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Parametric absolute stability of multivariable Lur'e systems
Automatica 36 (2000) 1365}1372
Brief Paper
Parametric absolute stability of multivariable Lur'e systems夽 Teruyo Wada , Masao Ikeda * , Yuzo Ohta, Dragoslav D. S[ iljak Department of Mechanical Systems Engineering, College of Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan Department of Mechanical Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan Department of Computer and Systems Engineering, Faculty of Engineering, Kobe University, Nada, Kobe 657-8501, Japan School of Engineering, Santa Clara University, Santa Clara, CA 95053-0569, USA Received 7 July 1997; revised 6 October 1999; received in "nal form 25 January 2000
Abstract The objective of this paper is to consider parametric absolute stability of multivariable Lur'e systems with uncertain parameters and constant reference inputs. Both state-space and frequency-domain conditions are formulated to guarantee that the system remains stable despite the shift of the equilibrium location caused by changes of uncertain parameters and reference inputs. The state-space condition can be tested using existing software tools for solving linear matrix inequalities. For testing the Popov-type frequencydomain condition, the value set software tool for the Interval Arithmetic can be used. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Absolute stability; Nonlinear control systems; Uncertain dynamical systems; Equilibrium; Lyapunov function; Linear matrix inequality; Popov criterion
1. Introduction Parametric absolute stability has been introduced by Wada, Ikeda, Ohta and S[ iljak (1998) to present a new framework for dealing simultaneously with the existence and stability of the steady state of Lur'e systems subject to uncertain parameters and a scalar reference input. The proposed framework o!ers a departure from the standard robust absolute stability analysis of uncertain Lur'e systems, where it has been assumed that the equilibrium is "xed at the origin of the underlying state space (S[ iljak, 1969; GrujicH & Petkovski, 1987; Dahleh, Tesi & Vicino,
夽
A part of this paper was presented at the 13th IFAC World Congress, which was held in San Francisco, CA, USA, during 30 June}5 July 1996. This paper was recommended for publication in revised form by Associate Editor C.V. Hollot under the direction of Editor R. Tempo. * Corresponding author. Tel.: #81-6-6879-7335; fax: #81-6-68797247. E-mail address: [email protected] (M. Ikeda). Partially supported by the Ministry of Education, Culture, and Sports, Japan, under the Grant-in-Aid for Scienti"c Research (A) 07305014. Partially supported by the National Science Foundation, USA, under the grant ECS-9526142.
1993; Bhattacharyya, Chapellat & Keel, 1995). By adding a constant reference input, the location of the equilibrium changes and becomes uncertain (Wada et al., 1998); it may even disappear. For this reason, equilibrium analysis is needed, and cannot be avoided, as well as stability analysis. Then, the concept of parametric stability (Ikeda, Ohta & S[ iljak, 1991) becomes relevant. The notion of parametric stability has been introduced in the context of competitive equilibrium systems (S[ iljak, 1978) and Lotka-Volterra models of multispecies communities (Ikeda & S[ iljak, 1980), where uncertainty of system parameters almost always induces uncertainty of equilibrium locations. In Lur'e control systems with uncertain parameters and reference inputs, the study of parametric stability becomes particularly di$cult because of the inherent uncertainty in the nonlinear elements; the shape of the nonlinear functions are not known other than they satisfy the sector conditions. The purpose of this paper is to provide state-space and frequency-domain criteria for parametric absolute stability of multivariable Lur'e systems with reference inputs. Both criteria will be obtained in two steps. In the "rst step, conditions are presented to guarantee the existence of an equilibrium, the location of which, however, cannot be known exactly. Then, to provide possibly a less conservative sector condition on the nonlinearity at the
0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 5 2 - 2
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equilibrium point for stability analysis, the region of the existence of the uncertain equilibrium is estimated. In the second step, conditions for global asymptotic stability of the uncertain equilibrium are derived using parameterdependent Liapunov functions, which capture more faithfully the e!ect of uncertain parameters on system stability. The state-space condition is formulated in terms of parametric linear matrix inequalities (LMI), and the frequency-domain condition which contains parameters is obtained by applying a multivariable Popov condition (Wada & Ikeda, 1993). The frequency-domain analysis is an extension of the scalar nonlinearity case (Wada et al., 1998). In the case when the uncertain parameters are the coe$cients of the system matrices and the uncertainty set in the parameter space is a polytope, a "nite number of LMIs, which do not contain parameters, are su$cient to conclude stability. The non-parametric LMIs can easily be solved (e.g., Gahinet & Nemirovskii, 1993; Boyd, El Ghaoui, Feron & Balakrishnan, 1994). On the other hand, when the uncertain parameter set is characterized by intervals of individual parameters, the Popov-type condition can be tested by the Interval Arithmetic (e.g., Ohta, Gong & Haneda, 1990; Malan, Milanese, Taragna & Garlo!, 1992).
Fig. 1. Multivariable Lur'e system S.
utilize a Lur'e}Postnikov-type Liapunov function (Wada & Ikeda, 1993). Since the reference input r changes the operating point of the system S, we consider a neighborhood E of e"0, which is supposed a possible region of the equilibrium point of the control error e determined by r. Then, we assume a sector condition for the multivariable vectorvalued function u(e) at each point e( in E as 04(e!e( )2+u(e)!u(e( ),4(e!e( )2K(e( )(e!e( ), ∀e( 3E, ∀e3RK,
where K(e( ) is a positive-de"nite matrix continuously depending on e( 3E. Under the assumption (2), the condition (3) is equivalent to +u(e)!u(e( ),2K\(e( )+u(e)!u(e( ), 4(e!e( )2+u(e)!u(e( ),, ∀e( 3E, ∀e3RK,
2. Parametric absolute stability
(3)
(4)
(2)
which will be used extensively in this paper. Using (4) at e( "0 and the system description (1), we later estimate a region in which the actual equilibrium point of e exists, to con"rm that the sector condition (3) de"ned for E provides enough information about the nonlinearity u(e) for stability analysis of S. If it is not ensured, we need to restrict the allowable region R of the reference input r in a smaller set so that the estimated region of the equilibrium point, which will be given by Lemmas 1}3, is included in E, or we identify the sector condition (3) at e( in a region larger than E, including the estimated region. When the reference input r is 0, the origin x"0 is an equilibrium state of S for any p3P. For rO0, however, the location of the equilibrium state is not the origin and unknown due to uncertainty of the linear and nonlinear parts. Thus, stability properties of S become crucially dependent on the constant reference input r as well as the parameter vector p, as illustrated by Wada et al. (1998). For this reason, we treat the pair (r, p) as a parameter vector. To deal simultaneously with the existence of an equilibrium and absolute stability of the system under parametric uncertainty, the authors have introduced the concept of parametric absolute stability (Wada et al., 1998).
where Du(e) is the Jacobian matrix of u(e). The assumption (2) is required in stability analysis of S, where we
De5nition 1. The Lur'e system S of Fig. 1 is said to be parametrically absolutely stable if for any (r, p)3R;P
The Lur'e system is a feedback control system which is composed of a dynamic linear part and a static nonlinear part. In this paper, we consider a multivariable Lur'e system S illustrated by Fig. 1, where the linear part contains an uncertain constant parameter vector p. That is, S is described as S: x "A(p)x#B(p)u, y"C(p)x, u"u(e), e"r!y,
(1)
where x3RL, u3RK, y3RK are the state, input, output of the linear part, respectively, and e3RK is the control error. The reference input r3RK is a constant vector in a compact and simply connected region R including r"0. In (1), A(p), B(p), C(p) are matrices of appropriate dimensions, which depend continuously on the parameter vector p3RJ. We assume that the parameter vector p belongs to a compact and simply connected set P, and that for all p3P, A(p) is a stable matrix. We assume that the nonlinear function u : RKPRK is continuously di!erentiable, and satis"es the conditions u(0)"0 and +Du(e),2"Du(e), ∀e3RK,
T. Wada et al. / Automatica 36 (2000) 1365}1372
and any nonlinearity u(e) satisfying (2) and (3), the following conditions hold. (i) There exists a unique equilibrium state x(r, p) of S, and the corresponding control error e(r, p) is contained in E. (ii) x(r, p) is globally asymptotically stable.
