H Controller Design for the Lateral Dynamics of a Boeing 747 200 Edición Única

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(1)INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY DIVISION OF ENGINEERING AND ARCHITECTURE GRADUATE PROGRAM IN ENGINEERING. H∞ CONTROLLER DESIGN FOR THE LATERAL DYNAMICS OF A BOEING 747-200 BY. MANUEL GIACOMÁN ZARZAR IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF. MASTER OF SCIENCE IN AUTOMATION. MONTERREY, NUEVO LEÓN, MÉXICO. DECEMBER 2006.

(2) c Copyright by Manuel Giacomán Zarzar, 2006 All Rights reserved. ii.

(3) INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY. Thesis committee members recommend the present Thesis of Ing. Manuel Giacomán Zarzar to be accepted in partial fulfillment of the requirements for the academic degree Master of Science in AUTOMATION. Comité:. ————————————————– Dr. Ricardo Ramirez Supervisor ITESM, CAMPUS MONTERREY. ————————————————– Prof. Peter J. Fleming Cosupervisor UNIVERSITY OF SHEFFIELD. ————————————————– Dr. José de Jesús Rodriguez Examiner ITESM, CAMPUS MONTERREY. ————————————————– Dr. Arturo Galván Examiner ITESM, CAMPUS MONTERREY. ————————————————– Dr. Francisco Angel Bello Chairman of the Graduate Master Programs ITESM, CAMPUS MONTERREY. MONTERREY, NUEVO LEÓN, MÉXICO. iii. DECEMBER 2006.

(4) Acknowledgements. I would like to thank to all the people who supports me during my stay in Mexico and in Sheffield. To my supervisor Dr. Ricardo Ramirez Mendoza, for his invaluable support and encouraging me in this research. To my co-supervisor Prof. Peter J. Fleming and all the University Technology Centre (UTC) group, for giving me the privilege to work with tem. To Dr. Jose de Jesus Rodriguez and Dr. Arturo Galvan, for their comments and participation as examiners. To Dr. Robert Harrison, Alex Shenfield and specially to Dr. Arturo Molina-Cristobal and Dr. Ian Griffin for their technical advice. To my teachers, for sharing with me their knowledge. To my friends and classmates for building a comfortable environment and making this work an enjoyable experience. To Teresita del Castillo. Thanks to her support and motivation I was able to participate in different international exchange programs.. iv.

(5) Dedication To God for guiding, protecting and being with me in each moment of my life. To my parents Manuel Giacomán Murra and Marı́a Clara Zarzar Charur for inculcating me the values and growing the person that I am today. Your love and support always gave me the willing to do all my projects. To my sister Elizabeth for being always present at my side. To my little sister Patricia, for fullfilling me with joy. To my grandmother City; thanks to her blessings I was able to reach my goals and maintain the right path.. A Dios por guiarme, protegerme y estar conmigo en cada momento de mi vida. A mis Papas Manuel Giacomán Murra y Marı́a Clara Zarzar Charur por inculcarme los valores y formar la persona que soy el dia de hoy. Su amor y soporte fueron los pilares que me proporcionaron ánimo para realizar todos mis proyectos. A mi hermana Elizabeth por estar siempre presente a mi lado A mi hermanita Patricia, por emanar y llenarme siempre de alegrı́a. A mi abuelita City, ya que gracias a sus bendiciones he podido alcanzar mis metas y mantenerme en el correcto camino. Gracias por creer en mi Con cariño Manuel. v.

(6) Abstract. The airplane dynamics is by nature a nonlinear process, which can be represented as a rigid body with some translational and angular velocities and forces and moments acting on it. The control system for the flight dynamics primary consists on sensors which provide the information about the aircraft condition and atmospheric data; then this information is used to provide the commands for the control surfaces and engines, and feedback is incorporated to assure a good tracking behavior and to reject disturbances such as wind gust and turbulence. The main function of the control systems is to add safety and facility such that the flight missions can be accomplished. In this document it is presented the control of the lateral dynamics for a civil aircraft. The main concern in commercial aircraft is the performance of the airplane, that is, the capacity to reject the disturbances and the ability to auto-stabilize to improve the quality of the flight. Different techniques have been studied and applied to the control of the lateral dynamics, some of them are nonlinear controller, LQG controller, H∞ controller and the mixed H2 /H∞ approach. In this work the attention is focussed in the design of a robust H∞ controller based on the Mixed Sensitivity and Loop Shaping approach. It is also considered the application of Linear Matrix Inequalities and Multiobjective Genetic Algorithms (MOGA) as an optimization tool applied to the problem of H∞ mixed sensitivity and loop shaping procedures. At the end the differences between these approaches are detailed.. vi.

(7) Contents 1 Introduction 1.1 Motivation of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background 2.1 Introduction . . . . . . . . . . . . . . . . . . 2.2 Flight Dynamics . . . . . . . . . . . . . . . . 2.2.1 Linearization . . . . . . . . . . . . . 2.2.2 Aircraft Equations of Motion . . . . . 2.2.3 Performance Parameters . . . . . . . 2.3 Multivariable Systems . . . . . . . . . . . . 2.3.1 Internal Stability . . . . . . . . . . . 2.3.2 Multivariable System Gain . . . . . . 2.4 Robustness . . . . . . . . . . . . . . . . . . 2.4.1 Uncertainty . . . . . . . . . . . . . . 2.4.2 SISO Robustness . . . . . . . . . . . 2.4.3 Multivariable Robustness . . . . . . . 2.5 H Infinity Controller . . . . . . . . . . . . . 2.5.1 H Infinity Norm . . . . . . . . . . . 2.5.2 General Control Problem Formulation 2.5.3 Mixed Sensitivity Design Controller . 2.5.4 H Infinity Loop Shaping . . . . . . . 2.5.5 Applications of H Infinity Controllers 2.6 Optimization . . . . . . . . . . . . . . . . . 2.6.1 Linear Matrix Inequalities . . . . . . 2.6.2 Multiobjective Optimization . . . . . 3 Optimal Controller 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Linear Quadratic Controller . . . . . . . . 3.2.1 LQR Design for a Boeing 747-100 3.3 Linear Quadratic Gaussian Controller . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. vii. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . .. 2 2 3 3. . . . . . . . . . . . . . . . . . . . . .. 5 5 5 6 8 12 15 16 17 19 19 21 25 26 27 27 28 30 32 32 32 34. . . . .. 38 38 38 39 44.

(8) 3.4. 3.3.1 LQG Design for a Boeing 747-100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference Tracking Optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Aircraft Lateral Dynamics Control Through the Design of an H infinity Controller 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Poles and Zeros, Controllability and Observability . . . . . . . . . . . . 4.3.2 Bode plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Singular Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Procedure to Implement an H Infinity Controller . . . . . . . . . . . . . . . . . . 4.5 Mixed Sensitivity Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Selection the Weighting Matrices W1 and W2 . . . . . . . . . . . . . . . 4.5.2 Selecting the Weighting Matrices W1 and W3 . . . . . . . . . . . . . . . 4.5.3 Mixed Sensitivity Controller by LMI Techniques . . . . . . . . . . . . . 4.6 H Infinity Loop Shaping Controller . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 First design without scaling . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Second design without scaling and using the derivative kick configuration 4.6.3 Third design with scaling and using the derivative kick configuration . . . 4.7 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46 48. . . . . . . . . . . . . . . . . .. 50 50 50 52 52 52 54 56 56 57 58 59 60 61 61 62 62 66. . . . . . .. 68 68 68 70 76 77 79. . . . .. 81 81 81 82 82. A Geometric, Mass and Aerodynamics characteristics of Airplanes A.1 Boeing 747-100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Boeing 747-200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Lateral Directional Derivatives and Stability Coefficients . . . . . . . . . . . . . . . . . . . . . . .. 87 87 88 90. B Turbulence Models. 91. C Small disturbance Theory. 95. 5 Optimization 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Optimization Process of H infinity Controller with MOGA . . . 5.3 Optimization of H infinity Loop Shaping Controller with MOGA 5.4 Incorporation of an Integrator to the Mixed Sensitivity Design . 5.5 Design of Mixed H2/Hinf Synthesis aided by MOGA . . . . . . 5.6 Comparison between Mixed Sensitivity and Loop Shaping . . . 6 Conclusions and Future Work 6.1 Optimal Controller . . . . 6.2 H Infinity Controller . . . 6.3 Optimization . . . . . . . 6.4 Future Work . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. viii. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. . . . . . .. . . . ..

