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Parametric absolute stability of multivariable lur'e systems: A popov-type condition and application of polygon interval arithmetic
Nonlinear
Analysis,
Theory,
Methods
PII:SO362-546X(97)00260-5
PARAMETRIC
& Applications. Vol. 30, No. 6. pp. 3713-3723, 1991 Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Elsevier Science Ltd Printed in Great Britain. All tights reserved 0362-546X/97 $17.00 + 0.00
STABILITY OF MULTIVARIABLE LUR’E SYSTEMS: A POPOV-TYPE CONDITION AND APPLICATION OF POLYGON INTERVAL ARITHMETIC
TERUYO
ABSOLUTE
WADAt,
MASAO
IKEDAS,
YUZO
OHTAS,
and DRAGOSLAV
D. SILJAK]]
1Dept.
Mechanical Systems Eng., Osaka Prefecture University, Sakai, Osaka 593, Japan SDept. Mechanical Eng., Osaka University, Suita, Osaka 565, Japan §Dept. Computer and Systems Eng., Kobe University, Nada, Kobe 657, Japan 11B & M Swig Professor, Santa Clara University, Santa Clara, CA 95053, U.S.A.
Key words and terval arithmetic
phrases:
(PIA),
Parametric absolute parameter-dependent
stability, Liapunov
Popov-type function,
condition, Lur’e systems, nonlinear systems.
polygon
in-
1. INTRODUCTION Parametric absolute stability has been introduced in [l-3] to deal simultaneously with feasibility and stability of equilibria in Lur’e systems with parametric uncertainties in the linear part and changes of the constant reference inputs. While in the case of single-input-single-output systems [l], the stability test was a Popov-like graphical construction, in multivariable systems [2] the test was based upon the recent Linear Matrix Inequalities (LMI) approach (4, 51. The objective of this paper is to pursue the Popov approach in the multivariable case, and develop a new frequency criterion for parametric absolute stability. The notion of parametric absolute stability is a natural outgrowth of the robust absolute stability of unforced Lur’e systems with uncertain parameters, which has been studied for quite some time [6, 71. By adding a constant reference input to a Lur’e system, the location of the equilibrium becomes uncertain with a possibility of the equilibrium disappearing altogether. In this case, the concept of parametric stability [8] b ecomes relevant. It has been introduced in the context of commodels of multispecies communities [lo], where petitive equilibrium systems [9] and Lotka-Volterra uncertainty of system parameters almost always induces the shifts of equilibrium states. In Lur’e systems with uncertain parameters and reference inputs, the study of parametric stability becomes particularly difficult because of the inherent uncertainty in the nonlinear elements; the shape of the nonlinear functions are not known other than that they satisfy sector inequalities. For this reason, computation of existence regions of equilibrium states and subsequent stability analysis require a more refined analysis. In particular, we shall use the recent notion of parameter-dependent Liapunov
functions
[ll,
121 to capture
more
faithfully
the
effects
of uncertain
parameters.
we shall provide a numerical example to illustrate the intricacies of parametric stability well as the testing of stability conditions using the algorithms [13] of Polygon Interval (PIA). 2. PARAMETRIC
ABSOLUTE
Finally,
analysis, as Arithmetic
STABILITY
The Lur’e system is a feedback control system which consists of a dynamical linear part and a static nonlinear part. Here, we consider a multi-input-multi-output Lur’e system S illustrated by Fig. 1, whose linear time-invariant part includes an uncertain parameter vector p. That is, S is given
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1. Multivariable
Lur’e
Analysts
System
S
by i = A(p)x
+ B(p)u,
2~ = v(e),
(2.1) (2.2)
Y = C(Pb
e=r--y,
where z E Rn, IL E R”, y E R”L are the state, input, output of the linear part, respectively, and e E R” is the control error. The reference input T E R m is a constant vector in a compact and simply connected region R, and its nominal value is 0. In (2.1), A(p), B(p), C(p) are matrices of appropriate dimensions, which depend continuously on the parameter vector p E R”. The parameter p belongs to a compact and simply connected set uncertainty P. We assume that for all p E P, .4(p) is a stable matrix. We also assume that the nonlinear function cp : Rm + R”’ is continuously differentiable and satisfies the conditions {Dq$e)}T {(p(e
+ 2;) -
do)
= f-4
= Dp(e), (p(e)}TK-‘(e){(p(e
Ve E Rm
(2.3) + C) - p(e)}
5 ?{(p(e
+ E) - p(e)},
Ve E E,
Vi? E Rm (2.4)
(2.5)
in a neighborhood E of e = 0, where ZQ(e) is the Jacobian matrix of cp(e), and K(e) is a positive definite matrix continuously depending on e E E. The assumption (2.3) is required in the stability analysis to utilize a Lur’e-Postnikov type Liapunov function [14]. The assumption (2.4) is a sector condition for the multivariable vector-valued function p at each point e in E. The total system is described as s : 2 = A(p)s
+ B(p)cp[r
- C(p)z].
(2.6)
When the reference input T equals the nominal value 0, the origin is an equilibrium state for any p E P. For T # 0, however, the equilibrium state can change due to uncertainty in p and changes of T, and stability properties of S depend on the constant reference input T as well as the parameter p, as illustrated in [l, 21. Thus, we treat the pair (r,p) as a parameter vector. With this augmented parameter, the concept of the parametric absolute stability has been defined in [2], which deals simultaneously with the existence of an equilibrium and absolute stability of the system under parametric uncertainty. (2.7) DEFINITION. The Lur’e system S of Fig. 1 is said to be parametrically absolutely stable if for any (r,p) E R x P and any nonlinearity satisfying (2.3)-(2.5), the following conditions hold: (i) There exists a unique equilibrium state z.F(r,p) of S and the corresponding control error ee(r,p) is included in E. (ii) cc:“(r,p) is globally asymptotically stable. Our objective is to formulate Popov-type conditions which are sufficient for parametric absolute stability of the multivariable Lur’e system S. We first derive a matrix inequality which ensures the existence of an equilibrium of S, and compute the region of the corresponding control error, which is a prerequisite for stability analysis. Then, we present a parametric matrix inequality in the frequency domain that guarantees the global asympt,otic stability of the equilibrium.
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ANALYSIS
We start with the obvious fact that if an equilibrium state exists and it is globally asymptotically stable, then the equilibrium state is unique. This means that we do not have to mention explicitly the uniqueness of an equilibrium, which is required in (i) of Definition (2.7), since we consider its stability in the next section. For any (r,p) E R x ‘P, equilibrium states of the system S are solutions of the equation
4~1~ + B(p)cpb- Cbh-4= 0, which is obtained
by setting k = 0 in (2.6). This equation x + A-‘(PMPM
and can be represented
(3.1)
is equivalent
- C(p)4
to the equation (3.2)
= 0,
by two equations x + A-l(p)B(p)cp(e) e = T - C(p)x.
= 0
(3-3) (3.4)
Thus, if the equation e - T + Go(p)cp(e) with respect to e has a solution x = xe(r, p), where
e = ee(r,p),
= 0
then from (3.3),
the equation
(3.5) (3.1) has a solution
GO(P) = -C(PM-‘(PMPJ. (3.6) This means that the discussion on the existence of an equilibrium state 9(~-,p) is reduced to that on the existence of t?(r,p). Furthermore, an estimate of the region where ee(r,p) exists is required for computation of an upper bound of the sector of the nonlinear function ‘p in the subsequent stability analysis. Therefore, we provide a condition for the existence of a solution e = ee(r,p) of (3.5) and the corresponding estimate under the assumption
$@)K-'(O)(p(C) which
is the sector condition
(3.8) LEMMA.