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O is an n;m zero matrix, and I is an m-dimensional L"K K identity matrix, then for any (r, p)3R;P and any nonlinear function u(e) satisfying (6), there exists a solution
x e
x(r, p)
"
e(r, p)
of (5) and e(r, p) is in the region
3. LMI conditions in state space We present LMI su$cient conditions for parametric absolute stability of the Lur'e system S. We "rst formulate an LMI which ensures the existence of an equilibrium of S, then compute the region of the corresponding control error e, and provide LMI conditions for global asymptotic stability of the equilibrium.
2""X2 (p)r"" E(r, p)" e( 3RK : ""e( ""4 , (9) j [R(p)]
where X (p) is a submatrix consisting of m bottom rows of X(p), R(p) is the left side of (7), "" ) "" denotes the Euclidean norm, and j [ ) ] denotes the minimum eigenvalue.
The outline of the proof of Lemma 1 is given in the appendix.
3.1. Equilibrium analysis
3.2. Stability analysis
We recall the fact that if an equilibrium state exists and it is globally asymptotically stable, then the equilibrium is unique. For this reason, we do not mention the uniqueness of equilibrium here, which is required in (i) of De"nition 1, since we consider its global asymptotic stability in next subsection. For any (r, p)3R;P, an equilibrium state x(r, p) of S is computed as a solution of the equation
We now derive LMIs to be satis"ed for global asymptotic stability of an equilibrium state under the assumption that it exists. Let (r, p) be arbitrarily "xed in R;P, and x(r, p) be an equilibrium state of the system S. Then, S is equivalent to the deviation system
A(p)x#B(p)u(e)"0, e"r!C(p)x,
u (e )"u[e #e(r, p)]!u[e(r, p)], e "!C(p)x
(5)
which is obtained by setting x "0 in (1). We derive a condition for the existence of a solution of (5) under the assumption u2(e)K\(0)u(e)4e2u(e), ∀e3RK,
(6)
which is the sector condition for u(e) at the origin, that is, the case when e( "0 in (4). We give also an estimate of the region where the equilibrium point e(r, p) of the control error is located. The estimate is required for computation of an upper bound of the sector of the nonlinear function u(e) at e( "e(r, p), which is needed in the subsequent stability analysis. Lemma 1. If, for any p3P, there exists an (n#m); (n#m) matrix X"X(p) such that the LMI
M2(p)X#X2M(p) X2¸(p)#J
(7)
'0
2K\(0)
¸2(p)X#J2
holds, where
M(p)"
A(p) O L"K , C(p) I K
B(p)
¸(p)"
O K"K
, J"
O L"K , !I K (8)
SI : x "A(p)x #B(p)u [!C(p)x ],
(10)
where x "x!x(r, p), (11)
and e "e!e(r, p). The nonlinear function u (e ) satis"es u (0)"0 and +Du (e ),2"Du (e ), ∀e 3RK
(12)
which follows (2). Furthermore, if e(r, p)3E, then u (e ) satis"es the condition u (e )2K\[e(r, p)]u (e )4e 2u (e ), ∀e 3RK,
(13)
which is equivalent to the sector condition (4) at e( " e(r, p). We can consider stability of the equilibrium state x(r, p) of S by analyzing stability of the equilibrium state x "0 of SI . Since SI is also a Lur'e system, we use a parameterdependent Lur'e}Postnikov-type Liapunov function <(x )"x 2H(r, p)x
[!C(p)x ]2u [!hC(p)x ] dh, (14) where H(r, p) is a positive-de"nite matrix, and l(r, p) is a real number satisfying #l(r, p)
H(r, p)#l(r, p)C2(p)K[e(r, p)]C(p)'0. (15) For such H(r, p) and l(r, p), <(x ) is positive de"nite.
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Under (12), we calculate the total derivative of <(x ) with respect to SI (Wada & Ikeda, 1993) as
l(r, p) #2x 2 H(r, p)B(p)! A2(p)C2(p) u [!C(p)x ] 2 !l(r, p)u 2[!C(p)x ]C(p)B(p)u [!C(p)x ].
(16)
We then apply (13) to obtain
u [!C(p)x ]
.
(17)
Thus, the existence of H(r, p) and l(r, p) such that the last quadratic form is negative de"nite and (15) holds, is a su$cient condition for global asymptotic stability of the origin of SI . This implies that under the condition of Lemma 1, the existence of such H(r, p) and l(r, p) for each parameter (r, p)3R;P is su$cient for the Lur'e system S to be parametrically absolutely stable. To investigate their existence, we need to compute e(r, p) for each (r, p)3R;P since (15) and (17) contain e(r, p). However, the computation of e(r, p) is impossible because the nonlinearity u(e) is any function satisfying (2) and (3), and it is not given explicitly. To overcome such a problem, we utilize the region E(r, p) of e(r, p), which has been provided in Lemma 1, and employ the following upper bound matrix K (r, p) for K[e(r, p)] in (15) and (17): K (r, p)5K(e( ), ∀e( 3E(r, p).
(19)
A2(p)H#HA(p) l 1 B2(p)H! C(p)A(p)! C(p) 2 2 l 1 HB(p)! A2(p)C2(p)! C2(p) 2 2 (0 l !K\(r, p)! [B2(p)C2(p)#C(p)B(p)] 2
(20)
3.3. A Lur'e system with a polytopic linear part
l(r, p) 1 H(r, p)B(p)! A2(p)C2(p)! C2(p) 2 2 l(r, p) !K\[e(r, p)]! [B2(p)C2(p)#C(p)B(p)] 2 x
l H# C2(p)K (r, p)C(p)'0 2
hold. Then, the Lur'e system S is parametrically absolutely stable.
