How can I specify less oschillatory behaviour for PID

Question

JAMES KEEN 2021-3-2

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/760386-how-can-i-specify-less-oschillatory-behaviour-for-pid

评论： JAMES KEEN 2021-3-8

How can I speicfy that I do not want a heavily underdamped response as seen in the image attached? The addition of the derivative term makes it difficult to implement in simulink enviroment.

--------------------------------------------------------------First script------------------------------------------------------------

clear all, close all, clc

dt = 0.000005; % this is 10^-6 0.000001

PopSize = 40;

MaxGenerations = 25;

s = tf('s');

G =(1.44e09)/((s*s)+5333*s+9.6e07)

options = optimoptions(@ga,'PopulationSize',PopSize,'MaxGenerations',MaxGenerations,'OutputFcn',@myfun);

[x,fval] = ga(@(K)pidtest(G,dt,K),2,-eye(2),zeros(2,1),[],[],[],[],[],options);

--------------------------------------------------------------Second script--------------------------------------------------------

function J = pidtest(G,dt,parms)

s = tf('s');

K = parms(1)+ parms(2)/s

Loop = series(K,G);

ClosedLoop = feedback(Loop,1);

t = 0:dt:0.005; % this indicates length of time to show

e = 1 - step(ClosedLoop,t);

J = sum(t'.*abs(e)*dt);

step(1*ClosedLoop,t)

h = findobj(gcf,'type','line');

set(h,'linewidth',1);

drawnow

---------------------------------------------------------------end of script------------------------------------------------------

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JAMES KEEN 2021-3-4

编辑：JAMES KEEN 2021-3-4

The image I have just shown you has gained PID controller values from the following code. It has an offset of 0.05v which I do not like though so I have been trying various codes.

----------------------------1st script-----------------------------------

clear all, close all, clc

dt = 0.000001; % this is 10^-6 0.000001

PopSize = 40; % was 40

MaxGenerations = 25; % was 25

s = tf('s');

G =(1.44e09)/((s*s)+5333*s+9.6e07)

options = optimoptions(@ga,'PopulationSize',PopSize,'MaxGenerations',MaxGenerations,'OutputFcn',@myfun);

[x,fval] = ga(@(K)pidtest(G,dt,K),3,-eye(3),zeros(3,1),[],[],[],[],[],options);

----------------------------------------------------------------------2nd script-----------------------------

function J = pidtest(G,dt,parms)

s = tf('s');

K = parms(1)+ parms(2)/s + parms(3)*s/(0.000001*s+1)%/(1+0.000001*s)% this is 10^-6 0.000001

Loop = series(K,G);

ClosedLoop = feedback(Loop,1);

t = 0:dt:0.005; % this indicates length of time to show

[y,t] = step(ClosedLoop,t);

u = lsim(K,1-y,t);

Q = 1;

R = 0.000005; % 5*10^-6 change these to effect rise, overshoot, settling time? 10^-8 rn

J = dt*sum(Q*(1-y(:)).^2 + R*u(:).^2)

step(5*ClosedLoop,t)

h = findobj(gcf,'type','line');

set(h,'linewidth',1);

drawnow

----------------------------------------end of script------------------------------------------

-----------------------------------------------------end of code used on that simulink model-----------------------------------------------------------------------------------------------------------------------------------------------------------

I have a code which shows to eliminate the steady state error but it does not translate onto the simulink model shown. The new code which gets rid of the steady state error is shown below.

----------------------------1st script-----------------------------------

clear all, close all, clc

dt = 0.000001; % this is 10^-6 0.000001

PopSize = 40; % was 40

MaxGenerations = 25; % was 25

s = tf('s');

G =(1.44e09)/((s*s)+5333*s+9.6e07)

options = optimoptions(@ga,'PopulationSize',PopSize,'MaxGenerations',MaxGenerations,'OutputFcn',@myfun);

[x,fval] = ga(@(K)pidtest1(G,dt,K),3,-eye(3),zeros(3,1),[],[],[],[],[],options);

----------------------------------------------------------------------2nd script-----------------------------

function J = pidtest(G,dt,parms)

s = tf('s');

K = parms(1)+ parms(2)/s + parms(3)*s/(0.000001*s+1) %/(1+0.000001*s)% this is 10^-6 0.000001

Loop = series(K,G);

ClosedLoop = feedback(Loop,1);

t = 0:dt:0.005; % this indicates length of time to show

[y,t] = step(ClosedLoop,t);

S=stepinfo(ClosedLoop);

rt1=S.RiseTime;

rt2=S.SettlingTime;

rt3=S.Overshoot;

%%% J = (dt*sum((1-y(:)).^2))+rt1+rt2+rt3;

u = lsim(K,1-y,t);

Q = 1;

R = 0.000005; % 5*10^-6 change these to effect rise, overshoot, settling time? 10^-8 rn

J = dt*sum(Q*(1-y(:)).^2 + R*u(:).^2)+rt1+rt2+rt3;

step(5*ClosedLoop,t)

h = findobj(gcf,'type','line');

set(h,'linewidth',1);

drawnow

----------------------------------------end of script------------------------------------------

Mathieu NOE 2021-3-5

在 MATLAB Online 中打开

hi again

I tried this PID tuning code and found it pretty effective

V-Tiger - File Exchange - MATLAB Central (mathworks.com)

results :

kp = 0.1926

ki = 1.9297e+03

kd = 2.2012e-05

attached the code (demo file modified for your purpose)

