Class Of Nonlinear Time-varying Systems Research Articles

Robust output feedback stabilization of a class of time-varying non-linear systems

This paper deals with the problem of robust output feedback stabilization of a class of time-varying non-linear systems. This class of systems involves two kinds of time-varying uncertainties: those norm-bounded and those bounded by a smooth non-linear function of the output. Under the assumption that the zero dynamics of the system are uniformly asymptotically stable and some additional mild conditions, we show via a Lyapunov function approach that the uncertain system can be robustly stabilized by a time-varying non-linear output feedback controller. The order of this controller turns out to be one less than the relative degree of the uncertain system. A systematic design procedure is given for constructing the controller. Illustrative examples are given. Note that the results generalize several previous results on robust output feedback stabilization.

Robust control for nonlinear time-varying systems with application to a robotic manipulator

Sliding mode control methods have been used widely because they provide robustness against parameter variations and disturbances. This paper focuses on the problem of a robust output-sliding control design for a class of nonlinear time-varying systems. Output signals are used for the switching function definition. The control law formulation is emphasized. Input and output linearization is used. Output tracking can be achieved against a class of time-varying parameter variations and external disturbances. An application to a two-degree-of-freedom manipulator indicates that the proposed switching control law drives the system state trajectories onto the chosen sliding mode in finite time and the output tracking is guaranteed.

Comments on "A unified approach for stability analysis of a class of time-varying nonlinear systems"

A unified approach for constructing strictly passive systems has been proposed by Feng and Fei (1999). However, we have found that contradictory cases exist and additional prerequisites might be required for the proofs of the lemmas. Illustrations and proofs are provided in this paper to act as counterexamples.

An analytic approach to the design of nonlinear time-varying control systems

This paper addresses a controller design approach for a specific class of nonlinear time-varying systems based on an appropriate choice of a nonlinear time-varying dynamic feedback. A general form of the nonlinear time-varying dynamic controller is proposed. Lyapunov function theory was extensively used to prove global stability of closed systems. Conditions of local and global asymptotic stability of the nonlinear time-varying system are given. An analytic algorithm of controller design is presented. It is shown that the proposed design technique can successfully be applied to the specific class of the systems, the state variables of which are not fully observed. Simulation results show that the proposed approach to the nonlinear time-varying controller design is effective, and the closed-loop system is robust to small parametrical variations.

An analytic approach to the design of nonlinear time-varying control systems

This paper addresses a controller design approach for a specific class of nonlinear time-varying systems based on an appropriate choice of a nonlinear time-varying dynamic feedback. A general form of the nonlinear time-varying dynamic controller is proposed. Lyapunov function theory was extensively used to prove global stability of closed systems. Conditions of local and global asymptotic stability of the nonlinear time-varying system are given. An analytic algorithm of controller design is presented. It is shown that the proposed design technique can successfully be applied to the specific class of the systems, the state variables of which are not fully observed. Simulation results show that the proposed approach to the nonlinear time-varying controller design is effective, and the closed-loop system is robust to small parametrical variations.

Technical stability of nonlinear time-varying systems with small parameters

Technical stability allowing quantitative estimation of trajectory behavior of a dynamical system over a given time interval was considered. Based on a differential comparison principle and a basic monotonicity condition, technical stability relative to certain prescribed state constraint sets of a class of nonlinear time-varying systems with small parameters was analyzed by means of vector Liapunov function method. Explicit criteria of technical stability are established in terms of coefficients of the system under consideration. Conditions under which the technical stability of the system can be derived from its reduced linear time-varying (LTV) system were further examined, as well as a condition for linearization approach to technical stability of general nonlinear systems. Also, a simple algebraic condition of exponential asymptotic stability of LTV systems is presented. Two illustrative examples are given to demonstrate the availability of the presently proposed method.

A unified approach for stability analysis of a class of time-varying nonlinear systems

It is shown that an integral with negative feedback from its output terminal through a strictly passive system is still strictly passive. By using successive negative feedback around integrals, the conditions of strict passivity and a unified approach for constructing strictly passive systems are obtained for time-varying nonlinear systems of different orders. A new concept of dissipation factor is defined in this paper. The stability of a class of time-varying nonlinear systems is examined by using this passivity analysis.

