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doi: 10.3934/mcrf.2022018
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Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling

 Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382055, India

* Corresponding author: Shivam Dhama

Received  September 2021 Revised  February 2022 Early access April 2022

Fund Project: The second author acknowledges research support from DST SERB Project No. EMR/2015/000904

In this paper, we study the dynamics of a linear control system with given state feedback control law in the presence of fast periodic sampling at temporal frequency $1/\delta$ ($0 < \delta \ll 1$), together with small white noise perturbations of size $\varepsilon$ ($0< \varepsilon \ll 1$) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters $\varepsilon,\delta$, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as $\varepsilon,\delta \searrow 0$. The effective fluctuation process is found to vary, depending on whether $\delta \searrow 0$ faster than/at the same rate as/slower than $\varepsilon \searrow 0$. The most interesting case is found to be the one where $\delta, \varepsilon$ are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a diffusive term due to the small noise and an effective drift term which captures the cumulative effect of the fast sampling. In this regime, our results yield a time-inhomogeneous Markov process which provides a strong (pathwise) approximation of the original non-Markovian process, together with estimates on the ensuing error. A simple example involving an infinite time horizon linear quadratic regulation problem illustrates the results.

Citation: Shivam Dhama, Chetan D. Pahlajani. Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022018
References:
 [1] M. Adès, P. E. Caines and R. P. Malhamé, Stochastic optimal control under Poisson-distributed observations, IEEE Trans. Automat. Control, 45 (2000), 3-13. [2] D. J. Antunes, J. P. Hespanha and C. J. Silvestre, Volterra integral approach to impulsive renewal systems: Application to networked control, IEEE Trans. Automat. Control, 57 (2012), 607-619.  doi: 10.1109/TAC.2011.2166300. [3] D. Antunes, J. Hespanha and C. Silvestre, Stability of networked control systems with asynchronous renewal links: An impulsive systems approach, Automatica J. IFAC, 49 (2013), 402-413.  doi: 10.1016/j.automatica.2012.11.033. [4] R. P. Anderson, D. Milutinović and D. V. Dimarogonas, Self-triggered sampling for second-moment stability of state-feedback controlled SDE systems, Automatica J. IFAC, 54 (2015), 8–15. https://www.sciencedirect.com/science/article/pii/S0005109815000308?via%3Dihub doi: 10.1016/j.automatica.2015.01.020. [5] A. Arapostathis, A. Biswas and V. S. Borkar, Controlled equilibrium selection in stochastically perturbed dynamics, Ann. Probab., 46 (2018), 2749-2799.  doi: 10.1214/17-AOP1238. [6] S. R. Athreya, V. S. Borkar, K. S. Kumar and R. Sundaresan, Simultaneous small noise limit for singularly perturbed slow-fast coupled diffusions, Appl. Math. Optim., 83 (2021), 2327-2374.  doi: 10.1007/s00245-019-09630-w. [7] P. Billingsley, Convergence of Probability Measures, John Wiley and Sons Inc., second edition, 1999. doi: 10.1002/9780470316962. [8] D. P. Borgers, V. S. Dolk, G. E. Dullerud, A. R. Teel and W. P. M. H. Heemels, Time-regularized and periodic event triggered control for linear systems, in Control Subject to Computational and Communication Constraints, Springer, 2018,121–149. https://link.springer.com/chapter/10.1007/978-3-319-78449-6_7 [9] S. Bourguin, S. Gailus and K. Spiliopoulos, Typical dynamics and fluctuation analysis of slow–fast systems driven by fractional Brownian motion, Stoch. Dyn., (2021), 2150030, 30 pp. doi: 10.1142/S0219493721500301. [10] S. Cerrai, A Khasminskii type averaging principle for stochastic reaction–diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948.  