Stabilization of a Class of Uncertain Systems

  • M. S. ZakharenkovEmail author


We consider the problem to synthesize a stabilizing control u synthesis for systems \(\frac{{dx}}{{dt}} = Ax + Bu\) where A ∈ ℝn×n and B ∈ ℝn×m, while the elements αi,j(·) of the matrix A are uniformly bounded nonanticipatory functionals of arbitrary nature. If the system is continuous, then the elements of the matrix B are continuous and uniformly bounded functionals as well. If the system is pulse-modulated, then the elements of the matrix B are differentiable uniformly bounded functions of time. It is assumed that k isolated uniformly bounded elements \({\alpha _{{i_l},{j_l}}}\left( \cdot \right)\) satisfying the condition \(\mathop {\inf }\limits_{\left( \cdot \right)} \left| {{\alpha _{{i_l},{j_l}}}\left( \cdot \right)} \right|{\alpha _ - } > 0,\quad l \in \overline {1,k}\) are located above the main diagonal of the matrix A(·), where Gk is the set of all isolated elements of the system, J1 is the set of indices of rows of matrix A(·) containing isolated elements, and J2 is the set of indices of its rows free of isolated elements. It is assumed that other elements located above the main diagonal are sufficiently small provided that their row indices belong to J1, i.e., \(\mathop {\sup }\limits_{\left( \cdot \right)} \left| {{\alpha _{i,j}}\left( \cdot \right)} \right| < \delta ,\quad {\alpha _{i,j}} \notin {G_k},\quad i \in {J_1},\quad j > i\). All other elements located above the main diagonal are uniformly bounded. The relation u = S(·)x is satisfied in the continuous case, while the relation u = ξ(t) is satisfied in the pulse-modulated case; here the components of the vector ξ are outputs of synchronous pulse elements. Constructing a special quadratic Lyapunov function, one can determine a matrix S(·) such that the closed system becomes globally exponentially stable in the continuous case. In the pulse-modulated case, input pulses are synthesized such that the system becomes globally asymptotically stable.


uncertain pulse-modulated systems stabilization of uncertain systems 


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  1. 1.
    O. Kwon and J. H. Park, “Matrix inequality approach to a novel stability criterion for time-delay systems with nonlinear uncertainties,” J. Optim. Theory Appl. 126, 643–656 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    W. Qian, S. Cong, Y. Sun, and S. Fei, “Novel robust stability criteria for uncertain systems with time-varying delay,” Appl. Math. Comput. 215, 866–872 (2009).MathSciNetzbMATHGoogle Scholar
  3. 3.
    J. Li, Ch. Qian, and Sh. Ding, “Global finite-time stabilization by output feedback for a class of uncertain nonlinear systems,” Int. J. Control 83, 2241–2252 (2010).CrossRefzbMATHGoogle Scholar
  4. 4.
    L. Liu and J. Huang, “Global robust output regulation of output feedback systems with unknown high-frequency gain sign,” IEEE Trans. Autom. Control 51, 625–631 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ju. Zhai, W. Li, and Sh. Fei, “Global output feedback stabilization for a class of uncertain non-linear systems,” IET Control Theory Appl. 7, 305–313 (2013).MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Zakharenkov, I. Zuber, and A. Gelig, “Stabilization of a new classes of uncertain systems,” IFAC Proc. 48, 1034–1037 (2015).Google Scholar
  7. 7.
    I. E. Zuber, T. V. Voloshinova, and A. Kh. Gelig, “An extended class of stabilizable uncertain systems,” Vestn. St. Petersburg Univ.: Math. 49, 238–242 (2016). Scholar
  8. 8.
    A. Kh. Gelig and I. E. Zuber, “Using the direct and indirect control to stabilize some classes of uncertain systems. II. Pulse and discrete systems,” Autom. Remote Control 73, 1498–1510 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. E. Andeen, “The principle of equivalent areas,” Trans. Am. Inst. Electr. Eng., Part 2 79, 332–336 (1960).Google Scholar
  10. 10.
    A. Kh. Gelig and A. N. Churilov, Stability and Oscillations of Nonlinear Pulse-Modulated Systems (S.-Peterb. Gos. Univ., St. Petersburg, 1993; Birkhäuser, Boston, 1998), p. 362.CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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