noun-1‘... "pug-.uuru « This is to certify that the dissertation entitled ROBUST OUTPUT REGULATION OF MINIMUM PHASE NONLINEAR SYSTEMS USING CONDITIONAL SERVOCOMPENSATORS presented by SRIDHAR SESHAGIRI has been accepted towards fulfillment of the requirements for the Ph.D. degree in Electrical and Computer Engg. , ' .v / mm “45/ Major Professor's ' 1 3/08/2003 Date MSU is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c:/ClRC/DateDue.p65-p.15 ROBUST OUTPUT REGULATION OF MINIMUM PHASE NONLINEAR SYSTEMS USING CONDITIONAL SERVOCOMPENSATORS By Sm'dhar Seshagz’m' A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical and Computer Engineering 2003 ABSTRACT ROBUST OUTPUT REGULATION OF MINIMUM PHASE NONLINEAR SYSTEMS USING CONDITIONAL SERVOCOMPENSATORS By Sridhar Seshagz'm' The design of robust output feedback controllers for output regulation of minimum- phase nonlinear systems with well-defined normal form is considered, with emphasis on their transient performance. Previous work has shown how to design such con- trollers by incorporating a linear servocompensator in a continuous sliding mode design, but the asymptotic error regulation is usually achieved at the expense of poor transient performance. In this work, we present an approach to improve the transient performance. The servocompensator is designed as a “conditional” one that provides servocompensation only inside the boundary layer, effectively eliminating the perfor- mance degradation. We give both regional as well as semi-global results for error convergence, and show that the output feedback continuous sliding mode controller with conditional servocompensator can be tuned to recover the performance of a state feedback ideal sliding mode control. To my family iii ACKNOWLEDGEMENTS I would like to express my deepest appreciation to my advisor Prof. Hassan K. Khalil, for his insight, guidance and invaluable assistance, without which this thesis would not have been possible. His dedication to the subject and personal integrity have served as an example and an inspiration to me. Not only did he direct my research as a student, but he also played the very important role of a mentor and helped me grow as a person. He knew how to plod me on with his infinite patience when the going was rough and stood by me through some testing personal times. I leave with the great pleasure of knowing that he was recently awarded the title of University Distinguished Professor, while I was still his student. I would also like to thank the members of my Ph.D. committee, Dr. Fathi Salem, Dr. Steve Shaw, Dr. Ranjan Mukherjee and Dr. Charles MacCluer for their willing- ness to serve on my committee and their assistance and encouragement. Thanks are also due to my other instructors, specially Mr. Krishnamurthy, my undergraduate project advisor, Dr. William Brown, Dr. Ronald Fintushel, and Dr. John McCarthy who taught me the basics of algebra and topology with their skillful instruction in the classroom, but more importantly, for their constant encouragement and friendship. I am also thankful to my group members, Leonid, Jeff, Hyon, Nazir, Bader, and Ahmad Dabroom (the last three from my MS years), other friends in the department, John, Wes, Meng and Ali Khurram, the department staff, Marilyn, Pauline, Vanessa and Sheryl, and my colleagues and friends from my brief stint at Ford, specially Kathy and Suresh, for making my stay in the US such a pleasant experience. iv It would be an understatement to say that my stay in MSU has been the mem- orable experience that it has because of my friends; Kundi, Uday, Josh, Ustaad, Durgesh and Arun from the old days, and Amit, Bil, Srivats, Srividya, Srikanth, Skanth, and Prasanna from more recent times. To them I owe the pleasure of many things, both small and large, that contributed to my Ph.D., often in unpredictable ways. To my family I owe the maximum gratitude, for years of their unflinching support, patience and understanding. They provided me with the impetus to finish with their generous encouragement (“when are you going to finish ?” ), and infinite patience (“are you going to finish at all ?”). To all of them, Amma, Appa, Perippa, Manimma, Mani Perippa, Sriram, Kalyani, Rajiv, Srikanth, Ambi Anna, Malini Manni, Raja Anna, Geeta Manni, Amrith, Ranjana, Sripriya, Sama mama and family, Kannan vaathiyar and family, and many many others I owe my heartfelt thanks for always being there for me, and to Rajiv simply for being one of the nicest persons I’ve ever known ! And lastly, I’d like to thank Gunds for being such a good friend, for the many conversations that we had over the years that kept me (us ?) sane in most part, and from whose dissertation portions of this acknowledgement have been shamelessly lifted :-) WW3W afiarfirmarfimrsfirafi? I mwmfi: rrfirfi‘nfififie'é’rafi? II szrfizfizwfinfiwm u Table of Contents List of Figures 1 Introduction 1.1 Background and Motivation ....................... 1.2 Preliniminaries .............................. 1.2.1 Input-output Linearizable Systems and the Normal Form . . . 1.2.2 Integral Control of Nonlinear Systems .............. 1.2.3 Output Regulation of Nonlinear Systems ............ 1.2.4 Sliding Mode Control ....................... 1.2.5 High-gain Observers ....................... 1.3 Overview of the Thesis .......................... 2 Regulation Using Conditional Integrators 2.1 Introduction ................................ 2.2 Motivating Example ........................... 2.3 Problem Statement ............................ 2.4 Controller Design ............................. vi ix 10 12 13 13 14 22 26 2.4.1 Partial State Feedback Design .................. 2.4.2 Output Feedback Design ..................... 2.5 Closed-Loop Analysis ........................... 2.5.1 Boundedness and Convergence .................. 2.5.2 Performance ............................ 2.6 Universal Integral Regulator ....................... 2.7 Conclusions ................................ Appendix A Derivation of (2.22) ........................ Appendix B Derivation of (2.25) ........................ Global Regulation under State Feedback 3.1 Introduction ................................ 3.2 Problem Statement ............................ 3.3 Controller Design ............................. 3.4 Closed—Loop Analysis ........................... 3.5 Conclusions ................................ Tracking Using Conditional Servocompensators 4. 1 Introduction ................................ 4.2 Problem Statement ............................ 4.3 Controller Design ............................. 4.4 Closed-Loop Analysis ........................... vii 26 30 31 32 37 40 43 45 49 49 49 54 56 59 59 60 66 68 4.5 Internal Model Perturbation ....................... 4.6 Simulation Example ........................... 4.7 Conclusions ................................ Appendix A Derivation of (4.18) ........................ Appendix B Derivation of (4.21) ........................ 5 Applications 5.1 Introduction ................................ 5.2 Continuous Stirred Tank Reactor .................... 5.2.1 System Model ........................... 5.2.2 PI controller design ........................ 5.3 Permanent Magnet Stepper Motor .................... 5.3.1 System Model ........................... 5.3.2 Integral Control — State Feedback ................ 5.3.3 Integral Control - Output Feedback ............... 5.3.4 Servocompensator Design .................... 5.4 Conclusions ................................ 6 Conclusions 6.1 Summary of Results ........................... 6.2 Future Work ................................ Bibliography viii 77 86 88 91 91 92 94 95 99 100 103 105 108 113 115 115 117 119 1.1 1.2 2.1 2.2 2.3 2.4 2.5 4.1 4.2 5.1 5.2 List of Figures The general setup for the solution of the robust output regulation prob- lem. .................................... Chattering due to delay in control switching ............... Asymptotic error regulation with improved transient performance us- ing the “conditional integrator” ...................... Performance recovery under the output feedback conditional integrator design .................................... Effect of disturbance on the conventional and the conditional integrator designs. .................................. Effect of time delay on the ideal SMC and the conditional integrator design .................................... Universal regulator for relative degree one systems : PI controller with anti-windup, followed by saturation; K, = kko/p, K p = k/ H, and L = p/k ................................... Performance improvement over the conventional servocompensator de- sign using a conditional servocompensator. ............... Effect of internal model perturbation on the tracking error ....... Block schematic of a CSTR ....................... Temperature control of CSTR using a PI controller with conditional integrator ................................. 10 17 18 20 21 41 81 83 96 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Comparison of performance with the conditional and the conventional integrator designs. ............................ Effect of uncertainties on the response of the PI controller with condi- tional integrator .............................. Schematic of a two phase PMSM. .................... Tracking error performance under state—feedback integral control. Tracking error performance under the output-feedback MIMO univer- sal integral control. ............................ Effect of sampled-time implementation on ideal SMC and CSMC with conditional integrator. .......................... Sinusoidal reference tracking using MIMO servocompensators. 98 99 101 106 108 Chapter 1 Introduction 1.1 Background and Motivation The output regulation problem is one of the most fundamental problems in con- trol theory, dealing with the design of a controller to make the output of a fixed plant asymptotically track a reference signal, while asymptotically rejecting a disturbance signal, both of which are produced by an autonomous system called the exosystem. For multivariable, time-invariant, finite-dimensional, linear systems, an exhaustive ac- count of the available theory can be found, for instance, in the works of Davison [19] and Francis and Wonham [27]. It was established in these papers that the solvability of the problem requires the solvability of a system of two linear matrix equations, called the regulator equations. This, in turn, is equivalent, as illustrated by Hautus [30], to a certain property of the transmission polynomials of a composite system which incorporates the plant and the exosystem. A striking result of the theory is the observation that any controller which solves the problem can always be viewed as the interconnection of two components, called the servocompensator and the stabiliz- ing compensator. The former is a device that incorporates an internal model of the exosystem, i.e., a model capable of generating the reference and disturbance signals produced by the exosystem. The role of the latter is then to stabilize the augmented system formed of the plant and the servocompensator. The generic setup is shown in Fig 1.1. The above mentioned property is known as the internal model principle, and reduces in the special case of constant references and disturbances to the well known idea of integral control. In fact, one of the earliest formal acknowledgements of the internal model principle can be found in the work of Minorsky [62], where he observes that his “second class” of controllers, popularly known today as proportional-integral (PI) controllers, ‘has the remarkable result that such a constant disturbance has no influence upon the device’. *@ Servcccm ensator , Stabilizing 2‘ 1 + p compensator P ant Measured signals Figure 1.1: The general setup for the solution of the robust output regulation problem. In this thesis, we concentrate on the design of controllers to solve the output regulation problem for a class of nonlinear systems, robustly with respect to parameter uncertainties, with emphasis on their transient performance. Specifically, the class of systems that we consider are mimimum—phase nonlinear systems transformable to the normal form, uniformly in a compact set of the unknown system parameters. For this class of systems, robust continuous feedback control techniques like min-max control, or sliding mode control (SMC) can be used to ensure convergence of the tracking error to a small ball, while rejecting bounded disturbances. However, making the error arbitrarily small requires the use of high-gain feedback near the origin; see, for example, [7, 18, 21]. The classical idea of servocompensator + stabilizing compensator design has been used by Khalil and co-workers in [43, 45, 46, 57, 58] to achieve asymptotic output regulation. For the case of constant references and disturbances [46, 57], the servocompensator is simply an integrator driven by the tracking error; and its inclusion creates an equilibrium point at which the tracking error is zero. For the more general case [43, 45, 58], a linear internal model is identified, which generates the trajectories of the exosystem and, along with them, a number of higher-order harmonics generated by the system nonlinearities. 1 This is then used to synthesize a servocompensator, the inclusion of which creates an invariant manifold on which the error is zero. In order to achieve nonlocal stabilization of the disturbance dependent equilibrium point or zero-error manifold, the stabilizing compensator is designed using the robust control techniques mentioned in the previous paragraph. Furthermore, the controller uses only error feedback, and is designed using the separation approach of Esfandiari and Khalil [24], where a state feedback controller is first designed and then a saturated high-gain observer is used to recover the performance of the state feedback design. While the above mentioned designs achieve robust output regulation, they do not address the issue of transient performance. In fact, the steady-state performance achieved in these papers often happens at the expense of degradation of the transient performance. This is due in part to the increase in system order as a result of the servocompensator, and in part to the interaction of the servocompensator with the control saturation, which in the case of integral control leads to the well-known problem of windup [4]. The goal of this dissertation is to address the issue of transient performance degradation in the “conventional” integrator and servocompensator designs of [43, lSee Sections 1.2.3 and 4.2 for further discussion of this point. 45, 46, 57, 58]. To that end, the main contribution of the dissertation is a new ap- proach to introducing integral and servo action, done within the continuous sliding mode control (CSMC) framework of [45, 46]. In the new approach, the integrator or servocompensator is “active” only inside the boundary layer, resulting in “condi- tional” integrators and servocompensators that effectively eliminate the degradation in transient performance. Analytical results for the stability of the closed-loop system and asymptotic output regulation are provided, and the improvement in transient per- formance is confirmed by showing that the output feedback CSMC with conditional integrator/servocompensator can be tuned to recover the performance of a state feed- back ideal SMC. The rest of this chapter is organized as follows. In the next section, we briefly review some of the main elements of this dissertation. These include H . Input-output linearizable systems and the normal form [\3 . Integral control of nonlinear systems 3. Output regulation of nonlinear systems A . Sliding mode control, and O1 . Output feedback using high-gain observers. The normal form can be considered the starting point of the controller designs in Chapters 2 to 4, which use the technique of sliding mode control to design the stabi- lizing compensator. Integral control is the topic of Chapters 2 and 3, and its extension to the output regulation problem is the topic of Chapter 4. Output feedback using high-gain observers is used in Chapters 2 and 4. We conclude this chapter with an overview of the thesis in Section 1.3. 1 .2 Preliniminaries 1.2.1 Input-output Linearizable Systems and the Normal Form The single-input single-output (SISO) nonlinear system :i: = f(~'v) + 9(30) u, y = h(96) (1-1) with state a: E R", input a E R, and output y E R, where f and g are sufficiently smooth vector fields on a domain D E R" and h : D —> R is a sufficiently smooth func- tion, is said to have relative degree p, 1 __<_ p S n, in a region Do 6 D if LgL}‘1h(z) = 0, for i = 1,2, - -- ,p - 1 and LgL’fkthr) 76 0 for all :1: 6 Do, where th(.r) = g—Eflx) is the Lie derivative of h with respect to f, and L}h(:r) = L f(Lif”1h(x)). Remark 1.1 It follows from the definition that the relative degree is the smallest integer p for which the control it appears in the expression for y("). It can be shown (see, for example, [37]) that the relative degree condition guarantees the existence of a local change of variables [nT éT]T = T(x), 6 E R”, n E R""’, that transforms (1.1) to the normal form 6 = A66 + Bc 7(1):) [21. - a(a:)] (1.2) y = Cca: where the pair (Ac, BC, Cc) is a canonical form representation of a chain of p inte- grators, 7(3) = LgLfi’lh(:r), and a(m) = —L?h(a:) /L9L’f"1h(a:). The state feedback control a = (a(x)7(x) + v) /ry(a‘) reduces the input-output map to 5,, 2 gm = v, i.e., renders it linear, and consequently makes the component 77 unobservable from the output. The equation 1'7 = 45(17, 6) is called the internal dynamics of the system; when 6 = 0, it is called the zero dynamics. For the controller designs in this thesis, we ensure the boundedness and conver- gence of 5 by a robust control design. Consequently, in order to show stability of the full system, some sort of a stability condition has to be imposed on the internal dynamics. The assumption that we make is that the internal dynamics 7'] = ¢(n,§) is input-to—state stable (ISS) with 6 as the driving input, which guarantees that 17 is bounded whenever E is, and that 7) tends to zero as g tends to zero [47]. Extensions of the above concepts to multi-input multi-output (MIMO) systems can be found, for example, in [37, 65]. 1.2.2 Integral Control of Nonlinear Systems Integral control achieves robust output regulation for the case of constant or asymptotically constant exogenous signals. As mentioned in Section 1.1, the inclusion of an integrator creates an equilibrium point at which the error is zero. While external disturbances or uncertainties in the system model shift the equilibrium point, the integrator ensures that the tracking error is zero, as long as the controller stabilizes this equilibrium point. Integral control of nonlinear systems has been studied by several researchers. It was shown in Hepburn and Wonham [31] that integral control suffices to guarantee structurally stable regulation, i.e., regulation in the presence of “small” parameter variations. Similar results were given by Desoer and Lin [22], for exponentially stable plants having a strictly increasing dc steady-state input-output map. Local results were also given in Huang and Rugh [35] using the method of extended linearization. Semi-global results for fully linearizable systems using state-feedback were reported in Freeman and Kokotovic [28]. Regional and semi-global results for input-output linearizable systems with asymptotically stable zero dynamics using output-feedback were given by Mahmoud and Khalil [57] and Khalil [46]. 1.2.3 Output Regulation of Nonlinear Systems The integral control idea of the the previous section can be generalized to the case of exogenous signals generated by a neutrally stable system under some additional assumptions. In particular, as mentioned in Section 1.1, once an internal model is identified, an appropriate servocompensator can be designed and augmented with the plant. The servocompensator is the generalization of the integrator, and its inclusion creates an invariant manifold on which the error is zero. As was the case with the equilibrium point, the zero-error manifold is disturbance dependent, but the servocompensator ensures that the tracking error is zero, as long as the controller stabilizes this zero—error manifold. The output regulation problem for nonlinear systems has been studied by many researchers in the past two decades, among whom we specially mention Isidori and co—workers [12, 13, 14, 38, 39, 67, 68, 69], Huang and co—workers [16, 32, 33, 34, 35, 36] and Khalil and co—workers [43, 45, 58]. The work was initiated, to the best of our knowledge, by Hepburn and Wonham [31], who presented an extension of the notion of the internal model for nonlinear systems defined on differentiable manifolds. The pioneering work of Isidori and Byrnes [39] showed how the results of FTancis and Wonham [27] could be extended to nonlinear plants and nonlinear neutrally stable exosystems with their formulation of the nonlinear regulator equations. They also showed that the transmission polynomial interpretation of Hautus [30] had its natural extension in terms of the zero dynamics, which is the nonlinear analog of the notion of transmission zeros. The results in [39] were local, and required smallness of both the exogenous signals and the initial states. A different “computational approac ” to the problem, involving a power series expansion of the solution of the regulator equations, was pursued by Huang and Rugh [35, 36]. Their method allowed for large exogenous signals, but was still local in the initial states. Regional and semi-global results first appeared, for the case of fully linearizable systems, in the work of Khalil [43]. An important contribution of [43] was the observation that in the nonlinear case the internal model must be able to generate not only the trajectories of the exosystem, but also a number of their higher-order harmonics. This idea was also independently elaborated by Huang and Lin [34, 32] and by Delli Priscoli [63]. Appealing to the concept of immersion, Isidori [37] further refined the idea of an internal model, and provided a complete set of necessary and sufficient conditions for the existence of a solution to the local output regulation problem. Extensions of the design of [43] to the case of nontrivial zero dynamics were given in Mahmoud and Khalil [58]. By combining the structurally stable output regulation approach of [37, Chapter 8] with the robust control approach of [43], an interesting generalization of the results of [43] was given in Isidori [38]. A succinct overview of the output regulation problem for nonlinear systems, summarizing most the results discussed above, can be found in [12, 13]. In the next paragraph, we summarize some of the more recent results. A simplification of the robust servocompensator design of [58] is given in Khalil [45], where the only precise information that is required in the design of the controller is the relative degree of the plant, the sign of its high-frequency gain and the linear internal model. A semi-global controller that relaxes the assumption of input-to- state stability of the zero dynamics made in [45, 58] can be found in Serrani et al [68], and one that allows the frequencies of the exosystem to be unknown, making use of an adaptive internal model, can be found in [69]. A recent result that relaxes an assumption made in almost all previous works, including [45, 58, 68, 69], that the solution of the regulator equations be a polynomial in the exogenous signals, can be found in Chen and Huang [16]. 