Single-step full-state feedback control design for nonlinear hyperbolic PDEs

The present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation.     

Non-fragile guaranteed cost fuzzy control for nonlinear first-order hyperbolic partial differential equation systems

This paper concerns the non-fragile guaranteed cost control for nonlinear first-order hyperbolic partial differential equations (PDEs), and the case of hyperbolic PDE systems with parameter uncertainties is also addressed. A Takagi–Sugeno (T–S) fuzzy hyperbolic PDE model is presented to exactly represent the nonlinear hyperbolic PDE system. Then, the state-feedback non-fragile controller distributed in space is designed by the parallel distributed compensation (PDC) method, and some sufficient conditions are derived in terms of spatial differential linear matrix inequalities (SDLMIs) such that the T–S fuzzy hyperbolic PDE system is asymptotically stable and the cost function keeps an upper bound. Moreover, for the nonlinear hyperbolic PDE system with parameter uncertainties, using the above-design approach, the robust non-fragile guaranteed cost control scheme is obtained. Furthermore, the finite-difference method is employed to solve the SDLMIs. Finally, a nonlinear hyperbolic PDE system is presented to illustrate the effectiveness and advantage of the developed design methodology.     

Convergence Acceleration for Hyperbolic Systems Using Semicirculant Approximations

The iterative solution of systems of equations arising from systems of hyperbolic, time-independent partial differential equations (PDEs) is studied. The PDEs are discretized using a finite volume or finite difference approximation on a structured grid. A convergence acceleration technique where a semicirculant approximation of the spatial difference operator is employed as preconditioner is considered. The spectrum of the preconditioned coefficient matrix is analyzed for a model problem. It is shown that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of grid points and the artificial viscosity parameter. By linearizing the Euler equations around an approximate solution, a system of linear PDEs with variable coefficients is formed. When utilizing the semicirculant (SC) preconditioner for this problem, which has properties very similar to the full nonlinear equations, numerical experiments show that the favorable convergence properties hold also here. We compare the results for the SC method to those of a multigrid (MG) scheme. The number of iterations and the arithmetic complexities are considered, and it is clear that the SC method is more efficient for the problems studied. Also, the MG scheme is sensitive to the amount of artificial dissipation added, while the SC method is not.     

Nonlinear stabilization in infinite dimension

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infinite-dimensional setting. First we present designs for nonlinear ODEs with constant, time-varying or state-dependent input delays, which arise in numerous applications of control over networks. Second, we present a design for nonlinear ODEs with a wave (string) PDE at its input, which is motivated by the drilling dynamics in petroleum engineering. Third, we present a design for systems of (two) coupled nonlinear first-order hyperbolic PDEs, which is motivated by slugging flow dynamics in petroleum production in off-shore facilities. Our design and analysis methodologies are based on the concepts of nonlinear predictor feedback and nonlinear infinite-dimensional backstepping. We present several simulation examples that illustrate the design methodology.     

A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

In this paper, we study the effect of the choice of mesh quality metric, preconditioner, and sparse linear solver on the numerical solution of elliptic partial differential equations (PDEs). We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and radius ratio metrics in addition to conditioning-based scale-invariant and interpolation-based size-and-shape metrics. We employ the Jacobi, SSOR, incomplete LU, and algebraic multigrid preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric, preconditioner, and linear solver combination for the numerical solution of various elliptic PDEs with isotropic coefficients. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. In this paper, we consider Poisson’s equation, general second-order elliptic PDEs, and linear elasticity problems.     

On control design for PDEs with space-dependent diffusivity or time-dependent reactivity

In this paper the recently introduced backstepping method for boundary control of linear partial differential equations (PDEs) is extended to plants with non-constant diffusivity/thermal conductivity and time-varying coefficients. The boundary stabilization problem is converted to a problem of solving a specific Klein-Gordon-type linear hyperbolic PDE. This PDE is then solved for a family of system parameters resulting in closed-form boundary controllers. The results of a numerical simulation are presented for the case when an explicit solution is not available.     

Online policy iteration algorithm for optimal control of linear hyperbolic PDE systems

In this paper, the linear quadratic (LQ) optimal control problem is considered for a class of linear distributed parameter systems described by first-order hyperbolic partial differential equations (PDEs). Reinforcement learning (RL) technique is introduced for adaptive optimal control design from the design-then-reduce (DTR) framework. Initially, a policy iteration (PI) algorithm is proposed, which learns the solution of the space-dependent Riccati differential equation (SDRDE) online without requiring the internal system dynamics of the PDE system. To prove its convergence, the PI algorithm is shown to be equivalent to an iterative procedure of a sequence of space-dependent Lyapunov differential equations (SDLDEs). Then, the convergence is established by showing that the solutions of SDLDEs are a monotone non-increasing sequence that converges to the solution of the SDRDE. For implementation purpose, an online least-square method is developed for the approximation of the solutions of the SDLDEs. Finally, the proposed design method is applied to the distributed control of a steam-jacketed tubular heat exchanger to illustrate its effectiveness.     

