Tibor Takács

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.

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.

A remark on abstract multiplier conditions for robustness problems

Abstract This paper presents a generalization of recent results of Acikmese and Corless [B. Acikmese, M. Corless, Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions, Systems Control Letters 57 (2008) 78–94] concerning multiplier conditions of quadratic stability of uncertain/nonlinear systems. Abstract full block S-procedure results are formulated, extending the results of Scherer [C.W. Scherer, LPV control and full block multipliers, Automatica 37 (2001) 361–375] for such cases when the sets representing the uncertainties do not necessarily have subspace structure. The main contribution of the present work is the investigation of the conditions under which the results of Scherer and Acikmese and Corless can be formulated in a unified framework.

Budget balancing in a two-dimensional macroeconomic model

A two-person nonlinear dynamic game is presented to model the governments strategy to decrease the budget deficit, where Player 1 is the government using fiscal control and Player 2 represents the private sector. In our macroeconomic model the growth rate of the labour force is not known, but its lower and upper bounds are given a priori. This means that the system is uncertain, which makes the determination of an optimal solution (in a Nash, Stackelberg, etc. sense) impossible. Therefore, only a guaranteeing cost control is determined for Player 1. It is shown that the balancing by a guaranteeing cost control is possible even in the most unfavourable situation, when the governmental debt is higher and the volume of fixed capital stock is lower than the equilibrium value.