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performance results for singularly perturbed systems
Pergamon
PII: sooo5-1098(%)ooo11-8
Aukmuric~, Vol. 32, No. 6, PP. W-818, 19% Copyright Q 19% Elswier Science Ltd Printed in Great Britain. Ail rights reserved ixws-1098/% s15.00
t 0.00
New Stability/Performance Results for Singularly Perturbed Systems* LAHCEN Performance
SAYDYt
of singularly perturbed systems is investigated.
Key Words-Linear systems; singular perturbations; robustness; stability and performance; composite feedback; state space; stability upper bound; generalized stability.
Almlraet-The guardian map theory of generalized stability of parametrized linear time-invariant systems is used to prove new results on stability/performance of linear time-invariant singularly perturbed systems with an exogenous parameter, i.e. systems that contain, in addition to a singular parameter E, another uncertain parameter CL.The results give necessary and sufficient conditions for generalized stability for all sutIiciently small values of c and all values of p in a given interval [pi, pr]. In addition, explicit expressions for the largest upper bound l* on c for guaranteed stability and performance are given and the cases leading to finite or infinite E* are clearly delineated. Thus the results represent a significant addition to the classical Klimushev-Krasovskii theorem, while at the same time providing closed-form expressions for the maximum parameter range for stability and performance. Copyright 0 1996 Elsevier Science Ltd.
Klimushev and Krasovskii below is a classical result giving a condition for the asymptotic stability of (l), (2) for all sufficiently small values of the singular perturbation parameter E, and has been derived by several authors (Klimushev and Krasovskii, 1962; Desoer and Shensa 1970): Theorem 1.1. Let D be nonsingular. If the matrices A,:= A - PD-‘C and D are Hurwitzstable then the overall system (l), (2) is Hurwitz-stable for all sufficiently small E > 0, i.e. there exists an E> 0 such (l), (2) is Hurwitzstable for all E E (0, E].
1. INTRODUCTION
Consider the singularly perturbed
Note that this result only yields a sufficient condition (subsequently referred to as the KK condition) for stability of (l), (2) for all sufficiently small values of 6. In fact, the condition of KK is necessary and sufficient for Hurwitz stability of (l), (2) for E > 0 small enough when D has no eigenvalues on the j, axis.
linear system
i=Ax+Bz, E.?= Cx + Dz
(1) (2)
in which E > 0 is a small real parameter, x and z are vectors in R” and R” respectively. Owing to the presence of the small parameter l, the dynamics of (l), (2) naturally separates into slow and fast dynamics thus leading to the consideration of the reduced-order (slow) subsystem (D is assumed invertible) 1, = (A - BD-‘C)x,
=:Ags,
(3)
which evolves in R” instead of R”‘“. one seeks to infer some aspects of behavior of (l), (2) from similar the reduced-order system (3). The interest to us here are the stability the performance of (l), (2). The
Generally, the general aspects of aspects of as well as theorem of
Theorem 1.2. If D is nonsingular with no pure imaginary eigenvalues then the system (l), (2) is Hurwitz-stable for all E > 0 small enough if and only if D and A,, are Hurwitz-stable matrices. This follows from the fact that the spectrum of the
matrix
J(E) = (C;e
D:E >
approaches
{AI(&), . . . , MAO), &(D)le . . . , L(D)l~~
as
E -+ 0 (see Kokotovic et al., 1986). Clearly these results shed no light on what appears in the critical situation when D is nonsingular but merely marginally Hurwitzstable. Furthermore (and more importantly), they give no precise quantitative meaning to how small E should be for the above conclusions to hold.
*Received 11 July 1994, revised 1 June 1995; received in final form 1 June 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor J. M. Dion under the direction of Editor Ruth F. Curtain. Corresponding author Professor Lahcen Saydy. Tel. +l 514 340 4233; Fax +l 514 340 407% E-mail [email protected]. t Department of Electrical and Computer Engineering. l?cole Polytechnique de Montreal, Canada. go7
808
L. Saydy
From a design viewpoint, one considers singularly perturbed control system i =A,,x
+A,g
+B,u,
EZ = A2,x + Az2z + B2u,
the
(4) (5)
which for small E > 0 leads to the consideration of two reduced-order subsystems evolving on different time scales, namely, the slow subsystem 1, = &x, + &us,
(6)
where &=A,,
-A,,&;&,
B. = B, - A,2A;;B2,
(7)
and the fast subsystem ir = A22zf + Bzuf.
(8)
This two-time-scale separation into slow and fast subsystems makes it possible for the designer to procede by designing separate feedback laws for (6) and (8) and using a composite one to control the overall system (4), (5). More precisely, if u, = G#, and uf = G2zf are such that a(A,, BOG,) = {a,, . . . , a,} and a(A,, - B2G2) = {p,, . , pm} then it is well known that the composite feedback law (Kokotovic et al., 1986) u = (Go + G,)x +
Gz, (9)
G, = GzA;2’&Go
assigns the eigenvalues of the closed-loop system i=Ax+Bz,
(10)
6.j = Cx + Dz
(11)
(with A = A0 + B,(Go + G,), B = Al2 + B1G2, C = A21 + B,(Go + G,) and D = AZ2 - B2G2) to the locations {(Y,,. . . , a,, PI/e,. . . , &/E} for E small enough. In particular, if AZ2 is Hurwitzstable (stable fast dynamics) and {/?,, . . , p,,,} = g(Az2) then the feedback law u = G,x (obtained by setting G2 = 0 in (9)) accomplishes the goal above (using the slow variables only). Suppose that for practical considerations
Fig. 1. The region
$2.
relating to damping ratio, stability margin etc. it is desired to place the eigenvalues of the closed-loop system {al, . . . , a,, PI/e,. . . , &JE} in a specific subregion R of the open left-half complex plane (e.g. the one in Fig. 1). Then the composite feedback law (9) guarantees that the performance of the closed-loop loop system (lo), (11) is specified by that of the slow subsystem when E is small enough, since as ~-0 the closed-loop loop eigenvalues tend to {(Y,, , a,} (the remaining eigenvalues becoming unbounded in R along the directions specified by {PI, . , pm}). Thus for this two-time-scale composite design to work it is expected that E be small enough. The qualitative nature of this requirement represents quite a drawback from a design viewpoint, particularly when E is unknown. It is therefore important to be able to ascertain how small E should be for the overall system to retain the desired performance, namely that the eigenvalues of the closed-loop loop system (lo), (11) remain within R. Furthermore, it is not clear in the light of the above discussion what happens when mere stabilization of the closed-loop loop system is sought (i.e. Q coincides with open left-half complex plane) and the uncontrollable modes of the fast subsystem (A**, B2) happens to be purely imaginary. Since the parameter E typically represents a small physical quantity over which one has little or no control, it is of practical significance to find explicit, nonconservative, upper bounds lo on E ensuring (under minimal assumptions) that the above stability/performance conclusions are valid for all E E (0, eo] or, if possible, find the largest such upper bound E* : = sup {e. > 0: the system (l), (2) is stable in a generalized sense for all E E (0, ~~1). (12) Here generalized stability means confinement of matrix eigenvalues or polynomial zeros to general open subsets of the complex plane, and includes Hurwitz stability as a special case. Practical considerations relating to damping ratio, bandwidth etc. are commonly expressed in terms of the generalized stability formulation, with respect to a suitable domain in the complex plane. Previous results on upper-bound estimates lo for Hurwitz stability have been obtained by several authors, including Zien (1973), Javid (1978), Sandell (1979), Chow (1978), Khalil (1981) and Abed (1985). Feng (1988) used frequency-domain stability analysis to characterize the maximum parameter range on E for Hurwitz stability of (l), (2) under the hypotheses
New results for singularly perturbed and Krasovskii’s result. A of Klimushev necessary and sufficient condition for (general-
ized) stability of (1) for all sufficiently small values of the singular perturbation parameter E together with the closed-form computation of the largest upper bound E* on E for guaranteed stability was given for the first time in Abed et al. (1989). These results were obtained using the guardian map approach to the study of generalized stability of parametrized families of linear systems (Saydy, 1988; Saydy et al., 1990, 1991). In the present work, we present some new results pertaining to singularly perturbed systems given by (l), (2), and, utilizing guardian and semiguardian maps, investigate similar issues for ‘uncertain’ singularly perturbed linear systems with a regular perturbation parameter p (subsequently referred to as the exogenous parameter), i.e. singularly perturbed systems of the form i = A(p)x
+ B(p)z,
(13)
Ei = C&)x
+ D(p)z,
(14)
where p is an uncertain (possibly large) parameter known only to lie within an interval, say I_LE [pl, ~~1 with pl, p2 given scalars, and A(p), . . . , D(p) are polynomial functions of CL, i.e. A(p) = C;I” Aipi, . . . , D(p) = E:;f” Dip’. The approach presented can handle more general models (13), (14) for which these matrices are allowed to depend polynomially on E as well. This, however, is not done explicitly in the sequel for the sake of clarity. In this article, we shall give definite complete answers to the following two problems: (P,) obtain necessary and sufficient conditions for (generalized) stability of (13), (14) for all sufficiently small values of E > 0 and all p E [CL17 1121;~ (P2) obtain an explicit expression for the largest upper bound E* such that (13), (14) is stable for all E E (0, E*) and all p E [p,, p2], and clearly delineate those cases for which E* is infinite from those for which it is finite. The results in this article make it possible to analyse the stability and performance of very general closed-loop singularly perturbed systems. In particular, they apply to singularly perturbed systems in standard form as well as those not in standard form. Systems in the latter category have received increased attention in the literature recently (see e.g. Krishnan and McClamroch, 1994). Finally, note that since we seek the maximum upper bound E* for the closed-loop system to
809
systems
retain stability and performance of some kind, the values of E* with which we end up might turn out to be not so small (sometimes even infinite). As such, the calculation of E* serves only one purpose, namely it guarantees that the closed-loop system (13), (14) is well behaved in the generalized stability sense for all E E (0, E*) (and in no larger interval) and hence that the composite feedback (9), which we know works well for small enough 6, does actually work for all E E (0, E*). In particular, no information is given as to whether or not the two-time-scale separation is preserved in closed loop as E approaches E*. It is the view of this author that this issue can in fact also be tackled using the guardian map framework. This question will not, however, be pursued here. The paper is organized as follows. In Section 2, the concept of guardian and semiguardian maps is recalled and relevant results are given. In Section 3, the case of singularly perturbed systems with no exogenous parameter is considered and an example is given. In Section 4, the main results relating to problems (Pi) and (P2) above are given, together with an example. Finally, Section 5 contains the proofs of the results given in Sections 3 and 4. The following notation is used throughout the article. @ denotes the set of complex numbers, 6 the open left-half plane, G the closure of set 9, Q a generic open subset of the complex plane, symmetric with respect to the real axis, aR its boundary, Yn(Q) the set of all n x n real matrices (or nth-order real polynomials, depending on the context) with eigenvalues (zeros) in R, A 0 B the bialternate product of matrices A and B, 6!&the set of &h-order polynomials with real coefficients, Z(p) the zeros of a polynomial p, B(p, q) the Bezoutian matrix of polynomials p and q, finally P[r](.) (or simply p [r]) denotes a polynomial in (m) whose coefficients are themselves polynomials in r. 2. GUARDIAN
MAPS
The guardian map approach was introduced in Saydy (1988) and Saydy et al. (1990, 1991) as a unifying tool for the study of generalized stability of parameterized families of matrices or polynomials. For the purpose of making the presentation self-contained, a basic review of the essentials now follows (for more on the subject, the reader is referred to Saydy et al. (1991); see also the recent book by Barmish (1994) for an introduction to the subject). Definition 1. Let X be the set of all n x n real
matrices, or the set of all polynomials
of degree
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L. Saydy
n with real coefficients, and let Y be an open subset of X. Let Y map X into @. The map Y is said to be a semiguardian map for Y if for all x E 9, the implication Y(X) = 0
whenever
x E a,!+
holds. If the converse also holds, v is said to be a guardian map for Y. Furthermore, we say that the map v is polynomic if it is a polynomial function in the entries/coefficients of its argument. For the purposes of this paper, the set .Y will be a (generalized) stability set, i.e. a set of the form Y(Q) := {A E lRnx”: u(A) c !G?}, for matrix stability problems, and (with a slight abuse of notation) Y(Q):= {p E Pn:%(p) =Q} for polynomial stability problems. Here Q is an open subset of the complex plane that is symmetric with respect to the real axis, and an element of Y(Q) is said to be stable relative to CL Two examples of interest in later sections are given next. For more examples of guarded sets Y(Q) and systematic techniques to construct the associated guardian maps, the reader is referred to Saydy et al. (1991), where it is shown that in fact, many stability sets of practical interest enjoy the guardedness property with polynomic corresponding guardian or semiguardian maps. For example, any stability set for which Q={s =x +jy:p(x,y)
v: A H det (A) det 2(A 0 I)
(15)
guards the set of n x n Hurwitz-stable real matrices S(c -). For example, for a 2 X 2 matrix A, the bialternate product A 0 I = ;(a,, + u12), so that v(A) = det A trace (A), and vanishes if and only if A is singular or has opposite eigenvalues (trace (A) being equal to h,(A) + h,(A)). A similar reasoning holds for general matrices, since det [2(A 0 Z)] vanishes if and only eigenvalues (see the if A has opposite Appendix).
