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Please note that the recommended version of Scilab is 2024.1.0. This page might be outdated.
See the recommended documentation of this function
abinv
AB invariant subspace
Calling Sequence
[X,dims,F,U,k,Z]=abinv(Sys,alpha,beta,flag)
Arguments
- Sys
syslin
list containing the matrices[A,B,C,D]
.- alpha
(optional) real number or vector (possibly complex, location of closed loop poles)
- beta
(optional) real number or vector (possibly complex, location of closed loop poles)
- flag
(optional) character string
'ge'
(default) or'st'
or'pp'
- X
orthogonal matrix of size nx (dim of state space).
- dims
integer row vector
dims=[dimR,dimVg,dimV,noc,nos]
withdimR<=dimVg<=dimV<=noc<=nos
. Ifflag='st'
, (resp.'pp'
),dims
has 4 (resp. 3) components.- F
real matrix (state feedback)
- k
integer (normal rank of
Sys
)- Z
non-singular linear system (
syslin
list)
Description
Output nulling subspace (maximal unobservable subspace) for
Sys
= linear system defined by a syslin list containing
the matrices [A,B,C,D] of Sys
.
The vector dims=[dimR,dimVg,dimV,noc,nos]
gives the dimensions
of subspaces defined as columns of X
according to partition given
below.
The dimV
first columns of X
i.e V=X(:,1:dimV)
,
span the AB-invariant subspace of Sys
i.e the unobservable subspace of
(A+B*F,C+D*F)
. (dimV=nx
iff C^(-1)(D)=X
).
The dimR
first columns of X
i.e. R=X(:,1:dimR)
spans
the controllable part of Sys
in V
, (dimR<=dimV)
. (dimR=0
for a left invertible system). R
is the maximal controllability
subspace of Sys
in kernel(C)
.
The dimVg
first columns of X
spans
Vg
=maximal AB-stabilizable subspace of Sys
. (dimR<=dimVg<=dimV)
.
F
is a decoupling feedback: for X=[V,X2] (X2=X(:,dimV+1:nx))
one has
X2'*(A+B*F)*V=0 and (C+D*F)*V=0
.
The zeros od Sys
are given by : X0=X(:,dimR+1:dimV); spec(X0'*(A+B*F)*X0)
i.e. there are dimV-dimR
closed-loop fixed modes.
If the optional parameter alpha
is given as input,
the dimR
controllable modes of (A+BF)
in V
are set to alpha
(or to [alpha(1), alpha(2), ...]
.
(alpha
can be a vector (real or complex pairs) or a (real) number).
Default value alpha=-1
.
If the optional real parameter beta
is given as input,
the noc-dimV
controllable modes of (A+BF)
"outside" V
are set to beta
(or [beta(1),beta(2),...]
). Default value beta=-1
.
In the X,U
bases, the matrices
[X'*(A+B*F)*X,X'*B*U;(C+D*F)*X,D*U]
are displayed as follows:
[A11,*,*,*,*,*] [B11 * ] [0,A22,*,*,*,*] [0 * ] [0,0,A33,*,*,*] [0 * ] [0,0,0,A44,*,*] [0 B42] [0,0,0,0,A55,*] [0 0 ] [0,0,0,0,0,A66] [0 0 ] [0,0,0,*,*,*] [0 D2]
where the X-partitioning is defined by dims
and
the U-partitioning is defined by k
.
A11
is (dimR x dimR)
and has its eigenvalues set to alpha(i)'s
.
The pair (A11,B11)
is controllable and B11
has nu-k
columns.
A22
is a stable (dimVg-dimR x dimVg-dimR)
matrix.
A33
is an unstable (dimV-dimVg x dimV-dimVg)
matrix (see st_ility
).
A44
is (noc-dimV x noc-dimV)
and has its eigenvalues set to beta(i)'s
.
The pair (A44,B42)
is controllable.
A55
is a stable (nos-noc x nos-noc)
matrix.
A66
is an unstable (nx-nos x nx-nos)
matrix (see st_ility
).
Z
is a column compression of Sys
and k
is
the normal rank of Sys
i.e Sys*Z
is a column-compressed linear
system. k
is the column dimensions of B42,B52,B62
and D2
.
[B42;B52;B62;D2]
is full column rank and has rank k
.
If flag='st'
is given, a five blocks partition of the matrices is
returned and dims
has four components. If flag='pp'
is
given a four blocks partition is returned. In case flag='ge'
one has
dims=[dimR,dimVg,dimV,dimV+nc2,dimV+ns2]
where nc2
(resp. ns2
) is the dimension of the controllable (resp.
stabilizable) pair (A44,B42)
(resp. ([A44,*;0,A55],[B42;0])
).