2 ETp(E),
of cp(e) at the origin
t/C E Rm,
and is obtained
(3.7)
by setting
e = 0 in (2.4).
If ;{WP)
+ G;(P))
+ K-‘(O)
holds for all p E ‘P, then for any (~,p) E R x ‘P, a so&ion
> 0
(3.9)
of (3.5) exists in the region
e E R" : l/e11 5 - lb-II
(3.10)
HO
where
PO(P)
(3.11)
= IlGo(p)
YO(P) = hnin [ ;{Go(p)
+ G;(p)}]
PO(P)
+ G;f(pH
= hnin
[+o(P)
+ (1 - ~o(P))K-‘UM
‘Or
““)
’
’
(3.12) + {I-
Eo(PW-~(O)]
>
(3.13)
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/I,,[-] and A,,[.] denote the minimum is a positive number satisfying
eigenwalue
and the maximum
+o(P) + G;f(dl + (1 - hop-’ the vector norm norm.
11 11 denotes the Euclidean
norm,
eigenvalue,
> 0,
and the matrix
respectively,
60(p)
(3.14)
norm is induced from the vector
The proof of Lemma (3.8) is given in Appendix. If ye(p) > 0, then (3.9) is automatically satisfied. Thus the condition for the existence of ee(r,p) and the region Ee(r,p) are independent of the matrix K(0). Only when “/o(p) 5 0, the condition (3.9) is required to ensure the existence of ee(r,p). In the case when Lur’e system S is a single-input-single-output system, the condition (3.9) becomes the condition of (21. In this case, using the equality ys(p) = Go(p), the region where all of the equilibria ee(r,p) for all r E R exist is obtained from (3.10) as {e E R : le/ < P}
E%(P) = IJ Ee(r,p) rE‘R
(e E R : lel <
=
for Go(p) > 0
&I
for GO(P) IO
(3.15)
where r = max{]r] : r E R}. When Go(p) > 0, the region E%(p) is equivalent to the region of [2]. On the other hand, when Go(p) < 0, the region obtained from (3.15) by replacing 60(p) with its maximal value is the same as the result in [2]. From the above discussion, an equilibrium state of the Lure system S exists and its region of existence is obtained from (3.9). Furthermore, we note that we can calculate the region where the equilibrium state zf(r,p) exists by using (3.3). However, in the stability analysis, we do not need the region of P(r,p) and only the region of ee(*r,p) is required. All we have to know about zF(r,p) is its existence, which is ensured by condition (3.9). 4.
STABILITY
ANALYSIS
We first show the multivariable Popov-type condition which is sufficient for global asymptotic stability of an existing equilibrium. Then, combining this condition with the one given in Lemma (3.8), we present a condition for parametric absolute stability. Let (r,p) be an arbitrary fixed vector in ‘R x P. Suppose that the assumption of Lemma (3.8) holds and thus, there exists an equilibrium state P(T,~) of the system S. Then, S is transformed to the equivalent error system s : i = A(p)2 + B(p)@[-C(p)%], (4.1) where 2 = x - z?(r,p),
and
de) = cplee(r, p) + 4 - vi@(r,PI e = -C(p)%. The nonlinear
function
(4.2)
(p satisfies {D+(t?)}‘r
which follows from (2.3). Furthermore,
= D@(E),
if ee(r,p)
cp(~)TK-‘[ee(,,p)]~(~)
VE E Rm
(4.3)
E E, then (p satisfies the condition 5 ET@(E), VE E Rm,
(4.4)
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which is equivalent to the sector condition (2.4) at e = ee(r,p). We consider stability of the equilibrium state z”(r,p) of S by analyzing stability of the equilibrium state Z = 0 of the equivalent error system S. Since S is also a Lur’e system, we use a Lur’e-Postnikov type Liapunov function V(2)
= eI(r,p)5
+ v(r,p)
~1[-C(p)5]7~[-BC(p)f]dB.
(4.5)
where H(r,p) is a positive definite matrix, and v(r,p) is a real number. In (4.5), H(r,p) and v(r,p) depend on the parameter vector (~,p). Thus, V(-)z is a parameter-dependent Liapunov function. The Popov condition for multivariable Lur’e systems ensures that V is a Liapunov function, that is, it is positive definite and its total derivative along the behavior of S is negative definite [14]. Under the assumption (4.3) and (4.4), the Popov condition is as follows: If there exists a real number v = Y(T,~) such that the matrix inequality i[(l
+jz~)G(jw,p)
+ (1 - jz~)G~(---j~,p)]
is satisfied, then the zero solution of S is globally function matrix of the linear part G(s,p)
+ K-‘[ee(r,p)]
asymptotically
= C(p)lsI
> 0, VW E R+
stable, where G(s,p)
(4.6)
is the transfer
- A(P)]-‘B(P),
(4.7)
and R+ denotes R+ = [0, +oo) U {+~a}. If we can ensure the existence of v = v(T,I)) satisfying (4.6), then we can conclude the global asymptotic stability of the equilibrium state x”(r,p) of the Lur’e system S. To check the condition, we need to compute e”(r,p) for each (rip) E R x P since (4.6) includes ee(T,p). However, the computation of ee(r,p) is impossible because the nonlinearity ‘p is any function satisfying (2.3))(2.5) and it is not known explicitly. To overcome such a problem, we utilize the region F(r,p) of ee(r,p), which is given in Lemma (3.8). Hence, we employ the following upper bound matrix K,(r,p) for K[ee(r,p)] in (4.6) when Ee(r,p) C E: K(T,P)
2 K(e),
(4.8)
Ye E Ee(r,p).
Prom Lemma (3.8) and the above discussion, parametric absolute stability of S.
we obtain
(4.9)
p E P, the matrix
THEOREM.
Suppose that for any parameter ;{G(O,p)
holds, and for any reference input
T
+ GT(O,p)}
assume that for any (r,p) f[(l
+j~w)G(jw,p)
condition
sufficient
for
inequality
> 0
(3.9)
E R, we have F(r,p)
In addition,
+ K-‘(O)
a Popov-type
(4.10)
c E.
E ‘R x P, there exists a real number
+ (1 - jvw)GT(-jw,p)]
Then, the LuT’e system S is parametrically
absolutely
+ K,-‘(7,~) stable.
> 0,
v = v(~,p) VW E i?,.
such that (4.11)
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In Theorem (4.9), we use the notation G(O,p) instead of Go(p) which appeared in (3.9) of Lemma (3.8) since G(O,p) = Go(p). Thus, F(r,p) in (4.10) is the region defined in Lemma (3.8). From (4.8), the smaller the region F(r,p) is, the less conservative the condition (4.11) is. In other words, the larger 70(p) and pe(p) are, the smaller region Ee(r, p) can be. In the case when S is a single-input-single-output system, K,(r,p) in (4.8) can be chosen as the maximum value of K(e) in the region F(r,p). In this case, let us fix p arbitrarily in P. Then, the existence of v = v(p) satisfying (4.11) for all T E R is equivalent to the existence of Y = v(p) satisfying the inequality which is obtained by replacing &(r,p) with KR(P) = max{K(e) : e E E&(p)} in (4.11), where E%(p) is the region given in (3.15). The condition obtained from the above discussion for the parametric absolute stability of a single-input-single-output Lur’e system is the same as the result in [2], when G(O,p) > 0. On the other hand, when G(O,p) 5 0, by replacing 60(p) with the maximal value of be(p) in (3.15), the condition is equivalent to the result of [2]. Therefore the obtained condition is an extension of the result of [2] to multivariable Lur’e systems. 5.
