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I don't see the reason for underlining. Besides, it is not done everywhere. Please post your explanation on the talk page. |
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[[linear system]], such as
:<math>\frac{\mathrm{d}}{\mathrm{d}t}x=
▲:<math>y = \underline{C} x.</math>
One common cost functional used together with this system description is
:<math>J=\int_0^\infty ( x^T(t)
where the [[matrix (mathematics)|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
:<math>u(t)=-
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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