Invariance Conditions for Nonlinear Dynamical Systems

Recently, Horv\'ath, Song, and Terlaky [\emph{A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation}] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions

Robust positive invariance and ultimate boundedness of nonlinear systems

In this article the problem of characterizing sets, described by vector nonlinear inequalities of the form v(x) = w, as robustly positively invariant and targets of uniformly ultimate bounded nonlinear systems is investigated. The class of general parameter uncertain continuous-time dynamical systems affected by exogenous disturbances is considered. The approach is based on establishing an associated monotone nonlinear comparison system. A numerical example is presented to illustrate the approach

Invariance and contractiveness via comparison systems

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Further Results on the Linear Constrained Regulation Problem

International audienceThe problem of constrained regulation of linear systems around an equilibrium situated in the interior of a domain of attraction has been extensively investigated. In many engineering problems however, like obstacle avoidance problems, the regulation around an equilibrium situated on the boundary of the domain of attraction is necessary. For this kind of problems, the classical methods cannot be applied and design control methods are missing. Using invariant set techniques, the present paper proposes design methods for guaranteeing convergence to an equilibrium situated on the boundary of the feasible region, all by respecting the state constraints. A collision avoidance numerical example is presented for illustrating the theoretical results of the paper

Unconstrained and constrained stabilization of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established

Unconstrained and constrained stabilization of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established

Robust positive invariance and ultimate boundedness of nonlinear systems

In this article the problem of characterizing sets, described by vector nonlinear inequalities of the form v(x) = w, as robustly positively invariant and targets of uniformly ultimate bounded nonlinear systems is investigated. The class of general parameter uncertain continuous-time dynamical systems affected by exogenous disturbances is considered. The approach is based on establishing an associated monotone nonlinear comparison system. A numerical example is presented to illustrate the approach

Feedback Stabilization of Networked Control Systems

In this paper the stability analysis and control synthesis problems for networked control systems (NCS) with bounded transmission delays (constant and unknown or time-varying) are investigated. First, stability conditions for NCS described by ARMA models are established and a method for the determination of admissible delay range is developed. Then, a linear programming method for the design of linear state-feedback controllers guaranteeing the stability of the system for any delay belonging to a prespecified range is developed. Contrary to the usual approaches based on the use of quadratic Lyapunov functions, a polyhedral Lyapunov approach is adopted for both analysis and synthesis. A control synthesis numerical example is given to illustrate the reduction of conservatism of the tolerable delay range when compared to former results

Constrained stabilization of a two-inputs Buck-Boost DC/DC converter using a set theoretic method

This paper considers the problem of constrained stabilization of a two-input buck-boost DC/DC converter by linear state-feedback. It is demonstrated that, via an appropriate change of coordinates, a recent synthesis technique for constrained bilinear discrete-time systems can be applied to an averaged nonlinear model of the converter. Moreover, it is proven that the synthesis method yields a polyhedral constrained control invariant set for the converter model in the original coordinate system. The synthesis algorithm requires solving a single linear program off-line. An extensive simulation case study along with a preliminary, successful real-time experiment, demonstrate the effectiveness of the proposed methodology