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A Way to Stabilize Linear Systems With Delay
Copyright © IFAC Control Science and Technologv (8th Triennial World Congress) Kyoto. Japan. 1981
A W AY TO STABILIZE LINEAR SYSTEMS WITH DELAY T. Mori and M. Kuwahara Automatz'on Research Laboratory, Kyoto Unz'versz'ty, Ujz' 611, japan
Abstract. Existing schemes of stabilizing linear control systems with timedelay by linear feedback might be by no means easy one, since they often require solving the nonlinear matrix equation or the transcendental equation, or other cumbersome procedures. In this paper, we propose simple stabilizing methods of linear systems with delay. Required are checking positivity of a matrix containing two arbitrary parameters and solving the matrix Lyapunov equation with thes e parameters. We then consider the class of systems which fulfills the stabilizability conditions of the methods. We also show a stabilizability theorem of the system, and a simple stability theorem for the open-loop system, which are the direct consequences of th e main results. Keywords. Feedback; linear systems; Lyapunov equation; Stabilization; time lag systems.
1. INTRODUCTION
be the characteristi c roots of a matrix AE Rnxn and Amax(A) and \ min(A) give the
In analyses of linear control system with time-delay, mainly concerned subject has been stabilization by linear feedback with or without memory (delay). Many authors have reported this matter and contrived a variety of stabilizing schemes. However, existing methods of stabilizing systems with delay seem to be too involved in contrast with systems without delay from practical view points. For instance, some methods require solving the matrix Riccati equation or the transcendental equation, and others require transforming the system equations into canonical expressions or choosing many arbitrary parameters.
maXimum and minimum values of Ai (A) , respectively when they arE real.
Let I x I and
il AII be the Euclidean norms of a vector x and matrix A, respectively, i.e. x=(x'x)1/2 and 11
Ail =\ max 1/2 (A' A) .
The symbo I IJ (A) repre-
sents the matrix measure of A induced by this matrix norm, i.e. IJ (A)=1/2· Amax (A'+A). For X=X'E Rnxn, y=y'E- Rnxn, X>Y mean X-Y>Q (positive definite), also X2Y indicates X-Y : Q (nonnegative definite) where Q denotes the null matrix. We define a class of matrix pair (A, B) for A-= Rn xn, BE-Rnxm as follows;
In this paper, we derive simple stabilizing methods of linear systems with delay by memoryless (without delay) linear feedback. Stabilizability is checked by inspecting nagativity of the symmetric matrix containing two positive arbitrary parameters. After this, stabilizing law is ~onstructed by the solution of the matrix Lyapunov equation containing these parameters. The composition of this paper is as follows. In the next section, we give system description and show main theorems and their proofs. In section 3, we consider the assumptions of the theorems and the class of systems which can be stabilized by this approach. Section 4 presents a stabilizability theorem of the system and a stability theorem for the openloop system, which are the direct consequences of the main theorems. Frequently used symbols and notations will be cited in the following. (') denotes the symbol which transposes a vector or a matrix. Let ~ i(A)
-YL
~
{ (A, B) ! 3 K fc Rm xn, IJ (A+BK)
That is, ~ is considered to correspond to the class of systems x=Ax+Bu such that there exists a feedback law u=Kx satisfying W(A)
2. CONSTRUCTION OF STABILIZING FEEDBACK LAW We consider the following differentialdifference system; x(t)
Ax(t) + AOX(t-T) + Bu(t)
x(t)
4> (t),
t
( 1)
EO [- T, 0]
where x '" Rn, u c Rm, A EO- Rnxn, AD E Rnxn, T2:0 and 165
~ (t)
is a continuous vector-valued
166
T. Mori and M. Kuwahara
initial function.
d V\ -(x'(t)x'(t<)) dt (3) -
We will make an attempt to stabilize the above system by "memoryless" linear feedback,
P- 1
(2)
u(t) = Kx(t),
From (1) and (2), the closed-loop system can be written as ~(t) = Ax(t) + AOx(t- l ),
-
A = A +
Aoll
- i:I
J
x (t)
( 10)
x(t - c )j
Positivity (negativity) of the above quadratic form is equivalent to that of the matrix;
BK. (3)
Our main results are th e following theorems .
I
p-
[:
1Aol [P 0J
-~I
[Theor em 1]
J
I
Q
_'DJ
System (1) can be stabil i zed by the feedback (2), if there exist two positive numbers s ,
( 1 1)
B( >iI AII ) such th at Hl (s , (3 »0 for the matrix; Since vex, x t ) satisfies Hl ( i: , :3 ) = {
s4 + 211 (A) s 2 \ I (- Il (-A)+ B)2 Il (A)+ B J
(4) where I denotes the unit matrix. In thi s case , the feedback vector K is given by K = -B'p-l
(5)
where P=P'( E Rn xn) is the so luti on of the matrix Lyapunov equation; -(A+ SOP - p{ (A+ SO },
- 2s I.
(6)
1
sup IIX( t+s ) I: 2 ,
- 1
s· 0
remained requir ement for asymptotic stab ilit y of the system (3) is negative definiteness of the matrix (11). Conditions for sign de finiteness of partitioned matrices (Kreindler & Jameson, 1972) show that negative de finiteness of ( 11 ) is equivalent to PA' + AP + ~ p2 + ±AOAO ' < 0, which leads t o;
[Theorem 2]
S2p2 + sPA ' + s AP + AOAO' < Q.
