In this paper we study stability radii of positive polynomial matrices under affine ...... and R. J. Plemmons, Nonnegative Matrices in Mathematical Sciences,. Acad.
Explicit formulae for stability radii of positive polynomial matrices D.Hinrichsen Institut f¨ ur Dynamische Systeme Universit¨at Bremen 28334 Bremen, Germany
Nguyen Khoa Son Institute of Mathematics P.O. Box 631, Bo Ho 10000 Hanoi,Vietnam
Pham Huu Anh Ngoc∗ Department of Mathematics University of Hue, 32 Le Loi Str. Hue City, Vietnam
Abstract In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.
Keywords : Positive system, polynomial matrix, robust stability, stability radius .
1
Introduction
A dynamical system with state space Rn is called positive if any trajectory of the system starting at an initial state in the positive orthant Rn+ remains forever in Rn+ . Positive dynamical systems play an important role in the modeling of dynamical phenomena whose variables are restricted to be nonnegative. This model class is used in many areas such as economics, populations dynamics and ecology, see [2], [9]. The mathematical theory of positive systems is based on the theory of nonnegative matrices founded by Perron and Frobenius at the beginning of the past century. As references we mention [1], [3]. In this paper we will study robust stability of discrete time linear systems described by higher order difference equations of the form Aν y(t + ν) = Aν−1 y(t + ν − 1) + · · · + A1 y(t + 1) + A0 y(t),
t∈N
(1)
where we suppose det Aν = 0. Such a system is called positive if every solution y(·) whose first ν values y(0), . . . , y(ν − 1) lie in the positive orthant, remains there for all ∗
The research was carried out while on the visit at the Institut f¨ ur Dynamische Systeme, University of Bremen, under grant from the German Academic Exchange Service (DAAD), for which this author is most grateful.
1
later times. It is easily seen that this condition is equivalent to the nonnegativity of the −1 matrices A−1 ν A0 , . . . Aν Aν−1 . The above difference equation is called asymptotically stable if all the solutions y(t) of (1) tend to zero as t → ∞. An equivalent condition is that the associated polynomial matrix P (z) = Aν z ν − Aν−1 z ν−1 · · · − A0 is Schur stable in the sense that all the roots of p(z) = det P (z) lie in the complex open unit disk. Now assume that the coefficient matrices Ai , i ∈ ν := {0, 1, . . . , ν} are subjected to affine perturbations of the form Ai Ai (∆) = Ai + Di ∆i Ei ,
i∈ν
(2)
where the matrices Di , Ei (determining the structure of the perturbations) are given and the matrices ∆i ∈ Cli ×qi are unknown disturbance matrices. The object of this paper is to determine the stability radii [5] of positive Schur stable polynomial matrices or, equivalently, positive higher order difference equations (1), under perturbations of the above form with nonnegative structure matrices Di , Ei . The complex (resp. real) stability radius of the system (1) under perturbations of the form (2) is by definition the supremal value of r > 0 such that all the difference equations with perturbed coefficients Ai (∆) are stable whenever the overall perturbation ∆ = (∆0 , . . . , ∆ν ) is of norm γ(∆) = νi=0 ∆i < r (and real). It is known that for positive state space systems which are perturbed by affine perturbations with nonnegative structure matrices, the real and complex stability radii coincide and can be computed by a simple formula [6]. In the present paper we will extend these results to higher order difference equations of the above form. Since the modelling of discrete time systems often results in higher order difference equations [11], it is useful to have results available which can be directly applied without prior transformation into state-space form. Stability criteria for higher order difference and differential equations can be found in [12]. The complex stability radius of such systems was first investigated in [10] under the condition that the leading coefficient matrix Aν is the identity and remains unperturbed (monic case). Real and complex stability radii of matrix polynomials P (z) under unstructured perturbations of all the coefficient matrices including the leading one have recently been analyzed in [4]. In the present paper we will consider structured perturbations of all the coefficient matrices, but will mainly deal with the special case of positive systems. We proceed as follows. After recalling some preliminary results on nonnegative matrices in the next section, we first deal with arbitrary asymptotically stable difference equations of the form (1) and derive a computable formula for their complex stability radii under structured perturbations of the coefficient matrices (section 3). In section 4 we specialize to the positive case and show that for arbitrary affine perturbations with nonnegative structure matrices the real and the complex stability radii coincide and can be computed by simple formulae. The results are illustrated by two examples.