A2(p)H(r, p)#H(r, p)A(p) ; 1 l(r, p) C(p)A(p)! C(p) B2(p)H(r, p)! 2 2
;
LMIs
(18)
From Lemma 1 and the above discussion, we obtain the following LMI condition for parametric absolute stability of S. Theorem 1. Suppose that for any parameter p3P, there exists an (n#m);(n#m) matrix X"X(p) such that the LMI (7) holds, and for any reference input r3R, E(r, p)-E is satisxed. In addition, suppose that for any (r, p)3R;P, there exist a positive dexnite matrix H"H(r, p) and a real number l"l(r, p) such that the
When we apply the condition of Theorem 1, we generally have to test the LMIs for in"nitely many parameter values (r, p)3R;P. However, if the coe$cient matrices of the linear part of the parametric Lur'e system S are expressed as J J J A(p)" p A , B(p)" p B , C(p)" p C , (21) G G G G G G G G G we can obtain a "nite number of LMIs which guarantee parametric absolute stablity of S, where (A , B , G G C ), i"1, 2,2, l, are constant matrices and the region of G the parameter vector p"[p p 2 p ]2 is J
J P" p3RJ: p "1, p 50, i"1, 2,2, l . G G G
(22)
We call such a Lur'e system a polytopic Lur'e system. We "rst consider the existence of an equilibrium and the region of e(r, p). In Lemma 1 we replace X(p) with X common to all p3P. Although the resulting condition is more conservative, the LMI (7) becomes linear with respect to p and it can be reduced to a "nite number of LMIs as follows. Lemma 2. If there exists an (n#m);(n#m) matrix X which satisxes all LMIs
R" G
M2X#X2M G G ¸2X#J2 G
X2¸ #J G '0, i"1, 2,2, l, 2K\(0) (23)
where
A M" G G C G
O B L"K , ¸ " G , G I O K K"K
(24)
T. Wada et al. / Automatica 36 (2000) 1365}1372
then for any (r, p)3R;P, there exists an equilibrium state of the polytopic Lur'e system S and the equilibrium point e(r, p) is in the region
2""X "" " e( 3RK: ""e( ""4 E R max ""r"" , min j [R ] R G G PZ
(25)
where X is the submatrix consisting of the m bottom rows of X, and the matrix norm is that induced from the Euclidean vector norm. Lemma 2 is derived from Lemma 1 by using R(p)" J p R and j [R(p)]5min j [R ] for all p3P. G G G
G G Now we consider the stability condition in Theorem 1. given by (25) satis"es EM R -E. In We assume that E R Theorem 1, we replace K (r, p) with a positive de"nite M R which is independent of p and satis"es matrix K M R 5K(e( ), K
. ∀e( 3EM R
(26)
In addition, we replace H(r, p) with J H(p)" p H , H '0, i"1, 2,2, l G G G G
(27)
which is common to all r3R, and l(r, p) with l which is common to all (r, p)3R;P. Then, (19) and (20) are expressed as J J J p p ; " p; G I GI G GG G I G J\ J # p p +; #; ,'0, (28) G I GI IG G IG> J J J p p Q " pQ G I GI G GG G G I J\ J # p p +Q #Q ,(0, G I GI IG G IG>
(29)
where l M C , ; "H # C2K GI G 2 G R I Q " GI
A2H #H A G I I G l 1 B2H ! C A ! C G I 2 I G 2 I
(30)
l 1 H B ! A2C2! C2 I G 2 G I 2 I . l M \ !K R ! [B2C2#C B ] I G 2 G I
(31) Noting that (28) and (29) are in quadratic forms of p with nonnegative elements, we obtain the following theorem.
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Theorem 2. Suppose that there exists a matrix X satisfying all LMIs in (23), and for the given region R of the reference -E holds. In addition, suppose that there exist input r, EM R positive dexnite matrices H , i"1, 2,2, l, a real number l, G positive semidexnite matrices = , R , i(k, satisfying GI GI the LMIs ; #; 5!2= , i"1, 2,2, l!1, GI IG GI k"i#1, i#2,2, l,
; != $
!= ; $
2 != J 2 != J '0, $
!= J
!= J
2
Q #Q 42R , GI IG GI 2, l
Q R $ R
R Q $
;
JJ
i"1, 2,2, l!1,
(32)
k"i#1, i#2,
2 R J 2 R J (0. $
(33)
R 2 Q J J JJ Then, the polytopic Lur'e system S is parametrically absolutely stable. A numerical example which demonstrates this theorem was given in the original version (Wada, Ikeda, Ohta & S[ iljak, 1996) of this paper. 4. Popov-type conditions in frequency domain In this section, we provide su$cient conditions for parametric absolute stability in the frequency domain in terms of the transfer function matrix of the linear part. The conditions can be obtained similarly as in the scalar nonlinear case (Wada et al., 1998), although mathematical treatments are much complicated in the present multivariable case. Due to the space limitation, we omit the proofs of the lemma and theorem given below. We "rst give a matrix inequality which ensures the existence of an equilibrium. As seen in the previous section, an equilibrium x(r, p) of S is obtained as a solution of (5). Since A(p) is stable from the assumption and thus A\(p) exists, (5) can be transformed to x#A\(p)B(p)u(e)"0, e"r!C(p)x.
(34)
Substituting the "rst equation to the second one in (34), we obtain e!r#G (p)u(e)"0, (35) where G (p)"!C(p)A\(p)B(p). If (35) has a solution e"e(r, p), then from the "rst equation of (34), (5) has
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a solution x"x(r, p)"!A\(p)B(p)u[e(r, p)]. This means that the discussion on the existence of an equilibrium state x(r, p) can be reduced to that on the existence of e(r, p). We recall that we also need an estimate of the region of e(r, p) for the stability analysis. Thus, here we provide a condition for the existence of a solution e"e(r, p) of (35) and the corresponding estimate under the assumption (6) for the nonlinearity u(e). Lemma 3. If (36)
holds for all p3P, then for any (r, p)3R;P, a solution of (35) exists in the region
""r"" E(r, p)" e( 3RK : ""e( ""4 , k (p)
(37)
where
k (p)"
[+1#jl(r, p)u,G( ju, p)#+1!jl(r, p)u,G2(!ju, p)] #K\[e(r, p)]'0, ∀u3R (42) > holds, then the origin of S I is globally asymptotically stable, where G(s, p) is the transfer function matrix of the linear part G(s, p)"C(p)[sI!A(p)]\B(p)
+G (p)#G2 (p),#K\(0)'0
follows. If there exists a real number l(r, p) such that the matrix inequality
c (p) ""G (p)""
Theorem 3. Suppose that for any parameter p3P, the matrix inequality (38) (39)
o (p)"j [+G (p)#G2 (p),#+1!d (p),K\(0)],
(40) d (p) is a positive number satisfying +G (p)#G2 (p),#+1!d (p),K\(0)'0
(44)
for K[e(r, p)] in (42) under the assumption E(r, p)-E. Thus, we obtain a Popov-type su$cient condition for parametric absolute stability of S.
j [K(0)] d (p)o (p)
j [K(0)] ""G (p)#+1!d (p),K\(0)""
c (p)"j [+G (p)#G2 (p),],
and R "[0,#R)6+#R,. > If we can ensure the existence of l(r, p) satisfying (42), then we can conclude the global asymptotic stability of the equilibrium state x(r, p) of the original Lur'e system S. However, for the same reason as stated when we derived the condition of Theorem 1 from (15) and (17), it is not possible to test (42) directly. Therefore, we utilize the region E(r, p) of e(r, p), which has been given in Lemma 3, and employ the upper-bound matrix K (r, p) satifying K (r, p)5K(e( ), ∀e( 3E(r, p),
when c (p)'0
when c (p)40,
(43)
(41)
and j [ ) ] denotes the maximum eigenvalue.
