Assignment 4 - Control Theory 2

Second Half 2003

Problem 1

Let G(s) be the plant transfer function. The closed-loop transfer function is

T(s) =

G(s)

1+G(s)

Let w_p be such that |G(jw_p)| = 1 and ĞG(jw_p) = f, then prove that

|T(jw_p)| =

2sin

f_m

where phase margin f_m = 180 + f, -180^° £ f £ 180^°.

Problem 2

Let

C(s) =

1+s/w_z

1+s/w_p

; w_p > w_z

and w_l be the frequency at which ĞC(jw_l) is maximum over all w > 0. Show that

w_l =

w_z w_p

f_l

D
=

ĞC(jw_l) = sin^-1

æ
è

w_p/w_z - 1

w_p/w_z + 1

ö
ø

, and

20log₁₀|C(jw_l)| = 10log₁₀(w_p/w_z).

The ratio w_p/w_z is often expressed as m. Make a table of a few important values of m and f_l.

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, l_m(jw) = |L(jw)|. The closed-loop system sensitivity function is such that

ê
ê

1+tildeG(jw)

ê
ê

p_m(j w)

Prove that for a stable feedback system to satisfy the sensitivity function constraint the following inequality should be satisfied:

ê
ê

G(jw)

1+G(jw)

ê
ê

l_m(jw) +

ê
ê

1+G(jw)

ê
ê

p_m(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,

The closed-loop system has zero steady-state error to unit step reference input;
The crossover frequency is above 500 rad/s;
There is a minimum 60^° phase margin.

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File translated from T_EX by T_TH, version 3.40.
On 08 Aug 2003, 16:46.