This dissertation focuses on the high performance robust control of nonlinear systems in the presence of parametric uncertainties and uncertain nonlinearities ...
Adaptive Robust Control of Nonlinear Systems with Application to Control of Mechanical Systems by Bin Yao B.Eng. (Beijing University of Aeronautics and Astronautics, P.R.China ) 1987 M.Eng. (Nanyang Technological University, Singapore) 1992 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA at BERKELEY Committee in charge: Professor Masayoshi Tomizuka , Chair Professor Karl J. Hedrick Professor S. Shankar Sastry
1996
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Abstract Adaptive Robust Control of Nonlinear Systems with Application to Control of Mechanical Systems by Bin Yao Doctor of Philosophy in Mechanical Engineering University of California at Berkeley Professor Masayoshi Tomizuka , Chair This dissertation focuses on the high performance robust control of nonlinear systems in the presence of parametric uncertainties and uncertain nonlinearities (e.g., disturbances) and its application to the control of mechanical systems. A new approach, adaptive robust control (ARC), is proposed. The approach e ectively combines the design techniques of adaptive control (AC) and deterministic robust control (DRC) and improves performance by preserving the advantages of both AC and DRC. Speci cally, the approach guarantees a superior performance in terms of both transient error and nal tracking accuracy in the presence of parametric uncertainties and uncertain nonlinearities. This result overcomes the drawbacks of AC and makes the approach attractive to real applications. Through parameter adaptation, the approach achieves asymptotic tracking in the presence of parametric uncertainties without using a high-gain in the feedback loop, which implies that the control input is smooth. In this sense, ARC has a better tracking performance than DRC. The design is conceptually simple and amenable to implementation. A general framework of the proposed ARC is formulated in terms of adaptive robust control (ARC) Lyapunov functions. Through backstepping design, ARC Lyapunov functions can be successfully constructed for a large class of multi-input multi-output (MIMO) nonlinear systems transformable to a semi-strict feedback form. The method is applied to the control of robot manipulators in several applications. For trajectory tracking control, two ARC algorithms are developed: adaptive sliding mode control (ASMC) and desired compensation ARC (DCARC). ASMC is based on the sliding mode control (SMC) and the conventional adaptation law that uses the actual state variables in the regressor. DCARC uses the desired trajectory information in the regressor. Three di erent adaptive or robust control schemes are also derived for comparison: a simple nonlinear PID type robust control, a gain-based nonlinear PID type adaptive control, which requires no model information, and a combined parameter and gainbased adaptive robust control. All the algorithms are implemented and compared on a two-link direct drive robot. Comparative experimental results show the importance of the controller structure and the parameter adaptation. The proposed DCARC is found to provide the best tracking performance without increasing the control bandwidth and the control e ort. For a constrained robot manipulator, the end-e ector of which is in contact with rigid surfaces, a new constrained dynamic model is obtained to account for the e ect of contact surface friction. The ARC scheme utilizes a PI type force feedback control structure with a low proportional gain to avoid the
2 acausality problem. Possible impact problems caused by losing contact are alleviated by the guaranteed transient performance. An adaptation law driven by both motion and force tracking errors guarantees asymptotic motion and force tracking without any persistent excitation conditions. Simulation results verify the e ectiveness of the method. For the coordinated control of multiple robot manipulators handling a constrained object, a set of transformed dynamic equations are obtained in the joint space. In the transformed domain, internal force and external contact force have the same form and can be dealt with in the same way as in the constrained motion problem. A coordinated motion and force ARC controller is developed. It possesses the same nice properties as the ARC constrained motion controller mentioned above. Motion and force tracking control of robot manipulators in contact with unknown sti ness environments is formulated. An ARC motion and force controller is developed to deal with unknown robot parameters and surface parameters, such as sti ness and friction coecients, as well as uncertain nonlinearities caused by modeling errors.
Professor Masayoshi Tomizuka Dissertation Committee Chair
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Contents List of Figures
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List of Tables
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1 Introduction
1.1 Control of Uncertain Nonlinear Dynamics . . . . 1.2 Previous Work . . . . . . . . . . . . . . . . . . . 1.2.1 Adaptive Control (AC) . . . . . . . . . . 1.2.2 Deterministic Robust Control (DRC) . . . 1.3 Motivations and Contributions of the Dissertation 1.3.1 General Methodology . . . . . . . . . . . 1.3.2 General Form . . . . . . . . . . . . . . . 1.3.3 Applications . . . . . . . . . . . . . . . . 1.4 Outline of the Dissertation . . . . . . . . . . . .
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I Adaptive Robust Control - Theory
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2 Control of a First-order Uncertain System
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2.1 Deterministic Robust Control (DRC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Adaptive Control (AC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Adaptive Robust Control (ARC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Adaptive Robust Control of SISO Nonlinear Systems in a Semi-Strict Feedback form 3.1 Problem Formulation . . . . . . . . . . . . . . . . . 3.2 Smooth Projection and Positive De nite Function V 3.3 Backstepping Design Procedure . . . . . . . . . . . . 3.3.1 Step 1 . . . . . . . . . . . . . . . . . . . . . 3.3.2 Step 2 . . . . . . . . . . . . . . . . . . . . . 3.3.3 Step i . . . . . . . . . . . . . . . . . . . . . 3.3.4 Step n . . . . . . . . . . . . . . . . . . . . . 3.4 Guaranteed Transient Performance . . . . . . . . . . 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . .
4 General Framework of Adaptive Robust Control
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24 24 25 25 26 27 28 29 30 33 34 38
4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 ARC Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
iv 4.3 Adaptive Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Backstepping Design via ARC Lyapunov Functions 5.1 5.2 5.3 5.4
Initial MIMO Nonlinear Systems . . . . Augmented MIMO Nonlinear Systems I Backstepping Design I . . . . . . . . . . Augmented MIMO Nonlinear Systems II
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6 Adaptive Robust Control of MIMO Nonlinear Systems 6.1 MIMO Semi-Strict Feedback Form . . . . 6.2 Backstepping Design Procedure . . . . . . 6.2.1 Step 1 . . . . . . . . . . . . . . . 6.2.2 Step 2 . . . . . . . . . . . . . . . 6.2.3 Step i . . . . . . . . . . . . . . . 6.2.4 Step r . . . . . . . . . . . . . . . 6.2.5 Guaranteed Transient Performance 6.3 Simulation Results . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . .
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54 54 55 56 57 57 60 61 62 63
II Adaptive Robust Control - Applications
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7 Trajectory Tracking Control of Robot Manipulators
69 69 70 70 74 77 78 80 82 84 84 84 86 90
7.1 Dynamic Model of Robot Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Adaptive Sliding Mode Control (ASMC) . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Desired Compensation Adaptive Robust Control (DCARC) . . . . . . . . . . . . 7.2.3 Nonlinear PID Robust Control (NPID) . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Nonlinear PID Adaptive Control (PIDAC) . . . . . . . . . . . . . . . . . . . . . 7.2.5 Desired Compensation Adaptive Robust Control with Adjustable Gains (ARCAG) 7.3 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Performance Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Controller Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Comparative Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Other Applications
8.1 Constrained Motion and Force Control of Robot Manipulators . . . . . . . . . . . . . 8.1.1 Dynamic Model of Constrained Robots . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Adaptive Robust Control of Constrained Manipulators . . . . . . . . . . . . . . 8.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Coordinated Control of Multiple Robot Manipulators . . . . . . . . . . . . . . . . . . 8.2.1 Dynamic Model of Robotic Systems . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Adaptive Robust Control of Coordinated Manipulators . . . . . . . . . . . . . 8.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Motion and Force Tracking Control of Robot Manipulators in Contact With Unknown Sti ness Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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v 8.3.1 Dynamic Model of a Manipulator in Contact with a Sti Environment . . . . . . 114 8.3.2 ARC Motion and Force Tracking Control . . . . . . . . . . . . . . . . . . . . . . 116 8.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9 Conclusion
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Bibliography
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9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9.2 Suggested Ideas for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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List of Figures 2.1 Nondecreasing n-th smooth projection map . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8
Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Estimated parameters in the presence of parametric uncertainties . . . . . . . . . . . . Control input in the presence of parametric uncertainties . . . . . . . . . . . . . . . . . Tracking errors in the presence of parametric uncertainties and small disturbances (d1 = d2 = 0:02) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated parameters in the presence of parametric uncertainties and small disturbances (d1 = d2 = 0:02) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input in the presence of parametric uncertainties and small disturbances (d1 = d2 = 0:02) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracking errors in the presence of parametric uncertainties and large disturbances (d1 = d2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated parameters in the presence of parametric uncertainties and large disturbances (d1 = d2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input in the presence of parametric uncertainties and large disturbances (d1 = d2 = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 35 35 36 36 36 37 37 37 63 64 64 64 65 65 65
6.9 6.10 6.11 6.12 6.13
Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Estimated parameters in the presence of parametric uncertainties . . . . . . . . . . . . Control inputs in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Control inputs in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2) Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2) Estimated parameters in the presence of parametric uncertainties and disturbances (d1=d2=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control inputs in the presence of parametric uncertainties and disturbances (d1=d2=2) Control inputs in the presence of parametric uncertainties and disturbances (d1=d2=2) Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Tracking errors in the presence of parametric uncertainties . . . . . . . . . . . . . . . . Estimated parameters in the presence of parametric uncertainties . . . . . . . . . . . .
66 66 66 67 67 67
7.1 7.2 7.3 7.4 7.5
Berkeley/NSK Two-Link Direct-Drive Manipulator Experimental Setup . . . . . . . . . . . . . . . . . Transient Performance . . . . . . . . . . . . . . . Final Tracking Accuracy . . . . . . . . . . . . . . Average Tracking Errors . . . . . . . . . . . . . .
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7.6 7.7 7.8 7.9 7.10 7.11
Control E ort . . . . . . . . . . . . . . . . . . Control Chattering . . . . . . . . . . . . . . . Estimated payloads approach their true values Estimated Feedback Gains K^ . . . . . . . . . Joint Tracking Errors . . . . . . . . . . . . . Joint Control Torque . . . . . . . . . . . . . .
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11
Con guration of the Robot Moving on a Semi-circle Surface . . . . . . . . . . . . . . . 103 Position Tracking Error ep in the Presence of Parametric Uncertainties . . . . . . . . . 105 Force Tracking Error ef in the Presence of Parametric Uncertainties . . . . . . . . . . . 105 Interaction Force in the Presence of Parametric Uncertainties . . . . . . . . . . . . . . 106 Estimated Parameters in the Presence of Parametric Uncertainties . . . . . . . . . . . . 106 Position Tracking Error ep in the Presence of Parametric Uncertainties and Disturbances 106 Force Tracking Error ef in the Presence of Parametric Uncertainties and Disturbances . 107 Estimated Parameters in the Presence of Parametric Uncertainties and Disturbances . . 107 Joint Torque in the Presence of Parametric Uncertainties and Disturbances . . . . . . . 107 Con guration of a Robotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A Manipulator in Contact With a Sti Environment . . . . . . . . . . . . . . . . . . . 114
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List of Tables 7.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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Acknowledgements I would like to express my deepest gratitude to Professor Masayoshi Tomizuka for in uencing my way of thinking and for helping and supporting my research. I would also like to thank Professor Karl Hedrick and Professor Shankar Sastry for their invaluable comments as members of my dissertation committee. I would like to thank all my friends in Berkeley and all members of Professor Tomizuka's research group, with whom I shared many good and bad moments during my study here. I would especially like to thank Professor Hui Peng, Dr. George T. C. Chiu, Dr. Yean-Ren Hwang, Dr. LiangJong Huang, Dr. Satyajit Patwardhan, Professor Addisu Tesfaye, Dr. Wei-Hsin Yao, Dr. Eugene David Tung, Dr. Thomas M. Hessburg, Dr. Perry Li, Rob Bickel, Mohammed Al-Majed, Prabhakar Pagilla, Chieh Chen, and Lin Guo for their inspiring discussions and warm friendship, and Victor Chu and Carlos Osorio for their help with computer software in the laboratory. Finally, I would like to thank all my family members for their encouragement and my dearest Dorothy for her love and support.
1
Chapter 1
Introduction 1.1 Control of Uncertain Nonlinear Dynamics Although linear control theory has evolved a variety of powerful methods and has had a long history of successful industrial applications, it has found to be inadequate in many applications, for many reasons such as increasingly stringent performance requirements and large operating range, which invalidate the use of linearized models. Many physical systems have so-called "hard nonlinearities", such as Coulomb friction, saturation, dead zones, backlash, and hysteresis. These nonlinearities are nonsmooth or discontinuous, and do not allow linear approximations. They often cause undesirable behavior in the control system, such as instability and limit cycles if not properly handled. It may be necessary to apply nonlinear control to obtain acceptable performance. The design of nonlinear controllers is not necessarily complex. For example, in robot control, it is easier to design a stabilizing nonlinear controller than a stabilizing linear controller. Also, with the advances of low-cost microprocessors, it is neither dicult nor costly to implement nonlinear controllers. All these factors have made nonlinear control increasingly more popular, and the eld has grown quickly during the past twenty years. Earlier results in nonlinear control [47] required exact knowledge of the system dynamics. In reality, though we may apply physical laws to model the system and nd the shapes of the nonlinear functions, parameters of the system (e.g., the inertia parameters of a new object grasped by a robot) may depend on operational conditions and may not be precisely known in advance. Because of factors such as aging e ect, the parameters may also be slowly time-varying. These types of uncertainties are called parametric uncertainties and may cause the control law designed based on the nominal model unstable or degrade its performance. In mechanical systems, nonlinearities such as nonlinear friction force and backlash as well as external disturbances cannot be modeled exactly. These types of uncertain nonlinearities can be classi ed as unknown nonlinear functions. On the whole, the system may be subjected to both parametric uncertainties and unknown nonlinear functions. Control of uncertain nonlinear dynamics is, thus, essential for successful applications. In fact, during the past twenty years, the control of uncertain dynamics has been very popular. Numerous algorithms have been proposed, which can basically be classi ed into two classes: adaptive control and deterministic robust control.
2
1.2 Previous Work 1.2.1 Adaptive Control (AC) Biological systems cope easily and eciently with changes in their environments. As interests in control theory have shifted over the years to the control of systems with large uncertainty, e orts are naturally made to incorporate in them characteristics similar to those in living systems; numerous words, such as adaptation, learning, pattern recognition and self-organization, were introduced into the control literature. Among those words, adaptive control, which was born in the late 1950s to deal with parametric uncertainties, was the rst introduced. Since then, it has remained to be a mainstream research activity, with hundreds papers published on it every year, and has become a wellformed discipline, especially for linear systems. Earlier results in adaptive control were developed for linear time-invariant (LTI) systems [68, 4, 85, 104] described by x_ = A()x + B()u (1.1) y = C ()x + D()u where represents the vector of parameters that are unknown but constant. For LTI systems with relative degree one (the relative degree r of a LTI system is equal to the number of poles minus the number of zeros of its transfer function), a stable adaptive controller was proposed in [88] with the concept of positive realness playing an important role. With the concept of the augmented error introduced by Monopoli [77], the general problem with r 2 was nally solved around 1980 [87, 78, 34, 68]. These breakthrough results made researchers feel that the era of practical adaptive control had nally arrived. However, it was soon realized that the above adaptive control derived for the ideal case would result in the parameter error growing without bound and destabilizing the system when bounded disturbances were present [27]. It was also shown, primarily by simulations, that other perturbations, such as time-varying parameters and un-modeled dynamics [101], could result in instability. All of these clearly indicated that new approaches were needed to assure the boundedness of all the signals in the system and led to a body of work referred to as the robust adaptive control theory. Two distinct approaches were taken to achieve robustness. One is to use the appropriate reference input. The other is to modify the adaptation law. It was realized even in the 1960s [3] that for parameter convergence the reference input should satisfy certain conditions, generally referred to as persistent excitation (PE) conditions. Narendra and Annaswamy [83] demonstrated that the degree of persistent excitation would determine whether or not the system would be robust in the presence of speci ed disturbances, i.e., it was shown that in the absence of disturbances but with a persistently exciting input, an adaptive system is uniformly asymptotically stable. In view of the importance of PE in adaptive systems, Boyd and Sastry [6, 104] used frequency domain methods to show that if the spectral measure of the input was concentrated at at least n points, the state of an n-th order dynamical system would be persistently exciting. For most applications such as trajectory tracking control, the reference input (or desired output trajectory) is speci ed by the task and normally does not satisfy the PE condition. Thus, the rst approach has limitations in practical problems. As to the modi cation of adaptation laws to achieve robustness, several approaches were proposed. One was the use of a dead zone [27, 93] in the adaptation law. In this approach, it has to be assumed that the magnitude of the disturbance was known and asymptotic stability was lost. By introducing an additional term in the adaptation law (referred to as -modi cation), Ioannou and Kokotovic [46] achieved the uniform stability at large. However, when the disturbance was not present,
3 the error would no longer tend to zero and asymptotic stability was lost. To overcome this drawback, -modi cation was proposed in [84]. More recently, by assuming that parameters lie in a known compact set, projection methods presented by Sastry in [104] and by Goodwin and Mayne in [33] have become popular to achieve robustness. Other recent developments in adaptive control of linear systems include relaxing the assumptions under which stable adaptive control is possible. Another drawback of adaptive control is that the transient performance is not clear. It was shown in [171] that poor initial parameter estimates may result in unacceptable poor transient behavior. The design of adaptive controllers with improved transient performance is a current research topic. Fu [30] introduced a variable structure control (VSC) design for a relative degree two plants and Narendra and Boskovic [86] proposed a combined direct, indirect, variable structure method. However, transient performance under these methods is still not guaranteed and the resulting controllers are discontinuous, which leads to control chattering. With a known high frequency gain, an L1 formulation was used in [24] to improve transient performance of continuous model reference adaptive control (MRAC). The assumption of known high frequency gain was relaxed in [92] and a di erent interpretation using modi ed high order tuning was given in [134]. However, in all of these controllers, only parametric uncertainties were considered and robustness was not discussed. In trying to extend the above adaptive schemes from linear systems to nonlinear systems, one was faced with considerable obstacles. One important factor was the lack of a systematic design methodology for nonlinear feedback. As such, adaptive nonlinear control started with speci c problems, e.g., trajectory tracking control of rigid robot manipulators. A robot arm is constructed to simulate a human being's arm to accomplish a variety of tasks and has been widely used in industry to increase
exibility and productivity. Thus, high performance control of robots is of practical signi cance. Since robot dynamics are described by a set of highly coupled nonlinear di erential equations, control of such a system is challenging, and has been extensively studied during the past decade. Earlier results, such as the computed torque method [119, 81], which utilized the feedback linearization method [47], required exact knowledge of the robot dynamics. It was soon found that such methods could not perform well in practical application because of parametric uncertainties such as the payload. A nonlinear adaptive method that guarantees asymptotic stability without any approximation of nonlinear dynamics was rst developed by Craig, Hsu, and Sastry [23] around 1986. The requirement of acceleration measurements and invertibility of the estimate of the inertia matrix was later removed by Slotine and Li [110, 111], Wen and Bayard [135], Sadegh and Horowitz [103], and Middleton and Goodwin [75]. Sadegh and Horowitz presented an adaptive scheme [103] which used reference trajectory information rather than actual state information, and a locally exponentially stable adaptive algorithm [102] under the assumption of (semi) persistent excitation. Recently, Whitcomb, et al. [138] presented comparative experiments for di erent adaptive control algorithms. Motivated by the initial success of the adaptive control of robot manipulators, the adaptive control of general nonlinear systems has also undergone rapid developments during the past ten years [65, 105, 95], leading to global stability and tracking results for reasonably large classes of nonlinear systems [59, 62, 52]. Earlier results [82, 105, 123, 95, 96, 121, 9, 54, 26] were based on the feedback linearization method. Because of the parameter-dependent forms of feedback linearization conditions and the "certainty-equivalence" implementation, restrictions had to be imposed either on the location of unknown parameters or on the type of nonlinearities. Accordingly, the earlier results could be classi ed into two categories: the nonlinearity-constrained schemes [82, 105, 123, 95, 96], which do not restrict the location of unknown parameters but impose restrictions on the nonlinearities of the original system
4 as well as on those appearing in the transformed error system, and the uncertainty-constrained schemes [121, 9, 54], which impose restrictions on the location of unknown parameters but can handle all types of nonlinearities. Speci cally, in the rst category, as long as the norm of perturbing nonlinear terms was dominated by an ane function, for all initial estimates lying in some open neighborhood of the true values in the parameter space, global convergence results were obtained in [82] for purefeedback systems by updating estimates of both the feedback terms and the coordinate transformations that were required to linearize the system. Sastry and Isidori [105] solved the problem of adaptive asymptotic tracking of feedback linearizable minimum phase nonlinear systems (including pure-feedback systems). Overparametrization was required and some restrictive assumptions on nonlinearities, such as the change of coordinates being globally Lipschitz in terms of states, were made. An indirect scheme (indirect adaptive control di ers from direct adaptive control in that it relies on an observation error to update the parameters rather than relying on the output error) was proposed in [123] to overcome the overparametrization problem. The restrictive PE condition, an additional assumption required by the indirect scheme, was then eliminated by a "semi-indirect" scheme [123], which combined parameter estimation elements from both the direct and the indirect approaches. Global stabilization was achieved in [95] for feedback stabilizable nonlinear systems, a larger class of nonlinear systems than feedback linearizable nonlinear systems. For the uncertainty-constrained schemes, assuming that the matching condition (loosely speaking, the matching condition implies that control and uncertainty enter the system dynamics via the same channel) was satis ed, a feedback control scheme was developed in [121] for stable regulation of a class of nonlinear plants with parametric and dynamic uncertainties, and the estimate of stability region was given. The matching condition was relaxed to the extended-matching condition in [9, 54]. Praly, et al. [96] uni ed and generalized most of the earlier results by introducing a novel Lyapunov function for the design of direct schemes and by generalizing equation error ltering and regressor ltering for the design of indirect schemes. The key assumption in this approach was that a Lyapunov-like function existed and depended on unknown parameters in a particular way. Depending on the properties of this function, various designs were possible, including feedback linearization designs when this function was quadratic in the transformed coordinates. Output-feedback designs were studied in [55, 71, 72]. It soon became clear that the "certainty-equivalence" adaptive controllers based on the feedback linearization technique were unable to achieve stability without restrictions on nonlinearities. New thinking was needed for the systematic design of adaptive nonlinear controllers, resulting in the exciting era of adaptive nonlinear control [65]. The new thinking employed a recursive design methodology | backstepping. With this methodology, the construction of feedback control laws and associated Lyapunov functions became systematic. Strong properties, such as global or regional stability and tracking, were built into the nonlinear system in multiple steps, never higher than the system order. In contrast to feedback linearization methods that required cancelation of all nonlinearities, the backstepping design avoided wasteful cancelations and retained useful nonlinearities. Backstepping designs were exible and allowed a choice of design tools for dominating, or adapting to, uncertain nonlinearities. Speci cally, Kanellakopoulos, et al. [56, 60] presented a systematic design of globally stable and asymptotically tracking adaptive controllers for a class of nonlinear systems transformable to a parametric strict-feedback canonical form (local results for parametric pure-feedback systems). The number of overparametrization was reduced in half in [50], and the overparametrization problem was soon eliminated by Krstic, et al. [62] by elegantly introducing the concept of tuning function. Recently, the nonlinear damping was introduced by Kanellakopoulos [52, 53] to improve transient performance.
5 Generalization to output-feedback design was presented in [66, 67]. The nonlinear design method was also applied to linear systems in [63, 64]. Compared to the previous traditional adaptive control schemes for linear systems, which could not resolve the con ict between their linear form and their nonlinear nature, the new nonlinear design achieved stronger stability and convergence properties with a much more transparent and straightforward design procedure. These improvements o ered new insights into the eld of adaptive control.
1.2.2 Deterministic Robust Control (DRC) One of the earliest approaches to the control of uncertain systems was sliding mode control (SMC) or variable structure control (VSC) [48, 127, 128, 129, 114, 174, 165, 166, 28, 90, 31, 143], which was rst studied in the Soviet Union in the 1960's [48] and was introduced to western researchers by Utkin [127, 128, 129]. The central feature of SMC is sliding mode, in which the dynamic motion of the system is e ectively constrained to lie within a certain subspace of the full state space. The sliding mode is achieved by altering the system dynamics along some sliding surfaces in the state space so that the system state is rst brought to these surfaces or their intersection surface and is made to stay on them thereafter. During the sliding mode, the system is equivalent to an unforced system of lower order, termed the equivalent system, which is insensitive to both parametric uncertainties and unknown nonlinear functions when the matching condition is satis ed. The design of a SMC system consists of two stages. In the rst stage, sliding surfaces are selected so that the equivalent system is asymptotically stable and has a desired dynamic response. This stage may be completed without any assumptions about the form of the control functions. The static design of sliding surfaces was presented in [25] and dynamic sliding mode design was studied in [143, 12, 167, 154]. In the second stage, a control law is determined depending on the speci c plant and the chosen sliding surfaces to ensure that the chosen sliding mode is attained. Among SMC schemes for robot manipulators, there have been proposals to make each sliding surface attractive. This approach makes the problem complicated, resulting in a control law de ned implicitly by a set of fairly complicated algebraic inequalities [165, 166, 106]. By exploiting the passivity of robot dynamics, other researchers obtained simple control laws, which made the system state attracted to the intersection of the surfaces without necessarily reaching each individual one [90, 161, 141, 117]. Recently, a dynamic sliding mode controller, in which a dynamic compensator is introduced in forming the sliding surfaces, was employed in [143] to ensure that the system achieved a desired second-order model to realize several control purposes, such as impedance control, hybrid motion/force control, and constrained motion control. Reaching transients were also eliminated so that the system was maintained in the sliding mode all the time. Robust sliding mode control in the form of MIMO input-output (I/O) linearization was considered by Fernandez and Hedrick in [28]. Hedrick, et al., applied SMC to the control of automotive engines [79, 16, 37], aircraft ight control [40], electronics suspension control [1] and "platoon control" in automated highway systems [41]. Observers based on SMC were discussed in [113]. One of the drawbacks of the SMC is that, in general, it only applies to the uncertain systems which satisfy the matching condition. The most severe drawback of the SMC is that the control law is discontinuous across sliding surfaces. Such control laws lead in practice to control chattering, which involves high frequency control activity and may excite neglected high-frequency dynamics. To remove control chattering, smoothing techniques, such as a boundary layer [106, 112], have to be employed. However, such a modi cation can guarantee the tracking error only within a prescribed
6 precision. Although transient performance is still preserved at large, asymptotic stability is lost and a trade-o exists between control bandwidth and tracking precision. Another general deterministic robust control (DRC) technique has been developed based on Lyapunov's second method originally by Leitmann, et al. [39, 69, 21]. For uncertain systems satisfying the matching condition, a stabilizing discontinuous min-max control law was developed in [39, 38]. Like smoothed SMC, a continuous approximation of the min-max control law that guaranteed globally, uniformly, ultimately bounded (GUUB) stability instead of asymptotic stability was presented in [22]. Although the matching condition is met in many important applications, such as mechanical systems, it is still very restrictive. Subsequently, much e ort has been devoted to loosening the restrictions imposed by the matching condition. Two main approaches have been used to tackle this issue. The rst one studies the robustness of the controlled system against the mismatched uncertainty. In this approach, the uncertainty is rst decomposed into two categories, the matched and the mismatched. The controller is designed assuming no mismatched uncertainty. A passive stability analysis is then made for mismatched uncertainty. The framework was rst introduced by Barmish and Leitmann [5] for linear systems. Subsequent results were presented in [15, 13, 99]. Since this approach is based on the stability margin of the stabilized nominal system, certain restrictions on the mismatched uncertainty have to be made and the design procedure is not systematic. The second approach looks for a structural condition under which a systematic robust control design may be applied. This approach imposes restrictions on the location of uncertainty as in uncertainty-constrained adaptive nonlinear schemes. Along this line, Thorp and Barmish [124] presented a robust control design for linear uncertain systems satisfying a generalized matching condition. In extending the results to uncertain nonlinear systems, once, the backstepping procedure played an important role. Marino and Tomei [73] solved the robust stabilization problem of nonlinear systems with vanishing uncertainties and satisfying the strict feedback condition ( similar to the parametric-strict feedback condition). The case of nonvanishing uncertainties, which allows bounded disturbances and tracking, was solved by Freeman and Kokotovic in [29] by extending the results of [73]. A di erent approach, multiple surface sliding mode control, was presented by Won and Hedrick in [140]. The approach used a series of simple Lyapunov functions instead of the whole Lyapunov function in the backstepping design and made each sliding surface attractive outside a userde ned boundary layer thickness. Based on backstepping, Qu [97] presented the generalized matching condition for nonlinear systems in a pure-feedback form.
