# 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 input-output data in storage. This leads to, for example, computational methods for general nonlinear systems, and a necessary and sufficient condition from only one input-output data sample for linear time-invariant systems.

**Contact Persons:** Anne Romer, Julian Berberich, Johannes Köhler, 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.

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.

In control, linear models are preferable over nonlinear because of their simple structure and the well-investigated linear control theory. However, linear models are only valid for systems with mild nonlinear behaviour. Therefore, nonlinearity measures are introduced to quantify the nonlinearity of dynamical systems. These measures give an intuition whether a linear control design is valid. Implicit from the calculation of the nonlinearity measures, a best linear approximation of the nonlinear system behaviour is provided. Since the nonlinearity measures quantify the error of this approximation, results from robust control are applicable. Hence, this linear surrogate model might be preferable over a linear model by Jacobi-linearization or by some approaches from linear system identification. Motivated by the connection of nonlinearity measure to conicity of dynamical systems, a characterization of stability for feedback interconnections using nonlinearity measures can be derived by the concept of graph separation and IQC analysis. By obtaining guaranteed bounds for nonlinearity measures from input-output samples of a system and the characterization of closed-loop stability, nonlinearity measures can be used as a control-theoretic system property for data-driven system analysis and control.

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

**Selected Publications:**

- T. Martin and F. Allgöwer.
**Nonlinearity measures for data-driven system analysis and control.**

In Proc. of the 58th IEEE Conference on Decision and Control, Nice, France, 2019. - T. Schweickhardt and F. Allgöwer
**On Systems Gains, Nonlinearity measures: definition, computation and applications.**

IEEE Transactions on Automatic Control, 10:113-123, 2000.

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. An important open problem is the extension of this approach to output-feedback design problems, in particular for the case that no state-measurements are available.

**Contact Persons:** Julian Berberich, Anne Koch, Frank Allgöwer

**Selected Publications:**

- J. Berberich, A. Koch, C. W. Scherer, F. Allgöwer
**Robust data-driven state-feedback design***American Control Conference*, 2020, accepted.**[preprint]**

**Cooperations:**

- Carsten W. Scherer, University of Stuttgart, 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 and robustness of the closed loop, also in the case of noisy output measurements. Currently, we are investigating several open research directions, which include less conservative stability conditions, extensions to nonlinear systems, as well as practical aspects.

**Contact Persons:** Julian Berberich, Johannes Köhler, Frank Allgöwer

**Selected Publications:**

- J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer.
**Robust constraint satisfaction in data-driven MPC***IEEE Conference on Decision and Control, 2020,*submitted.**[preprint]** - J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer.
**Data-driven tracking MPC for changing setpoints***IFAC World Congress, 2020,*accepted.**[preprint]** - J. Berberich, F. Allgöwer
**A trajectory-based framework for data-driven system analysis and control***European Control Conference,*2020, accepted.**[preprint]** - J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer
**Data-driven model predictive control with stability and robustness guarantees***Transactions on Automatic Control*, 2019, submitted.**[preprint]**

**Cooperations:**

- Matthias A. Müller, Leibniz University Hannover, Germany