Creating Symbolic state space model and transforming to canonical form?

Hi all,

Is it possible to create a state space model based on symbolic parameters?... and then convert this model into controllable cannonical form?

Thanks,

Johnathon Street

Answer 1

[Paul]

Yes, at least in principle. For example:

>> A=sym('a',3)
 
A =
 
[ a1_1, a1_2, a1_3]
[ a2_1, a2_2, a2_3]
[ a3_1, a3_2, a3_3]
 
>> B=sym('b',[3 1])
 
B =
 
 b1
 b2
 b3
 
>> C=[B A*B A^2*B];  % controllability matrix
>> t3 = [0 0 1]/C;
>> Tinv=[t3;t3*A;t3*A^2]; % state transformation
>> Ac = simplify(Tinv*A/Tinv,100)
 
Ac =
 
[                                                                                                   0,                                                                     1,                  0]
[                                                                                                   0,                                                                     0,                  1]
[ a1_1*a2_2*a3_3 - a1_1*a2_3*a3_2 - a1_2*a2_1*a3_3 + a1_2*a2_3*a3_1 + a1_3*a2_1*a3_2 - a1_3*a2_2*a3_1, a1_2*a2_1 - a1_1*a2_2 - a1_1*a3_3 + a1_3*a3_1 - a2_2*a3_3 + a2_3*a3_2, a1_1 + a2_2 + a3_3]
 
>> Bc = simplify(Tinv*B)
 
Bc =
 
 0
 0
 1

[Johnathon Street]

what matrix did you use to create your t3? just out of curiousity

[Johnathon Street]

Actually would like to replicate it as my matrix is 2x2

[Paul]

Edited the answer to show t3.

Should be easily adaptable to your case.

[Johnathon Street]

Thank you for the help Paul.

I feel like i should know how to do that but i cant seem to find an answer anywhere.

well until now.

Thanks again

[Walter Roberson]

A=sym('a',3)

A = 

B=sym('b',[3 1])

B = 

C=[B A*B A^2*B];  % controllability matrix
t3 = [0 0 1]/C;
Tinv=[t3;t3*A;t3*A^2]; % state transformation
Ac = simplify(Tinv*A/Tinv,100)

Ac = 

Bc = simplify(Tinv*B)

Bc = 

Works for me. I just copied Paul's code exactly.

[Johnathon Street]

Yeah I just now got it to work. I dont know what i was doing...

It works just fine.

Thanks again