11.5 Conclusions In this chapter we have developed direct and indirect adaptive MIMO control schemes which use Takagi±Sugeno fuzzy systems or a class of neural networks. We have proven stability of the methods and shown that they guarantee asymptotic convergence of the tracking errors to zero, as well as boundedness of all the signals and parameter errors, regardless of any initialization constraints. Both methods allow for the inclusion of previous knowledge or expertise in form of linguistics regarding what the control input should be, in the direct case, or what the plant dynamics are, in the indirect case. We show that with or without such knowledge the stability and tracking properties of the controllers hold, and present two simulations for direct adaptive control that illustrate the method. References [1] Procyk, T. and Mamdani, E. (1979). `A Linguistic Self-organizing Process Controller' Automatica, 15(1), 15±30. [2] Driankov, D., Hellendoorn, H. and Reinfrank, M. M. (1993). An Introduction to Fuzzy Control. Springer-Verlag, Berlin Heidelberg. [3] Layne, J. R., Passino K. M. and Yurkovich, S. (1993). `Fuzzy Learning Control for Anti-skid Braking Systems', IEEE Trans. Control Systems Tech., 1(2) 122±129, June. [4] Layne, J. R., and Passino, K. M. (1993). `Fuzzy Model Reference Learning Control for Cargo Ship Steering', IEEE Control Systems Magazine, 13(6), 23±34, Dec. [5] Kwong, W. A., Passino, K. M., Lauknonen, E. G. and Yurkovich, S. (1995). `Expert Supervision of Fuzzy Learning Systems for Fault Tolerant Aircraft Control', Proc. of the IEEE, Special Issue on Fuzzy Logic in Engineering Applications, 83(3) 466±483, March. [6] Moudgal, V. G., Kwong, W. A., Passino K. M. and Yurkovich, S. (1995). `Fuzzy Learning Control for a Flexible-link Robot', IEEE Transactions on Fuzzy Systems, 3(2), 199±210, May. [7] Kwong, W. A. and Passino, K. M. (1996). `Dynamically Focused Fuzzy Learning Control', IEEE Trans. on Systems, Man, and Cybernetics, 26(1) 53±74, Feb. 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[24] Shankar Sastry, S. and Isidori, A. (1989). `Adaptive Control of Linearizable Systems', IEEE Transac. Autom. Contr., 34(11), 1123±1131, November. [25] Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press, Cambridge (Cambridgeshire); New York. Adaptive Control Systems 307 12 Adaptive robust control scheme with an application to PM synchronous motors J X. Xu, Q W. Jia and T H. Lee Abstract This chapter presents a new adaptive robust control scheme for a class of nonlinear uncertain dynamical systems. To reduce the robust control gain and widen the application scope of adaptive techniques, the system uncertainties are classi®ed into two dierent categories: the structured and non-structured uncertainties. The structured uncertainty can be separated and expressed as the product of known functions of states and a set of unknown constants. The upper bounding functions of the non structured uncertainties to be addressed in this chapter is only partially known with unknown parameters. Moreover, the bounding function is convex to the set of unknown parameters, i.e. the bounding function is no longer linear in parameters. The structured uncer- tainty is estimated with adaptation and compensated. Meanwhile, the adaptive robust method is applied to deal with the non structured uncertainty by estimating unknown parameters in the upper bounding function. The - modi®cation scheme [1] is used to cease parameter adaptation in accordance with the adaptive robust control law. The backstepping method [2] is also adopted in this chapter to deal with a system not in the parametric±pure feedback form, which is usually necessary for the application of backstepping control scheme. The new control scheme guarantees the uniform boundedness of the system and at the same time, the tracking error enters an arbitrarily designated zone in a ®nite time. The eectiveness of the proposed method is demonstrated by the application to PM synchronous motors. 