Abstract
The behavior of trajectories of multidimensional linear discrete-time systems with nonzero initial conditions is considered in two cases as follows. The first case is the systems with infinite degree of stability (the processes of a finite duration); the second case is the stable systems with a spectral radius close to 1. It is demonstrated that in both cases, large deviations of the trajectories from the equilibrium may occur. These results are applied to accelerated unconstrained optimization methods (such as the Heavy-ball method) for explaining the nonmonotonic behavior of the methods, which is observed in practice.
Similar content being viewed by others
References
Tsypkin, Ya.Z. Perekhodnye i ustanovivshiesya protsessy v impul'snykh tsepyakh (Transient Response and Steady-State Processes in Impulse Circuits), Moscow: Gosenergoizdat, 1951.
Tsypkin, Ya.Z., Theory of Discontinuous Control. III Avtom. Telemekh., 1950, no. 5, pp. 300–319.
Feldbaum, A.A., On the Arrangement of Roots for Characteristic Equation of Control Systems Avtom. Telemekh., 1948, no. 4, pp. 253–279.
Izmailov, R.N., The “Peak” Effect in Stationary Linear Systems with Scalar Inputs and Outputs Autom. Remote Control, 1987, vol. 48, no. 8, pp. 1018–1024.
Smirnov, G., Bushenkov, V., and Miranda, F., Advances on the Transient Growth Quantification in Linear Control Systems Int. J. Appl. Math. Statist., 2009, vol. 14, pp. 82–92.
Polyak, B. and Smirnov, G., Large Deviations for Non-Zero Initial Conditions in Linear Systems Automatica, 2016, vol. 74, no. 12, pp. 297–307.
Polyak, B.T., Tremba, A.A., Khlebnikov, M.V., Shcherbakov, P.S., and Smirnov, G.V., Large Deviations in Linear Control Systems with Nonzero Initial Conditions Autom. Remote Control, 2015, vol. 76, no. 6, pp. 957–976.
Kozyakin, V.S., Kuznetsov, N.A., and Pokrovskii, A.V., Transients in Quasi-Controllable Systems. Overshooting, Stability and Instability IFAC Proc., 1993, vol. 26, no. 2, part 4, pp. 871–874.
Kogan, M.M. and Krivdina, L.N., Synthesis of Multipurpose Linear Control Laws of Discrete Objects under Integral and Phase Constraints Autom. Remote Control, 2011, vol. 72, no. 7, pp. 1427–1439.
Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S. Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems with External Perturbations: The Technique of Linear Matrix Inequalities), Moscow: LENAND, 2014.
Hinrichsen, D., Plischke, E., and Wurth, F., State Feedback Stabilization with Guaranteed Transient Bounds Proc. 15 Int. Symp. Math. Theory Networks Syst. (MTNS), August, 2002, CDROM, paper 2132.
Shcherbakov, P., On Peak Effects in Discrete Time Linear Systems Proc. 25 Mediterranean Conf. Control Automat. (MED 2017), 2017, Valletta, Malta, pp. 376–381.
Polyak, B.T., Shcherbakov, P.S., and Smirnov, G.V., Peak Effects in Stable Linear Difference Equations J. Differ. Equat. Appl., 2018, vol. 27, no. 9, pp. 1488–1502.
Kalman, R. and Bertram, J., General Synthesis Procedure for Computer Control of Single-Loop and Multiloop Linear Systems (An Optimal Sampling System) Trans. Am. Inst. Electr. Eng., II. Appl. Industry, 1959, vol. 77, no. 6, pp. 602–609.
Kalman, R.E., Falb, P.L., and Arbib, M.A. Topics in Mathematical System Theory, New York: McGraw Hill, 1969. Translated under the title Ocherki po matematicheskoi teorii sistem, Moscow: Mir, 1977.
Elaydi, S. An Introduction to Difference Equations, New York: Springer, 2005.
Polyak, B.T., Some Methods of Speeding up the Convergence of Iteration Methods USSR Comp. Math. Math. Phys., 1964, vol. 4, no. 5, pp. 1–17.
Polyak, B.T. Vvedenie v optimizatsiyu, Moscow: Nauka, 1983. Translated under the title Introduction to Optimization, Translations Series in Mathematics and Engineering, New York: Optimization Software, 1987.
Danilova, M., Kulakova, A., and Polyak, B., Non-asymptotic Behaviour of Multi-Step Iterative Methods 24 Int. Conf. Difference Equat. Appl. (ICDEA 2018), Dresden, Germany, July 2018.
Acknowledgments
The work of B.T. Polyak was supported by the Russian Science Foundation, project no. 16-11-10015. We are grateful to P.S. Shcherbakov, M. Danilova and A. Kulakova for careful reading of the manuscript and helpful remarks.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 9, pp. 112–121.
Rights and permissions
About this article
Cite this article
Polyak, B.T., Smirnov, G.V. Transient Response in Matrix Discrete-Time Linear Systems. Autom Remote Control 80, 1645–1652 (2019). https://doi.org/10.1134/S0005117919090066
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117919090066