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Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria
Available online at www.sciencedirect.com
Systems & Control Letters 53 (2004) 89 – 105
www.elsevier.com/locate/sysconle
Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria Ying Yang∗ , Rao Fu, Lin Huang Department of Mechanics and Engineering Science, Center for Systems and Control, Peking University, Beijing 100871, PR China Received 21 June 2003; received in revised form 14 December 2003; accepted 26 February 2004
Abstract This paper focuses on a class of uncertain nonlinear systems which are subject to norm-bounded parameter uncertainty in the forward path and a vector-valued periodic nonlinearity in the feedback path, and addresses robust analysis and synthesis problems for such systems. Su5cient conditions for global asymptotic stability are derived in terms of linear matrix inequalities (LMIs) and a technique for the estimation of the uncertainty bound is proposed by solving a generalized eigenvalue minimization problem. The problem of robust synthesis is concerned with designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. It is shown that a solution to the robust synthesis problem for the uncertain system can be obtained by solving a synthesis problem for an uncertainty free system. A concrete example is presented to demonstrate the applicability and validity of the proposed approach. c 2004 Elsevier B.V. All rights reserved. Keywords: Nonlinear systems; Multiple equilibria; Norm-bounded parameter uncertainties; Analysis and synthesis; LMI
1. Introduction The past decades has witnessed signi?cant advances in stability analysis and synthesis for nonlinear feedback systems, see [7,13,14,17,18] and the references therein. But most of the results were applied and developed within the frames of absolute stability theory which was constructed for investigation of global stability of systems with single equilibrium. Meanwhile engineers often have to deal with systems with multiple equilibria, types of equilibria sets being rather diverse. In this case, new stability problems appeared. Since these problems cannot be investigated by means of classical Lyapunov or Popov tools, the necessity arises to develop the classical theory in such a way that it should be possible to use it for stability investigation of systems with multiple equilibria. Motivated by these facts, the qualitative theory of systems with multiple equilibria has been established to investigate global behavior of solutions for systems with multiple equilibria. This theory starts from the paper of Moser [12] and has been developed in a comprehensive way by Yakubovich, Leonov and their co-workers [5,10]. Interesting work in this ?eld, in the context of integral and integro-diDerential equations, can be seen in [2,11]. ∗
This work was supported by the National Science Foundation of China under Grant 60334030,10272001. Corresponding author. Tel.: +86-1062751815; fax: +86-1062764044. E-mail addresses: [email protected] (Y. Yang), [email protected] (R. Fu), [email protected] (L. Huang).
c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.02.024
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The behavior of solutions with respect to entire set of equilibria has not only purely theoretical interest for the qualitative theory of diDerential equations, but also important applied signi?cance. Note that a broad class of diDerential equations with angular coordinates in mechanics, phase-synchronization and other ?elds can be written in the form x˙ = f(x)
(1.1) n
n
n
with the vector ?eld f : R → R having mthe property f(x) = f(x + d) for every d ∈ and x ∈ R , where is a subgroup of Rn de?ned by := { i=1 ki di |ki ∈ Z}; m 6 n and {di }m i=1 are linearly independent vectors in Rn . Such systems are often called systems with cylindrical phase space, and are also called pendulum-like systems with respect to in [10,16]. In the previous literature [5,9,10,16,21], some important global asymptotic properties which characterizes the behavior of solutions for systems with multiple equilibria have been intensively studied and frequency-domain inequalities conditions have been established. However, compared with the richness of the exiting results on robust analysis and synthesis for nonlinear systems with single equilibrium, there is almost no robustness results or synthesis results of designing feedback controller to ensure those properties for systems with multiple equilibria. The aim of this paper is to bridge this gap by investigating a class of nonlinear feedback systems with periodic nonlinearities and provide a more complete picture of the behavior of solutions with respect to entire set of equilibria. In this paper, we will consider the robust analysis and synthesis problem for a class of uncertain nonlinear systems which are subject to norm-bounded parameter uncertainty in the forward path and a vector-valued periodic nonlinearity in the feedback path. The system under consideration will be described by a state-space model which contains parameter uncertainties in both the state and input matrices. Based on the Kalman– Yakubovich–Popov lemma connecting the frequency-domain inequality and linear matrix inequality [15], suf?cient conditions of global asymptotic stability for uncertain systems are given in terms of the existence of solutions of a set of LMIs. Meanwhile, the robust synthesis problem is addressed by designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. It will be shown that the robust synthesis problem can be converted into a synthesis problem for an uncertainty free system. Parallel results in H∞ control and positive real control for linear systems can be seen in [6,20], but the results in this paper are believed to be new in the context of global asymptotic analysis and synthesis for uncertain nonlinear systems with multiple equilibria. Furthermore, the formulation diDers from [6,20] in that it involves no parameter searching by using the linear matrix inequality method. With this LMI approach, the largest allowable magnitude of the admissible uncertainty can also be explicitly computed by solving a generalized eigenvalue minimization problem which is essentially a convex optimization problem and numerically e5cient. It should be pointed out here that the methodology used in this paper is not restricted to the class of systems under consideration. In fact, the existing formulations of criteria for stability, dichotomy and instability of the class of Lur’e systems in terms of the frequency response of the linear part of the system can also be recast into the framework of LMIs and the corresponding robust analysis and synthesis results can be derived in a similar way. The rest of this paper is organized as follows: Section 2 presents some basic results necessary for the successive development, while in Section 3, we give some LMI-based global asymptotic stability results for systems with norm-bounded parameter uncertainties; Section 4 presents both the static state and dynamic output feedback controller existence conditions for such uncertain systems and shows that the robust synthesis problem can be converted into a synthesis problem for an uncertainty free system. Section 5 gives a concrete example showing the application of the proposed method and we end this paper by a few concluding remarks in Section 6. Notation: Rn×n is the set of n × n real matrices. For a matrix A, AT denotes its transpose, A∗ its complex conjugate transpose. The matrix inequality A ¿ B (A ¿ B) means that A and B are square Hermitian matrices and A − B is positive (semi-)de?nite. He is Hermit operator with He A = A + AT .
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2. Preliminaries In this section, we review some useful results which will play crucial roles in this paper. First let us consider a nonlinear feedback system of the form x˙ = Ax + B’(z); n×n
z˙ = Cx + D’(z); n×m
m×n
(2.1) m×m
where A ∈ R ; B ∈ R ; C ∈R ; D∈R . We suppose that (A; B) is controllable, (A; C) is observable and ’ : Rm → Rm is a vector-valued function having the components ’i (z) = ’i (zi ) with z = (z1 ; z2 ; : : : ; zm )T . We assume that every component ’i : R → R is i periodic, satis?es a local Lipschitz condition and possesses a ?nite number of zeroes on [0; i ), and for each component ’i there exists z0i such that ’i (z0i ) = 0 and ’˙ i (z0i ) = 0. Let us introduce the vector di = (0; : : : ; 0; i ; 0;: : : ; 0), where i is the (n + i)th component of di , m then system (2.1) is pendulum-like with respect to = { i=1 ki di ; ki ∈ Z}. Assume that i i i = ’i (z) d z |’i (z) |d z; i ∈ m 0 0 and denotes = diag{1 ; 2 ; : : : ; m }. The transfer function of the linear part of (2.1) from the input ’ to the output −z˙ is given by K(s) = C(A − sI )−1 B − D: Suppose that K(0) is nonsingular and det A = 0. Any equilibrium of (2.1) satis?es Axeq = −B’(zeq );
Cxeq = −D’(zeq )
(2.2)
and consequently we have (D − CA−1 B)’(zeq ) = 0: By the assumption above, we can get that ’(zeq ) = 0 and every component (zeq )i is a zero of ’i ; i = m. From the ?rst equation of (2.2) it follows that xeq = 0. Since ’i is i -periodic, system (2.1) has in?nitely many isolated equilibria. Denition 2.1 (Leonov et al. [8]). System (2.1) is said to be globally asymptotically stable if each solution of it tends to a certain equilibrium as t → +∞. Remark 2.1. Note that in De?nition 2.1 it is not implied that all equilibria of (2.1) are Lyapunov stable. Moreover, it can be shown that in the case of global asymptotic stability of system (2.1) there exists at least one equilibrium that is not asymptotically stable [8]. In the following sections, we consider the global asymptotic stability in the sense of De?nition 2.1 for system (2.1) with parameter uncertainties. To derive our main results, we also need the following lemmas. Lemma 2.1 (Leonov et al. [10]). Suppose K(s) is stable and there exist diagonal matrices =diag{1 ; 2 ; : : : ; m }, = diag{1 ; 2 ; : : : ; m } and = diag{1 ; 2 ; : : : ; m } with ¿ 0 and ¿ 0 satisfying the following conditions: 1. 12 He[K(i!) − K(i!)K ∗ (i!)] − ¿ 0; 2. 4 ¿ ()2 then system (2:1) is globally asymptotically stable.
