OUTPUT FEEDBACK CONTROL IN THE PRESENCE OF UNCERTAINTIES: USING EXTENDED HIGH-GAIN OBSERVERS WITH DYNAMIC INVERSION By Joonho Lee A Dissertation Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering – Doctor of Philosophy 2014 Abstract OUTPUT FEEDBACK CONTROL IN THE PRESENCE OF UNCERTAINTIES: USING EXTENDED HIGH-GAIN OBSERVERS WITH DYNAMIC INVERSION By Joonho Lee Control design for uncertain nonlinear systems is an important issue. Uncertainties always reside in nonlinear systems due to incomplete mathematical model description or intended approximation factors in system models, e.g. linearization for system models. Furthermore, unexpected external disturbances and unmeasured system states increase the uncertainties in the systems. In this dissertation, we consider an uncertain nonlinear systems that takes the form of a chain of integrators and introduce control design methodologies based on output feedback control: using extended high-gain observers and dynamic inversion. The proposed output feedback controller results in a closed-loop system with a threetime-scale structure; an extended high-gain observer estimates unmeasured states and uncertainties in the fastest time scale and dynamic inversion is used to deal with nonaffine control inputs or input uncertainties in the intermediate time scale whereas the plant dynamics evolves in the slowest time scale. The dynamic inversion algorithm, based on sector conditions, results in fast convergence into inputs under state feedback control. Together with the extended high-gain observer, dynamic inversion results in performance recovery of a target system. The time-scale-separation approach is well-suited to underactuated mechanical systems to overcome the lack of the number of inputs. Since the time separation is created between subsystems in plant dynamics, subsystem dynamics are viewed as virtual inputs for the other subsystems. In this dissertation, we apply the time-scale separation strategy to two examples of underactuated mechanical systems in the presence of uncertainties, the inverted pendulum on a cart and the autonomous helicopter. Copyright by Joonho Lee 2014 Acknowledgments I would like to express my deepest and sincere gratitude to my two advisors, Professor Hassan Khalil and Professor Ranjan Mukherjee for their invaluable advice, guidance, constant support and encouragement. Without their help and support, this dissertation would not have been possible. I am especially thankful to my principle advisor, Dr. Khalil, for his wisdom, endless patience, and boundless knowledge on control theory. I have learned from him not only scholarly knowledge but also wisdom how to live as a good researcher. I would like to thank the members of my Ph.D. committee, Professor Guoming Zhu and Professor Brian Feeny for their willingness to serve on the committee as well as for their assistance and valuable input. To my wife, S.M., my two children, Ryan Lee and Isla Lee, my father, and mother, from the bottom of my heart, I appreciate their endless support. Especially, to my lovely wife, I would like to appreciate her endless support and patience. About her support, I cannot put into words. At every moment, she was with me bringing joy, love, happiness, and hope. v Table of Contents LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Introduction . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . 1.2 Dynamic Inversion . . . . . . . . . 1.3 High-Gain Observers and Extended 1.3.1 High-Gain Observers . . . . . . . 1.3.2 Extended High-Gain Observers . . . . . . . . . . . . . . . . . . . High-Gain . . . . . . . . . . . . . . . . . . 1 2 4 5 7 8 Recovery in the Presence of Un. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10 12 16 19 26 27 28 32 Chapter 3 Output Feedback Stabilization of Inverted Pendulum on a Cart in the Presence of Uncertainties . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stabilization in the Absence of Uncertainties . . . . . . . . . . . . . . . . . . . . 3.2.1 Dynamics of an inverted pendulum on a cart . . . . . . . . . . . . . . . . . . . 3.2.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Closed-loop system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Analysis using singular perturbations . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stabilization in the Presence of Uncertainties . . . . . . . . . . . . . . . . . . . . 3.3.1 Dynamic inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Extended High-Gain Observers (EHGOs) . . . . . . . . . . . . . . . . . . . . 3.3.3 Output feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Stability analysis in the presence of uncertainties . . . . . . . . . . . . . . . . 3.4 Simulation and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 34 37 37 39 40 41 44 45 46 47 48 56 56 59 63 Chapter 4 Output Feedback Control for an Autonomous Helicopter in the Presence of Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 66 Chapter 2 Output Feedback Performance certainties . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . 2.2 Problem Statements . . . . . . . . . . . . 2.3 Control Design . . . . . . . . . . . . . . . 2.4 Main Result . . . . . . . . . . . . . . . . . 2.5 Simulations . . . . . . . . . . . . . . . . . 2.5.1 Example 1 . . . . . . . . . . . . . . . . . 2.5.2 Example 2 . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 Dynamics of a Helicopter . . . . . . . . . . . . . . . 4.2.1 Rotational Dynamics of a Helicopter . . . . . . . 4.2.2 Translational Dynamics of a Helicopter . . . . . . 4.3 Stability Analysis in the Absence of Disturbances . 4.3.1 State feedback control . . . . . . . . . . . . . . . 4.3.2 Stability analysis . . . . . . . . . . . . . . . . . . 4.4 Stability Analysis in the Presence of Disturbances . 4.4.1 Design of Extend High-Gain Observers (EHGOs) 4.4.2 Output feedback control . . . . . . . . . . . . . . 4.4.3 Stability analysis in the presence of disturbances 4.5 Simulation Results . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 69 71 73 74 75 78 84 85 86 87 95 99 Chapter 5 Conclusions and Future 5.1 Concluding Remarks . . . . . . . . 5.1.1 Main contributions . . . . . . . . 5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 107 108 109 Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Appendix A Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Appendix B Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 vii List of Tables Table 4.1 Parameters in a helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . viii 70 List of Figures Figure 2.1 Figure 2.2 The trajectory starting from (e0 , z0 , s0 ) ∈ Ωa and η0 ∈ / {Vη ≤ b3 (ε/µ)2} 2 converges into (e, z, s, η) ∈ Ωb × {Vη ≤ b3 (ε/µ) }. . . . . . . . . . . . . . 22 The solid and dashed lines represent trajectories xi for i = 1, 2 driven by the proposed controller in (2.53) and reference trajectories xri for i = 1, 2 in (2.52), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Input trajectories u1 and u2 in (2.53) are shown and both u1 and u2 cross the values u1 = 2, 3 and u2 = 2, 3 which make the Jacobian matrix (∂f /∂u) singular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 The solid and dotted lines represent the system states x1 and x2 , and estimates xˆ1 and xˆ2 , respectively. . . . . . . . . . . . . . . . . . . . . . 31 The solid and dotted lines represent the system states x˙ 1 and x˙ 2 , and estimates σ ˆ1 and σ ˆ2 , respectively. . . . . . . . . . . . . . . . . . . . . . 32 −1 The solid, dashed, and dotted lines represent the inputs ur = Rm x, u with µ = 0.02 and ε/µ = 0.01, and u with µ = 0.2 and ε/µ = 0.01, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Figure 3.1 Inverted pendulum on a cart . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 3.2 The two-time-scale structure of the inverted pendulum on a cart system 44 Figure 3.3 Multi-time-scale structure for the closed-loop system . . . . . . . . . . 48 Figure 3.4 Trajectories of x1 and x2 for state feedback (solid line), output feedback with (ε2 , ε3 ) = (0.02, 0.002) (dotted line), and output feedback with (ε2 , ε3 ) = (0.01, 0.0001) (dashed line) . . . . . . . . . . . . . . . . . . . 59 Trajectories of α1 and α2 for state feedback (solid line), output feedback with (ε2 , ε3 ) = (0.02, 0.002) (dotted line), and output feedback with (ε2 , ε3 ) = (0.01, 0.0001) (dashed line). . . . . . . . . . . . . . . . . . . . 60 Trajectories of x1 and α1 (solid lines), and x2 and α2 (dashed lines) are shown in the top and middle subfigures. In the bottom subfigure, a trajectory of the input F in (3.26) is shown. . . . . . . . . . . . . . . . 61 Experimental testbed for the inverted pendulum on a cart - a product of Quanser [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Trajectories of x1 and its estimate xˆ1 , and α1 and its estimate α ˆ 1 are shown. The estimated values are indistinguishable from their true (measured) values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 ix Figure 3.9 Trajectories of x1 and α1 are shown with different control schemes. Solid lines driven by our control scheme, converge into the origin. Dotted lines generated by a LQR controller, have ultimate boundedness. Dash-dot lines provided by the control algorithm in [63] have the biggest ultimate boundedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Trajectories of inputs F in (3.26) (the top subfigure), LQR (the middle subfigure), and [63] (the bottom subfigure) are shown. . . . . . . . . . 65 Figure 4.1 Side and top view of the helicopter . . . . . . . . . . . . . . . . . . . . 71 Figure 4.2 The block diagram is shown for rotational dynamics (inner-loop) and translational dynamics (outer-loop) control structure via two time-scale separation. The blocks PR , PT are rotational dynamics and translational dynamics, respectively. The blocks CR , CT are controllers for rotational and translational dynamics, respectively. χ is the position of the helicopter and χr and Θr are reference trajectories for the translational and rotational dynamics, respectively. . . . . . . . . . . . . . . . . . . . . . 72 Trajectories x1 , y1 , and z1 (solid-lines) under the state feedback in the presence of disturbances and reference states xr , yr , and zr (dished-lines) 97 Trajectories of x1 , y1 , and z1 (solid-lines) under the output feedback in (4.53) in the presence of disturbances, and references xr , yr , and zr (dashed-lines) for rx (t) = 5 sin t, ry (t) = 5 cos t, and rz (t) = 5 sin t in (4.14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Trajectories φd , θd , and ψr (dashed-lines) are references for the states φ1 , θ1 , ψ1 of the rotational dynamics in the presence of disturbances. . . . 99 Figure 3.10 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Trajectories x1 , y1 and z1 (solid-lines) and the estimates xˆ1 , yˆ1 , and zˆ1 (dashed-lines) by the EHGO . . . . . . . . . . . . . . . . . . . . . . . . 100 Figure 4.7 Trajectories x2 , y2 and z2 (solid-lines) and the estimates xˆ2 , yˆ2 , and zˆ2 (dashed-lines) by the EHGO . . . . . . . . . . . . . . . . . . . . . . . . 101 Figure 4.8 Trajectories φ1 , θ1 and ψ1 (solid-lines) and the estimates φˆ1 , θˆ1 , and ψˆ1 (dashed-lines) by the EHGO . . . . . . . . . . . . . . . . . . . . . . . . 102 Figure 4.9 Trajectories φ2 , θ2 and ψ2 (solid-lines) and the estimates φˆ2 , θˆ2 , and ψˆ2 (dashed-lines) by the EHGO . . . . . . . . . . . . . . . . . . . . . . . . 103 Figure 4.10 Plots for sum of the actual terms of acceleration (without approximation in model, Fp in (4.7)) and external disturbances σΘ : dashed-lines and plots for sum of nominal terms of acceleration (i.e., FO in (4.11)) the estimate of external disturbances σχ by the EHGO: solid-lines . . . . . 104 Figure 4.11 Plots for the external disturbances σΘ (solid-lines) and plots for the estimates σ ˆΘ (dashed-lines) . . . . . . . . . . . . . . . . . . . . . . . . . 105 x Figure 4.12 Plots for the helicopter actual control inputs, TM , TT , a1s , and b1s under the output feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 xi Chapter 1 Introduction Control design of uncertain nonlinear systems has been a challenging problem. Mathematical models cannot capture entire features of system dynamics or unexpected external disturbances. Even if system dynamics were precise, nonaffine control inputs add difficulties to the design. Furthermore, systems states are not always measurable. All these factors make control design a challenging task. In this dissertation, we consider a class of nonlinear systems in the presence of uncertainties, which takes the form of a chain of integrators, e.g., a single-input-single-output nonlinear system, x˙ i = xi+1 , for i = 1, . . . , n − 1 x˙ n = f (x, z, u) + δ(x, u, t) (1.1) z˙ = f0 (x, z) y = x1 where x = [x1 , . . . , xn ]T ∈ Rn and z ∈ Rm are the system states, u ∈ R is the control input, δ(x, u, t) ∈ R is the uncertainty, and y ∈ R is the measured output. The chain of integrators is commonly used to describe dynamics of mechanical systems. If f (x, z, u) in (1.1) has affine control, i.e., f (x, z, u) = fn (x, z) + g(x)u, the system of (1.1) become a standard 1 normal form [30], [33]. This dissertation provide a way to deal with a class of uncertain nonlinear systems using extended high-gain observers and dynamic inversion. The extended high-gain observer is used to estimate both unmeasured states and uncertainties and the dynamic inversion deals with the nonaffine control or input uncertainties. In other words, using the extended high-gain observers and the dynamic inversion, a control design problem for uncertain nonlinear systems with the nonaffine control input, is viewed as the control design for the standard normal form in the absence of uncertainties. The dissertation is organized as follows. In Chapter 1, motivation examples, the dynamic inversion, high-gain observers, and the extended high-gain observers are introduced. Chapter 2 presents the performance recovery in the presence of uncertainties using the extended high-gain observer and dynamic inversion. In Chapter 3, the stabilization problem is introduced for the inverted pendulum on a cart in the presence of uncertainties. In Chapter 4, the output feedback control is designed for an autonomous helicopter in the presence of external disturbances. Lastly, conclusions and future works are provided in Chapter 5. 1.1 Motivation First, we consider the example of translational dynamics for x-direction in a helicopter with bounded uncertainty δ(t), i.e., an uncertain nonlinear system with nonaffine x˙ 1 = x2 1 x˙ 2 = − (cos φ1 sin θ1 cos ψ1 + sin φ1 sin ψ1 )TM + δ(t) m (1.2) where x1 and x2 are a position and velocity respectively, φ1 is viewed as an input given appropriate values of θ1 , ψ1 , and TM . One approach to deal with the nonaffine control in (1.2), is to introduce an additional integrator into the state space equation for the control 2 variable φ1 , i.e., x3 = φ1 , and x˙ 3 = ν = φ˙ 1 . We obtain the new system equation x˙ 1 = x2 x˙ 2 = − 1 (cos x3 sin θ1 cos ψ1 + sin x3 sin ψ1 )TM + δ(t) m (1.3) x˙ 3 = ν However, this approach changes the matched uncertainty in (1.2) into the unmatched uncertainty δ(x1 , t) in (1.3). The matching condition plays an important role in robust nonlinear control design as in Sections 14.1 and 14.2 of [33]. The uncertainty is matched when it enters the system equation at the same point as the control input. As the second motivation, we will consider uncertainties in the system. Consider the example of the inverted pendulum on a cart, x˙ 1 = x2 x˙ 2 = δ1 (α1 ) − δ2 (α1 )u (1.4) α˙ 1 = α2 α˙ 2 = u y = [x1 , α1 ]T where x = [x1 , x2 ]T ∈ R2 and α = [α1 , α2 ]T ∈ R2 are the system states; y ∈ R2 is the measured output; the domain of α1 is −π/2 < α1 < π/2; δ1 and δ2 are uncertainties; and we assume that δ2 > 0 and the sign of δ2 is known. The control objective is to stabilize the system at x = 0 and α = 0. In this dissertation, using the extended high-gain observer and the dynamic inversion, we will deal with the control design for the two cases: one is for control design for stabilization of the inverted pendulum on a cart at the upper equilibrium in Chapter 3 and the other is for control design for an autonomous helicopter with nonaffine control input in Chapter 4. 3 1.2 Dynamic Inversion In this section, we introduce a control algorithm to find approximate solutions for nonlinear maps, which is called dynamic inversion. In [52, 53], observers were used to invert nonlinear maps. In [25], two dynamic inversion algorithms were introduced: one is based on a Newton method and the other uses a gradient-decent algorithm. In [28, 29], using a twotime-scale approach together with a gradient decent algorithm, a dynamic inversion scheme was generated. In [64], a second-order sliding mode controller with saturation for a singleinput-single-output nonaffine systems was used to deal with uncertainties and nonaffine input forms. In [24], neural-networks and the mean value theorem were used to produce a dynamic inversion algorithm. Hovakimyan et al [27] also used neural-networks with a two-time-scale approach to deal with a single-input-single-output uncertain nonaffine system. To briefly explain the principle of dynamic inversion, the example in (1.2) is reconsidered. Now, it is assumed that δ(x, t) in (1.2) is known. For the stabilization of the system in (1.2), a controller form of the dynamic inversion is given by εφ˙ 1 = −K(x, φ1 ) − 1 (cos φ1 sin θ1 cos ψ1 + sin φ1 sin ψ1 )TM − uc m (1.5) where uc ∈ R is a reference input (e.g. uc = −kp x1 − kv x2 + rx (t) with kp , kv > 0), rx (t) is a reference trajectory, K(x, φ1 ) ∈ R is satisfies K(x, φ1 ) ≥ k0 > 0 with the positive constant k0 , over the domain of interest. With a sufficiently small positive ε ≃ 0 in (1.5), we obtain the quasi steady-state equation −K(x, φ1 ) − 1 (cos φ1 sin θ1 cos ψ1 + sin φ1 sin ψ1 )TM − uc m =0 (1.6) Using the assumption that K(x, φ1 ) ≥ k0 > 0, we have − 1 (cos φ1 sin θ1 cos ψ1 + sin φ1 sin ψ1 )TM = uc = −kp x1 − kv x2 + rx (t) m 4 (1.7) Since in view of a multi-time-scale approach, the system of (1.5) is fast and the system of (1.2) is slow, the fast system of (1.5) reaches the quasi-steady state while the variable x in 1 the slow system is almost frozen. This means that − (cos φ1 sin θ1 cos ψ1 + sin φ1 sin ψ1 )TM m is replaced with −kp x1 − kv x2 + rx (t) and then we obtain x˙ 1 = x2 (1.8) x˙ 2 = −kp x1 − kv x2 + rx (t) With a reference system x˙ r1 = xr2 (1.9) x˙ r2 = −kp xr1 − kv xr2 + rx (t) and error variables e1 = x1 − xr1 and e2 = x2 − xr2 , the error dynamics are e˙ 1 = e2 (1.10) e˙ 2 = −kp e1 − kv e2 which means that x1 and x2 track asymptotically the reference system in (1.9). In the next section, to realize the controller in (1.5) for the case of the unknown δ(x, t) and unmeasured states, the extended high-gain observers will be introduced. We start by introducing high-gain observers. 