Abstract. For analyzing the stability of discrete-time systems containing a feedback nonlinearity the Tsypkin criterion is the closest analog to the. Popov criterion ...
Proceedings of the 33rd Conference on Decision and Control Lake Buena V i , FL - December 1994
WM-13 1150
A Multivariable Extension of the Tsypkin Criterion Using a Lyapunov Function Approach Vikram Kapila and
Wassim M. Haddad
School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332-0150 Abstract For analyzing the stability of discrete-time systems containing a feedback nonlinearity the Tsypkin criterion is the closest analog to the Popov criterion which is used for analyzing such systems in continuoustime. Traditionally the proof of this criterion is based upon inputoutput properties and function analytic methods. In this paper we extend the Tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector-bounded memoryless time-invariant nonlinearities, along with providing a Lyapunov function proof for this classical result.
1. Introduction Ever since Popov derived a frequency domain condition for the absolute stability of continuous-time nonlinear systems, considerable effort has been devoted to deriving similar criteria for discrete-time systems (see [l] and references therein). Specifically, Tsypkin [5]gives a frequency domain condition for slope-restricted monotonic functions in the nonlinear element while Szegci, Jury and Lee, and Pearson and Gibson derive slightly different criteria for locally slope-bounded nonlinearities [I]. As in the continuous-time case the proof of the discrete-time Popov criterion, for systems containing locally slope-restricted nonlinearities, is based upon a Lur'e-Postnikov Lyapunov function involving the integral of the nonlinearity [I], i.e., V ( z ) = z T P z J,' +(U) do, where y = Cz and #(.) is a time-invariant, sector-bounded memorylessnonlinearity. However, as noted in [l] the Tsypkin criterion for systems with only a monotone restriction on the feedback nonlinearity is the closest analog of Popov's result in the discrete-time setting since in this case the criterion provides a simple graphical interpretation involving a straight line exclusion of a modified Nyquist plane. The proof of this classical result is based upon input-output properties and function-analytic methods [5]. The main contribution of this paper is to extend the T s y p kin criterion to multivariable systems containing an arbitrary number of monotonic sector-bounded memoryless time-invariant nonlinearities, along with providing a Lyapunov function proof with the corresponding Yakubovich-Kalman-Popov conditions needed to present a concise statement of this classical result. Specifically, an extended notion of a kinetic Lyapunov function [3] is used to show asymptotic stability of the nonlinear feedback system. That is, instead of finding the condition for the state variable z(k)to approach the zero equilibrium point, sufficient conditions for z(k 1) and y(k) to approach zero are found. Obviously, if the system is observable and z(k+ 1) and y(k) approach zero, the system arrives at one of its equilibrium points, and the two results are equivalent if the equilibrium point is unique.
+
+
2. Mathematical Preliminaries In this section we establish definitions, notation, and two key lemmas. Let R and C denote the real and complex numbers, let N denote { 1,2,3,.. .}, let ()= and ()* denote transpose and complex conjugate transpose, let I,, or Z denote the n x n identity matrix, and let 0, denote the n x n zero matrix. Furthermore, M L 0 (M > 0) denotes the fact that the Hermitian matrix M is nonnegative (positive) definite. Let n(z) and d ( z ) be polynomials in z with real coefficients. A function g ( z ) of the form g ( z ) = n(z)/d(z) is called a rational function. The function g ( z ) is called proper (resp., strictly proper) if, deg n(z) 5 deg d ( z ) (resp., deg n(z) < deg d ( z ) ) , where "deg" denotes the degree of the respective polynomials. In this paper a real-mtional mat& function is a matrix whose elements are rational functions with real coefficients. Furthermore, a transfer function G ( z ) is called proper
(resp., strictly proper) if, every element of G ( z ) is proper (rasp., strictly proper). Finally, an osymploticolly stable tmnsfer function is a transfer function each of whose poles is in the open unit disk. Let G ( z ) denote a state space realization of a transfer function
"e
G(z), that is, G ( z ) = C(z1 - A)-'B + D. The notation is used to denote a minimal realization. A square transfer function G ( z ) is called positive real if 1) all poles of G ( z ) are in the closed unit disk and 2) G(z)+ G * ( z ) is nonnegative definite for IzI > 1. A square transfer function G ( z ) is called strictly positive real [2] if 1) G(z) is asymptotically stable and 2) G ( e J e ) Go(.'') is positive definite for all 0 E [0,2~].Recall that a minimal realization of a positive real transfer function is stable in the sense of Lyapunov, while a minimal realization of a strictly positive real transfer function is asymptotically stable. For notational convenience we will omit all matrix dimensions throughout the paper and assume that all quantities have compatible dimensions. Furthermore, in this paper, G ( z ) will denote an m x m transfer function with input U E Rm,output y E P,and internal state z E 92". Next, we state the discrete-time strict positive real lemmaused to characterize strict positive realness in the state-space setting. Lemma 2.1 ( S t r i c t P o s i t i v e Real Lemma [2]). G ( z )'?
