Class Of Nonlinear Differential-algebraic Equations Research Articles

Index-Aware Model-Order Reduction for a Special Class of Nonlinear Differential-Algebraic Equations

We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term mathbf{f}(mathbf{E}mathbf{x}), where mathbf{E} is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and system size. Model-order reduction aims to eliminate this problem by generating reduced-order models that have lower computational cost to simulate, yet accurately represent the original large-scale system behavior. However, direct reduction and simulation of nonlinear differential-algebraic equations is difficult due to hidden constraints which affect the choice of numerical integration methods and model-order reduction techniques. We propose an extension of index-aware model-order reduction methods to a special class of nonlinear differential-algebraic equations without any kind of linearization. The proposed model-order reduction approach involves automatic decoupling of nonlinear differential-algebraic equations into nonlinear ordinary differential equations and algebraic equations, based on the decoupling of the linear differential equations obtained by ignoring the nonlinear term, thanks to an additional structural condition. This allows applying standard model-order reduction techniques to both parts without worrying about the index. The same procedure can also be used to simulate nonlinear differential-algebraic equations by standard integration schemes.

Some Runge-Kutta Algorithms with Error Control and Variable Stepsizes for Solving a Class of Differential-Algebraic Equations

In this paper, we propose and discuss numerical algorithms for solving a class of nonlinear differential-algebraic equations (DAEs). These algorithms are based on half-explicit Runge-Kutta methods (HERK) that have been studied recently for solving strangeness-free DAEs. The main idea of this work is to use the half-explicit variants of some well-known embedded Runge-Kutta methods such as Runge-Kutta-Fehlberg and Dormand-Prince pairs. Thus, we can estimate local errors and choose suitable stepsizes accordingly to a given tolerance. The cases of unstructured and structured DAEs are investigated and compared. Finally, some numerical experiences are given for illustrating the efficiency of the algorithms.

On convergence of continuous half-explicit Runge-Kutta methods for a class of delay differential-algebraic equations

In this paper, we propose and investigate continuous Runge-Kutta methods for solving a class of nonlinear differential-algebraic equations (DAEs) with constant delay. Real-life processes that involve simultaneously time-delay effect and constraints are usually described by delay DAEs. Solving delay DAEs is more complicated than solving non-delay ones since we should focus on both the time-delay and DAE aspects. Recently, we have revisited linear multistep methods and Runge-Kutta methods for a class of nonlinear DAEs (without delay) and shown the advantages of appropriately modified discretizations. In this work, we extend the use of half-explicit Runge-Kutta methods to a similar class of structured strangeness-free DAEs with constant delay. Approximation of solutions at delayed time is obtained by continuous extensions of discrete solution, i.e., continuous output formulas. Convergence analysis for continuous Runge-Kutta methods is presented. It is shown that order reduction that may happen with DAEs is avoided if we discretize an appropriately reformulated delay DAE (DDAE) instead of the original one. Difficulties arising in the implementation are discussed as well. Finally, numerical experiments are given for illustration.

Runge-Kutta methods revisited for a class of structured strangeness-free differential-algebraic equations

Numerical methods for a class of nonlinear differential-algebraic equations (DAEs) of the strangeness-free form are investigated. Half-explicit and implicit Runge-Kutta methods are revisited as they are applied to a reformulated form of the original DAEs. It is shown that the methods preserve the same convergence order and the same stability properties as if they were applied to ordinary differential equations (ODEs). Thus, a wide range of explicit Runge-Kutta methods and implicit ones, which are not necessarily stiffly accurate, can efficiently solve the class of DAEs under consideration. Implementation issues and a perturbation analysis are also discussed. Numerical experiments are presented to illustrate the theoretical results.

A General Approach to Waveform Relaxation Solutions of Nonlinear Differential-Algebraic Equations: The Continuous-Time and Discrete-Time Cases

For a general class of nonlinear differential-algebraic equations of index one, we develop and unify a convergence theory on waveform relaxation (WR). Convergence conditions are achieved for the cases of continuous-time and discrete-time WR approximations. Most of known convergence results in this field can be easily derived from the new theory established here.

Asymptotic output tracking of nonlinear differential-algebraic control systems

The problem of asymptotic output tracking is considered for control systems described by a class of nonlinear differential-algebraic equations. Assumptions are introduced, and an algorithm is developed. Based on the algorithm, a nonlinear feedback control law is then constructed which ensures, under appropriate conditions, that the closed-loop system has a unique bounded differentiable solution and the tracking error in the closed-loop system approaches zero when time goes to infinity.

Noninteracting Control for a Class of Nonlinear Differential-Algebraic Systems

This paper concerns nonlinear singular systems defined by a class of nonlinear differential-algebraic equations. Assumptions are given first for which initial value problem and the procedure of nonlinear coordinates transformation are well posed. The problem of noninterating control with internal stability via static state feedback is stated. Noninteracting control feedback law is then presented. An algorithmic procedure is developed to obtain the manifold on which there exists a unique differentiate solution to the closed-loop system. The conditions for local internal stability of a single equilibrium solution are indicated. Nonintercating control problem of the DAE systems is solved based on the framework.

Tracking in nonlinear differential-algebraic control systems with applications to constrained robot systems

We consider the design of a feedback control law for control systems described by a class of nonlinear differential-algebraic equations so that certain desired outputs track given reference inputs. The nonlinear differential-algebraic control system being considered is not in state variable form. Assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential-algebraic equations is expressed in a familiar normal form. A nonlinear feedback control law is then proposed which ensures, under appropriate assumptions, that the tracking error in the closed loop differential-algebraic system approaches zero exponentially. Applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints, constrained robot systems with joint flexibility, and constrained robot systems with significant actuator dynamics are discussed.

Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations

A theoretical framework for the stabilization of control systems defined by a class of nonlinear differential-algebraic equations is established. Assumptions are stated which guarantee that the nonlinear differential-algebaric equations can be described by a nonlinear control system defined on a smooth manifold. A procedure for obtaining a local state realization is developed. Conditions for local stabilization of a single equilibrium solution, including one set of conditions which can easily be checked using standard computations, are indicated.