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H∞ controller synthesis for pendulum-like systems
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Systems & Control Letters 50 (2003) 263 – 276 www.elsevier.com/locate/sysconle
H∞ controller synthesis for pendulum-like systems Y. Yang∗ , L. Huang Department of Mechanics and Engineering Science, Center for Systems and Control, Peking University, Beijing 100871, People’s Republic of China Received 8 February 2002; received in revised form 14 November 2002; accepted 11 April 2003
Abstract This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satis7ed, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs). c 2003 Elsevier B.V. All rights reserved. Keywords: Pendulum-like systems; Lagrange stabilizability; H∞ control; LMI
1. Introduction In this paper we consider the pendulum-like systems of the form x˙ = Ax + b’(t; );
= cT x;
(1.1)
where A ∈ Rn×n ; b; c ∈ Rn and ’ : R+ × R → R is continuous and locally Lipschitz continuous in the second argument, Det A = 0;
(1.2)
(∃ ∈ R)(∀t ∈ R+ )(∀ ∈ R): ’(t; + ) = ’(t; ):
(1.3)
Suppose there exist two constants 1 and 2 such that ’(t; ) (∀t ∈ R+ )(∀ = 0): 1 6 6 2 : It follows from (1.3) that 1 2 ¡ 0 (the trivial case 1 = 2 = 0 we exclude).
(1.4)
This work is supported by the National Key Basic Research Special Funds (No. G1998020302) and the National Science Foundation of China under Grant 10272001. ∗ Corresponding author. E-mail address: [email protected] (Y. Yang).
c 2003 Elsevier B.V. All rights reserved. 0167-6911/03/$ - see front matter doi:10.1016/S0167-6911(03)00159-2
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
In the past few years, pendulum-like systems, initially introduced by Tricomi [14], have received a strong renewed attention. There are various reasons for this. First, pendulum-like systems represent a class of nonlinear systems whose nonlinear functions are periodic and consequently the systems have in7nite equilibria sets. Such systems are widely applied in various 7elds of mechanics and engineering, for example, systems of phase synchronization [8,9]. Second, the feedback analysis and design for systems with multiple equilibria are rather diJerent from those for systems with single equilibrium. The classical absolute stability theory was constructed for investigation of global stability of systems with single equilibrium and the standard Lyapunov functions which are exploited in control theory are aimed at such systems. In this way new types of stability problems appeared and 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. Since pendulum-like systems always have an unbounded set of equilibrium points, they cannot be asymptotically stable. A natural analog of global asymptotical stability for such a system is Lagrange stability, i.e., boundedness of all solutions on [0; ∞). Usually, if Lagrange stability is established for a system, its gradient-like behavior (i.e., convergence of all solutions) may be easily proved with the help of an additional frequency domain condition [7]. In the previous literature [5,6,11], the problem of Lagrange stability of pendulum-like systems has been intensively studied. A frequency-domain criterion for Lagrange stability of system (1.1) – (1.4) is presented by Leonov in [6]. However, to date no controller synthesis results have been given. In this paper, we de7ne a new notion of Lagrange stabilizability, which extends the Lagrange stability notion considered by Leonov to a corresponding stabilizability notion for output feedback control. We present some Lagrange stabilization results for pendulum-like systems and design a Lagrange stabilizing output control law for uncertainty-free systems and systems with multiplicative norm-bounded uncertainty in the linear part, respectively. The approach taken in this paper is to convert the Lagrange stabilization problem into an equivalent of H∞ control. A similar idea was taken by Savkin and Petersen [12,13]. However in those papers, H∞ control methods were applied and developed only within the frames of absolute stability theory for investigation of global asymptotical stability of the single equilibrium. In this paper, we consider the Lagrange stability of pendulum-like systems with multiple equilibria. The frequency domain conditions of Lagrange stability given by Leonov is converted into a H∞ norm bound requirement, which enables us to give controller synthesis results by using the popular H∞ sub-optimal control theory. The linear matrix inequality approach is used to derive the controller existence conditions. With this LMI approach, we are free from the computational diLculties. Furthermore, it would be useful for extending the results to take account of the norm-bounded uncertainty in the linear part of the system. This paper is organized as follows. Section 2 presents problem statement and preliminary results necessary for successive development. In Section 3, we derive the controller existence conditions for uncertainty-free systems and systems with multiplicative norm-bounded uncertainty in the linear part respectively, and illustrate them on numerical examples. A brief concluding remark is given in Section 4. The following notations will be used in this paper. Let X T and X H indicate the transpose and the complex conjugate transpose of X . We denote by X1 ⊕ X2 the direct sum of matrix X1 and X2 , i. e. X1 X1 ⊕ X2 = : X2 For a full-row rank matrix X ∈ Rm×n , denoted by X † and X ⊥ full-column matrices satisfying X [X † X ⊥ ] = [Im 0m×(n−m) ]. 2. Preliminaries and problem statement First we present some de7nitions and preliminary results related to pendulum-like systems.
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
-
P
265
1/s
K (t, ) Fig. 1. Feedback pendulum-like system.
Denition 2.1. Systems (1.1)–(1.4) is said to be Lagrange stable if all its solutions are bounded on [0; ∞). System (1.1)–(1.4) is characterized by a transfer function of its linear part from the input ’ to the output − G(s) , cT (A − sI )−1 b:
(2.1)
Suppose G(s) is non-degenerate, i.e. its numerator and denominator are co-prime polynomials. By (1.2), G(s) can be written in the following frequency-domain form: G(s) =
1 P(s): s
Obviously, P(s) is also non-degenerate. Theorem 2.1 (Leonov et al. [6]). Suppose that there exists a number ¿ 0 such that the following conditions for systems (1.1)–(1.4) are ful8lled: 1. the matrix A + I has n − 1 eigenvalues with negative real parts, 2.
1−1 2−1 + ( 1−1 + 2−1 )Re G(j! − ) + |G(j! − )|2 6 0
∀! ∈ R:
(2.2)
Then systems (1.1)–(1.4) is Lagrange stable. In this paper, we consider the stabilization problem of systems (1.1)–(1.4). The feedback con7guration is shown in Fig. 1. This control system structure describes a wide class of systems that include integrators, such as servo-motor systems where accelerometers are used for feedback. Denition 2.2. Systems (1.1)–(1.4) is said to be Lagrange stabilizable if there exists a linear output feedback controller K(s) such that the closed loop system in Fig. 1 is Lagrange stable. The main purpose of this paper is to derive the Lagrange stabilizability condition for the pendulum-like systems (1.1)–(1.4) and then extend the results to the case where there is multiplicative norm-bounded uncertainty in the linear part. To derive our main results, we also need the following lemmas. Lemma 2.2 (Schur complement [1]). For any blocked matrix 11 12 ; T 12 22
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
the following conditions are equivalent: 11 12 ¿ 0, (i) T 12 22 −1 T (ii) 11 ¿ 0; 22 ¿ 12 11 12 , and −1 T (iii) 22 ¿ 0; 11 ¿ 12 22 12 .
