On delay-dependent passivity - Automatic Control, IEEE Transactions on
664
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
On Delay-Dependent Passivity E. Fridman and U. Shaked
Abstract—Sufficient conditions for passivity (positive realness) are obtained for continuous-time, linear, retarded, and neutral-type systems. A delay-dependent solution is given in terms of linear matrix inequalities (LMIs) by using a descriptor model transformation of the system and by applying Park’s inequality for bounding cross terms. A memoryless state-feedback solution is derived. Numerical examples are given which illustrate the effectiveness of the new theory. Index Terms—Delay-dependent criteria, linear matrix inequalities (LMIs), positive-real lemma, time-delay systems.
I. INTRODUCTION Positive realness (passivity) theory plays an important role in both electrical network and control systems (see, e.g., [1], [2]) and it has roots in circuit theory ([3], [4]). For systems with delay of retardedtype, positive realness has been studied by [5]–[7]. In [5], delay-independent sufficient conditions in terms of LMIs have been derived. In [6] necessary and sufficient conditions are given in terms of positivity of some kernel matrix constructed via transition matrix. In [7] frequency domain approach is applied and sufficient conditions are obtained. For infinite-dimensional systems, a positive-real lemma has been obtained in terms of Riccati operator equations (see [2] and the references therein). In the present note we give delay-dependent sufficient conditions for passivity of neutral type systems. We apply descriptor-type Lyapunov–Krasovskii functionals that were recently introduced in [10]–[12] for delay-dependent stability and control and Park’s inequality for bounding cross terms [13]. We also present a memoryless state-feedback controller via LMIs, such that the resulting closed-loop system is passive. Notation: Throughout the note, the superscript T stands for matrix transposition, Rn denotes the n dimensional Euclidean space, Rn2m is the set of all n 2 m real matrices, and the notation P > 0, for P 2 Rn2n means that P is symmetric and positive–definite. Denote xt () = x(t + )( 2 [0h; 0]). II. PASSIVITY AND POSITIVE REALNESS FOR LINEAR TIME-DELAY SYSTEMS Given the following system
x_ (t) 0 F x_ (t 0 g ) =
2
i=0
Ai x (t 0 hi ) + B1 w(t)
z (t) =Cx(t) + Dw(t)
dimensions. Denote h = maxfh1 ; h2 g. For simplicity only we consider a single delay g and two delays h1 and h2 . The results of this note can be easily applied to the case of multiple delays g1 ; . . . ; gm , h1 ; . . . ; hm and a distributed delay. Equation (1a) describes a system of neutral type since it contains a derivative with delay. In the case of F = 0 (1a) is a retarded type system (see, e.g., [8]). Neutral systems are encountered in modeling of lossless transmission lines, or in dynamical processes including steam and water pipes (see, e.g., [8] and the references therein). Unlike retarded systems, linear neutral systems may be destabilized by small changes of the delay and may be unstable even in the case when all the roots of the characteristic equation have negative real parts [8]. We are looking for a criterion for passivity that depends on the delays hi and does not depend on g . Delay-independence with respect to g guarantees that small changes in g do not destabilize the system [8]. To guarantee robustness of the results with respect to small changes of delay, we assume that the difference equation Dxt = x(t) 0 F x(t 0 g ) = 0 is asymptotically stable for all values of g or, equivalentaly, that A1 F is a Schur–Cohn stable matrix, i.e., all the eigenvalues of F are inside the unit circle. The transfer function of (1a-b) from w to z is given by
G(s) = C s I 0 F e0sg
0
2
i=0
Ai e0sh
01 B1 + D:
Definition 1: [1] The system (1a-b) is called passive if t
2
0
wT (t)z (t)dt 0
(2)
for all t1 0 and for all solution of (1a-b) with x0 = 0. Another less restrictive definition of passivity is given by [14]. Definition 2: [14] The system (1a-b) is called passive if there exists
0 such that t
2
0
t
wT (t)z (t)dt 0
0
wT (s)w(s)ds:
(3)
for all t1 0 and for all solution of (1a-b) with x0 = 0. Different model transformations were used in the past for delay-dependent stability (see, e.g., [9] and [13]). Recently, a new (descriptor) model transformation has been introduced [10]. Unlike previous transformations, the descriptor model leads to a system which is equivalent to the original one, it does not depend on additional assumptions for the stability of the transformed system and it requires bounding of fewer crossterms. It was shown in [10] and [12] that the latter transformation leads to less conservative conditions for stability and H control. Following [10], we represent (1a-b) in the equivalent descriptor form
1
(1a-b)
where x(t) 2 Rn is the system state vector, w(t) 2 Rq is the exogenous input, which can be either a control input or a reference signal and z (t) 2 Rq is the output of the system. The time delays 0 = h0 , 0 < hi , i = 1; 2 and g > 0 are assumed to be known. The matrices Ai , i = 0; . . . ; 2, F , B1 and C are constant matrices of appropriate Manuscript received May 9, 2001; revised September 24, 2001. Recommended by Associate Editor V. Balakrishnan. This work was supported by the Ministry of Absorption of Israel, and also by the C & M Maus Chair at Tel Aviv University. The authors are with the Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(02)03748-0.
