Implicit Euler Discretization Research Articles

Discretization of prescribed-time observers in the presence of noises and perturbations

An implicit Euler discretization scheme of the prescribed-time converging observer from Holloway and Krstic, (2019) is proposed, which preserves all main properties of the continuous-time counterpart, and it is investigated how to apply recursively this discretization on any interval of time. In addition, it is demonstrated that the estimation error is robustly stable with respect to the measurement noise with a linear gain (the property that does not hold in the continuous-time setting). The efficiency of the suggested discrete-time observer is illustrated through numeric experiments.

Root Mean Square Error Estimates for the Projection-Difference Method for the Approximate Solution of a Parabolic Equation with a Periodic Condition for the Solution

Using the projection-difference method, we construct an approximate solution of an abstract linear parabolic equation in a separable Hilbert space with a periodic condition for a solution. We use the Galerkin method for the spatial variables and the implicit Euler discretization for time. We obtain root mean square estimates of the error of approximate solutions that are effective both in time and spatial variables; these estimates imply the convergence of approximate solutions to an exact solution and allow one to find the convergence rate.

Discrete-time Flatness-based Controller Design using an Implicit Euler-discretization

In this contribution, we present a constructive method to derive flat sampled-data models for continuous-time flat systems through an implicit Euler-discretization. We show how the sampled-data model can be used subsequently for a flatness-based controller design, and illustrate our results with the well-known planar VTOL aircraft example.

Semi-Implicit Scheme to Solve Allen-Cahn Equation with Different Boundary Conditions

The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.

Dynamics modeling of gear transmissions with asymmetric load-dependent friction

This paper proposes a new concise mathematical model of gear transmission dynamics with asymmetric load-dependent friction. It is built only upon the rigid-contact mechanics and the Coulomb friction law, but provides a new theoretical perspective on dynamic and static characteristics of gear transmissions. The presented model is of the form of a differential–algebraic inclusion (DAI) characterized by some parameters, including input-side and output-side asymmetry coefficients. The presented model properly captures the static friction and even the non-backdrivability. It is applicable to different classes of transmissions, such as leadscrew transmissions, worm gear transmissions, and spur gear transmissions. The DAI representation is extended into a multi-dimensional representation for articulated rigid-body systems driven through joint transmissions. Moreover, simulation algorithms are derived through the implicit Euler discretization. Some simulation examples illustrate the capability of the presented simulation algorithm to reproduce load-dependent asymmetric frictional behaviors, which cannot be reproduced by conventional load-independent joint friction models.

Monolithic, non-iterative and iterative time discretization methods for linear coupled elliptic-parabolic systems


 We compare four numerical methods for the time discretization of linear coupled elliptic-parabolic systems.
 The monolithic method arising from an implicit Euler discretization is the primary method for solving the coupled system.
 Accelerated solution via non-iterative decoupling is possible by the semi-explicit Euler discretization, using a novel methodology from related delay differential equations. For poroelasticity, the fixed-stress splitting and undrained splitting methods enable iterative decoupled solves.
 We present formulations for the iterative methods in an abstract form and compare through numerical experiments the a priori convergence results for the four methods.

Solving phase-field models in the tensor train format to generate microstructures of bicontinuous composites

Phase-field models permit accounting for the underlying physics of the microstructure evolution process when simulating the emergent microstructure of a variety of multiscale materials. As an inherent characteristic, the individual phases (or states) of the material are not known in advance and only emerge upon evolution. To account for this matter, structured grids are typically used when solving phase-field models. In particular for large three-dimensional grids and long time scales, the used memory and computational time can be considerable, instigating research on alternative approaches.The work at hand studies the Tensor Train (TT) format as a possible remedy for phase-field simulations. More precisely, the TT format is a specific tensor format which permits – small so-called TT rank provided – to store huge arrays of data and comes with specific algorithms replacing the linear algebra operations familiar for fully stored arrays.We investigate the Cahn-Hilliard equation and use a semi-implicit discretization in time which is based on solving a single linear system per time step. We compare this strategy with a classic implicit Euler discretization using a Newton scheme as a nonlinear solver. Upon a finite-difference discretization in space, the resulting equations are recast in the Quantics Tensor Train (QTT) format and solved by dedicated linear solvers. We study the performance of the algorithms and investigate the effective properties of the resulting bicontinuous composites via computational homogenization. We show that the Cahn-Hilliard equation can be solved using the TT format and that runtimes scale well with the grid size.