in the case of polynomials. Theorems 1 and 2 below give fundamental necessary and sufficient conditions for stability of parametrized families of matrices or polynomials relative to domains of the complex plane corresponding to guarded and semiguarded stability sets respectively. Let r= (r,, . . , rk) E U be a vector of unknown parameters, where I!/ is a pathwiseconnected subset of R“, and let x(r) be a matrix in R”“” or a polynomial in P?,, that depends continuously on the parameter vector r. Given an open subset Q with guarded or semiguarded stability set Y’(Q), we seek basic conditions for x(r) to lie within Y(Q) for all values of r in U. Theorem 1. Let Y(R) be guarded by the map v. Then the family {x(r):r E U} is stable relative to R if and only if
(i) it is nominally stable, i.e. x(r”) E Y(Q) some r0 E U; and (,ii) v(x(r)) #O for all r E U.
Theorem 2. Let Y(Q) be semiguarded by the map v. The family {x(r):r E U} is stable relative to Q if and only if
(i) it is nominally stable, i.e. x(r’) E Y(Q) for some r0 E U; and (ii) x(r) E 9’(Q) for all r E UC,, where Uo:= {r E II: v(x(r))
= O}.
The above two results are illustrated in Figs 2 and 3. In the guarded case, a guardian map v evaluated along any given family {x(r): r E U} remains nonzero as long as that family is stable, and vanishes precisely when it exits the stability set Y(Q). In the semiguarded case, the vanishing of v no longer indicates unequivocally that a given family has exited the stability set for there could exist stable elements (so-called blind spots; see Saydy et al., 1991) for which v vanishes. In the case of @\ [a, p], it is seen from (16) that these
Example 2.2. Let Q = @\[a, p] (i.e. the set of all complex numbers that do not lie in the closed interval [a, p]). Then the set Y(Q) is semiguarded by
v: p ~p(~)p(P)
det B(p(s),
$,(s)j (16)
for
Fig. 2. The guarded
case.
New results for singularly perturbed
811
systems
3. SINGULARLY PERTURBED SYSTEMS WITH NO EXOGENOUS PARAMETER
For the system described by (l), (2), define the matrix
J(r) =
Fig. 3. The semiguarded case.
are any polynomials with no zeros in [CX,p] (i.e. stable) and at least one multiple zero, since in this case the Bezoutian of p and its derivative is singular (see the Appendix). For the specific domain @\]P1, Pzl, subsequently denoted by 5, we have the following corollary. Corollary 1. Let E := @\[pl, ~~1. Then the family of polynomials {p[r]:r E U} is stable relative to Z if and only if
(i) it is nominally stable, i.e. p[r’] E 9’(Z) for some r” E U; and (ii) I& = 0 where I-&:={r e
and p[r] E 9’(Z)
~:Arl(dWcL2)
for
=O}.
Remark
1. Condition (ii) above requires testing the stability of p[r] relative to B for all critical values of r E 172. This is feasible only when cr is a finite set. In degenerate cases, the polynomial det B(p[r](s), (d/ds)p[r](s)) vanishes identically, leading to such a situati0n.t This indicates that the polynomials p[r](s) and (dlds)p[r](s) have a common zero for all r. To prevent such a situation from happening, all one needs to do is extract the greatest common factor of p[r](s) and (d/ds)p[r](s), and then proceed. This is equivalent to computing Uz, according to Uzr:=[r
E U:detB(lj[r](s),%p[r](s))
v(r) = v. + vlr + . . . + v4-,r4-’
+ vqr9,
(19)
where q denotes the degree of the polynomial.
= 01, -$p[r](s))
(18)
9
>
where r:= l-’ is large when E is small. The stability of (l), (2) for all sufficiently small values of E is identical to the stability of the matrix J(r) for all sufficiently large values of r. Theorem 3 below presents necessary and sufficient conditions for the system (l), (2) to be stable for all sufficiently small values of the parameter of E >O relative to a given open subset R of the complex plane for which .S@(Q)is endowed with a polynomic guardian map vn. Furthermore, Theorem 3 also gives the largest upper bound E* strictly below of which stability is still guaranteed (no prior assumptions are made on the system; in particular, the conditions of Klimushev and Krasovskii need not hold, and the results apply to singularly perturbed systems in nonstandard form as well). Let v(r):= v&(r)). Since vn is polynomial in its argument, and J(r) depends linearly on r, we can write v(r) as a polynomial in r:
all r E IJf,,
(17) Uzr:= {r E U:det B(p[r](s),
(pCrt
=O),
where p denotes the square-free version of p and is given by B =plGCF(p, dplds). A method for extracting GCFs when the scalar coefficients involved belong to a unique factorization domain is given in Bose (1982). t This would happen if redundant guardian maps are used; e.g. det (A CBA) in lieu of det (A) det (A OA).
Theorem 3. Let &2be such that Y”(Q) is guarded by a polynomic map v, and let v(r) be as in (19).
(4
If v(r) vanishes identically then the singularly perturbed system (l), (2) is unstable relative to Q for all e > 0. v(r) does not vanish identically then the singularly perturbed system (l), (2) is stable relative to R for all sufficiently small E > 0 if and only if it is stable relative to R for an arbitrarily chosen O< e < l/r*, where r* denotes the largest positive real root (if any, otherwise set r* = 0) of v(r). In this case l* = l/r*.
(b) If
Proof.
see Section 5.
Noting that E* = +w if v(r) has no positive roots, it is easily seen that case (b) in the above corollary is in fact a condensed statement of the following two cases: (b,) If v(r) does not vanish identically and has no positive real zeros then (l), (2) is stable relative to &2for all sufficiently small e > 0 if and only if it is stable relative to R for an
812
L. Saydy arbitrarily E* = +m.
(b)
chosen E > 0. In the latter case,
Let v(r) have a largest positive real zero (Y/. Then (l), (2) is stable relative to Q for all sufficiently small E > 0 if and only if it is stable relative to Q for an arbitrarily chosen E < Icy,. In this case E* is finite and is given by E* = llcx,.
Remark 2. Different expressions for the guardian maps corresponding to a given region Q lead to different expression for v(r) in (19), but these will all, by definition, vanish for the same real values of r. The largest upper bound E* will not therefore depend on which particular guardian map is used for a given region. For example, when Q is the open left-half plane, both v,(A)=det(A@A) and q(A) = det (A) det (A 0 1) guard Y(Q). It can be shown, however, that det (A $ A) = det (A)[det (A 0 Z)]“; hence v2(r) will be a factor of v,(r). A one-shot test
The results of the previous section are conceptually simple and can be implemented easily. The question arises whether or not a simple one-shot test exists by which one can immediately ascertain stability of (1) (2) for all sufficiently small l or the lack thereof, Such a test, recorded as Theorem 4, is given next. It gives necessary and sufficient condition for stability in the form of a one-shot test, namely that the system (l), (2) is stable for all sufficiently small E > 0 if and only if it is stable for one judiciously picked value of E (no assumptions are made on A, B, C and D). This represents a significant complementary addition to Klimushev and Krasovskii’s classical result given earlier. Theorem 4. Let Y(Q) be guarded by a polynomic map Y$)and assume that vn(J(r)) is not identically zero.t Then there exists an F > 0 such that the system (l), (2) is stable relative to Sz for all l E (0, E) if and only if it is stable relative to Q for
E,,: =
1 1 + max Iv,I~~yyl. i<
(20)
Note that Theorem 4 settles the question of stability of the system (l), (2) for E > 0 small enough and yields an estimate of E, namely Q, t Since if it is then, according to Theorem unstable relative to R for all E >O.
3, the system
is
which it turns out is always less than 1. To compute the largest such estimate (i.e. E*), it is required to find the real roots of v(r) as stated in Theorem 3. Nevertheless, both theorems necessitate the computation of v(r). Even though a computer algebra package exists to that end, it is of interest to derive the analogous result of Theorem 4 directly in terms of the data A, B, C and D. i.e. without explicitly computing v(r) (for related work, see Mustafa, 1994). For simplicity, we shall restrict attention to the case of Hurwitz stability, i.e. R coincides with the open left-half complex plane.* Rewrite J(r) = : Jo + rJ, , where Jo = /,=
O , and let j(r):=J(r)OZ cO D C 1
A
(
o
B
o and >
= (&OZ) +
r(J, 0 I) =:JO + r.7,. Theorem 5. Suppose that J(r) is not singular and/or has no opposite eigenvalues for all value of r > O.§ Pick r” > 0 such that J(r’) is nonsingular with no opposite eigenvalues, and let r* := r” + max {(a-‘(M)
U (T -‘(fi)
fl
[-r”, +m)}, (21)
where M:= -J-‘(r’)J, and &I:=J-‘(r’)J,. Then for all the system (l), (2) is Hurwitz-stable sufficiently small values of E > 0 if and only if it is so for an arbitrarily chosen value of E E (0, l/r*). In this case, E* = l/r*. Note that checking the hypothesis of Theorem 5 can be done generally in finitely many steps and generically in one step. Indeed, in the case of Hurwitz stability, vC1 is given by v&(r)] = det J(r) det [2(J(r) 0 Z], and it vanishes precisely when J(r) is singular and/or has opposite eigenvalues (this follows from the fact that the eigenvalues of the matrix J(r) 0 Z are the sums of those of J(r); see the Appendix). Furthermore, yC1is a polynomic guardian map, as evidenced by the defining equation for the bialternate product of two matrices (see the Appendix). It therefore follows that v(r):= v&(r)] is a polynomial in r. Thus the hypothesis of Theorem 5 fails to hold precisely when v(r) vanishes identically, an assertion that can be settled quite easily by picking a randomly generated, say positive, value r” of r, and testing whether v(r’) z 0 or, equivalently, that J(r”) is singular and/or has $ Results analogous to Theorem 5 and Corollary 2 below involving more general regions R will be presented elsewhere. 5 If J(r) is singular and/or has opposite eigenvalues for all r > 0 then the system (l), (2) is Hurwitz-unstable for all E :,
0.
New results for singularly perturbed
813
systems
open-lc-apsystem r__________‘____________________,
Fig. 4. The chemical reactor model.
opposite eigenvalues. If J(r’) is nonsingular with no opposite eigenvalues then certainly the hypothesis of Theorem 5 holds; if not then, with probability one (i.e. generically), the hypothesis fails (Y = 0). Naturally, to be sure, one has to do at most degree (Y) + 1 checks involving different values of r” to (conceptually) completely settle the question (degree (Y) + 1 such tests checks are necessary to prove that Y = 0.) Corollary 2. Let D and A0 be Hurwitz matrices so that the system (l), (2) is Hun&z-stable for all sufficiently small E > 0. Pick lo > 0 such that J(E’) is Hurwitz-stable (or merely nonsingular with no opposite eigenvalues) and let r”:= l/e’. Then the largest stability upper bound is given by E* = l/r*, with r* as in (21).
