Simulink PID Tuner can't find good solution

Question

George Nehma 2023년 8월 2일

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https://kr.mathworks.com/matlabcentral/answers/2003477-simulink-pid-tuner-can-t-find-good-solution

답변: Sam Chak 2023년 8월 19일

I have a 6DOF Aircraft dynamics model and am trying to control it with a few PID controllers for each aspect as shown in the image

I am trying to use the PID Tuner to find some good values for my gains, but everytime the solution just blows up as you can see. I have been able to manually tune the gains to control certain maneuovers, but am wondering why the PID Tuner App can't find some good values?

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Sam Chak 2023년 8월 2일

Hi @George Nehma

I don't think it's possible to tune all 5 PID controller blocks simultaneously. If each subsystem is fully decoupled, then I believe you can tune the corresponding PID controller one at a time.

I remember that the 6-DOF Aircraft dynamics are highly nonlinear, and the states are strongly coupled. If the 6-DOF Aircraft dynamics are linearized, could you please show the model?

George Nehma 2023년 8월 2일

편집: George Nehma 2023년 8월 2일

HI @Sam Chak,

I just clicked into one of the controllers to tune, so I think I was trying to tune just one controller at a time.

You're correct, it is highly nonlinear and strongly coupled, but this is the State Space plant model the lineariser developed:

Plant =

A =

q_e2b q_e2b q_e2b q_e2b X_ECEF X_ECEF X_ECEF V_ECEF

q_e2b 3.922e-06 -4.374e-06 -1.362e-05 2.882e-05 0 0 0 0

q_e2b 6.998e-06 -7.805e-06 1.215e-05 -1.362e-05 0 0 0 0

q_e2b -2.247e-06 -3.395e-05 7.805e-06 4.374e-06 0 0 0 0

q_e2b -3.444e-05 -2.247e-06 -6.998e-06 -3.922e-06 0 0 0 0

X_ECEF 0 0 0 0 0 0 0 1

X_ECEF 0 0 0 0 0 0 0 0

V_ECEF 36.66 -23.95 15.67 4.609 2.626e-06 -2.518e-05 1.362e-05 -0.07975

V_ECEF -85.19 -79.31 -22.86 42.48 1.493e-05 -9.18e-05 4.881e-05 0.04115

V_ECEF 72.41 -5.426 26.81 -7.521 6.389e-06 -3.864e-05 1.935e-05 -0.103

w_b 2.498 -14.11 2.514 4.886 6.064e-06 -3.658e-05 1.98e-05 -0.04206

w_b 1.146 0.6344 0.3719 -0.4382 8.646e-07 -5.229e-06 2.827e-06 -0.0002576

w_b -2.667 4.153 -1.422 -1.213 -2.684e-06 1.623e-05 -8.777e-06 0.01655

Integrator 0 0 0 0 -0.0004318 0.002611 -0.001412 0

Alt Controll 0 0 0 0 -0.002879 0.01741 -0.009414 0

Phi Controll 0.03005 0.02694 -0.05243 0.09356 -5.77e-09 -3.997e-09 -5.628e-09 0

V_ECEF V_ECEF w_b w_b w_b Integrator Alt Controll Phi Controll

q_e2b 0 0 0.1311 0.4082 0.2288 0 0 0

q_e2b 0 0 0.1176 0.2288 -0.4082 0 0 0

q_e2b 0 0 -0.2288 0.1176 0.1311 0 0 0

q_e2b 0 0 0.4082 -0.1311 0.1176 0 0 0

X_ECEF 0 0 0 0 0 0 0 0

X_ECEF 1 0 0 0 0 0 0 0

X_ECEF 0 1 0 0 0 0 0 0

V_ECEF 0.02421 -0.1031 -0.07382 0.9576 0.3644 0 0 -0.0009028

V_ECEF -0.4573 0.1663 -0.05092 -6.056 0.2514 0 0 -0.0006227

V_ECEF 0.1807 -0.1861 -0.07159 3.32 0.3534 0 0 -0.0008755

w_b -0.05897 -0.03856 -1.856 0 0.1926 0 0 0.02236

w_b 0.006346 -0.00436 0 -0.4747 0 0 0 0

w_b 0.01162 0.01981 0.004665 0 -0.2395 0 0 0.001105

Integrator 0 0 0 0 0 0 0 0

Alt Controll 0 0 0 0 0 0 -0.01 0

Phi Controll 0 0 0 0 0 0 0 -0.1

B =

Elevator

q_e2b 0

X_ECEF 0

V_ECEF 0.01426

V_ECEF -0.08832

V_ECEF 0.04811

w_b 0

w_b -0.05363

w_b 0

Integrator 0

Alt Controll 0

Phi Controll 0

C =

q_e2b q_e2b q_e2b q_e2b X_ECEF X_ECEF X_ECEF V_ECEF

Theta Contro -93.56 -52.43 -26.94 30.05 -0.004312 0.02611 -0.01413 0

V_ECEF V_ECEF w_b w_b w_b Integrator Alt Controll Phi Controll

Theta Contro 0 0 0 0 0 1 -0.01 0

D =

Elevator

Theta Contro 0

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Answer 1

Sam Chak 2023년 8월 19일

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이 답변에 대한 바로 가기 링크

https://kr.mathworks.com/matlabcentral/answers/2003477-simulink-pid-tuner-can-t-find-good-solution#answer_1289852

MATLAB Online에서 열기

Hi @George Nehma

This is a very high 16th-order coupled system. I'm not sure if it can be stablized by the PID controller. So I tried the LQR approach, and the closed-loop feedback appears to be stable.

A1 = [3.922e-06 -4.374e-06 -1.362e-05 2.882e-05 0 0 0 0; ...

6.998e-06 -7.805e-06 1.215e-05 -1.362e-05 0 0 0 0; ...

-2.247e-06 -3.395e-05 7.805e-06 4.374e-06 0 0 0 0; ...

-3.444e-05 -2.247e-06 -6.998e-06 -3.922e-06 0 0 0 0; ...

0 0 0 0 0 0 0 1; ...

0 0 0 0 0 0 0 0; ...

36.66 -23.95 15.67 4.609 2.626e-06 -2.518e-05 1.362e-05 -0.07975; ...

-85.19 -79.31 -22.86 42.48 1.493e-05 -9.18e-05 4.881e-05 0.04115; ...

72.41 -5.426 26.81 -7.521 6.389e-06 -3.864e-05 1.935e-05 -0.103; ...

2.498 -14.11 2.514 4.886 6.064e-06 -3.658e-05 1.98e-05 -0.04206; ...

1.146 0.6344 0.3719 -0.4382 8.646e-07 -5.229e-06 2.827e-06 -0.0002576; ...

-2.667 4.153 -1.422 -1.213 -2.684e-06 1.623e-05 -8.777e-06 0.01655; ...

0 0 0 0 -0.0004318 0.002611 -0.001412 0; ...

0 0 0 0 -0.002879 0.01741 -0.009414 0; ...

0.03005 0.02694 -0.05243 0.09356 -5.77e-09 -3.997e-09 -5.628e-09 0];

A2 = [0 0 0.1311 0.4082 0.2288 0 0 0; ...

0 0 0.1176 0.2288 -0.4082 0 0 0; ...

0 0 -0.2288 0.1176 0.1311 0 0 0; ...

0 0 0.4082 -0.1311 0.1176 0 0 0; ...

0 0 0 0 0 0 0 0; ...

1 0 0 0 0 0 0 0; ...

0 1 0 0 0 0 0 0; ...

0.02421 -0.1031 -0.07382 0.9576 0.3644 0 0 -0.0009028; ...

-0.4573 0.1663 -0.05092 -6.056 0.2514 0 0 -0.0006227; ...

0.1807 -0.1861 -0.07159 3.32 0.3534 0 0 -0.0008755; ...

-0.05897 -0.03856 -1.856 0 0.1926 0 0 0.02236; ...

0.006346 -0.00436 0 -0.4747 0 0 0 0; ...

0.01162 0.01981 0.004665 0 -0.2395 0 0 0.001105; ...

0 0 0 0 0 0 0 0; ...

0 0 0 0 0 0 -0.01 0; ...

