Question 1-

The charge-coupled device (CCD) which is used in video movie cameras to convert the image into electrical signals can be used as part of an automatic focussing system in 35-mm cameras as shown in Figure 1a. The automatic focus system is a position control, where the desired position of the lens is an input selected by pointing the camera at the subject. The output is the actual position of the lens. The open-loop transfer function of this focussing system is

G_p(s) = 10/s(s+3)(s+9)

1) Determine a phase-lead controller G_c(s) of the form

G_c(s) = K(s+z₀)/(s+p₀)

for a closed-loop feedback control system as shown in Fig. 1b so that the system has a time constant of 0.5 s and a damping ratio of 0.707.

2) Determine the stability of the system. Hint: Matlab commands can be used; the command and the results must be given in the report.

3) Obtain the unit step response and the corresponding control effort. Hint: Matlab commands and/Simulink simulations can be used; the commands and Simulink models and the results must be given in the report.

Question 2-

2.1 Consider a gas fired industrial water boiler. The internal temperature y(t) of the system is controlled by providing an appropriate plant excitation signal u(t) which drives a gas flow control valve. The temperature measurement from a linearized thermocouple signal detects the radiative temperature of the gas combustion.

The open loop transfer function of this system is found from the least squares fit analysis

G_p(s) = Y(s)/U(s) = (2/(1+50s))e^-10s

The transportation lag in the above open-loop transfer function can be simplified using the following Pade' approximation:

e^-1d^s = (1/(1+t_ds+(t_d²s²/2!)))

Using the Ziegler-Nichols' ultimate-cycle method, design a PID controller G_c(s) of the form

G_c(s) = K(s+a)²/s

For a closed loop system as shown in Figure 2a.

2.2 Approximate Gc(s) by a proper transfer function.

2.3 Assume this water boiler system can be represented by a state-variable model

Design a state-variable feedback controller Kc for a closed-loop system shown in Fig. 2b. The controlled system is required to have a time constant of 10 s with a damping ratio of 0.707. Determine the corresponding loop transfer function, and derive the expression of the step response in time domain.