Stability Of Nonlinear Control Systems Research Articles

Feedback Stabilization of Quasi-One-Sided Lipschitz Nonlinear Discrete-Time Systems with Reduced-Order Observer

The feedback stabilization problem for nonlinear discrete-time systems with a reduced-order observer is investigated, in which the nonlinear terms of the systems satisfy the quasi-one-sided Lipschitz condition. First, a discrete-time reduced-order observer for nonlinear systems is designed. Then, a feedback controller with a reduced-order observer is designed for realizing the stabilization of nonlinear discrete-time systems. We prove that the design of a feedback controller and reduced-order observer of systems can be carried out independently in the case of discrete-time with nonlinear terms, which largely reduces the computational complexity of the observer and controller. The introduction of the quasi-one-sided Lipschitz condition simultaneously enhances the robustness and stability of nonlinear control systems. Finally, the feasibility and effectiveness of the proposed design approach is verified by a numerical simulation.

Continuous feedback stabilization of nonlinear control systems by composition operators

The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications. In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions. To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting. Employing the obtained characterizations in the more general context, we establish relationships between continuous stabiliza|bility in the conventional sense and in the generalized composition operator sense. This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system. That is, we reduce the question of continuous stabilizability to that of stability. Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.

Linear openness and feedback stabilization of nonlinear control systems

It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and continuously differentiable stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.

Generalized sampling solutions for discontinuous stabilization of nonlinear systems

Sample-and-hold solution and sample stability are useful tools for analyzing stability of nonlinear control systems with discontinuous state feedbacks; for nonlinear control systems on Euclidean spaces, open-loop asymptotic controllability and sample stability of closed-loop systems are equivalent. Moreover, several sample stabilizing state feedback controllers based on nonsmooth control Lyapunov functions (CLFs) are proposed.However, analyzing sample stability based on the nonsmooth CLFs are too complex tasks in general. This is a serious obstacle to introduce more advanced properties such as invariance principle and its application to nonlinear adaptive control or finite-time stability for the sample stability framework.In this paper, we introduce the concept of generalized sampling solution, which is defined as the uniform limit of a sequence of sample-and-hold solutions. Moreover, we define asymptotic stability for generalized sampling solutions.The main contribution of this paper is to prove the existence and the asymptotic stability under the assumption that a locally semiconcave practical CLF (LS-PFCLF) is given.We also briefly mention an invariance principle based on a generalized sampling solution and an LS-PCLF.

On the Interval Stability of Weak-Nonlinear Control Systems with Aftereffect.

Sufficient conditions of interval absolute stability of nonlinear control systems described in terms of systems of the ordinary differential equations with delay argument and also neutral type are obtained. The Lyapunov-Krasovskii functional method in the form of the sum of a quadratic component and integrals from nonlinearity is used at construction of statements.

Stabilization of Nonlinear Control Systems through Using Zobov’s Theorem and Neural Networks

Stabilization of Nonlinear Control Systems through Using Zobov’s Theorem and Neural Networks

Input-output triggered control using Lp-stability over finite horizons

Author(s): Tolic, D; Sanfelice, RG; Fierro, R | Abstract: This paper investigates stability of nonlinear control systems under intermittent information. Following recent results in the literature, we replace the traditional periodic paradigm, where the up-to-date information is transmitted and control laws are executed in a periodic fashion, with the event-triggered paradigm. Building on the small gain theorem, we develop input-output triggered control algorithms yielding stable closed-loop systems. In other words, based on the currently available (but outdated) measurements of the outputs and external inputs of a plant, a mechanism triggering when to obtain new measurements and update the control inputs is provided. Depending on the noise in the environment, the developed algorithm yields stable, asymptotically stable, and Lp-stable (with bias) closed-loop systems. Control loops are modeled as interconnections of hybrid systems for which novel results on Lp-stability are presented. The prediction of a triggering event is achieved by employing Lp-gains over a finite horizon. By resorting to convex programming, a method to compute Lp-gains over a finite horizon is devised. Finally, our approach is successfully applied to a trajectory tracking problem for unicycles. © 2014 John Wiley a Sons, Ltd.

LUT Controller Design with Piecewise Bilinear Systems Using Estimation of Bounds for Approximation Errors

We propose the stabilization of nonlinear control systems approximated by Piecewise Bilinear (PB) models. The approximated model is fully parametric and a Look-Up-Table (LUT) represents its controller. Input-Output (I/O) feedback linearization is applied to stabilize PB control systems. We further propose PB modeling combined with conventional feedback linearization as a very powerful tool for analyzing and synthesizing nonlinear control systems. We also propose a method for designing robust stabilization controllers taking modeling error into consideration. Examples confirm the feasibility of our proposals.

