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A new parameter estimate in singular perturbations
Systems & Control Letters 6 (1985) 193-198 North-Holland
August 1985
A new parameter estimate in singular perturbations Eyad H. ABED * Department of Electrical Engineering, Uniuersiry of Maryland, College Park, MD 20742, USA
in singular perturbation theory and for references to the early literature. Notation. By ‘a := b’ one intends that a is defined
Received 6 March 1985 Revised 17 April 1985 Abstract: A new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system. General time-invariant systems with a single small parameter are considered. The paper employs a Riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics. An application of the Brouwer fixed point theorem to the Riccati equation and of Liapunov’s direct method to the fast and slow subsystems results in the desired upper bound. Computation of the estimate requires only the solution of two Liapunov matrix equations. Keywords: Singular perturbation, Robustness of stability, Asymptotic stability, Fixed point methods, Liapunov’s direct method.
1. Introduction
as b. The transpose of a matrix G is denoted by G’. If G is a square matrix, a(G) denotes the spectrum (set of eigenvalues) of G. The vector norm employed in the paper is the Euclidean norm, indicated by ] 1. The matrix norm, also indicated by ] 1, is any symmetric matrix norm (i.e. ]G]=]G’]) which is compatible with the Euclidean norm; examplesinclude the spectral (or L2) norm and the Frobenius norm. This minor restriction on allowed norms is for simplicity and does not represent a limitation of the method. (Recall that the spectral norm of a real matrix P=(pij) is given by ]P]‘=A,,(P’P), where A,,,,,( P’P) denotes the largest eigenvalue of P’P. The Frobenius norm of P is given by ] P I 2 = Civjp,‘i [26].) Finally, J(E) will denote the Jacobian matrix
(2)
Consider the system ~=Ax+ By, ej=Cx+Dy,
(1)
in which E > 0 is a small real parameter, x and y are vectors in R” and R”’ respectively, and, A, B, C, D are matrices of appropriate dimension. System (1) is referred to as a singularly perturbed system since its dimension drops from n + m to n when the parameter E is formally set to 0. Theorem 1 below gives sufficient conditions for the asymptotic stability of (1) for all sufficiently small values of the singular perturbation parameter E and has been derived by several authors. See the excellent survey articles [21,22,24] for a discussion of results * Supported in part by the NSF under Grant No. ECS-8404275, and by the Power Engrg. Program, Univ. of Maryland, through grants from BG&E, PEPCO, VEPCO, Bechtel and General Electric. 0167-6911/85/$3.30
of (1). Recall that a matrix is a stability matrix (or simply stable) if its spectrum lies in the open left half complex plane [5]. Theorem 1 [19,14]. Let D be nonsingular. If the
matrices A, := A - BD-‘C and D are stable, then there exists an E > 0 such that the null solution of system (1) is asymptoticallystablefor all E E (0, E). Since the parameter typically representsa small physical quantity over which one has little or no control, it is of practical significance to find explicit upper bounds E,, on E ensuring that the conclusion of the theorem above is valid for all E E (0, E,,). The aim of this paper is to derive one such upper bound. The derivation consists of two main steps. First, the Brouwer fixed point theorem is used to obtain an initial upper bound on E
Q 1985, Elsevier Science Publishers B.V. (North-Holland)
193
which ensures the existence of a solution to an associated algebraic matrix Riccati equation. This initial upper bound is parametrized by the magnitude of a norm constraint imposed on a solution to the Riccati equation. The referenced Riccati equation (Eq. (6) below) is useful since its solution(s) can be used to produce a nonsingular similarity transformation whose effect is to explicitly separate the fast and slow dynamics of (1). Next, Liapunov’s direct method is applied to obtain a bound on the norm of the solution to the Riccati equation which ensures asymptotic stability of the fast and slow subsystems.An explicit upper bound on the singular perturbation parameter E readily follows. It appears that Zien’s 1973 paper [28] was the first to obtain concrete results on this issue. More recently, Javid [16] derived an upper bound valid for linear time-varying systems.Since [28] and [16] used similar contraction mapping arguments on function space, it is not surprising that the upper bound of [16] reduces to that of [28] in the time-invariant case. Sandell [25] has studied the problem using the singular value decomposition, obtaining an inequality involving E which was then used to yield explicit upper bounds in specific examples, though no generally appicable formula for an upper bound was derived, Chow [lo] contains a stability computation for a class of nonlinear autonomous singularly perturbed systems which can be directly applied to yield an explicit upper bound for E. Khalil [18] studies a class of nonlinear systems in which several singular perturbation parameters &r,. . . , Ed may appear. An upper bound on (a,.sz. . . aW)‘jM is obtained by a Liapunov function argument under the assumption that the mutual ratios ei/ej are bounded from above and below by known positive constants. In recent independent work, Balas [3,4] obtains an upper bound which applies to distributed parameter systems. The approach taken in [3,4] resembles that of the present work in that both begin by ensuring the existence of a transformation which exactly separates the fast and slow modes of (1).
