- Email: [email protected]
Output feedback stabilization of linear systems with delayed input*
IFAC
[:0[>
Copyright ~ IFAC Linear Time Delay Systems, Ancona, Italy, 2000
Publications www.elsevier.com ilocateiifac
OUTPUT FEEDBACK STABILIZATION OF LINEAR SYSTEMS WITH DELAYED INPUT* Xi Lil,
Carlos E. de Souza 2
1 Husky Oil Operations Ltd. 707, 8th Avenue S. w., Box 6525, Station D Calgary , Alberta T2P 3G7, Canada E-mail: [email protected]
2 Department of Systems and Control Laboratorio NacioTwl de Computat;iio Cientifica - LNCC Av. Getulio Vargas 333, 25651-070 Petropolis , RJ, Brazil E-mail: csouza@lncc .br
Abstract. This paper addresses the problem of output feedback stabilization of linear continuous time systems with time-delay in the control input. Attention is focused on the design of delaydependent output feedback controllers that ensure global asymptotic stability of the closed-loop system. Two controller design methods have been developed . The first one is concerned with the design of a control law which comprises a static feedback of the system output with the addition of a term involving the integral of past values of the input signal on a window of the size of the time-delay. On the other hand , the second method provides an observer-based controller. Copyright
©
2000 [FAC
Key Words. Output feedback stabilization , time-delay systems , observer-based controller, linear matrix inequalities.
1
INTRODUCTION
feedback have been considered. For example, stabilization techniques which are independent of the size of the time-delay have been proposed in Shen et al. (1991), Xie and de Souza (1993) , Lee et al. (1994), Choi and Chung (1996) , and Jeung et al. (1996) , whereas delay-dependent stabilization methods have been developed in Kojima and Ishijima (1995) , Li and de Souza (1997a,b, 1998) and de Souza and Li (1999).
One of the main problems encountered in many practical control designs is the existence of timedelay in either the control input , the states or the measurements; see, e.g. Malek-Zavarei and Jamshidi (1987) and Dugard and Verriest (1998). Time-delay is inherent in many process control systems (Morari and Zafiriou , 1989) and it widely appears in aircraft control design (Etkin , 1972). The problem of stabilization of dynamic systems with time-delays are, therefore, of theoretical and practical importance and has attracted considerable attention for several decades . Various techniques of stabilization of linear time-delay systems have been proposed over the past few years. where memory less state feedback and dynamic output
This paper considers the problem of output feedback stabilization of linear continuous-time with a single, constant, time-delay in the control input. Attention is focused on design methods of output feedback controllers which are dependent on the time-delay and two design methods are proposed. The first one deals with the design of a control law which comprises a static feedback of the system output and a term involving the integral of past values of the input signal on a window of the size of the time-delay. This method involve
-This work was supported in part by 'Conselho Nacional de Desenvolvimento Cientifico e Tecnologico - C!'>Pq ', Brazil , under grant 301653/ 96-8 / PQ for the second author.
39
the solvability of a rank minimization problem, which can be solved via existing techniques such as the cone complementary linearization algorithm of El Ghaoui et al. (1997). The second method is concerned with the design of an observer-based controller and is in terms of linear matrix inequalities (LMls).
Remark 1. Note that if either the matrix Y or Z has rank n, then the first or second inequality in (3) disappears. 0
Notation. Rn denotes the n dimensional Euc1idean space, Rnxm is the set of n x m real matrices, Amin (.) and Amax (.) stand for the minimum and maximum eigenvalue of a symmetric matrix, respectively, Tr{ . } denotes the matrix trace, and the notation X > 0, for a square matrix X, means that X is symmetric and positive definite.
In this section we will present two delay-dependent methods of designing output feedback stabilizing controllers for the time-delay system (1). The proposed methods have the advantage that can be implemented numerically via LMI techniques.
2
3
The first theorem deals with a control law which comprises a static feedback of the system output with the addition of a term involving the integral of the input signal over the last T seconds.
SYSTEM DEFINITION AND PRELIMIN ARIES
Theorem 1. Consider the system (l) satisfying Assumption 1. Then this system is stabilizable via an output feedback control law if the following equivalent conditions hold:
Consider linear time-delay systems described by the following state equation: x(t) yet)
= Ax(t) + Bu(t -
T)
(1)
= Cx(t)
MAli" RESULTS
(i) There exist matrices P that
where x(t) E Rn is the state, u(t) E Rnu is the control input , yet) E Rn.p is the measured output, T 2': 0 is a constant time-delay, and A, Band C are known real constant matrices of appropriate dimensions.
