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On the Stabilization and Observation of Linear Systems with Time-Varying Delay
STABILITY AND STABILIZATION II
Copyright © IFAC Contro l SCience and Technology (8th Triennial World Congress ) Kvoto . Japan . 198 1
ON THE STABILIZATION AND OBSERVATION OF LINEAR SYSTEMS WITH TIME-VARYING DELAY K. Yasuda*, M. Maekawa** and K. Hirai* "Department of Systems Engineering , Kobe University, Kobe , japan ""Central Research Institute, Sanyo Electric Company, Moriguchi, japan
Abstract. This oaper concerned with the stabilization and observation of a 1inear svsteJ:l v:ith time-varvinfl delav in the state. First, a sufficient condition for stabilization ' bv means of linear state feedback without delay is shown. ~ext, a sufficien(condition for the possibility of constructin~ a state estimator is found. Finally, it is shown that under these conditions and an additional one, a linear ti~e-varying delay systeJ:l can also be stabilized by apolying a linear feedback law where the output of the estiJ:lator is used instead of the state. Keywords. Stabilization; observer; tiJ:le-varyinq state feedback; Lyapunov methods. INTRODUCTION
syste~;
time-varying delay;
differential-difference equation: 1 x(t) A(t)x(t) + D(t)x(t-h(t)) S : 1 + B(t)u(t) (la) y(t) = C'(t)x(t) (lb) r n for tE[t O' 00 ), v!here X(t)ER , U(t)ER , and y(t)ER m are the state, the input, and the output of the system respectively; h(t) denotes the time delay; and A(t), B(t), C(t), and D(t) are matrices with appropriate dimensions. Suppose that every element of A(·), R(')' C(·), and D(·) is continuous, and there are positive numbers ha and hb such that
TiJ:le-varying delay appears in various real systems (Johnson, 1972; Seddon, 1968; Zadeh, 1952), and it freo,uently causes instability of the systems; nevertheless, so far the stabilization problem for linear time-varyinq delay systems have been treated in only a few studies (Ikeda, 1979). This paper is concerned with the stabilization and observation of a linear system with timevarying delay. The objective- is to establish for the time-varying delay systeJ:l the stabilizing method by a feedback compensator composed of a state estimator and an amnlifier, which is the standard stabilizing method for linear time-invariant systems without delay.
o ~ h(t) ~ ha
<
00 ,
tilt) ~ hb
<
1
(2)
for all tE[t O' 00). Let q, (.) denote the initial function of (la), that is, x(t) = q, (t) for tE[tO-h(t O)' tOJ·
First, the problem of stabilizing the system is considered, and a sufficient condition for stabilization by means of linear state feedback without delay is shown. Next, the problem of observing the state of the system is considered, which is formulated as the oroblem of designing a state estimator which nroduces the state estimate from only information on inputs and outputs of the system, and a sufficient condition for the possibility of construction is found. Finally, the probleJ:l of stabilizing the system by applyinq a linear feedback law where the outout of the state estimator is used instead of the state is discussed.
If u(·) is measurable, bounded on every finite subinterval of time on [to' 00) , and q, (.) measurable, then there exists a unique, continuous solution with the initial function q, (.) (Oquztoreli, 1966). Let the system of the form (1) (or (la)) with u(t) = 0 be called zero-input system, and define the stability of the system as follows : A zero-input system So is said to be exponentially stable, if there exist oositive numbers a and c such that any solution x(·) of So satisfies
Definition .
SYSTH1 DESCRIPTION AND DEFINITION OF STABILITY
(3)
Let us consider a linear system with timevarying delay described by the follovling 153
K. Yasuda, M. Maekawa and K. Hirai
154
for all tl and t2
( ~tl)'
where K' (t)
11 xt ll H, SUPt_h(t) ';;'T,;;,t lx(T) 1 , Ix(T) 1 fl (x'(T)x(T))1/2 . In this case, a is called stability degree.
= -
~(I +
-& L' (t) L(t ) ) B' (t) P(t ),(10)
\~here
62
~
(1-h b)exp(-2aha ) ,
6 > O.
