-
Comparing Model-based Control Strategies for a Quadruple Tank System: Decentralized PID, LMPC, and NMPC
Authors:
Anders H. D. Christensen,
Tobias K. S. Ritschel,
Jan Lorenz Svensen,
Steen Hørsholt,
Jakob Kjøbsted Huusom,
John Bagterp Jørgensen
Abstract:
This paper compares the performance of a decentralized proportional-integral-derivative (PID) controller, a linear model predictive controller (LMPC), and a nonlinear model predictive controller (NMPC) applied to a quadruple tank system (QTS). We present experimental data from a physical setup of the QTS as well as simulation results. The QTS is modeled as a stochastic nonlinear continuous-discret…
▽ More
This paper compares the performance of a decentralized proportional-integral-derivative (PID) controller, a linear model predictive controller (LMPC), and a nonlinear model predictive controller (NMPC) applied to a quadruple tank system (QTS). We present experimental data from a physical setup of the QTS as well as simulation results. The QTS is modeled as a stochastic nonlinear continuous-discrete-time system, with parameters estimated using a maximum-likelihood prediction-error-method (ML-PEM). The NMPC applies the stochastic nonlinear continuous-discrete-time model, while the LMPC uses a linearized version of the same model. We tune the decentralized PID controller using the simple internal model control (SIMC) rules. The SIMC rules require transfer functions of the process, and we obtain these from the linearized model. We compare the controller performances based on systematic tests using both the physical setup and the simulated QTS. We measure the performance in terms of tracking errors and rate of movement in the manipulated variables. The LMPC and the NMPC perform better than the decentralized PID control system for tracking pre-announced time-varying setpoints. For disturbance rejection, the MPCs perform only slightly better than the decentralized PID controller. The primary advantage of the MPCs is their ability to use the information of future setpoints. We demonstrate this by providing simulation results of the MPCs with and without such information. Finally, the NMPC achieves slightly improved tracking errors compared to the LMPC but at the expense of having a higher input rate of movement.
△ Less
Submitted 14 September, 2025;
originally announced September 2025.
-
On Erlang ODE approximations of differential equations with distributed time delays
Authors:
Tobias K. S. Ritschel
Abstract:
In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that we 1) propose an Erlang mixture approximation of the kernel in the DDEs and 2) use the linear chain trick to transform the resulting approximate DDEs to ODEs.…
▽ More
In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that we 1) propose an Erlang mixture approximation of the kernel in the DDEs and 2) use the linear chain trick to transform the resulting approximate DDEs to ODEs. We refer to this as the Erlang ODE approximation of the DDEs, and we prove that the Erlang mixture approximation converges for continuous and bounded kernels if the number of terms increases sufficiently fast. Furthermore, we show that if the kernel is also exponentially bounded, the Erlang ODE approximation can be used to assess the stability of the steady states of the original DDEs and that the solution to the ODE approximation converges. Additionally, we propose an approach based on bisection and least-squares estimation for determining optimal parameter values in the approximation. Finally, we present numerical examples that demonstrate the accuracy and convergence rates of the approximations and the efficacy of the proposed approach for bifurcation analysis and Monte Carlo simulation. The numerical examples involve a modified logistic equation, chemotherapy-induced myelosuppression, and a point reactor kinetics model of a molten salt nuclear fission reactor.
△ Less
Submitted 19 July, 2026; v1 submitted 18 February, 2025;
originally announced February 2025.
-
Numerical optimal control for distributed delay differential equations: A simultaneous approach based on linearization of the delayed variables
Authors:
Tobias K. S. Ritschel
Abstract:
Time delays are ubiquitous in industrial processes, and they must be accounted for when designing control algorithms because they have a significant effect on the process dynamics. Therefore, in this work, we propose a simultaneous approach for numerical optimal control of delay differential equations with distributed time delays. Specifically, we linearize the delayed variables around the current…
▽ More
Time delays are ubiquitous in industrial processes, and they must be accounted for when designing control algorithms because they have a significant effect on the process dynamics. Therefore, in this work, we propose a simultaneous approach for numerical optimal control of delay differential equations with distributed time delays. Specifically, we linearize the delayed variables around the current time, and we discretize the resulting implicit differential equations using Euler's implicit method. Furthermore, we transcribe the infinite-dimensional optimal control problem into a finite-dimensional nonlinear program, which we solve using Matlab's fmincon. Finally, we demonstrate the efficacy of the approach using a numerical example involving a molten salt nuclear fission reactor.
△ Less
Submitted 19 October, 2024;
originally announced October 2024.
