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BOOKSHELF
Applications and Algorithms
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n this issue of IEEE Control Systems Magazine, we bring you reviews of three books. First, Kathryn (“Katie”) Johnson reviews the book by Bianchi, De Battista, and Mantz on wind turbines. Next, B.N. Datta tells us about the book by Bhaya and Kaszkurewicz on the application of control ideas to numerical algorithms. Finally, Julie Thienel reviews the book by Crassidis and Junkins on estimation. As usual, each review is a minitutorial on what’s what in the field. As always, I welcome your suggestions for books to be reviewed, as well as volunteers to serve as reviewers. Scott Ploen
[email protected]
Wind Turbine Control Systems by FERNANDO D. BIANCHI, HERNÁN DE BATTISTA, and RICARDO J. MANTZ.
Recent concern about climate change has led to rapidly increasing interest in wind and other renewable energy sources. According to the Springer, 2007, American Wind Energy AssoISBN-13: 978-1-84628-492-2, US$119, 205 pages. ciation, the wind industry . installed more than 2400 MW of wind turbines in the United States in 2006, bringing the total U.S.-installed wind capacity to 11,600 MW [1]. This rapid growth has caught the interest of the controls research community, whose members see applications for advanced control technology in wind energy’s large, flexible structures. As a result, the time is ripe for a book on wind turbine control systems. Wind turbines extract kinetic energy from the wind and convert it to electrical energy, which can then be fed directly into a utility grid or used in standalone systems such as a water pump on a remote ranch. The most common type of modern wind turbine is a horizontal-axis wind turbine (HAWT), which has an axis of rotation that is parallel to Digital Object Identifier 10.1109/MCS.2007.903682
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the ground. In other words, a HAWT looks similar to a propeller or a fan. HAWTs typically have two or three blades attached to an oblong nacelle supported by a tubular or lattice tower. Energy is transferred from the rotating blades, or rotor, to the generator through a drivetrain that may consist of a single shaft or multiple shafts connected through one or more gear boxes. Wind turbines can be either variable or fixed speed. Fixed-speed turbines are usually configured to connect directly to the utility grid, so the electricity they produce must have the same frequency as the grid. Variable-speed turbines use power-processing equipment such as power electronics to convert their variable-frequency electricity to the fixed electrical grid frequency. Since wind-turbine aerodynamic efficiency is a function of wind speed, variablespeed turbines are designed to maximize aerodynamic efficiency over a wide range of wind speeds, unlike fixedspeed turbines. Variable-speed turbines can also absorb the energy in sudden wind gusts by speeding up and thus usually experience smaller loads than fixed-speed turbines. Excessive structural loads, such as blade bending, drivetrain torsion, and tower bending and torsion, can reduce turbine reliability and lifespan. In the worst case, these components can break or the blades can collide with the tower. However, fixed-speed turbines usually have lower initial costs than variable-speed turbines because they do not require special power-processing equipment. Although wind is one of the most well-developed and economical sources of renewable energy, advances in control can still improve wind turbine technology. Increases in wind turbine size have led to increases in both cost and structural flexibility compared to turbines of past decades. Increased flexibility creates new control problems. Perhaps more importantly, as individual turbines become more expensive, the cost of the actuators required for advanced control often makes up a smaller percentage of the total wind turbine system cost as compared to smaller, less expensive turbines. Since the primary goal sought by wind-turbine engineers is to produce the most electricity at the lowest cost, control schemes that reduce cost by alleviating structural loads or by increasing the efficiency of the turbine are beneficial to the wind-energy industry. The three most common actuators used for utility-scale wind turbine control are the yaw drive, the blade-pitch drives, and the generator. The yaw drive aligns the rotor and
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nacelle with the wind direction at any given time but typically operates slowly on large turbines to prevent gyroscopic loading. Due to its slowness, the yaw drive is not of particular interest to control engineers. The blade pitch actuators can be electromechanical or hydraulic in nature and are used to position the blades to a desired angular position to achieve various control objectives. For example, the blades are frequently positioned to achieve maximum aerodynamic efficiency in low winds and to limit efficiency, and thus power, in high winds. Pitch control can be used both to alleviate structural loads and to increase efficiency to maximize energy capture. Finally, generator torque control opposes the aerodynamic torque and thus indirectly controls the turbine speed. Like pitch control, generator torque control, which is often achieved using power electronics on modern utilityscale turbines, is also a good candidate for alleviating drivetrain bending loads and increasing efficiency. Smaller turbines may be equipped with active yaw, pitch, and torque control but typically rely more on passive control mechanisms such as a tail to align the turbine to the proper yaw position. The authors of Wind Turbine Control Systems are knowledgeable about the subject, having published several papers in this area [2]–[6]. These papers tackle various aspects of wind turbine control, including blade-pitch control and generator torque control, using linear parameter varying (LPV) gain scheduling and sliding mode control. Wind Turbine Control Systems provides a good introduction to wind energy for control engineers but is too short to serve as the sole source of information about the many complexities of wind turbine operation. Also, some sections contain jarring language and grammatical errors that make parts of the text difficult to read. Wind Turbine Control Systems is part of the series Advances in Industrial Control, which features control developments in industrial applications. The target audience for this text is members of the control research community who are interested in wind energy applications.
