Averaging method for impulsive differential inclusions with fuzzy right-hand side

In this paper the substantiation of the partial scheme of the averaging method for impulsive differential inclusions with fuzzy right-hand side in terms of R - solutions on the finite interval is considered.Consider the impulsive differential inclusion with the fuzzy right-hand side $$\dot x \in \varepsilon F(t,x) ,\ t \not= t_i,\ x(0)\in X_0,\quad\Delta x \mid _{t=t_i} \in \varepsilon I_i (x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)$$ where $t\in \mathbb{R}_+ $ is time, $x \in \mathbb{R}^n $ is a phase variable, $\varepsilon > 0 $ is a small parameter,$ F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n,$ $I_i \colon \mathbb{R}^n \to \mathbb{E}^n $ are fuzzy mappings, moments $t_i$ are enumerated in the increasing order.Associate with inclusion (1) the following partial averaged differential inclusion $$\dot\xi \in \varepsilon \widetilde F (t, \xi ),\ t \not= s_j ,\ \xi (0) \in X_0,\quad \Delta \xi \vert _{t=s_j} \in \varepsilon K_j (\xi ),\qquad\qquad\qquad\qquad\qquad\qquad\quad (2),$$ where the fuzzy mappings $ \widetilde F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n ; \quad K_j \colon \mathbb{R} \to \mathbb{E}^n $ satisfy the condition $$\lim _{T \to \infty } \frac 1T D \Big( \int\limits_t^{t+T} F(t,x) dt + \sum_{t \leq t_i < t+T} I_i (x),\int\limits_t^{t+T} \widetilde F(t,x)dt +\sum_{t \leq s_j < t+T} K_j (x) \Big) = 0,\quad\quad (3)$$ moments $s_j$ are enumerated in the increasing order. In the paper is proved the following main theorem:{\sl Let in the domain $ Q = \lbrace t \geq 0 , x \in G\subset \mathbb{R}^n \rbrace $ the following conditions fulfill:$1)$ fuzzy mappings $ F (t,x), \widetilde F(t,x), I_i(x),K_j(x) $are continuous, uniformly bounded with constant $M$, concave in $x,$ satisfy Lipschitz condition in $x$ with constant $ \lambda ;$$2)$ uniformly with respect to $t, x$ limit (3) exists and $\frac 1T i(t,t+T) \leq d < \infty ,\ \frac 1T j(t,t+T) \leq d < \infty,$where $i(t,t+T)$ and $j(t,t+T)$ are the quantities of impulse moments $t_i$ and $s_j$ on the interval$ [ t, t+T ] $;$3)$ {\rm R}-solutions of inclusion (2) for all $ X_0 \subset G^{\prime} \subset G $for $ t \in [0,L^{\ast} \varepsilon ^{-1} ] $ belong to the domain $G$ with a $ \rho $- neighborhood.Then for any $\eta > 0 $ and $L \in (0,L^{\ast}]$ there exists $\varepsilon _0 (\eta,L) \in (0,\sigma ] $ such that for all $\varepsilon \in (0, \varepsilon _0 ]$ and $t \in [0,L \varepsilon ^{-1}] $ the inequality holds:$D(R(t, \varepsilon ), \widetilde R (t, \varepsilon)) < \eta,$ where $R(t, \varepsilon), \widetilde R(t, \varepsilon ) $ are the {\rm R-} solutions of inclusions (1) and (2), $R(0, \varepsilon ) = \widetilde R (0, \varepsilon).$

On representation of functions that satisfy Lipschitz condition as convolution of functions from Lorentz spaces

Any $2\pi$-periodic function from the Lipschitz space $\Lambda_b^{\alpha}$ can be represented by way of the convolution of the functions from the Lorentz spaces $L_{p,r}$ and $L_{b,r'}$ in the case when $1 \leqslant b < \infty$, $0 < 1 - p^{-1} < \alpha < 1$ and the numbers $r$, $r'$ are picked in the corresponding way.

Fault-tolerant controller design with fault estimation capability for a class of nonlinear systems using generalized Takagi-Sugeno fuzzy model

In this paper, a new design approach to construct a fault-tolerant controller (FTC) with fault estimation capability is proposed using a generalized Takagi-Sugeno (T-S) fuzzy model for a class of nonlinear systems in the presence of actuator faults and unknown disturbances. The generalized T-S fuzzy model consists of some local models with multiplicative nonlinear terms that satisfy Lipschitz condition. Besides covering a very wide range of nonlinear systems with a smaller number of local rules in comparison with the conventional T-S fuzzy model and hence having less computational burden, the existence of the multiplicative nonlinear term solves the uncontrollability issues that the other generalized T-S fuzzy models with additive nonlinear terms dealt with. A state/fault observer designed for the considered generalized T-S fuzzy model and then, a dynamic FTC law based on the estimated fault information is proposed and sufficient design conditions are given in terms of linear matrix inequalities (LMIs). It can be shown that the number of LMIs are less than that of previously proposed methods and then feasibility of our method is more likely. The effectiveness of the proposed FTC approach is verified using a nonlinear mass-spring-damper system.

ROBUST H∞ FILTERING FOR A CLASS OF NONLINEAR UNCERTAIN SINGULAR SYSTEMS WITH TIME-VARYING DELAY

This paper focuses on the problem of robust H∞ filtering for a class of nonlinear uncertain singular systems with time-varying delay. First of all, the definition of robust H∞ filter is given. Considering the nonlinear disturbance link to uncertain singular systems with time-varying delay effects, the design idea of full-order robust H∞ filter based on the Lyapunov stability theory is presented. Under the condition that nonlinear uncertain functions satisfy Lipschitz condition, the sufficient condition under which nonlinear uncertain delayed filtering error singular systems are asymptotically stable and satisfy the robust H∞ performance is obtained by Lyapunov stability theory and linear matrix inequality (LMI) methods. Finally, two numerical examples are given to show the applicability of the proposed method.

LMI Enhanced Observer Design for Lipschitz Nonlinear Systems

Observer design for nonlinear systems has been an important and complex issue for decades. In this paper, considering a class of nonlinear systems which satisfy Lipschitz condition, a method for observer design is investigated based on Linear Matrix Inequality (LMI). This study focuses on the selection of gain matrices using LMI for two kinds of Lipschitz nonlinear systems, which are classified by the relationship between output and state. Simulation studies are made with Matlab/Simulink in this paper, and the simulation results verify the effectiveness of the proposed method.