-
Numerical optimal control for delay differential equations: A simultaneous approach based on linearization of the delayed state
Authors:
Tobias K. S. Ritschel,
Søren Stange
Abstract:
Time delays are ubiquitous in industry, and they must be accounted for when designing control strategies. However, numerical optimal control (NOC) of delay differential equations (DDEs) is challenging because it requires specialized discretization methods and the time delays may depend on the manipulated inputs or state variables. Therefore, in this work, we propose to linearize the delayed states…
▽ More
Time delays are ubiquitous in industry, and they must be accounted for when designing control strategies. However, numerical optimal control (NOC) of delay differential equations (DDEs) is challenging because it requires specialized discretization methods and the time delays may depend on the manipulated inputs or state variables. Therefore, in this work, we propose to linearize the delayed states around the current time. This results in a set of implicit differential equations, and we compare the steady states and the corresponding stability criteria of the DDEs and the approximate system. Furthermore, we propose a simultaneous approach for NOC of DDEs based on the linearization, and we discretize the approximate system using Euler's implicit method. Finally, we present a numerical example involving a molten salt nuclear fission reactor.
△ Less
Submitted 19 October, 2024; v1 submitted 3 October, 2024;
originally announced October 2024.
-
Optimal price signal generation for demand-side energy management
Authors:
Seyed Shahabaldin Tohidi,
Henrik Madsen,
Davide Calì,
Tobias K. S. Ritschel
Abstract:
Renewable Energy Sources play a key role in smart energy systems. To achieve 100% renewable energy, utilizing the flexibility potential on the demand side becomes the cost-efficient option to balance the grid. However, it is not trivial to exploit these available capacities and flexibility options profitably. The amount of available flexibility is a complex and time-varying function of the price s…
▽ More
Renewable Energy Sources play a key role in smart energy systems. To achieve 100% renewable energy, utilizing the flexibility potential on the demand side becomes the cost-efficient option to balance the grid. However, it is not trivial to exploit these available capacities and flexibility options profitably. The amount of available flexibility is a complex and time-varying function of the price signal and weather forecasts. In this work, we use a Flexibility Function to represent the relationship between the price signal and the demand and investigate optimization problems for the price signal computation. Consequently, this study considers the higher and lower levels in the hierarchy from the markets to appliances, households, and districts. This paper investigates optimal price generation via the Flexibility Function and studies its employment in controller design for demand-side management, its capability to provide ancillary services for balancing throughout the Smart Energy Operating System, and its effect on the physical level performance. Sequential and simultaneous approaches for computing the price signal, along with various cost functions are analyzed and compared. Simulation results demonstrate the generated price/penalty signal and its employment in a model predictive controller.
△ Less
Submitted 31 July, 2024;
originally announced July 2024.
-
Stability analysis of nonlinear stochastic flexibility function in smart energy systems
Authors:
Seyed Shahabaldin Tohidi,
Tobias K. S. Ritschel,
Georgios Tsaousoglou,
Uffe Høgsbro Thygesen,
Henrik Madsen
Abstract:
Demand-side management provides a great potential for improving the efficiency and reliability of energy systems. This requires a mechanism to connect the market level and the demand side. The flexibility function is a novel approach that bridges the gap between the markets and the dynamics of physical assets at the lower levels of the energy systems and activates demand-side flexibility with the…
▽ More
Demand-side management provides a great potential for improving the efficiency and reliability of energy systems. This requires a mechanism to connect the market level and the demand side. The flexibility function is a novel approach that bridges the gap between the markets and the dynamics of physical assets at the lower levels of the energy systems and activates demand-side flexibility with the purpose of decision-making as well as for offering a new framework for balancing and grid services. Employing this function as a key for many decision-making and control algorithms reveals that a mathematically rigorous stability analysis is required for it. In this paper, we investigate the stability properties of two nonlinear flexibility functions, as a dynamic mapping between electricity price and power consumption. Specifically, we analyze the stability of a deterministic flexibility function and an Itô stochastic flexibility function. Simulation results are also provided to demonstrate the dynamics of the flexibility functions and to show that the analytical results hold.
△ Less
Submitted 23 May, 2024;
originally announced May 2024.
-
An algorithm for distributed time delay identification based on a mixed Erlang kernel approximation and the linear chain trick
Authors:
Tobias K. S. Ritschel,
John Wyller
Abstract:
Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a model-based control strategy. However, identifying delays in differential equations is not straightforward and requires specialized methods. Therefore, we propose an algo…
▽ More
Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a model-based control strategy. However, identifying delays in differential equations is not straightforward and requires specialized methods. Therefore, we propose an algorithm for identifying distributed delays in delay differential equations (DDEs) that only involves simulation of ordinary differential equations (ODEs). Specifically, we 1) approximate the kernel in the DDEs (also called the memory function) by the probability density function of a mixed Erlang distribution and 2) use the linear chain trick (LCT) to transform the resulting DDEs into ODEs. Finally, the parameters in the kernel approximation are estimated as the solution to a dynamical least-squares problem, and we use a single-shooting approach to approximate this solution. We demonstrate the efficacy of the algorithm using numerical examples that involve the logistic equation and a point reactor kinetics model of a molten salt nuclear fission reactor.
△ Less
Submitted 12 May, 2024;
originally announced May 2024.
-
Adaptive flexibility function in smart energy systems: A linearized price-demand mapping approach
Authors:
Seyed Shahabaldin Tohidi,
Henrik Madsen,
Georgios Tsaousoglou,
Tobias K. S. Ritschel
Abstract:
This paper proposes an adaptive mechanism for price signal generation using a piecewise linear approximation of a flexibility function with unknown parameters. In this adaptive approach, the price signal is parameterized and the parameters are changed adaptively such that the output of the flexibility function follows the reference demand signal provided by the involved aggregator. This is guarant…
▽ More
This paper proposes an adaptive mechanism for price signal generation using a piecewise linear approximation of a flexibility function with unknown parameters. In this adaptive approach, the price signal is parameterized and the parameters are changed adaptively such that the output of the flexibility function follows the reference demand signal provided by the involved aggregator. This is guaranteed using the Lyapunov stability theorem. The proposed method does not require an estimation algorithm for unknown parameters, that eliminates the need for persistency of excitation of signals, and consequently, simplifies offering the flexibility services. Furthermore, boundedness of the price signal is ensured using a projection algorithm in the adaptive system. We present simulation results that demonstrate the price generation results using the proposed approaches.
△ Less
Submitted 15 February, 2024; v1 submitted 6 December, 2023;
originally announced December 2023.
-
Mathematical meal models for simulation of human metabolism
Authors:
Tobias K. S. Ritschel,
Asbjørn Thode Reenberg,
Peter Emil Carstensen,
Jacob Bendsen,
John Bagterp Jørgensen
Abstract:
We present and critically discuss five commonly used mathematical models of the meal glucose rate of appearance in humans. Such models are key to simulation of the metabolism in healthy people, people with diabetes, and obese people, and they are central to developing effective treatments and prevention strategies. Furthermore, we discuss important aspects of systematic mathematical modeling of hu…
▽ More
We present and critically discuss five commonly used mathematical models of the meal glucose rate of appearance in humans. Such models are key to simulation of the metabolism in healthy people, people with diabetes, and obese people, and they are central to developing effective treatments and prevention strategies. Furthermore, we discuss important aspects of systematic mathematical modeling of human metabolism, including meal consumption modeling, stoichiometry and reaction kinetics, and general-purpose model components.
△ Less
Submitted 31 July, 2023;
originally announced July 2023.
-
Quantitative modeling and simulation of biochemical processes in the human body
Authors:
Jacob Bendsen,
Peter Emil Carstensen,
Asbjørn Thode Reenberg,
Tobias K. S. Ritschel,
John Bagterp Jørgensen
Abstract:
We present a whole-body model of human metabolism that utilizes a system of organs and blood vessels to simulate the enzymatic reactions. The model focuses on key organs, including the brain, heart and lungs, liver, gut, and kidney, as well as muscle and adipose tissue. The model equations are formulated using stoichiometry and Michaelis-Menten kinetics to describe the enzymatic reactions. We demo…
▽ More
We present a whole-body model of human metabolism that utilizes a system of organs and blood vessels to simulate the enzymatic reactions. The model focuses on key organs, including the brain, heart and lungs, liver, gut, and kidney, as well as muscle and adipose tissue. The model equations are formulated using stoichiometry and Michaelis-Menten kinetics to describe the enzymatic reactions. We demonstrate how the model can be used to simulate the effects of prolonged fasting and intermittent fasting on selected metabolite concentrations and glucose flux. Furthermore, by simulating intermittent fasting the effect on the carbohydrate, the protein and the lipid storage is examined. We propose this method as a simple and intuitive approach for modeling the human metabolism, which is general, systematic and easy to incorporate. This could have potential applications in PK/PD drug development and in understanding metabolic disorders.
△ Less
Submitted 30 July, 2023;
originally announced July 2023.
-
A Newton-like Method based on Model Reduction Techniques for Implicit Numerical Methods
Authors:
Tobias K. S. Ritschel
Abstract:
In this paper, we present a Newton-like method based on model reduction techniques, which can be used in implicit numerical methods for approximating the solution to ordinary differential equations. In each iteration, the Newton-like method solves a reduced order linear system in order to compute the Newton step. This reduced system is derived using a projection matrix, obtained using proper ortho…
▽ More
In this paper, we present a Newton-like method based on model reduction techniques, which can be used in implicit numerical methods for approximating the solution to ordinary differential equations. In each iteration, the Newton-like method solves a reduced order linear system in order to compute the Newton step. This reduced system is derived using a projection matrix, obtained using proper orthogonal decomposition, which is updated in each time step of the numerical method. We demonstrate that the method can be used together with Euler's implicit method to simulate CO$_2$ injection into an oil reservoir, and we compare with using Newton's method. The Newton-like method achieves a speedup of between 39% and 84% for systems with between 4,800 and 52,800 state variables.
