Bounded Real Lemma and Stability of the Time Lag Nonlinear Control Systems

Copyrighl © l fAC ~hh Triennia l World Congress Budapesl, Hungary. 1984

BOUNDED REAL LEMMA AND STABILITY OF THE TIME LAG NONLINEAR CONTROL SYSTEMS Li Xunjing Department 01 Mathematics, Fudal/ UI/it'ersit)', SlllIIIghai 201903, TIll' People', Republic olChil/a

Abstract dx(t) 0

the stability of the time lag nonlinear control system

T , = Ax(t) + bf(c x(t - r».It 1S proved that the following are equivalent:

~

1

This paper studies

The trivial solution x=O of the system is globally asymptotically stable for every

r~

0 ,

2

f € ~;

0 and

stable for every

I

The 11near system

dx(t)

T Ax(t) + .1" be x(t-r) is asymptotically

-d~t-~

r ~ 0 and every .l" sr,tisfying

1 - h c T (i c.) I-A) - 1b

I

~

0 holds for every real

0 ~~< h;

fA)

3

0

The jnequality

. These results are extended to other

cases . Key words Bounded real lemma, nonlinear control systems , time lag systems , absolute stability, distributed parameter systems, Popov criterion.

INTRODUCTION

the discussion of the sufficient conditions on

Let the time lag nonlinear control system be dx(t) dt

the absolute stability of (1) for r = O.

Ax(t) + bf(z(t-r»,

in the feedback loop. And we do not

z(t) = cTx(t), where

exact value of r . This

x, band c are all

transpose of c,

n- vectors, c

T

is the

effect of time lag

A is an n X n- matrix whose 0

~

is the time lag, and the nonlinear function z

paper will consider the

r on the stability of (1).

stable if it is absolutely stable for ever y

f

r

h > 0,

is continuous.

denote

and

0

~fI-

~

0

was discussed by Popov and Halanay ( 1962 ) and

< h

Li ( 1963 ).

2

.{,p. z.

0 "; zf(z)

O.

systems was first considered by Chin ( 1960 ). The

f E ~ i f and only i f

there exists a constant p. such that

~

The unconditional stability of the time lag linear absolute stability of (1) for given time la g r

For given

0

know the

The system (1) is said to be h-uneonditionally

eigenvalues all have negative real parts, r of the real variable

~

In practice, there always exists a time lag r (1)

In this paper, it is proved that the s ystem (1) With a fixed

r

~

0, the system (1) is said to he

absolutely stable if the trivial solution

x

is

o

h-unconditionally stable if and only if the

linear system

of (1) is globally asymptotically Etable for every

f

dx(t)

~

E ~.

are asymptotically stable for every

When r = 0, the absolute stability of (1) was discussed by many authors

ever y

(Lurie, 1951; Letov,

1955; Popov, 1961; Zames, 1966 ). The positive real lemma

fL

satisfying

(2)

r

~

0

for real

and Vongpanitlerd, 1973

~

and

O$i:fL<. h . The inequality

T -1 1 - h l c (i Col I-A) b l~ O

(Yakubovic, 1962; Kalman, 1963;

Meyer, 1965; Anderson

T = Ax(t) + }'bc x(t-r)

(3)

is also a necessary and sufficient

ccndition of the h-unconditional stability of (1) .

shows that the Lurie ' s method and Popov criterion

The bounded real lemma ( Anderson & Vongpanitlerd,

are equiva l ent. All of them were restricted to

1973) 67

plays

an

importan t

role

jn the

Li Xunjing

68

dV(~~t»

proof of the above r es ults. These results are

__ /cTx(t)/ 2 _/dTx(t)j2

extended to other cases .

T -2d x(t) f ( z (t- r» /h _ - z 2 (t) - { d Tx(t) + f( z(t- r» / h} 2

MAIN THEOREMS THEOREM I 1 2

0 0

0

O ~ zf(

But

The system (I) is h-unconditionally stable ;

0

2 /h .

2

z):!i:j-L z . Hence, f

2

2

(z) ~l"' zf(z ~z) .

The ref o re,

fI-

sa ti sfying 0

~I'-<

dV(x(t» dt

h;

The inequality (3) h Qld s for every real ~

.

- z

2

(t)

z

2

2 (t- r) / h .

t

V(x(t» ~ V(x(O»

~

-

z 2(s)ds

o

=> 2 0 .

Obvious.

20 ~ 3°. Assume that there exist c.>o

2

I-}L c (i "V-A) then x(t) =

0

is true. If

+I'-

OS!,-< h and

2 rt

and ro"'O such that

T

.

(l ~. I-A)

-I

-I

be

be

-W r • 0 = 0,

i~ t

•

. 1S

~ V(x(O»

th e linear system in the case of

T

1- fl c (i fe) I-A)

-I

be

-iwr

+

jJ- 2

the sol uti on of

) z 2 (s o

satisfy ing

t-

CA)

r

~

T 2 2 I c x( s ~ d s / h

r = ro' This 2

T -I -i c (i GJ I-A) be

and constant

r'" 0 such tha t r

2

0

g2(x ( ')= o

1~1I2{V(X(0» +fL2)

2 f = "~U

2 (I-,J-/h ).

