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Simultaneous H∞ Stabilization for a Set of Nonlinear Systems
Simultaneous H ∞ Stabilization for a Set of Nonlinear Systems CAI Xiu-Shan1
HU Hui1
LV Gan-Yun1
Abstract Simultaneous H ∞ stabilization for a set of nonlinear systems in canonical form is dealt with by using control storage functions (CSF) in this paper. Necessary and sufficient conditions for the existence of common quadratic CSFs for such systems are given. A systematically method for constructing common quadratic CSFs is developed. Based on common CSFs, continuous state feedback laws are designed to simultaneously H ∞ stabilize the single-input systems and multi-input systems, respectively. An example is given to show the effectiveness of the method. Key words Nonlinear systems, disturbance rejection, simultaneous H ∞ stabilization, control storage functions (CSF), state feedback Citation Cai Xiu-Shan, Hu Hui, Lv Gan-Yun. Simultaneous H ∞ stabilization for a set of nonlinear systems. Acta Automatica Sinica, 2012, 38(3): 473−481 DOI 10.1016/S1874-1029(11)60309-1
Simultaneous stabilization problem arises frequently in practice. Due to parameter uncertainty, variation and various modes of the plant, a real system may hold several models and is switched from one to another during operation. Blondel[1] proved that the simultaneous stabilizability of more than two linear systems is rationally undecidable based on the frequency domain approach. Petersen[2] gave a nonlinear state feedback law to stabilize a set of singleinput linear systems simultaneously. For nonlinear systems, the simultaneous stabilization is generally more difficult. Ho-Mock-Qai et al.[3] established some relevant results for nonlinear systems. Wu[4] provided a method for designing a controller to simultaneously stabilize a set of single-input nonlinear systems. In Cai et al.[5] , we designed a continuous state feedback law to simultaneously stabilize a set of multi-input nonlinear systems with uncertain parameters. The simultaneous H ∞ stabilization problem is an extension of the simultaneous stabilization problem by incorporating an H ∞ performance criterion, such as in disturbance rejection. Specifically, a single controller is sought such that it simultaneously renders a set of systems being internally stable and satisfying an L2 - gain specification. In this sense, the simultaneous H ∞ stabilization problem is much more difficult than the problem of simultaneous stabilization without a performance criterion. For linear systems case, Savkin[6] gave necessary and sufficient conditions for the simultaneous H ∞ stabilization via nonlinear digital output feedback control. Miller et al.[7] considered the use of linear periodically time-varying controllers to solve the simultaneous H ∞ stabilization problem. Lee et al.[8] studied the simultaneous H ∞ stabilization problem via the chain scattering framework. Till now, few results have been published about simultaneous H ∞ stabilization nonlinear systems. In Wu[9] , a control storage function (CSF) method was developed for designing simultaneous H ∞ controller for a set of single-input nonlinear systems. Necessary and sufficient conditions for the existence of simultaneous H ∞ controller were acquired. CSF method is motivated by the control Lyapunov function (CLF) method. The concept of a
CLF was introduced by Artstein[10] , which made a tremendous impact on stabilization theory. It converted stability descriptions into tools for solving stabilization tasks. Sontag[11] presented a universal feedback scheme by using the CLF. The CLF has been widely adopted in various design problems[4, 9, 12−15] . One difficulty in applying CLFs or CSFs for solving control problems is how to construct them. It is very important to identify those nonlinear systems in which corresponding CSFs exist and can be constructed systematically. In this paper, we restrict our attention to a set of nonlinear systems in the canonical form. Necessary and sufficient conditions for the existence of common quadratic CSFs for such systems are given. A systematic method for constructing common quadratic CSFs is developed. Based on common CSFs, continuous state feedback laws are designed to simultaneously H ∞ stabilize the single-input systems and multi-input systems, respectively.
1
System description Consider a set of q nonlinear systems described by x) + Gs (x x)u uu + Hs (x x)w w ], x + B[F F s (x x˙ = Ax x) + ηs (x x)w w , s = 1, 2, · · · , q y = ξ s (x
where x ∈ Rn , u ∈ Rm , w ∈ Rν , and y ∈ Rr are the state, the input, the disturbance input and the output, respectively. We denote the s-th possible model in (1) as x) system Qs . Suppose that for every s ∈ {1, 2, · · · , q}, F s (x x) fs2 (x x) · · · fsl (x x)]T with fsi (0) = 0, i = 1, = [fs1 (x x) = (gsij (x x))l×m is of full row rank, 2, · · · , l, and Gs (x x) = (hsij (x x))l×ν , ξ s (x x) = [ξs1 (x x) ξs2 (x x) · · · and Hs (x x)]T with ξsσ (0) = 0, σ = 1, 2, · · · , r, and ηs (x x) = ξsr (x x))r×ν . The functions fsi (x x), gsij (x x), hsij (x x), ξsσ (x x), (ηsσj (x x), s = 1, 2, · · · , q; i = 1, 2, · · · , l; j = 1, 2, · · · , m; and ηsσj (x σ = 1, 2, · · · , r, are continuous. It is assumed that A and B take the following Brounovsky canonical forms: ⎡
Manuscript received January 13, 2011; revised April, 27, 2011 Supported by National Natural Science Foundation of China (61074011, 60774011) Recommended by Associate Editor LIU Yun-Gang 1. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
(1)
⎢ ⎢ A=⎢ ⎣
A1 0 .. . 0
0 A2 .. . 0
··· ··· .. . ···
0 0 .. . Al
⎤
⎡
⎢ ⎥ ⎢ ⎥ ⎥ , Ai = ⎢ ⎣ ⎦
0 .. . 0 0
1 .. . 0 0
··· .. . ··· ···
0 .. . 1 0
⎤ ⎥ ⎥ ⎥ ⎦
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ACTA AUTOMATICA SINICA Vol. 38 ⎤ b1 0 · · · 0 0 x) > 0. E and b take the following Brounovsky assume gs (x ⎢ 0 b2 · · · 0 ⎥ ⎢ . ⎥ canonical forms, respectively: . ⎢ ⎢ ⎥ ⎥ B=⎢ . (2) .. .. ⎥ , b i = ⎢ . ⎥ .. ⎡ ⎤ ⎡ ⎤ ⎣ .. ⎣ ⎦ ⎦ . . 0 . 0 1 ··· 0 0 1 r ×1 0 0 · · · bl ⎢ . . . ⎥ ⎢ . ⎥ i . . ... ⎥ ⎢ . . ⎢ . ⎥ E=⎢ . . (8) ⎥, b = ⎢ . ⎥ ⎣ ⎦ ⎣ 0 ⎦ where r1 + · · · + rl = n. 0 0 ··· 1 The objective of this paper is to establish a common 0 0 ··· 0 1 CSF for the systems in (1), and to find a continuous state feedback law u such that it internally stabilizes the systems We present a method of constructing common CSFs for x) + Gs (x x)u u], x + B[F F s (x in (1) simultaneously (i.e., x˙ = {Ax the systems in (7). A continuous feedback law is also obs = 1, 2, · · · , q are all asymptotically stable); and moreover, tained by the CSF such that all the closed-loop systems are given γ > 0, the resultant closed-loop systems, starting internally stable and satisfy the L2 - gain requirement (3). from the initial state x (0) = 0 , satisfy 2.1 Construction of a common CSF T T Pn−1 P 12 T 2 T be a symmetric matrix where Let P = w (t)dt y (t)yy (t)dt ≤ γ w (t)w (3) PT p22 12 0 0 (n−1)×(n−1) , P 12 ∈ Rn−1 , and p22 ∈ R. Denote Pn−1 ∈ R −1 T P = [β , · · · , β p for all T > 0 and for each w (·) ∈ L2 [0, T ]. 11 1n−1 ] 12 22 Define C1β as follows: Suppose that the systems in (1) satisfy the following assumption. ⎡ ⎤ 0 1 ··· 0 Assumption 1. Given γ > 0, for all s ∈ {1, 2, · · · , q} ⎢ ⎥ .. .. .. .. x)ηs (x x) < γ 2 I, and and for any x ∈ Rn , suppose that ηsT (x ⎢ ⎥ . . . . C = (9) T T 2 T −1 T ⎢ ⎥ 1β x)ξξ s (x x) + ξ s (x x)ηs (x x)(γ I −ηs (x x)ηs (x x)) ηs (x x)ξξ s (x x) ≤ ξ s (x ⎣ 0 ⎦ 0 ··· 1 x where Γ is a positive semidefinite matrix. x T Γx −β11 −β12 · · · −β1n−1 The following definition was presented in [9]. T Definition 1. A smooth, radially unbounded and posiC1β is called companion matrix of p−1 22 P 12 , and the charactive definite function Vs (·) : Rn → R is a CSF of Qs in (1) teristic polynomial of C1β is if for any x ∈ Rn /{00} and w ∈ Rν , it holds that λ1 (β) = λn−1 + β1n−1 λn−2 + · · · + β12 λ + β11 ∂V s x) + Gs (x x)u u + Hs (x x)w w )) + x + B(F F s (x inf (Ax u x ∂x Theorem 1. Consider the set of systems in (7). Sup
