Robust Stability Evaluation For Control Systems with Time-Invariant Nonlinearlty in Gain-Phase Plane

Copyright © IFAC Computer Aided Control Systems Design, Gent, Belgium, 1997

ROBUST STABILITY EVALUATION FOR CONTROL SYSTEMS WITH TIME-INVARIANT NONLINEARITY IN GAIN-PHASE PLANE Yoshifumi OKUYAMA, Hong CHEN and Fumiaki TAKEMORI

Department of Information and Knowledge Engineering, Tottori University, 4-101 , Koyama-cho Minami, Tottori 680, Japan E-mail: [email protected]

Abstract: In this paper, by applying the small gain theorem that concerns L2 gain in regards to a non linear subsystem with free parameter q ~ 0, a stability condition for control systems with time-invariant nonlinearity (equivalent to Popov's criterion) is presented. Using this concept, a robust stability condition in a gain-phase diagram for an open loop system (modified contour M) which corresponds to the Nichols chart is proposed . The stability margin is discussed in regards to the gain margin of a nominal system. This type of gain-phase diagram may be used in the design of a robust control system. Keywords: Robust stability, nonlinearity, Popov criterion, gain margins, Nichols charts.

1. INTRODUCTION In this paper , by applying a small gain theorem that concerns L2 gain to a nonlinear subsystem with free parameter q ~ 0, a stability condition for control systems with time-invariant nonlinearity (equivalent to Popov 's criterion) is given. Then , a robust stability condition in a gain-phase diagram for an open loop system (modified contour M) which corresponds to the Nichols chart is presented based on this concept . Relationships between gain margin and stability margin are discussed . This gain-phase diagram may be used for the design of a robust control system.

A small gain theorem with a L2 norm is generally applied for the robust stability condition in a frequency domain for control systems with uncertainty and nonlinearity. It can also be applied to time-varying nonlinearity or frequency dependent uncertainty and has been widely used as a design technique for H 00 robust control systems (Vidyasagar , 1985 ; Francis , 1987) . This concept has been extended in order to solve design problems associated with several uncertainties and/or nonlinearities (Packerd and Doyle, 1993) . However , stability theory with a L2 norm which was proposed by I.W .Sandberg (Sandberg, 1964) , consequently, imposes more conservative restrictions on the frequency response characteristics of the linear parts of a control system (Zames , 1966; Holtzman, 1970 ; Desoer and Vidyasagar, 1975; Harris and Valenca, 1983) . When the uncertainty in question is a multiplicative perturbation , the restriction is imposed on the peak value of the gain characteristic of a complementary sensitivity function . However, the relationship between the robust stability condition (which is based on a gain characteristic) and the stability condition for control systems with time-invariant linear characteristic (interval gain) or time-invariant nonlinear characteristic (sector nonlinearity) is not always clear.

r

+

e

y

N(e)

G(s)

v

u

+ +

d

Fig . 1 Nonlinear feedback control system 2. CONTROL SYSTEM WITH TIME-INVARIANT NONLINEARITY Consider a nonlinear feedback system as shown in Fig . 1. Here , G( s) indicates the time-invariant linear characteristic, the frequency response of which

109

is known. Even if G(s) has an uncertainty, when the band of both the real and the imaginary parts of frequency response characteristics (or characteristic curves of in a worst case), are taken into considerarion, the following discussion is still applicable.

is obtained . Here, the inner product and the norm is defined as

Assume that N(e) is a time-invariant zeromemory-type nonlinearity characteristic, which can be written as follows :

N(e)

= K(e + n(e)) = K(1 + O'(e))e, Iwl = In(e)1 = 100(e)I ' lel ~ aiel.

When written as (" -) or 11 · 11, it denotes the case when T -+ 00. Moreover, wT(t) and ej.(t) denote the truncation functions of w*(t) and e*(t) at t = T, respectively. ( Proof) From Eqs . (3) and (4), the following equation holds:

(1)

(2)

where 0'( e) is a nonlinear term relative to nominallinear gain I< (in other words , a multiplicative perturbation expression.)

a211e*1I} - IIw*ll} =

+

r

y

G(s)

u

2

~~ 11: ~: ) T

Based on sector nonlinear characteristics in Eq. (2) and the premise of this Lemma, i.e.,

+ d +

IIwll} ( w

Fig. 2 Equivalent nonlinear system

as for q 2: 0 and

For this nonlinear element n(e), a nonlinear subsystem as shown in Fig. 3 is assumed .

