Rotational Mast System - Transfer Function, PID Design

[YoshiMoshi]

Homework Statement

Given the system above, were a motor is rotating a mast that is attached to a second second mast by a spring with a spring constant K and a damping coefficient D.

The motor has a current I applied to it with coefficient K

Below are the provided values.

K_s is the spring constant. The other K is the machine constant. PO is the desired percent overshoot. T_s is the desired settling time. e_ss is the desired steady state error. I is the current applied to the motor.

Homework Equations

The Attempt at a Solution

I derive the ODE for the first mast, and solve for the angular acceleration, I get this.

I derive the ODE for the second mast, and solve for the angular acceleration, I get this.

I then put this into state space model A, B, C, D form

I put this into MATLAB to get the transfer function of the system.

A = [0, 0, 1, 0; 0, 0, 0, 1; -1/112500, 1/112500, -1/90000 1/90000; 4, -4, 1/75000, -1/75000];

B = [0; 0; 1250; 0];

C=[0, 0, 0, 1];

d=0;

[b, a]=ss2tf(A,B,C,d);

Tofs=tf(b, a)

Tofs =

0.01667 s^2 + 5000 s - 1.826e-12
-----------------------------------------------------
s^4 + 2.444e-05 s^3 + 4 s^2 - 6.099e-18 s - 2.055e-20

Continuous-time transfer function.

I then attempt to find the characteristics with MATLAB of the uncompensated system. This yields.

stepinfo(Tofs)
ans =
struct with fields:
RiseTime: NaN
SettlingTime: NaN
SettlingMin: NaN
SettlingMax: NaN
Overshoot: NaN
Undershoot: NaN
Peak: Inf
PeakTime: Inf

Is this a problem? The uncompensated system has a percent overshoot, and settling time of N/A. I can design a PID controller for this system so that way it has the desired characteristics?

I feel like maybe I have done something wrong, but I'm not completely sure. Any help would be greatly appreciated.

I really think that I did something wrong because I got that the system is unstable in MATLAB.

isstable(Tofs)
ans =
logical
0

 

[mark726]

Hi there,

It seems like you have correctly derived the ODEs and put them into state space form. However, I'm not sure if your values for the coefficients and constants are correct. For example, the damping coefficient D is missing from your state space model and the values for the spring constant K and the machine constant K are not specified. It would be helpful to have more information about these values in order to accurately model the system.

Also, it is possible that your system is unstable, which is why you are not getting any values for the desired characteristics. In that case, designing a PID controller would be necessary to stabilize the system and achieve the desired characteristics. I would recommend double checking your equations and values, and if everything seems correct, then designing a controller would be the next step.