This example shows how to examine the pole
and zero locations of dynamic systems both graphically using
Examining the pole and zero locations can be useful for tasks such as stability analysis or identifying near-canceling pole-zero pairs for model simplification. This example compares two closed-loop systems that have the same plant and different controllers.
Create dynamic system models representing the two closed-loop systems.
G = zpk(,[-5 -5 -10],100); C1 = pid(2.9,7.1); CL1 = feedback(G*C1,1); C2 = pid(29,7.1); CL2 = feedback(G*C2,1);
C2 has a much higher proportional
gain. Otherwise, the two closed-loop systems
Graphically examine the pole and zero locations of
pzplot plots pole and zero locations on the
complex plane as
respectively. When you provide multiple models,
the poles and zeros of each model in a different color. Here, there
poles and zeros of
CL1 are blue, and those of
The plot shows that all poles of
in the left half-plane, and therefore
CL1 is stable.
From the radial grid markings on the plot, you can read that the damping
of the oscillating (complex) poles is approximately 0.45. The plot
also shows that
CL2 contains poles in the right
half-plane and is therefore unstable.
Compute numerical values of the pole and zero locations
z = zero(CL2); p = pole(CL2);
pole return column
vectors containing the zero and pole locations of the system.