Criteria for dichotomy and gradient-like behavior of ... - Semantic Scholar
Automatica 43 (2007) 1583 – 1589 www.elsevier.com/locate/automatica
Brief paper
Criteria for dichotomy and gradient-like behavior of a class of nonlinear systems with multiple equilibria夡 Zhisheng Duan ∗ , Jinzhi Wang, Lin Huang State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, PR China Received 5 December 2005; received in revised form 24 November 2006; accepted 2 February 2007
Abstract This paper considers global properties of a class of nonlinear systems with infinite equilibria. Time-domain and frequency-domain criteria for dichotomy and gradient-like behavior are established. Similar to the sector condition in absolute stability problems, the bounded derivative condition of nonlinear functions can be used to reduce the conservativeness of the given criteria. Compared with the frequency-domain conditions, the time-domain inequalities can be solved easily by linear matrix inequality (LMI) toolbox and used to discuss controller design problems. Several examples are given to illustrate the effectiveness of the results. 䉷 2007 Published by Elsevier Ltd. Keywords: Dichotomy; Gradient-like behavior; Multiple equilibria; Linear matrix inequality (LMI)
1. Introduction The absolute stability of Lur’e systems has been studied extensively for several decades. The well-known circle and Popov criteria based-on frequency-domain method and matrix inequality conditions based-on time-domain method have been established (Khalil, 1996; Lozano, Brogliato, Egeland, & Maschke, 2000; Narendra & Taylor, 1973; Popov, 1973; Yakubovich, 1962). And according to the remarkable Kalman–Yakubovich–Popov (KYP) lemma Rantzer (1996), the frequency-domain and time-domain conditions are equivalent to each other, see Arcak, Larsen, and Kokotovic (2003) for the application of time and frequency-domain methods. Furthermore, by combining the sector restriction and slope restriction, less conservative criteria for absolute stability were also established, see Park (2002) and Suykens, Vandewalle, and De Moor (1998) and references therein. The study of 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Dragan Nesic under the direction of Editor Hassan Khalil. Work supported by the National Science Foundation of China under grants 60674093, 60334030, 10472001. ∗ Corresponding author. E-mail address: [email protected] (Z. Duan).
0005-1098/$ - see front matter 䉷 2007 Published by Elsevier Ltd. doi:10.1016/j.automatica.2007.02.003
absolute stability is of fundamental importance, e.g., in many problems of control and electrical circuits. Lur’e systems play important roles in neural networks (Guzelis & Chua, 1993) and synchronization theory (Curran & Chua, 1997). The absolute stability means the global stability of the single equilibrium which is an important property of nonlinear systems. It has been playing a significant role in dynamical systems. Besides global stability of the unique equilibrium, some other global properties such as dichotomy, Lagrange stability, gradient-like behavior, etc., are also very important for the study of nonlinear systems. Frequency-domain criteria of such properties have been established for a class of nonlinear system with multiple equilibria which was called pendulum-like system in Leonov, Ponomarenko, and Smirnova (1996). The pendulum-like system is a generalization of mathematical pendulum equations. It has many important applications in phaselocked loops and oscillation theory, see Leonov, Burkin, and Shepeljavyi (1992), Leonov et al. (1996) and references therein for a detailed discussion. Recently, analysis and control of such systems have got a new interest. Controller design and robustness analysis were studied in Duan, Huang, and Wang (2004), Wang, Huang, and Duan (2004), Yang, Fu, and Huang (2004). The nonexistence of bounded oscillating solutions in coupled systems was analyzed in Duan, Wang, and Huang (2005).
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Generally, it is hard to know the existence of chaotic attractors. Therefore, it is interesting to know the nonexistence of chaotic solutions by analyzing the property of dichotomy. This paper is devoted to the study of dichotomy and gradient-like behavior of pendulum-like systems. Though some frequency-domain criteria for such systems have been established in Leonov et al. (1996), this paper presents timedomain conditions directly by Lyapunov method. Then the corresponding frequency-domain interpretations are given based-on KYP lemma. In this way, the theorems are easier to be understood, and linear matrix inequality (LMI)-based conditions can be solved easily. The rest of this paper is organized as follows. In Section 2, time-domain and frequency-domain criteria of dichotomy are given for one of the canonical forms of pendulum-like systems. In Section 3, conditions of gradientlike behavior are presented similarly. In Section 4, the other canonical form of pendulum-like systems is considered and the corresponding criteria of dichotomy are provided. Examples are given to illustrate the results in Section 5. Throughout this paper, Re{Y } means 21 (Y + Y H ) for any real or complex square matrix Y . 2. Preliminaries Motivated by practical considerations in electric circuits, mechanics, and phase-synchronization theory, a class of nonlinear systems with infinite equilibria were studied in Leonov et al. (1996). And frequency-domain conditions for global properties such as dichotomy, gradient-like behavior and Bakaev stability were presented. In this paper we consider time and frequency domain conditions simultaneously for dichotomy and gradientlike behavior of such systems. First introduce some necessary preliminaries. Consider the following multi-input and multioutput system, ⎧ dx ⎪ = Ax + B(y), ⎨ dt ⎪ dy ⎩ = Cx + D(y), dt
(1)
where A ∈ Rn×n , B ∈ Rn×m , C ∈ Rm×n , D ∈ Rm×m , (y) = (1 (y1 ), . . . , m (ym ))T . Viewing y˙ as the output of the system, then the transfer function from (y) to y˙ is K(s) = C(sI − A)−1 B + D. Generally, the following assumptions for the system above are made: Assumption 1. A has no imaginary eigenvalues, (A, B) is controllable, (A, C) is observable and K(0) is nonsingular. Assumption 2. i : R → R is i -periodic, local Lipschitz continuous and possesses a finite number of zeroes on [0, i ), i = 1, . . . , m. In order to analyze the characteristics of system (1), define a set P as follows: P = {d = (0, . . . , 0, k1 1 , . . . , km m )T | ki ∈ Z},
where Z is the set of integers. Rewriting (1) as ˙ = f (t, ),
(2)
where =(x T y T )T and f : R+ ×Rn+m → Rn+m is continuous and locally Lipschitz continuous in the second argument, by the periodicity of i , (2) satisfies f (t, + d) = f (t, ), t 0, d ∈ P. This characteristic is just like the pendulum characteristic of mathematical pendulum equation, so system (1) is called pendulum-like system in Leonov et al. (1996). And because of the periodicity of i , system (1) is a typical nonlinear system with infinite isolated equilibria. The equilibrium set of (1) can be analyzed simply as follows. Any equilibrium (xeq , yeq ) of (1) satisfies Ax eq = −B(yeq ) and Cx eq = −D(yeq ). By the nonsingularity of A, it follows that (−CA−1 B + D)(yeq ) = 0. Since K(0) = D − CA−1 B is nonsingular (Assumption 1), (yeq ) = 0. Then one gets xeq = 0. Therefore, the equilibrium set E of (1) is E = {(xeq , yeq ) | xeq = 0, (yeq ) = 0}. Besides Assumptions 1 and 2, in this paper, suppose that i is continuously differentiable and i
di () i , −∞ < i , i < + ∞, i = 1, . . . , m. d
(3)
Let = diag(1 , . . . , m ), = diag(1 , . . . , m ). Before studying the global properties of system (1), first we introduce KYP lemma. This lemma will be used repeatedly in this paper to get conversions between time-domain and frequency-domain inequalities. Generally, in KYP lemma (Rantzer, 1996), the state matrix A is supposed to have no imaginary eigenvalues. In fact, for the non-strict inequality version of this lemma, it is admittable that A has zero eigenvalues other than pure imaginary eigenvalues, see Yakubovich–Kalman theorem given in Leonov et al. (1996) and Huang (2003). Noticing this point, we give the following lemma. Lemma 1. Given A ∈ Rn×n , B ∈ Rn×m , M = M T ∈ R(n+m)×(n+m) . Suppose that A has no pure imaginary eigenvalues and zero eigenvalues with multiplicity larger than 1 (there may be zero eigenvalue with multiplicity 1 for A), and (A, B) is controllable. Then the following two statements are equivalent: (i)
(jwI − A)−1 B
H
I
M
(jwI − A)−1 B I
0,
∀w ∈ R, (ii) there is a real symmetric matrix P such that M+
PA + AT P
PB
BTP
0
0.
Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
3. Dichotomy In some practical systems, we often care about the nonexistence of bounded oscillating solutions. At this time the convergence of bounded solutions is a nice property. Such kind of problems have been studied in Leonov et al. (1996) and Duan et al. (2005). Throughout this paper, system (1) is called to be dichotomous if every bounded solution is convergent to a certain equilibrium. System (1) is called to be gradient-like if every solution is convergent to a certain equilibrium. Remark 1. In this paper gradient-like behavior means that every solution of the corresponding system is convergent to a certain equilibrium. Since system (1) is with multiple equilibria, the convergence of every solution does not mean every equilibrium is Lyapunov stable. In fact, it can be shown that in the case of gradient-like behavior of system (1) there exists at least one equilibrium that is not Lyapunov asymptotically stable Leonov et al. (1992). Let x A B 0 z= , A˜ = , L= and (y) 0 0 Im C˜ = (C D), then system (1) can be rewritten as ˜ + L (y), z˙ = Az
˜ y˙ = Cz,
(4)
where (y) = d(y)/dt = (y)y. ˙ Using the method of Leonov et al. (1996), we can establish the LMI conditions for the property of dichotomy of system (1). Theorem 1. Under Assumptions 1 and 2, system (1) is dichotomous for all with condition (3), if there exist diagonal matrices P , and R with R 0, a symmetric matrix W and a scalar > 0 such that the following LMI is feasible WL + 21 C˜ T R( + ) 0, (5) LT W + 21 ( + )R C˜ −R ˜ where = W A˜ + A˜ T W − C˜ T R C˜ + LPC˜ + C˜ T PLT + C˜ T C. Proof. Take W (t) = zT (t)W z(t) and V (t) = dW (t)/dt + ˜ + L (y)) + 2T (y)P y˙ + y˙ T y. ˙ Note that V (t) = 2zT W (Az T T T ˜ ˜ ˜ 2z LP Cz + z C Cz. For all zi = 0, the bounded derivative condition (3) is equivalent to ui (xi ) =: ( i (yi ) − i y˙i )( i (yi ) − i y˙i ) 0, where i (yi ) = di (yi )/dt. Noticing that for any ri 0, i = 1, . . . , m, we have m
˜ ri ui (xi ) = T (y)R (y) − T (y)( + )R Cz
i=1
˜ 0, + z C˜ T R Cz T
(6)
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where R = diag(r1 , . . . , rm ). (5) guarantees that V (t) T (y) ˜ + zT C˜ T R Cz, ˜ ∀(z, (y)) = R (y) − T (y)( + )R Cz 0. By (6), it follows that V (t) 0. From 0 to t (t 0), inte t yi (t) grating V (t) gives 0 y˙ T y˙ dt − m i=1 pi yi (0) i (yi ) dyi − W (t) + W (0), where pi is the ith diagonal element of P . For any bounded solution of (1), y(t) and W (t) are also bounded. Therefore, it follows from the inequality above that y˙i ∈ L2 [0, +∞),
i = 1, . . . , m.
(7)
Since i (yi ), i = 1, . . . , m, has a bounded derivative for almost all t 0, y˙i has a bounded derivative too. Hence, y˙ is uniformly continuous on [0, +∞). Together with (7), we have y˙ → 0
as t → +∞.
(8)
Repeating the similar discussions as in Leonov et al. (1996), one can complete the proof easily. Remark 2. Compared with the results of Leonov et al. (1996), we established the LMI condition of dichotomy directly by time-domain method. In this way, the proof of the theorem would be much easier to be understood and the inequality (5) can be solved easily by the powerful toolbox (Gahinet, Nemirovski, & Laub, 1995). Since the frequency-domain inequality needs to be solved to test the criteria of global properties, generally only examples of single-input and single-output systems were considered in Leonov et al. (1996). However Theorem 1 can be easily used to study multi-input and multi-output systems, see the forthcoming examples. And obviously, Theorem 1 can also be used to consider controller design problems as studied in Duan et al. (2004). By Lemma 1, an equivalent frequency-domain condition for (5) can be given, which is just the frequency-domain result given in Leonov et al. (1996). Theorem 2. With the conditions of Theorem 1, the LMI (5) is feasible if, and only if, the following frequency domain inequality holds Re{PK(jw) − (jwI − K(jw))H R(jwI − K(jw))} + K H (jw)K(jw)0,
∀w ∈ R.
(9)
˜ ˜ −1 L = Proof. By simple computation, we have C(jwI − A) −1 T ˜ L=(1/jw)I . Taking an appropri(1/jw)K(jw), L (jwI− A) ate matrix M, then by Lemma 1 we know that (5) holds if, and only if, Re{(2/w 2 )PK(jw) − (1/w 2 )(jwI − K(jw))H R(jwI − K(jw))} + ( /w 2 )K H (jw)K(jw)0, ∀w ∈ R. The equivalence between (9) and the inequality above completes the proof. Obviously, when R = 0, the frequency-domain inequality in Theorem 2 reduces to Re{PK(jw)} + K H (jw)K(jw) 0, ∀w ∈ R. And correspondingly, the LMI (5) reduces to a simple inequality as shown in the following corollary. Corollary 1. Under Assumptions 1 and 2, system (1) is dichotomous for all with condition (3), if there exist a diagonal
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matrix P , a symmetric matrix W and a scalar > 0 such that the following LMI is feasible
The time derivative of v(t) along any trajectory of system (4) is given by
˜ + L (y)) + zT LP Cz ˜ − |(y)|T P V Cz, ˜ v(t) ˙ = 2zT W (Az
WA + AT W + C T C
WB + C T P + C T D
B T W + PC + D T C
D T D + PD + D T P
0.
(10)
Remark 3. The condition (3) plays a role in Theorems 1 and 2 as the canonical sector condition in the absolute stability of Lur’e systems. The forthcoming example shows that Theorem 1 is generally less conservative than Corollary 1. This means that the condition (3) can be used to improve the condition of dichotomy. On the other hand, system (1) can be considered as an uncertain nonlinear system, where the uncertain set is described by (3). If i − i is smaller for i = 1, . . . , m, i.e., the uncertain set is smaller, Theorem 1 is generally less conservative. This will also be shown by the forthcoming example. In addition, for the problem of dichotomy A does not need to be stable, so W is generally not positive definite in (10). Similarly, W in (5) is not necessarily positive definite. Obviously, if Re{P K(jw)} < 0, ∀w ∈ R, which is just like a strictly positive real condition in control theory, then the inequalities (5) and (10) hold.
(12) where |(y)|T =: (|1 (y1 )|, . . . , |m (ym )|), P =: diag(p1 , . . . , ˜ + L (y)) + zT LPCz ˜ − pm ). Furthermore, v(z) ˙ = 2zT W (Az T T T T T ˜ ˜ ˜ |(y)| PVCz − (y)E(y) − z C QCz + (y)E(y) + ˜ zT C˜ T QCz, where Q =: diag(q1 , . . . , qm ) and E =: diag(e1 , . . . , em ). By the condition (ii), we know that there exist Q0 =: diag(q01 , . . . , q0m ) and E0 =: diag(e01 , . . . , e0m ) such T (y)E(y)+zT C ˜ ˜ T (y)E0 (y)+ ˜ T QCz that |(y)|T PVCz+ T T ˜ ˜ z C Q0 Cz. Hence, we have ˜ v(z) ˙ + T (y)E0 (y) + zT C˜ T Q0 Cz ˜ + 2zT W AL (y) ˜ ˜ 2zT W Az + zT LPCz ˜ + T (y)E(y) + zT C˜ T QCz.
