We study the robustness of strong stability of the homogeneous difference equation via the concept of strong stability radii: complex, real and positive radii in this paper. We also show that in the case of positive systems, these radii coincide. Finally, a simple example is given.

1. Introduction

Motivated by many applications in control engineering, problems of robust stability of dynamical systems have attracted a lot of attention of researchers during the last twenty years. In the study of these problems, the notion of stability radius was proved to be an effective tool, see [1–5]. In this paper, we study the robustness of strong stability of the homogeneous difference equation under parameter perturbations.

The organization of this paper is as follows. In Section 2, we recall some results on nonnegative matrices and present preliminary results on homogeneous equations for later use. In Section 3, we study a complex strong stability radius under multiperturbations. Next, we present some results on strong stability radii of the positive class equations under parameter perturbations. It is shown that complex, real, and positive strong stability radii of positive systems coincide. More important, estimates and computable formulas of these stability radii are also derived. Finally, a simple example is given.

2. Preliminaries

2.1. Nonnegative Matrices

We first introduce some notations. Let n,⁢ l,⁢ q be positive integers, a matrix P = [p_ij] ∈ ℝ^l×q is said to be nonnegative (P ≥ 0) if all its entries p_ij are nonnegative. It is said to be positive (P > 0) if all its entries p_ij are positive. For P,Q ∈ ℝ^l×q, P > Q means that P − Q > 0. The set of all nonnegative l × q-matrices is denoted by . A similar notation will be used for vectors. Let 𝕂 = ℂ or ℝ, then for any x ∈ 𝕂ⁿ and P ∈ 𝕂^l×q, we define and by |x | = (|x_i|), | P | = [|p_ij|]. For any matrix A ∈ 𝕂^n×n the spectral radius and the spectral abscissa of A is defined by r(A) = max⁡{|λ | : λ ∈ σ(A)} and μ(A) = max⁡{ℜλ : λ ∈ σ(A)}, respectively, where σ(A) is the spectrum of A. We recall some useful results, see [6].

A norm ∥⋅∥ on 𝕂ⁿ is said to be monotonic if it satisfies

(2.1)

It can be shown that a vector norm ∥⋅∥ on 𝕂ⁿ is monotonic if and only if ∥x∥ = ∥|x | ∥ for all x ∈ 𝕂ⁿ, see [7]. All norms on 𝕂ⁿ we use in this paper are assumed to be monotonic.

Theorem 2.1 (Perron-Frobenius). Suppose that . Then

(i)
r(A) is an eigenvalue of A and there is a nonnegative eigenvector x ≥ 0,⁢ x ≠ 0 such that Ax = r(A)x.
(ii)
If λ ∈ σ(A) and |λ | = r(A) then the algebraic multiplicity of λ is not greater than the algebraic multiplicity of the eigenvalue r(A).
(iii)
Given α > 0, there exists a nonzero vector x ≥ 0 such that Ax ≥ αx if and only if r(A) ≥ α.
(iv)
(tI − A) ⁻¹ exists and is nonnegative if and only if t > r(A).

Theorem 2.2. Let . If |A | ≤ B then

(2.2)

2.2. Homogeneous Difference Equations

Consider the neutral differential difference equation of the following form:

(2.3)

where D(r,A) : C([−h; 0], ℝⁿ) → ℝⁿ is linear continuous defined by

(2.4)

Here each A_i is an n × n-matrix, each r_i is a constant satisfying r_i > 0 and

and y_t ∈ C([−h; 0], ℝⁿ) is defined by y_t(s) = y(t + s), s ∈ [−h; 0], t ≥ 0. Recall that there is a strictly close relation between the asymptotic behavior of solutions of (2.3) and that of associated linear homogeneous difference equations

(2.5)

or equivalently,

(2.6)

A study of the asymptotic behavior of solutions of system (2.6) plays a fundamental role in understanding the asymptotic behavior of solutions of linear neutral differential equations of the form (2.3), see [8].

We recall the definition in [8]: the operator D(r,A) or system (2.6) is called stable if the zero solution of (2.6) with y₀ ∈ C_D(r,A) = {ϕ ∈ C([−h,0], ℝⁿ) : D(r,A)ϕ = 0} is uniformly asymptotically stable.

Associated with system (2.6) we define the quasipolynomial

(2.7)

For s ∈ ℂ, if det⁡H(s) = 0, then s is called a characteristic root of the quasipolynomial matrix (2.7). Then, a nonzero vector x ∈ ℂⁿ satisfying H(s)x = 0 is called an eigenvector of H(⋅) corresponding to the characteristic root s. We set σ(H(⋅)) = {λ ∈ ℂ : detH(λ) = 0}, the spectral set of (2.7), and a_H = sup⁡{ℜλ : λ ∈ σ(H(⋅))}, the spectral abscissa of (2.7). The following lemma is a well-known result in [8].