If H2( S , B)
H2 (S , B) = {
.l'max(P- 1) Ix(t) 12 +
Using (2), (4), (5) and (6), we get
s4
+ 211 (A) s 2 } I ( B- ll (A))2 6- lJ (A) - 2 s BB' + AOAO'
s 2(p2+2I) - 2s(BP+BB ' ) + AOAO ' < (7)
for some 0 0, S (>II Al/) , th en the system (1) can be stabilized by the feedback (2) with (5) and the solution P of the following equation; (A- SOP + P (A- SI)' = - 2s I.
(12)
Q.
(13)
Now, turing to (6), we will make use of the following lower and upper bounds for its solution (Mori, 1980). ( 14)
(8)
- u (-A)+ 3
( 1 5)
The proofs will be shown by verifying asymptotic stability (El ' sgol'ts & Norkin, 1973; Bellman & Cooke, 1963; Hale, 1971) of the closed-loop system (3) for the corresponding feedback law.
Here, we use th e property of the matrix measure, lJ(A+ 31)=Il (A)+ 3 for any scalar value S. Considering the inequalities; ~ max(P)1 ~
(Proof of Theorem 1)
the condition;
Suppose the following Lyapunov functional for the closed-l oop system (3); vex, x t ) = x 'p- 1x + sf t x'xds (9) t- l where P is th e solution of (6) and XtE C[[- l , 0], Rn]. Observe that since s>IIAII, -(A+SI) i s a stable matrix and therefore P is positive definite. Taking time derivative along the solution of (3) yields,
P ~ Amin(P)I, (14) and (15) , it is easy to see
H ( S , 3) = 1
'-
4
t (-;'; (- A)+ 2)2 - 2i: BB ' + AOAO' < 0
is a sufficient condition for (13) . Thus, we can obtain a result for asymptotic stability of the closed -l oop system . Notice that Il(A) +6> 0 and - 11 (- A) +:3> 0 because of 6> 11 A" ~ 11 (A) ;;; 11 (-A) ~ - 11 A " . Q. E.D.
A Way to Stabilize Linear Systems with Delay You will easily be able to verify Theorem 2 in the same way as Theorem 1. Then the proof is omitted here. 3. CONSIDERATIONS ON THE MAIN RESULTS In this section we will give some notes concerning the theorems at the outset . Then we will consider the conditions in the theorems and the class of systems which could be stabilized by our methods. i) If the matrix BB' is non-singular, the system (1) can always be stabilized by feedback of the form (2). Because in either Hi( £ , 5), i=l, 2, we can choose £ such that the sum of the second and the third term is negative definite and, for this £ , can make the first term arbitrarily close to zero by choosing B. ii) If BB' is singular and at least AOAO' non-singular, ~ (A) < O is necessary for satisfying the assumptions of both theorems. Though this condition seems to be fairly strict, we can apply the theorems for systems belonging to ~~ , which includes controllable systems within transformations of the variables as shown later. For, if (A, B)107>t.-, there exists Kl such that ~ (A l)
Upon finding K2' the
control law may be constructed as u=(K 1+K2)x. iii) The theorems require negativity check of the matrix including two positive parameters . This procedure might be performed without difficulties because of the form of Hi( £ , B). Since, noting that the first term of Hi( E, S) is a scalar- multiple of the unit matrix, the conditions in the two theorems can be written, respectively as follows;
E4 ( B- ~ (A»2
+ 2 ~ (A) E 2 + fe E) < 0 B- ~ (A)
where fe E) is defined by feE) ~ \nax(AOAO' - 2EBB').
where
~ .(A)
167
denotes any matrix measure de-
riv ed from matrix norm Il AII . and K. is the constant such that II Atl2 ~ K . I I A I I. (e.g., K1 = Koo= hi) . Using these inequalities yields more general form of Hi( E, S) involving
~ .(.)
inst ead of ~ (.) in Theorem 1 or Theorem 2. However, throughout this paper we will adopt ~(A) = ~2(A) (i.e., K =1) for the sake of 2 simplicity. As shown in ii), in most cases ~ (A) < O or (A, B) 1O:7rr.. is an essential point of our stabilizing scheme. Hence, we will consider more precisely the class of system where (A, B) belongs to::irt. If (A, B) ""7'" , ~(Al) < O for A1=A+BK 1 and then we get Al '+A 1< Q by definition of ~ (.) . This shows the class::irL is also a class of stabilizable pair. However,7~ is not included by or includes the class of controllable pair, which is a typical subclass of stabilizable pair. To see this, it is enough to pick up an simple example;
A
[:
:],
B'
r:]
( 16)
which is a controllable pair but not belongs to 71'1-. An example having counter membership will be shown later as an application of the theorem. Attention should be payed to the fact that ~ depends on the selected bases of the space in contrast to stabilizability and controllability, which are both regardless of them. Namely, even though (A, B)~7~ we may choose T such that (T-IAT, T-IB)~~ for some classes of systems. Especially, controllable pair can always be made to be in ~ with an appropriate transformation matrix T. This can be shown as follows. Let (A, B) be a controllable pair. As is known well, there exists Kl such that A1=A+BK 1 has arbitrarily specified characteristic roots (restricted to those which are distributed symmetrically with respect to the real axis). Hence, we can choose such Kl that Al has distinct negative real characteristic roots. Then the transformation matrix T which diagonalizes Al can always be found.
Using
this T to transform Al yields Considering that fe E) is a function of E alone, we would be able to find a pair (E, B) satisfying the above scalar inequalities sys tematically. iv) Scrutinizing the proofs of inequalities (14) and (15) shows that the following relations are also valid correspondingly; >-.nin(P) () 2K. E ) "mal{ P ' -_~ -;-(--A...,,~):"';'-~ ':"';(7---A-:-)-+2-:-B::-
~(T- IAT
+ T-IBK 1T)
Amax (A 1) < O. Thus, we have (T-IAT, T-IB)E7~ In this way for the system (1) with (A, B) controllable, there always exists some nonsingular transformation matrix T and a feedback vector Kl such that we have ~(T-IAIT)
T. Mori a nd M. Kuwahara
168
y(t) + T-IBu .
