Stability Of Linear Periodic Systems Research Articles

Fixed-time stabilization of periodic linear systems and its application to the elliptical spacecraft rendezvous

A smooth periodic delayed feedback (SPDF) control scheme is proposed for the fixed-time stabilization problem of linear periodic systems subject to input delay. By investigating the monodromy matrix of the periodic system, it is proved that the SPDF controller can achieve the fixed-time stabilization of linear periodic systems with arbitrarily long yet bounded input delays under the condition that the original system is uniformly completely controllable. The proposed controller is continuously differentiable and smooth. The SPDF control scheme is then applied to the elliptical spacecraft rendezvous problem. The effectiveness of the established method is verified on numerical simulations.

Methods for Studying the Stability of Linear Periodic Systems Depending on a Small Parameter

Methods for Studying the Stability of Linear Periodic Systems Depending on a Small Parameter

Stability and stabilization of linear periodic systems: A commutator calculus approach

In this paper, methods of commutator calculus are used to study stability and stabilization for periodic linear systems. Asymptotic stability is investigated using a quadratic Lyapunov function, whose matrix is a piecewise-exponential approximation of the solution of the differential Lyapunov matrix equation. Algorithm for constructing a stabilizing control in the form of linear feedback, dependent on time, the matrix of which is periodic and piecewise constant is proposed. Examples of stability studies of a periodic linear system are given and an algorithm for stabilizing parametric oscillations of a linear oscillator is constructed.

Robust Stability of Linear Periodic Systems

A new method for analyzing the robust stability of linear periodic systems is proposed, which is based on the ideas of commutator calculus in combination with Lyapunov’s direct method. The stability analysis of a linear nonautonomous system of ordinary differential equations is reduced to the stability analysis of a linear system of differential equations with impulse action, for which Lyapunov’s direct method is used. New sufficient conditions for the robust stability of a linear periodic system with non-periodic perturbations are established. Some illustrative examples on the robust stability analysis of linear systems are given.

Global stabilization of periodic linear systems by bounded controls with applications to spacecraft magnetic attitude control

We study in this paper the three-axis magnetic attitude control of small spacecraft by considering the actuator saturation. By noticing that the linearized dynamics is a neutrally stable linear periodic system, a general problem of global stabilization of linear periodic systems with bounded feedback is formulated and solved by both state feedback and observer based output feedback. Explicit saturated linear feedback control laws are established by using solutions to some Lyapunov differential equations associated with the open-loop systems and global stability of the closed-loop system is proved by constructing an explicit Lyapunov function. To apply the theoretical results on the three-axis magnetic attitude control systems, necessary and sufficient conditions are established to guarantee that the linearized open-loop system is neutrally stable and explicit solutions to the associated Lyapunov equations are obtained consequently. The controllers for the three-axis magnetic attitude control systems are then given in closed-form. Simulations show the effectiveness of the proposed approach.

On semi-global stabilization of linear periodic systems with control magnitude and energy saturations

This paper establishes a systematic approach to solve the L∞(l∞) and L2(l2) semi-global stabilization problem of linear periodic systems with controls having bounded magnitude and energy, respectively. The developed approach will be referred to as L∞(l∞) and L2(l2) low gain feedback. Definitions, properties, and characterizations of this new concept are also provided, and particularly, the characterizations are based upon differential (difference) Lyapunov inequalities. Design of L∞(l∞) and L2(l2) low gain feedback by solving differential (difference) Riccati equations is proposed. Both continuous-time and discrete-time linear periodic systems are studied and both state feedback and observer based output feedback are considered. In the discrete-time setting, a linear matrix inequalities (LMIs) based solution to the l∞ and l2 semi-global stabilization problem is also established and the LMIs conditions are shown to be always solvable. Applications of the proposed approach to the elliptical spacecraft rendezvous system show the effectiveness of the established theory.