1.3 Motivations and Contributions of the Dissertation 1.3.1 General Methodology In spite of the recent rapid advances in adaptive nonlinear control, one problem remains unsolved, i.e., unknown nonlinear functions have not been considered. All the adaptive nonlinear controllers mentioned in Section 1.2.1 dealt with the ideal case of parametric uncertainties only. Nonlinearities of the system were assumed known and unknown parameters were assumed to appear linearly with respect to these known nonlinear functions. The integral adaptation laws developed for linear systems may lose stability when even a small disturbance is present. Considering that every real system has some sorts of disturbances, we wonder if we can safely implement such adaptive controllers. This is more serious for nonlinear systems, as shown in [100] for the adaptive control of robot manipulators. As in the adaptive control of linear systems, one may apply similar remedies to nonlinear systems. For
7 example, the adaptation law may be modi ed to achieve stability for bounded disturbances [100]. However, such modi cations do not guarantee tracking accuracy since the steady state tracking error can only be shown to stay within an unknown ball, whose size depends on the disturbances Furthermore, transient performance is unknown. In [94], by using a variant of the -modi cation and backstepping procedure, Polycarpou and Ioannou presented a robust adaptive control design for a class of single input single output (SISO) nonlinear systems in a "semi-strict" feedback form, which allowed both parametric uncertainties and unknown nonlinear functions. However, transient performance was not guaranteed and asymptotic stability was lost even in the presence of parametric uncertainties only. Despite the above drawbacks of adaptive control, one should realize that the main advantage of adaptive control lies in the fact that, through on-line parameter adaptation, parametric uncertainties can be eliminated and, thus, asymptotic stability or zero nal tracking error can be achieved in the presence of parametric uncertainties without using high-gain feedback. New thinking should be adopted to utilize this advantage judiciously. On the other hand, the deterministic robust control (DRC) mentioned in Section 1.2.2 employs proper controller structures to attenuate the e ect of the model uncertainties coming from both parametric uncertainties and unknown nonlinear functions. In general, it can guarantee transient performance and certain nal tracking accuracy. However, DRC does not discriminate between parametric uncertainties and unknown nonlinear functions and the control law uses xed parameters. Model uncertainties coming from parametric uncertainties cannot be reduced. In order to reduce tracking errors, the feedback gains must be increased, resulting in high-gain feedback and increased bandwidths of closed-loop systems. Theoretically, SMC can use discontinuous control laws and some of the so-called continuous DRC schemes [98, 97] can use in nite gain feedback control to achieve asymptotic tracking. However, those are impractical and unachievable solutions because of nite bandwidths of physical systems. In view of the above drawbacks and advantages of both adaptive control (AC) and DRC, this dissertation will propose a new approach, adaptive robust control (ARC). which uses both means | proper controller structure and parameter adaptation | to reduce tracking errors. The DRC technique will be used to design a baseline control law (proper controller structure) to guarantee transient performance and certain nal tracking accuracy. On top of it, parameter adaptation will be used to reduce the model uncertainties coming from parametric uncertainties (as in AC) and to improve tracking performance. In other words, the robust control problem is formulated under the general setting of DRC, but the di erence between parametric uncertainties and unknown nonlinear functions is recognized and parameter adaptation is used to reduce the parametric uncertainties. In general, DRC design needs the modeling uncertainties to be bounded by some functions with known shapes but the estimated parameters by AC design may be unbounded in the presence of unknown nonlinear functions. By formulating the robust control problem under the general setting of DRC, when one designs either the baseline robust control law or the parameter adaptation law, one always keeps in mind the above con icts between the DRC design and the AC design and solves the con icts at the beginning. In such a way, stronger stability results can be obtained. Such a formulation has several advantages: it can naturally eliminate the transient problem and robustness problem of adaptive control, while at the same time, improve the tracking performance of DRC by reducing model uncertainties. The qualitative results obtained [154, 153, 152, 159, 157, 158, 156, 160] well re ect this philosophy. In general, in the presence of both parametric uncertainties and unknown nonlinear functions, the same qualitative results as DRC are achieved. Furthermore, if the model is accurate | i.e., in the presence of parametric uncertainties
8 only | asymptotic tracking is achieved without using high-gain feedback as in AC. The above idea is simple and natural. In fact, during the past several years, some researchers in the two elds have been trying to achieve that goal. However, they all failed in one way or another. Researchers in the robust adaptive control eld [100, 94] tended to formulate the problem for parametric uncertainties rst and then to robustify the schemes. This approach inevitably complicated the problem because it lost the whole picture and leaded to conservative results | only stability was achieved and nothing could be obtained about performance. On the other hand, researchers in the DRC eld realized that parameter adaptation could reduce the control e ort [107, 110, 35] but did not consider its destabilizing e ect and the main advantages of the AC and DRC methods. Thus, when parameter adaptation was introduced in DRC design, as in the adaptive sliding mode control in [107], transient performance was lost and a discontinuous control law had to be used, since the traditional proof in AC was used. Furthermore, unlike the original SMC schemes for which smoothing techniques have been developed for the discontinuous control law, the scheme in [107] cannot directly employ the smoothing techniques since it is not robust to any approximation errors. This robustness problem was corrected in [107] by stopping adaptation inside the boundary layer. However, transient performance was not guaranteed and asymptotic stability could not be achieved in the presence of parametric uncertainties only. Finally, we would like to di erentiate our algorithms from other adaptive robust control algorithms that have appeared in the literature [14]. Instead of true parameter adaptation, those algorithms in [14] used adaptation to adjust some of the feedback gains to achieve stability when the bounds of modeling uncertainties were unknown. So, their main purpose was to relax the conditions under which stabilization was possible. In general, those schemes do not provide better performance than their DRC counterparts when the bounds of modeling uncertainties are known. By contrary, our algorithms use true parameter adaptation to improve performance instead of relaxing the stabilizing conditions. These claims are veri ed by the experimental results shown in chapter 7.
1.3.2 General Form The proposed ARC is formulated for general MIMO nonlinear systems in terms of the concept of adaptive robust control (ARC) Lyapunov functions. The formulation reduces the ARC of a system to the problem of nding an ARC Lyapunov function for the system. By using backstepping design procedure, we may successfully construct ARC Lyapunov functions for a class of MIMO nonlinear systems in a semi-strict feedback form. The form is very general and includes mechanical systems, such as robot manipulators. In the absence of unknown nonlinear functions, the form reduces to a parametric-strict feedback form, which extends the parametric-strict feedback form used in general adaptive nonlinear control [62, 65] in several ways. First, it is a MIMO version. A MIMO parametric-strict feedback form was also presented in [65] but it allowed coupling among di erent input channels of each layer at the last step only. Second, the form allows parametric uncertainties at each layer's input channels also, which increases the diculty in the design of a pure adaptive control law considerably. Third, the last layer's state equations do not have to be completely linearly parametrized (linear parametrization is a requirement in [62, 65]). This extension is vital for applications, such as control of robot manipulators, where the dynamics cannot be linearly parametrized in the state equations.
9
1.3.3 Applications below.
The proposed ARC is applied to the control of robot manipulators in several ways as explained
Trajectory Tracking Control of Robot Manipulators Industrial manipulators are commonly used in tasks such as painting, welding and material handing. In these tasks, their end-e ectors are required to move from one place to another in a free workspace or to follow desired trajectories. In order to meet increased productivity requirement as well as tight tolerance requirements, it is essential for the manipulator to follow a desired trajectory as close as possible at fast speed. Thus, trajectory tracking control of robot manipulators is of practical signi cance. It is also the simplest but most fundamental task in robot control [81]. Because of these factors, during the past decade, numerous adaptive algorithms and DRC algorithms have been proposed. In addition to those schemes, there are also some adaptive schemes [17, 116, 115] termed as performance-based (or direct) adaptive control in [18], in which adaptation laws are used to adjust controller gains instead of true parameters. Thus, these gain-based schemes share the same properties as the adaptive robust scheme in [14]. They are claimed to be simple, computationally ecient and require very little model information. Robustness to bounded disturbances is also guaranteed. However, they can only guarantee tracking errors within certain bounds even when the system is subject to parameter uncertainties only. Some comparative experiments were carried out in [138] to test some of the model-based (or parameter-based) adaptive algorithms. However, the tested algorithms belonged to the same class. Facing so many algorithms and so many qualitatively di erent approaches, one has diculty choosing a suitable one for a particular application since each algorithm has its own claim. Thus, it is of practical signi cance to test qualitatively di erent approaches on the same machine to understand their fundamental advantages and drawbacks. To work toward that direction, in addition to the proposed ARC, several typical robust and adaptive control algorithms are also developed for comparison. Speci cally, two ARC schemes, one based on the conventional adaptation law structure [110] and one using the idea of desired compensation adaptation law [103], are rst developed by applying the proposed ARC. Then, a very simple nonlinear PID scheme is proposed, which can guarantee the stability and requires little model information. By adjusting the feedback gains on-line, a simple gain-based adaptive control is also suggested to remove the requirements in choosing feedback gains in the nonlinear PID scheme. By combining the design techniques of the gain-based adaptive control with the proposed ARC, a new adaptive robust scheme is also proposed to remove the conditions on the selection of the controller gains. Finally, all schemes, as well as two benchmark adaptive control schemes [110, 103], are implemented and compared. Experimental results are presented to show the advantages and the drawbacks of each method. Comparative experimental results show that importance of using both proper controller structure and parameter adaptation in designing high-performance robust controllers. It is observed that the proposed ARC achieves the best tracking performance in the experiments. Detailed conclusions are given in chapter 7.
Constrained Motion and Force Control Another important class of tasks requires the robot end-e ector to make contact with its environment. Typical examples of such tasks are contour following, grinding, scrubbing, deburring, as
10 well as those related to assembly and with multi-arm robot systems. In these applications, the contact force between the end-e ector and the environment is generated, which modi es the dynamics of robot manipulator and creates some problems that do not exist in the free motion of robot systems. Research in this area has focused on simultaneous control of motion and force. Depending on the contact environment, di erent approaches [139, 143] have been proposed. The rst type of motion and force control considers the robot whose end-e ector is in contact with rigid surfaces [81, 132, 133, 74, 76, 163, 44, 162, 20, 146, 148, 147, 142, 168, 118, 45, 49, 108, 8, 70, 10]. In many cases the contact surface sti ness is so large that the surfaces must, in practical terms, be viewed as rigid. Such a view may be appropriate to prevent damage of either the workpiece or the end-e ector. Typical example of constrained motion is contour following, in which the robot end-e ector is required to move along a very sti or rigid contact surface. In the normal direction of the surface, the end-e ector's motion is restricted by the surface, and the robot can only move along the tangent direction of the surface. Correspondingly, contact force exists in the normal direction of the surface and no force but that of friction occurs along the tangent direction. This unique duality will be used in the subsequent formulation of constrained motion. When the robot moves on rigid surfaces, holonomic kinematic constraints are imposed on the robot motion that correspond to some algebraic constraints among the manipulator state variables. Dynamics of such a robot system is described by a set of nonlinear di erential-algebraic equations, which is called singular system [74] or descriptor system [76]. The objective is to control both the motion on the constraint surfaces and the generalized constrained force. A general theoretical framework of constrained motion control was rigorously developed by McClamroch and Wang [74]. The proposed controller was based on a modi cation of the computed torque method. A Lyapunov's direct method was utilized by Wang and McClamroch [133, 132] to develop a class of decentralized position and force controllers. Mill and Goldenberg [76] applied descriptor theory to constrained motion control. The controller was derived based on a linearized dynamic model of the manipulator. State feedback control and dynamic state feedback control were utilized to linearize the robot dynamics with respect to motion and contact force in [163], and [168], respectively. The above methods are based on the exact model of constrained robot dynamics. As in the case of free motion, robust control methods are needed. There are many papers applying the two robust control methods to constrained motion of robot manipulators: adaptive constrained motion control [10, 118, 49, 149, 2] for parametric uncertainties only, and SMC motion and force control [148, 143, 142, 145]. Basically, adaptive constrained motion control methods proposed in [108, 10, 118, 49] are all based on the reduced dynamic model proposed in [74], which enable motion and force controllers to be designed separately. It should be noted that this model is only valid for frictionless contact surfaces, while most real contact surfaces have friction. Furthermore, the previous parameter adaptation laws proposed are only driven by motion tracking error. Thus, the force tracking error can be guaranteed to be only bounded unless some persistent excitation conditions are satis ed | these are dicult to verify and depend on speci c desired motion trajectories. Although, theoretically, the force tracking error can be made small by using a large proportional force feedback gain [10, 49], the gain for the proportional force feedback is severely limited in applications because of the acausality that arises from the rigid body dynamics assumed in the modeling of the robot [91]. In fact, recent one-dimensional force experimental results presented by [130] and [91] suggest that the best force tracking performance is achieved by integral (I) force feedback or PI force feedback control. Considering these factors, we propose a new
11 transformed constrained dynamic model that is suitable for controller design and is also valid for friction surfaces with unknown friction coecients in [149, 155]. The resulting adaptive controller guarantees asymptotic motion and force tracking without persistent excitation, and has the expected PI type force feedback control structure with a low proportional force feedback gain. It should be noted that all the above force controllers are synthesized based on the assumption that the robot keeps contact with the surface when the controller is applied. This assumption is valid only if the controller has good transient performance since, otherwise, drastic transient response may cause the robot to lose contact with the surface, thus voiding the obtained result. Therefore, it is important to design a motion and force controller with a guaranteed transient performance. This goal is achieved by applying the proposed ARC and using our previous general formulation of constrained motion in [149, 155]. Dynamic motion sliding mode and ltered force tracking error are used to enhance the dynamic response of the system. The suggested control law can achieve asymptotic motion and force tracking without persistent excitation condition in the presence of parametric uncertainties, and has a guaranteed transient performance with a prescribed nal tracking accuracy in the presence of both parametric uncertainties and external disturbances or modeling errors. Simulation results illustrate the proposed motion and force controller.
Coordinated Control of Multiple Robot Manipulators For assembly-related tasks, such as heavy material handling, several manipulators are required to grasp a common object. In these applications, a set of homogeneous constraints are imposed on the positions of the manipulators. As a result, degrees of freedom (DOF) of the whole system decrease, and internal forces exerted on the object by the manipulators are generated. These internal forces do not a ect the motion of the object. To control the robots cooperatively, a number of control methods have been proposed. In closed kinematic chain methods [120, 80, 42, 172, 173, 32, 146], only the position of the whole system is controlled. Hence, the joint torque for a particular load of the object cannot be uniquely determined and load distribution is required. In hybrid position/force control methods [61, 125, 170, 137, 136, 126, 131], both position and internal force of the whole system are controlled. The DOF lost in the arm con guration from the imposed kinematic constraints is introduced as the DOF needed to control the internal forces of the system [61]. This scheme is important especially when the object is fragile or needs operations such as compression, tension, and torsion. The problem of a constrained object grasped by multiple manipulators has been considered in [169, 164, 43, 148]. The methods in [169, 164] were based on the exact model of the system, and the adaptive law derived from Popov hyperstability theory [43] needs the measurements of acceleration and force derivative. A set of transformed dynamic equations of the robotic system were obtained in the joint space in [148], in which internal force and constrained force had the same form and could be controlled in the same method. The VSC method was used to deal with the problem of parameter uncertainties as well as external disturbances. However, the e ect of friction force on the object was not considered, and the transformation was basically formed by a position relationship but may not be easily obtained. In this dissertation, we apply the ARC to the robust control of motion, internal force, and external contact force control of multiple manipulators handling a constrained object in the presence of both parametric uncertainties and disturbances. Parametric uncertainties may exist in the manipulators and in the object and in the friction coecients of contact surfaces. A set of transformed dynamic
12 equations are obtained in the joint space, in which internal force and external contact force have the same form [150]. Thus, internal force control and external contact force control can be dealt with in the same way as in constrained motion and force control. The resulting controller possesses those nice properties mentioned in the above subsubsection.
Motion and Force Tracking Control of Robot Manipulators In Unknown Sti ness Environments In addition to constrained motion, another important class of contact tasks is when the robot end-e ector comes in contact with surfaces that are not so rigid and can be modeled as sti ness environments. Typical examples include the deburring process. The objective in these applications is the same as that in constrained motion | i.e., control of motion along the tangent direction of the surface and control of force along the normal direction of the surface. There are only a few published papers addressing motion and force tracking control in the presence of unknown environmental sti ness. Carelli, et. al. proposed an adaptive force control method to estimate unknown parameters of the robot and the environmental sti ness in [11]. The inertia matrix of the robot is assumed to remain constant. Because of the highly nonlinear and coupled nature of the robot dynamics and the wide working range of the robot, this assumption is usually not satis ed. Recently, a variable structure adaptive (VSA) method was developed by Yao, et. al. in [144] to solve this problem. This method resulted in a two-loop control system. VSC method was used in the inner-loop that forced the system to reach and be maintained on a dynamic sliding mode provided by the outer-loop design. In the outer loop, the adaptive control method was used to estimate environmental sti ness and provide the system with good force tracking property. However, the resulting VSC control law was inherently discontinuous and the associated chattering problem had not been analyzed. In [151], we developed an adaptive motion and force control algorithm to eliminate the chattering problem. However, transient performance was not guaranteed when disturbances appeared. The e ect of timevarying equilibrium position was not considered. In this dissertation, we show that motion and force tracking control of such a system falls nicely into the proposed semi-strict feedback form with a relative degree two. The proposed ARC is applied and the resulting controller needs measurements of position, velocity and interaction force only. Transient performance is also guaranteed when disturbances appear.
1.4 Outline of the Dissertation The dissertation is organized into two parts. Part one deals with general theory and Part two talks about applications. Part one consists of the following chapters:
Chapter 2 uses a simple rst-order system to illustrate the general idea of the proposed ARC. An adaptive controller and a DRC controller are also constructed for comparison.
Chapter 3 generalizes the proposed ARC to a class of SISO nonlinear systems with arbitrary known relative degrees and transformable to a semi-strict feedback form.
Chapter 4 introduces the concept of ARC Lyapunov functions and presents a general framework of ARC for general nonlinear systems via ARC Lyapunov functions.
13
Chapter 5 talks about the systematic construction of ARC Lyapunov functions via the backstepping design procedure.
Chapter 6 solves the ARC of a class of MIMO nonlinear systems with arbitrary known relative degrees and transformable to a semi-strict feedback form. Part two consists of the following chapters:
Chapter 7 applies the proposed ARC to trajectory tracking control of robot manipulators. Several
conceptually di erent adaptive and robust control algorithms are also developed for comparison. Comparative experimental results on a SCARA robot are presented.
Chapter 8 applies the proposed ARC to constrained motion and force control, coordinated control of multiple manipulators, and motion and force control of robot manipulators in unknown sti ness environments.
Finally, Chapter 9 concludes the dissertation and discusses possible future research directions.
14
Part I
Adaptive Robust Control - Theory
15
Chapter 2
Control of a First-order Uncertain System In this chapter, we consider the tracking control of a rst-order nonlinear system to illustrate the basic ideas of the proposed adaptive robust control (ARC) scheme. The results can also be used later in the backstepping design for general nonlinear systems. The rst-order nonlinear system under consideration is described by
x_ = f (x; t) + u
x; u 2 R
(2.1)
where u is the control input. Normally, it is very hard to determine the exact form of the nonlinear function f (x; t). In this chapter, we describe it in two parts. The rst part represents all the terms that can be modeled and linearly parametrized: i.e., this part normally represents the terms derived by physical laws or certain kinds of approximation and its form or base shape is usually available but its magnitude may not be known in advance. For the rst order system (2.1), it is assumed to be described by (x; t), where (x; t) is a known shape function and is an unknown magnitude parameter. The second part is used to represent terms that cannot be modeled or linearly parametrized as well as those which may be due to external disturbances and modeling simpli cations, such as neglecting Columb friction. This part is denoted by (x; t). Therefore,
f (x; t) = '(x; t) + (x; t)
(2.2)
For controller design, it is necessary to make some reasonable assumptions about the prior knowledge of the plant. The more we know about the plant | i.e., the more strict the assumptions are | the better the nominal performance of the resulting controller will likely be. However, if the assumptions are too strict, the actual plant may not satisfy them and, thus, the obtained nominal performance may likely be useless. Although the exact value of the parameter and the modeling error (x; t) may not be known, the extent of the parametric uncertainty and modeled errors can often be predicted in advance. For example, when a robot picks up an object, although we may not know the exact mass property of the object, we know the maximum payload the robot is going to pick up. Thus, we can make the following reasonable and practical assumptions that and (x; t) are bounded by some known parameters or known functions, i.e., 2 = (min ; max ) (2.3) j(x; t)j (x; t)
16 where min , max , and (x; t) are the known scalars and the known function respectively. In this dissertation, all functions involved in the design are assumed to be bounded with respect to (w.r.t.) time t (e.g., for (x; t), there exists a function p (x) such that 8t; j (x; t)j p (x)), and have nite value when all their variables except t are nite (e.g., x 2 L1 =) (x; t) 2 L1 ). Let xd (t) be the desired output, which is assumed to be bounded with bounded derivatives up to a sucient order. The control problem can be formulated as that of designing a control law for u such that, under assumption (2.3), the system is either globally, ultimately, uniformly bounded (GUUB) stable or asymptotically stable, and the output x tracks xd (t). To illustrate what we want to do, two popularly used nonlinear synthesis methods, deterministic robust control (e.g. sliding mode control) and adaptive control, are rst applied. After that, the proposed new adaptive robust control is naturally introduced by e ectively combining the two methods.
2.1 Deterministic Robust Control (DRC) Since the system (2.1) has relative degree one, sliding mode control (SMC) can be applied. A dynamic sliding mode is employed to enhance the dynamic response of the system in sliding mode and eliminate the unpleasant reaching transient [154]. Let a dynamic compensator be x_c = Acxc + Bc e xc 2 Rnc Ac 2 Rncnc Bc 2 Rnc (2.4) yc = Cc xc yc 2 R Cc 2 R1nc where e = x , xd (t) is the tracking error and constant matrices (Ac ; Bc ; Cc) are chosen to ensure that the resulting dynamic sliding mode exhibits the desired dynamics. (Ac ; Bc ; Cc) is controllable and observable. The sliding mode controller is designed to make the following quantity remain zero. z = e + yc (2.5) = x , xr xr = xd (t) , yc Transfer function from z to e is (2.6) e = G,z 1(s)z where (2.7) Gz (s) = 1 + Gc(s) Gc(s) = Cc(sInc , Ac),1 Bc , 1 and In represents an n n identity matrix. From (2.7), Gz (s) can be arbitrarily assigned by suitably choosing dynamic compensator transfer function Gc (s) as long as G,z 1 (s) has relative degree zero. During the sliding mode, z = 0 and the system response is governed by the free response of transfer function G,z 1 (s). Therefore, as long as G,z 1 (s) is stable, the resulting dynamic sliding mode will be stable and is invariant to various modeling errors. Furthermore, the sliding mode can be arbitrarily shaped to possess any exponentially fast converging rate since poles of G,z 1(s) can be freely assigned. In addition to these results, G,z 1 (s) can be chosen to minimize the e ect of z on e when ideal sliding mode fz = 0g cannot be exactly achieved in practice. The rest of the design is to construct a control law such that the sliding mode is reached. The control law is suggested as u = uf + us (2.8) uf = x_ r (t) , ^'(x; t) us = ,kz , h(x; t)sgn(z)
17 where ^ 2 is the estimate of , sgn(:) denotes the discontinuous sign function de ned as sgn(z ) = 1 if z > 0 and sgn(z ) = ,1 if z < 0, and h(x; t) is any known bounding function satisfying (2.9) h(x; t) j , ~'(x; t) + (x; t)j 8^ 2 ^ where ~ = , is the estimation error. The required known function h(x; t) exists since the extent of uncertainties is known. For example, let
h(x; t) = (max , min )j'(x; t)j + (x; t)
(2.10)
h(x; t) can also be chosen in other ways to simplify the on-line computation time. Theorem 1 The control law (2.8) guarantees that the system (2.1) is exponentially stable and its output tracks the desired trajectory asymptotically. 4
Proof. Choose a positive de nite (p.d.) function as Vs = 12 z 2
(2.11)
From (2.1), (2.5), (2.8) and (2.9), its time derivative is V_ s = zz_ = z[,~'(x; t) + (x; t) + us ] (2.12) jzjj , ~'(x; t) + (x; t)j , kz2 , h(x; t)jzj 2 ,kz = ,2kVs Therefore, (2.13) Vs (t) exp(,2kt)Vs (0) which implies that z exponentially decays to zero. This result leads to the theorem 1 since the sliding mode is exponentially stable. 4
Corollary 1 If the initial value xc (0) of the dynamic compensator (2.4) can be chosen to satisfy Ccxc(0) = ,e(0) (2.14) then the system is maintained in the sliding mode all the time and reaching transient is eliminated, i.e., z (t) = 0; 8t. 4
Proof. If (2.14) is satis ed, z(0) = 0 and Vs(0) = 0. From (2.13), z(t) = 0; 8t.
4
Usually, the control law (2.8) is discontinuous across the sliding surface since it contains sgn(z). Such a control law leads to control chattering in practice. To overcome this problem, the above ideal SMC law can be smoothed by replacing discontinuous robust control term h sgn(z ) by a continuous function h (h sgn(z )). h (h sgn(z )) is required to satisfy the following two conditions: i: z h (h sgn(z )) 0 (2.15) ii: hjz j , z h (h sgn(z )) "(t) where "(t) is any bounded time-varying positive scalar, i.e., 0 < "(t) "max . "(t) is used to measure the approximation error. The SMC law is thus smoothed to
u = uf + us where uf is the same as before.
us = ,kz , h (h sgn(z))
(2.16)
18
Theorem 2 If the smoothed SMC control law (2.16) is applied, the system is exponentially stable at large with a guaranteed transient performance and nal tracking accuracy.