On the other hand, condition (i) of the theorem guarantees that ˜ + L (y)) + zT LPCz ˜ + T (y)E(y) + zT C˜ T QCz ˜ 2zT W (Az ˜ T (y)R (y) − T (y)( + )R Cz ˜ + zT C˜ T R Cz,
4. Gradient-like behavior The property of dichotomy is related to the convergence of bounded solutions. We can also discuss the global convergence (gradient-like behavior) of system (1) by the method above. Let vi = 0 i i ( ) d / 0 i |i ( )| d , V = diag(v1 , . . . , vm ), Fi ( ) = i ( ) − vi |i ( )|. Note that 0 i Fi ( ) d = 0, i = 1, . . . , m. Combining the periodic Lyapunov function method of Leonov et al. (1996) with the method of Theorem 1, we have Theorem 3. Under Assumptions 1 and 2, system (1) is gradient-like, if A is stable and there exist diagonal matrices P , Q, E and R with R 0, E > 0 and Q > 0, and a symmetric matrix W such that the following LMIs are feasible
WL + 21 C˜ T R( + )
i=1
LT W + 21 ( + )R C˜
−R
0,
˜ A˜ T W − C˜ T R C˜ + 1 LPC˜ + 1 C˜ T PLT + where =W A+ 2 2 T C˜ QC˜ + LELT . (ii)
2E
PV
VP
2Q
> 0.
v(t) =: z Wz +
m
i=1
pi
0
i = 0, . . . , m.
(16)
Repeating the similar discussions as in Leonov et al. (1996), one complete the proof easily. Similar to Theorem 2, a frequency-domain interpretation for the condition (i) of Theorem 3 can be given as follows. Theorem 4. Under conditions of Theorem 3, condition (i) of Theorem 3 is feasible if, and only if, the following frequencydomain inequality holds Re{PK(jw) − (jwI − K(jw))H R(jwI − K(jw))} + K H (jw)QK(jw) + E 0,
Proof. Take Lyapunov function candidate T
(14)
∀t 0. In addition, v(t) is bounded. This property follows from the facts that A is stable (and consequently z(t) is bounded), the functions Fi ( ) have mean value zero (and con y (t) sequently 0 i Fi ( ) d are bounded). Then it follows that +∞ 2 i (yi (t)) dt < + ∞, i = 0, . . . , m. Together with the fact 0 that i (yi (t)) are uniformly continuous, we have t→+∞
∀(z, (y)) = 0.
Combining (13), (14) and (6), we know that v(x) ˙ + ˜ 0. T (y)E0 (y) + zT C˜ T Q0 Cz From 0 to t integrating the two sides of the inequality above gives m t
v(t) − v(0) − (e0i 2i (yi (t)) + q0i y˙i2 (t)) dt, (15)
lim i (yi (t)) = 0,
(i)
(13)
yi 0
Fi ( ) d .
(11)
∀w ∈ R.
(17)
Proof. By the method of Theorem 2, (5) holds if, and only if, Re{(1/w 2 )PK(jw) − (1/w 2 )(jwI − K(jw))H R(jwI − K(jw))} + (1/w 2 )K H (jw)QK(jw) + (1/w 2 )E 0, ∀w ∈ R.
Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
The equivalence of (17) and the inequality above completes the proof. Similar to the discussions in the section above, when R = 0 in (17), the inequality (17) reduces to Re{PK(jw)} + K H (jw)QK(jw) + E 0, ∀w ∈ R. And correspondingly, Theorem 2 reduces to Corollary 2. Under Assumptions 1 and 2, system (1) is gradient-like, if A is stable and there exist diagonal matrices P , Q, E, and a symmetric matrix W such that (ii) of Theorem 3 and the following LMI are feasible
WA + AT W
+ C T QC
WB + C T P
B T W + PC + D T QC
where
= PD + D T P
+ C T QD
0,
1996). Then take a nonsingular matrix T such that ˆ −1 = (C1 C2 ) and CT B1 A1 0 ˆ −1 = T AT , T Bˆ = , 0 0 B2
Remark 4. From Corollary 2 we know that the stability of A and the observability of (A, C) imply > 0 in (18). that W yi However, because of the existence of m p F i ( ) d and i=1 i 0 ˜ A having zero eigenvalues, in Theorem 3 for the gradient-like behavior of system (1) the Lyapunov function v(t) in (11) is not necessarily positive definite. And in the proof of Theorem 3, the positive definiteness of v(t) is not required. This is different from the canonical absolute stability problems of Lur’e systems in which the Lyapunov function is positive definite. From Theorem 3, we can see that there is a parameter matrix coupling in the condition (ii). If V = 0, the condition (ii) disappears. Then we have Corollary 3. With the assumptions of Theorem 3, if the nonlinear function i ( ), i = 1, . . . , m, have zero mean values (vi = 0), then system (1) is gradient-like when condition (i) of Theorem 3 or (17) holds. System (1) studied above is one of the two canonical forms of pendulum-like systems. With the results above, we can also study the other canonical form of such systems.
Tx =
,
where A1 ∈ R(n−m)×(n−m) , B2 ∈ Rm×m , C2 ∈ Rm×m and B1 , C1 are matrices with compatible dimensions. Noticing the transformation, differentiating y in (19) gives the following system ˙ = A1 + B1 (y), y˙ = C1 A1 + (C1 B1 + C2 B2 )(y),
(18)
+ D T QD + E.
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(20)
which is just in the form of (1). Let the transfer function from ˆ Comparing (19) and ˆ ˆ −1 B. (y) to y in (19) be (s)= C(sI − A) (20), we know that (s)=(1/s)K(s), where K(s)=C1 A1 (sI − A1 )−1 B1 + (C1 B1 + C2 B2 ). Similar to Assumption 1, we suppose the following assumption holds for system (19). ˆ B) ˆ is controllable, (A, ˆ C) ˆ is observable and Assumption 3. (A, besides m zero eigenvalues (each of them is with multiplicity one), Aˆ has no pure imaginary eigenvalues, and s(s)|s=0 = K(s)|s=0 = K(0) is nonsingular. Obviously, by the controllability and observability of ˆ B, ˆ C), ˆ C2 and B2 are nonsingular. (A, By Theorem 2, we can establish the following frequencydomain criterion of dichotomy expressed in (s) for system (19). Theorem 5. Under Assumptions 2 and 3, system (19) is dichotomous for all with condition (3), if there exist diagonal matrices P and R with R 0, and a scalar > 0 such that 1 Re P (jw) − (I − (jw))∗ R(I − (jw)) −jw + ∗ (jw)(jw)0,
∀w ∈ R.
(21)
Let C0 = (0m×(n−m) B2−1 ). Then similar to Theorem 1, we can also establish LMI-based criterion of dichotomy expressed in ˆ B, ˆ C). ˆ (A,
5. The other canonical form of pendulum-like systems Consider the following nonlinear system, ⎧ dx ˆ + B(y), ˆ ⎨ = Ax dt ⎩ ˆ y = Cx,
(19)
where Aˆ ∈ ∈ ∈ m (ym ))T , i are periodic functions as discussed in the sections above. Suppose that Aˆ has m zero eigenvalues and the elementary factor of each zero eigenvalue is 1. Eq. (19) is the other canonical form for pendulum-like systems (Leonov et al., Rn×n , Bˆ
Theorem 6. Under Assumptions 2 and 3, system (19) is dichotomous for all with condition (3), if there exist diagonal matrices P and R with R 0, and a symmetric matrix W and a scalar > 0 such that the following LMI is feasible
Rn×m , Cˆ
Rm×n , (y)=(1 (y1 ), . . . ,
Bˆ T W + 21 ( + )R Cˆ
W Bˆ + 21 Cˆ T R( + ) −R
0, (22)
ˆ Aˆ T W − Cˆ T R Cˆ +T T C T P Cˆ + Cˆ T PC0 T + where =W A+ 0 ˆ Cˆ T C.
