A Study on the Motion Control of a Stabilizer System Using an Adaptiv e F uzzy Controller 2001 2
A bs tract 1 1 2 3 2.1 3 2.2 6 2.3 9 2.4 11 2.5 13 3 16 3.1 16 3.2 18 3.2.1 19 3.2.2 20 3.2.3 21 3.2.4 24
3.3 27 3.3.1 1 27 3.3.2 2 29 3.3.3 2/ 1 30 4 32 4.1 32 4.2 34 4.3 37 4.3.1 37 4.3.2 39 5 48 5.1 48 5.2 50 5.3 57 6 58 59 A ppendix 61
A Study on the Motion Control of a Stabilizer Sy stem U sing an A daptiv e Fuzzy Controller Tae-H oon K im Department of Control & Instrumentation Engineering, Graduate School, Korea Maritime University Abstract A tracking sy stem equipped on a fixed body needs the position al inform ation of the t arg et an d the control appar atu s to follow the azimuth angle and the elev ation angle of the moving object, when the tracking system is equipped on the moving v ehicle like a ship, it requires a st abilizin g sy stem t o flat the tracking system against the moving vehicle as w ell as the position al inform ation and the contr ol equipm ent. T he st abilizer sy stem compens ates the tr acking sy stem for the v ertical, horizont al an d direction al deviation s bet w een the tracking system and reference frame. T his st abilizer sy stem can be applied to a s atellite antenna on ships, a sun tracking sy stem on moving vehicles, and a camera
serv o control loop to t ake a st able im age ag ain st the vibr ation. In this paper, a stabilizer system using an active stabilization method is composed. An adaptive fuzzy controller is also sugg est ed, which is applicable t o sy st em s with structur al and parameter uncertainty. It is the 2nd/ 1st- type adaptive fuzzy control algorithm using adv antages of 1st - type and 2nd- type adaptiv e fuzzy alg orithm. Sev er al simulation s ar e ex ecut ed for v erifying the perform ance of the sugg est ed m ethod. T hrough experiments using a composed stabilizer system, tracking perform ances are ev aluated.
1, (Azimuth ) (Elevation ).,,. (Surge), (Sway), (Heave) (Roll), (Pitch ), (Yaw ) 6- (Six - degree of freedom movements).,., (Stabilizer system) [1-3]., T V,.,,,, [4,5]. (Active Stabilization method). - 1 -
X, Y 2,. (Piezo- Electric Gyro Sensor ), DC [6]. (Adaptive fuzzy controller ) [7,8]. (Adaptive Law ) 1 (First - type Adaptive fuzzy algorithm ) (Fuzzy rule) 2 (Second- type Adaptive fuzzy algorithm ) 2/ 1 (Second/ fir st - type Adaptive fuzzy algorithm ).. 2. 3 2/ 1, 4,. 5. 6. - 2 -
2 2.1 6-. 2.1 6-3 3., X, Y, Z 3 X, Y, Z 3,, 3. 2.1 6- Figure 2.1 Component of six - degree of freedom movements - 3 -
6-.,,,.,.,. (Stabilization). (Stabilizer ),.. T V,.,,,. (Passive Stabilization Method) (Active Stabilization Method).. - 4 -
..,.. 2.2 (Gimbal) 2, 2 DC. DC PWM (Pulse Width Modulation ). 2.2 2 Figure 2.2 Structure of a two- axis stabilizer system - 5 -