x is a x) = x T Px pose that Assumption 1 holds. Then V (x (4) y Ty − γ 2w Tw < 0 common CSF for the systems in (7) if P satisfies the following conditions H1 and H2 : H1 : p22 > 0 and λ1 (β) is a Hurwitz polynomial. x) is a CSF of (1) is precisely the The condition that Vs (x T T H2 : (Pn−1 − p−1 statement that for any x = 0 , w ∈ Rν 22 P 12P 12 )C n−1 − 1β + C1β (P
In−1 −1 −1 T p22 P 12P 12 )+ In−1 − P 12 p22 Γ is neg∂Vs −1 PT −P x) = 0 ⇒ BGs (x 12 p22 x ∂x ative definite where Γ is given in Assumption 1. ∂Vs Proof. Note thatΓ is positive x) + BHs (x x)w w ) + y Ty − γ 2w Tw < 0 x + BF F s (x (Ax semidefinite; then x ∂x
In−1 −1 In−1 − P 12 p22 Γ (5) is positive semidefi−1 PT −P 12 p22 Definition 2. A CSF Vs (·) : Rn → R of Qs satisfies the nite. If conditions H1 and H2 hold, then it is easy to deduce L2 - gain small control property if for each ε > 0 there are that this P is positive definite. x < δ1 and w δ1 > 0, δ2 > 0 such that if x = 0 satisfies x x, u, w ) for Qs as Define the Hamiltonian function φs (x u < ε such w < δ2 , then there is some u with u satisfies w that x) ∂V(x x, u, w ) = x) + gs (x x)u + h s (x x)w w )) + x + b (fs (x (Ex φs (x ∂Vs x ∂x x) + Gs (x x)u u + Hs (x x)w w )) + x + B(F F s (x (Ax T 2 T x ∂x (10) y y −γ w w (6) y Ty − γ 2w Tw < 0 After some manipulations, we have: ⎡
2
⎤
⎡
Single-input systems
Consider a set of q single-input nonlinear systems as follows: x) + gs (x x)u + h s (x x)w w ], x + b [fs (x x˙ = Ex x) + ηs (x x)w w , s = 1, 2 · · · , q y = ξ s (x n
ν
r
(7)
where x ∈ R , u ∈ R, w ∈ R , y ∈ R are the state, the input, the disturbance input, and the output, respectively. fs (·): Rn → R, gs (·) : Rn → R, hs (·) : Rn → R1×ν , ξs (·) : Rn → Rr , ηs (·) : Rn → Rr×ν are continuous functions. Suppose that for every s ∈ {1, 2, · · · , q}, fs (0) = 0, ξ s (00) = x) holds the same sign. Without loss of generality, 0 , and gs (x
x, u, w ) = x T (P E + E T P )x x+ φs (x x) + gs (x x)u] + ξ T x)ξξ s (x x) + (x xT Pbbhs (x x) + xT Pbb[fs (x 2x s (x x)ηs (x x)) × (γ 2 I − ηsT (x x)ηs (x x))−1 × ξT s (x x) + ξ T x)ηs (x x))T − (xT Pbbhs (x s (x x)ηs (x x))−1 × w − (γ 2 I − ηsT (x [w x) + ξ T x)ηs (x x))T ]T × (γ 2 I − ηsT (x x)ηs (x x)) × xT Pbbhs (x (x s (x x)ηs (x x))−1 (x xT Pbbhs (x x) + ξ T x)ηs (x x))T ] w − (γ 2 I − ηsT (x [w s (x (11)
Cai Xiu-Shan et al.: Simultaneous H ∞ Stabilization for a Set of · · ·
No. 7 Let
475
In view of Assumption 1, and from (13), (15), and (16), we have:
x) = x T (P E + E T P )x x + 2x xT Pbbfs (x x) + ξ T x)ξξ s (x x) + αs (x s (x x) + ξ T x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × xT Pbbh s (x (x s (x
x)ηs (x x)(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T ≤ ξT s (x s (x
x) + ξ T x)ηs (x x))T xT Pbbh s (x (x s (x x) = 2x xT Pbbgs (x x) βs (x Since for all s ∈ {1, 2, · · · , q} and for any x ∈ Rn , it holds x) > 0, it can be deduced that that gs (x x) = 0 ⇔ x T Pbb = 0 x T Pbbgs (x
x) = αs (x x + ξsT (x x)ξξ s (x x) + x T (P E + E T P )x
(12)
n
for any x ∈ R , and for all s ∈ {1, 2, · · · , q}. Thus
x + x T Γx x= x T (P E + E T P )x −1 −1 T T XT n−1 (Pn−1 − p22 P 12P 12 )(G11 − G12 p22 P 12 ) + −1 T T T (G11 − G12 p−1 22 P 12 ) (Pn−1 − p22 P 12P 12 ) +
In−1 In−1 − P 12 p−1 X n−1 −1 22 Γ PT −P 12 p22
(17) when x T P b = 0. Since
x) = x T (P E + E T P )x x + ξT x)ξξ s (x x) + αs (x s (x x)ηs (x x)(γ 2 I − ηsT (x x) etas (x x))−1 (ξξ T x)ηs (x x))T ξT s (x s (x (13) T
x) = 0. when x Pbbgs (x Divide E, b , and x into their block forms G11 G12 0 E= , b= 0 0 1
−1 T G11 − ⎡ G12 p22 P 12 = 0 1 ⎢ .. .. ⎢ . . ⎢ ⎣ 0 0 −β11 −β12
··· .. . ··· ···
0 .. . 1
⎤ ⎥ ⎥ ⎥ = Cβ ⎦
−β1n−1
from condition H2 , there exists a positive definite matrix ϕ such that
with ⎡
⎡ ⎤ ⎤ ··· 0 0 ⎢ ⎢ . ⎥ . ⎥ .. ⎢ ⎢ . ⎥ . .. ⎥ G11 = ⎢ ⎥ , G12 = ⎢ . ⎥ ⎣ ⎣ 0 ⎦ ⎦ ··· 1 1 ··· 0 x1 · · · xn−1 . and x T = X T xn , X T n−1 = n−1 By computing, we obtain 0 .. . 0 0
1 .. . 0 0
x T Pbb = X T n−1P 12 + xn p22 and x + x T E T Px x = x T P Ex XT xn × n−1 Pn−1 G11 + GT 11 Pn−1 T PT G + G 11 12 12 Pn−1 X n−1 xn
Pn−1 G12 + GT 11P 12 T PT G + G 12 12 12P 12
×
(14) Then x T Pbb = 0 implies −1 XT xn = −X n−1 p22 P 12
(15)
Substituting (15) to (14), we obtain that x + x T E T Px x= x T P Ex −1 −1 T T XT n−1 [(Pn−1 − p22 P 12P 12 )(G11 − G12 p22 P 12 ) + −1 T T T X n−1 (G11 − G12 p−1 22 P 12 ) (Pn−1 − p22 P 12P 12 )]X (16) when x T P b = 0.
−1 T T T (Pn−1 − p−1 22 P 12P 12 )C1β + C 22 P 12P 12 ) + 1β (Pn−1 − p
I n−1 In−1 − P 12 p−1 = −ϕ −1 22 Γ PT −P 12 p22 (18) x, u, w ) ≤ αs (x x) < 0, for Then it can be deduced that φs (x x) = 0, x = 0. Thus V (x x) = x T Px x is a common x T Pbbgs (x CSF for the systems in (7). An algorithm for construction of common quadratic CSFs for the systems in (7) is as follows: Step 1. Take an arbitrary positive number p22 and a Hurwitz polynomial λ1 (β) = λn−1 + β1n−1 λn−2 + · · · + β12 λ + β11 . Step 2. Let P T 12 = p22 [β11 , · · · , β1n−1 ]. Step 3. Take a positive definite matrix ϕ ∈ R(n−1)×(n−1) , and solve the Lyapunov equation M C1β +
In−1 −1 T C1β M = − In−1 − P 12 p22 Γ − ϕ. −1 PT −P 12 p22 −1 T Step 4. Calculate Pn−1 = M + P 12 p22 P 12 . Pn−1 P 12 Step 5. Let P = . PT p22 12 x is a common CSF of the systems in x) = xT Px Then V (x (7). The necessary conditions of a common quadratic CSF of the systems in (7) are given in the following. Corollary 1. Consider the set of systems in (7). Supx is a x) = x T Px pose that Assumption 1 holds. Then V (x common CSF for the systems in (7) only if P satisfies condition H1 and the following condition: −1 T T T H3 : (Pn−1 − p−1 22 P 12P 12 )C1β + C1β (Pn−1 − p22 P 12P 12 ) is negative definite. x is a common CSF for the x) = x T Px Proof. Since V (x systems in (7), P is positive definite. Hence p22 > 0. And for any x = 0 , w ∈ Rν , it holds that
476
ACTA AUTOMATICA SINICA
xT Pbbgs (x x) = 0 ⇒
Vol. 38
two functions as follows
xT P (Ex x + b fs (x x) + b hs (x x)w) + y Ty − γ 2w Tw < 0 2x x) > 0, s = After some manipulations and in view of gs (x 1, 2, · · · , q, for any x = 0 , w ∈ Rν , we have:
x + max[2x xTρ fs (x x) + ξ T x)ξξ s (x x) + x) = x T (P E + E T P )x α(x s (x s
x) + ξ T x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × xTρh s (x (x s (x x) + ξ T x)ηs (x x))T ] xTρh s (x (x s (x x) x) = 2x xT ρ min gs (x β(x s
T
x) = 0 x Pbbgs (x T
⇔ x Pbb = 0 x + ξT x)ξs (x x) − ⇒ xT (P E + E T P )x s (x x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T − (ξsT (x s (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T )T × w − (γ 2 I − ηsT (x (w s (x
(23) In order to give Theorem 2, we first prove the following Lemma1. Lemma 1. Under assumption, let P be the symmetric x), β(x x) be matrix constructed by Step 1 to Step 5 and α(x x ≤ δ}. x ∈ Rn |0 < x defined as (23). Denote N (δ) = {x Then it implies
x)ηs (x x)) × (γ 2 I − ηsT (x 2
w − (γ I − (w
x)ηs (x x))−1 (ξξ T x)ηs (x x))T ) ηsT (x s (x
lim sup
δ→0 x ∈N (δ)
<0
Then for any x = 0 , w ∈ Rν , it holds that
(19)
x T Pbb = 0 ⇒ x + ξT x)ξξ s (x x) + x T (P E + E T P )x s (x x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T < 0 (ξξ T s (x s (x (20) It can be seen from (19) that if (20) does not hold, we x, u, w ) < 0 in the presence of cannot find u such that φs (x x)ηs (x x))−1 × the worst case disturbance w = (γ 2 I − ηsT (x T T x x x x (x )η (x )) . Thus V (x ) = x Px is a CSF of the systems (ξξ T s s in (7) only if (20) holds. By Assumption 1, given γ > 0, for all s ∈ {1, 2, · · · , q}, x)ηs (x x) < γ 2 I. In view of and for any x ∈ Rn , holds ηsT (x x 0 (20), we deduce that for any = , x T Pbb = 0 ⇒ x T (P E + E T P ) x < 0
(21)
From (15) and (16), we have: x + x T E T Px x= x T P Ex −1 T −1 T XT n−1 [(Pn−1 − p22 P 12P 12 )(G11 − G12 p22 P 12 ) + −1 T T T X n−1 (G11 − G12 p−1 22 P 12 ) (Pn−1 − p22 P 12 P12 )]X
