Controller Design for Time-Delay Systems Using Discretized Lyapunov Functional Approach* Keqin GU† and Qing-Long HAN Department of Mechanical and Industrial Engineering Southern Illinois University at Edwardsville Edwardsville, Illinois 62026-1805, U.S.A.
Abstract This paper investigates the controller synthesis of uncertain linear time-delay systems. Stabilizability criteria are derived based on a discretized Lyapunov functional approach. A new controller design method is developed. Numerical examples show that the results using the proposed method are less conservative than some existing ones.
1. Introduction Many practical systems are subject to uncertainty and time-delay. Therefore, robust control of uncertain timedelay systems has received significant attentions recently. Some researchers have focused on the problem of designing robust memoryless controllers for uncertain time-delay systems. A number of results on this topic have been reported. Most of the early Lyapunov function (or functional) based approaches yield delay-independent stabilization criteria (Han and Mehdi, 1999a; Phoojaruenchanachai and Furuta, 1992; Shen et al., 1991). These results are often overly conservative and can not deal with systems whose stabilizability dependents on the size of the time delay. Razumikhin type of stabilizability criteria can indeed yield delay-dependent stabilizability results (Han and Mehdi, 1999b; Li and de Souza 1997; Niculescu et al., 1994). They can also allow time-varying delay. However, experience shows that the results are still rather conservative for many practical applications. This can be seen by applying these type of criteria to a constant time-delay system without uncertainty and comparing them with the analytical result (Gu, 1997). Gu (1997, 1999a) has proposed a discretized Lyapunov functional approach to check the stability of linear uncertain systems with constant time-delay. The criteria have shown significant improvements over the existing * This work was supported in part by the U.S. National Science Foundation under Grant INT-9818312. † Corresponding author. Tel: (618)650-2803; Fax: (618)6502555; E-mail:
[email protected]
results even under very coarse discretization. For uncertainty-free systems, the analytical results can be approached with fine discretization. Gu and Han (2000) has extended the result in Gu (1997) to the case where the delay may be time-varying. The result in Gu and Han (2000) has been improved in Han and Gu (2000) by using a generalized discretized Lyapunov functional approach. In this paper, based on the discretized Lyapunov functional approach in Gu (1997), we address the problem of robust control design for uncertain linear time-delay systems. The case of a single, constant time-delay is considered. We develop a design algorithm of a linear memoryless static state feedback controller for uncertain time-delay systems. Notations > 0 : symmetric positive definite matrix; < 0 : symmetric negative definite matrix; ≥ 0 : symmetric positive semi-definite matrix; ≤ 0 : symmetric negative semi-definite matrix; n ^ : the set of continuous ℜ valued functions on [ − r , 0] ; r : time-delay;
W W W W
2. Problem Statement Consider the uncertain time-delay system Σ (1) : x (t ) = A(1) (t ) x(t ) + Ad(1) (t ) x(t − r ) + B (1) (t )u (t ) (2.1) where x(t ) ∈ ℜn is the state vector, u (t ) ∈ ℜm is the control
vector,
A(1) (t ) ∈ ℜn×n ,
Ad(1) (t ) ∈ ℜ n×n
and
n×m
are uncertain matrices, which are unknown B (t ) ∈ ℜ and possibly time-varying, but known to be bounded by some compact set Ω , i.e. (1)
[ A(1) (t ), Ad(1) (t ), B (1) (t )] ∈ Ω ⊂ ℜn×(2 n + m ) , for all t ∈ [0, ∞) For a t ∈ [0, ∞) define xt ∈ ^ , xt (θ ) = x(t + θ ) , θ ∈ [− r , 0]
(2.2)
The objective is to design a static memoryless state feedback controller to stabilize the above system. For this purpose, consider an initial system of system (2.1) Σ
(2)
: x (t ) = A x(t ) + (2)
Ad(2) x(t
− r ) + B u (t ) (2)
Let the delay interval [−r , 0] be divided into N segments [δ i −1 , δ i ] of equal length h, where
δ i = −rm + ih , i = 0, 1, 2, h=r/N
(2.3)
which can be stabilized by a static memoryless state feedback controller.