It is well known [8,9,20,24,22,11,1] that if D is nonsingular the Riccati equation for the m X n matrix 4,
194
+eLBL-
C=O,
can be used to exactly separate the fast and slow dynamics of (1) for those values of E for which (3) has a solution. Moreover, one has the following representation for the spectrum of J(E): a(J(e))=u(A-BL)u+-‘D+LB).
(3)
(4)
If L exists, (4) implies that the eigenvalues of A - BL are the slow modes and those of E-‘D + LB are the fast modes of (1). In this paper Eq. (4) is all that is required from the results on explicit separation of fast and slow modes and the reader is referred to the survey articles mentioned above for further details. Note that by Eq. (3), L= D-‘C+O(
1~1).
It will prove convenient in the computations below to use the matrix r:=D-‘C-L
(5)
in favor of L, as F = 0( ] E1). In terms of F, (3) becomes [l]
-d-m- + ED--‘CBI’
= 0,
(6)
where A,, = A - BD-‘C is as defined in Theorem 1 above. Rewrite Eq. (6) in the fixed-point form
c(r) = r
(7)
where the parametrized mapping F,: R”‘x” + R n’Xn is defined for any real E by F,(r):=eD-‘{(I=
D-‘C)A,
+rm - D-‘CBr}.
(8)
A paramatrized upper bound s,(o) on E will now be derived such that for any LY> 0 and for all E E (0, &i(a)], ] F] G (Y. Proceeding, let (Y> 0 be given and let FE R”lx” satisfy ] F ] ( 1~.From (8) one has
IF,~~~I~~I~-‘l{(l~l+l~-l~I)I~~l +IBllrl*+
2. Separation of fast and slow modes
DL-ELA
August1985
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Volume6, Number3
d&ID-‘I{(Q+
ID-‘CBIlrl} ID-‘CI)IA,I
+a*IBI+aID-‘CBI}.
(9)
From (9) follows immediately that for ] F,(r) I Q a to hold it suffices that EG e,(a) where ~,(a) is
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Volume 6, Number 3
3. Perturbational stability analysis
given by
3.1. A basic perturbarional result
+a( p-‘CBl+lA,J)+a21BI}.
(10)
Thus for any LY> 0, F, is a continuous map of the closed ball BP:= {~ER”‘~“:
Irl
into itself whenever 0 f E Q e,(a). Note that B, is homeomorphic to the closed unit ball in R”“‘. The Brouwer fixed point theorem ([6], p. 54 and [15], p. 10) may now be invoked to establish the existence of a solution r(e) to (7) in B, for any E E (0,
El(a)lTheorem 2. For any LY> 0 and for all E < E,(a), the Riccati equation (6) has at least one solution r(e) with I r(e) I < CL Remark 1. Note that for any system(1) with B f 0 (the generic and nontrivial case), the upper bound E,(a) approaches0 as (Y+ co. It is clear, however, that a solution of (6) in the ball B, for a given (Y= (Yewill also belong to any ball B, with OL> (Ye. Therefore the upper bound of Theorem 2 can be made less conservative by using instead of E,(a) the revised estimate
The point of departure of this section is Brockett’s proof of a basic perturbational result on stability of linear systems([7], p. 205). In [7] one is concerned with the persistenceof exponential stability of time-varying linear systems under sufficiently small linear perturbations. The specialization of this result So the time-invariant case is elementary and is easily stated. Proposition 1. Let F be a constant square matrix and suppose that 2 = Fx is asymptotically stable. Then there is a p > 0 such that if A is a constant matrix with I Al < ~1, then i = (F+ A)x is also asymptotically stable.
Note that Proposition 1 follows from continuity of the eigenvaluesof a matrix in its parameters (Kato [17], p. 107). The proof of the time-varying version of Proposition 1 in [7] will now be specialized to yield an explicit upper bound CL*for I A I. Since F is stable there is a unique, positive definite matrix Q which solves the Liapunov matrix equation [7,27] F’Q + QF=
-I.
03)
Moreover, Q is given by the convergent integral Q =imeF’feFfdl.
04)
Remark 2. It is natural to attempt to factorize the
quadratic appearing in the denominator of the expression(10) for Ed. In this regard, note that replacing I D-‘CB I in (10) by I D-‘C I I B I results in the new (in general more conservative) upper bound E*: given by
Using X’QX as a candidate Liapunov function for the perturbed system i = (F + A)x, one computes $x’Qx=x’(F’Q+QF+QA+A’Q)x = -x’x
E*:(cr):=a/{
ILr’I(
ID-‘Cl
~(I~ol+4BI)}.
