- Email: [email protected]
Uncertainty and Feedback—H∞ Loop-shaping and the ν -Gap Metric, Glenn Vinnicombe, Imperial College Press, London, 2001, ISBN: 1-86094-163-X, 316pp., US$
Automatica 41 (2005) 1105 – 1108 www.elsevier.com/locate/automatica
Book reviews
Uncertainty and Feedback—H∞ Loop-shaping and the -Gap Metric, Glenn Vinnicombe, Imperial College Press, London, 2001, ISBN: 1-86094-163-X, 316pp., US$ Uncertainty and feedback have been at the centrestage of control science and engineering for a long time simply because uncertainty is unavoidable in any practical control design and there is a need for a system to be insensitive to any uncertainty and disturbance. In particular, since the 1980s, we have witnessed the significant development of H∞ control from theory to practice. Indeed, the robust and H∞ control has been very well documented by a number of books (Francis, 1987; Zhou, Doyle, & Glover, 1996; Green & Limebeer, 1995). One of the critical questions in robust control is how much knowledge is required in order to design a feedback controller that will give rise to a closed-loop system which is insensitive to what we do not know about the system. The well-known small gain theorem has characterized a set of stable uncertain dynamics for which a control system is able to preserve its stability/performance. The question is to what extent we go beyond the small gain theorem, for example, to allow some unstable uncertainty. The book by Vinnicombe is aimed at introducing tools that will help differentiate between uncertainties that are crucial to the feedback problem and uncertainties that are automatically taken care of by any reasonable feedback design. In this regard, the -gap metric, is introduced as a measure of uncertainty and homotopy arguments that only require uncertainties that are bounded on the imaginary axis are applied. The H∞ control paradigm is adopted, and in particular, the H∞ loop-shaping performance of McFarlance and Glover (1990) is used as an indicator of performance/stability margin. The book establishes a precise relationship between the H∞ loop-shaping performance of a given system and the robust performance of a family of systems centred around the given plant with a distance measured by the -gap metric. The strength of the book lies in its mathematical rigorousness and precise exposition of theoretical development. To help readers understand the basic concepts hiding behind the mathematical development, illustrative examples are given. Notes and references are provided in each chapter. In my view, the book, on the whole, is well written. It is a welcome addition to existing postgraduate level of textbooks by Green and Limebeer (1995) and Zhou et al. (1996) and is a
valuable reference for postgraduate students and researchers who pursue an in-depth understanding of uncertainty and feedback control. While the book is mainly focused on robust analysis, some design examples are given for illustration purpose. A notable limitation is that the book provides little detailed account of how the controller designs are carried out or why the controllers are chosen in many examples given. Weight selection, being an important issue in a practical control design, is not sufficiently explained in Chapters 2–4. For example, Chapter 4 discusses the issue of selecting weights to maximize the H∞ loop-shaping performance. Examples are expected to give a better illustration of how the guidelines proposed can be implemented and how the selection helps achieve better system performance. Following is a brief overview of individual chapters. Chapter 1 starts with a review on the robust stability analysis of feedback systems and the solution to the H∞ control. Stability analysis results using the two important tools, namely, the small gain theory and homotopy arguments, are given for feedback systems with various forms of uncertainties. The concept of the -gap metric is introduced. Chapter 2 is dedicated to the development of tight bounds on closed-loop sensitivity and complementary sensitivity functions of the loop-shaping design (McFarlance & Glover, 1990). A geometric interpretation of the loop-shaping performance is given through the Riemann sphere. The optimal H∞ loop-shaping controller is shown to be maximally robust to perturbations measured by the gap and -gap metrics. The relationship between the loop-shaping performance and the classical gain and phase margins is also established. Chapter 3 gives an in-depth study of the -gap metric, a natural dual of stability margin. It concentrates on establishing a fundamental relationship between the performance/stability robustness measure and the -gap metric. Examples are used to demonstrate how the -gap metric is useful in differentiating uncertainties that really matter while ignoring those that any reasonable feedback controller will be able to take care of. Chapter 4 focuses on the study of the problem of how to select proper weights to maximize the stability margin. First, an explicit calculation of the optimal stability margin as well as the corresponding optimal controller is given.