Robust control of nonlinear continuous-time systems with parameter uncertainties and input bounds

New results and innovative procedures are presented to design nonlinear robust controllers for a class of nonlinear time-varying systems with uncertain parameters and control constraints. The straightforward techniques for designing the robust bounded controllers are developed by utilizing the theoretical foundations of the HamiltonJacobi theory and L yapunov's method. The major contribution is the development of feasible and computationally efficient algorithms to stabilize uncertain nonlinear systems. Various aspects of recent developments concerning robust control and nonlinear optimization are discussed and further researched. Necessary and sufficient conditions for optimality and robust stability are studied. By applying robust design, this paper allows one to solve practical problems in nonlinear robust control and optimization by using nonlinear models with parameter variations and input bounds. To overcome the gap between the theory and its application, two examples are thoroughly studied. Robust control of the third-order chaotic attractor with uncertain parameters is performed to illustrate the efficiency of the proposed procedures. In addition, the goal of this article is to demonstrate the feasibility of the developed concept for multivariable flight control, and a bounded controller is designed for the F-18 fighter. This robust algorithm is synthesized by using the coupled nonlinear aerodynamics for the longitudinal and lateral axes.

Adaptive feedforward and feedback control of non-linear time-varying uncertain systems

An adaptive regulation scheme is proposed for a class of non-linear time-varying systems with parametric uncertainties. The proposed approach is based upon a combination of the adaptive backstepping design method and a feedforward control scheme to design a non-linear adaptive feedforward and feedback controller, such that robust output tracking can be achieved even in the presence of structured uncertainties, as well as time-varying, measurable disturbances. Although the systematic design procedure does not a priori satisfy the feedback linearizable system with triangular structures, however, the constructed condition must be satisfied to ensure that the control scheme has a stable inversion. Under the feasibility condition, the states of the resulting closed-loop system would be guaranteed boundedness and converge to a bounded set. Finally, the proposed methodology is illustrated by a chemical reactor example.

A discrete iterative learning control for a class of nonlinear time-varying systems

A discrete iterative learning control is presented for a class of discrete-time nonlinear time-varying systems with initial state error, input disturbance, and output measurement noise. A feedforward learning algorithm is designed under a stabilizing controller and is updated by more than one past control data in the previous trials. A systematic approach is developed to analyze the convergence and robustness of the proposed learning scheme. It is shown that the learning algorithm not only solves the convergence and robustness problems but also improves the learning rate for discrete-time nonlinear time-varying systems.

Time-varying continuous nonlinear systems: uniform asymptotic stability

The framework of the presented research is a large class of time-varying nonlinear systems with continuous motions. The study of the uniform asymptotic stability of the zero equilibrium state developed here goes back to, and relies on, the very foundations of the Lyapunov stability concept and the (second) Lyapunov method. Stability domains are first defined and examined. Then, their qualitative features are used to establish complete solutions to the problem of uniform asymptotic stability of the equilibrium for various subclasses of the systems. The solutions present conditions that are both necessary and sufficient for: (1) the uniform asymptotic stability, (2) an exact direct one-shot construction of a system Lyapunov function and (3) for a direct accurate one-shot determination of the asymptotic stability domain. In addition, the solutions establish a novel Lyapunov-method based approach to the asymptotic stability analysis. This enables an arbitrary selection of a function p(·) from a defined functi...

Robust tracking control of non-linear systems with uncertain dynamics Part 1

A robust tracking control algorithm based mainly on the theory of variable structure systems and Lyapunov's direct method is developed in this paper, to treat the tracking control problem of a class of non-linear time-varying systems with uncertain dynamics. A nominal model is used instead of an exact one. together with a description of the bounds of plant uncertainty. The objective is to design a control law such that all the plants lying within a specified band of uncertainly will be driven to reach the goal. The modelling error, time-varying system dynamics, system uncertainties, and disturbances are all treated as general disturbances to the nominal system. This algorithm greatly eases the difficulties present in requiring a precise model of the physical system for the control purpose and for handling the non-linear time-varying system. The non-linear time-varying system is first linearized to a linear time-varying system (with time-varying coefficients completely defined) via the method of variable s...

