Function: SFB_PROJ

Purpose. Computes a state feedback by projection of the open-loop eigenstructure.

Synopsis.

[k,V,W,pol] = sfb_proj(sys,Tr,Damp[,key[,C_ratio]]);

Description. The closed-loop poles are automatically chosen from the design parameters which are the minimum damping ratio (Damp) and the settling time (Tr). The right eigenvectors corresponding to eigenvalues satisfying the settling time and damping ratio requirements are re-assigned as they are. For other open-loop poles that need to be shifted, assignment of right eigenvectors is made by projection of the open-loop ones.

Input arguments

`sys`	LTI system (see `ss.m`). The matrices `sys.c` and `sys.d` are ignored.
`Tr`	Settling time.
`Damp`	Minimum damping ratio for closed-loop eigenvalues (0 < `Damp` < 1).
`key`	Is a string equal to `'p'` for orthogonal projection of all eigenvectors, or equal to `'m'` for minimum energy assignment (default: `key = 'm'`).
`C_ratio`	If the controllability degree of some eigenvalue is less than the maximum controllability degree divided by `C_ratio`, this eigenvalue is not shifted. Default value `C_ratio = 100`.

Output arguments

`k`	State feedback.
`V,W,pol`	Assigned eigenstructure.

See also: fb_prop, sob_proj, sfb_ins

Example. Design of two state feedback gains such that the damping ratio becomes larger than 0.707 and the settling time less than 15 seconds. First design by open-loop projection, second by minimum energy assignment.

   randn('seed',0); rand('seed',0);
   sys = rss(6,6,3);

   Tr = 15; Damp = 0.707;
   kp = sfb_proj(sys,Tr,Damp,'p');
   km = sfb_proj(sys,Tr,Damp,'m');
   norm(km)
   norm(kp)

The norm of the gain obtained with option 'm' is usually (not always) smaller than with option 'p'. For this example we obtain norm(km) = 5.79 and norm(kp) = 12.8.