Lemma 2.3 (Bounded real and only if there exists X XA + AT X XB BT X −1 C D
lemma [3]). Suppose G(s) = D + C(sI − A)−1 B; A is stable, then G(s) ∞ ¡ 1 if = X T ¿ 0, such that CT DT ¡ 0: −1
Lemma 2.4 (Projection lemma [4]). Let matrices B ∈ Rn×m ; C ∈ Rk×n ; Q ∈ Rn×n be given, suppose rank(B) ¡ n, rank(C) ¡ n, and Q = QT , then there is a matrix K of compatible dimensions such that BKC + (BKC)T + Q ¡ 0 if and only if B⊥ QB⊥T ¡ 0 and C T⊥ QC T⊥T ¡ 0. Lemma 2.5. The following two conditions are equivalent: XA + AT X + XNN T X + M T M XB C T T T (i) B X −I D ¡ 0; C D −I and
(ii)
XA + AT X
XB
XN
CT
BT X
−I
0
DT
N TX
0
−I
0
C
D
0
−I
M
0
0
0
MT
0 0 ¡ 0: 0
−I
Proof. By Lemma 2.2, we can prove this lemma easily.
3. Main results In this section, we derive controller existence conditions of the pendulum-like system for the uncertainty-free case and the case with multiplicative norm-bounded uncertainty in the linear part, respectively. First, we give a theorem which establishes the connection between the conditions of Lagrange stability given by Leonov and a H∞ norm bound of the modi7ed transfer function of the linear part of the system.
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
267
Theorem 3.1. Suppose that there exists ¿ 0 such that 1. P(s − ) is stable. 2. rP(s − ) − 2# − # s+
∞
¡ 1;
(3.1)
where r = 2=( 2−1 − 1−1 ), # = ( 2−1 + 1−1 )=( 2−1 − 1−1 ). Then systems (1.1)–(1.4) is Lagrange stable. Proof. It is obvious that assumption (1) of this theorem is equivalent to condition (1) of Theorem 2.1. Note that inequality (2.2) in Theorem 2.1 allows a simple geometrical interpretation. The hodograph of the “shifted” frequency response G(s − ) lies completely in the complex plane inside the circle with radius ( 2−1 − 1−1 )=2 and centered at (−( 1−1 + 2−1 )=2; 0). That means the longest distance from any points of G(s − ) to the center of the circle is no more than ( 2−1 − 1−1 )=2. It suLces to check −1 −1 1
1−1 + 2−1 ¡ 2 − 1 ; · P(s − ) + s − 2 2 ∞ i.e.
rP(s − ) s − + # ¡ 1: ∞
(3.2)
Rewrite (3.2) as s + rP(s − ) + #(s − ) s − · s+
∞
¡ 1:
(3.3)
Note that for any s ∈ C
s +
s − = 1; then (3.3) holds if and only if rP(s − ) − 2# + # ¡ 1: s+ ∞
So the theorem is proved. Remark 3.1. The signi7cance of this theorem is that, by regarding the linear part of the system as a cascade connection of an integral element and a plant P(s), we convert condition (2) of Theorem 2.1 into a H∞ norm bound requirement. From this H∞ condition, it is possible to derive an output feedback control law, which renders the closed-up system Lagrange stable within the framework of H∞ control. 3.1. Uncertainty-free case Consider the feedback pendulum-like system in Fig. 1. Suppose P(s) is a strictly proper plant and A −b ; P(s) = T c 0
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
K(s) is a controller to be designed and A k Bk : K(s) = C k Dk
The nonlinear function ’(t; ) satis7es (1.3) and (1.4). Theorem 3.2. If there exists ¿ 0 such that LD = ∅, where
R I n×n n×n LD = (R; S) ∈ R × R : R ∈ LB ; S ∈ L C ; ¿0 ; I S ⊥ 0 b LB = R ∈ Rn×n : R = RT ¿ 0; and 0 0
−
−rb U = S V =
0 A + I
R+R
−
−rbT
0
AT + I
0
1
0
⊥ 0 c LC = S ∈ Rn×n : S = S T ¿ 0; and 0 0
1
⊥T 0 0 b U 0 0 0
1 0
1
0
R
−2#
−rb
A + I
+
−
−rbT
0
AT + I
S
S
c #
0 1
1
0
0 ¡0 ; 0 1
0
[1 0]
1 0
⊥T 0 0 c V 0 0
−1
0
0
[ − 2# cT ]R
−
0 ¡0 ; 0 1
1 0 ; # −1
−2#
0
c
[1 0]S
−1
#
[ − 2# cT ]
#
−1
;
then there is a full-order controller K(s) such that the closed loop system in Fig. 1 is Lagrange stable. Proof. According to Theorem 3.1, the closed loop system in Fig. 1 is Lagrange stable if there exists ¿ 0 such that Gc (s − ) , P(s − )=(1 + K(s − )P(s − )) is stable and rGc (s − ) − 2# ¡ 1: + # (3.4) s+ ∞
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
269
Suppose that the state space realization of (rGc (s − ) − 2#)=(s + ) + # is − 0 0 1 −rb A + I + bDk cT bCk 0 Acl Bcl = 0 Bk c T Ak 0 : Ccl Dcl −2# cT 0 # By Lemma 2.3, (3.4) holds and Acl is stable if and only if there exists Xcl ¿ 0 such that T Xcl Acl + ATcl Xcl Xcl Bcl Ccl T T Bcl Xcl −1 Dcl ¡0 Ccl Dcl −1
(3.5)
Note that Acl is stable guarantees that condition (1) of Theorem 3.1 is satis7ed. We rewrite the matrix inequality (3.5) in the form ˆ T + Q ¡ 0; ˆ Cˆ + (BK ˆ C) BK where
Bˆ Q ∗ Cˆ
Xcl Bˆ 2 Xcl Aˆ + Aˆ T Xcl 0 Bˆ T1 Xcl = ˆ Cˆ 1 D12 Cˆ 2 ∗
Aˆ ˆ C1 Cˆ 2
(3.6)
Bˆ 1 Dˆ 11 Dˆ 21
− −rb Bˆ 2 0 Dˆ 12 = −2# KT 0 0
0 A + I 0 cT cT 0
Partitioning Xcl and Xcl−1 as S N R −1 ; Xcl , Xcl , T ∗ N MT
Cˆ T1 # ; −1 0
Xcl Bˆ 1 −1 # Dˆ 21 0 0 0 0 0 Ik
M ∗
1 0 0 # 0 0
0 b 0 0 DkT CkT
0 0 Ik : 0 BkT ATk
;
R; S; M; N ∈ Rn×n :
Using Lemma 2.4 on (3.6), we easily prove the theorem. We can solve the linear matrix inequalities in Theorem 3.2 by using LMI Toolbox [2]. According to Theorem 3.2, a procedure for controller design can be given as follows: 1. Solve the LMIs in Theorem 3.2 and get one solution (R; S). 2. From (R; S), compute two invertible matrices M; N ∈ Rn×n via singular value decomposition (SVD) such that MN T = I − RS:
