In this issue of IEEE Control Systems Magazine, we bring

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1 » BOOKSHELF Innovations in Education and Practice In this issue of IEEE Control Systems Magazine, we bring you reviews of three books. The first book, by Kay, presents a highly pedagogical approach to teaching probability theory. The second book, by Wang, Lin, and Lee, is devoted to the use of a feedback relay for identifying points on the Nyquist plot in support of controller design. Finally, the third book, by Codrons, focuses on robust control for process applications. As usual, I welcome your suggestions for books to be reviewed, as well as volunteers to serve as reviewers. Scott Ploen Scott.R.Ploen@jpl.nasa.gov Springer, 2006, ISBN , US$59.95, 883 pages. INTUITIVE PROBABILITY AND RANDOM PROCESSES USING MATLAB by STEVEN KAY About a year ago, I was asked to teach a graduate course on probability and random processes at the New Jersey Institute of Technology (NJIT) and to consider adopting the new book by Steven Kay for the course. Knowing the author had already written student-friendly texts on estimation [1] and detection [2], I welcomed the idea. Nonetheless, a careful decision was still necessary before turning my back on standard engineering treatments such as [3] and [4]. Kay s book occupies a unique place in the overcrowded market of textbooks on probability and random processes. The philosophy of the book is defined by the two keywords framing the title on the cover, namely, Intuitive and Matlab. From the first chapter on, it is apparent that the use of intuition is explored through many examples and remarks, with ample use of Matlab simulations. Moreover, the approach is complemented by a gradual exposition of the usual subject matter, wherein entire chapters are devoted to topics that other books such as [1] and [2] typically spend much less time discussing. In particular, the author dedicates one chapter to each of the following fundamental topics: discrete random variables, expected values for discrete random variables, multiple discrete random variables, conditional probability mass functions, and discrete random vectors. The above sequence of topics is then repeated in as many chapters for the case of continuous-time random variables. TEACHING EXPERIENCES After using the book for two semesters at NJIT, I reached the conclusion that, due to personal tastes, the style of the book may not appeal to everyone. What is unquestionable is that the book is aimed as a textbook for an undergraduate or firstyear graduate course and not as a reference for researchers. In fact, as opposed to standard treatments of this material, the book combines theory and examples without explicitly drawing the line between the two. Furthermore, this book relies less heavily on analysis and does not cover advanced topics such as queuing theory and spectral estimation [5]. As an instructor, I am delighted by the rigorous, yet down-to-earth, introduction of basic concepts that are conventionally taken for granted, such as the relationship between a probabilistic model and computer-based Monte Carlo simulations. I admire the wide variety of carefully thought out examples, and I find the idea of introducing Matlab from the very first chapter rewarding. Notation-wise, the author strikes a good balance between consistency and intuition, even if the notation used in the discussion of sample spaces is unconventional. On the negative side, I soon realized that the highly fragmented presentation requires that the instructor synthesize a careful selection of topics from the myriad examples and remarks to avoid confusion. In my experience, the initial reaction that students have to this book is puzzlement over its unconventional style, which gives equal weight to both theoretical and practical aspects. However, students soon recognize that the approach taken is ultimately highly pedagogical. I have also observed that the large variety of applications presented in the book tend to give students the impression that the subject is more complex than it really is. In this sense, a clear summary at the end of each chapter pointing out connections between various topics would be helpful. I have also found that the overview at the beginning of each chapter is often not well received by students. As discussed above, this problem can be overcome by an appropriate synthesis of topics in class by the instructor. 100 IEEE CONTROL SYSTEMS MAGAZINE» JUNE X/07/$ IEEE

2 CONCLUSIONS This new textbook is a breath of fresh air in the market of books devoted to probability and random processes. The book lives up to its ambition of setting a new standard for a modern, computer-based treatment of the subject. Despite the issues discussed above, I fully recommend its use in undergraduate and first-year graduate courses. REFERENCES [1] S. Kay, Fundamentals of Statistical Signal Processing, Volume 1, Estimation Theory. Englewood Cliffs, NJ: Prentice Hall, [2] S. Kay, Fundamentals of Statistical Signal Processing, Volume 2, Detection Theory. Englewood Cliffs, NJ: Prentice Hall, [3] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd ed. Reading, MA: Addison-Wesley, [4] A. Papoulis and U. Pillai, Probability, Random Variables and Stochastic processes, 4th ed. New York: McGraw Hill, [5] S. Kay, Modern Spectral Estimation: Theory and Practice. Englewood Cliffs, NJ: Prentice Hall, Osvaldo Simeone REVIEWER INFORMATION Osvaldo Simeone is currently an adjunct professor and postdoctoral researcher at the New Jersey Institute of Technology, Newark. He received his Ph.D. in information engineering from Politecnico di Milano, Milan, Italy, in His current research interests are in information theory and signal processing aspects of wireless systems with emphasis on cooperative communications, MIMO systems, ad hoc wireless networks, cognitive radio, and distributed synchronization Springer, 2003, ISBN , US$171, 385 pages. RELAY FEEDBACK: ANALYSIS, IDENTIFICATION AND CONTROL by Q.-G. Wang, C. Lin, and T.H. Lee Oscillation is a fundamental property of many technological systems. Two essential components for structurally sustainable oscillation are nonlinearity and feedback. A simple example of a system that generates a periodic signal consists of a relay in feedback with a dynamical system. Since such systems are easy to implement with analog or digital devices, they have been widely used in many applications for more than a century. Analysis of relay feedback systems is therefore a classical topic in control theory. Early work was motivated by relays in electromechanical systems and simple models for dry friction. The classical textbook [1] discusses phaseplane analysis illustrated by several examples. Self-oscillating adaptive controllers based on relay feedback were developed in the 1960s. More recent applications include - modulators for analog-to-digital conversion, power electronic dc-dc converters, and various control systems such as variable structure control and hybrid control. In 1984, an auto-tuner for automatically tuning proportionl-integral-differential (PID) controllers through a relay feedback experiment was considered in [2] and subsequently tested in several industrial applications [3], [4]. This technique triggered substantial efforts in developing practical experiments and identification methods for tuning low-order control laws as well as interest in the analysis of relay feedback systems [5]. A linear system with relay feedback can be described as ẋ = Ax + Bu, (1) y = Cx, (2) u = sgn y, (3) where x is an n-dimensional vector, u and y are scalars, and A, B, and C are constant matrices. The relay is modeled as { 1, y > 0, sgn y = 1, y < 0. Since the sign function is discontinuous at y = 0, existence of solutions does not follow from the theory of ordinary differential equations. Instead, we rely on an abstract representation of (1) (3) given by the differential inclusion ẋ F(x), where the set-valued right-hand side is Ax B, Cx > 0, F(x) = Ax + B[ 1, 1], Cx = 0, Ax + B, Cx < 0. The interpretation of F(x) is that when x belongs to the switching plane {x : Cx = 0}, the time derivative of x can take any value in the set {Ax + Bu : u [ 1, 1]}. The particular choice of ẋ is made in such a way that the solution x :[0, ) R n has some desirable property, such as piecewise-continuous differentiability. There is an extensive literature on the relation between solutions of differential equations with discontinuous right-hand sides and their corresponding differential inclusions. A classical reference on these generalized solutions is [6] X/07/$ IEEE JUNE 2007 «IEEE CONTROL SYSTEMS MAGAZINE 101

3 If the solutions to the differential equation always traverse the switching plane, the solutions can, in many cases, be considered in the classical sense. However, if the solutions approach the switching plane tangentially, more care needs to be taken in the definition of the solution [7], [8]. For example, the classical solution of ẋ = sgn (x), x(0) = 1, does not extend beyond the time instant t when x(t) = 0. Furthermore, the two-dimensional example ẋ 1 = sgn (x 1 ) + 2sgn (x 2 ), ẋ 2 = 2sgn (x 1 ) sgn (x 2 ) has a classical solution that spirals toward the origin in finite time [6]. Relay feedback systems often give rise to limit cycles. The traditional approach to analyzing oscillations in relay feedback systems is through frequencydomain or state-space methods [9]. The describing function approach is a frequency-domain method that in many cases gives approximate conditions for stable limit cycles. Rigorous results can be obtained by considering the Poincaré map, which describes the state evolution of the system between two consecutive intersections of the switching plane. However, it is well known that relay feedback systems can exhibit complex limit cycles that require alternative mathematical tools [8], [10] [12]. The frequency and amplitude