1106
Book reviews / Automatica 41 (2005) 1105 – 1108
A frequency response interpretation of the optimal stability margin leads to a guideline for choosing the weights. The problem of robust tracking is also examined with a tracking error bound derived in terms of the stability margin and the -gap metric. Chapter 5 discusses the issues of complexity and robustness of control design. The problem of concern is how to choose one with the lowest complexity in certain sense. In the chapter, the complexity of a controller is measured in terms of the complexity of its frequency response. It is noted that the optimization of controller under the constraint of a given measure of complexity is difficult. Chapter 6 presents two case studies. Chapter 7 examines the quantitative differences between the -gap metric and the gap metric. Chapter 8 studies approximation in the graph topology. It starts with the justification that the -gap is an appropriate measure for model or controller approximation (reduction) and shows that the optimally robust controller in the -gap metric is the same as that for the optimal H∞ loop-shaping, and no nonlinear, time-varying controller can do better. Two approximation schemes based on Hankel norm approximation are given together with lower and upper bounds on the smallest achievable error. Chapter 9 discusses the problem of characterizing the largest possible uncertainty set for a given controller that achieves an H∞ norm bound for a given plant. Using the homotopy arguments, the largest set, which includes possible unstable perturbations, is given in a computable manner. A precise condition for a perturbation to be admissible is given, which leads to the known results for the cases of additive and multiplicative uncertainties.
Finally, two appendices provide state-space formulae for calculating a coprime factor and the -gap metric, proofs of lemmas from Chapter 7, and singular value inequalities. References Francis, B. A. (1987). A course in H∞ control. Lecture notes in control and information science. Green, M., & Limebeer, D. J. N. (1995). Linear robust control. Englewood Cliffs, NJ: Prentice-Hall. McFarlance, D. C., & Glover, K. (1990). Robust controller design using normalized coprime factor plant descriptions. Lecture notes in control and information sciences. Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control. Upper Saddle River, NJ: Prentice-Hall.
Lihua Xie School of Electrical and Electronic Engineering, BLK S2, Nanyang Technological University, Singapore 639798 E-mail address: [email protected] About the Reviewer Lihua Xie is a Professor in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He received the B.E. and M.E. degrees in Electrical Engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and the Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia, in 1992. His current research interests include robust control, networked control systems, control of disk drive systems and smart structures, and estimation theory with applications in autonomous vehicles. He authored and co-authored one book and many articles. He is currently an Associate Editor of IEEE Transactions on Automatic Control, International Journal of Control, Automation and Systems and Journal of Control Theory and Applications. He served as an Associate Editor of the Conference Editorial Board, IEEE Control Systems Society, from 2000 to 2004.
doi:10.1016/j.automatica.2005.01.006
Nonsmooth Mechanics, Bernard Brogliato, Springer, New York, Heidelberg, Berlin, 1999, Communications and Control Engineering, ISBN 1-85233-143-7 124.95 Euro; 552pp. Nonsmooth mechanics is the study of classical mechanics, i.e., the mechanics of particles and rigid bodies moving subject to Newton’s and Euler’s laws, where one allows occurrences of impacts and other instantaneous changes in the dynamical behaviour of a system. This book is a survey of this intricate subject. In this review, I will attempt to describe the sorts of problems that are addressed in the book, as well as say something about the style in which they are presented. Since nonsmooth mechanics involves impacts, one needs a theory of differential equations able to handle these. The basic approach to this taken by Brogliato is to use differential equations where the nonsmoothness in time is modelled by the derivative, in the sense of the theory of distributions of Schwartz (1996), of a function of bounded variation. Of course, in a mechanical model, one does not usually know
the time of the impacts beforehand, but rather they arise as governed by the dynamical equations modelling the physical system. This is presented in this book in a formulation using unilateral constraints, as these are what arise most commonly in mechanics. For models of this type, some of the useful developments of smooth differential equations, e.g., existence and uniqueness of flows and continuity with respect to initial conditions, are adapted to this nonsmooth framework. The standard theorems do not transfer verbatim to the nonsmooth case. For example, one needs extra conditions to ensure uniqueness of solutions. In Chapter 1, where this theory of differential equations with impacts is presented, Brogliato initiates a pattern that recurs throughout the book. Namely, he points out connections to or from nonsmooth mechanics from or to other fields. For example, in Chapter 1, approaches involving hybrid systems, descriptor variable systems, the theory of outputs for control systems, and Filippov’s methods are touched upon. While ultimately beneficial to the book’s purpose, these “diversions” do take some getting used to.