(22) for x T Pbb = 0. By (21) and (22), then H3 is satisfied. Pn−1 P 12 On the other hand, P = is positive defiPT p22 12 −1 T nite, then Pn−1 −p22 P 12P 12 is positive definite too. Hence, T the Lyapunov theorem shows that G11 − G12 p−1 22 P 12 is sta−1 T ble, i.e., G11 − G12 p22 P 12 is a Hurwitz matrix. H1 is also satisfied. 2.2
Simultaneous H ∞ stabilization
This subsection will present a continuous feedback control law such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3) by using the common CSF established in Subsection 3.1. Let P be the symmetric matrix constructed by Step 1 to Step 5 and ρ be the last column of P . Then Pbb = ρ . Define
x) α(x ≤0 x |β(x )|
(24)
Proof. Let δ → 0. For any a sequence {ζζ n } such x), that {ζζ n } ∈ N (δ), because for all s ∈ {1, 2, · · · , q}, fs (x x), hs (x x), ξ s (x x), ηs (x x) are continuous with fs (0) = 0, gs (x x) > 0 for any x ∈ Rn , and P is ξ s (00) = 0 , and gs (x the symmetric matrix constructed by Step 1 to Step 5 x), β(x x) are defined as (23), we can deduce that and α(x α(ζζ n ) = o(β(ζζ n )), if β(ζζ n ) = 0, and δ → 0; on the other hand, if β(ζζ n ) = 0, from (15), (17), and (18), XT it can be deduced that α(ζζ n ) = −X n−1 ϕXn−1 , with −1 T T T X n−1 p22 P 12 and ζn = [Xn−1 , xn ], so α(ζζ n ) = xn = −X X n−1 < 0, if β(ζζ n ) = 0, and ζ n = 0. In conXT −X n−1 ϕX x) α(x clusion, we have: limδ→0 supx ∈N (δ) |β(x x)| ≤ 0. The limit may well be −∞. Theorem 2. Suppose that the systems in (7) satisfy Assumption 1. If there is some matrix P satisfying conditions H1 and H2 , then there exists a continuous control law such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3). Proof. Constructing matrix P by Step 1 to Step 5; then x is a common CSF of the q systems in (7) by x) = x T Px V (x Theorem 1. The state feedback law is taken as ⎧ x))2 + (β(x x))4 x) + (α(x α(x ⎨ , x Tρ = 0 − x) = u(x x β(x ) ⎩ 0, x Tρ = 0 (25) x), β(x x) are defined as (23). where α(x x) satisfies the L2 - gain small Since (24) implies that V (x control property by [9], the feedback control law u is a continuous function on Rn . x, u, w ) ≤ α(x x) + 2x xT ρgs (x x)u, if we can show Due to φs (x that x)u < 0 x) + 2x xT ρgs (x (26) α(x for any x = 0 and for all s ∈ {1, 2, · · · , q}, then all the closed-loop systems satisfy the requirement (3). In fact, if (26) holds, and in view of (10), then given γ > 0, we have: x, u, w ) = φs (x
x) ∂V (x x) + gs (x x)u + h s (x x)w w ]) + x + b[fs (x (Ex x ∂x y Ty − γ 2w Tw < 0
(27)
for any x = 0 , for all s ∈ {1, 2, · · · , q}, and for each w (·) ∈ L2 [0, T ]. For all T > 0, starting from the initial state x (0) =
Cai Xiu-Shan et al.: Simultaneous H ∞ Stabilization for a Set of · · ·
No. 7
0, integrating both sides of (27), we have: x(T )) − V (x x(0)) + V (x
T
y T (t)yy (t)dt − γ 2
0
T
w (t)dt < 0 w T (t)w
0
Consider the systems in (1). Divide Ai and Bi into their block forms as follows: Ai−1 A i2 0 Ai = , Bi = 1 0 0 with ⎡
x is a positive function and x (0) = 0 , we x) = x T Px Since V (x x(T )) ≥ 0, V (x x(0)) = 0 . So have V (x
T
y T (t)yy (t)dt < γ 2
0
T
w (t)dt w T (t)w
0
for all T > 0 and for each w (·) ∈ L2 [0, T ], that is, all the closed-loop systems satisfy the requirement (3). Now, we verify that (26) holds in the following three cases. Case 1. x Tρ > 0 x) > 0, which implies that u(x x) < If x Tρ > 0, then β(x xT ρgs (x x) ≥ β(x x). Therefore, 0. From (23), we have 2x x) − β(x x))u(x x) ≤ 0. It leads to xTρ gs (x (2x x) + 2x xT ρgs (x x)u(x x) ≤ α(x x) + β(x x)u(x x) ≤ − α(x
x) + β 4 (x x) < 0 α2 (x
Case 2. x Tρ = 0, x = 0 If x Tρ = 0, x = 0 and by (20), then we have: x) + 2x xTρ gs (x x)u = α(x x + ξT x)ξξ s (x x) + max x T (P E + E T P )x s (x s
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × (ξξ T s (x x)ηs (x x))T < 0 (ξξ T s (x Case 3. x Tρ < 0 x) < 0 By a similar discussion to Case 1, we can obtain β(x xTρ gs (x x) − β(x x))u ≤ 0. Thus and u > 0. We also have (2x x)u ≤ x) + 2x xTρ gs (x α(x x) + β(x x)u ≤ − α(x
x) + β 4 (x x) < 0 α2 (x
From the previous discussions, we know that the feedback control law (25) renders all the closed-loop systems satisfying the L2 - gain requirement (3). The following proves internal stability. Letting w = 0 and u in (25) and noting (26) yields x, u, 0) ≤ α(x x) + 2x xT P bgs (x x)u < 0 φs (x for any x = 0, and for all s ∈ {1, 2, · · · , q}. Therefore, all the closed-loop systems are internally stable.
3
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Multi-input systems
This section turns to consider the muti-input case. A common CSF is established for the q systems in (1). Then a continuous feedback law is obtained by the CSF such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3).
⎢ ⎢ Ai−1 = ⎢ ⎣
0 .. . 0 0
1 .. . 0 0
··· .. . ··· ···
⎡
⎤
0 .. . 1 0
⎢ ⎢ A i2 = ⎢ ⎣
⎥ ⎥ ⎥, ⎦
0 .. . 0 1
⎤ ⎥ ⎥ ⎥ ⎦
Assume that βi1 , βi2 , · · · , βi,ri−1 are the coefficients of the following polynomial: λi (β) = λri −1 + βi,ri −1 λri −2 + · · · + βi2 λ + βi1 Denote Λ= ⎡ Ir1 −1 ⎢ ⎢ ⎢ ⎣
P 12 p−1 −P 13
(28)
⎤ ..
.
Γ1 = ΛΓΛT
Irl −1
P l2 p−1 −P l3
⎥ ⎥ ⎥ ⎦
Since Γ is positive semidefinite, Γ1 is positive semidefinite too. Assume that λmax (Γ1 ) is the maximum eigenvalue of Γ1 . Piri −1 P i2 Let Pi = , i = 1, 2, · · · , l be symmetric T P i2 pi3 (ri −1)×(ri −1) , P i2 ∈ Rri −1 , and matrices, where Piri −1 ∈ R pi3 ∈ R. It is required that T βi1 βi2 · · · βi,ri −1 p−1 i3 P i2 = Then T Ai−1 − A i2 p−1 i3 Pi2 = Ciβ =
⎡ ⎢ ⎢ ⎢ ⎣
0 .. . 0 −βi1
1 .. . 0 −βi2
··· .. . ··· ···
0 .. . 1
⎤ ⎥ ⎥ ⎥ ⎦
−βi,ri −1
Let Pi , i = 1, 2, · · · , l, satisfy the following conditions: H4 : pi3 > 0 and λi (β) is a Hurwitz polynomial. T T −1 T H5 : (Piri −1 −p−1 i3 P i2P i2 )Ciβ +Ciβ (Piri −1 −pi3 P i2P i2 ) + λmax (Γ1 )Iri −1 is negative definite. Denote ⎤ ⎡ P1 ⎥ ⎢ P2 ⎥ ⎢ x x) = x T Px (29) P =⎢ ⎥ , V (x . .. ⎦ ⎣ Pl Theorem 3. Consider the set of systems in (1). Suppose x) = x T Px x is a common that Assumption 1 holds. Then V (x CSF for the systems in (1) if P satisfies conditions H4 and H5 . Proof. If Pi , i = 1, 2, · · · , l, satisfies conditions H4 and H5 , then it is easy to deduce that Pi is positive definite. It x is positive definite. x) = x T Px follows that V (x
478
ACTA AUTOMATICA SINICA
s (x x, u, w ) for Qs in (1) Define the Hamiltonian function φ as x) ∂V (x s (x x, u , w ) = x)+Gs (x x)u u +Hs (x x)w w ]) + x +B[F F s (x φ (Ax x ∂x T
2
T
y y −γ w w
(30)
After some manipulations, we have:
T T X i,ri −1 (Piri −1 − p−1 Ciβ i3 P i2P i2 )]X
x)ηs (x x))−1 (x xT P BHs (x x) ηsT (x
w − (γ I − (w
when x T i Pi Bi = 0. In view of Assumption 1, and from (33) and (37), we have:
x)ηs (x x))−1 ηsT (x
x)ηs (x x))T + ξT s (x
−
⎡
×
x)ηs (x x))(w w − (γ 2 I − ηsT (x x)ηs (x x))−1 × (γ 2 I − ηsT (x x) + ξ T x)ηs (x x))T ) xT P BHs (x (x s (x T
T T XT 1,r1 −1 , X 2,r2 −1 , · · · , X l,rl −1
l
(31)
Let T
−1 T XT i,ri −1 [(Piri −1 − pi3 P i2P i2 )Ciβ +
T T X i,ri −1 + Ciβ (Piri −1 − p−1 i3 P i2P i2 )]X
x) + ξ T x)ηs (x x))T )T × xT P BHs (x (x s (x
⎢ ⎢ Γ1 ⎢ ⎣
X 1,r1 −1 X 2,r2 −1 .. . X l,rl −1
x)ηs (x x))−1 (x xT P BHs (x x) + ξ T x)ηs (x x))T (γ 2 I − ηsT (x s (x x) = 2x xT P BGs (x x) βs (x x) Since for each x ∈ Rn , and for all s ∈ {1, 2, · · · , q}, Gs (x is of full row rank, we have that x) = 0 ⇔ x T P B = 0 x T P BGs (x
(32)
Thus x) = x T (P A + AT P )x x + ξT x)ξξ s (x x) + α s (x s (x x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T (ξξ T s (x s (x (33) x) = 0. when x T P BGs (x Use block matrix to express x T , that is T T xT = xT , xT · · · xT xi,ri , i = Xi,ri −1 1 x 2 l xi1 xi2 · · · xi,ri −1 , i = 1, 2, · · · , l XT i,ri −1 = Then xT 1 P1 B1
xT 2 P2 B2
and xT i Pi Bi =
XT i,ri −1
xi,ri
···
xT l Pl Bl
Piri −1
P i2
PT i2
pi3
XT i,ri −1P i2 + pi3 xi,ri ,
⎥ ⎥ ⎥≤ ⎦
i=1 T T X i,ri −1 Ciβ (Piri −1 − p−1 i3 P i2P i2 ) + λmax (Γ1 )Iri −1 ]X
T
x)ξξ s (x x) + (x xT P BHs (x x) + ξ T x)ηs (x x)) × ξT s (x s (x
⎤
−1 T XT i,ri −1 [(Piri −1 − pi3 P i2P i2 )Ciβ +
x) = x (P A + A P )x x + 2x x P BF F s (x x) + α s (x
xTP B =
(37)
i=1
x)ξξ s (x x) + (x xT P BHs (x x) + ξ T x)ηs (x x)) × ξT s (x s (x
2
T −1 T xT 2x i Pi Aix i = X i,ri −1 [(Piri −1 − pi3 P i2P i2 )Ciβ +
l