△ Less
Submitted 13 March, 2023;
originally announced March 2023.
-
Software principles and concepts applied in the implementation of cyber-physical systems for real-time advanced process control
Authors:
Anders H. D. Andersen,
Zhanhao Zhang,
Steen Hørsholt,
Tobias K. S. Ritschel,
John Bagterp Jørgensen
Abstract:
Cyber-physical systems (CPSs) for real-time advanced process control (RT-APC) are a class of control systems using network communication to control industrial processes. In this paper, we use simple examples to describe the software principles and concepts used in the implementation of such systems. The key software principles are 1) shared data in the form of a database, files, or shared memory,…
▽ More
Cyber-physical systems (CPSs) for real-time advanced process control (RT-APC) are a class of control systems using network communication to control industrial processes. In this paper, we use simple examples to describe the software principles and concepts used in the implementation of such systems. The key software principles are 1) shared data in the form of a database, files, or shared memory, 2) timers and threads for concurrent periodic execution of tasks, and 3) network communication between the control system and the process, and communication between the control system and the internet, e.g., the cloud to enable remote monitoring and commands. We show how to implement such systems for Linux operating systems applying the C programming language and we also comment on the implementation using the Python programming language. Finally, we present a complete simulation experiment using a real-time simulator.
△ Less
Submitted 5 July, 2023; v1 submitted 27 February, 2023;
originally announced February 2023.
-
Modeling, scientific computing and optimal control for renewable energy systems with storage
Authors:
Nicola Cantisani,
Tobias K. S. Ritschel,
Christian A. Thilker,
Henrik Madsen,
John Bagterp Jørgensen
Abstract:
This paper presents models for renewable energy systems with storage, and considers its optimal operation. We model and simulate wind and solar power production using stochastic differential equations as well as storage of the produced power using batteries, thermal storage, and water electrolysis. We formulate an economic optimal control problem, with the scope of controlling the system in the mo…
▽ More
This paper presents models for renewable energy systems with storage, and considers its optimal operation. We model and simulate wind and solar power production using stochastic differential equations as well as storage of the produced power using batteries, thermal storage, and water electrolysis. We formulate an economic optimal control problem, with the scope of controlling the system in the most efficient way, while satisfying the power demand from the electric grid. Deploying multiple storage systems allows flexibility and higher reliability of the renewable energy system.
△ Less
Submitted 17 December, 2022;
originally announced December 2022.
-
Model-based control algorithms for the quadruple tank system: An experimental comparison
Authors:
Anders H. D. Andersen,
Tobias K. S. Ritschel,
Steen Hørsholt,
Jakob Kjøbsted Huusom,
John Bagterp Jørgensen
Abstract:
We compare the performance of proportional-integral-derivative (PID) control, linear model predictive control (LMPC), and nonlinear model predictive control (NMPC) for a physical setup of the quadruple tank system (QTS). We estimate the parameters in a continuous-discrete time stochastic nonlinear model for the QTS using a prediction-error-method based on the measured process data and a maximum li…
▽ More
We compare the performance of proportional-integral-derivative (PID) control, linear model predictive control (LMPC), and nonlinear model predictive control (NMPC) for a physical setup of the quadruple tank system (QTS). We estimate the parameters in a continuous-discrete time stochastic nonlinear model for the QTS using a prediction-error-method based on the measured process data and a maximum likelihood (ML) criterion. In the NMPC algorithm, we use this identified continuous-discrete time stochastic nonlinear model. The LMPC algorithm is based on a linearization of this nonlinear model. We tune the PID controller using Skogestad's IMC tuning rules using a transfer function representation of the linearized model. Norms of the observed tracking errors and the rate of change of the manipulated variables are used to compare the performance of the control algorithms. The LMPC and NMPC perform better than the PID controller for a predefined time-varying setpoint trajectory. The LMPC and NMPC algorithms have similar performance.
△ Less
Submitted 9 December, 2022;
originally announced December 2022.
