Strong stability of neutral equations with dependent delays

Strong stability of neutral equations with dependent delays W Michiels, T Vyhlidal, P Zitek, H Nijmeijer D Henrion 1 Introduction We discuss stability properties of the linear neutral equation p 2 ẋt + H k ẋt τ k = A 0 xt + A k xt υ k, 1 where xt R n is the state variable, τ := τ 1,, τ p1 R + 0 p 1 and υ := υ 1,, υ p2 R + 0 p2 are time-delays, and H k and A k are real matrices An important aspect in the stability theory of neutral equations is the possible fragility of stability, in the sense that the asymptotic stability of the null solution of 1 may be sensitive to arbitrarily small perturbations of the delays τ This has led to the introduction of the notion of strong stability in [3], which explicitly takes into account the effect of small delay perturbations In [3] a necessary and sufficient condition for the strong stability of the null solution of 1 has been described for the case where the delays τ 1,, τ m can vary independently of each other, and in [6] related spectral properties have been discussed, along with a stabilization procedure for systems with an external input Note that robustness against delay perturbations is of primary interest in control applications, as parametric uncertainty and feedback delays are inherent features of control systems In the existing literature on the stability of neutral equations, subjected to delay perturbations, the delays, τ k, 1 k p 1, in 1 are almost exclusively as- Department of Computer Science, KULeuven E-mail: WimMichiels@cskuleuvenbe Centre of Applied Cybernetics, Czech Technical University in Prague Faculty of Mechanical Engineering, Eindhoven University of Technology LAAS-CNRS & Faculty of Electrical Engineering, Czech Technical University in Prague

sumed to be either mutually independent or commensurate all multiples of the same parameter -an exception is formed by [8] where a problem with three delays depending on two independent parameters is analyzed This simplifies the analysis considerably However, real systems might give rise to a model of the form 1 exhibiting a delay dependency caused by physical or other interactions in the system s dynamics This is explained with a lossless transmission line example in Chapter 96 of [2], where it is shown that a parallel transmission line which consists of a current source, two resistors and a capacitor, gives rise to a system of a neutral type with three delays in the difference part, which are integer combinations of two physical parameters In [1] boundary controlled partial differential equations are described that lead to a closed-loop system of neutral type, where the delays in the model are particular linear combinations of physical feedback delays and delays induced by propagation phenomena In [9] the robustness against small feedback delays of linear systems controlled with state derivative feedback is addressed, motivated by vibration control applications There, the closed-loop system can again be written in the form 1, where the delays τ k are combinations of actuator and sensor delays in input and output channels All these application motivate a study of the dependence of the stability properties of 1 on the delay parameters, under the assumption that the delays τ k, 1 k p 1, are not necessarily independent of each other, but are linear functions of m independent parameters r = r 1,, r m R + 0 m, as described in the following relation: τ k = γ k r, k = 1,, p 1, 2 with γ k := γ k,1,, γ k,m N m \ { 0}, for k = 1,, p 1 Note that the cases of mutually independent delays τ, respectively commensurate delays, appear in this framework as extreme cases m = p 1 and τ k = r k, k = 1,, p 1, respectively m = 1 Note also that no assumptions need to be made on the interdependency of the delays ν, because, as we shall see, this interdependency does not affect the stability robustness wrt small delay perturbations, unlike the interdependency of the delays r Whilst the aim of the paper is to develop the stability theory for neutral equations with dependent delays subjected to delay perturbations, the emphasis will be on the derivation of explicit strong stability criteria and on related spectral properties The criteria available in the literature are only applicable to special cases where severe restrictions are put on the delay dependency structure To obtain a general solution and, in this way, complete the theory, some type of intermediate lifting step may be necessary, where a delay difference equations with dependent delays is transformed into an equation with independent delays with the same spectral properties Hereto, the main step will boil down to the representation of a multivariable polynomial as the determinant of a pencil Such a representation will follow from arguments of realization theory, more precisely, from the construction of lower fractional representations LFRs See, for instance, [10] and the manual of the LFR toolbox [5] for an introduction The structure of the paper is as follows: after the introduction of some basic notions and results, the spectral properties of the neutral equation 1-2 and of the associated delay difference equation are addressed in Section 2 The main results