1.3 High-Gain Observers and Extended High-Gain Observers High-gain observers started from the earlier work in [21]. In [21], a fully linearizable nonlinear system was dealt with, which is a special case of the normal form. It was shown that the high-gain observers recovered the performance of state feedback controllers when the 5 observer gain is sufficiently high and the control input is globally bounded. In [7], the performance recovery of state feedback controllers was accomplished using saturated inputs and a separation principle for nonlinear systems was shown. More recent results on high-gain observers are available in [35]. We consider a multi-input-multi-output nonlinear system with multiple chains of integral, given by x˙ = Ax + Bφu (x, u, t) (1.11) y = Cx where x ∈ Rρ is system states, y ∈ Rm is a measured system output, u ∈ Rp is the control input, φu = [φu1 , · · · , φum ]T is assumed to be locally Lipschitz in its arguments over the domain of interest, and matrices A, B, C are  A = block diag[A1 , . . . , Am ], B = block diag[B1 , . . . , Bm ], C = block diag[C1 , . . . , Cm ],      Ai =       0 0 .. . 1 0 ··· ··· 0 1 0 ··· ··· ··· 0 .. . 0 1 0 ··· ··· ··· 0        ∈ Rρi ×ρi      (1.12) Bi = [0, 0, · · · , 0, 1]T ∈ Rρi ×1 Ci = [1, 0, · · · , · · · , 0] ∈ R1×ρi with 1 ≤ i ≤ m and ρ = ρ1 + · · · + ρm . It is assumed that a state feedback controller, u = Γ(x, t) is designed to asymptotically stabilize the system in (1.11) at the origin, where Γ(x, t) is locally Lipschitz in its arguments over the domain of interest. 6 1.3.1 High-Gain Observers We design a high-gain observer for the system in (1.3) as xˆ˙ = Aˆ x + Bφn (ˆ x, u, t) + H(y − C xˆ) (1.13) where φn (x, u, t) = [φn1 , · · · , φnm ]T is a nominal model of φu in (1.11) and φn (0, 0, t) = 0, and the observer gain, H, is chosen as  H = block diag[H1 , . . . , Hm ], hi1 /ε    hi2 /ε2   .. Hi =  .     hiρi −1 /ερi −1  hiρi /ερi             (1.14) with a small positive constant, 0 < ε ≪ 1. The components hij of H are chosen such that the polynomials λρi + hi1 λρi −1 + · · · + hiρi −1 λ + hiρi = 0, for i = 1, · · · , m (1.15) are Hurwitz. An important phenomenon in high-gain observers is peaking [7]. The scaled T T error variable is defined by η = [η1T , η2T , · · · , ηm ] , ηi = [ηi1 , · · · , ηiρi ]T , ηij = xij − xˆij , ερi +1−j for 1 ≤ i ≤ m, 1 ≤ j ≤ ρi (1.16) To show the peaking phenomenon, a SISO second-order subsystem with i = 1 and ρ1 = 2 is considered as x˙ 11 = x12 , x˙ 12 = φ11 7 (1.17) and the scaled error variables are η11 = x11 − xˆ11 , ε η12 = x12 − xˆ12 (1.18) The error dynamics are εη˙ 11 = −h11 η11 + η12 (1.19) εη˙ 12 = −h12 η11 + ε(φu,1 − φn,1 ) With sufficiently small ε, the effect of the difference, (φu,1 − φn,1), on the right-hand side of (1.19) is small enough that the behavior of the differential equation of (1.19) becomes a linear system. The solution of such linear systems contains a term of the form a exp(−at/ε), ε with a > 0 (1.20) As ε → 0, peaking of a/ε degrades the system performance and may destabilize the system [7]. One approach to overcome the peaking phenomenon is to design the control as a globally bounded function of the state, which can be achieved by saturating the control inputs or the state estimates [33]. The saturations are chosen such that they are not active over the domain of interest under state feedback. 1.3.2 Extended High-Gain Observers By adding one more integrator into the high-gain observer, an extended high-gain observer is created to estimate both unmeasured system states and uncertainties [23]. Consider a multi-input-multi-output nonlinear system in the presence of uncertainties σ(x, t) = 8 [σ1 , · · · , σm ]T ∈ Rm , given by x˙ = Ax + B[φn (x, u, t) + σ(x, t)] (1.21) y = Cx where x, y, φu (x, u, t), A, B and C are given below (1.11). An extended high-gain observer is designed as xˆ˙ = Aˆ x + B[φn (ˆ x, u, t) + σ ˆ (t)] + H(y − C xˆ) (1.22) σ ˆ˙ = He (y − C xˆ) where φn and H are same in (1.13) and He is He = [h1ρ1 /ερ1 +1 , · · · , hmρm /ερm +1 ]T (1.23) The observer gains, hi1 , · · · , hiρi +1 are chosen that the polynomials λρi +1 + hi1 λρi + · · · + hiρi λ + hiρi +1 for i = 1, · · · , m (1.24) are Hurwitz. Extended high-gaion observers have been used for several applications. In [51], robust stabilization of non-minimum phase nonlinear system was considered using a continuous sliding-mode control and an extended high-gain observer. Using fast estimation speed of the extended high-gain observer, the derivative of system output was estimated and one of unknown functions in the last chain of integrator was also estimated by the extended highgain observer. Then estimates were used for the sliding mode control. In [13], an extended high-gain observer as a fast time-scale was combined with an extended kalman filter as a slow time-scale to estimate states of internal dynamics. Estimates of the extended high-gain observer were used as a virtual measurement output for the extended kalman filter. 9 Chapter 2 Output Feedback Performance Recovery in the Presence of Uncertainties 2.1 Introduction Control of dynamical systems is challenging in the presence of uncertainties. Uncertainties may arise from inaccurate description of the dynamics by the mathematical model used, or can be due to external disturbances that are not accounted for in the model. Additional challenges are posed when the states of the system are not available for measurement and the control variables do not appear linearly in the mathematical model. To achieve desired control objectives, many of these challenges have been addressed by several researchers. To consider uncertain, nonaffine systems with external disturbances, Chakrabortty et al. [16, 17] designed a time-scale separation method. Two filers were used in [17] to deal with system uncertainties and nonaffine input forms; one filter was designed to estimate the uncertainties and the other filter was used to deal with nonaffine input forms. The work in [16] is an extension of Chakrabortty et al. [17] where unmodeled dynamics in the control 10 inputs are additionally taken into account. Hovakimyan et al. [28] proposed a fast gradient algorithm for dynamic inversion to deal with Multi-Input-Multi-Output (MIMO) nonaffine systems. The control approaches developed by Chakrabortty et al. [16, 17] and Hovakimyan et al. [28] are based on state feedback. Tanelli et al. [64] designed a state feedback control scheme for uncertain, Single-InputSingle-Output (SISO) nonaffine systems. A second-order sliding mode controller with saturation was used to deal with uncertainties and nonaffine input forms. The system states were estimated using the first-order differentiator proposed by Lavant [40] but the stability analysis was based on state feedback control. In [23], Extended High-Gain Observers (EHGOs) were designed to estimate unmeasured states and uncertainties by combining the High-Gain Observers (HGOs) proposed by Atassi and Khalil [7] with an additional integrator. Back and Shim [9] developed controllers for uncertain, SISO nonline systems using a time-scale separation approach and the circle criterion; the unmeasured states and external disturbances were estimated using an observer. These results were extended in [10] to deal with uncertain MIMO systems using the multi-variable circle criterion. The results in [23] and [9, 10] are applicable to systems with affine inputs. Hovakimyan et al. [27] proposed an output feedback controller for uncertain, SISO nonaffine input systems using neural network observers together with dynamic inversion. Ge and Zhang et al. [24] used neural networks to deal with SISO nonaffine systems and highgain observers were used to estimate the unmeasured states. Chen et al. [18] proposed state feedback control for uncertain, MIMO nonaffine systems using neural networks. Neural networks were used to model the system dynamics, uncertainties in the system, and input nonlinearities and uncertainties. A robust backstepping controller, combining backstepping with variable structure control, was used to deal with neural networks approximation errors. All of these methods based on neural networks require heavy computations and good prior knowledge of the system. In this chapter, we propose an output feedback control scheme for uncertain nonlinear 11 systems with nonaffine inputs and external disturbances; it is an extension of our earlier work [41, 42]. By operating an EHGO in the fastest time scale, unmeasured system states, model uncertainties, and external disturbance were estimated. For the fast recovery of target system inputs in the presence of uncertainties, dynamic inversion was used based on the estimates provided by the EHGO together with sector conditions for the inputs. Using a multi-time-scale controller, the performance of target system states and inputs is recovered in the presence of uncertainties. The singular perturbation method is used to analyze the closed-loop system behavior and establish stability and performance recovery. This chapter is organized as follows. In the Section 2.2, the problem is formulated for uncertain, MIMO, systems. In Section 2.3 the output feedback controller is presented along with the EHGO and the dynamic inversion algorithm. Section 2.4 provides the stability analysis and establishes performance recovery. Simulation results are presented in Section 2.5 and compared with results of other algorithms in the literature. Performance recovery is also verified through the simulations. Concluding remarks are provided in Section 2.6. 2.2 Problem Statements The goal of this chapter is to design an output feedback controller that can make an uncertain, MIMO, nonlinear system follow a target system. Consider a MIMO nonlinear system given by x˙ =Ax + Bf (x, z, u), z˙ =f0 (x, z), z(0) = z0 y =Cx, 12 x(0) = x0 (2.1) where A, B, and C represent n chains of integrators as A = block  0    0   . . Ai =   .    0  0 diag[A1 , . . . , An ], B = block diag[B1 , . . . , Bn ]    1 0 ··· 0 0        0  0 1 ··· 0      .   .. . . . . ..  .  , Bi =  . . .  .  .         0  0 ··· 0 1     0 ··· ··· 0 1 ρi ×ρi ρi ×1 C = block diag[C1 , . . . , Cn ], Ci = [1, 0, · · · , 0]1×ρi , and f (x, z, u) = fn (x, u) + σ(x, z, u) with   fn,1  . . fn (x, u) =   .  fn,n       n×1   σ1 (x, z, u)  .. , σ(x, z, u) =  .   σn (x, z, u)       n×1 1 ≤ i ≤ n, ρ = ρ1 + · · · + ρn , x ∈ Dx ⊂ Rρ, z ∈ Dz ⊂ Rm , and u ∈ Du ⊂ Rn . The domains Dx , Dz , and Du contains their respective origins. The output y ∈ Dy ⊂ Rn is measured, the nominal function fn (x, u) is known, and σ(x, z, u) is unknown. Assumptions for the system are made as follows. Assumption 1. The functions fn (x, u), σ(x, z, u), and f0 (x, z) are continuously differentiable. In this chapter, we are designing controllers for minimum-phase systems. Assumption 2, below, implies that the z-subsystem in (2.1) is bounded-input-bounded-state stable. Assumption 2. There exists a radially unbounded positive definite function Vz such that for all x ∈ Rρ and z ∈ Rm V˙ z ≤ 0 for z ≥ Wz (x) 13 (2.2) where Wz (x) is a nonnegative continuous function. A target system is defined by x˙ r = (A − BL)xr + Buc (t), xr (0) = xr0 , (2.3) where the matrix L is block diagonal such that the matrix (A − BL) is Hurwitz and uc (t) is a bounded command input belonging to the compact set Dr ⊂ Rn and its derivative u˙ c is chosen to be bounded. With the variable e = x − xr , the error dynamics are given by e˙ = (A − BL)e + BF (x, z, u, uc), (2.4) e0 = e(0) where F (x, z, u, uc) = f (x, z, u) + Lx − uc (t). With the error dynamics of (2.4), we have the following assumption. Assumption 3. • There is a unique continuously differentiable function φ(x, z, uc ) such that ur = φ(x, z, ur ) solves the equation (2.5) F (x, z, ur , uc ) = 0 The derivative u˙ r = φd (x, z, uc , u˙ c ) is bounded on compact sets of x and z. • There is a known matrix K(x, s + ur , uc ) such that the function F satisfies the sector condition sT K(x, s + ur , uc )F (x, z, s + ur , uc ) ≥ βsT s, β>0 (2.6) for all (x, z, ud ), with s = u − ur . Remark 1. When the system is affine in u, i.e., f (x, z, u) = h(x, z) + G(x, z)u, condition 14 (2.6) is equivalent to the existence of a known matrix K(x, s + ur , uc ) such that K(x, s + ur , uc )G(x, z) + GT (x, z)K T (x, s + ur , uc ) ≥ kI (2.7) with k > 0. This condition holds with K = I when G(x, z) satisfies G(x, z) + GT (x, z) ≥ kI, (2.8) k>0 This will be the case for single-input systems when G is positive and bounded away from zero. Condition (2.7) is less restrictive than (2.8) as it will be shown in Section 2.5 by an example. Remark 2. Earlier work on systems that are nonaffine in the input, in particular, [17] requires the Jacobian matrix (∂f /∂u) to satisfy the condition. ∂f ∂u + ∂f ∂u T ≥ kI, (2.9) k>0 and [27–29] require the Jacobian matrix (∂f /∂u) to satisfy either the condition (2.9) or the condition ζT ∂f T (x, u1 , uc ) ∂u ∂f (x, u2 , uc ) ζ ≥ 2kc ζ ∂u 2 ∀ζ ∈ Rn (2.10) where u1 and u2 are distinct variables and kc is a positive constant. The sector condition (2.6) is less restrictive than (2.9) and (2.10). For single-input systems, (2.9) requires the Jacobian to be positive for all u. The sector condition (2.6), on the other hand, allows the Jacobian to be negative as long as f (x, z, u) belongs to the sector [k, ∞), uniformly in x, z and uc . For multi-input systems, the sector condition does not require the Jacobian (∂f /∂u) to be nonsingular. The relation between K in (2.7) and the Jacobian matrix (∂f /∂u) will be 15 mentioned in Section‘2.3. In Section 2.5, it is shown that the sector condition in (2.7) is less conservative than the conditions (2.9) and (2.10) through numerical simulations. 2.3 Control Design We use dynamic inversion to deal with nonaffine and/ or uncertain functions. Had x and σ been available, the dynamic inversion would have been taken as µu˙ = −K(x, u, uc )F (x, z, u, uc ), u(0) = u0 (2.11) = −K(x, u, uc )[fn (x, u) + σ(x, z, u) + Lx − uc ] If the matrix K is chosen as the Jacobian matrix (∂f /∂u)T , i.e., K = (∂f /∂u)T (x, u, uc) and the function F in (2.11) is not a function of z, i.e., F = F (x, u, uc), the derivative of the Lyapunov function Vs = sT s/2 along the trajectories (2.1) and (2.3), is V˙ s = (u˙ − φd )T s + sT (u˙ − φd ) =− 1 2µ sT KF (x, u, uc) + F T (x, u, uc )K T s − φTd s (2.12) with s = u − ur and u˙ r = φd (x, uc , u˙ c ). By using the mean value theorem (Appendix B, [34]) with F (x, ur , uc ) = 0 in (2.5), F (x, u, uc) − F (x, ur , uc ) 1 = 0 ∂F x, (1 − σ)s + ur , uc dσs ∂u (2.13) for 0 ≤ σ ≤ 1, and ∂f /∂u = ∂F/∂u, V˙ s is rewritten as V˙ s = − 1 sT P T K T + KP s − φTd s 2µ 16 (2.14) where 1 P = 0 ∂f (x, (1 − σ)s + ur , uc )dσ ∂u (2.15) With sufficiently small µ and the condition sT (P T K T + KP )s ≥ kj s 2 (2.16) V˙ s ≤ −(ks /µ)Vs + δ with kj , ks > 0, and δ > 0 independent of µ. We note that the condition in (2.16) is similar to the condition in (2.10). In output feedback control, x and σ are estimated using the EHGO: xˆ˙ = Aˆ x + B[fn (ˆ x, u) + σˆ (t)] + H(ε)(y − C xˆ) σ ˆ˙ = Hn+1 (y − C xˆ), xˆ(0) = xˆ0 , (2.17) σ ˆ (0) = σ ˆ0 where σ ˆ (t) = [ˆ σ1 , · · · , σ ˆn ]T ∈ Rn is the estimate of σ(x, z, u), and H = block diag[H1 , · · · , Hn ], Hi (ε) = [αi,1 /ε, · · · , αi,ρi /ερi ]T , (2.18) Hn+1 = block diag[α1,ρ1 /ερ1 +1 , · · · , αn,ρn /ερn +1 ] The constants αi,1 , · · · , αi,ρi +1 are chosen such that the polynomials λρi +1 + αi,1 λρi + · · · + αi,ρi λ + αi,ρi +1 for i = 1, . . . , n are Hurwitz and the control parameter ε > 0 is small enough. We note that the small parameter ε is smaller than µ to make the dynamics of the EHGO faster than the dynamics of the dynamic inversion since the dynamic inversion uses estimates provided by the EHGO. Therefore, the control parameters ε and µ are chosen such that 0 < ε ≪ µ ≪ 1. 17 Using the EHGO in (2.17) together with the dynamic inversion in (2.11), the output feedback control is designed as µu˙ = −K[fn (ˆ xs , u) + σ ˆs + Lˆ xs − uc ] xˆs = [ˆ xTs1 , . . . , xˆTsn ]T , xˆsi = Mxi,1 sat xˆi,1 Mxi,1 σ ˆsi = Mσ1 sat σ ˆ1 Mσ1 σ ˆs = [ˆ σs1 , . . . , σ ˆsn ]T , . . . , Mxi,ρi sat , . . . , Mσn sat xˆi,ρi Mxi,ρi σˆn Mσn T (2.19) T for i = 1, . . . , n, where sat(·) is the saturation function defined by sat(k) = k |k| ≤ 1 (2.20) sign(k) |k| > 1 The saturation function is used to prevent peaking from degrading the system performance. The saturation levels Mxi,j and Mσi for j = 1, . . . , ρi and i = 1, . . . , n in (2.19), are determined outside of a compact set of interest, which is specified next. Under Assumption 3, the error dynamics of (2.4) with u = ur is exponentially stable at e = 0. Let P = P T > 0 be the solution of the Lyapunov equation P (A − BL) + (A − BL)T P = −I. With u = ur and Assumption 2, for any given positive constant cx and for all x(t) ∈ {Vx (x) ≤ cx } where Vx (x) = xT P x, and uc (t) ∈ Dr , the positively invariant set {Vz (z) ≤ cz + αz (cx )} can be chosen for the dynamics of z in (2.1), where αz (ce ) is a class K∞ function and cz > 0. Now, we can define the compact set Ωs = {Vx (x) ≤ cx } × {Vz (z) ≤ cz + αz (cx )} (2.21) By choosing cx sufficiently large, any compact subset of Rρ × Rm can be included in the interior of Ωs . Based on the compact set Ωs , the levels of saturation are determined as 18 follows. Mxi,j > Mσi > max x∈{Vx (x)≤cx } max (x,z)∈Ωs ,uc ∈Dr |xi,j |, (2.22) |σi (x, z, ur (x, z, uc ))| for j = 1, . . . , ρi and i = 1, . . . , n. 2.4 Main Result In this section, we will show that in the presence of uncertainties, the output feedback control (2.17) and (2.19) can recover the performance of both states and inputs of the target system T (2.3). Consider the fast variables η = [η1T , · · · , ηn+1 ]T with ηi = [ηi,1 , · · · , ηi,ρi ]T ∈ Rρi for error dynamics of the EHGO ηi,j = xi,j − xˆi,j , ηi,ρi +1 = σi (x, z, u) − σ ˆi (t) ερi +1−j (2.23) for 1 ≤ i ≤ n, 1 ≤ j ≤ ρi Using (2.4), the dynamics of z in (2.1), (2.17), and (2.19), the closed-loop system equations can be written in the standard singularly perturbed form e˙ = (A − BL)e + B[f (x, z, s + ur ) + Lx − uc (t)], (2.24) z˙ = f0 (x, z) (2.25) µs˙ = −K[fn (ˆ xs , s + ur ) + σ ˆs + Lˆ xs − uc ] − µφd (2.26) ¯1 ψ1 + (B ¯2 /µ)ψ2 ] εη˙ = Λη + ε[B (2.27) 19 where Λ = block diag[Λ1 , · · · , Λn ],  −αi,1 1 0 · · ·    −αi,2 0 1 . . .   .. .. . . . Λi =  . .. . .     −αi,ρi 0 · · · 0  −αi,ρi +1 0 · · · · · · 0    0  ..  .    