(11 )
Then the closed-loop system S can be described by c
STABILIZATION BY STATE FEEDBACK In this section, we consider the nroblem of stabilizing the system S represented by (la) by means of linear state feedback without delay, u(t) = K' (t)x(t) , (4 ) where K(·) has appropriate dimensions and is continuous. Applying (4) to (la), the resulting closed-looo system can be written as S : x(t) = (A(t)+B(t)K' (t))x(t) c + D(t)x(t-h(t)). (5) 1. There exists a state feedback of the form given by (4) such that the resultant closed-loop system Sc is exponentially stable, if (i) the pair (/\(.), B(')) is uniformly completely controllable (Kalman, 1960) and (fl) D(t) may be decomposed as (t) = 01 (t) + O2(t) (6) Theore~
°
for all t, where Dl (t) can be represented as linear combinations of the columns of B(t) at every t, and D2 (t) is small enouqh in terms of the norm defined as sUPt>t 11 D2(t) 11 in = 0 which 11 D2 (t) 11 denotes the matrix norm of D2 (t) derived from the Euclidian norm. The following lemma is necessary to prove Theorem 1. (Ikeda, 1972). If a pair (A(·), P.(.)) is uniformly completely controllable, then for any real numbers a and v (>-1) the Riccati equation P(t) + (A(t)+aI)'P(t) + P(t)(A(t)+aI) - P(t)B(t)B' (t)P(t) + (1+ v)I = 0 (7) has a continuous solution which satisfies
Lel707la 1
(8)
for all t, where a l and a 2 are positive numbers. 1) First observe that if of Theorem 1 are satisfied, then there exists a matrix L(·) being continuous such that (9) Dl(t) = B(t)L'(t)
(Proof of Theorem
(i) and (ii)
for every t and, from Lemma 1, there exists a matrix P(·) satisfyinq (7) and (8) for any v greater than -1. For a suitable positive constant a, we define the feedback gain K(') by
x(t) =
-&
[A(t) - ~ B(t) (I + L' (t)L(t) )R ' (t)P(t)]x(t) + B(t)L'(t)x(t-h(t)) + D2 (t)x(t-h(t)). (12) Let a candidate for the Lyapunov functional be given by V(t) = exp(2at) x'(t)P (t) x(t) + (1+ f)
f
t exp(2as)x'(s)x(s)ds , t-h(t) (13)
where v is an appropriate positive number \'Ihich satisfies 2
al lx(t) 1 exo(2 at)
~
V(t)
~ (a 2 + (1+ ~)ha ) 11 xt ll exp(2 at).
(14)
Then its time derivative alonq any solution of (12) is calculated as follows: V(t) = exp(2 at) {x'(t)[P(t) + (A(t)+aI)'P(t) + P(t)(A(t)+a I) - P(t)B(t)B'(t)P(t) P(t)B(t)L' (t)L(t)B' (t)P(t) + (1+f )I]x(t) + 2x'(t)(P(t)B(t)L'(t) + P(t)D 2 (t))x(t-h(t)) - (1+f)(1-h(t))exp(-2ah(t)) x x' (t-h(t))x(t-h(t))}
--&
exp(2at) {- I tL(t)B' (t)P(t)x(t)- 6x(t-h(t)) 12 - 2~62 1 2D2(t)P(t)x(t) - v6 2x(t-h(t)) 12 - x' (t)( f I - v~2 P(t) O2 (t) O2 (t) P(t) ) x (t) + (1+f)(02_(1-h(t))exp(-2ah(t)) x x'(t-h(t))x(t-h(t)) } (15) Using ine~ualities (8) and 02 ~ (1-h(t))exp(-2ah(t)) gives . v 2 2 2 V(t) ,;;, -exp(2at)( 2 - ~ a2 d2 lx' (t)x(t), (16) v!here d2 = SUflt>t 11 D2(t) 11 = 0 This imnlies that if, for given positive numbers a and v, ~is small enough to satisfy d ,;;, v6/2a 2 (17) 2 then V(t) 5- O. !'!hen V(t) 5- 0 from (14), it can be shown that the solution of (12)
Stabilization and Observation o f Linear Systems
satisfies Ix(t) 1 ~
C* ll xt
1
for all
tl ( ~tO )
Il ex;:J(- ex (T-t l )) T( ~tl)'
and
(18 )
Thus,
~Ie
2
2
I f the es t i ma t i on error e(t ) = w(t) - x(t) is bounded on [to' 00 ) and satisfies
a
independently of input u( · ) and the initi al functions , then Es is said to be the state estimator for system S.
~ sunt2-h(t2) ~T~t/* 11 xtl ll exp(- ex (T-t l )) C*exo(ex ha ) II xt II exr (- ex (t2-tl )) 1
for all tl and t 2 ( ~tl ) . proof is complete.
(21 )
UmIlet ll = 0
obta i n
II xt II = sUP t -h i t )
=
e(t) to be given are defined on [to-h(t O) ' tOJ and are measurable. De fini tion .
where
C* = [jL(a +(1+ ~ )h )Jl / 2 al
155
(19 )
Thi s shOl"s that the Q.E.D .
