- Email: [email protected]
Delay-dependent robust passivity control for uncertain time-delay systems
Journal of Systems Engineering and Electronics Vol. 18, No. 4, 2007, pp.879–884
Delay-dependent robust passivity control for uncertain time-delay systems Li Guifang1 , Li Huiying2 & Yang Chengwu2 1. Flight Coll. of Nanjing Univ. of Aeronautics and Astronautics, Nanjing 210006, P. R. China; 2. 810 Division, School of Power Engineering, Nanjing Univ. of Science and Technology, Nanjing 210094, P. R. China; (Received August 18, 2006)
Abstract: The robust passivity control problem is addressed for a class of uncertain delayed systems with timevarying delay. The parameter uncertainties are norm-bounded. First, the delay-dependent stability sufficient condition is obtained for the nominal system, and then, based-on the former, the delay-dependent robust passivity criteria is provided and the corresponding controller is designed in terms of linear matrix inequalities. Finally, a numerical example is given to demonstrate the validity of the proposed approach.
Keywords: passive control, time-varying delay, delay-dependent, parameter uncertainties, linear matrix inequality (LMI).
1. Introduction The passivity theory has its roots in the circuit theory and plays an important role in both electrical network and control systems. The fact that passivity and stability are closely related supplies a new approach to solve the problem of stabilization[1,2] . Moreover, passivity-based control has important robustness properties[3] . In addition, the passivity theory also provides an energy-based analysis approach, which is easier to perform when dealing with electrical circuits and mechanical systems[4] . On the other hand, time delay and parameter uncertainties often cause instability and bad performance in various engineering systems. The existing results for time-delay systems can be classified into two types; that is: delay-independent case and delaydependent case[5−12] . Generally, the latter are less conservative. To the bestx‘ of the authors’ knowledge, delay-dependent robust passivity control for uncertain time-delay systems has not been fully investigated. In this article, initially, the delay-dependent stability condition for a system with time-varying delay is obtained in terms of linear matrix inequality (LMI). Following which, based on the derived delay-
dependent stability condition, a delay-dependent robust passivity condition is proposed for delayed systems with norm-bounded uncertainties.
2. Problem statement Consider the following uncertain time-delayed system Σ∆ x(t) ˙ = [A + ∆A]x(t) + Ah x(t − h(t))+
(1a) [B + ∆B]u(t) + Bw w(t) x(t) = φ(t), t ∈ [−h(t), 0] (1b) z(t) = Cx(t) + Dw(t) (1c) n m Where x(t) ∈ R is the state, u(t) ∈ R is the control input, w(t) ∈ Rp is the disturbance input that belongs to L2 [0, ∞), z(t) ∈ Rp is the output, and φ(t) is the initial condition.A, Ah , B, Bw , C and D are the known, real constant matrices, with appropriate dimensions, which describe the nominal Σ∆ system, and ∆A and ∆B are the real matrix functions representing the time-varying parameter uncertainties. The admissible uncertainties are assumed by [∆A
∆B ] = E ∆(t) [Fa
Fb ]
(2)
where ∆(t) ∈ Rna ×nb is an unknown real time-varying matrix with the Lebesgue measurable element satisfy-
880
Li Guifang, Li Huiying & Yang Chengwu
ing
3.1 ∆T (t)∆(t) I
(3)
The unforced nominal system of system Σ∆ is Σ0
and E, Fa and Fb are the known, real constant matrices that characterize how uncertain parameters enter the nominal systems. And h(t) is a time-varying bounded delay satisfying ˙ ¯<1 0 h(t) h < +∞, h(t) h
(4)
The aim of this article is to design a controller u(t) = Kx(t), such that, the closed-loop systems are robustly strong and stable, with strict passivity for all admissible uncertainties, that is (1) The closed-loop systems are asymptotically stable for any delay that h(t) satisfies (4); (2) Under the zero initial condition, there exists a scalar α > 0, such that T T T 2 w(t) z(t)dt α w(t)T w(t)dt 0
x(t) ˙ = Ax(t) + Ah x(t − h(t)) + Bw w(t) z(t) = Cx(t) + Dw(t)
0
0
−h(t) 0
h(t)
ζ(s)ds
S
−h(t)
−h(t)
Lemma 2 For any real matrices Σ1 , Σ2 and Σ3 with appropriate dimensions, such that Σ3 = Σ3T , it follows that Σ3 + Σ1 F (t)Σ2 + Σ2T F (t)T Σ1T < 0 T
F (t) F (t) I
(6)
If and only if there exists a scalar ε > 0 Σ3 +
εΣ1 Σ1T
+ε
−1
(9) where A¯ = (A + Ah ), and ∗ means symmetrical elements. Proof Choose the candidate Lyapunov functional as follows t V (xt ) = xT P x + x(s)T Qx(s)ds+
Σ2T Σ2
<0
(7)
3. Main results First, the new delay-dependent stability condition is given, under which the unforced nominal system of system Σ∆ is asymptotically stable.
