Stability of Linear Tirne- Invariant Systems - ResearchGate

CORRESPONDENCE 335

Model Reference

1

+

^b16

+

^b2s

Fig. 3. Suggested Liapunov redesign.

not conclude that the system is asymptoti- cally stable if r ( t ) is not constant. However, it is stable and should tend to operate near e = 0, whereas the design using the Hermite matrix may not.

RICHARD V . MONOPOLI Dept. of Elec. Engrg.

University of Massachusetts Amherst. Mass.

Stability of Linear Tirne- Invariant Systems

INTRODUCTIOS

I t is well known that the stability of a linear time-invariant system can be deter- mined using a Liapunov function of the quadratic form. However, for establishing the asymptotic stability of such a system for all linear negative feedback gains in a given range, a Liapunov function which yields both necessary and sufficient conditions has not been available so far. This correspon- dence presents such a function and affirms a recent conjecture of Narendra.1 The in- terest in this problem is due to its being a limiting case of the Lur'e problem.

Manuscript received December 30, 1966.

1 K. S. h'arendra. 'A conjecture on the existence of a quadratic Liapunov function for linear time-in- variant feedback systems." Pvoc. 4fh A l k r b n C o ~ r f . ?n Circuit and System Tlzeory (Umvermty of Illinois.

Urbana). 1966.

1

t

-~

THE PROBLEM Consider the system defined by the equations

f = A x - b K u

u = h T x (1)

which represents a transfer function G(s) = h T ( d

-

A)-% (2) with a linear negative feedback gain K .

A is an ( n x n ) stable matrix, x, b, and h are real n vectors, and u is a scalar function of time.

I t is required to find a Liapunov function which yields necessary and sufficient condi- tions for the asymptotic stability of the sys- tem for all feedback gains

K

in (0,

a).

PRELIMINARIES

Let pk (k = 1,2, * * ) be frequencies such that

I m G(jpk) = 0

and

,

Let B k be a matrix associated with pk defined by

and

g k j(Bk

-

Bk*) (4)

where

*

denotes the complex conjugate.

Then

Yk

(. - 4)

hTR& = - B k

where

dk = G(jpJ

+

⁼ 1

>

0 - K - and

hTg& = 0. (5)

Further, from (3) and (4) it follows that - - R ~ , A f - - I f p d r ; = O ^{Y k}

B k

9 d

+

^{I& = 0. (6)}

In addition, it can be shown that2,3 Re [hTRk(iWI - A)-%]

for all real W,

THE LMPUXOV FUNCTION Consider as a possible Liapunov func- tion2.3

v

= +xTPx

+

^+r(jaKu2

+

_k

y

K ^[(hTR-)'

+

^{(hT4S)'2j (8)}

where P >O.

of the system (1) is given by

The derivative along the trajectories

p

= +xT( ATP

+

^{P A ) x}

- KuxT

[

P b - Bo ATh] - &hTb

+

S k R [ ( h T R d ( h T R d x )

+

(hTga) (hTSLAx)

]

- &Kz[(hTRkx)u(hTR&)

+

(hTgk~)u(hTg&)].

k

Using (5)

P

can be rewritten as, P = 3 x ( ATP

+

^{P A ) x} - BohTb(Ku)*

- KoxTIPb

-

^BoA'h]

-!- &K[(hrRs)(hTRkAX)

+

( h T g m ) ( h T i J d x ) ]

k

a class of differential equations with a single monotone

2 K. S. Narendra and C. P. Neuman. 'Stability of nonlinearity." J . S I A M on Control, vol. 4. pp. 295- 308 May 1966.

Ramapriyan. "Solution,of a modified Lur'e problem." i

M.

A. L. Thathachar, Y. D. Srinath, and H. K.

submitted for publicatlon In I E E E Trans. on Auto- matic Confrol.

336 IEEE TRANS-4CTIONS ON AUTOM4TIC CONTROL, JIJNE 1967 Adding and subtracting the following terms,

1) K6a2 (1

- $)

takes the form

iJ

⁼ +xT(ATP

+

^{P A ) x}

- ( s & T ~ + n ( K D ) ~

- K6a2 (1 -

$) .

Setting a k = & p ~ and using relations (6), f = QxT(ATP

+

P A ) x

- (

BohTb

+

^-

3

( K U ) ~

- K U X ~ P b

[

- Bo-ATh - 6h

-

^yk (1 - = K Kdk) RxTh]

K

-

K6a2 (1 -

$) .