-
State Estimation for Continuous-Discrete-Time Nonlinear Stochastic Systems
Authors:
Marcus Krogh Nielsen,
Tobias K. S. Ritschel,
Ib Christensen,
Jess Dragheim,
Jakob Kjøbsted Huusom,
Krist V. Gernaey,
John Bagterp Jørgensen
Abstract:
State estimation incorporates the feedback in optimization based advanced process control systems and is very important for the performance of model predictive control. We describe the extended Kalman filter, the unscented Kalman filter, the ensemble Kalman filter, and a particle filter for continuous-discrete time nonlinear systems involving stochastic differential equations. Continuous-discrete…
▽ More
State estimation incorporates the feedback in optimization based advanced process control systems and is very important for the performance of model predictive control. We describe the extended Kalman filter, the unscented Kalman filter, the ensemble Kalman filter, and a particle filter for continuous-discrete time nonlinear systems involving stochastic differential equations. Continuous-discrete time nonlinear systems is a natural way to model physical systems controlled by digital controllers. We implement the state estimation methods in Matlab, illustrate and evaluate their performance using simulations of the modified four-tank system. This system is non-stiff and the state estimation methods are implemented numerically using an explicit numerical integration scheme. We evaluate the accuracy of the state estimation methods in terms of the mean absolute percentage error over the simulation horizon. Each method successfully estimates the states and unmeasured disturbances of the simulated modified four-tank system. The key contribution is an overview and comparison of state estimation methods for continuous-discrete time nonlinear stochastic systems. This can guide efficient implementations.
△ Less
Submitted 5 December, 2022;
originally announced December 2022.
-
Estimating a Personalized Basal Insulin Dose from Short-Term Closed-Loop Data in Type 2 Diabetes
Authors:
Sarah Ellinor Engell,
Tinna Björk Aradóttir,
Tobias K. S. Ritschel,
Henrik Bengtsson,
John Bagterp Jørgensen
Abstract:
In type 2 diabetes (T2D) treatment, finding a safe and effective basal insulin dose is a challenge. The dose-response is highly individual and to ensure safety, people with T2D titrate by slowly increasing the daily insulin dose to meet treatment targets. This titration can take months. To ease and accelerate the process, we use short-term artificial pancreas (AP) treatment tailored for initial ti…
▽ More
In type 2 diabetes (T2D) treatment, finding a safe and effective basal insulin dose is a challenge. The dose-response is highly individual and to ensure safety, people with T2D titrate by slowly increasing the daily insulin dose to meet treatment targets. This titration can take months. To ease and accelerate the process, we use short-term artificial pancreas (AP) treatment tailored for initial titration and apply it as a diagnostic tool. Specifically, we present a method to automatically estimate a personalized daily dose of basal insulin from closed-loop data collected with an AP. Based on AP-data from a stochastic simulation model, we employ the continuous-discrete extended Kalman filter and a maximum likelihood approach to estimate parameters in a simple dose-response model for 100 virtual people. With the identified model, we compute a daily dose of basal insulin to meet treatment targets for each individual. We test the personalized dose and evaluate the treatment outcomes against clinical reference values. In the tested simulation setup, the proposed method is feasible. However, more extensive tests will reveal whether it can be deemed safe for clinical implementation.
△ Less
Submitted 25 November, 2022;
originally announced November 2022.
-
State Estimation Methods for Continuous-Discrete Nonlinear Systems involving Stochastic Differential Equations
Authors:
Marcus Krogh Nielsen,
Tobias K. S. Ritschel,
Ib Christensen,
Jess Dragheim,
Jakob Kjøbsted Huusom,
Krist V. Gernaey,
John Bagterp Jørgensen
Abstract:
In this work, we present methods for state estimation in continuous-discrete nonlinear systems involving stochastic differential equations. We present the extended Kalman filter, the unscented Kalman filter, the ensemble Kalman filter, and a particle filter. We implement the state estimation methods in Matlab. We evaluate the performance of the methods on a simulation of the modified four-tank sys…
▽ More
In this work, we present methods for state estimation in continuous-discrete nonlinear systems involving stochastic differential equations. We present the extended Kalman filter, the unscented Kalman filter, the ensemble Kalman filter, and a particle filter. We implement the state estimation methods in Matlab. We evaluate the performance of the methods on a simulation of the modified four-tank system. We implement the state estimation methods for non-stiff systems, i.e., using an explicit numerical integration scheme. The implementation of the extended Kalman filter utilises the Joseph stabilising form for numerical stability. We evaluate the accuracy of the state estimation methods in terms of the mean absolute percentage error over the simulation horizon. We show that each method successfully estimates the states and unmeasured disturbances of the simulated modified four-tank system. Finally, we present conclusions.
△ Less
Submitted 5 May, 2022;
originally announced May 2022.
