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# kpure

continuous SISO system limit feedback gain

### Syntax

K=kpure(sys [,tol])
[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, -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')

• evans — Evans root locus
• krac2 — continuous SISO system limit feedback gain

### History

 Version Description 6.0 Handling zpk representation