NJ;T(AQ
Q
=
and Q
>0
+ QAT)NBT < 0
N:! ~T (T)(PA
This paper addresses the problem of designing an output feedback control law for the system (1) such that the resulting closed-loop system is globally asymptotically stable. Attention is focused on design techniques of output feedback stabilizing controllers which are dependent on the timedelay.
>0
(4)
+ AT P)~(T)Ne <
0
p- 1
is
stabilizable
where NBT and life are any matrices whose columns form bases of the null spaces of BT and C, 7'espectively, and ~(t) = eAt. (ii) There exist matrices P
>0
and K such that
PA+ATp+PBKC~(-T)
(7) Moreover, a suitable output feedback control law is given by
and
Note that Assumption L which is necessary for output feedback stabilization of the system (1) in the absence of time-delay, is a necessary condition for the existence of a stabilizing output feedback control law for the system (1).
u(t)
=Ky(t) + KClOT~( -( T + 8) )Bu(t + 8)d8 (8)
Proof. (ii) Suppose that there exist matrices P > 0 and K satisfying the inequality of (7) . Let \'(t) = xT(t)Px(t) be a Lyapunov function candidate for the closed-loop system of (1) with the control law of (8). One readily obtains that the time-derivative of \/ (t) along the trajectory of this closed-loop system is given by
We conclude this section by recalling a lemma which will be used in the derivation of the results of this paper (see. e.g .. Gahinet and Apkarian, 1994).
Lemma 1. Given a symmetric matrix G E Rnxn and matrices Y E Rrxn and Z E Rsxn, consider the problem of finding some matrix e E Rrxs such that
\~'(t)
+
Denote by N y and .A/z any matrices whose columns form bases of the null spaces of Y and Z, respectively. Then (2) is solvable for if and only if
e
< 0,
NI G.A/z
< O.
= xT(t)(PA + ATp)x(t) + 2xT (t)P BKC[x(t - T)
(2)
.A/? GNy
(5) (6)
The system (1) is supposed to satisfy the following assumption:
Assumption 1. (A, B) (A, C) is detectable.
such
lOT ~(- (T + 8) )Bu(t -
T + 8)d8]
(9)
Note that since x(t)
(3)
40
= ~(T)X(t -
T)
+ !~T ~(t
- s)Bu(s - T)ds
then with the change of integration variable 8
=
back stabilization result for delay free systems (El Ghaoui and Gahinet ,1993; Iwasaki and Skelton, 1995) . In this case, the result turs out to be a necessary and sufficient condition for stabilizabil0 ity via static output feedback .
s - t one obtains that
x(t - T)
= ~(-T)X(t)
-loT~(-(T+8))BU(t-T+8)d8
(10)
The next theorem presents a method of designing an observer-based output feedback stabilizing controller for the system (1). In contrast to the result of Theorem 1, the controller design is in terms of the feasibility of LMIs and thus can be implemented numerically via existing convex optimization algorithms for solving LMIs; see, e.g. Boyd et al. (1994) .
Substituting (10) into (9), it follows that V(t)
=
xT(t)[PA+ATp+PBKC~(-T)
+ ~T (-T)C T KT BT p] x(t)
< 0 where the inequality follows from (7) . This implies that the closed-loop system of (1) with the control law (8) is globally asymptotically stable.
Theorem 2. Consider the system (1) satisfying Assumption 1. Then this system is stabilizable via an output feedback controller if there exist matrices P > 0, Q > 0 and Y satisfying the following LMls:
(i) By Lemma 1, inequality (7) is equivalent to N'£T p(PA N'f4>(-T)(PA
+ AT P)NBT P < 0
(11)
+ AT P)NC4>(-T ) < 0
(12)
N'£T(AQ PA
where NBT p and N C4>( -T) are any matrices whose columns form bases of the null spaces of BT P and C~( -T) , respectively.
x(t)
=Ax(t) + Bu(t -
u(t)=Kx(t)
T)
+ L[y(t)
- Cx(t)]
+ K lOT~( -(T + 8))Bu(t + 8)d8
where ~(t) = eAt, L following LMI:
Note that the matrix P and its inverse both occur affinely in (4) and (5) and thus the feasibility of these inequalities is a non-convex problem of rank-minimization type . This feasibility problem, however, can be considered as a cone complementary problem as follows ; see, e.g. El Ghaoui et al. (1997):
(14)
= p- 1 Y
(15)
(16)
and K satisfies the
with Q satisfying (1S) .