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
3. Get Xcl by Xcl
R
I
MT
0
=
I
S
0
NT
:
Once Xcl is determined, we can get a controller by using the following formula of all solutions to (3.6) given by Iwasaki and Skelton [4]. Suppose B ∈ Rn×m ; C ∈ Rk×n and Q = QT ∈ Rn×n are given. Let rb and rc be the ranks of B and C, respectively, and (BL ; BR ) and (CL ; CR ) be any full rank factors of B and C such that B = BL BR ; C = CL CR , then Kgen (B; C; Q) , {K ∈ Rm×k : ∃(Z; L; X ) ∈ Rm×k × Rrb ×rc × Rrb ×rb such that K = BR† .CL† + Z − BR† BR ZCL CL† ; . , −X −1 BLT CRT (CR CRT )−1 + X −1 Y 1=2 L(CR CRT )−1=2 ; , (BL X −1 BLT − Q)−1 ¿ 0; X ¿ 0; L ¡ 1; Y , X − BLT [ − CRT (CR CRT )−1 CR ]BL }:
(3.7)
Example 1. Let us consider systems (1.1)–(1.4) with n = 2 and the transfer function G(s) =
−0:25 : s(s + 1:3)
We assume that ’ satis7es the condition − 1:5 6
’(t; ) 6 2:5:
(3.8)
Take = 1 and solve the linear matrix inequalities corresponding to Theorem 3.2, we can get one solution 2:1303 −0:1253 1:7417 0:1673 R= ; S= : −0:1253 2:2556 0:1673 2:2556 From (R; S), take 0:1563 M= 2:0130
1:6321
−0:1267
;
N=
−0:1817
−1:6304
−2:0109
0:1473
such that MN T = I − RS. Then we have 1:7417 0:1673 −0:1817 0:1673 2:2556 −2:0109 Xcl = −0:1817 −2:0109 2:2336 −1:6304
0:1473
−0:1311
In virtue of (3.7), we can get a controller K(s) =
−0:3685s2 − 16:5s − 181:9 ; s2 + 42:99s + 447:5
−1:6304
0:1473 : −0:1311 2:1519
;
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
271
Nyquist Diagrams 0.5
Stability Boundary Closed Loop Open Loop
0.4
Imaginary Axis
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.4
-0.2
0
0.2 0.4 Real Axis
0.6
0.8
Fig. 2. Stability region and the shifted frequency response of the linear part.
∆ P
1/s K (t, )
Fig. 3. Pendulum-like system with multiplicative norm-bounded uncertainty.
such that the closed loop system in Fig. 1 is Lagrange stable. In Fig. 2, the boundary of Lagrange stability region and the hodograph of the shifted open loop and the closed loop frequency response of the linear part are shown and obviously, the frequency response lies within the stability region when the controller is applied. Remark 3.2. Note that in the above example, the existence of the solutions (R; S) is independent of the matrix A, which indicates that for a two-order system, where P(s) is strictly proper, if there exists a solution for a certain A, then for any other A, this solution is also a feasible solution to the corresponding LMIs. 3.2. Linear plant with norm-bounded uncertainty In the following, we consider Lagrange stabilization of a pendulum-like system with multiplicative normbounded uncertainty in the linear plant. Denote G(s) the linear part of the system and G(s)={(1=s)P(s)(1+ ); ∈ RH∞ ; ∞ 6 0}. The feedback con7guration is shown in Fig. 3. Denote Gc (s) ,
P(s)(1 + ) ; 1 + K(s)P(s)(1 + )
∞ 6 0
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
the closed loop transfer function of (P(1 + ); K). For single input single output systems, can be replaced by a complex number 1 + 2i; 12 + 22 6 02 , i.e. P(s)(1 + 1 + 2i) Gc (s) = (3.9) ∀1; 2; 12 + 22 6 02 : 1 + K(s)P(s)(1 + 1 + 2i) Suppose the state space realization of (rGc (s − ) − 2#)=(s + ) + # Acl Bcl ; Ccl Dcl − 0 0 T bCk Acl = −rb A + I + (1 + 1 + 2i)bDk c T 0 (1 + 1 + 2i)Bk c Ak − 0 0 0 0 T T bCk = −rb A + I + bDk c + 0 (1 + 2i)bDk c 0 (1 + 2i)Bk cT 0 Bk c T Ak 0 0 0 T 0 , Au + 0 (1 + 2i)bDk c ; T 0 0 (1 + 2i)Bk c 1 Bcl = 0 , Bu ; 0
is
0
0 0
(3.10)
Ccl = [ − 21 (1 + 1 + 2i)cT 0] = [ − 2# cT 0] + [0 (1 + 2i)cT 0] , Cu + [0 (1 + 2i)cT 0]; Dcl = #: Note that Au Bu Cu # is the state space realization of (rGc (s − ) − 2#)=(s + ) + # with Gc (s−) = P(s−)=(1+K(s−)P(s−)). Theorem 3.3. The system in Fig. 3 is Lagrange stable if there exist Xcl = XclT ¿ 0 and scalars ¿ 0; 5 ¿ 0 such that √ 0 ˆT Xcl Au + ATu Xcl Xcl Bu 5Xcl Bˆ CuT 5 C −1 0 # 0 BuT Xcl (3.11) 5Bˆ T Xcl 0 −1 5 0 ¡ 0; Cu # 5 −1 0 √ 0 ˆ C 0 0 0 −1 5
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
where
0
273
; bD Bˆ = k Bk
Cˆ = [0 cT 0]:
Proof. According to Theorem 3.1, the system in Fig. 3 is Lagrange stable if (i) Gc (s − ) is stable, and (ii) rGc (s−)−2# + # ¡ 1 ∀1; 2; 12 + 22 6 02 . s+ ∞
By Lemma 2.3, the above conditions are true if and only if there exists Xcl ¿ 0 such that H Xcl Bcl Ccl Xcl Acl + AH cl Xcl H H (Xcl ; 6cl ) , Bcl Xcl −1 Dcl ¡ 0: Ccl Dcl −1 Let
Xu =
Xcl Au + AH u Xcl
Xcl Bu
BuH Xcl
−1
Cu
#
CuH
# ; −1
0
(3.12)
Xcl bDk ; B X = k 0 I
Y = [[0
cT
0]
0
0]:
Then (3.12) leads to (Xcl ; 6cl ) = Xu + X (1 + 2i)Y + Y T (1 − 2i)X T ¡ 0: The above inequality is true if and only if there exists a scalar 5 ¿ 0 such that 0 Xu + 52 XX T + 2 Y T Y ¡ 0: 5 By Lemma 2.5, it is easy to verify that (3.13) is equivalent to (3.11). So the theorem follows.