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Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
Proof. By simple computation, we know that C0 T (sI − ˆ −1 T −1 T Bˆ = C0 (sI − T AT ˆ −1 )−1 T Bˆ = ˆ −1 Bˆ = C0 T (sI − A) A) (1/s)I . Then by Lemma 1, we know that the LMI (22) is equivalent to (21). Similarly, we can also establish conditions of gradient-like ˆ B, ˆ C) ˆ for system (19) by the behavior expressed in (s) or (A, method above.
2 1.5 1 0.5 0 -0.5
6. Examples Example 1. Consider the system in the form of (1) with the following data −0.4 3 1 0 A= , B= , −1 −0.5 −1.4 1 2 −1 1.2 C= , D= 0 1 −2 1 and 1 (y1 )=sin(y1 )−0.2, 2 (y2 )=sin(2y2 )−0.1. By Theorem 1, we know that this system is dichotomous when 1.9. If we take = −1.9, the condition (5) fails. At this time, we can see a solution as shown in Figs. 1 and 2 at the initial value x = (1, −0.5)T , y = (0.1, −5)T by computer simulation. From this solution, we can see that the solution x is something like a chaotic solution and y is unbounded when t → ∞ which show the complexity of solutions of pendulum-like systems. The chaotic phenomenon in pendulum-like systems is still a new topic which is worthy study.
-1 -1.5 -2
-3
-2
-1
0
1
2
3
Fig. 1. The solution x.
180 160 140 120 100 80 60 40 20
Example 2. Consider the system in the form of (1) with the following data ⎛ ⎞ ⎛ ⎞ −0.9 0 0.8 1 0 ⎜ ⎟ ⎜ ⎟ A=⎝ 0 0 1.1 ⎠ , B = ⎝ 0 0 ⎠ , 0 C=
0.3 0
−2.5 −1 0 −1 0 0 0.5 −0.1 , D= 0.2 0 −0.1 1
and 1 (y1 )=sin(y1 )−r, 2 (y2 )=sin(2y2 )−r, r is a parameter to be determined. For i given here, obviously we have =
−1
0
0
−2
,
=
1
0
0
2
.
By Theorem 3, we know that this system is gradient-like when r 0.82. However, if we take R = 0 in the condition (i) of Theorem 3, at this time Theorem 3 reduces to Corollary 2, then we can only know that the system is gradient-like when r 0.77. This shows the effects of (3) in reducing the conservativeness, i.e., Theorem 3 is generally less conservative than Corollary 2. Please refer to Figs. 3 and 4 for the gradient-like behavior of its solutions at three initial values x = (1, 0.5, 0.1)T , y = (2.1, 2)T ; x = (0.2, −2.5, 5)T , y = (3, 1)T ; x = (0.5, −1.5, 4.2)T , y = (3.5, 0.9)T .
0 -20 -20
0
20
40
60
80 100 120 140 160
Fig. 2. The solution y.
6 5 4 3 2 1 0 -1 -2 -3
0
2
4
6
8
10 12 14 16 18 20
Fig. 3. The solution x.
Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
3.5 3 2.5 2 1.5 1 0.5
0
2
4
6
8
10 12 14 16 18 20
Fig. 4. The solution y.
The bounds and given above are comparatively accurate. If we take −100 0 100 0 = , = , 0 −100 0 100 i.e., the uncertain set described by (3) is larger, at this time we test gradient-like behavior by Theorem 3, the parameter r reduces to r 0.77 which is just equal to the value we obtained by Corollary 2. This means that the condition (3) fails to reduce the conservativeness when the uncertain set is larger. References Arcak, M., Larsen, M., & Kokotovic, P. (2003). Circle and Popov criteria as tools for nonlinear feedback design. Automatic, 39, 643–650. Curran, P. F., & Chua, L. O. (1997). Absolute stability theory and synchronization problem. International Journal of Bifurcation and Chaos, 7, 1375–1383. Duan, Z. S., Huang, L., & Wang, L. (2004). Multiplier design for extended strict positive realness and its applications. International Journal of Control, 77(17), 1493–1502. Duan, Z. S., Wang, J. Z., & Huang, L. (2005). Input and output coupled nonlinear systems. IEEE Transactions on Circuits and Systems—I: Regular Papers, 52(3), 567–575. Gahinet, P., Nemirovski, A., & Laub, A. J. (1995). LMI control toolbox. Natick, MA: The Math Works, Inc. Guzelis, C., & Chua, L. O. (1993). Stability analysis of generalized cellular neural networks. International Journal of Circuit Theory and Applications, 21, 1–33. Huang, L. (2003). Fundamental theory on robustness and stability. Beijing, China: Scientific Publishing House, (in Chinese). Khalil, H. K. (1996). Nonlinear systems. (2nd ed.), Englewood Cliffs, NJ: Prentice-Hall, Inc. Leonov, G. A., Burkin, I. M., & Shepeljavyi, A. L. (1992). Frequency methods in oscillation theory. Dordrecht, MA: Kluwer Academic Publishers. Leonov, G. A., Ponomarenko, D. V., & Smirnova, V. B. (1996). Frequencydomain method for nonlinear analysis: Theory and applications. Singapore: World Scientific.
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Lozano, R., Brogliato, B., Egeland, O., & Maschke, B. (2000). Dissipative systems analysis and control: Theory and applications. London: Springer. Narendra, K. S., & Taylor, J. H. (1973). Frequency domain criteria for absolute stability. New York: Academic Press. Park, P. (2002). Stability criteria of sector and slope restricted Lur’e systems. IEEE Transactions on Automatic Control, 47(2), 308–313. Popov, V. M. (1973). Hyperstability of automatic control systems. New York: Springer. Rantzer, A. (1996). On the Kalman–Yakubovich–Popov lemma. Systems Control Letter, 28, 7–10. Suykens, J. A. K., Vandewalle, J., & De Moor, B. (1998). An absolute stability criterion for the Lur’e problem with sector and slope restricted nonlinearities. IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, 45(9), 1007–1009. Wang, J. Z., Huang, L., & Duan, Z. S. (2004). Design of controller for a class of pendulum-like system guaranteeing dichotomy. Automatica, 40(6), 1011–1016. Yakubovich, V. A. (1962). Solution of certain matrix inequalities in the stability theory of nonlinear control systems. Soviet Mathematics Doklady, 3, 620–623. Yang, Y., Fu, R., & Huang, L. (2004). Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria. Systems and Control Letters, 53(2), 89–105.
Zhisheng Duan received the M.S. degree in mathematics from Inner Mongolia University and the Ph.D. degree in control theory from Peking University in 1997 and 2000, respectively. From 2000 to 2002, he worked as a postdoctor in Peking University. He received 2001 Chinese Control Conference Guan-ZhaoZhi Award. Since 2003, he has been an associate professor with the Department of Mechanics and Engineering Science, Peking University. His research interests include robust control, stability of interconnected systems and frequencydomain methods of nonlinear systems, analysis and control of complex dynamical networks. Jinzhi Wang received the M.S. degree in mathematics from Northeast Normal University, China in 1988 and Ph.D. degree in control theory from Peking University in 1998. From July 1998 to February 2000 she was a post-doctor at Institute of Systems Science, the Chinese Academy of Sciences. From March 2000 to August 2000 she was a research associate in the University of Hong Kong. She is currently an associate professor at the Department of Mechanics and Engineering Science, Peking University. Her research interests include robust control, nonlinear control and control of systems with saturating actuators. Lin Huang received the B.S. and M.S. degrees in mathematics and mechanics from Peking University, in 1957 and 1961, respectively. In 1961, he joined the Department of Mechanics, Peking University, where he is a Professor of Control Theory. His research interests include stability of dynamical systems, robust control, nonlinear systems. He has authored three books, authored or coauthored more than 150 papers in the fields of stability theory and control. He is currently a member of the Chinese Academy of Science.