4
Proof. From (2.1) and (2.16), error dynamics is given by z_ + kz + h (h sgn(z)) = ,~'(x; t) + (x; t)
(2.17) Following the same steps as in (2.12) and noting (2.15), the time derivative of Vs is now given by V_ s jz jj , ~'(x; t) + (x; t)j , kz2 , zh (h sgn(z)) (2.18) ,kz2 + hjzj , zh (h sgn(z)) ,2kVs + "(t) so R Vs (t) exp(,2kt)Vs (0) + 0t exp(,2k(t , )"( )d (2.19) exp(,2kt)Vs(0) + "max 2k [1 , exp(,2kt)] This implies that the system is exponentially stable at large with a guaranteed transient performance and nal tracking accuracy since, qtheoretically, both the exponentially decaying rate 2k and the bound of the nal tracking error z (1), "max k , can be freely adjusted by the controller parameters k and ".
4
Remark 1 Two examples of the required continuous function h (h sgn(z )) are as follows.
Continuous Modi cation (CM).
First, as in most smoothed SMC schemes [112, 154], we use the continuous saturation function sat zz to replace sgn(z ). In order to take into account the time-varying nature of h, the strength of the discontinuity, we use a time-varying boundary layer thickness given by z = h+4""h , where "h is any small positive number to avoid the possible singularity in case that h = 0. Thus,
(2.20) h (h sgn(z )) = h(x; t) sat (h+4""h )z Obviously, (2.20) satis es condition i of (2.15). When jz j z , we have, hjz j , zhsat zz = 0. When jz j z , we have hjz j , zhsat zz hjzj , h42 "z2 = 1" [,( 12 hjzj , ")2 + "2 ] " (2.21) Thus, condition ii of (2.15) is satis ed. Smooth Modi cation (SM) . Later, when we extend the methodology from relative degree one systems to general relative degree n systems, we will use the backstepping design procedure, which recursively requires the derivatives of the control components at each step. In such a case, a smooth modi cation is preferred. For this purpose, similar to [94], a smooth approximation of sgn(:) by tanh(:) function is used by considering the following nice properties of tanh(:): tanh(0) = 0 tanh(1) = 1 tanh(,1) = ,1 (2.22) 0 juj , u tanh( "uz ) "z 8u 2 R and "z > 0 where = 0:2785. Letting "z = h" , we have
)z h (h sgn(z)) = h(x; t) tanh h("x;t (t) Noting (2.22), (2.23) satis es the conditions i and ii of (2.15).
(2.23)
4
19
2.2 Adaptive Control (AC) In this subsection, the conventional AC [112, 85] is applied. The adaptive control is formulated for parametric uncertainties only, i.e., for the case where (x; t) = 0. Let the control law be u = ufa + usa (2.24) ufa = x_ d (t) , ^'(x; t); usa = ,ke with ^ updated on-line by
^_ = '(x; t)e
(2.25)
Theorem 3 In presence of parametric uncertainties only ( = 0), if the adaptive control law (2.24)
with the update law (2.25) is applied, the system output follows the desired output asymptotically, i.e., e ,! 0 when t ,! 1. Additionally, if the desired trajectory satis es the following persistent excitation (PE) condition R t+T
t
j'(xd( ); )j2d "p
8t t0
(2.26)
where T; t0 and "p are some positive scalars, then, the estimated parameter ^ converges to its true value (i.e., ~ ,! 0 when t ,! 1). 4
Proof: Substituting (2.24) into (2.1), the error dynamics is e_ + ke = ,~'(x; t)
(2.27)
The time derivative of the positive de nite (p.d.) function
Va = 21 e2 + 21 ~2
(2.28)
is
(2.29) V_ a = e[,~'(x; t) , ke] + 1 ~^_ = ,ke2 which implies that e 2 L2 \ L1 and ~ 2 L1 . From (2.27), e_ 2 L1 and thus e is uniformly continuous. By Barbalat's lemma [112], e ,! 0 1 and asymptotic tracking is achieved. Furthermore, from (2.27), since all terms except e_ are uniformly continuous, e_ is uniformly continuous. Applying Barbalat's lemma again, e_ ,! 0. From (2.27), ~'(x; t) ,! 0. Thus, the PE condition (2.26) will guarantee that ~ ,! 0. 4
2.3 Adaptive Robust Control (ARC) The advantage of the adaptive control in section 2.2 is that, through on-line parameter adaptation, it can reduce the model uncertainty ~ ( in fact, ~ ,! 0). Thus, we can obtain asymptotic stability or a zero steady state tracking error without using high gain feedback (asymptotic stability is achieved for any gain k). However, there are two main drawbacks. First, transient performance of the system is not clear. Second, unknown nonlinear functions, such as external disturbance, are not 1 For a vector , which is a function of time, ,! 0 denotes the asymptotic convergence of
20 considered, and it is well known that the integral type adaptation law (2.25) may su er from parameter drifting and destabilize the system in the presence of even a small disturbance and measurement noise[100] when certain PE conditions are not satis ed. Considering that every real system is always subjected to some sorts of disturbances, it is natural to wonder if the above adaptive controller can be safely implemented. As contrast to adaptive control, transient performance and nal tracking accuracy are guaranteed in the smooth SMC design in section 2.1 for both parametric uncertainties and external disturbances. This result makes the SMC design attractive for applications. From (2.17), we can see that the SMC reduces the tracking error by attenuating the e ect of modeling uncertainties (the left side of (2.17) can be considered as a nonlinear lter and the right side represents modeling uncertainties). In order to reduce the tracking error, we have to use large feedback gains, i.e., large k or small ". However, since the bandwidth of any real system is limited, there will be a practical upper bound on the feedback gains that we can use. This fact limits the tracking accuracy that DRC can achieve in practice although theoretically it can achieve arbitrarily small nal tracking errors. For any chosen feedback gains, from (2.17), the real tracking error is proportional to the modeling uncertainty, ,~' + . Therefore, if we can introduce parameter adaptation in the DRC design to reduce the modeling uncertainty coming from the parametric uncertainty, ~', as in adaptive control, we may further improve the tracking accuracy. This is the rationale of the proposed adaptive robust control (ARC). The proposed ARC is to combine the design methodologies of DRC and AC to keep the advantages of the two methods while overcoming the previously mentioned drawbacks. In other words, we will try to use both means | proper controller structure and parameter adaptation | to reduce the tracking error. The way to do so is to use the DRC technique to design a baseline controller (proper controller structure) to guarantee transient performance and prescribed nal tracking accuracy for both parametric uncertainties and disturbances. On top of it, we will also use the adaptive control technique to update the parameters to obtain asymptotic output tracking in the presence of parametric uncertainties. To do so, we have to solve the con icts between the two design methodologies. DRC requires knowledge of the bounds of modeling uncertainties, but the estimated parameters by AC may not be bounded in the presence of unknown nonlinear functions. Thus, we have to modify the conventional adaptation law in such a way that it can guarantee that the estimated parameters stay in a prescribed uncertainty range all the time even in the presence of unknown nonlinear functions. Such a modi cation should not damage the correct estimation process for parametric uncertainties. In this dissertation, this modi cation is achieved by generalizing the smooth projection used in [122]. Let " be an arbitrarily small positive real number. There exists a real-valued, suciently smooth nondecreasing function (Fig.2.1) de ned by
( ) = 8 2 (2.30) ( ) 2 ^ = [min , " ; max + " ] 8 2 R with bounded derivatives up to order n , 1: i.e., there exist constants, ci > 0; i = 1; : : :; n , 1, such
that
j ddii ( )j ci Let
8 2 R;
i = 1; : : :; n , 1
V (~; ) = 1 0~( ( + ) , )d R
>0
(2.31) (2.32)
21 π (ν ) θimax
εθ
θimin
ν
θimax εθ
θimin
Figure 2.1: Nondecreasing n-th smooth projection map From assumption (2.3) and (2.30), ( + ) , is a nondecreasing function of that passes through the origin ( (0 + ) , = 0). Thus, V (~; ) is positive de nite w.r.t. ~. Furthermore, 1 @ ~ ^ @ ~V (; ) = ( () , )
(2.33)
which will later be used in the stability analysis. The suggested adaptive robust control law has the same structure as the smoothed SMC control (2.16) but with a projected parameter, ^ = (^), instead of a xed estimate. It is given by
u = uf + us uf = x_ r (t) , ^ '(x; t) us = ,kz , h (h(x; t)sgn(z ))
(2.34)
where, similar to (2.9), h(x; t) is any function satisfying
h(x; t) j , (^ , )'(x; t) + (x; t)j
8^ 2 ^
(2.35)
For example, let h(x; t) be
h(x; t) = (max , min + " )j'(x; t)j + (x; t)
(2.36)
^ is updated on-line by the following adaptation law ^_ = '(x; t)z
(2.37)
Theorem 4 If the control law (2.34) with (2.35) and (2.37) is applied to the system described by (2.1),
the following results hold:
22
A. In the presence of both parametric uncertainties and unknown nonlinear functions, the control input is bounded and Vs is bounded above by (2.19), in which the exponential converging rate and the
bound of nal tracking error can be freely adjusted by the controller parameters in a known form.
B. In the presence of parametric uncertainties only, (i.e., (x; t) = 0), in addition to the result in A, the system output tracks the desired output asymptotically. Furthermore, if the PE condition (2.26) is satis ed, the estimated parameter converges to its true value. 4
Proof. From (2.1) and (2.34), the error dynamics is given by z_ + kz + h (hsgn(z)) = ,~ '(x; t) + (x; t)
(2.38)
^ where ~ = , is bounded for any ^, which is an important property used later in the proof. In view of the similarity between the error dynamics (2.17) and (2.38) and the choice of the function h(x; t) by (2.35), the time derivative of Vs can be described by (2.18) with ~ replaced by ~ . Eq. (2.19) is still valid and the control input (2.34) is bounded. This fact proves the result A in Theorem 4. When (x; t) = 0, noting (2.33), (2.38), and condition i of (2.15), the time derivative of the p.d. function Vt = Vs + V is V_ t = z z_ + 1 ~ ^_ (2.39) = z [,~ '(x; t) , kz , h (hsgn(z ))] + ~ '(x; t)z 2 2 = ,kz , z h (hsgn(z )) ,kz
which implies that z 2 L2 \ L1 . From (2.38), z_ 2 L1 , which leads to z ,! 0 by Barbalat's lemma. Similar to the proof for the adaptive controller, the PE condition (2.26) will guarantee the convergence of the estimated parameter, which leads to the result B in Theorem 4. 4
Remark 2 The above theorem shows that the proposed ARC retains the results of both DRC and AC.
This fact naturally eliminates the drawbacks of each of the two methods. The main drawbacks of AC | the transient problem and the non-robustness to the unknown nonlinear functions | are overcome by A in the above theorem. The drawback of DRC | large nal tracking errors | is overcome by the improved performance in B. Therefore, the control law e ectively combines the DRC design with the AC design and achieves the expected goal. The analysis is qualitatively di erent from the adaptive robust control [100] for bounded disturbances in that not only is robustness obtained for a more general class of disturbances but performance robustness is also guaranteed by the suggested controller | i.e., arbitrarily fast exponential convergence can be provided and the nal tracking error can be adjusted by the controller parameters independent of the magnitude of disturbances. }
Remark 3 One of the good features of the propose ARC is that its underline control law is a DRC type
robust control law. The adaptation loop can be switched o at any time and the resulting control law is a DRC law. The result in A of the above theorem is still valid in such a case. }
Remark 4 In general, if we choose "(t) as a time-varying positive scalar converging to zero, i.e., "(1) = 0 (or exponentially converging to zero, i.e., "(t) "maxexp(," t) for some " > 0 and "max > 0), from the rst inequality of (2.19), Vs converges to zero asymptotically (or exponentially).
23 Thus, asymptotic output tracking (or exponential output tracking) can be obtained even in the presence of unknown nonlinear functions. The same is true for the smoothed SMC law (2.16) and the following analysis is also applied to it. Notice that although the control laws (2.34) are continuous for any nite time t, they tend to the ideal SMC law (2.8) as t ,! 1 (in nite gain feedback). Therefore, control chattering will appear when t ,! 1 and it is not surprising to see that the ideal performance of SMC law is obtained. This result corresponds to some of the continuous robust control techniques (e.g., in [98]). However, to truly remove control chattering, a large " and a small k have to be chosen to avoid very high gain caused by the limited bandwidth of the system in practice. Within the allowable limit in which control chattering is not excited, however, the larger the e ective gain, the smaller the nal tracking error. Since the system under consideration is nonlinear, it is not obvious how to choose k and "(t). Here, we roughly analyze the system in the following way. Normally, the dynamics around the sliding mode | i.e., dynamics about z given by (2.38) | is the fastest, and the system tracks the desired trajectory closely around the sliding mode, fz = 0g. This is true especially in the case when the initial tracking error is zero. Therefore, we can assume x xd (t) and treat the right hand side of (2.38) as slowly varying disturbances on the fast rst-order dynamics about z with the e ective proportional 2 gain @z@ [kz + h (h sgn(z ))] jz=0;x=xd . For the SM (2.23), the e ective gain is k + h(x"d((tt));t) . Suppose that the allowable limit is ka. Then we can choose a time varying " as
"(t) = h(kxad,(tk);t)
2
(2.40)
so that the e ective gain is at the allowable limit all the time to minimize the tracking error. A similar idea is used later in the experiments [154] to reduce the output tracking error. }
24
Chapter 3
Adaptive Robust Control of SISO Nonlinear Systems in a Semi-Strict Feedback form In chapter 2, we presented the adaptive robust control (ARC) technique for a simple rst-order nonlinear system (relative degree one). In this chapter, the ARC technique will be generalized to a class of SISO nonlinear systems with arbitrary known relative degrees and transformable to a semi-strictfeedback form. This generalization is achieved by combining the general deterministic robust control design technique with the well-known adaptive algorithms in [62, 59, 52] that were originally developed for SISO nonlinear systems in a parametric strict-feedback form.
3.1 Problem Formulation form
We consider the SISO nonlinear system transformable to the following semi-strict-feedback
xi+1 + T 'i(x1; : : :; xi; t) + i(x; t) 1 i n , 1 (3.1) (x)u + T 'n (x; t) + n (x; t) x1 'i (x1; : : :; xi; t) 2 Rp ; i = 1; : : :; n; are the known shape functions, which are assumed to be suciently smooth and, similar to (2.3), = [1 ; : : :; p]T 2 Rp and i (x; t) are the x_ i = x_ n = y = where x = [x1 ; : : :; xn ]T .
vector of unknown constant parameters and unknown nonlinear functions, respectively. For simplicity, denote xi = [x1 ; : : :; xi]T (in general, i;j denotes the j -th element of i, i denotes [T1 ; : : :; Ti ]T ). and i 's are assumed to satisfy 2 = f : min < < max; g (3.2) ji(x; t)j i(xi; t) i = 1; : : :; n where min = [1min ; : : :; pmin ]T 2 Rp, max = [1max; : : :; pmax]T 2 Rp, and i (xi ; t)'s are known 1 . The operation < for two vectors is performed in terms of the corresponding elements of the vectors (e.g., min < means that jmin < j ; 8j ). 1 If is a vector or matrix with elements being functions, is said to be known if its elements are known functions
of their variables
25 When i (x; t) = 0; 8i, i.e., in the absence of unknown nonlinear functions, if we treat xi+1 as the control input of the x_ i dynamics, the x_ i dynamics depends only on the states of its previous dynamics, i.e., x1; : : :; xi. In other words, only the feedback signals determine the dynamics. Such a form is called strict-feedback form and is studied in [62, 59, 52]. Eq. (3.1) is called a semi-strictfeedback form in that the bounding function i (xi ; t) is required to be the function of xj ; j i and t only, but i (x; t) may contain some bounded functions of xj ; j > i, thus violating the strict-feedback property. Some examples of (3.1) can be found in [94]. Let yd (t) be the desired output trajectory, which is assumed to be bounded with bounded derivatives up to n-th order. The control problem is stated as that of designing a bounded control law for the input u such that, under the assumption of (3.2), the system is stable and the output y tracks yd (t) asymptotically or in a GUUB fashion.
3.2 Smooth Projection and Positive De nite Function V To begin the controller design, we would like to de ne the multi-variable counterpart of the smooth projection (2.30) rst. De ne a smooth projection : Rp ! Rp by
( ) = [1(1); : : :; p(p)]T
(3.3)
for each vector 2 Rp with = [1; : : :; p ]T , in which each i : R ! R is de ned by (2.30) with
i = [imin ; imax] and i^ = [imin , "i ; imax + "i ]. Its j -th derivative is de ned by (j )( ) = [ ddjj 1(1 ); : : :; ddjpj p(p )]T . Thus, 1
( ) = 8 2 ( ) 2 ^ = f : i 2 i^ 8ig 8 2 Rp (j )( ) 2 j = f : jij ci j g 8 2 Rp j n
(3.4)
where ^ and j are compact sets and ^ is known. Let ^ be the smooth projection of ^, the estimate of , de ned by ^ = (^), and (i) = [ (^); : : :; (i)(^)]T . De ne ~ = ^ , , ~ = ^ , , and
V (~; ) = Ppi=1 1i
R ~
0
i i + i ) , i )di
i ( (
i > 0
(3.5)
From (3.2) and (3.4), V (~; ) is positive de nite w.r.t ~ for each 2 . Furthermore, @ 1 ~ ^ @ ~V (; ) = [ 1 (1(1 ) , 1 );
: : : ; 1p (p(^p ) , p)] = ~ T ,,1
(3.6)
where , = diag f 1; : : :; pg
3.3 Backstepping Design Procedure The design follows the recursive backstepping procedure in [62, 59, 52], which proceeds in the following steps.
26
3.3.1 Step 1
Let ~ 1 (x; t) = 1 (x; t). The rst equation of (3.1) can be rewritten as x_ 1 = x2 + T '1(x1 ; t) + ~ 1(x; t)
(3.7)
In (3.7), by viewing x2 as a virtual control, we can design for it a control law 1 such that x1 tracks its desired trajectory x1d(t). This design can be done by the ARC method presented in Chapter 2 by ~ 1(x; t)j ~(x1; t) = noticing that j (x1; t). From (2.34), the control law 1 is given by 2 . 1 (z1; ^ ; t) = 1f + 1s 1f (z1 ; ^ ; t) = x_ 1d (t) , ^ T '1(z1 + x1d (t); t) (3.8) ^ ;t)z1 h ( z ; 1 1 ^ ^ 1s(z1 ; ; t) = ,k1z1 , h1 (z1; ; t) tanh "1 (t) where x1 = z1 + x1d (t) has been used and h1 is any function with continuous partial derivatives up to (n , 1)-order. h1 satis es ~ x; t)j (3.9) h1 (z1 ; ^ ; t) sup2 j , ~ T '1(z1 + x1d ; t) + (
~ is known. For which is possible since is a known compact set and the bounding function of example, let h1 be any suciently smooth function satisfying (3.10) h1(z1; ^ ; t) Ppj=1 jM j'1j (z1 + x1d ; t)j + ~1 (z1 + x1d ; t) where jM = jmax , jmin + "j .
Remark 5 An easy way to obtain a smooth h1 is to use (3.10). Since the right hand side of (3.10) is
continuous but may not be suciently smooth because of the non-di erentiability of absolute operator j j at the origin, h1 can be chosen as equal to the right hand side of (3.10) by replacing operator j j by any suciently smooth operation As () satisfying As () j j ; 8 2 R. For example, a simple smooth operator As is given by
p As () = 2 + r
8 2 R
(3.11)
}
where r is any positive scalar.
The same as in [62], if x2 were the real control input, then the adaptation law would be given by (2.37) and the design would be nished. Since it is not the case, we postpone the choice of the adaptation law and use the rst tuning function [62]
1 (z1; t) = 1 (z1 ; t)z1
1 (z1; t) = '1 (z1 + x1d (t); t)
1 2 Rp
(3.12)
to denote the essential part of the adaptation law (2.37). De ne the di erence between the actual value of x2 and its desired value 1 in (3.8) to be the second error variable (3.13) z2 = x2 , 1 (z1; ^ ; t) 2 For simplicity, dynamic compensator (2.4) is dropped o | i.e., sliding surface z1 reduces to z1 = x1 , x1d (t).
The following design procedure will still be valid if a dynamic compensator is added. Furthermore, the smooth modi cation (2.23) is used.
27 Substituting (3.8) and (3.13) into (3.7), the rst error subsystem S1 becomes z_1 + k1 z1 + h1 tanh h" 1(tz)1 = z2 , ~T 1(z1 ; t) + ~ 1(x; t)
(3.14)
Choose Vs1 = 12 z12 . From (3.14), its time derivative is V_ s1 = z1z2 , [k1z12 + h1z1 tanh h" 1(tz)1 , ~ 1(x; t)z1] , ~T 1(z1 ; t)
(3.15)
1
1
3.3.2 Step 2 From (3.1) and (3.13), noticing (3.8) and (3.14), we have
@ 1 ^_ @ 1 1 z_2 = x3 + T '2(x2 ; t) + 2(x; t) , [ @ @z1 z_1 + @ ^ + @t ] = x3 + (^ , ~ )T '2(x2 ; t) + 2(x; t) (3.16) @ 1 ^_ @ 1 1 ~T ~ , @ @z1 [ 1s + z2 , '1 (z1 + x1d ; t) + 1(x; t)] , @ ^ , @t = x3 , 2f , 02s , ~ T 2 + ~ 2 (x; t) , p2 (^_ , , 2 ) in which by treating x1 and x2 as functions of z1 ; z2; ^ and t, i.e, x1 = z1 + x1d (t) and x2 = z2 + 1 (z1; (^); t), and noticing that ^ appears in 1 only in the form of (^), we have de ned the functions 2f , 02s , 2 , ~ 2(x; t) and p2 as 2f (z2; (^); t) = , (^)'2(x2; t) + @ @t1f @ 1s @ 1 1 02s (z2 ; (1); t) = @ @z1 [ 1s + z2 ] + @t + @ ^ , 2 1 (3.17) 2(z2 ; (^); t) = '2(x1; x2; t) , @ @z1 1(z1 ; t) @ 1 ~2 (x; t) ~ = 2(x; t) , @z1 1 (x; t) (1) p2 (z1; ; t) = @ @ ^1 (z1 ; (^); (1)(^); t) 1 where (1)(^) has appeared in the above expressions because of the term @ z2 ; (^); t) is the @ ^ . 2 ( second tuning function, which will be de ned later. From (3.2) and (3.17), 1 (3.18) j~ 2(x; t)j ~2(z2; ^ ; t) = 2(x2; t) + j @ @z j1 (x1; t) 1
Similar to (3.9) and (3.10), there exists a known function h2 (z2; ^ ; t) with continuous partial derivatives up to (n , 2) -order such that (3.19) h2(z2; ^ ; t) MT j 2 (z2; ^ ; t)j + ~2 (z2; ^ ; t) j , ~ T 2 + ~ 2 (x; t))j
where M = [1M ; : : :; pM ]T , iM is de ned in (3.10), and j j is in the sense of element operation if is a vector or matrix. Since x3 is the virtual control for (3.16), let z3 = x3 , 2 , where 2 is the desired control law for x3 , which will be speci ed later. Consider the augmented p.d. function
Vs2 = Vs1 + 12 w2 z22
where w2 > 0 is any weighting. From (3.15) and (3.16), its derivative is given by V_ s2 = z1 z2 , [k1z12 + h1z1 tanh h"1(tz)1 , ~ 1(x; t)z1] , ~ T 1 +w2 z2[z3 + 2 , 2f , 02s , ~ T 2 + ~ 2 (x; t) , p2 (^_ , , 2 )] = w2z2 z3 , [k1z12 + h1 z1 tanh h"11(tz)1 , ~ 1 (x; t)z1] , ~ T [ 1 + w2z2 2 ] +w2 z2 [ w12 z1 + 2 , 2f , 02s + ~ 2 (x; t)] , w2z2 p2 (^_ , , 2 )
(3.20)
(3.21)
28 where the term z1 z2 has been grouped together with 2 since it is going to be dealt with via 2 in this design step, and the term w2z2 z3 has been separated from the rest since it is going to be dealt with at the next step. De ne the second tuning function 2 and the function p~2 as 2 (z2; (^); t) = 1 (z1 ; t) + w2 z2 2(z2 ; (^); t) (3.22) p~2(z2 ; (1); t) = w2z2p2(z1 ; (1); t) With the choice of
2 (z2 ; (^); t) = 2f (z2; (^); t) + 2s (z2; (^); t) 2s = 02s + 002s 002s = , w12 z1 , k2z2 , h2 tanh h"22(tz)2
(3.23)
(3.21) becomes
V_ s2 = w2z2 z3 , P2j=1 wj zj [kj zj + hj tanh
hj zj , ~ j (x; t)] , ~ T 2 , p~2(^_ , , 2 ) "j
(3.24)
From (3.16) and (3.23), the second error subsystem becomes
z_2 = z3 + 002s , ~ T 2 + ~ 2(x; t) , p2(^_ , , 2)
(3.25)
3.3.3 Step i We will use mathematical induction to explain the remaining intermediate design steps. In the following, we treat xj ; j i as the function of z1 ; : : :; zj ; ^ and t, i.e., xj = zj + j ,1 (zj ,1 ; (j ,2) ; t) (for simplicity, denote 0 (t) = x1d (t)). Thus, we can recursively de ne the following functions for step j from those in the previous steps j (zj ; (j,2); t) = 'j (z1 + x1d; : : :; zj + j,1 ; t) , Pjk,=11 @ @zjk,1 k P ~ j (x; t) = j (x; t) , jk,=11 @ @zj,k 1 ~ k (x; t) (3.26) p1 = 0 pj (zj ,1 ; (j,1); t) = @ @j^,1 + Pjk,=11 @ @zjk,1 pk ( j , 1) p~j (zj ; ; t) = p~j ,1 + wj zj pj p~1 = 0 (j ,2) j (zj ; ; t) = j ,1 + wj zj j Let zj +1 = xj +1 , j and choose the desired control function j (zj ; (j ,1); t) as where
jf js 0js 00js
j = jf (zj ; (j,2); t) + js (zj ; (j,1); t)
(3.27)
= ,(^)'j (xj ; t) + @ (@tj,1)f = 0js + 00js P Pj ,1 @ j,1 00 = k=1 @zk [ ks + zk+1 ] + @ (@tj,1)s + jk,=11 @ @zj,k 1 pk ,( j , k ) + @ @j^,1 , j = , wwj,j 1 zj,1 , kj zj , hj tanh h"jj(tz)j , p~j,1, j
(3.28)
and hj (zj ; (j ,2); t) is any positive function that is speci ed in each step. Then, the j -th error subsystem may be assumed to be (3.29) z_ = z + 00 , ~ T + ~ (x; t) , p (^_ , , ) j
j +1
js
j
j
j
j
29 The augmented p.d. function is
Vsj = Vs(j ,1) + 21 wj zj2
wj > 0
and its derivative is given by V_sj = wj zj zj+1 , Pjk=1 wk zk [kk zk + hk tanh h"kk zk , ~ k (x; t)] ,~ T j , p~j (^_ , , j )
(3.30) (3.31)
It is easy to check that the rst two steps satisfy the above general forms. So we assume that they are valid for step j, 8j i , 1, and show that they are also true for step i to complete the induction process. From (3.2) and (3.26) (3.32) j~ i(x; t)j ~i(zi; (i,2); t) = i(xi; t) + Pij,=11 j @ @zi,1 j~j (zj ; (j,2); t) j
There exists a known function hi (zi ; (i,2); t) with continuous partial derivatives up to (n , i)-order such that (3.33) hi (zi ; (i,2); t) MT j i(zi; (i,2) ; t)j + ~i (zi; (i,2) ; t) From (3.1) and (3.26), noting (3.27) and (3.29) for step j < i, we have
z_i = xi+1 + T 'i (xi ; t) + j (x; t) Pi,1 @ i,1 _ _ @ @ i , 1 i , 1 00 T ^ ~ ^ ~ , j=1 @zj [zj+1 + js , j + j (x; t) , pj ( , , j )] + @^ + @t (3.34) = zi+1 + i , if , 0is , ~ T i + ~ i (x; t) , pi (^_ , , i ) where if and 0is satisfy the de nition (3.28), and i and pi are given by (3.26). If i is chosen to be in the form of (3.27), (3.34) reduces to the form (3.29). Furthermore, the derivative of Vsi is
V_ si = wi,1zi,1 zi , Pik,=11 wk zk [kk zk + hk tanh h"kk(tz)k , ~ k (x; t)] , ~ T i,1 ,p~i,1 (^_ , , i,1 ) + wizi[zi+1 + 00is , ~ T i + ~ i(x; t) , pi(^_ , , i) P = wizi zi+1 , i1 wk zk [kk zk + hk tanh h"kk(tz)k , ~ k ] , ~ T i , p~i (^_ , , i )
(3.35)
which agrees with the general form (3.31). This completes the induction process.