For stability analysis of the equiribrium whose existence is guaranteed by the above lemma, we recall the multivariable Popov condition (Wada & Ikeda, 1993). Let (r, p) be an arbitrarily "xed vector in R;P and x(r, p) be an equilibrium state of S. Then, in the same manner as it was done in the previous section, S is transformed to the equivalent deviation system SI of (10), which is also a Lur'e system. For this system, the Popov condition, which is based on the parameter-dependent Lur'e}Postnikov-type Liapunov function (14), is as
+G(0, p)#G2(0, p),#K\(0)'0
(36)
holds, and for any reference input r3R, E(r, p)-E is satisxed. In addition, suppose that for any (r, p)3R;P, there exists a real number l(r, p) such that [+1#jl(r, p)u,G( ju, p)#+1!jl(r, p)u,G2(!ju, p)] # K\(r, p)'0, ∀u3R . (45) > Then, the Lur'e system S is parametrically absolutely stable. It is generally di$cult to investigate the existence of l(r, p) satisfying the condition (45) for in"nitely many parameter values (r, p)3R;P. To reduce the di$culty, we may simplify (45) at some expense of sharpness of the parameter-dependent condition. For this, we compute which is independent of parameter values a region EM RP (r, p) and includes E(r, p) for all (r, p)3R;P. We introM RP such that duce a constant matrix K M RP 5K(e( ), ∀e( 3EM RP K
(46)
M RP in (45). Furthermore, we and replace K (r, p) with K consider l as being common to all (r, p)3R;P. Then, we
T. Wada et al. / Automatica 36 (2000) 1365}1372
examine the existence of a real number l satisfying M RP [(1#jlu)G( ju, p)#(1!jlu)G2(!ju, p)]#K \'0, ∀u3RM , ∀p3P. (47) > The reduced condition is more conservative than that of Theorem 3, but it can easily be checked by the Interval Arithmetic (e.g., Ohta et al., 1990; Malan et al., 1992) if the region of the parameter vector p is given as intervals of individual elements.
5. Conclusion Su$cient conditions for parametric absolute stability have been presented for multivariable Lur'e systems in terms of parametric LMIs and Popov-type conditions. For polytopic Lur'e systems, a "nite number of nonparametric LMIs have been obtained, which can be solved by a standard computation. For Lur'e systems in which uncertainty is described as intervals of individual elements, a parametric Popov-type condition has been obtained, which can be tested by the interval arithmetic approach.
Appendix A. Outline of Proof of Lemma 1 Lemma 1 can be proven by setting
f (z, r)"
A(p)x#B(p)u(e) e!r#C(p)x
z
"[M(p) ¸(p)]
u(e)
0
!
r
with z"[x2 e2]2, and utilizing the following lemma. Lemma A1. If, for some (zH, rH)3RL>K;R, f (zH, rH)"0 is satisxed and there exists a k'0 such that "" f (z, r)!f (zH, r)""5k""z!zH"", ∀z3RL>K, ∀r3R,
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Gahinet, P., & Nemirovskii, A. (1993). A general purpose LMI solver with benchmarks. Proceedings of the 32nd IEEE conference on decision and control (pp. 3162}3165). GrujicH , Lj. T., & Petkovski, Dj. (1987). On robustness of Lurie systems with multiple nonlinearities. Automatica, 23, 327}334. Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1991). Parametric stability. In G. Conte, A. M. Perdon, & B. Wyman, New trends in systems theory (pp. 1}20) Boston, MA: Birkhauser. Ikeda, M., & S[ iljak, D. D. (1980). Lotka-Volterra equations: Decomposition, stability, and structure. Journal of Mathematical Biology, 9, 65}83. Malan, S., Milanese, M., Taragna, M., & Garlo!, J. (1992). B algorithm for robust performances analysis in presence of mixed parametric and dynamic perturbations. Proceedings of the 31st IEEE conference on decision and control (pp. 128}133). Ohta, Y., Gong, L., & Haneda, H. (1990). Polygon interval arithmetic and design of robust control systems. Proceedings of the 29th IEEE conference on decision and control (pp. 1065}1067). S[ iljak, D. D. (1969). Nonlinear systems. New York: Wiley. S[ iljak, D. D. (1978). Large-scale dynamic systems: Stability and structure. New York: North-Holland. Wada, T., & Ikeda, M. (1993). Extended Popov criteria for multivariable Lur'e systems. Proceedings of the 32nd IEEE conference on decision and control (pp. 20}21). Wada, T., Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1996). Parametric absolute stability of multivariable Lur'e systems: An LMI condition and application to polytopic systems. Preprints of the 13th IFAC world congress, E (pp 19}24). Wada, T., Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1998). Parametric absolute stability of Lur'e systems. IEEE Transactions on Automatic Control, 43, 1649}1653.
Teruyo Wada was born in Osaka, Japan. She received the B. Eng. and M. Eng. degrees in Applied Physics from Osaka University in 1983, 1985, respectively, and the D. Eng. degree from Kobe University, Japan in 1995. From 1987 to 1990, she was a Research Associate at the Graduate School of Science and Technology, Kobe University. She moved to the College of Engineering, Osaka Prefecture University, Japan in 1990, where she is presently an Assistant Professor at the Department of Mechanical Systems Engineering. In 1997, she was a Visiting Research Associate at Santa Clara University, California, USA. Her research interests include robust stability, modeling of nonlinear systems, and system identi"cation.
then, for any r3R, there exists a solution z(r) of f (z, r)"0. This lemma is obtained from a more general existence condition (Ikeda et al., 1991) by applying the mean value theorem. References Bhattacharyya, S. P., Chapellat, H., & Keel, L. H. (1995). Robust control: The parametric approach. Upper Saddle River, NJ: Prentice-Hall. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM. Dahleh, M., Tesi, A., & Vicino, A. (1993). An overview of extremal properties for robust control of interval plants. Automatica, 29, 707}721.
Masao Ikeda was born in Kochi, Japan. He received the B. Eng., M. Eng., and D. Eng. degrees in Communication Engineering from Osaka University, Japan in 1969, 1971, and 1975, respectively. He joined the Department of Systems Engineering, Kobe University, Japan in 1973, where he became a Professor in 1990. Since 1995, he has been a Professor at the Department of Computer-Controlled Mechanical Systems, Osaka University. He has held visiting academic appointments at Santa Clara University, California, USA, Arizona State University, USA, and the National Aerospace Laboratory, Japan. His research interests include decentralized control, stabilization of nonlinear and/or time-varying systems, two-degrees-of-freedom servosystems, application of control theory to practical systems, and dynamic mass measurement.
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Yuzo Ohta was born in Hyogo, Japan. He received the B. Eng. and M. Eng. degrees in Electrical Engineering from Kobe University, Japan in 1972 and 1974, respectively, and the D. Eng. degree in Electronic Engineering from Osaka University, Japan in 1977. From 1977 to 1987, he was with Fukui University, Japan. In 1987 he joined Kobe University as an Associate Professor of the Department of Electronics Engineering, and presently he is a Professor at the Department of Computer and Systems Engineering. From 1981 to 1982, he was a Visiting Research Associate at Santa Clara University, California, USA. His research interest includes stability theory, robust control, nonlinear control, and computer aided analysis/design of control systems.
Dragoslav D. S[ iljak received the Ph.D. degree in 1963 from the University of Belgrade, Belgrade, Yugoslavia. Since 1964 he has been with Santa Clara University, Santa Clara, California, USA, where he is the B & M Swig University Professor at the School of Engineering and teaches courses in system theory and applications. His research interests are in the theory of large scale systems and its applications to problems in control engineering, power systems, economics, aerospace, and model ecosystems. He is the author of the monographs Nonlinear Systems (Wiley, 1969), Large-Scale Dynamic Systems (North Holland, 1978), and Decentralized Control of Complex Systems (Academic Press, 1991). He is an honorary member of the Serbian Academy of Sciences and Arts, Belgrade, Yugoslavia, and is a Fellow of IEEE.
