The Performance of Feedback Control Systems

Transcription

1 The Performace of Feedbac Cotrol Sytem Objective:. Secify the meaure of erformace time-domai the firt te i the deig roce Percet overhoot / Settlig time T / Time to rie / Steady-tate error e. ut igal uch a the te to tet the reoe of the cotrol ytem. 3. Correlatio betwee the ytem erformace ad the locatio of T.F. ole ad zero i the -lae 4. Relatiohi betwee the erformace ecificatio ad the atural frequecy ω ad damig ratio ζ for d-order ytem

2 R <Performace of a d-order y > - E Y Y R K Y R K ω ζω ω R ζ: damig ratio & ω: atural frequecy f we ca elect & K, we may be able to rovide deired ζ, ω, ad/or the combiatio deig a cotroller!, - ζ ω ± jω ζ σ ± jω Sig of the real art σ tability

3 Meaure of Performace d order y. for the te iut: R / π π Pea time T ω β ω ξ M t The ea reoe & The ercet overhoot e ξπ / ξ P. O. 00 e ξπ / ξ Settlig time % differece T 4 4τ τ ξω ξω Textboo: 5% T 3 3 τ ξω

4 Math. Derivatio ω f R, a uit te iut Y ξω ω L.T. Table ζω y t e t i ωβg θ β β ξ Pea: to derive the ea, differetiate yt, θ co ξ, 0 < ξ < L dy t dt ω Y ξω ω L.T. Table ω ξωt y t e iωβ t B To get the ea y t 0 ω β t π π π Pea time T ω β ω ξ The ea reoe M t The ercet overhoot P. O. 00 e e ξπ / ξ ξπ / ξ

5 Settlig time: Textboo: 5% 3 T ζω T ξω τ 3 3 β ζ β θ θ π θ β ω π β ω i ta i i i t t Therefore, % T T T e ξω τ ξω τ ζω ζω < θ β ξ ξ t w T t e e t y / ζ ξπ ξ β β ξ β 0, co < < ξ ξ θ

6 Cotrollability A ytem decribed by the matrice A, B ca be aid to be cotrollable if there exit a ucotraied cotrol u that ca trafer ay iitial tate x0 to ay other deired locatio xt. For the ytem A [ x ] matrix x& Ax Bu, B [ x m] matrix We ca determie whether the ytem i cotrollable by examiig the algebraic coditio. For a igle-iut, igle-outut ytem, the cotrollability matrix P c, i decribed i term of A ad B P c [B AB A B A - B], which i a [ x ] matrix. Therefore if the determiat P c i ozero, the ytem i cotrollable. Examle: coider a ytem rereeted by teady-tate equatio x x u, x 3x dx y x ad determie the coditio for cotrollability. We ca cofirm the requiremet o the arameter d by geeratig the matrix P c. Therefore we have P c 0 d Ad the determiate of P c i equal to d, which i a ozero oly whe d i ozero the ytem i cotrollable for d 0.

7 Obervability All the root of the characteritic equatio ca be laced where deired i the -lae if, ad oly if, a ytem i obervable ad cotrollable. Obervability refer to the ability to etimate a tate variable. Thu we ay a ytem may be obervable if the outut ha a comoet due to each tate variable. A ytem i obervable if, ad oly if, there exit a fiite time T uch that the iitial tate x0 ca be determied from the obervatio hitory yt give the cotrol ut. Coider the igle-iut, igle-outut ytem x Ax Bu ad y C x, where C i a row vector, ad x i a colum vector. Thi ytem i obervable whe the determiat of Q i ozero, where 3 which i a [ x ] matrix. Examle: Obervability of a two-tate ytem Coider the ytem give by order to chec ytem cotrollability ad Obervability, eed to evaluate P c ad Q matrice. The matrice for P c are Therefore we have Ad determiat P c 0. Thu the ytem i ot cotrollable. To determie Q we obtai

8 C [ ] ad [ ] CA Therefore we obtai Ad determiat Q 0. Therefore the ytem i ot obervable.

9 Deig of a d order Cotrol Sytem. Secify the meaure of erformace Percet overhoot P.O. / Settlig time T / Time to rie T r / Steady-tate error e Stability. Fid the characteritic equatio of the cotrol ytem 3. Comare it with C.E. of the geeral ecod order ytem ad elect PD cotrol gai to atify ytem requiremet Cotat roortioal gai K PD cotrol gai K P, K, & K D

10 R <Cotrol Deig: d order Sytem > - E Y Y R K Y R K ω ζω ω R ζ: damig ratio & ω: atural frequecy f we ca elect & K, we may be able to deig a cotroller to atify with erformace requiremet give by ζ, ω, ad/or the combiatio T 4 / ζ ω ice ζ ω & K ω. Fid C.E.: K. Comared with the geeral C.E.: ζ ω 3. Select K & to atify with the erformace requiremet ζ ω K ω <EX> Natural frequecy ω hould be 3 [Hz] K 9 Damig ratio hould be ζ 6 ω

11 EX R - E d Y Y R. Fid C.E.: d. Comared with the geeral C.E.: ζ ω 3. Select & d to atify with the erformace requiremet d ζ ω ζω ω ω ω <EX> d ζ ω Requiremet: T 4 / ζ ω [ec] ad ω 4 [Hz] From, 6 ad From, T 4 / ζ ω 4 / 4ζ ζ 0.5 d ζ ω 4 ω ω R

12 <Deig Examle> R Gc K G T where G G G c Uig Mao rule T K K P Q ω ζω ω C.E. Q 0 K 0 ζω ω K 0 By obervatio ω K. ζ ω ζ ω ad K ω Selectio of K baed o deig requiremet of ω T 4 ec Q T 4τ, τ ζω. Root ole of Q ± j S,, ζω ± jω ζ ± j ω 80 0 ± co - ζ

13 , - ζ ω ± jω ζ σ ± jω Q ζω ω K 0

14 <Deig Examle> Deig Requiremet: ζ hould be greater tha ad T hould be fater tha 4 [ec] C.E. of a d order y i give by Q d ζω ω d ζω, ω Proer choice of & d ca rovide required erformace ecified by ζ or/ad ω ettig time T 4 / ζω < 4 [ec] ζω > d ζω > ζω > ω > / ζ ω > REMARK: ζω ole go away to the left T fater Sice θ co - ζ co θ ζ, - ζ ω ± jω ζ σ ± jω R - E d Y Y R ω ζω ω R

15 <Domiat Pole> Sytem higher tha d -order The root of Q that caue the domiat traiet reoe of b the ytem if 5 0 or greater a * Examle Q 0 0 domiat ole -, - Q 3 a b c y 0 0 y t ae t be t ce 0t

16 PD cotrol R E G c U G U Cotroller G c D : cotat Cotrol iut U [ D ] E f G u t e t e t dt d de t dt. P cotrol: G c Gc G T G G c Q ζ w w. PD cotrol: G T c D D D Q D ζ w w

17 3. P cotrol: G c T 3 a 4. PD cotrol: G c P D D T D D 3 a D. Fid C.E.: H G c G c H 0. Comared with the geeral C.E.: ζ ω ω 3. Select K P, K, & K D to atify with the required erformace Select two ole domiat ole ear to the origi for the comario if there are more tha two ole comare the real art of ole EX> if ole are at -± j, -0 elect -± j 5 ζ ω 0 ω