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1 Chapter 12. DESIGN VIA STATE SPACE Puan National Univerity oratory
2 Table of Content v v v v v v v v Introduction Controller Deign Controllability Alternative Approache to Controller Deign Oberver Deign Obervability Alternative Approache to Oberver Deign SteadyState Error Deign via Integral Control.
3 Oberver Deign v Oberver Oberver (etimator) : Calculate tate variable that are not acceible from the plant Why we need to ue oberver In other application, ome of the tate variable may not be available. Too cotly to meaure tate or end tate to the controller So, etimated tate, rather than actual tate, are then feedback to the controller..
4 Oberver Deign Plant : & A Bu (12.57) y C Oberver : ˆ & Aˆ Bu yˆ Cˆ (12.58) (12.57) (12.58) Figure Statefeedback deign uing an oberver to etimate unavailable tate variable: a. openloop oberver & (12.59) ˆ A( ) ˆ y yˆ C( ) ˆ.
5 Oberver Deign (12.59) i the difference between the actual and etimated tate and i unforced. If the plant i table, thi difference approache to zero. The peed of convergence between the actual tate and the etimated tate i the ame. Becaue of the characteritic equation for (12.59) i the ame for (12.57) Need to eek a way to peed up the oberver and make it repone much fater. So, we ue the feedback to increae the peed of convergence..
6 Oberver Deign Figure Statefeedback deign uing an oberver to etimate unavailable tate variable: c. eploded view of a cloedloop oberver, howing feedback arrangement to reduce tatevariable etimation error.
7 Oberver Deign Plant : & A Bu (12.61) y C Oberver : ˆ & Aˆ Bu L( y yˆ ) yˆ Cˆ (12.60) ˆ i the error between the actual tate vector and the etimated tate vector y output yˆ i the error between the actual output and the etimated.
8 Oberver Deign (12.61) (12.60) ( & ˆ &) A( ) ˆ L( y yˆ ) (12.62) ( y yˆ ) C( ) ˆ & ˆ & ( A LC )( ) ˆ y yˆ C( ) ˆ (12.63) Let e ( ˆ ) & (12.64) e ( A LC) e y yˆ Ce If the eigenvalue of Eq. (12.64) are all negative, the etimated tate vector error will decay to zero..
9 Oberver Deign The characteritic equation i found from Eq. (12.64) Characteritic equation Deired characteritic equation det[ l I (A LC) ] 0 æ an L 0 ö æ l1 ö æ ( an 1 l1 ) L 0ö ç ç ç ç an L 0 ç l2 ç ( an2 l1 ) L 0 A LC ç M M M M M M ç M ( L 0) ç M M M M M M ç ç ç ç a L 1 ç ln 1 ç ( a1 ln 1) L 1 ç a L 0 ç l ç ( a l ) L 0 è 0 ø è n ø è 0 n ø ( a l ) ( a l ) L ( a l ) ( a l ) 0 n n1 n2 n1 1 n2 2 1 n1 0 n d d d d n n1 n2 n1 n2 L
10 Oberver Deign Figure Thirdorder oberver in oberver canonical form: a. before the addition of feedback; b. after the addition of feedback Oberver canonical form i ued to yield the eay olution for the oberver gain.
11 v Eample 12.5 Deign an oberver for the plant Oberver Deign ) 2)( 1)( ( 4) ( ) ( 2 3 G The oberver will repond 10 time fater than the controlled loop. )] ( 10 ) ( [4 )] ( 17 ) ( [ )] ( 8 [ ) ( ) ( ) ( ) 2)( 1)( ( 4) ( ) ( C R C R C C R C G
12 Oberver Deign 1. Repreent the plant in oberver canonical form 2. Ue the difference between actual output etimated output and add the feedback path.
13 Oberver Deign 3. Find the characteritic polynomial é 8 1 0ù é0ù ˆ & A ˆ Bu ˆ 1 u ë û ë 4û [ 1 0 0] y ˆ Cˆ ˆ The oberver error i e& ( A LC) e Characteritic equation: é ë ( 8 l1 ) ( 17 l2 ) ( 10 l ) ( ) ( ) ( ) ù 1 e 0û 8 l 17 l 10 l 0.
14 Oberver Deign 4. Evaluate the deired polynomial The cloed loop controlled ytem in Eample ± j2 To make thi oberver 10 time fater 10 ± j20 Select the third pole to be 10 time the real part of the dominant pole 100 Therefore, the deired characteritic equation become \ l 112, l 2483 and l
15 Obervability If any tate variable ha no effect upon the output, we cannot evaluate thi tate variable by oberving the output. Each tate variable can be oberved ince each i connected to the output All tate variable cannot be oberved ince any tate i not connected to the output.
16 Obervability If the initial tate vector, ( t ), can be found from u(t) and y(t) meaured over a finite interval of time from the ytem i aid to be unobervable 0 t 0, the ytem i aid to be obervable, otherwie [ ] y C obervable [ ] y C unobervable.
