System Identification, Environmental Modelling, and Control System Design

This chapter presents and discusses various aspects of what theory predicts in terms of accuracy of instrumental variable estimates. A general derivation of the covariance matrix of the parameter estimates is presented. This matrix is influenced by a number of user choices in the identification method, and it is further discussed how these user choices can be made in order to make the covariance matrix as small as possible in a well-defined sense. The chapter includes also a comparison with the prediction error method, and a discussion of in what situations an optimal instrumental variable method can be statistically efficient.

This chapter presents an estimation method for Hammerstein models under colored added noise conditions. The proposed method is detailed for both continuous-time and discrete-time models and is based on the refined instrumental variable method. In order to use a regression form, the Hammerstein model is reformulated as an augmented multi-input-single-output linear time invariant model. The performance of the proposed methods are exposed through relevant Monte Carlo simulation examples.

This chapter is aimed at helping readers decide whether identifiability and the closely connected property of distinguishability are theoretically important and practically relevant for their research or teaching. If this is so, they will find here methods that can be used to test models for these properties. The chapter also shows that measures of identifiability can be maximized, provided that there are some degrees of freedom in the procedure for data collection. If the model of interest cannot be made identifiable, all may not be lost, as we shall see. A large part of the presentation is tutorial in nature, with academic examples worked out in detail.

This chapter honors Peter, then, in recounting my career-long experience (1970–2010) of staring down the devilishly difficult: the problem of model structure identification—of using models for discovery. I still regard this matter as one of the grand challenges of environmental modeling (Beck et al., White Paper, 2009). If I appear modest about our progress in the presence of such enormity, so I am. But let no-one presume that I am therefore not greatly enthused by the progress I believe I and my students (now colleagues) have made over these four decades. It has been a privilege to be allowed the time to work on such a most attractive and engaging topic.

We consider the problem of identification of continuous time systems when the data is collected using non-uniform sampling periods. We formulate this problem in the context of Nonlinear Filtering. We show how a new class of nonlinear filtering algorithm (Minimum Distortion Filtering) can be applied to this problem. A simple example is used to illustrate the performance of the algorithm. We also compare the results with those obtained from (a particular realization) of Particle Filtering.

The chapter is inspired by the work of Peter Young who has made a life time of contributions to parameter estimation for real world systems.

Adaptive or learning algorithms have found wide use in control, signal processing and machine learning. Averaging analysis is a powerful tool for analysing the performance of such algorithms but is not as well known as it should be. This may be partly because it has been assumed to be an advanced method requiring considerable mathematical background such as weak convergence theory. But in Solo and Kong (Adaptive Signal Processing Algorithms, Prentice Hall, 1995) the averaging approach, which is an entirely separate technique , and is relatively straightforward to apply, was developed in a direct way that did not require weak convergence theory.

In this paper we recap that approach, developing averaging in an heuristic way and illustrating its use on a number of examples in a simple way.

In this contribution we describe measures for dependence and causality between component processes in multivariate time series in a stationary context. Symmetric measures, such as the partial spectral coherence, as well as directed measures, such as the partial directed coherence and the conditional Granger causality index, are described and discussed. These measures are used for deriving undirected and directed graphs (where the vertices correspond to the one-dimensional component processes), showing the inner structure of a multivariate time series. Our interest in these graphs originates from the problem of detecting the focus of an epileptic seizure, based on the analysis of invasive EEG data. An example for such an analysis is given in the last section of this chapter.

One of the most widely used contributions to forecasting methodology arising from the book by Box and Jenkins (Time Series Analysis, Forecasting and Control, Holden-Day, 1970) on Forecasting and Control, is their approach to modeling seasonal time series. Their models are used across the world, not least because they have been incorporated into standard software such as the X12-ARIMA seasonal adjustment package. This chapter will be an exposition of these methods, based on a selection of time series case studies, ranging from the airline series example presented by Box and Jenkins, to an example of half hourly electricity demand. The exposition will not, however, be limited to the approach advocated by Box and Jenkins for identifying these models, which is based on inspecting autocorrelation functions of seasonal and non-seasonal differences of the series. The characteristics of seasonality will first be considered, and other approaches to identifying the model suggested. These arise directly from Box and Jenkins (1970) arguments but were not explicitly promoted in their book. Other considerations of the chapter will be how seasonal ARIMA models characterize the relationship between fixed and variable seasonality, the incorporation of calendar effects and the extension of the Airline model to series with two seasonal periods.

