RESEARCH ARTICLE. Lyapunov-Based Exact Stability Analysis and Synthesis for Linear Single-Parameter Dependent Systems

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1 International Journal of Control Vol. 00, No. 00, Month 00x, 1 7 RESEARCH ARTICLE Lyapunov-Based Exact Stability Analysis and Synthesis for Linear Single-Parameter Dependent Systems Xiping Zhang Panagiotis Tsiotras Tetsuya Iwasaki School of Aerospace Engineering, Georgia Institute of Technology, Atlanta GA , USA. Department of Mechanical and Aerospace Engineering, University of California at Los Angeles, Los Angeles, CA , USA. Received 00 Month 00x; final version received 00 Month 00x We propose a class of polynomially parameter-dependent quadratic PPDQ Lyapunov functions for assessing the stability of single-parameter dependent linear, time-invariant, s-pdlti systems in a non-conservative manner. It is shown that stability of s-pdlti systems is equivalent to the existence of a PPDQ Lyapunov function. A bound on the degree of the polynomial dependence of the Lyapunov function in the system parameter is given explicitly. The resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex, finite-dimensional optimization problem. Extensions of the approach to state-feedback controller synthesis are also provided. Keywords: Parameter-dependent LTI systems; parameter-dependent Lyapunov functions; linear matrix inequalities. 1 Introduction Consider the single-parameter dependent linear, time-invariant s-pdlti system ẋ = A ρ x, A ρ := Aρ, ρ Ω R, x R n. 1 The stability of 1 for each parameter ρ Ω can be characterized via a parameter-dependent, quadratic Lyapunov function of the form Vx = x T Pρx := x T P ρ x. Since the explicit dependence of the matrix P ρ on the parameter ρ is not a priori evident, one typically starts by postulating a convenient functional dependence of P ρ on ρ, before proceeding to the derivation of the resulting stability conditions. The simplest choice is a constant that is, parameter-independent Lyapunovmatrix.UsingP ρ = P forallρ Ω,onethenobtainsthefollowing, infinite-dimensional family of matrix inequalities that ensure stability of the system 1. P > 0, A ρ P +PA T ρ < 0, ρ Ω The type of stability resulting from is the so-called quadratic stability and it ensures more than just Hurwitz stability of the matrix A ρ for each instance of ρ Ω: it actually guards against Corresponding author. tsiotras@gatech.edu, Tel: , Fax: ISSN: print/issn online c 00x Taylor & Francis DOI: /000717YYxxxxxxxx

2 arbitrarily fast time-variations of ρ. If it is known beforehand that ρ is constant, quadratic stability can thus be overly conservative. It is well known that in order to reduce conservatism, and derive conditions that are close to necessary, one needs to resort to the use of parameterdependent quadratic Lyapunov functions. In light of the previous discussion, given a parameter-dependent matrix P ρ, robust stability of 1 for all ρ Ω can be established if the following two matrix inequalities are satisfied P ρ > 0, A ρ P ρ +P ρ A T ρ < 0, ρ Ω. 3 In order to obtain exact that is, necessary and sufficient conditions for robust stability, the correct form of P ρ must be used in 3. Till very recently, even for the simplest case of affine dependence of the system matrix on the parameter ρ, the parameter dependence of the Lyapunov matrix leading to exact stability conditions for 1 has eluded researchers in the field. Notable exceptions and valid only for the special case when A ρ = A 0 + ρa 1 with ranka 1 = 1 are Narendra and Taylor 1973 and Dasgupta et al Recently, Bliman Bliman 004 showed that linear matrix inequalities LMI s of the form 3 have solutions that depend polynomially on the parameter ρ, provided they are feasible for each parameter value. The proof of this result hinges on Michael s Selection Theorem Michael 1956 in order to construct a continuous solution from the pointwise feasible solutions of 3. Polynomial dependence follows from the uniform approximation of continuous functions with polynomials over compact sets via Weirstrass s approximation theorem. Bliman later Bliman 004 used this result to show that pointwise Hurwitz stability of the affine-in-ρ system 1 ẋ = A ρ x, A ρ := A 0 +ρa 1, ρ Ω 4 is equivalent to the existence of a polynomially parameter-dependent quadratic PPDQ Lyapunov function of the form Vx = x T P ρ x, where P ρ = P 0 +ρp 1 +ρ P +...+ρ N P N = N ρ i P i 5 that satisfies inequalities 3. Moreover, as it was shown in Bliman 004, checking the infinitedimensional family of LMI s in 3 over a compact set can be cast as a finite-dimensional convex feasibility problem in terms of LMI s. This equivalence of the existence of a PPDQ Lyapunov matrix and the pointwise Hurwitz stability of 4 can be made precise: it means that the Lyapunov matrix depends on the parameter ρ via 5 in such a way that, for those values of the parameter for which the matrix A ρ is Hurwitz the stability conditions 3 are satisfied, while for the values of the parameter ρ for which the matrix A ρ is not Hurwitz, the stability conditions 3 fail. This result is in sharp contrast with all prior results, which use parameter-dependent Lyapunov functions to show stability of PDLTI systems Helmersson 1999, Bernstein and Haddad 1990, Haddad and Bernstein 1995, Apkarian 1998, Ramos and Peres 00, Henrion et al. 003, Leite and Peres 003, Ben-Tal and Nemirovski 00, Neto 1999, as these only deal with the sufficiency that is, the easy part of the robust stability problem. The main difficulty with the results in Bliman 004 is that the degree N of the polynomial dependenceof the matrix P ρ on ρ in 5 is not known a priori. What is it shown in Bliman 004 instead, is that there exists a sufficiently large integer N such that 5 will solve 3 if the matrix A ρ is Hurwitz for each instance of ρ Ω. In this paper we remove this restriction. We show that 1 The results in Bliman 004 are, in fact, more general than 4, as they are valid for a matrix A ρ having multi-affine dependence on a parameter vector.