(9) D Additional Performance Specifications. 99. E Controllability and Observability 101 E.1 Controllability and Observability Analysis for a Boeing 747-100 . . . . . . . . . . . . . . . . . . . 101 E.2 Controllability and Observability Analysis for a Boeing 747-200 . . . . . . . . . . . . . . . . . . . 102. ix.

(10) List of Tables 2.1 2.2 2.3 2.4 2.5. Nomenclature of forces, moments and velocities . Variation of lateral modes with speed and altitude Spiral mode flying qualities . . . . . . . . . . . . Roll mode (maximum time constant in seconds) . Dutch roll flying qualities . . . . . . . . . . . . .. . . . . .. 6 7 13 13 14. 3.1. Comparison between the open and closed loop response . . . . . . . . . . . . . . . . . . . . . . . .. 43. 4.1 4.2 4.3. Comparison of performances to a step response of 0.1 rad in the aileron. . . . . . . . . . . . . . . . Comparison of performance to different disturbances at the input. . . . . . . . . . . . . . . . . . . Comparison of robustness between mixed sensitivity and loop shaping . . . . . . . . . . . . . . . .. 67 67 67. 5.1 5.2 5.3 5.4. Comparison of performances to a step response of 0.1 rad in the aileron. . . . . . . . Comparison of performance to different disturbances at the input. . . . . . . . . . . Comparison of robustness between loop shaping and the design assisted with MOGA A comparative study between the mixed sensitivity and loop shaping synthesis . . . .. . . . .. 74 75 75 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90. A.1 Lateral stability coefficients. x. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . ..

(11) List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24. Definition of forces, moments and velocities in a reference frame, . . . . . . . . . . . . . . Types of solutions of the lateral dynamics equation . . . . . . . . . . . . . . . . . . . . . . Sideslip and roll angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aileron and rudder deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sideslip motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roll motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dutch roll motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular values of the open loop transfer function, to ensure noise and disturbance rejection. Coprime factor uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyquist diagram showing nominal stability . . . . . . . . . . . . . . . . . . . . . . . . . . Nyquist plot illustrating the nominal performance requirement . . . . . . . . . . . . . . . . Nyquist diagram showing the condition for Robust Stability . . . . . . . . . . . . . . . . . . Nyquist plot for robust performance under multiplicative uncertainty . . . . . . . . . . . . . General control configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal based representation of the mixed sensitivity problem . . . . . . . . . . . . . . . . . Mixed sensitivity in the general configuration . . . . . . . . . . . . . . . . . . . . . . . . . Left normalized coprime factorization of the plant . . . . . . . . . . . . . . . . . . . . . . . The shaped plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An implementation to avoid derivative kick . . . . . . . . . . . . . . . . . . . . . . . . . . a) A convex set. b) A Nonconvex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of GA process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto Optimal Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. 5 7 9 9 10 11 11 15 18 21 22 23 24 24 27 28 28 30 31 32 33 35 36 37. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8. LQ control system diagram . . . . . . . . . . . . . . . . . . Synthesis curves for Q . . . . . . . . . . . . . . . . . . . . Synthesis curves for R . . . . . . . . . . . . . . . . . . . . Response of the system to a disturbance input . . . . . . . . Manipulation signals send to the aileron and rudder actuators FFT of the control signals . . . . . . . . . . . . . . . . . . . LQG diagram . . . . . . . . . . . . . . . . . . . . . . . . . Response of the system in the presence of turbulence . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 39 41 41 43 44 44 45 47. xi. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . ..

(12) 3.9 Sensitivity Function of the LQG System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Block diagram of a reference tracking system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48 49. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18. Frequency response to aileron input . . . . . . . . . . . . . . . . . Frequency response to rudder input . . . . . . . . . . . . . . . . . . Singular values of the system . . . . . . . . . . . . . . . . . . . . . Bode plots of the Dryden filters . . . . . . . . . . . . . . . . . . . . Singular values indicating the performance of the system . . . . . . Response of the system to a step input of 5.729 in the aileron . . . . Response of the system to a step input of 5.729 in the aileron . . . . Response of the system at an altitude of 20,000 ft . . . . . . . . . . Response of the system at a see level altitude . . . . . . . . . . . . Singular values of PI plus gain weighted plant . . . . . . . . . . . . Singular values behavior during the design procedure . . . . . . . . Step response to aileron deflection and in the presence of turbulence Step response to aileron deflection and in the presence of turbulence Singular values behavior during the design procedure . . . . . . . . Step response to aileron deflection and in the presence of turbulence Performance and robustness indicators . . . . . . . . . . . . . . . . Step response to aileron deflection at an altitude of 20 000 ft . . . . Step response to aileron deflection at sea level . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 53 54 55 57 58 59 60 60 61 62 63 63 63 64 65 65 66 66. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15. Flow chart of the H infinity optimization process . . . . . . . . . . . . . . . . . . . . . . Step response to aileron deflection of 0.1 rad . . . . . . . . . . . . . . . . . . . . . . . . . Trade off graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trade off graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trade off graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trade off graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step response to aileron deflection with an altitude of 40,000 ft and M=0.9(nominal value) Step response to aileron deflection with an altitude of 20,000 ft and M=0.65 . . . . . . . . Step response to aileron deflection at sea level and M=0.25 . . . . . . . . . . . . . . . . . Performance of the system with multiplicative uncertainty . . . . . . . . . . . . . . . . . Mixed Sensitivity Configuration with an Integrator . . . . . . . . . . . . . . . . . . . . . Response of the systems to a step change of 0.1 rad in the aileron . . . . . . . . . . . . . . Response of the system to a disturbance input with magnitude of 0.01 . . . . . . . . . . . Multi-objective Plant Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Pareto Front Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 69 71 72 72 73 73 73 74 74 75 76 77 77 78 79. B.1 Spectra functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Forming filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Turbulence intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 93 94. xii. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ..

(13) 1. Nomenclature. p q r u,v,w L M N β φ δa δr b Hβ , Hp , Hr r y d u n σ σ̄ σ G K S T FFT RHP RGA κ SVD LQ LQR LQG H∞ H2 LMI MOGA. Roll rate (rad/s) Pitch rate (rad/s) Yaw rate (rad/s) Velocity components (m/s) Roll Moment Pitch Moment Yaw Moment Sideslip angle Roll angle Aileron deflection Rudder deflection Wingspan Dryden filters Reference signal System Outputs Disturbances Control signal Noise Singular Value Maximum Singular Value Minimum Singular Value Plant or System Controller Sensitivity Function Complementary Sensitivity Function Fast Fourier Transform Right Half Plane Relative Gain Array Condition number Singular Value Decomposition Linear Quadratic Control Linear Quadratic Regulator Linear Quadratic Gaussian Control Infinity norm Two norm Linear Matrix Inequality Multi-objective Genetic Algorithm.

