Stability of systems with nonlinear feedback through randomly time-varying delays

In a large class of multiloop control systems, many feedback loops are "closed" through a time-shared digital computer, using algorithms which require information from sources which are sampled at a rate which is not synchronized with the sampling of the individual "plants." This missynchronization, in conjunction with variations in the computer's task load caused by "interrupts," results in a randomly time-varying delay in the closing of the various feedback loops. Consequently, the dynamics of each controlled "plant" in such a system may be modeled by means of a stochastic delay-differential equation. This paper presents some new research results concerning the sample stability (as opposed to statistical, or ensemble stability) of nonlinear stochastic delay-differential equations.     

Input-output stability of time-varying nonlinear multiloop feedback systems

Sufficient conditions for the input-output stability (BIBO stability) of time-varying nonlinear multiloop feedback systems are established. The present objective is to analyze large-scale systems in terms of their lower order subsystems (subloops) and in terms of their interconnecting structure. Both time-domain and frequency-domain results are presented. In order to demonstrate the usefulness of the present approach, two specific examples are considered.     

Stability results for nonlinear feedback systems

This paper presents an approach towards deriving sufficient conditions for the stability of nonlinear feedback systems. The central features of the approach are twofold. Firstly, useful stability tests are obtained for the case when the subsystems have nonlinear dynamics; secondly, a unifying set of general stability criteria are given, from which known situations can be treated as special cases and new ones are handled with equal ease. The results are obtained by use of a recently developed theory of dissipative systems.     

Stability improvement of nonlinear systems by feedback

This paper is concerned with the property of stabilizing a nonlinear system to a specified equilibrium point arbitrarily fast by appropriate smooth feedback. Two closely related forms of this property are explored. One refers to the asymptotic transfer to the critical point with exponential decay, and it is shown that the systems possessing the above property are precisely those whose corresponding linearization is controllable. The second stabilizability form defined here amounts to driving the system back to an arbitrarily small neighborhood of the given point arbitrarily fast by smooth feedback. A global stabilization result is also included, finding application in Hamiltonian systems. Special emphasis is given to the bilinear case.     

Prescribed-time stabilization of p-normal nonlinear systems by bounded time-varying feedback

This article studies the prescribed-time stabilization problem of p-normal nonlinear systems by bounded time-varying high-gain feedback. A time-varying bounded controller, which can achieve prescribed-time stabilization, is designed via backstepping. Because of the usage of time-varying high-gain functions which grow unbounded as the time tends to the prescribed time, the usual separation technique is invalid. To solve this problem, a novel design scheme is proposed. It is shown that all of the closed-loop trajectories convergence to the origin in prescribed time. Finally, a numerical example verifies the effectiveness of the proposed method.     

Robust optimization of multivariable feedback systems with time-varying nonlinear uncertainties

A design criterion is developed to achieve the input-output decoupling of multivariable feedback systems and the robust stabilization of systems with time-varying nonlinear uncertainties. Moreover, an effective design algorithm is derived to achieve the robust optimization of multivariable feedback systems subjected to time-varying nonlnear uncertainties. The theory of minimum H^∞ -norm and the optimal interpolation technique are employed to solve this robust optimization problem. Since the requirements of internal stability are satisfied, this design algorithm performs appropriately, even if the plant is unstable and/or non-minimum phase. From the result of the robust optimization, we can predict the maximum sector bounds of nonlinear uncertainties that can be tolerated in the multivariable feedback system.     

On instability of feedback systems with a single nonlinear time-varying gain

Encirclement of the critical disk implies instability of a feedback system described by input-output relations just as in the case of systems with a finite-dimensional state representation.     

Stability of nonlinear feedback systems in a Volterra representation

Presented in this paper is a stability condition for a class of nonlinear feedback systems where the plant dynamics can be represented by a finite series of Volterra kernels. The class of Volterra kernels are limited to p‐linear stable operators and may contain pure delays. The stability condition requires that the linear kernel is non‐zero and that the closed loop characteristic equation associated with the linearized system is stable. Next, a sufficient condition is developed to upper bound the infinity‐norm of an external disturbance signal thereby guaranteeing that the internal and output signals of the closed loop nonlinear system are contained in L_∞. These results are then demonstrated on a design example. A frequency domain controller design procedure is also developed using these results where the trade‐off between performance and stability are considered for this class of nonlinear feedback systems. Copyright © 2000 John Wiley & Sons, Ltd.     

