Model predictive control (MPC) is among the most successful approaches for process control and has become a
de facto standard across the process industries. However, there remain some applications for which MPC is impractical due to the demand that an optimization problem is solved at every time step. For systems with high-dimensional state spaces or highly nonlinear dynamics, solving this optimization problem is prohibitive. “Explicit†MPC eliminates this demand by using an explicit formulation of a control law that is approximately equivalent to the typical online MPC controller, either by finding an analytical solution to the optimization problem (only available in some rare circumstances), or by interpolating via a “lookup table†of precomputed solutions (only valid for limited operating conditions, and unfeasible for high-dimensional problems). In this work, we present a new explicit MPC formulation using recent advances in data mining, and especially manifold learning, to build effective interpolation functions between the system states, process outputs, and the MPC-determined control policies. Because all variables of interest (states, outputs, control policies) lie on the same low dimensional embedding space, we can determine any one of these variables from either of the other two. We use the diffusion maps algorithm to discover such low dimensional embeddings, and then develop functions to interpolate between the embedding space, system states, process outputs, and control policies. Following this procedure, we can substantially expand the class of systems for which explicit MPC methods are practical.
First, we will provide a background on manifold learning, and particularly the diffusion maps algorithm. Then, we will demonstrate how to use diffusion maps to uncover the embedding space on which lies all the variables of the closed loop MPC system. Next, we will use geometric harmonics to construct interpolation functions between the different variables of interest, and show that these functions can be used online as feedback control laws. To demonstrate our approach, we will present three example systems: (1) a nonisothermal CSTR, (2) a mechanistic model of an optogenetic circuit (an emerging technology from synthetic biology), and (3) a PDE with low-dimensional long-term dynamics. Finally, we will compare our approach using diffusion maps to other data-driven methods for explicit MPC, including neural networks and deep Gaussian processes.
