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1 SICE -- FL6_21_1 1
2 ( ) (ptimal cntrl) Fig :, -V (inite hrizn) receding hrizn (mving hrizn) 2
3 (inite hrizn) Fig : 15 ASA (ininite hrizn) (t ) Fig : F-8 [2] (receding hrizn cntrl) Fig : 3
4 ( k 1) ( (, ) ( ) M ( q) q&& C( q, q& ) q& g( q) τ (,) ( k 1) A ( B ( ) & && k 1 d m m & 1 u m Fig : d u m k Fig : 4
5 U ( X u 2 U u 1 2 X U X ( ) X 1m 1m u 2 U ( k 1) A ( B 2 u 1 X 1 U X 1V 1V 9 9 ( k 1) ( (, ) 5
6 V (, u) 1 i l( (, ) F( ( )) l( (, ) F( ( )) (stage cst) (terminal cst) l( (, ) >, l(,), F( ( )) > l( (, ) ( Q( R l( (, ) F( ( )) F( ( )) ( ) P ( ) Q, R > 6
7 ptimal cntrl prblem P U () u { V (, u) u U ( )} ( ) : min ptimal u { u (; ), u (1; ),, u ( 1; ) } ( ) L (trajectry) { (; ), (1; ),, ( 1; ), ( ; ) } ( ) L (value unctin) V ( ) V (, u ( )) i ( ; ) 3 2 u () ( ( 1; ), 1; )) 1 V 1 () () 7
8 ( ) : u (; ) u ( ; ) receding hrizn (mving hrizn) u { u (; ), u (1; ),, u ( 1; )} ( ) L u () u 1 u 1 u 1 8
9 X {} X ( ) V F( ( )) F( ( )) ( ) P ( ) (, u) 1 i Kwn and Pearsn (1977) l( (, ) P () X F( ( )) ( ) X X X F( ( )) 9
10 ( ( Q( u ( R ) V (, u) i l ( (, ) u ( K( ( ) 1 K R B PB B PA V ( V ( (, u ( ) ( P( V V ( k l ( k ( k 1) A( B ( ) P A PA A PB R B PB B PA Q 1) V 1) P( k ( (, ) ( 1) ( P( V ( u ( K( 1 ( 1
11 ( k 1) A ( B U ( X ( ) X X X K ( R B P ( ) k,..., 1 k,..., 1 V (, u) i ( ) ( Q( u ( R ( ) P ( ) F( ( )) ( ) P ( ) F( ) l(, ( )) B) 1 B P O A ( ) K 11
12 X ( ) K ( k 1) ( A BK) ( U ( X k,..., 1 k,..., 3. ( ) X X ( ) K X k 1 K ( A BK) U k,1..., k ( A BK) X k X (),1..., O X,. k k { K( A BK) U,( A BK ) X r k,1,..., } O O 12
13 (terminal cnstraint) ( ) u (, u) : (1), ( ) (stabilizing cntrl) u X X X (terminal cnstraint set) () u φ(, u) φ(, u) : φ( ) φ( ) φ() φ(, u) φ( ) 13
14 A1 : X X, ( ) U, A2 : A3 : A4 : X, X (, ( )) X, [ F l](, ( )), X X X A3 : X (psitive invariant set) A4 : F() ( ) (cntrl Lyapunv unctin) A1-A3 : (easibility) 14
15 2 3 1 P ( ) u ~ ( ) ( ; ) ( 1; ) u ( ; ) u ( 1; ) ( 1; ) 1 u ( ; ) ( 1; ) 1 u ~ ( ) : L )) () { u (1; ),, u ( 1; ), ( ( ; } ( ( ; )) U ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U X X ( ( ; )) X 15
16 ( ; ) ( 1; ) ( 1; ) ( ; ) u () 1 () () u ~ ( ) u ( ; ) u ( 1; ) ( 1; ) 1 ( ; ) ( 1; ) u ( ; ) u ( 1; ) 1 u ( 1; ) u ( ; ) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U ( 1; ) 1 X ( ( ; )) 16
17 ( ; ) ( 1; ) ( 1; ) ( ; ) () u () 1 () u ~ ( ) u ( ; ) u ( 1; ) ( 1; ) 1 ( ; ) ( 1; ) u ( ; ) u ( 1; ) 1 u ( 1; ) u ( ; ) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U ( 1; ) 1 X ( ( ; )) 17
18 u ~ ( ) () ( ; ) 1 1 V (, u) l( (, ) V ( 1; ) u ( ; ) u ( 1; ) 1 i ( 1; ) u ( ; ) ( 1; ) 1 ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U X ( ( ; )) F( ( )) ~ ( (, u )) l( ( i; ), u ( i; )) F( ( ; )) i V ( (; ), u (; )) l( (; ), u (; )) F( ( ; )) l( ( ; ), ( ( ; ))) F( ( ( ; ), ( ( ; )))) 18
19 V u ~ ( ) () ( ; ) ( 1; ) u ( ; ) u ( 1; ) ( 1; ) u ( ; ) ( 1; ) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U ( ( ; )) X 1 1 ~ ( (, u )) V ( (; ), u (; )) l( (; ), u (; )) F( ( ; )) V () () l( ( ; ), ( ( ; ))) F( ( ( ; ), ( ( ; )))) V ( ) l(, ( )) F( ( ; ), ( ( ; ))) l( ( ; ), ( ( ; ))) Q F(, u) : F( ) F( ) 19