Next, an example dealing with composite control design for a two-tank chemical reactor is presented.? Example
3.1. In this example we consider the two-tank chemical reactor model (Kokotovic et al., 1986) in Fig. 4, where E is an unknown small time constant; C,, x1 and x2 denote the deviation in desired concentration and the deviations in concentration in the first and second tanks, respectively. Here we investigate the performance of the closed-loop system as typified by the stability region in Q in Fig. 1, with u = - 1 and CX= 1. We adopt the design viewpoint that a critically damped closed-loop system is sought in the ideal case of E = 0 but that a design leading to closed-loop eigenvalues within R is acceptable in a realistic situation of a not-so-small E. A state-space model is given by (4), (5), where
The slow subsystem (6) is specified in this case by A,, = AI, and B. = (0 4)‘. Since the fast dynamics is stable (hl(A22) = h2(A22) = -l), it is possible to synthetize a state feedback based solely on the slow variables xi and x2. It is easily checked that the gains K, = y and K2 = y lead to a critically damped reduced-order subsystem with eigenvalues at (-1, -1). With this choice of K, and K2, the dominant eigenvalues of the closed-loop system will be nearly critically damped if E happens to be sufficiently small. With u = K,xi + K2x2, the closed-loop system (lo), (11) is specified by
;
-f
j
0 -K2r
-KIr
;
-r
-2r I
E* computation.
For the present generalized problem, Y(Q) is guarded by (Saydy et al., 1991) v(A) = v,(A) ~264) dA), with q(A) := det (A + cd), v2(A) := det 2[(A + uZ) 0 I] and
stability
[t(l - a2)AOA
QA):=det
For this example, factors)$ v,(J(r))
- i(l+
a2)A20Z].
we have (modulo
constant
= $r2,
v2(J(r)) = 24 - 204r + 107r2 + 1190r3 - 2950r4 + lOOOr
and v3(J(r)) = r2( -116 - 1450r - 33271r2 - 190 880r3 - 341 200r4 + 957 OOO?+ 3 825 000r6 -6200000r7+1000000r8),
with the following zeros: %vi(J(r)))
= 101,
%(v,(J(r)))
= (-0.360,
%(v3(J(r)))
= (-0.231,
0.138, 0.355
f jO.265,2.462},
B, =
0 0
f j0.171, -0.012
f jO.O64,0.642 f jO.O27,5.471}. ,
A21
1 A22=(:;
-0.151
-2)
=
7 B2=
0
;.
t For an example comparing the present method with an existing one in a Hurwitz stability context see Abed et al. (1989).
It follows that (Ye= 5.4712; hence E* = l/c+ = 0.18. Thus we are guaranteed that the closedloop eigenvalues remain within Q for all #A symbolic manipulation package written in the computer algebra program. Mathematics was used to arrive at all the computations in all the examples presented.
814
L. Saydy
E E (0, 0.18) and exit Q precisely for E = E*. Indeed, the eigenvalues of J(E*) are (-7.7161, -0.7875, -1.5694 f j1.5694}, the last two of which belong to aQ. 4. SINGULARLY PERTURBED SYSTEMS AN EXOGENOUS PARAMETER
Denote
the matrix associated
WITH
with (13), (14)
bY
where r: = e-’ is positive. Theorem 6 below gives a necessary and sufficient condition for the (generalized) stability of the matrix J(r, p) for all p E [pi, ~~1 and all sufficiently large r > 0, or, equivalently, that of the matrix J(E, CL) for all p E [E_L,,p2] and all sufficiently small E >O. If J(E, p) is stable, Theorem 6 also provides the largest upper bound E* on E for which the stability of J(E, F) is guaranteed for all p E [p,, ~1~1. Let R be a subset of the complex plane of interest and suppose that the generalized stability set Y(Q) (for matrices) is endowed with a polynomic guardian map v<). In the case of Hurwitz stability, 52 is the open left-half plane, and one such map is given by (15) in Example 2.1. Let (23)
v(r, P):= v&(r+ P)I, 51(r):= v(r, pi)v(r, &(r):=det
~2)~
(24)
~(Qlrl(p), 2 ~[T)(p)), (25)
where B denotes the Bezoutian and 4 denotes the square-free version of the polynomial q (see Remark 1). Note that v, 5, and & are all polynomials of their arguments, since vi1 is a polynomic guardian map, J(r, p) is a matrix polynomial in r and p, and the Bezoutian is a bilinear function of the coefficients of its arguments. We also formally define the scalar quantities p=max(r:r
E.%,~(~,)}
(26)
6 = max{r E %‘([A: v[rl(k) @SF.)}, 10 otherwise,
(27)
r* = max {p, 6).
(28)
By ‘otherwise’ it is meant that either Z+(e2) = 0 or v[r](p) E P(Z) for all r E 5?(t2). Here .Z~(p):={%(p) n (0, m)} U 0, i.e. the set of real positive zeros of a polynomial p augmented with 0, and Z = C\ [pi, pz]. Thus 6 is the maximum positive zero of tr, say r,, for which v[r,](p), as a polynomial in p, is not in Y(Z), i.e. has zeros in [p,, ~~1. If either t2 has no positive zeros or v[r](p) as a polynomial in /I has no zeros in
[pi, ~~1 for every choice of r equal to a positive root of & then 6 = 0. We are now in position to state the main results of this section. Theorem 6. The following cases present themselves. Case l(i): V(E, p) -0. Then J(e, p) is not stable for all (E, II) E (0, =) X [p,, ~~1. Case l(ii):&,(r) = 0. Then J(E, pi) and/or J(E, pz) are unstable for all E > 0. Case 2: 5, f 0 (hence V(E, p) f 0). Pick q), p,, arbitrarily such that cl0 E [p,, wz] and lt1E (0, l/max (p, 6)). Then J(E, p) E P’(S) for all F E [F, , p2] and E > 0 sufficiently small if and only if J(E,), po) is stable relative to Q and the univariate polynomial v[Es’](~) has no roots in [p,, ~~1. We then have E” zz
1
max (P, 8) ’
(29)
Hemark 3.
(i) Note that, according to the definition of p and S, E* E R, i.e. it is allowed to take the value cc. (ii) In the particular important case of Hurwitz is stability, we point out that r =0 automatically a zero for 5, and t2; thus augmenting Z’([,) with (0) is futile. This follows from the fact that J(r, p) is singular for r = 0, which in turn implies that r is a factor of v(r, F). (iii) The following observation is also in order: strictly speaking, there exist situations whence it is not necessary to check that v[E,;‘](~) has no roots in [p,, ~~1; namely, suppose that ZF(&) n (0, x0>#0 and that max (p, S) = 6 with 6 < max {Z’(&)}; then the test involving V[E&A) may be dropped from the necessary and sufficient condition given above since in this case it is already fulfilled by construction of 6. In fact, Case 2 above is a condensed statement of three other cases, which when decomposed yields the more comprehensive statements below. Chsr 2. 5, f 0. 2(i). The univariate polynomials 5, and t2 have no positive real roots. In this case, J(E, P) is stable relative to Q for all sufficiently small E>O and all p E [F,, pL2] if and only if it is stable relative to SL for an arbitrary chosen pair (E, I*) E (0, a) x [p, , pLz] and the univariate polynomial v[E](~) has no roots in [pi, ~~1 for an arbitrary chosen value of E > 0. In this case, E* = +m. 2(ii). The univariate polynomials 5, has no
New results for singularly perturbed positive real roots, & has positive roots rl, rz,-.., r[ and the univariate polynomial v[r](p) has no roots in [pl, ~~1 for all values r = ri, i=l,... , 1. Then J(E, CL) is stable relative to Q for all sufficiently small E > 0 and all k E [pr, ~~1 if and only if it is stable relative to R for an arbitrary chosen pair (E, p) E (0, m) X [pI, pz]. In this case also E* = +m. 2(iii). Cases 2(i) and 2(ii) do not apply. In this case, J(e, CL) is stable relative to Q for all sufficiently small E > 0 and all p E [pl, ~~1 if and only if it is stable relative to &Y2 for an arbitrary chosen pair (e, P) E (0, l/max (P, 6)) x [CL,,~~1 and the univariate polynomial Y[E-‘I(p) has no roots in [pl, pZ] for an arbitrary chosen value of E E (0, l/max (p, 6)). In this case, E* is finite and is given by (29). Corollary 3 below clearly delineates the stable situations leading to an infinite E* from those leading to a finite one, and follows directly from the above statements. Corollary 3. Let Q be such that Y(Q) is endowed with a polynomic guardian map vn and let v be as in (23). Then J(E, CL)is stable relative to Q for all E > 0 and all p E [pr, pZ] if and only if
(i) it is stable relative to R for an arbitrary chosen pair (E, P) E (0, ml X [pl, CL*]; (ii) p = 6 = 0; and (iii) v[e-‘I(*) has no roots in [ccl, pL2]for some arbitrarily chosen value of E > 0.t Example 4.1. Let
f, = -X1 + (1+ /.Z)z, i2
=
(p
-
Ei
=
x1
+
2)x, x2
-
(1
+
p)x*
+
2,
(1+ j&)z,
where p E [-1, 11. In this case
J(r,p)=
-1
0
p-2
-(l+p)
r
r
(
1+ /.L* 1
,
-r(l + PL)
It can be easily verified that for the nominal value p = 0, this system is Hun&z-stable no matter how large E becomes. The question of interest is to see how the variations of p will impact on this fact. For the open left-half plane we obtain v(r, p) =: vl(r, p) v2(r, p), where vl(r, p) = r(1 + 2~* - 2~~) and v2(r, p) = 2 + 3~ + p22 + 2p(4 + p + p*)r + 3&l + F)r*. It is t The latter condition needs to be checked only in the event that T+(&) = 0, for otherwise if 6 = 0 and ST+(&) # 0 then it is already fultilled by construction of 6.
