ∗ := [Θ̃𝑇𝑓 (𝑡), Θ̃𝑇𝑔 (𝑡)]𝑇 is the parameter estimation error with Θ̃ 𝑓 (𝑡) := Θ̂ 𝑓 (𝑡) − Θ∗𝑓 , Θ̃𝑔 (𝑡) := Θ̂𝑔 (𝑡) − Θ∗𝑔 . To fulfill the fixed-time learning of the uncertainties 𝑓 (𝑥) and 𝑔(𝑥) in the system (3.4) without the requirement of the PE condition on the stream of data, a fixed-time CL method is presented next to guarantee that the parameter estimation error Θ̃(𝑡) dynamics are: 1) fixed-time stable for adaptive approximators with zero MFAE. 2) fixed-time attractive within a bounded set around zero for adaptive approximators with non-zero MFAE. 3.3 Fixed-time Concurrent Learning Identifier In this section, a novel fixed-time update law is presented to approximate the uncertainties of the system (3.4) that leverages the CL in its adaptive update law to eliminate the requirement of the PE condition. In this section, the introduced update law employs discontinuous gradient flows of the estimation errors to optimize the estimation error for current samples as well as samples collected in a recorded data stack, and the convergence analysis of the dynamics of the gradient update law is presented based on fixed-time Lyapunov stability. To employ the CL technique, which uses recorded experienced data along with current data in the update law, the past data is collected and stored in the memory stacks 𝑀 ∈ R(𝑝+𝑞)×𝑃 , 44 𝐿 ∈ R𝑛×𝑃 and 𝑋 ∈ R𝑛×𝑃 , at times 𝜏1 , ..., 𝜏𝑃 as, 𝑀 = [𝑑(𝜏 ¯ 1 ), 𝑑(𝜏 ¯ 2 ), ..., 𝑑(𝜏 ¯ 𝑃 )], 𝐿 = [𝑙(𝜏 ¯ 1 ), 𝑙(𝜏 ¯ 2 ), ..., 𝑙(𝜏 ¯ 𝑃 )], 𝑋 = [¯ 𝑥 (𝜏1 ), 𝑥¯(𝜏2 ), ..., 𝑥¯(𝜏𝑃 )], (3.18) where 𝑃 is the number of data points stored in every stack. The number of data points 𝑃 is chosen so that 𝑀 contains as many linearly independent elements as the dimension of 𝑑(𝑡) (i.e., the total number of linearly independent basis functions for 𝑓 (𝑥(𝑡)) and 𝑔(𝑥(𝑡))), given in (3.13). That is the rank of 𝑀 must be 𝑝 + 𝑞 which requires 𝑃 ≥ 𝑝 + 𝑞. Define the error 𝑒 ℎ (𝑡) for the ℎ𝑡ℎ recorded sample as 𝑒 ℎ (𝑡) = 𝑥ˆ¯ℎ (𝑡) − 𝑥¯(𝜏ℎ ), (3.19) where 𝑥ˆ¯ℎ (𝑡) =Θ̂𝑇 (𝑡)𝑑(𝜏 ¯ ℎ ) + 𝑒 −𝐶𝑡 𝑥¯(0), ¯ ℎ ) + 𝑐 𝑙(𝜏 (3.20) is the state estimation at time 0 ≤ 𝜏ℎ < 𝑡, ℎ = 1, ..., 𝑃, using the current estimated parameters matrix Θ̂(𝑡) and the recorded 𝑑(𝜏 ¯ ℎ ) and 𝑙(𝜏 ¯ ℎ ). Substituting 𝑥¯(𝜏ℎ ), from (3.15), in (3.19) leads to 𝑒 ℎ (𝑡) = Θ̃𝑇 (𝑡)𝑑(𝜏 ¯ ℎ ) − 𝜀¯(𝜏ℎ ). (3.21) In the proposed FxTCL method that is presented next, the stored data in 𝑀 is selected based on data recording algorithm in [34, 36] to maximize 𝜆 𝑚𝑖𝑛 (𝑆) 𝜆 𝑚𝑎𝑥 (𝑆) P𝑃 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ ). where 𝑆 = ℎ=1 𝑑(𝜏 Fixed-time concurrent learning update law The proposed fixed-time CL update law for the parameters in the system approximator (3.16) is given as X 𝑃 ¤̂ ¯ 𝑇 𝛾 1 𝑇 𝛾 2 ¯ ℎ )(⌊𝑒𝑇 (𝑡)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑡)⌉ 𝛾2 )], (3.22) Θ(𝑡) = −Γ[Ξ𝐺 𝑑(𝑡)(⌊𝑒 (𝑡)⌉ + ⌊𝑒 (𝑡)⌉ ) + Ξ𝐶 𝑑(𝜏 ℎ ℎ ℎ=1 45 where ⌊.⌉ 𝛾 := |.| 𝛾 𝑠𝑖𝑔𝑛(.) with |.| and 𝑠𝑖𝑔𝑛(.) understood in component-wise sense and 0 ≤ 𝛾1 < 1, 𝛾2 > 1. The matrices Γ, Ξ𝐺 , Ξ𝐶 ∈ R(𝑝+𝑞)×(𝑝+𝑞) , Γ > 0 is the positive definite learning rate matrix, Ξ𝐶 = 𝜉𝐶 𝐼 and Ξ𝐺 = 𝜉𝐺 𝐼 with positive constants 𝜉𝐶 > 0 and 𝜉𝐺 > 0. The above update law has two learning terms, the first term Ξ𝐺 𝑑(𝑡)(⌊𝑒¯ 𝑇 (𝑡)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑡)⌉ 𝛾2 ) containing the current state approximation error that is a nonlinear gradient descent term, and the second term, P ¯ ℎ )(⌊𝑒𝑇 (𝑡)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑡)⌉ 𝛾2 ), contains the experienced data, is the CL term. The Ξ𝐶 𝑃 ℎ=1 𝑑(𝜏 ℎ ℎ weights Ξ𝐶 and Ξ𝐺 do not need to be equal and by setting appropriate 𝜉𝐶 and 𝜉𝐺 , respectively, one of the two learning terms can be prioritized over the other. Remark 13 In the update law (3.22) by leveraging CL technique, discontinuous gradient flows of the current and stored identification errors are concurrently employed to, respectively, minimize the estimation error for the current stream of data and recorded memory samples. Discontinuous gradient-based adaptation of past data enables the update law (3.22) to converge to the optimal parameters in a fixed time regardless of the initial parameters’ estimation error. Therefore, given the recorded data, the fixed time of convergence can be computed a priori in this method. The convergence properties of the proposed method are investigated for adaptive approximators with zero and non-zero MFAEs in the following. Fixed-time Convergence Properties for Adaptive Approximators with Zero MFAEs (¯ 𝜀 (𝑡) = 0) The following theorem demonstrates the fixed-time convergence of the estimated parameters to their optimal values for the proposed FxTCL method (3.22), in adaptive approximators with zero MFAEs, i.e., 𝜀¯(𝑡) = 0. Theorem 3 Consider the approximator for nonlinear system (3.4) given in (3.16), whose parameters are adjusted according to the update law of (3.22) with 0 ≤ 𝛾1 < 1, 𝛾2 > 1 and a regressor given in (3.14). Let Assumptions 3-4 hold. Once the rank condition on 𝑀 is met, then for adaptive approximators with zero MFAEs, 𝜀¯(𝑡) = 0, the proposed update law (3.22) guarantees the fixed-time convergence of Θ̃(𝑡) to zero for 𝑡 > 𝑇 where the settling-time is bounded by 𝑇 ≤ 𝑇𝑚𝑎𝑥 46 and 2 2 𝑇𝑚𝑎𝑥 = 𝛾1 +1 + 𝛾2 +1 , (3.23) 𝛼1 𝑐 2 2 (1 − 𝛾1 ) 𝛼2 𝑐 2 2 (𝛾2 − 1) 𝛾1 +1 1−𝛾2 𝛾2 +1 such that 𝑐 2 = 2𝜆 𝑚𝑖𝑛 (Γ), 𝛼1 = 𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆), 𝛼2 = 𝜉𝐶 𝑛 2 𝜆 𝑚𝑖𝑛2 (𝑆). 2 Proof 5 Consider the Lyapunov function candidate 1 𝑉(𝑡) = 𝑡𝑟{Θ̃𝑇 (𝑡)Γ−1 Θ̃(𝑡)}. (3.24) 2 We know that 𝑐−1 2 −1 2 1 k Θ̃(𝑡)k ≤𝑉(𝑡) ≤ 𝑐 2 k Θ̃(𝑡)k , (3.25) where 𝑐 1 = 2𝜆 𝑚𝑎𝑥 (Γ), 𝑐 2 = 2𝜆 𝑚𝑖𝑛 (Γ). The time derivative of 𝑉 using (3.17), (3.21) and (3.22) yields, ¤̂ ¤ 𝑉(𝑡) =𝑡𝑟{Θ̃𝑇 (𝑡)Γ−1 Θ(𝑡)} X𝑃 ¯ =𝑡𝑟{ − Ξ𝐺 Θ̃𝑇 (𝑡)𝑑(𝑡)(⌊ 𝑑¯𝑇 (𝑡)Θ̃(𝑡)⌉ 𝛾1 + ⌊ 𝑑¯𝑇 (𝑡)Θ̃(𝑡)⌉ 𝛾2 ) − Ξ𝐶 Θ̃𝑇 (𝑡) ¯ ℎ )(⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)⌉ 𝛾1 𝑑(𝜏 ℎ=1 + ⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)⌉ 𝛾2 )}. (3.26) One knows that X𝑛 ¯ 𝑑¯𝑇 (𝑡)Θ̃(𝑡)⌉ 𝛾1 = Θ̃𝑇 (𝑡)𝑑(𝑡)⌊ ¯ 𝑖 | 𝛾1 +1 |(Θ̃𝑇 (𝑡)𝑑(𝑡)) 𝑖=1 ¯ 𝛾 +1 = k Θ̃𝑇 (𝑡)𝑑(𝑡)k 1 𝛾 +1 , (3.27) 1 and using Fact 1 ¯ k Θ̃𝑇 (𝑡)𝑑(𝑡)k≤ ¯ 𝛾 +1 , k Θ̃𝑇 (𝑡)𝑑(𝑡)k (3.28) 1 holds for 0 < 𝛾1 + 1 < 2. By using (3.27), (3.28) and 𝛾2 −1 𝑇 ¯ k Θ̃ (𝑡)𝑑(𝑡)k≤ 𝑛 2 +1) k Θ̃𝑇 (𝑡)𝑑(𝑡)k 2(𝛾 ¯ 𝛾 +1 , (3.29) 2 47 that holds based on Fact 1 for 0 < 𝛾1 + 1 < 2 < 𝛾2 + 1, one obtains, 1−𝛾2 X𝑃 ¤ 𝑉(𝑡) ¯ 𝛾1 +1 +𝑛 2 k Θ̃𝑇 (𝑡)𝑑(𝑡)k ≤ − 𝜉𝐺 (k Θ̃𝑇 (𝑡)𝑑(𝑡)k ¯ 𝛾2 +1 ) − 𝜉𝐶 ( ¯ ℎ )k 𝛾1 +1 k Θ̃𝑇 (𝑡)𝑑(𝜏 ℎ=1 1−𝛾2 X 𝑃 +𝑛 2 k Θ̃𝑇 (𝑡)𝑑¯𝑇 (𝜏ℎ )k 𝛾2 +1 ). ℎ=1 Thus, X𝑃 𝛾1 +1 1−𝛾2 X 𝑃 𝛾2 +1 ¤ 𝑉(𝑡) ≤ −𝜉𝐶 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)) 2 (Θ̃𝑇 (𝑡)𝑑(𝜏 − 𝜉𝐶 𝑛 2 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)) 2 . (Θ̃𝑇 (𝑡)𝑑(𝜏 ℎ=1 ℎ=1 (3.30) One can rewrite (3.30) as follows, ¤ 𝑉(𝑡) ≤ −𝛼1 k Θ̃(𝑡)k 𝛾1 +1 −𝛼2 k Θ̃(𝑡)k 𝛾2 +1 , (3.31) 𝛾1 +1 1−𝛾2 𝛾2 +1 P where 𝛼1 = 𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆), 𝛼2 = 𝜉𝐶 𝑛 2 𝜆 𝑚𝑖𝑛2 (𝑆) and since, 𝑆 = 𝑃 2 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ ) > 0, we 𝑑(𝜏 ℎ=1 have 𝛼1 > 0 and 𝛼2 > 0. Employing (3.25), (3.31) gives 𝛾1 +1 𝛾 +1 𝛾2 +1 𝛾 +1 1 2 ¤ 𝑉(𝑡) ≤ −𝛼1 𝑐 2 2 𝑉 2 (𝑡) − 𝛼2 𝑐 2 2 𝑉 2 (𝑡). (3.32) Let us introduce the following inequalities 𝑉 𝛾2 +1 (𝑡) ≤ 𝑉 𝛾1 +1 (𝑡) ≤ 𝑉(𝑡) , ∀𝑉(𝑡) ≤ 1, (3.33) 𝑉 𝛾2 +1 (𝑡) > 𝑉 𝛾1 +1 (𝑡) > 𝑉(𝑡) , ∀𝑉(𝑡) > 1. (3.34) Hence, from (3.25), (3.32), and (3.34), when 𝑉(𝑡) > 1, one obtains 𝛾2 +1 𝛾 +1 2 𝑉(𝑡) ≤ −𝛼2 𝑐 2 2 𝑉 2 (𝑡). ¤ (3.35) Thus, for any Θ̃(𝑡) such that 𝑉(Θ̃(0)) > 1, (3.35) ensures 𝑉(Θ̃(𝑡)) ≤ 1 for all 𝑡 ≥ 𝑇1 = 2 . 𝛾2 +1 𝛼2 𝑐 2 (𝛾2 −1) 2 48 Then when 𝑉(𝑡) ≤ 1, using inequality (3.33) for every 𝛾2 + 1 > 𝛾1 + 1 > 1, it follows from (3.32) that, 𝛾1 +1 𝛾 +1 1 𝑉(𝑡) ≤ −𝛼1 𝑐 2 2 𝑉 2 (𝑡). ¤ (3.36) and we derive 𝑉(Θ̃(𝑡)) = 0 for 𝑡 ≥ 𝑇2 where 𝑇2 = 2 . Therefore, 𝑉(Θ̃(𝑡)) = 0 for 𝛾1 +1 𝛼1 𝑐 2 (1−𝛾1 ) 2 ∀𝑡 ≥ 𝑇𝑚𝑎𝑥 where 2 2 𝑇𝑚𝑎𝑥 = 𝑇1 + 𝑇2 = 𝛾1 +1 + 𝛾2 +1 , 𝛼1 𝑐 2 2 (1 − 𝛾1 ) 𝛼2 𝑐 2 2 (𝛾2 − 1) and it implies that Θ̃(𝑡) = 0 for ∀𝑡 ≥ 𝑇𝑚𝑎𝑥 . Fixed-time Convergence Properties for Adaptive Approximators with Non-zero MFAEs 𝜀 (𝑡) 6= 0) (¯ The fixed-time convergence properties of the proposed FxTCL update law for adaptive approx- imators with non-zero MFAEs, 𝜀¯(𝑡) 6= 0, are given in the next theorem. Theorem 4 Consider the approximator for nonlinear system (3.4), given in (3.16), with param- eters adjusted by the update law of (3.22) with 0 ≤ 𝛾1 < 1, 𝛾2 > 1 and a regressor given in (3.14). Let Assumptions 3-4 and the rank condition on 𝑀 hold. Then, for adaptive approximators with non-zero MFAEs, the proposed update law (3.22) guarantees that 1) for 𝛾1 = 0, if s s 2𝜆 𝑚𝑎𝑥 (Γ) 𝜆 𝑚𝑎𝑥 (Γ) min{𝛼4 , 𝛼3 } max{ , }< (2𝜆 𝑚𝑖𝑛 (Γ))𝛾2 +1 𝜆 𝑚𝑖𝑛 (Γ) 𝜔 , then Θ̃(𝑡) is fixed-time convergent to zero for 𝑡 > 𝑇 and 𝑇 ≤ 𝑇𝑚𝑎𝑥 with 2 2 𝑇𝑚𝑎𝑥 = + √ √ , (3.37) 𝛼(𝛾2 − 1) 𝛼3 𝑐 2 − 𝜔 𝑐 1 r 2𝜆 𝑚𝑎𝑥 (Γ) 𝛼 2) for 0 < 𝛾1 < 1, if 𝛾 +1 < 𝜔4 , then Θ̃(𝑡) is fixed-time attractive with the following (2𝜆 𝑚𝑖𝑛 (Γ)) 2 bound s q 𝜆 𝑚𝑎𝑥 (Γ) k Θ̃(𝑡)k≤ 𝑚𝑖𝑛{ 2𝜆 𝑚𝑎𝑥 (Γ), 𝜇¯}, ∀𝑡 ≥ 𝑇, (3.38) 𝜆 𝑚𝑖𝑛 (Γ) 49 such that 𝑇 ≤ 𝑇𝑚𝑎𝑥 , √ 2 2(1 − (𝑐−0.52 min( 𝜇¯, 𝑐 1 ))1−𝛾1 ) 𝑇𝑚𝑎𝑥 = + 𝛾1 +1 , (3.39) 𝛼(𝛾2 − 1) 𝛼3 (1 − 𝛿)(1 − 𝛾1 )𝑐 2 2  1  𝑏 𝜀¯ ¯ 6= 0,    𝑚𝑎𝑥{ , ( 𝛼𝜔𝛿 ) 𝛾1 }, 𝑑(𝑡)    1 3 𝜇¯ = min 𝜆 2 (𝐷(𝑡)),𝜆¯ℎ (3.40)  𝑚𝑖𝑛  1  𝑏 𝜀¯ 𝜔 ¯ = 0,   𝑚𝑎𝑥{ 𝜆¯ , ( 𝛼 𝛿 ) 𝛾1 }, 𝑑(𝑡)  ℎ 3 where 1 𝛾2 +1 ¯ ¯𝑇 √ 𝜆¯ℎ = min 𝜆 𝑚𝑖𝑛 (𝐷(𝜏ℎ ))}, 𝐷(𝑡) = 𝑑(𝑡)𝑑 (𝑡), 𝛼 = 𝛼3 𝑐 2 2 − 𝜔 𝑐 1 , 2 ℎ=1,...,𝑃 𝛾1 +1 𝛾1 +1 1−𝛾2 𝛾2 +1 𝛾2 +1 𝛼3 = 𝜉𝐺 𝜆 𝑚𝑖𝑛2 (𝐷(𝑡)) + 𝜉𝐶 𝜆 𝑚𝑖𝑛2 (𝑆), 𝛼4 = 21−𝛾2 𝑛 2 (𝜉𝐺 𝜆 𝑚𝑖𝑛2 (𝐷(𝑡)) + 𝜉𝐶 𝜆 𝑚𝑖𝑛2 (𝑆)), 2−𝛾1 𝛾 𝛾 𝜔 = (𝜉𝐺 + 𝑃𝜉𝐶 )[𝑛 4 𝑏 𝜀¯1 + 𝑏 𝜀¯2 ]. Proof 6 Consider the Lyapunov function candidate (3.24) that satisfies (3.25). The time derivative of 𝑉 using (3.17), (3.21) and (3.22) yields, ¤ 𝑉(𝑡) ¯ =𝑡𝑟{ − Ξ𝐺 Θ̃𝑇 (𝑡)𝑑(𝑡)(⌊ 𝑑¯𝑇 (𝑡)Θ̃(𝑡) − 𝜀¯𝑇 (𝑡)⌉ 𝛾1 + ⌊ 𝑑¯𝑇 (𝑡)Θ̃(𝑡) − 𝜀¯𝑇 (𝑡)⌉ 𝛾2 ) X𝑃 − Ξ𝐶 Θ̃𝑇 (𝑡) ¯ ℎ )(⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡) − 𝜀¯𝑇 (𝜏ℎ )⌉ 𝛾1 + ⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡) − 𝜀¯𝑇 (𝜏ℎ )⌉ 𝛾2 )}. (3.41) 𝑑(𝜏 ℎ=1 Consider in the component-wise sense that |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 |≥ | 𝜀¯𝑖 (𝑡)|, for 𝑖 = 1, ..., 𝑛. It is worth mentioning that the last inequality is required when 𝑑(𝑡) ¯ 6= 0. Because, if 𝑑(𝑡) ¯ = 0 then the first term in (3.22) will be zero and in the second term of (3.22), called the CL term, the data collection algorithm ensures that 𝑑(𝜏 ¯ ℎ ) 6= 0, ℎ = 1, ..., 𝑃. Therefore, 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝑡)Θ̃(𝑡) − 𝜀¯𝑇 (𝑡)) = 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝑡)Θ̃(𝑡)) is implied. For any 𝑦 1 , 𝑦 2 ∈ R and 0 ≤ 𝛾1 < 1, the following inequality holds [129], |𝑦 1 + 𝑦 2 | 𝛾1 ≤ |𝑦 1 | 𝛾1 +|𝑦 2 | 𝛾1 . Therefore, defining 𝑦 1 = (𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡) and 𝑦 2 = 𝜀¯𝑖 (𝑡), one obtains that for all 𝑖 = 1, ..., 𝑛 |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 | 𝛾1 = |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡) + 𝜀¯𝑖 (𝑡)| 𝛾1 ≤|(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡)| 𝛾1 +| 𝜀¯𝑖 (𝑡)| 𝛾1 ⇒ |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 | 𝛾1 −| 𝜀¯𝑖 (𝑡)| 𝛾1 ≤|(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡)| 𝛾1 , 50 and then in the component-wise sense, −| 𝑑¯𝑇 (𝑡)Θ̃(𝑡) − 𝜀¯(𝑡)| 𝛾1 ≤ −| 𝑑¯𝑇 (𝑡)Θ̃(𝑡)| 𝛾1 +| 𝜀¯(𝑡)| 𝛾1 . (3.42) For 𝛾2 > 1, the following inequality [129] holds, |𝑦 1 + 𝑦 2 | 𝛾2 ≤ 2𝛾2 −1 (|𝑦 1 | 𝛾2 +|𝑦 2 | 𝛾2 ). Thus, for 𝑦 1 = (𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡) and 𝑦 2 = 𝜀¯𝑖 (𝑡), one has |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 | 𝛾2 = |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡) + 𝜀¯𝑖 (𝑡)| 𝛾2 ≤2𝛾2 −1 (|(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡)| 𝛾2 +| 𝜀¯𝑖 (𝑡)| 𝛾2 ) ⇒ |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 | 𝛾2 −2𝛾2 −1 | 𝜀¯𝑖 (𝑡)| 𝛾2 ≤2𝛾2 −1 |(𝑑¯𝑇 (𝑡)Θ̃(𝑡))𝑖 − 𝜀¯𝑖 (𝑡)| 𝛾2 , 𝑖 = 1, ..., 𝑛, and then component-wisely, for 𝛾2 > 1, it follows that −| 𝑑¯𝑇 (𝑡)Θ̃(𝑡) − 𝜀¯(𝑡)| 𝛾2 ≤ −21−𝛾2 | 𝑑¯𝑇 (𝑡)Θ̃(𝑡)| 𝛾2 +| 𝜀¯(𝑡)| 𝛾2 . (3.43) Now, using (3.42)-(3.43), 𝑉(𝑡) ¤ in (3.41) is upper bounded by ¤ 𝑉(𝑡) ≤ 𝑡𝑟{ − Ξ𝐺 Θ̃𝑇 (𝑡)𝑑(𝑡)(⌊ ¯ 𝑑¯𝑇 (𝑡)Θ̃(𝑡)⌉ 𝛾1 + 21−𝛾2 ⌊ 𝑑¯𝑇 (𝑡)Θ̃(𝑡)⌉ 𝛾2 ) ¯ + Ξ𝐺 Θ̃𝑇 (𝑡)𝑑(𝑡)(| 𝜀¯(𝑡)| 𝛾1 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝑡)Θ̃(𝑡)) + | 𝜀¯(𝑡)| 𝛾2 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝑡)Θ̃(𝑡))) X 𝑃 − Ξ𝐶 Θ̃𝑇 (𝑡) ¯ ℎ )(⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)⌉ 𝛾1 + 21−𝛾2 ⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)⌉ 𝛾2 ) 𝑑(𝜏 ℎ=1 X𝑃 + Ξ𝐶 Θ̃𝑇 (𝑡) ¯ ℎ )(| 𝜀¯(𝜏ℎ )| 𝛾1 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)) + | 𝜀¯(𝜏ℎ )| 𝛾2 𝑠𝑖𝑔𝑛(𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)))}. 𝑑(𝜏 ℎ=1 Therefore, using (3.27)-(3.29) and Fact 1, one obtains, X𝑃 ¤ 𝑉(𝑡) ¯ 𝛾1 +1 −𝜉𝐶 ≤ − 𝜉𝐺 k Θ̃𝑇 (𝑡)𝑑(𝑡)k ¯ ℎ )k 𝛾1 +1 +𝜉𝐺 k Θ̃𝑇 (𝑡)k(k| 𝜀¯(𝑡)| 𝛾1 k+k| 𝜀¯(𝑡)| 𝛾2 k) k Θ̃𝑇 (𝑡)𝑑(𝜏 ℎ=1 1−𝛾2 𝛾 1 + 𝜉𝐶 𝑃k Θ̃(𝑡)k(k| 𝜀¯(𝜏ℎ )| k+k| 𝜀¯(𝜏ℎ )| k) − 2 𝛾 2 1−𝛾 2 ¯ 𝛾2 +1 𝑛 2 (𝜉𝐺 k Θ̃𝑇 (𝑡)𝑑(𝑡)k X𝑃 + 𝜉𝐶 k Θ̃𝑇 (𝑡)𝑑¯𝑇 (𝜏ℎ )k 𝛾2 +1 ). (3.44) ℎ=1 q P𝑛 2𝛾1 = k 𝜀¯(𝑡)k 𝛾1 , and by using Fact 1, Since k| 𝜀¯(𝑡)| 𝛾1 k= 𝑖=1 | 𝜀¯𝑖 (𝑡)| 2𝛾1 1−𝛾1 k 𝜀¯(𝑡)k 2𝛾2 ≤ k 𝜀¯(𝑡)k, k 𝜀¯(𝑡)k 2𝛾1 ≤ 𝑛 2𝛾1 k 𝜀¯(𝑡)k, (3.45) 51 holds for all 0 < 2𝛾1 < 2 < 2𝛾2 . Using (3.45), (3.44) leads to 𝛾1 +1 X𝑃 𝛾1 +1 ¤ 𝑉(𝑡) ≤ −𝜉𝐺 (Θ̃𝑇 (𝑡)𝑑(𝑡)¯ 𝑑¯𝑇 (𝑡)Θ̃(𝑡)) 2 − 𝜉𝐶 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)) 2 (Θ̃𝑇 (𝑡)𝑑(𝜏 ℎ=1 1−𝛾1 1−𝛾1 + 𝜉𝐺 k Θ̃(𝑡)k(𝑛 2 k 𝜀¯(𝑡)k 𝛾1 +k 𝜀¯(𝑡)k 𝛾2 ) + 𝜉𝐶 𝑃k Θ̃(𝑡)k(𝑛 2 k 𝜀¯(𝜏ℎ )k 𝛾1 +k 𝜀¯(𝜏ℎ )k 𝛾2 ) 1−𝛾2 𝛾2 +1 X𝑃 𝛾2 +1 −2 1−𝛾 2 𝑛 2 𝑇 ¯ ¯𝑇 (𝜉𝐺 (Θ̃ (𝑡)𝑑(𝑡)𝑑 (𝑡)Θ̃(𝑡)) 2 + 𝜉𝐶 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑡)) 2 ). (Θ̃𝑇 (𝑡)𝑑(𝜏 ℎ=1 Therefore, ¤ 𝑉(𝑡) ≤ −𝛼3 k Θ̃(𝑡)k 𝛾1 +1 −𝛼4 k Θ̃(𝑡)k 𝛾2 +1 +𝜔k Θ̃(𝑡)k, (3.46) where 𝛾1 +1 𝛾1 +1 1−𝛾2 𝛾2 +1 𝛾2 +1 𝛼3 = 𝜉𝐺 𝜆 𝑚𝑖𝑛2 (𝐷(𝑡)) + 𝜉𝐶 𝜆 𝑚𝑖𝑛2 (𝑆), 𝛼4 = 21−𝛾2 𝑛 2 (𝜉𝐺 𝜆 𝑚𝑖𝑛2 (𝐷(𝑡)) + 𝜉𝐶 𝜆 𝑚𝑖𝑛2 (𝑆)), 1−𝛾1 𝛾 𝛾 𝜔 = (𝜉𝐺 + 𝑃𝜉𝐶 )[𝑛 2 𝑏 𝜀¯1 + 𝑏 𝜀¯2 ]. By using (3.25), (3.46) is written as, 𝛾1 +1 𝛾 +1 𝛾2 +1 𝛾 +1 1 2 √ 1 ¤ 𝑉(𝑡) ≤ − 𝛼3 𝑐 2 2 𝑉 2 (𝑡) − 𝛼4 𝑐 2 2 𝑉 2 (𝑡) + 𝜔 𝑐 1𝑉 2 (𝑡), (3.47) and employing inequality (3.34) when 𝑉(𝑡) > 1, one obtains 𝛾2 +1 𝛾 +1 2 √ 1 𝑉(𝑡) ≤ −𝛼4 𝑐 2 2 𝑉 2 (𝑡) + 𝜔 𝑐 1𝑉 2 (𝑡) ¤ 𝛾2 +1 𝛾 +1 𝛾2 +1 2 √ ≤ −𝛼4 𝑐 2 2 𝑉 2 (𝑡) + 𝜔 𝑐 1𝑉 2 (𝑡) 𝛾2 +1 ≤ −𝛼𝑉 2 (𝑡), (3.48) 𝛾2 +1 √ where 𝛼 = 𝛼4 𝑐 2 2 − 𝜔 𝑐 1 is positive if 𝛾2 +1 s √ 𝛼4 2𝜆 𝑚𝑎𝑥 (Γ) 𝛼4 𝑐 2 2 > 𝜔 𝑐 1 ⇒ > . (3.49) 𝜔 (2𝜆 𝑚𝑖𝑛 (Γ))𝛾2 +1 Thus, if (3.49) is met, for any Θ̃(𝑡) such that 𝑉(Θ̃(0)) > 1, (3.46) ensures 𝑉(Θ̃(𝑡)) ≤ 1 for all 𝑡 ≥ 𝑇3 = 𝛼(𝛾2 −1) . 2 52 1) If 𝛾1 = 0, for the case that 𝑉(𝑡) ≤ 1, using (3.33) and (3.47), one obtains √ 1 √ 1 ¤ 𝑉(𝑡) ≤ − 𝛼3 𝑐 2𝑉 2 (𝑡) + 𝜔 𝑐 1𝑉 2 (𝑡). (3.50) Therefore, satisfying s 𝛼3 𝜆 𝑚𝑎𝑥 (Γ) > , (3.51) 𝜔 𝜆 𝑚𝑖𝑛 (Γ) we have 𝑉(Θ̃(𝑡)) = 0 for 𝑡 ≥ 𝑇4 where 2 𝑇4 = √ √ . 𝛼3 𝑐 2 − 𝜔 𝑐 1 Therefore, for 𝛾1 = 0, once 𝑀 rank condition, (3.49) and (3.51) are satisfied then 𝑉(Θ̃(𝑡)) = 0 for 𝑡 ≥ 𝑇𝑚𝑎𝑥 where 2 2 𝑇𝑚𝑎𝑥 = 𝑇3 + 𝑇4 = + √ √ . 𝛼(𝛾2 − 1) 𝛼3 𝑐 2 − 𝜔 𝑐 1 This completes the proof of part 1. 2) If 0 < 𝛾1 < 1, for the case when 𝑉(𝑡) ≤ 1, using (46), one obtains ¤ 𝑉(𝑡) ≤ − 𝛼3 k Θ̃(𝑡)k 𝛾1 +1 +𝜔k Θ̃(𝑡)k ≤ − 𝛼3 (1 − 𝛿)k Θ̃(𝑡)k 𝛾1 +1 −𝛼3 𝛿k Θ̃(𝑡)k 𝛾1 +1 +𝜔k Θ̃(𝑡)k, where 0 < 𝛿 < 1. Hence, √ ¤ 𝑉(𝑡) ≤ −𝛼3 (1 − 𝛿)k Θ̃(𝑡)k 𝛾1 +1 , 𝜇¯ ≤ k Θ̃(𝑡)k≤ 𝑐 1 , (3.52) where  1  𝑏 𝜀¯ ¯ 6= 0,    𝑚𝑎𝑥{ , ( 𝛼𝜔𝛿 ) 𝛾1 }, 𝑑(𝑡)    1 3 𝜇¯ = min 𝜆 2 (𝐷(𝑡)),𝜆¯ℎ  𝑚𝑖𝑛  1  𝑏 𝜀¯ ¯ = 0.   𝑚𝑎𝑥{ 𝜆¯ , ( 𝛼𝜔𝛿 ) 𝛾1 }, 𝑑(𝑡)  ℎ 3 Using (3.25), (3.52) implies that 𝛾1 +1 𝛾 +1 1 𝑉(𝑡) ≤ −𝛼3 (1 − 𝛿)𝑐 2 2 𝑉 2 (𝑡), ¤ 53 and using comparison principle and (3.25), one obtains 𝛾1 +1 1−𝛾1 𝛼3 (1 − 𝛿)(1 − 𝛾1 )𝑐 2 2 2 𝑉(𝑡) ≤ (𝑉 2 (𝑇3 ) − 𝑡) 1−𝛾1 2 √ ≤ 𝑐−1 2 (𝑚𝑖𝑛( 𝜇¯, 𝑐 1 ))2 , then, the above inequality shows that Θ̃(𝑡) satisfies (3.38), for all 𝑡 > 𝑇𝑚𝑎𝑥 where 𝑇𝑚𝑎𝑥 = 𝑇3 + 𝑇5 and √ 2(1 − (𝑐−0.5 2 min( 𝜇¯, 𝑐 1 ))1−𝛾1 ) 𝑇5 = 𝛾1 +1 . 𝛼3 (1 − 𝛿)(1 − 𝛾1 )𝑐 2 2 Therefore, for 0 < 𝛾1 < 1, it is concluded that once 𝑀 rank condition and (3.49) are met then the solutions Θ̃(𝑡) are finite-time attractive by the bound given in (3.38) and √ 2 2(1 − (𝑐−0.5 2 min( 𝜇¯, 𝑐 1 ))1−𝛾1 ) 𝑇 ≤ + 𝛾1 +1 . 𝛼(𝛾2 − 1) 𝛼3 (1 − 𝛿)(1 − 𝛾1 )𝑐 2 2 This completes the proof. Remark 14 The convergence set (3.38) can be kept small by maximizing 𝜆 𝑚𝑖𝑛 (𝑆) that maximizes 𝛼3 and helps to minimize 𝜇¯ in (3.40). Furthermore, choosing 𝑃 = 𝑝 + 𝑞 (satisfying 𝑃 ≥ 𝑝 + 𝑞) 𝜆 (𝑆) helps to maximize 𝜆 𝑚𝑖𝑛 (𝑆) and minimize 𝜔. Moreover, maximizing 𝜆 𝑚𝑖𝑛 (𝑆) results in a faster 𝑚𝑎𝑥 convergence time for Θ̃(𝑡) as can be found in (3.23), (3.37), and (3.39). These results completely coincide with the obtained results in [34, 36]. Therefore, while applying the proposed FxTCL, the data recording algorithm in [34, 36] is used, where appropriate data is selected to maximize 𝜆 𝑚𝑖𝑛 (𝑆) 𝜆 𝑚𝑎𝑥 (𝑆) . Remark 15 In [81], for zero MFAE, it is shown that once the regressor satisfies the injectivity condition (analogous to the PE condition) for 𝛾1 = 0 and 𝛾2 > 1, the estimated parameters are fixed-time stable and for 𝛾1 ∈ (0, 1) and 𝛾2 > 1, the estimated parameters are just ultimately bounded. Moreover, the learning rate must satisfy some constraints and to check these constraints the online knowledge of the minimum and maximum singular values of the regressor and the upper bound of the unknown parameters are required which are hard to compute and check online. 54 However, by employing CL technique, in the proposed method for approximators with zero MFAE, it is shown that for 𝛾1 ∈ [0, 1) and 𝛾2 > 1, the estimated parameters are fixed-time stable and no constraint is imposed on the learning rates. Moreover, for both approximators with zero and nonzero MFAEs, no upper bound for the unknown parameters is required anymore. Above all, in sharp contrast to [81], employing the CL technique has eliminated the PE or injectivity requirement in the proposed FxTCL method for both approximators with zero and nonzero MFAEs. Comparison with other methods Three approaches will be considered in the next section for comparison. 