# Transient Response and Closed Loop Poles

Cuthbert Nyack
This applet shows how the closed loop poles can be calculated from the transient response. The transient response can be used to calculate the dominant closed loop poles. Since this is a third order system, all of the closed loop poles can be calculated. Parameter a is used to shift the imaginary s axis left or right.

The eg parameters (2.0, 1.0, 0.5, 1.0, 0.0, 2.0) show that both type 0 and 1 systems are stable.
(2.0, 1.0, 0.5, 1.0, ~-0.1, 2.0) show that the type 1 system is oscillatory at angular frequency ~1.19rad/s. This means that the dominant closed loop poles are at x ± jy = ~-0.1 ± 1.19j and the real pole r = -K/(T1*T2*(x*x+y*y)) = ~-2.8.
(2.0, 1.0, 0.5, 1.0, ~-0.5, 2.0) show that the type 0 system is oscillatory at angular frequency ~1.32rad/s. This means that the dominant closed loop poles are at x ± jy = ~-0.5 ± 1.32j and the real pole r = -(1 + K)/(T1*T2*T3(x*x+y*y)) = ~-3.0.

(10.0, 1.0, 1.0, 0.5, 0.0, 1.0) show that both type 0 and 1 systems are unstable.
(10.0, 1.0, 1.0, 0.5, ~0.43, 1.0) show that the type 1 system is oscillatory at angular frequency ~1.82rad/s. This means that the dominant closed loop poles are at x ± jy = ~+.43 ± 1.82j and the real pole r = -K/(T1*T2*(x*x+y*y)) = ~-2.87.
(10.0, 1.0, 1.0, 0.5, ~0.05, 1.0) show that the type 0 system is oscillatory at angular frequency ~2.31rad/s. This means that the dominant closed loop poles are at x ± jy = ~+0.05 ± 2.31j and the real pole r = -(1 + K)/(T1*T2*T3(x*x+y*y)) = ~-4.1.

Gif image below shows how applet should appear. 