x_ (t) = y (t);
y (t) = F y (t
0 g) +
2
i=0
Ai x (t
0 hi ) + B1 w(t):
(4)
The latter is equivalent to the following descriptor system with discrete and distributed delay in the variable y : x_ (t) =y (t) y (t) =F y (t
0018-9286/02$17.00 © 2002 IEEE
0
0 g) +
2
i=1
Ai
t t0h
2
i=0
Ai
x(t)
y ( )d + B1 w(t):
(5)
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the transfer matrix G of (1a-b) is positive real, i.e.,
A Lyapunov–Krasovskii functional for the system (5) has the form
+
2
t
i=1
t h
0
2
+
T
x
0
t
( )Si x( )d +
t
i=1 0h
G(i! )3 + G(i! ) 0:
x(t) y (t)
V (xt ; yt ) = xT (t)y T (t) EP
0
t g
y
T
( )U y( )d
y T (s)ATi Ri3 Ai y (s)d d
t+
Proof: (i) We note that
(6)
= I0n 00
=
P
d dt
0
P1 P2
91 =
i=0
+
2
i=1
2
P2 + P2
ATi
T
WiT3 Ai
+ 2
92 =P1 0 P2T +
i=0
93 = 0 P3 0 P3T +
i=0
ATi Wi3
P3 +
ATi
2
i=1
i=1
i=1
Si
0
91 92 3 93
3 3 3 3 3 3
3 3 3 3 3 1
i=0
9
T
0
2
3 3
i=1
PT
0
F
0 I 0 D 0 DT 0 3 0U xT (t 0 hi ) Si x (t 0 hi ) t
y T ( )ATi Ri3 Ai y ( )d
0
t h
t
0
t h
0 i
0
0
t h
0
Ai
y (s)ds:
P
(13)
0 and Mi , the following inequality
bT (s)a(s)ds
h1 811 h1 812
xT (t)y T (t) P T
For any 2n 2 2n-matrices Ri > holds [13]:
02
(10)
t
02
i (t)
6= 0
P2T B1 0 C T P3T B1 0 I 0 D 0 DT
3 3 3 1 1
0 0 CT 0 B1
PT
2 AT 0 I 0 i=0 i 2 Ai 0I + I 0I i=0 2 Si 0 i=1 2 Ui + hi AT Ri3 Ai 0 i i=1
PT
+
(9)
Ai e0i!h
:
colfx(t); y (t); w(t); y (t 0 g )g and
9
ATi Wi4
i3
i! I 0 F e0i!g
0 2zT w 0 wT w
=
where
Then, the following holds. i) The system (1a-b) is passive in the sense of Definition 2. ii) In the case of = 0 for all ! 2 R with
det
dV (xt ; yt ) dt
Ui + hi AiT Ri3 Ai
2
0
(12)
2 2
x_ (t)
Substituting (5) into (11), we obtain
Ai
+
xT (t)y T (t) P T
(11)
+
8i1 = [ WiT1 + P1 WiT3 + P2T ] 8i2 = [ WiT2 WiT4 + P3T ] Ri1 Ri2 Ri = : RT R i2
=2xT (t)P1x_ (t) =2
(7a-b) The first term of (6) corresponds to the descriptor system, the third—to the delay-independent conditions with respect to the discrete delays of y , while the second and the fourth terms correspond to the delay-dependent conditions with respect to the distributed delays (with respect to x). We obtain the following. Theorem 1: Assume A1. Consider the system of (1a-b). Let there exist n 2 n-matrices 0 < P1 ; P2 ; P3 ; Si = SiT , U = U T , Wi1 ; Wi2 ; Wi3 ; Wi4 , Ri1 = RiT1 ; Ri2 ; Ri3 = RiT3 ; i = 1; 2 and
0 that satisfy the linear matrix inequality (LMI), as shown in (8) at the bottom of the page, where
2
x(t) y (t)
xT (t)y T (t) EP
P1 > 0; U > 0; Si > 0:
P3
= xT P1 x
and, hence, differentiating the first term of (6) with respect to t we have
where
E
x y
xT y T EP
t
0
t h
a(s) b(s)
T
Ri MiT Ri
Ri Mi (2; 2)
a(s) ds (14) b(s)
for a(s) 2 R2n , b(s) 2 R2n given for s 2 [t 0 hi ; t]. Here, (2; 2) = (MiT Ri + I )Ri01(Ri Mi + I ).
h2 821 h2 822
0
0h1 R1 0 3 0h2 R2 3 3 3 3 3 3
0W13T A1 0W23T A2 0W14T A1 0W24T A2 0 0 0
0S1 3 3
0 0 0 0
0S2 3
P2T F P3T F
0 0 0 0 0
0U
0
(8)
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b
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Using this inequality for a(s)
colf0 Ai gy (s) and
=
= P colfx(t) y(t)g, we obtain
For
Wi
x i hi [ xT y T ] P T MiT Ri + I Ri01 (Ri Mi + I ) P y x + 2 xT (t) 0 xT (t 0 hi ) [ 0 ATi ] Ri Mi P y t 0 y(s)ds: + y T (s)[ 0 ATi ] Ri (15) A i t0h
Wi
Wi :
LMI (8) results from the latter LMI by expansion of the block matrices. (ii) Let ! be such that (10) holds and consider w(t) = ei!t w0 , w0 2 Rq . Define
x(t) = ei!t i! I 0 F e0i!g
0
2
i=0
x_ (t) 0 F x_ (t 0 gi ) =
B1 !0
and z (t) = Cx(t) + Dw(t). Then z (t) = (w; x; z ) satisfies (1a-b) and 2wT (t)z(t) = w0T [G3 (i!) + G(i!)] w0 :
0
9
PT
0 0 CT 0 B1
h1 81
0h R
0 0
3 3 3 3
3 3 3 3
3 3 3 3
0h R 3 3 3
AT0 P2 + P2T A0 +
3 3 3 3 3
0 1
2 =1
i
1
Si
2
1
0 0
0W T A0 2
1
3 3 3 3
Ad (s)x(t + s)ds + B1 w(t)
2
0S 3
2
P2T B1 0 C T PTB 3
1
(21)
(22)
0
PT
0 0
0 0
0S 3 3
P1 0 P2T + AT0 P3 0P3 0 P3T + U
2
0 0
0
2
0h
x_ (t) = A0 x(t) + A1 x(t 0 h) + B1 w z (t) = Cx(t)
1
0 I 0 D 0 DT 3
0
Ai x (t 0 hi ) i=0 + Ad1 v(t) + B1 w(t) t where v (t) = t0h eA (t0s) x(s)ds. Example 1 [5]: We consider the following system:
0W T A0
3 3
Ai x (t 0 hi )
x_ (t) 0 F x_ (t 0 g ) =
(17)
h2 82
i=0
and exponential matrix Ad (s) = Ad1 expf0Ad0 sg, Theorem 1 can be applied to the following augmented system with discrete delays: v_ (t) =x(t) 0 eA d x(t 0 d) + Ad0 v (t);
0:
Since w0 is arbitrary, this yields (ii). Remark 1: For = 0 and D = 0 LMI (8) implies that 0 = CT : PT 0 B1
2
+
ei!t G(i! )w0 , the triple
From (2), it follows that for all t1 0: t 2 wT (t)z(t)dt = t1 w03 (G3 (i!) + G(i!)) w0
(18)
and, thus, implies A1 [10]. Remark 3: In the case of system (1a-b), with distributed delay
01
Ai e0i!h
i
LMI (8) implies for " ! 0+ the delay-independent LMI shown in (19) at the bottom of the page. If LMI (19) is strictly feasible (i.e., holds with strict inequality) then (8) is feasible for a small enough " > 0 and for Ri and Wi that are given by (18). Thus, from Theorem 1 the following corollary holds. Corollary 1: Items (i) and (ii) of Theorem 1 hold if there exist 0 < P1 = P1T ; P2 ; P3 , U = U T and Si = SiT ; i = 1; 2 such that (19) is strictly feasible. Remark 2: As we have seen above, the delay-dependent conditions of Theorem 1 [with strict LMI (8)] are most powerful in the sense that they provide sufficient conditions for both the delay-dependent and the delay-independent cases (where (19) is strictly feasible). In the latter case, (8) is feasible for hi ! 1; i = 1; 2. Moreover, strict LMI (8) yields the following LMI: 0P3 0 P3T + U P3T F < 0 (20) 3 0U