A suboptimal feedback control for nonlinear time-varying systems with continuous inequality constraints

A feedback control law, which is constructed as a first-order approximation to the optimal control, is proposed for nonlinear time-varying systems subject to continuous inequality constraints on the control and state. This control law is effective under small state perturbations caused by changes on initial conditions and/or modeling uncertainty.     

Finite-time stabilization by state feedback control for a class of time-varying nonlinear systems

In this paper, finite-time stabilization is considered for a class of nonlinear systems dominated by a lower-triangular model with a time-varying gain. Based on the finite-time Lyapunov stability theorem and dynamic gain control design approach, state feedback finite-time stabilization controllers are proposed with gains being tuned online by two dynamic equations. Different from many existing finite-time control designs for lower-triangular nonlinear systems, the celebrated backstepping method is not utilized here. It is observed that our design procedure is much simpler, and the resulting control gains are in general not as high as those provided by the backstepping method. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

Inversion of nonlinear time-varying systems

A procedure for inverting nonlinear systems that presents some computational advantages when applied to flexible robot arms or, more generally, to mechanical structures, is given. The procedure is presented in a general setting, for nonlinear time-varying systems. An iterative algorithm that computes a smooth controlled invariant time-varying manifold, the mathematical properties of which are illustrated and exploited, is also proposed. An application to a flexible two-link manipulator illustrates the algorithm and its computational advantages     

Relative stability of a linear time-varying process withfirst-order nonlinear time-varying feedback

A method of establishing sufficient conditions for the stability of a system consisting of a time-varying linear process and a first-order nonlinear time-varying feedback is presented. Rate of decay of unforced response and BIBO stability can also be determined. The governing equation of the system is represented in the form of an integro-differential equation, so that a modified form of the Lyapunov-Razumikhin-type condition can be applied. The first part of the paper deals with stability of vector functional equations. The results obtained are applied to feedback systems. An example is worked out to show the applicability of the results obtained     

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input-output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.     

Stability analysis for nonlinear feedback control systems with linear actuators

Often, nonlinear feedback controllers are implemented with linear actuators, which have faster dynamics when compared with non-linear dynamics of the plant and controller models. However, how faster should the dynamics of the linear actuator be to ensure stability of the given nonlinear feedback control system? This note gives an analysis for the underlying stability issues and provides quantitative measures for the linear actuators employed in nonlinear feedback control systems.     

Reliable Memory Feedback Design for a Class of Nonlinear Fuzzy Systems with Time-varying Delay

This paper is concerned with the robust reliable memory controller design for a class of fuzzy uncertain systems with time-varying delay. The system under consideration is more general than those in other existent works. The controller, which is dependent on the magnitudes and derivative of the delay, is proposed in terms of linear matrix inequality (LMI). The closed-loop system is asymptotically stable for all admissible uncertainties as well as actuator faults. A numerical example is presented for illustration.     

Further results on output feedback stabilization for stochastic high-order nonlinear systems with time-varying delay

This article further discusses the output feedback stabilization problem for stochastic high-order nonlinear systems with time-varying delay. Under the weaker conditions on nonlinearities in drift and diffusion vector fields, by using the idea of homogeneous domination approach, skillfully choosing an appropriate Lyapunov–Krasoviskii functional, and successfully solving several troublesome obstacles in the design and analysis procedure, an output feedback controller is constructed to render the closed-loop system globally asymptotically stable in probability.     

Stability and robustness for nonlinear systems decoupled and linearized by feedback

We give some stability and robustness results for nonlinear systems which are transformed by feedback and immersion into decoupled linear systems. In particular we give a necessary condition involving the Euler-Poincaré characteristic of asymptotic unobservable submanifolds. This condition is successfully applied to a non-minimum-phase example (see Byrnes and Isidori [2]) borrowed from robotics which displays K-stability.     

Stability analysis of linear time-varying systems

This paper addresses the issue of the stability of linear time-varying systems. Recently developed elemental perturbation bound analysis is extended to obtain stability bounds on the parameters of linear time-varying systems. In particular, systems governed by the Mathieu equation are considered. The bounds obtained with the proposed method are compared with those of the circle criterion and matrix decomposition methods. The proposed method is computationally simple and the bounds obtained fare well with the other methods considered.