)L

2

W

rt

o

O!:j-L < h. 0, and choose

2

- (I- )L /h ) } z (s)ds.

¥' 0

0, real w

2

50

Denote holds for every

r )ds/h

- r

contradicts 20. So th e inEquality

Assume

2

+)1-

Thu s

PROOF We show 10~2 0 ~ 30 => lo. 1

r»

The linear system (2) is asymptotically stable

for every r 1> 0 and

3

+ f2 ( z (t -

The following are equivalent:

T -I I = I c (i w I - A) b.

2 2 T I c x( S)1 d s / h }, -r

WE ob tain

Therefore, the inequal it y 1- .,u../ cT(i W I-A)-l b I I

0

w 10 and I'- satisfying

holds for

/ cT(i W I-A)-lb l

O ~I'-<

is continuous in w

h. Since

and the equality

Hence

lim lcT( iWI-A)-l b l = 0 holds, we have

'''''~'''' for

1- }L/cT(i W I-A)-l bl :> 0

w I

O. Let

and

fl- h, we have

l-hIcT(i W I-A)-lb l~ O Thus 3

0

( w

1 0 ). Therefore,

is true.

0 0 0 3 ==> 1 . Assume that 3 is true. According to the bounded r eal lemma, there exist an n-ve c tor d and a positive definite

Pbh

s}~metric

matrix P such that

(4)

-d

(5) By applying the method used in Popov (1961) and Li (1963) , the inEqu8lity (5) and the following

Set

x(t)=e Then trere exists

m flxl2~

WP can prove that thE trivial solution of (I)

Let x(t) be the solution of (I). Then

is

globally as~ptotically stable for r ". 0 and f E ' \ . 0 is tn·. e. This completes the proof of

Thus 1 XT(t)(PA + ATp)x(t) + 2x T (t) Pbf (

By (4), we obtain

r t A( t- s) x(O)+ ) e bf( z (s- r »ds,

o

m :> 0 such that

Vex).

dV(x(t» dt

At

Z

Theorem I.

( t- r » . The s y stem (I) is called the direct control system . Now , we consider the indirect control system

69

Stability of the Nonlinear Control Systems

d~~t) = Ax(t) + bf( z(t-r )).

C (6)

dz (t) -- cTx(t) -d-twhere

and

f f( z(t-r )).

A. b. c are the same as in (1). Assume that

J'>O and f + c TA- 1b70. For given

h>O. denote

Then.

G(i CA»

is an

mxm-matrix. and

G*(i "' )

f t ~ i f and only i f there exists a constant ""

< zf(z

0 <)'-< hand 0

such that

)~)"

z

2

(dO).

THEOREM 3 THEOREM 2

There exists r

~

o

If the Hermitian matrix

0 such that the

(9)

linear system for real

Co)

then the trivial solution of (8) is

•

globally asymptotically stable for every

r. J

(7)

and

PROOF

j

Remark 1

=

fA) -

Since )C(O) < 0

J'l-If

of (8) is globally asymptotically stat·le for every

T + c (A_iWI)-l b l.

r.:!! 0

and ;-(+00 ) = + _ . there exists

Wo> 0 such that

r

o

~

f.~~

and

J

(j=l. 2 •...• m). j

J

Remark 2

We only prove that (9) is a sufficient

condition of the H-unconditional stability of (8).

O. Choose

The system (8) is said to be H-uncondi-

tionally stable if and only if the trivial solution

Define

JeW )

0

f. E ~ • J

is not asymptotically stable.

~

0 such that

PROOF

of Theorem 3.

By the bounded real lemma.

there exist real matrices

P and L with P positive

cefinite symmetric such that

then

PBH = - L.

iWo+}'-lr+

C

= i[W -

o

i. e . .A = i W o

T

(A-i W I) -l b } e o

,u.lf+

C

T

-iW r 0

0

(A-i W I)-l bl o

J

Let = O.

is a root of the characteristic

equation of (7) . Thus the linear system (7)

is

not asymptotically stable. This completes the proof

By the method used in Theorem 1. we can prove Theorem 3. Assume

of Theorem 2.

n = 1.

Let the system be

dx(t) dt

ax(t) +

m

Theorem 2 shows that the indirect control system (6)

has no

h-unconditional

sta~ility.

where

a
f.~~

THEOREM 4

What can happen when there are a number of non-

Denote

J

~

O.

f.E~ J

( j = 1,2,··. j

m ).

The system (10) is H-unconditionally

,ID

(8)

).

01 )

/

j=l PROOF

The

' only if' part was proved by Hale

(1977).

diag { h • 1 h2•

U

diag { )Ll·)L2· .. . ·)lm}

B

(l0)

J

m

H

F

J

(j=l. 2.

la/~L. hj/bjC j

dx(t) m T -d-- = Ax(t)+ L b.f.(c.x(t-r .)) . t j =l J J J J r.