3 3 the bound of the polynomial dependence of the Lyapunov matrix can be explicitly computed. We later extend this result to deal with the case of a matrix A ρ which depends polynomially on ρ, as follows ẋ = A ρ x, A ρ := ν a ρ i A i, ρ Ω. 6 Finally, we illustrate the benefits of the proposed approach for synthesizing parameter-dependent controllers for s-pdlti systems where both the state and the input matrices depend polynomially on the parameter. Extension of these results to the multi-parameter case is hindered by the computational complexity of the problem. An initial attempt towards this direction is reported in Zhang et al This paper offers several extensions and improvements over the results in Bliman 004 or other similar results in the literature Apkarian 1998, Leite and Peres 003, Ramos and Peres 00, Saydy et al. 1988, Henrion et al First, we provide an alternative much more direct derivation of the results in Bliman 004, at least for the single-parameter case. Our method is quite straightforward and it does not require advanced concepts beyond those of an introductory course in linear algebra. Second, and as an added benefit of this approach, we provide an explicit upper bound on the degree of the polynomial dependence of the quadratic Lyapunov function to ensure the stability of 4. This bounddepends on the rank of the matrix A 1. The computational complexityoftheconditionsisreducedastherankofa 1 decreases.finally,weshowthatchecking theconditionsforstability overacompactsetcanbecastasapairoflmi swithout conservatism. These results are extended in order to derive necessary and sufficient Lyapunov-based conditions for the stability of polynomial-in-ρ systems as in 6. We note here that although the stability of s-pdlti systems can also be checked using the guardian map techniques of Saydy et al. 1990, Fu and Barmish 1988 for similar results see also Zhang et al. 00 nonetheless, the Lyapunov-based stability conditions of this paper are also amenable to synthesis see Section 6. Such an extension to the synthesis problem is not directly evident from the use of guardian maps. Therefore, another major contribution of the paper is an approach for synthesizing parameter-dependent state-feedback controllers for polynomial s-pdlti systems. Crucial to our developments is a technical result of independent interest Proposition 3. which is used to test positive-definiteness of a polynomial matrix over a finite interval. Although convex characterizations for positive definiteness of polynomial matrices over infinite or semi-infinite intervals are available Genin et al. 003, Henrion et al. 003, the related problem over finite intervals is quite more involved. Only recently such conditions and for the scalar case only have been proposed Nesterov 000. It is noted that for the multi-variable case, the problem of characterizing the positivity of polynomials is related to the Positivstellensatz and is, in general, an NP-hard problem Parillo 003. In order to keep the derivations as terse as possible and avoid cluttering the paper with unnecessary notation we focus on the affine-in-ρ system 4. The results for the polynomial-in-ρ case 6 follow easily from those of 4. The paper is therefore organized as follows. In the first part of the paper we provide the main theorem Theorem.1 that states that Hurwitz stability of the matrix A ρ = A 0 +ρa 1 for each value of the parameter ρ Ω is equivalent to the existence of a polynomially parameter-dependent quadratic Lyapunov function of a known degree. In the second part of the paper we introduce a new lemma for checking the positive definiteness of a polynomial matrix over a finite interval and we then use this lemma to derive convex feasibility conditions for the pointwise Hurwitz stability of the matrix A ρ = A 0 + ρa 1 over a compact interval. A comparison between the proposed approach and the one given in Bliman 004 in terms of the computational complexity of the two is given next. In the third part of the paper

4 4 we extend these results in order to provide necessary and sufficient conditions for the polynomial s-pdlti system 6. The paper is concluded with an application of the proposed approach to controller synthesis for polynomial s-pdlti systems. For clarity, only the major results are given in the main body of the paper. For the interested reader, most secondary proofs and technical lemmas are given in the Appendix. Main Result In this section we consider LTI systems which depend affinely on a single real parameter, of the form ẋ = A ρ x, A ρ := A 0 +ρa 1, ρ Ω 7 where A 0,A 1 R n n and Ω R. Our main objective is to find computable, non-conservative conditions for checking the asymptotic stability of system 7. The parameter ρ is assumed to be constant and it is chosen from the set Ω. At this point we make no a priori assumptions on the set Ω that is, connected, bounded, compact, etc.. Definition.1 Mustafa 1995 Given a symmetric matrix P = P T R n n, define the vector vecp := P 11,...,P n1,p,...,p n,...,p nn T R 1 nn+1. Note that the usual vec operator Brewer 1978 that stacks the columns of a matrix P on top of each other consists of all the elements of vecp including some repetitions. For every symmetric matrix P = P T R n n, there exists a unique matrix D n of dimension n 1 nn+1 called the duplication matrix Magnus 1988, Mustafa 1995, which is independent of the matrix P and depends only on the dimension n of the matrix P, which satisfies vecp = D n vecp. 8 The Moore-Penrose inverse of D n satisfies the following properties Magnus 1988, Mustafa 1995 vecp = D + n vecp, D+ n D n = I1 nn+1, rankd n = rankd + n = 1 nn+1. Notice, in particular, that D n is always full column rank. Consequently, D + n = DT n D n 1 D T n. Thus, the mapping vec establishes a one-to-one correspondence between the symmetric matrices in R n n and vectors in R 1 nn+1. Definition. Mustafa 1995 Given A R n n, let  R1 nn+1 1 nn+1 be defined by  := D + na AD n, 9 where A A = I n A+A I n is the Kronecker sum of matrix A with itself. The results also hold when ρ varies very slowly so that a quasi-static point of view is valid; see, for example, Brockett 1970.

5 5 It is clear from Definition. that Âρ := A 0 +ρa 1 = Â0 +ρâ1. The first main result of the paper can now be stated as follows: Theorem.1: Given the matrices A 0, A 1 R n n with ranka 1 = r, let A ρ := A 0 +ρa 1 and let Then the following two statements are equivalent: { 1 m s := nr r +r, if r < n, 1 nn+1 1, if r = n. 10 i A ρ = A 0 +ρa 1 is Hurwitz for all ρ Ω. ii There exists a set of m s +1 real symmetric matrices {P i } 0 i ms, such that A ρ P ρ +P ρ A T ρ < 0, ρ Ω 11 ms P ρ = σ ρ ρ i P i > 0, ρ Ω 1 where σ ρ := signdetâ ρ with detâ ρ 0 for all ρ Ω. Proof See the Appendix. Theorem.1 states that checking the stability of 7 is equivalent to the existence of a Lyapunov matrix P ρ satisfying thetwo matrix inequalities 11 and 1 equivalently, 3. That is, if there exists a positive definite matrix P ρ for all ρ Ω such that 11 holds, the matrix A ρ is Hurwitz for each ρ Ω. By the same token and most importantly if for some ρ Ω the matrix A ρ is not Hurwitz, then there exists no positive-definite matrix that satisfies inequality 11. Remark.: Note that if the domain Ω is connected then σ ρ is constant via Corollary A.4 and the Lyapunov matrix in 1 reduces to m s P ρ = ρ i P i, ρ Ω. 13 Example.3 Consider the parameter-dependent matrix A ρ = A 0 +ρa 1, where A 0 = , A 1 = The exact stability domain for A ρ is , ,3.7973, which is computed with the method presented in Zhang et al. 00. In this example the stability domain is composed of two disjoint intervals. Using Theorem.1, we seek a polynomial Lyapunov matrix of degree m s = 1 nn+1 1= 5 since r = ranka 1 = n = 3. The parameter-dependent Lyapunov function satisfying A10 is given by P ρ = σ ρ P 0 +ρp 1 +ρ P +ρ 3 P 3 +ρ 4 P 4 +ρ 5 P 5

