Numerical solution of the problem of unsteady supersonic flow around the front part of the wings with a detached shock wave

The object of this paper is to describe a numerical method for the solution of the problem of unsteady supersonic flow around the front part of the wings with a detached shock wave. The planform and the airfoil of the wings must be constructible but otherwise arbitrary. The velocity vector of the free stream is directed in an arbitrary direction but without flow separation.The problem is formulated for the three-dimensional, hyperbolic system of equations of unsteady, inviscid, chemically reactive gas flow in a mixed subsonic and supersonic domain. It is solved by a finite difference, implicit, second-order method with the use of the time-dependent stationing principle.A curvilinear coordinate system is introduced in which the numerical domain becomes a simple rectangle. The coordinate system is time-dependent and adapted for the planform of the wing and the free-stream parameters. The planform of the wind and the velocity vector of the free stream may be changed in the time-dependent stationing process. The flow field is computed around the front part of wings which have an elliptical planform and an aspect ratio λ ? 2 for selected values of M_æ between 1.5 and 5.     

Further results on the L1 analysis of sampled-data systems via kernel approximation approach

This paper gives two methods for the L₁ analysis of sampled-data systems, by which we mean computing the L_∞-induced norm of sampled-data systems. This is achieved by developing what we call the kernel approximation approach in the setting of sampled-data systems. We first consider the lifting treatment of sampled-data systems and give an operator theoretic representation of their input/output relation. We further apply the fast-lifting technique by which the sampling interval [0, h) is divided into M subintervals with an equal width, and provide methods for computing the L_∞-induced norm. In contrast to a similar approach developed earlier called the input approximation approach, we use an idea of kernel approximation, in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions. Furthermore, it is shown that the approximation errors in the piecewise constant approximation or piecewise linear approximation scheme converge to 0 at the rate of 1/M or 1/M², respectively. In comparison with the existing input approximation approach, in which the input function (rather than the kernel function) of the input operator is approximated by piecewise constant or piecewise linear functions, we show that the kernel approximation approach gives improved computation results. More precisely, even though the convergence rates in the kernel approximation approach remain qualitatively the same as those in the input approximation approach, the newly developed former approach could lead to quantitatively improved approximation errors than the latter approach particularly when the piecewise linear approximation scheme is taken. Finally, a numerical example is given to demonstrate the effectiveness of the kernel approximation approach with this scheme.     

Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier-Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P_NP_M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P_NP_M schemes is that they contain classical high order accurate finite volume schemes (N=0) as well as standard discontinuous Galerkin methods (M=N) just as special cases, which therefore allows for a direct efficiency comparison.In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier-Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P_NP_M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M_s=10.     

H∞ model reduction for negative imaginary systems

This paper is concerned with the H_∞ model reduction for negative imaginary (NI) systems. For a given linear time-invariant system that is stable and NI, our goal is to find a stable reduced-order NI system satisfying a pre-specified H_∞ approximation error bound. Sufficient conditions in terms of matrix inequalities are derived for the existence and construction of an H_∞ reduced-order NI system. Iterative algorithms are provided to solve the matrix inequalities and to minimise the H_∞ approximation error. Finally, a numerical example is used to demonstrate the effectiveness of the proposed model reduction method.     

Improved stability and H∞ performance criteria for linear systems with interval time-varying delays via three terms approximation

This paper investigates the problem of delay dependent stability and H_∞ performance for a class of linear systems with interval time-varying delay. A new model transformation is first proposed by employing a three-terms approximation of delayed state variables, for a better approximation of delayed state. By using scaled small gain theorem and a simple Lyapunov–Krasovskii functional, new stability and H_∞ performance criteria are proposed in terms of linear matrix inequalities, which can be easily solved by using standard numerical packages. Finally, numerical examples are presented to illustrate the effectiveness of the proposed method.     

Sulla soluzione dell'equazione biarmonica con metodi di programmazione lineare

The solution of biharmonic equation is studied with linear programming methods to obtain an a posteriori estimate of truncation errors. For this the algorithm proposed by Barrodale and Young has been adopted, which adjusts to the peculiar problem's structure the simplex method, and a lowest memory occupation requires, avoiding any constraints duplication. Carrying out some different boundary points distribution, a comparison is made among the results obtained by minimizing the error according to two different norms. It's pointed out that withL ₁ norm, we can reach a simpler and shorter problem formulation, and a solution which offers a better punctual approximation than we could obtainwithL _∞ norm, which supplies at contrary a better approximation. It's important to point out that such algorithm, when we try to approximate withL _∞ norm the biharmonic equation solution generates a cyclic procedure; so the ‘lexicographic method’ variant has been introduced. Numerical tests carried out are exposed in the following tables.     

