Automatica 37 (2001) 1797}1802
Brief Paper
H controller reduction for nonlinear systems夽 Chee-Fai Yung *, He-Sheng Wang Department of Electrical Engineering, National Taiwan Ocean University, Keelung 202, Taiwan Center for Aviation and Space Technology, Bldg. 52, 195-12 Sec. 4, Chung Hsing Rd. Chutung, Hsinchu 310, Taiwan Received 15 October 1999; revised 11 October 2000; received in "nal form 4 April 2001
Abstract Su$cient conditions are proposed for the existence of reduced-order ("xed-order) controllers solving the standard nonlinear H output feedback control problem. State-space formulas for such reduced-order H controllers are also derived in terms of the solutions of two Hamilton}Jacobi inequalities. The development uses only elementary concepts of dissipativity and di!erential game, thus the proofs given are simple and clear. 2001 Elsevier Science Ltd. All rights reserved. Keywords: H control; Controller reduction; Hamilton}Jacobi inequality; Nonlinear systems
1. Introduction It has been shown that full-order H controllers can be constructed from two algebraic Riccati equations for linear systems or two Hamilton}Jacobi inequalities for nonlinear systems. The controllers thus obtained have a state dimension not less than that of the generalized plant (Ball, Helton, & Walker, 1993; Doyle, Glover, Khargonekar, & Francis, 1989; Isidori, 1994; Isidori & Kang, 1995; Lu & Doyle, 1994; Petersen, Anderson, & Jonckheere, 1991; Yung, Lin, & Yeh, 1996; Yung, Wu, & Lee, 1998). Since the generalized plant is built from the physical plant and some weighting functions that are used to re#ect performance and robustness requirements, the order of generalized plant may be very high. In this case, the full-order controllers may be of limited use in practical applications. Recently, a number of papers have appeared that deal with reduced-order (or "xed-order) H controller design for linear systems (see, e.g., DeShetler & Ridgely, 1992; Gahinet & Apkarian, 1994; Gu, Choi, & Postlethwaite, 1993; Haddad & Bernstein, 1990; Hsu, Yu, Yeh, & Banda, 1994; Hyland & Bernstein, 1984; Iwasaki & Skel夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Daizhan Cheng under the direction of Editor Hassan Khalil. * Corresponding author. Tel./fax: #886-2-2462-6993. E-mail addresses:
[email protected] (C.-F. Yung),
[email protected] (H.-S. Wang).
ton, 1993, 1994; Juang, Yeh, & Banda, 1996; Li & Chang, 1993; Pensar & Toivonen, 1993; Stoorvogel, Saberi, & Chen, 1991; Sweriduk & Calise, 1993; Xin, Guo, & Feng, 1996; Yeh, Rawson, & Banda, 1993; Yung, 2000). This paper continues this line of research to study the reduced-order H controller design problem for nonlinear systems. In terms of the two standard Hamilton}Jacobi inequalities (Isidori, 1994), we derive su$cient conditions for the existence of reduced-order ("xed-order) nonlinear H controllers, and give statespace formulas for such reduced-order nonlinear H controllers. The development uses only elementary concepts of dissipativity (Willems, 1972) and di!erential game (Basar & Bernhard, 1995), thus the proofs given are simple and clear.
