doi: 10.3934/mcrf.2021021

Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations

Graduate School of System Informatics, Kobe University, Nada, Kobe, Hyogo 657-8501, Japan

* Corresponding author: wakaiki@ruby.kobe-u.ac.jp

Received  June 2020 Revised  January 2021 Published  March 2021

Fund Project: The first author is supported by JSPS KAKENHI Grant Numbers JP20K14362

This paper addresses the following question: "Suppose that a state-feedback controller stabilizes an infinite-dimensional linear continuous-time system. If we choose the parameters of an event/self-triggering mechanism appropriately, is the event/self-triggered control system stable under all sufficiently small nonlinear Lipschitz perturbations?" We assume that the stabilizing feedback operator is compact. This assumption is used to guarantee the strict positiveness of inter-event times and the existence of the mild solution of evolution equations with unbounded control operators. First, for the case where the control operator is bounded, we show that the answer to the above question is positive, giving a sufficient condition for exponential stability, which can be employed for the design of event/self-triggering mechanisms. Next, we investigate the case where the control operator is unbounded and prove that the answer is still positive for periodic event-triggering mechanisms.

Citation: Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021021
References:
[1]

A. Anta and P. Tabuada, To sample or not to sample: Self-triggered control for nonlinear systems, IEEE Trans. Automat. Control, 55 (2010), 2030-2042.  doi: 10.1109/TAC.2010.2042980.  Google Scholar

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

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L. BaudouinS. Marx and S. Tarbouriech, Event-triggered damping of a linear wave equation, IFAC-PapersOnLine, 52 (2019), 58-63.  doi: 10.1016/j.ifacol.2019.08.011.  Google Scholar

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K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.  Google Scholar

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N. EspitiaA. GirardN. Marchand and C. Prieur, Event-based control of linear hyperbolic systems of conservation laws, Automatica J. IFAC, 70 (2016), 275-287.  doi: 10.1016/j.automatica.2016.04.009.  Google Scholar

[7]

N. EspitiaA. GirardN. Marchand and C. Prieur, Event-based boundary control of a linear $2\times 2$ hyperbolic system via backstepping approach, IEEE Trans. Automat. Control, 63 (2018), 2686-2693.  doi: 10.1109/TAC.2017.2774011.  Google Scholar

[8]

L. EtienneS. Di Gennaro and J.-P. Barbot, Periodic event-triggered observation and control for nonlinear Lipschitz systems using impulsive observers, Int. J. Robust Nonlinear Control, 27 (2017), 4363-4380.  doi: 10.1002/rnc.3802.  Google Scholar

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R. GoebelR. G. Sanfelice and A. R. Teel, Hybrid dynamical systems: Robust stability and control for systems that combine continuous-time and discrete-time dynamics, IEEE Control Syst. Mag., 29 (2009), 28-93.  doi: 10.1109/MCS.2008.931718.  Google Scholar

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W. P. M. H. HeemelsM. C. F. Donkers and A. R. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Automat. Control, 58 (2013), 847-861.  doi: 10.1109/TAC.2012.2220443.  Google Scholar

[11]

W. P. M. H. HeemelsJ. H. Sandee and P. O. J. Van Den Bosch, Analysis of event-driven controllers for linear systems, Int. J. Control, 81 (2008), 571-590.  doi: 10.1080/00207170701506919.  Google Scholar

[12]

A. IlchmannZ. Ke and H. Logemann, Indirect sampled-data control with sampling period adaptation, Int. J. Control, 84 (2011), 424-431.  doi: 10.1080/00207179.2011.557782.  Google Scholar

[13]

Z. JiangB. CuiW. Wu and B. Zhuang, Event-driven observer-based control for distributed parameter systems using mobile sensor and actuator, Comput. Math. Appl., 72 (2016), 2854-2864.  doi: 10.1016/j.camwa.2016.10.009.  Google Scholar

[14]

