实验二：调节器参数变化对控制过程的影响

选取一阶惯性环节加滞后环节建模，通过实验测试法获取传递函数参数。

tau=35;
T=37;
K=0.64;
s=tf('s');
W=exp(-s*tau)*K/(T*s+1);
Ts=1;
W=c2d(W,Ts);
%use the state-space representation for more efficient modeling of discrete-time delays
W=ss(W);

不同控制器比较

比较纯比例，PI，PD，PID控制器的控制效果

C={'开环','P','PI','PD','PID'};

ess=zeros(1,5);

figure()

hold on;

t=1:600;

stepplot(W,t);

y=step(W,t);

info=stepinfo(y,t);

s=repmat(info,1,5);

ess(1)=1-y(end);

%pid反馈控制

for i=2:5

Gc=pidtune(W,C{i});

Gc=pidstd(Gc);

Phi=feedback(series(Gc,W),1);

stepplot(Phi,t);

y=step(Phi,t);

s(i)=stepinfo(y,t);

ess(i)=1-y(end);

end

legend(C+"控制")

grid;xlabel("时间/s");ylabel("幅度")

title("过程对象控制效果仿真");

T = table( ...

[s.RiseTime]',[s.PeakTime]',[s.SettlingTime]', ...

[s.Overshoot]',ess',...

'VariableNames',{'上升时间','峰值时间','调节时间','超调量','稳态误差'},...

'RowNames',C');

display(T)

T = 5×5 table 

	上升时间	峰值时间	调节时间	超调量	稳态误差
1 开环	81.2981	600	179.7470	0	0.3600
2 P	14.0726	79	567.0263	73.3301	0.3914
3 PI	42.2625	115	237.6857	10.1351	0
4 PD	0.7138	69	594.6006	93.6699	0.3509
5 PID	40.3500	107	212.8011	8.4915	0

Gc1=pidtune(W,"P");

Phi1=feedback(series(Gc1,W),1);

%pid反馈控制

for i=3:4

figure()

hold on

stepplot(Phi1,t,'k-.');

Gc=pidtune(W,C{i});

Gc=pidstd(Gc)

Phi=feedback(series(Gc,W),1);

stepplot(Phi,t,'k-');

legend("纯比例控制",C(i)+"控制")

grid;xlabel("时间");ylabel("幅度");

title("纯比例控制与"+C(i)+"控制效果比较")

end

Gc =
 
             1       Ts  
  Kp * (1 + ---- * ------)
             Ti      z-1 

  with Kp = 1.3, Ti = 49.9, Ts = 1
 
Sample time: 1 seconds
Discrete-time PI controller in standard form

Gc =
 
                   z-1 
  Kp * (1 + Td * ------)
                   Ts  

  with Kp = 2.51, Td = 17, Ts = 1
 
Sample time: 1 seconds
Discrete-time PD controller in standard form

figure()

hold on;

specs="k"+["--","-.","-"];

pids=["P","PI","PID"];

for i=1:3

Gc=pidtune(W,pids(i));

Gc=pidstd(Gc);

Phi=feedback(series(Gc,W),1);

stepplot(Phi,t,specs(i));

end

legend("纯比例控制","PI控制","PID控制")

grid;xlabel("时间");ylabel("幅度");

title("控制效果比较")

不同控制器参数比较

使用自带函数整定PID控制器

Gpid=pidtune(W,"PID");
Gpid=pidstd(Gpid)
Gpid =
 
             1       Ts            z-1 
  Kp * (1 + ---- * ------ + Td * ------)
             Ti      z-1           Ts  

  with Kp = 1.46, Ti = 50.5, Td = 4.18, Ts = 1
 
Sample time: 1 seconds
Discrete-time PID controller in standard form

增大或减小比例度

Pi=1/Gpid.Kp;

Pi_inc=Pi*1.5;

Pi_dec=Pi*0.5;

figure();hold on;

Phi=feedback(series(Gpid,W),1);

Phi1=feedback(series(pidstd(1/Pi_inc,Gpid.Ti,Gpid.Td,Inf,Ts),W),1);