We substitute (15) into (12) and integrate the resulting inequality in t from 0 to t1 . We obtain (by Schur complements) that (3) holds if the LMI, as shown in (16) at the bottom of the page, is feasible, where for i = 1; 2
i1 Wi2 =Ri Mi P; Wi = W ; Wi3 Wi4 8i =WiT + P T ; 8i = [ 8i1 8i2 ] ; 2 2 0 ATi 9 =9 + WiT A0 00 + i i=1 i=1 0 0
= 0P; Ri = "Ih2n ; i = 1; . . . ; m
F
0
0 0 0
(16)
0U P2T A1 P3T A1
0 I 0 D 0 DT 0 3 0S 1 1 1 1
1
P2T A2 P3T A2
0 0
0S 3
2
P2T F P3T F
0 0 0
0U
0:
(19)
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where
= 00ak1 0ka ; 2 B1 = [ 0 1 ]T ; and
667
= 00c 00 C = [0 1]: In [5], it is shown that the system is passive with = 0 for a1 > 0 a2 > 0 c2 < a22 : By Theorem 1, choosing a1 = 1, a2 = 2, c = 2:3 and k = 2:2, which A0
A1
do not satisfy the conditions of [5], we find that the system is delay independently passive with = 0. Increasing c and taking c = 2:7 we obtain that the system is passive for 0 h < 9:8. The polar plot of the transfer function from w to z is depicted in Fig. 1. It resides entirely in the right half of the complex plane. We note that in the latter example we utilized (17) and the fact that P3T B1 = 0. Since B1 = [ 0 1 ]T and C = B1T P2 , we had that
P2
= 10 P2 + 01
C and P3
= P03
where P2 , P3 2 R122 . Thus, we solved (8) for P1 , P2 , P3 , S , Ri , i = 1, 2, 3, and Wj , j = 1 . . . 4. III. STATE-FEEDBACK CONTROL We apply the results of the previous section to the state-feedback design of passive systems. Given the system
x_ (t) 0 F x_ (t 0 g ) =A0 x(t) + A1 x (t 0 h1 ) + A2 x (t 0 h2 ) + B1 w(t) + B2 u(t) (t) + D12 u(t) + Dw(t) (23) z (t) =Cx
where x and w are defined in Section II, u 2 R` is the part of the control input that is used for feedback, F , A0 , A1 , A2 , B1 , B2 are
4
3 3 3 3 3 3 3 3 3 3 3
0 + QT B 1
CT
In
0 0 0 0 0 0
0
0S101 3 3 3 3
=
0
u(t) = Kx(t)
achieves passivity with
0
In
QT
0 0 0 0 0 0 0
0
0S201 3 3 3
A1
0 0
0h1 R1 3 3 3 3 3 3 3 3 3
0
QT
0 0 0 0 0 0 0 0
In
0U101 3 3
w
to z in Example 1 for
constant matrices of appropriate dimension, z is the objective vector, C 2 Rq2n , D12 2 Rq2` and D 2 Rq2q . We look for the state-feedback gain matrix K which, via the control law
h1 ("1 I + I ) h2 ("2 I + I ) "1
0
0 I 0 D 0 DT 3 3 3 3 3 3 3 3 3 3 QT
Fig. 1. The polar plot of the transfer function from h 9:7.
0h2 R2 3 3 3 3 3 3 3 3 h1 QT
0 of the closed-loop system.
"2
0 0 0
0 0 0 0 0 0 0 0 0 0 0
0
0S1 3 3 3 3 3 3 3
AT1
0h1 R101 3
0
0
F
A2
0 0 0 0
h2 QT
0 0 0 0 0
0S2 3 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0
(24)
0
0U1 3 3 3 3 3
AT2
0h2 R201
0
(27)
668
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=
+
=
A0
= A0 + B2 K; Ai = Ai ; i = 1; 2; C = C + D12 K:
=0
P 01
=1 = 1 ( + )
=0
=Q=
Q2
Q1 Q2
+ Q2T 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
0
Q3
(25)
41
0Q3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
QT 3
0 0 = 0 = =1
0
1 = diag fQ; I g
and
B1
0
"1 A1 S1 0 0 0 0S1
3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3
0
"2 A2 S2 0 0 0 0 0S2
3 3 3 3 3 3 3 3
Q1
0 0 0 0 0 0 0S1
3 3 3 3 3 3 3 3 3 3 3 3 3
Q1
0 0 0 0 0 0 0 3 0S2 3 3 3 3 3 3
3 3 3 3 3 3
0 0
0 0
+
0
0 = 0 = 0 =
(26a-b)
0
0
2
+
ATi
2
+ Y T B2T :
"i ATi
i=0 i=1 The state-feedback gain is then given by (24), where
K
h1 "1 h1 "1
0 I 0 D 0 DT
+
+ 0
41 = Q3 0 Q2T + Q1
( + 1) R11 ( + 1) R12T 0 0h1 R11
+ Y T D12T
Q1 C T
1
0
4=
Applying Theorem 1 to the above matrices, results in a nonlinear matrix inequality because of the terms P2T B2 K and P3T B2 K . We therefore consider another version of LMI condition which is derived from (16). In order to obtain an LMI, we have to restrict ourselves to the case "i P , i , 2, where "i 2 R is a scalar parameter. Note of Wi that for "i (8) implies the delay-dependent conditions of [10] (for ), while for "i 0 (8) yields the delay-independent condition F of Corollary 1. It is obvious, from the requirement of < P1 and the fact that in (8) the term 0 P3 P3T must be negative–definite, that P is nonsingular. Defining
=
1
we multiply (16) by T and , on the left and on the right, respectively. Applying the Schur formula to the quadratic term in Q, we obtain the inequality, as shown in (27) at the bottom of the previous page, where 2 T I i=0 Ai Q QT 2 I 0I i=0 Ai 0I 2 T i=1 "i Ai : Q QT 2 0I i=1 "i Ai We substitute (25) into (27), denote KQ1 by Y and obtain the following. Theorem 2: Assume A1. Consider the system of (23). The statefeedback law of (24) achieves passivity of the closed-loop system with
if for some prescribed scalars "1 , "2 2 R, there exist Q1 > , < S1 S101 , < S2 S201 , < U U 01 , Q2 ; Q3 ; 2 Rn2n , 0 1 0 1 2n22n < R1 R1 , < R2 R2 2 R , K 2 R`2n and ` 2 n Y 2 R that satisfy the LMI shown in (28) at the bottom of the page, where Ri1 , Ri2 and Ri3 are the (1,1), (1,2) and (2,2) blocks of , 2, and where Ri , i
Remark 4: The case where B1 B2 in the system description of (23) corresponds to the standard case where w is the external control command and the actual input to the plant is w Kx. The case where B1 6 B2 describes the situation where only a part of the control inputs is used for feedback. Substituting (24) into (23), we obtain the structure of (1a-b) with
0 F U 0 0 0 0 0 0 0 0U
( + 1) R12 ( + 1) R13 0 0h1 R12 0h1 R13
h1 "1 h1 "1
Q2T Q3T
0 0 0 0 0 0 0 0 3 0U
3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0
= Y Q101:
( + 1) R21 ( + 1) R22T 0 0 0 0h2 R21
h2 "2 h2 "2
h1 Q2T AT1 h1 Q3T AT1
0 0 0 0 0 0 0 0 0
3 3 3 3 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0 0 0
(29)
( + 1) R22 ( + 1) R23 0 0 0 0h2 R22 0h2 R23
h1 "2 h2 "2
3 3 3 3 3 3 3 3 3 3
h2 Q2T AT2 h2 Q3T AT2
0 0 0 0 0 0 0 0 0 0 0