1) Concurrent learning [36]: The asymptotically converging CL has the following update law, X𝑃 ¤̂ ¯ 𝑇 (𝑡) + Σ𝐶 ¯ ℎ )𝑒𝑇 (𝑡)), Θ(𝑡) = −Γ𝐶 (Σ𝐺 𝑑(𝑡)𝑒 𝑑(𝜏 ℎ (3.53) ℎ=1 where Γ𝐶 > 0, Σ𝐺 = 𝜎𝐺 𝐼, Σ𝐶 = 𝜎𝐶 𝐼 with positive constants 𝜎𝐺 > 0, 𝜎𝐶 > 0. In contrary to the proposed method (3.22), the update law (3.53) can just guarantee asymptotic convergence of the estimated parameters rather than the fixed-time convergence. 2) Fixed-time parameter estimation [77, 78, 81]: The fixed-time parameter estimation law [77, 78, 81] is as follows ¤̂ ¯ 𝑇 (𝑡)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑡)⌉ 𝛾2 ), Θ(𝑡) = −𝐾 𝑑(𝑡)(⌊𝑒 (3.54) for some 𝐾 > 0. In contrast to the proposed method (3.22), (3.54) requires the PE [77, 78] or injectivity [81] condition on the regressor to guarantee fixed-time convergence. However, the proposed FxTCL method (3.22) employs past recorded experienced data to obviate the PE requirement while update law (3.55) only employs current data and require PE or injectivity condition. 3) Finite-time concurrent learning [41]: The finite-time CL introduced in [41] uses the following update law, P𝑃 ¯ ¤̂ ′ X𝑃 ℎ=1 𝑑(𝜏ℎ )𝑒𝑇ℎ (𝑡) ¯ 𝑇 Θ(𝑡) = −Γ (𝐾1 𝑑(𝑡)𝑒 (𝑡) + 𝐾2 ¯ 𝑇 𝑑(𝜏ℎ )𝑒 ℎ (𝑡) + 𝐾3 P𝑃 ), (3.55) ℎ=1 k ℎ=1 𝑑(𝜏 ¯ ℎ )𝑒𝑇 (𝑡)k ℎ 55 where Γ′ > 0, 𝐾 𝑗 = 𝑘 𝑗 𝐼 > 0 with constant 𝑘 𝑗 > 0 for 𝑗 = 1, 2, 3. In contrast to the proposed method (3.22), the settling time of (3.55) depends on the initial parameter estimation error Θ̃(0) as follows p 2 𝛼′ k Θ̃(0)k+𝛽′ 2𝜆 𝑚𝑖𝑛 (Γ′) 𝑇 ≤ 𝑇𝑚𝑎𝑥 (Θ̃(0)) = ′ 𝑙𝑛 p , 𝛼 𝛽′ 2𝜆 𝑚𝑖𝑛 (Γ′) q 𝜆 𝑚𝑖𝑛 (𝑆) 𝛼′ = 2𝜆 𝑚𝑖𝑛 (Γ′)𝜆 𝑚𝑖𝑛 (𝑘 1 𝐷(𝑡) + 𝑘 2 𝑆), 𝛽′ = 𝑘 3 2𝜆 𝑚𝑖𝑛 (Γ′) . (3.56) 𝜆 𝑚𝑎𝑥 (𝑆) Remark 16 It is notable that to guarantee fixed-time convergence for update law (3.54), not only injectivity or PE condition is required but also its learning rate, 𝐾, must satisfy some constraints on the entire time of learning where the minimum and maximum singular values of the regressor and the upper bound of the unknown parameters are needed to be known. Moreover, CL (3.53) only guarantees the asymptotic convergence of the estimated parameters and finite-time CL (3.55) can only ensure that there is a finite-time of convergence that cannot be computed due to the dependence of the settling time on initial parameter estimation error. Therefore, the proposed fixed-time CL update law (3.22) that guarantees fixed-time convergence regardless of the initial parameter estimation error and does not require PE condition under the rank condition of recorded data, intuitively outperforms the previously mentioned methods. 3.4 Simulation Results In this section, the performance of the proposed fixed time CL is numerically examined in comparison with asymptotically converging CL, fixed-time parameter estimation, and finite-time CL, given by (3.53), (3.54) and (3.55), respectively. In the following examples, the 𝑥 domain is defined by D𝑥 = [𝑥 𝐿 , 𝑥 𝐻 ] where 𝑥 𝐿 = −2 and 𝑥 𝐻 = 2, initial values and the controllers are all set to zero and a small exponential sum of sinusoidal input is injected into the system controller to ensure the rank condition on the collected 𝜆 (𝑆) data where data selection procedure in [34, 36] for maximizing 𝜆 𝑚𝑖𝑛 (𝑆) is employed for CL, finite- 𝑚𝑎𝑥 time CL and the proposed fixed-time CL methods. To fairly compare the speed and precision of the mentioned online learning methods for approximating 𝑓ˆ(𝑥) and 𝑔ˆ(𝑥) on the whole domain of 56 𝑥 as time evolves, the following learning errors are computed online Z Z 𝐸 𝑓 (𝑡) = 𝑛 k𝑒 𝑓 (𝑥(𝑡))k𝑑 𝑥, 𝐸 𝑔 (𝑡) = k𝑒 𝑔 (𝑥(𝑡))k𝑑 𝑛 𝑥. D D The simulations are done in MATLAB with Euler integration with the sample time equal to 0.001 seconds. In the simulations, the results of the proposed fixed-time CL, asymptotically converging CL, fixed-time parameter estimation, and finite-time CL methods respectively given in (3.22), (3.53)-(3.55), are labeled by FxTCL, CL, FxT, and FTCL, respectively. Example 1: Adaptive approximators with zero MFAEs Consider the following system ¤ = 𝑝 1 𝑥(𝑡) + 𝑝 2 𝑥(𝑡) cos(𝑥(𝑡)) + 𝑝 3 𝑒 −𝑥(𝑡) 𝑢(𝑡), 𝑥(𝑡) (3.57) where the regressors are fully known as 𝑧(𝑥(𝑡), 𝑢(𝑡)) = [𝑥(𝑡), 𝑥(𝑡) cos(𝑥(𝑡)), 𝑒 −𝑥(𝑡) 𝑢(𝑡)] with 𝑝+𝑞 = 3. The unknown parameters are [𝑝 1 , 𝑝 2 , 𝑝 3 ] = [−0.5, 0.5, 0.5]. We set 𝑃 = 3 for CL, finite-time CL and FxTCL methods. Let 𝜉𝐺 = 𝜎𝐺 = 𝑘 1 = 3, 𝜉𝐶 = 𝜎𝐶 = 𝑘 2 = 1, 𝑘 3 = 0.1. Set Γ′ = Γ = Γ𝐶 = 𝐾 = 𝐼, 𝛾1 = 0.5 and 𝛾2 = 2. Fig. 3.1 depicts the true parameters and the approximated parameters for CL, fixed-time parameter estimation, finite-time CL, and the proposed fixed-time CL methods. In Fig. 3.1, CL, finite-time CL and the proposed fixed-time CL succeeded in convergence to true parameters where FxTCL resulted in faster convergence in comparison with CL and finite-time CL. As shown in Fig. 3.1, fixed-time method (3.54), did not succeed in convergence to the true parameters due to the lack of the PE condition. The online learning errors 𝐸 𝑓 (𝑡) and 𝐸 𝑔 (𝑡) are plotted in Fig. 3.2 where FxTCL shows faster converging to the origin in comparison with the other methods. The integral absolute errors (IAEs) of 𝐸 𝑓 (𝑡) and 𝐸 𝑔 (𝑡) for all methods are computed in Table 3.1 where FxTCL with IAEs 14.81 and 2.58, respectively, for 𝐸 𝑓 (𝑡) and 𝐸 𝑔 (𝑡) has resulted in the best precision of online learning in comparison with other mentioned methods. Using data selection algorithm 𝜆 (𝑆) in [34, 36] to maximize 𝜆 𝑚𝑖𝑛 (𝑆) , the maximum of 𝜆 𝑚𝑖𝑛 (𝑆) is obtained as 0.07. Therefore, using 𝑚𝑎𝑥 (3.23), the upper bound of the settling-time for the proposed FxTCL is obtained as 𝑇𝑚𝑎𝑥 = 57 seconds. Figs. 3.1 and 3.2 show that the settling-time for the FxTCL method satisfies the expected 57 0.6 0.4 True CL 0.2 FxT Parameters FTCL FxTCL 0 -0.2 -0.4 -0.6 0 10 20 30 40 50 60 70 80 Time (s) Figure 3.1: Estimated parameters for approximators with zero MFAE. 3 CL FxT 2 FTCL E f (t) FxTCL 1 0 0 10 20 30 40 50 60 70 80 Time (s) 4 CL 3 FxT E g (t) FTCL 2 FxTCL 1 0 0 10 20 30 40 50 60 70 80 Time (s) Figure 3.2: Online learning errors for approximators with zero MFAE. Table 3.1: Learning errors comparison Example 1 Example 2 IAE 𝐸 𝑓 (𝑡) IAE 𝐸 𝑔 (𝑡) IAE 𝐸 𝑓 (𝑡) IAE 𝐸 𝑔 (𝑡) CL 30.02 7.37 1448 6079 FxT 111.37 96.33 3252 10587 FTCL 19.11 5.40 1319 4022 FxTCL 14.81 2.58 907 2281 settling-time bound, however, the experienced stacks are not prerecorded and the experienced data is collected online during the learning. It is worth noting that the obtained bound for the FxTCL is valid for any initial condition in D𝑥 . Example 2: Adaptive approximators with non-zero MFAEs Now, consider the following 58 system ¤ = 𝑥(𝑡) sin(0.5𝑥(𝑡)) + (3 + cos(𝑥(𝑡)))𝑢(𝑡), 𝑥(𝑡) (3.58) where the associated 𝑓 (𝑥) and 𝑔(𝑥) are fully unknown uncertainties. In this example, since there is no knowledge about the exact regressors, it is expected that the learning errors are fixed-time attractive to a bound near zero. Therefore, radial basis function neural networks that are linearly parameterized universal approximators are used. Here, we consider 5 radial basis functions defined 2 𝑥−𝑐 𝑗 as exp(− ), 𝑗 = 1, 2, ..., 5, where the centroids 𝑐 𝑗 are uniformly picked on the interval 2𝜎 𝑗 2 [𝑥 𝐿 , 𝑥 𝐻 ] = [−2, 2] and the spreads are all fixed to 𝜎 𝑗 = 1.2. Therefore, the employed regressor is − k 𝑥(𝑡)−(−2) k 2 − k 𝑥(𝑡)−(2) k 2 − k 𝑥(𝑡)−(−2) k 2 − k 𝑥(𝑡)−(2) k 2 𝑧(𝑥(𝑡), 𝑢(𝑡)) = [𝑒 2(1.2)2 , ..., 𝑒 2(1.2)2 ,𝑒 2(1.2)2 𝑢(𝑡), ..., 𝑒 2(1.2)2 𝑢(𝑡)]𝑇 , with 10 independent basis functions that leads to setting 𝑃 = 10. Thus, the approximation of (3.58) is given as ¤ = Θ̂𝑇 (𝑡)𝑧(𝑥(𝑡), 𝑢(𝑡)) = [𝑝 1 , 𝑝 2 , ..., 𝑝 10 ]𝑧(𝑥(𝑡), 𝑢(𝑡)). 𝑥(𝑡) Let Γ = Γ′ = Γ𝐶 = 𝐾 = 𝐼, and 𝜉𝐺 = 𝜎𝐺 = 𝑘 1 = 1, 𝜉𝐶 = 𝜎𝐶 = 𝑘 2 = 0.2, 𝑘 3 = 0.02 and 𝛾1 = 0.5 and 𝛾2 = 2. It should be noted that in finite-time CL, increasing 𝑘 3 = 0.02 causes chattering in the approximation error. The mentioned learning methods led to the approximated parameters depicted in Fig. 3.3. In Fig. 3.3(a), it is shown that the fixed-time parameter estimation method cannot guarantee convergence of parameters to their true values, due to the lack of PE. Fig. 3.3(d) shows that the proposed FxTCL satisfying the rank condition, succeeded in convergence to the suitable parameters while CL and FTCL methods need more time for convergence as respectively shown in parts (b) and (c) of Fig. 3.3. The steady state approximations for the uncertainties 𝑓 (𝑥) and 𝑔(𝑥), on the x-domain D𝑥 , are depicted in Fig. 3.4 where the steady-state approximations of FxTCL could better match the true values of 𝑓 (𝑥) and 𝑔(𝑥) in comparison with other methods. As the comparison of the learning errors 𝐸 𝑓 (𝑡) and 𝐸 𝑔 (𝑡) in Fig. 3.5 shows, the fixed-time parameter estimation, did not perform well in learning the uncertainty due to the lack of the PE condition. 59 FxT CL 1 3 Parameters 2 0.5 1 0 0 -1 0 (a) 800 0 (b) 800 FTCL FxTCL 4 4 Parameters 2 2 0 0 -2 -2 0 (c) 800 0 (d) 800 Time (s) Time (s) Figure 3.3: Estimated parameters for approximators with non-zero MFAE: (a). Estimated parameters by FxT method, (b). Estimated parameters by CL method, (c). Estimated parameters by FTCL method, (d). Estimated parameters by FxTCL. 2 True 1.5 CL FxT f(x) 1 FTCL FxTCL 0.5 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 6 True 4 CL g(x) FxT FTCL 2 FxTCL 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x Figure 3.4: Steady state uncertainty approximations. However, the proposed FxTCL, FTCL and CL errors showed bounded convergence near zero in Fig. 3.5, where FxTCL method is faster in converging to a smaller bound near zero in comparison with others. Furthermore, based on the results of IAEs for 𝐸 𝑓 (𝑡) and 𝐸 𝑔 (𝑡) in Table 1, FxTCL resulted in the lowest learning errors during the whole time of online learning in comparison with other mentioned methods. In the presented numerical results, it is shown that the proposed FxTCL has outperformed other mentioned methods both in terms of precision and convergence speed. 60 6 CL FxT 4 FTCL E f (t) FxTCL 2 0 0 100 200 300 400 500 600 700 800 Time (s) 20 CL 15 FxT E g (t) FTCL 10 FxTCL 5 0 0 100 200 300 400 500 600 700 800 Time (s) Figure 3.5: Online learning errors for approximators with non-zero MFAE. 3.5 Conclusion In this chapter, a fixed-time concurrent learning system identification method is introduced without the persistence of excitation (PE) requirement. In this method, the concurrent learning relaxes the requirement of the PE condition to a rank condition on the memory stack of recorded data. It is shown that the richness of the recorded experienced data depends on the minimum eigenvalue properties of the stack of regressor’s data which influences the speed and precision of the proposed fixed-time concurrent learning method. Simulation results are given where it is shown that the proposed fixed-time concurrent learning has outperformed other mentioned methods in both terms of precision and convergence speed. 61 CHAPTER 4 ONLINE IDENTIFICATION OF NOISY FUNCTIONS VIA A DATA-REGULARIZED LEARNING APPROACH 4.1 Introduction This chapter presents online learning rule that paves the way to designing an active learning approach to collect informative data that improves the convergence rate as well as reduce the effect of the noise variance on the estimation. This is in sharp contrast to the existing online approaches that require independent and identically distributed (i.i.d) data samples for which there is no systematic approach to verify them. More specifically, it is shown that as the data is streaming, the fixed-size memory data can be updated to improve the strong convexity properties of the data-regularized loss function to reduce the ultimate bound and improve the convergence speed. The exponential convergence rate is also guaranteed under a rank condition on the matrix of the memory data. The rate of convergence to the ultimate bound also depends on the quality of the stored data. More specifically, the employed data-regularized loss function is strongly convex as long as a rank condition on the fixed-size memory data is satisfied, and does not impose any bias on the estimated parameters. The strong convexity parameter of the loss function depends on the maximum and minimum eigenvalues of the memory data matrix, which can be improved by replacing new samples with old ones. The presented online data-regularized CL-based SGD ensures a finite-sample performance guarantee by providing a bound of the estimated parameters for every time step. Moreover, it is shown how the function approximation with noisy measurements can be leveraged in system identification and RL applications. Simulation examples are also provided to verify the effectiveness of the proposed approach and the results are compared with the standard SGD. Notation R, N, and Z+ respectively show the set of real, natural and all nonnegative integers. k.k denotes the Euclidean norm for vectors and induced 2-norm for matrices. 𝑡𝑟(.) indicates trace of a matrix. The minimum and maximum eigenvalues of matrix 𝐴 are, respectively, denoted by 𝜆 𝑚𝑖𝑛 (𝐴) and 𝜆 𝑚𝑎𝑥 (𝐴). The matrix 𝐼 denotes the identity matrix of appropriate dimensions. We 62 use 𝛾1 ◦ 𝛾2 to denote the composition of two functions 𝛾1 and 𝛾2 where 𝛾𝑖 : R ↦→ R, for 𝑖 = 1, 2. A function 𝛼 : R > 0 ↦→ R > 0 is a K-function if it is continuous, strictly increasing and 𝛼(0) = 0; it is a K∞ -function if it is a K-function and also 𝛼(𝑠) → ∞ as 𝑠 → ∞; and it is a positive definite function if 𝛼(𝑠) > 0 for all 𝑠 > 0, and 𝛼(0) = 0. A function 𝛽 : R > 0× R> 0 → R> 0 is a KL-function if, for each fixed 𝑘 > 0, the function 𝛽(·, 𝑘) is a K-function, and for each fixed 𝑠 > 0, the function 𝛽(𝑠, ·) is decreasing and 𝛽(𝑠, 𝑘) → 0 as 𝑘 → ∞. All random variables are assumed to be defined on a probability space (Ω, F , P), with Ω as the sample space, F as its associated Borel 𝜎-algebra and P as the probability measure. For a random variable 𝑤 : Ω −→ R𝑛 defined on the probability space (Ω, F , P), with some abuse of notation, the statement 𝑤 ∈ R𝑛 is used to state the dimension of the random variable. Finally, E[𝑋] denotes the expected value of the random variable 𝑋 on the probability space (Ω, F , P). 4.2 Preliminaries Consider the following stochastic discrete-time (DT) dynamics system 𝑥(𝑘 + 1) = 𝐹(𝑥(𝑘), 𝑣(𝑘)), (4.1) where 𝑥(𝑘) ∈ D ⊂ R𝑛 is the measurable state vector and D is a compact set; 𝑣(𝑘) ∈ R𝑛 is a zero-mean independent white noise and 𝐹 : D × R𝑛 ↦→ R𝑛 . The following definition and lemmas are introduced for stability analysis and convergence of this stochastic system. Definition 5 [132] Consider the stochastic system (4.1) and fix 𝜖 ∈ (0, 1). The system is said to be practical stable in probability (PS-P) if there exist a positive constant 𝛾 and a class KL-function 𝛽(·, ·) such that  P 𝑥(𝑘) ≤ 𝛽 (k𝑥 0 k , 𝑘) + 𝛾 ≥ 1 − 𝜖 . The following lemma gives the criterion on practical stability in probability (PS-P) for the system (4.1). Lemma 3 [132] The system (4.1) is PS-P if there exist a positive definite function 𝑉(𝑥(𝑘)) and 63 real scalar 𝑑 ≥ 0 and K∞ -functions 𝛼1 , 𝛼2 , 𝛼3 such that 𝛼1 (k𝑥(𝑘)k) ≤ 𝑉(𝑥(𝑘)) ≤ 𝛼2 (k𝑥(𝑘)k), and   E 𝑉 𝑥(𝑘 + 1) − 𝑉(𝑥(𝑘)) ≤ −𝛼3 (k𝑥(𝑘)k) + 𝑑, (4.2) where 𝛼3 ◦ 𝛼2−1 is a convex function. Definition 6 [133] The origin of the system (4.1) is said to be exponentially bounded in mean square with exponent 𝑎 if there exist constants 0 < 𝑎 < 1, 𝑐 1 ≥ 0 and 𝑐 2 > 0 such that 𝐸 k𝑥(𝑘)k 2 ≤ 𝑐 1 + 𝑐 2 (1 − 𝑎) 𝑘 . (4.3) Remark 17 Definition 6 does not necessarily imply that 𝐸 k𝑥(𝑘)k 2 decreases monotonically for all 𝑘. It only implies that the bound on 𝐸 k𝑥(𝑘)k 2 decreases exponentially and as 𝑘 → ∞ the mean square of the process is bounded by 𝐸 k𝑥(∞)k 2 ≤ 𝑐 1 where 𝑐 1 depends on the noise disturbing the system. Lemma 4 [134] For the system (4.1), if there exists a function 𝑉(𝑥(𝑘)) with 𝑉(0) = 0 such that, ∀𝑘 ≥ 0, 1) E[𝑉(𝑥(𝑘))] ≥ 𝑐E[𝜌(k𝑥(𝑘)k)], and 2) E[𝑉(𝑥(𝑘 + 1))] − E[𝑉(𝑥(𝑘))] ≤ 𝑀 − 𝑎E[𝑉(𝑥(𝑘))], for some 𝜌(.) ∈ K, and constants 𝑐 > 0, 𝑀 ≥ 0, and 0 < 𝑎 < 1, then 𝑘−1 X 𝑐E[𝜌(k𝑥(𝑘)k)] ≤ E[𝑉(𝑥(𝑘))] ≤(1 − 𝑎) 𝑘 E[𝑉(𝑥(0))] + 𝑀 (1 − 𝑎)𝑖 , (4.4) 𝑖=0 and lim 𝑘→∞ E[𝜌(k𝑥(𝑘)k)] ≤ 𝑐𝑎 𝑀. The following definitions are also used throughout the chapter. Definition 7 [135] The sequence {𝑥(𝑘)}∞ 𝑘=1 converges to 𝑥 ∗ with a linear (exponentially fast) rate if k𝑥(𝑘 + 1) − 𝑥 ∗ k≤ 𝛾k𝑥(𝑘) − 𝑥 ∗ k, or equivalently if k𝑥(𝑘 + 1) − 𝑥 ∗ k≤ 𝛾 𝑘 k𝑥(0) − 𝑥 ∗ k for some 𝛾 ∈ (0, 1). The convergence is sublinear if 𝛾 = 1. For the linear (sublinear) convergence, the error rate is 𝑂(𝛾 𝑘 ) ( O(1 𝑘) ) . Definition 8 (Markov’s Inequality [136]). Let 𝑋 be a non-negative random variable. Then for all real positive constant 𝑎 > 0, E[X] P(𝑋 ≥ 𝑎) ≤ . (4.5) 𝑎 64 Definition 9 (Jason’s inequality [137]): If 𝑋 is a random variable and 𝜑 is a convex function, then 𝜑(E[𝑋]) ≤ E[𝜑(𝑋)]. Definition 10 (Persistently exciting (PE) [29]) The bounded vector signal 𝜑(𝑥(𝑘)) ∈ R𝑛 is PE if there exist a natural number 𝑁 and 𝛼 > 0 such that 𝜏+𝑁 X 𝜑(𝑥(𝑘))𝜑𝑇 (𝑥(𝑘)) ≥ 𝛼𝐼, ∀𝜏 ∈ Z+ . (4.6) 𝑘=𝜏 Definition 11 (Strongly Convex and Smooth Functions [138]) A convex function 𝑓 is said to be 𝛼-strongly convex if 𝛼 𝑓 (𝑦) ≥ 𝑓 (𝑥) + 𝑓 (𝑥)𝑇 (𝑦 − 𝑥) + ||𝑦 − 𝑥|| 2 (4.7) 2 Moreover, a continuously differentiable function 𝑓 is 𝛽-smooth if its gradient is 𝛽-Lipschitz. That is, if || 𝑓 (𝑥) − 𝑓 (𝑦)||≤ 𝛽||𝑥 − 𝑦|| (4.8) A twice differentiable function 𝑓 is 𝛼-strongly convex if for all 𝑥, one has ∇2 𝑓 (𝑥) ≥ 𝛼 𝐼 and is 𝛽-smooth if for all 𝑥, one has ∇2 𝑓 (𝑥) ≤ 𝛽 𝐼. 4.3 Problem Formulation and Motivation Stochastic Function Identification: Problem Formulation Consider the following DT func- tion 𝑦(𝑘) = 𝑓 (𝑥(𝑘)) + 𝑣(𝑘), (4.9) where 𝑥 ∈ D ⊂ R𝑛 is the measurable state vector and D is a compact set; 𝑓 : D ↦→ R𝑛 , 𝑣 ∈ R𝑛 is an additive zero-mean independent white noise. Assumption 5 The function 𝑓 (.) is unknown; however, its noisy measurements 𝑦(𝑘) as well as 𝑥(𝑘) are available for measurement. The noise 𝑣(𝑘) in system (4.9) satisfies the following assumption. 