State Dependent Parameter (SDP) modelling has been developed by Professor Peter Young in the 1990s to identify non-linearities in the context of dynamic transfer function models. SDP is a very efficient approach and it is based on recursive filtering and Fixed Interval Smoothing (FIS) algorithms. It has been applied successfully in many applications, especially to identify Data-Based Mechanistic models from observed time series data in environmental sciences. In this paper we highlight the role played by the SDP ideas, namely in the simplified State-Dependent Regression (SDR) form, in the context of sensitivity analysis and meta-modelling. Fruitful joint co-operation with Peter Young has led to a series of papers, where SDR has been applied to perform sensitivity analysis, to reduce model’s complexity and to build meta-models (or emulators) capable to reproduce the main features of large simulation models. Finally, we will describe how SDR algorithms can be effectively used in the context of the identification and estimation of tensor product smoothing splines ANOVA models, improving their performances.

This chapter describes an important generalisation of the State Dependent Parameter (SDP) approach to the modelling of nonlinear dynamic systems to include Multi-State Dependent Parameter (MSDP) nonlinearities. The recursive estimation of the MSDP model parameters in a multivariable state space occurs along a multi-path trajectory, employing the Kalman Filter and Fixed Interval Smoothing algorithms. The novelty of the method lies in redefining the concepts of sequence (predecessor, successor), allowing for its use in a multi-state dependent context, so producing efficient parameterisation for a fairly wide class of non-linear, stochastic dynamic systems. The format of the estimated model allows its direct use in control system design. Two worked examples in Matlab are included.

Band-pass, Kalman, and adaptive filters are used for removal of resuscitation artifacts from human ECG signals. The paper is tutorial and clarifies the rationale for applying these methods in the particular biomedical context. Novel aspects of the exposition are deterministic interpretation and comparative study of the methods. A database of separately recorded human ECG and animal resuscitation artifact signals is used for evaluation of the methods. The considered performance criterion is the signal-to-noise ratio (SNR) improvement, defined as the ratio of the SNRs of the filtered signal and the given ECG signal y. The empirical results show that for low SNR of y a band-pass filter yields the best performance while for high SNR an adaptive filter yields the best performance.

Recent work involving the use of robots in stroke rehabilitation has developed model-based algorithms to control the application of functional electrical stimulation to the upper limb of stroke patients with incomplete paralysis to assist in reaching tasks. This, in turn, requires the identification of the response of a human muscle to electrical stimulation. In this chapter an overview of the progress reported in the literature is given together with some currently open research questions.

The Data-Based Mechanistic (DBM) modelling of stochastic, dynamic systems is predominantly an inductive approach that attempts to extract as much mechanistic information as possible from the available time series data, normally in the form of an identifiable linear, or nonlinear, transfer function model that explains the data well. It recognises that, in contrast to most man-made dynamic systems, the nature of many natural systems, particularly at the holistic or macro-level (global climate, river catchment, macro-economy etc.), is still not well understood. ‘Reductionist’ approaches to modelling such systems, often based on the aggregation of hypothetico–deductive models formulated at the micro level, normally results in very large computer simulation models. In contrast to their DBM counterparts, such large models are not normally identifiable from the available data and so rely not only on the validity of the multiple hypotheses on which they are based, but also on how these hypotheses are perceived to combine, in order to produce the aggregate model. This chapter places DBM modelling in the historical context of ‘natural philosophy’; outlines the major stages in its analysis of time series data; and argues that it can help to bridge the gap between complex computer models and simple, ‘dominant mode’, data-based models. This is illustrated by an example which shows how the DBM approach has been used to evaluate models for the transport and dispersion of solutes in river systems.