6 6 where, P 0 = , P 1 = , P = , P 3 = , P 4 = , P 5 = These values of P 0,P 1,...,P 5 were computed from P i = vec N 1 i veci n, i = 0,1,...,m s were AdjÂ0+ρÂ1 = m s ρi N i, as in the proof of Theorem.1. It can be checked numerically that A ρ P ρ +P ρ A T ρ < 0forall ρ R. However, P ρ is positivedefiniteonlyfor ρ , , , as required by Theorem.1. Remark.4: Compared to other existing methods Barmish 1994, Mustafa and Davidson 1995, Mustafa 1995, Fu and Barmish 1988, Saydy et al. 1988, 1990, which can be only used to check the stability over a connected domain which includes the origin, Theorem.1 can be used to determine the whole stability domain of a PDLTI system, even if this domain is composed of several disjoint intervals of R, as in the case of Example.3. Moreover, the method of Barmish 1994, Fu and Barmish 1988, Saydy et al. 1988, 1990 using guardian maps, although it provides necessary and sufficient conditions for stability, it does not seem easily extendable to controller synthesis. The Lyapunov-base result of Theorem.1 on the other hand can be used to design parameter-varying controllers for s-pdlti plants; see Section 6. 3 A Convex Characterization of the Stability Conditions The analysis of the previous section shows that when the domain Ω is connected, the parameterdependent matrix A ρ = A 0 +ρa 1 is Hurwitz for any ρ Ω, if and only if there exists a Lyapunov matrix which depends polynomially on the parameter ρ, that is, m s P ρ = ρ i P i, 15 such that the corresponding two matrix inequalities P ρ > 0, A ρ P ρ +P ρ A T ρ < 0, 16 are satisfied for each ρ Ω. In order to be able to use the stability criterion of Theorem.1 in practice, we need a relatively simple method to determine the feasibility of the infinite matrix inequalities 16. In the sequel we provide computable, non-conservative conditions to test the matrix inequalities 16 over any compact and connected interval Ω. Without loss of generality we take Ω = [ 1, 1].

7 7 To this end, let the vector ρ [q] R q be defined by ρ [q] := 1 ρ ρ. ρ q 1, 17 and notice that the parameter-dependent matrix in 15 can be rewritten as P ρ = ρ [k] I n T P Σ ρ [k] I n, 18 where k := ms +1 and P Σ = PΣ T Rnk nk is a constant, symmetric matrix here ms denotes the smallest integer which is larger than or equal to ms. Note that the matrix P Σ is not unique. One possible choice is given by P 0 P 1 P P Σ := 1 1 P P 3. P 3 P.. 4, Pm 1 if m s is even, 19 P m 1 P m and P 0 P 1 P Σ := 1 P 1 P P 3 P 3 P P m P m 1 P m P m 0, if m s is odd. 0 On the other hand, for any given symmetric matrix P Σ one can construct a unique polynomial matrix P ρ of the form 15 using the expression 18. A similar result can be shown for the polynomial matrix in 16. This is shown in the following lemma. The following notation is adapted from Bliman 004 Ĵ k := [ I k 0 k 1 ], ˇJk := [ 0 k 1 I k ]. 1 Lemma 3.1 Bliman 004: Given a matrix A ρ = A 0 + ρa 1 R n n and a symmetric, parameter-dependent matrix P ρ R n n as P ρ = ρ [k] I n T P ρ [k] I n, let R ρ := A ρ P ρ +P ρ A T ρ. Then R ρ can be written as R ρ = ρ [k+1] I n T R ρ [k+1] I n

8 8 where, R := H T PF +F T PH 3 H := Ĵk I n 4 F := Ĵk A T 0 + ˇJ k A T 1 5 where Ĵk and ˇJ k as in 1. Notice that the matrix R depends linearly upon each of the matrices P, A 0 and A Exact LMI Conditions for Checking the Stability of PDLTI Systems Using 18 the first inequality in 16 can be rewritten as ρ [k] I n T P Σ ρ [k] I n > 0, ρ Ω. 6 where P Σ as in 19 or 0. Moreover, using Lemma 3.1 the second inequality in 16 can be rewritten as ρ [k+1] I n T R Σ ρ [k+1] I n < 0, ρ Ω, 7 where, R Σ := H T P Σ F +F T P Σ H 8 H := Ĵk I n 9 F := Ĵ k A T 0 + ˇJ k A T 1 30 Proposition 3.: Let Θ R nk nk. Then the matrix inequality ρ [k] I n T Θ ρ [k] I n < 0 31 holds for all ρ [ 1, 1] if and only if there exist matrices D R nk 1 nk 1 and G R nk 1 nk 1 such that D = D T > 0, G+G T = 0, Θ < ] T [ ] [Ĵk 1 I n D G ][Ĵk 1 I n. 3 ˇJ k 1 I n G T D ˇJ k 1 I n Proof The matrix inequality 31 is equivalent to the condition x T ρ [k] I n T Θ ρ [k] I n x < 0, ρ [ 1, 1], x R n 33 Let ζ = ρ [k] I n x and notice that ˇJ k 1 I n ρĵk 1 I n ρ [k] I n = ˇJ k 1 ρĵk 1 I n ρ [k] I n = ˇJ k 1 ρĵk 1ρ [k] I n = 0

