{ 8. { CHAPTER 8. Å (sampling time) x[k] =x(kå) u(ú) t t + Å (u[k]) x[k + 1] =A d x[k] +B d u[k] (8:) (diãerence equation) A d =e AÅ ; B d = Z Å 0 e A

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Chapter 8 ( _x = Ax +Bu; y = Cx) 8.1 8.1.1 x(t) =e At x(0) + _x =Ax +Bu (8:1) Z t 0 e A(tÄú) Bu(ú)dú u(t) x(0) t x(t) t =kå (k + 1)Å x(t) x[k + 1] =e AÅ x[k] + Z t+å t { 8.1 { e A(t+ÅÄú) Bu(ú)dú (8:)

{ 8. { CHAPTER 8. Å (sampling time) x[k] =x(kå) u(ú) t t + Å (u[k]) x[k + 1] =A d x[k] +B d u[k] (8:) (diãerence equation) A d =e AÅ ; B d = Z Å 0 e Aú dúb (y = Cx) t =kå y[k] =Cx[k] (8:4) (discretization) (8:); (8:4) (discrete-time model) (8:) (descrete-time state space model) (continuous-time model) 8. 1 (:4) A d =e AÅ = 6 4 1 (e p ãå +e ÄpãÅ ) 1 p ã (ep ãå Äe Ä p ãå ) p ã (e p ãå Äe ÄpãÅ ) 1 (e p ãå +e ÄpãÅ ) 7 5 b d = å 6 4 1 ã (ep ãå +e ÄpãÅ )Äã p1 (e p ãå Äe ÄpãÅ ) ã 7 5 Å = 1[msec] A d = 4 1:001 1:004Ç10Ä :485 1:001 5; b d = 4 Ä5:778Ç10Ä6 Ä1:156Ç10 Ä 5 (8:)

8.1. { 8. { 8.1. (8:) x[1] = A d x[0] +B d u[0] x[] = A d x[1] +B d u[1] =A dx[0] +A d B d u[0] +B d u[1].. x[n] 8.1. x[n] =A n X dx[0] + nä1 A näiä1 i=0 d B d u[i] (8:5) u[k] = 0 (k = 0; 1;ÅÅÅ) x[n] =A n dx[0] (8:6) A d (ï 1 ;ÅÅÅ;ï n ) T A d x[i] =Tz[i] z[n] = T Ä1 A n dtz[0] = (T Ä1 A d T) n z[0] = diagfï n 1 ;ÅÅÅ;ïn n gz[0] z[0] lim n!1 z[n] = 0 jï i j<1; i = 1; ;ÅÅÅ;n (8:7) T lim n!1 z[n] = 0 lim n!1 x[n] = 0 (8:6) (8:7) 1 A ï v A d =e AÅ A d v = e AÅ v = (I +AÅ + (AÅ)! = (1 +ïå + ï! +ÅÅÅ)v = e ïå v +ÅÅÅ)v (8:8)

{ 8.4 { CHAPTER 8. A d e ïå 8. 1 (Re(ï)<0) (je ïå j<1) ï 1 =õ+j!; ï i =õ+j(! + kô ) (k = 0;Ü1;Ü;ÅÅÅ) (8:9) Å 8. 8. 1 A d 1:0511; 0:9514 Ü49:84 e 49:84Ç0:001 = 1:0511; e Ä49:84Ç0:001 = 0:9514 A d 1 8.1.4 (8:) 8. 1 x[0] = 0 x[l] = x s u[k]; k = 0; 1;ÅÅÅ;lÄ1 x[0] = 0 (8:5) x[n] = [B d A d B d ÅÅÅA nä1 d B d ] 6 4 u[nä 1] u[nä ]. u[0] 7 5 rank([b d A d B d ÅÅÅA nä1 d B d ]) =n (8:10) k =n x[n] =x s

8.1. { 8.5 { 8. (8:) (8:10) 8. x[0] x[l] = 0 u[k]; k = 0; 1;ÅÅÅ;lÄ 1 A d = 0 B d (x[k + 1] =B d u[k] ) rank(b d ) =n det(a d ) = 0 (A d 0 ) (8:) (8:1) (8:10) :1 8. (1) (A d ;B d ) () (8:10) () z C rank([zi n ÄA d B d ]) =n (4) A d ÄB d K d K d (8:) u[k] =ÄK d x[k] K d (8:1) (8:) 8. 4 (8:1) A (8:9) (8:)

{ 8.6 { CHAPTER 8. Å 8. 8. 1 det([b d A d b d ]) =Ä1:57Ç 10 Ä7 det(a d )6= 0 k d ï 1 = 0:8; ï = 0:9 k d = [Ä5:407Ç10 Ä 40:77] 0( ) k d = [Ä8:64Ç10 4 Ä 10:0] A cd =A d Äb d k d A cd = 0 0 (ånite time settling control) 8.1.5 x[k + 1] = A d x[k] y[k] = Cx[k] 6 4 y[0] y[1]. y[n] = 7 6 5 4 C CA d. CA nä1 d x[0] 7 5