( 17)
Furthermore , ~ (T-1AIT) can be designed t o any pr escribed value by choosing an appropr i ate vector KI . I t should be noted that stabil it y properties a r e kept unchanged for any suc h transformations of the system ( I) . Thus, for any controllable pair (A, B) we always find T such that (T-I AT , T- 1B) 6 J>L.At present , we do not have any way to construct KI generally in case I"here (A , B) ", :'7rr- but un contollable . However, for instance , under a specified condition such as Range (BB ' ) ~
C+ (A+A' )
(18)
where C+ (A+A') deno t es the eigenspace of A+A' corresponding to nonnegative characteristic roots, a simple feedback law of t he form K= :, B' ex /: R) realizes )1 (A I )
Subs tituting E=1 a nd 3=2 as an example, Hl (l, 2)<9 can be eas ily checked a nd by so lving (6) for these values of ( E , B) , we can ob ta in the solu ti on , P=I. So, we have u2= -B' x. The f in al form of the stabilizing sheme will be given as u=ul+u2=-3B'x. 4. SOME CONSEQUENCES OF THE ~!AIN RESULTS in this section we show that there follows from the theorems of section 2 a stabili zabili t y condition of the sys t em in a specific situation and a stabil i ty condi ti on of the open- loop system. Imposing of a certain condition on AO ' we have a stabil izability result; (Theorem 3) Le~
th e following t wo conditions be sa ti sfi -
ed . (i)
(ii)
In conc luding this section , we wi ll present an illustrative example of the theorem.
(A,
B)~-7n-
Each column of AO is expressed by linear combination of columns of B.
Example:
Then the sys t em ( I ) can be stabilized by feedback of th e form (2).
Consider the problem of stabilizing the sys t em ;
(Proof)
.[:::
l [XI 01 o J (t--r»)
2/3
(t - -
x2
( 19)
by linear feedback, u = Kx(t) ,
x(t)
While either Theorem 1 o r Theorem 2 will bring ou t th e result, we wou l d prove on the base of Theorem 1, for convenience ' sake. Firs t, it should be noted that the transfor mations of variables preserve the condition (ii). Therefore, as stated in sec ti on 3 , con tr o ll able pair (A , B) satisfying (ii) can be transformed so as to mee t both (i) and (ii). From the condi ti on (i), we a l ways have W(A I ) < 0 for Al = A+BKl by choosing Kl appropriately . So, we assume W(A) < 0 without loss of generality in th e sequel . Now , from the condi t ion (ii), there exis t s a matr i x C E Rmxn such that
(20)
As men ti oned above , u (A)
AO = BC. Then the sum of the second a nd third term of Hl( E, 3 ) becomes
what has been stated, we first make W(A I )=-1 by s i mple feedback of the form, ul=K l x=0 B'x, ER. This value is the lower bound which co ul d be achieved by this form of feedback
.: t
law.
By choosi ng ~ =-2. we have Al=-I and
therefore W(Al)= - U( - Al)=-I, IIA1 11=1. HI( E, B) of Theorem I can be calculated as 4 (6_E ) 2 I
[
(~~~ ) -2E+~ .i.. 9
% E4
-2 ~ BB'
+ BCC'B'.
If we c hoose E so tha t 2EI ~ CC', th e above matrix is negative s emi -d ef inite. Furthermore , choosing 3 large enough so as to satisfy th e cond iti on ; (22)
(which is possible because of the form of the right hand side of (22» leads to H1 (E,
2E2 4 ---+-
( B-1) 2 ( B-1 ) 9
(2 1 )
B) < Q. For these values of (E, S ) we have a stabilizing l aw of the form (2) with (5) and
(6) .
Q.E.D.
A Way t o Stabilize Linear Systems with Delay We next consider the condition of Theorem 2 , that is, the existence of (S, S) satisfyi ng S4
I
+
(3 - ~(A»2
2\l(A) ;: 2
3- ~ (A)
t ra nsform t he system ( 1) wi t h an appropriate non- singular transformation matrix T and feedback u=Kl x into the form;
A
I
yet) = T- 1 1Ty(t) + aoy(t - : ) , iJ (T -
- 2 €BB' + AOAO ' < Q. Consider i ng BB' ;:; Q, a sufficien t condition for the above i s a scalar i nequality; ;: 4 + 2dA) ( 3- ~ (A»2 3- ~ (A)
~2
+
:~ AO :l 2
> ::
< Q.
(23)
Ao :1
(24)
[Theorem 4) The open- loop system in (I), that is, the system, ~(t) = Ax(t) + AOx(t - T)
(25 )
stable , if (24) holds.
In fact , the stability condition (24) for system (25) has already been reported ~n more general setting elsewhere (Mori , Fukuma and Kuwahara, 1980). But there is employed a quite different approach . The condition (24) shows that , intuitively speaking , if the system without delay is enough stable relative to the "magnitude" of the term of delay, then the system with delay is also stable . This condi t ion , though i t might be conservative, may be useful in many practical situations because of its extremely simple form. Theorem 4 teaches an another way to stabilize systems with delay . If we choose in some way the feedback vector K so that the condition (24) can be met for the closed - loop system, this feedback law stabilizes the system. However, we wi l l not pursue the stabilizing scheme along this l i n e h ere. We only show that in a specif i c situat i on Theorem 4 works well t o construct the stabil i zing feedback l aw. [Corollary ) If Ao = aO I
<
o.
