Stability of polynomials with conic uncertainty - Semantic Scholar

Stability of polynomials with conic uncertainty V. L. Kharitonov∗ Inst. of Applied Mathematics University of St.-Petersburg St.-Petersburg Russia

D. Hinrichsen Inst. f¨ ur Dynamische Systeme Universit¨at Bremen D-28334 Bremen Germany

Abstract In this paper, we describe a conic approach to the stability theory of uncertain polynomials. We present necessary and sufficient conditions for a conic set p0 + K of polynomials to be Hurwitz stable (K is a convex cone of polynomials of degree ≤ n and deg p0 = n). As analytical tools we derive an edge theorem and Rantzer type conditions for marginal stability (semi-stability). The results are applied to prove an extremal ray result for conic sets whose cone of directions is given by an interval polynomial.

Key words: Hurwitz stability, robust stability, interval polynomial, value set, convex cone.

1

Introduction

The stability of polynomials with uncertain parameters has received much attention over the past decade. The importance of this subject for robustness analysis of control systems has been emphasized in many papers, recent references are e.g. [Bar94], [BK91], [DTV93], [Ran92]. Two basic approaches can be discerned in the literature. The first approach studies the stability of convex polytopes of polynomials Π = conv {p1 , ..., pN }

(1)

where p1 , ..., pN are real polynomials of degree ≤ n. The aim is to find a minimal test set Θ ⊂ Π such that the stability of the polynomials in Θ implies the stability of all the polynomials in Π. This approach was initiated in [Kha78] where it was proved that there exists a test set of only four polynomials for the stability of a given interval polynomial Π=

n X

[aj , aj ]sj .

(2)

j=0

This author would like to thank the Deutsche Forschungsgemeinschaft (DFG) for its support during the writing of this paper. ∗

1

Over the past decade a great number of papers have contributed to the development of this approach. General polytopes of polynomials and different stability domains were considered and recently some of the available results for polynomials have been extended to quasipolynomials [KZ92]. Amongst the most important contributions in the polynomial field are the edge theorem [BHL88], the concept of value set [ADM89] and Rantzer’s condition [Ran92]. Mathematically, the approach is based on classical algebraic stability criteria for Hurwitz and Schur polynomials and the value set method. The object of the second approach is not a given set of polynomials but a parametrized family of sets of perturbed polynomials around a “nominal“ one. Suppose p(s) = p(s, a) = a0 + a1 s + . . . + an sn ∈ R[s]

(3)

is the nominal polynomial and the coefficient vector a = (a0 , a1 , ..., an ) ∈ R1×(n+1) is affinely perturbed: a ❀ a(∆) = a +

N X

δj ej

(4)

j=1

where the row vectors ej ∈ R1×(n+1) are given and ∆ = (δ1 , ..., δN ) is an unknown vector of parameter deviations. Then the parametrized sets of perturbed polynomials are P (ρ) = {p(s, a(∆)) ; ∆ ∈ RN , k∆k < ρ}

(5)

where k · k is a given norm on the parameter space RN . This approach reflects an uncertainty where a nominal polynomial and the structure of its parameter perturbations are a priori known but the size of the parameter deviations is unknown. The aim is to determine the size of the smallest perturbation ∆ which destabilizes the stable nominal polynomial p(s, a). This question was already discussed in the context of interval polynomials (i.e. with respect to a weighted maximum norm) in [Kha79]. First results for the 2-norm were obtained in [SBD85] (unstructured perturbations) and [BHB87] (structured perturbations). These two papers are based on results in [FM58] which describe the topological boundary of the set of real Schur polynomials in coefficient space. Computable formulae for the stability radius of a given stable polynomial with respect to arbitrary affine real or complex perturbations, arbitrary perturbation norms and arbitrary (open) stability domains in C were presented in [HP89]. These results were derived by applying the available theory of stability radii [HP86], [HP88] to polynomials via their companion matrices. In order to obtain non-conservative robustness measures in concrete applications it is important to choose an uncertainty model which accurately reflects the available a priori knowledge. In particular, it is important to choose appropriate perturbation structures and an adequate gauge function by which the size of a parameter perturbation is measured. In a recent paper [QD92] the use of Minkowsky functionals µC is proposed in order to obtain a more flexible tool for measuring the size of an affine perturbation. If C ⊂ RN is an absorbing compact convex set (i.e. containing the origin in its interior), µC satisfies the triangle inequality and is positive homogeneous. But it differs from a norm in that it does not necessarily assign equal values to ∆ and −∆. The shape of the compact convex set C has to be chosen according to the specific uncertainty in a concrete application. 2

In this paper we go a step further and analyse the stability of polynomials with conic uncertainty, i. e. we suppose that a priori a convex cone of directions is known within which the coefficient vector of the nominal polynomial is being perturbed. Thus the set of all possible perturbed polynomials is no longer an affine space but a conic set p0 + K. Here K is a given convex cone of polynomials representing the a priori knowledge about possible disturbances. In the framework of [QD92] this corresponds to dropping the requirement that the convex set C be absorbing, i.e. contain the origin in its interior. For an arbitrary nonempty compact convex set C we are interested in the stability radius of a stable nominal polynomial p0 with respect to perturbations in the direction of C: rR (p0 ; C) = inf{ρ > 0; ∃c ∈ C : p(s, a + ρc) unstable} .

(6)

Dropping the assumption that C be absorbing considerably complicates the theory. In particular, the stability radius of p0 may become infinite in certain directions C. In fact this is the only subject of the present paper. We will derive necessary and sufficient conditions for a polynomial p0 to have infinite robustness with respect to perturbations in the direction of C. In other words, if K denotes the convex cone generated by C we are looking for necessary and sufficient conditions for p0 + K to consist only of stable polynomials. Thus, although motivated by a stability radius problem we end up with a typical problem formulation of the first approach. Since a polytope Π is Hurwitz stable if and only if the closed convex cone generated by Π is Hurwitz stable, the stability theory of conic sets contains the stability theory of polytopes, but not vice versa. In contrast with the previous literature in this field we deal with unbounded convex sets of polynomials. This causes some difficulties. Stable conic sets may come arbitrarily close to the boundary of the set of Hurwitz polynomials whereas stable polytopes always have a positive distance to the boundary. This forces us to extend existing results on stability to marginal or semistability. We proceed as follows. In the next section we recall some notions from convex analysis and derive necessary conditions for a conic set to be stable. In Section 3 we prove some basic facts about the value set of (semi-)stable cones of polynomials, we establish an edge theorem for semi-stable polyhedral cones K and an extremal ray criterion for the stability of a conic set p0 + K. These results reduce the stability analysis of conic sets to the checking of certain segments and rays of polynomials for (semi-)stability. In Section 4 we derive Rantzer type conditions and a simple algebraic test for the (semi-)stability of segments and rays. Finally, in Section 5 the results are applied to conic sets p0 + K whose cone of directions K is generated by an interval polynomial. It is shown that the stability of these special conic sets can be checked by testing four extremal rays for stability and one interior polynomial for semi-stability.