+a)
+ x’(QA + A’Q)x.
(15)
Noting that there is a p such that I A I < p implies
(12)
Remark 3. Note that in the derivation of Theorem 2 only existence of a solution r to (6) neededto be
ensured. A further upper bound on E to ensure uniqueness could easily be derived by a contraction mapping argument as in Chow et al. ([ll], pp. 16-18) or Balas ([4], Appendix 1). However, this could only result in a more conservative final estimate and would thus be counterproductive.
IQA+A’QI
06)
completesthe proof of Proposition 1. To obtain an explicit upper bound CL*,use the triangle inequality (recall that 1A I = I A’l):
IQA+A’QI G lQl(l4+l~‘l)=21Qll4 Eq. (16) now yields the following result. 195
Propos#on 2. Let the assumptionsof Proposition 1 hold. Then F + A will be a stability matrix if 1A 1 < p* where
‘*:=m
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SYSTEMS & CONTROL LETTERS
Volume 6, Number 3
is stable. By Proposition 2, this will be the case if (i) D is stable, and (ii) the inequality IeD-‘CB-erBI
1
(17)
TQ’I
(22)
is valid, where Q’ is defined as
and where Q is given by Eq. (14). Remark 4. The upper bound (17) of Proposition 2 can be improved by repeating the foregoing computation using in place of (13) the more general Liapunov matrix equation F’Q + QF = -P and optimizing the resulting upper bound over a chosen set of positive definite matrices P. The required computational effort would, however, be significantly increased.The implications of this for the upper bound on E which is obtained below will be apparent.
Now suppose that I I’[ is constrained so that 1F 1 c CCThen the left side of (22) is not larger than E(I D-‘CBI +a[ BI). Defining 1’ &z(a) := 2lQ’)(ID-‘CBI+alBI)’
3.2. Stability of the slow subsystem
4. The upper bound
As noted earlier, (4) implies that the eigenvalues of the separatedslow subsystemare precisely the eigenvaluesof the matrix A - BL =: AS if a solution L to (3) exists. In terms of F (seeEq. (5)) this matrix is given by
It is now straightforward to give an upper bound e0 on E such that for 0
AS=A-BD-‘C+BT=A,+BT,
08)
with F a solution to (6). Proposition 2 now implies that AS is a stability matrix (i.e. the slow modes are stable) if (i) A, is stable, and (ii) there is a solution F of (6) with 1BT 1 < l/2 1Qs 1 where Qs
:=/gooe4f
eAd
dt.
This will be true if the hypotheses of Theorem 1 hold and if
is satisfied, where F is some solution to Eq. (6). 3.3. Stability of the fast subsystem By Eq. (4) and the definition (5) of r, the eigenvaluesof the fast subsystemare those of the matrix E-‘D + D-‘CB - TB. Since E> 0 the fast subsystemis stable if andlonly if the matrix iif := D + 196
ED-‘CB
-
ETB
(21)
it now follows that the fast subsystemis stable if (i) a solution I’ to (6) exists which satisfies the bound II’1 <(Y, and (ii) E < Ed.
Finally, the fast subsystem will be well defined and stable if the last remark holds and if E -Ce2(a) where e*(a) is given by Eq. (24). By Eq. (4) stability of the fast and slow subsystemsimplies that of the singularly perturbed system (1). These remarks are summarizedin the following theorem, in which the upper bound E,, of this paper is stated explicitly. Note that the freedom in a! is reflected in taking a maximum over 0 G (YG E in Eq. (26). Theorem 3. Let the assumptionsof Theorem1 hold, so that both A, = A - BD-‘C and D are stability matrices. Then the null solution of (1) is asymptotically stable for all E in the open interval (0, .eO) where e0 is given by
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and where $(a), ~(a) and Cuare given by Eqs. (ll), (24) and (25) respectively.
