Computation of the Lyapunov matrix for periodic time-delay systems and its application to robust stability analysis

doi:10.1016/j.sysconle.2019.104501

Systems & Control Letters

Volume 132, October 2019, 104501

https://doi.org/10.1016/j.sysconle.2019.104501 Get rights and content

Abstract

A new procedure for computing the delay Lyapunov matrix for periodic time-delay systems that is based on the numerical solution of a partial differential equations (PDE) system is presented. The introduction of a new set of boundary conditions that are satisfied by the PDE system allows us to propose a new methodology for computing the initial conditions required by the implemented numerical scheme. The potential of the presented results is demonstrated by obtaining robust stability conditions depending on the delay Lyapunov matrix with respect to the system parameters, the delay and the frequency. The theoretical results are applied to the widely known delayed Mathieu equation.

Introduction

Time-delay systems are known to be suitable for modeling a variety of physical phenomena (see, for instance, the books [1], [2], [3], [4]). A well known tool for analyzing their stability properties is the Lyapunov–Krasovskii approach. In the linear time-invariant case, the first results concerning the Lyapunov–Krasovskii functionals with negative prescribed derivatives were presented in [5]. Notable advances in this direction were then developed in [6] and [7]. In [8], this approach received a decisive impulse with the introduction of the so-called functionals of complete type. These functionals have two notable features: (i) they admit a quadratic lower bound if the system is stable and (ii) they are determined by the delay Lyapunov matrix (see [9] for a comprehensive study of this class of functionals).

Periodic delay systems have the complexity of both delay systems and time-varying systems. The stability properties of this class of equations have been analyzed by different methods, based mainly on the Floquet theory (see, for instance, [3] and [10]). A time-domain approach for studying periodic time-delay systems is presented in [11] by extending the Lyapunov–Krasovskii functionals of complete type framework developed for the invariant case [9]. This new perspective has made it possible to obtain robust stability conditions with respect to the parameters, exponential estimates [11], and more recently, to prove necessary stability conditions depending on the delay Lyapunov matrix [12].

These useful theoretical results depend on the knowledge of the delay Lyapunov matrix, which cannot be computed by the semianalytical method employed for time-invariant linear systems (see Section 2.10 in [9]). The computation of the delay Lyapunov matrix for periodic systems is a difficult task. In [11] and [13], it is shown that for the case of delay equal to the principal period, the Lyapunov matrix is a solution of a delay free partial differential equations (PDE) system that satisfies a set of boundary conditions. An iterative procedure based on the minimization of a function aiming at the computation of the initial conditions of a modified Runge–Kutta method is proposed. However, this methodology only works for some particular cases as it demands both the objective function to have a minimum and the minimization process to be convergent.

In this paper, we present an effective method for computing the delay Lyapunov matrix, which is based on the introduction of a new set of boundary conditions that are satisfied by the PDE system. This substantial modification enables us to propose a new procedure for computing the initial conditions. The main advantage of this new proposal, compared to the one presented in [11], is that it is not iterative and therefore it avoids the previously mentioned difficulties.

Having an algorithm for computing this matrix that defines the functional of complete type makes possible the robust stability analysis of periodic systems against time-varying uncertainties. In [11] and [13] this analysis is proposed by considering uncertain parameters. Here, the introduction of a new functional allows us to determine robust stability conditions for periodic systems with uncertain parameters, delay, and frequency.

Finally, we apply the results to the delayed Mathieu equation, which is a benchmark in the field. In particular, the delay Lyapunov matrix and robust stability bounds are obtained for the first time, up to our knowledge, for this equation.

The paper is organized as follows. The framework of Lyapunov–Krasovskii functionals is recalled in the next section. In Section 3, we present the new proposal to compute the delay Lyapunov matrix for time-delay periodic systems. A general robust stability analysis for this class of systems is carried out in Section 4. The robust stability results are applied to the well known delayed Mathieu equation in Section 5, and we end with some concluding remarks in Section 6.

Notation: $P C ([- h, 0], R^{n})$ represents the space of $R^{n}$ valued piecewise continuous functions on $[- h, 0]$ . The supremum norm is denoted by ${‖ φ ‖}_{h} = {sup}_{θ \in [- h, 0]} ‖ φ (θ) ‖$ , where $‖ \cdot ‖$ denotes the Euclidean norm for vectors. Inequality $Q > 0$ means that matrix $Q$ is positive definite and $λ_{m i n} Q$ and $λ_{m a x} Q$ represent the minimum and maximum eigenvalue of matrix $Q$ , respectively. Matrix $I_{n}$ denotes the identity matrix in $R^{n}$ .