A NUMERICAL
EXAMPLE
VIA
PIA
show a numerical example in which we apply Theorem (4.9) to check parametric absolute of Lur’e systems. In the system S, let the coefficient matrices of the linear part be given as
We now
stability
-P1 0 0
A(P) =
p*(p:-
Let the region P of the parameter P =
{P E R4
2)
-p*iilip3,]
1
IG)=
[ k
a]
vector p = (PI p2 p3 p41T be
: ~1
E
10.5,
and the region R of the reference input
11,
31,
~2 E [‘X
1.51,
~3 E [I,
p4
E [2,
5))
(5.2)
r be
R = {r E R2 : llrll < 1).
(5.3)
We assume that the nonlinear function ‘p satisfies the sector condition R2 : ]]e]] < 6) with the positive definite matrix 0.9 + 0.941lel( -0.4 - 0.41]]e]] [
\ I=
K(e)
Since the characteristic
4.66 -2.04 polynomial d(s)
and pl, pz, p3 are positive part is calculated as
-2.04 1
-0.4 - 0.41]]e]] 0.2 + 0.2]]e]] I
=
(s +pl)(s
A(p)
+ pZ)(s
is + P2P3
10s’
4
s+Pl
(5.5)
2)
function
matrix
of the linear
5s+p1+4pz (8 + Pl)(S
=
+
The transfer
A(p) is stable.
2
--
(5.4)
for 4 < ]]e]] < 6.
s+Pl
G(s,P)
E = {e E
for IIf4I.//i2 =
1
of the matrix
real numbers,
(2.4) in the region
-I- {2(p1
-+- p2p4(p2 3- PlP2P4(P2
+ 4p2 -
2)}S - 2)
+ P2)
+ 5p2p3 + 2(p1
(s+pl)(s+P2)(s+P2P3+2)
+ lo) + 4p2)(p2p3
+ 2)
(5.6)
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First, we check the existence of an equilibrium
Analysts
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state. Since we have
PIP2 2(Pl
+ 4PZ)(PZP3
+
PIP2
and use the intervals
(5.2) of pi (i = 1,2,3,4), -Ye
(pZP3
we obtain z 4.12,
2) + p1p2p*(p2 +
-
(5.7)
2)
‘4
I
the lower bound
of 70(p)
VP E p
(5.8)
from (3.12). This means that we have {G(O,p) -t GT(0,p)}/2 > 0. From K-l(O) > 0, this implies that (3.9) holds. Thus, for any (r,p) E R x P, there exists an equilibrium state of the Lur’e system S. Furthermore, computing the upper bound of (IG(O,p))( f rom (5.2), we have IlG(O,p)ll 5 20.2. Hence, from (3.10) and (5.3), we obtain the region of ee(r,p) for any (r,p) E R x P, F(r,p) Then, we investigate the stability above discussion for any (r,p) E ‘R and F(r,p) c Ek from (5.9), (4.10) (r,p) E R x P, it is difficult to check let us introduce the matrix ER which
c
{e E R2 : IJel( 5 4.91).
(5.9)
of the equilibrium state whose existence is ensured from the P. Let rR = {e E R2 : J(e]] 5 4.91). Since i$R c E is satisfied. Actually, for infinitely many parameter values the existence of v(r,p) satisfying the inequality (4.11). Thus, is independent of (r,p) and satisfies
x
ITa 2 K(e),
Ve E ER.
(5.10)
Next, we show the existence of V, common to all (r,p) E R x P, which satisfies the matrix inequality obtained by replacing K,(r,p) with -n K in (4.11). That is, we examine the existence of a real number Y satisfying Q(w,p,u)
= ;[(l
+jvw)G(jw,p)
+ (1 -jvw)GT(-jw,p)]
+R,’
> 0,
VW E R+, vp E P. The reduced condition is more conservative by the PIA. From (5.4), the matrix
than that of Theorem
(5.11)
(4.9), but it can be easily checked
satisfies (5.10). We show the existence of u satisfying (5.11). In the case when S is a singleinput-single-output Lur’e system, we can check the existence of such v by using the Popov locus [2]. However, since we deal with the multi-input-multi-output Lur’e system and (5.11) is a matrix inequality, we cannot show the existence of such v graphically. Thus, we substitute an appropriate value to the Y in (5.11) and examine the positive definiteness of Q(w,p,v). From (5.8), (5.11) is satisfied at w = 0. In addition, we can show that (5.11) is also satisfied at w --) -too. Hence, the continuity of Q(w,p, V) with respect to w ensures the matrix inequality (5.11) if det Q(w,p, v) # 0 holds for any nonnegative w and any parameter p E P. Fig. 2 shows the region which covers the value set of det Q(w,p, V) for all p E P at each w > 0 by applying the PIA [13], where v = 0.5 is adopted. From Fig. 2, we have detQ(w,p, V) # 0 and (5.11) is satisfied. Therefore the Lur’e system S is parametrically absolutely stable.
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w
Fig.
2. Value
6.
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&(w,p,
0.5)]
CONCLUSIONS
A sufficient condition for parametric absolute stability has been derived as a Popov-type condition. The condition is based on the existence of a real number which satisfies a parametric matrix inequality in the frequency domain. If the Lur’e system is not a single-input-single-output system, then there is no straightforward procedure to check if the condition is satisfied. In this paper, we show by a numerical example how the condition can be checked by setting an appropriate value to the real number and examining the satisfaction of the parametric matrix inequality using the PIA. ACKNOWLEDGEMENTS
The research reported the grant ECS-9526142.
herein
was supported
in part by the National
Science Foundation
under
REFERENCES 1.
2.
3.
4. 5.
WADA, T., SILJAK, D. D., OHTA, Y., and IKEDA, M., Parametric stability of control systems, in Recent Advances m Mathematical Theory of Systems, Control, Networks and Signal Processing II; (Edited by H. Kimura and S. Kodama), pp. 377-382. Mita Press (1992). WADA, T., IKEDA, M., OHTA, Y., and SILJAK, D. D., Parametric absolute stability of Lur’e systems, Proc. 34th IEEE Conference on Decisron and Control, pp. 1449-1454. New Orleans (1995). WADA, T., IKEDA, M., OHTA, Y., and SILJAK, D. D., Parametric absolute stability of multivariable Lur’e Proc. 13th IFAC World Congress Vol. E, systems: An LMI condition and application to polytopic systems, pp. 19-24. San Francisco, CA (1996). BOYD, S., GHAOUI, L. E., FERON, E., and BALAKRISHNAN, V., Zznear Matriz Inequalities in System and Control Theory, SIAM, Philadelphia (1994). GAHINET, P. A., NEMIROVSKII, A., LAUB, A. J. , and CHILALI, M., ZMI Control Toolbox, The MathWorks, Natick, MA (1995).
