Control Subspace of LTI System X = Ax + Bu: Nonzero Parameter Analysis

[ayham87]

Homework Statement

Given the LTI system X =Ax+Bu where

A =[ 0 1 0 0 0 0; a1 0 0 a2 0 0;0 0 0 1 0 0;0 a3 0 0 0 0; 0 0 0 0 0 1;0 0 0 0 a4 0];
B = [0 0 0 ;b1 0 0 ; 0 0 0 ;0 b2 0 ;0 0 0 ;0 0 b2];


and the parameters a1; a2; a3; a4; b1; b2 are all nonzero. Determine whether the system is completely controllable. If not, find the controllable subspace of the state space.

The Attempt at a Solution

I tried systematically but it seems to long so i try by MATLAB by the following code

syms a1 a2 a3 a4 b1 b2
A =[ 0 1 0 0 0 0; a1 0 0 a2 0 0;0 0 0 1 0 0;0 a3 0 0 0 0; 0 0 0 0 0 1;0 0 0 0 a4 0];
B = [0 0 0 ;b1 0 0 ; 0 0 0 ;0 b2 0 ;0 0 0 ;0 0 b2];

Co = ctrb(A,B)


Thanks in advanced

 

[n.karthick]

The variables A and B need not be symbolic variables. Just declare A and B as double floats and use ctrb function. It worked for me.

 

[ayham87]

Thanks for your response

actually "syms" not for A & B, its for the variables (a1 a2 b1 b2 a3 a4), I declare A & B as double float :

syms a1 b1 a2 a3 a4 b2
A =[ 0 1 0 0 0 0; a1 0 0 a2 0 0;0 0 0 1 0 0;0 a3 0 0 0 0; 0 0 0 0 0 1;0 0 0 0 a4 0];
B = [0 0 0 ;b1 0 0 ; 0 0 0 ;0 b2 0 ;0 0 0 ;0 0 b2];
A = float('double');
B = float('double');
Co = ctrb(A,B);


but i get the following error:

The following error occurred converting from struct to double:
Error using ==> double
Conversion to double from struct is not possible.

Error in ==> ctrb at 32
co(:,1:nu) = b;


can u help me again...thanks in advanced

 

[viscousflow]

the ctrb function doesn't handle symbolics. Just do the steps to find out controllability "by hand" via matlab. Its not that long, I've done it already, just create a few loops and such. I can provide the code I used to do it before I figured out MATLAB had the functionality, if you'd like.

 

[DeeNos]

for your response!

Based on the given system, the controllability matrix Co has a rank of 6, which is equal to the dimension of the state space. This indicates that the system is completely controllable. The nonzero parameters do not affect the controllability of the system.