are presented in Section 3, where computational expressions are presented that lead to explicit strong stability conditions An application is presented Section 4 Some concluding remarks end the paper Throughout the paper the notation r σ, respectively r e, stands for the spectral radius, respectively the radius of the essential spectrum The notation α stands for the spectral abscissa For the sake of conciseness proofs are omitted, which can be found in the full version of the paper [7] Preliminaries The initial condition for the neutral system 1-2 is a function segment ϕ C[ τ, 0], R n, where τ = max k {1,,p1 } τ k and C[ τ, 0], R n is the Banach space of continuous functions mapping the interval [ τ, 0] into R n and equipped with the supremum-norm The fact that the map N : C[ τ, 0], R n R n, defined by N ϕ = ϕ0 + H k ϕ τ k, is atomic at zero guarantees existence and uniqueness of solutions of 1 Let xϕ : t [ τ, xϕt R n be the unique forward solution with initial condition ϕ C[ τ, 0], R n, ie xϕθ = ϕθ, θ [ τ, 0] Then the state at time t is given by the function segment x t ϕ C[ τ, 0], R n defined as x t ϕθ = xϕt + θ, θ [ τ, 0] Denote with T t; r, υ the solution operator, mapping initial data onto the state at time t, ie T t; r, υϕθ = x t ϕθ = xϕt + θ, θ [ τ, 0] 3 This is a strongly continuous semi-group The associated delay difference equation of 1 is given by xt + H k xt γ k r = 0 4 For any initial condition ϕ C D [ τ, 0], R n, where C D [ τ, 0], R n = {ϕ C[ τ, 0], R n : N ϕ = 0}, a solution of 4 is uniquely defined Let T D t; r be the corresponding solution operator The asymptotic behavior of the solutions and, thus, the stability of the null solution of the neutral equation 1 is determined by the spectral radius r σ T t; r, υ, satisfying r σ T 1; r, υ = e c N r, υ, c N r, υ = sup {Rλ : det N λ; r, υ = 0}, 5 where the characteristic matrix N is given by p 2 N λ; r, υ = λ D λ; r A 0 A k e λυ k 6

and D λ; r = I + H k e λ γ k r For instance, the null solution is exponentially stable if and only if r σ T 1; r, υ < 1 or equivalently c N r, υ < 0 [3] see [2] for an overview of stability definitions and their relation to spectral properties In a similar way, the stability of the delay difference equation 4 is determined by the spectral radius where c D r = r σ T D 1; r = e c D r, 7 {, det D λ; r 0, λ C, sup {Rλ : det D λ; r = 0}, otherwise An important property in the stability analysis of neutral equations is the relation 8 r e T 1; r, υ = r σ T D 1; r, 9 see eg [3] From this follows the well known result that a necessary condition for the exponential stability of the null solution of 1-2 is given by the exponential stability of the null solution of the delay difference equation 4 In what follows we will call the solutions of det N λ; r, υ = 0 the characteristic roots of the neutral system 1 Analogously we will call the solutions of det D λ; r = 0 the characteristic roots of the delay difference equation 4 2 Spectral properties We discuss some spectral properties of the neutral equation 1, which are important for the rest of the paper These results are based on [6, 3] Difference equation It is well known that the spectral radius 7 is not continuous in the delays r, see eg [2, 3, 6], which carries over to 8 As a consequence, we are from a practical stability point of view led to the smallest upper bound on the real parts of the characteristic roots, which is insensitive to small delay changes: Definition 1 For r R + 0 m, let C D r R be defined as C D r = lim ɛ 0+ c ɛ r, where c ɛ r = sup {c D r + δ r : δ r R m and δ r ɛ} Clearly we have C D r c D r, and the inequality can be strict, as shown in [6] and illustrated later on We have the following results: Proposition 2 The following assertions hold: 1 the function r R + 0 m C D r is continuous;

2 CD r = c D r for rationally independent 1 r; 3 for all r 1, r 2 R + 0 m, we have sign CD r 1 = sign CD r 2 10 The property 10 leads us the following definition: Definition 3 Let Ξ := sign CD r, r R + 0 m A consequence of the non-continuity of c D wrt r is that arbitrarily small perturbations on the delays may destroy stability of the delay difference equation This observation has lead to the introduction of the concept of strong stability in [3]: Definition 4 The null solution of the delay difference equation 4 is strongly exponentially stable if there exists a number ˆr > 0 such that the null solution of xt + H k xt γ k r + δ r = 0 is exponentially stable for all δ r R + m satisfying δ r < ˆr and r k + δr k > 0, 1 k m The following condition follows from Proposition 2: Proposition 5 The null solution of 4 is strongly exponentially stable if and only if Ξ < 0 In the special case where all the delays τ can vary independently of each other, explicit strong stability conditions can be obtained from the following result, which corresponds to Theorem 6 of [6]: Proposition 6 If p 1 = m and τ i = r i, 1 i m, then the following assertions hold: 1 for all r R + 0 m, C D r is the unique zero of the strictly decreasing function c R fc; τ 1, with f given by fc; r = max α H k e cr k e iθ k 11 θ [0, 2π] m 1 The m components of r = r 1,, r m are rationally independent if and only if m n kr k = 0, n k Z implies n k = 0, k = 1,, m For instance, two delays r 1 and r 2 are rationally independent if their ratio is an irrational number

