Why does my Transfer function overshoots
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syms s real
zeros = roots([0.075, 1, 1]);
poles = roots([1, 3, 5, 0]);
Gs = zpk(zeros, poles,1);
rlocus(Gs)
As you can see, eventhough K=45.3 is on the root locus with damping coeff. 1 it still overshoots. What might cause this problem or am i doing something wrong ?
采纳的回答
Define the plant
zeros = roots([0.075, 1, 1]);
poles = roots([1, 3, 5, 0]);
Gs = zpk(zeros, poles,1);
As was pointed out by @Sam Chak below, the plant model used in the root locus analysis is not the same as the plant model used in the Simulink model.
tf(Gs)
However, proceeding with this model is instructive, in my opinion.
Define the closed loop system check the poles
Gcl = feedback(45.3*Gs,1)
damp(Gcl)
Keep in mind that the information in the rlocus data tip only applies to the selected pole or selected complex conjugate pair, so it's important to select the dominant pole(s), as has been done in this case becasue the pole at s = -1.083 is effectively cancelled by the zero at s = -1.089. Essentially, the closed loop system is well-approximated the pair of poles at s = -23 and the zero at s = -12.24.
Gclapproximate = tf(45.3*[1 12.24],[1 47.22 557.9]);
If we ignore the effect of the zero at s = -12.24 we'd have
Gclnozero = tf(45.3*12.24,[1 47.22 557.9]);
Compare all three
step(Gcl,Gclapproximate,Gclnozero);
legend('true','approx','nozero')
As expected, the no-zero case has no overhsoot because it's basically the response of two, identical, real poles. The effect of an additional, real zero in the left-half plane is to speed up the step response and potentially give rise to overshoot. Consult a good textbook to understand why that is the case.
zeros = roots([0.075, 1, 1]);
poles = roots([1, 3, 5, 0]);
Gs = zpk(zeros, poles,0.075);
tf(Gs)
Or more simply
Gs = tf([0.075, 1, 1],[1, 3, 5, 0])
figure
Gcl = feedback(45.3*Gs,1)
step(Gcl)
This system has zeros and poles at
zpk(Gcl)
damp(Gcl)
The dominant poles are lightly damped, which one would see on the root locus for this plant, and then we'd have to account for the effect of the zero at s = -12.24 on the response.
更多回答（1 个）
Hi @Berker
I believe the design objective is to determine the gain from the root locus such that the response avoids exhibiting oscillatory behavior or maintains the percentage overshoot as low as possible without becoming overdamped. If the plant is given as follows,
then the design procedure can be referred to:
%% Plant
zeros = roots([0.075, 1, 1]);
poles = roots([1, 3, 5, 0]);
Gp = zpk(zeros, poles, 0.075)
% Gp = tf([0.075, 1, 1], [1, 3, 5, 0]) % directly models the Plant
%% Check if the Gain is as accurate as shown in the data tip (rlocus)
[r, k] = rlocus(Gp);
idx = find(abs(imag(r(1,:)) < eps));
rlGain = k(idx(1))
%% True Control Gain
Gc = rlGain*(1/0.075);
%% Closed-loop system
Gcl = feedback(Gc*Gp, 1)
step(Gcl), grid on
1 个评论
If following your original plant's zpk transfer function, you will still find the root locus gain to be around 45.3, and the same design procedure remains applicable.
%% Plant
zeros = roots([0.075, 1, 1]);
poles = roots([1, 3, 5, 0]);
Gp = zpk(zeros, poles, 1) % <-- k is 1
% Gp = tf([0.075, 1, 1], [1, 3, 5, 0]) % directly models the Plant
%% Check if the Gain is as accurate as shown in the data tip (rlocus)
[r, k] = rlocus(Gp);
idx = find(abs(imag(r(1,:)) < eps));
rlGain = k(idx(1))
%% True Control Gain
Gc = rlGain*(1/0.075);
%% Closed-loop system
Gcl = feedback(Gc*Gp, 1)
step(Gcl), grid on
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