Brief paper
Criteria for dichotomy and gradient-like behavior of a class of nonlinear systems with multiple equilibria夡 Zhisheng Duan ∗ , Jinzhi Wang, Lin Huang State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, PR China Received 5 December 2005; received in revised form 24 November 2006; accepted 2 February 2007
Abstract This paper considers global properties of a class of nonlinear systems with infinite equilibria. Time-domain and frequency-domain criteria for dichotomy and gradient-like behavior are established. Similar to the sector condition in absolute stability problems, the bounded derivative condition of nonlinear functions can be used to reduce the conservativeness of the given criteria. Compared with the frequency-domain conditions, the time-domain inequalities can be solved easily by linear matrix inequality (LMI) toolbox and used to discuss controller design problems. Several examples are given to illustrate the effectiveness of the results. 䉷 2007 Published by Elsevier Ltd. Keywords: Dichotomy; Gradient-like behavior; Multiple equilibria; Linear matrix inequality (LMI)
1. Introduction The absolute stability of Lur’e systems has been studied extensively for several decades. The well-known circle and Popov criteria based-on frequency-domain method and matrix inequality conditions based-on time-domain method have been established (Khalil, 1996; Lozano, Brogliato, Egeland, & Maschke, 2000; Narendra & Taylor, 1973; Popov, 1973; Yakubovich, 1962). And according to the remarkable Kalman–Yakubovich–Popov (KYP) lemma Rantzer (1996), the frequency-domain and time-domain conditions are equivalent to each other, see Arcak, Larsen, and Kokotovic (2003) for the application of time and frequency-domain methods. Furthermore, by combining the sector restriction and slope restriction, less conservative criteria for absolute stability were also established, see Park (2002) and Suykens, Vandewalle, and De Moor (1998) and references therein. The study of 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Dragan Nesic under the direction of Editor Hassan Khalil. Work supported by the National Science Foundation of China under grants 60674093, 60334030, 10472001. ∗ Corresponding author. E-mail address: [email protected] (Z. Duan).
0005-1098/$ - see front matter 䉷 2007 Published by Elsevier Ltd. doi:10.1016/j.automatica.2007.02.003
absolute stability is of fundamental importance, e.g., in many problems of control and electrical circuits. Lur’e systems play important roles in neural networks (Guzelis & Chua, 1993) and synchronization theory (Curran & Chua, 1997). The absolute stability means the global stability of the single equilibrium which is an important property of nonlinear systems. It has been playing a significant role in dynamical systems. Besides global stability of the unique equilibrium, some other global properties such as dichotomy, Lagrange stability, gradient-like behavior, etc., are also very important for the study of nonlinear systems. Frequency-domain criteria of such properties have been established for a class of nonlinear system with multiple equilibria which was called pendulum-like system in Leonov, Ponomarenko, and Smirnova (1996). The pendulum-like system is a generalization of mathematical pendulum equations. It has many important applications in phaselocked loops and oscillation theory, see Leonov, Burkin, and Shepeljavyi (1992), Leonov et al. (1996) and references therein for a detailed discussion. Recently, analysis and control of such systems have got a new interest. Controller design and robustness analysis were studied in Duan, Huang, and Wang (2004), Wang, Huang, and Duan (2004), Yang, Fu, and Huang (2004). The nonexistence of bounded oscillating solutions in coupled systems was analyzed in Duan, Wang, and Huang (2005).
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Generally, it is hard to know the existence of chaotic attractors. Therefore, it is interesting to know the nonexistence of chaotic solutions by analyzing the property of dichotomy. This paper is devoted to the study of dichotomy and gradient-like behavior of pendulum-like systems. Though some frequency-domain criteria for such systems have been established in Leonov et al. (1996), this paper presents timedomain conditions directly by Lyapunov method. Then the corresponding frequency-domain interpretations are given based-on KYP lemma. In this way, the theorems are easier to be understood, and linear matrix inequality (LMI)-based conditions can be solved easily. The rest of this paper is organized as follows. In Section 2, time-domain and frequency-domain criteria of dichotomy are given for one of the canonical forms of pendulum-like systems. In Section 3, conditions of gradientlike behavior are presented similarly. In Section 4, the other canonical form of pendulum-like systems is considered and the corresponding criteria of dichotomy are provided. Examples are given to illustrate the results in Section 5. Throughout this paper, Re{Y } means 21 (Y + Y H ) for any real or complex square matrix Y . 2. Preliminaries Motivated by practical considerations in electric circuits, mechanics, and phase-synchronization theory, a class of nonlinear systems with infinite equilibria were studied in Leonov et al. (1996). And frequency-domain conditions for global properties such as dichotomy, gradient-like behavior and Bakaev stability were presented. In this paper we consider time and frequency domain conditions simultaneously for dichotomy and gradientlike behavior of such systems. First introduce some necessary preliminaries. Consider the following multi-input and multioutput system, ⎧ dx ⎪ = Ax + B(y), ⎨ dt ⎪ dy ⎩ = Cx + D(y), dt
(1)
where A ∈ Rn×n , B ∈ Rn×m , C ∈ Rm×n , D ∈ Rm×m , (y) = (1 (y1 ), . . . , m (ym ))T . Viewing y˙ as the output of the system, then the transfer function from (y) to y˙ is K(s) = C(sI − A)−1 B + D. Generally, the following assumptions for the system above are made: Assumption 1. A has no imaginary eigenvalues, (A, B) is controllable, (A, C) is observable and K(0) is nonsingular. Assumption 2. i : R → R is i -periodic, local Lipschitz continuous and possesses a finite number of zeroes on [0, i ), i = 1, . . . , m. In order to analyze the characteristics of system (1), define a set P as follows: P = {d = (0, . . . , 0, k1 1 , . . . , km m )T | ki ∈ Z},
where Z is the set of integers. Rewriting (1) as ˙ = f (t, ),
(2)
where =(x T y T )T and f : R+ ×Rn+m → Rn+m is continuous and locally Lipschitz continuous in the second argument, by the periodicity of i , (2) satisfies f (t, + d) = f (t, ), t 0, d ∈ P. This characteristic is just like the pendulum characteristic of mathematical pendulum equation, so system (1) is called pendulum-like system in Leonov et al. (1996). And because of the periodicity of i , system (1) is a typical nonlinear system with infinite isolated equilibria. The equilibrium set of (1) can be analyzed simply as follows. Any equilibrium (xeq , yeq ) of (1) satisfies Ax eq = −B(yeq ) and Cx eq = −D(yeq ). By the nonsingularity of A, it follows that (−CA−1 B + D)(yeq ) = 0. Since K(0) = D − CA−1 B is nonsingular (Assumption 1), (yeq ) = 0. Then one gets xeq = 0. Therefore, the equilibrium set E of (1) is E = {(xeq , yeq ) | xeq = 0, (yeq ) = 0}. Besides Assumptions 1 and 2, in this paper, suppose that i is continuously differentiable and i
di () i , −∞ < i , i < + ∞, i = 1, . . . , m. d
(3)
Let = diag(1 , . . . , m ), = diag(1 , . . . , m ). Before studying the global properties of system (1), first we introduce KYP lemma. This lemma will be used repeatedly in this paper to get conversions between time-domain and frequency-domain inequalities. Generally, in KYP lemma (Rantzer, 1996), the state matrix A is supposed to have no imaginary eigenvalues. In fact, for the non-strict inequality version of this lemma, it is admittable that A has zero eigenvalues other than pure imaginary eigenvalues, see Yakubovich–Kalman theorem given in Leonov et al. (1996) and Huang (2003). Noticing this point, we give the following lemma. Lemma 1. Given A ∈ Rn×n , B ∈ Rn×m , M = M T ∈ R(n+m)×(n+m) . Suppose that A has no pure imaginary eigenvalues and zero eigenvalues with multiplicity larger than 1 (there may be zero eigenvalue with multiplicity 1 for A), and (A, B) is controllable. Then the following two statements are equivalent: (i)
(jwI − A)−1 B
H
I
M
(jwI − A)−1 B I
0,
∀w ∈ R, (ii) there is a real symmetric matrix P such that M+
PA + AT P
PB
BTP
0
0.
Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
3. Dichotomy In some practical systems, we often care about the nonexistence of bounded oscillating solutions. At this time the convergence of bounded solutions is a nice property. Such kind of problems have been studied in Leonov et al. (1996) and Duan et al. (2005). Throughout this paper, system (1) is called to be dichotomous if every bounded solution is convergent to a certain equilibrium. System (1) is called to be gradient-like if every solution is convergent to a certain equilibrium. Remark 1. In this paper gradient-like behavior means that every solution of the corresponding system is convergent to a certain equilibrium. Since system (1) is with multiple equilibria, the convergence of every solution does not mean every equilibrium is Lyapunov stable. In fact, it can be shown that in the case of gradient-like behavior of system (1) there exists at least one equilibrium that is not Lyapunov asymptotically stable Leonov et al. (1992). Let x A B 0 z= , A˜ = , L= and (y) 0 0 Im C˜ = (C D), then system (1) can be rewritten as ˜ + L (y), z˙ = Az
˜ y˙ = Cz,
(4)
where (y) = d(y)/dt = (y)y. ˙ Using the method of Leonov et al. (1996), we can establish the LMI conditions for the property of dichotomy of system (1). Theorem 1. Under Assumptions 1 and 2, system (1) is dichotomous for all with condition (3), if there exist diagonal matrices P , and R with R 0, a symmetric matrix W and a scalar > 0 such that the following LMI is feasible WL + 21 C˜ T R( + ) 0, (5) LT W + 21 ( + )R C˜ −R ˜ where = W A˜ + A˜ T W − C˜ T R C˜ + LPC˜ + C˜ T PLT + C˜ T C. Proof. Take W (t) = zT (t)W z(t) and V (t) = dW (t)/dt + ˜ + L (y)) + 2T (y)P y˙ + y˙ T y. ˙ Note that V (t) = 2zT W (Az T T T ˜ ˜ ˜ 2z LP Cz + z C Cz. For all zi = 0, the bounded derivative condition (3) is equivalent to ui (xi ) =: ( i (yi ) − i y˙i )( i (yi ) − i y˙i ) 0, where i (yi ) = di (yi )/dt. Noticing that for any ri 0, i = 1, . . . , m, we have m
˜ ri ui (xi ) = T (y)R (y) − T (y)( + )R Cz
i=1
˜ 0, + z C˜ T R Cz T
(6)
1585
where R = diag(r1 , . . . , rm ). (5) guarantees that V (t) T (y) ˜ + zT C˜ T R Cz, ˜ ∀(z, (y)) = R (y) − T (y)( + )R Cz 0. By (6), it follows that V (t) 0. From 0 to t (t 0), inte t yi (t) grating V (t) gives 0 y˙ T y˙ dt − m i=1 pi yi (0) i (yi ) dyi − W (t) + W (0), where pi is the ith diagonal element of P . For any bounded solution of (1), y(t) and W (t) are also bounded. Therefore, it follows from the inequality above that y˙i ∈ L2 [0, +∞),
i = 1, . . . , m.
(7)
Since i (yi ), i = 1, . . . , m, has a bounded derivative for almost all t 0, y˙i has a bounded derivative too. Hence, y˙ is uniformly continuous on [0, +∞). Together with (7), we have y˙ → 0
as t → +∞.
(8)
Repeating the similar discussions as in Leonov et al. (1996), one can complete the proof easily. Remark 2. Compared with the results of Leonov et al. (1996), we established the LMI condition of dichotomy directly by time-domain method. In this way, the proof of the theorem would be much easier to be understood and the inequality (5) can be solved easily by the powerful toolbox (Gahinet, Nemirovski, & Laub, 1995). Since the frequency-domain inequality needs to be solved to test the criteria of global properties, generally only examples of single-input and single-output systems were considered in Leonov et al. (1996). However Theorem 1 can be easily used to study multi-input and multi-output systems, see the forthcoming examples. And obviously, Theorem 1 can also be used to consider controller design problems as studied in Duan et al. (2004). By Lemma 1, an equivalent frequency-domain condition for (5) can be given, which is just the frequency-domain result given in Leonov et al. (1996). Theorem 2. With the conditions of Theorem 1, the LMI (5) is feasible if, and only if, the following frequency domain inequality holds Re{PK(jw) − (jwI − K(jw))H R(jwI − K(jw))} + K H (jw)K(jw)0,
∀w ∈ R.
(9)
˜ ˜ −1 L = Proof. By simple computation, we have C(jwI − A) −1 T ˜ L=(1/jw)I . Taking an appropri(1/jw)K(jw), L (jwI− A) ate matrix M, then by Lemma 1 we know that (5) holds if, and only if, Re{(2/w 2 )PK(jw) − (1/w 2 )(jwI − K(jw))H R(jwI − K(jw))} + ( /w 2 )K H (jw)K(jw)0, ∀w ∈ R. The equivalence between (9) and the inequality above completes the proof. Obviously, when R = 0, the frequency-domain inequality in Theorem 2 reduces to Re{PK(jw)} + K H (jw)K(jw) 0, ∀w ∈ R. And correspondingly, the LMI (5) reduces to a simple inequality as shown in the following corollary. Corollary 1. Under Assumptions 1 and 2, system (1) is dichotomous for all with condition (3), if there exist a diagonal
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matrix P , a symmetric matrix W and a scalar > 0 such that the following LMI is feasible
The time derivative of v(t) along any trajectory of system (4) is given by
˜ + L (y)) + zT LP Cz ˜ − |(y)|T P V Cz, ˜ v(t) ˙ = 2zT W (Az
WA + AT W + C T C
WB + C T P + C T D
B T W + PC + D T C
D T D + PD + D T P
0.
(10)
Remark 3. The condition (3) plays a role in Theorems 1 and 2 as the canonical sector condition in the absolute stability of Lur’e systems. The forthcoming example shows that Theorem 1 is generally less conservative than Corollary 1. This means that the condition (3) can be used to improve the condition of dichotomy. On the other hand, system (1) can be considered as an uncertain nonlinear system, where the uncertain set is described by (3). If i − i is smaller for i = 1, . . . , m, i.e., the uncertain set is smaller, Theorem 1 is generally less conservative. This will also be shown by the forthcoming example. In addition, for the problem of dichotomy A does not need to be stable, so W is generally not positive definite in (10). Similarly, W in (5) is not necessarily positive definite. Obviously, if Re{P K(jw)} < 0, ∀w ∈ R, which is just like a strictly positive real condition in control theory, then the inequalities (5) and (10) hold.
