Optimizing PD Tuner Parameters for Improved Performance: A MATLAB Study

[JI567]

Homework Statement

So I have been asked to design a PD Controller on Matlab that satisfies certain performance criteria which are as follows :

1. The 10-90% rise time is less than 0.5 s
2. The percentage overshoot is less than 5%.
3. The settling time is less than 5 s

I was given the closed loop transfer function of the system which had two parameters as Kd and Kp where
Kp lies between a range of 0<Kp<1.5
and Kd lies between a range of 0<Kd<4.4

Question is to figure out how Kd and Kp influence the performance criterias.

Homework Equations

The closed loop transfer function = $G_{cl}(s) \$ = $\frac {K_Ds + K_P} {1.1{s}^2+K_Ds+K_P} \$

The Attempt at a Solution

I have acheived the performance criteria as stated on MATLAB when I put $K_D \$ as maximum of 4.39 and $K_P \$ as around the half of the range as 0.7. However, I really can't tell how exactly they affect the performance criterias because I have produced many graphs with different values of both parameters and I just can't figure out a relationship between them. Please HELP!

 

[Adrmay]

One thing I have noticed is that if either of the parameters was set to be too high, the response would become very unstable. Is there any way I can state a mathematical relationship between Kd and Kp and these performance criterias?