doi: 10.1214/08-AAP560. [11] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z. [12] G. Chen, C. Fan, J. Sun and J. Xia, Mean square exponential stability analysis for ito stochastic systems with aperiodic sampling and multiple time-delays, IEEE Transactions on Automatic Control, 2021. https://ieeexplore.ieee.org/abstract/document/9410357 [13] T. Chen and B. Francis, Optimal Sampled-Data Control Systems, Springer, 1996. [14] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, second edition, 1998. https://link.springer.com/book/10.1007/978-3-642-03311-7 [15] R. Dong and X. Mao, Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations, Math. Control Relat. Fields, 10 (2020), 715-734.  doi: 10.3934/mcrf.2020017. [16] V. Drǎgan and I. G. Ivanov, On the stochastic linear quadratic control problem with piecewise constant admissible controls, J. Franklin Inst., 357 (2020), 1532-1559.  doi: 10.1016/j.jfranklin.2019.10.036. [17] M. I. Freidlin and R. B. Sowers, A comparison of homogenization and large deviations, with applications to wavefront propagation, Stochastic Process. Appl., 82 (1999), 23-52.  doi: 10.1016/S0304-4149(99)00003-4. [18] M. I. Freidlin and A. D. Wentzell, Random perturbations of Hamiltonian systems, Mem. Amer. Math. Soc., 109 (1994), 523.  doi: 10.1090/memo/0523. [19] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, third edition, 2012. doi: 10.1007/978-3-642-25847-3. [20] R. Goebel, R. G. Sanfelice and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability and Robustness, Princeton University Press, 2012. [21] R. Z. Has'minskii, On stochastic processes defined by differential equations with a small parameter, Theory of Probability & Its Applications, 11 (1966), 211-228.  doi: 10.1137/1111018. [22] R. Z. Has'Minskii, On the principle of averaging the Ito's stochastic differential equations, Kybernetika, 4 (1968), 260-279. [23] W. P. M. H. Heemels, M. 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. [24] J. P. Hespanha, Linear Systems Theory, Princeton University Press, 2009. [25] J. P. Hespanha, Modeling and analysis of networked control systems using stochastic hybrid systems, Annual Reviews in Control, 38 (2014), 155–170. https://www.sciencedirect.com/science/article/pii/S1367578814000352 [26] J. P. Hespanha and A. R. Teel, Stochastic impulsive systems driven by renewal processes, Proc. 17th International Symposium on Mathematical Theory of Networked Systems, (2006), 606–618. https://scholar.google.co.in/scholar?q=Stochastic+impulsive+systems+driven+by+renewal+processes&hl=en&as_sdt=0&as_vis=1&oi=scholart [27] J. Higham Desmond, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [28] I. Karafyllis and C. Kravaris, Global stability results for systems under sampled-data control, Internat. J. Robust Nonlinear Control, 19 (2009), 1105-1128.  doi: 10.1002/rnc.1364. [29] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics, Springer-Verlag New York, second edition, 1991. doi: 10.1007/978-1-4612-0949-2. [30] H. K. Khalil, Performance recovery under output feedback sampled-data stabilization of a class of nonlinear systems, IEEE Trans. Automat. Control, 49 (2004), 2173-2184.  doi: 10.1109/TAC.2004.838496. [31] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5. [32] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, SIAM, 1999. https://epubs.siam.org/doi/book/10.1137/1.9781611971118 [33] D. S. Laila, D. Nesic and A. R. Teel, Open- and closed-loop dissipation inequalities under sampling and controller emulation, European Journal of Control, 8 (2002), 109–125. https://www.sciencedirect.com/science/article/pii/S0947358002702161 [34] D. Nesic and A. R. Teel, A framework for stabilization of non-linear sampled-data systems based on their approximate discrete-time models, IEEE Trans. Automat. Control, 49 (2004), 1103-1122.  