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions

AbstractThis paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.

Implicit Discrete-Time Adaptive First-Order Sliding Mode Control With Predefined Convergence Time

This brief proposes an adaptive first-order sliding mode control (SMC) with a predefined convergence time in a digital framework. It is based on an implicit Euler discretization of the set-valued signum function in SMC and a time-dependent adaptive gain scheme. Two typical cases are discussed and their stability analysis is provided, simple affine systems with first-order SMC and LTI systems with equivalent control-based sliding mode control (ECB-SMC), both in the presence of unknown perturbations. Numerical simulations show that, compared with the conventional SMC, the proposed implicit discrete-time SMC guarantees the convergence of the states within any predefined arbitrary time.

New finite volume element methods in the ALE framework for time-dependent convection–diffusion problems in moving domains

This paper develops new finite volume element methods (FVEMs) in the Arbitrary Lagrangian–Eulerian (ALE) framework for the time-dependent convection–diffusion problems on moving domains. In particular, we present two fully discrete schemes, one is based on the implicit Euler (IE) discretization and the other is based on the combination of the IE and geometric conservation laws (GCL). Stability and error estimation are analyzed for these two schemes. The second scheme satisfying the GCL demonstrates better on stability for long time simulations. Finally, numerical experiments are carried out to illustrate the theoretical results.

Feedback Linearization of Implicit Discrete Systems

Feedback linearization of a nonlinear system is an actively employed technique to control a nonlinear system. Feedback linearizability of continuous-time systems is generally not preserved under sampling and is dependent upon the choice of discretization method. This paper modifies the set of necessary and sufficient conditions for feedback linearization given by Grizzle (1986), so they can be employed for implicitly described discrete-time control system and using them establishes an equivalence between the discrete-time feedback linearizability of the Implicit and Explicit Euler discretization schemes.

Discrete-Time Adaptive Super-Twisting Observer With Predefined Arbitrary Convergence Time

This brief proposes an adaptive observer based on the super-twisting algorithm (STA) and its discrete-time realization with a predefined convergence time. In contrast to conventional adaptive STA that tries to adaptively reduces the gain sizes as much as possible in accordance with external disturbances, the proposed adaptive observer increases the gain sizes such that the convergence time is ensured to be within the predefined convergence-time period. The numerical chattering associated with these large gains is suppressed by employing the proposed discrete-time realization based on an implicit Euler discretization method. While keeping the property of predefined convergence time, the observation precision of the proposed discrete-time scheme is consistent with the STA, i.e., standard asymptotical second-order accuracy level. The superiority of the adaptive observer and its realization scheme is demonstrated through a circuit system example.

Discrete-Time Super-Twisting Fractional-Order Differentiator With Implicit Euler Method

This brief proposes a discrete-time fractional-order differentiator based on the super-twisting algorithm for second-order systems. The differentiator achieves higher performance with respect to the classical ones of integer order in terms of convergence time and robustness. It relaxes the classical boundedness condition required to be satisfied by the second-order derivatives of the functions in conventional differentiators. The numerical integration is performed by an implicit Euler discretization technique based on the Fractional Adams-Moulton method, which significantly suppresses the chattering. The significance of the proposed differentiator is demonstrated through a simulation example, comparing with the classical ones.

An Improved Sliding Mode Differentiator Combined with Sliding Mode Filter for Estimating First and Second-order Derivatives of Noisy Signals

This paper proposes a new sliding mode differentiator combined with a sliding mode filter for estimating first and second-order derivatives of noisy signals. The proposed differentiator can be seen as a version of Slotine et al.’s sliding mode observer extended with an additional non-Lipschitz property, which is intended to realize a faster reaching to the sliding mode. It behaves as a noise-reduction filter that is composed of first, second and third-order low-pass filter in the sliding mode, but also employs the filter that is composed of second, third and fourth-order low-pass filter out of the sliding mode. Moreover, the differentiator effectively removes impulsive noises by combining a sliding mode filter and its discrete-time implementation is based on the implicit (backward) Euler discretization, which does not result in chattering and realizes the exact sliding mode. Experiments show that the proposed algorithm has a better balance between the noise attenuation and small phase lag than the linear-filtered Euler differentiation and previous sliding mode differentiators. It was validated through experiments using optical encoder signals of industrial robots

