Function: FB_PROP

Purpose. Assigns eigenstructure (eigenvalues and eigenvectors) by proportional (state or output) feedback.

Synopsis.

k = fb_prop(sys,k0,pol,key,def_pb[,cc,dd[,ee]]);
k = fb_prop(sys,k0,pol,'p',def_pb,fb0);
k = fb_prop(sys,k0,pol,'p',def_pb,K,'s');
k = fb_prop(sys,k0,pol); <- single-input system

Description. See help defin_vw.

Input arguments

All input arguments are similar to those of the function defin_vw except:

k0 If the problem is under-specified, k is computed as being the feasible gain the closest to k0. (k0=0 means that k0 is the zero matrix with appropriate size).

Output argument

k Feedback gain.

See also: defin_vw, fb_dyn, fb_tun

Example 1. Assignment of { -0.1 + 0.1 j , -0.1 - 0.1 j , -2 , -4 }. The eigenvectors associated with -0.1 + 0.1 j and its conjugate value are obtained by projection of the open-loop eigenvectors associated to both open-loop eigenvalues { -1 , -0.5 }. Assignment of -2 and -4 are made with minimum energy from +2 and -4. The open-loop eigenvalues -3 and -6 are expected to be moderately perturbed by the designed feedback on account of the use of projection and minimum energy ideas.

   a=diag([-1;2;-3;-4;-0.5;-6]);
   b=[eye(2,2);eye(2,2);eye(2,2)];
   c=[eye(4,4) ones(4,2)];
   sys = ss(a,b,c,0);

   key  = [ 'p'               'm'    'm'  ];
   pol  = [ -.1+.1*i          -2     -4   ];
   def_pb=[[(-1)+i*(-0.5);1] [2;0]  [-4;0]];
   k=fb_prop(sys,0,pol,key,def_pb)

Example 2. Assignment of { -0.6 , -1.1 , -2 , -5.9 , -3.9}. The assignment of the 3 first eigenvalues is made respectively by (projection, projection, minimum energy) from open-loop eigenstructure (-0.5 , -1 , +2). The assignment of -5.9 is such that an additional eigenvalue -3.9 is also assigned.

   key    = [ 'p'      'p'     'm'    'e' ];
   pol    = [ -0.6     -1.1    -2     -5.9];
   def_pb = [ -0.5     -1       2     -3.9];
   k=fb_prop(sys,0,pol,key,def_pb)

Example 3. Assignment of { -1.1 , -0.6 , -2 , -5.9 }. The 2nd entry of the eigenvector associated with -1.1 must be zero. The 1st entry of the input direction associated with -0.6 must be zero. Minimum energy assignment for -2 and -5.9 from open-loop eigenvalues +2 and -6.

   key    = [ 'z'     'z'      'm'    'm'  ];
   pol    = [ -1.1    -0.6     -2     -5.9 ];
   def_pb = [  2       7        2     -6   ];
   k=fb_prop(sys,0,pol,key,def_pb)

See also the tutorial section (Example 1 and Example 2) for additional illustrative examples.