This missile utilises a thrust vectoring control system, whereby adjusting the thrust vector angle

can control the pitch $($

$)$ pitch angle. After derivation of nonlinear motion equations of the missile, calculation of parameters such as aerodynamics coefficients, thrust etc., and finally, linearisation of those equations, a Transfer Function $($TF$)$ can be extracted for each operation point. A TF between

$($or

$-$ pitch angle$),$ and

for one operational condition can be expressed by the following equation $[1]$:

$(1)$

As can be observed from equation $(1),$ the system is third order including two unstable ones and one zero. As a design engineer, you are asked to design a PID controller with rate feedback for this system. An embodiment of this controller can be like Fig. $2$:

PID.jpg

Fig. $2$: A PID controller with rate feedback

In this block diagram,

is the desired pitch angle,

$,$

$,$ and

are PID gains. Moreover, Actuator and Missile symbolise their corresponding transfer functions. Using MATLAB and Simulink $($or any other simulation tool that you prefer$),$ you must design a pitch controller that is able to satisfy a set of controller requirements. Moreover, you are asked to analyse the effect of an actuator on your system's closed$-$loop behaviour. This can be achieved by conducting the Closed Loop Step Response and Frequency Domain analysis and you need to report your design regarding the closed$-$loop system performance and stability. The following design activities should be carried out and reported in your final submission:

$1-$ Consider the actuator is ideal and its transfer function is:

$.$ Using the obtained knowledge in this module, design and simulate a stable controller $($for unit step response in $20$s$)$ with the best achievable performance, and report the results in such a way that the following requirements $(12$ design criteria$)$ are satisfied: