Periodic Reporting for period 2 - Cont4Med (Estimation and control under limited information with application to biomedical systems)
Okres sprawozdawczy: 2022-07-01 do 2023-12-31
The goal of this project is to develop estimation and control strategies for dynamical systems where only a very limited amount of information is available. This information includes current measurements of the system as well as a mathematical model. The main motivation for considering these problems are biomedical applications, where such a small amount of available information is often inherent. Examples include hormone concentration measurements when considering thyroidal diseases (which are typically only taken every several days/weeks) or monitoring the size of a tumour. Estimating the current state of the system and devising appropriate control actions (e.g. therapy or medication in the above biomedical examples) is very challenging in such applications.
The required methods are not covered by existing approaches in the literature yet, necessitating the development of novel methods and tools. Within this project, particular focus lies on the following aspects.
First, we will study under which conditions is it possible to reconstruct the full internal system information using only few output measurements. Sampling strategies and suitable nonlinear state estimators will be derived.
Second, state estimation and control strategies will be developed for situations with only partial or no knowledge of a mathematical description of the system in question. Again, this is of intrinsic importance in biomedical applications where often the underlying physical principles are only partially understood or too complex. This necessitates the design of so called data- and learning-based methods, for which desired guarantees can be given, even in case of few measurements.
Third, the developed tools will be extended to large-scale systems, where estimation and control has to be achieved in a distributed fashion.
The successful achievement of the project goals will
(i) enable estimation and control in systems with very few, sampled measurements,
(ii) constitute a big step towards a holistic data-based systems and control theory,
(iii) result in a new, data-driven, paradigm for the control of large-scale systems, and
(iv) enable the design of systematic, personalised, and optimal control strategies in biomedical applications.
The required methods are not covered by existing approaches in the literature yet, necessitating the development of novel methods and tools. Within this project, particular focus lies on the following aspects.
First, we will study under which conditions is it possible to reconstruct the full internal system information using only few output measurements. Sampling strategies and suitable nonlinear state estimators will be derived.
Second, state estimation and control strategies will be developed for situations with only partial or no knowledge of a mathematical description of the system in question. Again, this is of intrinsic importance in biomedical applications where often the underlying physical principles are only partially understood or too complex. This necessitates the design of so called data- and learning-based methods, for which desired guarantees can be given, even in case of few measurements.
Third, the developed tools will be extended to large-scale systems, where estimation and control has to be achieved in a distributed fashion.
The successful achievement of the project goals will
(i) enable estimation and control in systems with very few, sampled measurements,
(ii) constitute a big step towards a holistic data-based systems and control theory,
(iii) result in a new, data-driven, paradigm for the control of large-scale systems, and
(iv) enable the design of systematic, personalised, and optimal control strategies in biomedical applications.
Since the beginning of the project, several results regarding the estimation and control of dynamical systems under limited information have been obtained. One source of limited information here considered is the case of having only infrequent and irregular data measurements available. In the context of Work Package 1, conditions for estimating the internal state of the system from irregular output measurements have been studied. In particular, it was shown under which circumstances the infrequently measured data are enough to reconstruct the state of a linear discrete-time system.
The other class of missing information considered in this project regards systems with unknown mathematical model, as proposed in Work Package 2. Different results have been obtained in this field. First, a data-based state estimation scheme for linear systems was developed that relies solely on collected data from the system. Secondly, the machine learning technique known as Gaussian process regression was used to design a state estimator for unknown nonlinear systems. Rigorous theoretical performance guarantees for these estimators were provided. Furthermore, the case was studied in which both the model of the system is unknown and only irregular data measurements are available. Here, methods for obtaining data-based representations of unknown linear systems from a set of irregular measurements were developed. Moreover, a method to obtain data-based representations of linear continuous-time systems was developed. This differs from the previously existing tools that could only describe discrete-time systems. Also, various existing methods for data-based representation and control of linear systems were extended to some classes of nonlinear systems. Additionally, a systematic procedure to design persistently exciting inputs, which allow the collection of informative data from the system, was obtained. Finally, computational complexity concerns have been addressed by developing efficient algorithms to design data-based optimal controllers. Some first extensions of these developed methods to large-scale, distributed systems have already been achieved and will further be considered within Work Package 3.
Finally, progress in the context of Work Package 4 regarding the biomedical application that mainly motivates this project has also been achieved. A predictive control scheme for the hypothalamic-pituitary-thyroid axis was designed to determine optimal medication dosages for hypothyroid patients. Additionally, a mathematical model of the pituitary-thyroid feedback loop was extended, allowing an improved understanding of the origin of the Allan-Herndon-Dudley syndrome.
