Control System Steady State Error for VTOL

Question

Abdul Rahman Alam 2022-12-5

1
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1871097-control-system-steady-state-error-for-vtol

编辑： Sam Chak 2022-12-5

I am designing the control system that have a crossover frequency is wc=6 rad/s, 90 degrees least phase margin,and steady-state error must be zero.

but I am having steady state error equal to 1, I am trying to make it 0 but it is not working.

need guidance on what I am doing wrong.

Equations that I am using (values of a and b are random):

My code:

clc

clear

close

%%

wc=6;

G=tf(0.51,[0.23 0.38 1]);

[mag,pahse]=bode(G,wc);

Kp=1/mag;

L=Kp*G;

pzmap(G)

axis ([-1 1 -3 3])

%%

margin(G); grid;

hold on

margin(L);

legend('G(s)','L(s)')

hold off

%%

%klead

a=10;

klead1=tf([a wc],[1 a*wc]);

klead2=tf([a wc],[1 a*wc]);

Klead=klead1*klead2;

L=Kp*G*Klead;

figure(2)

margin(L); grid;

%%

%klag

b=1;

Klag=tf([1 b],[1 0]);

L=Kp*G*Klead*Klag;

figure(3)

margin(L); grid;

e=feedback(L,1)

figure(4)

step(e); grid;

legend('Kp=0.0669')

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Answer 1

Sam Chak 2022-12-5

2
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1871097-control-system-steady-state-error-for-vtol#answer_1120412

编辑：Sam Chak 2022-12-5

在 MATLAB Online 中打开

Hi @Abdul Rahman Alam

The transfer function between the error signal

and the input signal

is defined by

where

is the closed-loop system.

wc = 6;

Gp = tf(0.51, [0.23 0.38 1])

Gp = 0.51 --------------------- 0.23 s^2 + 0.38 s + 1 Continuous-time transfer function.

[mag, phase] = bode(Gp, wc);

Kp = 1/mag;

L = Kp*Gp;

pzmap(Gp)

axis ([-1 1 -3 3])

%%

margin(Gp); grid;

hold on

margin(L);

legend('G(s)','L(s)')

hold off

%%

%klead

a = 10;

klead1 = tf([a wc], [1 a*wc]);

klead2 = tf([a wc], [1 a*wc]);

Klead = klead1*klead2;

L = Kp*Gp*Klead;

figure(2)

margin(L); grid;

%%

%klag

b = 1;

Klag = tf([1 b], [1 0]);

L = Kp*Gp*Klead*Klag;

figure(3)

margin(L); grid;

Gcl = minreal(feedback(L, 1))

Gcl = 3317 s^3 + 7297 s^2 + 5174 s + 1194 ------------------------------------------------------------- s^5 + 121.7 s^4 + 7119 s^3 + 1.377e04 s^2 + 2.083e04 s + 1194 Continuous-time transfer function.

figure(4)

err = 1 - Gcl; % error signal

step(err); grid;

legend('Kp=0.0669')

It achieves zero steady state error for a unit step input.

yss = dcgain(Gcl)
yss = 1.0000

Another way to check is the steady-state output of

. If the

, then it is guaranteed to achieve zero steady state error.

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Abdul Rahman Alam 2022-12-5

编辑：Abdul Rahman Alam 2022-12-5

@Sam Chak can you enlighthen me how did you get the values of (kp, ki, kd, Tf)? we only studied it as equations but dind't apply it on actual example. and is Tf stand for closed loop transfer function?

Sam Chak 2022-12-5

编辑：Sam Chak 2022-12-5

在 MATLAB Online 中打开

Hi @Abdul Rahman Alam

If you find the explanation and MATLAB code helpful, please consider accepting ✔ and voting 👍 the Answer. Thanks a bunch! 🙏

Back to your query, clicking the link, you will find the PIDF controller, which stands for Proportional, Integral, and Derivative with first-order filter on derivative term.

https://www.mathworks.com/help/control/ug/pid-controller-types-for-tuning.html#bu0gy1l

So, the Tf is actually the time constant of the first-order filter, as shown in the Continuous-Time Controller Formula:

Note that Tf is usually a very small value so that it approximates ideal PID controller:

As mentioned previously, the values of (kp, ki, kd, Tf) were manually tuned until performance requirements were achieved. In fact, I used margin() iteratively to check that. Mine was the computer-assisted manual tuning.

If you are looking for the "Computer-tunes-it-for-me" approach, then try this powerful script:

Gp = tf(0.51, [0.23 0.38 1])

Gp = 0.51 --------------------- 0.23 s^2 + 0.38 s + 1 Continuous-time transfer function.

% Control Design

wc = 6; % desired crossover frequency

opts = pidtuneOptions('PhaseMargin', 90, 'DesignFocus', 'reference-tracking');

[Gc, info] = pidtune(Gp, 'PIDF', wc, opts)

Gc = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 4.34, Ki = 1.83, Kd = 2.43, Tf = 0.00146 Continuous-time PIDF controller in parallel form.

info = struct with fields:

Stable: 1 CrossoverFrequency: 6 PhaseMargin: 90

% Closed-loop transfer function

Gcl = minreal(feedback(Gc*Gp, 1))

Gcl = 3705 s^2 + 6607 s + 2785 ------------------------------------------ s^4 + 687.3 s^3 + 4842 s^2 + 9589 s + 2785 Continuous-time transfer function.

% Error transfer function

Ge = 1 - Gcl;

step(Ge); grid;

Gcp = Gc*Gp;

margin(Gcp)

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