Proof. By defining e(t) = x(t) - x(t) , the closedloop system of (1) with the controller (15)-(16) can be described by the following state-space model Ic{t)
Tr{QP}
[9
<0
Moreover, under the above conditions, a suitable control law has the following observer-based structure
Remark 2. Theorem 1 provides an output feedback stabilization result for linear systems with delayed control input. The first part of the theorem gives a sufficient condition for the existence of output feedback stabilizing controllers, whereas the second part supplies a method for constructing such controllers.
subject to (4) , (6) and
- Y C - CTyT
(13)
where NBT is any matrix whose columns form bases of the null space of BT.
NotethatsinceNBTp = P-1NBT and N C4>(-T) = ~-l(-T)Nc, it follows that (11) and (12) are equivalent to (4)-(6). This completes the proof. \7\7\7
minimize
+ AT P
+ QAT)NBT < 0
~] ~ o.
u(t)
= Axc{t) + Bu(t -
(18)
T)
= Kxc(t) + K lOT~( -T -
8)Bu(t
+ 8)d8
(19)
where To solve the above problem, the cone complementary linearization algorithm which was proposed in El Ghaoui et al. (1997) can be used.
A
o
Further, it follows from the proof of Theorem 1 that the matrix P which appears in the inequality of (7) is the same matrix P as in (4) and (5). Thus , once a solution P > 0 and Q > 0 to (4)-(6) is found as described above, the gain K of the controllaw (8) can be obtained by solving (7) , which is an LMI problem as the matrix P is known. 0
LC ] A. - LC •
(20)
(21)
eAt _ e(A-L C) t] e(A-L C) t
Remark 3. In the absence of time-delay, Theorem 1 reduces to an existing static output feed-
(22)
Applying Theorem 1 (ii) to the closed-loop system of (18)-(19) , it follows that the controller (15)-(16)
41
Conversely, suppose that there exist matrices Q > 0 satisfying the inequalities (17) and (25) , respectively. We now prove that there exists a positive real number p such that
solves the stabilization problem if there exist matrices P > 0 and k such that
PA +.F P + piJk4> (-T) + 4>T (-T )kTiJT P < O.
o and P >
(23)
_
P=
Using (20)-(22), inequality (23) can be rewritten as
p.4+.4Tp < 0
[pQ-l
o
~]
satisfies (24) . To this end, first notice that
(24)
where A=
[A
+ BKe- AT
o
LC+BK~ ] A-LC
where Ao = A - LC n = Q-l(A + BKe- AT ) + (A + BKe-ATfQ-l
A _ -AT -e -(A-LC)T . u-e
In the sequel, we will show that the condition of (24) is equivalent to the existence of matrices Q > 0 and P > 0 such that the LMIs (13) and (14) hold . Note that since in (17) the matrix ~(-T)Q is non-singular , by using Lemma 1, it follows that the inequality (17) is equivalent to condition of (13). On the other hand, with Y = PL , the inequality (14) is equivalent to the following LMI P(A - LC)
+ (A
- LC)Tp
< O.
1}1
= LC + BK (e- Ar _
e-(A-LC)T) .
Hence, considering (25) and using Schur's complements, the inequality (24) is satisfied if p > 0 is such that Q-l(A
+ BKe- AT ) + (A + BKe-Ar)TQ-l
- pQ-ll}1(PA o
(25)
+ A~p)-II}1TQ-l < O.
Observe that the latter condition holds for any > 0 such that
In view of the above , we shall show that the LMI (24) is feasible if and only if the LMIs (17) and (25) are feasible .
p
P)..min
To this end, first suppose that (24) has a solution P > O. With P partitioned in n x n blocks as follows
(I}1[P(A - LC)
+ (A
- LC)T prll}1T)
> )..max ((A + BKe-AT)Q + Q(A + BKe-Ar)T) Finally, in view of (17) and (25), such a real number p is guaranteed to exist. V'V'V'
it can be easily established that the inequality (24) implies
Remark 4. Theorem 2 provides a design method of observer-based controllers for the problem of output feedback stabilization of linear input delayed systems. The proposed method has the advantage that it is given in terms of the solvability of LMIs , and as such can be implemented numerically very efficiently using standard LMI 0 techniques.
