Numerical tools for the efficient steady-state optimization of discrete-time Lur'e-type control systems

Many problems arising in system identification, controller design, and model reduction can be formulated as nonlinear optimization problems, where the cost function depends on steady-state system responses. Solving such problems often requires the computation of numerous steady-state system responses and their gradients with respect to parameter changes, posing significant computational challenges. In this article, we focus on discrete-time nonlinear multiple-input, multiple-output Lur’e-type systems that exhibit unique, bounded, and globally exponentially stable steady-state solutions. Our main contribution is a computationally efficient mixed-time-frequency (MTF) algorithm for the computation of the steady-state response of Lur’e systems with guaranteed numerical convergence. Additionally, we propose a method for efficiently computing gradients of the cost function with respect to parameter changes, again leveraging the proposed MTF algorithm. The proposed tools offer substantial computational advantages as the underlying nonlinear optimization problems require extensive steady-state response calculations and gradient evaluations. We validate our approach with a benchmark system identification problem, demonstrating a 95% reduction in computation time.