3 0h1 R11 0h1 R12 3 3 0h1 R13 3 3 3 0h2 R21 0h2 R22 3 3 3 3 0h2 R23
0
(28)
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The above results refer to the case where F = 0. For F = diag f0:1; 0:2g we obtained, applying = 0:33 and = 0:2, that the state feedback gain
K = [ :0015
099:8821 ]
yields a passive system 8 h 2 [0 1:145]. Remark 5: Note that the products i Ai Si and i Rij (i = 1, 2, j = 1, 2, 3) in (28) are nonlinear in the unknown parameters. However, since i are scalars, we solve (28) for different values of i that lead to a minimum . For example, in Example 2 we calculated the minimum achievable for different values of . We have found there that the function ( ) is convex and this is how we obtained the optimal value of = 00:33. IV. CONCLUSION
Fig. 2. D
The polar plot of the closed-loop transfer function from
= 0 1 in Example 2.
w
to
z
for
:
The LMIs in Theorems 1 and 2 are affine in the system matrices. Thus, it can be applied also to the case where these matrices are uncertain and are known to reside within a given polytope. Example 2: We consider the system
x(t) _ 0 F x(t _ 0 g) =A0 x(t) + A1 x(t 0 h) + B1 w(t) + B2 u(t) z(t) =Cx(t) + D12 u(t) + Dw(t)
(30)
where
0 A0 = 0 0 B1 = 1 D12 =0:1;
0 2
A1 =
01 03
0 0
0 C = [0 1] 1 and F = 0: B2 =
(31)
This system describes a case where the external input w and the feedback input u are applied via the same input matrix B1 = B2 . Considering D = 0, we seek a state-feedback gain matrix K that will result in a passive closed-loop system. Applying Theorem 2 we obtained that for = 0:2, h = 1:26 and = 0:3, the closed-loop system (30) with
u = Kx(t) = [ :0143
099:4224 ] x(t)
is passive. The same state-feedback gain matrix makes the system positive real in the case of = 0 and D = 0:1. The polar plot of the resulting closed-loop transfer function, from the control input w to z (with D = :1) is depicted in Fig. 2. It is clearly seen that this plot resides entirely in the right half of the complex plane and its “distance” from the imaginary axis may be considered as the overdesign that stems from the fact that the condition provided by Theorem 2 is only sufficient.
A delay-dependent solution is proposed for the problem of passive state feedback control of linear time-invariant neutral and retarded type systems. The solution provides sufficient conditions in the form of LMIs. Although these conditions are not necessary, the overdesign entailed is minimal since it is based on an equivalent (descriptor) model transformation, which leads to the bounding of a smallest number of cross terms and since a new less conservative bounding is applied. One question that often arises when solving control problems for systems with time-delay is whether the solution obtained for certain i hi . delays hi will satisfy the design requirements for all delays h The answer is the affirmative, since the LMI in Theorem 2 is affine in the time delays. REFERENCES [1] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994, vol. 15, SIAM Studies in Applied Mathematics. [2] R. Curtain, “Old and new perspectives on the positive-real lemma in systems and control theory,” Z. Angew. Math. Mech., vol. 79, pp. 579–590, 1999. [3] V. Bevelevich, Classical Network Synthesis. New York: Van Nostrand, 1968. [4] C. Desoer and M. Vidyasagar, Feedback System: Input-Output Properties. New York: Academic, 1975. [5] S. I. Niculescu and R. Lozano, “On the passivity of linear delay systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 460–464, Mar. 2001. [6] V. Razvan, S. I. Niculescu, and R. Lozano, “Input-output passive framework for delay systems,” in Proc. Conf. Decision Control, 2000. [7] G. Lu, L. Yeung, D. Ho, and Y. Zheng, “Strict positive realness for linear time-invariant systems with time-delays,” in Proc. Conf. Decision Control, 2000. [8] J. Hale, Functional Differential Equations. New York: SpringerVerlag, 1977. [9] V. Kolmanovskii and J.-P. Richard, “Stability of some linear systems with delays,” IEEE Trans. Automat. Control, vol. 44, pp. 984–989, May 1999. [10] E. Fridman, “New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems,” Syst. Control Lett., vol. 43, pp. 309–319, 2001. [11] L. Xie, E. Fridman, and U. Shaked, “A robust H control of distributed delay systems with application to combustion control,” IEEE Trans. Automat. Contr., vol. 46, pp. 1930–1935, Dec. 2001. [12] E. Fridman and U. Shaked, “A descriptor system approach to H control of time-delay systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 253–279, Feb. 2002. [13] P. Park, “A delay-dependent stability criterion for systems with uncertain time-invariant delays,” IEEE Trans. Automat. Contr., vol. 44, pp. 876–877, Apr. 1999. [14] R. Lozano, B. Brogliato, O. Egeland, and B. Maschke, Dissipative Systems Analysis and Control. Theory and Applications. London, U.K.: Springer-Verlag, 2000.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
On Delay-Dependent Passivity E. Fridman and U. Shaked
Abstract—Sufficient conditions for passivity (positive realness) are obtained for continuous-time, linear, retarded, and neutral-type systems. A delay-dependent solution is given in terms of linear matrix inequalities (LMIs) by using a descriptor model transformation of the system and by applying Park’s inequality for bounding cross terms. A memoryless state-feedback solution is derived. Numerical examples are given which illustrate the effectiveness of the new theory. Index Terms—Delay-dependent criteria, linear matrix inequalities (LMIs), positive-real lemma, time-delay systems.
