Emeritus Laureate Professor Graham Goodwin

The advances in digital computer technology have led to its application in a very wide variety of areas. In particular, it has been used to replace the analog controller in many control systems. However, to use the digital computer as a controller one has to overcome the following problem: The input and output signals of the physical plant are analog, namely, continuous-time signals, and the digital computer can only accept and generate sequences of numbers, namely, discrete-time signals (we do not discuss here the quantization problem).

This paper presents a nonlinear observer-based controller configuration for a class of nonlinear single-input single-output systems with certain observability and stability properties. Particular attention is paid to the effect of input disturbances typically present in chemical processes. It is pointed out that conventional feedback linearizing internal model controllers reject output disturbances linearly, but that they can experience significant performance degradation in the presence of input disturbances. This problem is shown to originate in the lack of the linear superposition property. It is proven that the proposed control architecture rejects constant input disturbances from the control signal and then recovers linear reference tracking. The convenient tuning and improved performance are illustrated for pH neutralization.

Abstract. In time series analysis of data sequences, the estimation of the parameters of an identified autoregressive moving-average (ARMA) model is a well-known and straightforward exercise. However, if the parameters of the model are periodic (i.e. a periodic ARMA (PARMA) model) then the estimation process becomes more difficult. This paper describes an on-line parameter estimation technique, based on methods from automatic control, which is demonstrated to provide consistent estimates of PARMA model parameters. Copyright © 1995, Wiley Blackwell. All rights reserved

This paper cites two cases where the Kalman filter has proved its relevance to the real world of industrial control. The author shows how a Kalman filter based technique was used to achieve center-line gauge control in rolling mills, and how a new method of zinc coating mass estimation and control in a continuous galvanizing plant was developed with the aid of periodic Kalman filtering techniques. It is concluded that modern control techniques, in general, and the Kalman filter, in particular, have been successfully translated from the 'ivory towers' of academia to the real commercial world.

Accurate control of the ferro-static head in continuous casting is believed to be a key factor in improving the quality of the cast product. This paper describes an application of advanced control, that resulted in a complete re-commissioning of the investigated mould level control system. Careful physical modelling identified smooth as well as non-smooth nonlinearities in the process. The paper describes three different control strategies that were implemented and evaluated.

This paper proposes an iterative algorithm for achieving robust performance optimization for uncertain linear systems. The objective is to obtain a controller which minimizes the weighted infinity norm of sensitivity for the worst uncertainty scenario. The algorithm seeks the solution to this mini-max problem by solving the associated maximal and minimal problems in an iterative sequence. The paper describes the conceptual basis of the algorithm supported by simulation examples. © 1993 Taylor & Francis Group, LLC.

This paper investigates low to utilize data to predict a robust performance bandwidth for model-based controllers. The procedures relies on a stochastic embedding approach, in which unmodelled dynamics are viewed as the realization of a stochastic process in the frequency domain. This enhances estimation of a nominal model by an associated model error quantification. It is shown how this joint information can be used to predict a bandwidth of robust performance. For cases where appropriate model-and controller-order are not known a priori, an iterative identification/controller design procedure emerges.

There are three key elements that iterative estimation/controller design techniques require: a closed loop identification procedure to yield an improved model, an uncertainty quantification procedure that captures the validity range of the new model, and a design procedure that utilizes both the model and its uncertainty to generate an improved controller. This paper reports a novel iterative procedure based on estimation of multiplicative increments to generate the successive models, stochastic embedding for quantifying the associated uncertainties and IMC for controller design. The procedure is illustrated on a laboratory process.

This paper describes a case study of advanced control in continuous bloom casting. The study employed nonlinear identification techniques to improve physical understanding of the process. As a consequence, the primary cause of control difficulties in the existing design was found to be slip-stick friction in the hydraulic slide gate valve. A new controller was then designed to compensate for this key nonlinear effect. The new control strategy reduced the magnitude of the previously observed oscillations on the mould level by a factor of ten to one. Subsequent analysis revealed a considerable economic impact resulting in potential savings of many hundreds of thousands of dollars.