815
systems
easily verified that vr does not vanish for any (r, p) E (0, +m) X [p.,, p2], so that only v2 needs to be considered in the sequel in lieu of v. We have tl(r) = v2(r, -l)v2(r,
1) = -48r(r
+ l)*,
b(r) = det B( Q-lb-4 $ vz[rl(~)) = 4r*(l+
36r - 956r’ - 26283
- 4218r4 - 3108r5 - 540r6 + 324r’ + 81r*), L?Z(5,)= (0, - 1) (hence p = 0) and
%(5,) = (-3.95366,
-0.0188325, 0, 0, 0.0506569,
3.29928, -1.358 f 0.402j, -0.331 f 0.621j). To determine 6, we need to investigate the stability of v[r](*) for r E Z’(5,) = (0.0506569, 3.29928}=:{rI, r2} relative to E:=C\[-l,l]. The real zeros of v[r2](p) are (-3.03377, -3.03377, -0.0329318) (one of which is multiple, as expected). Since these intersect [-1, 11, it follows that v[r2](p) is not stable relative to 8; hence 6 = r2 = 3.29928 and max (p, 6) = 3.29928. From Theorem 6, it follows that the overall system is Hurwitz-stable for all E > 0 small enough and all I_LE [- 1, l] if and only if J(rO, pO) is Hurwitz-stable and v2[r0](p) has no zeros in [-1, l] for an arbitrary choice of (rO, pO) E (3.29928, +m) X [-1, 11. It turns out that while for r. = 5 and p. = 0 the eigenvalues of J(5,O) are (-6.95954, -0.0202285 f 0.847366j}, implying nominal Hurwitz stability, the zeros of v2[ro](p) = 2 + 118~ + 86~~ + 10~~ are given by {- 14.7524, -1.29239, -0.005245); hence the above system fails to be stable for small enough E > 0, since -0.005245 E [ - 1, 11.Note, however, that an entirely different conclusion is reached if p is restricted to [0,11,namely that the system is stable for all E > 0 and all p E [0, l] (i.e. E* = m). To see this, note that (i) J(5,O) is Hurwitz(ii) p = 0 still since t,(r) = stable, v2(r, O)v2(r, 1) = 2(6 + 12r + 6r*) has no positive zeros and S = 0 since neither v2[r2](p) nor v2[rI](p) have zeros in [0, l] (the latter polynomial having (-5, 0.9257, -5.09257, -0.76118) for zeros); hence the result follows from Corollary 3. 5. PROOFS
Proof of Theorem 3. Case 1: v identically zero. In this case, the matrix J(r) is unstable relative to Q for each r > 0. This follows immediately, since v guards
Y(Q)*
816
L. Saydy
Case 2: v not identically zero. It follows that v(r) has finitely many zeros. If the polynomial v(r) has no positive real zeros then J(r) does not cross X?‘(Q) as r varies in (0, +m). Thus the family {J(r) : r E (0, + 03)) lies entirely within either Y(Q) or Y(Q). To determine which situation prevails, it suffices to test whether J(r) E Y’(Q) or J(r) E Y’“‘(Q) for an arbitrarily chosen r in (0, +m). If, on the other hand, v(r) has 12 1 real positive zeros 0 < (Y,< . . < a/ then Theorem 1 implies that J(r) E 9’(Q) for all r > (Y[if and only if J(r) E Y(Q) for an arbitrarily chosen r > (Y,. It is also clear that, in this case, the largest neighborhood of +m in which J(r) E Y(Q) is ( (Y,,+m), which translates into (0, l/al) for J(E). Proof of Theorem
4. By a well-known theorem (see e.g. Marden, 1949), all the zeros of the polynomial (19) lie within the disc in the complex plane centered at the origin and of radius
R:=l
+maxJYil i+? Ivql’
implying that v(r) := v&(r)) does not vanish for all r > R. Theorem 1 therefore implies that generalized stability of the matrix J(r) at an arbitrary r > R is equivalent to its generalized stability for all sufficiently large r. 0 Proof of Theorem 5. It follows directly from the discussion following Theorem 5 that if J(r) is singular and/or has opposite eigenvalues for all r > 0 then the system (1) is Hurwitz-unstable for all E > 0, since, in that case, vJJ(r)] = 0. Next, letting n = r - r”, we can write v&(r)]
= det (Jo + rJ,) det (Jo + t-j,) = det [J(rO) + v./,] det [J(r”) + d,]
(the size of ii?; see the Appendix). It follows from (30) that the (finite) zeros of q(v) are given by 9’(q) = {A-’ : A E a(M) U a@); A # 0) = (8(M) u a-‘(fi))\{~}. By defining n* to be the largest real zero of q in [-r’,+m) (note that -r” E T(q), since *.= v,[J(O)] = det Jo det Jo = 0), ‘. . max {(a-‘(M) U fF’(ti)) fl [-r”, Ya)], iY ’ is seen that q(v) # 0 for all 77E (v*, co), while q(v*) = 0. In terms of vQ, this amounts to saying that v&(r)] ZO for all r E (r*, co), while v&l(r)] = 0 for r = r*. Combining this with Theorem 1, it is seen that J(r) is Hurwitz-stable for r large enough if and only if it is Hurwitz-stable for a particular value of r E (r*, m). Furthermore, r* is the largest value of r for which the above assertion fails to hold. The corresponding statement in terms of l yields the result of Theorem 5. 0 Proof
of Theorem 6. The result of Case l(i) follows from the fact that v, is a guardian map for Y(Q). Indeed, it is clear from Definition 1 that if ~$1 = 0 then J @Y(Q). Similarly to Case l(i), l,(r):= V(E, p,)v(e, pZ) =O implies that J(E, 11,) or/and J(E, p2) are unstable relative to !2 for all E > 0. Suppose now that t,(r) f 0. Then this implies that the polynomial 5, has finitely many zeros. It also follows that V(E, p) f 0. Thus & does not vanish identically, since D[r](p), being the square-free version of v[r](p), has no multiple roots for all values of r (recall that & is the determinant of the Bezoutian of Q and its derivative with respect to II, and is identically zero if and only if Q has multiple roots for all r; see the Appendix). Therefore 6 is well defined. Since Y(Q) is guarded by vn, Theorem 1 implies that J(r, p) E Y(R) for all r E (r*, 03) and p E [p,, puz] if and only if J(ro, po) E Y(Q) for some (ro, PO) E (r*, m) X [PI, 1.4 and
v(r, P) # 0 for all (c II) E (r*, WIX[h, 1.4.
= det J(rO) det j(r’) det (I - +4) xdet(Z-nfi) =:4(n), where, with a slight abuse of notation, I denotes the identity matrix with the appropriate dimension. Thus n E (-r”, + m) for r > 0, and equals 0 when r = r”; hence q(v) f0, since q(0) = det J(r”) det J(r”) is nonzero by hypothesis. Assuming now that n # 0, we have
(Note that r* < +m, by definition.) Clearly, saying that v(r, p) #O for all r E (r*, M) and p E [p,, p2] is equivalent to saying that v[r](p) E Y’(Z) for all r E (r*, ~0). Hence J(r, p) E Y(Q) for all r E (r*, 03) and F E [CL,,pZ] if and only if J(ro, po) E Y(Q) for some (ro, po) E (r*, ~0)X IPI, 1.4 and v[rI(cL) E 33 for all r E (r*, M). Recalling that the set Y(E) is semiguarded for polynomials by the map v: p *P(~i)&J
q(77) = 77N+n+mdet J(r’) det J(r’) X
where
A:= l/n
det (AZ - M) det (AZ - a), and
(30)
N = i(n + m)(n + m - 1)
det R(p(s), $p(.r))
we obtain by virtue of Corollary 1 that the one-parameter family of polynomials v[r](p) E Y(Q)
New results for singularly perturbed for all r E (r*, + m) if and only if;pl’(“’ ; Y(Ej for some r E r*, r0 E (r*, +m), m):&(r)=0}=0 and v[r](p) e’&‘(S) for all r E 12, = (r E (r *, + m) : &(r) = 0). By construction of p and S (and hence r*), Uk = 0 and v[r](p) E 9’(Z) for all r E Uzr. Therefore, in order for the statement v[r](p) E 9’(E) for all r E (r*, m) to hold, it is, owing to the special construction of r*, necessary and sufficient that v[ro](p) E Y(Z) for some r. E (r*, m) only. Combining the above arguments, we conclude that J(r, p) E 3’(Q) for all r E (r*, m) and P E [CL,,~~1 if and only if J(ro, po) E 9’(Q) for some (ro, PO) E (r*, m) X [PI, ~21and 4rol(~) E Y(E) for some r. E (r*, m). This establishes the first part of Case 2 in Theorem 6. To complete the proof, suppose that J(r0,
PO)
E WV
for
SOme
(r0,
po)
E (r*,
m)
X
and v[r](p) E 9’(Z) for some r E (r*, m), say r = r,, so that J(r, CL) (respectively J(e, P)) E y(Q) for all (r, P) E (r*, m) X [k, 14 (respectively (e, p) E (0, E*) X [pr, ~~1). Suppose momentarily that r* > 0 (i.e. E* < m). Then, owing to its construction, r*(e*) is the largest (smallest) positive value of r(e) for which the above conclusion does not hold; i.e. J(r*, p) # [pL1,
y(Q)
pcLz]
(J(E*, CL) e y(Q))
for SOme CLE [PI, 1.4,
while .J(r, p) E 9’(Q) (J(E, p) E Y(Q)) for all r > r* (E < E*) and p E [pI, ~~1. Finally, if r* = 0 then (26)-(28) imply that both p and 6 are 0, which in turn means, by virtue of Corollary 1, that v[r](p) E 9’(Q) for all r E (0, m), i.e. that v(r, P) # 0 for all (5 CL) E (0, m) X [h , 1.4. It follows from Theorem 1 that J(r, p) E Y(Q) for all (r, p) E (0, m) X [p,, p*J, or, in terms of E, that J(E, CL)E P’(Q) for all (E, CL)E (0, m) X [pI, pz], i.e. E* = +m and (29) applies in this case as well. 0 Acknowledgements-The author thanks Professors E. H. Abed and A. L. Tits of the Institute for Systems Research, University of Maryland and Harvard University for their help and Professor G. L. Blankenship of the same institution for raising a question that led to the consideration of this application of guardian maps. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant NSERC-OGPO122106 and le Fonds FCAR under Grant FCAR NC1 193. REFERENCES Abed, E. (1985). A new parameter estimate in singular perturbation. Syst. Control Lett., 3, 193-198. Abed, E., L. Saydy and A. Tits (1989). Generalized stability of linear singularly perturbed systems including calculation of maximal parameter range. In M. A. Kaashoek, J. H. van Schuppen and A. C. M. Rand (Eds), Robust Control of Linear Systems and Nonlinear Control, pp. 191-203. Birkhluser, Base]. Barmish, B. (1994). New Tools for Robustness of Linear Systems. Macmillan, New York. Bose, N. (1982). Applied Multidimensional Systems Theory. Van Nostrand Reinhold, New York. Chow J. H. (1978). Asymptotic stability of a class of
systems
817
non-linear singularly perturbed systems. J. Franklin Inst., 305,27X281. Desoer, C. and M. Shensa (1970). Networks with small and very large parasitics: natural frequencies and stability. Proc. IEEE, 58,1933-1938. Feng, W. (1988). Characterization and computation for the bound Q* in. linear time-invariant singuiarfy perturbed. Svst. Control Lett.. 11. 195-202. Javid, S. (1978). Uniform asymptotic stability of linear time-varying singularly perturbed systems. J. Franklin Inst., 30&U-37. Khahl, H. (1981). Asymptotic stability of multiparameter singularly perturbed systems. Automatica, 17, 797-804. Klimushev, A. and N. Krasovskii (1962). Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms. Appl. Math. Mech., 25, 1011-1025. Kokotovic, P., H. Khalil and J. O’Reilly (1986). Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London. Krishnan. H. and N. McCIamroch (1994). On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form. IEEE Trans. Autom. Control, AC-39,1079-1084. Marden, M. (1949). The geometry of the Zeros of a Polynomial in a Complex Variable. American Mathematical Society, Providence, RI. Mustafa, D. (1994). How much integral action can a system tolerate? Lin. Algebra Applies, 205/206,%5-970. Sandell, N. (1979). Robust stability of systems with application to singular perturbations. Automatica, 15, 467-470. Saydy, L. (1988). Studies in robust stability. PhD thesis, University of Maryland. Saydy, L., A. Tits and E. Abed (1990). Robust stability of linear systems relative to guarded domains. In P. Dorato and R. Yedavalli (Eds), Recent Advances in Robust Control, pp. 131-138. IEEE Press, New York. Saydy, L., A. Tits and E. Abed (1991). Guardian maps and the generalized stability of parametrized families of matrices and polynomial. Math. Control, Sig. Syst., 3, 345-371. Zien, L. (1973). An upper bound for the singular parameter in a stable, singularly perturbed system. J. Franklin Inst., 295,373-381.
6. APPENDIX In the following two sections we present the bialtemate product and the Bezoutian, and briefly discuss relevant properties (for more details the reader is referred to Saydy et al. (1991) and the references therein).
Bialternate product Let A and B be n X n matrices. To introduce the bialternate product of A and B, we first establish some notation. Let V” be the $n(n - 1)-tuple consisting of pairs of integers (p, q), p = 2,3,. . , n, q = 1,. , p - 1, listed lexicographically. That is, V” = [(2, l), (3, l), (3,2), (4, l), (4,2), (4,3),. . , (n, n - l)]. Denote by V: the ith entry of V”. Denote
where the dependence off on A and B is kept implicit simplicity. The bialternate product of A and B, denoted A 0 B, is a :n(n - 1)-dimensional square matrix whose entry is given by (A OB), = f(V:; V;). For example bialtemate product of a 3 X 3 matrix A = (a,-) with identity matrix is a 3 X 3 matrix given by
for by ijth the the
818
L. Saydy
The bialtemate product has some properties. In particular, if u(A) = {A,, a(2(AOI))={A,
+&,A,
+A, ,...
interesting spectral , A,} then ,A,
iA,,...,A,+A,,...,A,~,
+A,,A, +A,}
and (r(2(A201-AOA)) = {(A, - A#, (A, - A#,
, (A, -
A,,)'.
(A*-A1)2,...,(A2-A,)2,...,(A,~1
-A,,)'}.
2(A 0 I) (respectively 2(A2 0 I - A 0 A)) become singular if and only if A has opposite (respectively multiple) eigenvalues. Thus
The Bezoutian
Given any polynomial a(s) = a,? + + a,s + a”, a, # 0, define the polynomial a(s):=s”a(sc’) = a,,s” + + a,, ,s + a,andthenXnmatrixS(a)byS,(a)=a,+,_,ifi+j-lsn, 0 otherwise. The Bezoutian B(a, b) of two polynomials a and h may then be expressed as the n X n matrix, n being the largest-of the degrees of a and b, given by B(a, b):= S(a)S(b)P - S(b)S(li)P, where P is the permutation matrix specified by P,, = 1 for all i, j such that i + j - 1 = n, 0 otherwise. Our interest in the Bezoutian stems from the following fact. The polynomials a(s) and b(s) have no common zeros if and only if the associated_Bezoutian B(a, b) equivalently, the matrix S(s)S(b) - S(b)S(a^)) is (or, nonsingular.
