Chance-Constrained Model Predictive Control based on Deterministic Density Approximation and Homotopy Continuation

• Applicant: Professor Dr.-Ing. Uwe D. Hanebeck
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 267437392

In Model Predictive Control (MPC), control inputs are obtained by online solving an optimization problem. In order to be applicable in real-world applications, MPC has to provide closed-loop stability and robustness to endogenous and exogenous disturbances. For this purpose, constraints are introduced into the MPC problem. In standard MPC, the constraints are hard, because the disturbances are assumed to be bounded, i.e., they are contained within some closed set. Thus, constraint satisfaction is guaranteed for the worst case scenario ignoring probably available statistical properties of the disturbances. The desire to incorporate statistical properties of disturbances in order to achieve better performance and the additional desire to consider unbounded disturbances modeled as stochastic processes led to the development of chance-constrained MPC. In chance-constrained MPC, hard deterministic constraints are replaced by soft constraints that require that the constraints are satisfied with a specified probability. Chance-constrained MPC can be applied in chemical engineering, robotically-assisted surgery, tracking control, energy-efficient temperature control, etc.The main challenge of chance-constrained MPC is the tractability of the MPC solution process. This process consists of an optimization procedure that generally has to be evaluated numerically using iterative methods. For this purpose, in order to check the constraints at each iteration step, multivariate integrals of probability densities have to be computed. State-of-the-art methods therefore approximate these densities, either by means of conservative deterministic approximations or stochastic sample-based approximations. Both these approaches have their advantages and disadvantages. Conservative deterministic approximations yield low computational burden but they do not find optimal control policies due to their conservatism. Sample-based approximation methods on the other hand are far less conservative but they are not applicable in real-time applications due to their computational burden.In this proposal, we investigate a new MPC approach that combines the advantages of the conservative approximation methods and the stochastic sampling methods. For this purpose, we approximate the occurring densities using a deterministic sample-based method. This approximation method allows to tune the conservatism level by adapting the number of samples. In order to solve the MPC optimization problem, we will apply homotopy continuation methods. These methods first solve the unconstrained control problem for which analytical solutions are available if process and measurement dynamics are linear. Then, the constraints are introduced progressively into the control problem and search algorithms can be applied. We expect the proposed method to be computationally faster than state-of-the-art methods based on stochastic sampling yielding optimization results with the same quality.

• DFG Programme: Research Grants