(3.13)
Note that the criterion of (3.11), which is a bilinear matrix inequality in respect to the variable (Xcl ; K), cannot be reduced to a linear matrix inequality by using Lemma 2.4 as we have done for uncertainty-free case. Therefore, another approach proposed in [10] is considered here and a LMI-based controller existence condition is devised for system in Fig. 3. Take a parameter set p = {Pf ; Pg ; Ph ; Wf ; Wg ; Wh ; L}, Pf ; Pg ∈ Rn×n ; Ph ∈ R(nc −n)×(nc −n) , Wf ∈ R1×nc , Wg ∈ Rnc ×1 , Wh ∈ R; L ∈ Rnc ×nc , and Pf I ¿ 0; Ph ¿ 0: I Pg Denote that L=
L11
L12
L21
L22
;
Wf = [Wf1 Wf2 ];
where L11 ∈ Rn×n ; Wf1 ∈ R1×n ; Wg1 ∈ Rn×1 .
Wg =
Wg1 Wg2
;
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
De7ne the following Pf Mp (p) , In 0
matrix-valued aLne function: In 0 A Pf + BWf1 Pg 0 L11 ; MA (p) , 0 Ph L21
MC (p) , [C1 Pf C1 0];
B1 + BWh N1
A + BWh C Pg A + Wg1 C Wg2 C
BWf2
L12 ; L22
; P B + W N MB (p) , g 1 g1 1
MD (p) , D1 ;
Wg2 N1 ∗ (p) , Mp (p) ⊕
−MA (p) − MAT (p)
MB (p)
MBT (p)
1
MC (p)
−MD (p)
McT (p)
−MD (p) ; 1
where A = B1 = C1 =
−
0
−rb
A + I
1
0
0
0
;
B=
0 b
C = [0 cT ]
;
;
N1 = [0 5]; cT
−2#
√
0 T 5 c
0
;
D1 =
#
5
0
0
:
Using the results in [10], we have Theorem 3.4. For some nc ¿ n, if there exist p and ¿ 0; 5 ¿ 0 such that ∗ (p) ¿ 0; then there is a controller K(s) of order nc such that K(s) Lagrange stabilize the closed loop system in Fig. 3. Furthermore, the state space realization of K(s) is determined by
Dk Ck Bk A k
1 0 0 Wh Wf1 Wf2 = B −Pg−1 0 Wg1 L11 − Pg A Pf L12 Wg2 L21 L22 0 0 Inc −n 1 −CPf Sf−1 0 −1 ×0 Sf−1 0 ; Sf , Pf − Pg (¿ 0): 0 Ph−1 0
(3.14)
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
275
Example 2. Let us consider the system with the transfer function 0:1 G(s) = (1 + ); ∞ 6 0:925 s(s + 2:5) and ’ satisfying (3.8). For simplicity, we design a full order controller. If we set 5 = 0:925 and = 1, there exists a parameter set p 5:4256 −2:2872 2:2358 −10:1830 Pf = ; Pg = ; Wh = 82:2025; −2:2872 10:8851 −10:1830 126:2668 Wf = [ − 127:9 2718:8];
Wg =
0:0350 −2:9086
;
L=
−1:1658
0:3718
9:7586
−30:8936
;
which satis7es the condition in Theorem 3.4. The desired stabilization output feedback controller can be constructed by K(s) =
82:2s2 + 394:2s + 474:3 : s2 + 23:63s + 43:72
4. Conclusion This paper deals with the output feedback Lagrange stabilization problem for pendulum-like systems. The frequency domain conditions for Lagrange stability have been revisited with an H∞ norm bound. It is shown that the problem under study can be solved within the framework of H∞ control. Expressed in terms of a set of LMIs, the controller existence conditions for Lagrange stabilizability are obtained for both cases of uncertainty free and multiplicative norm-bounded uncertainty. Numerical examples illustrate the eLciency of the controller design methods proposed in this paper. Acknowledgements The authors express their sincere gratitude to the reviewers for their constructive suggestions which help improve the presentation of this paper. References [1] P. Gahinet, P. Apkarian, A linear matrix inequality approach to H∞ control, Int. J. Robust Nonlinear Control 4 (1994) 421–448. [2] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox User’s Guide, 1st Edition, The MathWorks, Inc., Natick, MA, 1995, pp. 1760 –1500. [3] M. Green, D.J.N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood CliJs, NJ, 1995. [4] Iwasaki, R.E. Skelton, All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica 30(8) (1994) 1307–1317. [5] G.A. Leonov, A. Noack, V. Reitmann, Asymptotic orbital stability conditions for Tows by estimates of singular values of the linearization, Nonlinear Anal. 44 (8) (2001) 1057–1085. [6] G.A. Leonov, D.V. Ponomarenko, V.B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis, World Scienti7c, Singapore, 1996. [7] G.A. Leonov, V.B. Smirnova, Stability and oscillations of solutions of integro-diJerential equations of pendulum-like systems, Math. Nachr. 177 (1996) 157–181. [8] G.A. Leonov, V.B. Smirnova, Analysis of frequency-of-oscillations-controlled systems, in: Proceedings of International Conference on Control of Oscillations and Chaos, Vol. 2, 1997, pp. 439 – 441.
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[9] G.A. Leonov, A.M. Tomayev, T.L. Chshiyeva, Stability of frequency-phase locked automatic frequency control systems, Soviet J. Commun. Technol. Electron. 37 (11) (1992) 1–9. [10] I. Masubuchi, A. Ohara, N. Suda, LMI-based controller synthesis: a uni7ed formulation and solution, Int. J. Robust Nonlinear Control 8 (8) (1998) 669–686. [11] V. Rasvan, Dynamical systems with several equilibria and natural Liapunov functions, Arch. Math. 34 (1998) 207–215. [12] A.V. Savkin, I.R. Petersen, Nonlinear versus linear control in the absolute stabilizability of uncertain systems with structured uncertainty, in: Proceedings of IEEE Conference on Decision and Control, San Antonio, TX, 1993, pp. 172–177. [13] A.V. Savkin, I.R. Petersen, A connection between H ∞ control and the absolute stabilizability of uncertain systems, Systems Control Lett. 23 (1994) 197–203. [14] F. Tricomi, Integrazione di unequazione diJerenziale presentatasi in electrotechnica, Ann. Roma Scuola Norm. Sup. Pisa: Scienza Phys. Mat. 2 (1933) 1–20.