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SILJAK, D. D., Polytopes of nonnegative polynomials, in Recent Advances in Robust Control (Edited by P. DORAT0 and R. K. YEDAVALLI), pp. 71-77. IEEE Press, New York (1990). DAHLEH, M., TESI, A., and VICINO, A., An overview of extremal properties for robust control of interval plants, Automatica, 29 707-721 (1993). IKEDA, M., OHTA, Y., and SILJAK, D. D., Parametric stability, in New fiends in Systems Theory (Edited by G. CONTE, A. M. PERDON, and B. WYMAN), pp. l-20. Birkhauser, Boston (1991). SILJAK, D. D., Large-Scale Dynamic Systems: Stabilzty and Structure, North-Holland, New York (1978). IKEDA, M., and SILJAK, D. D., Lotka-Volterra equations: Decomposition, stability and structure, J. Math. Biol., 9, 65-83 (1980). BARMISH, B. R., New Tools for Robustness of Linear Systems, Macmillan, New York (1994). GAHINET, P., APKARIAN, P., and CHILALI, M., Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Automat. Conk., 41 436-442 (1996). OHTA, Y., GONG, L., and HANEDA, H., Polygon interval arithmetic and design of robust control systems, Proc. 29th IEEE Conference on Decision and Control, pp. 106551067. Honolulu, HI (1990). WADA, T., and IKEDA, M., Extended Popov criteria for multivariable Lur’e systems, Proc. 32nd IEEE Conference err Decision and Control, pp. 2621. San Antonio, TX (1993).
7. 8. 9. 10. 11. 12. 13. 14. 15.
ORTEGA,
J. M.,
pp.
Academic
70-71.
and
TO prove Lemma equation
RHEINBOLDT, Press,
New
C., Iterative
W. York
of Nonlinear
Solution
Equations
(3.8), we describe
APPENDIX a sufficient condition f(Z,T)
for the existence
LEMMA.
If,f,for
(z*,T*)
some
Variables,
of a solution
= 0
differentiable with respect and compact in Rm.
E R’ x FL, f(z*, T*) = 0
is satisfied
and there exists a number Ilf(Z,T)
then,
for
any
T
- f(z*,T)ll
> CLllZ - z*ll,
this lemma from the following
lemma by applying
solution
det DDzf(z*,
‘d’t E R’L, VT E ‘?
T*)
#
of (A.3).
OF
LEMMA
(3.8): Let us arbitrarily f (e,r)
Then,
(3.5) is represented equation,
I n addition,
z~(T)
[15].
suppose that for some
64.6) VT
ER
(A.7)
with respect to z, and 82 of (A.l) in 2.
f(r,r)
is the boundary
of 2.
fix p in ‘P, and define
(A.8)
= e - T + Go(pMe).
by f (e,
For this algebraic
the mean value theorem
0
f (Z,T) # 0, t/Z E 82, holds, where Vzf is the Jacobian matrix of Then, for any T E R, there exists a sohtion
(A-4)
of (A.l).
Z"(T)
(A.5) LEMMA[~]. Suppose that Z* is a uniqae bounded open neighborhood 2 of z*,
PROOF
(A.31
p > 0 such that
E R, there exists a solution
We can obtain
of the (A-1)
for any T E ‘R, where f(z,r) is a function from R” x R to R”, continuously to z, and continuous with respect to T . The region R is simply connected (A.2)
in Several
(1970).
T)
= 0.
we show that the assumptions
(A.91 of Lemma
(A.2) are satisfied
for z = e.
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the equality m,
(A.10)
0) = 0
implies that (A.3) holds for e’ = 0, T* = 0. Thus, in the following, we show that the existence of p > 0 satisfying
(A.4).
Let (All) (A.12)
WO(P) = GO(P) + (1 - ~o(P))K-‘(0) d = e - (1 - Go(p)}K-‘(O)p(e), then, we have f(e,r) Since (3.14) implies
= 2 - T + Wo(p)cp(e).
(A.13)
that $~O(P)
there exists We-l(p).
+ WJVPH
(A.14)
> 0,
Thus, it follows from (3.7) and (3.13) that
IlWFT(~)41.
Ilf(e,r)
- f(O,~.)ll
2: ~TK1(PHf(e,r) - f(O,r)l > TW Avo-‘(p)i!
(A.15) This imnlies 1
that
Ilf(e, r) - f(%r)ll > - -IIEI(. IIwo~p~l, In the case when represented by
ye(p)
> 0, we can chose
(A.16)
the value So(p) = 1 from (3.14).
Then,
(A.16)
Ilf(e,r) - f(o,T)ll 2 *llt?ll I,Go~p~,,
(A.17)
via the equalities pe(p) = ye(p), We(p) = Go(p) and d = e. In the case when ye($) < 0, (3.14) implies that 60(p) < 1. Thus, from the sector condition we have
Ml = Ile - (1 = llK-1/2(0) 2 ,,Kl&O),, 2 ,,Kl~2(o)l,
~0b)lK-‘(OMe)ll [K1/2(0)e - (1 - Se(p)}K-1/2(0)y(e)] llK1/2(0)e
is
(3.7),
]]
- (1 - ~0(p)}K-1/2(0)cp(e)II
[~0b)II~1/2(Ok4
+ (1 - ~0(p)1~ll~‘~*W4I
- Il~-1~2(%4e)ll~]
> ~0b)ll~‘~2Wll -
11~1’2(0)11 (A.18)
Second World
From this inequality
and (A.16),
Congress
of Nonlinear
Analysts
3723
we obtain
“f(eyr) - f(“,T)” z hn[K(0)I Amu[K(0)]
~O(P)PO(P)
Ilell,
IIWo(p)ll (A.19)
(3.11) of PO(P) and (A.17),
Hence, from the definition
(A.19),
the inequality (A.20)
Ilf(e, T) - f@,r)ll 2 ~0b)llell
holds. This implies that (A.4) is satisfied for p = PO(P). Therefore, the assumptions of Lemma (A.2) are satisfied, and we conclude that for any (~,p) E R x ‘P, there exists a solution ee(r,p) of (3.5). Now, we compute the region where ee(r,p) exists. In (A.20), let e = ee(r,p), then we obtain the inequality (A.21) IId Z ~o(PW(~,p)ll. Therefore, (A.22)
the equilibrium
ee(r,p)
exists in P(r,p)
0
given by (3.10).
REMARK. In the above proof, we used the inequality (A.23)
llK’~2(0)41 Z IlK-‘i2(Oh4e)llNow, we show that this inequality for any e E Rm, the inequality
can be derived from (3.7). S’mce K(0)
is a positive
definite
matrix,
0 5 [e - K-‘(0)cp(e)lTK(O)[e - K-‘(0)9(e)] = eTK(0)e - 2eTq(e) + (pT(e)K-‘(O)(p(e) (A.24) holds. Thus, from (3.7), we have 0 5 eT+4e) - vT(eW1(Me) 5 eTK(0)e - eTp(e). This implies
(A.25)
that 0 5 eTp(e)
From the inequalities
(A.26)
and (3.7), we obtain ,pT(e)Kwl(0)cp(e)
which is equivalent
to (A.23).
5 eTK(0)e.