2
Preliminaries
Let K = C or R and n, l, q be positive integers. Inequalities between real matrices or vectors will be understood componentwise, i. e. for two real l × q-matrices A = (aij ) and B = (bij ), the inequality A ≥ B means aij ≥ bij for i = 1, · · · , l, j = 1, · · · , q. The 2
n l×q set of all nonnegative l × q-matrices is denoted by Rl×q we + . If x ∈ K and P ∈ K define |x| = (|xi |) and |P | = (|pij |) . Then |CD| ≤ |C||D|. For any matrix A ∈ Kn×n the spectral radius of A is denote by ρ(A) = max{|λ| : λ ∈ σ(A)}, where σ(A) := {z ∈ C : det(zIn − A) = 0} is the set of all eigenvalues of A. The spectral radius has the following monotonicity property, see e.g. [6].
∀ C, D ∈ Cn×n : |C| ≤ |D| ⇒ ρ(C) ≤ ρ(|C|) ≤ ρ(|D|).
(3)
A norm · on Kn is said to be monotonic if |x| ≤ |y| implies x ≤ y for all x, y ∈ Kn . Every p-norm on Kn , 1 ≤ p ≤ ∞, is monotonic. Throughout the paper, the norm M of a matrix M ∈ Kl×q is always understood as the operator norm defined by M = maxy=1 My where Kq and Kl are provided with some monotonic vector norms. Then, the operator norm · has the following monotonicity property, see e.g. [6], P ∈ Kl×q , Q ∈ Rl×q + , |P | ≤ Q ⇒ P ≤ |P | ≤ Q.
(4)
For later use, we summarize some basic properties of nonnegative matrices which will be used in the sequel (see [1], [9], [6]). Theorem 2.1. Let A ∈ Rn×n + , t ∈ R . Then (i) (Perron-Frobenius) ρ(A) is an eigenvalue of A and there exists a nonnegative eigenvector x ≥ 0, x = 0 such that Ax = ρ(A)x. (ii) Given α ∈ R, α > 0, there exists a nonzero vector x ≥ 0 such that Ax ≥ αx if and only if ρ(A) ≥ α. (iii) (tIn − A)−1 exists and is nonnegative if and only if t > ρ(A). n×n . Then (iv) Given B ∈ Rn×n + ,C ∈ C
|C| ≤ B
3
=⇒
ρ(A + C) ≤ ρ(A + B).
Complex stability radii of high order difference equations
Consider the ν-th order linear difference equation of the form (1) where A0 , A1 , ..., Aν ∈ Rn×n are given matrices. With this equation, we associate the polynomial matrix P (z) := Aν z ν − Aν−1 z ν−1 − · · · − A0 .
(5)
Denote by σ(P (·)) the set of all roots of the characteristic equation of (1), that is σ(P (·)) = {λ ∈ C : det P (λ) = 0}, and let ρ(P (·)) := sup{|λ| : λ ∈ σ(P (·))}. Then, σ(P (·)) and ρ(P (·)) are called the spectrum and spectral radius of the polynomial matrix P (·), respectively. If det P (z) ≡ 0 then σ(P (·)) = C, otherwise the spectrum of P (·) is a finite subset of C consisting of at most deg det P (z) “eigenvalues” of (1). 3
Definition 3.1. A polynomial matrix P (·) of the form (5) is called Schur stable if σ(P (·)) ⊂ C1 := {z ∈ C : |z| < 1}, or equivalently ρ(P (·)) < 1. Remark 3.2. If det P (z) ≡ 0 the difference equation (1) has a finite dimensional solution set (i.e. defines an autonomous behaviour, in terms of behavioural system theory [11]) and can be transformed equivalently into a first order difference equation x(t) = Ax(t) (see e.g. [13]). It is known that λ ∈ σ(A) if and only if det P (λ) = 0 (see [12]). Hence the state space system described by x(t) = Ax(t) is asymptotically stable if and only if the polynomial matrix P (z) is Schur stable. If det Aν = 0, an equivalent state space system is easily determined by introducing the state vector x(t) = [y(t − ν + 1)T , y(t − ν + 2)T , ..., y(t)T ]T : 0 In 0 ... 0 0 0 0 0 In ... 0 .. .. .. ... .. .. x(t + 1) = Ax(t), t ∈ N ; A = . (6) 0 0 0 ... 0 In −1 −1 −1 Aν Aν−1 Aν Aν−2 .. ... .. Aν A0 If det Aν = 0, the construction of an equivalent state space system is more complicate (see [13]). We now assume that the polynomial matrix (5) is Schur stable and the coefficient matrices A0 , A1 , ..., Aν are subjected to structured perturbations of the following type Ai Ai (∆) := Ai + Di ∆i Ei ,
∆i ∈ Cli ×qi ,
(7)
where Di ∈ Rn×li , Ei ∈ Rqi ×n , i ∈ ν are given matrices defining the structure of the perturbations and ∆i , i ∈ ν are unknown disturbance matrices. The linear space ∆K = Kl0 ×q0 ×...×Klν ×qν of all pertubation families ∆ = (∆0 , ..., ∆ν ), with ∆i ∈ Kli ×qi is endowed with the norm γ(∆) = γ(∆0 , ..., ∆ν ) = νi=0 ∆i , (8) where the norms ∆i are operator norms on Kli ×qi , induced by given monotonic vector norms on the spaces Kli , Kqi , i ∈ ν (K = R, C). We denote the perturbed polynomial matrices by P∆ (s) := (Aν + Dν ∆ν Eν )sν − (Aν−1 + Dν−1 ∆ν−1 Eν−1 )sν−1 − · · · − (A0 + D0 ∆0 E0 ). (9) Throughout the paper, we set inf ∅ = ∞, 0−1 = ∞, ∞−1 = 0. Definition 3.3. Let the polynomial matrix (5) be Schur stable. Then the complex stability radius of the polynomial matrix (5) with respect to perturbations of the form (7) is defined by rC = inf{γ(∆0 , ..., ∆ν ) : ∀i ∈ ν ∆i ∈ Cli ×qi ,
ρ(P∆ (·)) ≥ 1}.