",
N
Choose Q, R, S and T to be continuous piecewise linear, i.e. ∆
Q i (α ) = Q(δ i −1 + α h) = (1 − α )Qi −1 + α Qi
Now consider a class of systems
(3.3a)
∆
S i (α ) = S (δ i −1 + α h) = (1 − α ) Si −1 + α Si
Σ : = λΣ(1) + (1 − λ )Σ(2) , λ ∈ [0, 1]
(3.3b)
∆
T i (α ) = T (δ i −1 + α h) = (1 − α )Ti −1 + α Ti
That is
(3.3c)
∆
Tkji (α ) = Tkj (δ i −1 + α h) = (1 − α )Tkj ,i −1 + α Tkj ,i
Σ : x (t ) = [ A(2) + λ ( A(1) (t ) − A(2) )]x(t )
R i (α ) = R(δ i −1 + α h) = (1 − α ) Ri −1 + α Ri
+[ Ad(2) + λ ( Ad(1) (t ) − Ad(2) )]x(t − r ) +[ B (2) + λ ( B (1) (t ) − B (2) )]u (t )
(2.4)
Let the static state feedback control law be u (t ) = Fx(t )
(2.5)
ΣCL : x (t ) = {( A(2) + B (2) F ) + λ[( A(1) (t )
Lemma 1 (Gu,1997). For piecewise linear Q , S and R as described by (3.3), the Lyapunov functional satisfies
+[ Ad(2) + λ ( Ad(1) (t ) − Ad(2) )]x(t − r ) (2.6)
3. Stability of Time-Delay Systems
(3.1)
ℜ
0 1 V (φ ) = φ T (0) Pφ (0) + φ T (0) ∫ Q(ξ )φ (ξ ) d ξ − r 2 1 0 0 + ∫ [ ∫ φ T (ξ ) R (ξ − η )φ (η )dη ]d ξ 2 −r −r 1 0 + ∫ φ T (ξ ) S (ξ )φ (ξ )d ξ 2 −r
where P ∈ ℜ n×n , P = PT Q : [−r , 0] → ℜn×n
S : [−r , 0] → ℜn×n , S T (ξ ) = S (ξ ) R :[−r , r ] → ℜn×n , R(−ξ ) = RT (ξ )
Si ≥ 0 , i = 0,1, 2," , N Rˆ Qˆ T >0 ˆ Q P
(3.5)
R0 R−1 R R0 Rˆ = 1 # # RN RN −1 ˆ Q = [Q0 Q1
The stability of system (3.1) can be investigated by choosing a quadratic Lyapunov functional candidate 6
(3.4)
(3.6)
where
Consider the stability problem of time-delay system
V ( xt ) : ^
V (φ ) ≥ εφ T (0)φ (0) for some ε > 0 if
− A(2) ) + ( B (1) (t ) − B (2) ) F ]}x(t )
= A(t ) x(t ) + Ad (t ) x(t − r )
(3.3e)
for 0 ≤ α ≤ 1 . Then, the following results were provn in Gu (1997).
Then the closed-loop system is
x (t )
(3.3d)
∆
"
R− N R− N +1 # R0
"
QN ]
" " %
Lemma 2 (Gu, 1997). For piecewise linear Q, S, R and T as described by (3.3), the derivative of the Lyapunov functional along the solution of (3.1) satisfies
V (φ ) ≤ −εφ T (0)φ (0) (3.2)
(3.7)
for some ε > 0 if the following inequality is satisfied 1 ∆ + T11,i + j −1 2T T −∆ + T12,i + j −1 ∆3ijT for all i = 1, 2,
−∆ 2 + T12,i + j −1
",
S0 + T22,i + j −1 ∆ij4T
4 ∆ ij > 0 (3.8) 1 ( Si − Si −1 ) N ∆ 3ij
N , j = 0, 1 , [ A(t ), Ad (t )] ∈ Ω1 , where
∆1 = −[ PA(t ) + AT (t ) P + S N + QN + QNT ]
∆ Σ2 = PAd(2) + λ P[ Ad(1) (t ) − Ad(2) ] − Q0
∆ 2 = PAd − Q0
∆ 3Σij = h( A(2) + B (2) F )T Qi −1+ j + λ h[( A(1) (t ) − A(2) )
∆ 3ij = hAT (t )Qi −1+ j − (Qi − Qi −1 ) + hRiT− N −1+ j
+ ( B (1) (t ) − B (2) ) F ]T Qi −1+ j − (Qi − Qi −1 ) + hRiT− N −1+ j
∆ ij4 = hAdT Qi −1+ j − hRiT−1+ j and Ti ( i = 0, 1, 2,
",
∆ Σ4 ij = hAd(2)T Qi −1+ j + λ h[ Ad(1)T (t ) − Ad(2)T ]Qi −1+ j − hRiT−1+ j and Ti ( i = 0, 1, 2,
N ) satisfying N −1
T0 + TN + 2 ∑ Ti = 0
N −1
Now, by Corollaries 1 and 2, we have the stabilization result for system (2.4).