(26)
If _\l (T -1 A1T ) > l ao l can be achieved, the
closed - loop system (26)
~s
stable by Theorem
This is indeed possible because L (T- 1A 1T) can be set to any prescribed value .
5 . CONCLUS IONS Simple ways to stabilize linear systems with delay by feedback of the present values of the variables are derived. Stabilizability is checked by inspecting negativity of the symmetric matrix containing two arb i trar~ parameters. A stabilizing feed~ack law ~s . then easily implemented by solv~ng the matr ~ x Lyapunov equation involving these parameter: . Search for these parameters can be systemat~ cally performed and the matrix Lyapunov equation can be solved far easi e r compared with solving nonlinear matrix equation or transcendental equation as i n the cases of some earlier approaches. Some cons i derations on the class of the systems whi c h can be stabilized by this scheme a r e also shown. In most cases, an essentia l requirement is that the mat r ix measure of the closed - loop system matrix can be mad e negative by an appropriate feedback law . The class of th~ matrices, for which this can be performed ~s denoted ~~ and some discussions are given on this class. Fu r ther effort should be r equired to study the properties of the class and the method for obtaining the feed back law. A stabilizability theorem of the system and a simple stability theorem of open - loop system ar e derived based o? the main results. Especially Theor em 4 ~s useful in considering stabilizability property from slightly different viewpoint. Studies will have been continued on finding an easy way to stabilize and to ch e ck stabilizability of systems with delay . REFERENCES Armstrong , E. S. ( 1975). An Extension of Bass ' Algorithm for Stabilizing Linea r Continuous Constant Systems . I EEE Tran s. Autom. Co y.~ roZ , AC- 2C, 153- 154 . Bellman , R. and Cooke , K.L. (1963).
Di ffer entia l - Differ ence (Iao l < co , aO € R) and (A , B):
cont ro l lab l e in t he sys t em ( I ), t he feedback l aw of the form (2) s t ab i l i zes t he sys t em
(1) .
Eq ua~i o ns .
Academic Press , New York . Coppel, W. A. ( 1965) . StabiZ~ty and .
Asymptotic Behavior of D~ ffere nt" a l Equations . D. C. Hea t h , Bos t o n. El'sgol 't s , L . E. and Norkin , S.
( 1973) ..
I ntr oduc tion to t he Theor y and App l~ cation of Different ia l Equat i ons wi th De viating Arguments . Academic Pr ess ,
(Proof) As stated
A T) 1
Q. E.D .
Observe that this condition is irrespective of 6. Thus, (24) is the condition for e xi stence of ( s , 3 ) which satisfies (23) . Since (24) is also independent of B, it is a stability condition for B=Q, i.e., the openloop system. These arguments are summarized as follows .
~s
1-
4.
The existence condition of s satisfying (23) can be easily obtained as - \.! (A)
169
~n
the preced ing sect i o n, we can
New York .
170
T. Mori and M. Kuwahara (1966) . Diff el"ential Equations, Stability , Os cillat i ons , Time - Lags .
Halanay, A.
Academi c Press, New York. Hal e , J. (1971) . Fu nct i onal Di ffer entia l E~ ua=ic"s . Springer, New York . lked a , ' M. a nd Ohta, J . ( 1976). Stabili zatio n o f Linear Systems with Time Delay. Tl'cms . Soc . In s t rum . & Contr o l Eng . J 1 2, 637 - 641. Kreindler, E. and Jame son, A. ( 1972) . Conditi ons for No n- negativeness of Partiti oned Matrices . I EEE Tr ans.
Autol?l . COY!t l'ol , AC- 17, 147-148. Kwon, W.H. a nd Pears on, A.E. (1977). A Note o n Fee dba c k Stabilization of a Differ ential - Differ e n ce Syst em. IEEE Tr ans.
Auto,;,; . Con =l'ol , AC- 22, 468-470. Mo ri, T . ( 1980 ). Another Bounds for the Soluti o n t o the Ly apunov Matrix Equati on . Tr ans . Inst. Electr on. & Corrzl?lun . Eng . A, 63, 38- 40. Mo ri, T. Fukuma , N. and Kuwahara, M. (1980). A Simpl e Stabilit y Test f o r Linear Sy st ems with Delay . Tl"ans . Ins t .
Ele ct l'on . & Commun . Eng . A, 63,842-843. Nazar o ff, G. J. (197 3). Stability and Stabili zation of Linear Differential Delay Sys t ems. IEEE Trans . Autom .
Cont r ol, AC- 18,317-318. Nazaroff, G. J . and Hewe r, G.A. ( 19 73). Stabilization of Linear Differential Delay Sys t ems. IEEE Tr ans . Autom .
Cont r ol , AC- 18,673-674. Osipov, Y. S. ( 19 65) . Stabilization of Contro ll ed Syst ems with Delays. Di ffel' ential Equations , 1> 463-475. Ross, D.W . (1971). Controll e r Design for Time Lag Syst ems v ia a Quadratic Criteri on . I EEE Trans. Autom . Control, AC- 16, 664-672 . Sinha, A.S. C. (1972). Stability of Solutions of Differential Equations with Retarded Arguments. IEEE Trans . Autom.
Cont r ol, AC- 17, 241-242.
Discussion to Paper 6.2 V.M . Matrosov ( USSR ) : Besides t he timevarying de lay, what differences are there be tween your comparison principle and the comparison principle for the difference - differential system with the vector Lyapunov function. , W. Y. Oh t a (Japan): In Gromova s comparison principle, the problem is redu ce d to that of ordinary differential inequalit y . On the other hand, our c omparis on principl e deals directly with the functional differential inequality. Therefore the two comp a ri son princ ipl e s ar e different.