+
[M]
is strictly positive real if and only if there exist matrices
P,L, and W with P positive definite such that P =A ~ P + AL ~ L , o = B ~ P -A c + W = L , 0 = D +DT - B T P B - WTW,
(2.1) (2.2) (2.3)
are satisfied, the pair ( A ,L) is observable, and rank G ( z )
eI*[
= eJ',
=
m for
w E R, where G ( z ) * Finally, we state a key lemma involving controllability of an augmented pair. Lemma 2.2. Given a triple ( A ,B , C), if ( A ,B) is controllable and
z
det A
#
0, then
([
:] [ 1) ,
is controllable if and only if det
CA-'B # 0 .
3. An Extension of the Tsypkin Criterion
to Multiple Nonlinearities In this section we extend the Tsypkin criterion to the multivariable setting. For the statement of this result we consider the absolute s t b bility problem for a class 9 of time-invariant sector bounded monotonic nonlinearities 6 : 32'"
-+ R'".
Specifically, given G(z)
I*[
e
we derive conditions that guarantee global asymptotic stability of the negative feedback interconnection of G ( z ) and 6 for all 6 E 9. Note that the negative feedback interconnection of G ( z ) and 6(.) has the state-space representation
~ ( +k1) = A z ( k ) - B$(y(k)),
y(k)
= Cz(k).
(3.1,3.2)
To state our main result, the following definitions are needed. Let
M E Rmxmbe a given positive definite matrix. Next, define the set @ of allowable nonlinearities 9
This research was supported in part by the National Science Foundation under Research Grants ECS-9109558 and ECS-9350181.
0-7803-1968-0/94$4.0001994 IEEE
-
[e]
e {6
:P
6 by
+ W : 6T(y)[M-'6(y) - y] < 0, y E W, y # d ( ~= ) [6l(Yl), 6 ~ ( Y 2 ) , . . . , 6 m ( y ~ ) I Tand , < J-:- (e 1, o,+ E 9, i = l , . . . , m } . I
843
0,
(3.3)
In the special case when m = 1 the inequality condition characterizing 0 is equivalent to the more familiar sector condition 0 < b(y)y < Mya, y E 91. For convenience in stating the multivariable generalization of the Tsypkin criterion we define
and grouping terms yields AV(%.)
e
[.e] , letN:diag[NI,Na,
...,”I
+ +
+
P =A~PA. L~L, 0 = B T P A , - NCO- S W T L , o = ZM-’ - B ~ P B -- W ~ W .
(3.5) (3.6)
In this case,
+
-P l ~ ( k )
Now using (3.4)-(3.6) yields Av(zo)
(3.4)
+
z:(k)[A:PAo
+ + + 2bT(y(k + l))[M-’b(y(k + 1)) - ~ ( +k l)]. (3.13)
benonnegative-
definite, and assume det CA-’B # 0, then P ( z ) 2 M-’ [I (1 z-’)N]G(z), is strictlypositive real, if and only if there exists matrices P, L, and W with P positive definite satisfying
I
- bT(y(k 1))[BzPA. - N C O - S]za(k) - z:(k)[BTPA0 - NCO- SITb(y(k 1)) - bT(y(k 1))[2M-’ - B,TPB,Ib(Y(k + 1))
Theorem 3.1 (The Multivariable Tsypkin Criterion). Let G(z)
+
+
Next adding and subtracting 2bT(y(k l))M-’+(y(k 1 ) ) and 2 b T ( y ( k + l ) ) y ( k + l ) to and from (3.12), noting that y ( k + l ) = Sza(k),
5 -[Lz,(k)- W b ( ~ ( k +1))lT[Lzo(k) - W b ( ~ ( k+ I))] + 2bT[y(k + l))[M-’b(y(k + 1)) - ~ ( +k I)]. (3.14)
+
Since 2bT(y(k + 1))[M-’)(y(k + 1)) - y(k l)] 5 0, it follows that A V ( z , ) 5 0, which proves stability in the sense of Lyapunov. To show global asymptotic stability we need to show that A V ( z . ) = 0 implies z = 0. Note that A V ( z . ) = 0 implies that y(k 1 ) = 0, k E N and hence $(y(k 1)) = 0 and Lz,(k) = 0. Furthermore, in this case z,(k+ 1) = A.z.(k) - B,,b(y(k+ 1 ) ) = A,z.(k). Thus, using z,(k 1) = A.z,(k), Lz.(k) = 0, and the observability of ( A . , L ) , it follows from the PBH test that z.(k) = 0, k I? N ,which further implies, since ( A , C ) is observable, that ~ ( k = ) 0, k E N. Thus, the only solution satisfying A V ( z ( k + l ) , y ( k ) ) = 0 is the z ( k ) = 0, k E N, solution and hence it follows from the discrete-time version of LaSalle’s theorem that global asymptotic stability holds. CI Theorem 3.1 presents a generalization of the Tsypkin criterion [5] to the case of multivariable systems containing an arbitrary number of memoryless time-invariant sector-bounded monotonic nonlinearities. It is interesting to note that unlike other discrete-time extensions of the Popov criterion [I, 41, which require local slope restrictions on the nonlinear feedback function, the Tsypkin criterion holds for nonlinear functions possessing a monotonic characteristic. Furthermore, t.he Lyapunov function proofs of discrete-time Popov criterion for locally slope restricted nonlinearities given by Pearson and Gibson, SzegB, and Haddad and Bernstein [l, 41, do not require a system augmentation of the form z.(k). As noted in [5], in the SISO case, the criterion given by Theorem 3.1 has an interesting geometric interpretation. Specifically, setting z = ReG(z) and y = coswReG(z) + s i n w I m G ( z ) - ReG(z), and requirement that E ( z ) be strictly positive real is equivalent to
+
+
+
where y(k) = C z ( k ) ,is a Lyapunov function that guarantees that the negative feedback interconnection of G ( z ) and 9 is globally asymptotically stable for all 6 E @. Proof. First, define w ( k ) k z ( k + 1) so that
~ ( + k1) = A w ( ~ ) Bb(y(k
+ l)),
and z.(k+l) = A . z , ( k ) - B . / ( y ( k + l ) ) ,
y(k
+ 1) = C W ( ~ ) ,(3.8, 3.9)
[4 k )
wherez.(k):
y ( k ) ] . Fur-
+
Im]z.(k) and y(k l ) = Sz.(k). thermore, note that y(k) = [Om., Now, since ( A , B , C ) is minimal, ( A , B ) is controllable. Hence, it follows from Lemma 2.2 that if det C A - ’ B # 0 then (&,Bo) is also controllable. Next, note that O(z) has a minimal realization given by G ( z )
[ M I . Now
it follows from Lemma 2.1 that O ( z ) is
strictly positive real if and only if there exists P,L, and W with P positive-definite satisfying (3.4)-(3.6). Next, for 4 E 8 consider the Lyapunov function candidate (3.7). Since P is positive definite, 4 E @, and N;is a nonnegative scalar, it follows that V(z.) given by (3.7) is positive definite. The corresponding Lyapunov difference is given by
1 + z - N y > 0.
(3.15)
Condition (3.15) is a frequency domain stability criterion with a graphical interpretation in a modified Nyquist plane, involving z and y, in and slope 7$. terms of a straight line with a real axis intercept
4. Conclusion (3.10)
Now, using the fact that #;(.) is a monotonic nonlinear function, it follows that the integral term in (3.10) can be bounded from above by
In this paper we extended the Tsypkin criterion for monotonic sector-bounded nonlinearities developed in [5] to multivariable systems containing an arbitrary number of monotonic sector-bounded nonlinearities. Specifically, explicit Lyapunov functions along with the corresponding Yakubovich-Kalman-Popov conditions were obtained to pr* vide a concise statement of this extension. Using the framework developed in [4] the results developed in this paper can be used to synthesize robust feedback controllers.
References
so that
Furthermore, note that since y(k (3.11) becomes
C.z.(k)
+ 1) = Sz,(k) and y(k + 1) - y(k) =
[l] R.W. Brockett, “The Status of Stability Theory for Deterministic Systems,“ IEEE TAC., AC-12, pp. 596-606, 1967. [2] P.E. Cains, Linear Stochastic Systems, (Wiley), 1989. [3] S.S.L. Chang, “Kinetic Lyapunov Function for Stability Analysis of Nonlinear Control Systems,” JBE., Trans. A S M E . 83--D, pp. 91-94, 1961. [4] W.M. Haddad and D.S. Bernstein, “Parameter-Dependent Lya-
punov Functions and the Discrete-Time Popov Criterion for Robust Analysis,” Automotica, 30, pp. 1015-1021, 1994. [5] Y.Z.Tsypkin, “A Criterion for Absolute Stability of Automatic Pulse Systems with Monotonic Characteristics of the Nonlinear Element,” Sou. Phyr.-Doklndy, 9, pp. 263-266, 1964.
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