A Data-Driven Frequency-Domain Approach for Robust Controller Design via Convex Optimization

The objective of this dissertation is to develop data-driven frequency-domain methods for designing robust controllers through the use of convex optimization algorithms. Many of today's industrial processes are becoming more complex, and modeling accurate physical models for these plants using first principles may be impossible. Albeit a model may be available; however, such a model may be too complex to consider for an appropriate controller design. With the increased developments in the computing world, large amounts of measured data can be easily collected and stored for processing purposes. Data can also be collected and used in an on-line fashion. Thus it would be very sensible to make full use of this data for controller design, performance evaluation, and stability analysis. The design methods imposed in this work ensure that the dynamics of a system are captured in an experiment and avoids the problem of unmodeled dynamics associated with parametric models. The devised methods consider robust designs for both linear-time-invariant (LTI) single-input-single-output (SISO) systems and certain classes of nonlinear systems. In this dissertation, a data-driven approach using the frequency response function of a system is proposed for designing robust controllers with $\mathcal{H}_{\infty}$ performance. Necessary and sufficient conditions are derived for obtaining $\mathcal{H}_{\infty}$ performance while guaranteeing the closed-loop stability of a system. A convex optimization algorithm is formulated to obtain the controller parameters which ensure system robustness; the controller is robust with respect to the frequency-dependent uncertainties of the frequency response function. For a certain class of nonlinearities, the proposed method can be used to obtain a best-linear-approximation with an associated frequency-dependent uncertainty to guarantee the stability and performance for the underlying linear system that is subject to nonlinear distortions. The controller for this design scheme is presented as a ratio of two linearly-parameterized transfer functions; in this manner, the numerator and denominator of a controller are simultaneously optimized. With this construction, it can be shown that as the controller order increases, the solution to the convex problem converges to the global optimal solution of the $\mathcal{H}_{\infty}$ problem. This method is then extended to the 2-degree-of-freedom discrete-time controller where the necessary and sufficient conditions are imposed for multiple weighted sensitivity functions. The concepts behind these design methods are then used to devise necessary and sufficient conditions for ensuring the closed-loop stability of systems with sector-bounded nonlinearities. The conditions are simple convex feasibility constraints which can be used to stabilize systems with multi-model uncertainty. Additionally, a method is proposed for obtaining $\mathcal{H}_{\infty}$ performance for systems with uncertain gains within these sectors. By convexifying the $\mathcal{H}_{\infty}$ problem, the global optimal solution to an approximate problem is obtained. For low-order controllers, the solution to this approximate problem may lead to solutions far from the optimal solution of the true $\mathcal{H}_{\infty}$ problem. Thus two methods are proposed to address this issue for low-order controllers. In one method, a non-convex problem is formulated which optimizes the basis function parameters of a controller while guaranteeing the stability of the closed-loop systems. In another method, a set of convex problems are solved in an iterative fashion to obtain the desired performance (which also guarantees the closed-loop stability of the system). With both methods, the local solution to the $\mathcal{H}_{\infty}$ problem for fixed-structure controllers is obtained. However, the convex problem is computationally tractable and can also consider $\mathcal{H}_2$ performance. The effectiveness of the proposed method(s) is illustrated by considering several case studies that require robust controllers for achieving the desired performance. The main applicative work in this dissertation is with respect to a power converter control system at the European Organization for Nuclear Research (CERN) (which is used to control the current in a magnet to produce the desired field in controlling particle trajectories in particle accelerators). The proposed design methods are implemented in order to satisfy the challenging performance specifications set by the application while guaranteeing the system stability and robustness using data-driven design strategies.