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10.8 Designs for nonlinear dynamics The adaptive inverse approach developed in Sections 10.4±10.6 can also be applied to systems with nonsmooth nonlinearities at the inputs of smooth nonlinear dynamics [16, 17]. State feedback and output feedback adaptive inverse control schemes may be designed for some special cases of a nonlinear plant xtf xt gxtut; utNvt ythxt 10:128 where f xPR n , gxPR n , and hxPR are smooth functions of x P R n , NÁ represents an unknown nonsmooth actuator uncertainty as a dead-zone, backlash, hysteresis or piecewise-linear characteristic, and vtPR is the control input, while ut is not accessible for either control or measurement. The main idea for the control of such plants is to use an adaptive inverse vt NIu d t 10:129 to cancel the eect of the unknown nonlinearity NÁ so that feedback control schemes designed for (10.128) without NÁ, with the help of NIÁ, can be applied to (10.128) with NÁ, to achieve desired system performance. To illustrate this idea, we present an adaptive inverse design [16] for a third- order parametric-strict-feedback nonlinear plant [6] with an actuator non- linearity NÁ: x 1 x 2 ÃT s ' 1 x 1 x 2 x 3 ÃT s ' 2 x 1 ; x 2 x 3 ' 0 x ÃT s ' 3 x 0 xu; u Nv 10:130 where x x 1 ; x 2 ; x 3 T , x i , i 1; 2; 3, are the state variables available for measurement, à s P R n s is a vector of unknown constant parameters with a known upper bound M s : k à s k 2 < M s , for the Euclidean vector norm kÁk 2 , ' 0 P R, 0 P R and ' i P R p , i 1; 2; 3, are known smooth nonlinear functions, and b 1 < j 0 xj < b 2 , for some positive constants b 1 , b 2 and Vx P R 3 . To develop an adaptive inverse controller for (10.130), we assume that the nonlinearity NÁ is parametrized by à P R n as in (10.2) and its inverse NIÁ is parametrized by P R n , an estimate of à , as in (10.10), with the stated properties. Then, the adaptive backstepping method [6] can be combined with an adaptive inverse NIÁ to control the plant (10.130), in a three-step design: Step 1: Let the desired output be rt to be tracked by the plant output yt, with bounded derivatives r k t, k 1; 2; 3. De®ning z 1 x 1 À r and z 2 x 2 À 1 , where 1 is a design signal to be determined, we have z 1 z 2 1 ÃT s ' 1 À r 10:131 Adaptive Control Systems 281 to be stabilized by 1 with respect to the partial Lyapunov function V 1 1 2 z 2 1 1 2 s À à s T À À1 s s À à s 10:132 where s tPR n s is an estimate of the unknown parameter vector à s ,and À s À T s > 0. The time derivative of V 1 is V 1 z 1 z 2 1 T s ' 1 À r s À à s T À À1 s s À À s z 1 ' 1 10:133 Choosing the ®rst stabilizing function as 1 Àc 1 z 1 À T s ' 1 r 10:134 with c 1 > 0, and de®ning 1 À s z 1 ' 1 , we obtain V 1 Àc 1 z 2 1 z 1 z 2 s À à s T À À1 s s À 1 10:135 Step 2: Introducing z 3 x 3 À 2 with 2 to be determined, we have z 2 z 3 2 ÃT s ' 2 À @ 1 @x 1 x 2 ÃT s ' 1 À @ 1 @ s s À @ 1 @r r À @ 1 @ r r 10:136 To derive 2 , together with 1 in (10.134), to stabilize (10.131) and (10.136) with respect to V 2 V 1 1 2 z 2 2 , we express the time derivative of V 2 as V 2 Àc 1 z 2 1 z 2 z 1 z 3 2 À @ 1 @x 1 x 2 À @ 1 @ s s À @ 1 @r r À @ 1 @ r r T s ' 2 À @ 1 @x 1 ' 1 ! s À à s T À À1 s s À 2 10:137 where 2 1 À s z 2 ' 2 À @ 1 @x 1 ' 1 . Suggested by (10.137), we choose the second stabilizing function 2 as 2 Àz 1 À c 2 z 2 @ 1 @x 1 x 2 @ 1 @ s 2 À À s f s s s @ 1 @r r @ 1 @ r r À T s ' 2 À @ 1 1 ' 1 10:138 where c 2 > 0, and f s s 0ifk s k 2 M s f 0 1 À e À 0 k s k 2 ÀM s 2 if k s k 2 > M s @ 10:139 with f 0 > 0, 0 > 0, and M s > k à s k 2 . Then we can rewrite (10.137) as 282 Adaptive inverse for actuator compensation V 2 Àc 1 z 2 1 À c 2 z 2 2 z 2 z 3 À z 2 @ 1 @ s s À 2 À s f s s s s À à s T À À1 s s À 2 10:140 Step 3: Choose the feedback control u d t for (10.129) as u d 1 0 À z 2 À c 3 z 3 À ' 0 2 k1 @ 2 @x k x k1 @ 2 @ s s 3 k1 @ 2 @r kÀ1 r k À T s À z 2 @ 1 @ s À s ' 3 À 2 k1 @ 2 @x k ' k À À s f s s s z 2 @ 1 @ s ! 