0
−h(t)
(5)
ζ(s) Sζ(s)ds
(8b)
<0
ζ(s)ds
T
(8a)
Theorem 1 Consider the unforced nominal system Σ0 with w(t) ≡ 0. Then for the given positive scalars h, ¯h and α, the system Σ0 is asymptotically stable for any time-varying delay h(t) satisfying (4), if there exist symmetric positive-definite matrices P, Q and R such that the following LMI holds. ⎡ ⎤ 0 −P Ah hAT R P A¯ + A¯T P + Q ⎢ ⎥ ⎢ ⎥ ¯ R ∗ −(1 − h)Q 0 hAT ⎥ ⎢ h ⎥ ⎢ ¯ ⎥ ⎢ 1−h Θ= ⎢ R 0 ⎥ ∗ ∗ − ⎦ ⎣ h ∗ ∗ ∗ −hR
0
Which holds for all T > 0 and w(t) ∈ L2 [0, T ]. Before proceeding to the main results, the authors introduce the following two important lemmas. Lemma 1[5] If for any positive-definite matrix 0 < S = S T ∈ Rr×r and function h(t) > 0, the vector function ζ : [−h(t), 0] → Rr , such that, the concerned integration is well defined, then T
Delay-dependent stability
t−h(t)
t
t+α
(10)
T
x˙ (s)Rx(s)dsdα ˙
where P, Q and R are positive-definite matrices to be chosen. Then, by the Newton-Leibniz formula, it follows that t
x(t − h(t)) = x(t) − and
t−h(t)
¯ x(t) ˙ = Ax(t) − Ah
x(s)ds ˙
(11)
t
t−h(t)
x(s)ds ˙
(12)
Differencing V (xt ) along with the trajectory of Eq. (8a) gives T ˙ V˙ (xt ) = 2xT P x˙ + x(t)T Qx(t) − (1 − h(t))x h (t) Q×
˙ T Rx(t)− ˙ xh (t) + h(t)x(t) t ˙ (1 − h(t)) x(s) ˙ T Rx(s)ds ˙ t−h(t)
Delay-dependent robust passivity control for uncertain time-delay systems
T
¯ − Ah 2x P Ax(t)
t t−h(t)
T
⎡
x(s)ds ˙ +
T
¯ h (t) Qxh (t)+ x(t) Qx(t) − (1 − h)x t ¯ ˙ − (1 − h) x(s) ˙ T Rx(s)ds ˙ hx(t) ˙ T Rx(t)
⎢ ⎢ ⎢ ⎢ ⎢ Θ1 = ⎢ ⎢ ⎢ ⎢ ⎣
(13)
t−h(t)
P A¯ + A¯T P + Q
0
−P Ah
∗
¯ −(1 − h)Q
∗
∗
∗
∗
0 ¯ 1−h R − h ∗
∗ P Bw − C
∗ T
T
hA R
⎤
∗
⎥ ⎥ hAT hR ⎥ ⎥ ⎥<0 0 0 ⎥ ⎥ T R ⎥ αI − (D + D T ) hBw ⎦ ∗ −hR
By Lemma 1, it follows that
0
V˙ (xt ) xT (P A¯ + A¯T P + Q + hAT RA)x− t T 2x P Ah x(s)ds ˙ + hxh (t)T AT h RAh xh (t)+ t−h(t)
(16)
¯ h (t)T Qxh (t)− 2hx(t)T AT RAh xh (t) − (1 − h)x
T t t ¯ 1−h x(s)ds ˙ R x(s)ds ˙ h t−h(t) t−h(t) (14)
T
881
T
Denoting X(t)= x(t) xh (t)
T T
t
t−h(t)
,
x(s)ds ˙
by the Schur complement and applying (9) it follows that, V˙ (xt ) X(t)T ΘX(t) < 0
(15)
Proof By considering Theorem 1, for the stability of the unforced system Σ0 with w(t) ≡ 0, any time delay h(t) satisfying (4) follows from inequality (9). Moreover, LMI(16) implies LMI(9), hence the system Σ0 is asymptotically stable. To establish the strict passivity condition for system Σ0 , assume zero initial conditions and choose the same Lyapunov functional of form (10), following which the trajectory of Eq.(8a) satisfies t x(t) ˙ = (A + Ah )x(t) − Ah x(s)ds ˙ + Bw w(t) t−h(t)
when x(t) and xh (t) = x(t − h(t)) are not zero. This completes the proof. Remark 1 In fact, Theorem 1 is a delay dependent and delay-derivative dependent stability condition for system Σ0 . If the delay is constant, that is h(t) = d, then it reduces to a delay-dependent condition. 3.2
Delay-dependent passivity
In the sequel, based on Theorem 1, the authors focus on the problem of the strong stability and strict passivity of system Σ0 . ¯ Theorem 2 For the given positive scalars h, h and α, system Σ0 is strongly stable with strictly passivity for any delay h(t) satisfying (4), if there exist positive definite matrices P, Q and R, such that the following LMI holds.
