PSB500EN – Control of Mechatronics Systems Course Work 2 (CW2) – Modelling & Control of non-linear Systems BEng. Electromechanical Engineering

PROBLEM: non-linear state space model

The following diagram shows an idealised representation of a small underwater vehicle thruster.

The following assumptions are considered for this model:

• The resistor is ohmic and inductor La has ordinary linear relationship.
• The gyrator with modulus Km is a permanent magnet DC motor and, in this case, it is acting as a motor.
• The total inertia of the system (propeller, rotor) is captured in the lumped inertia J.
• The propeller produces a nonlinear, velocity dependent drags on the system:

T_b = aω + bω²      where ω > 0 is the shaft / propeller angular velocity

Assume that the voltage source V the only input, the angular velocity of the output shaft as the only output and they are known as the following signals:

u1 = V(t) as input
y1 = ω(t) as output

The motor relationship are T_m = K_m I_a and induced voltage e = K_m ω

Answer the following question/tasks:

1. Derive the differential equation which describe the system characteristics in time domain.
2. Derive the state space equation for this system, assuming that the state variables are:

z₁(t) = λ(t)     magnetic flux

z₂(t) = L_j(t)

The outcome of the state space model for this system is supposed to be:

z₁ = u1 – ^R_a/_La z₁ – ^k_m/_J z₂

z₂ = ^k_m/_La z₁ – ^a/_J z₂ – ^b/_Jz z₂²

y₁ = ¹/_J z₂

Show your workings in details and thereafter enter this model using matlab simulation (either m file and/or Simulink method)

Assuming that the following parameters are given, determine the state space model in terms of the two state variables z₁ andz₂

Knowing that z₁ = 0 and z₂ = 0 for equilibrium point determine the exact location of equilibrium point z

Perform linearisation about this equilibrium point and finally write down the state space equation with matrix A, B , C and D

You are required to prepare a technical report with the above analysis. Please split the various tasks to different group member and each one of you is taking ownership & responsibility to your parts and success of the whole project. Each one of you need to do brief presentation in the part that you are doing. Anyone caught, not doing anything/just take a ride, will fail in this CW2