Book reviews / Automatica 41 (2005) 1105 – 1108
One of the more challenging problems in nonsmooth mechanics concerns the modelling of impacts. Part of the difficulty is that one has to address matters concerning the materials of which a system is comprised; this is simply not a concern in mechanics without impacts, as long as a rigid approximation invalid. In Chapters 2 and 4, Brogliato addresses the matter of modelling impacts. The simplest situation, as far as understandability of the model is concerned, involves approximating an impact with a sequence of springs and dampers between the objects in contact. In the limit, provided one maintains some compatibility conditions between the parameters, the solutions converge to that of a possible impact model. By such means, one can model completely elastic impacts (i.e., those where all energy is instantaneously dissipated at impact), completely elastic impacts (i.e., those where no energy is dissipated during the impact), and intermediate situations as limits of smooth processes. Another method, called the method of penalising functions, considers the equations of motion with a sequence of forcing terms added. One wishes that the sequence of forcing terms have the property that the limiting solutions are well-defined and satisfy the unilateral constraint. These matters are addressed in Chapter 2. The matter of determining coefficients of restitution, and of incorporating friction models in impacts is considered in Chapter 4. Whenever one is dealing with mechanics, it is important to acknowledge that one is modelling actual systems whose behaviour can be observed. Therefore, it is essential that the mathematical/physical/metaphysical models one employs agree with observations. On the other hand, a purely empirical approach, where one cooks up a model to agree with experimental data, is ultimately unsatisfying. Thus, this bridge needs to be gapped. I found myself wishing that these issues were addressed more in the book. However, perhaps these is less a deficiency of the book, and more a reflection of the current state of the research. In mechanics, variational principles are seen by many as being King. Brogliato gives a chapter on including impacts in a variational analysis. Two main variational approaches are considered. One is Gauss’s principle, which is an instantaneous variational principle. The approach taken to Gauss’s Principle leads directly to the complementarity formulations for multiple unilateral constraints considered in Chapter 5 of the book. The second approach is the more standard one in variational mechanics, which is Hamilton’s principle. Here one needs to modify the spaces of curves one allows in the analysis. Brogliato also points out an interesting connection between variational methods for systems with unilateral constraints and the Pontryagin maximum principle. At the beginning of the chapter is a discussion of virtual displacements of the sort that I have always found unsatisfying; the concepts of “real” and “virtual” do not seem to me to be given a precise definition; at least, I cannot understand what they mean. However, apart from this, I found this chapter an interesting one.
1107
g
Fig. 1.
The matter of multiple unilateral constraints is the topic of Chapter 5, and this is a long chapter, almost 90 pages. The problems considered are best described by using the same simple example as used by Brogliato, and that I show in Fig. 1. Note that there are two unilateral constraints. The main problem of multiple unilateral constraints is, “Which of the constraints is active at any given moment?” In the example, we could ask whether one, both, or neither of the constraints is active. If the initial condition is as shown, with zero velocity, then it is clear that the “bottom” constraint will be active, and the “top” one inactive. Another simple example considered by Brogliato is that of three balls simultaneously in contact (perhaps known to the reader as “Newton’s cradle”). For more complicated systems, one needs a systematic way of figuring out which constraints are active. The first technique presented is the sweeping process of Moreau (1986). In this formulation, convexity plays a key rôle. The other formulation considered is the complementarity formulation, where one studies systems that are formed from differential equations, algebraic equations, and algebraic inequalities. In either of these formulations, one must develop theories of existence and uniqueness of solutions. The story here is quite a complicated one, as can be seen by the detailed treatment given to the example of a rod falling on the plane (called Painlevé’s example). In Chapter 6, Brogliato considers a configuration space formulation of impact modelling. Here, the system is thought of as being described by its configuration space, and thus the motion of the system is described by a point moving in configuration space. In this formulation, one needs to understand the rôle of the kinetic energy metric in impact phenomenon, since it is with respect to it that notions of orthogonality must be defined. Both single and multiple unilateral constraints are considered in this setting. The first six chapters in the book can be seen as having to do primarily with modelling, and with understanding the properties of the models. The final two chapters are
