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ss2ss — state-space to state-space conversion, feedback, injection
[Sl1,right,left]=ss2ss(Sl,T, [F, [G , [flag]]])
| 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) |
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
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| - | 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)))
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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))
| << ss2des | ss2tf >> |