A Study on Predctve Control Usng a Short-Term Predcton Method Based on Chaos Theory (Predctve Control of Nonlnear Systems Usng Plural Predcted Dsturbance Values) Noryasu MASUMOTO, Waseda Unversty, 3-4-1 Okubo, Shnjuku, Tokyo 169-8555, Japan Hrosh YAMAKAWA, Waseda Unversty A predctve control method for nonlnear mechancal systems s dscussed n ths study. values near future of dsturbance tme seres of the controlled system are predcted by a short-term predcton method based on chaos theory whch was proposed by the authors. The deas of the mum samplng perod and the forward horzon are ntroduced n the predcton method and methods to obtan them are shown n ths work. In order to show the effectveness of the proposed method when t s appled to nonlnear systems, numercal examples for a Duffng system are presented. Key Words : Vbraton Control, Predctve Control, Chaos Theory, Short-Term Predcton, Duffng System Duffng mx+ cx+ k 1 x + k 3 x 3 = a cos ω t Duffng a 5001 Duffng,, 64-65, C (1998), 3361-3368. Thompson, J.M.T. and Stewart, H.B., Nonlnear Dynamcs and Chaos, (1986), 3, Wley. t * Forward Horzon h f Ke * Backward Horzon h b Dsplacement x m 4.00 - -4.00 Uncontrolled Predctve control Predctve control + feedback control 5000 5100 500 5300 5400 Tme step Fg. A1 Control results of the dsplacement
Duffng Dsturbance Mechancal system Output {W} = (..., w,...) {Y} = (..., y,...) Fg. 1 Dsturbance-output relaton of a mechancal system Table 1 Classfcaton of predctve control methods Mechancal system (Model s known) Lnear Nonlnear Target tme seres of predcton Dsturbance Output Dsturbance Output The number of predcted values used for control forces n x (t)= f x(t) + B u (t)+cw(t) y (t)=dx(t) x y u w n m l f n B C D n 1 n l n m f B C D u u (t)=u p (t)+u f (t) u p u f u p h f 1 u p (k τ)= Ω T w ((k + ) τ) =0 h ^ f w h f Ω k τ τ τ h f h f Ω ( = 0, 1,..., h f -1) h 1 u p ( τ; h)= Ω T w (( + ) τ) =0 h h Ω ( = 0, 1,..., h -1) h o + J h f (τ; h)= x T (( + ) τ) Qx(( + ) τ) =1 h o 1 =0 Ru p (( + ) τ ; h) Q n n R h o
x ( τ ) = 0 Runge-Kutta τ h J h f ( τ ; h ) h h = 1,,..., h max h h - J h f ( τ ; h ) = 1, 1+1,..., τ - 1 +1-1 +1 h - J h f ( τ ; h ) 1 h - J h f ( τ ; h ) h h f h f h f Ω ( = 0, 1,..., h f -1) - 1 +1 = + h o -1 + h o -1 ( + h o -1) + h max -1 h o h max {X} { X}=, x 1, x, x +1, {X} { X * }= x 1, x,, x n η η x ^ n x n +1 δ {X} η δ +1 δ δ η Takens {X * } ^x n + λ ( λ = 1,,..., h f ) {X * } λ {X * } {X * λ} s λ {X * } s λ = mod ( n 1, λ )+1 {X * ^ λ} x n + λ λ {X * } ^x n + λ ( λ = 1,,..., h f )... {X * }=( x 1 x... x s......... x n -... x n -... x n -1 x n ) δ η η Gona {X * λ} {X * } (η r ) = x (η 1),, x 1, x T ( = η, η +1,, n ) η η r (η ) {X * } η η r (η ) x x x +1 x +1 = f (η ) (η r ) r (η ) f (η ) (η ) x +1 x j +1 (ε)= MAX (η ) (η ) (η r j B (ε) d r ) (η ), r j C B (η ) (ε ) r (η ) ε η d C (η ) (ε)=max x s... x n - x n - {X * }=( x s... x n - x n - x n ) Fg. Generaton of partal tme seres (η ) C (ε) C (η ) (ε ) C (η ) (ε ) η η =, 3, 4,... C (η ) (ε ) η {X * } {X * λ} Ke λ λ 1 λ h f h f h f η η Takens x n
Ke * = MAX Ke λ λ Ke * λ {X * λ} = n η = Ke * * ) C n (ε)= x n +λ x j +λ MAX * ) (ε) d r * ) j r j * ) B n * ) r * ) T = x * 1) λ,, x λ, x ( = s λ + * 1) λ,, n λ, n ) {X * λ} x n ^x n +λ r * ) j r * ) n r * ) n C * ) x (ε)= n +λ x n +λ d r * ) * ) n ^ C p C * ) (ε ) x n +λ x n +λ = x n +λ + C p d ( r * ) * ) n ) when xn x n +λ x n +λ C p d ( r * ) * ) n ) when xn > x n +λ x n x n +λ x n x n +λ {X * } ^x n + λ ( λ = 1,,..., h f ) C p h b h b C p E (µ,c 1 p )= 1 1 +1 = 1 x +λ (µ,c p ) x +λ x +λ 1 1 n - λ µ Ke * ~ x +λ ( µ, C p ) Ke* µ µ J hb ( µ, C p ) µ Table Optmum desgn problem for C p Desgn varable C p (Intal value) (1.00) Objectve functon J h b (, C p ) = ~ x + (, C p ) - x + x + Constrant condton C p C p 1 C p µ C p µ µ =, 3,... µ-e 1 ( µ, C p ) µ I mn = { µ MIN µ E ( µ,c 1 p )} I mn µ * h b = Ke * + µ * h f +1 µ * C p C p ^x n + λ ( λ = 1,,..., h f ) h b x n -(h b -1),..., x n -1, x n h b µ * C p x n Ke * r * ) n r * ) n µ * r * ) n r * ) n + {X λ * } t Ke * r * ) n - λ r * ) n t t t * t * f ( t ; Ke * )= = 1 * ) g * Ke r + λ * ) g * Ke r + λ g Ke * ( r * ) ) = x gke* r * ) Ke * r * ) ( 1 ) t t ( = 1,,... ) r * ) r * ) Ke * -dmensonal phase space r *) r *) n n + r *) n - r *) r *) n n + - λ Fg. 3 The nearest pont to the current pont n phase space