1108
Book reviews / Automatica 41 (2005) 1105 – 1108
concerned with stability and control theory. The central idea in the stability treatment is Lyapunov analysis. However, the presence of nonsmoothness, of course, requires one to change the approach significantly, introducing concepts from discrete dynamics to accommodate impacts. This appears to be a presently active area of research. Also considered in the stability treatment are some bifurcation theory and stability analysis for compliant impact models. After a brief discussion of controllability, the majority of Chapter 8 on feedback control is dedicated to stability and tracking using feedback. One of the central concerns in these problems is the use of feedback to ensure that impacts are handled smoothly. There appears to be a great deal of work to be done in this area. For example, Brogliato mostly addresses the fully actuated case, i.e., the common situation for robotic manipulators. Finally, there are four appendices in the book dealing with distribution theory, measure theory, functions of bounded variation, and convex analysis. One of the things that will make this book difficult to read for beginners is the unevenness of the mathematical background. Perhaps the author cannot be blamed for this; to have presented all of the mathematical background, even in a mostly incomplete form, would have rendered the book unmanageable. Nonetheless, the reader should be aware of some of the gaps arising in the mathematical treatment. Let me give three examples that occur in the appendices. Definition A.2 says that a distribution is a continuous linear function on the set D of infinitely differentiable functions with compact support. However, there is no hint at what “continuous” means, thus rendering the definition somewhat impotent. It is only three pages later, in another section, that the reader is warned in a side comment that the topology on D is relevant, although it is still not defined. In like manner, all that is said about a topological space is that it is a set, along with a collection of subsets, called open sets, “that satisfy certain properties.” Without knowing what these properties are, the idea of a measurable function, and therefore ultimately the idea of the Lebesgue integral, is not defined. And, the last nitpicky point I shall give, the Lebesgue measure is defined only on Borel sets. This is not logically incorrect, but it also does not give the usual Lebesgue theory, where one deals with the additional, but standard, complication of having to complete the measure. (Note that without a measure being complete, one loses the useful fact that a function that is a.e. equal to an integrable function is itself integrable.) These matters are all easily addressed by the reader going to other sources to fill in all of the details. But the point is that the reader will have to do this, particularly if they are initially unfamiliar with the mathematical background.
I shall close by making some general comments on the style of the book. To a reader, such as myself, who has grown accustomed to having theorems serve as waypoints in navigating technical material, Brogliato’s book is often difficult to read. There are relatively few precisely stated results, and many long text passages that required much effort for me to comprehend. However, this is certainly as much a reflection on my own deficiencies as a reader as they are on the writing style. I know that certainly many readers will be quite comfortable with the style of presentation. Nonetheless, when you add in my comments above concerning the treatment of the mathematical background, I think it is fair to say that the book’s style is not for everyone, nor for every purpose. However, as long as one understands the book’s rôle, I think it is a very useful contribution to the literature. The book’s two best features, in my view, are (1) its detailed survey of the literature (there are over 1000 items in the bibliography) and (2) its detailed presentation of many examples illustrating both the techniques and their limitations. And, as I mentioned above, I also like the way the book makes contact between the main subject matter, and other topics that are not obviously related. For readers interested in the field, this book will serve as an excellent introductory survey.
References Moreau, J.-J. (1986). Standard inelastic shocks and dynamics of unilateral constraints. In G. del Piero, & F. Maceri, F. (Eds.), Unilateral problems in structural analysis, CISM courses and lectures, (Vol. 288, pp. 173–221). New York, Heidelberg, Berlin: Springer. Schwartz, L. (1996). Théorie des distributions. Hermann, Paris: Publications de l’Institut de Mathématique de l’Université de Strasbourg.