of the output of a relay feedback experiment reflects the dynamics of the plant P(s) = C(sI A) 1 B. For a stable plant P with positive steady-state gain and damped frequency response, the oscillation corresponds typically to the first intersection point of the Nyquist curve P(jω) with the negative real axis [2]. Although this single point is a crude estimate of P, it gives information about the system in a frequency range important for control design. For simple control laws such as PI and PID controllers, this information is often sufficient to tune the controller parameters to obtain adequate closed-loop performance. The autotuner is consequently based on a scheme in which the controller is first replaced with a relay, the amplitude and frequency of the oscillation is measured, the controller parameters are derived from these measurements, and, finally, the controller replaces the relay in the control loop. By inserting a filter in series with the plant, additional oscillation frequencies can be obtained. In this way, multiple points on the Nyquist curve can be identified, and thus a more accurate model of P(jω) is found. The cost of obtaining a higher fidelity model is longer experiment time, which may have implications in practice. For more elaborate control-design schemes, the excitation signal should be optimized to maximize the benefit of each experiment. However, many process control loops in industry can be improved considerably by the simple auto-tuner with a single relay experiment. CONTENTS OF THE BOOK Relay Feedback: Analysis, Identification and Control is an extensive text covering the analysis of oscillations in relay feedback systems, system identification based on relay feedback experiments, and controller design based on the identified models. The book is divided into three parts: (I) analysis of relay feedback systems, (II) process identification from relay feedback tests, and (III) controller design. Each part is divided into four or five chapters. Part I presents fundamental properties of singleinput, single-output (SISO) linear systems with relay feedback. Relay feedback systems that include time delays and relay hysteresis are also treated. As a result, the model structure given in (1) (3) is only a special case. Chapter 1 discusses the existence of solutions. This topic is important since relay feedback systems do not always have solutions in the classical sense. The book avoids complications resulting from solutions converging to the switching surface by making appropriate technical assumptions. The remaining three chapters of Part I deal with limit cycles. Specifically, Chapter 2 presents conditions on the existence of limit cycles, while chapters 3 and 4 give results on local and global stability of limit cycles. Local stability is derived through linearization of the Poincaré map. A global stability result is obtained by applying the contraction mapping theorem. The presentation in Part I is well written and easy to follow. Although an overview is given at the beginning of each chapter, the rest of the material is quite technical and consists of a collection of recent results reported by the authors in various papers. Several examples and figures are also used to help illustrate the development. Part II discusses system identification based on relay feedback experiments. In Chapter 5 the authors review the basic relay feedback experiment and some of its variants. The introduction of an extra relay in the feedback loop to enhance the excitation of the closed-loop dynamics is discussed. A decentralized relay experiment for multivariable plants is briefly introduced as well. Instead of computing a single point on the Nyquist curve, a frequency-response analysis is executed directly on the input and output plant data. The authors discuss the advantages of this technique in Chapter 6 and also discuss experimental results on two pilot plants, namely, a water tank laboratory process and a heat exchanger. To utilize design methods based on parametric models of the plant, Chapter 7 discusses methods for approximating frequency responses with low-order transfer functions. Chapter 8, the last chapter of Part II, presents an alternative method for identifying the closed-loop plant model. The material in Part II is less technical compared to Part I and should be straightforward for the practitioner to apply. Some more 102 IEEE CONTROL SYSTEMS MAGAZINE» JUNE 2007

