Positivity-based robust control of flexible systems

Positivity-based control design for flexible structures provides closed-loop stability regardless of parametric variations and unmodelled dynamics. The present framework requires the plant to be square and the actuators colocated with rate sensors. These constraints severely limit achievable performance in control systems using the Positivity approach. In this thesis, a method is developed to eliminate the colocation constraint. A methodology to design robust control law for flexible structures is developed. The proposed approach eliminates the requirement for colocated actuator/sensor pairs thus enables achieving better performance characteristics. A dynamic embedding is constructed to render a nominal plant positive real. Noncolocated and nonsquare plants are allowed. A class of embeddings applicable to flexible structures is parametrized and several methods to compute the embeddings are developed. This technique is extended to discrete-time systems. The issue of maintaining positive real under plant perturbations was studied. The positive real robustness problem is shown to be equivalent to a stability robustness problem. An algorithm is developed to augment the embeddings to guarantee robust positive realness for all plant perturbations. A robust control law is synthesized based on positivity to meet performance requirements. The design process takes advantages of an efficient algorithm using the Linear Matrix Inequalities (LMI). The resulting controller is a multivariable extension of the single-loop Proportional-Integral-Derivative (PID) controller. The Draper Tetrahedral Truss structure and the Two-Mass Spring-Damper system are used to demonstrate the proposed approach. It is shown that the methodology proposed here yields better performance than the conventional colocation design while preserving the robustness guaranteed by the positivity approach.