Periodic Reporting for period 3 - Cont4Med (Estimation and control under limited information with application to biomedical systems)
Okres sprawozdawczy: 2024-01-01 do 2025-06-30
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 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 model 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 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 model 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, observability and detectability conditions for linear and nonlinear systems under infrequent/irregular output sampling have been studied. Moreover, an event-triggered moving horizon estimation scheme was developed.
The other class of missing information considered in this project regards systems with unknown mathematical model, as proposed in Work Package 2. In this field, a data-based state estimation scheme for linear systems was developed. Moreover, we designed a Gaussian process-based state estimator for unknown nonlinear systems. Rigorous theoretical performance guarantees for these estimators were provided. We also showed an approach to perform inference in Gaussian process regression for dynamical systems with noisy inputs. Furthermore, we developed methods for data-based representation and control of linear systems for which both the model of the system is unknown and only irregular data measurements are available. Moreover, a method to obtain data-based representations of linear continuous-time systems was obtained. This differs from the previously existing tools that could only describe discrete-time systems. In addition, novel data-driven output-feedback control strategies have been developed. 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, allowing the collection of informative data, was obtained. Finally, computational complexity concerns have been addressed by developing efficient algorithms to design data-based optimal controllers.
Within Work Package 3, we have extended data-driven system representation and control methods to large-scale systems by finding data-based distributed controllers solving the multiagent synchronization problem.
Finally, progress in the context of Work Package 4 regarding the biomedical application that mainly motivates this project has also been achieved. Predictive control schemes for the hypothalamic-pituitary-thyroid axis were designed to determine optimal medication dosages for different thyroidal diseases. 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. We also investigated the preservation of detectability in approximately discretized models, which is relevant for state estimation in biomedical systems.
The other class of missing information considered in this project regards systems with unknown mathematical model, as proposed in Work Package 2. In this field, a data-based state estimation scheme for linear systems was developed. Moreover, we designed a Gaussian process-based state estimator for unknown nonlinear systems. Rigorous theoretical performance guarantees for these estimators were provided. We also showed an approach to perform inference in Gaussian process regression for dynamical systems with noisy inputs. Furthermore, we developed methods for data-based representation and control of linear systems for which both the model of the system is unknown and only irregular data measurements are available. Moreover, a method to obtain data-based representations of linear continuous-time systems was obtained. This differs from the previously existing tools that could only describe discrete-time systems. In addition, novel data-driven output-feedback control strategies have been developed. 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, allowing the collection of informative data, was obtained. Finally, computational complexity concerns have been addressed by developing efficient algorithms to design data-based optimal controllers.
Within Work Package 3, we have extended data-driven system representation and control methods to large-scale systems by finding data-based distributed controllers solving the multiagent synchronization problem.
Finally, progress in the context of Work Package 4 regarding the biomedical application that mainly motivates this project has also been achieved. Predictive control schemes for the hypothalamic-pituitary-thyroid axis were designed to determine optimal medication dosages for different thyroidal diseases. 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. We also investigated the preservation of detectability in approximately discretized models, which is relevant for state estimation in biomedical systems.
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 mathematical models and allow to consider state constraints (like, 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 allowing the data-based representation of continuous-time systems was developed and used for the design of data-driven controllers. This result was also extended for the data-based representation and control of large scale multiagent continuous-time systems.
Moreover, our contributions allow to estimate the internal system state from only few and irregular output measurements. 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 will have developed robustly stable estimators for linear and nonlinear systems where only few and irregular measurements are available, both in a model-based and in a data-based setting. We will also obtain novel data-based controllers for nonlinear systems with provable stability guarantees. Moreover, we will continue our work on the design of distributed data-based controllers for large scale systems using a game-theoretic approach. 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 from only few and irregular output measurements. 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 will have developed robustly stable estimators for linear and nonlinear systems where only few and irregular measurements are available, both in a model-based and in a data-based setting. We will also obtain novel data-based controllers for nonlinear systems with provable stability guarantees. Moreover, we will continue our work on the design of distributed data-based controllers for large scale systems using a game-theoretic approach. 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.