(17) Then using the same technique as in the proof of Theorem 1, it can be easily shown that the time derivative of V (xt ) along with the solution of Eq.(8a) satisfies ¯ A¯T P +Q+hATRA)x−2xT P Ah × V˙ (xt ) xT (P A+ t T T x(s)ds+hx ˙ h (t) Ah RAh xh (t)+ t−h(t)
T 2hx(t)T AT RAh xh (t)+hw(t)T Bw RBw w(t)+
2hx(t)T AT RBw w(t)+2x(t)T P Bw w(t)+ T ¯ 2hxh (t)T AT h RBw w(t)−(1− h)xh (t) Qxh (t)−
T t t 1− ¯h x(s)ds ˙ R x(s)ds ˙ h t−h(t) t−h(t) (18) Next, the authors consider the following performance index t J(xt ) = [αw(t)T w(t)−2w(t)T z(t)]dt+V (xt ) (19) 0
882
Li Guifang, Li Huiying & Yang Chengwu ⎡
Then, using Eq.(8b) in J(xt ) yields ˙ t ) = V˙ (xt ) + αw(t)T w(t) − 2w(t)T z(t) = J(x V˙ (xt ) − 2w(t)T Cx(t)+
(20)
w(t)T [αI − (D + D T )]w(t) Setting
x(t)T xh (t)T
Y (t) =
t
t−h(t)
T
T x(s)ds ˙
w(t)T
and after some manipulations, the following is obtained ˙ t ) Y (t)T × J(x ⎡ ¯ A¯T P +Q+hATRA hAT RAh P A+ ⎢ ⎢ T ¯ ∗ −(1 − h)Q+hA ⎢ h RAh ⎢ ⎢ ∗ ∗ ⎣ ∗ ∗ ⎤ T T P Bw + hA RBw − C −P Ah ⎥ ⎥ 0 hAT ⎥ h RBw ⎥ Y (t) ⎥ 1−¯ h ⎥ R 0 − ⎦ h T ∗ hBw RBw + αI − (D + D T ) (21) By the Schur complement, (21) can be rewritten as ˙ t ) Y (t)T Θ1 Y (t) J(x
(22)
˙ t ) < 0, which imUsing LMI (16), it follows that J(x T T plies 2 w(t)T z(t)dt α w(t)T w(t)dt. That is, 0
0
system Σ0 is strongly stable with strict passivity. The proof is completed. 3.3
Robust passivity control
The authors are now in the position to design a controller which renders the resulting closed-loop systems robustly strongly stable with strictly passive for all admissible uncertainties. ¯ and Theorem 3 For given positive scalars h, h α, the uncertain delayed systems Σ∆ are robustly strongly stable with strictly passive for any delay h(t) satisfying (4), if there exist a scalar γ > 0, and positive definite matrices X, N, Z > 0 and matrix W such that the following LMI holds.