Choosing the same Lyapunov functional as (3.3) for system (2.6), by Lemmas 1 and 2, one can easily obtain the following results. Corollary 1. The Lyapunov functional satisfies
V (φ ) ≥ εφ T (0)φ (0)
(4.1)
Si ≥ 0 , i = 0,1, 2," , N Rˆ Qˆ T M (Z ) = >0 ˆ Q P
(4.2) (4.3)
where Z = [ P, Q, R, S , T ]T . Corollary 2. The derivative of the Lyapunov functional along the solution of (2.6) satisfies
≤ −εφ (0)φ (0) T
(4.4)
for some ε > 0 if the following inequality is satisfied H (Z , F , λ ) = + T12,i + j −1
S0 + T22,i + j −1 ∆ Σ4Tij
Theorem 1. System (2.4) is robustly stabilizable via a linear state feedback controller (2.5) if there exist real matrices P, Qˆ , and Rˆ satisfying (4.3) and S0 ≥ 0 , and Ti i = 0, 1, 2, " , N satisfying (4.6) such that (4.5) is satisfied. Remark 1. Condition (4.2) (i.e. Si ≥ 0 , i = 0,1, 2," , N ) is implied by S0 ≥ 0 and (4.5).
for some ε > 0 if
−∆ 2Σ
(4.6)
i =1
4. Controller Synthesis
1 ∆ Σ + T11,i + j −1 2T T −∆ Σ + T12,i + j −1 ∆ 3ΣTij
N ) satisfying
T0 + TN + 2 ∑ Ti = 0
(3.9)
i =1
V (φ )
",
4 ∆ Σij > 0 (4.5) 1 ( Si − Si −1 ) N ∆3Σij
for all i = 1, 2," , N , j = 0, 1 , [ A(1) (t ), Ad(1) (t ), B (1) (t )] ∈ Ω ,
Remark 2. From (2.4), it is apparent to see that if the parameter λ → 1 , system (2.4) approaches the original system (2.1). From Theorem 1 and Remark 2, for the robust stabilization of system (2.1)-(2.2), the following result is easily obtained. Theorem 2. System (2.1)-(2.2) is robustly stabilizable via a linear state feedback controller (2.5) if there exist real matrices P, Qˆ , and Rˆ satisfying (4.3) and S0 ≥ 0 , and Ti i = 0, 1, 2, " , N satisfying (4.6) such that the following inequality is satisfied H ( Z , F ,1) = 1 ∆ Σ1 + T11,i + j −1 2T T −∆ Σ1 + T12,i + j −1 ∆3ΣT1ij
−∆ Σ21 + T12,i + j −1 S0 + T22,i + j −1 ∆ Σ4T1ij
4 ∆ Σij > 0 (4.7) 1 ( Si − Si −1 ) N ∆ 3Σ1ij
where
for all i = 1, 2," , N , j = 0, 1 , [ A(1) (t ), Ad(1) (t ), B (1) (t )] ∈ Ω , where
∆1Σ = −{P( A(2) + B (2) F ) + λ P[( A(1) (t ) − A(2) )
∆1Σ1 = −{P[ A(1) (t ) + B (1) (t ) F ] + [ A(1)T (t ) + F T B (1)T (t )]P
+ ( B (1) (t ) − B (2) ) F ] + ( A(2)T + F T B (2)T ) P
+ S N + QN + QNT }
+ λ[( A(1)T (t ) − A(2)T ) + F T ( B (1)T (t ) − B (2)T )]P
∆ Σ21 = PAd(1) (t ) − Q0
+ S N + QN + QNT }
∆3Σ1ij = h[ A(1) (t ) + B (1) (t ) F ]T Qi −1+ j − (Qi − Qi −1 )
+ hRiT− N −1+ j ∆ Σ4 1ij = hAd(1)T (t )Qi −1+ j − hRiT−1+ j The LMIs derived need to hold for all the possible uncertain system matrices. Here, we assume that the uncertainty set is polytopic
Ω = Co{( A j , B j ) j = 1, 2,
",
nv }
( k )1 ∆ Σ + T11,i + j −1 ( k )2T T + T12, −∆ Σ i + j −1 ∆ (Σkij)3T
Then it is easy to see that one only needs to check the satisfaction of LMIs at all the vertices.