2

Preliminaries

Let Pn denote the n + 1-dimensional vector space of all polynomials of the form (3). By P0 we denote the set of constant polynomials including the zero polynomial. Throughout this paper we identify each polynomial (3) with its coefficient vector (a0 , . . . , an ) ∈ Rn+1 and provide Pn with the topology induced from Rn+1 via this identification. 3

In the following we will need some terminology and results from convex analysis and we refer the reader to [Roc70] for notational conventions and further details. Definition 2.1 A subset K of Pn is said to be a cone if it is closed under positive scalar multiplication, i.e. αp ∈ K when p ∈ K and α > 0. A convex cone is a cone which is a convex set. A convex cone is called pointed if it does not contain a line, i.e. K ∩ (−K) ⊂ {0}. Note that a cone may or may not contain the origin (the zero polynomial). K is a convex cone if and only if it is closed under addition and positive scalar multiplication. For any subset P ⊂ Pn we denote by conv P the convex hull of P . The convex hull of a finite set is called a polytope. The cone of nonnegative real numbers is denoted by R+ . Definition 2.2 If P ⊂ Pn then ray P = {αp; p ∈ P, α > 0} is called the cone generated by P and cone P = conv ray P is called the convex cone generated by P . A convex cone K is said to be polyhedral if there exist p1 , . . . , pN ∈ Pn such that  N X

K=

j=1

 

αj pj ; (α1 , . . . , αN ) ∈ RN +.

(7)

Every polyhedral cone is closed. Definition 2.3 A polynomial g in a convex set P ⊂ Pn is called extremal if P \ {g} is convex. If K is a convex cone, g ∈ K \ {0} is said to generate an extremal ray if K \ ray g is convex. The set of all generators of extremal rays in K is denoted by Ke . A set C is said to be a basis of a cone K if there is a hyperplane H, 0 6∈ H such that C = K ∩ H and C contains exactly one point in each ray of K. The following lemma collects some facts which will be used later. Lemma 2.4 Suppose that K is a closed convex pointed cone in Pn . Then (i) K = conv Ke . (ii) P ⊂ Pn generates K as a convex cone with zero (in the sense that K = cone P ∪{0}) if and only if Ke ⊂ ray P . (iii) K has a compact convex basis C. (iv) K is polyhedral if and only if it has a basis which is a polytope. 4

In this paper we will study convex sets of Hurwitz stable polynomials. A polynomial of the form (3) is called Hurwitz stable or simply stable if all its roots have negative real parts. It is called semi-stable if all its roots have nonpositive real parts. Nonzero constant polynomials are stable whereas the zero polynomial is not semi-stable. By Hn we denote the set of stable and by Hn the set of semi-stable polynomials of the form (3). Both Hn and Hn are cones; they are convex if and only if n ≤ 2. We say that a set P of polynomials (3) is stable (resp. semi-stable) if P \ {0} ⊂ Hn (resp. P \ {0} ⊂ Hn ). In particular, we are interested in convex sets of the form P = p0 + K

(8)

where p0 ∈ Pn and K ⊂ Pn is a convex cone. These sets represent polynomials with conic uncertainty (specified by K) and will be called conic sets starting at p0 . The set (8) is stable (resp. semi-stable) if and only if the convex cone P0 = ray(p0 + K)

(9)

is stable (resp. semi-stable). Note that this cone may not be closed even if K is closed. In particular, if K is polyhedral as in (7) then p0 + K is stable if and only if the bounded convex set Π0 =

 N X

αj pj ; α0 > 0, αj ≥ 0 for j = 1, . . . , N,



j=0

N X

j=0

αj = 1

 

(10)



is stable. For p0 = 0, Π0 is a polytope, but if p0 6= 0 then Π0 will in general not be closed. Note that Π0 may be stable although its closure is not stable. Geometrically speaking, a stable polyhedral set of the form (10) may touch the set Pn \ Hn of non-Hurwitz stable polynomials whereas a stable polytope always has a positive distance from the set Pn \Hn . If C ⊂ Pn is compact and convex, the stability of C is equivalent to the stability of the convex cone K = ray C. But the stability of the conic set p0 + K which does not start at the origin can in general not be reduced equivalently to the stability of a compact convex set. Hence the stability theory of conic sets in Pn is more general than the stability theory of convex compact sets in Pn . Remark 2.5 Given p ∈ Pn of the form (3), the coefficients ak , k = 0, . . . , deg p are said to be the proper coefficients of p. If p ∈ Hn , the proper coefficients of p are either all positive or all negative. If p ∈ Hn , the coefficients of p are either all nonnegative or all nonpositive. The following lemma shows that we may restrict our considerations to polynomials with only positive (resp. nonnegative) coefficients. Lemma 2.6 Suppose that P ⊂ Pn is convex and is not contained in a line through zero. If P is stable (resp. semi-stable) then all the polynomials in P have only positive (resp. nonnegative) or only negative (resp. nonpositive) proper coefficients. Proof: Assume that P is stable and that p0 ∈ P has positive proper coefficients and p1 ∈ P has negative proper coefficients. Then the convex combination pt = tp1 + (1 − t)p0 5

(11)

has only positive or only negative proper coefficients, for every t ∈ [0, 1]. But this is only possible if the segment between the two polynomials passes through zero. Fixing p0 we see that all the polynomials in P with negative coefficients belong to Rp0 . Fixing p1 it follows that all the polynomials in P with positive coefficients belong to Rp1 = Rp0 . This contradicts the assumption that P is not contained in a line through zero. The proof for the case that P is semi-stable proceeds similarly. The stability analysis of convex sets P which are contained in a line through zero is trivial. Therefore we may henceforth restrict our analysis to convex sets in the positive orthant + P+ n of Pn . Pn consists of all the polynomials in Pn whose coefficients are nonnegative. In the following lemma we summarize some topological properties of Hn . For any subset P ⊂ Pn we denote by cl (P ) the closure of P and by int (P ) the interior of P in Pn . Lemma 2.7

(i) The interior of Hn is the set of all polynomials in Hn of degree n.