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Ξ
0
−Ah Z
Bw − XC T
∗
¯ −(1 − h)N
0
∗
∗
∗
∗
0 ¯ (1 − h) Z − h ∗
αI − (D + D T )
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
h(XAT + W T B T )
γE
XHaT + W T HbT
hXAT h
0
0
0
0
0
T hBw
0
0
−hZ
γhE
0
∗
−γI
0
∗
∗
−γI
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(23) ¯ + X A¯T + BW + W T B T + N where Ξ = AX Moreover, the robust passivity controller is given by u(t) = W X −1 x(t). Proof Substituting K) + (∆A + ∆B · K) ing and postmultiplying diag P −1 P −1 R−1
matrix A with (A + B · in LMI(16), Premultiplythe resultant matrix with
−1 , and combining I R
it with the denotes X = P −1 , N = P −1 QP, W = KP −1 and Z = R−1 , the following LMI is obtained. ⎡ Ξ∆ 0 −Ah Z Bw − XC T ⎢ ⎢ ∗ −(1 − h)N ¯ 0 0 ⎢ ⎢ ⎢ (1 − ¯h) ⎢ ∗ Z 0 ∗ − ⎢ h ⎢ ⎢ ∗ ∗ ∗ αI − (D + D T ) ⎣ ∗
∗
∗ T
T
T
hX(A + ∆A) + hW (B + ∆B) hXAT h 0 T hBw
⎤
∗
⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎦
−hZ (24) where T Ξ ∆ = Ah X + XAT h + (A + ∆A)X + X(A + ∆A) +
(B + ∆B)W + W T (B + ∆B)T + N
Delay-dependent robust passivity control for uncertain time-delay systems Together with Lemma 2, LMI (24) is equivalent to the following LMI. ⎡
layed systems ⎡ A=⎣
⎤
E ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Π +⎢ ⎢ 0 ⎥ ∆ Ha X + H b W 0 0 0 0 + ⎥ ⎢ ⎢ 0 ⎥ ⎦ ⎣ hE ⎡ ⎤ XHaT + W T HbT ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎥ T ⎢ T T ⎥ ⎢ ∆ E 0 0 0 hE 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎦ ⎣ 0 <0 (25) where ⎡
Ξ 0 −Ah Z Bw − XC T ⎢ ⎢ ∗ −(1 − ¯ h)N 0 0 ⎢ ⎢ ⎢ (1 − ¯ h) Π =⎢ ∗ Z 0 ∗ − ⎢ h ⎢ ⎢ ∗ ∗ ∗ αI − (D + D T ) ⎣ ∗ ∗ ∗ ∗ ⎤ h(XAT + W T B T ) ⎥ ⎥ hXAT ⎥ h ⎥ ⎥ (26) 0 ⎥ ⎥ T ⎥ hBw ⎦ −hZ
Rearrange(25) gives the LMI (23) holds. By Theorem 2, it follows that the uncertain delayed system Σ∆ is robustly strongly stable with strict passivity for any delay h(t) satisfying (4). Remark 2 Theorem 3 utilizes a parameterized LMI to solve the robust passivity control problem for the uncertain delayed system Σ∆ .
4. Numerical example
883
−0.02
0
0
−0.09 ⎤
⎡ B=⎣ ⎡ E=⎣
0.04 −0.10
⎡
⎤
⎡
⎤ 0
0
−0.02
0
⎤
⎢ ⎥ ⎦ , Ah = ⎢ −0.01 −0.01 ⎥ , ⎣ ⎦
⎦ , Bw = ⎣
−0.03
−0.01
⎤ 0 −0.10 ⎡
⎦ , Ha = ⎣
⎦ , C = 0 1 , D = 1, ⎤
−0.40
0
0
−0.02
⎦,
⎡ ⎤ 1 Hb = ⎣ ⎦ 0 Applying Theorem 3 the authors obtained the following: h = 9.090 9, h1 = 0.6, α = 0.8 and F (t) = ∆ sin t, |∆| < 1 the solutions of LMI (23) are as follows ⎤ ⎡ 0.082 8 0.045 0 ⎦ N =⎣ 0.045 0 0.168 7 ⎤
⎡ X=⎣
4.688 0
0.043 4
0.043 4
0.030 8 ⎤
⎡ Z =⎣
⎦
5.910 0 0.114 2
⎦
0.114 2 5.464 4
0.492 9 3.421 8 K = −0.493 70 112.537 0 , γ = 34.380 4 W =
Therefore, the uncertain delayed systems Σ∆ are robustly strongly stable with strict passivity, with the controller u = Kx(t) = [−0.493 70 112.537 0]x(t).
5. Conclusions In this article a delay-dependent solution is proposed for the robust passivity control problem of a class of uncertain delayed systems with parameter uncertainties. The robust, strongly stable, with strict passivity sufficient condition is given in terms of linear matrix inequality (LMI). The corresponding controller is explicitly constructed.
References [1] Sepulchre R, Jankovic M, Kokotovic P V. Constructive
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Consider the following uncertain de-
nonlinear control. London: Springer-verlag, 1997.
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Li Guifang, Li Huiying & Yang Chengwu
[2] Hill D J, Moylan P J. The stability of nonlinear dissipative
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Li Guifang was born in 1978. She received Ph. D. in Control Theory and Control Engineering in Nanjing University of Science and Technology. Now she is working in Flight College of Nanjing University of Aeronautics and Astronautics.Her research interests include nonlinear robust control and passive systems theory. E-mail:[email protected].
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Li Huiying was born in 1979. She received Ph. D. in Control Theory and Control Engineering in Nanjing University of Science and Technology. Her research interests include filtering theory and time-delayed systems. E-mail:[email protected].
Trans. on Autom. Contr., 2002, 7(4): 664–669. [9] Xu S Y, James L. Improved delay-dependent stability criteria for time-delay systems. IEEE Trans.on Autom.Contr., 2005, 50(3): 384–387. [10] Li X, De Souza C E. Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica, 1997,33(9): 1657–1662.
