Submanifolds of dimension n-1 in [epsilon]n̳ with normals satisfying a Lipschitz condition.

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H∞ observer-based control for discrete-time one-sided Lipschitz systems with unknown inputs

This paper investigates the problem of robust observer-based stabilization for a class of one-sided nonlinear discrete-time systems subjected to unknown inputs. We propose a simple simultaneous state and input estimator. A nonlinear controller is then proposed to compensate for the effects of unknown inputs and to ensure asymptotic stability in a closed loop. Several mathematical artifacts are used to deduce stability conditions expressed in terms of linear matrix inequalities. To show high performances of the proposed technique, a relevant example is provided with comparisons to recent results.

Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay

In this paper, we address the problem of unknown input observer design, which simultaneously estimates state and unknown input, of a class of nonlinear discrete-time systems with time-delay. A novel approach to the state estimation problem of nonlinear systems where the nonlinearities satisfy the one-sided Lipschitz and quadratically inner-bounded conditions is proposed. This approach also allows us to reconstruct the unknown inputs of the systems. The nonlinear system is first transformed to a new system which can be decomposed into unknown-input-free and unknown-input-dependent subsystems. The estimation problem is then reduced to designing observer for the unknown-input-free subsystem. Rather than full-order observer design, in this paper, we propose observer design of reduced-order which is more practical and cost effective. By utilizing several mathematical techniques, the time-delay issue as well as the bilinear terms, which often emerge when designing observers for nonlinear discrete-time systems, are handled and less conservative observer synthesis conditions are derived in the linear matrix inequalities form. Two numerical examples are given to show the efficiency and high performance of our results.

Observer design for one-sided Lipschitz discrete-time systems subject to delays and unknown inputs

In this paper, we address the problem of observer design for a class of nonlinear discrete-time systems in the presence of delays and unknown inputs. The nonlinearities studied in this work satisfy the one-sided Lipschitz and quadratically inner-bounded conditions which are more general than the traditional Lipschitz conditions. Both H∞ observer design and asymptotic observer design with reduced-order are considered. The designs are novel compared to other relevant nonlinear observer designs subject to time delays and disturbances in the literature. In order to deal with the time-delay issue as well as the bilinear terms which usually appear in the problem of designing observers for discrete-time systems, several mathematical techniques are utilized to deduce observer synthesis conditions in the linear matrix inequalities form. A numerical example is given to demonstrate the effectiveness and high performance of our results.

Robust observer design for a class of one-sided Lipschitz systems subject to time delays and disturbances in both plants and outputs

In this paper, we consider the problem of designing state observers for a class of one-sided Lipschitz time-delay systems subject to disturbances. Not only the plants but also the outputs of the systems are subject to time delays and disturbances. The one-sided Lipschitz and quadraticallyinner-bounded conditions, which are more general than the traditional Lipschitz condition, are utilized to cover a broader class of nonlinearities. Based on the Lyapunov-Krasovskii approach, a robust observer design is proposed where synthesis condition is derived in terms of linear matrixinequality (LMI). A numerical example is given to show the effectiveness of the obtained results.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Approximate Controllability Of Semilinear Systems Using Integral Contractors

In this article, a non-autonomous (time-varying) semilinear system is considered and its approximate controllability is investigated. The notion of 'bounded integral contractor', introduced by Altman, has been exploited to obtain sufficient conditions for approximate controllability. This condition is weaker than Lipschitz condition. The main theorems of Naito [11, 12] are obtained as corollaries of our main results. An example is also given to show how our results weaken the conditions assumed by Sukavanam[17].

Sliding mode-like learning control for SISO complex systems with T-S fuzzy models

In this paper, a sliding mode-like learning control scheme is developed for a class of single input single output (SISO) complex systems. First, the Takagi-Sugeno (T-S) fuzzy modelling technique is employed to model the uncertain complex dynamical systems. Second, a sliding mode-like learning control is designed to drive the sliding variable to converge to the sliding surface, and the system states can then asymptotically converge to zero on the sliding surface. The advantages of this scheme are that: 1) the information about the uncertain system dynamics and the system model structure is not required for the design of the learning controller; 2) the closed-loop system behaves with a strong robustness with respect to uncertainties; 3) the control input is chattering-free. The sufficient conditions for the sliding mode-like learning control to stabilise the global fuzzy model are discussed in detail. A simulation example for the control of an inverted pendulum cart is presented to demonstrate the effectiveness of the proposed control scheme.

A robust learning control for SISO Nonlinear systems with T-S Fuzzy Model : C02-robust control

In this paper, a robust learning control is developed for a class of single input single output (SISO) nonlinear systems with T-S fuzzy model. It is seen that the proposed sliding mode learning control with the powerful Lipshitz-like condition can guarantee the stability, convergence and robustness of the closed-loop system without involving any assumptions on uncertain system dynamics. In addition, theconcept that the local system with the maximum membership function dominates the system dynamic behaviours helps to greatly simplify the control system design. It will be further seen that the continuous learning control ensures the advantage of chattering-free that may occur in conventional sliding mode systems. Simulation examples are presented to demonstrate the effectiveness of the proposed learning control through the comparison with the H-infinity control.

Bi-Lipschitz A-triviality of map germs and Newton filtrations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Bi-Lipschitz G-triviality and Newton polyhedra, G = R, C, K, R-V, C-V, K-V

In this work we provide estimates for the bi-Lipschitz G-triviality, G = C or K, for a family of map germs satisfying a Lojasiewicz condition. We work with two cases: the class of weighted homogeneous map germs and the class of non-degenerate map germs with respect to some Newton polyhedron. We also consider the bi-Lipschitz triviality for families of map germs defined on an analytic variety V. We give estimates for the bi-Lipschitz G(V)-triviality where G = R,C or K in the weighted homogeneous case. Here we assume that the map germ and the analytic variety are both weighted homogeneous with respect to the same weights. The method applied in this paper is based in the construction of controlled vector fields in the presence of a suitable Lojasiewicz condition. In the last section of this work we compare our results with other results related to this work showing tables with all estimates that we know, including ours.

Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation

We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type partial derivative u/partial derivative n + g( x, u) = 0 when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by partial derivative u/partial derivative n+gamma(x) g(x, u) = 0, where gamma(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary,. is a bounded function and we show the convergence of the solutions in H-1 and C-alpha norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.