It is imrortant to note that the stabilizinq method pro posed can be realized even if delay ti me hit) is not known nrecisel y , as lona as the upper bound ha and the up~er bound of its derivative hb are known . Remark 1. The limits of the stability de gree which can be desiqned depend on the magnitude of d2 , so that the system S is not always stabilized with pre-specified stability degree; while in the case of D2 (t) = 0 for all t, it is free from such a limitation. Remark 2. In comparison with the result (Ikeda, 1979), which is the same as Theorem 1 when D2 (t ) = 0, the stabilizing method obtained here has the advantage that a design taking account of perturbations of plant parameters is pos s ible. Remark 3. Let f( v) = v8 / 2a 2 , which is the right side of (17 ). It can be shown that f( v) is monotonousl y increasing and bounded so that the supremum of d2 exists for a given ex . STATE ESTItlATION BY ESTH1ATOR In this section, the observation problem is considered, and it is formulated as the problem of designing a state estimator . As a candidate for the state estimator associated with the system S, let us consider the linear time-delav system of the form z(t ) = (A(t )-H(t)C ' (t))z(t) E . + D(t)z(t-h(t)) + H(t)y(t) s . + B(t)u(t) (20a) \ wit) = z(t) (20b) where Z(t)ER n and W(t)ER n are the state and the output of Es res~ecti ve 1y, and fl ( .) whose elements are continuous is a matrix to be designed. Assume that the initial functions
From (1) and (20), it follows that the behavior of errors e (t) is gover ned by the homogeneous eC!uation E : e(t) = (A(t)-H(t )C' (t))e (t) r + D(t)e (t-h (t ) ) . (22) Utilizing thi s , we can obtain the followin g theorem. Theorem 2.
A matri x H(·) exists such that Es is the state esti mator for S, if ( i ) the pair (C' (.), A(·)) is uniforml y comnletel y observable (Kalman, 1960), (ii ) both C( ·) and 0(·) are bounded, and (iii) D(t) may be de comnosed as D(t) = D3(t) + D4 (t) (23)
where D3(t) can be represented as linear combinations of the rows of C(t-h (t)) at every t, and D4(t) is small enough in terms of the norm SUPt >t II D4(t) 11 · = 0 The constructibility of a state estimator for a linear time-varying system without delay has been derived by using the relation for the state transition matri x between the system and its dual system (Ikeda, 1975); however such a technique cannot be aoplied to a time-varyng delay system because the causality would not hold for the dual system which is a timevarying delay system of the advanced type . In order to avoid this difficulty, we employ the Lyapunov method . To verify Theorem 2, the fo 11 o~li ng 1emma is necessary. If a pair (C ' (·), A(·)) is uniforml y completely observable, then for any real numers ex and ~ ( > -l) the Riccati equation Q(t) + ~(t)(A(t)+ ex I)' + (A(t)+ex I)Q(t) - Q(t)C(t)C ' (t)Q(t) + (l+ ~ )I = 0 (24 ) has a continuous solution which satisfies (25) o < blI ~ Q(t) ~ b2 I for all t, where b and b2 are positive l numbers.
Lerrurza 2 .
(Proof of Theorem 2) Assume that (i), (ii) , and (iii) of Theorem 2 are satisfied . Then there exists a matrix M(·), continuous and bounded, such that D)(t) = C(t-h(t))M'(t-h(t)) (26)
156
K. Yasuda, M. Maekawa and K. Hirai
for every t and, from Lemma 2, there exists a symmetrical matrix Q(.) satisfying (24) and (25) for any ~ greater than -1. For a suitable H(·) as
~ositive
constant a , we give
From these inequalities, it can be shown that the solution of (28) satisfies
~Q(-t)C(t)(I + -tzW(t)M(t)). (27)
H(t)
Then (22) can be rewritten as e( t) = [A ( t) - ~ Q( - t ) C( t )( I + +
This immediately yields V ,;, 0 when d4 is small enough to satisfy d4 ,;, ~6 /2bZ (32)
~l(t-h(t))C'
for all tl( ~tO) and T( ~tl)' \vhere
-tz 11' (t Jrl ( t) )C' (t) Je (t )
(t-h(t) )e(t-h(t)) + D4 (t)e(t-h(t)) (28)
Assocoated with this, define a functional as
[l
b
l
(b + (d 2 +0 ) h / 62 )] 1/ Z Z 3 a
Hence we obtain 11 etz ll,;, COexp( aha) 11 etl ll ex[)(-a(t 2-t l )!34) for all tl and tZ( ~tl)' This [Tleans that (28) is the state estimator for the system S.
