Abstract
In this paper, we first provide a mathematical description and proof for the robust stability of systems with convex polytopic uncertainty through the construction of attractive invariant sets. Then, we present the problem of estimating the convergence time to arbitrarily tight over-approximations of an invariant set, from both a known initial condition and for initial conditions belonging to a convex polytopic set. We then propose various analytical and numerical methods for computing the aforementioned convergence time. Finally, a number of numerical examples, including a flexible link robotic manipulator, are given to illustrate the results.
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References
Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, New York (1998)
Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River (1996)
Battacharyya, S., Chapellat, H., Keel, L.: Robust Control: The Parametric Approach. Prentice-Hall, Upper Saddle River (1997)
Apkarian, P., Gahinet, P., Becker, G.: Self-scheduled H ∞ control of linear parameter-varying systems: a design example. Automatica 31(9), 1251–1261 (1995)
Daafouz, J., Bernussou, J.: Parameter dependent Lyapunov function for discrete time systems with time varying parametric uncertainties. Systems and Control Letters 43, 355–359 (2001)
de Oliveira, M., Bernussou, J., Geromel, J.: A new discrete-time robust stability condition. Systems and Control Letters 37, 261–265 (1999)
Blanchini, F.: Set invariance in control. Automatica 35, 1747–1767 (1999)
Khalil, H.: Nonlinear Systems, 3rd edn. Prentice-Hall, New Jersey (2002)
Haimovich, H., Seron, M.: Bounds and invariant sets for a class of discrete-time switching systems with perturbations. International Journal of Control (2013), doi:10.1080/00207179.2013.834536 (published online: September 13, 2013)
Olaru, S., De Doná, J.A., Seron, M.M., Stoican, F.: Positive invariant sets for fault tolerant multisensor control schemes. International Journal of Control 83(12), 2622–2640 (2010)
Seron, M.M., De Doná, J.A., Olaru, S.: Fault tolerant control allowing sensor healthy-to-faulty and faulty-to-healthy transitions. IEEE Transactions on Automatic Control 77(7), 1657–1669 (2012)
Franklin, G., Powell, J.D., Emami-Naeini, A.: Feedback Control of Dynamic Systems. Prentice Hall, Upper Saddle River (2002)
Veberic, D.: Having fun with Lambert W(x) function, CoRR, vol. abs/1003.1628 (2010), http://arxiv.org/abs/1003.1628
Veberic, D.: Lambert W function for applications in Physics. Computer Physics Communications 183(12), 2622–2628 (2012), http://www.sciencedirect.com/science/article/pii/S0010465512002366
Houari, A.: Additional applications of the Lambert W function in Physics. European Journal of Physics 34(3), 695 (2013), http://stacks.iop.org/0143-0807/34/i=3/a=695
Banwell, T.: Bipolar transistor circuit analysis using the Lambert W-function. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47(11), 1621–1633 (2000)
Hwang, C., Cheng, Y.-C.: Use of Lambert W function to stability analysis of time-delay systems. In: Proceedings of the 2005 American Control Conference, vol. 6, pp. 4283–4288 (June 2005)
Sira-Ramírez, H., Castro-Linares, R.: Sliding mode rest-to-rest stabilization and trajectory tracking for a discretized flexible joint manipulator. Dynamics and Control 10(1), 87–105 (2000)
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McCloy, R.J., De Doná, J.A., Seron, M.M. (2015). On the Estimation of Convergence Times to Invariant Sets in Convex Polytopic Uncertain Systems. In: Chalup, S.K., Blair, A.D., Randall, M. (eds) Artificial Life and Computational Intelligence. ACALCI 2015. Lecture Notes in Computer Science(), vol 8955. Springer, Cham. https://doi.org/10.1007/978-3-319-14803-8_5
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DOI: https://doi.org/10.1007/978-3-319-14803-8_5
Publisher Name: Springer, Cham
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