Linear Continuous-time Periodic Systems Research Articles

Cooperative Output Regulation for Continuous-Time Linear Periodic Systems

Cooperative Output Regulation for Continuous-Time Linear Periodic Systems

Optimal control of unknown continuous time linear periodic systems by adaptive dynamic programming with applications to magnetic attitude control

AbstractThis article studies the data‐driven controller design of continuous‐time linear periodic systems. By treating the optimal control problem as a two‐player zero‐sum game problem, the adaptive dynamic programming technique is utilized. Based on the Newton's method, a one‐loop model‐based policy‐iteration (PI) method is introduced. Since the initial matrix for starting the proposed model‐based PI method is not easy to choose, a data‐driven value iteration method without requiring the knowledge of the initial matrix is then proposed. The convergence of the proposed methods is analyzed and the numerical simulation verifies the effectiveness of the proposed scheme.

Continuous-Time Linear Periodic Systems: A Note on the Controllability Interval Length

Continuous-time linear systems with periodic coefficients of period T are considered in this paper. We study the question of the controllability interval length of periodic systems. It is known that a controllable T-periodic system is controllable over intervals of the length kT where k is the degree of the minimal polynomial of the monodromy matrix. It is proved that this boundary kT is exact and cannot be decreased. It is proved that controllability of a T-periodic system on some interval of the length sT depends on the initial moment, in general.

Reinforcement learning for adaptive optimal control of continuous-time linear periodic systems

This paper studies the infinite-horizon adaptive optimal control of continuous-time linear periodic (CTLP) systems, using reinforcement learning techniques. By means of policy iteration (PI) for CTLP systems, both on-policy and off-policy adaptive dynamic programming (ADP) algorithms are derived, such that the solution of the optimal control problem can be found without the exact knowledge of the system dynamics. Starting with initial stabilizing controllers, the proposed PI-based ADP algorithms converge to the optimal solutions under mild conditions. Application to the adaptive optimal control of the lossy Mathieu equation demonstrates the efficacy of the proposed learning-based adaptive optimal control algorithm.

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

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.

Stability analysis and stabilisation in linear continuous-time periodic systems by complex scaling

ABSTRACT By scaling in the complex domain (namely, complex scaling) the return difference relations of linear continuous-time periodic (LCP) feedback systems, we generalise the 2-regularised stability criteria for asymptotic stability in this paper. The stability conditions of the generalised criterion are necessary and sufficient, and involve neither contour and locus orientation specification, nor open-loop Floquet factorisation and its eigenvalues distribution. Finite-dimensional implementation of the suggested criteria is considered via a two-step truncation approach. The finite-dimensional criteria are implementable either graphically with locus plotting, or numerically without locus plotting, besides retaining the aforementioned technical advantages. Furthermore, also exploiting the complex scaling technique, stabilisation of LCP systems with static state feedback is worked out in the internal or external stability sense, whose alternative interpretations in terms of the small-gain theorem and the Gronwall inequality are explicated. To illustrate the main results, the lossy Mathieu differential equation is investigated.

Model reference tracking control of continuous-time periodic linear systems with actuator jumping fault and its applications in orbit maneuvering

This paper deals with the model reference tracking control problem of continuous-time periodic linear systems when the actuator occurs jumping fault. The main contribution is to formulate the parametric design algorithm for the systems by utilizing the parametric solution of the generalized Sylvester matrix equations. The existence condition of the controller is deduced based on the Lyapunov stability theory. The controller consists of the additive contribution of two terms: a feedback term and a feedforward term. The feedback term is the feedback control law which can stabilize the system with finite expected cost. The feedforward term is the complete parametric feedforward tracking compensator. The simulation for flying around mission is carried out about two spacecrafts in elliptical orbit. The simulation results show the effectiveness of the proposed approach.