CONTENTS Wind Turbine Control Systems is organized into six chapters and three appendices. The first four chapters are devoted to wind energy, wind-turbine modeling, and control challenges in wind turbines. The last two chapters and the appendices explain the LPV controllers developed for wind turbines. Chapter 1 is an introductory chapter that provides a brief overview of wind-energy conversion systems, gain scheduling, and robust control of wind turbines. The last section of the chapter gives an outline of the book. An introduction to wind energy and wind turbines is given in Chapter 2. An understanding of the nature of the wind inflow is critical in wind-turbine controller design. Accordingly, this chapter summarizes the main characteristics of the wind inflow, including the mean, turbulence,
and the wind’s probability distribution with respect to wind speed. The majority of the chapter is an overview of the aerodynamics of horizontal-axis wind turbines. Two commonly used aerodynamic models, the actuator-disc model and the blade-element model, are presented. The former is useful for deriving the maximum possible aerodynamic efficiency of a wind turbine, called the Betz limit, and the latter is useful in control design for load reduction. The brevity of Chapter 2 is such that only an overview of wind-turbine aerodynamics is presented. The interested reader might also wish to consult [7] and [8]. Chapter 3 develops models for wind turbine systems, also known as wind-energy conversion systems (WECSs), and for the wind input. The WECS models are divided into mechanical, aerodynamic, electrical, and pitch subsystems. The mechanical dynamics are derived using Lagrangian techniques, and the aerodynamic subsystem is built from the aerodynamic thrust and torque equations derived in Chapter 2. The electrical subsystem includes three different generators: a directly coupled squirrel-cage induction generator, a stator-controlled squirrel-cage induction generator, and a rotor-controlled doubly fed induction generator. Brief descriptions of the features of each generator type are included in the chapter. Finally, the pitch subsystem is described by a first-order differential equation representing the pitch actuator dynamics. These four subsystems are combined into a nonlinear model for the entire WECS. The main control objectives for WECS are presented in Chapter 4 along with strategies for achieving these objectives. Specifically, the main control objectives for wind turbines are maximizing aerodynamic efficiency when operating at low wind speeds and limiting power capture at high wind speeds, while limiting mechanical loads to safe levels and keeping the power quality at an acceptable level. The control strategies discussed include fixed speed, fixed pitch; fixed speed, variable pitch; variable speed, fixed pitch; and variable speed, variable pitch. The descriptions of each strategy are accurate, but the brevity of the chapter is such that novices in the area of wind turbine applications might want to supplement their reading with other literature. The final two chapters of the text present gain-scheduling techniques for wind-turbine control. Chapter 5 focuses on variable-speed, fixed-pitch wind turbines, while Chapter 6 tackles variable-speed, variable-pitch turbines. The gainscheduling technique used in each case depend on LPV systems. Speed and torque control loops are developed to achieve the control objectives presented in Chapter 4, and simulation results are presented for both simplified and realistic wind inputs. The controllers are also shown to be robust to uncertainties. Readers who are interested in applying control theory to wind turbines may wish to focus on Chapter 6, since the wind industry trend has been toward variable-speed, variable-pitch turbines rather than variablespeed, fixed-pitch turbines, especially at the utility scale.
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Readers unfamiliar with linear matrix inequalities, gainscheduling techniques, and LPV systems will be pleased to see that the authors provide more information on these topics in the appendices.
CONCLUSIONS Wind Turbine Control Systems provides an overview of wind turbines with an emphasis on various control objectives and LPV-based strategies for control. The book’s brevity does not allow for a thorough enough treatment to cover all aspects of the design of control laws for a wind turbine in the field. However, the text does provide an introduction that can be a useful starting point for such a design. Control engineers interested in more closely understanding the operation of wind turbines can consult [7] and [8]. Readers interested in other types of wind turbine control might wish to review the research conducted by control engineers at the National Renewable Energy Laboratory’s National Wind Technology Center [9]–[14].
