# Data-Driven System Analysis and Control

While data-driven methods for system analysis and control have become increasingly popular over the past years, only few such methods give rigorous guarantees. With this motivation, we develop control-theoretic approaches to data-driven analysis and synthesis problems with desirable theoretical properties in the absence of any model knowledge.

Finding rigorous and efficient ways to integrate data into control theory has been a problem of great interest for many decades. Since most of the classical contributions in control theory rely on model knowledge, the problem of finding such a model from measured data, i.e., system identification, has become a mature research field. More recently, learning controllers directly from data has received increasing interest, but theoretical guarantees have rarely been addressed. Our research aims at developing model-free system analysis and control methods, which are only based on measured data. One approach towards this goal is to extract control-theoretic system properties such as dissipativity or nonlinearity measures from data, which can then be used to design controllers via standard methods from the literature. Moreover, we develop controller design methods such as H_{∞}-controllers or model predictive control approaches, based directly on measured data.

With the rising amount of data, there has been an increasing interest in what is referred to as data-driven controller design. One complementary approach to this direct controller design from data is to learn and analyze certain dissipation inequalities from data first since they allow for the direct application of well-known feedback theorems for controller design. Hence, by learning such system-theoretic input-output properties from data, we obtain insights to the a-priori unknown system, we are not bound to a certain controller structure beforehand while still providing control theoretic guarantees for the closed-loop behavior. Therefore, in this research direction, we study methods to determine dissipation inequalities of the underlying system from available data in storage. This leads to, for example, a necessary and sufficient condition from only one input-output data sample for linear time-invariant systems. By sum-of-squares optimization, also computationally tractable conditions can be derived to verify incremental dissipativity and to find optimal IQCs (e.g. nonlinearity measures) for polynomial and general nonlinear systems.

**Contact Persons:** Julian Berberich, Tim Martin, Felix Brändle, Frank Allgöwer

**Selected Publications:**

- J. M. Montenbruck and F. Allgöwer.
**Some Problems Arising in Controller Design from Big Data via Input-Output Methods.***In Proc. of the 55th Conference on Decision and Control*, Las Vegas, USA, 2016. - A. Romer, J. M. Montenbruck and F. Allgöwer.
**Determining Dissipation Inequalities from Input-Output Samples.***In Proc. 20th IFAC World Congress*, Toulouse, France, 2017. - Anne Romer, Julian Berberich, Johannes Köhler, Frank Allgöwer
**One-shot verification of dissipativity properties from input-output data.***IEEE Control Systems Letters*, vol. 3, no. 3, 2019. - Anne Koch, Julian Berberich, Johannes Köhler, Frank Allgöwer
**Determining optimal input-output properties: A data-driven approach.***Automatica, 2020, 109906.* - Anne Koch, Julian Berberich, Frank Allgöwer
**Provably robust verification of dissipativity properties from data.***IEEE Trans. Automat. Control*, 67(8):4248-4255, 2020. - T. Martin and F. Allgöwer
**Dissipativity verification with guarantees for polynomial systems from noisy input-state data.***IEEE Control Systems Lett.*, 5(4):1399-1404, 2021. - T. Martin and F. Allgöwer.
**Data-driven inference on optimal input-output properties of polynomial systems with focus on nonlinearity measures.***IEEE Trans. Automat. Control, 68(5):2832 - 2847,*2022. - T. Martin and F. Allgöwer
**Data-driven system analysis of nonlinear systems using polynomial approximation.***IEEE Trans. Automat. Control, DOI:10.1109/TAC.2023.3321212**.*

In this research direction, we seek to determine system properties of an unknown input-output system by iteratively conducting (numerical) experiments. In contrast to systems analysis from offline data as introduced above, we hence assume one can apply probing signals to a system and measures the corresponding output. Under this premise, we provide sampling schemes for which we obtain convergence guarantees towards the respective system property for linear time-invariant systems. These sampling strategies to iteratively determine the operator gain, passivity measures and conicity of linear time-invariant systems, for example, are based on gradient dynamical systems and saddle point flows, where the respective gradients can be computed from only input-output data.