References
111E.H. Abed, Multiparameter singular perturbation probOne anticipates the most laborious task in computing .sOto be the computation of the matrices QS and Q’. Rather than relying on the explicit form of these matrices revealed in Eqs. (19), (23) it is advisable to use instead an efficient numerical algorithm for directly solving the (linear) Liapunov matrix equations associatedwith QS and Q’. Becauseof this flexibility, the upper bound proposed here may prove less expensiveto compute for large systemsthan that of [28], since the latter requires the evaluation of integrals on [0, 00) in which the matrix exponentials eAo’and e”’ appear. Davison and Man [12] and Davison [13] give an algorithm for the numerical solution of Liapunov matrix equations.This algorithm convergesin 0( n’) steps, where n is the number of rows (or columns) of the n X n unknown matrix Q. For further references and a discussionof recent advancesin the numerical solution of Liapunov matrix equations, see Laub ([23], Section IV-E). 5. Conclusions
The paper has presented a derivation of a new upper bound ~a on the small parameter E of a singularly perturbed system (1) such that if Theorem 1 applies the asymptotic stability of (1) is ensured if 0 < E < eO. The analysis consisted of applying the Brouwer fixed point theorem and Liapunov’s direct method to obtain explicit estimates for well known results on time scale separation and on regular perturbation of stable linear systems. An expression for et, was then obtained by considering the implications of these estimatesfor the stability of fast and slow subsystems associatedwith system (1). Computation of the new upper bound involves only the numerical solution ,of two Liapunov matrix equations, for which efficient algorithms are available. Acknowledgments
lems: Iterative expansions and asymptotic stability, Sys-
terns Control Len. 5 (1985) 279-282.
121E.H. Abed, Singularly perturbed Hopf bifurcation,
IEEE
Trans. Circuits and Systems (to appear). [31 M.J. Balas, Singular perturbations in stable feedback con-
trol of distributed parameter systems, in: M.D. Ardema, Ed., Singular Perturbations in Systems and Control (Springer, Wien-New York, 1983) 241-267. 141 M.J. Balas, Stability of distributed parameter systemswith finite-dimensional controller-compensators using singular perturbations, J. Math. Anal. Appl. 99 (1984) 80-108. 151R. Bellman, Introduction to Matrix Anabsis (McGraw-Hill, New York, 1960). 161M.S. Berger, Nonlinearity and Functional Analysis (Academic Press,New York, 1977). 171 R.W. Brockett. Finite Dimensional Linear Systems (John Wiley, New York, 1970). PI K.W. Chang, Remarks on a certain hypothesis in singular perturbations, Proc. Amer. Math. Sot. 23 (1969) 41-45. 191K.W. Chang, Singular perturbations of a general boundary value problem, SIAM J. Math. Anal. 3 (1972) 520-526. WI J.H. Chow, Asymptotic stability of a class of non-linear singularly perturbed systems, J. Franklin Inst. 305 (1978) 275-281. 1111J.H. Chow, Editor, Time-Scale Modeling of Dynamic Networks with Applications to Power Systems (Springer, Berlin, 1982). WI E.J. Davison and F.T. Man, The numerical solution of A ’Q + QA = - C, IEEE Trans. Automat. Control 13 (1968) 448-449. P31 E.J. Davison, The numerical solution of i = A, X + XA2 + D, X(0) = C, IEEE Trans. Automat. Control 20 (1975) 566-567. iI41 C.A. Desoer and M.J. Shensa,Networks with very small
and very large parasitics: Natural frequencies and stability, Proc. IEEE 58 (1970) 1933-1938. (151 J.K. Hale, Ordinary Dvferential Equations, 2nd ed. (Krieger, Malabar, FL, 1980). 1161S.H. Javid, Uniform asymptotic stability of linear timevarying singularly perturbed systems,J. Franklin Inst. 305 (1978) 27-37. [I71 T. Kato, Perturbation
Theory for
Linear
Operators
(Springer, New York, 1966). WI H.K. Khalil. Asymptotic stability of nonlinear multiparameter singularly perturbed systems, Automatica 17 (1981) 797-804. 1191A.I. Klimushev and N.N. Krasovskii, Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms, Appl. Math. Mech. 25 (1962) 1011-1025. WI P.V. Kokotovic, A Riccati equation for block-diagonalixation of ill-conditioned systems, IEEE Trans. Automat. Control 20 (1975) 812-814.
The author is grateful to Prof. I.D. Mayergoyz, Prof. P.S. Krishnaprasad and Mr. J.-H. Fu of the University of Maryland for helpful comments.The remarks of the reviewer are also appreciated.
1211P.V. Kokotovic, R.E. G’Malley, Jr. and P. Sannuti, Singular perturbations and order reduction in control theory An overview, Automatica 12 (1976) 123-132.
197
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[22] P.V. Kokotovic, Applications of singular perturbation techniques to control problems, SIAM Review 26 (1984) 501-550. [23] A.J. Laub, Numerical linear algebra aspects of control design computations, IEEE Trans. Automat. Control 30
(1985)97-108. [24] V.R. Saksena, J. O’ReiUy and P.V. Kokotovic, Singular perturbations and time-scale methods in control theory: , Survey 1976-1983, Automatica 20 (1984) 273-294.
LETTERS
August 1985
[25] N.R. Sandell, Jr., Robust stability of systems with application to singular perturbations, Automafica 15 (1979)
467-470. [26] G.W. Stewart,
Introduction to Matrix Computations (Academic Press, New York, 1973). [27] M. Vidyasagar, Nonlinear Systems Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1978). [28] L. Zien, An upper bound for the singular parameter in a stable, singularly perturbed system, J. Franklin Inst. 295 (1973) 373-381.