ℋ ∞ model reduction for continuous-time switched stochastic hybrid systems

This article deals with the problem of computing an approximation system for a continuous-time switched stochastic system, such that the ?_∞ gain of the error system is less than a prescribed scalar. By using the average dwell-time approach and the piecewise Lyapunov function technique, a sufficient condition is first proposed, which guarantees the error system to be mean-square exponentially stable with a weighted ?_∞ performance. Then, the model reduction is solved by using the projection approach, which casts the model reduction into a sequential minimisation problem subjected to linear matrix inequality constraints by employing the cone complementary linearisation algorithm. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory.     

Robust H ∞ filter design for time-delay systems with saturation

This paper investigates the H∞ filtering problem for a class of linear continuous-time systems with both time delay and saturation. Such systems have time delay in their state equations and saturation in their output equations, and their process and measurement noises have unknown statistical characteristics and bounded energies. Based on the Lyapunov-Krasovskii stability theorem and the linear matrix inequalities (LMIs) technique, a generalized dynamic filter architecture is proposed, and a filter design method is developed. The linear H∞ filter designed by the method can guarantee the H∞ performance. The parameters of the designed filter can be obtained by solving a kind of LMI. An illustrative example shows that the design method proposed in this paper is very effective.     

Intelligent mixed H2/H∞ adaptive tracking control system design using self-organizing recurrent fuzzy-wavelet-neural-network for uncertain two-axis motion control system

In this paper, an intelligent adaptive tracking control system (IATCS) based on the mixed H₂/H_∞ approach under uncertain plant parameters and external disturbances for achieving high precision performance of a two-axis motion control system is proposed. The two-axis motion control system is an X–Y table driven by two permanent-magnet linear synchronous motors (PMLSMs) servo drives. The proposed control scheme incorporates a mixed H₂/H_∞ controller, a self-organizing recurrent fuzzy-wavelet-neural-network controller (SORFWNNC) and a robust controller. The combinations of these control methods would insure the stability, robustness, optimality, overcome the uncertainties, and performance properties of the two-axis motion control system. The SORFWNNC is used as the main tracking controller to adaptively estimate an unknown nonlinear dynamic function that includes the lumped parameter uncertainties, external disturbances, cross-coupled interference and frictional force. Moreover, the structure and the parameter learning phases of the SORFWNNC are performed concurrently and online. Furthermore, a robust controller is designed to deal with the uncertainties, including the approximation error, optimal parameter vectors and higher order terms in Taylor series. Besides, the mixed H₂/H_∞ controller is designed such that the quadratic cost function is minimized and the worst case effect of the unknown nonlinear dynamic function on the tracking error must be attenuated below a desired attenuation level. The mixed H₂/H_∞ control design has the advantage of both H₂ optimal control performance and H_∞ robust control performance. The sufficient conditions are developed for the adaptive mixed H₂/H_∞ tracking problem in terms of a pair of coupled algebraic equations instead of coupled nonlinear differential equations. The coupled algebraic equations can be solved analytically. The online adaptive control laws are derived based on Lyapunov theorem and the mixed H₂/H_∞ tracking performance so that the stability of the proposed IATCS can be guaranteed. Furthermore, the control algorithms are implemented in a DSP-based control computer. From the experimental results, the motions at X-axis and Y-axis are controlled separately, and the dynamic behaviors of the proposed IATCS can achieve favorable tracking performance and are robust to parameter uncertainties.     

Mixed H2/H∞ state-feedback design for microsatellite attitude control

Owing to a great decrease in size and weight, the pointing accuracy of microsatellite is vulnerable to space environmental disturbances and the internal uncertainty of moment-of-inertia variation. Mixed H₂/H_∞ control, giving consideration to both stability robustness and root-mean-square (rms) performance, is particularly attractive for attitude controller design of microsatellites. By using linear matrix inequality method, the numerical solution of mixed H₂/H_∞ state-feedback controller can be efficiently solved. The performance differences between mixed H₂/H_∞ controller and its two extremes—pure H₂ controller and pure H_∞ controller—are discussed in detail. Mixed H₂/H_∞ controller shows the remarkable capability of achieving a balanced compromise between H₂ and H_∞ performances.     

