MA3005/MA3705 Control Theory Homework

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MA3005/MA3705 Control Theory

Homework (Due 06/11/22, 23:59)

Instructions

1.    Show all ofyour workings and drawings clearly.

2.    Submit your homework as a pdf. The homework can be typed, written digitally or via pen and paper. Do ensure that the orientation of your submitted homework is upright.

3.    You may use numerical software such as MATLAB for this homework. However, you have to sketch the root locus manually and not generate it via MATLAB or other software.

1. An electric bicycle is constrained to move along a straight line and its displacement can be represented by x (Fig. 1). Its back wheel can be actuated by a motor to produce a mechanical torque of T (t). As the wheels rotate, they will experience friction from the ground. This bicycle is also connected to a spring that has a spring constant of K and a damper that has a damping coefficient of C. It is given that the mass of the bicycle is M, while each wheel has a radius ofR and a moment of inertia ofJ. The angular displacements of the back wheel and front wheel can also be represented by e1  and e2 , respectively. Assuming no slip conditions and zero initial conditions, derive  where T(s) = L(T (t)) . Also, find the damping ratio, natural frequency and DC gain of this system.                                               (8 marks)

 

Fig. 1. An electric bicycle.

2. Draw the root locus for the unity feedback system shown in Fig. 2 by answering all the questions below.

s)

K

Fig. 2. A unity feedback system.

(a) Identify all the starting and ending points of the locus.                                           (4 marks)

(b) Find all the departure angles (if applicable).                                                           (2 marks)

(c) Find all the arrival angles (if applicable).                                                                (2 marks)

(d) Find all the break-in and break-out points (if applicable).                                      (3 marks)

(e) Manually sketch the root locus.                                                                               (4 marks)

3. The transfer function of another system is given as:

G(s) = R(s) = (s + 20)2 .

 

Find the steady-state output of the system, css (t), when:

r(t) = 20 sin (t +2(几)).






2022-11-04