2 we have Ξ = sign logδ 0, where δ 0 := max θ [0, 2π] m α H k e iθ k ; 12 Neutral equation From 7 it follows that c N r, υ c D r, r, υ R + 0 m R + p2 13 The next property deals with the continuity of the function r, υ c N r, υ: Proposition 7 The function is continuous r, υ R + 0 m R + p 2 max C D r, c N r, υ Note that this result implies that the interdependency of the delays υ, if any, does not affect the robustness of exponential stability wrt small delay perturbations Strong stability can be defined in the same way as for a delay difference equation: Definition 8 The null solution of the neutral equation 1-2 is strongly exponentially stable if there exists a number ˆr > 0 such that the null solution of p 2 ẋt + H k ẋt γ k r + δ r = A 0 + A k xt υ k + δυ k is exponentially stable for all δ r R + m and δ υ R + p 2 satisfying δ r < ˆr, δ υ < ˆr and r k + δr k > 0, ν l + δν l > 0, 1 k m, 1 l p 2, A combination of Proposition 7 and the relation 13 leads to: Proposition 9 The null solution of the neutral equation 1 is strongly exponentially stable if and only if c D r, ν < 0 and Ξ < 0 3 Computational expressions We derive computational formulae for the quantities C D r and Ξ, which, by Proposition 5 and Proposition 9, directly result in strong stability conditions Recall that the characteristic function of 4 is given by: D λ; r = det I + H k e λ γ k r 14

By formally setting x i = e λ ri, i = 1,, m, the function 14 can be interpreted as a multivariable polynomial px 1,, x m := det I + H k Π m l=1 x γ k,l l, 15 with some constraints on the variables In the context of realization theory, it is known that every multivariable polynomial can be lifted and expressed as the determinant of a linear pencil, see [10] We have the following result, for which we give a constructive proof: Proposition 10 There always exist real square matrices H 1,, H m of equal dimensions such that m px 1,, x m = det I + H k x k 16 Proof The proof is based on expressing the polynomial matrix I + H k Π m l=1 x γ k,l l as a lower linear fractional representation Let input w R n and output z R n be such that z = I + H k Π m l=1 x γ k,l l w It can be verified that this expression is equivalent to: [ ] [ ] z w = M, u = x y u 1,, x m y, 17 where [ M11 M 12 M = M 21 M 22 ] := I s 1 blocks {}}{ 0 0 H 1 I 0 0 0 I 0 I 0 sp 1 blocks {}}{ 0 0 H p1 I 0 0 0 I 0 I 0

and x 1,, x m = x 1 I n γ1,1 x mi n γ1,m x 1I n γp1,1, x m I n γ p 1,m with s k = m l=1 γ k,l, 1 k p 1 and I u, u N, denoting the u-by-u unity matrix From 17 we obtain It follows that z = I + M 12 x 1,, x m I M 22 x 1,, x m 1 M 21 y px 1,, x m = det I + M 12 x 1,, x m I M 22 x 1,, x m 1 M 21 = det I + I M 22 x 1,, x m 1 M 21 M 12 x 1,, x m = det I + M 21 M 12 M 22 x 1,, x m = det I + m H k x k, where H k = M 21 M 12 M 22 e k, and e k is the k-th unit vector in R m Returning to the original problem, by Proposition 10 there always exist matrices H 1,, H m such that m det D λ; r = det I + H k e λr k 18 Hence, D λ; r can be interpreted as the characteristic function of the lifted delay difference equation m zt + H k zt r k = 0 As this equation satisfies the assumptions of Proposition 6, the following, main result directly follows: Theorem 11 For the delay difference equation 4 we have Ξ = sign logδ 0, where δ 0 := max θ [0, 2π] m m α H k e iθ k

