1. Implementation of Intelligent Control on TITech Pendulum
M.Yamakita
Dept. of Mechanical and Control Systems Eng.
Tokyo Inst. Of Tech.
2018/11/20

2. TITech Pendulum Top View
TITech pendulum is a rational type pendulum which
was developed for control algorithm test bed by prof.
K.Furuta at Tokyo Institute of Technology.
Side View
Coordinate System
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3. Dynamic Equation and Parameters
Non-Linear Dynamic Equation
Linearlized Dynamic Equation at Upright Position
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4. Stabilizing Control at Upright Position
Standard Regulator Problem
Linear Quadratic Optimal Regulator(LQR)
Optimal Feedback Control
State Observer (Minimal Order Observer)
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5. Swing Up Control of TITech Pendulum
Assume Input Sequence (Bang-Bang Control)
Determination of T1, T2, T3, and T4
Constrained Successive Optimization
Open Loop Control
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6. Reduction of Degree of the System
Assumption
Reduced Model used for Non-Linear Control
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7. Non-Linear State Feedback Control
Controlled Vector Field of the System
U = 0
U = +Max
U = -Max
Combine !
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8. Control Result
Let’s use this controller
as a expert.
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9. Fuzzy Set Allocation to the ProblemZO NS NM
NB1
NB2 PS PM
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10. If-THEN Rule for the ProblemPM
If x is NM and y is ZO then z is PB.
If x is NB1 and y is PS then z is NB.
If x is NM and y is PS then z is NB.
If x is NS and y is PS then z is NB.
If x is NB1 and y is ZO then z is ZO.
If x is NS and y is ZO then z is ZO.
If x is NB2 then z is ZO.
If x is ZO then z is ZO.
If y is NS then z is ZO.
If y is PM the z is ZO. PS ZO NS
NB2
NB1 NM NS ZO
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11. Simulation Environment
MaTX/RtMaTX： Prof.Masanobu　KOGA
　　　　　Matrix Operation Language
Useful Functions for Control System Design
(Similar to Matlab)
Free Software
Running on Windows, Unix, DOS …
Download→
Scherzo：Mr.　Masaaki　HARA
　　　　　　３D　Animation　Software(Free)
X Window+Open GL(Mesa,glut)
Download→
2018/11/20

