Orthogonal Polynomial Bases for Data-Driven Analysis and Control of Continuous-Time Systems

• Published in: IEEE transactions on automatic control Vol. 69; no. 7; pp. 4307 - 4319
• Main Authors: Rapisarda, P., van Waarde, Henk J., Camlibel, M.K.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <named-content xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" content-type="math" xlink:type="simple"> <inline-formula> <tex-math notation="LaTeX"> mathcal {H}_{2}</tex-math> </inline-formula> </named-content>-performance; Approximation; Approximation error; Chebyshev approximation; Continuous time systems; continuous-time linear systems; Control systems; Controllability; Data analysis; data-driven control; informativity; polynomial orthogonal basis; Polynomials; quadratic stabilization; Representations; System dynamics; Time invariant systems; Time signals; Trajectory; Transforms
• ISSN: 0018-9286; 1558-2523; 2334-3303; 1558-2523
• DOI: 10.1109/TAC.2023.3321214

We use polynomial approximation theory to perform data-driven analysis and control of linear, continuous-time invariant systems. We transform the continuous-time input trajectories and state trajectories into discrete sequences consisting of the coefficients of their orthogonal polynomial bases representations. We show that the dynamics of the transformed input signals and state signals and those of the original continuous-time trajectories are described by the same system matrices. We investigate informativity, quadratic stabilization, and <inline-formula><tex-math notation="LaTeX">\mathcal {H}_{2}</tex-math></inline-formula>-performance problems for continuous-time systems. We deal with the case in which machine-precision accuracy in the representation of continuous-time signals can be achieved from the data using a finite number of basis elements, and the case in which the approximation error is nonnegligible.