V(t) = ex[)(2 at)e' (t)Q-l (-t)e(t) 1 +~
( 33)
Q. E. D.
Jt
eXD(2 as)e' (s) t-h(t) x [C(s)f1' (sH1(s)C' (s) + oI]e(s)ds (29) where ~ = v6 2/2b 22(>0) and v is a Dositive parameter. V(t) satisfies
Remark 4. The limits of a , the conver~ence deoree, depend on d4 ; eSDecially in case D4 (t) = 0, a can be s[)ecified arbitrarily.
bl le(t) 12 exp(2 at) ,;, V(t) ,;, (b 2+(d 32+o)h/ 62) 11 et 112 exp (2at), (30) where
The system S can be stabilized by using its estimate produced by the estimator instead of the real state which may not be available directly.
d3 = SUPt >t 11 D3 (t) 11 . = 0 Then its time derivative along any solution of (28) can be shown as follows:
Su[)[)ose all conditions in Theorems 1 and Z are satisfied. If 8(·) or C(·) is bounded, then the system S can be exponentially stabilized by ap~lying the feedback law of the forr.l u (t) = K' (t h~ (t ) , where w(t) is the output of the estimator for S, and K(·) is almost the same as the feedback gain selected as in (10).
U
STABILIZATION BY OUTPUT FEEDBACK
Theorem 3 .
V( t) = exp(Zat) {e'(t)Q-l(_t)[d(~t) Q(-t) + Q(-t)(A(t)+aI)' + (A(t)+aI)Q(-t) - Q(-t)C(t)C'(t)Q(-t) + (l-~)IJQ-l(-t)e(t) + e'(t)(izI - (1+ ~ )Q-2(-t))e(t)
The proof of this theorem is omitted.
+ Ze' (t)[ Q- 1( - t) t·l (t- h( t) ) C' (t- h( t) ) + Q-l(-t)D (t)]e(t-h(t)) 4 (l-h(t))exp(-Z ah(t))e'(t-h(t))[C'(t-h(t)) x ~l(t-h(t))r.1' (t-h(t))C(t-h(t)) + oIJe(t-h(t))}
COtKLUSIQ~J
--tz
We have obtained sufficient conditions for a linear time-varying delay system to be stabilized bv means of linear state feedback without del~Y and for a state estimator associated = exp(Z at) with thi system to be capable of construction. x { - 1Q- 1 ( - t ) e ( t ) - rl ( t - h ( t ) ) C' (t - h ( t ) ) e ( t - h ( t ) ) 12 These result mav be reqarded as a natural exnansion of tile results for linear time1 1 -1 1 - ~ D4(t)Q (-t)e(t)- oe(t-h(t)) 2 - e'(t) invariant delay systems (Ikeda, 1978; K\~on. 1977). It is easy to apply the proposed methx Q-l(_t)( * Q2(_t) +~D4(t)D4(t) _ ~ I)Q-l(-t)e(t) od of stabilization and observation to linear 1 . systems with multi-delays, and linear inter+ (l-62(l-h(t)) exp(-2ah(t))) connected delay systems. x e' (t-h(t))[C(t-h(t))rl' (t-h(t)) x r1(t-h(t))C' (t-h(t)) + oI]e(t-h(t)) } ACKN01/LEDGC1ENT
,;, exp(Zat) where d4
=
(ii bZ2 + ~d42 - ~ )e' (t)Q-2( -t)e(t§l '
SUPt~to [1 D4 (t) 11
(
.
)
The authors express their qratitude to Professor M. Ikeda of Kobe University, Japan, for his valuable comments and helpful discussions.
Stabilization and Observation of Linear Systems
REFERENCES Ikeda, ~., H. ~aeda, and S. Kodama (1972) . Stabilization of linear systems. SIN~ J. Control, 10, 716-729. Ikeda, 11., H.t1aeda, and S. Kodar:la (1975). Esti mation and feedbac k in linear timevaryin9 systems: a deterministic theory. SIA~ J . Control, 13, 3~4-326. Ikeda, :: . , K. : 1urakam~ and I. lIaneda (1978). Observers for linear time-delay systems. Trans . Soc . Instrum. & Control EnCl., 14, 1-6. (in Ja:.>anese) Ikeda, M., and T. Ashida (1979) . Stabilization of 1ineilr syster:ls v'ith time-varying delay . IEEE trans . Autom . Control, 24, 369-370. Johnson, R. A. (1972 ). Functional equations, a:.>pro xi mations, and dynami c resronse of systems ~ith variable ti me delay . IEEE Trans. Autom. Control, 17, 39 3-401. Kalman, R. E. (1960) . Contributions to the theory of o[ltimal control. Rol. Soc. ~~at. rle xicana, 5, 102-119. K~lon, H . H., and A.. E. Pearson (1977). A note on feedback stabilization of a differential-rlifference system . IEEE Trans . Autom. Control, 22, 468-470 . Oijuzt6reli, ~,~t! . (1966). Linear delaydifferential e~uations. Time-lan control systems, Academic Press, Ne~ York. P:'>. 76-11 2. Seddon, J . P., and R. A. Johnson (1968) . The simulation of variable delay. IEEE Trans. Comout., 17, 89-94. Zade~A. -r1952). Operational analysis of variable-delay syster:ls . Proc . IRE, 40, 564-568.
