Éva Gyurkovics

Stabilization of sampled-data nonlinear systems by receding horizon control via discrete-time approximations ☆

Abstract Results on stabilizing receding horizon control of sampled-data nonlinear systems via their approximate discrete-time models are presented. The proposed receding horizon control is based on the solution of Bolza-type optimal control problems for the parametrized family of approximate discrete-time models. This paper investigates both situations when the sampling period T is fixed and the integration parameter h used in obtaining approximate model can be chosen arbitrarily small, and when these two parameters coincide but they can be adjusted arbitrary. Sufficient conditions are established which guarantee that the controller that renders the origin to be asymptotically stable for the approximate model also stabilizes the exact discrete-time model for sufficiently small integration and/or sampling parameters.

A note on Wirtinger-type integral inequalities for time-delay systems

Different integral inequalities play important role when problems of stability analysis and controller synthesis for time-delay systems are considered. The connection of Jensen’s inequality and extended Jensen’s inequality is well understood now. To reduce the conservativeness introduced by the application of Jensen’s inequality, several versions of the Wirtinger integral inequality have been published recently. This note presents a comparison between some of these inequalities.

Sampled-Data Model Predictive Control for Nonlinear Time-Varying Systems: Stability and Robustness

We describe here a sampled-data Model Predictive Control framework that uses continuous-time models but the sampling of the actual state of the plant as well as the computation of the control laws, are carried out at discrete instants of time. This framework can address a very large class of systems, nonlinear, time-varying, and nonholonomic.

Quadratic stabilisation with H ∞ -norm bound of non-linear discrete-time uncertain systems with bounded control

The paper addresses the problem of quadratic stabilisability with H∞-norm bound of uncertain discrete-time control-affine systems by norm-bounded controls. Both structured parameter uncertainties and unstructured exogenous disturbances are taken into account. The given definition of quadratic stabilisability is a generalisation of that used for linear systems so far. A necessary condition of the stabilisability is formulated. A state feedback control satisfying an a priori constraint is proposed for the solution of the formulated H∞ problem. The proposed method may be applicable even in such cases when the linearisation technique cannot be used.

Application of a multiplier method to uncertain Lur’e-like systems

Abstract This paper investigates the conditions under which an abstract matrix multiplier method can be applied to determine guaranteeing cost controls for systems containing nonlinear/uncertain elements via linear matrix inequalities (LMIs). Quadratically constrained uncertainties and nonlinearities are considered, which comprehend the cases of norm-bounded, positive-real and sector-bounded uncertainties/nonlinearities. Both the discrete-time and the continuous-time cases are discussed. Necessary and sufficient conditions are formulated in the case of unstructured uncertainty. The conditions are sufficient in the structured case. The cost guaranteeing controls can be determined by solving LMIs. Numerical examples illustrate the effectiveness of the proposed method.

Stabilization of Sampled-Data Nonlinear Systems by Receding Horizon Control via Discrete-Time Approximations

Abstract Results on stabilizing receding horizon control of sampled-data nonlinear systems via their approximate discrete-time models are presented. The proposed receding horizon control is based on the solution of Bolza-type optimal control problems for the parametrized family of approximate discrete-time models. This paper investigates the situation when the sampling period and the integration parameter used in obtaining approximate model coincide and can be chosen arbitrarily small. Sufficient conditions are established which guarantee that the controller that renders the origin to be asymptotically stable for the approximate model also stabilizes the exact discrete-time model for sufficiently small sampling parameters.

Multiple summation inequalities and their application to stability analysis of discrete-time delay systems

Abstract This paper is devoted to stability analysis of discrete-time delay systems based on a set of Lyapunov–Krasovskii functionals. New multiple summation inequalities are derived that involve the famous discrete Jensen׳s and Wirtinger׳s inequalities, as well as the recently presented inequalities for single and double summation in [16] . The present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of LMIs establishing a rigorous theoretical basis for the comparison of conservatism of the investigated methods. Numerical examples illustrate the efficiency of the method.

Conditions for MPC Based Stabilization of Sampled-Data Nonlinear Systems Via Discrete-Time Approximations

This paper is devoted to the stabilization problem of nonlinear continuous- time systems with piecewise constant control functions. The controller is to be computed by the receding horizon control method based on discrete-time approximate models. Multi-rate — multistep control is considered and both measurement and computational delays are allowed. It is shown that the same family of controllers that stabilizes the approximate discrete-time model also practically stabilizes the exact discrete-time model of the plant. The conditions are formulated in terms of the original continuoustime models and the design parameters so that they should be veri.able in advance.

Guaranteeing cost strategies for infinite horizon difference games with uncertain dynamics

The paper addresses the problem of determination of guaranteeing cost control strategies for discrete-time two-persons zero-sum non-linear games over in.nite horizon. In the proposed approach the objective functional is appropriately modi.ed in order to cope with the uncertainties, and a su.cient condition is given to ensure that a given state-feedback is a guaranteeing cost control. The results are applied also for linear systems with uncertainties of linear fractional structure to derive guaranteeing cost strategies for both players. It is shown that this approach can successfully be applied in this case, when the method of introducing .ctitious games as proposed in previous papers may come up against a di.culty. The results are illustrated by numerical examples.

Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities

This paper is devoted to stability analysis of continuous-time delay systems with interval time-varying delays having known bounds on the delay derivatives. A parameterized family of Lyapunov–Krasovskii functionals involving multiple integral terms is introduced, and novel multiple integral inequalities are utilized to derive sufficient stability condition for systems with time-varying delays. The efficiency of the proposed method is illustrated by numerical examples.