Comment on the stability of the system and design a full

Question

Question 1:

The file 'winding.dat' contains input-output data from an industrial winding process. The file winding.txt contains a detailed description of the process. Make sure you place the 'winding.dat' file in the 'Matlab' folder in 'Documents'. Read 'winding.txt' carefully then:

1. Use the Matlab system identification toolbox to determine a best fit model for the process. Screen capture the best fit model fit.

2. import the model to the Matlab workspace: If the model is not in state-space form, convert it to state-space and answer the following:

(a) Is the system fully controllable?

(b) Is the system fully observable?

(c) Is the system stable?

(d) Plot the step response of the system Include screen shots of the relevant sections in your answers.

Question 2:

A linear time invariant system is described as:

with initial conditions x₁ = 2, x₂ = -2, w₁ = 0.1sin(cos(t)), and d(t) represents the unknown disturbance.

Design a robust sliding mode regulator with sliding mode gain, λ = 5 on the sliding mode surface. Simulate your designed controller with Matlab. Plot the control input, and system state vector. (Hint: In the simulation, use "rand" in Matlab to generate a single random number.)

Question 3

Given the an industrial sub-process described by the following LTI system:.

1. Comment on the stability of the system

2. Design a full state feedback controller for the system with poles at -3, and -2 ± j0.4 using the pole placement method.

3. Using Matlab, plot the step response of the controlled system.

Question 4

Given the state-equation of a linear system as

x^·(t) = Ax(t) + Bu(t)

with u(t) = constant for KT ≤ t ≤ (k+1)T.

The system is discretised resulting in the following discrete-time equation:

X[(k +1)T] = Φ (T)X(kT) + θ (T)u(kT)

if

find the matrices Φ (T) and θ (T).