Introducing the change of variable Q = P 1- 1 > 0, it results that the inequality (26) is equivalent to (17). Next, defining Q = is equivalent to
p- 1 >
0, the inequality (24)
4
CONCLUSIONS
(27) Partitioning
Q accordingly
to
.4,
This paper focused on the design of output feedback controllers for continuous-time linear systems with a constant time-delay in the control input. Two methods of designing output feedback controllers for solving the stabilization problem have been developed. The first method involves the solvability of a rank minimization problem, which can be solved via existing techniques such as the cone complementary linearization algorithm of El Ghaoui et al. (1997). On the other hand , the second design method, which provides an observerbased controller. is based on the feasibility of linear matrix inequalities.
namely
it can be easily verified that the inequality (27) implies
Now, defining P = Q;l > 0, it follows that the inequality (28) is equivalent to (25).
42
5
REFERENCES
[IJ Boyd, S., L. El Ghaoui , E . Feron and V. Balakrishnan (1994) . Linear Matrix Inequalities in Systems and Control Theory . Studies in Applied Mathematics, Vo!. 15, SIAM, Philadelphia, PA [2J Choi, H.H. and M.J. Chung (1996). Observerbased 11.00 controller design for state delayed linear systems. Automatica 32 , 1073-1075. [3J de Souza, C.E . and X. Li (1999) . Delaydependent 11.00 control of uncertain linear state-delayed systems. Automatica 35 , 13131321. [4J Dugard, L. and E .1. Verriest (eds.) (1998). Stability and Control of Time-Delay Systems . Springer-Verlag, Berlin , Germany. [5J El Ghaoui , L. and P. Gahinet (1993) . Rank minimization under LMI constraints: A framework for output feedback problems . In Proc. 1993 European Control ConI , Groningen, The Netherlands.
[15] Li, X. and C.E . de Souza (1998) . Output Feedback Stabilization of Linear Time-delay Systems . In L. Dugard and E.1. Verriest (eds.), Stability and Control of Time-Delay Systems , Springer-Verlag, Berlin, 241-258. [16J Malek-Zavarei, M. and M. Jamshidi (1987). Time Delay Systems: Analysis, Optimization and Applications. North-Holland , Amsterdam , The Netherlands. [17] Morari, M. and E . Zafiriou (1989) . Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ. [18] Shen, J.C ., B.-S . Chen and F .-C. Kung (1991) . Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach . IEEE Trans . Autom. Control 36 , 638640. [19] Xie, L. and C.E . de Souza (1993) . Robust stabilization and distrurbance attenuation for uncertain delay systems. In Proc. 1993 European Control ConI , Groningen, The Netherlands.
[6J El Ghaoui, L., F. Oustry and M.A. Ramit (1997) . A cone complementary linearization algorithm for static output feedback and related problems. IEEE Trans . Autom. Control 43 , 1171-1176.
[7J Etkin, B. (1972). Dynamics of Atmospheric Flight. Wiley, New York. [8J Gahinet, P. and P. Apkarian (1994) . A linear matrix inequality approach to 11.00 contro!' Int. J. Robust and Nonlinear Control 4, 421448. [9J Iwasaki, T. and R.E . Skelton (1995) . Parameterization of all stabilizing controllers via quadratic Lyapunov functions . J . Optimization Theory and Applications 85 , 291-307. [lOJ Kojima , A. and S. Ishijima (1995). Explicit formulas for operator Riccati equations arising in 11.00 control with delays . In Proc. 34th IEEE ConI Decision and Control , New Orleans, LA . [11J Jeung, E.T., D.C. Oh , J.H . Kim and H.B. Park (1996). Robust controller design for uncertain systems with time delays: LMI approach. A utomatica 32 , 1229- 1231. [12] Lee , J .H. , S.W . Kim and \V .H. Kwon (1994) . Memoryless 11.00 controllers for state delayed systems. IEEE Trans. Autom. Control 39 , 159-162. [13J Li, X. and C.E. de Souza (1997a) . LMI Delaydependent robust stability and stabilization of uncertain linear delay systems: A linear matrix inequality approach . IEEE Tmns. Autom . Control 42 , 1144- 1148. [14J Li , X. and C.E. de Souza (1997b). Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica 33 , 1657- 1662.