An LPV approach to the robust control of a class of quasi-linear propagation processes

In this paper the trajectory tracking control problem for a certain class of propagation processes modeled as quasi-linear parameter varying systems is considered. The propagation physical models are generally described by means of partial differential equations (PDEs). However in real world control problems the PDE models are usually converted into ordinary differential equations (ODEs) models adopting numerical and/or physical approximations. In many practical problems it happens that the propagation dynamics are linear, while the boundary conditions are described by nonlinear algebraic equations. A trajectory following control scheme is proposed for this class of systems together with a robust performance analysis based on the concept of quadratic stability with an H_∞ norm bound. An LMI based observer synthesis procedure is also proposed to increase the closed loop system performance.     

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

Singular PDEs and the single-step formulation of feedback linearization with pole placement

The present work proposes a new formulation and approach to the problem of feedback linearization with pole placement. The problem under consideration is not treated within the context of geometric exact feedback linearization, where restrictive conditions arise from a two-step design method (transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived, by using Lyapunov's auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization and pole-placement design objectives may be accomplished in a single step, effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input.     

Distributed Proportional-spatial Derivative control of nonlinear parabolic systems via fuzzy PDE modeling approach

In this paper, a distributed fuzzy control design based on Proportional-spatial Derivative (P-sD) is proposed for the exponential stabilization of a class of nonlinear spatially distributed systems described by parabolic partial differential equations (PDEs). Initially, a Takagi-Sugeno (T-S) fuzzy parabolic PDE model is proposed to accurately represent the nonlinear parabolic PDE system. Then, based on the T-S fuzzy PDE model, a novel distributed fuzzy P-sD state feedback controller is developed by combining the PDE theory and the Lyapunov technique, such that the closed-loop PDE system is exponentially stable with a given decay rate. The sufficient condition on the existence of an exponentially stabilizing fuzzy controller is given in terms of a set of spatial differential linear matrix inequalities (SDLMIs). A recursive algorithm based on the finite-difference approximation and the linear matrix inequality (LMI) techniques is also provided to solve these SDLMIs. Finally, the developed design methodology is successfully applied to the feedback control of the Fitz-Hugh-Nagumo equation.     

Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary

The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems.In [1], we have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as auto. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature.In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.     

Discrete Fundamental Solution Preconditioning for Hyperbolic Systems of PDE

We present a new preconditioner for the iterative solution of systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electromagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple one-stage iterative method. As an example of a more involved problem, we consider the steady state solution of the non-linear Euler equations in a two-dimensional, non-axisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.     

Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems.

This correspondence studies the problem of finite-dimensional constrained fuzzy control for a class of systems described by nonlinear parabolic partial differential equations (PDEs). Initially, Galerkin's method is applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, a systematic modeling procedure is given to construct exactly a Takagi-Sugeno (T-S) fuzzy model for the finite-dimensional ODE system under state constraints. Then, based on the T-S fuzzy model, a sufficient condition for the existence of a stabilizing fuzzy controller is derived, which guarantees that the state constraints are satisfied and provides an upper bound on the quadratic performance function for the finite-dimensional slow system. The resulting fuzzy controllers can also guarantee the exponential stability of the closed-loop PDE system. Moreover, a local optimization algorithm based on the linear matrix inequalities is proposed to compute the feedback gain matrices of a suboptimal fuzzy controller in the sense of minimizing the quadratic performance bound. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.     

Max-plus eigenvector representations for solution of nonlinear H/sub /spl infin// problems: basic concepts

The H/sub /spl infin// problem for a nonlinear system is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient. The solutions are typically nonsmooth, and further, there is nonuniqueness among the class of viscosity solutions. In the case where one tests a feedback control to see if it yields an H/sub /spl infin// controller, the PDE is a Hamilton-Jacobi-Bellman equation. In the case where the "optimal" feedback control is being determined as well, the problem takes the form of a differential game, and the PDE is, in general, an Isaacs equation. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. In this paper, we develop an entirely new class of methods for obtaining the "correct" solution of such PDEs. These methods are based on the linearity of the associated semigroup over the max-plus (or, in some cases, min-plus) algebra. In particular, solution of the PDE is reduced to solution of a max-plus (or min-plus) eigenvector problem for known unique eigenvalue 0 (the max-plus multiplicative identity). It is demonstrated that the eigenvector is unique, and that the power method converges to it. An example is included.     

Computationally efficient simultaneous policy update algorithm for nonlinear H∞ state feedback control with Galerkin's method

The main bottleneck for the application of H_∞ control theory on practical nonlinear systems is the need to solve the Hamilton–Jacobi–Isaacs (HJI) equation. The HJI equation is a nonlinear partial differential equation (PDE) that has proven to be impossible to solve analytically, even the approximate solution is still difficult to obtain. In this paper, we propose a simultaneous policy update algorithm (SPUA), in which the nonlinear HJI equation is solved by iteratively solving a sequence of Lyapunov function equations that are linear PDEs. By constructing a fixed point equation, the convergence of the SPUA is established rigorously by proving that it is essentially a Newton's iteration method for finding the fixed point. Subsequently, a computationally efficient SPUA (CESPUA) based on Galerkin's method, is developed to solve Lyapunov function equations in each iterative step of SPUA. The CESPUA is simple for implementation because only one iterative loop is included. Through the simulation studies on three examples, the results demonstrate that the proposed CESPUA is valid and efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

On Resistive MHD Models with Adaptive Moving Meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics.     

A multidomain spectral method for solving elliptic equations

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.