65   Assumption 6 For ∀𝑠, 𝑡 ∈ Z+ , E 𝑣(𝑠) = E 𝑣(𝑡) = 0, and  n o  𝜎 2 , 𝑠 = 𝑡,   E 𝑣(𝑠)𝑇 𝑣(𝑡) =   0, 𝑠 6= 𝑡.  In this chapter, we consider the problem of online DT function identification, 𝑓 (𝑥(𝑘)), from streaming noisy measurements and recorded experienced data. Here, linearly parameterized adaptive approximation models [10] are employed to represent 𝑓 (𝑥(𝑘)) as follows, 𝑓 (𝑥(𝑘)) = Θ∗𝑇 𝜑(𝑥(𝑘)) + 𝜀(𝑥(𝑘)), (4.10) where the matrix Θ∗ ∈ DΘ ⊂ R𝑞×𝑛 denotes the unknown optimal parameters of the approximater, given by Θ∗ = arg min { 𝑠𝑢 𝑝 𝜀(𝑥(𝑘)) }, (4.11) Θ∈DΘ 𝑥(𝑘)∈D where DΘ is a compact set. The measurable vector 𝜑 : D ↦→ R𝑞 denotes the basis functions, while 𝑞 is the number of linearly independent basis functions for approximating 𝑓 (𝑥(𝑘)). The quantity 𝜀(𝑥(𝑘)) ∈ R𝑛 is the minimum functional approximation error (MFAE) for 𝑓 (𝑥(𝑘)). If the unknown functions 𝑓 (𝑥(𝑘)) can be approximated exactly by the model, one has 𝜀(𝑥(𝑘)) = 0. Assumption 7 For the compact set D, the approximators’ basis functions are bounded i.e., 𝑏 1 ≤ k𝜑(𝑥(𝑘))k≤ 𝑏 2 , ∀𝑥 ∈ D, where 𝑏 1 ≥ 0 and 𝑏 2 > 0. Moreover, the approximation error 𝜀(𝑥(𝑘)) is bounded by 𝑏 𝜀 ≥ 0, i.e., sup k𝜀(𝑥)k≤ 𝑏 𝜀 . 𝑥∈D Now using (4.10), (4.9) can be written as 𝑦(𝑘) = Θ∗𝑇 𝜑(𝑥(𝑘)) + 𝜀(𝑥(𝑘)) + 𝑣(𝑘). (4.12) Let the approximator of (4.12) be of the form 𝑦ˆ(𝑘) = Θ̂𝑇 (𝑘)𝜑(𝑥(𝑘)), (4.13) 66 where Θ̂(𝑘) ∈ R𝑞×𝑛 is the estimation of parameter matrix Θ∗ at time 𝑘. The approximation error 𝑒(𝑘) is defined as 𝑒(𝑘) = 𝑦ˆ(𝑘) − 𝑦(𝑘) = Θ̃𝑇 (𝑘)𝜑(𝑘) − 𝜀(𝑘) − 𝑣(𝑘), (4.14) where Θ̃(𝑘) := Θ̂(𝑘) − Θ∗ is the parameter estimation error. The goal of function approximation is to learn the unknown parameters vector Θ∗ by Θ̂ by fitting (4.13) to data samples. This function approximation problem has many applications in machine learning. For example, identification of dynamic systems and learning the value function in reinforcement learning can be transformed into this regressor, as shown later. The following definition is needed to formalize the goal of this chapter, stated in Problem 1. Definition 12 [139] Let 0 < 𝜖 < 1. Let a learning algorithm 𝐴 is designed to iteratively learn the unknown parameters Θ∗ and let its output at iteration or time 𝑘 be Θ̂(𝑘). We say that the set 𝑆Θ (𝑘) is a probabilistic bound of the learning algorithm at time 𝑘, if P[Θ∗ − Θ̂(𝑘) ∈ 𝑆Θ (𝑘)] ≥ (1 − 𝜖) for time 𝑘 and after. We say 𝑆Θ is the probabilistic ultimate bound of the learning if P[lim 𝑘→∞ (Θ∗ − Θ̂(𝑘)) ∈ 𝑆Θ ] ≥ (1 − 𝜖). Problem 1 Consider the function (4.12) and let its approxixmator be (4.13). Let Assumptions 5- 7 be satisfied. Design an iterative learning algorithm 𝐴 such that Θ∗ − Θ̂(𝑘) converges exponentially fast to a probabilistic ultimate bound with minimum size. Moreover, the algorithm 𝐴 provides finite-sample guarantees for the estimation error. Remark 18 The probabilistic ultimate bound in Definition 12 can be achieved by assuring that the error dynamics of the learning, i.e, Θ̃, is PS-P (which is defined in Definition 5). In this case, the exponential convergence to the ultimate bound is characterized by the 𝛽(·, ·) function, and the ultimate bound is characterized by 𝛾. Moreover, the sets 𝑆Θ (𝑘) and 𝑆Θ are balls of radius with 𝛽(Θ(0), 𝑘) + 𝛾 and 𝛾, respectively. Remark 19 Note that due to the noise in (4.12), no point estimate Θ̂(𝑘) of the unknown parameters Θ∗ can entirely predict the outcome of the stochastic function (4.12). Therefore, a high-confidence set for the estimation is typically found to provide guarantees on how far the estimated parameters can be from the optimal parameters. In our proposed approach, we will show 67 that this set has a transient part that goes to zero as 𝑘 goes to infinity and the steady state part depends on the noise variance and the MFAE. An efficient algorithm is then the one that makes the transient response faster (i.e, faster convergence to the ultimate bound) and also makes the size of the ultimate bound smaller (more confidence and robust learning). Standard SGD algorithms are typically presented to reduce the computational complexity of the optimization for the case where the noise and MFAE are both identically zero in (4.12). To clarify this, consider the case where the noise and MFAE are both identically zero and 𝑁 samples {𝜙(𝑥(𝑘), 𝑦(𝑘))} 𝑁𝑘=1 are collected that span the entire space of the function. Then, optimizing the following finite sum of the function approximation error using either least squares or gradient decent guarantees convergence to the optimal parameter, X𝑁 ℓΘ̃ (𝑘) = 𝑒(𝑘)2 . (4.15) 𝑘=1 However, when 𝑁 is large, then, to reduce the computational complexity, the SGD randomly samples from the set of 𝑁 data samples and perform the following gradient descent only on the randomly selected sample, Θ̂(𝑘 + 1) = Θ̂(𝑘) − 𝜉 𝑘 𝜑(𝑘)𝑒𝑇 (𝑘), (4.16) where 𝜉 𝑘 is the learning rate. That is, the SGD optimizes the instantaneous loss function 1 ℓΘ̃ (𝑘) = 𝑒𝑇 (𝑘)𝑒(𝑘). (4.17) 2 Convergence of the SGD is shown to include a transient part and a steady-state part (depending on the variance of the estimation of the entire gradient with the gradient of one sample). However, in our setting, the ultimate bound of the estimation error depends on the inherent measurement noise and not on randomly selecting from available set of samples. Besides, for the time-varying regressor for which the data samples are streaming and not available at once, the signals 𝜑(𝑘) interacting with the parameter estimates must remain PE during the estimation procedure. If PE condition is not satisfied, the parameters converge to either a wrong value or do not converge at all. The following Lemma shows that if the PE condition is satisfied, then the optimization solution becomes unique due to strong convexity of the empirical average of the loss function. 68 Lemma 5 Consider the function approximation problem with noisy measurements (4.12) and the approximation error (4.14). Let the SGD algorithm (4.16) be used to learn the unknown parameters Θ∗ , where 𝜉 𝑘 is a time-varying learning rate satisfying 𝜉 𝑘 > 0, ∀𝑘 ∈ N and the P∞ sequence {𝜉 𝑘 }∞𝑘=1 converges to 0 and that 𝑘=1 𝑘 𝜉 = ∞. Then, at any time 𝑡 ≥ 1, the SGD optimizes the empirical average of the loss unction given by 1 X 𝑡 ℓΘ̃ (𝑡) = 𝑒𝑇 (𝑘)𝑒(𝑘). (4.18) 2𝑡 𝑘=1 Moreover, it converges to a unique solution if the signal 𝜑(𝑘) is PE. Proof 7 While the SGD (4.16) optimizes the instantaneous loss function defined in (4.17) at every time step, it is shown in [140] that using a learning rate that satisfies the conditions provided in the statement of the lemma, the update law (4.16), any any time 𝑡 ≥ 1, optimizes the empirical average of the observations in (4.18). The gradient and hessian of the instantaneous error ℓΘ̃ (𝑘) at Θ̃(𝑘) are, respectively, given as ∇ℓΘ̃ (𝑘) = 𝜑(𝑘)𝑒𝑇 (𝑘) = 𝜑(𝑘)𝜑𝑇 (𝑘)Θ̃(𝑘) − 𝜑(𝑘)𝜀𝑇 (𝑘) − 𝜑(𝑘)𝑣𝑇 (𝑘), (4.19) and ∇2 ℓΘ̃ (𝑘) = 𝜑(𝑘)𝜑𝑇 (𝑘). (4.20) Therefore, for the empirical average function ℓΘ̃ (𝑡), this becomes 1X 𝑡 ∇2 ℓΘ̃ (𝑡) = 𝜑(𝑥(𝑘))𝜑𝑇 (𝑥(𝑘)). (4.21) 𝑡 𝑘=1 If the signal 𝜑(𝑘) is PE (i.e., if it satisfies (4.6)), then ∇2 ℓΘ̃ (𝑡) ≥ 𝛼𝐼 and thus ℓΘ̃ (𝑡) is strongly convex after some time 𝑡 ≥ 𝑁. Therefore, due ot the strong convexity, a unique solution to the optimization problem is found. This completes the proof. On one hand, when the excitation of the signal 𝜙(𝑥) decays quickly, the online SGD cannot receive necessary amount of information about Θ∗ and fails to estimate it correctly. If the PE condition is not satisfied, the error terms (4.18) is only convex over time and not strongly convex, 69 and thus can become zero even if Θ∗ converges to a wrong value. On the other hand, even though gradient descent achieves linear convergence rate for a strongly convex function, SGD does not enjoy the linear convergence rate of gradient descent under strong convexity and only achieve sublinear convergence rate [141, 142]. Sublinear convergence rate, however, is not strong because it has the property that the longer you run the algorithm, the less progress it makes. A fundamental question is how to develop new online learning algorithms that achieve linear convergence rates and under relaxed PE condition that can be easily verified and improved. To fulfill the learning of the uncertainties 𝑓 (𝑥) in (4.9) without requiring the PE condition on the data stream, a data-regularized CL-based SGD based learning is proposed next to ensure that the parameter estimation error Θ̃(𝑘) dynamics is PS-P, and thus solve Problem 1. Before presenting our algorithm, the following subsection shows two applications of the function identification with noisy measurements. Motivation for Function Identification: Value Learning in Reinforcement Learning and System identification as Function Identifiers In this subsection, we show that the value learning in RL and the model learning in system identification can be formalized as instances of the problem of stochastic DT function identification. Value Function Learning Consider the system described by the following stochastic nonlinear difference equation 𝑥(𝑘 + 1) = 𝐺(𝑥(𝑘), 𝑢(𝑘)) + 𝑣(𝑘), (4.22) where 𝑥 ∈ D ⊂ R𝑛 and 𝑢 ∈ D𝑢 ⊂ R𝑚 are the system’s states and inputs, respectively; D and D𝑢 are compact sets, 𝐺 : D × D𝑢 ↦→ D is the dynamics function, and 𝑣(𝑘) is the zero mean white noise at time 𝑘 with covariance Σ. A stage cost or reward function for the state 𝑥 and action 𝑢 at time 𝑘 is considered as 𝑟(𝑥(𝑘), 𝑢(𝑘)). For a fixed control policy 𝜋 : D ↦→ D𝑢 , the cost-to-go for a single realization and the initial condition 𝑥(0) is defined by " # X𝑁 𝐽(𝑥(0), 𝜋)=E 𝑟 (𝑥(𝑘), 𝜋(𝑥(𝑘))) . (4.23) 𝑘=0 70 Defining the value function for the policy 𝜋 as 𝑉 𝜋 (𝑥) := 𝐽(𝑥 , 𝜋), the following Bellman equation is obtained 𝑉 𝜋 (𝑥(𝑘)) = 𝑟(𝑥(𝑘), 𝑢(𝑘)) + E[𝑉 𝜋 (𝑥(𝑘 + 1))], (4.24) where 𝑢(𝑘) = 𝜋(𝑥(𝑘)) and 𝑥(𝑘 + 1) is the system’s next state under the control action 𝑢(𝑘). Based on (4.24), consider the Bellman operator introduced below, 𝑇𝑉 𝜋 (𝑥(𝑘)) := 𝑟(𝑥(𝑘), 𝑢(𝑘)) + E[𝑉 𝜋 (𝑥(𝑘 + 1))]. (4.25) Then, the Bellman equation (4.24) becomes 𝑉 𝜋 (𝑥) = 𝑇𝑉 𝜋 (𝑥). (4.26) To solve (4.26) for value function, the value function is typically parametrized in the form of 𝑉Θ𝜋 (𝑥(𝑘)) = Θ∗𝑇 𝜙(𝑥(𝑘)) where Θ∗ are the unknown optimal parameters of the approximation model. The goal is to learn the unknown parameter Θ∗ using data. The fact that the Bellman equation (4.26) is a contraction map [143] is leveraged in many studies by the stochastic approximation to learn the parameters of the parametrized value function. The exact value of 𝑇𝑉 𝜋 (𝑥) is not available due to the expectation operator and only its noisy estimates are provided typically using the temporal difference approach. Defining 𝐿(Θ̂(𝑘), 𝑥(𝑘)) = [𝑉Θ𝜋 (𝑥(𝑘)) − 𝑇𝑉 𝜋 (𝑥(𝑘))]2 , then only its noisy measurements are available due to the expectation Θ operator in 𝑇𝑉Θ 𝜋 (𝑥(𝑘)). The goal is to learn the probabilistic ultimate bound 𝑆 , such that Θ P[lim 𝑘→∞ (Θ∗ − Θ̂(𝑘)) ∈ 𝑆Θ ] ≥ (1 − 𝜖) for 𝜖 ∈ (0, 1), while only noisy measurements are available for the loss function 𝐿(Θ̂(𝑘), 𝑥(𝑘)). Iteratively solving the Bellman’s equation by solving this optimization problem using noisy samples is at the heart of reinforcement learning algorithms such as policy iteration and value iteration. System Dynamics Identification Consider the following DT system, 𝑥(𝑘 + 1) =f (𝑥(𝑘)) + g (𝑥(𝑘))𝑢(𝑘) + 𝑣(𝑘), (4.27) 71 where 𝑥 ∈ D ⊂ R𝑛 and 𝑢 ∈ D𝑢 ⊂ R𝑚 are the system’s states and inputs, respectively; D and D𝑢 are compact sets; f : D ↦→ R𝑛 , and g : D ↦→ R𝑛×𝑚 are the unknown nonlinear drift and input terms,respectively, and 𝑣(𝑘) ∈ R𝑛 is a zero-mean independent white noise with covariance Σ. The system identification aim is to learn the unknown dynamics in (4.27), namely to approximate f (𝑥(𝑘)) and g (𝑥(𝑘)). Linearly parameterized adaptive approximation models are employed to, respectively, represent f (𝑥(𝑘)) and g (𝑥(𝑘)) as follows, f (𝑥(𝑘)) = Θ∗𝑇f 𝜓(𝑥(𝑘)) + 𝑒 f (𝑥(𝑘)), (4.28) g (𝑥(𝑘)) = Θ∗𝑇g 𝜒(𝑥(𝑘)) + 𝑒 g (𝑥(𝑘)), (4.29) where the matrices Θ∗f ∈ DΘ ⊂ R𝑟×𝑛 and Θ∗g ∈ DΘg ⊂ R𝑠×𝑛 denote the unknown optimal f parameters of the adaptive approximation models, where DΘ and DΘg are compact sets. The f 𝑟 𝑠 vectors 𝜓 : D ↦→ R and 𝜒 : D ↦→ R , are the computable basis functions; 𝑟 and 𝑠 are the number of linearly independent basis functions to, respectively, approximate f (𝑥(𝑘)) and g (𝑥(𝑘)). In (4.28) and (4.29), 𝑒 f (𝑥(𝑘)) ∈ R𝑛 and 𝑒 g (𝑥(𝑘)) ∈ R𝑛×𝑚 are, respectively, the MFAEs for f (𝑥(𝑘)) and g (𝑥(𝑘)). If f (𝑥) and g (𝑥) can be approximated exactly by the models Θ𝑇f 𝜓(𝑥) and Θ𝑇g 𝜒(𝑥), respectively, one has 𝑒 f (𝑥) = 𝑒 g (𝑥) = 0. Using (4.28)-(4.29), the system dynamics (4.27) is rewritten as 𝑥(𝑘 + 1) = Θ∗𝑇 𝑧(𝑥(𝑘), 𝑢(𝑘)) + 𝜀(𝑥(𝑘), 𝑢(𝑘)) + 𝑣(𝑘), (4.30) where Θ∗ = [Θ∗𝑇 , Θ∗𝑇 𝑇 (𝑟+𝑠)×𝑛 and 𝑧(𝑥(𝑘), 𝑢(𝑘)) = [𝜓𝑇 (𝑥(𝑘)), 𝑢𝑇 (𝑘)𝜒𝑇 (𝑥(𝑘))]𝑇 ∈ R(𝑟+𝑠) f g ] ∈R and 𝜀(𝑥, 𝑢) = 𝑒 f (𝑥) + 𝑒 g (𝑥)𝑢. Now, consider the approximator be 𝑥ˆ(𝑘 + 1) =Θ̂𝑇 (𝑘)𝑧(𝑥(𝑘), 𝑢(𝑘)), (4.31) where Θ̂(𝑘) = [Θ̂𝑇f (𝑘), Θ̂𝑇g (𝑘)]𝑇 ∈ R(𝑟+𝑠)×𝑛 , Θ̂f (𝑘) and Θ̂g (𝑘) are respectively the estimation for parameter matrices Θ∗ , Θ∗f and Θ∗g at time 𝑘. 72 Let hΘ (𝑥(𝑘)) be defined as hΘ (𝑥(𝑘)) = 𝑥ˆ(𝑘 + 1) − 𝑥(𝑘 + 1) = Θ̃𝑇 𝑧(𝑥(𝑘), 𝑢(𝑘)) − 𝜀(𝑘) − 𝑣(𝑘), (4.32) where Θ̃(𝑘) := Θ̂(𝑘) − Θ∗ := [Θ̃𝑇f (𝑘), Θ̃𝑇g (𝑘)]𝑇 is the parameter estimation error with Θ̃f (𝑘) := Θ̂f (𝑘) − Θ∗f , Θ̃g (𝑘) := Θ̂g (𝑘) − Θ∗g . The goal is to learn the probabilistic ultimate bound 𝑆Θ , such that P[lim 𝑘→∞ (Θ∗ − Θ̂(𝑘)) ∈ 𝑆Θ ] ≥ (1 − 𝜖) for 𝜖 ∈ (0, 1), while only noisy measurements are available for the loss function 𝐿(Θ̂(𝑘), 𝑥(𝑘)) = [hΘ (𝑥(𝑘))]2 . The availability of 𝑥(𝑘 + 1) at time 𝑘, which is required based on (4.32), can be relaxed either by employing regressor filtering [42] or using estimators [37]. 4.4 Data-regularized Concurrent Learning-based SGD for Function Identi- fier with noisy measurements In this section, a data-regularized CL-based SGD update law is presented to approximate the function given in (4.9). In sharp contrast to SGD and mini-batch SGD version, rather than estimating the gradient of the error using a single (current) sample or mini-batch of random samples, the presented approach approximates the gradient flows of the estimation errors using the current data as well as fixed samples collected in recorded data stacks. Leveraging a fixed-size memory of data, for which the data are selected based on an easy-to-verify data-richness condition, rather than random, allows us to not only eliminate the PE condition requirement on the stream of data, but also to improve the convergence rate and providing guarantees using the Lyapunov theory. The convergence analysis of the dynamics of the data-regularized CL-based SGD estimation law is given based on practical stability in probability. To employ the data-regularized CL-based SGD, which uses recorded experienced data along with current data, the previous data are collected and stored in the memory stacks M ∈ R𝑞×𝑃 and Y ∈ R𝑛×𝑃 , at times 𝜏1 , ..., 𝜏𝑃 as, M = [𝜑(𝜏1 ), 𝜑(𝜏2 ), ..., 𝜑(𝜏𝑃 )], Y = [𝑦(𝜏1 ), 𝑦(𝜏2 ), ..., 𝑦(𝜏𝑃 )], (4.33) 73 where 𝑃 is the number of data points stored in the history stacks. 