ss2ss

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

Calling Sequence

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

Arguments

: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) 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 linear system projection
• feedback feedback operation