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// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) INRIA - F. Delebecque
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
function [p1,r1]=lft(p,r,p#,r#)
//[p1,r1]=lft(p,r,p#,r#)
//linear fractional transform between two standard plants
//p and p# in state space form or in transfer form.
//r= size(p22) r#=size(p22#);
// lft(p,r, k) is the linear fractional transform between p and
// a controller k (k may be a gain or a controller in
// state space form or in transfer form);
// lft(p,k) is lft(p,r,k) with r=size of k transpose;
//!
[lhs,rhs]=argn(0);
if type(p)==1 then
p=syslin([],[],[],[],p);
else
if and(typeof(p)<>["rational","state-space"]) then
error(msprintf(gettext("%s: Wrong type for input argument #%d: Linear state space or a transfer function expected.\n"),"lft",1))
end
end
dom=p.dt
p#pos=3
if rhs==2 then
rhs=3;p#=r;
r=size(p#');
p#pos=2
end //rhs=2
if and(typeof(p#)<>["constant","rational","state-space"]) then
error(msprintf(gettext("%s: Wrong type for input argument #%d: Linear state space or a transfer function expected.\n"),"lft",p#pos))
end
pssdflag=%f;
if typeof(p)=="state-space" then
if typeof(p.D)=="polynomial" then
pssdflag=%t;
end
end
if typeof(p#)=="state-space" then
if typeof(p#.D)=="polynomial" then
pssdflag=%t;
end
end
if pssdflag then
if rhs==3 then
[p1,r1]=lftpssd(p,r,p#);return;
end
if rhs==4 then
[p1,r1]=lftpssd(p,r,p#,r#);return;
end
end
if rhs==3 then
if type(p)==1 then
// lft( gain, linear system in ss form)
[nl,nc]=size(p);
l1=1:nc-r(1);
l2=nc-r(1)+1:nc;
k1=1:nl-r(2);
k2=nl-r(2)+1:nl;
d11=p(l1,k1);d12=p(l1,k2);d21=p(l2,k1);d22=p(l2,k2);
if type(p#)==1 then
//p# is a gain
dk=p#,
id=inv(eye(d22*dk)-d22*dk),
dd=d11+d12*dk*id*d21,
p1=dd;
else
// p# is not a gain
p#1=p#(1)
if p#1(1)=="lss" then
[ak,bk,ck,dk]=p#(2:5);
id=inv(eye(d22*dk)-d22*dk),
aa= ak+bk*id*d22*ck,
bb=bk*id*d21,
cc=d12*(ck+dk*id*d22*ck),
dd=d11+d12*dk*id*d21;
p1=syslin(dom,aa,bb,cc,dd)
end
if p#1(1)=="r" then
p1=d11+d12*p#*invr(eye()-d22*p#)*d21;
end
end
end //type(p)=1
pof1=p(1);
if pof1(1)=="lss" then
// lft(standard in ss form,linear system in ss form)
[a,b1,b2,c1,c2,d11,d12,d21,d22]=smga(p,r);
if type(p#)==1 then
//p# is a gain
dk=p#,
id=inv(eye(d22*dk)-d22*dk),
aa=a+b2*dk*id*c2;
