After studying this chapter you will be able to

• describe the output-error model-estimation problem;

• parameterize the system matrices of a MIMO LTI state-space model of fixed and known order such that all stable models of that order are presented;

• formulate the estimation of the parameters of a given system parameterization as a nonlinear optimization problem;

• numerically solve a nonlinear optimization problem using gradient-type algorithms;

• evaluate the accuracy of the obtained parameter estimates via their asymptotic variance under the assumption that the signal-generating system belongs to the class of parameterized state-space models; and

• describe two ways for dealing with a nonwhite noise acting on the output of an LTI system when estimating its parameters.

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

• explain that the identification of an LTI model making use of real-life measurements is more then just estimating parameters in a user-defined model structure;

• identify an LTI model in a cyclic manner of iteratively refining data and models and progressively making use of more complex numerical optimization methods;

• explain that the identification cycle requires many choices to be made on the basis of cautious experiments, the user's expertise, and prior knowledge about the system to be identified or about systems bearing a close relationship with, or resemblance to, the target system;

• argue that a critical choice in system identification is the selection of the input sequence, both in terms of acquiring qualitative information for setting or refining experimental conditions and in terms of accurately estimating models;

• describe the role of the notion of persistency of excitation in system identification;

• use subspace identification methods to initialize prediction-error methods in identifying state-space models in the innovation form; and

• understand that the art of system identification is mastered by applying theoretical insights and methods to real-life experiments and working closely with an expert in the field.

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:

1. (i) Lecture 1: Introduction and review of linear algebra (Chapters 1 and 2)

2. (ii) Lecture 2: Review of system theory and probability theory (Chapters 3 and 4)

3. (iii) Lecture 3: Kalman filtering (Chapter 5)

4. (iv) Lecture 4: Estimation of frequency-response functions (Chapter 6)

5. (v) Lecture 5: Estimation of the parameters in a state-space model (Chapters 7 and 8)

6. (vi) Lecture 6: Subspace model identification (Chapter 9)

7. (vii) Lecture 7: From theory to practice: the system-identification cycle (Chapter 10).

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.

A telephone network consists of a network of exchanges (or routers, in more modern formulations). Many of these are themselves the centre of a local star network, in that they have direct connections to individual subscribers in the region. However, it is the exchanges that we shall regard as constituting the nodes of a network. The defining feature of a loss network is that a call is accepted only if a clear route to its destination is available, otherwise the call is lost.

Random variation can enter the system in various ways. Even if all the parameters of operation and loading are constant and equilibrium has been reached, there will be statistical variation of the numbers and types of calls in progress. This is the source that receives most attention in at least that part of the literature favoured by mathematicians. One might consider it secondary in comparison with the more radical type of uncertainty which faces the system when there are massive variations in load or major internal failures. It is these that determine system structure, by setting a premium on versatility and robustness.

Nevertheless, there will always be times when a system is working close to capacity, and real-time decisions on acceptance and routing must be made to accommodate normal statistical variation. Such circumstances are associated with the decentralised realisation of policies, and will be considered in the next chapter.

We shall then begin by considering a system operating deterministically with a fixed demand pattern. The variables (e.g. numbers of calls in progress, capacity assigned to a given link) will be regarded as continuous in this treatment.

The size of this appendix may seem excessive, in view of the limited mention of random graphs in the main body of the text. However, given that a review as meticulous and wide-ranging as that by Albert and Barabási (2002) regards certain central questions as open for which a full analysis exists in the literature, it seems appropriate to give a brief and self-contained account of that analysis. It cannot be long concealed that much of the analysis referred to is that developed by the author over some 30 years. Detailed references are given in the literature survey of Section A3.8; the principal advances are treatment of the so-called first-shell model, leading to the integral representions (A3.13), (A3.31) and (A3.35) of various generating functions, in which a raft of conclusions is implicit.

The random graph model can be expressed either in terms of graphs or in terms of polymers if we identify nodes with atoms, arcs with bonds, components (connected subgraphs) with polymer molecules and node ‘colours’ with types of atom. We shall for preference use the term ‘unit’ rather than atom, since the unit does occasionally have a compound (but fixed) form. The situation in which a ‘giant’ component forms (i.e. in which all but an infinitesimal proportion of the nodes form a single component) is identified with the phenomenon of gelation in the polymer context.

We come now to a most remarkable piece of work. This concerns the optimisation of load-bearing structures in engineering, such as frameworks consisting of freely jointed struts and ties – we give examples. The purpose of such structures is to communicate stress, a vector quantity, in an economic fashion from the points where external load is applied to the points where it can be off-loaded: the load-accepting foundation.

