Estabilidade Robusta

Esta an´alise considera o seguinte fato: mesmo garantindo a estabilidade em malha fechada do Modelo Linear, tem-se que a estabilidade n˜ao est´a para o sistema do mundo real G0_{, devido a dinˆamicas de alta freq¨uˆencia que s˜ao desprezadas.}

Apˆendice C

Recupera¸c˜ao do Ganho de Malha

na Entrada

Neste apˆendice apresenta-se o desenvolvimento alg´ebrico para ajustar o ganho do observador para recupera¸c˜ao da malha de controle LQR. Este ajuste ´e feito por meio de manipula¸c˜oes das matrizes de covariˆancia do ru´ıdos. Segundo Lewis (Lewis and Syrmos 1995) e Kwakernaak (Kwakernaak and Sivan 1972), mostra-se que sob as hip´oteses: v → 0 e Σ → 0, obt´em-se para as Equa¸c˜oes (5.31) e (5.32), ΣCT(v2Θo)−1CΣ → BBT, (C.1) e L → ΣCT_(v2_Θ o)−1, (C.2) mas L(v2_Θ o)LT = ΣCT(v2Θo)−1(v2Θo)[ΣCT(v2Θo)−1]T, = ΣCT(v2Θo)−1(v2Θo)(v2Θo)−1CΣ, = ΣCT_(v2_Θ o)−1CΣ. (C.3)

Logo, das Equa¸c˜oes (C.1) e (C.3):

L(v2_Θ

o)LT → BBT, (C.4)

APˆENDICE C. RECUPERAC¸ ˜AO DO GANHO DE MALHA NA ENTRADA 98 A solu¸c˜ao, L → 1 vBU Θ −1 2 o . (C.5)

sendo U uma matriz unit´aria. Definindo-se a matriz de realimenta¸c˜ao do compensador,

Φc(s) = [sI − (A − BK)]−1, (C.6)

o ganho de malha aberta na entrada,

Lr(s) = F (s)G(s) = K[sI − (A − BK − LC)]−1LCΦB,

= K[sI − (A − BK) + LC]−1_LCΦB,

= K[Φ−1_c + LC]LCΦB. (C.7)

Usando o lema da inversa, como feito anteriormente, Equa¸c˜ao (5.17), na Equa¸c˜ao (C.7), Lr(s) = K[Φc− ΦcL(I + CΦcL)−1CΦc]LCΦB, = KΦc[I − L(I + CΦcL)−1CΦc]LCΦB, = KΦc[L − L(I + CΦcL)−1CΦcL]CΦB, Fatorando [I + CΦcL]−1, Lr(s) = KΦcL[I − (I + CΦcL)−1CΦcL]CΦB, = KΦcL(I + CΦcL)−1[(I + CΦcL) − CΦcL]CΦB, = KΦcL(I + CΦcL)−1CΦB. (C.8)

Calculando-se L(I + CΦcL)−1, quando L → 1_vBU Θ−

1 2

APˆENDICE C. RECUPERAC¸ ˜AO DO GANHO DE MALHA NA ENTRADA 99 L(I + CΦcL)−1 → 1 vBU Θ −1 2 o (I + CΦc 1 vBU Θ −1 2 o )−1, → 1 vBU Θ −1 2 o (vI + CΦcBU Θ −1 2 o )−1v, → BU Θ−12 o (vI + CΦcBU Θ− 1 2 o )−1 | {z } v→0 , → BU Θ−12 o (CΦcBU Θ −1 2 o )−1, → BU Θ−12 o (UΘ− 1 2 o )−1(CΦcB)−1, L(I + CΦcL)−1 → B(CΦcB)−1. (C.9)

Assim, das Equa¸c˜oes (C.8) e (C.9),

Lr(s) → KΦcB(CΦcB)−1CΦB. (C.10)

Como

Φc= (Φ−1+ BK)−1, (C.11)

usando-se novamente o lema da inversa, na Equa¸c˜ao (C.11),

Φc = Φ − ΦB(KΦB + I)−1KΦ, = Φ[I − B(KΦB + I)−1KΦ]. Ent˜ao, ΦcB = ΦB[I − (KΦB + I)−1KΦB], (C.12) CΦcB = CΦB[I − (KΦB + I)−1KΦB], a matriz inversa de CΦcB, (CΦcB)−1 = [I − (KΦB + I)−1KΦB]−1(CΦB)−1. (C.13)

Substituindo as Equa¸c˜oes (C.12) e (C.13) na Equa¸c˜ao (C.10),

Referˆencias Bibliogr´aficas

Bailey, G.V.; J.M. (1990). LQG/LTR robust control system design for low- pressure feedwater heater train Murphy. Southeastcon ’90, Proceedings, IEEE Vol.3, 853 – 856.

B¨ack, Thomas, David B. Fogel and Zbigniew Michalewicz (1997). Handbook of

Evolutionary Computation. first ed.. Institute of Physics Publishing and Ox-

ford University press. Bristol and New York.

Bottura, C. P. and J.V. da Fonseca Neto (1999). Parallel genetic algorithm fitness function team for eigenstructure assignment via lqr designs. Congress on

Evolutionary Computation - CEC99. Washington, DC, USA. Vol 2, 1035–

1042.

Brito Filho, Joaquim Gomes, Jo˜ao Viana da Fonseca Neto, Marlon P. de Melo Wolf, Carlos C. Teixeira and Ivanildo Abreu (2006). A Genetic Algorithm for Eigenstructure Assignment via LQR Design - Simulations and Analysis. Relat´orio T´ecnico PPPGE - UFMA - DE.EE. pp. 1 – 8.