Andrew D. Lewis Department of Mathematics and Statistics, Queen’s University, Kingston, Ont., K7L 3N6, Canada K7L 3N6 E-mail address: [email protected] About the Reviewer Andrew Lewis received his undergraduate degree in Mechanical Engineering from the University of New Brunswick in 1987, and his M.Sc. and Ph.D. in 1988 and 1995, respectively, both in Applied Mechanics from the California Institute of Technology. From 1995 to 1996 he was a Postdoctoral Fellow in Control and Dynamical Systems at the California Institute of Technology, and from 1996–1998 he was a Postdoctoral Fellow in the Mathematics Department at the University of Warwick. He is now an Associate Professor in the Mathematics and Statistics Department at Queen’s University in Kingston, Ont. in Canada. His research interests include geometric mechanics and differential geometric control theory. doi:10.1016/j.automatica.2005.01.005
Book reviews
Uncertainty and Feedback—H∞ Loop-shaping and the -Gap Metric, Glenn Vinnicombe, Imperial College Press, London, 2001, ISBN: 1-86094-163-X, 316pp., US$ Uncertainty and feedback have been at the centrestage of control science and engineering for a long time simply because uncertainty is unavoidable in any practical control design and there is a need for a system to be insensitive to any uncertainty and disturbance. In particular, since the 1980s, we have witnessed the significant development of H∞ control from theory to practice. Indeed, the robust and H∞ control has been very well documented by a number of books (Francis, 1987; Zhou, Doyle, & Glover, 1996; Green & Limebeer, 1995). One of the critical questions in robust control is how much knowledge is required in order to design a feedback controller that will give rise to a closed-loop system which is insensitive to what we do not know about the system. The well-known small gain theorem has characterized a set of stable uncertain dynamics for which a control system is able to preserve its stability/performance. The question is to what extent we go beyond the small gain theorem, for example, to allow some unstable uncertainty. The book by Vinnicombe is aimed at introducing tools that will help differentiate between uncertainties that are crucial to the feedback problem and uncertainties that are automatically taken care of by any reasonable feedback design. In this regard, the -gap metric, is introduced as a measure of uncertainty and homotopy arguments that only require uncertainties that are bounded on the imaginary axis are applied. The H∞ control paradigm is adopted, and in particular, the H∞ loop-shaping performance of McFarlance and Glover (1990) is used as an indicator of performance/stability margin. The book establishes a precise relationship between the H∞ loop-shaping performance of a given system and the robust performance of a family of systems centred around the given plant with a distance measured by the -gap metric. The strength of the book lies in its mathematical rigorousness and precise exposition of theoretical development. To help readers understand the basic concepts hiding behind the mathematical development, illustrative examples are given. Notes and references are provided in each chapter. In my view, the book, on the whole, is well written. It is a welcome addition to existing postgraduate level of textbooks by Green and Limebeer (1995) and Zhou et al. (1996) and is a
valuable reference for postgraduate students and researchers who pursue an in-depth understanding of uncertainty and feedback control. While the book is mainly focused on robust analysis, some design examples are given for illustration purpose. A notable limitation is that the book provides little detailed account of how the controller designs are carried out or why the controllers are chosen in many examples given. Weight selection, being an important issue in a practical control design, is not sufficiently explained in Chapters 2–4. For example, Chapter 4 discusses the issue of selecting weights to maximize the H∞ loop-shaping performance. Examples are expected to give a better illustration of how the guidelines proposed can be implemented and how the selection helps achieve better system performance. Following is a brief overview of individual chapters. Chapter 1 starts with a review on the robust stability analysis of feedback systems and the solution to the H∞ control. Stability analysis results using the two important tools, namely, the small gain theory and homotopy arguments, are given for feedback systems with various forms of uncertainties. The concept of the -gap metric is introduced. Chapter 2 is dedicated to the development of tight bounds on closed-loop sensitivity and complementary sensitivity functions of the loop-shaping design (McFarlance & Glover, 1990). A geometric interpretation of the loop-shaping performance is given through the Riemann sphere. The optimal H∞ loop-shaping controller is shown to be maximally robust to perturbations measured by the gap and -gap metrics. The relationship between the loop-shaping performance and the classical gain and phase margins is also established. Chapter 3 gives an in-depth study of the -gap metric, a natural dual of stability margin. It concentrates on establishing a fundamental relationship between the performance/stability robustness measure and the -gap metric. Examples are used to demonstrate how the -gap metric is useful in differentiating uncertainties that really matter while ignoring those that any reasonable feedback controller will be able to take care of. Chapter 4 focuses on the study of the problem of how to select proper weights to maximize the stability margin. First, an explicit calculation of the optimal stability margin as well as the corresponding optimal controller is given.
