Function: CSTR_QUD

Purpose. Defines linear constraints for pole shifting into a quadrilateral.

Synopsis.

CSTR = cstr_qud(CSTR,sys,Quad,lambda,speed,K0[,target])

Description. The quadrilateral is defined by Quad. Only the closed-loop poles the closest to the entries of the vector lambda (if they are outside the quadrilateral) are considered for defining pole shifting constraints. Three constraints are defined (as illustrated below) for each pole to be shifted into the quadrilateral.

This function belongs to the tuning sub-toolbox.

Input arguments

`CSTR`	Matrix of existing constraints when `cstr_qud` is called.
`sys`	LTI system (see `ss.m`).
`Quad`	Defines a quadrilateral into which the closed-loop poles the closest to the entries of lambda must be shifted. `Quad` is a 2 by 2 complex matrix defining the corners of the quadrilateral (which must be inside the upper half complex plane including the real axis). In the above figure `Quad = [Q11 Q12;Q21 Q22]`.
`lambda`	Vector of the eigenvalues (with positive imaginary part) that must be shifted into the quadrilateral. If all eigenvalues are to be shifted set `lambda = eig_fb(sys,K0)`. See also `sort_ev`.
`speed`	Is a scalar number s.t. `0 < speed < = 0.1`: `speed` must be large enough to obtain fast convergence and small enough to avoid non-convergence.
`K0`	Initial proportional gain.
`target`	Complex number: pole target inside the quadrilateral (point `T` in the above figure).

Output argument

CSTR Used for communications between RMCT functions.

See also: fb_tun, sort_ev, dist_qud

Example: All the eigenvalues are shifted into the trapezium with upper left and right corners -10+10i and -1+j and lower left and right corners -10 and -1. Use ``trials and errors'' for the choice of the sixth input argument (speed). First, is defined the system and criterion:

  randn('seed',0); rand('seed',0);
  sys = rss(10,3,3);
  K0 = zeros(3,3);
  Quad = [-10+10*i -1+1*j;-10 -1];
  CRIT = crit_k(ones(3,3),zeros(3,3));

The tuning loop is:

   for ii = 1:60;
      lam = eig_fb(sys,K0);
      CSTR = cstr_qud([],sys,Quad,lam,0.01,K0);
      k = fb_tun(CSTR,CRIT);
      K0 = K0 + k;
  end;

and the results are ([eig_fb(sys) eig_fb(sys,K0)]):

  -1.8717 + 4.2640i  -4.3049
  -1.8717 - 4.2640i  -2.4739 + 2.9886i
  -3.2058 + 3.2339i  -2.4739 - 2.9886i
  -3.2058 - 3.2339i  -2.4571 + 2.4050i
  -3.4763            -2.4571 - 2.4050i
  -2.3909            -2.3384 + 0.9544i
  -1.3022            -2.3384 - 0.9544i
  -0.2355            -1.0303 + 0.1174i
  -0.5278            -1.0303 - 0.1174i
  -0.4960            -0.9222

All the eigenvalues have almost reached the trapezium.