1.2.4 Sliding Mode Control Following [45, 46], the robust control technique that we use in this thesis to design the stabilizing compensator is a continuous version of sliding mode control. Sliding mode control can guarantee asymptotic tracking with zero steady-state error for a wide class of nonlinear systems, and its design is accomplished through a two- step process. The first step is the design of a sliding surface function 8, so designed that when the system trajectories are on the sliding surface s = 0, the system has the desired behavior. The second step is the design of the control to force the tra- jectories to reach the surface s = 0 in finite time and remain on it thereafter. The motion thus consists of a reaching phase during which trajectories starting off the surface s = 0 move towards it and reach it in finite time, and a sliding phase during which motion is confined to the surface s = 0. In its original form, often referred to as ideal SMC, finite-time convergence to and invariance of the surface s = 0 is accomplished by a discontinuous control that switches sign across the surface. As a result of switching non-idealities such as time-delays and parasitic sensor/ actuator dynamics, the variable 3 does not remain identically at zero at the end of the reach- ing phase, but oscillates about it. Fig 1.2 shows a typical zig—zag motion about the surface due to a time-delay in implementing the control. The trajectory starts in the region 3 > 0 and heads towards the surface s = 0, first hitting it at the point P. In ideal SMC, the sliding phase should begin at this instant of time. However, in reality, there is a delay between the time the sign of 8 changes and the time the control switches. During this period, the trajectory crosses the surface into the region 3 < 0. When the control switches, the trajectory reverses its direction and the pro- cess repeats. The above phenomenon, known as chattering, is an important drawback of ideal SMC, and can excite unmodelled high-frequency dynamics, degrade system performance, and even result in instability. Various approaches have been proposed to reduce/ eliminate chattering; see, for example, Bartolini et al [8], Young et al [75] and the references therein. The most common one is to replace the discontinuous con- trol by a continuous approximation in a boundary layer of the sliding surface. This method can reduce chattering but often at the expense of a finite steady-state error. Asymptotic error regulation can be recovered in a CSMC design by augmenting the system with a servocompensator and including the output of the servocompensator as part of the sliding variable 3. Such an approach has been pursued for the case of constant exogenous signals by Chang [15], Khalil [46], and Baik et al. [6], and for the more general case by Khalil [45]. A s=0 s>0 s<0 Figure 1.2: Chattering due to delay in control switching. 1.2.5 High-gain Observers High-gain observers provide an important technique for the design of output feedback controllers for nonlinear systems. A high-gain observer is essentially an approximate differentiator that robustly estimates the derivatives of the output. Its gain depends on a small parameter c, which can be adjusted to guarantee that the estimation error decays to an 0(6) value within an arbitrarily small time interval. The use of high-gain observers in the design of output feedback control of input-output linearizable minimum phase nonlinear systems was studied by Esfandiari and Khalil in [24]. A key contribution of their study is the use of saturation nonlinearities to 10 overcome the peaking phenomenon associated with high-gain observers. The observer is designed to assign the observer eigenvalues at 0(1/6) values in the open left-half complex plane. This results in exponential modes of the form (1/e)m exp(—ta/e) for some positive constant a and positive integer m. While the decay of the exponential term exp(—-ta/e) can be made faster by decreasing c, the amplitude term (1 / c)” grows larger with such decrease, resulting in an impulse-like peaking in the estimates. For nonlinear systems, this peaking can destabilize the system because of the possibility of finite escape times. Esfandiari and Khalil showed that saturating the feedback control law outside a compact region of interest protects the plant from the effects of peaking. Because the peaking transients decay rapidly, the saturation period is small. During this period, the state of the plant is close to initial value, and the estimation error decays to an 0(1/6) value. As a result, the output feedback controller recovers the performance under state feedback as 6 tends to zero. Since its introduction, the technique in [24] has received a lot of attention in output-feedback designs and has been included in a few textbooks [37, 47, 64]. One of the important consequences of this technique is the ability to separate the design of output feedback control for nonlinear systems into a state feedback design followed by the design of the high-gain oberver. Teel and Praly [72] developed a generic separa- tion principle, which showed that (semi)global stabilizability via state feedback plus uniform observability imply semiglobal stabilizability via output feedback. Atassi and Khalil [5] provided a more comprehensive separation principle and showed that the output feedback controller recovers the performance of the state feedback controller in the sense of recovering asymptotic stability of an equilibrium point, its region of attraction, and its trajectories. An extensive survey of the use of high-gain observers in nonlinear control can be found in [44]. 11 1.3 Overview of the Thesis In Chapter 2, we consider the problem of robust output regulation for multi- input multi-output input-output linearizable minimum phase systems. For this case, we design a continuous sliding mode control with conditional integrators that achieve asymptotic error regulation with good transient performance. Analytical results are given for regional and semiglobal regulation, and we prove that the output feedback controller with conditional integrators recovers the performance of a state feedback ideal SMC that does not include integral action. In Chapter 3, we consider a state-feedback version of the design considered in Chapter 2, with the specific goal of achieving global regulation. In addition to the global regulation result, we also prove a performance recovery result that is slightly sharper than the one of Chapter 2. In Chapter 4, we consider the extension of the conditional integrator design of Chapter 2 to the more general output regulation problem, where the reference and disturbance signals are generated by a neutrally stable exosystem. For this case, we design a conditional servocompensator that provides servocompensation only inside the boundary layer, achieving asymptotic output regulation with good transient performance. As before, we give regional and semiglobal results for the regulation and also show the performance recovery of an ideal SMC design. A result on the effect of small internal model perturbations on the error is also provided. In Chapter 5, we apply the designs of Chapters 2 to 4 to two applications : temperature control of a continuous stirred tank reactor (CSTR) and position control of a permanent magnet stepper motor (PMSM). For the CSTR, we apply the design of Chapter 2, while for the PMSM, we consider the designs of Chapters 2 to 4. Finally we summarize our results and provide directions for future research in Chapter 6. 12 Chapter 2 Regulation Using Conditional Integrators 2.1 Introduction We consider the problem of robust output regulation for multi-input multi—output (MIMO) minimum phase nonlinear systems transformable into the normal form, uni- formly in a set of constant disturbances and uncertain parameters. For this class of systems, prior work has shown how to achieve zero steady-state error by introducing integral action in the controller [46, 57]. Integral control creates an equilibrium point at which the tracking error is zero. Robust control is designed to bring the trajecto- ries to a small neighborhood of the equilibrium point, within which the control acts as a high-gain feedback that stabilizes the equilibrium point. While [57] accomplishes this through continuous min—max control, [46] uses continuous sliding mode control (CSMC) to design the controller as a universal one, where the only precise information about the plant that is used is its relative degree and the sign of its high-frequency gain. The asymptotic regulation achieved by integral action happens at the expense 13 of degrading the transient performance. Even in the absence of control saturation, integral action makes the response more oscillatory. When the control saturates, in- tegrator buildup results, causing large overshoots and settling times. In this chapter, we present a new approach for introducing integral action to alleviate this transient performance degradation. This is done within the continuous SMC framework of [46]. The integrator is modified to provide integral action only inside the bound— ary layer, i.e., only “conditionally” , effectively eliminating the transient performance deterioration. Both regional as well as semi-global results for asymptotic regulation are provided. The improvement in transient performance is shown analytically by proving that the output feedback continuous SMC with the conditional integrator recovers the performance of a state feedback ideal SMC without an integrator. 2.2 Motivating Example Consider the second order system 5&1 = $2 1'32 = am? + bzg + c223 + u (2-1) y = $1 The constants a, b and c are assumed to be unknown, but bounded with known bounds. The control objective is to regulate the output y to a constant value r. In ideal SMC design, the sliding surface can be chosen as s = 19181 +e2, where el = y — r, 82 = él, and k1 > 0. This ensures that when motion is constrained to s = 0, the error e1 converges asymptotically to zero. Differentiating, one obtains s' = kleg +a(e1+r)2 +be2 +063 +u 14 Finite-time convergence to, and invariance of, s = 0 can be achieved by choosing u 2 ul + u2, where the equivalent control ul is designed to cancel known or nominal terms in the expression for s and can be taken as _ ~ 2 ‘ . 3 U1 — —k182 — 0(81 + 7‘) - b€2 — 082 where a, f), and 6 are nominal values of a, b, and c respectively. The switching control U2 is designed to handle the uncertain terms in the resulting expression for s and can be taken as U2 = —IO(€1 + 792 + filezl + 7|€2|3 + 5] sgn(s) where the positive constants a, 6, and '7 are upper bounds on [a - (1|, [b — 6|, and [c — 5|, respectively, and 6 > 0. This choice ensures that s converges to zero in finite time and stays there for all future time, which guarantees that el and e2 converge to zero asymptotically. However, as is well-known, this design suffers from chattering in the presence of switching nonidealities or unmodeled high-frequency dynamics. Various approaches have been proposed to reduce or eliminate chattering [75], the most common one being replacing the discontinuous term sgn(s) by its continuous approximation sat(s/u). This method can eliminate chattering but often at the cost of a non-zero steady-state error, that is proportional to u. In order to obtain smaller errors, it is therefore necessary to make it smaller, which in turn, leads to chattering again. It is possible to recover the asymptotic regulation achieved by ideal SMC by using integral control within a continuous SMC setting. Integral action is conventionally introduced by augmenting the system with an integrator driven by the tracking error, i.e., d 2 e1. In the case of the particular example, suppose we do this and also modify the sliding surface to s = koo + klel + e2, where now kg and k1 are chosen to ensure that the polynomial X" + klA + k0 is Hurwitz, which guarantees that when motion is 15 restricted to s = 0, the error el converges asymptotically to zero. The previous steps can then be repeated to design u. In particular, we take U1 = —k0€1 — k182 — C(Cl + 7‘)2 — b€2 —' C83 U2 = -[a(e1+r)2+filezl+ilezl3+6lsat(s/u) The presence of integral action guarantees that there is an equilibrium point, within 0(a) of the origin, at which el = 0. Now, to achieve asymptotic regulation we do not need it to be arbitrarily small; we only need it to be “small enough” to stabilize the equilibrium point. 1 However, while integral control, as designed above, can achieve asymptotic regulation, the transient response deteriorates when compared to that under ideal SMC. To address the transient response degradation with conventional integral control, we modify the integrator design as follows. Let s be as before, i.e., s = koa+k1e1 +82, but now k0 > 0 is arbitrary, k1 > 0 is retained from the ideal SMC design, and a is the output of 0" = -—k00 + u sat(s/u) (2.2) To see the relation of (2.2) to integral control, observe that inside the boundary layer {[s] g u}, (2.2) reduces to 6' = klel + e2 2 klel + él, which implies that el = 0 at equilibrium. Thus (2.2) represents a “conditional integrator” that provides integral action only inside the boundary layer. The control is taken as in the continuous approximation of ideal SMC, i.e., u = ul + Ug, where u1 = —k1e2 — 6(61 + r)2 — beg — éeg U2 = —[a(el + r)2 + ,Blezl + ’7[62[3 + 6] sat(s//.i) The simulation results are shown in Figure 2.1. Numerical values used in the simu- lWe naturally expect this fact to be of consequence when there are switching nonidealities, and will show so through simulation later on. 16 lationarea=0.6,b=2.5,c=0.1,a=1,h=2,c=0,r=1,a=0.5,fl=0.6, 7 = 0.1, 6 = 1, u = 0.1, and 221(0) = $2(0) = 0(0) = 0. The constant k1 = 5 in the ideal SMC case, its continuous approximation, and the conditional integrator design, with k0 = 1 in the conditional integrator design. The values of kc and In in the conventional integrator design are taken as 25 and 10 respectively. The following ob— servations can be made from Figure 2.1 : (i) the conventional integral control recovers the asymptotic regulation that is lost in the continuous approximation of ideal SMC (without integral control), but at the expense of degraded transient performance; in particular, the error convergence to zero is sluggish; (ii) the conditional integral de- sign also achieves the task of asymptotic regulation, but without any degradation in transient performance; in fact, in the subplot for the transient behavior, the responses of the ideal SMC and the conditional integrator design are almost indistinguishable. Steady-state response Transient response 0.4 g , 0.05 a, , Conventional Cénditiona| _ _. _ §_ _ _ .- r 02 .. . . integrator ......... integrator. . . I g \ g . x I . 0.04.,,‘ ........ : ....... .. I I I ‘ I I ,— — —:— — —- — A — — _ _ ConVentionjal , ' ' o ..... I . ....... 2. . . _ . . . . . . . . . : integrator; ‘ I I o 03 . . . . . . . .; ........ g ..... ‘ ................. a) I I . °’ 3 l . _0.2-..l .......... ; ..... ....; ............ . ; 3 3 .; g I 5 3 5 3 3 I 3 Continuous g) f Continuous g, 0.02 .. . . . . . . 5.. ; ...... ...... ‘ sapproximatiorl % _o 4 _. _' ................... apprexirnation. % figdgfigf'; ‘jof ideal SMC a ' _ of ideal SMC <3 9 : i: - I'— I ; j I'— 3 Ideal t ' ' 2 0.01 r. . . . . . f ....... SMC, . .\ ........ ...... .I _06...' ........... ....... ....... .1 I :\ : I 3 \ I ' ' \ I , Ideal -\_ ' g -08‘]' ........... j . . I ... _ -i _____________ I . —1 -0.01 i o 2 . 4 6 5 6 7 a 9 10 Time (sec) Time (sec) Figure 2.1: Asymptotic error regulation with improved transient performance using the “conditional integrator”. The transient performance recovery property of the conditional integrator design is also retained under output feedback, when the state e2 is replaced by its estimate 17 ég obtained from the high-gain observer (HGO) 61 = C2 + 011(81— él)/€ 62 = 02(61—él)/€2 The positive constants al and (12 are chosen to assign the roots of the Hurwitz polynomial A2 + a1/\ + 012, and e is chosen sufficiently small. In order to take care of the peaking phenomenon associated with high-gain observers [24], the control is saturated outside a compact set of interest. Simulation results are shown in Figure 2.2, with a1 = 15, 012 = 50, 61(0) = 62(0) 2 0, and a saturation level of 50 for the control. State feedback trajectories Output feedback trajectories as the high gain inside the BL increases as the HGO gain increases, u=0.1 0.05 l ' 0'18 T r l l " ' ‘ ' . : : : I- \\ 0.16,’,”‘,‘ ........ ......... 0'04»? lllll \ ...... llllllllllll : lllllllllllllll O 14p]: ..... ...... [ ........ ......... I ‘. 5 ”=0" ' .’ 5 i 5 9— e=ooo1 0 j \ ---- [1:05 0.12.] ....... i ........ ' . a owl‘s, m I |. _ 'E=0.01 o . . : . ................ : ................ . ....... .. O 0.024 ......... \ .............. i ............... I 008" ...... ....... | ........ ......... 2 l \ t : co , : ,: : : 2 ' : laL) 006 .- ............... I ......................... § 9 ‘a z 0 I . m- 0.01».' ............. \. ........... ...... 0.04s ............... t ........................ l 3 ‘. , . \. i l —o.o1 + -o.02 4 6 8 10 o 2 4 6 8 10 Time (sec) Time (sec) Figure 2.2: Performance recovery under the output feedback conditional integrator design. Performance recovery in Figure 2.2 is shown in two steps : (i) as It tends to zero, the response under state feedback continuous SMC approaches ideal SMC; (ii) for fixed a, the response under the output feedback continuous SMC approaches that 18 under state feedback continuous SMC as 6 tends to zero. For the general case, we will prove in Section 2.4.2 that the closed-loop trajectories under output feedback continuous SMC with the conditional integrator approach those of the state feedback ideal SMC as u, e tend to zero. The previous discussion showed how the design of the controller proceeds in gen- eral. Since our design requires that the control be bounded, a possible simplification of the controller is to choose the equivalent control to be zero and the coefficient of the switching component to be constant, i.e., u = —k sat(§/u) Since, in practical applications, a constraint on the control magnitude appears nat- urally as an actuator limit, one might simply choose k as the maximum permissible control magnitude. Since the only precise information about the plant that such a controller uses is its relative degree and the sign of its high-frequency gain, it is referred to as a universal integral regulator [46]. We will discuss the universal inte— gral regulator design further in Section 2.5 and show that the integrator modification (2.2) can be interpreted, in this case, as a special choice of a traditional anti-windup scheme. To continue with this discussion, note that the control magnitude required to accommodate a step change in r increases as the step increases. One way to deal with this when the control is constrained with an apriori specified bound is through trajectory planning schemes. 2 In order to illustrate this, we consider the following modification to the previous simulations. The control is replaced by u = —k sat(.§/u), with k: = 50. All other values are retained from the previous simulations, except 1‘, which is increased to 1.5. One can verify, for example, by simulation, that the sliding condition is not satisfied. However, when the constant reference r is “smoothed” 2A more detailed discussion on this issue can be found in the next section. 19 by passing it through the filter 1/(rs + 1)2, with r = 0.5, the control magnitude is now sufficient to overcome the uncertain terms and the sliding condition is satisfied. Furthermore, since 8(0) 2 0, the sliding condition ensures that s stays inside the boundary layer for all future time, which along with e1(0) = 0 implies that the error 81 itself is small for all time, under both the conventional as well as the conditional integrator designs. When the trajectory stays inside the boundary layer during the transient period, the conditional integrator acts as an integrator all the time; hence, we do not expect any significant difference between the transient responses of the two designs. The advantage of the conditional integrator design becomes clear when we consider an unexpected disturbance that causes an abrupt change in the state of the system. For example, consider an additive impulse-like disturbance d(t) of magnitude 75 acting at the input of the system between t = 5 and t = 5.1385 seconds. The response of the two designs are shown in Figure 2.3. Response to reference Response to and disturbance “04 reference 0.8 r ' r f 5 r ' . 0.6 ......... ...... . . . . . . . .CSMC WIIIL g g conditional 0.4 ......... E . "'“E ........ E...integrator. .. er 2 2 / z of § 02 ......... ........ . . . . . j ....... .. ........ + § 6 ' ' ] r ' a o: 0 '2 ‘ o: E I I: E ii I If i‘: S _0.2 ......... . ............. |. I i ........ ' ......... Q '— I :. I l f f '- _0.4. ...... CSMC. NINTH”; ....... 1 ......... cenvenfiOnal..E E -0.6 ........ integrator . . . ll ........ ......... : : II : ; -0.8 i A i 1 i I i . o 2 4 6 8 10 o 1 2 8 4 5 Time (sec) Time (sec) Figure 2.3: Effect of disturbance on the conventional and the conditional integrator designs. We see from the plot on the left that while the system responses are almost 20 identical (indistinguishable in that plot) before the onset of the disturbance, the response to the disturbance is significantly degraded with the conventional integrator design. The response before the disturbance is seen better in the plot on the right. Lastly, before we present the system description and the problem statement for the general case, we make a small digression. In order to highlight the issue of chattering, we repeat the first two simulations under the assumption that a time delay of T = 0.01 seconds precedes the control input. The results are shown in Figure 2.4. Effect of time delay on ideal SMC, .3 its continuous approximation and the conditional integrator design x 10 1 I I l l I | l l I 0) 9 B \ OI ' ' ‘ i . . . . . E \ 1 . . . fi Conditional integrator ' ' : CSMC (“=0 1): 9 _4 _ .................. _ ......... i ....... Idea] SMC ..................................... ,_ . , . CSMC (11:0. 01): Z 3 ‘5 '. ;'.'_'1”_‘."_':;1_'_“."_'j'_';: ;'.‘_".;‘;‘_':; 1'_".f_'1"_'.'; :;;'._ ‘‘‘‘ ;'_'.'; : '.".1‘. J; J; : _ '41:: -6 I I I I I i I I l 9 9.1 9.2 9.3 9.4 9. 5 9.6 9.7 9.8 9.9 10 TIme (sec) Conditional Ideal SMC CSMC (p.=0.1) CSMC (u:0.01) integrator (p.=0.1) 1 —o.5a 1 -0.6 . 0 i . _0'59 ......... ......... 0 3 f 72 /—~—3 E _1 . _0_59 ......... g ......... _1 o : o , _ I __2 ] . ‘ . h] . _0'59 ......... ......... _2 j ‘ —a l 0.59 ' -3 ’ —o.6 ' 9 9.5 10 9 9 5 10 9 9.5 10 9 9.5 10 Time (sec) Time (sec) Time (sec) Time (880) Figure 2.4: Effect of time delay on the ideal SMC and the conditional integrator design. 21 We see from the figure that there is chattering in the control and that the property of asymptotic regulation is lost with ideal SMC. Replacing the discontinuous control with its continuous approximation eliminates chattering when ,u = 0.1, but at the expense of a relatively large non-zero steady-state error. Reducing u to 0.01 results in chattering again. The non-zero steady-state error can be handled by the conditional integrator design, where as mentioned earlier, the value of It does not have to be made arbitrarily small and hence we can expect that this design will not suffer from chattering. This is validated by the simulation results of Figure 2.4. While we have included a simulation with a time delay to highlight a merit of this approach, we do not present any analysis for this case in the succeeding sections. 2.3 Problem Statement Consider an MIMO nonlinear system, modeled by :i: = f(rc, 0) + 2:11 91-(93, 9)[m + 5:“(13 9: wlla yi : hi(x16)1 13137” (2.3) where a: E R" is the state, it E R“ is the control input, y E R'” is the output, 0 is a vector of unknown constant parameters that belongs to a compact set 6 C R”, w(t) is a piecewise continuous exogenous signal that belongs to a compact set W C Rq, f () and g,(-) are smooth vector fields on D dg Dz x G, where D; is an open connected subset of R", h,(-) are smooth functions on D, and the disturbances 5,-(-) are continuous functions on D x W. The formulation in (2.3) allows for matched disturbances that may depend on time—varying exogenous signals. We will specify a restriction on w shortly, when we are ready to state the control objective. Our first assumption is that the disturbance-free system (2.3) has a well-defined normal form, possibly with zero dynamics [37]. 22 Assumption 2.1 The system 2': = f(zr, 6) + g(x, 6)u, y = h(a:, 0) has a strong vector relative degree {p1, p2, . . . ,pm} in D3, i.e., ngLfih,(:c,6)=0for0gk_<_p,—2,1~gigm,1gjgm and A(:I:, I9) (if! {ngLfi‘Tlhi} is nonsingular for all a: 6 D2 and 0 6 8. Furthermore, the distribution span{g1, - - - ,gm} is involutive, uniformly in t9, and there is a change of variables 7’ 711(3), 0) n- =T(:r,0)= ,neR P, reap (2.4) g T2($10) where§= {5‘}: With £3- = Lit—1hr} 1 Si S PI, 1 S i S m, andp = P1+P2+"‘+Pm; such that ngn, = 0 V 1 S j 5 m, 1 S i S n —- p, and T(:r,0) is a. difl‘eomorphism of Dr onto its image. The vector relative degree and involutivity of the distribution span{gl, ° - ' ,gm} guar- antee the existence of the change of variables (2.4) locally [37]. Assumption 2.1 goes beyond that by requiring (2.4) to hold on a given region, uniformly in 0. With the change of variables (2.4), we rewrite (2.3) in the normal form 1'7 = 4507.5,0) .. . (2.5) £1 Aigz + Bi [bi(771€1 6) + 23:1 aij (771 61 6) (“j + 6j(n1€161w))l where, for 1 S i S m, the pair (A,, B.) is a controllable canonical form that represents a chain of p,- integrators, b,(-) = L?‘h,~, and {a,j(-)} = A(). Our interest is in the regulation problem. To that end, we require that the exogenous signal w(t) approaches a constant limit was, i.e., limtnoow(t) = w,,. In a 23 similar vein, the reference r,(t) that the output y,- is required to asymptotically track has the following two properties: 0 r,(t) and its derivatives up to the p,th derivative are bounded, and r(p‘)(t) is i piecewise continuous, for all t Z 0 o limtnoor,(t) = r,,, and limtnoorfjkt) = 0 for 1 g j S p,. This class of signals includes constant signals as a special case. Formulating the problem with time-varying references, which are asymptotically constant, accommo- dates a common practice in many applications; for example, in “trajectory planning” schemes employed to achieve point-to-point motion in the control of robotic manipu- lators [66, Chapter 5], or in pre-filter smoothing of a step command in the control of electric drives [55, Chapter 15]. The formulation also takes advantage of the robust control approach to designing the stabilizing controller, in the following sense. When the reference satisfies the first property, the robust control design ensures ultimate boundedness of the tracking error. When the second property is satisfied as well, the integral action gaurantees that the error asymptotically converges to zero. Let r,, = {r,-,,}, w(t) = w -— w,,, V‘(t) = [r, — r,,,,r,-(1), - -- ,r,(”“1)]T, w(t) = {r,("‘)}, and u(t) = {V'}. By construction, w(t), u(t), and w(t) are bounded for alltZOandconvergetozeroastaoo. LetX C R“,AC Rp,andAoCR'” be compact sets such that r,, E X, V(t) 6 A, and w(t) 6 A0 for all t 2 0, and lo be a positive constant such that [[V” _<_ l0 for all u E A. Set (1 = (r,,,6,w,,) and D; = X x G x W. To solve the regulation problem, it is necessary that for every d E Dd, there exist an equilibrium point at which y = r,, and a control input that can maintain equilibrium. This is guaranteed by our next assumption. Assumption 2.2 For each d E Dd, there exist a unique equilibrium point 17: = f(d) E 24 D1 and a unique control {I = a(d) such that 0 = f(ifl)+9(i19llfi+5(i19.wss)l,and r3, = h(:E,6) With the change of variables (2.4), the equilibrium point f(d) maps into (17(d), {_(d)), where {‘(d) = [r,-,,,0, - -- ,0]T. Let 2 = 77 — 77, and ei = 5‘ — E‘ - V‘, and rewrite (2.5) as 2 = qb(z,e+u,d) 6‘ = A,e‘ + B,- [b,-(z, e + u, d) — rfp‘) + 2;, (1.3-(2,6 + V. d)(uj + M2. 6 + V. 61.13))1 (2.6) where, for convenience, we write the functions (f), b,-, a,,-, and 6,- in terms of the new variables. Since we do not necessarily require our assumptions to hold globally, we need to restrict our analysis in the (z,e) variables to a region that maps back into the domain D3. The following assumption states such a restriction. Assumption 2.3 There exist positive constants l1 and l2, independent of d, such thatforalldEDd,w€W,I/EA andwEAo, e6 s‘i—E‘weu <11} and 26 zté‘gjzn <12} =>x€ D, In the output feedback case, the only components of the state (2, e) that are available for feedback are 6‘, = y,- — r,~, 1 S i S m. The unavailability of the partial-state e is dealt with by using a high-gain observer to estimate its unmeasured components. The unavailability of z is not an issue because we will design the control u to regulate the error e to zero and then rely on a minimum-phaselike assumption, stated below, to guarantee boundedness of z. The assumption has two parts. The first part states that with (e + V) as the driving input, the system 2 = ¢(z, e + V, d) is input-to-state stable over a certain region [47], which implies that with e + l/ = 0, the origin of 25 i = ¢(z, 0, d) is asymptotically stable. This is strengthened in the second part of the assumption to local exponential stability of the origin. Assumption 2.4 (i) There exist a C'1 proper function Vz : Z -—> R+, possibly depen- dent on d, and class [C functions A,- : [0, l2) —> R+(i = 1,2,3) and’y : [0, lo+l1) —> R+, independent of d, such that ,\,(||z||) _<_ VAR/ad) S A2(||Z||) 3V: + 3V2 0t 02 ¢(Z,6+V,d) S -/\3(||ZI|)I V “Z” Z 7(||6+V||) for all e E 8, z E Z, 1/ E A, and d E D. Furthermore, 7(l0) < A;1(/\1(l2)). (ii) The equilibrium point z = 0 of i = 0(2, 0, d) is exponentially stable, uniformly in d. 2.4 Controller Design Relying on the separation principle [5] that is common to the output feedback designs of [46] and [57], we pursue the same procedure for designing the controller used in those papers. First, a globally bounded partial state-feedback controller that meets the design objectives is designed under the assumption that e is available for feedback. Next, a high-gain observer is used to estimate the derivatives of the measured outputs i 61. 2.4.1 Partial State Feedback Design The first step in the sliding mode design is to specify a sliding surface on which sliding motion occurs [75]. In the absence of integral action, we define the sliding 26 surface s,- = 0 by Pr-1 3,- = Z kj-e; + e; (2.7) 1:1 where the positive constants k], - - - rich—1 are chosen such that the polynomial AP1‘1+ k;,—1’\pi-2 + . . . + k: is Hurwitz, which guarantees that when motion is constrained to the surface s,- = 0, the tracking error e‘] and its derivatives converge to zero. Differentiating (2.7) and using (2.6), we have = a + Zaijl‘nuj + 6.01 (2.8) i=1 where E-() = b,(-) — rfp‘) + 3:11 ;C;+1. Let F(z, e, V, d, w) = {F}(z,e,I/,d,r§p‘))}. In the 8180 case, (2.8) reduces to 3 = F(') + a(°)l’u + 5(°)l and a standard assumption in this case is to require the sign of a(-) to be known and a(-) to be bounded away from zero. Our next assumption can be thought of as a straightforward extension to the MIMO case. Assumption 2.5 A(z, e+u, d) = I‘(z, e+u, d)A(e, V) where A is a known nonsingular matrix andI‘ = diag[71,- -- gym], with 7,(-) Z 70 > 0, 13 i g m, for alle E 8, z E Z, V E A, d E Dd, and some positive constant 70. In the ideal SMC case, the control it can then be taken as u = A‘l(e, V)[—13‘(e, u,w) + v], v,- = —fi,-(e, 14w) sgn(s,) (2.9) 27 where F,() is a nominal value of F,(-), which could be, but not restricted to Fl ) = hie?“ — rip" +8.0 b,() is a nominal value of b,-(-), and the component v,- is designed to handle uncer- tainties. Note that F,() = 0 is possible. The choice of 6,-(o) will be made clear shortly. Guided by the motivating example in Section 2.1, we introduce integral action as follows. First, the ideal sliding surface function 8,- of (2.7) is modified to Pi—l koo,+ +12; kge; + 82‘, (2.10) where o,- is the output of d.- = —kia.- + u.- sat (ff) , a.(o ) e [- Min/ks I (2.11) with It}, > 0, and u, a small positive parameter to be specified later. Furthermore, as was done in Section 2, the ideal SMC (2.9) is modified to the continuous control u = A’1(e, V)[—F(e, mm) + v], v,- = —fi,-(e, 11,13) sat(s,-/u,~) (2.12) Inside the boundary layer {[s,| S In}, Pi-l Zk’e'+e £8 8:, where the “augmented error” cf, is a linear combination of the tracking error 6‘1 and its derivatives up to order (p,- — 1). At equilibrium, cf, = 0, which implies that 6'] = 0. Hence, equation (2.11) represents a conditional integrator, which provides integral 28 action only inside the boundary layer. In the present case, the resulting equation for s»,- can be written as s, = A,(z, e, w, o, d, a) — 7,-(z, e + u, d)p,(e, u, w) sane/II.) (2.13) where A(-) = {AM} = F(-) - 1“(-)15‘(')+ A(-) 6(-) + {k3(—k30.- + u.- sans/ml} In order to specify how 31H is chosen, we make the following standard assumption. Assumption 2.6 Let A.(-) 71(‘l max] [3 g,(e,1/, w), 1 S i g m (2.14) for some known functions p,(-), where the maximization is taken over all (2, 8,0) 6 ‘I’c,dEDd,I/EA,wEAo, andwEW. The compact set III, will be defined in the next section using Lyapunov functions, and will serve as an estimate of the region of attraction. The functions ,8,- are chosen as fl,(-) = 9,-(-) + q,, where q,- > 0. From (2.13) and (2.14), it follows that inside \Ilc, 3,3,- 3 —'yo q, [3,], whenever [3,] 2 m. We note that the right-hand side of (2.14) is independent of z and 0, even though A,- may depend on .2 and o. The former, while restrictive, is necessiated by the fact that z is unavailable for feedback, and is justified since (2.14) is only required to hold over a compact set. The latter is done purely for convenience, and is not restrictive, since, as we shall see later on, [[0]] = O(max u,), so that the contribution of o is not significant, provided the constants u,- are sufficiently small. 29 2.4.2 Output Feedback Design The output feedback design uses the following high-gain observer to estimate e‘. é‘- = é§+1+a;(ei—é1)/(e.-)i. 13152—1 .. . _ . (2.15) é}... = ai...(ei-éi)/(€I)"" where c,- > 0 is a design parameter, and the positive constants a; are chosen such that the roots of A!" + 012.914 + - - - + 024). + a},i = 0 have negative real parts. In (2.15), 6; is an estimate of 6;, the (j — I)“ derivative of 6']. Let 91-1 3, = 1e30,- + Z 1:316; + 6;, (2.16) j=l be the corresponding estimate of 3,, 3 where 0,- is now the output of 0',- = —k30, + u,- sat (§,/u,) (2.17) We replace 8 and s with their estimates é and 3 in the control (2.12), and saturate the control outside a compact set of interest. In particular, rewrite the control (2.12) 88 u, = T,(e, V,w, 0), where T() = A"1(--)[—F() + v] Inside ‘11,, 6 belongs to A,, a compact subset of B”. Let S, be the maximum value of |T,-(e, V, w, 0)], where the maximization is taken over all V E A, w 6 A0, I01] S u,/k(‘, and e 6 Ace, where A6,, is a compact set that contains A6 in its interior. The control u is then taken as u, = S,- sat(T,-(é, V, w,0)/S,-) (2.18) 3We can take 6‘] as the estimate provided by (2.15) or the measured output e’]. 30 In summary, the output feedback controller is given by (2.15)-(2.18), where T(é,V,w,0) = A’1(é,V)[—F(é,V,w)+v] v,- = —fi,(é,V,w)sat(§,/u,) (2-19) [Bi(éauaw) = Qi(éaVaw)+Qi To complete the controller design, we must specify how u,- and c,- are chosen. The parameters ,u, result from replacing an ideal SMC with its continuous approximation, and hence should be chosen “sufficiently small” to recover the performance of the ideal SMC. Similarly, in order for the output-feedback controller to recover the performance under state-feedback, the high-gain observer parameters 6, should also be chosen “sufficiently small”. Therefore, one might view u,- and e,- as tuning parameters and first reduce u,- gradually until the transient response under partial state feedback is close enough to the ideal SMC, and then reduce e,- gradually until the transient response under output feedback is close enough to that under state feedback. The asymptotic theory of the next section guarantees that this tuning procedure will work. 2.5 Closed-Loop Analysis For i = 1, - -- ,m, define (‘ 6 RP"-1 by (BUT = [(CilT 6:1,] and write the closed- loop system in the standard singularly perturbed form 0'1 = —ki101+ #1 8at((31— Ni(€i) 0 and P, = P,T > 0 be the solutions of the Lyapunov equations Q,M,- + MiTQ, = —I and P,L,- + LfP, = —1 respectively. For the components 3, 0, C, and (p, we use the quadratic Lyapunov functions (Id-‘1”? V.°(o )=‘ 2 8.. 0?, V‘(<)“=‘"C‘T or. and V.“’(cp I'dé‘eflacp‘ .1. 2 respectively, and we use the Lyapunov function V (t, 2, d) for z. The sets \Ilc and 2, are defined by ‘11, ‘15—! Q, x Q”, Q, déf (11:10“),th [17:12,“ where Qc. = {V.-(C‘)S (61 + Mil2X1.V.-(81S) #3149071) S i0” III/k ) 2.} ch = {Vz(t.2.d) S A400 + l3|]c]|)}, (2'21) 26. {WW9 S 512191} c,- > II,- is a positive constant, c = {q}, A4 = A2 0 7 is a class KI function, and x,, l3, and 19, are positive constants independent of u,- and e, to be specified shortly. Before we show that III, x )3, serves as an estimate of the region of attraction, we need to ensure that (z, e, 0) E III, implies that (2,6) E Z x 8. It can be verified that Hell 3 l3||c|| in DC, where I3 is a positive constant independent of c. Using this fact, along with Assumption 2.4(i), it follows that choosing c to ensure that l3]]C]] < min{l1, A;1(A1(l2)) — lo} guarantees that (2,6) 6 Z x 8 for all (2,6, 0) E ‘11,. Since the boundaries of the set \IIC x E, are formed of Lyapunov surfaces, to show that this set is positively invariant, it suffices to show that the derivatives of the corresponding Lyapunov functions are non-positive on the respective boundaries. 33 Using the fact that |s,—| S c,-, |0,-| S ui/kf, in \Ilc, and the inequality Vic S -||Ci||2 + QIIC‘II llQiCz'll (lsil + kSIUiI) it is easy to ShOW that Vf S 0 0n the boundary V, = (c,- + M)?” for the choice Xi = 4HQiCiH2Amax(Qi). Since (no, 3 -kf,|a,~|"2 + u,|o,~| it follows that V," S 0 on the boundary V," = %(ui/k3)2. Next, we consider the 3,- equation, which differs from (2.13) only in the term A} Inside 26,, “90'” = 0(e,-), which can be used to show that, for sufficiently small 6,, the control is not saturated inside ‘11,; x 53¢- Using this, along with s, 2 §,- + N,(e,-)cp‘, it can be shown that A; is 0(||c||oo) inside \IIc x 2‘. Let c,- be small enough that |AI()| < 70%. On the boundary V,” = %c? we have sat(s,~/u,-) = sgn(si), so that Vi S -|8e|l7i(')fii(') — lAi(‘)| - IAI(°)Il Using (2.14), the definition of m, and the fact that |A3()| < "yogi, it follows that V,’ < O on the boundary V," = %c?. Assumption 2.4(i) shows that V,, S 0 on the boundary VZ = A4(lo + l3||c||). Finally, using the inequality V¢<_”(,0i”2 2 i M'B :3, i— _+ || 4 ||M,-B,~||2 Amaz(M,-) 3,2. 4 4Though we only require equality, i.e., 19.- = 4 ||M.-Bz'||2 Amax(M,-) 5,2 to show that V,“’ S 0, we replace it with the strict inequality above, in order to arrive at the succeeding inequality in the next step of the analysis. 34 Hence, ‘116 x 23, is positively invariant. Our next step is to show that for any bounded 6(0), and any (2(0), 6(0), 0(0)) E 0,, where 0 < b, < 6,, it is possible to choose 6, such that the trajectory enters the set \IIC x )3, in finite time. Since, for all (2, 6, 0) E (2,, the right-hand side of the slow equation of (2.20) is bounded uniformly in 6, for all (2(0), 6(0), 0(0)) 6 0,, there is a finite time To, independent of e, such that for all 0 S t S T0, (2(t),6(t),0(t)) E 9,. During this interval, using the definition of 19,, we have VP 3 —a:.nsoiu2 for V.-“’( 0:", where a‘ is a positive constant such that ”6” S llullooa" for all 6 E 0”. An argument similar to the one for \IIC x 2, can be used to show that \II,, x 2, is positively invariant. This completes the first part of the analysis. To prove the second part, note that when 12') = 0, 1/ = 0 and w = O, the system has a unique equilibrium point (2 = 0,6 = 0, 0, = 0,, (,0 = 0). Let 5, = [£30, be the corresponding equilibrium value of 3,, 0 = 0 — 0, and is” = s — 5. By the converse 35 Lyapunov theorem [47], Assumption 2.4(ii) implies that in some neighborhood of 2 = 0 there is a Lyapunov function sz(2, d) that satisfies A5H2||2 S sz < Asllzllg, (31/22/32)¢(Z,0, d) S —/\7||Z||2, and “(Wu/(92H S Asllle (2.23) for some positive constants A5 to A3, independent of d. Let Q = blockdiag[Q1, - . - , 62",], and P = blockdiag[P1, - - - ,Pm]. Using 1 ~ 1 - v = V,,(z, d) + Agr’l‘Qc + §A10||0||2 + Eus“? + cpTPcp (2.24) as a Lyapunov function candidate, where Ag, A10 > 0, it can be verified that (see Appendix B), by first taking /\9 large enough, then A10 large enough, then llulloo small enough, and lastly Hellco small enough, V satisfies an inequality of the form V S —/\11V + A12\/l7(||I/(t)|| + ”w(t)” + ||?D(t)||) (2-25) for some positive constants A11 and A12, uniformly in u and 6. Since 117, V(t), w(t) —* 0 as t —> 00, the preceeding inequality can be used to show that all trajectories approach the equilibrium point (2 = O, 6 = 0, 0 = 0, (p = 0) as t tends to infinity. If all assumptions hold globally, the controller can achieve semiglobal regulation. We summarize our conclusions in the following theorem. Theorem 2.1 Suppose Assumptions 2.1 through 2.6 are satisfied, the constants c,, x,, 19,, and 13 are chosen as described before, é(0) is bounded, and the initial states (z(0),e(0),0(0)) belong to the set \Ilb, where 0 < b, < c,. Then, there exists if > O, and for each u with ||u||oo E (0,u"], there exists 6* = 6(a) > 0, such that, for u, 6 (0,u"‘] and e, 6 (0,6*], all state variables of the closed-loop system under the output feedback controller (2.15)-(2.19) are bounded, and lim,_,ooe(t) = 0. If, in addition, all the assumptions hold globally, then, given compact sets N C R" and 36 M C R”, the foregoing conclusion holds for all (2(0),6(0)) E N and 6(0) 6 M, provided \II,’ is chosen large enough to include N. 2.5.2 Performance We saw in Section 2.1, via simulation, that the output feedback continuous SMC with a conditional integrator recovers the performance of the state feedback ideal SMC. The following theorem shows that the closed-loop trajectories under the two controllers can be made arbitrarily close. Theorem 2.2 Let X = (2,6) be part of the state of the closed-loop system for (2.6) under the output feedback continuous SMC (2.15)-{2.19), and X‘ = (2’,6") be the state of the closed-loop system under the state feedback ideal SMC control (2. 7), (2.9), with X (0) = X ‘(0). Then, under the hypotheses of Theorem 2.1, for every 7' > 0, there exists 0' > 0, and for each ,u with ||u||oo E (0,u*], there exists 6‘ = 6*(u) > 0, such that, for u, E (0, u‘] and e, 6 (0,6‘], ||X(t) — X*(t)|| S r V t 2 0. Proof. We prove the theorem in two parts. First, we look at the trajectories under state feedback continuous SMC with the conditional integrator. Let X l = (2*,6") be part of the state of the closed-loop system under the control (2.10)-(2.12), with X 1(0) = X " (0). For this case, we show that, for sufficiently small u,, X l(t) — X * (t) = 0(Ilulloo) V t 2 0. Let 3* and s“ be the corresponding sliding surface functions of the two systems. Let IM = {1, - -- ,m} and t1 = min{tL 1}, where t)r = min{t : Is:'(t)| S u,} and t; = min{t: |s:(t)| = 0} iEIM IEIM If t, > 0, using sat(s;'(t)/u,) = sgn(s,1'(t)) V 0 S t < t1, it can be shown that X"(t) = X‘(t) V 0 S t S t1. Next, we consider Xl(t) and X‘(t) in the time interval t 2 t1. LGt 11 = {i 3 l8l(t1)| S Mi} U {i 3 3:01) = 0} 37 Since Xl(t1) = X’(t,), |sI(t1) — s:(t1)| 2 WI 0I(t,)| S u,. Using this, along with the definition of I, and the fact that |sI(t)| and Is:(t)| monotonically converge to the positively invariant sets {ISI | S u,} and {0} respectively, it can be shown that for all i E 11, |sI(t) — s:(t)| S 3h, for all t 2 t1. It follows that for all i E 11, sI (t) — s; (t) = O(u,) V t 2 0. Since the equations for (‘1 and C" are identical stable linear equations, driven by inputs sI — k3 0I and s: respectively, where |kI, 0I I S u, and sI — s: = O(,u,), continuity of solutions on the infinite time interval [47, Theorem 9.1] can be used to show that for sufficiently small u,, C‘I(t) — ("(t) = 001,) and hence 6‘I(t) — e“(t) = 0(u,) for all i E I, and t 2 t1. In particular, if I, = IM, then d(t) — 6“(t) = O(||u||oo) for all t 2 t1, which can then be used to show that 2i(t) — 2"(t) = O(||u||°o) for all t _>_ t1, so that the result follows. If I, 75 IM, let t2 6 (t1, oo) = min{tI,t;), where tI= ' t: it <- dt“: ' t: ft :0 2 1.6111131} l8.