A W AY TO STABILIZE LINEAR SYSTEMS WITH DELAY T. Mori and M. Kuwahara Automatz'on Research Laboratory, Kyoto Unz'versz'ty, Ujz' 611, japan
Abstract. Existing schemes of stabilizing linear control systems with timedelay by linear feedback might be by no means easy one, since they often require solving the nonlinear matrix equation or the transcendental equation, or other cumbersome procedures. In this paper, we propose simple stabilizing methods of linear systems with delay. Required are checking positivity of a matrix containing two arbitrary parameters and solving the matrix Lyapunov equation with thes e parameters. We then consider the class of systems which fulfills the stabilizability conditions of the methods. We also show a stabilizability theorem of the system, and a simple stability theorem for the open-loop system, which are the direct consequences of th e main results. Keywords. Feedback; linear systems; Lyapunov equation; Stabilization; time lag systems.
1. INTRODUCTION
be the characteristi c roots of a matrix AE Rnxn and Amax(A) and \ min(A) give the
In analyses of linear control system with time-delay, mainly concerned subject has been stabilization by linear feedback with or without memory (delay). Many authors have reported this matter and contrived a variety of stabilizing schemes. However, existing methods of stabilizing systems with delay seem to be too involved in contrast with systems without delay from practical view points. For instance, some methods require solving the matrix Riccati equation or the transcendental equation, and others require transforming the system equations into canonical expressions or choosing many arbitrary parameters.
maXimum and minimum values of Ai (A) , respectively when they arE real.
Let I x I and
il AII be the Euclidean norms of a vector x and matrix A, respectively, i.e. x=(x'x)1/2 and 11
Ail =\ max 1/2 (A' A) .
The symbo I IJ (A) repre-
sents the matrix measure of A induced by this matrix norm, i.e. IJ (A)=1/2· Amax (A'+A). For X=X'E Rnxn, y=y'E- Rnxn, X>Y mean X-Y>Q (positive definite), also X2Y indicates X-Y : Q (nonnegative definite) where Q denotes the null matrix. We define a class of matrix pair (A, B) for A-= Rn xn, BE-Rnxm as follows;
In this paper, we derive simple stabilizing methods of linear systems with delay by memoryless (without delay) linear feedback. Stabilizability is checked by inspecting nagativity of the symmetric matrix containing two positive arbitrary parameters. After this, stabilizing law is ~onstructed by the solution of the matrix Lyapunov equation containing these parameters. The composition of this paper is as follows. In the next section, we give system description and show main theorems and their proofs. In section 3, we consider the assumptions of the theorems and the class of systems which can be stabilized by this approach. Section 4 presents a stabilizability theorem of the system and a stability theorem for the openloop system, which are the direct consequences of the main theorems. Frequently used symbols and notations will be cited in the following. (') denotes the symbol which transposes a vector or a matrix. Let ~ i(A)
-YL
~
{ (A, B) ! 3 K fc Rm xn, IJ (A+BK)
That is, ~ is considered to correspond to the class of systems x=Ax+Bu such that there exists a feedback law u=Kx satisfying W(A)
2. CONSTRUCTION OF STABILIZING FEEDBACK LAW We consider the following differentialdifference system; x(t)
Ax(t) + AOX(t-T) + Bu(t)
x(t)
4> (t),
t
( 1)
EO [- T, 0]
where x '" Rn, u c Rm, A EO- Rnxn, AD E Rnxn, T2:0 and 165
~ (t)
is a continuous vector-valued
166
T. Mori and M. Kuwahara
initial function.
d V\ -(x'(t)x'(t<)) dt (3) -
We will make an attempt to stabilize the above system by "memoryless" linear feedback,
P- 1
(2)
u(t) = Kx(t),
From (1) and (2), the closed-loop system can be written as ~(t) = Ax(t) + AOx(t- l ),
-
A = A +
Aoll
- i:I
J
x (t)
( 10)
x(t - c )j
Positivity (negativity) of the above quadratic form is equivalent to that of the matrix;
BK. (3)
Our main results are th e following theorems .
I
p-
[:
1Aol [P 0J
-~I
[Theor em 1]
J
I
Q
_'DJ
System (1) can be stabil i zed by the feedback (2), if there exist two positive numbers s ,
( 1 1)
B( >iI AII ) such th at Hl (s , (3 »0 for the matrix; Since vex, x t ) satisfies Hl ( i: , :3 ) = {
s4 + 211 (A) s 2 \ I (- Il (-A)+ B)2 Il (A)+ B J
(4) where I denotes the unit matrix. In thi s case , the feedback vector K is given by K = -B'p-l
(5)
where P=P'( E Rn xn) is the so luti on of the matrix Lyapunov equation; -(A+ SOP - p{ (A+ SO },
- 2s I.
(6)
1
sup IIX( t+s ) I: 2 ,
- 1
s· 0
remained requir ement for asymptotic stab ilit y of the system (3) is negative definiteness of the matrix (11). Conditions for sign de finiteness of partitioned matrices (Kreindler & Jameson, 1972) show that negative de finiteness of ( 11 ) is equivalent to PA' + AP + ~ p2 + ±AOAO ' < 0, which leads t o;
[Theorem 2]
S2p2 + sPA ' + s AP + AOAO' < Q.
If H2( S , B)
H2 (S , B) = {
.l'max(P- 1) Ix(t) 12 +
Using (2), (4), (5) and (6), we get
s4
+ 211 (A) s 2 } I ( B- ll (A))2 6- lJ (A) - 2 s BB' + AOAO'
s 2(p2+2I) - 2s(BP+BB ' ) + AOAO ' < (7)
for some 0 0, S (>II Al/) , th en the system (1) can be stabilized by the feedback (2) with (5) and the solution P of the following equation; (A- SOP + P (A- SI)' = - 2s I.
