Stability Of Linear Control Systems Research Articles

Approach to the synthesis of an aperiodic robust automatic control system based on the gradient-speed method of Lyapunov vector functions

One of the actual problems for the theory and practice of control of dynamic objects is the development of methods for research and synthesis of control systems of multidimensional objects. The paper proposes a universal approach to construct Lyapunov vector functions directly from the equation of state of control system and a new gradient-speed method of Lyapunov vector functions to study aperiodic robust stability of linear control system with m inputs and n outputs. The study of aperiodic robust stability of automatic control systems is based on the construction of Lyapunov vector functions and gradient-speed dynamic control systems. The basic statements of Lyapunov's theorem about asymptotic stability and notions of stability of dynamic systems are used. The representation of control systems as gradient systems and Lyapunov functions as potential functions of gradient systems from the catastrophe theory allow to construct the full-time derivative of Lyapunov vector functions always as a sign-negative function equal to the scalar product of the velocity vector on the gradient vector. The conditions of aperiodic robust stability are obtained as a system of inequalities on the uncertain parameters of the automatic control system, which are a condition for the existence of the Lyapunov vector-function. A numerical example of synthesis of aperiodic robustness of a multidimensional control object is given. The example shows the main stages of the developed synthesis method, the study of the system stability at different values of the coefficients k, confirming the consistency of the proposed method. Transients in the system satisfy all requirements

Leonov’s Method of Nonstationary Stabilization in the Theory of Linear Control Systems

A brief review on stabilization problem for linear control systems is presented. The statements of Pyragas’ and Brockett’s problems for stabilization of linear control systems by time-delayed feedback control and time-varying static output feedback are given. Leonov’s algorithms of nonstationary stabilization for solutions of Pyragas’ and Brockett’s problems are presented. Necessary and sufficient conditions for the stabilizability of two- and three-dimensional systems are given.

A note on the positive boundary robust stability of linear control systems

A note on the positive boundary robust stability of linear control systems

Linear Weakly Hard Real-Time Control Systems: Time- and Event-Triggered Stabilization

This article considers the stabilization of linear control systems under weakly hard real-time (WHRT) constraints. In the context of networked control systems, such WHRT constraints allow us to specify packet loss properties of a communication network over windows of a given length, e.g., at least n consecutive transmissions in every window of m consecutive transmission attempts are successful. Time- and event-triggered stabilization results are derived. The control design exploits a graph that captures the available information on possible packet losses. An algorithm for computing such a graph for given WHRT constraints is provided. Both stabilization results are illustrated and compared using a numerical example from literature.

Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey

Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey

Stabilization of linear control systems. Pole assignment problem. A survey

This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for solving Brockett’s stabilization problem are provided. The effective necessary and sufficient conditions for stabilization of two- and three-dimensional controllable linear systems are given in terms of the system parameters. The pole assignment problem and the related issues for linear systems are considered.

Regions of exponential stability in coefficient space for linear systems on nonuniform discrete domains

The paper addresses the problem of exponential stability of linear control systems defined on nonuniform discrete domains. The main goal of this research is to find necessary and sufficient stability conditions in terms of the coefficients of the characteristic polynomial, associated with a system, and to study the geometry of stability regions, determined by the derived conditions in the coefficient space. As the first step in this direction, this paper presents such conditions for a special case of second-order systems defined on a discrete time scale with two asymptotic graininesses. The geometry of the corresponding regions in the coefficient space is illustrated for different values of the asymptotic graininesses.

On Asymptotic Stability of Linear Control Systems

Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and problem of stabilization by linear feedback.

Stability analysis of sampled-data systems considering time delays and its application to electric power markets

This paper investigates the stability of linear control systems with aperiodic sampled data and communication delays. A systematic analysis method is presented and then it is applied to an electric power market. Firstly, the sampled-data system is transformed into a system with a special time-varying delay via the input delay method. Secondly, a less conservative stability criterion is derived based on Lyapunov theory. Several augmented terms and an extra integral term are introduced during the constructing of candidate Lyapunov–Krasovskii functional (LKF); and an improved free-weighting matrix approach is used to handle with the LKF itself and its derivative for obtaining the relaxed conditions ensuring the positive and decreasing requirements of the LKF. The benefit of those treatments on the conservativeness-reducing is analyzed and verified based on a simple numerical example. Finally, the application of the proposed method to a simplified electric power market is investigated, including modeling the system with market clearing time and communication delay, and determining the stability region. The application also shows the practical significance of the reducing of the conservativeness.

Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems

Two types of nonlinear control algorithms are presented for uncertain linear plants. Controllers of the first type are stabilizing polynomial feedbacks that allow to adjust a guaranteed convergence time of system trajectories into a prespecified neighborhood of the origin independently on initial conditions. The control design procedure uses block control principles and finite-time attractivity properties of polynomial feedbacks. Controllers of the second type are modifications of the second order sliding mode control algorithms. They provide global finite-time stability of the closed-loop system and allow to adjust a guaranteed settling time independently on initial conditions. Control algorithms are presented for both single-input and multi-input systems. Theoretical results are supported by numerical simulations.