17 Obervability v The obervability matri nth order plant & A Bu y C O M é C CA M ëca n1 ù û If the rank( O ) i n, O i called obervability matri M M.
18 Obervability v Eample 12.6 Determine if the ytem of Figure i obervable Figure Sytem of Eample
19 Obervability The tate and output equation for the ytem é ù é0ù & A Bu u ë 4 3 2û ë 1û y C Obervability matri i O M [ 0 5 1] é C ù é ù CA 2 ë CA û ë û Rank of O M i 3, the ytem i obervable..
20 Alternative Approache to Oberver Deign How to deign oberver for ytem not in Oberver Canonical Form. Match the coefficient of characteritic polynomial det [ I (A LC) ] with thoe of the deired Tranform the plant to oberver canonical form, deign in oberver canonical form, and tranform the deign back to the original repreentation. Aume a plant not repreented in oberver canonical form é C ù CA 2 CA O M Z y Cz M n2 obervabiliy matri CA (12.84) n1 (12.85) ëca û z & Az Bu.
21 Alternative Approache to Oberver Deign Aume that the ytem can be tranformed to the oberver canonical form z P with the tranformation (12.86) Subtituting (12.86) into (12.84) and premultiplying the tate equation by P Obervability matri 1 1 & P AP P Bu y CP (12.87) 1 é CP ù é C ù 1 CP(P AP) CA OM CP(P AP)(P AP) CA X M M n1 ëcp(p AP)(P AP)L(P AP) û ëca û P (12.88).
22 Alternative Approache to Oberver Deign Subtituting (12.85) into (12.88) and olving for P 1 P OM O M (12.89) Z the tranformation, P, can be found from the two obervability matrice. Uing Eq. (12.87) and the form uggeted by Eq. (12.64) X e& 1 ( P AP L CP)e z y yˆ CPe (12.90) Since, 1 P z 1 ˆ P zˆ 1 e ˆ P e z (12.90) e & (A PL C)e z z y yˆ Ce z (12.91).
23 Alternative Approache to Oberver Deign Comparing Eq. (12.91) to (12.64) oberver gain vector L PL (12.92) z v Eample 12.8 Deign an oberver for the plant G( ) ( 1)( 1 2)( 5) The cloedloop performance of the oberver i governed by the characteritic polynomial :
24 Alternative Approache to Oberver Deign Firt repreent the plant in it original cacade form é5 1 0 ù é0ù z & Az Bu z u ë 0 0 1û ë 1û y Cz [ 1 0 0] z Obervability matri O Mz i O Mz é C ù é 1 0 0ù CA 2 ë CA û ë û.
25 Alternative Approache to Oberver Deign Characteritic equation for the plant i 3 2 det( I A) Uing thi characteritic polynomial to form the oberver canonical form & A B y C u é 8 1 0ù ; ë û A [ 1 0 0] Obervabliliy matri for the oberver canonical form become, O C é C ù é 1 0 0ù C A 2 ë C û ë A û M.
26 Alternative Approache to Oberver Deign Deign the oberver for the oberver canonical form A LC é ù él1 ù é (8 l1 ) 1 0ù l (17 l ) 0 1 [ ] 2 2 ë û ë l 3 û ë (10 l3) 0 0û characteritic polynomial i det[ I (A L C )] (8 l ) (17 l ) (10 l ) The deired cloedloop oberver characteritic equation i é ù 2483 ë 49900û Oberver gain vector 112 L.
27 Alternative Approache to Oberver Deign Tranform the deign back to the original repreentation The tranformation matri P O 1 Mz O M é 1 0 0ù ë 1 1 1û Tranforming to the original repreentation L PL z é 112 ù 2147 ë 47619û.
28 SteadyState Error Deign via Integral Control Dicu how to deign ytem repreented in tate pace for teadytate error A feedback path from the output ha been added to form the error, e The integrator increae the ytem type and reduce the previou finite error to zero Figure Integral control for teadytate error deign.
29 SteadyState Error Deign via Integral Control Derive the form of the tate equation for the ytem of Fig & A Bu & (12.112) r y C r N y C é & ù é A 0ù é ù ébù é0ù u N 0 N 0 1 ë & û ëc û ë û ë û ë û é ù y [ C 0] ë N û r (12.113).
30 SteadyState Error Deign via Integral Control é ù u K KeN [ K Ke ] (12.114) ë N û Subtituting Eq.(12.114) into Eq.(12.113) ( A BK) é & ù é BK ù é ù é0 e ù r N 0 N 1 ë & û ë C û ë û ë û é ù y [ C 0] ë N û Ue the characteritic equation to deign K and tranient repone K e (12.115) to yield the deired.
31 Homework: Divide by 9 THANK YOU.
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