Notes on Data-Driven Output-Feedback Control of Linear MIMO Systems

Recent works have approached the data-driven design of dynamic output-feedback controllers for discrete-time linear time-invariant (LTI) systems by constructing nonminimal state vectors composed of past inputs and outputs. Depending on the system's complexity (order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula>, lag <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula>, and number of outputs <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula>), it was observed in several works that such an approach presents significant limitations. In particular, many works require to restrict the class of LTI systems to those satisfying the relation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p\ell =n$</tex-math></inline-formula>. In this note, we show how to address the general multi-input multi-output (MIMO) case (for which <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p\ell \geq n$</tex-math></inline-formula> in general) by constructing an alternative nonminimal state vector from data. Different from the existing literature, our method guarantees the satisfaction of certain rank conditions when the system is persistently excited, thereby facilitating the direct data-driven dynamic output-feedback control of MIMO systems by applying methods that were originally developed for the input-state data setting.