# On σ-entropy Analysis of Linear Stochastic Systems in State Space

## DOI:

https://doi.org/10.52846/stccj.2021.1.1.8## Keywords:

stochastic systems, linear systems, system sensitivity, entropy function## Abstract

In this paper the problem of random disturbance attenuation capabilities in linear continuous systems is studied. It is supposed that the system operates under random disturbances with bounded σ-entropy level. σ-entropy norm indicates a performance index of the continuous system on the set of the random signals with bounded σ-entropy. This paper presents a time-domain solution to the calculation of σ-entropy norm of the continuous linear time-invariant system. σ-entropy norm is defined after solving coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations.

## References

D. Mustafa and K. Glover, Minimum Entropy H∞ Control. Lecture Notes in Control and Information Sciences, vol. 146, Springer Verlag, 1990.

D.Z. Arov and M.G. Krein, “On calculation of entropy integrals and their minima in generalized extension problems,” Acta Sci. Math., vol. 45, pp. 33–50, 1983.

D. Mustafa, “On H∞ control, LQG control and minimum entropy,” in Robust Control of Linear Systems and Nonlinear Control, Series: Progress in Systems and Control Theory, vol. 4, M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran, Eds. Boston: Birkh¨auser, 1990, pp. 317–327.

P.A. Iglesias, D. Mustafa, and K. Glover, “Discrete-time H∞ controllers satisfying a minimum entropy criterion,” Systems & Control Letters, vol. 14, no. 4, pp. 275–286, April 1990.

P.A. Iglesias and D. Mustafa, “State-space solution of the discrete-time minimum entropy control problem via separation,” IEEE Transactions on Automatic Control, vol. 38, no. 10, pp. 1525–1530, October 1993.

I. Yaesh and U. Shaked, “Minimum entropy static output-feedback control with an H∞-norm performance bound,” IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 853–858, June 1997.

I.R. Petersen, M.R. James, and P. Dupuis, “Minimax optimal control of stochastic uncertain systems with relative entropy constraints,” IEEE Transactions on Automatic Control, vol. 45, no. 3, pp. 398–412, March 2000.

A.P. Kurdyukov, O.G. Andrianova, A.A. Belov, and D.A. Gol’din, “In between the LQG/H2- and H∞-control theories,” Automation and Remote Control, vol. 82, no. 4, pp. 565–618, April 2021.

A.P. Kurdyukov and V.A. Boichenko, “The spectral method of the analysis of linear control systems,” International Journal of Applied Mathematics and Computer Science, vol. 29, pp. 667–679, December 2019.

V.A. Boichenko and A.A. Belov, “On calculation of σ-entropy norm of continuous linear time-invariant systems,” Proc. 2020 15th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (STAB). Moscow, Russia, 2020, pp. 1–4.

V.A. Boichenko, “The new approach to the analysis of linear control systems,” Proc. 2018 14th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference). Moscow, Russia, 2018, pp. 1–4.

K. Zhou, K. Glover, B. Bodenheimer, and J. Doyle, “Mixed H2 and H∞ performance objectives. I. Robust performance analysis,” IEEE Transactions on Automatic Control, vol. 39, no. 8, pp. 1564–1574, August 1994.

K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control. Prentice Hall Inc., Upper Saddle River, 1996.

D.S. Bernstein, Matrix Mathematics. Princeton University Press, New Jersey, 2005.

W. Rudin, Real and Complex Analysis. McGraw-Hill, New York, 1986.

A.A. Belov and V.A. Boichenko, “S-entropy analysis of LTI continuoustime systems with stochastic external disturbance in time domain,” Proc. 2020 24th International Conference on System Theory, Control and Computing (ICSTCC). Sinaia, Romania, 2020, pp. 184–189.

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*Syst. Theor. Control Comput. J.*, vol. 1, no. 1, pp. 30–35, Jun. 2021, doi: 10.52846/stccj.2021.1.1.8.