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Lemma 2.2 (Rantzer [15]). Given A ∈ Rn×n ; B ∈ Rn×n ; M = M T ∈ R(n+m)×(n+m) , with det(j!I − A) = 0 for ! ∈ R and (A; B) controllable, the following two statements are equivalent: 1. ∗ (j!I − A)−1 B ( j!I − A)−1 B 6 0; M I I ∀! ∈ R ∪ {∞}; 2. there exists a matrix P = P T ∈ Rn×n such that T A P + PA PB M+ 6 0: 0 BT P The corresponding equivalence for strict inequalities holds even if (A; B) is not controllable. Lemma 2.3. Let T1 = T1T ; T2 ; T3 be real matrices of appropriate size, then the following statements are equivalent: 1. T1 + He(T2 T3 ) ¡ 0; ∀ : T 6 !2 I ; 2. there exists a positive number " ¿ 0 such that 1 T1 + "!2 T2 T2T + T3T T3 ¡ 0; " 3. there exists a positive number " ¿ 0 such that T3T T1 + "!2 T2 T2T ¡ 0; T3 −"I 4. there exists a positive number " ¿ 0 such that T1 + "!2 T3T T3 T2 ¡ 0: T2T −"I
(2.3)
(2.4)
(2.5)
Remark 2.2. The above result is not new and, in fact, is well known in the robustness analysis literatures. Here we give it in an alternative form of LMI. The advantage of this representation is that when there is LMI variable in T2 which can result in a product term with ", we can resort to using (2.5) for convex optimization and vice versa. This method does not involve any parameter searching so that there is no need to consider the convergence property of (2.5). 3. Robust analysis In this section, we derive the global asymptotical stability results for system (2.1) when there is norm-bounded parameter uncertainties in the linear part. First, we give a theorem which establishes the connection between the frequency-domain conditions given in Lemma 2.1 and an LMI-based criterion. Theorem 3.1. Suppose A is Hurwitz, then the conditions of Lemma 2:1 are satis
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
2
2
93
¿ 0:
(3.1b)
Proof. Let M=
C T C
C T (D + 12 )
(DT + 12 )C
+ DT D + 12 (D + DT )
and using Lemma 2.2, we can prove the equivalence of (3.1a) and condition (1) of Lemma 2.1. Note that the upper left corner of M is positive semide?nite, it follows from (3.1a) and Hurwitz stability of A that P ¿ 0. (3.1b) is directly derived from condition (2) of Lemma 2.1. Remark 3.1. The signi?cance of this theorem is that, by using Lemma 2.2 we convert the conditions of Lemma 2.1 into an equivalent LMI requirement. From this LMI condition, it is possible to extend the results to take into account the parameter uncertainty in the linear part of the system and derive feedback control law which renders the closed-up system globally asymptotically stable by using the e5cient numerical linear matrix inequalities methods. As an immediate consequence, we have a more convenient criterion as stated in the following. Corollary 3.1. If there exist P = P T ¿ 0 and diagonal matrices , ¿ 0 and ¿ 0 such that T C T C A P + PA PB C T (D + 12 ) + ¡ 0: 0 (DT + 12 )C + DT D + 12 (D + DT ) BT P
2
2
(3.2a)
¿ 0;
(3.2b)
then system (2:1) is globally asymptotically stable. Let us now consider the following uncertain system described by state-space models of the form: x˙ = (A + OA)x + B’(z);
z˙ = Cx + D’(z);
(3.3)
where OA stands for the parameter uncertainties which are norm-bounded and of the form OA = HFE
(3.4)
and H ∈ Rn×i , E ∈ Ri×m are known constant matrices and F ∈ Ri×i is an unknown matrix function satisfying F T F 6 !2 I
(3.5)
with ! ¿ 0 a given constant. From the de?nition of in Lemma 2.1 we know 6 %I , where % ¡ 1. In the following sections, we assume that the system is controllable and observable for all admissible uncertainties. Then we have the following result:
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Theorem 3.2. There exist diagonal matrices ¿ 0; ¿ 0 and such that (3:2) holds for the uncertain system (3:3) satisfying (3:4) if and only if there exists a scaling parameter " ¿ 0 such that (3:2) holds with 0 0 0 ˆ ˆ = ; ˆ = ; = (3.6) 0 aI 0 cI 0 dI for the system ˆ + Bˆ ’(z); x˙ = Ax ˜ where
ˆ + Dˆ ’(z); z˙ = Cx ˜
(3.7)
! ! ˆ ˆ H ; A=A+ √ HE; B = B (a + 1)" a+1 C D 0 Cˆ = √ ; Dˆ = "E 0 bI
and a; b; c; d satisfying a ¿ 0; b=−
c2 ¿ c+2 ; 2a
4(1 + a) ; 1 − %2 d=
b(2 − c) − 2 : 2
(3.8)
Proof. Eq. (3.2) holds for system (3.7) if there exists a positive de?nite solution P = P T ¿ 0 to the linear matrix inequality T Cˆ T ˆCˆ Cˆ T (ˆDˆ + 12 ) ˆ Aˆ P + P Aˆ P Bˆ + ¡0 (3.9) ˆ Bˆ T P 0 (Dˆ T ˆ + 12 ) ˆ Cˆ ˆ + Dˆ T ˆDˆ + 12 He(ˆD) with
2ˆ
ˆ˜
˜ˆ
2ˆ
¿ 0:
(3.10)
ˆ ˆ into (3.9) leads to ˆ B; ˆ D; ˆ C; ˆ ;ˆ ; Substituting A;
√ T ! C T C + a"E T E + He P 6 ft A + √a+1 HE PB + C T (D + 12 ) √ ! PH + ab + 2c "E (a+1)" T T T 1 1 ¡0 B P + (D + 2 )C + D D + 2 He(D) 0 √ T 2 ! c √ H P + ab + 2 "E 0 (d + ab + bc)I (a+1)"
(3.11) From (3.8) we have ab +
c = −1; 2
d + ab2 + bc = −1
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then (3.11) becomes
C T C + a"E T E + He P A +
T
T
√! a+1
HE
PB + C T (D + 12 ) T
1 2
B P + (D + )C √ √ ! H T P − "E
√
! PH (a+1)"
1 2
+ D D + He(D)
0
0
−I
(a+1)"
−
√
"E T
¡ 0:
Using Schur Complement, the above inequality is equivalent to
!2 C C + A P + PA + PHH T P + (a + 1)"E T E (a + 1)" BT P + (DT + 12 )C T
T
T
1 2
PB + C (D + ) ¡ 0: T 1 + D D + 2 He(D)
(3.12)
By Lemma 2.3, (3.12) holds if and only if for any F satisfying (3.5)
C T C
C T (D + 12 )
(DT + 12 )C
+ DT D + 12 He(D)
+
He(A + HFE)T P
PB
BT P
0
¡ 0:
As for (3.10), we can verify by straightforward manipulations that
2ˆ
ˆ˜
˜ˆ
2ˆ
¿0
is equivalent to
2
2
¿0
by noting that (3.8) implies 4ad ¿ %2 c2 : Thus completes the proof. Corollary 3.2. The uncertain system (3:3) is globally asymptotically stable by Lemma 2:1 if and only if there exists a scaling parameter " ¿ 0 such that the condition in Corollary 3:1 is satis
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Next we consider the uncertain system described by x˙ = (A + OA)x + (B + OB)’(z);
z˙ = Cx + D’(z);
(3.13)
where OA and OB have the form of [OA OB] = HF[E1 E2 ]
(3.14)
and H ∈ Rn×i , E1 ∈ Rj×n , E2 ∈ Rj×m are known constant matrices and F ∈ Ri×j is an unknown matrix function satisfying F T F 6 !2 I with ! ¿ 0 a given constant. Then we have the following result: Theorem 3.3. Suppose there exist P = P T ¿ 0, diagonal matrices ; ¿ 0; ¿ 0 and a positive number " ¿ 0 such that the following linear matrix inequalities hold: T C C + AT P + PA PB + C T (D + 12 ) PH "!E1T T B P + (DT + 1 )C + DT D + 1 (D + DT ) 0 "!E2T 2 2 (3.15a) ¡ 0; T H P 0 −"I 0 "!E1 "!E2 0 −"I
2
2
¿ 0;
(3.15b)
then system (3:13) is globally asymptotically stable. Proof. By Corollary 3.1, system (3:13) is globally asymptotically stable if there exist P ¿ 0 and diagonal matrices ; ¿ 0; ¿ 0 such that C T C C T (D + 12 ) (DT + 12 )C + DT D + 12 (D + DT ) (A + HFE1 )T P + P(A + HFE1 ) P(B + HFE2 ) + ¡0 (B + HFE2 )T P 0 and (3.15b) holds. Denote T C C + AT P + PA M= BT P + (DT + 12 )C
PB + C T (D + 12 ) + DT D + 12 (D + DT )
then (3.16) can be written in the form of PH M + He F[E1 E2 ] ¡ 0: 0
;
(3.16)