20 V () u ~ ( ) ( ; ) 1 1 ~ ( (, u )) V ( ) l(, ( )) F( ( ; ), ( ( ; ))) l( ( ; ), ( ( ; ))) V V ( ) l(, ( )) [ F l]( ( ; ), ( ( ; ))) ( ) l(, ( )) Q A4 : ( 1; ) u ( ; ) u ( 1; ) ( 1; ) u ( ; ) ( 1; ) [ F l](, ( )) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U ( ( ; )), X X 2
21 u ~ ( ) () ( ; ) ( 1; ) u ( ; ) u ( 1; ) ( 1; ) 1 1 ~ ( V (, u )) V ( ) l(, ( )) u ( ; ) ( 1; ) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U ( ( ; )) X u ~ ( ) : L )) V V { u (1; ),, u ( 1; ), ( ( ; } ( ) V (, u~ ( )) ~ ( ( ) V (, u )) V ( ) l(, ( )) 21
22 V u ~ ( ) () ( ; ) V ( 1; ) u ( ; ) u ( 1; ) ( 1; ) 1 1 ( ) V ( ) l(, ( )) (, ( )) l(, ( )) u ( 1; ) Q V ( ; ) ( ( ; ), ( ( ; ))) ( ( ; ), ( ( ; ))) X ( ( ; )) U ( ( ; )) (, u) : V ( ) V ( ) value unctin V ( ) > value unctin V ( ) ( ) V X 22
23 (ininite hrizn) (t ) (receding hrizn cntrl) l( (, ) V (, u) 1 i l( (, ) F( ( )) F( ( )) (terminal cst) F( ( )) 23
24 ( k 1) A( B y ( C( 1 V (, u) { ( Q( u ( R } i [ F l](, ( )) A4 : P ( ) k, X A P A Q P ( A k ) Q A k? ( ) P ( ) A4 A : A BK : Q K Q ( ) : K RK 24
25 1 V (, u) { ( Q( u ( R } i 1 i i { { F( ( ( Q( u ( Q( u ( R } i ( ) P ( ) { ( Q( u ( R } ( R } )) P ( A k k ) Q A k (ininite hrizn) A. Jadbabaie, J. Yu and J. Hauser, Uncnstrained Receding-Hrizn Cntrl nlinear Systems, IEEE rans. Autmatic Cntrl, Vl.46,.5, pp , 21. cntrl Lyapunv unctin 25
26 ( k 1) A( B V ( ( Q( u ( R ) (, u) i 1 V i (, u) 1 i ( ( Q( u ( R ) u ( K( ( ) ( ( Q( u ( R ( Q( u ( R ) U ( X ( ) X X k,..., 1 k,..., i K ~ ( ) 1 R B PB B PA ( ) P ( )! ( ) X ( ) X ( ) K( K ( ) 1 R B P B B P A 26
27 l( (, ) F ( ( )) (, u) l( (, ) V 1 i l( (, ) F( ( )) l( (, ) Hrizn G. Grimm, M. J. Messina, S. E. una and A. R. eel, Mdel Predictive Cntrl: Fr Want a Lcal Cntrl Lyapunv Functin, All is t Lst, IEEE rans. Autmatic Cntrl, Vl.5,.5, pp , 25. A. Jadbabaie and J. Hauser, On the Stability Receding Hrizn Cntrl With a General erminal ky Institute Cst, echnlgy IEEE rans. Autmatic Cntrl, Vl.5,.5, pp ,
28 Hver Crat ( ) Visual Feedback System () Fig : Hver Crat Fig : Visual Feedback System 28
29 Hver Crat (Hver Crat) Hver Crat, d cs dt 3 1/ L sin 3 cs 3 u 1/ L sin 3 [ ] 1, 2, 3 [ ] u u1, u2. Ohthuka and A. Kdama, Autmatin Cde Generatin System r nlinear Receding Hrizn Cntrl,, Vl.38,.7, pp Fig : Hver Crat Fig : Hver Crat ~htsuka/paper/cca2hp_hver.pd 29
30 (Hver Crat) U u ~ u ma ma X ~ ma ma t ( ( )) L( ( t ), t ) J ϕ ) dt ϕ t {( ) ( ) u u} 1 L p( t) ( t) Q p( t) ( t) 2 1 ( p( t) ( t) ) S ( p( t) ( t) ) 2 Q > > p(t) S π [ ] 2 ( t) (1 e αt ) (stage cst) (terminal cst) C/GMRES 3
31 (Hver Crat) 31
32 Visual Feedback Cntrl (Visual Feedback Cntrl) & 1 1 ξ M ( q) C( q, q& ) ξ M ( q) u & sλ R z w wc J ξ b sλ R z w wc J αj t J u, t) l ( τ ), τ ) dτ RHC b b R wc arget Object R wc Camera R& wc ( ( ) M ( ( t )) t 1 l( ( τ ), τ )) 4 Q u R u ( ( t )) 4V ( ( t )) Planar Manipulatr Fig : Visual Feedback System Q > R 1 > V () (stage cst) M (terminal cst) 32
33 (Visual Feedback Cntrl) 33
34 (nnlinear mdel predictive cntrl) (rbust mdel predictive cntrl) (ast systems) (hybrid systems) rajectry Generatin Path Planning 34
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