P, PD Optimal Control.

Cuthbert Nyack
Quadratic optimal control for P control can be examined by setting Td = 0 in the applet below.

Some examples for P control are
Parameters (1.0, 1.0, 1.0, 0.0, 1.0), J = 1.489952, z = 0.5
Parameters (1.0, 0.5, 1.0, 0.0, 1.0), J = 1.239999, z = 0.7071

Parameters (1.0, 2.0, 1.0, 0.0, 1.5), J = 1.983975, z = 0.3535

Parameters (1.58, 0.5, 0.4, 0.0, 1.5), J = 0.867455, z = 0.5625

For PD control both K and Td can be varied. Some examples for PD control obtained by fixing K and varying Td are:-
Parameters (4.0, 1.0, 1.0, 0.39, 0.6), J = 1.480138, z = 0.640.
Parameters (6.0, 1.0, 0.5, 0.36, 0.6), J = 1.008473, z = 0.645.

Parameters (6.0, 1.0, 1.0, 0.32, 0.6), J = 1.830311, z = 0.5960.
If Td is fixed at 0.32 and K varied, then J reaches a minimum of 1.137079 at K = 1.36 with z = 0.6153.
If K is now fixed at 1.36 and Td varied, then J reaches a minimum of 1.002893(Hg = 1.0) at K = 1.36 with z = 0.8893.
After successive adjustments of K and Td the following parameters (1.01, 1.0, 1.0, 1.03, 1.0) J = 0.979746, z = 0.1.0150 and closed loop poles at -0.844 , -1.195 are obtained.

Setting m = 0.1 gives the parameters
(3.47, 1.0, 0.1, 0.96, 0.6) J = 0.304381. z = 0.1.1625 are obtained.




Gif image below shows how applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.