Skip to main content
SpringerLink
Log in

Time Invariance of Optimal Control in a Stochastic Linear Controller Design with Dynamic Scaling of Coefficients

  • CONTROL IN STOCHASTIC SYSTEMS AND UNDER UNCERTAINTY CONDITIONS
  • Published:
Journal of Computer and Systems Sciences International Aims and scope

Abstract

This paper considers the design problem of a stochastic linear-quadratic controller over an infinite time-horizon with dynamic scaling of the coefficients in the state equation and the cost criterion. Dynamic scaling means multiplying the coefficients by a positive time-varying function. The optimality criteria used are extensions of the long-term average cost and pathwise long-term average cost. The integral of the scaling function is applied to normalize the performance indices. It is shown that, the optimal control law is time-invariant and can be obtained through a steady-state optimal strategy known for the autonomous system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. W. X. Zhong, Duality System in Applied Mechanics and Optimal Control (Springer, New York, 2006).

    Google Scholar 

  2. J. S. Arora, Optimization of Structural and Mechanical Systems (World Scientific, Singapore, 2007).

    Book  Google Scholar 

  3. A. Tewari, Aeroservoelasticity: Modeling and Control (Springer, New York, 2015).

    Book  Google Scholar 

  4. A. A. Lebedev, V. T. Bobronnikov, M. N. Krasil’shchikov, and V. V. Malyshev, Statistical Dynamics of Controlled Flight (Mashinostroenie, Moscow, 1978) [in Russian].

    Google Scholar 

  5. V. V. Malyshev, Optimization Methods in Problems of System Analysis and Control (MAI, Moscow, 2010) [in Russian].

  6. A. M. Letov, “Analytical design of regulators I,” Avtom. Telemekh. 21 (4), 436–441 (1960).

    Google Scholar 

  7. A. M. Letov, “Analytical design of regulators II,” Avtom. Telemekh. 21, 561–568 (1960).

    Google Scholar 

  8. A. M. Letov, “Analytical design of regulators III,” Avtom. Telemekh. 21, 661–665 (1960).

    Google Scholar 

  9. V. A. Yakubovich, “Optimization and invariance of linear stationary control systems,” Avtom. Telemekh., No. 8, 5–45 (1984).

  10. I. E. Kazakov and D. I. Gladkov, Optimization Methods for Stochastic Systems (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  11. A. I. Kibzun, Stochastic Control of Dynamic Systems, The School-Book (MAI, Moscow, 1991) [in Russian].

  12. D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems (Springer, New York, 2012).

    MATH  Google Scholar 

  13. J. C. Wu and J. N. Yang, “Control of lateral-torsional motion of Nanjing TV transmission tower,” in Computational Mechanics in Structural Engineering: Recent Developments (Elsevier, New York, 1999), pp. 43–56.

    Google Scholar 

  14. Z. Tan and P. M. Bainum, “Optimal linear quadratic Gaussian digital control of an orbiting tethered antenna/reflector system,” J. Guidance, Control Dyn. 17, 234–241 (1994).

    Article  Google Scholar 

  15. A. Lamperski and N. J. Cowan, “Time-changed linear quadratic regulators,” in Proceedings of the 2013 European Control Conference ECC (IEEE, New York, 2013), pp. 198–203.

  16. R. Singh and V. Gupta, “On LQR control with asynchronous clocks,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conferences CDC-ECC (IEEE, New York, 2011), pp. 3148–3153.

  17. D. Aadland and S. Shaffer, “Time compression and saving rates,” J. Neurosci. Psychol. Econ. 8, 217–240 (2015).

    Article  Google Scholar 

  18. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, New York, 1972).

    MATH  Google Scholar 

  19. I. Karafyllis and J. Tsinias, “Non-uniform in time stabilization for linear systems and tracking control for non-holonomic systems in chained form,” Int. J. Control 76, 1536–1546 (2003).

    Article  MathSciNet  Google Scholar 

  20. P. L. Smith, R. Ratcliff, and D. K. Sewell, “Modeling perceptual discrimination in dynamic noise: Time-changed diffusion and release from inhibition,” J. Math. Psychol. 59, 95–113 (2014).

    Article  MathSciNet  Google Scholar 

  21. H. Jiang, H. L. Gray, and W. A. Woodward, “Time-frequency analysis—G (λ)-stationary processes,” Comput. Stat. Data Anal. 51, 1997–2028 (2006).

    Article  MathSciNet  Google Scholar 

  22. C. P. C. Vijverberg, “Time deformation, continuous Euler processes and forecasting,” J. Time Series Anal. 27, 811–829 (2006).