(14) Chapter 1. Introduction 1.1 Motivation of the Research In the last years the aerospace industry in Mexico has grown at a constant rate and is expected to increase even more. Before, the aerospace industry in Mexico has concerned only with the manufacture of parts for the airplanes with some clusters situated in Sonora, Guaymas and Monterrey. However the aerospace industry is moving forward to the design and investigation, some companies already established are GE, Global Vantage, ICKTAR, Volare Engineering, ITR and Competitive Global. A great impulse to this sector has been brought by Bombardier, when decided to install a Manufacturing and Training center in Queretaro. This center is expected to produce commercial airplanes in five year from January 2006. Aircrafts demand highly reliable controllers for each of its parts: the jet engines, the aircraft lateral and longitudinal dynamics, the landing gear and braking systems, etc. Each of these subsystems incorporate many variables making the control of them a multivariable system. It is very common that in multivariable systems some of the established objectives are in conflict for example the rise time and overshoot or the performance and robustness. Moreover these objectives are also limited by constrains imposed by the actuators which can only handle certain range. Not been enough, multivariable systems introduces interaction making the control of the outputs more challenging. Different techniques such as H2 and H∞ have been well studied and provide very good results. Two problems have been found with these techniques: 1) They focuss principally on robustness H∞ or performance H2 ; 2) The selection of the weighting functions is not a straight forward process. To overcome these difficulties, in the last decade it has been proposed some multiobjective techniques like the mixed H2 /H∞ and the used of Genetic Algorithms as an approach to look for the optimal performance for a set of objectives. These techniques have probed to be efficient and are still under research. In this research it is considered the application of Linear Matrix Inequalities and Multiobjective Genetic Algorithms (MOGA), developed by Fonseca and Fleming [Fonseca and Fleming, 1993], as an optimization tool applied to the problem of H∞ mixed sensitivity and loop shaping procedures. At the end the differences between these. 2.

(15) Introduction. 3. approaches are detailed.. 1.2 Outline of the Thesis Chapter 2 provides the necessary background in which the thesis is based. First the lateral dynamics of an airplane, the derivation of the linear equation, the modes in which the lateral mode can be split and the performance or handling qualities are introduced. The airplane dynamics is explained through a multivariable system and in many cases, as illustrated in chapter 5, it presents trade off among some performances; this trade off can be known forehead by analyzing the properties of multivariable systems. The chapter also review the concepts of robustness and the design process of an H∞ controller via mixed sensitivity and loop shaping. The optimization is discussed with the incorporation of Genetic Algorithms and Linear Matrix Inequalities. Chapter 3 considers the problem of a Linear Quadratic Controller. It demonstrates through an example how do the poles of the closed loop system can be chosen by selecting the appropriate matrices Q and R that penalize the state vector and control signals. Next, an LQG controller is obtained for the same system and it is explained the main advantages and disadvantages with respect to the LGR. Finally the LQR design is modified for a tracking system, in which case, the number of adjusting parameters is increased with respect to the LQR. Chapter 4 explains the main problem which consists on the control of an aircraft lateral dynamics, specifically for a Boeing 747-200. Before the design of a controller is carried out, the system is analyzed to get the necessary information about its condition and the possible difficulties that could arise. Thereafter, an H∞ controller is designed with the mixed sensitivity approach, first by solving 2 Riccati equations and comparing the response with the solution achieved by using Linear Matrix Inequalities Techniques. Also an H∞ Loop Shaping Controller is proposed confirming its simplicity. The end of the chapter summarizes the efficiency of both types of controllers and the differences between them. Chapter 5 considers the same controller design presented in chapter 4, but an optimization is performed over them. The multiobjective genetic algorithm approach is used in which the performance and restrictions are set as the objectives and the program MOGA examines all the possible solutions, in a restricted space, that meet the requirements. At the end it is illustrated the improvements accomplished with MOGA with respect to those in chapter 4 and how the combination of LMIs and MOGA results in a powerful approach. The trade off among the established objectives is exposed in parallel coordinates plots. Finally it is discussed a comparison of the design methodologies carried out along this work. Chapter 6 presents the conclusion of the results undertaken an devises suggestions for future work.. 1.3 Contributions • A comparison between two of the most popular H∞ designs: the Mixed Sensitivity and the Loop Shaping Approach was carried out. The Loop Shaping Approach demonstrates more flexibility and simplicity since.

(16) 4 it requires just to choose 4 variables to obtain the desire controller and these variables lies in some specified range limited by the desire to shape the open loop singular values. On the other hand, the mixed sensitivity Approach demands the selection of 8 or more variables to shape the closed loop transfer functions, and the range of this variables is not seen straight forward as in the Loop Shaping Design Procedure. Moreover the mixed sensitivity has the disadvantage of not being possible to handle within the same design the introduction of an integrator and in some cases, this integrator produces a great overshoot when a step change in the reference is given. • Even though that the H∞ Loop Shaping Design does not allow to place directly the poles at the desire locations, it was shown how with the Multi-objective Genetic Algorithm (MOGA) one can set as an objective the region in which the poles should lie. It should be noted that in this case the pole placement is equivalent to specify as an objective the settling time of the response to a step input. • The Multi-objective Genetic Algorithm Method was extended to the design of the mixed H2 /H∞ synthesis, demonstrating that tighter bounds can be placed on both the H2 norm and the H∞ norm at the same time. • This work represents one of the first steps in the research of Flight Dynamics Control at ITESM. • There is a strong opportunity in Mexico to increase the research in aerospace control..

(17) Chapter 2. Background 2.1 Introduction This chapter is concerned to review the main concepts related to the design of a controller for an aircraft lateral dynamics. It begins with a description of the linearized equations of an airplane lateral dynamics, which completely describe it movements; these consist on spiral, roll and dutch roll modes. In section 2.3 multivariable systems are explained, aiming to review the necessary ideas to analyze the properties of the particular airplane. In section 2.4 the definition and restrictions to achieve robustness are introduced. This then leads to the design of a robust H∞ controller as explained in section 2.5. Finally Linear Matrix Inequalities and Multi-objective Optimization Genetic Algorithm are regarded as a complementary and helpful tool for the design of an H∞ optimal controller.. 2.2 Flight Dynamics The dynamics of an airplane can be split into two main motions: the longitudinal and the lateral. The longitudinal motion comprises the pitch and acceleration of an airplane while the lateral includes the roll and yaw movements. The airplane is represented as a rigid body with some translational and angular velocities and forces and moments acting on it. In Figure 2.1, [Nelson, 1998] and Table 2.1 it is shown the definition of forces, moments, moments of inertia and velocities in the body fixed coordinate.. Figure 2.1: Definition of forces, moments and velocities in a reference frame, 5.

(18) 6. Angular rates Velocity components Aerodynamic force components Aerodynamic moment components Moment of inertia about each axis. Roll axis p u X L Ix. Pitch axis q ν Y M Iy. Yaw axis r w Z N Iz. Table 2.1: Nomenclature of forces, moments and velocities. 2.2.1. Linearization. The airplane equations of motion are nonlinear; additionatlly, the lateral and longitudinal dynamics are cross coupled, which complicate the analysis of the airplane. However, a good approximation can be obtained when using the linearized model, which implies that the lateral and longitudinal dynamics can be studied independently. In the rest of the document, the study of the lateral stability and control is presented.. Assumptions For military aircrafts, the linearization process gives a poor result, because these aircrafts are submitted to large angles and very complex maneuvers; furthermore, they have been designed to be unstable. For the case of commercial aircrafts, the linearization process offers very good results, because they meet the following assumptions needed to use the linear equations. • The lateral and longitudinal dynamics are decoupled. • Small changes of angles (∠ < 15). • Moment of inertia about the ”xz” plane Ixz = 0. • Constant coefficients. – Constant speed. – Constant atmospheric parameters. • Rigid model (no airplane deformations). • Symmetry about the x, z plane. See figure 2.1 The former assumptions were made because as it is shown in Table 2.2, [Bernard Etkin, 1996], the spiral mode, roll mode and dutch roll mode (which compose the lateral dynamics) vary with the altitude and the aircraft’s velocity. From table 2.2: • Period T =. 2π ω. τ • thalf = 1.442 is the time that must be elapsed during which any disturbance quantity will double or halve itself, Figure 2.2..

