# Root Locus, Step and Sine Response For Multi Input System

Cuthbert Nyack
For a multi input system described by the state vector approach, the root locus can be obtained from the determinant of the system matrix and is shown in the following applet.

The closed loop poles can be determined from the applet where the parameters are simplified by having a2 = a3 etc.
for eg parameters (0.6, 1.0, 1.0, 2.0, 2.0, 0.01) the closed loop poles are at -0.68, -1.22, -2.92, -3.63.
for (0.6, 1.8, 1.0, 2.0, 2.0, 0.01) the poles are at -0.37, -1.37, -2.75, -3.9.
for (0.5, 3.0, 2.0, 1.0, 0.85, 0.76) the poles are at -3.8, -1.05, ~-0.9 ±0.76.
for (1.0, 1.0, 2.0, 1.0, 0.5, 0.97) the poles are at -3.0, -1.0, ~-1.25 ±0.97.
for (1.0, 3.0, 2.0, 1.0, 1.0, 0.86) 3 of the poles are at -1.2, -0.5 ± j0.86.
Although all 4 transfer functions are first order, the system can have complex conjugate poles eg for smaller values of b1, b4.

The applet below shows the the step response of the multi input system.
for parameters (1.0, 1.0, 1.4, 0.6, 3.0, 1.2, 3.0, 1.2, 1.4, 0.6, 5.0, 0.4) the response is a transient oscillation.
for (1.0, 1.0, 1.4, 0.4, 3.0, 1.2, 3.0, 1.2, 1.4, 0.4, 5.0, 0.4) the response is an oscillation. ie complex conjugate pole on the imaginary axis.
for (1.0, 1.0, 1.5, 1.0, 2.5, 1.0, 2.5, 1.0, 1.5, 1.0, 3.0, 0.1) the response is a ramp. ie real pole at the origin.
for (1.0, 1.0, 2.0, 1.0, 2.5, 1.0, 2.5, 1.0, 2.5, 1.0, 4.0, 1.0) the response is characteristic of real poles.

The applet below shows the system response to a sinusoidal input at r2 and illustrates the phase relation of signals at different points. For parameters (1.0, 1.0, 1.0, 1.0, 3.4, 2.0, 3.4, 2.0, 1.0, 1.0, 5.0, 0.7) the response is a sinusoid.
For parameters (1.0, 1.0, 1.0, 0.4, 3.4, 2.0, 3.4, 2.0, 1.0, 0.4, 2.0, 1.2) the response is a sinusoid with linearly increasing amplitude. ie cc poles on the imaginary axis.

Gif image below shows how applet should appear.