Local Stabilization Of Nonlinear Systems Research Articles

Robust Local Stabilization of Nonlinear Systems With Controller-Dependent Norm Bounds: A Convex Approach With Input-Output Sampling

This letter presents a framework for synthesizing a robust full-state feedback controller for systems with unknown nonlinearities. Our approach characterizes input-output behavior of the nonlinearities in terms of local norm bounds using available sampled data corresponding to a known region about an equilibrium point. A challenge in this approach is that if the nonlinearities have explicit dependence on the control inputs, an a priori selection of the control input sampling region is required to determine the local norm bounds. This leads to a “chicken and egg” problem, where the local norm bounds are required for controller synthesis, but the region of control inputs needed to be characterized cannot be known prior to synthesis of the controller. To tackle this issue, we constrain the closed-loop control inputs within the sampling region while synthesizing the controller. As the resulting synthesis problem is non-convex, three semi-definite programs (SDPs) are obtained through convex relaxations of the main problem, and an iterative algorithm is constructed using these SDPs for control synthesis. Two numerical examples are included to demonstrate the effectiveness of the proposed algorithm.

A Predictor Based Linearization Approach for Local Stabilization of Nonlinear Systems With Input Delay

In this paper, a predictor based linearization approach is presented to solve the local asymptotic stabilization (LAS) problem of time-invariant nonlinear systems with a large constant input delay. Under stabilizability and detectability of the linearized time-delay system, we prove that LAS is achievable by either predictor based state or output feedback controllers, which can be designed directly based on the linearization. The predictor based linearization technique developed in this research complements the work [1] and improves its result by removing the forward completeness condition and a number of technical requirements as well as by expanding the class of nonlinear systems with input delay to the general case. An example is given to demonstrate the application of the predictor based linearization technique.

Derivative feedback control of nonlinear systems with uncertain equilibrium states and actuator constraints

In this paper, we study the local stabilization of nonlinear systems with uncertain equilibrium states and in the presence of actuator constraints. We propose a derivative feedback control scheme to stabilize the nonlinear system, and to drive the system states to its true equilibrium state even when the location of such equilibrium is uncertain. Actuator constraints in the feedback control are also considered in this paper, and stability conditions are derived for the cases when the actuator output energy is bounded, and the actuator output is subject to saturation. Stability conditions are derived in the form of matrix inequalities for both cases of actuator constraints, and numerical methods are discussed to synthesize feasible control solutions. The effectiveness of the proposed method is illustrated by a numerical example, and experimentally demonstrated through a magnetic levitation test rig.

Local Stabilization of Nonlinear Systems Through the Reduction Model Approach

We study a general class of nonlinear systems with input delays of arbitrary size. We adapt the reduction model approach to prove local asymptotic stability of the closed loop input delayed systems, using feedbacks that may be nonlinear. Our Lyapunov-Krasovskii functionals make it possible to determine estimates of the basins of attraction for the closed loop systems.

On the Stability of Critical Point for Positive Systems and Its Applications to Biological Systems

The coexistence and extinction of species are important concepts for biological systems and can be distinguished by an investigation of stability. When determining local stability of nonlinear systems, Lyapunov indirect method based on the Jacobian linearization has been widely employed due to its simplicity. Despite such popularity, it is not applicable to singular systems whose Jacobian has at least one eigenvalue that is equal to zero. In such singular cases, an appropriate Lyapunov function should be sought to determine the stability of systems, which is rather difficult and quite involved. In this paper, we seek for a simple criterion to determine stability of the equilibrium that is located at the boundary of the positive orthant, when one of eigenvalues of the Jacobian is zero. The goal of the paper is to present a generalized condition for the equilibrium to attract all trajectories that starting from initial condition in the positive orthant and near the equilibrium. Unlike the Lyapunov direct method, the proposed method requires just a simple algebraic computation for checking the stability of the critical point. Our approach is applied to various biological systems to show the effectiveness of the proposed method.

A Constructive Approach to Local Stabilization of Nonlinear Systems by Dynamic Output Feedback

The note considers the problem of local stabilization of nonlinear systems by dynamic output feedback. A new concept, namely, local uniform observability of feedback control law, is introduced. The main result is that if a nonlinear system is Nth-order approximately stabilizable by a locally uniformly observable state feedback, then it is stabilizable by dynamic output feedback. Based on the approximate stability, a constructive method for designing dynamic compensators is presented. The design of the dynamic compensators is beyond the separation principle and can handle systems whose linearization might be uncontrollable and/or unobservable. An example of nonminimum phase nonlinear systems is presented to illustrate the utility of the results.

Decoupling normalizing transformations and local stabilization of nonlinear systems

A method of decoupling normalizing transformations has been developed. According to the method only the part of differential equations corresponding to the dynamic on a center manifold has to be modified by means of the normalizing transformations of a Poincare type. The existence of the normalizing transformation completely decoupling the stable dynamic from the center manifold dynamic has been proved. A numerical procedure for the calculation of asymptotic series for the decoupling normalizing transformation has been proposed. The developed method is especially important for the perturbation theory of center manifold and, in particular, for the local stabilization theory. In the paper some sufficient conditions for local stabilization have been given.

Approximation of decoupling normalizing transformations and local stabilization of nonlinear systems

A numerical procedure for the calculation of asymptotic series for the decoupling normalizing transformation has been proposed. The developed method has been applied to obtain some new sufficient conditions for local stabilization of nonlinear systems with noncontrollable linearizations.

On vanishing stability regions in nonlinear systems with high-gain feedback

When local stabilization of nonlinear systems is achieved by linear feedback, the resulting stability region may vanish as the feedback gains increase. This is demonstrated by examples in which neglected nonlinearities create an unstable limit cycle around an asymptotically stable equilibrium. As the feedback gains tend to infinity the unstable limit cycle shrinks to the equilibrium. If a feedback linearization design is applied, the same instability mechanism may occur when the nonlinearities are not precisely known.