10:141 where c 3 > 0, and the adaptive update law for s t is s À s 3 l1 z l ! l À À s f s s s 10:142 with ! i x 1 ; FFF; x i ; s ' i À iÀ1 k1 @ iÀ1 @x k ' k , for i 1; 2; 3 and 0 0. With this choice of u d , we have z 3 Àz 2 À c 3 z 3 À s À à s T ! 3 z 2 @ 1 @ s À s ! 3 10:143 Then, the overall system Lyapunov function V V 2 1 2 z 2 3 1 2 À à T À À1 À à 10:144 with À being diagonal and positive, has the time derivative V À 3 k1 c k z 2 k À f s s s À à s T s À à T À À1 Àz 3 0 !z 3 0 d n 10:145 where !tPR n is the known vector signal describing the signal motion of the inverse NIÁ in (10.10), and d n t is the bounded unparametrized nonlinearity error in (10.11). The expression (10.145), with the need of parameter projection for t for implementing an inverse NIÁ, suggests the adaptive update law for t: ÀÀz 3 0 ! f 10:146 initialized by (10.35) and projected with f t in (10.36) for gtÀÀz 3 0 !. This adaptive inverse control scheme consisting of (10.129), (10.141), (10.142) and (10.146) has some desired properties. First, the modi®cation (10.139) ensures that Àf s s s À à s T s 0, and the parameter projection (10.36) ensure that À à T À À1 f 0 and i tP a i ; b i , i 1; FFF; n , for Adaptive Control Systems 283 t 1 t; FFF; n t T and a i , b i de®ned in (10.31). Then, from (10.145) and (10.146), we have V À 3 k1 c k z 2 k À f s s s À à s T s À à T À À1 f z 3 0 d n 10:147 Since d n and 0 are bounded, and f s s s À à s T s grows unboundedly if s grows unboundedly, it follows from (10.144) and (10.147) that s ;P L I ,and z k P L I , k 1; 2; 3. Since z 1 x 1 À r is bounded, we have x 1 P L I and hence ' 1 x 1 PL I . It follows from (10.134) that 1 P L I .Byz 2 x 2 À 1 , we have x 2 P L I and hence ' 2 x 1 ; x 2 PL I . It follows from (10.138) that 2 P L I . Similarly, x 3 P L I , and u d P L I in (10.141). Finally, vt in (10.129) and ut in (10.128) are bounded. Therefore, all closed loop system signals are bounded. Similarly, an output feedback adaptive inverse control scheme can be developed for the nonlinear plant (10.128) in an output-feedback canonical form with actuator nonlinearities [17]. For such a control scheme, a state observer [6] is needed to obtain a state estimate for implementing a feedback control law to generate u d t as the input to an adaptive inverse: vt NIu d t, to cancel an actuator nonlinearity: utNvt. 10.9 Concluding remarks Thus far, we have presented a general adaptive inverse approach for control of plants with unknown nonsmooth actuator nonlinearities such as dead zone, backlash, hysteresis and other piecewise-linear characteristics. This approach combines an adaptive inverse with a feedback control law. The adaptive inverse is to cancel the eect of the actuator nonlinearity, while the feedback control law can be designed as if the actuator nonlinearity were absent. For parametrizable actuator nonlinearities which have parametrizable inverses, state or output feedback adaptive inverse controllers were developed which led to linearly parametrized error models suitable for the developments of gradient projection or Lyapunov adaptive laws for updating the inverse parameters. The adaptive inverse approach can be viewed as an algorithm-based compensation approach for cancelling unknown actuator nonlinearities caused by component imperfections. As shown in this chapter, this approach can be incorporated with existing control designs such as model reference, PID, pole placement, linear quadratic, backstepping and other dynamic compensa- tion techniques. An adaptive inverse can be added into a control system loop without the need to change a feedback control design for a known linear or nonlinear dynamics following the actuator nonlinearity. Improvements of system tracking performance by an adaptive inverse have 284 Adaptive inverse for actuator compensation been shown by simulation results. Some signal boundedness properties of adaptive laws and closed loop systems have been established. However, despite the existence of a true inverse which completely cancels the actuator non- linearity, an analytical proof of a tracking error convergent to zero with an adaptive inverse is still not available for a general adaptive