Transient Response in Matrix Discrete-Time Linear Systems
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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.
Keywordsdiscrete-time systems transient response stability large deviations infinite degree of stability multidimensional systems Heavy-ball method
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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.
- 1.Tsypkin, Ya.Z. Perekhodnye i ustanovivshiesya protsessy v impul'snykh tsepyakh (Transient Response and Steady-State Processes in Impulse Circuits), Moscow: Gosenergoizdat, 1951.Google Scholar
- 2.Tsypkin, Ya.Z., Theory of Discontinuous Control. III Avtom. Telemekh., 1950, no. 5, pp. 300–319.Google Scholar
- 4.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.Google Scholar
- 10.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.Google Scholar
- 11.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.Google Scholar
- 14.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.Google Scholar
- 19.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.Google Scholar