State feedback control for structured descriptor systems: a graph theoretic approach (MS)

Abstract

Many physical systems, such as electrical networks, micro-grids, space vehicles and constrained mechanical systems are often modeled appropriately by combination of differential and algebraic equations and the resulting model is known as Differential and Algebraic (DAE) model. The dynamic behavior of energy storage elements in the system is described using differential equations whereas the algebraic equations arise to satisfy conservation laws and boundary conditions. Such systems are known as Descriptor Systems. Often in practical applications, it is observed that the system matrices in these DAE models capture some inherent structure,that is, the entries in the system matrices are either a fixed zero/one or a free parameter and such Structured Descriptor Systems can naturally be represented by directed graphs. Graph theoretic approach is useful since it is possible to study the structural properties of the system without depending on the numerical parameters. State feedback control or pole placement has been widely used in industries since several decades. With the help of state feedback control, the dynamics of a controllable plant can be modified by assigning the closed loop poles at arbitrary locations of the complex plane. Since state feedback controllers are more reliable and less complex hence, it is easy to implement them in many practical applications. In this thesis, the problem of designing a static state feedback control for a LTI structured descriptor systems is considered. As only the structure of system is taken into consideration, the designed controller is robust to parametric perturbations. The preliminaries on descriptor systems and structured systems are discussed. The digraph representation for open loop and closed loop structured descriptor systems is defined and thereafter the problem is formulated. Corresponding to the closed loop system, a square matrix is defined and a result is proposed to compute the coefficients of the closed loop characteristic polynomial using graphs. The adjoint corresponding to the open loop system matrix is computed using graphs. A relation between the adjoint of open loop system matrix and the input to the system is derived which is used to compute the feedback gain vector. A graph theoretic sufficient condition is proposed which is based on the existence of spanning cycle family in the resulting digraph. The effectiveness of the proposed approach is verified by taking numerical example.