Stability analysis and stabilization of linear continuous-time periodic systems by variation-of-constant discretization

Abstract

By the variation-of-constant formula for solutions to differential equations, discretization remodeling of linear continuous-time periodic (LCP) systems is established, which provides us with variation-of-constant difference (VOCD) models for LCP systems with uniform convergence under mild assumptions. Involving neither Fourier analysis nor Floquet factorizations, the VOCD models are linear discrete-time periodic, and essentially time-invariant; this paves a way to apply analysis and synthesis techniques for linear discrete-time systems to LCP ones. More specifically, stability criteria for LCP systems are developed with their VOCD counterparts. Also, stabilizing LCP systems is addressed by periodically time-varying state feedback in the VOCD models via sequential pole assignment, which can be implemented as gain-scheduling and periodic piecewise constant state feedback in the continuous-time sense. Numerical examples are included to illustrate efficiency of the approach and applications.

Introduction

Linear continuous-time periodic (LCP) systems are frequently modeled in practical engineering problems; for instance, rolling motion of ships in waves [1] and flapping dynamics of helicopter rotor blades [8], [29] are typical. As a matter of fact, this is exactly the case when stability analysis and stabilization design are discussed, when periodic dynamics [4], [9], [11], [14], [21], [31], [37], complicated systems under periodic regulation control [33], control actions driven with sampling and holding [6], or vibrations rejection [24] are concerned. As special periodic time-varying systems, sampled-data systems are intensively studied [2], [12], [26]. Latest results are reported in [13], [15], [18], [19].

The motivation of the study is due to the following facts. Even without exhausting the literature, one will soon realize that Floquet factorizations [23], [38] of LCP systems are involved in Floquet–Lyapunov coordinates transformation to remodel them by their quasi-LTI counterparts so as to enable time-domain analysis and synthesis; Fourier expansion techniques are adopted for developing frequency-domain analysis and synthesis. However, Floquet factorizations are hard to obtain analytically, and Fourier expansions possess infinite-dimensional features and convergence issues are unavoidable [40]. By the Floquet factorization and/or Fourier expansion schemes, the Floquet theorem [22], [25] reflects asymptotic stability in LCP systems by the monodromy eigenvalues distribution. Absolute stability of LCP systems with integral quadratic nonlinearities is dealt with in [17] and [32] by the cutting plane algorithm and the Hamiltonian approach, respectively, while input/output stability and Youla-style parameterization of stabilizing controllers are discussed in [7] via the graph representation theory. Asymptotic stability is attacked by the Lyapunov method [5] and the harmonic analysis [34], [35], [36], [37].

The situation around stability analysis and stabilization remains unchanged when feedback configurations of LCP plants and/or controllers are concerned. For instance, generalized Nyquist criteria are claimed by Wereley [29], and a 2-regularized Nyquist criterion is established by Zhou and Hagiwara [36], which entail Floquet factorizations and Fourier expansions for open-loop poles distribution and operator truncation convergence. Lately, a complex scaling stability analysis method by Zhou [39] is claimed without Floquet factorizations but still with Fourier expansions.

As the major contributions of the study, LCP systems are remodeled and examined around stability analysis and stabilization, based on the variation-of-constant formula for solutions to differential equations. The discretization remodeling produces linear discrete-time periodic and essentially time-invariant difference equations, namely the VOCD models. The VOCD models are equivalent to the cyclic reformulation ones of periodic discrete-time systems [27], [28], and are in the pointwise LTI form [41]. Thus, the VOCD technique is also a pointwise modeling approach, and highly numerically tractable. Exploiting the advantages of the VOCD models, stability analysis and stabilization are attacked in a discrete-time LTI fashion. More precisely, stability criteria for LCP systems are developed with the VOCD counterparts, under necessary and/or sufficient conditions in terms of pointwise system matrices and structural features. The VOCD approach completely gets rid of Floquet factorization as well as Fourier analysis. By the VOCD models, stabilization is developed by sequential pole assignment, which is implementable as gain-scheduling and periodically piecewise constant state feedback.

The outline of the paper is as follows. Section 2 collects notations, preliminaries to LCP systems and their discretization modeling via the variation-of-constant formula. Stability analysis is discussed in Section 3, while stabilization is formulated and addressed in Section 4. Numerical examples are sketched in Section 5. Conclusions are included in Section 6.

The notations of the paper are standard. $\mathbb{R}$ (resp., $\mathbb{C}$) is the field of all real (resp., complex) numbers, while ${\mathbb{R}}^m$ is the m-dimensional Euclidean space. Euclidean vector norm and the induced matrix norm are denoted by ‖ · ‖. Also, $0<Q=Q^T\in {\mathbb{R}}^{m\times m}$ (resp., $0\le Q=Q^T\in {\mathbb{R}}^{m\times m}$) implies that Q is symmetric and positive definite (resp., positive semi-definite). I_n denotes the n × n identity matrix, and I_n,i denotes the (ni) × (ni) identity matrix (consisting of i identity matrices I_n, i.e., $I_{n,i}=I_{ni}$). λ(·) is the set of all eigenvalues of a matrix (·), whose ith eigenvalue is denoted by λ_i(·).