Stability and stabilizability of linear time-delay systems on homogeneous time scales

This paper provides necessary and sufficient conditions for the exponential stability of a linear retarded time-delay system defined on a homogeneous time scale. Conditions are formulated in terms of a characteristic equation associated with the system. This approach is then used to develop feedback stabilizability criteria.


INTRODUCTION
Traditionally dynamic systems are studied in a continuous time domain.Over the years numerous sophisticated techniques have been developed for an efficient analysis.At the same time the digital world motivates development of rigorous methods for analysis of systems evolving in discrete time, not necessarily uniform.A lot of efforts have been made to adopt the ideas from continuous settings to the discrete domain.However, it is still a great challenge for scientists how to effectively merge the two paradigms.One of the most successful attempts is based on the notion of time scales, which first appeared in [12].The main idea extends beyond the continuous and discrete domains.The theory of systems on time scales comprises nonuniformly sampled systems, interval models, hybrid systems, etc.This formalism has been successfully applied to solve various problems.A good introduction can be found in [4].
There exists a large class of systems naturally obeying the delay property, e.g.communication systems, long transmission lines, pipelines, remote control systems such as satellites, etc.Some of the available results can be found in classical monographs such as [1,10,11].The majority of works deal separately with the continuous-or discrete-time cases.Very few results are available for the study of time-delay systems on time scales.To address stability related problems, the approach based on the Lyapunov functions is usually used.In [20] the necessary and sufficient conditions for the asymptotic stability of linear positive systems with bounded time-varying delays were derived.A similar approach was applied in [14], where the stability of delay impulsive (at fixed times) systems on time scales was studied.A slightly larger class of impulsive hybrid systems was considered in [19].Furthermore, [13] proposes sufficient conditions for the practical stability of hybrid dynamic systems.Some more specific adaptations for the case of neural networks are reported in [8].
In this paper, the time scales calculus is used to unify stability studies of linear time-delay systems of retarded type (when delay enters the state) defined on homogeneous time scales.We rely on a functional representation of a time-delay system following the ideas presented in [11,16].In brief, contributions of this paper can be seen in the following.The Laplace transform is extended to operate with time-delay systems defined on a homogeneous time scale.The developed tools, given in terms of the characteristic equation of a system, are then used to derive necessary and sufficient conditions for the exponential stability.Furthermore, necessary and sufficient conditions for stabilizability are presented.Since the main results partly (in the discrete-time case) rely on the delay-free case, the necessary proofs for the systems without time delay are collected in Appendix A.
The rest of the paper is organized as follows.Section 2 provides a brief overview of the time scales calculus.Section 3 presents a unified stability definition of a linear time-delay system defined on a homogeneous time scale.It is accompanied by the necessary and sufficient exponential stability condition.The developed mathematical tools to operate with the Laplace transform on time scales are presented.The end of Section 3 is devoted to a stabilizability problem.

CALCULUS ON TIME SCALES
The following definitions and a general introduction to time scales calculus can be found in [4].A time scale T is an arbitrary nonempty closed subset of the set of real numbers R.This paper is focused on two cases most important for control theory, i.e. the continuous-time case T = R and the discrete-time case T = τZ := {τk | k ∈ Z} for τ > 0. For t ∈ T the forward jump operator σ : T → T is defined by σ (t) := inf {s ∈ T | s > t}.The graininess function µ : T → [0, ∞) is defined by µ(t) := σ (t) − t.If µ(t) ≡ const, then a time scale T is called homogeneous.In this paper we assume that the time scale T is homogeneous.The delta derivative of ξ : T → R is denoted by ξ ∆ (t) and the operators antiderivative is denoted by ∫ ξ (t)∆t.Table 1 illustrates the above-mentioned for two typical cases of T, where id means identity operator.
In order to simplify exposition of the paper, sometimes the time argument t is omitted, so ξ := ξ (t).

Exponential stability
Let T be a time scale unbounded above and 0 ∈ T. A matrix A ∈ R n×n is said to be regressive with respect to T when I + µ(t)A is invertible for all t ∈ T, where I denotes the identity matrix.Consider the following linear system of delta differential equations on time scales T where t ∈ T and x(t) ∈ R n .
(ii) for every T ∈ T there exists t ∈ T with t > T such that 1 + µ(t)λ = 0, where we use the convention log 0 = −∞ in (i).

Definition 3 ([17]). Define for arbitrary t
Then, the set of exponential stability for T is given by Next, the theorem from [17] is adopted, providing the spectral characterization of the region of exponential stability for a system defined on a homogeneous time scale.

MAIN RESULTS
Consider a linear control system defined on a homogeneous time scale T where T and is left-continuous at 0. To analyse the stability property, consider an autonomous version of system (2) For There exist several definitions of the unilateral Laplace transform on time scales, e.g.[4][5][6]17].For the case of the bilateral Laplace transform on time scales see [7].

Definition 5 ([5]
). Assume that x : T → C is regulated.Then the Laplace transform of x is defined by for z ∈ D{x}, where D{x} consists of all1 complex numbers z ∈ C for which the improper integral exists.
Let ξ > 0. Define the function g(t) as Proof.The proof is based on direct application of Definition 5, i.e.
According to (ii) and (iii) from [4, Theorem 2.36], the right-hand side of the above equation can be rewritten as Define u := t − ξ or equivalently t = u + ξ .Then u ∈ T := T − ξ .Since T is assumed to be a homogeneous time scale, using part (v) of [4, Theorem 2.36] and the fact from [4, Exercise 2.51] (see Appendix B for the proof), we get where Applying the Laplace transform to both sides of (3) and using Theorem 3 and Proposition 2, we get whose characteristic equation can be written as where I n is the n × n identity matrix.To prove the theorem it is necessary to consider two separate cases: continuous (which is proved in [16]) and discrete, which is addressed in the following subsection.

