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]` 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 .. _st_ility: st_ility.html .. _ssrand: ssrand.html .. _cainv: cainv.html .. _ss2ss: ss2ss.html .. _ddp: ddp.html