Semidefinite Relaxations for Stochastic Optimal Control Policies

Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation have led to the discovery of a formulation of the value function as a linear Partial Differential Equation (PDE) for stochastic nonlinear systems with a mild constraint on their disturbances. This has yielded promising directions for research in the planning and control of nonlinear systems. This work proposes a new method obtaining approximate solutions to these linear stochastic optimal control (SOC) problems. A candidate polynomial with variable coefficients is proposed as the solution to the SOC problem. A Sum of Squares (SOS) relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in Portland, Oregon. 7 pages, colo

Multi-agent model predictive control for transport phenomena processes

Throughout the last decades, control systems theory has thrived, promoting new areas of development, especially for chemical and biological process engineering. Production processes are becoming more and more complex and researchers, academics and industry professionals dedicate more time in order to keep up-to-date with the increasing complexity and nonlinearity. Developing control architectures and incorporating novel control techniques as a way to overcome optimization problems is the main focus for all people involved. Nonlinear Model Predictive Control (NMPC) has been one of the main responses from academia for the exponential growth of process complexity and fast growing scale. Prediction algorithms are the response to manage closed-loop stability and optimize results. Adaptation mechanisms are nowadays seen as a natural extension of prediction methodologies in order to tackle uncertainty in distributed parameter systems (DPS), governed by partial differential equations (PDE). Parameters observers and Lyapunov adaptation laws are also tools for the systems in study. Stability and stabilization conditions, being implicitly or explicitly incorporated in the NMPC formulation, by means of pointwise min-norm techniques, are also being used and combined as a way to improve control performance, robustness and reduce computational effort or maintain it low, without degrading control action. With the above assumptions, centralized (or single agent) or decentralized and distributed Model Predictive Control (MPC) architectures (also called multi-agent) have been applied to a series of nonlinear distributed parameters systems with transport phenomena, such as bioreactors, water delivery canals and heat exchangers to show the importance and success of these control techniques

Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model

We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior

Performance Characterization, Development, and Application of Artificial Potential Function Guidance Methods

The primary objective was to examine artificial potential function (APF) guidance performance when applied to systems with limited control authority in a dynamic environment and develop a hybrid guidance to improve algorithm convergence and computational cost. Performance with respect to both computation time and cost was improved by hybridizing the APF approach with receding horizon planning. Results showed that for the hybrid algorithm, computation time was improved from the optimal control solution while improving the convergence and cost from the APF solution. While the hybrid method greatly improved performance for a saturated system in dynamic environment, this was limited to a fully actuated system. When applied with indirect control, performance was improved, but did not converge. Based on this initial data, the hybrid approach shows promise in regard to implementation within a real-time guidance scheme, however, there is still work to be done before it will be fully effective. The secondary objective was to determine what classes of problems are well-suited to APFs or APFhybrids. The data suggests that APFs and the hybrid algorithm proposed are best applied to fully actuated systems. Additionally, if external dynamics or substantial saturation exist, APF guidance performs better when supplemented with an alternative method

On the Parameter Selection Problem in the Newton-ADI Iteration for Large Scale Riccati Equations

The numerical treatment of linear-quadratic regulator problems for parabolic partial differential equations (PDEs) on infinite time horizons requires the solution of large scale algebraic Riccati equations (ARE). The Newton-ADI iteration is an efficient numerical method for this task. It includes the solution of a Lyapunov equation by the alternating directions implicit (ADI) algorithm in each iteration step. On finite time intervals the solution of a large scale differential Riccati equation is required. This can be solved by a backward differentiation formula (BDF) method, which needs to solve an ARE in each time step. Here, we study the selection of shift parameters for the ADI method. This leads to a rational min-max-problem which has been considered by many authors. Since knowledge about the complete complex spectrum is crucial for computing the optimal solution, this is infeasible for th

Finite-Dimensional RHC Control of Linear Time-Varying Parabolic PDEs: Stability Analysis and Model-Order Reduction

This chapter deals with the stabilization of a class of linear time-varying parabolic partial differential equations employing receding horizon control (RHC). Here, RHC is finite-dimensional, i.e., it enters as a time-depending linear combination of finitely many indicator functions whose total supports cover only a small part of the spatial domain. Further, we consider the squared l1-norm as the control cost. This leads to a nonsmooth infinite-horizon problem which allows a stabilizing optimal control with a low number of active actuators over time. First, the stabilizability of RHC is investigated. Then, to speed-up numerical computation, the data-driven model-order reduction (MOR) approaches are adequately incorporated within the RHC framework. Numerical experiments are also reported which illustrate the advantages of our MOR approaches.Comment: 20 pages, 6 figure