1  0 (2.28) ρi ×ρi ¯1 = block diag[B1,1 , · · · , B1,n ], B ¯2 = block diag[B2,1 , · · · , B2,n ], B      Bi   0  , B2,i =  B1,i =    0 Bi (ρi +1)×1 (ρi +1)×1 for i = 1, . . . , n and the functions ψ1 and ψ2 are given by ψ1 =(1/ε)[fn (x, s + ur ) − fn (ˆ xs , s + ur )] ψ2 =µ ∂σ ∂x ∂σ + ∂z T [Ax + Bf (x, z, s + ur )] T ∂σ f0 (x, z) − ∂u (2.29) T KFs Fs = fn (ˆ xs , s + ur ) + σ ˆs + Lˆ xs − uc We note that ur and φd (x, z, s + ur ) in (2.24) and (2.25), respectively, are mentioned in the Assumption 3, the nominal function fn (x, s + ur ) is Lipschitz in its arguments so that the inequality fn (x, s + ur ) − fn (ˆ xs , s + ur ) ≤ εkψ1 η is satisfied with kψ1 independent of ε. we are going to show the stability of the boundary layer and reduced systems in the closedloop system (2.24)-(2.27). Since the z-subsystem of (2.25) with the input x is bounded-inputbounded-state stable, we are focusing on the subsystems (2.24), (2.26), and (2.27). Now, using the time-scale structure of the closed-loop system (2.24), (2.26), and (2.27), 20 the stability analysis of each subsystem will be shown. By considering the subsystem (2.27) as the fast system and the other subsystems (2.24) and (2.26) as the slow system, the boundary layer system in (2.27) can be obtained by εη˙ = Λη, (2.30) η(0) = η0 Since the matrix Λ is Hurwitz, the boundary layer system (2.30) is exponentially stable at the origin. After the fast variable η reaches its quasi-steady state, η = 0, the reduced system for (2.24) and (2.26) is obtained by setting η = 0 and ε = 0. In the reduced system, the dynamic inversion (2.26) is viewed as fast and the subsystem (2.24) is slow. The boundary layer system for (2.26) is given by µs˙ = −KF (x, z, s + ur , uc ), s(0) = s0 (2.31) To investigate the stability of the boundary layer system (2.31), the Lyapunov function Vs = (sT s)/2 is defined. With the sector condition (2.6), its derivative V˙ s along the trajectory (2.31) is µV˙ s = −sT KF ≤ −β s 2 (2.32) Therefore, the boundary layer system (2.31) is exponentially stable at s = 0. The reduced system for (2.24) is obtained by setting ε = 0, η = 0, µ = 0, and s = 0. With the input u = ur and x = xˆ, the reduced system for (2.24) is obtained as e˙ = (A − BL)e, 21 e(0) = e0 (2.33) Figure 2.1: The trajectory starting from (e0 , z0 , s0 ) ∈ Ωa and η0 ∈ / {Vη ≤ b3 (ε/µ)2 } converges into (e, z, s, η) ∈ Ωb × {Vη ≤ b3 (ε/µ)2 }. which is exponentially stable at e = 0. The following theorem shows that all trajectories will be bounded and (x, u) approach the trajectories of the target system (2.3) as µ, (ε/µ), and ε tend to zero. Theorem 1. Consider the closed-loop system (2.24) - (2.27) under Assumption 1, 2, and 3. Suppose the trajectories (x, z, u, x ˆ, σ ˆ ) start from the compact sets (x0 , z0 , u0 ) ∈ Sx ×Sz ×Su ⊂ Dx × Dz × Du and (ˆ x0 , σ ˆ0 ) ∈ Qx × Qs ⊂ Rρ × Rn . Then, there exists a constant ς > 0 such that for max{µ, (ε/µ), ε} < ς • all trajectories are bounded; • x − xr → 0 and u − ur → 0 as µ, (ε/µ), ε → 0 for all t ≥ 0. Proof. As the first part of the proof, we are going to show that all trajectories enter a positively invariant set. We define the Lyapunov functions Ve = eT P e and Vη = η T Pη η for the subsystems, (2.24) and (2.27), respectively, where Pη is the solutions for the Lyapunov equation Pη Λ + ΛT Pη = −I. We define the sets 22 Ωa = {Ve ≤ a1 }×{Vz ≤ cz + αz (cx )}×{Vs ≤ a2 } Ωb = {Ve ≤ b1 }×{Vz ≤ cz + αz (cx )}×{Vs ≤ b2 } (2.34) Ωc = {Ve ≤ c1 }×{Vz ≤ cz + αz (cx )} with the relations 0 < a1 < b1 < c1 , 0 < a2 < b2 (2.35) The constant c1 is chosen such as that e ∈ {Ve ≤ c1 } ⇒ x ∈ {Vx ≤ cx } (2.36) so that ∀e ∈ {Ve ≤ c1 }, z ∈ {Vz ≤ cz + αz (cx )}. The constant a1 and a2 are chosen such that Sx × Sz × Su ⊂ Ωa . Similar to earlier work on high-gain observers, e.g., [33], it can be shown that, for sufficiently small (ε/µ), the set Ωb × {Vη ≤ b3 (ε/µ)2 }, for some b3 > 0, is positively invariant. This is done by showing that the derivatives of Ve , Vz , and Vs are negative on the boundaries {Ve = a1 }, {Vz = cz + αz (cx )}, and {Vs = a2 }, respectively. Similarly, it can be shown that for sufficiently small λ = max{µ, ε/µ}, the set Ωc × {Vs ≤ c2 λ2 } × {Vη ≤ b3 (ε/µ)2}, for some c2 > 0, is positively invariant. We show boundedness of the trajectories in two steps: • firstly, the trajectories (e, z, s) and η starting from (e0 , z0 , s0 ) ∈ Ωa and η(0) ∈ / {Vη ≤ b2 (ε/µ)2} enter the set Ωb × {Vη ≤ b3 (ε/µ)2 } in finite time as depicted in Fig. 2.1; • secondly, the trajectories (e, z), s, and η starting from Ωb × {Vη ≤ b3 (ε/µ)2} enters the set Ωc × {Vs ≤ c2 λ2 } × {Vη ≤ b3 (ε/µ)2 } in finite time. In the first step, consider that the initial conditions are in (e0 , z0 , s0 ) ∈ Ωa and η(0) starting from outside the set {Vη ≤ b3 (ε/µ)2}. Due to the scaling (2.23), η(0) could be of the order of 23 ¯1 ψ1 + (B ¯2 /µ)ψ2 ] in (2.27) is bounded 1/ερm where ρm = maxi=1,...,n ρi . Because the term [B by k1 η + k2 for all (x, z, s) ∈ Ωb , for some positive constants k1 and k2 , it can be shown that εV˙ η ≤ − η 2 + εk3 η 2 + εk4 η (2.37) for some positive constants k3 and k4 . Hence, εV˙ η ≤ − 1 η 2 2 + ε η µ (2.38) for ε < 1/(2k3). It follows that there is b3 > 0 such that εV˙ η ≤ −γ1 Vη , for Vη ≥ ε µ 2 b3 (2.39) for some γ1 > 0. Taking into consideration that η(0) ≤ k5 /ερm for some positive constant k5 > 0, we obtain Vη ≤ k6 −γ1 t/ε e ε2ρm (2.40) for some positive constant k6 > 0. Since ε ≤ ε/µ, we can estimate the time the trajectory will enter {Vη ≤ b3 (ε/µ)2 } by the more conservative time, T1 (ε), when Vη = b3 ε2 . The time T1 (ε) is given by k6 −γ1 T1 /ε ε ln e = b3 ε2 ⇔ T1 = 2ρ m ε γ1 b3 k6 2(ρ ε m +1) (2.41) By L’Hopital’s rule, it can be shown that limε→0 T1 (ε) = 0. Because a1 < b1 , a2 < b2 , and the right-hand side functions of (2.24), (2.25), and (2.27) are bounded uniformly in ε, there is time T0 such that (x, z, s) ∈ Ωb for all t ∈ [0, T0 ]. By choosing ε small enough we can have 24 T1 (ε) = (1/2)T0 . In the second step, we repeat the same argument to show that s enters the set {Vs ≤ c2 λ2 }. The right-hand side of (2.26) is a perturbation of the right-hand side of (2.31) with the perturbation term bounded by k7 η + k8 µ for some positive constants k7 and k8 . Because η cannot leave the set {Vη ≤ b(ε/µ)2 }, k7 η + k8 µ ≤ k9 λ for some k9 > 0 (2.42) where λ = max{µ, ε/µ}. Hence µV˙ s ≤ −β s 2 + k10 λ s (2.43) Therefore, there is c2 > such that µV˙ s ≤ −γ2 Vs , for Vs ≥ c2 λ2 (2.44) This show that there is time T2 = T2 (µ) such that the trajectory enters {Vs ≤ c2 λ2 }. Once again by choosing µ small enough, (e, z) stay in the set {Ve ≤ c1 } × {Vz ≤ cz + α(cx )}. Thus, with the time T (ε, µ) = T1 (ε) + T2 (µ), the trajectory enters the positively invariant set Ωc × {Vs ≤ c2 λ2 } × {Vη ≤ b3 (ε/µ)2 }. Lastly, we are going to show the performance recovery for x and u. Since the proof is similar for both variables, we show it only for u. The nominal model of (2.26) is the system (2.31), which is exponential stable at s = 0. The difference between (2.26) and (2.31) is G = − K[fn (ˆ xs , s + ur ) − fn (x, s + ur ) (2.45) +σ ˆs (t) − σ(x, z, s + ur ) + L(ˆ xs − x)] + µu˙ r where u˙ r = φd (x, z, uc , u˙ c ) is bounded on the compact set of x, z, uc , and u˙ c , i.e., u˙ r ≤ urm , 25 with urm > 0, uniformly in µ and ε. The function G is bounded G ≤ Km [εL1 η + L2 η ] + µurm (2.46) for some positive constants Km , L1 , L2 , which can be made arbitrarily small by choosing sufficient small µ and ε/µ, for t ≥ T (ε, µ). Using Theorem 9.1 in [33], we conclude that u(t) − ur (t) ≤ δ(µ, ε/µ), ∀t ≥ T (µ, ε) > 0 (2.47) where δ(µ, ε) → 0 as µ, (ε/µ), ε → 0. For the time interval t ∈ [0, T (µ, ε)], since the trajectories u and ur are in the compact set, we have two inequalities u(t) − u(0) ≤ kt t, ur (t) − u(0) ≤ kt t (2.48) with kt > 0, during the time interval. Using the triangle inequality, we obtain the inequality u − ur ≤ 2kt T (µ, ε), ∀t ∈ [0, T (µ, ε)] (2.49) for all t ≥ 0 (2.50) Therefore, given any δ1 > 0, we can ensure that u(t) − ur (t) ≤ δ1 , by choosing ε, µ, and (ε/µ) sufficiently small. 2.5 Simulations In this section, we choose examples for the comparison with other papers, [28], and [17]. The first example, which has nonaffine input, considers the case that the Jacobian matrix is singular at some values. As a second example, which has affine input forms and system 26 uncertainties, the camera image coordinate system in [17] is modified to show that our proposed control methods provide less conservative conditions than ones in [17]. 2.5.1 Example 1 First, we are considering the case where the Jacobian matrix, (∂f /∂u) in (2.10), is singular at some values. In this example, we assume that the state x = [x1 , x2 ]T is available and there is no system uncertainty. The MIMO nonlinear nonaffine system is given by    2u31 15u21  − + 36u1   x˙ 1   x1 +  =  3 2 2u2 − 15u2 + 36u2 x˙ 2 (2.51) The target system is x˙ r = (A − BL)xr + uc , xr = [xr1 , xr2 ]T     0   −x1  100  uc =    , A − LB =  0 −x2 100 (2.52) Our proposed controller for the dynamic inversion is   3 2  x1 +2u1 −15u1 +36u1 +x1 − 100  µu˙ = −  , 3 2 2u2 −15u2 +36u2 +x2 − 100 (2.53) where u = [u1 , u2 ]T , the small constant µ = 0.1, and K in (2.6) is chosen as an identity matrix, i.e., K = I2×2 (a 2×2-identity matrix). In Fig. 2.2, the solid lines x1 and x2 generated by the proposed controller in (2.52), converge into the reference trajectories (dashed lines) in (2.52). Since the proposed controller in (2.53) is designed based on the sector condition, the controller is not affected by the singularity of the Jacobian matrix in Fig. 2.3. 27 120 100 x1 80 60 40 20 0 0 1 2 3 t (sec) 4 5 6 0 1 2 3 t (sec) 4 5 6 120 100 x2 80 60 40 20 0 Figure 2.2: The solid and dashed lines represent trajectories xi for i = 1, 2 driven by the proposed controller in (2.53) and reference trajectories xri for i = 1, 2 in (2.52), respectively. 2.5.2 Example 2 A modified model from [17] is given by x˙ = Rm u, y = [x1 , x2 ]T ,    cos φ sin φ  Rm =   sin φ − cos φ (2.54) with x = [x1 , x2 ]T , u = [u1 , u2 ]T , and φ = 45◦ . The target system with uc = [0, 0]T is the same as in (2.52). With the condition φ = 45◦ , the Jacobian condition (∂Rm /∂u)T + (∂Rm /∂u) ≥ kp I2×2 in [17] is not satisfied. Using the dynamic inversion, the control inputs are designed 28 6 u 1 4 2 0 −2 0 1 2 3 t (sec) 4 5 6 0 1 2 3 t (sec) 4 5 6 6 u 2 4 2 0 Figure 2.3: Input trajectories u1 and u2 in (2.53) are shown and both u1 and u2 cross the values u1 = 2, 3 and u2 = 2, 3 which make the Jacobian matrix (∂f /∂u) singular. by (2.55) µu˙ = −KF where    u1 cos φ + u2 sin φ + x1  KF =   −u1 sin φ + u2 cos φ − x2 29 (2.56) The sector condition in Assumption 3 is satisfied as follows. sT KF ≥ β s 2 (2.57) with the constant β > 0. Now, the EHGO is designed to estimate uncertainties i.e., the right-hand side of (2.54), Rm u, h11 xˆ˙ 1 = σ ˆ1 + (x1 − xˆ1 ), ε h21 (x2 − xˆ2 ), xˆ˙ 2 = σ ˆ2 + ε h12 σ ˆ˙ 1 = 2 (x1 − xˆ1 ) ε h 22 σ ˆ˙ 2 = 2 (x2 − xˆ2 ) ε (2.58) where σ ˆ1 and σ ˆ2 are the estimates of x˙ 1 and x˙ 2 , respectively. The constants hi,1 and hi,2 , i = 1, 2 are chosen such that the polynomials λ2 + hi,1 λ + hi,2 = 0, (2.59) for i = 1, 2 are Hurwitz. The output feedback control is µu˙ = −KFs ,   σ1 /Mσ1 ) + x1   Mσ1 sat(ˆ KFs =   −Mσ2 sat(ˆ σ2 /Mσ2 ) − x2 where the levels of saturation, Mσ1 and Mσ2 are chosen such that the saturations will not be activated in the range of state feedback control. For the simulation, the parameters are given by µ = 0.02, ε = 0.0002, hi,1 = 3, hi,2 = 1 (2.60) for i = 1, 2. In Fig. 2.4, the states x1 and x2 (solid lines) and their estimates xˆ1 and xˆ2 (dotted lines), respectively are plotted. The initial conditions of trajectories are given by x(0) = [2, 4]T , u(0) = [0, 0]T , the estimates xˆ1 (0) = 0 and xˆ2 (0) = 0, and σ ˆ1 (0) = 0 and 30 2.5 1 2 x , Est:x 1.5 1 1 0.5 0 −0.5 0 1 2 3 4 5 3 4 5 t(sec) 5 3 2 2 x , Est:x 2 4 1 0 0 1 2 t(sec) Figure 2.4: The solid and dotted lines represent the system states x1 and x2 , and estimates xˆ1 and xˆ2 , respectively. σ ˆ2 (0) = 0. In Fig. 2.4, the estimate trajectories quickly converge into the system states and both the states and estimates are indistinguishable. The results shown in Fig. 2.5, indicate the EHGO successfully estimates the uncertainties (i.e., the entire terms in the right-hand side of (2.54)) in a short period of time. At beginning of the simulations in Fig. 2.5, the peaking phenomena are shown and quickly disappear, which are overcome by the use of saturation functions. To illustrate the performances recovery for inputs, we choose the same rate of (ε/µ) = 0.01 with different values µ = 0.2 and µ = 0.02. The other parameters in −1 the EHGO are the same as in (2.60). In Fig. 2.6, the input ur = Rm x is solid lines, the dashed lines represents the inputs under the output feedback with parameters µ = 0.02 and ε/µ = 0.01, and the dotted lines are trajectories for the input under the output feedback 31 4 2 0 −2 0 1 2 3 4 5 3 4 5 t(sec) 6 4 2 0 −2 −4 0 1 2 t(sec) Figure 2.5: The solid and dotted lines represent the system states x˙ 1 and x˙ 2 , and estimates σ ˆ1 and σ ˆ2 , respectively. with parameters µ = 0.2 and ε/µ = 0.01. The input u with the control parameters µ = 0.02 and ε/µ = 0.01, has an faster convergence into ur than the input with control parameters µ = 0.2 and ε/µ = 0.01. 2.6 Conclusions Unmeasured states, uncertainties, and nonaffine inputs pose challenges in control design for nonlinear systems. An output feedback control design was proposed to address these challenges. The unmeasured states and uncertainties were estimated using an EHGO and sector conditions were utilized for dynamic inversion to deal with nonaffine and uncertain 32 1 Input u 1 0 −1 −2 −3 −4 −5 0 1 2 3 4 5 3 4 5 t (sec) 3 Input u 2 2 1 0 −1 0 1 2 t (sec) −1 Figure 2.6: The solid, dashed, and dotted lines represent the inputs ur = Rm x, u with µ = 0.02 and ε/µ = 0.01, and u with µ = 0.2 and ε/µ = 0.01, respectively. inputs. The EHGO and the dynamic inversion together result in exponential convergence of the states to those of a target system. The stability and performance of the system were analyzed using singular perturbation methods and the effectiveness of the proposed controller was verified through numerical simulations. Our future work will consider extension of our approach to non-minimum phase systems. 33 Chapter 3 Output Feedback Stabilization of Inverted Pendulum on a Cart in the Presence of Uncertainties 3.1 Introduction An inverted pendulum on a cart is a classical example of an underactuated mechanical system and its stabilization problem has been investigated by many researchers. Based on linearized system dynamics, controllers can be designed to stabilize the equilibrium but the size of the region of attraction is typically small. Furthermore, these controllers are not very effective in the presence of significant uncertainties in the system model. In this chapter we present an output feedback control design that can stabilize the equilibrium in the presence of significant uncertainties and provide a large region of attraction. One representative approach for stabilization of the inverted pendulum on a cart is based on the energy of the system. Spong and Praly [62] used partial feedback linearization to linearize the cart dynamics followed by transfer of energy from the cart to the pendulum. A stabilizing controller is invoked when the configuration of the system reaches a neighbour34 hood of the equilibrium. Astrom and Furuta [6] used a Lyapunov function based on the potential energy of the pendulum, and Lozano et al. [46] stabilized the pendulum around its homoclinic orbit prior to stabilization. Fradkov [22] developed a control method using an energy-based objective function and the speed-gradient, and Shiriaev et al. [61] proposed a modified controller using the idea of variable structure systems. Muralidharan et al. [50] designed a controller for the two-wheeled inverted pendulum using the interconnection and damping-passivity-based control (IDA-PBC) method proposed by Ortega et al. [55] for underactuated systems. Sarras et al. [59] combined the approach of the Immersion & Invariance proposed by Astolfi et al [5] with the Hamiltonian formulation to accommodate underactuation degree greater than one. Bloch et al. [12], [11] used the controlled Lagrangian approach to derive a desired closed-loop system dynamics for stabilization. The controller is designed by matching the dynamic equations for the uncontrolled and controlled Lagrangians. In [12], only the kinetic energy was shaped to obtain the desired dynamics whereas both kinetic and potential energies were shaped in [11]. Angeli [4] developed a smooth feedback law for almost-global stabilization based on the energy-shaping control strategy in [12]. Auckly [8] derived a stabilizing controller by solving a set of linear partial differential equations; these equations were obtained by matching the desired closed-loop system dynamics based on the potential energy with the original dynamics. Among other approaches, Mazenc et al. [49] and Teel [66] developed control methods based on the concept of interconnected systems. In [49], the stability analysis was carried out using a Lyapunov function whereas in [66] a nonlinear small gain theorem was used. OlfatiSaber [54] proposed a transformation to convert the system into cascade normal form, for which existing control methods can be used for stabilization. A two-time-scale approach was proposed by Getz et al. [26] and Srinivasan et al. [63]. In [26], the trajectories of the pendulum were rapidly converged to a reference trajectory and the reference trajectory was slowly varied to converge the cart to its desired position. In [63], low gains were used near the equilibrium for separation of time scales. All of the methods discussed above require 35 exact knowledge of the system dynamics and are unlikely to guarantee stabilization in the presence of significant uncertainties. To deal with uncertainties of the system model, Ravichandran et al. [57] used a two-timescale approach together with Lyapunov redesign. However, the transient behavior of the fast system was not analyzed. Park et al. [56] utilized two sliding surfaces for the pendulum and cart subsystems to stabilize the system in the presence of disturbances but uncertainties in system parameters were not considered. Adhikary et al. [2] used backstepping and sliding mode control to the normal form of the system. Both uncertainties and disturbances were considered but they were introduced after the system was converted into normal form. Xu et al. [68] used integral sliding-mode control [15] to deal with uncertainties in the two-wheeled mobile inverted pendulum but the size of the region of attraction of the equilibrium is small since the controller is designed based on the linearized system dynamics. In this chapter we present an output feedback controller to stabilize the inverted pendulum on a cart in the presence of significant uncertainties. Extended High-Gain Observers and dynamic inversion are combined together with a multi-time-scale structure to deal with model uncertainties. The stability analysis for the multi-time-scale structure is carried out using singular perturbation methods; the advantage of this approach is that the behavior of the system can be analyzed independently for each time scale. The multi-time-scale structure of the controller effectively provides a large region of attraction and this is illustrated through simulations. Output feedback control of the inverted pendulum on a cart has not been proposed earlier and it is shown here that it can recover the performance of the system under state feedback. The chapter is organized as follows. In section 3.2, a state feedback controller is designed using a two-time-scale structure; uncertainties are not considered. In section 3.3, the output feedback controller is designed in the presence of uncertainties. Simulation and experimental results are presented in section 3.4 and conclusions are provided in section 3.5. 