doi: 10.1109/TAC.2004.831175. [35] D. Nesic, A. R. Teel and D. Carnevale, Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems, IEEE Trans. Automat. Control, 54 (2009), 619–624. https://ieeexplore.ieee.org/document/4796269 doi: 10.1109/TAC.2008.2009597. [36] D. Nesic, A. R. Teel and P. V. Kokotovic, Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, Systems Control Lett., 38 (1999), 259-270.  doi: 10.1016/S0167-6911(99)00073-0. [37] B. Oksendal, Stochastic Differential Equations, Springer, sixth edition, 2003. doi: 10.1007/978-3-642-14394-6. [38] M. Röckner and L. Xie, Averaging principle and normal deviations for multiscale stochastic systems, Comm. Math. Phys., 383 (2021), 1889-1937.  doi: 10.1007/s00220-021-04069-z. [39] M. Röckner and L. Xie, Diffusion approximation for fully coupled stochastic differential equations, Ann. Probab., 49 (2021), 1205-1236.  doi: 10.1214/20-aop1475. [40] A. V. Skorokhod, F. C. Hoppensteadt and H. Salehi, Random Perturbation Methods with Applications in Science and Engineering, volume 150 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002. doi: 10.1007/b98905. [41] K. Spiliopoulos, Large deviations and importance sampling for systems of slow-fast motion, Appl. Math. Optim., 67 (2013), 123-161.  doi: 10.1007/s00245-012-9183-z. [42] K. Spiliopoulos, Fluctuation analysis and short time asymptotics for multiple scales diffusion processes, Stoch. Dyn., 14 (2014), 1350026, 22 pp. doi: 10.1142/S0219493713500263. [43] A. Tanwani, D. Chatterjee and D. Liberzon, Stabilization of deterministic control systems under random sampling: Overview and recent developments, Uncertainty in Complex Networked Systems. In T. Basar, editor, Springer, 2018,209–246. https://link.springer.com/chapter/10.1007/978-3-030-04630-9_6 [44] A. Tanwani and O. Yufereva, Error covariance bounds for suboptimal filters with Lipschitzian drift and Poisson-sampled measurements, Automatica J. IFAC, 122 (2020), 109280, 14 pp. doi: 10.1016/j.automatica.2020.109280. [45] W. Wang, R. Postoyan, D. Nesic and W. P. M. H. Heemels, Periodic event-triggered control for nonlinear networked control systems, IEEE Trans. Automat. Control, 65 (2020), 620-635.  doi: 10.1109/TAC.2019.2914255. [46] J. I. Yuz and G. C. Goodwin, Sampled-Data Models for Linear and Nonlinear Systems, Springer, 2014. [47] O. Zeitouni, Approximate and limit results for nonlinear filters with small observation noise: the linear sensor and constant diffusion coefficient case, IEEE Transactions on Automatic Control, 33 (1988), 595–599. https://ieeexplore.ieee.org/abstract/document/1262

show all references

References:
 [1] M. Adès, P. E. Caines and R. P. Malhamé, Stochastic optimal control under Poisson-distributed observations, IEEE Trans. Automat. Control, 45 (2000), 3-13. [2] D. J. Antunes, J. P. Hespanha and C. J. Silvestre, Volterra integral approach to impulsive renewal systems: Application to networked control, IEEE Trans. Automat. Control, 57 (2012), 607-619.  doi: 10.1109/TAC.2011.2166300. [3] D. Antunes, J. Hespanha and C. Silvestre, Stability of networked control systems with asynchronous renewal links: An impulsive systems approach, Automatica J. IFAC, 49 (2013), 402-413.  doi: 10.1016/j.automatica.2012.11.033. [4] R. P. Anderson, D. Milutinović and D. V. Dimarogonas, Self-triggered sampling for second-moment stability of state-feedback controlled SDE systems, Automatica J. IFAC, 54 (2015), 8–15. https://www.sciencedirect.com/science/article/pii/S0005109815000308?via%3Dihub doi: 10.1016/j.automatica.2015.01.020. [5] A. Arapostathis, A. Biswas and V. S. Borkar, Controlled equilibrium selection in stochastically perturbed dynamics, Ann. Probab., 46 (2018), 2749-2799.  doi: 10.1214/17-AOP1238. [6] S. R. Athreya, V. S. Borkar, K. S. Kumar and R. Sundaresan, Simultaneous small noise limit for singularly perturbed slow-fast coupled diffusions, Appl. Math. Optim., 83 (2021), 2327-2374.  