Systems & Control Letters 50 (2003) 263 – 276 www.elsevier.com/locate/sysconle
H∞ controller synthesis for pendulum-like systems Y. Yang∗ , L. Huang Department of Mechanics and Engineering Science, Center for Systems and Control, Peking University, Beijing 100871, People’s Republic of China Received 8 February 2002; received in revised form 14 November 2002; accepted 11 April 2003
Abstract This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satis7ed, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs). c 2003 Elsevier B.V. All rights reserved. Keywords: Pendulum-like systems; Lagrange stabilizability; H∞ control; LMI
1. Introduction In this paper we consider the pendulum-like systems of the form x˙ = Ax + b’(t; );
= cT x;
(1.1)
where A ∈ Rn×n ; b; c ∈ Rn and ’ : R+ × R → R is continuous and locally Lipschitz continuous in the second argument, Det A = 0;
(1.2)
(∃ ∈ R)(∀t ∈ R+ )(∀ ∈ R): ’(t; + ) = ’(t; ):
(1.3)
Suppose there exist two constants 1 and 2 such that ’(t; ) (∀t ∈ R+ )(∀ = 0): 1 6 6 2 : It follows from (1.3) that 1 2 ¡ 0 (the trivial case 1 = 2 = 0 we exclude).
(1.4)
This work is supported by the National Key Basic Research Special Funds (No. G1998020302) and the National Science Foundation of China under Grant 10272001. ∗ Corresponding author. E-mail address: [email protected] (Y. Yang).
c 2003 Elsevier B.V. All rights reserved. 0167-6911/03/$ - see front matter doi:10.1016/S0167-6911(03)00159-2
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
In the past few years, pendulum-like systems, initially introduced by Tricomi [14], have received a strong renewed attention. There are various reasons for this. First, pendulum-like systems represent a class of nonlinear systems whose nonlinear functions are periodic and consequently the systems have in7nite equilibria sets. Such systems are widely applied in various 7elds of mechanics and engineering, for example, systems of phase synchronization [8,9]. Second, the feedback analysis and design for systems with multiple equilibria are rather diJerent from those for systems with single equilibrium. The classical absolute stability theory was constructed for investigation of global stability of systems with single equilibrium and the standard Lyapunov functions which are exploited in control theory are aimed at such systems. In this way new types of stability problems appeared and 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. Since pendulum-like systems always have an unbounded set of equilibrium points, they cannot be asymptotically stable. A natural analog of global asymptotical stability for such a system is Lagrange stability, i.e., boundedness of all solutions on [0; ∞). Usually, if Lagrange stability is established for a system, its gradient-like behavior (i.e., convergence of all solutions) may be easily proved with the help of an additional frequency domain condition [7]. In the previous literature [5,6,11], the problem of Lagrange stability of pendulum-like systems has been intensively studied. A frequency-domain criterion for Lagrange stability of system (1.1) – (1.4) is presented by Leonov in [6]. However, to date no controller synthesis results have been given. In this paper, we de7ne a new notion of Lagrange stabilizability, which extends the Lagrange stability notion considered by Leonov to a corresponding stabilizability notion for output feedback control. We present some Lagrange stabilization results for pendulum-like systems and design a Lagrange stabilizing output control law for uncertainty-free systems and systems with multiplicative norm-bounded uncertainty in the linear part, respectively. The approach taken in this paper is to convert the Lagrange stabilization problem into an equivalent of H∞ control. A similar idea was taken by Savkin and Petersen [12,13]. However in those papers, H∞ control methods were applied and developed only within the frames of absolute stability theory for investigation of global asymptotical stability of the single equilibrium. In this paper, we consider the Lagrange stability of pendulum-like systems with multiple equilibria. The frequency domain conditions of Lagrange stability given by Leonov is converted into a H∞ norm bound requirement, which enables us to give controller synthesis results by using the popular H∞ sub-optimal control theory. The linear matrix inequality approach is used to derive the controller existence conditions. With this LMI approach, we are free from the computational diLculties. Furthermore, it would be useful for extending the results to take account of the norm-bounded uncertainty in the linear part of the system. This paper is organized as follows. Section 2 presents problem statement and preliminary results necessary for successive development. In Section 3, we derive the controller existence conditions for uncertainty-free systems and systems with multiplicative norm-bounded uncertainty in the linear part respectively, and illustrate them on numerical examples. A brief concluding remark is given in Section 4. The following notations will be used in this paper. Let X T and X H indicate the transpose and the complex conjugate transpose of X . We denote by X1 ⊕ X2 the direct sum of matrix X1 and X2 , i. e. X1 X1 ⊕ X2 = : X2 For a full-row rank matrix X ∈ Rm×n , denoted by X † and X ⊥ full-column matrices satisfying X [X † X ⊥ ] = [Im 0m×(n−m) ]. 2. Preliminaries and problem statement First we present some de7nitions and preliminary results related to pendulum-like systems.
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
-
P
265
1/s
K (t, ) Fig. 1. Feedback pendulum-like system.
Denition 2.1. Systems (1.1)–(1.4) is said to be Lagrange stable if all its solutions are bounded on [0; ∞). System (1.1)–(1.4) is characterized by a transfer function of its linear part from the input ’ to the output − G(s) , cT (A − sI )−1 b:
(2.1)
Suppose G(s) is non-degenerate, i.e. its numerator and denominator are co-prime polynomials. By (1.2), G(s) can be written in the following frequency-domain form: G(s) =
1 P(s): s
Obviously, P(s) is also non-degenerate. Theorem 2.1 (Leonov et al. [6]). Suppose that there exists a number ¿ 0 such that the following conditions for systems (1.1)–(1.4) are ful8lled: 1. the matrix A + I has n − 1 eigenvalues with negative real parts, 2.