2.2 2.3 X, Y 2,, 2 X, Y 2., X Y. 2.3 Figur e 2.3 Block diagr am of a st abilizer sy st em - 6 -
2.3 3.,,,. Murata. 2 X Y.. X, Y 2 2 DC. 2. 3.,,.. PWM DC.,, LCD [9,10]. - 7 -
2.1 Photo 2.1 Photograph of a stabilizer sy stem - 8 -
2.3. (Mechanical Gyros) (Optical Gyroscopes), (Coriolis Effect ) (Piezo- Electric Gyroscopes). (Drift Error ) Murata ENV - 05DB Gyrostar. 2.4 80[deg/ sec], 2.5[V]. 2.4 Figur e 2.4 Output char acteristic curv e of a gyr o sen sor - 9 -
(Low Pass Filter ) A/ D.. X, Y 2. 2.2 Photo 2.2 Photogr aph of a gyr o sen sor mounted - 10 -
2.4 2., X Y 2., DC. PWM.. A/ D. 2.5 Figur e 2.5 Structure of a m otion stabilizer - 11 -
(a) Roll motion (b ) Pitch motion 2.3 2 Photo 2.3 Phot ogr aph of a tw o- axis motion stabilizer - 12 -
2.5 80C196KC,, LCD. A/ D..., DC PWM. A/ D [11-16]. 2.6 Figure 2.6 Block diagram of a data controller - 13 -
Intel 16 80C196KC 8 A/ D DC PWM,. [17-19]. 20MHz 4 (HSO) 4 (HSI) 256 RAM 28 / 16 1.75 s 16 16 (16MHz) (Power down)/ (Idle Mode) 16 (W atchdog timer ) (Full duplex ) 8 / 16 16 / / 8/ 10 A/ D 232 - PT S (Peripheral T ransaction Server ) 8 I/ O PWM 16-14 -
2.4 Photo 2.4 Phot ogr aph of a data contr oller - 15 -
3 3.1 (Adaptive Fuzzy Controller ), [20]. (Fuzzy Logic System )., [7,8]. (Linguistic Information ) [2 1]..,.,. (Order ) (Bounds).. (Adaptive Law ).,. (Direct ) - 16 -
(Indirect )..,.. IF - T HEN. 1 (First- type) 2 (Second- type). (3.1) [22]. f (x ) = M l = 1 M l = 1 l [ [ n i = 1 F l i (x i) ] n i = 1 F l i (x i) ] (3.1), l, F l i (x i). l 1, l F l i (x i) 2. - 17 -
3.2, 3.1. (Fuzzy Basis Function) (Identification Model).. 3.1 Figure 3.1 Block diagram of the indirect adaptive fuzzy control sy stem - 18 -
3.2.1 n x 1 = x 2, x 2 = x 3, x n = p(x 1,, x n ) + q(x 1,, x n ) u, y = x 1 (3.2) x ( n ) = p (x, x,, x ( n - 1) ) + q(x, x,,x ( n - 1) ) u, y = x (3.3), p(x), q(x ) :, u R, y R :, x = (x 1, x 2,, x n ) T = (x, x,,x ( n - 1) ) T R N. p (x ) q(x ) (3.4) (3.5). p ( x p ) = T p p ( x ) (3.4) T q( x q ) = q q ( x ) (3.5), = ( 1,, M ) T : l( x ) = M l = 1 [ n i = 1 F l i (x i) n i = 1 F l i (x i) ], ( x) = ( 1 ( x ),, M ( x )) T : [23]. - 19 -
3.2.2 y ( t) y m ( t). e = y m - y. h ( s) = s n + k 1 s ( n - k = ( k n,, k 1 ) T 1) + + k n R N. u = 1 q(x ) [ - p (x ) + y ( n ) m + k T e] (3.6) (3.6) (3.3) e = ( e, e,, e ( n - 1) ) T e ( n ) + k 1 e ( n - 1) + + k n e = 0 (3.7). lim t e( t) = 0. (3.6) p(x) q(x) p ( x p ) q( x q ) (Certainty Equivalent Control) u c. u c = 1 q( x q ) [ - p( x p ) + y ( n ) m + k T e] (3.8) - 20 -
3.2.3 (3.3) (3.8). e ( n ) = - k T e + [ p( x p ) - p (x) ] + [ ( q( x q ) - q( x) ]u c (3.9) e = c e + b c [ ( p ( x p ) - p (x)) + ( q( x q ) - q(x )) u c ] (3.10), c =. 0 1 0 0 0 0 0 0 1 0 0 0, b c = 0 0 0 0 0 1 - k n - k n - 1 - k 1 0 0 1 c Lyapunov (Positive Definite Symmetric Matrix ) P., Q [24]. T c P + P c = - Q (3.11) Lyapunov V e = 1 2 e T P e (3.12) - 2 1 -
