Control Of Uncertain Dynamical Systems Research Articles

Quantization Effect on Sliding Mode Control of Uncertain Dynamical Systems

AbstractThis paper studies the quantization effect on dynamical behaviors of the sliding mode control system with matched uncertainties and disturbances. The sufficient condition for stability of the system is given. The performance of both the sliding mode and the system trajectories with quantized state feedback are discussed. We show that the sliding mode with quantization is piecewise constant and the trajectory chatters in the boundary layer of the designed sliding manifold with thickness O(μ), where μ is the quantization level. Simulation results are presented to show the effectiveness of the theoretical results.

A class of adaptive robust controllers for uncertain dynamical systems with unknown virtual control coefficients

The problem of backstepping approach-based adaptive robust controller design is considered for a class of nonlinear systems in the presence of uncertain system parameters, external disturbances, and unknown time-varying virtual control coefficients. It is supposed that the upper bounds of external disturbances and unknown time-varying virtual control coefficients are unknown. In particular, it is also supposed that the lower bounds of time-varying virtual control coefficients are unknown. Then, the improved adaptation laws with σ-modification are employed to estimate the unknown values containing all the unknown bounds of such uncertainties. With the help of the adaptation laws, a class of backstepping approach-based continuous adaptive robust state feedback controllers is developed. Furthermore, by employing the improved Lyapunov-Krasovskii functional method, it is also shown that the proposed adaptive robust backstepping controller can guarantee the uniform asymptotic stability of such uncertain dynamical systems. Finally, a numerical example is given to demonstrate the validity of the results.

Robust sampled-data control with random missing data scenario

The problem of robust sampled-data control for uncertain dynamic systems in the presence of missing data has been investigated. By using stochastic variables with a Bernoulli distributed white sequence to model missing data, along with time-varying norm-bounded uncertainties, and the input delay approach for sampled-data systems, two models for the considered sampled-data control system are proposed as the consecutive missing data scenario. Moreover, when data are missing, the control signals are held as the last received data. Sufficient conditions for the existence of desired robust sampled-data controllers are presented in the form of a linear matrix inequality. A numerical example is given to illustrate the validity of the proposed methods and to compare the results between the two proposed models of the sampled-data control system.

PI Sliding Mode Control for Mismatched Uncertain Systems

This paper focused on the proportional–integral sliding mode control for uncertain dynamic systems with mismatch uncertainties. First, the switching surface condition for the sliding mode control is synthesized. Then the control law is designed to drive the state trajectories of the system onto the sliding surface and the system remains in it thereafter. The proposed control law is applied to a numerical example and its performance is compared to the state variable feedback control system design methodology. Key words: sliding mode control; state variable feedback regulator; mismatched uncertainties

Sliding mode control of uncertain dynamical systems with time delay using the continuous time approximation method

A study of sliding mode control of uncertain dynamical systems with time delay is presented in this paper. The systems are assumed to have constant delay time and uncertain parameters with known upper and lower bounds. The method of continuous time approximation is applied to formulate the sliding mode control problem. The proposed treatment of the control delay leads to a higher-order control formulation for a system with additive uncertainties only. An optimal sliding surface is designed such that on the sliding surface, the controlled state variables act like a feedback control to the uncontrolled state variables. Examples of non-linear systems are presented to demonstrate the theoretical work.

Call for papers: Special Issue on Learning Issues in Feedback Control of Uncertain Dynamical Systems (SILIFC)

Call for papers: Special Issue on <i>Learning Issues in Feedback Control of Uncertain Dynamical Systems</i> (<i>SILIFC</i>)

Adaptive robust control of uncertain dynamical systems with multiple time-varying delays

The problem of adaptive robust stabilisation is considered for a class of dynamical systems with multiple time-varying delayed state perturbations, time-varying uncertain parameters, and external disturbances. It is assumed that the upper bounds of the delayed state perturbations, uncertainties and external disturbances are unknown, and that the time-varying delays are any non-negative continuous and bounded functions. In particular, it is not required that the derivatives of time-varying delays have to be less than one. For such a class of uncertain time-delay systems, a new method is presented whereby a class of memoryless continuous adaptive robust state feedback controllers is proposed. By employing a quasi-Lyapunov function, it is shown that the solutions of uncertain time-delay systems can be guaranteed to be uniformly exponentially convergent towards a ball which can be as small as desired. In addition, since the proposed adaptive robust state feedback controllers are completely independent of time delays, the results obtained in the study may be also applicable to a class of dynamical systems with uncertain time delays. Finally, a numerical example is given to demonstrate the validity of the results.