1−1 2−1 + ( 1−1 + 2−1 )Re G(j! − ) + |G(j! − )|2 6 0
∀! ∈ R:
(2.2)
Then systems (1.1)–(1.4) is Lagrange stable. In this paper, we consider the stabilization problem of systems (1.1)–(1.4). The feedback con7guration is shown in Fig. 1. This control system structure describes a wide class of systems that include integrators, such as servo-motor systems where accelerometers are used for feedback. Denition 2.2. Systems (1.1)–(1.4) is said to be Lagrange stabilizable if there exists a linear output feedback controller K(s) such that the closed loop system in Fig. 1 is Lagrange stable. The main purpose of this paper is to derive the Lagrange stabilizability condition for the pendulum-like systems (1.1)–(1.4) and then extend the results to the case where there is multiplicative norm-bounded uncertainty in the linear part. To derive our main results, we also need the following lemmas. Lemma 2.2 (Schur complement [1]). For any blocked matrix 11 12 ; T 12 22
266
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
the following conditions are equivalent: 11 12 ¿ 0, (i) T 12 22 −1 T (ii) 11 ¿ 0; 22 ¿ 12 11 12 , and −1 T (iii) 22 ¿ 0; 11 ¿ 12 22 12 .
Lemma 2.3 (Bounded real and only if there exists X XA + AT X XB BT X −1 C D
lemma [3]). Suppose G(s) = D + C(sI − A)−1 B; A is stable, then G(s) ∞ ¡ 1 if = X T ¿ 0, such that CT DT ¡ 0: −1
Lemma 2.4 (Projection lemma [4]). Let matrices B ∈ Rn×m ; C ∈ Rk×n ; Q ∈ Rn×n be given, suppose rank(B) ¡ n, rank(C) ¡ n, and Q = QT , then there is a matrix K of compatible dimensions such that BKC + (BKC)T + Q ¡ 0 if and only if B⊥ QB⊥T ¡ 0 and C T⊥ QC T⊥T ¡ 0. Lemma 2.5. The following two conditions are equivalent: XA + AT X + XNN T X + M T M XB C T T T (i) B X −I D ¡ 0; C D −I and
(ii)
XA + AT X
XB
XN
CT
BT X
−I
0
DT
N TX
0
−I
0
C
D
0
−I
M
0
0
0
MT
0 0 ¡ 0: 0
−I
Proof. By Lemma 2.2, we can prove this lemma easily.
3. Main results In this section, we derive controller existence conditions of the pendulum-like system for the uncertainty-free case and the case with multiplicative norm-bounded uncertainty in the linear part, respectively. First, we give a theorem which establishes the connection between the conditions of Lagrange stability given by Leonov and a H∞ norm bound of the modi7ed transfer function of the linear part of the system.
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
267
Theorem 3.1. Suppose that there exists ¿ 0 such that 1. P(s − ) is stable. 2. rP(s − ) − 2# − # s+
∞
¡ 1;
(3.1)
where r = 2=( 2−1 − 1−1 ), # = ( 2−1 + 1−1 )=( 2−1 − 1−1 ). Then systems (1.1)–(1.4) is Lagrange stable. Proof. It is obvious that assumption (1) of this theorem is equivalent to condition (1) of Theorem 2.1. Note that inequality (2.2) in Theorem 2.1 allows a simple geometrical interpretation. The hodograph of the “shifted” frequency response G(s − ) lies completely in the complex plane inside the circle with radius ( 2−1 − 1−1 )=2 and centered at (−( 1−1 + 2−1 )=2; 0). That means the longest distance from any points of G(s − ) to the center of the circle is no more than ( 2−1 − 1−1 )=2. It suLces to check −1 −1 1
1−1 + 2−1 ¡ 2 − 1 ; · P(s − ) + s − 2 2 ∞ i.e.
rP(s − ) s − + # ¡ 1: ∞
(3.2)
Rewrite (3.2) as s + rP(s − ) + #(s − ) s − · s+
∞
¡ 1:
(3.3)
Note that for any s ∈ C
s +
s − = 1; then (3.3) holds if and only if rP(s − ) − 2# + # ¡ 1: s+ ∞
So the theorem is proved. Remark 3.1. The signi7cance of this theorem is that, by regarding the linear part of the system as a cascade connection of an integral element and a plant P(s), we convert condition (2) of Theorem 2.1 into a H∞ norm bound requirement. From this H∞ condition, it is possible to derive an output feedback control law, which renders the closed-up system Lagrange stable within the framework of H∞ control. 3.1. Uncertainty-free case Consider the feedback pendulum-like system in Fig. 1. Suppose P(s) is a strictly proper plant and A −b ; P(s) = T c 0
268
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
K(s) is a controller to be designed and A k Bk : K(s) = C k Dk
The nonlinear function ’(t; ) satis7es (1.3) and (1.4). Theorem 3.2. If there exists ¿ 0 such that LD = ∅, where
R I n×n n×n LD = (R; S) ∈ R × R : R ∈ LB ; S ∈ L C ; ¿0 ; I S ⊥ 0 b LB = R ∈ Rn×n : R = RT ¿ 0; and 0 0
−
−rb U = S V =
0 A + I
R+R
−
−rbT
0
AT + I
0
1
0
⊥ 0 c LC = S ∈ Rn×n : S = S T ¿ 0; and 0 0
1
⊥T 0 0 b U 0 0 0
1 0
1
0
R
−2#
−rb
A + I
+
−
−rbT
0
AT + I
S
S
c #
0 1
1
0
0 ¡0 ; 0 1
0
[1 0]
1 0
⊥T 0 0 c V 0 0
−1
0
0
[ − 2# cT ]R
−
0 ¡0 ; 0 1
1 0 ; # −1
−2#
0
c
[1 0]S
−1
#
[ − 2# cT ]
#
−1
;
then there is a full-order controller K(s) such that the closed loop system in Fig. 1 is Lagrange stable. Proof. According to Theorem 3.1, the closed loop system in Fig. 1 is Lagrange stable if there exists ¿ 0 such that Gc (s − ) , P(s − )=(1 + K(s − )P(s − )) is stable and rGc (s − ) − 2# ¡ 1: + # (3.4) s+ ∞
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
269
Suppose that the state space realization of (rGc (s − ) − 2#)=(s + ) + # is − 0 0 1 −rb A + I + bDk cT bCk 0 Acl Bcl = 0 Bk c T Ak 0 : Ccl Dcl −2# cT 0 # By Lemma 2.3, (3.4) holds and Acl is stable if and only if there exists Xcl ¿ 0 such that T Xcl Acl + ATcl Xcl Xcl Bcl Ccl T T Bcl Xcl −1 Dcl ¡0 Ccl Dcl −1
(3.5)
Note that Acl is stable guarantees that condition (1) of Theorem 3.1 is satis7ed. We rewrite the matrix inequality (3.5) in the form ˆ T + Q ¡ 0; ˆ Cˆ + (BK ˆ C) BK where
Bˆ Q ∗ Cˆ
Xcl Bˆ 2 Xcl Aˆ + Aˆ T Xcl 0 Bˆ T1 Xcl = ˆ Cˆ 1 D12 Cˆ 2 ∗
Aˆ ˆ C1 Cˆ 2