August 1985
A new parameter estimate in singular perturbations Eyad H. ABED * Department of Electrical Engineering, Uniuersiry of Maryland, College Park, MD 20742, USA
in singular perturbation theory and for references to the early literature. Notation. By ‘a := b’ one intends that a is defined
Received 6 March 1985 Revised 17 April 1985 Abstract: A new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system. General time-invariant systems with a single small parameter are considered. The paper employs a Riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics. An application of the Brouwer fixed point theorem to the Riccati equation and of Liapunov’s direct method to the fast and slow subsystems results in the desired upper bound. Computation of the estimate requires only the solution of two Liapunov matrix equations. Keywords: Singular perturbation, Robustness of stability, Asymptotic stability, Fixed point methods, Liapunov’s direct method.
1. Introduction
as b. The transpose of a matrix G is denoted by G’. If G is a square matrix, a(G) denotes the spectrum (set of eigenvalues) of G. The vector norm employed in the paper is the Euclidean norm, indicated by ] 1. The matrix norm, also indicated by ] 1, is any symmetric matrix norm (i.e. ]G]=]G’]) which is compatible with the Euclidean norm; examplesinclude the spectral (or L2) norm and the Frobenius norm. This minor restriction on allowed norms is for simplicity and does not represent a limitation of the method. (Recall that the spectral norm of a real matrix P=(pij) is given by ]P]‘=A,,(P’P), where A,,,,,( P’P) denotes the largest eigenvalue of P’P. The Frobenius norm of P is given by ] P I 2 = Civjp,‘i [26].) Finally, J(E) will denote the Jacobian matrix
(2)
Consider the system ~=Ax+ By, ej=Cx+Dy,
(1)
in which E > 0 is a small real parameter, x and y are vectors in R” and R”’ respectively, and, A, B, C, D are matrices of appropriate dimension. System (1) is referred to as a singularly perturbed system since its dimension drops from n + m to n when the parameter E is formally set to 0. Theorem 1 below gives sufficient conditions for the asymptotic stability of (1) for all sufficiently small values of the singular perturbation parameter E and has been derived by several authors. See the excellent survey articles [21,22,24] for a discussion of results * Supported in part by the NSF under Grant No. ECS-8404275, and by the Power Engrg. Program, Univ. of Maryland, through grants from BG&E, PEPCO, VEPCO, Bechtel and General Electric. 0167-6911/85/$3.30
of (1). Recall that a matrix is a stability matrix (or simply stable) if its spectrum lies in the open left half complex plane [5]. Theorem 1 [19,14]. Let D be nonsingular. If the
matrices A, := A - BD-‘C and D are stable, then there exists an E > 0 such that the null solution of system (1) is asymptoticallystablefor all E E (0, E). Since the parameter typically representsa small physical quantity over which one has little or no control, it is of practical significance to find explicit upper bounds E,, on E ensuring that the conclusion of the theorem above is valid for all E E (0, E,,). The aim of this paper is to derive one such upper bound. The derivation consists of two main steps. First, the Brouwer fixed point theorem is used to obtain an initial upper bound on E
Q 1985, Elsevier Science Publishers B.V. (North-Holland)
193
which ensures the existence of a solution to an associated algebraic matrix Riccati equation. This initial upper bound is parametrized by the magnitude of a norm constraint imposed on a solution to the Riccati equation. The referenced Riccati equation (Eq. (6) below) is useful since its solution(s) can be used to produce a nonsingular similarity transformation whose effect is to explicitly separate the fast and slow dynamics of (1). Next, Liapunov’s direct method is applied to obtain a bound on the norm of the solution to the Riccati equation which ensures asymptotic stability of the fast and slow subsystems.An explicit upper bound on the singular perturbation parameter E readily follows. It appears that Zien’s 1973 paper [28] was the first to obtain concrete results on this issue. More recently, Javid [16] derived an upper bound valid for linear time-varying systems.Since [28] and [16] used similar contraction mapping arguments on function space, it is not surprising that the upper bound of [16] reduces to that of [28] in the time-invariant case. Sandell [25] has studied the problem using the singular value decomposition, obtaining an inequality involving E which was then used to yield explicit upper bounds in specific examples, though no generally appicable formula for an upper bound was derived, Chow [lo] contains a stability computation for a class of nonlinear autonomous singularly perturbed systems which can be directly applied to yield an explicit upper bound for E. Khalil [18] studies a class of nonlinear systems in which several singular perturbation parameters &r,. . . , Ed may appear. An upper bound on (a,.sz. . . aW)‘jM is obtained by a Liapunov function argument under the assumption that the mutual ratios ei/ej are bounded from above and below by known positive constants. In recent independent work, Balas [3,4] obtains an upper bound which applies to distributed parameter systems. The approach taken in [3,4] resembles that of the present work in that both begin by ensuring the existence of a transformation which exactly separates the fast and slow modes of (1).