Identification, Estimation, and Control of Uncertain Dynamic Systems: A Nonparametric Approach

This article is devoted to a presentation of the author' practice of the non-parametric estimation theory for the estimation, filtering, and control of uncertain dynamic systems. The fundamental advantage of this approach is a weak dependency on prior modeling assumptions about uncertain dynamic components. This approach appears to be of great interest for the control of general discrete-time processes, and in particular, biotechnological processes, which are emblematic of nonlinear uncertain and partially observed systems.

Kernel-Based Least Squares Policy Iteration for Reinforcement Learning

In this paper, we present a kernel-based least squares policy iteration (KLSPI) algorithm for reinforcement learning (RL) in large or continuous state spaces, which can be used to realize adaptive feedback control of uncertain dynamic systems. By using KLSPI, near-optimal control policies can be obtained without much a priori knowledge on dynamic models of control plants. In KLSPI, Mercer kernels are used in the policy evaluation of a policy iteration process, where a new kernel-based least squares temporal-difference algorithm called KLSTD-Q is proposed for efficient policy evaluation. To keep the sparsity and improve the generalization ability of KLSTD-Q solutions, a kernel sparsification procedure based on approximate linear dependency (ALD) is performed. Compared to the previous works on approximate RL methods, KLSPI makes two progresses to eliminate the main difficulties of existing results. One is the better convergence and (near) optimality guarantee by using the KLSTD-Q algorithm for policy evaluation with high precision. The other is the automatic feature selection using the ALD-based kernel sparsification. Therefore, the KLSPI algorithm provides a general RL method with generalization performance and convergence guarantee for large-scale Markov decision problems (MDPs). Experimental results on a typical RL task for a stochastic chain problem demonstrate that KLSPI can consistently achieve better learning efficiency and policy quality than the previous least squares policy iteration (LSPI) algorithm. Furthermore, the KLSPI method was also evaluated on two nonlinear feedback control problems, including a ship heading control problem and the swing up control of a double-link underactuated pendulum called acrobot. Simulation results illustrate that the proposed method can optimize controller performance using little a priori information of uncertain dynamic systems. It is also demonstrated that KLSPI can be applied to online learning control by incorporating an initial controller to ensure online performance.

Reliability‐based robust control for uncertain dynamical systems using feedback of incomplete noisy response measurements

AbstractA reliability‐based output feedback control methodology is presented for controlling the dynamic response of systems that are represented by linear state‐space models. The design criterion is based on a robust failure probability for the system. This criterion provides robustness for the controlled system by considering a probability distribution over a set of possible system models with a stochastic model of the excitation so that robust performance is expected. The control command signal can be calculated using incomplete response measurements at previous time steps without requiring state estimation. Examples of robust structural control using an active mass driver on a shear building model and on a benchmark structure are presented to illustrate the proposed method. Copyright © 2003 John Wiley &amp; Sons, Ltd.

On the convergence properties of a 2-sliding control algorithm for non-linear uncertain systems

The second order sliding mode control of uncertain dynamical systems can be expressed in terms of the stabilization of uncertain second-order non-linear systems with incomplete state availability. The existence of the solution strictly depends on the assumptions made on the uncertainties, and global results are not always attainable. In this paper the class of systems for which a suitable variable structure control algorithm ensures finite time global convergence to the sliding manifold is identified. Simulation results highlight the effectiveness of the proposed method.

Variable Structure Control Design for Uncertain Dynamic Systems with Sector Nonlinearities

This paper presents a new robust control for uncertain dynamic systems with sector nonlinear input. It is implemented by using variable structure control. The proposed variable structure controller ensures the global reaching condition of the sliding mode of the uncertain dynamic system. However, in the sliding mode, the investigated uncertain system still bears the insensitivity to the uncertainties and disturbances as the systems with linear input.