(12)
Q.
(13)
Now, turing to (6), we will make use of the following lower and upper bounds for its solution (Mori, 1980). ( 14)
(8)
- u (-A)+ 3
( 1 5)
The proofs will be shown by verifying asymptotic stability (El ' sgol'ts & Norkin, 1973; Bellman & Cooke, 1963; Hale, 1971) of the closed-loop system (3) for the corresponding feedback law.
Here, we use th e property of the matrix measure, lJ(A+ 31)=Il (A)+ 3 for any scalar value S. Considering the inequalities; ~ max(P)1 ~
(Proof of Theorem 1)
the condition;
Suppose the following Lyapunov functional for the closed-l oop system (3); vex, x t ) = x 'p- 1x + sf t x'xds (9) t- l where P is th e solution of (6) and XtE C[[- l , 0], Rn]. Observe that since s>IIAII, -(A+SI) i s a stable matrix and therefore P is positive definite. Taking time derivative along the solution of (3) yields,
P ~ Amin(P)I, (14) and (15) , it is easy to see
H ( S , 3) = 1
'-
4
t (-;'; (- A)+ 2)2 - 2i: BB ' + AOAO' < 0
is a sufficient condition for (13) . Thus, we can obtain a result for asymptotic stability of the closed -l oop system . Notice that Il(A) +6> 0 and - 11 (- A) +:3> 0 because of 6> 11 A" ~ 11 (A) ;;; 11 (-A) ~ - 11 A " . Q. E.D.
A Way to Stabilize Linear Systems with Delay You will easily be able to verify Theorem 2 in the same way as Theorem 1. Then the proof is omitted here. 3. CONSIDERATIONS ON THE MAIN RESULTS In this section we will give some notes concerning the theorems at the outset . Then we will consider the conditions in the theorems and the class of systems which could be stabilized by our methods. i) If the matrix BB' is non-singular, the system (1) can always be stabilized by feedback of the form (2). Because in either Hi( £ , 5), i=l, 2, we can choose £ such that the sum of the second and the third term is negative definite and, for this £ , can make the first term arbitrarily close to zero by choosing B. ii) If BB' is singular and at least AOAO' non-singular, ~ (A) < O is necessary for satisfying the assumptions of both theorems. Though this condition seems to be fairly strict, we can apply the theorems for systems belonging to ~~ , which includes controllable systems within transformations of the variables as shown later. For, if (A, B)107>t.-, there exists Kl such that ~ (A l)
Upon finding K2' the
control law may be constructed as u=(K 1+K2)x. iii) The theorems require negativity check of the matrix including two positive parameters . This procedure might be performed without difficulties because of the form of Hi( £ , B). Since, noting that the first term of Hi( E, S) is a scalar- multiple of the unit matrix, the conditions in the two theorems can be written, respectively as follows;
E4 ( B- ~ (A»2
+ 2 ~ (A) E 2 + fe E) < 0 B- ~ (A)
where fe E) is defined by feE) ~ \nax(AOAO' - 2EBB').
where
~ .(A)
167
denotes any matrix measure de-
riv ed from matrix norm Il AII . and K. is the constant such that II Atl2 ~ K . I I A I I. (e.g., K1 = Koo= hi) . Using these inequalities yields more general form of Hi( E, S) involving
~ .(.)
inst ead of ~ (.) in Theorem 1 or Theorem 2. However, throughout this paper we will adopt ~(A) = ~2(A) (i.e., K =1) for the sake of 2 simplicity. As shown in ii), in most cases ~ (A) < O or (A, B) 1O:7rr.. is an essential point of our stabilizing scheme. Hence, we will consider more precisely the class of system where (A, B) belongs to::irt. If (A, B) ""7'" , ~(Al) < O for A1=A+BK 1 and then we get Al '+A 1< Q by definition of ~ (.) . This shows the class::irL is also a class of stabilizable pair. However,7~ is not included by or includes the class of controllable pair, which is a typical subclass of stabilizable pair. To see this, it is enough to pick up an simple example;
A
[:
:],
B'
r:]
( 16)
which is a controllable pair but not belongs to 71'1-. An example having counter membership will be shown later as an application of the theorem. Attention should be payed to the fact that ~ depends on the selected bases of the space in contrast to stabilizability and controllability, which are both regardless of them. Namely, even though (A, B)~7~ we may choose T such that (T-IAT, T-IB)~~ for some classes of systems. Especially, controllable pair can always be made to be in ~ with an appropriate transformation matrix T. This can be shown as follows. Let (A, B) be a controllable pair. As is known well, there exists Kl such that A1=A+BK 1 has arbitrarily specified characteristic roots (restricted to those which are distributed symmetrically with respect to the real axis). Hence, we can choose such Kl that Al has distinct negative real characteristic roots. Then the transformation matrix T which diagonalizes Al can always be found.
Using
this T to transform Al yields Considering that fe E) is a function of E alone, we would be able to find a pair (E, B) satisfying the above scalar inequalities sys tematically. iv) Scrutinizing the proofs of inequalities (14) and (15) shows that the following relations are also valid correspondingly; >-.nin(P) () 2K. E ) "mal{ P ' -_~ -;-(--A...,,~):"';'-~ ':"';(7---A-:-)-+2-:-B::-
~(T- IAT
+ T-IBK 1T)
Amax (A 1) < O. Thus, we have (T-IAT, T-IB)E7~ In this way for the system (1) with (A, B) controllable, there always exists some nonsingular transformation matrix T and a feedback vector Kl such that we have ~(T-IAIT)
T. Mori a nd M. Kuwahara
168
y(t) + T-IBu .
