Stability Analysis of Piecewise Affine Systems with Multi-model Model Predictive Control
For control engineers, this provides a less conservative stability analysis framework for multi-model MPC with PWA models, addressing a known bottleneck in ensuring closed-loop stability under uncertainty.
This work proposes an input-to-output stability analysis for closed-loop systems using piecewise affine models and multi-model linear MPC under unstructured uncertainty, employing integral quadratic constraints to assess robustness. The method reduces conservatism via static multipliers and is validated on two engineering case studies.
Constrained model predictive control (MPC) is a widely used control strategy, which employs moving horizon-based on-line optimisation to compute the optimum path of the manipulated variables. Nonlinear MPC can utilize detailed models but it is computationally expensive; on the other hand linear MPC may not be adequate. Piecewise affine (PWA) models can describe the underlying nonlinear dynamics more accurately, therefore they can provide a viable trade-off through their use in multi-model linear MPC configurations, which avoid integer programming. However, such schemes may introduce uncertainty affecting the closed loop stability. In this work, we propose an input to output stability analysis for closed loop systems, consisting of PWA models, where an observer and multi-model linear MPC are applied together, under unstructured uncertainty. Integral quadratic constraints (IQCs) are employed to assess the robustness of MPC under uncertainty. We create a model pool, by performing linearisation on selected transient points. All the possible uncertainties and nonlinearities (including the controller) can be introduced in the framework, assuming that they admit the appropriate IQCs, whilst the dissipation inequality can provide necessary conditions incorporating IQCs. We demonstrate the existence of static multipliers, which can reduce the conservatism of the stability analysis significantly. The proposed methodology is demonstrated through two engineering case studies.