PII: sooo5-1098(%)ooo11-8
Aukmuric~, Vol. 32, No. 6, PP. W-818, 19% Copyright Q 19% Elswier Science Ltd Printed in Great Britain. Ail rights reserved ixws-1098/% s15.00
t 0.00
New Stability/Performance Results for Singularly Perturbed Systems* LAHCEN Performance
SAYDYt
of singularly perturbed systems is investigated.
Key Words-Linear systems; singular perturbations; robustness; stability and performance; composite feedback; state space; stability upper bound; generalized stability.
Almlraet-The guardian map theory of generalized stability of parametrized linear time-invariant systems is used to prove new results on stability/performance of linear time-invariant singularly perturbed systems with an exogenous parameter, i.e. systems that contain, in addition to a singular parameter E, another uncertain parameter CL.The results give necessary and sufficient conditions for generalized stability for all sutIiciently small values of c and all values of p in a given interval [pi, pr]. In addition, explicit expressions for the largest upper bound l* on c for guaranteed stability and performance are given and the cases leading to finite or infinite E* are clearly delineated. Thus the results represent a significant addition to the classical Klimushev-Krasovskii theorem, while at the same time providing closed-form expressions for the maximum parameter range for stability and performance. Copyright 0 1996 Elsevier Science Ltd.
Klimushev and Krasovskii below is a classical result giving a condition for the asymptotic stability of (l), (2) for all sufficiently small values of the singular perturbation parameter E, and has been derived by several authors (Klimushev and Krasovskii, 1962; Desoer and Shensa 1970): Theorem 1.1. Let D be nonsingular. If the matrices A,:= A - PD-‘C and D are Hurwitzstable then the overall system (l), (2) is Hurwitz-stable for all sufficiently small E > 0, i.e. there exists an E> 0 such (l), (2) is Hurwitzstable for all E E (0, E].
1. INTRODUCTION
Consider the singularly perturbed
Note that this result only yields a sufficient condition (subsequently referred to as the KK condition) for stability of (l), (2) for all sufficiently small values of 6. In fact, the condition of KK is necessary and sufficient for Hurwitz stability of (l), (2) for E > 0 small enough when D has no eigenvalues on the j, axis.
linear system
i=Ax+Bz, E.?= Cx + Dz
(1) (2)
in which E > 0 is a small real parameter, x and z are vectors in R” and R” respectively. Owing to the presence of the small parameter l, the dynamics of (l), (2) naturally separates into slow and fast dynamics thus leading to the consideration of the reduced-order (slow) subsystem (D is assumed invertible) 1, = (A - BD-‘C)x,
=:Ags,
(3)
which evolves in R” instead of R”‘“. one seeks to infer some aspects of behavior of (l), (2) from similar the reduced-order system (3). The interest to us here are the stability the performance of (l), (2). The
Generally, the general aspects of aspects of as well as theorem of
Theorem 1.2. If D is nonsingular with no pure imaginary eigenvalues then the system (l), (2) is Hurwitz-stable for all E > 0 small enough if and only if D and A,, are Hurwitz-stable matrices. This follows from the fact that the spectrum of the
matrix
J(E) = (C;e
D:E >
approaches
{AI(&), . . . , MAO), &(D)le . . . , L(D)l~~
as
E -+ 0 (see Kokotovic et al., 1986). Clearly these results shed no light on what appears in the critical situation when D is nonsingular but merely marginally Hurwitzstable. Furthermore (and more importantly), they give no precise quantitative meaning to how small E should be for the above conclusions to hold.
*Received 11 July 1994, revised 1 June 1995; received in final form 1 June 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor J. M. Dion under the direction of Editor Ruth F. Curtain. Corresponding author Professor Lahcen Saydy. Tel. +l 514 340 4233; Fax +l 514 340 407% E-mail [email protected]. t Department of Electrical and Computer Engineering. l?cole Polytechnique de Montreal, Canada. go7
808
L. Saydy
From a design viewpoint, one considers singularly perturbed control system i =A,,x
+A,g
+B,u,
EZ = A2,x + Az2z + B2u,
the
(4) (5)
which for small E > 0 leads to the consideration of two reduced-order subsystems evolving on different time scales, namely, the slow subsystem 1, = &x, + &us,
(6)
where &=A,,
-A,,&;&,
B. = B, - A,2A;;B2,
(7)
and the fast subsystem ir = A22zf + Bzuf.
(8)
This two-time-scale separation into slow and fast subsystems makes it possible for the designer to procede by designing separate feedback laws for (6) and (8) and using a composite one to control the overall system (4), (5). More precisely, if u, = G#, and uf = G2zf are such that a(A,, BOG,) = {a,, . . . , a,} and a(A,, - B2G2) = {p,, . , pm} then it is well known that the composite feedback law (Kokotovic et al., 1986) u = (Go + G,)x +
Gz, (9)
G, = GzA;2’&Go
assigns the eigenvalues of the closed-loop system i=Ax+Bz,
(10)
6.j = Cx + Dz
(11)
(with A = A0 + B,(Go + G,), B = Al2 + B1G2, C = A21 + B,(Go + G,) and D = AZ2 - B2G2) to the locations {(Y,,. . . , a,, PI/e,. . . , &/E} for E small enough. In particular, if AZ2 is Hurwitzstable (stable fast dynamics) and {/?,, . . , p,,,} = g(Az2) then the feedback law u = G,x (obtained by setting G2 = 0 in (9)) accomplishes the goal above (using the slow variables only). Suppose that for practical considerations
Fig. 1. The region
$2.
relating to damping ratio, stability margin etc. it is desired to place the eigenvalues of the closed-loop system {al, . . . , a,, PI/e,. . . , &JE} in a specific subregion R of the open left-half complex plane (e.g. the one in Fig. 1). Then the composite feedback law (9) guarantees that the performance of the closed-loop loop system (lo), (11) is specified by that of the slow subsystem when E is small enough, since as ~-0 the closed-loop loop eigenvalues tend to {(Y,, , a,} (the remaining eigenvalues becoming unbounded in R along the directions specified by {PI, . , pm}). Thus for this two-time-scale composite design to work it is expected that E be small enough. The qualitative nature of this requirement represents quite a drawback from a design viewpoint, particularly when E is unknown. It is therefore important to be able to ascertain how small E should be for the overall system to retain the desired performance, namely that the eigenvalues of the closed-loop loop system (lo), (11) remain within R. Furthermore, it is not clear in the light of the above discussion what happens when mere stabilization of the closed-loop loop system is sought (i.e. Q coincides with open left-half complex plane) and the uncontrollable modes of the fast subsystem (A**, B2) happens to be purely imaginary. Since the parameter E typically represents a small physical quantity over which one has little or no control, it is of practical significance to find explicit, nonconservative, upper bounds lo on E ensuring (under minimal assumptions) that the above stability/performance conclusions are valid for all E E (0, eo] or, if possible, find the largest such upper bound E* : = sup {e. > 0: the system (l), (2) is stable in a generalized sense for all E E (0, ~~1). (12) Here generalized stability means confinement of matrix eigenvalues or polynomial zeros to general open subsets of the complex plane, and includes Hurwitz stability as a special case. Practical considerations relating to damping ratio, bandwidth etc. are commonly expressed in terms of the generalized stability formulation, with respect to a suitable domain in the complex plane. Previous results on upper-bound estimates lo for Hurwitz stability have been obtained by several authors, including Zien (1973), Javid (1978), Sandell (1979), Chow (1978), Khalil (1981) and Abed (1985). Feng (1988) used frequency-domain stability analysis to characterize the maximum parameter range on E for Hurwitz stability of (l), (2) under the hypotheses
New results for singularly perturbed and Krasovskii’s result. A of Klimushev necessary and sufficient condition for (general-
ized) stability of (1) for all sufficiently small values of the singular perturbation parameter E together with the closed-form computation of the largest upper bound E* on E for guaranteed stability was given for the first time in Abed et al. (1989). These results were obtained using the guardian map approach to the study of generalized stability of parametrized families of linear systems (Saydy, 1988; Saydy et al., 1990, 1991). In the present work, we present some new results pertaining to singularly perturbed systems given by (l), (2), and, utilizing guardian and semiguardian maps, investigate similar issues for ‘uncertain’ singularly perturbed linear systems with a regular perturbation parameter p (subsequently referred to as the exogenous parameter), i.e. singularly perturbed systems of the form i = A(p)x
+ B(p)z,
(13)
Ei = C&)x
+ D(p)z,
(14)
where p is an uncertain (possibly large) parameter known only to lie within an interval, say I_LE [pl, ~~1 with pl, p2 given scalars, and A(p), . . . , D(p) are polynomial functions of CL, i.e. A(p) = C;I” Aipi, . . . , D(p) = E:;f” Dip’. The approach presented can handle more general models (13), (14) for which these matrices are allowed to depend polynomially on E as well. This, however, is not done explicitly in the sequel for the sake of clarity. In this article, we shall give definite complete answers to the following two problems: (P,) obtain necessary and sufficient conditions for (generalized) stability of (13), (14) for all sufficiently small values of E > 0 and all p E [CL17 1121;~ (P2) obtain an explicit expression for the largest upper bound E* such that (13), (14) is stable for all E E (0, E*) and all p E [p,, p2], and clearly delineate those cases for which E* is infinite from those for which it is finite. The results in this article make it possible to analyse the stability and performance of very general closed-loop singularly perturbed systems. In particular, they apply to singularly perturbed systems in standard form as well as those not in standard form. Systems in the latter category have received increased attention in the literature recently (see e.g. Krishnan and McClamroch, 1994). Finally, note that since we seek the maximum upper bound E* for the closed-loop system to
809
systems
retain stability and performance of some kind, the values of E* with which we end up might turn out to be not so small (sometimes even infinite). As such, the calculation of E* serves only one purpose, namely it guarantees that the closed-loop system (13), (14) is well behaved in the generalized stability sense for all E E (0, E*) (and in no larger interval) and hence that the composite feedback (9), which we know works well for small enough 6, does actually work for all E E (0, E*). In particular, no information is given as to whether or not the two-time-scale separation is preserved in closed loop as E approaches E*. It is the view of this author that this issue can in fact also be tackled using the guardian map framework. This question will not, however, be pursued here. The paper is organized as follows. In Section 2, the concept of guardian and semiguardian maps is recalled and relevant results are given. In Section 3, the case of singularly perturbed systems with no exogenous parameter is considered and an example is given. In Section 4, the main results relating to problems (Pi) and (P2) above are given, together with an example. Finally, Section 5 contains the proofs of the results given in Sections 3 and 4. The following notation is used throughout the article. @ denotes the set of complex numbers, 6 the open left-half plane, G the closure of set 9, Q a generic open subset of the complex plane, symmetric with respect to the real axis, aR its boundary, Yn(Q) the set of all n x n real matrices (or nth-order real polynomials, depending on the context) with eigenvalues (zeros) in R, A 0 B the bialternate product of matrices A and B, 6!&the set of &h-order polynomials with real coefficients, Z(p) the zeros of a polynomial p, B(p, q) the Bezoutian matrix of polynomials p and q, finally P[r](.) (or simply p [r]) denotes a polynomial in (m) whose coefficients are themselves polynomials in r. 2. GUARDIAN
MAPS
The guardian map approach was introduced in Saydy (1988) and Saydy et al. (1990, 1991) as a unifying tool for the study of generalized stability of parameterized families of matrices or polynomials. For the purpose of making the presentation self-contained, a basic review of the essentials now follows (for more on the subject, the reader is referred to Saydy et al. (1991); see also the recent book by Barmish (1994) for an introduction to the subject). Definition 1. Let X be the set of all n x n real
matrices, or the set of all polynomials
of degree
810
L. Saydy
n with real coefficients, and let Y be an open subset of X. Let Y map X into @. The map Y is said to be a semiguardian map for Y if for all x E 9, the implication Y(X) = 0
whenever
x E a,!+
holds. If the converse also holds, v is said to be a guardian map for Y. Furthermore, we say that the map v is polynomic if it is a polynomial function in the entries/coefficients of its argument. For the purposes of this paper, the set .Y will be a (generalized) stability set, i.e. a set of the form Y(Q) := {A E lRnx”: u(A) c !G?}, for matrix stability problems, and (with a slight abuse of notation) Y(Q):= {p E Pn:%(p) =Q} for polynomial stability problems. Here Q is an open subset of the complex plane that is symmetric with respect to the real axis, and an element of Y(Q) is said to be stable relative to CL Two examples of interest in later sections are given next. For more examples of guarded sets Y(Q) and systematic techniques to construct the associated guardian maps, the reader is referred to Saydy et al. (1991), where it is shown that in fact, many stability sets of practical interest enjoy the guardedness property with polynomic corresponding guardian or semiguardian maps. For example, any stability set for which Q={s =x +jy:p(x,y)
v: A H det (A) det 2(A 0 I)
(15)
guards the set of n x n Hurwitz-stable real matrices S(c -). For example, for a 2 X 2 matrix A, the bialternate product A 0 I = ;(a,, + u12), so that v(A) = det A trace (A), and vanishes if and only if A is singular or has opposite eigenvalues (trace (A) being equal to h,(A) + h,(A)). A similar reasoning holds for general matrices, since det [2(A 0 Z)] vanishes if and only eigenvalues (see the if A has opposite Appendix).
in the case of polynomials. Theorems 1 and 2 below give fundamental necessary and sufficient conditions for stability of parametrized families of matrices or polynomials relative to domains of the complex plane corresponding to guarded and semiguarded stability sets respectively. Let r= (r,, . . , rk) E U be a vector of unknown parameters, where I!/ is a pathwiseconnected subset of R“, and let x(r) be a matrix in R”“” or a polynomial in P?,, that depends continuously on the parameter vector r. Given an open subset Q with guarded or semiguarded stability set Y’(Q), we seek basic conditions for x(r) to lie within Y(Q) for all values of r in U. Theorem 1. Let Y(R) be guarded by the map v. Then the family {x(r):r E U} is stable relative to R if and only if
(i) it is nominally stable, i.e. x(r”) E Y(Q) some r0 E U; and (,ii) v(x(r)) #O for all r E U.