𝑃 is determined such that M contains as many linearly independent elements as the dimension of 𝜑(𝑘) (i.e., the number of linearly independent basis functions for 𝑓 (𝑥(𝑘))). That is, 𝑃 ≥ 𝑞. The error 𝑒 ℎ (𝑘) for the ℎ𝑡ℎ recorded sample, but using the current estimation of the function parameters at time 𝑘, is 𝑒 ℎ (𝑘) = 𝑦ˆℎ (𝑘) − 𝑦(𝜏ℎ ), (4.34) where 𝑦ˆℎ (𝑘) = Θ̂𝑇 (𝑘)𝜑(𝜏ℎ ), (4.35) is the estimation at time 0 ≤ 𝜏ℎ < 𝑘, ℎ = 1, ..., 𝑃, using the current estimated parameters matrix Θ̂(𝑘) and the recorded 𝜑(𝜏ℎ ). Substituting 𝑦(𝜏ℎ ), from (4.12), in (4.34) leads to 𝑒 ℎ (𝑘) = Θ̃𝑇 (𝑘)𝜑(𝜏ℎ ) − 𝜀(𝜏ℎ ) − 𝑣(𝜏ℎ ). (4.36) Now, the following data-regularized loss function is considered 1 1X 𝑃 ℓΘ̃ (𝑘) = 𝑒𝑇 (𝑘)𝑒(𝑘) + 𝑒𝑇 (𝑘)𝑒 ℎ (𝑘). (4.37) 2 2 ℎ=1 ℎ The following lemma guarantees that this data-regularized objective function is strongly convex in the absence of PE condition when the rank condition on M is satisfied. Lemma 6 The loss function (4.37) is strongly convex if the matrix M in (4.33) is full-row rank. Proof 8 The gradient and hessian of ℓΘ̃ (𝑘) in (4.37) at Θ̃(𝑘) are respectively defined as X𝑃 ∇ℓΘ̃ (𝑘) = 𝜑(𝑘)𝑒𝑇 (𝑘) + 𝜑(𝑘)𝑒𝑇ℎ (𝑘) ℎ=1 = 𝜑(𝑘)𝜑𝑇 (𝑘)Θ̃(𝑘) − 𝜑(𝑘)𝜀𝑇 (𝑘) − 𝜑(𝑘)𝑣𝑇 (𝑘) X𝑃 + {𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ )Θ̃(𝑘) − 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ ) − 𝜑(𝜏ℎ )𝑣𝑇 (𝜏ℎ )}, (4.38) ℎ=1 and 𝑃 X ∇2 ℓΘ̃ (𝑘) = 𝜑(𝑘)𝜑𝑇 (𝑘) + 𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ ). (4.39) ℎ=1 74 Figure 4.1: Data-regularized CL-based SGD for noisy function identification P In (4.39), the satisfaction of M rank condition keeps 𝑆 = 𝑃 ℎ=1 𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) > 0 which ensures the strong convexity of ℓΘ̃ (𝑘), i.e., ∇2 ℓΘ̃ (𝑘) > 0. The data-regularized CL-based SGD parameter estimation law presented in the next subsection, which is obtained by minimizing (4.37), results in the linear convergence of the approximated parameters’ error to a probabilistic ultimate bound and thus solves Problem 1. Moreover, it is shown how the selection of stored data in M which 𝜆 (𝑆) P maximizes the condition number 𝜆 𝑚𝑖𝑛 (𝑆) with 𝑆 = 𝑃 ℎ=1 𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ ), improves the convergence 𝑚𝑎𝑥 rate and reduces the parameters’ estimation error bound. This is in contrast to standard SGD for which there is no systematic approach for data selection to reduce the convergence bound and improve the convergence rate. The proposed data-regularized CL-based SGD update law for the approximator (4.13) is given as X𝑃 Θ̂(𝑘 + 1) = Θ̂(𝑘) − [Ξ𝐺 𝜑(𝑘)𝑒𝑇 (𝑘) + Ξ𝐶 𝜑(𝜏ℎ )𝑒𝑇ℎ (𝑘)]. (4.40) ℎ=1 The matrices Ξ𝐺 , Ξ𝐶 ∈ R𝑞×𝑞 , are the positive definite learning rate matrices for, respectively, P the gradient descent term (𝜑(𝑘)𝑒𝑇 (𝑘)) and concurrent learning term ( 𝑃 ℎ=1 𝜑(𝜏ℎ )𝑒𝑇ℎ (𝑘)) where Ξ𝐶 = 𝜉𝐶 𝐼 and Ξ𝐺 = 𝜉𝐺 𝐼 with constants 𝜉𝐶 > 0 and 𝜉𝐺 > 0. Fig. 4.1 shows the introduced data-regularized CL-based SGD for noisy function identification. The stochastic convergence properties of the proposed data-regularized CL-based SGD are 75 investigated next. Stochastic Convergence Properties for Data-regularized Concurrent Learning-based SGD Update Law In this section, first PS-P of the estimated parameters in convergence to their optimal values is ensured using the proposed data-regularized CL-based SGD method (4.40). Then, it is discussed that the proposed data-regularized CL-based SGD method (4.40) guarantees the finite-sample boundedness in probability of the estimated parameters’ error in every step 𝑘. Exponential Probabilistic Ultimate Boundedness of Parameters’ Estimation Error Theorem 5 Consider the approximator of nonlinear function in (4.9) given in (4.13), whose parameters are adjusted according to the update law of (4.40). Let Assumptions 5-7 hold. Once the rank condition on M is satisfied and 𝜉𝐶 > 0 and 𝜉𝐺 > 0 are chosen such that 1 (𝜉𝐺 𝑏 22 + 𝜉𝐶 𝜆 𝑚𝑎𝑥 (𝑆))2 < 𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆) + 𝜉𝐺 𝑏 21 < 1, (4.41) 2 then, the update law (4.40) guarantees PS-P and exponential probabilistic ultimate boundedness of Θ̃(𝑘) = Θ̂(𝑘) − Θ∗ . That is, for 𝜖 ∈ (0, 1), r 𝑏 P{k Θ̃(𝑘)k≤ } ≥ 1 − 𝜖, (4.42) 𝜖 where 𝑏 = 𝜆 2𝑑(𝑄) , 𝑚𝑖𝑛 2𝐵2 𝑏 2𝜀 𝑑= + 𝐶(𝜎), (4.43) 𝜆 𝑚𝑖𝑛 (𝑄) 𝜆 𝑚𝑖𝑛 (𝑄) = 2𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆) + 2𝜉𝐺 𝑏 21 − (𝜉𝐺 𝑏 22 + 𝜉𝐶 𝜆 𝑚𝑎𝑥 (𝑆))2 , (4.44) 𝐵 = 𝑏 2 [𝜉𝐺 + 𝜉𝐶 𝑃 + 𝜉𝐺 2 𝑏2 + 𝜉 𝜉 𝑏2 𝑃 + 𝜉 𝜉 𝜆 2 2 𝐺 𝐶 2 𝐺 𝐶 𝑚𝑎𝑥 (𝑆) + 𝜉𝐶 𝜆 𝑚𝑎𝑥 (𝑆)𝑃], (4.45) 𝐶(𝜎) =𝑏 2𝜀 𝑏 22 (𝜉𝐺 + 𝑃𝜉𝐶 )2 + 𝑏 22 𝜎 2 (𝜉𝐺 2 + 𝜉 2 𝑃 2 ). 𝐶 (4.46) Proof 9 Consider the Lyapunov function candidate 𝑉(𝑘) = 𝑡𝑟{Θ̃𝑇 (𝑘)Θ̃(𝑘)}. (4.47) 76 One knows that 𝑉(Θ̃(𝑘)) ≤ k Θ̃(𝑘)k 2 . (4.48) For the given 𝑉(𝑘) one has E[𝑉(𝑘 + 1)] − 𝑉(𝑘) = E[𝑡𝑟{Θ̃𝑇 (𝑘 + 1)Θ̃(𝑘 + 1)}] − 𝑡𝑟{Θ̃𝑇 (𝑘)Θ̃(𝑘)} = 𝑡𝑟{E[Θ̃𝑇 (𝑘 + 1))Θ̃(𝑘 + 1)] − Θ̃𝑇 (𝑘)Θ̃(𝑘)}. (4.49) Using Θ̃(𝑘) = Θ̂(𝑘) − Θ∗ in (4.40) gives X 𝑃 Θ̃(𝑘 + 1) =Θ̃(𝑘) − [Ξ𝐺 𝜑(𝑘)𝑒𝑇 (𝑘) + Ξ𝐶 𝜑(𝜏ℎ )𝑒𝑇ℎ (𝑘)]. (4.50) ℎ=1 Substituting (4.50) in (4.49) gives X𝑃 E[𝑉(𝑘 + 1)] − 𝑉(𝑘) = 𝑡𝑟{E[(Θ̃𝑇 (𝑘) − [Ξ𝐺 𝑒(𝑘)𝜑𝑇 (𝑘) + Ξ𝐶 𝑒 ℎ (𝑘)𝜑𝑇 (𝜏ℎ )])(Θ̃(𝑘) ℎ=1 𝑃 X − [Ξ𝐺 𝜑(𝑘)𝑒𝑇 (𝑘) + Ξ𝐶 𝜑(𝜏ℎ )𝑒𝑇ℎ (𝑘)])] − Θ̃𝑇 (𝑘)Θ̃(𝑘)}. (4.51) ℎ=1 Using (4.14) and (4.36), one has E[𝑉(𝑘 + 1)] − 𝑉(𝑘) = 𝑡𝑟{E[(Θ̃𝑇 (𝑘) − [Ξ𝐺 Θ̃𝑇 (𝑘)𝜑(𝑘)𝜑𝑇 (𝑘) − Ξ𝐺 𝜀(𝑘)𝜑𝑇 (𝑘) − Ξ𝐺 𝑣(𝑘)𝜑𝑇 (𝑘) X𝑃 X𝑃 X𝑃 + Ξ𝐶 Θ̃𝑇 (𝑘)𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) − Ξ𝐶 𝜀(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) − Ξ𝐶 𝑣(𝜏ℎ )𝜑𝑇 (𝜏ℎ )])(Θ̃(𝑘) ℎ=1 ℎ=1 ℎ=1 X𝑃 − [Ξ𝐺 𝜑(𝑘)𝜑𝑇 (𝑘)Θ̃(𝑘) − Ξ𝐺 𝜑(𝑘)𝜀𝑇 (𝑘) − Ξ𝐺 𝜑(𝑘)𝑣𝑇 (𝑘) + Ξ𝐶 𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ )Θ̃(𝑘) ℎ=1 X𝑃 X𝑃 − Ξ𝐶 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ ) − Ξ𝐶 𝜑(𝜏ℎ )𝑣𝑇 (𝜏ℎ )])] − Θ̃𝑇 (𝑘)Θ̃(𝑘)}. (4.52) ℎ=1 ℎ=1 P𝑃 By employing 𝑆 = ℎ=1 𝜑(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) and 𝜙(𝑘) = 𝜑(𝑘)𝜑𝑇 (𝑘), (4.52) is written as, E[𝑉(𝑘 + 1)] − 𝑉(𝑘) = 𝑡𝑟{E[(Θ̃𝑇 (𝑘) − [Ξ𝐺 Θ̃𝑇 (𝑘)𝜙(𝑘) − Ξ𝐺 𝜀(𝑘)𝜑𝑇 (𝑘) − Ξ𝐺 𝑣(𝑘)𝜑𝑇 (𝑘) X𝑃 X𝑃 + Ξ𝐶 Θ̃𝑇 (𝑘)𝑆 − Ξ𝐶 𝜀(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) − Ξ𝐶 𝑣(𝜏ℎ )𝜑𝑇 (𝜏ℎ )])(Θ̃(𝑘) − [Ξ𝐺 𝜙(𝑘)Θ̃(𝑘) ℎ=1 ℎ=1 𝑃 X − Ξ𝐺 𝜑(𝑘)𝜀𝑇 (𝑘) − Ξ𝐺 𝜑(𝑘)𝑣𝑇 (𝑘) + Ξ𝐶 𝑆 Θ̃(𝑘) − Ξ𝐶 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ ) ℎ=1 X𝑃 − Ξ𝐶 𝜑(𝜏ℎ )𝑣𝑇 (𝜏ℎ )])] − Θ̃𝑇 (𝑘)Θ̃(𝑘)}. (4.53) ℎ=1 77 Based on the independency of noise 𝑣(𝑘) and Assumption 6, the expectation of cross terms multiplication of 𝑣(𝑘) with Θ̃(𝑘) and 𝜀(𝑘), for every 𝑘 ∈ Z+ , is equal to zero and hence omitted. Therefore, (4.53) is rewritten as follows E[𝑉(𝑘 + 1)] − 𝑉(𝑘) = 𝑡𝑟{E[Θ̃𝑇 (𝑘)Θ̃(𝑘) − 2Ξ𝐺 Θ̃𝑇 (𝑘)𝜙(𝑘)Θ̃(𝑘) − 2Ξ𝐶 Θ̃𝑇 (𝑘)𝑆 Θ̃(𝑘) + Ξ𝐺 2 Θ̃𝑇 (𝑘)𝜙𝑇 (𝑘)𝜙(𝑘)Θ̃(𝑘) + 2Ξ Ξ Θ̃𝑇 (𝑘)𝑆𝜙(𝑘)Θ̃𝑇 (𝑘) + Ξ2 Θ̃𝑇 (𝑘)𝑆𝑇 𝑆 Θ̃𝑇 (𝑘) 𝐺 𝐶 𝐶 X𝑃 + 2Ξ𝐺 𝜀(𝑘)𝜑𝑇 (𝑘)Θ̃(𝑘) + 2Ξ𝐶 𝜀(𝜏ℎ )𝜑𝑇 (𝜏ℎ )Θ̃(𝑘) − 2Ξ𝐺 2 𝜀(𝑘)𝜑𝑇 (𝑘)𝜙𝑇 (𝑘)Θ̃(𝑘) ℎ=1 X 𝑃 + Ξ𝐺 2 𝜀(𝑘)𝜑𝑇 (𝑘)𝜑(𝑘)𝜀𝑇 (𝑘) − 2Ξ Ξ 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ )𝜙𝑇 (𝑘)Θ̃(𝑘) − 2Ξ𝐺 Ξ𝐶 𝜀(𝑘)𝜑𝑇 (𝑘)𝑆 Θ̃(𝑘) 𝐺 𝐶 ℎ=1 X𝑃 X𝑃 − 2Ξ𝐶 2 Θ̃𝑇 (𝑘)𝑆𝑇 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ ) + 2Ξ𝐺 Ξ𝐶 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ )𝜑(𝑘)𝜀𝑇 (𝑘) ℎ=1 ℎ=1 X𝑃 X𝑃 + Ξ𝐺 2 𝑣(𝑘)𝜑𝑇 (𝑘)𝜑(𝑘)𝑣𝑇 (𝑘) + Ξ2 𝜀(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) 𝜑(𝜏ℎ )𝜀𝑇 (𝜏ℎ ) 𝐶 ℎ=1 ℎ=1 X 𝑃 X𝑃 + Ξ𝑇𝐶 Ξ𝐶 𝑣(𝜏ℎ )𝜑𝑇 (𝜏ℎ ) 𝜑(𝜏ℎ )𝑣𝑇 (𝜏ℎ )] − Θ̃𝑇 (𝑘)Θ̃(𝑘)}. (4.54) ℎ=1 ℎ=1 Now using 𝑄 = 2Ξ𝐶 𝑆 + 2Ξ𝐺 𝜙(𝑘) − Ξ𝐺 2 𝜙𝑇 (𝑘)𝜙(𝑘) − 2Ξ Ξ 𝜙(𝑘)𝑆 − Ξ2 𝑆𝑇 𝑆, (4.55) 𝐺 𝐶 𝐶 one obtains the upper bound of (4.54) as E[𝑉(𝑘 + 1)] − 𝑉(𝑘) ≤ −Θ̃𝑇 (𝑘)𝜆 𝑚𝑖𝑛 (𝑄)Θ̃(𝑘) + 2k Θ̃(𝑘)k𝐵𝑏 𝜀 + 𝐶(𝜎), (4.56) where 𝜆 𝑚𝑖𝑛 (𝑄), 𝐵 and 𝐶(𝜎) are given in (4.44)-(4.46). Knowing 1 2𝐵2 𝑏 2𝜀 2k Θ̃(𝑘)k𝐵𝑏 𝜀 − 𝜆 𝑚𝑖𝑛 (𝑄)k Θ̃(𝑘)k 2 ≤ , 2 𝜆 𝑚𝑖𝑛 (𝑄) one rewrites (4.56) as 1 2𝐵2 𝑏 2𝜀 E[𝑉(𝑘 + 1)] − 𝑉(𝑘) ≤ − 𝜆 𝑚𝑖𝑛 (𝑄)k Θ̃(𝑘)k 2 + + 𝐶(𝜎) 2 𝜆 𝑚𝑖𝑛 (𝑄) = −𝛼3 (k Θ̃(𝑘)k) + 𝑑, (4.57) 78 where 𝑑 is given in (4.43) and 1 𝛼3 (k Θ̃(𝑘)k) = 𝜆 𝑚𝑖𝑛 (𝑄)k Θ̃(𝑘)k 2 . 2 Using (4.48), one can rewrite (4.57) as follows 1 E[𝑉(Θ̃(𝑘 + 1))] − 𝑉(Θ̃(𝑘)) ≤ − 𝜆 𝑚𝑖𝑛 (𝑄)k Θ̃(𝑘)k 2 +𝑑 2 1 ≤ − 𝜆 𝑚𝑖𝑛 (𝑄)𝑉(Θ̃(𝑘)) + 𝑑. (4.58) 2 Taking expectation on both sides of the above equation, and using E[ 12 𝜆 𝑚𝑖𝑛 (𝑄)𝑉(Θ̃(𝑘))] ≥ 1 2 𝜆 𝑚𝑖𝑛 (𝑄)E[𝑉(Θ̃(𝑘))] derived from Jason’s inequality, one has 1 E[𝑉(Θ̃(𝑘 + 1))] − E[𝑉(Θ̃(𝑘))] ≤ − 𝜆 𝑚𝑖𝑛 (𝑄)E[𝑉(Θ̃(𝑘))] + 𝑑. (4.59) 2 Now using Lemma 4 and (4.59), in order to show that the mentioned bound in (4.42) is exponential bounded in probability, one needs 1 0 < 𝜆 𝑚𝑖𝑛 (𝑄) < 1. (4.60) 2 In order to satisfy 0 < 12 𝜆 𝑚𝑖𝑛 (𝑄), using (4.44), one needs to choose 𝜉𝐶 > 0 and 𝜉𝐺 > 0 such that 1 (𝜉𝐺 𝑏 22 + 𝜉𝐶 𝜆 𝑚𝑎𝑥 (𝑆))2 < 𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆) + 𝜉𝐺 𝑏 21 , (4.61) 2 and to meet 21 𝜆 𝑚𝑖𝑛 (𝑄) < 1 or 𝜆 𝑚𝑖𝑛 (𝑄) < 2, using (4.44), one obtains 2𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆) + 2𝜉𝐺 𝑏 21 − (𝜉𝐺 𝑏 22 + 𝜉𝐶 𝜆 𝑚𝑎𝑥 (𝑆))2 < 2 ⇒ (𝜉𝐺 𝑏 22 + 𝜉𝐶 𝜆 𝑚𝑎𝑥 (𝑆))2 + 2(1 − 𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆) − 𝜉𝐺 𝑏 21 ) > 0 ⇒ 0 < 𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆) + 𝜉𝐺 𝑏 21 < 1. (4.62) Since 𝜉𝐶 > 0 and 𝜉𝐺 > 0 are chosen such that (4.41) is met, (4.61) and (4.62) are also satisfied and this leads to (4.60). Thus, if E[𝑉(Θ̃(𝑘))] > 𝜆 2𝑑(𝑄) , then E[𝑉(Θ̃ 𝑘+1 )]−E[𝑉(Θ̃(𝑘))] < 0, whereas, after E[𝑉(Θ̃(𝑘))] 𝑚𝑖𝑛 enters the set DΘ̃ = {Θ̃(𝑘) : E[𝑉(Θ̃(𝑘))] ≤ 𝑏}, (4.63) 79 it is possible to have E[𝑉(Θ̃ 𝑘+1 )]−E[𝑉(Θ̃(𝑘))] ≥ 0 where 𝑏 = 𝜆 2𝑑(𝑄) . However, E[𝑉(Θ̃(𝑘))] stays 𝑚𝑖𝑛 within the positive invariant set DΘ̃ . Thus, for E[𝑉(Θ̃(0))] > 𝑏, ultimately one has E[𝑉(Θ̃(𝑘))] ≤ 𝑏. Based on the Markov’s inequality, for any 𝜖 ∈ (0, 1), one has 𝑏 𝜖E[𝑉(Θ̃(𝑘))] P{𝑉(Θ̃(𝑘)) > }≤ ≤ 𝜖. (4.64) 𝜖 𝑏 Thus, using (4.48), it yields that 𝑏 𝜖E[𝑉(Θ̃(𝑘))] P{k Θ̃(𝑘)k 2 > }≤ ≤ 𝜖, (4.65) 𝜖 𝑏 which leads to r 𝑏 P{k Θ̃(𝑘)k≤ } ≥ 1 − 𝜖. (4.66) 𝜖 Therefore, based on Lemma 3 and Definition, Θ̃ is PS-P and exponential probabilistic ultimate bounded to the bound given in (4.42). This completes the proof. Finite-sample Boundedness of the Parameters’ Estimation Error in probability The pro- posed data-regularized CL-based SGD update rule (4.40) guarantees the exponential probabilistic ultimate boundedness of the parameter estimation error as 𝑘 → ∞. Moreover, the following lemma ensures that the proposed method can also guarantee finite-sample boundedness in probability of the parameters’ estimation error Θ̃(𝑘) at any time 𝑘 where 𝑘 > 𝑃. Therefore, it solves Problem 1. Lemma 7 Consider the approximator of nonlinear function in (4.9) given in (4.13), whose parameters are adjusted according to the update law of (4.40). Let Assumptions 5-7 hold. Once the rank condition on M is satisfied and 𝜉𝐶 > 0 and 𝜉𝐺 > 0 are chosen such that (4.41) is met, then the proposed update law (4.40) guarantees finite-sample boundedness in probability at every time 𝑘 > 𝑃 for parameter estimation error Θ̃(𝑘) = Θ̂(𝑘) − Θ∗ . That is, r 𝑏𝑘 P{k Θ̃(𝑘)k≤ } ≥ 1 − 𝜖, (4.67) 𝜖 where 1 𝑑 𝑏 𝑘 = (1 − 𝜆 𝑚𝑖𝑛 (𝑄)) 𝑘 k Θ̃(0)k 2 + 1 , (4.68) 2 𝜆 𝑚𝑖𝑛 (𝑄) 2 80 is a constant for every time 𝑘 > 𝑃, 𝜆 𝑚𝑖𝑛 (𝑄) and 𝑑 are respectively given in (4.44) and (4.43). Proof 10 Using (4.59) and (4.4) in Lemma 4, one has 1 𝑘−1 X 1 E[𝑉(Θ̃(𝑘))] ≤(1 − 𝜆 𝑚𝑖𝑛 (𝑄)) 𝑘 E[𝑉(Θ̃(0))] + 𝑑 (1 − 𝜆 𝑚𝑖𝑛 (𝑄))𝑖 ⇒ 2 𝑖=0 2 1 𝑑 E[k Θ̃(𝑘)k 2 ] ≤(1 − 𝜆 𝑚𝑖𝑛 (𝑄)) 𝑘 k Θ̃(0)k 2 + 1 . (4.69) 2 𝜆 𝑚𝑖𝑛 (𝑄) 2 Now, using Markov’s inequality, for every 0 < 𝜖 < 1, one has 𝑏 𝜖E[𝑉(Θ̃(𝑘))] P{𝑉(Θ̃(𝑘)) > 𝑘 } ≤ ≤ 𝜖, (4.70) 𝜖 𝑏𝑘 which implies (4.67) with 𝑏 𝑘 is given in (4.68). Therefore, (4.67) represents the finite-sample q 𝑏 bound in probability of 𝜖𝑘 for k Θ̃(𝑘)k at every finite-time 𝑘 > 𝑃. This completes the proof. Remark 20 As discussed in [124], the concurrent learning approach is based on the combination of a gradient descent algorithm with an auxiliary static feedback update law, which can be viewed as a type of 𝜎-modification [10] and ensures bounded exponential convergence without the PE requirement by keeping enough measurements in memory. Here, the same extension is applied to the proposed data-regularized CL-based SGD in (4.40). Remark 21 The parameter estimation law (4.40) converges exponentially fast to a bound which depends on the noise variance. Employing the memory data selection algorithm [144] which 𝜆 (𝑆) maximizes 𝜆 𝑚𝑖𝑛 (𝑆) helps to shrink the convergence bound of parameters’ estimation error in 𝑚𝑎𝑥 𝜆 (𝑆) (4.42). Moreover, leveraging rich memory data in terms of maximizing 𝜆 𝑚𝑖𝑛 (𝑆) leads to a narrower 𝑚𝑎𝑥 𝜆 (𝑆) finite-sample bound in (4.67) for the parameters’ estimation error. Intuitively, maximizing 𝜆 𝑚𝑖𝑛 (𝑆) 𝑚𝑎𝑥 provides a higher convexity parameter for the introduced data-regularized loss function (4.37). Remark 22 The analysis of this section shows that Based on (4.69), the error E[k Θ̃(𝑘)k 2 ] rate at any time 𝑘 is 𝑂((1 − 12 𝜆 𝑚𝑖𝑛 (𝑄)) 𝑘 ) + 𝑂( 𝜆 𝑑 (𝑄) ). Therefore, the parameter estimation error with a 𝑚𝑖𝑛 linear rate of 𝑂((1 − 21 𝜆 𝑚𝑖𝑛 (𝑄)) 𝑘 ) converges to the bound 1 𝑑 shrinking by rich memory data 2 𝜆 𝑚𝑖𝑛 (𝑄) 𝜆 (𝑆) selection through maximizing 𝜆 𝑚𝑖𝑛 (𝑆) that maximizes 𝜆 𝑚𝑖𝑛 (𝑄). Note that 𝜆 𝑚𝑖𝑛 (𝑆) amounts to the 𝑚𝑎𝑥 𝛼-strong convexity and 𝜆 𝑚𝑎𝑥 (𝑆) amounts to the 𝛽-smoothness. Therefore, through selecting data to reuse, the condition number of the function under optimization is improved and consequently the learning rate is improved. 81 4.5 Simulations In this section, the performance of the presented data-regularized CL-based SGD for online approximators with zero and non-zero MFAEs is compared with SGD [145] whose estimation law is given as follows, Θ̂(𝑘 + 1) = Θ̂(𝑘) − Γ𝐺 [𝜑(𝑘)𝑒𝑇 (𝑘)], (4.71) where Γ𝐺 = 𝛾𝐺 𝐼, with positive constants 𝛾𝐺 > 0. In the examples, the simulation time span is [𝑘 0 , 𝑘 𝑓 ] with 𝑘 0 = 0 and 𝑘 𝑓 = 10000, and the 𝑥 domain is defined by D = [𝑥 𝐿 , 𝑥 𝐻 ] with 𝑥 𝐿 < 𝑥 𝐻 , 𝑥 𝐿 , 𝑥 𝐻 ∈ R and D is quantized by 𝑥 −𝑥 [𝑥 𝐿 : 𝑘𝐻 −𝑘 𝐿 : 𝑥 𝐻 ]. In the proposed data-regularized CL-based SGD method, 𝜉𝐶 and 𝜉𝐺 are 𝑓 0 chosen such that (4.41) is satisfied and setting 𝜉𝐺 > 𝜉𝐶 , the current data is prioritized over recorded data. For SGD (4.71), let 𝛾𝐺 = 0.1. In all cases, the initial parameters’ values are all set to zero. The additive measurement noise 𝑣(𝑘) is a zero-mean independent white noise with uniform distribution in the interval [−¯ 𝑣 , 𝑣¯], i.e., 𝑣(𝑘) = −¯ 𝑣 + 2¯ 𝑣 (𝑟𝑎𝑛𝑑), where 𝑟𝑎𝑛𝑑 is used to generate pseudorandom scalar with uniform distribution on the interval (0, 1). Two different values of 𝑣¯ = 0.01 and 𝑣¯ = 0.1 which respectively lead to the variances 𝜎 2 = 3 × 10−5 and 𝜎 2 = 3 × 10−3 for the defined noise are respectively employed for Examples 1 and 2. To have a fair comparison between the mentioned methods for approximating 𝑓 (𝑥) on the whole domain of 𝑥, the learning error, given below, is calculated. Z 𝐸(𝑘) = E[ k𝑒(𝑥(𝑘))k𝑑 𝑛 𝑥], D where the expected value is estimated by averaging over several realizations of the learning algo- rithms, starting from the same initial condition. In the simulations, the results of the proposed data-regularized CL-based SGD and SGD methods are respectively labeled by CL-SGD and SGD. Example 1: Approximator with zero MFAE (𝜀(𝑘) = 0) Consider the following function 𝑦(𝑘) = 𝑝 1 𝑒 −𝑥(𝑘) + 𝑝 2 𝑒 −𝑥(𝑘) sin(𝑥(𝑘)) + 𝑣(𝑘), 82 where the parameters [𝑝 1 , 𝑝 2 ] are unknown and the regressors are known as 𝑧(𝑥(𝑘)) = [𝑒 −𝑥(𝑘) , 𝑒 −𝑥(𝑘) sin(𝑥(𝑘))], with 𝑞 = 2. The unknown parameters are [𝑝 1 , 𝑝 2 ] = [−0.5, 0.5] and D is given with 𝑥 𝐿 = 0 and 𝑥 𝐻 = 2. We set 𝑃 = 2 for data-regularized CL-based SGD methods. Let 𝜉𝐺 = 0.1, 𝜉𝐶 = 0.01 for data-regularized CL-based SGD method. Based on the obtained results, the rank condition on M matrix is satisfied in the first 𝑞 = 2 steps. Therefore, 𝑃 is chosen as 𝑃 = 2 which satisfies 𝑃 ≥ 𝑞. After 𝑃 steps, the data selection algorithm [144] is employed to improve the richness of the recorded memory data. Fig. 4.2 depicts the true parameters and the approximated parameters for data-regularized CL- based SGD and SGD methods for two different noise variances 𝜎 2 = 3 × 10−5 and 𝜎 2 = 3 × 10−3 . For both noise variances, Fig. 4.2 shows that while SGD could not converge to the vicinity of true parameters, but data-regularized CL-based SGD succeeded in convergence to the vicinity of true parameters. The online learning error 𝐸(𝑘) of data-regularized CL-based SGD and SGD for different noise variances 𝜎 2 = 3 × 10−5 and 𝜎 2 = 3 × 10−3 are plotted in Fig. 4.3 where data-regularized CL-based SGD shows converging to the vicinity of the origin while SGD could not approach zero. However, in Figs. 4.2 and 4.3, the converged values for the case with higher noise variance 𝜎 2 = 3 × 10−3 show larger variations in comparison with lower noise variance 𝜎 2 = 3 × 10−5 , as expected from (4.42). The integral absolute errors (IAEs) of 𝐸(𝑘) for data- regularized CL-based SGD and SGD methods are computed in Table 4.1 where data-regularized CL-based SGD with 𝐸(𝑘) IAEs, 348.41 and 480.97, respectively, for noise variances 𝜎 2 = 3×10−5 and 𝜎 2 = 3 × 10−3 , has resulted in better precision of online learning compared with SGD. In this example, using data selection algorithm [144], one obtains 𝜆 𝑚𝑖𝑛 (𝑆) = 0.24, 𝜆 𝑚𝑎𝑥 (𝑆) = 1.01, 𝑏 1 = 0.2 and 𝑏 2 = 1. Since in this example 𝑏 𝜀 = 0, for 𝜖 = 0.2, the probabilistic bounds in (4.42) for the defined noise with variances 𝜎 2 = 3 × 10−5 and 𝜎 2 = 3 × 10−3 is, respectively, obtained as 0.06 and 0.6. Fig. 4.4 shows that for 20 different implementations of the data-regularized CL-based SGD method for noise variances 𝜎 2 = 3 × 10−5 and 𝜎 2 = 3 × 10−3 , the parameter estimation error Θ̃(𝑘) stays within the specified bounds. Example 2: Approximators 83 Figure 4.2: Parameters’ estimation for approximators with zero MFAE. 0.2 0.15 E(k) 0.1 0.05 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 k (time steps) 0.3 0.2 E(k) 0.1 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 k (time steps) Figure 4.3: Online learning errors for approximators with zero MFAE. Table 4.1: IAE 𝐸(𝑘) learning errors comparison Example 1 Example 2 𝜎 2 = 3 × 10−5 𝜎 2 = 3 × 10−3 𝜎 2 = 3 × 10−5 𝜎 2 = 3 × 10−3 CL-SGD 348.41 480.97 15052 15369 SGD 1342.9 1347.9 42210 42213 84 Figure 4.4: Online parameter estimation error of approximators with zero MFAE for 20 different implementations. with non-zero MFAE (𝜀(𝑘) 6= 0) Now, consider the following function, 𝑦(𝑘) = 2 + cos(𝑥(𝑘)) + 𝑣(𝑘), (4.72) where the associated 𝑓 (𝑥) = 2 + cos(𝑥(𝑘)) is fully unknown. For this example, a radial basis function neural network is used with 5 radial basis functions − k 𝑥(𝑘)−𝑐𝑖 k 2 𝑒 2𝜎𝑖 2 , 𝑖 = 1, 2, ..., 5, where the centroids 𝑐𝑖 are uniformly picked on D = [𝑥 𝐿 , 𝑥 𝐻 ] = [−2, 2] and the spreads 𝜎𝑖 = 1.2 for all basis functions. The rank condition on M matrix is satisfied in 𝑞 = 5 steps; therefore, 𝑃 is chosen as 𝑃 = 𝑞 = 5 satisfying 𝑃 ≥ 𝑞. The data selection algorithm in [144] is employed after the first 5 steps to improve the richness of the recorded data. The approximation of (4.72) is given as − k 𝑥(𝑘)+2 k 2 − k 𝑥(𝑘)−2 k 2 𝑦ˆ(𝑘) = Θ̂𝑇 (𝑘)𝜑(𝑥(𝑘)) = [𝑝 1 , 𝑝 2 , ..., 𝑝 5 ][𝑒 2(1.2)2 , ..., 𝑒 2(1.2)2 ]𝑇 . Employing 𝜉𝐺 = 0.1, 𝜉𝐶 = 0.05 for data-regularized CL-based SGD method for noise variances 𝜎 2 = 3×10−5 and 𝜎 2 = 3×10−3 leads to the approximated parameters shown in Fig. 4.5. The SGD parameters in Fig. 4.5 did not converge to the suitable parameters, while data-regularized CL-based 85 Figure 4.5: Parameters’ estimation for approximators with non-zero MFAE. 5 4 f(x) 3 2 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 5 4 f(x) 3 2 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x Figure 4.6: Steady-state uncertainty approximations. 