43
[:0[>
Copyright ~ IFAC Linear Time Delay Systems, Ancona, Italy, 2000
Publications www.elsevier.com ilocateiifac
OUTPUT FEEDBACK STABILIZATION OF LINEAR SYSTEMS WITH DELAYED INPUT* Xi Lil,
Carlos E. de Souza 2
1 Husky Oil Operations Ltd. 707, 8th Avenue S. w., Box 6525, Station D Calgary , Alberta T2P 3G7, Canada E-mail: [email protected]
2 Department of Systems and Control Laboratorio NacioTwl de Computat;iio Cientifica - LNCC Av. Getulio Vargas 333, 25651-070 Petropolis , RJ, Brazil E-mail: csouza@lncc .br
Abstract. This paper addresses the problem of output feedback stabilization of linear continuous time systems with time-delay in the control input. Attention is focused on the design of delaydependent output feedback controllers that ensure global asymptotic stability of the closed-loop system. Two controller design methods have been developed . The first one is concerned with the design of a control law which comprises a static feedback of the system output with the addition of a term involving the integral of past values of the input signal on a window of the size of the time-delay. On the other hand , the second method provides an observer-based controller. Copyright
©
2000 [FAC
Key Words. Output feedback stabilization , time-delay systems , observer-based controller, linear matrix inequalities.
1
INTRODUCTION
feedback have been considered. For example, stabilization techniques which are independent of the size of the time-delay have been proposed in Shen et al. (1991), Xie and de Souza (1993) , Lee et al. (1994), Choi and Chung (1996) , and Jeung et al. (1996) , whereas delay-dependent stabilization methods have been developed in Kojima and Ishijima (1995) , Li and de Souza (1997a,b, 1998) and de Souza and Li (1999).
One of the main problems encountered in many practical control designs is the existence of timedelay in either the control input , the states or the measurements; see, e.g. Malek-Zavarei and Jamshidi (1987) and Dugard and Verriest (1998). Time-delay is inherent in many process control systems (Morari and Zafiriou , 1989) and it widely appears in aircraft control design (Etkin , 1972). The problem of stabilization of dynamic systems with time-delays are, therefore, of theoretical and practical importance and has attracted considerable attention for several decades . Various techniques of stabilization of linear time-delay systems have been proposed over the past few years. where memory less state feedback and dynamic output
This paper considers the problem of output feedback stabilization of linear continuous-time with a single, constant, time-delay in the control input. Attention is focused on design methods of output feedback controllers which are dependent on the time-delay and two design methods are proposed. The first one deals with the design of a control law which comprises a static feedback of the system output and a term involving the integral of past values of the input signal on a window of the size of the time-delay. This method involve
-This work was supported in part by 'Conselho Nacional de Desenvolvimento Cientifico e Tecnologico - C!'>Pq ', Brazil , under grant 301653/ 96-8 / PQ for the second author.
39
the solvability of a rank minimization problem, which can be solved via existing techniques such as the cone complementary linearization algorithm of El Ghaoui et al. (1997). The second method is concerned with the design of an observer-based controller and is in terms of linear matrix inequalities (LMls).
Remark 1. Note that if either the matrix Y or Z has rank n, then the first or second inequality in (3) disappears. 0
Notation. Rn denotes the n dimensional Euc1idean space, Rnxm is the set of n x m real matrices, Amin (.) and Amax (.) stand for the minimum and maximum eigenvalue of a symmetric matrix, respectively, Tr{ . } denotes the matrix trace, and the notation X > 0, for a square matrix X, means that X is symmetric and positive definite.
In this section we will present two delay-dependent methods of designing output feedback stabilizing controllers for the time-delay system (1). The proposed methods have the advantage that can be implemented numerically via LMI techniques.
2
3
The first theorem deals with a control law which comprises a static feedback of the system output with the addition of a term involving the integral of the input signal over the last T seconds.
SYSTEM DEFINITION AND PRELIMIN ARIES
Theorem 1. Consider the system (l) satisfying Assumption 1. Then this system is stabilizable via an output feedback control law if the following equivalent conditions hold:
Consider linear time-delay systems described by the following state equation: x(t) yet)
= Ax(t) + Bu(t -
T)
(1)
= Cx(t)
MAli" RESULTS
(i) There exist matrices P that
where x(t) E Rn is the state, u(t) E Rnu is the control input , yet) E Rn.p is the measured output, T 2': 0 is a constant time-delay, and A, Band C are known real constant matrices of appropriate dimensions.