The other class of missing information considered in this project regards systems with unknown mathematical model, as proposed in Work Package 2. Different results have been obtained in this field. First, a data-based state estimation scheme for linear systems was developed that relies solely on collected data from the system. Secondly, the machine learning technique known as Gaussian process regression was used to design a state estimator for unknown nonlinear systems. Rigorous theoretical performance guarantees for these estimators were provided. Furthermore, the case was studied in which both the model of the system is unknown and only irregular data measurements are available. Here, methods for obtaining data-based representations of unknown linear systems from a set of irregular measurements were developed. Moreover, a method to obtain data-based representations of linear continuous-time systems was developed. This differs from the previously existing tools that could only describe discrete-time systems. Also, various existing methods for data-based representation and control of linear systems were extended to some classes of nonlinear systems. Additionally, a systematic procedure to design persistently exciting inputs, which allow the collection of informative data from the system, was obtained. Finally, computational complexity concerns have been addressed by developing efficient algorithms to design data-based optimal controllers. Some first extensions of these developed methods to large-scale, distributed systems have already been achieved and will further be considered within Work Package 3.
Finally, progress in the context of Work Package 4 regarding the biomedical application that mainly motivates this project has also been achieved. A predictive control scheme for the hypothalamic-pituitary-thyroid axis was designed to determine optimal medication dosages for hypothyroid patients. Additionally, a mathematical model of the pituitary-thyroid feedback loop was extended, allowing an improved understanding of the origin of the Allan-Herndon-Dudley syndrome.
Among the most important results obtained so far in this project are the developed novel methods for purely data-based system analysis, state estimation and control design. In particular, a theoretical result known as Willems' fundamental lemma, which allows for a purely data-based system representation, has been exploited within this project to design robust state estimators that, different from most existing results, do not require the availability of a mathematical model and furthermore allow to consider state constraints (such as, e.g. nonnegativity constraints of hormone concentrations). Similarly, extensions of the fundamental lemma were developed and used to design novel control methods for some classes of nonlinear systems. To this end, systematic methods to design input signals that persistently excite certain classes of nonlinear systems were proposed for the first time. Furthermore, a novel extension of Willems' lemma that allows the data-based representation of continuous-time systems was developed, paving the way for data-based control design also for continuous-time systems.
Moreover, our contributions allow to estimate the internal system state and/or to obtain a data-based system description when only few and potentially irregular measurements from the system are available. These results are of intrinsic importance for the biomedical applications that motivate the theoretical developments of this project.
Finally, in the context of the hypothalamic-pituitary-thyroid (HPT) axis, which serves as the main biomedical application within this project, optimal medication strategies have been developed for different thyroidal diseases and the impact of different medication strategies were analysed using modern control-theoretic tools.
By the end of the project, we expect to have developed robustly stable estimators and/or controllers for nonlinear systems where only few and irregular measurements are available, both in a model-based and in a data-/learning-based setting. For all of the proposed methods, provable theoretical guarantees regarding stability and robustness against external disturbances will be studied. Furthermore, we will work on the design of distributed data-based controllers and estimators for large scale systems where, again, only sparse measurements are available. Finally, the developed methods will be applied and validated on biomedical systems, in particular in the context of the HPT axis, in order to, e.g. develop optimal medication strategies.
Moreover, our contributions allow to estimate the internal system state and/or to obtain a data-based system description when only few and potentially irregular measurements from the system are available. These results are of intrinsic importance for the biomedical applications that motivate the theoretical developments of this project.
Finally, in the context of the hypothalamic-pituitary-thyroid (HPT) axis, which serves as the main biomedical application within this project, optimal medication strategies have been developed for different thyroidal diseases and the impact of different medication strategies were analysed using modern control-theoretic tools.
By the end of the project, we expect to have developed robustly stable estimators and/or controllers for nonlinear systems where only few and irregular measurements are available, both in a model-based and in a data-/learning-based setting. For all of the proposed methods, provable theoretical guarantees regarding stability and robustness against external disturbances will be studied. Furthermore, we will work on the design of distributed data-based controllers and estimators for large scale systems where, again, only sparse measurements are available. Finally, the developed methods will be applied and validated on biomedical systems, in particular in the context of the HPT axis, in order to, e.g. develop optimal medication strategies.