Abstract
This paper discusses the data-driven design of linear quadratic regulators, i.e., to obtain the regulators directly from experimental data without using the models of plants. In particular, we aim to improve an existing design method by reducing the amount of the required experimental data. Reducing the data amount leads to the cost reduction of experiments and computation for the data-driven design. We present a simplified version of the existing method, where parameters yielding the gain of the regulator are estimated from only part of the data required in the existing method. We then show that the data amount required in the presented method is less than half of that in the existing method under certain conditions. In addition, assuming the presence of measurement noise, we analyze the relations between the expectations and variances of the estimated parameters and the noise. As a result, it is shown that using a larger amount of the experimental data might mitigate the effects of the noise on the estimated parameters. These results are verified by numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Our data are available in the sense that the same (or similar) data can be generated by simulations based on the mathematical contents of this article and references.
Notes
Although two design methods were proposed in [9], we introduce one of them called Algorithm 1.
References
Campi, M. C., Lecchini, A., & Savaresi, S. M. (2002). Virtual reference feedback tuning: A direct method for the design of feedback controllers. Automatica, 38(8), 1337–1346.
Soma, S., Kaneko, O., & Fujii, T. (2004). A new method of controller parameter tuning based on input-output data—Fictitious Reference Iterative Tuning (FRIT). IFAC Proceedings Volumes, 37(12), 789–794.
Yan, P., Liu, D., Wang, D., & Ma, H. (2016). Data-driven controller design for general MIMO nonlinear systems via virtual reference feedback tuning and neural networks. Neurocomputing, 171, 815–825.
Nicoletti, A., Martino, M., & Karimi, A. (2019). A robust data-driven controller design methodology with applications to particle accelerator power converters. IEEE Transactions on Control Systems Technology, 27(2), 814–821.
Baggio, G., Katewa, V., & Pasqualetti, F. (2019). Data-driven minimum-energy controls for linear systems. IEEE Control Systems Letters, 3(3), 589–594.
Berberich, J., Koch, A., Scherer, C.W., & Allgöwer, F. (2020). Robust data-driven state-feedback design. In: Proceedings of the 2020 American Control Conference, pp. 1532–1538.
Bisoffi, A., De Persis, C., & Tesi, P. (2022). Data-driven control via Petersen’s lemma. Automatica, 145, 110537.
Lewis, F. L., Vrabie, D. L., & Syrmos, V. L. (2012). Optimal Control. Hoboken, NJ: Wiley.
Gonçalves da Silva, G. R., Bazanella, A. S., Lorenzini, C., & Campestrini, L. (2019). Data-driven LQR control design. IEEE Control Systems Letters, 3(1), 180–185.
van Waarde, H. J., & Mesbahi, M. (2020). Data-driven parameterizations of suboptimal LQR and \(H_2\) controllers. IFAC-PapersOnLine, 53(2), 4234–4239.
Rotulo, M., De Persis, C., & Tesi, P. (2020). Data-driven linear quadratic regulation via semidefinite programming. IFAC-PapersOnLine, 53(2), 3995–4000.
De Persis, C., & Tesi, P. (2021). Low-complexity learning of linear quadratic regulators from noisy data. Automatica, 128, 109548.
Aangenent, W., Kostić, D., de Jager, B., Molengraft, R., & Steinbuch, M. (2005). Data-based optimal control. In: Proceedings of the 2005 American Control Conference, pp. 1460–1465.
Lim, R.K., Phan, M.Q., & Longman, R.W. (1998). State estimation with ARMarkov models. Princeton University Department of Mechanical and Aerospace Engineering Technical Report (3046).
Sturm, J. F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11(1–4), 625–653.
Guo, T., Al Makdah, A. A., Krishnan, V., & Pasqualetti, F. (2023). Imitation and transfer learning for LQG control. IEEE Control Systems Letters, 7, 2149–2154.
Acknowledgements
The first author would like to thank Mr. A. Takeuchi for discussing the proposed design method.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors have no conflicts of interest to declare that are relevant to the contents of this paper.
Appendices
Proof of Theorem 1
By using (24) and the linearity of expectation, \(\textsf{E}[\hat{\varTheta }]\) is calculated as
where the fourth equality is derived from \(Y_0^*=\varTheta \bar{Z}^*\) due to (21). Therefore, (26) holds.
A direct calculation using (1), (2), (24), (26), and the linearity of expectation yields
where the fourth equality is given by \(Y_0^*=\varTheta \bar{Z}^*\). Hence, we obtain (27) due to (3). This completes the proof.
Proof of Theorem 2
We first prove (i). By using the linearity of expectation and \((\bar{Z}^*+W_x)(\bar{Z}^*+W_x)^\dagger = I_{n+mN}\), (26) can be rewritten as
Because the plant (4) is stable from the assumption and the input u(t) and the noises \(w_x(t)\) and \(w_y(t)\) satisfy \(u(t)\in \mathbb {R}^m\), \(w_x(t)\in \mathbb {R}^n\), and \(w_y(t)\in \mathbb {R}^\ell \), respectively, the magnitudes of the elements of the matrices \(\varTheta \), \(W_x\), \(W_{y0}\), and \(Y_0^*\) in (B3) and (27) are finite. Hence, applying \((\bar{Z}^*+W_x)^\dagger \rightarrow 0_{L\times (n+mN)}\) to (B3) and (27) proves (i).
Next, we prove (ii). Under \(L\ge n+mN\) shown in Table 1, the singular value decomposition of \(\bar{Z}^*+W_x\) gives
where \(T\in \mathbb {R}^{L\times L}\) and \(V\in \mathbb {R}^{(n+mN)\times (n+mN)}\) are orthogonal matrices and
for the singular values \(\sigma _i\) \((i=1,2,\ldots ,n+mN)\) of \(\bar{Z}^*+W_x\). Note in (B5) that \(\sigma _i\ne 0\) holds for every \(i\in \{1,2,\ldots ,n+mN\}\) because \(\bar{Z}^*+W_x\) is assumed to be full row rank. From (B4) and the orthogonality of T and V, if \((\bar{Z}^*+W_x)^\dagger \rightarrow 0_{L\times (n+mN)}\) holds, \(\sigma _i\rightarrow \infty \) holds for every \(i\in \{1,2,\ldots ,n+mN\}\). This implies that the Frobenius norm \(\Vert \bar{Z}^*+W_x\Vert _F\) of \(\bar{Z}^*+W_x\) approaches infinity because its square is equal to the sum of \(\sigma _i^2\) \((i=1,2,\ldots ,n+mN)\). By the definition of the Frobenius norm, if \(\Vert \bar{Z}^*+W_x\Vert _F\) approaches infinity, the number of the elements of \(\bar{Z}^*+W_x\), i.e., \(L(n+mN)\), approaches infinity because the magnitudes of the elements are finite by a discussion similar to the above. This, together with the assumption that the parameter N is fixed, proves (ii), which completes the proof.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Izumi, S., **n, X. Reduction of data amount in data-driven design of linear quadratic regulators. Control Theory Technol. (2024). https://doi.org/10.1007/s11768-024-00220-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11768-024-00220-y