Stabilization of uncertain systems on rings

M. V. Lomonosov Moscow State University. Translated from Matematicheskoe Modelirovanie. Published by Moscow University, Moscow, 1993, pp. 279–295.     

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 active fault‐tolerant control systems by uncertain nonhomogeneous markovian jump models

This article investigates the stabilization and control problems for a general active fault‐tolerant control system (AFTCS) in a stochastic framework. The novelty of the research lies in utilizing uncertain nonhomogeneous Markovian structures to take account for the imperfect fault detection and diagnosis (FDD) algorithms of the AFTCS. The underlying AFTCS is supposed to be modeled by two random processes of Markov type; one characterizing the system fault process and the other describing the FDD process. It is assumed that the FDD algorithm is imperfect and provides inaccurate Markovian parameters for the FDD process. Specifically, it provides uncertain transition rates (TRs); the TRs that lie in an interval without any particular structures. This framework is more consistent with real‐world applications to accommodate different types of faults. It is more general than the previously developed AFTCSs because of eliminating the need for an accurate estimation of the fault process. To solve the stabilizability and the controller design problems of this AFTCS, the whole system is viewed as an uncertain nonhomogeneous Markovian jump linear system (NHMJLS) with time‐varying and uncertain specifications. Based on the multiple and stochastic Lyapunov function for the NHMJLS, first a sufficient condition is obtained to analyze the system stabilizability and then, the controller gains are synthesized. Unlike the previous fault‐tolerant controllers, the proposed robust controller only needs to access the FDD process, besides it is easily obtainable through the existing optimization techniques. It is successfully tested on a practical inverted pendulum controlled by a fault‐prone DC motor. © 2016 Wiley Periodicals, Inc. Complexity 21: 318–329, 2016     

Robust H∞ output dynamic observer-based control of uncertain time-delay systems

In this paper, the robustness and H_∞ control problems of output dynamic observer-based control for a class of uncertain linear systems with time delay are considered. Under no disturbance input, the asymptotic stabilization for uncertain time-delay systems will be guaranteed. Linear matrix inequality (LMI) optimization approach is used to design three classes of the H_∞ output dynamic controls. Based on the results of this paper, the constraint of matrix equality is not necessary for designing the observer-based controls.     

Robust output feedback sliding mode control for uncertain discrete time systems

This paper proposes a robust output feedback controller for a class of uncertain discrete-time, multi-input multi-output, linear, systems. This method, which is based on the combination of discrete-time sliding mode control (DTSMC) and Kalman estimator, ensures the stability, robustness and an output tracking against the modeling uncertainties at large sampling periods. For this purpose, an appropriate structure is considered for sliding surface and the Lyapunov theory for the mismatched uncertain system is then used to design its parameter. This problem leads to solve a set of linear matrix inequalities. A new method is then proposed to reach the quasi-sliding mode and stay thereafter. Simulation studies show the effectiveness of the proposed method in the presence of parameter uncertainties and external disturbances at large sampling periods.     

Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems

This paper addresses the problem of semi-global finite-time decentralized output feedback control for large-scale systems with both higher-order and lower-order terms. A new design scheme is developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. Specifically, we first design a homogeneous observer and an output feedback control law for each nominal subsystem without the nonlinearities. Then, based on the homogeneous domination approach, we relax the linear growth condition to a polynomial one and construct decentralized controllers to render the nonlinear system semi-globally finite-time stable.     

Stabilization of linear systems by rotation

We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.     

Robust output feedback stabilization for two-dimensional continuous systems in roesser form

This paper considers the problem of robust stabilization via dynamic output feedbackcontrollers for uncertain two-dimensional continuous systems described by the Roesser's state space model. The parameter uncertainties are assumed to be norm-bounded appearing in all the matrices of the system model. A sufficient condition for the existence of dynamic output feedback controllers guaranteeing the asymptotic stability of the closed-loop system for all admissible uncertainties is proposed. A desired dynamic output feedback controller can be constructed by solving a set of linear matrix inequalities. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the proposed method.     

Stabilization of discrete systems by dynamic regulator

We consider the system

Stabilization and parameter estimation for an Euler–Bernoulli beam equation with uncertain harmonic disturbance under boundary output feedback control

This paper is concerned with the boundary stabilization and parameter estimation of an Euler–Bernoulli beam equation with one end fixed, and control and uncertain amplitude of harmonic disturbance at another end. A high-gain adaptive regulator is designed in terms of measured collocated end velocity. The existence and uniqueness of the classical solution as well as smooth solution of the closed-loop system are justified. It is shown that the state of the system approaches the standstill as time goes to infinity and meanwhile the estimated parameter converges to the unknown parameter.     

The guaranteed cost control for uncertain nonlinear large-scale stochastic systems via state and static output feedback

The guaranteed cost control (GCC) problem involved in decentralized robust control of a class of uncertain nonlinear large-scale stochastic systems with high-order interconnections is considered. After determining the appropriate conditions for the stochastic GCC controller, a class of decentralized local state feedback controllers is derived using the linear matrix inequality (LMI). The extension of the result of the study to the static output feedback control problem is discussed by considering the Karush-Kuhn-Tucker (KKT) conditions. The efficiency of the proposed design method is demonstrated on the basis of simulation results.     

Adaptive fuzzy output feedback control for a class of uncertain nonlinear systems with unknown backlash-like hysteresis

An output feedback controller is proposed for a class of uncertain nonlinear systems preceded by unknown backlash-like hysteresis, where the hysteresis is modeled by a differential equation. The unknown nonlinear functions are approximated by fuzzy systems based on universal approximation theorem, where both the premise and the consequent parts of the fuzzy rules are tuned with adaptive schemes. The proposed approach does not need the availability of the states, which is essential in practice, and uses an observer to estimate the states. An adaptive robust structure is used to cope with lumped uncertainties generated by state estimation error, approximation error of fuzzy systems and external disturbances. Due to its adaptive structure the bound of lumped uncertainties does not need to be known and at the same time the chattering is attenuated effectively. The strictly positive real (SPR) Lyapunov synthesis approach is used to guarantee asymptotic stability of the closed-loop system. In order to show the effectiveness of the proposed method simulation results are illustrated.     

Stability of pulse-modulated systems with a monotone equivalent nonlinearity in the modulator

A pulse-modulated system described by a functional differential equation is considered. Sufficient frequency conditions for global stability are obtained by using the averaging method and a frequency theorem.