Robust Domain Of Attraction Research Articles

Data-driven robust stabilization with robust domain of attraction estimate for nonlinear discrete-time systems

Nonlinear robust control is pursued by overcoming the drawback of linear robust control that it ignores available information about existing nonlinearities and the resulting controllers may be too conservative, especially when the nonlinearities are significant. However, most existing nonlinear robust control approaches just consider the affine nonlinear nominal model and thereby ignore available information about existing non-affine nonlinearities. When the general nonlinear nominal model is considered, the robust domain of attraction (RDOA) of closed-loops requires extensive investigation because it is hard to achieve the global stabilization. In this paper, we propose a new nonlinear robust control method based on Lyapunov function to stabilize a discrete-time uncertain system and to estimate the RDOA of closed-loops. First, a sufficient condition for robust stabilization of all plants in a plant set and estimation of the RDOA of all closed-loops is proposed. Then, to tackle the non-affine nonlinearities, a data-driven method of estimating the robust negative-definite domains (RNDD) is presented, and based on it the estimation of the RDOA of closed-loops and the resulting controller design are also given.

Data-Driven Robust Stabilization with Robust DOA Enlargement for Nonlinear Systems

This paper present a method based on simulation data to optimize Lyapunov functions to stabilize nonlinear systems such that an estimation of the domain of attraction (DOA) is maximized. For non-affine nonlinear system, our previous work proposes an approach to estimate robust closed-loop DOA for uncertain nonlinear systems by sampling the state- and input-space. However, the main drawback is that the Lyapunov function is given and does not consider the problem of finding a good Lyapunov function to enlarge the estimate of the robust closed-loop DOA. The motivation of this paper is to enlarge the estimate of the closed-loop DOA in order to reduce conservatism of the DOA estimate. To achieve this goal, a solvable optimization problem is formulated to use sum-of-squares techniques to evaluate the cost for a given Lyapunov function and then optimizing over Lyapunov functions via existing meta-heuristic optimization methods. The effectiveness of proposed method is verified by numerical results.

On the domain of attraction and local stabilization of nonlinear parameter‐varying systems

SummaryIn this paper, the domain of attraction (DA) and the local stabilization problem are investigated under the nonlinear parameter‐varying (NPV) framework. Specifically, to take into account the time‐varying parameter, we generalize the definition of robust DA (RDA) and propose the concept of parameter‐dependent DA (PDA) for NPV systems, together with the sum‐of‐squares (SOS) conditions for their estimations. Differently from the existing DA‐related works, the theoretical results in this paper can be applied to a large class of nonlinear polynomial systems including the time‐invariant, the parameter‐dependent, and the uncertain ones. Moreover, the commonly used iterative/coordinate‐wise search is avoided because we eliminate the bilinear product terms between the Lyapunov functional and the SOS multipliers. We also consider the local stabilization problem of NPV systems, in which the RDA can be specified as a control performance index. Finally, several examples are given for illustration purposes.

Rational Lyapunov functions for estimating and controlling the robust domain of attraction

This paper addresses the estimation and control of the robust domain of attraction (RDA) of equilibrium points through rational Lyapunov functions (LFs) and sum of squares (SOS) techniques. Specifically, continuous-time uncertain polynomial systems are considered, where the uncertainty is represented by a vector that affects polynomially the system and is constrained into a semialgebraic set. The estimation problem consists of computing the largest estimate of the RDA (LERDA) provided by a given rational LF. The control problem consists of computing a polynomial static output controller of given degree for maximizing such a LERDA. In particular, the paper shows that the computation of the best lower bound of the LERDA for chosen degrees of the SOS polynomials, which requires the solution of a nonconvex optimization problem with bilinear matrix inequalities (BMIs), can be reformulated as a quasi-convex optimization problem under some conditions. Moreover, the paper provides a necessary and sufficient condition for establishing tightness of this lower bound. Lastly, the paper discusses the search for optimal rational LFs using the proposed strategy.

Enlarging the guaranteed region of attraction in nonlinear systems with bounded parametric uncertainty

A novel approach to enlarge the guaranteed region of attraction in nonlinear systems with bounded parametric uncertainties based on the design of a nonlinear controller is proposed. The robust domain of attraction (RDA) is estimated using the parameter-dependent quadratic Lyapunov function and enlarged by the optimal controlling parameters. The problem of extending the RDA is indicated in a form of three-layer optimization problem. Some examples illustrate the efficiency of the proposed strategy in enlarging RDA.

Estimating the robust domain of attraction and directional enlargement of attraction domain via Markov models

SUMMARYThis paper proposes a new approach for estimating the robust domain of attraction (RDA) and directional enlargement of the DA for dynamical systems. The proposed method analyzes stability of dynamical systems by Markov modeling and employs invariant measure as the stability indicator. Markov chains analysis focuses on asymptotic behaviors of systems and ignores the transient ones. The proposed method expresses the problem of estimating RDA and directional enlargement of DA as an infinite dimensional linear problem. The resulting linear problem is converted to a finite dimensional optimization problem using approximated Markov transition function. As a novel application, the directional enlargement of DA is used in order to increase the critical clearing time of power systems. The efficiency of proposed methods is shown via simulations. Copyright © 2012 John Wiley & Sons, Ltd.

Estimating the domain of attraction for uncertain polynomial systems

Estimating the Domain of Attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been proposed for the case of known polynomial systems allowing one to find the Largest Estimate of the DA (LEDA) for a given Lyapunov Function (LF). However, the problem of estimating the Robust DA (RDA), that is the DA guaranteed for all possible uncertainties in an uncertain system, it is still an unsolved problem. In this paper, LMI methods are proposed for estimating the RDA in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope. Specifically, the issue of computing the Robust LEDA (RLEDA), that is the intersection of all LEDAs, is considered for common and parameter-dependent LFs, providing constant and parameter-dependent lower bounds. The computation of approximations with simple shape of the RLEDA in the case of parameter-dependent LFs is also discussed.

The estimation of a robust domain of attraction using Gershgorin theorem

The estimation of a robust domain of attraction using Gershgorin theorem