This paper addresses the issue of identifying a state dependent input nonlinearity in a Data Based Mechanistic (DBM) flood forecasting model based on the data rather than some prior conceptualisation of nonlinearity in the system response. Four forms of nonlinear function are presented. A power law may be useful when the input non-linearity is simple. The Radial Basis Function (RBF) network method is appropriate for systems that exhibit well defined but complex input non-linearities. The Piecewise Cubic Hermite Data Interpolation (PCHIP) method also provides the flexibility to map complex input non-linearity shapes while providing the ability to maintain a natural curve. Overfit to the calibration data is a risk in both RBF and PCHIP methods when a large number of knots are used. The Takagi-Sugeno Fuzzy Inference method, together with interactive tuning, provides an alternative approach that allows human-in-the-loop interaction during the parameter estimation process but is not optimal in any statistical sense. Future work will explore the use of these methods with continuous time transfer functions and optimisation of the nonlinear function at the same time as the transfer function.

The Aggregated Dead Zone (ADZ) model developed by Peter Young and Tom Beer is here applied to the analysis of tracer data from larger rivers. The model provides excellent fits to the observed concentrations, with a dispersive fraction parameter that varies relatively little with discharge. It is also shown how the information on transport and dispersion at different discharges can be augmented by pollution incident and continuously logged water quality data. The model can then be applied to predict the downstream dispersion of pollutants at any arbitrary discharge, taking account of the uncertainty in estimating the ADZ model parameters. Further work remains to be done on relating the parameters of the model to the physical and hydraulic characteristics of river reaches and in gathering data on the gain factor for different pollutants.

In order for water resources management to effectively cope with all the key drivers of global change (climate, demographic, economic, social, policy/law/institutional, and technology changes), it is essential that the traditional sector-by-sector management approach to water resources is transformed into a new paradigm, where water is considered as the principal and cross cutting medium for balancing food, energy security, and environmental sustainability. One major technical challenge in expanding the scope of water resources management across sectors and to the river basin level is to develop new methodologies and tools to cope with the increasing complexity of water systems. When dealing with large water resources systems, particularly with water reservoir networks, traditional non-linear, stochastic control approaches, like Stochastic Dynamic Programming (SDP), suffer from the curse of dimensionality that makes them essentially unusable. In this chapter we review the most advanced, general approaches available to overcome, or at least effectively mitigate, SDP limits in dealing with large water reservoir networks. Depending on the strategy adopted to alleviate the dimensionality burden we distinguish two classes of approaches: methods based on the restriction of the degrees of freedom of the control problem and methods based on the simplification of the water system model. Emphasis is given to the technical implications of the high dimension and highly non-linear nature of water reservoir systems and to the human dimension component involved in their operation. For each approach a real world numerical application is briefly presented.

River basins are key components of water supply grids. River basin operators must handle a complex set of objectives including runoff storage, flood control, supply for consumptive use, hydroelectric power generation, silting management, and maintenance of river basin ecology. At present, operators rely on a combination of simulation and optimization tools to help make operational decisions. The complexity associated with this approach, however, makes it unsuitable for real-time (daily or hourly) operation. The consequence is that between longer-term optimized operating points, river basins are largely operated in open loop. This leads to operational inefficiencies most notably wasted water and poor ecological outcomes. This chapter proposes a systematic approach for the real-time operation of entire river basin networks, using optimal control and employing simple low order models.

Agriculture is the world wide biggest consumer of water. However, a large portion of the water is wasted due to inefficient distribution from lakes and reservoirs via rivers to farms. More efficient water distribution can be achieved with the help of improved control and decision support systems, but in order to design such systems, river models are required. Traditionally, the Saint Venant equations which are partial differential equations, have been used for modelling rivers. They are however difficult to use for control design and thus, simpler alternative models are sought. In this paper, system identification techniques are used to obtain models which are useful for control design. We show through experimental validation and actual control design that simple time-delay and integrator-delay models are sufficient for control design.