Uniform discrete-time case
To prove Theorem 4 in the discrete-time case we need several technical results.Recall that the classical Z -transform is defined as where x : N 0 → C. Note that Eq. ( 6) can be extended on τZ as The following proposition defines the relation between the Z -transform and the Laplace transform on a homogeneous discrete-time scale.Proposition 3. Let T = τZ and x : T → C be regulated.Then Proof.According to (ii) and (iii) from [4, Theorem 2.36], (4) can be rewritten as Using the definition of the improper integral on τZ (see [4, Theorem 1.79]), (9) becomes for those values of z ̸ = −1/τ for which this series converges.Next, using [4, p. 118] and (7), relation ( 10) yields (8), which concludes the proof.
Remark 2. The results of [5] follow as a special case from Proposition 3. In addition, note that (8) coincides with that obtained in [3, Definition 4.1], but the proof is different.

Lemma 1. The Laplace transform of a function g(t) = x(t − h) on
Proof.The proof follows from Proposition 2 by observing that e z (t, 0) = (1 + τz) t τ on τZ.However, further an alternative proof is presented.Using relation (8) and the fact that x(t) = 0 for t < 0, one can write
Proof.Consider system (12).According to (13), the characteristic equation can be found as which, due to the specific structure of the matrix, transforms to det Multiply both sides of ( 14) by τ r /(1 + τz) r to get det Now it is necessary to show that the obtained characteristic equation is equivalent to (5) specified on τZ.Recall that ( 5) is given by det Since T = τZ, using [15,Corollary 3.4] and the fact that e ⊖z (−ξ , 0) = 1/(1+τz) −ξ τ , Eq. ( 5) can be rewritten as det The same result can be obtained by direct application of Lemma 1to system (11).Observe that ( 13) and ( 15) coincide.Therefore, χ e (z) = 0 and χ s (z) = 0 are equivalent on τZ.Proof of Theorem 4. Let first T = τZ.From Proposition 4 it follows that the extended system ( 12) is exponentially stable if and only if spec(A e ) ⊂ S (T) or equivalently if the solutions of the characteristic equation χ e (z) = 0 lie in S (T).Then, according to Theorem 5, both characteristic equations, χ e (z) = 0 and χ s (z) = 0, are equivalent on τZ.Therefore, by Proposition 5, the exponential stability of ( 11) means the exponential stability of ( 12).This concludes the proof of Theorem 4 for τZ.Finally, recall that for T = R this equivalence was shown in [16].

Stabilizability of linear time-delay systems
Recall several facts of linear systems without delay defined on a homogeneous time scale T Definition 6. System ( 16) is stabilizable if there is a matrix K such that the system x ∆ = (A − BK)x is exponentially stable.
The pair (A, B) is called controllable if rank[B, AB, . . ., A n−1 B] = n.The following theorem was stated in [2] without a proof.
For the completeness of the paper a proof of Theorem 6 is given in Appendix A. Recall that the timedelay system defined on a homogeneous time scale T is given by (2) as 2) is said to be stabilizable if there exists a feedback of the form u(t is exponentially stable.
Remark 3. Observe that if h = 0, then Definition 7 reduces to Definition 6.
Proof.Assume that system (2) is stabilizable, i.e. by Definition 7 there exists a feedback of the form ) such that the closed-loop system ( 17) is exponentially stable.Let T = τZ.Then the stabilizing feedback becomes u can be rewritten as ψ ∆ = A e ψ + B e u (18) or equivalently as ψ ∆ = (A e − B e F)ψ, (19) where and u := Fψ with F = [F r , . . ., F 1 , F 0 ].By Proposition 5 system ( 19) is exponentially stable, because we assumed stabilizability of system (2), which by Definition 6 implies exponential stability.Hence, the feedback u = Fψ indeed stabilizes (19).The presented arguments can be applied in the opposite way to finalize the proof.Finally, it should be mentioned that the continuous-time case is addressed in [9] and [16].

Theorem 7. System 2 is stabilizable on a homogeneous time scale T if and only if rank
for z / ∈ S (T).

CONCLUSION
Although theories of differential and difference equations are inherently different, time scales based formalism allows an effective unification of the two coexisting paradigms.Furthermore, such formalism is expected to be a right tool for further extension, since time scales incorporate other cases (such as nonuniformly sampled systems, interval models, hybrid systems, etc.).In this paper, linear retarded timedelay systems, defined on a homogeneous time scale, are addressed.Necessary and sufficient conditions for exponential stability are formulated in terms of the characteristic equation.This approach is further used to present feedback stabilizability criteria.The future work will be devoted to extend obtained results to the case of more general time scales.APPENDIX B Exercise 2.51 from [4]: Show that if T has constant graininess τ ≥ 0 and if α is constant with 1 + ατ ̸ = 0, then e α (t + s, 0) = e α (t, 0)e α (s, 0) for all s,t ∈ T.

Theorem 4 .
System (3)  is exponentially stable if and only if all the solutions of the characteristic equation χ s (z) = 0 lie in S (T).