Floquet theory: a useful tool for understanding nonequilibrium dynamics

Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers.

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.

On Reliable Stabilization of Linear Periodic Systems

An algorithm is proposed to synthesize a reliable controller with a given stability margin for linear and periodic systems optimized with respect to a quadratic performance criterion. A reliable controller synthesized by the algorithm guarantees the stability margin and is close to the linear-quadratic requlator. The importance of ensuring the stability margin is demonstrated. The proposed algorithm is based on methods of linear matrix inequalities and can be implemented using standard MATLAB routines. As an example, a reliable controller that stabilizes the program motion of a hopping machine is synthesized

Algebraic and Persistency-of-Excitaion Conditions for Stability of Linear Periodic Systems

We present sufficient conditions for asymptotic stability of a class of periodic linear systems. The type of systems that we study contain as a particular case, the system ẋ = F(t)x with F(t) positive semidefinite and symmetric. Our results are stated on the basis of structural conditions on the system (an algebraic check) and a condition of persistency of excitation. Our proofs rely on Center-manifold theorem.

Quadratic Lyapunov Functions for the Linear Integro-Differential Equations with Periodic Coefficients

Necessary and sufficient conditions for stability of linear periodic systems with mixed discrete/distributed delays are obtained in the form of solvability of PDE on a square. Linear-quadratic and H-inf infinite horizon control problem with general quadratic cost function are solved for such retarded systems. The solution is expressed in terms of ODE on a square.

Lyapunov Equations, Inequalities and Stability Theorems for Periodic Systems

Lyapunov equations and inequalities are studied in connection with the stability of linear periodic systems. First, as preliminaries, existing results on Lyapunov equations in time-invariant systems are summarized in the form of equivalence theorems. Then, based on those results, existence conditions are derived for positive-definite or nonnegative-definite periodic solutions of periodic Lyapunov differential equations and inequalities with reference to the detectability and observability. Such conditions are stated all as equivalent conditions for the system to be either asymptotically stable or stable. Brief discussions are included also for periodic Lyapunov difference equations in discrete-time periodic systems.

On the use of reachability Gramians for the stabilization of linear periodic systems

This paper deals with the development of explicit formulas, based on reachability Gramians, for the stabilization of linear periodic systems. In contrast with the ‘pointwise’ strategy already known in the literature, a different ‘intervalwise’ scheme is proposed, involving reduced computational effort. Stability results are provided in both continuous and discrete time under the sole controllability assumption without requiring system reversibility.

Efficient numerical treatment of periodic systems with application to stability problems

AbstractTwo efficient numerical methods for dealing with the stability of linear periodic systems are presented. Both methods combine the use of multivariable Floquet–Liapunov theory with an efficient numerical scheme for computing the transition matrix at the end of one period. The numerical properties of these methods are illustrated by applying them to the simple parametric excitation problem of a fixed end column. The practical value of these methods is shown by applying them to some helicopter rotor blade aeroelastic and structural dynamics problems. It is concluded that these methods are numerically efficient, general and practical for dealing with the stability of large periodic systems.

Stability of linear periodic systems

A method for calculating the solution, in Floquet form, to a system of linear differential equations with periodic parameters is developed. As a result, both the periodic and exponential parts of the solution are developed as power series in a small parameter, ϵ. From this solution, approximate expressions for the characteristic exponents are derived and hence the stability of the system can be determined. The technique is analogous and parallels that developed by Cesari, Hale, Gambill, et al. By returning to the basis of the underlying process of casting out the secular terms, a method is developed which has advantages from the technical and computational point of view. The expressions for the characteristic exponents are explicit, namely, series of powers of ϵ, whose terms can be easily written, and which converge for ϵ sufficiently small. More importantly, from a practical standpoint, an estimate of the error which results when the series is truncated is derived. Using such an estimate, we draw a circle on the complex plane, centered at the approximate characteristic exponents, within which the exact characteristic exponents must lie.