Phi2=feedback(series(pidstd(1/Pi_dec,Gpid.Ti,Gpid.Td,Inf,Ts),W),1);

stepplot(Phi,"k-",t);

stepplot(Phi1,"k-.",t);

stepplot(Phi2,"k:",t);

grid;xlabel("时间");ylabel("幅度");

title("增大或减小比例度");

label=[

sprintf("比例度Pi=%.2f",Pi);

sprintf("增大比例度至Pi=%.2f",Pi_inc);

sprintf("减小比例度至Pi=%.2f",Pi_dec)

];

legend(label)

T=[struct2table(stepinfo(Phi));struct2table(stepinfo(Phi1));struct2table(stepinfo(Phi2))];

T.Properties.RowNames=label;

T = 3×9 table 

	RiseTime	TransientTime	SettlingTime	SettlingMin	SettlingMax	Overshoot	Peak	PeakTime
1 比例度Pi=0.68	41	213	213	0.9088	1.0849	8.4910	1.0849	107
2 增大比例度至Pi=1.03	97	258	258	0.9012	1	0	1	709
3 减小比例度至Pi=0.34	17	952	952	0.3912	1.7586	75.8570	1.7586	82

增大或减小积分时间常数

Ti=Gpid.Ti;

Ti_inc=Ti*1.5;

Ti_dec=Ti*0.5;

figure();hold on;

Phi=feedback(series(Gpid,W),1);

Phi1=feedback(series(pidstd(Gpid.Kp,Ti_inc,Gpid.Td,Inf,Ts),W),1);

Phi2=feedback(series(pidstd(Gpid.Kp,Ti_dec,Gpid.Td,Inf,Ts),W),1);

stepplot(Phi,"k-",t)

stepplot(Phi1,"k-.",t)

stepplot(Phi2,"k:",t);

grid;xlabel("时间");ylabel("幅度");

title("增大或减小积分时间常数")

label=[

sprintf("积分时间常数Ti=%.2f",Ti);

sprintf("增大积分时间常数至Ti=%.2f",Ti_inc);

sprintf("减小积分时间常数至Ti=%.2f",Ti_dec)];

legend(label)

T=[struct2table(stepinfo(Phi));struct2table(stepinfo(Phi1));struct2table(stepinfo(Phi2))];

T.Properties.RowNames=label;

T = 3×9 table 

	RiseTime	TransientTime	SettlingTime	SettlingMin	SettlingMax	Overshoot	Peak	PeakTime
1 积分时间常数Ti=50.50	41	213	213	0.9088	1.0849	8.4910	1.0849	107
2 增大积分时间常数至Ti=75.75	59	362	362	0.8484	0.9997	0	0.9997	849
3 减小积分时间常数至Ti=25.25	30	609	609	0.6512	1.6300	63.0028	1.6300	110

增大或减小微分时间常数

Td=Gpid.Td;

Td_inc=Td*1.5;

Td_dec=Td*0.5;

figure();hold on;

Phi=feedback(series(Gpid,W),1);

Phi1=feedback(series(pidstd(Gpid.Kp,Gpid.Ti,Td_inc,Inf,Ts),W),1);

Phi2=feedback(series(pidstd(Gpid.Kp,Gpid.Ti,Td_dec,Inf,Ts),W),1);

stepplot(Phi,"k-",t)

stepplot(Phi1,"k-.",t)

stepplot(Phi2,"k:",t);

grid;xlabel("时间");ylabel("幅度");

title("增大或减小微分时间常数")

label=[

sprintf("微分时间常数Td=%.2f",Td);

sprintf("增大微分时间常数至Td=%.2f",Td_inc);

sprintf("减小微分时间常数至Td=%.2f",Td_dec)

];

legend(label);

T=[struct2table(stepinfo(Phi));struct2table(stepinfo(Phi1));struct2table(stepinfo(Phi2))];

T.Properties.RowNames=label;

T = 3×9 table 

	RiseTime	TransientTime	SettlingTime	SettlingMin	SettlingMax	Overshoot	Peak	PeakTime
1 微分时间常数Td=4.18	41	213	213	0.9088	1.0849	8.4910	1.0849	107
2 增大微分时间常数至Td=6.26	41	186	186	0.9021	1.0417	4.1742	1.0417	107
3 减小微分时间常数至Td=2.09	38	225	225	0.9117	1.1304	13.0360	1.1304	108