Abstract -The concepts of guardian and semiguardian maps were recently introduced as tools for assessing robust generalized stability of parametrized families ...
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Express Letters On Robust Eigenvalue Configuration A. L. Tits and L. Saydy Abstract -The concepts of guardian and semiguardian maps were recently introduced as tools for assessing robust generalized stability of parametrized families of matrices or polynomials. Necessary and sufficient conditions were obtained for stability of parametrized families with respect to a large class of open subsets of the complex plane, namely those with which one can associate a polynomic pardian or semiguardian map. This note focuses on a class of disconnected subsets of the complex plane, of interest in the context of dominant pole assignment and filter design. It is first observed that the robust stability conditions originally put forth are in fact necessary and sufficient for the number of eigenvalues (matrices) or zeros (polynomials) in any given connected component to be the same for all the members of the given family. Polynomic semiguardian maps are then identified for a class of disconnected regions of interest. These maps are in fact “essentially guarding with respect to one-parameter families.” Keywords: Robust stability, eigenvalue clustering, root clustering, parametrized families, dominant pole assignment, filter design.
I. INTRODUCTION In [I] and [2] the concepts of guardian and semiguardian maps were introduced as tools for assessing robust generalized stability of parametrized families of real matrices or polynomials. Necessary and sufficient conditions leading to computational tests were obtained for stability of one- (resp. two-) parameter families with respect to a large class of open subsets R of the complex plane, namely those with which a corresponding polynomic guardian o r semiguardian (resp. guardian) map can b e associated. The techniques and results of [2] were subsequently extended to the case of complex matrices and polynomials [31. This note focuses on a class of disconnected regions R, of interest in the context of dominant pole assignment or filter design [4], [5].In Section I1 below, after reviewing the essentials of the approach introduced in [1]-[3], we observe that results somewhat stronger than those given in those papers hold when the region R is disconnected. Then, in Section 111, polynomic semiguardian maps are identified for a class of disconnected regions of interest. These maps are in fact “essentially guarding with respect to one-parameter families.” Section IV is devoted to concluding remarks. 11. GUARDIAN AND SEMIGUARDIAN MAPS
The guardian map approach was introduced in [1]-[3] as a unifying tool for the study of generalized stability of parametrized families of matrices or polynomials. We first recall some of the basic concepts used in this approach. The following notation is used: 23 and d 9 denote the closure and the boundary of a given cr(A) the spectrum of matrix A , A H the Hermitian set 9, transpose of A , 8 and @ the Kronecker product and sum of matrices [6],[2]. Manuscript received June 28, 1990. This letter was recommended by Editor R. Liu. The authors are with the Systems Research Center and Department of Electrical Engineering, University of Maryland, College Park, MD 20742.
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Definition 1 [2J Let X denote the set of all n X n complex matrices or the set of all monk polynomials of degree n with complex coefficients, and let 9 be an open subset of X.Let v map X into C.We say that v is semiguarding for 9 if the implication x E d 9 ”(X) = 0
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holds. The map v guaids 9 if the converse implication also If v is semiguarding for 9, any holds whenever x €9. x E 9 for which u ( x ) = 0 is said to be a blind spot for ( U , 9). The map v is said to be polynomic if it is a polynomial function of the entries (matrix case) or coefficients (polynomial case) of its argument, and of their complex conjugates. Of special interest are sets 9 of the form 9(R)c Z,where 4(R) denotes the set of all complex matrices (polynomials) with eigenvalues (zeros) in 0, a given open subset of the complex plane. In the sequel, we focus on sets of this type. Let Q, := { x ( r ) : r E U ] , U c R k pathwise connected, be a family of n x n matrices or nth-order monic polynomials depending continuously on a parameter vector r , and let R be an open subset of the complex plane. In [2], the following necessary and sufficient condition for robust stability was obtained. Theorem 1 [2J Let v be a semiguardian map for 9 ( R ) and assume that the family Q, is nominally stable, i.e., x ( r o )E /(R) for some r o E U . Then the whole family Q, is stable relative to s1 if and only if x ( r ) E /(Cl) for all r E U for which Y ( x ( Y ) ) = 0. If v guards /(Cl), then is stable relative to R if and only if v ( x ( r ) )+ 0 for all r E U.
Suppose now that R is the union of several connected components. Clearly, continuity of x(.) implies that if the entire family @ is stable relative to R then, in fact, for any r E U, x ( r ) has the same number of eigenvalues or zeros (counting multiplicity) in each component of R as x ( r o ) does. Specifically, let 9 = (a,:i = 1;. .,l], with I > 1, be a family of finitely many connected components of R, let n = (n,; . ., n,) be an 1-tuple of nonnegative integers, and denote by /(an, 9, n) c 9(R) the set of all complex matrices (polynomials) of / ( C l ) having exactly n, eigenvalues (zeros) in n,, i = 1;. .,1. The following holds. Theorem 2: Let v be a semiguardian map for 9(Q) and suppose that x ( r o ) E 9 ( R , 5 P , n ) for some r o E U. Then Q, c / ( O n ,9, n) if and only if x ( r ) E /(a, 9, n ) for all r E U for and 9 entirely covers R, which v ( x ( r ) )= 0. If v guards this holds if and only if v ( x ( r ) )# 0 for all r E U.
In [2] and [3] it is shown that for a large class of regions of interest, a polynomic guardian or semiguardian map is available. In such a case, if r is a scalar parameter and x depends polynomially on r , Theorems 1 and 2 provide tests involving the computation of the zeros of a univariate polynomial, namely v ( x ( . ) ) . If v guards J ( R ) , the test amounts to verifying that v ( x ( . ) ) has no zeros in a specific real interval, and this can be done by means of Sturm sequences (see, e.g., [7]).
111. CONSTRUCTING SEMIGUARDIAN MAPSFOR OF DISCONNECTED REGIONS
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For the sake of simplicity of exposition, we now focus on families of matrices. Similar results hold for the case of polynomials. W e also assume that x depends polynomially on r .
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Consider an open set R C C such that a polynomic semii.e., such guardian map v : C n x n+ C is available for ./’(C\dR), that v ( A ) = 0,
whenever
A~ds(C\dR)
whenever
U(
(1)
or, equivalently, U(
A ) = 0,
A ) n dR
#
0.
(2)
Note that, since
d/(R)
Cd/(C\dR)
(3)
(1) implies that v is semiguarding for J(Cl), but not conversely;
also note that (1) neither implies nor is implied by guardedness of S(R) by v. In [2] and [3],semiguardian maps are obtained for /(a) whenever R can be expressed in the form
R = { s : r ( R e ( s ) , I m ( s ) ) 0 centered at C E C , i.e.,
R = ( s : Is-cl