(10)
If the disturbance matrices in (10) are restricted to the real spaces Rli ×qi , i ∈ ν, then we obtain the real stability radius rR . We first analyse under which condition rK = 0. 4
Proposition 3.4. Let P (z) be Schur stable and p∆ (z) := det P∆ (z), p(z) := det P (z). Then (i) rK = 0 ⇐⇒ ∃ ∆ ∈ ∆K : deg p∆ (z) > deg p(z). (ii) If the perturbation (7) are unstructured, i.e Di = Ei = In for i ∈ ν, then rC = rR = 0 ⇐⇒ det Aν = 0. Proof. (i) To prove the implication ⇐, let p∆ (z) = det P∆ (z) = anν (∆)z nν + anν−1 (∆)z nν−1 + ... + a0 (∆) ˜ ν ) ∈ ∆K is chosen such that ˜ = (∆ ˜ 0 , ..., ∆ and suppose that ∆ ˜ β + aβ−1 (∆)z ˜ β−1 + ... + a0 (∆), ˜ p∆˜ (z) = aβ (∆)z
˜ = 0 aβ (∆)
is a perturbed polynomial of maximum degree among all p∆ (z), ∆ ∈ ∆K . By assumption α := deg p(z) < deg p∆˜ (z) = β. Note that aβ (∆) is a multivariate polynomial ˜ = 0, it is not identical of the elements of the matrices ∆0 , ∆1 , ..., ∆ν . Since aβ (∆) zero. Hence for every ε > 0, there exists ∆(ε) = (∆0 (ε), ..., ∆ν (ε)) ∈ ∆K such that aβ (∆(ε)) = 0 and γ(∆(ε)) < ε. Consider the following polynomials q(s) := aα (0)sβ−α + aα−1 (0)sβ−α+1 + ... + a0 (0)sβ , qε (s) := aβ (∆(ε)) + aβ−1 (∆(ε))s + ... + a0 (∆(ε))sβ . Since ai (∆(ε)) → ai (0) for i ∈ β as → 0 and q(0) = 0, it follows from Rouch´e’s Theorem that the polynomials qε (s) have zeros sε satisfying sε → 0 as ε → 0. Then, zε = s1ε is a zero of p∆(ε) (z) and so the perturbations ∆(ε) are destabilizing if ε is sufficiently close to 0. It follows that rK < ε for every ε > 0, i.e. rK = 0. Conversely, assume that deg p∆ (z) ≤ deg p(z) for all ∆ ∈ ∆K . Since the coefficients ai (∆), i ∈ ν are continuous in ∆, it follows from the continuity property of polynomial roots and the Schur stability of P (z) that rK > 0, see e.g. [11, Thm. 7.3.5]. (ii) Suppose that Di = Ei = In , i ∈ ν. From the explicit expression for the determinant we see that deg p∆ (z) ≤ νn and the coefficient of z νn in the expansion of det P∆ (z) = p∆ (z) is det(Aν +∆ν ) for all ∆ ∈ ∆K . Hence there exists ∆ ∈ ∆K such that deg p∆ (z) > deg p(z) if and only if det Aν = 0. Thus (ii) is a direct consequence of (i). In the rest of this paper, we always assume that Aν is a regular matrix. Remark 3.5. The assumption det Aν = 0 is quite restrictive for structured perturbations. We believe that for perturbation of the form (7), robust stability criteria can be obtained under the weaker assumption det P (z) ≡ 0. However, such an analysis will require more subtle algebraic considerations connecting algebraic systems theory with perturbation theory and robust stability analysis. This will be a subject of future research. Proposition 3.6. Suppose that P (z) is of the form (5) with det Aν = 0. If ∆ = (∆0 , ..., ∆ν ) ∈ ∆C is a perturbation satisfying γ(∆) < rC then Aν + Dν ∆ν Eν is a regular matrix. 5
Proof. Itsuf f icestoshowthat β := inf{X : X ∈ Clν ×qν , det(Aν + Dν XEν ) = 0} ≥ rC . Now suppose β < rC . Since det(Aν + Dν ∆ν Eν ) = 0 if and only if det(In + 1 Eν A−1 ν Dν ∆ν ) = 0, it follows that β = Eν Aν −1 Dν is finite. So there exists a matrix X ν ) = 0. Let ∆(ξ) = (0, ..., 0, Xξ), 0 ≤ ξ ≤ 1. It = β and det(Aν + Dν XE such that X ν ) is a regular matrix for ξ < 1. If is clear that γ(∆(ξ)) = ξβ < rC and (Aν + Dν XξE we write det P∆(ξ) (z) == bv (ξ)z v + bv−1 (ξ)z v−1 + · · · + b0 (ξ) =: qξ (z),
