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Recent work, Goodwin et al. (2015, 2018), has established that, subject to an assumption regarding the relative blood glucose response times of meal and insulin, it is optimal for the minimisation of the peak blood glucose excursion to apply a bolus of insulin at the time of (or slightly prior to) meal ingestion. A key assumption underlying this result is that the impact of an impulse of insulin “lasts longer” than the impact of an impulse of food. The result is consistent with clinical trials reported in Lopez et al. (2014) . In the current paper we consider the converse situation where the response of the meal lasts longer than the response of an insulin impulse. The latter situation is known to arise in certain circumstances e.g., when low glycaemic index and/or hi-fat/hi-protein meals are consumed (Bell et al., 2015; Lodefalk et al., 2008; van der Hoogt et al., 2017). We consider the same set-up as in Goodwin et al. (2015). However, here we show that, when the response of the meal lasts longer than the response of an insulin impulse, it is optimal to use an open-loop policy combining an insulin bolus (applied with the meal) and a specific form of decaying insulin flow thereafter. This gives support to results for clinical trials, reported in Lopez et al. (2017). However, as far as we are aware, no earlier work has given a methodology to quantify the best insulin flow after the initial bolus nor the best split between bolus and post-bolus insulin flow. In support of the theory, we apply the results to a model obtained from clinical trials on a real subject with Type 1 diabetes and include a discussion of open-loop versus closed-loop policies.

A fundamental performance limit is derived for a class of positive nonlinear systems. The performance limit describes the achievable output response in the presence of a positive disturbance and subject to a sign constraint on the allowable input. An explicit optimal input is derived which minimises the maximum output response whilst ensuring that the minimum output response does not fall below a pre-specified lower bound. The result provides a fundamental performance standard against which all control policies, including closed loop schemes, can be compared. Implications of the result are examined in the context of blood glucose regulation for Type 1 Diabetes.

Relay autotuners are widely adopted in practice due to their inherent robustness and ease of use. The core idea has been in existence for three decades. Various practical embellishments have been proposed in the literature to deal with such issues as static loads, chattering and low frequency drift. The current paper is motivated by the use of relay autotuning to build and identify models for the control of blood glucose levels in type 1 diabetics. The blood glucose level is affected by large time varying disturbances with broad spectral content arising from carbohydrate consumption and exercise. Consequently, existing strategies for relay autotuning were found to be inappropriate for this problem. To deal with this class of problem we propose a modified relay autotuner incorporating a highly resonant filter. The paper gives theoretical support to the idea, briefly discusses associated implementation issues, suggests several extensions and embellishments and demonstrates its features via simulation studies. The method is also applied to the modelling of the insulin to blood glucose response in a type 1 diabetic.

Open-loop poles on, or near, the imaginary axis in the complex plane are a common occurrence in feedback systems. They can occur as part of the model in systems containing lightly damped resonances or they can be deliberately introduced so as to ensure perfect tracking for sinusoidal or periodic signals. This paper establishes both time and frequency domain implications of such poles on system performance. Dual results for zeros on, or near, the imaginary axis are also discussed. Many practitioners are, at least heuristically, aware of the broad implications of these results but, to the best of the authors’ knowledge, they have not previously been explicitly articulated nor quantified.

This paper establishes a set of performance properties associated with a general class of linear, time-invariant observers for linear systems having predictable disturbances irrespective of how the observer is designed. The concept of the class of linear, unbiased observers is embellished. The implications of the ideas in the context of feedback, integral action and feedforward control are also briefly examined.

This paper presents a fundamental design trade-off applicable to a class of linear positive systems. The result connects the maximum and minimum output response peaks due to a disturbance for all feasible inputs. A core consequence of the result is that, when the disturbance pulse response peaks faster than the input pulse response, then attempts to minimise the maximum peak response to the disturbance are necessarily accompanied by unavoidable undershoot at a later time. The result has potential application in many areas. For example, it provides a defensible benchmark for comparison of all possible insulin treatment strategies for Type 1 diabetes patients. This includes any control strategy implemented in the context of an Artificial Pancreas (AP). The lack of such a benchmark has been a deficiency in the existing AP literature.

