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Measure of quality of finite-dimensional linear systems: a frame-theoretic view. (English) Zbl 1478.93060

Summary: A measure of quality of a control system is a quantitative extension of the classical binary notion of controllability. In this article we study the quality of linear control systems from a frame-theoretic perspective. We demonstrate that all LTI systems naturally generate a frame on their state space, and that three standard measures of quality involving the trace, minimum eigenvalue, and the determinant of the controllability Gramian achieve their optimum values when this generated frame is tight. Motivated by this, and in view of some recent developments in frame-theoretic signal processing, we propose a natural measure of quality for continuous time LTI systems based on a measure of tightness of the frame generated by it and then discuss some properties of this frame-theoretic measure of quality.

MSC:

93B05 Controllability
93C05 Linear systems in control theory
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[1] Müller, P. C.; Weber, H. I., Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems, Automatica, 8, 3, 237-246 (1972) · Zbl 0242.93014
[2] Lions, J.-L., Measures of controllability, Georgian Math. J., 1, 1, 47-52 (1994) · Zbl 0812.93046
[3] Pasqualetti, F.; Zampieri, S.; Bullo, F., Controllability metrics, limitations and algorithms for complex networks, IEEE Trans. Control Netw. Syst., 1, 1, 40-52 (2014) · Zbl 1370.93164
[4] Duffin, R. J.; Schaeffer, A. C., A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72, 2, 341-366 (1952) · Zbl 0049.32401
[5] Daubechies, I.; Grossmann, A.; Meyer, Y., Painless nonorthogonal expansions, J. Math. Phys., 27, 5, 1271-1283 (1986) · Zbl 0608.46014
[6] Benedetto, J. J.; Fickus, M., Finite normalized tight frames, Adv. Comput. Math., 18, 2-4, 357-385 (2003) · Zbl 1028.42022
[7] Casazza, P. G.; Fickus, M.; Kovačević, J.; Leon, M. T.; Tremain, J. C., A physical interpretation of tight frames, (Harmonic Analysis and Applications (2006), Springer), 51-76 · Zbl 1129.42418
[8] Shen, L.; Papadakis, M.; Kakadiaris, I. A.; Konstantinidis, I.; Kouri, D.; Hoffman, D., Image denoising using a tight frame, IEEE Trans. Image Process., 15, 5, 1254-1263 (2006)
[9] Zhou, W.; Yang, S.; Zhang, C.; Fu, S., Adaptive tight frame based multiplicative noise removal, SpringerPlus, 5, 1, 122 (2016)
[10] Sheriff, M. R.; Chatterjee, D., Optimal dictionary for least squares representation, J. Mach. Learn. Res., 18, 107, 1-28 (2017) · Zbl 1442.15007
[11] Summers, T. H.; Cortesi, F. L.; Lygeros, J., On submodularity and controllability in complex dynamical networks, IEEE Trans. Control Netw. Syst., 3, 1, 91-101 (2016) · Zbl 1370.93055
[12] Zhao, Y.; Cortes, J., Gramian-based reachability metrics for bilinear networks, IEEE Trans. Control Netw. Syst., 4, 3, 620-631 (2017) · Zbl 06988982
[13] Christensen, O., (An Introduction to Frames and Riesz Bases. An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis (2016), Birkhäuser/Springer: Birkhäuser/Springer Cham) · Zbl 1348.42033
[14] Clarke, F., Functional Analysis, Calculus of Variations and Optimal Control, Vol. 264 (2013), Springer Science & Business Media
[15] Luenberger, D. G., Optimization by Vector Space Methods (1997), John Wiley & Sons
[16] Willcox, K.; Peraire, J., Balanced model reduction via the proper orthogonal decomposition, AIAA J., 40, 11 (2002)
[17] Marshall, A. W.; Olkin, I.; Arnold, B. C., (Inequalities: Theory of Majorization and Its Applications. Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics (2011), Springer-Verlag: Springer-Verlag New York) · Zbl 1219.26003
[18] Antezana, J.; Massey, P.; Ruiz, M.; Stojanoff, D., The Schur-Horn theorem for operators and frames with prescribed norms and frame operator, Illinois J. Math., 51, 2, 537-560 (2007) · Zbl 1137.42008
[19] Casazza, P. G.; Leonhard, N., Classes of finite equal norm Parseval frames, Contemp. Math., 451, 11-32 (2008) · Zbl 1210.42047
[20] M.R. Sheriff, D. Chatterjee, On a frame theoretic measure of quality of lti systems, in: Proceedings of the 56th IEEE Conference on Decision & Control, Melbourne, Australia, 2017, pp. 4012-4017.
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