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On the assignability of LTI systems with arbitrary control structures

ABSTRACT In this paper, the assignability of linear time-invariant (LTI) systems with respect to arbitrary control structures is addressed. It is well established that the closed-loop spectrum of an LTI system with an arbitrary control structure is confined to the set containing the fixed-modes of the system with respect to that control structure. However, the assignment of the closed-loop spectrum is not merely limited by the existence of fixed-modes in practical scenarios. The pole assignment may require excessive control effort or even become infeasible due to the presence of small perturbations in the system dynamics. To offer more insights in such more realistic scenarios, a continuous measure known as fixed-mode radius is developed. However, its evaluation is confronted with a highly non-convex optimization problem combined with the need for a combinatorial search. This study utilises properties of positive-definite cones and duality theory to formulate the assignability assessment as an optimization problem with linear matrix inequality (LMI) constraints. Based on the suggested formulation, three alternative methods are proposed to evaluate the distance to unassignability. The first two methods offer alternative non-iterative and convex programs. The third method proposes an iterative convex optimization while updating the binary variables based on the dual variables. All the proposed methods rely on convex optimization, do not involve gridding over the complex plane and circumvent the combinatorial nature of the problem by using properties of positive definite cones. Simulation results confirm the effectiveness of the proposed methods in the assessment of fixed-mode radius with respect to arbitrary control structures.