Robust Model Predictive Control via Scenario Optimization (1206.0038v1)

Abstract: This paper discusses a novel probabilistic approach for the design of robust model predictive control (MPC) laws for discrete-time linear systems affected by parametric uncertainty and additive disturbances. The proposed technique is based on the iterated solution, at each step, of a finite-horizon optimal control problem (FHOCP) that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances, followed by a specific command selection rule implemented in a receding horizon fashion. The scenario FHOCP is always convex, also when the uncertain parameters and disturbance belong to non-convex sets, and irrespective of how the model uncertainty influences the system's matrices. Moreover, the computational complexity of the proposed approach does not depend on the uncertainty/disturbance dimensions, and scales quadratically with the control horizon. The main result in this paper is related to the analysis of the closed loop system under receding-horizon implementation of the scenario FHOCP, and essentially states that the devised control law guarantees constraint satisfaction at each step with some a-priori assigned probability p, while the system's state reaches the target set either asymptotically, or in finite time with probability at least p. The proposed method may be a valid alternative when other existing techniques, either deterministic or stochastic, are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance-constrained optimization problem.

• The paper presents a scenario-based methodology that leverages random uncertainty samples to formulate a convex finite-horizon optimal control problem.
• It ensures probabilistic guarantees for constraint satisfaction and state convergence, reducing the conservativeness typical of worst-case designs.
• Numerical results confirm high reliability with a modest number of scenarios, enhancing computational efficiency for robust control design.

Analysis of "Robust Model Predictive Control via Scenario Optimization"

The paper presented by G.C. Calafiore and Lorenzo Fagiano offers a comprehensive discussion on a novel approach for designing robust Model Predictive Control (MPC) laws for discrete-time linear systems facing parametric uncertainties and additive disturbances. The methodology leverages scenario optimization, a probabilistic technique that enhances the robust MPC framework by incorporating random uncertainty scenarios into a finite-horizon optimal control problem (FHOCP).

MPC is an established technique in the control community, enabling explicit handling of constraints in control design processes. While robust MPC approaches traditionally ensure stability and constraint satisfaction against uncertainties, they often rely on deterministic assumptions requiring convex optimization problems, which may not always be feasible or computationally tractable. Existing methods tend to optimize for worst-case scenarios, adding to their conservativeness.

The approach in this paper shifts from deterministic algorithms to a randomized scenario-based one, allowing the consideration of randomly sampled uncertainty scenarios. The scenario FHOCP retains convexity regardless of the uncertainty or disturbance characteristics, thus mitigating the intractability issues found in non-convex uncertainty sets. This is significant as it maintains the problem's compatibility with conventional convex optimization solutions. Notably, the proposed method's computational complexity grows quadratically with the control horizon but remains insensitive to the dimensionality of uncertainties or disturbances, making it efficient and scalable.

A key contribution of the paper is the introduction of probabilistic guarantees in terms of robustness and constraint satisfaction. The control law, under a receding-horizon implementation, aims to guarantee constraint adherence with a predefined reliability level, termed probability p, thus balancing the robustness with computational efficiency. The method also ensures that the system state converges to a target set with probability at least p, either asymptotically or within finite time, providing a formal probabilistic characterization absent in traditional techniques.

Practically, the scenario-based method becomes a competitive alternative when deterministic MPC approaches fail due to non-convexity or excessive conservativeness. The technique's adaptability makes it suitable for complex systems where uncertainties influence the system matrices in non-linear or non-affine manners—a situation commonly encountered in advanced industrial applications.

The paper also highlights an intriguing aspect: the scenario optimization methodology does not require a perfectly characterized statistical profile of uncertainties. This flexibility is beneficial in applications with partially known disturbance distributions, broadening the scope and applicability of the approach to real-world problems.

Numerical simulations validate the effectiveness of the method, demonstrating high success probabilities even with a small number of scenarios. The paper offers probabilistic bounds on the number of scenarios required to achieve a desired reliability level, simplifying its adoption in practical MPC implementations by reducing computational burden while maintaining robustness.

In conclusion, the research bridges a critical gap in MPC design for systems with complex uncertainties. By marrying probabilistic techniques with robust control design, it offers a rigorous framework that enhances both the theoretical understanding and practical feasibility of robust MPC. Future work in this domain could explore extensions to more complex systems, such as those involving non-linear dynamics or those operating under tighter real-time constraints. The implications for AI and control systems are significant, suggesting a potential for more robust and adaptable autonomous systems deployed in dynamic and uncertain environments.