36 3.2 Stabilization in the Absence of Uncertainties We present a control strategy to stabilize the desired equilibrium of the inverted pendulum on a cart system, in the absence of uncertainties. The controller is based on the designs proposed by [25] and [63]; here we cast the closed-loop system dynamics in two-time scale format for the purpose of stability analysis. The stability analysis is done by transforming the system into a standard singularly perturbed one. Remark 3. As an intermediate step for the output feedback controller in Section 3.3, we design a controller in this section in the absence of uncertainties. 3.2.1 Dynamics of an inverted pendulum on a cart The dynamics of an inverted pendulum on a cart are given by       2  mp + mc ℓmp cos α   x¨   ℓmp α˙ sin α   F  +    =  α ¨ gℓmp sin α ℓmp cos α ℓ2 mp 0  Figure 3.1: Inverted pendulum on a cart 37 (3.1) where mp , mc are masses of the pendulum and the cart, respectively; g is the acceleration due to gravity; and ℓ is the length of the pendulum - see Fig. 3.1. The variables x and α denote the position of the cart and the angular displacement of the pendulum, respectively; α is measured clockwise from the vertical following the notation in [25]. The variable F denotes the force applied on the cart and is the control input. With the choice of state variables x1 = x, x2 = x, ˙ α1 = α, α2 = α˙ the system equations of (3.1) take the form x˙ 1 = x2 , x˙ 2 = fx (α1, α2 , F ), α˙ 1 = α2 , α˙ 2 = fα (α1, α2 , F ) (3.2) 1 F + Gx (mp + mc − mp cos2 α1 ) − cos α1 fα = F + Gα ℓ(mp + mc − mp cos2 α1 ) (ℓmp α22 sin α1 − mp g cos α1 sin α1 ) Gx = (mp + mc − mp cos2 α1 ) cos α1 g sin α1 − Gx Gα = ℓ ℓ (3.3) where fx = We consider equations in (3.2) over the domain x = [x1 , x2 ]T ∈ Dx and α = [α1 , α2 ]T ∈ Dα where Dx = {−ax1 < x1 < ax1 } × {−ax2 < x2 < ax2 } ⊂ R2 and Dα = {−aα1 < α1 < aα1 } × {−aα2 < α2 < aα2 } ⊂ R2 are bounded. The constants, ax1 , ax2 , aα1 , and aα2 are positive numbers and aα1 < π/2. 38 3.2.2 Control Design The choice of the control input F = (mc + mp − mp cos2 α1 )(u − Gx ) (3.4) with u = g tan α1 − ℓ cos α1 νd (3.5) results in x˙ 1 = x2 (3.6) ℓ cos α1 x˙ 2 = g tan α1 − νd α˙ 1 = α2 (3.7) α˙ 2 = νd We choose νd = −β1 (α1 − αr ) − β2 α2 (3.8) where αr is a reference trajectory for α1 , which will be chosen using the concept of equilibrium manifold [25]. The desired dynamics of the x-subsystem is x˙ 1 =x2 x˙ 2 =vext − 39 ℓ cos α1 (3.9) νd where vext is chosen as (3.10) vext = −γ1 x1 − γ2 x2 and (ℓ/ cos α1 )νd is considered to be a vanishing perturbation. By comparing the actual x-subsystem in (3.6) with the desired x-subsystem in (3.9), the desired reference trajectory for αr can be determined as follows g tan αr = vext ⇔ αr = tan−1 vext g (3.11) The control input νd in (3.8) can now be designed as follows: vext g νd = −β1 α1 − tan−1 3.2.3 − β2 α2 (3.12) Closed-loop system Using (3.4), (3.5), and (3.8), the closed-loop system dynamics can be represented as x˙ 1 =x2 x˙ 2 =g tan α1 + ℓ cos α1 [β2 α2 + β1 (α1 − αr )] (3.13) α˙ 1 =α2 α˙ 2 = − β2 α2 − β1 (α1 − αr ) The above system is comprised of two subsystems: the cart dynamics described by states x1 and x2 , and the pendulum dynamics described by states α1 and α2 . Through proper choice of β1 and β2 , we can ensure that α1 converges to αr quickly and α2 converges to 0. Then, 40 the cart dynamics is described by x˙ 1 =x2 (3.14) x˙ 2 =g tan αr = vext = −γ1 x1 − γ2 x2 which guarantees asymptotic stability of the origin (x, x) ˙ = (0, 0). 3.2.4 Analysis using singular perturbations To make the cart dynamics slower than the pendulum dynamics, we choose low gains for the cart controller: γ1 = ε21 k1 and γ2 = ε1 k2 , where ε1 is a small positive parameter and positive constants k1 and k2 are independent of ε1 . We note that instead of using the low gains in the cart controller, high gains can be used in the pendulum controller. However, the use of high gains results in peaking in the slow dynamics. The change of variables y = [y1 , y2 ]T , y1 = ε21 x1 , y2 = ε1 x2 (3.15) θ = [θ1 , θ2 ]T , θ1 = α1 − αr , θ2 = α2 (3.16) transforms the system (3.13) into the singularly perturbed form y˙ 1 =ε1 y2 y˙ 2 =ε1 [−k1 y1 − k2 y2 + hx (θ, αr )] θ˙1 =θ2 + ε1 hα (y, θ, αr , F ) θ˙2 = − β1 θ1 − β2 θ2 41 (3.17) where hx and hα are given by the expressions hx =g [tan(θ1 + αr ) − tan αr ] + ℓ (β1 θ1 + β2 θ2 ) , cos(θ1 + αr ) g (k1 y2 + k2 fx ) hα = 2 g + (−k1 y1 − k2 y2 )2 (3.18) and fx is defined in (3.3), except that α1 should now be replaced by (θ1 + αr ) in accordance with (3.15). We note that hx and hα are bounded by constants independent of ε1 for all ε1 ≪ 1 over the domains Dx and Dα . The boundary layer system is obtained by setting ε1 = 0 in (3.17):   θ˙ = Aθ θ, Aθ =  0 1 −β1 −β2    (3.19) where β1 and β2 are chosen such that Aθ is Hurwitz. The reduced system is obtained by setting ε1 = 0:   y˙ = ε1 Ay y, Ay =  0 1 −k1 −k2    (3.20) where k1 and k2 are chosen such that Ay is Hurwitz. The two time-scale structure of the system is depicted in Fig. 3.2. It follows from Theorem 11.4 of [33] that there exists a positive constant ε∗1 such that for ε1 ∈ (0, ε∗1) the origin of the closed-loop system (3.17) is exponentially stable. The design of β1 and β2 should ensure that α1 stays in the set |α1 | < aα1 where aα1 < π/2. Since θ1 = α1 − αr = α1 − tan−1 (vext /g), by choosing ε1 small enough we can constrain θ1 to the set |θ1 | ≤ bθ1 with bθ1 < π/2. The initial state θ(0) belongs to a compact set {|θ1 | ≤ aθ1 , |θ2 | ≤ aθ2 } where aθ1 < π/2 and aθ2 is some positive constant. We are going to 42 design β1 and β2 to obtain a Lyapunov function Vθ for the system: θ˙1 = θ2 , θ˙2 = −β1 θ1 − β2 θ2 such that the compact set {Vθ ≤ cθ } contains the set {|θ1 | ≤ aθ1 , |θ2 | ≤ aθ2 } and is contained in the strip |θ1 | ≤ bθ1 with aθ1 < bθ1 < π/2. By showing that V˙ θ is negative definite we ensure that the set {Vθ ≤ cθ } is positively invariant and all trajectories starting in {|θ1 | ≤ aθ1 , |θ2 | ≤ aθ2 } stay in the strip |θ1 | ≤ bθ1 for all t ≥ 0. The gains β1 and β2 are taken as β1 = (βc1 /µ) and β2 = (βc2 /µ) with positive constants βc1 and βc2 , and a small positive constant µ. The Lyapunov function is taken as  ˜ θ= ˜  Vθ = θ˜T Pα θ,  β1 β2 1 θ1  2 , Pα= θ1 + θ2 0    0 d 2 (3.21) 1 By choosing cθ < b2θ1 we have 2 V θ ≤ cθ ⇒ 1 2 1 θ1 ≤ cθ < b2θ1 ⇒ |θ1 | < bθ1 2 2 Over the set {|θ1 | ≤ aθ1 , |θ2 | ≤ aθ2 }, 1 d Vθ ≤ a2θ1 + 2 2 β1 β2 2 aθ1 + aθ2 Therefore by choosing d< (2cθ − a2θ1 ) [(β1 /β2 )aθ1 + aθ2 ]2 we ensure that {|θ1 | ≤ aθ1 , |θ2 | ≤ aθ2 } ⊂ {Vθ ≤ cθ }. As in standard analysis of singularly perturbed systems (Theorem 2.1 of Chapter 7.2 of [36]), the derivative V˙ θ will be negative 43 Normal Speed System Slow System Cart Pendulum (3.8) (3.11) (3.6) (3.7) (3.10) Figure 3.2: The two-time-scale structure of the inverted pendulum on a cart system definite by choosing µ sufficiently small. 3.3 Stabilization in the Presence of Uncertainties In the presence of parameter uncertainties, fx and fα of (3.2) are unknown. We however note that the following conditions (the sign of the input coefficients in (3.3)) hold in Dx and Dα : 1 mp + mc − mp cos2 α1 cx cos α1 sign(cα ) > 0, cα = ℓ sign(cx ) > 0, cx = (3.22) Extended High-Gain Observers will be used to estimate the uncertain terms in fx and fα in addition to the states; and dynamic inversion will be used to compute the inputs F and u, described by (3.4) and (3.5), using the estimates of fx and fα . 44 3.3.1 Dynamic inversion With the knowledge of the sign of the input coefficients, as shown in (3.22), we use a dynamic inversion scheme which is different from ones in [25], [29], and [28], in order to compute the control inputs F and u. The proposed dynamic inversion scheme in the chapter is an extension in our earlier work [42] since our earlier work dealt with SISO systems. In particular, had fx and fα been known, we could have used     ˙  F   −fx + u  ε2  =  u˙ fα − νd (3.23) to solve for F and u, starting from arbitrary initial conditions. In the above equation, ε2 is a small positive number; its relationship with ε1 will be discussed later. As ε2 → 0 and the stability of (3.23) is guaranteed, fx ≈ u and fα ≈ νd . The equation (3.23) is a singularly perturbed system of the form         ˙ Gx F   F    −cx 1  ε2   = Az   +   , Az =   u˙ u Gα − νd −cα 0 (3.24) whose quasi-steady-state solution is given by (3.4) and (3.5). Since the foregoing equation is linear in F and u and the matrix Az with frozen x and α is Hurwitz, it can be seen that for sufficiently small ε2 , F and u converge fast to their values that satisfy (3.4) and (3.5). The stability analysis for the system in (3.24) will be discussed in Section 3.3. Remark 4. The dynamic inversion scheme used in this section is different from ones in [25], [29], and [28]. • In [25] and [28], the dynamic inversion schemes were developed by the Jacobian with respect to inputs whereas the proposed one in the chapter is based on the stability of the fast dynamics in (3.24), which only requires the knowledge on the signs of inputs. 45 • In [25], the stability analysis for the dynamic inversion is limited to a local neighborhood of the equilibrium, whereas our stability analysis is based on Lyapunov functions, which covers a larger domain than one in [25]. • In [29], the dynamic inversion is only for SISO systems whereas our dynamic inversion scheme is able to deal with MIMO systems since the dynamic inversion has the inputs F and u in (3.24). 3.3.2 Extended High-Gain Observers (EHGOs) Now, we assume that velocity and acceleration terms, x2 , α2 , σx (x, α, u), and σα (x, α, u), which are used in the dynamic inversion, are unknown. EHGOs are designed to estimate the acceleration and velocity terms. The EHGOs for the cart and pendulum systems are designed as xˆ˙ 1 =ˆ x2 + h11 ε3 (x1 − xˆ1 ) xˆ˙ 2 =f¯x (α ˆ1, α ˆ2 , F ) + σ ˆx + σ ˆ˙ x = h13 ε33 α ˆ˙ 1 =α ˆ2 + h23 ε33 (x1 − xˆ1 ) (x1 − xˆ1 ) h21 ε3 (3.25) (α1 − α ˆ1) α ˆ˙ 2 =f¯α (α ˆ1, α ˆ2, F ) + σ ˆα + σ ˆ˙ α = h12 ε23 h22 ε23 (α1 − α ˆ1) (α1 − α ˆ1) where f¯x and f¯α are the nominal values of fx and fα in (3.3); σˆx and σ ˆα denote the estimates of σx and σα , which are the uncertainties in the values of fx and fα , respectively, i.e. fx = f¯x +σx and fα = f¯α + σα . The constants hij for i = 1, 2 and j = 1, 2, 3 are chosen such that the 46 following polynomials s3 + hi1 s2 + hi2 s + hi3 , for i = 1, 2 are Hurwitz and ε3 is a small positive number. Remark 5. The parameters ε1 , ε2 and ε3 should satisfy ε1 ≪ 1, ε2 ≪ 1, and (ε3 /ε2) ≪ 1. This requirement can be intuitively explained as follows. Since the EHGOs’ estimates σ ˆx and σ ˆα are used in dynamic inversion, the observer dynamics should be faster than the dynamic inversion algorithm; hence, (ε3 /ε2 ) ≪ 1. Since the dynamic inversion computes u and F , which are used to implement the controller, it’s dynamics should be faster than the dynamics of the closed-loop system with no uncertainty (3.13); hence ε2 ≪ 1. Since the x-dynamics is much slower than the α-dynamics, ε1 ≪ 1. 3.3.3 Output feedback control Using the dynamic inversions together with the EHGOs, the output feedback control is     α ˆ2 σ ˆx ¯ −fx α1, Mθ sat , F −Mx sat +u  ˙ F   Mθ Mx   ε2  =  σ ˆα α ˆ2 ¯ u˙ , F +Mα sat − νˆd fα α1, Mθ sat Mθ Mα (3.26) where νˆd = −β1 (α1 − α ˆ r ) − β2 Mθ sat α ˆ r = tan−1 vˆext g , vˆext = −γ1 x1 − γ2 xˆ2 47 α ˆ2 Mθ , (3.27) Entire System Fast System EHGOs (3.31) Slow System Fast System Dynamic Inversion (3.30) Slow System Fast System Pendulum Dynamics (3.29) Slow System Cart Dynamics (3.28) Figure 3.3: Multi-time-scale structure for the closed-loop system To protect the system from peaking, the saturation function sat(·) e, sat(e) = if |e| ≤ 1 sign(e), if |e| > 1 is used. The saturation limits Mx , Mα , and Mθ are determined such that the saturation functions will not be invoked under state feedback. 3.3.4 Stability analysis in the presence of uncertainties The closed-loop system is represented in the singularly perturbed form y˙ 1 =ε1 y2 (3.28) y˙ 2 =ε1 fx (θ1 + αr , θ2 , F ) 48 θ˙1 =θ2 + ε1 hα (y, θ, αr , F ) (3.29) θ˙2 =fα (θ1 + αr , θ2 , F ) ε2 z˙ =Az z + ψ(·) − ε2 φ(·) ¯1 ∆1 + B ¯2 ∆2 + ε3 η˙ =Aη η + ε3 B (3.30) 1 ε2 ¯2 ∆3 B (3.31) ¯1 and B ¯2 are given in Appendix and z = [zF , zu ]T with where Aη , B zF = F − F ∗ , zu = u − u∗ With the variables F ∗ and u∗ , the conditions fx (θ1 + αr , θ2 , F ∗ ) − u∗ = 0 and fα (θ1 + αr , θ2 , F ∗) − νd = 0 in (3.23) hold, and ψ(·) = 0 when η = 0, and φ(·) is bounded uniformly in ε2 . The fast variables η = [ηx , ηα ]T , ηx = [ηx1 , ηx2 , ηx3 ]T and ηα = [ηα1 , ηα2 , ηα3 ]T are defined by ηx1 = x2 − xˆ2 x1 − xˆ1 , ηx2 = , 2 ε3 ε3 ηx3 = σx(θ1 + αr ,θ2 , F )− σ ˆx ηα1 = α1 − α ˆ1 α2 − α ˆ2 , ηα2 = , 2 ε3 ε3 ηα3 = σα(θ1 + αr ,θ2 , F )− σ ˆα , We note that fx and fα of (3.28) and (3.29) are bounded uniformly in ε2 and ε3 . The stability analysis for the each subsystem will be done by starting from the fastest one, i.e. the error dynamics of the two EHGOs (3.31) to the slowest one, i.e. the cart dynamics (3.28). The singularly perturbed system can be viewed as a two-time-scale structure if the error dynamics of the two EHGOs (3.31) are the fast subsystem, while other subsystems (3.28), (3.29), (3.30) are the slow one as depicted in Fig.3.3. The boundary layer system, 49 which is obtained by setting ε3 = 0 in (3.31), ε3 dη dt = Aη η is exponentially stable. We note that in the error dynamics of the EHGOs, the matrix Aη ¯1 , B ¯2 , and B ¯3 are uniformly is Hurwitz, and ∆1 , ∆2 , and ∆3 and the constant matrix B bounded in ε3 whose definitions are given in the Appendix. Next, the dynamic inversion (3.30) is slow relative to the EHGOs and fast relative to the systems (3.28) and (3.29) as depicted in Fig.3.3. Setting ε3 = 0 and η = 0, which yields x = xˆ, α = α, ˆ fx = f¯x + σ ˆx , fα = f¯α + σ ˆα , results in the boundary layer system     ∗ ∗ ˙∗  F   −fx (θ1 + αr , θ2 , F ) + u  ε2 z˙ = Az z +    − ε2  ∗ ∗ u˙ fα (θ1 + αr , θ2 , F ) − νd with fx (θ1 + αr , θ2 , F ∗) = cx F ∗ + Gx , fα (θ1 + αr , θ2 , F ∗ ) = −cα F ∗ + Gα Since fx (θ1 + αr , θ2 , F ∗) − u∗ = 0 and fα (θ1 + αr , θ2 , F ∗ ) − νd = 0 with the inputs F ∗ and u∗ , we have  ˙∗   F  ε2 z˙ = Az z − ε2   ∗ u˙ 50 Setting ε2 = 0 yields exponential stability of z = 0. After the EHGOs and dynamic inversion reach quasi-steady state, hierarchically, i.e., ε3 = 0, η = 0, ε2 = 0, z = 0 we have the reduced system for the inverted pendulum on a cart which is the same as the system in (3.17). The reduced system also has a time-scale structure and its stability analysis is given in Section 3.2.4. Typically, a slow variable in a multi-time-scale structure is assumed to be constant although it is evolving slowly. We consider the behavior of the slow variable and define sets for stability analysis of our multi-time-scale structure using Lyapunov functions. Lyapunov functions for three of the four subsystems are defined by; Vy = y T Py y, Vz = z T Pz z, and Vη = η T Pη η, where Py , Pz , and Pη are solutions of Lyapunov equations with right-hand sides equal to the negative identity matrix. The Lyapunov function Vθ is defined in (3.21). The fastest variable η converges quickly into the set {Vη ≤ ρ(ε3 /ε2 )2 } with a positive constant ρ, while the variables y, θ, and z move relatively slowly. We define a set (y, θ, z) ∈ {Vy ≤ a1 } × {Vθ ≤ a2 } × {Vz ≤ a3 } with positive constants a1 , a2 , and a3 . Although a short convergence time period T1 (ε3 ) exists for the fastest variable η, the trajectories of variables y, θ, and z can leave the set {Vy ≤ a1 } × {Vθ ≤ a2 } × {Vz ≤ a3 }. Therefore, we define the superset {Vy ≤ b1 }×{Vθ ≤ b2 }×{Vz ≤ b3 }, where bi , (bi > ai ) can be arbitrarily close to ai , i = 1, 2, 3, (for a sufficiently small ε3 ) that satisfies the condition for the constrained domain for θ. After the variable η converges into the set {Vη ≤ ρ(ε3 /ε2 )2 }, we consider the time period T2 (ε2 ) for convergence of the second fastest variable z. During the time period T2 (ε2 ), the trajectories of y and θ can leave the set {Vy ≤ b1 } × {Vθ ≤ b2 }. To guarantee that the condition for the constrained domain for θ is satisfied during both time periods T1 (ε3 ) and T2 (ε2 ), we define the superset (y, θ) ∈ {Vy ≤ A1 } × {Vθ ≤ A2 } where Ai (Ai > bi ) can be arbitrarily close to bi , i = 1, 2, (for a sufficiently small ε2 ). 51 The main result of this chapter is presented next with the help of the following theorem. Theorem 2. Let X1 be any compact set of (x, α) in the domain Dx × Dα , X2 any compact subset of R2 , and X3 any compact subset of R6 . There exists ε∗ such that for all ε1 < ε∗ , ε2 < ε∗ , ε3 < ε∗ , ε3 < ε∗ ε2 and for all initial states (x(0), α(0)) ∈ X1 , (F (0), u(0)) ∈ X2 , and (ˆ x(0), α ˆ (0)) ∈ X3 , the closed-loop system (3.28) - (3.31) has an exponentially stable equilibrium point, at which x = 0 and α = 0, and the set X1 × X2 × X3 is a subset of the region of attraction. Proof. It is shown in the Appendix that the closed-loop system (3.28) - (3.31) can be written as y˙ = ε1 Ay y + B hx (θ, αr ) + cx zF (3.32) θ˙ = Aθ θ + ε1 Ehα (y, θ, αr , F ) − Bcα zF (3.33) ε2 z˙ = Az z + ψ(·) − ε2 φ(·) ¯1 ∆1 + B ¯2 ∆2 + ε3 η˙ = Aη η + ε3 B (3.34) 1 ε2 ¯2 ∆3 B (3.35) For the first part of the proof, all trajectories starting from (x(0), α(0), F (0), u(0)) ∈ X1 ×X2 , will converge into the desired equilibrium while α1 satisfies the condition for the constrained domain. We are going to show that all trajectories converge to a positive invariant set {Vy ≤ d1 } × {Vθ ≤ (ε∗θ )2 d2 } × {Vz ≤ (ε∗z )2 c3 } × {Vη ≤ (ε3 /ε2 )2 ρ} with the positive constants d1 , d2 , c3 , ρ, ε∗θ = max{ε1, ε∗z }, and ε∗z = max{ε2 , (ε3 /ε2 )}. Note that it is shown that in section 3.2.4, the condition for the constrained domain for α1 with sufficiently small ε1 > 0, is satisfied even though change of variables is used. We are going to use the following hierarchical, repeated process • First, we are going to prove that a subset {Vz ≤ b3 } × {Vη ≤ (ε3 /ε2 )2 ρ} of {Vy ≤ b1 } × {Vθ ≤ b2 }×{Vz ≤ b3 }×{Vη ≤ (ε3 /ε2 )2 ρ} is positively invariant by properly choosing ρ, 52 while the variable η is viewed as fast and the other variables y, θ, and z are considered as slow. The time period T1 (ε3 ) for η to converge into the set {Vη ≤ (ε3 /ε2 )2 ρ}, can be arbitrarily small as ε3 → 0. • Second, it will be shown that the subset {Vθ ≤ c2 } × {Vz ≤ (ε∗z )2 c3 } × {Vη ≤ (ε3 /ε2 )2 ρ} of {Vy ≤ c1 } × {Vθ ≤ c2 } × {Vz ≤ (ε∗z )2 c3 } × {Vη ≤ (ε3 /ε2 )2 ρ} with b1 < c1 < A1 and b2 < c2 < A2 is positively invariant by properly choosing c3 . In this stage, the variable η is already in the set {Vη ≤ (ε3 /ε2 )2 ρ} and the variable z converges rapidly into the set {Vz ≤ (ε∗z )2 c3 } with the convergence time period T2 (ε2 ). The time T2 (ε2 ) can be arbitrarily small as ε2 → 0. • Lastly, it will be claimed that the set of {Vy ≤ d1 } × {Vθ ≤ (ε∗θ )2 d2 } × {Vz ≤ (ε∗z )2 c3 } × {Vη ≤ (ε3 /ε2 )2 ρ} with c1 < d1 < A1 is positively invariant by properly choosing d2 . In this stage, the variables z and η are in the set {Vz ≤ (ε∗z )2 c3 } × {Vη ≤ (ε3 /ε2 )2 ρ} and the fast variable θ converges into {Vθ ≤ (ε∗θ )2 d2 }. In this proof, we are going to show only the first bullet out of three since proofs for the others are the same as the first bullet. There is an upper bound η ≤ (ε3 /ε2 ) ρ/λmin (Pη ), where symbols λmax (N) and λmin (N) are used to denote maximum and minimum eigenvalues of a matrix N, respectively, and the variables y and θ are in a compact set. The derivative of Vz along the trajectory (3.34) is 1 V˙ z≤− ε2 z 2+ 2 ε2 Pz m k ψ η +2Pzm kφ1 y +kφ2 θ +kφ3 z for z ≥ ε∗z 4Pzm az z z ≤− az z 2ε2 2 (3.36) k zb for all (y, θ, z, η) ∈ {Vy ≤ b1 } × {Vθ ≤ b2 } × {Vz = b3 } × {Vη ≤ (ε3 /ε2 )2 ρ}, where az = 1 − 2ε2 kφ1 Pzm , Pzm = λmax (Pz ), and the positive constants kzb , kψ , kφ1 , kφ2 , and kφ3 are 53 independent of ε∗z . The derivative of Vη along the trajectory (3.35) is 1 η 2 + 2Pηm (kη1 y + kη2 θ + kη3 z ) V˙ η ≤ − ε3 1 + (kη4 z +kη5 η )+kη6 η η ε2 aη V˙ η ≤ − η 2, 2ε3 for η ≥ ε3 ε2 4Pηm bη aη (3.37) for all (y, θ, z, η) ∈ {Vy ≤ b1 } × {Vθ ≤ b2 } × {Vz ≤ (ε∗z )2 b3 } × {Vη = ρ(ε3 /ε2 )2 }, where aη = 1 − 2Pηm (ε3 /ε2 )kη5 + ε3 kη6 , Pηm = λmax (Pη ), and the positive constants bη and kη1 to kη6 are independent of (ε3 /ε2 ). By choosing ρ = 16Pη3m (bη /aη )2 and using (3.36) and (3.37), we conclude that the set {Vz ≤ b3 } × {Vη ≤ (ε3 /ε2)2 ρ} is positively invariant. Now, it will be shown that for sufficiently small ε3 , trajectories starting from (F, u) ∈ X2 and (ˆ x, α ˆ ) ∈ X3 enter the corresponding the invariant set of {Vz ≤ b3 } × {Vη ρ(ε3 /ε2 )2 } in the finite time T1 (ε3 ), where limε3 →0 T1 (ε3 ) = 0. There exists the error bound η(0) ≤ kt /ε23 ˙ and z, with a non-negative constant kt . Due to the continuity and boundedness of y, ˙ θ, ˙ we have y(t)−y(0) ≤kf t, θ(t)−θ(0) ≤kf t, z(t)−z(0) ≤kf t with the constant kf > 0. Instead of computing the time Tn when the trajectory η enters into the set {Vη ≤ ρ(ε3 /ε2)2 }, we will find the longer time T1 than Tn to ensure that the trajectory enters the set in a finite time. Using the bound for the initial condition of η and the property of exponential stable Lyapunov function Vη in (3.37), we have Vη ≤ σ2 ε43 exp(−σ1 t/ε3 ) 54 where σ1 = aη /(2Pηm ) and σ2 = Pηm kt2 . Due to ε3 < (ε3 /ε2 ), we obtain ε23 ρ = σ2 ε43 exp(−σ1 T1 /ε3 ) The bound for the time T1 (ε3 ) ∈ (0, T0 ] is obtained T1 (ε3 ) = ε3 σ1 ln( σ2 1 ) ≤ T0 6 ρ4 ε3 2 As ε3 → 0, T1 (ε3 ) → 0. As the second part of the proof, we are going to show that the closed-loop system (3.28) - (3.31) has an exponentially stable equilibrium point, at which x = 0, α = 0. Consider the derivative of the Lyapunov functions Vθ and Vz along the trajectories (3.33) and (3.34), respectively, are V˙ θ≤− km θ 2+ ε1kh1 y + ε1 kh2 θ +(ε1 kh3 + cαm) z V˙ y ≤ ε1 [− y 2 + 2kθ Pym θ (3.38) θ y + 2cxm Pym z y ] (3.39) where Pym = λmax (Py ), cx ≤ cxm , cα ≤ cαm and the positive constants km , kh1 , kh2 , kh3 , and kθ are independent of ε1 , ε2 , ε3 , and (ε3 /ε2 ). √ √ By defining Wy = Vy , Wθ = Vθ , Wz = Vz , and Wη = 55 Vη and using (3.36), (3.37), (3.38), and (3.39), we have D + W ≤ −MW, W = [Wy , Wθ , Wz , Wη ]T   −ε1 k12 −ε1 k13 0   ε1 k11   −ε k (1 − ε k )k ∗ −(ε k + k ∗ ) 0  1 22 22 1 23   1 21 23  M= k k  33 34  ∗ −k32 ( )(1 − ε2 k33 )−   −k31  ε2 ε2    k∗ kM −k41 −k42 −(k43+ 43 ) ε2 1 ε3 ∗ 1 − ε3 k44 − k44 and D + W (·) denotes the upper right-hand derivative, ε3 ε2 and kij∗ for i, j = 1, . . . , 4 are positive constants independent of ε1 , ε2 , ε3 , and where kM = and kij (ε3 /ε2 ). Consider the differential equation U˙ = −MU with U = [Uy , Uθ , Uz , Uη ]T and the same initial condition U(0) = W (0), whose origin is exponentially stable since the leading principal minors of the matrix M can be all positive (i.e., the matrix M can be Hurwitz) by choosing ε1 ≪ 1, ε2 ≪ 1, ε3 ≪ 1, and ε3 /ε2 ≪ 1 small enough. Using a vectorial comparison method in Chapter IX of [58], we conclude that W ≤ U for all t ≥ 0. Therefore, the closedloop system (3.28) - (3.31) has an exponentially stable equilibrium point, at which x = 0, α = 0. 3.4 3.4.1 Simulation and Experiment Simulation results For simulations, the system parameters were assumed to be mc = 0.94 kg, mp = 0.23 kg 2 g = 9.8 m/s , ℓ = 0.3206 m 56 (3.40) The state feedback controller described by (3.4), (3.5), (3.10), and (3.12) was implemented using the following parameter values; The control parameters for the state feedback are γ1 = ε21 k1 , γ2 = ε1 k2 , β1 = 5, β2 = 3 (3.41) where ε1 , k1 , and k2 were chosen as ε1 = 0.2, k1 = 2, k2 = 1 (3.42) For the output feedback controller, we assume that the system dynamics is completely unknown except for the sign conditions in (3.22). The output feedback controller with dynamic inversion described by (3.25), (3.26), and (3.27) was implemented by setting f¯x (·) = 0 and f¯α (·) = 0 in (3.25) and (3.26). The following parameter values were used hi1 = 5, hi2 = 5, hi3 = 4, i = 1, 2 The parameters ε1 and βi , γi , i = 1, 2, are the same as those used in state feedback control - see (3.41) and (3.42). The saturation limits Mx , Mα , and Mθ are chosen to be slightly greater than the maximum absolute values of fx , fα , and α2 , respectively, observed in state feedback control simulations. For both state feedback and output feedback, the initial states x1 (0), x2 (0), α1 (0), and α2 (0) were chosen as x1 (0) = 0 m, x2 (0) = 0 m/s, α1 (0) = 0.8727 rad (50◦ ), 57 α2 (0) = 0 rad/s The initial conditions used for the dynamic inversion and the EHGOs were F (0) = 0, u(0) = 0, xˆ1 (0) = 0.1, xˆ2 (0) = 0.1, fˆx (0) = 0, α ˆ 1 (0) = 0.1, α ˆ 2 (0) = 0.1, fˆα (0) = 0 To investigate the performance of output feedback vis-a-vis state feedback, we simulate two cases with: (ε2 , ε3 ) = (0.02, 0.002) and (ε2 , ε3 ) = (0.01, 0.0001). The results are shown in Figs.3.4 and 3.5. The plots of x1 and x2 are shown in Fig.3.4 and the plots of α1 and α2 are shown in Fig.3.5; these plots have different time horizons since the dynamics of x1 and x2 are slower than the dynamics of α1 and α2 . Both Figs.3.4 and 3.5 indicate that the states converge to the desired values and the output feedback controller is able to recover the performance of the state feedback controller when ε2 and ε3 are chosen small enough. We present results from a second simulation where the initial configuration of the pendulum is almost horizontal with different initial conditions. The initial conditions were assumed to be x1 (0) = 0 m, x2 (0) = −3 m/s, α1 (0) = 1.3963 rad (80◦ ), α2 (0) = π rad/s 2 The time-scale control parameters ε1 , ε2 , and ε3 were chosen as ε1 = 0.05, ε2 = 0.002 and ε3 = 0.0001, and the control parameters β1 and β2 were chosen as β1 = 15, β2 = 10. The remaining control parameters and initial conditions were chosen to be identical to the first simulation. The results, shown in Fig.3.6, indicate that the pendulum and the cart are both successfully stabilized to their desired configuration. 58 −4 1 x (m) 15 −5 10 −6 5 15 16 17 0 −5 0 10 20 30 t (sec) 40 4 2 60 −2 6 x (m/s) 50 −3 2 8 0 10 −2 −4 0 10 20 30 t (sec) 40 50 60 Figure 3.4: Trajectories of x1 and x2 for state feedback (solid line), output feedback with (ε2 , ε3 ) = (0.02, 0.002) (dotted line), and output feedback with (ε2 , ε3 ) = (0.01, 0.0001) (dashed line) 3.4.2 Experimental results The experimental testbed for the inverted pendulum on a cart is shown in Fig.3.7. A 6V-DC motor with a planetary gearhead (reduction ratio 3.71:1) drives the cart on the racks. The angle of the pendulum and the position of the cart are measured by optical encoders that have a resolution of 1024 lines per revolution. The experimental hardware was interfaced with a dSPACE board and the output feedback controller was implemented in the Matlab/Simulink environment with a sampling interval of 0.0006 sec. The dynamics of the inverted pendulum on a cart is described by (3.2) and (3.3) and the nominal parameter values are given by (3.40). The dynamic inversion based output feed59 0.8 −0.1 −0.15 −0.2 −0.25 1 α (rad) 0.6 0.4 0.2 1 2 3 0 −0.2 0 5 10 15 20 25 t (sec) 0 −0.6 −0.5 2 α (rad/s) 0.5 −0.8 −1 −1 −1.5 0 5 10 15 0.5 20 1 25 t (sec) Figure 3.5: Trajectories of α1 and α2 for state feedback (solid line), output feedback with (ε2 , ε3 ) = (0.02, 0.002) (dotted line), and output feedback with (ε2 , ε3 ) = (0.01, 0.0001) (dashed line). back controller described by (3.25), (3.26), and (3.27) was implemented using the following parameter values k1 = 9, k2 = 5, γ1 = ε21 k1 , γ2 = ε1 k2 , β1 = 50, β2 = 30, hi1 = 3, hi2 = 3, hi3 = 1, for i = 1, 2 ε1 = 0.2, ε2 = 0.01, ε3 = 0.005 60 x1 (m), x2 (m/s) 200 100 0 −100 50 100 150 200 250 t (sec) 2 2 α (rad), α (rad/s) 0 F (N) 1 0 −2 0 10 20 30 40 t (sec) 50 60 70 80 300 200 100 0 0 0.5 1 t (sec) 1.5 2 Figure 3.6: Trajectories of x1 and α1 (solid lines), and x2 and α2 (dashed lines) are shown in the top and middle subfigures. In the bottom subfigure, a trajectory of the input F in (3.26) is shown. The initial conditions were chosen as follows x1 (0) = −0.38 m, x2 (0) = 0 m/sec, α1 (0) = 0.19 rad (10.9◦ ), α2 (0) = 0 rad/sec The initial angle of the pendulum was chosen close to the upright configuration such that the cart position did not exceed the physical limit of the racks and the motor did not exceed its torque limit. To reduce the effect of measurement noise, the encoder signals were passed through low-pass filters of bandwidth 1000 Hz. The experimental results are shown in Fig.3.8. Until around 0.5 sec, the pendulum on the 61 Figure 3.7: Experimental testbed for the inverted pendulum on a cart - a product of Quanser [1] cart is held manually while the power switch is off. At around 0.5 sec, the power switch is turned on. The trajectories of x1 and its estimate xˆ1 , and α1 and its estimate α ˆ 1 , all converge to the origin. Within an allowable operation range of the system, we compared experimentally results generated by our control algorithm with ones provided by an LQR controller and the control algorithm in [63]. The LQR controller is designed by following instructions in manufacturer manuals. The stabilization control scheme proposed by [63] was also implemented to check the effectiveness of estimates of uncertainties by the EHGOs. In Fig. 3.9, we show the ultimate boundedness results from system uncertainties, which could be due to friction between cart’s pion and racks, mass of the cart and pendulum, etc. In Fig. 10, the effectiveness of fast estimates by the EHGOs with the dynamic inversion results in the appropriate control input F whereas the other two controllers use high gains to stabilize the system at the equilibrium in the presence of uncertainties. 62 x 1, x ˆ 1(m) 0.2 0 −0.2 −0.4 0 1 2 3 4 t (sec) 5 6 7 8 0 1 2 3 4 t (sec) 5 6 7 8 α1 , α ˆ1 (r ad) 0.2 0.1 0 −0.1 Figure 3.8: Trajectories of x1 and its estimate xˆ1 , and α1 and its estimate α ˆ 1 are shown. The estimated values are indistinguishable from their true (measured) values. 3.5 Conclusion An output feedback controller for stabilization of an inverted pendulum on a cart was presented. From a practical point of view, this is an important contribution since all states of the system are typically not accessible and uncertainties reside in the system. To estimate the unmeasured states and to compensate for the uncertain dynamics, Extended High-Gain Observers were used. To deal with uncertainties in the input coefficients, dynamic inversion was used. Both Extended High-Gain Observers and dynamic inversion introduce fast time scales and this required the controller to be designed using a multi-time-scale structure. The multi-time-scale structure is well-suited for control of underactuated systems, and for the 63 0.4 1 x (m) 0.2 0 −0.2 −0.4 0 5 10 15 20 25 15 20 25 t (sec) 0.2 0 1 α (rad) 0.1 −0.1 −0.2 0 5 10 t (sec) Figure 3.9: Trajectories of x1 and α1 are shown with different control schemes. Solid lines driven by our control scheme, converge into the origin. Dotted lines generated by a LQR controller, have ultimate boundedness. Dash-dot lines provided by the control algorithm in [63] have the biggest ultimate boundedness. inverted pendulum on a cart, additional time scale separation was used to first converge the pendulum to a reference trajectory and then converge the cart to its desired configuration. Using singular perturbation methods, the stability of the closed-loop system was analyzed and exponential stability of the equilibrium was established. Numerical simulations were used to show that the output feedback controller recovers the performance of state feedback and to demonstrate a large region of attraction of the equilibrium. Experimental results were used to demonstrate the feasibility of practical implementation with uncertainties in system parameters. Our future work will focus on extending our approach to output feedback stabilization of other underactuated mechanical systems. 64 F (N) in (3.26) 100 50 0 0 5 10 15 20 25 15 20 25 15 20 25 F (N) in LQR t (sec) 100 50 0 0 5 10 t (sec) F (N) in Ref 100 50 0 0 5 10 t (sec) Figure 3.10: Trajectories of inputs F in (3.26) (the top subfigure), LQR (the middle subfigure), and [63] (the bottom subfigure) are shown. 65 Chapter 4 Output Feedback Control for an Autonomous Helicopter in the Presence of Disturbances 4.1 Introduction In recent years, autonomous helicopter operation has been used in various areas such as above-ground transportation, forest fire monitoring, monitoring criminal activity, and multiagent, multi-objective UAV mission in [60, 67]. However designing a control system for an autonomous helicopter is a challenging task. Since helicopter dynamics have nonaffine control inputs and are underactuated mechanical systems, it is difficult to control and it can easily become unstable compared to other mechanical systems like ground vehicles. In [3, 20, 38, 39], controllers were proposed for helicopters without considering unmeasured states and uncertainties. In [38], a dynamic extension concept from [30] was used to eliminate internal dynamics in an approximate model of a helicopter. In [39], a differential flatness method was proposed through an approximation model. In [39], using the concept of a natural two-time-scale separation, it was possible to design outer-loop (position dynam66 ics) and inner-loop (the rest of the system) controls separately and overcome the feature of underactuated mechanical systems. In [3], while considering dynamics of actuators, the approximated (input-affine) model was combined with backstepping to control a helicopter. Since an approximate model was used in both [38, 39] and [3], and neither disturbance nor model uncertainties were considered, the proposed methods are not valid in the presence of disturbance. In [31, 47, 48], using high gains a two-time-scale approach for helicopter dynamics was proposed considering the helicopter system parameter uncertainties. In [31], based on a two time-scale separation approach between rotational and translational dynamics, a controller with an affine control input model was designed to track the vertical reference trajectory which has unknown phase, amplitude, and frequency, while stabilizing the lateral, longitudinal, and attitude dynamics. In [31], the robustness of the controller to uncertainties was considered through numerical simulations. In [47, 48], a state feedback controller robust to uncertain aerodynamical parameters of the helicopter was proposed, which is based on the linear approximation of control inputs. In [48], high gains were used to dominate uncertain parameters and to render the helicopter rotational dynamics quickly converge into desired trajectories which are control inputs for the translational dynamics. Moreover, nested saturation control was used to prevent the controller having singularities. In [47], the systematic control design process was presented, based on the earlier work of [48]. For the three papers, it was assumed that states of the system were measurable and external disturbances were not considered. Nonaffine control inputs were approximated to affine control inputs. In [14, 19, 65], neural networks were proposed to deal with nonaffine control inputs and uncertainties. In [14], neural networks were used to deal with uncertain, input-nonaffine, nonlinear systems (for example, attitude dynamics of a helicopter). In [19], using neural networks and backstepping scheme, uncertain system parameters and external disturbances were dealt with under state feedback control. In [65], robust adaptive neural networks control was designed in the absence of uncertainties for vertical flight of helicopters, i.e., a 67 single-input-single-output nonaffine system. The system states were estimated by high-gain observers [33] and adaptive nenural networks were used to deal with a nonaffine control input. However, neural networks require training, and selection of the basis and weights often requires significant computation. Disturbance estimators were used to consider uncertainties in helicopter dynamics in [41, 45]. In [45], nonlinear model predictive control with disturbance observers was used to deal with parameter uncertainties and external disturbances under the assumption that states of the system were measurable and control inputs were affine. In [41], output feedback control design for an unmanned helicopter in the presence of uncertainties was developed for the rotational dynamics; the dynamic inversion scheme was used to deal with nonaffine control inputs and an extended high-gain observer estimated unmeasured states, system parameter uncertainties, and external disturbances. In this chapter, the output feedback control of a helicopter is proposed as an extension of [41] from a Single-Input-Single-Output (SISO) systems to MIMO systems. We propose to use an Extended High-Gain Observer (EHGO) to estimate the system states and disturbances of a helicopter instead of a neural network. In order to deal with nonaffine control inputs in a helicopter, the EHGO is used together with the method of dynamic inversion. The combined system has five time scales: two-time scales are required by plant dynamics between translational and rotational dynamics; the third time-scale is required by dynamic inversion for the translational dynamics; the fourth time-scale is required by the dynamic inversion for the rotational dynamics; and the fifth, fastest, time scale is required by the EHGO for estimation of the states, uncertain system parameter, and external disturbances. This chapter is organized as follow. In Section 4.2, a helicopter model is given. In Section 4.3, we define the problem and design state feedback control in the absence of uncertainties. The stability analysis for the closed-loop system is conducted under state feedback. In Section 4.4, output feedback control for full helicopter dynamics is designed in the presence of uncertainties using the EHGO and dynamic inversion. Based on the singular perturbation 68 method, the stability for the multi-time-scale closed-loop system is analyzed. The effectiveness of the proposed control scheme is verified through numerical simulations in Section 4.5. Section 4.6 presents concluding remarks. 