, (3.10) (3.12). V e = = - 1 2 e T P e + 1 2 e T P e (3.13) 1 2 e T Qe + e T P b c [ ( p ( x p ) - p(x )) + ( q( x q ) - q(x )) u c ] e 0 V e. V e 0 V e V. (3.13) V e 0 0. u c (Supervisory Control) u s V e > V, V e 0. u. u = u c + u s (3.14) (3.14). e = c e + b c [ ( p ( x p ) - p(x)) + ( q( x q ) - q(x )) u c - q(x ) u s ] (3.15) (3.15) (3.12). - 22 -
V e = - 1 2 e T Qe + e T P b c [ ( p ( x p ) - p (x)) (3.16) + ( q( x q ) - q(x )) u c - q(x ) u s ] - 1 2 e T Qe + e T P b c [ p ( x p ) + p (x) + q( x q) u c + q(x )) u c ] - e T P b c q(x) u s, p U, q U, q L (Bounds). p U, q U, q L, (3.16) u s. u s = I * 1 sg n ( e T P b c ) 1 q L ( x ) [ p ( x p ) + p U ( x) + q( x q) u c + q U ( x) u c ] (3.17), V e > V I * 1 = 1, V e V I * 1 = 0. y 0 sg n (y ) = 1, y < 0 sg n (y ) = - 1. V e > V (3.17) (3.16) (3.18) V e 0. V e - 1 2 e T Qe 0 (3.18) - 23 -
3.2.4 u u c u s..,. * p = arg min p p [ s up x U c p( x p ) - p(x) ] (3.19) * q = arg min q q [ s up x U c q( x q) - q(x) ] (3.20), p q p q (Constraint Set)., p q p q. p = { p : p M p } (3.21) q = { q : q M q, y l } (3.22), M p, M q,. (3.1) y l, y l > 0. (Minimum Approximation Error ) = ( p( x * p ) - p( x )) + ( q( x * q ) - q( x)) u c (3.23) (3.15). - 24 -
e = c e - b c q(x) u s + b c [ ( p( x p ) - p ( x p * )) + ( q( x q ) - q( x q * )) u c + ] (3.24) p ( x p ) q( x q ) (3.4) (3.5), (3.24). e = c e - b c q(x) u s + b c + b c [ p T p ( x ) + T q q( x ) u c ] (3.25), p = p - * p, q = q - * q. Lyapunov V = 1 2 e T P e + 1 T p 2 1 p + 1 T q 2 2 q (3.26), 1, 2. (3.25) (3.26) V. V = - 1 2 e T Qe - q(x ) e T P b c u s + e T P b c + 1 1 p T [ p + 1 e T P b c p ( x) ] + 1 2 T q [ q + 2 e T P b c q ( x) u c ] (3.27) - 25 -
, p = p, q = q. (3.27) (3.17) q(x) >0 q(x) e T P b c u s 0.. p = - 1 e T P b c p ( x ) (3.28) q = - 2 e T P b c q ( x ) u c (3.29) (3.27) (3.28) (3.29). V - 1 2 e T Qe + e T P b c (3.30) (3.30) Q 0. e T P b c = 0, p(x ), g (x ) p( x p ), q( x q ), V 0. (Universal Approximation T heorem ). - 26 -
3.3 3.3.1 1 1 p( x p ) q( x q ) IF.,.. 3.2 Figure 3.2 1 Adjustable parameters of a fir st - type indirect adaptiv e fuzzy contr oller - 27 -
U c.., 3.3 U c p ( x p ) q( x q )., U c p ( x p ) q( x q ). 1. 3.3 2 Figur e 3.3 Fuzzy rules for a 2nd- order sy stem - 28 -
3.3.2 2 1. 2 IF. 3.4 Figur e 3.4 2 Adju stable param eter s of a second- type indirect adaptiv e fuzzy contr oller. p, l q F. i. p ( x p ) q( x q ). - 29 -
3.3.3 2/ 1 2/ 1 1 2. 1. 2 IF. 3.5 2/ 1 Figur e 3.5 Adju st able parameter s of a second/fir st - type indirect adaptiv e fuzzy contr oller - 30 -
2/ 1 2 1., 2 IF. 1.., 2/ 1 p, q. p (0) q (0)... 2/ 1 2 1. - 3 1 -