v = nν, 0 ≤ ξ ≤ 1,
ν ) = 0 for ξ ∈ [0, 1) and bν (1) = 0. Arguing as in Lemma then bν (ξ) = det(Aν + Dν XξE 3.4, it follows that the polynomial qξ (z) has a zero with modulus greater than an arbitrary given positive number if ξ is sufficiently close to 1. So, ∆(ξ) is a destabilizing perturbation for ξ sufficiently close to 1. However, this conflicts with γ(∆(ξ)) ≤ β < rC . This completes the proof. The partitioned transfer matrix G(z) = (Gij (z))i,j∈ν associated with the polynomial matrix (5) and the perturbation structure (7) is defined by Gij (z) := Ei P (z)−1 Dj ∈ Cqi×lj (z),
i, j ∈ ν.
(11)
Lemma 3.7. Let the polynomial matrix (5) be Schur stable. Then, for every z0 ∈ C, |z0 | = 1, there exists a perturbation ∆ := (∆0 , ..., ∆ν ) such that γ(∆0 , ..., ∆ν ) =
1 and det P∆ (z0 ) = 0. maxi∈ν Gii (z0 )
(12)
Moreover, if Gii (1) ∈ Rq+i×li for every i ∈ ν then there exists a real perturbation ∆ satisfying (12) with z0 = 1. Proof. Given z0 ∈ C, |z0 | = 1, suppose that maxi∈ν Gii (z0 ) = Gi0 i0 (z0 ) for some i0 ∈ ν. By the definition of Gi0 i0 (z0 ), there exists a vector u0 ∈ Cli0 , u0 = 1 such that Gi0 i0 (z0 )u0 = Gi0 i0 (z0 ). Then, by the Hahn-Banach Theorem, there exists y0 ∗ ∈ (Cqi0 )∗ , y0∗ = 1 (where y0∗ is the dual norm) satisfying y0∗ Gi0 i0 (z0 )u0 = Gi0 i0 (z0 )u0 ˜ i0 := Gi0 i0 (z0 )−1 u0 y ∗ ∈ Cli0 ×qi0 . It is clear that ∆ ˜ i0 = Gi0 i0 (z0 )−1 and Define ∆ 0 ˜ i0 Ei0 x0 = ∆ ˜ i0 Gi0 i0 (z0 )u0 = ˜ i0 Gi0 i0 (z0 )u0 = u0 . Setting x0 := P (z0 )−1 Di0 u0 , we have ∆ ∆ ˜ i0 Ei0 x0 , i.e. u0 . Thus x0 = 0 and P (z0 )x0 = Di0 u0 = Di0 ∆ ˜ i0 Ei0 )x0 = 0. (P (z0 ) − Di0 ∆ ˜ ν z −ν . Otherwise, we set If i0 = ν, we set ∆i = 0 for i ∈ {0, 1, ..., ν − 1} and ∆ν := −∆ 0 ˜ i0 z0−i0 . Then, we get ∆i = 0, i = i0 and ∆i0 = ∆
(Aν + Dν ∆ν Eν )z0ν − (Aν−1 + Dν−1 ∆ν−1 Eν−1 )z0ν−1 − · · · − (A0 + D0 ∆0 E0 ) x0 = 0. Therefore ∆ = (∆0 , ..., ∆ν ) satisfies (12). q ×l If Gi0 i0 (z0 ) ∈ R+i0 i0 for z0 = 1 then we have Gi0 i0 (1) = max li
li
u∈R+0 ,u=1
Gi0 i0 (1)u,
see [7]. Thus we can choose u0 ∈ R+0 such that u0 = 1 and Gi0 i0 (1)u0 = Gi0 i0 (1). Since Gi0 i0 (1)u0 ≥ 0 there exists by a theorem of Krein and Rutman [8] a positive linear form y0∗ ∈ (Cqi0 )∗ of dual norm y0∗ = 1 such that y0∗ Gi0 i0 (1)u0 = Gi0 i0 (1)u0 . Hence ˜ i0 and ∆ constructed as above are nonnegative. This completes the perturbations ∆ the proof. 6
Using this lemma we obtain the following estimates for the complex stability radius. Theorem 3.8. Let G(z) = (Gij (z))i,j∈ν be the associated transfer matrix of the polynomial matrix (5) with perturbation (7). If P (z) is Schur stable then 1 1 ≤ rC ≤ . maxz∈C,|z|=1 max{Gij (z) : i, j ∈ ν} maxz∈C,|z|=1 max{Gii (z) : i ∈ ν} (13) In particular, if Di = Dj for all i, j ∈ ν rC =
or
Ei = Ej for all i, j ∈ ν then
1 maxz∈C,|z|=1 max{Gii (z) : i ∈ ν}
(14)
Proof. To prove (13), we note that by the continuity of ρ(P∆ (·)) in ∆ = (∆0 , ..., ∆ν ), it follows that rC = inf{γ(∆0 , ..., ∆ν ) : ∀i ∈ ν ∆i ∈ Cli ×qi ,
ρ(P∆ (·)) = 1}.
