Question

Question 3 The block diagram of a digital control system is given in Fig. 2. R(Z) + VT CZ 0.52 0.24(2 +0.5) (2-0.4) (z-1) Fig. 2 (a) Write an expression for the closed-loop z-transfer function of the system, C(z)/R(z). Given that one of the poles is located at z=0.235, complete a pole-zero diagram for the system. [8 marks] (b) Using a sampling interval, T = 0.5 s, map the poles and zeros onto the primary strip in the s-plane. [8 marks] (c) Explain why you would be confident that predictions of the system response based on the location of dominant s-plane poles would be accurate. [4 marks]

Answer 1

Question a:

the loop transfer function is given by L(z) = 0.5*0.24*z*(z+0.5)/((z-1)*(z-0.4)^2)

L(z) =   (0.12 z^2 + 0.06 z) / (z^3 - 1.8 z^2 + 0.96 z - 0.16)

The closed loop transfer function is given by

T(z) = L(z) / (1+L(z)) = (0.12 z^2 + 0.06 z) / (z^3 - 1.8 z^2 + 0.96 z - 0.16 + 0.12 z^2 + 0.06 z))

T(z) =   (0.12 z^2 + 0.06 z) / (z^3 - 1.68 z^2 + 1.02 z - 0.16)

The poles of the closed loop transfer function are given by: z^3 - 1.68 z^2 + 1.02 z - 0.16 = 0

The poles are given below.

0.7224 + 0.3979i
0.7224 - 0.3979i
0.2353

MATLAB is used to plot the pole zero diagram.

code to plot:

figure;zplane([0.12 0.06],[1 -1.68 1.02 -0.16]);

Imaginary Part ...... ......... ........ o to -0.5 0.5 Real Part

With T = 0.5, and zero order hold method to convert from discrete to continuous domains, we have,

Tc(s) {continuous counterpart of discrete domain transfer function T(z)} will be equal to

Tc(s) =

0.07413 s^2 + 0.8756 s + 3.364
---------------------------------
s^3 + 3.665 s^2 + 3.394 s + 3.364

The poles and zeros are given by

Poles:
-2.8942
-0.3855 + 1.0069i
-0.3855 - 1.0069i

Zeros:

-5.9058 + 3.2412i
-5.9058 - 3.2412i

The polez-zero diagram is plotted below.

Pole-Zero Map EX Imaginary Axis (seconds) EX Real Axis (seconds)

Question (c):

The dominant poles are the closest poles to the jw axis in s plane. Basically these are are poles that dictate the most of the response of the system in most cases. This is because, the poles which are closer to jw axis have higher time constant {time constant and location of pole are inversely related: closer the pole to jw axis larger its tie constant.} which have larger amount of time to settle and therefore the response of the overall system with dominant pole pair and a bunch of non-dominant poles will largely look like the {close match} response of the dominant pole pair it self.

In the above case, we have poles located at   -2.8942, -0.3855 + 1.0069i, -0.3855 - 1.0069i. The real part of the complex poles {this is their distance from jw axis, a factor that indicates how close they are} is -0.3855 which is 7.5 times as close as the other pole to jw axis { usually this greater than 5 times considered}.Therefore, 0.3855 + 1.0069i, -0.3855 - 1.0069i are the dominant pole pairs and -2.8942 is non dominant pole.