4 discussion on the relation to the system identification literature would have been desirable. For example, how does the presented MIMO identification scheme compare to existing techniques for identification based on frequency-domain or time-domain data? Part III focuses on the design of linear controllers. As pointed out by the authors, there is a vast domain of applicable techniques, and thus there is no attempt in the book to cover them all. The text reviews internal model control (IMC) for SISO systems in Chapter 9 and MIMO systems in Chapter 10. Chapter 11 discusses a variant of IMC for unstable plants. Chapter 12 is on decentralized control. There is no specific link between Part III and Parts I or II other than the fact that decentralized relay experiments are briefly mentioned at the end of the last chapter. In fact, the focus on IMC is motivated by the general opinion that it is a popular control architecture in the process control industry. It would have been nice to see some further connection with the material of previous chapters. For example, can the model uncertainty introduced by a simpler relay experiment be compensated for by closed-loop design? Is there a tradeoff between experiment time and achievable control performance? WHO SHOULD READ THIS BOOK This book is suitable for both researchers and workers interested in obtaining an in-depth understanding of relay feedback systems and their application to automatic tuning of controllers. The book is a research monograph and consists of a detailed survey of recent papers by the authors on the analysis of relay feedback systems and their application to identification and control design. The book does not compare alternative approaches but rather is focused along a particular line of research. As a consequence, the book is probably not suitable as a stand-alone textbook for a graduate control course. Instead it is better suited as a complement to a nonlinear control course, a system identification course, or a course on control design. For the next edition of this book, it would be desirable to include some discussion on the limitations of the approach taken. Although a few related methodologies are listed in the introduction of each chapter, little discussion of the relative pros and cons of various methodologies is presented. For example, the extensive recent literature on iterative identification and control design [13] is not mentioned. Additional questions can also be raised, such as why relay experiments for MIMO plants, or SISO systems with multiple relays, are not treated in Part I. For example, systems with more than one relay can lead to interesting extensions of the results presented in Part I, with possible connections to the literature on hybrid systems. To conclude, the authors have produced a well-written and detailed text on relay feedback with applications to system identification and control design. The book presents an ambitious project that takes the reader from advanced mathematical theory on nonsmooth dynamical systems to tuning techniques directly applicable to industrial control systems. The book appeals to a diverse audience, from researchers in nonlinear control to practicing control engineers. REFERENCES [1] A.A. Andronov, S.E. Khaikin, and A.A. Vitt, Theory of Oscillators. Oxford: Pergamon Press, [2] K.J. Åström and T. Hägglund, Automatic tuning of simple regulators with specifications on phase and amplitude margins, Automatica, vol. 20, no. 5, pp , [3] K.J. Åström and T. Hägglund, Advanced PID Control. Research Triangle Park, NC: Instrument Society of America, [4] C. Knospe, PID control: Introduction to the Special Section, IEEE Contr. Sys. Mag., vol. 26, no. 1, pp , [5] C.-C. Yu, Autotuning of PID Controllers: Relay Feedback Approach. New York: Springer-Verlag, [6] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. New York: Kluwer Academic, [7] M. di Bernardo, K.H. Johansson, and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, Int. J. Bifurcations Chaos, vol. 11, no. 4, pp , Apr [8] K.H. Johansson, A. Rantzer, and K.J. Åström, Fast switches in relay feedback systems, Automatica, vol. 35, no. 4, pp , Apr [9] Y.Z. Tsypkin, Relay Control Systems. Cambridge, U.K.: Cambridge Univ. Press, [10] K.H. Johansson, A. Barabanov, and K.J. Åström, Limit cycles with chattering in relay feedback systems, IEEE Trans. Automt. Contr., vol. 47, no. 9, pp , [11] S. Varigonda and T.T. Georgiou, Dynamics of relay relaxation oscillators, IEEE Trans. Automat. Contr., vol. 46, no. 1, pp , [12] J. Goncalves, A. Megretski, and M. Dahleh, Global stability of relay feedback systems, IEEE Trans. Automat. Contr., vol. 46, no. 4, pp , [13] H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin, Iterative feedback tuning: Theory and applications, IEEE Contr. Sys. Mag., vol. 18, no. 4, pp , Karl H. Johansson REVIEWER INFORMATION Karl H. Johansson received a Ph.D. in electrical engineering in 1997 from Lund University in Sweden. He is an associate professor at the School of Electrical Engineering, Royal Institute of Technology, Sweden, and holds a senior researcher position at the Swedish Research Council. His research interests are in networked control systems, hybrid and embedded control, and control applications in automotive, automation, and communication systems. He received the Young Author Prize from IFAC in 1996 and the Peccei Award from the International Institute of System Analysis, Austria, in JUNE 2007 «IEEE CONTROL SYSTEMS MAGAZINE 103