157
Copyright © IFAC Contro l SCience and Technology (8th Triennial World Congress ) Kvoto . Japan . 198 1
ON THE STABILIZATION AND OBSERVATION OF LINEAR SYSTEMS WITH TIME-VARYING DELAY K. Yasuda*, M. Maekawa** and K. Hirai* "Department of Systems Engineering , Kobe University, Kobe , japan ""Central Research Institute, Sanyo Electric Company, Moriguchi, japan
Abstract. This oaper concerned with the stabilization and observation of a 1inear svsteJ:l v:ith time-varvinfl delav in the state. First, a sufficient condition for stabilization ' bv means of linear state feedback without delay is shown. ~ext, a sufficien(condition for the possibility of constructin~ a state estimator is found. Finally, it is shown that under these conditions and an additional one, a linear ti~e-varying delay systeJ:l can also be stabilized by apolying a linear feedback law where the output of the estiJ:lator is used instead of the state. Keywords. Stabilization; observer; tiJ:le-varyinq state feedback; Lyapunov methods. INTRODUCTION
syste~;
time-varying delay;
differential-difference equation: 1 x(t) A(t)x(t) + D(t)x(t-h(t)) S : 1 + B(t)u(t) (la) y(t) = C'(t)x(t) (lb) r n for tE[t O' 00 ), v!here X(t)ER , U(t)ER , and y(t)ER m are the state, the input, and the output of the system respectively; h(t) denotes the time delay; and A(t), B(t), C(t), and D(t) are matrices with appropriate dimensions. Suppose that every element of A(·), R(')' C(·), and D(·) is continuous, and there are positive numbers ha and hb such that
TiJ:le-varying delay appears in various real systems (Johnson, 1972; Seddon, 1968; Zadeh, 1952), and it freo,uently causes instability of the systems; nevertheless, so far the stabilization problem for linear time-varyinq delay systems have been treated in only a few studies (Ikeda, 1979). This paper is concerned with the stabilization and observation of a linear system with timevarying delay. The objective- is to establish for the time-varying delay systeJ:l the stabilizing method by a feedback compensator composed of a state estimator and an amnlifier, which is the standard stabilizing method for linear time-invariant systems without delay.
o ~ h(t) ~ ha
<
00 ,
tilt) ~ hb
<
1
(2)
for all tE[t O' 00). Let q, (.) denote the initial function of (la), that is, x(t) = q, (t) for tE[tO-h(t O)' tOJ·
First, the problem of stabilizing the system is considered, and a sufficient condition for stabilization by means of linear state feedback without delay is shown. Next, the problem of observing the state of the system is considered, which is formulated as the oroblem of designing a state estimator which nroduces the state estimate from only information on inputs and outputs of the system, and a sufficient condition for the possibility of construction is found. Finally, the probleJ:l of stabilizing the system by applyinq a linear feedback law where the outout of the state estimator is used instead of the state is discussed.
If u(·) is measurable, bounded on every finite subinterval of time on [to' 00) , and q, (.) measurable, then there exists a unique, continuous solution with the initial function q, (.) (Oquztoreli, 1966). Let the system of the form (1) (or (la)) with u(t) = 0 be called zero-input system, and define the stability of the system as follows : A zero-input system So is said to be exponentially stable, if there exist oositive numbers a and c such that any solution x(·) of So satisfies
Definition .
SYSTH1 DESCRIPTION AND DEFINITION OF STABILITY
(3)
Let us consider a linear system with timevarying delay described by the follovling 153
K. Yasuda, M. Maekawa and K. Hirai
154
for all tl and t2
( ~tl)'
where K' (t)
11 xt ll H, SUPt_h(t) ';;'T,;;,t lx(T) 1 , Ix(T) 1 fl (x'(T)x(T))1/2 . In this case, a is called stability degree.
= -
~(I +
-& L' (t) L(t ) ) B' (t) P(t ),(10)
\~here
62
~
(1-h b)exp(-2aha ) ,
6 > O.