In case flag='st'
one has dims=[dimR,dimVg,dimVg+nc,dimVg+ns]
and in case flag='pp'
one has dims=[dimR,dimR+nc,dimR+ns]
.
nc
(resp. ns
) is here the dimension of the controllable
(resp. stabilizable) subspace of the blocks 3 to 6 (resp. 2 to 6).
This function can be used for the (exact) disturbance decoupling problem.
DDPS: Find u=Fx+Rd=[F,R]*[x;d] which rejects Q*d and stabilizes the plant: xdot = Ax+Bu+Qd y = Cx+Du+Td DDPS has a solution if Im(Q) is included in Vg + Im(B) and stabilizability assumption is satisfied. Let G=(X(:,dimVg+1:$))'= left annihilator of Vg i.e. G*Vg=0; B2=G*B; Q2=G*Q; DDPS solvable iff [B2;D]*R + [Q2;T] =0 has a solution. The pair F,R is the solution (with F=output of abinv). Im(Q2) is in Im(B2) means row-compression of B2=>row-compression of Q2 Then C*[(sI-A-B*F)^(-1)+D]*(Q+B*R) =0 (<=>G*(Q+B*R)=0)
Examples
nu=3;ny=4;nx=7; nrt=2;ngt=3;ng0=3;nvt=5;rk=2; flag=list('on',nrt,ngt,ng0,nvt,rk); Sys=ssrand(ny,nu,nx,flag);my_alpha=-1;my_beta=-2; [X,dims,F,U,k,Z]=abinv(Sys,my_alpha,my_beta); [A,B,C,D]=abcd(Sys);dimV=dims(3);dimR=dims(1); V=X(:,1:dimV);X2=X(:,dimV+1:nx); X2'*(A+B*F)*V (C+D*F)*V X0=X(:,dimR+1:dimV); spec(X0'*(A+B*F)*X0) trzeros(Sys) spec(A+B*F) //nr=2 evals at -1 and noc-dimV=2 evals at -2. clean(ss2tf(Sys*Z))
nx=6;ny=3;nu=2; A=diag(1:6);A(2,2)=-7;A(5,5)=-9;B=[1,2;0,3;0,4;0,5;0,0;0,0]; C=[zeros(ny,ny),eye(ny,ny)];D=[0,1;0,2;0,3]; sl=syslin('c',A,B,C,D);//sl=ss2ss(sl,rand(6,6))*rand(2,2); [A,B,C,D]=abcd(sl); //The matrices of sl. my_alpha=-1;my_beta=-2; [X,dims,F,U,k,Z]=abinv(sl,my_alpha,my_beta);dimVg=dims(2); clean(X'*(A+B*F)*X) clean(X'*B*U) clean((C+D*F)*X) clean(D*U) G=(X(:,dimVg+1:$))'; B2=G*B;nd=3; R=rand(nu,nd);Q2T=-[B2;D]*R; p=size(G,1);Q2=Q2T(1:p,:);T=Q2T(p+1:$,:); Q=G\Q2; //a valid [Q;T] since [G*B;D]*R + [G*Q;T] // is zero closed=syslin('c',A+B*F,Q+B*R,C+D*F,T+D*R); // closed loop: d-->y ss2tf(closed) // Closed loop is zero spec(closed('A')) //The plant is not stabilizable! [ns,nc,W,sl1]=st_ility(sl); [A,B,C,D]=abcd(sl1);A=A(1:ns,1:ns);B=B(1:ns,:);C=C(:,1:ns); slnew=syslin('c',A,B,C,D); //Now stabilizable //Fnew=stabil(slnew('A'),slnew('B'),-11); //slnew('A')=slnew('A')+slnew('B')*Fnew; //slnew('C')=slnew('C')+slnew('D')*Fnew; [X,dims,F,U,k,Z]=abinv(slnew,my_alpha,my_beta);dimVg=dims(2); [A,B,C,D]=abcd(slnew); G=(X(:,dimVg+1:$))'; B2=G*B;nd=3; R=rand(nu,nd);Q2T=-[B2;D]*R; p=size(G,1);Q2=Q2T(1:p,:);T=Q2T(p+1:$,:); Q=G\Q2; //a valid [Q;T] since [G*B;D]*R + [G*Q;T] // is zero closed=syslin('c',A+B*F,Q+B*R,C+D*F,T+D*R); // closed loop: d-->y ss2tf(closed) // Closed loop is zero spec(closed('A'))
See Also
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