NJ;T(AQ
Q
=
and Q
>0
+ QAT)NBT < 0
N:! ~T (T)(PA
This paper addresses the problem of designing an output feedback control law for the system (1) such that the resulting closed-loop system is globally asymptotically stable. Attention is focused on design techniques of output feedback stabilizing controllers which are dependent on the timedelay.
>0
(4)
+ AT P)~(T)Ne <
0
p- 1
is
stabilizable
where NBT and life are any matrices whose columns form bases of the null spaces of BT and C, 7'espectively, and ~(t) = eAt. (ii) There exist matrices P
>0
and K such that
PA+ATp+PBKC~(-T)
(7) Moreover, a suitable output feedback control law is given by
and
Note that Assumption L which is necessary for output feedback stabilization of the system (1) in the absence of time-delay, is a necessary condition for the existence of a stabilizing output feedback control law for the system (1).
u(t)
=Ky(t) + KClOT~( -( T + 8) )Bu(t + 8)d8 (8)
Proof. (ii) Suppose that there exist matrices P > 0 and K satisfying the inequality of (7) . Let \'(t) = xT(t)Px(t) be a Lyapunov function candidate for the closed-loop system of (1) with the control law of (8). One readily obtains that the time-derivative of \/ (t) along the trajectory of this closed-loop system is given by
We conclude this section by recalling a lemma which will be used in the derivation of the results of this paper (see. e.g .. Gahinet and Apkarian, 1994).
Lemma 1. Given a symmetric matrix G E Rnxn and matrices Y E Rrxn and Z E Rsxn, consider the problem of finding some matrix e E Rrxs such that
\~'(t)
+
Denote by N y and .A/z any matrices whose columns form bases of the null spaces of Y and Z, respectively. Then (2) is solvable for if and only if
e
< 0,
NI G.A/z
< O.
= xT(t)(PA + ATp)x(t) + 2xT (t)P BKC[x(t - T)
(2)
.A/? GNy
(5) (6)
The system (1) is supposed to satisfy the following assumption:
Assumption 1. (A, B) (A, C) is detectable.
such
lOT ~(- (T + 8) )Bu(t -
T + 8)d8]
(9)
Note that since x(t)
(3)
40
= ~(T)X(t -
T)
+ !~T ~(t
- s)Bu(s - T)ds
then with the change of integration variable 8
=
back stabilization result for delay free systems (El Ghaoui and Gahinet ,1993; Iwasaki and Skelton, 1995) . In this case, the result turs out to be a necessary and sufficient condition for stabilizabil0 ity via static output feedback .
s - t one obtains that
x(t - T)
= ~(-T)X(t)
-loT~(-(T+8))BU(t-T+8)d8
(10)
The next theorem presents a method of designing an observer-based output feedback stabilizing controller for the system (1). In contrast to the result of Theorem 1, the controller design is in terms of the feasibility of LMIs and thus can be implemented numerically via existing convex optimization algorithms for solving LMIs; see, e.g. Boyd et al. (1994) .
Substituting (10) into (9), it follows that V(t)
=
xT(t)[PA+ATp+PBKC~(-T)
+ ~T (-T)C T KT BT p] x(t)
< 0 where the inequality follows from (7) . This implies that the closed-loop system of (1) with the control law (8) is globally asymptotically stable.
Theorem 2. Consider the system (1) satisfying Assumption 1. Then this system is stabilizable via an output feedback controller if there exist matrices P > 0, Q > 0 and Y satisfying the following LMls:
(i) By Lemma 1, inequality (7) is equivalent to N'£T p(PA N'f4>(-T)(PA
+ AT P)NBT P < 0
(11)
+ AT P)NC4>(-T ) < 0
(12)
N'£T(AQ PA
where NBT p and N C4>( -T) are any matrices whose columns form bases of the null spaces of BT P and C~( -T) , respectively.
x(t)
=Ax(t) + Bu(t -
u(t)=Kx(t)
T)
+ L[y(t)
- Cx(t)]
+ K lOT~( -(T + 8))Bu(t + 8)d8
where ~(t) = eAt, L following LMI:
Note that the matrix P and its inverse both occur affinely in (4) and (5) and thus the feasibility of these inequalities is a non-convex problem of rank-minimization type . This feasibility problem, however, can be considered as a cone complementary problem as follows ; see, e.g. El Ghaoui et al. (1997):
(14)
= p- 1 Y
(15)
(16)
and K satisfies the
with Q satisfying (1S) .
