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1 () 2014,,,,, - (ISS) Sontag Hamilton-Jacobi H L 2-, Hamilton-Jacobi-Issacs H,, ISS, ISS Strict Feedback Form,, / / / /154

2 3 Aine System ( ) m ẋ = (x)g(x)u = (x) g i (x)u i i=1 y = h(x) 3 x n C 1 C C 1 - φ ψ φ 1 x, u( R m ), y( R l ) General Nonlinear System ẋ = (x, u) y = h(x) x, u( R m ), y( R l ) ψ C 1 - M R n (= n ) C ( ) C () / /154 (3 ) 3 x ẋ? p R n T p M R n T M x ẋ n x M, (x, ẋ) T M ẋ x ẋ x ẋ T M U = M U R n 3 3 () R SO(3) 3 : R T R = I, det R =1 Ṙ = S(ω)R R Ṙ 3 ω =(ω 1,ω 2,ω 3 ) T S(ω) = 0 ω 3 ω 2 ω 3 0 ω 1 ω 2 ω 1 0 ω R Ṙ / /154

3 3 1 Aine System x n ẋ x n ẋ = (x) (x) Lipschitz Lipschitz / / 154 Lipschitz Lipschitz : ẋ = (x), x R n x(0) = x 0 x(t) (t 0)?? Lipschitz Lipschitz x ẋ = { 1 (x 0) 1 (x<0) Time x =0 ẋ =0 ẋ = / / 154

4 Lipschitz Lipschitz Lipschitz ẋ = sgn(x) 3 x x Lipschitz Lipschitz : (x) U Lipschitz (x 1 ) (x 2 ) M x 1 x 2 M(> 0) Time ( R n ) Lipschitz Lipschitz (x) Lipschitz x =0 x U x U x Lipschitz (x) Lipschitz ( U x M ) / / 154 Lipschitz Lipschitz Lipschitz Lipschitz = Lipschitz Lipschitz Lipschitz Lipschitz (Picard-Lindelö ) (x) Lipschitz ẋ = (x), x(0) = x 0 x(t) T (x 0 ) 0 t T () (x) Lipschitz ẋ = (x), x(0) = x 0 : ẋ = x 3 ( Lipschitz) Finite time blowup Lipschitz Lipschitz (y = x 2 ) Lipschitz Time / / 154

5 Lipschitz Lipschitz (Peano existence theorem) Picard-Lindelö (x) t Peano existence theorem Carathéodory s existence theorem Coddington & Levinson (1955) E.A. Coddington, N. Levinson: Theory o Ordinary Dierential Equations, McGraw-Hill (1955) / / 154 (): () 2 M(θ) θ c(θ, θ)g(θ) =u () u = c(θ, θ)g(θ)m(θ)v θ = v? / / 154

6 : Lie Lie 1 Lg h 0 y (SISO ) MIMO Lie Lie 1 Lg h 0 y (SISO ) MIMO ẋ = (x)g(x)u y = h(x) : u = α(x)β(x)v v y v u y x / / 154 Lie Lie Lie Lie 1 Lg h 0 y (SISO ) MIMO tool = Lie () h(x): (x): M R(x) M TM (x n = ) (L h)(x) = n i=1 h x i i (x) = h x (x)(x) Lie Lie 1 Lg h 0 y (SISO ) MIMO : ẋ = (x) x(t) x y = h(x) dy dt = h(x) dx x dt = h(x) x (x) =(L h)(x) L h x h(x) ẋ = (x) (L g L h)(x) =(L g (L h))(x) (L k h)(x) =(L (L ( (L h) )))(x) }{{} k times / / 154

7 t 2? Lie Lie 1 Lg h 0 y (SISO ) MIMO 1 (m =1)1 (l =1) t ẋ = (x)g(x)u y = h(x) ẏ = h x dx dt = h x ((x)g(x)u) =(L gu h)(x, u) =L h(x)l g h(x)u L gu x h(x) h(x) Lie Lie 1 Lg h 0 y (SISO ) MIMO d k y dt k? NO = Lk guh 2 1 ÿ = d dt {(L guh)(x(t),u(t))} x x u ÿ = d dt (L guh)(x, u) =L gu L h L gu L g h u u L g h ẏ = C(Ax Bu) =CAx CBu L g h u y 2 y 2 L g h = / / 154 L g h 0 y 2 : L g h =0 y 2 ẏ = L h(x)l g h(x) u : L g h =0 Lie Lie 1 Lg h 0 y (SISO ) MIMO u (L g h)(x) u = L h(x)v L g h(x) ẏ = v v y = L g h Lie Lie 1 Lg h 0 y (SISO ) MIMO L g L h(x) 0 ÿ = L gu L h = L 2 h(x)l g L h(x) u u = L2 h(x)v L g L h(x) ÿ = v / / 154