12. 2018/11/20

13. 2018/11/20

14. 2018/11/20

15. Matrix Element Operation Using MaTX at Glance
(Matrix) A, B;
A = Z(2,2); B = I(2,2);
A(1,2) = 3.0;
B(2,2) = A(2,2);
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16. Matrix Operation Using MaTX at Glance
Matrix A, B,C;
1. Addition, Subtraction, Multiplication, Division
C = A+B; C = A-B; C= A*B; C = A\B (C = A~*B)
2. Transpose, Complex Conjugate Transpose
C = A’; C = B#;
3. Eignevalue, Eignevector
eigenvalues(A); {D,X} = eig(A) ;
4. Determinant, Inversion, Rnak
det(A); C = A~; rank(A);
5. Singular Decomposition, Schur Decomposition
{U,S,V} = svd(A); {UT} = schur(A);
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17. Structure of the Simulator
diff_eqs(dx,t,u,x)
link_eqs(u,t,x) t x u
Logging Data
Simulation Functions:
{t,x,u} = OdeAuto(ts,te,x0,diff_eqs,link_eqs);
{t,x,u} = OdeHybrid(ts,te,T,x0,diff_eqs,link_eqs);
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18. Fuzzy Set Allocation on Real Line (Matrix Representation) 1h1 l2 C2 h2 … x S1 S2
Sx(:,1) stands for fuzzy set S1
Sx(:,2) stands for fuzzy set S2 …
S1’
S2’ 1
C1’
l1’
h1’
l2’
C2’
h2’ y 1
C1’’
C2’’ z
(S1’’)
(S2’’)
(Singletons)
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19. IF-THEN Rules and Function Realization
R1:If x is S1 and y is S1’ then z is S1’’’.
R2:If x is S2 and y is S2’ then z is S2’’’. …
We assume system output and control input are real values not are fuzzy sets. w2 w1 z
C1’’
C2’’
(Normalized weighted sum)
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20. Functions and Variables in the Sample Program
FIsMember(x0, Sx,1)
Fand(a,b)
w(1,i)
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21. 2018/11/20 Real I0,L0,C0,m1,J1,l1,C1, g; Matrix f; Real Umax;
Real MF, PF;
Integer NO_OF_RULES, MAX_DIV;
Matrix FSet1, FSet2, FSet3;
Func void main() {
　　　Matrix x0;
　　　Array tc, xc, uc;
　　　Real T, h;
　　　Integer i, N;
　　　Matrix A, B, D, C, G, Q, r, P;
　　　void diff_eqn(), link_eqn(), Set_FSets();
　　　Integer IsNear();
/* Setting for Fuzzy Control */
　　　Set_FSets();
　　　Umax = 1.25;
L0 = 0.215;
I0 = 1.75e-2;
C0 = 0.118;
m1 = 5.38e-2;
J1 = 1.98e-4;
l1 = 0.113;
C1 = 8.3e-5;
g = 9.81;
D = [[I0+m1*L0^2, m1*l1*L0]
[m1*l1*L0, J1+m1*l1^2]];
C = [[C0, 0]
[0, C1]];
G = [[0, 0]
[0, -m1*l1*g]];
A = Z(4,4);
A(1,3)= 1.0;
A(2,4)= 1.0;
A(3:4,1:4) = D~*[-G, -C];
B = Z(4,1);
B(3:4,1) = D~*[1.0,0]';
r = I(1,1);
Q = [[30,0,0,0][0,2000,0,0][0,0,0.1,0][0,0,0,10]];
{f,P} = lqr(A, B, Q, r);
x0 = Z(4,1);
x0(2,1) = -3.14;
h = 5.0e-3;
{tc, xc, uc}
　　　= OdeHybrid(0.0, T, h, x0, diff_eqn, link_eqn,h,1/30.0);
N =Cols(tc);
for(i=1;i<=N;i++) {
　　　　xc(2,i) = -xc(2,i) + PI; ｝
print [[tc] [xc] [uc]] >> "results.mat"; }
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22. Func void diff_eqn(dx,t,x,u)
Matrix dx,x,u;
Real t; {
Matrix D,C,G,th,dth;
Real c1,s1, s2;
th = x(1:2,1);
dth= x(3:4,1);
s1 = sin(th(2,1));
c1 = cos(th(2,1));
s2 = sin(2*th(2,1));
D = [[I0 +m1*(L0^2+l1^2*s1*s1), m1*l1*L0*c1]
[m1*l1*L0*c1, J1+m1*l1^2]];
C = [[C0+m1*l1^2*s2*dth(2,1)/2.0,-m1*l1*L0*s1+m1*l1^2*s2*dth(2,1)/2.0]
[-m1*l1^2*s2*dth(2,1), C1]];
G = [0, -m1*l1*g*s1]';
dx = [[dth] [D~*([1.0,0.0]'*u - C*dth-G)]]; }
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23. Func void link_eqn(u, t, x)
Real t;
Matrix u, x; {
Matrix FControl();
if(IsNear(x)) {
u = -f*(x - [0, 0, 0, 0]');
} else {
u = FControl(x); }
if(u(1,1) > Umax) {
u(1,1) = Umax;
if(u(1,1) < -Umax) {
u(1,1) = -Umax;
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24. 2018/11/20 Func Matrix FControl(xx) Matrix xx; { Func Real Defuzzy(S)
Integer i;
Real x,y, z, m;
Real FIsMember(), FAnd();
Matrix FSet3d;
Real Defuzzy();
void FOr();
NO_OF_RULES = 9;
x = xx(2,1);
y = xx(4,1);
FSet3d = FSet3;
for(i=1;i<=Cols(FSet3d);i++) {
FSet3d(2,i) = 0.0; }
FOr(FAnd(FIsMember(x, FSet1, 3), FIsMember(y, FSet2, 2)), FSet3d, 1);
FOr(FAnd(FIsMember(x, FSet1, 2), FIsMember(y, FSet2, 3)), FSet3d, 3);
FOr(FAnd(FIsMember(x, FSet1, 3), FIsMember(y, FSet2, 3)), FSet3d, 3);
FOr(FAnd(FIsMember(x, FSet1, 4), FIsMember(y, FSet2, 3)), FSet3d, 3);
FOr(FAnd(FIsMember(x, FSet1, 2), FIsMember(y, FSet2, 2)), FSet3d, 2);
FOr(FIsMember(y, FSet2, 1), FSet3d, 2);
FOr(FIsMember(y, FSet2, 4), FSet3d, 2);
FOr(FIsMember(x, FSet1, 1), FSet3d, 2);
FOr(FIsMember(x, FSet1, 5), FSet3d, 2);
z = Defuzzy(FSet3d);
return([z]);
Func Real Defuzzy(S)
Matrix S; {
Integer i, n;
Real z, w;
n = Cols(S);
z = 0.0;
w = 0.0;
for(i=1;i<=n;i++) {
z = z + S(1,i)*S(2,i);
w = w + S(2,i); }
return(z/w);
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25. Open Loop Control
Crisp Logic Control
Fuzzy Logic Control
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26. 2018/11/20 Func void Set_FSets() { Integer i,j; MF = -1e+8; PF = 1e+8;
MF = -1e+8;
PF = 1e+8;
/* Fuzzy Set */
FSet1 = Z(3,5);
FSet1(:,1) = [MF, -6.5, -3.85]';
FSet1(:,2) = [-5.3, -3.85, -3.3]';
FSet1(:,3) = [-3.5, -3.3, -2.66]';
FSet1(:,4) = [-3.3, -2.66, -1.11]';
FSet1(:,5) = [-2.66, 0.0, PF]';
FSet2 = Z(3,4);
FSet2(:,1) = [MF, -26.0, 3.18]';
FSet2(:,2) = [-17.53, -1.82, 5.42]';
FSet2(:,3) = [3.0, 12.17, 26.0]';
FSet2(:,4) = [17.17, 26, PF]';
FSet3 = Z(1,3);
FSet3(1,1) = -Umax;
FSet3(1,2) = 0.0;
FSet3(1,3) = Umax; }
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27. Simulation Results (1)
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28. Simulation Result (2)
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29. References
K.Furuta, M.Yamakita, S.Kobayashi:Swing-up control of inverted pendulum using pseudo-state feedback, Proc. Inst. Mech. Engrs. Vol.206 (1992)
M.Koga: MaTX reference manual (1995)
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