Yang Chengwu was born in 1938. He is a professor in Control Theory and Control Engineering in Nanjing University of Science and Technology. His research interests include nonlinear control theory and 3D systems theory. E-mail:[email protected].
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Luan Xiaoli was born in 1979. She is a doctoral student of the Institute of Automation, Southern Yangtze University. Her research interests include robust control and optimization of complex nonlinear systems. E-mail: xiaoli [email protected]
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Liu Fei was born in 1965. He received the Ph. D. de-
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[10] Tanaka K. An approach to stability criteria of neuralnetwork control systems. IEEE Trans. on Neural Networks, 1996, 7 (3): 629–642. [11] Limanond S, Si J. Neural-network-based control design: an LMI approach. IEEE Trans. on Neural Networks, 1998, 9 (6): 1422–1429.
control engineering. Now he is a professor of the Institute of Automation, Southern Yangtze University. His research interests include advanced control theory and application, industrial systems monitoring and diagnosis, and intelligent technique with emphasis on fuzzy and neural systems.
Delay-dependent robust passivity control for uncertain time-delay systems Li Guifang1 , Li Huiying2 & Yang Chengwu2 1. Flight Coll. of Nanjing Univ. of Aeronautics and Astronautics, Nanjing 210006, P. R. China; 2. 810 Division, School of Power Engineering, Nanjing Univ. of Science and Technology, Nanjing 210094, P. R. China; (Received August 18, 2006)
Abstract: The robust passivity control problem is addressed for a class of uncertain delayed systems with timevarying delay. The parameter uncertainties are norm-bounded. First, the delay-dependent stability sufficient condition is obtained for the nominal system, and then, based-on the former, the delay-dependent robust passivity criteria is provided and the corresponding controller is designed in terms of linear matrix inequalities. Finally, a numerical example is given to demonstrate the validity of the proposed approach.
Keywords: passive control, time-varying delay, delay-dependent, parameter uncertainties, linear matrix inequality (LMI).
1. Introduction The passivity theory has its roots in the circuit theory and plays an important role in both electrical network and control systems. The fact that passivity and stability are closely related supplies a new approach to solve the problem of stabilization[1,2] . Moreover, passivity-based control has important robustness properties[3] . In addition, the passivity theory also provides an energy-based analysis approach, which is easier to perform when dealing with electrical circuits and mechanical systems[4] . On the other hand, time delay and parameter uncertainties often cause instability and bad performance in various engineering systems. The existing results for time-delay systems can be classified into two types; that is: delay-independent case and delaydependent case[5−12] . Generally, the latter are less conservative. To the bestx‘ of the authors’ knowledge, delay-dependent robust passivity control for uncertain time-delay systems has not been fully investigated. In this article, initially, the delay-dependent stability condition for a system with time-varying delay is obtained in terms of linear matrix inequality (LMI). Following which, based on the derived delay-
dependent stability condition, a delay-dependent robust passivity condition is proposed for delayed systems with norm-bounded uncertainties.
2. Problem statement Consider the following uncertain time-delayed system Σ∆ x(t) ˙ = [A + ∆A]x(t) + Ah x(t − h(t))+
(1a) [B + ∆B]u(t) + Bw w(t) x(t) = φ(t), t ∈ [−h(t), 0] (1b) z(t) = Cx(t) + Dw(t) (1c) n m Where x(t) ∈ R is the state, u(t) ∈ R is the control input, w(t) ∈ Rp is the disturbance input that belongs to L2 [0, ∞), z(t) ∈ Rp is the output, and φ(t) is the initial condition.A, Ah , B, Bw , C and D are the known, real constant matrices, with appropriate dimensions, which describe the nominal Σ∆ system, and ∆A and ∆B are the real matrix functions representing the time-varying parameter uncertainties. The admissible uncertainties are assumed by [∆A
∆B ] = E ∆(t) [Fa
Fb ]
(2)
where ∆(t) ∈ Rna ×nb is an unknown real time-varying matrix with the Lebesgue measurable element satisfy-
880
Li Guifang, Li Huiying & Yang Chengwu
ing
3.1 ∆T (t)∆(t) I
(3)
The unforced nominal system of system Σ∆ is Σ0
and E, Fa and Fb are the known, real constant matrices that characterize how uncertain parameters enter the nominal systems. And h(t) is a time-varying bounded delay satisfying ˙ ¯<1 0 h(t) h < +∞, h(t) h
(4)
The aim of this article is to design a controller u(t) = Kx(t), such that, the closed-loop systems are robustly strong and stable, with strict passivity for all admissible uncertainties, that is (1) The closed-loop systems are asymptotically stable for any delay that h(t) satisfies (4); (2) Under the zero initial condition, there exists a scalar α > 0, such that T T T 2 w(t) z(t)dt α w(t)T w(t)dt 0
x(t) ˙ = Ax(t) + Ah x(t − h(t)) + Bw w(t) z(t) = Cx(t) + Dw(t)
0
0
−h(t) 0
h(t)
ζ(s)ds
S
−h(t)
−h(t)
Lemma 2 For any real matrices Σ1 , Σ2 and Σ3 with appropriate dimensions, such that Σ3 = Σ3T , it follows that Σ3 + Σ1 F (t)Σ2 + Σ2T F (t)T Σ1T < 0 T
F (t) F (t) I
(6)
If and only if there exists a scalar ε > 0 Σ3 +
εΣ1 Σ1T
+ε
−1
(9) where A¯ = (A + Ah ), and ∗ means symmetrical elements. Proof Choose the candidate Lyapunov functional as follows t V (xt ) = xT P x + x(s)T Qx(s)ds+
Σ2T Σ2
<0
(7)
3. Main results First, the new delay-dependent stability condition is given, under which the unforced nominal system of system Σ∆ is asymptotically stable.