-
A whole-body multi-scale mathematical model for dynamic simulation of the metabolism in man
Authors:
Peter Emil Carstensen,
Jacob Bendsen,
Asbjørn Thode Reenberg,
Tobias K. S. Ritschel,
John Bagterp Jørgensen
Abstract:
We propose a whole-body model of the metabolism in man as well as a generalized approach for modeling metabolic networks. Using this approach, we are able to write a large metabolic network in a systematic and compact way. We demonstrate the approach using a whole-body model of the metabolism of the three macronutrients, carbohydrates, proteins and lipids. The model contains 7 organs, 16 metabolit…
▽ More
We propose a whole-body model of the metabolism in man as well as a generalized approach for modeling metabolic networks. Using this approach, we are able to write a large metabolic network in a systematic and compact way. We demonstrate the approach using a whole-body model of the metabolism of the three macronutrients, carbohydrates, proteins and lipids. The model contains 7 organs, 16 metabolites and 31 enzymatic reactions. All reaction rates are described by Michaelis-Menten kinetics with an addition of a hormonal regulator based on the two hormones insulin and glucagon. We incorporate ingestion of food in order to simulate metabolite concentrations during the feed-fast cycle. The model can simulate several days due to the inclusion of storage forms (glycogen, muscle protein and lipid droplets), that can be depleted if food is not ingested regularly. A physiological model incorporating complex cellular metabolism and whole-body mass dynamics can be used in virtual clinical trials. Such trials can be used to improve the development of medicine, treatment strategies such as control algorithms, and increase the likelihood of a successful clinical trial.
△ Less
Submitted 3 May, 2022;
originally announced May 2022.
-
Large-scale Virtual Clinical Trials of Closed-loop Treatments for People with Type 1 Diabetes
Authors:
Tobias K. S. Ritschel,
Asbjørn Thode Reenberg,
John Bagterp Jørgensen
Abstract:
We propose a virtual clinical trial for assessing the safety and efficacy of closed-loop diabetes treatments prior to an actual clinical trial. Such virtual trials enable rapid and risk-free pretrial testing of algorithms, and they can be used to compare different treatment variations for large and diverse populations. The participants are represented by multiple mathematical models, consisting of…
▽ More
We propose a virtual clinical trial for assessing the safety and efficacy of closed-loop diabetes treatments prior to an actual clinical trial. Such virtual trials enable rapid and risk-free pretrial testing of algorithms, and they can be used to compare different treatment variations for large and diverse populations. The participants are represented by multiple mathematical models, consisting of stochastic differential equations, and we use Monte Carlo closed-loop simulations to compute detailed statistics of the closed-loop treatments. We implement the virtual clinical trial using high-performance software and hardware, and we present an example trial with two mathematical models of one~million participants over 52~weeks (i.e., two~million simulations), which can be completed in 2~h 9~min.
△ Less
Submitted 3 May, 2022;
originally announced May 2022.
-
Nonlinear Model Predictive Control and System Identification for a Dual-hormone Artificial Pancreas
Authors:
Asbjørn Thode Reenberg,
Tobias K. S. Ritschel,
Emilie B. Lindkvist,
Christian Laugesen,
Jannet Svensson,
Ajenthen G. Ranjan,
Kirsten Nørgaard,
John Bagterp Jørgensen
Abstract:
In this work, we present a switching nonlinear model predictive control (NMPC) algorithm for a dual-hormone artificial pancreas (AP), and we use maximum likelihood estimation (MLE) to identify model parameters. A dual-hormone AP consists of a continuous glucose monitor (CGM), a control algorithm, an insulin pump, and a glucagon pump. The AP is designed with a heuristic to switch between insulin an…
▽ More
In this work, we present a switching nonlinear model predictive control (NMPC) algorithm for a dual-hormone artificial pancreas (AP), and we use maximum likelihood estimation (MLE) to identify model parameters. A dual-hormone AP consists of a continuous glucose monitor (CGM), a control algorithm, an insulin pump, and a glucagon pump. The AP is designed with a heuristic to switch between insulin and glucagon as well as state-dependent constraints. We extend an existing glucoregulatory model with glucagon and exercise for simulation, and we use a simpler model for control. We test the AP (NMPC and MLE) using in silico numerical simulations on 50 virtual people with type 1 diabetes. The system is identified for each virtual person based on data generated with the simulation model. The simulations show a mean of 89.3% time in range (3.9-10 mmol/L) and no hypoglycemic events.
△ Less
Submitted 28 February, 2022;
originally announced February 2022.
-
High-performance Uncertainty Quantification in Large-scale Virtual Clinical Trials of Closed-loop Diabetes Treatment
Authors:
Asbjørn Thode Reenberg,
Tobias K. S. Ritschel,
Bernd Dammann,
John Bagterp Jørgensen
Abstract:
In this paper, we propose a virtual clinical trial for assessing the performance and identifying risks in closed-loop diabetes treatments. Virtual clinical trials enable fast and risk-free tests of many treatment variations for large populations of fictive patients (represented by mathematical models). We use closed-loop Monte Carlo simulation, implemented in high-performance software and hardware…
▽ More
In this paper, we propose a virtual clinical trial for assessing the performance and identifying risks in closed-loop diabetes treatments. Virtual clinical trials enable fast and risk-free tests of many treatment variations for large populations of fictive patients (represented by mathematical models). We use closed-loop Monte Carlo simulation, implemented in high-performance software and hardware, to quantify the uncertainty in treatment performance as well as to compare the performance in different scenarios or of different closed-loop treatments. Our software can be used for testing a wide variety of control strategies ranging from heuristical approaches to nonlinear model predictive control. We present an example of a virtual clinical trial with one million patients over 52 weeks, and we use high-performance software and hardware to conduct the virtual trial in 1 h and 22 min.