r * ) t * Duffng mx+ cx+ k 1 x + k 3 x 3 = a cos ω t a a = 7.50 N Dynamcs a (t) = 7.30 + 4.00 10 1 sn (5.00 10 1 t + π 4 ) 6.90 a 7.70 1.00 10-1 N f (t)=a (t) cos ω t + nose Dsplacement x m Ampltude a N.50 -.50 5.00 7.00 8.00 9.00 10.0 Ampltude of exctaton a N Fg. 4 Bfurcaton dagram of a Duffng system n ths study 10.0 5.00-5.00 Table 3 Parameters of the Duffng system Mass of partcle m Kg 1.00 Dampng coeffcent c Ns/m 5.00 10 - Stffness coeffcent of x k 1 N/m Stffness coeffcent of x 3 k 3 N/m 3 1.00 Angular frequency of exctaton rad/s 1.00-10.0 10.0 0.0 30.0 40.0 50.0 Tme t sec Fg. 5 Tme seres of dsturbance ( t = 1.30 10-1 sec ) Estmaton value Objectve functon 10.0 9.00 8.00 7.00 Ke = 5 Ke = 7 Ke = 9 5.00 10.0 15.0 0.0 5.0 30.0 Samplng perod t sec 10 - Fg. 6 Calculaton results of the mum samplng perod 10.0 9.90 9.80 Ke 0 5 10 15 0 Horzon parameter h Fg. 7 h - J h ( τ ; h) plots f Table 4 Embeddng dmensons 1 3 4 5 6 7 8 9 10 6 9 6 6 6 9 7 8 9 8 t * t = 1.00 10 - sec t = 1.00 10-1 sec t * = 1.30 10-1 sec h - J h f ( τ ; h ) h f = 10 λ = 1,,..., h f = 10 λ Ke * = 9 µ-e 1 ( µ, C p ) µ-c p 349 µ 435 C p µ * = 400 µ * C p =.13 10-1 h f = 4071 5001 8001
Error rate C p 10-1 1.50 1.00 0.50 I mn = { 349 435 } } 0 100 00 300 400 500 The number of Ke * -dmensonal ponts Fg. 8 - E (, C p ) plots 1 Dsplacement x m Dsplacement x m 10-1 3.00 1.00 0 100 00 300 400 500 The number of Ke * -dmensonal ponts Fg. 9 - C p plots 4.00 - Uncontrolled Predctve control Predctve control + feedback control -4.00 5000 5100 500 5300 5400 Tme step Fg. 10 Control results (controlled from the 5001st step) 4.00 - Uncontrolled Predctve control Predctve control + feedback control -4.00 8000 8100 800 8300 8400 Tme step Fg. 11 Control results (controlled from the 8001st step) Duffng,,, 11-6, (1975), 663-668. Farmer, J. D. and Sdorowch, J., Predctng chaotc tme seres, Phys. Rev. Lett., 59-8 (1987), 845-848. Casdagl, M., Nonlnear predcton of chaotc tme seres, Physca D, 35 (1989), 335-356. Sughara, G. and May, R. M., Nonlnear forecastng as a way of dstngushng chaos from measurement error n tme seres, Nature, 344 (1990), 734-741. Gona, M., Cmagall, V., Morgav, G. and Perrone, A., Local predcton of Chaotc tme seres, Proc. the 33rd Mdwest Symposum on Crcuts and Systems, (1990), 894-897.,,,, No. 97-1-I (1997-3), 14-15. Masumoto, N. and Yamakawa, H., A Study on Predctve Control Usng Short-Term Predcton Method Based on Chaos Theory, Proc. the 1997 ASME Internatonal Mechancal Engneerng Congress and Exposton, DE- 95/AMD-3, (1997), 97-105.,,,, No. 98-1-I (1998-3), 143-144.,,, 64-65, C (1998), 3361-3368. Masumoto, N. and Yamakawa, H., A Study on Predctve Control Usng Short-Term Predcton Method Based on Chaos Theory (Determnaton of the Control Force Usng Plural Predcted Dsturbances), Proc. the 1998 ASME Internatonal Mechancal Engneerng Congress and Exposton, DE-97/DSC-65, (1998), 179-184.,,, No. 99-7 (1999-11), 374-377. Takens, F., Detectng strange attractors n turbulence, Dynamcal Systems and Turbulence, Warwck 1980, Lecture Notes n Mathematcs, 898, (1981), 366-381, Sprnger-Verlag. Thompson, J.M.T. and Stewart, H.B., Nonlnear Dynamcs and Chaos, (1986), 3, Wley. Nusse, H.E. and Yorke, J.A., Dynamcs: Numercal Exploratons, nd Ed., (1998), Sprnger-Verlag.