PID simulink block for MIMO system

Pls I need help... I have this satellite system with 3axial control and i want use PID controller.. So i need simulink block for PID controller for MIMO system. Pls help as this is needed for my Thesis

[Sam Chak]

Hi Enejor Emmanuel

Please note that dynamics is decoupled. In other words, you can apply the PID controller directly on the θ axis.

If you study the stability of the system, and you select the parameters to ensure that , then you use two PD controllers for stabilize the ϕ–ψ motion. Still you cannot tune the gains in the PD Controller blocks because the auto-tuning algorithm requires a SISO model, though you can enter the values of the control gains manually.

[Enejor Emmanuel]

Thank you professor Sam ... I really appreciate your response... But pls can you help with the PID solutions I already tried the lqr and I want to compare the results from PID with the solutions from the lqr... I will really appreciate if you can take a little time from your busy schedule to help... I await your response

[Sam Chak]

Hi Enejor Emmanuel

I hope you read the answer carefully. A method of implementing the PID controllers was suggested together with the equations. I'm merely suggesting control strategies for you to deal with the control problem, which is about implementing the SISO-based PID controllers in the MIMO system.

Another way to implement the PID controllers is through the designed LQR. Recall the gains from the LQR to control ϕ:

.

You can group them and make 3 sets of PD controllers for :

.

In theory, you should obtain more or less the same performances.

If these strategies are not what you seek to learn for solving the control problem in Simulink, please consult with your academic supervisor to find out exactly what he or she expects.

Answer 1

[Sam Chak]

Hi Enejor Emmanuel

This simple simulation is just to show you that it is possible to design and implement the PID controllers based on the control strategy that I have described previously. The initial conditions in this simulation are , , .

Answer 2

[Sam Chak]

Hi Enejor Emmanuel

The PID control tuning requires the plant model that must be single input, single output (SISO). Because the dynamics are coupled, if you want to tune the PID controller for each axis, you could first use the feedback linearization technique to decouple the dynamics with respect to the control inputs, and then tune the PID controller accordingly. For example,

.

With that, you can construct a transfer function in Simulink and then use the PID controller block to tune it

.

Else, the Linear Quadratic Regulator (LQR) is a popular control method that can handle linear MIMO systems.

% parameters
Ix = 3*9.8194;
Iy = 3*9.7030;
Iz = 3*9.7309;
wo = 0.00106496; 	% calculated exactly at altitude 686 km
a1 = -4*(wo^2)*(Iy - Iz)/Ix; 
a2 = -3*(wo^2)*(Ix - Iz)/Iy; 
a3 = -1*(wo^2)*(Iy - Ix)/Iz; 
a4 = (Ix - Iy + Iz)*wo;
b1 = 1/Ix; 
b2 = 1/Iy;
b3 = 1/Iz;

% state space matrices
A = [0 0 0 1 0 0; 0 0 0 0 1 0; 0 0 0 0 0 1; a1 0 0 0 0 a4/Ix; 0 a2 0 0 0 0; 0 0 a3 -a4/Iz 0 0];
B = [zeros(3); diag([b1, b2, b3])]; 
C = eye(6); 
D = zeros(6, 3);

% design of LQR
Q = 1e0*C'*C
R = 1e0*eye(rank(B))
[K, S, e] = lqr(A, B, Q, R)
sysC = ss(A-B*K, B, C, D)
step(sysC)