( 17)
Furthermore , ~ (T-1AIT) can be designed t o any pr escribed value by choosing an appropr i ate vector KI . I t should be noted that stabil it y properties a r e kept unchanged for any suc h transformations of the system ( I) . Thus, for any controllable pair (A, B) we always find T such that (T-I AT , T- 1B) 6 J>L.At present , we do not have any way to construct KI generally in case I"here (A , B) ", :'7rr- but un contollable . However, for instance , under a specified condition such as Range (BB ' ) ~
C+ (A+A' )
(18)
where C+ (A+A') deno t es the eigenspace of A+A' corresponding to nonnegative characteristic roots, a simple feedback law of t he form K= :, B' ex /: R) realizes )1 (A I )
Subs tituting E=1 a nd 3=2 as an example, Hl (l, 2)<9 can be eas ily checked a nd by so lving (6) for these values of ( E , B) , we can ob ta in the solu ti on , P=I. So, we have u2= -B' x. The f in al form of the stabilizing sheme will be given as u=ul+u2=-3B'x. 4. SOME CONSEQUENCES OF THE ~!AIN RESULTS in this section we show that there follows from the theorems of section 2 a stabili zabili t y condition of the sys t em in a specific situation and a stabil i ty condi ti on of the open- loop system. Imposing of a certain condition on AO ' we have a stabil izability result; (Theorem 3) Le~
th e following t wo conditions be sa ti sfi -
ed . (i)
(ii)
In conc luding this section , we wi ll present an illustrative example of the theorem.
(A,
B)~-7n-
Each column of AO is expressed by linear combination of columns of B.
Example:
Then the sys t em ( I ) can be stabilized by feedback of th e form (2).
Consider the problem of stabilizing the sys t em ;
(Proof)
.[:::
l [XI 01 o J (t--r»)
2/3
(t - -
x2
( 19)
by linear feedback, u = Kx(t) ,
x(t)
While either Theorem 1 o r Theorem 2 will bring ou t th e result, we wou l d prove on the base of Theorem 1, for convenience ' sake. Firs t, it should be noted that the transfor mations of variables preserve the condition (ii). Therefore, as stated in sec ti on 3 , con tr o ll able pair (A , B) satisfying (ii) can be transformed so as to mee t both (i) and (ii). From the condi ti on (i), we a l ways have W(A I ) < 0 for Al = A+BKl by choosing Kl appropriately . So, we assume W(A) < 0 without loss of generality in th e sequel . Now , from the condi t ion (ii), there exis t s a matr i x C E Rmxn such that
(20)
As men ti oned above , u (A)
AO = BC. Then the sum of the second a nd third term of Hl( E, 3 ) becomes
what has been stated, we first make W(A I )=-1 by s i mple feedback of the form, ul=K l x=0 B'x, ER. This value is the lower bound which co ul d be achieved by this form of feedback
.: t
law.
By choosi ng ~ =-2. we have Al=-I and
therefore W(Al)= - U( - Al)=-I, IIA1 11=1. HI( E, B) of Theorem I can be calculated as 4 (6_E ) 2 I
[
(~~~ ) -2E+~ .i.. 9
% E4
-2 ~ BB'
+ BCC'B'.
If we c hoose E so tha t 2EI ~ CC', th e above matrix is negative s emi -d ef inite. Furthermore , choosing 3 large enough so as to satisfy th e cond iti on ; (22)
(which is possible because of the form of the right hand side of (22» leads to H1 (E,
2E2 4 ---+-
( B-1) 2 ( B-1 ) 9
(2 1 )
B) < Q. For these values of (E, S ) we have a stabilizing l aw of the form (2) with (5) and
(6) .
Q.E.D.
A Way t o Stabilize Linear Systems with Delay We next consider the condition of Theorem 2 , that is, the existence of (S, S) satisfyi ng S4
I
+
(3 - ~(A»2
2\l(A) ;: 2
3- ~ (A)
t ra nsform t he system ( 1) wi t h an appropriate non- singular transformation matrix T and feedback u=Kl x into the form;
A
I
yet) = T- 1 1Ty(t) + aoy(t - : ) , iJ (T -
- 2 €BB' + AOAO ' < Q. Consider i ng BB' ;:; Q, a sufficien t condition for the above i s a scalar i nequality; ;: 4 + 2dA) ( 3- ~ (A»2 3- ~ (A)
~2
+
:~ AO :l 2
> ::
< Q.
(23)
Ao :1
(24)
[Theorem 4) The open- loop system in (I), that is, the system, ~(t) = Ax(t) + AOx(t - T)
(25 )
stable , if (24) holds.
In fact , the stability condition (24) for system (25) has already been reported ~n more general setting elsewhere (Mori , Fukuma and Kuwahara, 1980). But there is employed a quite different approach . The condition (24) shows that , intuitively speaking , if the system without delay is enough stable relative to the "magnitude" of the term of delay, then the system with delay is also stable . This condi t ion , though i t might be conservative, may be useful in many practical situations because of its extremely simple form. Theorem 4 teaches an another way to stabilize systems with delay . If we choose in some way the feedback vector K so that the condition (24) can be met for the closed - loop system, this feedback law stabilizes the system. However, we wi l l not pursue the stabilizing scheme along this l i n e h ere. We only show that in a specif i c situat i on Theorem 4 works well t o construct the stabil i zing feedback l aw. [Corollary ) If Ao = aO I
<
o.
