Complexity Reduction in Explicit Model Predictive Control
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The thesis is focused on low complexity constrained control systems. Model Predictive Control (MPC) has been huge success for constrained control due to its ability to handle input, state and output constraints. It is not only applicable to single-input single-output (SISO) systems but it also handles multiple-input multiple-output systems (MIMO). However, due to its on-line computational complexity MPC is limited to the systems with slow dynamics and low sampling rates. Furthermore, high computational complexity makes the code difficult to verify, thus it is also limited to applications that are not safety critical. Explicit Model Predictive Control overcomes these limitations of the Model Predictive Control by formulating the problem as a multi-parametric quadratic programming (MPQP) problem. Thus, the optimization problem is solved off-line and the control law is given as a piecewise affine (PWA) function of a present state. Thereby, the online problem is transformed into evaluating simple look up table. However, as the problem size increases, the explicit solution in terms of the number of regions becomes more complex. In consequence, the off-line calculations become excessively demanding, and the memory requirements to store explicit solution increase significantly. An alternative approach to formulate explicit MPC design is to use a controlled contractive set. In that case, the complexity of the explicit solution depends on the complexity of the controlled contractive set. This thesis focuses on developing the low complexity constrained control based on low complexity controlled contractive sets. It proposes a novel optimization based approach to formulate a polytopic controlled contractive set. The formulated problem is a highly non-convex problem. In order to solve the problem, particle swarm optimization (PSO) is used. The formulation is transformed into an unconstrained problem by appending suitable penalty functions as PSO is applicable only for unconstrained systems. Different solutions obtained from PSO are merged together to obtain a larger contractive set which is then simplified by a number of techniques. The method used to obtain a low complexity contractive set is applied on a motion cueing algorithm (MCA) to illustrate the extent of complexity reduction by the proposed method. MCA is the algorithm which computes a control law to reproduce the motion of a real car in a driving simulator. A polytopic set (with the origin in its interior) can be described as the level set of a Piecewise Linear (PWL) function. Thus, the use of polytopic contractive sets corresponds to use of PWL Lyapunov functions. An alternative would be to find controlled contractive sets described by level sets of higher order functions, corresponding to the use of higher order Lyapunov functions. However, the size of the controlled contractive set is limited due to quadratic nature of the Lyapunov function and the corresponding linear control law in case of already developed ellipsoidal sets. The thesis proposes a new approach to obtain a larger contractive set by allowing a flexible degree of the Lyapunov function and the control law. It formulates an approximate problem which is solved by sum-of-squares programming. The method results in significantly larger contractive sets as illustrated in the thesis.