△ Less
Submitted 28 February, 2022;
originally announced February 2022.
-
A one-size-fits-all artificial pancreas for people with type 1 diabetes based on physiological insight and feedback control
Authors:
Tobias K. S. Ritschel,
Asbjørn Thode Reenberg,
Emilie B. Lindkvist,
Christian Laugesen,
Jannet Svensson,
Ajenthen G. Ranjan,
Kirsten Nørgaard,
Bernd Dammann,
John Bagterp Jørgensen
Abstract:
We propose a model-free artificial pancreas (AP) for people with type 1 diabetes. The algorithmic parameters are tuned to a virtual population of 1,000,000 individuals, and the AP repeatedly estimates the basal and bolus insulin requirements necessary for maintaining normal blood glucose levels. Therefore, the AP can be used without healthcare personnel or engineers customizing the algorithm to ea…
▽ More
We propose a model-free artificial pancreas (AP) for people with type 1 diabetes. The algorithmic parameters are tuned to a virtual population of 1,000,000 individuals, and the AP repeatedly estimates the basal and bolus insulin requirements necessary for maintaining normal blood glucose levels. Therefore, the AP can be used without healthcare personnel or engineers customizing the algorithm to each user. The estimates are based on bodyweight, measurements from a continuous glucose monitor (CGM), and estimates of the meal carbohydrate contents. In a virtual clinical trial with all 1,000,000 individuals (i.e., a Monte Carlo closed-loop simulation), the AP achieves a mean time in range of more than 87% and almost 89% of the participants satisfy several glycemic targets.
△ Less
Submitted 27 February, 2022;
originally announced February 2022.
-
Model Predictive Control Tailored to Epidemic Models
Authors:
Philipp Sauerteig,
Willem Esterhuizen,
Mitsuru Wilson,
Tobias K. S. Ritschel,
Karl Worthmann,
Stefan Streif
Abstract:
We propose a model predictive control (MPC) approach for minimising the social distancing and quarantine measures during a pandemic while maintaining a hard infection cap. To this end, we study the admissible and the maximal robust positively invariant set (MRPI) of the standard SEIR compartmental model with control inputs. Exploiting the fact that in the MRPI all restrictions can be lifted withou…
▽ More
We propose a model predictive control (MPC) approach for minimising the social distancing and quarantine measures during a pandemic while maintaining a hard infection cap. To this end, we study the admissible and the maximal robust positively invariant set (MRPI) of the standard SEIR compartmental model with control inputs. Exploiting the fact that in the MRPI all restrictions can be lifted without violating the infection cap, we choose a suitable subset of the MRPI to define terminal constraints in our MPC routine and show that the number of infected people decays exponentially within this set. Furthermore, under mild assumptions we prove existence of a uniform bound on the time required to reach this terminal region (without violating the infection cap) starting in the admissible set. The findings are substantiated based on a numerical case study.
△ Less
Submitted 4 May, 2022; v1 submitted 12 November, 2021;
originally announced November 2021.
-
How to coordinate vaccination and social distancing to mitigate SARS-CoV-2 outbreaks
Authors:
Sara Grundel,
Stefan Heyder,
Thomas Hotz,
Tobias K. S. Ritschel,
Philipp Sauerteig,
Karl Worthmann
Abstract:
Most countries have started vaccinating people against COVID-19. However, due to limited production capacities and logistical challenges it will take months/years until herd immunity is achieved. Therefore, vaccination and social distancing have to be coordinated. In this paper, we provide some insight on this topic using optimization-based control on an age-differentiated compartmental model. For…
▽ More
Most countries have started vaccinating people against COVID-19. However, due to limited production capacities and logistical challenges it will take months/years until herd immunity is achieved. Therefore, vaccination and social distancing have to be coordinated. In this paper, we provide some insight on this topic using optimization-based control on an age-differentiated compartmental model. For real-life decision making, we investigate the impact of the planning horizon on the optimal vaccination/social distancing strategy. We find that in order to reduce social distancing in the long run, without overburdening the healthcare system, it is essential to vaccinate the people with the highest contact rates first. That is also the case if the objective is to minimize fatalities provided that the social distancing measures are sufficiently strict. However, for short-term planning it is optimal to focus on the high-risk group.
△ Less
Submitted 20 April, 2021; v1 submitted 22 December, 2020;
originally announced April 2021.