Hydrological models have an important role to play in supporting water management. In this paper we consider the insights developed from Pater Young’s research in the context of the particularly challenging problem of predicting the effects of land use and land management change across multiple scales. The strengths and weaknesses of alternative modelling approaches are reviewed. We then consider the utility of physics-based models, firstly applied to issues of upland grazed pasture and the effect of tree shelter belts, supported by an extensive multi-scale field experimental programme at Pontbren, Wales. The models provide useful quantification of local (field-scale) effects; we use meta-modelling emulation to extend the simulations to catchment scale. Secondly we consider the problem of upland peatland management, in the absence of detailed local data. Physics-based modelling provides generic guidance on the effects of drainage and drain blocking on flood risk. We then consider the potential of conceptual models, conditioned by regionalised indices—in this case a Base Flow Index and the US SCS Curve Number. BFI has considerable power in constraining ungauged catchment simulations, and with speculative adjustment of soil categorisation, catchment-scale effects of land management can be simulated. The CN approach is applied through subjective mapping of US to UK soils; although derived for the US, results show that it has considerable utility for UK regional application. Finally, we reflect on the role of Data Based Modelling, and, in application to our detailed experimental data, show its usefulness in identifying appropriate model structures to guide hydrological application.

Catchment classification remains a significant challenge for hydrologists, with available schemes not providing a sufficient basis for consistently distinguishing between different types of hydrological behavior. We analyze 278 catchments distributed across the Eastern USA using a data-based mechanistic (DBM) strategy. We attempt to understand the catchment similarity that can be found with respect to both model parameters (if the same model structure is applied) and with respect to model structures identified as most suitable. Finally, we relate the identified structures and parameters to available physical and climatic catchment-scale characteristics to see whether a further generalization of our result is possible. A significant regional pattern emerged, reflecting the influences of aridity, elevation (steepness) and temperature. In terms of parameter estimates, the most interesting variability between catchments is seen in the response nonlinearity. Significant regional patterns in the non-linearity parameters emerged, and reasonable physical explanations were proposed. Overall, the results of our preliminary study provided here give the impression that the DBM method could be fruitfully applied towards the objective of catchment classification.

Water quality and pollutant transport modelling play an important role in assessing the risk from pollution incidents. They are also important for the sustainable management of water resources. An ability to predict the concentrations of a pollutant travelling along the river is necessary in assessing the ecological impact of the pollutant and to plan a remedy against possible damage to humans and the environment. The risk from a pollutant at a given location along the river depends on the maximum concentration of any toxic component, the travel times of the pollutant from the release point and the duration over which its concentration exceeds feasible threshold levels. These tasks require modelling pollutant transport under varying flow conditions, i.e. unsteady flow. The aim is to outline on-going research on data-based models and to compare the results with physically-based approaches using worked case examples from pollutant transport modelling. In addition to steady-state examples, a Stochastic Transfer pollutant transport model in varying flow conditions is presented. A case study on water quality modelling has the form of a tutorial on the application of a multi-rate Stochastic Transfer Function to the identification of environmental processes.

Phloem vasculature within higher plants functions at very high hydrostatic pressure (10 atmospheres). When the pressure is disrupted, there is a surge of flow that almost immediately results in blockage, making experimentation difficult. Therefore, there are few reliable measurements of sap contents and limited understanding of the biophysics of its flow. Consequently we still do not know how partitioning between competing sinks is controlled. In vivo measurement using radioactive tracers is an important tool in the study of phloem function, but has rarely been quantitatively analysed. A detailed time sequence of phloem sap movement through a plant is possible with in vivo measurement of 11C tracer, which is ideal for input-output analysis. Input-output analysis provides a parsimonious description of tracer movement. The only estimates of transport distribution times, pathway leakage, and partitioning between competing sinks that have been reported are based upon input-output analysis of 11C-labelled photosynthate. These quantitative measurements have led to the first mechanistic understanding of phloem partitioning between competing sinks, from which sink priority has been shown to be an emergent property.

