MODULE CODE: MEC08106    

MODULE TITLE: Control Measurement and Instrumentation

EXAM DURATION:  2 hours

 

Question 1

Figure Q1.1 

Considering  the vehicle dynamics shown in the free-body diagram shown in Figure Q1.1. The vehicle, of mass m, is acted on by a control force, u. The force u represents the force generated at the road/tire interface. For this simplified model, assume that we can control this force directly and neglect the dynamics of the powertrain, tires, etc., that go into generating the force. The resistive forces, bv, due to rolling resistance and wind drag, are assumed to vary linearly with the vehicle velocity, v, and act in the direction opposite the vehicle's motion.

1. Find the system dynamic model (differential equation) shown in Figure Q1.1
   4. marks]
2. The dynamic model obtained in step (a) can be represented by a general first-order differential equation below

                                                          ?????????(????)+????(????)=????????(????)                                                                               (Q1b) 

Find the representations of time constant τ  and gain k  for the given m and b, respectively. 4marks] 

2. For τ=10 and k=1 in equation (Q1b), and a unit step input (u(t)=1), sketch the unit step response, y(t), from the zero-initial condition of this first-order system  .[11 marks]
3. The general form of the first-order system can be described as

?????????(????)+????(????) =????????(????)

The unit step response of a first-order system is shown in Figure Q1.2

 

Figure Q1.2

Find time constant τ  and gain k based on Figure Q1.2, respectively. [6 marks]                                                               Total[25] marks.

ANSWER(PURCHASE FULL PAPER TO GET ALL THE SOLUTIONS)

1a)

The input function is u(t) and the  function is y(t). by  newton’s law, the  differential  equation is obtained as:

  

Where m is the mass, u(t) is  the control force input function of time. b is the resistive force (which varies due to the velocity).

The  dynamic model is given as: 

1b)

given:  

τyt+yt=kut 

By comparing with  the dynamic model.

 

 

 ,                  making τ  the subject of formula.

Therefore,

 

Also,

 

 ,  recal

 

1c)

 

                    substituting .

 

If u(t) = 1, unit step function.

 

 

Solving  the differential equation.

Assume

 

Therefore, for  the complementary part

 

 

Thus,

 

The complementary solution:

 

The particular part = 1

Therefore,   the general solution is:

 

The initial is:   

 

 

Therefore,

 

The sketch:

1d)

Recall,

 

 

 

Note:

0.06τ+2=k                                                   (1)

                                               (2)

 

 sec,

 

Question 2

1. Briefly explain the following terminologies: 
   1. Transient response
   2. Disturbance signal
   3. Error signal iv.    Steady-state error

[8 marks]

2. For a control system shown in Figure Q2.1, find the transfer functions  and , respectively.

 

Figure Q2.1

 

Find the overshoot and the peak time based on Figure Q2.2. Using the overshoot and the peak time you identified from Figure Q2.2, compute the damping ratio and the natural frequency.9marks Total [25] mark  

 

(c) The unit step response of a second-order system is shown in Figure Q2.2.  [8 Marks]

Figure Q2.2

Question 3

Consider the electrical system shown in Figure Q3.1, where R is the resistence, C is the capacitance, L is inductance, u(t) is the input voltage, y(t) is the output voltage.

 

 

Figure Q3.1

 

1. Find the differential equation describing the system shown in Figure Q3.1.                     

[4 marks] 

2. Compute the transfer function of this system relating the input u(t) to the output y(t). The initial conditions are assumed to be zero.         

[3 marks] 

3. Calculate the values of the natural frequency and the damping ratio of this system.

[6 marks]

4. Using a PD controller to control the second-order system, the control system block diagram is shown in Figure Q3.2

 

 

Figure 3.2

 

Find the close-loop system transfer function Y(s)/(U(s) for the system shown in Figure 3(d). For a₁=-1, a₀=-4 and k=2, determine if the system is stable. If not, determine how many poles are on the right-hand side of the s plane. Can the system be stabilised by a proportional plus derivative controller as shown in Figure 3.2? If yes, find the ranges of K_p and K_D which make the steady state error for a unit step input less than 0.1.

[13 marks] Total [25] marks. 

Question 4

A second-order plant is described by the equation

                                                       

                                  (4.1)

where u is the input and y is the output.

1. PID controller consists of three terms, namely proportional, integral, and derivative control. Briefly explain the function of each of the three PID terms. 6marks]
2. Compute the transfer function of the plant shown in equation (4.1) and determine if it is stable. The initial conditions are assumed to be zero.  6marks]
3. Can this plant be stabilised by a proportional controller 

u=K_p(r-y)

where r is the desired (reference) signal? If not, explain why not. If yes, find the range of values of K_p for which the feedback system is stable. Draw a block diagram of the feedback system. [6 marks]   

(d) Can the system be stabilised by a proportional plus derivative controller, G_c(s)=K_p+K_Ds? If yes, find the ranges of K_p and K_D. If not, explain briefly your reasoning.     7marks                                                                                                                                         Total [25] marks. 

Question 5

Figure 5.1

 

Consider the armature-controlled DC servomotor system shown in Figure 5.1. In the system,

R = armature resistance = 2 ohms

L = armature inductance =0.2 henry

K_m = motor-torque constant = 0.2 N-m/ampere

K = back emf constant = 0.02 volts-s/rad

ω = θ = velocity angular of the motor shaft, radian J = moment of inertia of the motor = 0.02 kg-m²

b = viscous-friction coefficient of the motor = 0.02 N-m/rad/s v = applied armature voltage, volt i = armature current, ampere e = back electromotive force (emf), volt

T = torque developed by the motor, N-m

T_l = load torque applied to the motor shaft ( we consider it as a disturbance here)

1. Show that the differential equations of the system are 

iR+Ldi/dt=V-Kω  and Jω+bω=T-Tl                                               

Find the transfer functions Ω(s)/V(s) and Ω (s)/T_l(s) (where Ω (s) and V(s) are the      Laplace ω and v)

[6 marks]

2. Calculate the values of the natural frequency and the damping ratio of the system represented the transfer functions Ω (s)/V(s).

[4 marks] 

3. By using the numerical values given above, develop a proportional controller, v=K_pe=K_p(ω _d- ω), where ω _d is the desired speed] and for a unit step input choose the value of K_p to make the steady state error < 5% and for a unit step disturbance the steady-state error <10%.

[10 marks]

4. Briefly explain why the proportional gain K_p cannot be chosen to be very big and propose a way to reduce or eliminate the steady-state error.

[5 marks]  

Total [25] marks.

Purchase full paper by adding to cart

Last updated: Aug 08, 2021 10:48 AM

add to cart ← Back