(ii) cl (Hn ) = Hn ∪ {0} . (iii) The boundary of Hn in Pn consists of the zero polynomial, all stable polynomials of degree < n and all semi-stable polynomials in Pn having at least one root on the imaginary axis. The assertions of the lemma follow from the continuous dependence of polynomial roots on the coefficient vector. We conclude this section with necessary conditions for a conic set to be stable or semi-stable. A proof of the following lemma can be found in [Won79]. Lemma 2.8 Let p, q ∈ P+ n be polynomials with m = deg p > deg q = ℓ. Then as ε ↓ 0, ℓ zeros of qε = εp + q tend to the roots of q while m − ℓ tend asymptotically to infinity along the rays through the (m − ℓ)-th roots of −1. More precisely, if ζ0 ∈ C is any root of ζ m−ℓ + 1 = 0, there exists, for every small ε > 0 a root λ(ε) of the polynomial qε such that λ(ε) lim 1/(m−ℓ) = ζ0 . ε→0 ε For any nonempty set P ⊂ P+ n we denote by deg P the maximum of the degrees of the polynomials in P . Proposition 2.9

(i) If P ⊂ P+ n is convex and semi-stable then, for all p ∈ cl (P ) \ {0}, deg p ≥ deg P − 2 .

+ (ii) If p0 ∈ P+ n \ {0}, K ⊂ Pn is a cone and P = p0 + K is semi-stable then p0 and K are semi-stable.

Proof: (i) Suppose that q ∈ P is such that m = deg q = deg P and p ∈ cl (P ) \ {0}. Assume by way of contradiction that ℓ = deg p < m − 2. Then d = q − p is of degree m and p + εd ∈ P for ε ∈ (0, 1). Since m − ℓ > 2 there exists a root ζ0 of sm−ℓ + 1 = 0 with positive real part. By Lemma 2.8 it follows that there is a root of p + εd approximating 6

the asymptote R+ ζ0 . This is in contradiction with the semi-stability of P . (ii) By assumption, for every q ∈ K \ {0} and every ε > 0, the polynomials pε = p0 + εq and qε = εp0 + q = ε(p0 + ε−1 q) are semi-stable. If deg p0 ≥ deg q the roots of pε converge to the roots of p0 as ε ↓ 0 (since deg pe = deg p0 ) while deg q roots of qε converge to the roots of q by Lemma 2.8. Similarly, if deg q > deg p0 then deg p0 roots of pε converge to the roots of p0 and the roots of qε converge to the roots of q. Hence the assertion (ii) follows. Note that if P = p0 + K is stable then K is semi-stable but not necessarily stable. Although we are mainly interested in the stability of conic sets, this fact forces us to investigate semi-stability criteria for convex cones.

3

Value set criteria for the stability and semistability of conic sets

In this section we will investigate the value set of conic sets of polynomials and establish necessary and sufficient criteria for stability and semi-stability. If P ⊂ P+ n and z ∈ C, then VP (z) = {p(z); p ∈ P \ {0}}

(12)

is called the value set of P at z. VP (z) is the image of P \ {0} by the (linear) evaluation map p 7→ p(z). Since P ⊂ P+ n the set P \ {0} is a cone or a convex set if and only if P has the same property. Hence VP (z) is a cone or a convex set if P is a cone or a convex set, respectively. Note that the closed convex cones in C are of very simple form: They are either reduced to the origin (i) or a ray emanating from the origin (ii) or a line through the origin (iii) or a sector with vertex at the origin (iv) or a closed half-plane through the origin (v) or the whole complex plane (vi). The value set of a polytope Π is in general more complicated than the value set of the convex cone generated by Π. Therefore the value set method is particularly attractive in the context of our conic approach to stability. Remark 3.1 If P is bounded then z 7→ VP (z) is a continuous set valued map. However, if P is not bounded this is no longer the case. For instance, the value set of a convex cone may be the whole plane at every point z 6= z0 in a neighbourhood of z0 ∈ C but may collapse to the origin at z0 . The following proposition is well known for sets of polynomials of the same degree. Proposition 3.2 (zero exclusion principle) Suppose that a subset P ⊂ P+ n is convex and contains a stable polynomial of degree ≥ deg P − 1. Then P is stable if and only if the value set VP (ıω) does not contain the origin for any ω ≥ 0. Proof: If 0 ∈ VP (ıω) then P contains a polynomial p 6= 0 with the root ıω, hence P is unstable. Conversely, suppose that P contains a nonzero polynomial q with a root z ∈ C, Re z > 0. By assumption there is a stable polynomial p ∈ P of degree ≥ deg P −1. Consider the polynomials pt = tq + (1 − t)p, t ∈ [0, 1]. If m = deg p ≥ deg q = ℓ then 7