Let ∆ = (∆0 , ..., ∆ν ) be a destabilizing disturbance such that ρ(P∆ (·)) = 1. Then, there exist a complex number λ, |λ| = 1 and a non-zero vector x ∈ Cn such that
ν−1 P∆ (λ)x = (Aν + Dν ∆ν Eν )λν − (Ai + Di ∆i Ei )λi x = 0. i=0
Since the polynomial matrix (5) is stable, we get
ν−1 P (λ)−1 −Dν ∆ν Eν λν + (Di ∆i Ei )λi x = x. i=0
Let k be an index such that Ek x = max{Ei x, i ∈ ν} then from the last equality it follows that Ek x = 0. Multiplying the last equation with Ek from the left, we obtain ν
Gkj (λ)∆j Ej x ≥ Ek x.
j=0
Therefore, max max{Gij (z) : i, j ∈ ν}
z∈C,|z|=1
ν
∆j
≥ 1.
j=0
Hence γ(∆0 , ..., ∆ν ) =
ν j=0
∆j ≥
1 , maxz∈C,|z|=1 max{Gij (z) : i, j ∈ ν}
and this proves the first inequality in (13). The second inequality in (13) follows from Lemma 3.7 and the definition of rC . Finally (14) follows from (13) and the fact that if Di = Dj (resp. Ei = Ej ) for all i, j ∈ ν then all the block entries Gij (z) of G(z) in the same block row (resp. block column) are identical. 7
4
Stability radii of positive polynomial matrices
We now restrict ourselves to positive polynomial matrices. Definition 4.1. The polynomial matrix (5) with det Aν = 0 is called positive if the −1 matrices A−1 ν Aν−1 , ..., Aν A0 are nonnegative. Remark 4.2. ¿From the above definition, it follows that P (z) in (5) with det Aν = 0 is a positive polynomial matrix if and only if the associated discrete state space system of the form (6) is positive. It is clear from the Definition 3.3, that the stability radii of the polynomial matrix (5) with respect to perturbations of the type (7) satisfy rC ≤ rR .
(15)
We will show that, for positive polynomial matrices, equality in (15) holds and moreover, these stability radii are easily computed. We need the following property of the spectral radius of positive polynomial matrices. Lemma 4.3. Let the polynomial matrix (5) be stable and positive. Then, ρ(Aν −1 Aν−1 + Aν −1 Aν−2 + ... + Aν −1 A0 ) < 1.
(16)
Proof. Assume to the contrary that ρ(Aν −1 Aν−1 + Aν −1 Aν−2 + ... + Aν −1 A0 ) ≥ 1. For the continuous real function f (θ) = θν − ρ(Aν −1 Aν−1 θν−1 + Aν −1 Aν−2 θν−2 + · · · + Aν −1 A0 ), θ ∈ [1, +∞), we have f (1) = 1 − ρ(Aν −1 Aν−1 + Aν −1 Aν−2 + · · · + Aν −1 A0 ) ≤ 0 and lim f (θ) = +∞. θ→+∞
Hence f (θ0 ) = 0 for some θ0 ≥ 1, and so θ0 ν = ρ(Aν −1 Aν−1 θ0 ν−1 + · · · + Aν −1 A0 ). It follows, by Theorem 2.1 (i), that θ0 ν is an eigenvalue of the nonnegative matrix (Aν −1 Aν−1 θ0 ν−1 +· · ·+Aν −1 A0 ) or, equivalently, det(Aν θ0 ν −Aν−1 θ0 ν−1 −· · ·−A0 ) = 0, with θ0 ≥ 1 and this yields a contradiction with our assumption that ρ(P (·)) < 1. Thus (16) follows. We are now in a position to derive the first main result of the paper. Theorem 4.4. Let the polynomial matrix (5) be stable and positive. Assume that the matrices Ai are subjected to parameter perturbations of the form (7), where A−1 ν Di ∈ qi ×n i Rn×l , E ∈ R , i ∈ ν. If D = D or E = E (i, j ∈ ν), then i i j i j + + rC = rR =
1 maxi∈ν Ei (Aν − Aν−1 − · · · − A0 )−1 Di
8
.