(A.26)
the inequality 5 eTK(0)e
(A.27)
Analysis,
Theory,
Methods
PII:SO362-546X(97)00260-5
PARAMETRIC
& Applications. Vol. 30, No. 6. pp. 3713-3723, 1991 Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Elsevier Science Ltd Printed in Great Britain. All tights reserved 0362-546X/97 $17.00 + 0.00
STABILITY OF MULTIVARIABLE LUR’E SYSTEMS: A POPOV-TYPE CONDITION AND APPLICATION OF POLYGON INTERVAL ARITHMETIC
TERUYO
ABSOLUTE
WADAt,
MASAO
IKEDAS,
YUZO
OHTAS,
and DRAGOSLAV
D. SILJAK]]
1Dept.
Mechanical Systems Eng., Osaka Prefecture University, Sakai, Osaka 593, Japan SDept. Mechanical Eng., Osaka University, Suita, Osaka 565, Japan §Dept. Computer and Systems Eng., Kobe University, Nada, Kobe 657, Japan 11B & M Swig Professor, Santa Clara University, Santa Clara, CA 95053, U.S.A.
Key words and terval arithmetic
phrases:
(PIA),
Parametric absolute parameter-dependent
stability, Liapunov
Popov-type function,
condition, Lur’e systems, nonlinear systems.
polygon
in-
1. INTRODUCTION Parametric absolute stability has been introduced in [l-3] to deal simultaneously with feasibility and stability of equilibria in Lur’e systems with parametric uncertainties in the linear part and changes of the constant reference inputs. While in the case of single-input-single-output systems [l], the stability test was a Popov-like graphical construction, in multivariable systems [2] the test was based upon the recent Linear Matrix Inequalities (LMI) approach (4, 51. The objective of this paper is to pursue the Popov approach in the multivariable case, and develop a new frequency criterion for parametric absolute stability. The notion of parametric absolute stability is a natural outgrowth of the robust absolute stability of unforced Lur’e systems with uncertain parameters, which has been studied for quite some time [6, 71. By adding a constant reference input to a Lur’e system, the location of the equilibrium becomes uncertain with a possibility of the equilibrium disappearing altogether. In this case, the concept of parametric stability [8] b ecomes relevant. It has been introduced in the context of commodels of multispecies communities [lo], where petitive equilibrium systems [9] and Lotka-Volterra uncertainty of system parameters almost always induces the shifts of equilibrium states. In Lur’e systems with uncertain parameters and reference inputs, the study of parametric stability becomes particularly difficult because of the inherent uncertainty in the nonlinear elements; the shape of the nonlinear functions are not known other than that they satisfy sector inequalities. For this reason, computation of existence regions of equilibrium states and subsequent stability analysis require a more refined analysis. In particular, we shall use the recent notion of parameter-dependent Liapunov
functions
[ll,
121 to capture
more
faithfully
the
effects
of uncertain
parameters.
we shall provide a numerical example to illustrate the intricacies of parametric stability well as the testing of stability conditions using the algorithms [13] of Polygon Interval (PIA). 2. PARAMETRIC
ABSOLUTE
Finally,
analysis, as Arithmetic
STABILITY
The Lur’e system is a feedback control system which consists of a dynamical linear part and a static nonlinear part. Here, we consider a multi-input-multi-output Lur’e system S illustrated by Fig. 1, whose linear time-invariant part includes an uncertain parameter vector p. That is, S is given
3713
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Congress
of Nonlinear
1. Multivariable
Lur’e
Analysts
System
S
by i = A(p)x
+ B(p)u,
2~ = v(e),
(2.1) (2.2)
Y = C(Pb
e=r--y,
where z E Rn, IL E R”, y E R”L are the state, input, output of the linear part, respectively, and e E R” is the control error. The reference input T E R m is a constant vector in a compact and simply connected region R, and its nominal value is 0. In (2.1), A(p), B(p), C(p) are matrices of appropriate dimensions, which depend continuously on the parameter vector p E R”. The parameter p belongs to a compact and simply connected set uncertainty P. We assume that for all p E P, .4(p) is a stable matrix. We also assume that the nonlinear function cp : Rm + R”’ is continuously differentiable and satisfies the conditions {Dq$e)}T {(p(e
+ 2;) -
do)
= f-4
= Dp(e), (p(e)}TK-‘(e){(p(e
Ve E Rm
(2.3) + C) - p(e)}
5 ?{(p(e
+ E) - p(e)},
Ve E E,
Vi? E Rm (2.4)
(2.5)
in a neighborhood E of e = 0, where ZQ(e) is the Jacobian matrix of cp(e), and K(e) is a positive definite matrix continuously depending on e E E. The assumption (2.3) is required in the stability analysis to utilize a Lur’e-Postnikov type Liapunov function [14]. The assumption (2.4) is a sector condition for the multivariable vector-valued function p at each point e in E. The total system is described as s : 2 = A(p)s
+ B(p)cp[r
- C(p)z].
(2.6)
When the reference input T equals the nominal value 0, the origin is an equilibrium state for any p E P. For T # 0, however, the equilibrium state can change due to uncertainty in p and changes of T, and stability properties of S depend on the constant reference input T as well as the parameter p, as illustrated in [l, 21. Thus, we treat the pair (r,p) as a parameter vector. With this augmented parameter, the concept of the parametric absolute stability has been defined in [2], which deals simultaneously with the existence of an equilibrium and absolute stability of the system under parametric uncertainty. (2.7) DEFINITION. The Lur’e system S of Fig. 1 is said to be parametrically absolutely stable if for any (r,p) E R x P and any nonlinearity satisfying (2.3)-(2.5), the following conditions hold: (i) There exists a unique equilibrium state z.F(r,p) of S and the corresponding control error ee(r,p) is included in E. (ii) cc:“(r,p) is globally asymptotically stable. Our objective is to formulate Popov-type conditions which are sufficient for parametric absolute stability of the multivariable Lur’e system S. We first derive a matrix inequality which ensures the existence of an equilibrium of S, and compute the region of the corresponding control error, which is a prerequisite for stability analysis. Then, we present a parametric matrix inequality in the frequency domain that guarantees the global asympt,otic stability of the equilibrium.
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ANALYSIS
We start with the obvious fact that if an equilibrium state exists and it is globally asymptotically stable, then the equilibrium state is unique. This means that we do not have to mention explicitly the uniqueness of an equilibrium, which is required in (i) of Definition (2.7), since we consider its stability in the next section. For any (r,p) E R x ‘P, equilibrium states of the system S are solutions of the equation
4~1~ + B(p)cpb- Cbh-4= 0, which is obtained
by setting k = 0 in (2.6). This equation x + A-‘(PMPM
and can be represented
(3.1)
is equivalent
- C(p)4
to the equation (3.2)
= 0,
by two equations x + A-l(p)B(p)cp(e) e = T - C(p)x.
= 0
(3-3) (3.4)
Thus, if the equation e - T + Go(p)cp(e) with respect to e has a solution x = xe(r, p), where
e = ee(r,p),
= 0
then from (3.3),
the equation
(3.5) (3.1) has a solution
GO(P) = -C(PM-‘(PMPJ. (3.6) This means that the discussion on the existence of an equilibrium state 9(~-,p) is reduced to that on the existence of t?(r,p). Furthermore, an estimate of the region where ee(r,p) exists is required for computation of an upper bound of the sector of the nonlinear function ‘p in the subsequent stability analysis. Therefore, we provide a condition for the existence of a solution e = ee(r,p) of (3.5) and the corresponding estimate under the assumption
$@)K-'(O)(p(C) which
is the sector condition
(3.8) LEMMA.
