Course Notes for MIT 6.832

This video discusses optimal nonlinear control using the Hamilton Jacobi Bellman (HJB) equation, and how to solve this using dynamic programming.

Little has been done in the study of these intriguing questions, and I do not wish to give the impression that any extensive set of ideas exists that could be called a "theory." What is quite surprising, as far as the histories of science and philosophy are concerned, is that the major impetus for the fantastic growth of interest in brain processes, both psychological and physiological , has come from a device, a machine, the digital computer. In dealing with a human being and a human society, we enjoy the luxury of being irrational, illogical, inconsistent, and incomplete, and yet of coping. In operating a computer, we must meet the rigorous requirements for detailed instructions and absolute precision. If we understood the ability of the human mind to make effective decisions when confronted by complexity, uncertainty, and irrationality, then we could use computers a million times more effectively than we do. Recognition of this fact has been a motivation for the spurt of research in the field of neurophysiology. The more we study the information-processing aspects of the mind, the more perplexed and impressed we become. It will be a very long time before we understand these processes sufficiently to reproduce them. In any case, the mathematician sees hundreds and thousands of formidable new problems in dozens of blossoming areas, puzzles galore, and challenges to his heart's content. He may never resolve some of these, but he will never be bored. What more can he ask?

This overview paper reviews numerical methods for solution of optimal control problems in real-time, as they arise in nonlinear model predictive control (NMPC) as well as in moving horizon estimation (MHE). In the first part, we review numerical optimal control solution methods, focussing exclusively on a discrete time setting. We discuss several algorithmic ''building blocks'' that can be combined to a multitude of algorithms. We start by discussing the sequential and simultaneous approaches, the first leading to smaller, the second to more structured optimization problems. The two big families of Newton type optimization methods, Sequential Quadratic Programming (SQP) and Interior Point (IP) methods, are presented, and we discuss how to exploit the optimal control structure in the solution of the linear-quadratic subproblems, where the two major alternatives are ``condensing'' and band structure exploiting approaches. The second part of the paper discusses how the algorithms can be adapted to the real-time challenge of NMPC and MHE. We recall an important sensitivity result from parametric optimization, and show that a tangential solution predictor for online data can easily be generated in Newton type algorithms. We point out one important difference between SQP and IP methods: while both methods are able to generate the tangential predictor for fixed active sets, the SQP predictor even works across active set changes. We then classify many proposed real-time optimization approaches from the literature into the developed categories.