9 9 Therefore, ζ satisfies the constraint ˇJ k 1 ρĵk 1 I n ζ = 0 for all real ρ such that ρ 1. Let J = ˇJ k 1 I n and C = Ĵk 1 I n. From Lemma A.7 in the Appendix it follows that any ζ that satisfies the constraint ˇJ k 1 ρĵk 1 I n ζ = J ρcζ = 0 is of the form ζ = ρ [k] I n x for some x R n and ρ [ 1,+1]. Condition 31 is therefore equivalent to the condition ζ T Θζ < 0 with ζ = ρ [k] I n x, x R n. Applying now Lemma A.8, one has that ζ T Θζ < 0 if and only if there exist matrices D R nk 1 nk 1 and G R nk 1 nk 1 such that 3 holds. Example 3.3 Let P ρ = 1+ǫI n ρ I n. It is clear that if ǫ > 0, P ρ is positive definite for all ρ [ 1, 1]. If, on the other hand, ǫ < 0, P ρ is not positive definite for all ρ [ 1, 1]. Rewriting P ρ in the form 18, one obtains P ρ = ρ [] T I n P Σ ρ [] I n [ ] T [ ][ ] In 1+ǫIn 0 In = ρi n 0 I n ρi n 34 and applying Proposition 3., with k =, the condition P ρ > 0 for all ρ [ 1, 1] is equivalent to the existence of matrices D = D T > 0 and G+G T = 0 such that [ ] T [ ] C D G C P Σ < J G D][ T J where J = [0 n n I n ] and C = [I n 0 n n ]. Using 34 the matrix inequality 35 is equivalent to the existence of matrices D = D T > 0 and G+G T = 0 such that 35 [ ] D 1+ǫIn G < G T I n D A necessary condition for the existence of D in 36 is I n < D < 1+ǫI n. When ǫ > 0, such a D always exists and choosing G = 0 the LMI 35 is feasible. When ǫ < 0, no D can satisfy 36 and the LMI is infeasible. For both cases, the result of Proposition 3. agrees with the direct stability analysis. The following is a direct consequence of Proposition 3.. It provides convex conditions in terms of LMI s for checking the robust stability of the parameter dependent matrix A ρ = A 0 +ρa 1 for ρ [ 1,+1]. Theorem 3.4: Let the parameter-dependent matrix A ρ = A 0 + ρa 1, where A 0,A 1 R n n with rank A 1 = r and let k = ms +1 where, { 1 m s := nr r +r, if r < n, 1 nn+1 1, if r = n. 37 Then, A ρ is Hurwitz for all ρ 1 if and only if there exist a structured symmetric matrices P Σ R nk nk as in 19 or 0, a symmetric matrix D R nk nk and a skew-symmetric matrix G R nk nk, such that P 0 > 0, D = D T > 0, G+G T = 0, R Σ < where R Σ = R Σ P Σ as in ] T [ ] [Ĵk I n D G ][Ĵk I n, 38 ˇJ k I n G T D ˇJ k I n

10 10 Proof According to Theorem.1, A ρ is Hurwitz for all ρ 1 if and only if there exists a matrix P ρ which depends polynomially on the parameter ρ of degree m s, where m s as in 37, such that the matrix inequalities 16 are satisfied. From Lemma 3.1 these inequalities can be written in the form 6 and 7, and Proposition 3. shows that the inequalities 6 and 7 are equivalent to the feasibility of the LMI conditions D = D T > 0, G+ GT = 0, P Σ < D = D T > 0, G+G T = 0, R Σ < for some matrices D, G R nk 1 nk 1 and D,G R nk nk. ] T [ ] [Ĵk 1 I n D G ][Ĵk 1 I n, 39 ˇJ k 1 I n G T D ˇJ k 1 I n ] T [ ] [Ĵk I n D G ][Ĵk I n, 40 ˇJ k I n G T D ˇJ k I n Notice now that when A ρ is nominally stable, that is, when the matrix A 0 is Hurwitz, inequality 39isnotnecessary.ThisisduetothefactthatA 0 Hurwitzalongwithinequality40guarantees that P 0 > 0. By the same token, the condition P 0 > 0 along with 40 implies nominal stability A 0. These two conditions also imply that P ρ > 0 for all ρ 1 since A ρ P ρ +P ρ A T ρ < 0 implies nonsingularity of P ρ for all ρ 1; see Iwasaki and Shibata 001. Assuming therefore nominal stability, one can replace inequality 39 with the condition P 0 > 0, and thus 38 follows. Example 3.5 Let A ρ = 1+ǫI +ρi. Here A 0 = 1+ǫI and A 1 = I. It is clear that if ǫ > 0, A ρ is Hurwitz for all ρ [ 1, 1] whereas if ǫ < 0, A ρ is not Hurwitz for all ρ [ 1, 1]. Applying Theorem 3.4 with n = and m s = 1 nn+1 1 = one has P ρ = P 0 +ρp 1 +ρ P = ρ [] I T P Σ ρ [] I R ρ = A ρ P ρ +P ρ A T ρ = ρ [3] I T R Σ ρ [3] I where, P Σ = [ ] P0 0.5P 1 0.5P 1 P 41 and A 0 P 0 +P 0 A T 0 R Σ = 0.5A 0 P 1 +P 1 A T 0 +A 1P 0 0.5A 1 P 1 +P 1 A T 1 +P A T 0 +A 0P A 1 P 1 A 1 P 0 Using the MATLAB TM LMI Toolbox Gahinet et al one can solve 38 with a small positive value of ǫ say, ǫ = to obtain the solution [ ] [ ] [ ] P 0 =, P =, P =, D = , G = On the other hand, for any small negative small value of ǫ say, ǫ = no solution to the inequalities 38 exists. Theorem 3.4 thus gives the same results as the direct stability analysis.

11 11 Example 3.6 Let A ρ = A 0 +ρa 1 where A 0 = , A 1 = Using the method of Zhang et al. 00, one can show that the matrix A ρ is Hurwitz if and only if ρ , In this example, n = 4, r = ranka 1 = and m s = 1 nr r + r = 7. The parameter-dependent Lyapunov matrix P ρ = 7 ρi P i where the matrices P i, i = 0,1,...,7 are given by P 0 = , P 1 = , P = , P 3 = , P 4 = , P 5 = P 6 = , P 7 = 0 4 4, satisfies the matrix inequality A ρ P ρ + P ρ A T ρ < 0 for all ρ R, but it is positive-definite only when ρ , On the other hand, the matrix inequalities 38 are infeasible. This is expected, since [ 1, 1] is not a subset of , Let now A ρ = A 0 + ρa 1 where A 1 = 0.5A 1. The exact stability domain for this matrix is , Applying the algorithm of Theorem 3.4, and using the MATLAB TM LMI Toolbox Gahinet et al. 1995, it can be verified that the inequalities 38 are indeed feasible. This result agrees with the direct analysis, since [ 1, 1] , and thus A 0 +ρa 1 is Hurwitz for all ρ [ 1, 1]. Remark 3.7: The results of Example 3.6 indicate that the bound on the degree of the polynomial dependence for the Lyapunov function given in Theorem.1 is not tight. This is also evident from the results of Narendra and Taylor 1973 for the case when r = 1. According to Narendra and Taylor 1973, p. 79 a Lyapunov matrix with a linear dependence characterizes stability whenranka 1 = 1. Theorem.1 on theother handrequires a Lyapunov matrix of degree n. Even when r = ranka 1 > 1, the polynomial dependence of the Lyapunov matrix that proves stability may be smaller than the bound given in 37. To see this, let T ρ be a parameterdependent transformation such that the matrix Ãρ := T 1 ρ A ρt ρ is in companion form 3, that 3 The transformation to the companion form imposes the restriction that the matrix A ρ is nonderogatory for all ρ Ω.