2 ETp(E),
of cp(e) at the origin
t/C E Rm,
and is obtained
(3.7)
by setting
e = 0 in (2.4).
If ;{WP)
+ G;(P))
+ K-‘(O)
holds for all p E ‘P, then for any (~,p) E R x ‘P, a so&ion
> 0
(3.9)
of (3.5) exists in the region
e E R" : l/e11 5 - lb-II
(3.10)
HO
where
PO(P)
(3.11)
= IlGo(p)
YO(P) = hnin [ ;{Go(p)
+ G;(p)}]
PO(P)
+ G;f(pH
= hnin
[+o(P)
+ (1 - ~o(P))K-‘UM
‘Or
““)
’
’
(3.12) + {I-
Eo(PW-~(O)]
>
(3.13)
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/I,,[-] and A,,[.] denote the minimum is a positive number satisfying
eigenwalue
and the maximum
+o(P) + G;f(dl + (1 - hop-’ the vector norm norm.
11 11 denotes the Euclidean
norm,
eigenvalue,
> 0,
and the matrix
respectively,
60(p)
(3.14)
norm is induced from the vector
The proof of Lemma (3.8) is given in Appendix. If ye(p) > 0, then (3.9) is automatically satisfied. Thus the condition for the existence of ee(r,p) and the region Ee(r,p) are independent of the matrix K(0). Only when “/o(p) 5 0, the condition (3.9) is required to ensure the existence of ee(r,p). In the case when Lur’e system S is a single-input-single-output system, the condition (3.9) becomes the condition of (21. In this case, using the equality ys(p) = Go(p), the region where all of the equilibria ee(r,p) for all r E R exist is obtained from (3.10) as {e E R : le/ < P}
E%(P) = IJ Ee(r,p) rE‘R
(e E R : lel <
=
for Go(p) > 0
&I
for GO(P) IO
(3.15)
where r = max{]r] : r E R}. When Go(p) > 0, the region E%(p) is equivalent to the region of [2]. On the other hand, when Go(p) < 0, the region obtained from (3.15) by replacing 60(p) with its maximal value is the same as the result in [2]. From the above discussion, an equilibrium state of the Lure system S exists and its region of existence is obtained from (3.9). Furthermore, we note that we can calculate the region where the equilibrium state zf(r,p) exists by using (3.3). However, in the stability analysis, we do not need the region of P(r,p) and only the region of ee(*r,p) is required. All we have to know about zF(r,p) is its existence, which is ensured by condition (3.9). 4.
STABILITY
ANALYSIS
We first show the multivariable Popov-type condition which is sufficient for global asymptotic stability of an existing equilibrium. Then, combining this condition with the one given in Lemma (3.8), we present a condition for parametric absolute stability. Let (r,p) be an arbitrary fixed vector in ‘R x P. Suppose that the assumption of Lemma (3.8) holds and thus, there exists an equilibrium state P(T,~) of the system S. Then, S is transformed to the equivalent error system s : i = A(p)2 + B(p)@[-C(p)%], (4.1) where 2 = x - z?(r,p),
and
de) = cplee(r, p) + 4 - vi@(r,PI e = -C(p)%. The nonlinear
function
(4.2)
(p satisfies {D+(t?)}‘r
which follows from (2.3). Furthermore,
= D@(E),
if ee(r,p)
cp(~)TK-‘[ee(,,p)]~(~)
VE E Rm
(4.3)
E E, then (p satisfies the condition 5 ET@(E), VE E Rm,
(4.4)
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which is equivalent to the sector condition (2.4) at e = ee(r,p). We consider stability of the equilibrium state z”(r,p) of S by analyzing stability of the equilibrium state Z = 0 of the equivalent error system S. Since S is also a Lur’e system, we use a Lur’e-Postnikov type Liapunov function V(2)
= eI(r,p)5
+ v(r,p)
~1[-C(p)5]7~[-BC(p)f]dB.
(4.5)
where H(r,p) is a positive definite matrix, and v(r,p) is a real number. In (4.5), H(r,p) and v(r,p) depend on the parameter vector (~,p). Thus, V(-)z is a parameter-dependent Liapunov function. The Popov condition for multivariable Lur’e systems ensures that V is a Liapunov function, that is, it is positive definite and its total derivative along the behavior of S is negative definite [14]. Under the assumption (4.3) and (4.4), the Popov condition is as follows: If there exists a real number v = Y(T,~) such that the matrix inequality i[(l
+jz~)G(jw,p)
+ (1 - jz~)G~(---j~,p)]
is satisfied, then the zero solution of S is globally function matrix of the linear part G(s,p)
+ K-‘[ee(r,p)]
asymptotically
= C(p)lsI
> 0, VW E R+
stable, where G(s,p)
(4.6)
is the transfer
- A(P)]-‘B(P),
(4.7)
and R+ denotes R+ = [0, +oo) U {+~a}. If we can ensure the existence of v = v(T,I)) satisfying (4.6), then we can conclude the global asymptotic stability of the equilibrium state x”(r,p) of the Lur’e system S. To check the condition, we need to compute e”(r,p) for each (rip) E R x P since (4.6) includes ee(T,p). However, the computation of ee(r,p) is impossible because the nonlinearity ‘p is any function satisfying (2.3))(2.5) and it is not known explicitly. To overcome such a problem, we utilize the region F(r,p) of ee(r,p), which is given in Lemma (3.8). Hence, we employ the following upper bound matrix K,(r,p) for K[ee(r,p)] in (4.6) when Ee(r,p) C E: K(T,P)
2 K(e),
(4.8)
Ye E Ee(r,p).
Prom Lemma (3.8) and the above discussion, parametric absolute stability of S.
we obtain
(4.9)
p E P, the matrix
THEOREM.
Suppose that for any parameter ;{G(O,p)
holds, and for any reference input
T
+ GT(O,p)}
assume that for any (r,p) f[(l
+j~w)G(jw,p)
condition
sufficient
for
inequality
> 0
(3.9)
E R, we have F(r,p)
In addition,
+ K-‘(O)
a Popov-type
(4.10)
c E.
E ‘R x P, there exists a real number
+ (1 - jvw)GT(-jw,p)]
Then, the LuT’e system S is parametrically
absolutely
+ K,-‘(7,~) stable.
> 0,
v = v(~,p) VW E i?,.
such that (4.11)
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In Theorem (4.9), we use the notation G(O,p) instead of Go(p) which appeared in (3.9) of Lemma (3.8) since G(O,p) = Go(p). Thus, F(r,p) in (4.10) is the region defined in Lemma (3.8). From (4.8), the smaller the region F(r,p) is, the less conservative the condition (4.11) is. In other words, the larger 70(p) and pe(p) are, the smaller region Ee(r, p) can be. In the case when S is a single-input-single-output system, K,(r,p) in (4.8) can be chosen as the maximum value of K(e) in the region F(r,p). In this case, let us fix p arbitrarily in P. Then, the existence of v = v(p) satisfying (4.11) for all T E R is equivalent to the existence of Y = v(p) satisfying the inequality which is obtained by replacing &(r,p) with KR(P) = max{K(e) : e E E&(p)} in (4.11), where E%(p) is the region given in (3.15). The condition obtained from the above discussion for the parametric absolute stability of a single-input-single-output Lur’e system is the same as the result in [2], when G(O,p) > 0. On the other hand, when G(O,p) 5 0, by replacing 60(p) with the maximal value of be(p) in (3.15), the condition is equivalent to the result of [2]. Therefore the obtained condition is an extension of the result of [2] to multivariable Lur’e systems. 5.