12.1 Introduction Numerous adaptive robust control algorithms for systems containing uncer- tainties have been developed [1]±[11]. In [3] variable structure control with an adaptive law is developed for an uncertain input±output linearizable nonlinear system, where linearity-in-parameter condition for uncertainties is assumed. The unknown gain of the upper bounding function is estimated and updated by adaptation law so that the sliding condition can be met and the error state reaches the sliding surface and stays on it. To deal with a class of nonlinear systems with partially known uncertainties, in [4] an adaptive law using a dead zone and a hysteresis function is proposed to guarantee both the uniform boundedness of all the closed loop signals and uniform ultimate boundedness of the system states. In both control schemes, it is assumed that the system uncertainties are bounded by a bounding function which is a product of a set of known functions and unknown positive constants. The objective of adaptation is to estimate these unknown constants. In [1], a new adaptive robust control scheme is developed for a class of nonlinear uncertain systems with both parameter uncertainties and exogenous disturbances. Including the categories of system uncertainties in [3] and [4] as its subsets, it is assumed that the disturbances are bounded by a known upper bounding function. Furthermore, the input distribution matrix is assumed to be constant but unknown. In this chapter we proposed a continuous adaptive robust control scheme which is the extension of [1] in the sense that more general classes of nonlinear uncertain dynamical systems are under consideration. The unknown input distribution matrix of the system input can be state dependent here instead of being a constant matrix in [1]. To reduce the robust control gain and widen the application scope of adaptive techniques, the system uncertainties are supposed to be composed of two dierent categories: the ®rst can be separated and expressed as the product of known function of states and a set of unknown constants, and the other category is not separable but with partially known bounding functions. It is further assumed that the bounding function is convex to the set of unknown parameters, i.e. the bounding function is no longer linear in parameters. The ®rst category of uncertainties is dealt with by means of the well-used adaptive control method. Meanwhile an adaptive robust method is applied to deal with the second category of uncertainties, where the unknown parameters in the upper bounding function are estimated with adaptation. It should also be noted that the backstepping method [2] is adopted in this chapter to deal with a system not in the parametric±pure feedback form, which is usually necessary for the application of a backstepping control scheme. The proposed method is further applied to a permanent magnet synchronous (PMS) motor, which is a typical nonlinear control system. The dynamics of the PM synchronous motor can be presented by a dynamic electrical subsystem Adaptive Control Systems 309 and a mechanical subsystem, which are nonlinear dierential equations. Strictly speaking, most control methods for permanent magnet synchronous motors are only locally stable because the d-axis current is assumed to be zero and the design procedure is based on the reduced model. In this chapter, instead of only zeroing d-axis current, the extra d-axis control input voltage is used to deal with the nonlinear coupling part of the dynamics as well. This chapter is organized as follows. Section 12.2 describes the class of nonlinear uncertain systems to be controlled. Section 12.3 gives the design procedure of the adaptive robust control and the stability analysis. Section 12.4 describes the application of the proposed control method to the PM synchro- nous motors. 