A fuzzy clustering based technique for piecewise affine approximation of a class of nonlinear systems

In recent years piecewise affine (PWA) modeling has developed as an attractive tool for the approximation of various complex nonlinear systems. In spite of the wide application of PWA modeling, the optimal approximation of a continuous time nonlinear system with scalar functions by the minimum number of affine systems has not been addressed properly in literature. This paper deals with a fuzzy clustering based approach for the optimal PWA approximation of a class of continuous time nonlinear systems. The technique is based on the trade-off between increasing the approximation accuracy of the various nonlinear functions and simplifying the approximation by the minimum number of subsystems. As an application, the technique is utilized to obtain a PWA approximation of the glucose regulation system. Numerical simulations depicted that, for a given number of subsystems, the derived glucose regulation model provides an optimal approximation of the original nonlinear system. The model also provided some biological insight about the interactions involved in glucose regulation.     

Analytical approximation of weakly nonlinear continuous systems using renormalization group method

A direct method based on renormalization group method (RGM) is proposed for determining the analytical approximation of weakly nonlinear continuous systems. To demonstrate the application of the method, we use it to analyze some examples. First, we analyze the vibration of a beam resting on a nonlinear elastic foundation with distributed quadratic and cubic nonlinearities in the cases of primary and subharmonic resonances of the nth mode. We apply the RGM to the discretized governing equation and also directly to the governing partial differential equations (PDE). The results are in full agreement with those previously obtained with multiple scales method. Second, we obtain higher order approximation for free vibrations of a beam resting on a nonlinear elastic foundation with distributed cubic nonlinearities. The method is applied to the discretized governing equation as well as directly to the governing PDE. The proposed method is capable of producing directly higher order approximation of weakly nonlinear continuous systems. It is shown that the higher order approximation of discretization and direct methods are not in general equal. Finally, we analyze the previous problem in the case that the governing differential equation expressed in complex-variable form. The results of second order form and complex-variable form are not in agreement. We observe that in use of RGM in higher order approximation of continuous systems, the equations must not be treated in second order form.     

Robust tube-based MPC of constrained piecewise affine systems with bounded additive disturbances

In this article, we propose a robust tube-based MPC formulation for a class of hybrid systems, namely autonomously switched PWA systems, with bounded additive disturbances. The term tube-based refers to those control techniques whose objective is to maintain all possible trajectories of the uncertain system inside a tube which is a set around the nominal (or reference) system trajectory, that is free from disturbances. Common methods in tube-based control systems consider an error dynamical system as the difference between the state of the nominal system and the state of the perturbed system. However, this definition of the error dynamical system leads to a complicated switched affine system for PWA systems. Therefore, we use a new notion of the reference system similar to the nominal system except that the switching between the various modes of the PWA system is driven by the state of the real system. Using this reference system instead of the nominal system leads us to an error dynamical system that can be modeled as a switched linear system. We employ a switched linear controller to stabilize this error system under arbitrary switching. This auxiliary controller forces the states of the uncertain system to remain in a tube confined to the invariant set around the state of the reference system. We add new constraints and tighten some other constraints of the nominal hybrid MPC for the reference system, in order to ensure convergence of the uncertain system and to guarantee robust exponential stability of the closed-loop system.     

Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models: a new nonlinear model and the Bhatnagar-Gross-Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar-Gross-Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases: the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar-Gross-Krook model.     

Numerical analysis to discontinuous Galerkin methods for the age structured population model of marine invertebrates

In this article we consider the age structured population growth model of marine invertebrates. The problem is a nonlinear coupled system of the age‐density distribution of sessile adults and the abundance of larvae. We propose the semidiscrete and fully‐discrete discontinuous Galerkin schemes to the nonlinear problem. The DG method is well suited to approximate the local behavior of the problem and to easily take the locally refined meshes with hanging nodes adaptively. The simple communication pattern between elements makes the DG method ideal for parallel computation. The global existence of the approximation solution is proved for the nonlinear approximation system by using the broken Sobolev spaces and the Schauder's fixed point theorem, and error estimates are obtained for both the semidiscrete scheme and the fully‐discrete scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

Numerical solution of a nonlocal problem arising in plasma physics

In this paper, we present a numerical method to approximate the equilibrium of a plasma magnetically confined in a stellarator device. We consider a two-dimensional nonlocal free-boundary problem which involves both relative and decreasing rearrangements. We use a finite-element discretization to the differential operator and an iterative algorithm in the nonlinear terms. An approximation scheme for the computation of the rearrangement has been implemented and tested. Finally we give numerical results for an helical system.     