W. Kang and E. Fridman, Distributed sampled-data control of Kuramoto-Sivashinsky equation, Automatica J. IFAC, 95 (2018), 514-524.  doi: 10.1016/j.automatica.2018.06.009.  Google Scholar

[15]

I. Karafyllis and M. Krstic, Sampled-data boundary feedback control of 1-D parabolic PDEs, Automatica J. IFAC, 87 (2018), 226-237.  doi: 10.1016/j.automatica.2017.10.006.  Google Scholar

[16]

I. KarafyllisM. Krstic and K. Chrysafi, Adaptive boundary control of constant-parameter reaction-diffusion PDEs using regulation-triggered finite-time identification, Automatica J. IFAC, 103 (2019), 166-179.  doi: 10.1016/j.automatica.2019.01.028.  Google Scholar

[17]

Z. KeH. Logemann and R. Rebarber, Approximate tracking and disturbance rejection for stable infinite-dimensional systems using sampled-data low-gain control, SIAM J. Control Optim., 48 (2009), 641-671.  doi: 10.1137/080716517.  Google Scholar

[18]

Z. KeH. Logemann and R. Rebarber, A sampled-data servomechanism for stable well-posed systems, IEEE Trans. Automat. Control, 54 (2009), 1123-1128.  doi: 10.1109/TAC.2009.2013032.  Google Scholar

[19]

Z. KeH. Logemann and S. Townley, Adaptive sampled-data integral control of stable infinite-dimensional linear systems, Systems Control Lett., 58 (2009), 233-240.  doi: 10.1016/j.sysconle.2008.10.015.  Google Scholar

[20]

D. Lehmann and J. Lunze, Event-based control with communication delays and packet losses, Int. J. Control, 85 (2012), 563-577.  doi: 10.1080/00207179.2012.659760.  Google Scholar

[21]

P. LinH. Liu and G. Wang, Output feedback stabilization for heat equations with sampled-data controls, J. Differential Equ., 268 (2020), 5823-5854.  doi: 10.1016/j.jde.2019.11.019.  Google Scholar

[22]

H. Logemann, Stabilization of well-posed infinite-dimensional systems by dynamic sampled-data feedback, SIAM J. Control Optim., 51 (2013), 1203-1231.  doi: 10.1137/110850396.  Google Scholar

[23]

H. LogemannR. Rebarber and S. Townley, Stability of infinite-dimensional sampled-data systems, Trans. Amer. Math. Soc., 355 (2003), 3301-3328.  doi: 10.1090/S0002-9947-03-03142-8.  Google Scholar

[24]

H. LogemannR. Rebarber and S. Townley, Generalized sampled-data stabilization of well-posed linear infinite-dimensional systems, SIAM J. Control Optim., 44 (2005), 1345-1369.  doi: 10.1137/S0363012903434340.  Google Scholar

[25]

H. Logemann and S. Townley, Discrete-time low-gain control of uncertain infinite-dimensional systems, IEEE Trans. Automat. Control, 42 (1997), 22-37.  doi: 10.1109/9.553685.  Google Scholar

[26]

A. Mironchenko and C. Prieur, Input-to-state stability of infinite-dimensional systems: Recent results and open questions, SIAM Review, 62 (2020), 529-614.  doi: 10.1137/19M1291248.  Google Scholar

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[28]

R. Postoyan, R. G. Sanfelice and W. P. M. H. Heemels, Inter-event times analysis for planar linear event-triggered controlled systems, 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 1662–1667. doi: 10.1109/CDC40024.2019.9028888.  Google Scholar

[29]

R. Rebarber and S. Townley, Generalized sampled data feedback control of distributed parameter systems, Systems Control Lett., 34 (1998), 229-240.  doi: 10.1016/S0167-6911(98)00011-5.  Google Scholar

[30]

R. Rebarber and S. Townley, Nonrobustness of closed-loop stability for infinite-dimensional systems under sample and hold, IEEE Trans. Automat. Control, 47 (2002), 1381-1385.  doi: 10.1109/TAC.2002.801189.  Google Scholar