0
−h(t)
(5)
ζ(s) Sζ(s)ds
(8b)
<0
ζ(s)ds
T
(8a)
Theorem 1 Consider the unforced nominal system Σ0 with w(t) ≡ 0. Then for the given positive scalars h, ¯h and α, the system Σ0 is asymptotically stable for any time-varying delay h(t) satisfying (4), if there exist symmetric positive-definite matrices P, Q and R such that the following LMI holds. ⎡ ⎤ 0 −P Ah hAT R P A¯ + A¯T P + Q ⎢ ⎥ ⎢ ⎥ ¯ R ∗ −(1 − h)Q 0 hAT ⎥ ⎢ h ⎥ ⎢ ¯ ⎥ ⎢ 1−h Θ= ⎢ R 0 ⎥ ∗ ∗ − ⎦ ⎣ h ∗ ∗ ∗ −hR
0
Which holds for all T > 0 and w(t) ∈ L2 [0, T ]. Before proceeding to the main results, the authors introduce the following two important lemmas. Lemma 1[5] If for any positive-definite matrix 0 < S = S T ∈ Rr×r and function h(t) > 0, the vector function ζ : [−h(t), 0] → Rr , such that, the concerned integration is well defined, then T
Delay-dependent stability
t−h(t)
t
t+α
(10)
T
x˙ (s)Rx(s)dsdα ˙
where P, Q and R are positive-definite matrices to be chosen. Then, by the Newton-Leibniz formula, it follows that t
x(t − h(t)) = x(t) − and
t−h(t)
¯ x(t) ˙ = Ax(t) − Ah
x(s)ds ˙
(11)
t
t−h(t)
x(s)ds ˙
(12)
Differencing V (xt ) along with the trajectory of Eq. (8a) gives T ˙ V˙ (xt ) = 2xT P x˙ + x(t)T Qx(t) − (1 − h(t))x h (t) Q×
˙ T Rx(t)− ˙ xh (t) + h(t)x(t) t ˙ (1 − h(t)) x(s) ˙ T Rx(s)ds ˙ t−h(t)
Delay-dependent robust passivity control for uncertain time-delay systems
T
¯ − Ah 2x P Ax(t)
t t−h(t)
T
⎡
x(s)ds ˙ +
T
¯ h (t) Qxh (t)+ x(t) Qx(t) − (1 − h)x t ¯ ˙ − (1 − h) x(s) ˙ T Rx(s)ds ˙ hx(t) ˙ T Rx(t)
⎢ ⎢ ⎢ ⎢ ⎢ Θ1 = ⎢ ⎢ ⎢ ⎢ ⎣
(13)
t−h(t)
P A¯ + A¯T P + Q
0
−P Ah
∗
¯ −(1 − h)Q
∗
∗
∗
∗
0 ¯ 1−h R − h ∗
∗ P Bw − C
∗ T
T
hA R
⎤
∗
⎥ ⎥ hAT hR ⎥ ⎥ ⎥<0 0 0 ⎥ ⎥ T R ⎥ αI − (D + D T ) hBw ⎦ ∗ −hR
By Lemma 1, it follows that
0
V˙ (xt ) xT (P A¯ + A¯T P + Q + hAT RA)x− t T 2x P Ah x(s)ds ˙ + hxh (t)T AT h RAh xh (t)+ t−h(t)
(16)
¯ h (t)T Qxh (t)− 2hx(t)T AT RAh xh (t) − (1 − h)x
T t t ¯ 1−h x(s)ds ˙ R x(s)ds ˙ h t−h(t) t−h(t) (14)
T
881
T
Denoting X(t)= x(t) xh (t)
T T
t
t−h(t)
,
x(s)ds ˙
by the Schur complement and applying (9) it follows that, V˙ (xt ) X(t)T ΘX(t) < 0
(15)
Proof By considering Theorem 1, for the stability of the unforced system Σ0 with w(t) ≡ 0, any time delay h(t) satisfying (4) follows from inequality (9). Moreover, LMI(16) implies LMI(9), hence the system Σ0 is asymptotically stable. To establish the strict passivity condition for system Σ0 , assume zero initial conditions and choose the same Lyapunov functional of form (10), following which the trajectory of Eq.(8a) satisfies t x(t) ˙ = (A + Ah )x(t) − Ah x(s)ds ˙ + Bw w(t) t−h(t)