It is well known [8,9,20,24,22,11,1] that if D is nonsingular the Riccati equation for the m X n matrix 4,
194
+eLBL-
C=O,
can be used to exactly separate the fast and slow dynamics of (1) for those values of E for which (3) has a solution. Moreover, one has the following representation for the spectrum of J(E): a(J(e))=u(A-BL)u+-‘D+LB).
(3)
(4)
If L exists, (4) implies that the eigenvalues of A - BL are the slow modes and those of E-‘D + LB are the fast modes of (1). In this paper Eq. (4) is all that is required from the results on explicit separation of fast and slow modes and the reader is referred to the survey articles mentioned above for further details. Note that by Eq. (3), L= D-‘C+O(
1~1).
It will prove convenient in the computations below to use the matrix r:=D-‘C-L
(5)
in favor of L, as F = 0( ] E1). In terms of F, (3) becomes [l]
-d-m- + ED--‘CBI’
= 0,
(6)
where A,, = A - BD-‘C is as defined in Theorem 1 above. Rewrite Eq. (6) in the fixed-point form
c(r) = r
(7)
where the parametrized mapping F,: R”‘x” + R n’Xn is defined for any real E by F,(r):=eD-‘{(I=
D-‘C)A,
+rm - D-‘CBr}.
(8)
A paramatrized upper bound s,(o) on E will now be derived such that for any LY> 0 and for all E E (0, &i(a)], ] F] G (Y. Proceeding, let (Y> 0 be given and let FE R”lx” satisfy ] F ] ( 1~.From (8) one has
IF,~~~I~~I~-‘l{(l~l+l~-l~I)I~~l +IBllrl*+
2. Separation of fast and slow modes
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SYSTEMS & CONTROLLETTERS
Volume6, Number3
d&ID-‘I{(Q+
ID-‘CBIlrl} ID-‘CI)IA,I
+a*IBI+aID-‘CBI}.
(9)
From (9) follows immediately that for ] F,(r) I Q a to hold it suffices that EG e,(a) where ~,(a) is
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3. Perturbational stability analysis
given by
3.1. A basic perturbarional result
+a( p-‘CBl+lA,J)+a21BI}.
(10)
Thus for any LY> 0, F, is a continuous map of the closed ball BP:= {~ER”‘~“:
Irl
into itself whenever 0 f E Q e,(a). Note that B, is homeomorphic to the closed unit ball in R”“‘. The Brouwer fixed point theorem ([6], p. 54 and [15], p. 10) may now be invoked to establish the existence of a solution r(e) to (7) in B, for any E E (0,
El(a)lTheorem 2. For any LY> 0 and for all E < E,(a), the Riccati equation (6) has at least one solution r(e) with I r(e) I < CL Remark 1. Note that for any system(1) with B f 0 (the generic and nontrivial case), the upper bound E,(a) approaches0 as (Y+ co. It is clear, however, that a solution of (6) in the ball B, for a given (Y= (Yewill also belong to any ball B, with OL> (Ye. Therefore the upper bound of Theorem 2 can be made less conservative by using instead of E,(a) the revised estimate
The point of departure of this section is Brockett’s proof of a basic perturbational result on stability of linear systems([7], p. 205). In [7] one is concerned with the persistenceof exponential stability of time-varying linear systems under sufficiently small linear perturbations. The specialization of this result So the time-invariant case is elementary and is easily stated. Proposition 1. Let F be a constant square matrix and suppose that 2 = Fx is asymptotically stable. Then there is a p > 0 such that if A is a constant matrix with I Al < ~1, then i = (F+ A)x is also asymptotically stable.
Note that Proposition 1 follows from continuity of the eigenvaluesof a matrix in its parameters (Kato [17], p. 107). The proof of the time-varying version of Proposition 1 in [7] will now be specialized to yield an explicit upper bound CL*for I A I. Since F is stable there is a unique, positive definite matrix Q which solves the Liapunov matrix equation [7,27] F’Q + QF=
-I.
03)
Moreover, Q is given by the convergent integral Q =imeF’feFfdl.
04)
Remark 2. It is natural to attempt to factorize the
quadratic appearing in the denominator of the expression(10) for Ed. In this regard, note that replacing I D-‘CB I in (10) by I D-‘C I I B I results in the new (in general more conservative) upper bound E*: given by
Using X’QX as a candidate Liapunov function for the perturbed system i = (F + A)x, one computes $x’Qx=x’(F’Q+QF+QA+A’Q)x = -x’x
E*:(cr):=a/{
ILr’I(
ID-‘Cl
~(I~ol+4BI)}.
