Semi-Definite Programming (SDP) Approach for Robust State PD Control Design for Descriptor Systems

Abstract

Various practical systems, such as robotic, aerospace, and electronic systems, appear to be singular by nature and cannot be represented by the explicit state-space formulation. Consequently, the descriptor state-space formulation, also known as the implicit state-space formulation, is used for representing such singular systems. This thesis addresses the stabilization problem for linear time-invariant (LTI) descriptor systems. Techniques are proposed for synthesizing state Proportional-Derivative (PD) controllers that ensure the stability of the resulting closed-loop system and further guarantee desired decay rates. The results are then extended to handle the robust stabilization problem for uncertain polytopic descriptor systems that have an affine dependence on the parametric uncertainties. Using Schur Complement-based lemmas, the synthesis problems are cast as semi-definite programs (SDPs) to be solved via linear matrix inequality (LMI) techniques or as nonlinear problems to be solved using the bisection method along with LMI techniques. Furthermore, the proposed formulation of the synthesis problems is augmented to account for energy optimization by minimizing the sought control gains. Illustrative examples are provided to demonstrate the efficacy of the proposed control strategies. The results show how the synthesized state PD controllers may allow for asymptotic stabilization, decay rate maximization, gain minimization, or a combination of the latter two objectives. The simulation results also demonstrate that the robust controller can handle system failures modeled as parametric uncertainties, unlike a controller designed for nominal parameter values.