I. INTRODUCTION Positive realness (passivity) theory plays an important role in both electrical network and control systems (see, e.g., [1], [2]) and it has roots in circuit theory ([3], [4]). For systems with delay of retardedtype, positive realness has been studied by [5]–[7]. In [5], delay-independent sufficient conditions in terms of LMIs have been derived. In [6] necessary and sufficient conditions are given in terms of positivity of some kernel matrix constructed via transition matrix. In [7] frequency domain approach is applied and sufficient conditions are obtained. For infinite-dimensional systems, a positive-real lemma has been obtained in terms of Riccati operator equations (see [2] and the references therein). In the present note we give delay-dependent sufficient conditions for passivity of neutral type systems. We apply descriptor-type Lyapunov–Krasovskii functionals that were recently introduced in [10]–[12] for delay-dependent stability and control and Park’s inequality for bounding cross terms [13]. We also present a memoryless state-feedback controller via LMIs, such that the resulting closed-loop system is passive. Notation: Throughout the note, the superscript T stands for matrix transposition, Rn denotes the n dimensional Euclidean space, Rn2m is the set of all n 2 m real matrices, and the notation P > 0, for P 2 Rn2n means that P is symmetric and positive–definite. Denote xt () = x(t + )( 2 [0h; 0]). II. PASSIVITY AND POSITIVE REALNESS FOR LINEAR TIME-DELAY SYSTEMS Given the following system
x_ (t) 0 F x_ (t 0 g ) =
2
i=0
Ai x (t 0 hi ) + B1 w(t)
z (t) =Cx(t) + Dw(t)
dimensions. Denote h = maxfh1 ; h2 g. For simplicity only we consider a single delay g and two delays h1 and h2 . The results of this note can be easily applied to the case of multiple delays g1 ; . . . ; gm , h1 ; . . . ; hm and a distributed delay. Equation (1a) describes a system of neutral type since it contains a derivative with delay. In the case of F = 0 (1a) is a retarded type system (see, e.g., [8]). Neutral systems are encountered in modeling of lossless transmission lines, or in dynamical processes including steam and water pipes (see, e.g., [8] and the references therein). Unlike retarded systems, linear neutral systems may be destabilized by small changes of the delay and may be unstable even in the case when all the roots of the characteristic equation have negative real parts [8]. We are looking for a criterion for passivity that depends on the delays hi and does not depend on g . Delay-independence with respect to g guarantees that small changes in g do not destabilize the system [8]. To guarantee robustness of the results with respect to small changes of delay, we assume that the difference equation Dxt = x(t) 0 F x(t 0 g ) = 0 is asymptotically stable for all values of g or, equivalentaly, that A1 F is a Schur–Cohn stable matrix, i.e., all the eigenvalues of F are inside the unit circle. The transfer function of (1a-b) from w to z is given by
G(s) = C s I 0 F e0sg
0
2
i=0
Ai e0sh
01 B1 + D:
Definition 1: [1] The system (1a-b) is called passive if t
2
0
wT (t)z (t)dt 0
(2)
for all t1 0 and for all solution of (1a-b) with x0 = 0. Another less restrictive definition of passivity is given by [14]. Definition 2: [14] The system (1a-b) is called passive if there exists
0 such that t
2
0
t
wT (t)z (t)dt 0
0
wT (s)w(s)ds:
(3)
for all t1 0 and for all solution of (1a-b) with x0 = 0. Different model transformations were used in the past for delay-dependent stability (see, e.g., [9] and [13]). Recently, a new (descriptor) model transformation has been introduced [10]. Unlike previous transformations, the descriptor model leads to a system which is equivalent to the original one, it does not depend on additional assumptions for the stability of the transformed system and it requires bounding of fewer crossterms. It was shown in [10] and [12] that the latter transformation leads to less conservative conditions for stability and H control. Following [10], we represent (1a-b) in the equivalent descriptor form
1
(1a-b)
where x(t) 2 Rn is the system state vector, w(t) 2 Rq is the exogenous input, which can be either a control input or a reference signal and z (t) 2 Rq is the output of the system. The time delays 0 = h0 , 0 < hi , i = 1; 2 and g > 0 are assumed to be known. The matrices Ai , i = 0; . . . ; 2, F , B1 and C are constant matrices of appropriate Manuscript received May 9, 2001; revised September 24, 2001. Recommended by Associate Editor V. Balakrishnan. This work was supported by the Ministry of Absorption of Israel, and also by the C & M Maus Chair at Tel Aviv University. The authors are with the Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(02)03748-0.