This paper addresses the problem of providing bounds on estimated plant frequency responses in a form suitable for robust control design. Our approach is to consider the undermodelling as a particular realisation of a random variable and to derive bounds based on averages over all possible noise realisations and over all possible undermodelling realisations. We critically examine the performance of these bounds relative to those that would be obtained by fitting a high order model to the data and then truncating to a low order model. We also show that the parameter in the distribution for the undermodelling can be estimated from the data analagously to the way measurement noise variance is estimated from prediction errors. We propose several new estimators and examine their finite data and asymptotic properties.

Modern military aircraft are fitted with Engine Monitoring Systems (EMS), which have the potential to provide maintenance personnel with valuable information for diagnosing engine faults and assessing engine condition. In this study, analytical redundancy methods have been applied to gas turbine engine transient data with the view to extracting the desired fault information. The basic idea is to use mathematical models to interrelate the measured variables and then monitor the effects of fault conditions on the new estimates of the model parameters. In most of the existing literature the models used are assumed to be perfect with the primary source of error arising from the measurement noise. In the technique to be described, a new method of quantifying the effects of changes in the operating conditions is presented when simplified models are employed. The technique accounts for undermodeling effects and errors arising from linearization of an inherently nonlinear system. Results obtained show a marked improvement over those obtained with traditional methods. © 1991 ASME.

The demand for wider, thinner rolled steel has led to an increasing interest in the control of the strip 'shape'. Mechanisms for controlling shape include roll bending, roll tilt (or mill steer) and spray pattern control. This paper will principally focus on an analysis of the spray control system and the associated interaction between adjacent zones. This paper describes a new approach to shape control, a mathematical model is derived for the dynamic thermal expansion of the rolls of the a tandem cold rolling mill. The model structure has been specifically designed to facilitate control studies. The system is highly non-linear but the model suggests how the system can be linearized by an appropriate input signal transformation. The linearized system then represents a coupled multi-input multi-output dynamical system linking the cooling water sprays to the measured exit strip shape. This linearized model leads to a dynamical decoupling procedure which reduces the system to a set of single-input single-output control systems. Apart from the application to shape control, the method proposed here has more general applicability since the same kind of problem arises in a variety of cross-machine problems.

The development of a workstation for experiments in adaptive control is described. This workstation, which provides the user with a 'tool box' of resources, is unique in that it implements algorithms using a new discrete time operator, the delta operator, which replaces the shift operator q.

A method is presented to improve the control of cold rolled strips shape by using the coolant sprays. This control strategy is further enhanced by decoupling the zonal interactions and linearising the control strategy. We also discuss an alternative method of coolant valve design to assist coolant flow.

It is shown that the problem of quantifying the effect of model uncertainty on estimation has a simple and intuitively appealing solution when the problem is set up in a Bayesian framework. The results indicate that in the case of Gaussian prior distributions, the optimal estimator, accounting for undermodeling, is particularly straightforward. In the non-Gaussian case, the estimator provides the best linear unbiased estimate. In both cases, the method gives a quantification of the effect of noise and undermodeling on the estimates. The results presented are believed to be of considerable importance in estimation since undermodeling is a generic problem and the results seem to have widespread application. For example, they seem to offer a mechanism for combining robust control and adaptive control. They also have impact on other areas, such as fault detection and diagnosis and experimental design.

This paper illustrates the application of an integrated approach to adaptive control by reference to the case study problem proposed by Masten and Cohen. The underlying philosophy of our approach includes the integration of robust and adaptive control and the unification of continuous and discrete systems theory. A feature of the examples presented below is that they have been run in real time using a general-purpose adaptive control toolbox. Copyright © 1990 John Wiley & Sons, Ltd.

