Continuous-time linear-quadratic regulator
For linear (or linearized) systems described by a state-space model, state-feedback controllers can be designed by minimizing a quadratic cost function which takes into account the error and the input. The problem with arbitrary fixed weights can be written as an algebraic Riccati equation; the function care gives its solution if it exists.
When you open LQR_ct.sq from Sysquake (menu File/Open), four figures are
displayed: the closed-loop eigenvalues, the weights on the state and the input
as sliders, and the time responses of the output and input of the controlled system
with one of the state initial values set to 1 and the others to
0
The diagonal elements of the state and input weights are displayed as sliders. You can change them with the mouse. Non-diagonal weights are zero.
Output time response of the controlled system. The initial value of all states is zero, except for one of them which is 1. You can change which one is 1 by double-clicking the figure.
Input time response of the controlled system. The initial value of all states is zero, except for one of them which is 1. You can change which one is 1 by double-clicking the figure.
Closed-loop frequency response between a state disturbance and the output, as a singular value plot. The singular value plot is the equivalent of the Bode diagram for single-input single-output systems.
A continuous-time model can be given as four matrices A, B, C, and D:
s X(s) = A X(s) + B U(s) Y(s) = C X(s) + D U(s)
When selected, moving the mouse above a sensitivity plot will display a corresponding line in other sensitivity and eigenvalue plots.