Vibrational stabilization and the Brockett problem

The present paper deals with the exposition of methods for solving the Brockett problem on the stabilization of linear control systems by a nonstationary feedback. The paper consists of two parts. We consider continuous linear control systems in the first part and discrete systems in the second part. In the first part, we consider two approaches to the solution of the Brockett problem. The first approach permits one to obtain low-frequency stabilization, and the second part deals with high-frequency stabilization. Both approaches permit one to derive necessary and sufficient stabilization conditions for two-dimensional (and three-dimensional, for the first approach) linear systems with scalar inputs and outputs. In the second part, we consider an analog of the Brockett problem for discrete linear control systems. Sufficient conditions for low-frequency stabilization of linear discrete systems are obtained with the use of a piecewise constant periodic feedback with sufficiently large period. We obtain necessary and sufficient conditions for the stabilization of two-dimensional discrete systems. In the second part, we also consider the control problem for the spectrum (the pole assignment problem) of the monodromy matrix for discrete systems with a periodic feedback.

Stabilization of linear control systems in Hilbert space with nonlinear feedback

This paper considers the problem of stabilization of linear control systems in Hilbert space with nonlinear feedback. Extensions of known results in the linear feedback case to the nonlinear one are given.

Operator Valued Fourier Multipliers and Stability of Strongly Continuous Semigroups

Stability for strongly continuous semigroups on Banach spaces is described in terms of L p -Fourier multiplier properties of the resolvent of the generator. A discrete version of this theory is developed, that lead to a de- scription of the spectra of the semigroups in terms of discrete L p -Fourier mul- tipliers. Applications to stability of linear control systems on Banach spaces are given. Mathematics Subject Classification (2000). Primary 47D03, 34G10; Secondary 93D25.

Analytic criteria for the stability of linear control systems with time delay

AbstractThe influence of time delay in linear control systems is often modeled by approximations. Here analytic criteria for determining the stability of linear differential delay systems are described. The characteristic function of such a system is a quasipolynomial, i.e. a polynomial Q(z) multiplied by powers of the exponential function exp(‐τz) plus another polynomial P(z). Using concepts of the theory of functions it has been shown that by evaluating a sequence of path integrals or the integral over the derivative of the logarithm of the characteristic function along the positive part of the imaginary axis it can be determined if the system is stable. As new result it is derived that by estimating the maximal modulus of the zeros of the characteristic function this can be reduced to the calculation of a path integral of this function over a finite contour.

Global guidance stabilization of linear control systems

Global guidance stabilization of linear control systems

An algebraic criterion for robust stability of linear control systems

A new robust stability test for linear control systems is described. The condition at which robust stability is violated is transformed into an equivalent problem in which the existence of a real root of a multivariable polynomial is investigated. This multivariable problem is reduced to that of the solvability of a set of univariable polynomial equations in real numbers, for which a number of efficient numerical methods are available. The use of the method is illustrated in the design of feedback control for an open-loop unstable batch chemical reactor.

A factorization principle for stabilization of linear control systems

AbstractBy introducing a fictitious signal y0 if necessary we define a transform magnified image which generalizes the passage from the scattering to the chain formalism in circuit theory. Given a factorization 𝒫˜ = ⊖R of 𝒫˜ where R is a block matrix function with a certain key block equal to a minimal phase (or outer) matrix function, we show that a given compensator u = Ky is internally stabilizing for the system 𝒫 if and only if a related compensator K′ is stabilizing for ⊖. Factorizations 𝒫˜ = ⊖R with ⊖ having a certain block upper triangular form lead to an alternative derivation of the Youla parametrization of stabilizing compensators. Factorizations with ⊖ equal to a J‐inner matrix function (in a precise weak sense) lead to a parametrization of all solutions K of the H∞ problem associated with 𝒫. This gives a new solution of the H∞ problem completely in the transfer function domain. Computation of the needed factorization 𝒫˜ = ⊖R in terms of a state‐space realization of 𝒫 leads to the state‐space formulas for the solution of the H∞ problem recently obtained in the literature.

Linear compensator designs based exclusively on input-output information are never robust with respect to unmodeled dynamics

The effects of unmodeled, higher-order dynamics or parasitics on the stability of linear control systems are investigated. A description is first given of a class of perturbations of a given state equation that cannot be distinguished from the original on the basis of input-output measurements alone. Then it is shown that given any plant-compensator pair, perturbations of each system of this type can always be found that destabilize the closed-loop configuration. The effect of destabilizing perturbations on output behavior is explored. >

Stability and controllability

Abstract The paper unifies proofs of several results on the stability of linear control systems. The proofs are based on a criterion of controllability derived in the paper. This criterion suggests an efficient test of controllability.

Note on the Stability of Linear Control System with Distributed-Lag

Note on the Stability of Linear Control System with Distributed-Lag