How to PID tuning to meet conditions for settling time and overshoot

PID tuning to meet conditions for settling time and overshoot while a stable system with minimum peak time and zero velocity error.

So I am trying to find the gain values for a PI control system that would give me a settling time not exceeding 6 seconds, and an maximum overshoot not going over 5% while ensuring that the peaktime is the lowest it can be, and that the system is stable, and also has zero velocity error.

I have written the following code. Starting with a kp and ki value of 1 each, I get a system that gives desirable overshoot and settling time, but I am wondering if the peaktime can be even lower while still having settling time <= 6 and overshoot <= 5.

I am using the following toolboxes: Control System Toolbox

Questions

1. Using the rlocus function, I have also shown that the real parts of the poles are negative, so this demonstrates that my system is stable right?
2. Also am I using Lsim correctly to determine if velocity error is zero? The resultant graph has a gray line showing the time response, and a blue line that is parallel to it. This means zero velocity error right?
3. What is the best way to fine-tune the gain values kp and ki to minimize peak time while ensuring the above conditions are still met?

I would like to use matlab only and no simulink for this please.

clc
clear all

% plant transfer function
G = tf([1], [0.5 1.5 1]) 
kp = 1
ki = 1
% PI controller
C = tf([kp ki], [1 0])
% closed loop transfer function
T = feedback(C*G, 1)
rlocus(T)
% Find the poles
poles = pole(T)
% step response
figure;
step(T);
title('Step Response');
grid on;
% Step analysis
info = stepinfo(T)
% Ramp Input
t = 0:0.01:10;
ramp = t;
% System response to ramp
figure;
lsim(T, ramp, t);
title('Ramp Response')
legend
grid on

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Here is the solution using pidtune(). There is no direct way to input the desired settling time and overshoot percentage; however, you can enter the desired phase margin. This has been a concern for me in MATLAB for many years. Nevertheless, based on the desired overshoot percentage, you can apply the formula from your lecture notes to determine the desired phase margin.

%% The Plant
Gp  = tf([1], [0.5 1.5 1])

Gp =
 
           1
  -------------------
  0.5 s^2 + 1.5 s + 1
 
Continuous-time transfer function.

%% Using pidtune
Pm  = 68.2;     % desired Phase Margin
opt = pidtuneOptions('PhaseMargin', Pm, 'DesignFocus', 'balanced');
[Gc, info] = pidtune(Gp, 'pidf', opt)

Gc =
 
             1            s

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