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According to Lemma 2.3, the above inequality holds if and only if there exists a positive number " ¿ 0 such that T E1 1 PH 2 M + "! [H T P 0] ¡ 0 [E1 E2 ] + " 0 E2T it can be easily proved that the above inequality is equivalent to (3.15a). Remark 3.3. The above result show that assessing global asymptotic stability of the uncertain system (3.13) satisfying (3.14) can be carried out by solving two LMIs which is essentially a convex optimization problem and numerically e5cient. From the above results, we can also derive the following global asymptotic stability conditions based on the determination of the largest allowable magnitude of the admissible uncertainty which will not destabilize the system. The signi?cance of this result is that it provides a basis to evaluate the quality of the design and presents an e5cient way to access the robustness of a feedback system in engineering practice. Corollary 3.3. The uncertain system (3:13) with respect to (3:14) for F T F 6 − *I is globally asymptotically stable, where * is the global minimum of the following generalized eigenvalue minimization problem with respect to P = P T ¿ 0 and diagonal matrices ¿ 0, ¿ 0, and a positive number " ¿ 0: min * s:t:
T1 − *"T3T T3
T2
T2T
−"I
2
2
T1 =
¡ 0;
¿ 0;
C T C + AT P + PA
BT P + (DT + 12 )C PH T2 = ; 0
PB + C T (D + 12 )
+ DT D + 12 (D + DT )
T3 = [E1 E2 ]:
;
(3.17)
4. Robust synthesis In this section, we consider the robust synthesis problem for uncertain systems. It is concerned with designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. Let us ?rst consider the uncertain system of the form x˙ = (A + OA)x + B1 + + (B2 + OB)u; z˙ = Cx + D11 + + D12 u; y = x; + = ’(z);
(4.1)
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
where x ∈ Rn is the state, u ∈ Rp is the control input, + ∈ Rm is the nonlinear input, z˙ ∈ Rm is the controlled output and ’(·) : Rm → Rm is a vector-valued nonlinear mapping. OA and OB represent the parameter uncertainty which belongs to certain bounded compact set. First, we will show that if there exists a dynamic output feedback controller such that system (4.1) is globally asymptotically stable according to Lemma 2.1, there also exists a static state feedback controller to realize the same purpose. Theorem 4.1. For uncertain system (4:1) satisfying (3:14), if there exists a dynamic output feedback controller K(s) such that the resulting closed-loop system is globally asymptotically stable by Lemma 2:1 for all admissible uncertainties, there must exists a static state feedback controller that achieves the same result as well. Furthermore, if the dynamic output feedback controller is x˙k = Ak xk + Bk y;
u = C k xk + D k y
(4.2)
and P is the positive de
Proof. Denote Aˆ = A + OA and Bˆ = B + OB. The closed-loop system of (4.1) with (4.2) is c + B+; x˙c = Ax
c + D11 +; z˙ = Cx
where xc = [xT xkT ]T and ˆ k BC ˆ k Aˆ + BD = A ; Ak Bk
= B
(4.3)
B1 0
;
= [C + D12 Dk D12 Ck ]: C
According to Corollary 3.1, system (4.3) is globally asymptotically stable if there exist a positive de?nite matrix P ¿ 0 and diagonal matrices ; ¿ 0; ¿ 0 such that T C +A T P + P A T (D11 + 1 ) +C C PB 2 ¡ 0; (4.4a) T T T + D11 T P + (D11 B + 12 )C D11 + 12 (D11 + D11 )
2
2
¿ 0:
Multiplying diag(P −1 ; I ) on the left- and right-hand side of (4.4a) we get T CQ + QA T + AQ T (D11 + 1 ) + QC QC B 2 ¡ 0: T T T T + (D11 B + 12 )CQ + D11 D11 + 12 (D11 + D11 )
(4.4b)
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B; into the above inequality we have C Substituting Q; A;
T T [Q11 (C + D12 Dk )T + Q12 CkT D12 ][(C + D12 Dk )Q11 + D12 Ck Q12 ]
∗
T [Q11 (C + D12 Dk )T + Q12 CkT D12 ](D11 + 12 )
∗
∗
∗
∗
T T + D11 D11 + 12 (D11 + D11 )
T + (D11
+
1 2
T )[(C + D12 Dk )Q11 + D12 Ck Q12 ]
ˆ k )T + Q12 CkT BˆT + (Aˆ + BD ˆ k )Q11 + BC ˆ k Q12 Q11 (Aˆ + BD
∗
∗
∗
∗ ¡ 0;
B1T
∗
0
B1
where ∗ is the matrix that need not to be written out. Note that the above inequality guarantees
T T [Q11 (C + D12 Dk )T + Q12 CkT D12 ][(C + D12 Dk )Q11 + D12 Ck Q12 ] T + (D11
+
1 2
T [Q11 (C + D12 Dk )T + Q12 CkT D12 ](D11 +
T )[(C + D12 Dk )Q11 + D12 Ck Q12 ]
1 2
)
T T + D11 D11 + 12 (D11 + D11 )
ˆ k )T + Q12 CkT BˆT + (Aˆ + BD ˆ k )Q11 + BC ˆ k Q12 Q11 (Aˆ + BD
B1
B1T
0
¡ 0:
−1 Multiplying diag(Q11 ; I ) on the left- and right-hand side of the above inequality leads to
−1 T T −1 [(C + D12 Dk )T + Q11 Q12 CkT D12 ][(C + D12 Dk ) + D12 Ck Q12 Q11 ] T (D11 +
+
1 2
−1 T [(C + D12 Dk )T + Q11 Q12 CkT D12 ](D11 +
T −1 )[(C + D12 Dk ) + D12 Ck Q12 Q11 ]
1 2
)
T T + D11 D11 + 12 (D11 + D11 )
ˆ k )T Q−1 + Q−1 Q12 CkT BˆT Q−1 + Q−1 (Aˆ + BD ˆ k Q12 Q−1 ˆ k ) + Q−1 BC (Aˆ + BD 11 11 11 11 11 11
−1 Q11 B1
−1 B1T Q11
0
¡ 0:
−1 we have Denote Y = Q11
(C + D12 K)T (C + D12 K)
(C + D12 K)T (D11 + 12 )
T T T (D11 + 12 )(C + D12 K) + D11 D11 + 12 (D11 + D11 )
+
ˆ ˆ T Y + Y (Aˆ + BK) (Aˆ + BK)
YB1
B1T Y
0
¡ 0:
Note that the above inequality with (4.4b) guarantees global asymptotical stability for the closed-loop system corresponding to the system (4.1) with the state feedback u = Kx ˆ x˙ = (Aˆ + BK)x + B1 +;
z˙ = (C + D12 K)x + D11 +:
Thus completes the theorem. In view of Theorem 4.1, we develop the following conditions for the existence of a static state feedback K such that system (4.1) satisfying (3.14) is globally asymptotically stable. Theorem 4.2. Consider the uncertain system (4:1) satisfying (3:14), then there exists a state feedback controller u = Kx such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties, if and only if for some " ¿ 0 this controller achieves the same property for the
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scaled system +B 2 u; 1 + + B x˙ = Ax +D 11 + + D 12 u; z˙ = Cx y = x; + = ’(z);
(4.5)
where ! ! 2 = B2 + ! H B √ ; B A=A+ HE1 ; B1 = HE2 ; 1 (a + 1)" a+1 (a + 1)" C D11 0 D12 C= √ ; D11 = ; D12 = √ 0 bI "E1 "E2
(4.6)
with the diagonal matrices such that (3:2) holds for the closed-loop system having the forms of ˆ =
0
0
aI
;
ˆ =
0
0
cI
ˆ =
;
0
0
dI
;
where a; b; c; d satisfying (3:8). Proof. Let xc = [xT xk ]T , the closed-loop system (4.1) with the state feedback u = Kx can be described by the following state-space equations S c + B1 +; x˙c = (AS + HF E)x
S c + D11 +; z˙ = Cx
where AS = A + B2 K;
ES = E1 + E2 K;
CS = C + D12 K
and the closed-loop system of (4.5) with the state feedback u = Kx has the form of ! 1 +; S S H E xc + B x˙c = A + √ a+1 CS 11 +; z˙ = √ xc + D "ES
S E; S CS are as in (4.7). Then the desired result can be get immediately from Theorem 3.2 where A;
(4.7)
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In the following, we will consider the output feedback controller synthesizing to achieve global asymptotic stability for uncertain system: x˙ = (A + OA)x + B1 + + (B2 + OB)u; z˙ = C1 x + D11 + + D12 u; y = C2 x + D21 +;
(4.8)
where OA; OB satisfying (3.14). Theorem 4.3. There exists a linear dynamic output feedback controller K(s) such that the resulting closed-loop system is globally asymptotically stable by Lemma 2:1 for all admissible uncertainties, if and only if for some " ¿ 0 this controller achieve the same property for the following scaled system: +B 1 + + B 2 u; x˙ = Ax 1 x + D 11 + + D 12 u; z˙ = C y = C2 x + [D21
0]+;
(4.9)
B 1 ; D 1 ; B 2 ; C 11 ; D 12 are given in (4:6) and the diagonal matrices such that (3:2) holds for where matrices A; the closed-loop system have the form of ˆ =
0
0
aI
;
ˆ =
0
0
cI
ˆ =
;
0
0
dI
;
where a; b; c; d satisfying (3:8). Proof. Without the loss of generality, we assume that K(s) is strictly proper and has the following state– space realization x˙k = Ak xk + Bk y;
u = C k xk :
(4.10)
Let xc = [xT xk ]T , the closed-loop system (4.8) with the controller (4.10) is given by the state-space equations S c + BS 1 +; x˙c = (AS + HS F E)x