~

a2I1ell},

+ ae, ~;) T 2: 0, a > 0 the following inequaliy :

that is,

r-'-'-'-'-'-'-'-'-'-'-'-'- '-'-'-'-'-'-',

i

aq

2 = a l1ell} - IIwll} + 2aq ( w + ae,

v

I<

a 11 e + q ~~ 11: -11 w -

w

e'

i i

can be obtained for any truncation time T .

i i i

Incidentally, Eq. (6) in this Lemma may be appropriate for T -+ 00 . The details were described in our previous results (Okuyama, 1988) for a continuous-time system and (Okuyama and Takemori, 1996) for a discrete time system.

L._._._._._._._._._._._._._._._._._._._ n(e) Fig. 3 Nonlinear subsystem

As is obvious from the figure , the following equation is obtained: * de e = e + q dt'

w

*

de = w- _O'q. dt

0

(3)

2. ROBUST STABILITY CONDITION

(4)

By placing nonlinear subsystem n( e) of Fig. 3 inside nonlinear part n( e) of the control system in Fig . 2, the loop transfer function from w* to e* can be expressed as follows:

Hence, the following lemma can be given.

[ Lemma-!] If the following inequality is satisfied:

_ . (1 + qs)KG(s) H(O', q, s) = 1 + (1 + O'qs)KG(s/

(7)

Hence, by applying the small gain theorem in regards to L2 gains, the following robust stability condition can be obtained:

for any q 2: 0

(1 + jqw)KG(jw) 1 + (1 + jO'qw)KG(jw)

I

(6)

110

I

<

1 Cl '

(8)

4. VERIFICATION OF STABILITY J( G(jw)

is ex-

= p(w)ei8 (w) = U(w) + jV(w),

(9)

If the open loop transfer function pressed as

J(G(jw)

IN A GAIN-PHASE PLANE As is obvious, the following curve on the gainphase plane:

Eq. (8) is also written as

e(O, w) = M,

I

p(l + jqw)ei8 1 + p(l + jOtqw )ei8 (l+jqw)(U+jV) 1 = l+(l+jOtqw)(U+jV) < a'

I

I

I

M = const.

(18)

in Eq. (15), corresponds to the contours of the constant M in the Nichols diagram. Therefore, the following curve based on Eq. (11):

(10)

e(q,w)

The robust stability condition for Eq. (10) can be rewritten as the following theorem.

={(q,w,B(w),p(w)) = M

(19)

becomes the modified contour for contour M of the Nichols chart. The modified contours are given by the following lemma.

[ Theorem-l ] For any q 2: 0, if nonlinear sector a satisfies the following inequality, the control system of Fig. 1 is robust stable:

[ Lemma-2] (Modified Contour) The contour is drawn by first obtaining phase B of e (q, w) = ---:--::--r=:::::::::==:=P:;:;=:===::====== the open loop characteristic for each wand next 2 2 2 -qw sin B + J q w sin B + p2 + 2p cos B + 1 calculating gain p according to the following equal tions:

< -=-, Vw. Ot

(11)

( Proof) From the square of both sides of Eq. (10),

When M

a 2p2(1+q 2w2) < (1+pcosB-aqwpsinB)2 +(aqwp cos B + psinB)2 . (12)

p

a 2p2 < -2aqwpsinB + (1

= - M~- 1 (M cos B -

±

Thus, the following quadratic inequality can be obtained:

> 1,

M

~-1 J(M cosB -

qw sin B)

qw sin B)2 - (M2 - 1). (20)

When M = 1,

+ pCOSB)2 + p 2 sin 2 B.