It is well known that integral action can be added to linear systems to achieve zero steady-state error for constant reference inputs and disturbances. The result has also been extended [3] to exponentially stable nonlinear systems. This note shows that a similar property holds for robust adaptive control systems. © 1989 IEEE

This paper addresses the problem of time delay modelling using rational approximations. Two classes of models are discussed based on all-pole and all-pass transfer functions. A detailed analysis is carried out comparing the respective merits of these approximate models. The estimation of the parameters in these models is investigated and the associated control design problem using rational models for the delay is studied. Simulation results are presented which support the theoretical analysis.

This paper examines several issues which arise in state and parameter estimation for linear systems having stochastic disturbances. A key objective will be to ensure that the discrete time results are formulated in such a way that they are consistent with the corresponding continuous time results when fast sampling is used. Some issues in stochastic systems theory with particular emphasis on cases when fast sampling is used are examined. By fast sampling we mean that the sampling frequency, fs, should be greater than the bandwidth of any signal of interest.

This paper illustrates an integrated indirect adaptive design technique using a case study example. Key features of this approach are the separate optimization of procedures for parameter estimation and control law synthesis as well as a mechanism for integrating robust and adaptive control design. Copyright © 1989 John Wiley & Sons, Ltd.

This paper describes an experimental investigation of adaptive control applied to a coupled tank apparatus. The study is intended to explore several practical considerations which arise in the application of adaptive control to realistic processes. In the experiments, the apparatus was configured so as to have several difficulties found in practical systems, including unmodelled time delay, unmodelled high-frequency lag and unmodelled non-linearities. These features are studied both theoretically and experimentally. The adaptive controller is shown to perform satisfactorily in the presence of these practical difficulties. Also a comparison with other strategies for control system design is given. Copyright © 1989 John Wiley & Sons, Ltd.

This paper presents a new approach to the understanding and treatment of the effects of undermodelling in the estimation of models for dynamical systems. There already exists a substantial body of knowledge dealing with the implications of undermodelling on robust control system design. However, the corresponding problem for model estimation has received only scant attention. This paper presents a general conceptual and philosophical framework which adds the important dimension of undermodelling to the estimation problem. The philosophy of the approach is to embed the nominal model structure into a larger class of models which includes a generic characterization of undermodelling. This broader perspective allows the effects of limited model complexity on estimation to be quantified. The implications of these results are believed to be wide-ranging and include new insights into the role of user choices and experimental conditions in estimation, integration of robust and adaptive control, new strategies for fault detection and diagnosis, and so on. Copyright © 1989 John Wiley & Sons, Ltd.

A method to quantify uncertainty due to undermodeling is presented. The unmodeled dynamics are embedded in a general class of systems that is defined using realistic a priori information. This embedding principle can be formalized in several different ways; the one presented involves the setting of a stochastic framework, where the unmodeled dynamics are taken to be a particular realization of a stochastic embedding process. A priori knowledge is used to choose suitable statistics for this process. This approach allows quantification of the effect of the modeling errors on the estimated transfer function in the frequency domain. The principal advantage of this approach is that it allows robust and adaptive control to be considered within the same conceptual framework.

An application of an adaptive control scheme to the problem of controlling fluid level in a pair of coupled tanks is presented. The study is intended to explore several practical issues arising in the use of adaptive control to realistic processes. The controlled system exhibits different time constants for filling and emptying as well as unmodeled static nonlinearity, time delay, high-frequency lag, and D.C. offset. The control scheme has a proportional-integral (PI) structure and is designed using a certainty-equivalence form of pole assignment. The adaptive controller is shown to perform satisfactorily in the presence of these practical difficulties and has characteristics which are superior to those obtained from a well-tuned proportional-integral-derivative (PID) controller.

The authors present a stability result for a decentralized adaptive controller applied to a multivariable linear plant. Robustness is achieved by running the estimators for the loops in parallel with dead-zone functions which take account of the interactions. The essential stability condition is that a certain matrix be an M-matrix. This restricts the interactions to be small relative to the stability of each loop.