In this paper we establish a relationship between the complementary sensitivity function of output feedback control and two other sensitivity functions, namely a (scaled) complementary sensitivity function associated with feedback control based on an auxiliary variable, and a complementary sensitivity function associated with an observer. Implications of the results in related problem areas, e.g. robust model predictive control, are briefly discussed.

Finite data and moving horizon estimation schemes are increasingly being used for a range of practical problems. However, both schemes suffer from potential conceptual difficulties. In the case of finite data, most of the methods in common use, excluding Bayesian strategies, depend upon asymptotic results. On the other hand, in the case of moving horizon estimation, there are two associated problems, namely (i) estimation error quantification is typically not available as a part of the solution and (ii) one needs to provide some form of prior state estimate (the so-called arrival cost). The current paper proposes a combined MAP–Bayesian scheme which, inter alia, addresses the finite data and moving horizon problems described above. The scheme combines MAP and Bayesian strategies. The efficacy of the method is illustrated via numerical examples.

This paper addresses the following problem in networked control: “If a control law is implemented over a channel that supports a certain fixed bit rate what is the best choice for the control update rate and, consequently, the number of bits carried in each sample?” A restricted architecture in which linear filters are used for the encoder/decoder is considered and a quantizer with linear feedback is deployed. Subject to these restrictions, a procedure for designing the controller and associated filters is presented. These filters are then deployed to choose the best number of bits per control update. It is shown, subject to the above restrictions, that it is generally best to use one bit per sample, in which case, the control update rate is equal to the bit rate. Our analysis has two points of departure from contemporary literature in this area. Firstly, we focus on bits per unit time, as opposed to bits per sample. Secondly we use a fixed number of bits in every time period as opposed to an average bit rate.

In spite of its easy implementation, ability to handle constraints and nonlinearities, etc., model predictive control (MPC) does have drawbacks including tuning difficulties. In this paper, we propose a refinement to the basic MPC strategy by incorporating a tuning parameter such that one can move smoothly from an existing controller to a new MPC strategy. Each change of this tuning parameter leads to a new stabilising control law, therefore, allowing one to gradually move from an existing control law to a new and better one. For the infinite horizon case without constraints and for the general case with state and input constraints, stability results are established. We also examine the practical applicability of the proposed approach by employing it in the nominal prediction model of the tube-based output feedback robust MPC method. The merits of the proposed method are illustrated by examples.

One of the most commonly used tools in systems science is that of nonlinear filtering. Applications can be found in control engineering, telecommunications, radar tracking, environmental systems, economics and many other areas. The goal of this paper is to contribute to the application of nonlinear filtering theory by presenting insights into the role of temporal sampling especially the use of up-sampling and down-sampling.

In this paper, we study the accuracy of linear multiple-input multiple-output (MIMO) models obtained by maximum likelihood estimation. We present a frequency-domain representation for the information matrix for general linear MIMO models. We show that the variance of estimated parametric models for linear MIMO systems satisfies a fundamental integral trade-off. This trade-off is expressed as a multivariable 'water-bed' effect. An extension to spectral estimation is also discussed.

In this paper a real-time method for generating signals of constrained amplitude and a given (arbitrary) spectrum is presented. This technique is based on the concatenation of sinusoidal signals of suitably chosen frequencies in order to obtain a signal with the desired sample autocovariance sequence as the number of samples tends to infinity. The effectiveness of the method is demonstrated theoretically and via simulations.

In this paper, we present a novel algorithm for estimating the parameters of a linear system when the observed output signal is quantized. This question has relevance to many areas including sensor networks and telecommunications. The algorithms described here have closed form solutions for the SISO case. However, for the MIMO case, a set of pre-computed scenarios is used to reduce the computational complexity of EM type algorithms that are typically deployed for this kind of problem. Comparisons are made with other algorithms that have been previously described in the literature as well as with the implementation of algorithms based on the Quasi-Newton method.