The use of stochastic analytic center for yield maximization of systems with general distributions of component values

This paper presents a new method for maximizing manufacturing yield when the realizations of system components are dependent random variables with general distributions. The method uses a new concept of stochastic analytic center introduced herein to design the unknown parameters of component values. Design specifications define a feasible region which, in the nonlinear case, is linearized using a first-order approximation. The resulting problem becomes a convex optimization problem. Monte Carlo simulation is used to evaluate the actual yield of the optimal designs of a tutorial example.     

Numerical method for solving diffusion-wave phenomena

We find solutions for the diffusion-wave problem in 1D with n-term time fractional derivatives whose orders belong to the intervals (0,1),(1,2) and (0,2) respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. This method transfers the diffusion-wave problem into the corresponding infinite system of linear algebraic equations through the coefficients, which are uniquely solvable under some relations between the coefficients with index zero.The method is applicable for nonlinear problems too.     

Greedy expansions in convex optimization

This paper is a follow-up to the author’s previous paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there the three most popular greedy algorithms in nonlinear approximation in Banach spaces-Weak Chebyshev Greedy Algorithm, Weak Greedy Algorithm with Free Relaxation, and Weak Relaxed Greedy Algorithm-for solving convex optimization problems. We continue to study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. In this paper we concentrate on greedy algorithms that provide expansions, which means that the approximant at the mth iteration is equal to the sum of the approximant from the previous, (m ? 1)th, iteration and one element from the dictionary with an appropriate coefficient. The problem of greedy expansions of elements of a Banach space is well studied in nonlinear approximation theory. At first glance the setting of a problem of expansion of a given element and the setting of the problem of expansion in an optimization problem are very different. However, it turns out that the same technique can be used for solving both problems. We show how the technique developed in nonlinear approximation theory, in particular, the greedy expansions technique, can be adjusted for finding a sparse solution of an optimization problem given by an expansion with respect to a given dictionary.     

Optimal convergence rates to nonlinear diffusion waves for the compressible Euler-Poisson system with damping

In this work, we study the 1-D isentropic bipolar hydrodynamic model. This model takes the form of compressible Euler-Poisson system with nonlinear damping added to the momentum equations. Under some smallness conditions, the solutions to the Cauchy problem of the system globally exist and convergence to the nonlinear diffusion waves, which are the corresponding solutions of nonlinear parabolic equations given by the Darcy's law with a specified initial data. The optimal convergence rates are obtained by Green function method when the initial perturbation is in L¹-space.     

ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR A PHASE FIELD BENDING ELASTICITY MODEL OF VESICLE MEMBRANE DEFORMATION

In this paper, we study numerical approximations of a recently proposed phase fieldmodel for the vesicle membrane deformation governed by the variation of the elastic bend-ing energy. To overcome the challenges of high order nonlinear differential systems and thenonlinear constraints associated with the problem, we present the phase field bending elas-ticity model in a nested saddle point formulation. A mixed finite element method is thenemployed to compute the equilibrium configuration of a vesicle membrane with prescribedvolume and surface area. Coupling the approximation results for a related linearized prob-lem and the general theory of Brezzi-Rappaz-Raviart, optimal order error estimates for thefinite element approximations of the phase field model are obtained. Numerical results areprovided to substantiate the derived estimates.     