An algorithm that extracts the disturbance model from a minimal plant model having a different signal is introduced. The minimal model is used for adaptive pole placement incorporating the internal model principle. The closed-loop system is shown to reject the disturbances asymptotically. A number of simulated examples are given to demonstrate the performance of the algorithm.

Adaptive control has been a topic of interest to control engineers for at least four decades. However, initial attempts to use these methods in practice were largely unsuccessful due to inadequate computer hardware and to lack of a suitable supporting theory. In recent years this situation has changed somewhat due to advances in microprocessor technology and due to the development of a better theoretical understanding of adaptive control. The aim of this paper will be to survey some of the recent results in digital adaptive control and to relate them to practical design issues where possible.

It is shown that the above identification algorithm is convergent in the sense that the plant-model output error asymptotically tends to zero provided a slight modification is added to the original algorithm as described by Landau [1]. © 1985 IEEE.

This note shows how a continuous-time adaptive control law can be constructed which automatically includes integral action. The control law is based on the internal model principle. Global convergence can be readily established for the algorithm using standard methods. © 1985 IEEE.

This paper describes an improved controller design for a conventional metal strip rolling mill with two work-rolls supported by larger diameter backup-rolls. The strategy is based on a model which represents the effects of backup-roll eccentricity, the plastic deformation process and elastic strains in the mill stand components. The strategy overcomes many of the fundamental difficulties inherent in earlier approaches. An important feature of the design is its ability to track an arbitrary backup-roll induced eccentricity disturbance whose waveform changes dynamically and whose period is not known accurately because speed and rotation measurements are only available for the work-rolls. The feedback controller design compensates for force, hysteresis, mill dependent parameters and for the inherently nonlinear response of the force adjusting mechanism. Simulation studies are presented showing the advantages of the new design and its insensitivity to plant parameter variations.

There has been much discussion in recent literature on the question of robustness properties of adaptive control algorithms. The point of view adopted in this paper is that a 'robust' adaptive control algorithm is simply a combination of a 'robust' control law with a 'robust' parameter estimator. Since the constituent algorithms are themselves robust, we have coined the term adaptive robust control rather than the usual term robust adaptive control.

This paper is concerned with the problem of robust estimation and control. The treatment is not comprehensive but instead a number of specific results are presented aimed at (a) gaining a unified description of certain aspects of modern estimation and control theory and (b) isolating those features of particular algorithms which might give enhanced robustness properties especially when implemented digitally. The paper describes preliminary progress towards these long term goals.

The potential benefits of compatibility checking of aircraft dynamic flight data has, in the past, not been fully realized when applied to real data. It is suggested that this is partly due to the presence of errors in the real data which are not usually accounted for in computer simulation studies. When factors such as instrument offset from centre of gravity, measurement time delays, sinusoidal disturbances and non-linearities are accounted for, good results can be achieved with moderate quality instrumentation. The effect of these factors on the identified instrument errors are studied using the extended Kalman Filter approach and the maximum likelihood method. The relative merits of each approach are studied.

It is shown that the standard stochastic adaptive control algorithms for time-invariant systems have an inherent robustness property which renders them applicable, without modification, to time-varying systems whose parameters converge exponentially. One class of systems satisfying this requirement is the class having non-steady-state Kalman filter or innovations representations. This allows the usual assumption of a stationary ARMAX representation to be replaced by a more general state space model.

An indirect adaptive control scheme is presented for deterministic plants which are not necessarily minimum phase. Global convergence is established for the scheme in the sense that the closed loop poles are asymptotically assigned and the system input and output remain bounded for all time. A key feature of the scheme is that no persistency of excitation condition is required. The algorithm has been designed with time-varying problems in mind and uses recursive least squares with variable forgetting factor, normalized regression vectors, and a matrix gain with constant trace.

