Periodic Reporting for period 2 - Cont4Med (Estimation and control under limited information with application to biomedical systems)
Periodo di rendicontazione: 2022-07-01 al 2023-12-31
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 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.
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.