()l_u.}an 2 1.6111131} Is.()| } Let XI be the part of the state X i with the components ei,i E 11, deleted and XI be the corresponding part of X *. For i E IM\II, we have sat(sI/u,) = sgn(s?) = sign(sI ) V t, S t < t2, so that, during this period, the right-hand side of the equations for XI and X f are Lipschitz functions of their arguments. Viewing XI and XI“ as states of systems driven by inputs (0I,e‘I) and 6" respectively, i E 11, and using the fact that lkf, 0I| S u,, and e"f — e" = 0(a,), the results of [47, Theorem 3.4], dealing with continuity of solutions on compact time-intervals, can be used to show that, for sufficiently small u,, XI(t) -Xf (t) = 0(||u||oo) V t, S t S t2. Using this, the previous arguments involving [47, Theorem 9.1] can then be repeated to show that, for sufficiently small u,, 6‘I(t) — e“(t) = O(u,) V t _>_ t2 and all i E 12, where 12 = {2 E IM\1133I(t2)|S Ills} U {2 E IM\1123:(t2)= 0} 38 In particular, if I, U 12 = IM, then 6I(t) — 6*(t) = O(||u||oo) V t 2 t2, which can then be used to show that 2*(t) — 2*(t) = O(||,u||oo) V t 2 t2, so that the result follows. If I, U 12 79 1M, the result follows by an inductive argument that uses [47, Theorem 3.4] and [47, Theorem 9.1] alternately. In particular, this completes the first part of the proof, which shows that there exists u“ > 0 such that Hulloo 6 (0, u‘] => IIX*(t) -X”(t)|l s 7/2 v t 2 0. In the second part of the proof, we use the idea in [5] to show that the trajectories X of the system under output feedback approach the trajectories X i under state feedback as e —> 0. In particular, we show that there exists 6” = 6"(u) such that for all e, S 6*, ||X(t)—Xi(t)|| S r/2 Vt Z 0. This is done by dividing the time interval [0, 00) into three sub-intervals [0, T(e)], [T (6), T3] and [T3, 00) and showing that the inequality “X (t) —Xi (t) H S r / 2 holds over each of these sub-intervals. From asymptotic stability of the two systems, we know that there exists a finite time T3, independent of e, such that ”X (t) — XI(t)|| S r/2 V t 2 T3. Also, as mentioned in Section 5.1, there is a time interval [0,T(e)], with T(e) —I 0 as e -—> 0, during which the fast variable 4p decays to an 0(||e||oo) value. It can be shown that global boundedness of the controls implies that over this interval, “X (t) — X Wt)“ S A0T(e), for some positive constant A0 that is independent of 6. Since T(e) —> 0 as e —> 0, for small enough ”cum, ||X(t) - XI(t)” S 7/2 V t G [0,T(e)]. Lastly, noting that X(T(e)) — XI(T(€)) —+ 0 as e —-> 0 and (p is O(||e||oo), and using the continuous dependence of the solutions of differential equations on compact time intervals [47, Theorem 3.4], one can show that it is possible to choose 6 to satisfy the inequality ”X (t) — X I(t)|| S r/ 2 over the time interval [T(e),T3]. This shows that ”X (t) — X i(t)|| S 7/2 V t Z 0. The conclusion of Theorem 2 then follows from the triangle inequality. 39 2.6 Universal Integral Regulator For SISO systems, the flexibility that is available in the choice of the functions F and H can be exploited to simplify the controller to k k k“ “ u=—k sat(§/u)=_k sat( 00+ 161+ 262+ +6p) u (2.26) As mentioned in Section 2.1, this particular design, while having a simple structure, is also natural since the control is required to be bounded. It is clear from (2.26) that the only precise knowledge about the plant that is used is its relative degree and the sign of its high-frequency gain LgL’flh. This “universal” design was first presented in [46], for the conventional integrator, where was shown that the structure of the universal integral regulator coincides with the classical PI and PID controllers, followed by saturation, for relative degree one and two plants, respectively. In the present case, when p = 1, the integrator equation can be rewritten as ('1 = 61 + (it/16M - U) where ,=_,,,,(&Im) ,m,? (121421.) u u The term a is the “unsaturated version” of the control u, so that the controller (2.26) has the structure shown in Figure 2.5. It is a PI controller with “anti-windup” [25], followed by saturation. In the relative degree p case, the integrator equation can be rewritten as a = ea + (u/k)(il - V) where 6,, = 23’: kJ-ej +ep is the augmented error that was defined in Section 2.3. It is 40 Plantly L Figure 2.5: Universal regulator for relative degree one systems : PI controller with anti-windup, followed by saturation; K 1 = kko/p, K p = k/u, and L = p/ k. clear from the expression for 0 that the anti-windup structure of Figure 2.5 is retained in this case as well. The control (2.26) now represents a “PIN”1 controller”, with a conditional (anti-windup) integrator, followed by saturation. This interpretation of the conditional integrator as a specially tuned version of an anti-windup scheme for the universal integral regulator design was presented in [70]. 2.7 Conclusions We have presented a new approach to introducing integral action in the control of nonlinear systems, which captures the regional and semi-global asymptotic regulation results of [46] and [57], while improving the transient response. In the new approach, the integrator is designed in such a way that it provides integral action only “con- ditionally”, effectively eliminating the performance degradation. The improvement in performance is demonstrated analytically by showing that the output-feedback continuous sliding mode controller, with conditional integrator, recovers the perfor- mance of an ideal state-feedback sliding mode controller, without integral action, as the controller parameters u, and 6, tend to zero. In view of this result, the con- trol design can start with the ideal state-feedback sliding mode control, where the 41 parameters of the sliding surface s: = 0 are chosen to meet the transient response specifications. Then, integral action is introduced by modifying s; to s, = k?0, + 33’, with 0, = —k,90, + a, sat(s,/u,). The discontinuous term sgn(sz‘) in the ideal SMC is replaced by sat(s,/u,). The parameters a, are reduced gradually until the transient response is close enough to the ideal case. Finally, a high—gain observer is brought in to estimate the derivatives of the tracking error. The observer parameters 6, are gradually reduced until the transient performance is close enough to the ideal case. Note that in the ideal SMC design, the inequality that corresponds to (2.14) will have A, terms that do not account for the 0, variables. However, since 0, is 0(u,), the fl,’s of the ideal SMC design will still work in the presence of the conditional integrator, provided the u,’s are sufficiently small. While modifying the integral control designs of [46] and [57] from conventional to conditional integrators, we have also extended the problem statement to MIMO systems and allowed time-varying matched disturbances. Moreover, we proved that the trajectories under output feedback approach those under state feedback as e -+ 0. This property also holds for [46] and [57], but was not proved there. 42 Appendix A Derivation of (2.22) The arguments of Section 2.4.1 show that V,’ < 0 whenever |s,| 2 u,. Suppose ”1(1— Ci) S '31" S Hi, 50 that 's ’Yi(')3i(')|3i|2 .. Vi S - u + lAi(')H3i| + lAi (“)HSil i Using Assumption 2.6 and the definition of fi,(-), it can be shown that _7i(')5i(‘)lsil2 + lAi(')H3i| < _VOQiISiI Hi _ 2 for u,(1 - <,) S |s,| S u, , provided c, is small enough that max 9, S q,/(2§,) — q,. Choosing 6, small enough that |A;()| < yoq,/2, it follows that V,“ < 0 whenever |s,| Z u,(1 — c,), which shows that s,(t) reaches the set {|s,| S u,(1 — <,)} in finite time and stays there for all future time. Thereafter, |81|+ kbiail S 2m which along with the inequality Vf S -||C’I||2 + 2IICIII ”QiCi” (lsil + kélail) and the definition of x,, can be used to show that V,C S —-||("||2 / 2, whenever V,-C _>_ 16,23)“. This shows that (f(t) reaches the set {ViC S 16ufx,} in finite time and stays therein for all future time. Next, we note that e 6 (2,, implies that ”e“ S IIuIIOOa', where oz" is a positive constant independent of u. Since lim,_.oou(t) = 0, it follows that there is a finite time after which He + u” S r"||,u||oo, where r“ is any positive constant that satisfies r" > a“. From Assumption 2.4(i) and the definition of A4, it follows that V, S -)\3(||2||) for V, > A4(||,u||oor*), which shows that 2(t) reaches the 43 set {V, S A4(||u||oor*)} in finite time and stays therein. Lastly, the fact that w(t) reaches the set {l/I‘p(<,o‘) S ||e||§oi9,} in finite time and stays therein was already shown in Section 2.4.1. This completes the proof of the statement that every trajectory starting inside the set \IIC x 2, enters the set \II,, x 2, in finite time and stays therein for all future time. 44 Appendix B Derivation of (2.25) The derivative of V of (2.24) is given by V = aizflae + V,Cl) — nut”? + MEN 62,0 —k60,) i=1 — A10 214,5? + /\10 Z 5i (52' - N45086:.) (2,27) i=1 i=1 + 7:1“: ¢I|————7‘(')5:(':si + A,(.) + A319] i: m ,p‘ITBB, [b,-() r?" +Za,,()( )(uj+5( ))l i=1 We arrange (2.27) in a quadratic form of II = [”2” ”(ll H0“ “59'“ ||<,o||]T. From (2.23), we have 81/2, _ 8V2: 22 82 ¢(2,6+u,d)— 02 (2, 0,)+d S —/\7||»ZI|2 + ’\8L¢”Z“(“e” + ||V||)) z[¢( ,6 + V, d) — ¢(2, 0, d)] where L, is the Lipschitz constant of d(z, ., d). Using 6 2 {8i}, where 6i 2 [(CilTBLJT 8:", = 5, — k30, — K‘C‘, and K‘ = [kI, - -- ’khi-ll’ it can be verified that ”6“ S Alslléll + A14||5|| + Alsllé‘ll (228) for some positive constants A,;, to M5, so that the first term of (2.27) satisfies 8V,, . - 372w + u, d) s —I7IIzII2 + A8L¢IIZII(/\13IICII+ A14IIUII+ Imus“ + IIVII) (2.29) 45 It is easily verified that the second and third terms of (2.27) satisfy —A9ncn2 + 2A9 2 Nae-(é.- — zeta-I 5 Blur”? + 2A9II Am by making ||e||oo sufficiently small, so that by redefining A13, the right hand side of inequality (2.31) can be replaced simply by —A,0k0||0||2+)\,0||0||(||§||+A,8||<,o||), and we use this idea in the definition of some of the constants later on. For the fourth term of (2.27), the idea is to rearrange expressions by adding and subtracting appropriate terms. We show this in detail for one expression, and a similar procedure applies to the others. For example 7,(2, e + u, d)fi,(6, 11,13) = 7,-(2, e + u, d)fi,(6, u, w) — 7,-(0, e + u, d)fl,(e, u, w) + 7,-(0, e + V,d)fi,(6,1/,w) — 7,-(0, mam-(0, 11,01) + 7,-(0, VI d)/3i(0, 14 w) — “rt-(0, 0, (1)5400, 10*) + 7,-(0, 0, d)fl,(0, o, w) — 7,-(0, o, d)fi,(0, 0, 0) + 71(0) 0, d)fi,'(0, OI 0) Consequently, using (2.28), it can be verified that ”Yd/EC) ‘13 7,(2, 6+V, d)fl,(6, V, w)- ’Yz'(0, 0, d)fi,(0, O, 0) satisfies |7i(')5i(')| S A19||Z||+ A20||C|| + /\21||5|| + Azzllgll + A23HV|| + A24Hw|l 46 for some positive constants A,g through A24. Likewise, A,(-) dre-f A,(2, 6, w, 0, (1,213) — A,(0, 0,0, 0, d,0) satisfies an inequality similar to the one above, except that ”V” is replaced by ”an. Rewriting _7i(')5i(')3i + M) = _ mom-(Is.- _ Mia-(Is + M) #i z #i Iii I _ 7i“), 0’ dlfii (0’ 0'! 0)§i + ”i A,(0, 0, 0, 0, d, 0) and using, from (2.13), the fact that 7,(0,O,d)fi,(0,0,0)(§,/u,) = A,(0,0,0,0,d,0), we have ~ g —,Yi(.)'81(.)8i+A,(-)+Af(') S _W _ 7i('):6i(')§i§i +Ai(')§i+Ai(')~ i_— 3i m u, u,- Finally, using the facts that |§,-| S u,, 'y,(-) 2 70, fl() 2 q,, déf min q,, and that inside I the set ‘11,, x E, A’I'() _—_- —kI,N,-(cs,)<,oi + Z a,,-(2, 6 + V, d) [T,~(6, V, w, 0) — TJ-(e, V, w, 0)] i=1 it can be verified that the fourth term of (2.27) satisfies i§,I—1)%(—ls— + A.-(-) + Am] 3 -- (ff-fi- -- A25) IIS'II2 i=1 (2.32) + ”g” (Azsllzll + A27HC“ + A28W” + 329“?” + AaollV” + A31|lwll+ AMIMI”) for some positive constants A25 through A32. Noting that M“) — Tip” + 20:50le + 5j(')) = 31' — 1635:" IVA—K52" - (€350 — RIC i=1 where R‘ = k;i_1Ki — [O kI - - - kin—2], and using the results of the previous discussion, 47 it can be verified that the final term of (2.27) satisfies m “SDI“2 m i T (Pi) m 1 2 Z-_€i +2Z(99) PIBI [bi—TV +Za,j( (uj+6j)] —H;—A33 “(0“ i=1 i=1 j=l + ||\2c —)\2d 73 : koAm ~43, _A3d Tail—(II: — A25 "Md . rem—V3- where the positive constants A33, and AM to A4,, are independent of 6; A25 and A“, to A3, are independent of u; A,,, and A2,, are independent of Am; and Au, is independent of A9. Therefore, by choosing A9 large enough, then Am large enough, then Ilulloo small enough, and lastly ||e||oo small enough, we can make ’P positive definite. Given this fact, the equivalence between (2.34) and (2.25) follows easily from (2.23) and (2.24). 48 Chapter 3 Global Regulation under State Feedback 3. 1 Introduction In the previous chapter, we showed that the output feedback continuous sliding mode controller with a conditional integrator could be tuned to achieve semi-global regulation when all the conditions hold globally. However, it does not achieve global regulation. The semi-global result is a limitation of the high-gain observer based design, which requires that the control be globally bounded. In this chapter, under certain additional assumptions, we show that the semi-global result of the output feedback design can be extended to a global result under state feedback. 3.2 Problem Statement Consider a MIMO nonlinear system, modeled by i = fo($)+Af(x,9)+lelgi(%9)lui+5i($I9vw)li y, = hi($),1SiSm (3.1) 49 where x E R" is the state, it E Rm is the control input, y E Rm is the output, 0 is a vector of unknown constant parameters that belongs to a compact set 6 C R", w(t) is a piecewise continuous exogenous signal that belongs to a compact set W C R", fo(~) and h,(-) are smooth functions on R", A f () and g,(-) are smooth functions on R" x G, and the disturbances 6,(-) are continuous functions on R" x G x W. The function f0(x) is assumed to be precisely known, so that the uncertainty is lumped into the term A f (x, 6). The output functions h,(x) are assumed to be precisely known. The uniform vector relative degree assumption of Assumption 2.1 is modified as follows. Assumption 3.1 The system :1": = f0(x) + g(x, 6)u, y = h(x) has a strong vector relative degree {p,,p2, . . . ,pm} in D,, i.e., ngL’IOh,(x) = O for 0SkSp,—2,1SiSm,1SjSm, cmdA(a:,e)d=ef {ng L?;_1h,} is nonsingular for all x E D, and 6 E 9. Furthermore, the distribution span{g,, - -- , gm} is involutive, uniformly in 6, and there is a change of variables 77 711(1), 6) n— =T(a:,0)= ,neR ”.6612” (3.2) E T205) whereé = {6"}, withé} -—- 143ml 52' s p..1 s 2' s m, andp=p1+p2+---+pm, such that ngn, == 0 V 1 S j S m, 1 S i S n—p, and T(x, 6) is a global diffeomorphism ofR" into R". Remark 3.1 The transformation T, no longer depends on the unknoum parameter 9, i.e., it is known. Consequently, since the state x is available for feedback, it follows that E is available for feedback. 50 Assumption 3.2 The uncertain term A f (x, 6) satisfies Af(x,6) e Ker[dh,,deoh,-, . .- egg—212,], 1 Si 3 m (3.3) Assumption 3.2 places a restriction on A f (x, 6). It can be verified that the state model (3.1), along with Assumptions 3.1 and 3.2 allow one to work with a class of systems that includes those in which the uncertainty satisfies the generalized matching condition of [65, Chapter 9], as defined in Remark 3.2 below. Remark 3.2 Consider the MIMO system 3': = f0($) + Af(x, 6) + 2:,[gi($) + Agi($, 9)]111', y,- = hi(x),1SiSm where the nominal system it = f0(x) + g(x)u, y = h(x) has a strong vector relative degree {p,, p2, . . . , pm}. The uncertain terms A f (x, 6) and Ag, (x, 6) are said to satisfy the generalized matching condition [65, Chapter .9], if, for 1 S i S m, LAfLJf'oh, = 0for0SjSp,—2, LAgijoh, = 0for0SjSp,—1,1SkSm. The generalized matching condition of Remark 3.2 is weaker than the matching con- dition, and therefore the class of sytems in Remark 3.2 includes those where the uncertain terms satisfy the matching condition. It is also clear that the class of sys- tems described by (3.1), and Assumptions 3.1 and 3.2 include the ones described in Remark 3.2. A class of systems and accompanying assumptions, similar to the ones in our work, for the problem of robust output tracking of MIMO systems using sliding mode control, can be found in [23]. 51 Assumptions 3.1 and 3.2 allow (3.1) to be rewritten in the normal form 7'7 $07, 6, 9) .. . (3.4) 5' = A? + Bi [h(x) + 13bit”: 6) + 2;; 00(33 9)(“j + 5j(l‘,9,w))] where, for 1 S i S m, the pair (A,, B,) is a controllable canonical form that represents . . I. .-1 a chain of p, integrators, b,() = L’f’oh,, Ab,(-) = LAfL’f’; h,, and {a,,-(-)} = A(-.) The output y is required to asymptotically track a reference signal r(t). As before, we assume that the exogenous signal w(t) and the reference signal r(t) satisfy the following properties: 0 lim,_.oow(t) = w,, o r,(t) and its derivatives up to the p,th derivative are bounded, and rIp‘)(t) is piecewise continuous, for all t Z 0 o lim,_.oor,(t) = r,,,, and lim,_.oorI")(t) = 0 for 1 Sj S p,. Define rs, = {r,,,}, w(t) = w — wss, V(t) = In - rissiri(l)a"' ,Ti(p‘—l)]T, w(t) = {r,(P‘)}, V(t) 2 {Vi}, and let X C R”, A C R”, and A0 C Rm be compact sets such that r,, E X, V(t) E A, and w(t) E A0 for all t _>_ 0. Set d = (r,,,6,w,,) and D, = X x 9 x W. Assumption 2.2 is modified as follows. Assumption 3.3 For each d E Dd, there exist a unique equilibrium point :7: = f(d) and a unique control a = u(d) such that o = My?) +Af(x,6)+g(x,6)[u+6(x,6,wss)], and r,, = h(x) 52 With the change of variables (3.2), the equilibrium point f(d) maps into (17(d), (_(d)), where {‘(d) = [r,,,,0, - -- ,0]T. Let 2:77—17and e"=£‘I-§_'I—Vi and rewrite (3.4) as 2 = ¢(2,6+V,d) 65‘ = Aiei + Bi [h(x) + Abih‘: 6) - TIM + 27:1 00(53: 9X“, + (SJ-(127,640)» (3.5) Remark 3.3 Since 6i = 6‘ - E‘ — Vi, where 6‘, 5‘, and Vi are known, it follows that e is available for feedback. Assumption 2.3 is clearly irrelevant for the global problem, and Assumption 2.4 is modified to hold with class [Coo functions A,(-), (i 2: 1, 2, 3) and y(-), i.e., Assumption 3.4 There exist a C1 proper function V, : Z —-> R+, possibly dependent on d, and class [Coo functions A, : [0,oo) —-> R+(i = 1,2,3) and 7 : [0,oo) -> R,, independent of d, such that A1(IIZ||)S Vz(t,z,d) S /\2(IIZH) €9Vz + 3V; at 62 ¢(2,e+ Vad) S ->\3(||Z|l), V “Z” 2 7(l|€+ 1/II) for all 6 E E, 2 E Z, V E A, and d E D. Furthermore, the equilibrium point 2 = 0 of 2 = ¢(2, 0, d) is exponentially stable, uniformly in d. Assumption 3.4 implies that the system 2' = 45(2, 6+V, d) is input-to—state stable (ISS) with (6 + V) as the driving input [47]. 53 3.3 Controller Design The control design proceeds exactly as before, i.e., let Pi—l s, = 16,0, + Z 1:36:31 + e}, (3 6) j=l where the positive constants kI, - ~ - ,kfkl are chosen such that the polynomial Apt—1 + k2.-1A”“2 + - - . + kI is Hurwitz, k6 > 0, and 0, is the output of 03' = —kI,0, + Iii 8at(5i/#i), 02(0) 6 [_Mi/kfiifli/kfl] (3-7) This results in 53' = [Vii—193m + u,- Sat(3i/Hi)l + F413, 6’" Tip”, 9) + iaijklluj + 52“] j=l where Pi - 1 F,(-) = b,(x) + Ab,(x, 6) - 7‘?” + Z I6;-+1 i=1 Assumption 3.5 A(x, 6) = I‘(x,6)/l(x) where A(x) is a known nonsingular matrix andI‘ = diag['y,, - -- ,ym], with 7,-(-) Z 70 > 0, 1S i S m, for allx E R", and6 E 9, for some positive constant '70. The control u is taken as u = A’1(x)[—F(x,e,w) + v], v, = -B,(x, e, w) sat(s,/,u,) (3.8) 54 where, as before, F() = {F,()} is a nominal value of F(-) = {F,()}, which could be, but not restricted to, Pi-l so = b. 0 As before, the parameters a, should be chosen sufficiently small, in order to recover the performance of the ideal SMC. This completes the design of the controller. 55 3.4 Closed-Loop Analysis For i = 1, - -- ,m, define C‘ E 12““ by (6i)T = [(C‘)T 62‘] and write the closed- loop system in the form (I, = —k(i)0',' + H,“ sat(s,/u,) “I = M, z.‘f'C', Sg—kIO', C C ( 0 ) (3.9) s, = A,(x, 0, 6, w, a) - '7,(x, 6) 6,(x, w) sat(s,/u,) 2 = ¢(Z,€+V,d) where M, and C, are as defined in Section 2.5, and note that M, is Hurwitz by design. Let Q, = Q? > 0 be the solution of the Lyapunov equation Q,M, + MTQ, = -I and def i iT i V.‘(< I = c ac. Our first step is to show that for any initial state x(0), all trajectories of the closed-loop system are bounded. To that end, let c,- be a positive constant such that c, > m, and define the set I11, ‘13} o, x 9,, n, dé‘ (Hg, 9,), 9c, = {V,-CK") S (Ci +Mi)2XiI |3i| S Ci: lail S Iii/k3}, 962 = {W(t,2,d) S A400 + l3HC”)} c = {c,-}, A, = A2 0 'y is a class [Coo function, x, = 4||Q,C,||2/\m,,,(Q,), and 13 is a positive constant such that He” S l3||c|| in 51,. It can be verified that 0,0, < O on the boundary |0,| = u,/kI,, VIC S O on the boundary V,-C = (c, + u,)2x,, s,s, S 0 on the boundary |s,| = c,-, and that V, S O on the boundary V, = A4(l0 + l3||c||). Since the set \Ilc can be chosen large enough to include any state 23(0) and the control is independent of c, the preceeding argument shows that all states of the closed-loop system (3.9) are bounded. In fact, the preceeding argument can be extended to show that for any bounded x(0), the trajectories of the system reach the positively invariant 56 set ‘11,, in finite time, where ‘1’” = $2,. >< {ICU/32,61) S A4(ll#||oor‘)} def m 1' ,' i all : Hi=1{(e I01") 2 '31" S [1,, lail S 1%, Vic(€ ) S IGILEXi} u = {u,}, r” > 01", and 02* is a positive constant such that “6“ S Hullooa" for all 6 E Q”. Inside this set, the system has a unique equilibrium point (2 = 0, e = 0, 0, = 0,) when a = 0, V = 0 and w = 0. The equilibrium analysis of the preceeding chapter can be repeated almost verbatim to show that for sufficiently small u,, all trajectories starting inside ‘11,, approach the equilibrium point (2 = 0,6 = 0,0 = 0, (,0 = 0) as t tends to infinity. In particular, we have the following result. Theorem 3.1 Suppose Assumptions 3.1 through 3.6 are satisfied, then, for any bounded initial state x(0), the state x(t) of the closed-loop system under the state feedback con- troller (3.6)-(3.8) is bounded for all t Z 0. Moreover, there exists u‘ > 0, such that, for u, E (0,u‘], lim,_.ooe(t) = 0. In order to state the analogous result of Theorem 2.2 of Section 2.4.2, dealing with the closeness of trajectories of the CSMC to ideal SMC, consider the ideal SMC control i-1 i i i Si = 5:1 kjej + ep‘. - . (3.10) u = A‘1(x)[—F(x,e,w) + v], v, = —6,(x,6,w) sgn(si/ur) The arguments of Theorem 2.2 can be repeated to prove the following result. Theorem 3.2 Let x“ be the state of the closed-loop system under the ideal SMC control (3.10) and x be part of the state of the closed-loop system under the continuous SMC (3.6)-(3.8). Then, under the hypotheses of Theorem 3.1, given any compact subset S of R", with x"(0) = 2(0) E 8, there exists u‘ > 0, such that, ,u, E (0, [2’] => 3‘0) - x(t) = 0(IluooII) V t 2 0. 