A NUMERICAL
EXAMPLE
VIA
PIA
show a numerical example in which we apply Theorem (4.9) to check parametric absolute of Lur’e systems. In the system S, let the coefficient matrices of the linear part be given as
We now
stability
-P1 0 0
A(P) =
p*(p:-
Let the region P of the parameter P =
{P E R4
2)
-p*iilip3,]
1
IG)=
[ k
a]
vector p = (PI p2 p3 p41T be
: ~1
E
10.5,
and the region R of the reference input
11,
31,
~2 E [‘X
1.51,
~3 E [I,
p4
E [2,
5))
(5.2)
r be
R = {r E R2 : llrll < 1).
(5.3)
We assume that the nonlinear function ‘p satisfies the sector condition R2 : ]]e]] < 6) with the positive definite matrix 0.9 + 0.941lel( -0.4 - 0.41]]e]] [
\ I=
K(e)
Since the characteristic
4.66 -2.04 polynomial d(s)
and pl, pz, p3 are positive part is calculated as
-2.04 1
-0.4 - 0.41]]e]] 0.2 + 0.2]]e]] I
=
(s +pl)(s
A(p)
+ pZ)(s
is + P2P3
10s’
4
s+Pl
(5.5)
2)
function
matrix
of the linear
5s+p1+4pz (8 + Pl)(S
=
+
The transfer
A(p) is stable.
2
--
(5.4)
for 4 < ]]e]] < 6.
s+Pl
G(s,P)
E = {e E
for IIf4I.//i2 =
1
of the matrix
real numbers,
(2.4) in the region
-I- {2(p1
-+- p2p4(p2 3- PlP2P4(P2
+ 4p2 -
2)}S - 2)
+ P2)
+ 5p2p3 + 2(p1
(s+pl)(s+P2)(s+P2P3+2)
+ lo) + 4p2)(p2p3
+ 2)
(5.6)
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First, we check the existence of an equilibrium
Analysts
3719
state. Since we have
PIP2 2(Pl
+ 4PZ)(PZP3
+
PIP2
and use the intervals
(5.2) of pi (i = 1,2,3,4), -Ye
(pZP3
we obtain z 4.12,
2) + p1p2p*(p2 +
-
(5.7)
2)
‘4
I
the lower bound
of 70(p)
VP E p
(5.8)
from (3.12). This means that we have {G(O,p) -t GT(0,p)}/2 > 0. From K-l(O) > 0, this implies that (3.9) holds. Thus, for any (r,p) E R x P, there exists an equilibrium state of the Lur’e system S. Furthermore, computing the upper bound of (IG(O,p))( f rom (5.2), we have IlG(O,p)ll 5 20.2. Hence, from (3.10) and (5.3), we obtain the region of ee(r,p) for any (r,p) E R x P, F(r,p) Then, we investigate the stability above discussion for any (r,p) E ‘R and F(r,p) c Ek from (5.9), (4.10) (r,p) E R x P, it is difficult to check let us introduce the matrix ER which
c
{e E R2 : IJel( 5 4.91).
(5.9)
of the equilibrium state whose existence is ensured from the P. Let rR = {e E R2 : J(e]] 5 4.91). Since i$R c E is satisfied. Actually, for infinitely many parameter values the existence of v(r,p) satisfying the inequality (4.11). Thus, is independent of (r,p) and satisfies
x
ITa 2 K(e),
Ve E ER.
(5.10)
Next, we show the existence of V, common to all (r,p) E R x P, which satisfies the matrix inequality obtained by replacing K,(r,p) with -n K in (4.11). That is, we examine the existence of a real number Y satisfying Q(w,p,u)
= ;[(l
+jvw)G(jw,p)
+ (1 -jvw)GT(-jw,p)]
+R,’
> 0,
VW E R+, vp E P. The reduced condition is more conservative by the PIA. From (5.4), the matrix
than that of Theorem
(5.11)
(4.9), but it can be easily checked
satisfies (5.10). We show the existence of u satisfying (5.11). In the case when S is a singleinput-single-output Lur’e system, we can check the existence of such v by using the Popov locus [2]. However, since we deal with the multi-input-multi-output Lur’e system and (5.11) is a matrix inequality, we cannot show the existence of such v graphically. Thus, we substitute an appropriate value to the Y in (5.11) and examine the positive definiteness of Q(w,p,v). From (5.8), (5.11) is satisfied at w = 0. In addition, we can show that (5.11) is also satisfied at w --) -too. Hence, the continuity of Q(w,p, V) with respect to w ensures the matrix inequality (5.11) if det Q(w,p, v) # 0 holds for any nonnegative w and any parameter p E P. Fig. 2 shows the region which covers the value set of det Q(w,p, V) for all p E P at each w > 0 by applying the PIA [13], where v = 0.5 is adopted. From Fig. 2, we have detQ(w,p, V) # 0 and (5.11) is satisfied. Therefore the Lur’e system S is parametrically absolutely stable.
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1
10
Analysts
102
103
w
Fig.
2. Value
6.
set of 20 log[det
&(w,p,
0.5)]
CONCLUSIONS
A sufficient condition for parametric absolute stability has been derived as a Popov-type condition. The condition is based on the existence of a real number which satisfies a parametric matrix inequality in the frequency domain. If the Lur’e system is not a single-input-single-output system, then there is no straightforward procedure to check if the condition is satisfied. In this paper, we show by a numerical example how the condition can be checked by setting an appropriate value to the real number and examining the satisfaction of the parametric matrix inequality using the PIA. ACKNOWLEDGEMENTS
The research reported the grant ECS-9526142.
herein
was supported
in part by the National
Science Foundation
under
REFERENCES 1.
2.
3.
4. 5.
WADA, T., SILJAK, D. D., OHTA, Y., and IKEDA, M., Parametric stability of control systems, in Recent Advances m Mathematical Theory of Systems, Control, Networks and Signal Processing II; (Edited by H. Kimura and S. Kodama), pp. 377-382. Mita Press (1992). WADA, T., IKEDA, M., OHTA, Y., and SILJAK, D. D., Parametric absolute stability of Lur’e systems, Proc. 34th IEEE Conference on Decisron and Control, pp. 1449-1454. New Orleans (1995). WADA, T., IKEDA, M., OHTA, Y., and SILJAK, D. D., Parametric absolute stability of multivariable Lur’e Proc. 13th IFAC World Congress Vol. E, systems: An LMI condition and application to polytopic systems, pp. 19-24. San Francisco, CA (1996). BOYD, S., GHAOUI, L. E., FERON, E., and BALAKRISHNAN, V., Zznear Matriz Inequalities in System and Control Theory, SIAM, Philadelphia (1994). GAHINET, P. A., NEMIROVSKII, A., LAUB, A. J. , and CHILALI, M., ZMI Control Toolbox, The MathWorks, Natick, MA (1995).