4.2 Dynamics of a Helicopter In this section, the rigid body dynamics of a helicopter are presented. It is assumed that the actuator dynamics are sufficiently fast compared to the rigid body dynamics and can be ignored in the mathematical model. The rigid body dynamics are based on [38] and written as   b   b b   b   M 0   v˙   ω × mv   f  , = +   τb ω b × Iω b ω˙ b 0 I where M = diag[m, m, m] and I = diag[Ixx , Iyy , Izz ] are mass matrix and the inertia matrix of the helicopter, respectively; m is the mass of a helicopter; Ixx , Iyy , and Izz are the principle moments of inertia; v b = vxb , vyb , vzb T is the body velocity vector; vib for i = x, y, z are the linear velocities in the x, y, and z directions; τ b = [τ1b , τ2b , τ3b ]T is the torque, specified later in next Subsection 4.2.1; and ω b = [ω1 , ω2 , ω3 ]T is the angular velocity vector where ωj for j = 1, 2, 3 are the angular velocities about x, y, and z axes. The input force matrix is expressed as   XM  fb =   YM + YT  ZM     0       + RT (Θb )  0  ,       mg 69 where the set of forces (XM , YM , YT , and ZM ) or moments (RM , MM , NM , and MT ) acting on a helicopter is given by XM = −TM sin a1s , YM = TM sin b1s , ZM = −TM cos a1s cos b1s , RM ⋍ YT = −TT , ∂RM b1s − QM sin a1s , ∂b1s NM ⋍ −QM cos a1s cos b1s , MM ⋍ ∂MM a1s + QM sin b1s , ∂a1s MT = −QT . Q 1.5 Q QM = CM TM + DM and QT = CTQ TT1.5 + DTQ are the approximate rotor torque equations Q for main and tail rotors, respectively (we follow the model in [38]). CM and CTQ are the Q thrust coefficients of TM and TT , respectively, and DM and DTQ are the lift drag coefficients of TM and TT , respectively. The system parameters are given in Section 4.5. a1s and b1s are longitudinal and lateral tilts of the tip path plane of the main rotor with respect to the shaft, respectively; and TM and TT are main rotor thrust and tail rotor thrust, respectively. The gravitational acceleration is g = 9.8 m/s2 and the rotation matrix R(Θb ) is defined by   cθcψ sφsθcψ − cθsψ cφsθcψ + sφsψ  R(Θb ) =   cθsψ sφsθsψ + cφcψ cφsθsψ − sφcψ  −sθ sφcθ cφcθ    ,   where Θb = [φ, θ, ψ]T , φ (roll), θ (pitch), ψ (yaw) are the Euler angels and s(·) = sin(·), c(·) = cos(·). Control inputs, TM , TT , a1s , and b1s are used for a helicopter control under the assumption that the dynamics of a1s and b1s are sufficiently fast. In Table. 4.1, Table 4.1: Parameters in a helicopter hM hT lM lT yM Distance from COM(Center of Mass) to the main rotor along the z axis Distance from COM to the tail rotor along the z axis Distance from COM to the main rotor along the x axis Distance from COM to the tail rotor along the x axis Distance from COM to the main rotor along the y axis 70 parameters are given for the helicopter. The side and top view of the helicopter is shown with the parameters in Fig. 4.1. 4.2.1 Rotational Dynamics of a Helicopter The helicopter dynamics can be divided into two parts, an inner-loop and an outer-loop due to a natural time-scale separation [38] as shown in Fig. 4.2. In Fig. 4.2, φ, θ, ψ are actual Euler angle values of a helicopter, and Θr = [φr , θr , ψr ]T is the reference trajectory which is generated from the controller of translational dynamics, CT . The inner-loop is rotational dynamics of a helicopter whereas the outer-loop is translational dynamics. The inner-loop dynamics of a helicopter (attitude dynamics) are given by ˙ b = Ψ(Θb )ω b Θ b ω˙ = −I −1 b (4.1) b b ω × Iω + Bτ , Figure 4.1: Side and top view of the helicopter 71 Figure 4.2: The block diagram is shown for rotational dynamics (inner-loop) and translational dynamics (outer-loop) control structure via two time-scale separation. The blocks PR , PT are rotational dynamics and translational dynamics, respectively. The blocks CR , CT are controllers for rotational and translational dynamics, respectively. χ is the position of the helicopter and χr and Θr are reference trajectories for the translational and rotational dynamics, respectively. where B = diag[1/Ixx , 1/Iyy , 1/Izz ]; and   1 sin φ tan θ cos φ tan θ  Ψ(Θb ) =  cos φ − sin φ  0  0 sin φ sec θ cos φ sec θ    ,    0 − sin θ  1  Ψ−1 (Θb ) =   0 cos φ sin φ cos θ  0 − sin φ cos φ cos θ       (4.2) In order to make (4.1) compatible to extended high-gain observer method, the following coordinates are used as ˙b =Ω Θ (4.3) Ω˙ = FI (Θb , Ω, τ b ), where Ω = Ψ(Θb )ω b = ˙ θ, ˙ ψ˙ φ, T is a vector of the Euler angle rates. In (4.3), Ω˙ = ¨ θ, ¨ ψ] ¨ T is obtained from (4.1) as FI (Θb , Ω, τ b ) = [φ, φ¨ = (ω2 cos φ tan θ − ω3 sin φ tan θ)φ˙ + (ω2 sec2 θ sin φ + ω3 cos φ sec2 θ)θ˙ (Izz − Iyy ) (Izz − Ixx ) (Iyy − Ixx ) ω2 ω3 + ω1 ω3 sin φ tan θ − ω1 ω2 cos φ tan θ Ixx Iyy Izz 1 b sin φ tan θ b cos φ tan θ b τ + τ2 + τ3 , + Ixx 1 Iyy Izz − 72 (4.4) (Iyy − Ixx ) (Izz − Ixx ) ω1 ω3 cos φ + ω1 ω2 sin φ θ¨ = −(ω2 sin φ + ω3 cos φ)φ˙ + Iyy Izz cos φ b sin φ b τ − τ + Iyy 2 Izz 3 (4.5) ψ¨ = (ω2 cos φ sec θ − ω3 sec θ sin φ)φ˙ + (ω2 sec θ sin φ tan θ + ω3 cos φ sec θ tan θ)θ˙ (Izz − Ixx ) (Iyy − Ixx ) ω1 ω3 sec θ sin φ − ω1 ω2 cos φ sec θ Iyy Izz sec θ sin φ b cos φ sec θ b τ2 + τ3 , + Iyy Izz + (4.6) and τ = [τ1b , τ2b , τ3b ]T is given by ∂RM b1s − QM sin(a1s ) + TM sin(b1s )hM − TM cos(a1s ) cos(b1s )yM − TT hT , ∂b1s ∂MM τ2b = a1s + QM sin(b1s ) − QT + TM sin(a1s )hM − TM cos(a1s ) cos(b1s )lM , ∂a1s τ1b = τ3b = − QM cos(a1s ) cos(b1s ) − TM sin(b1s )lM + TT lT , 4.2.2 Translational Dynamics of a Helicopter The translational dynamics of a helicopter (i.e., position dynamics) are given by P˙ =v b 1 v˙ = R(Θ)f b = FP , m b 73 (4.7) where P = [x, y, z]T , v b = [x, ˙ y, ˙ z] ˙ T are a position vector and a velocity vector in North-EastDown orientation, respectively, and FP = [fx , fy , fz ]T is fx = 1 m {−TM cos θ cos ψ sin a1s + (sin φ sin θ cos ψ − cos φ sin ψ)[TM sin b1s − TT ] − TM (cos φ sin θ cos ψ + sin φ sin ψ) cos a1s cos b1s } fy = 1 m {−TM cos θ sin ψ sin a1s (4.8) + (sin φ sin θ sin ψ + cos φ cos ψ)[TM sin b1s − TT ] − TM (cos φ sin θ sin ψ − sin φ cos ψ) cos a1s cos b1s } fz = 1 m {TM sin θ sin a1s + (sin φ cos θ)[TM sin b1s− TT ] − TM cos φ cos θ cos a1s cos b1s } + g 4.3 Stability Analysis in the Absence of Disturbances In this section, we describe a control strategy for a helicopter in the absence of uncertainties, based on a time-scale separation strategy between the translational and rotational dynamics. The control objective is to follow given reference trajectories xr , yr , zr , and ψr with inputs TM , TT , a1s and b1s . In this section we assume that all system states and dynamic models are known. For the translational dynamics, we neglect flapping dynamics a1s and b1s and the tail rotor thrust TT in y-direction in [38], [31], [48], [47] and the translational dynamics are given by    1 P¨ = R(Θ)   m  0 0 −TM 74      0     + 0        g (4.9) We are going to design a controller which renders the rotational dynamics faster than the translational dynamics while dealing with nonaffine control inputs. Based on the time-scale separation between the rotational and translation dynamics, new control inputs ur = [TT , a1s , b1s ]T for the rotational dynamics and ut = [TM , φd , θd ]T for the translational dynamics are designed, where φd and θd in dynamic inversion control will be specified later. As depicted in Fig. 4.2, an inner-loop controller for the rotational dynamics is designed to force the rotational dynamics to follow the desired reference trajectories φr , θr , and ψr with a fast convergence rate. An outer-loop controller for the translational dynamics is designed to provides desired reference trajectories φr , θr , and ψr for the rotational dynamics and to obtain slower translation dynamics than rotational dynamics. A dynamic inversion controller will be designed to deal with nonaffine input forms. 4.3.1 State feedback control With state variables χx = [x1 , x2 ]T = [x, x] ˙ T, χy = [y1 , y2 ]T = [y, y] ˙ T, χz = [z1 , z2 ]T = [z, z] ˙T ˙ T, Θφ = [φ1 , φ2] = [φ, φ] ˙ T, Θθ = [θ1 , θ2 ] = [θ, θ] ˙T Θψ = [ψ1 , ψ2 ] = [ψ, ψ] T T (4.10) T the translational dynamics of (4.9) are rewritten as (4.11) χ˙ = Aχ + BFO where χ = [χTx , χTy , χTz ]T ∈ Dχ , Dχ ⊂ R6 is bounded, A = block diag[A1 , A1 , A1 ], B = block diag[B1 , B1 , B1 ] with    0 1  A1 =  , 0 0 75    0  B1 =   1 (4.12) and     fx        FO =  f  y =    fz 1 − (cos φ1 sin θ1 cos ψ1 + sin φ1 sin ψ1 )TM m 1 − (cos φ1 sin θ1 sin ψ1 − sin φ1 cos ψ1 )TM m 1 − (cos φ1 cos θ1 )TM + g m       (4.13) We note that in the translational dynamics in (4.11), control inputs are φ1 , θ1 , and TM where φ1 and θ1 are viewed as the virtual control provided by the rotational dynamics as the fast time scale. A target system for the translational dynamics is given by χ˙ r = (A − BL)χr + Buc (4.14) where χr = [xr , x˙ r , yr , y˙ r , zr , z˙r ]T ∈ Dχr , Dχr ⊂ R6 is bounded, the matrix L = block diag[Lχ , Lχ , Lχ ] with Lχ = [kp , kv ] is block diagonal such that the matrix (A − BL) is Hurwitz, and uc = [rx , ry , rz ]T is a bounded commend input and continuously differentiable. To track the reference trajectories for the translational dynamics, it is required to find φr , θr and Tm∗ given ψr such that  ∗ fx (φr , θr , ψr , TM )   ∗ Fχ =   fy (φr , θr , ψr , TM )  ∗ fz (φr , θr , TM )     + Lχ − uc (t) = 0   (4.15) The dynamic inversion in the translational dynamics is used to obtain approximated solutions φd , θd , and TM for (4.15). With the state variables in (4.10), the rotational dynamics of a helicopter in (4.3) can be represented as ˙ = AΘ + BFI , Θ 76 (4.16) where Θ = [ΘTφ , ΘTθ , ΘTψ ]T ∈ DΘ ⊂ R6 and FI = [fφ , fθ , fψ ]T = FI (Θb , Ω, τ b ) ∈ R3 in (4.3). The domain DΘ is given by DΘ = DΘφ ×DΘθ ×DΘψ , DΘp = {−ap ≤ p ≤ ap }×{bp ≤ p˙ ≤ bp }, for p = φ, θ, ψ with bounded 0 < ap < π/2 and b > 0. There are two tasks of a controller for the rotational dynamics: one is to deal with nonaffine input forms which is required to find solutions a∗1s , b1s , and TT∗ such that  fφ (Θ, TM , a∗1s , b∗1s , TT∗ )+kφ1 (φ1 −φd )+kφ2 φ2 )       ∗ ∗ ∗ FΘ =  fθ (Θ, TM , a1s , b1s , TT )+kθ1 (θ1 −θd ) + kθ2 θ2 )   = 0,   fψ (Θ, TM , a∗1s , b∗1s , TT∗ )+kψ1 (ψ1 −ψr )+kψ2 ψ2 ) (4.17) where fφ , fθ , and fψ given in in (4.13), and kφi , kθi , kψi for i = 1, 2 are high gains specified later; and the other is to make the rotational dynamics converge quickly into the desired reference trajectories φd , θd , and ψr provided by a dynamic inversion controller. For the rotational dynamics, the dynamic inversion is used to find approximated solutions a1s , b1s , and TT for (4.17). Two dynamic inversion controllers in [43, 44] are designed to deal with nonaffine inputs ut = [TM , φd, θd ]T and ur = [TT , a1s , b1s ]T . The dynamic inversion for the translational dynamics is ε2 u˙ t = − T ∂Fχd ∂ut (4.18) Fχd (Θ, χ, ut, uc ) where ut ∈ Dt ⊂ R3 ; (∂Fχd /∂ut ) is given in Appendix Appendix B;   fx (φd , θd , ψr , TM )  Fχd (Θ, χ, ut, uc ) =   fy (φd , θd , ψr , TM )  fz (φd , θd , TM )     + Lχ − uc (t),   77    rx (t)     uc (t) =  r (t)   y   rz (t) (4.19) fx , fy and fz are given in (4.13); and rx (t), ry (t), and rz (t) are bounded reference commends. The dynamic inversion for the rotational dynamics T ∂FΘ ε3 u˙ r = − ∂ur (4.20) FΘ (Θ, ut , ur , ψr ) where ur ∈ Dr ⊂ R3 , (∂FΘ /∂ur ) is given in Appendix Appendix B, and    fφ (Θ, ut, ur )+kφ1 (φ1 −φd )+kφ2 φ2 )     FΘ (Θ, ut, ur , ψr ) =  f (Θ, u , u )+k (θ −θ ) + k θ ) θ t r θ 1 d θ 2   1 2   fψ (Θ, ut , ur )+kψ1 (ψ1 −ψr )+kψ2 ψ2 ) (4.21) To obtain the fast rotational dynamics, kφi , kθi , kψi for i = 1, 2 are chosen as kφ1 = kθ1 = kψ1 = k1 , ε21 kφ2 = kθ2 = kψ2 = k2 ε1 (4.22) where kr,1 and kr,2 are positive constants independent of ε1 and ε2 . The assumption for the Jacobian matrices (∂Fχd /∂ut ) and (∂FΘ /∂ur ) are as follows. Assumption 4. In the domains Dχ , Dχr , DΘ , Dt and Dr , • the Jacobian matrices (∂Fχd /∂ut ) and (∂FΘ /∂ur ) are nonsingular; • λmin (∂Fχd /∂ut )(∂Fχd /∂ut )T > a with a > 0 and λmin (∂FΘ /∂ur )(∂FΘ /∂ur )T >b with b > 0; • Fχ and FΘ are continuously differentiable, where λmin (P ) denotes the minimum eigenvalue of the matrix P . 4.3.2 Stability analysis Using the systems (4.11), (4.14), and (4.16) with the control (4.18) and (4.20), the standard singularly perturbed form for the closed-loop system is derived. Error variables for the 78 translational dynamics are ex1 = x1 − xr , ex2 = x2 − x˙ r ey1 = y1 − yr , ey2 = y2 − y˙ r ez1 = z1 − zr , ez2 = z2 − z˙r (4.23) Time-scaled variables for the rotational dynamics are eφ2 = ε1 (φ2 − φ˙ r ) eφ1 = φ1 − φr , eθ2 = ε1 (θ2 − θ˙r ) eθ1 = θ1 − θr , (4.24) eψ2 = ε1 (ψ2 − ψ˙ r ) eψ1 = ψ1 − ψr , Error variables of the dynamic inversion for the translational and rotational dynamics are st = ut − u∗t and sr = ur − u∗r with    TM     ut =  φ  d ,   θd   ∗  TM   u∗t =   φr  θr  ,      TT     ur =   a1s  ,   b1s   ∗  TT    ∗  u∗r =   a1s    b∗1s (4.25) u∗t and u∗r are satisfied with FΘ (Θ, ut, u∗r , ψr ) = 0, Fχ (Θ, χ, u∗t , uc ) = 0   ∗  fx (φr , θr , ψr , TM )     ∗ Fχ (Θ, χ, u∗t , uc ) =  f (φ , θ , ψ , T )  y r r r M  + Lχ − uc (t),   ∗ fz (φr , θr , TM ) (4.26) In view of Assumption 4 and the Lipschitz property of Fχd in its arguments, it is reasonable to have the following assumption on the relation between Fχd and s : Assumption 5. kl st ≤ Fχd (Θ, χ, ut , uc ) − Fχd (Θ, χ, u∗t , uc ) ≤ kl s with some constants 79 kl , kp > 0 on the domains Dχ , Dχr , DΘ , Dt , and Dr So, u∗r is a function, u∗r = u∗r (Θ, ut , ψr ) and u∗t a function, u∗t = u∗t (Θ, χ, uc). Error dynamics of translational dynamics are represented as (4.27) e˙ χ = Aχ eχ + BFχ (Θ, χ, ut, uc ) with eχ = [eTx , eTy , eTz ]T and Aχ = A − BL. With the time-scaled variables in (4.24), the closed-loop standard singular perturbed form for error dynamics of the rotational dynamics is ¨ r − ε1 k2 Θ ˙ r] ε1 e˙ Θ = Aθ eΘ + B[k1 (Θd − Θr ) + ε21 FΘ (Θ, ut , ur , ψr ) − ε21 Θ (4.28) with Aθ = A − BLθ , eΘ = [eTφ , eTθ , eTψ ]T , eφ = [eφ1 , eφ2 ]T , eθ = [eθ1 , eθ2 ]T , eψ = [eψ1 , eψ2 ]T , Lθ = block diag[L1 , L1 , L1 ], L1 = [k1 , k2 ], Θd = [φd , θd , ψr ]T , and Θr = [φr , θr , ψr ]T . The error dynamics for the variable st are ε2 s˙ t = − ∂Fχd ∂ut T Fχd (Θ, χ, ut , uc ) − ε2 u˙ ∗t (4.29) where u˙ ∗t is u˙ ∗t = ∂ut ∂Θ ˙ + Θ ∂ut ∂χ χ˙ + ∂ut ∂uc u˙ c (4.30) The error dynamics for sr is ∂FΘ ε3 s˙ r = − ∂ur T FΘ (Θ, ut, ur , ψr ) − ε3 u˙ ∗r 80 (4.31) where u˙ ∗r is u˙ ∗r = ∂ur ∂Θ ˙ − Θ ∂ur ∂ut 1 ε2 ∂Fχd ∂ut T Fχd + ∂ut ∂ψr ψ˙ r (4.32) Using (4.27), (4.28), (4.29), and (4.31), the singularly perturbed form is e˙ χ = Aχ eχ + BFχ (eΘ + Θer , eχ + χr , s + u∗ , uc ) (4.33) ¨ r − ε1 k2 Θ ˙ r] ε1 e˙ Θ = Aθ eΘ + B[k1 (Θd − Θr ) + ε21 FΘ (Θ, s + u∗ , ψr ) − ε21 Θ ε2 s˙ t = − ∂Fχd ∂ut ε3 s˙ r = − ∂FΘ ∂ur (4.34) T Fχd (Θ, eχ + χr , st + u∗t , uc) − ε2 u˙ ∗t (4.35) FΘ (Θ, eχ + χr , st + u∗t , sr + u∗r , ψr ) − ε3 u˙ ∗r (4.36) T We have a three-time-scale structure in (4.33), (4.34), (4.35), and (4.36). We note that the small parameters ε1 , ε2 , and ε3 have the relation, 0 < ε3 ≪ ε2 ≪ ε1 ≪ 1. Since u˙ ∗t in (4.35) has the term (1/ε1), it is required to have 0 < ε2 ≪ ε1 ≪ 1. ε3 is required to be 0 < ε3 ≪ ε2 ≪ 1 since u˙ r in (4.36), has the term (1/ε2 ). Stability analysis starts from the fastest boundary layer system of the dynamic inversion in (4.36). The boundary layer system can be obtained setting ε3 = 0 on the right-hand side of (4.36) ε3 s˙ r = − ∂FΘ ∂ur T FΘ (Θ, eχ + χr , st + u∗t , sr + u∗r , ψr ) (4.37) A Lyapunov function for the boundary layer system is Vr = FΘT FΘ /2. The derivatives of the Lyapunov function along trajectories in (4.37) and under Assumption 4. is V˙ r = − + FΘT 1 ε3 FΘT ∂FΘ ∂Θ ∂FΘ ∂ur ˙ + Θ ∂FΘ ∂ur ∂FΘ ∂χ T χ˙ − 81 FΘ 1 ε2 ∂FΘ ∂ut ∂Fχd ∂ut (4.38) T Fχd Then, V˙ r ≤ − 1 δ1 δ2 + + δ3 ε1 ε2 Vr + 2ε3 λmin(Pr ) (4.39) where Pr = (∂FΘ /∂ur )(∂FΘ /∂ur )T ; and δ1 , δ2 , and δ3 are positive constants related to the ˙ F T (∂FΘ /∂ut )(∂Fχ /∂ut )T Fχ , and F T (∂FΘ /∂χ)χ˙ upper bounds for the terms FΘT (∂FΘ /∂Θ)Θ, Θ Θ and independent of ε3 . With ε3 ≪ ε2 ≪ ε1 ≪ 1, V˙ r is negative with Vr = 0. Now the fastest variable ur reached in the quasi-steady state, i.e. ur = u∗r and the other subsystems (4.33), (4.34), and (4.35) are viewed as the reduced system with ε3 = 0 and sr = 0. The reduce system has a multi-time-scale structure in which (4.35) is fast and the other two, (4.33) and (4.34) are slow. By setting ε2 = 0, and sr = 0 (i.e., the fastest system reaches the quasi-steady state) on the right-hand side of (4.35), the boundary layer systems for the second fastest dynamic inversion is obtained by ∂Fχd ∂ut ε2 s˙ t = − T Fχd (Θ, eχ + χr , st + u∗t , uc ) (4.40) A Lyapunov function Vt = (FχTd Fχd )/2 is defined and the derivative of the Lyapunov functiono is V˙ t = − + FχTd 1 ε2 FχTd ∂Fχd ∂Θ V˙ t ≤ − ∂Fχd ∂ut ∂Fχd ∂ut ˙ + Θ ∂Fχd ∂χ 1 2ε2 λmin (Pt ) T Fχd χ˙ + Vt + ∂Fχd ∂uc δ4 + δ5 ε1 (4.41) u˙ c (4.42) where Pt = (∂Fχd /∂ut )(∂Fχd /∂ut )T , δ4 and δ5 are positive constants related to the upper ˙ FχT (∂Fχ /∂χ)χ, bounds for the terms FχTd (∂Fχd /∂Θ)Θ, ˙ and FχTd (∂Fχd /∂uc )u˙ c , and independ d dent of ε2 . With ε2 ≪ ε1 ≪ 1, V˙ t is negative with Vt = 0. 82 Two fast variables ur and ut reach in the quasi-steady state, i.e., ur = u∗r and ut = u∗t  TT∗  TT −  ∗ sr = ur − u∗r =   a1s − a1s  b1s − b∗1s     = 0,    ∗ TM  TM −  st = ut − u∗t =   φd − φr  θd − θr    =0   (4.43) With Θr = [φd , θd , ψr ]T and Θr = [φr , θr , ψr ]T given below (4.28) and (4.43), we obtain (Θd − Θr ) = 0. Setting εi = 0 for i = 1, 2, st = 0, and sr = 0 on the right-hand side of both (4.33) and (4.34), we obtain the reduce system e˙ χ = Aχ eχ , (4.44) ¨ r − ε1 k2 Θ ˙r ε1 e˙ Θ = AΘ eΘ − ε21 Θ (4.45) Now, in the reduced system, (4.44) and (4.45), the rotational dynamics are faster than the translational dynamics. The boundary layer system for (4.45) is obtained setting ε1 = 0 on the right-hand side of (4.45), ε1 e˙ Θ = AΘ eΘ (4.46) which has asymptotic stability at eΘ = 0. Setting ε1 = 0 and eΘ = 0, the reduced system for (4.44) is e˙ χ = Aχ eχ (4.47) which is asymptotically stable at eχ = 0. By using a composite Lyapunov function, the effect of the interconnections for the closed-loop system (4.33), (4.34), (4.35), and (4.36), can be considered. The procedure for the effect of the interconnections are similar to the output feedback control stability analysis. So, we omit the procedure which will be shown 83 in Section 4.4. 