M p = - 2(M cosB - qw sin B)'

(13)

(21)

Consequently, as a solution of Eq. (13)

a< -

qw sin B p

(Proof) From Eqs. (11) and (19),

+

p

(14)

-qw sinB + J q2w2 sin 2 B+ p2

is given . 0

p = -M qw sin B+M Vq 2W 2 sin 2 B + p2

(M2 - 1)p2

w

(15)

The point that these modified contours M are different from the Nichols chart contours is that angular frequency w is contained explicitly in Eqs. (20) and (21). In this paper, for each w the phase angle B is determined from

(16)

that is, if the following is satisfied:

Mo <

1

-=-, Ot

+ 2pM(M cosB - qwsinB) + M2 =

0 (22) can be obtained. Eqs.(20) and (21) are given by the solution for Eq (22). 0

Eq. (11) in Theorem-1 is for all w considered and a certain q. Therefore, if a min ·max of e (q, w) is obtainable, then Eq. (11) in Theorem-1 can also be written as q

+ 2p cos () + 1.

Then, the following quadratic equation,

yr=cU'""2-+---::"CV=2

Mo = e(qo,wo) = min· maxe(q,w),

=M

and

As is obvious when q = 0, the left side of Eq. (11) becomes the absolute value of the complementary sensitivity function, i.e., p e(O, w) = -r======== Vp2 + 2pcosB + 1

. J(l + U)2 + V2 = IT(Jw)l·

+ 2p cos B + 1

B = B(w) (17)

= LJ(G(jw) = LG(jw),

and then p is calculated based on Eqs. (20) and (21). It should be noted that won a modified contour M is the same value only for horizontal axis B.

the control system is robust stable.

III

In other words, modified contours M are invariant only for a gain change in open loop characteristic

sector in which non linear characteristics are permitted. That is, it is an example that shows Aizerman's conjecture to be valid.

KG(jw). The following theorem is given under the abovementioned premise and notes.

On the other hand, GP 2 is a gain-phase curve for J{ 0.57 by which the peak value becomes M = 1.5 . It corresponds to a limit of the robust stability condition which can be applied to a timevarying nonlinear control system. Although the robust control system is usually designed based on this concept, the robust stability condition is more conservative . Here, the gain margin becomes g2 = 10.8 dB.

=

[ Theorem-2 ] If gain-phase curve B(w )-p(w) exists in the following area as determined by q = qo: -

e(qo,w) =e(qo,w,B(w),p(w»

~

Mo <

1 -=-, et

Vw

(23) the nonlinear control system as shown in Fig. 1 is robust stable. (Proof) Obviously, as for a certain gain-phase curve B(w) -pew) of open loop system KG(jw),

Vw

Figure 6 shows the calculation result of modified contour M as to q ~ 0 for

(q,w,B(w),p(w» = M = 2.0,

that is, et = 0.5 and gain-phase curves GP 1 and GP 2 •

(24)

is valid in general, because the right side of this inequality is a peak value for angular frequency w. Furthermore, Wo is a peak frequency. However , Eq. (24) is also valid for q = qo. Therefore,

[ Example-2 ] Consider the following controller and controlled system: 2(1

Vw

(25)

s(1

+ s)(1 - 0.5s) + 5s)(1 + 0.2s)'

Mo

[ Example-! ] Consider the following controller and controlled system: 1 J( = 1.2. (26)

e(q, wo) .

Mo

-201og 1o Mo

(27)

dB,

On the other hand, GP 2 is a gain-phase curve for J{ 0.67 by which the peak value becomes Mo = 2.1. It is a limit for a robust stability condition which can be applied to a time-varying nonlinear control system . The gain margin in this case is g2 = 14.6 dB .

=

In this example, q by which e(q ,wo) is minimized is qo = 2.0 and the gain-phase curve contacts with a modified contour by B = -180 degrees. Here, the gain margin is g1 = 4.44 dB and equals

+1=

+ 1 = 3.36

determined by the point where contour M becomes -180 degrees.

that is, a = 0.667 and gain-phase curve GP 1 for the controlled system.

Mo

(30)

In this example, q by which e(q,wo) is minimized is qo = 0.5. Obviously, the gain-phase curve contacts with the modified contour except on B = -180 degrees. The gain margin is g1 = 5.15dB, which is different from

is obtained. Figure 5 shows calculation results of modified contour M as to q ~ 0 for

-201og 1o Mo

= M = 2.1

that is , et = 0.476 and gain-phase curve GP 1 for the controlled system.