0 0 0 0 0 0 0 -0.1];

A = [A1 A2];

B = [0;

0;

0.01426;

-0.08832;

0.04811;

0;

-0.05363;

0;

0];

C = [-93.56 -52.43 -26.94 30.05 -0.004312 0.02611 -0.01413 0 0 0 0 0 0 1 -0.01 0];

% LQR design

Q = 1e4*eye(length(A));

R = 1;

K = lqr(A, B, Q, R)

K = 1×16

1.0e+07 * 2.5596 -3.8716 0.0389 2.5615 -0.0000 0.0003 -0.0001 0.2496 0.1813 0.3216 0.4398 0.0563 6.6404 0.0000 -0.0000 0.0581

sys = ss(A-B*K, B, C, 0);

isstable(sys)

ans = logical

1

N = 1/dcgain(sys)

N = 100.0000

% closed-loop system

sys2 = ss(A-B*K, B*N, C, 0)

sys2 = A = x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x1 3.922e-06 -4.374e-06 -1.362e-05 2.882e-05 0 0 0 0 0 0 0.1311 0.4082 0.2288 0 x2 6.998e-06 -7.805e-06 1.215e-05 -1.362e-05 0 0 0 0 0 0 0.1176 0.2288 -0.4082 0 x3 -2.247e-06 -3.395e-05 7.805e-06 4.374e-06 0 0 0 0 0 0 -0.2288 0.1176 0.1311 0 x4 -3.444e-05 -2.247e-06 -6.998e-06 -3.922e-06 0 0 0 0 0 0 0.4082 -0.1311 0.1176 0 x5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 x6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 x7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 x8 -3.65e+05 5.521e+05 -5530 -3.653e+05 3.062 -43.23 15.23 -3.559e+04 -2.585e+04 -4.586e+04 -6.272e+04 -8028 -9.469e+05 -1.426 x9 2.261e+06 -3.419e+06 3.433e+04 2.262e+06 -18.96 267.7 -94.32 2.204e+05 1.601e+05 2.84e+05 3.885e+05 4.972e+04 5.865e+06 8.832 x10 -1.231e+06 1.863e+06 -1.868e+04 -1.232e+06 10.33 -145.8 51.38 -1.201e+05 -8.72e+04 -1.547e+05 -2.116e+05 -2.709e+04 -3.195e+06 -4.811 x11 2.498 -14.11 2.514 4.886 6.064e-06 -3.658e-05 1.98e-05 -0.04206 -0.05897 -0.03856 -1.856 0 0.1926 0 x12 1.373e+06 -2.076e+06 2.086e+04 1.374e+06 -11.51 162.6 -57.27 1.338e+05 9.721e+04 1.725e+05 2.359e+05 3.02e+04 3.561e+06 5.363 x13 -2.667 4.153 -1.422 -1.213 -2.684e-06 1.623e-05 -8.777e-06 0.01655 0.01162 0.01981 0.004665 0 -0.2395 0 x14 0 0 0 0 -0.0004318 0.002611 -0.001412 0 0 0 0 0 0 0 x15 0 0 0 0 -0.002879 0.01741 -0.009414 0 0 0 0 0 0 0 x16 0.03005 0.02694 -0.05243 0.09356 -5.77e-09 -3.997e-09 -5.628e-09 0 0 0 0 0 0 0 x15 x16 x1 0 0 x2 0 0 x3 0 0 x4 0 0 x5 0 0 x6 0 0 x7 0 0 x8 0.5824 -8280 x9 -3.607 5.128e+04 x10 1.965 -2.793e+04 x11 0 0.02236 x12 -2.19 3.114e+04 x13 0 0.001105 x14 0 0 x15 -0.01 0 x16 0 -0.1 B = u1 x1 0 x2 0 x3 0 x4 0 x5 0 x6 0 x7 0 x8 1.426 x9 -8.832 x10 4.811 x11 0 x12 -5.363 x13 0 x14 0 x15 0 x16 0 C = x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 y1 -93.56 -52.43 -26.94 30.05 -0.004312 0.02611 -0.01413 0 0 0 0 0 0 1 -0.01 0 D = u1 y1 0 Continuous-time state-space model.

dcgain(sys2)

ans = 1.0000

stepinfo(sys2)

ans = struct with fields:

RiseTime: 3.2197e+05 TransientTime: 5.9513e+05 SettlingTime: 6.7426e+05 SettlingMin: 0.9000 SettlingMax: 0.9969 Overshoot: 0 Undershoot: 71.6491 Peak: 0.9969 PeakTime: 9.4624e+05

step(sys2, 1e6), grid on

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