8 3... Lie Lie 1 Lg h 0 y (SISO ) MIMO : L g h =0, L g L h =0 d 3 y dt 3 L g L 2 h(x) 0 = L gul 2 h = L 3 h(x)l g L 2 h(x) u u = L3 h(x)v L g L 2 h(x) d3 y dt 3 = v Lie Lie 1 Lg h 0 y (SISO ) MIMO : x 0 y ρ x 0 U x0 (L g L i h)(x) =0, i =0,...,ρ 2, x U x0 (L g L ρ 1 h)(x 0 ) 0 ρ ρ ẏ = L h(x) ÿ = L 2 h(x). d ρ 1 y dt ρ 1 d ρ y dt ρ = Lρ 1 h(x) = Lρ h(x)l gl ρ 1 h u ρ u / / 154 (SISO ) Lie Lie 1 Lg h 0 y (SISO ) MIMO ẋ = Ax bu y = cx (x) =Ax, g(x) =b, h(x) =cx = ρ cb = cab = ca 2 b = = ca ρ 2 b =0, ca ρ 1 b 0 () Lie Lie 1 Lg h 0 y (SISO ) MIMO ρ ρ : : d ρ y dt ρ = Lρ h(x)l gl ρ 1 h(x) u u = Lρ h(x)v L g L ρ 1 h(x) d ρ y dt ρ = v y = h(x), ẏ = L h(x),...,d ρ 1 y/dt ρ = L ρ 1 h(x) (x ) / / 154

9 MIMO Lie Lie 1 Lg h 0 y (SISO ) MIMO MIMO (l m) : x 0 (ρ 1,...,ρ l ) x 0 U x0 (L gk L i h j )(x) =0, j =1,...,l,i=0,...,ρ j 2, k =1,...,m, x U x0 L g1 L ρ1 1 h 1 (x 0 ) L gm L ρ1 1 h 1 (x 0 ) rank. = l L g1 L ρl 1 h l (x 0 ) } {{ L gm L ρl 1 h l (x 0 ) } =G(x) d ρ 1 y 1 dt ρ 1. d ρ l y l dt ρ l = L ρ1 h 1(x). L ρl h l(x) G(x)u Lie Lie 1 Lg h 0 y (SISO ) MIMO L ρ1 h 1(x) u = G T (x)(g(x)g T (x)) 1. v L ρl h l(x) d ρ 1 y 1 dt ρ 1. d ρ l y l dt ρ l = v / / 154 (1) Lie Lie 1 Lg h 0 y (SISO ) MIMO Lie Lie 1 Lg h 0 y (SISO ) MIMO : ẋ 1 = u 1 cos x 3 ẋ 2 = u 1 sin x 3 ẋ 3 = u 2 (x 1,x 2 ) x 3 u 1 ( 1) u 2 ( 2) (x 1, x 2 ) (G(x) ) ( ) x1 d cos x y = 3 x 2 d sin x 3 x 3 (x 1 d cos x 3, x 2 d sin x 3 ) / / 154

10 (2) Lie Lie 1 Lg h 0 y (SISO ) MIMO r =(1, 1) : [ cos x3 d sin x ẏ = G(x)u = 3 sin x 3 d cos x 3 d 0 G(x) : u = ]( u1 [ cos x3 sin x 3 ) ](ṙx k{r x (x 1 d cos x 3 )} sin x 3 /d cos x 3 /d ṙ y k{r y (x 2 d sin x 3 )} (r x,r y ) u 2 ) Lie Lie 1 Lg h 0 y (SISO ) MIMO / / 154 Normal Form n ρ n ρ? Φ(x): x (z T,ξ T ) T Normal Form Normal Form z 1 = h(x),z 2 = L h(x),...,z ρ = L ρ 1 h(x) ξ Normal Form: y = z 1 ż 1 = z 2.. ż ρ = L ρ h(φ 1 (z,ξ)) L g L ρ 1 h(φ 1 (z,ξ)) u ξ = γ(z,ξ)ζ(z,ξ)u SISO ζ(z,ξ) = / / 154

11 ζ( ) =0 Normal Form ζ( ) =L g ξ =0 ξ : L g ξ = ξ x g =0 n 1 ( = ) z 1,...,z ρ 1 z 1,...,z ρ 1 n ρ ξ Normal Form y 0 y t 0 z =0 z =0 u = L gl ρ 1 h(φ 1 (0,ξ)) L ρ h(φ 1 (0,ξ)) n ρ ξ = γ(0,ξ) ζ(0,ξ) L gl ρ 1 h(φ 1 (0,ξ)) L ρ h(φ 1 (0,ξ)) ζ(z,ξ) =0 y 0 y y (exo system) / / 154 Normal Form : ẋ = y = ( 0 [ ] 0 1 x ) x ( ) 1 u 1 G(s) = s 1 s 2 s 1 u = x 1 x 2 v : [ ] ( ) ẋ = x v y = ( 0 1 ) G(s) = s 1 s(s 1) x Normal Form : () ( ) / / 154

12 z Normal Form Vidyasagar : [ ] ẋ = (x) ż = g(z)γ(x, z)x ẋ = (x) ż = g(z) γ(x, z) ẋ = (x), ż = g(z) ẋ = x ż = z z 3 x x Normal Form [ ] [ 0 ] [ ] : ẋ 1 = x 2 u ẋ 2 = x 1 x 2 1x 3 2 u y = x 1 : ẋ 2 = x 2 () : u = x 1 x 2 : ẋ 1 = x 1 () ẋ 1 = x 1 ẋ 2 = x 2 x 2 1x / / 154 Normal Form z? 2 z : 2 () Normal Form () () / / 154

