Large deviations in linear control systems with nonzero initial conditions
- 122 Downloads
Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work  as early as in 1948. However later, studies in this direction have faded down, and since then, the notion of transient process basically means the response of the system with zero initial conditions to the unit step input. A breakthrough in this direction is associated with the paper  by R.N. Izmailov, where large deviations of the trajectories from the origin were shown to be unavoidable if the poles of the closed-loop system are shifted far to the left in the complex plane.
In this paper we continue the analysis of this phenomenon for systems with nonzero initial conditions. Namely, we propose a more accurate estimate of the magnitude of the peak and show that the effect of large deviations may be observed for different root locations. We also present an upper bound on deviations by using the linear matrix inequality (LMI) technique. This same approach is then applied to the design of a stabilizing linear feedback aimed at diminishing deviations in the closed-loop system. Related problems are also discussed, e.g., such as analysis of the transient response of systems with zero initial conditions and exogenous disturbances in the form of either unit step function or harmonic signal.
KeywordsRemote Control Linear Matrix Inequality Lyapunov Equation Companion Form Root Location
Unable to display preview. Download preview PDF.
- 7.Akunov, T.A., Dudarenko, N.A., Polinova, N.A., and Ushakov, A.V., Analysis of Processes in Continuous Time Systems with Multiple Complex Conjugate Eigenvalues of the State Matrices, Nauchn.-Tekhn. Vestn. Inform. Tekhnol., Mekh., Optiki, 2013, no. 4(86), pp. 25–33.Google Scholar
- 8.Akunov, T.A., Dudarenko, N.A., Polinova, N.A., and Ushakov, A.V., Degree of Proximity of Simple and Multiple Eigenvalue Structures: Minimization of the Trajectory Peaks in Unperturbed Motion of Aperiodic Systems, Nauchn.-Tekhn. Vestn. Inform. Tekhnol., Mekh., Optiki, 2014, no. 2(90), pp. 39–46.Google Scholar
- 10.Polyak, B.T. and Smirnov, G.V., Large Deviations in Continuous-Time Linear Single-Input Control Systems, Proc. 19 IFAC World Congr., Cape Town, Aug. 24–29, 2014, pp. 5586–5591.Google Scholar
- 13.Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems Subject to Exogenous Disturbances: The Linear Matrix Inequalitiy Technique), Moscow: LENAND, 2014.Google Scholar
- 14.Hinrichsen, D., Plischke, E., and Wurth, F., State Feedback Stabilization with Guaranteed Transient Bounds, Proc. 15 Int. Symp. Math. Theory Networks & Syst. (MTNS-2002), Notre Dame, Indiana, Aug. 2002, paper no. 2132 (CDROM).Google Scholar
- 21.Bushenkov, V. and Smirnov, G., Stabilization Problems with Constraints: Analysis and Computational Aspects, Amsterdam: Gordon and Breach, 1997.Google Scholar
- 22.Polyak, B.T. and Tremba, A.A., Closed-Form Solution of Linear Differential Equations with Equal Roots of the Characteristic Equation, Proc. XII All-Russia Workshop on Control Problems (VSPU-2014), Moscow, Jun. 2014, pp. 212–217.Google Scholar
- 23.Grant, M. and Boyd, S., CVX: Matlab Software for Disciplined Convex Programming (web page and software), URL http://stanford.edu/boyd/cvx.Google Scholar
- 26.A Letter by N.G. Chebotarev on a Mathematical Problem Related to the Evaluation of the Regulated Coordinate When the Disturbing Force is Bounded in Absolute Value, Avtom. Telemekh., 1948, vol. vn9, no. sn4, pp. 331–334.Google Scholar