In Octave, you can perform the requested tasks step by step for the given transfer functions and system. Here's how to approach each part:

Given Data:

• $G_p = G_d = \frac{1}{2s^2 + 9s + 2}$

• $G_f = G_m = 1$

• $G_c = 2$

• The closed-loop transfer function is:

  $\frac{dy(s)}{ds} = \frac{G_p G_f G_c}{1 + G_p G_f G_c G_m} y_{sp}'(s)$

This expression represents the closed-loop response, and we can solve it in Octave.

Step-by-step instructions:

1. Closed-loop response for a unit step change and inverse response:

You need to form the closed-loop transfer function and then compute the step response.

1. First, define the transfer function in Octave.

octaveCopy code
pkg load control  % Load the control package if necessary

% Define G_p as a transfer function
s = tf('s');
G_p = 1 / (2*s^2 + 9*s + 2);
G_f = 1; % Given G_f = 1
G_m = 1; % Given G_m = 1
G_c = 2; % Given G_c = 2

% Closed-loop transfer function
sys_cl = (G_p * G_f * G_c) / (1 + G_p * G_f * G_c * G_m);

2. Calculate the step response of the closed-loop system for a unit step input.

octaveCopy code
% Step response
step(sys_cl);
title('Closed-loop Step Response');

For the inverse response, you can compute and plot the impulse response to examine the system behavior under such conditions.

octaveCopy code
% Impulse response (for inverse response)
impulse(sys_cl);
title('Closed-loop Impulse Response');

2. Find the poles and zeros of the closed-loop transfer function:

You can use the following commands to find the poles and zeros.

octaveCopy code
% Find poles and zeros of the closed-loop system
poles = pole(sys_cl)
zeros = zero(sys_cl)

3. Plot the Pole-Zero Map (pzmap), Impulse, and Step Responses:

To plot the pole-zero map, impulse response, and step response, use:

octaveCopy code
% Plot pole-zero map
pzmap(sys_cl);
title('Pole-Zero Map');

% Plot step response
figure;
step(sys_cl);
title('Closed-loop Step Response');

% Plot impulse response
figure;
impulse(sys_cl);
title('Closed-loop Impulse Response');

4. Find the partial fraction and final inverse function:

To find the partial fraction expansion of the closed-loop transfer function:

octaveCopy code
% Partial fraction expansion
[r, p, k] = residue([num], [den]);  % Replace [num] and [den] with the numerator and denominator of sys_cl

% Display the partial fractions
[r, p, k]

Finally, you can apply inverse Laplace to the partial fractions to obtain the time-domain function:

octaveCopy code
% Inverse Laplace transform (manually from the partial fraction result, using standard formulas)

Tips:

• Use tfdata to extract the numerator and denominator for the partial fraction expansion if necessary.
• You can use symbolic computation for the inverse Laplace transform if you prefer a symbolic approach.

Questions for further exploration:

1. How do the poles affect the system's stability in the closed-loop response?
2. What is the significance of the pole-zero plot for understanding system dynamics?
3. How would the system change if $G_c$ had a different value?
4. Can you explain the relationship between the impulse and step response in this system?
5. How can we use the partial fraction decomposition to better understand system behavior?

Tip:

When analyzing control systems, pay close attention to pole locations, as they directly impact the system's stability and transient behavior.