EEL 205 Control Engineering
Practice Problem Set 4
1. The series combination of a resistor R and capacitor C is fed with input voltage
u(t)=u(kT);kT <t[(k+1)T
Find the difference equation for the voltage across the capacitor, when sampled at each k, if vc(0) = vc0.
[vc(k+1)=e−T/RCvc(k)+(1−e−T/RC)u(k)]
2. For the discrete time systems shown in the
Fig. P2, obtain the relevant difference
equations.
a. y(k)+3y(k−1)+3y(k−2)=r(k−2)
b. y(k)+y(k−1)+y(k−2)=
r(k)+2r(k−1)+2r(k−2)
3. Obtain the z-transforms for the following
impulse response sequences:
a. g(k) = sin k
b. g(k) = k2
zsin1
z2−2zcos 1 +1;z(z+1)
(z−1)3
4. Obtain the inverse of the following z-transforms as impulse response sequences:
G(z)= z
(z−1)(z−1/2);G(z)= z
(z−1/2)2;G(z)= z2+z4
(z−1)(z+1)(z−1/2)2
2−2(1/2)k;k(1/2)k+1;4+(4/9)(−1)k−(31/9)(1/2)k−(5/3)(1/2)k+1
5. Obtain the plant transfer function for the following single-input/single-output systems, as well as the open loop
pole-zero plots for them:
a. x1(k+1)= 3
2x1(k)+x2(k)+u(k)
x2(k+1)=−1
2x1(k)
y(k)=x1(k)
b. y(k+2)=y(k+1)− 2
5u(k)+ 3
5u(k+1)
z
z2−3/2z+1/2 ,zero at origin, poles at 1, 1/2 ; 3
5z−2/3
z(z−1), zero at 2/3, poles at origin, 1
6. For the continuous time system shown in Fig.
P6, obtain the matrices A, B, C, D, and the
resolvent matrix Φ(s).
EEL 205/PS4
_ _ _
Delay
T Delay
T
2
+ + +
r k
( ) y k
( )
Delay
T Delay
T
y k
( )
r k
( )
+
+
+
+
_ _
P2
(a)
(b)
1/s
1/s
3
Y
_
P6
1/s
2
4
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