INDEX

Function: OB_GENE

Purpose. Observer design. Dual to defin_vw.

Synopsis.

[U,T,Pi] = ob_gene(sys,pol,key,def_pb[,bb,dd]);

Description. Observer design by juxtaposition of elementary observers i.e. computation of the matrices U, T and Pi. These matrices are defined in the following figure.

Each elementary observer design is considered as a dual right eigenvector assignment problem (see help1 defin_vw). The options are 'z' and 'n' for decoupling, 'p', 'v' for projection or 'm' for ``dual minimum energy''.

Input arguments.

 sys LTI system (see syslin).
 pol Vector (of length q) of assigned eigenvalues. Do not repeat conjugate values.
 key Vector (of length q) defining options for elementary observer designs. key(i) is one of the strings 'z', 'i', 'p', 'n' or 'v'.
 def_pb Matrix having q columns. Non significant entries are set to zero. If key(i) =

'z': def_pb(:,i) contains the indices of the entries of the eigenvector corresponding to pol(i) that must be set to zero.

'm': minimum energy assignment, def_pb(1,i) contains the open-loop poles that must be shifted to pol(i) with ``minimum energy''.

'p': for projection of open-loop eigenvector. Three cases

  • one open-loop real (resp. nonreal) pole replaced by a closed-loop real (resp. nonreal) one: pol(i) is the assigned (closed-loop) eigenvalue def_pb(1,i) is the concerned open-loop eigenvalue.
  • two open-loop real poles OL_pole_1 and OL_pole_2 replaced by one closed-loop non real pole CL_pole. Then, pol(i) = CL_pole and def_pb(1,i) = OL_pole_1 + j OL_pole_2, def_pb(2,i) = 1.
  • one open-loop non real OL_pole replaced by two closed-loop real CL_pole_1 and CL_pole_2. Then, pol(i) = CL_pole_1 + j CL_pole_2, def_pb(1,i) = OL_pole, def_pb(2,i) = 2.

'n': The vectors ui (left eigenvector) and ti (output direction) corresponding to pol(i) will be such that ui*bb(:,def_pb(:,i)) + ti*dd(:,def_pb(:,i)) = 0. Default bb = sys.b , dd = sys.d.

'v': def_pb(:,i) is n by 1 (n number of states), it is the transconjugate of the desired vector ui (that will be assigned by least squares minimization).

Output arguments.

 U,T,Pi Matrices defining an observer (z = U x is observed.)

See also: fb_prop, defin_vw, sob_proj, ob_ins, add_obs