deg pt = m, t ∈ [0, 1) and there are continuous functions t 7→ λi (t), i = 1, . . . , m on [0, 1) such that λi (t), . . . , λm (t) are precisely the m roots of pt (counting multiplicities). Since one of these roots converges to z as t ↑ 1 (Lemma 2.8) there exists by continuity t ∈ [0, 1] such that pt has a purely imaginary root ıω, ω ≥ 0. If deg p < deg q, then deg p = deg q − 1 and deg p roots of pt , t ∈ (0, 1] tend to the roots of p as t ↓ 0, starting at roots of q = p1 . The remaining root tends to −∞ by Lemma 2.8. Again it follows that there exists t ∈ (0, 1] such that pt has a purely imaginary root. Thus in both cases we have 0 ∈ VP (ıω) for some ω ≥ 0. As a special case, we obtain the following necessary and sufficient conditions for conic sets to be stable. Corollary 3.3 Suppose that K ⊂ P+ n is a convex cone containing the zero polynomial + and p0 ∈ Pn is a stable polynomial of degree ≥ deg K − 1. Then the conic set P = p0 + K is stable if and only if −p0 (ıω) 6∈ VK (ıω), ω ≥ 0. (13) Proof: Since VP (ıω) = p0 (ıω) + VK (ıω), the value set VP (ıω) does not contain the origin if and only if condition (13) holds. We now discuss some properties of the value set VK (ıω) if K is a stable or semi-stable convex cone. Lemma 3.4 Suppose that K is a stable convex cone in P+ n . Then the value set VK (ıω) is a pointed convex cone in the complex plane for all ω ≥ 0. Proof: VK (ıω) is a convex cone for all ω ≥ 0. If it would contain a line Rz, z 6= 0 for some ω ≥ 0 then z = p(ıω) and −z = q(ıω) for some p, q ∈ K and so p(ıω) + q(ıω) = 0. Since p + q ∈ K \ {0} this contradicts the stability of K. The converse of this corollary does not hold in general. For instance, K = cone {s, s+1} ⊂ P+ n is a closed convex cone with pointed value set VK (ıω) = cone {ıω, ıω + 1} for ω ≥ 0; but K is not stable. However, under an additional condition, the pointedness of the value sets VK (ıω) for all ω ≥ 0 implies the stability of the convex cone K. Proposition 3.5 Suppose that K is a closed convex cone in P+ n containing a stable polynomial of degree ≥ deg K − 1. If all the extremal generators g ∈ Ke of K have no roots on the imaginary axis and the value set VK (ıω) is pointed for all ω ≥ 0 then K is stable. Proof: Suppose that K is not stable. Then there exists by the zero exclusion principle a nonzero polynomial p ∈ K and some ω ≥ 0 such that p(ıω) = 0. By Lemma 2.4, p can be respresented as positive linear combination of extremal generators gi ∈ Ke . Now assume that p=

N X

j=1

8

αj gj

is such a representation with N minimal. Then N > 1, since no extremal generator has roots on the imaginary axis. It follows that −g1 (ıω) =

N X

α1−1 αj gj (ıω) 6= 0

j=2

and hence VK (ıω) must contain a line, in contradiction with our assumption. This concludes the proof. Remark 3.6 A closed convex cone K ⊂ P+ n containing a stable polynomial of degree ≥ deg K − 1 satisfies the assumptions of Proposition 3.5 (and hence is stable) if and only if p(ıω)q(ıω) 6= 0, | arg p(ıω) − arg q(ıω)| < π, ω ≥ 0, p, q ∈ Ke (14) where the argument function is defined continuously along the paths ω 7→ p(ıω) and ω 7→ q(ıω), respectively. We now consider value sets of semi-stable convex cones. Proposition 3.7 Suppose that K is a semi-stable convex cone in P+ n . Then VK (ıω) 6= C,

ω ≥ 0.

Proof: Assume that there is ω0 ≥ 0 such that VK (ıω0 ) = C. Choose nonzero complex numbers si , i = 0, 1, 2 such that s2 6∈ Rs1

and s0 + s1 + s2 = 0.

There are polynomials pi ∈ K, i = 0, 1, 2 such that pi (ıω0 ) = si . Then −p0 (ıω0 ) ∈ int cone {p1 (ıω0 ), p2 (ıω0 )}. By continuity there exists a neigbourhood U of ıω0 in C such that −p0 (s) ∈ int cone {p1 (s), p2 (s)} ,

s ∈ U.

Choosing some z ∈ U with Re z > 0 we see that there exist α, β > 0 such that p0 (z) + αp1 (z) + βp2 (z) = 0. Since p0 + αp1 + βp2 ∈ K \ {0}, this contradicts the semi-stability of K. For arbitrary convex sets of polynomials we obtain the following counterpart of Proposition 3.7. Corollary 3.8 Suppose that C is a semi-stable convex set in P+ n . Then 0 6∈ int VC (ıω),

9

ω ≥ 0.

Proof: The cone K = ray C is convex and semi-stable with value set ω ≥ 0.

VK (ıω) = ray VC (ıω),

If 0 ∈ int VC (ıω) for some ω > 0 then VK (ıω) = C and this is in contradiction to Proposition 3.7. By this corollary, if C ⊂ P+ n is a semi-stable convex set and p ∈ C satisfies p(ıω) ∈ int VC (ıω) for all ω > 0 and p(0) > 0 then p(ıω) does not have a root on the imaginary axis, hence it is stable. Therefore we obtain the following nP

o

N N Corollary 3.9 Suppose that K = is a semi-stable polyj=1 αj pj ; (α1 , . . . , αN ) ∈ R+ + hedral cone in Pn such that VK (0) is not reduced to the origin and

int VK (ıω) 6= ∅, Then every polynomial p =

PN

j=1 αj pj

ω > 0.

(15)

with αj > 0 for all j = 1, . . . , N is stable.

The following proposition is a conic version of the Edge Theorem, see [BHL88]. Proposition 3.10 Suppose that K ⊂ P+ n is a polyhedral cone and {p1 , . . . , pN } is any finite set of generators of K. Then K is stable if and only if all the one-parameter families tpj + (1 − t)pk , t ∈ [0, 1] corresponding to edges of the polytope Π = conv {p1 , . . . , pN } are stable. Proof: This is an immediate consequence of the Edge Theorem since K is stable if and only if Π is stable. We now prove an edge theorem for semi-stable polyhedral cones. Proposition 3.11 Suppose that K ⊂ P+ n is a polyhedral cone and {p1 , . . . , pN } is any finite set of generators of K. Then K is semi-stable if and only if all the one-parameter families tpj + (1 − t)pk , t ∈ [0, 1] (16) corresponding to edges of the polytope Π = conv {p1 , . . . , pN } are semi-stable. Proof: The necessity of the condition is trivial. Now suppose that K satisfies the condition and contains a polynomial q having a root z0 with positive real part δ. Consider the left shift operator on Pn : S δ : Pn −→ Pn , 2

p(s) 7→ p(s + δ/2).

(17)

For every polynomial p ∈ P+ n , the shifted polynomial S δ2 (p) has a root z ∈ C if and only if p has a root z + δ/2. Hence S δ maps the segment of polynomials (16) onto the segment 2 of stable polynomials tS δ (pj ) + (1 − t)S δ (pk ) ⊂ Hn+ , 2

t ∈ [0, 1].