z˙ = CS 1 xc + D11 +:
where AS =
A
B 2 Ck
Bk C2
Ak
;
HS =
CS 1 = [C1 D12 Ck ];
BS 1 =
H
;
0
B1 Bk D21
:
ES = [E1 E2 Ck ];
(4.11)
102
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
While the closed-loop system of (4.9) with the controller (4.10) has the form of ! ! S S S S S H B +; x˙c = A + √ H E xc + 1 (a + 1)" a+1 D11 0 CS 1 z˙ = √ +; xc + 0 bI "ES S BS 1 ; HS ; E; S CS 1 are as in (4.11). Then we can get the result from Theorem 3.2. where A; 5. Numerical example Let us consider an example of concrete systems studied in the theory of phase-locked loops (PLL) and show how the results derived in this paper can be used to investigate the locking-in phenomenon of PLL with parameter uncertainty. The locking-in of an arbitrary solution to an equilibrium point is just a synonym of the global asymptotic stability for PLL, the dynamics of which is described by (2.1). The locking-in property is extremely signi?cant for PLL. Many papers and monographs are devoted to it, for example [1,4,19]. The block diagram shown in Fig. 1 is close to the actual implementation of a PLL. Then with some assumptions the model in Fig. 1 can be simpli?ed to the one in Fig. 2 and the voltage-controlled oscillator (VCO) can be considered as an integrator [1]. The model shown in Fig. 2 is rearranged so that all linear components appear in the forward path and the nonlinear component appears in the feedback path as shown in Fig. 3. Both the reference phase signal and the VCO control signal are set to zero in Fig. 3. From Fig. 3, the transfer function from the input sin(·) to the output −1 is 1 G(s) = K(s); s where K(s) is the transfer function from sin(·) to −1˙ and described by state–space model of the form −63 −20 8 x˙ = x+ +; 32 0 0 1˙ = [2 − 2]x − 0:5+; + = sin 1:
(5.1)
System (5.1) describes the dynamics of an autonomous phase-locked loop with a second-order ?lter of type “2/2” [10] and a 23-periodic input nonlinearity. Now, we suppose that the state and input matrices of system (5.1) are subject to norm-bounded parameter uncertainty of the form (3.14) with F T F 6 I and the weighting functions are given as 1 0 1 1 0 ; E2 = : H= ; E1 = 0 −1 1 0 1 Solving the linear matrix inequalities in Theorem 3.3 by using LMI Toolbox [3], we get 944:4 531:4 P= ; = 1165:8; 531:4 1946:6 = 101:7725;
= 7100:3;
" = 301:5821:
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
sin( ω1 t + θ )
103
Low-Pass Filter VCO
cos( ω2 t + φ )
control voltage Fig. 1. Phase-locked loop.
θ
Low-Pass Filter
sin(·) φ
Kv/s γ
Fig. 2. Model of Phase-locked loop.
G(s) GFilt (s)
Kv /s
φ
sin(·) Fig. 3. Rearrangement into a linear component with a feedback nonlinear component.
Thus the system (5.1) satisfying (3.14) is globally asymptotically stable for all admissible uncertainty, i.e. the PLL achieves robust locking-in property. Solve the generalized eigenvalue problem corresponding to (3.17), we get the largest allowable uncertainty bound for system (5.1) is * = −1:5666: From Corollary 3.3, this result guarantees that the uncertain pendulum-like system (5.1) with (3.14) will be robustly stale for ∀F; F T F 6 1:5666I . This estimation can be veri?ed by Fig. 4 where the numerical experiment results of system (5.1) with 20 randomly generated initial value x0 ; 10 and F(F = 1:2516) are given. The result presented in Fig. 4 where all of xi converge to 0 and 1 converges to 2k3 shows that the systems perturbed by those F are all global asymptotically stable. This observation coincides with Theorem 3.3 and Corollary 3.3 and con?rms the robust stability of system (5.1). 6. Conclusion In this paper, robust analysis and synthesis results guaranteeing the global asymptotic stability for a class of uncertain nonlinear systems with multiple equilibria have been presented. Using the Kalman–Yakubovich– Popov lemma in terms of linear matrix inequality as the analytical framework, early work performed by Leonov et al for nominal systems is extended to take account of system uncertainties and derive feedback control law to ensure stability for robust case. A concrete example of phase-locked loop illustrates the e5ciency of the proposed approach. Under this LMI-based framework, other global asymptotic properties of systems with
104
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105 Simulation Results 8 x1 x2 φ
6
4
x and φ
2
0
-2
-4
-6
-8
0
1
2
3
4
5
6
7
8
9
10
Time(s) Fig. 4. Simulation for F = 1:2516.
multiple equilibria can be investigated as well as synthesizing corresponding feedback controller to ensure those properties. These will be the subjects of further study. References [1] D.Y. Abramovitch, Lyapunov redesign of analog phase-lock loops, IEEE Trans. Comm. 38 (12) (1990) 2197–2202. [2] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. [3] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox User’s Guide, 1st Edition, The MathWorks, 24 Prime Park Way, Natick, MA 01760-1500, 1995. [4] F.M. Gardner, Phaselock Techniques, Wiley, New York, 1979. [5] A.K. Gelig, G.A. Leonov, V.A. Yakubovich, The Stability of Nonlinear Systems with Nonunique Equilibrium, Nauka, Moscow, 1978 (in Russian). [6] K. Gu, H∞ control of systems under norm bounded uncertainties in all system matrices, IEEE Trans. Automat. Control 39 (6) (1994) 1320–1322. [7] W.M. Haddad, V. Kapila, Fixed-architecture controller synthesis for systems with input–output time-varying nonlinearities, Int. J. Robust Nonlinear Control 7 (7) (1997) 675–710. [8] G.A. Leonov, I.M. Burkin, A.L. Shepeljavyi, Frequency Methods in Oscillation Theory, Kluwer Academic Publishers, Dordrecht, 1992. [9] G.A. Leonov, A. Noack, V. Reitmann, Asymptotic orbital stability conditions for Wows by estimates of singular values of the linearization, Nonlinear Anal. 44 (8) (2001) 1057–1085. [10] G.A. Leonov, D.V. Ponomarenko, V.B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis, World Scienti?c, Singapore, 1996. [11] G.A. Leonov, V.B. Smirnova, Stability and oscillations of solutions of integro-diDerential equations of pendulum-like systems, Math. Nachr. 177 (1996) 157–181. [12] J. Moser, On nonoscillating networks, Quart. Appl. Math. 25 (1967) 1–9. [13] K.S. Narendra, J.H. Taylor, Frequency Domain Criteria for Absolute Stability, Academic Press, New York, 1973.
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105
V. Popov, Absolute stability of non-linear systems of automatic control, Automation Remote Control 22 (1961) 857–875. A. Rantzer, On the Kalman–Yakubovich–Popov lemma, Systems Control Lett. 28 (1) (1996) 7–10. V. Rasvan, Dynamical systems with several equilibria and natural Liapunov functions, Arch. Math. 34 (1998) 207–215. J.A.K. Suykens, J. Vandewalle, B.D. Moor, An absolute stability criterion for the Lur’e problem with sector and slope restricted nonlinearities, IEEE Trans. Circuits Systems I 45 (9) (1998) 1007–1009. M. Vidyasagar, Nonlinear System Analysis, Prentice-Hall, Englewood CliDs, NJ, 1993. N.E. Wu, Analog phaselock loop design using Popov criterion, in: Proceedings of the American Control Conference, 2002, pp. 16–18. L. Xie, Y.C. Soh, Positive real control problem for uncertain linear time-invariant systems, Systems Control Lett. 24 (4) (1999) 265–271. Y. Yang, L. Huang, H∞ controller synthesis for pendulum-like systems, Systems Control Lett. 50 (4) (2003) 263–276.
