# Stabilization of a Class of Uncertain Systems

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## Abstract

We consider the problem to synthesize a stabilizing control *u* synthesis for systems \(\frac{{dx}}{{dt}} = Ax + Bu\) where *A* ∈ ℝ^{n×n} and *B* ∈ ℝ^{n×m}, while the elements α_{i,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 *G*_{ k } is the set of all isolated elements of the system, *J*_{1} is the set of indices of rows of matrix *A*(·) containing isolated elements, and *J*_{2} 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 *J*_{1}, 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.

## Keywords

uncertain pulse-modulated systems stabilization of uncertain systems## Preview

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## References

- 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.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.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.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.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.M. Zakharenkov, I. Zuber, and A. Gelig, “Stabilization of a new classes of uncertain systems,” IFAC Proc.
**48**, 1034–1037 (2015).zbMATHGoogle Scholar - 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). https://doi.org/10.3103/S1063454116030158.MathSciNetCrossRefzbMATHGoogle Scholar - 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.R. E. Andeen, “The principle of equivalent areas,” Trans. Am. Inst. Electr. Eng., Part 2
**79**, 332–336 (1960).Google Scholar - 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