International audience; In this paper, a new dynamic model describing the epileptic seizure initiation through transition from interictal to ictal state in a brain predisposed to epilepsy is suggested. The model follows Freeman's approach where the brain is viewed as a network of interconnected oscillators. The proposed nonlinear model is experimentally motivated and relies on changes in synaptic strength in response to excitatory spikes. This model exhibits a threshold beyond which a bifurcation toward a short-term plasticity state occurs leading to seizure onset. A resulting explanatory assumption is that when considering epilepsy, brain regions are characterized by abnormally low thresholds toward short-term synaptic plasticity. It is shown by simulation that the proposed model enables some experimentally observed qualitative features to be reproduced. Moreover, a preliminary discussion on the impact of the underlying assumptions on the fundamental issue of seizure control is proposed through an EEG based feedback control scheme

Nonlinear inverse problems arising in dynamic system state/parameter estimation are generally non-convex, and possess multiple local minima that may threaten the convergence of global optimization routines. This problem is generally addressed by multiple starting point based algorithms. An alternative approach is to use the recently proposed concept of safe redundancy in order to derive an algorithm that crosses singularities by exploiting the particular nature of dynamic inverse problems. These techniques might seem similar, at least at first. The present communication aims to provide insight into the difference between these two approaches. This is accomplished through a simple illustrative example.

In this paper we establish the equivalence between least costly and traditional experiment design for control. We consider experiment design problems for both open and closed loop systems. In open loop, equivalence is established for three specific cases, relating to different parametrisations of the covariance expression (i.e. finite and high order approximations) and model structure (i.e. dependent and independently parameterised plant and noise models). In the closed loop setting, we consider only finite order covariance expressions. H"~ performance specifications for control are used to determine the bounds on the covariance expression for both the open and closed loop cases.

In this paper, we present novel results that parameterize a broad class of robust output-feedback model predictive control (MPC) policies for discrete-time systems with constraints and unstructured model uncertainty. The MPC policies we consider employ: (i) a linear state estimator, (ii) a pre-determined feedback gain (iii) a set of ''tighter constraints'' and (iv) a quadratic cost function in the degrees of freedom and the estimated state. Contained within the class, we find both well-known control policies and policies with novel features. The unifying aspect is that all MPC policies within the class satisfy a robust stability test. The robust stability test is suited to synthesis and incorporates a novel linear matrix inequality (LMI) condition which involves the parameters of the cost function. The LMI is shown to always be feasible under an appropriate small-gain condition on the pre-determined feedback gain and the state estimator. Moreover, we show, by means of both theoretical and numerical results, that choosing the cost function parameters subject to the proposed condition often leads to good nominal performance whilst at the same time guaranteeing robust stability.

In networked control systems (NCSs) achievable performance is limited by the communication links employed to transmit signals in the loop. In the present work, we characterise LTI coding systems which optimise performance for various NCS architectures. We study NCSs where the communication link is situated between plant output and controller, and NCSs where the communication link is located between controller and actuator. Furthermore, we present a novel NCS architecture, which is based upon the Youla parameterisation. We show that, which of these architectures gives best performance depends, inter alia, upon characteristics of a related non-networked design, plant disturbances and reference signal. A key aspect of our work, resides in the utilisation of fixed signal-to-noise ratio channel models which give rise to parsimonious designs, where channel utilisation is kept low. The results are verified with simulations utilising bit-rate limited channels.

This paper develops the idea of min-max robust experiment design for dynamic system identification. The idea of min-max experiment design has been explored in the statistics literature. However, the technique is virtually unknown by the engineering community and, accordingly, there has been little prior work on examining its properties when applied to dynamic system identification. This paper initiates an exploration of these ideas. The paper considers linear systems with energy (or power) bounded inputs. We assume that the parameters lie in a given compact set and optimise the worst case over this set. We also provide a detailed analysis of the solution for an illustrative one parameter example and propose a convex optimisation algorithm that can be applied more generally to a discretised approximation to the design problem. We also examine the role played by different design criteria and present a simulation example illustrating the merits of the proposed approach.

We show that the Lagrangian dual of a constrained linear estimation problem is a particular nonlinear optimal control problem. The result has an elegant symmetry, which is revealed when the constrained estimation problem is expressed as an equivalent nonlinear optimisation problem. The results extend and enhance known connections between the linear quadratic regulator and linear quadratic state estimation problems.

Saturating actuators are present in all real control systems. Their effect on system performance clearly depends on the range of control action required relative to the saturation bounds. Much of the prior work on this topic has centred on how to switch linear controllers so as to avoid saturation occurring. This, however, has meant that the full input authority has not been exploited in the control law. Recently, two alternative methods have been proposed for switching linear controllers so as to force the input into saturation. They achieve this goal by scaling the controls or by allowing over-saturation in the switching scheme. In this paper the two methods are combined into a more general scheme. It is also shown that the combined scheme is capable of achieving superior performance. A robust version of the algorithm is also described which is applicable to a class of uncertain systems.