[6] H. De Battista, P.F. Puleston, R.J. Mantz, and C.F. Christiansen, “Sliding mode control of wind energy systems with DOIG-power efficiency and torsional dynamics optimization,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 728–734, 2000. [7] J.F. Manwell, J.G. McGowan, and A.L. Rogers, Wind Energy Explained: Theory, Design, and Application. West Sussex, U.K.: Wiley, 2002. [8] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind Energy Handbook. New York: Wiley, 2001. [9] M.J. Balas, A. Wright, M.M. Hand, and K. Stol, “Dynamics and control of horizontal axis wind turbines,” in Proc. American Control Conf., Denver, CO, 2003. vol. 5, pp. 3781–3793. [10] K.A. Stol and M.J. Balas, “Periodic disturbance accommodating control for blade load mitigation in wind turbines,” J. Solar Energy Eng., vol. 125, no. 4, pp. 379–385, 2003. [11] K.A. Stol, “Disturbance tracking control and blade load mitigation for variable-speed wind turbines,” J. Solar Energy Eng., vol. 125, no. 4, pp. 396–401, 2003. [12] A.D. Wright, L.J. Fingersh, and M.J. Balas, “Testing state-space controls for the controls advanced research turbine,” J. Solar Energy Eng., vol. 128, no. 4, pp. 506–515, 2006. [13] B. Boukjezzar, L. Lupu, H. Siguerdidjane, and M.M. Hand, “Multivariable control strategy for variable speed, variable pitch wind turbines,” Renewable Energy, vol. 32, no. 8, pp. 1273–1287, 2007. [14] K. Johnson, L. Pao, M. Balas, and L. Fingersh, “Control of Variable-Speed Wind Turbines: Standard and adaptive techniques for maximizing energy capture,” IEEE Control Systems Mag., Vol. 26, No. 3, pp. 70–81, June 2006.
Kathryn E. Johnson
REFERENCES
REVIEWER BIOGRAPHY
[1] “Annual U.S. wind power rankings track industry’s rapid growth,” (May 7, 2007) [Online]. Available: http://www.awea.org/newsroom/releases/ Annual_US_Wind_Power_Rankings_041107.html [2] F.D. Bianchi, R.J. Mantz, and C.F. Christiansen, “Control of variablespeed wind turbines by LPV gain scheduling,” Wind Energy, vol. 7, no. 1, pp. 1–8, 2004. [3] H. De Battista, and R.J. Mantz, “Dynamical variable structure controller for power regulation of wind energy conversion systems,” IEEE Trans. Energy Conversion, vol. 19, no. 4, pp. 756–763, 2004. [4] F.D. Bianchi, R.J. Mantz, and C.F. Christiansen, “Power regulation in pitch-controlled variable-speed WECS above rated wind speed,” Renewable Energy, vol. 29, no. 11, pp. 1911–1922, 2004 [5] H. De Battista, R.J. Mantz, and C.F. Christiansen, “Dynamical sliding mode power control of wind driven induction generators,” IEEE Trans. Energy Conversion, vol. 15, no. 4, pp. 451–457, 2000.
Control Perspectives on Numerical Algorithms and Matrix Problems by A. BHAYA and E. KASZKUREWICZ
Although control theory has been a major source of numerical linear algebra problems [2], [3], this book SIAM, 2006, takes the opposite viewpoint, ISBN 0-89871-602-0, US$97.00. where basic concepts of control theory, such as feedback control, optimal control, and realization theory, are interpreted in terms of iterative methods, numerical analysis, and optimization. Most importantly, it is
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Kathryn E. Johnson received the B.S. degree in electrical engineering from Clarkson University in 2000 and the M.S. and Ph.D. degrees in electrical engineering from the University of Colorado in 2002 and 2004, respectively. In 2005 she held a postdoctoral research position at the National Renewable Energy Laboratory’s National Wind Technology Center. In October 2005, she was appointed Clare Boothe Luce assistant professor at the Colorado School of Mines in the Division of Engineering. Her research interests are in control systems and control applications, specifically wind energy.
shown that control-theoretic connections can lead to better numerical algorithms.