10 E(k) 5 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 k (time steps) 10 E(k) 5 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 k (time steps) Figure 4.7: Identification errors for approximators with non-zero MFAE. 86 SGD succeeded in convergence to the appropriate parameters. The steady state approximations for the function 𝑓 (𝑥) is given in Fig. 4.6 where for the two different values of the noise variances, it is depicted that the data-regularized CL-based SGD could better identify the unknown function compared with SGD. As the comparison of the learning error 𝐸(𝑘) in Fig. 4.7 shows, the SGD method could not perform well in the unknown function identification, however data-regularized CL-based SGD error showed ultimate bounded convergence near zero. Moreover, based on the IAE for 𝐸(𝑘) in Table 1, data-regularized CL-based SGD results in 15052 and 15369 for noise variances 𝜎 2 = 3 × 10−5 and 𝜎 2 = 3 × 10−3 , respectively, which are lower in comparison with SGD. 4.6 Conclusion This chapter presents a data-regularized concurrent learning-based stochastic gradient descent (CL-based SGD) method that leverages recorded data to guarantee linear (exponential) bounded convergence of the estimated parameters’ error. It is shown that the richness of the memory data improves the speed of convergence and reduces the probabilistic bound of convergence. Lyapunov analysis guaranteed that the proposed data-regularized CL-based SGD method not only ensures the practical stability in probability of the estimated parameters’ error but can ensure a finite-sample boundedness in probability of the estimated parameters’ error. Simulation results verified that the employed data-regularized CL-based SGD could improve the speed and precision of convergence for the estimated parameters in comparison with SGD. 87 CHAPTER 5 DETERMINISTIC AND STOCHASTIC FIXED-TIME STABILITY OF DISCRETE-TIME AUTONOMOUS SYSTEMS 5.1 Introduction In this chapter, we develop fixed-time stability conditions for both deterministic and stochastic DT autonomous nonlinear systems. First, fixed-time stability for equilibria of deterministic DT autonomous systems is defined. That is, a settling-time function is defined with a fixed upper bound independent of the initial condition. We then present Lyapunov theorems for fixed-time stability of both unperturbed and perturbed deterministic DT systems. Moreover, the sensitivity of fixed-time stability properties to perturbations of systems is investigated under the assumption of the existence of a locally Lipschitz discrete Lyapunov function. It is ensured that fixed-time stability is preserved under perturbations in the form of fixed-time attractiveness. Furthermore, sufficient Lyapunov conditions for fixed-time stability in probability of stochastic DT systems and their stochastic settling-time function are presented. The presented framework will pave the way for designing control laws with guaranteed satisfaction of a given performance measure in fixed time. Moreover, the presented stability results can be leveraged to develop fixed-time observers and identifiers for deterministic and stochastic DT systems, which are of great importance in control of safety-critical systems that highly rely on a system model and a state estimator to make less-conservative and feasible decisions. This is because fixed-time stability allows the system to preview and quantify probable errors in state estimators and identifiers considerably fast, which can be employed by the control system to avoid conservatism. Notations: In this chapter, the following notations are employed. R, R+ , Z, N+ , and N represent, respectively, the set of real numbers, non-negative real numbers, integer numbers, natural numbers except zero, and natural numbers. Moreover, R𝑛 represents the set of 𝑛 × 1 real column vectors. k.k is used to denote induced 2-norm for matrices and the Euclidean norm for vectors. The trace of a matrix 𝐴 is indicated with 𝑡𝑟(𝐴). |.| denotes the absolute value of any scalar 𝑥. ⌊.⌋ : R ↦→ Z is the floor function. ∆(.) is the DT difference operator for deterministic systems 88 and is defined for a function 𝑉(𝑦(𝑘)) : R𝑛 ↦→ R+ as ∆𝑉(𝑦(𝑘 + 1)) = 𝑉(𝑦(𝑘 + 1)) − 𝑉(𝑦(𝑘)). All random variables are assumed to be defined on a probability space (Ω, F , P), with Ω as the sample space, F as its associated Borel 𝜎-algebra and P as the probability measure. For a random variable 𝜈 : Ω −→ R𝑛 defined on the probability space (Ω, F , P), with some abuse of notation, the statement 𝜈 ∈ R𝑛 is used to state the dimension of the random variable. E[𝑋] denotes the expected value of the random variable 𝑋 on the probability space (Ω, F , P). It is assumed that the probability space (Ω, F , P) admits a sequence of mutually independent identically distributed random vectors 𝜈(𝑘), 𝑘 ∈ N. 5.2 Fixed-time Stability for Deterministic Discrete-time Systems In this section, the fixed-time stability of autonomous unperturbed deterministic DT systems is defined and the Lyapunov theorem specifying the sufficient conditions for their fixed-time stability is presented. Consider the following nonlinear DT system, 𝑦(𝑘 + 1) = 𝐹(𝑦(𝑘)), (5.1) where 𝐹 : D 𝑦 ↦→ D 𝑦 , 𝐹(0) = 0 is a nonlinear function on D 𝑦 , and D 𝑦 is an open set with 0 ∈ D 𝑦 . Moreover, 𝑦(𝑘) ∈ D 𝑦 ⊆ R𝑛 , 𝑘 ∈ N is the system state vector. For an initial condition 𝑦(0), define the solution sequence 𝑦(𝑘), 𝑘 ∈ N 𝑦(0) ⊆ N, where N 𝑦(0) is the maximal interval of existence of 𝑦(𝑘) after which the solution may cease outside the domain of 𝐹(.). Then, the solution sequence 𝑦(𝑘), 𝑘 ∈ N 𝑦(0) ⊆ N is uniquely defined in forward time for every initial condition 𝑦(0) ∈ D 𝑦 irrespective of whether or not the function 𝐹(.) is a continuous function [97]. Before proceeding, the following definitions are needed. Definition 13 (Locally Lipschitz function) A function 𝑓 (𝑥) is locally Lipschitz on a domain Ω ⊂ R𝑛 if for each point in Ω there exist a neighborhood Ω0 and a positive constant 𝐿 such that || 𝑓 (𝑥) − 𝑓 (𝑦)||≤ 𝐿 ||𝑥 − 𝑦||, ∀𝑥 ∈ Ω0 , 𝑦 ∈ Ω0 . (5.2) Moreover, 𝐿 is called the Lipschitz constant of 𝑓 (𝑥). 89 The following definition extends the fixed-time stability definition presented in [65] for CT systems to DT systems. Definition 14 (Fixed-time stability) Consider the DT nonlinear system (5.1). The zero solution of 𝑦(𝑘) = 0 to the system (5.1) is said to be fixed-time stable, if there exist an open neighborhood N𝑦 ⊆ D 𝑦 of the origin and a settling time function 𝐾 : N𝑦 \{0} ↦→ N+ , such that: 1) The system (5.1) is Lyapunov stable. That is, for every 𝜖 > 0, there exists a 𝛿 > 0 such that if ||𝑦(0)||≤ 𝛿, then ||𝑦(𝑘)||≤ 𝜖 for all 𝑘 ∈ {0, ..., 𝐾(𝑦(0)) − 1}. 2) For every initial condition 𝑦(0) ∈ N𝑦 \{0}, the solution sequence 𝑦(𝑘) of (5.1) reaches the equilibrium point and remains there after 𝑘 > 𝐾(𝑦(0)) and ∀𝑦(0) ∈ N𝑦 , where 𝐾 : N𝑦 \{0} ↦→ N+ . 3) The settling-time function 𝐾(𝑦(0)) is bounded, i.e., ∃𝐾𝑚𝑎𝑥 ∈ N+ : 𝐾(𝑦(0)) ≤ 𝐾𝑚𝑎𝑥 , ∀𝑦(0) ∈ N𝑦 \{0}. DT nonlinear system (5.1) is globally fixed-time stable if it is fixed-time stable with N𝑦 = D 𝑦 = R𝑛 . Remark 23 If only conditions 1) and 2) of the above definitions are satisfied, the finite-time stability [59] is resulted. In contrast, the fixed-time stability imposes the additional condition 3). This requirement makes the upper bound of the settling time in the fixed-time stability independent of the initial condition, in contrast to the finite-time stability. Therefore, the fixed-time stability is a stronger type of stability than the finite-time stability. The following theorem provides sufficient conditions under which the system (5.1) is fixed-time stable. Theorem 6 Consider the nonlinear DT system (5.1). Suppose there is a Lyapunov function 𝑉 : D 𝑦 ↦→ R+ where D 𝑦 is an open neighborhood around the origin and there exist a neighborhood Ω 𝑦 ⊂ D 𝑦 of the origin such that 𝑉(𝑦(0)) = 0, (5.3) 𝑉(𝑦(𝑘)) > 0, 𝑦(𝑘) ∈ Ω 𝑦 \{0}, (5.4) 𝑉(𝑦(𝑘)) ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼 min{ , max{𝑉 𝑟 1 (𝑦(𝑘)), 𝑉 𝑟 2 (𝑦(𝑘))}}, 𝑦(𝑘) ∈ Ω 𝑦 \{0}, (5.5) 𝛼 for some positive constants 0 < 𝛼 < 1, 0 < 𝑟 1 < 1, and 𝑟 2 > 1. Then, the system (5.1) is fixed-time 90 stable and has a settling-time function 𝐾 : N𝑦 ↦→ N+ that satisfies 1 1 1 𝐾(𝑦(0)) ≤ ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 −1 1−𝑟 )⌋ + ⌊𝛼 (𝛼 2 − 1)⌋ + 3, (5.6) for all 𝑦(0) ∈ N𝑦 \{0} where N𝑦 is an open neighborhood of the origin. Moreover, if D 𝑦 = R𝑛 , 𝑉(.) is radially unbounded and (5.5) holds on R𝑛 , then system (5.1) is globally fixed-time stable. Proof The Lyapunov stability of the system (5.1) can be concluded using similar arguments as of [97]. The proof of fixed-time stability consists of three parts. In the first part, we show that 1 1−𝑟 for 𝑉(𝑦(0)) ≥ 𝛼 2 , the settling time function is 𝐾(𝑦(0)) = 1. In the second part, we show 1 1 that if 𝛼 1−𝑟 1 1−𝑟 < 𝑉(𝑦(0)) < 𝛼 2 , there exists a settling-time function with a fixed upper bound 𝐾 ∗ (i.e., 𝐾(𝑦(0)) ≤ 𝐾 ∗ ) such that one has 𝑉(𝑦(𝑘)) = 0, ∀𝑘 > 𝐾 ∗ . Finally, in the third part, 1 for 𝑉(𝑦(0)) ≤ 𝛼 1−𝑟 1 , the Lyapunov function reaches 𝑉(𝑘) = 0 with the settling-time function 𝐾(𝑦(0)) = 1. Since 0 < 𝑟 1 < 1 and 𝑟 2 > 1, one has 𝑉 𝑟 2 (𝑦(𝑘)) ≤ 𝑉 𝑟 1 (𝑦(𝑘)), ∀𝑉(𝑦(𝑘)) ≤ 1, (5.7) 𝑉 𝑟 1 (𝑦(𝑘)) < 𝑉 𝑟 2 (𝑦(𝑘)), ∀𝑉(𝑦(𝑘)) > 1. (5.8) 1 1 We first prove part 1 where 𝑉(𝑦(0)) ≥ 𝛼 1−𝑟 2 . In this case, since 𝛼 1−𝑟 2 > 1, using (5.8), (5.5) leads to 𝑉(𝑦(𝑘)) 𝑟 ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼 min{ , 𝑉 2 (𝑦(𝑘))}. (5.9) 𝛼 1 1−𝑟 Moreover, since 𝑉(𝑦(0)) ≥ 𝛼 2 , the above inequality for 𝑘 = 0 yields ∆𝑉(𝑦(1)) ≤ −𝑉(𝑦(0)). (5.10) 1 1−𝑟 Now, (5.10) implies that the settling time function is 𝐾(𝑦(0)) = 1, for 𝑉(𝑦(0)) ≥ 𝛼 2 . For part 2 where 1 1 𝛼 1−𝑟 1 1−𝑟 < 𝑉(𝑦(𝑘)) < 𝛼 2 , based on (5.5), first we show that 𝑉(𝑘) reduces to 𝑉(𝑦(𝑘)) ≤ 1 after some time where this time is upper bounded by a fixed constant 𝐾1∗ . 91 1 Note that for 1 < 𝑉(𝑦(𝑘)) < 𝛼 1−𝑟 2 , using (5.8), one has 𝑉(𝑦(𝑘)) min{ , max{𝑉 𝑟 1 (𝑦(𝑘)), 𝑉 𝑟 2 (𝑦(𝑘))}} = 𝑉 𝑟 2 (𝑦(𝑘)), (5.11) 𝛼 Then, (5.11) and (5.5), lead to 𝑉(𝑦(𝑘 + 1)) ≤ 𝑉(𝑦(𝑘)) − 𝛼𝑉 𝑟 2 (𝑦(𝑘)). (5.12) 1 The condition (5.12) holds for 𝑘 = 0, ..., 𝐾1∗ − 1 where 1 < 𝑉(𝑦(𝑘)) < 𝛼 1−𝑟 2 . Therefore, using (5.12) for 𝑘 = 0, 1, ..., 𝐾1∗ − 1, one has 𝑉(𝑦(1)) − 𝑉(𝑦(0)) ≤ −𝛼𝑉 𝑟 2 (𝑦(0)), 𝑉(𝑦(2)) − 𝑉(𝑦(1)) ≤ −𝛼𝑉 𝑟 2 (𝑦(1)), .. . 𝑉(𝑦(𝐾1∗ − 1)) − 𝑉(𝑦(𝐾1∗ − 2)) ≤ −𝛼𝑉 𝑟 2 (𝑦(𝐾1∗ − 2)), 𝑉(𝑦(𝐾1∗ )) − 𝑉(𝑦(𝐾1∗ − 1)) ≤ −𝛼𝑉 𝑟 2 (𝑦(𝐾1∗ − 1)), (5.13) where summing up the left and right-hand-side terms leads to 𝐾 ∗ −1 X1 𝑉(𝑦(𝐾1∗ )) − 𝑉(𝑦(0)) ≤ −𝛼𝑉 𝑟 2 (𝑦(𝑘)). (5.14) 𝑘=0 Using the fact that 𝑉(𝑦(𝑘)) < 𝑉(𝑦(𝑘 − 1)) and (5.14), one has 𝐾 ∗ −1 𝐾 ∗ −1 X1 X1 𝑉 𝑟 2 (𝑦(𝑘) ≥ 𝐾1∗𝑉 𝑟 2 (𝑦(𝐾1∗ − 1)), ⇒ −𝛼 𝑉 𝑟 2 (𝑦(𝑘) ≤ −𝛼𝐾1∗𝑉 𝑟 2 (𝑦(𝐾1∗ − 1)). (5.15) 𝑘=0 𝑘=0 Employing (5.14) and (5.15) leads to 𝑉(𝑦(𝐾1∗ )) − 𝑉(𝑦(0)) ≤ −𝛼𝐾1∗𝑉 𝑟 2 (𝑦(𝐾1∗ − 1)), (5.16) which implies 𝑉(𝑦(0)) − 𝑉(𝑦(𝐾1∗ )) 𝐾1∗ ≤ . (5.17) 𝛼𝑉 𝑟 2 (𝑦(𝐾1∗ − 1)) 92 Using 𝑉(𝑦(𝑘)) < 𝑉(𝑦(𝑘 − 1)), one can rewrite (5.17) as follows 𝑉(𝑦(0)) − 𝑉(𝑦(𝐾1∗ )) 𝐾1∗ ≤ . (5.18) 𝛼𝑉 𝑟 2 (𝑦(𝐾1∗ )) 1 1−𝑟 Having 1 < 𝑉(𝑦(0)) < 𝛼 2 for 𝑘 < 𝐾1∗ and 𝑉(𝑦(𝐾1∗ )) ≤ 1, (5.18) implies 1 𝛼 1−𝑟 2 − 1 𝐾1∗ ≤ , (5.19) 𝛼 which leads to the integer upper bound for 𝐾1∗ as follows 1 𝐾1∗ ≤ ⌊𝛼−1 (𝛼 1−𝑟 2 − 1)⌋ + 1. (5.20) 1 ∗ 1−𝑟 Note that since for 𝑘 > 𝐾1 one has 𝑉(𝑦(𝑘)) ≤ 1. Thus, for 𝛼 1 < 𝑉(𝑦(𝑘)) ≤ 1, using (5.7) one has 𝑉(𝑦(𝑘)) min{ , max{𝑉 𝑟 1 (𝑦(𝑘)), 𝑉 𝑟 2 (𝑦(𝑘))}} = 𝑉 𝑟 1 (𝑦(𝑘)), 𝛼 which leads to rewriting (5.5) as follows 𝑉(𝑦(𝑘 + 1)) ≤ 𝑉(𝑦(𝑘)) − 𝛼𝑉 𝑟 1 (𝑦(𝑘)). (5.21) 1 There exists a fixed positive integer 𝐾2∗ and time 𝑘 > 𝐾2∗ such that 𝑉(𝑘) reaches 𝑉(𝑦(𝑘)) ≤ 𝛼 1−𝑟 1 and using (5.21) for 𝑘 = 𝐾1∗ , 𝐾1∗ + 1, ..., 𝐾2∗ − 1 one obtains 𝑉(𝑦(𝐾1∗ + 1)) − 𝑉(𝑦(𝐾1∗ )) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝐾1∗ )), 𝑉(𝑦(𝐾1∗ + 2)) − 𝑉(𝑦(𝐾1∗ + 1)) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝐾1∗ + 1)), .. . 𝑉(𝑦(𝐾2∗ − 1)) − 𝑉(𝑦(𝐾2∗ − 2)) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝐾2∗ − 2)), 𝑉(𝑦(𝐾2∗ )) − 𝑉(𝑦(𝐾2∗ − 1)) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝐾2∗ − 1)), (5.22) Summation of the left and right-half-side terms in (5.22) gives 𝐾 ∗ −𝐾 ∗ −1 2 X1 𝑉(𝑦(𝐾2∗ )) − 𝑉(𝑦(𝐾1∗ )) ≤ −𝛼 𝑉 𝑟 1 (𝑦(𝐾1∗ + 𝑖)). (5.23) 𝑖=0 93 Using the fact that 𝑉(𝑘) ≤ 𝑉(𝑘 −1) and (5.23), by employing a similar procedure as in (5.14)-(5.18), one obtains, 𝑉(𝐾1∗ ) − 𝑉(𝐾2∗ − 1) 𝐾2∗ − 𝐾1∗ ≤ . (5.24) 𝛼𝑉 𝑟 1 (𝑦(𝐾2∗ − 1)) 1 Since 𝛼 1−𝑟 1 < 𝑉(𝑦(𝑘)) < 1 for 𝑘 = 𝐾1∗ , 𝐾1∗ + 1, ..., 𝐾2∗ − 1, (5.24) reduces to 1 1 𝐾2∗ ≤ 𝐾1∗ + ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 )⌋ + 1. (5.25) Using (5.20), (5.25) is rewritten as follows 1 1 1 𝐾2∗ ≤ ⌊𝛼−1 (𝛼 1−𝑟 2 − 1)⌋ + ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 )⌋ + 2. (5.26) 1 ∗ 1−𝑟 At time 𝑘 > 𝐾2 for which 𝑉(𝑦(𝑘)) ≤ 𝛼 1 , (5.5) reduces to ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝑉(𝑦(𝑘)), (5.27) which leads to 𝑉(𝑦(𝑘 + 1)) = 0 for 𝑘 ≥ 𝐾2∗ + 1. This completes the proof of part 2. 1 The proof of part 3 where 𝑉(𝑦(0)) ≤ 𝛼 1−𝑟 1 is also derived based on (5.27) where 𝑉(𝑘) reaches zero with 𝐾(𝑦(0)) = 1. Hence, the Lyapunov function reaches 𝑉(𝑦(𝑘)) = 0 with the settling-time function 𝐾(𝑦(0)) such that 1 1 𝐾(𝑦(0)) = 1, 𝑉(𝑦(0)) ≥ 𝛼 1−𝑟 2 𝑎𝑛𝑑 𝑉(𝑦(0)) ≤ 𝛼 1−𝑟 1 , (5.28) and 1 1 1 1 1 𝐾(𝑦(0)) ≤ ⌊𝛼−1 (𝛼 1−𝑟 2 − 1)⌋ + ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 )⌋ + 3, 𝛼 1−𝑟 1 < 𝑉(𝑦(0)) < 𝛼 1−𝑟 2 . (5.29) Therefore, the system is fixed-time stable, and the system trajectory converges to the origin with the settling-time function given in (5.6). This completes the proof. 94 Moreover, if N𝑦 = D 𝑦 = R𝑛 and 𝑉(.) is radially unbounded, the global fixed-time stability follows using the same procedure.  Remark 24 Based on the definitions of fixed-time [65] and finite-time [59] stabilities, an autonomous DT fixed-time stable system is also finite-time stable. Fixed-time stability, which requires stronger conditions in comparison with finite-time stability, needs to represent a fixed upper bound for the settling-time function. However, in finite-time stability, the settling-time is a function of the initial conditions and no fixed upper bound is provided. Therefore, fixed-time stability is a stronger type of stability than asymptotic, exponential and finite-time stabilities for DT systems. 5.3 Sensitivity to Deterministic Perturbation for Fixed-time Stable Discrete-time Systems The system (5.1) usually describes a nominal model of the system that works under ideal conditions. Nevertheless, many real-world systems are under uncertainties and disturbances that affect the system’s behavior. To account for these uncertainties, a more accurate representation of the system can be given by the following deterministic perturbed model 𝑦(𝑘 + 1) = 𝐹(𝑦(𝑘)) + 𝑔(𝑘, 𝑦(𝑘)), (5.30) where 𝑔 represents a perturbation caused by disturbances, uncertainties, or modeling errors. This section investigates the solution behavior of the deterministic perturbed system (5.30) in a neigh- borhood of the fixed-time stable equilibrium of the nominal system (5.1). Assumption 8 The perturbation term 𝑔 is bounded, i.e., sup k𝑔(𝑘, 𝑦(𝑘))k< 𝛿0 , (5.31) N+ ×D 𝑦 for some 𝛿0 < ∞. The following definition extends the fixed-time attractiveness definition presented in [65] and [146] for CT systems to DT systems. Definition 15 (Fixed-time attractiveness) The perturbed system (5.30) is said to be fixed-time attractive by a bounded set Y around the equilibrium point, if ∀𝑦(0) ∈ N𝑦 the solution sequence 95 𝑦(𝑘) of (5.30) reaches Y in finite time 𝑘 > 𝐾(𝑦(0)) and remains there for all 𝑘 > 𝐾(𝑦(0)), where 𝐾 : N𝑦 \{0} ↦→ N+ is the settling-time function and the settling-time function 𝐾(𝑦(0)) is bounded, i.e., ∃𝐾𝑚𝑎𝑥 ∈ N+ : 𝐾(𝑦(0)) ≤ 𝐾𝑚𝑎𝑥 , ∀𝑦(0) ∈ N𝑦 . The following lemma is required in the proof of Lyapunov-based fixed-time attractiveness of perturbed deterministic systems. Lemma 8 Let 𝑉(𝑦(𝑘)) : D 𝑦 ↦→ R+ be a fixed-time Lyapunov function for the the nominal (unperturbed) system (5.1), i.e., 𝑉(𝑦(𝑘)) satisfies conditions (5.3)-(5.5) for the system (5.30) when 𝑔 = 0. Let also 𝑉(𝑦(𝑘)) be locally Lipschitz continuous on D 𝑦 with Lipschitz constant 𝐿𝑉 and Assumption 8 hold. Then, for the perturbed deterministic system (5.30), 𝑉(𝑘) satisfies 𝑉(𝑦(𝑘)) ∆𝑉(𝑦(𝑘 + 1)) ≤ − 𝛼 min{ , max{𝑉 𝑟 1 (𝑦(𝑘)), 𝑉 𝑟 2 (𝑦(𝑘))}} + 𝐿𝑉 k𝑔(𝑘, 𝑦(𝑘))k, (5.32) 𝛼 where ∆𝑉(𝑦(𝑘 + 1)) is computed along the solution of the unperturbed deterministic system. Proof The proof is similar to [147], which is developed for exponential stability, and is thus omitted.  