Systems & Control Letters 53 (2004) 89 – 105
www.elsevier.com/locate/sysconle
Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria Ying Yang∗ , Rao Fu, Lin Huang Department of Mechanics and Engineering Science, Center for Systems and Control, Peking University, Beijing 100871, PR China Received 21 June 2003; received in revised form 14 December 2003; accepted 26 February 2004
Abstract This paper focuses on a class of uncertain nonlinear systems which are subject to norm-bounded parameter uncertainty in the forward path and a vector-valued periodic nonlinearity in the feedback path, and addresses robust analysis and synthesis problems for such systems. Su5cient conditions for global asymptotic stability are derived in terms of linear matrix inequalities (LMIs) and a technique for the estimation of the uncertainty bound is proposed by solving a generalized eigenvalue minimization problem. The problem of robust synthesis is concerned with designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. It is shown that a solution to the robust synthesis problem for the uncertain system can be obtained by solving a synthesis problem for an uncertainty free system. A concrete example is presented to demonstrate the applicability and validity of the proposed approach. c 2004 Elsevier B.V. All rights reserved. Keywords: Nonlinear systems; Multiple equilibria; Norm-bounded parameter uncertainties; Analysis and synthesis; LMI
1. Introduction The past decades has witnessed signi?cant advances in stability analysis and synthesis for nonlinear feedback systems, see [7,13,14,17,18] and the references therein. But most of the results were applied and developed within the frames of absolute stability theory which was constructed for investigation of global stability of systems with single equilibrium. Meanwhile engineers often have to deal with systems with multiple equilibria, types of equilibria sets being rather diverse. In this case, new stability problems appeared. Since these problems cannot be investigated by means of classical Lyapunov or Popov tools, the necessity arises to develop the classical theory in such a way that it should be possible to use it for stability investigation of systems with multiple equilibria. Motivated by these facts, the qualitative theory of systems with multiple equilibria has been established to investigate global behavior of solutions for systems with multiple equilibria. This theory starts from the paper of Moser [12] and has been developed in a comprehensive way by Yakubovich, Leonov and their co-workers [5,10]. Interesting work in this ?eld, in the context of integral and integro-diDerential equations, can be seen in [2,11]. ∗
This work was supported by the National Science Foundation of China under Grant 60334030,10272001. Corresponding author. Tel.: +86-1062751815; fax: +86-1062764044. E-mail addresses: [email protected] (Y. Yang), [email protected] (R. Fu), [email protected] (L. Huang).
c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.02.024
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
The behavior of solutions with respect to entire set of equilibria has not only purely theoretical interest for the qualitative theory of diDerential equations, but also important applied signi?cance. Note that a broad class of diDerential equations with angular coordinates in mechanics, phase-synchronization and other ?elds can be written in the form x˙ = f(x)
(1.1) n
n
n
with the vector ?eld f : R → R having mthe property f(x) = f(x + d) for every d ∈ and x ∈ R , where is a subgroup of Rn de?ned by := { i=1 ki di |ki ∈ Z}; m 6 n and {di }m i=1 are linearly independent vectors in Rn . Such systems are often called systems with cylindrical phase space, and are also called pendulum-like systems with respect to in [10,16]. In the previous literature [5,9,10,16,21], some important global asymptotic properties which characterizes the behavior of solutions for systems with multiple equilibria have been intensively studied and frequency-domain inequalities conditions have been established. However, compared with the richness of the exiting results on robust analysis and synthesis for nonlinear systems with single equilibrium, there is almost no robustness results or synthesis results of designing feedback controller to ensure those properties for systems with multiple equilibria. The aim of this paper is to bridge this gap by investigating a class of nonlinear feedback systems with periodic nonlinearities and provide a more complete picture of the behavior of solutions with respect to entire set of equilibria. In this paper, we will consider the robust analysis and synthesis problem for a class of uncertain nonlinear systems which are subject to norm-bounded parameter uncertainty in the forward path and a vector-valued periodic nonlinearity in the feedback path. The system under consideration will be described by a state-space model which contains parameter uncertainties in both the state and input matrices. Based on the Kalman– Yakubovich–Popov lemma connecting the frequency-domain inequality and linear matrix inequality [15], suf?cient conditions of global asymptotic stability for uncertain systems are given in terms of the existence of solutions of a set of LMIs. Meanwhile, the robust synthesis problem is addressed by designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. It will be shown that the robust synthesis problem can be converted into a synthesis problem for an uncertainty free system. Parallel results in H∞ control and positive real control for linear systems can be seen in [6,20], but the results in this paper are believed to be new in the context of global asymptotic analysis and synthesis for uncertain nonlinear systems with multiple equilibria. Furthermore, the formulation diDers from [6,20] in that it involves no parameter searching by using the linear matrix inequality method. With this LMI approach, the largest allowable magnitude of the admissible uncertainty can also be explicitly computed by solving a generalized eigenvalue minimization problem which is essentially a convex optimization problem and numerically e5cient. It should be pointed out here that the methodology used in this paper is not restricted to the class of systems under consideration. In fact, the existing formulations of criteria for stability, dichotomy and instability of the class of Lur’e systems in terms of the frequency response of the linear part of the system can also be recast into the framework of LMIs and the corresponding robust analysis and synthesis results can be derived in a similar way. The rest of this paper is organized as follows: Section 2 presents some basic results necessary for the successive development, while in Section 3, we give some LMI-based global asymptotic stability results for systems with norm-bounded parameter uncertainties; Section 4 presents both the static state and dynamic output feedback controller existence conditions for such uncertain systems and shows that the robust synthesis problem can be converted into a synthesis problem for an uncertainty free system. Section 5 gives a concrete example showing the application of the proposed method and we end this paper by a few concluding remarks in Section 6. Notation: Rn×n is the set of n × n real matrices. For a matrix A, AT denotes its transpose, A∗ its complex conjugate transpose. The matrix inequality A ¿ B (A ¿ B) means that A and B are square Hermitian matrices and A − B is positive (semi-)de?nite. He is Hermit operator with He A = A + AT .
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
91
2. Preliminaries In this section, we review some useful results which will play crucial roles in this paper. First let us consider a nonlinear feedback system of the form x˙ = Ax + B’(z); n×n
z˙ = Cx + D’(z); n×m
m×n
(2.1) m×m
where A ∈ R ; B ∈ R ; C ∈R ; D∈R . We suppose that (A; B) is controllable, (A; C) is observable and ’ : Rm → Rm is a vector-valued function having the components ’i (z) = ’i (zi ) with z = (z1 ; z2 ; : : : ; zm )T . We assume that every component ’i : R → R is i periodic, satis?es a local Lipschitz condition and possesses a ?nite number of zeroes on [0; i ), and for each component ’i there exists z0i such that ’i (z0i ) = 0 and ’˙ i (z0i ) = 0. Let us introduce the vector di = (0; : : : ; 0; i ; 0;: : : ; 0), where i is the (n + i)th component of di , m then system (2.1) is pendulum-like with respect to = { i=1 ki di ; ki ∈ Z}. Assume that i i i = ’i (z) d z |’i (z) |d z; i ∈ m 0 0 and denotes = diag{1 ; 2 ; : : : ; m }. The transfer function of the linear part of (2.1) from the input ’ to the output −z˙ is given by K(s) = C(A − sI )−1 B − D: Suppose that K(0) is nonsingular and det A = 0. Any equilibrium of (2.1) satis?es Axeq = −B’(zeq );
Cxeq = −D’(zeq )
(2.2)
and consequently we have (D − CA−1 B)’(zeq ) = 0: By the assumption above, we can get that ’(zeq ) = 0 and every component (zeq )i is a zero of ’i ; i = m. From the ?rst equation of (2.2) it follows that xeq = 0. Since ’i is i -periodic, system (2.1) has in?nitely many isolated equilibria. Denition 2.1 (Leonov et al. [8]). System (2.1) is said to be globally asymptotically stable if each solution of it tends to a certain equilibrium as t → +∞. Remark 2.1. Note that in De?nition 2.1 it is not implied that all equilibria of (2.1) are Lyapunov stable. Moreover, it can be shown that in the case of global asymptotic stability of system (2.1) there exists at least one equilibrium that is not asymptotically stable [8]. In the following sections, we consider the global asymptotic stability in the sense of De?nition 2.1 for system (2.1) with parameter uncertainties. To derive our main results, we also need the following lemmas. Lemma 2.1 (Leonov et al. [10]). Suppose K(s) is stable and there exist diagonal matrices =diag{1 ; 2 ; : : : ; m }, = diag{1 ; 2 ; : : : ; m } and = diag{1 ; 2 ; : : : ; m } with ¿ 0 and ¿ 0 satisfying the following conditions: 1. 12 He[K(i!) − K(i!)K ∗ (i!)] − ¿ 0; 2. 4 ¿ ()2 then system (2:1) is globally asymptotically stable.
