Recently, there has been renewed interest in data-driven control, that is, the design of controllers directly from observed data. In the case of linear time-invariant (LTI) systems, several approaches have been proposed that lead to tractable optimization problems. On the other hand, the case of nonlinear dynamics is considerably less developed, with existing approaches limited to at most rational dynamics and requiring the solution to a computationally expensive Sum of Squares (SoS) optimization. Since SoS problems typically scale combinatorially with the size of the problem, these approaches are limited to relatively low order systems. In this paper, we propose an alternative, based on the use of state-dependent representations. This idea allows for synthesizing data-driven controllers by solving at each time step an on-line optimization problem whose complexity is comparable to the LTI case. Further, the proposed approach is not limited to rational dynamics. The main result of the paper shows that the feasibility of this on-line optimization problem guarantees that the proposed controller renders the origin a globally asymptotically stable equilibrium point of the closed-loop system. These results are illustrated with some simple examples. The paper concludes by briefly discussing the prospects for adding performance criteria.
A Loewner Matrix Based Convex Optimization Approach to Finding Low Rank Mixed Time/Frequency Domain Interpolants
We consider the problem of finding the lowest order stable rational transfer function that interpolates a set of given noisy time and frequency domain data points. Our main result shows that exploiting results from rational interpolation theory allows for recasting this problem as minimizing the rank of a matrix constructed from the frequency domain data (the Loewner matrix) along with the Hankel matrix of time domain data, subject to a semidefinite constraint that enforces stability and consistency between the time and frequency domain data. These results are applied to a practical problem: identifying a system from noisy measurements of its time and frequency responses. The proposed method is able to obtain stable low order models using substantially smaller matrices than those reported earlier and consequently in a fraction of the computation time.
- Publication Date:
- NSF-PAR ID:
- 10186506
- Journal Name:
- 2020 American Control Conference
- Page Range or eLocation-ID:
- 5169 to 5174
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Publisher's Version of Record
- https://doi.org/10.23919/ACC45564.2020.9147366