2. Problem formulation and preliminaries Consider a smooth (i.e. C) nonlinear system described by the state equations x "f (x)#g (x)w#g (x)u, (1a) z"h (x)#k (x)u, (1b) y"h (x)#k (x)w, (1c) where x represents the state de"ned on a neighborhood of the origin in 1L. Throughout, we assume that the origin is an equilibrium, i.e. f (0)"0; without loss of
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 3 2 - 7
1798
C.-F. Yung, H.-S. Wang / Automatica 37 (2001) 1797}1802
generality, we assume also that h (0)"0 and h (0)"0. There are two inputs to the system: u31K is the control input and w31K represents a set of exogenous inputs which includes disturbances to be rejected and/or reference commands to be tracked. Eq. (1b) de"nes the controlled variable z31N expressed as cost of the state x and the input u required to achieve the prescribed performance speci"cations. y31N is the measured variable which is a function of the state x and the exogenous input w. In this paper, we restrict ourselves to the consideration of systems satisfying the following standing assumption considered in, e.g., Isidori (1994). Assumption A1. The matrices
is negative dexnite near x"0, where
1 < 2 w (x)" g2(x) (x), H 2 x
(H2) There exists a smooth positive dexnite function Q(x), locally dexned on a neighborhood of the origin in 1L, such that the function Q > (x)" (x)( fI (x)!g (x)k2 (x)R\(x)hI (x)) x 1 Q #u2 (x)R (x)u (x)# (x)g (x) H H 4 x
R (x) " : k2 (x)k (x) and R (x) " : k (x)k2 (x) are nonsigular for all x near x"0. Our aim in this paper is to "nd a reduced-order ("xedorder) output feedback controller of the form Q "F()#G()y, u"H(),
(2)
where 31P (r)n) is de"ned on a neighborhood of the origin, with F(0)"0 and H(0)"0, such that the resulting closed-loop system has a locally asymptotically stable equilibrium at the origin (x, )"(0,0), and has ¸-gain ), or equivalently, such that there exists a neighborhood of the origin (x, )"(0,0) such that for all ¹'0 and for each input w( ) )3¸[0, ¹], the state trajectory of the closed-loop system starting from the initial state (x(0), (0))"(0,0) remains in the neighborhood for all t3[0, ¹], and the response z( ) ) of the closed-loop system satis"es
2
z(t) dt)
2
Q 2 ;(I!k2 (x)R\(x)k (x))g2(x) (x) x ! hI 2(x)R\(x)hI (x) is negative dexnite near x"0 and its Hessian matrix is nonsingular at x"0, where fI (x)"f (x)#g (x)w (x), H hI (x)"h (x)#k (x)w (x). H Then the nonlinear H output feedback control problem is solved by the output feedback x( "fK (x( )#g( (x( )y, u"hK (x( ),
(4)
where x( 31L is dexned on a neighborhood of the origin, f K (x( )"fI (x( )#g (x( )u (x( )!g( (x( )hI (x( ), H hK (x( )"u (x( ), H
w(t) dt.
For details, see Van der Schaft (1992). We conclude this section by recalling from (Isidori, 1994) the following results which provides an output feedback controller solving the problem in question. Proposition 1. Consider system (1) and suppose Assumption A1 is satisxed. Suppose the following hypotheses hold. (H1) There exists a smooth positive dexnite function <(x), locally dexned on a neighborhood of the origin in 1L, such that the function < > (x)" (x) f (x)#h2(x)h (x) x #w2 (x)w (x)!u2 (x)R (x)u (x) H H H H
1 < 2 u (x)"!R\(x) g2(x) (x)#k2 (x)h (x) . H 2 x
(3)
and g( (x( ) satisxes Q Q (x( )g( (x( )"(2hI 2(x( )# (x( )g (x( )k2 (x( ))R\(x( ). x( x(
3. Main results In this section, we will propose a reduced-order controller of the form (2) that locally asymptotically stabilizes the resulting closed-loop system and renders its ¸-gain ). For this purpose, we "rst assume that there exists a smooth function : 1LP1P de"ned around the origin x"0 in 1L with (0)"0 and rank /x(0)"r. The rank condition implies that the restriction of to some neighborhood of x"0 is an surjection (Bartle,
C.-F. Yung, H.-S. Wang / Automatica 37 (2001) 1797}1802
1976). Then we make a change of variables K "!(x)#,
(5)
where K 31P and 31P are de"ned on a neighborhood of the origin. In terms of these variables the resulting closed-loop system is x "f (x )#g (x )w C C C C C z"h (x ) C C where x "col(x, K ), C
(6)
1799
Proof. Consider the candidate Lyapunov function =(x ). It is easy to see that, along the trajectories of the C closed-loop system,
d= = 2 K x , #z!w. ,w " C x dt C
(8)