4 4.1 4.1..,.. p (0), q( 0). p (0), q (0),.. p, q. p, q. 1, 2. 0,. - 32 -
g g m e u c u s : ( ) [rad/ sec] : ( ) [rad] : ( ) [rad] : ( ) [rad] : : u :, u = u c + u s p (x), p (0), q(x ) : q(0) : p, q : : (1 :, 2 : ) 4.1 Figure 4.1 Block diagram of a designed indir ect adaptiv e fuzzy control sy st em for motion contr ol - 33 -
4.2,, 3. s n + k 1 s n - 1 + + k n = 0 k 1,, k n. Q (3.11) Lyapunov P >0. M p, M q,, V. x ( t) u. k, M p, M q,, V. y m, y n m, p U ( x), q U ( x), q L ( x ). F l i. p( x p ) q( x q ). IF i = 1, 2,, n F l i i m 1 m 2 m n. p( x p ) q( x q ). - 34 -
R ( l p1,, l pn ) p : IF x 1 is F l p1 1 and and x n is F l pn n THE N p ( x p ) is G ( l p1,, l pn ) (4.1) R ( l q1,, l qn ) q : IF x 1 is F l q1 1 and and x n is F l qn n THE N q ( x q ) is H ( l q1,, l qn ) (4.2) (4.1) (4.2). (4.3). ( l 1,, l n ) ( x) = m 1 l 1 = 1 n i = 1 m n ( l n = 1 F l i (x i i ) n i = 1 F l i(x i i )) (4.3) p ( 0) q (0) p( x p ) q( x q ) (3.4) (3.5). u. u c (3.8) u s (3.17). p ( x p ) q( x q ) (3.4) (3.5).. 1 p q. p q (3.28) (3.29). - 35 -
4.2 Figur e 4.2 Flow chart of a indirect adaptiv e fuzzy contr ol pr ogram - 36 -
4.3 4.3.1 DC. e a ( t) = R a i a ( t) + L a i a ( t) + e b ( t) (4.4) T m ( t) = K i i a ( t) (4.5) e b ( t) = K b m ( t) (4.6) T m ( t) = J m m ( t) + B m m ( t) + T L ( t) (4.7) m ( t) = m ( t) (4.8) m ( t) = - K i K b + R a B m R a J m m ( t) + K i R a J m e a ( t) - 1 J m T L ( t) (4.9), R a :, L a :, J m :, B m :, K i :, K b :, e a ( t) :, i a ( t) :, e b ( t) :, - 37 -
m ( t) :, m ( t) :, T m ( t) :, T L ( t) :.. 4.1, (4.8) (4.9). x 1 ( t) = x 2 ( t) (4.10) x 2 ( t) = - 3.625 x 2 ( t) + 6.25u ( t) - 50 T L ( t) (4.11) y ( t) = x 1 ( t) (4.12), x 1 ( t), x 2 ( t), u ( t), y ( t), T L ( t). 4.1 DC T able 4.1 Parameter s of the DC motor Parameter Value Unit R a 4 ohm L a 0 hen ry J m 0.02 kg m 2 B m 0.01 N m / rad/ sec K i 0.5 N m / A K b 0.5 V/ rad/ sec - 38 -
4.3.2 (4.11) T L ( t). x ( 2) ( t) = p (x) + q(x ) u (4.13) p (x) = - 3.625x 2 + 4.5 sin (x 1 ) cos (x 2 ) + a 1 ( t) x 1 x 2 (4.14) q(x) = 6.25 + a 2 ( t) (4.15). - 6 x 1 6, - 4 9 x 2 49, 0 u 25 p(x ) - 3.625x 2 + 4.5 sin (x 1 ) cos (x 2 ) + a 1 ( t) x 1 x 2 11.02 q(x ) 6.25 + a 2 ( t) 8.25 q(x ) 6.25 - a 2 ( t) 4.25. k 1 = 4, k 2 = 4 Q (3.11) P. Q = [ 32 0 0 32 ], P = 36 4 [ 4 5] - 39 -
. 1 3.6. m 1 = 5, m 2 = 5 m = m 1 m 2 = 25. 4.3 1 Figur e 4.3 Member ship function s of a fir st - type fuzzy logic sy stem 1 25 1. p (0) [- 5 5], q (0) [5 7.5]. - 40 -
2.. (4.16) (4.17) m = 15.,. R ( l m ) p : IF x 1 is F l pm 1 and x 2 is F l pm 2 THE N p ( x p ) is G ( l pm ) (4.16) R ( l m ) q : IF x 1 is F l qm 1 and x 2 is F l qm 2 THE N q ( x q ) is H ( l qm ) (4.17) 15 1, 1. 2/ 1 2 1., 2 1 (4.16) (4.17). 1 25 15.. 1 2. 2/ 1 2. - 4 1 -