and the matrices H k are such that 18 holds Furthermore, for all r R + 0 m, C D r is the unique zero of the strictly decreasing function m c R fc; r = max α H k e cr k e iθ k θ [0, 2π] m The main step, the lifting procedure for the computation of the matrices H k, 1 k m, is now illustrated with an example: Example 12 If p 1 = 3, m = 2, γ 1 = 1, 0, γ 2 = 0, 1 and γ 3 = 1, 1, then the delay difference equation 4 becomes: xt + H 1 t r 1 + H 2 xt r 2 + H 3 xt r 1 + r 2 = 0 19 The characteristic equation of 19 is given by det I + H 1 e λr1 + H 2 e λr2 + H 3 e λr1+r2 = 0 The lifting technique described in the proof of Theorem 11 leads to the equivalent expression: H 1 0 0 0 0 H 2 0 H 3 H det I + 1 0 0 0 H 1 0 0 0 e λr 1 0 H + 2 0 H 3 0 H 2 0 H e λr 2 = 0 3 0 0 I 0 0 0 0 0 4 Application The following model from [4] see also [1] for a simplified version, w tt x, t w xx x, t + 2aw t x, t + a 2 wx, t = 0, t 0, x [0, 1], 20 w0, t = 0, w x 1, t = kw t 1, t h, 21 describes the transversal movement of a beam clamped at one side and stabilized by applying a force at the other side The variable wx, t describes the transversal movement at position x at time t The parameter h 0 represents a small delay in the velocity feedback, k 0 is the controller gain, and a 0 represents a damping constant When substituting a solution of the form wx, t = e λt zx in 20-21 the following characteristic equation is obtained, 1 + e 2a e λ2 + ke λh ke 2a e λ2+h = 0 22 Note that this equation can be interpreted as the characteristic equation of a delay difference equation of the form 4, exhibiting three delays τ 1, τ 2, τ 3 = 2, h, 2+h, that depend on two independent delays r 1, r 2 = 2, h

If h = 0, the characteristic roots are λ = 1 2 log 1 + k 1 k a + i πl + π 4 1 + signk 1, l Z As for all k 1, ck := 1 2 log 1 + k 1 k a < 0, 23 the system with h = 0 is stable for all k 1 As k approaches 1, the real parts of the characteristic roots move off to, which indicates superstability at k = 1 meaning that perturbations disappear in a finite time This is indeed the case and can be explained as follows: the general solution of 20 can be written as a combination of two traveling waves, a solution φx te at moving to the right and a solution ψx + te at moving to the left If k = 1, then φx te at satisfies the second boundary condition, and thus the reflection coefficient at x = 1 is zero; at x = 0 the wave φx + t is reflected completely Consequently all perturbations of the zero solution disappear in a finite time at most 2 time-units Next, we look at the effect of a small feedback delay h in the application of the boundary control If the delays r 1, r 2 = 2, h are rationally independent, which occurs if h is an irrational number, then we have c D r = C D r Proposition 2, and the stability condition is given by Ξ < 0 which also guarantees stability for all h > 0 To compute Ξ, we apply Theorem 11, based on the lifting?? This yields: [ ] [ ] e 2a 1 0 0 Ξ = max θ [0, 2π] r σ + e iθ It follows that { = max λ : = 1 2 0 0 } k 1+ e 2a λ λ e 2a = 1, λ C e 2a + k + e 2a + k 2 + 4ke 2a Ξ < 1 k < tanha, 2ke 2a where < can be replaced with >, = We conclude the following: 1 if k < tanha, then the system 20-21 is exponentially stable for all h 0 2 if k > tanha, then the system 20-21 is exponentially unstable for all irrational values of h Consequently, there exist arbitrarily small values of h that destroy the exponential stability of the system without delay in the boundary control In Figure 1 we show the quantities CD 2, h for a = 1 and different values of h, as a function of the feedback gain k The function k ck, with ck defined in 23, is also displayed and the critical values of k, k = tanh1, is indicated The displayed results suggest that lim h 0+ CD 2, h = + if k 1 This is indeed the case and a consequence of the fact that some, but not all delays vanish in 22 as h 0+ k

4 3 1/40 1/20 1/10 h=1/5 2 1 0 1 2 3 4 5 0 05 1 15 2 k Figure 1 The solid curves correspond to C D 2, h for different values of h The dashed curve corresponds to 23 5 Conclusions The stability theory for neutral equations and delay difference equation subjected to delay perturbations has been developed for the case where the delays have an arbitrary dependency structure, with a particular emphasis on spectral properties and computational expressions for C D and Ξ, that, among others, lead to explicit strong stability conditions Instrumental to this, it has been shown that a general delay difference equation with dependent delays can be transformed, without changing the characteristic equation, into a delay difference equation with possibly larger dimension but with independent delays, such that the stability theory for systems with independent delays can be applied to complete the theory An essential step of this lifting procedure consists of representing a multivariate polynomial as the determinant of a pencil

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