13 Lie 1 2 λ (A) (B) Lie 1 2 λ (A) (B) n λ(x) n ρ =0 λ(x) λ(x)? / / 154 λ(x) Lie (Lie bracket) Lie 1 2 λ (A) (B) : : u = α(x)β(x)v (β(x) 0) z =Φ(x) ż = a 0 a n 1 z. 0 v 1 z 1 n Φ 1 (x)(= z 1 ) n : 1 1 n (= ) λ(x) Lie 1 2 λ (A) (B) Lie bracket : (x), g(x): M TM ( ) [,g](x) = g (x) x x g(x) 2 g x = ( x) T sec. x x = g ( x) T sec. T sec. x = g ( x) x = ( x) T sec. x1 x2 [,g](x) = lim T 0 1 T 2 (x 1(x, T ) x 2 (x, T )) / / 154

14 Lie (2) Lie 1 2 λ (A) (B) (a 1, a 2 ) [,g] = [g, ] [a 1 1 a 2 2,g]=a 1 [ 1,g]a 2 [ 2,g] [,a 1 g 1 a 2 g 2 ]=a 1 [,g 1 ]a 2 [,g 2 ] [,[g, p]][g, [p, ]] [p, [,g]]=0 (Jacobi ) [α, βg] =αβ[,g]α (L β) g (L g α) β L [,g] h = L L g h L g L h () Lie 1 2 λ (A) (B) n : 1 (n 1) 2 (L g λ)(x) =0 (L g L λ)(x) =0. (L g L n 2 λ)(x) =0 n (L g L n 1 λ)(x) 0 Lie bracket Lie : L [,g] λ = L L g λ L g L λ / / ad Lie 1 2 λ (A) (B) L g λ =0 L g L λ = L [,g] λ L L g λ =0 }{{} =0 L g L 2 λ = L [,g] L λ L L g L λ }{{} =0 = L [,[,g]] λ L L [,g] λ =0 }{{} =0. L g L n 2 λ =( 1) n L [,[ [,g] ]] λ =0 Lie 1 2 λ (A) (B) : 0 = g ad g =[,g] ad k g =[,[ [,g] ]] }{{} k times ad 0 g = g / / 154

15 1 2 Lie 1 2 λ (A) (B) Lie Bracket 1 : (L g λ)(x) =0 (L ad gλ)(x) =0. (L ad n 2 λ)(x) =0 g Lie 1 2 λ (A) (B) 1 (L g L n 1 λ)(x) = (L [,g] L n 2 λ)(x)(l L g L n 2 λ)(x) }{{} =0 = L ad 2 λ L L [,g] L n 3 λ gln 3 = L ad 3 gln 4 λ L L ad 2 = =( 1) n 1 L ad n 1 λ 0 g gln 4 λ L L g L n 2 λ L 2 L g L n 3 λ / / 154 λ Lie 1 2 λ (A) (B) : (L g λ)(x) =0 (L ad λ)(x) =0 g (L ad 2 gλ)(x) =0 λ(x). (L ad n 2 λ)(x) =0 g (L ad n 1 λ)(x) 0 g Lie 1 2 λ (A) (B) : g, ad g,...,ad n 1 g ad k g g, ad g,...,ad k 1 g ad k g(x) =c 0 (x)g(x)c 1 (x)ad g(x) c k 2 (x)ad k 1 g(x) ad k1 g(x) =c 0 (x)ad g(x)(l c 0 )(x)g(x) c k 3 (x)ad k 1 g(x)(l c k 3 )(x)ad k 2 g(x) c k 2 (x){c 0 (x)g(x)c 1 (x)ad g(x) c k 2 (x)ad k 1 g(x)} (L c k 2 )(x)ad k 1 g(x) ad k1 g(x) / / 154

16 (A) (1) Lie 1 2 λ (A) (B) λ(x) (A): n g, ad g,...,ad n 1 g (=) Lie 1 2 λ (A) (B) 1 (n 1) (L g λ)(x) =0 (L ad gλ)(x) =0 (L ad 2 gλ)(x) =0. (L ad n 2 λ)(x) =0 g ( 2 ) [ λ λ x,p(x) =,..., λ ] p(x) =0, x 1 x n p = g, ad g,...,ad n 2 g λ/ x g, ad g,...,ad n 2 g / / 154 (2) Lie 1 2 λ (A) (B) n (n 1) g, ad g,...,ad n 2 g 0 ω(x) s(x)( λ/ x) =ω(x) λ(x) s(x) ( 0)? s(x) Lie 1 2 λ (A) (B) x R n q L p1 λ =0,...,L pq λ =0 (p 1 (x),...,p q (x) ) : n q λ 1 (x),...,λ n q (x) (=) Δ(x) =span{p 1 (x),...,p q (x)} Δ(x) [δ 1,δ 2 ] Δ, δ 1 Δ, δ 2 Δ / / 154