+a)
+ x’(QA + A’Q)x.
(15)
Noting that there is a p such that I A I < p implies
(12)
Remark 3. Note that in the derivation of Theorem 2 only existence of a solution r to (6) neededto be
ensured. A further upper bound on E to ensure uniqueness could easily be derived by a contraction mapping argument as in Chow et al. ([ll], pp. 16-18) or Balas ([4], Appendix 1). However, this could only result in a more conservative final estimate and would thus be counterproductive.
IQA+A’QI
06)
completesthe proof of Proposition 1. To obtain an explicit upper bound CL*,use the triangle inequality (recall that 1A I = I A’l):
IQA+A’QI G lQl(l4+l~‘l)=21Qll4 Eq. (16) now yields the following result. 195
Propos#on 2. Let the assumptionsof Proposition 1 hold. Then F + A will be a stability matrix if 1A 1 < p* where
‘*:=m
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is stable. By Proposition 2, this will be the case if (i) D is stable, and (ii) the inequality IeD-‘CB-erBI
1
(17)
TQ’I
(22)
is valid, where Q’ is defined as
and where Q is given by Eq. (14). Remark 4. The upper bound (17) of Proposition 2 can be improved by repeating the foregoing computation using in place of (13) the more general Liapunov matrix equation F’Q + QF = -P and optimizing the resulting upper bound over a chosen set of positive definite matrices P. The required computational effort would, however, be significantly increased.The implications of this for the upper bound on E which is obtained below will be apparent.
Now suppose that I I’[ is constrained so that 1F 1 c CCThen the left side of (22) is not larger than E(I D-‘CBI +a[ BI). Defining 1’ &z(a) := 2lQ’)(ID-‘CBI+alBI)’
3.2. Stability of the slow subsystem
4. The upper bound
As noted earlier, (4) implies that the eigenvalues of the separatedslow subsystemare precisely the eigenvaluesof the matrix A - BL =: AS if a solution L to (3) exists. In terms of F (seeEq. (5)) this matrix is given by
It is now straightforward to give an upper bound e0 on E such that for 0
AS=A-BD-‘C+BT=A,+BT,
08)
with F a solution to (6). Proposition 2 now implies that AS is a stability matrix (i.e. the slow modes are stable) if (i) A, is stable, and (ii) there is a solution F of (6) with 1BT 1 < l/2 1Qs 1 where Qs
:=/gooe4f
eAd
dt.
This will be true if the hypotheses of Theorem 1 hold and if
is satisfied, where F is some solution to Eq. (6). 3.3. Stability of the fast subsystem By Eq. (4) and the definition (5) of r, the eigenvaluesof the fast subsystemare those of the matrix E-‘D + D-‘CB - TB. Since E> 0 the fast subsystemis stable if andlonly if the matrix iif := D + 196
ED-‘CB
-
ETB
(21)
it now follows that the fast subsystemis stable if (i) a solution I’ to (6) exists which satisfies the bound II’1 <(Y, and (ii) E < Ed.
Finally, the fast subsystem will be well defined and stable if the last remark holds and if E -Ce2(a) where e*(a) is given by Eq. (24). By Eq. (4) stability of the fast and slow subsystemsimplies that of the singularly perturbed system (1). These remarks are summarizedin the following theorem, in which the upper bound E,, of this paper is stated explicitly. Note that the freedom in a! is reflected in taking a maximum over 0 G (YG E in Eq. (26). Theorem 3. Let the assumptionsof Theorem1 hold, so that both A, = A - BD-‘C and D are stability matrices. Then the null solution of (1) is asymptotically stable for all E in the open interval (0, .eO) where e0 is given by
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and where $(a), ~(a) and Cuare given by Eqs. (ll), (24) and (25) respectively.