Structured H∞‐control of infinite‐dimensional systems

We develop a novel frequency‐based H_∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H_∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H_∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H_∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H_∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H_∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for linear infinite-dimensional systems

In this paper, we consider mixed H ₂/H _∞ control problems for linear infinite-dimensional systems. The first part considers the state feedback control for the H ₂/H _∞ control problems of linear infinite-dimensional systems. The cost horizon can be infinite or finite time. The solutions of the H ₂/H _∞ control problem for linear infinitedimensional systems are presented in terms of the solutions of the coupled operator Riccati equations and coupled differential operator Riccati equations. The second part addresses the observer-based H ₂/H _∞ control of linear infinite-dimensional systems with infinite horizon and finite horizon costs. The solutions for the observer-based H ₂/H _∞ control problem of linear infinite-dimensional systems are represented in terms of the solutions of coupled operator Riccati equations. The first-order partial differential system examples are presented for illustration. In particular, for these examples, the Riccati equations are represented in terms of the coefficients of first-order partial differential systems.     

A meshless method for numerical solution of the coupled Schrödinger-KdV equations

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L ₂ and L _∞ error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank–Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme.     

Approximation of time-dependent,multi-component,viscoelastic fluid flow

In this article we analyse a fully discrete approximation to the time dependent viscoelasticity equations allowing for multicomponent fluid flow. The Oldroyd B constitutive equation is used to model the viscoelastic stress. For the discretization, time derivatives are replaced by backward difference quotients, and the non-linear terms are linearized by lagging appropriate factors. The modeling equations for the individual fluids are combined into a single system of equations using a continuum surface model. The numerical approximation is stabilized by using an SUPG approximation for the constitutive equation. Under a small data assumption on the true solution, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of the mesh parameter h, the time discretization parameter Δt, and the SUPG coefficient ν are also derived. Numerical simulations of viscoelastic fluid flow involving two immiscible fluids are also presented.     

Approximating nonstationary ph(t)⧸ph(t)⧸1⧸c queueing systems

A state space partitioning and surrogate distribution approximation (SDA) approach for analyzing the time-dependent behavior of queueing systems is described for finite-capacity, single server queueing systems with time-dependent phase arrival and service processes. Regardless of the system capacity, c, the approximation requires the numerical solution of only k₁ + 3k₁k₂ differential equations, where k₁ is the number of phases in the arrival process and k₂ is the number of phases in the service process, compared to the k₁ + ck₁ k₂ Kolmogorov-forward equations required for the classic method of solution. Time-dependent approximations of mean and standard deviation of the number of entities in the system are obtained. Empirical test results over a wide range of systems indicate that the approximation is extremely accurate.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Robust Finite-time <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Linear Time-varying Delay Systems with Bounded Control via Riccati Equations

In this paper, we will present new results on robust finite-time H_∞ control for linear time-varying systems with both time-varying delay and bounded control. Delay-dependent sufficient conditions for robust finite-time stabilization and H_∞ control are first established to guarantee finite-time stability of the closed-loop system via solving Riccati differential equations. Applications to finite-time H_∞ control to a class of linear autonomous time-delay systems with bounded control are also discussed in this paper. Numerical examples are given to illustrate the effectiveness of the proposed method.     

Adaptive Reliable <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Uncertain Affine Nonlinear Systems

For general input affine nonlinear systems, robust reliable control designs are commonly available that compensate the actuator faults in pure outage mode. In this paper, a more general and complex problem is considered and an adaptive reliable H_∞ controller is designed for a class of uncertain input affine nonlinear systems in the presence of actuators fault. The key element of the work is the introduction of a novel adaptive mechanism that estimates the faults which are modeled as an outage or loss of effectiveness and stabilizes the overall system. Incorporating with the parameter projection algorithm and the solution of Hamilton-Jacobi-Inequality (HJI), the proposed method combines adaptive reliable control and robust H_∞ control techniques. A numerical approach is developed based on the Taylor series expansion for solving the HJI. Various simulation examples are given to illustrate the effectiveness of the proposed adaptive reliable H_∞ control scheme over the conventional H_∞ control and reliable H_∞ control method.     

Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ω _j ⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.