    Article  MathSciNet  Google Scholar 

  23. C. P. C. Vijverberg, “A time deformation model and its time-varying autocorrelation: An application to US unemployment data,” Int. J. Forecast. 25, 128–145 (2009).

    Article  Google Scholar 

  24. E. S. Palamarchuk, “Stabilization of linear stochastic systems with a discount: Modeling and estimation of the long-term effects from the application of optimal control strategies,” Math. Models Comput. Simul. 7, 381 (2015).

    Article  MathSciNet  Google Scholar 

  25. M. H. A. Davis, Linear Estimation and Stochastic Control (Chapman and Hall, London, 1977).

    MATH  Google Scholar 

  26. E. Palamarchuk, “On infinite time linear-quadratic Gaussian control of inhomogeneous systems,” in Proceedings of the 2016 European Control Conference ECC (IEEE, New York, 2016), pp. 2477–2482.

  27. E. S. Palamarchuk, “Analysis of criteria for long-run average in the problem of stochastic linear regulator,” Autom. Remote Control 77, 1756 (2016).

    Article  MathSciNet  Google Scholar 

  28. T. A. Belkina and E. S. Palamarchuk, “On stochastic optimality for a linear controller with attenuating disturbances,” Autom. Remote Control 74, 628 (2013).

    Article  MathSciNet  Google Scholar 

  29. T. A. Belkina, Yu. M. Kabanov, and E. L. Presman, “On a stochastic optimality of the feedback control in the LQG-problem,” Teor. Probab. Appl. 48, 592–603 (2003).

    Article  Google Scholar 

  30. M. Y. Wu and A. Sherif, “On the commutative class of linear time-varying systems,” Int. J. Control 23, 433–444 (1976).

    Article  MathSciNet  Google Scholar 

  31. V. I. Kalenova and V. M. Morozov, “On the control of linear time-varying systems of a special form,” J. Comput. Syst. Sci. Int. 52, 333 (2013).

    Article  MathSciNet  Google Scholar 

  32. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications (Springer, Berlin, 1985).

    Book  Google Scholar 

  33. F. L. Lewis, D. Vrabie, and V. L. Syrmos, Optimal Control (Wiley, New York, 2012).

    Book  Google Scholar 

  34. J. H. Stock, “Hysteresis and the evolution of postwar US and UK unemployment,” in Economic Complexity: Chaos, Sunspots, Bubbles, and Nonlinearity, Proceedings of the 4th International Symposium in Economic Theory and Econometrics, Ed. by W. A. Barnett, J. Geweke, and K. Shell (Cambridge Univ. Press, Cambridge, MA, 1989), pp. 361–382.

  35. S. J. Turnovsky, Macroeconomic Analysis and Stabilization Policy (Cambridge Univ. Press, Cambridge, MA, 1977).

    MATH  Google Scholar 

  36. J. K. Sengupta, “Optimal stabilization policy with a quadratic criterion function,” Rev. Econ. Studies 37, 127–145 (1970).

    Article  Google Scholar 

  37. B. Sack, “Does the Fed act gradually? A VAR analysis,” J. Monet. Econ. 46, 229–256 (2000).

    Article  Google Scholar 

  38. G. Loewenstein and D. Prelec, “Anomalies in intertemporal choice: Evidence and an interpretation,” Quart. J. Econ. 107, 573–597 (1992).

    Article  Google Scholar 

  39. B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Courier, New York, 2007).

    Google Scholar 

  40. E. K. Bonkas and Z. K. Liu, “Suboptimal design of regulators for jump linear system with time-multiplied quadratic cost,” IEEE Trans. Autom. Control 46, 131–136 (2001).

    Article  MathSciNet  Google Scholar 

  41. X. Xie, J. Lam, and C. Fan, “Robust time-weighted guaranteed cost control of uncertain periodic piecewise linear systems,” Inform. Sci. 460, 238–253 (2018).

    Article  MathSciNet  Google Scholar 

  42. A. Czornik, “On time-varying LQG,” IFAC Proc. 31, 411–415 (1998).

Download references

Funding

This study was carried out within the research project at Central Economics and Mathematics Institute, Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, Russia

    E. S. Palamarchuk

Authors
  1. E. S. Palamarchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. S. Palamarchuk.

Additional information

Translated by A. Mazurov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Palamarchuk, E.S. Time Invariance of Optimal Control in a Stochastic Linear Controller Design with Dynamic Scaling of Coefficients. J. Comput. Syst. Sci. Int. 60, 202–212 (2021). https://doi.org/10.1134/S1064230721020106

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064230721020106

Search

Navigation