LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of 𝒟_R-regions

Abstract

This paper introduces an approach for the design of a state-feedback controller for LPV systems that achieves pole clustering in a union of DR-regions. The design conditions, obtained using a partial pole placement theorem, are eventually expressed in terms of linear matrix inequalities, which can be solved efficiently using available solvers. In addition, it is shown that the approach can be modified in a shifting sense, which means that the controller gain is computed such that different values of the varying parameters imply different regions of the complex plane where the closed-loop poles are situated, thus enabling online modification of the closed-loop performance. The effectiveness of the proposed method is demonstrated by means of simulations. KEYWORDSLinear parameter varying (LPV) systems, pole placement, union of regions, linear matrix inequalities (LMIs), state-feedback control.

On the State-Feedback Controller Design for Polynomial Linear Parameter-Varying Systems with Pole Placement within Linear Matrix Inequality Regions

“…The result, finally, corresponds to a state-feedback controller whose gains are presented as a polynomial function of the time-varying parameter allocating the closed-loop eigenvalues within an LMI region. Let us point out this contribution concerning [12], research close to that presented within this paper. The proposed method rendering the results shown throughout the present paper is related to polynomial LPV systems, while that contained in [12] is applied to affine LPV systems without considering time-varying parameters with high-order polynomial dependency.…”

“…Let us point out this contribution concerning [12], research close to that presented within this paper. The proposed method rendering the results shown throughout the present paper is related to polynomial LPV systems, while that contained in [12] is applied to affine LPV systems without considering time-varying parameters with high-order polynomial dependency. In other words, the proposal in [12] considers LPV systems with time-varying dependence in an affine form, computing results by solving its proposal at subsystems formed by the combination of the time-varying parameters' maximum and minimum values.…”

“…The previous can be viewed as the main contribution of LPV systems to control systems theory. There exist several representations with different characteristics in the context of LPV systems: descriptor LPV systems [7][8][9], a polytopic LPV representation which interpolates solutions for linear subsystems with their summation viewed as the LPV system's dynamical behavior [10][11][12][13][14], a quasi-LPV representation from which one can perceive the time-varying dependency as a part of the system state [15,16], and affine LPV systems [17,18]. Within the affine representation, in turn, if the time-varying parameter is defined in polynomial functions of two or greater degrees, a polynomial LPV (PLPV) system should be addressed [19][20][21][22].…”

“…The proposed method rendering the results shown throughout the present paper is related to polynomial LPV systems, while that contained in [12] is applied to affine LPV systems without considering time-varying parameters with high-order polynomial dependency. In other words, the proposal in [12] considers LPV systems with time-varying dependence in an affine form, computing results by solving its proposal at subsystems formed by the combination of the time-varying parameters' maximum and minimum values. In general, in the case of two time-varying parameters, a set of four subsystems is that from which the referred results are presented, with its solution associated with the overall LPV system.…”