Describes the development of an instrumentation technique for the measurement of oil film thickness in the bottom end bearings of reciprocating piston engines. (from paper)

This study compares and contrasts the characteristics of the least squares parameter estimation algorithm applied to problems in filtering, prediction and control. The authors discuss the underlying assumptions, the specific form that the algorithm takes and comment on the similarities and differences in the analysis of convergence.

An examination is made of the asymptotic properties of a least squares algorithm for adaptively calculating a d-step ahead prediction of a time series. It is shown that, with probability one, the sample mean square difference between the recursive prediction and the optimal linear prediction converges to zero. Relatively weak assumptions are required regarding the underlying model of the time series.

This study shows that previous results on multi-input multi-output discrete time deterministic and stochastic adaptive control can be extended to a wider class of systems. It is shown that global convergence can be established for an adaptive control algorithm applied to the general class stably invertible systems.

This paper establishes the global convergence (in the sense that the control input, plant output, and all adapting controller parameters remain bounded as the input and output asymptotically approach their respective optimal values) of adaptive implementation of one-step-ahead optimal controllers based on input matching and applied to all minimum phase and some nonmimimum phase, linear, discrete-time, deterministic, time-invariant, single-input, single-output systems. The key assumption is that the one-step ahead optimal controller designed using the true system parameters results in a stable closed-loop system.

Adaptive control of discrete, deterministic, single-input single-output systems is considered using a ModelReference type approach. A sufficient condition upon the adaptation mechanism is obtained. Any algorithm satisfying it solves the adaptive control problem. In particular, constant gain algorithms are shown to verify it without requiring auxiliary signals or anticipative knowledge of the desired output, as generally required. Recent papers have established global convergence for stochastic adaptive control and prediction algorithms for discrete time linear systems. The general delay-white noise case and the unit delay-colored noise case have been treated. This paper extends the results to the general delay-colored noise case. Also, a slightly different proof technique is used which relies only upon elementary analysis and is thus of pedagogical interest. Copyright © 1980 by The Institute of Electrical and Electronics Engineers. Inc.

This paper establishes the global convergence (in the sense that the control input, plant output, and all adapting controller parameters remain bounded as the input and output asymptotically approach their respective optimal values) of adaptive implementation of one-step-ahead optimal controllers based on input matching and applied to all minimum phase and some nonmimimum phase, linear, discrete-time, deterministic, time-invariant, single-input, single-output systems. The key assumption is that the one-step ahead optimal controller designed using the true system parameters results in a stable closed-loop system.

In adaptive control, 3 orders are relevant. These are the system order, model order and controller order. In much of control theory and identification, the system and model orders are assumed to be equal. However, in principle, it would be possible to independently assign the order of the system, model and controller. This paper discusses the rationale behind such an assignment and the resulting implications. A new result on adaptive control of non-minimum phase systems is also presented.

A description is given of recent results concerning global convergence of a stochastic adaptive control algorithm for discrete time linear systems. It is shown that, with probability one, the algorithm will ensure the system inputs and outputs are sample mean square bounded and the mean square output tracking error achieves its global minimum possible value for linear feedback control. Thus, asymptotically, the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known.

This work establishes global convergence for a class of adaptive control algorithms applied to discrete time multi-input multi-output deterministic linear system. It is shown that the algorithms will ensure that the system inputs and outputs remain bounded for all time and that the output tracking error converges to zero.

The design of an optimal experiment for determining the thermal diffusivity and radiation constant for a heat diffusion process is considered. The design variables considered are the boundary perturbation and the spatial location of a temperature measuring sensor. It is shown that suitable choice of these variables leads to improved estimation accuracy. This example illustrates an approach to design of optimal experiments for distributed parameter systems and the results are further illustrated by a hybrid computer simulation. It is also shown that the design philosophy can be applied to a wide class of systems described by partial differential equations.