The following theorem provides the behavior of deterministic fixed-time stable DT systems under bounded deterministic perturbations. Theorem 7 Suppose there exists a Lyapunov function 𝑉 : Ω 𝑦 ↦→ R+ which is locally Lipschitz on an open neighborhood Ω 𝑦 of the origin with Lipschitz constant 𝐿𝑉 and satisfies (5.3)-(5.5) for the nominal system (5.1) for some real positive numbers 𝛼, 𝑟 1 , 𝑟 2 > 0 such that 0 < 𝛼 < 1, 0 < 𝑟 1 < 1, and 𝑟 2 > 1. Let Assumption 8 hold. Then, around the origin, the system (5.30) is fixed-time attractive to the following bound 𝑏 𝑦 = {𝑦 ∈ Ω 𝑦 : 𝑉(𝑦) ≤ B}, (5.33) where 1   𝑚 𝐿 𝛿 1  ( 1 𝛼𝑉 0 ) 𝑟 2 ,   1 < 𝑉(𝑦(0)) < 𝛼 1−𝑟 2 , B= 1 (5.34)  𝑚 2 𝐿𝑉 𝛿0 𝑟1  1−𝑟  ( 𝛼 ) 1,  𝛼 1 < 𝑉(𝑦(0)) ≤ 1, 96 and its fixed-time bounded settling-time function is 𝐾(𝑦(0)) ≤ 𝐾 ∗ where  1 1   −1 1−𝑟 1−𝑟  ⌊𝛼𝑐 (𝛼 2 − 1)⌋ + 1,  1 < 𝑉(𝑦(0)) < 𝛼 2 , 𝐾∗ = 𝑟1 1 (5.35)     ⌊𝛼𝑑−1 (𝛼 𝑟 1 −1 − 𝛼)⌋ + 1, 𝛼 1−𝑟 1 < 𝑉(𝑦(0)) ≤ 1, 𝛼𝑐 = (1 − 𝑚1 )𝛼, 𝛼𝑑 = (1 − 𝑚1 )𝛼. The constants 𝑚 1 > 1 and 𝑚 2 > 1 are selected such that 1 2  1   1−𝑟  𝛼B 𝑟 2 − 𝑚 1 𝐿𝑉 𝛿0 > 0,  1 < 𝑉(𝑦(0)) < 𝛼 2 , 1 (5.36)    𝛼B 𝑟 1 − 𝑚 2 𝐿𝑉 𝛿0 > 0,  𝛼 1−𝑟 1 < 𝑉(𝑦(0)) ≤ 1 . Proof According to Theorem 6, the origin is the fixed-time stable equilibrium for the unper- turbed or nominal system (5.1). Lemma 8 and (5.31) imply that 𝑉(𝑦(𝑘)) ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼 min{ , max{𝑉 𝑟 1 (𝑦(𝑘)), 𝑉 𝑟 2 (𝑦(𝑘))}} + 𝐿𝑉 𝛿0 . (5.37) 𝛼 1 For 1 < 𝑉(𝑦(0)) < 𝛼 1−𝑟 2 , (5.37) leads to ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼𝑉 𝑟 2 (𝑦(𝑘)) + 𝐿𝑉 𝛿0 . (5.38) 1 1−𝑟 Having 1 < 𝑉(𝑦(0)) < 𝛼 2 and 𝑉(𝑦(0)) > B, and using (5.36) and 𝑚 1 > 1, one has 𝛼B 𝑟 2 − 𝑚 1 𝐿𝑉 𝛿0 > 0 ⇒ −𝛼B 𝑟 2 + 𝑚 1 𝐿𝑉 𝛿0 < 0, ⇒ −𝛼B 𝑟 2 + 𝐿𝑉 𝛿0 < 0, (5.39) which results in 1 𝐿𝑉 𝛿 0 < 𝛼B 𝑟 2 . (5.40) 𝑚1 1 For 𝑦(0) ∈/ 𝑏 𝑦 (𝑉(𝑦(0)) > B) and 1 < 𝑉(𝑦(0)) < 𝛼 1−𝑟 2 , (5.38) and (5.40) imply that 1 ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼𝑉 𝑟 2 (𝑘) + 𝛼B 𝑟 2 . (5.41) 𝑚1 Using 𝑉(𝑦(𝑘)) > B, (5.41) is upper bounded as follows ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼𝑐𝑉 𝑟 2 (𝑦(𝑘)), (5.42) 97 such that 𝛼𝑐 = (1 − 𝑚1 )𝛼 is positive. Using the results of part 2 in the proof of Theorem 6, (5.42) 1 1 implies that for 𝑦(0) ∈/ 𝑏 𝑦 and 1 < 𝑉(𝑦(0)) < 𝛼 1−𝑟 2 with 𝛼 < 𝑚 1 𝐿𝑉 𝛿0 , 𝑦(𝑘) reaches the invariant 1 set (5.33) within the fixed time steps 𝐾 ∗ = ⌊𝛼𝑐−1 (𝛼 1−𝑟 2 − 1)⌋ + 1 and remains there after. 1 Using (5.37), for 𝛼 1−𝑟 1 < 𝑉(𝑦(0)) ≤ 1, one has ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝑘)) + 𝐿𝑉 𝛿0 . (5.43) 1 Having 𝛼 1−𝑟 1 < 𝑉(𝑦(0)) ≤ 1 and 𝑉(𝑦(0)) > B, and using (5.36) and 𝑚 2 > 1, one has 𝛼B 𝑟 1 − 𝑚 2 𝐿𝑉 𝛿0 > 0 ⇒ −𝛼B 𝑟 1 + 𝑚 2 𝐿𝑉 𝛿0 < 0, ⇒ −𝛼B 𝑟 1 + 𝐿𝑉 𝛿0 < 0. (5.44) From (5.44), one obtains 1 𝐿𝑉 𝛿 0 < 𝛼B 𝑟 1 . (5.45) 𝑚2 1 1−𝑟 For 𝑦(0) ∈/ 𝑏 𝑦 (𝑉(𝑦(0)) > B) and 𝛼 1 < 𝑉(𝑦(0)) ≤ 1, then (5.43) and (5.45) imply that 1 ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼𝑉 𝑟 1 (𝑦(𝑘)) + 𝛼B 𝑟 1 . (5.46) 𝑚2 Using 𝑉(𝑦(𝑘)) > B, (5.46) is upper bounded as follows ∆𝑉(𝑦(𝑘 + 1)) ≤ −𝛼𝑑 𝑉 𝑟 1 (𝑦(𝑘)), (5.47) such that 𝛼𝑑 = (1 − 𝑚1 )𝛼 is positive. Using the results of part 2 in Theorem 6 proof, (5.47) implies 2 1 that for 𝑦(0) ∈/ 𝑏 𝑦 and 𝛼 1−𝑟 1 < 𝑉(𝑦(0)) < 1 with 𝑚 2 𝐿𝑉 𝛿0 < 𝛼, 𝑦(𝑘) reaches the invariant set 𝑟1 ∗ −1 (5.33) within the fixed time steps 𝐾 = ⌊𝛼𝑑 (𝛼 1 −1 − 𝛼)⌋ + 1 and remains in 𝑏 𝑦 ever after. This 𝑟 completes the proof.  Remark 25 In (5.33), the bound B is either a function of 𝑚 1 or 𝑚 2 , as given is (5.34). Notice that the fixed-time attractive bound (5.34) increases by choosing large values for 𝑚 1 or 𝑚 2 and accordingly the fixed-time of convergence given in (5.35) decreases. Therefore, the bigger we choose the bounded set B, the shorter the fixed-time of convergence will be. 98 5.4 Fixed-time Stability in Probability for Stochastic Discrete-time Systems Consider the DT nonlinear stochastic system given by a.s. y(𝑘 + 1) =f (y(𝑘)) + g (y(𝑘))𝜈(𝑘) , 𝐹(y(𝑘), 𝜈(𝑘)), y(0) ≡ y0 , 𝑘 ∈ N, (5.48) where, for every 𝑘 ∈ N, y(𝑘) ∈ D ⊆ R𝑛 is a D-valued stochastic process with y0 ∈ D, and 𝜈(𝑘) ∈ R𝑛 , 𝑘 ∈ N, is the independent and identically distributed zero-mean stochastic process on (Ω, F , P). f : D → D and g : D → R𝑛×𝑛 are continuous functions with f (0) = 0 and g (0) = 0 where y𝑒 = 0 is the equilibrium of the system (5.48), if and only if y(.) is P-almost surely (a.s.) a.s. equal to zero (i.e., y(.) ≡ 0) and is a solution of (5.48). A stochastic process y : [0, 𝜅] × Ω → D is a solution sequence of (5.48) on the discrete-time a.s. interval [0, 𝜅] with initial condition y(0) ≡ y0 if y(𝑘) satisfies (5.48) almost surely. a.s. The following definitions are given for stability in probability for the zero solution y(𝑘) ≡ 0 of the DT nonlinear stochastic system (5.48). Definition 16 [119, 148] a.s. 1) The zero solution y(𝑘) ≡ 0 to (5.48) is Lyapunov stable in probability, if for every 𝜀 > 0 and 𝜌 ∈ (0, 1), there exist 𝛿 = 𝛿(𝜀, 𝜌) > 0 such that, for all ||y0 ||< 𝛿,   P sup ky(𝑘)k> 𝜀 ≤ 𝜌. 𝑘∈N a.s. 2) The zero solution y(𝑘) ≡ 0 to (5.48) is asymptotically stable in probability if it is Lyapunov stable in probability and, for every 𝜌 ∈ (0, 1), there exists 𝛿 = 𝛿(𝜌) > 0 such that if ||y0 ||< 𝛿, then   P lim ky(𝑘)k= 0 ≥ 1 − 𝜌. 𝑘→∞ a.s. 3) The zero solution y(𝑘) ≡ 0 to (5.48) is globally asymptotically stable in probability if it is Lyapunov stable in probability and, for all y0 ∈ R𝑛 ,   P lim ky(𝑘)k= 0 = 1. 𝑘→∞ a.s. 4) The zero solution y(𝑘) ≡ 0 to (5.48) is exponentially stable in probability if for some 0 < 𝛾 < 1 independent of 𝜈, it is Lyapunov stable in probability and, for every 𝜌 ∈ (0, 1), there 99 exists 𝛿 = 𝛿(𝜌) > 0 such that if ||y0 ||< 𝛿, then   P lim k𝛾 𝑘 y(𝑘)k= 0 ≥ 1 − 𝜌. 𝑘→∞ a.s. 5) The zero solution y(𝑘) ≡ 0 to (5.48) is globally exponentially stable in probability if for some 0 < 𝛾 < 1 independent of 𝜈, it is Lyapunov stable in probability and, for all y0 ∈ R𝑛 ,   P lim k𝛾 𝑘 y(𝑘)k= 0 = 1. 𝑘→∞ Definition 17 [119] For the DT stochastic dynamical system (5.48) and 𝑉 : D → R+ , the difference operator ∆𝑉 of y is given as follows, ∆𝑉(y) = E[𝑉(𝐹(y, 𝜈))] − 𝑉(y), y ∈ D. Note that the difference operator in Definition 5 is a deterministic function and does not involve the expectation of the system state trajectory and only involves the expectation over the random noise variable 𝜈. Moreover, the random vectors 𝜈(𝑘), 𝑘 ∈ N, all have the same distribution. In the following, sufficient conditions for Lyapunov, asymptotic and exponential stability in probability for the system (5.48) are given. Lemma 9 [118,148]: Consider the discrete-time nonlinear stochastic system (5.48) and assume that there exists a continuous function 𝑉 : D → R+ such that 𝑉(0) = 0, 𝑉(y) > 0, y ∈ D, y 6= 0, ∆𝑉(y) ≤ 0, y ∈ D. a.s. Then, the zero solution y(𝑘) ≡ 0 to (5.48) is Lyapunov stable in probability. Moreover, if ∆𝑉(y) < 0, y ∈ D, y 6= 0, a.s. then the zero solution y(𝑘) ≡ 0 to (5.48) is asymptotically stable in probability. Furthermore, if ∆𝑉(y) < −𝛾𝑉(y), 0 < 𝛾 < 1, y ∈ D, y 6= 0, 100 a.s. then the zero solution y(𝑘) ≡ 0 to (5.48) is exponentially stable in probability. If D = R𝑛 and a.s. 𝑉(·) is radially unbounded, then the zero solution y(𝑘) ≡ 0 to (5.48) is globally asymptotically or exponentially stable in probability under the defined Lyapunov conditions. The following definition provides the characteristics of stochastic DT systems under which they are fixed-time stable in probability. Definition 18 (Fixed-time stability in probability) Consider the stochastic DT nonlinear system a.s. (5.48). The zero solution of y(𝑘) ≡ 0 to the system (5.48) is said to be fixed-time stable in probability, if there exist a stochastic process called stochastic settling time function 𝐾(y, ·), such that: 1) The system (5.48) is Lyapunov stable in probability. That is, for every 𝜖 > 0 and 𝜌 ∈ (0, 1), a.s. there exists a 𝛿 = 𝛿(𝜖, 𝜌) > 0 such that for all y(0) ≡ y0 ∈ D\{0}, if ||y(0)||≤ 𝛿, then ! P sup y(𝑘) > 𝜀 ≤ 𝜌. 𝑘∈ [ 0,𝐾 ( y0 ,𝜈 )) a.s. 2) For every initial condition y(0) ≡ y0 ∈ D\{0}, the solution sequence y(𝑘) is defined on     0, 𝐾 y0 , 𝜈 , 𝜈 ∈ Ω, y(𝑘) ∈ D\{0}, 𝑘 ∈ 0, 𝐾 y0 , 𝜈 , and   P y 𝐾 y0 , 𝜈 = 0 = 1. 3) The stochastic settling-time function 𝐾(y, ·), for all y ∈ D, is finite almost surely and there exist a fixed-time upper bound for the stochastic settling-time 𝐾(y, ·), i.e., E[𝐾(y0 , 𝜈)] ≤ 𝐾𝑚𝑎𝑥 where 𝐾𝑚𝑎𝑥 is a positive integer. a.s. The zero solution y(𝑘) ≡ 0 to (5.48) is globally fixed-time stable in probability if it is fixed time stable in probability with D = R𝑛 . Lemma 10 Consider the nonlinear stochastic DT system (5.48) and the scalar system 𝑉(𝑥(𝑘 + 1)) = 𝛾(𝑉(𝑥(𝑘))), 𝑥(𝑘) ∈ R𝑛 , (5.49) where 𝑉(𝑥(𝑘)) 𝛾(𝑉(𝑥(𝑘))) = 𝑉(𝑥(𝑘)) − 𝛼 min{ , max{𝑉 𝑟 1 (𝑥(𝑘)), 𝑉 𝑟 2 (𝑥(𝑘))}}, (5.50) 𝛼 101 such that 0 < 𝛼 < 1, 0 < 𝑟 1 < 1, and 𝑟 2 > 1. If there exists a continuous positive-definite function 𝑉 : R𝑛 → R+ and the nondecreasing function 𝛾 : R+ → R+ such that  E 𝑉(𝐹(y, 𝜈)] ≤ 𝛾(𝑉(y)), 𝑦 ∈ R𝑛 , then  𝑉 y0 ≤ 𝑥 0 , 𝑥 0 ∈ R+ implies E[𝑉(y(𝑘))] ≤ 𝑥(𝑘), 𝑘 ∈ N, where the sequence 𝑥(𝑘), 𝑘 ∈ N, satisfies (5.49). Proof. This Lemma is an extension of finite-time stability conditions [119], which is provided for fixed-time stability conditions. The proof is similar and is omitted.  The following theorem represents the sufficient Lyapunov conditions for fixed-time stability in probability for stochastic DT nonlinear systems. Theorem 8 Consider the nonlinear stochastic system (5.48). If there exists a continuous and radially unbounded function 𝑉 : R𝑛 → R+ such that 𝑉(0) = 0, (5.51) 𝑉(y) > 0, y ∈ R𝑛 \{0}, (5.52) E[𝑉(𝐹(y, 𝜈))] ≤ 𝛾(𝑉(y)), y ∈ R𝑛 \{0}, (5.53) a.s. where 𝛾(.) is given in (5.50), then the zero solution y(𝑘) ≡ 0 to (5.48) is globally fixed-time stable in probability. Moreover, there exists a stochastic settling-time 𝐾 : R𝑛 → N such that   E 𝐾 y0 ≤ 𝐾ˆ (𝑥 0 ) < 𝐾𝑚𝑎𝑥 , (5.54) where 𝐾(·) is almost surely finite stochastic settling-time function and 𝐾ˆ (𝑥 0 ) is the finite settling-   time function of (5.49) and 𝐾𝑚𝑎𝑥 is the fixed upper bound for 𝐾ˆ (𝑥 0 ) and E 𝐾 y0 . Proof Based on (5.50) and (5.53), one has E[𝑉(𝐹(y, 𝜈))] − 𝑉(y) ≤ 𝛾(𝑉(y)) − 𝑉(y) < 0, y ∈ R𝑛 \{0}, (5.55) 102 a.s. and hence, it follows from Lemma 9 that the zero solution y(𝑘) ≡ 0 to (5.48) is globally asymptotically stable in probability. Now, consider the nonlinear DT system (5.49) and note that, by Theorem 6, the zero solution 𝑥(𝑘) ≡ 0 to (5.49) is globally fixed-time stable and there exists 1 1 1 𝐾ˆ (𝑥 0 ) < ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 )⌋ + ⌊𝛼−1 (𝛼 1−𝑟 2 − 1)⌋ + 3, such that 𝑥(𝑘) = 0, 𝑘 ≥ 𝐾ˆ (𝑥 0 ) , 𝑥 0 ∈ R+ .  a.s. Now, let 𝑉 y0 < 𝑥 0 , y(0) ≡ y0 ∈ R𝑛 , and it follows from Lemma 10 that E[𝑉(y(𝑘))] = 0, 𝑘 ≥ 𝐾ˆ (𝑥 0 ) . a.s. Since 𝑉(y(𝑘)), 𝑘 ∈ N, is a nonnegative random variable, it follows that 𝑉(y(𝑘)) ≡ 0 for all a.s. 𝑘 ≥ 𝐾ˆ (𝑥 0 ). Then, it follows from (5.51) and (5.52) that y(𝑘) ≡ 0 for all 𝑘 ≥ 𝐾ˆ (𝑥 0 ). Therefore,   there exists a stochastic settling-time E[𝐾 y0 ] ≤ 𝐾ˆ (𝑥 0 ) such that y(𝑘) = 0, 𝑘 ≥ 𝐾 y0 . Finally,  since E[𝐾 y0 ] ≤ 𝐾ˆ (𝑥 0 ), it follows that   1 1 1 E 𝐾 y0 ≤ 𝐾ˆ (𝑥 0 ) < ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 )⌋ + ⌊𝛼−1 (𝛼 1−𝑟 2 − 1)⌋ + 3, and hence, Definition 6 is satisfied.  5.5 Example Illustration and Simulation This sections provides examples to verify the correctness of the presented fixed-time stability results. Examples 1 and 2 are, respectively, presented for deterministic scalar and higher-order systems without uncertainties and perturbations. Examples 3 and 4 are counterexamples that show that if the Lyapunov conditions for a deterministic scalar or higher-order system guarantee its fixed-time stability, by adding noise to the system, the same Lyapunov candidate only guarantees exponential stability in probability, and not fixed-time stability in probability. These examples clearly show that moving from a fixed-time stable deterministic system to a stochastic system with the same dynamics, one might look for new Lyapunov function candidates than the one used for the deterministic system to show its fixed-time stability in probability, if there exists one. 103 Example 1. (Fixed-time stable scalar deterministic discrete-time system) Consider the scalar nonlinear DT system given as follows 𝑟 ′ 𝑟 ′ 𝑦(𝑘 + 1) =𝑎𝑦(𝑘) − 𝛼′ 𝑠𝑖𝑔𝑛(𝑦(𝑘)) min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}, (5.56) where 𝑦(𝑘) ∈ R , 𝑘 ∈ N, 12 < 𝑎 ≤ 1, 𝛼′ ∈ (0, 1), 𝑟 1′ ∈ (0, 1) and 𝑟 2′ > 1. Now, using Theorem 6, it is shown that the zero solution 𝑦(𝑘) = 0 to (5.56) with 𝑎 = 1 is globally fixed-time stable. 1 1 Consider 𝑉(𝑦(𝑘)) = 𝑦 2 (𝑘) and 𝑦 𝐿 < 𝑦(0) < 𝑦 𝐻 where 𝑦 𝐿 = 𝛼′ 1−𝑟 1 and 𝑦 𝐻 = 𝛼′ 1−𝑟 2 (Note that if 𝑦(0) > 𝑦 𝐻 or 𝑦(0) < 𝑦 𝐿 , then the zero solution 𝑦(𝑘) = 0 for (5.56) with 𝑎 = 1 is fixed-time stable with 𝐾(𝑦(0)) = 1). The difference of 𝑉(𝑦(𝑘)) = 𝑦 2 (𝑘) is as follows, 𝑟 ′ 𝑟 ′ ∆𝑉(𝑦(𝑘)) =[𝑎𝑦(𝑘) − 𝛼′ 𝑠𝑖𝑔𝑛(𝑦(𝑘)) min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}]2 − 𝑦 2 (𝑘) 𝑟 ′ 𝑟 ′ =(𝑎𝑦(𝑘))2 − 2𝑎𝛼′ |𝑦(𝑘)|min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }} 𝑟 ′ 𝑟 ′ + (𝛼′ min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }})2 − 𝑦 2 (𝑘) 𝑟 ′ 𝑟 ′ =(𝑎 2 − 1)𝑦 2 (𝑘) + 𝛼′ min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}× 𝑟 ′ 𝑟 ′ (−2𝑎|𝑦(𝑘)|+𝛼′ min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}). (5.57) Using the fact that 𝑟′ 𝑟′ |𝑦(𝑘)|> 𝛼′ min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}, (5.58) one has 𝑟 ′ 𝑟 ′ − 2𝑎|𝑦(𝑘)|+𝛼′ min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }} < 𝑟 ′ 𝑟 ′ (1 − 2𝑎)𝛼′ min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}. (5.59) Therefore, using (5.59), (5.57) leads to, 2𝑟 ′ 2𝑟 ′ ∆𝑉(𝑦(𝑘)) ≤ (𝑎 2 − 1)𝑦 2 (𝑘) + (1 − 2𝑎)𝛼′2 min{𝑦 2 (𝑘)/𝛼′2 , max{𝑦 1 (𝑘), 𝑦 2 (𝑘)}}, (5.60) and using 𝑉(𝑦(𝑘)) = 𝑦 2 (𝑘) one can rewrite (5.60) as follows, 𝑟 ′ 𝑟 ′ ∆𝑉(𝑦(𝑘)) ≤ (𝑎 2 − 1)𝑉(𝑦(𝑘)) + (1 − 2𝑎)𝛼′2 min{𝑉(𝑘)/𝛼′2 , max{𝑉 1 (𝑘), 𝑉 2 (𝑘)}}. (5.61) 104 Table 5.1: Parameters 𝛼′, 𝑟 1′ , 𝑟 2′ , and fixed-time upper bound of settling-time function (𝐾 ∗ ) for (5.56) with 𝑎 = 1 and the initial condition 𝑦(0) = 20. 𝛼′ 𝑟 1′ 𝑟 2′ 𝐾∗ 𝑦𝐿 𝑦𝐻 Case 1 0.4 0.2 1.2 59601 0.31 97.6 Case 2 0.7 0.9 1.1 2558 0.02 35.4 Case 3 0.3 0.6 1.3 34002 0.04 55.3 Case 4 0.7 0.9 10 3 0.02 1.04 Since 21 < 𝑎 ≤ 1, (5.61) is rewritten as 𝑟 ′ 𝑟 ′ ∆𝑉(𝑦(𝑘)) ≤ −𝛽𝛼′2 min{𝑉(𝑘)/𝛼′2 , max{𝑉 1 (𝑘), 𝑉 2 (𝑘)}}, (5.62) where 𝛽 = (2𝑎 − 1) and for 12 < 𝑎 ≤ 1, 0 < 𝛽 ≤ 1. For 𝑎 = 1, (5.62) leads to 𝑟 ′ 𝑟 ′ ∆𝑉(𝑦(𝑘)) ≤ −𝛼′2 min{𝑉(𝑘)/𝛼′2 , max{𝑉 1 (𝑘), 𝑉 2 (𝑘)}}, (5.63) which is analogous to (5.5) where 𝛼 = 𝛼′2 , 𝑟 1 = 𝑟 1′ and 𝑟 2 = 𝑟 2′ , and all the parameters conditions mentioned in Theorem 6 are satisfied. Therefore, it is shown that system (5.56) with 𝑎 = 1 is globally fixed-time stable. Based on (5.6), the fixed upper bound for the settling-time function of system (5.56) with 𝑎 = 1 is 2 2 2 ∗ ′ 1−𝑟 ′ ′ 1−𝑟 ′ ′−2 ′ 1−𝑟 ′ 𝐾 = ⌊𝛼 1 (1 − 𝛼 1 )⌋ + ⌊𝛼 (𝛼 2 − 1)⌋ + 3. (5.64) The state trajectory of the system (5.56) with 𝑎 = 1 is simulated in Fig. 5.1 for 4 different values of parameters 𝛼′, 𝑟 1′ and 𝑟 2′ to verify the fixed-time convergence of the system (5.56) with 𝑎 = 1, and 𝑦(0) = 20 such that 𝑦 𝐿 < 𝑦(0) < 𝑦 𝐻 in Cases 1-3 and 𝑦(0) > 𝑦 𝐻 for Case 4. As depicted in Fig. 5.1, the settling-time is less than 𝐾 ∗ for Cases 1-3 where 𝐾 ∗ is calculated using (5.64) and given in Table 5.5, and as mentioned in (5.28), for Case 4, 𝐾(𝑦(0)) = 1. In Fig. 5.2, the state trajectory of system (5.56) with 𝑎 = 1 and Case 1 parameters (𝛼′ = 0.4, 𝑟 1′ = 0.2, 𝑟 2′ = 1.2) is simulated for 4 different initial conditions, 𝑦(0) = 0.1, (𝑦(0) < 𝑦 𝐿 ), 𝑦(0) = 8, (𝑦 𝐿 < 𝑦(0) < 𝑦 𝐻 ), 𝑦(0) = 80, (𝑦 𝐿 < 𝑦(0) < 𝑦 𝐻 ) and 𝑦(0) = 8000, (𝑦 𝐻 < 𝑦(0)) where as expected for 𝑦(0) = 0.1 and 𝑦(0) = 8000, the settling-time is 𝐾(𝑦(0)) = 1, and for 𝑦(0) = 8 and 𝑦(0) = 80 the convergence to zero is achieved in few steps which ensures 𝐾(𝑦(0)) ≤ 𝐾 ∗ . 105 Case 1 Case 2 20 20 15 15 y(k) 10 y(k) 10 5 5 0 0 0 5 10 15 0 5 10 15 k (time steps) k (time steps) Case 3 Case 4 20 20 15 15 y(k) 10 y(k) 10 5 5 0 0 0 5 10 15 0 5 10 15 k (time steps) k (time steps) Figure 5.1: Different fixed times of convergence for system (5.56) with 𝑎 = 1 and different values of 𝛼′, 𝑟 1′ and 𝑟 2′ . Case 1, y(0)=0.1 Case 1, y(0)=8 0.1 8 6 y(k) 0.05 y(k) 4 2 0 0 0 5 10 0 5 10 k (time steps) k (time steps) Case 1, y(0)=80 Case 1, y(0)=8000 80 8000 60 6000 y(k) 40 y(k) 4000 20 2000 0 0 0 5 10 0 5 10 k (time steps) k (time steps) Figure 5.2: Fixed-time convergence for Case 1 of system (5.56) with 𝑎 = 1 for different initial values. 