4.4 Stability Analysis in the Presence of Disturbances In this section, output feedback control will be designed using the extended high-gain observer to estimate unmeasured system states and external disturbances. Using the singular perturbation method, stability analysis for the closed-loop system will be conducted. Let us consider the case where the helicopter dynamics in (4.11) and (4.16) have external disturbances as follows. χ˙ = Aχ + B[FO (Θ, ut) + σχ (t)] yχ = Cχ (4.48) ˙ = AΘ + B[FI (Θ, ut , ur ) + σΘ (t)] Θ yΘ = CΘ where χ and FO (Θ, ut ) for the translational dynamics and Θ and FI (Θ, ut , ur ) for the rotational dynamics are given right after below of (4.11) and (4.16), respectively; yχ and yΘ are measurements; and C = block diag[C1 , C1 , C1 ] is with C1 = [1, 0]. It is assumed that the external disturbances, σχ (t) = [σx , σy , σz ]T and σΘ (t) = [σφ , σθ , σψ ]T have the following properties. Assumption 6. The functions σχ (t) and σΘ (t) are continuously differentiable. 84 4.4.1 Design of Extend High-Gain Observers (EHGOs) Unknown external disturbances σχ and σΘ , and unmeasured states will be estimated using the EHGO ˆ ut ) + σˆχ (t)] + Hχ (ε4 )(yχ − C χ) χ ˆ˙ = Aχˆ + B[FO (Θ, ˆ ˆ σ ˆ˙ χ = Hχe (yχ − C χ) ˆ˙ = AΘ ˆ + B[FI (Θ, ˆ ut , ur ) + σˆΘ (t)] + HΘ (ε4 )(yΘ − C Θ) ˆ Θ (4.49) ˆ σ ˆ˙ Θ = HΘe (yΘ − C Θ) ˆ = [Θ ˆT,Θ ˆT,Θ ˆ T ]T , respectively, where the estimates of χ and Θ are χˆ = [χˆTx , χ ˆTy , χ ˆTz ]T and Θ φ θ ψ with χ ˆx = [ˆ x1 , xˆ2 ]T , χˆy = [ˆ y1 , yˆ2 ]T , ˆ φ = [φˆ1 , φˆ2 ]T , Θ ˆ θ = [θˆ1 , θˆ2 ]T , Θ χˆz = [ˆ z1 , zˆ2 ]T (4.50) ˆ ψ = [ψˆ1 , ψˆ2 ]T Θ and σ ˆχ = [ˆ σx , σ ˆy , σ ˆz ]T and σ ˆΘ = [ˆ σφ , σ ˆθ , σ ˆψ ]T are estimates of σχ (t) and σΘ (t), respectively. The observer gains, Hχ , Hχe , HΘ and HΘe are given by Hχ = block diag[H1 , H2 , H3 ], Hi = [hi1 /ε4, hi2 /ε24 ]T , HΘ = block diag[H4 , H5 , H6 ] (4.51) for i = 1, . . . , 6 Hχe = block diag[h13 /ε34 , h23 /ε34 , h33 /ε34 ], HΘe = block diag[h43 /ε34 , h53 /ε34 , h63 /ε34 ] where the components hi1 , hi2 , and hi3 of the high gains are chosen such that the polynomials λ3 + hi,1 λ2 + hi,2 λ + hi,3 for i = 1, . . . , 6 (4.52) are Hurwitz and the control parameter ε4 > 0 is small enough. The relation between ε3 and ε4 is ε4 ≪ ε3 ≪ 1 since the dynamic inversion uses estimates provided by the EHGO. 85 4.4.2 Output feedback control With the EHGO in (4.49), the output feedback control, based on the dynamic inversion, is designed as T ∂Fχd ε2 u˙ t = − ∂ut ∂FΘ ε3 u˙ r = − ∂ur ˆ χs ) Fχs (Θs , χs , ut , uc , σ (4.53) T ˆ Θs ) FΘs (Θs , ut , ur , ψr , σ where Θs = [ΘTφs , ΘTθs , ΘTψs ]T χs = [χTxs , χTys , χTzs ]T       θ1 ψ1 φ1             = = , Θ , Θ Θ φs =  ˆ ˆ ˆ θ ψ s s θ2 ψ2 φ2      Mθ sat Mψ sat Mφ sat (4.54) Mφ Mθ Mψ       y1 x1 z1       = , χ χxs =  = , χ   ,   y z s s yˆ2 xˆ2 zˆ2 My sat Mx sat Mz sat My Mx Mz T T σ ˆs = [ˆ σχTs , σ ] ˆΘ s σ ˆ χs  σx /M1 )  M1 sat(ˆ  = σy /M2 )  M2 sat(ˆ  M3 sat(ˆ σz /M3 )    ,   σ ˆ Θs  σφ /M4 )  M4 sat(ˆ  = σθ /M5 )  M5 sat(ˆ  M6 sat(ˆ σψ /M6 )    ,   (4.55) and Fχs and FΘs are Fχs  σx /M1 ) + kx1 x1 + kx2 Mx sat(ˆ x2 /Mx ) − rx (t)  fx (φd , θd , ψ1 , TM ) + M1 sat(ˆ  = σy /M2 ) + ky1 y1 + ky2 My sat(ˆ y2 /My ) − ry (t)  fy (φd , θd , ψ1 , TM ) + M2 sat(ˆ  fz (φd , θd , ψ1 , TM ) + M3 sat(ˆ σz /M3 ) + kz1 y1 + kz2 Mz sat(ˆ z2 /Mz ) − rz (t) 86     (4.56)   FΘs  ˆ σφ /M4 ) + kφ1 (φ1 − φd ) + kφ2 Mφ sat(φ2 /Mφ )   fφ (Θ, ut, ur ) + M4 sat(ˆ   = σθ /M5 ) + kθ1 (θ1 − θd ) + kθ2 Mθ sat(θˆ2 /Mθ )    fθ (Θ, ut , ur ) + M5 sat(ˆ   ˆ fψ (Θ, ut , ur ) + M6 sat(ˆ σψ /M6 ) + kψ1 (ψ1 − ψd ) + kψ2 Mψ sat(ψ2 /Mψ )  (4.57) The saturation function, sat(·) is defined by sat(k) = k |k| ≤ 1 (4.58) sign(k) |k| > 1 The saturation levels Mj for j = φ, θ, ψ, x, y, z, 1, 2, . . . , 6 are determined such that the saturation function will not be activated under the state feedback. 4.4.3 Stability analysis in the presence of disturbances T T ] for the EHGO are given by The fast error variables η = [ηχT , ηΘ ηχ = [ηxT , ηyT , ηzT ]T , ηΘ = [ηφT , ηθT , ηψT ]T ηx = [ηx1 , ηx2 , ηx3 ]T , ηy = [ηy1 , ηy2 , ηy3 ]T , ηz = [ηz1 , ηz2 , ηz3 ]T ηφ = [ηφ1 , ηφ2 , ηφ3 ]T , ηθ = [ηθ1 , ηθ2 , ηθ3 ]T , ηψ = [ηψ1 , ηψ2 , ηψ3 ]T (4.59) where x1 − xˆ1 x2 − xˆ2 , ηx2 = , ηx3 = σx (t) − σ ˆx (t) 2 ε4 ε4 y1 − yˆ1 y2 − yˆ2 = , ηy2 = , ηy3 = σy (t) − σ ˆy (t) 2 ε4 ε4 z1 − zˆ1 z2 − zˆ2 = , ηz3 = σz (t) − σ ˆz (t) , ηz2 = 2 ε4 ε4 ηx1 = ηy1 ηz1 87 (4.60) and φ2 − φˆ2 φ1 − φˆ1 , ηφ2 = , ηφ3 = σφ (t) − σ ˆφ (t) 2 ε4 ε4 θ1 − θˆ1 θ2 − θˆ2 ηθ1 = , η = , ηθ3 = σθ (t) − σ ˆθ (t) θ2 ε24 ε4 ψ2 − ψˆ2 ψ1 − ψˆ1 , η = , ηψ3 = σψ (t) − σ ˆψ (t) ηψ1 = ψ 2 ε24 ε4 ηφ1 = (4.61) Error dynamics for the EHGO are ¯1 ∆1 + B ¯2 ∆2 ] ε4 η˙ = Λη + ε4 [B (4.62) where Λ = block diag[A, A] ∈ R18×18 , ¯1 = block diag[Be1 , · · · , Be1 ] ∈ R18×6 , B Be1 = [0, 1, 0]T ¯2 = block diag[Be2 , · · · , Be2 ] ∈ R18×6 , B T  ˆ ˆ ˆ  (fx (φ1 , θ1 , ψ1 , TM ) − fx (φ1 , θ1 , ψ1 , TM ))/ε4   (f (φ , θ , ψ , T ) − f (φˆ , θˆ , ψˆ , T ))/ε y 1 1 1 M 4  y 1 1 1 M   (fz (φ1 , θ1 , TM ) − fz (φˆ1 , θˆ1 , TM ))/ε4  ∆1 =   ˆ u))/ε4  (fφ (Θ, u) − fφ (Θ,    ˆ u))/ε4 (fθ (Θ, u) − fθ (Θ,   ˆ u))/ε4 (fψ (Θ, u) − fψ (Θ, Be2 = [0, 0, 1]         ,                ∆2 =         (4.63)  σ˙ x   σ˙ y     σ˙ z    σ˙ φ    σ˙ θ    σ˙ ψ (4.64) ˆ u)) for i = x, y, z, φ, θ, ψ, have the We note that the components of ∆1 , i.e., (fi (Θ, u) − fi (Θ, ˆ u) ≤ ε4 η , and ∆2 ≤ kδ with kδ > 0. bound fi (Θ, u) − fi (Θ, Using the target system of (4.14), the plant dynamics (4.48), and (4.53) with the error 88 variables eχ , eΘ , and s, the closed-loop system is presented in the singularly perturbed form e˙ χ = Aχ eχ + BFχ (Θ, eχ + χr , ut , uc , σχ (t)) (4.65) ¨ r − ε1 k2 Θ ˙ r] ε1 e˙ Θ = Aθ eΘ + B[k1 (Θd − Θr ) + ε21 FΘ (Θ, ut , ur , ψr , σΘ (t)) − ε21 Θ ε2 s˙ t = − ∂Fχd ∂ut ε3 s˙ r = − ∂FΘ ∂ur (4.66) T Fχd (Θ, eχ + χr , st + u∗t , uc , σχ ) + ∆σχ + ∆χ − ε2 u˙ ∗t (4.67) FΘ (Θ, st + u∗t , sr + u∗r , ψr , σΘ ) + ∆σΘ + ∆Θ − ε3 u˙ ∗r (4.68) T ¯1 ∆1 + B ¯2 ∆2 ] ε4 η˙ = Λη + ε4 [B (4.69) where Fχ and FΘ are given in (4.26) and (4.21), respectively and u∗t and u∗r are satisfied with Fχ (Θ, χ, u∗t , uc , σχ ) = 0,  ∆σχ    =    σ ˆx M1 sat M1 σ ˆy M2 sat M2 σ ˆz M3 sat M3 xˆ2 kx2 Mx sat Mx   yˆ2  ∆χ =  ky2 My sat  My   zˆ2 kz2 Mz sat Mz  FΘ (Θ, ut, u∗r , ψr , σΘ ) = 0,   − σx    − σy  ,   − σz  − x2     − y2  ,    − z2 ∆σΘ  σ ˆφ  M4 sat M4   σ ˆy =  M5 sat M5   σ ˆz M6 sat M6  kφ2 Mφ sat     ∆Θ =   kθ2 Mθ sat     kψ2 Mψ sat φˆ2 Mφ θˆ2 Mθ ψˆ2 Mz (4.70)  − σφ    − σθ     − σψ (4.71)  − φ2      − θ2       − ψ2 (4.72) Now, the next theorem states the stability analysis for the closed-loop system (4.65), (4.66), (4.67), (4.67), and (4.69). Theorem 3. Consider the closed-loop system (4.65), (4.66), (4.67), (4.67), and (4.69) under 89 the Assumption 4, 5, and 6. There exists ε∗ such that all ε1 < ε∗ , ε2 < ε∗ , ∗ (ε2 /ε1 ) < ε , ε3 < ε∗ , ∗ (ε3 /ε2 ) < ε , ε4 < ε∗ (ε4 /ε3 ) < ε (4.73) ∗ ˆ and for all initial states (χ(0), Θ(0)) ∈ X1 , (ut (0), ur (0)) ∈ X2 , and (χ(0), ˆ Θ(0)) ∈ X3 , where X1 is a compact set of (χ, Θ) in the domain Dχ × DΘ , X2 is any compact subset of R2 , and X3 is any compact subset of R18 , all trajectories are bounded and the size of the ultimate boundedness for error state variables in the error dynamics can be arbitrarily small with sufficiently small εi for i = 1, . . . , 4 and (ε2 /ε1 ), (ε3 /ε2 ), (ε4 /ε3 ). Proof. We consider Lyapunov functions Vχ = eTχ Pχ eχ , VΘ = eTΘ Pχ eΘ , and Vη = η T Pη η where Pχ , PΘ , and Pη are solutions of the Lyapunov equations, ATχ Pχ +Pχ Aχ = −I, ATΘ PΘ +PΘ AΘ = −I, and ΛT Pη + Pη Λ = −I. Since we are going to use a time-scale separation approach between subsystems, sets are defined by Ωa = {Vχ ≤ a1 } × {VΘ ≤ a2 } × {Vt ≤ a3 } × {Vr ≤ a4 } Ωb = {Vχ ≤ b1 } × {VΘ ≤ b2 } × {Vt ≤ b3 } × {Vr ≤ b4 } (4.74) Ωc = {Vχ ≤ c1 } × {VΘ ≤ c2 } × {Vt ≤ c3 } Ωd = {Vχ ≤ d1 } × {VΘ ≤ d2 } with 0 < a1 < b1 < c1 < d1 , 0 < a2 < b2 < c2 < d2 , 0 < a3 < b3 < c3 , o < a4 < b4 (4.75) To consider relations between the trajectories χ and eχ , and Θ and eΘ , the constants m1 and m2 are chosen such that (eχ , eΘ ) ∈ {Vχ ≤ a1 } × {VΘ ≤ a2 } =⇒ (χ, Θ) ∈ X1 90 (4.76) Now, we briefly describe the process of the proof for the boundedness of trajectories using the next steps: • initially, the trajectories (eχ , eΘ , st , sr ) and η starting from (eχ (0), eΘ (0), st (0), sr (0)) ∈ Ωa and η(0) from the outside of the set of {Vη ∈ ρε24 }, enter the set Ωb × {Vη ≤ ρ1 ε24 }; • secondly, the trajectories (eχ , eΘ , st , sr ) and η starting from Ωb × {Vη ≤ ρε24 } enter the set Ωc × {Vr ≤ ρ2 µ21 } × {Vη ≤ ρ1 ε24 } with µ1 = (ε3 /ε2 ); • thirdly, the trajectories (eχ , eΘ , st , sr ) and η starting from Ωc × {Vr ≤ ρ2 µ21 } × {Vη ≤ ρ1 ε24 } enter the set Ωd × {Vt ≤ ρ3 µ22 } × {Vr ≤ ρ2 µ21 } × {Vη ≤ ρ1 ε24 } with µ2 = (ε2 /ε1 ) ; • lastly, the trajectories (eχ , eΘ , st , sr ) and η starting from Ωd × {Vt ≤ ρ3 µ22 } × {Vr ≤ ρ2 µ21 } × {Vη ≤ ρ1 ε24 } enter the set {Vχ ≤ e1 } × {VΘ ≤ e2 ε21 } × {Vt ≤ ρ3 µ22 } × {Vr ≤ ρ2 µ21 } × {Vη ≤ ρ1 ε24 }, where e1 and e2 are positive constants. Since these above four steps are similar, we are going to show only the first bullet and the others will be omitted. In the first step, initial trajectories (eχ , eΘ , st , sr ) and η start from the set (eχ (0), eΘ (0), st (0), sr (0)) ∈ Ωa and η ∈ / {Vη ∈ ρε24 } with η(0) ≤ (k/ε24 ). The derivative of Vη along the trajectories (4.65), (4.66), (4.67), (4.68), and (4.69) is V˙ η = − 1 ε4 ¯1 ∆1 + B ¯2 ∆2 )T Pη η η T η + (B (4.77) ¯1 ∆1 + B ¯2 ∆2 ) in (4.69) for all (eχ , eΘ , st , sr ) ∈ Ωa , i.e., Using the bound of the term (B ko1 η + ko2 for some positive constants ko1 and ko2 , we obtain 1 η 2 + ko3 η 2 + ko4 η V˙ η ≤ − ε4 1 1 ≤− η 2 + ko4 η for ε4 < 2ε4 2ko3 91 (4.78) where koi for i = 1, . . . , 4 are positive constants. With ε4 < 1/(2ko3 ), γ1 V˙ η ≤ − ε4 Vη , for Vη ≥ ρ1 ε24 (4.79) where ρ1 = Pη2m γ22 , some γ1 > 0, γ2 > 0, and Pηm = λmax (Pη ). As previous works of the high-gain observers, the trajectory η starts from the outside of the set {Vη ≤ ρε24 } with η(0) ≤ (k/ε24 ) and enters into the set Ωb × {Vη ≤ ρ1 ε24 } in a finite time T (ε4). As ε4 → 0, T (ε4 ) → 0. Since the proof of the finite time convergence is similar to previous Chapter 2 and 3, the proof is omitted. For the second bullet, since the trajectory η cannot leave the set {Vη ≤ ρ1 ε24 }, η has the upper bound, η ≤ 4ko4 ε4 . With this upper bound, the similar procedure can be used to prove the second bullet so that the proof for the rest of them will be omitted. All trajectories enter the set, {Vχ ≤ e1 } × {VΘ ≤ e2 ε21 } × {Vt ≤ ρµ22 } × {Vr ≤ ρµ21 } × {Vη ≤ ρ1 ε24 }, which can be taken as a positively invariant set. We are going to show that the size of the ultimate boundedness can be arbitrarily small with sufficient small control parameters, 0 < ε4 ≪ ε3 ≪ ε2 ≪ ε1 ≪ 1. Consider the derivative of the Lyapunov function Vχ along the trajectories (4.65), (4.66), (4.67), (4.68), and (4.69) is V˙ χ = −eTχ eχ + 2FχT (Θ, eχ + χr , ut , uc , σχ )χ B T Pχ eχ ≤ − eχ 2 (4.80) + 2Pχm (kχ1 eΘ + kχ2 st ) eχ where Pχ ≤ Pχm and some positive constants kχ1 > 0 and kχ2 > 0. The derivative of the Lyapunov function VΘ along the trajectories (4.65), (4.66), (4.67), (4.68), and (4.69), is V˙ Θ = − 1 ε1 1 ≤− ε1 ¨ T ) − k2 Θ ˙T + eTΘ eΘ + 2 ε1 (FΘT − Θ r r eΘ 2 2k1 ε1 + ε1 kΘ1 ( FΘ + Θr1 ) + kΘ2 Θr2 + 92 (Θd − Θr )T B T PΘ eΘ k Θ3 ε1 st eΘ (4.81) where kΘi for i = 1, 2, 3 are positive constants. The derivative of the Lyapunov function Vt along the trajectories (4.65), (4.66), (4.67), (4.68), and (4.69), is V˙ t = − 1 ε2 FχTd + FχTd ∂Fχd ∂Θ + FχTd ∂Fχd ∂χ ∂Fχd ∂ut ∂Fχd ∂ut 1 ε1 T Fχd − 1 ε2 FχTd ∂Fχd ∂ut ∂Fχd ∂ut T (Fχs − Fχd ) ¨ r) + Θ ˙r+Θ ˜ r (t)] [AΘ eΘ + k1 B(Θd − Θr )] + [ε1 (FΘ − Θ ∂Fχd ∂uc [Aχ eχ + Fχ + χr ] + FχTd u˙ c + ∂Fχd ∂σχ σ˙ χ , (4.82) Using the bounds, V˙ t is V˙ t ≤ − kt1 ε2 1 ε1 + Fχd 2 + kt2 ε2 Fχd (∆σχ + ∆χ ) ¯ 1 (t) (kt3 eΘ + kt4 st ) + ε1 kt5 ( FΘ + Θr1 (t)) + kt6 Θ Fχd (4.83) + [kt7 eχ + kt8 (kχ1 eΘ + kχ2 st )] Fχd + kt9 ∆t (t) Fχd where kti for i = 1, . . . , 9 are positive constants, the bounds for ∆σχ and ∆χ are ∆σχ ≤ kt10 η with kt10 > 0 and ∆χ ≤ ε3 η after the saturation active period for the EHGO. The derivative of the Lyapunov function Vr along the trajectories (4.65), (4.66), (4.67), (4.68), and (4.69), is V˙ r = − − 1 ε2 + FΘT 1 ε3 FΘT FΘT ∂FΘ ∂Θ ∂FΘ ∂ur ∂FΘ ∂ut ∂Fχd ∂ut 1 ε1 T ∂FΘ ∂ur 1 ε3 FΘ − T Fχd + FΘT FΘT ∂FΘ ∂ur ∂FΘ ∂ψr ψ˙ r + ∂FΘ ∂ur ∂FΘ ∂σΘ T [FΘs − FΘ ] σ˙ Θ ¨ r) + Θ ˙r+Θ ˜ r (t)] [AΘ eΘ + k1 B(Θd − Θr )] + [ε1 (FΘ − Θ 93 (4.84) Then, inequality for V˙ r is k r1 V˙ r ≤ − ε3 1 ε1 + FΘ 2 + k r2 ε3 FΘ (∆σΘ + ∆Θ ) + k r3 ε2 (kr4 eΘ + kr5 st ) + ε1 kr6 ( FΘ + kr7 Θr1 ) FΘ Fχd (4.85) ¯ 2 (t) FΘ FΘ + Θ where kri for i = 1, . . . , 7 are positive constants, ∆σΘ and ∆Θ become ∆σΘ ≤ kΘ8 η and ∆Θ ≤ ε4 kΘ9 η after passing the transient period for the EHGOs. Using the method in Vχ , Section 9.3 of in [33] with (4.78), (4.80), (4.81), (4.83), (4.85) and choosing W1 = √ √ √ W2 = VΘ , W3 = Vt , W4 = Vr , and W5 = Vη , we obtain D + W1 ≤ −k¯a1 W1 + k¯a2 W2 + k¯a3 W3 D + W2 ≤ − D + W3 ≤ − k¯b1 k¯b3 W2 + ε1 k¯b2 W4 + W3 + ε1 k¯b4 δ¯1 (t) + k¯b5 δ¯2 (t) ε1 ε1 k¯c k¯c4 + ε4 k¯c5 k¯c1 − 2 − k¯c3 W3 + W5 + ε1 k¯c6 W4 ε2 ε1 ε2 k¯c7 ¯ + kc8 W2 + k¯c9 W1 + ε1 k¯c10 δ¯3 (t) + k¯c11 δ¯4 (t) ε1 k¯d1 k¯d5 k¯d6 k¯d3 + k¯d4 ε4 D + W4 ≤ − W5 + − ε1 k¯d2 W4 + + ε3 ε3 ε2 ε1 + W3 + k¯d7 ε1 W2 + ε1 k¯d8 δ¯5 (t) + k¯d9 δ¯6 (t) k¯e1 D + W5 ≤ − W5 + k¯e2 W5 + k¯e3 δ¯7 (t) ε4 (4.86) where D + W (·) denotes the upper right-hand derivative; the notation related to k¯pi for p = a, b, c, d, e and i = 1, 2, . . . , 11, denotes the positive constants independent on ε1 , ε2 , ε3 , and ε4 ; and δ¯i for i = 1, . . . , 7 are nonvanishing perturbations. The matrix form of (4.86) is D + W ≤ −HW + ε1 Γ1 + Γ2 (4.87) where D + W = [D + W1 , D + W2 , D + W3 , D + W4 , D + W5 ]T , W = [W1 , W2 , W3 , W4 , W5 ]T ; Γ1 and 94 Γ2 are Γ1 = [0, k¯b4 δ¯1 (t), k¯c10 δ¯3 (t), k¯d8 δ¯5 (t), 0]T , Γ2 = [0, k¯b5 δ¯2 (t), k¯c11 δ¯4 (t), k¯d9 δ¯6 (t), k¯e3 δ¯7 (t)]T (4.88) Since the off-diagonal components of H are positive, H is quasi-monotone increasing [58] with the condition 0 < ε4 ≪ ε3 ≪ ε2 ≪ ε1 ≪ 1 and is given in Appendix Appendix B. Consider the differential equation U˙ = −HU + ε1 Γ1 + Γ2 (4.89) with U = [U1 , U2 , U3 , U4 ]T and the same initial conditions U(0) = W (0). Using the vectorial comparison method in Chapter IX of [58], it is concluded that W ≤ U for all t ≥ 0 and the steady state of U(t) is H −1(ε1 Γ1 + Γ2 ). The computation of the size of the ultimate boundedness is given in Appendix Appendix B. Since the size of ultimate boundedness is dependent on εi for i = 1, . . . , 4, i.e., as εi → 0 for i = 1, . . . , 4 with 0 < ε4 ≪ ε3 ≪ ε2 ≪ ε1 ≪ 1, the size of the ultimate boundedness can be made arbitrarily small. 4.5 Simulation Results The performances of the proposed controller are illustrated through dynamics of a helicopter. The inertial, geometric, and aerodynamic parameters from [37] are listed below Ix = 0.142413 Iy = 0.271256 Iz = 0.271492 lM = −0.015 yM = 0 hM = 0.2943 Q CM = 0.004452 Q DM = 0.6304 (∂RM /∂b1s ) = 25.23 CTQ = 0.005066 DTQ = 0.008488 (∂MM /∂a1s ) = 25.23. With full dynamics of a helicopter, results from the state feedback are compared to results from the output feedback to show the important role and benefit of the EHGO in presence 95 of uncertainties. The translational and rotational dynamics of a helicopter given in (4.7) and (4.1), respectively, were considered in presence of disturbances, σχ = [3 sin t, 3 sin t, 3 sin t]T and σΘ = [cos t, cos t, cos t]T like wind gusts. The control objective is to track the reference u(t) = [rx , ry , rz ]T = [5 sin t, 5 cos t, 5 sin t]T and ψr = 0.1 rad in the presence of the external disturbances σχ and σΘ . In numerical simulations, we used a helicopter model in (4.1) and (4.7) without approximations. For the state feedback controller in (4.18) and (4.20), and the output feedback controller (4.53), the common control parameters are given by kp = 8, kv = 4, ε1 = 0.1, k1 = 2 k2 = 4, ε2 = 0.001, (4.90) ε3 = 0.0007 For the EHGO, the observer gains of Hi , Hχe , and HΘe in (4.49), are ε4 = 0.0001, hi1 = 3, hi2 = 3, hi3 = 1, for i = 1, . . . , 6 (4.91) The saturation levels for the estimates by the EHGO are chosen not to be activated under the state feedback. For both the state feedback and output feedback, the initial states for the plant and reference dynamics were chosen as χ = [1, 0.2, 1, −0.1, 1, 0.1]T , Θ = [0, 0, 0, 0, 0, 0]T , χr = [0.1, 1, 0, 0, 1]T (4.92) The initial conditions for the dynamic inversion controllers and the EHGO were ut (0) = [TM (0), φd (0), θd (0)]T = [48, 0.5, 0.5]T , ur (0) = [TT (0), a1s (0), b1s (0)]T = [3, 0, 0]T ˆ Θ(0) = [1, 0, 1, 1, 1, 0]T , T χ(0) ˆ = [0, 0, 0, 0, 0, 0] , σ ˆχ = [0, 0, 0]T , σ ˆΘ = [0, 1, 0]T 96 (4.93) For the comparison with the output feedback, the state feedback controllers (4.18) and (4.20) 2 0 −2 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 2 0 −2 0 2 4 t (sec) 2 0 −2 0 2 4 t (sec) Figure 4.3: Trajectories x1 , y1 , and z1 (solid-lines) under the state feedback in the presence of disturbances and reference states xr , yr , and zr (dished-lines) with the control parameters (4.90) and the initial conditions in (4.93), was simulated, which is shown in Fig. 4.3. In Fig. 4.3, we can see tracking errors between trajectories x1 , y1 , and z1 driven by the state feedback controller and the reference states xr , yr , and zr due to the effects of external disturbances. In Fig. 4.4 and 4.5, under the proposed output feedback, system states, χ and Θ and the references xr , yr , zr , and ψr are almost indistinguishable. In Fig. 4.5, the references are φd and ψd which are provided by the dynamic inversion. In Fig. 4.6 and 4.7, the system states xi , yi , and zi for i = 1, 2 are plotted with solid-lines. The estimates xˆi , yˆi , and zˆi for i = 1, 2 are dashed-lines. At begin of the simulations, the peaking due to high gains and the difference of initial conditions, is saturated to prevent 97 2 0 −2 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 2 0 −2 0 2 4 t (sec) 2 0 −2 0 2 4 t (sec) Figure 4.4: Trajectories of x1 , y1 , and z1 (solid-lines) under the output feedback in (4.53) in the presence of disturbances, and references xr , yr , and zr (dashed-lines) for rx (t) = 5 sin t, ry (t) = 5 cos t, and rz (t) = 5 sin t in (4.14) from degrading the system performance. In Fig. 4.8 and 4.9, trajectories of the state states φi , θi , and ψi , and the estimate φˆi , θˆi , and ψˆi for i = 1, 2 are shown. At the beginning of the simulation, peaking is saturated. Expect at the beginning of the simulation, the systems states and estimates are indistinguishable. In Fig. 4.10, the components of sum of Fp in (4.7) and σχ are plotted with solid-lines and the components of sum of FO in (4.11) and σ ˆχ are plotted with dashed-lines. At the first part of simulations, the peaking is saturated. In Fig. 4.11, the external disturbance σΘ (solid lines) and tis estimate σ ˆΘ (dashed-lines) are shown. At the first part of simulations, the peaking is saturated. The actual helicopter control inputs TM , TT , a1s , and b1s are shown in Fig. 4.12. 98 1 φ ,φ d 1 0 −1 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 1 θ ,θ d 1 0 −1 0 2 4 t (sec) 1 ψ ,ψ r 1 0 −1 0 2 4 t (sec) Figure 4.5: Trajectories φd , θd , and ψr (dashed-lines) are references for the states φ1 , θ1 , ψ1 of the rotational dynamics in the presence of disturbances. 