Proof. By defining e(t) = x(t) - x(t) , the closedloop system of (1) with the controller (15)-(16) can be described by the following state-space model Ic{t)
Tr{QP}
[9
<0
Moreover, under the above conditions, a suitable control law has the following observer-based structure
Remark 2. Theorem 1 provides an output feedback stabilization result for linear systems with delayed control input. The first part of the theorem gives a sufficient condition for the existence of output feedback stabilizing controllers, whereas the second part supplies a method for constructing such controllers.
subject to (4) , (6) and
- Y C - CTyT
(13)
where NBT is any matrix whose columns form bases of the null space of BT.
NotethatsinceNBTp = P-1NBT and N C4>(-T) = ~-l(-T)Nc, it follows that (11) and (12) are equivalent to (4)-(6). This completes the proof. \7\7\7
minimize
+ AT P
+ QAT)NBT < 0
~] ~ o.
u(t)
= Axc{t) + Bu(t -
(18)
T)
= Kxc(t) + K lOT~( -T -
8)Bu(t
+ 8)d8
(19)
where To solve the above problem, the cone complementary linearization algorithm which was proposed in El Ghaoui et al. (1997) can be used.
A
o
Further, it follows from the proof of Theorem 1 that the matrix P which appears in the inequality of (7) is the same matrix P as in (4) and (5). Thus , once a solution P > 0 and Q > 0 to (4)-(6) is found as described above, the gain K of the controllaw (8) can be obtained by solving (7) , which is an LMI problem as the matrix P is known. 0
LC ] A. - LC •
(20)
(21)
eAt _ e(A-L C) t] e(A-L C) t
Remark 3. In the absence of time-delay, Theorem 1 reduces to an existing static output feed-
(22)
Applying Theorem 1 (ii) to the closed-loop system of (18)-(19) , it follows that the controller (15)-(16)
41
Conversely, suppose that there exist matrices Q > 0 satisfying the inequalities (17) and (25) , respectively. We now prove that there exists a positive real number p such that
solves the stabilization problem if there exist matrices P > 0 and k such that
PA +.F P + piJk4> (-T) + 4>T (-T )kTiJT P < O.
o and P >
(23)
_
P=
Using (20)-(22), inequality (23) can be rewritten as
p.4+.4Tp < 0
[pQ-l
o
~]
satisfies (24) . To this end, first notice that
(24)
where A=
[A
+ BKe- AT
o
LC+BK~ ] A-LC
where Ao = A - LC n = Q-l(A + BKe- AT ) + (A + BKe-ATfQ-l
A _ -AT -e -(A-LC)T . u-e
In the sequel, we will show that the condition of (24) is equivalent to the existence of matrices Q > 0 and P > 0 such that the LMIs (13) and (14) hold . Note that since in (17) the matrix ~(-T)Q is non-singular , by using Lemma 1, it follows that the inequality (17) is equivalent to condition of (13). On the other hand, with Y = PL , the inequality (14) is equivalent to the following LMI P(A - LC)
+ (A
- LC)Tp
< O.
1}1
= LC + BK (e- Ar _
e-(A-LC)T) .
Hence, considering (25) and using Schur's complements, the inequality (24) is satisfied if p > 0 is such that Q-l(A
+ BKe- AT ) + (A + BKe-Ar)TQ-l
- pQ-ll}1(PA o
(25)
+ A~p)-II}1TQ-l < O.
Observe that the latter condition holds for any > 0 such that
In view of the above , we shall show that the LMI (24) is feasible if and only if the LMIs (17) and (25) are feasible .
p
P)..min
To this end, first suppose that (24) has a solution P > O. With P partitioned in n x n blocks as follows
(I}1[P(A - LC)
+ (A
- LC)T prll}1T)
> )..max ((A + BKe-AT)Q + Q(A + BKe-Ar)T) Finally, in view of (17) and (25), such a real number p is guaranteed to exist. V'V'V'
it can be easily established that the inequality (24) implies
Remark 4. Theorem 2 provides a design method of observer-based controllers for the problem of output feedback stabilization of linear input delayed systems. The proposed method has the advantage that it is given in terms of the solvability of LMIs , and as such can be implemented numerically very efficiently using standard LMI 0 techniques.