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
Lemma 2.2 (Rantzer [15]). Given A ∈ Rn×n ; B ∈ Rn×n ; M = M T ∈ R(n+m)×(n+m) , with det(j!I − A) = 0 for ! ∈ R and (A; B) controllable, the following two statements are equivalent: 1. ∗ (j!I − A)−1 B ( j!I − A)−1 B 6 0; M I I ∀! ∈ R ∪ {∞}; 2. there exists a matrix P = P T ∈ Rn×n such that T A P + PA PB M+ 6 0: 0 BT P The corresponding equivalence for strict inequalities holds even if (A; B) is not controllable. Lemma 2.3. Let T1 = T1T ; T2 ; T3 be real matrices of appropriate size, then the following statements are equivalent: 1. T1 + He(T2 T3 ) ¡ 0; ∀ : T 6 !2 I ; 2. there exists a positive number " ¿ 0 such that 1 T1 + "!2 T2 T2T + T3T T3 ¡ 0; " 3. there exists a positive number " ¿ 0 such that T3T T1 + "!2 T2 T2T ¡ 0; T3 −"I 4. there exists a positive number " ¿ 0 such that T1 + "!2 T3T T3 T2 ¡ 0: T2T −"I
(2.3)
(2.4)
(2.5)
Remark 2.2. The above result is not new and, in fact, is well known in the robustness analysis literatures. Here we give it in an alternative form of LMI. The advantage of this representation is that when there is LMI variable in T2 which can result in a product term with ", we can resort to using (2.5) for convex optimization and vice versa. This method does not involve any parameter searching so that there is no need to consider the convergence property of (2.5). 3. Robust analysis In this section, we derive the global asymptotical stability results for system (2.1) when there is norm-bounded parameter uncertainties in the linear part. First, we give a theorem which establishes the connection between the frequency-domain conditions given in Lemma 2.1 and an LMI-based criterion. Theorem 3.1. Suppose A is Hurwitz, then the conditions of Lemma 2:1 are satis
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
2
2
93
¿ 0:
(3.1b)
Proof. Let M=
C T C
C T (D + 12 )
(DT + 12 )C
+ DT D + 12 (D + DT )
and using Lemma 2.2, we can prove the equivalence of (3.1a) and condition (1) of Lemma 2.1. Note that the upper left corner of M is positive semide?nite, it follows from (3.1a) and Hurwitz stability of A that P ¿ 0. (3.1b) is directly derived from condition (2) of Lemma 2.1. Remark 3.1. The signi?cance of this theorem is that, by using Lemma 2.2 we convert the conditions of Lemma 2.1 into an equivalent LMI requirement. From this LMI condition, it is possible to extend the results to take into account the parameter uncertainty in the linear part of the system and derive feedback control law which renders the closed-up system globally asymptotically stable by using the e5cient numerical linear matrix inequalities methods. As an immediate consequence, we have a more convenient criterion as stated in the following. Corollary 3.1. If there exist P = P T ¿ 0 and diagonal matrices , ¿ 0 and ¿ 0 such that T C T C A P + PA PB C T (D + 12 ) + ¡ 0: 0 (DT + 12 )C + DT D + 12 (D + DT ) BT P
2
2
(3.2a)
¿ 0;
(3.2b)
then system (2:1) is globally asymptotically stable. Let us now consider the following uncertain system described by state-space models of the form: x˙ = (A + OA)x + B’(z);
z˙ = Cx + D’(z);
(3.3)
where OA stands for the parameter uncertainties which are norm-bounded and of the form OA = HFE
(3.4)
and H ∈ Rn×i , E ∈ Ri×m are known constant matrices and F ∈ Ri×i is an unknown matrix function satisfying F T F 6 !2 I
(3.5)
with ! ¿ 0 a given constant. From the de?nition of in Lemma 2.1 we know 6 %I , where % ¡ 1. In the following sections, we assume that the system is controllable and observable for all admissible uncertainties. Then we have the following result:
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
Theorem 3.2. There exist diagonal matrices ¿ 0; ¿ 0 and such that (3:2) holds for the uncertain system (3:3) satisfying (3:4) if and only if there exists a scaling parameter " ¿ 0 such that (3:2) holds with 0 0 0 ˆ ˆ = ; ˆ = ; = (3.6) 0 aI 0 cI 0 dI for the system ˆ + Bˆ ’(z); x˙ = Ax ˜ where
ˆ + Dˆ ’(z); z˙ = Cx ˜
(3.7)
! ! ˆ ˆ H ; A=A+ √ HE; B = B (a + 1)" a+1 C D 0 Cˆ = √ ; Dˆ = "E 0 bI
and a; b; c; d satisfying a ¿ 0; b=−
c2 ¿ c+2 ; 2a
4(1 + a) ; 1 − %2 d=
b(2 − c) − 2 : 2
(3.8)
Proof. Eq. (3.2) holds for system (3.7) if there exists a positive de?nite solution P = P T ¿ 0 to the linear matrix inequality T Cˆ T ˆCˆ Cˆ T (ˆDˆ + 12 ) ˆ Aˆ P + P Aˆ P Bˆ + ¡0 (3.9) ˆ Bˆ T P 0 (Dˆ T ˆ + 12 ) ˆ Cˆ ˆ + Dˆ T ˆDˆ + 12 He(ˆD) with
2ˆ
ˆ˜
˜ˆ
2ˆ
¿ 0:
(3.10)
ˆ ˆ into (3.9) leads to ˆ B; ˆ D; ˆ C; ˆ ;ˆ ; Substituting A;
√ T ! C T C + a"E T E + He P 6 ft A + √a+1 HE PB + C T (D + 12 ) √ ! PH + ab + 2c "E (a+1)" T T T 1 1 ¡0 B P + (D + 2 )C + D D + 2 He(D) 0 √ T 2 ! c √ H P + ab + 2 "E 0 (d + ab + bc)I (a+1)"
(3.11) From (3.8) we have ab +
c = −1; 2
d + ab2 + bc = −1
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
95
then (3.11) becomes
C T C + a"E T E + He P A +
T
T
√! a+1
HE
PB + C T (D + 12 ) T
1 2
B P + (D + )C √ √ ! H T P − "E
√
! PH (a+1)"
1 2
+ D D + He(D)
0
0
−I
(a+1)"
−
√
"E T
¡ 0:
Using Schur Complement, the above inequality is equivalent to
!2 C C + A P + PA + PHH T P + (a + 1)"E T E (a + 1)" BT P + (DT + 12 )C T
T
T
1 2
PB + C (D + ) ¡ 0: T 1 + D D + 2 He(D)
(3.12)
By Lemma 2.3, (3.12) holds if and only if for any F satisfying (3.5)
C T C
C T (D + 12 )
(DT + 12 )C
+ DT D + 12 He(D)
+
He(A + HFE)T P
PB
BT P
0
¡ 0:
As for (3.10), we can verify by straightforward manipulations that
2ˆ
ˆ˜
˜ˆ
2ˆ
¿0
is equivalent to
2
2
¿0
by noting that (3.8) implies 4ad ¿ %2 c2 : Thus completes the proof. Corollary 3.2. The uncertain system (3:3) is globally asymptotically stable by Lemma 2:1 if and only if there exists a scaling parameter " ¿ 0 such that the condition in Corollary 3:1 is satis
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
Next we consider the uncertain system described by x˙ = (A + OA)x + (B + OB)’(z);
z˙ = Cx + D’(z);
(3.13)
where OA and OB have the form of [OA OB] = HF[E1 E2 ]
(3.14)
and H ∈ Rn×i , E1 ∈ Rj×n , E2 ∈ Rj×m are known constant matrices and F ∈ Ri×j is an unknown matrix function satisfying F T F 6 !2 I with ! ¿ 0 a given constant. Then we have the following result: Theorem 3.3. Suppose there exist P = P T ¿ 0, diagonal matrices ; ¿ 0; ¿ 0 and a positive number " ¿ 0 such that the following linear matrix inequalities hold: T C C + AT P + PA PB + C T (D + 12 ) PH "!E1T T B P + (DT + 1 )C + DT D + 1 (D + DT ) 0 "!E2T 2 2 (3.15a) ¡ 0; T H P 0 −"I 0 "!E1 "!E2 0 −"I
2
2
¿ 0;
(3.15b)
then system (3:13) is globally asymptotically stable. Proof. By Corollary 3.1, system (3:13) is globally asymptotically stable if there exist P ¿ 0 and diagonal matrices ; ¿ 0; ¿ 0 such that C T C C T (D + 12 ) (DT + 12 )C + DT D + 12 (D + DT ) (A + HFE1 )T P + P(A + HFE1 ) P(B + HFE2 ) + ¡0 (B + HFE2 )T P 0 and (3.15b) holds. Denote T C C + AT P + PA M= BT P + (DT + 12 )C
PB + C T (D + 12 ) + DT D + 12 (D + DT )
then (3.16) can be written in the form of PH M + He F[E1 E2 ] ¡ 0: 0
;
(3.16)
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