(26)
If _\l (T -1 A1T ) > l ao l can be achieved, the
closed - loop system (26)
~s
stable by Theorem
This is indeed possible because L (T- 1A 1T) can be set to any prescribed value .
5 . CONCLUS IONS Simple ways to stabilize linear systems with delay by feedback of the present values of the variables are derived. Stabilizability is checked by inspecting negativity of the symmetric matrix containing two arb i trar~ parameters. A stabilizing feed~ack law ~s . then easily implemented by solv~ng the matr ~ x Lyapunov equation involving these parameter: . Search for these parameters can be systemat~ cally performed and the matrix Lyapunov equation can be solved far easi e r compared with solving nonlinear matrix equation or transcendental equation as i n the cases of some earlier approaches. Some cons i derations on the class of the systems whi c h can be stabilized by this scheme a r e also shown. In most cases, an essentia l requirement is that the mat r ix measure of the closed - loop system matrix can be mad e negative by an appropriate feedback law . The class of th~ matrices, for which this can be performed ~s denoted ~~ and some discussions are given on this class. Fu r ther effort should be r equired to study the properties of the class and the method for obtaining the feed back law. A stabilizability theorem of the system and a simple stability theorem of open - loop system ar e derived based o? the main results. Especially Theor em 4 ~s useful in considering stabilizability property from slightly different viewpoint. Studies will have been continued on finding an easy way to stabilize and to ch e ck stabilizability of systems with delay . REFERENCES Armstrong , E. S. ( 1975). An Extension of Bass ' Algorithm for Stabilizing Linea r Continuous Constant Systems . I EEE Tran s. Autom. Co y.~ roZ , AC- 2C, 153- 154 . Bellman , R. and Cooke , K.L. (1963).
Di ffer entia l - Differ ence (Iao l < co , aO € R) and (A , B):
cont ro l lab l e in t he sys t em ( I ), t he feedback l aw of the form (2) s t ab i l i zes t he sys t em
(1) .
Eq ua~i o ns .
Academic Press , New York . Coppel, W. A. ( 1965) . StabiZ~ty and .
Asymptotic Behavior of D~ ffere nt" a l Equations . D. C. Hea t h , Bos t o n. El'sgol 't s , L . E. and Norkin , S.
( 1973) ..
I ntr oduc tion to t he Theor y and App l~ cation of Different ia l Equat i ons wi th De viating Arguments . Academic Pr ess ,
(Proof) As stated
A T) 1
Q. E.D .
Observe that this condition is irrespective of 6. Thus, (24) is the condition for e xi stence of ( s , 3 ) which satisfies (23) . Since (24) is also independent of B, it is a stability condition for B=Q, i.e., the openloop system. These arguments are summarized as follows .
~s
1-
4.
The existence condition of s satisfying (23) can be easily obtained as - \.! (A)
169
~n
the preced ing sect i o n, we can
New York .
170
T. Mori and M. Kuwahara (1966) . Diff el"ential Equations, Stability , Os cillat i ons , Time - Lags .
Halanay, A.
Academi c Press, New York. Hal e , J. (1971) . Fu nct i onal Di ffer entia l E~ ua=ic"s . Springer, New York . lked a , ' M. a nd Ohta, J . ( 1976). Stabili zatio n o f Linear Systems with Time Delay. Tl'cms . Soc . In s t rum . & Contr o l Eng . J 1 2, 637 - 641. Kreindler, E. and Jame son, A. ( 1972) . Conditi ons for No n- negativeness of Partiti oned Matrices . I EEE Tr ans.
Autol?l . COY!t l'ol , AC- 17, 147-148. Kwon, W.H. a nd Pears on, A.E. (1977). A Note o n Fee dba c k Stabilization of a Differ ential - Differ e n ce Syst em. IEEE Tr ans.
Auto,;,; . Con =l'ol , AC- 22, 468-470. Mo ri, T . ( 1980 ). Another Bounds for the Soluti o n t o the Ly apunov Matrix Equati on . Tr ans . Inst. Electr on. & Corrzl?lun . Eng . A, 63, 38- 40. Mo ri, T. Fukuma , N. and Kuwahara, M. (1980). A Simpl e Stabilit y Test f o r Linear Sy st ems with Delay . Tl"ans . Ins t .
Ele ct l'on . & Commun . Eng . A, 63,842-843. Nazar o ff, G. J. (197 3). Stability and Stabili zation of Linear Differential Delay Sys t ems. IEEE Trans . Autom .
Cont r ol, AC- 18,317-318. Nazaroff, G. J . and Hewe r, G.A. ( 19 73). Stabilization of Linear Differential Delay Sys t ems. IEEE Tr ans . Autom .
Cont r ol , AC- 18,673-674. Osipov, Y. S. ( 19 65) . Stabilization of Contro ll ed Syst ems with Delays. Di ffel' ential Equations , 1> 463-475. Ross, D.W . (1971). Controll e r Design for Time Lag Syst ems v ia a Quadratic Criteri on . I EEE Trans. Autom . Control, AC- 16, 664-672 . Sinha, A.S. C. (1972). Stability of Solutions of Differential Equations with Retarded Arguments. IEEE Trans . Autom.
Cont r ol, AC- 17, 241-242.
Discussion to Paper 6.2 V.M . Matrosov ( USSR ) : Besides t he timevarying de lay, what differences are there be tween your comparison principle and the comparison principle for the difference - differential system with the vector Lyapunov function. , W. Y. Oh t a (Japan): In Gromova s comparison principle, the problem is redu ce d to that of ordinary differential inequalit y . On the other hand, our c omparis on principl e deals directly with the functional differential inequality. Therefore the two comp a ri son princ ipl e s ar e different.