Vibration Control of Active Structures

I was introduced to structural control by Raphael Haftka and Bill Hallauer during a one year stay at the Aerospace and Ocean Engineering department of Virginia Tech., during the academic year 1985-1986. At that time, there was a tremendous interest in large space structures in the USA, mainly because of the Strategic Defense Initiative and the space station program. Most of the work was theoretical or numerical, but Bill Hallauer was one of the few experimen talists trying to implement control systems which worked on actual structures. When I returned to Belgium, I was appointed at the chair of Mechanical Engi neering and Robotics at ULB, and I decided to start some basic vibration control experiments on my own. A little later, smart materials became widely available and offered completely new possibilities, particularly for precision structures, but also brought new difficulties due to the strong coupling in their constitutive equations, which requires a complete reformulation of the classical modelling techniques such as finite elements. We started in this new field with the sup port of the national and regional governments, the European Space Agency, and some bilateral collaborations with European aerospace companies. Our Active Structures Laboratory was inaugurated in October 1995.

Texte du rabat

This text is an introduction to the feedback control of lightly damped flexible structures; the emphasis is placed on basic issues such as actuator and sensor selection, placement and dynamics, and actual implementation for solving practical problems. The book consists of 11 chapters; in chapters 2 to 5, the open-loop transfer functions of various active structures are derived from their constitutive equations; the discussion includes a truss and sandwich beams and plates with embedded piezoelectric actuators and sensors. The virtues of collocated actuator-sensor configurations are pointed out and used to develop active damping with guaranteed stability. Chapters 6 to 8 are devoted to the model-based control of SISO systems; optimal control is developed graphically using the symmetric root locus; the gain-phase relationship is discussed and the design tradeoffs are explained in the frequency domain. The issues of robustness with respect to the parametric uncertainty and the spillover instability are examined. After two short chapters on controllability (ch. 9) and stability (ch. 10), the book concludes with a set of applications to active damping and precision positioning of a set of aerospace, mechanical and civil engineering structures. The book is intended for structural engineers who want to acquire some background in vibration control; it can be used as a textbook for a graduate course on vibration control or active structures. The text is supplemented with 98 problems.

Résumé
`The book is highly recommended to students, practising structural engineers and to researchers who are interested in vibration control and active structures.'
Zentralblatt MATH, 910 (1999)
`The book is written in a very easy and interesting style and has a good set of problems at the end of each chapter. This reviewer enjoyed reading the book and would recommend it for any graduate course on control of structures. This book occupies a unique place in the set of references in this important field.'
Mechatronics, 10 (2000)

Contenu
1 Introduction.- 1.1 Active versus passive.- 1.2 Smart materials and structures.- 1.3 Control strategies.- 1.4 The various steps of the design.- 1.5 Organization of the book.- 1.6 References.- 1.7 Problems.- 2 Some concepts of structural dynamics.- 2.1 Equation of motion of a discrete system.- 2.2 Vibration modes.- 2.3 Modal decomposition.- 2.4 Transfer function of collocated systems.- 2.5 Continuous structures.- 2.6 Guyan reduction.- 2.7 References.- 2.8 Problems.- 3 Actuators, piezoelectric materials, and active structures.- 3.1 Introduction.- 3.2 Proof-mass actuator.- 3.3 Reaction wheels and gyrostabilizers.- 3.4 Piezoelectric actuators.- 3.5 Passive damping with piezoceramics.- 3.6 Active cantilever beam.- 3.7 Active truss.- 3.8 Active plate with piezo strips.- 3.9 References.- 3.10 Problems.- 4 Collocated versus non-collocated control.- 4.1 Introduction.- 4.2 Pole-zero flipping.- 4.3 Collocated control.- 4.4 Non-collocated case.- 4.5 Notch filter.- 4.6 Pole-zero flipping in the structure.- 4.7 Effect on the Bode plots.- 4.8 Relation to the mode shapes.- 4.9 The role of damping.- 4.10 References.- 4.11 Problems.- 5 Active damping with collocated pairs.- 5.1 Introduction.- 5.2 Direct Velocity Feedback.- 5.3 Acceleration feedback.- 5.4 Positive Position Feedback.- 5.5 Integral Force Feedback.- 5.6 Remarks.- 5.7 References.- 5.8 Problems.- 6 State space approach.- 6.1 Introduction.- 6.2 State space description.- 6.3 System transfer function.- 6.4 Pole placement by state feedback.- 6.5 Linear Quadratic Regulator.- 6.6 Observer design.- 6.7 Kalman Filter.- 6.8 Reduced order observer.- 6.9 Separation principle.- 6.10 Transfer function of the compensator.- 6.11 References.- 6.12 Problems.- 7 Analysis and synthesis in the frequency domain.- 7.1 Gain and phase margins.- 7.2Nyquist criterion.- 7.3 Nichols chart.- 7.4 Feedback specification for SISO systems.- 7.5 Bode gain-phase relationships.- 7.6 The Bode Ideal Cutoff.- 7.7 Non-minimum phase systems.- 7.8 Usual compensators.- 7.9 References.- 7.10 Problems.- 8 Optimal control.- 8.1 Introduction.- 8.2 Quadratic integral.- 8.3 Deterministic LQR.- 8.4 Stochastic response to a white noise.- 8.5 Stochastic LQR.- 8.6 Asymptotic behaviour of the closed-loop.- 8.7 Prescribed degree of stability.- 8.8 Gain and phase margins of the LQR.- 8.9 Full state observer.- 8.10 Kalman-Bucy Filter (KBF).- 8.11 Linear Quadratic Gaussian (LQG).- 8.12 Duality.- 8.13 Spillover.- 8.14 Loop Transfer Recovery (LTR).- 8.15 Integral control with state feedback.- 8.16 Frequency shaping.- 8.17 References.- 8.18 Problems.- 9 Controllability and Observability.- 9.1 Introduction.- 9.2 Controllability and observability matrices.- 9.3 Examples.- 9.4 State transformation.- 9.5 PBH test.- 9.6 Residues.- 9.7 Example.- 9.8 Sensitivity.- 9.9 Controllability and observability Gramians.- 9.10 Relative controllability and observability.- 9.11 Model reduction.- 9.12 References.- 9.13 Problems.- 10 Stability.- 10.1 Introduction.- 10.2 Linear systems.- 10.3 Liapunov's direct method.- 10.4 Liapunov functions for linear systems.- 10.5 Liapunov's indirect method.- 10.6 An application to controller design.- 10.7 Energy absorbing controls.- 10.8 References.- 10.9 Problems.- 11 Applications.- 11.1 Digital implementation.- 11.2 Active damping of a truss structure.- 11.3 Active damping of a plate.- 11.4 Active damping of a stiff beam.- 11.5 The HAC/LAC strategy.- 11.6 Tendon control of cable structures.- 11.7 References.- 11.8 Problems.