Setting w"0 in the above equality and noting (7) shows that d=/dt is negative de"nite near x "0. This proves C that the equilibrium x "0 of the closed-loop system C is locally asymptotically stable. Furthermore, since K (x ))0 for all x near x "0 by hypothesis, (8) H C C C
f (x)#g (x)H(K #(x)) f (x )" , C C ! (x) f (x)! (x)g (x)H(K #(x))#F(K #(x))#G(K #(x))h (x) x x
g (x) g (x )" C C ! (x)g (x)#G(K #(x))k (x) x and
implies that d= #z!w)0 dt from which we conclude that the closed-loop system (6) has ¸-gain ). This completes the proof. 䊐
h (x )"h (x)#k (x)H(K #(x)). C C The problem of choosing a control law (2) in such a way that the ¸-gain of the closed-loop system (6) from the exogenous input w to the penalty output z is less than or equal to can be viewed as a game problem of rendering the so-called Hamiltonian function K: 1L>P;1L>P; 1K P1 de"ned as K(x , p, w)"p2( f (x )#g (x )w)#h (x )!w C C C C C C C nonpositive for each x and each (p, w). It is easy to verify C that the Hamiltonian function can be rewritten as
1 K(x , p, w)"K (x , p)! w! g2(x )p , C C 2 C C
(7)
where 1 K (x , p)"p2f (x )# p2g (x )g2(x )p#h2(x )h (x ). C C C C C C C C C C C 4
We are now in the position to state our main result. Theorem 3. Suppose that Assumption A1 is satisxed and that Hypotheses H1 and H2 of Proposition 1 hold. Suppose that there exists a smooth function : 1LP1P, locally dexned on a neighborhood of the origin x"0 in 1L, with (0)"0 and /x(0) (/x)2(0)"I. Suppose also that there exists a smooth positive dexnite function ;, locally dexned on a neighborhood of the origin "0 in 1P, which satisxes ;/(0) /x(0)"/x(0) Q/x(0). Then, if F, G, and H satisfy F((x))" (x) fK ( x), x
(9)
G((x))" (x)g( (x) x
(10)
and respectively
A preliminary lemma will be needed in the sequel.
H((x))"hK (x),
Lemma 2. Consider (1), (2) and (5). Suppose that Assumption A1 is satisxed. Suppose also that there exists a smooth positive dexnite function =(x ), locally dexned on a neighC borhood of the origin in 1L>P, such that the function K (x ) " : K (x , (=/x )2(x )) is negative for all nonH C C C C zero x around x "0. Then the controller (2) locally C C asymptotically stabilizes the resulting closed-loop system (6) and renders its ¸-gain ).
the rth order controller (2) locally asymptotically stabilizes the resulting closed-loop system (6) and renders its ¸-gain ).
(11)
Proof. With =(xC)"<(x)#;(K ), which is positive de"nite by construction, it can be shown by using (9)}(11) that the function K (x )"K (x,K ) satis"es H C H K (x,0)"> (x), K /K (x,0)"0, and K /K (0,0)" H H H
1800
C.-F. Yung, H.-S. Wang / Automatica 37 (2001) 1797}1802
/x(0)(S/x)(0)(/x)2(0), where the function S is de"ned as Q S(x)" (x)( fI (x)!g( (x)hI (x))#u2 (x)R (x)u (x) H H x 1 Q # (x)(g (x)!g( (x)k (x)) 4 x
Q 2 ;(g (x)!g( (x)k (x))2 (x). x By Taylor expansion theorem around K "0, we have K 1 H (x,0)K #h.o.t., K (x,K )"> (x)# K 2 H 2 K where `h.o.t.a means higher order terms. Note that S/x(0) is negative de"nite under the hypothesis (H2) (see Isidori, 1994) and > (x) is also negative de"nite by the hypothesis (H1). This shows that K (x,K ) is H negative for all nonzero x . This completes the proof by C Lemma 2. 䊐 Remark. (a) Currently, it is pretty di$cult to say how the order r of the reduced controller can be determined, in particular the minimal order r . As a matter of fact,
even for the linear case it is still an open problem (see Yung, 2000). However, we give a simple observation that may be used to deduce a possible minimal order of the reduced controller. To do this, we need the following proposition. See Isidori (1995) for details. Proposition 4. Let be a nonsingular involutive distribution of dimension r and assume that is invariant under the vector xelds f,K K , K ,2, K . (Here g( ,[K , K ,2, K ].) N N Moreover, suppose that the codistribution span dhK ,2, dhK (here hK ,[hK ,2, hK ]) is contained in the K K codistribution ,. Then for each point x it is possible to xnd a neighborhood ; of x and a local coordinates transformation zO (x) dexned on ; such that, in the new coordinates, the control system (4) is represented by equations of the form
Q "fK ( , )#g( ( , )y, K
Q "f ( )#g( ( )y, (12) y"hK ( ), where "(z ,2, z )2, "(z ,2, z )2. P P> L Therefore, if the conditions in Proposition 4 hold, the controller (4) can be decomposed in the form (12). In the system's input}output viewpoint, the system (12) is equivalent to the following system.