Brief Paper
Parametric absolute stability of multivariable Lur'e systems夽 Teruyo Wada , Masao Ikeda * , Yuzo Ohta, Dragoslav D. S[ iljak Department of Mechanical Systems Engineering, College of Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan Department of Mechanical Engineering, Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan Department of Computer and Systems Engineering, Faculty of Engineering, Kobe University, Nada, Kobe 657-8501, Japan School of Engineering, Santa Clara University, Santa Clara, CA 95053-0569, USA Received 7 July 1997; revised 6 October 1999; received in "nal form 25 January 2000
Abstract The objective of this paper is to consider parametric absolute stability of multivariable Lur'e systems with uncertain parameters and constant reference inputs. Both state-space and frequency-domain conditions are formulated to guarantee that the system remains stable despite the shift of the equilibrium location caused by changes of uncertain parameters and reference inputs. The state-space condition can be tested using existing software tools for solving linear matrix inequalities. For testing the Popov-type frequencydomain condition, the value set software tool for the Interval Arithmetic can be used. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Absolute stability; Nonlinear control systems; Uncertain dynamical systems; Equilibrium; Lyapunov function; Linear matrix inequality; Popov criterion
1. Introduction Parametric absolute stability has been introduced by Wada, Ikeda, Ohta and S[ iljak (1998) to present a new framework for dealing simultaneously with the existence and stability of the steady state of Lur'e systems subject to uncertain parameters and a scalar reference input. The proposed framework o!ers a departure from the standard robust absolute stability analysis of uncertain Lur'e systems, where it has been assumed that the equilibrium is "xed at the origin of the underlying state space (S[ iljak, 1969; GrujicH & Petkovski, 1987; Dahleh, Tesi & Vicino,
夽
A part of this paper was presented at the 13th IFAC World Congress, which was held in San Francisco, CA, USA, during 30 June}5 July 1996. This paper was recommended for publication in revised form by Associate Editor C.V. Hollot under the direction of Editor R. Tempo. * Corresponding author. Tel.: #81-6-6879-7335; fax: #81-6-68797247. E-mail address: [email protected] (M. Ikeda). Partially supported by the Ministry of Education, Culture, and Sports, Japan, under the Grant-in-Aid for Scienti"c Research (A) 07305014. Partially supported by the National Science Foundation, USA, under the grant ECS-9526142.
1993; Bhattacharyya, Chapellat & Keel, 1995). By adding a constant reference input, the location of the equilibrium changes and becomes uncertain (Wada et al., 1998); it may even disappear. For this reason, equilibrium analysis is needed, and cannot be avoided, as well as stability analysis. Then, the concept of parametric stability (Ikeda, Ohta & S[ iljak, 1991) becomes relevant. The notion of parametric stability has been introduced in the context of competitive equilibrium systems (S[ iljak, 1978) and Lotka-Volterra models of multispecies communities (Ikeda & S[ iljak, 1980), where uncertainty of system parameters almost always induces uncertainty of equilibrium locations. In Lur'e control systems with uncertain parameters and reference inputs, the study of parametric stability becomes particularly di$cult because of the inherent uncertainty in the nonlinear elements; the shape of the nonlinear functions are not known other than they satisfy the sector conditions. The purpose of this paper is to provide state-space and frequency-domain criteria for parametric absolute stability of multivariable Lur'e systems with reference inputs. Both criteria will be obtained in two steps. In the "rst step, conditions are presented to guarantee the existence of an equilibrium, the location of which, however, cannot be known exactly. Then, to provide possibly a less conservative sector condition on the nonlinearity at the
0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 5 2 - 2
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equilibrium point for stability analysis, the region of the existence of the uncertain equilibrium is estimated. In the second step, conditions for global asymptotic stability of the uncertain equilibrium are derived using parameterdependent Liapunov functions, which capture more faithfully the e!ect of uncertain parameters on system stability. The state-space condition is formulated in terms of parametric linear matrix inequalities (LMI), and the frequency-domain condition which contains parameters is obtained by applying a multivariable Popov condition (Wada & Ikeda, 1993). The frequency-domain analysis is an extension of the scalar nonlinearity case (Wada et al., 1998). In the case when the uncertain parameters are the coe$cients of the system matrices and the uncertainty set in the parameter space is a polytope, a "nite number of LMIs, which do not contain parameters, are su$cient to conclude stability. The non-parametric LMIs can easily be solved (e.g., Gahinet & Nemirovskii, 1993; Boyd, El Ghaoui, Feron & Balakrishnan, 1994). On the other hand, when the uncertain parameter set is characterized by intervals of individual parameters, the Popov-type condition can be tested by the Interval Arithmetic (e.g., Ohta, Gong & Haneda, 1990; Malan, Milanese, Taragna & Garlo!, 1992).
Fig. 1. Multivariable Lur'e system S.
utilize a Lur'e}Postnikov-type Liapunov function (Wada & Ikeda, 1993). Since the reference input r changes the operating point of the system S, we consider a neighborhood E of e"0, which is supposed a possible region of the equilibrium point of the control error e determined by r. Then, we assume a sector condition for the multivariable vectorvalued function u(e) at each point e( in E as 04(e!e( )2+u(e)!u(e( ),4(e!e( )2K(e( )(e!e( ), ∀e( 3E, ∀e3RK,
where K(e( ) is a positive-de"nite matrix continuously depending on e( 3E. Under the assumption (2), the condition (3) is equivalent to +u(e)!u(e( ),2K\(e( )+u(e)!u(e( ), 4(e!e( )2+u(e)!u(e( ),, ∀e( 3E, ∀e3RK,
2. Parametric absolute stability
(3)
(4)
(2)
which will be used extensively in this paper. Using (4) at e( "0 and the system description (1), we later estimate a region in which the actual equilibrium point of e exists, to con"rm that the sector condition (3) de"ned for E provides enough information about the nonlinearity u(e) for stability analysis of S. If it is not ensured, we need to restrict the allowable region R of the reference input r in a smaller set so that the estimated region of the equilibrium point, which will be given by Lemmas 1}3, is included in E, or we identify the sector condition (3) at e( in a region larger than E, including the estimated region. When the reference input r is 0, the origin x"0 is an equilibrium state of S for any p3P. For rO0, however, the location of the equilibrium state is not the origin and unknown due to uncertainty of the linear and nonlinear parts. Thus, stability properties of S become crucially dependent on the constant reference input r as well as the parameter vector p, as illustrated by Wada et al. (1998). For this reason, we treat the pair (r, p) as a parameter vector. To deal simultaneously with the existence of an equilibrium and absolute stability of the system under parametric uncertainty, the authors have introduced the concept of parametric absolute stability (Wada et al., 1998).
where Du(e) is the Jacobian matrix of u(e). The assumption (2) is required in stability analysis of S, where we
De5nition 1. The Lur'e system S of Fig. 1 is said to be parametrically absolutely stable if for any (r, p)3R;P
The Lur'e system is a feedback control system which is composed of a dynamic linear part and a static nonlinear part. In this paper, we consider a multivariable Lur'e system S illustrated by Fig. 1, where the linear part contains an uncertain constant parameter vector p. That is, S is described as S: x "A(p)x#B(p)u, y"C(p)x, u"u(e), e"r!y,
(1)
where x3RL, u3RK, y3RK are the state, input, output of the linear part, respectively, and e3RK is the control error. The reference input r3RK is a constant vector in a compact and simply connected region R including r"0. In (1), A(p), B(p), C(p) are matrices of appropriate dimensions, which depend continuously on the parameter vector p3RJ. We assume that the parameter vector p belongs to a compact and simply connected set P, and that for all p3P, A(p) is a stable matrix. We assume that the nonlinear function u : RKPRK is continuously di!erentiable, and satis"es the conditions u(0)"0 and +Du(e),2"Du(e), ∀e3RK,
T. Wada et al. / Automatica 36 (2000) 1365}1372
and any nonlinearity u(e) satisfying (2) and (3), the following conditions hold. (i) There exists a unique equilibrium state x(r, p) of S, and the corresponding control error e(r, p) is contained in E. (ii) x(r, p) is globally asymptotically stable.
