Abstract
In this chapter we propose a method for the analysis of sampled-data systems with sampling jitter. We consider that the sampling interval is unknown and time-varying and we provide a method for estimating the Lyapunov exponent. The proposed method is hybrid, in the sense that it combines continuous-time models (based on time delay systems) with polytopic embedding methods, specific to discrete-time approaches. The approach exploits the fact that the command is a piecewise constant signal and leads to less conservative stability conditions with respect to the existing literature. Using geometrical arguments, a lower bound of the Lyapunov exponent can be expressed as a generalized eigenvalue problem.
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References
Balluchi, A., Murrieri, P., Sangiovanni-Vincentelli, A.L.: Controller Synthesis on Non-uniform and Uncertain Discrete–Time Domains. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 118–133. Springer, Heidelberg (2005)
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)
Cloosterman, M.B.G., Hetel, L., van de Wouw, N., Heemels, W.P.M.H., Daafouz, J., Nijmeijer, H.: Controller synthesis for networked control systems. Automatica 46(10), 1584–1594 (2010)
Donkers, M.C.F., Hetel, L., Heemels, W.P.M.H., van de Wouw, N., Steinbuch, M.: Stability Analysis of Networked Control Systems Using a Switched Linear Systems Approach. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 150–164. Springer, Heidelberg (2009)
Felicioni, F.E., Junco, S.J.: A Lie algebraic approach to design of stable feedback control systems with varying sampling rate. In: Proceedings of the 17th IFAC World Congress, Seoul, Korea, pp. 4881–4886 (July 2008)
Fridman, E., Seuret, A., Richard, J.-P.: Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40, 1441–1446 (2004)
Fujioka, H.: Stability analysis for a class of networked / embedded control systems: Output feedback case. In: Proceedings of the 17th IFAC World Congress, Seoul, Korea, pp. 4210–4215 (July 2008)
Gielen, R., Olaru, S., Lazar, M.: On polytopic embeddings as a modeling framework for networked control systems. In: Proc. 3rd Int. Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control, Pavia, Italy (2008)
Hetel, L., Daafouz, J., Iung, C.: Stabilization of arbitrary switched linear systems with unknown time varying delays. IEEE Transactions on Automatic Control (2006)
Hetel, L., Daafouz, J., Iung, C.: LMI control design for a class of exponential uncertain systems with application to network controlled switched systems. In: Proceedings of 2007 IEEE American Control Conference (2007)
Hetel, L., Kruszewski, A., Perruquetti, W., Richard, J.P.: Discrete and intersample analysis of systems with aperiodic sampling. IEEE Transactions on Automatic Control (2011)
Kao, C.-Y., Lincoln, B.: Simple stability criteria for systems with time-varying delays. Automatica 40(8), 1429–1434 (2004)
Lofberg, J.: Yalmip: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004), http://control.ee.ethz.ch/~joloef/yalmip.php
Mirkin, L.: Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Transactions on Automatic Control 52(6), 1109–1112 (2007) ISSN 0018-9286
Naghshtabrizi, P., Hespanha, J.P.: Stability of network control systems with variable sampling and delays. In: Singer, A., Hadjicostis, C. (eds.) Proc. of the Forty-Fourth Annual Allerton Conference on Communication, Control, and Computing (September 2006)
Naghshtabrizi, P., Teel, A., Hespanha, J.P.: Exponential stability of impulsive systems with application to uncertain sampled-data systems. Systems and Control Letters 57(5), 378–385 (2008)
Oishi, Y., Fujioka, H.: Stability and stabilization of aperiodic sampled-data control systems using robust linear matrix inequalities. Technical Report of the Nanzan Academic Society Information Sciences and Engineering (2009)
Sala, A.: Computer control under time-varying sampling period: An LMI gridding approach. Automatica 41(12), 2077–2082 (2005)
Skaf, J., Boyd, S.: Analysis and synthesis of state-feedback controllers with timing jitter. IEEE Transactions on Automatic Control 54(3), 652–657 (2009) ISSN 0018-9286
Walsh, G.C., Ye, H., Bushnell, L.: Stability analysis of networked control systems. IEEE Transactions on Control Systems Technology, 2876–2880 (1999)
Wittenmark, B., Nilsson, J., Torngren, M.: Timing problems in real-time control systems. In: Proceedings of the 1995 American Control Conference, Seattle, WA, USA (1995)
Zhang, W., Branicky, M.S., Phillips, S.M.: Stability of networked control systems. IEEE Control Systems Magazine, 84–99 (2001)
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Hetel, L., Kruszewski, A., Richard, JP. (2012). A Hybrid Method for the Analysis of Non-uniformly Sampled Systems. In: Sipahi, R., Vyhlídal, T., Niculescu, SI., Pepe, P. (eds) Time Delay Systems: Methods, Applications and New Trends. Lecture Notes in Control and Information Sciences, vol 423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25221-1_19
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DOI: https://doi.org/10.1007/978-3-642-25221-1_19
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