The problem of optimising such structures was considered in a paper, remarkable for its brevity and penetration as much as for its prescience, published as early as 1904 by the Australian engineer A. G. M. Michell. He derived the dual form of the problem and exhibited its essential role in determining the optimal design. This was at a time when there was no general understanding of the role or interpretation of the dual variable – Michell uses the concept of a ‘virtual displacement’, a term that we shall see as justified. He then went on to derive the continuous form of the dual, corresponding to the unrestricted optimisation of structures on the continuum. This opened the way to the study of structures made of material with directional properties (e.g. the use of resin-bonded fibre-glass matting in yacht hulls, the laying down of bone in such a way as to meet the particular pressures and tensions to which it is subjected). It also turned out to offer an understanding of materials that behave plastically rather than elastically under load – i.e. that yield catastrophically when load reaches a critical value.

Whither? Why?

The contents list gives a fair impression of the coverage attempted. Networks, both deterministic and stochastic, have emerged as objects of intense interest over recent decades. They occur in communication, traffic, computer, manufacturing and operational research contexts, and as models in almost any of the natural, economic and social sciences. Even engineering frame structures can be seen as networks that communicate stress from load to foundation.

We are concerned in this book with the characterisation of networks that are optimal for their purpose. This is a very natural ambition; so many structures in nature have been optimised by long adaptation, a suggestive mechanism in itself. It is an ambition with an inbuilt hurdle, however: one cannot consider optimisation of design without first considering optimisation of function, of the rules by which the network is to be operated. On the other hand, optimisation of function should find a more natural setting if it is coupled with the optimisation of design.

The mention of communication and computer networks raises examples of areas where theory barely keeps breathless pace with advancing technology. That is a degree of topicality we do not attempt, beyond setting up some basic links in the final chapters. It is well recognised that networks of commodity flow, electrical flow, traffic flow and even mechanical frame structures and biological bone structures have unifying features, and Part I is devoted to this important subclass of cases.

In Chapter 1 we consider a rather general model of commodity distribution for which flow is determined by an extremal principle.

By ‘distributional networks’ we mean networks that carry the flow of some commodity or entity, using a routing rule that is intended to be effective and even optimal. The chapter titles give examples. Operational research abounds in such problems (see Ball et al., 1995a,b and Bertsekas, 1998), but the versions we consider are both easier and harder than these. In operational research problems one is usually optimising flow or operations upon a given network, whereas we aim also to optimise the network itself. Even the Bertsekas text, although entitled Network Optimization, is concerned with operation rather than design. The design problem is more difficult, if for no other reason than that it cannot be settled until the operational rules are clarified. On the other hand, there may be a simplification, in that the optimal network is not arbitrary, but has special properties in its class.

The ‘flow’ may not be a material one – see the Michell structures of Chapters 7–9, for which the entity that is communicated through the network is stress. The communication networks of Part IV are also distributional networks, but ones that have their own particular dynamic and stochastic structure.

The setting

We are daily and directly convinced of the importance of traffic networks, and more particularly of road networks, and of the desirability that the factors that ensure their smooth running should be well understood. There is a vast literature on the subject, and a substantial body of theory, which must indeed take account of new factors.

The road network is a graph, whose streets constitute the arcs and whose junctions the nodes. However, traffic does not consist of just a single ‘commodity’ to be shunted around the network, but breaks up into classes labelled by their destination, and possibly by other characters as well. For simplicity, we shall suppose that destinations coincide with exit nodes.

This seems then just a particular case of the destination-specific distribution net of Chapter 2. However, we saw there that a naive optimisation led to a totally unrealistic solution, in which every origin–destination pair for which there was demand was connected by a direct route, of rating matched to this demand. Such a solution ignores a multitude of issues, arguable only in the order in which one should list them. From the aesthetic or environmental point of view, one could never allow the countryside to suffer this death of a thousand cuts. From the utilitarian point of view, land, and substantial blocks of it, is needed for other purposes (e.g. homes, offices, shopping, industry, agriculture, leisure activities, plus access to some sites and total protection of others), and its usage must be planned.

The Jackson model of Chapter 13 was a natural one, whose behaviour was easily analysed. However, it allowed no state feedback, and so could not be induced to respond to current network state. Various attempts have been made to modify the model to achieve such feedback (see Section 13.5), but these have not led to what one might regard as the natural successor in the next generation of models.

