Digital Root Locus in Scilab

Question

I have the following digital plant:

\$G(z)=\cfrac{z+1}{(z-0.3)(z-1)}\$

I want to use root locus to design a system with:

(1) position error=0;

(2) overshoot \$\leq 10\%\$

(3) settling time \$\leq\$ 5 seconds

(4) rise time as small as possible

I made the design manually:

Calculations done so far:

The overshoot requirement leads to \$ \zeta \geq 0.6 \$,

\$t_s=\cfrac{4.5}{|\sigma|}\Rightarrow |\sigma|\geq \cfrac{4.5}{5}=0.9\$

Unitary circle centered in the origin with radius of \$e^{-0.9T}\$. If T=1, the radius is \$e^{-0.9}\approx 0.41\$.

\$-\cfrac{1}{h}=\cfrac{z+1}{(z-0.3)(z-1)}\$

Zero: z=-1

Poles: z=0.3; z=1

Break point:

\$N'D-ND'=0 \Leftrightarrow -z^2-2z+1.6=0\leftrightarrow z\approx 0.612 \vee z=-2.612\$

I need to determine \$h\$, so I have to determine \$z_h\$. How could I determine \$h\$ in Scilab?

Answer 1

Using the following code we can obtain the path the poles go as you change k.

G = poly([-1],'z')/poly([0.3 1],'z');

figure;
evans(G,10)
zgrid;

y = [];
for k = logspace(-2,1,1000)
    y = [y abs(roots((k*G/(1+k*G)).den))];
end

figure;
plot2d("ll",logspace(-2,1,1000),y')
legend('a1','a2')

//pick here your optimal gain
L = 0.5*G;
L = L/(1+L);
t = 0:21;
figure;
plot(t,dsimul(tf2ss(L),ones(1,1+max(t))),'x')

module of the poles in function of k.

root locus of the system.

If you manually/algorithmically search the path in \$ y\$ you will be able to find the \$ k \$ that minimizes the time, for that you don't have to just to minimize \$ |\sigma|\$ but also get it to be near the minimum damping \$ \zeta = 0.6\$. And get a result similar to