Design of robust sub-optimal nonlinear H∞ controllers with some applications

The problem of nonlinear sub-optimal H_∞ controller design with some applications is addressed in the paper. Nonlinear H_∞ control has robust performance in response to external disturbances and parameter uncertainty as well as capability in dealing with nonlinear systems. In order to obtain the nonlinear H_∞ control law, some partial differential inequalities so-called Hamilton–Jacobi–Isaacs (HJI) should be solved. There are some approximate solutions, which are generally based on the approximation of nonlinear parts of HJI inequalities. Using the Taylor series expansion, a sub-optimal solution for the HJI inequalities will be obtained. To assess the performance of the method, two applications are considered: the tracking problem of a two-degree-of-freedom rigid robot manipulator and speed control in a permanent magnet synchronous (PMS) motors. Simulation results show superior performance for higher order approximate controllers than that of lower order ones.     

Model reduction for large-scale dynamical systems via equality constrained least squares

In this paper, we present a new method of model reduction for large-scale dynamical systems, which belongs to the SVD-Krylov based method category. It is a two-sided projection where one side reflects the Krylov part and the other side reflects the SVD (observability gramian) part. The reduced model matches the first r+i Markov parameters of the full order model, and the remaining ones approximate in a least squares sense without being explicitly computed, where r is the order of the reduced system, and i is a nonnegative integer such that 1≤i<r. The reduced system minimizes a weighted ?₂ error. By the definition of a shift operator, the proposed approximation is also obtained by solving an equality constrained least squares problem. Moreover, the method is generalized for moment matching at arbitrary interpolation points. Several numerical examples verify the effectiveness of the approach.     

Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation

We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution. In this paper we study two specific examples: the Suliciu-type and the Kerr-Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr-Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.     

A numerical approach to solve the model for HIV infection of CD4+T cells

In this study, we will obtain the approximate solutions of the HIV infection model of CD4⁺T by developing the Bessel collocation method. This model corresponds to a class of nonlinear ordinary differential equation systems. Proposed scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are computed using the matrix operations of derivatives together with the collocation method. The reliability and efficiency of the proposed approach are demonstrated in the different time intervals by a numerical example. All computations have been made with the aid of a computer code written in Maple 9.     

Robust L ∞ reliable control for impulsive switched nonlinear systems with state delay

This paper investigates the problem of robust L _∞ reliable control for a class of uncertain impulsive switched nonlinear systems with time-delay in the presence of actuator failure. Based on the dwell time approach, we firstly obtain a sufficient condition of exponential stability for the impulsive switched nonlinear system with time-delay, and L _∞ performance for the considered system is also analyzed. Then, based on above results, a state feedback controller, which guarantees the exponential stability with L _∞ performance of the corresponding closed-loop system, is constructed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

基于Fuzzy推理的时变系统建模

提出一种基于Fuzzy推理的时变系统建模方法,其基本思想是:对时间维度进行分割,在每个较短的时间间隔内用时不变模型代替时变模型,将这些时不变模型组合在一起,最终获得一个整体非线性时变的微分方程模型.分别研究了输入输出型时变系统和状态空间型时变系统的模型建立方法,除了从理论上保证了所获得的模型对系统的逼近性,还从仿真实验验证了用该方法建立的模型对非线性时变系统有很好的逼近效果.     

New approach for the nonlinear programming with transient stability constraints arising from power systems

This paper presents a new approach for solving a class of complicated nonlinear programming problems arises from optimal power flow with transient stability constraints (denoted by OTS) in power systems. By using a functional transformation technology proposed in Chen et al. (IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48:327–339, [2001]), the OTS problem is transformed to a semi-infinite programming (SIP). Then based on the KKT (Karush-Kuhn-Tucker) system of the reformulated SIP problem and the finite approximation technology, an iterative method is presented, which develops Wu-Li-Qi-Zhou’ (Optim. Methods Softw. 20:629–643, [2005]) method. In order to save the computing cost, some typical computing technologies, such as active set strategy, quasi-Newton method for the subproblems coming from the finite approximation model, are addressed. The global convergence of the proposed algorithm is established. Numerical examples from power systems are tested. The computing results show the efficiency of the new approach.