[31]

R. Rebarber and S. Townley, Robustness with respect to sampling for stabilization of Riesz spectral systems, IEEE Trans. Automat. Control, 51 (2006), 1519-1522.  doi: 10.1109/TAC.2006.880797.  Google Scholar

[32]

A. Selivanov and E. Fridman, Distributed event-triggered control of diffusion semilinear PDEs, Automatica J. IFAC, 68 (2016), 344-351.  doi: 10.1016/j.automatica.2016.02.006.  Google Scholar

[33]

P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Automat. Control, 52 (2007), 1680-1685.  doi: 10.1109/TAC.2007.904277.  Google Scholar

[34]

T. J. TarnJr. R. Zavgern and X. Zeng, Stabilization of infinite-dimensional systems with periodic gains and sampled output, Automatica J. IFAC, 24 (1988), 95-99.  doi: 10.1016/0005-1098(88)90012-X.  Google Scholar

[35]

M. Tucsnak and G. Weiss, Observation and Control of Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[36]

M. Wakaiki and H. Sano, Event-triggered control of infinite-dimensional systems, SIAM J. Control Optim., 58 (2020), 605-635.  doi: 10.1137/18M1179717.  Google Scholar

[37]

M. Wakaiki and H. Sano, Sampled-data output regulation of unstable well-posed infinite-dimensional systems with constant reference and disturbance signals, Math. Control Signals Systems, 32 (2020), 43-100.  doi: 10.1007/s00498-019-00252-9.  Google Scholar

[38]

M. Wakaiki and Y. Yamamoto, Stability analysis of perturbed infinite-dimensional sampled-data systems, Systems Control Lett., 138 (2020), 104652, 8 pp. doi: 10.1016/j.sysconle.2020.104652.  Google Scholar

[39]

X. Wang and M. D. Lemmon, Self-triggered feedback control systems with finite-gain $\mathcal{L}_2$ stability, IEEE Trans. Automat. Control, 54 (2009), 452-467.  doi: 10.1109/TAC.2009.2012973.  Google Scholar

show all references

References:
[1]

A. Anta and P. Tabuada, To sample or not to sample: Self-triggered control for nonlinear systems, IEEE Trans. Automat. Control, 55 (2010), 2030-2042.  doi: 10.1109/TAC.2010.2042980.  Google Scholar

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[3]

L. BaudouinS. Marx and S. Tarbouriech, Event-triggered damping of a linear wave equation, IFAC-PapersOnLine, 52 (2019), 58-63.  doi: 10.1016/j.ifacol.2019.08.011.  Google Scholar

[4]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[5]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.  Google Scholar

[6]

N. EspitiaA. GirardN. Marchand and C. Prieur, Event-based control of linear hyperbolic systems of conservation laws, Automatica J. IFAC, 70 (2016), 275-287.  doi: 10.1016/j.automatica.2016.04.009.  Google Scholar

[7]

N. EspitiaA. GirardN. Marchand and C. Prieur, Event-based boundary control of a linear $2\times 2$ hyperbolic system via backstepping approach, IEEE Trans. Automat. Control, 63 (2018), 2686-2693.  doi: 10.1109/TAC.2017.2774011.  Google Scholar

[8]

L. EtienneS. Di Gennaro and J.-P. Barbot, Periodic event-triggered observation and control for nonlinear Lipschitz systems using impulsive observers, Int. J. Robust Nonlinear Control, 27 (2017), 4363-4380.  doi: 10.1002/rnc.3802.  Google Scholar

[9]

R. GoebelR. G. Sanfelice and A. R. Teel, Hybrid dynamical systems: Robust stability and control for systems that combine continuous-time and discrete-time dynamics, IEEE Control Syst. Mag., 29 (2009), 28-93.  doi: 10.1109/MCS.2008.931718.  Google Scholar

[10]

