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On Optimal Control
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I attempt below a popularized account on what I do for a living.
The desire to “ improve” things and to be in “control” of one’s life and surroundings are fundamental to human existence. Without these desires, civilization certainly will not be where it is today. Thus, one can argue “optimal control theory” is a very basic branch of science/mathematics.
However despite this early start, the science of optimization and control did not began as a discipline until the invention of calculus and the theory of maxima and minima. Even then, the real world control problems that can be solved are few and far in between. It wasn’t until WWII and the need for weapons fire control that the study of control and servomechanism began in earnest. A key publication immediately after WWII is the famed
Hubert M. James, Nathaniel B. Nichols, and Ralph S. Phillips. Theory of Servomechanisms, volume 25 of MIT Radiation Laboratory Series. McGraw-Hill, New York, 1947
followed by G.S. Brown and D. Campbell, Principles of Servomechanism, Wiley 1948. At the more philosophical level we also have the Norbert Wiener, Cybernetics; or, Control and communication in the animal and the machine. MIT Technology Press, 1948.
These three books pretty much defined the field of control at that time and made MIT the center of the control world in the Fifties except for a largely ignored and poorly understood (at that time) book by H.S. Tsien, Engineering Cybernetics, McGraw Hill 1955. Tsien was five years ahead of his time. It wasn’t until the late Fifties and early Sixties where aerospace and the race to the moon became important national goals that optimal control theory enters a new phase. Large bodies of new applied mathematical technique such as, linear algebra, calculus of variation, stochastic processes, functional analysis etc. besides Laplace Transforms and complex variables were added to our tool chest. It is not an exaggeration to say that “optimal control theory” made possible the Apollo moon landing in 1969. It was the golden age for the control discipline. The succeeding decades see further broadening of application areas and deepening of theoretical insights for the field. I touched upon these in my blog articles
and won’t repeat them here. The crowning glory of the discipline arguably is the Draper award for the invention of the Kalman filter (see http://www.sciencenet.cn/m/user_content.aspx?id=14253 )
Let me discuss below briefly here for the lay audience what I think as the fundamental ingredients and contributions of “control theory”
1. The idea of FEEDBACK.
The problem of control is fundamentally straight forward. You determine “what to do” in order to accomplish certain goals under a given set of circumstances. This is what is known as open-loop control. The idea of “feedback” is not present. It comes in only because of a practical concern. Because the circumstance under which we need to solve “optimal control” is forever changing, it is not practical to re-solve the problem every time the circumstances changes. We want an automatic way of accomplishing this. This is how the terminology “automatic control” or “feedback control” arose. The earliest example of this is of course the well known speed governors for steam engines. We measure the current set of circumstances (information input) and determine or re-solve for the appropriate optimal control (decision output) for the current instant. We do this continuously. The mapping from current information to current decision is a multivariable functions commonly denoted as the feedback control law or strategies. I have discussed the concept of strategy and its computational difficulty elsewhere. http://www.sciencenet.cn/m/user_content.aspx?id=26889
2. The idea of DYNAMICS
We are all familiar since childhood with the lesson of “planning for the future” and the story of the hard working ants and the fun-loving cicada. In other words, current decision has implications for the future. In financial terms, “Saving for a rainy day” is another golden rule. One can argue that the central problem of control and system theory is to properly account for the future through current decisions. This is easily said than done. Just think for the moon landing, the Apollo vehicle must have the correct direction and speed at the starting instant so that gravity can take it to the correct place at the correct time some quarter of a million miles and days in the future. But through understanding of the dynamics of motion and gravitational law, we can do it. But the dynamics for each problem (e.g., guiding the national economy or planning for your own retirement) is different. The contribution of control theory is that we devised a universal way of doing it. Basically, we discovered a general way to reflect future consequences to the current instant so that current decision can be made only based on current information which now includes the reflected consequences. We can do this reflection so long as we have a model of the dynamics involved. This way we only need to solve an ordinary (static) optimization problem every time a decision is to be made. This universal method of reflection is known as “the Principle of Optimality in Dynamic Programming”, “the Maximum Principle of Pontriagin”, “the Co-state or multiplier equations of the Calculus of Variations”, etc.
3. Dealing with Uncertainty.
The famous saying that “in life only death and tax are certain” is the recognition that almost everything in this world is uncertain. In fact, the need for feedback control discussed in #1 above is due to the fact problem circumstances changes constantly. However, rather than react after things have changed, why not deal with possible changes before hand. Consider the following simple optimal control problem – fly from point A to point B in minimal time. The original (open loop) optimal solution is simply “fly with maximum speed in the direction of AB”. However, in the real world there may be a cross wind with varying strength blowing along the path AB. Using the original solution will almost certainly cause you to miss the goal. Thus, we devise the feedback (closed loop) solution “fly with maximum speed in the direction of current position and B”. But a further level of sophistication is to say “fly with feedback control and with the added rudder correction proportional to the average cross wind speed”. This is known in our discipline as stochastic optimal control. Other terminologies such as adaptive control and learning control are nothing but more sophisticated ways of accounting for uncertainties in the problem. Nothing can be more general than using the current information (whatever they may be) to produce the current optimal decision.
4. Extensions of the discipline
Originally, control theory deals only with control of mechanical systems such as motors, missiles, etc. Nowadays, systems include everything natural as well as manmade, for example, national economy, electric power grid, communication networks, biological systems, manufacturing and traffic systems, and e-commerce. Identification and modeling of such diverse systems also became part of control system study and research. Centralized control from one source is now generalized to decentralized control and control against opposing and cooperating agents (e.g., differential games. See http://www.sciencenet.cn/m/user_content.aspx?id=21891 ). The intellectual underpinnings of the discipline now overlap with that of Operation Research and theoretical Computer Science.
As Vannevar Bush said famously in his classic paper on science – the frontier is endless. I see good things to come in the future.
https://blog.sciencenet.cn/blog-1565-209522.html
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