In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

We present a direct parametrization for continuous-time stochastic state-space models that ensures external stability via the stochastic bounded-real lemma. Our formulation facilitates the construction of probabilistic priors that enforce almost-sure stability, which are suitable for sampling-based Bayesian inference methods. We validate our work with a simulation example and demonstrate its ability to yield stable predictions with uncertainty quantification.

The article introduces a procedure for determining an approximation of the optimal amount of photovoltaics (PVs) for powering water distribution networks (WDNs) through grid-connected PVs. The procedure aims to find the PV amount, minimizing the total expected cost of the WDN over the lifespan of the PVs. The approach follows an iterative process, starting with an initial estimate of the PV quantity and then calculating the total cost of WDN operation. To calculate the total cost of the WDN, we sample PV power profiles that represent the future production based on a probabilistic PV production model. Simulations are conducted assuming these sampled PV profiles power the WDN, and pump flow rates are determined using a control method designed for PV-powered WDNs. Following the simulations, the overall WDN cost is calculated. Since we lack access to derivative information, we employ the derivative-free Nelder-Mead method for iteratively adjusting the PV quantity to find an approximation of the optimal value. The procedure is applied for the WDN of Randers, a Danish town. By determining an approximation of the optimal quantity of PVs, we observe a 14.5% decrease in WDN costs compared to the scenario without PV installations, assuming a 25-year lifespan for the PV panels.

This article presents a novel and computationally efficient approach for determining long-acting insulin doses in individuals with type-2 diabetes (T2D) without relying on a specific physiological model. The main objective is maintaining safe blood glucose (BG) levels within a targeted range. To achieve this, the strategy optimizes the parameters of a designated control law through an online optimization method combined with a gradient estimation technique that relies on noisy evaluations of an objective function. Two distinct control laws are employed and compared, along with two gradient estimation methods: recursive least squares (RLSs) and a one-point residual feedback method. Moreover, we explore the potential advantages of integrating continuous glucose monitoring (CGM) readings into the proposed framework for basal insulin dosing in T2D individuals. This approach is compared with the use of self-monitored BG (SMBG) devices alone. Through simulations with three different models, the proposed insulin calculation framework’s performance surpasses existing strategies from the literature. The results also show that using CGM readings as feedback for long-acting insulin dose calculations significantly lowers the risk of hypoglycemia. Furthermore, the one-point residual feedback method outperformed RLS while. Finally, through the simulation results, we highlight the importance of the choice of the control law in the online optimization framework.

This paper presents a novel approach for improving the performance of digital-to-analog converters (DACs) using moving-horizon optimal quantisation (MHOQ) with optimal noise-shaping filters and digital calibration. DACs have reduced performance due to non-linearities; principally quantisation and element mismatch. The proposed method integrates an optimal noise-shaping filter for a given reconstruction filter with MHOQ to minimise quantisation error at the filter output. In addition, by including a digital calibration model of the measured element mismatch, the modified MHOQ is able to mitigate the distortion caused by this effect. Simulation results are provided that demonstrate that the approach significantly improves the signal-to-noise-and-distortion ratio (SINAD).

This paper proposes a Robust Model Predictive Control (RMPC) method for energy-efficient and reliable pump scheduling in Water Distribution Networks (WDNs), accounting for model uncertainties and demand forecast errors. Building on a previous robust control approach, this extended method uses a linear model with bounded disturbances and optimizes affine disturbance-based pump schedules over a receding horizon. The optimization complexity is reduced from O(N-6) to O(N-3) via a sparse reformulation. When applied to the Randers WDN in Denmark, the method surpasses traditional MPC variants in meeting constraints while maintaining comparable economic performance.

A method for shaping the power spectral density (PSD) of the total error due to uniform quantisation is proposed. It utilises non-subtractive dithering, generated using a joint specification of the probability density function (PDF) and the PSD by way of stochastic minimisation (SM). The output of the quantiser can be made linear and continuous in the mean and the variance of the error can be made independent of the quantiser input by using a dither with a triangular PDF. However, shaping the error PSD to a desired form for reconstruction at the quantiser output remains input dependent. An adaptive dithering approach is implemented to address this dependency. By exploiting symmetry properties of uniform quantisation, it is possible to use SM to generate a limited number of dither sequences and reuse them to shape the total error PSD for arbitrary inputs. The approach is implemented using a look-up table (LUT). When optimised for reconstruction filtering, simulation results demonstrate an improved PSD shaping performance over the state-of-the-art feed-forward method of over two orders of magnitude within a given bandwidth.