8.. { 8.7 { rank([c T A T dc T ÅÅÅ(A nä1 d ) T C T ]) =n (8:11) A d ÄH d C H d A d ÄH d C ( ) H d ^x[k + 1] =A d^x[k] +B d u[k] +H d (y[k]äc ^x[k]) (8:1) e[k] =x[k]ä ^x[k] 0 8. _x = y = Ax +Bu Cx (8:1) z[k + 1] = A dk z[k] +B dk y[k] u[k] = C dk z[k] +D dk y[k] (8:14) (8:14) A=D D=A D=A () (0 ) (8:) ( )

{ 8.8 { CHAPTER 8. _z = A k z +B k y u = C k z (8:14) (8:15) 8..1 v(t) ( ) z(t) Z z(t) = v(t)dt (8:16) z[k + 1] =z[k] + Åv[k] Å z[k] =z(kå) [kå; (k+1)å] v(t) _z! z[k + 1]Äz[k] Å (8:15) z[k + 1] = (I +A k Å)z[k] + ÅB k y[k] u[k] = C k z[k] (8:17) (forward diãerence) 8.. (backward diãerence) _z! z[k]äz[kä 1] Å (8:15) (IÄA k Å)z[k + 1] =z[k] +B k Åy[k + 1] ^z[k] := (IÄA k Å)z[k]ÄB k Åy[k] ^z[k]

8.. { 8.9 { ^z[k + 1] = A db^z[k] +B db y[k] u[k] = C db^z[k] +D db y[k] (8:18) A db = B db = C db = D db = (IÄA k Å) Ä1 (IÄA k Å) Ä1 B k Å C k (IÄA k Å) Ä1 C k (IÄA k Å) Ä1 B k Å 8.. 0 ( ) z[k + 1] = A dz z[k] +B dz y[k] u[k] = C dz z[k] (8:19) A dz = B dz = C dz = e A kå Z Å 0 C k e A kú dúb k A k Z Å 0 e A kú dú=a Ä1 k (ea kå ÄI) A dz = B dz = C dz = e A kå A Ä1 k (e A kå ÄI)B k C k

{ 8.10 { CHAPTER 8. 8..4 k;k + 1 z;v z[k];z[k + 1];v[k];v[k + 1] z[k + 1] =z[k] + Å (v[k + 1] +v[k]) q (u[k + 1] = qu[k]) z[k]=v[k] z[k] v[k] = Å q + 1 qä 1 (8:0) (8:16) z(s)=v(s) z(s) v(s) = 1 s (8:1) (8:0) (8:1) ( s) s! qä 1 Åq + 1 (8:15) (8:) qä 1 Åq + 1 z[k] =A kz[k] +B k y[k] (8:) (IÄ Å A k)z[k + 1]Ä Å B ky[k + 1] = (I + Å A k)z[k] + Å B ky[k] ^z[k] := (IÄ Å A k)z[k]ä Å B ky[k] ^z[k] ^z[k + 1] = A dt^z[k] +B dt y[k] u[k] = C dt^z[k] +D dt y[k] A dt = B dt = C dt = D dt = (I + Å A k)(iä Å A k) Ä1 ÅPB k C k P C k P Å B k (8:) P = (IÄ Å A k) Ä1

8.. { 8.11 { e x 1 Padìe e x = 1 +x= 1Äx= A d 8..5 A d B d C d D d I +A k Å ÅB k C k 0 Q ÅQB k C k Q C k QB k Å 0 e A kå A Ä1 k (e AkÅ ÄI)B k C k 0 (I +A k Å=)P ÅPB k C k P C k PB k Å= P = Q = (IÄ Å A k) Ä1 (IÄA k Å) Ä1 e At =I+At + (At)! + (At)! +ÅÅÅ Q = (IÄA k Å) Ä1 = I +A k Å + (A k Å) + (A k Å) +ÅÅÅ (I +A k Å=)P = I +A k Å + (A kå) + (A kå) 8 +ÅÅÅ (8:4) A d =I+A k Å +A 0 ; B d =B k Å +B 0 A 0 ;B 0 A 0 B 0 0 0 (A k Å) +ÅÅÅ A k B k Å +A k B kå +ÅÅÅ 0 (A kå)! (A kå) + (A kå)! + (A kå) 4 +ÅÅÅA k B k Å +A kb k Å 6 +ÅÅÅ +ÅÅÅA k B k Å +A kb k Å 4 +ÅÅÅ

{ 8.1 { CHAPTER 8. Å Å Å C d ;D d A d = I +A k Å (8:5) B d = ÅB k C d C k 0 C k + ÅC k A k + Å C k A k +ÅÅÅ ÅC kb k + Å C k A k B k +ÅÅÅ 0 C k 0 C k + Å C ka k + Å 4 C ka k +ÅÅÅ Å C kb k + Å 4 C ka k B k +ÅÅÅ D d

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