x + 2x xT P B[F F s (x x) + Gs (x x)u u] + x T (P A + AT P )x
(γ I −
0 at least. Moreover, we have: and there exists an xi =
x) ≤ x T (P A + AT P )x x + x T Γx x= α s (x
s (x x, u , w ) = φ
2
Vol. 38
0 1
when x T P B = 0. By condition H5 , it can be deduced that s (x x) ≤ α x) < 0, when xT P BGs (x x) = 0, x = 0. Thus s (x φ x) = x T Px x is a common CSF for the systems in (1). V (x In the following, we give the necessary conditions of a common quadratic CSF of the systems in (1). Corollary 2. Consider the set of systems in (1). Under x defined by (29) is a common x) = xT Px assumption, V (x CSF for the systems in (1) if and only if P satisfies condition H4 and the following condition: T H6 : for each i ∈ {1, 2, · · · , l}, (Piri −1 − p−1 i3 P i2P i2 )Ciβ T −1 T + Ciβ (Piri −1 − pi3 P i2P i2 ) is negative definite. Proof. Using the arguments as Corollary 1, we can deduce it. x) are Suppose that for all s ∈ {1, 2, · · · , q}, matrices Gs (x described by x) = G(x x)Ds (x x) (39) Gs (x x) is an l×m matrix, has full row rank, and Ds (x x) where G(x x), ds2 (x x), · · · , dsm (x x)}. It is further supposed = diag{ds1 (x x) are continuous with x) and dsj (x that the elements of G(x x) = 0 dsj (x
for all s ∈ {1, 2, · · · , q}, j ∈ {1, 2, · · · , m}, x ∈ R . Moreover, we assume for each j, j ∈ {1, 2, · · · , m}, the q funcx), s = 1, 2, · · · , q hold the same sign. We now tions dsj (x define a matrix which is independent of s as follows:
i = 1, 2, · · · , l
x), ν2 (x x), · · · , νm (x x)} x) = diag{ν1 (x Ω(x where
=
x) = νj (x
x), min dsj (x
x) > 0 if dsj (x
x), max dsj (x
x) < 0 if dsj (x
s s
(35)
From (34) and (35), when x T P B = 0, x = 0, we have: T xT i Pi Bi = X i,ri −1P i2 + pi3 xi,ri = 0, i = 1, 2, · · · , l (36)
(40) n
(34)
(38)
(41)
(42)
x)dsj (x x) − νj2 (x x) ≥ 0 for j = By (42), we have νj (x 1, 2, · · · , m, and s = 1, 2, · · · , q, or equivalently, the diagonal matrices x)Ω(x x) − Ω2 (x x), s = 1, 2, · · · , q Ds (x
(43)
Cai Xiu-Shan et al.: Simultaneous H ∞ Stabilization for a Set of · · ·
No. 7
479
are positive semidefinite. Theorem 4. Consider the systems in (1). If under x) satisfy (39) and (40), Assumption 1 and matrices Gs (x and there is some matrix P satisfying conditions H4 and H5 , then there exists a continuous control law such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3). x) = x T Px x defined by (29) is Proof. By Theorem 3, V (x a common CSF for the q systems in (1). Denote
Consider (46) now. From (45) and (48), we have:
x+ x) = x T (P A + AT P )x α (x F s (x x)+ ξ T x)ξξ s (x x)+ (x xT P BHs (x x) + xT P BF max 2x s (x
when x T P B = 0. From the previous discussions, we know that the feedback control law (45) renders all the closed-loop systems satisfying the L2 - gain requirement (3). The following proves internal stability. Letting w = 0 and u in (45) and noting (46), (47), and (49), yields
s
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × ξT s (x x) + ξ T x)ηs (x x))T xT P BHs (x (x s (x
x, u , 0) ≤ x T (P A + AT P )x x+ φs (x
x) = 2Ω(x x)B T Px x x)GT (x β(x
s (x x, u , w ) ≤ α x) + 2x xT P BGs (x x)u u≤ s (x φ x)u u x) + 2x xT P BGs (x α (x
(46)
T
x)u u< x)+2x x P BGs (x Now let us consider (45), and prove α (x 0, x = 0 by in following two cases. x) = 0, x = 0 Case 1. β(x x) = 0 is equivx) and G(x x), β(x By the assumptions on Ω(x alent to x T P B = 0. From (38), (44) and (46), and in view of Assumption 1, we have: s (x x, u , w ) ≤ α x) = (x φ x + max ξ T x)ξξ s (x x) + x T (P A + AT P )x s (x s
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T ≤ (ξξ T s (x s (x x + x T Γx x<0 x T (P A + AT P )x
(47)
when x T P B = 0, x = 0 . x) = 0 Case 2. β(x By (43), we obtain that
2
T
T x) − βT (x x) × x)β(x x)β(x x P BGs (x x)2 x))2 + (βT (x x)β(x ( α(x x) x)β(x βT (x
for any x = 0, and for all s ∈ {1, 2, · · · , q}. Therefore, all the closed-loop systems are internally stable.
4
Example
Example 1. Consider the following set of 3 singleinput nonlinear systems: ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎨ x˙ = x 2 3 (50) Qs : ⎪ x) + gs (x x)u u + hs (x x)w w x˙ 3 = fs (x ⎪ ⎪ ⎩ x) + ηs (x x)w w , s = 1, 2, 3 y = ξs (x where x = [x1 x2 , x3 ]T , and x) = x1 cosx2 f1 (x
x) = x21 f2 (x
x) = x1 + x3 f3 (x x) = 2 + sin2 x2 g2 (x x) = 1 − cosx2 h1 (x x) = x1 − x3 h3 (x x) = x1 + x2 sinx3 ξ2 (x x) = 1 η1 (x x) = 1 + sinx1 η3 (x
x) = e(x1 +x2 ) g1 (x x) = 1 + x23 g3 (x x) = x2 x3 sinx1 h2 (x x) = x2 + x3 ξ1 (x x) = x1 − x2 ξ3 (x x) = 1 − cosx2 η2 (x
x)ξs (x x) ≤ x T Γx x ηsT (x
T
x)[Ds (x x)Ω(x x) − Ω (x x)]G (x x)B Px x≥0 x P BG(x
x)u u<0 x) + 2x xT P BGs (x α (x
x)ξs (x x) + ξsT (x x)ηs (x x)(γ 2 I − ηsT (x x)ηs (x x))−1 × ξsT (x
x) − βT (x x) = x)β(x x)β(x x P BGs (x T
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × (ξξ T x)ηs (x x))T ≤ (ξξ T s (x s (x
Let γ = 3. It can be shown that
T
x) − − α(x
F s (x x) + Gs (x x)u u] + ξ T x)ξξ s (x x) + xT P B[F 2x s (x
(44)
The feedback law is taken as ⎧ x))2 ⎪ x))2 +(βT (x x)β(x x)+ ( α(x α (x ⎨ x) , xTP B = 0 u = −β(x T (x x x ) β(x ) β ⎪ ⎩ 0, xTP B = 0 (45) By a similar proof to that of Theorem 2, u is continuous on Rn . The detail is omitted. From (30), we have:
Thus
s (x x, u , w ) ≤ α x) + 2x xT P BGs (x x)u u≤ (x φ x))2 x))2 +(βT (x x)β(x x) − ( α(x − α(x x) x)β(x x)+ βT (x ≤ α (x x) x)β(x βT (x x))2 < 0 x))2 + (βT (x x)β(x − ( α(x (49)
x)ηs (x x) < 9, s = 1, 2, 3. with Γ = diag{4, 4, 2}, and ηsT (x By Theorem 1, there exists a common CSF for the systems in (50). Let p22 = 1 and λ1 (β) = λ2 + λ + 1. Then PT 12 = [1 1]
≤0
(48)
I2
− P 12 p−1 22 Γ
I2
−1 PT −P 12 p22
=
6 2
2 6
480
ACTA AUTOMATICA SINICA
Choose ϕ =
−1 0
0 −1
Vol. 38
. Thus ⎡
9.5 x) = x T ⎣ 4.5 V (x 1
4.5 8 1
⎤ 1 1 ⎦x 1
is a common CSF for the systems in (50). Define ⎤ ⎡ 0 9.5 4.5 T⎣ x) = x 9.5 9 9 ⎦x + α(x 4.5 9 2 x) + ξsT (x x)ξs (x x) + max[2(x1 + x2 + x3 )fs (x s
x) + ξsT (x x)ηs (x x)) × ((x1 + x2 + x3 )hs (x x)ηs (x x))−1 × (9 − ηsT (x
Fig. 2 State trajectories, control law, output, and disturbance input of system Q2 (The initial state is [2, 1, 1]T , and w(t) = x1 sin(30t) + x22 + 2x3 .)
x) + ξsT (x x)ηs (x x))T ] ((x1 + x2 + x3 )hs (x x) = 2(x1 + x2 + x3 ) min gs (x x) β(x s
By Theorem 2, the feedback law ⎧ x) + α2 (x x) + β 4 (x x) α(x ⎨ − , u= x β(x ) ⎩ 0,
if x1 + x2 + x3 = 0
if x1 + x2 + x3 = 0 (51) is such that all the closed-loop systems in (50) and (51) are internally stable and satisfy the L2 - gain requirement (3). Figs. 1 ∼ 3 show the state trajectories, the control law, the output and the disturbance input of the 3 systems in (50) with the state feedback law (51), respectively.
5
Conclusion
This paper presents a novel method for constructing common quadratic CSFs for a set of nonlinear systems in canonical form. An algorithm is given to obtain common quadratic CSFs for such systems. Then continuous state feedback laws are designed to simultaneously H ∞ stabilize the single-input systems and multi-input systems, respectively. Finally, the effectiveness of the proposed schemes is illustrated by a simulation example.
Fig. 3 State trajectories, control law, output, and disturbance input of system Q3 (The initial state is [2, −1, 2]T , and w(t) = x21 e−t − x2 cos(10t) − x3 sin(20t).)
References 1 Blondel V. Simultaneous Stabilization of Linear Systems. New York: Springer, 1994
2 Petersen I. A procedure for simultaneously stabilizing a collection of single input linear systems using non-linear state feedback control. Automatica, 1987, 23(1): 33−40 3 Ho-Mock-Qai B, DayawansaW. Simultaneous stabilization of linear and nonlinear systems by means of nonlinear state feedback. SIAM Journal on Control and Optimization, 1999, 37(6): 1701−1725 4 Wu J. Simultaneous stabilization for a collection of singleinput nonlinear systems. IEEE Transactions on Automatic Control, 2005, 50(3): 328−337
Fig. 1 State trajectories, control law, output, and disturbance input of system Q1 (The initial state is [1, 1, 2]T , and w(t) = cost − sin2 (25t).)
