PSB500EN Control of Mechatronics Systems Coursework 2 Assignment Questions | CU

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 the inductor La has an 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 drag on the system:

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

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

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

The motor relationship is Tm = Km Ia, and the induced voltage e = Km ω

Answer the following questions/tasks:

Derive the differential equation that describes the system characteristics in the time domain.

Derive the state space equation for this system, assuming that the state variables are:

z1(t) = λ(t)     magnetic flux

z2(t) = Lj(t)

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

z1 = u1 – Ra/La z1 – km/J z2

z2 = km/La z1 – a/J z2 – b/Jz z22

y1 = 1/J z2

Show your workings in detail 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 and z2

Knowing that z1 = 0 and z2 = 0 for the equilibrium point, determine the exact location of the 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 among different group members, and each one of you is taking ownership & responsibility for your part and the success of the whole project. Each one of you needs to do a brief presentation on the part that you are doing. Anyone caught not doing anything/just taking a ride will fail in this CW2.