= mine(q ,wo) = e(qO,wo) = 1.5 q

= M = 1.5 ,

(29)

= mine(q ,wo) = e(qO,wo) = 2.1 q

[(q,w,B(w),p(w»

s(1+s)2'

(q,w,B(w),p(w»

= 2.0 .

is obtained. Figure 8 shows calculation results of modified contour M as to q ~ 0 for

NUMERICAL EXAMPLES

Mo

J(

Figure 7 shows calculation results for e(q ,wo). When nominal linear gain J( = 2.0,

is obtained. It can be shown that Eq . (23) in Theorem-2 is equivalent to Eqs. (16) and (17) . 0

Figure 4 shows calculation results for When nominal linear gain J( = 1.2,

(28)

4.44 dB,

Figure 9 shows calculation results of modified contour M as to q ~ 0 for

which is determined by the point where contour M is -180 degrees . It corresponds to the size of a

[(q,w,B(w),p(w»

112

= M = 2.5,

(31)

that is, GP 2 .

a=

0.4 and gain-phase curves GP l and

Packerd, A. and J . Doyle (1993) . The Complex Structured Singular Value. Automatica, pp . 71-109.

Figure 10 is an example of broken (polygonal) line nonlinear characteristic N(e). Figure 11 shows the time response for the control system. Because the stability region for a linear characteristic is o < K < 3.63, the response in Fig. 11 is a counter example to Aizerman's conjecture.

Sandberg, 1. W. (1964). A Frequency Domain Condition for the Stability of Feedback Systems Containing a Single Time Varying Nonlinearity. Bell Systems Tech . J., Vol. 43, pp . 1601-1608. Vidyasagar, M . (1985). Control System Synthesis: A Factorization Approach, MIT Press.

CONCLUSIONS Zames, G. (1966). On the Input-Output Stability of Time Varying Nonlinear Feedback Systems. IEEE Trans . AC-ll, Pt I, pp . 228238, Pt Il, pp . 465-475 .

In this paper, by applying the small gain theorem concerning L2 gain to a nonlinear subsystem with free parameter q, a stability criterion in the frequency domain of control systems with the timeinvariant nonlinearity (equivalent to Popov's criterion) was given . Then, a robust stability conditon on a gain-phase diagram of the open loop system which corresponds to the Nichols chart was suggested based on this concept. As for stability margin, it was discussed in regards to the gain margin of a nominal system. The evaluation of the robust performance in a gain-phase diagram for this kind of non linear control system may be used in the design of a robust control system.

APPENDIX

!{('lO.Wo) =

2.3

REFERENCES Desoer, C. A. and M . Vidyasagar (1975) . Feedback System: Input-Output Properties, Academic Press.

{( 'lOo wo) = 1.0

o

Francis, B. A. (1987). A Course in Hoc Control Theory, Springer-Verlag.

2

3

q 5

Fig. 4 e(q,wo) curves for Example-l

Grujic, L. T. (1981) . On Absolute Stability and the Aizerman Conjecture. A utomatica, pp . 335-349 . Harris, C. J . and J . M. E . Valenca (1983) . The Stability of Input-Output Dynamical Systems,

Academic Press. Holtzman, J. M . (1970). Nonlinear System Theory, Prentice-Hall . Okuyama, Y. (1988). Robust Stability of Feedback Control Systems Containing an Uncertain Nonlinearity. Trans . of the Institute of Systems, Control and Information Engineers,

pp. 9-16 ( in Japanese ) . Okuyama, Y and F . Takemori (1996). Robust Stability Evaluation of Sampled-Data Control Systems with a Sector Nonlinearity. Preprints of 13th IFAC World Congress, Vol. H, pp 41-46.

Fig. 5 Modified contour and gain-phase curves for Example-l (M = 1.5, 0::; w ::; 20)

113

(J deg

Fig . 9 Modified contour and gain-phase curves 2.5, 0 ~ w ~ 20) for Example-2 (M

Fig. 6 Modified contour and gain-phase curves for Example-1 (M = 2.0, 0 ~ w ~ 20)

=

Fig. 10 Broken line nonlinearity Fig. 7 ';(q ,wo) curves for Example-2

e

0.5

e(t)

Step input r = 0.4

t----~-\---t~~I,f)I------.!!....(} doe

-0.5

Fig. 11 Time response for Example-2

Fig. 8 Modified contour and gain-phase curves for Example-2 (M = 2.1, 0 ~ w ~ 20)

114