1106
Book reviews / Automatica 41 (2005) 1105 – 1108
A frequency response interpretation of the optimal stability margin leads to a guideline for choosing the weights. The problem of robust tracking is also examined with a tracking error bound derived in terms of the stability margin and the -gap metric. Chapter 5 discusses the issues of complexity and robustness of control design. The problem of concern is how to choose one with the lowest complexity in certain sense. In the chapter, the complexity of a controller is measured in terms of the complexity of its frequency response. It is noted that the optimization of controller under the constraint of a given measure of complexity is difficult. Chapter 6 presents two case studies. Chapter 7 examines the quantitative differences between the -gap metric and the gap metric. Chapter 8 studies approximation in the graph topology. It starts with the justification that the -gap is an appropriate measure for model or controller approximation (reduction) and shows that the optimally robust controller in the -gap metric is the same as that for the optimal H∞ loop-shaping, and no nonlinear, time-varying controller can do better. Two approximation schemes based on Hankel norm approximation are given together with lower and upper bounds on the smallest achievable error. Chapter 9 discusses the problem of characterizing the largest possible uncertainty set for a given controller that achieves an H∞ norm bound for a given plant. Using the homotopy arguments, the largest set, which includes possible unstable perturbations, is given in a computable manner. A precise condition for a perturbation to be admissible is given, which leads to the known results for the cases of additive and multiplicative uncertainties.
Finally, two appendices provide state-space formulae for calculating a coprime factor and the -gap metric, proofs of lemmas from Chapter 7, and singular value inequalities. References Francis, B. A. (1987). A course in H∞ control. Lecture notes in control and information science. Green, M., & Limebeer, D. J. N. (1995). Linear robust control. Englewood Cliffs, NJ: Prentice-Hall. McFarlance, D. C., & Glover, K. (1990). Robust controller design using normalized coprime factor plant descriptions. Lecture notes in control and information sciences. Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control. Upper Saddle River, NJ: Prentice-Hall.
Lihua Xie School of Electrical and Electronic Engineering, BLK S2, Nanyang Technological University, Singapore 639798 E-mail address: [email protected] About the Reviewer Lihua Xie is a Professor in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He received the B.E. and M.E. degrees in Electrical Engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and the Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia, in 1992. His current research interests include robust control, networked control systems, control of disk drive systems and smart structures, and estimation theory with applications in autonomous vehicles. He authored and co-authored one book and many articles. He is currently an Associate Editor of IEEE Transactions on Automatic Control, International Journal of Control, Automation and Systems and Journal of Control Theory and Applications. He served as an Associate Editor of the Conference Editorial Board, IEEE Control Systems Society, from 2000 to 2004.
doi:10.1016/j.automatica.2005.01.006
Nonsmooth Mechanics, Bernard Brogliato, Springer, New York, Heidelberg, Berlin, 1999, Communications and Control Engineering, ISBN 1-85233-143-7 124.95 Euro; 552pp. Nonsmooth mechanics is the study of classical mechanics, i.e., the mechanics of particles and rigid bodies moving subject to Newton’s and Euler’s laws, where one allows occurrences of impacts and other instantaneous changes in the dynamical behaviour of a system. This book is a survey of this intricate subject. In this review, I will attempt to describe the sorts of problems that are addressed in the book, as well as say something about the style in which they are presented. Since nonsmooth mechanics involves impacts, one needs a theory of differential equations able to handle these. The basic approach to this taken by Brogliato is to use differential equations where the nonsmoothness in time is modelled by the derivative, in the sense of the theory of distributions of Schwartz (1996), of a function of bounded variation. Of course, in a mechanical model, one does not usually know
the time of the impacts beforehand, but rather they arise as governed by the dynamical equations modelling the physical system. This is presented in this book in a formulation using unilateral constraints, as these are what arise most commonly in mechanics. For models of this type, some of the useful developments of smooth differential equations, e.g., existence and uniqueness of flows and continuity with respect to initial conditions, are adapted to this nonsmooth framework. The standard theorems do not transfer verbatim to the nonsmooth case. For example, one needs extra conditions to ensure uniqueness of solutions. In Chapter 1, where this theory of differential equations with impacts is presented, Brogliato initiates a pattern that recurs throughout the book. Namely, he points out connections to or from nonsmooth mechanics from or to other fields. For example, in Chapter 1, approaches involving hybrid systems, descriptor variable systems, the theory of outputs for control systems, and Filippov’s methods are touched upon. While ultimately beneficial to the book’s purpose, these “diversions” do take some getting used to.
