Bounds and Invariant Sets for a Class of Switching Systems with Delayed-state-dependent Perturbations

We present a novel method to compute componentwise transient bounds, ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. We give conditions for the derived bounds to be of local or semi-global nature. In addition, we deal with the case of perturbation bounds whose dependence on a delayed state is of affine form as a particular case of nonlinear dependence for which the bounds derived are shown to be globally valid. A sufficient condition for practical stability is also provided. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components.

💡 Research Summary

The paper addresses the analysis and synthesis problem for continuous‑time linear systems that switch among a finite set of modes while being subjected to disturbances whose bounds may depend nonlinearly on a delayed version of the state. The authors propose a component‑wise (element‑wise) framework that yields explicit transient bounds, ultimate (steady‑state) bounds, and invariant sets for each state component, without resorting to a vector norm or a scaling matrix.

The system model is
  ẋ(t)=A_{σ(t)}x(t)+w(t), σ(t)∈{1,…,N},
where each A_i∈ℝ^{n×n} is a constant system matrix and w(t)∈ℝ^{n} is a disturbance. For every component i the disturbance satisfies
  |w_i(t)| ≤ δ_i(|x(t−τ)|),
with τ≥0 a known constant delay and δ_i:ℝ_{\ge0}^{n}→ℝ_{\ge0} a continuous, non‑decreasing function. The special case of affine dependence, δ_i(y)=α_i+∑jβ{ij}y_j (α_i,β_{ij}≥0), is treated in detail because it leads to globally valid results.

The core idea is to construct a comparison system that bounds the absolute values of the state components. By taking the element‑wise absolute value of the dynamics and over‑approximating the switching matrices with a common Metzler matrix M (i.e., M has non‑negative off‑diagonal entries and negative diagonal entries), the authors obtain the differential inequality

  β̇(t) ≤ Mβ(t)+γ(β(t−τ)),

where β(t)∈ℝ_{\ge0}^{n} is the vector of component‑wise upper bounds and γ_i(·)=δ_i(·). If M is Hurwitz (all eigenvalues have negative real parts) and γ belongs to class K (continuous, zero at the origin, and strictly increasing), standard results for monotone delayed systems guarantee the existence of a unique equilibrium β*≥0 satisfying

  0 = Mβ* + γ(β*).

Moreover, for any initial bound β(θ)∈