3.3.4 Step n This is the nal design step. By letting xn+1 = (x)u, the last equation of (3.1) has the same form as the intermediate step i n , 1. Therefore, the general form ((3.26) to (3.31)) applies to Step n. Since u is the real control, we can choose it as (3.36) u = (1x) n (zn ; (n,2) ; t) where n is given by (3.27), in which hn is determined from (3.33) with j = n. By doing so, zn+1 = 0. Specify the adaptation law as (3.37) ^_ = , n Then, the n-th error subsystem (3.29) becomes (3.38) z_n = 00ns , ~ T n + ~ n (x; t)
30 and the derivative of the augmented p.d. function Vsn is given by
V_sn = , Pnk=1 wk zk [kk zk + hk tanh
hk zk "k
, ~ k (x; t)] , ~ T n
(3.39)
Theorem 5 With the control law (3.36) and the adaptation law (3.37), the following results hold for
the system (3.1) if the assumption (3.2) is satis ed:
A . In the presence of both parametric uncertainties and unknown nonlinear functions, the control
input is bounded. The system is stable and the output tracking error exponentially converges to a ball whose size can be freely adjusted by controller parameters in a known form. Vsn is bounded above by R Vsn (t) exp(,2kv t)Vsn(0) + 0t exp(,2kv (t , )"v ( )d (3.40) exp(,2kv t)Vsn(0) + "vmax 2kv [1 , exp(,2kv t)] P where kv =minfk1; : : :; kng, "v (t)= nk=1 "k (t), and "vmax = maxt "v (t).
B . In the presence of parametric uncertainties only (i.e., i (x; t) = 0, 8i), in addition to the result in A, the system output tracks the desired output asymptotically.
4
Proof. From (3.26), n = Pnk=1 wk zk k . Noticing (3.33), (3.32) and the condition ii of
(2.15), (3.39) becomes
V_sn = , Pnk=1 wk zk [kk zk + hk tanh h"kk zk , ~ k (x; t) + ~ T k ] , Pnk=1fwk [kkzk2 + hk zk tanh h"kk(tz)k , hk jzk j] , Pnk=1 wk kk zk2 + Pnk=1 "k ,2kv Vsn + "v (t)
(3.41)
which leads to (3.40). Since (j )(^); 0 j n , 1, are bounded for any ^ and all the terms involved are bounded functions w.r.t. t, it is easy to check that (3.40) also guarantees that all the variables in (3.26) to (3.28) are bounded for j , which implies that the state x is bounded and so is the control input (3.36). Since key k2 = kz1k2 2Vsn , A of the Theorem is thus proved. When i (x; t) = 0; 8i, from (3.26), ~ i (x; t) = 0; 8i. Choose a p.d. function Van as Van = Vsn + V (~; ). Noticing (3.6), (3.39), and (3.37), we have ~ ) ^_ V_ an = V_ sn + @V@(~; Pn hk zk ] , ~ T + ~ T ,,1 ^_ (3.42) = , k=1 wk zk [kk zk + hk tanh " n ( t ) k Pn P hk zk ] , n w k z 2 = , k=1 wk zk [kk zk + hk tanh " k=1 k k k k (t) Therefore, z = [z1; : : :; zn ]T 2 Ln2 2 Ln1 . It is also easy to check that z_ is bounded. So, z ,! 0 by Barbalat's lemma, and B of the Theorem is proved. 4
3.4 Guaranteed Transient Performance In Theorem 5, the exponentially converging rate, 2kv , can be any q value by adjusting the T controller gains k = [k1; : : :; kn] and the nal tracking accuracy, kz (1)k "vmax kv , can be made T T arbitrarily small by increasing k = [k1; : : :; kn ] and decreasing " = ["1; : : :; "n ] . However, Vsn (0) also depends on k and ", and thus the transient behavior of the error system may not be improved
31 by increasing k and reducing " if Vsn (0) increases. To deal with this problem, the idea of trajectory initialization in [53] will be used to render z (0) = 0 independent of the choice of k and ". Let x1d (t) be the trajectory created by the following n-th order stable system
x(1nd) + 1 x(1nd,1) + : : : + n x1d = yd(n) + 1yd(n,1) + : : : + n yd
(3.43)
where yd (t) is the desired output. Recursively de ne the following functions
h1 (x1; ; t) = T '1 (x1; t) P (xi,1;;t) hi (xi; ; t) = T 'i (xi; t) + ik,=11 @hi,1@x [xk+1 + T 'k (xk ; t)] k i = 2; : : :; n , 1 + @hi,1 (@txi,1 ;;t)
(3.44)
Lemma 1 Each part of the control functions j = jf + js can be written in the following forms A: jf = x(1jd) , hj (xj ; ^ ; t)+ fbj (zj ; (j ,2); t) where every term in fbj contains either zk or tanh h"kk zk as a factor for some k j: B: every term in js contains either zk or tanh h"kk zk as a factor for some k j
(3.45)
Proof. We prove the lemma by induction. First, from (3.8), 1f satis es A for fb1 = 0 and B is obviously satis ed. Thus, we assume that A and B are valid for 8jl i , 1, and we prove that they are also valid for i. From B, 8j i , 1 and 8l, all the terms in @@t ljs contain either zk or tanh hk ("zkk(;tt)); zk as a factor for some k j . Thus, from (3.26) and (3.28), is satis es B. So (B) k,2 is true for i. From (3.28) and A for 8j i , 1 and noticing @x@tk = @ k,1(zk@t,1 ; ;t) , we have (i,3) ;t)
xi,1 ;^ ;t) @ k,1 , @hi,1 (xi,1 ;^ ;t) + @fb(i,1) (zj ; if = ,^ 'i(xi; t) + x(1id) , Pik,=11 @hi,1 (@x @t @t @t k (i) (i,2) ^ = x1d , hi (xi ; ; t) + fbi (zi ; ; t)
where
(3.46)
(i,3)
xi,1 ;^ ;t) [x + ^T ' , @ k,1 ] + @fb(i,1) (zj ; ;t) fbi = , Pik,=11 @hi,1 (@x k+1 k @t @t k Pi,1 @hi,1 @fb(i,1) (zj ;(i,3) ;t) @ @ k , 1 f k , 1 s T ^ = , k=1 @xk [xk+1 + 'k , @t , @t ] + @t (3.47) Pi,1 @hi,1 @fb(i,1) (zj ;(i,3) ;t) @ k , 1 s = , k=1 @xk [xk+1 , kf , @t ] + @t Pi,1 @hi,1 @fb(i,1) (zj ;(i,3) ;t) @ k , 1 s = , k=1 @xk [zk+1 + ks , @t ] + @t k,1s contains either z or It is thus obvious that A is satis ed for j = i since every term in ks and @ @t l h z l l tanh "l as a factor for some l k. 4
Lemma 2 If the initial conditions x1d (0); : : :; x(1nd,1) (0) of the ltered trajectory x1d (t) are chosen as
x1d (0) = x1 (0) x(1id,1)(0) = xi(0) + hi,1 (xi,1 (0); ^(0); 0); then z (0) = 0.
i = 2; : : :; n
(3.48)
4
32
Proof. We use induction to prove the above lemma. It is obvious that z1(0) = 0. So we assume that x1d(0); : : :; and x(1id,2) (0) have been chosen to render zj (0) = 0; 8j i , 1. Then, from (3.45) and noticing that zj (0) = 0; 8j i , 1, we have i,1f (0) = x(1id,1)(0) , hi,1 (xi,1(0); ^ (0); 0) (3.49) i,1s (0) = 0
By choosing x(1id,1) (0) according to (3.48), we have zi (0) = xi (0) , i,1 (0) = 0, which completes the proof. 4 Remark 6 From (3.1), in the absence of unknown nonlinear functions, the i-th derivative of the output is (3.50) y (i)jj =0 = xi + hi (xi; ; t) Thus the above trajectory initialization (3.48) can be considered as placing the initial condition x(1id) (0) at the best initial estimate of y (i) (0) by substituting ^ (0) for . A similar implication is rst observed in [65] for the adaptive control of parametric strict-feedback systems. Also, from this implication, trajectory initialization can be performed independently from the choice of controller parameters such as k and ". 4 Theorem 6 Given the desired trajectory x1d (t) generated by (3.43) with the initial conditions (3.48), the following results hold for the system (3.1) if the control law (3.36) and the adaptation law (3.37) are applied and the assumption (3.2) is satis ed: A . In the presence of both parametric uncertainties and unknown nonlinear functions, the control input is bounded and Vsn is bounded above by R Vsn (t) 0t exp(,2kv (t , )"v ( )d "2vmax (3.51) kv [1 , exp(,2kv t)] Transient performance and nal tracking accuracy of the output tracking can be freely adjusted by controller parameters in a known form.
B . In the presence of parametric uncertainties only, in addition to the result in A, the system output
tracks the desired output asymptotically. 4 Proof. Noting Theorem 5, we only have to show that the output tracking error has a guaranteed transient performance. Since the initial conditions x1d (0); : : :; x(1nd,1)(0) chosen by (3.48) are independent of the choice of the controller parameters k and ", the trajectory planning error, ed (t) = x1d (t) , yd (t), can be guaranteed to possess any good transient behavior by suitably choosing the Hurwitz polynomial Gd (s) = sn + 1 sn,1 + : : : + n without being a ected by k and ". On the other hand, such a desired trajectory initialization renders z (0) = 0 by Lemma 2. From (3.40), (3.51) is true, which indicates that z1 can be made arbitrarily small by increasing k and decreasing ". Therefore, any good transient performance of the output tracking error e = y , yd = z1 (t) + ed (t) can be guaranteed by the choice of the controller gains k and " in a known form. 4 Remark 7 Similar comments as in remarks 2 to 4 in Chapter 2 can be made for the above ARC law. The above design method can be extended to the case of bounded time-varying parameters with bounded derivatives up to a suciently high order. Results similar to the one in B of Theorem 6 can be obtained. Robustness to the neglected high frequency dynamics may also be obtained since our controller guarantees exponential stability at large. }
33
Remark 8 If wi = 1; 8i; and the parameter projection is not used, the adaptation law (3.37) reduces to the one used in [62, 53]. The reason of introducing the weighting wi is to gain more freedom in shaping the transient performance since only the tracking error z1 is of our concern. }
3.5 Simulation Results Consider the following relative degree 2 nonlinear system:
x_ 1 = x2 + 1 x21 + 1 (x2) 1 (x2) = d1 sin(r1x2 ) x_ 2 = u + 2 (x21 + x22) sin3(t) + 2(x) 2(x) = d2 cos(r2x1x2 ) y = x1
(3.52)
where 1 and 2 are unknown parameters satisfying (3.2) in which 1min = ,3; 1max = 0; 2min = ,4 and 2max = 0, r1 and r2 are assumed to be unknown, d1 and d2 are also unknown but are bounded by d1 d1M = 2 and d2 d2M = 2 respectively. It is observed that (3.52) is not in a strict feedback form but satis es the semi-strict feedback form (3.1) since
j1(x2)j 1 = d1M j2(x)j 2 = d2M
(3.53)
From (3.10), we choose h1 (z1; t) = 1M (z1 + x1d )2 + 1 where = [1; 1]T . 1 is then determined by (3.8), where k1 = 5 and "1 = 0:5, and the projection i in de ning ^ is speci ed by 8 1 ^ ^i > imax > < imax + i [1 , exp(, i (i , imax ))] if i (^i ) = > ^i (3.54) ^i 2 [imin ; imax ] : 1 ^ ^ imin , i [1 , exp( i (i , imin ))] i < min which is monotone increasing and has a continuous rst derivative. From (3.17), 2f ; 02s ; 2 and p2 can be obtained. 2 is formed from (3.22) where w2 = 0:5. h2 is determined by (3.19) and is given by 1 h2 = MT j 2j + 2 + 1j @ @z1 j
(3.55)
Therefore, 2 can be determined by (3.23), where k2 = 5 and "2 = , and the control law is given by (3.36). Estimated parameters are updated by (3.37) where , = diag f500000; 150000g. The desired output is yd = 0:1sin(0:5t) and xd and x_ d are calculated by (3.43) with initial values given by (3.48) where 1 = 80 and 2 = 1600. Actual plant parameters are 1 = ,2; 2 = ,3; r1 = 2 and r2 = 3 with initial estimates ^1(0) = 0 and ^2(0) = 0. Sampling time is 1ms. Three controllers are run for comparison:
ARC : The proposed adaptive robust controller as described in the above. DRC : The same control law as in ARC but without using the parameter adaptation law (3.37). In such a case, the proposed control law is equivalent to the conventional DRC law.
AC : By setting hi = 0 and without using parameter projection, i.e., letting ( ) = , the suggested control law is equivalent to the nonlinear adaptive control law in [62].
34 To test the nominal performance, simulations were run for parametric uncertainties only, i.e.,
d1 = d2 = 0. The tracking error z1 is shown in Fig. 3.1, from which we can see that all three controllers
have very good tracking ability. ARC has a much better nal tracking accuracy than DRC since the estimated parameters approach their desired values as shown in Fig. 3.2, and has a better transient response than AC. This result substantiates the necessity of using parameter adaptation to improve tracking performance. As shown in Fig.3.3, control inputs do not exhibit chattering.3 To test the performance robustness, small disturbances were rst added to the system, i.e., d1 = d2 = 0:02 in (3.52). Fig. 3.4 shows the tracking error z1 . Roughly speaking, the tracking performance of DRC remains unchanged but the AC's performance degrades a lot because of the wrong fast-changing parameter estimates shown in Fig. 3.5. Although the ARC's performance also degrades, it still achieves the best tracking performance. Control inputs shown in Fig. 3.6 do not exhibit chattering. Large disturbances were also added to the system, i.e., d1 = d2 = 2 in (3.52). As shown in Fig. 3.7, ARC achieves the same tracking performance as DRC although its estimated parameters vary quite wildly ( Fig. 3.8). AC has the worst tracking ability and a large control e ort since its estimated parameters are unbearably large. Fig. 3.9 presents the control inputs for ARC and DRC, which do not exhibit control chattering. All these results illustrate the e ectiveness of the proposed ARC.
3.6 Conclusions In this chapter, we have presented a systematic design of adaptive robust controllers for a class of SISO nonlinear systems transformable to a semi-strict feedback form in the presence of both parametric uncertainties and unknown nonlinear functions. By utilizing prior knowledge of the bounds of parametric uncertainties and the bounding function of unknown nonlinear functions, we use a simple smooth projection of the estimated parameters updated by an adaptation law similar to that in [62, 52] in designing the robust control law to combine the backstepping adaptive control [62, 52] with conventional DRC techniques. This approach preserves the advantages of the two methods while eliminating their drawbacks. Simulation results validate the analysis.
3 Only the input during the rst 0.8 second is presented in order to show the large transient of the initial input
clearly
35 −3
1.5
x 10
1
Tracking error z1
0.5
0
−0.5
−1
−1.5
−2 0
Solid: ARC
0.5
1
Dashed: DRC
1.5
2 Time(s)
Dashdot: AC
2.5
3
3.5
4
Figure 3.1: Tracking errors in the presence of parametric uncertainties 1
0
Estimated parameters
−1
−2
−3
−4
Solid: theta1 (ARC) Dashed: theta2 (ARC) Dashdot: theta1 (AC) Dotted: theta2 (AC)
−5
−6 0
0.5
1
1.5
2 Time(s)
2.5
3
3.5
4
Figure 3.2: Estimated parameters in the presence of parametric uncertainties 14
12
10 Solid: ARC
Dashed: DRC
Dotted: AC
Control input
8
6
4
2
0
−2 0
0.1
0.2
0.3
0.4 Time(s)
0.5
0.6
0.7
0.8
Figure 3.3: Control input in the presence of parametric uncertainties
36 −3
2
x 10
1.5 Solid: ARC
Dashed: DRC
Dashdot: AC
Tracking error z1
1
0.5
0
−0.5
−1
−1.5
−2 0
0.5
1
1.5
2 Time(s)
2.5
3
3.5
4
Figure 3.4: Tracking errors in the presence of parametric uncertainties and small disturbances (d1 = d2 = 0:02) 3 2
Solid: theta1 (ARC)
Dashed: theta2 (ARC)
Dashdot: theta1 (AC) Dotted: theta2 (AC)
Estimated parameters
1 0 −1 −2 −3 −4 −5 −6 0
0.5
1
1.5
2 Time(s)
2.5
3
3.5
4
Figure 3.5: Estimated parameters in the presence of parametric uncertainties and small disturbances (d1 = d2 = 0:02) 14
12
10 Solid: ARC
Dashed: DRC
Dotted: AC
Control input
8
6
4
2
0
−2 0
0.1
0.2
0.3
0.4 Time(s)
0.5
0.6
0.7
0.8
Figure 3.6: Control input in the presence of parametric uncertainties and small disturbances (d1 = d2 = 0:02)
37 0.12
0.1
0.08 Solid: ARC
Dashed: DRC
Dashdot: AC
Tracking error z1
0.06
0.04
0.02
0
−0.02
−0.04 0
0.5
1
1.5
2 Time(s)
2.5
3
3.5
4
Figure 3.7: Tracking errors in the presence of parametric uncertainties and large disturbances (d1 = d2 = 2) 350
300 Solid: theta1 (ARC)
Estimated parameters
250
Dashed: theta2 (ARC)
Dashdot: theta1 (AC) Dotted: theta2 (AC)
200
150
100
50
0
−50 0
0.5
1
1.5
2 Time(s)
2.5
3
3.5
4
Figure 3.8: Estimated parameters in the presence of parametric uncertainties and large disturbances (d1 = d2 = 2) 15 10 5
Control input
0 −5 −10 −15 Solid: ARC
Dashed: DRC
Dotted: AC
−20 −25 −30 0
0.1
0.2
0.3
0.4 Time(s)
0.5
0.6
0.7
0.8
Figure 3.9: Control input in the presence of parametric uncertainties and large disturbances (d1 = d2 = 2)
38
Chapter 4
General Framework of Adaptive Robust Control In this chapter, we will present a general framework of the adaptive robust control of nonlinear systems using the ARC Lyapunov functions.
4.1 Problem Formulation Considering the following general MIMO nonlinear system: x_ = f (x; ; t) + B (x; ; t)u + D(x; t)(x; ; u; t) (4.1) y = h(x; t) where y 2 Rm and u 2 Rm are the output and input vectors respectively, x 2 Rn is the state vector, 2 Rp is the vector of unknown parameters, h(x; t); f (x; ; t); B(x; ; t), and D(x; t) 2 Rnld are known, and (x; ; u; t) 2 Rld represents the vector of unknown nonlinear functions such as disturbances and modeling errors. Similar to (2.3), we make the following reasonable and practical assumptions:
Assumption 1 Parametric uncertainties and the unknown nonlinear functions satisfy
2 = f : min < < max g (4.2) (x; ; u; t) 2 = f : k(x; ; u; t)k (x; t) g where min ; max and (x; t) are known. Let yd (t) 2 Rm be the desired outputs at t, and let the output tracking errors be denoted as ey = y , yd (t). The adaptive robust control problem can now be formulated as that of designing a control law for the inputs u such that, under the assumption of (4.2), the system is globally stable and
the output tracking has a prescribed transient performance and nal tracking accuracy. Furthermore, in the presence of parametric uncertainties only, asymptotic output tracking should be achieved.
4.2 ARC Lyapunov Functions Since almost all adaptive nonlinear controllers and deterministic robust controllers were synthesized by Lyapunov functions, it is natural that Lyapunov functions will be utilized here to formulate
39 the general problem. In addition, to solve the con ict between DRC and AC, the smooth projection presented in Chapter 3 will be utilized. Namely, we only use the projected parameter estimates ^ and the derivatives of the projection, (j ) (^), as they belong to those compact sets in (3.4) for any ^. Let V (x; ; (lV ) ; t) be a positive semi-de nite (p.s.d.) function with continuous partial derivatives (lV is any index), which satis es the following assumptions:
Assumption 2 Bounded V means bounded x, and guaranteed transient performance of V (t) is equivalent to the guaranteed transient performance of output tracking error ey (e.g., guaranteed exponential convergence of V (t) ,! 0 means guaranteed exponential convergence of ey ,! 0). Basically, Assumption 2 says that the stability and performance of the nonlinear system (4.1) can be converted to the study of the stability and performance of the scalar function V , which is much easier to deal with. The control law we seek consists of two parts given by
(4.3) u(x; (lu) ; t) = ua (x; (lu); t) + us (x; (lu); t) where lu is an index, ua functions as an adaptive control law and us a robust control law to be designed within an allowable set u .
Assumption 3 There exists a continuous control law ua (x; lu ; t) such that 8us 2 u : @V [f (x; ; t) + B (x; ; t)(ua + us )] + @V ,W (x; ; (lr ) ; t) + ~T (x; (lr ); u; t) + @V , @x @t @ ^ or, equivalently, V_ j=0 ,W + ~T + @V (^_ + , )
@ ^
(4.4) (4.5)
where lr = maxflV + 1; lug, (x; (lr ); u; t) is a known function, V_ j=0 represents the derivative of V under the condition that = 0, and W (x; ; (lr ); t) is any continuously di erentiable p.s.d. function which satis es the condition that asymptotic convergence of W means asymptotic output tracking, i.e., W ,! 0 =) ey ,! 0. Assumption 3 guarantees that there exists an adaptive control law to achieve asymptotic output tracking in the presence of parametric uncertainties only as shown later.
Remark 9 In [65], the adaptive control Lyapunov function (aclf) is introduced for the following single input plant
x_ = f (x) + F (x) + b(x)u
(4.6) A smooth function V (x; ) , positive de nite (p.d.) and radially unbounded in x for each , is called an aclf for (4.6) if there exists a s.p.d. , such that n h io @V T inf u2R @V @x f (x) + F (x) + ,( @ ) + b(x)u < 0
(4.7)
It can be proved that there exists an aclf V for (4.6) i there exists a control u = (x; ^) such that @V (x;^) hf (x) + F (x) + B (x) (x; ^)i ,W (x; ^) + ~T (x; ^) + @V (x;^) , (x; ^) @x @ ^
(4.8)
for some W 0 and (x; ^). Comparing (4.8) with Assumption 3 in (4.4) for the system (4.6), we can see that they have a similar structure except that we use the projected estimated parameters and add a robust term. 4
40 In the presence of unknown nonlinear functions, principally, there is no way that we can estimate the unknown parameters accurately since the model is not accurate. The best we can do is to use DRC to synthesize a robust control law.
Assumption 4 . There exists a us (x; (lu) ; t) 2 u such that 8 2 and 8 2 : @V @x
[f (x; ; t) + B (x; ; t)(ua + us ) + D(x; ; u; t)] + @V @t (lr ) , @V , ( x; ; u; t ) , V + c ( t ) V V @ ^
(4.9)
or, equivalently,
^_ (4.10) V_ ,V V + cV (t) + @V @ ^ ( + , ) where V > 0 and cV (t) is a bounded positive scalar, i.e., 0 cV (t) cV max . Both V and cV (t) are supposed to be freely adjusted by some controller parameters in a known form without a ecting the initial value of V, V (0).
Normally, Assumption 4 can be satis ed since all the unknown terms involved belong to some known compact sets or a bounded range.
De nition 1 A continuously di erentiable p.s.d. function V (x; ; (lV ) ; t) is called an adaptive robust
control (ARC) Lyapunov function for (4.1) if it satis es Assumptions 2-4 for some continuous control functions ua (x; (lu) ; t) and us (x; (lu) ; t) and the adaptation function (x; (lr ); u; t).
4.3 Adaptive Robust Control In the above section, we introduced the concept of ARC Lyapunov functions. In this section, we will utilize this idea to solve the ARC of (4.1).
Theorem 7 If there exists an ARC Lyapunov function V for (4.1), then, by using the control law (4.3) and the following adaptation law
^_ = ,, (x; (lr ); u(x; (lu); t); t)
(4.11)
the following results hold if Assumption 1 is satis ed:
A. In general, the control input and the system state are bounded with V bounded above by R
V (t) exp(,V t)V (0) + 0t exp(,V (t , ))cV ( )d exp(,V t)V (0) + cVmax V [1 , exp(,V t)]
(4.12)
Output tracking is guaranteed to have arbitrary good transient performance and nal tracking accuracy.
B. If, after a nite time, there are no unknown nonlinear functions, i.e., (x; ; u; t) = 0; 8t t0 ; for some nite t0 , in addition to the result in A, the system outputs track the desired outputs asymptotically.