THE STABILITY CRITERION A direct application of the LIeyer-Kal- man-Yakubovich lemma2 gives the cri- terion for asymptotic stability as

+ 1

^{y k (1 - K f}

dX)

^h'Rr]

. [ i d

- A ] - ' b 2 0 (12) for all real a, m-here 6>0.

the form

Use of property (7) yields the criterion in Re Z Go)G (ju)

2

0

(13)

for all real o, where Z(S) = 6

+ Bos

Since &>_0 and K < X , one can always choose 71; and p k to get

where 6>0 and Bo, Ck are any non-negative numbers. If only neutral stability is de- sired, 620.

INTERPRETATION OF THE CRITERION Brockett and WillemsL have shown that a necessary and sufficient condition for a transfer function G(5) to be stable f o r all linear negative feedback gains in (0, R ) is that there should exist a multiplier Z ( s ) such that

Z(s) [ W )

+ f ]

is positive real, and

P

where Xi and pk are frequencies a t which

and

d Arg [GGu)

+

⁼ K

'I

du

<

^{0 a t o = hi}

This assumes that the first nonzero value of Arg GCjo) is negative.

A slight extension of the result yields the form of multiplier for asymptotic stability as

where 6>0 can be arbitrarily small.

By expanding (18) in partial fractions, Z(s) can be put in the same form as (15).

Thus the criterion (13) developed using the Liapunov function (8) is equivalent to the criterion of Brockett and IVillems and is hence necessary and sufficient.

If the phase angle of GGo) starts in the positive direction, one can consider l/(GCjw)

+

¹

/ X )

in the place of GGu)

+

¹

/x

^{and pro-}

ceed as before.

The results can be extended to the case where G(s) is the ratio of equal-order poly- nomials. L-sing a well-known transforma- tion: the stability of such a system is equic- alent to that of a system having a transfer function G ( 5 ) -G( m ) with negative feed- backgains in [0,

KT/(l+EG(

m))], and hence the results derived earlier can be applied.

CONCLKSION

It has been shown that a necessary and sufficient condition for the asymptotic stabil- ity of a linear time-invariant system for ail linear negative feedback gains K in (0, IC)

is the existence of a Liapunov function of the form

V = xTPx

+

^KxTAfx

where P is positive definite, and A€ = &hhT

+

&[RkThhTRk

+

9 k T h h T g k ]

Is

is positive semidefinite.

Thus the Liapunov function is the sum of a positive-definite quadratic form and a positive-semidefinite quadratic form de- pending linearly on

K

as conjectured by Narendra.1

$1. A. L. THATHACHAR

?VI. D. SRINATH Dept. of Elec. Engrg.

Indian Inst. of Sci.

Bangalore 12, India

-4ufkor's Reply6

Thathachar and Srinath are to be con- gratulated on deriving a simple Liapunov function for proving the stability of a linear time-invariant feedback system in the entire range 0

<

K

<

^m (or

E )

of the feedback gain.

The proof is based on the fact that a Z L C ( S ) function always exists so that G ( s ) Z x ( s ) is positive real if the feedback system is stable for all feedback gains in the range 0

<

K

<

^m

.

The authors' contribution lies in choosing the terms of the Liapunov function, par- ticularly the matrices RK and Ig in (8) so that the desired multiplier Z L C ( S ) can be ob- tained. Since the existence of the multiplier Z L C ( S ) is both necessary and sufficient t o prove the stability of the feedback system, the existence of the Liapunov function of the form derived is both necessary and suf- ficient for stability in the range 0 <K

<

^m

(or

E ) .

On the basis of the results obtained, one can proceed to determine, for the time-in- variant case, a Liapunov function having the general form

V(r) = XTPZ

+

K X T Q Z (19) where P is a positive-definite matrix and Q a semidefinite matriv which can be ex- pressed as

Q =

1

&HihltTHiT

where

Hi

is an ( E Xn)

matrix.

The Liapunov function (Sj derived in the correspondence can be used for the most restrictive case t o obtain an LC function Z L C ( S ) , for which G(s)Zx(s) is positive real. In most cases, however, where Z R C ( S ) , Z R L ( S ) , or Z R L C ( S ) multipliers exist, simpler Liapunov func- tions of the general form (19) can be deter- mined. In such cases, the Liapunov func- tion for the linear time-invariant case can be suitably modified for different nonlinear situations.2

The form of the 0 matrix is closely re-

CORRESPONDENCE

TABLE I

>ft!ltiplier ~ ( s )

'

Nonlinear Gain for which System Is Stable

Q GW a s ) P.R.

i

1) Q = 0 (Null Matrix) Konlinear and Time-Varying Gains Constant

o < - - < m

f ( C . 0 c

2) Q = uhhT ^{f ( C )}

(Popor)

(1 + a d O < - < m

0

3) Q = ahhT

+

t i T i T i T ZRL or ZRC Monotonic Increasing Function f(u)

I

^I

riT = hT(qiI

+

A1-l o < - < = ^f(u)

C

- v i is a real zero of G(s)

4) Q [in Equation (S)l ^{ZLC :} Constant Gain K. 0

<

K

<

m

gain functions for which the system is also stable. Examples of this relationship are in- dicated in Table I.