5 Cai X S, Han Z Z, Zhang W. Simultaneous stabilization for a collection of multi-input nonlinear systems with uncertain parameter. Acta Automatica Sinica, 2009, 35(2): 206−209
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6 Savkin A V. Simultaneous H∞ control of a finite collection
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of linear plants with a single nonlinear digital controller. Systems and Control Letters, 1998, 33(5): 281−289
ear differential inclusions subject to disturbance. Nonlinear
7 Miller D E, Chen T. Simultaneous stabilization with near
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optimal H∞ performance. IEEE Transactions on Automatic
clusion systems subject to disturbances. Acta Automatica
Control, 2002, 47(12): 1986−1998
Sinica, 2010, 36(9): 1327−1331
8 Lee P H, Soh Y C. Simultaneous H∞ stabilization. International Journal of Control, 2004, 77(2): 111−117 9 Wu J. Simultaneous H∞ control for nonlinear systems. IEEE Transactions on Automatic Control, 2009, 54(3): 606 −610 10 Artstein Z. Stabilization with relaxed controls. Nonlinear Analysis, 1983, 7(11): 1163−1173 11 Sontag E. A “universal” construction of Artstein s theorem on nonlinear stabilization. Systems and Control Letters, 1989, 13(2): 117−123 12 Cai X S, Han Z Z, Wang X. An analysis and design method for systems with structural uncertainty. International Journal of Control, 2006, 79(12): 1647−1653
Dynamics, 2009, 58(3): 487−496
15 Cai X S, Han Z Z, Huang J. Stabilization for a class of non-linear systems with uncertain parameter based on center manifold. IET Control and Applications, 2010, 4(9): 1558− 1568 CAI Xiu-Shan Professor at the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. Her research interest covers nonlinear control and its applications. Corresponding author of this paper. E-mail: [email protected] HU Hui Master student at the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. His research interest covers nonlinear control and its applications. E-mail: [email protected] LV Gan-Yun Associate professor at the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. His research interest covers nonlinear control and its applications. E-mail: [email protected]
HU Hui1
LV Gan-Yun1
Abstract Simultaneous H ∞ stabilization for a set of nonlinear systems in canonical form is dealt with by using control storage functions (CSF) in this paper. Necessary and sufficient conditions for the existence of common quadratic CSFs for such systems are given. A systematically method for constructing common quadratic CSFs is developed. Based on common CSFs, continuous state feedback laws are designed to simultaneously H ∞ stabilize the single-input systems and multi-input systems, respectively. An example is given to show the effectiveness of the method. Key words Nonlinear systems, disturbance rejection, simultaneous H ∞ stabilization, control storage functions (CSF), state feedback Citation Cai Xiu-Shan, Hu Hui, Lv Gan-Yun. Simultaneous H ∞ stabilization for a set of nonlinear systems. Acta Automatica Sinica, 2012, 38(3): 473−481 DOI 10.1016/S1874-1029(11)60309-1
Simultaneous stabilization problem arises frequently in practice. Due to parameter uncertainty, variation and various modes of the plant, a real system may hold several models and is switched from one to another during operation. Blondel[1] proved that the simultaneous stabilizability of more than two linear systems is rationally undecidable based on the frequency domain approach. Petersen[2] gave a nonlinear state feedback law to stabilize a set of singleinput linear systems simultaneously. For nonlinear systems, the simultaneous stabilization is generally more difficult. Ho-Mock-Qai et al.[3] established some relevant results for nonlinear systems. Wu[4] provided a method for designing a controller to simultaneously stabilize a set of single-input nonlinear systems. In Cai et al.[5] , we designed a continuous state feedback law to simultaneously stabilize a set of multi-input nonlinear systems with uncertain parameters. The simultaneous H ∞ stabilization problem is an extension of the simultaneous stabilization problem by incorporating an H ∞ performance criterion, such as in disturbance rejection. Specifically, a single controller is sought such that it simultaneously renders a set of systems being internally stable and satisfying an L2 - gain specification. In this sense, the simultaneous H ∞ stabilization problem is much more difficult than the problem of simultaneous stabilization without a performance criterion. For linear systems case, Savkin[6] gave necessary and sufficient conditions for the simultaneous H ∞ stabilization via nonlinear digital output feedback control. Miller et al.[7] considered the use of linear periodically time-varying controllers to solve the simultaneous H ∞ stabilization problem. Lee et al.[8] studied the simultaneous H ∞ stabilization problem via the chain scattering framework. Till now, few results have been published about simultaneous H ∞ stabilization nonlinear systems. In Wu[9] , a control storage function (CSF) method was developed for designing simultaneous H ∞ controller for a set of single-input nonlinear systems. Necessary and sufficient conditions for the existence of simultaneous H ∞ controller were acquired. CSF method is motivated by the control Lyapunov function (CLF) method. The concept of a
CLF was introduced by Artstein[10] , which made a tremendous impact on stabilization theory. It converted stability descriptions into tools for solving stabilization tasks. Sontag[11] presented a universal feedback scheme by using the CLF. The CLF has been widely adopted in various design problems[4, 9, 12−15] . One difficulty in applying CLFs or CSFs for solving control problems is how to construct them. It is very important to identify those nonlinear systems in which corresponding CSFs exist and can be constructed systematically. In this paper, we restrict our attention to a set of nonlinear systems in the canonical form. Necessary and sufficient conditions for the existence of common quadratic CSFs for such systems are given. A systematic method for constructing common quadratic CSFs is developed. Based on common CSFs, continuous state feedback laws are designed to simultaneously H ∞ stabilize the single-input systems and multi-input systems, respectively.
1
System description Consider a set of q nonlinear systems described by x) + Gs (x x)u uu + Hs (x x)w w ], x + B[F F s (x x˙ = Ax x) + ηs (x x)w w , s = 1, 2, · · · , q y = ξ s (x
where x ∈ Rn , u ∈ Rm , w ∈ Rν , and y ∈ Rr are the state, the input, the disturbance input and the output, respectively. We denote the s-th possible model in (1) as x) system Qs . Suppose that for every s ∈ {1, 2, · · · , q}, F s (x x) fs2 (x x) · · · fsl (x x)]T with fsi (0) = 0, i = 1, = [fs1 (x x) = (gsij (x x))l×m is of full row rank, 2, · · · , l, and Gs (x x) = (hsij (x x))l×ν , ξ s (x x) = [ξs1 (x x) ξs2 (x x) · · · and Hs (x x)]T with ξsσ (0) = 0, σ = 1, 2, · · · , r, and ηs (x x) = ξsr (x x))r×ν . The functions fsi (x x), gsij (x x), hsij (x x), ξsσ (x x), (ηsσj (x x), s = 1, 2, · · · , q; i = 1, 2, · · · , l; j = 1, 2, · · · , m; and ηsσj (x σ = 1, 2, · · · , r, are continuous. It is assumed that A and B take the following Brounovsky canonical forms: ⎡
Manuscript received January 13, 2011; revised April, 27, 2011 Supported by National Natural Science Foundation of China (61074011, 60774011) Recommended by Associate Editor LIU Yun-Gang 1. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
(1)
⎢ ⎢ A=⎢ ⎣
A1 0 .. . 0
0 A2 .. . 0
··· ··· .. . ···
0 0 .. . Al
⎤
⎡
⎢ ⎥ ⎢ ⎥ ⎥ , Ai = ⎢ ⎣ ⎦
0 .. . 0 0
1 .. . 0 0
··· .. . ··· ···
0 .. . 1 0
⎤ ⎥ ⎥ ⎥ ⎦
474