Book reviews / Automatica 41 (2005) 1105 – 1108
One of the more challenging problems in nonsmooth mechanics concerns the modelling of impacts. Part of the difficulty is that one has to address matters concerning the materials of which a system is comprised; this is simply not a concern in mechanics without impacts, as long as a rigid approximation invalid. In Chapters 2 and 4, Brogliato addresses the matter of modelling impacts. The simplest situation, as far as understandability of the model is concerned, involves approximating an impact with a sequence of springs and dampers between the objects in contact. In the limit, provided one maintains some compatibility conditions between the parameters, the solutions converge to that of a possible impact model. By such means, one can model completely elastic impacts (i.e., those where all energy is instantaneously dissipated at impact), completely elastic impacts (i.e., those where no energy is dissipated during the impact), and intermediate situations as limits of smooth processes. Another method, called the method of penalising functions, considers the equations of motion with a sequence of forcing terms added. One wishes that the sequence of forcing terms have the property that the limiting solutions are well-defined and satisfy the unilateral constraint. These matters are addressed in Chapter 2. The matter of determining coefficients of restitution, and of incorporating friction models in impacts is considered in Chapter 4. Whenever one is dealing with mechanics, it is important to acknowledge that one is modelling actual systems whose behaviour can be observed. Therefore, it is essential that the mathematical/physical/metaphysical models one employs agree with observations. On the other hand, a purely empirical approach, where one cooks up a model to agree with experimental data, is ultimately unsatisfying. Thus, this bridge needs to be gapped. I found myself wishing that these issues were addressed more in the book. However, perhaps these is less a deficiency of the book, and more a reflection of the current state of the research. In mechanics, variational principles are seen by many as being King. Brogliato gives a chapter on including impacts in a variational analysis. Two main variational approaches are considered. One is Gauss’s principle, which is an instantaneous variational principle. The approach taken to Gauss’s Principle leads directly to the complementarity formulations for multiple unilateral constraints considered in Chapter 5 of the book. The second approach is the more standard one in variational mechanics, which is Hamilton’s principle. Here one needs to modify the spaces of curves one allows in the analysis. Brogliato also points out an interesting connection between variational methods for systems with unilateral constraints and the Pontryagin maximum principle. At the beginning of the chapter is a discussion of virtual displacements of the sort that I have always found unsatisfying; the concepts of “real” and “virtual” do not seem to me to be given a precise definition; at least, I cannot understand what they mean. However, apart from this, I found this chapter an interesting one.
1107
g
Fig. 1.
The matter of multiple unilateral constraints is the topic of Chapter 5, and this is a long chapter, almost 90 pages. The problems considered are best described by using the same simple example as used by Brogliato, and that I show in Fig. 1. Note that there are two unilateral constraints. The main problem of multiple unilateral constraints is, “Which of the constraints is active at any given moment?” In the example, we could ask whether one, both, or neither of the constraints is active. If the initial condition is as shown, with zero velocity, then it is clear that the “bottom” constraint will be active, and the “top” one inactive. Another simple example considered by Brogliato is that of three balls simultaneously in contact (perhaps known to the reader as “Newton’s cradle”). For more complicated systems, one needs a systematic way of figuring out which constraints are active. The first technique presented is the sweeping process of Moreau (1986). In this formulation, convexity plays a key rôle. The other formulation considered is the complementarity formulation, where one studies systems that are formed from differential equations, algebraic equations, and algebraic inequalities. In either of these formulations, one must develop theories of existence and uniqueness of solutions. The story here is quite a complicated one, as can be seen by the detailed treatment given to the example of a rod falling on the plane (called Painlevé’s example). In Chapter 6, Brogliato considers a configuration space formulation of impact modelling. Here, the system is thought of as being described by its configuration space, and thus the motion of the system is described by a point moving in configuration space. In this formulation, one needs to understand the rôle of the kinetic energy metric in impact phenomenon, since it is with respect to it that notions of orthogonality must be defined. Both single and multiple unilateral constraints are considered in this setting. The first six chapters in the book can be seen as having to do primarily with modelling, and with understanding the properties of the models. The final two chapters are
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Book reviews / Automatica 41 (2005) 1105 – 1108