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According to Lemma 2.3, the above inequality holds if and only if there exists a positive number " ¿ 0 such that T E1 1 PH 2 M + "! [H T P 0] ¡ 0 [E1 E2 ] + " 0 E2T it can be easily proved that the above inequality is equivalent to (3.15a). Remark 3.3. The above result show that assessing global asymptotic stability of the uncertain system (3.13) satisfying (3.14) can be carried out by solving two LMIs which is essentially a convex optimization problem and numerically e5cient. From the above results, we can also derive the following global asymptotic stability conditions based on the determination of the largest allowable magnitude of the admissible uncertainty which will not destabilize the system. The signi?cance of this result is that it provides a basis to evaluate the quality of the design and presents an e5cient way to access the robustness of a feedback system in engineering practice. Corollary 3.3. The uncertain system (3:13) with respect to (3:14) for F T F 6 − *I is globally asymptotically stable, where * is the global minimum of the following generalized eigenvalue minimization problem with respect to P = P T ¿ 0 and diagonal matrices ¿ 0, ¿ 0, and a positive number " ¿ 0: min * s:t:
T1 − *"T3T T3
T2
T2T
−"I
2
2
T1 =
¡ 0;
¿ 0;
C T C + AT P + PA
BT P + (DT + 12 )C PH T2 = ; 0
PB + C T (D + 12 )
+ DT D + 12 (D + DT )
T3 = [E1 E2 ]:
;
(3.17)
4. Robust synthesis In this section, we consider the robust synthesis problem for uncertain systems. It is concerned with designing a feedback controller such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties. Let us ?rst consider the uncertain system of the form x˙ = (A + OA)x + B1 + + (B2 + OB)u; z˙ = Cx + D11 + + D12 u; y = x; + = ’(z);
(4.1)
98
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
where x ∈ Rn is the state, u ∈ Rp is the control input, + ∈ Rm is the nonlinear input, z˙ ∈ Rm is the controlled output and ’(·) : Rm → Rm is a vector-valued nonlinear mapping. OA and OB represent the parameter uncertainty which belongs to certain bounded compact set. First, we will show that if there exists a dynamic output feedback controller such that system (4.1) is globally asymptotically stable according to Lemma 2.1, there also exists a static state feedback controller to realize the same purpose. Theorem 4.1. For uncertain system (4:1) satisfying (3:14), if there exists a dynamic output feedback controller K(s) such that the resulting closed-loop system is globally asymptotically stable by Lemma 2:1 for all admissible uncertainties, there must exists a static state feedback controller that achieves the same result as well. Furthermore, if the dynamic output feedback controller is x˙k = Ak xk + Bk y;
u = C k xk + D k y
(4.2)
and P is the positive de
Proof. Denote Aˆ = A + OA and Bˆ = B + OB. The closed-loop system of (4.1) with (4.2) is c + B+; x˙c = Ax
c + D11 +; z˙ = Cx
where xc = [xT xkT ]T and ˆ k BC ˆ k Aˆ + BD = A ; Ak Bk
= B
(4.3)
B1 0
;
= [C + D12 Dk D12 Ck ]: C
According to Corollary 3.1, system (4.3) is globally asymptotically stable if there exist a positive de?nite matrix P ¿ 0 and diagonal matrices ; ¿ 0; ¿ 0 such that T C +A T P + P A T (D11 + 1 ) +C C PB 2 ¡ 0; (4.4a) T T T + D11 T P + (D11 B + 12 )C D11 + 12 (D11 + D11 )
2
2
¿ 0:
Multiplying diag(P −1 ; I ) on the left- and right-hand side of (4.4a) we get T CQ + QA T + AQ T (D11 + 1 ) + QC QC B 2 ¡ 0: T T T T + (D11 B + 12 )CQ + D11 D11 + 12 (D11 + D11 )
(4.4b)
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
99
B; into the above inequality we have C Substituting Q; A;
T T [Q11 (C + D12 Dk )T + Q12 CkT D12 ][(C + D12 Dk )Q11 + D12 Ck Q12 ]
∗
T [Q11 (C + D12 Dk )T + Q12 CkT D12 ](D11 + 12 )
∗
∗
∗
∗
T T + D11 D11 + 12 (D11 + D11 )
T + (D11
+
1 2
T )[(C + D12 Dk )Q11 + D12 Ck Q12 ]
ˆ k )T + Q12 CkT BˆT + (Aˆ + BD ˆ k )Q11 + BC ˆ k Q12 Q11 (Aˆ + BD
∗
∗
∗
∗ ¡ 0;
B1T
∗
0
B1
where ∗ is the matrix that need not to be written out. Note that the above inequality guarantees
T T [Q11 (C + D12 Dk )T + Q12 CkT D12 ][(C + D12 Dk )Q11 + D12 Ck Q12 ] T + (D11
+
1 2
T [Q11 (C + D12 Dk )T + Q12 CkT D12 ](D11 +
T )[(C + D12 Dk )Q11 + D12 Ck Q12 ]
1 2
)
T T + D11 D11 + 12 (D11 + D11 )
ˆ k )T + Q12 CkT BˆT + (Aˆ + BD ˆ k )Q11 + BC ˆ k Q12 Q11 (Aˆ + BD
B1
B1T
0
¡ 0:
−1 Multiplying diag(Q11 ; I ) on the left- and right-hand side of the above inequality leads to
−1 T T −1 [(C + D12 Dk )T + Q11 Q12 CkT D12 ][(C + D12 Dk ) + D12 Ck Q12 Q11 ] T (D11 +
+
1 2
−1 T [(C + D12 Dk )T + Q11 Q12 CkT D12 ](D11 +
T −1 )[(C + D12 Dk ) + D12 Ck Q12 Q11 ]
1 2
)
T T + D11 D11 + 12 (D11 + D11 )
ˆ k )T Q−1 + Q−1 Q12 CkT BˆT Q−1 + Q−1 (Aˆ + BD ˆ k Q12 Q−1 ˆ k ) + Q−1 BC (Aˆ + BD 11 11 11 11 11 11
−1 Q11 B1
−1 B1T Q11
0
¡ 0:
−1 we have Denote Y = Q11
(C + D12 K)T (C + D12 K)
(C + D12 K)T (D11 + 12 )
T T T (D11 + 12 )(C + D12 K) + D11 D11 + 12 (D11 + D11 )
+
ˆ ˆ T Y + Y (Aˆ + BK) (Aˆ + BK)
YB1
B1T Y
0
¡ 0:
Note that the above inequality with (4.4b) guarantees global asymptotical stability for the closed-loop system corresponding to the system (4.1) with the state feedback u = Kx ˆ x˙ = (Aˆ + BK)x + B1 +;
z˙ = (C + D12 K)x + D11 +:
Thus completes the theorem. In view of Theorem 4.1, we develop the following conditions for the existence of a static state feedback K such that system (4.1) satisfying (3.14) is globally asymptotically stable. Theorem 4.2. Consider the uncertain system (4:1) satisfying (3:14), then there exists a state feedback controller u = Kx such that the resulting closed-loop system is globally asymptotically stable for all admissible uncertainties, if and only if for some " ¿ 0 this controller achieves the same property for the
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
scaled system +B 2 u; 1 + + B x˙ = Ax +D 11 + + D 12 u; z˙ = Cx y = x; + = ’(z);
(4.5)
where ! ! 2 = B2 + ! H B √ ; B A=A+ HE1 ; B1 = HE2 ; 1 (a + 1)" a+1 (a + 1)" C D11 0 D12 C= √ ; D11 = ; D12 = √ 0 bI "E1 "E2
(4.6)
with the diagonal matrices such that (3:2) holds for the closed-loop system having the forms of ˆ =
0
0
aI
;
ˆ =
0
0
cI
ˆ =
;
0
0
dI
;
where a; b; c; d satisfying (3:8). Proof. Let xc = [xT xk ]T , the closed-loop system (4.1) with the state feedback u = Kx can be described by the following state-space equations S c + B1 +; x˙c = (AS + HF E)x
S c + D11 +; z˙ = Cx
where AS = A + B2 K;
ES = E1 + E2 K;
CS = C + D12 K
and the closed-loop system of (4.5) with the state feedback u = Kx has the form of ! 1 +; S S H E xc + B x˙c = A + √ a+1 CS 11 +; z˙ = √ xc + D "ES
S E; S CS are as in (4.7). Then the desired result can be get immediately from Theorem 3.2 where A;
(4.7)
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
101
In the following, we will consider the output feedback controller synthesizing to achieve global asymptotic stability for uncertain system: x˙ = (A + OA)x + B1 + + (B2 + OB)u; z˙ = C1 x + D11 + + D12 u; y = C2 x + D21 +;
(4.8)
where OA; OB satisfying (3.14). Theorem 4.3. There exists a linear dynamic output feedback controller K(s) such that the resulting closed-loop system is globally asymptotically stable by Lemma 2:1 for all admissible uncertainties, if and only if for some " ¿ 0 this controller achieve the same property for the following scaled system: +B 1 + + B 2 u; x˙ = Ax 1 x + D 11 + + D 12 u; z˙ = C y = C2 x + [D21
