Numerical experiment of measurement-integrated simulation to reproduce turbulent flows with feedback loop to dynamically compensate the solution using real flow information

Reproduction of the exact structure of real turbulent flows is crucial in many applications. Four Dimensional variation (4D-VAR) is widely used in numerical weather forecasting, but it requires huge computational power to repeatedly solve flow dynamics and its adjoint, and, therefore, is not suitable to apply to problems of real-time flow reproduction such as feedback flow control. Kalman filter and observer, in which numerical solution converges to the real state asymptotically by means of the feedback signal proportional to the difference between the calculated state and the real state, requiring much less computational load than the variational method, are potential candidates to solve the problem. By comparing Kalman filter and observer, the latter has simpler structure retaining essential part of the state estimation. This study deals with a special type of observer, or measurement-integrated simulation (MI simulation), in which a SIMPLER-based flow solver is used as the mathematical model of the system in place of approximate small dimensional linear differential equations usually used in observers. Reproduction of the exact structure of a turbulent flow was investigated by a MI simulation. A numerical experiment was performed for a fully developed turbulent flow in a pipe with a square cross section. The MI simulation was performed with the feedback from the standard solution in the flow domain for the cases using: (1) all velocity components at all grid points, (2) partial velocity components at all grid points, or (3) all velocity components at partial grid points. Convergence of the MI simulation to the standard solution was investigated using the steady error norm for the convergent state and the time constant for the transient state. The result of the MI simulation using all the velocity information exponentially converges to the standard solution with a steady state error reduced from that of the ordinary simulation in a range of the feedback gain. Decreasing the feedback gain reduces the effect of feedback, and a feedback gain which is too large destabilizes the closed loop system, resulting in large error. The time constant decreases almost inversely proportional to the feedback gain as long as the feedback system is stable. For the MI simulation with the feedback using limited information, feedback using two velocity components by omitting one transverse velocity component showed a good result, although the other results were not satisfactory. For the MI simulation with the feedback using limited grid points, the result of the MI simulation applying the feedback at the grid points on every 20th plane in the x₁ direction was almost the same as that using all grid points at some feedback gain, while the result with the feedback on the planes skipped in the x₂ direction requires 10 times more planes to achieve the same reduction rate.     

Multi-objective control for stochastic model reference systems based on LMI approach and sliding mode control concept

In this article, a multi-objective control u(t) is designed for stochastic model reference systems to achieve the following three objectives simultaneously: the pole placement constraint, H _∞-norm constraint and individual error state variance constraint. Using the invariance property of sliding mode control, the reference model input and the plant error term will disappear on the sliding mode of the error system. By combining the upper bound covariance control theory, pole placement skill and H _∞-norm control theory, a controller, in which the control feedback gain matrix is synthesised utilising linear matrix inequality (LMI) approach, is derived to achieve the above multiple objectives. Furthermore, a practical example for the problem of ship yaw-motion systems is adopted to illustrate the proposed method.     

Optimal stabilization of bodies in electromagnetic suspensions without measurements of their location

The optimal stabilization problem is considered for bodies in electromagnetic suspensions. To solve this problem, we form a linear stationary control law for the linearized system. This law is based on the feedback principle and uses the measuring of the current intensity in the circuit of the electromagnet, while the location of the body and its velocity are not measured. The optimality criterion is the generalized H _∞-norm of the linearized system: it characterizes the extinguishing level for perturbations generated by external actions and unknown initial conditions. To compute the feedback parameters, the technique of linear matrix inequalities is applied. We provide mathematical simulation examples for the dynamics of a body in an electromagnetic suspension.     

Global stability of trajectory tracking of robot underPD control

It is shown for the first time that, even if there exist nonlinear unknown dynamics, aPD feedback control without higher-order nonlinear compensation can guarantee global stability for the trajectory following problem of a robot manipulator. ThePD control under investigation is a position and velocity feedback control with a time-varying gain, and does not contain any higher-order nonlinearity. The proposed control is in general continuous and does not require any knowledge of robotic systems except size bounding function on nonlinear dynamics. Asymptotic stability of velocity tracking error and arbitrarily small position tracking error are guaranteed. Another novel and interesting result shown in this paper is that a measure on protection against saturation of actuators has been incorporated into consideration of control design and robustness analysis.This work is supported in part by U.S. National Science Foundation under grant MSS-9110034.     