(3.6)
Bˆ 1 Dˆ 11 Dˆ 21
− −rb Bˆ 2 0 Dˆ 12 = −2# KT 0 0
0 A + I 0 cT cT 0
Partitioning Xcl and Xcl−1 as S N R −1 ; Xcl , Xcl , T ∗ N MT
Cˆ T1 # ; −1 0
Xcl Bˆ 1 −1 # Dˆ 21 0 0 0 0 0 Ik
M ∗
1 0 0 # 0 0
0 b 0 0 DkT CkT
0 0 Ik : 0 BkT ATk
;
R; S; M; N ∈ Rn×n :
Using Lemma 2.4 on (3.6), we easily prove the theorem. We can solve the linear matrix inequalities in Theorem 3.2 by using LMI Toolbox [2]. According to Theorem 3.2, a procedure for controller design can be given as follows: 1. Solve the LMIs in Theorem 3.2 and get one solution (R; S). 2. From (R; S), compute two invertible matrices M; N ∈ Rn×n via singular value decomposition (SVD) such that MN T = I − RS:
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
3. Get Xcl by Xcl
R
I
MT
0
=
I
S
0
NT
:
Once Xcl is determined, we can get a controller by using the following formula of all solutions to (3.6) given by Iwasaki and Skelton [4]. Suppose B ∈ Rn×m ; C ∈ Rk×n and Q = QT ∈ Rn×n are given. Let rb and rc be the ranks of B and C, respectively, and (BL ; BR ) and (CL ; CR ) be any full rank factors of B and C such that B = BL BR ; C = CL CR , then Kgen (B; C; Q) , {K ∈ Rm×k : ∃(Z; L; X ) ∈ Rm×k × Rrb ×rc × Rrb ×rb such that K = BR† .CL† + Z − BR† BR ZCL CL† ; . , −X −1 BLT CRT (CR CRT )−1 + X −1 Y 1=2 L(CR CRT )−1=2 ; , (BL X −1 BLT − Q)−1 ¿ 0; X ¿ 0; L ¡ 1; Y , X − BLT [ − CRT (CR CRT )−1 CR ]BL }:
(3.7)
Example 1. Let us consider systems (1.1)–(1.4) with n = 2 and the transfer function G(s) =
−0:25 : s(s + 1:3)
We assume that ’ satis7es the condition − 1:5 6
’(t; ) 6 2:5:
(3.8)
Take = 1 and solve the linear matrix inequalities corresponding to Theorem 3.2, we can get one solution 2:1303 −0:1253 1:7417 0:1673 R= ; S= : −0:1253 2:2556 0:1673 2:2556 From (R; S), take 0:1563 M= 2:0130
1:6321
−0:1267
;
N=
−0:1817
−1:6304
−2:0109
0:1473
such that MN T = I − RS. Then we have 1:7417 0:1673 −0:1817 0:1673 2:2556 −2:0109 Xcl = −0:1817 −2:0109 2:2336 −1:6304
0:1473
−0:1311
In virtue of (3.7), we can get a controller K(s) =
−0:3685s2 − 16:5s − 181:9 ; s2 + 42:99s + 447:5
−1:6304
0:1473 : −0:1311 2:1519
;
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
271
Nyquist Diagrams 0.5
Stability Boundary Closed Loop Open Loop
0.4
Imaginary Axis
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.4
-0.2
0
0.2 0.4 Real Axis
0.6
0.8
Fig. 2. Stability region and the shifted frequency response of the linear part.
∆ P
1/s K (t, )
Fig. 3. Pendulum-like system with multiplicative norm-bounded uncertainty.
such that the closed loop system in Fig. 1 is Lagrange stable. In Fig. 2, the boundary of Lagrange stability region and the hodograph of the shifted open loop and the closed loop frequency response of the linear part are shown and obviously, the frequency response lies within the stability region when the controller is applied. Remark 3.2. Note that in the above example, the existence of the solutions (R; S) is independent of the matrix A, which indicates that for a two-order system, where P(s) is strictly proper, if there exists a solution for a certain A, then for any other A, this solution is also a feasible solution to the corresponding LMIs. 3.2. Linear plant with norm-bounded uncertainty In the following, we consider Lagrange stabilization of a pendulum-like system with multiplicative normbounded uncertainty in the linear plant. Denote G(s) the linear part of the system and G(s)={(1=s)P(s)(1+ ); ∈ RH∞ ; ∞ 6 0}. The feedback con7guration is shown in Fig. 3. Denote Gc (s) ,
P(s)(1 + ) ; 1 + K(s)P(s)(1 + )
∞ 6 0
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Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
the closed loop transfer function of (P(1 + ); K). For single input single output systems, can be replaced by a complex number 1 + 2i; 12 + 22 6 02 , i.e. P(s)(1 + 1 + 2i) Gc (s) = (3.9) ∀1; 2; 12 + 22 6 02 : 1 + K(s)P(s)(1 + 1 + 2i) Suppose the state space realization of (rGc (s − ) − 2#)=(s + ) + # Acl Bcl ; Ccl Dcl − 0 0 T bCk Acl = −rb A + I + (1 + 1 + 2i)bDk c T 0 (1 + 1 + 2i)Bk c Ak − 0 0 0 0 T T bCk = −rb A + I + bDk c + 0 (1 + 2i)bDk c 0 (1 + 2i)Bk cT 0 Bk c T Ak 0 0 0 T 0 , Au + 0 (1 + 2i)bDk c ; T 0 0 (1 + 2i)Bk c 1 Bcl = 0 , Bu ; 0
is
0
0 0
(3.10)
Ccl = [ − 21 (1 + 1 + 2i)cT 0] = [ − 2# cT 0] + [0 (1 + 2i)cT 0] , Cu + [0 (1 + 2i)cT 0]; Dcl = #: Note that Au Bu Cu # is the state space realization of (rGc (s − ) − 2#)=(s + ) + # with Gc (s−) = P(s−)=(1+K(s−)P(s−)). Theorem 3.3. The system in Fig. 3 is Lagrange stable if there exist Xcl = XclT ¿ 0 and scalars ¿ 0; 5 ¿ 0 such that √ 0 ˆT Xcl Au + ATu Xcl Xcl Bu 5Xcl Bˆ CuT 5 C −1 0 # 0 BuT Xcl (3.11) 5Bˆ T Xcl 0 −1 5 0 ¡ 0; Cu # 5 −1 0 √ 0 ˆ C 0 0 0 −1 5
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
where
0
273
; bD Bˆ = k Bk
Cˆ = [0 cT 0]:
Proof. According to Theorem 3.1, the system in Fig. 3 is Lagrange stable if (i) Gc (s − ) is stable, and (ii) rGc (s−)−2# + # ¡ 1 ∀1; 2; 12 + 22 6 02 . s+ ∞
By Lemma 2.3, the above conditions are true if and only if there exists Xcl ¿ 0 such that H Xcl Bcl Ccl Xcl Acl + AH cl Xcl H H (Xcl ; 6cl ) , Bcl Xcl −1 Dcl ¡ 0: Ccl Dcl −1 Let
Xu =
Xcl Au + AH u Xcl
Xcl Bu
BuH Xcl
−1
Cu
#
CuH
# ; −1
0
(3.12)
Xcl bDk ; B X = k 0 I
Y = [[0
cT
0]
0
0]:
Then (3.12) leads to (Xcl ; 6cl ) = Xu + X (1 + 2i)Y + Y T (1 − 2i)X T ¡ 0: The above inequality is true if and only if there exists a scalar 5 ¿ 0 such that 0 Xu + 52 XX T + 2 Y T Y ¡ 0: 5 By Lemma 2.5, it is easy to verify that (3.13) is equivalent to (3.11). So the theorem follows.