(17)
Proof. Since the matrices Aν −1 Aj , j ∈ ν are nonnegative matrices, we have for all z ∈ C, |z| = 1 ν−1 ν−2 −1 −1 −1 + A−1 + ... + A−1 |A−1 ν Aν−1 z ν Aν−2 z ν A0 | ≤ Aν Aν−1 + Aν Aν−2 + ... + Aν A0
It follows from Lemma 4.3 and Theorem 2.1 (iv) that for all z ∈ C, |z| = 1 ν−1 ν−2 + A−1 + ... + Aν −1 A0 ) < 1. < 1. ρ(A−1 ν Aν−1 z ν Aν−2 z
As a consequence, we obtain the following expansion for P (z)−1 on the unit circle in C ∞ k −1 −1 −1 ν−1 ν−2 A A z + A A z + · · · + A A ν ν−1 ν ν−2 ν 0 A−1 P (z)−1 = ν . ν(k+1) z k=0 Hence, on the unit circle k −1 |P (z)−1 Aν | ≤ ∞ Aν−1 + Aν −1 Aν−2 + · · · + Aν −1 A0 ) = P (1)−1 Aν . k=0 (Aν
(18)
It follows that for z ∈ C, |z| = 1 −1 |Gii(z)| = |Ei P (z)−1 Di | = |Ei P (z)−1 Aν A−1 ν Di | ≤ Ei P (1) Di = Gii (1).
(19)
By the monotonicity property (4) of the operator norms, we get Gii (z) ≤ Gii(1),
z ∈ C, |z| = 1.
By (14), this implies rC =
1 . maxi∈ν Gii (1)
(20)
On the other hand, (19) implies that Gii (1) ∈ Rl+i ×qi for every i ∈ ν. By definition of the real stability radius rR and Lemma 3.7, we have rR ≤
1 maxi∈ν Gii (1)
(21)
¿From inequalities (15), (20), (21), it follows that rC = rR =
1 1 = . maxi∈ν Gii (1) maxi∈ν Ei (Aν − Aν−1 − · · · − A0 )−1 Di
We now turn to a different perturbation structure and assume that the polynomial matrix (5) is subjected to perturbations of the following kind : Ai Ai (δ) := Ai +
N
δij Bij ,
(22)
j=1
where Bij ∈ Cn×n , i ∈ ν, j ∈ N := {1, 2, ..., N} are given matrices and δij ∈ C are unknown scalar parameters. Arbitrary affine perturbation of the Ai can be represented in this way. For δ = (δij ) ∈ Cν×N , we define ν−1 N ν i (23) Pδ (z) := (Aν + N j=1 δνj Bνj )z − i=0 (Ai + j=1 δij Bij )z . 9
Definition 4.5. Let the polynomial matrix (5) be Schur stable. Then, the complex stability radius of the polynomial matrix (5) with respect to perturbations of the form (22) is defined by rCa = inf{maxi∈ν,j∈N |δij | : δ = (δij ) ∈ Cp×N , ρ(Pδ (·)) ≥ 1}.
(24)
If the scalar disturbances δij in (24) are restricted to be real, then we obtain the real stability radius rRa . Let
In Ei := ... ∈ CnN ×n In
Di := [Bi1 · · · BiN ] ∈ Cn×nN ,
(25)
and Di := {diag(δi1 In , . . . , δiN In ) : δij ∈ C} ⊂ CnN ×nN , i ∈ ν. Then it is easy to see that the perturbations (22) are equivalently represented in the form Ai
Ai (δ) = Ai + Di ∆i Ei ,
We define
∆i ∈ Di .
Eν D(z) := [Dν z ν , Dν−1 z ν−1 , ..., D0 ] ∈ Cn×(ν+1)nN , E := ... ∈ C(ν+1)nN ×n E0
(26)
(27)
and ∆ := diag(∆ν , −∆ν−1 , ..., −∆0 ) ∈ D, D := {∆ : ∆i ∈ Di , i ∈ ν} ⊂ C(ν+1)nN ×(ν+1)nN . Then, we have ν
D(z)∆E = Dν ∆ν Eν z −
ν−1
Di ∆i Ei z i ,
i=0
and so by (23) Pδ (z) = P (z) + D(z)∆E,
∆ ∈ D.
