Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of X_{c}, X_{\bar{o}} , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

Principal component analysis in linear systems: Controllability, observability, and model reduction

There is an increasing demand for dynamic systems to become safer and more reliable This requirement extends beyond the normally accepted safety-critical systems such as nuclear reactors and aircraft, where safety is of paramount importance, to systems such as autonomous vehicles and process control systems where the system availability is vital It is clear that fault diagnosis is becoming an important subject in modern control theory and practice Robust Model-Based Fault Diagnosis for Dynamic Systems presents the subject of model-based fault diagnosis in a unified framework It contains many important topics and methods; however, total coverage and completeness is not the primary concern The book focuses on fundamental issues such as basic definitions, residual generation methods and the importance of robustness in model-based fault diagnosis approaches In this book, fault diagnosis concepts and methods are illustrated by either simple academic examples or practical applications The first two chapters are of tutorial value and provide a starting point for newcomers to this field The rest of the book presents the state of the art in model-based fault diagnosis by discussing many important robust approaches and their applications This will certainly appeal to experts in this field Robust Model-Based Fault Diagnosis for Dynamic Systems targets both newcomers who want to get into this subject, and experts who are concerned with fundamental issues and are also looking for inspiration for future research The book is useful for both researchers in academia and professional engineers in industry because both theory and applications are discussed Although this is a research monograph, it will be an important text for postgraduate research students world-wide The largest market, however, will be academics, libraries and practicing engineers and scientists throughout the world

Robust Model-Based Fault Diagnosis for Dynamic Systems

In this paper we propose a new dynamic model for friction. The model captures most of the friction behavior that has been observed experimentally. This includes the Stribeck effect, hysteresis, spring-like characteristics for stiction, and varying break-away force. Properties of the model that are relevant to control design are investigated by analysis and simulation. New control strategies, including a friction observer, are explored, and stability results are presented. >

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A new model for control of systems with friction

A survey of models, analysis tools and compensation methods for the control of machines with friction

Presents a guide to sliding mode control for practicing control engineers. It offers an accurate assessment of the so-called chattering phenomenon, catalogs implementable sliding mode control design solutions, and provides a frame of reference for future sliding mode control research.

A control engineer's guide to sliding mode control

From the Publisher:
One thesis of this book is that state-space methods can be presented in a style that can be grasped by the engineer who is more interested in using the results than in proving them. Another thesis is that results are useful. The book is addressed not only to students but also to a general audience of engineers and scientists who are interested in becoming familiar with state-space methods either for direct application to control system design or as a background for reading the periodical literature. The author has tried to keep the chapters reasonably independent and to use customary symbols wherever practical. He has also selected fifteen or so examples and weaved them into the fabric of the text and the homework problems.

Control System Design: An Introduction to State-Space Methods

The problem of estimating the state x of a linear process in the presence of a constant but unknown bias vector b is considered. This bias vector influences the dynamics and/or the observations. It is shown that the optimum estimate \hat{x} of the state can be expressed as \hat{x} = x + V_{x}\hat{b} (1) where \tilde{x} is the bias-free estimate, computed as if no bias were present, \hat{b} is the optimum estimate of the bias, and V x is a matrix which can be interpreted as the ratio of the covariance of \tilde{x} and \hat{b} to the variance of \hat{b} . Moreover, \hat{b} can be computed in terms of the residuals in the bias-free estimate, and the matrix V x depends only on matrices which arise in the computation of the bias-free estimates. As a result, the computation of the optimum estimate \tilde{x} is effectively decoupled from the estimate of the bias \hat{b} , except for the final addition indicated by (1).

Treatment of bias in recursive filtering

Two new models for "slip-stick" friction are presented. One, called the "bristle model," is an approximation designed to capture the physical phenomenon of sticking. This model is relatively inefficent numerically. The other model, called the "reset integrator model," does not capture the details of the sticking phenomenon, but is numerically efficient and exhibits behavior similar to the model proposed by Karnopp in 1985. All three of these models are preferable to the classical model which poorly represents the friction force at zero velocity. Simulation experiments show that the new models and the Karnopp model give similar results in two examples. In a closed-loop example, the classical model predicts a limit cycle which is not observed in the laboratory. The new models and the Karnopp model, on the other hand, agree with the experimental observation.

On the Modeling and Simulation of Friction

A new method of approximating the transfer function of a high-order linear system by one of lower order is proposed. Called the "Routh approximation method" because it is based on an expansion that uses the Routh table of the original transfer function, the method has a number of useful properties: if the original transfer function is stable, then all approximants are stable; the sequence of approximants converge monotonically to the original in terms of "impulse response" energy; the approximants are partial Pade approximants in the sense that the first k coefficients of the power series expansions of the k th-order approximant and of the original are equal; the poles and zeros of the approximants move toward the poles and zeros of the original as the order of the approximation is increased. A numerical example is given for the calculation of the Routh approximants of a fourth-order transfer function and for illustration of some of the properties.

Routh approximations for reducing order of linear, time-invariant systems

From the Publisher:
Stressing the importance of simulation and performance evaluation for effective design, this new text looks at the techniques engineers use to design control systems that work. It covers qualitative behavior and stability theory; graphical methods for nonlinear stability; saturating and discontinuous control; discrete-time systems; adaptive control; and more. For electrical engineers working in modern control system design.

Bernard Friedland

Papers

Control System Design: An Introduction to State-Space Methods

Treatment of bias in recursive filtering

On the Modeling and Simulation of Friction

Routh approximations for reducing order of linear, time-invariant systems

Advanced control system design