CONTENTS Chapter 2 establishes connections between iterative methods for root finding and feedback controllers. Specifically, it is shown how an iterative root-finding algorithm can be interpreted as a feedback controller applied to a dynamical system defined by the data of the problem. Of particular interest is the study of regions of convergence of Newton's method as well as the study of the effect of disturbances on Newton’s method by means of Lyapunov theory. A derivation of the classical conjugate gradient method from a proportional-derivative controller is also presented. In this context, it should be noted that the conjugate gradient method is a Krylov subspace method, while the relationship between Krylov subspaces and the controllability subspace is well known [1].
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Chapter 3 connects optimal control with structural iterative methods. This chapter begins with simple examples showing how certain dynamical systems with associated cost functions and boundary conditions lead to iterative methods for root finding. As stated in the chapter, “Of course, the proof of the pudding is in the eating, so whatever the choices are, the end result should be a robust, easily implementable algorithm.” After showing that the Kokotovic-Siljak method [4] for finding the zeros of a polynomial with complex coefficients is a type of variable-structure Newton’s method, the main topic of the chapter is discussed, namely, optimal control approaches for solving unconstrained optimization problems. It is shown how the problem of minimizing a convex function leads to an optimal control problem where the control enters into both the objective and constraint functions. Next, the algorithm used to compute the minimum of an unconstrained function is then interpreted as an inverse Lyapunov function problem. Several examples are given to support these statements. The chapter concludes with a discussion of an interesting fact given in [5], namely, that a class of unconstrained nonlinear programming problems can be transcribed into a class of discrete-time multistage optimal control problems, which can be solved using techniques of differential dynamic programming. Chapter 4 discusses neural networks and gradient dynamical systems (GDSs), which can be used to solve linear and quadratic programming problems. It is well known that constrained optimization problems can be transformed into unconstrained optimization problems using exact penalty functions and then solved by means of a GDS method. The key idea is to adjust the penalty parameters, interpreted as control parameters, in the resulting GDS, using a control Lyapunov function approach. Special GDS solvers with discontinuous right-hand sides, which arise in this formulation, are analyzed using a Persidiski-type control Lyapunov function. Some simple GDS solvers, such as support vector machines and k -winners-take-all networks used by the neural-network and pattern-recognition communities, are described in the above context. Chapter 5 explores the relationship between control and numerical methods for solving ordinary differential equations. Specifically, it is shown that proportional-integral control can be used to design adaptive step-size control algorithms, while shooting methods for boundary value problems can be recast as feedback control problems. A connection between shooting methods and iterative learning is also established. The chapter concludes with discussions of how matrix preconditioners for iterative methods can be related to decentralized control and also to Dstability arising in economics. The last topic is somewhat outside the scope of this chapter.
The last chapter shows several additional applications of control and systems theory to problems in numerical algorithms. These applications include the relationships between optimal control and Bezier curves, between optimal control and least-squares, and between feedback control and the QR iteration algorithm with shifts. Additional topics covered include the relationship between the stability of Runge-Kutta integration and the KalmanYakubovich-Popov lemma, probability and estimationtheoretic application of systems theory, and quantum mechanics and control theory. The purpose of this chapter is to convince the reader that the scope of the approach is wide and is not confined to the topics discussed in detail in this book.
OBSERVATIONS This book is beautifully written. Distinguished features include numerous examples to facilitate a clear understanding of the concepts; diagrams and tables; a summary along with notes and references at the end of each chapter; an interesting quotation from a famous researcher at the beginning of each chapter, which sets the tone of that chapter; and several hundred references. The book contains a wealth of material from several areas of science and engineering, including numerical linear algebra, numerical analysis, optimization theory, and systems theory. The book will be delightful reading by control theorists with modest background in numerical analysis and optimization. On the other hand, numerical analysts may require some background in control theory to appreciate the contents. B.N. Datta
REFERENCES [1] D. Boley and G.H. Golub, “The Lanczos-Arnoldi algorithm and Controllability,” Sys. Contr. Lett., vol 4, pp. 317–327, 1984. [2] B.N. Datta, Numerical Methods for Linear Control Systems—Design and Analysis. Boston, MA: Elsevier, 2004. [3] R.V. Patel, A. Laub, and P. Van Dooren, Eds., Numerical Linear Algebra Techniques for Systems and Control. New York: IEEE Press, 1993. [4] P. Kokotovic and D. Siljak, “Automatic analog solution of algebraic equations and plotting of root-loci by generalized Mitrovic method,” IEEE Trans. Ind. Applicat., vol. 83, pp. 324–328, 1964. [5] D.M. Murray and S.J. Yakowitz, “The application of optimal control methodology to nonlinear programming problems,” Math. Programming, vol. 21, pp. 331–347, 1981.