106 However, for 21 < 𝑎 < 1, based on (5.62), Lemma 9 and a similar procedure to Theorem 6 proof, one can show that the system (5.56) with 12 < 𝑎 < 1 is exponentially stable. It is worth to note that the autonomous system given in (5.56) with 𝑎 = 1 can be considered as the following closed-loop system 𝑦(𝑘 + 1) = 𝐴𝑦(𝑘) + 𝐵𝑢(𝑘), (5.65) 𝑟 ′ 𝑟 ′ where 𝐴 = 1, 𝐵 = 1 and 𝑢(𝑘) = −𝛼′ 𝑠𝑖𝑔𝑛(𝑦(𝑘)) min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }} with 𝑦(𝑘) ∈ R , 𝑘 ∈ N, 𝛼′ ∈ (0, 1), 𝑟 1′ ∈ (0, 1) and 𝑟 2′ > 1. In other words, in this example we have presented a fixed-time feedback controller 𝑟 ′ 𝑟 ′ 𝑢(𝑘) = −𝛼′ 𝑠𝑖𝑔𝑛(𝑦(𝑘)) min{|𝑦(𝑘)|/𝛼′, max{|𝑦(𝑘)| 1 , |𝑦(𝑘)| 2 }}, which could stabilize the linear system (5.65) in a fixed amount of time. Example 2. (Fixed-time stable deterministic discrete-time higher-order system) Consider the nonlinear DT system of order 3 given as follows 1 𝑦 1 (𝑘 + 1) = 𝑦 1 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(𝑦 1 (𝑘)) min{|𝑦 1 (𝑘)|/¯𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , 3 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}, (5.66) 1 𝑦 2 (𝑘 + 1) = 𝑦 2 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(𝑦 2 (𝑘)) min{|𝑦 2 (𝑘)|/¯𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , 3 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}, (5.67) 1 𝑦 3 (𝑘 + 1) = 𝑦 3 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(𝑦 3 (𝑘)) min{|𝑦 3 (𝑘)|/¯𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , 3 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}, (5.68) where 𝑦(𝑘) = [𝑦 1 (𝑘), 𝑦 2 (𝑘), 𝑦 3 (𝑘)]𝑇 ∈ R3 , 𝑘 ∈ N, 𝛼 ¯ ∈ (0, 1), 𝑟¯1 ∈ (0, 1) and 𝑟¯2 > 1. Now, using Theorem 6, it is shown that the zero solution 𝑦(𝑘) = 0 to the above higher-order system is globally fixed-time stable. Consider 𝑉(𝑦(𝑘)) = |𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|, (5.69) 1 1 and 𝑉𝐿 < 𝑉(0) < 𝑉𝐻 where 𝑉𝐿 = 𝛼 ¯ 1−¯𝑟 1 and 𝑉𝐻 = 𝛼 ¯ 1−¯𝑟 2 (Note that if 𝑉(0) > 𝑉𝐻 or 𝑉(0) < 𝑉𝐿 , then the zero solution 𝑦(𝑘) = 0 of the above higher-order system is fixed-time stable 107 with 𝐾(𝑦(0)) = 1). The difference of (5.69) is as follows, ∆𝑉(𝑦(𝑘)) = |𝑦 1 (𝑘 + 1)|−|𝑦 1 (𝑘)|+|𝑦 2 (𝑘 + 1)|−|𝑦 2 (𝑘)|+|𝑦 3 (𝑘 + 1)|−|𝑦 3 (𝑘)|, (5.70) where using (5.66)-(5.68) leads to 1 ∆𝑉(𝑦(𝑘)) = |𝑦 1 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(𝑦 1 (𝑘)) min{|𝑦 1 (𝑘)|/¯ 𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , 3 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}|−|𝑦 1 (𝑘)|+|𝑦 2 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(𝑦 2 (𝑘)) min{|𝑦 2 (𝑘)|/¯ 𝛼, 1 max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}|−|𝑦 2 (𝑘)| 3 1 + |𝑦 3 (𝑘) − 𝛼¯ 𝑠𝑖𝑔𝑛(𝑦 3 (𝑘)) min{|𝑦 3 (𝑘)|/¯ 𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , 3 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}|−|𝑦 3 (𝑘)|. (5.71) Consider a =𝑦𝑖 (𝑘), 1 b =¯ 𝛼 𝑠𝑖𝑔𝑛(𝑦𝑖 (𝑘)) min{|𝑦𝑖 (𝑘)|/¯ 𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , 3 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}, (5.72) where one knows that a and b have the same sign and for 𝑖 = 1, 2, 3, |a |≥ |b |. Therefore one has |a − b |= |a |−|b |. (5.73) Thus using (5.72)-(5.73), (5.71) is rewritten as follows ∆𝑉(𝑦(𝑘)) = 1 −𝛼¯ min{|𝑦 1 (𝑘)|/¯ 𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }} 3 1 −𝛼¯ min{|𝑦 2 (𝑘)|/¯ 𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }} 3 1 −𝛼¯ min{|𝑦 3 (𝑘)|/¯ 𝛼, max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}, 3 (5.74) where (5.74) leads to |𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)| ∆𝑉(𝑦(𝑘)) ≤ − 𝛼 ¯ min{ , max{[|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯1 , ¯ 𝛼 [|𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|]𝑟¯2 }}. (5.75) 108 Using 𝑉(𝑦(𝑘)) = |𝑦 1 (𝑘)|+|𝑦 2 (𝑘)|+|𝑦 3 (𝑘)|, (5.75) is rewritten as follows 𝑉(𝑦(𝑘)) ∆𝑉(𝑦(𝑘)) ≤ − 𝛼 ¯ min{ , max{𝑉(𝑦(𝑘))𝑟¯1 , 𝑉(𝑦(𝑘))𝑟¯2 }}, (5.76) ¯ 𝛼 which is analogous to (5.5) where 𝛼 = 𝛼 ¯ , 𝑟 1 = 𝑟¯1 and 𝑟 2 = 𝑟¯2 , and all the parameters conditions mentioned in Theorem 6 are satisfied. Therefore, it is shown that the higher-order system specified in (5.66)-(5.68) is globally fixed-time stable. Based on (5.6), the fixed upper bound for the settling- time function of this higher-order system is 1 1 1 𝐾 ∗ = ⌊𝛼¯ 1−¯𝑟 1 (1 − 𝛼 ¯ −1 (¯ ¯ 1−¯𝑟 1 )⌋ + ⌊ 𝛼 𝛼 1−¯𝑟 2 − 1)⌋ + 3. (5.77) The state trajectories for the system (5.66)-(5.68) with 𝛼 ¯ = 0.47, 𝑟¯1 = 0.2 and 𝑟¯2 = 1.2 are simulated in Figs. 5.3-5.5 for three different values of initial conditions. Based on the given 1 1 parameters, one has 𝑉𝐿 = 𝛼 ¯ 1−¯𝑟 1 = 0.3892 and 𝑉𝐻 = 𝛼 ¯ 1−¯𝑟 2 = 43.60. Fig. 5.3 shows the system (5.66)-(5.68) trajectories reached the origin with 𝐾(𝑦(0)) = 1 for initial conditions 𝑦 1 (0) = 0.1, 𝑦 2 (0) = 0.01, 𝑦 3 (0) = 0.001 which imply that 𝑉(0) < 𝑉𝐿 . Fig. 5.4 shows that for the 3 states’ trajectories, it took several steps to reach the origin because the initial conditions are 𝑦 1 (0) = 9, 𝑦 2 (0) = 9.5, 𝑦 3 (0) = 10 and 𝑉𝐻 < 𝑉(0) < 𝑉𝐿 . Moreover, in this case, using (6), the fixed upper bound for the convergence time is obtained as 93 steps where the convergence is achieved before this time. In Fig. 5.5 where the initial conditions are 𝑦 1 (0) = 90, 𝑦 2 (0) = 900, 𝑦 3 (0) = 9000 and 𝑉(0) < 𝑉𝐻 , the state trajectories reached the origin with 𝐾(𝑦(0)) = 1. Example 3. (Lyapunov function candidate: from deterministic fixed-time stable scalar systems to their stochastic counterparts) In this counterexample we show that the deterministic global fixed-time stable system may not preserve its fixed-time stability under the same Lyapunov function candidate after it is exposed to stochastic noise. Consider the scalar stochastic nonlinear DT system as follows 𝑟 ′ 𝑟 ′ y(𝑘 + 1) = 𝑎y(𝑘) − 𝛼′ 𝑠𝑖𝑔𝑛(y(𝑘)) min{|y(𝑘)|/𝛼′, max{|y(𝑘)| 1 , |y(𝑘)| 2 }} + 𝑏y(𝑘)𝜈(𝑘), (5.78) 109 State trajectories for V(0) 1, 𝜈(𝑘) ∈ R is a zero-mean stochastic r 2 noise with E[𝜈(𝑘)] = 0 and E[𝜈 (𝑘)] = 𝜎 , 2 < 𝑎 ≤ 1 and 𝑏 < 1−𝑎2 . 2 2 1 𝜎 a.s. Now, using Theorem 8 and the results of Example 1, it is shown that the zero solution y(𝑘) ≡ 0 to (5.78) (the stochastic version of (5.56)) does not show global fixed-time stability in probability for 𝑎 = 1 but preserves its exponential stability in probability for 21 < 𝑎 < 1, using the same Lyapunov function as in Example 1. State trajectories for VL 1, 𝜈(𝑘) ∈ R is a √ zero-mean stochastic noise with E[𝜈(𝑘)] = 0 and E[|𝜈(𝑘)|] = 𝑐2 , 0 < 𝑐 < 𝛼 ¯ . Now, using Theorem a.s. 8 and the results of Example 2, it is shown that the zero solution y(𝑘) ≡ 0 to (5.86)-(5.88) (the stochastic version of (5.66)-(5.68)) does not show global fixed-time stability in probability, using the same Lyapunov function as in Example 2. Consider 𝑉(y(𝑘)) = |y1 (𝑘)|+|y2 (𝑘)|+|y3 (𝑘)|, (5.89) such that for the system (5.86)-(5.88), one has ∆𝑉(y(𝑘)) = E[|y1 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(y1 (𝑘)) min{|y1 (𝑘)|/¯ 𝛼, 1 max{[|y1 (𝑘)|+|y2 (𝑘)|+|y3 (𝑘)|]𝑟¯1 , 3 [|y1 (𝑘)|+|y2 (𝑘)|+|y3 (𝑘)|]𝑟¯2 }} + y1 (𝑘)𝜈(𝑘)|] − |y1 (𝑘)|+E[|y2 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(y2 (𝑘)) min{|y2 (𝑘)|/¯ 𝛼, 1 max{[|y1 (𝑘)|+|y2 (𝑘)|+|y3 (𝑘)|]𝑟¯1 , 3 [|y1 (𝑘)|+|y2 (𝑘)|+|y3 (𝑘)|]𝑟¯2 }} + y2 (𝑘)𝜈(𝑘)|] − |y2 (𝑘)|+E[|y3 (𝑘) − 𝛼 ¯ 𝑠𝑖𝑔𝑛(y3 (𝑘)) min{|y3 (𝑘)|/¯ 𝛼, 1 max{[|y1 (𝑘)|+|y2 (𝑘)|+|y3 (𝑘)|]𝑟¯1 , [|y1 (𝑘)|+|y2 (𝑘)| 3 + |y3 (𝑘)|]𝑟¯2 }} + y3 (𝑘)𝜈(𝑘)|] − |y3 (𝑘)|, (5.90) 113 Using triangle inequality for a, b (a and b are defined in (5.72)) and c = y𝑖 (𝑘)𝜈(𝑘), for 𝑖 = 1, 2, 3, as |a − b + c |≤ |a − b |+|c |, (5.91) and employing (5.89), (5.72)-(5.73), and the noise properties, one can rewrite (5.90) as below, ∆𝑉(y(𝑘)) ≤ − 𝛼 ¯ min{𝑉(𝑘)/¯ 𝛼, max{𝑉(𝑘)𝑟¯1 , 𝑉(𝑘)𝑟¯2 }} + 𝑐2𝑉(𝑘), (5.92) One knows that (5.92) can not support the global fixed-time stability in probability of the system (5.86)-(5.88), due to the injected noise stochasticity, while in Example 2 it was shown that the same system without noise is fixed-time stable. By using Lemma 9, one can show that the system √ (5.86)-(5.88) preserved its exponential stability in probability for 0 < 𝑐 < 𝛼 ¯. 5.6 Conclusion and Future work This chapter addressed the fixed-time stability for deterministic and stochastic discrete-time (DT) autonomous systems based on fixed-time Lyapunov stability analysis. Novel Lyapunov conditions are derived under which the fixed-time stability of autonomous DT deterministic and stochastic systems is certified. The sensitivity to perturbations for fixed-time stable DT systems is analyzed and the analysis shows that fixed-time attractiveness can be resulted from the presented Lyapunov conditions. For both cases of fixed-time stable and fixed-time attractive systems, the fixed upper bounds of the settling-time functions are given. For the future work, we plan to employ the presented fixed-time stability analysis to develop fixed-time identifiers and controllers for DT systems. 114 CHAPTER 6 DISCRETE-TIME NONLINEAR SYSTEM IDENTIFICATION: A FIXED-TIME CONCURRENT LEARNING APPROACH 6.1 Introduction The overarching objective of this chapter is to present a fixed-time concurrent learning (FxTCL) algorithm for discrete-time systems to 1) ensure fixed-time parameter convergence independent of the initial estimation errors and 2) relax the PE condition to a rank condition on the recorded data using CL. In the presented FxTCL, the settling-time upper bound is independent of the initial parameter estimation error. To achieve this goal, a modified gradient-descent update law is presented for learning the unknown system parameters. This update law reuses past collected data at every time instance and leverages discontinuous and non-integer powers of the identification errors. The Lyapunov analysis presented in our previous work in [128] is then leveraged to guarantee fixed-time convergence of the system parameters to their true values. The main contributions of this chapter include the following. First, a novel discrete-time update law is presented using the CL technique that identifies system uncertainties in a fixed amount of time. Fixed-time convergence is guaranteed under a rank condition on recorded memory data, which is weaker than the standard PE condition. The rigorous analysis using fixed-time Lyapunov stability guarantees the convergence of estimated parameters’ error to zero for adaptive approximators with parametric uncertainties under a condition on the learning rate. Third, a fixed-time upper bound for the parameters’ estimation error settling-time function, independent of the initial parameter estimation error, is computed. Notation R, Z, and N+ , respectively, denote the sets of real, integer, and natural numbers without zero. k.k denotes the Euclidean and induced 2 norms for vectors and matrices, respectively. 𝑡𝑟(.) shows the trace of a matrix. 𝜆 𝑚𝑖𝑛 (𝐴) and 𝜆 𝑚𝑎𝑥 (𝐴), respectively, show the minimum and maximum eigenvalues of matrix 𝐴. 𝐼 is the identity matrix of appropriate dimensions. ⌊.⌋ : R ↦→ Z denotes the floor function. P𝑛 1 In general, for a vector 𝑥 = [𝑥 1 , 𝑥 2 , ..., 𝑥 𝑛 ]𝑇 ∈ R𝑛 , the 𝑝-norm is defined as k𝑥k 𝑝 = ( 𝑖=1 |𝑥𝑖 | 𝑝 ) 𝑝 . 115 Moreover, for positive constants 𝑟 and 𝑠, if 0 < 𝑟 < 𝑠, based on Hölder inequality [129], one has 1 1 k𝑥k 𝑠 ≤ k𝑥k 𝑟 ≤ 𝑛 𝑟 − 𝑠 k𝑥k 𝑠 . q Frobenius norm of matrix 𝐴 ∈ R𝑚×𝑛 defined as k 𝐴k 𝐹 = 𝑡𝑟(𝐴𝑇 𝐴), implies k 𝐴k≤ k 𝐴k 𝐹 ≤ p min (𝑚, 𝑛)k 𝐴k. 6.2 Problem Formulation Consider a nonlinear discrete-time system as below, 𝑥(𝑘 + 1) = 𝑓 (𝑥(𝑘)) + 𝑔(𝑥(𝑘))𝑢(𝑘), (6.1) where 𝑥 ∈ D𝑥 ⊂ R𝑛 and 𝑢 ∈ D𝑢 ⊂ R𝑚 are, respectively, the system state and control input vectors, D𝑥 and D𝑢 are compact sets; the drift and input functions 𝑓 : D𝑥 ↦→ R𝑛 and 𝑔 : D𝑥 ↦→ R𝑛×𝑚 are functions with parametric uncertainty. The functions 𝑓 (𝑥) and 𝑔(𝑥) with parametric uncertainties are represented as 𝑓 (𝑥(𝑘)) = Θ∗𝑇 𝑓 𝜑(𝑥(𝑘)), (6.2) 𝑔(𝑥(𝑘)) = Θ∗𝑇 𝑔 𝜒(𝑥(𝑘)), (6.3) where Θ∗𝑓 ∈ D 𝑓 ⊂ R 𝑝×𝑛 and Θ∗𝑔 ∈ D𝑔 ⊂ R𝑞×𝑛 are the optimal unknown parameters, and D 𝑓 and D𝑔 are compact sets. 𝜑 : D𝑥 ↦→ R 𝑝 and 𝜒 : D𝑥 ↦→ R𝑞 , are the basis functions, where 𝑝 and 𝑞 are, respectively, the number of linearly independent basis functions to approximate 𝑓 (𝑥(𝑘)) and 𝑔(𝑥(𝑘)). Using (6.2)-(6.3), (6.1) is written as 𝑥(𝑘 + 1) = Θ∗𝑇 𝑧(𝑥(𝑘), 𝑢(𝑘)), (6.4) where Θ∗ = [Θ∗𝑇 , Θ∗𝑇 𝑇 (𝑝+𝑞)×𝑛 , and 𝑧(𝑥(𝑘), 𝑢(𝑘)) = [𝜑𝑇 (𝑥(𝑘)), 𝑢𝑇 (𝑘)𝜒𝑇 (𝑥(𝑘))]𝑇 ∈ 𝑓 𝑔 ] ∈ R R(𝑝+𝑞) . The measurements of 𝑥(𝑘 + 1) are not accessible. Therefore, regressor filtering [10, 39, 42] of the system (6.4), gives 𝑥(𝑘) = Θ∗𝑇 𝑑(𝑘) − 𝑙(𝑘) + 𝐶 𝑘 𝑥(0), (6.5) 𝑑(𝑘 + 1) = 𝑐𝑑(𝑘) + 𝑧(𝑥(𝑘), 𝑢(𝑘)), 𝑑(0) = 0, 𝑙(𝑘 + 1) = 𝐶𝑙(𝑘) + 𝐶𝑥(𝑘), 𝑙(0) = 0, (6.6) 116 P 𝑘−1 𝑘−ℎ where 𝐶 = 𝑐𝐼, −1 < 𝑐 < 1, 𝑙(𝑘) = ℎ=0 𝐶 𝑥(ℎ) is the filtered regressor of 𝑥(𝑘), and P 𝑘−1 𝑘−ℎ−1 𝑑(𝑘) = ℎ=0 𝑐 𝑧(𝑥(ℎ), 𝑢(ℎ)) is the filtered regressor of 𝑧(𝑥(𝑘), 𝑢(𝑘)). By dividing (6.5) to 𝑛 𝑠 := 1 + 𝑑𝑇 (𝑘)𝑑(𝑘) + 𝑙 𝑇 (𝑘)𝑙(𝑘), one has the normalized form of (6.5) given below, 𝑥¯(𝑘) =Θ∗𝑇 𝑑(𝑘) ¯ − 𝑙(𝑘)¯ + 𝐶 𝑘 𝑥¯(0), (6.7) where 𝑑¯ = 𝑛𝑑𝑠 , 𝑙¯ = 𝑛𝑙𝑠 , and 𝑥¯ = 𝑛𝑥𝑠 . Consider the approximator of (6.7) as follows, ¯ − 𝑙(𝑘) 𝑥ˆ¯(𝑘) =Θ̂𝑇 (𝑘)𝑑(𝑘) ¯ + 𝐶 𝑘 𝑥¯(0), (6.8) where Θ̂(𝑘) = [Θ̂𝑇𝑓 (𝑘), Θ̂𝑇𝑔 (𝑘)]𝑇 ∈ R(𝑝+𝑞)×𝑛 , Θ̂ 𝑓 (𝑘) and Θ̂𝑔 (𝑘) are, respectively, the estimated parameters’ matrices for Θ∗ , Θ∗𝑓 and Θ∗𝑔 at time 𝑘. The state estimation error is given as ¯ 𝑒(𝑘) = 𝑥ˆ¯(𝑘) − 𝑥¯(𝑘) = Θ̃𝑇 (𝑘)𝑑(𝑘), (6.9) where Θ̃(𝑘) := Θ̂(𝑘) − Θ∗ := [Θ̃𝑇𝑓 (𝑘), Θ̃𝑇𝑔 (𝑘)]𝑇 is the parameter estimation error such that Θ̃ 𝑓 (𝑘) := Θ̂ 𝑓 (𝑘) − Θ∗𝑓 , Θ̃𝑔 (𝑘) := Θ̂𝑔 (𝑘) − Θ∗𝑔 . Problem 1: Consider the system (1), or equivalently (7). Let the system model (8) be used for identifying the unknown parameters of (7). Design a fixed-time update law to ensure that the parameter estimation error Θ̃(𝑘) dynamics are fixed-time stable. Remark 26 To our knowledge, fixed-time system identification for discrete-time systems has not been investigated in the literature. We present for the first time a solution to Problem 1 by developing a modified gradient descent-based update law that leverages the recorded past data to relax the PE condition. 6.3 Preliminaries Definition 19 [29] The signal 𝑑(𝑘) is called persistently exciting if there are positive scalars P 𝜐1 , 𝜐2 and 𝑇 ∈ N+ where ∀𝜏 ∈ N+ , 𝜐1 𝐼 ≤ 𝜏+𝑇 𝑘=𝜏 𝑑(𝑘)𝑑𝑇 (𝑘) ≤ 𝜐2 𝐼. Definition 20 (Fixed-time stability [59]) Consider the system 𝑧(𝑘 + 1) = 𝐹(𝑧(𝑘)), (6.10) 117 where 𝑧 ∈ D𝑧 , 𝐹 : D𝑧 ↦→ R𝑛 and D𝑧 is an open neighborhood of the origin which is the equilibrium point of (6.10). The nonlinear system (6.10) is fixed-time stable, if there is an open neighborhood N𝑧 ⊆ D𝑧 of the origin and a settling time function 𝐾 : N𝑧 \{0} ↦→ N+ , such that: 1) The system (6.10) is Lyapunov stable, i.e., for every 𝜖 > 0, there exists a 𝛿 > 0 such that if ||𝑧(0)||≤ 𝛿, then ||𝑧(𝑘)||≤ 𝜖 for all 𝑘 ∈ {0, ..., 𝐾(𝑧(0)) − 1}. 2) For every initial condition 𝑧(0) ∈ N𝑧 \{0}, the solution sequence 𝑧(𝑘) of (6.10) reaches the equilibrium point and remains there after 𝑘 > 𝐾(𝑧(0)) and ∀𝑧(0) ∈ N𝑧 , where 𝐾 : N𝑧 \{0} ↦→ N+ . 3) The settling-time function 𝐾(𝑧(0)) is bounded, i.e., ∃𝐾𝑚𝑎𝑥 ∈ N+ : 𝐾(𝑧(0)) ≤ 𝐾𝑚𝑎𝑥 , ∀𝑧(0) ∈ N𝑧 \{0}. Lemma 11 [128] Consider the nonlinear discrete-time system (6.10). Consider there is a continuous Lyapunov function 𝑉 : D𝑧 ↦→ R where D𝑧 is an open neighborhood around the origin and there exists a neighborhood Ω𝑧 ⊂ D𝑧 of the origin such that 𝑉(𝑧(0)) = 0, 𝑉(𝑧(𝑘)) > 0, 𝑧(𝑘) ∈ Ω𝑧 \{0} and 𝑉(𝑧(𝑘)) ∆𝑉(𝑧(𝑘 + 1)) ≤ −𝛼 min{ , max{𝑉 𝑟 1 (𝑧(𝑘)), 𝑉 𝑟 2 (𝑧(𝑘))}}, 𝑧(𝑘) ∈ Ω𝑧 \{0}, (6.11) 𝛼 for constants 0 < 𝛼 < 1, 0 < 𝑟 1 < 1, and 𝑟 2 > 1. Then, system (6.10) is fixed-time stable and has a settling time function 𝐾 : N𝑧 ↦→ N+ that for all 𝑧(0) ∈ N𝑧 \{0} satisfies 1 1 1 𝐾(𝑧(0)) ≤ ⌊𝛼 1−𝑟 1 (1 − 𝛼 1−𝑟 1 )⌋ + ⌊𝛼−1 (𝛼 1−𝑟 2 − 1)⌋ + 3, (6.12) where N𝑧 is an open neighborhood of the origin. 6.4 Fixed-time Concurrent Learning of the Unknown Discrete-time Dynam- ics To employ CL for the approximation (6.8), the past data of (6.5)-(6.6) is recorded in the memory matrices 𝑀 ∈ R(𝑝+𝑞)×𝑃 , 𝐿 ∈ R𝑛×𝑃 and 𝑋 ∈ R𝑛×𝑃 , at time steps 𝜏1 , ..., 𝜏𝑃 , ¯ 1 ), 𝑑(𝜏 𝑀 = [𝑑(𝜏 ¯ 2 ), ..., 𝑑(𝜏 ¯ 𝑃 )], 𝐿 = [𝑙(𝜏 ¯ 1 ), 𝑙(𝜏 ¯ 2 ), ..., 𝑙(𝜏 ¯ 𝑃 )], 𝑋 = [¯𝑥 (𝜏1 ), 𝑥¯(𝜏2 ), ..., 𝑥¯(𝜏𝑃 )], (6.13) 118 where 𝑃 (the number of stored data in each memory matrice) is chosen such that 𝑀 is full-row rank, which is called 𝑀 rank condition and requires 𝑃 ≥ 𝑝 + 𝑞. Now, for the ℎ𝑡ℎ stored data, the error 𝑒 ℎ (𝑘) is defined as 𝑒 ℎ (𝑘) = 𝑥ˆ¯ℎ (𝑘) − 𝑥¯(𝜏ℎ ), (6.14) where 𝑥ˆ¯ℎ (𝑘) =Θ̂𝑇 (𝑘)𝑑(𝜏¯ ℎ ) − 𝑙(𝜏 ¯ ℎ ) + 𝐶 𝑘 𝑥¯(0), (6.15) is the state estimation at time step 0 ≤ 𝜏ℎ < 𝑘, ℎ = 1, ..., 𝑃, using the recorded 𝑑(𝜏 ¯ ℎ ) and 𝑙(𝜏 ¯ ℎ ), and the current estimated parameters matrix Θ̂(𝑘). Substituting 𝑥¯(𝜏ℎ ) into (6.14), one obtains 𝑒 ℎ (𝑘) = Θ̃𝑇 (𝑘)𝑑(𝜏¯ ℎ ). (6.16) The proposed FxTCL law for estimating the parameters of the system approximator is presented as follows X𝑃 Θ̂(𝑘 + 1) = Θ̂(𝑘) − Γ[Ξ𝐺 𝑑(𝑘)𝑒 ¯ 𝑇 (𝑘) + Ξ ( 𝐶 ¯ ℎ )(⌊𝑒𝑇 (𝑘)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑘)⌉ 𝛾2 ))], 𝑑(𝜏 (6.17) ℎ ℎ ℎ=1 where ⌊.⌉ 𝛾 := |.| 𝛾 𝑠𝑖𝑔𝑛(.) with |.| and 𝑠𝑖𝑔𝑛(.) understood in component-wise sense and 0 < 𝛾1 < 1, 𝛾2 > 1. Γ = 𝛾𝐼 is the learning rate with constant 𝛾 > 0. Ξ𝐶 = 𝜉𝐶 𝐼 and Ξ𝐺 = 𝜉𝐺 𝐼 are weight matrices with constants 𝜉𝐶 > 0 and 𝜉𝐺 > 0, which can be set to prioritize one of the two learning ¯ terms (i.e. 𝑑(𝑘)𝑒 𝑇 (𝑘) and P𝑃 𝑑(𝜏 ¯ ℎ )(⌊𝑒𝑇 (𝑘)⌉ 𝛾1 +⌊𝑒𝑇 (𝑘)⌉ 𝛾2 )) in (6.17) over the other. Moreover, ℎ=1 ℎ ℎ before the few 𝑃 steps of learning, required for filling the data stacks in (6.13) and satisfying the rank condition, we set Ξ𝐶 = 0 such that (6.17) only employs current data to update the estimated parameters. 6.5 Fixed-time Convergent Analysis In this section, the convergence analysis of the gradient update law dynamics is given based on fixed-time Lyapunov stability. Theorem 9 Let the system (6.1) be approximated by (6.8), whose parameters are adjusted using (6.17) with 0 < 𝛾1 < 1, 𝛾2 > 1 and a regressor given in (6.6). Let the rank condition on 𝑀 is 119 satisfied. If 𝛾 satisfies 2 max{𝑎 𝑢 , 𝑏 𝑢 } < 𝛾 < min{ , 𝑏 𝑢 , 𝑎 𝑢 }, 𝑓 𝑜𝑟 1 < Υ, (6.18) 𝑛𝜉𝐺 2 max{𝑎 𝑙 , 𝑏 𝑙 } < 𝛾 < min{ , 𝑎 , 𝑏 }, 𝑓 𝑜𝑟 0 < Υ ≤ 1, (6.19) 𝑛𝜉𝐺 𝑙 𝑙 𝛾 = 0, 𝑓 𝑜𝑟 Υ = 0, (6.20) P where Υ = k 𝑃 ℎ=1 ℎ 𝑒𝑇 (𝑘 − 1)k. Then, the update law (6.17) ensures the fixed-time convergence of Θ̃(𝑘) to zero for 𝑘 > 𝐾(Θ̃0 ) (i.e., it solves Problem 1). Besides, the settling time of convergence is given by 2 2 2 1−𝛾1 1−𝛾1 −1 1−𝛾2 𝐾(Θ̃0 ) ≤ max{⌊𝛼𝑖 (1 − 𝛼𝑖 )⌋ + ⌊𝛼𝑖 (𝛼𝑖 − 1)⌋} + 3, (6.21) 𝛼𝑖 >0 such that 𝛼𝑖 = min{𝑎𝑖 , 𝑏𝑖 }, 𝑖 = 1, 2. For 1 < Υ, one has 𝑖 = 1, 𝛾 𝛾1 +1 𝛾 𝛾2 +1 𝑎1 = 𝑎𝑢 ( ) 2 , 𝑏1 = 𝑏𝑢 ( ) 2 , 𝑛 𝑛 and 𝜂1 ≥ Υ𝛾2 −1 ; for 0 < Υ ≤ 1, one has 𝑖 = 2, 𝛾 𝛾1 +1 𝛾 𝛾2 +1 𝑎2 = 𝑎𝑙 ( ) 2 , 𝑏2 = 𝑏𝑙 ( ) 2 , 𝑛 𝑛 and 𝜂2 ≥ 1 ; where Υ 1−𝛾 1 𝛾1 +1 𝛾1 +1 1−𝛾1 𝑎𝑢 2 =𝜉𝐶 [2𝜆 𝑚𝑖𝑛 (𝑆) − 𝑛𝛾𝜆 𝑚𝑎𝑥 2 (𝑆)(2𝜉 (𝑛 2 ) + 𝜉 (𝑛1−𝛾1 )], (6.22) 𝐺 𝐶 1−𝛾2 𝛾2 +1 𝛾2 +1 1−𝛾1 𝑏𝑢 =2𝜉𝐶 [(𝑛 2 )𝜆 𝑚𝑖𝑛2 (𝑆) − 𝑛𝛾𝜆 𝑚𝑎𝑥 2 (𝑆)(𝜉 + 𝜉 (𝑛 2 ) + 0.5𝜉 𝜂 )], 𝐺 𝐶 𝐶 1 (6.23) 𝛾1 +1 𝛾1 +1 1−𝛾1 1−𝛾1 𝑎𝑙 2 =2𝜉𝐶 [𝜆 𝑚𝑖𝑛 (𝑆) − 𝑛𝛾𝜆 𝑚𝑎𝑥 (𝑆)𝑛 2 2 (𝜉𝐺 + 𝜉𝐶 ) + 0.5𝜉𝐶 (𝑛 2 )𝜂2 ], (6.24) 1−𝛾2 𝛾2 +1 𝛾2 +1 𝑏𝑙 =𝜉𝐶 [2(𝑛 2 2 )𝜆 𝑚𝑖𝑛 (𝑆) − 𝑛𝛾𝜉𝐺 𝜆 𝑚𝑎𝑥 2 (𝑆)(2𝜉 + 𝜉 )], (6.25) 𝐺 𝐶 1−𝛾2 1−𝛾2 a−1 a (𝑛 2 )a − 1 (𝑛 2 )a 𝑎𝑢 = , 𝑎𝑢 = , 𝑏𝑢 = , 𝑏𝑢 = , (6.26) b b c c 1−𝛾2 1−𝛾2 a−1 a (𝑛 2 )a − 1 (𝑛 2 )a 𝑎𝑙 = , 𝑎𝑙 = , 𝑏𝑙 = , 𝑏𝑙 = , (6.27) d d e e 120 𝛾1 +1 𝛾2 +1 2 a =2𝜉𝐶 𝜆 𝑚𝑖𝑛 (𝑆), e = 𝜆 𝑚𝑎𝑥 2 (𝑆)𝑛[4𝜉 𝜉 + 𝜉 2 ], 𝐶 𝐺 𝐶 𝛾1 +1 1−𝛾1 b =𝜆 𝑚𝑎𝑥 2 (𝑆)𝑛[2𝜉 𝜉 (𝑛 2 ) + 𝜉 2 (𝑛1−𝛾2 )], 𝐶 𝐺 𝐶 𝛾2 +1 1−𝛾1 c =𝜆 𝑚𝑎𝑥2 (𝑆)𝑛[4𝜉 𝜉 + 2𝜉 2 (𝑛 2 ) + 𝜉 2 𝜂 (𝑛1−𝛾1 )], 𝐶 𝐺 𝐶 𝐶 1 𝛾1 +1 1−𝛾1 1−𝛾1 2 d =𝜆 𝑚𝑎𝑥 (𝑆)𝑛[(𝑛 2 )2𝜉𝐶 𝜉𝐺 + 2𝜉𝐶 (𝑛 2 ) + 𝜉𝐶2 𝜂2 (𝑛1−𝛾1 )], 2 X𝑃 𝑆= ¯ ℎ )𝑑¯𝑇 (𝜏ℎ ). 