Introducing the change of variable Q = P 1- 1 > 0, it results that the inequality (26) is equivalent to (17). Next, defining Q = is equivalent to
p- 1 >
0, the inequality (24)
4
CONCLUSIONS
(27) Partitioning
Q accordingly
to
.4,
This paper focused on the design of output feedback controllers for continuous-time linear systems with a constant time-delay in the control input. Two methods of designing output feedback controllers for solving the stabilization problem have been developed. The first method involves the solvability of a rank minimization problem, which can be solved via existing techniques such as the cone complementary linearization algorithm of El Ghaoui et al. (1997). On the other hand , the second design method, which provides an observerbased controller. is based on the feasibility of linear matrix inequalities.
namely
it can be easily verified that the inequality (27) implies
Now, defining P = Q;l > 0, it follows that the inequality (28) is equivalent to (25).
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5
REFERENCES
[IJ Boyd, S., L. El Ghaoui , E . Feron and V. Balakrishnan (1994) . Linear Matrix Inequalities in Systems and Control Theory . Studies in Applied Mathematics, Vo!. 15, SIAM, Philadelphia, PA [2J Choi, H.H. and M.J. Chung (1996). Observerbased 11.00 controller design for state delayed linear systems. Automatica 32 , 1073-1075. [3J de Souza, C.E . and X. Li (1999) . Delaydependent 11.00 control of uncertain linear state-delayed systems. Automatica 35 , 13131321. [4J Dugard, L. and E .1. Verriest (eds.) (1998). Stability and Control of Time-Delay Systems . Springer-Verlag, Berlin , Germany. [5J El Ghaoui , L. and P. Gahinet (1993) . Rank minimization under LMI constraints: A framework for output feedback problems . In Proc. 1993 European Control ConI , Groningen, The Netherlands.
[15] Li, X. and C.E . de Souza (1998) . Output Feedback Stabilization of Linear Time-delay Systems . In L. Dugard and E.1. Verriest (eds.), Stability and Control of Time-Delay Systems , Springer-Verlag, Berlin, 241-258. [16J Malek-Zavarei, M. and M. Jamshidi (1987). Time Delay Systems: Analysis, Optimization and Applications. North-Holland , Amsterdam , The Netherlands. [17] Morari, M. and E . Zafiriou (1989) . Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ. [18] Shen, J.C ., B.-S . Chen and F .-C. Kung (1991) . Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach . IEEE Trans . Autom. Control 36 , 638640. [19] Xie, L. and C.E . de Souza (1993) . Robust stabilization and distrurbance attenuation for uncertain delay systems. In Proc. 1993 European Control ConI , Groningen, The Netherlands.
[6J El Ghaoui, L., F. Oustry and M.A. Ramit (1997) . A cone complementary linearization algorithm for static output feedback and related problems. IEEE Trans . Autom. Control 43 , 1171-1176.
[7J Etkin, B. (1972). Dynamics of Atmospheric Flight. Wiley, New York. [8J Gahinet, P. and P. Apkarian (1994) . A linear matrix inequality approach to 11.00 contro!' Int. J. Robust and Nonlinear Control 4, 421448. [9J Iwasaki, T. and R.E . Skelton (1995) . Parameterization of all stabilizing controllers via quadratic Lyapunov functions . J . Optimization Theory and Applications 85 , 291-307. [lOJ Kojima , A. and S. Ishijima (1995). Explicit formulas for operator Riccati equations arising in 11.00 control with delays . In Proc. 34th IEEE ConI Decision and Control , New Orleans, LA . [11J Jeung, E.T., D.C. Oh , J.H . Kim and H.B. Park (1996). Robust controller design for uncertain systems with time delays: LMI approach. A utomatica 32 , 1229- 1231. [12] Lee , J .H. , S.W . Kim and \V .H. Kwon (1994) . Memoryless 11.00 controllers for state delayed systems. IEEE Trans. Autom. Control 39 , 159-162. [13J Li, X. and C.E. de Souza (1997a) . LMI Delaydependent robust stability and stabilization of uncertain linear delay systems: A linear matrix inequality approach . IEEE Tmns. Autom . Control 42 , 1144- 1148. [14J Li , X. and C.E. de Souza (1997b). Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica 33 , 1657- 1662.
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