A new control method and its robustness for a class of nonlinear systems

For a class of time-varying nonlinear systems described by the equation $$\dot x = f(x,t) + g(x,t)u$$ , the precalculating control is not available if the input matrixg(x,t) is not invertible. With Lyapunov's second method, a stabilizing controller which makes the system practically stable is constructed in this paper. It is shown that the implementation of this scheme depends on some so-called posi-invertibility conditions forg(x,t). In case the system is partly stable, the method, named part-calculating control, can simplify the on-line computations. Without the assumption that the nominal system is asymptotically stable, the method is applied to the problems of control for the corresponding uncertain system that satisfies the matching condition. When the matching condition is not satisfied, the mismatching control problem is also studied with Lyapunov's second method.

Identification of linear periodically time-varying systems using white-noise test inputs

A nonparametric technique for the identification of a class of nonlinear time-varying systems using white-noise test inputs published previously (Marmarelis, 1981, IEE Proc.-D, 128, 211–214) is applied to computer simulated linear periodically time-varying systems. This identification technique avoids the necessity of ensemble averaging through a separation of system dynamics and time-varying characteristics. The purpose of this paper is to present the results of these simulations and discuss some practical issues that arise in the actual implementation of the method. The simulation results corroborate the ability of this technique to practically identify periodically time-varying systems (of unknown periodicity) by use of a single input-output data-record.

Right factorization of a class of time-varying nonlinear systems

A class of nonlinear continuous-time time-varying plants with a state-space description which has a uniformly completely controllable linear part is considered. For this class, a right factorization is obtained. For the case in which the state is available for feedback, a normalized right-coprime factorization is realized.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Microprocessor-Based Novel Variable Structure Control for Robot Manipulator

Abstract A new variable structure control scheme which consists of continuous adaptive gain feedback(PID) and feedforward controls is developed to achieve accurate decoupled model following in a class of nonlinear timevarying systems in the presence of disturbances, parameter variations and non 1inear dynamic interact ions. The method is based on modified model following control with an improved sliding mode, which results in reducing the undesirable chattering remarkably. In addition an external torque observer with which the effects of external disturbances including parameter variations are roughly canceled is constructed. Then the developed method is practically applied to decoupled model following motion control for a two-degrees-of-freedom manipulator powered by PWM transistor converter-fed servo motors. The overall control strategies are implemented with both NEC PC-9801F (Intel 8086 Microprocessor) and TI TMS32020(DSP). The DSP is a very attractive hardware for the realization of the robust and fast observer algorithm. It is confirmed by experiments that the position trajectories are smooth and track the desired trajectories (output models) accurately. It is found t hat this new controller is simple and easy to be designed, performs extremely satisfactorily and is superior to the “conventional” PID controller for the robot manipulator

Non-linear model following

Model following as a controller design principle is applied to a class of non-linear time-varying systems, using a non-linear time-varying model. The conditions for the existence of a stabilizing model matching controller are derived and transformed into frequency domain criteria, suitable for the practical design.

A general approach to the stability analysis of adaptive systems

Many of the problems in adaptive control can be formulated as stability problems of a special class of non-linear time-varying systems. The paper discusses a general approach developed in recent years exemplified by several recent contributions to the field. The approach is based on order relationships between inputs and outputs of linear systems, with particular reference to a basic structure used in adaptive control.

Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators†

A methodology is presented of feedback control to achieve accurate tracking for a class of nonlinear time-varying systems in the presence of disturbances and parameter variations. The methodology uses in its idealized form piecewise continuous feedback control laws, resulting in the state trajectory `sliding' along a discontinuity surface in the state space. The idealized form of the methodology results in perfect tracking of the required signals; however certain non-idealities associated with its implementation cause the trajectory to 'chatter' along the sliding surface resulting in the generation of an undesirable high-frequency component which may excite high-frequency unmodelled dynamics of the control systems. To rectify this situation, it is shown how continuous control laws which approximate the discontinuous control law may be used to obtain disturbance and parameter variation insensitive tracking. At the same time, the continuous control laws decrease the extent of unwanted high-frequency signals.

On the controllability of nonlinear systems with time-variable delays

This note deals with the controllability, of a certain class of nonlinear time-varying systems with time-variable delays in state and implicit derivative. The notion of measure of noncompactness of a set and methods for its determination in the spaces of continuous and differentiable functions are considered. By using the measure of noncompactness of a set and the fixed-point theorem of Darbo, a theorem is proved on the existence of solutions of a certain type of nonlinear time-varying systems with multiple time-variable delays in state with derivative of the state-vector on the right-hand side. Sufficient conditions are given for global and local controllability of such systems.