17 (B) Lie 1 2 λ (A) (B) λ(x) (B): span{g(x), ad g,...,ad n 2 g} Lie 1 2 λ (A) (B) Δ n = span{g, ad g,...,ad n 1 g} n Δ n 1 = span{g, ad g,...,ad n 2 g} / / 154 (1) (2) Lie 1 2 λ (A) (B) L δ λ(x) =0(δ Δ n 1 ) λ(x) 1 λ x [g, ad g,...,ad n 1 g] =[0,...,0, L ad n 1 λ] g 0 L g λ = L g L λ = = L g L n 2 λ =0 L g L n 1 λ 0 λ(x) n Lie 1 2 λ (A) (B) : z =Φ(x) : ż = z 0 0 : z 1 = λ(x) z 2 =(L λ)(x). z n =(L n 1 λ)(x) 0. 0 L n λ L gl n 1 λ u u = Ln λ(x) L g L n 1 λ(x) v L g L n 1 λ(x) / / 154

18 (1) (2) Lie 1 2 λ (A) (B) : ( ) 2 i M z = MG K z z 0 e = Ri d dt {L(z)i} L(z) = 2K z z 0 L 0 z () i e M G z M Magnetic levitation system z 0 R () L(z) inductance (z ) L 0 inductance() K (= μ 0N 2 S/4) () R L i e Lie 1 2 λ (A) (B) e = e s () : z s Kes /(R MG) z 0 ż s = 0 i s e s /R : x =(z z s, ż,i i s ) T : u = e e s : x 2 ẋ = G K(x 3 i s ) 2 0 M(x 1 z s z 0 ) 2 0 u 1/L(x φ(x) 1 z s ) ( 1 φ(x) = Rx 3 2Kx ) 2(x 3 i s ) LL(x 1 z s ) (x 1 z 0 z s ) / / 154 (3) (4) g(x) = 0 0 1/L(x 1 z s ) ad g =[,g] = ad 2 g =[,[,g]] = (A) 0 2K(x 3 i s ) M(x 1 z 0 z s ) 2 L(x 1 z s ) R L(x 1 z s ) 2 2K(x 3 i s ) M(x 1 z 0 z s ) 2 L(x 1 z s ) rankδ 3 = rank{,[,g], [,[,g]]} =3 () / 154 Lie 1 2 λ (A) (B) (B) Δ 2 = span 0 1, [g, [,g]] = Δ 2 involutive 0 2K M(x1z 0 z s ) 2 L(x1x s ) 2 0 Δ 2 λ = x / 154

19 Lie 1 2 λ (A) (B) n 2 3 Lyapunov Lyapunov V Lyapunov / / 154 (1) Lyapunov Lyapunov V Lyapunov : ẋ = (x) (x 0 )=0 x 0 (equilibrium (point), ) x x =0 ẋ =0 Lyapunov Lyapunov V Lyapunov : Boundedness ẋ = (x) U x(0) K(x(0)) x(t) K(x(0)) (t 0) () : (Local) Stability LS ẋ = (x) x =0 () ɛ>0 δ(ɛ) > 0 x(0) <δ(ɛ) x(t; x(0)) <ɛ,t 0 () () ( ) () Lyapunov () / / 154

20 (2) (3) Lyapunov Lyapunov V : Attractiveness U x(0) x(t; x(0)) 0 (t ) U () : (Local) Asymptotical Stability LAS ẋ = (x) x =0 () x =0 Lyapunov Lyapunov V : Global Stability GS ẋ = (x) x =0 : Global Asymptotical Stability GAS ẋ = (x) x =0 Lyapunov Lyapunov Lyapunov / / 154 Lyapunov Lyapunov V (x) Lyapunov : V (x) : V (x) : V (0) = 0 V (x) > 0, x 0 LS: V 0 LAS: V < 0(x 0) Lyapunov Lyapunov V Lyapunov x 2 x 1 V (x) =x 2 1 2x 1 x 2 2x 2 2 =(x 1 x 2 ) 2 x 2 2 Lyapunov Lyapunov V Lyapunov () GS: V 0 V (x) () GAS: V < 0(x 0) V (x) V (x) x V (x) < 0 (x 0) (Radially unbounded)? V (x) ( x ) / / 154

21 V ẋ = (x) (x)? V (x) (x) : 2 1 Lyapunov Lyapunov V Lyapunov V (x) = V x dx dt V (x) = x (x)(= L V (x)) V / x V x (x) = ( V x 1,..., V x n L V L L V ) Lyapunov Lyapunov V Lyapunov (= Separatrix) / / 154 Lyapunov Lyapunov Lyapunov V Lyapunov V (x) S a = {x V (x) a} (a >0) (= ) a S a = {x V (x) =a} V V (x) p(a) < 0, x S a, a > 0 V p(v ) < 0 V 0 Lyapunov Lyapunov V Lyapunov Lyapunov : V < 0 (x 0) V Lyapunov : V 0 V Lyapnov V =0 Lyapunov (Lyapunov ) Lyapunov? / / 154

22 Yoshizawa La Salle = Lyapunov Lyapunov V Lyapunov Ω x(0) t>0 x(t) Ω Ω Yoshizawa La Salle : Ω Ω E( Ω) E M Ω M Ω E M Lyapunov : V (x) Lyapunov E = {x V (x) =0} x =0 x =0 M = {0} Lyapunov Lyapunov V Lyapunov : [ ] 0 1 ẋ = Ax = x 1 1 [ ] V (x) =x T Px = x T 1 0 x = x x 2 2 Lyapunov V (x) =x T (PA A T P )x = 2x 2 2 E = {x x 2 =0} E ẋ 2 =0 x E ẋ 2 = x 1 x 2 =0 E / / 154 V (x) V (x) Lyapunov Lyapunov V Lyapunov / / 154

