M.Yamakita

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Implementation of Intelligent Control onTITech
Pendulum

• M.Yamakita
• Dept. of Mechanical and Control Systems Eng.
• Tokyo Inst. Of Tech.

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Top View

• TITech Pendulum

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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Dynamic Equation and Parameters
Non-Linear Dynamic Equation
Linearlized Dynamic Equation at Upright Position
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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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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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Reduction of Degree of the System
Assumption
Reduced Model used for Non-Linear Control
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Non-Linear State Feedback Control
Controlled Vector Field of the System
U 0
U Max
U -Max
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Control Result
Lets use this controller as a expert.
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Fuzzy Set Allocation to the Problem
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If-THEN Rule for the Problem
PM
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
ZO
NS
NM
NB1
NB2
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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?http//www.matx.org/machines
ScherzoMr. Masaaki HARA 3D Animation
Software(Free) X WindowOpen
GL(Mesa,glut) Download?http//www.matx.org/machin
es
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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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Matrix Operation Using MaTX at Glance
Matrix A, B,C 1. Addition, Subtraction,
Multiplication, Division C AB C A-B
C AB C A\B (C AB) 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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Structure of the Simulator
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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Fuzzy Set Allocation on Real Line (Matrix
Representation)

x
Sx(,1) stands for fuzzy set S1
Sx(,2) stands for fuzzy set S2

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(S1)
(S2)
(Singletons)
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IF-THEN Rules and Function Realization
R1If x is S1 and y is S1 then z is S1. R2If
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
C2
C1
(Normalized weighted sum)
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Functions and Variables in the Sample Program
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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 I0m1L02,
m1l1L0 m1l1L0, J1m1l12 C C0,
0 0, C1 G 0, 0 0,
-m1l1g   A Z(4,4) A(1,3) 1.0 A(2,4)
1.0 A(34,14) D-G, -C B
Z(4,1) B(34,1) D1.0,0' r
I(1,1) Q 30,0,0,00,2000,0,00,0,0.1,00,
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(i1iltNi) xc(2,i)
-xc(2,i) PI   print tc xc uc
gtgt "results.mat"  
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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(12,1) dth x(34,1)   s1
sin(th(2,1)) c1 cos(th(2,1)) s2
sin(2th(2,1))   D I0 m1(L02l12s1s1),
m1l1L0c1 m1l1L0c1, J1m1l12 C
C0m1l12s2dth(2,1)/2.0,-m1l1L0s1m1l12
s2dth(2,1)/2.0 -m1l12s2dth(2,1),
C1 G 0, -m1l1gs1'   dx dth
D(1.0,0.0'u - Cdth-G)
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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) gt Umax) u(1,1)
Umax if(u(1,1) lt -Umax) u(1,1)
-Umax  
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Func Matrix FControl(xx) Matrix xx 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(i1iltCols(FSe
t3d)i) FSet3d(2,i) 0.0   FOr(FAnd(FI
sMember(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(i1iltni) z z
S(1,i)S(2,i) w w S(2,i) return(z/w)

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Open Loop Control
Crisp Logic Control
Fuzzy Logic Control
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Func void Set_FSets() Integer i,j   MF
-1e8 PF 1e8   / 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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Simulation Results (1)
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Simulation Result (2)
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References

1. K.Furuta, M.Yamakita, S.KobayashiSwing-up
   control of inverted pendulum using pseudo-state
   feedback, Proc. Inst. Mech. Engrs. Vol.206 (1992)
2. M.Koga MaTX reference manual (1995)