2

S δ is a linear isomorphism of Pn onto itself. Therefore the shifted polynomials S δ (pi ), i = 2 2 1, . . . , N generate the polyhedral cone S δ (K) and the edges of S δ (Π) are exactly the 2 2 images of the edges of Π via S δ . Hence it follows from Proposition 3.10 that S δ (K) is 2 2 stable. But S δ (K) contains the unstable polynomial S δ (q) which is a contradiction. 2

2

10

Edge theorems reduce the stability of a polytope or polyhedral cone to the stability of a collection of one-parameter families of polynomials. The question arises which collection of one-parameter families yields the most efficient stability test. The answer to this question strongly depends on the concrete polyhedral set at hand. The set of edges of a polytope constitutes a universally applicable and geometrically appealing selection of segments which suffices as a test set for stability, according to the edge theorem. However, as the results in [Bar94] demonstrate, far more efficient selections of segments may be found for specific classes of polytopes. The method of value sets provides us with a guideline for constructing such selections. Definition 3.12 Given a polytope Π = conv {p1 , . . . , pN } ⊂ P+ n , a set S of segments in Π is said to be covering if the union of the segments [p(ıω), q(ıω)], [p, q] ∈ S covers the boundary of the value set VΠ (ıω) for all ω ≥ 0: [

[p(ıω), q(ıω)] ⊃ ∂VΠ (ıω),

ω ≥ 0.

[p,q]∈S

An easy argument shows that in the situation of Definition 3.12 deg( S) = deg Π (consider the behaviour of VΠ (ıω) as ω → ∞). Hence we obtain the following corollary of the zero exclusion principle. S

Corollary 3.13 Suppose that S is a covering collection of segments of a polytope Π = conv {p1 , . . . , pN } ⊂ P+ n . Then Π is stable if and only if all the segments [p, q] ∈ S are stable. A conic counterpart of Definition 3.12 is Definition 3.14 Given a closed convex cone K ⊂ P+ n , a subset S ⊂ K is said to be a covering selection of polynomials for K if, for every ω ≥ 0, the value set VK (ıω) is the smallest convex cone containing the value set VS(ıω). If K is an arbitrary closed convex cone and S ⊂ K is a covering selection of polynomials for K then VK (ıω) = Vcone S(ıω), ω ≥ 0 . (18) Therefore the zero exclusion principle implies Corollary 3.15 Suppose that K is a closed convex cone containing a stable polynomial of degree ≥ deg K − 1 and S ⊂ K is a covering selection of polynomials for K. Then K is stable if and only if the convex cone generated by S is stable. The following theorem gives an extremal ray stability criterion for polyhedral conic sets provided it is known that the cone of directions of this set is semi-stable (a necessary condition). For later applications we state the theorem in a slightly more general form for closed convex cones with a finite covering selection. + Theorem 3.16 Suppose that p0 ∈ P+ n , K is a semi-stable closed convex cone in Pn , deg p0 ≥ deg K − 1 and S ⊂ K is a finite covering selection of polynomials for K. Then the conic set p0 + K is stable if and only if the rays

p0 + R+ g, are stable. 11

g∈S

(19)

Proof: The ’only if’ statement is trivial. To prove the converse, we first consider the case when K is stable. Assume that K and the rays (19) are stable, but p0 + K is not stable. By Corollary 3.3 there exists ω ≥ 0 such that −p0 (ıω) ∈ VK (ıω). Set ω0 = inf{ω ∈ R+ ; −p0 (ıω) ∈ VK (ıω)} . There exists a sequence (ωk ) converging to ω0 from above such that −p0 (ıωk ) ∈ VK (ıωk ). Since VK (ıωk ) is pointed, there exist, for every k ∈ N, two polynomials pk , qk ∈ S such that VK (ıωk ) = cone {pk (ıωk ), qk (ıωk )}. Replacing (ωk ) by a suitable subsequence we may suppose without restriction of generality that all pk = p, qk = q for suitable p, q ∈ S and all k ∈ N. It follows that −p0 (ıωk ) = αk p(ıωk ) + βk q(ıωk ),

k∈N

for some αk , βk ≥ 0, k ∈ N. Now assume for a moment that the sequence (γk ) = (max{αk , βk }) is not bounded. Working with a subsequence if necessary we can suppose that limk→∞ γk = ∞. Then h

i

lim (γk−1 αk )p(ıωk ) + (γk−1 βk )q(ıωk ) = lim −γk−1 p0 (ıωk ) = 0 .

k→∞

k→∞

Without restriction we may assume that the two sequences (γk−1 αk ), (γk−1 βk ) are convergent, with limits a and b, respectively. Then a + b > 0. Taking limits we obtain ap(ıω0 ) + bq(ıω0 ) = 0 . But this is a contradiction since ap + bq is nontrivial, belongs to K and hence is stable. We conclude that the two sequences (αk ), (βk ) are bounded. Without restriction we may assume that they are convergent, with limits α and β, respectively. Passing to the limit it follows that −p0 (ıω0 ) = αp(ıω0 ) + βq(ıω0 ) . Clearly ω0 > 0 (since p0 , p, q ∈ P+ n are stable). By continuity −p0 (ıω0 ) cannot belong to the interior of VK (ıω0 ) since otherwise −p0 (ıω) ∈ VK (ıω) for some ω ∈ (0, ω0). But this implies that −p0 (ıω0 ) lies on an extremal ray of the pointed convex cone VK (ıω0 ), i.e. −p0 (ıω0 ) ∈ ray p(ıω0 ) or −p0 (ıω0 ) ∈ ray q(ıω0 ). But this contradicts assumption (19) so that the theorem is proved in the case where K is stable. Now assume that the rays (19) are stable and K is semi-stable, but p0 + K is not stable. As before there exists ω ≥ 0 such that −p0 (ıω) ∈ VK (ıω). If −p0 (ıω) ∈ ray g(ıω) for some g ∈ S, we obtain a contradiction to assumption (19). Thus −p0 (ıω) ∈ int cone {p(ıω), q(ıω)} for some p, q ∈ S. Using the shift operator described in the proof of Proposition 3.11 there exists, by continuity, ε > 0 such that −Sε (p0 )(ıω) ∈ int cone {Sε (p)(ıω), Sε(q)(ıω)} .

