Q1)A simplified model of a dc motor, is given by

where i(t) = armature motor current, R = armature resistance (5 ohms),  u(t) = input voltage, L = armature inductance (200mH),  Ω(t) = motor speed,

J = motor inertia (0.02 kgm²),  k₁= back-emf constant (0.1 V/rad/s)   and k₂ = torque constant (0.1 Nm/A).

a) By setting x₁(t) = i(t) and x₂(t) = Ω(t) write the system in state-space form by using the above numerical values.

b) Assuming that the speed is measured, that is the output y(t) =W(t), state whether the system is state and/or output controllable?

c) Design an open-loop controller such that the speed has a constant value of 0.5 rad/s. Show your result via simulation using Simulink. Explain why open-loop control is not recommendable in practice.

d) We wish to design an output feedback controller of the form u(t) =ky +v in order to regulate the speed at a constant value. Give the range of values of k for which the closed-loop system is stable.

e) Show your result by carrying a simulation of your result using Simulink.

f) Now, we wish to design an state feedback controller of the form u(t) =kx +v in order to regulate the speed at a constant value. Show that the range of values of k for which the closed-loop system is stable larger than that obtained in Question d) for the output feedback. Give the advantage for using the state feedback over the output feedback.

g) Compare the state feedback to the output feedback through simulation using Simulink.

Q2) The transfer function of a linear system is given by

H(s) = Y(s)/U(s) = 50/s(s+250)

a) Simulate the step response of the above system using Simulink. Comment on the figure obtained.

b) Give a state space representation of the above system.

c) Give a discrete-time representation of the above system using a zero-order hold.

d) Analyse the stability and controllability of the discrete time model and comment on the result obtained.

e) Assuming T=0.1s, design a state feedback of the form u(n)=Kx(n)+v(n) so that the closed-loop system is stable. Confirm your result using Simulink.

f) For T=0.1s, calculate the value of v(n) such that the value of y(n) converges to 3 in steady state. Validate your result using Simulink.