(11 )
Then the closed-loop system S can be described by c
STABILIZATION BY STATE FEEDBACK In this section, we consider the nroblem of stabilizing the system S represented by (la) by means of linear state feedback without delay, u(t) = K' (t)x(t) , (4 ) where K(·) has appropriate dimensions and is continuous. Applying (4) to (la), the resulting closed-looo system can be written as S : x(t) = (A(t)+B(t)K' (t))x(t) c + D(t)x(t-h(t)). (5) 1. There exists a state feedback of the form given by (4) such that the resultant closed-loop system Sc is exponentially stable, if (i) the pair (/\(.), B(')) is uniformly completely controllable (Kalman, 1960) and (fl) D(t) may be decomposed as (t) = 01 (t) + O2(t) (6) Theore~
°
for all t, where Dl (t) can be represented as linear combinations of the columns of B(t) at every t, and D2 (t) is small enouqh in terms of the norm defined as sUPt>t 11 D2(t) 11 in = 0 which 11 D2 (t) 11 denotes the matrix norm of D2 (t) derived from the Euclidian norm. The following lemma is necessary to prove Theorem 1. (Ikeda, 1972). If a pair (A(·), P.(.)) is uniformly completely controllable, then for any real numbers a and v (>-1) the Riccati equation P(t) + (A(t)+aI)'P(t) + P(t)(A(t)+aI) - P(t)B(t)B' (t)P(t) + (1+ v)I = 0 (7) has a continuous solution which satisfies
Lel707la 1
(8)
for all t, where a l and a 2 are positive numbers. 1) First observe that if of Theorem 1 are satisfied, then there exists a matrix L(·) being continuous such that (9) Dl(t) = B(t)L'(t)
(Proof of Theorem
(i) and (ii)
for every t and, from Lemma 1, there exists a matrix P(·) satisfyinq (7) and (8) for any v greater than -1. For a suitable positive constant a, we define the feedback gain K(') by
x(t) =
-&
[A(t) - ~ B(t) (I + L' (t)L(t) )R ' (t)P(t)]x(t) + B(t)L'(t)x(t-h(t)) + D2 (t)x(t-h(t)). (12) Let a candidate for the Lyapunov functional be given by V(t) = exp(2at) x'(t)P (t) x(t) + (1+ f)
f
t exp(2as)x'(s)x(s)ds , t-h(t) (13)
where v is an appropriate positive number \'Ihich satisfies 2
al lx(t) 1 exo(2 at)
~
V(t)
~ (a 2 + (1+ ~)ha ) 11 xt ll exp(2 at).
(14)
Then its time derivative alonq any solution of (12) is calculated as follows: V(t) = exp(2 at) {x'(t)[P(t) + (A(t)+aI)'P(t) + P(t)(A(t)+a I) - P(t)B(t)B'(t)P(t) P(t)B(t)L' (t)L(t)B' (t)P(t) + (1+f )I]x(t) + 2x'(t)(P(t)B(t)L'(t) + P(t)D 2 (t))x(t-h(t)) - (1+f)(1-h(t))exp(-2ah(t)) x x' (t-h(t))x(t-h(t))}
--&
exp(2at) {- I tL(t)B' (t)P(t)x(t)- 6x(t-h(t)) 12 - 2~62 1 2D2(t)P(t)x(t) - v6 2x(t-h(t)) 12 - x' (t)( f I - v~2 P(t) O2 (t) O2 (t) P(t) ) x (t) + (1+f)(02_(1-h(t))exp(-2ah(t)) x x'(t-h(t))x(t-h(t)) } (15) Using ine~ualities (8) and 02 ~ (1-h(t))exp(-2ah(t)) gives . v 2 2 2 V(t) ,;;, -exp(2at)( 2 - ~ a2 d2 lx' (t)x(t), (16) v!here d2 = SUflt>t 11 D2(t) 11 = 0 This imnlies that if, for given positive numbers a and v, ~is small enough to satisfy d ,;;, v6/2a 2 (17) 2 then V(t) 5- O. !'!hen V(t) 5- 0 from (14), it can be shown that the solution of (12)
Stabilization and Observation o f Linear Systems
satisfies Ix(t) 1 ~
C* ll xt
1
for all
tl ( ~tO )
Il ex;:J(- ex (T-t l )) T( ~tl)'
and
(18 )
Thus,
~Ie
2
2
I f the es t i ma t i on error e(t ) = w(t) - x(t) is bounded on [to' 00 ) and satisfies
a
independently of input u( · ) and the initi al functions , then Es is said to be the state estimator for system S.
~ sunt2-h(t2) ~T~t/* 11 xtl ll exp(- ex (T-t l )) C*exo(ex ha ) II xt II exr (- ex (t2-tl )) 1
for all tl and t 2 ( ~tl ) . proof is complete.
(21 )
UmIlet ll = 0
obta i n
II xt II = sUP t -h i t )
=
e(t) to be given are defined on [to-h(t O) ' tOJ and are measurable. De fini tion .
where
C* = [jL(a +(1+ ~ )h )Jl / 2 al
155
(19 )
Thi s shOl"s that the Q.E.D .