12 1 is, Ã ρ = a n ρ a n 1 ρ a n ρ a 1 ρ It can be shown that a i ρ is a polynomial in ρ of degree r for all i = 1,...,n. The results of Parks 196 and Mori and Kokame 1995 see also Narendra and Taylor 1973, p. 81 as well as Henrion 000 and Henrion et al. 004 can be used to show that a Lyapunov matrix, say P ρ, of degree no larger than r characterizes Hurwitz stability of 44 for each ρ. Notice, however, that this Lyapunov matrix does not characterize stability of the original matrix A ρ. Such a matrix is given by P ρ := T ρ Pρ T T ρ which, in general, may not be polynomial in ρ. Even if the Lyapunov matrix P ρ is polynomial, its degree will most likely be higher than the degree of P ρ. If, on the other hand, A ρ is in companion form to begin with, then this approach provides a quadratic-in-ρ Lyapunov matrix since r = ranka 1 = 1 in this case although a linear-in-ρ Lyapunov matrix will suffice as mentioned previously. Thus, a general solution for the upper bound on the polynomial degree of the Lyapunov matrix that characterizes stability of the matrix A ρ = A 0 +ρa 1 seems to remain open. 4 Computational Complexity of the Proposed LMI Conditions In this section we investigate the computational complexity of the LMI conditions of Theorem 3.4. In particular, we compare the LMI conditions 38 with those of Bliman 004 and show that, in general, the LMI conditions proposed herein are more efficient than the ones given in Bliman 004. We also propose a modification to the LMI conditions in Bliman 004 to make them competitive with those of Theorem 3.4. Let us consider again the s-pdlti system 7, rewritten below for convenience ẋ = A ρ x := A 0 +ρa 1 x, ρ In the context of parameter-dependent Lyapunov function analysis, Bliman Bliman 004 proposed a nice method to convert the robust stability problem45 with real parametric uncertainty to a robust stability problem with a complex uncertainty. The approach of Bliman 004 hinges on the observation that { ρ R : ρ 1 } = { z + z : z C, z = 1 }. 46 Substitution of ρ = z + z/ into 45 yields an n th order auxiliary system of the form z + z ẋ = Ax, A = A 0 + A 1, z C, z = 1, 47 where z is a complex parameter. Clearly, system 45 is robustly stable against the real uncertainty ρ 1 if and only if the system 47 is robustly stable with respect to the complex uncertainty z = 1. For the single real parameter case the main result in Bliman 004 can be briefly summarized

13 13 as follows. In the following, we assume for simplicity that N is an odd integer and we let q := N Also, for a square matrix M, we define HeM := M +M, where M is the complex conjugate of M. Bliman Bliman 004 developed a robust stability condition for 47, and hence also for 45, in terms of the following polynomially parameter-dependent quadratic PPDQ Lyapunov function Vx := x T Px, P := z [N+1] I n Pz [N+1] I n 49 where P R qn qn is a constant symmetric matrix. Note that [ HeAP = He A 0 + where, R := He z + z ] A 1 z [q] I n Pz [q] I n = z [q+1] I n Rz [q+1] I n [Ĵq I n T PĴq A T0 + 1 Ĵq I n T PˇJ q A T1 + 1 ˇJ q I n T PĴq A T1 ] 50 The condition HeAP < 0 for all z = 1 can be converted to an LMI see, for instance Iwasaki and Hara 003, and a robust stability condition can be given as follows. Theorem 4.1: Consider the uncertain system in 47. Let a positive integer N, and a positive definite matrix P R qn qn be given, and define R by 50. The following statements are equivalent. i The function defined in 49 is a Lyapunov function that proves stability of 47, i.e., P > 0, AP +PA T < 0, z C, z = 1. ii There exists a symmetric matrix D R qn qn such that R < ] T [ ] [Ĵq I n D 0 ][Ĵq I n ˇJ q I n 0 D ˇJ q I n 51 Bliman s robust stability condition Bliman 004, Th. 4.3 for the single parameter case can then be stated as follows. Corollary 4. Bliman 004: If for some non-negative integer N there exist symmetric matrices P,D R qn qn satisfying 51 and P > 0 5 then the system 47 is robustly stable. Moreover, if the system 47 is robustly stable then there exist symmetric matrices P,D R qn qn satisfying 51 and P > 0, for sufficiently large N. Our result of Theorem 3.4, on the other hand, considers the following PPDQ Lyapunov function

14 14 for the uncertain system in 45: Vx := x T Px, P := N ρ i P i. 53 Theorem 3.4 shows that the satisfaction of the LMI s 38 are sufficient for robust stability of 45 for any non-negative integer N. It is also necessary for N greater than the bound given in 10. At this point, it is temptingto say that theboundobtained in Theorem.1 also provides abound on the degree N of the PPDQ Lyapunov function 49 such that the condition of Corollary 4. is not only sufficient but also necessary for robust stability of 45. However, this does not seem to be the case due to the constraint P > 0 in 5. In the sequel, we shall elaborate on this point and show how our result of Theorem 3.4 relates to Corollary 4.. By Theorem 3.4, the system 45 is robustly stable if and only if there exists a PPDQ Lyapunov function53 ofdegreen > m s wherem s isaknownintegergivenby10.withthereal/complex conversion ρ := z + z/ discussed earlier, this Lyapunov function can be represented as P = N z + z i P P i = z [N+1] I n z [N+1] I n 54 for some symmetric matrix P R qn qn. It is not difficult to give an explicit representation of P in terms of P i ; see, for example, 19 or 0. Note that P is now structured and the number of independent variables is the same as that of the P i, i.e., N +1nn+1/ = qnn+1, which is smaller than the number qnqn+1 for the unstructured case of equation 49. Since the Lyapunov matrix in 53 can be represented as 49 with structured P, after applying Bliman s real/complex conversion, a new robust stability condition is obtained as follows. Theorem 4.3: The system 45 is robustly stable if and only if there exist symmetric matrices P i = P T i R n i = 0,1,...,m s and D = D T R qn qn such that P 0 > 0, R < where P is defined by 54 in terms of P i. ] T [ ] [Ĵq I n D 0 ][Ĵq I n ˇJ q I n 0 D ˇJ q I n We now have three robust stability conditions: Bliman s original condition Corollary 4., Bliman s structured Theorem 4.3, and ours Theorem 3.4. For any nonnegative integer N, each of the latter two is equivalent to the existence of a PPDQ Lyapunov function 53 that proves robust stability of 45. Moreover, Theorem 3.4 and Theorem 4.3 give necessary and sufficient conditions for robust stability of 45 if the degree N is greater than the known bound m s from Theorem.1. However, it is not clear whether Bliman s original condition in Corollary 4. is also necessary whenever N > m s, due to the positivity constraint P > 0. Without P > 0, the condition is weaker than the one of Theorem 4.3 and hence it is necessary, but the additional constraint P > 0 may destroy thenecessity unlessn is chosen sufficiently large as in Bliman 004. For the unstructured PPDQ Lyapunov function in 49, it seems difficult to relax the constraint P > 0 to be P 0 > 0 as in Theorem 3.4 due to the fact that the nominal case z = 0 is not included in the set of uncertainties characterized by z = 1. However, one may choose the nominal case 55