ACTA AUTOMATICA SINICA Vol. 38 ⎤ b1 0 · · · 0 0 x) > 0. E and b take the following Brounovsky assume gs (x ⎢ 0 b2 · · · 0 ⎥ ⎢ . ⎥ canonical forms, respectively: . ⎢ ⎢ ⎥ ⎥ B=⎢ . (2) .. .. ⎥ , b i = ⎢ . ⎥ .. ⎡ ⎤ ⎡ ⎤ ⎣ .. ⎣ ⎦ ⎦ . . 0 . 0 1 ··· 0 0 1 r ×1 0 0 · · · bl ⎢ . . . ⎥ ⎢ . ⎥ i . . ... ⎥ ⎢ . . ⎢ . ⎥ E=⎢ . . (8) ⎥, b = ⎢ . ⎥ ⎣ ⎦ ⎣ 0 ⎦ where r1 + · · · + rl = n. 0 0 ··· 1 The objective of this paper is to establish a common 0 0 ··· 0 1 CSF for the systems in (1), and to find a continuous state feedback law u such that it internally stabilizes the systems We present a method of constructing common CSFs for x) + Gs (x x)u u], x + B[F F s (x in (1) simultaneously (i.e., x˙ = {Ax the systems in (7). A continuous feedback law is also obs = 1, 2, · · · , q are all asymptotically stable); and moreover, tained by the CSF such that all the closed-loop systems are given γ > 0, the resultant closed-loop systems, starting internally stable and satisfy the L2 - gain requirement (3). from the initial state x (0) = 0 , satisfy 2.1 Construction of a common CSF T T Pn−1 P 12 T 2 T be a symmetric matrix where Let P = w (t)dt y (t)yy (t)dt ≤ γ w (t)w (3) PT p22 12 0 0 (n−1)×(n−1) , P 12 ∈ Rn−1 , and p22 ∈ R. Denote Pn−1 ∈ R −1 T P = [β , · · · , β p for all T > 0 and for each w (·) ∈ L2 [0, T ]. 11 1n−1 ] 12 22 Define C1β as follows: Suppose that the systems in (1) satisfy the following assumption. ⎡ ⎤ 0 1 ··· 0 Assumption 1. Given γ > 0, for all s ∈ {1, 2, · · · , q} ⎢ ⎥ .. .. .. .. x)ηs (x x) < γ 2 I, and and for any x ∈ Rn , suppose that ηsT (x ⎢ ⎥ . . . . C = (9) T T 2 T −1 T ⎢ ⎥ 1β x)ξξ s (x x) + ξ s (x x)ηs (x x)(γ I −ηs (x x)ηs (x x)) ηs (x x)ξξ s (x x) ≤ ξ s (x ⎣ 0 ⎦ 0 ··· 1 x where Γ is a positive semidefinite matrix. x T Γx −β11 −β12 · · · −β1n−1 The following definition was presented in [9]. T Definition 1. A smooth, radially unbounded and posiC1β is called companion matrix of p−1 22 P 12 , and the charactive definite function Vs (·) : Rn → R is a CSF of Qs in (1) teristic polynomial of C1β is if for any x ∈ Rn /{00} and w ∈ Rν , it holds that λ1 (β) = λn−1 + β1n−1 λn−2 + · · · + β12 λ + β11 ∂V s x) + Gs (x x)u u + Hs (x x)w w )) + x + B(F F s (x inf (Ax u x ∂x Theorem 1. Consider the set of systems in (7). Sup
x is a x) = x T Px pose that Assumption 1 holds. Then V (x (4) y Ty − γ 2w Tw < 0 common CSF for the systems in (7) if P satisfies the following conditions H1 and H2 : H1 : p22 > 0 and λ1 (β) is a Hurwitz polynomial. x) is a CSF of (1) is precisely the The condition that Vs (x T T H2 : (Pn−1 − p−1 statement that for any x = 0 , w ∈ Rν 22 P 12P 12 )C n−1 − 1β + C1β (P
In−1 −1 −1 T p22 P 12P 12 )+ In−1 − P 12 p22 Γ is neg∂Vs −1 PT −P x) = 0 ⇒ BGs (x 12 p22 x ∂x ative definite where Γ is given in Assumption 1. ∂Vs Proof. Note thatΓ is positive x) + BHs (x x)w w ) + y Ty − γ 2w Tw < 0 x + BF F s (x (Ax semidefinite; then x ∂x
In−1 −1 In−1 − P 12 p22 Γ (5) is positive semidefi−1 PT −P 12 p22 Definition 2. A CSF Vs (·) : Rn → R of Qs satisfies the nite. If conditions H1 and H2 hold, then it is easy to deduce L2 - gain small control property if for each ε > 0 there are that this P is positive definite. x < δ1 and w δ1 > 0, δ2 > 0 such that if x = 0 satisfies x x, u, w ) for Qs as Define the Hamiltonian function φs (x u < ε such w < δ2 , then there is some u with u satisfies w that x) ∂V(x x, u, w ) = x) + gs (x x)u + h s (x x)w w )) + x + b (fs (x (Ex φs (x ∂Vs x ∂x x) + Gs (x x)u u + Hs (x x)w w )) + x + B(F F s (x (Ax T 2 T x ∂x (10) y y −γ w w (6) y Ty − γ 2w Tw < 0 After some manipulations, we have: ⎡
2
⎤
⎡
Single-input systems
Consider a set of q single-input nonlinear systems as follows: x) + gs (x x)u + h s (x x)w w ], x + b [fs (x x˙ = Ex x) + ηs (x x)w w , s = 1, 2 · · · , q y = ξ s (x n
ν
r
(7)
where x ∈ R , u ∈ R, w ∈ R , y ∈ R are the state, the input, the disturbance input, and the output, respectively. fs (·): Rn → R, gs (·) : Rn → R, hs (·) : Rn → R1×ν , ξs (·) : Rn → Rr , ηs (·) : Rn → Rr×ν are continuous functions. Suppose that for every s ∈ {1, 2, · · · , q}, fs (0) = 0, ξ s (00) = x) holds the same sign. Without loss of generality, 0 , and gs (x
x, u, w ) = x T (P E + E T P )x x+ φs (x x) + gs (x x)u] + ξ T x)ξξ s (x x) + (x xT Pbbhs (x x) + xT Pbb[fs (x 2x s (x x)ηs (x x)) × (γ 2 I − ηsT (x x)ηs (x x))−1 × ξT s (x x) + ξ T x)ηs (x x))T − (xT Pbbhs (x s (x x)ηs (x x))−1 × w − (γ 2 I − ηsT (x [w x) + ξ T x)ηs (x x))T ]T × (γ 2 I − ηsT (x x)ηs (x x)) × xT Pbbhs (x (x s (x x)ηs (x x))−1 (x xT Pbbhs (x x) + ξ T x)ηs (x x))T ] w − (γ 2 I − ηsT (x [w s (x (11)
Cai Xiu-Shan et al.: Simultaneous H ∞ Stabilization for a Set of · · ·
No. 7 Let
475
In view of Assumption 1, and from (13), (15), and (16), we have:
x) = x T (P E + E T P )x x + 2x xT Pbbfs (x x) + ξ T x)ξξ s (x x) + αs (x s (x x) + ξ T x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × xT Pbbh s (x (x s (x
x)ηs (x x)(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T ≤ ξT s (x s (x
x) + ξ T x)ηs (x x))T xT Pbbh s (x (x s (x x) = 2x xT Pbbgs (x x) βs (x Since for all s ∈ {1, 2, · · · , q} and for any x ∈ Rn , it holds x) > 0, it can be deduced that that gs (x x) = 0 ⇔ x T Pbb = 0 x T Pbbgs (x
x) = αs (x x + ξsT (x x)ξξ s (x x) + x T (P E + E T P )x
(12)
n
for any x ∈ R , and for all s ∈ {1, 2, · · · , q}. Thus
x + x T Γx x= x T (P E + E T P )x −1 −1 T T XT n−1 (Pn−1 − p22 P 12P 12 )(G11 − G12 p22 P 12 ) + −1 T T T (G11 − G12 p−1 22 P 12 ) (Pn−1 − p22 P 12P 12 ) +
In−1 In−1 − P 12 p−1 X n−1 −1 22 Γ PT −P 12 p22
(17) when x T P b = 0. Since
x) = x T (P E + E T P )x x + ξT x)ξξ s (x x) + αs (x s (x x)ηs (x x)(γ 2 I − ηsT (x x) etas (x x))−1 (ξξ T x)ηs (x x))T ξT s (x s (x (13) T
x) = 0. when x Pbbgs (x Divide E, b , and x into their block forms G11 G12 0 E= , b= 0 0 1
−1 T G11 − ⎡ G12 p22 P 12 = 0 1 ⎢ .. .. ⎢ . . ⎢ ⎣ 0 0 −β11 −β12
··· .. . ··· ···
0 .. . 1
⎤ ⎥ ⎥ ⎥ = Cβ ⎦
−β1n−1
from condition H2 , there exists a positive definite matrix ϕ such that
with ⎡
⎡ ⎤ ⎤ ··· 0 0 ⎢ ⎢ . ⎥ . ⎥ .. ⎢ ⎢ . ⎥ . .. ⎥ G11 = ⎢ ⎥ , G12 = ⎢ . ⎥ ⎣ ⎣ 0 ⎦ ⎦ ··· 1 1 ··· 0 x1 · · · xn−1 . and x T = X T xn , X T n−1 = n−1 By computing, we obtain 0 .. . 0 0
1 .. . 0 0
x T Pbb = X T n−1P 12 + xn p22 and x + x T E T Px x = x T P Ex XT xn × n−1 Pn−1 G11 + GT 11 Pn−1 T PT G + G 11 12 12 Pn−1 X n−1 xn
Pn−1 G12 + GT 11P 12 T PT G + G 12 12 12P 12
×
(14) Then x T Pbb = 0 implies −1 XT xn = −X n−1 p22 P 12
(15)
Substituting (15) to (14), we obtain that x + x T E T Px x= x T P Ex −1 −1 T T XT n−1 [(Pn−1 − p22 P 12P 12 )(G11 − G12 p22 P 12 ) + −1 T T T X n−1 (G11 − G12 p−1 22 P 12 ) (Pn−1 − p22 P 12P 12 )]X (16) when x T P b = 0.
−1 T T T (Pn−1 − p−1 22 P 12P 12 )C1β + C 22 P 12P 12 ) + 1β (Pn−1 − p
I n−1 In−1 − P 12 p−1 = −ϕ −1 22 Γ PT −P 12 p22 (18) x, u, w ) ≤ αs (x x) < 0, for Then it can be deduced that φs (x x) = 0, x = 0. Thus V (x x) = x T Px x is a common x T Pbbgs (x CSF for the systems in (7). An algorithm for construction of common quadratic CSFs for the systems in (7) is as follows: Step 1. Take an arbitrary positive number p22 and a Hurwitz polynomial λ1 (β) = λn−1 + β1n−1 λn−2 + · · · + β12 λ + β11 . Step 2. Let P T 12 = p22 [β11 , · · · , β1n−1 ]. Step 3. Take a positive definite matrix ϕ ∈ R(n−1)×(n−1) , and solve the Lyapunov equation M C1β +
In−1 −1 T C1β M = − In−1 − P 12 p22 Γ − ϕ. −1 PT −P 12 p22 −1 T Step 4. Calculate Pn−1 = M + P 12 p22 P 12 . Pn−1 P 12 Step 5. Let P = . PT p22 12 x is a common CSF of the systems in x) = xT Px Then V (x (7). The necessary conditions of a common quadratic CSF of the systems in (7) are given in the following. Corollary 1. Consider the set of systems in (7). Supx is a x) = x T Px pose that Assumption 1 holds. Then V (x common CSF for the systems in (7) only if P satisfies condition H1 and the following condition: −1 T T T H3 : (Pn−1 − p−1 22 P 12P 12 )C1β + C1β (Pn−1 − p22 P 12P 12 ) is negative definite. x is a common CSF for the x) = x T Px Proof. Since V (x systems in (7), P is positive definite. Hence p22 > 0. And for any x = 0 , w ∈ Rν , it holds that
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ACTA AUTOMATICA SINICA
xT Pbbgs (x x) = 0 ⇒
Vol. 38
two functions as follows
xT P (Ex x + b fs (x x) + b hs (x x)w) + y Ty − γ 2w Tw < 0 2x x) > 0, s = After some manipulations and in view of gs (x 1, 2, · · · , q, for any x = 0 , w ∈ Rν , we have:
x + max[2x xTρ fs (x x) + ξ T x)ξξ s (x x) + x) = x T (P E + E T P )x α(x s (x s
x) + ξ T x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × xTρh s (x (x s (x x) + ξ T x)ηs (x x))T ] xTρh s (x (x s (x x) x) = 2x xT ρ min gs (x β(x s
T
x) = 0 x Pbbgs (x T
⇔ x Pbb = 0 x + ξT x)ξs (x x) − ⇒ xT (P E + E T P )x s (x x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T − (ξsT (x s (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T )T × w − (γ 2 I − ηsT (x (w s (x