It is imrortant to note that the stabilizinq method pro posed can be realized even if delay ti me hit) is not known nrecisel y , as lona as the upper bound ha and the up~er bound of its derivative hb are known . Remark 1. The limits of the stability de gree which can be desiqned depend on the magnitude of d2 , so that the system S is not always stabilized with pre-specified stability degree; while in the case of D2 (t) = 0 for all t, it is free from such a limitation. Remark 2. In comparison with the result (Ikeda, 1979), which is the same as Theorem 1 when D2 (t ) = 0, the stabilizing method obtained here has the advantage that a design taking account of perturbations of plant parameters is pos s ible. Remark 3. Let f( v) = v8 / 2a 2 , which is the right side of (17 ). It can be shown that f( v) is monotonousl y increasing and bounded so that the supremum of d2 exists for a given ex . STATE ESTItlATION BY ESTH1ATOR In this section, the observation problem is considered, and it is formulated as the problem of designing a state estimator . As a candidate for the state estimator associated with the system S, let us consider the linear time-delav system of the form z(t ) = (A(t )-H(t)C ' (t))z(t) E . + D(t)z(t-h(t)) + H(t)y(t) s . + B(t)u(t) (20a) \ wit) = z(t) (20b) where Z(t)ER n and W(t)ER n are the state and the output of Es res~ecti ve 1y, and fl ( .) whose elements are continuous is a matrix to be designed. Assume that the initial functions
From (1) and (20), it follows that the behavior of errors e (t) is gover ned by the homogeneous eC!uation E : e(t) = (A(t)-H(t )C' (t))e (t) r + D(t)e (t-h (t ) ) . (22) Utilizing thi s , we can obtain the followin g theorem. Theorem 2.
A matri x H(·) exists such that Es is the state esti mator for S, if ( i ) the pair (C' (.), A(·)) is uniforml y comnletel y observable (Kalman, 1960), (ii ) both C( ·) and 0(·) are bounded, and (iii) D(t) may be de comnosed as D(t) = D3(t) + D4 (t) (23)
where D3(t) can be represented as linear combinations of the rows of C(t-h (t)) at every t, and D4(t) is small enough in terms of the norm SUPt >t II D4(t) 11 · = 0 The constructibility of a state estimator for a linear time-varying system without delay has been derived by using the relation for the state transition matri x between the system and its dual system (Ikeda, 1975); however such a technique cannot be aoplied to a time-varyng delay system because the causality would not hold for the dual system which is a timevarying delay system of the advanced type . In order to avoid this difficulty, we employ the Lyapunov method . To verify Theorem 2, the fo 11 o~li ng 1emma is necessary. If a pair (C ' (·), A(·)) is uniforml y completely observable, then for any real numers ex and ~ ( > -l) the Riccati equation Q(t) + ~(t)(A(t)+ ex I)' + (A(t)+ex I)Q(t) - Q(t)C(t)C ' (t)Q(t) + (l+ ~ )I = 0 (24 ) has a continuous solution which satisfies (25) o < blI ~ Q(t) ~ b2 I for all t, where b and b2 are positive l numbers.
Lerrurza 2 .
(Proof of Theorem 2) Assume that (i), (ii) , and (iii) of Theorem 2 are satisfied . Then there exists a matrix M(·), continuous and bounded, such that D)(t) = C(t-h(t))M'(t-h(t)) (26)
156
K. Yasuda, M. Maekawa and K. Hirai
for every t and, from Lemma 2, there exists a symmetrical matrix Q(.) satisfying (24) and (25) for any ~ greater than -1. For a suitable H(·) as
~ositive
constant a , we give
From these inequalities, it can be shown that the solution of (28) satisfies
~Q(-t)C(t)(I + -tzW(t)M(t)). (27)
H(t)
Then (22) can be rewritten as e( t) = [A ( t) - ~ Q( - t ) C( t )( I + +
This immediately yields V ,;, 0 when d4 is small enough to satisfy d4 ,;, ~6 /2bZ (32)
~l(t-h(t))C'
for all tl( ~tO) and T( ~tl)' \vhere
-tz 11' (t Jrl ( t) )C' (t) Je (t )
(t-h(t) )e(t-h(t)) + D4 (t)e(t-h(t)) (28)
Assocoated with this, define a functional as
[l
b
l
(b + (d 2 +0 ) h / 62 )] 1/ Z Z 3 a
Hence we obtain 11 etz ll,;, COexp( aha) 11 etl ll ex[)(-a(t 2-t l )!34) for all tl and tZ( ~tl)' This [Tleans that (28) is the state estimator for the system S.
