Projeto de Compensador de Avanço-Atraso para Controle de Sistemas - Análise e Cálculos

g1s 1 s5 4 Tr 03b wp ex1x2π ξπ ln01 ξπ 1ξ223025 1ξ2 ξπ2 530181 ξ2 ξπ2 5301ξ2 53018 ξ2π2 5301 53018 ξ2 53018 15175 ξ 0591 Τraub 03 0938 314 0938 Wn π arcosξ 909 rads 03110349 93Tr1ξ2 Pol desejado g2 2ξWns Wn2 0 S2 20591909 9092 s2 10745s 8262 δ 534 733j tg1 β 733 534 5392 tg1 733 134 7664 B 12608 φ 180 10336 12608 φ 4944 733 534 10336 β T xsen180 B φ 2 T xsen3832 062 909xsen8476 9090995 Wn xsen180 B φ 2 T 00682 α xsen180 B φ 2 p xsen8476 p 2599 WnTsen180 B φ 2 909 00682 xsen3832 P 1 1 5641 Td 00682 2599 Zero 1 1 1466 T 00682 P 1466 2 5641 Kc S Zo 1 g P0 S5 4 s 534 733j Kc S 1466 S5 4 g 5641 1 s 534 733j 10922 Compensador Gcs Gcs GCs 10922 S 5641 1 s 1466 s 5 4 10922s 616110 s3 4S2 1466s 5864s 10922s 616110 S3 4s2 733s Malha fechada 10922s 616110 S3 4s2 13252s 616110 Ku lim s0 GcsGs Ku lim s0 10922 s 5641 1 5 1466 8s 4 61611 5864 Ku 10506 Constante de erro estático de velocidade Ku para o sistema projetado g1s 4 S2S6 P 2 6 b Mp 20 e Tr Ta Para encontrar ξ desejado ξπ ln018 ξπ2 2940 1 ξ2 9859 ξ2 2940 2940 ξ2 9859 ξ2 2940 ξ2 2940 12799 ξ2 2940 ξ 04792 Tr 009 pencontrar Wn wn π arcosξ Tr1ξ2 wn 314 10710 2069 298440 0091 02296 009 07703 Com isso temos nosso polinômio desejado S2 2 ξ wn s w2 S2 2 04792 29844 s 2984402 S2 286024 s 890664 S 143012 261946j 261946 β 64849 φ 180 10758 11516 φ 4274 X 1430 6 2 83 241 10758 β 11516 T usen180 β Φ 2 T usen3612 wn usen180 β Φ 2 298440 usen7895 T 0589 292907 00201 α usen180 β Φ 2 usen7895 wn T usen180 β Φ 2 298440 00201 usen3612 α 09814 03536 2775 Pólo 1 Tα 1 00201 2775 17928 zero 1 T 49751 ρ Para encontrar kc kc s zo 4 s po s 2s 6 s 143012 261946j kc s po s2s6 s zo kc s 49751 s 2s 6 s 17928 4 1 33136 s 143012 261946 gcs 33136 s 17928 s 49751 s2s6 4 132544s 2376248 s³ 57751s² 4100086s 59701 Em malha fechada 132544s 2376248 s³ 57751s² 1735448s 2435949 gps 200 ss 10s 20 Tp 063s e Ts 1s Tp π 063 π wn wn π 063 wn 4981 Ts 4 ζwn 1 4 ζ4981 ζ 4 4981 ζ 0803 Polinômio desejado s² 2ζwns wn² s² 2 4981 0803s 4981² s² 7999s 24810 s₁ 3999 2968j Q2 gs 0005 ss 005 Mp 003 tr 18s ζ 035 wc 011 rads Φm 35 5 40 kp cosΦ gjwc kd senΦ wc gjwc Φ 16 lg gjwcdB 853 dB lg gjwc 037 cs kp kds sp 1 p 100 wc p 11 cs 2 1219s 1 683s 1 cs 26 675 5 1 eno 1 c0 zp 1 kp kp g0 kp2 c0 g0 cs s zs cs kp s zs kp kp2 s gs 001s 005s 007 requisito Mp 01 Tr 15 s ess 0 ζ 059 PM 59 66 65 ωc 018 cs k 1 Kd s Kp sp 1 s zs Gjωc 112 dB Kp 31 gjωc 144 Kd 98 PM 36 p 100 ωc 18 φ 65 144 180 29 z ωc 10 018 Q3 Ex 3 gs 1s 5s 10 s2 15s 50 requisitos Mp 01 tr 01 s ess 005 ζ 059 Tr π arccosζωn 1 ζ2 ωn ωc 27 rads PM 59 gjωc 159 gjωc dB 578 dB PM 159 180 21 PM 69 φ PM gjωc 180 φ 48 a 014 τ 0098 k dB 10 log a gjωc dB k 10 kdB20 29045 c1s k τ s 1 a τ s 1 29045 0098 s 1 0013 s 1 kp c10 g0 58 ess 11 kp 11 58 015 ess 15 ess2 ess 5 c20 zp z c20 p z 0052 p 11 kp2 002 ess2 kp2 323 tilibra Q 4 gs k s 1 s 2 s 6 PWp 10 ζ 05012 φm 23 147412ζ2 φm 0922 rads ωm 586 Calculando tempo de picos ζp 01 tp 1 ωd ωn 1ζ2 ωd 080 rad s PL kω 30n1 kω lim s0 gs kω lim s0 k s 1s s 2 s6 kω o k72 k 12 30 360 gs 360 s1 s s2 s 6 φ 211 ωp 185 rads ωg wcg Es k c s 1as 1 0 a 1 φ max 586 211 0 455 a 1 sen φ max a 0163 p φ max 455 1 sen φ max 1 20 log10 k 20 log10 k2 20 log k605 1τ 1α C jωmax 1 γa 2444 rad c jωmax 776 dB C jωn dB gjωn dB 0 dB ωn 1at ωn 101673 1 ωn 294 rad Cs 00832 s 10013 s 1 T 00832 Wn 2970 tilibra Q5 Matriz controlável a 1 o 14 s14 50 x x x 14 x 1 u 14s 1 o 14 5 7 z y1 4 x b 55s1 5s5 2 2 25 s 5 25 s 50 x1o x 1 u S10 y 200 x1 25 u c 2s1 x1 0 1 x1 0 u s2 3s2 2 3 x2 1 y1 1 2 x1 0 u d 53 s3 ss22s2 s3 2s 2 2s x1 0 1 0 0 5 1 x1 0 0 2 2 x3 1 u y 3 1 0 x1 0 u e 510s2 5 25 s2 s3 s2 5 36 Q6 0 0 0 2 1 0 50 xt 1 ut 0 1 15 yt 0 0 1 realimentação autovalores em 8 6j e 30 U K x A BK 0 0 0 2k1 2k2 2k3 k1 k2 k3 2 K1 2 k2 2 k3 0 0 0 sI A 2 k1 2 k2 2 k3 1 k1 k2 50 k3 0 15 det sI A s2 2k1 k2 15s2 30 k1 17 k2 k3 50s 30 k2 2k3 100 k1 Pol desejado s8 6j s 8 6j s 50 s2 16s 100 5 30 s3 16s2 100s 30s2 480s 3000 s3 46s2 580s 3000 Comparando 2 k1 k2 15 46 30k1 17 k2 k3 50 580 100 k1 30 k2 2 k3 3000 ajustando 2 k1 k2 31 30k1 17 k2 k3 530 100 k1 30 k2 2 k3 3000 k1 k2 k3 43 55 175 c Erro nulo em regime p entrada ao degrau w 1 A BK 86 110 350 42 55 225 0 1 15 C A BK 1 B A BK 1 cof A BK T det A BK 265 1425 0 115 115 1 100 0 15 C11 C21 C31 2 2C13 C23 C12 C22 C32 0 0 1 C A BK1 B C cof A BK T B 2C13 C23 det A BK det A BK C13 1 13 42 55 42 0 1 C23 1 23 86 110 86 0 1 det A BK 3000 n 1 3000 1500 242 86 2 3000 observador autovalores S12 4 8 6 j 32 24 j s 30 Pd0s s3 94s2 3520 s 48000 det sI A LC s3 l5 l3s2 L1 50 L2s L1 L2 L3 0 0 1 0 0 21 0 0 L2 0 0 L3 sI A LC 5 0 L1 7 5 50 L2 0 1 s l5 L3 Comparando 15 L3 94 L3 79 50 L2 3520 L2 3470 L1 4800 To modo co cancelar o azoro com autovalores em 6 8j xt 0 1 xt 0 ut yt 1 0 10 xt 50 15 10 gs C sI A1 B 100 s1 s2 15s 50 Polo desejado s 6 6j s 6 6j s 1 s3 13s2 112s 100 Matriz aumentada Aa A 0 Ba B C 0 0 Ac 0 1 0 Bc 0 50 15 0 10 10 10 0 0 detsI Aa Ba Ka Pd Aa Ba Ka 0 1 0 0 k1 k2 k3 50 15 0 10 10 10 0 0 0 0 0 0 5010k1 1510k2 10k3 s 0 5010k1 1510k2 10k3 10 10 0 10 10 0 det s3 15 10k2 s2 k1 50 100k3 s 100k3 15 10k2 13 k1 50 100k3 112 100k3 100 ka 162 02 4 k1 k2 k3 a determinar matriz de ganhos do observador xt 0 1 0 xt 0 ut yt 2 4 0 xt 0 0 0 0 0 8 5 5 4 Observável ca 2 4 0 kern CA2 0 2 4 CA3 32 20 14 c observável pd Ob detsI A LC pcalculadora s 1 0 L1 2 4 0 0 5 1 2L1 4L1 0 s2L1 1 4L1 0 8 5 s3 2L2 4L2 0 54L2 1 22 3 4L2 0 82L3 54L3 s3 s 2L1s2 4L2s 3 L1 4L18 2L3 5 4L3s 2L1 5 31 4L12L2 s2 4L2 2L1 3s3 8 2L3 32L1 8L3 5s 10L1 4L3 s 8L1 5 4L1 s 3 12L12L2 s3 s2 4L2 2L1 3 s6L1 10L2 4L3 5 42L1 6L2 L32 8 APP 4 4 L2 4L2 2L1 3 21 2L1 4L2 18 6L1 10L2 4L3 5 108 6L1 10L2 4L3 103 42L1 6L2 2L3 8 270 42L1 6L2 2L3 262 L1 698 L2 1 L3 1275