57 3.5 Conclusions In this chapter, we considered the design of robust sliding mode control with conditional integrators for uncertain MIMO nonlinear systems, with the specific ob- jective of achieving global regulation when all the assumptions hold globally. We showed that, under certain additional assumptions, the semi-global output feedback result of the previous chapter can be extended to a global result, by means of a com- bined of full state/ error feedback design. The result is global with respect to the initial state of the plant and bounded reference signals, with the control being inde- pendent of the bound on the reference signal; however, it is dependent on the bound on the exogenous (disturbance) signal. Analytical results for the improvement in per- formance over conventional integral control were provided. Since we deal with state feedback, the performance recovery result, dealing with closeness of the closed-loop system trajectories to those under ideal sliding mode control, is sharper than the one obtainable under output feedback. 58 Chapter 4 Tracking Using Conditional Servocompensators 4. 1 Introduction We consider the design of a robust controller for the output regulation problem, where the exogenous signals are generated by a neutrally stable exosystem. Previous work [45] has shown how to do this by incorporating a servocompensator in a sliding mode design, but the transient performance is degraded when compared to ideal SMC. Extending the conditional integrator idea of Chapter 2, we design the servocompen- sator as a conditional one that is active only inside the boundary layer, achieving asymptotic output regulation, but with improved transient performance. Both re- gional as well as semi-global asymptotic results are provided, and we analytically show that the controller can be tuned to recover the performance of an ideal SMC. A result from [45] regarding ultimate boundedness of the tracking error under internal model perturbation is recalled, and a simulation example is included to demonstrate the improvement in transient performance over the conventional servocompensator design of [45], and the effect of the internal model perturbation. 59 4.2 Problem Statement Consider a single-input single-output nonlinear system, modeled by x = f(x, 6) + g(x, 6)u + 6(x, d, 6), h(x, 6) + 7(d, 6) (4.1) a: II where x E R" is the state, u E R is the control input, y E R is the measured output, d E R1" is a time-varying disturbance input. The functions f, g, 6, h and 7 depend continuously on 6, a vector of unknown constant parameters, which belongs to a compact set 8 C R’. We assume that, for all 6 E 9, the functions are sufficiently smooth on U9, an open connected subset of R” that could depend on 6, for all d in a compact set of interest. The functions 6 and 7 vanish at d = 0, i.e., 6(x, 0, 6) = 0 and 7(0, 6) = 0 for all 6 E G and x E U9. Our first assumption is that the disturbance-free system has a well-defined normal form, possibly with zero dynamics. Assumption 4.1 The system {4 .1 ), with d = 0, has a uniform relative degree p S n for all x e U, and e e e; i.e., Lgh(x,6) = L,L,h(e,e) = = L,L§’:2h(e~,e) = o and ILgL?-1h(x, 6)| 2 go > 0 where go is independent of 6. Moreover, there exists a difi'eomorphism 7' .—. T(x,6) (4.2) E of U9 onto its image that transforms (4.1), with d = 0, into the normal form 1 7'7 $07.50) 6i = §i+1,1SiSP—1 Sp z (307,5,9) + a(n,£16)u y=€i 1For p = n, r) and the r'requation are dropped. 60 Assumption 4.2 In the presence of disturbance, the change of variables (4.2) trans- forms the system into the form 2 7) Z ¢a(na€1w”9€mida6) €==su+wkesaeengigm_1 éi : €i+l + ‘Ili(n)€l) ’°')€31d)0))m S i S P _ 1 I (4'4) 6.: Mmev+amrsm+wxmeav. y = {i +7(d26) J where 1 S m S p — 1. The functions ‘11, vanish at d = 0. Examples of physical systems which are transformable into the normal form in As- sumption 4.1, uniformly in a compact set of system parameters, can be found, for example in [37, Section 4.10]. Geometric conditions under which a system can be transformed into the form in Assumption 4.2 can be found, for example, in [60]. Assumption 4.3 Let p0 be the disturbance relative degree and i5 = p — p0. Th6 disturbance and reference signals d(t) and r(t) have the following properties for all tZO: o d(t) and its derivatives up to the 6th derivative are bounded, and d(‘3l(t) is piecewise continuous o r(t) and its derivatives up to the pth derivative are bounded, and r(”) (t) is piece— wise continuous o lim,_.oo[’D(t) —- 15(t)] = 0 and lim,_,oo[y(t) -— 37(t)] = 0, where DTU) = [d(t), ' ' ' I d(fi)(t)], 327‘“) = [T(t), ' ' ° ,7‘(p)(t)] 2For m = 1, the first frequation is dropped. 61 and D(t) and 37(t) are generated by the known exosystem ’ll} 1' Saw, 1’) (4.5) _ = Pow )7 where So has distinct eigenvalues on the imaginary axis and w(t) belongs to a compact set W. Let D and Y be compact subsets of ROB“)? and RV’“) respectively, such that ’D E D and y E Y, and d(w) and r(w) denote the steady-state values of d and r as determined by the exosystem (4.5). Define r,(w, 6) to 7rm (w, 6) by 71'1 = F—7(d,6), 071’, 7ri+1 = Esow—‘I’i(7r1,"w7ri,g,9), 1SiSm—1. Assumption 4.4 There exists a unique mapping A(w, 6) that solves the partial dif- ferential equation EA 56301“ = (1)0(A3 7f], ' ' ' 3 Wm, 7rm-i-l) J, 6) for all w E W, where arr", — 7rm+1 = 31:80“) _ \pm()‘a 7T1, ' ' ' 7 Wm, d1 0) Let 7ri+1 = 90%Sow_q’i(Ai7rlan° 77rit(i)6)1 m+1 SiSp_1 62 The steady-state value of the control u on the zero—error manifold {17 = A(w, 6),£ = r(w, 6)} is given by 1 8n, x(w,6) — a(/\,7r,6) [Ow Sow — b()\,7r, 6) — \Ilp()\, 7r,d, 6) Assumption 4.5 There exists a set of real numbers c0, - ~ ,cq._,, independent of 6, such that x(w, 6) satisfies the identity LgX = CoX + CiLsX + ' " + Cq—1L3_1X (4-6) for all (w, 6) E W x 9, where L,X = (gig-Wow and the characteristic polynomial pq_cq_lpq_l_...—CO has distinct roots on the imaginary axis. Motivation for Assumption 4.5 comes from the nonlinear counterpart of the inter- nal model principle, which recognizes that in the nonlinear case, the controller must be able to reproduce not only the trajectories generated by the exosystem, but also a number of higher-order nonlinear deformations thereof, an idea that was elaborated independently in [34], [43], [63]. Assumption 4.5, along with the notion of immersion [37, Chapter 8], allow the construction of a finite dimensional linear internal model, as we will soon show. However, before we do so, a couple of remarks are in order. Note that, among other things, the matrix So in Assumption 4.3, and hence the frequencies of the exosystem need to be precisely known. For the case where the frequencies of the exosystem are unknown, an alternate design, that makes use of an adaptive internal model whose “natural frequencies” are automatically tuned to match those of the unknown exosystem, can be found in [69]. A recent result, which relaxes Assumption 4.5, thereby removing the restriction that the solution of 63 the regulator equations be a polynomial in the exogenous signals, and allows for a nonlinear internal model, can be found in [16]. Defining 0 1 0 x 0 0 1 0 L.x S = , T = 0 0 1 LII-2X _CO ... ... Cq-1_ _Lg-lxd and P = [1 0 - « - 0] ,xq, it can be shown that x(w, 6) is generated by the internal model 8r(w, 6) 6w x(w, 6) = I‘r(w, 6) Sow = Sr(w, 6), To tackle the tracking problem, we apply the change of variables 2 = 17 — /\(w, 6) and 6, : y“_1)— r(I'I), 1 S i S p VT(t) = [DT(t) - fiT(t), V(t) - V(t)] and note that V(t) belongs to a compact set A and lim,_.ooV(t) = 0. With this change of variables, system (4.4) can be rewritten as 2' = ¢0(Z, 6, V,w, 6) 6, = 6, ,1 S 2 S — 1 +1 P (4.7) 6,, = b0(2, 6, V, w, 6) + 00(2, 6, V, w, 6)u, ym = 61 64 where ym is the measured tracking error. The functions ¢0(-), ao(o) and b0(-) satisfy 60(0,0,0,w,6) = 0 00(0,0,0,w,6) = a(/\(w,6),7r(w,6),6) b0(0,0,0,w,6) = —X(w,6)a()\(w,6),7r(w,6),6) In the new variables, the zero-error manifold is given by {2 = 0, 6 = 0}. The next two assumptions of similar to corresponding ones in Chapter 2. Assumption 4.6 There exist positive constants r, and r2, independent of (V, w, 6), such that for all (V,w, 6) E A x W x 9, Hell < r, and “2” < 7‘, => x E U9 Define the balls 8 = {6 E R”: ”e” < r,} and Z = {2 E R”"’ : [[2]] < 7‘2} Since A is compact, there exists r;, > 0 such that “V” < r3 for all V E A. Therefore, “(6T9 VT)” < 7‘,+ T3 for all e E 8 and V E A. Assumption 4.7 (i) There exist a C1 proper function V, : Z x W —I R,, possibly dependent on 6, and class IC functions a, : [0,r2) ——> R+(i = 1,2, 3) and 6 : [0, r, + r3) -—> R+, independent of (w, 6), such that 01(llzll) S Vz(z,’w.9) S a2(||Z||)a 3V, 3V, 0 ¢0(Z,e,l/,w,0) + a 3011) S -(13(]]Z]|) 82 w for all ”2” 2 6(||(6T,VT)||), ”(6T,VT)“ < r, +r3 and (2,w,6) E Z x W x 8. Further- more, 6(r3) < a§1(a,(r2)). 65 (ii) There exists a Lyapunov function V,,(2,w,6), defined in some neighborhood of 2 = 0, and positive constants A, to A4, independent of (w, 6), such that A1||Z||2 S sz(z,w.9) S A2IIZ||2 avzz av'zz 82' ¢0(Z,0,0,w,0) + aw Sow S —A3”Z“2 (9V 22 < A “—82 _ 4H3“ 4.3 Controller Design Our design of the conditional servocompensator follows very closely that of the conditional integrator in [70]. Basically, it involves modifying the servocompensator 0=S0+J6,, JT= [0,---0,1] in [45] to make it “active” only inside the boundary layer. Assume for the present that the state 6 is available for feedback. To simplify the notation in what is to come, we define CT = [6, 62 e,,_,] and K2 = [k, k, k,.,] In the absence of the servocompensator, one could take the sliding surface as S = K2< + 6p with K2 chosen such that the polynomial A”‘1 + k,,_,A“’”2 + - - - + k2A + k, is Hurwitz. This guarantees that when motion is confined to the manifold s = 0, the error 6, converges to zero asymptotically. Servocompensation is then introduced by modifying the sliding surface to S = K10 + K2C + 8p (48) 66 where 0 is the output of the conditonal servocompensator 0 = (S — JK,)0 + quat(s/u) (4.9) with u > 0 being the width of the boundary layer and K, chosen such that S — JK, is Hurwitz, which is always possible since the pair (S, J) is controllable. Equation (4.9) represents a perturbation of the exponentially stable system 0 = (S — J K ,)0, with the norm of the perturbation bounded by the small parameter p. Inside the boundary layer, i.e., when |s| S [1, equation (4.9) reduces to 0 = S0 + .160 (4.10) where the “augmented error” 6,, = K2C + 6,, is a linear combination of the track- ing error and its derivatives up to order p — 1. Equation (4.10) coincides with the servocompensator of [45] only in the case when p = 1. Since the state 6 is unavailable for feedback, we use the following linear high-gain observer to robustly estimate the derivatives of 6,: 61' = éH—l +9i(€1 — é1)/€i, 1 S i S P-1 (4.11) ép = gp(8, - é1)/5p where e > 0 is a design parameter to be specified, and the positive constants g,, - - - , g, are chosen such that the polynomial A” + g,A”‘1 + + gp_,A + g, is Hurwitz. We replace 5 by its estimate s, given by p-l s = K10 +k,e, +222, +6, (4.12) i=2 67 where 0 is the the output of 0 = (S — JK,)0 + ,quat(.§/u) (4.13) The control is taken as u = —k sign(LgL’f’_1h) sat(.§/u) (4.14) To complete the design, we need to specify the design parameters k, u and e. The constant k is an upper bound on the control u. Since in typical applications the control has to satisfy magnitude constraints, we can simply choose k to be the maximum permissible control magnitude. The parameters a and 6 should be chosen sufficiently small. In particular, we show in the next section that there exists [1‘ > 0, and for each u E (0, u“), there is an 6* = 6*(u) > 0 such that asymptotic tracking is achieved for each 0 < u < u“ and 0 < e < 6*. The only precise knowledge about the plant that is required to calculate and implement the control (4.14) is its relative degree p, the sign of its high-frequency gain LgL’I-lh, and the characteristic polynomial of Assumption 4.5. 4.4 Closed-LOOp Analysis The current analysis shares many points in common with the ones in [45], [70], and the previous chapter, so we only outline the idea, taking care to highlight any differences. Analogous to the case in Chapter 2, the main difference between the analysis in this section and that in [45] is treating 0 and C separately in the current work, while in [45], they are lumped together in one vector. 68 We write the closed-loop system in the singularly perturbed form , w = Sow 2 = ¢0(Z,8,V,w,6) 0 2 A00 + quat(s — N(6)(p)/)u) I (4.15) C = A u, define the compact set Q, by :2. ‘33 { 0 and 7,(c) = max|A(-)|/|ao(-)|, with the maximization taken over all (2,6,0) E (2,, V E A, w E W, and 6 E G, we have 35‘ < 0 on the boundary |s| = c. Assumption 4.7 shows that V, S 0 on the boundary V, = a4(cp4 + r;,). Finally, using the inequality - 1 V. 3 7161)? + 2n,“ IlPeB2||72(C) 3Inequality (4.17) can be viewed in two ways. Given 6 > 0, it is a constraint on the minimum value k. Alternatively, given k, it is a constraint on the estimate of the region of attraction. 71 where 7,(c) = maxlb0(-) — kIaO(-)|sat((s - N(e) 4MP, B2 ||27§(c)Amax(P,p). It follows that the set (2, x 2, is positively invariant. Our next step is to show that for any bounded 6(0), and any (2(0), 6(0), 0(0)) E 0,, where 0 < b < c, it is possible to choose 6 such that the trajectory enters the set it, x E, in finite time. Using the fact that for (2,6,0) E 0,, the right-hand side of the slow equation of (4.15) is bounded uniformly in 6, it follows that for all (2(0),6(0),0(0)) E 0,, there is a finite time To, independent of e such that for all 0 S t S To, (2(t), 6(t), 0(t)) E (2,. During this interval, using the definition of p,, we have V.» S —p7||sp||2. for Ve(<.0) 2 62105 for some p7 > 0. This inequality can be used to show that (4.19) C = ACC+B,(s—K,0) s = A(-)-k|ao(-)l (ted—r) ab = Ate+eBs [boo —klao(')| (WM , Next, we claim that there exists a unique matrix M such that SM=MS and —K,M=I‘ To see this, note that since A, is Hurwitz and S has eigenvalues on the imaginary axis, the Sylvester equation AUX — XS = J1" has a unique solution. That this solution satisfies SX — XS = 0 and K,X + I‘ = 0 is shown in [49]. Thus M = X is the required matrix. Defining M,,={2=0,0=0,e=0, 0 as t —> 00, the preceeding inequality can then be used to show that every trajectory inside ‘11,,, approaches M“ as t —I 00. Our conclusions can be summarized in the following theorem. Theorem 4.1 Suppose Assumptions 4.1 through 4.7 are satisfied and consider the closed-loop system formed of the plant (4.7) and the output feedback controller (4.11)- (4.14). Suppose 6(0) is bounded and the initial states (2(0),6(0),0(0)) E (2,, where 0 < b < c, and 6 satisfies (4.16) and (4.17). Then, there exists u“ > O and for each u E (0,,u’], there exists 6" = e"(u), such that for all u E (0,u*] and e E (0, 6"], all the state variables of the closed-loop system are bounded and lim,_,ooe(t) = 0. The estimate of the region of attraction (2,, is limited only by two factors: the region of validity of our assumptions, and the control level k. If all the assumptions hold globally and k can be chosen arbitrarily large, the controller can achieve semi-global regulation. Corollary 4.1 Suppose Assumptions 4.1 through 4.7 are satisfied globally, i.e., U9 = R”, the functions a,(-) (i = 1,2,3) and 6() are class [Coo functions, and k can be chosen arbitrarily large. Given compact sets N C R" and .C C R”, choose c > b > 0 such that $1,, is large enough to contain N , and choose k large enough to satisfy (4 .17) 74 Then, there exists if > 0 and for each u E (0,;1‘], there exists 6* = 6*(u), such that for all u E (0,u‘] and 6 E (0,6‘], and for all initial states (2(0),6(0)) E N and 6(0) E L, all the state variables of the closed-loop system formed of the plant (4.7) and the output feedback controller (4.11)-(4.14) are bounded and lim,_.ooe(t) = 0. We conclude this section with the following theorem on the performance of the controller, which states that the controller recovers the performance of ideal state- feedback SMC, without servocompensation. Consider the ideal SMC u = —ksign(LgL’;"1h)sgn(s) (4.22) s = 2:11 k,6,+6,, Theorem 4.2 Let X = (2,6) be part of the state of the closed-loop system for the system (4.7) with the output feedback control (4.11)-(4.14) and X " = (2*,6“) b6 the state of the closed-loop system with the state feedback control (4.22), with X (0) = X ‘(O). Then, under the hypotheses of Theorem 4.1, for every p > 0, there exists ,u“ > 0 and for each u E (0,,a*], there exists 6* = 6“(u), such that for all u E (0,u*] and 6 E (0,6‘], ||X(t) — X*(t)|| S g V t 2 0. Proof. We prove the theorem in two parts. First, we compare the trajectories under ideal SMC with those under state feedback continuous SMC with the conditional servocompensator. Let X I = (2", BI) be part of the state of the closed-loop system under the control (4.8), (4.9) and u = ——k sign(LgL’f’_lh) sat(s/,a) with X 1(0) = X ‘(0). For this case, we show that, for sufficiently small u, X i(t) — X *(t) = 0(u) V t 2 0. Let s'r and s* be the corresponding sliding surface functions 75 of the two systems and to = min{t : |sI(t)| S ,u(1+ p2) V t Z to} If to > 0, then since |K,0i(t)| S up, V t, it follows that sat(sI(t)/,u) = sgn(sI(t)) = sgn(sl(t) — K,0I(t)) V 0 S t S to which can then be used to show that X T(t) = X *(t) V 0 S t < to. The result holds trivially if to = 0. We now consider X I(t) and X "(t) in the time interval t 2 to. Since X I(t0) = X“ (to), we have ISI(to) - 8*(toll = |K1 0i(to)| S #102 Using this, along with the fact that |si(t)| and [3* (t)| monotonically converge to the positively invariant sets {lsll S u} and {0}, respectively, it can be shown that |8I(t) - S"(0| S 214(1 + p2) V t Z to It follows that sl(t) — s*(t) = 0(a) V t Z 0. Since the equations for (I and C‘ are identical stable linear equations, driven by inputs 3* — K, 0'r and s“ respectively, where IK, 0"] S up, and si — s“ = 0(u), continuity of solutions on the infinite time interval [47 , Theorem 9.1] can be used to show that for sufficiently small u, (1(t) — (‘(t) = 0(u) and hence ei(t) - 6*(t) = 0(u) for all t Z to, which can then be used to show that 2"(t) — 2*(t) = 0(u) for all t _>_ to, so that the first part of the proof follows. In particular, there exists u“ > 0 such that M 6 (0,61 => “w(t)-X715)” S 7'/2 Viz 0 76 In the second part of the proof, the idea in [5], that was used in Theorem 2.2 in Chapter 2, can be repeated to show that the trajectories X of the system under output feedback approach the trajectories X I under state feedback as 6 —+ 0. In particular, there exists 6" = 6“(u) > 0 such that s 6 (0,6) => ||X(t) — Xi(t)|| g r/2 v t 2 0 The conclusion of Theorem 4.2 then follows from the triangle inequality. 4.5 Internal Model Perturbation As mentioned in the concluding remarks in Section 4.3, our design requires that the constants co to 6,, in Assumption 4.5, and hence the frequencies of the exosystem, be precisely known. In addition, as noted in the remarks following it, Assumption 4.5 is equivalent to requiring that the control input, when restricted to the zero-error manifold, be a polynomial function of the exogenous signals [13]. A violation of either of these conditions results in a perturbation of the internal model. To make the idea precise, let x(w, 6) = —bo(0, 0, 0, w, 6)/ao(0, 0, 0, w, 6) and suppose 2(w, 6) is a nominal value of x(w, 6) that satisfies (4.6). As mentioned above, there are two sources for this perturbation. First, the frequencies of the exosystem are unknown, so that x satisfies (4.6) with unknown coefficients co to cq, while )2 does so with nominal coefficients 6,, to 59, which are used to construct the internal model. Second, the assumption that the control input, when restricted to the zero—error manifold, is a polynomial function of the exogenous signals, does not hold, so that X does not satisfy (4.6), but an approximation )2 of it does. In this case, we note that since any continuous function can be approximated to arbitrary accuracy 77 on compact sets by polynomials, a linear internal model that generates )2 can be used to approximate x arbitrarily closely. Regardless of the source of the perturbation, using the results of [45, Section 5], it can be shown that provided the perturbation is small, the controller of the previous sections achieves ultimate boundedness of the tracking error, with the bound being proportional to the size of the perturbation. In particular, let )2(w, 6) = )2(w, 6) — )2(w, 6), and suppose [520116)] S 5x V (w,6) E W x 6 The analysis proceeds exactly as in Section 4.4 upto the point of showing that the trajectories enter the set ‘11,“. Using the fact that b0(0, 0, 0, w, 6) = —X(w, 6)a(A, 7r, 6) = —)2(w, 6)a(A, 7r, 6) + )2(w, 6)a(A, 7r, 6) it can be shown that inside the set \IIM, when V = 0, the closed-loop equation (4.19) can be written as w = Sow I 2 = ¢0(2, e, 0, w, 6) 0 2 A00 + J(s — N(6) C = ACC+B,(s—K,0) s = A(2, 0, 6, 1p, 0,w, 6) — k|ao(2, 0, 6, 0, w, 6)] 312623) + )2(w, 6)a(A,7r, 6) 6,0 = A¢go + 6B2 [bo(z,0, e, 0, w, 6) — klao(-, 0, )I (tflpifl) + )2(w,6)a(A,7r, 6)] , (4.23) where A(2, 0, 6, cp, V, w, 6) = A(2, 0, 6, ,0, V, w, 6) — b0(0, 0, 0, 0,w, 6) — )2(w, 6)a(A, 7r, 6) 50(2, 0, 6, V, w, 6) = b0(2, 0, 6, V, w, 6) — b0(0, 0, 0, 0,w, 6) — )2(w, 6)a(A, 7r, 6) 78 It can be verified that the system (4.23) has M“ as an invariant manifold when )2 = 0. Equation (4.23) takes the form of [45, Equation (A1)], and satisfies all the assumptions of [45, Lemma 2], so that the results of [45, Lemma 2] can be applied to show that (4.23) has an exponentially attractive manifold 191,, that is 0(6x) close to M ,,, and on which 6 = 0(6x). The Lyapunov analysis of the final part of the proof of Theorem 4.1 can be repeated to show that all trajectories inside \IIM approach M“ as t tends to infinity. Our results can be summarized in the following theorem. Theorem 4.3 Under the hypotheses of Theorem 4.1, there exists a“ > 0 and for each ,u E (0,;1‘], there exists 6* = 6"(u) > 0 and (5; = 6;;(6) > 0, such that for all 5x E [0, 6;], u E (0, u‘], and 6 E (0, 6“], all the state variables of the closed-loop system are bounded and converge to an invariant manifold where 6 = 0(6X). 