I t is this author's opinion that the same form (19j of the Liapunov function can also be used to derive conditions for the stability of linear time-varying systems b y replacing IC b y K ( t ) i? (19). In such cases, the time derivative K ( t ) has to be constrained in some manner.

K. S. SARENDR~

Yale University Xew Haven, Conn.

Explicit Stability Criteria for the Damped Mathieu Equation

INTRODUCTION

T h e purpose of this correspondence is t o establish explicit stability criteria for the Mathieu equation with damping term 6>0:

1-

+

^66.i-

+

^(w2

+

^{E cos 2)s = 0. (1)}

Since the time-invariant part of (1) is uni- formly asymptotically stable, it is thus knoxvn t h a t for E sufficiently small the per- turbed equation will also be stable. An ap- plication of Liapunov's direct method en- ables one t o obtain a bound on the parame- ter e within which bound stability is as- sured. In particular, it will be shown t h a t under the reasonable assumptions of large w and small 6, the condition for stability of (1 j becomes a very simple inequality relation- ship involving only e and 6, Le., independent of the frequency w .

-~PPLICATIOS OF LIAPUSOV'S DIRECT METHOD Equation (l), the damped hlathieu

equation with damping term 6>0, fre- quency ^{U ,} and small parameter E , is equic- alent to the system

x 1 = x2

x 2 = - (w'

+

^{E cos 2 t h l - 46x2.}

Manuscript received January 3, 1967.

One can then write

X = [A

+

e P ( t ) ] x ( 2 ) where

x = ( a f?)'

A = (-w? O

-qs ' >

P ( t ) =

(

_-1 ^{0 0} ₀

)

^{cos 2t.}

T h e "prime" is used t o denote the transpose of a vector or a matrix. The eigenvalues XI,? for the time-invariant part of (2) are easily determined to be

- 6

+

^{4 6 2 - 1 6 w 2}

4

__- X l , Z =

from which it is seen that Re (X1.2] < O for all 6 >0, w #O. Hence, the time-invariant part of (2) is uniformly asymptotically stable. This ensures that for E sufficiently small, the perturbed system will also be stable. I t is now desired t o obtain explicit stability criteria involving e, 6, and W.

Toward that end, choose as a tentative Liapunov function for (2) the quadratic form

I:(x! 1) = V(x) = x'Bx

where B ' = B = ( b i , j . This choice of V ( x j leads to

F(x,

tj = - X ' [ C - ~ ( P ' B

+

B P ~ ] X

where

A ' B + BA = - C (3) with the matrix C restricted to be positive definite. Then, noting t h a t b12 =bZ1, and choosing C equal to the identity matrix, (3) becomes

( b l 1 - 36b12 - w2b2? 2 ( b l l - $6b12) -2w2b12 b l l - +6bln - w2b?,>

= -

(; Y).

337 The matrix B determined in this manner is the matrix of a positive definite quadratic form.

\\-ith B now given by (41, one has

i

¹

-1

^w2

+

¹

and thus

p ( x , t ) = - x'Dx where

!1 r + > c o s 2 t e -E

D =

I

6w2

I w ' + 1

l e ~ C O S 2 t

i.

In order that V ( x ) be a Liapunov func- tion, those values of E , 6, and w are sought such that D is positive semidefinite. .A suf- ficient condition that the real, symmetric, time-varying matrix D be positive semi- definite is t h a t its principal determinants be strictly greater than zero. This condition leads to the following constraints on E , 6, and w :

1

+

^{cos 21}

>

0 w2

€ p 2 )

^{cos? 2:} - f cos 2t - 1

<

0.

-2

The above inequalities are satisfied for all t when

1

^{e l} <us ( 3

and

B-

<

E

<

ET ( 6 4 where

\$'henever (51 and (6a) are satisfied, then V ( x ) is positive definite with negative semidefinite time derivative. Hence, V ( x ) qualifies as a Liapunov function and the

system (2) is stable. If all the principal de- terminants of D were greater than some number -y, Le.,

1

+

^{_ f _ cos 2 f}

>

⁷

0 2

where y is a positive constant independent of X and f , then p ( x , t ) would be negative definite, and the system ( 2 ) would therefore be asymptotically stable.

SIMPLIFICATION OF THE STABILITY CRITERI.4

Consider the case when w is large and 6 relatively small, i.e.,