EE207/PS1

EE207 Control Engineering

Practice Problem Set 1

1. Obtain a set of simultaneous first order system equations to describe

P1 R
i (t ) the dynamics for the circuit shown in Fig. P1, given that the output
of interest is the current i(t) for the applied voltage v(t) as the input.
L dv c (t) di(t) dv(t)
C = 1 i(t); = − 1 v c (t) − 1 i(t) + 1 v(t) + 1
v(t ) dt C dt L RC L R dt

2. In the mechanical system shown in Fig. P2, the

force fext(t) (input) affects two mass velocities v1(t) P2 v (t)
D 1 v (t)
and v2(t) of interest. Evolve the system dynamic 2
equations (not necessarily of first order) for this K
system. What is peculiar about the steady state fext M1 M2
condition for this system ?

d 2 v 1 (t) df ext dv 1 (t) d 2 v 2 (t)

= 1 − D − K [v 1 (t) − v 2 (t) ] ; = K [v 1 (t) − v 2 (t) ]
dt 2 M 1 dt M 1 dt M1 dt 2 M2

3. A mechanical damped inertial coupling is used to

springy shaft connect a load to a motor across a shaft, as shown in
J1 J2 Fig. P3. The input to this system is therefore the motor
K speed ωext(t), while the output is the load speed Ω(t).
ωext ( t ) Ω (t) The shaft has a torsional spring constant K, while the
D
P3 inertial coupling is in two halves of inertia J1 and J2
with a damping constant D that applies in between.
Obtain a single differential equation relating the input and output for the system.
J 1 J 2 d 3 (t) ( d 2 (t) KJ 2 d (t)
+ J1 + J2 ) + + K $ (t) = K $ * ext (t)
D dt 3 dt 2 D dt

4. An accelerometer is a device used to monitor speed variations y (t)

P4 x (t)
in a vehicle. One configuration for such a meter is shown in 0
D/2
Fig. P4 (read the formulation around Fig. 2.6 of the text for
better understanding, probably on pp. 28-29). The frame is P
mounted in a vehicle moving on horizontal plain, and its K/2 K/2
position at any instant is given by x(t) from some known
reference. A moving mass M, supported by springs of D/2
constant K/2, and damped on the two lateral surfaces by D/2
has a pointer P that coincides with a graduation 0 when the vehicle runs at constant speed. As it accelerates, the
mass moves by a distance y(t) between P and 0. If
K − D 2<0
M 2M
find y(t) for t m 0 if the vehicle suddenly accelerates at a fixed rate starting with zero acceleration initially. (The
double derivative of x(t) can be used as an input, and y(t) as the output for the system.)
d 2 x(t) 2 K 2 d 2 x(t)
y(t) = K 1 e  1 t + K 2 e  2 t + K ; 1, 2 = − D ! D − K ; K1 å − $ ; etc.
M dt 2 2M 2M M M( 2 −  1 ) dt 2
 EE207/PS1
5. The accelerometer of Problem 4 is now installed in an aircraft. When the aircraft takes off at an angle φ to the
horizontal, the meter as in Fig. P4 can be expected to do the same, so that an additional force attributable to the
weight of M acts as an input to the system. Obtain the system equation for this modified situation. If y(t) is
known to modify marginally with φ according to a fixed derivative

Øy(t)/Ø& = (t)
determine the linearised equation in α(t) for φ = 0.
d 2 y(t) D dy(t) K d 2 x(t) d 2 (t) D d(t) K 2
Ø d x(t) − g
+ + y(t) = − g $ sin & ; + + (t) =
dt 2 M dt M dt 2 dt 2 M dt M Ø& dt 2

6. The circuit in Fig. P6 includes a mutually coupled pair

of coils with resistance R1 and R2, self inductance L1 and P6
R1 R2
L2, and mutual inductance M. R3 is an external load
M
resistance. If the variables of interest are the currents
i1(t) and i2(t), with the voltage v(t) as an input variable, i (t ) i (t )
v(t ) 1 L L 2 R
set up the dynamic equations for the system in the form 1 2 3
di 1 (t)
= a 11 $ i 1 (t) + a 12 $ i 2 (t) + b 1 $ v(t)
dt
di 2 (t)
= a 21 $ i 1 (t) + a 22 $ i 2 (t) + b 2 $ v(t)
dt

R1L2 (R + R 3 )M R1M (R + R 3 )L 1 L2 M
a 11 = ;a = 2 ;a = ; a = 22 ; b1 = ;b =
M 2 − L 1 L 2 12 M 2 − L 1 L 2 21 M 2 − L 1 L 2 22 M − L1L2 L1L2 − M2 1 L1L2 − M2