Theorem 2. Let Y(Q) be semiguarded by the map v. The family {x(r):r E U} is stable relative to Q if and only if
(i) it is nominally stable, i.e. x(r’) E Y(Q) for some r0 E U; and (ii) x(r) E 9’(Q) for all r E UC,, where Uo:= {r E II: v(x(r))
= O}.
The above two results are illustrated in Figs 2 and 3. In the guarded case, a guardian map v evaluated along any given family {x(r): r E U} remains nonzero as long as that family is stable, and vanishes precisely when it exits the stability set Y(Q). In the semiguarded case, the vanishing of v no longer indicates unequivocally that a given family has exited the stability set for there could exist stable elements (so-called blind spots; see Saydy et al., 1991) for which v vanishes. In the case of @\ [a, p], it is seen from (16) that these
Example 2.2. Let Q = @\[a, p] (i.e. the set of all complex numbers that do not lie in the closed interval [a, p]). Then the set Y(Q) is semiguarded by
v: p ~p(~)p(P)
det B(p(s),
$,(s)j (16)
for
Fig. 2. The guarded
case.
New results for singularly perturbed
811
systems
3. SINGULARLY PERTURBED SYSTEMS WITH NO EXOGENOUS PARAMETER
For the system described by (l), (2), define the matrix
J(r) =
Fig. 3. The semiguarded case.
are any polynomials with no zeros in [CX,p] (i.e. stable) and at least one multiple zero, since in this case the Bezoutian of p and its derivative is singular (see the Appendix). For the specific domain @\]P1, Pzl, subsequently denoted by 5, we have the following corollary. Corollary 1. Let E := @\[pl, ~~1. Then the family of polynomials {p[r]:r E U} is stable relative to Z if and only if
(i) it is nominally stable, i.e. p[r’] E 9’(Z) for some r” E U; and (ii) I& = 0 where I-&:={r e
and p[r] E 9’(Z)
~:Arl(dWcL2)
for
=O}.
Remark
1. Condition (ii) above requires testing the stability of p[r] relative to B for all critical values of r E 172. This is feasible only when cr is a finite set. In degenerate cases, the polynomial det B(p[r](s), (d/ds)p[r](s)) vanishes identically, leading to such a situati0n.t This indicates that the polynomials p[r](s) and (dlds)p[r](s) have a common zero for all r. To prevent such a situation from happening, all one needs to do is extract the greatest common factor of p[r](s) and (d/ds)p[r](s), and then proceed. This is equivalent to computing Uz, according to Uzr:=[r
E U:detB(lj[r](s),%p[r](s))
v(r) = v. + vlr + . . . + v4-,r4-’
+ vqr9,
(19)
where q denotes the degree of the polynomial.
= 01, -$p[r](s))
(18)
9
>
where r:= l-’ is large when E is small. The stability of (l), (2) for all sufficiently small values of E is identical to the stability of the matrix J(r) for all sufficiently large values of r. Theorem 3 below presents necessary and sufficient conditions for the system (l), (2) to be stable for all sufficiently small values of the parameter of E >O relative to a given open subset R of the complex plane for which .S@(Q)is endowed with a polynomic guardian map vn. Furthermore, Theorem 3 also gives the largest upper bound E* strictly below of which stability is still guaranteed (no prior assumptions are made on the system; in particular, the conditions of Klimushev and Krasovskii need not hold, and the results apply to singularly perturbed systems in nonstandard form as well). Let v(r):= v&(r)). Since vn is polynomial in its argument, and J(r) depends linearly on r, we can write v(r) as a polynomial in r:
all r E IJf,,
(17) Uzr:= {r E U:det B(p[r](s),
(pCrt
=O),
where p denotes the square-free version of p and is given by B =plGCF(p, dplds). A method for extracting GCFs when the scalar coefficients involved belong to a unique factorization domain is given in Bose (1982). t This would happen if redundant guardian maps are used; e.g. det (A CBA) in lieu of det (A) det (A OA).
Theorem 3. Let &2be such that Y”(Q) is guarded by a polynomic map v, and let v(r) be as in (19).
(4
If v(r) vanishes identically then the singularly perturbed system (l), (2) is unstable relative to Q for all e > 0. v(r) does not vanish identically then the singularly perturbed system (l), (2) is stable relative to R for all sufficiently small E > 0 if and only if it is stable relative to R for an arbitrarily chosen O< e < l/r*, where r* denotes the largest positive real root (if any, otherwise set r* = 0) of v(r). In this case l* = l/r*.
(b) If
Proof.
see Section 5.
Noting that E* = +w if v(r) has no positive roots, it is easily seen that case (b) in the above corollary is in fact a condensed statement of the following two cases: (b,) If v(r) does not vanish identically and has no positive real zeros then (l), (2) is stable relative to &2for all sufficiently small e > 0 if and only if it is stable relative to R for an
812
L. Saydy arbitrarily E* = +m.
(b)
chosen E > 0. In the latter case,
Let v(r) have a largest positive real zero (Y/. Then (l), (2) is stable relative to Q for all sufficiently small E > 0 if and only if it is stable relative to Q for an arbitrarily chosen E < Icy,. In this case E* is finite and is given by E* = llcx,.
Remark 2. Different expressions for the guardian maps corresponding to a given region Q lead to different expression for v(r) in (19), but these will all, by definition, vanish for the same real values of r. The largest upper bound E* will not therefore depend on which particular guardian map is used for a given region. For example, when Q is the open left-half plane, both v,(A)=det(A@A) and q(A) = det (A) det (A 0 1) guard Y(Q). It can be shown, however, that det (A $ A) = det (A)[det (A 0 Z)]“; hence v2(r) will be a factor of v,(r). A one-shot test
The results of the previous section are conceptually simple and can be implemented easily. The question arises whether or not a simple one-shot test exists by which one can immediately ascertain stability of (1) (2) for all sufficiently small l or the lack thereof, Such a test, recorded as Theorem 4, is given next. It gives necessary and sufficient condition for stability in the form of a one-shot test, namely that the system (l), (2) is stable for all sufficiently small E > 0 if and only if it is stable for one judiciously picked value of E (no assumptions are made on A, B, C and D). This represents a significant complementary addition to Klimushev and Krasovskii’s classical result given earlier. Theorem 4. Let Y(Q) be guarded by a polynomic map Y$)and assume that vn(J(r)) is not identically zero.t Then there exists an F > 0 such that the system (l), (2) is stable relative to Sz for all l E (0, E) if and only if it is stable relative to Q for
E,,: =
1 1 + max Iv,I~~yyl. i<
(20)
Note that Theorem 4 settles the question of stability of the system (l), (2) for E > 0 small enough and yields an estimate of E, namely Q, t Since if it is then, according to Theorem unstable relative to R for all E >O.
3, the system
is
which it turns out is always less than 1. To compute the largest such estimate (i.e. E*), it is required to find the real roots of v(r) as stated in Theorem 3. Nevertheless, both theorems necessitate the computation of v(r). Even though a computer algebra package exists to that end, it is of interest to derive the analogous result of Theorem 4 directly in terms of the data A, B, C and D. i.e. without explicitly computing v(r) (for related work, see Mustafa, 1994). For simplicity, we shall restrict attention to the case of Hurwitz stability, i.e. R coincides with the open left-half complex plane.* Rewrite J(r) = : Jo + rJ, , where Jo = /,=
O , and let j(r):=J(r)OZ cO D C 1
A
(
o
B
o and >
= (&OZ) +
r(J, 0 I) =:JO + r.7,. Theorem 5. Suppose that J(r) is not singular and/or has no opposite eigenvalues for all value of r > O.§ Pick r” > 0 such that J(r’) is nonsingular with no opposite eigenvalues, and let r* := r” + max {(a-‘(M)
U (T -‘(fi)
fl
[-r”, +m)}, (21)
where M:= -J-‘(r’)J, and &I:=J-‘(r’)J,. Then for all the system (l), (2) is Hurwitz-stable sufficiently small values of E > 0 if and only if it is so for an arbitrarily chosen value of E E (0, l/r*). In this case, E* = l/r*. Note that checking the hypothesis of Theorem 5 can be done generally in finitely many steps and generically in one step. Indeed, in the case of Hurwitz stability, vC1 is given by v&(r)] = det J(r) det [2(J(r) 0 Z], and it vanishes precisely when J(r) is singular and/or has opposite eigenvalues (this follows from the fact that the eigenvalues of the matrix J(r) 0 Z are the sums of those of J(r); see the Appendix). Furthermore, yC1is a polynomic guardian map, as evidenced by the defining equation for the bialternate product of two matrices (see the Appendix). It therefore follows that v(r):= v&(r)] is a polynomial in r. Thus the hypothesis of Theorem 5 fails to hold precisely when v(r) vanishes identically, an assertion that can be settled quite easily by picking a randomly generated, say positive, value r” of r, and testing whether v(r’) z 0 or, equivalently, that J(r”) is singular and/or has $ Results analogous to Theorem 5 and Corollary 2 below involving more general regions R will be presented elsewhere. 5 If J(r) is singular and/or has opposite eigenvalues for all r > 0 then the system (l), (2) is Hurwitz-unstable for all E :,
0.
New results for singularly perturbed
813
systems
open-lc-apsystem r__________‘____________________,
Fig. 4. The chemical reactor model.
opposite eigenvalues. If J(r’) is nonsingular with no opposite eigenvalues then certainly the hypothesis of Theorem 5 holds; if not then, with probability one (i.e. generically), the hypothesis fails (Y = 0). Naturally, to be sure, one has to do at most degree (Y) + 1 checks involving different values of r” to (conceptually) completely settle the question (degree (Y) + 1 such tests checks are necessary to prove that Y = 0.) Corollary 2. Let D and A0 be Hurwitz matrices so that the system (l), (2) is Hun&z-stable for all sufficiently small E > 0. Pick lo > 0 such that J(E’) is Hurwitz-stable (or merely nonsingular with no opposite eigenvalues) and let r”:= l/e’. Then the largest stability upper bound is given by E* = l/r*, with r* as in (21).
