Robust model predictive control under dynamic uncertainties : an integral quadratic constraints framework
Date
2025
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Abstract
In this thesis, we consider the problem of controlling and stabilizing uncertain systems such that constraints are satisfied at each point in time. Model uncertainties and disturbances occur in almost all real-world applications due to unpredictable external perturbations, measurement noise, modeling or linearization errors, uncertain time delays, and uncertain parameters. These uncertainties are often nonlinear, time-varying, and dynamic. The presence of uncertainties deteriorates performance and significantly complicates the satisfaction of constraints such as the avoidance of collisions, overheating, or safety shutdown protocols. This thesis presents a systematic framework to solve such challenging problems. In particular, we present a novel robust model predictive control (MPC) framework for linear systems subject to general dynamic uncertainties. To describe the influence of the uncertainties on the controlled system and the constraints, we leverage integral quadratic constraints (IQCs). IQCs provide a unified framework to represent a broad class of structured uncertainties, including linear time-invariant, time-varying, frequency-dependent dynamic, and even nonlinear elements.
The thesis contains two pivotal contributions. In the first part, a systematic algorithm is developed to design discrete-time robust controllers that optimize the worst case performance over all possible uncertainties. The performance measures we consider are the $\mathcal{H}_\infty$-norm, energy-to-peak gain, peak-to-peak gain, or a multiobjective mix of these performance criteria. The proposed algorithm iterates between an analysis step and a synthesis step, similar to the D-K-iteration in $\mu$-synthesis. In the analysis step, these worst-case performance criterion is analyzed for a fixed controller based on IQCs by optimizing the involved IQC multiplier. In the synthesis step, the controller parameters are optimized for a fixed IQC multiplier. Both steps can be executed by solving a convex semi-definite program. Furthermore, we provide guarantees that the algorithm is monotone and improves the performance in every iteration until convergence. The robustness of the resulting controller is demonstrated in numerical examples.
In the second part, we propose a novel robust output-feedback MPC framework based on IQCs. The method can deal with all uncertainties that can be described using IQCs and uses a tube-based approach, where the tube size is dynamic and grows or shrinks depending on the nominal excitation of the uncertainty. We show that this approach reduces the conservatism of the bounds significantly compared to static tube approaches. Furthermore, we provide rigorous guarantees for robust stability, constraint satisfaction, and recursive feasibility. Finally, we reduce the conservatism of the proposed MPC algorithm further by using state or output measurements to initialize the prediction at each time step.
Overall, this thesis bridges a gap between classical robust control and modern MPC methods by addressing constraint satisfaction under dynamic uncertainties in a unified framework.
The result is a theoretically sound and computationally viable solution for robust control of constrained and safety-critical systems with model uncertainties and disturbances.