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O is an n;m zero matrix, and I is an m-dimensional L"K K identity matrix, then for any (r, p)3R;P and any nonlinear function u(e) satisfying (6), there exists a solution
x e
x(r, p)
"
e(r, p)
of (5) and e(r, p) is in the region
3. LMI conditions in state space We present LMI su$cient conditions for parametric absolute stability of the Lur'e system S. We "rst formulate an LMI which ensures the existence of an equilibrium of S, then compute the region of the corresponding control error e, and provide LMI conditions for global asymptotic stability of the equilibrium.
2""X2 (p)r"" E(r, p)" e( 3RK : ""e( ""4 , (9) j [R(p)]
where X (p) is a submatrix consisting of m bottom rows of X(p), R(p) is the left side of (7), "" ) "" denotes the Euclidean norm, and j [ ) ] denotes the minimum eigenvalue.
The outline of the proof of Lemma 1 is given in the appendix.
3.1. Equilibrium analysis
3.2. Stability analysis
We recall the fact that if an equilibrium state exists and it is globally asymptotically stable, then the equilibrium is unique. For this reason, we do not mention the uniqueness of equilibrium here, which is required in (i) of De"nition 1, since we consider its global asymptotic stability in next subsection. For any (r, p)3R;P, an equilibrium state x(r, p) of S is computed as a solution of the equation
We now derive LMIs to be satis"ed for global asymptotic stability of an equilibrium state under the assumption that it exists. Let (r, p) be arbitrarily "xed in R;P, and x(r, p) be an equilibrium state of the system S. Then, S is equivalent to the deviation system
A(p)x#B(p)u(e)"0, e"r!C(p)x,
u (e )"u[e #e(r, p)]!u[e(r, p)], e "!C(p)x
(5)
which is obtained by setting x "0 in (1). We derive a condition for the existence of a solution of (5) under the assumption u2(e)K\(0)u(e)4e2u(e), ∀e3RK,
(6)
which is the sector condition for u(e) at the origin, that is, the case when e( "0 in (4). We give also an estimate of the region where the equilibrium point e(r, p) of the control error is located. The estimate is required for computation of an upper bound of the sector of the nonlinear function u(e) at e( "e(r, p), which is needed in the subsequent stability analysis. Lemma 1. If, for any p3P, there exists an (n#m); (n#m) matrix X"X(p) such that the LMI
M2(p)X#X2M(p) X2¸(p)#J
(7)
'0
2K\(0)
¸2(p)X#J2
holds, where
M(p)"
A(p) O L"K , C(p) I K
B(p)
¸(p)"
O K"K
, J"
O L"K , !I K (8)
SI : x "A(p)x #B(p)u [!C(p)x ],
(10)
where x "x!x(r, p), (11)
and e "e!e(r, p). The nonlinear function u (e ) satis"es u (0)"0 and +Du (e ),2"Du (e ), ∀e 3RK
(12)
which follows (2). Furthermore, if e(r, p)3E, then u (e ) satis"es the condition u (e )2K\[e(r, p)]u (e )4e 2u (e ), ∀e 3RK,
(13)
which is equivalent to the sector condition (4) at e( " e(r, p). We can consider stability of the equilibrium state x(r, p) of S by analyzing stability of the equilibrium state x "0 of SI . Since SI is also a Lur'e system, we use a parameterdependent Lur'e}Postnikov-type Liapunov function <(x )"x 2H(r, p)x
[!C(p)x ]2u [!hC(p)x ] dh, (14) where H(r, p) is a positive-de"nite matrix, and l(r, p) is a real number satisfying #l(r, p)
H(r, p)#l(r, p)C2(p)K[e(r, p)]C(p)'0. (15) For such H(r, p) and l(r, p), <(x ) is positive de"nite.
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Under (12), we calculate the total derivative of <(x ) with respect to SI (Wada & Ikeda, 1993) as
l(r, p) #2x 2 H(r, p)B(p)! A2(p)C2(p) u [!C(p)x ] 2 !l(r, p)u 2[!C(p)x ]C(p)B(p)u [!C(p)x ].
(16)
We then apply (13) to obtain
u [!C(p)x ]
.
(17)
Thus, the existence of H(r, p) and l(r, p) such that the last quadratic form is negative de"nite and (15) holds, is a su$cient condition for global asymptotic stability of the origin of SI . This implies that under the condition of Lemma 1, the existence of such H(r, p) and l(r, p) for each parameter (r, p)3R;P is su$cient for the Lur'e system S to be parametrically absolutely stable. To investigate their existence, we need to compute e(r, p) for each (r, p)3R;P since (15) and (17) contain e(r, p). However, the computation of e(r, p) is impossible because the nonlinearity u(e) is any function satisfying (2) and (3), and it is not given explicitly. To overcome such a problem, we utilize the region E(r, p) of e(r, p), which has been provided in Lemma 1, and employ the following upper bound matrix K (r, p) for K[e(r, p)] in (15) and (17): K (r, p)5K(e( ), ∀e( 3E(r, p).
(19)
A2(p)H#HA(p) l 1 B2(p)H! C(p)A(p)! C(p) 2 2 l 1 HB(p)! A2(p)C2(p)! C2(p) 2 2 (0 l !K\(r, p)! [B2(p)C2(p)#C(p)B(p)] 2
(20)
3.3. A Lur'e system with a polytopic linear part
l(r, p) 1 H(r, p)B(p)! A2(p)C2(p)! C2(p) 2 2 l(r, p) !K\[e(r, p)]! [B2(p)C2(p)#C(p)B(p)] 2 x
l H# C2(p)K (r, p)C(p)'0 2
hold. Then, the Lur'e system S is parametrically absolutely stable.
