# Large deviations in linear control systems with nonzero initial conditions

## Abstract

Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work [1] 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 [2] 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.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.Feldbaum, A.A., On the Root Location of Characteristic Equations of Control Systems,
*Avtom. Telemekh.*, 1948, no. 4, pp. 253–279.MathSciNetGoogle Scholar - 2.Izmailov, R.N., The “Peak” Effect in Stationary Linear Systems with Scalar Inputs and Outputs,
*Autom. Remote Control*, 1987, vol. 48, no. 8, part 1, pp. 1018–1024.MathSciNetGoogle Scholar - 3.Luenberger, D.G., An Introduction to Observers,
*IEEE Trans. Autom. Control*, 1971, vol. 35, pp. 596–602.CrossRefGoogle Scholar - 4.Liberzon, D.,
*Switching in Systems and Control*, Boston: Birkh¨auser, 2003.MATHCrossRefGoogle Scholar - 5.Moler, C. and Van Loan, C., Nineteen Dubious Ways to Compute the Exponential of a Matrix,
*SIAM Rev.*, 1978, vol. 20, pp. 801–836.MATHMathSciNetCrossRefGoogle Scholar - 6.Moler, C. and Van Loan, C., Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,
*SIAM Rev.*, 2003 vol. 45, no. 1, pp. 3–49.MATHMathSciNetCrossRefGoogle Scholar - 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 - 9.Smirnov, G., Bushenkov, V., and Miranda, F., Advances on the Transient Growth Quantification in Linear Control Dystems,
*Int. J. Appl. Math. Statist.*, 2009, vol. 14, pp. 82–92.MathSciNetGoogle 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 - 11.van Dorsselaer, J.L.M., Kraaijevanger, J.F.B.M., and Spijker, M.N., Linear Stability Analysis in the Numerical Solution of Initial Value Problems,
*Acta Numer.*, 1993, vol. 2, pp. 199–237.MathSciNetCrossRefGoogle Scholar - 12.Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in Systems and Control Theory, Philadelphia: SIAM, 1994.CrossRefGoogle 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 - 15.Whidborne, J.F. and McKernan, J., On Minimizing Maximum Transient Energy Growth,
*IEEE Trans. Autom. Control*, 2007, vol. 52, no. 9, pp. 1762–1767.MathSciNetCrossRefGoogle Scholar - 16.Balandin, L.V. and Kogan, M.M., Lyapunov Function Method for Control Law Synthesis under One Integral and Several Phase Constraints,
*Differ. Equat.*, 2009, vol. 45. no. 5, pp. 670–679.MATHMathSciNetCrossRefGoogle Scholar - 17.Whidborne, J.F. and Amar, N., Computing the Maximum Transient Energy Growth,
*BIT Numer. Math.*, 2011, vol. 51, no. 2, pp. 447–557.MATHMathSciNetCrossRefGoogle Scholar - 18.Polotskii, V.N., On the Maximal Errors of an Asymptotic State Identifier,
*Autom. Remote Control*, 1978, vol. 39, no. 8, part 1, pp. 1116–1121.MathSciNetGoogle Scholar - 19.Polotskii, V.N., Estimation of the State of Single-Output Linear Systems by Means of Observers,
*Autom. Remote Control*, 1980, vol. 41, no. 12, part 1, pp. 1640–1648.MathSciNetGoogle Scholar - 20.Sussmann, H.J. and Kokotovic, P.V., The Peaking Phenomenon and the Global Stabilization of Nonlinear Systems,
*IEEE Trans. Autom. Control.*, 1991, vol. 36, no. 4, pp. 424–439.MATHMathSciNetCrossRefGoogle 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
- 24.Bulgakov, A.Ja., An Efficiently Calculable Parameter for the Stability Property of a System of Linear Differential Equations with Constant Coefficients,
*Siberian Math. J.*, 1980, vol. 21, no. 3, pp. 339–347.MathSciNetCrossRefGoogle Scholar - 25.Nechepurenko, Yu.M., Bounds for the Matrix Exponential Based on the Lyapunov Equation and Limits of the Hausdorff Set,
*Comput. Math. Math. Phys.*, 2002, vol. 42, no. 2, pp. 125–134.MathSciNetGoogle 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
- 27.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.MATHMathSciNetCrossRefGoogle Scholar