# Data-driven linear quadratic tracking control

Data-driven control is an alternative to the classical model-based control paradigm that does not identify a model of the plant. In this paper, we show that linear quadratic tracking control is equivalent to errors-in-variables Kalman filtering. Alternatively, data-driven tracking control is identification with two data sets—the identification data and the tracking trajectory. Contrary to the identification problem however the data-driven control problem is fitting a model to a signal that is (in general) not a (noisy) trajectory of the data generating system. We show links between structured low-rank approximation and subspace data-driven control. The basic subspace method is the one from [1]. This method is developed under the assumption that the data is exact. The main question that we study is how to modify the method so that it performs well under noise. We compare empirically the structured low-rank approximation method and the following modifications of the following modifications of the basic subspace method: 1. pseudo-inverse, 2. $$\ell_1$$-norm regularization, 3. unstructured low-rank approximation, 4. structured low-rank approximation, and 5. nuclear norm regularization.

## Reference:

1. I. Markovsky and P. Rapisarda. Data-driven simulation and control. Int. J. Contr., 81(12):1946-1959, 2008.