12.2 Problem formulation Consider a class of uncertain dynamical system described by  x 0  f 0 tB 0 pg 0 p; tx 1 Rg 0 x; p;!;tx 2 12:1  x 1  f 1 x; tB 1 pfI  E 1 x; p; tu 1 tg 1 x; p; tRg 1 x; p;!;tg 12:2  x 2  f 2 x; tB 2 pfI  E 2 x; p; tu 2 tg 2 x; p; tRg 2 x; p;!;tg 12:3 where x i x i1 ; x i2 ; ; x in i  b P n i ; i  0; 1; 2; are the measurable state vec- tors of the system, where n 0  n 1 and n 0  n 1  n 2  n; x P n is de®ned as x x b 0 ; x b 1 ; x b 2  b ; u i u i1 ; u i2 ; ; u in i  b P n i , i  1; 2, are the control inputs of the system; p Pis an unknown system parameter vector and  is the set of admissible system parameters; f i P n i , i  0; 1; 2, are nonlinear function vectors; g i P n i , i  0; 1; 2, and Rg 0 P n 0 Ân 2 , Rg i P n i , i  1; 2, are non- linear uncertain function vectors of the state x, unknown parameter p, time t as well as a set of random variables !. Here we make the following assumptions: (A1) f 0 t, f 1 x; t and f 2 x; t are known nonlinear function vectors. The matrices B i p, i  0; 1; 2, are unknown but positive de®nite. (A2) For E i P n i Ân i , i  1; 2 Vt P0; I Vx Ph Vp P r max i !  1 2 E i  1 2 E b i !r min i > À1 12:4 where Á indicates the eigenvalues of `Á'. (A3) The structured uncertainty g i P n i , i  0; 1; 2, are nonlinear function vectors which can be expressed as 310 Adaptive robust control scheme: an application to PM synchronous motors g 0 p; t 0 p 0 t g 1 x; p; t H 1 p H 1 x; t g 2 x; p; t 2 p 2 x; t  i  diag  b i1 ; ;  b in i   i  i1 ; i2 ; ; in i  b ; i  0; 1; 2 12:5 where  0 ,  H 1 and  2 are unknown parameter matrices and  0 ,  H 1 and  2 are known function vectors. The nonstructured uncertainty Rg i x; p;!;t, i  0; 1; 2, are bounded such that Vt P  Vx Ph Vp P jjRg i x; p;!;tjj  d i x; q i ; t12:6 where jj Á jj represents the Euclidean norm for vectors and the spectral norm for matrices; h is a compact subset of  n in which the solution of (12.1)±(12.3) uniquely exists with respect to the given desired state trajectory x d t.  d i x; q i ; t, i  0; 1; 2, are upper bounding functions with unknown parameter vectors q i P. Here  d i x; q i ; t is dierentiable and convex to q i , that is  d i x; q i2 ; tÀ d x; q i1 ; t q i2 À q i1  b @ d @q i     q i1 12:7 The control objective is to ®nd suitable control inputs u 1 and u 2 for the state x 0 to track the desired trajectory x d tP n 0 , where x d is continuously dierentiable. Remark 2.1 The sub system (12.1) has x 1 as its input. However, it is not in the parametric±pure feedback form due to the existence of the nonlinear uncertain term Rg 0 x; p;!;tx 2 . Thus the well-used backstepping design needs to be revised to deal with the dynamical system (12.1)±(12.3). Remark 2.2 It should be noted that g i x; p; t can be absorbed into Rg i x; p;!;t. However, it is obviously more conservative. This can be clearly shown through the following example. Assume that the structured uncertainty is g   1  1   2  2 with actual values  1  a,  2 Àa and a is an unknown constant. Assume that the nonlinear function  2   1 R, where jjRjj ( jj 1 jj. Then jjgjj jj 1 jj Ájj 1 jj jj 2 jj Ájj 2 jj   b jj 1 , where   jj 1 jj; jj 2 jj b  jjajj; jjajj b and jj 1  jj 1 jj; jj 2 jj b . The upperbound parameter to be estimated is   jjajj; jjajj