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Approximate Explicit Model Predictive Control


C.N. Jones

University of British Columbia

Constrained finite time optimal control problems can be expressed as mathematical programs parameterized by the current state of the system: the so-called multi-parametric programs. These problems have received a great deal of attention in the control community during the last few years because solving the parametric program is equivalent to synthesizing the optimal state-feedback controller. For many cases of interest, the resulting synthesized controllers are simple piecewise-affine functions, which enables receding horizon control to be used not only in slowly sampled systems requiring powerful computers but now also in high-speed embedded applications. The primary limitation of these optimal `explicit solutions' is that the complexity can grow quickly with problem size. In this talk I will introduce new methods to compute approximate explicit control laws that can trade-off time and space complexity against sub-optimality. The first class of methods generate control laws by approximating the complex epigraph of the optimal cost function with a simpler polyhedron. The proposed approaches are based on extensions to classic convex hull and vertex enumeration algorithms: beneath/beyond and double description. These methods are incremental in nature, meaning that for a specific piece of computational hardware a controller can be computed to meet either storage or hard real-time specifications. In the second half of the talk, I will discuss a new approach that allows an extra dimension of flexibility by trading sub-optimality for computation time and/or storage space. A fixed number of online optimization steps are executed after warm-starting from a feasible approximate piecewise affine control law computed using one of the aforementioned algorithms. The key is a preprocessing method that provides hard real-time, stability and performance guarantees for the proposed controller. This framework allows the design engineer to explicitly specify the available online computational power and storage resources, and then synthesizes an approximate controller that guarantees both system stability and feasibility while maximizing performance within the allocated fixed time and space complexities.


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