Set Of Differential Algebraic Equations Research Articles

A differential-algebraic extended Kalman filter with exact constraint satisfaction

This work proposes an Augmented Manifold Differential-Algebraic Extended Kalman Filter for state and input estimation for multibody systems described by a set of differential algebraic equations. The proposed Kalman Filter allows for exact constraint satisfaction and does not require reformulation of the multibody system equations. The approach consists of two main steps: (i) elimination of the Lagrange multipliers from the system state via a nullspace projection and (ii) an additional correction step that constrains the a-posteriori estimated system state to lie on the constraint manifold. The latter is achieved by solving a constrained optimization problem using a Euclidean descent approach and projection on the constraint manifold, i.e. manifold optimization. The proposed approach is numerically validated on both open and closed kinematic chain rigid multibody models and is shown to be robust and indifferent to the type of kinematic chain being considered. Furthermore, an experimental validation is carried out where the forces and torques acting on the wheel center of a McPherson car suspension test setup are successfully estimated using a flexible multibody model and the proposed Kalman Filter.

Application of the Constrained Formulation to the Nonlinear Sloshing Problem Based on the Arbitrary Lagrangian–Eulerian Method

Abstract This study deals with an application of constrained formulation to a nonlinear sloshing problem based on the arbitrary Lagrangian–Eulerian finite element method (ALE). The ALE method incorporates a discretized form of equations of motion with mesh updating algorithms in order to prevent a problem of mesh distortion. This paper focuses on an analytical aspect of such treatments as constrained systems in the formulation of the ALE method. Since the mesh updating algorithms give algebraic relations for nodal coordinates, this study treats these relations as constraints. Then, we introduce formulation for constrained systems based on the method of Lagrange multipliers. As a result of this formulation, equations of motion are given by differential algebraic equations (DAEs) consisting of differential equations for time evolution of physical quantities and algebraic equations (constraints). The present method can be classified into a kind of augmented formulation. Moreover, we present a size-reduction technique used in the Newton–Raphson method in order to remove a part of the redundant degrees-of-freedom in the iterative procedures, because the resulting set of DAEs involves a larger number of unknowns than the minimal number of degrees-of-freedom due to the introduction of the constrained formulation. The derived reduced system still holds the physically essential part of equations of motion for the sloshing problem. Even though it is not reduced to the minimal degrees-of-freedom, it does not involve algebraically complicated and inefficient procedures in computation. In addition, this study presents a method to introduce damping effects defined in the modal space into the FEM models. The proposed approach is validated by comparisons with experimental data in the time domain analysis.

Numerical and experimental investigation on the synthesis of extended Kalman filters for cable-driven parallel robots modeled through DAEs

Cable-driven parallel robots are parallel robots where light-weight cables replace rigid bodies to move an end-effector. Their peculiar design allows obtaining large workspaces, high-dynamic handlings, ease of reconfigurability and, in general, low-cost architecture. Knowing the full state variables of a cable robot may be essential to implement advanced control and monitoring strategies and imposes the development of state observers. In this work a general approach to develop nonlinear state observers based on an extended Kalman filter (EKF) is proposed and validated both numerically and experimentally by referring to a cable-suspended parallel robot. The state observer is based on a system model obtained by converting a set of differential algebraic equations into ordinary differential equations through different formulations: the penalty formulation, the Udwadia–Kalaba formulation, and the Udwadia–Kalaba–Phohomsiri formulation, which have been chosen since they can handle the presence of redundant constraints as often happens in cable-driven parallel robots. In the numerical investigation, the EKF is validated simulating encoders heavily affected by quantization errors to demonstrate the filtering capabilities of EKF. In the experimental investigation, a very challenging validation is proposed: only two sensors measuring the rotations of two motors are used to estimate the actual position and velocity of the end-effector. This result cannot be achieved by sole forward kinematics and clearly proves the effectiveness of the proposed observer.

Multiphase Porous Electrode Theory for the Next Generation of SOFC/SOEC Electrodes

The mixed-ionic electronic conduction (MIEC) of gadolinium doped ceria (CGO) under reduced oxygen conditions makes it an excellent fuel electrode material for SOFC/SOEC applications. As part of a composite electrode (Ni/CGO), the nickel phase offers a fast electronic conduction pathway to the current collector and may act as an electrocatalyst at the three-phase boundary. Although the Ni/CGO cermet provides superior electrochemical performance compared with older technologies such as Ni/YSZ, the models used to study MIEC materials do not capture the unique transport and kinetic physics which makes them an excellent choice for the next generation of fuel electrodes. Within the framework of multiphase porous electrode theory, this work provides a novel set of differential-algebraic equations which captures the effects of the activation overpotential on electronic defect concentration, electrostatic surface potential and ionic transport to accurately predict the current-voltage behaviour of the Ni/CGO electrode. Moreover, through concerted electron and proton tunnelling events, we unify the theory of the electrostatic surface potential with proton-coupled electron transfer kinetics.