x_ (t) = y (t);
y (t) = F y (t
0 g) +
2
i=0
Ai x (t
0 hi ) + B1 w(t):
(4)
The latter is equivalent to the following descriptor system with discrete and distributed delay in the variable y : x_ (t) =y (t) y (t) =F y (t
0018-9286/02$17.00 © 2002 IEEE
0
0 g) +
2
i=1
Ai
t t0h
2
i=0
Ai
x(t)
y ( )d + B1 w(t):
(5)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
665
the transfer matrix G of (1a-b) is positive real, i.e.,
A Lyapunov–Krasovskii functional for the system (5) has the form
+
2
t
i=1
t h
0
2
+
T
x
0
t
( )Si x( )d +
t
i=1 0h
G(i! )3 + G(i! ) 0:
x(t) y (t)
V (xt ; yt ) = xT (t)y T (t) EP
0
t g
y
T
( )U y( )d
y T (s)ATi Ri3 Ai y (s)d d
t+
Proof: (i) We note that
(6)
= I0n 00
=
P
d dt
0
P1 P2
91 =
i=0
+
2
i=1
2
P2 + P2
ATi
T
WiT3 Ai
+ 2
92 =P1 0 P2T +
i=0
93 = 0 P3 0 P3T +
i=0
ATi Wi3
P3 +
ATi
2
i=1
i=1
i=1
Si
0
91 92 3 93
3 3 3 3 3 3
3 3 3 3 3 1
i=0
9
T
0
2
3 3
i=1
PT
0
F
0 I 0 D 0 DT 0 3 0U xT (t 0 hi ) Si x (t 0 hi ) t
y T ( )ATi Ri3 Ai y ( )d
0
t h
t
0
t h
0 i
0
0
t h
0
Ai
y (s)ds:
P
(13)
0 and Mi , the following inequality
bT (s)a(s)ds
h1 811 h1 812
xT (t)y T (t) P T
For any 2n 2 2n-matrices Ri > holds [13]:
02
(10)
t
02
i (t)
6= 0
P2T B1 0 C T P3T B1 0 I 0 D 0 DT
3 3 3 1 1
0 0 CT 0 B1
PT
2 AT 0 I 0 i=0 i 2 Ai 0I + I 0I i=0 2 Si 0 i=1 2 Ui + hi AT Ri3 Ai 0 i i=1
PT
+
(9)
Ai e0i!h
:
colfx(t); y (t); w(t); y (t 0 g )g and
9
ATi Wi4
i3
i! I 0 F e0i!g
0 2zT w 0 wT w
=
where
Then, the following holds. i) The system (1a-b) is passive in the sense of Definition 2. ii) In the case of = 0 for all ! 2 R with
det
dV (xt ; yt ) dt
Ui + hi AiT Ri3 Ai
2
0
(12)
2 2
x_ (t)
Substituting (5) into (11), we obtain
Ai
+
xT (t)y T (t) P T
(11)
+
8i1 = [ WiT1 + P1 WiT3 + P2T ] 8i2 = [ WiT2 WiT4 + P3T ] Ri1 Ri2 Ri = : RT R i2
=2xT (t)P1x_ (t) =2
(7a-b) The first term of (6) corresponds to the descriptor system, the third—to the delay-independent conditions with respect to the discrete delays of y , while the second and the fourth terms correspond to the delay-dependent conditions with respect to the distributed delays (with respect to x). We obtain the following. Theorem 1: Assume A1. Consider the system of (1a-b). Let there exist n 2 n-matrices 0 < P1 ; P2 ; P3 ; Si = SiT , U = U T , Wi1 ; Wi2 ; Wi3 ; Wi4 , Ri1 = RiT1 ; Ri2 ; Ri3 = RiT3 ; i = 1; 2 and
0 that satisfy the linear matrix inequality (LMI), as shown in (8) at the bottom of the page, where
2
x(t) y (t)
xT (t)y T (t) EP
P1 > 0; U > 0; Si > 0:
P3
= xT P1 x
and, hence, differentiating the first term of (6) with respect to t we have
where
E
x y
xT y T EP
t
0
t h
a(s) b(s)
T
Ri MiT Ri
Ri Mi (2; 2)
a(s) ds (14) b(s)
for a(s) 2 R2n , b(s) 2 R2n given for s 2 [t 0 hi ; t]. Here, (2; 2) = (MiT Ri + I )Ri01(Ri Mi + I ).
h2 821 h2 822
0
0h1 R1 0 3 0h2 R2 3 3 3 3 3 3
0W13T A1 0W23T A2 0W14T A1 0W24T A2 0 0 0
0S1 3 3
0 0 0 0
0S2 3
P2T F P3T F
0 0 0 0 0
0U
0
(8)
666
b
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Using this inequality for a(s)
colf0 Ai gy (s) and
=
= P colfx(t) y(t)g, we obtain
For
Wi
x i hi [ xT y T ] P T MiT Ri + I Ri01 (Ri Mi + I ) P y x + 2 xT (t) 0 xT (t 0 hi ) [ 0 ATi ] Ri Mi P y t 0 y(s)ds: + y T (s)[ 0 ATi ] Ri (15) A i t0h
Wi
Wi :
LMI (8) results from the latter LMI by expansion of the block matrices. (ii) Let ! be such that (10) holds and consider w(t) = ei!t w0 , w0 2 Rq . Define
x(t) = ei!t i! I 0 F e0i!g
0
2
i=0
x_ (t) 0 F x_ (t 0 gi ) =
B1 !0
and z (t) = Cx(t) + Dw(t). Then z (t) = (w; x; z ) satisfies (1a-b) and 2wT (t)z(t) = w0T [G3 (i!) + G(i!)] w0 :
0
9
PT
0 0 CT 0 B1
h1 81
0h R
0 0
3 3 3 3
3 3 3 3
3 3 3 3
0h R 3 3 3
AT0 P2 + P2T A0 +
3 3 3 3 3
0 1
2 =1
i
1
Si
2
1
0 0
0W T A0 2
1
3 3 3 3
Ad (s)x(t + s)ds + B1 w(t)
2
0S 3
2
P2T B1 0 C T PTB 3
1
(21)
(22)
0
PT
0 0
0 0
0S 3 3
P1 0 P2T + AT0 P3 0P3 0 P3T + U
2
0 0
0
2
0h
x_ (t) = A0 x(t) + A1 x(t 0 h) + B1 w z (t) = Cx(t)
1
0 I 0 D 0 DT 3
0
Ai x (t 0 hi ) i=0 + Ad1 v(t) + B1 w(t) t where v (t) = t0h eA (t0s) x(s)ds. Example 1 [5]: We consider the following system:
0W T A0
3 3
Ai x (t 0 hi )
x_ (t) 0 F x_ (t 0 g ) =
(17)
h2 82
i=0
and exponential matrix Ad (s) = Ad1 expf0Ad0 sg, Theorem 1 can be applied to the following augmented system with discrete delays: v_ (t) =x(t) 0 eA d x(t 0 d) + Ad0 v (t);
0:
Since w0 is arbitrary, this yields (ii). Remark 1: For = 0 and D = 0 LMI (8) implies that 0 = CT : PT 0 B1
2
+
ei!t G(i! )w0 , the triple
From (2), it follows that for all t1 0: t 2 wT (t)z(t)dt = t1 w03 (G3 (i!) + G(i!)) w0
(18)
and, thus, implies A1 [10]. Remark 3: In the case of system (1a-b), with distributed delay
01
Ai e0i!h
i
LMI (8) implies for " ! 0+ the delay-independent LMI shown in (19) at the bottom of the page. If LMI (19) is strictly feasible (i.e., holds with strict inequality) then (8) is feasible for a small enough " > 0 and for Ri and Wi that are given by (18). Thus, from Theorem 1 the following corollary holds. Corollary 1: Items (i) and (ii) of Theorem 1 hold if there exist 0 < P1 = P1T ; P2 ; P3 , U = U T and Si = SiT ; i = 1; 2 such that (19) is strictly feasible. Remark 2: As we have seen above, the delay-dependent conditions of Theorem 1 [with strict LMI (8)] are most powerful in the sense that they provide sufficient conditions for both the delay-dependent and the delay-independent cases (where (19) is strictly feasible). In the latter case, (8) is feasible for hi ! 1; i = 1; 2. Moreover, strict LMI (8) yields the following LMI: 0P3 0 P3T + U P3T F < 0 (20) 3 0U