inverse control design. Moreover, adaptive inverse control designs for systems with unknown multivariable or more general nonlinear dynamics are still open issues under investigation. References [1] Elliott, H. and Wolovich, W. A. (1984) `Parametrization Issues In Multivariable Adaptive Control', Automatica, Vol. 20, No. 5, 533±545. [2] Goodwin, G. C. and Sin, K. S. (1984) Adaptive Filtering Prediction and Control. Prentice-Hall. [3] Hatwell, M. S., Oderkerk, B. J., Sacher, C. A. and Inbar, G. F. (1991) `The Development of a Model Reference Adaptive Controller to Control the Knee Joint of Paraplegics', IEEE Trans. Autom. Cont., Vol. 36, No. 6, 683±691. [4] Ioannou, P. A. and Sun, J. (1995) Robust Adaptive Control. Prentice-Hall. [5] Krasnoselskii, M. A. and Pokrovskii, A. V. (1983) Systems with Hysteresis. Springer-Verlag. [6] Krstic  , M., Kanellakopoulos, I. and Kokotovic  , P. V. (1995) Nonlinear and Adaptive Control Design. John Wiley & Sons. [7] Merritt, H. E. (1967) Hydraulic Control Systems. John Wiley & Sons. [8] Physik Instrumente (1990) Products for Micropositioning. Catalogue 108±12, Edition E. [9] Rao, S. S. (1984) Optimization: Theory and Applications. Wiley Eastern. [10] Recker, D. (1993) `Adaptive Control of Systems Containing Piecewise Linear Nonlinearities', Ph.D. Thesis, University of Illinois, Urbana. [11] Tao, G. and Ioannou, P. A. (1992) `Stability and Robustness of Multivariable Model Reference Adaptive Control Schemes', in Advances in Robust Control Systems Techniques and Applications, Vol. 53, 99±123. Academic Press. [12] Tao, G. and Kokotovic  , P. V. (1996) Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley & Sons. [13] Tao, G. and Ling, Y. (1997) `Parameter Estimation for Coupled Multivariable Error Models', Proceedings of the 1997 American Control Conference, 1934±1938, Albuquerque, NM. [14] Tao, G. and Tian, M. (1995) `Design of Adaptive Dead-zone Inverse for Nonminimum Phase Plants', Proceedings of the 1995 American Control Conference, 2059±2063, Seattle, WA. [15] Tao, G. and Tian, M. (1995) `Discrete-time Adaptive Control of Systems with Multi-segment Piecewise-linear Nonlinearities', Proceedings of the 1995 American Control Conference, 3019±3024, Seattle, WA. Adaptive Control Systems 285 [16] Tian, M. and Tao, G. (1996) `Adaptive Control of a Class of Nonlinear Systems with Unknown Dead-zones', Proceedings of the 13th World Congress of IFAC, Vol. E, 209±213, San Francisco, CA. [17] Tian, M. and Tao, G. (1997) `Adaptive Dead-zone Compensation for Output- Feedback Canonical Systems', International Journal of Control, Vol. 67, No. 5, 791±812. [18] Truxal, J. G. (1958) Control Engineers' Handbook. McGraw-Hill. 286 Adaptive inverse for actuator compensation 11 Stable multi-input multi- output adaptive fuzzy/neural control R. Ordo  n Ä ez and K. M. Passino Abstract In this chapter, stable direct and indirect adaptive controllers are presented which use Takagi±Sugeno fuzzy systems, conventional fuzzy systems, or a class of neural networks to provide asymptotic tracking of a reference signal vector for a class of continuous time multi-input multi-output (MIMO) square nonlinear plants with poorly understood dynamics. The direct adaptive scheme allows for the inclusion of a priori knowledge about the control input in terms of exact mathematical equations or linguistics, while the indirect adaptive controller permits the explicit use of equations to represent portions of the plant dynamics. We prove that with or without such knowledge the adaptive schemes can `learn' how to control the plant, provide for bounded internal signals, and achieve asymptotically stable tracking of the reference inputs. We do not impose any initialization conditions on the controllers, and guarantee convergence of the tracking error to zero. 11.1 Introduction Fuzzy systems and neural networks-based control methodologies have emerged in recent years as a promising way to approach nonlinear control problems. Fuzzy control, in particular, has had an impact in the control community because of the simple approach it