The last two decades have witnessed a significant momentum in chaos theory applications to environmental systems. The outcomes are certainly encouraging, especially considering the still fairly exploratory stage of the theory in the field. Nevertheless, there have also been persistent skepticisms and criticisms on these studies, motivated by the potential limitations in chaos identification methods. The goal of this chapter is to offer a balanced perspective of chaos studies in environmental systems: between the philosophy of chaos theory and the down-to-earth pragmatism needed in its applications. After a presentation of the development of chaos theory, some basic identification methods are described and their reliability for determining system properties demonstrated. A brief review of chaos theory studies in environmental systems as well as the progress and pitfalls is then made. Analysis of four river flow series presents support to the contention that environmental systems are neither deterministic nor stochastic, but a combination of the two, and that chaos theory can offer a middle-ground approach to our extreme deterministic and stochastic views.

This tutorial chapter uses case studies based on recent engineering applications, to re-examine the non-minimal, state variable feedback approach to control system design. We show how the non-minimal state space (NMSS) representation seems to be the natural description of a discrete-time Transfer Function, since its dimension is dictated by the complete structure of the model. This is in contrast to minimal state space descriptions, which only account for the order of the denominator and whose state variables, therefore, usually represent combinations of input and output signals. The resulting control algorithm can be interpreted as a logical extension of the conventional Proportional-Integral (PI) controller, facilitating its straightforward implementation using a standard hardware-software arrangement. Finally, the basic NMSS approach is readily extended into multivariable, model-predictive and nonlinear control systems, hence the chapter briefly discusses these areas and gives pointers to the latest research results.

Complex industrial processes are most often investigated by the formulation of mathematical models that combine the various static and dynamic unit processes into a computer-based ‘simulation model’ that is normally very large, with many parameters characterising numerous, interconnected linear and nonlinear components. By contrast, the multifarious model-based methods available for the design of control systems for such processes are normally based on much simpler models that reflect the ‘dominant modal’ behaviour of the system and so they cannot be applied directly to the large simulation model. This has given rise to a large literature on methods of ‘dynamic model reduction’, where a reduced order representation of the high order simulation model is obtained in various different ways. This chapter considers a recent approach of this kind, where the reduced order ‘emulation model’ is inferred by the application of advanced statistical identification and estimation methods to data obtained from planned experiments on the large simulation model. The practical utility of this Data-Based Mechanistic (DBM) approach to emulation modelling is illustrated by its application to the modelling and control of a multivariable, electrical power generation process. In this case, the ‘nominal’ emulation model is identified as a third order, three input-three output transfer function model and this is used as the basis for the successful Proportional-Integral-Plus (PIP) multivariable control of the large simulation model.

One of the key components in a renewable energy system such as wind energy generator is a three-phase regenerative PWM converter. This component is nonlinear and time-varying by nature. However, with the classical synchronous frame transformation, the nonlinear model is linearized to obtain a continuous-time state-space model. Based on the linearized model, in this paper, a continuous-time model predictive control system (Laguerre function based) for a three-phase regenerative PWM converter is designed and implemented on a laboratory scaled test-bed that was built by the authors. In particular, a prescribed degree of stability is applied to provide a simple tuning parameter to the closed-loop performance.

This chapter illustrates the utility of, and provides the basic documentation for, SSpace, a recently developed MATLAB toolbox for the analysis of State Space systems. The key strength of the toolbox is its generality and flexibility, both in terms of the particular state space form selected and the manner in which generic models are straightforwardly translated into MATLAB code. With the help of a relatively small number of functions, it is possible to fully exploit the power of state space systems, performing operations such as filtering, smoothing, forecasting, interpolation, signal extraction and likelihood estimation. The chapter provides an overview of SSpace and demonstrates its usage with several worked examples.