abinv

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] 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)

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))

// 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'))

See Also

• cainv — Dual of abinv
• st_ility — stabilizability test
• ssrand — random system generator
• ss2ss — state-space to state-space conversion, feedback, injection
• ddp — disturbance decoupling

Authors

F.D.