Measurement-integrated simulation of wall pressure measurements using a turbulent model for analyzing oscillating orifice flow in a circular pipe

The development of a highly functionalized orifice flowmeter with high accuracy under realistic conditions is desired. This paper presents a method for analyzing oscillating air flow through an orifice in a circular pipe. A measurement-integrated (MI) simulation using a standard k–ε model was used to reduce the computation time. In a previous study, the feedback law of the MI simulation was determined by considering the effect of the computational fluid dynamics (CFD) grid on contracted flow. However, the previous method required the measurement of inlet flow rate, which is not feasible in many applications. Therefore, an MI simulation was proposed that only requires wall pressures, which are much simpler to measure than flow rate. In this MI simulation, the wall pressure downstream of an orifice was measured, and a new proportional–integral controller feedback algorithm was developed to control the inlet flow rate in the computed flow field. The proposed MI simulations were performed for steady and oscillatory flow rates up to 10 Hz. It was found that this MI simulation provides accurate solutions at a significantly shorter computation time than conventional CFD analysis.     

Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems

We present in this paper an analysis of a semi-Lagrangian second order Backward Difference Formula combined with hp-finite element method to calculate the numerical solution of convection diffusion equations in ℝ². Using mesh dependent norms, we prove that the a priori error estimate has two components: one corresponds to the approximation of the exact solution along the characteristic curves, which is O(Dt²+h^m+1(1+\frac\mathopen|logh|Dt))O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t})); and the second, which is O(Dt^p+|| [(u)\vec]-[(u)\vec]_h||_L^￥)O(\Delta t^{p}+\| \vec{u}-\vec{u}_{h}\|_{L^{\infty}}), represents the error committed in the calculation of the characteristic curves. Here, m is the degree of the polynomials in the finite element space, [(u)\vec]\vec{u} is the velocity vector, [(u)\vec]_h\vec{u}_{h} is the finite element approximation of [(u)\vec]\vec{u} and p denotes the order of the method employed to calculate the characteristics curves. Numerical examples support the validity of our estimates.     

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det (D ² u ⁰)=f (>0) based on the vanishing moment method which was developed by the authors in Feng and Neilan (J. Sci. Comput. 38:74–98, 2009) and Feng (Convergence of the vanishing moment method for the Monge-Ampère equation, submitted). In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −εΔ² u ^ε +det D ² u ^ε =f accompanied by appropriate boundary conditions. This new approach enables us to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ^ε of the regularized problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error u^e-u^e_hu^{\varepsilon}-u^{\varepsilon}_{h}. Due to the strong nonlinearity of the underlying equation, the standard error estimate technique, which has been widely used for error analysis of finite element approximations of nonlinear problems, does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Argyris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u⁰-u_h^eu^{0}-u_{h}^{\varepsilon}, and numerically examine what is the “best” mesh size h in relation to ε in order to achieve these rates.     

Stability analysis and robust composite controller synthesis for flexible joint robots

In this paper the control of flexible joint manipulators is studied in detail. The model of N-axis flexible joint manipulators is derived and reformulated in the form of singular perturbation theory and an integral manifold is used to separate fast dynamics from slow dynamics. A composite control algorithm is proposed for the flexible joint robots, which consists of two main parts. Fast control, u _f, guarantees that the fast dynamics remains asymptotically stable and the corresponding integral manifold remains invariant. Slow control, u _s, consists of a robust PID designed based on the rigid model and a corrective term designed based on the reduced flexible model. The stability of the fast dynamics and robust stability of the PID scheme are analyzed separately, and finally, the closed-loop system is proved to be uniformly ultimately bounded (UUB) stable by Lyapunov stability analysis. Finally, the effectiveness of the proposed control law is verified through simulations. The simulation results of single- and two-link flexible joint manipulators are compared with the literature. It is shown that the proposed control law ensures robust stability and performance despite the modeling uncertainties.     

Output feedback stabilisation of single-input single-output linear systems with I/O network-induced delays. An eigenvalue-based approach

This work addresses the output feedback stabilisation problem for a class of linear single-input single-output systems subject to I/O network delays. More precisely, we are interested in the characterisation of the set of delay and gain parameters guaranteeing the stability of the closed-loop system. To perform such an analysis, we adopt an eigenvalue perturbation based approach. Various illustrative numerical examples complete the presentation.     

State feedback for non-linear control systems

A theory of static state feedback for non-linear discrete-time systems is developed. The theory applies to non-linear systems possessing a recursive representation of the form x _k+1 =?(x _k , u_k ), where ? is a continuous function, and it deals with the construction of continuous state feedback functions that internally stabilize a given system. The theory yields an explicit method for the computation of stabilizing feedback functions, and several examples of the computation of such functions are provided.     