Theorem 2.3. System (2.6) is stable if and only if a_H < 0.

It is well known that a_H is not continuous in the delays (r₁, … ,r_N), see [9]. One consequence of the noncontinuity is that arbitrarily small perturbations on the delays may destroy stability of the difference equation. This has led to the introduction of the concept of strong stability in Hale and Verduyn Lunel [10].

Definition 2.4. System (2.6) is strongly stable in the delays if it is stable for each .

The concept of strong stability has interested many researchers as in [8–13] and references therein. Now we recall a result in [10].

Theorem 2.5. The following statements are equivalent:

(i)
system (2.6) is strongly stable,
(ii)
.

We set ℂ₁ = {z ∈ ℂ:|z | < 1} and ∂ℂ₁ = {z ∈ ℂ:|z | = 1}. Since r(⋅) is continuous in ℂ^n×n, we imply the continuity of the following function g : (∂ℂ₁) ^N → ℝ defined by

(2.8)

Moreover, by the compactness of the set

, there exists

such that

(2.9)

By the above result, we can get the following statement: system (2.6) is strongly stable if and only if

(2.10)

3. Main Results

3.1. Complex Strong Stability Radius

Suppose that system (2.6) is strongly stable. Now we assume that each matrix A_i is subjected to the perturbation of the form

(3.1)

where

are given matrices defining the structure of the perturbations and

are unknown matrices,

. We write the perturbed system

(3.2)

Definition 3.1. Let system (2.6) be strongly stable. The complex, real, and positive strong stability radii of system (2.6) under perturbations of the form (3.1) are defined by

(3.3)

respectively, we set inf⁡∅ = +∞.

If system (2.6) is strongly stable, we define a function

. It is easy to see that H(⋅,⋅) is well-defined. For any

, we set

(3.4)

Theorem 3.2. Let system (2.6) be strongly stable. Then we have

(i)

(3.5)
(ii)
in particular, if D_i = D_j (or E_i = E_j) for all , then we have
(3.6)

Proof. Let be a destabilizing disturbance. Then there exists such that . This means that there exists a nonzero vector x satisfying

(3.7)

This follows that

(3.8)

or equivalently,

(3.9)

Choose

such that

. Multiplying the above equation with E_q, we obtain

(3.10)

This implies that

(3.11)

From this inequality and the definition of r_ℂ, the left-hand inequality of (i) follows:

(3.12)

Now it remains to prove the right-hand inequality of (i):

(3.13)

Indeed, for any

, and

, there exists nonzero vector

such that ∥x∥ = 1 and ∥G_ii(λ,z)x∥ = ∥G_ii(λ,z)∥. By Hahn-Banach theorem, there exists

satisfying ∥y^*∥ = 1 and y^*(G_ii(λ,z)x) = ∥G_ii(λ,z)x∥. We define a matrix

by setting

(3.14)

Now we construct the disturbance Δ = (Δ₁, … ,Δ_N) defined by

(3.15)

It is easy to check that

. Moreover, we have

(3.16)

where

. This means that Δ is a destabilizing disturbance. Thus,

(3.17)

The proof of (i) is complete, and (ii) can be obtained directly from (i).

In general, the complex, real, and positive radius are distinct, see [4, 5]. Theorem 3.2 reduces the computation of the complex strong stability radius to a global optimization problem with many variations while the problem for the real stability radius is much more difficult, see [5]. It is therefore natural to investigate for which kind of systems these three radii coincide. The answer will be found in the next section.

3.2. Strong Stability Radii of Positive Systems

In this section, we restrict system (2.6) to be positive, that is, A_i are nonnegative for all .

Lemma 3.3. Let . Then we have

(i)
;
(ii)
,

Proof. (i) By Theorem 2.2, we have

(3.18)

(ii) the positivity of can be implied by Theorem 2.1. The right-hand inequality can be obtained by the following formula:

(3.19)

This completes the proof.

It is important to note from above lemma that under positivity assumptions, system (2.6) is strongly stable if and only if r(A₁ + ⋯+A_N) < 1.