We study the Continuous-Discrete Kalman Filter (CD-KF) for State-Space Models (SSMs) where continuous-time dynamics are observed via multiple sensors with discrete, irregularly timed measurements. Our focus extends to scenarios in which the measurement process is coupled with the states of an auxiliary SSM. For instance, higher measurement rates may increase energy consumption or heat generation, while a sensor’s accuracy can depend on its own spatial trajectory or that of the measured target. Each sensor thus carries distinct costs and constraints associated with its measurement rate and additional constraints and costs on the auxiliary state. We model measurement occurrences as independent Poisson processes with sensor-specific rates and derive an upper bound on the mean posterior covariance matrix of the CD-KF along the mean auxiliary state. The bound is continuously differentiable with respect to the measurement rates, which enables efficient gradient-based optimization. Exploiting this bound, we propose a finite-horizon optimal control framework to optimize measurement rates and auxiliary-state dynamics jointly. We further introduce a deterministic method for scheduling measurement times from the optimized rates. Empirical results in state-space filtering and dynamic temporal Gaussian process regression demonstrate that our approach achieves improved trade-offs between resource usage and estimation accuracy.

A method for shaping the power spectral density (PSD) of the total error due to uniform quantisation is proposed. It utilises non subtractive dithering, generated with a joint specification of the probability density function (PDF) and the PSD. If the dither PDF has certain properties, the expected value of the quantiser output can be linearised and the variance of the error (the error power) can be made independent of the quantiser input. It is demonstrated that making the error power independent of the input enables shaping the error PSD as a separate problem. The method relies on the use of a linear noise colouring filter and a static non-linear transform, which imposes some restrictions on the specification of the error PSD. However, it is possible to synthesise a range of filters that can be used to shape the PSD of the error. Simulations are provided to verify and illustrate the operation of the method.

Hydronic Heating, Ventilation, and Air Conditioning (HVAC) systems, where water is used as a media for cooling energy transport, are often used in large buildings. Distributed Air Handling Units (AHUs) condition the air for the cooling and ventilation needs in the building by controlling the chilled water flow. A distributed pump setup, where local pump controllers control the exhaust air temperature, is considered. Commissioning of HVAC is important for the operation of the HVAC and is the focus of this paper. Specifically, a method for local pump controller design, that enables individual operation of the local control loops, as well as operation of the fully connected system. This controller design is expected to fulfill the need for flexibility when setting the building into operation, and thereby ensure better building performance in the end. The theoretical findings are supported by numerical studies of a chilled water HVAC system.

Digital-to-analog converters (DACs) are used to reconstruct digital quantised signals in the analog domain. Practical DACs exhibit several non-ideal effects; principally integral non-linearity (INL). INL is caused by element mismatch, meaning actual output levels deviate from the ideal, causing distortion. To reduce this error, a method employing moving horizon optimal control is proposed, where INL has been integrated into the model. The model is built by precisely measuring the INL of the physical device and organising the data into a lookup table. Previous attempts at using moving horizon optimal quantiser (MHOQ) have assumed ideal quantisation and are therefore unable to reduce the impact of INL. Several other methods exist that can linearise and mitigate the impact of INL but have significant drawbacks, such as a lack of guaranteed stability, complexity, and the need to use specialised and custom circuit topologies that cannot be replicated with off-the-shelf equipment. The effectiveness of the proposed method is demonstrated via simulations and experiments on an off-the-shelf DAC. Copyright (C) 2024 The Authors.

This paper explores the development and application of both linear and nonlinear model predictive control (MPC) strategies for insulin titration in type 2 diabetes (T2D) subjects. By utilizing daily blood glucose measurements, alongside information on insulin injections and meal intake from the previous day, we adjust the insulin sensitivity parameter of the internal model of the controller. This adjustment is based on the steady-state glucose error between the internal model and the plant model. The performance of these strategies was assessed using a high-fidelity T2D model, demonstrating their potential in enhancing the management of T2D. Copyright (C) 2024 The Authors.