J J

stable if and only if

linear feedbacks in the system

where

b.f.(c.x(t-r.)).

j

J

EXTENSIONS

l: j=l

hm}

Now we prove the 'if' part. Set

Vex) =

x

2

,

then dV(x(t)) = 2ax2(t) dt m + 2 b.x(t)·f.(c.x(t-r.)).

z:

J=1 J

J

J

J

Li Xunjing

70

Let X be a Hilbert space, <','

Hence dV(x(t» dt

product, and

, 2ax2(t)

> be

the inner

A be the infinitesimal generator of

an analytic semigroup

eAt

such that

o"

j=l with the constants

M> 0 and

0(

t

(13)

)

> 0, and band

c are elements of X. Consider the distributed parameter system dx(t) = Ax(t) + bf(z(t-r», dt When

z(t) = <: c,x(t» ~ ( 1 + 2d )V(x(t»

V(x(t+s»

THEOREM 6

Le.

(-max r

j

~

~

s

0),

dV(x(t» dt

The following are equivalent:

0 1 The system (14) is h-unconditionally stable; 0 2 The linear system

d > 0 is a constant, we have

where

dx(t)

, _ 2Ia/x2(t)

~=

Ax(t) +

JL b
is asyrr.ptotically stable wjth every

3

0

1 - hl
~ -2( a -(1-d)L h . /b . c./ )V(x(t», j=l

]

~

0

and

The inequality

m

dV(x(t» dt

r

JL(O~;t
Hence

d

(15)

m

2 + 2(1+d) L)'./b . c.lx (t). j=l ] ] ]

if

(14)

.

]]

>1 ~

0

holds fer every real W •

> 0 is sufficiently small. PROOF The proof of

By the Razumikhin's theorem (1956), we can prove

1°==>20~3°

is the same as

that of Theorem 1.

Thecrem 4. Since we lack the bounded real lemma in the 0 3 ::>10 is to be

Hilbert space, the proof of

Remark If The time lags rj are dependent on t, and

r.(t)

satisfy

]

0

~

r.(t) ]

~

globally asymptotical stability of (10) for

f .E ]

~ .h

and

r.(t) satisfying 0 ]

j

modified. We may use the method of Li (1963) to

r, then the

~

prove it.

holds r.(t) ]

~

r. CONCLUSIONS In studying the nonlinear contr ol s ystems, it is

THEOREM 5

Assume that the continuous function f

of two variables

t and z satisfies

0;:; zf(t,z) with

0

~)lz

2

~)l<

h, and the inequality

1 -

hlc T (i",I_A)-l b / ~ 0

possible to obtain the necessary and sufficient conditions for their stability by introducing a time lag in the nonlinear feedback function. bOl
The

real lemma and Popov criterion are useful.

holds for real '" . Then the trivial solution of the s ystem

REFERENCES Anderson, B.D.O. and S. Vongpanitlerd (1973),

dx(t) = Ax(t) + bf(t,z(t-r», dt (12) z(t) = cTx(t)

Network Analysis and Synthesis, PrenticeHall, Inc.

is globally asymptotically stable.

Chin Yuan-shun (1960), Unccndjtional stability of systems with time lags, Acta Math. Sinica,

PROOF

Its proof is the same as that of Theorem 1.

10, 125-142. Hale, J.K. (1977), Theory of functional differen-

Remark

A similar result to the system

t.

dx(t) = Ax(t) + b].f]. (t,cT].x(t-r » j dt j=l may be proved.

tial equations, Springer-Verlag. Kalman, R.E. (1963), Lyapunov functions for the problem of Lurie in automatic control, Proc. Nat. Acad. Sci. (USA), 49, 201-205.

Stability of the Non1inear Control Systems Letov, A.M. (1955), Stability of automatic controls, Gosizdat, Tekh-teoret, lit. L1 Xunj1ng (1963), On the absolute stability of systems with time lags, Acta Math. Sinica, ~,

558-573.

Lurie, A.I. (1951), Some nonlinear problems in the theory of automatic controls, Gosizdat, Tekh-teoret, lit. Meyer, K.L. (1965), SIAM J. Control, Ser. A,

1,

373-383. Popov, V.M. (1961), Absolute stability of nonlinear systems of automatic control, Avtom. & Telekh.,

~2,

961-979.

Popov, V.M. & A. Halanay (1962), On the stability of nonlinear automatic control systems with lagging argument, Avtom. & Telekh.,

ll,

842-851. Razumikhin, S.S. (1956), On the stability of systems with a delay, Prikl. Mat. & Mekh., 20, 500-512. Yakubovic, V.A. (1962), The solution of certain matrix inequalities in automatic control theory, Dokl. Akad. Nauk. SSSR,

~,

1304-

1307. Zames, G. (1966), On the input-output stability of time-varying nonlinear feedback systems, IEEE Trans. Autom. Control, AC-11, 228-238; 405-476.

71