Next, an example dealing with composite control design for a two-tank chemical reactor is presented.? Example
3.1. In this example we consider the two-tank chemical reactor model (Kokotovic et al., 1986) in Fig. 4, where E is an unknown small time constant; C,, x1 and x2 denote the deviation in desired concentration and the deviations in concentration in the first and second tanks, respectively. Here we investigate the performance of the closed-loop system as typified by the stability region in Q in Fig. 1, with u = - 1 and CX= 1. We adopt the design viewpoint that a critically damped closed-loop system is sought in the ideal case of E = 0 but that a design leading to closed-loop eigenvalues within R is acceptable in a realistic situation of a not-so-small E. A state-space model is given by (4), (5), where
The slow subsystem (6) is specified in this case by A,, = AI, and B. = (0 4)‘. Since the fast dynamics is stable (hl(A22) = h2(A22) = -l), it is possible to synthetize a state feedback based solely on the slow variables xi and x2. It is easily checked that the gains K, = y and K2 = y lead to a critically damped reduced-order subsystem with eigenvalues at (-1, -1). With this choice of K, and K2, the dominant eigenvalues of the closed-loop system will be nearly critically damped if E happens to be sufficiently small. With u = K,xi + K2x2, the closed-loop system (lo), (11) is specified by
;
-f
j
0 -K2r
-KIr
;
-r
-2r I
E* computation.
For the present generalized problem, Y(Q) is guarded by (Saydy et al., 1991) v(A) = v,(A) ~264) dA), with q(A) := det (A + cd), v2(A) := det 2[(A + uZ) 0 I] and
stability
[t(l - a2)AOA
QA):=det
For this example, factors)$ v,(J(r))
- i(l+
a2)A20Z].
we have (modulo
constant
= $r2,
v2(J(r)) = 24 - 204r + 107r2 + 1190r3 - 2950r4 + lOOOr
and v3(J(r)) = r2( -116 - 1450r - 33271r2 - 190 880r3 - 341 200r4 + 957 OOO?+ 3 825 000r6 -6200000r7+1000000r8),
with the following zeros: %vi(J(r)))
= 101,
%(v,(J(r)))
= (-0.360,
%(v3(J(r)))
= (-0.231,
0.138, 0.355
f jO.265,2.462},
B, =
0 0
f j0.171, -0.012
f jO.O64,0.642 f jO.O27,5.471}. ,
A21
1 A22=(:;
-0.151
-2)
=
7 B2=
0
;.
t For an example comparing the present method with an existing one in a Hurwitz stability context see Abed et al. (1989).
It follows that (Ye= 5.4712; hence E* = l/c+ = 0.18. Thus we are guaranteed that the closedloop eigenvalues remain within Q for all #A symbolic manipulation package written in the computer algebra program. Mathematics was used to arrive at all the computations in all the examples presented.
814
L. Saydy
E E (0, 0.18) and exit Q precisely for E = E*. Indeed, the eigenvalues of J(E*) are (-7.7161, -0.7875, -1.5694 f j1.5694}, the last two of which belong to aQ. 4. SINGULARLY PERTURBED SYSTEMS AN EXOGENOUS PARAMETER
Denote
the matrix associated
WITH
with (13), (14)
bY
where r: = e-’ is positive. Theorem 6 below gives a necessary and sufficient condition for the (generalized) stability of the matrix J(r, p) for all p E [pi, ~~1 and all sufficiently large r > 0, or, equivalently, that of the matrix J(E, CL) for all p E [E_L,,p2] and all sufficiently small E >O. If J(E, p) is stable, Theorem 6 also provides the largest upper bound E* on E for which the stability of J(E, F) is guaranteed for all p E [p,, ~1~1. Let R be a subset of the complex plane of interest and suppose that the generalized stability set Y(Q) (for matrices) is endowed with a polynomic guardian map v<). In the case of Hurwitz stability, 52 is the open left-half plane, and one such map is given by (15) in Example 2.1. Let (23)
v(r, P):= v&(r+ P)I, 51(r):= v(r, pi)v(r, &(r):=det
~2)~
(24)
~(Qlrl(p), 2 ~[T)(p)), (25)
where B denotes the Bezoutian and 4 denotes the square-free version of the polynomial q (see Remark 1). Note that v, 5, and & are all polynomials of their arguments, since vi1 is a polynomic guardian map, J(r, p) is a matrix polynomial in r and p, and the Bezoutian is a bilinear function of the coefficients of its arguments. We also formally define the scalar quantities p=max(r:r
E.%,~(~,)}
(26)
6 = max{r E %‘([A: v[rl(k) @SF.)}, 10 otherwise,
(27)
r* = max {p, 6).
(28)
By ‘otherwise’ it is meant that either Z+(e2) = 0 or v[r](p) E P(Z) for all r E 5?(t2). Here .Z~(p):={%(p) n (0, m)} U 0, i.e. the set of real positive zeros of a polynomial p augmented with 0, and Z = C\ [pi, pz]. Thus 6 is the maximum positive zero of tr, say r,, for which v[r,](p), as a polynomial in p, is not in Y(Z), i.e. has zeros in [p,, ~~1. If either t2 has no positive zeros or v[r](p) as a polynomial in /I has no zeros in
[pi, ~~1 for every choice of r equal to a positive root of & then 6 = 0. We are now in position to state the main results of this section. Theorem 6. The following cases present themselves. Case l(i): V(E, p) -0. Then J(e, p) is not stable for all (E, II) E (0, =) X [p,, ~~1. Case l(ii):&,(r) = 0. Then J(E, pi) and/or J(E, pz) are unstable for all E > 0. Case 2: 5, f 0 (hence V(E, p) f 0). Pick q), p,, arbitrarily such that cl0 E [p,, wz] and lt1E (0, l/max (p, 6)). Then J(E, p) E P’(S) for all F E [F, , p2] and E > 0 sufficiently small if and only if J(E,), po) is stable relative to Q and the univariate polynomial v[Es’](~) has no roots in [p,, ~~1. We then have E” zz
1
max (P, 8) ’
(29)
Hemark 3.
(i) Note that, according to the definition of p and S, E* E R, i.e. it is allowed to take the value cc. (ii) In the particular important case of Hurwitz is stability, we point out that r =0 automatically a zero for 5, and t2; thus augmenting Z’([,) with (0) is futile. This follows from the fact that J(r, p) is singular for r = 0, which in turn implies that r is a factor of v(r, F). (iii) The following observation is also in order: strictly speaking, there exist situations whence it is not necessary to check that v[E,;‘](~) has no roots in [p,, ~~1; namely, suppose that ZF(&) n (0, x0>#0 and that max (p, S) = 6 with 6 < max {Z’(&)}; then the test involving V[E&A) may be dropped from the necessary and sufficient condition given above since in this case it is already fulfilled by construction of 6. In fact, Case 2 above is a condensed statement of three other cases, which when decomposed yields the more comprehensive statements below. Chsr 2. 5, f 0. 2(i). The univariate polynomials 5, and t2 have no positive real roots. In this case, J(E, P) is stable relative to Q for all sufficiently small E>O and all p E [F,, pL2] if and only if it is stable relative to SL for an arbitrary chosen pair (E, I*) E (0, a) x [p, , pLz] and the univariate polynomial v[E](~) has no roots in [pi, ~~1 for an arbitrary chosen value of E > 0. In this case, E* = +m. 2(ii). The univariate polynomials 5, has no
New results for singularly perturbed positive real roots, & has positive roots rl, rz,-.., r[ and the univariate polynomial v[r](p) has no roots in [pl, ~~1 for all values r = ri, i=l,... , 1. Then J(E, CL) is stable relative to Q for all sufficiently small E > 0 and all k E [pr, ~~1 if and only if it is stable relative to R for an arbitrary chosen pair (E, p) E (0, m) X [pI, pz]. In this case also E* = +m. 2(iii). Cases 2(i) and 2(ii) do not apply. In this case, J(e, CL) is stable relative to Q for all sufficiently small E > 0 and all p E [pl, ~~1 if and only if it is stable relative to &Y2 for an arbitrary chosen pair (e, P) E (0, l/max (P, 6)) x [CL,,~~1 and the univariate polynomial Y[E-‘I(p) has no roots in [pl, pZ] for an arbitrary chosen value of E E (0, l/max (p, 6)). In this case, E* is finite and is given by (29). Corollary 3 below clearly delineates the stable situations leading to an infinite E* from those leading to a finite one, and follows directly from the above statements. Corollary 3. Let Q be such that Y(Q) is endowed with a polynomic guardian map vn and let v be as in (23). Then J(E, CL)is stable relative to Q for all E > 0 and all p E [pr, pZ] if and only if
(i) it is stable relative to R for an arbitrary chosen pair (E, P) E (0, ml X [pl, CL*]; (ii) p = 6 = 0; and (iii) v[e-‘I(*) has no roots in [ccl, pL2]for some arbitrarily chosen value of E > 0.t Example 4.1. Let
f, = -X1 + (1+ /.Z)z, i2
=
(p
-
Ei
=
x1
+
2)x, x2
-
(1
+
p)x*
+
2,
(1+ j&)z,
where p E [-1, 11. In this case
J(r,p)=
-1
0
p-2
-(l+p)
r
r
(
1+ /.L* 1
,
-r(l + PL)
It can be easily verified that for the nominal value p = 0, this system is Hun&z-stable no matter how large E becomes. The question of interest is to see how the variations of p will impact on this fact. For the open left-half plane we obtain v(r, p) =: vl(r, p) v2(r, p), where vl(r, p) = r(1 + 2~* - 2~~) and v2(r, p) = 2 + 3~ + p22 + 2p(4 + p + p*)r + 3&l + F)r*. It is t The latter condition needs to be checked only in the event that T+(&) = 0, for otherwise if 6 = 0 and ST+(&) # 0 then it is already fultilled by construction of 6.