Section snippets

Lyapunov–Krasovskii framework

We consider the following dynamical equation with one delay: $\dot{x} (t) = A_{0} (t) x (t) + A_{1} (t) x (t - h),$ where $A_{0} (t)$ and $A_{1} (t)$ are matrices of continuous coefficients with period $T$ , i.e. $A_{j} (t) = A_{j} (t + T)$ , $j = 0, 1$ , with range in $R^{n \times n}$ , and $h$ is the delay. We denote the solution of system (1) with initial condition $x (t_{0} + θ) = φ (θ), θ \in [- h, 0], t_{0} \in R, φ \in P C ([- h, 0], R^{n}),$ by $x (t, t_{0}, φ)$ , and its restriction to the interval $[t - h, t]$ by $x_{t} (t_{0}, φ)$ .

We say that system (1) is exponentially stable if there exist constants $γ > 0$ and $σ > 0$ such that $‖ x$

Computation of the delay Lyapunov matrix

In this section, a delay free PDE system for the computation of the delay Lyapunov matrix is presented. The introduction of a new set of boundary conditions allows us to give a procedure for computing the initial conditions that is simpler than the one in [11].

Robust stability analysis

In this section, we present a general robustness result for system (1), which covers uncertainties in the parameters, delays and frequency. We consider system (1) with a general perturbation term $f (t, z_{t})$ : $\dot{z} (t) = A_{0} (t) z (t) + A_{1} (t) z (t - h) + f (t, z_{t}) .$ Here, we assume that the functional $f (t, φ)$ is continuous with respect to both arguments, it is Lipschitz with respect to the second argument and it satisfies $‖ f (t, φ) ‖ \leq β_{0} ‖ φ (0) ‖ + β_{1} ‖ φ (- h) ‖ + sign (η) (γ_{0} \int_{- h - η}^{- h} ‖ φ (θ) ‖ d θ + γ_{1} \int_{- 2 h - 2 η}^{- 2 h - η} ‖ φ (θ) ‖ d θ),$ with $β_{i}, γ_{i} \geq 0$ , $i = 0, 1$ ,

Case study: delayed Mathieu equation

We consider the delayed Mathieu equation $\ddot{y} (t) + ρ \dot{y} (t) + (δ + ε cos (ω t)) y (t) = b y (t - T),$ where $ω = 2 π ∕ T$ . This interesting equation combines delays and the parametric excitation phenomenon, and it is the starting point for the stability analysis of some engineering models, such as the milling process [3].

The state space representation of Eq. (24) is given by system (1) with $\begin{matrix} A_{0} (t) = [\begin{bmatrix} 0 & 1 \\ - δ - ε cos (ω t) & - ρ \end{bmatrix}] and A_{1} = [\begin{bmatrix} 0 & 0 \\ b & 0 \end{bmatrix}], \end{matrix}$ with the parameter numerical values $ρ = 0.1, δ = 7, ε = 1, b = - 1, T = 2 .$

By constructing the stability map with the Matlab

Conclusion

We propose a new procedure for computing the delay Lyapunov matrix for periodic time-delay systems, and present a general robust stability analysis concerning uncertain parameters, delay and frequency by introducing a functional constructed by using the so-called functional of complete type. The new proposal for the computation of the delay Lyapunov matrix in combination with the robustness analysis enables us to compute robustness bounds on uncertainties of the well known delayed Mathieu

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Project CONACYT, Mexico 180725 and Project SEP-CINVESTAV, Mexico 155.

References (19)

RepinI.M.
Quadratic Liapunov functionals for systems with delay
J. Appl. Math. Mech.
(1965)
InfanteE.F. et al.
A Liapunov functional for a matrix difference-differential equation
J. Differential Equations
(1978)
KharitonovV.L. et al.
Lyapunov–Krasovskii approach to the robust stability analysis of time-delay systems
Automatica
(2003)
ZhabkoA.P. et al.
A numerical method for the construction of Lyapunov matrices for linear periodic systems with time delay
OchoaG. et al.
Critical frequencies and parameters for linear delay systems: A Lyapunov matrix approach
Systems Control Lett.
(2013)
KolmanovskiiV. et al.
Introduction to the Theory and Applications of Functional Differential Equations
(1999)
NiculescuS.I.
Delay Effects on Stability: A Robust Control Approach
(2001)
InspergerT. et al.
Semi-discretization for Time-delay Systems: Stability and Engineering Applications, Vol. 178
(2011)
FridmanE.
Introduction to Time-delay Systems: Analysis and Control
(2014)

There are more references available in the full text version of this article.

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Computation of the Lyapunov matrix for periodic time-delay systems and its application to robust stability analysis

Abstract

Introduction

Section snippets

Lyapunov–Krasovskii framework

Computation of the delay Lyapunov matrix

Robust stability analysis

Case study: delayed Mathieu equation

Conclusion

Declaration of Competing Interest

Acknowledgments

J. Appl. Math. Mech.

J. Differential Equations

Automatica

Systems Control Lett.

Introduction to the Theory and Applications of Functional Differential Equations

Delay Effects on Stability: A Robust Control Approach

Semi-discretization for Time-delay Systems: Stability and Engineering Applications, Vol. 178

Introduction to Time-delay Systems: Analysis and Control