15 15 of 45 to be ρ = 1, or z = 1, and replace P > 0 by TP 1 [N+1] I n 1 [N+1] I n > With this substitution, the conditions of Corollary 4. become necessary and sufficient whenever N > m s. Assuming the same bound N = m s for all cases, the computational complexity of the three robust stability conditions can now be assessed in terms of the number of variables and the LMI dimensions. The result is summarized below, where p := qn = N +1n/. Table 1. Comparison of computational complexity of available methods. Stability Condition no. of variables LMI dimensions Theorem 3.4 pn+1+p p+n, n, p Theorem 4.3 pn+1+pp+1 p+n, n Corollary 4. pp+1+pp+1 p+n, p Thus the LMI conditions 38 of Theorem 3.4 seem computationally most efficient among all the three, as they have the least variables and the smallest LMI dimension considering two LMI s of dimensions n 1 and n are smaller than one LMI of dimension n 1 +n. In summary, we have the following observations relating our stability conditions to the ones in Bliman 004, for the single real parameter case: i our conditions do not seem to provide an explicit lower bound on the degree N of the unstructured PPDQ Lyapunov function 49 such that the conditions 51 and 5 are not only sufficient but also necessary for robust stability of 45; ii our conditions give a bound on the degree of Bliman s unstructured PPDQ Lyapunov function 49 if P > 0 in 5 is replaced by 56; iii our conditions give a bound on the degree of Bliman s PPDQ Lyapunov function 49 if P is structured as in 54 and P > 0 in 5 is replaced by P 0 > 0 as in 55; iv finally, our proposed robust stability conditions are of lower computational complexity. The complexity reduction is achieved due to two main factors: one is the degree bound on the PPDQ Lyapunov function, and the other is the direct application of the D, G-scaling, rather than the real/complex conversion followed by application of the D-scaling. 5 Stability Conditions for s-pdlti Systems with Polynomial Parameter Dependence In this section we deal with a system having a polynomial dependence on a single parameter as follows ẋ = A ρ x, A ρ := ν a ρ i A i, ρ Ω 57 where A i R n n for i = 0,1,...,ν a. Theorem.1 can be extended to the system 57 as follows. Theorem 5.1: Consider the polynomially parameter-dependent matrix 57 and assume that dim[nâ1 NÂ NÂν a ] = l. Then the following two statements are equivalent: i A ρ := ν a ρi A i is Hurwitz for all ρ Ω.

16 16 ii There exists a set of m p +1 real symmetric matrices {P i } 0 i mp, such that A ρ P ρ +P ρ A T ρ < 0, ρ Ω 58 mp P ρ = σ ρ ρ i P i > 0, ρ Ω 59 where m p := ν a min{ 1 nn+1 1, 1 nn+1 l} 60 and where σ ρ = signdetâρ with detâρ 0 for all ρ Ω. Moreover, if Ω is connected, without loss of generality we can take σ ρ = +1 for all ρ Ω. Proof It easy toshow thatâρ = ν a ρi  i. ThenusingLemmaA.6 intheappendixone obtains that ν a m p Adj ρ i  i = ρ i N i, for some constant matrices N i, i = 1,,...,m p where m p as in 60. The rest of the proof now follows as in Theorem.1. In the sequel we assume that Ω is compact and connected. Without loss of generality we may take Ω = [ 1,+1]. Using the fact that the polynomial matrix P ρ in 59 can be written as P ρ = ρ [k] T I n P Σ ρ [k] I n where k = m p + 1 for some P Σ R nk nk, we can easily generalize Lemma 3.1 to show that the matrix in 58 can be written compactly as follows where, R ρ := A ρ P ρ +P ρ A T ρ = ρ [k+νa] I n T R Σ ρ [k+ν a] I n R Σ := H T P Σ F +F T P Σ H 6 H := ĴkĴk+1 Ĵk+ν a 1 I n 63 ν a F := ĴkĴk+1 Ĵk+ν a 1 A T 0 + ˇJ k ˇJk ˇJ k+i 1 Ĵ k+i Ĵk+ν a 1 A T 1 64 i=1 61 We are now ready to provide a necessary and sufficient condition for the stability of 57 for ρ 1 in terms of LMIs. Corollary 5.: Let the parameter-dependent matrix A ρ = ν a ρi A i, where A i R n n with dim[nâ1 N NÂν a ] = l and let κ := mp +ν a where, m p := ν a min{ 1 nn+1 1, 1 nn+1 l} 65 Then, A ρ is Hurwitz for all ρ 1 if and only if there exist symmetric matrices P Σ

17 17 R nκ νa+1 nκ νa+1, D R nκ nκ and a skew-symmetric matrix G R nκ nκ, such that P 0 > 0, D = D T > 0, G+G T = 0, R Σ < where R Σ = R Σ P Σ as in ] T [ ] [Ĵκ I n D G ][Ĵκ I n, 66 ˇJ κ I n G T D ˇJ κ I n 6 State-Feedback Controller Synthesis The results of the previous section can be used to design parameter-dependent state-feedback controllers for the controlled s-pdlti system ẋ = A ρ x+b ρ u, ρ Ω 67 where A ρ R n n and B ρ R n m are parameter-dependent matrices n a n b A ρ := ρ i A i, B ρ := ρ i B i 68 and Ω is a compact and connected set. This is shown in the next theorem. Theorem 6.1: The following statements are equivalent: i There exists a polynomially parameter-dependent state feedback controller u = K ρ x that stabilizes the system for all ρ Ω. ii There exist an integer m p and symmetric matrices P i, i = 1,...,m p such that m p P ρ := ρ i P i, P ρ > 0, A ρ P ρ +P ρ A T ρ < B ρbρ T, ρ Ω. 69 If statement i holds for a gain matrix K ρ R m n of degree n k, then statement ii holds with m p ν a 1 nn+1 1 where ν a := max{n a,n b +n k }. Conversely, if ii holds, then a stabilizing state feedback gain in i can be given by K ρ = µ ρ B T ρp 1 ρ, µ ρ := detp ρ /ǫ, ǫ := min ρ Ω detp ρ 70 which is a polynomial matrix of degree n k n b +m p n a 1. Proof Suppose i holds for a matrix K ρ of degree n k. Then the closed-loop system is described by 57 with A ρ := A ρ + B ρ K ρ. The matrix A ρ has degree ν a := max{n a,n b + n k }. Then Theorem 5.1 implies satisfaction of for some P ρ of degree m p ν 1 a nn+1 1. Let now N ρ be the null space of Bρ T. Then multiplying the Lyapunov inequality in 58 by N ρ from the right and by Nρ T from the left, one obtains N T ρa ρ P ρ +P ρ A T ρn ρ < 0. By Finsler s theorem Skelton et al. 1997, there exists τ ρ > 0 such that A ρ P ρ +P ρ A T ρ < τ ρb ρ Bρ T

