After studying this chapter you will be able to

Introduction

In Chapter 3 the response of an LTI system to various deterministic signals, such as a step input, was considered. A characteristic of a deterministic signal or sequence is that it can be reproduced exactly. On the other hand, a random signal, or a sequence of random variables, cannot be exactly reproduced. The randomness or unpredictability of the value of a certain variable in a modeling context arises generally from the limitations of the modeler in predicting a measured value by applying the “laws of Nature.” These limitations can be a consequence of the limits of scientific knowledge or of the desire of the modeler to work with models of low complexity. Measurements, in particular, introduce an unpredictable part because of their finite accuracy.

After studying this chapter you will be able to

Introduction

In this chapter we review some basic topics from linear algebra. The material presented is frequently used in the subsequent chapters.

Since the 1960s linear algebra has gained a prominent role in engineering as a contributing factor to the success of technological breakthroughs.

Linear algebra provides tools for numerically solving system-theoretic problems, such as filtering and control problems. The widespread use of linear algebra tools in engineering has in its turn stimulated the development of the field of linear algebra, especially the numerical analysis of algorithms. A boost to the prominent role of linear algebra in engineering has certainly been provided by the introduction and widespread use of computer-aided-design packages such as Matlab (MathWorks, 2000b) and SciLab (Gomez, 1999). The user-friendliness of these packages allow us to program solutions for complex system-theoretic problems in just a few lines of code.

After studying this chapter you will be able to

Introduction

This chapter continues the discussion started in Chapter 7, on estimating the parameters in an LTI state-space model. It addresses the determination of a model of both the deterministic and the stochastic part of an LTI model.

The objective is to determine, from a finite number of measurements of the input and output sequences, a one-step-ahead predictor given by the stationary Kalman filter without using knowledge of the system and covariance matrices of the stochastic disturbances. In fact, these system and covariance matrices (or alternatively the Kalman gain) need to be estimated from the input and output measurements. Note the difference from the approach followed in Chapter 5, where knowledge of these matrix quantities was used. The restriction imposed on the derivation of a Kalman filter from the data is the assumption of a stationary one-step-ahead predictor of a known order.

After studying this chapter you will be able to

Introduction

The problem of identifying an LTI state-space model from input and output measurements of a dynamic system, which we analyzed in the previous two chapters, is re-addressed in this chapter via a completely different approach. The approach we take is indicated in the literature (Verhaegen, 1994; Viberg, 1995; Van Overschee and De Moor, 1996b; Katayama, 2005) as the class of subspace identification methods. These methods are based on the fact that, by storing the input and output data in structured block Hankel matrices, it is possible to retrieve certain subspaces that are related to the system matrices of the signal-generating state-space model. Examples of such subspaces are the column space of the observability matrix, Equation (3.25) on page 67, and the row space of the state sequence of a Kalman filter.

Making observations through the senses of the environment around us is a natural activity of living species. The information acquired is diverse, consisting for example of sound signals and images. The information is processed and used to make a particular model of the environment that is applicable to the situation at hand. This act of model building based on observations is embedded in our human nature and plays an important role in daily decision making.

Model building through observations also plays a very important role in many branches of science. Despite the importance of making observations through our senses, scientific observations are often made via measurement instruments or sensors. The measurement data that these sensors acquire often need to be processed to judge or validate the experiment, or to obtain more information on conducting the experiment. Data are often used to build a mathematical model that describes the dynamical properties of the experiment. System-identification methods are systematic methods that can be used to build mathematical models from measured data. One important use of such mathematical models is in predicting model quantities by filtering acquired measurements.

A milestone in the history of filtering and system identification is the method of least squares developed just before 1800 by Johann Carl Friedrich Gauss (1777–1855). The use of least squares in filtering and identification is a recurring theme in this book. What follows is a brief sketch of the historical context that characterized the early development of the least-squares method.

After studying this chapter you will be able to

Introduction

This chapter deals with two important topics: signals and systems. A signal is basically a value that changes over time. For example, the outside temperature as a function of the time of the day is a signal. More specifically, this is a continuous-time signal; the signal value is defined at every time instant. If we are interested in measuring the outside temperature, we will seldom do this continuously. A more practical approach is to measure the temperature only at certain time instants, for example every minute. The signal that is obtained in that way is a sequence of numbers; its values correspond to certain time instants. Such an ordered sequence is called a discrete-time signal.

After studying this chapter you will be able to

Introduction

In this chapter the problem of determining a model from input and output measurements is treated using frequency-domain methods. In the previous chapter we studied the estimation of the state given the system and measurements of its inputs and outputs. In this chapter we do not bother about estimating the state. The models that will be estimated are input–output models, in which the state does not occur. More specifically, we investigate how to obtain in a simple and fast manner an estimate of the dynamic transfer function of an LTI system from recorded input and output data sequences taken from that system. We are interested in estimating the frequency-response function (FRF) that relates the measurable input to the measurable output sequence. The FRF has already been discussed briefly in Section 3.4.4 and its estimation is based on Lemma 4.3 via the estimation of the signal spectra of the recorded input and output data. Special attention is paid to the case of practical interest in which the data records have finite data length.

After studying this chapter you will be able to

Introduction

After the treatment of the Kalman filter in Chapter 5 and the estimation of the frequency-response function (FRF) in Chapter 6, we move another step forward in our exploration of how to retrieve information about linear time-invariant (LTI) systems from input and output measurements. The step forward is taken by analyzing how we can estimate (part of) the system matrices of the signal-generating model from acquired input and output data. We first tackle this problem as a complicated estimation problem by attempting to estimate both the state vector and the system matrices. Later on, in Chapter 9, we outline the so-called subspace identification methods that solve such problems by means of linear least-squares problems.

Nonparametric models such as the FRF could also be obtained via the simple least-squares method or the computationally more attractive fast Fourier transform.

After studying this chapter you will be able to

Introduction

In the previous chapters, it was assumed that time sequences of input and output quantities of an unknown dynamical system were given. The task was to estimate parameters in a user-specified model structure on the basis of these time sequences.

This book is intended as a first-year graduate course for engineering students. It stresses the role of linear algebra and the least-squares problem in the field of filtering and system identification. The experience gained with this course at the Delft University of Technology and the University of Twente in the Netherlands has shown that the review of undergraduate study material from linear algebra, statistics, and system theory makes this course an ideal start to the graduate course program. More importantly, the geometric concepts from linear algebra and the central role of the least-squares problem stimulate students to understand how filtering and identification algorithms arise and also to start developing new ones. The course gives students the opportunity to see mathematics at work in solving engineering problems of practical relevance.

The course material can be covered in seven lectures:

The authors are of the opinion that the transfer of knowledge is greatly improved when each lecture is followed by working classes in which the students do the exercises of the corresponding classes under the supervision of a tutor.