4.6 Simulation Example To show the performance improvement with the conditional servocompensator, we consider a second-order system modelled by the equations 2:, = x2, :6, = —6,(x, — xI/3l) + 62v, y = x, (4.24) with the reference signal r(t) = rosin(wt), which is generated by the exosystem 0 w u') = w, wT(0) = [0, r0], r(t) = w, —w 0 It can easily be verified that 1 X = a—[—c.f"w, + 6,(w, — wI/3!)] 2 79 and that )2 satisfies the identity Lgx = cox + c,L,X + - -- + cq_,L3‘l)2 with q = 4, co = —9w4, c, = 0, c2 = -10w2 and c3 = 0. We show the performance of four designs: the first is an ideal SMC, the second is a continuous approximation that does not use a servocompensator, the third uses the fourth order conventional servocompensator 0 = S0 + J6,, and the last design uses the conditional servocompensator (4.13). In the first two designs, 6 = k,6, + 62, while 6 = K ,0 + k,6, + 62 in the last two designs. For all designs except the conventional servocompensator, the scalar k, is chosen as any positive constant. In the conditional servocompensator design, K, is chosen to make (S — JK,) Hurwitz. For the conventional servocompensator, k, and K, are chosen to make the matrix S J —K, —k, Ah: Hurwitz (see [45]). The estimate 62 is provided by the high-gain observer 61: é2 +91(€1— é1)/€, 62 = 92(61— é1)/€2 with g, and g2 chosen such that the polynomial A2 + g,A + 92 is Hurwitz. The control is taken as u = —k sat(§/p.). We use the following numerical values in the simulation: 6, = 1, 62 = 3, w = 0.5 rad/s, r0 = 1, k = 10, a = 0.1, k, = 5 in the first, second and last designs, with K, chosen to assign the eigenvalues of S — JK, at -0.5, -1, -1.5 and -2. For the third design, we choose k, and K, to assign the eigenvalues of AI, at -0.5, -1, -1.5, -2 and -3. The observer parameters are chosen as g, = 6, g, = 5 and 6 = 0.01. The results of the simulation are shown in Fig 4.1, and the improvement in the transient performance with the conditional servocompensator is clear. In particular, the transient response of this design is close (almost indistinguishable in the figure) to that of the ideal SMC design. As expected, the transient response of the CSMC 80 design without a servocompensator is also close to that of the ideal SMC, but does not result in asymptotic error convergence, while the conditional servocompensator design does. Zero steady-state error is also achieved with the third design, which employs a conventional servocompensator, but at the expense of degraded transient performance. Performance improvement with conditional servocompensator design 0-4 ! ! I i ! i F ! ! ,\ : I . . . . . . . 1 \ I I I I I I I I I 0-2 r ‘ 1‘ ' ' V ‘1 """" Design'3 :‘CSMC with conventional ServocompensatOT """" i """" ‘ ‘5 u \/ ; I s s 2 s 2 ; £2 0,... ...... .\ ' r-l“*—_' 2 i ,'\ ’7’ ; ,~---—:___.'__——'———-1——-—.‘-—"" 9 l l " :P_02_4. \ ........ .......... .......... .......................................................... .. B I ; DeSign 1 : Ideal SMC . j j : j g _0, _I , . . . . . .3 . .Desighz :.CSMC without servoicompenSator ..... 3 .......... 1.. . . . . . .. .1 . . ..... _ a, ‘ Design 4 : CSMC with Conditional servocompenSator 1 C I I I I I ? {3‘06— ......... .......... .......................... _ Q : i- E -0.8 .................. .’ ............................................................................... .. _1 L i 1 1 1 l 1 1 1 o 1 2 3 4 5 6 7 8 9 10 Time(sec) x10“ 5 I F i i A 4)— .................. '_\ ......... ...... Designz ............. /.H".-.. .................. _. 0’ i’ ' (without servocom ehsation) " T‘- s 3_ .................. ./ ........... .\\ ...... : ................. p......... ..I ......... ...\ .............. —i ‘6', l: - : : I : ‘\ I 2...... ........... [bl ............. \ ..... i ................... i ........ I ............ i ..... ' ............. _ § ,- E \, / i ,I 3 ‘. 91_, ................. \ ..... I ............. : ....... \ ........... _ 3’ I : \E I \ 8 : \ I \ t—1_ ................. - ......... / .................................................................. _. a) l-’ : .\. ..’ . \ oa_ -........ ........ : ............ :..\ ................. t./ ................... : ............ \ ..... .. .E 2 , Desngn1zldealSMC : - r : . x - ' ' : \. I: : \ g4- ...... . I....and..DesugnsS.and.4.: ..... - \ ........... I .............. -.\..._ ,: ,./ (with servocompensation) \. , .v ; g \, ~ _4_’.‘ ....... . ...................................... ... _5 i i i i 25 30 35 4o 45 50 Time (sec) Figure 4.1: Performance improvement over the conventional servocompensator design using a conditional servocompensator. 81 Equation (4.24) represents an approximation of the pendulum equation. To show the effect of internal model perturbations, suppose that the system in question is the simple pendulum, described by the equation in = $2, $2 = —615in(a:1) + 62u, y = 2:1 (4.25) With the control objective the same as that in the previous simulation, it can be verified that, in the current case X = 5— —w2w1 + Blsin(w1)] 2 so that Assumption 4.5 does not hold. Suppose that sin(w1) is approximated by the successively higher order polynomials p1(w1) 2 ml, p2(w1) = wl — will/3!, and p3(w1) = 1111 — w? / 3! +w‘i’ / 5! respectively, to which correspond the perturbed nominal values of the steady-state control 1 Xi = El—U‘flwl + 01Pi(w1)l1 Z: 13 213 2 It can be verified that X1 satisfies (4.6) with q = 2, c0 = —w2, and 01 = 0, while 5&3 does so with q = 6, co = —225w6, cl = O, Cg = —259w4, c3 = 0, c4 = —35w2, and c5 = O. The constants for )22 are as specified in the previous simulation. We compare the performance of three conditional servocompensator designs, of orders 2, 4, and 6, corresponding to the polynomial approximations p1(-), p2(-), and p3(-) respectively. For the servocompensator of order 2, K1 is chosen to assign the eigenvalues of S — J K 1 at -O.5 and -1, for that of order 4, at -0.5, -1, -1.5 and -2, and for that of order 6, at - 0.5, -1, -1.5, -2, -2.5 and -5. All other values are retained from the previous simulation, except k, which is chosen as 20. The results are shown in Fig 4.2a. For comparison, we also show the performance of the conventional servocompensator design of [45], 82 with the eigenvalues of Ah placed as in [45]. As expected from the results of Theorem 4.3, for both designs, there is a reduction in the steady state tracking error going from the lowest order approximation to the highest. Fig 4.2a : Steady state tracking error |e1| (absolute value) Conditional servocompensator 4 Conventional servocompensator 95 96 97 98 99 100 95 96 97 98 99 100 Time (sec) Time (sec) Fig 4.2b : Tracking error 91 during the transient period 1 I I I I I o 8 _ ............. 5. . . Cenventional.seryoeompensater (9&6). ................. ............. _ 0.6 _ ........... / . Conventional servocompensator (q=4 . .) ................ § ............... .. 0.4 _ .............. Z. . ./ .......... ................. Conventional servocompensator (q=.2) ._ 0.2_ ......... 7" ....... \ ..\.’.\.r.,...-.-:..;-..".‘;. ,,,, —..H/“ ................ ............... _ / ' /’ \\‘ : '--‘.~;_ : .- x m~ , - . __ _L- ______ 0 o _ ..... / ........ /. ./ ............ \A .... _ _- ....- “ =7. _______ / 'i Z ‘ T 7 E “““ 3 3 __O.25. ..... I ..... / .......... . ................ ..... ...........: ........... ............... - I /' E E S 3 E _O.4 — - . . . ./; ....... ................... . ............. - ................................. ............... ‘l -05 _ ..... / .......... E ................ .C enditienal. servocompensator ............................. _ .’ 5 ’ (q=2, q=4 and q=6) . 3 -0.8"/ ............ .............. ............... _ _1 i 1 i i i o 05 1 1.5 2 25 3 Time (sec) Figure 4.2: Effect of internal model perturbation on the tracking error. Fig 4.2b shows the transient response of the controllers. We see that while the 83 transient responses are almost identical for the three designs in the case of the condi- tional servocompensator (indistinguishable in the figure), they become progressively degraded as the order of the approximation increases in the case of the conventional servocompensator design. 4.7 Conclusions In this chapter, we extended the conditional integrator design of Chapter 2 to that of a conditional servocompensator, also within a sliding mode control framework for minimum-phase nonlinear systems. As before, in the new approach, servocompen- sation is provided only “conditionally”, i.e., inside the boundary layer, thus effectively eliminating the transient performance degradation brought about by the conventional servocompensator design. Analytical results are provided for regional and semi-global asymptotic tracking, and the improvement in performance is shown analytically by proving that the performance of the output feedback continuous sliding mode con- troller, with a conditional servocompensator, can be tuned to recover the performance of an ideal state feedback sliding mode controller, without a servocompensator. We also studied the effect of internal model perturbations on the tracking error, and showed that in the presence of perturbation, the tracking error is ultimately bounded, with a bound that depends on the magnitude of the perturbation. In the case of such perturbations resulting from the approximation of a continuous function by polynomials, the magnitude of the perturbation can be made arbitrarily small, by increasing the order of the approximating polynomial. However, doing so increases the order of the internal model, and hence the system order. While in general, the transient response of a system becomes worse as its order increases, such is not the case with the conditional servocompensator. In particular, our result shows that the performance with the conditional servocompensator is always “close” to that with an 84 ideal sliding mode controller of fixed order, regardless of the order of the conditional servocompensator. This shows as an advantage of the conditional servocompensator design over the conventional one, in which the transient reponse becomes progressively degraded as the order of the approximating internal model increases. Extensions to relax the design to include an equivalent control component and allow the coefficient of the switching component to be error and/ or time dependent should be straightforward, as should be extensions to the multi-input, multi—output case. Such extensions are carried out in the special case of integral control in [71]. 85 Appendix A Derivation of (4.18) The arguments of section 4.4 show that 53' S —psgo|s| whenever |§| Z 11, provided 6 is small enough that IN (e)cp| < 11. Next we consider 3 in the set {Isl 2 110- 60)} fl {lél S u} where 60 < p6/(4k) and e is chosen small enough that IN (e) can be used to show that V; S —||C||2/2 whenever V; 2 112,08 for the choice p3 = 4p3(1 + p2)2. This shows that C (t) reaches the set {V; S ung} in finite time and stays therein for all future time. It can be verified that inside Q", ”e” < 2p4p. Since 86 limtsoou(t) = 0, it follows that there is a finite time after which ||(eT, VT)” 3 2pm. From Assumption 4.7 and the definition of 04, it follows that V; S —a3(||z||) for V; 2 014(11p9), where p9 = 2m. This shows that 2(t) reaches the set {Vz _<_ a4(up9)} in finite time and stays therein. Lastly, the fact that w(t) reaches the set {V,p(cp) S 62/)5} in finite time and stays therein was already shown in Section 4.3. This completes the proof of the statement that every trajectory starting inside the set QC x E. enters the set \IIW, in finite time and stays therein for all future time. 87 Appendix B Derivation of (4.21) We consider each term in the derivative of V of (4.20) separately and then arrange the derivative in a quadratic form of H = [||z|| “(II ”6” |§| ||\15l§| + Alellwll) for some positive constants A15 and A16. 88 The expression for 3' is given by § 3' - K1(11/k) sign(LgL‘f’_lh) MST(w, 9) AI) — klao(-)l (#40 =%H+Kfiw+KMG-NMW+KH®%”’%T ) — K1(11/k) sign(LgL‘;_lh) MST(w,6) — kIaOI-n (4%92) — Imp/k) sign(LgL‘}'1h) MSTW) It can be verified that KlAao + 1(le = KlAaé + K1J§ + K186. Using this, along with the fact that K156 = KIS(n/k) sign(LgL’f’—lh) MT(w,0), and SM 2 MS, we have . —N S 2 b0() + KIAOC +K1JS — KIJN(E)<,0— k|a0()| ("S—MEE) +K2 [62 C3 8p]T = K4C + K55 + kps — KIJN(6)(0 + bo(-) — k|a0(-)| (__s _ 11(6)(p) In arriving at the last line, we have used the property that Kg [62 e3 - - - ep]T can be rewritten as K4C + kp_1(.§ — K16) for a suitable K4, and then appropriately defined K5 and k,,. Using the property that b0(0,0,0,w,l9) + (10(0, 0,0,w, 0) 11 = 0, where 17. = —k sign(LgL’f"1h) (Klé/u), and rewriting b0(-) — k|a0(-)| (ZZZ—((1?) = b0(z, e, V,w,B) — b0(0,0,0,w,6) + bo(0, O, 0, w, 0) + ao(0, 0, 0, w, 6) 1‘1 ” — N + K ‘ + k |a0(0,0,0,w,0)| (Klé/u) — k |a0(0,0,0,w,6)| (8 (6:20 10) +kWMQQQwflN(i:%EE)—kMMaamwflfl(i:%gfi) it can be verified that the fourth term in the derivative of V satisfies an inequality of 89 ~; kg .. ~ ~ 88 S _(712 - A17) ISI2 + |8|(/\18||Z|| + /\19||CH + A2o||0|| + A21||90|| + A22||V||) for some positive constants A17 to A22, where A17 is independent of 11. A similar argument can then be used to show that the final term satisfies an inequality of the form . 1 ~ ~ K. s — (; — A23) Ilwllz + l|w|l(/\24||Z|| + A2511\1b —)‘1c ->\1a A5 -A2b —)‘2c _A2d 77 = A6 "A3c -A3d 11: fl - M —/\4.1 H l — — A23 - 6 _ where the positive constants A23, and AM to A4,; are independent of 6; A17 and Ale to Age are independent of p; A11; and A21, are independent of A6; and Ala is independent of A5. Therefore, by choosing A5 large enough, then A6 large enough, then 11 small enough, and lastly 6 small enough, we can make ’P positive definite. The equivalence between (4.26) and (4.21) follows from Assumption 4.7 and (4.20). 90 Chapter 5 Applications 5. 1 Introduction In this chapter, we apply the controller designs of Chapters 2 through 4 to two applications : temperature control of a continuous stirred tank reactor (CSTR) and position control of a permanent magnet stepper motor (PMSM). For the CSTR, our interest is in the regulation problem where the references and disturbances are asymptotically constant and the control magnitude is constrained, so that the universal integral regulator of Section 2.6 is a natural choice for the controller. Since the particular class of CSTR systems that we study in this chapter is relative degree one, the universal integral regulator is simply a PI controller with anti-windup. Control of the CSTR problem under input constraints and varying assumptions on the class of CSTR, available measurements, and control objectives has been widely studied; see, for example, [1, 2, 40, 41, 73, 74] and the references therein. Some of these references deal specifically with P1 controllers [1, 2, 40] and the issue of anti-windup. For the PMSM, we examine both the integral control design of Chapters 2 and 3 for the case of asymptotically constant references and disturbances, as well as the 91 servocompensator design of Chapter 4 for the case of references and disturbances generated by a neutrally stable exosystem. Whereas the design in Chapter 4 was done for SISO systems, as mentioned in the concluding remarks of the chapter, the results could have been extended to MIMO systems, and we do so by simulation for the PMSM. For the output feedback design of Chapter 2, we show that the controller reduces to a MIMO version of the universal integral regulator design of Section 2.6. The problem of position control of a PMSM that we study in this chapter has been widely studied, under varying assumptions, by several authors; see, for example, [9, 10, 11, 17, 20, 48, 51, 52, 53, 59, 61, 76, 77]. Also, while we specifically apply our designs to the PMSM, the results can be extended to other types of motors, such as permanent magnet synchronous motors [6, 54], and brushed and brushless dc motors, studied, for example, in [20]. 5.2 Continuous Stirred Tank Reactor The continuous stirred tank reactor is widely used in the chemical process indus- try. It consists of a well-stirred tank into which there is a continuous flow of reacting material and from which the reacted or partially reacted material passes continu- ously. It is generally assumed to be homogeneous and therefore modelled as having no spatial variations in concentration, temperature or reaction rate throughout the vessel. In fact, the main assumption concerning the CSTR dynamics is that perfect mixing occurs inside the reactor. A schematic of such a chemical reactor, with typical process symbolism [3], is shown in Fig 5.1. While a wide variety of methodologies, including input-output feedback lineariza- tion [7 4] and adaptive versions of feedback linearization [26, 50] have been explored for the stabilization of chemical reactors, in virtually all present day industrial ap- plications the problem is efficiently solved using PI controllers. Whereas traditional 92 Reactants __.. \ Mixer l'CD P . _______'_\ Heating/ ‘ L’ Cooling /// 4* Reactor H In (>'<3 / __.. l 4”” if T:- T Jacket / k \\___. Q/ T 5‘0“ [Control ] l l Out Products > Figure 5.1: Block schematic of a CSTR analysis of CSTRs controlled with PI control algorithms resorted to linear system analysis together with linearized models, recently, there have been few works which address the nonlinear problem directly. The stabilization of chemical reactors by out— put feedback with PI—type controllers has been reviewed and treated in detail in the Ph.D. thesis of Jadot [40]. A robust control scheme in the face of uncertain kine- matics for a class of CSTRs has been proposed by Alvarez-Ramirez et al [2], where it has been shown that the proposed controller has the structure of PI control. A more recent result, which goes beyond just the analysis of closed-loop stability, and focuses on transient performance, has been discussed in Alvarez-Ramirez et al [1]. In this chapter, we focus our attention on the same class of CSTRs studied in [1, 2]. The design of Section 2.6, when applied to this class of CSTRs, yields a PI controller with an anti-windup structure. 93 5.2.1 System Model We consider the first-order, irreversible, exothermal chemical reaction, which occurs in a constant volume continuous stirred tank reactor. The process dynamics are given by [1, 2] é : g(cin — C) _ R(C’T)) . (5.1) T = 6(T,-,, — T) + AH,R(c, T) + 7(Tj — T) where c is the reactor concentration, c,-,, is the inlet concentration, T is the reactor temperature, T1,, is the inlet temperature, T]- is the jacket temperature, AH, is the reaction enthalpy, 6 is the dilution rate, 7 > 0 is the heat transfer coefficient, and R(c, T) is the reaction rate, given by 73(01T) = Cfioexp(-EA/RT) The control objective is to regulate the reactor temperature T via manipulation of the jacket temperature Tj. The system (5.1) has relative degree one uniformly in the system parameters, and satisfies Assumption 2.2 for all physically allowable values of the parameters, i.e., given any desired reactor temperature Teq > 0, there is a unique set of equilibrium values egg and Tjeq of the reactor concentration c and the jacket temperature T,- respectively, such that 0 = 0(c,-,, " Ceq) “ R(0691Teq)1 O = 6(T, — Teq) + AHrR(CeQ1 Teq) + V(Tjeq - T60) We assume the following values for the constants, taken from [2], 30 = exp(25), EA/R =104,AH, = 200,19 = 1, 7 = 1, c," = 1, T," = 350 and T,- = 350, where T, is the nominal value of the jacket temperature. Due to limitations in cooling/ heating 94 equipment, the jacket temperature T,- is subject to saturation constraints. Similar to [2], we assume that T,- 6 [300,400]. 5.2.2 PI controller design Defining the states 77 = c, 5 = T, input 11 2 AT, = T,- — T, and output 3; = T, we rewrite (5.1) in the form of (2.5), where the functions (f(), b() and a(-) are given by 1 $0715) = l-n-nexp(2 -”§1) (5.2) 11(7), g) = 200 17 exp (25 — lg‘i) — 2g + 700, a(17,§) = 1 Note that for the specified numerical values for the constants, the CSTR has an unstable open-loop equilibrium point (ceq, T6,) = (0.5, 400) and that the control 11 is bounded in magnitude by 50. For the purpose of simulation, we assume that the desired temperature at which the CSTR is to be stabilized corresponds to the unstable equilibrium point (c, T) = (0.5, 400). Therefore, we take the reference Tm, as a step of magnitude 400. Since the system is relative degree one, the sliding surface function is taken as s=koa+e,e=£—r (5.3) where k0 > 0 and a is the output of the conditional integrator (I = —k00 + ,u sat(s//1) (5.4) The control is taken as u = —k sat(s/u) (5.5) which, as mentioned in Section 2.6, is a PI controller with anti-windup. 1Even though we assume that all the constants in (5.1) are known, so that ¢(-), b(-) and a(-) do not depend on an unknown parameter, it is clear that it is possible to allow for such unknown terms. 95 E 550 , . , , , I I 1 . 3 :' \ : g 500 P- ........................... I]. . .\ ..... .............................. . ......... . .......... . ........ d a) 11 \ \I ; I ' .. ' . ' g. 450 l I\ 3 3 — Conditional integrator 93 4 : \ \ \ E 3 - -, Conventional integrator h /: ‘\ . : ' ' 9 400 _ ........................ A . ‘ . . _______ O ‘ \ ' __ _J " _- — 8 : : \ -:-—- " : m 350 l l l l l l l l I 0 0 5 1 1 5 2 2.5 3 3 5 4 4 5 5 H- l I l I l l l l I 9 400 ' - 3 S 8 E 350 " .9 2'3 . . 0 300... ............................ .......__.._____ _. .............................................. _i g 1 1 1 1 1 1 1 1 1 0 0 5 1 1 5 2 2.5 3 3 5 4 4 5 5 150 I I I I I I I r i in . 2 . D . m I 'c , (U . > . a, . .E « _‘Q 3 (7) : _100 l l l l l l l l O 0 5 1 1 5 2 2.5 3 3 5 4 4 5 5 b 40 I l l l I I, -_ \ l l l I ‘5 3 ’ s ‘zx . . g- 20— ...................................... ./.../ ...... ............... \.\..\.; ...................... —l /I I ‘ ‘~ _. 2 0 / : ; ‘‘‘‘‘‘ 9 / : 9 \ \ . : / : _ _ ...... \.'