Direct data-driven control : from linear systems to piecewise affine systems

Abstract

Motivated by the challenges of obtaining accurate system models and the convenience of collecting system data, direct data-driven control (DDC) is becoming increasingly popular in the control community. Compared with model-based control methods, DDC approaches aim to design controllers from data directly, bypassing the construction of the system model. This thesis addresses several new challenges in developing DDC methods for unknown systems, ranging from linear time-invariant (LTI) to piecewise affine (PWA) systems.

Firstly, this thesis studies the DDC problem for LTI systems with unmeasurable states. A data-driven output feedback controller is designed using input-output data. Specifically, a novel dimension reduction method is proposed to construct a state using the original input and compressed output. The state-space model based on this new state is proven to be controllable, which enables the construction of a data-based state-space representation with the same input-output relationship as the original system using the pre-collected input-output data. This representation is then used to form a group of data-dependent linear matrix inequalities (LMIs), which are further used to calculate the output feedback control gain.

Secondly, this thesis studies the robust DDC problem for LTI systems with unknown and bounded disturbances. A robust data-driven controller is designed to guarantee internal stability and prescribed $H_\infty$ control performance using input-state-output or input-output data, depending on whether the state is measurable. This method first constructs a set containing all systems that can generate the pre-collected input-state-output data and then designs a controller for all systems in the set. To reduce the conservativeness caused by a single dataset, multiple datasets are used in the controller design. Additionally, the obtained results are extended to the autoregressive exogenous (ARX) systems using only input-output data.

Thirdly, this thesis studies the data-driven predictive control (DPC) problem for LTI systems. For systems without disturbances, a robust data-driven predictive control (RDPC) scheme is proposed using input-state-output or input-output data, which meets the fundamental stability requirement and achieves specific optimal performance. Unlike existing DPC schemes based on behavioral system theory, this approach requires less stringent conditions for the pre-collected data, facilitating easier implementation. For systems with disturbances, an RDPC scheme is also proposed using input-state-output or input-output data. Unlike the former, the latter uses noisy measurement data and employs multiple datasets.

Lastly, inspired by the usefulness of PWA systems in addressing control problems of practical systems using linear control theory, this thesis studies the DDC problem for PWA systems. For systems without disturbances, a data-driven output feedback controller is proposed using input-state-output or input-output data, which can achieve exponential stabilization. Particularly, the partition information of PWA systems is incorporated into the controller design to mitigate the conservativeness of the controller, and multiple datasets are employed to reduce the difficulty in data collection. Moreover, the concept of left coprime is introduced to ensure the controllability of the new state-space system constructed using input-output data when the state is unmeasurable. For systems with disturbances, a data-driven $H_\infty$ controller is designed using input-state-output data.