(28)
Thus, according to Definition 4.5, the complex and real stability radii of the polynomial matrix (5) subjected to affine parameter perturbations (22) can be rewritten in the following form (29) rCa = inf{∆ : ∆ ∈ D, ρ(P (·) + D(·)∆E) ≥ 1}, a (ν+1)nN ×(ν+1)nN rR = inf{∆ : ∆ ∈ DR , ρ(P (·) + D(·)∆E) ≥ 1}, DR := D ∩ R where the perturbation norm ∆ is given by ∆ = max |δij |, i∈ν, j∈N
∆ = diag(δν1 In , ..., δνN In ; ......; δ01 In , ..., δ0N In ) ∈ D. 10
(30)
Theorem 4.6. Suppose the polynomial matrix (5) is Schur stable, positive and subn×n jected to affine perturbations of the form (22), where A−1 ν Bij ∈ R+ , i ∈ ν, j ∈ N . Then rCa = rRa = where B :=
ν
i=0
1 , ρ (P (1)−1 B)
(31)
N
j=1 Bij .
Proof. Setting I := I(ν+1)nN ×(ν+1)nN , we obtain from (29) rRa = inf{∆ : ∆ ∈ DR , ∃z ∈ C |z| ≥ 1 and det(I + EP (z)−1 D(z)∆) = 0} where E, D(z) are defined by (27). Now EP (1)−1D(1) = EP (1)−1 Aν A−1 ν D(1) ≥ 0, hence rCa ≤ rRa ≤ inf{∆ : ∆ ∈ DR ,
det(I + EP (1)−1 D(1)∆) = 0}
det(I + αEP (1)−1D(1)) = 0} =
≤ inf{|α| : α ∈ R,
1 . ρ (EP (1)−1D(1))
(32)
On the other hand, we have by (29) and the continuity of the spectrum rCa = inf{∆ : ∆ ∈ D, =
inf
z∈C,|z|=1
inf{∆ : ∆ ∈ D,
ρ(P (·) + D(·)∆E) ≥ 1} det(In + P (z)−1 D(z)∆E) = 0}.
(33)
Now let ∆ ∈ D be any perturbation such that det(I + P (z)−1 D(z)∆E) = 0 for some z ∈ C, |z| = 1. Then det(I + EP (z)−1 D(z)∆) = 0 and since ∆ is diagonal and ∆ = maxi∈ν,j∈N |δij | we get ∆I ≥ |∆|. Hence, by (3) ∆ρ(|EP (z)−1 D(z)|) ≥ ρ(EP (z)−1 D(z)∆) ≥ 1.
(34)
In the proof of Theorem 4.4, we have seen that |P (z)−1 Aν | ≤ P (1)−1 Aν for every z ∈ C, |z| = 1 (see (18)). Moreover, since A−1 ν Bij ≥ 0 for all i ∈ ν, j ∈ N by assumption, −1 we have |A−1 D(z)| = A D(1) on the unit circle. It follows that ρ (|EP −1 (z)D(z)|) ≤ ν ν ρ (EP (1)−1D(1)) for every z ∈ C, |z| = 1. So by (33) and (34) rCa ≥
1 1 ≥ . −1 z∈C,|z|=1 ρ(|EP (z) D(z)|) ρ (EP (1)−1 D(1)) inf
Combining (32) and (35) we obtain, making use of [7, Thm.1.3.20], rCa = rRa = Since D(1)E =
ν
i=0
N
1 1 = . −1 −1 ρ (EP (1) D(1)) ρ (P (1) D(1)E)
j=1 Bij
= B by (27) and (25), formula (31) follows.
