An indirect Lyapunov approach to robust stabilization for a class of linear fractional-order system with positive real uncertainty

The paper is concerned with the problem of the robust stabilization for a class of fractional order linear systems with positive real uncertainty under Riemann–Liouville (RL) derivatives. Firstly, by utilizing the continuous frequency distributed model of the fractional integrator, the fractional order system is expressed as an infinite dimensional integral order system. And via using indirect Lyapunov approach and linear matrix inequality techniques, sufficient condition for robust asymptotic stability of the fractional order systems and design methods of the state feedback controller are presented. Secondly, by using matrixs singular value decomposition technique the static output feedback controller and observer-based controller for asymptotically stabilizing the fractional order systems are derived. Finally, the validity of the proposed methods are demonstrated by numerical examples.     

Output stabilization in uncertain many input-many output systems

The article considers robust output stabilization of a class of uncertain many input-many output systems. The problem is solved by the method of asymptotic invariance in the class of continuous feedbacks. Transients in the closed-loop control systems are estimated and the dependence of transient performance on observer and feedback parameters is investigated. Translated from Nelineinaya Dinamika i Upravlenie, pp. 159–172, 1999.     

Stabilization of infinite-dimensional undamped second order systems by using a parallel compensator

In this paper stabilization of infinite-dimensional undamped second-order systems is considered in the case where the input and output operators are collocated. The systems have an infinitenumber of poles and zeros on the imaginary axis. In the casewhere only position feedback is available, a parallel compensatoris effective. The stabilizer is constructed by a P-controllerfor the augmented system which consists of the controlled systemand a parallel compensator. The asymptotic stability of theaugmented system is proved by LaSalle's invariance principleunder compactness of the resolvent.     

Invariance Principles for Autonomous Infinite Delay Difference Systems

The invariance principles for autonomous difference systems with infinite delay are established.As applications of the obtained invariance principles, criteria for asymptotic stability and asymptotic constancy of solutions are also given.     

Stability and stabilization of a class of nonlinear impulsive hybrid systems based on FSM with MDADT

The issue of stability and stabilization for a class of nonlinear impulsive hybrid systems based on finite state machine (FSM) with mode-dependent average dwell time (MDADT) is investigated in this paper. The concepts of global asymptotic stability and global exponential stability are extended for the systems, and the multiple Lyapunov functions (MLFs) are constructed to prove the sufficient conditions of global asymptotic stability and global exponential stability, respectively. Furthermore, the method of stabilization is also given for the hybrid systems. The application of MLFs and MDADT leads to a reduction of conservativeness in contrast with classical Lyapunov function. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed approach.     

多时滞非线性中立型反馈系统的鲁棒稳定性

The stabilization of a class of neutral systems with multiple time-delays is considered. To stabilize the neutral system with nonlinear uncertainty, a state feedback control law via compound memory and memoryless feedback is derived. by constructed Lyapunov functional, delay-independent stability criteria are proposed that are sufficient to ensure a uniform asymptotic stability property. Finally, two concise examples are provided to illustrate the feasibility of our results.     

Using asymptotic invariance methods for output stabilization of a class of uncertain systems with stable zero dynamics

Conclusion We have introduced the notion of exponential invariance and have shown that output stabilization of uncertain systems of first relative order with a stable zero dynamics reduces to the design of an exponentially invariant (in the generalized sense) system. We have derived sufficient conditions of exponential invariance and applied them to calculate the parameters of large-gain stabilizing algorithms and nonlinear dynamic feedbacks. Transient bounds have been derived. It has been shown that the properties of a closed-loop control system depend only on the exponential invariance parameters and can be specified in advance, before the control is chosen. Examples have been examined that demonstrate the constructivity of the proposed approach. The approach is shown to be applicable to a wide class of uncertain controlled systems, including both phase-minimal and phase-nonminimal plants. Translated from Metody Analiza Nelineinykh Sistem, Moscow University, pp. 87–101, 1997.     

On a new approach to asymptotic stabilization problems

A numerical algorithm for solving the asymptotic stabilization problem by the initial data to a fixed hyperbolic point with a given rate is proposed and justified. The stabilization problem is reduced to projecting the resolving operator of the given evolution process on a strongly stable manifold. This approach makes it possible to apply the results to a wide class of semidynamical systems including those corresponding to partial differential equations. By way of example, a numerical solution of the problem of the asymptotic stabilization of unstable trajectories of the two-dimensional Chafee-Infante equation in a circular domain by the boundary conditions is given.     