In this paper, we derive a PAC-Bayes bound on the generalisation gap, in a supervised time-series setting for a special class of discrete-time non-linear dynamical systems. This class includes stable recurrent neural networks (RNN), and the motivation for this work was its application to RNNs. In order to achieve the results, we impose some stability constraints, on the allowed models. Here, stability is understood in the sense of dynamical systems. For RNNs, these stability conditions can be expressed in terms of conditions on the weights. We assume the processes involved are essentially bounded and the loss functions are Lipschitz. The proposed bound on the generalisation gap depends on the mixing coefficient of the data distribution, and the essential supremum of the data. Furthermore, the bound converges to zero as the dataset size increases. In this paper, we 1) formalize the learning problem, 2) derive a PAC-Bayesian error bound for such systems, 3) discuss various consequences of this error bound, and 4) show an illustrative example, with discussions on computing the proposed bound. Unlike other available bounds the derived bound holds for non i.i.d. data (time-series) and it does not grow with the number of steps of the RNN.

In this paper we derive a PAC-Bayesian error bound for a class of stochastic dynamical systems with inputs, namely, for linear time-invariant stochastic state-space models (stochastic LTI systems for short). This class of systems is widely used in control engineering and econometrics, in particular, they represent a special case of recurrent neural networks. In this paper we 1) formalize the learning problem for stochastic LTI systems with inputs, 2) derive a PAC-Bayesian error bound for such systems, and 3) discuss various consequences of this error bound.

A model describing glitch disturbances is introduced and analysed. The model is used to find the response of glitch disturbances when adding a periodic dither to the input and applying low-pass filtering to the output. This averaged response is used to find a criterion to determine a sufficiently large dither amplitude that will make the averaged glitch disturbances appear as a constant on the output of the low-pass filter. An important resulting property is that the averaged response to glitch disturbances is independent of the model input. By fitting the model to the measured response of a common off-the-shelf digital-to-analogue converter, simulations are used to verify the analytical results.

Digital-to-analog conversion is essential in digital signal processing applications, including closed-loop control schemes. Noise and distortion in digital-to-analog converters result in reduced performance for high-precision mechatronics such as nano-positioning. Glitches are common in practical switched systems such as digital-to-analog converters; observed as an output disturbance. Due to the wide-bandwidth, impulse-like behavior, control law bandwidth is generally too low to provide adequate attenuation; deteriorating open and closed-loop performance. This article demonstrates how large-amplitude high-frequency periodic dither mitigates the effect of glitches in a nano-positioning system under closed-loop control. Simulations are performed using a model that includes significant non-linearities with a response fitted to an off-the-shelf commercial device, as well as using standard linear time-invariant models for other system components fitted to the responses of common, commercially available devices. The results highlight the significance of reconstruction filter design when applying dithering in this setting.

In this paper, we present two simple and novel methods for automatic personalization of target blood glucose concentration values for individuals with Type 2 Diabetes (T2D). The methods can be integrated with any insulin dosing algorithm, or used to provide an individualized reference BG concentration value for medical professionals to consider when determining long-acting insulin doses and other oral medications. The proposed methods were tested in three different simulation models, with different long-acting insulin dosing strategies, and were found to reduce instances of hypoglycemia.

In this paper we derive a Probably Approxilmately Correct(PAC)-Bayesian error bound for linear time-invariant (LTI) stochastic dynamical systems with inputs. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. In particular, with the bound derived in this paper relates future average prediction errors with the prediction error generated by the model on the data used for learning. In turn, this allows us to provide finite-sample error bounds for a wide class of learning/system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

This paper deals with the control of pumps in large-scale water distribution networks with the aim of minimizing economic costs while satisfying operational constraints. Finding a control algorithm in combination with a model that can be applied in real-time is a challenging problem due to the nonlinearities presented by the pipes and the network sizes. We propose a predictive control algorithm with a periodic horizon. The method provides a way for the economic operation of large water networks with a small linear model. Economic Predictive control with a periodic horizon and a terminal state constraint is constructed to keep the state trajectories close to an optimal periodic trajectory. Barrier terms are also included in the cost function to prevent constraint violations. The proposed method is tested on the EPANET implementation of the water network of a medium size Danish town (Randers) and shown to perform as intended under varying conditions.