18 18 for each ρ Ω. Since Ω is compact and τ ρ is a continuous function of ρ see Lemma A.9 in the Appendix, τ max := max ρ Ω τ ρ > 0 is well defined. Then, redefining P ρ to be P ρ /τ max, we have 69. Thus we have i ii. The converse can be proved by direct substitution. Specifically, with the control gain matrix 70 one obtains A ρ +B ρ K ρ P ρ +P ρ A ρ +B ρ K T ρ = A ρ P ρ +P ρ A T ρ µ ρ B ρ B T ρ < 0 for all ρ Ω, where it is noted that µ ρ 1 by definition. This proves that K ρ is stabilizing. Since degµ ρ nm p see A16 from the expression K ρ P ρ = µ ρ Bρ T it follows that n k +m p n a m p +n b. Therefore, n k n b +m p n 1. The synthesis condition 69 is a pair of polynomially-parameterized LMI s and hence it can be converted exactly to finite dimensional LMI s by eliminating the parameter ρ using Proposition 3.. The details are left to the reader. Example 6. Consider the following controlled polynomial s-pdlti system A ρ = [ ] 1 +ρ 1 [ ] 0, B 0 1 ρ = [ ] +ρ 0 [ ] The uncontrolled system is stable for ρ < and unstable otherwise. Using the Theorem 6.1 we can design the following parameter-dependent feedback gain that will ensure that the closed-loop system is stable for all ρ [ 1,+1] K ρ = [ ] +ρ [ ] +ρ [ ] The polynomial Lyapunov matrix that ensures the stability of the closed-loop system is computed by the solution of the LMI problem 69 and is given by P = [ ] [ ] ρ for this example ǫ = The closed-loop system matrix is third order and is given by A cl = [ ] [ ] [ ] [ ] ρ +ρ ρ It can be easily verified that this matrix is Hurwitz for all ρ [ 1,+1]. 7 Conclusions In this paper we propose a class of parameter-dependent Lyapunov matrices P ρ = Pρ, which can be used to test the stability of single parameter-dependent linear, time-invariant, s-pdlti systems of the form ẋ = ν a ρi A i x where ρ Ω. The proposed Lyapunov matrix has polynomial dependence on the parameter ρ of a known degree and can be used to derive exact that is, necessary and sufficient conditions for the stability of s-pdlti systems. We show that checking these conditions over a compact interval can be cast as a finite-dimensional convex programming problem in terms of linear matrix inequalities without conservatism. Finally, using these results we provide an extension to state-feedback control design for s-pdlti systems using parameter-varying gains. Specifically, our synthesis result Theorem 6.1 can be useful for designing gain-scheduled controllers for LPV systems with a slowly varying parameter. This is

19 REFERENCES 19 because conditions for each frozen parameter ρ Ω are necessary and sufficient for robust stability of 57 with respect to an arbitrarily slowly varying parameter ρt Ω. Furthermore, the result of Theorem 6.1 can be used for on-site parameter tuning for plants containing a constant parameter whose value is unknown but its range Ω is known at the time of control design. Then one may design a class of controllers parameterized by ρ Ω and then tune the parameter ρ on site when the controller is actually implemented. Acknowledgement: The authors would like to acknowledge insightful discussions with Dr. P.-A. Bliman on the stability of parameter-dependent LTI systems. References Apkarian, P. 1998, Advanced Gain-Scheduling Techniques for Uncertain Systems, IEEE Transactions on Control Systems Technology, 6, 1 3. Barmish, B.R., New Tools for Robustness of Linear Systems, Macmillan Publishing Company Ben-Tal, A., and Nemirovski, A. 00, On Tractable Approximations of Uncertain Linear Matrix Inequalities Affected by Interval Uncertainty, SIAM Journal on Optimization, 1, Bernstein, D.S., and Haddad, W.M. 1990, Robust Stability and Performance Analysis for State Space System via Quadratic Lyapunov Bounds, SIAM Journal of Matrix Analysis and Application, 11, Bliman, P.A.004, A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters, SIAM Journal on Control and Optimization, 4, Bliman, P.A. 004, An Existence Result for Polynomial Solutions of Parameter-Dependent LMIs, Systems & Control Letters, 51, Brewer, J.W.1978, Kronecker Products and Matrix Calculus in System Theory, IEEE Transactions on Circuits and Systems, 5, Brockett, R.W., Finite Dimensional Linear Systems, John Wiley & Sons, Inc Dasgupta, S., Anderson, B.D.O., and Fu, M. 1994, Lyapunov Functions for Uncertain Systems with Applications to the Stability of Time Varying Systems, IEEE Transactions on Circuits and Systems, 41, Fu, M., and Barmish, B.R. 1988, Maximal Unidirectional Perturbation Bounds for Stability of Polynomials and Matrices, Systems and Control Letters, 11, Gahinet, P., Nemirovskii, A., Laub, A., and Chilali, M., LMI Control Toolbox, The Mathworks, Inc Genin, Y., Hachez, Y., Nesterov, Y., and van Dooren, P. 003, Optimization Problems over Positive Pseudopolynomial Matrices, SIAM Journal of Matrix Analysis and Applications, 5, Haddad, W.M., and Bernstein, D.S. 1995, Parameter-dependent Lyapunov Functions and the Popov Criterion in Robust Analysis and Synthesis, IEEE Transactions on Automatic Control, 40, Helmersson, A. 1999, Parameter Dependent Lyapunov Functions Based on Linear Fractional Transformation, in 14th World Congress of IFAC, July, Vol. F, Beijing, China: IFAC, pp Henrion, D. Dec. 000, Quadratic Matrix Inequalities and Stability of Polynomials, Technical Report LAAS-CNRS 0054, Laboratoire d Analyse et d Architecture des Systèmes, Toulouse, France. Henrion, D., Arzelier, D., and Peaucelle, D. 003, Positive Polynomial Matrices and Improved LMI Robustness Conditions, Automatica, 39, Henrion, D., Arzelier, D., Peaucelle, D., and Lasserre, J.B. Feb. 004, On Parameter-

