3S(~ Book Reviews both as a course text and its a source of reference, but its ~alue in both respects is somewhat diminished by the omission of a number of significant dcvelopmems which ha~e occurred during the past decade. R t!lel'ellces Chen, C.-T. {1970). introduction to Linear 5"ystem I lwory. Holt, Rinehart and Winston, New York. Kailath, T. (1980). Linear Systems. Prentice Hall, Englewood Cliffs, New Jersey. Ku~era. V. (1974). Discrete Linear Control: 7he Polynomial Equation Approach. Wiley, Chichester. O'Reilly, J. ( 1983 i. Observers.lot Linear Systems. Academic Press, London. Patel, R. V. and Munro, N. 119821. ~l,tri~armbh: ";~ ,,',, ; ",',' amt Design. Pergamon. Oxford. ,,Ibout the rel iewcr Stephen Barnett is Proli~ssor ol Applied Mathematics at the: University of Bradford. He holds a Ph.l). from Loughborough University. and B.Sc., M.Sc. and D,Sc. degrees from Manchestel University, all in mathematics. He has published o~e~ /00 papers and five books in control theor~ and related area,,,, including Polynonlials and Linear Control Sy.~tems (Dekker, 1983 ). ~latrh e.~ in Control 7heot 3" (2nd edn, Krieger. 1984} and h~tr,dlatiu, l,, Mathenlatical Control 7heow with R. G. Cameron 12nd edn. Oxford University Press, 1985i. tie has held visiting pro- fessorships at Queen's University. ('anada and North ('arolina State University. Raleigh. Discrete Systems: Analysis, Control and Optimization* M. S. Mahmoud and M. G. Singh Reviewer: GENE F. FRANKLIN Stanford University, Stanford, CA 94305, U.S.A. THIS BOOKdiscusses many topics in the theory of discrete systems in a way mainly of interest to control engineers. The examples, and there are many of them, include data taken from control of temperature, power systems, turbofan engines, inventory and river pollution. Chapters on analysis discuss the problems of dynamic response, z-transforms, model reduction, controllability observability and stability via the direct method of Lyapunov. Pole placement (especially for dead-beat response), the construction of prediction and current observers and a two stage design method for systems having fast and slow components is discussed under control. The chapters on optimization treat state and parameter estimation, adaptive control and linear quadratic regulators. Each chapter contains a number of itlusffative examples but there is no continuity of examples from one chapter to the next. Because of the control orientation of the book, it is worthwhile to discuss it in the context of control engineering. The field of control engineering is very broad and the tasks of the control engineer are varied. At the core, control engineering is the design of logic to generate input signals that will control the response of some dynamic system. Usually the inputs are generated via actuators and the logic depends on having output variables obtained via sensors so the system includes feedback. A possible classification of the tasks is as follows. 1. Problem definition and specification. 2. Sensor and actuator selection and placement. 3. Controller technology selection. 4. Modelling, identification, and model simplification. 5. Analysis of dynamic response: dynamic specifications. 6. Control law design and optimization. 7. Simulation and verification of robustness. 8. Realization and testing. When the selection of the controller technology is digital logic, the control action is usually sampled and the system is discrete. Also, when the data is available only in discrete time, as happens with economic data, data from chemical analytical instruments and from some other digital sensors, the system must be analyzed as discrete. This book, as indicated in the title, is restricted to discrete systems. Tasks 5 and 6 in the list above require relatively little knowledge of the physical laws of the object being controlled and often draw heavily on contributions from mathematics and statistics. As indicated in the subtitle, this book is also restricted to these topics plus those aspects of identification and model reduction that can be done without reference to the phy=ical system. The fact that the authors have chosen to concentrate on the theoretical parts of discrete systems alone is reasonable, for there * Discrete Systems: Analysis, Control and Optimization by M. S. Mahmoud. Springer. Berlin 11984). 669pp. DM 152. is substantial material there. However, it is very disappointing that a book clearly aimed toward control engineers should fail to point out the connections between the topics they discuss and the other tasks of control. For example, in the design of digital systems that are discrete because the controller is discrete, one of the most critical problems is the selection of the sampling period. Not only is this problem not discussed in the book, neither are methods for obtaining a discrete model from continuous equations. Models for the examples are given in discrete form and the reader is told that these are obtained after suitable normalizing and sampling operations. There is no material to give the reader insight into the physical problem or the design specifications. For example, in the problem on river pollution in section 8.5.4. the states are given as "'D.O." and ~'B.O.D." with no definitions of what the variables stand for. There is more to this point than the definition of the variables; an important part of control is the selection of variables to be measured and the choice and placement of the actuators and the sensors. In the currently widely studied problem of robotics, for example, the issue of sensor design and what to sense is among the most important of problems. The control of a mechanism with collocated actuator and sensor is very different from control with non-collocated devices. Even with concentration on the theoretical aspects of the problem, it seems only reasonable to expect a book on control to connect the topics to the underlying physical problem at least to the extent of helping the reader understand where the problem comes from and which are the important factors. This book is particularly lacking in this respect. Within the scope of theoretical topics in discrete control, there are several of great recent interest that are not adequately mentioned in the book. One of these is robustness. ]'his issue has several dimensions, including robustness of design algorithms. robustness of the control laws and robustness of the implementation to computer accuracy and architecture, The book discusses the issue of robustness of design algorithms but seems to depend overly much on the use of the Jordan form for computing various results. This form is known to be particularl~ difficult to compute. A mote surprising omission is the stochastic separation theorem. The book has a rather complete derivation of the Kalman filter and of the linear, quadratic regulator. That these two are not completed with the separation theorem is very surprising indeed. t:rom the point of ~iew of the production ot the book itself, it contains a distressing number of typographic errors. These do not generally interfere with understanding the material but they do slow the reading and cause considerable annoyance. In one case, example 2 of section 7.4.7, there is a numerical error since the matrix given does not have the cigenvalues given tbr it, the system is stable although the text calls it unstable based on thc given eigenvalues. This reviewer did not run the examples and thus did not resolve this particular discrepancy, but this case indicates that those using the book should study the examples with care. As was said earlier, the field of control is a vast one and it is not