(12) where |(y)|T =: (|1 (y1 )|, . . . , |m (ym )|), P =: diag(p1 , . . . , ˜ + L (y)) + zT LPCz ˜ − pm ). Furthermore, v(z) ˙ = 2zT W (Az T T T T T ˜ ˜ ˜ |(y)| PVCz − (y)E(y) − z C QCz + (y)E(y) + ˜ zT C˜ T QCz, where Q =: diag(q1 , . . . , qm ) and E =: diag(e1 , . . . , em ). By the condition (ii), we know that there exist Q0 =: diag(q01 , . . . , q0m ) and E0 =: diag(e01 , . . . , e0m ) such T (y)E(y)+zT C ˜ ˜ T (y)E0 (y)+ ˜ T QCz that |(y)|T PVCz+ T T ˜ ˜ z C Q0 Cz. Hence, we have ˜ v(z) ˙ + T (y)E0 (y) + zT C˜ T Q0 Cz ˜ + 2zT W AL (y) ˜ ˜ 2zT W Az + zT LPCz ˜ + T (y)E(y) + zT C˜ T QCz.
On the other hand, condition (i) of the theorem guarantees that ˜ + L (y)) + zT LPCz ˜ + T (y)E(y) + zT C˜ T QCz ˜ 2zT W (Az ˜ T (y)R (y) − T (y)( + )R Cz ˜ + zT C˜ T R Cz,
4. Gradient-like behavior The property of dichotomy is related to the convergence of bounded solutions. We can also discuss the global convergence (gradient-like behavior) of system (1) by the method above. Let vi = 0 i i ( ) d / 0 i |i ( )| d , V = diag(v1 , . . . , vm ), Fi ( ) = i ( ) − vi |i ( )|. Note that 0 i Fi ( ) d = 0, i = 1, . . . , m. Combining the periodic Lyapunov function method of Leonov et al. (1996) with the method of Theorem 1, we have Theorem 3. Under Assumptions 1 and 2, system (1) is gradient-like, if A is stable and there exist diagonal matrices P , Q, E and R with R 0, E > 0 and Q > 0, and a symmetric matrix W such that the following LMIs are feasible
WL + 21 C˜ T R( + )
i=1
LT W + 21 ( + )R C˜
−R
0,
˜ A˜ T W − C˜ T R C˜ + 1 LPC˜ + 1 C˜ T PLT + where =W A+ 2 2 T C˜ QC˜ + LELT . (ii)
2E
PV
VP
2Q
> 0.
v(t) =: z Wz +
m
i=1
pi
0
i = 0, . . . , m.
(16)
Repeating the similar discussions as in Leonov et al. (1996), one complete the proof easily. Similar to Theorem 2, a frequency-domain interpretation for the condition (i) of Theorem 3 can be given as follows. Theorem 4. Under conditions of Theorem 3, condition (i) of Theorem 3 is feasible if, and only if, the following frequencydomain inequality holds Re{PK(jw) − (jwI − K(jw))H R(jwI − K(jw))} + K H (jw)QK(jw) + E 0,
Proof. Take Lyapunov function candidate T
(14)
∀t 0. In addition, v(t) is bounded. This property follows from the facts that A is stable (and consequently z(t) is bounded), the functions Fi ( ) have mean value zero (and con y (t) sequently 0 i Fi ( ) d are bounded). Then it follows that +∞ 2 i (yi (t)) dt < + ∞, i = 0, . . . , m. Together with the fact 0 that i (yi (t)) are uniformly continuous, we have t→+∞
∀(z, (y)) = 0.
Combining (13), (14) and (6), we know that v(x) ˙ + ˜ 0. T (y)E0 (y) + zT C˜ T Q0 Cz From 0 to t integrating the two sides of the inequality above gives m t
v(t) − v(0) − (e0i 2i (yi (t)) + q0i y˙i2 (t)) dt, (15)
lim i (yi (t)) = 0,
(i)
(13)
yi 0
Fi ( ) d .
(11)
∀w ∈ R.
(17)
Proof. By the method of Theorem 2, (5) holds if, and only if, Re{(1/w 2 )PK(jw) − (1/w 2 )(jwI − K(jw))H R(jwI − K(jw))} + (1/w 2 )K H (jw)QK(jw) + (1/w 2 )E 0, ∀w ∈ R.
Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
The equivalence of (17) and the inequality above completes the proof. Similar to the discussions in the section above, when R = 0 in (17), the inequality (17) reduces to Re{PK(jw)} + K H (jw)QK(jw) + E 0, ∀w ∈ R. And correspondingly, Theorem 2 reduces to Corollary 2. Under Assumptions 1 and 2, system (1) is gradient-like, if A is stable and there exist diagonal matrices P , Q, E, and a symmetric matrix W such that (ii) of Theorem 3 and the following LMI are feasible
WA + AT W
+ C T QC
WB + C T P
B T W + PC + D T QC
where
= PD + D T P
+ C T QD
0,
1996). Then take a nonsingular matrix T such that ˆ −1 = (C1 C2 ) and CT B1 A1 0 ˆ −1 = T AT , T Bˆ = , 0 0 B2
Remark 4. From Corollary 2 we know that the stability of A and the observability of (A, C) imply > 0 in (18). that W yi However, because of the existence of m p F i ( ) d and i=1 i 0 ˜ A having zero eigenvalues, in Theorem 3 for the gradient-like behavior of system (1) the Lyapunov function v(t) in (11) is not necessarily positive definite. And in the proof of Theorem 3, the positive definiteness of v(t) is not required. This is different from the canonical absolute stability problems of Lur’e systems in which the Lyapunov function is positive definite. From Theorem 3, we can see that there is a parameter matrix coupling in the condition (ii). If V = 0, the condition (ii) disappears. Then we have Corollary 3. With the assumptions of Theorem 3, if the nonlinear function i ( ), i = 1, . . . , m, have zero mean values (vi = 0), then system (1) is gradient-like when condition (i) of Theorem 3 or (17) holds. System (1) studied above is one of the two canonical forms of pendulum-like systems. With the results above, we can also study the other canonical form of such systems.
Tx =
,
where A1 ∈ R(n−m)×(n−m) , B2 ∈ Rm×m , C2 ∈ Rm×m and B1 , C1 are matrices with compatible dimensions. Noticing the transformation, differentiating y in (19) gives the following system ˙ = A1 + B1 (y), y˙ = C1 A1 + (C1 B1 + C2 B2 )(y),
(18)
+ D T QD + E.
1587
(20)
which is just in the form of (1). Let the transfer function from ˆ Comparing (19) and ˆ ˆ −1 B. (y) to y in (19) be (s)= C(sI − A) (20), we know that (s)=(1/s)K(s), where K(s)=C1 A1 (sI − A1 )−1 B1 + (C1 B1 + C2 B2 ). Similar to Assumption 1, we suppose the following assumption holds for system (19). ˆ B) ˆ is controllable, (A, ˆ C) ˆ is observable and Assumption 3. (A, besides m zero eigenvalues (each of them is with multiplicity one), Aˆ has no pure imaginary eigenvalues, and s(s)|s=0 = K(s)|s=0 = K(0) is nonsingular. Obviously, by the controllability and observability of ˆ B, ˆ C), ˆ C2 and B2 are nonsingular. (A, By Theorem 2, we can establish the following frequencydomain criterion of dichotomy expressed in (s) for system (19). Theorem 5. Under Assumptions 2 and 3, system (19) is dichotomous for all with condition (3), if there exist diagonal matrices P and R with R 0, and a scalar > 0 such that 1 Re P (jw) − (I − (jw))∗ R(I − (jw)) −jw + ∗ (jw)(jw)0,
∀w ∈ R.
