Question: A continuous time plant with transfer function G(s) = 3/s(s+7) is driven by a digital compensator of gain K involving sampling at a rate of 0.25s with zero order hold. The system has unity feedback. (a) Obtain the discrete system closed loop transfer function. (b) Apply Jury’s test to analyse the range of controller gain K for which the system would be stable.
Jury’s Stability Test
If the characteristic equation of a sampled data system is as follows:
Q(z) = an zn + an-1 zn-1 + ……+ a1 z + a0
Then the following Jury’s array is formed
a0 a1 a2 --------------- an-1 an
an an-1 an-2 --------------- a1 a0
b0 b1 b2 --------------- bn-1
bn-1 bn-2 bn-3 --------------- b0
c0 c1 c2 ---------------
cn-2 cn-3 cn-4 ---------------
- -
- -
Repeat rows until 3 elements in a row ie 2n – 3 rows for an nth order system
The elements of the 2nd, 4th rows etc are the elements of the row previous to it in reverse order.
The elements of the remaining rows are calculated from the determinants of elements from the 2 rows above it as follows:
Eg: bk = determinant of a0 an-k
an ak
The tests for the system to be stable ie for Q(z) to have no roots on or outside the unit circle are as follows:
- an > 0
- [ Q(z)] z = 1 > 0
3.[ Q(z)] z = -1 > 0 if n is even
[ Q(z)] z = -1 < 0 if n is odd
4. |a0| < |an|
|b0| > |bn-1|
|c0| > |cn-2|
|d0| > |dn-3|
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Question:
A continuous time plant with transfer function G(s) = 3/s(s+7) is driven by a digital compensator of gain K involving sampling at a rate of 0.25s with zero order hold. The system has unity feedback.
(a) Obtain the discrete system closed loop transfer function.
(b) Apply Jury’s test to analyse the range of controller gain K for which the system would be stable.
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