**Contact Persons:** Anne Romer, Tim Martin, Frank Allgöwer

**Selected Publications:**

- Anne Romer, Jan Maximilian Montenbruck, Frank Allgöwer
**Sampling strategies for data-driven inference of passivity properties.***In Proc. of the 56th IEEE Conference on Decision and Control*, Melbourne, Australia, 2017. - Anne Romer, Jan Maximilian Montenbruck, Frank Allgöwer

Data-driven inference of conic relations via saddle-point dynamics.*In Proc. 9th IFAC Symposium on Robust Control Design*, Florianopolis, Brazil, 2018. - Anne Koch, Jan Maximilian Montenbruck, Frank Allgöwer

Sampling strategies for data-driven inference of input-output system properties.*IEEE Tansactions on Automatic Control*, 2021.

Although system identification is a well-established research field, there are still only few methods which are computationally tractable and yield guarantees on the identification error from noisy data of finite length. This research project circumvents the system identification step by directly characterizing the closed-loop behavior under state- or output-feedback, using only measured data. Our goal is to translate model-based controller design methods to this data-driven framework, while retaining desirable guarantees for the closed loop.

Recently, we developed a purely data-driven parametrization of the closed loop under state-feedback, based on a single noisy open-loop trajectory without any model knowledge. Using known results from robust control, this parametrization can be employed to design, e.g., data-driven H_{∞}-controllers with end-to-end-guarantees for the closed loop, thus providing a promising alternative to sequential system identification and robust control. Further, we have developed systematic tools to combine data with available prior knowledge in order to reduce conservatism and improve performance in robust controller design.

Moreover, we considered the controller design for general nonlinear systems. To this end, we investigate two different approaches:

1) A controller design by SOS optimization and our data-based representation of general nonlinear systems by Taylor polynomials.

2) Koopman-based feedback design which guarantees closed-loop stability by lifting the nonlinear system to an infinite dimensional system while accounting for the approximation error due to computational restrictions to finite dimensions.

**Contact Persons:** Julian Berberich, Tim Martin, Robin Strässer, Felix Brändle, Frank Allgöwer

**Selected Publications:**

- J. Berberich, C. W. Scherer, F. Allgöwer
**Combining prior knowledge and data for robust controller design***Transactions on Automatic Control*, 2022, doi:10.1109/TAC.2022.3209342.

- J. Berberich, A. Koch, C. W. Scherer, F. Allgöwer
**Robust data-driven state-feedback design***Proc. American Control Conference*, pp. 1532-1538, 2020. - R. Strässer, J. Berberich, F. Allgöwer
**Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics***Proc. 60th IEEE Conference on Decision and Control (CDC), pp. 4344–4351, 2021.* - T. Martin, T. B. Schön, F. Allgöwer
**Gaussian inference for data-driven state-feedback design of nonlinear systems***Proc. 22nd IFAC World Congress**, pp. 4796-4803, 2023.* - T. Martin, T. B. Schön, F. Allgöwer
**Guarantees for data-driven control of nonlinear systems using semidefinite programming: A survey***Annual Reviews in Control**, DOI: 10.1016/j.arcontrol.2023.100911, 2023.* - R. Strässer, J. Berberich, F. Allgöwer
**Robust data-driven control for nonlinear systems using the Koopman operator***Proc. 22nd IFAC World Congress. pp. 2257–2262, 2023.* - R. Strässer, M. Schaller, K. Worthmann, J. Berberich, F. Allgöwer
**Koopman-based feedback design with stability guarantees.***Submitted, preprint: arxiv:2312.01441, 2023.*

**Cooperations:**

- Carsten W. Scherer, University of Stuttgart, Germany
- Thomas B. Schön, Uppsala University, Sweden
- Karl Worthmann, Technische Universität Ilmenau, Germany
- Manuel Schaller, Technische Universität Ilmenau, Germany

Model predictive control (MPC) is a powerful control method, which can handle nonlinear systems and constraints. For the implementation of MPC, an accurate model of the plant is required. In this project, we developed an MPC approach, which uses only measured input-output data to control an unknown system, without identifying a model. This novel framework for data-driven MPC relies on behavioral system theory, which provides an implicit system description based on past measured data. It is appealing both in terms of theoretical properties as well as practical aspects. The main advantage over existing adaptive or learning-based methods is that it requires only an initially measured, persistently exciting data trajectory as well as an upper bound on the system order, but no (set-based) model description and no online estimation process.