This paper presents a consistent framework for the quantification of noise and undermodelling errors in transfer function model estimation. We use the, so-called, ''stochastic embedding'' approach, in which both noise and undermodelling errors are treated as stochastic processes. In contrast to previous applications of stochastic embedding, in this paper we represent the undermodelling as a multiplicative error characterised by random walk processes in the frequency domain. The benefit of the present formulation is that it significantly simplifies the estimation of the parameters of the embedded process yielding a closed-form expression for the model error quantification. Simulation and experimental examples illustrate how the random walks effectively capture typical cases of undermodelling found in practice, including underdamped modes. The examples also show how to use the method as a tool in the determination of model order and pole location in fixed denominator model structures.

Requiring a controlled system to be decoupled may increase some performance costs, however, decoupling may be desirable from an applied perspective. This paper gives an explicit quantification of decoupling costs in terms of the average quadratic tracking error. The analysis exploits the parametrisation of all decoupling controllers, together with Wiener-Hopf frequency domain techniques.

This paper shows that fundamental limitations arise from the presence of stable zeros on or near the j@w-axis. The stability of the closed-loop system is used in conjunction with the Laplace transform to show that time domain integral constraints must be satisfied by the system response. The presence of stable zeros near the j@w axis together with upper limits on the permissible overshoot of the output signal, place an effective lower bound on the settling time of the closed loop system. The constraints are illustrated by an application to the control of a single stand reversing mill.

This paper develops integral constraints on sensitivity vectors for multivariable feedback systems due to either unstable poles or non-minimum-phase zeros of the plant. The results presented cover the continuous (rational or distributed) case, and are based on the use of directions corresponding to zeros or poles of the plant. The integral constraints are in vector form, and give new insights into the inherent trade-offs in sensitivity reduction for multivariable systems and the cost of decoupling.

This paper examines trade-offs arising in the design of linear filters. These show that there exist fundmental limits on filter performance—limits that are independent of the method of design. These limits may be employed to determine a priori whether desired design specifications are achievable. Analogous constraints are well known in the area of feedback control systems, but, to the best of our knowledge, they have not been previously articulated for the filtering problem.

This paper presents results on the frequency response of sampled data systems. In particular, two new frequency domain sensitivity functions are defined which give qualitative and quantitative information regarding the continuous-time response of these systems. The functions allow a quantification of intersample behaviour.

The identification of an undermodeled transfer function from input-output data is stated as a constrained optimization problem. The constraints determine the identification procedure, the residual error and whether on average the magnitude of the frequency response is L2-overbiased, L2-underbiased or L2-unbiased, as measured by a certain weighted L2-bias integral. The L2-unbiased solutions are linear combinations of L2-overbiased and L2-underbiased solutions, which are precisely the classical least squares estimates. They can be obtained from the solution of certain eigenvalue problems.

This brief paper examines qualitative aspects of the distribution of errors in least squares estimation when undermodelling is present. A key conclusion is that the system gain tends to be underestimated. This has important implications in robust control design using estimated models.

This paper suggests how nonlinear adaptive control might lead to improved control of the dissolved oxygen (DO) concentration in the aerator of a wastewater treatment plant. The DO dynamics can be represented by a bilinear model for which we are interested in both parameter identification and control. The estimation of key parameters of the process model is important because the values of these parameters cannot be obtained from direct measurement. Hence a least-squares procedure for obtaining unique parameter estimates is developed and then combined with a minimum variance control algorithm to obtain an adaptive controller which is used both to generate useful parameter estimates and to control the process. Extensions to the case where the parameters vary at the same rate as the DO are also discussed.

Recently sufficient conditions have been presented for robust stability of adaptive control algorithms. The purpose of this paper is to further examine these conditions, and to study the question of robustness and performance for adaptive and non-adaptive controllers. In particular, we compare an adaptive controller with a fixed linear controller for a representative system including structured and unstructured modelling errors. In the case of unstructured modelling errors alone, we show that the theoretical sufficient conditions for signal boundedness of the adaptive controller are comparable with those for the fixed linear controller designed to achieve equivalent performance. When structured modelling errors (i.e. parametric uncertainties) are added, then simulation studies of a second-order system suggest that the adaptive controller achieves superior performance to the fixed linear controller.

