ss2ss - state-space to state-space conversion, feedback, injection

Calling Sequence

[Sl1,right,left]=ss2ss(Sl,T, [F, [G , [flag]]])

Parameters

• Sl : linear system (syslin list) in state-space form
• T : square (non-singular) matrix
• Sl1, right, left : linear systems (syslin lists) in state-space form
• F : real matrix (state feedback gain)
• G : real matrix (output injection gain)

Description

Returns the linear system Sl1=[A1,B1,C1,D1] where A1=inv(T)*A*T, B1=inv(T)*B, C1=C*T, D1=D.

Optional parameters F and G are state feedback and output injection respectively.

For example, Sl1=ss2ss(Sl,T,F) returns Sl1 with:

and right is a non singular linear system such that Sl1=Sl*right.

Sl1*inv(right) is a factorization of Sl.

Sl1=ss2ss(Sl,T,0*F,G) returns Sl1 with:

and left is a non singular linear system such that Sl1=left*Sl (right=Id if F=0).

When both F and G are given, Sl1=left*Sl*right.

- When flag is used and flag=1 an output injection as follows is used

and then a feedback is performed, F must be of size (m+p,n)

( x is in R^n , y in R^p, u in R^m ).

right and left have the following property:

  Sl1 =  left*sysdiag(sys,eye(p,p))*right 
   

- When flag is used and flag=2 a feedback (F must be of size (m,n)) is performed and then the above output injection is applied. right and left have the following property:

  Sl1 = left*sysdiag(sys*right,eye(p,p)))
   

Examples

Sl=ssrand(2,2,5); trzeros(Sl)       // zeros are invariant:
Sl1=ss2ss(Sl,rand(5,5),rand(2,5),rand(5,2)); 
trzeros(Sl1), trzeros(rand(2,2)*Sl1*rand(2,2))
// output injection [ A + GC, (B+GD,-G)]
//                  [   C   , (D   , 0)]
p=1,m=2,n=2; sys=ssrand(p,m,n);

// feedback (m,n)  first and then output injection.

F1=rand(m,n);
G=rand(n,p);
[sys1,right,left]=ss2ss(sys,rand(n,n),F1,G,2);

// Sl1 equiv left*sysdiag(sys*right,eye(p,p)))

res=clean(ss2tf(sys1) - ss2tf(left*sysdiag(sys*right,eye(p,p))))

// output injection then feedback (m+p,n) 
F2=rand(p,n); F=[F1;F2];
[sys2,right,left]=ss2ss(sys,rand(n,n),F,G,1);

// Sl1 equiv left*sysdiag(sys,eye(p,p))*right 

res=clean(ss2tf(sys2)-ss2tf(left*sysdiag(sys,eye(p,p))*right))

// when F2= 0; sys1 and sys2 are the same 
F2=0*rand(p,n);F=[F1;F2];
[sys2,right,left]=ss2ss(sys,rand(n,n),F,G,1);

res=clean(ss2tf(sys2)-ss2tf(sys1))
 

See Also

projsl  feedback