Modelling and robust control of uncertain dynamical systems using fuzzy set theory

New modelling and control approaches for uncertain nonlinear dynamical systems using fuzzy set theory are developed and proposed in this paper. The philosophy of the proposed modelling approach is to provide a new fuzzy-set based representation of uncertain systems so that robust control design is made feasible without resorting to model simplification or imposing restrictive conditions on system uncertainty (heretofore required by most traditional control design approaches). Based on the new generated fuzzy model representation of uncertain systems, a fuzzy control design approach is also proposed. The latter consists of extending design techniques of the sliding mode control method to fuzzy systems. Accordingly, new control design concepts using fuzzy sets are introduced. The stability analysis of the proposed fuzzy-model based control system is also provided in this paper. The benefits of the present study are illustrated through numerical simulations of the fuzzy modelling and control of a single-link robot arm with uncertain dynamics.

Variable Structure Control of Uncertain Dynamical Systems with Time-Varying Delay

The problem of variable structure control of uncertain time-delay dynamical systems is discussed. Two types of sliding surfaces are considered, and a class of variable structure control strategies are proposed. It is shown that for the uncertainties satisfying the matching conditions, the resulting variable structure control systems are invariant, i.e. the systems are completely insensitive to such uncertainties. Some sufficient conditions are also derived such that the robust stability of the resulting variable structure control systems with unmatched uncertainties can be guaranteed. In addition, the time-varying delay is assumed only to be any bounded and continuous function, and all of the control laws, the sliding surfaces, and the sufficient conditions are independent of the delay. Therefore, the results obtained in the paper are also applicable to the systems with perturbed time-delay.

A design scheme of variable structure adaptive control for uncertain dynamic systems

A design scheme of variable structure adaptive control for linear time-invariant system with uncertain dynamics is proposed. Both additive and multiplicative unmodeled dynamics are taken into consideration. The transfer function of the modeled part of the plant may have unstable zeros and unstable poles. A sign-following system and logic switchings are introduced into the control system. The global stability of the overall system is proved. Simulation results show the effectiveness of the proposed method.

One approach to the control of uncertain dynamical systems

We present a controller design methodology for uncertain systems which is based on the constructive use of Lyapunov stability theory. The uncertainties, which are deterministic, are characterized by certain structural conditions and known as well as unknown bounds. As a consequence of the Lyapunov approach, the methodology is not restricted to linear or time-invariant systems. The robustness of these controllers in the presence of singular perturbations is considered. The situation in which the full state of the system is not available for measurement is also considered as are other generalizations.

A variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties

In deterministic design of feedback controllers for uncertain dynamical systems, upper bounds on the norm of the uncertainties are an important clue to guarantee the stability of the closed-loop system. Sometimes, however these upper bounds may not be easily obtained because of the complexity of structure of the uncertainties. Simple adaptation laws for upper bounds on the norm of the uncertainties are proposed, and these adaptive upper bounds are used to design a variable-structure control system that guarantees the asymptotic stability of uncertain dynamical systems. >

Design of robust controllers for uncertain dynamical systems

The general design approach for robust controllers of uncertain dynamical systems is considered. The design approach, utilizing only the bound of uncertainty, can be used to construct various classes of controllers which render the systems practically stable. No statistical information of the uncertainty is required, and the system performance is described in a deterministic manner.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A variable structure approach to feedback regulation of uncertain dynamical systems

Feedback control of uncertain dynamical systems is considered. Adopting a deterministic approach based on the theory of variable structure systems, a feedback regulator is developed which guarantees global asymptotic stability or global ultimate boundedness under certain hypotheses on the bounded system uncertainties and disturbances.

Linear ultimate boundedness control of uncertain dynamical systems

Given a state equation whose description includes uncertain parameters, it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded. When facing such a problem, one often has available or assumes an upper bound for the excursions of the uncertain parameters. Within this framework, the objective is to guarantee, no matter how ‘nature’ selects the uncertainty, that the state will eventually end up and remain within some pre-specified region. When this region is a small neighborhood about the origin, the concept of uniform ultimate boundedness is tantamount to ‘practical’ asymptotic stability. The objectives of this paper are twofold: first and foremost, our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control, can in point of fact be solved via linear control. This is the main result and is given in Theorem 4.1. Secondly, we show that some of the so-called matching assumptions (which have been required in the earlier literature) can be weakened somewhat. This enables one to design controllers for a larger class of dynamical systems than previously considered.