Assignment 4 - Control Theory 2

Problem 1

Let G(s) be the plant transfer function. The closed-loop transfer function is
 T(s) = G(s) 1+G(s) .
Let wp be such that |G(jwp)| = 1 and ÐG(jwp) = f, then prove that
|T(jwp)| =  1

 2sin fm 2
where phase margin fm = 180 + f,      -180° £ f £ 180°.

Problem 2

Let
 C(s) = 1+s/wz 1+s/wp ;       wp > wz
and wl be the frequency at which ÐC(jwl) is maximum over all w > 0. Show that
 wl = Ö wz wp ,

 fl D = ÐC(jwl) = sin-1 æè wp/wz - 1 wp/wz + 1 öø , and

 20log10|C(jwl)| = 10log10(wp/wz).
The ratio wp/wz is often expressed as m. Make a table of a few important values of m and fl.

Problem 3

Figure 1: Block Diagram: Problem 3
Let the true system transfer function in the above Figure 1 be modelled using a multiplicative uncertainty as [G\tilde](s) = G(s)(1 + L(s)). The model we have available for design is G(s) and let the uncertainty magnitude be, lm(jw) = |L(jw)|. The closed-loop system sensitivity function is such that
 êê 1 1+tildeG(jw) êê < 1 pm(j w) .
Prove that for a stable feedback system to satisfy the sensitivity function constraint the following inequality should be satisfied:
 êê G(jw) 1+G(jw) êê lm(jw) + êê 1 1+G(jw) êê pm(jw) < 1.

Problem 4

Figure 2: Block Diagram: Problem 4
In the above Figure 2, let
 G(s) = 100 (s/10+1)(s/20+1) .
Design controller C(s) such that,
1. The closed-loop system has zero steady-state error to unit step reference input;

2. The crossover frequency is above 500 rad/s;

3. There is a minimum 60° phase margin.

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On 08 Aug 2003, 16:46.