Development of a constraint stabilization method of multibody systems based on fuzzy logic control

The numerical solution of multibody systems is not a straightforward problem. The formulation of the equations of motion is augmented with the constraint equations that lead to a set of differential algebraic equations (DAEs). These constraints govern the relative motion between the system’s components at the position level (geometric constraints) and may restrict the velocity of particular components (rolling constraints). There are several factors that determine the effectiveness of numerical integration methods and the extent of their applicability owing to the various motion circumstances. These factors include numerical stability throughout the integration and computation time, as well as allowable error percentage and the length of simulation time. In this regard, this research examines existing approaches for constraint stabilization during numerical integration and introduces a new methodology based on fuzzy control algorithm, whose coefficients are independent of the dynamic characteristics of different systems. Schematics of the new methodology are presented; two examples of spatial multibody systems with holonomic and nonholonomic constraints are solved to evaluate the effectiveness of the proposed method. It can be concluded that fuzzy control contributes an excellent solution for generic system configuration and is suitable for lengthy simulations with minimal computation time.

Modeling of closed-circuit ball milling of cement clinker via a PBM with a variable Tromp curve for classification

Full-scale closed-circuit cement ball milling was modeled using a true unsteady-state simulator (TUSSIM), based on a transient cell-based population balance model (PBM) with a set of differential algebraic equations (DAEs). As a major novelty, the PBM for the mill was coupled with a dust load-dependent, variable Tromp curve for the air classifier. Results from the dynamic simulations suggest that lower air flow rate or higher rotor speed of the classifier not only led to a finer product but also increased the dust load of the classifier feed. When the dust load was too high, operational failure due to overloading of the whole circuit was detected. Finally, TUSSIM was used for process optimization with a global optimizer-DAE solver to identify the air classifier's parameters that yielded desirable cement quality while maximizing production rate. We have demonstrated that the optimization could increase production rate by 7% compared to the baseline process.

Modeling and Simulation of a Single Catalytic Fixed Bed in an Ammonia Reactor

We present a dynamical model of a single catalytic fixed bed in a quench-type ammonia reactor. This model consists of thermodynamics, kinetics, pressure loss, and mass and energy balances, which upon spatial discretisation results in a set of differential-algebraic equations. Compared to similar models in the literature, the main novelties in the model are: 1) a rigorous thermodynamic model using a thermodynamic software library; and 2) a correlation between the gas velocity and the pressure gradient throughout the bed, i.e. the pressure loss is included in the model and used to compute the gas velocity. We investigate steady state solutions and dynamical simulations to understand the behaviour of the single catalytic fixed bed. We determine the optimal steady state operational area and inverse response dynamics are observed. The model can be used to develop model-based control, which is relevant for dynamic and flexible operation of ammonia reactors in a Power-to-Ammonia plant.

Joint–coordinate adjoint method for optimal control of multibody systems

This paper presents a joint–coordinate adjoint method for optimal control of multi-rigid-body systems. Initially formulated as a set of differential-algebraic equations, the adjoint system is brought into a minimal form by projecting the original expressions into the joint’s motion and constraint force subspaces. Consequently, cumbersome partial derivatives corresponding to joint-space equations of motion are avoided, and the approach is algorithmically more straightforward. The analogies between the formulation of Hamilton’s equations of motion in a mixed redundant-joint set of coordinates and the necessary conditions arising from the minimization of the cost functional are demonstrated in the text. The observed parallels directly lead to the definition of a joint set of adjoint variables. Through numerical studies, the performance of the proposed approach is investigated for optimal control of a double pendulum on a cart. The results demonstrate a successful application of the joint-coordinate adjoint method. The outcome can be easily generalized to optimal control of more complex systems.

SimCOVID: Open-Source Simulation Programs for the COVID-19 Outbreak.

This paper presents open-source computer simulation programs developed for simulating, tracking, and estimating the COVID-19 outbreak. The programs consist of two separate parts: one set of programs built in Simulink with a block diagram display, and another one coded in MATLAB as scripts. The mathematical model used in this package is the SIR, SEIR, SEIRD, and SEIRV models represented by a set of differential–algebraic equations. It can be easily modified to develop new models for the problem. A generalized method is adopted to simulate worldwide outbreaks in an efficient, fast, simple, and visualized way. To get a good tracking of the virus spread, a sum of sigmoid steps was proposed to capture any dynamic changes in the data. The parameters used for the input (infection and recovery rate functions) are computed using the parameter estimation tool in MATLAB. Several statistic methods were applied for the rate function including linear, mean, root-mean-square, and standard deviation. In addition, an adaptive neuro-fuzzy inference system is employed and proposed to train the model and predict its output. Another program is presented for visualizing the COVID-19 data for each country worldwide in different dimensional views. The programs can be used as a teaching tool and for research studies.