Recent work on the estimation of plant dynamics using data collected under closed loop conditions is surveyed. This problem is of practical importance since it is often not feasible to operate a plant in open loop. Feedback may be required to ensure stability or to maintain acceptable product quality. It is shown that the presence of feedback can lead to ambiguities in the estimation of open loop models from closed loop data. It is also shown that these ambiguities can be avoided provided the feedback system satisfies certain structural constraints. Examples illustrating various aspects of closed loop estimation are presented.

Preliminary studies are described related to the problem of designing adaptive regulators when the structure of the control law is restricted. The design of restricted complexity regulators is of practical importance. An examination is made of several different approaches to the design of restricted complexity adaptive regulators. It is shown that restricted complexity regulators can be readily designed and may offer several advantages over conventional regulators.

Problems associated with the identification of systems operating in closed loop are investigated. These problems include: (1) Conditions under which unique identification is possible. (2) Identification procedures which are applicable in closed loop. (3) Achievable accuracy when there are constraints on the allowable input or output variations. The above ideas are applied to the identification of a 200 Kw synchronous generating plant driven by a diesel prime mover. A simple model of the diesel prime mover is recovered from experimental data collected during operation in closed loop.

Experiment design procedures have been described in the chapter for a number of diverse modeling problems, including agriculture, economics, social sciences, chemical engineering, industrial engineering and biological sciences. One of the earliest publications on optimal test signal design appeared in 1960, the major result being that an input having impulsive autocorrelation function is optimal for the estimation of the parameters of a moving average process in the presence of white measurement noise. For those problems where the choice of sampling instants and presampling filters does arise, it has been recognized that this choice often has a significant effect on the information return from the experiment. The choice is particularly critical when the available computer storage and analysis time are limited. In these cases, the experiment should be designed to maximize the average information content of each sample. In the sampling problem, four cases can be distinguished depending upon whether the samples are uniformly or non-uniformly spaced in time and whether the test signal is prespecified or is to be jointly designed. The chapter presents the state of the art in the design of sampling intervals in each of these cases. © 1976, Academic Press Inc.

This note gives the necessary conditions for local identifiability of innovations models of linear dynamic systems. It is shown that the information matrix is singular if the input power density spectrum contains less than p/2m frequency components where p is the number of input gain parameters (elements of B and D) and m is the number of outputs. © 1974 Taylor and Francis Group, LLC.

This paper discusses the problem of optimal design of experimental conditions for linear system identification. It is demonstrated that, in general, to achieve maximal return from an experiment, coupled design of all the experimental conditions, namely the test signal, sampling intervals and filters, should be carried out simultaneously. For the case of uniform sampling it is shown that joint design of the presampling filter, sampling rate and input can be carried out in the frequency domain. For the case of nonuniform sampling a sequential design procedure is developed which optimizes the information increment between samples. © 1974, IEEE. All rights reserved.

In this paper the research work currently in progress on computer aided control system design at Imperial College is described. Programs have been developed for the various facets of control work and these programs have been combined into a unified design facility.

A sufficient condition for consistent estimates of parametera in linear S.I.S.O. system identification is that the input signal should be ¿ persistently exciting ¿, However., in practice the exact nature of the test signal has a direct bearing upon the identification accuracy. This paper shows how a-priori knowledge of the approximate system and noise model parameters may be used to design a test signal which will minimize the variance of the parameter estimates. The design procedure is computationally efficient as full advantage is taken of the special structure of linear S.I.S.O. systems. The optimal test signals are shown to result in a significant reduction in the parameter variances. This is of considerable practical importance as either the experiment time may be reduced for the same model accuracy or for fixed experiment time a model of much higher accuracy may be obtained. © 1973 Taylor and Francis Group, LLC.

An interactive collection of programs for optimal test signal design and closed loop system identification is described. The optimal test signals are shown to lead to improved identification accuracy. The closed loop test signal design algorithm has been implemented as an interactive program on a PDP-15 using a storage display for graphical output. Several examples showing the application of the design collection are presented. One example shows that models for open loop unstable systems may be obtained from closed loop measurements.

