##
**Adaptive filtering prediction and control.**
*(English)*
Zbl 0653.93001

“Adaptive Filtering Prediction and Control” is a very good book. This book is well written, and succeeds in describing the body of material pertaining to both deterministic and stochastic identification and control. The models considered are all discrete-time models, mostly linear, but some material is devoted to special classes of nonlinear or time-varying models. This book is enjoyable to read, and contains an excellent reference list.

The preface of the book states its objective is to present the theory of adaptive prediction and control for linear discrete-time systems. I believe the book succeeds in this goal. Many technical variations of adaptive control and estimation problems are discussed; pertinent theorems are stated, and for the most part, proved. The introductory chapter states that parameter estimation is a key issue in prediction and control, and appropriately, much of the book is devoted to solutions to a variety parameter estimation problems. Also covered are topics in adaptive prediction and control for both deterministic and stochastic systems.

It is also claimed in the preface that this book is self-contained, and that at least the deterministic material could be handled by an undergraduate with exposure to systems theory. I disagree. The book is almost self-contained, with an excellent appendix, and an excellent list of references. I believe the reading of this text (even the deterministic part) requires more than the standard undergraduate exposure to systems theory, which, in many cases, is quite minimal. For the easiest reading of this book, one should have had at least a graduate course in linear system theory (for the deterministic part) and some exposure to stochastic processes (for the stochastic part). I would say this is my major complaint about the book, which by no means discredits the material covered. I have some more specific comments which I will list below:

The notation used on p. 23 of superscript prime for the inverse of a unimodular matrix is slightly confusing.

The discussion (p. 38) on discrete-time nonlinear systems should really include general state affine models, rather than only bilinear ones, since state affine models may be used to approximate all nonlinear state space models (arbitrarily closely).

The exercises seem more or less straight forward, and doable for a reader with only the book to refer to. There is not a lot of extra talk in this book. One is introduced to new concepts one after the other, but these are carefully organized in each (sub) outline, and the concepts progress smoothly.

A few more comments here on how readable this text is for the unexposed:

1. Lagrange multipliers are introduced for optimization (p. 51) and never explained.

2. Finding the eigenvalue and eigenvector of a matrix is not explained, although one is told how to find generalized eigenvectors in the appendix, (if one knew how to find eigenvectors).

3. A reader really must know how to read this text. For instance, on p. 277, maximum likelihood estimation is mentioned, without reference. It is described in the appendix on p. 505, but a reader would have to be informed enough to think that such a topic might exist in the appendix.

Further comments:

The version of multivariable parameter estimation given is very limiting and confusing. At first, it limits variability of the coefficients of each of the outputs to be the same \((y_ i=\Phi\) \(T_ i\theta_ 0\), for all i). A general multivariable least squares algorithm can be derived, in which one doesn’t need special conditions (p. 95, 96) to have a scalar type least squares algorithm. What’s confusing about this section is that it appears on p. 96 that \(\theta_ 0\) went from being a vector to matrix, but this is never stated.

Theorems in chapter 7.3 (proved in the appendix) mention a convergence rate of certain filtering parameters, and the proof is left to a reference given in the appendix, rather than shown in the text. The issue of significance of convergence rates of algorithms is not discussed.

Issues of convergence of stochastic recursive least squares algorithms are thoroughly discussed. Implications of some of the sufficient assumptions such as persistence of excitation and passivity conditions are not fully discussed. The theory present is just that, and it works well as a theory. It may not be so simple to find systems which satisfy these assumptions, let alone verify that they are satisfied, although this fact does not discredit the contents of the text as a treatise on the theory of adaptive prediction and control.

The preface of the book states its objective is to present the theory of adaptive prediction and control for linear discrete-time systems. I believe the book succeeds in this goal. Many technical variations of adaptive control and estimation problems are discussed; pertinent theorems are stated, and for the most part, proved. The introductory chapter states that parameter estimation is a key issue in prediction and control, and appropriately, much of the book is devoted to solutions to a variety parameter estimation problems. Also covered are topics in adaptive prediction and control for both deterministic and stochastic systems.

It is also claimed in the preface that this book is self-contained, and that at least the deterministic material could be handled by an undergraduate with exposure to systems theory. I disagree. The book is almost self-contained, with an excellent appendix, and an excellent list of references. I believe the reading of this text (even the deterministic part) requires more than the standard undergraduate exposure to systems theory, which, in many cases, is quite minimal. For the easiest reading of this book, one should have had at least a graduate course in linear system theory (for the deterministic part) and some exposure to stochastic processes (for the stochastic part). I would say this is my major complaint about the book, which by no means discredits the material covered. I have some more specific comments which I will list below:

The notation used on p. 23 of superscript prime for the inverse of a unimodular matrix is slightly confusing.

The discussion (p. 38) on discrete-time nonlinear systems should really include general state affine models, rather than only bilinear ones, since state affine models may be used to approximate all nonlinear state space models (arbitrarily closely).

The exercises seem more or less straight forward, and doable for a reader with only the book to refer to. There is not a lot of extra talk in this book. One is introduced to new concepts one after the other, but these are carefully organized in each (sub) outline, and the concepts progress smoothly.

A few more comments here on how readable this text is for the unexposed:

1. Lagrange multipliers are introduced for optimization (p. 51) and never explained.

2. Finding the eigenvalue and eigenvector of a matrix is not explained, although one is told how to find generalized eigenvectors in the appendix, (if one knew how to find eigenvectors).

3. A reader really must know how to read this text. For instance, on p. 277, maximum likelihood estimation is mentioned, without reference. It is described in the appendix on p. 505, but a reader would have to be informed enough to think that such a topic might exist in the appendix.

Further comments:

The version of multivariable parameter estimation given is very limiting and confusing. At first, it limits variability of the coefficients of each of the outputs to be the same \((y_ i=\Phi\) \(T_ i\theta_ 0\), for all i). A general multivariable least squares algorithm can be derived, in which one doesn’t need special conditions (p. 95, 96) to have a scalar type least squares algorithm. What’s confusing about this section is that it appears on p. 96 that \(\theta_ 0\) went from being a vector to matrix, but this is never stated.

Theorems in chapter 7.3 (proved in the appendix) mention a convergence rate of certain filtering parameters, and the proof is left to a reference given in the appendix, rather than shown in the text. The issue of significance of convergence rates of algorithms is not discussed.

Issues of convergence of stochastic recursive least squares algorithms are thoroughly discussed. Implications of some of the sufficient assumptions such as persistence of excitation and passivity conditions are not fully discussed. The theory present is just that, and it works well as a theory. It may not be so simple to find systems which satisfy these assumptions, let alone verify that they are satisfied, although this fact does not discredit the contents of the text as a treatise on the theory of adaptive prediction and control.

Reviewer: C.Schwartz

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93C55 | Discrete-time control/observation systems |

93C40 | Adaptive control/observation systems |

93B30 | System identification |

93C05 | Linear systems in control theory |

93E10 | Estimation and detection in stochastic control theory |

93E11 | Filtering in stochastic control theory |

93E12 | Identification in stochastic control theory |

93E20 | Optimal stochastic control |

62M20 | Inference from stochastic processes and prediction |

93E25 | Computational methods in stochastic control (MSC2010) |