. ......... . .......... I. . / ............... i .................................................. c \g g _ , : —' _40 1 1 1 1 1 1 1 1 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (sec) Pl control of a continuously stirred thermal reactor Figure 5.2: Temperature control of CSTR using a PI controller with conditional integrator 96 Numerical values for the controller parameters are chosen as k = 50, p. = 0.5 and k0 = 1. The initial conditions of the states are taken as 71(0) = 0.5, {(0) = 350, and 0(0) 2 0. Fig 5.2 shows the reactor temperature T, the jacket temperature T,-, the sliding variable 3 and the integrator output a. For comparison, we also plot the variables for the corresponding conventional integrator design, which uses {7 = 81, with expressions for s and u and other numerical values retained from the conditional design. The following observations can be made from the figure. The inclusion of the conventional integrator causes the sliding condition 35‘ < 0 to be violated, since the control needs to be large enough to overcome the term kolc'rl = kolel, which can be large if |e(0)| is large. The subplot for the sliding variable 3 shows that the sliding condition is not satisfied, and the integrator winds up, causing the control to be in saturation for a longer period of time. The large buildup in the integrator (windup) with the conventional integrator design can be observed in the last subplot, and the extended period of saturation in the controller as a consequence is clear from the subplot of the jacket temperature Tj. On the other hand, with the conditional integrator, koldl = k0|-koO’+/J sat(s/p)|, which is “small” even with large k0, provided p is small. From the subplot for s, we see that the sliding condition is satisfied for the conditional integrator design. It is clear from the figure that the tracking performance with the conventional integrator design is significantly degraded over the conditional integrator design. Suppose that, in order to have the sliding condition satisfied, we choose k0 small in the conventional integrator design. To that end, we let [to = 0.01 in the conven- tional integrator design, but retain the value of k0 = 1 for the conditional integrator design. It can be verified that the sliding condition is satisfied for both designs for the specified values. The results are shown in Fig 5.3, and we see from the figure that the convergence of the error to zero is sluggish with the conventional integrator design when compared to the conditional integrator design. 97 Comparison of performance with the conditional and conventional integrator designs 06 l I l l l 1 . .l : —- Conditional integrator 33 0.5 \. ................................................................. _ _ Conventional integrator l _...\..\.. ........... ............ 3 ............ E. ........... i ............ i .......... - l,— 0-4 $-~_.~.’__--__.. . . . . L _______________________ g 03. ........... ............ ............ ............ ............ ............ ........... .. U) ................................................................................................ _ E. x ................................................................... .. O ........................... CU L . l- : _0.1 l l l l l l I 2 2.5 3 3.5 4 4.5 5 5.5 6 Time (sec) Figure 5.3: Comparison of performance with the conditional and the conventional integrator designs. As mentioned in Section 2.6, the only precise information about the CSTR system that the control (5.5) uses is its relative degree and the sign of the high frequency gain. It does not use precise information about any system parameters. To show that the controller is robust to uncertainties, we consider a -10% step disturbance in T," at t = 6 and a setpoint change in Tm, from 400 to 410 at t = 8, which are the same uncertainties considered in [2] to demonstrate the robustness of their PI controller. The results are shown in Fig 5.4, and we see that the response is not degraded in the face of such uncertainties. In fact, the effect of the step disturbance in Tin at t = 6 is almost negligible, as can be seen from the subplot of the reactor temperature T, and can be explained as follows. When the parameter Tm changes at t = 6, it changes the equilibrium point. Prior to this change, 3 is inside the boundary layer {ISI 3 p}. The magnitude of the change is not large enough to force 3 outside the boundary layer, as can be inferred from the fact that the control does not reach its saturation limits. Since 6 = s — had, from Isl S ,u, and ko|a| g u, it follows that |e| 3 2p in this case. For the step change in Tref at t = 8, we have e(8+) = —10, which forces 3 outside the boundary layer. The trajectory re—enters the boundary layer at t r: 8.3, after which 98 the error 6 satisfies |e| S 2,11 and asymptotically approaches zero. Our results are comparable to the ones reported in [2]. Response of the PI controller to disturbances and setpoint changes 4‘2 F l ! l 1 I 1 l P 410 . . . . . . . . . §§§ 402 Reactor temperature T .5 8 p— .— —.-. .. I.— 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 420 r l I 1 I I I i I 8 o l i ................................................................... . ............................................................... '23 ................................................................ Jacket temperature 'l'j é E; _- P _ c I l I a O — _- _ 5 5.5 6 6.5 7 7.5 8 Time (sec) Figure 5.4: Effect of uncertainties on the response of the PI controller with conditional integrator 5.3 Permanent Magnet Stepper Motor Permanent magnet stepper motors (PMSM) have become a popular alternative to the traditionally used brushed DC motors (BDCM) for many high performance 99 motion control applications for several reasons : better reliability due to the elimi- nation of mechanical brushes, better heat dissipation as there are no rotor windings, higher torque-to—inertia ratio due to a lighter rotor, lower price, and easy interfacing with digital systems [20, 77]. They are now widely used in numerous motion con- trol applications such as robotics, printers, process control systems etc. Some of the drawbacks of PM machines when operated in open loop are the occurrence of large overshoots and settling times, especially when the load inertia is high, and the fact that microstepping is not possible in the open loop mode of operation. As a result, over the years, many control algorithms that can improve the performance of PMSMs in a closed loop operation have been examined. Zribi and Chiasson [76] used the technique of exact feedback linearization using full state feedback, with extensions to the partial state feedback case in [10, 17], and experimental validation of the controller in [10]. Adaptive solutions to the problem, under varying assumptions on the measurable states and on what parameters in the system are partially or wholly known, have appeared, for example, in the works of Dawson and co—workers [11, 20], Khorrami and co-workers [48, 51, 52, 53, 61] and oth- ers [59]. A sliding mode controller along with implementation results was reported in Zribi et al. [77]. In order to avoid the chattering problem associated with the “static” discontinuous SMC, a “dynamic” or “second-order” SMC was proposed, where the discontinuitites were relegated to the derivatives of the control input. Integral ac- tion occurs in the second-order SMC as a result of designing with the derivative of the input, so that the controller in [77] is the closest in similarity to the SMC with (conditional) integral action that we design. 5.3.1 System Model A schematic of a PMSM that has a slotted stator with two phases, and a PM rotor is shown in Fig 5.5. 100 AA D [1A Figure 5.5: Schematic of a two phase PMSM. The mathematical model of the PMSM is given below [10, 76, 77] El... ~1—va( -Rz'a + me sin(N, 43)) In), — Rib + me cos(Nr 45)) (5.6) l(- J sz'a sin(N, 43) + sz’b cos(N, ¢)— Bw — TL) 9:13- le‘ a]? ng't- Li.) where in is the current in winding A, ii, is the current in winding B, 43 is the angular displacement of the shaft of the motor, w is the angular velocity of the shaft of the motor, 12,, is the voltage across winding A, vb is the voltage across winding B, N, is the number of rotor teeth, J is the rotor and load inertia, B is the viscous friction coefficient, L and R are the inductance and resistance respectively of the phase windings, Km is the motot torque (back—emf) constant, and TL is the load torque. The model neglects the slight magnetic coupling between the phases, the small change in inductance as a function of the rotor position, the detent torque [48], and the variation in inductance due to magnetic saturation. A nonlinear transformation, known as the 101 direct-quadrature (DQ) transformation can be used to transform these equations into a form more suitable for designing nonlinear controllers, i.e., one to which feedback linearization techniques can be applied [10, 76, 77]. This transformation changes the frame of reference from the fixed phase axes to axes that are moving with the rotor. The DQ transformation from the fixed axes variables ($0,331,) to the dq axes variables (rd, mg) is defined by and (,3, cos(N,¢) sin(N,¢) 1:0 33,, —sin(N,¢) cos(N,¢) mg, The direct current id corresponds to the component of the stator magnetic field along the axis of the rotor magnetic field, while the quadrature current 2', corresponds to the orthogonal component. Defining the states, inputs, and outputs as 2:; = id, :52 = iq, 2:3 2 w, 334 = (1), ul = p,,, 1L2 = 1),, y1 = id, and y2 = (15, it can be verified that (5.6) can be rewritten in the form of the following state model for the PMSM . l {131 = —k11‘1 + [$532233 '1' ksul i2 = —k1.’L‘2 — k5$1$3 — [$21133 '1' [6611.2 5333 = k3$2 — 134333 — do i (5-7) 5&4 = 133 y1 = $1 $12 = 1154 J where the constants k1 to k6 and do are related to N,, J, B, L, R, Km and TL by R Km k1 L, 2 L , 3 102 5.3.2 Integral Control - State Feedback It is desired that the rotor angular position and direct-axis current track given references w(t) and [w(t) that tends to constant values as t —+ 00. It can be verified that the system has full vector relative degree p = {1,3}, globally in R4. Consequently the sliding surface functions 31 and 32 are given by 81 = k601+ 61) el déf yl — Idda (5 8) . .. def ' 82 = [9302 + kiez + [€362 + 82, 82 = 92 - ¢d where 01 and 02 are the outputs of the conditional integrators 0: = —k‘10'1 + lsat 81 [1.1 1 0 fl ( / ) (59) 0:2 = —k30’2 + flgSflt(Sg/fl2) k3, k3 > O, and k? and kg are chosen such that the polynomial 2:2 +k§x+kf is Hurwitz. Since 62 = 1133232 — k4x3 — do - @5512) is required to construct so, the parameters k3, k4 and do will need to be known. We assume that k1, kg and k6 are unknown, corresponding to uncertainties in the resistance R and inductance L of the phase windings, while k5, being the number of rotor teeth, is precisely known. Consequently, the vector of unknown parameters is taken as 6 = [k1, k2, ko]T. It can be verified that the expressions for s, of Section 3.3 take the form .81 = [fix—[€601 + ulsat(31/u1))+ F1($16111c(&11)’6)+ 011(2), 6’)u1 (5 10) s, = kg(—k302 + ugsat(so/u2)) + F2(a:, 62, (E2, (9'2, (155,3), 0) + a22(:r, 6)u2 with F1(-) = —61:1:1+ k5x2x3 — 1‘”, F2(-) = kfég +k§é2 — 5,3) — k4(k3:c2 - k4x3 — do) — 1:361:02 — koksxlxo - k362r3, a11(-) = 63, and a22(-) = 1:393. Specifically, the matrix A(x,0) in Assumption 3.5 is given by A($,6) = diag[03,k303], so that the natural choices for F(x, 9) and A(x) become I‘(:z:,0) = (63/63)12x2 and A(x) = diag[é3, k3é3], 103 where 93 > 0 is a nominal value of 03. With this choice, the control u in (3.8) becomes where (3T 2 [61, 82, ég, éo], and wT = [1531), $23)]. We will choose the nominal control component F,(-) to cancel all known/ nominal terms in F,(-), i.e., F1() = —é1$1+ k5$2$3 — [($31) F20 = k¥é2 + kgéz — ((9:13) — (CM/C3332 — ($41133 — do) — k3é1$2 - k3k51‘1x3 — kaéfla where 61 > 0 and 62 > O are nominal values of 61 and 62 respectively. To make the choice of fi,(-) precise, suppose that 0,- E [65",6f‘4], where 0 < 6;" < 0:” and 9:” and 6M 1' are known. Let A,(-) = Rf) - (93/53)“) and p,(:l:, e, to) be such that (93A,(-) sup 3 9,-(111, e, w) where the supremum is taken over all :1: E R“, e E R4, 6,- E [63",63‘4] and w E R2. The functions 6,- are chosen as [3,-(-) = g,(-) + q,, where q,- > 0. Note that we have not included the term {k3(—k30,~ + u,- sat(s,-/1i,-)} in the definition of A,(-) as was done in Section 3.3. This is done purely as a matter of convenience. The contribution of this term to the left hand side of the inequality in Assumption 3.6 is 001,-) and can therefore be accounted for by the term q,, provided a,- is sufficiently small. A remark to this effect was also made in Section 2.7. The results of Theorem 3.1 allow us to conclude that the controller (5.11) achieves global regulation, provided pl, #2 are sufficiently small. For the purpose of simulation, we use the following values for the system pa- 104 rameters, obtained from [77] : Km = 0.1349 N m/A, J = 4.1295 x 10“4 kgm”, B = 0.0013 Nm/rad/s, N, = 50 and TL = 0.2 kg. The resistance R and inductance L are assumed to be unknown, with nominal values of R = 20 Q and L = 35 mH re- spectively. Also, R 6 [R'", RM], and L E [Lm, LM], with L'” = 30 mH, LM = 40 mH, Rm = 19 Q, and RM = 21 Q. The actual values of the resistance and inductance are taken as R = 19 Q and L = 40 mH. The current reference is taken as [w(t) = 0A [54]. The values of the controller parameters are taken as k}, = 20, kg = 100, It? = 7.5 x 104, kg = 550, p1 = 0.1, and p2 = 50. Initial values for all the states are taken as zero. The switching terms 54') are taken as 510) = [(RM — Rllmll + (LM — iller$2$3| + (Ill/1:: (11 = 2°1kllH1LM p2(-) = [(RM — R)k3|:c,| + (LM — fol/shag + kgég — k4(k32:2 — 1:423 — do) —k3k5$1$3[ + QQ]/f/, (12 = 2.1kgfl2LM The desired angular position is taken as ¢d(t) = 0.03142 [u(t) + u(t — 0.5)], where u(t) is the unit step function. The results of the simulation are shown in Fig 5.6. We focus our attention on the error 62, since it corresponds to the variable (I) of physical interest and also because it is the harder of the two outputs to control. However, our observations regarding 82 also hold for 61. From the figure, it is clear that good tracking performance with very little overshoot is achieved, independent of the magnitude of ¢d(t). 5.3.3 Integral Control - Output Feedback Suppose that the angular velocity w of the motor shaft is unavailable for feedback. It is easy to verify that for the conditions of Assumption 2.1 to be satisfied, i.e., for the system to have a uniform vector relative degree and be transformable to normal form, none of the positive constants k,- and do need to be exactly known. Accordingly, we will assume that in the present case, in addition to the resistance R and inductance 105 Error in angular position ¢(t) with state feedback CSMC 0.005 7 7 0.005 0 0 ‘0'005 -0.005 mm “0.01 m“ (:3 0015 § -001 B " 5 c c -0.015 g -0.02 .g "7’ "7’ -o.02 a? -0.025 a? _003 -0.025 -0.035~ -0.00 -0.04 * i -0.035 ‘ i i 0 0.1 0.2 0.3 0.5 0.6 0.7 Time (sec) Time (sec) Figure 5.6: Tracking error performance under state-feedback integral control. L, the parameters K m, B, J and the load torque TL are all unknown, and take the vector 6 of unknown parameters as 0 = [[01, kg, 1:3, 10,, k6, do]T. Since 0) is unavailable for feedback, so is ég = w — dd. Furthermore, even if 02 were available for feedback, since 6'2 = koiq —k4w—do — $3, and kg, 104 and do are unknown, {2'2 would be unavailable for feedback. Therefore, we estimate éz and ég in (5.8) using the high-gain observer (HGO) 21 = 22+a1(eo—zl)/e 2'2 = 23 + 02(62 — zl)/e2 (5.12) is = a3(eg — 21)/e3 where the positive constants a1, 02, and also are chosen such that the polynomial A3 + m)? + a2). + a3 is Hurwitz. We replace ég, E2, and also so in (5.9) by their estimates 22, 23, and 62 = [0302 + kfez + 10322 + 23 respectively. Finally, motivated in part by the goal of simplifying the design, and in part by the need to work with 106 saturated controls, we (i) make use of the flexibility in choosing If}, and let F,- = 0, and (ii) choose (if) as a constant M,- (say), which is equal to the maximum physically allowable value for the control component |u,-|. As we previously pointed out, the matrix corresponding to A() in Assumption 2.5 is diagonal, so that A can be taken as the identity. With this choice, the final expression for the control (2.18), (2.19) then becomes u, = —M1 sat(sl/,u1) (5.13) “2 = —M2 sat(§2/u2) which can be considered the MIMO version of the universal integral control design of Section 2.6. For the purpose of simulation, we let Km, J, B, N,, TL, 103, kg, kf, kg, #1, [J2 and [dd retain their values from the previous simulation. Also we take R = 19.5 0, L = 30 mH, and (pd = 0.03142 rods. The HGO gains are taken as 011 = 17, 02 = 80 and 013 = 100, and the saturation levels for the controls as M1 = M2 = 50. We compare the performance of the output feedback controller with the partial state feedback design u,- = -M,- sat(s,- / 11,-), which makes use of measurements of 61, co, ég and éo. This could, for instance, be the case when the full state a: is available for feedback and the parameters k3, k4 and do are known. Fig 5.7(a) shows the results of the simulation for e = 10‘“, and we see that good tracking performance is achieved by the output feedback controller, which uses minimal information about the system. Fig 5.7(b) shows the effect of e on the performance recovery of the state feedback design, and it is clear from the figure that the error 62 under output feedback approaches the error 62 under state feedback as 6 tends to zero. Lastly, to show the merit of the proposed scheme versus ideal SMC, consider a sampled-data implementation of the above controller, i.e., we assume that the inputs to the controller are sampled and held signals, with a zero-order hold, and likewise for the controller outputs. We redo the previous simulation (with the state feed- 107 Position error e2 0-01 t t f T 0-04 T t t I 0.005, — State feedback _4 0035”,” — — Output feedback, 8:10 0 : : : : 0.03 -0.005 0.025 e -0.01 08 0.02 '9 -0.015 8 0.015»- d) -0.02 » - ~ 0.01 -0.025 -- - - 0.005 -0.03 0 -0.035 * i i ‘ -0.005 ‘ ‘ '5 i 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Time (sec) Time (sec) Fig 5.7(b) : Performance recovery of slb Fig 5.7(a) : MIMO universal integral controller as 8 tends to zero Figure 5.7: Tracking error performance under the output-feedback MIMO universal integral control. back CSMC) and compare the results versus the ideal sliding mode control 11.,- = —M,- sgn(si). The sampling period is assumed to be T = 0.1ms. The results are shown in Fig 5.8, and we see that asymptotic regulation is lost with the ideal SMC, and there is considerable chattering in the control 12,. Asymptotic regulation is re- tained with the CSMC with conditional integrator, and there is no chattering in the control. 5.3.4 Servocompensator Design The integral control designs of the previous subsections were done with the goal of point-to-point motion of the PMSM. In many positioning applications, the desired trajectory for the position is a sinusoid [20]. Specifically, suppose that the desired 108 Effect of sample-and-hold element on the performance of the controller x10 60 0W5”: ........ ......... IIIIIIIIII'IIWIIIIIIII“II'|I'!I‘1I‘II|I"II1I_I'I'I 05 I l I I i'iiihiifk'ik'ifkl""1 ' "illil '. ........................................... IIIIIII] H i] I--, .. z 2 . : . 1.1." !.1:.!.,!1_!.1:_l_!. h .‘m ' ,‘i in} 0’ _1 .. — CSMC wrth conditional integrator . > 20“ ll “if I- I IIII'II. I !!I:i_ |:il"1'”! l 5 — - ldealSMC 5 "! " 1'1" '! i"-Ii' '! "'I' !" 'l' " '- i *- ' * s o. i'Ii.Ii!"i."i-." ' ' 'i-lli. a -1.5*'-----~; ................................... C ."-. '~-i"-|"-3|!‘Il|'ll'll'll ""1 C : OIIIIIIIIIIIIIIIIIII .Ii Ii.“ III,“ II III"! .9 _21 .......................................... 9 l'i 11 I II: I ii I. iI:I. 'i" I! Ii“ 1! IIIII I! l !I l1 1 r: ... -,'1-.:'I-I1 I ll [1 |--.' m c I1..|.i..1..|.|i.1,i _ I .......................................... , o | ||.|l..|.|.. III. -3+ .......................................... .. II..;I! -, II -_II 1.] III ll I l- -. l I!ll.|l:.'i.'li;.'! ' 'III. 35. ........................................... 1' " Il ": 1! I I! [2|] 1! l' 1' l' 1' "[3 "1'5," I " .4014]! 1Il...I.II.I.I|_..I.,I.I.II .-I I-I: I 'I III-1 : : : : II-III.II:|.II-I,I~'I.I-| 'I-II IIIilIIIIII -4L-—-’~—— ........ I ........ l 'l-.]':.]'.J-|.]'IUIJI'l?'lt-1 . . - s—~ -— 4— .. a : ' _4I5 i L i i '60 i i i 0.09 0.092 0.094 0.096 0.098 0.1 0.09 0.092 0.094 0.096 0.098 0.1 Time (sec) Time (see) Figure 5.8: Effect of sampled-time implementation on ideal SMC and CSMC with conditional integrator. trajectory asymptotically converges to ¢d(t) = To sin(wot). 2 We show that the servocompensator design of Chapter 4 can be applied to this case. To that end, our first goal is to identify a suitable linear internal model that generates the steady-state values of the control inputs 2),; and 2),. As before, the desired reference for the current id is a constant Ida. It can be verified that with steady-state values $1,,(t) = Ida and 2:4,,(t) = To sin(wot) respectively for 2:1 and $4, the steady-state values of $2 and x3 are given by $2,, = [do + k4rowo cos(wot) — rowg sin(wot)]/ k3 and $3,, = rowo cos(wot) respectively, and the steady state values of the control inputs '1), and v, are given by 111,, = ’71 + '72 cos(wot) + 73 sin(2wot) + '74 cos(2wot) (5 14) u2,, = ’75 + 76 sin(wot) + 77 cos(wot) 2More general reference trajectories can be considered, as long as they satisfy the conditions of Assumption 4.3. 109 for some constants '71 to '77. The steady state value of the control 211,, satisfies (4.6) of Assumption 4.5 with q = 5, c0 = 0, cl = —4w3, c2 = 0, c3 = —5w3, and c4 = 0, while u2,, does so with q = 3, co = 0, cl = flag and C2 = 0. Let 1 0 O O 1 0 CDC 0 I—L 0 SI 0001 0—w30 '0 0 0 0 0 1 0 _0 —4w3 0 —5w3 0. be the internal model matrices corresponding to um, and u2,,, and J1=[0 0 0 011T,J2=[0 0 1]T be the corresponding J matrices, as defined in Section 4.3. We take 01 and 02 as outputs of the conditional servcompensators a, = (S.- — J,Kg)a, + u,J,-sat(§,-/p,-) (5.15) where 31 = K601 + 81) el (ii-{91 _ Lid: (5 16) 32 = K302 + kfez + 193532 + 352, 62 4;! 312 — ¢d The matrices K6 are chosen such that S,- — JIKI‘I are Hurwitz, the scalars k? and kg are chosen such that the polynomial :52 + 103:1: + kf is Hurwitz, 3‘1 = 31, g, = K302 + kfeg + kgzg + .23, where 22 and :53 are estimates of é2 and (9'2 respectively, provided by the high-gain observer (5.12). The control is taken as in (5.13), i.e., u,- = —M,- sat(§,~/uI) (5.17) 110 This completes the design of the controller. For the purposes of simulation, all values are retained from the one in the previous subsection, except for the reference mm and the matrices K (I. The former is chosen as ¢d(t) = r0 cos(wot) 3, for which two sets of values of (r0,w0) are used, (r0,w0) = (7r/2, 2) and (ro,wo) = (7r/10, 5). The latter are chosen to place the eigenvalues of SI — JIKé and 5'2 — JgKg at {—1,—2:l:j, —3:l:j} and {—1, —2 :l: j} respectively. We compare the performance against a CSMC design that does not include a servocompensator, i.e., $1 = 61, 3'2 = kfeg +k§22 +23, where 22 and .23 are as defined above, and u,- = —M,- sat(.