REVIEWER INFORMATION B.N. Datta is a professor of mathematical sciences and an adjunct professor of electrical and mechanical engineering at Northern Illinois University. His research interests blend linear and numerical linear algebra with control and systems theory as well as vibration and structural engineering. He is the author of two books and more than 100 papers.
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Optimal Estimation of Dynamic Systems by JOHN L. CRASSIDIS and JOHN L. JUNKINS.
Optimal Estimation of Dynamic Systems provides a comprehensive overview of various estimation methods and their application to aerospace vehicles. Although there is a great deal of focus in aerospace engineering curricula on controlling aerospace vehicles, Chapman & Hall/ CRC, 2004, less emphasis is placed on issues ISBN:1-58488-391-X, relating to estimation. This imbalUS$140. ance is undesirable because all control systems rely on some form of measurement or state information to perform properly. For example, aerospace vehicles are governed by nonlinear equations of motion, and typically the relationship between the states of the vehicle and the available measurements is also nonlinear. Once the nonlinear equations are linearized to design a linear control law, it then becomes critically important to understand the closed-loop coupling between the control law and the state estimator. To this end, Optimal Estimation of Dynamic Systems provides the aerospace engineer with the tools required to perform this analysis. This book fills a void in the available literature on estimation methods as applied to aerospace vehicles. Although several books develop estimation theory [1]–[4], as far as I am aware there are no books in print that focus on the application of modern estimation techniques to aircraft and spacecraft. Furthermore, many of the advances made in estimation theory in the area of attitude determination are available to students and practitioners only through the research literature. Much of that development is now available through this new text.
CONTENTS Optimal Estimation of Dynamic Systems covers various methods for solving both static parameter estimation problems and dynamic state estimation problems. The authors progress from basic least squares methods, through dynamic state filtering, and end with a discussion of the relationship between optimal control and estimation theory. Chapter 1 presents a review of basic least squares techniques, including linear, sequential, and nonlinear least squares. The terminology for the rest of the book is established, and several examples are provided. Chapter 2 introduces statistical properties in the least squares problem. Optimal estimation based on both minimum variance and maximum likelihood methods is presented. Chapters 1 and 2 provide generic presentations of least squares techniques without yet making specific applications to aerospace systems. As a result, these chapters serve as a
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general introduction to least squares for engineers working in various disciplines. Chapter 3 provides a basic overview of linear systems theory. Attitude kinematics and rigid body dynamics for a spacecraft are presented along with an overview of attitude representations. Orbital mechanics and aircraft flight dynamics are also briefly discussed. Although the notation used in Chapter 3 is, for the most part, clear and consistent, one exception is that the notation used to denote vector derivatives with respect to a particular coordinate system is the same as the notation used to identify the coordinate system in which a particular vector is resolved. For beginners, it is often difficult to understand the difference between frame-dependent differentiation and the resolution of vectors in a particular coordinate system. Care must be taken to distinguish these subtle concepts. Chapter 4 applies the techniques discussed in the first two chapters to several specific aerospace estimation problems. First, nonlinear least squares is applied to position estimation using GPS pseudorange measurements. Next, an example of spacecraft attitude estimation from star tracker observations is given. The terminology used in this example is somewhat confusing because the alignment angles of the onboard camera, which typically represent the orientation of the camera relative to the spacecraft, are not clearly distinguished from the Euler angles of the spacecraft. Next, Davenport’s q-method is derived, and an example is given demonstrating its use in spacecraft attitude estimation. Chapter 4 also includes the development of a nonlinear least squares method for spacecraft orbit determination along with an example. Maximum likelihood estimation is applied to aircraft parameter estimation, again with an example to demonstrate the method. Finally, Chapter 4 concludes with the eigensystem realization algorithm, which can be used to determine the mass, stiffness, and damping matrices for vibrational systems. Chapter 5 provides a comprehensive overview of sequential estimation. The chapter starts with a simple example of a first-order filter and continues to build complexity as more advanced filtering topics are introduced. The Kalman filter is developed in both continuous and discrete time. Stability is addressed as well as steady-state applications for time-invariant systems. The extended Kalman filter is developed, again in both continuous and discrete time. The chapter concludes with several advanced filtering topics. Each method presented in the chapter is nicely summarized in table form, providing a quick at-aglance reference for implementing the various methods. Several examples are also included. However, as the reader progresses through the book, the examples become less detailed in terms of the implementation of the algorithm but instead are focused on demonstrating the performance associated with each method. Chapter 6 presents advanced smoothing concepts. Fixed-interval, fixed-point, and fixedlag smoothers are all discussed in discrete and continuous