(23) In order to give Theorem 2, we first prove the following Lemma1. Lemma 1. Under assumption, let P be the symmetric x), β(x x) be matrix constructed by Step 1 to Step 5 and α(x x ≤ δ}. x ∈ Rn |0 < x defined as (23). Denote N (δ) = {x Then it implies
x)ηs (x x)) × (γ 2 I − ηsT (x 2
w − (γ I − (w
x)ηs (x x))−1 (ξξ T x)ηs (x x))T ) ηsT (x s (x
lim sup
δ→0 x ∈N (δ)
<0
Then for any x = 0 , w ∈ Rν , it holds that
(19)
x T Pbb = 0 ⇒ x + ξT x)ξξ s (x x) + x T (P E + E T P )x s (x x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T < 0 (ξξ T s (x s (x (20) It can be seen from (19) that if (20) does not hold, we x, u, w ) < 0 in the presence of cannot find u such that φs (x x)ηs (x x))−1 × the worst case disturbance w = (γ 2 I − ηsT (x T T x x x x (x )η (x )) . Thus V (x ) = x Px is a CSF of the systems (ξξ T s s in (7) only if (20) holds. By Assumption 1, given γ > 0, for all s ∈ {1, 2, · · · , q}, x)ηs (x x) < γ 2 I. In view of and for any x ∈ Rn , holds ηsT (x x 0 (20), we deduce that for any = , x T Pbb = 0 ⇒ x T (P E + E T P ) x < 0
(21)
From (15) and (16), we have: x + x T E T Px x= x T P Ex −1 T −1 T XT n−1 [(Pn−1 − p22 P 12P 12 )(G11 − G12 p22 P 12 ) + −1 T T T X n−1 (G11 − G12 p−1 22 P 12 ) (Pn−1 − p22 P 12 P12 )]X
(22) for x T Pbb = 0. By (21) and (22), then H3 is satisfied. Pn−1 P 12 On the other hand, P = is positive defiPT p22 12 −1 T nite, then Pn−1 −p22 P 12P 12 is positive definite too. Hence, T the Lyapunov theorem shows that G11 − G12 p−1 22 P 12 is sta−1 T ble, i.e., G11 − G12 p22 P 12 is a Hurwitz matrix. H1 is also satisfied. 2.2
Simultaneous H ∞ stabilization
This subsection will present a continuous feedback control law such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3) by using the common CSF established in Subsection 3.1. Let P be the symmetric matrix constructed by Step 1 to Step 5 and ρ be the last column of P . Then Pbb = ρ . Define
x) α(x ≤0 x |β(x )|
(24)
Proof. Let δ → 0. For any a sequence {ζζ n } such x), that {ζζ n } ∈ N (δ), because for all s ∈ {1, 2, · · · , q}, fs (x x), hs (x x), ξ s (x x), ηs (x x) are continuous with fs (0) = 0, gs (x x) > 0 for any x ∈ Rn , and P is ξ s (00) = 0 , and gs (x the symmetric matrix constructed by Step 1 to Step 5 x), β(x x) are defined as (23), we can deduce that and α(x α(ζζ n ) = o(β(ζζ n )), if β(ζζ n ) = 0, and δ → 0; on the other hand, if β(ζζ n ) = 0, from (15), (17), and (18), XT it can be deduced that α(ζζ n ) = −X n−1 ϕXn−1 , with −1 T T T X n−1 p22 P 12 and ζn = [Xn−1 , xn ], so α(ζζ n ) = xn = −X X n−1 < 0, if β(ζζ n ) = 0, and ζ n = 0. In conXT −X n−1 ϕX x) α(x clusion, we have: limδ→0 supx ∈N (δ) |β(x x)| ≤ 0. The limit may well be −∞. Theorem 2. Suppose that the systems in (7) satisfy Assumption 1. If there is some matrix P satisfying conditions H1 and H2 , then there exists a continuous control law such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3). Proof. Constructing matrix P by Step 1 to Step 5; then x is a common CSF of the q systems in (7) by x) = x T Px V (x Theorem 1. The state feedback law is taken as ⎧ x))2 + (β(x x))4 x) + (α(x α(x ⎨ , x Tρ = 0 − x) = u(x x β(x ) ⎩ 0, x Tρ = 0 (25) x), β(x x) are defined as (23). where α(x x) satisfies the L2 - gain small Since (24) implies that V (x control property by [9], the feedback control law u is a continuous function on Rn . x, u, w ) ≤ α(x x) + 2x xT ρgs (x x)u, if we can show Due to φs (x that x)u < 0 x) + 2x xT ρgs (x (26) α(x for any x = 0 and for all s ∈ {1, 2, · · · , q}, then all the closed-loop systems satisfy the requirement (3). In fact, if (26) holds, and in view of (10), then given γ > 0, we have: x, u, w ) = φs (x
x) ∂V (x x) + gs (x x)u + h s (x x)w w ]) + x + b[fs (x (Ex x ∂x y Ty − γ 2w Tw < 0
(27)
for any x = 0 , for all s ∈ {1, 2, · · · , q}, and for each w (·) ∈ L2 [0, T ]. For all T > 0, starting from the initial state x (0) =
Cai Xiu-Shan et al.: Simultaneous H ∞ Stabilization for a Set of · · ·
No. 7
0, integrating both sides of (27), we have: x(T )) − V (x x(0)) + V (x
T
y T (t)yy (t)dt − γ 2
0
T
w (t)dt < 0 w T (t)w
0
Consider the systems in (1). Divide Ai and Bi into their block forms as follows: Ai−1 A i2 0 Ai = , Bi = 1 0 0 with ⎡
x is a positive function and x (0) = 0 , we x) = x T Px Since V (x x(T )) ≥ 0, V (x x(0)) = 0 . So have V (x
T
y T (t)yy (t)dt < γ 2
0
T
w (t)dt w T (t)w
0
for all T > 0 and for each w (·) ∈ L2 [0, T ], that is, all the closed-loop systems satisfy the requirement (3). Now, we verify that (26) holds in the following three cases. Case 1. x Tρ > 0 x) > 0, which implies that u(x x) < If x Tρ > 0, then β(x xT ρgs (x x) ≥ β(x x). Therefore, 0. From (23), we have 2x x) − β(x x))u(x x) ≤ 0. It leads to xTρ gs (x (2x x) + 2x xT ρgs (x x)u(x x) ≤ α(x x) + β(x x)u(x x) ≤ − α(x
x) + β 4 (x x) < 0 α2 (x
Case 2. x Tρ = 0, x = 0 If x Tρ = 0, x = 0 and by (20), then we have: x) + 2x xTρ gs (x x)u = α(x x + ξT x)ξξ s (x x) + max x T (P E + E T P )x s (x s
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × (ξξ T s (x x)ηs (x x))T < 0 (ξξ T s (x Case 3. x Tρ < 0 x) < 0 By a similar discussion to Case 1, we can obtain β(x xTρ gs (x x) − β(x x))u ≤ 0. Thus and u > 0. We also have (2x x)u ≤ x) + 2x xTρ gs (x α(x x) + β(x x)u ≤ − α(x
x) + β 4 (x x) < 0 α2 (x
From the previous discussions, we know that the feedback control law (25) renders all the closed-loop systems satisfying the L2 - gain requirement (3). The following proves internal stability. Letting w = 0 and u in (25) and noting (26) yields x, u, 0) ≤ α(x x) + 2x xT P bgs (x x)u < 0 φs (x for any x = 0, and for all s ∈ {1, 2, · · · , q}. Therefore, all the closed-loop systems are internally stable.
3
477
Multi-input systems
This section turns to consider the muti-input case. A common CSF is established for the q systems in (1). Then a continuous feedback law is obtained by the CSF such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3).
⎢ ⎢ Ai−1 = ⎢ ⎣
0 .. . 0 0
1 .. . 0 0
··· .. . ··· ···
⎡
⎤
0 .. . 1 0
⎢ ⎢ A i2 = ⎢ ⎣
⎥ ⎥ ⎥, ⎦
0 .. . 0 1
⎤ ⎥ ⎥ ⎥ ⎦
Assume that βi1 , βi2 , · · · , βi,ri−1 are the coefficients of the following polynomial: λi (β) = λri −1 + βi,ri −1 λri −2 + · · · + βi2 λ + βi1 Denote Λ= ⎡ Ir1 −1 ⎢ ⎢ ⎢ ⎣
P 12 p−1 −P 13
(28)
⎤ ..
.
Γ1 = ΛΓΛT
Irl −1
P l2 p−1 −P l3
⎥ ⎥ ⎥ ⎦
Since Γ is positive semidefinite, Γ1 is positive semidefinite too. Assume that λmax (Γ1 ) is the maximum eigenvalue of Γ1 . Piri −1 P i2 Let Pi = , i = 1, 2, · · · , l be symmetric T P i2 pi3 (ri −1)×(ri −1) , P i2 ∈ Rri −1 , and matrices, where Piri −1 ∈ R pi3 ∈ R. It is required that T βi1 βi2 · · · βi,ri −1 p−1 i3 P i2 = Then T Ai−1 − A i2 p−1 i3 Pi2 = Ciβ =
⎡ ⎢ ⎢ ⎢ ⎣
0 .. . 0 −βi1
1 .. . 0 −βi2
··· .. . ··· ···
0 .. . 1
⎤ ⎥ ⎥ ⎥ ⎦
−βi,ri −1
Let Pi , i = 1, 2, · · · , l, satisfy the following conditions: H4 : pi3 > 0 and λi (β) is a Hurwitz polynomial. T T −1 T H5 : (Piri −1 −p−1 i3 P i2P i2 )Ciβ +Ciβ (Piri −1 −pi3 P i2P i2 ) + λmax (Γ1 )Iri −1 is negative definite. Denote ⎤ ⎡ P1 ⎥ ⎢ P2 ⎥ ⎢ x x) = x T Px (29) P =⎢ ⎥ , V (x . .. ⎦ ⎣ Pl Theorem 3. Consider the set of systems in (1). Suppose x) = x T Px x is a common that Assumption 1 holds. Then V (x CSF for the systems in (1) if P satisfies conditions H4 and H5 . Proof. If Pi , i = 1, 2, · · · , l, satisfies conditions H4 and H5 , then it is easy to deduce that Pi is positive definite. It x is positive definite. x) = x T Px follows that V (x
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s (x x, u, w ) for Qs in (1) Define the Hamiltonian function φ as x) ∂V (x s (x x, u , w ) = x)+Gs (x x)u u +Hs (x x)w w ]) + x +B[F F s (x φ (Ax x ∂x T
2
T
y y −γ w w
(30)
After some manipulations, we have:
T T X i,ri −1 (Piri −1 − p−1 Ciβ i3 P i2P i2 )]X
x)ηs (x x))−1 (x xT P BHs (x x) ηsT (x
w − (γ I − (w
when x T i Pi Bi = 0. In view of Assumption 1, and from (33) and (37), we have:
x)ηs (x x))−1 ηsT (x
x)ηs (x x))T + ξT s (x
−
⎡
×
x)ηs (x x))(w w − (γ 2 I − ηsT (x x)ηs (x x))−1 × (γ 2 I − ηsT (x x) + ξ T x)ηs (x x))T ) xT P BHs (x (x s (x T
T T XT 1,r1 −1 , X 2,r2 −1 , · · · , X l,rl −1
l
(31)
Let T
−1 T XT i,ri −1 [(Piri −1 − pi3 P i2P i2 )Ciβ +
T T X i,ri −1 + Ciβ (Piri −1 − p−1 i3 P i2P i2 )]X
x) + ξ T x)ηs (x x))T )T × xT P BHs (x (x s (x
⎢ ⎢ Γ1 ⎢ ⎣
X 1,r1 −1 X 2,r2 −1 .. . X l,rl −1