provides to use heuristic control knowledge for nonlinear control problems. However, in the more complicated situations where the plant parameters are subject to perturbations, or when the dynamics of the system are too complex to be characterized reliably by an explicit mathematical model, adaptive schemes have been introduced that gather data from on-line operation and use adaptation heuristics to auto- matically determine the parameters of the controller. See, for example, the techniques in [1]±[7]; to date, no stability conditions have been provided for these approaches. Recently, several stable adaptive fuzzy control schemes have been introduced [8]±[12]. Moreover, closely related neural control approaches have been studied [13]±[18]. In the above techniques, emphasis is placed on control of single-input single- output (SISO) plants (except for [4], which can be readily applied to MIMO plants as it is done in [5, 6], but lacks a stability analysis). In [19], adaptive control of MIMO systems using multilayer neural networks is studied. The authors consider feedback linearizable, continuous-time systems with general relative degree, and utilize neural networks to develop an indirect adaptive scheme. These results are further studied and summarized in [20]. The scheme in [19] requires the assumptions that the tracking and neural network param- eter errors are initially bounded and suciently small, and they provide convergence results for the tracking errors to ®xed neighbourhoods of the origin. In this chapter we present direct [21] and indirect [22] adaptive controllers for MIMO plants with poorly understood dynamics or plants subjected to param- eter disturbances, which are based on the results in [8]. We use Takagi±Sugeno fuzzy systems or a class of neural networks with two hidden layers as the basis of our control schemes. We consider a general class of square MIMO systems decouplable via static nonlinear state feedback and obtain asymptotic con- vergence of the tracking errors to zero, and boundedness of the parameter errors, as well as state boundedness provided the zero dynamics of the plant are exponentially attractive. The stability results do not depend on any initializa- tion conditions, and we allow for the inclusion in the control algorithm of a priori heuristic or mathematical knowledge about what the control input should be, in the direct case, or about the plant dynamics, in the indirect case. Note that while the indirect approach is a fairly simple extension of the corresponding single-input single-output case in [8], the direct adaptive case is not. The direct adaptive method turns out to require more restrictive assumptions than the indirect case, but is perhaps of more interest because, as far as we are aware, no other direct adaptive methodology with stability proof for the class of MIMO systems we consider here has been presented in the literature. The results in this chapter are nonlocal in the sense that they are global whenever the change of coordinates involved in the feedback lineariza- tion of the MIMO system is global. The chapter is organized as follows. In Section 11.2 we introduce the MIMO direct adaptive controller and give a proof of the stability results. In Section 11.3 we outline the MIMO indirect adaptive controller, giving just a short sketch of the proof, since it is a relatively simple extension of the results in [8]. In Section 11.4 we present simulation results of the direct adaptive method 288 Stable multi-input multi-output adaptive fuzzy/neural control applied, ®rst, to a nonlinear dierential equation that satis®es all controller assumptions, as an illustration of the method, and then to a two-link robot. The robot is an interesting practical application, and it is of special interest here because it does not satisfy all assumptions of the controller; however, we show how the method can be made to work in spite of this fact. In Section 11.5 we provide the concluding remarks. 