(3.13)
Note that the criterion of (3.11), which is a bilinear matrix inequality in respect to the variable (Xcl ; K), cannot be reduced to a linear matrix inequality by using Lemma 2.4 as we have done for uncertainty-free case. Therefore, another approach proposed in [10] is considered here and a LMI-based controller existence condition is devised for system in Fig. 3. Take a parameter set p = {Pf ; Pg ; Ph ; Wf ; Wg ; Wh ; L}, Pf ; Pg ∈ Rn×n ; Ph ∈ R(nc −n)×(nc −n) , Wf ∈ R1×nc , Wg ∈ Rnc ×1 , Wh ∈ R; L ∈ Rnc ×nc , and Pf I ¿ 0; Ph ¿ 0: I Pg Denote that L=
L11
L12
L21
L22
;
Wf = [Wf1 Wf2 ];
where L11 ∈ Rn×n ; Wf1 ∈ R1×n ; Wg1 ∈ Rn×1 .
Wg =
Wg1 Wg2
;
274
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
De7ne the following Pf Mp (p) , In 0
matrix-valued aLne function: In 0 A Pf + BWf1 Pg 0 L11 ; MA (p) , 0 Ph L21
MC (p) , [C1 Pf C1 0];
B1 + BWh N1
A + BWh C Pg A + Wg1 C Wg2 C
BWf2
L12 ; L22
; P B + W N MB (p) , g 1 g1 1
MD (p) , D1 ;
Wg2 N1 ∗ (p) , Mp (p) ⊕
−MA (p) − MAT (p)
MB (p)
MBT (p)
1
MC (p)
−MD (p)
McT (p)
−MD (p) ; 1
where A = B1 = C1 =
−
0
−rb
A + I
1
0
0
0
;
B=
0 b
C = [0 cT ]
;
;
N1 = [0 5]; cT
−2#
√
0 T 5 c
0
;
D1 =
#
5
0
0
:
Using the results in [10], we have Theorem 3.4. For some nc ¿ n, if there exist p and ¿ 0; 5 ¿ 0 such that ∗ (p) ¿ 0; then there is a controller K(s) of order nc such that K(s) Lagrange stabilize the closed loop system in Fig. 3. Furthermore, the state space realization of K(s) is determined by
Dk Ck Bk A k
1 0 0 Wh Wf1 Wf2 = B −Pg−1 0 Wg1 L11 − Pg A Pf L12 Wg2 L21 L22 0 0 Inc −n 1 −CPf Sf−1 0 −1 ×0 Sf−1 0 ; Sf , Pf − Pg (¿ 0): 0 Ph−1 0
(3.14)
Y. Yang, L. Huang / Systems & Control Letters 50 (2003) 263 – 276
275
Example 2. Let us consider the system with the transfer function 0:1 G(s) = (1 + ); ∞ 6 0:925 s(s + 2:5) and ’ satisfying (3.8). For simplicity, we design a full order controller. If we set 5 = 0:925 and = 1, there exists a parameter set p 5:4256 −2:2872 2:2358 −10:1830 Pf = ; Pg = ; Wh = 82:2025; −2:2872 10:8851 −10:1830 126:2668 Wf = [ − 127:9 2718:8];
Wg =
0:0350 −2:9086
;
L=
−1:1658
0:3718
9:7586
−30:8936
;
which satis7es the condition in Theorem 3.4. The desired stabilization output feedback controller can be constructed by K(s) =
82:2s2 + 394:2s + 474:3 : s2 + 23:63s + 43:72
4. Conclusion This paper deals with the output feedback Lagrange stabilization problem for pendulum-like systems. The frequency domain conditions for Lagrange stability have been revisited with an H∞ norm bound. It is shown that the problem under study can be solved within the framework of H∞ control. Expressed in terms of a set of LMIs, the controller existence conditions for Lagrange stabilizability are obtained for both cases of uncertainty free and multiplicative norm-bounded uncertainty. Numerical examples illustrate the eLciency of the controller design methods proposed in this paper. Acknowledgements The authors express their sincere gratitude to the reviewers for their constructive suggestions which help improve the presentation of this paper. References [1] P. Gahinet, P. Apkarian, A linear matrix inequality approach to H∞ control, Int. J. Robust Nonlinear Control 4 (1994) 421–448. [2] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox User’s Guide, 1st Edition, The MathWorks, Inc., Natick, MA, 1995, pp. 1760 –1500. [3] M. Green, D.J.N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood CliJs, NJ, 1995. [4] Iwasaki, R.E. Skelton, All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica 30(8) (1994) 1307–1317. [5] G.A. Leonov, A. Noack, V. Reitmann, Asymptotic orbital stability conditions for Tows by estimates of singular values of the linearization, Nonlinear Anal. 44 (8) (2001) 1057–1085. [6] G.A. Leonov, D.V. Ponomarenko, V.B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis, World Scienti7c, Singapore, 1996. [7] G.A. Leonov, V.B. Smirnova, Stability and oscillations of solutions of integro-diJerential equations of pendulum-like systems, Math. Nachr. 177 (1996) 157–181. [8] G.A. Leonov, V.B. Smirnova, Analysis of frequency-of-oscillations-controlled systems, in: Proceedings of International Conference on Control of Oscillations and Chaos, Vol. 2, 1997, pp. 439 – 441.
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