5 Springer, 2005, ISBN , US$129.00, 229 pages. PROCESS MODELLING FOR CONTROL A UNIFIED FRAMEWORK USING STANDARD BLACK-BOX TECHNIQUES by BENOIT CODRONS Process Modelling for Control addresses a key issue in modelbased control design, namely, how one identifies both a nominal model and an uncertainty region associated with the model that can be used to design a high-performance controller. Although model identification has been addressed in classical textbooks [1], to my knowledge this book is unique in that it presents the first comprehensive study on identification of models for robust control design and validation. However, industrial control practitioners may find the mathematical concepts difficult to apply. Thus, the target audience for this book is mostly academics in the control field. The main message conveyed in this book is that identification and validation of process models used for control design is most effective in a closed-loop setting. The author shows, through theoretical arguments and examples, that closed-loop identification generally results in the identification of smaller uncertainty sets as compared to sets identified in open loop, resulting in less conservative designs. On the other hand, several cases are discussed where closed-loop identification must be used with caution since it can lead to inaccurate results due to singularities in the closed-loop transfer functions. The book presents several numerical examples to illustrate the topics discussed and to confirm expected results. In general, I found the examples to be excellent in referring to practical situations that illustrate the theoretical arguments presented in the book. However, one cannot ignore the fact that these are simulation studies; examples with real experimental data based on industrial operation are not given. CONTENTS The book consists of seven chapters. The introduction poses the key question addressed in the book, namely, how to identify a model that is geared toward control design. While this chapter presents a brief summary of the relevant literature, it misses some classical work on the structured singular value [2], [3]. Chapter 2 covers the theoretical background necessary for understanding the later chapters. Among the topics covered are linear fractional transformations, coprime factorizations, robust stability tests based on the gap metric, identification theory, and model-reduction methodologies. For practitioners, this section can be skipped and referred back to as needed. Chapter 3 presents the rationale for identifying closedloop models and model errors by using bias and variance formulas. Equation (3.16) for calculating the variance is of key importance. This equation shows that, due to the weighting of the transfer function estimates by the sensitivity function, the estimates identified in closed-loop experiments have smaller variance than estimates obtained in the open loop. This fact is especially true near the crossover frequency, which is of critical importance for control design. The author notes that the same model accuracy can be obtained in open-loop identification by providing more power to the excitation signal near the crossover frequency. However, the merit in closed-loop identification is that it relieves the practitioner from designing a special excitation signal different from a standard white noise input. Chapter 4 discusses the specific problem of closedloop identification in the presence of singularities of the transfer function near specific frequencies. The chapter shows that closed-loop identification produces more accurate models. Caution must be applied in selecting excitation signals in the presence of singularities. Chapter 5 addresses model and controller validation, which is one of the main topics of the book. A key element of this chapter is the identification of uncertainty sets. As a chemical process control researcher and practitioner, I found that one drawback in the text is that it does not address process nonlinearity as a source of model error in process control. Nonlinearity is a key issue in chemical processes due to nonlinear reaction kinetics, multiplicity of steady states, and highly nonlinear dependencies of thermo-physical properties with respect to state variables. Consequently, many researchers in the chemical process control field approximate nonlinear models by nominal linear models with some associated model uncertainty to account for nonlinear behavior. However, the book is deficient with respect to the identification of uncertainty sets when the source of model uncertainty is due to nonlinearities rather than noise or truncation errors due to insufficient data. The issue of nonlinearity is addressed in the literature by identifying a different linear model with associated uncertainty around each operating condition. A larger uncertainty set is then constructed as the union of the identified models and uncertainty sets [4]. Also, the techniques discussed in this book are applicable only to linear time-invariant systems. This restriction is a major limitation in chemical process control since time-varying models are widely used to model chemical processes. At the end of Chapter 5, the model validation steps are summarized. Step 5 consists of verifying the worstcase stability and performance with a newly designed controller K. If the validation is satisfactory, the controller is implemented in the process. However, the uncertainty must be re-evaluated with K to guarantee that the uncertainty identified with K does not change 104 IEEE CONTROL SYSTEMS MAGAZINE» JUNE X/07/$ IEEE

