Housekeeping

ZEROS IN THE LHP:

Consider the transfer function with a single ZERO in the left-half-plane. By taking the Inverse Laplace of we see that the system behavior will be governed by where δ represents the dirac-delta function. To better understand the effect of this zero we can employ a similar method to the poles to derive the following differential equation. . We can observe from this equation that the output will be proportional to the input (scaled by the magnitude of the zero) as well as the first derivative of the input.This indicates that the initial output response will be in the same direction as the input.

Due to the example provided being improper, observer the below system with a single pole at 0.

time = 0:0.01:10;

 

f = figure();

for z = -1*(1:1:5)

plant = (s-z)/(s);

y = step(plant, time);

legendlabel=string(z);

plot(time,y, 'DisplayName','zero = '+legendlabel);

hold on

end

f.Position = [0 0 1300 500];

title('Step Response Based on Zero');

xlabel('time (s)'); xlim([0,10]);

ylabel('magnitude'); ylim([0,20]);

legend('FontSize',15, 'Location', 'northwest')

hold off

ZEROS IN THE RHP:

Consider the transfer function with a single ZERO in the left-half-plane. By taking the Inverse Laplace of we see that the system behavior will be governed by where δ represents the dirac-delta function. To better understand the effect of this zero we can employ a similar method to the poles to derive the following differential equation. . We can observe from this equation that the output will be proportional to the negative input (scaled by the magnitude of the zero) as well as the first derivative of the input.This indicates that the initial output response will be in the opposite direction as the input.

Due to the example provided being improper, observer the below system with a single pole at 0.

time = 0:0.01:10;

 

f = figure();

for z = 1*(1:1:5)

plant = (s-z)/(s);

y = step(plant, time);

legendlabel=string(z);

plot(time,y, 'DisplayName','zero = '+legendlabel);

hold on

end

f.Position = [0 0 1300 500];

title('Step Response Based on Zero');

xlabel('time (s)'); xlim([0,10]);

ylabel('magnitude'); ylim([-20,5]);

legend('FontSize',15, 'Location', 'northeast')

hold off

WHAT EXACTLY DO ZEROS DO?:

One function of zeros can be seen in the root locus wherein an increasing gain will push closed-loop poles towards zeros, and a zero in the RHP will guarantee a limit to the gain that can be applied without forcing the system into instability.

Another function of zeros is on the initial response of a system given an input. With a zero in the LHP, the derivative term will be positive and induce a positive initial response, with a zero in the RHP causing a negative initial response.