MA3005 Control Theory

Homework 1

Consider the following model of an electric motor. 

The variables are (Laplace transforms in brackets):

• v (V): input voltage
• i (I): motor current
• e (E): back EMF
• ? (T): motor torque
• ω (Ω): output shaft angular velocity

The constant parameters are:

• L: electric inductance
• R: electric resistance
• b: motor viscous friction
• J: moment of inertia of the rotor

Q1 [5 marks] 

Consider the electrical sub-system. Find the relationship between V, E, and I. Assuming that V and E are the inputs and I is the output, draw the resulting block diagram.

Notes:? Electrical circuits will n? ot? be tested in the CA/exam.

Q2 [3 marks]  

Consider the mechanical sub-system. Find the relationship between T and Ω. Assuming that T is the input and Ω is the output, draw the resulting block diagram.

Q3 [5 marks] 

The couplings between the electrical and mechanical systems are given by:

• e = k_e? ω_? [counter-electromotive force]
• ? =  k_t? i_?

Connect the block diagrams found in Q1 and Q2 and compute the resulting transfer function Ω/V.

Q4 [7 marks]

In this question, we use the following numerical values

• L = 3 x 10^-? ⁶
• R = 4
• b = 4 x 10^-? ⁶
• J = 3 x 10^-? ⁶
• k_e? ? = 3 x 10^-? ²
• k_t? ? = 3 x 10^-? 2

1. [3 marks] Determine the stability of the system
2. [4 marks] Sketch the step response of the system for t ∈ [0, 0.2] (using e.g.

Python/Matlab)