Q "fK ( )#g( ( )y, y"hK ( ).
De"ne a injection map by " (z) and a composite map O . Then the local coordinates trans formation O(x) will bring the controller (4) into the following reduced-order system. Q "f K ()#g( ()yOF()#G()u, y"hK ()OH(). (b) It has been shown in Theorem 3 that the achievement of closed-loop asymptotic stability is implied by the ful"llment of the negative de"niteness of > (x). If > (x) is just negative semide"nite near x"0, then closed-loop asymptotic stability can still be achieved if the equilibrium "0 of the system Q "F() is locally asymptotically stable (i.e. the controller (2) itself is internally stable) and system (1) satis"es the following standard assumption usually considered in the study of full-order H controller design problem (see Isidori (1994) for details). Assumption A2. Any bounded trajectory x(t) of the system x (t)"f (x(t))#g (x(t))u(t) satisfying h (x(t))#k (x(t))u(t)"0 for all t*0, is such that lim x(t)"0. R To see this, we observe that along any trajectory (x( ) ),K ( ) )) of the closed-loop system (6) with w"0 we have d=(x(t), K (t)) )!h (x(t))#k (x(t))H(K (t)#(x(t))) dt )0. This proves that the equilibrium (x,K )"(0,0) of the closed-loop system is stable. To prove asymptotic stability, we observe that any trajectory (x( ) ),K ( ) )) such that d=(x(t),K (t))/dt"0 for all t*0 is necessarily a trajectory of x "f (x)#g (x)H(K #(x)),
(13a)
QK "! (x)( f (x)#g (x)H(K #(x)))#F(K #(x)) x #G(K #(x))h (x) such that x(t) and K (t) are bounded and h (x(t))#k (x(t))H(K (t)#(x(t)))"0
(13b)
C.-F. Yung, H.-S. Wang / Automatica 37 (2001) 1797}1802
1801
H2 H Q #2 (0)R (0) (0)" (0) (0) K K x x
for all t*0. Thus, we have H(K (t)#(x(t)))"!R\(x(t))k2 (x(t))h (x(t))
for all t*0. As a result, (12) becomes x "f (x)!g (x)R\(x)k2 (x)h (x), QK "! (x)( f (x)!g (x)R\(x)k2 (x)h (x)) x
;
fK 2 hK (0)!g (0) (0) (0)# (0) x x x x
;
2Q 2 hK fK (0)!g (0) (0) (0) (0) x x x x
1 Q (0) (0)[g (0)!g( (0)k (0)] # 2 x x
#F(K #(x))#G(K #(x))h (x). Moreover, by Assumption A2 it is concluded that lim x(t)"0. Consequently, the hypothesis that the R equilibrium "0 of the system Q "F() is asymptotically stable and a well-known stability property of cascade connected systems (see, e.g., Isidori (1995, Appendix B.2)) imply that lim K (t)"0. Hence we can R conclude closed-loop asymptotic stability by LaSalle's invariance principle.