when x(t) and xh (t) = x(t − h(t)) are not zero. This completes the proof. Remark 1 In fact, Theorem 1 is a delay dependent and delay-derivative dependent stability condition for system Σ0 . If the delay is constant, that is h(t) = d, then it reduces to a delay-dependent condition. 3.2
Delay-dependent passivity
In the sequel, based on Theorem 1, the authors focus on the problem of the strong stability and strict passivity of system Σ0 . ¯ Theorem 2 For the given positive scalars h, h and α, system Σ0 is strongly stable with strictly passivity for any delay h(t) satisfying (4), if there exist positive definite matrices P, Q and R, such that the following LMI holds.
(17) Then using the same technique as in the proof of Theorem 1, it can be easily shown that the time derivative of V (xt ) along with the solution of Eq.(8a) satisfies ¯ A¯T P +Q+hATRA)x−2xT P Ah × V˙ (xt ) xT (P A+ t T T x(s)ds+hx ˙ h (t) Ah RAh xh (t)+ t−h(t)
T 2hx(t)T AT RAh xh (t)+hw(t)T Bw RBw w(t)+
2hx(t)T AT RBw w(t)+2x(t)T P Bw w(t)+ T ¯ 2hxh (t)T AT h RBw w(t)−(1− h)xh (t) Qxh (t)−
T t t 1− ¯h x(s)ds ˙ R x(s)ds ˙ h t−h(t) t−h(t) (18) Next, the authors consider the following performance index t J(xt ) = [αw(t)T w(t)−2w(t)T z(t)]dt+V (xt ) (19) 0
882
Li Guifang, Li Huiying & Yang Chengwu ⎡
Then, using Eq.(8b) in J(xt ) yields ˙ t ) = V˙ (xt ) + αw(t)T w(t) − 2w(t)T z(t) = J(x V˙ (xt ) − 2w(t)T Cx(t)+
(20)
w(t)T [αI − (D + D T )]w(t) Setting
x(t)T xh (t)T
Y (t) =
t
t−h(t)
T
T x(s)ds ˙
w(t)T
and after some manipulations, the following is obtained ˙ t ) Y (t)T × J(x ⎡ ¯ A¯T P +Q+hATRA hAT RAh P A+ ⎢ ⎢ T ¯ ∗ −(1 − h)Q+hA ⎢ h RAh ⎢ ⎢ ∗ ∗ ⎣ ∗ ∗ ⎤ T T P Bw + hA RBw − C −P Ah ⎥ ⎥ 0 hAT ⎥ h RBw ⎥ Y (t) ⎥ 1−¯ h ⎥ R 0 − ⎦ h T ∗ hBw RBw + αI − (D + D T ) (21) By the Schur complement, (21) can be rewritten as ˙ t ) Y (t)T Θ1 Y (t) J(x
(22)
˙ t ) < 0, which imUsing LMI (16), it follows that J(x T T plies 2 w(t)T z(t)dt α w(t)T w(t)dt. That is, 0
0
system Σ0 is strongly stable with strict passivity. The proof is completed. 3.3
Robust passivity control
The authors are now in the position to design a controller which renders the resulting closed-loop systems robustly strongly stable with strictly passive for all admissible uncertainties. ¯ and Theorem 3 For given positive scalars h, h α, the uncertain delayed systems Σ∆ are robustly strongly stable with strictly passive for any delay h(t) satisfying (4), if there exist a scalar γ > 0, and positive definite matrices X, N, Z > 0 and matrix W such that the following LMI holds.