doi: 10.1007/s00245-019-09630-w. [7] P. Billingsley, Convergence of Probability Measures, John Wiley and Sons Inc., second edition, 1999. doi: 10.1002/9780470316962. [8] D. P. Borgers, V. S. Dolk, G. E. Dullerud, A. R. Teel and W. P. M. H. Heemels, Time-regularized and periodic event triggered control for linear systems, in Control Subject to Computational and Communication Constraints, Springer, 2018,121–149. https://link.springer.com/chapter/10.1007/978-3-319-78449-6_7 [9] S. Bourguin, S. Gailus and K. Spiliopoulos, Typical dynamics and fluctuation analysis of slow–fast systems driven by fractional Brownian motion, Stoch. Dyn., (2021), 2150030, 30 pp. doi: 10.1142/S0219493721500301. [10] S. Cerrai, A Khasminskii type averaging principle for stochastic reaction–diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948.  doi: 10.1214/08-AAP560. [11] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z. [12] G. Chen, C. Fan, J. Sun and J. Xia, Mean square exponential stability analysis for ito stochastic systems with aperiodic sampling and multiple time-delays, IEEE Transactions on Automatic Control, 2021. https://ieeexplore.ieee.org/abstract/document/9410357 [13] T. Chen and B. Francis, Optimal Sampled-Data Control Systems, Springer, 1996. [14] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, second edition, 1998. https://link.springer.com/book/10.1007/978-3-642-03311-7 [15] R. Dong and X. Mao, Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations, Math. Control Relat. Fields, 10 (2020), 715-734.  doi: 10.3934/mcrf.2020017. [16] V. Drǎgan and I. G. Ivanov, On the stochastic linear quadratic control problem with piecewise constant admissible controls, J. Franklin Inst., 357 (2020), 1532-1559.  doi: 10.1016/j.jfranklin.2019.10.036. [17] M. I. Freidlin and R. B. Sowers, A comparison of homogenization and large deviations, with applications to wavefront propagation, Stochastic Process. Appl., 82 (1999), 23-52.  doi: 10.1016/S0304-4149(99)00003-4. [18] M. I. Freidlin and A. D. Wentzell, Random perturbations of Hamiltonian systems, Mem. Amer. Math. Soc., 109 (1994), 523.  doi: 10.1090/memo/0523. [19] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, third edition, 2012. doi: 10.1007/978-3-642-25847-3. [20] R. Goebel, R. G. Sanfelice and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability and Robustness, Princeton University Press, 2012. [21] R. Z. Has'minskii, On stochastic processes defined by differential equations with a small parameter, Theory of Probability & Its Applications, 11 (1966), 211-228.  doi: 10.1137/1111018. [22] R. Z. Has'Minskii, On the principle of averaging the Ito's stochastic differential equations, Kybernetika, 4 (1968), 260-279. [23] W. P. M. H. Heemels, M. 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. [24] J. P. Hespanha, Linear Systems Theory, Princeton University Press, 2009. [25] J. P. Hespanha, Modeling and analysis of networked control systems using stochastic hybrid systems, Annual Reviews in Control, 38 (2014), 155–170. https://www.sciencedirect.com/science/article/pii/S1367578814000352 [26] J. P. Hespanha and A. R. Teel, Stochastic impulsive systems driven by renewal processes, Proc. 17th International Symposium on Mathematical Theory of Networked Systems, (2006), 606–618. https://scholar.google.co.in/scholar?q=Stochastic+impulsive+systems+driven+by+renewal+processes&hl=en&as_sdt=0&as_vis=1&oi=scholart [27] J. Higham Desmond, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [28] I. Karafyllis and C. Kravaris, Global stability results for systems under sampled-data control, Internat. J. Robust Nonlinear Control, 19 (2009), 1105-1128.  doi: 10.1002/rnc.1364. [29] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics, Springer-Verlag New York, second edition, 1991. doi: 10.1007/978-1-4612-0949-2. [30] H. K. Khalil, Performance recovery under output feedback sampled-data stabilization of a class of nonlinear systems, IEEE Trans. Automat. Control, 49 (2004), 2173-2184.  