-
How much testing and social distancing is required to control COVID-19? Some insight based on an age-differentiated compartmental model
Authors:
Sara Grundel,
Stefan Heyder,
Thomas Hotz,
Tobias K. S. Ritschel,
Philipp Sauerteig,
Karl Worthmann
Abstract:
In this paper, we provide insights on how much testing and social distancing is required to control COVID-19. To this end, we develop a compartmental model that accounts for key aspects of the disease: 1) incubation time, 2) age-dependent symptom severity, and 3) testing and hospitalization delays; the model's parameters are chosen based on medical evidence, and, for concreteness, adapted to the G…
▽ More
In this paper, we provide insights on how much testing and social distancing is required to control COVID-19. To this end, we develop a compartmental model that accounts for key aspects of the disease: 1) incubation time, 2) age-dependent symptom severity, and 3) testing and hospitalization delays; the model's parameters are chosen based on medical evidence, and, for concreteness, adapted to the German situation. Then, optimal mass-testing and age-dependent social-distancing policies are determined by solving optimal control problems both in open loop and within a model predictive control framework. We aim to minimize testing and/or social distancing until herd immunity sets in under a constraint on the number of available intensive care units. We find that an early and short lockdown is inevitable but can be slowly relaxed over the following months.
△ Less
Submitted 2 November, 2020;
originally announced November 2020.
-
Nonlinear model reduction of dynamical power grid models using quadratization and balanced truncation
Authors:
Tobias K. S. Ritschel,
Frances Weiß,
Manuel Baumann,
Sara Grundel
Abstract:
In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to reduce the models in order to utilize them in real-…
▽ More
In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to reduce the models in order to utilize them in real-time. We reformulate the nonlinear power grid models as quadratic systems and reduce them using balanced truncation based on approximations of the reachability and observability Gramians. Finally, we present examples involving numerical simulation of reduced EN and SM models of the IEEE 57 bus and IEEE 118 bus systems.
△ Less
Submitted 26 May, 2020;
originally announced May 2020.
-
Filtering and Prediction of the Blood Glucose Concentration using an Android Smart Phone and a Continuous Glucose Monitor
Authors:
Zeinab Mahmoudi,
Dimitri Boiroux,
Tobias K. S. Ritschel,
John Bagterp Jørgensen
Abstract:
In this paper we numerically assess the performance of Java linear algebra libraries for the implementation of nonlinear filters in an Android smart phone (Samsung A5 2017). We implemented a linear Kalman filter (KF), an extended Kalman filter (EKF), and an unscented Kalman filter (UKF). These filters are used for state and parameter estimation, as well as fault detection and meal detection in an…
▽ More
In this paper we numerically assess the performance of Java linear algebra libraries for the implementation of nonlinear filters in an Android smart phone (Samsung A5 2017). We implemented a linear Kalman filter (KF), an extended Kalman filter (EKF), and an unscented Kalman filter (UKF). These filters are used for state and parameter estimation, as well as fault detection and meal detection in an artificial pancreas. We present the state estimation technologies used for glucose estimation based on a continuous glucose monitor (CGM). We compared three linear algebra libraries: The Efficient Java Matrix Library (EJML), JAMA and Apache Common Math. Overall, EJML provides the best performance for linear algebra operations. We demonstrate the implementation and performance of filtering (KF, EKF and UKF) using real CGM data.
△ Less
Submitted 27 August, 2019;
originally announced August 2019.
-
Dynamic Optimization of Thermodynamically Rigorous Models of Multiphase Flow in Porous Subsurface Oil Reservoirs
Authors:
Tobias K. S. Ritschel,
John Bagterp Jørgensen
Abstract:
In this paper, we consider dynamic optimization of thermal and isothermal oil recovery processes which involve multicomponent three-phase flow in porous media. We present thermodynamically rigorous models of these processes based on 1) conservation of mass and energy, and 2) phase equilibrium. The conservation equations are partial differential equations. The phase equilibrium problems that are re…
▽ More
In this paper, we consider dynamic optimization of thermal and isothermal oil recovery processes which involve multicomponent three-phase flow in porous media. We present thermodynamically rigorous models of these processes based on 1) conservation of mass and energy, and 2) phase equilibrium. The conservation equations are partial differential equations. The phase equilibrium problems that are relevant to thermal and isothermal models are called the UV and the VT flash, and they are based on the second law of thermodynamics. We formulate these phase equilibrium problems as optimization problems and the phase equilibrium conditions as the corresponding first order optimality conditions. We demonstrate that the thermal and isothermal flow models are in a semi-explicit differential-algebraic form, and we solve the dynamic optimization problems with a previously developed gradient-based algorithm implemented in C/C++. We present numerical examples of optimized thermal and isothermal oil recovery strategies and discuss the computational performance of the dynamic optimization algorithm in these examples.
△ Less
Submitted 24 July, 2018;
originally announced July 2018.