A generalized sector‐bound approach to feedback stabilization of nonlinear control systems

SUMMARYThis paper proposes and develops a generalized sector‐bound approach to feedback stabilization of nonlinear control systems described by state–space models. This approach is inherited from the methodology of the sector‐bounded or passive nonlinearities and influenced by the concept of absolute and quadratic stability. It aims not only to regionally stabilize the nonlinear dynamics asymptotically but also to maximize the estimated region of quadratic attraction and to ensure nominal performance at each equilibrium. More importantly, it has a close connection to gain scheduling and switching control. A path of equilibria is programmed on the basis of the assumption of centered‐ ε‐cover, which leads to a sequence of linear controllers that regionally stabilize the desired equilibrium asymptotically. Simulation results are worked out to illustrate our proposed design method. Copyright © 2012 John Wiley & Sons, Ltd.

A New Approach to Popov Criterion for Stability Analysis of Nonlinear Control Systems

The Popov criterion for the stability of nonlinear control systems is considered. The Popov criterion gives sufficient conditions for stability of nonlinear systems in the frequency domain. It has a direct graphical interpretation and is convenient for both design and analysis. In the article presented, a table of transfer functions of linear parts of nonlinear systems is constructed. The tables includes frequency response functions and offers solutions to the stability of the given systems. The table makes a direct stability analysis of selected nonlinear systems possible. The stability analysis is solved analytically and graphically. Then it is easy to find out if the nonlinear system is or is not stable; the task that usually ranks among the difficult task in engineering practice.

On problems of Aizerman and Kalman

The problems of the stability of nonlinear control systems posed by Aizerman and Kalman have stimulated the development of methods for detecting hidden periodic oscillations in multidimensional dynamical systems.

Экспоненциальная стабилизируемость нелинейных управляемых систем

Экспоненциальная стабилизируемость нелинейных управляемых систем

Integral input-to-state stability of nonlinear control systems with delays

Abstract Integral input-to-state stability is an interesting concept that has been recently introduced to nonlinear control systems. This paper generalizes this concept to nonlinear control systems with delays. These delays can be bounded, unbounded, and even infinite. Theorems for integral input-to-state stability are derived by developing the method of Razumikhin technique in the theory of functional differential equations.

Nonlinear discrete-time control of systems with a Naimark–Sacker bifurcation

In this paper we study the problem of the stabilization of nonlinear control system with one complex uncontrollable mode. We find normal forms and quadratic invariants; then we compute center manifolds, and use bifurcation theory to synthesize quadratic controllers.

Quadratic Stabilization of Nonlinear Control Systems with a Double-Zero Bifurcation

In this paper, we determine the normal forms for control systems with a double-zero bifurcation. Based on the normal forms, we find invariants, stabilizability conditions and synthesize a quadratic stabilizing controller.

On the Stability of Nonlinear Control Systems with Uncertainty

In this paper, we study the stability problem of nonlinear dynamical control systems. We consider continuous-time dynamical systems whose nominal part is stable and whose perturbed part (uncertainties) is norm-bounded by a positive function. Under some conditions on the perturbation, by using Lyapunov techniques, we show that the system can be uniformly asymptotically stable by a continuous controller.

Stabilization of nonlinear systems by use of semidefinite Lyapunov functions

We give a sufficient condition for smooth stabilization of nonlinear control systems. This condition generalizes the well-known Jurdjevic-Quinn results in the sense that it uses nonnegative semidefinite Lyapunov functions rather than positive definite ones, and that it applies to nonaffine control systems. Stabilizing feedback is explicitly computed.

Solution of the problem of absolute stability of nonlinear controlled system

It is proved that the properties of a nonlinear system are at the interface of two theories - deterministic and stochastic. Hence, one has to consider a steady random process at the output of the system when a similar random process forms an input of the system. In this case one has a determining factor which is the symmetry. It is the symmetry that makes a dynamic system as a frequency amplifier which allows determing the angle of absolute stability.

Asymptotic Controllability and Exponential Stabilization of Nonlinear Control Systems at Singular Points

We discuss the relation between exponential stabilization and asymptotic controllability of nonlinear control systems with constrained control range at singular points. Using a discounted optimal control approach, we construct discrete feedback laws minimizing the Lyapunov exponent of the linearization. Thus we obtain an equivalence result between uniform exponential controllability and uniform exponential stabilizability by means of a discrete feedback law.

Smooth stabilization of nonlinear control systems

This paper provides the framework for a general approach for constructing smooth stabilizing feedback for smooth nonlinear systems when it is possible to construct such a feedback. Algorithms for determining the local smooth asymptotic stabilizability of two specific classes of smooth nonlinear control systems are developed. These two classes of systems are identified with systems whose linearizations possess either one zero eigenvalue or a pair of purely imaginary eigenvalues. The local stabilizability of any critical nonlinear system in either of these classes may be tested once the system's linearization is block diagonalized with respect to the critical and stable eigenvalues.