815
systems
easily verified that vr does not vanish for any (r, p) E (0, +m) X [p.,, p2], so that only v2 needs to be considered in the sequel in lieu of v. We have tl(r) = v2(r, -l)v2(r,
1) = -48r(r
+ l)*,
b(r) = det B( Q-lb-4 $ vz[rl(~)) = 4r*(l+
36r - 956r’ - 26283
- 4218r4 - 3108r5 - 540r6 + 324r’ + 81r*), L?Z(5,)= (0, - 1) (hence p = 0) and
%(5,) = (-3.95366,
-0.0188325, 0, 0, 0.0506569,
3.29928, -1.358 f 0.402j, -0.331 f 0.621j). To determine 6, we need to investigate the stability of v[r](*) for r E Z’(5,) = (0.0506569, 3.29928}=:{rI, r2} relative to E:=C\[-l,l]. The real zeros of v[r2](p) are (-3.03377, -3.03377, -0.0329318) (one of which is multiple, as expected). Since these intersect [-1, 11, it follows that v[r2](p) is not stable relative to 8; hence 6 = r2 = 3.29928 and max (p, 6) = 3.29928. From Theorem 6, it follows that the overall system is Hurwitz-stable for all E > 0 small enough and all I_LE [- 1, l] if and only if J(rO, pO) is Hurwitz-stable and v2[r0](p) has no zeros in [-1, l] for an arbitrary choice of (rO, pO) E (3.29928, +m) X [-1, 11. It turns out that while for r. = 5 and p. = 0 the eigenvalues of J(5,O) are (-6.95954, -0.0202285 f 0.847366j}, implying nominal Hurwitz stability, the zeros of v2[ro](p) = 2 + 118~ + 86~~ + 10~~ are given by {- 14.7524, -1.29239, -0.005245); hence the above system fails to be stable for small enough E > 0, since -0.005245 E [ - 1, 11.Note, however, that an entirely different conclusion is reached if p is restricted to [0,11,namely that the system is stable for all E > 0 and all p E [0, l] (i.e. E* = m). To see this, note that (i) J(5,O) is Hurwitz(ii) p = 0 still since t,(r) = stable, v2(r, O)v2(r, 1) = 2(6 + 12r + 6r*) has no positive zeros and S = 0 since neither v2[r2](p) nor v2[rI](p) have zeros in [0, l] (the latter polynomial having (-5, 0.9257, -5.09257, -0.76118) for zeros); hence the result follows from Corollary 3. 5. PROOFS
Proof of Theorem 3. Case 1: v identically zero. In this case, the matrix J(r) is unstable relative to Q for each r > 0. This follows immediately, since v guards
Y(Q)*
816
L. Saydy
Case 2: v not identically zero. It follows that v(r) has finitely many zeros. If the polynomial v(r) has no positive real zeros then J(r) does not cross X?‘(Q) as r varies in (0, +m). Thus the family {J(r) : r E (0, + 03)) lies entirely within either Y(Q) or Y(Q). To determine which situation prevails, it suffices to test whether J(r) E Y’(Q) or J(r) E Y’“‘(Q) for an arbitrarily chosen r in (0, +m). If, on the other hand, v(r) has 12 1 real positive zeros 0 < (Y,< . . < a/ then Theorem 1 implies that J(r) E 9’(Q) for all r > (Y[if and only if J(r) E Y(Q) for an arbitrarily chosen r > (Y,. It is also clear that, in this case, the largest neighborhood of +m in which J(r) E Y(Q) is ( (Y,,+m), which translates into (0, l/al) for J(E). Proof of Theorem
4. By a well-known theorem (see e.g. Marden, 1949), all the zeros of the polynomial (19) lie within the disc in the complex plane centered at the origin and of radius
R:=l
+maxJYil i+? Ivql’
implying that v(r) := v&(r)) does not vanish for all r > R. Theorem 1 therefore implies that generalized stability of the matrix J(r) at an arbitrary r > R is equivalent to its generalized stability for all sufficiently large r. 0 Proof of Theorem 5. It follows directly from the discussion following Theorem 5 that if J(r) is singular and/or has opposite eigenvalues for all r > 0 then the system (1) is Hurwitz-unstable for all E > 0, since, in that case, vJJ(r)] = 0. Next, letting n = r - r”, we can write v&(r)]
= det (Jo + rJ,) det (Jo + t-j,) = det [J(rO) + v./,] det [J(r”) + d,]
(the size of ii?; see the Appendix). It follows from (30) that the (finite) zeros of q(v) are given by 9’(q) = {A-’ : A E a(M) U a@); A # 0) = (8(M) u a-‘(fi))\{~}. By defining n* to be the largest real zero of q in [-r’,+m) (note that -r” E T(q), since *.= v,[J(O)] = det Jo det Jo = 0), ‘. . max {(a-‘(M) U fF’(ti)) fl [-r”, Ya)], iY ’ is seen that q(v) # 0 for all 77E (v*, co), while q(v*) = 0. In terms of vQ, this amounts to saying that v&(r)] ZO for all r E (r*, co), while v&l(r)] = 0 for r = r*. Combining this with Theorem 1, it is seen that J(r) is Hurwitz-stable for r large enough if and only if it is Hurwitz-stable for a particular value of r E (r*, m). Furthermore, r* is the largest value of r for which the above assertion fails to hold. The corresponding statement in terms of l yields the result of Theorem 5. 0 Proof
of Theorem 6. The result of Case l(i) follows from the fact that v, is a guardian map for Y(Q). Indeed, it is clear from Definition 1 that if ~$1 = 0 then J @Y(Q). Similarly to Case l(i), l,(r):= V(E, p,)v(e, pZ) =O implies that J(E, 11,) or/and J(E, p2) are unstable relative to !2 for all E > 0. Suppose now that t,(r) f 0. Then this implies that the polynomial 5, has finitely many zeros. It also follows that V(E, p) f 0. Thus & does not vanish identically, since D[r](p), being the square-free version of v[r](p), has no multiple roots for all values of r (recall that & is the determinant of the Bezoutian of Q and its derivative with respect to II, and is identically zero if and only if Q has multiple roots for all r; see the Appendix). Therefore 6 is well defined. Since Y(Q) is guarded by vn, Theorem 1 implies that J(r, p) E Y(R) for all r E (r*, 03) and p E [p,, puz] if and only if J(ro, po) E Y(Q) for some (ro, PO) E (r*, m) X [PI, 1.4 and
v(r, P) # 0 for all (c II) E (r*, WIX[h, 1.4.
= det J(rO) det j(r’) det (I - +4) xdet(Z-nfi) =:4(n), where, with a slight abuse of notation, I denotes the identity matrix with the appropriate dimension. Thus n E (-r”, + m) for r > 0, and equals 0 when r = r”; hence q(v) f0, since q(0) = det J(r”) det J(r”) is nonzero by hypothesis. Assuming now that n # 0, we have
(Note that r* < +m, by definition.) Clearly, saying that v(r, p) #O for all r E (r*, M) and p E [p,, p2] is equivalent to saying that v[r](p) E Y’(Z) for all r E (r*, ~0). Hence J(r, p) E Y(Q) for all r E (r*, 03) and F E [CL,,pZ] if and only if J(ro, po) E Y(Q) for some (ro, po) E (r*, ~0)X IPI, 1.4 and v[rI(cL) E 33 for all r E (r*, M). Recalling that the set Y(E) is semiguarded for polynomials by the map v: p *P(~i)&J
q(77) = 77N+n+mdet J(r’) det J(r’) X
where
A:= l/n
det (AZ - M) det (AZ - a), and
(30)
N = i(n + m)(n + m - 1)
det R(p(s), $p(.r))
we obtain by virtue of Corollary 1 that the one-parameter family of polynomials v[r](p) E Y(Q)
New results for singularly perturbed for all r E (r*, + m) if and only if;pl’(“’ ; Y(Ej for some r E r*, r0 E (r*, +m), m):&(r)=0}=0 and v[r](p) e’&‘(S) for all r E 12, = (r E (r *, + m) : &(r) = 0). By construction of p and S (and hence r*), Uk = 0 and v[r](p) E 9’(Z) for all r E Uzr. Therefore, in order for the statement v[r](p) E 9’(E) for all r E (r*, m) to hold, it is, owing to the special construction of r*, necessary and sufficient that v[ro](p) E Y(Z) for some r. E (r*, m) only. Combining the above arguments, we conclude that J(r, p) E 3’(Q) for all r E (r*, m) and P E [CL,,~~1 if and only if J(ro, po) E 9’(Q) for some (ro, PO) E (r*, m) X [PI, ~21and 4rol(~) E Y(E) for some r. E (r*, m). This establishes the first part of Case 2 in Theorem 6. To complete the proof, suppose that J(r0,
PO)
E WV
for
SOme
(r0,
po)
E (r*,
m)
X
and v[r](p) E 9’(Z) for some r E (r*, m), say r = r,, so that J(r, CL) (respectively J(e, P)) E y(Q) for all (r, P) E (r*, m) X [k, 14 (respectively (e, p) E (0, E*) X [pr, ~~1). Suppose momentarily that r* > 0 (i.e. E* < m). Then, owing to its construction, r*(e*) is the largest (smallest) positive value of r(e) for which the above conclusion does not hold; i.e. J(r*, p) # [pL1,
y(Q)
pcLz]
(J(E*, CL) e y(Q))
for SOme CLE [PI, 1.4,
while .J(r, p) E 9’(Q) (J(E, p) E Y(Q)) for all r > r* (E < E*) and p E [pI, ~~1. Finally, if r* = 0 then (26)-(28) imply that both p and 6 are 0, which in turn means, by virtue of Corollary 1, that v[r](p) E 9’(Q) for all r E (0, m), i.e. that v(r, P) # 0 for all (5 CL) E (0, m) X [h , 1.4. It follows from Theorem 1 that J(r, p) E Y(Q) for all (r, p) E (0, m) X [p,, p*J, or, in terms of E, that J(E, CL)E P’(Q) for all (E, CL)E (0, m) X [pI, pz], i.e. E* = +m and (29) applies in this case as well. 0 Acknowledgements-The author thanks Professors E. H. Abed and A. L. Tits of the Institute for Systems Research, University of Maryland and Harvard University for their help and Professor G. L. Blankenship of the same institution for raising a question that led to the consideration of this application of guardian maps. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant NSERC-OGPO122106 and le Fonds FCAR under Grant FCAR NC1 193. REFERENCES Abed, E. (1985). A new parameter estimate in singular perturbation. Syst. Control Lett., 3, 193-198. Abed, E., L. Saydy and A. Tits (1989). Generalized stability of linear singularly perturbed systems including calculation of maximal parameter range. In M. A. Kaashoek, J. H. van Schuppen and A. C. M. Rand (Eds), Robust Control of Linear Systems and Nonlinear Control, pp. 191-203. Birkhluser, Base]. Barmish, B. (1994). New Tools for Robustness of Linear Systems. Macmillan, New York. Bose, N. (1982). Applied Multidimensional Systems Theory. Van Nostrand Reinhold, New York. Chow J. H. (1978). Asymptotic stability of a class of
systems
817
non-linear singularly perturbed systems. J. Franklin Inst., 305,27X281. Desoer, C. and M. Shensa (1970). Networks with small and very large parasitics: natural frequencies and stability. Proc. IEEE, 58,1933-1938. Feng, W. (1988). Characterization and computation for the bound Q* in. linear time-invariant singuiarfy perturbed. Svst. Control Lett.. 11. 195-202. Javid, S. (1978). Uniform asymptotic stability of linear time-varying singularly perturbed systems. J. Franklin Inst., 30&U-37. Khahl, H. (1981). Asymptotic stability of multiparameter singularly perturbed systems. Automatica, 17, 797-804. Klimushev, A. and N. Krasovskii (1962). Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms. Appl. Math. Mech., 25, 1011-1025. Kokotovic, P., H. Khalil and J. O’Reilly (1986). Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London. Krishnan. H. and N. McCIamroch (1994). On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form. IEEE Trans. Autom. Control, AC-39,1079-1084. Marden, M. (1949). The geometry of the Zeros of a Polynomial in a Complex Variable. American Mathematical Society, Providence, RI. Mustafa, D. (1994). How much integral action can a system tolerate? Lin. Algebra Applies, 205/206,%5-970. Sandell, N. (1979). Robust stability of systems with application to singular perturbations. Automatica, 15, 467-470. Saydy, L. (1988). Studies in robust stability. PhD thesis, University of Maryland. Saydy, L., A. Tits and E. Abed (1990). Robust stability of linear systems relative to guarded domains. In P. Dorato and R. Yedavalli (Eds), Recent Advances in Robust Control, pp. 131-138. IEEE Press, New York. Saydy, L., A. Tits and E. Abed (1991). Guardian maps and the generalized stability of parametrized families of matrices and polynomial. Math. Control, Sig. Syst., 3, 345-371. Zien, L. (1973). An upper bound for the singular parameter in a stable, singularly perturbed system. J. Franklin Inst., 295,373-381.
6. APPENDIX In the following two sections we present the bialtemate product and the Bezoutian, and briefly discuss relevant properties (for more details the reader is referred to Saydy et al. (1991) and the references therein).
Bialternate product Let A and B be n X n matrices. To introduce the bialternate product of A and B, we first establish some notation. Let V” be the $n(n - 1)-tuple consisting of pairs of integers (p, q), p = 2,3,. . , n, q = 1,. , p - 1, listed lexicographically. That is, V” = [(2, l), (3, l), (3,2), (4, l), (4,2), (4,3),. . , (n, n - l)]. Denote by V: the ith entry of V”. Denote
where the dependence off on A and B is kept implicit simplicity. The bialternate product of A and B, denoted A 0 B, is a :n(n - 1)-dimensional square matrix whose entry is given by (A OB), = f(V:; V;). For example bialtemate product of a 3 X 3 matrix A = (a,-) with identity matrix is a 3 X 3 matrix given by
for by ijth the the
818
L. Saydy
The bialtemate product has some properties. In particular, if u(A) = {A,, a(2(AOI))={A,
+&,A,
+A, ,...
interesting spectral , A,} then ,A,
iA,,...,A,+A,,...,A,~,
+A,,A, +A,}
and (r(2(A201-AOA)) = {(A, - A#, (A, - A#,
, (A, -
A,,)'.
(A*-A1)2,...,(A2-A,)2,...,(A,~1
-A,,)'}.
2(A 0 I) (respectively 2(A2 0 I - A 0 A)) become singular if and only if A has opposite (respectively multiple) eigenvalues. Thus
The Bezoutian
Given any polynomial a(s) = a,? + + a,s + a”, a, # 0, define the polynomial a(s):=s”a(sc’) = a,,s” + + a,, ,s + a,andthenXnmatrixS(a)byS,(a)=a,+,_,ifi+j-lsn, 0 otherwise. The Bezoutian B(a, b) of two polynomials a and h may then be expressed as the n X n matrix, n being the largest-of the degrees of a and b, given by B(a, b):= S(a)S(b)P - S(b)S(li)P, where P is the permutation matrix specified by P,, = 1 for all i, j such that i + j - 1 = n, 0 otherwise. Our interest in the Bezoutian stems from the following fact. The polynomials a(s) and b(s) have no common zeros if and only if the associated_Bezoutian B(a, b) equivalently, the matrix S(s)S(b) - S(b)S(a^)) is (or, nonsingular.