4
41
Proof. Noting (4.11), from Assumption 4, we have V_ ,V V + cV (t)
(4.13)
which leads to (4.12). From Assumption 2, the system state x is bounded and thus the control input is bounded. Since the exponentially converging rate, V , and the bound of the nal tracking errors, cV max , can be freely adjusted by controller parameters in a known form without a ecting V (0), any V prescribed good transient performance of V can be guaranteed. Therefore, from Assumption 2, output tracking can be guaranteed to have any good transient performance and nal tracking accuracy. This proves A of the Theorem. Now consider the situation that (x; ; u; t) = 0; 8t t0 ; for some nite t0 . Since x is bounded as shown in A, from (4.11), ^_ (t) 2 Lp1 ; 8t. Thus, ^(t0 ) is bounded. Choose a p.s.d. function as (4.14) Va(x; ; ^; t) = V (x; ; (lV ) ; t) + V (~; ) Then, Va(t0 ) is bounded. 8t t0 , noticing (3.6), (4.5), and (4.11), the derivative of Va along (4.1) is (4.15) V_ = V_ j + @ V (~; )^_ ,W; 8t t a
=0
@ ~
0
_ n p Therefore, W 2 L1 and Va 2 L1 . Since u 2 Lm 1 , x_ 2 L1 and ^ 2 L1 . These imply that W is uniformly continuous. By Barbalat's lemma [112], W ,! 0 and thus, from Assumption 3, asymptotic output tracking is achieved. 4
Remark 10 In the absence of adaptation (i.e., , = 0), the proposed ARC law reduces to a DRC law and Result A of Theorem 7 still holds. Therefore, the adaptation loop can be switched o at any time without a ecting the stability. 4
Remark 11 Although ^ is not guaranteed to be bounded in the presence of unknown nonlinear functions,
the stability and the performance of the controller is not a ected since only the bounded projection and its bounded derivatives are used in the design. Furthermore, since ^_ is bounded, for any nite time, ^(t) is bounded. In application, the execution time is always nite, so the issue of boundedness of ^ is not essential here. In addition, in view of Remark 10, either reinitialization or switching o the adaptation can be used in case that unbearable wrong adaptation is observed. }
Corollary 2 For the system (4.1) under Assumption 1, if there exists an ARC Lyapunov function V (x; ; t), which is not a function of ^, then, by using the control law (4.3) and the following modi ed
adaptation law
^_ = ,,[l (^) + (x; (lr ) ; u(x; (lu) ; t); t)]
(4.16)
where l (^) is any vector of functions that satis es the following conditions i. ii. we have the results in Theorem 7.
l (^) = 0 ~T l (^) 0
if if
^ 2 ^ 62
(4.17)
4
42 The reason for using (4.16) is that by suitably choosing l (^), we can make the parameter estimation process more robust and guarantee that ^ is bounded since l (^) acts like a nonlinear damping term. Proof. Since V is not a function of ^, @V @ ^ = 0. Thus, (4.13) and (4.12) are not a ected, and the results in A of Theorem 7 remain valid. When 8t t0 ; = 0, from Assumption 3 and (4.17), following the same proof as in (4.15), we have (4.18) V_a ,W , ~T l (^) ,W Thus, the results in B of Theorem 7 remain valid. 4
Remark 12 Sometimes, the right hand side of adaptation law (4.16) can be discontinuous since it only causes ^_ to be discontinuous, and ^ is still continuous which is normally used in the control law. Therefore, the discontinuous modi cation law l (^) may be allowed. In such a case, the widely used
projection method in adaptive systems [104, 33] can be employed, which is de ned for any bounded open convex set as described in the following. 1 De ne a set, 0 = ,, 2 ( ), which is a bounded open convex set. Let @ denote the () the projection of the vector onto the hyperplane tangent to @ 0 at ,, 12 ^, boundary of , Pr and perp the unit vector perpendicular to the tangent hyperplane of at ^ 2 @ , pointing outward. Then, l (^) is given by 8 ( > ^ 2 > < 0 if T , 0 ^ 2 @ and perp l (^) = > (4.19) 1 1 > T , < 0 : , , ,, 2 Pr ,, 2 ^ 2 @ and perp Let be the closure of . The above choice of l (^) guarantees that ^ 2 no matter what the control law is and what the error dynamics are as long as the initial estimate is in . This is because the resulting ^_ in (4.16) always points inside or along the tangent plane of when ^ 2 @ . Clearly, condition i of (4.17) is satis ed. When ^ 2 @ , by the de nition of perp , 1 1 T ~ 0 (4.20) (, 2 perp )T ,, 2 ~ = perp 8 2 1
1
1
,, 2 is given by Thus, , 2 perp is along the outward normal direction of @ 0 at ,, 2 ^, and Pr
,, 12 Pr where scalar cn is
= ,, 12 , cn , 12 perp
1 1 T , ,perp )T (,,2 ) = cn = (, 2 kperp 12 1 , k2 k, 2 k2 perp
perp
(4.21) (4.22)
T , < 0, from (4.22), cn > 0, and, from (4.19) and (4.21), Thus, when ^ 2 @ and perp h
,, 12 ~l (^) = ~ , , ,, 12 Pr = cn ~T perp 0
i
(4.23)
Noting (4.19), (4.23), and the fact that 8^ 2 ; (^) = ^, we have that 8^ 2 @ ; ~ l (^) 0. Since 8t; ^(t) 2 and is open, ^ 62 is equivalent to ^ 2 @ . Thus, condition ii of (4.17) is satis ed and (4.19) satis es all the required conditions.
43 For the open convex set given by (4.2), with the modi cation (4.19) in which , is a diagonal matrix, the adaptation law (4.16) becomes 8 > > > > > >
,(, )i > > > > > :
0
if
^ = imax and (, )i < 0 imin < ^i < imax ^ = imax and (, )i 0 > i : ^i = imin and (, )i 0 ^i = imin and (, )i > 0 } 8i >
i and thus damage the strict-feedback property. For SISO systems (i.e., mr = 1), when i ( ; t) = 0, Bi1 = 0; 8i; (i.e., in the absence of unknown nonlinear functions and parametric uncertainties in the input channels) and does not present in the r-th subsystem, (6.3) reduces to the parametric strict-feedback form in [62] where an adaptive controller was developed. For the SISO parametric-strict feedback form, the case of Bi ( i ; ; t) being an unknown positive scalar bi is also studied in [65], where over-parametrization about bi is used, i.e., two parameter estimates are needed for one bi .
6.2 Backstepping Design Procedure In this section, the backstepping design procedure in Chapter 5 will be applied to solve the ARC of the system (6.3). In the following recursive design, we need to use trajectory initialization to satisfy the compatibility conditions of the connections like (5.9) to obtain the guaranteed transient performance as T ]T directly, in Chapter 3. For this purpose, instead of tracking the desired outputs yd (t) = [ydT1b ; : : :; ydrb
56 T ]T , in which the i-th block we will deign the controller to track the ltered outputs yt (t) = [ytT1b; : : :; ytrb m , m of outputs, ytib 2 R i i,1 , are created by the following (r , i + 1)-th order stable system (r,i+1) (r,i+1) (r,i) (r,i) (ytib , ydib ) + i1 (ytib , ydib ) + : : : + i(r,i+1) (ytib , ydib ) = 0
(6.5)
(r,i) Such a procedure enables us to choose the initial conditions ytib (0); : : :; ytib (0) freely. Also, we would like to know the explicit dependence of the control law on yt and its derivatives so that we can perform trajectory initialization. In the general development in Chapter 5, yt and its derivatives are considered as known functions of t and thus only t appears in the expression of the control law. In the following, they will appear as variables instead of functions of t in any function. Let Gri ( i ; ; t) and Gli( i ; ; t) denote the right and the left substitution matrices of the matrix Bi1 ( i ; ; t), Grr ( ; ; t) and Glr ( ; ; t) for the matrix Br ( ; ; t), Grr ( ; ; t) and Glr ( ; ; t) for the matrix Br ( ; ; t), and GrM ( r,1 ; ; t) and GlM ( r,1 ; ; t) for the matrix M ( r,1 ; ; t). Also, denote (k ) T ; : : :; (y (k))T ]T ; 8k. Recursively de ne yju by y1u = yt1b ; : : :; yju = [y (1)T ; y T ]T = ytib = [ytib j ,1u tib tjb (j ,1)T T T m T j [yt1b ; : : :; ytjb] 2 R , and Lju by L1u = L1 ; : : :; Lju = [Uj Lj ,1u Nj ]Lj . The design proceeds in the following steps:
6.2.1 Step 1 The rst system is de ned as the rst subsystem of (6.3), which is described by
x_ 1 = f10 ( 1; t) + F1 ( 1 ; t) + B1 ( 1 ; ; t)x2;m1 + D1( 1; t)1 _1 = 01 ( 1 ; t) + 11( 1 ; t) y1 = y1b = x1
(6.6)
By treating x2;m1 as the control for (6.6), comparing (6.6) with the system (5.8), and noting Assumptions 14, 16 and 17, we can see that the rst system can be considered as a special case of (5.8) with mI = 0. x1 ; 1; x2;m1 ; f10; F1 ; B1 ; D1, and 1 in (6.6) correspond to xe ; ; u; fe0 ; Fe ; Be ; De ; and e in (5.8), respectively. Therefore, we can apply the backstepping design procedure in section 5.3 to nd an ARC Lyapunov function V1 for the rst system. V1 is given by (5.11), i.e.,
V1(x1; yt1b(t)) = 12 z1T E1z1 z1(x1; yt1b) = x1 , 0(yt1b)
0(yt1b) = yt1b
(6.7)
with the associated control functions given by (5.13) and (5.23). For simplicity, in the following, us is chosen according to (5.29). Noting (5.19), (5.14), and (5.15), after some tedious substitutions and calculations, we obtain the nal form of the control as 1 ( 1 ; ^ ; yt(1) 1b (t); t) = 1a + 1s ,1 (1) ,1 0 0^ (6.8) 1a ( 1; ^ ; yt(1) 1b (t); t) = L1 [yt1b , f1 , 1 , E1 Q1 z1 ] (1) , 1 , 1 1 2 1s ( 1; ^ ; yt1b (t); t) = , 4(1,u1 )"e1 h1L1 E1 z1 where
01( 1; t) = F1 L1( 1 ; ^ ; t) = B^1 = B1( 1; ^ ; t) 0 r h1( 1; ^ ; yt(1) 1b (t); t) M kE1[ 1 + G1 ( 1 ; 1a; t)k + kE1D1k1 ( 1 ; t) u1( 1; ^ ; t) sup2 kE1B11 ( 1 ; ~ ; t)L,1 1E1,1k
(6.9)
57 The adaptation function is given by (5.14) where T 0 r 1 ( 1; ^ ; yt(1) 1b ; t) = , 1 + G1 ( 1; 1; t) E1z1
(6.10)
yt1b(0) = x1(0)
(6.11)
z1 (0) = 0
(6.12)
,
By choosing from (6.7), we have
From (6.8), the control law has the following structure
^ (1) ^ 1 = L,1u1 ( 1; ^ ; t)yt(1) 1b + 1y ( 1 ; ; yt1b ; t) , 1p( 1 ; ; yt1b; t) , 1 , 1 1y = 1s , L1u E1 Q1z1 1p = L,1u1 [f10 + 01 ^ ] with the property that every element of 1y contains z1 as a factor.
(6.13)
6.2.2 Step 2 Now, augment the rst system by the second subsystem in the same way as in section 5.2 to obtain the second system. Then, the second system has the state vector 2 , the input vector x3;m2 , and the output vector y2 = [y1Tb ; y2Tb]T . From (6.12) and (6.13), 1y (0) = 0. By choosing
^ yt(1) x2;m1(0) + 1p( 1 (0); ^(0); yt1b(0); 0)] 1b (0) = L1u ( 1 (0); (0); 0)[ ^ = B ( 1 (0); (0); 0)[x2;m1 (0) + f10 ( 1 (0); 0) + 01 (0)^ (0)]
(6.14)
we have,
(6.15) x2;m1 (0) = 1(0) Thus, with 1 ; x2 ; 2; x3;m2 , and y2 corresponding to xI ; xe ; ; u, and y in (5.8) respectively, the second system satis es all the assumptions in section 5.2, and we can apply the backstepping design results again to obtain an ARC Lyapunov function V2 with the associated control law 2 and the adaptation function 2 . Detailed expressions will be obtained from the general expressions in the following.
6.2.3 Step i
In general, 8i r , 1, the i-th system of (6.3) is the i -system with the input vector xi+1;mi 2 Rmi and the output vector yi 2 Rmi , i.e.,
x_ j = fj0 ( j ; t) + Fj ( j ; t) + Bj ( j ; ; t)xj+1;mj + Dj ( j ; t)j ( ; ; t) _j = 0j ( j ; t) + 1j ( j ; t) 1ji T T T T T T yi = [yi,1; yib] = [y1b ; : : :; yib ]
(6.16)
58 The (i , 1)-th system can be rearranged as 2
f10 + B10 ( 1 ; t)x2;m1 6 6 01( 1 ; t) 6
3
2
7 7 7 7 7 7 7 5
6 6 6 6 6 6 6 4
F1 ( 1; t) + Gr1 ( 1; x2;m1 ; t) 11 ( 1 ; t)
.. + . 0 fi,1 ( i,1 ; t) 0i,1 ( i,1 ; t) 3 2 2 0 6 7 6 6 7 6 0 6 7 6 6 7 6 . .. + 66 + 7x 6 i;m i , 1 7 6 6 7 6 i,1 ; ; t) 5 4 Bi,1 ( 4 0 yi,1 = [y1Tb; : : :; y(Ti,1)b ]T
_ i,1 =
6 6 6 6 4
.. . Fi,1( i,1 ; t) 1i,1 ( i,1 ; t)
D1 0
...
0
7 7 7 7 7 7 7 5
3
72 7 76 76 74 7 7 i,1 5
D
3
1 .. . i,1
3
(6.17)
7 7 5
which has the form of the xI -subsystem in (5.1) with i,1 corresponding to xI in (5.1). Thus, the i-th system can be considered as the system obtained by augmenting the i , 1-th system by the i-th subsystem in the same way as in section 5.2. xi ; i ; xi+1;mi ; fi0 ; Fi ; Bi ; Di, and i in (6.16) correspond to xe ; ; u; fe0 ; Fe ; Be ; De ; and e in (5.8) respectively.
Lemma 9 8i < r, the backstepping design results in section 5.3 can be recursively applied to nd an (0) (1) ARC Lyapunov function, Vj ( j ; (j ,1) ; yt(1jb,1); : : :; ytjb ; t), an associated control law j ( j ; (j ); yt(1jb); : : :; ytjb ; t), ( j ) ( j ) (1) and an adaptation function j ( j ; ; yt1b; : : :; ytjb ; t) for each j -th system where j i. At each step j , by de ning T ]T zj = xj , j,1; j ,1 = [ Tj,1; ytib (6.18) zj (0) = 0 is achieved by choosing the initial values of the ltered reference trajectories, yju (0). Furthermore, j has the following structure (0) (1) (0) (1) j = L,ju1 ( j ; (j) ; yt(1jb,1); : : :; ytjb ; t)yju + jy ( j ; (j ); yt(1jb,1); : : :; ytjb ; yju ; t) ( j ) ( j , 1) (0) , jp( j ; ; yt1b ; : : :; ytjb ; t)
with the property that every term in jy contains a zk as a factor for some k j .
(6.19)
}
Proof. We proceed to prove the lemma by induction. We assume that Lemma 9 is true for
i , 1 and are going to show that it is true for i-th system to complete the proof. Since zk (0) = 0; 8k i , 1; and every term in i,1y contains a zl as a factor for some l i , 1, we have i,1y (0) = 0. Thus by choosing yi(1) i,1 (0); (i,1)(0); yt(1i,b 2)(0); : : :; yi(0) ,1u (0) = Li,1u ( ,1t(0); 0)[xi;mi,1 (0) (i,1) (i,2) + jp ( i,1 (0); (0); yt1b (0); : : :; yi(0) ,1t ; t)] from (6.19), we have
(6.20)
(6.21) xi;mi,1 (0) = i,1 (0) Thus, all Assumptions in section 5.2 are satis ed by the i-th system and the backstepping design results in section 5.3 can be applied. An ARC Lyapunov function Vi can be found by (5.11), which is rewritten
59 as
Vi ( i ; (i,1); yt(1i,b 1); : : :; ytib; t) = Vi,1 + 21 ziT Eizi = Pij =1 21 zjT Ej zj zi ( i; (i,1); yt(1i,b 1); : : :; ytib; t) = xi , i,1 T ]T : i,1 ( i; (i,1); yt(1i,b 1); : : :; ytib; t) = [ Ti,1 ( i; (i,1); yt(1i,b 1); : : :; yti(1),1b ; t); ytib
(6.22)
The associated control is given by (5.13) and (5.29). By noting (5.19), (5.14), and (5.15), and after some tedious substitutions and calculations, we obtain the nal form of the control as (1) i ( i ; (i); nyt(1i)b; : : :; ytib ; t) = ia + is h , 1 , 1 1 @ i,1 (f 0 + B 0 x ia = Li ,Ei Qi zi ," Ei,1UiT Bi0,T1 Ei,1zi,1 + UiT Pij,=1 j j +1;mj ) @xj j # i
i,1 y (k+1) + @ i,1 + N y (1) , f 0 + @ @i,j 1 0j + UiT ij,=11 ik,=0j @ i tib (k) tjb i @t @ytjb h i o P @ , 0i ^ , , ij,=12 ( @^j )T Uj Ej+1zj+1 , UiT @ @i^,1 ,( i,1 , 0i T Eizi) is = , 4(1,1ui )"ei h2i L,i 1 Ei,1zi where n 0i = Ei,1UiT GlTi,1( i,1 ; Ei,1zi,1 ; t) ,o UiT Pij,=11 @ @xi,j 1 [Fj +Grj ( j ; xj +1;mj ; t)] + @ @i,j 1 1j + Fi j T T Li = 2B^i + Bi1 ( i; ,, Pij,=12 ( @ +1 zj +1 ; t) , Ui ZBi ^ ) Uj Ej3 @ ziT EiBi1 ( i ; ,( @ @i,^1;1 )T ; t) 7 6 .. 7 ZBi = 664 7 . 5 @ i , 1 ;m ziT Ei Bi1 ( i; ,( @ ^ i,1 )T ; t) hi M kEi [ 0i + Gri( i; ia ; t)k + Pij,=11 kEiUiT @ @xi,j 1 Dj kj + kEiDi ki ui sup2 kEiBi1( i; ~ ; t)L,i 1Ei,1k The adaptation function is given by (5.14) , i = i,1 , 0i + Gri ( i ; i; t) T Eizi Noting (6.19) for i,1 , the terms in i which contain yiu(1) are (i,j +1) (1) (1) L,i 1 [UiT Pij,=11 @y@ (ii,,j1) ytjb + Niytib ] + is ( i ; (i); yt(1i)b ; : : :; ytib ; t) tjb @ i,1 y (2) + N y (1)] + = L,i 1 [UiT @y i tib is (1) i,1u i,1u (1) ,1 @ i,1y (2) = L,i 1 [UiT L,i,11u yi(2) ,1u + Niyit ] + Li @yj(1),1u yj ,1u + is = L,iu1 yiu(1) + iy where i,1y y (2) , kzi h2 E ,1 z ] iy = L,i 1 [UiT @ @y(1) i,1u 1,ui i i i P
P
i,1u (1) yi,1u and
(6.23)
(6.24)
(6.25)
(6.26)
(6.27)
Since zk ; 8k i , 1, does not depend on every term of i,1y has a zk as a factor, so does @ i,1y . It is thus clear from (6.27) that every term of has a z as a factor for k i. Thus, has iy k i @y(1) i,1u
T T T the form (6.19). Obviously, by choosing yiu (0) = [yi(1) ,1u (0); ytib(0)] in terms of (6.20) and (6.28) ytib(0) = NiT xi (0) = [xi;mi,1 +1 (0); : : :; xi;mi (0)]T we have zi (0) = 0 by (6.22). (6.22), (6.23), (6.25), (6.20), (6.28), and (6.26) agree with the general conjectures about the k-th system for k i , 1 and, thus, are true for every system by induction. 4
60
6.2.4 Step r By using the general formula in step i, we can recursively nd the ARC Lyapunov function for each system until the r , 1-th system. Then, by augmenting the r , 1th system by the r-th subsystem in the same way as in the augmented system (5.33), we obtain the r-th system, which is the entire system (6.3). Similar to Step i, if we choose yr(1) ,1u (0) in the same way as in (6.20), the compatibility condition (5.9) will be satis ed and it is easy to check that all Assumptions for the augmented system (5.33) are satis ed. Thus, we can apply the backstepping design results in section 5.4 to obtain an ARC Lyapunov function for the system (6.3). The nal form of the ARC Lyapunov function Vr is given by ,1 1 z T Ej zj + 1 z T M ,1 ( r,1 ; ; t)zr Vr( ; (r,1) ; ; yt(1rb,1); : : :; ytrb; t) = Prj =1 2 j 2 r ( r , 1) ( r , 1) T T ]T : zr ( ; ; yt1b ; : : :; ytrb; t) = xr , r,1 r,1 = [ r,1; ytrb
The associated control law is (1) r ( ; (r); ynt(1rb); : : :; ytrb ; t) = ra + rs h ,2 ( @ j )T U E z i , 1 ra = Lr ,Qr zr , UrT Br0,T 1 Er,1zr,1 , 0 ^ , , Prj=1 j jo+1 j +1 @ ^ @ r , 1 0 0 0 T 0 ^ ^ ^ , , ## + M0fB , fr , MUr @^ , , dM ( r,1; zr ; t) rs = , 4(1,1ur )"er h2r L,r 1zr where
(6.29)
(6.30)
n
,1 @ r,1 [Fj 0 = UrT GlTr,1( r,1 ; Er,1zr,1 ; t) , Mo0UrT Prj =1 @xj @ r , 1 r 1 +Gj ( j ; xj +1;mj ; t)] + @j j + Fr + DM ( r,1 ; zr ; t) 0 = r,1 , 0Thzr i ,1 @ r,1 (f 0 + B 0 xj +1;m ) + @ r,1 0 fB = UrT P" rj=1 j j j @xj j #@j P P ,1 r,j @ r,1 y (k+1) + @ r,1 + Nr y (1) +UrT rj =1 k=0 @y(k) tjb trb @t tjb 0 = F , GrM ( r,1; fB ; t) + DM ( r,1 ; zr ; t) 0# = DM#( r,1; zr; tP ) , Dp#( ; t) ,2 ( @ j )T U E z ; t) , MU ^ rT ZBr Lr = B^2r + Br1 ( ; ,, rj =1 j j3+1 j +1 ^ @ zrT Br1 ( ; ,( @ @r,^1;1 )T ; t) 7 6 .. 7 ZBr = 664 7 . 5 @ r , 1 ;m zrT Br1 ( ; ,( @ ^ r,1 )T ; t) and the bounding functions hr and ur satisfy the following conditions ,1 kU T @ r,1 Dj kj + kDr kr + M ( ; zr ; t) hr eM k e( ; (r); ra; t)k + kM Prj=1 r @xj , ~ ~ ~ ur sup2 k(Br + Br , M Ur ZBr )L 1 k in which the form of the function e is de ned by
( ; (r); ; t) = 0 ( ; (r); t) + Grr ( ; ; t) ( ; (r); ; t) = 0( ; (r); t) , GrT r ( ; ; t)zr ( r ) 0 r ( ; ; ; t) = + GM ( r,1 ; UrT @ @r^,1 , ( ; (r); ; t) ; t) + Grr ( ; ; t) e ( ; (r); ; t) = [ T ; T ; 0#T ]T
(6.31)
(6.32)
(6.33)
61 The forms of the vectors or matrices dM ; DM ; DM ; DM#; M and Dp# used in the above are obtained from (5.38), (5.39) and (5.41). The adaptation function is 2 3 (r); r ; t) ( ; 6 7 e = 64 , T ( ; (r); r; t)zr 75 (6.34) ( r ) , 0#T ( ; ; t)zr Theorem 10 If the actual control law (6.35) u = r ( ; (r); ^; #^ ; yt(1rb); : : :; yrt(1); t) with the adaptation law , (6.36) ^_ e = ,,e e( ; r ; ^ ; #^ ; yt(1rb); : : :; yrt(1); r; t) is applied to the system (6.3) under the Assumptions 14-17, the following results hold A. In general, the control input and the state are bounded with Vr bounded above by R [1 , exp(,Vr t)] Vr (t) 0t exp(,Vr (t , ))cVr ( )d cVrVmax (6.37) r
min(Qi ) g; 2min (Qi ) g and c = r " . Output tracking is where Vr = minfminir,1 f 2max Vr i=1 ei (Ei ) kM guaranteed to have arbitrary good transient performance and nal tracking accuracy. B. If, after a nite time, there are no unknown nonlinear functions, i.e., i = 0; 8t t0 ; for some nite t0 , in addition to the result in A, the system outputs track the desired outputs asymptotically. 4 Proof. By using the trajectory initialization (6.20) and (6.28), zi (0) = 0 for all i's. Thus, Vr(0) = 0. The above theorem is then a Direct application of Theorem 7. 4 P
6.2.5 Guaranteed Transient Performance From (6.37, any good transient performance about the Vr can be guaranteed by adjusting controller parameters Qi and "ei . This result in turn guarantees any good transient performance about tracking error vector et (t) = y , yt since Vr is a p.s.d. quadratic function of zi , and etib = yib , ytib , the i-th block of et , is the vector of last mi , mi,1 elements of zi . If the above trajectory initialization is independent of Qi and "ei , from (6.5), the trajectory planning error, ed (t) = yt (t) , yd (t); converges to zero exponentially and can be guaranteed to possess any good transient behavior when one suitably chooses the Hurwitz polynomials Gid (s) = s(r,i+1) + i1 sr,i + : : : + i(r,i+1) without being a ected by Qj and "ei . Thus, the actual output tracking error, e = et + ed , can be guaranteed to have any good transient performance. So, in the following, the same as in the SISO case in section 3.4, we (k ) illustrate that the above trajectory initialization actually places ytib (0) at the estimated value of yib(k) (0) by neglecting all unknown nonlinear functions and using ^ (0) for and thus the initialization process is independent of the controller gains. From (6.11), yt1b (0) = y1b (0). From (6.3), y1(1)b = h1( 1; x2; ; t) + D1( 1 ; t)1 (6.38) h1 = f10( 1 ; t) + F1 ( 1 ; t) + B1( 1; ; t)x2 From (6.14) and (6.38), (6.39) yt(1) 1 (0); x2(0); ^(0); 0) = y1(1)b (0) j=0; =^ (0) 1b (0) = h1 ( which supports the above claim. The general proof is very tedious and is omitted.
62
6.3 Simulation Results Consider the following time-varying system with two inputs u = [u1 ; u2] and two outputs
x_ 1 = "1 x21 + 2 x2;1 + 1(x#2 ) " # 2 + x2 ) sin3(t) ( x x 1 1 2 ; 1 x_ 2 = 1 + u + x 2(x) x31 + 5x2;2 2;2 y = [x1; x2;2]T
(6.40)
where
1 (x2) = d1 sin(r1x2 ) (6.41) 2 (x) = d2 cos(r2x1 x2 ) 1 2 [,1; 1], and 2 2 [10; 15]. r1, r2, d1 2 [0; d1M ], and d2 2 [0; d2M ] are assumed to be unknown. It can be observed that (6.40) has input channel uncertainty for x1 dynamics (term 2 x2;1) and are not in a strict feedback form, but it does satisfy the MIMO semi-strict feedback form (6.3) since j1(x2)j 1 = d1M and j2(x)j 2 = d2M . Therefore, we can apply the results in section 6.2 to obtain an ARC law. In simulation, actual plant parameters are 1 = 1, 2 = 14, r1 = 2, and r2 = 3. Sampling time is 1ms. The same projections as in (3.54) are used. The desired outputs are yd = [sin(0:5t); 0:5sin(1:2t)]T , which have two frequency components. The lter output trajectories are created by (6.5) where 11 = 80, 12 = 1600, and 21 = 40. Controller parameters are E1 = 1, E2 = I2 , Q1 = 10; Q2 = diag f100; 100g, , = diag f100; 500g, kz1 = 0:1, and kz2 = 0:1. Three controllers are run for comparison: ARC : the proposed adaptive robust control as described above; DRC : Deterministic Robust Control | the same control law as in ARC but without the using parameter adaptation law; AC : Adaptive Control, which is obtained by setting hi = 0 and without using parameter projection | i.e., letting ( ) = in ARC. To test the nominal performance, a simulation is run for the parametric uncertainties only (i.e., d1 = d2 = 0). Tracking errors z1 = y1 , yt1 and z2;2 = y2 , yt2 are shown in Fig. 6.1 and Fig. 6.2 respectively. We can see that all the three controllers have very good tracking ability. The estimated parameters in ARC and AC approach their desired values as shown in Fig. 6.3. The proposed ARC has a better transient and a much better nal tracking accuracy than RC. AC also has a good nal tracking accuracy but has the worst transient response. These results substantiate the necessity of using parameter adaptation to improve nal tracking accuracy and using robust control to improve transient performance. Control inputs of all three controllers are smooth and more or less the same as shown in Fig. 6.4 and Fig. 6.5. To test the performance robustness, large disturbances are added to the system (i.e., let d1 = d2 = 2 in (3.52)). The tracking errors z1 and z2;2 are shown in Fig. 6.6 and Fig. 6.7, and the estimated parameters are shown in Fig. 6.8. The ARC still achieves the best tracking performance although its estimated parameters run quite wildly. AC has a very large tracking error (actually diverging) since its estimated parameters run wildly and have a diverging trend. Control inputs are smooth as shown in Fig. 6.9 and Fig. 6.10. All these results illustrate the e ectiveness of the proposed ARC.