(19) Background. 7. Altitude (ft) 0 0 20000 20000 20000 40000 40000 40000. Mach No. 0.45 0.65 0.5 0.65 0.8 0.7 0.8 0.9. Spiral Mode thalf(s) 35.7 34.1 76.7 64.2 67.3 -296 94.9 -89.2. Rolling convergence thalf(s) 0.56 0.44 0.94 0.76 0.85 1.5 1.23 1.45. Lateral oscillation (Dutch roll) Period (s) Nhalf(cycles) 5.98 0.87 4.54 0.71 7.3 1.58 5.89 1.33 4.82 1.12 7.99 1.93 6.64 3.15 6.19 1.18. Table 2.2: Variation of lateral modes with speed and altitude. Figure 2.2: Types of solutions of the lateral dynamics equation • Nhalf = is the number of cycles that it takes to the perturbation to half or double. When the modes (roots) are real the only parameter is thalf . When the modes are oscillatory: thalf. Nhalf. 0.693 ζωn p 0.11 1 − ζ 2 = |ζ|. or double. or double. =. (2.1). (2.2). If the characteristics roots of the airplane lateral dynamics are analyzed, it can be noted that there are 4 roots: two real, and two imaginary. The real roots are related to the spiral and roll mode, while the imaginary to the Dutch roll. If we look again to Table 1, we will see that thalf is closely related with the time constant or pole of the spiral and roll mode, and the Nhalf , thalf depend on the natural frequency and damping of the dutch roll mode. This will be deeply discussed later..

(20) 8. 2.2.2. Aircraft Equations of Motion. Based on the linearization assumptions and the small disturbance theory described in Appendix C, the following equation of motion for the lateral movement of the airplane is obtained.. ẋ = Ax + Bu + F w y. (2.3). = Cx. where matrices A,B,C,F and the input, output and state variables are defined as . Yβ u0. Yp u0.   Lβ A=   Nβ 0. Lp Np 1 .   B=  .   C=  . .   X= . β ρ r φ.     .   F =  u=. ". δa δr. #. Y. − 1− Lr Nr 0. 1 0 0 0. 0 0 1 0 Y. − up0 −Lp −Np 0. βw   w =  ρw  rw. g cos θ0 u0.      . 0 0 0. . Lδa Nδa 0. 0 1 0 0. . . Yδr u0. 0 Lδa Nδa 0. − uβ0 −Lβ −Nβ 0 . Yr u0.    . 0 0 0 1. (2.5).     . − Yur0 −Lr −Nr 0. (2.6).     . .   Y = . β = sideslip angle, Figure 2.3, [Nelson, 1998] ρ = roll rate, Figure 2.1 r = yaw rate, Figure 2.1 φ = roll angle, Figure 2.3 δa = aileron deflection, Figure 2.4, [Nelson, 1998] δr = rudder deflection, Figure 2.4 ∆βw = change in sideslip angle due to horizontal wind perturbances ∆ρw = roll rate perturbations due to vertical winds ∆rr = yaw rate changes due to wind disturbances. (2.4). (2.7). β ρ r φ.     .

(21) Background. 9. Figure 2.3: Sideslip and roll angle. Figure 2.4: Aileron and rudder deflection The coefficients of the matrices A, B and F are calculated from the lateral directional derivatives, which depend on the lateral stability coefficients; for the formulas, refer to appendix A. The lateral equations of motion of an airplane could be split into three simpler equations, which represent the airplane’s movements. From the state space equation stated above, the spiral, roll and dutch roll approximation equations are obtained.. Spiral Mode The spiral mode is characterized by changes in heading and direction of travel as illustrated in Figure 2.5, [Nelson, 1998]. For conventional airplanes, this root is real but is usually only slowly convergent or somewhat divergent; therefore, a slightly divergent spiral mode is acceptable provided that the time required to double amplitude is not too short. Since the lateral motion is uncoupled from the longitudinal motion only for small deviations from equilibrium, as the spiral deviation becomes large, longitudinal motion is also induced, [Phillips, 2004]. This is the reason why in.

(22) 10 this case of study, we are considering that only small changes in angles are presented.. Figure 2.5: Sideslip motion The heading angle ∆ψ = −∆β is equal to a negative change of the sideslip angle. Nevertheless, the sideslip angle will be used since the aerodynamics moments do not depend on the heading angle. In order to obtain an approximation of the spiral mode, the side force equation and roll angle are neglected from equation 2.3, [Nelson, 1998].. ṙ +. Lβ β + Lr r = 0. (2.8). ṙ = Nβ β + Nr r. (2.9). Lr Nβ − Lβ Nr =0 Lβ. (2.10). Lβ Nr − Lr Nβ Lβ. (2.11). The characteristic root is λ= Roll Mode. This mode is the consequence of the aileron deflection, see Figure 2.6, [Nelson, 1998]. ”The roll mode can be obtained by neglecting the rolling and yawing moments that result from the sideslip and yawing rate compared to those generated from the rolling rate. Also, the side force derivatives with respect to sideslip and yawing rate are neglected, since they have little effect on the roll motion”, [Phillips, 2004]. Thus, ∂L ∂L ∆δa + ∆ρ = Ix ∆φ̈ ∂δa ∂ρ Substituting ρ = φ̇ and obtaining the homogenous response, the root of this mode is found.. (2.12).

(23) Background. 11. τ ∗ ρ̇ + ρ = 0 λ=−. 1 ∂L/∂ρ = Lρ = τ Ix. (2.13) (2.14). Figure 2.6: Roll motion. Dutch Roll Mode Is a damped oscillatory motion that is characterized by a combination of rolling, yawing and sideslip, Figure 2.7, [Nelson, 1998]. Again, the lateral equations become coupled to the longitudinal equations for all, except for small lateral oscillations.. Figure 2.7: Dutch roll motion The Dutch roll approximation is found by assuming the motion consists primary of sideslip and yaw. The rolling rate, bank angle, rolling momentum equation and the side force with respect to yawing rate are usually neglected, [Phillips, 2004]..

(24) 12. ". β̇ ṙ. #. =. Yβ u0. ". Mβ. − 1−. Yr u0. Nr. #". r. Yβ Nr − Yr Nβ + u0 Nβ u0 1 Yβ + u0 Nr ζ=− 2ωn u0. ωn =. 2.2.3. β r. #. (2.15). (2.16). (2.17). Performance Parameters. All the concepts presented so far are closely related to each other. Analyzing Equation 2.3, when the characteristic roots of the closed loop transfer function are found Ẋ = AX + Bu. (2.18). sX(s) = AX(s) + Bu(s). (2.19). G(s) = C(sI − A)−1 B. (2.20). four roots are obtained. Usually there are two negative and real roots and two complex roots. In these eigenvalues is contained the information about the airplane, that is, one of the real root represents the spiral mode, and the other one the roll mode; while the two complex roots refer to the dutch roll motion. Because the roots depend on the lateral directional derivatives any change in the velocity or altitude will change them as it was illustrated before in Table 2.2. The evaluation of the stability is obtained simply from the signs of the real parts of the eigenvalues, however, this information is not sufficient, it should also be analyzed the handling qualities of an airplane. The handling qualities are some performance parameters which are specified with the help of the pilots’ opinions; they are closely related to how easily it is for the pilots to control and maneuver the airplane. These handling qualities have been unified and are established by the military specification MIL-F-8785B in terms of the root of the airplane modes.. Spiral Mode The time constant for this mode is large compared to pilot reaction time, therefore this motion is controlled with pilot’s command. However if the spiral divergence is too fast, pilot’s attention is required and this will degrade the pilot’s opinion of the aircraft’s performance. The spiral mode requirements are shown in the following table, which indicate the minimum time to double amplitude T2 , [Phillips, 2004]; the relation between the time to double amplitude and the time constant is: Ts =. T2 ln(2).

