EE207/PS4

EE207 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.
[v c (k + 1) = e −T/RCv c (k) + (1 − e −T/RC )u(k) ]
P2
+ + Delay + Delay 2. For the discrete time systems shown in the
r (k ) T y (k )
_ _ T _ Fig. P2, obtain the relevant difference
equations.
a. y(k) + 3y(k − 1) + 3y(k − 2) = r(k − 2)
2
(a) b. y(k) + y(k − 1) + y(k − 2) =
r(k) + 2r(k − 1) + 2r(k − 2)
+
+
y (k ) 3. Obtain the z-transforms for the following
+ impulse response sequences:
+ Delay Delay a. g(k) = sin k
r (k )
_ _ T T b. g(k) = k2
(b) z(z + 1 )
z sin 1 ;
z 2 − 2z cos 1 + 1 (z − 1 ) 3

4. Obtain the inverse of the following z-transforms as impulse response sequences:

G(z) = z ; G(z) = z ; G(z) = z2 + z4
(z − 1 )(z − 1/2 ) (z − 1/2 ) 2 (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. x 1 (k + 1) = 3 x 1 (k) + x 2 (k) + u(k)
2
x 2 (k + 1) = − 1 x 1 (k)
2
y(k) = x 1 (k)

b. y(k + 2) = y(k + 1) − 2 u(k) + 3 u(k + 1)

5 5

z , zero at origin, poles at 1, 1/2 ; 3 z − 2/3 , zero at 2/3, poles at origin, 1

z 2 − 3/2z + 1/2 5 z(z − 1 )

P6
6. For the continuous time system shown in Fig.
_ 1/ s 1/ s 1/ s Y P6, obtain the matrices A, B, C, D, and the
_ _ resolvent matrix Φ(s).
3
2
4
 EE207/PS4

0 1 0
0 0 1 , [0 ], 1 0 0 , [0 ],  11 (s) = s 2 + 3s + 2
s 3 + 3s 2 + 2s + 4
−4 −2 −3

7. For the state representation

d x1 0 1 x1 1 0 u1
= +
dt x2 −5 −6 x2 0 1 u2

y1 1 0 x1 0 0 u1
= +
y2 0 1 x2 0 1 u2
the initial state is x(0) = [3 2]T, and the input is u(t) = [1 1]T for t m 0. Determine the dynamics of the output
vector, and the state transition matrix.
y1 28/5 + 11e −t − 23/5e −5t 5e −t − e −5t e −t − e −5t
= 1 ; (t) = 1
y2 4 −4 − 11e −t + 23e −5t 4 −t
−5e + 5e −5t −e −t + 5e −5t

8. It is desired to realise a transfer function

R (s )
G(s) = s 2 + 6s + 5
(s + 2 )(s + 3 )(s + 4 ) b3 b2 b1
in the three state form shown in Fig. P8.
Determine parameters a1-a3 and b1-b3. + X3(s ) + X2(s )+ X1(s )
1/ s 1/ s 1/ s Y (s )
_ + _ + _

P8 a2
a3 a1

[a1=−2, a2=−3, a3=−4, b1=1, b2=−1, b3=−3]