Nonlinear Model Predictive Control Reduction Strategies for Real-time Optimal Control

Abstract

This thesis presents a variety of strategies to accelerate the turnaround times (TATs) of nonlinear and hybrid model predictive controllers (MPCs). These strategies are unified by the themes of symbolic computing, nonlinear model reduction and automotive control.

The first contribution of this thesis is a new MPC problem formulation, called symbolic single shooting (symSS), that leverages the power of symbolic computing to generate an optimization problem of minimal dimension. This formulation is counter to the recent trend of introducing and exploiting sparsity of the MPC optimization problem for tailored solvers to exploit. We make use of this formulation widely in this thesis.

The second contribution of this thesis is a novel application of proper orthogonal decomposition (POD) to MPC. In this strategy we construct a dimensionally-reduced optimization problem by restricting the problem Lagrangian to a subspace. This subspace is found by running simulations offline from which we extract the important solution features. Using this restricted Lagrangian we are able to reduce the problem dimension dramatically, thus simplifying the linear solve. This leads to TAT accelerations of more than two times with minimal controller degradation.

The third contribution of this thesis is an informed move blocking strategy. This strategy exploits the features extracted in the restricted Lagrangian subspace to derive a sequence of increasingly blocked move blocking strategies. These move blocking strategies can then be used to reduce the dimension of the optimization problem in a sparse manner, leading to even greater acceleration of the controller TAT .

The fourth contribution of this thesis is a new quasi-Newton method for MPC. This method utilizes ideas similar to singular perturbation-based model reduction to truncate the expression for the problem Hessian at the symbolic level. For nonlinear systems with a modest Lipschitz constant, we can identify the timestep as a `small' parameter about which we can do a perturbative expansion of the Lagrangian and its derivatives. Truncating to first order in the timestep, we are able to find a good approximation of the Hessian leading to TAT acceleration.

The fifth contribution of this thesis is controller integration strategy based on nested MPCs. Using the symSS formulation we can construct an explicit model of a controlled plant that includes the full model as well as the MPC's action. This form of the controlled plant model allows us to generate exact derivatives so that fast solvers can be used for real time application. We focus here on the problem of planning and motion control integration for autonomous vehicles but this strategy can be extended for other problems that require accurate models of a controlled plant.

The sixth contribution of this thesis is a strategy to handle integer controls in MPC based on a few reasonable assumptions: our predictions over the horizon are almost perfect and the future is inevitable. These assumptions enforce a degree of continuity, in the integer controls, between solutions over different timesteps that allow us to mitigate chatter and enforce a hard upper bound on solution complexity. This strategy constrains the integer solution of one timestep to be related to that of the previous timestep. Our results show that this strategy provides acceptable control performance while achieving TATs that are orders of magnitude smaller than those for conventional MINLP-based methods, thereby opening the door to new real-time applications of hybrid MPC.