(25) Background. 13 Class I and IV II and III. Category All B and C All. Level 1 12 s 20 s 20 s. Level 2 12 s 12 s 12 s. Level 3 4s 4s 4s. Table 2.3: Spiral mode flying qualities Roll Mode The roll mode is characterized to be a heavily overdamped motion: When a wing aircraft responds to a change in aileron input, the time constant is very short and the airplane quickly approaches a steady rolling rate that is proportional to aileron deflection. If the time constant of the roll mode is much longer than a second approximately, there exists a lag between the time the pilot’s maneuver and the airplane to start rolling; the consequence is that the pilot feels that he is commanding the roll acceleration instead of the rolling rate. This will degrade the pilot’s opinion, [Phillips, 2004]. The next table shows the handling qualities for the roll motion, [Phillips, 2004].. Class I, IV II, III All I, IV II, III. Category A A B C C. Level 1 1 1.4 1.4 1 1.4. Level 2 1.4 3 3 1.4 3. Level 3 10 10 10 10 10. Table 2.4: Roll mode (maximum time constant in seconds). Dutch Roll Mode The Dutch roll motion can be very annoying to both pilots and passengers if the damping is light. Due to the fact that the oscillation of the dutch roll can be very close to the response time of a human pilot, it can be very difficult for a pilot to repress it. ”If the dutch roll damping is extremely light and the period is short, the airplane can be quite difficult to handle”, [Phillips, 2004]. Hence, the following requirements must be met. Robert Nelson [Nelson, 1998] classifies the performance parameters as follows: Level 1 Flying qualities clearly adequate for this mission flight phase. Level 2 Flying qualities adequate to accomplish the mission flight phase but with some increase in pilot workload and/or degradation in mission effectiveness or both. Level 3 Flying qualities such that the airplane can be controlled safely but pilot workload is excessive and/or mission effectiveness is inadequate or both. Category A flight phases can be terminated safely and category B and C flight phases can be completed..

(26) 14 Level 1 1 1 1 1 2 3. Category A A B C C All All. Class I, IV II, III All I, II, IV II, III All All. Min ζ 0.19 0.19 0.08 0.08 0.08 0.02 0.02. Min ζωn rad/s 0.35 0.35 0.15 0.15 0.15 0.05. Min ωn rad/s 1 0.4 0.4 1 0.4 0.4 0.4. Table 2.5: Dutch roll flying qualities Classification of airplanes Class I Small, light airplanes, such as light utility, primary trainer, and light observation craft. Class II Medium weight, low to medium maneuverability airplanes, such as heavy utility/search and rescue, light or medium transport/cargo/tanker, reconnaissance, tactical bomber, heavy attack and trainer for Class II. Class III Large, heavy, low to medium maneuverability airplanes, such as heavy transport/cargo/tanker, heavy bomber and trainer for Class III. Class IV High maneuverability airplanes, such as fighter/interceptor, attack, tactical reconnaissance, observation and trainer for class IV.. Flight phase categories Category A Nonterminal flight phase that requires rapid maneuvering, precision tracking, or precise flight path control. Included in the category are air to air combat, ground attack, weapon delivery/launch, aerial recovery, reconnaissance, in flight refueling, terrain following, antisubmarine search, and close formation flying. Category B Nonterminal flight phases that are normally accomplished using gradual maneuvers and without precision tracking, although accurate flight path control may be required. Included in the category are climb, cruise, loiter, in flight refueling, descent, emergency descent, emergency deceleration, and aerial delivery. Category C Terminal flight phases are normally accomplished using gradual maneuvers and usually require accurate flight path control. Included in this category are takeoff, catapult takeoff, approach, wave off/go around and landing. Others performance parameters shown in some papers are gathered in appendix D. As shown before, the airplane lateral dynamics consists in the spiral, roll and dutch roll mode and it must be handled as a multivariable problem, since each of these modes interact with the others..

(27) Background. 15. 2.3 Multivariable Systems In multivariable systems the inputs and outputs are vectors, which have magnitude and direction; furthermore these systems also exhibit interaction between the inputs and outputs. These are the main differences with respect to a SISO system and why it is more difficult to handle them. Consider figure 2.8 and assume that the transfer functions are multivariable. Therefore the following equations can be obtained. The order of the matrix multiplication is important.. Figure 2.8: Feedback Configuration. y = To (r − n) + So Gdi + So do. (2.21). r − y = So (r − do ) + To n − So Gdi. (2.22). u = KSo (r − n − do ) − Ti di. (2.23). up = KSo (r − n − do ) + Si di. (2.24). where So = (I + GK)−1 = output sensitivity function Si = (I + KG)−1 = input sensitivity function To = GK(I + GK)−1 = output complementary sensitivity function Ti = KG(I + KG)−1 = input complementary sensitivity function It can be verified that So + To = I Si + Ti = I T (s) = SGK(s) = GKS(s) Si K(s) = KS(s) To simplify the nomenclature S = So and T = To ..

(28) 16. 2.3.1. Internal Stability. In order to assure internal stability, the transfer function matrices Si , KS, SG and S from equation 2.21 must be stable, assuming that there are no RHP zero-pole cancellation between G and K. It can be observed that for an internally stable feedback loop: • If G has a RHP zero at s=z, then GK, T, SG, KG and Ti will each have a RHP zero at s=z. • If G has a RHP pole at s=p, then GK and KG also have a RHP pole at s=p, but S, KS and Si will have a RHP zero s=p. Trade off From equations 2.21 and 2.24, the effect of the disturbances do and di on the plant output can be made small by making the output sensitivity function small. Also, if S is small, then T will become one because S + T = I; this is what we want for a good tracking signal, but it also means that the noise will be amplified. Now looking to the control signal up , it is wanted to make it small to reduce wear on components, to avoid saturation, to reduce energy consumption and because the linear approximation works well when the control signal is small. Therefore KS must be small at all frequencies and Si must be small for the range in which the disturbances have enough energy to affect the system. However constrains in the control signal can limit the ability to track references and reject disturbances. It is clearly seen that there is a trade off between disturbance and noise rejection and in the control signals. Luckily, a disturbance normally appears at low frequencies while the noise at high frequencies. Therefore the design objective states as follow: • σ̄(S(jω)) ≤ εd ,. 0 ≤ ω ≤ ωd ,. • σ̄(T (jω)) ≤ εn ,. ωn ≤ ω ≤ ∞,. εd ≪ 1 εn ≪ 1. disturbance rejection noise rejection. • KS(jω) tends to zero at low frequencies, for small control signals. • K(jω) tends to zero at high frequencies, for small control signals. Here ωd and ωn indicate the upper limit of disturbances and the lower limit of noise and σ is the maximum singular value. It should be noted that ωd < ωn otherwise the system could not reject disturbances and noise at the same time. An easy way to achieve the former conditions is to design a controller so that the gain at low frequencies is high, while the gain at high frequencies is low. More details about this approach will be discussed in the McFarlane Glover Loop Shaping design..