We substitute (15) into (12) and integrate the resulting inequality in t from 0 to t1 . We obtain (by Schur complements) that (3) holds if the LMI, as shown in (16) at the bottom of the page, is feasible, where for i = 1; 2
i1 Wi2 =Ri Mi P; Wi = W ; Wi3 Wi4 8i =WiT + P T ; 8i = [ 8i1 8i2 ] ; 2 2 0 ATi 9 =9 + WiT A0 00 + i i=1 i=1 0 0
= 0P; Ri = "Ih2n ; i = 1; . . . ; m
F
0
0 0 0
(16)
0U P2T A1 P3T A1
0 I 0 D 0 DT 0 3 0S 1 1 1 1
1
P2T A2 P3T A2
0 0
0S 3
2
P2T F P3T F
0 0 0
0U
0:
(19)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
where
= 00ak1 0ka ; 2 B1 = [ 0 1 ]T ; and
667
= 00c 00 C = [0 1]: In [5], it is shown that the system is passive with = 0 for a1 > 0 a2 > 0 c2 < a22 : By Theorem 1, choosing a1 = 1, a2 = 2, c = 2:3 and k = 2:2, which A0
A1
do not satisfy the conditions of [5], we find that the system is delay independently passive with = 0. Increasing c and taking c = 2:7 we obtain that the system is passive for 0 h < 9:8. The polar plot of the transfer function from w to z is depicted in Fig. 1. It resides entirely in the right half of the complex plane. We note that in the latter example we utilized (17) and the fact that P3T B1 = 0. Since B1 = [ 0 1 ]T and C = B1T P2 , we had that
P2
= 10 P2 + 01
C and P3
= P03
where P2 , P3 2 R122 . Thus, we solved (8) for P1 , P2 , P3 , S , Ri , i = 1, 2, 3, and Wj , j = 1 . . . 4. III. STATE-FEEDBACK CONTROL We apply the results of the previous section to the state-feedback design of passive systems. Given the system
x_ (t) 0 F x_ (t 0 g ) =A0 x(t) + A1 x (t 0 h1 ) + A2 x (t 0 h2 ) + B1 w(t) + B2 u(t) (t) + D12 u(t) + Dw(t) (23) z (t) =Cx
where x and w are defined in Section II, u 2 R` is the part of the control input that is used for feedback, F , A0 , A1 , A2 , B1 , B2 are
4
3 3 3 3 3 3 3 3 3 3 3
0 + QT B 1
CT
In
0 0 0 0 0 0
0
0S101 3 3 3 3
=
0
u(t) = Kx(t)
achieves passivity with
0
In
QT
0 0 0 0 0 0 0
0
0S201 3 3 3
A1
0 0
0h1 R1 3 3 3 3 3 3 3 3 3
0
QT
0 0 0 0 0 0 0 0
In
0U101 3 3
w
to z in Example 1 for
constant matrices of appropriate dimension, z is the objective vector, C 2 Rq2n , D12 2 Rq2` and D 2 Rq2q . We look for the state-feedback gain matrix K which, via the control law
h1 ("1 I + I ) h2 ("2 I + I ) "1
0
0 I 0 D 0 DT 3 3 3 3 3 3 3 3 3 3 QT
Fig. 1. The polar plot of the transfer function from h 9:7.
0h2 R2 3 3 3 3 3 3 3 3 h1 QT
0 of the closed-loop system.
"2
0 0 0
0 0 0 0 0 0 0 0 0 0 0
0
0S1 3 3 3 3 3 3 3
AT1
0h1 R101 3
0
0
F
A2
0 0 0 0
h2 QT
0 0 0 0 0
0S2 3 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0
(24)
0
0U1 3 3 3 3 3
AT2
0h2 R201
0
(27)
668
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002
=
+
=
A0
= A0 + B2 K; Ai = Ai ; i = 1; 2; C = C + D12 K:
=0
P 01
=1 = 1 ( + )
=0
=Q=
Q2
Q1 Q2
+ Q2T 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
0
Q3
(25)
41
0Q3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
QT 3
0 0 = 0 = =1
0
1 = diag fQ; I g
and
B1
0
"1 A1 S1 0 0 0 0S1
3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3
0
"2 A2 S2 0 0 0 0 0S2
3 3 3 3 3 3 3 3
Q1
0 0 0 0 0 0 0S1
3 3 3 3 3 3 3 3 3 3 3 3 3
Q1
0 0 0 0 0 0 0 3 0S2 3 3 3 3 3 3
3 3 3 3 3 3
0 0
0 0
+
0
0 = 0 = 0 =
(26a-b)
0
0
2
+
ATi
2
+ Y T B2T :
"i ATi
i=0 i=1 The state-feedback gain is then given by (24), where
K
h1 "1 h1 "1
0 I 0 D 0 DT
+
+ 0
41 = Q3 0 Q2T + Q1
( + 1) R11 ( + 1) R12T 0 0h1 R11
+ Y T D12T
Q1 C T
1
0
4=
Applying Theorem 1 to the above matrices, results in a nonlinear matrix inequality because of the terms P2T B2 K and P3T B2 K . We therefore consider another version of LMI condition which is derived from (16). In order to obtain an LMI, we have to restrict ourselves to the case "i P , i , 2, where "i 2 R is a scalar parameter. Note of Wi that for "i (8) implies the delay-dependent conditions of [10] (for ), while for "i 0 (8) yields the delay-independent condition F of Corollary 1. It is obvious, from the requirement of < P1 and the fact that in (8) the term 0 P3 P3T must be negative–definite, that P is nonsingular. Defining
=
1
we multiply (16) by T and , on the left and on the right, respectively. Applying the Schur formula to the quadratic term in Q, we obtain the inequality, as shown in (27) at the bottom of the previous page, where 2 T I i=0 Ai Q QT 2 I 0I i=0 Ai 0I 2 T i=1 "i Ai : Q QT 2 0I i=1 "i Ai We substitute (25) into (27), denote KQ1 by Y and obtain the following. Theorem 2: Assume A1. Consider the system of (23). The statefeedback law of (24) achieves passivity of the closed-loop system with
if for some prescribed scalars "1 , "2 2 R, there exist Q1 > , < S1 S101 , < S2 S201 , < U U 01 , Q2 ; Q3 ; 2 Rn2n , 0 1 0 1 2n22n < R1 R1 , < R2 R2 2 R , K 2 R`2n and ` 2 n Y 2 R that satisfy the LMI shown in (28) at the bottom of the page, where Ri1 , Ri2 and Ri3 are the (1,1), (1,2) and (2,2) blocks of , 2, and where Ri , i
Remark 4: The case where B1 B2 in the system description of (23) corresponds to the standard case where w is the external control command and the actual input to the plant is w Kx. The case where B1 6 B2 describes the situation where only a part of the control inputs is used for feedback. Substituting (24) into (23), we obtain the structure of (1a-b) with