11
(35)
Remark 4.7. (31) implies that the stability radii rCa = rRa do not depend, in the positive case, on the specific perturbation structure for the individual coefficient matrix Ai but only on the accumulated effect represented by the sum B of all the structure matrices Bij . To make this more precise, consider the special case where only one matrix coefficient, say A0 is perturbed while all the other matrix coefficients remain unperturbed. If A0 (δ0 ) = A0 + δ0 B,
Ai (δ0 ) ≡ Ai ,
i ∈ {1, 2, ..., ν},
then the perturbed polynomial matrix has the form Pδ0 (z) = Aν z ν + · · · + A1 z + (A0 + δ0 B) and the correspondent stability radius is equal to ρ(P (1)1 −1 B) . Thus concentrating all perturbations on a single coefficient matrix leaves the stability radius invariant. We conclude the paper by two examples illustrating Theorems 4.4 and 4.6. Example 4.8. Consider the linear difference system A2 y(t + 2) = A1 y(t + 1) + A0 y(t), where
1 −1 1 −1 , A2 = A2 = 0 −2 0
−1/2 , −1/2
t∈N
0 1 A1 = , 0 0
3/4 1 A0 = . 0 −1
The associated polynomial matrix is
z 2 − 3/4 −z 2 − z − 1 . P (z) = A2 z − A1 z − A0 = 0 −2z 2 + 1 2
(36)
Clearly P (z) is Schur stable. Moreover, it is easily verified that A−1 2 A1 ≥ 0 and A ≥ 0, so that P (z) is positive. Now suppose that P (z) is perturbed as follows A−1 0 2 1 0 1 1 2 ∆2 )z − (A1 + ∆1 )z − (A0 + ∆0 ) (37) P∆ (z) = (A2 + 0 −1 0 −1 where ∆2 ∈ K2×2 and ∆1 , ∆0 ∈ K1×2 . The perturbations are of the form (7) with 1 1 1 0 , D1 = , D2 = . E0 = E1 = E2 = I2 and D0 = −1 0 0 −1 −1 −1 Since Ei ≥ 0 and A−1 4.4 2 D0 ≥ 0, A2 D1 ≥ 0, A2 D2 ≥ 0, the assumptions of Theorem 4 −12 −1 −1 are satisfied. We have Gii (1) = (A2 − A1 − A0 ) Di and (A2 − A1 − A0 ) = 0 −1 so that 16 4 4 12 G00 (1) = , G11 (1) = , G22 (1) = . 1 0 0 1
If we provide K2 with the maximum norm and K1 with | · |, we obtain G00 (1) = 16, G11 (1) = 4, G22 (1) = 16. Hence rC = rR = 1/16. If ∆0 +∆1 +∆2 < 1/16, 12
P∆ (z) is Schur stable. On the other hand a destabilizing perturbation ∆ = (∆0 , ∆1 , ∆2 ) of minimal norm γ(∆) = 1/16 can be constructed as in the proof of Lemma 3.7. Here we may choose i0 = 0 or i0 = 2. Let e.g. i0 = 2. Then we may take u0 = [1 1] to obtain G22 (1)u0 = G22 (1) = 16. The dual norm on (K2 )∗ = K1×2 is the
∗ ∗ 1-norm. To obtain y0∗G22 (1)u0 = G22 (1) with y 1 = 1 we choose y0 = 1 0 . 0
1 0 −1 1 1 0 = −1/16 is a minimal Hence ∆ = (0, 0, ∆2 ) with ∆2 = −G22 (1) 1 0 1 norm destabilizing perturbation. In fact, one easily verifies that det P∆ (1) = 0. Note that minimum norm destabilization is obtained here by only perturbing the leading coefficient matrix A2 . In the following example we deal with the same perturbed polynomial as in the previous one, but represent it in the form (23) by an appropriate choice of the structure matrices Bij . Since the corresponding perturbation norm (30) is different from γ(∆) used in the previous example we obtain different stability radii rCa = rRa . Example 4.9. Let P (z) and P∆ (z) be as in (36) and (37), respectively. Writing
δ δ ∆0 = δ01 δ02 , ∆1 = δ11 δ12 , ∆2 = 21 22 and setting δ23 δ24 1 0 0 1 0 0 0 0 B21 = , B22 = , B23 = , B24 = , B11 = B21 , B12 = B22 , 0 0 0 0 −1 0 0 −1 1 0 0 1 0 0 B01 = , B02 = , B03 = B04 = B13 = B14 = −1 0 0 −1 0 0 the perturbed polynomial P∆ (z) (37) can be represented in the form Pδ (z) = (A2 +
4
2
δ2j B2j )z − (A1 +
j=1
4
δ1j B1j )z − (A0 +
j=1
4
δ0j B0j )
j=1
It is easily verified that A−1 2 Bij ≥ 0 for i = 0, 1, 2, j = 1, . . . , 4. Hence we can apply Theorem 4.6 and obtain 2 4 3 3 a a −1 −1 rC = rR = ρ(P (1) B) where B = Bij = . −2 −2 i=0 j=1
−1
Since P (1)
4 −12 = (A2 − A1 − A0 ) = (see Example 4.8) we get 0 −1 36 36 a a ) = 1/38. rC = rR = ρ( 2 2 −1
Thus Pδ (z) is Schur stable if |δij | < 1/38 for all i ∈ 2, j ∈ 3. On the other hand it is easily verified that if we set δˆij = 1/38 for all i ∈ 2, j ∈ 3 then δˆ = (δˆij ) is a minimum norm destabilizing perturbation satisfying det Pδˆ(1) = 0.
13
stability
2 3 4 Ref erences
2
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