W. P. M. H. HeemelsM. C. F. Donkers and A. R. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Automat. Control, 58 (2013), 847-861.  doi: 10.1109/TAC.2012.2220443.  Google Scholar

[11]

W. P. M. H. HeemelsJ. H. Sandee and P. O. J. Van Den Bosch, Analysis of event-driven controllers for linear systems, Int. J. Control, 81 (2008), 571-590.  doi: 10.1080/00207170701506919.  Google Scholar

[12]

A. IlchmannZ. Ke and H. Logemann, Indirect sampled-data control with sampling period adaptation, Int. J. Control, 84 (2011), 424-431.  doi: 10.1080/00207179.2011.557782.  Google Scholar

[13]

Z. JiangB. CuiW. Wu and B. Zhuang, Event-driven observer-based control for distributed parameter systems using mobile sensor and actuator, Comput. Math. Appl., 72 (2016), 2854-2864.  doi: 10.1016/j.camwa.2016.10.009.  Google Scholar

[14]

W. Kang and E. Fridman, Distributed sampled-data control of Kuramoto-Sivashinsky equation, Automatica J. IFAC, 95 (2018), 514-524.  doi: 10.1016/j.automatica.2018.06.009.  Google Scholar

[15]

I. Karafyllis and M. Krstic, Sampled-data boundary feedback control of 1-D parabolic PDEs, Automatica J. IFAC, 87 (2018), 226-237.  doi: 10.1016/j.automatica.2017.10.006.  Google Scholar

[16]

I. KarafyllisM. Krstic and K. Chrysafi, Adaptive boundary control of constant-parameter reaction-diffusion PDEs using regulation-triggered finite-time identification, Automatica J. IFAC, 103 (2019), 166-179.  doi: 10.1016/j.automatica.2019.01.028.  Google Scholar

[17]

Z. KeH. Logemann and R. Rebarber, Approximate tracking and disturbance rejection for stable infinite-dimensional systems using sampled-data low-gain control, SIAM J. Control Optim., 48 (2009), 641-671.  doi: 10.1137/080716517.  Google Scholar

[18]

Z. KeH. Logemann and R. Rebarber, A sampled-data servomechanism for stable well-posed systems, IEEE Trans. Automat. Control, 54 (2009), 1123-1128.  doi: 10.1109/TAC.2009.2013032.  Google Scholar

[19]

Z. KeH. Logemann and S. Townley, Adaptive sampled-data integral control of stable infinite-dimensional linear systems, Systems Control Lett., 58 (2009), 233-240.  doi: 10.1016/j.sysconle.2008.10.015.  Google Scholar

[20]

D. Lehmann and J. Lunze, Event-based control with communication delays and packet losses, Int. J. Control, 85 (2012), 563-577.  doi: 10.1080/00207179.2012.659760.  Google Scholar

[21]

P. LinH. Liu and G. Wang, Output feedback stabilization for heat equations with sampled-data controls, J. Differential Equ., 268 (2020), 5823-5854.  doi: 10.1016/j.jde.2019.11.019.  Google Scholar

[22]

H. Logemann, Stabilization of well-posed infinite-dimensional systems by dynamic sampled-data feedback, SIAM J. Control Optim., 51 (2013), 1203-1231.  doi: 10.1137/110850396.  Google Scholar

[23]

H. LogemannR. Rebarber and S. Townley, Stability of infinite-dimensional sampled-data systems, Trans. Amer. Math. Soc., 355 (2003), 3301-3328.  doi: 10.1090/S0002-9947-03-03142-8.  Google Scholar

[24]

H. LogemannR. Rebarber and S. Townley, Generalized sampled-data stabilization of well-posed linear infinite-dimensional systems, SIAM J. Control Optim., 44 (2005), 1345-1369.  doi: 10.1137/S0363012903434340.  Google Scholar

[25]

H. Logemann and S. Townley, Discrete-time low-gain control of uncertain infinite-dimensional systems, IEEE Trans. Automat. Control, 42 (1997), 22-37.  doi: 10.1109/9.553685.  Google Scholar

