Thread Subject: Working with state-space representation of multivariable models

Hello,
I'm trying to figure out how to convert from transfer function
representation to state-space representation.
For example:
g1=tf(1,[1 1]);
g2=tf(2,[1 2]);

G = [g1,g2;g2,-g1]
will give this:

Transfer function from input 1 to output...
        1
 #1: -----
      s + 1

        2
 #2: -----
      s + 2

Transfer function from input 2 to output...
        2
 #1: -----
      s + 2

       -1
 #2: -----
      s + 1

Now, I'm trying to find a way how to convert this to state-space
model. It seems that tf2ss is not working with MIMO systems, and yet
state-space representation supports MIMO models.
If there is no MATLAB function, can you tell how to approach this
problem?
Thank you

Use the ss command to cast the transfer function to a state-space model.

ss(G)

-craig


"Micik" <brobigi@yahoo.com> wrote in message
news:43928588-e29d-48b8-8fed-40e030c712ca@f63g2000hsf.googlegroups.com...
> Hello,
> I'm trying to figure out how to convert from transfer function
> representation to state-space representation.
> For example:
> g1=tf(1,[1 1]);
> g2=tf(2,[1 2]);
>
> G = [g1,g2;g2,-g1]
> will give this:
>
> Transfer function from input 1 to output...
> 1
> #1: -----
> s + 1
>
> 2
> #2: -----
> s + 2
>
> Transfer function from input 2 to output...
> 2
> #1: -----
> s + 2
>
> -1
> #2: -----
> s + 1
>
> Now, I'm trying to find a way how to convert this to state-space
> model. It seems that tf2ss is not working with MIMO systems, and yet
> state-space representation supports MIMO models.
> If there is no MATLAB function, can you tell how to approach this
> problem?
> Thank you

Thank you Craig.
When browsing through help I skiped ss function because I saw:SYS =
SS(A,B,C,D) in function prototype. Help about this could be better.
That solves my problem

Micik <brobigi@yahoo.com> wrote in message <43928588-e29d-48b8-8fed-40e030c712ca@f63g2000hsf.googlegroups.com>...
> Hello,
> I'm trying to figure out how to convert from transfer function
> representation to state-space representation.
> For example:
> g1=tf(1,[1 1]);
> g2=tf(2,[1 2]);
>
> G = [g1,g2;g2,-g1]
> will give this:
>
> Transfer function from input 1 to output...
> 1
> #1: -----
> s + 1
>
> 2
> #2: -----
> s + 2
>
> Transfer function from input 2 to output...
> 2
> #1: -----
> s + 2
>
> -1
> #2: -----
> s + 1
>
> Now, I'm trying to find a way how to convert this to state-space
> model. It seems that tf2ss is not working with MIMO systems, and yet
> state-space representation supports MIMO models.
> If there is no MATLAB function, can you tell how to approach this
> problem?
> Thank you
============================================================

As I know, the number of state variables should be the same as the number of poles in the transfer function. If there are repetitive poles in the transfer function, SS command can not generate the minimum number of state variables for the problem. Let's look at this example:

Transfer function is:

H11=tf([-6.4 12.8],conv([0.5 1],[16.7 1]));
H12=tf([28.35 -18.9],conv([1.5 1],[21 1]));
H21=tf([-23.1 6.6],conv([3.5 1],[10.9 1]));
H22=tf([29.1 -19.4],conv([1.5 1],[14.4 1]));

H=[H11,H12;H21,H22];

The pole at -2/3 is repeated twice so that we have 7 poles. let's solve it by SS function the result is

sys=ss(H)

a =
            x1 x2 x3 x4 x5 x6 x7 x8
   x1 -2.06 -0.479 0 0 0 0 0 0
   x2 0.25 0 0 0 0 0 0 0
   x3 0 0 -0.3775 -0.2097 0 0 0 0
   x4 0 0 0.125 0 0 0 0 0
   x5 0 0 0 0 -0.7143 -0.127 0 0
   x6 0 0 0 0 0.25 0 0 0
   x7 0 0 0 0 0 0 -0.7361 -0.1852
   x8 0 0 0 0 0 0 0.25 0
 
b =
       u1 u2
   x1 2 0
   x2 0 0
   x3 1 0
   x4 0 0
   x5 0 2
   x6 0 0
   x7 0 2
   x8 0 0
 
c =
            x1 x2 x3 x4 x5 x6 x7 x8
   y1 -0.3832 3.066 0 0 0.45 -1.2 0 0
   y2 0 0 -0.6055 1.384 0 0 0.6736 -1.796
 
d =
       u1 u2
   y1 0 0
   y2 0 0
 
Continuous-time model.


Which has 8 variables instead of 7. But the minimum number of variables is 7 and the beautiful 7 variable configuration is as follows:

A=diag([-2 -0.0599 -0.6667 -0.0476 -0.2857 -0.0917 -0.0694]);
B=[1 0;1 0;0 1; 0 1; 1 0; 1 0; 0 1];
C=[-1.5802 0.8138 1.9385 -1.0385 0 0 0 ; 0 0 3.0078 0 -1.7838 1.1783 -1.6605];

You can check both of the results by:

I8=eye(8);
syms s;
trans1=sys.c*(s*I8-sys.a)^(-1)*sys.b;
simplify(trans1)

OR

I7=eye(7);
trans2=C*(s*I7-A)^(-1)*B;
simplify(trans2);

Both of them can generate the transfer function.

My question is that how we can generate the minimum number of variables required for state space realization for this problem using MATLAB easily.