Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 360–366 | Cite as

Synthesis of Stabilization Control on Outputs for a Class of Continuous and Pulse-Modulated Undefined Systems

  • I. E. ZuberEmail author
  • A. Kh. Gelig


Consider system
$$\left\{ {\begin{array}{*{20}{c}} {{{\dot x}_1} = {\varphi _1}(.) + {\rho _1}{x_{l + 1}},} \\ {{{\dot x}_m} = {\varphi _m}(.) + {\rho _m}{x_n},} \\ {{{\dot x}_{m + 1}} = {\varphi _{m + 1}}(.) + {\mu _1},} \\ {{{\dot x}_n} = {\varphi _n}(.) + {\mu _1},} \end{array}} \right.$$
where x1, …, and xn is the state of the system, u1, …, and ul are controls, n/l is not an integer, and l ≥ 2. It is supposed that only outputs x1, …, and xl are measurable, (l > n) and ϕi(·) are non-anticipating arbitrary functionals, and 0 < ρ–≤ ρi (t, x1, …, and xl) ≤ ρ+. Using the backstepping method, we construct the square Lyapunov function and stabilize the control for the global exponential stability of the closed loop system. The stabilization by means of synchronous modulators with a sufficiently high impulsion frequency is considered as well.


uncertain systems output stabilization global exponential stability 


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  1. 1.
    R. Jia C. Qian and J. Zhai “Semi-global stabilisation of uncertain non-linear systems by homogeneous output feedback controllers,” IET Control Theory Appl. 6, 165–172 (2012).MathSciNetCrossRefGoogle Scholar
  2. 2.
    J.-Y. Zhai W.-G. Li and S.-M. Fei “Global output feedback stabilization for a class of uncertain non-linear systems,” IET Control Theory Appl. 7, 305–313 (2013).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Y. Man and Y. Liu “Global output-feedback stabilization for a class of uncertain time-varying nonlinear systems,” Syst. Control Lett. 90, 20–30 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I. E. Zuber and A. Kh. Gelig, “Stabilization by output of continuous and pulse-modulated uncertain systems,” Vestn. St. Petersburg Univ.: Math. 50, 342–348 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Kh. Gelig and I. E. Zuber “Stabilization by multi-dimensional output for certain class of undefined systems,” Avtom. Remote Control 79, 1545–1557 (2018).CrossRefzbMATHGoogle Scholar
  6. 6.
    V. A. Yakubovich G. A. Leonov and A. Kh. Gelig, Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities (Nauka, Moscow, 1978; World Sci., London, 2004).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia
  2. 2.St. Petersburg State UniversitySt. PetersburgRussia

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