11.2 Direct adaptive control Consider the MIMO square nonlinear plant (i.e. a plant with as many inputs as outputs [23, 24]) given by X f Xg 1 Xu 1 FFF g p Xu p y 1 h 1 X F F F y p h p X 11:1 where X x 1 ; FFF; x n T P R n is the state vector, U Xu 1 ; FFF; u p T P R p is the control input vector, Y Xy 1 ; FFF; y p T P R p is the output vector, and f ; g i ; h i ; i 1; FFF; p are smooth functions. If the system is feedback linearizable [24] by static state feedback and has a well-de®ned vector relative degree r X r 1 ; FFF; r p T , where the r i 's are the smallest integers such that at least one of the inputs appears in y r i i , input±output dierential equations of the system are given by y r i i L r i j h i p j1 L gj L r i À1 f h i u j 11:2 with at least one of the L g j L r i À1 f h i T 0 (note that L f hX X R n 3 R is the Lie derivative of h with respect to f , given by L f hX @h @X f X. De®ne, for convenience, i X X L r i f h i and ij X X L g j Lf r i À1 h i . In this way, we may rewrite the plant's input±output equation as y r 1 1 F F F y r p p P T T R Q U U S |{z} Y r t 1 F F F p P T T R Q U U S |{z} AX;t 11 ÁÁÁ 1p F F F F F F F F F p1 ÁÁÁ pp P T T R Q U U S |{z} BX;t u 1 F F F u p P T T R Q U U S |{z} Ut 11:3 Consider the ideal feedback linearizing control law, U à u à 1 ; FFF; u à p T , Adaptive Control Systems 289 U à B À1 ÀA m11:4 (note that, for convenience, we are dropping the references to the independent variables except where clari®cation is required), where the term m 1 ; FFF; p T is an input to the linearized plant dynamics. In order for U à to be well de®ned, we need the following assumption: (P1) Plant Assumption The matrix B as de®ned above is nonsingular, i.e. B À1 exists and has bounded norm for all X P S x ; t ! 0, where S x P R n is some compact set of allowable state trajectories. This is equivalent to assuming p B! min > 0 11:5 kBk 2 1 B max < I11:6 where p B and 1 B are, respectively, the smallest and largest singular values of B. In addition, in order to be able to guarantee state boundedness under state feedback linearization, we require: (P2) Plant Assumption The plant is feedback linearizable by static state feedback; it has a general vector relative degree r r 1 ; FFF; r p T , and its zero dynamics are exponen- tially attractive (please refer to [24] for a review on the concept of zero dynamics and static state feedback of square MIMO systems). We also assume the state vector X to be available for measurement. Our goal is to identify the unknown control function (11.4) using fuzzy systems. Here we will use generalized Takagi±Sugeno (T±S) fuzzy systems with centre average defuzzi®cation. To brie¯y present the notation, take a fuzzy system denoted by ~ f X; W (in our context, X could be thought of as the state vector, and W as a vector of possibly exogenous signals). Then, ~ f X; W R i1 c i i R i1 i . Here, singleton fuzzi®cation of the input vectors X x 1 ; FFF; x n T , W w 1 ; FFF; w q T is assumed; the fuzzy system has R rules, and i is the value of the membership function for the premise of the ith rule given the inputs X, W. It is assumed that the fuzzy system is constructed in such a way that 0 i 1 and R i1 i T 0 for all X P R n , W P R q . The parameter c i is the consequent of the ith rule, which in this chapter will be taken as a linear combination of Lipschitz continuous functions, z k XPR; k 1; FFF; m À 1, so that c i a i;0 a i;1 z 1 XÁÁÁ a i;mÀ1 z mÀ1 X; i 1; FFF; R. De®ne 290 Stable multi-input multi-output adaptive fuzzy/neural control [...]... techniques An adaptive inverse can be added into a control system loop without the need to change a feedback control design for a known linear or nonlinear dynamics following the actuator nonlinearity Improvements of system tracking performance by an adaptive inverse have Adaptive Control Systems 285 been shown by simulation results Some signal boundedness properties of adaptive laws and closed loop systems... Model Reference Adaptive Control Schemes', in Advances in Robust Control Systems Techniques and Applications, Vol 53, 99±123 Academic Press  [12] Tao, G and Kokotovic, P V (1996) Adaptive Control of Systems with Actuator