On semi-global stabilization of minimum phase nonlinear systems without vector relative degrees

SUMMARYIn a recent paper, we developed a structural decomposition for multiple input multiple output nonlinear systems that are affine in control but otherwise general. In this paper, we exploit the structural properties of such a decomposition in the application of the backstepping design technique on multiple input nonlinear systems. In particular, this decomposition simplifies the conventional backstepping design and motivates new backstepping design procedures that are able to stabilize systems on which the conventional backstepping procedure is not applicable. Copyright © 2011 John Wiley & Sons, Ltd.

New criteria of almost sure exponential stability and instability of nonlinear stochastic systems with a generalization to stochastic coupled systems

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

It is can identify three main classes of problems broadly characterizing stability of a dynamical systems, viz., (1) stability with respect to a part of the variables of the zero equilibrium position (Lyapunov-Rumyantsev stability problem), (2) stability of the partialzero equilibrium position, and (3) stability with respect to a part of the variables of the partialzero equilibrium position. In the problem of stability with respect to a part of the variables of the zero equilibrium position of systems of ordinary differential equations with continuous right-side assumes the domain of initial perturbations to be a sufficiently small neighborhood of the zero equilibrium position. Along with this statement, the case then initial perturbations can be large with respect to one part of non-controlled variables and arbitrary with respect to their other part is also considered. On the other hand, for stability problem of partial zero equilibrium positions of systems of ordinary differential equations also naturally assume that initial perturbations of variables that do not define the given equilibrium position can be large with respect to one part of the variables and arbitrary with respect to their other part. Contrary the assumptions that initial perturbations of this variables are either only arbitrary or only large the combined assumption made it possible an admissible trade-off between the meaning sense for notion of stability and the respective requirements on the Lyapunov functions. The article studies the problem of stability for nonlinear discrete-time systems: stability with respect to a part of the variables of partial equilibrium position. Initial perturbations of variables that do not define the given equilibrium position can be large (belonging to an arbitrary compact set) with respect to one part of the variables and arbitrary with respect to their other part. A conditions of stability of this type are obtained in the context of a discrete analog of the Lyapunov functions method, which generalize a number of existing results. Example is given. The problem of unification of process of studying stability problems of stationary and non-stationary nonlinear discrete-time systems is also discussed

This paper focuses on the stability and stabilization problems of nonlinear fractional-order systems with fractional-order : 0 < α < 1. A sufficient condition for the stability and stabilization of nonlinear fractional-order systems is presented by Lyapunov direct approach. All results are obtained in term of linear matrix inequalities (LMI). Finally, two numerical examples are given to demonstrate the validity of the proposed approach.

In this paper finite-time stabilization of nonlinear affine systems, is considered. First we provide a formula for a finite-time stabilizing state feedback control law, for the last class of system. Then we extend the result to the case of a norm-bounded uncertain system and a polytopic uncertain system stabilization are given. Some numerical examples are given to illustrate the main result.

The problem of feedback linearization (FL) of continuous and discrete nonlinear MIMO systems is considered. The idea of FL method consists in converting the original nonlinear system into a linear one by means of feedback. Then, the methods of control theory for linear systems are used for system design. A widespread approach to FL design is based on the method of normal form, that uses a nonsingular transformation of system state variables z = T(x). In order to obtain a normal form of the nonlinear system in the neighbor of a some point, it is necessary to determine a special function - the system virtual output, for which a relative degree (in the case of single input) or a vector relative degree (in the case of multiple input) is determined. Applicability of the normal form method for FL is provided by the conditions of controllability and involutivity for the considered nonlinear system, which are not always true. Moreover, when developing a linearizing control law, the main difficulty lies in the transition from transformed variables z to state variables x of the original system. In this paper, we propose another approach, based on representing the original nonlinear system into a state-dependent coefficient form and applying the canonical similarity transformation z = T(x)x, that allow getting the system to canonical form, that considerably simplifies the FL problem. Such similarity transformation allow accomplishing linearization of system without determining of the virtual system output. Another advantage of the proposed method is that the technique of the transition from the transformed variables z to the state variables x of the original system is simpler. The results are illustrated by examples for continuous and discrete nonlinear systems.

This note is devoted to study the stabilization problem of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. By introducing the notation of input-to-state practical stability and an event-triggered strategy, we establish the input-to-state practically exponential mean-square stability of the suggested system. Moreover, we investigate the stabilization result by designing the feedback gain matrix and the event-triggered feedback controller, which is expressed in terms of linear matrix inequalities. Also, the lower bounds of interexecution times by the proposed event-triggered control method are obtained. Finally, an example is given to show the effectiveness of the proposed method. Compared with a large number of results for discrete-time stochastic systems, only a few results have appeared on the event-triggered control for continuous-time stochastic systems. In particular, there have been no published papers on the event-triggered control for continuous-time stochastic delay systems. This note is a first try to fill the gap on the topic.

This paper is concerned with the stabilization problem for nonlinear stochastic delay systems with Markovian switching by feedback control based on discrete-time state and mode observations. By constructing an efficient Lyapunov functionals, we establish the sufficient stabilization criteria not only in the sense of exponential stability (both the mean-square stability and the almost sure stability) but also in other sense – that of stability and asymptotic stability. Meanwhile, the upper bound on the duration τ between two consecutive state and mode observations is obtained. Numerical examples are provided to demonstrate the effectiveness of our theoretical results.

The robust stability problem for nonlinear distributed delay stochastic systems with polytopic-type uncertainties is discussed. Based on parameter-dependent Lyapunov-krasovs-kii functional and free-weighting matrix method, a delay-dependent and parameter-dependent stability condition is presented in terms of linear matrix inequalities. The result in this paper improves the existing stability criteria and has less conservativeness. The simulation example shows that the result is effective.

Observer-based stabilization of sector-bounded nonlinear stochastic systems in the presence of intermittent measurements

On observer-based fault detection for nonlinear systems

We study the problem of output feedback stabilization for stochastic nonlinear systems. We consider a class of stochastic nonlinear systems in observer canonical form with stable zero-dynamics. We introduce a sequence of state transformations that transform the system into a lower triangular structure that is amenable for integrator backstepping design. Then we design the output feedback controller and prove that the closed-loop system is bounded in probability. Under an infinite-horizon risk-neutral cost criterion, the controller designed can guarantee an arbitrarily small long term average cost. Furthermore, when the disturbance vector field vanishes at the origin, the closed-loop system is asymptotically stable in the large, and the risk-neutral cost is guaranteed to be zero. With special care, the controller preserves the equilibrium of the nonlinear system.

Decentralized stabilization of large-scale stochastic nonlinear systems with time-varying powers

In this paper, the problem of exponential stability for nonlinear systems with time delay on time scales is investigated. Based on the Wirtinger-based inequality on time scales, and the Lyapunov-Krasovskii functional approach, a sufficient criterion for exponential stability for nonlinear time-delay systems on time scales is derived. It is shown that the idea of this paper provides a unified approach to study the exponential stability problems for continuous-time systems and their discrete-time counterparts simultaneously. To illustrate the effectiveness of our result, two simulation examples are given.