A collection of programs has been prepared, primarily for industrial computer-aided design, which aims at providing all features commonly grouped under the heading now called ¿classical control design methods¿ . Existing algorithms were used where possible, but where they were unnecessarily restrictive new algorithms were developed. Two added features of the design collection are a common controlling program and a common data base, so that the various design algorithms can be quickly used in succession on a particular problem. The individual design programs available are: root locus; Nyquist, Nichols; Bode; step-response; bilinear transformations; continuous-to-discrete transformations; Routh criterion; polynomial root solution. Each program is briefly described. A simple example illustrates some features of the collection.

It is shown that the optimal test signal for linear single-input single-output parameter estimation is characterized by its autocorrelation function. This fact has been shown to lead to an efficient algorithm for test signal design. Furthermore, a procedure has been described for arriving at the most appropriate design criteria for any given ultimate use of the model.

Saturation of the input transducer is a frequently occurring difficulty with minimum-variance controllers. This saturation can lead to serious deterioration in the performance. In the letter, a minimum-variance control law is derived for the case when the input amplitude is constrained. © 1972, The Institution of Electrical Engineers. All rights reserved.

When fitting a continuous model to p.r.b.n. results, there are several sources of error. The letter deals with the incorporation of interclock sampling and folding-error corrections for the normal m sequence, the maximally orthogonal sequence and the modified m sequence. It further shows how to correct for d.c. offset errors and for bias and ripple in the autocorrelation function where they occur. A comparison is made between the three sequences using an analogue model and the advantages of each sequence are pointed out. © 1970, The Institution of Electrical Engineers. All rights reserved.

Nearly all the computational time required in digital calculation of the gradient of an error index with respect to initial state is taken up by integration of differential equations. By direct application of the variational calculus, it is shown that the gradient may be calculated by integration of the state equations and costate equations, namely 2n equations. Previously, it has been thought necessary to integrate the n state equations and n2 state-sensitivity equations, namely n2 + n equations. © 1970, The Institution of Electrical Engineers. All rights reserved.

Results on the calculation of gradient in state and parameter estimation, contained in a recent letter, are extended to curvature calculation. © 1970, The Institution of Electrical Engineers. All rights reserved.

An input signal is synthetised which minimises the covariance of the error in the estimates of the parameters and states of a dynamic process. The method is applicable as long as the noise is approximately white and initial estimates of parameters and states are available. © 1969, The Institution of Electrical Engineers. All rights reserved.

Crosscorrelation using pseudorandom binary noise as the test signal is a well established technique for the identification of linear systems, but there are several practical difficulties. We show that two of these difficulties, namely the necessity to have fast clock rates and a long p.r.b.n. period can be eliminated by using interclock sampling and by analytically removing folding errors. Further, if a maximally orthogonal test sequence is used, bias or ripple difficulties are overcome, and the analysis is particularly simple and practically tractable. © 1969, The Institution of Electrical Engineers. All rights reserved.

An adjoint variable technique is developed which simplifies the calculation of the second derivatives of an error index with respect to parameters and initial states. The adjoint variable technique requires the numerical integration of only a fraction of the number of equations required by existing techniques. © 1968 Taylor and Francis Group, LLC.

A new class of cyclic binary sequences is shown to exist, having an autocorrelation function representable by a series of periodically spaced alternate positive and negative spikes, with zero elsewhere. © 1967, The Institution of Electrical Engineers. All rights reserved.

The quasilinearisation method (or, equivalently, the parameter-influence-coefficient method) is currently preferred to the adjoint-variable method in solving the identification problem because it leads to a one-point boundary-value problem. It is shown that hybrid use of the two methods greatly reduces the number of first-order differential equations to be numerically integrated. © 1967, The Institution of Electrical Engineers. All rights reserved.

A modified msequence is proposed to simplify the instrumentation in using the Gardiner method of identifying the linear channel of a nonlinear process. The class of sequence has properties suited to multivariate crosscorrelation. © 1966, The Institution of Electrical Engineers. All rights reserved.