(29) Background. 2.3.2. 17. Multivariable System Gain. In multivariable systems, as in the case of SISO systems, the transfer function tell us how much each element of a vector input is amplified and phase shifted at a given frequency. However the gain in MIMO systems is defined as follows: gain=. kG(jω)u(jω)k2 ky(jω)k2 = ku(jω)k2 ku(jω)k2. (2.25). where k...k2 is the two norm and is evaluated as. v um uX |yi (jω)|2 ky(jω)k2 = t. (2.26). i=1. Note that the value of kG(jω)u(jω)k2 at each frequency does not depend on the magnitude of u(jω), but in its direction. That is why it is not possible to cancel u in the numerator with the denominator as in the SISO case. Furthermore, the classical techniques for measuring the gain margin and phase margin, are not useful for MIMO systems since they do not take into consideration the interaction presented; this was explained in 1978 by Doyle [Doyle, 1979]; instead, the gain of the system is measured through the singular values.. Singular Values If an input vector is chosen so that the maximum gain is attained at each frequency, then, the maximum possible gain denoted by σ(jω) is the maximum singular value. Conversely, if the direction is chosen to obtain the minimum gain in the range of frequencies, the smallest possible gain σ(jω) is called the minimum singular value. The maximum and minimum singular values are defined as follows: q σ(G(jω)) = max λi (GH (jω)G(jω)) q σ(G(jω)) = min λi (GH (jω)G(jω)). (2.27) (2.28). where λi is the ith eigenvalue of GH (jω)G(jω). GH (jω) denotes the conjugate transpose. Therefore the maximum and minimum singular values delimit the gain of the system. 0 ≤ σ(G(jω)) ≤. kG(jω)u(jω)k2 ≤ σ(G(jω)) < ∞ ku(jω)k2. (2.29). Recall that it is wanted to have σ̄(S(jω)) < ǫd for ω < ωd and σ̄(T (jω)) < ǫn for ω > ωn . Therefore, shaping the open loop as in figure 2.9, the requirements are fulfill since 1 1 + GK GK σ(T (jω)) = 1 + GK σ(S(jω)) =. for large GK σ(S(jω)) tends to zero. for small GK σ(T (jω)) tends to GK..

(30) 18. Figure 2.9: Singular values of the open loop transfer function, to ensure noise and disturbance rejection. In conclusion high loop gain is needed in low frequency ranges to reject disturbances, while low gain is needed at high frequencies to reject noise. It is important to notice that the transition between the high gain and the low gain should not be with a slope higher than -20 dB/decade; the reason for that is to keep the phase lag to less than -180 inside the control loop bandwidth. Besides the singular value there are some other tools that help to analyze the condition of the plant. These are the condition number and the relative gain array.. Condition Number The condition number and the relative gain array are commonly used to quantify the degree of directionality and the level of interaction in multivariable systems. The condition number is given by κ(G(jω)) =. σ(G(jω)) σ(G(jω)). (2.30). and is bounded to 1 ≤ κ < ∞. If the condition number is large (greater than 10), then the transfer function matrix is ill-conditioned, which means that the system has significantly different gains in different directions. Due to the fact that the condition number depends strongly on the scaling of inputs and outputs, one could consider minimize the condition number by choosing an appropriate scaling. A small condition number indicates that the plant is not sensitive to uncertainty, but the contrary does not hold; for that reason, it is important to have another measure as the relative gain array. Relative Gain Array The relative gain array (RGA) of a nonsingular transfer function matrix G(s) is given by. Λ(G(jω)) = G(jω). O. (G−1 (jω))T. (2.31).

(31) Background where. N. 19. denote element by element product.. The RGA has the following properties specified in [Skogestad and Postlethwaite, 1996]: • It is independent of the input and output scaling. • Its rows and columns sum one. • The sum norm of the RGA, kΛksum , is very close to the minimized condition number. The sum norm is the sum of the magnitudes of the elements of the matrix G. • It is the identity matrix if G is upper or lower triangular. It is useful for: • Pairing the best input output relation. • If there are more inputs than outputs or viceversa, which ones will be best to omit in order to form a square matrix. • Plants with large RGA elements (above 1) around the crossover frequency are difficult to control because of sensitivity to input uncertainty (neglecting actuator dynamics). It is, small changes in the individual elements of a matrix G could lead to a singular matrix. • To check how sensitive is the transfer function matrix to a diagonal input uncertainty.. 2.4 Robustness The objective in robust multivariable feedback control system design is to synthesize a control law which maintains the system response and error signals within the specified tolerances despite the effects of uncertainty on the system.. 2.4.1. Uncertainty. The 2 ways in which a system can become unstable is due to the presence of disturbances and uncertainty. The disturbances are the unwanted inputs from the environment, while the uncertainty in a model can have several origins: • Some parameters in the linear model are only know approximately. • The parameters in the linear model may vary due to nonlinearities or changes in the operating conditions. • Measurement devices have imperfections. • At high frequencies the order of the model is not know and the uncertainties can exceed 100% at some frequencies. • When choosing a simpler model to deal with. • The controller implemented may be a model reduction of the designed one..

(32) 20 These sources of uncertainty can be grouped into parametric uncertainty, when the parameters of the model are uncertain and neglected and unmodelled dynamics uncertainty, when there are some missing dynamics. The parametric uncertainty perturbations occur at low frequencies, while the unmodelled dynamics generate perturbations at high frequencies, [Skogestad and Postlethwaite, 1996]. A way to include the uncertainty in the model is with additive and multiplicative uncertainty. A multivariable system with additive uncertainty can be written as: Gp (jω) = G(jω) + ∆a (jω). (2.32). where the upper bound of the uncertainty is defined by σ(∆a (jω)) ≤ ρa (ω). (2.33). It can also be modelled as multiplicative uncertainty. Gp (jω) = G(jω)(1 + ∆m (jω)). σ(∆m (jω)) ≤ ρm (ω). (2.34). It is assume that G contains all the RHP poles and zeros, ∆(jω) is stable and bounded by ∆(jω) ≤ ρ(ω). In SISO systems the additive uncertainty is changed to multiplicative because it is easier to deal with when designing a robust controller. However in multivariable systems, it is not convenient to switch the additive to multiplicative uncertainty, because there is a factor of σ(G(jω)) which can increase in a higher value the amount of uncertainty. The same case happens when trying to move the output multiplicative uncertainty Gp (s) = (1 + ∆o )G, to the input multiplicative uncertainty Gp (s) = G(1 + ∆i ); here the factor is the condition number. A practical way to deal with the uncertainty in the design procedure, is to rewrite the upper bound in terms of a weighting function as defined below. ˜ ∆(s) = w(s)∆(s) where. |w(jω)| = ρ(ω). and. ˜ σ̄(w(jω)∆(jω)) ≤ ρ(ω). (2.35) ˜ σ̄(∆(jω)) ≤1 ∀ω. (2.36). and ω(s) is normally represented as ω(s) =. τ s + r0 (τ /r∞ )s + 1. (2.37). where r0 → is the relative uncertainty at steady state 1/τ → is the frequency where the relative uncertainty reaches 100% r∞ → is the magnitude of the weight at higher frequencies In most of the cases the effect of the uncertainty in the system can be diminished with the feedback, but careful should be taken at the crossover frequency when the sensitivity function could take values over 1. From now on, it.

(33) Background. 21. is assumed that the uncertainty models are stable and represented by equation 2.35. Another mean to represent the uncertainty is by using the coprime factor uncertainty, as illustrated in figure 2.10. It is easily probed that the closed transfer function of figure 2.10 is given by the following equation in which M and N refer to the denominator and numerator polynomial of the transfer function. A more detailed information about coprime factorization is given forward when it comes with the Loop Shaping design controller. G = (M + ∆M )−1 (N + ∆N ). (2.38). Figure 2.10: Coprime factor uncertainty The uncertainty is always present when modelling mathematically real systems, and the objective is to get a controller that could work for the mathematical model and also for the real system. It is, we want the controller to be robust. A robust control guarantees the stability and performance of a set of models M if and only if the following characteristics are met. • Nominal Stability • Nominal Performance • Robust Stability • Robust Performance First it will be described the robustness for the SISO systems in order to have a better insight by looking at the Nyquist diagrams; after that it will be generalized for the multivariable case.. 2.4.2. SISO Robustness. Nominal Stability A system has nominal stability if the closed loop has internal stability. Refering to the Nyquist theorem, this means, that the number of times that the curve GK encircles the -1+0j point in the counterclockwise direction, must be equal to the number of right half plane poles of the open loop plant and controller. See figure 2.11..