Cruz, Jos´e Jaime da (1996). Controle Robusto Multivari´avel. EDUSP.

Davies, R. and T. Clarke (1995). Parallel implementation of a genetic algorithm.

IEEE. 3(1), 1–9.

Doyle, J. C. and G. Stein (1979). Robustness with Observers. IEEE Transactions

on Automatic Control 24(4), 607–611.

Doyle, J. C. and G. Stein (1981). Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis. IEEE Transactions on Automatic Control 26, 4– 16.

REFERˆENCIAS BIBLIOGR ´AFICAS 101

Ferreira, Carlos C. T. e Fonseca Neto, J. V. (2003). Projeto do Filtro de Kalman via Aloca¸c˜ao de Autoestrutura e Algoritmo Gen´etico. VI Simp´osio Brasileiro

de Automa¸c˜ao Inteligente.

Ferreira e Fonseca, Carlos C. T., J. V. Fonseca Neto and F. A. Torrico (2003). Aloca¸c˜ao de Autoestrutura via Controle LQG/LTR e Computa¸c˜ao Evolutiva.

VI Simp´osio Brasileiro de Automa¸c˜ao Inteligente.

Fonseca Neto, Jo˜ao Viana da (2000). Aloca¸c˜ao Computacional Inteligente de Au- toestruturas para Controle Multivari´avel. PhD thesis. UNICAMP.

Fonseca Neto, Jo˜ao Viana da (2003). An Inequalities Method for Multivariable

System’s Eigenstructure Assignment via Genetic Multi-Objective Optimiza- tion. 2003 American Control Conference. Denver-Colorado.

Gasem, C. Zhou; J.R. Whiteley; E.A. Misawa; K.A.M. (1995). Application of en- hanced LQG/LTR for distillation control. Control Systems Magazine, IEEE 15(4), 56 – 63.

Goldberg, David Edward (1989). Genetic Algorithms in search, Optimization, and

Machine Learning. Addison-Wesley Publishing Company Inc.. USA.

Holland, J.H. (1975). Adaptation in Natural and Artificial Systems. University of Michigan Press . Ann Arbor-Michigan-USA.

Jafar, Zarei, Montazeri Allahyar, Motlagh Mohmmad Reza Jahed and Javad Posh- tan (2006). Design and comparison of LQG/LTR and Hinf controllers for a VSTOL Flight control System. IEEE pp. 1–18.

Johnson, Prof.M.J Grimble; M.A (1987). Recent trends in linear optimal quadratic multivariable control system design. IEEE.

Jos´e Paulo F., Garcia, Bortolo Edson S., Ribeiro Jean Marcos S. and Garcia Lizete Maria C. F. (2004). Active Noise attenuation Using LQG/LTR Con- trol. Eletrˆonica de Potˆencia 9(2), 23–26.

Keller, J.P. (2005). Interactive Control System Design. Control Eneginering and

REFERˆENCIAS BIBLIOGR ´AFICAS 102

Kishor, Nand, Saini R.P. and Dr.S.P.Singh (2004). LQG/LTR Controller for Speed Governing of Hydro-Turbine. IEEE. pp. 1125–1128.

Kwakernaak, H. (1969). Optimal Low-Sensitivity Linear Feedback Systems. Au-

tomatic.

Kwakernaak, H. and R. Sivan (1972). Linear Optimal Control Systems. Wiley- Interscience. New York.

Lahdhiri, T.; Alouani, A.T. (1993). LQG/LTR pitch attitude control of an earth- orbiting spacecraft. Decision and Control, 1993., Proceedings of the 32nd

IEEE Conference on vol.1, Pages:445 – 446.

Lee, Myoen-Song Choi; Jeong-Woo Lee; K.Y. (1996). An auxiliary LQG/LTR robust controller design for cogeneration plants Young-Moon Park. Energy

Conversion, IEEE Transactions on 1(2), 407 – 413.

Lewis, Frank L. and Vassilis L. Syrmos (1995). Optimal Control. John Wiley and Sons, Inc.. USA.

Liu, G.P. and R.J Patton (1998). Eigenstructure Assignment for Control System

Design. John Willey & Sons.

Melo Wolf, Marlon P. de, Jo˜ao Viana da Fonseca Neto, Carlos C. Teixeira, Ivanildo Abreu and Joaquim Gomes Brito Filho (2006). A Genetic Algorithm for Eigenstructure Assignment via LQR Design - Theoretical Development. Re-

lat´orio T´ecnico PPPGE - UFMA - DE.EE. pp. 1–7.

M.Maciejowski, J. (1989). Multivariable Feedback Design. Addison - Wiley. Paschall, J.M.; D.B. Ridgely; R.N. (1994). Autopilot Design for a Tail-Controlled

Missile Using LQG/LTR With Eigenstructure Reassignment Brown. Ameri-

can Control Conference Vol.3, 3278 – 3282.

Rab-Alibeik, H.; Setayeshi, S. (2003). Improved temperature control of a PWR nuclear reactor using an LQG/LTR based controller. Nuclear Science, IEEE

REFERˆENCIAS BIBLIOGR ´AFICAS 103

Weerasooriya, S.; Phan, D.T. (1995). Discrete-time LQG/LTR design and mod- eling of a disk drive actuator tracking servo system. Industrial Electronics,

IEEE Transactions on Volume: 42(Issue: 3), Pages:240 – 247.

Wu, Ying-Yu Tzou. Hsiang-Jui (1995). LQG/LTR control of an AC induction servo drive. IEEE Transactions on 10, 214 – 221.