20 0 REFERENCES Dependent Lyapunov Functions for Robust Stability of Linear Systems, Technical Report, Laboratoire d Analyse et d Architecture des Systèmes, Toulouse, France. Horn, R.A., and Johnson, C.R., Matrix Analysis, Cambridge, United Kingdom: Cambridge University Press Iwasaki, T., and Hara, S. August 003, Generalized KYP Lemma: Unified Characterization of Frequency Domain Inequalities with Applications to System Design, Technical Report METR-003-7, School of Information Science and Technology, The University of Tokyo, Japan. Iwasaki, T., Meinsma, G., and Fu, M. 000, Generalized S-procedure and Finite Frequency KYP Lemma, Mathematical Problems in Engineering, 6, Iwasaki, T., and Shibata, G. 001, LPV System Analysis via Quadratic Separator for Uncertain Implicit Systems, IEEE Transactions on Automatic Control, 46, Kailath, T., Linear Systems, Englewood Cliffs, New Jersey: Prentice-Hall Leite, V., and Peres, P.003, An Improved LMI Condition for Robust D-Stability of Uncertain Polytopic Systems, IEEE Transactions on Automatic Control, 48, Magnus, J.R., Linear Structures, London: Charles Griffin Michael, E. 1956, Continuous selections I, Annals of Mathematics, 63, Mori, T., and Kokame, H. 1995, Polytopes of Lyapunov Functions for Polytopes of Polynomials, Syst. Control Lett., 5, Mustafa, D. 1995, Block Lyapunov Sum with Applications to Integral Controllability and Maximal Stability of Singularly Perturbed Systems, International Journal of Control, 61, Mustafa, D., and Davidson, T.N. 1995, Block Bialternate Sum and Associated Stability Formulae, Automatica, 31, Narendra, K.S., and Taylor, J.H., Frequency Domain Criteria for Absolute Stability, New York, NY: Academic Press Nesterov, Y. 000, Squared Functional Systems and Optimization Problems, in High Performance Optimization eds. H. Frenk, K. Roos and T. Terlaky, Dordrecht, the Netherlands: Kluwer Academic Publishers, pp Neto, A.T. 1999, Parameter-Dependent Lyapunov Functions for a Class of Uncertain Linear Systems: an LMI Approach, in Proceedings of the IEEE 38th Conference on Decision and Control, Dec., Phoenix, AZ, pp Parillo, P. 003, Semidefinite Programming Relaxations for Semianalytic Problems, Mathematical Programming, Series B, 96, Parks, P.C. 196, A New Proof of the Routh-Hurwitz Stability Criterion Using the Second Method of Lyapunov, Cambridge Philosophical Society, 58, Ramos, D., and Peres, P. 00, An LMI Condition for the Robust Stability of Uncertain Continuous-Time Linear Systems, IEEE Transactions on Automatic Control, 47, Saydy, L., Tits, A.L., and Abed, E.H. 1988, Robust Stability of Linear Systems Relative to Guarded Domains, in Proceedings of the 7th IEEE Conference on Decision and Control, Austin, TX, pp Saydy, L., Tits, A.L., and Abed, E.H. 1990, Guardian Maps and the Generalized Stability of Parametrized Families of Matrices and Polynomials, Mathematics of Control, Signals and Systems, 3, Skelton, R.E., Iwasaki, T., and Grigoriadis, K.M., A Unified Algebraic Approach to Linear Control Design, New York: Taylor & Francis Zhang, X., Lanzon, A., and Tsiotras, P. 00, On Robust Stability of LTI Parameter Dependent Systems, in 10th Mediterranean Conference on Control and Automation, July, Lisbon, Portugal. Zhang, X., Tsiotras, P., and Bliman, P.A. 005, Multi-Parameter Dependent Lyapunov Functions for the Stability Analysis of LTI Parameter-Dependent Systems, in Proceedings of the 13th IEEE Mediterranean Conference on Control and Automation, Limassol, Cyprus, pp.

21 REFERENCES 1 Appendix A: Proofs Before providing the proof of Theorem.1, we need to first establish a series of technical lemmas. The following lemma is a direct consequence of the properties of the determinant of a matrix. It can be found, for instance in Kailath 1980, p Lemma A.1: Let matrices A,B R n n with rankb = r and let ρ R. Then degdeta + ρb r. The following lemma states that the adjoint of the parameter-dependent matrix A + ρb is a matrix polynomial in ρ of a certain maximal degree which depends on the rank of the matrix B. Recall that given an invertible matrix A R n n, its inverse can be calculated from A 1 = AdjA/detA where AdjA is the adjoint of A. Lemma A.: Given matrices A,B R n n with rankb = r and ρ R, the adjoint of the matrix A+ρB is a matrix polynomial in ρ of degree at most min{r,n 1}, i.e., AdjA+ρB = min{r,n 1} ρ i N i. A1 Proof From the definition of the adjoint of a matrix Horn and Johnson 1991 it follows that [ AdjA+ρB ] ij = 1i+j deta+ρb [ji], 1 i,j n where [ji] is the n 1 n 1 submatrix of in which the j-th row and the i-th column are eliminated and [ ]ij is the ij th element of the matrix [ ]. Since rank B [ji] rankb = r it follows that rankb [ji] min{r,n 1}, 1 i,j n From Lemma A.1, and since A+ρB [ji] R n 1 n 1, it follows that deg deta+ρb [ji] = deg deta[ji] +ρb [ji] rankb[ji] min{r,n 1}, 1 i,j na3 From A and A3 it follows that AdjA + ρb is a matrix-valued polynomial of degree min{r,n 1} and hence, there exist constant matrices {N i },1,...,min{r,n 1} such that A1 holds. We also note that the matrices N i in A1 can be calculated explicitly from the matrices A and B. The details are left to the reader. Lemma A.3 Magnus 1988, Mustafa 1995: Given A R n n and  as in Definition., the eigenvalues of  are the 1 nn + 1 numbers λ i + λ j, 1 j i n where λ i, λ j are the eigenvalues of A. The following is immediate from Lemma A.3. A