23 1 : V (x) : s(u, y) u y : () () () () ( 0) 1 : V (x(t 1 )) V (x(t 0 )) t1 t 0 s(u(t),y(t))dt V (x)... V (0) = 0, V(x) 0 s( )... V ( ) V s(u, y) / / () : u Required supply: ( t1 ) V r (x(t 1 )) in s(u, y) dτ, x(t 0 )=0 u,t 1 t 0 t1 t 0 s(u, y)dt 0, x(t 0 )=0, u( ) 1 V r available storage ( V a (x(t 0 )) = sup u,t 1 t1 t 0 ) s(u, y)dt 1 : x(t 0 )=0 : required supply V r V (x) V a (x) V (x) V r (x) / / 154

24 1 s(u, y) =γ 2 u 2 y 2 : u y L 2 - γ s(u, y) =u T y : s(u, y) =u T y a u 2 b y 2 : IFP OFP IFP, OFP Hill & Moylan / / 154 IFP OFP IFP, OFP Hill & Moylan (passivity) : u T y V (x) V (x(t 1 )) V (x(t 0 )) V (x) V u T y t1 t 0 u T ydt IFP OFP IFP, OFP Hill & Moylan LCR 2 : [ ] T H ẋ =(J R) g(x)u x y = g(x) T [ H x H J R [ ] [ ] T H H Ḣ = R y T u u T y x x ] T / / 154

25 IFP OFP IFP, OFP Hill & Moylan 2 u System 1 System 2 y 1 y y 2 2 u y 2 u 1 System 1 System 2 y 1 u 2 y IFP OFP IFP, OFP Hill & Moylan Augmented System u u y y M(x) System 1 M(x) T V (x(t 1 )) V (x(t 0 )) = t1 t1 t 0 u T ydt = t1 t 0 u T y dt t 0 u T M(x) T ydt / / 154 IFP OFP IFP OFP IFP OFP IFP, OFP Hill & Moylan OFP(Output Feedback Passivity): IFP(Input Feedback Passivity): s(u, y) =u T y ρy T y OFP(ρ) Passive System ρi OFP(ρ) s(u, y) =u T y νu T u IFP(ν) Passive System νi IFP(ν) IFP OFP IFP, OFP Hill & Moylan α Σ OFP(ρ) ασ OFP(ρ/α) Σ IFP(ν) ασ IFP(αν) OFP( ρ) IFP(ρ) OFP( ρ) ρi ρi IFP(ρ) / / 154

26 IFP OFP IFP, OFP Hill & Moylan Lyapunov : V (0) = 0, V(x) 0, V 0 Lyapunov V (x) E = {x V (x) =0} x =0 E x =0 : ẋ = (x, u), y = h(x, u) 1. u =0 2. u =0u =0 3. y = h(x) u = ky (k >0) IFP OFP IFP, OFP Hill & Moylan ẋ = (x, u), y = h(x, u) : Zero-State Detectability (ZSD): y 0 x(t) 0 (t ) : Zero-State Observability (ZSO): y 0 x(t) / / 154 IFP, OFP 1. u =0 0 V (x) V (x) 0 2. V (x) 0 V (x) =0 u u 1 System 1 y 1 y System 1 u T 1 y 1 ρ 1 y T 1 y 1 ν 1 u T 1 u 1 IFP OFP IFP, OFP Hill & Moylan 0 V (x) u T h(x, u), or u h(x, u) h(x, u) =h(x, 0) η(x, u)u V (x) =0 u u T h(x, 0) u T η(x, u)u 0 h(x, 0) = 0 {x V (x) =0} ẋ = (x, 0) {x h(x, 0) = 0} {x V (x) =0} x 0 x = V (x) =0y = h(x) =0 V kh(x) T h(x) {x h(x) =0} 0 x =0 IFP OFP IFP, OFP Hill & Moylan y 2 u 2 V 1 (x 1 ) System 2 System 2 u 1 =0, u 2 =0 System1,2 u T 2 y 2 ρ 2 y2 T y 2 ν 2 u T 2 u 2 u =0 V 2 (x 2 ) 1. ν 1 ρ 2 0 ν 2 ρ ν 1 ρ 2 > 0 ν 2 ρ 1 > 0 V 1, V : V 1 V 2 Lyapunov / / 154

