kpure continuous SISO system limit feedback gain

## Syntax

                [K,R] = kpure(sys)
                [K,R] = kpure(sys, tol)

## Arguments

sys

  A SISO linear dynamical system, in state space, transfer function or zpk representations, in continuous or discrete time.

tol

  a positive scalar. tolerance used to determine if a root is imaginary or not. The default value is `1e-6`

K

  Real vector, the vector of gains for which at least one closed loop pole is imaginary.

R

  Complex vector, the imaginary closed loop poles associated with the values of `K`.

## Description

`K=kpure(sys)` computes the gains `K` such that the system `sys` feedback by `K(i)` (`sys/.K(i)`) has poles on imaginary axis.

## Examples

    num=real(poly([-1+%i, -1-%i, -1+8*%i  -1-8*%i],'s'));
    den=real(poly([0.5 0.5  -6+7*%i  -6-7*%i  -3 -7 -11],'s'));
    h=num/den;

    [K,Y]=kpure(h)
    clf(); evans(h)
    plot(real(Y),imag(Y),'+r')

num=real(poly([-1+%i, -1-%i, -1+8*%i -1-8*%i],'s')); den=real(poly([0.5 0.5 -6+7*%i -6-7*%i -3 -7 -11],'s')); h=num/den; [K,Y]=kpure(h) clf();evans(h) plot(real(Y),imag(Y),'+r')

    num=real(poly([-1+%i*1, -1-%i*1, 2+%i*8  2-%i*8 -2.5+%i*13 -2.5-%i*13],'s'));
    den=real(poly([1 1 3+%i*3 3-%i*3 -15+%i*7  -15-%i*7  -3 -7 -11],'s'));
    h=num/den;

    [K,Y]=kpure(h)
    clf(); evans(h,100000)
    plot(real(Y),imag(Y),'+r')

num=real(poly([-1+%i*1, -1-%i*1, 2+%i*8 2-%i*8 -2.5+%i*13 -2.5-%i*13],'s')); den=real(poly([1 1 3+%i*3 3-%i*3 -15+%i*7 -15-%i*7 -3 -7 -11],'s')); h=num/den; [K,Y]=kpure(h) clf();evans(h,100000) plot(real(Y),imag(Y),'+r')

## See also

- evans
- krac2

## History

6.0

Handling zpk representation