Lemma 3.4. Suppose that system (2.6) is positive and strongly stable. Then, for any , we have

(3.20)

Proof. For any , we have . Thus, for an arbitrary vector x ∈ ℝ^p,

(3.21)

By Lemma 3.3, we have

. Thus, we imply

(3.22)

Theorem 3.5. Let system (2.6) be strongly stable and positive. Assume that all are nonnegative matrices. If D_i = D_j or , then

(3.23)

where

Proof. By Theorem 3.2, we have

(3.24)

Moreover, using Lemma 3.4, we get

(3.25)

Since r_ℂ ≤ r_ℝ ≤ r₊, we only need to prove that

(3.26)

Indeed, for any

, since G_ii(1, 1) is a nonnegative matrix, there exists nonnegative vector

such that ∥x∥ = 1 and ∥G_ii(1, 1)x∥ = ∥G_ii(1, 1)∥. Using Krein-Rutman theorem, see [14], there exists

satisfying ∥y^*∥ = 1 and y^*(G_ii(1, 1)x) = ∥G_ii(1, 1)x∥. We define a nonnegative matrix

by setting

(3.27)

Now we construct the positive disturbance Δ = (Δ₁, … ,Δ_N) defined by

(3.28)

It is easy to check that

. Moreover, we have

(3.29)

where

. It means that Δ is a destabilizing disturbance. Thus

(3.30)

The proof is complete.

Now we turn to a different perturbation structure and assume that each matrix A_i is subjected to perturbations of the form

(3.31)

where B_ij are given matrices defining the structure of the perturbations and δ_ij are unknown scalars representing parameter uncertainties. So we can write the perturbed system

(3.32)

Definition 3.6. Let system (2.6) be strongly stable. The complex, real, and positive strong stability radii of system (2.6) under perturbations of the form (3.31) are defined by

(3.33)

respectively, we set inf⁡∅ = +∞, and

, where

Lemma 3.7. Suppose system (2.6) is strongly stable, positive and . Then

(3.34)

Proof. Because , we only need to prove that Indeed, for a destabilizing disturbance , there exist a and a nonzero vector x ∈ 𝕂ⁿ such that

(3.35)

This yields

(3.36)

By Theorem 2.1, we get

(3.37)

It means that

is also a destabilizing disturbance. Thus, by the definition of complex and real radii,

. The proof is complete.

Theorem 3.8. Suppose system (2.6) is strongly stable, positive and . Then

(3.38)

where B = ∑_i,jB_ij.

Proof. By Lemma 3.7, we only need to prove that

(3.39)

To do it, taking arbitrary destabilizing disturbance

, by Lemma 3.3 and Theorem 2.1, there exist a λ ≥ 1 and a nonzero vector

such that

(3.40)

or equivalently,

(3.41)

This yields

(3.42)

Then, we have

(3.43)

Using Theorem 2.1 again, we obtain

(3.44)

or equivalently,

(3.45)

Thus, from the definition of

, one has

(3.46)

On the other hand, setting

. Then, by Theorem 2.1, there exists a nonnegative vector

satisfying

(3.47)

This is equivalent to

(3.48)

Hence,

(3.49)

This means that

is a destabilizing disturbance and thus,

. The proof is complete.

Now we consider the following example to illustrate the obtained results.

Example 3.9. Consider system

(3.50)

where

(3.51)

Then we have

(3.52)

Thus system (3.50) is strongly stable.

Assume that the matrices A₁,A₂ are subjected to perturbations of the form A₁ → A₁ + D₁Δ₁E₁,⁢ ⁢ A₂ → A₂ + D₂Δ₂E₂, where

(3.53)

Then

(3.54)

If ℝ² is provided with the norm defined by ∥(x,y)∥ = |x | +|y|, then by Theorem 3.5, we have

(3.55)

Assume that the given two matrices A₁,⁢ A₂ are subjected to perturbations of the form A₁ → A₁ + δ₁₁B₁₁ + δ₁₂B₁₂,⁢ A₂ → A₂ + δ₂₁B₂₁ + δ₂₂B₂₂, where

(3.56)

Then

(3.57)

By Theorem 3.8, we get

(3.58)

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The Robustness of Strong Stability of Positive Homogeneous Difference Equations

Abstract

1. Introduction

2. Preliminaries

2.1. Nonnegative Matrices

2.2. Homogeneous Difference Equations

3. Main Results

3.1. Complex Strong Stability Radius

3.2. Strong Stability Radii of Positive Systems

References

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The Robustness of Strong Stability of Positive Homogeneous Difference Equations

Abstract

1. Introduction

2. Preliminaries

2.1. Nonnegative Matrices

2.2. Homogeneous Difference Equations

3. Main Results

3.1. Complex Strong Stability Radius

3.2. Strong Stability Radii of Positive Systems

References

Citing Literature

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