LQR_ct.sq

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.

First contact

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. You can change the weights with the mouse. To change the state whose initial value is 1, double-click its figure; a dialog box will be displayed where you can enter the state number. The state number is associated to the figure; you can display different responses if you add other time response figures.

Figures

State and Input Weights

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.

Time Response to Initial Condition y(t)

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.

Time Response to Initial Condition u(t)

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.

Sensitivity

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.

Settings

Model

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)

Display Frequency Line

When selected, moving the mouse above a sensitivity plot will display a corresponding line in other sensitivity and eigenvalue plots.


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