bb=b1+b2*dk*id*d21,
cc=c1+d12*dk*id*c2,
dd=d11+d12*dk*id*d21,
p1=syslin(dom,aa,bb,cc,dd)
else
// p# is not a gain
[ak,bk,ck,dk]=p#(2:5);
id=inv(eye(d22*dk)-d22*dk);
aa=[a+b2*dk*id*c2, b2*(ck+dk*id*d22*ck);
bk*id*c2, ak+bk*id*d22*ck],
bb=[b1+b2*dk*id*d21;bk*id*d21],
cc=[c1+d12*dk*id*c2, d12*(ck+dk*id*d22*ck)],
dd=d11+d12*dk*id*d21;
p1=syslin(dom,aa,bb,cc,dd)
end
end
if pof1(1)=="r" then
//lft(standard plant in tf form,linear system in tf form)
[p11,p12,p21,p22]=smga(p,r);
p1=p11+p12*p#*invr(eye()-p22*p#)*p21
end //p(1)=='lss'
end //rhs=3
if rhs==4 then
if type(p)==1 then
//lft(gain,standard plant )
[nl,nc]=size(p);
l1=1:nc-r(1);
l2=nc-r(1)+1:nc;
k1=1:nl-r(2);
k2=nl-r(2)+1:nl;
d11=p(l1,k1);d12=p(l1,k2);d21=p(l2,k1);d22=p(l2,k2);
if type(p#)==1 then
//p# is a gain
[nl,nc]=size(p#);
l1=1:nc-r#(1);
l2=nc-r#(1)+1:nc;
k1=1:nl-r#(2);
k2=nl-r#(2)+1:nl;
d#11=p#(l1,k1);d#12=p#(l1,k2);d#21=p#(l2,k1);d#22=p#(l2,k2);
g=inv(eye()-d22*d#11);
g#=inv(eye()-d#11*d22);
dd11=d11+d12*g#*d#11*d21;
dd12=d12*g#*d#12;
dd21=d#21*g*d21;
dd22=d#22+d#21*g*d22*d#12;
p1=[dd11,dd12;dd21,dd22];
r1=size(dd22);
end
p#1=p#(1);
if p#1(1)=="lss"
//p# in state form
[a#,b#1,b#2,c#1,c#2,d#11,d#12,d#21,d#22]=smga(p#,r#);
g=inv(eye()-d22*d#11);
g#=inv(eye()-d#11*d22);
aa=a#+b#1*g*d22*c#1;
bb1= b#1*g*d21;
bb2= b#2+b#1*g*d22*d#12;
cc1= d12*g#*c#1;
cc2= c#2+d#21*g*d22*c#1;
dd11=d11+d12*g#*d#11*d21;
dd12=d12*g#*d#12;
dd21=d#21*g*d21;
dd22=d#22+d#21*g*d22*d#12;
p1=syslin(dom,aa,[bb1,bb2],[cc1;cc2],[dd11,dd12;dd21,dd22]);
r1=size(dd22);
end
if p#1(1)=="r" then
[j11,j12,j21,j22]=smga(p#,r#);
g=invr(eye()-d22*j11);
g#=invr(eye()-j11*d22);
ptfg11=d11+d12*j11*g*d21;
ptfg12=d12*g#*j12;
//ptfg21=j21*g*p21;
ptfg21=j21*(eye()+d22*g#*j11)*d21;
ptfg22=j22+j21*d22*g#*j12;
p1=[ptfg11,ptfg12;ptfg21,ptfg22];
r1=size(ptfg22)
end
end //type(p)=1
pof1=p(1);
if pof1(1)=="lss" then
//lft(standard plant in ss form,standard plant in ss form)
[a ,b1,b2,c1,c2,d11,d12,d21,d22]=smga(p,r);
if type(p#)==1 then
//p# is a gain
[nl,nc]=size(p#);
l1=1:nc-r#(1);
l2=nc-r#(1)+1:nc;
k1=1:nl-r#(2);
k2=nl-r#(2)+1:nl;
d#11=p#(l1,k1);d#12=p#(l1,k2);d#21=p#(l2,k1);d#22=p#(l2,k2);
g=inv(eye()-d22*d#11);
g#=inv(eye()-d#11*d22);
aa=a+b2*g#*d#11*c2;
bb1=b1+b2*g#*d#11*d21;
bb2=b2*g#*d#12;
cc1=c1+d12*g#*d#11*c2;
cc2=d#21*g*c2;
