University College Dublin
School of Electrical, Electronic & Communications Eng.
EEEN40010 Control Theory

Experiment 4CT2

Digital Control

1. Objective:

To investigate the design of digital controllers.

2. Background Information:

See earlier handouts.

The commands c2d or c2dm permit evaluation of an equivalent discrete-time system for any given continuous-time system. You may also wish to use the command zgrid. Use the help or doc commands for more information. Commands ss and tf may be useful when employed with an additional input equal to the sampling period.

3. Problem

1. The fan and plate is a classic control teaching problem. It comprises a DC motor driving a fan which creates an air stream sufficient to push a hanging plate. We adopt the following very approximate model:

L di(t)/dt + Ri(t) + kω(t) = e(t)

T(t) = ki(t)

J(dω(t))/dt = T(t) - μω(t) - T_s if ω(t) > 0

F(t) = v(T(t) - T_U)

Id²θ(t)/dt² = F(t -t_L)cos(θ(t)) - mgsin(θ(t)) - η.dθ(t)/dt

where i is the DC motor current, ω is the angular velocity of the fan, θ is the angle which the plate makes with the downward vertical. Note that ω is the angular velocity of the fan, not of the plate. tL is latency, indicating that there is a short time lag as the air moves from the fan to the plate. It is better to consider this system to be made up of three separate systems: (i) the fan, being a system with input e(t) a motor voltage and output T(t) a torque;

(ii) the air, being a system with input F(t) a force and output F(t-t_L) a delayed force; (iii) the plate, being a system with input F(t-L) and output θ(t) the plate angle. Given R = 0.2 Ω and T_U = 2T_S show that (e, i, ω, θ) = (10 V, 20A, 180 rpm, 0 rad) is a valid operating point provided μ = 10/(6Π²) kgm²/s, k = 1/Π Vs and TS = 10/Π kgm²/s². Choosing these values show that the linearisation about the selected operating point is:

For remaining parameter values we will take: t_L = 0.2 sec, m = 0.284 kg, g = 9.81 m/s², L = 1 mH, J = 0.013 kgm², η = 0.3076 kgm/s, I = 0.0195 kgm and v = 0.11 m^-1.

2. Using Simulink characterise the unit step response of the linearised plant in terms of steady- state error, 2% settling time and percentage overshoot. Select a sampling period so that there are at least 20 samples before 2% settling time and the latency tL is an integer number of sampling periods.

3. Find the associated discrete-time system assuming that a zero order hold circuit is employed. Is the discrete-time system non-minimum phase?

4. Design a discrete-time linear state feedback controller with observer to achieve a zero steady-state error to a step input, a 2% settling time not exceeding 1.75 sec and a percentage overshoot not more than 10% Check the behaviour of the resulting closed loop system using both the linearised local model and the original global model.

5. Design a discrete-time PID/lead/lag controller using the root locus method to meet the same specifications as in question 4. Check the behaviour of the resulting closed loop system using both the linearised local model and the original global model.

6. Design also a discrete-time PID/lead/lag controller using loop-shaping to meet the same specifications as in question 4. Check the behaviour of the resulting closed loop system using both the linearised local model and the original global model.

Attachment:- DIGITAL CONTROL.rar