Spurious caustics of dispersion-relation-preserving schemes

A linear dispersive mechanism leading to a burst in the L _∞ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated with interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.     

Keller's Box-Scheme for the One-Dimensional Stationary Convection-Diffusion Equation

u ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu _x −ɛu _xx =f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG finite element method. Received June 7, 2001; revised October 2, 2001     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.     

Computational Hydrodynamic Stability and Flow Control Based on Spectral Analysis of Linear Operators

This paper considers the analysis and control of fluid flows using tools from dynamical systems and control theory. The employed tools are derived from the spectral analysis of various linear operators associated with the Navier?CStokes equations. Spectral decomposition of the linearized Navier-Stokes operator, the Koopman operator, the spatial correlation operator and the Hankel operator provide a means to gain physical insight into the dynamics of complex flows and enables the construction of low-dimensional models suitable for control design. Since the discretization of the Navier-Stokes equations often leads to very large-scale dynamical systems, matrix-free and in some cases iterative techniques have to be employed to solve the eigenvalue problem. The common theme of the numerical algorithms is the use of direct numerical simulations. The theory and the algorithms are exemplified on flow over a flat plate and a jet in crossflow, as prototypes for the laminar-turbulent transition and three-dimensional vortex shedding.     

Bang-bang strategies using interrupted observations for steering a random motion of a point

Consider the random motion in the plane of a point M, whose velocity u = (u₁, u₂) is perturbed by an R²-valued Gaussian white noise. The point is equipped with an observation mechanism which provides interrupted noisy observations of a given target. The interruptions are characterized by a jump Markov process taking on the values 0 or 1, and using these observations the point M wishes to steer itself into the target. Sufficient conditions on optimal velocity laws (strategies) u are derived, and suboptimal strategies computed. The numerical results obtained show that the suboptimal strategies obey the separation principle of optimal stochastic control theory.     

Improved Controller Design for Uncertain Positive Systems and its Extension to Uncertain Positive Switched Systems

This paper is concerned with the controller design of uncertain positive systems. First, we decompose the feedback gain matrix K_m×n into m×n non‐negative components and m×n non‐positive components. For the non‐negative components, each component contains only one positive element and the other elements are zero. Similarly, each non‐positive component contains only one negative element and the other ones are zero. Then, a simple and effective controller design approach of uncertain positive systems is proposed by incorporating the decomposed feedback gain matrix into the resulting closed‐loop systems and further applied to uncertain positive switched systems. It is shown that the designed controller is less conservative compared with those in the literature. Finally, a numerical example is provided to verify the validity of the proposed design.     

L1 synthesis of a static output controller for positive systems by LMI iteration

An approach to find a static output feedback gain that makes the feedback system positive and minimizes the L₁ gain is proposed. The problem of finding a static output feedback gain has 3 aspects: stabilizing the system, making the system positive, and then minimizing the L₁ gain. Each subproblem is described by bilinear matrix inequality with respect to the feedback gain and the Lyapunov matrix or vector. Linear matrix inequality (LMI) that is sufficient to satisfy bilinear matrix inequality is derived using a convex‐concave decomposition, and the feedback gain sequence is calculated by an iterative solution of LMI. The sequence of the upper bounds on the design parameter is guaranteed to be monotonically nonincreasing for each algorithm. Similarly, 2 other LMIs are derived for each subproblem using another convex‐concave decomposition and PK iteration. The effectiveness of these algorithms is illustrated via several numerical examples.     

Dyadic output feedback laws generated as Kronecker products : optimal and suboptimal solutions

The problem of eigenvalue assignment in the system dx/dt equals; Ax + Bu, y = Dx, using the dyadic output feedback law u = u₀ + q · p^Ty is considered via a formulation developed earlier by the author, in which p and q occur in the Kronecker product vector p?q. The equations governing the values of p and q which give an optimum approximation to a prescribed spectrum of eigenvalues are derived, and a special case is solved. Various facets of the problem of generating suboptimal solutions are discussed.     

A penalty method for the approximation of projections over cones in Hilbert spaces

Given a closed convex coneK in a Hilbert spaceH and a vectoru ₀ ∈H, a penalty method is built up in order to approximate the projection ofu ₀ over the polar coneK ^* ofK, without making use of the inverse transform of the canonical mapping ofH into its dual spaceH′. Such method is outlined in n⁰ 1, 2. In n⁰3 a complete analysis of the errors of the method is explained. In n⁰4 the method is applied to find error bounds for the numerical approximation of the projection ofu ₀ onK.