63 The simulation is also run to track a less rich output trajectory, yd = [(1 , exp(,4t)); 0:5(1 , exp(,4t))], which exponentially decays to a constant value and thus does not have much frequency content after a short time. In the presence of parametric uncertainties, the tracking errors z1 and z2;2 are shown in Fig. 6.11 and Fig. 6.12 . It can be seen that the ARC still achieves perfect tracking and has a pretty good parameter estimation as shown in Fig. 6.13. It is interesting to observe that AC diverges quickly during the initial transient because of its wrong parameter adaptation while it is supposed to achieve asymptotic stability in theory. This result veri es the non-robustness nature of pure adaptive control since the closed loop system may be unstable under the e ect of sampling only (in this simulation, the only approximation comes from the discrete implementation of the continuous control laws). This observation further substantiates the necessity of using robust control in the design of a baseline control law.
6.4 Conclusions In this chapter, by using the backstepping design procedures in Chapter 5, we have solved the ARC of a class of MIMO nonlinear systems in a semi-strict feedback form. The form allows both parametric uncertainties and unknown nonlinear functions. Parametric uncertainties are also allowed in the input channels of each layer and the linear parametrization requirement is relaxed to include mechanical systems. Simulation results show that the proposed ARC has a better tracking performance than its DRC counterpart and a better performance robustness than AC.
0.05 0.04 Solid: ARC
Dotted: DRC
Dashdot: AC
0.03
Tracking error of z1
0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05 0
1
2
3
4 Time (sec)
5
6
7
8
Figure 6.1: Tracking errors in the presence of parametric uncertainties
64 0.025 0.02
Tracking error of z2
0.015 0.01 0.005 0 −0.005 −0.01 Solid: ARC
Dotted: DRC
Dashdot: AC
−0.015 −0.02 0
1
2
3
4 Time (sec)
5
6
7
8
Figure 6.2: Tracking errors in the presence of parametric uncertainties
Estimated parameters
15
10 Solid: theta1 (ARC) Dashdot: theta2 (ARC) Dashed: theta1 (AC) Dotted: theta2 (AC)
5
0 0
1
2
3
4 Time (sec)
5
6
7
8
Figure 6.3: Estimated parameters in the presence of parametric uncertainties 14
12
Control input u1
10
8 Solid: ARC
Dotted: DRC
Dotted: AC
6
4
2
0
−2 0
1
2
3
4 Time (sec)
5
6
7
8
Figure 6.4: Control inputs in the presence of parametric uncertainties
65 5 4 Solid: ARC
Dotted: DRC
Dotted: AC
3
Control input u2
2 1 0 −1 −2 −3 −4 0
1
2
3
4 Time (sec)
5
6
7
8
Figure 6.5: Control inputs in the presence of parametric uncertainties 0.1 0.08 0.06
Tracking error of z1
0.04 0.02 0 −0.02 −0.04 −0.06 Solid: ARC
Dashed: DRC
Dashdot: AC
−0.08 −0.1 0
1
2
3
4 Time
5
6
7
8
Figure 6.6: Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2) 0.08
0.06 Solid: ARC
Dashed: DRC
Dashdot: AC
Tracking error of z2
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08 0
1
2
3
4 Time
5
6
7
8
Figure 6.7: Tracking errors in the presence of parametric uncertainties and disturbances (d1=d2=2)
66 70
60 Solid: theta1 (ARC)
Estimated parameters
50
Dashdot: theta2 (ARC)
Dashed: theta1 (AC) Dotted: theta2 (AC)
40
30
20
10
0
−10 0
1
2
3
4 Time
5
6
7
8
Figure 6.8: Estimated parameters in the presence of parametric uncertainties and disturbances (d1=d2=2) 14 12 10
Control input u1
8 Solid: ARC
Dashed: RC
Dotted: AC
6 4 2 0 −2 −4 0
1
2
3
4 Time
5
6
7
8
Figure 6.9: Control inputs in the presence of parametric uncertainties and disturbances (d1=d2=2) 6
4
Control input u2
2
0
−2
−4 Solid: ARC −6 0
Figure 6.10: (d1=d2=2)
1
2
Dashed: RC
3
4 Time
Dotted: AC
5
6
7
8
Control inputs in the presence of parametric uncertainties and disturbances
67 0.08 0.07 0.06
Tracking error of z1
0.05 Solid: ARC
Dotted: DRC
Dashdot: AC
0.04 0.03 0.02 0.01 0 −0.01 0
1
2
3
4 Time
5
6
7
8
Figure 6.11: Tracking errors in the presence of parametric uncertainties 0.06
0.05
Tracking error of z2
0.04 Solid: ARC
Dotted: DRC
Dashdot: AC
0.03
0.02
0.01
0
−0.01 0
1
2
3
4 Time
5
6
7
8
Figure 6.12: Tracking errors in the presence of parametric uncertainties 16 14
Estimated parameters
12 10 8 Solid: theta1 (ARC) 6
Dashdot: theta2 (ARC)
Dashed: theta1 (AC) Dotted: theta2 (AC)
4 2 0 −2 0
1
2
3
4 Time
5
6
7
8
Figure 6.13: Estimated parameters in the presence of parametric uncertainties
68
Part II
Adaptive Robust Control - Applications
69
Chapter 7
Trajectory Tracking Control of Robot Manipulators In this chapter, the proposed ARC is applied to the trajectory tracking control of robot manipulators. Two schemes are developed: adaptive sliding mode control (ASMC) is based on SMC and the conventional adaptation law structure in which the regressor uses the actual state feedback information; desired compensation adaptive robust control (DCARC) is based on the desired compensation adaptation law structure, in which the regressor uses the desired trajectory information only. In addition, three conceptually di erent adaptive and robust control schemes | a simple nonlinear PID type robust control, a gain-based nonlinear PID type adaptive control, and a combined parameter and gain based adaptive robust control | are derived for comparison. All algorithms, as well as two other established adaptive schemes, are implemented and compared on the UCB/NSK SCARA direct drive robot.
7.1 Dynamic Model of Robot Manipulators A dynamic equation of a general rigid link manipulator having n degrees of freedom in free space can be written as [81]
M (q; )q + C (q; q;_ )q_ + G(q; ) + f~(q; q;_ t) = u
(7.1)
where q 2 Rn is the joint displacement vector, 2 Rl is the vector of a suitably selected set of the robot parameters, u 2 Rn is the applied joint torque, M (q; ) 2 Rnn is the inertia matrix, C (q; q;_ )q_ 2 Rn is the Coriolis and centrifugal force, G(q; ) 2 Rn is the gravitational force, and f~(q; q;_ t) 2 Rn is the vector of unknown nonlinear functions such as external disturbances and joint friction. Equation (7.1) has the following properties that will facilitate the controller design [81, 109, 89, 103, 98].
Property 1 . M (q; ) is a symmetric positive de nite (s.p.d.) matrix, and there exists km > 0 such that km Inn M (q; ). Furthermore, for the robot with all joints revolute or prisma, there exists kM > 0 so that M (q; ) kM Inn . For a general robot, M (q; ) kM Inn is valid for any nite workspace q = fq : kq , q0 k qmax g where q0 and qmax are some constants. Property 2 . The matrix N (q; q;_ ) = M_ (q; ) , 2C (q; q;_ ) is a skew-symmetric matrix.
70
Property 3 . M (q; ); C (q; q_; ), and G(q; ) can be linearly parametrized in terms of . Therefore, we can write
M (q; )qr + C (q; q;_ )q_r + G(q; ) = f0(q; q;_ q_r ; qr ) + Y (q; q;_ q_r ; qr ) where Y 2 Rnl , q_r and qr are any reference vectors.
(7.2)
We assume that Assumption 11 in section 5.4 is satis ed and the disturbance f~(q; q;_ ; t) can be bounded by kf~(q; q;_ t)k hf (q; q;_ t) (7.3) where hf (q; q;_ t) is a known scalar function. We can now formulate the trajectory tracking control of robot manipulators as follows: Suppose qd (t) 2 Rn is given as the desired joint motion trajectory. Let e = q (t) , qd (t) 2 Rn be the motion tracking error. For the robot manipulator described by (7.1), design a control law u so that the system is stable and q tracks qd (t) as close as possible.
7.2 Control Algorithms 7.2.1 Adaptive Sliding Mode Control (ASMC) Let x1 = q and x2 = q_. In state space, (7.1) can be rewritten as
x_ 1 = x2 x_ 2 = M ,1 (x1; )[,C (x1; x2; )x2 , G(x1; ) + u , f~(x1; x2; t)] y = x1
(7.4)
Noting (7.3) and Properties 1 and 3, we can see that (7.4) is in the semi-strict feedback form (6.3) with a relative degree r = 2 and satis es all assumptions in section 6.1. Thus, we can apply the general results in section 6.2 to obtain an ARC controller. However, since the rst equation of (7.4) does not have any modeling uncertainties, (7.4) satis es the matching condition. The controller design can, thus, be simpli ed and treated as a relative degree one design, which is much easier to deal with, and powerful SMC techniques can be used in designing the baseline robust control law. The detailed design procedure follows. Similar to the DRC controller in section 2.1, a dynamic sliding mode is employed to eliminate the unpleasant reaching transient and to enhance the dynamic response of the system in sliding mode. Let a dynamic compensator be z_ = Az z + Bz e z 2 Rnc Az 2 Rnc nc Bz 2 Rncn (7.5) yz = Cz z + Dz e yz 2 Rn Cz 2 Rnnc Dz 2 Rnn where (Az ; Bz ; Cz ; Dz ) is controllable and observable. The sliding mode controller is designed to make the following quantity remain zero. = e_ + yz 2 Rn (7.6) = q_ , q_r q_r = q_d (t) , yz Transfer function from to e is e = G, 1(s) (7.7)
71 where
Gc(s) = Cz (sInc , Az ),1 Bz + Dz
G (s) = sIn + Gc (s)
(7.8)
The state space realization of (7.7) is
x_ = A x + B
y = Cx
(7.9)
where x = [z T ; eT ]T 2 Rnc +n and "
A = ,ACz ,BDz z z
#
"
0
B = I n
#
C = [0 ; In ]
(7.10)
From (7.7), G, 1 (s) can be arbitrarily assigned by suitably choosing a dynamic compensator transfer function Gc (s) as long as G, 1 (s) has relative degree one. Since during sliding mode, = 0, the system response is governed by the free response of transfer function G, 1 (s). Therefore, as long as G, 1 (s) is stable, the resulting dynamic sliding mode will be stable and is invariant to various modeling errors. Furthermore, the sliding mode can be arbitrarily shaped to possess any exponentially fast converging rate, since poles of G, 1 (s) can be freely assigned. In addition, G, 1 (s) can be chosen to minimize the e ect of on e when the ideal sliding mode f = 0g cannot be exactly achieved in practice. The equivalent results in state space can be stated as follows: there exists an s.p.d. solution P for any s.p.d. matrix Q for the following Lyapunov equation,
AT P + P A = ,Q
(7.11)
(Q ) Furthermore, min max (P ) can be arbitrarily shaped by assigning the poles of A to the far left plane to obtain any exponentially fast converging rate. In addition, when Cz is of full column rank, the initial value z (0) of the dynamic compensator (7.5) can be chosen to satisfy
Cz z (0) = ,e_ (0) , Dz e(0)
(7.12)
then (0) = 0. It is shown in [154] that choosing the initial value z (0) in such a way guarantees that the system is maintained in the sliding mode all the time and the reaching transient is eliminated when ideal sliding mode control is applied. Therefore, in the following, such a choice is made and (0) = 0 is used whenever dynamic sliding mode is used. Noting (7.6) and Property 3, (7.1) can be rewritten as
M (q; )_ + C (q; q;_ ) + f0 (q; q;_ q_r ; qr ) + Y (q; q;_ q_r ; qr ) + f~(q; q;_ t) = u
(7.13)
Let h (q; q;_ q_r ; qr ) be a bounding function satisfying
kY (q; q;_ q_r; qr ) ~k = kY ^ , Y k h (q; q;_ q_r ; qr)
8 ^ 2 ^
For example, choose
h (q; q;_ q_r ; qr) = kY (q; q;_ q_r ; qr)k M where M = k max , min + " k. De ne hs(q; q;_ q_r ; qr ; t) = hf (q; q;_ t) + h (q; q;_ q_r; qr ; t)
(7.14) (7.15) (7.16)
72 The following continuous control law is suggested
u = ua + h (,hs kk ) ua = f0(q; q;_ q_r ; qr ) + Y (q; q;_ q_r; qr ) ^ , K
(7.17)
where K is any s.p.d. matrix and h(,hs kk ) is a continuous approximation of the ideal SMC control, ,hs kk with an approximation error "(t). where h is a positive scalar function and is a De nition 2 For any discontinuous vector like ,h kk ), with an approximation error "(t) is de ned vector of functions, its continuous approximation, h(,h kk
to be a vector of functions that satis es the following two conditions: )0 i. T h (,h kk ) "(t) ii. hk k + T h(,h kk
(7.18)
}
Remark 18 A natural generalization of the concept of boundary layer [106] to multiple input/output cases is given by
) = ,(1 + 1 h)h h (,h kk (7.19) kk+ (t) where 1 > 0 is any positive scalar, and (t) is any bounded time-varying positive scalar, i.e., 0 (t) max, which has the role of boundary layer thickness. It is easy to show [154] that (7.18) is satis ed for " = 4 (t1) . }
) = [h1; : : :; hn ]T is given by Remark 19 A smooth h(,h kk
hi = ,htanh hi(ti) From (2.22), condition i of (7.18) is satis ed and
T h = , Pni=1 h i tanh
hi i
(7.20)
Pni=1( i , hj i j) Pni=1 i , hk k P Thus, condition ii of (7.18) is satis ed for " = ni=1 i .
(7.21)
}
Remark 20 The same as in Remark 4 in Chapter 2, in order to achieve a good tracking accuracy, a time varying (t) similar to (2.40) has to be employed, which is quite complicated and is not easily implemented. To overcome this problem, the following modi cation is suggested: 8 > >
,(1 , c1)Ks , c1 hs kk > : ,h s kk
1
kk h h = hs(q;q;_ q _r(;tq)r ;t)+"1 h kk (1 + "2 ) h kk (1 + "2) h
(7.22)
h where Ks is any s.p.d. matrix, c1 = k"k, 2 h , and "1 and "2 are any positive scalars. It can be shown [154] that (7.18) is satis ed for " = (1 + "2 ) (t). The above modi cation is quite simple and yet it provides the desired properties { namely, around sliding mode fk k = 0g, a xed feedback gain matrix is employed all the time and thus can be 1 is replaced by here to make the presentation clear.
73 chosen near its allowable limit without inducing control chattering. We can also tune the gain around each joint separately since it is a gain matrix instead of a nonlinear scalar gain. When the system is away from sliding surfaces, the original nonlinear feedback control law is employed to guarantee the stability at large. It is shown in [154] by both simulation results and experimental results that the above modi cation can achieve a better tracking performance than (7.19). }
Lemma 10 The following p.s.d. function V
V = 21 T M (q; )
(7.23)
is an ARC Lyapunov function for (7.1) with the control (7.17) and the adaptation function given by
= Y T (q; q;_ q_r; qr )
(7.24)
}
Proof. From Property 1, Assumption 2 in section 4.2 is satis ed by V for (7.1). From
_ = T C . Noting (7.13) and (7.17), the derivative of V is Property 2, 21 T M
_ = T [M (q; )_ + C (q; q;_ ) ] V_ = T M _ + 12 T M = T [Y (q; q;_ q_r ; qr ) ~ , f~(q; q;_ t) , K + h(,hs kk )]
(7.25)
When f~ = 0, noting condition i of (7.18), we have
V_ jf~=0 , T K + ~T Y T (q; q;_ q_r; qr )
(7.26)
Thus, Assumption 3 (4.4) is satis ed for W = T K . In general, when f~ 6= 0, from (7.3), (7.14), (7.16), (7.25), and condition ii of (7.18), V_ k k[kY (q; q;_ q_r ; qr ) ~ k + kf~(q; q;_ t)k] , T K + T h(,hs kk ) (7.27) kkhs , T K + T h (,hs kk ) ,T K + " ,V V + " (K ) where V = 2min 4 kM . Thus, Assumption 4 (4.9) is satis ed, which completes the proof. Noting that V is not a function of ^, we can use the adaptation law (4.16) (replacing with ) with discontinuous modi cation (4.19) | i.e., (4.24), which is rewritten here as 8 > > > > > >
,(, )i > > > > > :
0
8^i = imax and (, )i < 0 ^ > < imin < i < imax ^i = imax and (, )i 0 > : ^ i = imin and (, )i 0 ^i = imin and (, )i > 0
(7.28)
The above results are summarized in the following theorem.
Theorem 11 If the control law (7.17) with the adaptation law (7.28) is applied to the manipulator
described by (7.1), the following results hold:
74
A. In general, all the signals in the system remain bounded and tracking errors, e and e_, exponentially converge to some balls with size proportional to ". Furthermore, the tracking error is bounded by
k(t)k2
2 [exp(, t)V (0) + Z t exp(, (t , ))"( )d ] V V k m
0
(7.29)
In addition, if (7.12) is satis ed, then V (0) = 0 in (7.29). B. If after a nite time, f~ = 0, then the following are true:
a) ,! 0;
e ,! 0
e_ ,! 0
when t ,! 1 i.e., the robot follows the desired motion
trajectories asymptotically. Additionally, if the desired motion trajectory satis es the following persistent excitation condition Z t+T Y T (qd ; q_d ; q_d ; qd)Y (qd; q_d; q_d ; qd)d "d IkE 8t t0 (7.30) t
where T; t0 and "d are some positive scalars, and qd(3)(t) is bounded, then b) ~ ,! 0 when t ,! 1: i.e., estimated parameters converge to their true values.
4
Remark 21 The extra freedom in choosing dynamic sliding mode G, 1 (s) can be utilized to minimize the e ect of a non-zero on the tracking error e. For example, if the system is mainly subject to some constant disturbances, a constant steady state may appear. By including a di erentiator in the numerator of G, 1(s), e.g., G, 1 (s) = s2 +ksp s+ki In , which can be realized by choosing the dynamic compensator parameter as Cz = In ; Az = 0 ; Bz = ki In ; Dz = kp In , a zero steady state tracking error e(1) can be obtained. } Remark 22 By setting us = 0 in (7.17), without using parameter projection and any modi cation to the adaptation law, and taking o the dynamic compensator (i.e., letting Cz = 0; Az = 0; Bz = 0; Dz > 0
in (7.5)), the control law (7.17) reduces to Slotine and Li's well-known adaptive algorithm (SLAC), which is also tested later for comparison. }
7.2.2 Desired Compensation Adaptive Robust Control (DCARC) The regressor Y in adaptation function (7.24) depends on the actual state. In [103], Sadegh and Horowitz proposed a desired compensation adaptation law (DCAL), in which the regressor is calculated by reference trajectory information only. By doing so, one obtains a resulting adaptation law that is less sensitive to noisy velocity signals and has a better robustness as well as a signi cantly reduced amount of on-line computation. Comparative experiments in [138] demonstrated the superior tracking performance of the DCAL. Inspired by these results, a desired compensation adaptive robust control (DCARC) is proposed in this subsection. It is shown in Appendix 1 that there are known non-negative bounded scalars 1(t), 2(t),
3(t), and 4(t), which depend on the reference trajectory and A only, such that the following inequality is satis ed
kf0(q; q;_ q_r; qr ) + Y (q; q;_ q_r ; qr) , f0(qd ; q_d; q_d; qd) , Y (qd; q_d; q_d; qd) k 1kx k + 2kk + 3kkkxk + 4kx k2
(7.31)
75
1; 2; 3, and 4 can be determined o -line. Similar to (7.14), there exists a known scalar function h (qd; q_d; qd) such that kY (qd; q_d; q_d; qd) ~ k = kY ^ , Y k h (qd; q_d; qd) 8 ^ 2 ^ (7.32) Since h (qd ; q_d ; qd) depends on reference trajectory only, it can be determined o -line, one of the
advantages of this scheme. Similar to (7.16), de ne
hs (q; q;_ t) = hf (q; q;_ t) + h (qd ; q_d; qd ) The following continuous robust control law is suggested u = ua + h (,hs kk ) ua = f0(qd ; q_d ; q_d ; qd) + Y (qd ; q_d; q_d ; qd) ^ , K , Kx x , 5kx k2
(7.33) (7.34)
where K > 0 is an s.p.d. matrix, 5 is a positive scalar, Kx = BT P , in which P is determined by (7.11), and h is a continuous approximation of ,hs kk with an approximation error ".
Lemma 11 The following p.s.d. function V = 21 T M (q; ) + 21 xT P x
(7.35)
is an ARC Lyapunov for (7.1) with the control law (7.34) and the adaptation law
= Y T (qd; q_d ; q_d ; qd)
Proof. Noting Property 1, 1 k k k2 + 1 (P )kx k2 V 1 k k k2 + 1 (P )kx k2 2 m 2 min 2 M 2 max
(7.36) (7.37)
Thus, Assumption 2 is satis ed by V . Noting (7.13), (7.9) and (7.11), di erentiating V with respect to time yields V_ = T [M (q; )_ + C (q; q;_ ) ] + 21 xT (AT P + P A )x + xT P B (7.38) = T [u , f0 (q; q;_ q_r ; qr ) , Y (q; q;_ q_r ; qr ) , f~ + BT P x ] , 12 xT Q x Substituting the control law (7.34) into (7.38) and noting (7.31), we can obtain V_ = T [Y (qd; q_d ; q_d; qd ) ~ + f0(qd ; q_d; q_d; qd ) , f0 (q; q;_ q_r ; qr ) + Y (qd ; q_d; q_d ; qd) ,Y (q; q;_ q_r ; qr) , f~ , K , 5kx k2 + h ] , 12 xT Q x (7.39) ,T K , 5kx k2kk2 , 21 xT Q x + T Y (qd; q_d; q_d; qd) ~ , T f~ + T h + 1k kkx k + 2k k2 + 3k k2kx k + 4kx k2k k Applying the inequality
w1jy1 jjy2j w2y12 + w3y22
8y1; y2 2 R w1; w2; w3 0
where 4w2w3 = w12 to (7.39), we have, V_ ,T K , 5 kxk2kk2 , 12 xT Q x + T Y (qd; q_d ; q_d; qd ) ~ , T f~ + T h + 6k k2 + 7kx k2 + 2k k2 + 8k k2 + 9k k2kx k2 + 10kx k2 + 11kx k2k k2 = , T [K , ( 2 + 6 + 8 )In ] , xT [ 12 Q , ( 7 + 10)In+nc ]x ,[ 5 , ( 9 + 11)]kxk2kk2 + T Y (qd; q_d; q_d; qd) ~ , T f~ + T h
(7.40)
(7.41)
76 where
6 7 = 41 12
8 9 = 14 32
10 11 = 14 42 By choosing controller parameters K ; Q , and 5 as min (K) "3 + 2 + 6 + 8 min (Q) 2("3 + 7 + 10)
5 9 + 11
(7.42)
(7.43)
where "3 is any positive scalar, (7.41) becomes
V_ ,"3(kk2 + kx k2 ) + T Y (qd; q_d ; q_d; qd) ~ , T f~ + T h
(7.44)
When f~ = 0, noting condition i of (7.18), (7.44) becomes
V_ jf~=0 ,"3(kk2 + kx k2 ) + ~T Y T (qd ; q_d ; q_d; qd)
(7.45)
Thus, Assumption 3 (4.4) is satis ed for W = "3 (k k2 + kx k2). In general, when f~ 6= 0, from (7.3), (7.32), (7.33), (7.37), and condition ii of (7.18), (7.44) becomes (7.46) V_ ,"3 (kk2 + kx k2) + hs kk + T h ,V V + " where V is a positive scalar satisfying (7.47) V maxfk 2; "3 (P )g M max Thus, Assumption 4 (4.9) is satis ed if V can be freely adjusted, which is shown in the following remark. This completes the proof. 4
Remark 23 In (7.46), the exponential convergence rate V can be any large value by choosing the controller parameters as follows. Noting V is bounded below by (7.47) and kM is a xed constant, V can be any large value as long as we can arbitrarily choose "3 and max"3(P ) . Therefore, rst set "3 to its desired value and let Q satisfy (7.43), in which 7 and 10 can be any xed values. Then, choose the dynamic compensator parameter A such that the solution P of (7.11) makes max"3(P ) big enough. 1; 2; 3, and 4 in (7.31) can then be determined, and 6 ; 8; 9 and 11 can be calculated to satisfy (7.42). Finally, choose K and 5 such that (7.43) is satis ed. In this way, theoretically, any fast exponential convergence rate can be achieved.
}
Noting that V is not a function of ^, we can use the adaptation law (7.28) to achieve ARC of the robot manipulator (7.1), which is summarized in the following theorem.
Theorem 12 If the control law (7.34) and the adaptation law (7.28) with given by (7.36) is applied to the manipulator described by (7.1), the same results as Theorem 11 can be obtained.
4
77
Remark 24 In the control law (7.34) and the adaptation function (7.36), the regressor Y (qd ; q_d; q_d; qd ) is a function of the reference trajectory only and, thus, can be calculated o -line.
