Continuity and Monotonicity of the MPC Value Function with respect to Sampling Time and Prediction Horizon

The digital implementation of model predictive control (MPC) is fundamentally governed by two design parameters; sampling time and prediction horizon. Knowledge of the properties of the value function with respect to the parameters can be used for developing optimisation tools to find optimal system designs. In particular, these properties are continuity and monotonicity. This paper presents analytical results to reveal the smoothness properties of the MPC value function in open- and closed-loop for constrained linear systems. Continuity of the value function and its differentiability for a given number of prediction steps are proven mathematically and confirmed with numerical results. Non-monotonicity is shown from the ensuing numerical investigation. It is shown that increasing sampling rate and/or prediction horizon does not always lead to an improved closedloop performance, particularly at faster sampling rates

Analytical results for the multi-objective design of model-predictive control

In model-predictive control (MPC), achieving the best closed-loop performance under a given computational resource is the underlying design consideration. This paper analyzes the MPC design problem with control performance and required computational resource as competing design objectives. The proposed multi-objective design of MPC (MOD-MPC) approach extends current methods that treat control performance and the computational resource separately -- often with the latter as a fixed constraint -- which requires the implementation hardware to be known a priori. The proposed approach focuses on the tuning of structural MPC parameters, namely sampling time and prediction horizon length, to produce a set of optimal choices available to the practitioner. The posed design problem is then analyzed to reveal key properties, including smoothness of the design objectives and parameter bounds, and establish certain validated guarantees. Founded on these properties, necessary and sufficient conditions for an effective and efficient solver are presented, leading to a specialized multi-objective optimizer for the MOD-MPC being proposed. Finally, two real-world control problems are used to illustrate the results of the design approach and importance of the developed conditions for an effective solver of the MOD-MPC problem

Analysis of unconstrained nonlinear MPC schemes with time varying control horizon

For discrete time nonlinear systems satisfying an exponential or finite time controllability assumption, we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints. Based on our formula, we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop