2 n dx (x)+g(x)u () x n u (x), g(x) x n () +2 -a -b -b -a 3 () x,u dx x () dx () + x x + g()u + O 2 (x, u) x x x + g()u + O 2 (x, u) (2) x O 2 (x, u) x u 2 x(x,x 2,,x n ) T, (x) ( (x), 2 (x),, n (x)) T x x 2 2 x n x x 2 x 2 n x 2 x n 2 x n n x n (3) g() x x A x, B g() (4) x (2) x u O 2 (x, u) dx Ax + Bu (5) u Fx (6) 2 3
線形システムと思いこむ 非線形システム 線形システム (a) F 2: 接線 非線形システム (b) 3: : 2
4 M(θ) θ + k(θ, θ) τ (7) θ M(θ) θ k(θ, θ) τ M(θ) M(θ) (7) 厳密に線形化されたシステム v u 非線形 θ,θ M(θ) システム入力変換 k(,) 線形化フィードバック F 線形コントローラコントローラ 4: 非線形システム τ k(θ, θ)+m(θ)v (8) v θ v (9) 線形化フィードバック 線形化されたシステム ( ) ( )( ) ( d θ I θ + θ θ I ) v () (θ T, θ T ) T v θ () ( ) θ v F () θ (7) ( ) τ k(θ, θ)+m(θ)f θ (2) θ 4 5 5: M(θ, θ, θ (r ) )θ (r) + k(θ, θ, θ (r ) ) N(θ, θ, θ (r ) )u (3) θ (i) 3
d i θ M( ) N( ) i (3) 厳密に線形化されたシステム v β() u 非線形システム x T() ξ 5 入力変換 α(,) () [,2] [-5] () v ξ T (x) (4) u α(x)+β(x)v (5) T (x) T () () dξ ξ dx x T x {(T (ξ)) + g(t (ξ))α(t (ξ))} + T x g(t (ξ))β(t (ξ))v (ξ)+ḡ(ξ)v (6) (ξ) Aξ, ḡ(ξ) B (7) (A, B) (4) (5) ξ v Fξ (8) 線形化フィードバック F 線形コントローラ コントローラ 6: u α(x)+β(x)fξ α(x)+β(x)ft(x) (9) (4) ξ x (8) (9) () 6 7 5 ([,2,4]) () (4), (5) (7) 2 a) {ad g, ad g, ad2 g,,adn g}(x) x b) {ad g, ad g, ad2 g,,adn 2 g}(x) ad i g(x) ad g g(x) (2) 4
元の座標系 非線形システム 新しい座標系 元の座標系 ad i+ g [,ad i g] (2) [,g] g (x) g(x) (22) x x [,g](x) Lie bracket { (x), 2 (x),, r (x)} γ j,k i (x) [ j, k ] r i γ j,k i (x) i (x) (23) (a) {B, AB, A 2 B,, A n B} (b) n 2 [ad g, ad g] {ad g} involutive n 2 Frobenius [,2] L ad i g φ(x), i,,,n 2 (24) L ad n gφ(x) (25) φ(x) L h(x) Lie L h(x) h (x) x (26) L h(x) L h(x) (27) L i+ h(x) L {L i h(x)} (28) 線形化されたシステム新しい座標系 7: (24) (25) φ(x) ξ ξ φ(x) ξ 2 L φ(x) ξ ξ 3 L 2 φ(x) (29) ξ n L n φ(x) (5) u Ln φ(x) L g L n φ(x) + L g L n φ(x) v (3) 5
dξ ξ + v (3) [6] d x x 2 x 3 tan(x 3 ) tan(x2) cos(x 3) tan(x 2) cos(x 3) + cos(x 2)cos(x 3) u (32) (32) ad g(x) g(x) cos(x 2)cos(x 3) ad g(x) [,g](x) g (x) x x g(x) cos(x 2)cos(x 3)+sin(x 2)sin(x 3) cos 2 (x 2)cos 3 (x 3) cos 3 (x 2)cos 2 (x 3) ad 2 g(x) [,ad g](x) [ ] x ad g (x) x ad g(x) cos 3 (x 2)cos 4 (x 3), {ad g(), ad g(), ad2 g()} (a) [ad g, ad g](x) [ad g, ad g](x) [ad g, ad g](x) [g, ad [ g](x) ] x ad g g(x) g x ad g(x) span{ad g, ad g} [ad g, ad g](x) [ad g, ad g](x) x {ad g(x),ad g(x)} Lie bracket ad g(x) ad g(x) {ad g(x), ad g(x)} involutive (b) (32) (29) (3) (24)(25) φ(x) (24) ( φ L ad g φ(x), φ, φ ) ad g(x) x x 2 x 3 φ x 2 cos(x 2 )cos(x 3 ) L ad g φ(x) ( φ, φ, φ ) ad x x 2 x g(x) 3 φ cos(x 2 )cos(x 3 )+sin(x 2 )sin(x 3 ) x 2 cos 2 (x 2 )cos 3 (x 3 ) φ x 3 cos 3 (x 2 )cos 2 (x 3 ) (24) φ(x) φ x 2, φ x 3 (25) φ(x) φ/ x φ x φ(x) φ(x) x (29) (3) ξ φ(x) x ξ ξ 2 L φ(x) tan(x 3 ) (33) ξ 3 L 2 φ(x) tan(x 2) cos 3 (x 3) u L3 φ(x) L g L 2 φ(x) + L g L 2 φ(x) v cos(x 2 ){3sin 2 (x 2 )tan(x 3 ) tan(x 2 )} +cos 3 (x 2 )cos 4 (x 3 )v (34) Lie L φ(x) φ x (x) tan(x 3) 6
[ ] L 2 φ(x) L {L φ(x)} x tan(x 3) (x) tan(x 2) cos 3 (x 3 ) L 3 φ(x) L {L 2 φ(x)} 3sin2 (x 2 )tan(x 3 ) tan(x 2 ) cos 2 (x 2 )cos 4 (x 3 ) L g L 2 φ(x) cos 3 (x 2 )cos 4 (x 3 ) (33) (34) (32), (33), (34) dξ dξ 2 dξ 3 dx tan(x 3 ) ξ 2 d tan(x 3) { tan(x 3 )} dx 3 x 3 tan(x 2 ) cos 2 tan(x 2) (x 3 ) cos(x 3 ) cos 3 (x 3 ) ξ 3 d tan(x 2 ) cos 3 (x 3 ) { tan(x 2 ) x 2 cos 3 (x 3 ) } dx 2 + { tan(x 2 ) x 3 cos 3 (x 3 ) } dx 3 3sin2 (x 2 )tan(x 3 ) tan(x 2 ) cos 2 (x 2 )cos 4 (x 3 ) + cos 3 (x 2 )cos 4 (x 3 ) u v dξ ξ + v (33) (34) (33) tan(x 2 ) tan(x 3 ) π/2 <x 2 <π/2, π/2 <x 3 <π/2 v Fξ ( ) 2 3 ξ (33),(34) u cos(x 2 ){3sin 2 (x 2 )tan(x 3 ) tan(x 2 )} +cos 3 (x 2 )cos 4 (x 3 )v v x + 2 tan(x 3 )+ 3 tan(x 2 ) cos 3 (x 3 ) 6 LQ H LQ () (5) LQ u F x LQ LQ (8) F (9) { α u x + β()f T } x x x + O 2 (x) (35) x () α() T () O 2 (x) LQ u F x F (9) LQ LQ 7 ρ 7
ρ ρ [,7] ρ (6) (4) (5) dξ Aξ + Bv + Oρ+ (ξ,v) (36) ρ [,7] 8 [6] ρ extended linearization[8 - ] pseudolinearization[] [6] :,, 3-8, 85/858 (992) [7] AJKrener: Approximate Linearization by State Feedbackand Coordinate Change, Systems and Control Letters, vol5, 8/85 (984) [8] WTBaumann and WJRugh: FeedbackControl o Nonlinear Systems by Extended Linearization, IEEE Trans on Automatic Control, volac-3, no, 4/46 (986) [9] WJRugh: An Extended Linearization Approach to Nonlinear System Inversion, IEEE Trans on Automatic Control, volac-3, no8, 725/733 (986) [] WTBaumann and WJRugh: FeedbackControl o Analytic Nonlinear Systems by Extended Linearization, SIAM J o Control and Optimization, vol25, no5, 34/352 (987) [] CReboulet and CChampetier: A New Method or Linearizing Non-Linear System: the Pseudolinearization, Int J o Control, vol4, no4, 63/638 (984) [],,,,, : (993) [2] AIsidori: Nonlinear Control Systems, Springer- Verlag (st ed 985, 2nd ed 989) [3] BJakubczyk and WRespondek: On Linearization o Control Systems, Bull Acad Polonaise Sci Ser Sci Math, vol28, 57/522 (98) [4] RSu: On the Linear Equivalents o Nonlinear Systems, Systems and Control Letters, vol2, no, 48/52 (982) [5] LRHunt, RSu and GMeyer: Design or Multi- Input Nonlinear Systems, in RWBrockett et al Eds, Dierential Geometric Control Theory, Birkhauser (982) 8
A () ξ ξ 2 ξ d 2 ξ 3 + u (37) ξ n ξ n a(ξ) b(ξ) ξ n Luenberger 2 b(ξ) u a(ξ) b(ξ) + b(ξ) v (38) (37) dξ ξ + v (39) () (37) () (37) (4) ξ T (x) ξ 2 T 2 (x) (4) T n (x) ξ n n T i (x) (37) T 2 (x) ξ 2 dξ dt T 3 (x) ξ 3 dξ 2 dt 2 T n (x) ξ n dξ n dt n (4) T 2 (x),t 3 (x),,t n (x) T (x) (37) T (x) (37) T (x) T (x) T (x) t dt T x T x dx T x {(x)+g(x)u} T x (x)+ T g(x)u (42) x ( T x, T x 2,, ) T x n (43) (4) dt / T 2 (x) x u T x g(x), T 2(x) T (x) (44) x Lie (28) L g T (x), T 2 (x) L T (x) (45) T (x) (42) Lie dt L T (x)+ul g T (x) (46) Lie x (x) dx (x) (47) φ(x) Lie dφ L φ(x) (48) T 2 (x) dt 2 L T 2 (x)+ul g T 2 (x) L 2 T + ul g L T 2 (49) (4) dt 2 / T 3 (x) x L g L T (x), T 3 (x) L 2 T (x) (5) 9
n T (x) L g L i T (x), i,,,n 2 (5) T k (x) L k T (x), k 2, 3,,n (52) ξ n T n (x) L n T (x) dξ n Ln T (x)+ul g L n T (x) (53) (37) L n T (x) a(x) (54) L g L n T (x) b(x) (55) (38) u Ln T (x) L g L n T (x) + L g L n T (x) v (56) T (x) (24)(25) φ(x) (3) () (5)(55) T (x) (5)(55) Lie T (x) (5)(55) Lie bracket(22) Lie bracket L L g φ(x) L g L φ(x) L [,g] φ(x) (57) (5) i L g L T (x) L L g T (x) L [,g] T (x) L ad g T (x) (58) (5) i L g T (x) i 2 L g L 2 T (x) L L g L T (x) L [,g] L T (x) L L [,g] T (x)+l [,[,g]] T (x) L L ad g T (x)+l ad 2 g T (x) L ad 2 g T (x) (59) (5) L g L T (x) (58) L ad g T (x) (5) (55) Lie Lie bracket Lie () (37) ad g(x),ad g(x),,ad n g(x) x (62) () (37) (62) (6)(6) T (x) (62) (6)(6) T (x) Frobenius Frobenius { (x), 2 (x),, r (x)} x L i φ j (x), i, 2,,r (63) n r φ j (x), j, 2,, n r { (x), 2 (x),, r (x)} involutive φ j (x) φ i / x x { (x), 2 (x),, r (x)} g(x) L g φ k (x) (64) k, ( k n r) (62) (6)(6) T (x) {ad g(x),ad g(x),,ad n 2 g(x)} involutive (65) L ad i g T (x), i,,,n 2 (6) L ad n g T (x) (6)