b . This implies that the actual uncertainty g  aR has been ampli®ed to its normed product jjajj Ájj 1 jj jjajjÁjj 1 Rjj, which is obviously much larger than jjajj ÁjjRjj even if the estimates converge to the true values. On the contrary, if the uncertainty is expressed by (12.5), the unknown parameters to be Adaptive Control Systems 311 estimated is a; Àa b . This means that, when the estimated parameters are near the true values, the estimated uncertainty of g will be able to approach the actual uncertainty aR. 12.3 Adaptive robust control with l-modi®cation The adaptive robust technique is combined with backstepping method in this section to develop a controller which guarantees the global boundedness of the system. The design procedures are presented in detail as follows. De®ne the measured state tracking error vector as e 0  x 0 À x d 12:8 and the parameter matrices as  i XB À1 i ; i  0; 1; 2 12:9 We further denote z 0  e 0 , z 1  x 1 À x ref 1 , z 2  x 2 , z z b 0 ; z b 1 ; z b 2  b , and the auxiliary control x ref 1 is de®ned as x ref 1 ÀK 0 z 0 À   0  0 À   0 f 0 À  x d 0 12:10 where K 0 is a gain matrix.   0 and   0 are the estimates of  0 and  0 respectively. The ®rst order derivative of x ref 1 is derived as follows:  x ref 1 ÀK 0  z 0 À    0  0 À   0   0 À    0 f 0 À  x d 0 À   0   f 0 À  x d 0  Àf HH 1 À  H 0  H 0 12:11 where f HH 1  K 0 f H 0     0  0    0   0     0 f 0 À  x d    0   f 0 À  x d  f HH 0  f 0 À  x d  H 0 K 0 B 0  0 ; K 0 B 0   H 0  0 ; z 1  x ref 1  b 12:12 Then the plant (12.1)±(12.3) can be rearranged as follows:  z 0  f H 0 tB 0 pg 0 p; tRg 0 z; p; x ref 1 ;!;tz 2  z 1  x ref 1 12:13  z 1  f H 1 z; x ref 1 ; tB 1 pfI  E 1 z; p; x ref 1 ; tu 1 tg H 1 z; p; x ref 1 ; t Rg 1 z; p; x ref 1 ;!;tg 12:14  z 2  f 2 z; x ref 1 ; tB 2 pfI  E 2 z; p; x ref 1 ; tu 2 tg 2 z; p; x ref 1 ; t Rg 2 z; p; x ref 1 ;!;tg 12:15 312 Adaptive robust control scheme: an application to PM synchronous motors where f H 1  f 1  f HH 1 g H 1   1  1  1  H 1 ; B À1 1  H 0   1  H 1 ; H 0  b 12:16 The adaptive robust control law. De®ne the parameter error matrices as  q i  q i À  q i ; 12:17   i   i À   i ; 12:18   i   i À   i ; i  0; 1; 2 12:19 where  q i ;   i ;   i are the estimates of q i ;  i ;  i , i  0; 1; 2, respectively. The control law u i , i  1; 2, are chosen to be u i  u c i  u v i 12:20 u c i ÀK i z i i À 2z 0 À   i  i À   i f H i À v d i Ài À 1v d 0 u v i À r 2 max i jju c i jj 2 z i 1 r min i r max i jjz b i u c i jj " v i v d i    2 d i   d i jjz i jj " d i z i v d 0    2 d 0 jjz 0 jj 2   d 0 jjz 0 jjjjz 2 jj  " d 0 z 2 12:21 where K i P n i Ân i , i  1; 2, is a gain matrix; " v i , i  1; 2, and " d i , i  0; 1; 2, are positive constants; f H 2 Xf 2 ;   d i X d i z;  q i ; x ref 1 ; t; The corresponding adaptive laws are de®ned as    i   i1 z i  b i À  i1   i     i   i2 z i f H b i À  i2     q i   i3 jjz i jj @ d i @q i     ^ q i À i3  q i  ; i  0; 1; 2 12:22 where  ij ; j  1; 2; 3 are positive de®nite matrices chosen to be  ij  diag  1 ij ; 2 ij ; ; n i ij 12:23 Adaptive Control Systems 313  ij ; j  1; 2; 3, which constitute the -modi®cation scheme, are de®ned as  ij  k ij " 0 Àjjzjj z P E 0 0 elsewhere & 12:24 where k ij , i  0; 1; 2, j  1; 2; 3, are positive constants. E 0 Xfe  jjejj <" 0 g12:25 where " 0 is a positive constant specifying the desired tracking error bound. Convergence analysis. For the above adaptive robust controller, we have the following theorem. Theorem 3.1 By properly choosing the control gain matrix, the proposed adaptive robust control law (12.20)±(12.24) ensures that the system trajectory enters the set E 0 in a ®nite time. Moreover, the tracking errors as well as the parameter estimation errors are bounded by the set D  & z;   i ;   i ;  q i ; i  0; 1; 2  z b z   2 i0 trace f   b i   i gtrace f   b i   i g  q b i  q i  < 1 2 k H  2 i0 k i1 " 0 trace f b i  i gk i2 " 0 trace f b i  i gk i3 " 0 q b i q i 2" !' 