Quick Updates for the Perturbed Static Output Feedback Control Problem in Linear Systems with Applications to Power Systems

Abstract

This paper introduces a method for efficiently updating a nominal stabilizing static output feedback (SOF) controller in perturbed linear systems. As operating points and state-space matrices change in dynamic systems, accommodating updates to the SOF controller are necessary. Traditional methods address such changes by re-solving for the updated SOF gain, which is often (i) computationally expensive due to the NP-hard nature of the problem or (ii) infeasible due to the limitations of its semidefinite programming relaxations. To overcome this, we leverage the concept of minimum destabilizing real perturbation (MDRP) to formulate a norm minimization problem that yields fast, reliable controller updates. This approach accommodates a variety of known perturbations, including abrupt changes, model inaccuracies, and equilibrium-dependent linearizations. We remark that the application of our proposed approach is limited to the class of SOF controllers in perturbed linear systems. We also introduce geometric metrics to quantify the proximity to instability and rigorously define stability-guaranteed regions. Extensive numerical simulations validate the efficiency and robustness of the proposed method. Moreover, such extensive numerical simulations corroborate that although we utilize a heuristic optimization method to compute the MDRP, it performs quite well in practice compared to an existing approximation method in the literature, namely the hybrid expansion-contraction (HEC) method. We demonstrate the results on the SOF control of multi-machine power networks with changing operating points, and demonstrate that the computed quick updates produce comparable solutions to the traditional SOF ones, while requiring orders of magnitude less computational time.