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Ξ
0
−Ah Z
Bw − XC T
∗
¯ −(1 − h)N
0
∗
∗
∗
∗
0 ¯ (1 − h) Z − h ∗
αI − (D + D T )
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
h(XAT + W T B T )
γE
XHaT + W T HbT
hXAT h
0
0
0
0
0
T hBw
0
0
−hZ
γhE
0
∗
−γI
0
∗
∗
−γI
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(23) ¯ + X A¯T + BW + W T B T + N where Ξ = AX Moreover, the robust passivity controller is given by u(t) = W X −1 x(t). Proof Substituting K) + (∆A + ∆B · K) ing and postmultiplying diag P −1 P −1 R−1
matrix A with (A + B · in LMI(16), Premultiplythe resultant matrix with
−1 , and combining I R
it with the denotes X = P −1 , N = P −1 QP, W = KP −1 and Z = R−1 , the following LMI is obtained. ⎡ Ξ∆ 0 −Ah Z Bw − XC T ⎢ ⎢ ∗ −(1 − h)N ¯ 0 0 ⎢ ⎢ ⎢ (1 − ¯h) ⎢ ∗ Z 0 ∗ − ⎢ h ⎢ ⎢ ∗ ∗ ∗ αI − (D + D T ) ⎣ ∗
∗
∗ T
T
T
hX(A + ∆A) + hW (B + ∆B) hXAT h 0 T hBw
⎤
∗
⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎦
−hZ (24) where T Ξ ∆ = Ah X + XAT h + (A + ∆A)X + X(A + ∆A) +
(B + ∆B)W + W T (B + ∆B)T + N
Delay-dependent robust passivity control for uncertain time-delay systems Together with Lemma 2, LMI (24) is equivalent to the following LMI. ⎡
layed systems ⎡ A=⎣
⎤
E ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Π +⎢ ⎢ 0 ⎥ ∆ Ha X + H b W 0 0 0 0 + ⎥ ⎢ ⎢ 0 ⎥ ⎦ ⎣ hE ⎡ ⎤ XHaT + W T HbT ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎥ T ⎢ T T ⎥ ⎢ ∆ E 0 0 0 hE 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎦ ⎣ 0 <0 (25) where ⎡
Ξ 0 −Ah Z Bw − XC T ⎢ ⎢ ∗ −(1 − ¯ h)N 0 0 ⎢ ⎢ ⎢ (1 − ¯ h) Π =⎢ ∗ Z 0 ∗ − ⎢ h ⎢ ⎢ ∗ ∗ ∗ αI − (D + D T ) ⎣ ∗ ∗ ∗ ∗ ⎤ h(XAT + W T B T ) ⎥ ⎥ hXAT ⎥ h ⎥ ⎥ (26) 0 ⎥ ⎥ T ⎥ hBw ⎦ −hZ
Rearrange(25) gives the LMI (23) holds. By Theorem 2, it follows that the uncertain delayed system Σ∆ is robustly strongly stable with strict passivity for any delay h(t) satisfying (4). Remark 2 Theorem 3 utilizes a parameterized LMI to solve the robust passivity control problem for the uncertain delayed system Σ∆ .
4. Numerical example
883
−0.02
0
0
−0.09 ⎤
⎡ B=⎣ ⎡ E=⎣
0.04 −0.10
⎡
⎤
⎡
⎤ 0
0
−0.02
0
⎤
⎢ ⎥ ⎦ , Ah = ⎢ −0.01 −0.01 ⎥ , ⎣ ⎦
⎦ , Bw = ⎣
−0.03
−0.01
⎤ 0 −0.10 ⎡
⎦ , Ha = ⎣
⎦ , C = 0 1 , D = 1, ⎤
−0.40
0
0
−0.02
⎦,
⎡ ⎤ 1 Hb = ⎣ ⎦ 0 Applying Theorem 3 the authors obtained the following: h = 9.090 9, h1 = 0.6, α = 0.8 and F (t) = ∆ sin t, |∆| < 1 the solutions of LMI (23) are as follows ⎤ ⎡ 0.082 8 0.045 0 ⎦ N =⎣ 0.045 0 0.168 7 ⎤
⎡ X=⎣
4.688 0
0.043 4
0.043 4
0.030 8 ⎤
⎡ Z =⎣
⎦
5.910 0 0.114 2
⎦
0.114 2 5.464 4
0.492 9 3.421 8 K = −0.493 70 112.537 0 , γ = 34.380 4 W =
Therefore, the uncertain delayed systems Σ∆ are robustly strongly stable with strict passivity, with the controller u = Kx(t) = [−0.493 70 112.537 0]x(t).
5. Conclusions In this article a delay-dependent solution is proposed for the robust passivity control problem of a class of uncertain delayed systems with parameter uncertainties. The robust, strongly stable, with strict passivity sufficient condition is given in terms of linear matrix inequality (LMI). The corresponding controller is explicitly constructed.
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