% vtiger_demo demonstrates V-Tiger.
    % Parameter setting of Plant G(s), sampling time ts[s], and so on.
    ts=1e-5;    % sampling time [s] (original value was 0.01)
    s=tf('s');  % complex variable of Laplace transform
    z=tf('z',ts);   % shift operator
    p=(1-1/z)/ts;   % differential operator based on backward Euler's rule
%     Gs = 5/(0.01*s^2+0.2*s+10)*exp(-0.1*s);   % plant G(s) to be controlled
%     Gs = 1.44e09/(s^2+5333*s+9.6e07);   % plant G(s) to be controlled
    Gs = 1.44e09/(s^2+5333*s+9.6e07);   % plant G(s) to be controlled
    
    G = c2d(Gs,ts); % G(z) is derived by discretizing G(s) with zero order holder
    % Design initial controller K0 using  The Ziegler-Nichols rule
    [Ku,Pm,Wu,Wcp] = margin(G);    % get Gaim margin Ku at Wu[rad/s]
    Tu = 1/(Wu/2/pi);   % When K=Ku, self-excited vibration with period Tu[s] will occur.
    kp0=0.6*Ku; ki0=kp0/(0.5*Tu);    kd0=kp0*0.125*Tu;  % ZN classical parameters
    K0 = kp0 + ki0/p + kd0*p;   % ZN PID controller K0(z)
    
    % Measuring one-shot data y00(t) and u00(t)
    u00=ones(300,1);    % input u00(t) is a step function
    u00(1)=0;   % initial value must be different from the other values <-- IMPORTANT!
    y00=lsim(G,u00);    % y00 is simulated
    r=u00;  % reference input to feedback system
    
    % Step 1) Make step responses to cyclic, and get frequency data.
    freq.y0jw = fft4step(y00);  % y0(j w) from y00(t)
    freq.u0jw = fft4step(u00);  % u0(j w) from u00(t)
    freq.r0jw = fft4step(r);    % r0(j w) from r(t)
    freq.p = fft4tf(p,length(u00)*2);   % p(j w) from differential operator p
    freq.r = r;   % r(t) is reference input to feedback system
    freq.wST=0.05;   % Error band of settling time for cost function (Settling Time Threshold)
    freq.OVr=5;   % Overshoot [%] for constraints
    freq.GMr=6;   % Gain margin [dB] for constraints:  Regulator 3-10dB, 20-inf deg
    freq.PMr=45;  % Phase margin [deg] for constraints:Servo    10-20dB, 40-60  deg
    
    % Step 2) Optimize PID gains by evaluating overshoot, settling time,
    %         and stability margins using virtual time response.
    [kp,ki,kd] = vtigerPID(freq,[kp0 ki0 kd0]);   % Get optimum PID gains. [kp0 ki0 kd0] is initial value for optimization
    K = kp + ki/p + kd*p;  % PID controller by V-Tiger
    
    disp('-----------------------------------------------------')
    % Verify the controller of V-Tiger and ZN by simulations
    [y,u] = freq2yu(freq,K); % Virtual time responses predicted by V-Tiger when K is used
    Gcl = feedback(ss(G*K),1);  % closed loop transfer function. feedback(a,b)=a/(1+a*b). ss(G) is ss(Gcl);  % State space representation from G
    yt=lsim(Gcl,r); % yt is true y(t) simulated using true plant model G(z)
    yZN=lsim(feedback(ss(G*K0),1),r);% y(t) by K0 (ZN) is simulated using true plant model G(z)
    figure(1),
    p1=plot([yt yZN y-yt y00]);
    hold on
    vtigerPID(freq,[kp ki kd],1);
    hold off; grid; xlabel('sample number k (0.01k [sec])')
    legend(p1,'y (V-Tiger)','y (ZN)','error of true/virtual y','y_{00}','Location','southeast');
    title('PID control result. V-Tiger is better than Ziegler-Nichols rule')
    disp('V-Tiger has completed controller design using y00 instead of the plant model.')
    disp(['Plant G(s) to be discretized with sampling time ts=' num2str(ts) '[s] is as follows:']), 
    [yZN,iZN]=max(yZN); text(iZN,yZN,'\leftarrow y (ZN)','Color','red','FontSize',14);
    [yvt,ivt]=max(yt); text(ivt,yvt+0.06,['y (V-Tiger)';'\downarrow '],'Color','blue','FontSize',14);
    text(length(y)*0.16,y00(end)*0.8,['\uparrow                                      '; ...
                                      'y_{00} (used by V-Tiger instead of model G(z))'],'Color','magenta','FontSize',14);
    text(length(y)*0.25,max(y)*0.58,[ ...
    'V-Tiger optimization is as follows:                       '; ...
    '  Cost function: Settling time (error band is \pm 3%)     '; ...
    '  Constraints: Overshoot<3%, Stability margins> 3dB, 20deg'])
    Gs, disp('');
    disp('Fig.1 shows step resonses. V-Tiger is better than ZN method.'),
    disp('press any key to type code of "vtiger_demo.m".'), 
    pause
    disp('-----------------------------------------------------')
    figure(2), margin(G*K);    text(10,-1000,'Open-loop by V-Tiger')
    dbtype vtiger_demo 1:37
    disp('-----------------------------------------------------')
    
    % one more figure for myself 
     figure(3),plot(yt)
    title('PID control result. V-Tiger is better than Ziegler-Nichols rule')
    xlabel('sample number k (0.01k [sec])')
    ylabel('step response')
    kp
    ki
    kd
    