Q 2 (0) (0) ;[g (0)!g( (0)k (0)]2 x x hK 2 hK 2 #2 (0) (0)R (0) (0) (0). x x x x
(14)
Here in the second equality, we use the fact that H hK 2 (0)" (0) (0), K x x
4. Conclusions A method has been proposed for designing reducedorder H controllers of nonlinear systems. It has been shown that reduced-order nonlinear H controllers can be constructed from the solutions of two Hamilton}Jacobi inequalities and three auxiliary equations.
F fK 2 (0)" (0) (0) (0). K x x x The second equality in (14) is implied by the identity that ; Q (0) (0)" (0) (0). K x x x Next, it can be shown that
Acknowledgements The work was supported by the National Science Council of the Republic of China under Grant NSC-882115-M-019-001.
; Appendix Proof of K /K (0,0)"/x(0)(S/x(0))2/x(0). H First, observe that
K ; F H H (0,0)" (0) (0)! (0)g (0) (0) K K K K x
#
1 ; # (0) (0)g (0)!G(0)k (0) 2 K x
2; ; (0)g (0)!G(0)k (0) (0) x K
hI u2 u fI Q (0)!g( (0) (0) (0)#2 H (0) R (0) H (0) x x x x x
1 Q # (0)[g (0)!g( (0)k (0)] 2 x Q ;[g (0)!g( (0)k (0)]2 (0). x Then use the identity that
2; H F (0)! (0)g (0) (0) (0) K x K K
S Q fI hI (0)" (0) (0)!g( (0) (0) x x x x
f K (x)!g (x)u (x)"fI (x)!g( (x)hI (x), H it is easy to see that
S 2 K H (0,0)" (0) (0) (0). x x K x This completes the proof.
1802
C.-F. Yung, H.-S. Wang / Automatica 37 (2001) 1797}1802
References Ball, J. A., Helton, J. W., & Walker, M. L. (1993). H control for enonlinear systems with output feedback. IEEE Transactions on Automatic Control, 38, 546}559. Bartle, R. G. (1976). The elements of real analysis (2nd ed.). New York: Wiley. Basar, T., & Bernhard, P. (1995). H-optimal control and related minimax design problems: A dynamic game approach (2nd ed.). Boston: Birkhauser. DeShetler, D. R., & Ridgely, D. B. (1992). Reduced-order H and H compensation via gradient techniques, Proceedings of the 31th IEEE CDC, Tuscon, AZ, USA (pp. 2262}2267). Doyle, J. C., Glover, K., Khargonekar, P. P., & Francis, B. A. (1989). State-space solutions to standard H and H control problems. IEEE Transactions on Automic Control, AC-34, 831}847. Gahinet, P., & Apkarian, P. (1994). A linear matrix inequality approach to H control. International Journal of Robust and Nonlinear Control, 4, 421}448. Gu, D. W., Choi, B. W., & Postlethwaite, I. (1993). Low-order H suboptimal controllers. Proceedings of the 12nd world congress of IFAC, Sydney, Australia, Vol. 3, 347}350. Haddad, W. M., & Bernstein, D. S. (1990). Generalized Riccati equations for the full and reduced-order mixed-norm H /H standard problem. Systems and Control Letters, 14, 185}197. Hsu, C. S., Yu, X., Yeh, H. H., & Banda, S. S. (1994). H compensator design with minimal-order observers. IEEE Transactions on Automatic Control, 39, 1679}1681. Hyland, D. C., & Bernstein, D. S. (1984). The optimal projection equations for "xed-order dynamic compensation. IEEE Transactions on Automatic Control, 29, 1034}1037. Isidori, A. (1994). H control via measurement feedback for a$ne nonlinear systems. International Journal of Robust and Nonlinear Control, 4, 553}574. Isidori, A. (1995). Nonlinear control systems (3rd ed.). Berlin: Springer. Isidori, A., & Kang, W. (1995). H control via measurement feedback for general nonlinear systems. IEEE Transactions on Automatic Control, 40, 466}472. Iwasaki, T., & Skelton, R. E. (1993). All low order H controllers with covariance upper bound. Proceedings of the ACC, San Francisco, CA, USA (pp. 2180}2184). Iwasaki, T., & Skelton, R. E. (1994). All controllers for the general H control problem: LMI existence conditions and state space formulas. Automatica, 30, 1307}1317. Juang, J. C., Yeh, H. H., & Banda, S. S. (1996). Observer-based compensators for "xed-order H control. International Journal of Control, 64, 