§,-/p,-). The results of the simulation are shown in Fig 5.9. As before, we see that good tracking performance is achieved by the output feedback MIMO servocompensator design, which uses minimal information about the system. The transient performance of the CSMC without servocompensator is close to the one with a servocompensator (indistinguishable in the figure), but the steady-state error is non-zero. Before we present our conclusions, we make the following observation. Through- out, we have assumed that the load torque TL is constant. It is not hard to see that this assumption can be relaxed. For the integral control design of sections 5.3.2 and 5.3.3, it can be verified that all that is required is that the load torque be asymptot- ically constant. Since in this case, the angular position ¢(t) and the angular velocity w(t) asymptotically approach a. constant and zero respectively, we can allow position or velocity dependent load torques, i.e., TL = f,(¢(t), w(t)), where f.,(-) is a suficiently smooth function of its arguments. For the servocompensator design of this section, the situation is slightly more restrictive on account of Assumption 4.5. It can be verified that in this case, for Assumption 4.5 to be satisfied, f,(-) will have to be a 3This is conceptually not different from the (MU) = r0 sin(wot) that we designed for, just phase- shifted from it by 1r/ 2, and was chosen this way to have a non-zero initial error 62 (0). 111 Position error 92 Position error e2 Output feedback MIMO servocompensator = r ,(0 =1t/2,2 (r0100) (TL/2,2) “04 (0 o) ( ) 0.2 7 V ' 1 v I 1 0h 4. ; L ........................................... 09°. _1 "1 5 — With conditional servocompensator ......................................... at) — — Without sevrvocompensatorfi c _2 ......... ........ '........: ................ .4 ............................................. _g 5 .g I ......................................... .I 0- _3I..I _4».l_..’.”’-r‘\:~L.-r”’: . . . 1 1 _5 1 ; 4 1' 0 0.1 0.2 0.3 0.4 0.5 5 6 7 8 9 Time (sec) Time (sec) (r0,mo)=(rr/10,5) x 10“ (ro,0)0)=(1r/10,5) 0.05 I 1 I I r QN _I........' ........ '........ ........ I ....... .1 ............................................. 5 _ With conditional serv nsator at) - - Without servocompensator ........................................ I C _2 .9 E ........................................... 'ICTI - a _3 ......... .................................. _4 r. r __.; ..’. _‘.\..j..../.. it} .’. ..... 5. x. -035 . i i A _5 i i l i 0.1 0.2 0.3 0.4 0.5 5 6 7 8 9 Time (sec) Time (sec) Figure 5.9: Sinusoidal reference tracking using MIMO servocompensators. 112 polynomial function of its inputs, and that its form must be known. For example, if fT(¢,w) = 2 new iEI,jEJ where I, J C Z20, the sets I and J will have to be known, but not aij. When the polynomial condition is violated, for example, when the load torque is of the form TL = N sin(qS) say, then, as mentioned in Chapter 4, polynomial approximations may be used to achieve practical regulation of the error. 5.4 Conclusions In this chapter, we considered the application of the controller designs presented in Chapters 2 through 4 to two applications : temperature control of a continuous stirred tank reactor (CSTR) and position control of a permanent magnet stepper motor (PMSM). For the CSTR, we considered the problem of constant references and distur- bances, to which the design of Section 2.6 could be applied. Since the class of CSTRs considered in this chapter is relative degree one, the resulting controller is a PI con- troller with anti-windup. Such a structure was also considered in [1, 2]. The design was initiated in [2], based on modelling error estimation ideas, and with the goal of handling uncertain parameters, and the resulting structure was shown to be a PI con- troller with anti-windup. Furthermore, it was shown in [1] that the controller could recover the performance of a saturated inverse feedback control. The corresponding result in our case is the recovery of the performance an ideal sliding mode control. Even though a direct analytic comparison of the two controllers is not possible, it was observed via simulations that the results in [2] and our work are comparable. For the PMSM, we considered both constant as well as sinusoidal references. In the constant references case, we looked at both the state feedback design of Chapter 3 113 for the global regulation problem, and the output feedback design of Chapter 2 for the regional problem. The output feedback controller was designed as the MIMO version of the universal integral regulator of Section 2.6. Good tracking performance was achieved with both designs, in spite of partial knowledge of the machine parameters. The same was also seen to be true with sinusoidal references, in which case the controller is the MIMO version of the universal servocompensator of Chapter 4. While we specifically considered PMSMS in this chapter, our results can be extended to other types of motors, such as permanent magnet synchronous and dc motors. 114 Chapter 6 Conclusions In this thesis, we have addressed the problem of robust output regulation by out- put feedback for input-output linearizable minimum-phase nonlinear systems, with emphasis on the transient performance. As mentioned in Chapter 1, the issue of transient performance needs to be addressed because conventional approaches to de- signing servocompensators often result in poor transient performance. To that end, we have presented a new approach to introducing servo action that results in improved transient performance over the conventional approach. Analytical results have been provided for regional and semi-global regulation and also for the performance recovery of a state feedback ideal SMC design. A summary of results by chapter is provided below. 6.1 Summary of Results In Chapter 2, we considered the output regulation problem for the special case of (asymptotically) constant exogenous signals. For this case, we designed a continuous sliding mode controller with a conditional integrator that achieves asymptotic error regulation with good transient performance. The conditional integrator is so designed that it provides integral action only inside the boundary layer. Analytical results were 115 given for regional and semiglobal regulation, and we proved that the output feedback controller with conditional integrators recovers the performance of a state feedback ideal SMC that does not include integral action. Advantages of the proposed method over the conventional approach as regards the transient performance and over ideal SMC as regards the problem of chattering were shown by simulation. I In Chapter 3, we considered the extension of the semi-global regulation result of Chapter 2 to a global one under full state feedback. In Chapter 4, we considered an extension of the design of Chapter 2 for the more general output regulation problem. To that end, we designed a conditional servocompensator that provides servocompensation only inside the boundary layer. As before, regional and semiglobal results for regulation were given and also for performance recovery of a state feedback ideal SMC design. We also studied the effect of internal model perturbations on the tracking error and showed that, in the presence of such perturbation, the tracking error is ultimately bounded by a bound that depends on the magnitude of the perturbation. In Chapter 5, we applied the designs of Chapters 2 through 4 to two application examples, temperature control of a continuous stirred tank reactor (CSTR) and po- sition control of a permanent magnet stepper motor (PMSM). The controller for the CSTR problem was based on the design in Section 2.6, and for the particular class of CSTR systems that we considered was simply a PI controller with anti-windup. For the PMSM, we considered simulations involving all of the designs in Chapters 2 to 4. Whereas the results of Section 2.6 and Chapter 4 were presented for the single-input single-output case, their application to the PMSM shows how such results can be extended to a multi-input multi-output problem. For both the CSTR as well as the PMSM example, the simulation results show that the controller designs of this thesis result in good performance. 116 6.2 Future Work The conditional servocompensator design of this thesis is based on the idea of providing servocompensation only conditionally, inside the boundary layer of a slid- ing mode design. However, it is clear that any controller that has the structure of a saturated high-gain feedback is a promising candidate for the application of the con- ditional servocompensator design. Identifying previous works that use servocompen- sators within such a controller structure and applying the conditional servocompen- sator idea to see if the performance of such designs can be improved is an interesting line of future work. A second promising direction in which research can be pursued is understanding whether the servocompensator design can be modified to be used in conjunction with other controller designs that extend the class of systems or relax the assumptions made in this thesis. Examples of two such designs are the ones by Serrani et al (i) for the output regulation of nonminimum phase nonlinear systems [68] and (ii) using adaptive internal models when the frequencies of the exosystem are unknown [69]. It will also be useful to evaluate other techniques for the output regulation of systems with constraints on the control, and see how they compare to the results of this thesis. One such technique [1] was mentioned when we discussed the CSTR example in Chapter 5. Related to [1] is the work of Kapoor and Daoutidis [42], which extends the results of [1] along two directions : (i) it allows the system relative degree to be greater than one, and (ii) deals with the general output regulation problem, not just the case of constant exogenous signals. Another design for the output regulation of constrained linear systems, but with the goal of semi-global regulation when the control level is fixed apriori, can be found in Lin et al [56]. Finally, further work needs to be done on understanding how to tune the con- troller parameters. For example, it is not hard to see that when the level of the control is fixed apriori, there is a trade-off between the region of attraction and the speed 117 of convergence, which is dictated by the choice of the sliding surface parameters. Identifying such trade-offs will offer insight into how the parameters can be tuned to achieve specific objectives, and also identify possible limitations on the achievable performance. A partial account of such details can be found in the Ph.D. thesis of Grognard [29]. 118 Bibliography [1] J. Alvarez-Ramirez and A. Morales. PI control of continuously stirred tank reactors : Stability and performance. Chem. Engg. Science, 55:5497—5507, 2000. [2] J. Alvarez-Ramirez, A. Morales, and I. Cervantes. Robust proportional-integral control. Ind. Eng. Chem. Res, 37:4740—4747, 1998. [3] R. Antonelli and A.Astolfi. Continuous stirred tank reactors: easy to stabilise? Automatica (submitted), 2002. [4] K.J. Astrom and L. Rundqwist. Integrator windup and how to avoid it. In Proceedings of the American Control Conference, pages 1693—1698, 1989. [5] AN. Atassi and H.K. Khalil. A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Automat. Contr., 44(9):1672—1687, 1999. [6] I-C. Baik, K-H. Kim, and M-J Youn. Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique. IEEE Trans. Contr. Sys. Tech, 8(1):47—54, 2000. [7] BR. Barmish, M. Corless, and G. Leitman. A new class of stabilizing controllers for uncertain dynamical systems. SIAM Journal of Control, 21(2):246—265, 1983. [8] G. Bartolini, A. Ferrara, E. Usai, and V.I. Utkin. On multi-input chattering free second order sliding mode control. IEEE Trans. Automat. Contr., 45(9):1711— 1717, 2000. [9] M. Bodson and J .N. Chiasson. Application of nonlinear control methods to the positioning of a permanent magnet stepper motors. In Proceedings of the IEEE Conference on Decision and Control, pages 531-532, 1989. [10] M. Bodson, J.N. Chiasson, R.T. Novotnak, and RB. Rekowski. High perfor- mance nonlinear feedback control of a permanent magnet stepper motor. IEEE Trans. Contr. Sys. Tech, 1(1):5—14, 1993. [11] TC. Burg, J. Hu, D.M. Dawson, and P. Vedagarbha. A global exponential position tracking controller for a permanent magnet stepper motor via output feedback. In Proceedings of the 3rd IEEE Conference on Control Applications, pages 213—218, 1994. 119 [12] CI. Byrnes and A. Isidori. Output regulation for nonlinear systems: an overview. Int. J. Robust and Nonlinear Contr., 10:323—337, 2000. [13] CI. Byrnes, D.F. Priscoli, and A.Isidori. Output Regulation of Uncertain Non- linear Systems. Birkhauser, Boston, 1997. [14] CI. Byrnes, D.F. Priscoli, A.Isidori, and W. Kang. Structurally stable output regulation of nonlinear systems. Automatica, 33:369—385, 1997. [15] L.W. Chang. Dynamics of a sliding control with a first-order plus integral sliding condition. Dynamics and Control, 2(2):201—-219, 1992. [16] Z. Chen and J. Huang. Nonlinear internal model and the robust output regulation problem. In Proceedings of the American Control Conference, pages 1560—1565, 2003. [17] J .N. Chiasson and RT. Novotnak. Nonlinear speed observer for the PM stepper motor. IEEE Trans. Automat. Contr., 38:1584—1588, 1993. [18] M. Corless and G. Leitman. Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Trans. Automat. Contr., 26(5):1139—1144, 1981. [19] E.J. Davison. The robust control of a servomechanism problem for linear time- invariant multivariable systems. IEEE Trans. Automat. Contr., 21:25—34, 1976. [20] D.M. Dawson, J. Hu, and TC. Burg. Nonlinear Control of Electric Machinery. Control Engineering Series, Marcel Dekker, Inc., New York, 1998. [21] RA. DeCarlo, S.H. Zak, and GP. Matthews. Variable structure control of non- linear multivariable systems : A tutorial. Proceedings of the IEEE, 76:212—232, 1988. [22] C. Desoer and C—A. Lin. Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans. Automat. Contr., 30(9):861—867, 1985. [23] H. Elrnali and N. Olgac. Robust output tracking control of nonlinear MIMO systems via sliding mode technique. Automatica, 28(1):145—151, 1992. [24] F. Esfandiari and H.K. Khalil. Output feedback stabilization of fully linearizable systems. Int. J. Contr., 56:1007—1037, 1992. [25] HA. Fertik and CW. Ross. Direct digital control algorithm with anti-windup feature. ISA Transactions, 6(4):63-68, 1967. [26] A. Hadkov, R. Ortega, and G. Bastin. Semiadaptive control of convexity parametrized systems with application to temperature regulation of chemical reactors. In Proceedings of the IFAC-ADCHEM, 1997. 120 [27] BA. Francis and W.M. Wonham. The internal model principle of control theory. Automatica, 12:457—465, 1976. [28] RA. Fmeman and RV. Kokotovic. Optimal nonlinear controllers for feedback linearizable systems. In Proceedings of the American Control Conference, pages 2722—2726, 1995. [29] F. Grognard. Control of Constrained Systems: Closed-loop, Open-loop, and Hy- brid Solutions. PhD. Thesis, Universit catholique de Louvain, Louvain-la—Neuve, Belgium, 2001. [30] M. Hautus. Linear matrix equations with applications to the regulator problem. Outils and Modeles Mathematique pour l’Automatique, pages 399—412, 1983. [31] J .S.A. Hepburn and W.M. Wonham. Error feedback and internal model on differentiable manifolds. IEEE Trans. Automat. Contr., 29:397—403, 1984. [32] J. Huang. Asymptotic tracking and disturbance rejection in uncertain nonlinear systems. IEEE Trans. Automat. Contr., 40:1118—1122, 1995. [33] J. Huang and Z. Chen. A general framework for tacking output regulation prob- lem. In Proceedings of the American Control Conference, pages 102—109, 2002. [34] J. Huang and CF. Lin. On a robust nonlinear multivariable servomechanism problem. IEEE Trans. Automat. Contr., 39:1510—1513, 1994. [35] J. Huang and W.J . Rugh. On a nonlinear multivariable servomechanism problem. Automatica, 26(6):963—972, 1990. [36] J. Huang and W.J . Rugh. Stabilization on zero-error manifolds and the nonlinear servomechanism problem. IEEE Trans. Automat. Contr., 37(7):1009—1013, 1992. [37] A. Isidori. Nonlinear Control Systems. Springer Verlag, New York, third edition, 1995. [38] A. Isidori. A remark on the problem of semiglobal nonlinear output regulation. IEEE Trans. Automat. Contr., 42(12):1734—1738, 1997. [39] A. Isidori and OJ. Byrnes. Output regulation of nonlinear systems. IEEE Trans. Automat. Contr., 35(2):131—140, 1990. [40] F. J adot. Dynamics and robust nonlinear PI control of stirred tank reactors. PhD. Thesis, CESAME, Universit Catholique de Louvain, Louvain-larNeuve, Belgium, 1996. [41] F. Jadot, J.F. Veil, and G. Bastin. Robust global stabilization of stirred tank reactors by saturated output feedback. European Journal of Control, 5:361—371, 1999. 121 [42] N. Kapoor and P. Daoutidis. An observer-based anti-windup scheme for non- linear systems with input constraints. Int. J. Contr., 72(1):18—29, 1999. [43] H.K. Khalil. Robust servomechanism output feedback controllers for feedback linearizable systems. Automatica, 30:1587—1599, 1994. [44] H.K. Khalil. High-gain observers in nonlinear feedback control. In New Directions in Nonlinear Observer Design, volume 244, pages 249—268. Lecture Notes in Control and Information Sciences, Springer, London, 1999. [45] H.K. Khalil. On the design of robust servomechanisms for minimum phase non- linear systems. Int. J. Robust and Nonlinear Contr., 10:339—361, 2000. [46] H.K. Khalil. Universal integral controllers for minimum phase nonlinear systems. IEEE Trans. Automat. Contr., 45(3):490—494, 2000. [47] H.K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, New Jersey, third edition, 2002. [48] F. Khorrami, P. Krishnamurthy, and H. Melkote. Modeling and Adaptive Non- linear Control of Electric Motors. Springer-Verlag, New York, 2003. [49] H. Knobloch, A.Isidori, and D. Flockerzi. Topics in Control Theory. DMV- Seminar Series, Birkhauser, Boston, 1993. [50] K.A. Kosanovich, M.J. Piovoso, V. Rokhlenko, and A. Guez. Nonlinear adaptive control with parameter estimation of a cstr. Journal Proc. Control, 5:133—148, 1995. [51] P. Krishnamurthy and F. Khorrami. Adaptive control of stepper motors without current measurements. In Proceedings of the American Control Conference, pages 1563—1568, 2001. [52] P. Krishnamurthy and F. Khorrami. Permanent magnet stepper motor control via position-only feedback. In Proceedings of the American Control Conference, pages 3180—3185, 2002. [53] P. Krishnamurthy and F. Khorrami. Robust adaptive voltage-fed permanent magnet stepper motor control without current measurements. IEEE Trans. Contr. Sys. Tech, 11:415—425, 2003. [54] S. Laghrouche, F. Plestan, A. Glumineau, and R. Boisliveau. Robust second order sliding mode control for a permanent magnet synchronous motor. In Pro- ceedings of the American Control Conference, pages 4071—4076, 2003. [55] W. Leonhard. Control of Electrical Drives. Springer, Berlin, 1985. [56] Z. Lin, R. Mantri, and A. Saberi. Semi-global output regulation for linear systems subject to input saturation - a low-and—high gain design. In Proceedings of the American Control Conference, pages 3214—3218, 1995. 122 [57] NA. Mahmoud and H.K. Khalil. Asymptotic regulation of minimum phase nonlinear systems using output feedback. IEEE Trans. Automat. Contr., 41(10):1402—1412, 1996. [58] NA. Mahmoud and H.K. Khalil. Robust control for a nonlinear servomechanism problem. Int. J. Contr., 66(6):779—802, 1997. [59] R. Marino, S.Peresada, and P. Tomei. Nonlinear adaptive control of permanent magnet stepper motors. Automatica, 31(11):1595—1604, 1995. [60] R. Marino and P. Tomei. Nonlinear Control Design : Geometric, Adaptive and Robust. Prentice Hall International (UK) Limited, 1995. [61] H. Melkote and F. Khorrami. Nonlinear output feedback control for stepper motors : a robust adaptive approach. In Proceedings of the IEEE Conference on Control Applications, pages 755—760, 1999. [62] N. Minorsky. Directional stability of automatically steered bodies. Journal Amer. Society of Naval Engg., 34:280—309, 1922. [63] DP. Priscoli. Robust tracking for polynomial plants. In Proceedings of the European Control Conference, pages 369—373, 1993. [64] Z. Qu. Robust Control of Nonlinear Uncertain Systems. Wiley Series in Nonlinear Science, Wiley, New York, 1998. [65] S. Sastry. Nonlinear Systems: Analysis, Stability, and Control. Springer-Verlag, New York, 1999. [66] L. Sciavicco and B. Siciliano. Modelling and Control of Robotic Manipulators. McGraw-Hill, New York, 1996. [67] A. Serrani and A. Isidori. Global robust output regulation for a class of nonlinear systems. Systems and Control Letters, 39:133—139, 2000. [68] A. Serrani, A. Isidori, and L. Marconi. Semiglobal robust output regulation of nonminimum—phase nonlinear systems. Int. J. Robust and Nonlinear Contr., 10:379—396, 2000. [69] A. Serrani, A. Isidori, and L. Marconi. Semiglobal nonlinear output regulation with adaptive internal model. IEEE Dans. Automat. Contr., 46:1178—1194, 2001. [70] S. Seshagiri and H.K. Khalil. Universal integral controllers with anti-reset windup for minimum phase nonlinear systems. In Proceedings of the IEEE Conference on Decision and Control, pages 4186—4191, 2001. [71] S. Seshagiri and H.K. Khalil. On introducing integral action in sliding mode control. In Proceedings of the IEEE Conference on Decision and Control, 2002. 123 [72] A. Teel and L. Praly. Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems and Control Letters, 22(5):313-325, 1994. [73] J .F. Veil and G. Bastin. Robust stabilization of chemical reactors. IEEE Trans. Automat. Contr., 42:473—481, 1997. [74] J.F. Veil, F.Jadot, and G. Bastin. Global stabilization of exothermic chemical reactors under input constraints. Automatica, 33(8):1437—1448, 1997. [75] K.D. Young, V.I. Utkin, and U. Ozguner. A control engineer’s guide to sliding mode control. IEEE Trans. Contr. Sys. Tech, 7(3):328—342, 1999. [76] M. Zribi and J .N. Chiasson. Position control of a PM stepper motor by exact linearization. IEEE Trans. Automat. Contr., 36:620-625, 1991. [77] M. Zribi, H. Sira-Ramirez, and A. Ngai. Static and dynamic sliding mode control schemes for a PMSM. Int. J. Contr., 74(2):103—117, 2001. 124 MICHIGAN STATE UNIVERSITY LIBR [[1]/2]] 11]] ill/illiiil] [iii] 3 1293 02504 9382