0 F U 0 0 0 0 0 0 0 0U
( + 1) R12 ( + 1) R13 0 0h1 R12 0h1 R13
h1 "1 h1 "1
Q2T Q3T
0 0 0 0 0 0 0 0 3 0U
3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0
= Y Q101:
( + 1) R21 ( + 1) R22T 0 0 0 0h2 R21
h2 "2 h2 "2
h1 Q2T AT1 h1 Q3T AT1
0 0 0 0 0 0 0 0 0
3 3 3 3 3 3 3 3 3 3 3
0 0 0 0 0 0 0 0 0 0 0 0 0
(29)
( + 1) R22 ( + 1) R23 0 0 0 0h2 R22 0h2 R23
h1 "2 h2 "2
3 3 3 3 3 3 3 3 3 3
h2 Q2T AT2 h2 Q3T AT2
0 0 0 0 0 0 0 0 0 0 0
3 0h1 R11 0h1 R12 3 3 0h1 R13 3 3 3 0h2 R21 0h2 R22 3 3 3 3 0h2 R23
0
(28)
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The above results refer to the case where F = 0. For F = diag f0:1; 0:2g we obtained, applying = 0:33 and = 0:2, that the state feedback gain
K = [ :0015
099:8821 ]
yields a passive system 8 h 2 [0 1:145]. Remark 5: Note that the products i Ai Si and i Rij (i = 1, 2, j = 1, 2, 3) in (28) are nonlinear in the unknown parameters. However, since i are scalars, we solve (28) for different values of i that lead to a minimum . For example, in Example 2 we calculated the minimum achievable for different values of . We have found there that the function ( ) is convex and this is how we obtained the optimal value of = 00:33. IV. CONCLUSION
Fig. 2. D
The polar plot of the closed-loop transfer function from
= 0 1 in Example 2.
w
to
z
for
:
The LMIs in Theorems 1 and 2 are affine in the system matrices. Thus, it can be applied also to the case where these matrices are uncertain and are known to reside within a given polytope. Example 2: We consider the system
x(t) _ 0 F x(t _ 0 g) =A0 x(t) + A1 x(t 0 h) + B1 w(t) + B2 u(t) z(t) =Cx(t) + D12 u(t) + Dw(t)
(30)
where
0 A0 = 0 0 B1 = 1 D12 =0:1;
0 2
A1 =
01 03
0 0
0 C = [0 1] 1 and F = 0: B2 =
(31)
This system describes a case where the external input w and the feedback input u are applied via the same input matrix B1 = B2 . Considering D = 0, we seek a state-feedback gain matrix K that will result in a passive closed-loop system. Applying Theorem 2 we obtained that for = 0:2, h = 1:26 and = 0:3, the closed-loop system (30) with
u = Kx(t) = [ :0143
099:4224 ] x(t)
is passive. The same state-feedback gain matrix makes the system positive real in the case of = 0 and D = 0:1. The polar plot of the resulting closed-loop transfer function, from the control input w to z (with D = :1) is depicted in Fig. 2. It is clearly seen that this plot resides entirely in the right half of the complex plane and its “distance” from the imaginary axis may be considered as the overdesign that stems from the fact that the condition provided by Theorem 2 is only sufficient.
A delay-dependent solution is proposed for the problem of passive state feedback control of linear time-invariant neutral and retarded type systems. The solution provides sufficient conditions in the form of LMIs. Although these conditions are not necessary, the overdesign entailed is minimal since it is based on an equivalent (descriptor) model transformation, which leads to the bounding of a smallest number of cross terms and since a new less conservative bounding is applied. One question that often arises when solving control problems for systems with time-delay is whether the solution obtained for certain i hi . delays hi will satisfy the design requirements for all delays h The answer is the affirmative, since the LMI in Theorem 2 is affine in the time delays. REFERENCES [1] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994, vol. 15, SIAM Studies in Applied Mathematics. [2] R. Curtain, “Old and new perspectives on the positive-real lemma in systems and control theory,” Z. Angew. Math. Mech., vol. 79, pp. 579–590, 1999. [3] V. Bevelevich, Classical Network Synthesis. New York: Van Nostrand, 1968. [4] C. Desoer and M. Vidyasagar, Feedback System: Input-Output Properties. New York: Academic, 1975. [5] S. I. Niculescu and R. Lozano, “On the passivity of linear delay systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 460–464, Mar. 2001. [6] V. Razvan, S. I. Niculescu, and R. Lozano, “Input-output passive framework for delay systems,” in Proc. Conf. Decision Control, 2000. [7] G. Lu, L. Yeung, D. Ho, and Y. Zheng, “Strict positive realness for linear time-invariant systems with time-delays,” in Proc. Conf. Decision Control, 2000. [8] J. Hale, Functional Differential Equations. New York: SpringerVerlag, 1977. [9] V. Kolmanovskii and J.-P. Richard, “Stability of some linear systems with delays,” IEEE Trans. Automat. Control, vol. 44, pp. 984–989, May 1999. [10] E. Fridman, “New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems,” Syst. Control Lett., vol. 43, pp. 309–319, 2001. [11] L. Xie, E. Fridman, and U. Shaked, “A robust H control of distributed delay systems with application to combustion control,” IEEE Trans. Automat. Contr., vol. 46, pp. 1930–1935, Dec. 2001. [12] E. Fridman and U. Shaked, “A descriptor system approach to H control of time-delay systems,” IEEE Trans. Automat. Contr., vol. 47, pp. 253–279, Feb. 2002. [13] P. Park, “A delay-dependent stability criterion for systems with uncertain time-invariant delays,” IEEE Trans. Automat. Contr., vol. 44, pp. 876–877, Apr. 1999. [14] R. Lozano, B. Brogliato, O. Egeland, and B. Maschke, Dissipative Systems Analysis and Control. Theory and Applications. London, U.K.: Springer-Verlag, 2000.