441}461. Li, X. P., & Chang, B. C. (1993). A parameterization approach to reduced-order H controller design. Proceedings of the 32nd IEEE CDC, San Antonio, TX, USA (pp. 2909}2184). Lu, W. M., & Doyle, J. C. (1994). H control of nonlinear system via output feedback: controller parameterization. IEEE Transactions on Automatic Control, 39, 2517}2521. Pensar, J. A., & Toivonen, H. T. (1993). On the design of "xed-structure H optimal controllers. Proceedings of the 32nd IEEE CDC, San Antonio, TX, USA (pp. 668}673). Petersen, I. R., Anderson, B. D. O., & Jonckheere, E. A. (1991). A "rst principles solution to the nonsigular H control problem. Interna tional Journal of Robust Nonlinear Control, 1, 171}185. Stoorvogel, A. A., Saberi, A., & Chen, B. M. (1991). A reduced-order observer-based controller design for H optimization. Proceedings
of the AIAA guidance, navigation, and control conference, New Orleans, LA, USA (pp. 716}722). Sweriduk, D., & Calise, A. J. (1993). Robust "xed-order dynamic compensation: a di!erential game approach. Proceedings of the IEEE conference on aerospace control systems, West Lake Village, CA, USA (pp. 458}462). Van der Schaft, A. J. (1992). ¸-gain analysis of nonlinear systems and nonlinear state feedback H control. IEEE Transactions on Automatic Control, 37, 770}784. Willems, J. C. (1972). Dissipative dynamical systems, part I: General theory. Archives for Rational Mechanics and Analysis, 45, 321}351. Xin, X., Guo, L., & Feng, C. B. (1996). Reduced-order controllers for continuous and discrete time singular H control problems based on LMI. Automatica, 32, 171}185. Yeh, H. H., Rawson, J. L., & Banda, S. S. (1993). Robust control design with real-parameter uncertainties, Proceedings of the 32nd IEEE CDC, Chicago, IL, USA (pp. 3249}3256). Yung, C. F., Lin, Y. P., & Yeh, F. B. (1996). A family of nonlinear H output feedback controllers. IEEE Transactions on Automatic Control, 41, 232}236. Yung, C. F., Wu, J. L., & Lee, T. T. (1998). H control for more general nonlinear systems. IEEE Transactions on Automatic Control, 43, 1724}1727. Yung, C. F. (2000). Reduced-order H controller design*An algebraic Riccati equation approach. Automatica, 36, 923}926.
Chee-Fai Yung was born in Taiwan, Republic of China in 1960. He received his BS degree in Electrical Engineering from National Taiwan University, Taipei in 1982, and the Ph.D. degree in Electrical Engineering from National Cheng-Kung University, Tainan, Taiwan in 1986. From 1986 to 1993, he was an Associate Researcher at the Chung-Shan Institute of Science and Technology, Taiwan. In January 1993, he became an Associate Professor in the Department of Mathematics, Tunghai University, Taichung, Taiwan. Since August 1993, he has been with the Department of Electrical Engineering, National Taiwan Ocean University, where he is currently a Professor. He has also been an Associate Professor with the Department of Electrical Engineering, National Taiwan Institute of Technology from 1988 to 1999. He received the Excellent Research Award in 2000 from National Science Council. His main research interests are robust control, nonlinear control, H control, descriptor systems theory, PC-based real-time control and applications.
He-Sheng Wang was born in Taiwan, Republic of China in 1970. He received the BS degree in Electrical Engineering from National Central University in 1993, and the MS and Ph.D. degrees in Electrical Engineering from National Taiwan University in 1995 and 1999, respectively. He is currently a research engineer at the Center for Aviation and Space Technology, Industrial Technology Research Institute (CAST/ITRI). His research interests include H control, linear and nonlinear control of descriptor systems, and mathematical theory of satellite navigation.