12

(20)

˜ = cone {Sε (p), Sε (q)} ⊂ Sε (K) are stable. Moreover, the rays But Sε (p0 ) and the cone K Sε (p0 ) + ray Sε (p) = Sε (p0 + R+ p)

and Sε (p0 ) + R+ Sε (q) = Sε (p0 + R+ q)

˜ = ˜ = cone {Sε (p), Sε (q)}, S are stable. Applying the first part of the proof to Sε (p0 ), K {Sε (p), Sε (q)} we conclude that Sε (p0 ) + cone {Sε (p), Sε (q)} is stable. But this contradicts (20). Therefore −p0 (ıω) 6∈ VK (ıω) for all ω ≥ 0 and hence p0 + K is stable. If K = cone {p1 , . . . , pN } ⊂ P+ n is a polyhedral cone, then S = {p1 , . . . , pN } is a covering selection of polynomials for K and the previous theorem shows that p0 + K is stable if and only if K is semi-stable and the rays p0 + R+ pj , j = 1, ..., N are stable.

4

(Semi-)stability of segments and rays of polynomials

The above results show that for the stability analysis of conic sets it is important to have semi-stability and stability criteria for closed segments [p0 , p1 ] and for rays p0 + R+ p1 of polynomials. Because of p0 + αp1 = (1 + α)



1 α p0 + p1 , 1+α 1+α 

α≥0

a ray p0 + R+ p1 is (semi-)stable if and only if the semi-open segment [p0 , p1 ) = {(1 − t)p0 + tp1 ; t ∈ [0, 1)} is (semi-)stable. We first study segments of polynomials by Rantzer’s method. Proposition 4.1 Suppose that p0 , p1 ∈ P+ n are semi-stable polynomials such that one of the following four conditions is satisfied. (i) The difference d = p1 − p0 satisfies ∂ arg(d(ıω)) < 0, ∂ω

ω ∈ {w > 0; d(ıw) 6= 0} .

(21)

(ii) Each of the polynomials p0 , p1 has at least one root in the open left halfplane and



sin(2 arg(d(ıω))) ∂ arg(d(ıω)) , < ∂ω 2ω

ω ∈ {w > 0; d(ıw) 6= 0} .

(22)

(iii) One of the polynomials p0 , p1 has at least one root in the open left halfplane and ∂ arg(d(ıω)) ≤ 0, ∂ω

ω ∈ {w > 0; d(ıw) 6= 0} .

(23)

(iv) Each of the polynomials p0 , p1 has at least two roots in the open left halfplane and



sin(2 arg(d(ıω))) ∂ arg(d(ıω)) , ≤ ∂ω 2ω

13

ω ∈ {w > 0; d(ıw) 6= 0} .

(24)

Then no polynomial p in the open segment (p0 , p1 ) = {αp1 + (1 − α)p0 ; α ∈ (0, 1)}

(25)

has roots on the imaginary axis which are not joint roots of p0 and p1 . Proof: Semistability of the polynomials p0 , p1 implies (see [Ran92]) ∂ arg(pi (ıω)) ≥ 0, ∂ω

ω ∈ {w > 0; pi (ıw) 6= 0},

i = 0, 1

(26)

If additionally they have at least one root in the open left halfplane then

and



sin(2 arg(p (ıω))) ∂ arg(pi (ıω)) i , ≥ ∂ω 2ω

ω ∈ {w > 0; pi (ıw) 6= 0},

i = 0, 1

(27)

∂ arg(pi (ıω)) > 0, ω ∈ {w > 0; pi (ıw) 6= 0}, i = 0, 1 (28) ∂ω If they have at least two roots in the open left halfplane then strict inequality holds in (27). Now suppose that 0 ∈ (p0 (ıω0 ), p1 (ıω0 )) and pi (ıω0 ) 6= 0, i = 0, 1 for some ω0 ≥ 0. Then (see [Ran92]) ∂ arg(d(ıω0 )) ≥ min ∂ω

(

∂ arg(p0 (ıω0 )) ∂ arg(p1 (ıω0 )) , ∂ω ∂ω

)

(29)

Equality takes place only when the two derivatives on the right hand side coincide. Moreover ω0 > 0, since p0 (0) + p1 (0) > 0 implies (p0 (0), p1 (0)) ⊂ (0, ∞). Hence we obtain a contradiction in each of the four cases (i) - (iv). We conclude that for no ω ≥ 0 we have |p0 (ıω)| + |p1 (ıω)| > 0 and 0 ∈ (p0 (ıω), p1(ıω)) . This means that if p(ıω) = 0 for some p ∈ (p0 , p1 ) and ω ≥ 0, then necessarily p0 (ıω) = p1 (ıω) = 0. Corollary 4.2 Suppose that p0 , p1 ∈ P+ n are semi-stable and satisfy one of the conditions (i) - (iv) of Proposition 4.1. Then (a) If p0 and p1 do not have joint roots on the imaginary axis and the open segment (p0 , p1 ) contains a semi-stable polynomial p then all polynomials in (p0 , p1 ) are stable. (b) If the segment [p0 , p1 ] contains a stable polynomial (in particular, if one of the polynomials p0 , p1 is stable) then all polynomials in the open segment (p0 , p1 ) are stable. (c) If p0 , p1 satisfy conditions (iii) or (iv) and the open segment (p0 , p1 ) contains a semi-stable polynomial then all polynomials in (p0 , p1 ) are semi-stable.

14

Proof: (a) (p0 , p1 ) cannot contain an unstable polynomial q since otherwise the segment [p, q] ⊂ (p0 , p1 ) would contain a polynomial with a root on the imaginary axis, and this contradicts the previous proposition. (b) Suppose that [p0 , p1 ] contains a stable polynomial. Then (p0 , p1 ) contains a stable polynomial and there cannot exist a joint root of p0 , p1 on the imaginary axis. Hence (a) implies (b) (c) Let u be the maximal common divisor of p0 , p1 which has only roots on the imaginary axis. Then pi = uqi , i = 0, 1 where q0 , q1 have no common roots on ıR and the open segment (q0 , q1 ) contains a semi-stable polynomial. Now arg u(ıω) is a constant function of ω between two successive joint roots of p0 , p1 on the imaginary axis so that ∂ arg(d(ıω)) ∂ arg(q1 (ıω) − q0 (ıω)) = , ∂ω ∂ω

ω ∈ {w > 0; d(ıw) 6= 0} .

