Elliposoidal estimates of trajectory sensitivity of multi-dimensional processes based on generalized singular values problems

The problem of studying the sensitivity of control processes to parameter variations is considered. To solve the problem, the trajectory sensitivity apparatus was used, the use of which, together with the state-space method, made it possible to construct sensitivity models. Based on the models, ellipsoidal estimates of the trajectory sensitivity functions in terms of the state, output, and error of linear multidimensional continuous systems in the form of majorants and minorants are determined. Calculations are performed using the generalized singular value decomposition of matrices composed of trajectory parametric sensitivity functions. The resulting ellipsoidal estimates, due to the meaningful possibilities of the generalized singular value decomposition of matrices, have the property of minimal sufficiency. The estimation method makes it possible, using the left singular basis corresponding to the extremal generalized singular values, to determine subspaces in the state, output, and error spaces that are characterized at each moment of time by the largest and smallest additional motion in terms of the norm. The right singular basis allows us to determine subspaces in the parameter space that generate the largest and smallest additional motion in the norm. The proposed approach solves the problem of “optimal nominal”, that is, the problem of choosing the nominal value of the parameter vector of the plant units that provide the multidimensional controlled process with the smallest value of the ellipsoidal estimates of the trajectory sensitivity functions, as well as to compare the flow of multidimensional controlled processes according to the ellipsoidal estimates of the trajectory parametric sensitivity.

1. Rozenvasser E.N., Iusupov R.M. Sensitivity of Automatic Control Systems. Moscow, Jenergija Publ., 1981, 464 p. (in Russian)
2. Tomovic R., Vucobratovic M. General Sensitivity Theory. North-Holland, 1972, 258 p.
3. Omorov R.O. The modal sensitivity, robustness and roughness of dynamic systems (review article). Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2021, vol. 21, no. 2, pp. 179–190. (in Russian). https://doi.org/10.17586/2226-1494-2021-21-2-179-190
4. PerkinsW.R., CruzI.B., GonsalesR.L. Design of minimum sensitivity systems. IEEETransactionson Automatic Control, 1968, vol. AC-13, no. 6, pp. 159–167. https://doi.org/10.1109/TAC.1968.1098853
5. Akunov T.A., Ushakov A.V. Estimates of the trajectory parametric sensitivity functions for control systems. Russian Electromechanics, 1992, no. 1, pp. 87–92. (in Russian)
6. Akunov T.A. Analisys and Synthesis of Multidimensional Systems Using the Tools of Ellipsoidal Assessments of the Processes Quality. Dissertation for the degree of candidate of technical sciences. St. Petersburg, Saint Petersburg Institute of Fine Mechanics and Optics, 1993, 185 с. (in Russian)
7. Horn R.A., Johnson Ch.R. Matrix Analysis. Cambridge University Press, 1985, 561 p.
8. Akunov T.A., Alisherov S., Omorov R.O., Ushakov A.V. Modal Estimations of Quality of Processes in Linear Multivarible Systems. Bishkek, Ilim Publ., 1991, 59 p. (in Russian)
9. Voevodin V.V., Kuznetcov Iu.A. Matrices and Сalculations. Moscow, Nauka Publ., 1984, 160 p. (in Russian)
10. Gantmacher F. R. The Theory of Matrices. AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000, 660 p.
11. Forsythe G.E., Malcolm M.A., Moler C.B. Computer Methods for Mathematical Computations. Prentice-Hall, 1977, 259 p.
12. Golub G.H., Van Loan Ch.F. Matrix Computations. JHU Press, 1996, 756 p.
13. Grigorev V.V., Drozdov V.N., Lavrentev V.V., Ushakov A.V. Synthesis of Discrete Controllers Using Computers. Leningrad, Mashinostroenie Publ., 1983, 245 p. (in Russian)
14. AkunovT.A., SudarchikovS.A., UshakovA.V. Analysisofalgorithmicproblemsatresearchofsensitivityoflumpedparametersystems. Journal of Instrument Engineering, 2008, vol. 51, no. 7, pp. 17–21. (in Russian)
15. Omorov R.O., Akunov T.A. Robustness of interval dynamic systems: stability and ellipsoid quality estimates. Automation and Control Problems, 2021, no. 3(42), pp. 47–57. (in Russian)
16. Andrievskii B.R., Fradkov A.L. Elements of Mathematical Modeling in MATLAB 5 and Scilab Computing Environments. St. Petersburg, Nauka Publ., 2001, 286 p. (in Russian)

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License