Robust Output Regulation for Continuous-Time Periodic Systems

We consider controller design for robust output tracking and disturbance rejection for continuous-time periodic linear systems with periodic reference and disturbance signals. As our main results we present four different controllers: A feedforward control law and a discrete-time dynamic error feedback controller for output tracking and disturbance rejection, a robust discrete-time feedback controller, and finally a discrete-time feedback controller that achieves approximate robust output tracking and disturbance rejection. The presented constructions are also new for time-invariant finite and infinite-dimensional systems. The results are illustrated with two examples: A periodically time-dependent system of harmonic oscillators and a nonautonomous two-dimensional heat equation with boundary disturbance.

Controllability discrepancy and irreducibility/reducibility of Floquet factorisations in linear continuous-time periodic systems

The paper reports interesting but unnoticed facts about irreducibility (resp., reducibility) of Flouqet factorisations and their harmonic implication in term of controllability in finite-dimensional linear continuous-time periodic (FDLCP) systems. Reducibility and irreducibility are attributed to matrix logarithm algorithms during computing Floquet factorisations in FDLCP systems, which are a pair of essential features but remain unnoticed in the Floquet theory so far. The study reveals that reducible Floquet factorisations may bring in harmonic waves variance into the Fourier analysis of FDLCP systems that in turn may alter our interpretation of controllability when the Floquet factors are used separately during controllability testing; namely, controllability interpretation discrepancy (or simply, controllability discrepancy) may occur and must be examined whenever reducible Floquet factorisations are involved. On the contrary, when irreducible Floquet factorisations are employed, controllability interpretation discrepancy can be avoided. Examples are included to illustrate such observations.

Average Floquet factorisations in linear continuous-time periodic systems

The paper examines existence and properties about average Floquet factorisations, determined via average integration instead of matrix logarithms, for the transition matrices of finite-dimensional linear continuous-time periodic (FDLCP) systems. We talk about the following aspects about average Floquet factorisations: existence conditions, properties and relationships with the conventional Floquet factorisations. We reveal that average scaling and modelling may be incomplete or even incorrect for reflecting dynamics in general FDCLP systems, compared to the Floquet factorisations defined through matrix logarithms.

H∞ norm computation of linear continuous-time periodic systems by a structure-preserving algorithm

A new reliable structure-preserving algorithm for computing H∞ norm of linear continuous-time periodic systems is proposed in this paper. In the computation of the H∞ norm, no Riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic Hamiltonian system will be evaluated. First, the monodromy matrix is expressed as the product of state transition matrices of the Hamiltonian system. Second, these state transition matrices, which have been proved to be symplectic matrices, are evaluated by a structure-preserving Magnus series method. Then, in order to preserve the standard symplectic form of the monodromy matrix, the structure-preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix. At last, the effectiveness and the high accuracy of the proposed structure-preserving algorithm are demonstrated by numerical examples.

L ∞ and L 2 semi-global stabilisation of continuous-time periodic linear systems with bounded controls

This article is concerned with L ∞ and L 2 semi-global stabilisation of continuous-time periodic linear systems with bounded controls. Two problems, namely L ∞ semi-global stabilisation with controls having bounded magnitude and L 2 semi-global stabilisation with controls having bounded energy, are solved based on solutions to a class of periodic Lyapunov differential equations (PLDEs) resulting from the problem of minimal energy control with guaranteed convergence rate. Under the assumption that the open-loop system is (asymptotically) null controllable with constrained controls, periodic feedback are established to solve the concerned problems. The proposed PLDE-based approaches possess the advantage that the resulting controllers are easy to implement since the designers need only to solve a linear differential equation. A numerical example is worked out to illustrate the effectiveness of the proposed approach.

CONTRAPOSITION 2-REGULARIZED NYQUIST STABILITY CRITERIA FOR LINEAR CONTINUOUS-TIME PERIODIC SYSTEMS

By employing the 2-regularized determinant technique about Hilbert-Schmidt operators and creating what we call the contraposition return difference relationship for finite-dimensional linear continuous-time periodic (FDLCP) feedback systems, a contraposition 2-regularized Nyquist stability criterion is developed for asymptotic stability analysis in this study. The stability conditions of the contraposition 2-regularized Nyquist stability criterion are necessary and sufficient, which can be implemented graphically in a way similar to what we do in linear time-invariant cases but without involving open-loop poles distribution and encircling orientation counting of the Nyquist loci. A numeric implementation algorithm for the contraposition 2-regularized Nyquist criterion is also proposed via truncating the harmonic transfer operator of the FDLCP systems.

