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.TH ss2ss 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
ss2ss - state-space to state-space conversion, feedback, injection
.SH CALLING SEQUENCE
.nf
[Sl1,right,left]=ss2ss(Sl,T, [F, [G , [flag]]])
.fi
.SH PARAMETERS
.TP 15
Sl
: linear system (\fVsyslin\fR list) in state-space form
.TP
T
: square (non-singular) matrix
.TP
Sl1, right, left
: linear systems (syslin lists) in state-space form
.TP
F
: real matrix (state feedback gain)
.TP
G
: real matrix (output injection gain)
.SH DESCRIPTION
Returns the linear system \fVSl1=[A1,B1,C1,D1]\fR
where \fVA1=inv(T)*A*T, B1=inv(T)*B, C1=C*T, D1=D\fR.
.LP
Optional parameters \fVF\fR and \fVG\fR are state feedback
and output injection respectively.
.LP
For example,
\fVSl1=ss2ss(Sl,T,F)\fR returns \fVSl1\fR with:
.LP
.IG
.nf
| inv(T)*(A+B*F)*T , inv(T)*B |
Sl1 = | |
| (C+D*F)*T , D |
.fi
.FI
.LA \[ \mbox{\verb+Sl1+} = \left( \begin{array}{cc} T^{-1}(A+BF)T & T^{-1} (B) \\
.LA (C+DF)T & D \end{array} \right) \]
and \fVright\fR is a non singular linear system such that \fVSl1=Sl*right\fR.
.LP
\fVSl1*inv(right)\fR is a factorization of \fVSl\fR.
.LP
.LP
\fVSl1=ss2ss(Sl,T,0*F,G)\fR returns \fVSl1\fR with:
.IG
.nf
| inv(T)*(A+G*C)*T , inv(T)*(B+G*D) |
Sl1 = | |
| C*T , D |
.fi
.FI
.LA \[ \mbox{\verb+Sl1+} = \left( \begin{array}{cc} T^{-1}(A+GC)T & T^{-1} (B+GD) \\
.LA CT & D \end{array} \right) \]
.LP
and \fVleft\fR is a non singular linear system such that \fVSl1=left*Sl\fR (\fVright=Id\fR if \fVF=0\fR).
When both \fVF\fR and \fVG\fR are given, \fVSl1=left*Sl*right\fR.
.TP
-
When \fVflag\fR is used and \fVflag=1\fR an output injection
as follows is used
.IG
.nf
| inv(T)*(A + GC)*T , inv(T)*(B+GD,-G) |
| C*T , (D , 0) |
.fi
.FI
.LA \[ \mbox{\verb+Sl1+} = \left( \begin{array}{cc} T^{-1}(A+GC)T &
.LA T^{-1} \left(B+GD,-G\right) \\
.LA CT & \left(D , 0 \right)\end{array} \right) \]
and then a feedback is performed, \fVF\fR must be of size \fV(m+p,n)\fR
.IG
( x is in R^n , y in R^p, u in R^m ).
.FI
.LA ( $x$ is in $R^n$, $y$ in $R^p$, $u$ in $R^m$ ).
.LA
\fVright\fR and \fVleft\fR have the following property:
.nf
Sl1 = left*sysdiag(sys,eye(p,p))*right
.fi
.TP
-
When \fVflag\fR is used and \fVflag=2\fR a feedback (\fVF\fR must be of
size \fV(m,n)\fR) is performed and then the above output injection is applied.
\fVright\fR and \fVleft\fR have
the following property:
.nf
Sl1 = left*sysdiag(sys*right,eye(p,p)))
.fi
.SH EXAMPLE
.nf
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))
.fi
.SH SEE ALSO
projsl, feedback
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