A Brush-Type Tire Model with Nonsmooth Representation

This paper proposes a brush-type tire model with a new mathematical representation. The presented model can be seen as a generic model that describes the distributed viscoelastic force and Coulomb-like friction force, which are balancing each other at each point, in the contact patch. The model is described as a partial differential algebraic inclusion (PDAI), which involves the set-valuedness to represent the static friction. A numerical integration algorithm for this PDAI is derived through the implicit Euler discretization along both space and time. Some numerical comparisons with Magic Formula and a LuGre-based tire model are presented. The results show that, with appropriate choice of parameters, the proposed model is capable of producing steady-state characteristics similar to those of Magic Formula. It is also shown that the proposed model realizes a proper static friction state, which is not realized with a LuGre-based tire model.

Kronecker product based preconditioners for boundary value method discretizations of space fractional diffusion equations

This paper is concerned with the construction of efficient preconditioners for systems arising from boundary value methods time discretization of space fractional diffusion equations. The boundary value methods lead to a coupled block system which is in the form of the sum of two Kronecker products. Our approach is based on an alternating Kronecker product splitting technique which leads to a splitting iteration method. We show that the splitting iteration converges to the unique solution of the linear system and derive the optimal values of the involved iteration parameters. The splitting iteration is then accelerated by a Krylov subspace method like GMRES. One component of the Kronecker product preconditioners has the same structure as the matrix derived from implicit Euler discretization of the problem. Therefore, we can reuse the available high performance of implicit Euler discretization preconditioners as the building block for our preconditioners. Several numerical experiments are presented to show the effectiveness of our approaches.

Torque-Bounded Admittance Control Realized by a Set-Valued Algebraic Feedback

This paper proposes a new admittance controller that realizes safe behavior even under torque saturation. The new controller is analytically equivalent to a conventional admittance controller as long as the actuator torque is not saturated, but is free from unsafe behaviors such as snapping back, oscillation, or overshoots, which may happen with conventional admittance controllers after torque saturation. The new controller is described by a differential algebraic inclusion, and can be understood as a conventional admittance controller expanded with an additional algebraic loop through a normal-cone operator. Its continuous-time representation involves a nonsmooth, set-valued function, but its discrete-time implementation is free from set-valuedness and given as a closed-form algorithm as a result of the use of implicit (backward) Euler discretization. The controller is tested with one joint of an industrial manipulator equipped with a force sensor.

Efficient preconditioners for Radau-IIA time discretization of space fractional diffusion equations

This paper is concerned with the construction of efficient preconditioners for systems arising from implicit Runge-Kutta time discretization methods for one-dimensional and two-dimensional space fractional diffusion equations. We consider the third-order, two-stage Radau-IIA method, which leads to a coupled 2 × 2 block system. Our approach is based on a Schur complement formulation for the unknown at the second stage. We present an approximate factorization of the Schur complement and derive the eigenvalue bounds of the preconditioned system. The components of the approximate factorization have the same structure as the system derived from implicit Euler discretization of the problem. Therefore, we reuse the available high performance of implicit Euler discretization preconditioners as the building block for our preconditioners. Several numerical experiments are presented to show the effectiveness of our approaches.

A New Quick-Response Sliding Mode Tracking Differentiator With its Chattering-Free Discrete-Time Implementation

This paper presents a new quick-response sliding mode tracking differentiator (new TD) for feedback control of mechatronic systems. The new TD is an extension of Jin et al.'s sliding mode tracking differentiator (TD-J) by employing an exponential reaching term for balancing the trade-off between noise attenuating efficiency and convergence speed. The discrete-time algorithm of the new TD is derived by using the implicit-Euler discretization and an equivalent between a set-valued signum function and a saturation function, and it does not produce chattering, which has been one major challenge of implementing sliding mode technique in discrete-time. Simulations and experiments are conducted for validating the effectiveness of the new TD.

A First-Order Differentiator with First-Order Sliding Mode Filtering

This paper proposes a sliding mode differentiator for estimating the first-order derivatives of noisy signals. The proposed differentiator can be seen as a version of Slotine et al.’s sliding mode observer extended with additional non-Lipschitzness. It behaves exactly as a first-order low-pass filter in the sliding mode and is globally convergent. Its discrete-time implementation is based on the implicit (backward) Euler discretization, which does not result in chattering. The differentiator is validated through some numerical examples.