doi: 10.1109/TAC.2004.838496. [31] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5. [32] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, SIAM, 1999. https://epubs.siam.org/doi/book/10.1137/1.9781611971118 [33] D. S. Laila, D. Nesic and A. R. Teel, Open- and closed-loop dissipation inequalities under sampling and controller emulation, European Journal of Control, 8 (2002), 109–125. https://www.sciencedirect.com/science/article/pii/S0947358002702161 [34] D. Nesic and A. R. Teel, A framework for stabilization of non-linear sampled-data systems based on their approximate discrete-time models, IEEE Trans. Automat. Control, 49 (2004), 1103-1122.  doi: 10.1109/TAC.2004.831175. [35] D. Nesic, A. R. Teel and D. Carnevale, Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems, IEEE Trans. Automat. Control, 54 (2009), 619–624. https://ieeexplore.ieee.org/document/4796269 doi: 10.1109/TAC.2008.2009597. [36] D. Nesic, A. R. Teel and P. V. Kokotovic, Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, Systems Control Lett., 38 (1999), 259-270.  doi: 10.1016/S0167-6911(99)00073-0. [37] B. Oksendal, Stochastic Differential Equations, Springer, sixth edition, 2003. doi: 10.1007/978-3-642-14394-6. [38] M. Röckner and L. Xie, Averaging principle and normal deviations for multiscale stochastic systems, Comm. Math. Phys., 383 (2021), 1889-1937.  doi: 10.1007/s00220-021-04069-z. [39] M. Röckner and L. Xie, Diffusion approximation for fully coupled stochastic differential equations, Ann. Probab., 49 (2021), 1205-1236.  doi: 10.1214/20-aop1475. [40] A. V. Skorokhod, F. C. Hoppensteadt and H. Salehi, Random Perturbation Methods with Applications in Science and Engineering, volume 150 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002. doi: 10.1007/b98905. [41] K. Spiliopoulos, Large deviations and importance sampling for systems of slow-fast motion, Appl. Math. Optim., 67 (2013), 123-161.  doi: 10.1007/s00245-012-9183-z. [42] K. Spiliopoulos, Fluctuation analysis and short time asymptotics for multiple scales diffusion processes, Stoch. Dyn., 14 (2014), 1350026, 22 pp. doi: 10.1142/S0219493713500263. [43] A. Tanwani, D. Chatterjee and D. Liberzon, Stabilization of deterministic control systems under random sampling: Overview and recent developments, Uncertainty in Complex Networked Systems. In T. Basar, editor, Springer, 2018,209–246. https://link.springer.com/chapter/10.1007/978-3-030-04630-9_6 [44] A. Tanwani and O. Yufereva, Error covariance bounds for suboptimal filters with Lipschitzian drift and Poisson-sampled measurements, Automatica J. IFAC, 122 (2020), 109280, 14 pp. doi: 10.1016/j.automatica.2020.109280. [45] W. Wang, R. Postoyan, D. Nesic and W. P. M. H. Heemels, Periodic event-triggered control for nonlinear networked control systems, IEEE Trans. Automat. Control, 65 (2020), 620-635.  doi: 10.1109/TAC.2019.2914255. [46] J. I. Yuz and G. C. Goodwin, Sampled-Data Models for Linear and Nonlinear Systems, Springer, 2014. [47] O. Zeitouni, Approximate and limit results for nonlinear filters with small observation noise: the linear sensor and constant diffusion coefficient case, IEEE Transactions on Automatic Control, 33 (1988), 595–599. https://ieeexplore.ieee.org/abstract/document/1262
Sample paths for the components $X_1\triangleq X_1^{ \varepsilon,\delta}(t),\; X_2 \triangleq X_2^{ \varepsilon,\delta}(t)$ of the sde defined by (10) and $S_1 \triangleq S_1^{ \varepsilon}(t) = x_1(t)+ \varepsilon Z_1(t),\; S_2 \triangleq S_2^{ \varepsilon}(t) = x_2(t)+ \varepsilon Z_2(t)$ with $\varepsilon = 2^{-5},\; \delta = 2^{-4},$ and $T = 2^3$. Here $Z_1(t)$ and $Z_2(t)$ are the components of $Z(t)$ defined in (14)
For $1 \le i \le 7$, let $e_i$ be the vector $(e_{i,1},e_{i,2})$, where, for $j = 1,2$, the quantity $e_{i,j}$ is the mean of $|X^{ \varepsilon,\delta}_j(T)-S^ \varepsilon_j(T)|$ over 1000 sample paths generated by the Euler-Maruyama method, with $T = 2^3$, $\delta = 2^{-4}$. On a $\log_2$-$\log_2$ scale, we see that as $\varepsilon\triangleq 2^{-i}$ decreases, the corresponding error $e_{i,j}$ also decreases
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