Hence d˜ = q1 −q0 satisfies one of the two conditions (23) and (24) for ω ∈ {w > 0; d(ıw) 6= ˜ 0} and thus, by continuity, for ω ∈ {w > 0; d(ıw) 6= 0}. Therefore we can apply (a) to the semi-stable polynomials q0 , q1 and obtain that all polynomials in the open segment (q0 , q1 ) are stable. Since (p0 , p1 ) = u(q0 , q1 ) statement (c) is proved. Remark 4.3 As a consequence of Corollary 4.2 the ray p0 + R+ p1 (or equivalently the semi-open segment [p0 , p1 )) is stable if p0 is stable, p1 is semi-stable and one of the conditions (i)-(iv) of Proposition 4.1 is satisfied. After these Rantzer type sufficiency conditions we now derive necessary and sufficient stability criteria for rays of the form (19). The stability of the ray p0 + R+ p1

(30)

can be expressed via two classical tools of control theory: the Nyquist plot and the root locus. In fact, the root locus of the rational function g(s) = p1 (s)/p0 (s) is given by Rootlocus (g) = {λ ∈ C; ∃α ≥ 0 : p0 (λ) + αp1 (λ) = 0} .

(31)

Hence the ray (30) is stable if and only if the roots locus of g(s) lies in the open left halfplane. Now let us suppose that p0 is stable. By the zero exclusion principle (30) is stable if and only if p0 (ıω) + αp1 (ıω) 6= 0, ω ≥ 0, α ≥ 0 . (32) (32) is equivalent to the condition that the Nyquist curve of the rational function g(s) = p1 (s)/p0 (s) does not intersect the negative real axis R− = (−∞, 0): {p1 (ıω)/p0(ıω); ω ≥ 0} ∩ R− = ∅ .

(33)

Write pk (ıω) = fk (−ω 2 ) + ıωhk (−ω 2 ),

ω ≥ 0, k = 0, 1

(34)

where fk , hk are polynomials with positive (resp. nonnegative) real coefficients for k = 0 (resp. k = 1). (33) is equivalent to Im(p1 (ıω)/p0 (ıω)) = 0 =⇒ Re(p1 (ıω)/p0 (ıω)) ≥ 0, 15

ω≥0

(35)

or, for all ω ≥ 0, h

i

h

i

ω f0 (−ω 2 )h1 (−ω 2 ) − h0 (−ω 2 )f1 (−ω 2 ) = 0 ⇒ f0 (−ω 2 )f1 (−ω 2 ) + ω 2 h0 (−ω 2 )h1 (−ω 2 ) ≥ 0. (36) Since p0 , p1 ∈ P+ this implication holds automatically for ω = 0. Hence we obtain n Lemma 4.4 Suppose that p0 , p1 ∈ P+ n , p0 is stable and fj , hj are defined by (34). Then the ray p0 + R+ p1 is stable if and only if H(ξ) = f0 (ξ)f1 (ξ) − ξh0 (ξ)h1 (ξ) is nonnegative at all the negative roots of F (ξ) = f0 (ξ)h1 (ξ) − f1 (ξ)h0 (ξ). To check the stability of the ray p0 + R+ p1 for a stable polynomial p0 via this criterion we only have to solve one algebraic equation of degree ≤ max{deg p0 , deg p1 }. The following figures illustrate how stability and instability of the ray p0 + R+ p1 is reflected in the root locus and the Nyquist plot of g = p1 /p0 . In Figure 1 deg p0 − deg p1 = 3, hence the ray p0 + R+ p1 is unstable because of Proposition 2.9. In Figure 2 deg p0 − deg p1 = 2 and the ray is stable although p1 is not (only semi-stable). In Figure 3 deg p0 − deg p1 = 1, p0 and p1 are stable, but the ray is unstable. -3

1

x 10

10

0 8

-1

6 4

-2

Imag Axis

2

-3 0 -2

-4

-4

-5

-6

-6 -8 -10 -10

-8

-6

-4

-2

0 Real Axis

2

4

6

8

-7 -4

10

-2

0

2

4

6

8

10 -3

x 10

Figure 1: Root locus and Nyquist plot for

p1 (s) p0 (s)

=

s2 +s+1 (s+1)(s+2)...(s+5)

4 0.06

3 0.04

2 0.02

Imag Axis

1 0

0 -0.02

-1 -0.04

-2 -0.06

-3 -0.08

-4 -10

-9

-8

-7

-6

-5 -4 Real Axis

-3

-2

-1

0

-0.1 -0.02

Figure 2: Root locus and Nyquist plot for

16

0

p1 (s) p0 (s)

0.02

=

0.04

0.06

s ((s+1)2 +1)(s+8)

0.08

0.1

0.12

4

200

3

0 -200

2

-400

Imag Axis

1

-600 0

-800 -1

-1000 -2

-1200 -3

-1400 -4 -6

-5

-4

-3

-2 Real Axis

-1

0

1

-1600 -500

Figure 3: Root locus and Nyquist plot for

5

0

p1 (s) p0 (s)

500

=

1000

1500

2000

(s+1)(s+2) (s+0.1)3

Application to interval polynomials

In this section we will apply the previous results to cones generated by interval polynomials (“interval cones”) and to conic sets which are translates of such cones. In view of the necessary conditions for a conic set to be stable (see Proposition 2.9) we are particularly interested in finite test sets for the semi-stability of interval polynomials. Via Theorem 3.16 this will give us an algebraic stability test for conic sets which are translates of interval cones. Consider the interval polynomial Π=

n X

[aj , aj ]sj ⊂ P+ n

(37)

j=0

where aj ≤ aj , j = 0, . . . , n are given nonnegative real numbers. It is well known that the value set of Π is given by the rectangle VΠ (ıω) = conv {p1 (ıω), p2 (ıω), p3(ıω), p4 (ıω)},

ω≥0

(38)

where pi , i = 1, . . . , 4 are the polynomials p1 (s) p2 (s) p3 (s) p4 (s)

= = = =

a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + . . . a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + . . . a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + . . . a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + . . . .

(39)

The edges of the rectangle are parallel to the real and imaginary axes, for all ω ≥ 0, and the corner poynomials pi satisfy Re p1 (ıω) = Re p4 (ıω) ≤ Re p2 (ıω) = Re p3 (ıω) Im p1 (ıω) = Im p2 (ıω) ≤ Im p3 (ıω) = Im p4 (ıω),

ω ≥ 0.