concerned with stability and control theory. The central idea in the stability treatment is Lyapunov analysis. However, the presence of nonsmoothness, of course, requires one to change the approach significantly, introducing concepts from discrete dynamics to accommodate impacts. This appears to be a presently active area of research. Also considered in the stability treatment are some bifurcation theory and stability analysis for compliant impact models. After a brief discussion of controllability, the majority of Chapter 8 on feedback control is dedicated to stability and tracking using feedback. One of the central concerns in these problems is the use of feedback to ensure that impacts are handled smoothly. There appears to be a great deal of work to be done in this area. For example, Brogliato mostly addresses the fully actuated case, i.e., the common situation for robotic manipulators. Finally, there are four appendices in the book dealing with distribution theory, measure theory, functions of bounded variation, and convex analysis. One of the things that will make this book difficult to read for beginners is the unevenness of the mathematical background. Perhaps the author cannot be blamed for this; to have presented all of the mathematical background, even in a mostly incomplete form, would have rendered the book unmanageable. Nonetheless, the reader should be aware of some of the gaps arising in the mathematical treatment. Let me give three examples that occur in the appendices. Definition A.2 says that a distribution is a continuous linear function on the set D of infinitely differentiable functions with compact support. However, there is no hint at what “continuous” means, thus rendering the definition somewhat impotent. It is only three pages later, in another section, that the reader is warned in a side comment that the topology on D is relevant, although it is still not defined. In like manner, all that is said about a topological space is that it is a set, along with a collection of subsets, called open sets, “that satisfy certain properties.” Without knowing what these properties are, the idea of a measurable function, and therefore ultimately the idea of the Lebesgue integral, is not defined. And, the last nitpicky point I shall give, the Lebesgue measure is defined only on Borel sets. This is not logically incorrect, but it also does not give the usual Lebesgue theory, where one deals with the additional, but standard, complication of having to complete the measure. (Note that without a measure being complete, one loses the useful fact that a function that is a.e. equal to an integrable function is itself integrable.) These matters are all easily addressed by the reader going to other sources to fill in all of the details. But the point is that the reader will have to do this, particularly if they are initially unfamiliar with the mathematical background.
I shall close by making some general comments on the style of the book. To a reader, such as myself, who has grown accustomed to having theorems serve as waypoints in navigating technical material, Brogliato’s book is often difficult to read. There are relatively few precisely stated results, and many long text passages that required much effort for me to comprehend. However, this is certainly as much a reflection on my own deficiencies as a reader as they are on the writing style. I know that certainly many readers will be quite comfortable with the style of presentation. Nonetheless, when you add in my comments above concerning the treatment of the mathematical background, I think it is fair to say that the book’s style is not for everyone, nor for every purpose. However, as long as one understands the book’s rôle, I think it is a very useful contribution to the literature. The book’s two best features, in my view, are (1) its detailed survey of the literature (there are over 1000 items in the bibliography) and (2) its detailed presentation of many examples illustrating both the techniques and their limitations. And, as I mentioned above, I also like the way the book makes contact between the main subject matter, and other topics that are not obviously related. For readers interested in the field, this book will serve as an excellent introductory survey.
References Moreau, J.-J. (1986). Standard inelastic shocks and dynamics of unilateral constraints. In G. del Piero, & F. Maceri, F. (Eds.), Unilateral problems in structural analysis, CISM courses and lectures, (Vol. 288, pp. 173–221). New York, Heidelberg, Berlin: Springer. Schwartz, L. (1996). Théorie des distributions. Hermann, Paris: Publications de l’Institut de Mathématique de l’Université de Strasbourg.
Andrew D. Lewis Department of Mathematics and Statistics, Queen’s University, Kingston, Ont., K7L 3N6, Canada K7L 3N6 E-mail address: [email protected] About the Reviewer Andrew Lewis received his undergraduate degree in Mechanical Engineering from the University of New Brunswick in 1987, and his M.Sc. and Ph.D. in 1988 and 1995, respectively, both in Applied Mechanics from the California Institute of Technology. From 1995 to 1996 he was a Postdoctoral Fellow in Control and Dynamical Systems at the California Institute of Technology, and from 1996–1998 he was a Postdoctoral Fellow in the Mathematics Department at the University of Warwick. He is now an Associate Professor in the Mathematics and Statistics Department at Queen’s University in Kingston, Ont. in Canada. His research interests include geometric mechanics and differential geometric control theory. doi:10.1016/j.automatica.2005.01.005