Optimal control

Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimization algorithms.

Linear quadratic control

It is very common, when designing proper control systems, to model reality as a

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linear system, such as

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:rac{mathrm{d}}{mathrm{d}t}x=A x + B u

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:y = C x. ,

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One common cost functional used together with this system description is

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:J=int_0^infty ( x^T(t)Qx(t) + u^T(t)Ru(t) ),mathrm{d}t

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where the matrices Q and R are positive-semidefinite and positive-definite, respectively. Note that this cost functional is thought in terms of penalizing the control energy (measured as a quadratic form) and the time it takes the system to reach zero-state. This functional could seem rather useless since it assumes that the operator is driving the system to zero-state, and hence driving the output of the system to zero. This is indeed right, however the problem of driving the output to the desired level can be solved after the zero output one is. In fact, it can be proved that this secondary problem can be solved in a very straightforward manner. The optimal control problem defined with the previous functional is usually called the state regulator problem and its solution the linear quadratic regulator (LQR) which is no more than a feedback matrix gain of the form

Related Topics:
Matrices - State regulator problem

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:u(t)=-K(t)cdot x(t)

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where K is a properly dimensioned matrix and solution of the continuous time dynamic Riccati equation. This problem was elegantly solved by R. Kalman (1960).

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