Since neither a model nor state-measurements of the system are available, the analysis of this MPC is challenging. Using terminal equality constraints, we proved stability, robustness, and constraint satisfaction of the closed loop, also in the case of noisy output measurements. Further, we have extended these results to stability guarantees using terminal set constraints and a terminal cost function, guarantees without any terminal ingredients for a sufficiently long prediction horizon, and distributed data-driven MPC for dynamically coupled linear systems with guaranteed stability. Finally, we have developed an MPC scheme to control nonlinear systems using only measured input-output data, which are updated online. Under suitable assumptions on the data, system, and design parameters, we prove that this scheme practically exponentially stabilizes the closed loop.

**Contact Persons:** Julian Berberich, Yifan Xie, Frank Allgöwer

**Selected Publications:**

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer
**Stability in data-driven MPC: an inherent robustness perspective***Proc. Conference on Decision and Control,*2022, pp. 1105-1110.

- D. Müller, J. Feilhauer, J. Wickert, J. Berberich, F. Allgöwer, O. Sawodny
**Data-driven predictive disturbance observer for quasi continuum manipulators***Proc. Conference on Decision and Control,*2022, pp. 1816-1822.

- M. Köhler, J. Berberich, M. A. Müller, F. Allgöwer
**Data-driven distributed MPC of dynamically coupled linear systems***Proc. International Symposium on Mathematical Theory of Networks and Systems (MTNS),*2022, pp. 365-370.

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer
**Linear tracking MPC for nonlinear systems Part I: the model-based case***Transactions on Automatic Control*, vol. 67, no. 9, pp. 4390-4405, 2022.

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer
**Linear tracking MPC for nonlinear systems Part II: the data-driven case***Transactions on Automatic Control*, vol. 67, no. 9, pp. 4406-4421, 2022.

- J. Bongard, J. Berberich, J. Köhler, F. Allgöwer
**Robust stability analysis of a simple data-driven model predictive control approach***Transactions on Automatic Control*, 2022, doi: 10.1109/TAC.2022.3163110.

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer.
**On the design of terminal ingredients for data-driven MPC***IFAC Conference on Nonlinear Model Predictive Control,*2021, pp. 257-263.

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer
**Data-driven model predictive control with stability and robustness guarantees***Transactions on Automatic Control*, vol. 66, no. 4, pp. 1702-1717, 2021.

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer.
**Robust constraint satisfaction in data-driven MPC***Proc. Conference on Decision and Control,*2020, pp. 1260-1267. - J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer.
**Data-driven tracking MPC for changing setpoints***IFAC PapersOnLine,*vol. 53, no. 2, pp. 6923-6930, 2020. - J. Berberich, F. Allgöwer
**A trajectory-based framework for data-driven system analysis and control***European Control Conference,*2020, pp. 1365-1370

**Cooperations:**

- Matthias A. Müller, Leibniz University Hannover, Germany
- Johannes Köhler, ETH Zürich, Switzerland
- Oliver Sawodny, University of Stuttgart, Germany

Neural networks are successfully applied in many fields, e.g., classification, segmentation, and reinforcement learning. However, given that they lack rigorous stability and robustness guarantees and can be easily fooled by adversarial attacks, they are unsuitable for safety-critical applications like autonomous driving or medical applications. By overapproximating neural networks exploiting the fact that the underlying activation functions are slope-restricted and sector-bounded, we can formulate linear matrix inequalities that serve as certificates for robustness and closed-loop stability. These certificates can then be used in neural network analysis, or they can also be incorporated into neural network training as constraints. The latter results in neural networks that fulfill desired properties, e.g., an upper bound on the Lipschitz constant chosen by the user or closed-loop stability of a system that includes a neural network nonlinearity.

**Contact Persons:** Patricia Pauli, Frank Allgöwer

**Selected Publications:**

- P. Pauli, N. Funcke, D. Gramlich, M. A. Msalmi, F. Allgöwer.
**Neural network training under semidefinite constraints**In*61st IEEE Conference on Decision and Control (CDC)*, 2022 (accepted). [preprint]

- P. Pauli, D. Gramlich, J. Berberich, F. Allgöwer.
**Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers**In*60th IEEE Conference on Decision and Control (CDC)*, 2021. [available online] - P. Pauli, J. Köhler, J. Berberich, A. Koch, F. Allgöwer.
**Offset-free setpoint tracking using neural network controllers**

In*Learning for Dynamics & Control Conference**,*2021. [available online] - P. Pauli, A. Koch, J. Berberich, P. Kohler, F. Allgöwer.
**Training robust neural networks using Lipschitz bounds**

In*IEEE Control Systems Lett.**,*2021, doi: 10.1109/LCSYS.2021.11.3050444. [available online]