Dual control theory

Dual control theory is a branch of control theory that deals with the control of systems whose characteristics are initially unknown. It is called dual because in controlling such a system the controller's objectives are twofold:

(1) Action: To control the system as well as possible based on current system knowledge
(2) Investigation: To experiment with the system so as to learn about its behavior and control it better in the future.

These two objectives may be partly in conflict.

Dual control theory was developed by Alexander Aronovich Fel'dbaum in 1960. He showed that in principle the optimal solution can be found by dynamic programming, but this was often believed to be impractical; as a result a number of methods for designing sub-optimal dual controllers have been devised.

However, Jinbae Choi finds the optimal dual controller for the multiple model linear systems with unknown parameters in 2016;^[1] The Bellman equation in Dynamic programming is modified to an equivalent functional equation including the system mode of the unknown parameters. With the modified Bellman equation, the optimal cost is calculated and then, separated into the cost corresponding to certainty equivalent and the subtractive cost called dual cost. The dual cost quantifies the investigation aspects of the controller, and the optimal dual controller has the dual property of action and investigation for the discrete-time multiple model linear systems with unknown parameters.

Example

To use an analogy: if you are driving a new car you want to get to your destination cheaply and smoothly, but you also want to see how well the car accelerates, brakes and steers so as to get a better feel for how to drive it, so you will do some test manoeuvers for this purpose. Similarly a dual controller will inject a so-called probing (or exploration) signal into the system that may detract from short-term performance but will improve control in the future.

References

^ Choi, Jinbae (2016). Closed-Loop Optimal Control of Discrete-Time Multiple Model Linear Systems with Unknown Parameters (PhD). Case Western Reserve University.

Feldbaum, A.A. (April 1961) [September 1960 (in Russian, pp. 1240–1249)]. "Dual control theory, Part I". Automation and Remote Control. 21 (9): 874–880.
Feldbaum, A.A. (May 1961) [November 1960 (in Russian, pp. 1453–1464)]. "Dual control theory, Part II". Automation and Remote Control. 21 (11): 1033–1039.
Wittenmark, B. "Adaptive Dual Control" (Seminar handout). Lund University. {{cite journal}}: Cite journal requires |journal= (help)
Wittenmark, B. "Adaptive Dual Control Methods: An Overview". Lund University. CiteSeerX 10.1.1.25.7446. {{cite journal}}: Cite journal requires |journal= (help)

[1] Choi, Jinbae (2016). Closed-Loop Optimal Control of Discrete-Time Multiple Model Linear Systems with Unknown Parameters (PhD). Case Western Reserve University.

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