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SILJAK, D. D., Polytopes of nonnegative polynomials, in Recent Advances in Robust Control (Edited by P. DORAT0 and R. K. YEDAVALLI), pp. 71-77. IEEE Press, New York (1990). DAHLEH, M., TESI, A., and VICINO, A., An overview of extremal properties for robust control of interval plants, Automatica, 29 707-721 (1993). IKEDA, M., OHTA, Y., and SILJAK, D. D., Parametric stability, in New fiends in Systems Theory (Edited by G. CONTE, A. M. PERDON, and B. WYMAN), pp. l-20. Birkhauser, Boston (1991). SILJAK, D. D., Large-Scale Dynamic Systems: Stabilzty and Structure, North-Holland, New York (1978). IKEDA, M., and SILJAK, D. D., Lotka-Volterra equations: Decomposition, stability and structure, J. Math. Biol., 9, 65-83 (1980). BARMISH, B. R., New Tools for Robustness of Linear Systems, Macmillan, New York (1994). GAHINET, P., APKARIAN, P., and CHILALI, M., Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Automat. Conk., 41 436-442 (1996). OHTA, Y., GONG, L., and HANEDA, H., Polygon interval arithmetic and design of robust control systems, Proc. 29th IEEE Conference on Decision and Control, pp. 106551067. Honolulu, HI (1990). WADA, T., and IKEDA, M., Extended Popov criteria for multivariable Lur’e systems, Proc. 32nd IEEE Conference err Decision and Control, pp. 2621. San Antonio, TX (1993).
7. 8. 9. 10. 11. 12. 13. 14. 15.
ORTEGA,
J. M.,
pp.
Academic
70-71.
and
TO prove Lemma equation
RHEINBOLDT, Press,
New
C., Iterative
W. York
of Nonlinear
Solution
Equations
(3.8), we describe
APPENDIX a sufficient condition f(Z,T)
for the existence
LEMMA.
If,f,for
(z*,T*)
some
Variables,
of a solution
= 0
differentiable with respect and compact in Rm.
E R’ x FL, f(z*, T*) = 0
is satisfied
and there exists a number Ilf(Z,T)
then,
for
any
T
- f(z*,T)ll
> CLllZ - z*ll,
this lemma from the following
lemma by applying
solution
det DDzf(z*,
‘d’t E R’L, VT E ‘?
T*)
#
of (A.3).
OF
LEMMA
(3.8): Let us arbitrarily f (e,r)
Then,
(3.5) is represented equation,
I n addition,
z~(T)
[15].
suppose that for some
64.6) VT
ER
(A.7)
with respect to z, and 82 of (A.l) in 2.
f(r,r)
is the boundary
of 2.
fix p in ‘P, and define
(A.8)
= e - T + Go(pMe).
by f (e,
For this algebraic
the mean value theorem
0
f (Z,T) # 0, t/Z E 82, holds, where Vzf is the Jacobian matrix of Then, for any T E R, there exists a sohtion
(A-4)
of (A.l).
Z"(T)
(A.5) LEMMA[~]. Suppose that Z* is a uniqae bounded open neighborhood 2 of z*,
PROOF
(A.31
p > 0 such that
E R, there exists a solution
We can obtain
of the (A-1)
for any T E ‘R, where f(z,r) is a function from R” x R to R”, continuously to z, and continuous with respect to T . The region R is simply connected (A.2)
in Several
(1970).
T)
= 0.
we show that the assumptions
(A.91 of Lemma
(A.2) are satisfied
for z = e.
3722
Second World
First,
Congress
of Nonlinear
Analysts
the equality m,
(A.10)
0) = 0
implies that (A.3) holds for e’ = 0, T* = 0. Thus, in the following, we show that the existence of p > 0 satisfying
(A.4).
Let (All) (A.12)
WO(P) = GO(P) + (1 - ~o(P))K-‘(0) d = e - (1 - Go(p)}K-‘(O)p(e), then, we have f(e,r) Since (3.14) implies
= 2 - T + Wo(p)cp(e).
(A.13)
that $~O(P)
there exists We-l(p).
+ WJVPH
(A.14)
> 0,
Thus, it follows from (3.7) and (3.13) that
IlWFT(~)41.
Ilf(e,r)
- f(O,~.)ll
2: ~TK1(PHf(e,r) - f(O,r)l > TW
(A.15) This imnlies 1
that
Ilf(e, r) - f(%r)ll > - -IIEI(. IIwo~p~l, In the case when represented by
ye(p)
> 0, we can chose
(A.16)
the value So(p) = 1 from (3.14).
Then,
(A.16)
Ilf(e,r) - f(o,T)ll 2 *llt?ll I,Go~p~,,
(A.17)
via the equalities pe(p) = ye(p), We(p) = Go(p) and d = e. In the case when ye($) < 0, (3.14) implies that 60(p) < 1. Thus, from the sector condition we have
Ml = Ile - (1 = llK-1/2(0) 2 ,,Kl&O),, 2 ,,Kl~2(o)l,
~0b)lK-‘(OMe)ll [K1/2(0)e - (1 - Se(p)}K-1/2(0)y(e)] llK1/2(0)e
is
(3.7),
]]
- (1 - ~0(p)}K-1/2(0)cp(e)II
[~0b)II~1/2(Ok4
+ (1 - ~0(p)1~ll~‘~*W4I
- Il~-1~2(%4e)ll~]
> ~0b)ll~‘~2Wll -
11~1’2(0)11 (A.18)
Second World
From this inequality
and (A.16),
Congress
of Nonlinear
Analysts
3723
we obtain
“f(eyr) - f(“,T)” z hn[K(0)I Amu[K(0)]
~O(P)PO(P)
Ilell,
IIWo(p)ll (A.19)
(3.11) of PO(P) and (A.17),
Hence, from the definition
(A.19),
the inequality (A.20)
Ilf(e, T) - f@,r)ll 2 ~0b)llell
holds. This implies that (A.4) is satisfied for p = PO(P). Therefore, the assumptions of Lemma (A.2) are satisfied, and we conclude that for any (~,p) E R x ‘P, there exists a solution ee(r,p) of (3.5). Now, we compute the region where ee(r,p) exists. In (A.20), let e = ee(r,p), then we obtain the inequality (A.21) IId Z ~o(PW(~,p)ll. Therefore, (A.22)
the equilibrium
ee(r,p)
exists in P(r,p)
0
given by (3.10).
REMARK. In the above proof, we used the inequality (A.23)
llK’~2(0)41 Z IlK-‘i2(Oh4e)llNow, we show that this inequality for any e E Rm, the inequality
can be derived from (3.7). S’mce K(0)
is a positive
definite
matrix,
0 5 [e - K-‘(0)cp(e)lTK(O)[e - K-‘(0)9(e)] = eTK(0)e - 2eTq(e) + (pT(e)K-‘(O)(p(e) (A.24) holds. Thus, from (3.7), we have 0 5 eT+4e) - vT(eW1(Me) 5 eTK(0)e - eTp(e). This implies
(A.25)
that 0 5 eTp(e)
From the inequalities
(A.26)
and (3.7), we obtain ,pT(e)Kwl(0)cp(e)
which is equivalent
to (A.23).
5 eTK(0)e.
(A.26)
the inequality 5 eTK(0)e
(A.27)