(34) 22. Figure 2.11: Nyquist diagram showing nominal stability After nominal stability has been obtained, the second step is to verify whether or not the system performs according to the required specifications. A way to measure the performance of the system is to establish some bounds at different frequencies; this is done by introducing weighting functions.. Weighting Functions A weighting function is a stable minimum phase system by which the requirements for the closed loop performance are defined. Some examples of weighting functions are WS, WT, WKS, each adds a constrain to the transfer function matrix that is being multiplied by it. When using synthesis methods to design the controller such as H∞ it is very common to specify the requirements as weighting functions; this matrices specify the upper bound of the sensitivity and complementary sensitivity functions. A very easy way to choose the performance weighting functions is to use the following equations for a first and second order filter. Wp (s) =. s Ms. + ωb. s + ωb ess. Wp2. √s ( Ms. + ωb )2 = √ (s + ωb ess )2. where: ωb = minimum bandwidth for disturbance rejections. Ms = maximum peak magnitude of S. ess = maximum tracking error. [Skogestad and Postlethwaite, 1996]. (2.39).

(35) Background. 23. Nominal Performance Nominal performance demands the plant to fulfill all the requirements for the specific model. The condition for nominal performance is |Wp S| < 1 |Wp | <. ∀ω. 1 = |1 + GK| |S|. (2.40) ∀ω. (2.41). where Wp is the performance weighting matrix. A graphical way can be represented as in figure 2.12.. Figure 2.12: Nyquist plot illustrating the nominal performance requirement Wp places an upper bound in the sensitivity function. For the purpose of nominal performance, the nyquist plot of GK must not cross the circle drawn by Wp at each frequency.. Robust Stability Consider a system Gp (s) = G(s)(1 + △m ); the system has robust stability if the number of encirclements in the counterclockwise direction of the critical point by the nyquist diagram of Gp K(s) is equal to the number of open loop RHP poles of Gp K(s).This means that a set of systems belonging to Gp must be stable. See figure 2.13 At every frequency a representation of a robust model is a circle of radio GK(jω)ρ(ω) centered at each frequency of the nominal model. It is clear that to fulfill the requirement of robust stability |1 + GK(jω)| > |GK(jω)|ρm (ω). ∀ω. (2.42). and the condition for robust stability with multiplicative uncertainty is kT (jω)ρm (ω)k∞ < 1. (2.43).

(36) 24. Figure 2.13: Nyquist diagram showing the condition for Robust Stability The condition for robust stability with additive uncertainty can be found by taking the last equation an converting the multiplicative to additive uncertainty. Then kKS(jω)ρa (ω)k∞ < 1. (2.44). Robust Performance Robust performance requires robust stability plus nominal performance. The way to achieve it, is to avoid that the circle of the multiplicative uncertainty touches the circle of the performance at each frequency, as shown in figure 2.14.. Figure 2.14: Nyquist plot for robust performance under multiplicative uncertainty.

(37) Background. 25. Condition for robust performance with multiplicative uncertainty k|Wp S(jω)| + |T (jω)|ρm (ω)k∞ < 1. (2.45). Condition for robust performance with additive uncertainty k|Wp S(jω)| + |KS(jω)|ρm (ω)k∞ < 1. 2.4.3. (2.46). Multivariable Robustness. As in the SISO case we must achieve nominal stability, nominal performance, robust stability and robust performance in order to obtain a robust controller.. Nominal stability To get nominal stability the transfer function matrices Si , KS, SG and S must be internally stable, as stated before in the section multivariable transfer function matrices.. Nominal Performance For SISO systems a scalar weighting function was used as a performance weighting function. In multivariable systems, it could be used a scalar weighting function, a diagonal weighting function matrix or a full weighting function matrix. The first two are most commonly used due to the facility to design them. A multivarible system achieve nominal peformance if: • the loop is nominal stable. • kWp S(jω)k∞ < 1 Robust Stability A system has robust stability with input multiplicative uncertainty if • the loop is nominal stable • kKSG(jω)ρi (ω)k∞ < 1 A system has robust stability with output multiplicative uncertainty if • the loop is nominal stable • kT (jω)ρo (ω)k∞ < 1 A system has robust stability with additive uncertainty if • the loop is nominal stable • kKS(jω)ρa (ω)k∞ < 1.

(38) 26 Robust Performance A system has robust performance with input multiplicative uncertainty if • the loop is nominal stable • kWp S(jω) + KSG(jω)ρi (ω)k∞ < 1 A system has robust performance with output multiplicative uncertainty if • the loop is nominal stable • kWp S(jω) + T (jω)ρo (ω)k∞ < 1 A system has robust performance with additive uncertainty if • the loop is nominal stable • kWp S(jω) + KS(jω)ρa (ω)k∞ < 1 In the next section it is explained how to incorporate the robustness in the design process of a controller.. 2.5 H Infinity Controller The H∞ theory was first introduced by Zames [Zames, 1981] when he emphasized the subject of plant uncertainty; in some problems the disturbance is not known, which limits the applicability of the optimal control. It was until 1988 when a simple state space H∞ controller was announced by Glover and Doyle, [Doyle, 1989]. As the H∞ continued to develop the relation between the H2 and the H∞ norm was seen to be closer and it started the idea of designing a performance controller with H2 and robustifying it with H∞ . This proposal involved multiobjetive optimization and was investigated by Scherer [Scherer, 1995] with Linear Matrix Inequalities (LMI) Techniques, with which he has been working through for the H∞ optimization. However, the introduction of LMI was reported first by Willems [Willems, 1971] in 1971, and since then it has been used in problems of control systems. For a broader overview of the development of LMI refer to [Cristbal, 2005]. The H∞ robust control is a frequency domain optimization in which the main objective is to attain a controller with which the infinity norm is minimized. It is a synthesis methodology because the designer only has to select the weighting matrices to get the desire gain of the closed loop response, while the synthesis method takes care by itself about the phase of the closed loop system. Among the most important properties of the H∞ is that it is capable to deal with plant modelling errors and unknown uncertainties; it can handle multivariable systems and the design process is an extension of the classical control theory which facilitates its implementation..

(39) Background. 2.5.1. 27. H Infinity Norm. The H∞ space is one of the member of the spaces introduced by the mathematician Hardy and is defined as follows. kSk∞ = sup|S(jω)|. ωǫℜ. (2.47). where sup indicated the peak value of S. In the case of multivariable systems, the peak value is indicated by the maximum singular value of the transfer function matrix. kSk∞ = max σ̄(S(jω)) = max here y is the output and u is the input of the system.. ky(t)k2 ku(t)k2. ωǫℜ. (2.48). The H∞ problem can be solved using different procedures like the mixed sensitivity or the loop shaping; however the structure of the problem formulation has always the same structure as presented in the next section.. 2.5.2. General Control Problem Formulation. It is a standard control problem formulation in which any particular problem can be represented. This structure is presented in figure 2.15.. Figure 2.15: General control configuration Here G is the augmented plant, K is the controller and there is a positive feedback. The inputs and outputs to the system are defined as, [Skogestad and Postlethwaite, 1996]: ω = input vector of the plant z =output vector of the plant υ = input vector to the controller u = output vector of the controller G can be represented in the state space form as   A ẋ     z  =  C1 v C2 . and must satisfy the following conditions. B1 D11 D21.   B2 x   D12   w  v D22. (2.49).

