On centralized optimal control

It can be argued that the Holy Grail of control theory is the determination of the optimal feedback control law or simply the feedback control law. This is understandable given the huge success of the linear quadratic Gaussian (LQG) theory and applications for the past half-century. It is not an exaggeration to say that the entire aerospace industry, from the Apollo moon landing to the latest global positioning system (GPS), owe a debt to this control-theoretic development in the late 1950s and early 1960s. As a result, the curse of dimensionality notwithstanding, finding the optimal control law for more general dynamic systems remains an idealized goal for all problem solvers. We continue to hope that with each advance in computer hardware and mathematical theory, we will move one step closer to this ultimate goal. Efforts such as feedback linearization and multimode adaptive control can be viewed as such successful attempts. It is the thesis of this note to argue that this idealized goal of control theory is somewhat misplaced. We have been seduced by our early successes with the LQG theory and its extensions. The simple but often not emphasized fact is this: It is extremely difficult to specify and impossible to implement a general multivariable function even if the function is known.