0]+;
(4.9)
B 1 ; D 1 ; B 2 ; C 11 ; D 12 are given in (4:6) and the diagonal matrices such that (3:2) holds for where matrices A; the closed-loop system have the form of ˆ =
0
0
aI
;
ˆ =
0
0
cI
ˆ =
;
0
0
dI
;
where a; b; c; d satisfying (3:8). Proof. Without the loss of generality, we assume that K(s) is strictly proper and has the following state– space realization x˙k = Ak xk + Bk y;
u = C k xk :
(4.10)
Let xc = [xT xk ]T , the closed-loop system (4.8) with the controller (4.10) is given by the state-space equations S c + BS 1 +; x˙c = (AS + HS F E)x
z˙ = CS 1 xc + D11 +:
where AS =
A
B 2 Ck
Bk C2
Ak
;
HS =
CS 1 = [C1 D12 Ck ];
BS 1 =
H
;
0
B1 Bk D21
:
ES = [E1 E2 Ck ];
(4.11)
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Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
While the closed-loop system of (4.9) with the controller (4.10) has the form of ! ! S S S S S H B +; x˙c = A + √ H E xc + 1 (a + 1)" a+1 D11 0 CS 1 z˙ = √ +; xc + 0 bI "ES S BS 1 ; HS ; E; S CS 1 are as in (4.11). Then we can get the result from Theorem 3.2. where A; 5. Numerical example Let us consider an example of concrete systems studied in the theory of phase-locked loops (PLL) and show how the results derived in this paper can be used to investigate the locking-in phenomenon of PLL with parameter uncertainty. The locking-in of an arbitrary solution to an equilibrium point is just a synonym of the global asymptotic stability for PLL, the dynamics of which is described by (2.1). The locking-in property is extremely signi?cant for PLL. Many papers and monographs are devoted to it, for example [1,4,19]. The block diagram shown in Fig. 1 is close to the actual implementation of a PLL. Then with some assumptions the model in Fig. 1 can be simpli?ed to the one in Fig. 2 and the voltage-controlled oscillator (VCO) can be considered as an integrator [1]. The model shown in Fig. 2 is rearranged so that all linear components appear in the forward path and the nonlinear component appears in the feedback path as shown in Fig. 3. Both the reference phase signal and the VCO control signal are set to zero in Fig. 3. From Fig. 3, the transfer function from the input sin(·) to the output −1 is 1 G(s) = K(s); s where K(s) is the transfer function from sin(·) to −1˙ and described by state–space model of the form −63 −20 8 x˙ = x+ +; 32 0 0 1˙ = [2 − 2]x − 0:5+; + = sin 1:
(5.1)
System (5.1) describes the dynamics of an autonomous phase-locked loop with a second-order ?lter of type “2/2” [10] and a 23-periodic input nonlinearity. Now, we suppose that the state and input matrices of system (5.1) are subject to norm-bounded parameter uncertainty of the form (3.14) with F T F 6 I and the weighting functions are given as 1 0 1 1 0 ; E2 = : H= ; E1 = 0 −1 1 0 1 Solving the linear matrix inequalities in Theorem 3.3 by using LMI Toolbox [3], we get 944:4 531:4 P= ; = 1165:8; 531:4 1946:6 = 101:7725;
= 7100:3;
" = 301:5821:
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105
sin( ω1 t + θ )
103
Low-Pass Filter VCO
cos( ω2 t + φ )
control voltage Fig. 1. Phase-locked loop.
θ
Low-Pass Filter
sin(·) φ
Kv/s γ
Fig. 2. Model of Phase-locked loop.
G(s) GFilt (s)
Kv /s
φ
sin(·) Fig. 3. Rearrangement into a linear component with a feedback nonlinear component.
Thus the system (5.1) satisfying (3.14) is globally asymptotically stable for all admissible uncertainty, i.e. the PLL achieves robust locking-in property. Solve the generalized eigenvalue problem corresponding to (3.17), we get the largest allowable uncertainty bound for system (5.1) is * = −1:5666: From Corollary 3.3, this result guarantees that the uncertain pendulum-like system (5.1) with (3.14) will be robustly stale for ∀F; F T F 6 1:5666I . This estimation can be veri?ed by Fig. 4 where the numerical experiment results of system (5.1) with 20 randomly generated initial value x0 ; 10 and F(F = 1:2516) are given. The result presented in Fig. 4 where all of xi converge to 0 and 1 converges to 2k3 shows that the systems perturbed by those F are all global asymptotically stable. This observation coincides with Theorem 3.3 and Corollary 3.3 and con?rms the robust stability of system (5.1). 6. Conclusion In this paper, robust analysis and synthesis results guaranteeing the global asymptotic stability for a class of uncertain nonlinear systems with multiple equilibria have been presented. Using the Kalman–Yakubovich– Popov lemma in terms of linear matrix inequality as the analytical framework, early work performed by Leonov et al for nominal systems is extended to take account of system uncertainties and derive feedback control law to ensure stability for robust case. A concrete example of phase-locked loop illustrates the e5ciency of the proposed approach. Under this LMI-based framework, other global asymptotic properties of systems with
104
Y. Yang et al. / Systems & Control Letters 53 (2004) 89 – 105 Simulation Results 8 x1 x2 φ
6
4
x and φ
2
0
-2
-4
-6
-8
0
1
2
3
4
5
6
7
8
9
10
Time(s) Fig. 4. Simulation for F = 1:2516.
multiple equilibria can be investigated as well as synthesizing corresponding feedback controller to ensure those properties. These will be the subjects of further study. References [1] D.Y. Abramovitch, Lyapunov redesign of analog phase-lock loops, IEEE Trans. Comm. 38 (12) (1990) 2197–2202. [2] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. [3] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox User’s Guide, 1st Edition, The MathWorks, 24 Prime Park Way, Natick, MA 01760-1500, 1995. [4] F.M. Gardner, Phaselock Techniques, Wiley, New York, 1979. [5] A.K. Gelig, G.A. Leonov, V.A. Yakubovich, The Stability of Nonlinear Systems with Nonunique Equilibrium, Nauka, Moscow, 1978 (in Russian). [6] K. Gu, H∞ control of systems under norm bounded uncertainties in all system matrices, IEEE Trans. Automat. Control 39 (6) (1994) 1320–1322. [7] W.M. Haddad, V. Kapila, Fixed-architecture controller synthesis for systems with input–output time-varying nonlinearities, Int. J. Robust Nonlinear Control 7 (7) (1997) 675–710. [8] G.A. Leonov, I.M. Burkin, A.L. Shepeljavyi, Frequency Methods in Oscillation Theory, Kluwer Academic Publishers, Dordrecht, 1992. [9] G.A. Leonov, A. Noack, V. Reitmann, Asymptotic orbital stability conditions for Wows by estimates of singular values of the linearization, Nonlinear Anal. 44 (8) (2001) 1057–1085. [10] G.A. Leonov, D.V. Ponomarenko, V.B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis, World Scienti?c, Singapore, 1996. [11] G.A. Leonov, V.B. Smirnova, Stability and oscillations of solutions of integro-diDerential equations of pendulum-like systems, Math. Nachr. 177 (1996) 157–181. [12] J. Moser, On nonoscillating networks, Quart. Appl. Math. 25 (1967) 1–9. [13] K.S. Narendra, J.H. Taylor, Frequency Domain Criteria for Absolute Stability, Academic Press, New York, 1973.
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