Erratum to: Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent ti
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E R R AT U M
Erratum to: Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays Huabin Chen · Chuanxi Zhu · Peng Hu · Yong Zhang
© Springer Science+Business Media B.V. 2012
Erratum to: Nonlinear Dyn (2012) 69:1023–1039 DOI 10.1007/s11071-012-0324-3
1 Introduction
In Ref. [2] [Nonlinear Dyn. 69:1023–1039 (2012)], since the Lyapunov functionals V2 (t, xt , i) and V3 (t, xt , i) are inappropriate, the proofs of Theorems 3.1– 3.2 and Theorems 4.1–4.2 need some important improvements. Besides, the results of Examples 5.1–5.3 are necessarily modified.
In [2] [Nonlinear Dyn. 69:1023–1039 (2012)], the authors mainly studied the robustly exponential stabilization for the uncertain Markovian jump systems with mode-dependent time-varying state delays. Firstly, by constructing a modified Lyapunov functional and using the free-weighting matrices technique, some delay-dependent robustly exponential stability criteria on such systems can be obtained in terms of linear matrix inequalities (LMIs). Then, based on the obtained stability criteria, a state-feedback controller is designed, which can guarantee the robustly exponential stability of the uncertain closed-loop systems. In [2] [Nonlinear Dyn. 69:1023–1039 (2012)], although the exponential stability of such systems can be obtained by constructing the following Lyapunov functional:
The online version of the original article can be found under doi:10.1007/s11071-012-0324-3. H. Chen () · C. Zhu Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi Province, P.R. China e-mail: [email protected] C. Zhu e-mail: [email protected] P. Hu School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, Hubei Province, P.R. China e-mail: [email protected] Y. Zhang College of Information Science and Technology, Wuhan University of Science and Technology, Wuhan 430081, Hubei Province, P.R. China e-mail: [email protected]
V (t, xt , i) = V1 (t, xt , i) + V2 (t, xt , i) + V3 (t, xt , i), where V1 (t, xt , i) = eκt x T (t)Pi x(t), V2 (t, xt , i) t = eκ(s+h) x T (s)Q1 x(s) ds t−hi (t)
+
t
t−h
eκ(s+h) x T (s)Q1i x(s) ds,
H. Chen et al.
V3 (t, xt , i) 0 =
t
−h t+θ
+
0
eκ(s−θ) x T (s)(Q2 + Q3 )x(s) ds dθ t
−h t+θ
eκ(s−θ) x˙ T (s)Q4 x(s) ˙ ds dθ,
the state-feedback controller is very difficult to design since there are two important problems to be considered. Firstly, the inequalities in (4.1) of [2] [Nonlinear Dyn. 69:1023–1039 (2012)] are bilinear matrix inequalities, in which the bilinear constraints are not convex and are very hard to be solved. Thus, there is a confusion in Example 5.3 [Nonlinear Dyn. 69:1023– 1039 (2012)]; secondly, in order to design the statefeedback controller, two variables Ri = XiT Q1 Xi ,
(1.1)
and Ti = XiT Q2 Xi ,
(1.1)
defined in Theorem 4.1 of [2] [Nonlinear Dyn. 69:1023– 1039 (2012)] are not rational, where Xi , Ri and Ti are mode-dependent design parameters. These two equalities are impossible for each
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