A2(p)H(r, p)#H(r, p)A(p) ; 1 l(r, p) C(p)A(p)! C(p) B2(p)H(r, p)! 2 2
;
LMIs
(18)
From Lemma 1 and the above discussion, we obtain the following LMI condition for parametric absolute stability of S. Theorem 1. Suppose that for any parameter p3P, there exists an (n#m);(n#m) matrix X"X(p) such that the LMI (7) holds, and for any reference input r3R, E(r, p)-E is satisxed. In addition, suppose that for any (r, p)3R;P, there exist a positive dexnite matrix H"H(r, p) and a real number l"l(r, p) such that the
When we apply the condition of Theorem 1, we generally have to test the LMIs for in"nitely many parameter values (r, p)3R;P. However, if the coe$cient matrices of the linear part of the parametric Lur'e system S are expressed as J J J A(p)" p A , B(p)" p B , C(p)" p C , (21) G G G G G G G G G we can obtain a "nite number of LMIs which guarantee parametric absolute stablity of S, where (A , B , G G C ), i"1, 2,2, l, are constant matrices and the region of G the parameter vector p"[p p 2 p ]2 is J
J P" p3RJ: p "1, p 50, i"1, 2,2, l . G G G
(22)
We call such a Lur'e system a polytopic Lur'e system. We "rst consider the existence of an equilibrium and the region of e(r, p). In Lemma 1 we replace X(p) with X common to all p3P. Although the resulting condition is more conservative, the LMI (7) becomes linear with respect to p and it can be reduced to a "nite number of LMIs as follows. Lemma 2. If there exists an (n#m);(n#m) matrix X which satisxes all LMIs
R" G
M2X#X2M G G ¸2X#J2 G
X2¸ #J G '0, i"1, 2,2, l, 2K\(0) (23)
where
A M" G G C G
O B L"K , ¸ " G , G I O K K"K
(24)
T. Wada et al. / Automatica 36 (2000) 1365}1372
then for any (r, p)3R;P, there exists an equilibrium state of the polytopic Lur'e system S and the equilibrium point e(r, p) is in the region
2""X "" " e( 3RK: ""e( ""4 E R max ""r"" , min j [R ] R G G PZ
(25)
where X is the submatrix consisting of the m bottom rows of X, and the matrix norm is that induced from the Euclidean vector norm. Lemma 2 is derived from Lemma 1 by using R(p)" J p R and j [R(p)]5min j [R ] for all p3P. G G G
G G Now we consider the stability condition in Theorem 1. given by (25) satis"es EM R -E. In We assume that E R Theorem 1, we replace K (r, p) with a positive de"nite M R which is independent of p and satis"es matrix K M R 5K(e( ), K
. ∀e( 3EM R
(26)
In addition, we replace H(r, p) with J H(p)" p H , H '0, i"1, 2,2, l G G G G
(27)
which is common to all r3R, and l(r, p) with l which is common to all (r, p)3R;P. Then, (19) and (20) are expressed as J J J p p ; " p; G I GI G GG G I G J\ J # p p +; #; ,'0, (28) G I GI IG G IG> J J J p p Q " pQ G I GI G GG G G I J\ J # p p +Q #Q ,(0, G I GI IG G IG>
(29)
where l M C , ; "H # C2K GI G 2 G R I Q " GI
A2H #H A G I I G l 1 B2H ! C A ! C G I 2 I G 2 I
(30)
l 1 H B ! A2C2! C2 I G 2 G I 2 I . l M \ !K R ! [B2C2#C B ] I G 2 G I
(31) Noting that (28) and (29) are in quadratic forms of p with nonnegative elements, we obtain the following theorem.
1369
Theorem 2. Suppose that there exists a matrix X satisfying all LMIs in (23), and for the given region R of the reference -E holds. In addition, suppose that there exist input r, EM R positive dexnite matrices H , i"1, 2,2, l, a real number l, G positive semidexnite matrices = , R , i(k, satisfying GI GI the LMIs ; #; 5!2= , i"1, 2,2, l!1, GI IG GI k"i#1, i#2,2, l,
; != $
!= ; $
2 != J 2 != J '0, $
!= J
!= J
2
Q #Q 42R , GI IG GI 2, l
Q R $ R
R Q $
;
JJ
i"1, 2,2, l!1,
(32)
k"i#1, i#2,
2 R J 2 R J (0. $
(33)
R 2 Q J J JJ Then, the polytopic Lur'e system S is parametrically absolutely stable. A numerical example which demonstrates this theorem was given in the original version (Wada, Ikeda, Ohta & S[ iljak, 1996) of this paper. 4. Popov-type conditions in frequency domain In this section, we provide su$cient conditions for parametric absolute stability in the frequency domain in terms of the transfer function matrix of the linear part. The conditions can be obtained similarly as in the scalar nonlinear case (Wada et al., 1998), although mathematical treatments are much complicated in the present multivariable case. Due to the space limitation, we omit the proofs of the lemma and theorem given below. We "rst give a matrix inequality which ensures the existence of an equilibrium. As seen in the previous section, an equilibrium x(r, p) of S is obtained as a solution of (5). Since A(p) is stable from the assumption and thus A\(p) exists, (5) can be transformed to x#A\(p)B(p)u(e)"0, e"r!C(p)x.
(34)
Substituting the "rst equation to the second one in (34), we obtain e!r#G (p)u(e)"0, (35) where G (p)"!C(p)A\(p)B(p). If (35) has a solution e"e(r, p), then from the "rst equation of (34), (5) has
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a solution x"x(r, p)"!A\(p)B(p)u[e(r, p)]. This means that the discussion on the existence of an equilibrium state x(r, p) can be reduced to that on the existence of e(r, p). We recall that we also need an estimate of the region of e(r, p) for the stability analysis. Thus, here we provide a condition for the existence of a solution e"e(r, p) of (35) and the corresponding estimate under the assumption (6) for the nonlinearity u(e). Lemma 3. If (36)
holds for all p3P, then for any (r, p)3R;P, a solution of (35) exists in the region
""r"" E(r, p)" e( 3RK : ""e( ""4 , k (p)
(37)
where
k (p)"
[+1#jl(r, p)u,G( ju, p)#+1!jl(r, p)u,G2(!ju, p)] #K\[e(r, p)]'0, ∀u3R (42) > holds, then the origin of S I is globally asymptotically stable, where G(s, p) is the transfer function matrix of the linear part G(s, p)"C(p)[sI!A(p)]\B(p)
+G (p)#G2 (p),#K\(0)'0
follows. If there exists a real number l(r, p) such that the matrix inequality
c (p) ""G (p)""
Theorem 3. Suppose that for any parameter p3P, the matrix inequality (38) (39)
o (p)"j [+G (p)#G2 (p),#+1!d (p),K\(0)],
(40) d (p) is a positive number satisfying +G (p)#G2 (p),#+1!d (p),K\(0)'0
(44)
for K[e(r, p)] in (42) under the assumption E(r, p)-E. Thus, we obtain a Popov-type su$cient condition for parametric absolute stability of S.
j [K(0)] d (p)o (p)
j [K(0)] ""G (p)#+1!d (p),K\(0)""
c (p)"j [+G (p)#G2 (p),],
and R "[0,#R)6+#R,. > If we can ensure the existence of l(r, p) satisfying (42), then we can conclude the global asymptotic stability of the equilibrium state x(r, p) of the original Lur'e system S. However, for the same reason as stated when we derived the condition of Theorem 1 from (15) and (17), it is not possible to test (42) directly. Therefore, we utilize the region E(r, p) of e(r, p), which has been given in Lemma 3, and employ the upper-bound matrix K (r, p) satifying K (r, p)5K(e( ), ∀e( 3E(r, p),
when c (p)'0
when c (p)40,
(43)
(41)
and j [ ) ] denotes the maximum eigenvalue.