dd11=d11+d12*g#*d#11*d21;
dd12=d12*g#*d#12;
dd21=d#21*g*d21;
dd22=d#22+d#21*g*d22*d#12;
p1=syslin(dom,aa,[bb1,bb2],[cc1;cc2],[dd11,dd12;dd21,dd22]);
r1=size(dd22);
else
//p# in state form
[a#,b#1,b#2,c#1,c#2,d#11,d#12,d#21,d#22]=smga(p#,r#);
g=inv(eye()-d22*d#11);
g#=inv(eye()-d#11*d22);
aa=[a+b2*g#*d#11*c2 b2*g#*c#1;
b#1*g*c2 a#+b#1*g*d22*c#1];
bb1=[b1+b2*g#*d#11*d21;
b#1*g*d21];
bb2=[b2*g#*d#12;
b#2+b#1*g*d22*d#12];
cc1=[c1+d12*g#*d#11*c2 d12*g#*c#1];
cc2=[d#21*g*c2 c#2+d#21*g*d22*c#1];
dd11=d11+d12*g#*d#11*d21;
dd12=d12*g#*d#12;
dd21=d#21*g*d21;
dd22=d#22+d#21*g*d22*d#12;
p1=syslin(dom,aa,[bb1,bb2],[cc1;cc2],[dd11,dd12;dd21,dd22]);
r1=size(dd22);return;
end // type(p#)=1
end //p(1)=='lss'
pof1=p(1);
if pof1(1)=="r" then
//lft(standard plant in tf form,standard plant in tf form)
[p11,p12,p21,p22]=smga(p,r);
[j11,j12,j21,j22]=smga(p#,r#);
g=invr(eye()-p22*j11);
g#=invr(eye()-j11*p22);
ptfg11=p11+p12*j11*g*p21;
ptfg12=p12*g#*j12;
//ptfg21=j21*g*p21;
ptfg21=j21*(eye()+p22*g#*j11)*p21;
ptfg22=j22+j21*p22*g#*j12;
p1=[ptfg11,ptfg12;ptfg21,ptfg22];
r1=size(ptfg22)
end //p(1)='r'
end //rhs=4
endfunction
function [lf,r1]=lftpssd(p,r,k,r2)
//lft for pssd (inria report #1827, dec. 1992)
[lhs,rhs]=argn(0);
if rhs==2 then
rhs=3;k=r;r=size(k');
end
if rhs==3 then
[pp,qq]=size(p);
l1=pp-r(1);k1=qq-r(2);
l2=r(1);k2=r(2);
[l3,k4]=size(k);
l4=k1;k3=l1;
psi=[p,[-eye(l1,k3),zeros(l1,k4);
zeros(l2,k3),-eye(l2,k4)];
zeros(l3,k1),-eye(l3,k2),zeros(l3,k3),k;
eye(l4,k1),zeros(l4,k2+k3+k4)];
w=clean(ss2tf(psi));
lf=[zeros(k3,k1+k2),eye(k3,k3),zeros(k3,k4)]*inv(psi)*...
[zeros(l1+l2+l3,l4);eye(l4,l4)];
r1=[l1,k1];
return;
end
if rhs==4 then
[pp,qq]=size(p);
l1=pp-r(1);k1=qq-r(2);
l2=r(1);k2=r(2);
[pk,qk]=size(k);
l3=pk-r2(1);k5=qk-r2(2);
l4=r2(1);k6=r2(2);
l5=k1;k3=l4;k4=l1;l6=k6;
psi=...
[p,[zeros(l1,k3),-eye(l1,k4);zeros(l2,k3+k4)],[zeros(l1,k5+k6);-eye(l2,k5),zeros(l2,k6)];
[zeros(l3,k1),-eye(l3,k2);zeros(l4,k1+k2)],[zeros(l3,k3+k4);-eye(l4,k3),zeros(l4,k4)],k;
[eye(l5,k1),zeros(l5,k2+k3+k4+k5+k6);zeros(l6,k1+k2+k3+k4+k5),eye(l6,k6)]];
lf=...
[zeros(k4,k1+k2+k3),eye(k4,k4),zeros(k4,k5+k6);
zeros(k3,k1+k2),eye(k3,k3),zeros(k3,k4+k5+k6)]*inv(psi)*...
[zeros(l1+l2+l3+l4,l5+l6);eye(l5+l6,l5+l6)];
r1=[k3,l6];
end
endfunction
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