12:26 where k H is de®ned to be k H  max fk H ij ; i  0; 1; 2 j  1; 2; 3g k H ij  1 k HH ij min f min B À1 i ; min  À1 ij g k HH ij  2 minf min K; k ij g max f max B À1 i ; max  À1 ij g  max A and  min A indicate the maximum and minimum eigenvalues of the matrix A respectively, and " and  are positive values to be de®ned later. Proof The following positive de®nite function V is selected V  V 1  V 2  V 2 12:27 where V 1  1 2 z b 0 B À1 0 z 0  1 2 trace f   b 0  À1 01   0 g 1 2 trace f   b 0  À1 02   0 g 1 2  q b 0  À1 03  q 0 ; 12:28 V 2  1 2 z b 1 B À1 1 z 1  1 2 trace f   b 1  À1 11   1 g 1 2 trace f   b 1  À1 12   1 g 1 2  q b 1  À1 13  q 1 ; 12:29 V 3  1 2 z b 2 B À1 2 z 2  1 2 trace f   b 2  À1 21   2 g 1 2 trace f   b 2  À1 22   2 g 1 2  q b 2  À1 23  q 2 12:30 314 Adaptive robust control scheme: an application to PM synchronous motors Take the derivatives of V 1 , V 2 and V 3 along the trajectory of the dynamic system (12.13)±(12.15), we have (See Appendices A±C)  V 1 Àz b 0 K 0 z 0  z b 0 z 1    d 0 jjz 0 jj Á jjz 2 jjÀ 1 2  01 trace f   b 0   0 gÀ 1 2  02 trace f   b 0   0 g  1 2  01 trace f b 0  0 g 1 2  02 trace f b 0  0 g12:31  V 2 Àz b 1 K 1 z 1 À z b 1 z 0 À 1 2  11 trace f   H b 1   H 1 gÀ 1 2  12 trace f   b 1   1 gÀ 1 2  13  q b 1  q 1  1 2  11 trace f H b 1  H 1 g 1 2  12 trace f b 1  1 g 1 2  13 q b 1 q 1  " v 1  " d 1 12:32  V 3 Àz b 2 K 2 z 2 À z b 2 v d 0 À 1 2  21 trace f   b 2   2 gÀ 1 2  22 trace f   b 2   2 gÀ 1 2  23  q b 2  q 2  1 2  21 trace f b 2  2 g 1 2  22 trace f b 2  2 g 1 2  23 q b 2 q 2  " v 2  " d 2 12:33 Then we can easily get that  V   V 1   V 2   V 3 Àz b Kz    d 0 jjz 0 jj Ájjz 2 jjÀz b 2 v d 0  1 2  2 i0 À i1 trace f   b i   i gÀ i2 trace f   b i   i g À  i3  q b i  q i   i1 trace f b i  i g i2 trace f b i  i g i3 q b i q i   2 i1 " v i  " d i  Àz b Kz    d 0 jjz 0 jj Ájjz 2 jj À   2 d 0 jjz 0 jj 2 jjz 2 jj 2   d 0 jjz 0 jjjjz 2 jj " d 0  1 2  2 i0 À i1 trace f   b i   i g À  i2 trace f   b i   i gÀ i3  q b i  q i  i1 trace f b i  i g i2 trace f b i  i g i3 q b i q i    2 i1 " v i  " d i  Àz b Kz  1 2  2 i0 À i1 trace f   b i   i gÀ i2 trace f   b i   i gÀ i3  q b i  q i   i1 trace f b i  i g i2 trace f b i  i g i3 q b i q i " 12:34 where K  diag K 0 ; K 1 ; K 2 ,and"   2 i0 " d i   2 i1 " v i . By choosing K such that  min K! " c 1 " 2 0 12:35 where c 1 is an arbitrary positive constant, then from (12.24) we have  V Àz b Kz  " Àc 1 ; V z P n À E 0 12:36 Note that, in terms of the adaptive robust control law (12.20)±(12.24),  V is a Adaptive Control Systems 315 [...]... Backstepping methodology Adaptive control algorithms, 23±5 robustness problems, 81 Adaptive control Lyapunov functions (aclf), 185, 188 Adaptive internal model control schemes, 7±21 330 Index certainty equivalence control laws, 9± 12, 15±16 adaptive H2 optimal control, 11, 13± 14, 18±19 adaptive H optimal control, 11±12, 13, 14, 18±19 model reference adaptive control, 10±11, 13, 14, 18 partial adaptive pole placement,... 