27 Hill & Moylan IFP OFP IFP, OFP Hill & Moylan : System 1 u 1 =0 V 1 System 2 y 2 = Ku 2 K K λ min λ max V 2 =0 λ min ρ 2 λ 2 max ν 2 > 0 ρ 2 > 0, ν 2 u T 2 y 2 ρ 2 y T 2 y 2 ν 2 u T 2 u 2 (λ min ρ 2 λ 2 max ν 2 )u T 2 u 2 0 λ min ρ 2 λ 2 max ν 2 > 0 ρ 2 > 0, ν 2 ν 1 ρ 2 > 0 ν 2 ρ 1 > 0 OFP(ρ 1 ) System 1 (ν 1 =0) K System 1 (ρ 1 =0, ν 1 =0) K IFP OFP IFP, OFP Hill & Moylan : ẋ = (x)g(x)u y = h(x)j(x)u Hill & Moylan, 1976: V (x) s(u, y) =u T y ρy T y νu T u k q : R n R k, W : R n R k m L V = 1 2 q(x)t q(x) ρh(x) T h(x) L g V (x) =h(x) T 2ρh(x) T j(x) q T (x)w (x) W (x) T W (x) = 2νI j(x)j(x) T 2ρj(x) T j(x) / / 154 Hill & Moylan...(1) Hill & Moylan...(2) IFP OFP IFP, OFP Hill & Moylan Hill & Moylan IFP : ν IFP(ν) j(x) 0 : ρ =0j(x)j(x) T =2νI W (x) T W (x) (): V (x) L V 0 L g V (x) =h(x) T j(x) =0 u =0 Lyapunov : ρ = ν =0W (x) =0 Hill & Moylan IFP OFP IFP, OFP Hill & Moylan : j(x) =0 V (x) m 1 (L g h)(0) : V/ x(0) = 0 ( ) h x (0) = g T 2 V x 2 (0) 2 V/ x 2 (0) R T R h/ x rank Rg(0) = m rank (L g h)(0) = rank {g(0) T R T Rg(0)} = m / / 154

28 (1) (2) IFP OFP IFP, OFP Hill & Moylan Hill & Moylan : ẋ = Ax Bu, y = Cx Du V (x) =x T Px/2 (P > 0) L, W D =0 PA A T P = L T L PB = C T L T W W T W = D D T PA A T P 0 PB = C T IFP OFP IFP, OFP Hill & Moylan (Positive Real): H(s) =C(sI A) 1 B D ( ) (positive real ) 1. Re (λ i (A)) 0, i =1,...,n 2. H(jω)H( jω) T 0, ω / λ i (A) 3. A s i lim (s s i )H(s) s si Positive Real Lemma: H(s) / / 154 (3) IFP OFP IFP, OFP Hill & Moylan (Strictly Positive Real): H(s) =C(sI A) 1 B D ( ) 1. Re (λ i (A)) < 0, i =1,...,n 2. H(jω)H( jω) T > 0, ω / λ i (A) 3. H( ) H( ) T > 0 lim ω ω2(m q) det[h(jω) H( jω) T ] > 0 q = rank[h( )H( )] Kalman-Yakubovich-Popov Lemma: H(s) P >0, L, W, ɛ>0 PA A T P = L T L ɛp PB = C T L T W W T W = D D T FB FB positive real IFP/OFP D =0PA A T P<0, PB = C T / / 154

29 Vidyasagar y 2 = φ(u 2 ) y 2 αβ 2 u 2 2 < β α 2 u 2 2 (u 2 0) y 2 =0 (u 2 =0) { αu 2 2 <u 2 y 2 <βu 2 2 (u 2 0) y 2 =0 (u 2 =0) y = β 2 u 2 y 2 FB FB positive real IFP/OFP (α, β) β = { u T 2 y 2 >αu T 2 u 2 (u 2 0) y 2 =0 (u 2 =0) FB FB positive real IFP/OFP y = α 2 u 2 u / / 154 FB FB positive real IFP/OFP (α, β) 1 (α, β) 0 System 1 FB FB positive real IFP/OFP 1 (α, β) (β >0) 1 (1/β)I V (x) OFP( k) k = αβ/(β α) 0 u 1 y 2 ( 1/ β)i System 1 y y 1 1 u 2 u 2 ( 1/ β)i / / 154

30 (1) FB FB positive real IFP/OFP OFP( k) V ȳ T 1 (u 1 kȳ 1 )= ū 2 (y 2 kū 2 ) ū 2 = u 2 y 2 /β ū 2 (y 2 kū 2 ) > 0 (ū 2 0) V < 0 (y 1 0) 0 u 1 y 2 ( 1/ β)i System 1 ( 1/ β)i y y 1 1 u 2 u 2 FB FB positive real IFP/OFP u ' αβ /( β α) ( 1/ β)i System 1 u1 y1 u y u = β/(β α) (u 1 αy 1 ), y = u 1 /β y 1 u '' ( 1/ β)i System 1 u1 y 1 αi y' y' / / 154 (2) (1) FB FB positive real IFP/OFP V 1 u T y =(u 1 αy 1 ) T (u 1 /β y 1 ) = α β { u T 1 y 1 1 β α β ut 1 u 1 αβ } α β yt 1 y 1 1 u T y 1 α β ut u αβ α β yt y FB FB positive real IFP/OFP SISO G 0 (s) G 0 (s) p (Gain Margin): (α, β) 1/κ j0 ( κ (α, β)) p 1 α 1 β Im Re (Sector Margin): (α, β) (α, β) / / 154