4.4 1 Figur e 4.4 Step respon se of the fir st - type adaptiv e fuzzy contr ol sy stem - 42 -
4.5 1 Figure 4.5 Sinusoidal response of the first- type adaptiv e fuzzy contr ol sy stem - 43 -
4.6 2 Figure 4.6 Step r esponse of the second- type adaptiv e fuzzy contr ol sy stem - 44 -
4.7 2 Figur e 4.7 Sinu soidal respon se of the second- type adaptiv e fuzzy contr ol sy stem - 45 -
4.8 2/ 1 Figure 4.8 Step respon se of the second/fir st - type adaptiv e fuzzy contr ol sy stem - 46 -
4.9 2/ 1 Figur e 4.9 Sinu soidal respon se of the second/fir st - type adaptiv e fuzzy contr ol sy stem - 47 -
5 5.1 Murata ENV - 05DB Gyrostar, DC SPECT ROL 10K., 80C196KC 10bit A/ D.,. 5.1. 1[degree], - 0.005[degree/ sec].,.,, A/ D.,. - 48 -
5.1 Figure 5.1 Output angle and err or of gyro sen sor and potentiomet er - 49 -
5.2 X, Y,.,. 1... 5.2 m 1 = 3, m 2 = 2 5.3 ( m = m 1 m 2 = 6 ), 5.4 m 1 = 2, m 2 = 3 5.5 ( m = m 1 m 2 = 6). p(x ) g (x ) p (0) [- 3 3], q ( 0) [3.5 5] 6 1. A/ D.,, 20[ms] 80C196KC. - 50 -
5.2 Figure 5.2 Membership functions of a fuzzy logic system for a roll plant R 1 rolling : IF x 1 is F 1 1 and x 2 is F 1 2 THEN p is G 1 11 and q is H 1 11 R 2 rolling : IF x 1 is F 1 1 and x 2 is F 2 2 THEN p is G 2 12 and q is H 2 12 R 3 rolling : IF x 1 is F 2 1 and x 2 is F 1 2 THEN p is G 3 2 1 and q is H 3 2 1 R 4 rolling : IF x 1 is F 2 1 and x 2 is F 2 2 THEN p is G 4 22 and q is H 4 22 R 5 rolling : IF x 1 is F 3 1 and x 2 is F 1 2 THEN p is G 5 3 1 and q is H 5 3 1 R 6 rolling : IF x 1 is F 3 1 and x 2 is F 2 2 THEN p is G 6 32 and q is H 6 32 5.3 Figur e 5.3 Fuzzy rules for a roll plant - 5 1 -
5.4 Figure 5.4 Membership functions of a fuzzy logic system for a pitch plant R 1 tch in g : IF x 1 is F 1 1 and x 2 is F 1 2 THE N p is G 1 11 and q is H 1 11 R 2 tch in g : IF x 1 is F 1 1 and x 2 is F 2 2 THE N p is G 2 12 and q is H 2 12 R 3 tch in g : IF x 1 is F 1 1 and x 2 is F 3 2 THE N p is G 3 13 and q is H 3 13 R 4 tch in g : IF x 1 is F 2 1 and x 2 is F 1 2 THE N p is G 4 21 and q is H 4 21 R 5 tch in g : IF x 1 is F 2 1 and x 2 is F 2 2 THE N p is G 5 22 and q is H 5 22 R 6 tch in g : IF x 1 is F 2 1 and x 2 is F 3 2 THE N p is G 6 23 and q is H 6 23 5.5 Figur e 5.5 Fuzzy rules for a pitch plant - 52 -
5.6 Figure 5.6 Roll respon se of a plant - 53 -
5.7 Figure 5.7 Roll respon se of a adaptive fuzzy control sy stem - 54 -
5.8 Figure 5.8 Pit ch r espon se of a plant - 55 -
5.9 Figur e 5.9 Pitch respon se of a adaptiv e fuzzy control sy st em - 56 -
5.3 5.6 5.7.. 5[degree]... 15[m s].. 2[degree]. 5.8 5.9.,..., 5.5[degree] 2.5[degree].. - 57 -
6,...,.,, 2/ 1.,.. - 58 -
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A ppendix A. 80C 196KC M ain Circu it - 6 1 -
A ppendix B. I/ O Interf ace Circu it - 62 -