determined. Second for the F and λ given in the previous step, one can construct a generalized eigenvalue problem to determine new Z ( k +1) and ∆λ ( k )1 > 0 . If 1 [λ ( k ) , λ ( k )1 ] , belongs to the interval where (k )
( k +1)
Third if it is not the case, fixing the new derived Z and λ ( k )1 , one can find new F ( k +1) and ∆λ ( k )2 > 0 that solve a generalized eigenvalue problem. If 1 belongs to the new interval [λ ( k )1 , λ ( k +1) ] , where λ ( k +1) = λ ( k )1 + ∆λ ( k )2 , the static state feedback controller u (t ) = F ( k ) x(t ) for system (2.1)-(2.2) is obtained. Otherwise repeat the above process until one get a static state feedback controller. If the interval [λ ( k ) , λ ( k )1 ] or [λ ( k )1 , λ ( k +1) ] that includes 1 is not obtained, the procedure fails. Based on Remark 2, Theorem 2 and the above discussion, we will give a design procedure. First of all, we introduce some notations as follows. Let λ ( k +1) − λ ( k ) = ∆λ ( k ) = ∆λ ( k )1 + ∆λ ( k )2 , where ∆λ ( k )1 > 0 and ∆λ ( k )2 > 0 Denote
G ) = G ( k )2T ( k )3T Gij ( k )1
G(Z
,F
(k )
,λ
(k )
S0( k ) + T22,i + j −1 ∆ (Σkij)4T
R0( k ) (k ) R = 1 # RN( k )
R−( k1 )
"
R0( k )
"
#
%
RN( k−)1
"
Qˆ ( k ) = [Q0( k )
Q1( k )
"
R−( kN) R−( kN) +1 # R0( k )
QN( k ) ]
and
∆ (Σk )1 = −{P ( k ) ( A(2) + B (2) F ( k ) ) + λ ( k ) P ( k ) [( A(1) (t ) − A(2) ) + ( B (1) (t ) − B (2) ) F ( k ) ] + ( A(2)T + F ( k )T B (2)T ) P ( k ) + λ ( k ) [( A(1)T (t ) − A(2)T ) + F ( k )T ( B (1)T (t ) − B (2)T )]P ( k ) + S N( k ) + QN( k ) + QN( k )T } ∆ (Σk )2 = P ( k ) Ad(2) + λ ( k ) P ( k ) [ Ad(1) (t ) − Ad(2) ] − Q0( k ) ∆ (Σkij)3 = h( A(2) + B (2) F ( k ) )T Qi(−k1)+ j + λ ( k ) h[( A(1) (t ) − A(2) ) + ( B (1) (t ) − B (2) ) F ( k ) ]T Qi(−k1)+ j − (Qi( k ) − Qi(−k1) ) + hRi(−kN)T−1+ j ∆ (Σkij)4 = hAd(2)T Qi(−k1)+ j + λ ( k ) h[ Ad(1)T (t ) − Ad(2)T ]Qi(−k1)+ j −hRi(−k1)+T j and
G ( k )1 = P ( k ) [( A(1) (t ) − A(2) ) + ( B (1) (t ) − B (2) ) F ( k ) ] +[( A(1)T (t ) − A(2)T ) + F ( k )T ( B (1)T (t ) − B (2)T )]P ( k ) G ( k )2 = P ( k ) [ Ad(1) (t ) − Ad(2) ] Gij( k )3 = −h[( A(1) (t ) − A(2) ) + ( B (1) (t ) − B (2) ) F ( k ) ]T Qi(−k1)+ j Gij( k )4 = − h[ Ad(1)T (t ) − Ad(2)T ]Qi(−k1)+ j Now we state the algorithm for designing the static state feedback controller u (t ) = Fx(t ) as follows. Algorithm 4.1
Rˆ ( k ) M (Z (k ) ) = ˆ (k ) Q
(k )
Rˆ ( k )
(k )
λ ( k )1 = λ ( k ) + ∆λ ( k )1 , the static state feedback controller u (t ) = F ( k ) x(t ) robustly stabilizes system (2.1)-(2.2).
∆ Σ( kij)3
where
(4.8)
Now we give the idea for designing the static state feedback controller u (t ) = Fx(t ) . First we set λ = 0 . From the initial system (2.3) (it is supposed to be stabilized) one obtains a proper gain matrix F (0) such that the closed-loop system of system (2.3) with u (t ) = F (0) x(t ) is asymptotically stabilized. From (4.2)-(4.3) and (4.5), a group of matrices Z (0) = [ P (0) , Q(0) , R (0) , S (0) , T (0) ]T are
( k )4 ∆ Σij 1 (k ) ( Si − Si(−k1) ) N
−∆ Σ( k )2 + T12,i + j −1
Qˆ ( k )T P ( k ) G
( k )2
0 Gij( k )4T
H (Z (k ) , F (k ) , λ (k ) ) =
Step 1. Let λ (0) = 0 , design F (0) to stabilize Σ (2) . From 0 ≤ S0 , 0 < M (Y (0) ) and 0 < H ( Z (0) , F (0) , λ (0) ) , one Gij( k )3 Gij( k )4 0
obtains Z (0) = [ P (0) , Q(0) , R (0) , S (0) , T (0) ]T . Step 2. For F ( k ) and λ ( k ) given in the previous step, find new Z ( k +1) and ∆λ ( k )1 that solve the generalized eigenvalue problem 1 subject to: min (k ) Z ∆λ ( k )1
0 ≤ S0 ; 0 < M ( Z ( k ) ) ; 0 < H ( Z ( k ) , F ( k ) , λ ( k ) ) G (Z (k ) , F (k ) , λ (k ) )