10, 13, 14, 18 design of robust adaptive law, 7±9 direct adaptive control with dead zone, 27±8 robustness analysis, 12±18 simulation examples, 18±19 stability analysis, 12±18 See also Individual model control schemes Adaptive nonlinear control, see Nonlinear systems Adaptive passivation, 119±21, 123±4, 127±38 adaptively feedback passive (AFP) systems, 128±30, 132 examples and extensions, 135±8 adaptive. .. and Qing-Wei Jia, (1997) `An adaptive Robust Control Scheme for a Class of Nonlinear Uncertain Systems', International Journal of Systems Science, Vol 28, No 4, 429±434 [2] Kokotovic, P V., (1992) `The Joy of Feedback: Nonlinear and Adaptive' , IEEE Control systems, Vol 12, No.6, 7±17 [3] Teh-Lu Liao, Li-Chen Fu and Chen-Fa Hsu, (1990) `Adaptive Robust Tracking of Nonlinear Systems and With an Application... to a Robotic Manipulator', Systems & Control Letters, Vol 15, 339±348 [4] Brogliato, B and Tro®no Neto, A (1995) `Practical Stabilization of a Class of Nonlinear Systems with Partially Known Uncertainties', Automatica, Vol 31, No 1, 145 ±150 [5] Ioannou, P A and Jing Sun, (1996) Robust Adaptive Control Prentice-Hall, Inc  [6] Ioannou, P A and Kokotovic, P V (1983) Adaptive Systems with Reduced Models... Basar, T., 186 Blaschke product, 4±5, 11 Bounded-input bounded-output (BIBO) stable systems, 125 Bounded-input bounded-state (BIBS) stable systems, 132 Byrnes, C.I., 123, 136 Certainty equivalence control laws, 9±12, 15±16, 72, 185, 216 adaptive H2 optimal control, 11, 13 14, 18±19 adaptive HI optimal control, 11±12, 13, 14, 18±19 ... adaptively feedback passive (AFP) systems, 128±30, 132 examples and extensions, 135±8 adaptive output feedback passivation, 136±8 controlled Dung equations, 135±6 recursive adaptive passivation, 131±5 Adaptive robust control, see Robust adaptive control Adaptive stabilizability, 127±8 Adaptive stabilization, 81±2 direct localization principle, 89±101 localization in the presence of unknown disturbance, 99±101... Narendra, K S (1982) `Bounded Error Adaptive Control' , IEEE Transac Autom Contr., Vol 27, No 6, 1161±1168  [8] Taylor, D G Kokotovic, P V Marino, R and Kanellakopoulos, I (1989) `Adaptive Regulation of Nonlinear Systems with Unmodeled Dynamics', IEEE Trans Autom Control, Vol 34, No 1, 405±412 [9] Sastry, S S and Isidori, A (1989) `Adaptive Control of Linearizable Systems', IEEE Transac Autom Contr.,... reference frame The control objective is to develop a link position tracking controller for the electromechanical dynamics of (12.40)± (12.43) despite parametric uncertainty In this chapter we assume that all the motor parameters are unknown Remark 4.1 In most existing control schemes for PM synchronous motors, the controllers are designed based on the following reduced model 318 Adaptive robust control scheme:... chapter, an adaptive robust control scheme is developed for a class of uncertain systems with both unknown parameters and system disturbances The uncertainties are assumed to be composed of two categories: the structured category and the nonstructured category with partially known bounding functions The structured uncertainty is estimated with the adaptive method Meanwhile, the adaptive robust method is... with adaptation It is shown that the control scheme developed here can guarantee the uniform boundedness of the system and assure that the tracking error enters the arbitrarily designated zone in a ®nite time The e€ectiveness of the control scheme is veri®ed by theoretical analysis, as well as applications to a permanent magnet synchronous motor Adaptive Control Systems 321 a† e time (sec) …b† ! time . typical nonlinear control system. The dynamics of the PM synchronous motor can be presented by a dynamic electrical subsystem Adaptive Control Systems 309 and a mechanical subsystem, which are. New York. Adaptive Control Systems 307 12 Adaptive robust control scheme with an application to PM synchronous motors J X. Xu, Q W. Jia and T H. Lee Abstract This chapter presents a new adaptive. Multi-input Multi-output Adaptive Fuzzy Control& apos;, In IEEE Conference on Decision and Control, 610±615, Kobe, Japan, September. [23] Shankar Sastry, S. and Bodson, M. (1989). Adaptive Control: Stability, Convergence,