Feedback command for MIMO systems in MATLAB

I have a state space model with 12 outputs and 4 inputs. I want to close the loop of 8th output with 3 input. then in outer loop i want to close outer loop for 1 output and 3 input again (Quadrotor Pitch and X position states). I have designed PD controller for both loops which is in feedforward.

states = {'x' 'y' 'z' 'u' 'v' 'w' 'Phi' 'Theta' 'Psi' 'p' 'q' 'r'};
inputs = {'Thrust' 'Roll Torque' 'Pitch Torque' 'Yaw Torque'};
outputs = {'x' 'y' 'z' 'u' 'v' 'w' 'Phi' 'Theta' 'Psi' 'p' 'q' 'r'};
STATE_SPACE_MODEL = ss(A,B,C,D,'statename',states,'inputname',inputs,'outputname',outputs);

Now, I can close the first inner loop this way

Pitch_State=STATE_SPACE_MODEL('Theta','Pitch Torque');
PITCH_CL=feedback(Pitch_State*P_PD,1);

now, i want to close the outer loop that contains 'x' state and have a seperate PD controller. I cant find out how to close the outer loop using feeback command can anyone help me out?

Answer 1

[Paul]

I think this does what you want (assuming negative feedback)

Pitch_State = STATE_SPACE_MODEL({'x';'Theta'},'Pitch Torque')
PITCH_CL = feedback(Pitch_State*P_PDinner,1,2);
PITHC_CL = feedback(PITCH_CL,P_PDOuter,1,1);

PITCH_CL will have one input and two outputs still named 'Pitch Torque' and 'x' / 'Theta' respectively. You'll probably want to rename that input.

[Muhammad Adeel Ahsan Awan]

let suppose

N=100;
Pinner_PD=0.15+(0.32*N/(1+(N/s)));
P_outter=0.15+(0.32*N/(1+(N/s)));

and if we run the code below

Pitch_State = STATE_SPACE_MODEL({'x';'Theta'},'Pitch Torque')
PITCH_CL = feedback(Pitch_State*P_PDinner,1,2);
PITHC_CL = feedback(PITCH_CL,P_PDOuter,1,1);

then i get this error

Error using * (line 80)

Model I/O dimensions must agree.

Error in Untitled3 (line 3)

PITCH_CL = feedback(Pitch_State*P_PD,1,2);

[Paul]

Transcription and typo errors on my part:

Pitch_State = STATE_SPACE_MODEL({'x';'Theta'},'Pitch Torque')
PITCH_CL = feedback(Pitch_State*P_PDinner,1,1,2);
PITCH_CL = feedback(PITCH_CL*P_PDOuter,1,1,1);

[Muhammad Adeel Ahsan Awan]

Thanks alot!!!

It worked!

[Muhammad Adeel Ahsan Awan]

can you tell me how can i plot the margins/bode of open loop (X outer)?

[Paul]

What have you tried so far?

After you close the inner loop, how would you form the open loop transfer function for the outer loop?

[Muhammad Adeel Ahsan Awan]

I tried and i obtained the required bode plot