[26]

A. Mironchenko and C. Prieur, Input-to-state stability of infinite-dimensional systems: Recent results and open questions, SIAM Review, 62 (2020), 529-614.  doi: 10.1137/19M1291248.  Google Scholar

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[28]

R. Postoyan, R. G. Sanfelice and W. P. M. H. Heemels, Inter-event times analysis for planar linear event-triggered controlled systems, 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 1662–1667. doi: 10.1109/CDC40024.2019.9028888.  Google Scholar

[29]

R. Rebarber and S. Townley, Generalized sampled data feedback control of distributed parameter systems, Systems Control Lett., 34 (1998), 229-240.  doi: 10.1016/S0167-6911(98)00011-5.  Google Scholar

[30]

R. Rebarber and S. Townley, Nonrobustness of closed-loop stability for infinite-dimensional systems under sample and hold, IEEE Trans. Automat. Control, 47 (2002), 1381-1385.  doi: 10.1109/TAC.2002.801189.  Google Scholar

[31]

R. Rebarber and S. Townley, Robustness with respect to sampling for stabilization of Riesz spectral systems, IEEE Trans. Automat. Control, 51 (2006), 1519-1522.  doi: 10.1109/TAC.2006.880797.  Google Scholar

[32]

A. Selivanov and E. Fridman, Distributed event-triggered control of diffusion semilinear PDEs, Automatica J. IFAC, 68 (2016), 344-351.  doi: 10.1016/j.automatica.2016.02.006.  Google Scholar

[33]

P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Automat. Control, 52 (2007), 1680-1685.  doi: 10.1109/TAC.2007.904277.  Google Scholar

[34]

T. J. TarnJr. R. Zavgern and X. Zeng, Stabilization of infinite-dimensional systems with periodic gains and sampled output, Automatica J. IFAC, 24 (1988), 95-99.  doi: 10.1016/0005-1098(88)90012-X.  Google Scholar

[35]

M. Tucsnak and G. Weiss, Observation and Control of Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[36]

M. Wakaiki and H. Sano, Event-triggered control of infinite-dimensional systems, SIAM J. Control Optim., 58 (2020), 605-635.  doi: 10.1137/18M1179717.  Google Scholar

[37]

M. Wakaiki and H. Sano, Sampled-data output regulation of unstable well-posed infinite-dimensional systems with constant reference and disturbance signals, Math. Control Signals Systems, 32 (2020), 43-100.  doi: 10.1007/s00498-019-00252-9.  Google Scholar

[38]

M. Wakaiki and Y. Yamamoto, Stability analysis of perturbed infinite-dimensional sampled-data systems, Systems Control Lett., 138 (2020), 104652, 8 pp. doi: 10.1016/j.sysconle.2020.104652.  Google Scholar

[39]

X. Wang and M. D. Lemmon, Self-triggered feedback control systems with finite-gain $\mathcal{L}_2$ stability, IEEE Trans. Automat. Control, 54 (2009), 452-467.  doi: 10.1109/TAC.2009.2012973.  Google Scholar

Figure 1.  Event-triggered control system
Figure 2.  Self-triggered control system
Figure 3.  State norm $ \|x(t)\| $ of self-triggered control system
Figure 4.  Input $ u(t) $ of self-triggered control system
Figure 5.  Inter-event times $ t_{k+1} - t_k $ of self-triggered control system
Figure 6.  Bound on threshold $ \varepsilon $ and lower bound $ \tau_{\min} $ of inter-event times of event-triggering mechanism (27)
Figure 7.  Comparison of state norm $ \|x(t)\| $ between event-triggered control system and periodic sampled-data system
Figure 8.  Comparison of input $ u(t) $ between event-triggered control system and periodic sampled-data system
Figure 9.  Comparison of inter-event times $ t_{k+1}-t_k $ between event-triggered control system and periodic sampled-data system
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