In addition to the reduction of on-line computation time, this result also removes the problem of noise correlation between the estimation error and the adaptation signals, especially when the velocity measurement is noisy in implementation [103], and, thus, enhances the performance robustness of the resulting adaptive control law. }
Remark 25 By setting us = 0 in (7.34), without using parameter projection and any modi cation to the adaptation law, and taking o the dynamic compensator (i.e., letting Cz = 0; Az = 0; Bz = 0; Dz > 0
in (7.5)), the control law (7.34) reduces to the well-known desired compensation adaptation law (DCAL) by Sadegh and Horowitz [103], which is also implemented for comparison. }
7.2.3 Nonlinear PID Robust Control (NPID) In this subsection, a simple robust control with nonlinear PID feedback structure is designed. We assume that only bounded disturbances appear | i.e., hf in (7.3) is a constant instead of a function of states. The following simple control structure is suggested (7.48) u = fc , (K + 5kx k2) , Kx x where fc is any constant vector that is used to cancel the low frequency component, K > 0 is a s.p.d. matrix, 5 is a positive scalar, and Kx = BT P , in which P is determined by (7.11). For the p.s.d. function V given by (7.35), its derivative is given by (7.38). Substituting control law (7.48) into (7.38) and following similar steps as in (7.39) and (7.41), we have V_ = T [fc , K , 5kx k2 , f0 (q; q;_ q_r ; qr ) , Y (q; q;_ q_r ; qr ) , f~] , 12 xT Q x T [fc , f0(qd; q_d; q_d; qd) , Y (qd; q_d; q_d; qd) , K , 5kx k2 , f~] , 12 xT Qx + 1kkkxk + 2kk2 + 3kk2kxk + 4kxk2kk (7.49) kk [kfc , f0(qd; q_d; q_d; qd) , Y (qd; q_d; q_d; qd) k + hf ] , T K
, 5kxk2kk2 , 12 xT Qx + 6k k2 + 7kx k2 + 2k k2 + 8 k k2 + 9k k2kx k2 + 10kx k2 + 11kx k2 k k2 where 6; 7; 8; 9; 10, and 11 satisfy (7.42). De ne c0(t) = kfc , f0 (qd; q_d; q_d ; qd) , Y (qd; q_d ; q_d; qd) k + hf Noting that c0 is bounded, we can choose the controller parameters K ; Q , and 5 as 2 min (K ) "3 + 2 + 6 + 8 + 4"c(0t) min (Q ) 2("3 + 7 + 10)
5 9 + 11 Then, (7.49) becomes
2
V_ kkc0 , 4c0" kk2p, "3 (k k2 + kx k2) " , ( 2cp0" kk , " , "3(kk2 + kxk2) ,V V + "
where V satis es (7.47). This leads to the following theorem.
(7.50)
(7.51)
(7.52)
78
Theorem 13 If the simple control law (7.48) with controller parameters satisfying (7.51) is applied to the robot manipulator described by Eq. (7.1) with bounded modeling error (7.3), then all signals in the system remain bounded and tracking errors, e(t) and e_ (t), exponentially converge to some balls, the sizes of which are proportional to ". }
Remark 26 By choosing the dynamic compensator as an integrator, x consists of e and
Rt
0 e;
thus, control law (7.48) may be considered as a nonlinear PID feedback control, which is quite easy to implement since it does not require any model information, except some bounds in choosing controller parameters. }
7.2.4 Nonlinear PID Adaptive Control (PIDAC) Feedback gains in the nonlinear PID robust controller are required to satisfy the condition (7.51), in which the lower bounds are not quite straightforward to calculate. Although analytic formulas exist to calculate them, as given in the above development, often the calculated lower bounds are so conservative and so large that they actually may not be used in implementation because of the limited bandwidth of physical systems. Also, the constant feedforward control term fc may not quite match the low frequency component of the feedforward term because of parametric uncertainties. In this subsection, a gain-based nonlinear PID adaptive controller is proposed to solve these diculties. First, choose any Q > 2"3I and thus determine Kx = BT P by (7.11). There exist 7 and
10 satisfying (7.51), and 6 and 11 satisfying (7.42). This means that there exist constant K and 5 such that (7.51) is satis ed. In the following, we do not need to calculate K and 5 , but only need to know their existence. The following control law is suggested:
u = f^c , (K^ + ^5kx k2) , Kx x
(7.53)
Let K be the independent components of K . For example, if we want a diagonal K , K consists of the n diagonal elements only. ^K represents its estimate. Then we can write K = YK () K K^ = YK ( ) ^K (7.54) K~ = (K^ , K ) = YK ( ) ~K ~K = ^K , K where YK ( ) is a known function. The gain adaptation law is chosen as f_^c = ,0f [,,00f (f^c , fc0) , ] ^_ K = ,0 K [,,00 K ( ^K , K 0) + YK ( )T ]
^_ 5 = ,0 [,,00 (^ 5 , 50) + kx k2k k2]
(7.55)
where ,0f ; ,00f ; ,0 K ; ,00 K ; ,0 , and ,00 are any constant s.p.d. matrix or scalars; fc0 ; K 0, and 50 are the corresponding initial estimates. Choose a p.d. function as Va = V + 12 f~cT ,0f ,1f~c + 12 ~KT ,0 K ,1 ~K + 12 ~5T ,0 ,1 ~5 (7.56) where f~c = f^c , fc ; ~5 = ^5 , 5, and V is as de ned by (7.35). Rewrite (7.53) as u = u + f~c , YK ( ) ~K , ~5kx k2 (7.57) u = fc , (K + 5kx k2) , Kx x
79 Noting that ~_ = ^_ with adaption law (7.55), and following similar derivations as in (7.49) and (7.52), we can obtain V_a = T [f~c , YK () ~K , ~5 kx k2 ] + T [u , Y (q; q;_ q_r ; qr ) , f~ + BT P x ] , 21 xT Q x + f~cT [,,00f (f^c , fc0) , ] + ~KT [,,00 K ( ^K , K0) + YK ()T ] +~ 5T [,,00 (^ 5 , 50) + kx k2k k2] 2 " , T [K , ( 2 + 6 + 8 + c40" )In] , xT [ 21 Q , ( 7 + 10)In+nc ]x (7.58) ,[ 5 , ( 9 + 11)]kxk2kk2 , f~cT ,00f f~c , f~cT ,00f (fc , fc0) , ~KT ,00 K ~K , ~KT ,00 K ( K , K0) , ~5T ,00 ~5 , ~5T ,00 ( 5 , 50) " , "3(kk2 + kxk2) , f~cT ,00f f~c , f~cT ,00f (fc , fc0) , ~KT ,00 K ~K , ~KT ,00 K ( K , K0) , ~5T ,00 ~5 , ~5T ,00 ( 5 , 50)
Case I : First, consider the case that the initial estimates K0 and 50 satisfy the condition (7.51) for
fc = fc0 . Since the only condition in choosing fc ; K ; and 5 is that they should satisfy the condition (7.51), we can choose fc = fc0 ; K = K 0 and 5 = 50. In such a case, (7.58) becomes V_ a " , "3(kk2 + kx k2 ) , f~cT ,00f f~c , ~KT ,00 K ~K , ~5T ,00 ~5 (7.59) ,0V Va + " where So,
0V 2
minf"3; min (,00f ); min (,00 K ); ,00 g 1 0,1 0 maxfkM ; max(P ); max(,0, f ); max (, K ); 1=, g
Va exp(,0V t)Va(0) +
Z
0
t
exp(,0V (t , )) ( )d
(7.60)
(7.61)
Case II : Now, consider the general case that the initial estimates K0 and 50 may not
satisfy the condition (7.51). From (7.58): V_a ,f~cT (,00f , 14In )f~c , ~KT (,00 K , 15I ) ~K , ~5T (,00 , 16)~ 5 + 17 ,00V Va + 17 where 14; 15, and 16 are any positive scalars such that 14 < min (,00f );
16 < ,00 , and 00 2 00 2 00 2
17 = " + k,f (f4c ,14fc0 )k + k, K ( 4K 15, K 0 )k + k, ( 45 ,16 50 )k minf"3 ; min (,00f ), 14 ; min (,00 K ), 15 ; ,00 , 16 g 00V 2 max fkM ; max (P ); max (,0,1 ); max (,0,1 ); 1=,0 g
So,
f
Va
K
(7.62)
15 < min (,00 K ); (7.63)
Z t 00 00 exp(,V t)Va(0) + exp(,V (t , )) 17( )d 0
(7.64)
Cases I and II lead to the following theorem by taking (7.56) into consideration.
Theorem 14 If the control law (7.53) with the adaptation law (7.55) is applied to the robot manipulator
described by Eq. (7.1) with bounded modeling error (7.3), all signals in the system remain bounded. Furthermore, A . If the initial estimates K 0 and 50 satisfy the condition (7.51) for fc = fc0 , the tracking errors are bounded by (7.61), i.e., tracking errors exponentially converges to some balls whose sizes are proportional to controller parameter ".
80
B . In general, the tracking errors are bounded by (7.64).
4
Remark 27 The above adaptive controller does not require any model information and has a simple nonlinear PID feedback structure. Thus, it can be easily implemented and costs little computation time, however, bounded disturbances are assumed in the development, and asymptotic stability is not guaranteed even in the presence of parameter uncertainties only. Also, when the initial estimates do not satisfy the condition (7.51), the error bound 17 in (7.64) is not guaranteed to be reduced by suitably choosing controller gains and theoretical performance may not be guaranteed. }
7.2.5 Desired Compensation Adaptive Robust Control with Adjustable Gains (ARCAG) The DCARC scheme in subsection 7.2.2 requires that feedback gains satisfy condition (7.43), which has the same drawback as the nonlinear PID robust control (NPID) scheme, as pointed out in the above subsection. In this subsection, by incorporating a gain-based adaptive control synthesis technique into the design of the DCARC scheme, a new adaptive robust controller is proposed to overcome this diculty. As in the above subsection, choosing any Q > 2"3 I and obtaining Kx = BT P by (7.11), there exist constant K and 5 such that (7.43) is satis ed. Since K and 5 are unknown, instead of using constant feedback gains K and 5 in (7.34), we will adjust them as in the above gain-based adaptive control. The resulting control law is given by
u = ua + h (,hs kk ) (7.65) ua = f0(qd ; q_d ; q_d ; qd) + Y (qd ; q_d; q_d ; qd) ^ , K^ , Kx x , ^5kx k2 in which the parameter adaptation law for is the same as in DCARC, and the gain adaptation laws
are suggested as
^_ K = ,0 K [,,00 K ( ^K , K 0) + YK ( )T ]
^_ 5 = ,0 [,,00 (^ 5 , 50) + kx k2k k2]
Choose a positive de nite (p.d.) function as Vp = V + 12 ~KT ,0 K ,1 ~K + 12 ~5T ,0 ,1 ~5 where V is de ned by (7.35). Rewrite (7.65) as
u = u , YK () ~K , ~5kx k2 u = f0 (qd; q_d ; q_d; qd) + Y (qd; q_d ; q_d; qd ) ^ , (K + 5kx k2) , Kxx + h
(7.66)
(7.67)
(7.68)
and de ne V_ ju as (actually the derivative of V under the control u as shown later)
V_ ju = T [u , f0 (q; q;_ q_r ; qr) , Y (q; q;_ q_r ; qr ) , f~ + BT P x ] , 21 xT Q x
(7.69)
Noting (7.38) and (7.66), we have
V_ p = V_ ju +T [,YK () ~K , ~5 kx k2 ] + ~KT ,0 K ,1 ^_ K + ~5T ,0 ,1 ^_ 5 = V_ ju , ~KT ,00 K ~K , ~KT ,00 K ( K , K 0 ) , ~5T ,00 ~5 , ~5T ,00 ( 5 , 50)
(7.70)
81 Noting that V_ ju has the same form as the V_ in (7.38) with u replaced by u and that u is the same as the control (7.34) used in DCARC with gains satisfying (7.43), all the derivations from (7.38) to (7.46) remain valid if we replace V_ by V_ ju . Thus, in general, from (7.46), (7.71) V_ ju ,"3(k k2 + kx k2) + " and when f~ = 0, from (7.45), V_ ju ,"3 (kk2 + kx k2) + ~T Y T (qd ; q_d; q_d ; qd) From (7.71) and (7.70), V_p ,"3(k k2 + kx k2 ) + " , ~KT ,00 K ~K , ~KT ,00 K ( K , K0) , ~5T ,00 ~5 , ~5T ,00 ( 5 , 50)
(7.72) (7.73)
From (7.73), following similar arguments as in case I and case II of subsection 7.2.4, we have the following theorem.
Theorem 15 If the control law (7.65) with the adaptation law (7.28) and (7.66) is applied to the robot manipulator described by Eq. (7.1), all signals in the system remain bounded. Furthermore,
A. If the initial estimates K 0 and 50 satisfy the condition (7.43), then,
Vp
Z t 0 0 exp(,Vp t)Vp (0) + exp(,Vp (t , ))"( )d 0
(7.74)
minf"3; min (,00 K ); ,00 g 1 0 maxfkM ; max (P ); max(,0, K ); 1=, g
(7.75)
where 0Vp is a scalar satisfying
0Vp 2
and, thus, tracking errors exponentially converge to some balls whose sizes are proportional to the controller parameter ".
B. In general, the tracking errors are bounded by
Vp where
Z t 00 exp(,Vp t)Vp(0) + exp(,Vp (t , )) 18( )d 0
00
2
2
00
18 = " + k, K ( 4K 15, K0)k + k, ( 45 ,16 50 )k f"3 ; min (,00 K ), 15 ; ,00 , 16 g 00Vp 2 maxmin 00 fk ; max (P ); max (,0,1 ); 1=,0 g
(7.76)
K
(7.77)
4
In the following, we will show that this controller can actually do more than what stated in the above theorem, a reasonable assertion in view of the great performance o ered by its counterpart DCARC. We consider the nominal case of no disturbances | i.e., f~ = 0. From (7.72) and (7.70), V_p ,"3(kk2 + kx k2) + ~T Y T (qd ; q_d; q_d; qd ) , ~KT ,00 K ~K (7.78) , ~KT ,00 K ( K , K0) , ~5T ,00 ~5 , ~5T ,00 ( 5 , 50)
82 If the initial gain estimates satisfy the condition (7.43), we can set K = K 0 and 5 = 50. Thus, Vp satis es Assumption 2 (i.e., (4.4)) with given by (7.36) and (7.79) W = ,"3 (k k2 + kx k2) , ~KT ,00 K ~K , ~5T ,00 ~5 Thus, the adaptation law (7.28) guarantees that W ,! 0 and asymptotic tracking is achieved. So we have the following theorem.
Theorem 16 In the absence of disturbances (i.e., f~ = 0), if the initial gain estimates K 0 and 50
4
satisfy condition (7.43), asymptotic tracking is achieved.
7.3 Experimental Set-up Experiments are conducted on the planar UCB/NSK two axis SCARA direct drive manipulator system. The robot (Fig. 7.1) consists of four major mechanical parts, two NSK direct drive motors (Model 1410 for the rst axis with maximum torque 245 Nm and Model 608 for the second axis with maximum torque 39.2 Nm), and two aluminum links. The actual link lengths between the centers of joints are 0.36m and 0.24m respectively. MASS INERTIA LENGTH
4.85 kg 2 0.099kgm 380mm
Link 2
NSK RS 608 MAX TORQUE 39.2Nm MAX SPEED 1.1rps ENCODER RES. 153,600cpr 2 ROTOR INERTIA 0.0077kgm Motor 2
Link 1 MASS INERTIA LENGTH
10.6kg 2 0.565kgm 610mm
Motor 1 NSK RS 1410 MAX TORQUE MAX SPEED ENCORDER RES. ROTOR INERTIA
245Nm 1.1rps 153,600cpr 2 0.267kgm
Figure 7.1: Berkeley/NSK Two-Link Direct-Drive Manipulator Fig. 7.2 shows the experimental set-up. A 486 PC equipped with IBM Data Acquisition and Control Adapters (DACA) board is used to control the entire setup. Each DACA board contains two 12 bit D/A and four 12 bit A/D converters. The three-phase sensor feedback signal is fed through a 10-bit Resolver to Digital Converter (RDC), which provides a motor position resolution of 153,600 pulses per revolution (or 4:09 10,5 rad). The velocity signal is then obtained by the di erence of two consecutive position measurements with a rst-order lter 2 . At each sampling time, based on the digital feedback signals of the position and velocity, the torque control input for each joint is calculated in the 486 PC and sent to the DACA board. The real-time code is written in C language. The analog torque inputs from the DACA board are used to drive each motor through two NSK ampli ers. The NSK motors 2 The robot is equipped with tachometers to measure the joint rotation velocities, which are fed to the 486 PC
through the A/D channels of the IBM DACA board, but the signals are too noisy and not used.
83 IBM DACA BOARD NSK TWO LINK DIRECT DRIVE MANIPULATOR
NSK SERIES 1.5 AMPLIFIER
D/A1 TORQUE CMD A/D1 VELOCITY 1 D/A0 TORQUE CMD
POWER LINE FEEDBACK SIGNALS
486 PC
A/D0 VELOCITY 2
REAL-TIME CONTROL
POSITION DECODER
FEEDBACK SIGNALS POWER LINE
NSK SERIES 1.0 AMPLIFIER
Figure 7.2: Experimental Setup are variable reluctance motors. The ampli ers contain digital communication circuits that convert the torque commands into the necessary three-phase communication current to drive the motors in a torque mode. Multiple poles are used within the motor to produce high torque output. To make such direct drive motors behave like conventional DC motors, internal nonlinear feedback is used. Details of the experimental setup and modeling can be found in [57]. The matrices in dynamic equation (7.1) are given by [57] "
#
M (q) = pp1 ++2pp3CCq2 p2 +pp3 Cq2 3 q2 2 " 2 , p 3 q_2 Sq2 ,p3 (q_1 + q_2 )Sq2 C (q; q_) = p q_ S 0 3 1 q2 G(q) = 0
#
(7.80)
where Cq2 = cos(q2 ); Sq2 = sin(q2 ); p1, p2 , and p3 , the combined robot and payload parameters, are given by p1 = pa1 + 0:194mp; p2 = pa2 + 0:0644mp; and p3 = pa3 + 0:0864mp; respectively, mp is the payload mass, and pa1 = 3:1623; pa2 = 0:1062; and pa3 = 0:17285 are the robot parameters. The friction term Ff (q; q_) is lumped into f~(q; q;_ t) and is bounded by (7.3), where hf = 9. In the experiment, only payload mass mp is unknown with the maximum payload, mpmax = 10kg . Thus, letting = mp and = ( ,0:00001; mpmax + 0:00001), (7.2) can be formed. Since all the controllers are supposed to deal with model uncertainties, the initial estimate of the payload is set to 9kg , with an actual value in experiments being around 1kg . All experiments are conducted at a sampling time T = 1ms.
84
7.4 Experimental Results All schemes presented before were implemented and compared. In addition, Slotine and Li's adaptive algorithm [110] and Sadegh and Horowitz's DCAL [103], which achieves the best tracking performance in the experiments reported by Whitcomb, et al [138], are also implemented for comparison.
7.4.1 Performance Indexes Since we are interested in tracking performance, sinusoidal trajectories with a smoothed initial starting phase are adopted for each joint. In this experiment, the desired joint trajectories are qd = [1:5(1:181 , 0:3343exp(,5t) , cos(t , 0:561)) ; 1:3045 , 0:538exp(,5t) , cos( 34 t , 0:697))]T (rad), which are reasonably fast. Zero initial tracking errors are used and each experiment is run for ten seconds, i.e, Tf = 10s. Commonly used performance measures, such as the rising time, damping and steady state error, are not adequate for nonlinear systems like robots. In [138], the scalar valued L2 norm given R T by L2 [e(t)] = ( T1f 0 f ke(t)k2dt)1=2 is used as an objective numerical measure of tracking performance for an entire error curve e(t). However, it is an average measure, and large errors during the initial transient stage cannot be predicted. Thus, the sum of the maximal absolute value of tracking error of each joint, eM = e1M + e2M , is used as an index of measure of transient performance, in which eiM = maxt2[0;Tf ]fjei (t)jg. The maximal absolute value and the average tracking error of each joint R during the last three seconds are de ned by eiF = maxt2[Tf ,3;Tf ] fjei (t)jg and L[eif ] = 13 TTff,3 jei jdt respectively. Then, eF = e1F + e2F and L[ef ] = L[e1f ] + L[e2f ] are used as indexes to measure the R steady state tracking error, The average control input of each joint, L[ui ] = T1f 0Tf jui jdt, is used to evaluate the amountPof control e ort. The average of control input increments of each joint is de ned 10000 ju (kT ) , u ((k , 1)T )j. The sum of the normalized control variations 1 by L[ui] = 10000 i k=1 i P2 L[ui ] of each joint, cu = i=1 L[ui ] , is used to measure the degree of control chattering.
7.4.2 Controller Gains The choice of feedback gains is crucial to achieve a good tracking performance for all controllers. A discussion of the gain tuning processes for each controller follows in detail. In general, the larger the feedback gains (especially, the gain K ), the smaller the tracking errors. However, if the gains are too big, the robot will be subject to severe control chattering and a large noisy sound can be heard. After the gains exceed certain limits, the structural resonance is excited because of severe control chattering and the system goes unstable. Thus, in order to achieve a fair comparison, we will try to tune gains of each controller such that the tracking errors of each controller are minimized while maintaining the same degree of control chattering for all controllers.
ASMC: Adaptive Sliding Mode Control .
As explained in Remark 21 in Section 7.2.1, a dynamic compensator (nc = 2) is formed by (7.5), in which Az = 0I2 ; Bz = 400I2; Cz = I2 ; Dz = 40I2 with initial values calculated on-line by (7.12). Such a choice of gains guarantees that the resulting sliding mode is critically damped with corner frequency w = 20.
85 The adaptation law is given by (7.28) where , = 10. Thus, ^ = and h (q; q;_ q_r ; qr ) can be determined by (7.15). The control law is then formed by (7.17), in which K = diag f40; 5g and us is determined by (7.22) where Ks = diagf60; 4g; "1 = 1, "2 = 0:5, and = 300.
SLAC: Slotine and Li's Adaptive Algorithm .
The control law is formed as explained in Remark 22 in Section 7.2.1, in which Dz = 20I2 is used to provide the same corner frequency w for the sliding mode as in ASMC. A large K = diag f180; 15g is used to produce roughly the same degree of control chattering as ASMC. This gain is slightly larger than the combined feedback gain for , K + Ks in ASMC.
DCARC: Desired Compensation Adaptive Robust Control . The same dynamic compensator as ASMC is used. Letting Q = diag f105; 104g, P is calculated from (7.11) and the resulting gain matrix Kx is [120; 0; 1250; 0; 0; 12; 0; 125]. The control law is given by (7.34), in which K = diag f100; 8g and 5 = 1000. us is given by (7.22), in which Ks = diag f60; 4g; "1 = 1; "2 = 0:5, = 200, and hs is calculated by (7.33). The adaptation law is given by (7.28) with given by (7.36) and , = 10. DCAL: Sadegh and Horowitz's Desired Compensation Adaptation Law .
The control law is formed as explained in Remark 25 in Section 7.2.2, in which Dz = 20I2 as in SLAC. By using the same Q as in DCARC, the resulting Kx is [0; 0; 2500; 0; 0; 0; 0; 250]. A large K = diag f170; 14g is used to produce roughly the same degree of control chattering as DCARC and the rest of controller parameters are the same as in DCARC.
DCRC: Desired Compensation Robust Control .
The control law is the same as in DCARC except not to use the adaptation law. In such a case, the proposed DCARC reduces to a robust control (termed as DCRC(I) in the following). To verify the e ect of using a dynamic compensator, the same control law is applied, but without using the dynamic compensator, i.e., without the integrator, which is obtained by setting Cz = 0; Az = 0; Bz = 0; Dz = 20I . Correspondingly, Kx = [0; 0; 2500; 0; 0; 0; 0; 250] by using the same Q (termed DCRC(NI) in the following).
NPID: Nonlinear PID Robust Control .
The control law is given by (7.48) with the same 5 and Kx as in DCRC. fc = 0. A large K = diag f160; 12g is used. NPID(I) stands for integrator case and NPID(NI) for no integrator case as in DCRC.
PIDAC: Nonlinear PID Adaptive Control .
The control law is given by (7.53) with the same Kx as DCRC and a diagonal K = diag f K 1; K 2g. The gain adaptation law is given by (7.55), where K 0 = [10; 1]T , 50 = 1000, fc0 = 0; ,f 0 = diagf10; 2g; ,f 00 = diagf0:1; 0:1g; ,0 K = diag f1000; 10g; ,00 K = diag f0:0002; 0:02g; ,0 = 104, and ,00 = 2 10,5 .
ARCAG: Desired Compensation Adaptive Robust Control with Adjustable Gains
86 The control law is given by (7.65) with the same Kx, us , and the parameter adaptation law as DCARC and a diagonal K = diag f K 1; K 2g. The gain adaptation law is given by (7.66) where K 0 = [20; 3]T , 50 = 500. ,0 K = diag f1000; 10g, ,00 K = diag f0:00003; 0:003g, ,0 = 104, and ,00 = 10,4 .
7.4.3 Comparative Experimental Results As in [138], we rst test the reliability of the results by running the same controller several times. It is found that the standard deviation of the error from di erent runs is negligible. The experimental results are shown in the following table (unit is rad for tracking errors and Nm for control input torques).