time. As with Chapter 5, the methods are summarized in table form. Chapter 7 extends the applications from Chapter 4 to sequential estimation algorithms, beginning with a GPS example. The authors provide a brief introduction to sidereal time and geodetic coordinates and then apply an extended Kalman filter to spacecraft position and velocity estimation using GPS pseudorange measurements. An example demonstrating the performance of the algorithm is then given. Next, an extended Kalman filter is developed for spacecraft attitude estimation, utilizing the multiplicative quaternion formulation. Orbit estimation is presented next, with the reader referred to the nonlinear fixed-point smoother in Chapter 6. Finally, target tracking and parameter estimation of aircraft are presented. Two target-tracking filters are developed; the first is a simple filter that estimates position and velocity, while the second expands the target-tracking state to include the acceleration. Again, an example highlights the application of the target-tracking filter. Parameter estimation is implemented through an extended Kalman filter. The book concludes with Chapter 8, which ties together optimal control and estimation. Following an overview of optimal control, the authors address the issues faced when coupling estimation and control. A section on loop transfer recovery, a methodology for achieving stability margins, is also included. The chapter concludes with an example of controlling a spacecraft during a large-angle slew maneuver. Every chapter of the book includes numerous homework problems, some of which are mysteriously marked with a cloverleaf. There are four appendices. The first appendix, which provides an overview of linear algebra, includes many useful matrix identities. The second appendix covers basic probability concepts, and the third appendix covers parameter optimization. Finally, the fourth appendix directs the reader to a Web site containing Matlab code. The availability of code for each example in the book is a welcome addition to the theoretical material. The book serves as a valuable reference for the practicing aerospace engineer working in the area of guidance,
navigation, and control (GNC). The book is also suitable as a textbook for graduate-level courses in estimation. Students should have a background in linear algebra, probability, and linear systems theory to take full advantage of the topics covered. The appendices offer a nice review of the necessary background material but likely would not serve as a replacement for the above-mentioned courses. I believe the book is too advanced for undergraduates, although the first part of the book on parameter estimation could possibly be used in an elective senior undergraduate course.
CONCLUSIONS Optimal Estimation of Dynamic Systems is an excellent, muchneeded text in estimation for aerospace vehicles. The authors apply their experience in the fields of estimation and control to provide a thorough reference for the practicing GNC engineer. The book will advance the understanding of estimation techniques for aerospace graduate students as well. The authors provide numerous examples, helpful tables showing the implementation of the various algorithms, and insights into the unique aspects of estimating the state parameters for both satellites and aircraft. It was a pleasure to review such a well-written book. Julie Thienel
REFERENCES [1] A. Gelb, Ed., Applied Optimal Estimation. Cambridge, MA: MIT Press, 1988; New York: Wiley, 2001. [2] R.G. Brown, and P.Y. Hwang, Introduction to Random Signals and Applied Kalman Filtering, 3rd ed. New York: Wiley, 1997. [3] P.S. Maybeck, Stochastic Models, Estimation, and Control, Volume 1. New York: Academic, 1979. [4] J.R. Wertz, Ed., Spacecraft Attitude Determination and Control. Norwell, MA: Kluwer, 2002.
REVIEWER INFORMATION Julie Thienel is an assistant professor at the U.S. Naval Academy in the aerospace engineering department. Previously, she worked at the NASA Goddard Space Flight Center for 20 years in spacecraft guidance, navigation, and control.
Unevenly Odd
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t was Vannevar Bush in electrical engineering who was Norbert's chief sponsor. Bush was a hands-on engineer with originality and drive in constructing what today would be called analogue computers—as for example Bush's MIT differential analyzer that serendipitously was available to grind out ballistic tables during World War II. Few geniuses in pure mathematics had the feel for physical models that Wiener did, and this made the Wiener-Bush team a fruitful one. Otherwise, they were something of an odd couple: Bush, the practical and laconic Yankee tinkerer, and Wiener the myopic parody of an absent-minded professor. It helped that electrical engineering in those pre-Laplace-transform days was unevenly heuristic and overdue for an infusion of Wiener's kind of rigor. — From P. A. Samuelson, “Some Memories of Norbert Wiener,” in The Legacy of Norbert Wiener: A Centennial Symposium, American Mathematical Society, 1997, p. 37.
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