4.6 Conclusions An output feedback controller for a helicopter system was presented. In the presence of uncertainties, the output feedback controller is able to track the given reference trajectories xr , yr , zr , and ψr . The states and uncertainties in the helicopter dynamics were estimated using the EHGO and dynamic inversion was subsequently used for design of the controller to deal with nonaffine control inputs. In the time-scale structure the EHGO estimated unmeasurable system states and uncertain system parameters and external disturbances and the estimates were utilized in the two dynamic inversion controllers. There is also a time-scale structure between the two dynamic inversion controllers, in which the rotational 99 2 0 −2 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 2 0 −2 0 2 4 t (sec) 2 0 −2 0 2 4 t (sec) Figure 4.6: Trajectories x1 , y1 and z1 (solid-lines) and the estimates xˆ1 , yˆ1 , and zˆ1 (dashedlines) by the EHGO dynamic inversion controller is faster than the translational dynamic inversion controller. By using a time scale between the two dynamic inversion controller, we were able to design an efficient controller with less dimensions than one dynamic inversion without a time scale. The dynamic inversion controllers were designed to render the rotational dynamics faster than the translational dynamics to overcome underactuated system structures. Using the multitime-scale separation approach, the proposed controller was able to control the full degree of freedom (i.e. 6 degrees of freedom) for an unmanned helicopter. The singular perturbation method was used to design controllers and analyze the multi-time-scale structure. This is confirmed through numerical simulations. 100 2 0 −2 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 2 0 −2 0 2 4 t (sec) 2 0 −2 0 2 4 t (sec) Figure 4.7: Trajectories x2 , y2 and z2 (solid-lines) and the estimates xˆ2 , yˆ2 , and zˆ2 (dashedlines) by the EHGO 101 1 0 −1 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 1 0 −1 0 2 4 t (sec) 1 0 −1 0 2 4 t (sec) Figure 4.8: Trajectories φ1 , θ1 and ψ1 (solid-lines) and the estimates φˆ1 , θˆ1 , and ψˆ1 (dashedlines) by the EHGO 102 1 0 −1 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 5 0 −5 0 2 4 t (sec) 1 0 −1 0 2 4 t (sec) Figure 4.9: Trajectories φ2 , θ2 and ψ2 (solid-lines) and the estimates φˆ2 , θˆ2 , and ψˆ2 (dashedlines) by the EHGO 103 10 0 −10 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) 10 0 −10 0 2 4 t (sec) 10 0 −10 0 2 4 t (sec) Figure 4.10: Plots for sum of the actual terms of acceleration (without approximation in model, Fp in (4.7)) and external disturbances σΘ : dashed-lines and plots for sum of nominal terms of acceleration (i.e., FO in (4.11)) the estimate of external disturbances σχ by the EHGO: solid-lines 104 , 2 0 −2 0 2 4 6 8 10 6 8 10 6 8 10 t (sec) , 2 0 −2 0 2 4 t (sec) , 2 0 −2 0 2 4 t (sec) Figure 4.11: Plots for the external disturbances σΘ (solid-lines) and plots for the estimates σ ˆΘ (dashed-lines) 105 100 50 0 0 2 4 6 8 10 6 8 10 6 8 10 6 8 10 t (sec) 10 5 0 0 2 4 t (sec) 1 0 −1 0 2 4 t (sec) 1 0 −1 0 2 4 t (sec) Figure 4.12: Plots for the helicopter actual control inputs, TM , TT , a1s , and b1s under the output feedback 106 Chapter 5 Conclusions and Future Works 5.1 Concluding Remarks This dissertation is considering a class of uncertain nonlinear systems which have the form of a chain of integrators focusing on output feedback. The uncertain nonlinear systems were governed by a multi-time-scale structure control design. In the output feedback design, the Extended High-Gain Observers were used to estimate unmeasured systems states, uncertain system parameters, and external disturbances as well. Using the Extended High-Gain Observers, the uncertain factors were eliminated in the uncertain nonlinear systems. The estimates were provided to the dynamic inversion. The dynamic inversion was able to deal with nonaffine control inputs, system parameter uncertainties, and disturbances using the estimates. The proposed control design created a multi-time-scale structure in the uncertain nonlinear systems, in which the plant dynamics were forced to have a time-scale structure by the controller. The time-scale structure was well-suited to underactuated mechanical systems where the number of inputs are less than the number of the degrees of freedom since the fast dynamics in the plant are considered as virtual inputs to the slow dynamics. This time-scale structure approach were verified using two examples of underactuated mechanical systems: the 107 inverted pendulum on a cart and the autonomous helicopter. The multi-time scale structures were analyzed through the singular perturbation method. Moreover, the stability for the close-loop systems was guaranteed by the proposed control schemes. The proposed controllers were verified through the numerical simulations and experimental tests. 5.1.1 Main contributions The main contributions of this dissertation are as follow as. 1. In Chapter 2, • this dissertation provided a novel way to deal with nonlinear systems which have the form of chains of integrators, nonaffine control inputs, unmeasured system states, and uncertainties; • to deal with the uncertain, nonaffine, nonlinear systems, the extended high-gain observer and the dynamic inversion were combined using a multi-time-scale separation approach; • the time-scale approach control design was devised and the stability of the proposed controller was conducted using the singular perturbation method. 2. In Chapter 3, • the output feedback stabilization control design for the inverted pendulum on a car in the presence of uncertainties was proposed through a multi-time-scale approach; • the combination of the Extended High-Gain Observer and the dynamic inversion provided a new way to deal with unmeasured systems states and uncertain system parameters; 108 • the stability analysis for the closed-loop system of the inverted pendulum on a cart was conducted using the singular perturbation method; • the proposed control design was verified through both numerical simulations and experimental tests. 3. In Chapter 4, • output feedback control design for tracking given reference of the autonomous helicopter in the presence of uncertainties was proposed through a multi-timescale separation approach; • Using the extended high-gain observers and two dynamic inversion controllers with a multi-time-scale separation, the nonaffine control inputs and uncertainties were considered and a time-scale structure between the translational and rotational dynamics were constructed; • The stability analysis for the multi-time-scale structure in the closed-loop system was conducted through the singular perturbation method; • the proposed controller was verified through numerical simulations. 5.2 Future Works Extensions of this dissertation as future works are given as follows. • In Chapter 2, the future work will consider extension of our approach to non-minimum phase systems. The main issue of this work would be how to deal with unstable zero dynamics in the absence of measurements in the zero dynamics; • in Chapter 3, our future work will focus on extending our approach to output feedback stabilization of other underactuated mechanical systems with two degrees of freedom. Expected difficulties would be how to create a time-scale structure in the underacuated mechanical systems with the two degrees of freedom; 109 • in Chapter 4, extension of our work will be implementing the proposed control algorithm into miniature helicopters and develop new control algorithms for the other types of unmanned vehicles. 110 Appendices 111 Appendix A Appendix for Chapter 2 In this section, we are going to derive the standard singularly perturbed systems (3.32), (3.33), (3.34), and (3.35) with time-scaled variables. Error dynamics of the EHGOs With the fast variables for EHGOs are ηx1 = x1 − xˆ1 x2 − xˆ2 , ηx2 = , 2 ε3 ε3 ηx3 = σx(θ1 + αr ,θ2 , F )− σ ˆx ηα1 α1 − α ˆ1 α2 − α ˆ2 , = , ηα2 = 2 ε3 ε3 (1) ηα3 = σα(θ1 + αr ,θ2 , F )− σ ˆα , with θ1 = α1 − αr , θ2 = α2 , where xˆi and α ˆ i for i = 1, 2, and σ ˆx and σ ˆα , are given in (3.25). The derivatives of ηx1 and ηα1 along the trajectories of (3.2) and (3.25) are ε3 η˙ x1 = −h11 ηx1 + ηx2 (2) ε3 η˙ α1 = −h21 ηα1 + ηα2 The derivatives of ηx2 and ηα2 along the trajectories of (3.2) and (3.25) are ε3 η˙ x2 = − h12 ηx1 + σx (α1 , α2 , F ) − σ ˆx (t) + f¯x (α1 , α2 , F ) − f¯x (α ˆ1 , α ˆ2, F ) =− h12 ηx1+ηx3 + f¯x (α1 , α2 , F )− f¯x (α ˆ1 , α ˆ2, F ) ε3 η˙ α2 = − h22 ηα1 + σα (α1 , α2 , F ) − σ ˆα (t) + f¯α (α1 , α2 , F ) − f¯α (α ˆ1, α ˆ2, F ) =− h22 ηα1+ηα3 + f¯α(α1 ,α2,F )− f¯α (ˆ α1,α ˆ 2, F ) 112 (3) The derivatives of ηx3 and ηα3 along the trajectories of (3.2), (3.25) and (3.26) are ε3 η˙ x3 = − h13 ηx1 + ε3 σ˙ x (α1 , α2 , F ) ∂σx α˙ 2 ∂α2 α ˆ2 [f¯x (α1 , Mθ sat( ), F ) Mθ ∂σx ∂α1 = − h13 ηx1 + ε3 − 1 · ε2 ∂σx ∂F α2 + + Mx sat(ˆ σx /Mx ) − u] (4) ε3 η˙ α3 = − h23 ηα1 + ε3 σ˙ α (α1 , α2 , F ) ∂σα ∂α1 ∂σα α˙ 2 ∂α2 α ˆ2 [f¯x (α1 , Mθ sat( ), F ) Mθ = − h23 ηα1 + ε3 − 1 · ε2 ∂σα ∂F α2 + + Mx sat(σˆx /Mx ) − u] The error dynamics for the EHGOs are ¯1 ∆1 + B ¯2 ∆2 + ε3 η˙ = Aη η + ε3 [B 1 ε2 ¯2 ∆3 ] B (5) T T T T T where η = [η x , ηα ] with ηx = [ηx1 , ηx2 , ηx3 ] and ηα = [ηα1 , ηα2 , ηα3 ] ; the matrices Aη =   Aη1 03×3  ¯ ¯2 are given by , B1 and B  03×3 Aη2      −h21 1 0   −h11 1 0      ,  , Aη2 = −h Aη1= 0 1 −h 0 1 22 12         −h23 0 0 −h13 0 0  ¯j =  B  Bj 03×1  03×1   , for j = 1, 2 Bj B1 = [0, 1, 0]T , B2 = [0, 0, 1]T 113 with the zero matrix 0i×j ∈ Ri×j , B = [0, 1]T ; and ∆i for i = 1, 2, 3 are   ∆1=    ∆2=    ∆3=  where ∆1 ≤ k∆ η  f¯x (α1 , α2 , F ) − f¯x (α ˆ1 , α ˆ2, F )  ε3  ¯ ¯ fα (α1 , α2 , F ) − fα (α ˆ1, α ˆ2 , F )  ε3  ∂σx ∂σx α2 + fα (α1 , α2 , F )  ∂α1 ∂α2   ∂σα ∂σα α2 + fα (α1 , α2 , F ) ∂α1 ∂α2  ∂σx ¯ α ˆ2 σ ˆx [fx (α1, Mθsat( ), F)+Mx sat( )−u]  ∂F Mθ Mx   α ˆ2 σ ˆx ∂σα ¯ [fx (α1, Mθsat( ), F)+Mx sat( )−u] ∂F Mθ Mx with a positive constant k∆ due to the Lipschitz conditions ˆ1, α ˆ 2 , F ) ≤ ε 3 k ∆α η f¯x (α1 , α2 , F ) − f¯x (α ˆ 1 , αˆ2 , F ) ≤ ε3 k∆x η and f¯α (α1 , α2 , F ) − f¯α (α with positive constants k∆x and k∆α . Error dynamics of the dynamic inversion With the change of variables for the dynamic inversion, zF = F − F ∗ , zu = u − u∗ (6) the derivative of zF along the trajectories of (3.2), (3.25), and (3.26) with multiplication of ε2 , is ε2 z˙F = ε2 F˙ − ε2 F˙ ∗ α ˆ2 σ ˆx = −f¯x α1 , Mθ sat( +u−ε2F˙ ∗ ), F −Mxsat Mθ Mx (7) By adding and subtracting f¯x (α1 , α2 , F ) + σx (α1 , α2 , F ) and adding f¯x (α1 , α2 , F ∗ ) + σx (α1 , α2 , F ∗ ) − u∗ (i.e., f¯x (α1 , α2 , F ∗ ) + σx (α1 , α2 , F ∗ ) − u∗ = 0) to the right-hand side 114 of ε2 z˙F , we obtain ε2 z˙F = −f¯x (α1 , α2 , F ) − σx (α1 , α2 , F ) + f¯x (α1 , α2 , F ∗ ) + σx (α1 , α2 , F ∗) + f¯x (α1 , α2 , F ) − f¯x α1 , Mθ sat + σx (α1 , α2 , F ) − Mx sat α ˆ2 Mθ ,F (8) σ ˆx +u−u∗ −ε2 F˙ ∗ Mx Since f¯x (α1 , α2 , F ∗ ) + σx (α1 , α2 , F ∗ ) (9) − f¯x (α1 , α2 , F ) − σx (α1 , α2 , F ) = −cx zF , it is simplified to ε2 z˙F = −cx zF + zu α ˆ2 Mθ + f¯x (α1 , α2 , F ) − f¯x α1 , Mθ sat + σx (α1 , α2 , F ) − Mx sat σ ˆx Mx ,F (10) − ε2 F˙ ∗ Since F ∗ = F ∗ (α1 , α2 , νd ) is F∗ = [u∗ − Gx ] , cx u∗ = g tan α1 − ℓ cos α1 νd , (11) with cx in (3.22), Gx in (3.3), and νd in (3.12), F˙ ∗ is F˙ ∗ = ∂F ∗ ∂α1 α˙ 1 + ∂F ∗ ∂α2 α˙ 2 + ∂F ∗ ∂νd ν˙ d = F1 (θ1 + αr , θ2 , F ∗ ) + ε1 F2 (θ1 + αr , θ2 , y, F ∗) 115 (12) where F1 and F2 are ∂F ∗ ∂F ∗ ∂F ∗ ∂F ∗ α2+ fα (·) − β1 − β2 ∂α1 ∂νd ∂α2 ∂νd ∂F ∗ g F2=−β1 [k1 y2 + k2 fx (·)] 2 ∂νd g 2 + vext F1= (13) By differentiating zu and multiplying z˙u by ε2 , we obtain ε2 z˙u = ε2 u˙ − ε2 u˙ ∗ σα α ˆ2 ,F +Mαsat − νˆd −ε2 u˙ ∗ =f¯α α1, Mθsat Mθ Mα (14) With the similar procedure of the derivation for ε2 z˙F , adding and subtracting f¯α (α1 , α2 , F )+ σα (α1 , α2 , F ) and subtracting f¯α (α1 , α2 , F ∗ ) + σα (α1 , α2 , F ∗ ) − νd (i.e., f¯α (α1 , α2 , F ∗ ) + σα (α1 , α2 , F ∗) − νd = 0), we obtain ε3 z˙u=−cαzF + f¯α α1 ,Mθsat α ˆ2 ,F − f¯α (α1 ,α2 ,F ) Mθ σ ˆα −σα (α1 ,α2 ,F )+νd − νˆd −ε2 u˙ ∗ + Mα sat Mα (15) where νd − νˆd=−β1 (αr − α ˆ r)−β2 Mθ sat α ˆ2 − α2 Mθ (16) with α ˆ r in (3.27) and u∗ = u∗ (α1 , νd ) in (11), u˙ ∗ is ∗ u˙ = ∂u∗ ∂α1 α˙ 1 + ∂u∗ ∂νd ν˙ d (17) = u1 (θ1 + αr , θ2 , F ) + ε1 u2 (θ1 + αr , θ2 , y, F ) ∂u∗ ∂u∗ ∂u∗ α2 − β2 fα (·) − β1 ∂α1 ∂νd ∂νd ∂u∗ g u2 = −β1 [k1 y2 + k2 fx (·)] 2 2 ∂νd g + vext u1 = 116 (18) With (10) and (15), we obtain (19) ε2 z˙ = Az z + ψ(·) − ε2 φ(·) where Az is given in (3.24), z = [zF , zu ]T , φ = [F˙ ∗ , u˙ ∗]T , ψ= [ψ1 , ψ2 + νd − νˆd ]T is ψ1 = f¯x (α1 , α2 , F ) − f¯x α1 , Mθ sat + σx (α1 , α2 , F ) − Mx sat α ˆ2 Mθ σ ˆx Mx α ˆ2 ψ2 = f¯α α1 ,Mθsat ,F − f¯α (α1 ,α2 ,F ) Mθ σ ˆα −σα (α1 ,α2 ,F ) + Mα sat Mα ,F (20) and ψ(·)|η=0 = 0 and when saturation is not effective ψ ≤ kψ η with a positive constant kψ . Error dynamics of the plant The change of variables θ1 = α1 − αr with αr in (3.11) and θ2 = α2 for the pendulum dynamics, and y1 = ε21 x1 and y2 = ε1 x2 for the cart dynamics, is used. The derivative of θ1 along the trajectories of (3.2), (3.25), and (3.26) is θ˙1 = α˙ 1 − α˙ r = θ2 + ε1 hα (21) with hα in (3.18) and the derivative of θ2 is θ˙2 = fα (α1 , α2 , F ) = −cα F + Gα 117 (22) with cα in (3.22) and Gα in (3.3). Adding and subtracting fα (α1 , α2 , F ∗ ) = −cα F ∗ + Gα with F ∗ and u∗ in (11), we obtain θ˙2= fα (α1 ,α2,F ∗ )+fα (α1 ,α2 ,F )−fα (α1 ,α2 ,F ∗ ) = −cα F ∗ + Gα +fα (α1 ,α2 ,F )−fα (α1 ,α2 ,F ∗ ) (23) = −β1 θ1 − β2 θ2 − cα zF With (21) and (23), the pendulum dynamics is θ˙ = Aθ θ + ε1 Ehα (y, θ, αr , F ) − Bcα zF (24) where Aθ is given in (3.19), θ = [θ1 , θ2 ]T , E = [1, 0]T , and B = [0, 1]T . With the slow variables of y1 = ε21 x1 and y2 = ε1 x2 for the cart dynamics, the derivatives of y1 and y2 along the trajectories of (3.2), (3.25), and (3.26) are y˙ 1 = ε1 y2 (25) y˙ 2 = ε1 fx (α1 , α2 , F ) With F ∗ and u∗ in (11), adding and subtracting ε1 fx (α1 , α2 , F ∗) = ε1 [cx F ∗ + Gx ] to y˙ 2 , it is obtained y˙ 2 = ε1 [fx (α1 , α2 , F ∗ ) + fx (α1 , α2 , F ) − (α1 , α2 , F ∗)] = ε1 [cx F ∗ + Gx + cx zF ] (26) = ε1 [−k1 y1 − k2 y2 + hx + cx zF ] with hx (θ, αr ) in (3.18) and hx (0, αr ) = 0. With (25) and (26), we obtain y˙ = ε1 Ay y + B hx (θ, αr ) + cx zF 118 (27) where Ay is given in (3.20) and y = [y1 , y2 ]T . 119 Appendix B Appendix for Chapter 4 Jacobian matrix Jacobian matrix for the translational dynamics The Jacobian for the translational dynamics (∂Fχd /∂ut ) is given by (∂Fχ /∂ut ) is   ∂Fχ   =  ∂ut  ∂fx ∂TM ∂fy ∂TM ∂fz ∂TM ∂fx ∂φd ∂fy ∂φd ∂fz ∂φd ∂fx ∂θd ∂fy ∂θd ∂fz ∂θd        (28) The components of the matrix are as follows. 1 ∂fx (cos φd sin θd cos ψ1 + sin φd sin ψ1 ) =− ∂TM m 1 ∂fx = (sin φd sin θd cos ψ1 − cos φd sin ψ1 )TM ∂φd m ∂fx 1 (cos φd cos θd cos ψ1 )TM =− ∂θ m (29) ∂fy 1 (cos φd sin θd sin ψ1 − sin φd cos ψ1 ) =− ∂TM m 1 ∂fy (sin φd sin θd sin ψ1 + cos φd cos ψ1 )TM = ∂φd m ∂fy 1 =− (cos φd cos θd sin ψ1 )TM ∂θd m (30) 1 ∂fz =− (cos φd cos θd ) ∂TM m ∂fz 1 = (sin φd cos θd )TM ∂φd m ∂fz 1 (cos φd sin θd )TM = ∂θd m (31) 120 Jacobian matrix for the rotational dynamics The Jacobian matrix (∂FΘ /∂ur ) is   ∂FΘ   =  ∂ur  ∂fφ ∂TT ∂fθ ∂TT ∂fψ ∂TT ∂fφ ∂a1s ∂fθ ∂a1s ∂fψ ∂a1s ∂fφ ∂b1s ∂fθ ∂b1s ∂fψ ∂b1s        (32) The components of the matrix are as follows. ∂fφ = ∂TT ∂fφ = ∂a1s ∂fφ = ∂b1s 1 Ixx 1 Ixx 1 Ixx ∂τ1b sin φ1 tan θ1 + ∂TT Iyy b ∂τ1 sin φ1 tan θ1 + ∂a1s Iyy b ∂τ1 sin φ1 tan θ1 + ∂b1s Iyy ∂fθ = ∂TT ∂fθ = ∂a1s ∂fθ = ∂b1s ∂fψ = ∂TT ∂fψ = ∂a1s ∂fψ = ∂b1s cos φ1 Iyy cos φ1 Iyy cos φ1 Iyy sec θ1 sin φ1 Iyy sec θ1 sin φ1 Iyy sec θ1 sin φ1 Iyy ∂τ2b cos φ1 tan θ1 + ∂TT Izz b ∂τ2 cos φ1 tan θ1 + ∂a1s Izz b ∂τ2 cos φ1 tan θ1 + ∂b1s Izz ∂τ2b sin φ1 − ∂TT Izz b ∂τ2 sin φ1 − ∂a1s Izz b ∂τ2 sin φ1 − ∂b1s Izz ∂τ3b ∂TT ∂τ3b ∂a1s ∂τ3b ∂b1s ∂τ2b cos φ1 sec θ1 + ∂TT Izz b ∂τ2 cos φ1 sec θ1 + ∂a1s Izz b ∂τ2 cos φ1 sec θ1 + ∂b1s Izz 121 ∂τ3b ∂TT ∂τ3b ∂a1s ∂τ3b ∂b1s (33) (34) ∂τ3b ∂TT ∂τ3b ∂a1s ∂τ3b ∂b1s (35) ∂τ1b ∂TT ∂τ1b ∂b1s ∂τ2b ∂TT ∂τ2b ∂b1s ∂τ3b ∂TT ∂τ3b ∂b1s = − hT , ∂τ1b = −QM cos a1s + TM sin a1s cos b1s yM , ∂a1s ∂RM + TM hM cos b1s + TM yM cos a1s sin b1s , ∂b1s ∂τ2b ∂MM = − 1.5CTQ TT0.5 , = + TM hM cos(a1s ) + TM sin(a1s ) cos(b1s )lM , ∂a1s ∂a1s = (36) =QM cos(b1s ) + TM cos(a1s ) sin(b1s )lM , =lT , ∂τ3b = QM sin(a1s ) cos(b1s ), ∂a1s =QM cos(a1s ) sin(b1s ) − TM cos(b1s )lM . Computation for the size of ultimate boundedness The matrix H The matrix H is given by   H4 H = 01×4   H4 =  H3 H42 H41 k¯d1 − ε1 k¯d2 ε3   H51  ¯ ke1 ¯  , − ke2 ε4    , H41 H51  0    0   ¯ = kd3 + k¯d4 ε4  −  ε3  ¯  kc4 + ε4 k¯c5 − ε2 0   =  −ε1 k¯b2  −ε1 k¯c6  ¯ −k¯a2  ka1  k¯b1  H3 =  0 ε1  ¯c  k 7 −k¯c9 − + k¯c8 ε1 122        , H42 =                  0 k¯d − 7 ε1 k¯d5 k¯d6 − + ε2 ε1  −k¯a3 k¯b − 3 ε1 k¯c2 ¯ k¯c1 − − kc 3 ε2 ε1       (37) T       (38) (39) The size of ultimate boundedness By multiplying ε4 by both left-hand and right-hand sides of the last inequality, D + W5 in (4.86), the size of the upper bound W5 is approximated to W5 ≤ k¯e3 δ¯7 (t)/(k¯e1 + ε4 k¯e2 ) (40) The sizes of the boundedness for Wi for i = 1, . . . , 4, are computed by using the inverse of block matrices in Appendix A.20 of [32] as follows.   H4−1 =  H3 H42 −1 H41   ¯ kd1 − ε1 k¯d2 ε3   = H3−1 + E4 ∆−1 4 F4 −∆−1 4 F4 −E4 ∆−1 4 ∆−1 4  (41)   where ∆4 = (k¯d1 − ε3 ε1 k¯d2 )/ε3 − H42 H3−1H41 , E4 = H3−1 H41 and F4 = H42 H3−1 . The matrix H4−1 is rewritten as   H4−1 =  H3−1 + O(ε3 )H41 ¯ 43 O(ε3)H  ¯ O(ε3)H42   O(ε3) (42) where 0 < O(εp ) ≤ kp εp with positive numbers kp , εp , 0 < εp ≪ kp . The matrix H3−1 can be computed as   H3−1 =  H2 H32 H31 k¯c k¯c1 − 2 − k¯c3 ε2 ε1 −1     = H2−1 + E3 ∆−1 3 F3 −∆−1 3 F3 −E3 ∆−1 3 ∆−1 3    (43) where  ¯ ¯  ka1 −ka2  H2 =  k¯b1  , 0 ε1  H31  ¯  −ka3  =  k¯  , b − 3 ε1  123 H32  −k¯c9  = k¯c7 ¯ + kc 8 − ε1 T   (44) ∆3 = ε2 k¯a2 k¯c1 − − ε2 k¯c3 ε1 1 ε2 − H32 H2−1 H31 (45) E3 = H2−1 H31 and F3 = H32 H2−1 . The matrix H3−1 is rewritten as   H3−1 =   ¯ 32 + O(ε2 )H31 O(ε2)H   ¯ 33 O(ε2)H O(ε2) H2−1 (46) With (42), (46), and (44), the upper bounds of Wi for i = 1, 2, 3, 4 can be computed by using H4−1 (ε1 Γ3 + Γ4 ) = Γ = [Γ11 , Γ22 , Γ33 , Γ44 ]T Γ3 = [0, k¯b4 δ¯1 (t), k¯c10 δ¯3 (t), k¯d8 δ¯5 (t)]T , (47) Γ4 = [0, k¯b5 δ¯2 (t), k¯c11 δ¯4 (t), k¯d9 δ¯6 (t)]T Then the upper bounds for each component of Γ is Γ11 ≤ [ε1 kf1 + O(ε2 )][ε1 k¯b4 δ¯1 (t) + k¯b5 δ¯2 (t)] + O(ε2)[ε1 k¯c10 δ¯3 (t) + k¯c11 δ¯4 (t)] + O(ε3)[ε1 k¯d8 δ¯5 (t) + k¯d9 δ¯6 (t)] Γ22 ≤ O(ε2)[ε1 k¯b4 δ¯1 (t) + k¯b5 δ¯2 (t)] + [ε1 kf2 + O(ε2)][ε1 k¯c10 δ¯3 (t) + k¯c11 δ¯4 (t)] + O(ε3)[ε1 k¯d8 δ¯5 (t) + k¯d9 δ¯6 (t)] Γ33 (48) ≤ O(ε2)[ε1 k¯b4 δ¯1 (t) + k¯b5 δ¯2 (t)] + O(ε2 )[ε1 k¯c10 δ¯3 (t) + k¯c11 δ¯4 (t)] + O(ε3)[ε1 k¯d8 δ¯5 (t) + k¯d9 δ¯6 (t)] Γ44 ≤ O(ε2)O(ε3)[ε1 k¯b4 δ¯1 (t) + k¯b5 δ¯2 (t)] + O(ε2)O(ε3)[ε1 k¯c10 δ¯3 (t) + k¯c11 δ¯4 (t)] + O(ε3)[ε1 k¯d8 δ¯5 (t) + k¯d9 δ¯6 (t)] with k¯f1 > 0 and k¯f2 > 0. 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