and Sensor Nonlinearities John Wiley & Sons [13] Tao, G and Ling, Y (1997) `Parameter Estimation for Coupled Multivariable Error Models', Proceedings of the 1997 American Control Conference,... [16] Tian, M and Tao, G (1996) `Adaptive Control of a Class of Nonlinear Systems with Unknown Dead-zones', Proceedings of the 13th World Congress of IFAC, Vol E, 209± 213, San Francisco, CA [17] Tian, M and Tao, G (1997) `Adaptive Dead-zone Compensation for OutputFeedback Canonical Systems', International Journal of Control, Vol 67, No 5, 791±812 [18] Truxal, J G (1958) Control Engineers' Handbook McGraw-Hill... convergent to zero with an adaptive inverse is still not available for a general adaptive inverse control design Moreover, adaptive inverse control designs for systems with unknown multivariable or more general nonlinear dynamics are still open issues under investigation References [1] Elliott, H and Wolovich, W A (1984) `Parametrization Issues In Multivariable Adaptive Control' , Automatica, Vol 20,... (1984) Adaptive Filtering Prediction and Control Prentice-Hall [3] Hatwell, M S., Oderkerk, B J., Sacher, C A and Inbar, G F (1991) `The Development of a Model Reference Adaptive Controller to Control the Knee Joint of Paraplegics', IEEE Trans Autom Cont., Vol 36, No 6, 683±691 [4] Ioannou, P A and Sun, J (1995) Robust Adaptive Control Prentice-Hall [5] Krasnoselskii, M A and Pokrovskii, A V (1983) Systems... conditions on the controllers, and guarantee convergence of the tracking error to zero 11.1 Introduction Fuzzy systems and neural networks-based control methodologies have emerged in recent years as a promising way to approach nonlinear control problems Fuzzy control, in particular, has had an impact in the control community because of the simple approach it provides to use heuristic control knowledge... Kanellakopoulos, I and Kokotovic, P V (1995) Nonlinear and Adaptive Control Design John Wiley & Sons [7] Merritt, H E (1967) Hydraulic Control Systems John Wiley & Sons [8] Physik Instrumente (1990) Products for Micropositioning Catalogue 108±12, Edition E [9] Rao, S S (1984) Optimization: Theory and Applications Wiley Eastern [10] Recker, D (1993) `Adaptive Control of Systems Containing Piecewise Linear Nonlinearities',... and Tian, M (1995) `Design of Adaptive Dead-zone Inverse for Nonminimum Phase Plants', Proceedings of the 1995 American Control Conference, 2059±2063, Seattle, WA [15] Tao, G and Tian, M (1995) `Discrete-time Adaptive Control of Systems with Multi-segment Piecewise-linear Nonlinearities', Proceedings of the 1995 American Control Conference, 3019±3024, Seattle, WA 286 Adaptive inverse for actuator... results In Section 11.3 we outline the MIMO indirect adaptive controller, giving just a short sketch of the proof, since it is a relatively simple extension of the results in [8] In Section 11.4 we present simulation results of the direct adaptive method Adaptive Control Systems 289 applied, ®rst, to a nonlinear dierential equation that satis®es all controller assumptions, as an illustration of the method,... understood dynamics The direct adaptive scheme allows for the inclusion of a priori knowledge about the control input in terms of exact mathematical equations or linguistics, while the indirect adaptive controller permits the explicit use of equations to represent portions of the plant dynamics We prove that with or without such knowledge the adaptive schemes can `learn' how to control the plant, provide . `Discrete-time Adaptive Control of Systems with Multi-segment Piecewise-linear Nonlinearities', Proceedings of the 1995 American Control Conference, 3019±3024, Seattle, WA. Adaptive Control Systems. error convergent to zero with an adaptive inverse is still not available for a general adaptive inverse control design. Moreover, adaptive inverse control designs for systems with unknown multivariable. Reference Adaptive Control Schemes', in Advances in Robust Control Systems Techniques and Applications, Vol. 53, 99±123. Academic Press. [12] Tao, G. and Kokotovic  , P. V. (1996) Adaptive Control