Table 7.1: Experimental Results Controller eM eF L[ef ] L2 [e] L[u1] ASMC 0.0301 0.0167 0.0058 0.0058 32.1 SLAC 0.0520 0.0325 0.0160 0.0133 32.8 DCARC 0.0201 0.0134 0.0039 0.0039 30.6 DCAL 0.0353 0.0199 0.0092 0.0081 30.3 DCRC(I) 0.0256 0.0227 0.0077 0.0081 30.3 DCRC(NI) 0.0690 0.0486 0.0175 0.0209 29.6 NPID(I) 0.0202 0.0195 0.0066 0.0061 30.5 NPID(NI) 0.0386 0.0346 0.0151 0.0145 29.8 PIDAC 0.0705 0.0158 0.0057 0.0070 30.4 ARCAG 0.0364 0.0119 0.0035 0.0045 30.2
L[u2] cu 6.2 6.2 6.4 6.3 6.3 6.1 6.4 6.3 6.3 6.3
0.54 0.55 0.41 0.43 0.42 0.40 0.41 0.40 0.44 0.42
The above results are also displayed in Fig. 7.3 to Fig. 7.4. Based on the above experimental data, the following general results can be concluded: a . Parameter Adaptation Improves Tracking Accuracy If we compare the parameter-based adaptive controllers with their robust counterparts, i.e., DCARC versus DCRC(I), DCAL versus DCRC(NI), then we can see that, in terms of both nal tracking accuracy (Fig. 7.4) and average tracking errors (Fig. 7.5), parameter adaptation reduces the tracking errors around a factor of 2. The parameter-based adaptive controllers also have better transient performance (Fig. 7.3). The improvement comes from the fact that the estimated payloads approach their true values, which is shown in Fig. 7.8. This result veri es the importance of introducing parameter adaptation. All controllers use almost the same amount of control e ort and have the same degree of control chattering, as shown in Fig. 7.6 and Fig. 7.7, and thus the comparison is fair. b . Dynamic Compensator Improves Tracking Accuracy Comparing the controllers having dynamic compensators with their counterparts not employing dynamic compensators, i.e., DCRC(I) versus DCRC(NI) and NPID(I) versus NPID(NI), we can see that introducing dynamic compensators reduces the tracking errors by more than a factor of 2 in terms of all the performance indexes, as shown in Fig. 7.3 to Fig. 7.5. The comparison is
0.04
0.035
Figure 7.4: Final Tracking Accuracy
ARCAG
PIDAC
NPID(NI)
NPID(I)
DCRC(NI)
DCRC(I)
DCAL
DCARC
SLAC
ASMC
Final Tracking Error (rad)
ARCAG
PIDAC
NPID(NI)
NPID(I)
DCRC(NI)
DCRC(I)
DCAL
DCARC
SLAC
ASMC
Maximum Tracking Error (rad)
87
0.08
0.07
0.06 eM
0.05
0.04
0.03
0.02
0.01
0
Figure 7.3: Transient Performance
0.05
0.045
eF L[ef]
0.03
0.025
0.02
0.015
0.01
0.005
0
Figure 7.6: Control E ort ARCAG
PIDAC
NPID(NI)
NPID(I)
DCRC(NI)
DCRC(I)
DCAL
DCARC
SLAC
ASMC
Tracking Error (rad) 0.02
ARCAG
PIDAC
NPID(NI)
NPID(I)
DCRC(NI)
DCRC(I)
DCAL
DCARC
SLAC
ASMC
Control Input (Nm)
88
0.025
L2[e]
0.015
0.01
0.005
0
Figure 7.5: Average Tracking Errors
35
30
25
20 L[u1] L[u2]
15
10
5
0
89
0.6 Cu
0.5 0.4 0.3 0.2 0.1
PIDAC
ARCAG
NPID(NI)
NPID(I)
DCRC(NI)
DCRC(I)
DCAL
DCARC
SLAC
ASMC
0
8
9
Figure 7.7: Control Chattering
10
8
Solid: DCARC Dashed: DCAL
Dashdot: ASMC Dotted: SLAC
Estimated Payload
6
4
2
0
−2 0
1
2
3
4
5 Time
6
7
10
Figure 7.8: Estimated payloads approach their true values
90 fair, as shown by the control e ort in Fig. 7.6, and the degree of control chattering in Fig. 7.7. This result supports the importance of employing proper controller structure. c . Desired Compensation Improves Tracking Accuracy Comparing the controllers having desired compensation with their counterparts using actual state in model compensation design, i.e., DCARC versus ASMC and DCAL versus SLAC, we can see that, in terms of all performance indexes (Fig. 7.3 to Fig. 7.5), the controllers with desired compensation have a better tracking performance. They also have a less degree of control chattering, as shown in Fig. 7.7. d . Gain-based Adaptive Controllers via Robust Controllers If we compare the gain-based adaptive controllers with their robust counterparts, i.e., PIDAC versus NPID(I) and ARCAG versus DCARC, we can see that gain-based adaptive controllers can have a large stability margin for the choice of feedback gains since they can use small initial gain estimates. Because of the small initial estimates, they have larger initial tracking errors or poorer transient response, as seen from Fig. 7.3. The estimated feedback gains (e.g., K^ shown in Fig. 7.9) increase quickly to some values that are slightly larger than the xed feedback gains used in their robust counterparts (e.g., when t = 10s, K^ (t) = diag f180; 12:6g for PIDAC but K = diag f160; 12g for NPID(I)). This is the reason that they achieve a slightly better nal tracking accuracy, as shown in Fig. 7.4. We should keep in mind, however, that this advantage comes from the slightly increased degree of control chattering, as shown in Fig. 7.7. Therefore, in practice, gain-based adaptive controllers do not o er any advantage in improving tracking performance. They may be used in the initial gain-tuning process to obtain the lower bound of the stabilizing feedback gains instead of using a troublesome and conservative theoretical formula like (7.51). However, caution should be taken. Large dampings (e.g., ,00 K and ,00 in (7.55)) should be used; otherwise, the resulting nal estimates may be too big that they may exceed the practical limits and destabilize the system because of their gain adaptation nature. Since the proposed DCARC possesses all the desirable good qualities | parameter adaptation, dynamic compensator, and desired compensation | it is natural that it achieves the best tracking performance, as seen from Fig. 7.3 to Fig. 7.5, by using the same amount of control e ort (Fig. 7.6) and control chattering (Fig. 7.7). These facts show again the importance of using the both means, parameter adaptation and proper controller structure, in designing high performance controllers, which is the main theme of the proposed ARC. Using either one of them alone is not enough | in fact, in these experiments, probably because the e ect of link dynamics is not so severe and the disturbances and measurement noise are not so small, the simple NPID robust controller out-performs DCAL, the adaptive controller that achieves the best tracking performance among existing adaptive controllers. The tracking errors of DCARC are plotted in Fig. 7.10 and the control inputs are shown in Fig. 7.11. Those spikes of the tracking errors after the initial transient occur at the time when the joint velocities change their directions. Thus, they are mainly caused by the discontinuous Columb friction.
7.5 Conclusions In this chapter, the proposed ARC is applied to the trajectory tracking control of robot manipulators. Two schemes are developed: ASMC is based on the conventional adaptation structure and
91 200 180
Estimated Feedback Gains
160 140 120 100 80 Solid: Kxi1 (PIDAC) 60
Dashdot: Kxi1 (ARCAG)
Dashed: Kxi2 (PIDAC)
Dotted: Kxi2 (ARCAG)
40 20 0 0
1
2
3
4
5 Time
6
7
8
9
10
Figure 7.9: Estimated Feedback Gains K^ DCARC is based on the desired compensation adaptation structure. A dynamic sliding mode is used to enhance the system response. In addition, several conceptually di erent robust and adaptive controllers are also constructed for comparison | a simple nonlinear PID type robust control, and a simple gainbased adaptive control, which requires almost no model information, and a combined parameter and gain-based adaptive robust control. All algorithms, as well as two existing adaptive control algorithms, SLAC and DCAL, are implemented on a two-link SCARA type robot manipulator. Comparative experimental results show the importance of using the both means, proper controller structure and parameter adaptation, in designing high performance controllers. It is observed that in these experiments, the proposed DCARC achieves the best tracking performance without increasing control e ort.
92
0.02
0.015 Solid: Joint 1
Dashed: Jpint 2
Tracking Errors (rad)
0.01
0.005
0
−0.005
−0.01
−0.015 0
1
2
3
4
5 Time
6
7
8
9
10
Figure 7.10: Joint Tracking Errors 100 Solid: Joint 1
Dashed: Jpint 2
80 60
Joint Torque (Nm)
40 20 0 −20 −40 −60 −80 0
1
2
3
4
5 Time
6
7
8
9
Figure 7.11: Joint Control Torque
10
93
Chapter 8
Other Applications 8.1 Constrained Motion and Force Control of Robot Manipulators In this section, the proposed ARC is applied to solve the motion and force control of constrained robot manipulators.
8.1.1 Dynamic Model of Constrained Robots When the robot end-e ector comes in contact with its environment, interaction forces/moments develop between the end-e ector and the environment. In a Cartesian coordinate system, let x 2 Rn0 denote the vector of the position/orientation of the robot end-e ector and F 2 Rn0 the vector of interaction forces/moments on the environment exerted by the robot at the end-e ector. Forces are decomposed along the Cartesian axes and moments are decomposed along the rotation axes de ning the angles of the orientation, which may not be orthogonal. For example, the three axes de ning the three Euler angles are not orthogonal. To account for the e ect of the interaction forces, dynamic equation (7.1) is modi ed to [81] M (q; )q + C (q; q_; )q_ + G(q; ) + f~(q; q;_ t) + J T (q)F = u (8.1) where J (q ) = @x(q )=@q 2 Rn0 n is the Jacobian matrix. In this section, it is assumed that the robot is nonredundant 1 (i.e., n0 = n) and the position, velocity, and constrained force measurements are all available. J is assumed to be nonsingular in a nite work space q . The robot end-e ector in contact with rigid constraint surfaces is considered. It is assumed that the end-e ector is initially in contact with the constraint surfaces, and the control exercised over the constrained force is such that the force will always hold the end-e ector on the constraint surfaces. Later, we will show how to choose controller parameters to satisfy this assumption. Suppose that the environment is described by a set of m rigid hypersurfaces [81, 74, 163]
(x) = 0 (x) = [ 1 (x); : : :; m (x)]T m n (8.2) which are mutually independent, and i (x) is assumed to be twice di erentiable with respect to x. The interaction force F can be written as
F = Fn + Ft = DT (x) + At ft (; vend; );
D(x) = @ (@xx)
1 The assumption of the robot being nonredundant can be easily removed as shown in the next section
(8.3)
94 where 2 Rm is a vector of Lagrange multipliers associated with the constraints which usually represent normal contact force components, Fn = DT (x) represents the constraint force (i.e., the normal contact force in the Cartesian space), and Ft = At ft (; vend ; ) is the vector of friction forces, the directions of which are speci ed by At , the unit tangent directions of the surfaces, with opposite sign to the end-e ector velocity vend . The magnitude ft (; vend ; ) is linearly proportional to the normal contact force Fn or . Thus, we can write
Ft = [LT (; x; x_ ) + L~ Tf (x; x_ )];
L ; L~ f 2 Rmn
(8.4)
in which LT (; x; x_ ) is used to describe the modeling part of the friction{L is linear with respect to the unknown friction coecients 2 Rk with known shape function{ and L~ Tf (x; x_ ) represents the modeling error. In general, L and L~ f are di erentiable except at points where vend changes direction on the surfaces, i.e., vend = 0. Those points are not considered. In the assumption of frictionless contact surfaces, i.e., Ft = 0, (8.3) reduces to the form given by [74, 76]. When motion of the robot is constrained to be on the surfaces (8.2), only (n , m) coordinates of the position vector can be speci ed independently [81, 142]. Control of all position coordinates of the robot is unnecessary, and only (n , m) position coordinates need to be controlled in the constrained motion of the robot. Therefore, motion control is in the (n , m) mutually independent curvilinear coordinates, (x) = [ 1(x); : : :; n,m (x)]T . (x) are assumed to be twice continuously di erentiable and independent of (x) in the nite workspace q . Thus, once (x) is regulated to the desired value d (t), combining with the constraints (8.2), the con guration of robot is uniquely determined. The generality of choosing (x) gives us great exibility in implementation. It can be selected as some joint angles qi , some end-e ector coordinates xi , or some task space coordinates, in which the resulting controller will be implemented in the joint space, Cartesian space, or task space respectively. For example, since D(x) is of full rank m, without the loss of generality, we can assume that the rst m columns of D(x) are independent. In this case, we can choose (x) = [xm+1 ; : : :; xn]T . De ne a set of curvilinear coordinates as [143, 163]
r = [rfT ; rpT ]T
rf = [ 1 (x); : : :; m(x)]T rp = [ 1(x); : : :; n,m (x)]T
(8.5)
Di erentiating (8.5), we obtain
r_ = Jxx_ = Jq q_
where
(8.6)
T ]T Jx = [D(x)T Jxp Jx = @r@x(x) @r (x(q)) J = J (x(q ))J (q ) Jq = @q q x
x) (n,m)n Jxp = @ ( @x 2 R (8.7) Jq ; Jx 2 Rnn Using transformations (8.5) and (8.6) in (8.1) and multiplying both sides by Jq,T , the dynamic equation (8.1) with the constraints (8.2) and the interaction force (8.3) can be expressed in terms of r as
M (r;" )r +# C (r; r_; )"r_ + G(#r; ) + B 0 (; r; r_) + f~r (r; r_; ; t) = ur r = r0 B 0 = I0m + B(; r; r_ ) p
(8.8)
95 or where
M12(r; )rp + C12(r; r_; )r_p + G1(r; ) + (Im + B1 ) + f~r1 (r; r_; ; t) = ur1 M22(r; )rp + C22(r; r_; )r_p + G2(r; ) + B2 + f~r2(r; r_ ; ; t) = ur2 "
(8.9)
#
11(r; ) M12(r; ) = Jq,T (q )M (q; )Jq,1(q ) M (r; ) = M M ( r; ) M ( r; ) " 21 # 22 C 11 C12 C (r; r_ ; ) = C C = Jq,T C (q; q;_ )Jq,1 , Jq,T M (q; )Jq,1 J_q Jq,1 21 " #22 G ( r; ) G(r; ) = G1(r; ) = Jq,T (q)G(q; ) "2 # B 1 B(; r; r_) = B = Jx,T LT (; x; x_ ) "2 # f r 1 ~ fr (r; r_; ; t) = f = f~rp (r; r_; t) + F~r (r; r_ ; t)
(8.10)
r2 , T f~rp = Jq (q )f~q (q; q;_ t) F~r ="Jx,T L~#Tf (x; x_ ) ur = uur1 = Jq,T (q )u r2
In (8.8), the constraints are simply described by rf = 0. The robot motion is thus uniquely determined by the coordinates rp. Also, the constraint force Fn has a simple structure in the new coordinate system, i.e., Jq,T Fn = [Im 0]T . In the absence of the surface friction forces and the unknown nonlinear functions, B1 = 0; B2 = 0, and f~r = 0. The constraint force does not appear in the second equation of (8.9). Therefore, motion control can be designed based on the reduced order equation without considering force control. This is the basic strategy adopted by most previous researchers in this area [74, 133, 76, 58, 163, 168, 7, 19, 36]. Clearly, in the presence of the surface friction forces, motion and force equations are coupled and a new strategy should be adopted. Let Kf = diag f kf 1; : : :; kfm g and Gf = diag f gf 1; : : :; gfm g be constant diagonal matrices with kfi > 0 and gfi 0; i = 1; : : :; m. By adding Gf to both sides of the rst equation of (8.9), adding and subtracting M21(r; )Kf to the right hand of the second equation of (8.9), and noting = Kf,1 Kf , Eq. (8.9) can be rewritten in a concise form as
H (rp; )v + Ch(rp; r_ p; )r_ + G(rp; ) + Bm (; ; rp; r_p) + f~r (rp ; r_ p; ; t) = ur + G f where
(8.11)
96 "
#
v = Krf p " ,1 M12 (r; ) # ( I + G ) K m f f H (rp; ) = M (r; ) # M22(r; ) " 21 Ch(rp; r_p; ) = C0 CC12 21
22
Bm (; ; rp; r_ p) = B (; r; r_) + Bm0 (rp; ) "
G f = G0f
#
Bm0 (rp; ) = ,
"
0
#
(8.12)
M21(r; )Kf
Equation (8.11), which possesses some nice properties introduced in the following, is the basic equation for our controller design. The physical meaning of introducing Kf and Gf in (8.11) will become apparent later in the controller design. The following properties are obtained for Eq. (8.11) in Appendix 2.
Property 4 . For the nite work space q in which Jq is nonsingular, H (rp; ) is a s.p.d. matrix for suciently small max(Kf ) = maxi fkfig. Furthermore, for max(Kf ) k1r00 , we have kr0 In max (Gf ) . 2 H (rp; ) kh00In where kh00 = kr00 , kr0 + 1+min (Kf ) Property 5 . The matrix Nh (rp; r_p; ) = H_ (rp; ) , 2Ch(rp; r_p; ) is a skew-symmetric matrix. Property 6 H (rp; ); Ch(rp; r_ p; ); G(rp; ); and Bm (; ; rp; r_ p) are linear w.r.t. the combined robot parameters and surface friction coecients, c = [ T ; T ]T 2 Rk where k = l + k , i.e., H (rp; )zv + Ch(rp; r_p; )zr + G(rp; ) + Bm (; ; rp; r_p) (8.13) = fc (rp; r_ p; zr ; zv ; ) + Yc (rp; r_ p; zr ; zv ; ) c where zr and zv are any reference values, and fc and Yc are known. Noting that f~r (rp; r_ p; ; t) is linear w.r.t , we make the following assumption: (8.14) jf~r (rp; r_p; ; t)j (rp; r_p; ; t) = p(rp; r_p; t) + (rp; r_p; t)kk where p (rp; r_ p; t) and (rp; r_ p; t) are known functions. Let rpd (t) = (x(qd (t))) 2 Rn,m be the desired robot motion trajectory and d(t) 2 Rm be the desired constrained force trajectory, which are suciently smooth. Let ep (t) = rp(t) , rpd(t) and ef (t) = (t) , d (t) be tracking errors of the motion and constrained force respectively. The constrained
motion and force control problem can now be stated as follows, under parametric uncertainties and the modeling error (8.14), design a control law for the actuator torque u or ur such that the robot manipulator described by (8.11) is stable and the motion and the constrained force of the robot track their desired values as close as possible.
8.1.2 Adaptive Robust Control of Constrained Manipulators In this subsection, by using a dynamic sliding mode, ARC of constrained manipulators is presented. 2 kr0 and kr00 are some positive constants de ned in Appendix 2
97
Dynamic Motion Sliding Mode The same strategy as in Chapter 7 is used to design a dynamic motion sliding mode controller. The dynamic motion sliding mode controller is designed to make the following quantity remain zero.
p 2 R(n,m)
p = e_p + yp ;
(8.15)
where yp is the output of a np -th order dynamic compensator given by
z_p = Ap zp + Bp ep; zp 2 Rnp (8.16) yp = Cpzp + Dpep ; yp 2 R(n,m) Constant matrices (Ap ; Bp; Cp; Dp) are chosen in the same way as in Chapter 7 to guarantee that the resulting motion sliding mode has prescribed good qualities. The initial value zp (0) of the dynamic compensator (8.16) is chosen as
Cpzp(0) = ,e_p (0) , Dp ep (0)
(8.17)
when Cp has rank n , m.
Dynamic Force Sliding Mode In (8.11), the relationship between the constraint force and the control input ur is static instead of dynamic. This static relationship poses some diculties in the dynamic control of the constraint force since the force tracking error eRf cannot be used to form force switching functions. In [149], the integral of force tracking error, If = 0t ef ( )d , was used to form force switching functions. Instead of controlling the constraint force directly, we stabilize If to control the constraint force indirectly. Here, in order to broaden the generality of the force controller and possibly to speed up the force response in the sliding mode, we also use the following ltered force tracking error in forming force switching functions: (8.18) z_f = ,Af zf + Af ef ; zf 2 Rm ; Af 2 Rmm where Af = diag f f 1; : : :; fm g is any s.p.d. diagonal matrix. The force switching functions are design as (8.19) f = Cf zf + Df If f 2 Rm; Cf ; Df 2 Rmm where Cf and Df are any p.s.d. diagonal matrices satisfying Cf Af +Df = Kf . Let Cf = diag fcf 1; : : :; cfmg with cfi 0 and Df = diag fdf 1; : : :; dfmg with dfi 0. Transfer function from f to If and zf are
If = Gf (s)f zf = Gzf (s)f
1
s+1 Gf (s) = diag f k fifis+d ; i = 1; : : :; mg fi fi s Gzf (s) = diag f kfi s+d ; i = 1; : : :; mg fi fi
(8.20)
From (8.20), both Gf (s) and Gzf (s) are stable. Thus during the force sliding mode, the ltered force tracking error zf and the integral force tracking error If will converge to zero. When Cf = 0, Df = Kf . The force switching functions (8.19) reduce to those used in [149], in which only the integral of force tracking errors is used in forming switching functions. In such a case, Gf (s) = diag f k1fi ; i = 1; : : :; mg, and Gzf (s) = diag f kfi( 1s s+1) ; i = 1; : : :; mg, which are stable. fi
98
Adaptive Robust Control Law By using the dynamic force sliding mode, the constrained force is regulated indirectly by controlling If and the ltered force tracking error zf . The state of the entire system thus includes the state variables of the original dynamics (8.11), the state variables of the dynamic compensators (8.16) and (8.18) and If , i.e., (8.21) x = [rpT ; r_pT ; zpT ; zfT ; IfT ]T From (8.15) and (8.19), the switching functions and their derivatives are "
#
"
#
= f = r_0 , zr p p _ = v , zv
(8.22)
where v is de ned in (8.12) and the reference velocity zr and the acceleration zv are given by "
#
"
#
zr (x) = zzfr = r_ ,,f y p " pr # " pd z K + zv (x) = zvf = f rd ,Cfy_Af zf vp pd p
#
(8.23)
Note that zv 6= z_r as opposite to the case in adaptive motion control [109, 103, 51, 89, 154]. Both zr and zv are calculable feedback signals. Let hs be a bounding function satisfying
hs (x; ^c; ; t) sup 2 fkYc (rp; r_p; zr ; zv ; ) ~c , f~r (rp; r_ p; ; t)kg
(8.24)
For example, let
hs = kY (rp; r_p; zr ; zv ; )k cM + p (rp ; r_p; t) + (rp; r_ p; t)kk
(8.25)
where cM = k( cmax , cmin + "c )k. By the de nition (8.13), Yc is linear w.r.t. . Thus, from (8.24) or (8.25), hs can be a linear function of kk, i.e.,
hs = hp(x; ^c ; t) + h(x; ^c ; t)kk for some positive functions hp and h. The control torque is suggested to be ^ v + C^h zr + G^ + (B^m , G f ) , K + h(,hs kk ) ur = Hz = fc (rp; r_ p; zr ; zv ; ) + Yc (rp; r_p; zr ; zv ; ) ^c , G f , K + h (,hs kk )
(8.26)
(8.27)
where K is a s.p.d. matrix, , zr ; and zv are de ned by (8.22) and (8.23), respectively, and hbar is a continuous approximation of the ideal SMC control ,hs kk with an approximation error ". Substituting the control law (8.27) into (8.11) and noting (8.22), the error dynamics are obtained as ~ v + C~h zr + G~ + B~m , f~r (rp; r_ p; ; t) , K + h H _ + Ch = Hz (8.28) = Yc (rp; r_p; zr ; zv ; ) ~c , f~r (rp; r_ p; ; t) , K + h
99 Noting Property 4, a p.s.d. function is chosen as V = 21 T H (rp; ) (8.29) with 1 k0 k k2 V 1 k00k k2 (8.30) 2 r 2 h From Property 5, 12 T H_ (rp; ) = T Ch (rp; r_ p; ) . Noting (8.28), di erentiating V with respect to time yields _ = T H _ + T Ch V_ = T H _ + 21 T H (8.31) T = [Yc (rp; r_p; zr ; zv ; ) ~c , f~r ] , T K + T h Noting (8.14), (8.24), and condition ii of (7.18),
V_ kkhs , T K + T h ,V V + "
(8.32)
where V is a positive scalar satisfying
V mink(00K ) h
(8.33) n
q
o
Noting (8.30), k k exponentially converges to a known set : k (1)k 2k"r0max V , and the expoq nentially converging rate V and the bound of the nal tracking errors, 2k"r0max V , can be freely adjusted by the controller parameters " and K in a known form. Since the sliding mode designed by (8.15) and (8.20) are exponentially stable with predetermined transient performances, ep ; e_ p; zp; zf , and If exponentially converge to some known sets whose size can be freely adjusted by the controller parameters in a known form. Therefore, the system is exponentially stable at large with a guaranteed transient performance. In the above, we have shown the exponential stability of the state x. However, we have not shown that the force tracking error ef is bounded and thus the control torque may be in nite or illde ned as it contains . In the following, this condition will be examined as it reveals some consequences of the causality problem important in the force control of constrained motion. The relationship between the constraint force and the control input is static and a small integral force tracking error or the ltered force tracking error does not necessarily mean a small force tracking error. This point has been neglected by most previous researchers. From (8.19) and (8.18), _f = Kf ef , Cf Af zf . Noting the de nition of H in (8.12) and lumping the terms containing or ef together, Eq. (8.28) can be rewritten as "
where
Im + Gf M12 M21 Kf M22
#"
ef _p
#
= w(x; t; c ; ^c ) + B~F (rp; r_p; ~c )ef + h
(8.34)
B~F (x; t; ~c) = B~m" (rp; r_p; ^c ) , F~r#(rp; r_ p; t) ,1 ~ v + C~h zr + G~ w(x; t; c; ^c )= (Im +MGf )Kf Cf Af zf , Ch + Hz 21 ,f~rp , K + B~F d
(8.35)
100 Multiplying both sides of (8.34) by MI = [Im ; ,M12 M22,1 ] , we can eliminate _p to obtain ef : (8.36) [Im + Gf , M12M22,1 M21 Kf ]ef = MI [w(x; t; c ; ^c ) + B~F ef + h]
If the above equation has a nite solution ef all the time, the control torque (8.27) is nite since all terms except ef are bounded as shown before. Noting (8.26), in general, h is a function of also since it is an approximation of ,hs kk . Thus, it is not so easy to gain insight into how to choose controller parameters to guarantee Eq. (8.36) to have a nite solution ef . However, noting that Im + Gf , part of the coecient matrix of ef at the left hand side of Eq. (8.36), is an s.p.d. matrix, we proceed in the following way. Since h is an approximation of ,hs kk , we can assume that
khk p() + hs
(8.37)
for some non-negative function p( ) 0. For example, for the smooth approximation like (7.20), p( ) = 0, and for the continuous approximation like (7.22), p() = kKs k. Noting (8.36 ), (8.26), and (8.37), we have
min (Im +Gf )kef k2 eTf (Im + Gf ) ef = eTf M12M22,1Mh21Kf ef + eTf MI [w + B~F ef + h ] kM12M22,1 M21Kf kkef k2 + kef k kMI wk + kMI B~F kkef k +kMI k(p( ) + hp + h k (kef k + kdk)] De ne If
(8.38)
=min(Im + Gf ) , kM12 M22,1M21Kf k , kMI B~F k , kMI kh
(8.39)
"
(8.40)
kef k 1 fkMI wk + kMI k (p() + hp + hkdk)g
(8.41)
for some " > 0, then, from (8.38),
Thus kef k is bounded since the right hand side of (8.41) depends on the bounded state x only. How to choose controller gains to guarantee (8.40) will be given later. In the above, we have shown that the suggested control law guarantees a certain transient performance and nal tracking accuracy with a bounded control. In the following, we will show that if the dynamic model is accurate, i.e., in the presence of parametric uncertainties only (f~r = 0 in Eq. (8.8)), asymptotic motion and force tracking can be obtained through parameter adaptation without using a high-gain in the feedback loop. From (8.31) and condition ii of (7.18), (8.42) V_ = T Yc (rp; r_p; zr ; zv ; ) ~c , T K Thus, Assumption 2 (4.4) is satis ed for W = T K and
= YcT (rp; r_p; zr ; zv ; ) Since V is not a function of ^c , we can use an adaptation law like (4.16), i.e., ^_ = ,,[l ( ^ ) + ] c
c
(8.43) (8.44)
101 where the adaptation function is given by (8.43) and l satis es the conditions like (4.17). Then, asymptotic output tracking can be obtained, i.e., ,! 0. Thus, x ,! 0. Since x_ is bounded, x is uniformly continuous. So, all terms in the right hand side of (8.41) are uniformly continuous, and thus ef is uniformly continuous. Since If ,! 0, by applying Barbalat's lemma, ef ,! 0. The above results can be summarized in the following theorem.
Theorem 17 For the constrained robot manipulator described by Eq. (8.11) with the modeling error (8.14), in the nite workspace q , with a suciently small Kf and (8.40) being satis ed, the following results hold if the control law (8.27) with the adaptation law (8.44) is applied:
a). In general, the system is exponentially stable at large with a guaranteed transient performance and nal tracking accuracy, i.e., ep ; e_ p; zp; zf and If exponentially converge to some balls whose sizes can be freely adjusted by controller parameters in a known form. The control is bounded and ef is bounded by (8.41).
b). In the presence of parametric uncertainties only, i.e, f~r = 0, the system is asymptotically stable in the sense that ep ; e_ p; zp; zf ; If ,! 0. Furthermore, ef ,! 0, i.e., asymptotic force tracking is
4
achieved.
Remark 28 There are several ways to guarantee > 0 as required by (8.40). We classify them in the following three cases:
Case 1 . Consider the case that the friction coecient vector is known and there is no contact friction modeling uncertainty (i.e., L~ f = 0). From (8.10), F~r = 0 and f~r = f~rp (rp; r_p; t). Thus we can set = 0 in (8.14). By setting ^ = , we have, B~ = 0. From (8.12), B~m = B~m0 (rp; ~ ). Noting that the only term containing in Yc ~c is B~m , from (8.24) and (8.26), we can choose h = sup kM~ 21kkKf k sup kB~m0 k = sup kB~m k (8.45) 2
2
2
From (8.39),
min (Im + Gf ) , kM12 M22,1 M21kkKf k , 2kMI k sup 2 fkM~ 21kgkKf k
(8.46)
By choosing a small weighing matrix Kf such that
max(Kf )