In view of Corollary 3.9 it is of interest to know when int VΠ (ıω) 6= ∅ for all ω > 0. 17

(40)

Lemma 5.1 Assume: There exist an odd index 0 ≤ j1 ≤ n and an even index 0 ≤ j2 ≤ n such that aj1 < aj1 and aj2 < aj2 . (41) Then int VΠ (ıω) 6= ∅ for all ω > 0. Proof: By definition of the corner polynomials (39) assumption (41) implies that strict inequalities hold in (40) for ω > 0. Hence the lemma follows from (38). To find a finite test set for semi-stability of interval polynomials we have to analyse the segments between the four corner polynomials pi which are mapped onto the edges of the rectangular value set VΠ (ıω) by the evaluation map p 7→ p(ıω) (Corollary 3.13). Applying Proposition 4.1 to the corner polynomials (39) we obtain: Corollary 5.2 Assume that the corner polynomials p1 , p2 (39) are semi-stable and that one of them has at least one root in the open left halfplane. If condition (41) is satisfied then no polynomial in the open segment (p1 , p2 ) has a purely imaginary root ıω, ω > 0. Proof: By definition (39) and assumption (41) we have d(ıω) = p2 (ıω) − p1 (ıω) = (a0 − a0 ) + (a2 − a2 )ω 2 + (a4 − a4 )ω 4 + . . . > 0,

ω > 0 (42)

whence

∂ arg(d(ıω)) = 0, ω > 0 . ∂ω Moreover it follows from (42) that p1 , p2 do not have joint roots on the imaginary axis except possibly at zero. Thus the corollary follows from Proposition 4.1 (iii). Remark 5.3 Analogous results to Corollary 5.2 can be proved for the corner segments (p2 , p3 ), (p4 , p3 ), and (p1 , p4 ). In each of these cases d(ıω) is either in (0, ∞) or in (0, ∞)ı for all ω > 0 so that the same arguments as in the proof of the previous corollary can be applied. The relative interior [Roc70] of the interval polynomial Π is given by relint Π = {p(s, a); ai ∈ (ai , ai ), i = 0, ..., n}

(43)

where (ai , ai ) = {ai } if ai = ai . By a slight abuse of terminology we call the elements of this set the interior polynomials of Π. Theorem 5.4 Assume that condition (41) is satisfied for the interval polynomial Π (37). Then Π is semi-stable if and only if the four corner polynomials pj , j = 1, . . . , 4 and some interior polynomial p0 ∈ relint Π are semi-stable. Moreover, in this case, all nonzero roots of every interior polynomial are in the open left halfplane and, if a0 > 0, then relint Π is stable.

18

Proof: Assume that p1 , ..., p4 are semi-stable. We first show that 0 6∈ int VΠ (ıω),

ω > 0.

(44)

Suppose that this is not the case and that ω0 = inf{ω ∈ R+ ; 0 ∈ int VΠ (ıω)}. By continuity we have 0 ∈ VΠ (ıω0 ) \ int VΠ (ıω0 ) so that 0 lies on the boundary of the value set VΠ (ıω0 ), say on the (horizontal or vertical) closed segment [pi (ıω0 ), pj (ıω0 )], i, j ∈ {1, . . . , 4}. By Corollary 5.2 and Remark 5.3 the origin cannot lie on the open segment (pi (ıω0 ), pj (ıω0 )) if at least one of the two polynomials has a root in the open left halfplane. If both have only roots on ıR then they both admit either only real or only purely imaginary values on the imaginary axis, and then 0 ∈ (pi (ıω), pj (ıω)) for some ω < ω0 by continuity. Since this contradicts the definition of ω0 we get pℓ (ıω0 ) = 0 for some ℓ ∈ {1, . . . , 4}. Suppose, for instance, that p1 (ıω0 ) = 0. Then p2 (ıω0 ) > 0 and arg p2 (ıω0 ) = 0, but there exist ωk > ω0 converging to ω0 such that 0 ∈ int VΠ (ıωk ) and consequently arg p2 (ıωk ) < 0. This contradicts the fact that, by semi-stability, ∂ arg(p2 (ıω)) ≥ 0, ∂ω

ω ∈ {w > 0; p2 (ıw) 6= 0} .

Similar arguments can be used to obtain a contradiction if pℓ (ıω0 ) = 0, ℓ = 2, 3, 4. This proves (44). Now consider the set of interior polynomials of Π. We have [Roc70, Thm. 6.6] q(ıω) ∈ int VΠ (ıω),

ω > 0,

q ∈ relint Π .

By (44) none of these polynomials has a root on ıR \ {0}. Let us assume for a moment that p2 (0) = a0 > 0. Then no interior polynomial q has a zero root. If q were unstable there would exist a polynomial in the segment [q, p0 ] ⊂ relint Π with a root on the imaginary axis. But we have just shown that this cannot be the case. Hence all the interior polynomials q are stable, and consequently all polynomials in Π = cl relint Π are semi-stable. It remains to treat the case where a0 = 0. Then all four corner polynomials have zero roots. Let m be the smallest multiplicity of these roots, i.e. m = min{k; ak > 0}. Define the interval polynomial n X ˜ = Π [aj , aj ]sj−m . j=m

˜ and its corner polynomials p˜j (s) = pj (s)/sm , j = 1, . . . , 4 satisfy the hypotheses of the Π theorem with p˜2 (0) > 0. Hence we can apply the above considerations to conclude that ˜ is semi-stable and relint Π ˜ is stable. Since Π = sm Π ˜ the theorem is proved. Π Combining the previous theorem with Theorem 3.16 we obtain an algebraic test for the stability of conic sets whose cone of directions is given by an interval polynomial.

19

Corollary 5.5 Suppose that p0 ∈ P+ n and K = ray Π is a cone generated by an interval + polynomial Π (37) in Pn , with corner polynomials p1 , ..., p4 (39). Then the conic family p0 + K is stable if and only if the four rays p0 + R+ pj ,

j = 1, . . . , 4

(45)

are stable and the interior polynomial p1 + ... + p4 is semi-stable. Proof: The necessity of the conditions is clear, see Proposition 2.9. Conversely suppose that the conditions are satisfied. Then the four corner polynomials are semi-stable and the interior polynomial (p1 + ... + p4 )/4 of Π is also semi-stable. Hence, by the previous theorem, the cone K is semi-stable. By (38) we have VK (ıω) = ray VΠ (ıω) = ray conv {p1 (ıω), p2(ıω), p3(ıω), p4 (ıω)},

ω≥0

Therefore S = {p1 , ..., p4 } is a covering selection of polynomials for the convex cone K = ray Π and the stability of p0 + K follows from Theorem 3.16. To determine the stability of the four rays (45) requires the solution of four algebraic equations, see Section 4.

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21