ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Proof of the Stabilization Theorem for a Linear Dynamic System of Unknown Order with Variable Parameters


Boris Vasilievich Ulanov

Togliatti State University,
Dotsent of the Department of Higher Mathematics and Mathematical Education,
Togliatti, Belorusskaya Street, 14
Candidate of physical and mathematical sciences, dotsent


The proof of the stabilization theorem for a linear dynamic system described by one differential equation of unknown order which does not exceed a given natural number, and with parameters that vary within any given limits is presented. For the synthesis of the control, it is necessary to know the limits within which the parameters of the differential equation and all derivatives of these parameters (up to the order less by one than a given natural number) are changed.



  1. Ulanov, B. V. [Control of dynamic systems with incomplete information about their parameters, state, dimension]. Doklady Akademii Nauk SSSR, 1989; V. 308, № 4. P. 803-806. (In Russ. )
  2. Ulanov, B. V. [On the control of non-stationary dynamic objects with unknown dimension]. Izvestiya vuzov. Matematika, 1990; №6. P. 83-85. (In Russ. )
  3. Kalman, R., Falb, P., Arbib, M. Ocherki po matematicheskoy teorii system [Topics in mathematical system theory]. Moscow, Mir Publ., 1971. 398 p. (In Russ. )
  4. Furasov, V. D. Ustoychivost' dvizheniya, otsenki i stabilizatsiya [Stability of movement, assessment and stabilization]. Moscow, Nauka Publ., 1977. 247 p. (In Russ. )
  5. Fomin, V. N., Fradkov, A. L., Yakubovich, V. A. Adaptivnoye upravleniye dinamicheskimi ob" yektami [Adaptive control of dynamic systems]. Moscow, Nauka Publ., 1981. 448 p.
  6. Kalman, R. E. A New Approach to Linear Filtering and Prediction Problems. Trans. ASME. 1960. Vol. 82, Ser. D. P. 35-45.
  7. Kalman, R. Ye., B'yusi R. New Results in Linear Filtering and Prediction Theory. Trans. ASME. 1961. Vol. 83, Ser. D. P. 95-108.
  8. Krasovskiy, N. N. [On the stabilization of unstable movements by additional forces with incomplete feedback]. Prikl. mat. i mekh. 1963; V. 27, Issue 4. P. 641-663. (In Russ. )
  9. Luyenberger, D. G. Observing the State of Linear System. IEEE Trans. Mil. Electron. 1964. Vol. Mil-8, №1. P. 74-80.
  10. Luyenberger, D. G. An Introduction to Observers. IEEE Trans. on Automatic Control. 1971. Vol. AC-16, №6. P. 596-602.
  11. Brash, F. M., Pirson, Dzh. B. Pole Placement Using Dynamic Compensator. IEEE Trans. on Automatic Control. 1970. Vol. AC-15, №1. P. 34-43.
  12. Brayson, A., Kho, Yu-shi. Prikladnaya teoriya optimal'nogo upravleniya [Applied theory of optimal control]. Moscow, Nauka Publ., 1972. 544 p. (In Russ. )
  13. Uonem, M. Lineynyye mnogomernyye sistemy upravleniya: Geometricheskiy podkhod [Linear multivariable control: a geometric approach]. Moscow, Nauka Publ., 1980. 375 p. (In Russ. )
  14. Gantmakher, F. R. Teoriya matrits [Matrix theory]. Moscow, Nauka Publ., 1966. 576 p. (In Russ. )

Full text (pdf)