For stability analysis of the equiribrium whose existence is guaranteed by the above lemma, we recall the multivariable Popov condition (Wada & Ikeda, 1993). Let (r, p) be an arbitrarily "xed vector in R;P and x(r, p) be an equilibrium state of S. Then, in the same manner as it was done in the previous section, S is transformed to the equivalent deviation system SI of (10), which is also a Lur'e system. For this system, the Popov condition, which is based on the parameter-dependent Lur'e}Postnikov-type Liapunov function (14), is as
+G(0, p)#G2(0, p),#K\(0)'0
(36)
holds, and for any reference input r3R, E(r, p)-E is satisxed. In addition, suppose that for any (r, p)3R;P, there exists a real number l(r, p) such that [+1#jl(r, p)u,G( ju, p)#+1!jl(r, p)u,G2(!ju, p)] # K\(r, p)'0, ∀u3R . (45) > Then, the Lur'e system S is parametrically absolutely stable. It is generally di$cult to investigate the existence of l(r, p) satisfying the condition (45) for in"nitely many parameter values (r, p)3R;P. To reduce the di$culty, we may simplify (45) at some expense of sharpness of the parameter-dependent condition. For this, we compute which is independent of parameter values a region EM RP (r, p) and includes E(r, p) for all (r, p)3R;P. We introM RP such that duce a constant matrix K M RP 5K(e( ), ∀e( 3EM RP K
(46)
M RP in (45). Furthermore, we and replace K (r, p) with K consider l as being common to all (r, p)3R;P. Then, we
T. Wada et al. / Automatica 36 (2000) 1365}1372
examine the existence of a real number l satisfying M RP [(1#jlu)G( ju, p)#(1!jlu)G2(!ju, p)]#K \'0, ∀u3RM , ∀p3P. (47) > The reduced condition is more conservative than that of Theorem 3, but it can easily be checked by the Interval Arithmetic (e.g., Ohta et al., 1990; Malan et al., 1992) if the region of the parameter vector p is given as intervals of individual elements.
5. Conclusion Su$cient conditions for parametric absolute stability have been presented for multivariable Lur'e systems in terms of parametric LMIs and Popov-type conditions. For polytopic Lur'e systems, a "nite number of nonparametric LMIs have been obtained, which can be solved by a standard computation. For Lur'e systems in which uncertainty is described as intervals of individual elements, a parametric Popov-type condition has been obtained, which can be tested by the interval arithmetic approach.
Appendix A. Outline of Proof of Lemma 1 Lemma 1 can be proven by setting
f (z, r)"
A(p)x#B(p)u(e) e!r#C(p)x
z
"[M(p) ¸(p)]
u(e)
0
!
r
with z"[x2 e2]2, and utilizing the following lemma. Lemma A1. If, for some (zH, rH)3RL>K;R, f (zH, rH)"0 is satisxed and there exists a k'0 such that "" f (z, r)!f (zH, r)""5k""z!zH"", ∀z3RL>K, ∀r3R,
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Gahinet, P., & Nemirovskii, A. (1993). A general purpose LMI solver with benchmarks. Proceedings of the 32nd IEEE conference on decision and control (pp. 3162}3165). GrujicH , Lj. T., & Petkovski, Dj. (1987). On robustness of Lurie systems with multiple nonlinearities. Automatica, 23, 327}334. Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1991). Parametric stability. In G. Conte, A. M. Perdon, & B. Wyman, New trends in systems theory (pp. 1}20) Boston, MA: Birkhauser. Ikeda, M., & S[ iljak, D. D. (1980). Lotka-Volterra equations: Decomposition, stability, and structure. Journal of Mathematical Biology, 9, 65}83. Malan, S., Milanese, M., Taragna, M., & Garlo!, J. (1992). B algorithm for robust performances analysis in presence of mixed parametric and dynamic perturbations. Proceedings of the 31st IEEE conference on decision and control (pp. 128}133). Ohta, Y., Gong, L., & Haneda, H. (1990). Polygon interval arithmetic and design of robust control systems. Proceedings of the 29th IEEE conference on decision and control (pp. 1065}1067). S[ iljak, D. D. (1969). Nonlinear systems. New York: Wiley. S[ iljak, D. D. (1978). Large-scale dynamic systems: Stability and structure. New York: North-Holland. Wada, T., & Ikeda, M. (1993). Extended Popov criteria for multivariable Lur'e systems. Proceedings of the 32nd IEEE conference on decision and control (pp. 20}21). Wada, T., Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1996). Parametric absolute stability of multivariable Lur'e systems: An LMI condition and application to polytopic systems. Preprints of the 13th IFAC world congress, E (pp 19}24). Wada, T., Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1998). Parametric absolute stability of Lur'e systems. IEEE Transactions on Automatic Control, 43, 1649}1653.
Teruyo Wada was born in Osaka, Japan. She received the B. Eng. and M. Eng. degrees in Applied Physics from Osaka University in 1983, 1985, respectively, and the D. Eng. degree from Kobe University, Japan in 1995. From 1987 to 1990, she was a Research Associate at the Graduate School of Science and Technology, Kobe University. She moved to the College of Engineering, Osaka Prefecture University, Japan in 1990, where she is presently an Assistant Professor at the Department of Mechanical Systems Engineering. In 1997, she was a Visiting Research Associate at Santa Clara University, California, USA. Her research interests include robust stability, modeling of nonlinear systems, and system identi"cation.
then, for any r3R, there exists a solution z(r) of f (z, r)"0. This lemma is obtained from a more general existence condition (Ikeda et al., 1991) by applying the mean value theorem. References Bhattacharyya, S. P., Chapellat, H., & Keel, L. H. (1995). Robust control: The parametric approach. Upper Saddle River, NJ: Prentice-Hall. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM. Dahleh, M., Tesi, A., & Vicino, A. (1993). An overview of extremal properties for robust control of interval plants. Automatica, 29, 707}721.
Masao Ikeda was born in Kochi, Japan. He received the B. Eng., M. Eng., and D. Eng. degrees in Communication Engineering from Osaka University, Japan in 1969, 1971, and 1975, respectively. He joined the Department of Systems Engineering, Kobe University, Japan in 1973, where he became a Professor in 1990. Since 1995, he has been a Professor at the Department of Computer-Controlled Mechanical Systems, Osaka University. He has held visiting academic appointments at Santa Clara University, California, USA, Arizona State University, USA, and the National Aerospace Laboratory, Japan. His research interests include decentralized control, stabilization of nonlinear and/or time-varying systems, two-degrees-of-freedom servosystems, application of control theory to practical systems, and dynamic mass measurement.
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Yuzo Ohta was born in Hyogo, Japan. He received the B. Eng. and M. Eng. degrees in Electrical Engineering from Kobe University, Japan in 1972 and 1974, respectively, and the D. Eng. degree in Electronic Engineering from Osaka University, Japan in 1977. From 1977 to 1987, he was with Fukui University, Japan. In 1987 he joined Kobe University as an Associate Professor of the Department of Electronics Engineering, and presently he is a Professor at the Department of Computer and Systems Engineering. From 1981 to 1982, he was a Visiting Research Associate at Santa Clara University, California, USA. His research interest includes stability theory, robust control, nonlinear control, and computer aided analysis/design of control systems.
Dragoslav D. S[ iljak received the Ph.D. degree in 1963 from the University of Belgrade, Belgrade, Yugoslavia. Since 1964 he has been with Santa Clara University, Santa Clara, California, USA, where he is the B & M Swig University Professor at the School of Engineering and teaches courses in system theory and applications. His research interests are in the theory of large scale systems and its applications to problems in control engineering, power systems, economics, aerospace, and model ecosystems. He is the author of the monographs Nonlinear Systems (Wiley, 1969), Large-Scale Dynamic Systems (North Holland, 1978), and Decentralized Control of Complex Systems (Academic Press, 1991). He is an honorary member of the Serbian Academy of Sciences and Arts, Belgrade, Yugoslavia, and is a Fellow of IEEE.