AB invariant subspace
[X,dims,F,U,k,Z]=abinv(Sys,alpha,beta,flag)
:Sys syslin list containing the matrices [A,B,C,D]. : :alpha (optional) real number or vector (possibly complex, location
of closed loop poles)
: :X orthogonal matrix of size nx (dim of state space). : :dims integer row vector dims=[dimR,dimVg,dimV,noc,nos] with
dimR<=dimVg<=dimV<=noc<=nos. If flag=’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) :
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:$))'= left annihilator of Vg i.e. G*Vg=0;
B2=G*B; Q2=G*Q; DDPS solvable iff [B2;D]*R + [Q2;T] =0 has a solution.
The pair F,R is the solution (with F=output of abinv).
Im(Q2) is in Im(B2) means row-compression of B2=>row-compression of Q2
Then C*[(sI-A-B*F)^(-1)+D]*(Q+B*R) =0 (<=>G*(Q+B*R)=0)
nu=3;ny=4;nx=7;
nrt=2;ngt=3;ng0=3;nvt=5;rk=2;
flag=`list`_('on',nrt,ngt,ng0,nvt,rk);
Sys=`ssrand`_(ny,nu,nx,flag);my_alpha=-1;my_beta=-2;
[X,dims,F,U,k,Z]=abinv(Sys,my_alpha,my_beta);
[A,B,C,D]=`abcd`_(Sys);dimV=dims(3);dimR=dims(1);
V=X(:,1:dimV);X2=X(:,dimV+1:nx);
X2'*(A+B*F)*V
(C+D*F)*V
X0=X(:,dimR+1:dimV); `spec`_(X0'*(A+B*F)*X0)
`trzeros`_(Sys)
`spec`_(A+B*F) //nr=2 evals at -1 and noc-dimV=2 evals at -2.
`clean`_(`ss2tf`_(Sys*Z))
// 2nd Example
nx=6;ny=3;nu=2;
A=`diag`_(1:6);A(2,2)=-7;A(5,5)=-9;B=[1,2;0,3;0,4;0,5;0,0;0,0];
C=[`zeros`_(ny,ny),`eye`_(ny,ny)];D=[0,1;0,2;0,3];
sl=`syslin`_('c',A,B,C,D);//sl=ss2ss(sl,rand(6,6))*rand(2,2);
[A,B,C,D]=`abcd`_(sl); //The matrices of sl.
my_alpha=-1;my_beta=-2;
[X,dims,F,U,k,Z]=abinv(sl,my_alpha,my_beta);dimVg=dims(2);
`clean`_(X'*(A+B*F)*X)
`clean`_(X'*B*U)
`clean`_((C+D*F)*X)
`clean`_(D*U)
G=(X(:,dimVg+1:$))';
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'))