(40) 28 • A, B2 , C2 is stabilizable and detectable • D12 has full column rank and D21 has full row rank The overall objective of this configuration is to design a controller that minimizes the effect of the norm from ω to z.. 2.5.3. Mixed Sensitivity Design Controller. When the H∞ norm is applied to a transfer function, it minimizes the gain between an input and an output signal. Another convenient approach is to minimize the effect of the disturbance in certain fictitious outputs as shown in the following figure.. Figure 2.16: Signal based representation of the mixed sensitivity problem Figure 2.16 can be reorganized and presented in the general problem configuration, see figure 2.17.. Figure 2.17: Mixed sensitivity in the general configuration From figure 2.17 the following relations are found between the input d and the fictitious outputs y1, y2 and y3. Performance Robustness to additive uncertainty and control effort. y1 = W1 Sd y2 = W2 KSd.

(41) Background. 29. Robustness to multiplicative uncertainty and noise. y 3 = W3 T d. Therefore minimizing the effect on the signals y1 /d, y2 /d, y3 /d, comprises the objective of robustness and performance; the objective of the mixed sensitivity can be written to minimized the transfer function matrices of y1 /d, y2 /d, y3 /d,.. min. W1 S(jω) W2 KS(jω) W3 T (jω). (2.50) ∞. This is the formulation of the mixed sensitivity problem. The way to evaluate equation 2.50 is by taking the squared root of the sum of the square of each element.. min. W1 S(jω) W2 KS(jω) W3 T (jω). = min ∞. p |W1 S(jω)|2 + |W2 KS(jω)|2 + |W3 T (jω)|2. (2.51) ∞. It is assumed that the weighting functions are stable and minimal phase transfer functions, if not the general algorithm of H∞ is not applicable. Some tricks to specify minimal phase transfer functions, according to [Skogestad and Postlethwaite, 1996], are the following: 1 + τ1 s 1 (2.52) ǫ≪1 τ2 ≪ τ1 s+ǫ 1 + τ2 s The first one helps to introduce an integrator to avoid the steady state error, whilst the second one to assure a small K outside the system bandwidth. During the design it should be noticed that the crossover frequency of W1 must be below enough of the crossover frequency of W3 ; otherwise the performance requirements will not be achievable. In other words, if in the same range disturbances and noise are presented, it is not possible to make S and T small at the same frequency. The advantage of this procedure is that one can shape the closed loop behavior by choosing the weighting matrices; nevertheless, the relation between the change in the weighting matrices and the output response is not direct and trying to solve this problem can be a little bit difficult when the 3 objectives are specified. Furthermore, in the mixed sensitivity problem formulation the H∞ controller tends to cancel the stable poles with its transmission zeros and any unstable poles is shifted to its jw axis mirror image, when the feedback closed loop is formed. A suboptimal controller is normally found by solving the Riccati equations for which W1 S(jω) W2 KS(jω) W3 T (jω). <γ ∞. (2.53). If an optimal controller is required then the algorithm can be used iteratively, reducing γ until the minimum is reached within a given tolerance, [Skogestad and Postlethwaite, 1996]. Nevertheless to find an optimal controller is complicated and in some cases is not needed, since the information that the designer has is just an approximation. Therefore achieving an optimal controller will make not a big difference with respect to the suboptimal..

(42) 30. 2.5.4. H Infinity Loop Shaping. The H∞ Loop Shaping design uses the coprime factorization to represent the transfer function matrix. G(s) = Nr (s)Mr−1 (s) for a right coprime factorization G(s) = Ml−1 (s)Nl (s) for a left coprime factorization where N and M are the numerator and denominator of the transfer function matrix. The coprimeness implies that there are not RHP poles and zeros cancellation between N and M. The advantage of using this representation for the uncertainty is that a tighter bound can be obtained. This is due to the fact that the uncertainty blocks enter from the same location in the block diagram and can be stacked on top of each other in an overall uncentainty which is full matrix, [Skogestad and Postlethwaite, 1996], see figure 2.18.. Figure 2.18: Left normalized coprime factorization of the plant. McFarlane Glover Loop Shaping Design Procedure The H∞ Loop Shaping technique was proposed by McFarlane and Glover, [McFarlane and Glover, 1988] and consists on shaping the singular values of the open loop response by pre and postmultiplying the transfer function matrix; then the augmented plant is robustly stabilized using H∞ techniques. This methodology uses the normalize coprime factorization of the plant as shown in figure 2.18; where the plant transfer function can be written as Gp = (M + ∆M )−1 (N + ∆N ). (2.54). where ∆M , ∆N are stable transfer functions that represent uncertainty in the nominal plant G. The objective of robust stability is γ= Assuming. KSM −1 (jω) SM −1 (jω). < ∞. 1 ǫ. (2.55).

(43) Background. 31. ∆M (jω) ∆N (jω). <ǫ. (2.56). ∞. then the controller is given by: K = −B T X(sI − A + BF + γ 2 (LT )−1 ZC T (C + DF ))−1 γ 2 (LT )−1 ZC T + DT F = −Σ−1 (DT C + B T X). L = (1 − γ 2 )I + XZ. (2.57) (2.58). and Z and X are the solutions of the followings Riccati equations (A − BΣ−1 DT C)Z + Z(A − BΣ−1 DT C)T − ZC T R−1 CZ + BΣ−1 B T = 0. (2.59). (A − BΣ−1 DT C)T X + X(A − BΣ−1 DT C) − XBΣ−1 B T X + C T R−1 C = 0. (2.60). with R = I + DDT. Σ = I + DT D. (2.61). Skogestad and Postlethwaite [Skogestad and Postlethwaite, 1996] present the following procedure to design an H∞ loop shaping controller. 1. Scale the plant inputs and outputs. Normally the scale is done with respect to a given percentage (10%) of its expected range. 2. Order the inputs and outputs so that the plant is as diagonal as possible. 3. Select the weights to pre and postmultiply the plant as illustrated in figure 2.19, to obtain the desirable singular values of Gs = W2 GW1 . (High gains at low frequencies, low gains at high frequencies and a roll off rate of -20dB). W2 is usually constant and W1 has the dynamic shaping. 4. Obtain the robust controller. If the margin ǫ < 0.25 then select another weighting functions. Good values of γ = 1/ǫ are between 1 and 3. 5. If the final design presents considerable overshot the configuration presented in figure 2.20 should be used. This prevents the references to affect directly the controller.. Figure 2.19: The shaped plant.

(44) 32. Figure 2.20: An implementation to avoid derivative kick. 2.5.5. Applications of H Infinity Controllers. H∞ Controllers has been applied successfully in a wide range of fields like ill condition distillations process [Petter Lundstrm and Doyle, 1999], satellites communications [Aiguo Ming, 2005], etc. Here it will be focussed in the control of an airplane lateral dynamics.. • In [Reichert, 1990], an H∞ control was applied to a lateral control system for an aircraft, using the rudder, aileron and vertical canard as the control inputs. The objective was to provide decouple signals between the command channels and robustness to variation in the plant model dynamics. • In [Sang Yee, 2001] a reliable and robust H∞ flight controller and mixed H2 /H∞ was obtained based on the iterative linear matrix inequality approach. The response of both controllers was compared in the case of lost of effectiveness in the inputs.. 2.6 Optimization Many control problems and design specifications have LMI formulations. This LMI formulation reduce the controller design problem into a convex optimization problem.. 2.6.1. Linear Matrix Inequalities. A Linear Matrix Inequality (LMI) is an expression of the form A(x) = A0 +. n X. xi Ai > 0. i=1. where. xǫℜn is the variable. Ai ǫℜmxm are symmetrical matrices. The A > 0 inequality means that A is positive definite, and defines a convex constrain in x.. (2.62).