6 significantly with respect to the uncertainty sets identified with the controller used for model identification. Chapter 6 discusses and compares model-reduction methodologies. Section compares control designs based on reduced models obtained from high-order models identified from data with control designs based on loworder models identified directly from closed-loop experiments. Direct identification of low-order models is favored by practitioners in the chemical process control field due to its simplicity. Identification of high-order models with subsequent reduction can be prohibitive when dealing with the large multi-input, multi-output (MIMO) problems prevalent in the chemical industry. The author correctly points out that a disadvantage of loworder models directly identified from data is that one cannot accurately simulate and compare the response of the system with different controllers. However, a robust control design can be obtained by identifying a low-order model with a corresponding uncertainty description. Some of the numerical results presented in Chapter 6 are unclear. For instance, the stability or offset cancellations shown in Figure 6.11 using controller K 4 are not obvious. The response seems to be unstable. Similarly, the results in Figure 6.14 seem to indicate that the controller K 2 does not have integral action. However, I believe that integral action is imposed through the choice of the weight transfer function W out given on p It is interesting that the performance of various controllers given in the examples is evaluated using timedomain metrics such as overshoot and offset for control designs based on frequency-dependent weights. Although one can relate frequency-dependent behavior to timeresponse characteristics by assuming second-order behavior, this connection may not be accurate for higher order systems. Thus, despite the approach to model reduction presented in Chapter 6, trial-and-error iteration is needed to choose the frequency-dependent performance weights that result in the desired time-response characteristics. Chapter 7 summarizes the book and suggests future challenges, referred to as missing links. For example, validation tools for multivariable systems are classified as a missing link. The structured singular value, although not mentioned in this context, may be useful in this regard. Additional missing links refer to the fact that the models and uncertainty sets discussed are used for validation but not for control design. I believe that these validation tools can be used to find both better controllers and corresponding models by conducting successive iterations of modelplus-uncertainty identification and control design. Alternatively, as mentioned above, the system can be identified in open loop and then, based on the open-loop identified model, one can design an optimal controller using robust control design techniques. However, as shown in the book, open-loop identification produces larger uncertainty sets, leading to potentially more conservative designs. REFERENCES [1] L. Ljung, System Identification: Theory for the User, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, [2] R. Smith, Model validation for robust control: An experimental process control application, Automatica, vol. 31, no. 11, pp , [3] R. Smith and J. Doyle, Model validation: A connection between robust control and identification, IEEE Trans. Automat. Contr., vol. 37, no. 7, pp , [4] C. Webb, H. Budman, and M. Morari, Identification of uncertainty bounds for robust control with applications to a fixed bed reactor, Ind. Eng. Chem. Res., vol. 34, pp , Hector M. Budman REVIEWER INFORMATION Hector M. Budman is a professor of chemical engineering at the University of Waterloo, Ontario, Canada. He received his B.S., M.A.Sc., and Ph.D. degrees in mechanical engineering at the Technion, Israel Institute of Technology, Haifa, Israel. From 1988 to 1990 he was a postdoctoral fellow at the California Institute of Technology, and from 1990 to 1992 he was a project manager at the Noranda Research Centre in Montreal, Canada. During 1999 he held a visiting professor fellowship in the Chemical Engineering Department at the Technion, Israel. His main areas of research are in modeling and robust control of chemical systems. JUNE 2007 «IEEE CONTROL SYSTEMS MAGAZINE 105