Stability and Stabilization Criteria for a Class of Uncertain Neutral Systems with Time-Varying Delays

In this paper, the robust asymptotic stability and stabilization for a class of uncertain neutral system with time-varying delays are considered. Based on the Lyapunov-Krasovskii functional theory, some stability and stabilization criteria are derived. Delay-dependent and delay-independent criteria are proposed for the stability and stabilization of the considered systems. State and output feedbacks are considered to stabilize the uncertain neutral system. A linear matrix inequality approach and a genetic algorithm are used to solve the stability and stabilization problems. Finally, some numerical examples are shown to illustrate the use of the obtained results.The research reported here was supported by the National Science Council of Taiwan under Grant NSC 91-2213-E-214-016.Communicated by C. T. Leondes     

Robust Stabilization for Dynamic Systems with Multiple Time-Varying Delays and Nonlinear Uncertainties

In this paper, the problem of the robust stabilization for a class of uncertain linear systems with multiple time-varying delays is investigated. The uncertainty is nonlinear time-varying and does not require a matching condition. A memoryless state-feedback controller for the robust stabilization of the system is proposed. Based on the Lyapunov method and the linear matrix inequality (LMI) approach, two sufficient conditions for the stability are derived. Two numerical examples are given to illustrate the proposed method.     

一类非线性系统的输出反馈控制及全局渐近稳定性研究

考虑不可测状态是非线性的非严格三角形非线性系统的全局渐近稳定性问题.提出了一种新的反馈控制设计方法,构造一个线性动态输出补偿器,并全局稳定所控制的非线性系统.     

Stabilization of switched linear systems by a controller of variable structure

We study the state stabilization problem for switched linear systems operating under parametric uncertainty and bounded coordinate disturbances. To solve the problem, we suggest an algorithmfor constructing a controller of variable structure on the basis of methods of simultaneous stabilization theory.     

Neural network hybrid adaptive control for nonlinear uncertain impulsive dynamical systems

A neural network hybrid adaptive control framework for nonlinear uncertain hybrid dynamical systems is developed. The proposed hybrid adaptive control framework is Lyapunov-based and guarantees partial asymptotic stability of the closed-loop hybrid system; that is, asymptotic stability with respect to part of the closed-loop system states associated with the hybrid plant states. A numerical example is provided to demonstrate the efficacy of the proposed hybrid adaptive stabilization approach.     

Generalization of the Jurdjevic-Quinn theorem and stabilization of bilinear control systems with periodic coefficients: I

The Jurdjevic-Quinn theorem on the global asymptotic stabilization of the origin is generalized to nonlinear time-varying affine control systems with periodic coefficients. The proof is based on the Krasovskii theorem on the global asymptotic stability for periodic systems and the introduced notion of “commutator” for two vector fields one of which is time-varying. The obtained sufficient conditions for stabilization are applied to bilinear control systems with periodic coefficients. We construct a control periodic in t in the form of a quadratic form in x that asymptotically stabilizes the zero solution of a bilinear periodic system with a time-invariant drift.     

Stabilization of stationary affine control systems with discrete time

We obtain new sufficient conditions for the local and global asymptotic stabilization of the zero solution of a nonlinear affine control system with discrete time and with constant coefficients by a continuous state feedback. We assume that the zero solution of the free system is Lyapunov stable. For systems with linear drift, we construct a bounded control in the problem of global asymptotic state and output stabilization. Corollaries for bilinear systems are obtained.     

Algebraic Approach to the Stabilization of a Differential System of Retarded Type

For a spectrally controllable linear autonomous systems with commensurable delays, we construct state feedbacks ensuring the complete damping of the original system (finite stabilization) as well as the complete damping of the original system and the asymptotic stability of the closed-loop system (complete stabilization). The spectral reduction and asymptotic stabilization problems are considered as auxiliary problems. The argument is constructive, and the results are illustrated by an example.     

Robust Stabilization for a Class of Uncertain Nonlinear Systems with Time-Varying Delay: Razumikhin-Type Approach

In this paper, using a Razumikhin-type approach, the stabilization of a class of uncertain nonlinear systems with time-varying delay is considered. The proposed controller is based on a specific optimal control problem. Global asymptotic stability is guaranteed for the proposed control if some algebraic condition is met. An example illustrates the use of the main result.     

Dynamics of the thermohaline circulation under uncertainty

The ocean thermohaline circulation under uncertainty is investigated by a random dynamical systems approach. It is shown that the asymptotic dynamics of the thermohaline circulation is described by a random attractor and by a system with finite degrees of freedom.     

Viability with Probabilistic Knowledge of Initial Condition, Application to Optimal Control

In this paper we provide an extension of the Viability and Invariance Theorems in the Wasserstein metric space of probability measures with finite quadratic moments in ? ^d for controlled systems of which the dynamic is bounded and Lipschitz. Then we characterize the viability and invariance kernels as the largest viability (resp. invariance) domains. As application of our result we consider an optimal control problem of Mayer type with lower semicontinuous cost function for the same controlled system with uncertainty on the initial state modeled by a probability measure. Following Frankowska, we prove using the epigraphical viability approach that the value function is the unique lower semicontinuous proximal episolution of a suitable Hamilton Jacobi equation.     

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.