However, one can indeed find a model with a full theory if one goes to the other extreme: of a model that is completely centralised, in that the optimiser can deploy all processing resources freely and instantaneously to any part of the network. There is then a full and exact theory for the optimisation of this deployment, in the light of current network state. Of course, such assumptions are unrealistically extreme; processing units can be deployed only at the work-station to which they are attached. Moreover, one would wish for a control rule of a decentralised nature, in that operators make local decisions largely on the basis of local information.

Nevertheless, this model (based on the multi-armed bandit) does mark a genuine advance, and does suggest policies for the case when resource deployment is free only within a work-station (see Section 14.5).

The model is similar to that of Chapter 13, in that there is a queue at every node (or a multi-class queue at every work-station, in the more explicit version). However, we shall speak of it as a time-sharing network rather than a queueing network, to emphasise that processing effort is allocated centrally rather than locally.

These last two chapters are intended as something of a corrective: to give significant examples of newer types, concepts and models of communication networks. This chapter continues the preoccupation of Chapter 16 with operation rather than design: to find rules that make effective use of poor and local information to achieve smooth operation of a rather diffuse system. The design aspect makes a natural return in Chapter 18; both the Internet and the Worldwide Web achieve their own design, partially at least, by a kind of directed evolution.

For loss networks a call was not accepted unless an open path from source to destination could be guaranteed. Once the connection was established, delivery of the message could begin. The Internet operates by packet-switching rather than circuit-switching. The ‘message’ is seen as a prepared file rather than a real-time conversation. This file is broken into ‘packets’ which are launched successively into the system, with no guarantee of an open path along the required route. The role of an ‘exchange’ is replaced by that of a ‘router’: a device that sorts packets as they arrive on incoming links and directs them on to the outgoing link appropriate for their destination. Each of these outgoing links is prefaced by a ‘buffer’, a queue that will store packets until there is free capacity on the link. The buffer is finite, however; once its capacity has been exceeded the overflow of packets is lost.

Final loss is not permitted, however, because the packets must be assembled at the destination to reconstitute the original file. If any packet is missing then transmission of the file is unacceptably impaired.

The setting

By ‘simple flow’ we mean those cases in which a single divisible commodity is to be transferred from the source nodes of a network to the sink nodes, and the routing of this transfer is determined by some extremal principle. For example, one might be sending oil from production fields to refineries in different countries, and would wish to achieve this transfer at minimal cost. (For the ideas of this chapter to be applicable one would have to assume that all oil is the same – if different grades of crude oil are to be distinguished then the more general models of Chapter 3 would be needed.) This is the classical single-commodity transportation problem of operations research (see e.g. Luenberger, 1989). However, we mean to take it further: to optimise the network as well as the routing. Further, in Chapters 4 and 5 we consider the radical effect when the design must be optimised to cope with several alternative loading patterns (i.e. patterns of supply and demand) or with environmental pressures.

Another example would concern the flow of electrical current through a purely resistive network. This has virtually nothing to do with the practical distribution of electrical power, which is achieved by sophisticated alternating-current networks, but the model is a very natural one, having practical implications. For given input/output specifications the flow through the network is determined physically: by balance relations and by Ohm's law. However, Ohm's law can be seen as a consequence of a ‘minimal dissipation’ criterion, so that one again has an extremal principle, this time a natural physical principle rather than an imposed economic one.

In Chapters 2, 4 and 7–9 we considered the evolution of a network towards optimality by, essentially, following the steepest-descent path of a prescribed criterion function, even if sometimes this calculation took a simple and intuitive form. However, one generally thinks of ‘evolution’ as something much less premeditated, in which some simple and plausible response rule induces structural changes in the system, whose effect and whose effective guiding principle (if any) become evident only with time. In the case of Darwinian evolution, the effective guiding principle is that of survival. The degree of sophistication induced by this stark criterion still lies beyond our full comprehension, and is comparable in scale only to the time and tirelessness of trial needed to achieve it.

Interest in these matters is demonstrated by the growing literature on evolving automata, artificial neural networks and the like. However, the Worldwide Web now presents an example, if simple, of spontaneous evolution in a technological context.

Random graphs and the Web

Real-life response rules will in general be local, in that a change is induced in some part of the system in response to stimuli or information manifest in that part. The execution of the rule and the information on which it is based will also show natural variability, i.e. be ‘random’. In the case of networks we are thus considering the generation of a random graph from local rules, and are interested to see whether the network thus generated can be seen as a statistical solution to an implied optimisation problem, and what its character may be.

There is now a large literature on random graphs.