(21)
Let C0 = (0m×(n−m) B2−1 ). Then similar to Theorem 1, we can also establish LMI-based criterion of dichotomy expressed in ˆ B, ˆ C). ˆ (A,
5. The other canonical form of pendulum-like systems Consider the following nonlinear system, ⎧ dx ˆ + B(y), ˆ ⎨ = Ax dt ⎩ ˆ y = Cx,
(19)
where Aˆ ∈ ∈ ∈ m (ym ))T , i are periodic functions as discussed in the sections above. Suppose that Aˆ has m zero eigenvalues and the elementary factor of each zero eigenvalue is 1. Eq. (19) is the other canonical form for pendulum-like systems (Leonov et al., Rn×n , Bˆ
Theorem 6. Under Assumptions 2 and 3, system (19) is dichotomous for all with condition (3), if there exist diagonal matrices P and R with R 0, and a symmetric matrix W and a scalar > 0 such that the following LMI is feasible
Rn×m , Cˆ
Rm×n , (y)=(1 (y1 ), . . . ,
Bˆ T W + 21 ( + )R Cˆ
W Bˆ + 21 Cˆ T R( + ) −R
0, (22)
ˆ Aˆ T W − Cˆ T R Cˆ +T T C T P Cˆ + Cˆ T PC0 T + where =W A+ 0 ˆ Cˆ T C.
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Proof. By simple computation, we know that C0 T (sI − ˆ −1 T −1 T Bˆ = C0 (sI − T AT ˆ −1 )−1 T Bˆ = ˆ −1 Bˆ = C0 T (sI − A) A) (1/s)I . Then by Lemma 1, we know that the LMI (22) is equivalent to (21). Similarly, we can also establish conditions of gradient-like ˆ B, ˆ C) ˆ for system (19) by the behavior expressed in (s) or (A, method above.
2 1.5 1 0.5 0 -0.5
6. Examples Example 1. Consider the system in the form of (1) with the following data −0.4 3 1 0 A= , B= , −1 −0.5 −1.4 1 2 −1 1.2 C= , D= 0 1 −2 1 and 1 (y1 )=sin(y1 )−0.2, 2 (y2 )=sin(2y2 )−0.1. By Theorem 1, we know that this system is dichotomous when 1.9. If we take = −1.9, the condition (5) fails. At this time, we can see a solution as shown in Figs. 1 and 2 at the initial value x = (1, −0.5)T , y = (0.1, −5)T by computer simulation. From this solution, we can see that the solution x is something like a chaotic solution and y is unbounded when t → ∞ which show the complexity of solutions of pendulum-like systems. The chaotic phenomenon in pendulum-like systems is still a new topic which is worthy study.
-1 -1.5 -2
-3
-2
-1
0
1
2
3
Fig. 1. The solution x.
180 160 140 120 100 80 60 40 20
Example 2. Consider the system in the form of (1) with the following data ⎛ ⎞ ⎛ ⎞ −0.9 0 0.8 1 0 ⎜ ⎟ ⎜ ⎟ A=⎝ 0 0 1.1 ⎠ , B = ⎝ 0 0 ⎠ , 0 C=
0.3 0
−2.5 −1 0 −1 0 0 0.5 −0.1 , D= 0.2 0 −0.1 1
and 1 (y1 )=sin(y1 )−r, 2 (y2 )=sin(2y2 )−r, r is a parameter to be determined. For i given here, obviously we have =
−1
0
0
−2
,
=
1
0
0
2
.
By Theorem 3, we know that this system is gradient-like when r 0.82. However, if we take R = 0 in the condition (i) of Theorem 3, at this time Theorem 3 reduces to Corollary 2, then we can only know that the system is gradient-like when r 0.77. This shows the effects of (3) in reducing the conservativeness, i.e., Theorem 3 is generally less conservative than Corollary 2. Please refer to Figs. 3 and 4 for the gradient-like behavior of its solutions at three initial values x = (1, 0.5, 0.1)T , y = (2.1, 2)T ; x = (0.2, −2.5, 5)T , y = (3, 1)T ; x = (0.5, −1.5, 4.2)T , y = (3.5, 0.9)T .
0 -20 -20
0
20
40
60
80 100 120 140 160
Fig. 2. The solution y.
6 5 4 3 2 1 0 -1 -2 -3
0
2
4
6
8
10 12 14 16 18 20
Fig. 3. The solution x.
Z. Duan et al. / Automatica 43 (2007) 1583 – 1589
3.5 3 2.5 2 1.5 1 0.5
0
2
4
6
8
10 12 14 16 18 20
Fig. 4. The solution y.
The bounds and given above are comparatively accurate. If we take −100 0 100 0 = , = , 0 −100 0 100 i.e., the uncertain set described by (3) is larger, at this time we test gradient-like behavior by Theorem 3, the parameter r reduces to r 0.77 which is just equal to the value we obtained by Corollary 2. This means that the condition (3) fails to reduce the conservativeness when the uncertain set is larger. References Arcak, M., Larsen, M., & Kokotovic, P. (2003). Circle and Popov criteria as tools for nonlinear feedback design. Automatic, 39, 643–650. Curran, P. F., & Chua, L. O. (1997). Absolute stability theory and synchronization problem. International Journal of Bifurcation and Chaos, 7, 1375–1383. Duan, Z. S., Huang, L., & Wang, L. (2004). Multiplier design for extended strict positive realness and its applications. International Journal of Control, 77(17), 1493–1502. Duan, Z. S., Wang, J. Z., & Huang, L. (2005). Input and output coupled nonlinear systems. IEEE Transactions on Circuits and Systems—I: Regular Papers, 52(3), 567–575. Gahinet, P., Nemirovski, A., & Laub, A. J. (1995). LMI control toolbox. Natick, MA: The Math Works, Inc. Guzelis, C., & Chua, L. O. (1993). Stability analysis of generalized cellular neural networks. International Journal of Circuit Theory and Applications, 21, 1–33. Huang, L. (2003). Fundamental theory on robustness and stability. Beijing, China: Scientific Publishing House, (in Chinese). Khalil, H. K. (1996). Nonlinear systems. (2nd ed.), Englewood Cliffs, NJ: Prentice-Hall, Inc. Leonov, G. A., Burkin, I. M., & Shepeljavyi, A. L. (1992). Frequency methods in oscillation theory. Dordrecht, MA: Kluwer Academic Publishers. Leonov, G. A., Ponomarenko, D. V., & Smirnova, V. B. (1996). Frequencydomain method for nonlinear analysis: Theory and applications. Singapore: World Scientific.
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Zhisheng Duan received the M.S. degree in mathematics from Inner Mongolia University and the Ph.D. degree in control theory from Peking University in 1997 and 2000, respectively. From 2000 to 2002, he worked as a postdoctor in Peking University. He received 2001 Chinese Control Conference Guan-ZhaoZhi Award. Since 2003, he has been an associate professor with the Department of Mechanics and Engineering Science, Peking University. His research interests include robust control, stability of interconnected systems and frequencydomain methods of nonlinear systems, analysis and control of complex dynamical networks. Jinzhi Wang received the M.S. degree in mathematics from Northeast Normal University, China in 1988 and Ph.D. degree in control theory from Peking University in 1998. From July 1998 to February 2000 she was a post-doctor at Institute of Systems Science, the Chinese Academy of Sciences. From March 2000 to August 2000 she was a research associate in the University of Hong Kong. She is currently an associate professor at the Department of Mechanics and Engineering Science, Peking University. Her research interests include robust control, nonlinear control and control of systems with saturating actuators. Lin Huang received the B.S. and M.S. degrees in mathematics and mechanics from Peking University, in 1957 and 1961, respectively. In 1961, he joined the Department of Mechanics, Peking University, where he is a Professor of Control Theory. His research interests include stability of dynamical systems, robust control, nonlinear systems. He has authored three books, authored or coauthored more than 150 papers in the fields of stability theory and control. He is currently a member of the Chinese Academy of Science.