References
111E.H. Abed, Multiparameter singular perturbation probOne anticipates the most laborious task in computing .sOto be the computation of the matrices QS and Q’. Rather than relying on the explicit form of these matrices revealed in Eqs. (19), (23) it is advisable to use instead an efficient numerical algorithm for directly solving the (linear) Liapunov matrix equations associatedwith QS and Q’. Becauseof this flexibility, the upper bound proposed here may prove less expensiveto compute for large systemsthan that of [28], since the latter requires the evaluation of integrals on [0, 00) in which the matrix exponentials eAo’and e”’ appear. Davison and Man [12] and Davison [13] give an algorithm for the numerical solution of Liapunov matrix equations.This algorithm convergesin 0( n’) steps, where n is the number of rows (or columns) of the n X n unknown matrix Q. For further references and a discussionof recent advancesin the numerical solution of Liapunov matrix equations, see Laub ([23], Section IV-E). 5. Conclusions
The paper has presented a derivation of a new upper bound ~a on the small parameter E of a singularly perturbed system (1) such that if Theorem 1 applies the asymptotic stability of (1) is ensured if 0 < E < eO. The analysis consisted of applying the Brouwer fixed point theorem and Liapunov’s direct method to obtain explicit estimates for well known results on time scale separation and on regular perturbation of stable linear systems. An expression for et, was then obtained by considering the implications of these estimatesfor the stability of fast and slow subsystems associatedwith system (1). Computation of the new upper bound involves only the numerical solution ,of two Liapunov matrix equations, for which efficient algorithms are available. Acknowledgments
lems: Iterative expansions and asymptotic stability, Sys-
terns Control Len. 5 (1985) 279-282.
121E.H. Abed, Singularly perturbed Hopf bifurcation,
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Trans. Circuits and Systems (to appear). [31 M.J. Balas, Singular perturbations in stable feedback con-
trol of distributed parameter systems, in: M.D. Ardema, Ed., Singular Perturbations in Systems and Control (Springer, Wien-New York, 1983) 241-267. 141 M.J. Balas, Stability of distributed parameter systemswith finite-dimensional controller-compensators using singular perturbations, J. Math. Anal. Appl. 99 (1984) 80-108. 151R. Bellman, Introduction to Matrix Anabsis (McGraw-Hill, New York, 1960). 161M.S. Berger, Nonlinearity and Functional Analysis (Academic Press,New York, 1977). 171 R.W. Brockett. Finite Dimensional Linear Systems (John Wiley, New York, 1970). PI K.W. Chang, Remarks on a certain hypothesis in singular perturbations, Proc. Amer. Math. Sot. 23 (1969) 41-45. 191K.W. Chang, Singular perturbations of a general boundary value problem, SIAM J. Math. Anal. 3 (1972) 520-526. WI J.H. Chow, Asymptotic stability of a class of non-linear singularly perturbed systems, J. Franklin Inst. 305 (1978) 275-281. 1111J.H. Chow, Editor, Time-Scale Modeling of Dynamic Networks with Applications to Power Systems (Springer, Berlin, 1982). WI E.J. Davison and F.T. Man, The numerical solution of A ’Q + QA = - C, IEEE Trans. Automat. Control 13 (1968) 448-449. P31 E.J. Davison, The numerical solution of i = A, X + XA2 + D, X(0) = C, IEEE Trans. Automat. Control 20 (1975) 566-567. iI41 C.A. Desoer and M.J. Shensa,Networks with very small
and very large parasitics: Natural frequencies and stability, Proc. IEEE 58 (1970) 1933-1938. (151 J.K. Hale, Ordinary Dvferential Equations, 2nd ed. (Krieger, Malabar, FL, 1980). 1161S.H. Javid, Uniform asymptotic stability of linear timevarying singularly perturbed systems,J. Franklin Inst. 305 (1978) 27-37. [I71 T. Kato, Perturbation
Theory for
Linear
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(Springer, New York, 1966). WI H.K. Khalil. Asymptotic stability of nonlinear multiparameter singularly perturbed systems, Automatica 17 (1981) 797-804. 1191A.I. Klimushev and N.N. Krasovskii, Uniform asymptotic stability of systems of differential equations with a small parameter in the derivative terms, Appl. Math. Mech. 25 (1962) 1011-1025. WI P.V. Kokotovic, A Riccati equation for block-diagonalixation of ill-conditioned systems, IEEE Trans. Automat. Control 20 (1975) 812-814.
The author is grateful to Prof. I.D. Mayergoyz, Prof. P.S. Krishnaprasad and Mr. J.-H. Fu of the University of Maryland for helpful comments.The remarks of the reviewer are also appreciated.
1211P.V. Kokotovic, R.E. G’Malley, Jr. and P. Sannuti, Singular perturbations and order reduction in control theory An overview, Automatica 12 (1976) 123-132.
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[22] P.V. Kokotovic, Applications of singular perturbation techniques to control problems, SIAM Review 26 (1984) 501-550. [23] A.J. Laub, Numerical linear algebra aspects of control design computations, IEEE Trans. Automat. Control 30
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August 1985
[25] N.R. Sandell, Jr., Robust stability of systems with application to singular perturbations, Automafica 15 (1979)
467-470. [26] G.W. Stewart,
Introduction to Matrix Computations (Academic Press, New York, 1973). [27] M. Vidyasagar, Nonlinear Systems Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1978). [28] L. Zien, An upper bound for the singular parameter in a stable, singularly perturbed system, J. Franklin Inst. 295 (1973) 373-381.