x)ηs (x x))−1 (x xT P BHs (x x) + ξ T x)ηs (x x))T (γ 2 I − ηsT (x s (x x) = 2x xT P BGs (x x) βs (x x) Since for each x ∈ Rn , and for all s ∈ {1, 2, · · · , q}, Gs (x is of full row rank, we have that x) = 0 ⇔ x T P B = 0 x T P BGs (x
(32)
Thus x) = x T (P A + AT P )x x + ξT x)ξξ s (x x) + α s (x s (x x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T (ξξ T s (x s (x (33) x) = 0. when x T P BGs (x Use block matrix to express x T , that is T T xT = xT , xT · · · xT xi,ri , i = Xi,ri −1 1 x 2 l xi1 xi2 · · · xi,ri −1 , i = 1, 2, · · · , l XT i,ri −1 = Then xT 1 P1 B1
xT 2 P2 B2
and xT i Pi Bi =
XT i,ri −1
xi,ri
···
xT l Pl Bl
Piri −1
P i2
PT i2
pi3
XT i,ri −1P i2 + pi3 xi,ri ,
⎥ ⎥ ⎥≤ ⎦
i=1 T T X i,ri −1 Ciβ (Piri −1 − p−1 i3 P i2P i2 ) + λmax (Γ1 )Iri −1 ]X
T
x)ξξ s (x x) + (x xT P BHs (x x) + ξ T x)ηs (x x)) × ξT s (x s (x
⎤
−1 T XT i,ri −1 [(Piri −1 − pi3 P i2P i2 )Ciβ +
x) = x (P A + A P )x x + 2x x P BF F s (x x) + α s (x
xTP B =
(37)
i=1
x)ξξ s (x x) + (x xT P BHs (x x) + ξ T x)ηs (x x)) × ξT s (x s (x
2
T −1 T xT 2x i Pi Aix i = X i,ri −1 [(Piri −1 − pi3 P i2P i2 )Ciβ +
l
x + 2x xT P B[F F s (x x) + Gs (x x)u u] + x T (P A + AT P )x
(γ I −
0 at least. Moreover, we have: and there exists an xi =
x) ≤ x T (P A + AT P )x x + x T Γx x= α s (x
s (x x, u , w ) = φ
2
Vol. 38
0 1
when x T P B = 0. By condition H5 , it can be deduced that s (x x) ≤ α x) < 0, when xT P BGs (x x) = 0, x = 0. Thus s (x φ x) = x T Px x is a common CSF for the systems in (1). V (x In the following, we give the necessary conditions of a common quadratic CSF of the systems in (1). Corollary 2. Consider the set of systems in (1). Under x defined by (29) is a common x) = xT Px assumption, V (x CSF for the systems in (1) if and only if P satisfies condition H4 and the following condition: T H6 : for each i ∈ {1, 2, · · · , l}, (Piri −1 − p−1 i3 P i2P i2 )Ciβ T −1 T + Ciβ (Piri −1 − pi3 P i2P i2 ) is negative definite. Proof. Using the arguments as Corollary 1, we can deduce it. x) are Suppose that for all s ∈ {1, 2, · · · , q}, matrices Gs (x described by x) = G(x x)Ds (x x) (39) Gs (x x) is an l×m matrix, has full row rank, and Ds (x x) where G(x x), ds2 (x x), · · · , dsm (x x)}. It is further supposed = diag{ds1 (x x) are continuous with x) and dsj (x that the elements of G(x x) = 0 dsj (x
for all s ∈ {1, 2, · · · , q}, j ∈ {1, 2, · · · , m}, x ∈ R . Moreover, we assume for each j, j ∈ {1, 2, · · · , m}, the q funcx), s = 1, 2, · · · , q hold the same sign. We now tions dsj (x define a matrix which is independent of s as follows:
i = 1, 2, · · · , l
x), ν2 (x x), · · · , νm (x x)} x) = diag{ν1 (x Ω(x where
=
x) = νj (x
x), min dsj (x
x) > 0 if dsj (x
x), max dsj (x
x) < 0 if dsj (x
s s
(35)
From (34) and (35), when x T P B = 0, x = 0, we have: T xT i Pi Bi = X i,ri −1P i2 + pi3 xi,ri = 0, i = 1, 2, · · · , l (36)
(40) n
(34)
(38)
(41)
(42)
x)dsj (x x) − νj2 (x x) ≥ 0 for j = By (42), we have νj (x 1, 2, · · · , m, and s = 1, 2, · · · , q, or equivalently, the diagonal matrices x)Ω(x x) − Ω2 (x x), s = 1, 2, · · · , q Ds (x
(43)
Cai Xiu-Shan et al.: Simultaneous H ∞ Stabilization for a Set of · · ·
No. 7
479
are positive semidefinite. Theorem 4. Consider the systems in (1). If under x) satisfy (39) and (40), Assumption 1 and matrices Gs (x and there is some matrix P satisfying conditions H4 and H5 , then there exists a continuous control law such that all the closed-loop systems are internally stable and satisfy the L2 - gain requirement (3). x) = x T Px x defined by (29) is Proof. By Theorem 3, V (x a common CSF for the q systems in (1). Denote
Consider (46) now. From (45) and (48), we have:
x+ x) = x T (P A + AT P )x α (x F s (x x)+ ξ T x)ξξ s (x x)+ (x xT P BHs (x x) + xT P BF max 2x s (x
when x T P B = 0. From the previous discussions, we know that the feedback control law (45) renders all the closed-loop systems satisfying the L2 - gain requirement (3). The following proves internal stability. Letting w = 0 and u in (45) and noting (46), (47), and (49), yields
s
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × ξT s (x x) + ξ T x)ηs (x x))T xT P BHs (x (x s (x
x, u , 0) ≤ x T (P A + AT P )x x+ φs (x
x) = 2Ω(x x)B T Px x x)GT (x β(x
s (x x, u , w ) ≤ α x) + 2x xT P BGs (x x)u u≤ s (x φ x)u u x) + 2x xT P BGs (x α (x
(46)
T
x)u u< x)+2x x P BGs (x Now let us consider (45), and prove α (x 0, x = 0 by in following two cases. x) = 0, x = 0 Case 1. β(x x) = 0 is equivx) and G(x x), β(x By the assumptions on Ω(x alent to x T P B = 0. From (38), (44) and (46), and in view of Assumption 1, we have: s (x x, u , w ) ≤ α x) = (x φ x + max ξ T x)ξξ s (x x) + x T (P A + AT P )x s (x s
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 (ξξ T x)ηs (x x))T ≤ (ξξ T s (x s (x x + x T Γx x<0 x T (P A + AT P )x
(47)
when x T P B = 0, x = 0 . x) = 0 Case 2. β(x By (43), we obtain that
2
T
T x) − βT (x x) × x)β(x x)β(x x P BGs (x x)2 x))2 + (βT (x x)β(x ( α(x x) x)β(x βT (x
for any x = 0, and for all s ∈ {1, 2, · · · , q}. Therefore, all the closed-loop systems are internally stable.
4
Example
Example 1. Consider the following set of 3 singleinput nonlinear systems: ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎨ x˙ = x 2 3 (50) Qs : ⎪ x) + gs (x x)u u + hs (x x)w w x˙ 3 = fs (x ⎪ ⎪ ⎩ x) + ηs (x x)w w , s = 1, 2, 3 y = ξs (x where x = [x1 x2 , x3 ]T , and x) = x1 cosx2 f1 (x
x) = x21 f2 (x
x) = x1 + x3 f3 (x x) = 2 + sin2 x2 g2 (x x) = 1 − cosx2 h1 (x x) = x1 − x3 h3 (x x) = x1 + x2 sinx3 ξ2 (x x) = 1 η1 (x x) = 1 + sinx1 η3 (x
x) = e(x1 +x2 ) g1 (x x) = 1 + x23 g3 (x x) = x2 x3 sinx1 h2 (x x) = x2 + x3 ξ1 (x x) = x1 − x2 ξ3 (x x) = 1 − cosx2 η2 (x
x)ξs (x x) ≤ x T Γx x ηsT (x
T
x)[Ds (x x)Ω(x x) − Ω (x x)]G (x x)B Px x≥0 x P BG(x
x)u u<0 x) + 2x xT P BGs (x α (x
x)ξs (x x) + ξsT (x x)ηs (x x)(γ 2 I − ηsT (x x)ηs (x x))−1 × ξsT (x
x) − βT (x x) = x)β(x x)β(x x P BGs (x T
x)ηs (x x))(γ 2 I − ηsT (x x)ηs (x x))−1 × (ξξ T x)ηs (x x))T ≤ (ξξ T s (x s (x
Let γ = 3. It can be shown that
T
x) − − α(x
F s (x x) + Gs (x x)u u] + ξ T x)ξξ s (x x) + xT P B[F 2x s (x
(44)
The feedback law is taken as ⎧ x))2 ⎪ x))2 +(βT (x x)β(x x)+ ( α(x α (x ⎨ x) , xTP B = 0 u = −β(x T (x x x ) β(x ) β ⎪ ⎩ 0, xTP B = 0 (45) By a similar proof to that of Theorem 2, u is continuous on Rn . The detail is omitted. From (30), we have:
Thus
s (x x, u , w ) ≤ α x) + 2x xT P BGs (x x)u u≤ (x φ x))2 x))2 +(βT (x x)β(x x) − ( α(x − α(x x) x)β(x x)+ βT (x ≤ α (x x) x)β(x βT (x x))2 < 0 x))2 + (βT (x x)β(x − ( α(x (49)
x)ηs (x x) < 9, s = 1, 2, 3. with Γ = diag{4, 4, 2}, and ηsT (x By Theorem 1, there exists a common CSF for the systems in (50). Let p22 = 1 and λ1 (β) = λ2 + λ + 1. Then PT 12 = [1 1]
≤0
(48)
I2
− P 12 p−1 22 Γ
I2
−1 PT −P 12 p22
=
6 2
2 6
480
ACTA AUTOMATICA SINICA
Choose ϕ =
−1 0
0 −1
Vol. 38
. Thus ⎡
9.5 x) = x T ⎣ 4.5 V (x 1
4.5 8 1
⎤ 1 1 ⎦x 1
is a common CSF for the systems in (50). Define ⎤ ⎡ 0 9.5 4.5 T⎣ x) = x 9.5 9 9 ⎦x + α(x 4.5 9 2 x) + ξsT (x x)ξs (x x) + max[2(x1 + x2 + x3 )fs (x s
x) + ξsT (x x)ηs (x x)) × ((x1 + x2 + x3 )hs (x x)ηs (x x))−1 × (9 − ηsT (x
Fig. 2 State trajectories, control law, output, and disturbance input of system Q2 (The initial state is [2, 1, 1]T , and w(t) = x1 sin(30t) + x22 + 2x3 .)
x) + ξsT (x x)ηs (x x))T ] ((x1 + x2 + x3 )hs (x x) = 2(x1 + x2 + x3 ) min gs (x x) β(x s
By Theorem 2, the feedback law ⎧ x) + α2 (x x) + β 4 (x x) α(x ⎨ − , u= x β(x ) ⎩ 0,
if x1 + x2 + x3 = 0
if x1 + x2 + x3 = 0 (51) is such that all the closed-loop systems in (50) and (51) are internally stable and satisfy the L2 - gain requirement (3). Figs. 1 ∼ 3 show the state trajectories, the control law, the output and the disturbance input of the 3 systems in (50) with the state feedback law (51), respectively.
5
Conclusion
This paper presents a novel method for constructing common quadratic CSFs for a set of nonlinear systems in canonical form. An algorithm is given to obtain common quadratic CSFs for such systems. Then continuous state feedback laws are designed to simultaneously H ∞ stabilize the single-input systems and multi-input systems, respectively. Finally, the effectiveness of the proposed schemes is illustrated by a simulation example.
Fig. 3 State trajectories, control law, output, and disturbance input of system Q3 (The initial state is [2, −1, 2]T , and w(t) = x21 e−t − x2 cos(10t) − x3 sin(20t).)
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