31 (2) positive real FB FB positive real IFP/OFP (Disc Margin): D(α, β) 1 ( 1 2 α 1 ) j0 1 ( 1 β 2 α 1 ) β () p D(α, β) 1 α 1 β ( ) ( ) () Im Re FB FB positive real IFP/OFP β>0 positive real : G 0 (s) D(α, β) Ḡ(s) = G 0(s)(1/β) αg 0 (s)1 strictly positive real G 0 (s) D(α, β) D(α, β) p Ḡ(s) positive real u '' ( 1/ β)i System 1 u1 y 1 αi y' System 1 G 0 (s) L[y ] L[u ] = G 0(s)(1/β) αg 0 (s) / / 154 positive real(2) IFP/OFP positive real = (Disc Criterion): G 0 (s) D(α, β) (α, β) OFP : 3 1. ɛ OFP( α ɛ) 2. D(α, ) 3. IFP(ν) (ν α) FB FB positive real IFP/OFP FB FB positive real IFP/OFP IFP : ɛ IFP( 1/β ɛ) D(0,β) 1 / α 1 / β / / 154

32 Hamiltonian FB Hamiltonian FB : q =(q 1,...,q n ) T : u =(u 1,...,u n ) T : T (q, q) : W (q) Lagrangian (Lagrangean ): L = T W Euler-Lagrange : d dt ( L q i d dt ) L q i = u i, [ ] T L q i =1,...,n [ ] T L = u q / / 154 Hamiltonian T (q, q) = 1 2 qt M(q) q M(q) Euler-Lagrange : [ L : p = q : q = φ(p, q) ] T Hamiltonian: H(p, q) = [ q T p L(q, q) ] q=φ(p,q) Hamiltonian FB M(q) q c(q, q)g(q) =u c(q, q) =c 1 (q, q)c 2 (q, q) [ ] (M(q) q) c 1 (q, q) = q ( ), q c 2 (q, q) = 1 [ ( q T ] T M(q) q) () 2 q [ ] T W g(q) = () q Hamiltonian FB T = q T M(q) q/2 : p = M(q) q Hamiltonian: H = 1 2 pt M(q) 1 p W (q) H q, q H = 1 2 qt M(q) q W (q) = T W / / 154

33 Hamiltonian FB L : dl = [ ] L d q q [ ] L dq q p, q, L H : dh = q T dp p T d q dl dl p p T d q [ ] L dh = q T dp dq q [ ] H = q T p [ ] [ ] H L = q q H p, q L q, q Legendre Hamiltonian FB Euler-Lagrange ṗ = [ ] T L u q Hamilton : p, q [ ] T H q = p [ ] T H ṗ = u q / / 154 Hamiltonian system Hamiltonian FB Port controlled Hamiltonian system: Hamiltonian system [ ] T H q = p [ ] T H ṗ = u q [ ] T H y = (= q) p Ḣ = u T y Port controlled Hamiltonian system H Hamiltonian FB Port-Controlled Hamiltonian System : T = q T M(q) q/2 =p T M(q) 1 p/2 (M(q) M 0 > 0) W (q) 0 u = ky = k q Ḣ = kyt y = k q T q p 0(t ) W q W/ q 0(x 0) u = k q (D ): W (q) =0 W / / 154

34 Hamiltonian FB : W (q) (q W/ q 0(x 0)) Hamiltonian: H(p, q) =H(p, q) W (q)w(q) = T (p, q)w (q) [ ] T H = p [ ] T H, p [ ] T H = q [ ] T H q = p [ ] T H ṗ = q [ ] T H y = (= q) p [ ] T H q [ ] T W q [ ] T W q [ ] T W u q [ ] T W q Hamiltonian FB Hamiltonian [ ] T W (): u = g(q) ū q port controlled Hamiltonian system: [ ] T H q = p [ ] T H ṗ = ū q y = [ ] T H (= q) p port controlled Hamiltonian system ū = ky / / 154 (1) (2) : W = k 1 2 (q q 0) T (q q 0 ) u = g(q) k 2 q k 1 (q q 0 ) Hamiltonian FB k 1 > 0 q 0 q u = g(q) k 1 (q q 0 )ū ū = k 2 y = k 2 q (k 2 > 0) W (q) =0 ( q 0 ) Hamiltonian FB g(q): k 2 q: D()- k 1 (q q 0 ): P()- PD (dynamic-based control) / / 154

35 (1) (2) Hamiltonian FB P τ O θ m z m 2 K O OP L m O m J G τ LK mg Hamiltonian FB P m z OP θ q =(q 1,q 2 ) T =(θ, z) T, q =( q 1, q 2 ) T =( θ, ż) q p u = τ, x =(q T, q T ) T =(θ, z, θ, ż) T x =(q T,p T ) T : : T = J 2 θ 2 m 2 {(z2 L 2 ) θ 2 2L θż ż 2 } = 1 2 qt M(q) q = 1 [ ] J m(l 2 z 2 ) ml 2 qt q ml m U = K 2 z2 mg(l y m )= K 2 z2 mg{l(1 cos θ)z sin θ} / / 154 (3) (4) : L = T U : ( ) τ M(q) q c(q, q) = 0 ( ) 2mz θż mg(z cos θ L sin θ) c(q, q) = mz θ 2 Kz mg sin θ : H = H k 1 2 q2 1 H( x) k 1 >m 2 G 2 /K : u = k 1 q 1 v : Hamiltonian FB : p = M(q) q : H = 1 2 pt M(q) 1 p U(q) : q = H p (= q) ṗ = H q ( τ 0 ) Hamiltonian FB q = H p (= q) ṗ = H q y = H p 1 = q 1 ( v 0 ) / / 154