Identification of multi-input multi-output (MIMO) linear dynamic systems is considered for the case when the measurements are obtained during closed loop operation. Both noise free and noisy feedback situations are analysed. For the case where the disturbances in the feedback path are a full rank stochastic process it is shown that, under certain mild conditions, physically meaningful models for the forward and reverse paths can be uniquely determined. For the case where the feedback path is noise free it is shown that the forward path model can be uniquely determined provided the regulator satisfies certain minimal complexity requirements.

In the literature to date there is a dichotomy between results on continuous- and discrete-time model reference control. This is highlighted in the case of continuous-time systems having relative degree greater than one. It is known that, for rapid sampling, these systems always give rise to a non-stably invertible discrete-time system and thus discrete model reference control is ruled out. On the other hand, there are many results pertaining to continuous-time model reference control of such systems. This apparent paradox can be resolved by a slight modification to the discrete-time model format as shown later. This alternative model is used to develop a new discrete model reference adaptive control law and a convergence analysis for the algorithm is presented.

This paper is concerned with the extension of the well-known minimum variance control strategy to the multi-input-multi-output case. It is shown that this extension is straightforward when the system interactor matrix is diagonal but presents some unexpected difficulties in the general case. We develop a suitable stochastic controller for the general case as a logical extension of the single input algorithm. We also explore the properties of the algorithm in detail. We also address the question of adaptive control of multivariable stochastic systems and investigate one possible strategy for overcoming the requirement of knowing the system interactor matrix a priori.

The problem of adaptive control of continuous time deterministic dynamic systems is re-examined. It is shown that the convergence proofs for these algorithms may be decomposed into “modules” dealing with estimation and control, yielding a “key technical lemma” analogous to that used successfully in the study of discrete time systems. The extra freedom provided by the modular structure is used to formulate existing algorithms in a common framework and to derive several new algorithms. It is also shown how least squares, as opposed to gradient, estimation can be used in continuous time adaptive control.

This paper establishes global convergence for an adaptive prediction algorithm. Key features of the algorithm are (i) a priori residuals are used in the regression vector, (ii) the input signal need not be persistently exciting, (iii) only an upper bound is required on the system order, (iv) no monitoring or projection procedure is required to guarantee stability and (v) arbitrary feedback is allowed between output and input.

Recent papers on stochastic adaptive control have established global convergence for algorithms using a stochastic approximation iteration. However, to date, global convergence has not been established for algorithms incorporating a least squares iteration. This paper establishes global convergence for a slightly modified least squares stochastic adaptive control algorithm. It is shown that, with probability one, the algorithm will ensure that 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.

The problem of optimal experiment design for parameter estimation in linear dynamic systems is studied. Results relating to both constrained input and output variances are established. For the case of constrained input variance, it is shown that a D-optimal experiment exists in which the system input is generated externally provided the system and noise transfer functions have no common parameters. For the case of constrained output variance, it is shown that an experiment in which the system input is generated by a combination of a minimum variance control law together with an external set point perturbation is D-optimal for certain classes of systems. Other related results are also presented which illustrate the role of feedback in optimal experiment design.

This paper presents an algorithm for adaptive control of linear systems having unknown time delay. A rational approximation is used for the delay and it is argued that within this framework, the analysis of the properties of the corresponding adaptive control algorithm essentially reduces to the issue of robustness to unmodelled dynamics. A global convergence theory is presented using recent methods for analyzing robustness properties of adaptive control algorithms. The analysis is supported by simulation results.

This paper presents an overview of the current status of convergence theory for adaptive control algorithms. Rather than giving a comprehensive survey, the paper aims to emphasize the conceptual common ground between different approaches. Possible areas for future research are also discussed.

This paper considers the problem of joint determination of input spectra and sampling rate for linear system identification. The concept of an 'input spectrum preserving sampler' is introduced and it is shown that, for this sampler, the usual anti-aliasing filter is optimal, and that joint optimal design of input spectrum and sampling rate may be readily performed in the frequency domain.