V(t) = ex[)(2 at)e' (t)Q-l (-t)e(t) 1 +~
( 33)
Q. E. D.
Jt
eXD(2 as)e' (s) t-h(t) x [C(s)f1' (sH1(s)C' (s) + oI]e(s)ds (29) where ~ = v6 2/2b 22(>0) and v is a Dositive parameter. V(t) satisfies
Remark 4. The limits of a , the conver~ence deoree, depend on d4 ; eSDecially in case D4 (t) = 0, a can be s[)ecified arbitrarily.
bl le(t) 12 exp(2 at) ,;, V(t) ,;, (b 2+(d 32+o)h/ 62) 11 et 112 exp (2at), (30) where
The system S can be stabilized by using its estimate produced by the estimator instead of the real state which may not be available directly.
d3 = SUPt >t 11 D3 (t) 11 . = 0 Then its time derivative along any solution of (28) can be shown as follows:
Su[)[)ose all conditions in Theorems 1 and Z are satisfied. If 8(·) or C(·) is bounded, then the system S can be exponentially stabilized by ap~lying the feedback law of the forr.l u (t) = K' (t h~ (t ) , where w(t) is the output of the estimator for S, and K(·) is almost the same as the feedback gain selected as in (10).
U
STABILIZATION BY OUTPUT FEEDBACK
Theorem 3 .
V( t) = exp(Zat) {e'(t)Q-l(_t)[d(~t) Q(-t) + Q(-t)(A(t)+aI)' + (A(t)+aI)Q(-t) - Q(-t)C(t)C'(t)Q(-t) + (l-~)IJQ-l(-t)e(t) + e'(t)(izI - (1+ ~ )Q-2(-t))e(t)
The proof of this theorem is omitted.
+ Ze' (t)[ Q- 1( - t) t·l (t- h( t) ) C' (t- h( t) ) + Q-l(-t)D (t)]e(t-h(t)) 4 (l-h(t))exp(-Z ah(t))e'(t-h(t))[C'(t-h(t)) x ~l(t-h(t))r.1' (t-h(t))C(t-h(t)) + oIJe(t-h(t))}
COtKLUSIQ~J
--tz
We have obtained sufficient conditions for a linear time-varying delay system to be stabilized bv means of linear state feedback without del~Y and for a state estimator associated = exp(Z at) with thi system to be capable of construction. x { - 1Q- 1 ( - t ) e ( t ) - rl ( t - h ( t ) ) C' (t - h ( t ) ) e ( t - h ( t ) ) 12 These result mav be reqarded as a natural exnansion of tile results for linear time1 1 -1 1 - ~ D4(t)Q (-t)e(t)- oe(t-h(t)) 2 - e'(t) invariant delay systems (Ikeda, 1978; K\~on. 1977). It is easy to apply the proposed methx Q-l(_t)( * Q2(_t) +~D4(t)D4(t) _ ~ I)Q-l(-t)e(t) od of stabilization and observation to linear 1 . systems with multi-delays, and linear inter+ (l-62(l-h(t)) exp(-2ah(t))) connected delay systems. x e' (t-h(t))[C(t-h(t))rl' (t-h(t)) x r1(t-h(t))C' (t-h(t)) + oI]e(t-h(t)) } ACKN01/LEDGC1ENT
,;, exp(Zat) where d4
=
(ii bZ2 + ~d42 - ~ )e' (t)Q-2( -t)e(t§l '
SUPt~to [1 D4 (t) 11
(
.
)
The authors express their qratitude to Professor M. Ikeda of Kobe University, Japan, for his valuable comments and helpful discussions.
Stabilization and Observation of Linear Systems
REFERENCES Ikeda, ~., H. ~aeda, and S. Kodama (1972) . Stabilization of linear systems. SIN~ J. Control, 10, 716-729. Ikeda, 11., H.t1aeda, and S. Kodar:la (1975). Esti mation and feedbac k in linear timevaryin9 systems: a deterministic theory. SIA~ J . Control, 13, 3~4-326. Ikeda, :: . , K. : 1urakam~ and I. lIaneda (1978). Observers for linear time-delay systems. Trans . Soc . Instrum. & Control EnCl., 14, 1-6. (in Ja:.>anese) Ikeda, M., and T. Ashida (1979) . Stabilization of 1ineilr syster:ls v'ith time-varying delay . IEEE trans . Autom . Control, 24, 369-370. Johnson, R. A. (1972 ). Functional equations, a:.>pro xi mations, and dynami c resronse of systems ~ith variable ti me delay . IEEE Trans. Autom. Control, 17, 39 3-401. Kalman, R. E. (1960) . Contributions to the theory of o[ltimal control. Rol. Soc. ~~at. rle xicana, 5, 102-119. K~lon, H . H., and A.. E. Pearson (1977). A note on feedback stabilization of a differential-rlifference system . IEEE Trans . Autom. Control, 22, 468-470 . Oijuzt6reli, ~,~t! . (1966). Linear delaydifferential e~uations. Time-lan control systems, Academic Press, Ne~ York. P:'>. 76-11 2. Seddon, J . P., and R. A. Johnson (1968) . The simulation of variable delay. IEEE Trans. Comout., 17, 89-94. Zade~A. -r1952). Operational analysis of variable-delay syster:ls . Proc . IRE, 40, 564-568.
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