36 (5) (6) Hamiltonian FB y =0, u =0q 1 = θ = θ 0 (const.), θ =0, θ =0 ml z mg(z cos θ 0 L sin θ 0 )k 1 θ 0 =0 m z Kz mg sin θ 0 =0 z k 1 θ 0 = z(lk mg cos θ 0 ) LK mg cos θ 0 0z z z 0 z = z 0, ż =0, z =0 z 0 Kk 1 θ 0 mg(kl mg cos θ 0 )sinθ 0 = Kk 1 θ 0 mgkl sin θ 0 m2 G 2 sin 2θ 0 =0 2 k 1 > max{m 2 G 2 /K, mg(kl mg)/4} θ 0 θ = θ 0 =0θ 0 z 0 z = z 0 =0 Hamiltonian FB LK mg cos θ 0 =0 z θ 0 =0 LK = mg () LK mg cos θ 0 0 y = k 2 y u = k 1 q 1 k 2 q 1 k 1 > max{m 2 G 2 /K, mg(kl mg)/4}, k 2 > / / 154 (1) : : ẋ = (x)g(x)u u = α(x) Lyapunov V (x) : ẋ = (x) =(x)g(x)α(x) Lyapunov Sontag-type Sontag Sontag Sontag-type cl Lyapunov Sontag-type Sontag Sontag Sontag-type cl V (x) / / 154

37 (2) Lyapunov Sontag-type Sontag Sontag Sontag-type cl [ ] V V = {(x)g(x)α(x)} x = L V (x)(l g V (x))α(x) < 0, (x 0) α(x) L g V V L g V (x) =0 V u = α(x) L V < 0 (x 0) V (x) : L g V (x) =0 x 0 x L V (x) < 0. α(x) Lyapunov Sontag-type Sontag Sontag Sontag-type cl (Control Lyapunov unction, cl): V (x) ẋ = (x) g(x)u (cl) V (x) L g V (x) =0 x 0 L V (x) < 0 cl ( ) α s (x) / / 154 Sontag-type Sontag-type Cl V (x) Sontag-type : u = α s (x) = L V (L V ) 2 (L g V (L g V ) T ) 2 L g V (L g V ) T (L g V ) T, L g V 0 0, L g V =0 Sontag-type Sontag-type cl V (x) L g V 0: V = L V L g Vα s (x) { } = L V L V (L V ) 2 (L g V (L g V ) T ) 2 = (L V ) 2 (L g V (L g V ) T ) 2 < 0 Lyapunov Sontag-type Sontag Sontag Sontag-type cl cl Lyapunov Sontag-type Sontag Sontag Sontag-type cl L g V =0, x 0: V = L V < 0 V / / 154

38 Sontag-type Sontag-type () Sontag-type L g V 0 α s (x) L g V =0? : 0, i b =0and a<0 φ(a, b) = a a 2 b 2, elsewhere b b a : V (x) Sontag-type α s (x) : a = L V, b = L g V (L g V ) T α s (x) = { 0, x =0 φ(l V,L g V (L g V ) T )(L g V ) T, x 0 Lyapunov Sontag-type Sontag Sontag Sontag-type cl S = {(a, b) R 2 b>0 or a<0} : p 2 F (a, b, p) =bp 2 2ap b =0 S p = φ(a, b) (b =0, a<0 ) F p (a, b, φ(a, b)) = 2 a 2 b 2 0, (a, b) S φ(a, b) Lyapunov Sontag-type Sontag Sontag Sontag-type cl α s (x) / / 154 Sontag-type Sontag-type L g V = 0 α s (x)???? L V < 0 α s (x) 0 (Small Control Property, scp): Cl V (x) α c (x) (α c (0) = 0) L V (x)l g V (x)α c (x) < 0, x 0 Lyapunov Sontag-type Sontag Sontag Sontag-type cl α s (x) 0 α s (x)???? L V > 0 α s (x) (L g V L V x ) α s Lyapunov Sontag-type Sontag Sontag Sontag-type cl Scp Sontag-type : Scp V (x) Lyapunov 1 Sontag-type / / 154

39 () cl Lyapunov Sontag-type Sontag Sontag Sontag-type cl : L V L g V α c, L V 0 α s α c α c 2 L g V 2, L V 0 α s L g V, L V 0 α c, L g V α s Scp cl V (x) Sontag-type Lyapunov Sontag-type Sontag Sontag Sontag-type cl : ẋ = (x)g(x)u (0) = 0 scp cl V (x) cl scp cl V (x) Sontag-type / / 154

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( ) 2014 2014 1/119 = (ISS) ISS ISS ISS iss-clf iss-clf ISS = (ISS) FB 2014 2/119 = (ISS) ISS ISS ISS iss-clf iss-clf ISS R + : 0 K: γ: R + R + K γ γ(0) = 0 K : γ: R + R + K γ K γ(r) (r ) FB K K K K R