H_∞-optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

This article investigates the (Formula presented.) -optimal estimation problem of a class of linear system with delays in states, disturbance input, and outputs. The estimator uses an extended Luenberger estimator format which estimates both the present and history states. The estimator is designed using an equivalent Partial Integral Equation (PIE) representation of the coupled nominal system. The advantage of the resulting PIE representation is compact and delay free—obviating the need for commonly used bounding technique such as integral inequalities which typically introduces conservatism into the resulting optimization problem. The (Formula presented.) -optimal estimator synthesis problem is then reformulated as a Linear Partial Inequality (LPI)—a form of convex optimization using operator variables and inequlities. Such LPI-based optimization problems can be solved using semidefinite programming via the PIETOOLS toolbox in Matlab. Compared with previous work, the proposed method simplifies the analysis and computation process and resulting in observers which are non-conservtism to 4 decimal places when compared with Pad (Formula presented.) -based ODE observer design methodologies. Numerical examples and simulation results are given to illustrate the effectiveness and scalability of the proposed approach.