𝑑(𝜏 ℎ=1 Proof 11 Consider the Lyapunov function, 𝑉(𝑘) as follows 𝑉(𝑘) = 𝑡𝑟{Θ̃𝑇 (𝑘)Γ−1 Θ̃(𝑘)}. (6.28) where its change rate, ∆𝑉(𝑘) = 𝑉(𝑘) − 𝑉(𝑘 − 1), is given below, ∆𝑉(𝑘) = 𝑡𝑟{Θ̃𝑇 (𝑘)Γ−1 Θ̃(𝑘) − Θ̃𝑇 (𝑘 − 1)Γ−1 Θ̃(𝑘 − 1)} = 𝑡𝑟{(Θ̃(𝑘) − Θ̃(𝑘 − 1))𝑇 Γ−1 (Θ̃(𝑘) + Θ̃(𝑘 − 1))}. (6.29) Using (6.17), (6.29) gives, 𝑃 X ∆𝑉(𝑘) = 𝑡𝑟{(−Γ[Ξ𝐺 𝑑(𝑘 ¯ − 1)𝑒𝑇 (𝑘 − 1) + Ξ𝐶 ( ¯ ℎ )(⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾2 ))])𝑇 × 𝑑(𝜏 ℎ ℎ ℎ=1 X𝑃 Γ−1 (2Θ̃(𝑘 − 1) − Γ[Ξ𝐺 𝑑(𝑘 ¯ − 1)𝑒𝑇 (𝑘 − 1) + Ξ𝐶 ( ¯ ℎ )(⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾1 + ⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾2 ))])}, 𝑑(𝜏 ℎ ℎ ℎ=1 (6.30) 121 ¯ and using 𝐷(𝑘) = 𝑑¯𝑇 (𝑘)𝑑(𝑘), ¯ (6.30) is rewritten as, X 𝑃 ∆𝑉(𝑘) = 𝑡𝑟{ − 2Ξ𝐺 𝑒(𝑘 − 1)𝑒𝑇 (𝑘 − 1) − 2Ξ𝐶 [ ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾1 𝑒𝑇ℎ (𝑘 − 1) ℎ=1 X𝑃 + ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾2 𝑒𝑇ℎ (𝑘 − 1)] + ΓΞ𝐺 2 𝑒(𝑘 − 1) 𝐷(𝑘 ¯ − 1)𝑒𝑇 (𝑘 − 1) ℎ=1 X𝑃 + 2ΓΞ𝐶 Ξ𝐺 ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾1 𝑑¯𝑇 (𝜏ℎ )𝑑(𝑘 ¯ − 1)𝑒𝑇 (𝑘 − 1) ℎ=1 X𝑃 + 2ΓΞ𝐶 Ξ𝐺 ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾2 𝑑¯𝑇 (𝜏ℎ )𝑑(𝑘 ¯ − 1)𝑒𝑇 (𝑘 − 1) ℎ=1 X 𝑃 X 𝑃 + ΓΞ𝐶 2 ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾1 𝑑¯𝑇 (𝜏ℎ ) ¯ ℎ )⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾1 𝑑(𝜏 ℎ ℎ=1 ℎ=1 X𝑃 X𝑃 + 2ΓΞ𝐶 2 ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾1 𝑑¯𝑇 (𝜏ℎ )( ¯ ℎ )⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾2 ) 𝑑(𝜏 ℎ ℎ=1 ℎ=1 X 𝑃 X𝑃 + ΓΞ𝐶 2 ⌊𝑒 ℎ (𝑘 − 1)⌉ 𝛾2 𝑑¯𝑇 (𝜏ℎ )( ¯ ℎ )⌊𝑒𝑇 (𝑘 − 1)⌉ 𝛾2 )}. 𝑑(𝜏 (6.31) ℎ ℎ=1 ℎ=1 𝛾 Using k ⌊ 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑘 − 1)⌉ 𝛾𝑖 k= k 𝑑¯𝑇 (𝜏ℎ )Θ̃(𝑘 − 1)k 2𝛾𝑖 for 𝑖 = 1, 2, and Facts 1-2, one rewrites 𝑖 (6.31) as follows X 𝑃 X 𝑃 𝛾 +1 𝛾 +1 ∆𝑉(𝑘) ≤ −2𝜉𝐺 k𝑒𝑇 (𝑘 − 1)k 2 −2𝜉𝐶 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾1 +1 −2𝜉𝐶 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾2 +1 1 2 ℎ=1 ℎ=1 X𝑃 2 k𝑒𝑇 (𝑘 − 1)k 2 +2𝜉 𝜉 𝛾 + 𝑛𝛾[𝜉𝐺 𝐶 𝐺 k𝑒𝑇ℎ (𝑘 − 1)k 2𝛾1 k𝑒𝑇 (𝑘 − 1)k 1 ℎ=1 X𝑃 X𝑃 𝛾 𝛾 + 2𝜉𝐶 𝜉𝐺 k𝑒𝑇ℎ (𝑘 − 1)k 2𝛾2 kk𝑒𝑇 (𝑘 − 1)k+𝜉𝐶2 ( k𝑒𝑇ℎ (𝑘 − 1)k 2𝛾1 )2 2 1 ℎ=1 ℎ=1 1−𝛾1 X 𝑃 X𝑃 X𝑃 𝛾 + 2𝜉𝐶2 𝑛 2 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾1 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾2 +𝜉𝐶2 ( k𝑒𝑇ℎ (𝑘 − 1)k 2𝛾2 )2 ]. (6.32) 2 ℎ=1 ℎ=1 ℎ=1 P𝑃 P P Using ℎ=1 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾𝑖 ≤ ( 𝑃 ℎ=1 ℎ k𝑒𝑇 (𝑘 − 1)k)𝛾𝑖 , k𝑒𝑇ℎ (𝑘 − 1)k≤ 𝑃 ℎ=1 ℎ k𝑒𝑇 (𝑘 − 1)k, 𝑖 = 1, 2, and 122 Fact 1, one has X𝑃 ∆𝑉(𝑘) ≤ −(2𝜉𝐺 − 𝑛𝛾𝜉𝐺 2 )k𝑒𝑇 (𝑘 − 1)k 2 −2𝜉 𝐶 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾1 +1 ℎ=1 1−𝛾2 X 𝑃 1−𝛾1 X 𝑃 − 2𝜉𝐶 (𝑛 2 ) k𝑒𝑇ℎ (𝑘 − 1)k 𝛾2 +1 +𝑛𝛾[2𝜉𝐶 𝜉𝐺 𝑛 2 ( k𝑒𝑇ℎ (𝑘 − 1)k)1+𝛾1 ℎ=1 ℎ=1 𝑃 X 𝑇 𝑃 X 𝑇 + 2𝜉𝐶 𝜉𝐺 ( k𝑒 ℎ (𝑘 − 1)k)1+𝛾2 + 𝜉𝐶2 (𝑛1−𝛾1 )( k𝑒 ℎ (𝑘 − 1)k)2𝛾1 ℎ=1 ℎ=1 1−𝛾1 X 𝑃 𝑃 X 𝑇 + 2𝜉𝐶2 𝑛 2 ( k𝑒𝑇ℎ (𝑘 − 1)k)𝛾1 +𝛾2 + 𝜉𝐶2 ( k𝑒 ℎ (𝑘 − 1)k)2𝛾2 ]. (6.33) ℎ=1 ℎ=1 P For Υ > 1 with Υ = k 𝑃 ℎ=1 ℎ 𝑒𝑇 (𝑘 − 1)k, and knowing 0 < 2𝛾1 < 𝛾1 + 1, 𝛾1 + 1 < 𝛾1 + 𝛾2 < 𝛾2 + 1 < 2𝛾2 , one has Υ2𝛾1 < Υ𝛾1 +1 , Υ𝛾1 +𝛾2 < Υ𝛾2 +1 . (6.34) Moreover, for Υ > 1 one obtains Υ2𝛾2 ≤ 𝜂1 Υ𝛾2 +1 𝑓 𝑜𝑟 𝜂1 ≥ Υ𝛾2 −1 . (6.35) Therefore, for Υ > 1, using (6.34) and (6.35), (6.33) is rewritten as follows, X𝑃 ∆𝑉(𝑘) ≤ −(2𝜉𝐺 − 𝑛𝛾𝜉𝐺 2 )k𝑒𝑇 (𝑘 − 1)k 2 −2𝜉 ( k𝑒𝑇ℎ (𝑘 − 1)k 𝛾1 +1 𝐶 ℎ=1 1−𝛾2 X𝑃 X𝑃 + (𝑛 2 ) k𝑒𝑇ℎ (𝑘 − 1)k 𝛾2 +1 ) + 𝑛𝛾[𝜂1 𝜉𝐶2 ( k𝑒𝑇ℎ (𝑘 − 1)k)𝛾2 +1 ℎ=1 ℎ=1 1−𝛾1 𝑃 X 𝑇 X 𝑃 + 2(𝑛 2 )𝜉𝐶 𝜉𝐺 ( k𝑒 ℎ (𝑘 − 1)k)1+𝛾1 + 2𝜉𝐶 𝜉𝐺 ( k𝑒𝑇ℎ (𝑘 − 1)k)1+𝛾2 ℎ=1 ℎ=1 X𝑃 1−𝛾1 X 𝑃 + 𝜉𝐶2 (𝑛1−𝛾1 )( k𝑒𝑇ℎ (𝑘 − 1)k)𝛾1 +1 + 2𝜉𝐶2 𝑛 2 ( k𝑒𝑇ℎ (𝑘 − 1)k)𝛾2 +1 ], (6.36) ℎ=1 ℎ=1 For the first term of (6.36) to have 2𝜉𝐺 − 𝑛𝛾𝜉𝐺 2 ≥ 0, one needs 2 𝛾≤ . (6.37) 𝑛𝜉𝐺 P𝑃 Hence, for Υ > 1 and 𝛾 ≤ 𝑛𝜉2 , using (6.9), (6.16), and 𝑆 = ℎ=1 ¯ ℎ )𝑑¯𝑇 (𝜏ℎ ), (6.36) is rewritten 𝑑(𝜏 𝐺 as ∆𝑉(𝑘) ≤ −𝑎 𝑢 k Θ̃(𝑘 − 1)k 𝛾1 +1 −𝑏 𝑢 k Θ̃(𝑘 − 1)k 𝛾2 +1 , (6.38) 123 where 𝑎 𝑢 and 𝑏 𝑢 are given in (6.22) and (6.23), respectively. In order to have 0 < 𝑎 𝑢 < 1 and 0 < 𝑏 𝑢 < 1, 𝛾 should, respectively, satisfy the following inequalities 𝑎𝑢 < 𝛾 < 𝑎𝑢 , 𝑏𝑢 < 𝛾 < 𝑏𝑢 , (6.39) where 𝑎 𝑢 , 𝑎 𝑢 , 𝑏 𝑢 and 𝑏 𝑢 are given in (6.26). For the case where 0 < Υ ≤ 1, and using 0 < 2𝛾1 < 𝛾1 + 1, 𝛾1 + 1 < 𝛾1 + 𝛾2 < 𝛾2 + 1 < 2𝛾2 , one obtains Υ𝛾1 +𝛾2 < Υ𝛾1 +1 , Υ2𝛾2 < Υ𝛾2 +1 . (6.40) Furthermore, for 0 < Υ ≤ 1, one has 1 Υ2𝛾1 ≤ 𝜂2 Υ𝛾1 +1 𝑓 𝑜𝑟 𝜂2 ≥ . (6.41) Υ1−𝛾1 Therefore, for 0 < Υ ≤ 1 and 𝛾 satisfying (6.37), using (6.40) and (6.41), (6.33) is rewritten as follows, X 𝑃 1−𝛾2 X𝑃 ∆𝑉(𝑘) ≤ −2(𝜉𝐶 k𝑒𝑇ℎ (𝑘 − 1)k 𝛾1 +1 +(𝑛 2 ) k𝑒𝑇ℎ (𝑘 − 1)k 𝛾2 +1 ) ℎ=1 ℎ=1 𝑃 X 𝑇 1−𝛾 1 X𝑃 + 𝑛𝛾[𝜂2 𝜉𝐶 𝑛 2 1−𝛾 1 ( k𝑒 ℎ (𝑘 − 1)k) 𝛾 1 +1 + 2𝑛 2 𝜉𝐶 𝜉 𝐺 ( k𝑒𝑇ℎ (𝑘 − 1)k)1+𝛾1 ℎ=1 ℎ=1 𝑃 X 𝑇 1−𝛾 𝑃 1 X 𝑇 + 2𝜉𝐶 𝜉𝐺 ( k𝑒 ℎ (𝑘 − 1)k)1+𝛾2 + 2𝜉𝐶2 𝑛 2 ( k𝑒 ℎ (𝑘 − 1)k)𝛾1 +1 ℎ=1 ℎ=1 X 𝑃 + 𝜉𝐶2 ( k𝑒𝑇ℎ (𝑘 − 1)k)𝛾2 +1 ], (6.42) ℎ=1 Thus, for 0 < Υ ≤ 1 and 𝛾 ≤ 𝑛𝜉2 , (6.42) is rewritten as 𝐺 ∆𝑉(𝑘) ≤ −𝑎 𝑙 k Θ̃(𝑘 − 1)k 𝛾1 +1 −𝑏 𝑙 k Θ̃(𝑘 − 1)k 𝛾2 +1 , (6.43) where 𝑎 𝑙 and 𝑏 𝑙 are, respectively, given in (6.24) and (6.25). In order to have 0 < 𝑎 𝑙 < 1 and 0 < 𝑏 𝑙 < 1, 𝛾 should, respectively, satisfy the following inequalities 𝑎𝑙 < 𝛾 < 𝑎𝑙 , 𝑏𝑙 < 𝛾 < 𝑏𝑙 , (6.44) 124 where 𝑎 𝑙 , 𝑎 𝑙 , 𝑏 𝑙 and 𝑏 𝑙 are given in (6.27). Therefore, in order to satisfy (6.37) and the inequalities (6.39) and (6.44) for Υ > 1 and 0 < Υ ≤ 1, respectively, 𝛾 needs to satisfy the following inequalities, respectively, 2 max{𝑎 𝑢 , 𝑏 𝑢 } <𝛾 < min{ , 𝑏 𝑢 , 𝑎 𝑢 }, 𝑓 𝑜𝑟 1 < Υ, (6.45) 𝑛𝜉𝐺 2 max{𝑎 𝑙 , 𝑏 𝑙 } <𝛾 < min{ , 𝑎 , 𝑏 }, 𝑓 𝑜𝑟 0 < Υ ≤ 1. (6.46) 𝑛𝜉𝐺 𝑙 𝑙 One obtains from (6.28) and Fact 2 that r 𝑛 𝛾 1 𝑉(Θ̃(𝑘)) ≤ k Θ̃(𝑘)k 2 ⇒ 𝑉 2 (Θ̃(𝑘)) ≤ k Θ̃(𝑘)k. (6.47) 𝛾 𝑛 Using (6.47), one rewrites (6.38) and (6.43) as follows 𝛾1 +1 𝛾2 +1 ∆𝑉(𝑘) ≤ −𝑎𝑖 𝑉 2 (𝑘 − 1) − 𝑏𝑖 𝑉 2 (𝑘 − 1), 𝑖 = 1, 2, (6.48) where for Υ > 1, 𝑖 = 1, 𝛾 𝛾1 +1 𝛾 𝛾2 +1 𝑎1 = 𝑎𝑢 ( ) 2 , 𝑏1 = 𝑏𝑢 ( ) 2 , 𝑛 𝑛 for 0 < Υ ≤ 1, 𝑖 = 2, 𝛾 𝛾1 +1 𝛾 𝛾2 +1 𝑎2 = 𝑎𝑙 ( ) 2 , 𝑏2 = 𝑏𝑙 ( ) 2 . 𝑛 𝑛 One can rewrite (6.48) as follows, 𝛾1 +1 𝛾2 +1 ∆𝑉(𝑘) ≤ −𝛼𝑖 max{𝑉 2 (𝑘 − 1), 𝑉 2 (𝑘 − 1)}, (6.49) where 𝛼𝑖 = min{𝑎𝑖 , 𝑏𝑖 }, 𝑖 = 1, 2. One knows that 𝛾1 +1 𝛾2 +1 min{𝑉(𝑘 − 1), 𝛼𝑖 max{𝑉 2 (𝑘 − 1), 𝑉 2 (𝑘 − 1)}} 𝛾1 +1 𝛾2 +1 ≤ 𝛼𝑖 max{𝑉 2 (𝑘 − 1), 𝑉 2 (𝑘 − 1)}, (6.50) implies 𝛾1 +1 𝛾2 +1 − 𝛼𝑖 max{𝑉 2 (𝑘 − 1), 𝑉 2 (𝑘 − 1)} ≤ 𝑉(𝑘 − 1) 𝛾1 +1 𝛾2 +1 −𝛼𝑖 min{ , max{𝑉 2 (𝑘 − 1), 𝑉 2 (𝑘 − 1)}}. (6.51) 𝛼𝑖 125 Therefore, using (6.51), for 𝑖 = 1, 2, (6.49) leads to 𝑉(𝑘 − 1) ∆𝑉(𝑘) ≤ −𝛼𝑖 min{ , 𝛼𝑖 𝛾1 +1 𝛾2 +1 max{𝑉 2 (𝑘 − 1), 𝑉 2 (𝑘 − 1)}}. (6.52) Lemma 11 and (6.52) imply that Θ̃(𝑘) converges to zero and a settling-time function is obtained as given in (6.21). By convergence of Θ̃(𝑘) to zero which results in Υ = 0, no further learning is required and one sets 𝛾 = 0 as given in (6.20). This completes the proof. Remark 27 The settling-time fixed upper bound in (6.21) certifies that the richer the recorded 𝜆 (𝑆) data in terms of having a bigger ratio of 𝜆 𝑚𝑖𝑛 (𝑆) (which leads to the bigger values of 𝛼𝑖 ), the 𝑚𝑎𝑥 smaller would be the upper bound for the settling-time in (6.21) and the identification converges in a faster fixed amount of time. Moreover, the fixed upper bound of the settling-time function can be computed along with the learning procedure and is obtained before Θ̃(𝑘) convergence to zero. Remark 28 Since the recorded regressors’ data are normalized, the lower bounds of learning rate 𝛾, max{𝑎 𝑢 , 𝑏 𝑢 } and max{𝑎 𝑙 , 𝑏 𝑙 }, respectively, given in (6.18) and (6.19), are usually negative for 𝜉𝐶 < 1. In order to ensure that a positive 𝛾 is chosen that satisfies (6.18)-(6.19) for Υ 6= 0, 𝛾 can be chosen as follows, 2 𝛾 = max{min{ , 𝑏 𝑢 , 𝑎 𝑢 } − 𝜖, max{𝑎 𝑢 , 𝑏 𝑢 } + 𝜖 }, 1 < Υ, 𝑛𝜉𝐺 2 𝛾 = max{min{ , 𝑎 , 𝑏 } − 𝜖, max{𝑎 𝑙 , 𝑏 𝑙 } + 𝜖 }, 0 < Υ ≤ 1, 𝑛𝜉𝐺 𝑙 𝑙 where 𝜖 is a very small positive constant (such as 𝜖 = 0.01 min{ 𝑛𝜉2 , 𝑏 𝑢 , 𝑎 𝑢 } for 1 < Υ or 𝐺 2 𝜖 = 0.01 min{ 𝑛𝜉 , 𝑎 𝑙 , 𝑏 𝑙 } for 0 < Υ ≤ 1). Moreover, to satisfy 𝜂1 ≥ Υ𝛾2 −1 and 𝜂2 ≥ 1−𝛾 1 , 𝐺 Υ 1 𝛾 −1 1 one can choose 𝜂1 = Υ 2 and 𝜂2 = 1−𝛾 , respectively. Υ 1 Remark 29 The learning rate 𝛾 extracted from either (6.18) or (6.19), is not a fixed constant and it is an adaptive time-varying scalar due to employing time-varying adaptive constants 𝜂1 = Υ𝛾2 −1 and 𝜂2 = 1−𝛾 1 , (respectively, satisfying (6.35) and (6.41)) where Υ = k P 𝑃 𝑒𝑇 (𝑘 − Υ 1 ℎ=1 ℎ P𝑃 ¯𝑇 1)k= k ℎ=1 𝑑 (𝜏ℎ )Θ̃(𝑘 − 1)k depends on the parameter estimation error Θ̃(𝑘) at every time 𝑘. Therefore, (6.17) is not the explicit Euler discretization of the continuous finite-time method given 126 in [43]. Moreover, in this chapter, the adaptive time-varying 𝛾 is different from the time-varying discretization for continuous finite-time systems in [149], which preserves finite-time and fixed- time proprieties but guarantees convergence in infinite time. Furthermore, employing an adaptive time-varying learning rate matches with the concepts of other discrete and finite-time learning studies [109, 122]. 6.6 Simulation Results and Discussion In this section, the performance of the presented fixed-time concurrent learning is examined in comparison with traditional gradient descent (GD) [10], asymptotically converging concurrent learning (CL) [48] and finite-time concurrent learning (FTCL) [42, 109] with, respectively, the following estimation laws, ¯ Θ̂(𝑘 + 1) = Θ̂(𝑘) − Γ𝐺 𝑑(𝑘)𝑒 𝑇 (𝑘), X𝑃 Θ̂(𝑘 + 1) = Θ̂(𝑘) − Γ𝐶 [Σ𝐺 𝑑(𝑘)𝑒¯ 𝑇 (𝑘) + Σ 𝐶 ¯ ℎ )𝑒𝑇 (𝑘)], 𝑑(𝜏 ℎ ℎ=1 𝑃 P𝑃 ¯ ℎ )𝑒𝑇 (𝑘) X ℎ=1 𝑑(𝜏 ℎ Θ̂(𝑘 + 1) = Θ̂(𝑘) − Γ′[𝐾1 𝑑(𝑘)𝑒 ¯ 𝑇 (𝑘) + 𝐾 ( 2 ¯ ℎ )𝑒𝑇 (𝑘) + 𝑑(𝜏 ℎ P𝑃 ¯ )], ℎ=1 𝜅 + k ℎ=1 𝑑(𝜏ℎ )𝑒𝑇ℎ (𝑘)k (6.53) where Γ𝐺 = 𝛾𝐺 𝐼, Γ𝐶 = 𝛾𝐶 𝐼, Γ′ = 𝛾′ 𝐼, Σ𝐺 = 𝜎𝐺 𝐼, Σ𝐶 = 𝜎𝐶 𝐼, 𝐾1 = 𝑘 1 𝐼 and 𝐾2 = 𝑘 2 𝐼 with constants 𝛾𝐺 > 0, 𝛾𝐶 > 0, 𝜎𝐺 > 0, 𝜎𝐶 > 0, 𝛾′ > 0, 𝑘 1 > 0, 𝑘 2 > 0 and 𝜅 > 0. The time interval for the simulation is given as [𝑘 0 , 𝑘 𝑓 ] with 𝑘 0 = 0 and 𝑘 𝑓 = 1000, and 𝑥 −𝑥 D𝑥 = [𝑥 𝐿 , 𝑥 𝐻 ] where 𝑥 𝐿 = 0, 𝑥 𝐻 = 2 and D𝑥 is discretized by [𝑥 𝐿 : 𝑘𝐻 −𝑘 𝐿 : 𝑥 𝐻 ]. In the presented 𝑓 0 FxTCL, 𝛾 is chosen to meet either (6.18) or (6.19) based on the value of Υ. We choose 𝜉𝐺 > 𝜉𝐶 to prioritize current data over recorded data. For gradient descent, 𝛾𝐺 = 0.6, for CL according to [48], 𝛾𝐶 is chosen as 𝛾𝐶 = 2𝜎 +𝜎 1𝜆 and for finite-time CL according to [109], 𝛾′ is 𝐺 𝐶 𝑚𝑎𝑥 (𝑆) chosen as ¯ − 1)) + 2𝑘 2𝜆 𝑚𝑖𝑛 (𝑆) 2𝑘 1𝜆 𝑚𝑖𝑛 ( 𝐷(𝑘 𝛾′ = . (6.54) ¯ − 1)) + 𝑘 2𝜆 𝑚𝑎𝑥 (𝑆)(1 + 1 ))2 (𝑘 1𝜆 𝑚𝑎𝑥 ( 𝐷(𝑘 𝜅 The controllers and initial values for all methods are zero. To ensure the rank condition on the 127 recorded data, a very small exponential decaying sum of sinusoidal input is added to the controller for the data selection procedure [144] employed in FxTCL, FTCL, and CL methods. For the speed and precision comparison of the mentioned methods for approximating 𝑓 (𝑥) and 𝑔(𝑥) on the entire domain of 𝑥, the following online learning errors are computed. Z Z 𝐸 𝑓 (𝑘) = 𝑛 k𝑒 𝑓 (𝑥(𝑘))k𝑑 𝑥, 𝐸 𝑔 (𝑘) = k𝑒 𝑔 (𝑥(𝑘))k𝑑 𝑛 𝑥. D𝑥 D𝑥 Consider the nonlinear system given below, 𝑝3 𝑥(𝑘 + 1) = 𝑝 1 𝑒 −𝑥(𝑘) + 𝑝 2 𝑒 −𝑥(𝑘) cos(𝑥(𝑘)) + 𝑢(𝑘), 1 + 𝑥(𝑘) where [𝑝 1 , 𝑝 2 , 𝑝 3 ] are the unknown parameters and the regressor is fully known as 𝑢(𝑘) 𝑧(𝑥(𝑘), 𝑢(𝑘)) = [𝑒 −𝑥(𝑘) , 𝑒 −𝑥(𝑘) cos(𝑥(𝑘)), ], 1 + 𝑥(𝑘) where 𝑝 + 𝑞 = 3. The values of unknown parameters are [𝑝 1 , 𝑝 2 , 𝑝 3 ] = [−1, 1.5, 1]. We choose 𝑃 = 3 for FxTCL, FTCL and CL methods. Let 𝜎𝐺 = 0.6, 𝜎𝐶 = 0.3 for CL method, and 𝑘 1 = 0.6, 𝑘 2 = 0.3 and 𝜅 = 0.4 for FTCL method, and 𝜉𝐺 = 0.6, 𝜉𝐶 = 0.3, 𝛾1 = 0.8 and 𝛾2 = 1.1 for FxTCL method. Fig. 6.1 shows the true and the approximated parameters for FxTCL, FTCL, CL, and GD approaches. As depicted in Fig. 6.1, while GD did not succeed to converge to the true parameters, FxTCL, FTCL, and CL converged to true values. However, FxTCL converged faster to the true values compared with the other mentioned methods. The online learning errors 𝐸 𝑓 (𝑘) and 𝐸 𝑔 (𝑘) for the FxTCL, FTCL, CL, and GD are shown in Fig. 6.2 where FxTCL converged faster to zero in comparison with other mentioned methods. 1.5 1 0.5 Parameters 0 True FxTCL FTCL -0.5 CL GD -1 -1.5 0 100 200 300 400 500 600 700 800 900 1000 k (time steps) Figure 6.1: Estimated parameters. 128 3 FxTCL FTCL 2 CL GD Ef(k) 1 0 0 100 200 300 400 500 600 700 800 900 1000 k (time steps) 4 FxTCL FTCL 3 CL Eg(k) GD 2 1 0 0 100 200 300 400 500 600 700 800 900 1000 k (time steps) Figure 6.2: Online learning errors. Table 6.1: Learning errors comparison IAE 𝐸 𝑓 (𝑘) IAE 𝐸 𝑔 (𝑘) FxTCL 33.84 50.93 FTCL 35.67 72.57 CL 45.01 101.85 GD 297.10 1295.6 The integral absolute errors (IAEs) of 𝐸 𝑓 (𝑘) and 𝐸 𝑔 (𝑘) for FxTCL, FTCL, CL, and GD methods are computed and given in Table 6.1 where FxTCL with IAEs 33.84 and 50.93 respectively for 𝐸 𝑓 (𝑘) and 𝐸 𝑔 (𝑘) has the lowest learning error compared with the other methods. 6.7 Conclusion This chapter presented a fixed-time learning method for discrete-time system dynamics’ iden- tification where concurrent learning is used to relax the persistence of excitation requirement on the regressor to an easy-to-check rank condition of the recorded data. The learning rate conditions are achieved for fixed-time convergence based on discrete fixed-time analysis. The richness of the memory data in terms of the data spectral proprieties affects the speed of convergence for the presented fixed-time learning method. Simulations verify that the presented fixed-time concurrent learning convergence speed and precision have outperformed the other methods. 129 CHAPTER 7 CONCLUSION AND FUTURE WORK In conclusion, first we introduced a finite-time distributed concurrent learning method for inter- connected systems’ identification in finite time. Leveraging local state communication among interconnected subsystems’ identifiers enabled them to identify every subsystem’s own dynamics as well as its interconnections’ dynamics. In this method, distributed concurrent learning relaxed the regressors’ persistence of excitation (PE) conditions to rank conditions on the recorded dis- tributed data in the memory stack of the subsystems. It is shown that the precision and convergence speed of the proposed finite-time distributed learning method depends on the spectral properties of the distributed recorded data. Simulation results show that the proposed finite-time distributed concurrent learning has outperformed the finite-time distributed gradient descent in both terms of precision and convergence speed. For future work, we aim to develop finite-time distributed identifiers and observers to be employed in appropriate distributed controllers for interconnected systems. Then we presented a fixed-time concurrent learning system identification method without the persistence of excitation (PE) requirement. In this method, the concurrent learning relaxes the requirement of the PE condition to a rank condition on the memory stack of recorded data. It is shown that the richness of the recorded experienced data depends on the minimum eigenvalue properties of the stack of regressor’s data which influences the speed and precision of the proposed fixed-time concurrent learning method. Simulation results are given where it is shown that the proposed fixed-time concurrent learning has outperformed other mentioned methods in both terms of precision and convergence speed. For future work, it is intended to extend the existing results for discrete-time systems. We also proposed a data-regularized concurrent learning-based stochastic gradient descent (CL-based SGD) method that leverages recorded data to guarantee linear (exponential) bounded convergence of the estimated parameters’ error. It is shown that the richness of the memory data improves the speed of convergence and reduces the probabilistic bound of convergence. Lyapunov 130 analysis guaranteed that the proposed data-regularized CL-based SGD method not only ensures the practical stability in probability of the estimated parameters’ error but can ensure a finite-sample boundedness in probability of the estimated parameters’ error. Simulation results verified that the employed data-regularized CL-based SGD could improve the speed and precision of convergence for the estimated parameters in comparison with SGD. Furthermore, we presented the fixed-time stability for deterministic and stochastic discrete- time (DT) autonomous systems based on fixed-time Lyapunov stability analysis. Novel Lyapunov conditions are derived under which the fixed-time stability of autonomous DT deterministic and stochastic systems is certified. The sensitivity to perturbations for fixed-time stable DT systems is analyzed and the analysis shows that fixed-time attractiveness can result from the presented Lyapunov conditions. For both cases of fixed-time stable and fixed-time attractive systems, the fixed upper bounds of the settling-time functions are given. Finally, we proposed a fixed-time learning method for discrete-time system dynamics’ identifi- cation where concurrent learning is used to relax the persistence of excitation requirement on the regressor to an easy-to-check rank condition of the recorded data. The learning rate conditions are achieved for fixed-time convergence based on discrete fixed-time analysis. The richness of the memory data in terms of the data spectral proprieties affects the speed of convergence for the presented fixed-time learning method. Simulations verify that the presented fixed-time concurrent learning convergence speed and precision have outperformed the other methods. For future work, we plan to employ the presented fixed-time stability analysis to develop fixed-time identifiers based on the dynamic regressor extension and mixing (DREM) technique. Moreover, the ideas of this dissertation can be easily extended to fixed-time controllers and identi- fiers for CT and DT systems. 131 BIBLIOGRAPHY [1] T. 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