Periodic Lyapunov Equation Based Approaches to the Stabilization of Continuous-Time Periodic Linear Systems

This note is concerned with stabilization of continuous-time periodic linear (CPL) systems with state feedback. The design is based on solutions to a class of parametric periodic Lyapunov differential equations (PLDEs) resulting from the problem of minimal energy control with guaranteed convergence rate. By carefully studying the properties of the PLDEs and their solutions, a continuous periodic state feedback is designed. The PLDE based approach is effective in designing stabilizing controller for CPL systems as the designers need only to solve a linear differential equation whose solution can be obtained analytically if the system is relative simple and can be computed numerically by integration in general cases. Necessary and sufficient conditions on the free parameter in the PLDE are proposed to guarantee the stability of the closed-loop system whose characteristic multiplier set can even be exactly computed accordingly. A numerical example is worked out to show the effectiveness of the proposed approach.

Floquet Factorization Variants in Linear Continuous-Time Periodic Systems

Floquet Factorization Variants in Linear Continuous-Time Periodic Systems

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H ∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.

Classification and characteristics of Floquet factorisations in linear continuous-time periodic systems

This article has two purposes. Firstly, the article is devoted to collecting basic facts about the Floquet theorem and various Floquet factorisation algorithms claimed for finite-dimensional linear continuous-time periodic (FDLCP) systems. Secondly, the article presents a unified representation framework for various Floquet factorisations in FDLCP systems, while structural and analytic characteristics about them are examined as well. More precisely, the following aspects are considered: (i) algorithms for Floquet factorisations; (ii) characteristics of Floquet factors; (iii) relationships and properties among Floquet factorisations. Most results are reported for the first time, while the others are generalised versions of existing ones.

Zeros and Poles of Linear Continuous-Time Periodic Systems: Definitions and Properties

The paper deals with definitions of zeros and poles and their features in finite-dimensional linear continuous-time periodic (FDLCP) systems under a harmonic framework. More precisely, system and transfer zeros and poles in the harmonic wave-to-wave sense are defined on what we call the regularized harmonic system operators and the harmonic transfer operators of FDLCP systems by means of regularized determinants; then their composition and properties related to system structures are examined via the Floquet theory and controllability/observability decompositions of FDLCP systems. The study shows that under mild assumptions, the harmonic transfer operators of FDLCP systems are analytic and meromorphic, on which zeros and poles are well-defined. Basic zero/pole relationships are established, which are similar to their linear time-invariant counterparts and in particular explicate some interesting harmonic wave-to-wave behaviors of FDLCP systems. The results are significant in analysis and synthesis of FDLCP systems when the harmonic approach is adopted.

Harmonic Lyapunov equations in continuous-time periodic systems: solutions and properties

First, the harmonic Lyapunov equations claimed on the Hilbert space I 2 are restricted to the Banach space I 1 for asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems. Second, solutions to the harmonic Lyapunov equations are scrutinised. Third, the harmonic Lyapunov equations are connected with periodic matrix differential (PMD) Lyapunov equations. The connection sheds new light on periodic solutions to the PMD Lyapunov equations. Fourth, the trace formula of the H 2 norm for FDLCP systems is shaped with periodic solutions to the PMD Lyapunov equations. Finally, an algorithm for computing periodic solutions to the PMD Lyapunov equations is proposed, which involves only solutions to algebraic Lyapunov equations. There are numerical examples to illustrate the results.

連続時間線形周期システムの可制御・不可制御部分への分解可能性について

連続時間線形周期システムの可制御・不可制御部分への分解可能性について