Modeling and analysis of a geometrically nonlinear joined wing under thrust force

The joined-wing aircraft is an alternative configuration compared to general cantilevered configuration because of its superiority. In the present work, an efficient modeling method for flexible joined wing is proposed. The fully intrinsic formulation is applied for the geometrically nonlinear effect and the static indeterminacy of joined wing is addressed based on the force method. To avoid mathematically over-determined problem, an orientation compatibility condition with profound physical meaning is established and subsequently a strict mathematical proof is given for the existence and uniqueness of rotation. By adopting Peters finite inflow theory, the unsteady aerodynamic model for joined wing is built up. The joined-wing aeroelastic model is eventually represented by a set of differential-algebraic equations. Following validation studies show that although the displacements and rotations are excluded, contributing to higher computing speed than the mixed formulation, the numerical accuracy of present method is close to mixed formulation and superior to the incremental method. Finally, the static aeroelastic analysis for a planar joined wing under a thrust force at wingtip is implemented within flutter boundary. Results show that the thrust force can lead to an axial compressive load in front wing and the co-existing solutions are revealed because of the buckling of front wing when the thrust force exceeds a threshold value. Further studies indicate that although either solution path is possible theoretically, the “upward-bending solution” is a more likely practical solution for the planar joined wing.

An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

Navier-Stokes equations are well known in modeling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performance of our proposed method compared to other known methods.

Modeling of Memristors under Periodic Signals of Different Parameters

In this paper, the problem of modeling memristors is studied. Two types of memristors with carbon and tungsten doping fabricated by the Knowm Inc. are tested. The memristors have been examined with either sinusoidal or triangle voltage wave periodic excitation. Some different frequencies, amplitudes and signal shapes have been applied. The collected data have been averaged and subjected to high frequency filtering. The quality of measurement data has also been discussed. The averaged measurement has been modeled using three popular memristor models: Strukov, Biolek and VTEAM. Some additional feathers to the considered models have been proposed and tested. Memristor is usually modeled by a set of algebraic-differential equations which link both electrical values (i.e., voltage and current) and the internal variable(s) responsible for the element dynamics. The interior-point with box constrains optimization method has been used to obtain the optimal parameters of the memristor model that fit best to the collected data. The results of the optimization process have been discussed and compared. The sensitivity to the different frequency range has been also examined and reviewed. Some conclusions and future work ideas have been postulated.

Estimating algorithms for prediction and spread of a factor as a pandemic: a case study of global COVID-19 prevalence

Background: This paper presents open-source computer simulation programs developed for simulating, tracking, and estimating the COVID-19 outbreak. Methods: The programs consisted of two separate parts: one set of programs built in Simulink with a block diagram display, and another one coded in MATLAB as scripts. The mathematical model used in this package was the SIR, SEIR, and SEIRD models represented by a set of differential-algebraic equations. It can be easily modified to develop new models for the problem. A generalized method was adopted to simulate worldwide outbreaks in an efficient, fast, and simple way. Results: To get a good tracking of the virus spread, a sum of sigmoid functions was proposed to capture any dynamic changes in the data. The parameters used for the input (infection and recovery rate functions) are computed using the parameter estimation tool in MATLAB. Several statistic methods were applied for the rate function including linear, mean, root-mean-square, and standard deviation. In addition, an adaptive neuro-fuzzy inference system (ANFIS) is employed and proposed to train the model and predict its output. Conclusion: Due to the fact that COVID-19 disease has not been detected in a large part of the passengers, because they do not have any symptoms yet. For this purpose, in setting our forecast model, two scenarios are considered for the probability of a symptomatic person to travel until 1) the first medical visit 2) hospitalization. Therefore, in the present study, an open source toolbox for modeling, simulation and estimating the prevalence of coronavirus was presented. This procedure is presented in such a way that it can be generalized and applied in other parts and applications of estimating the scenarios of an event, including the potential of several models, including SIR, which is sensitive to pollution, etc. In addition, several statistical measures were used to determine the optimal time parameters for sigmoid functions. also, an adaptive fuzzy neural inference system was used to generate model output based on some of the training tasks applied to the system. This program can be used as an educational tool or for research studies and this article promises some lasting contributions to field of COVID-19.

Reduced-order interval observer design for continuous-time descriptor LPV systems with uncertainties

In this paper, a reduced-order interval observer (R-IO) design technique is developed for continuous-time descriptor linear-parameter-varying systems with unknown-but-bounded uncertainties. First, by introducing an intermediate variable, the R-IO design amounts to estimating the linear functional of the system states. Then, with a parameter-dependent Luenberger-like structure, the R-IO existence condition is eventually formulated into a set of differential-algebraic equations, not involving any linear transformation process. Further, a parametric solution to such an R-IO is derived through solving the equation set, which clearly shows the design degrees of freedom. Finally, the correctness of the proposed results is verified by a numerical system and an electrical circuit system.

A modular approach for dynamic modeling of multisegment continuum robots

The accurate dynamic model for continuum robots is necessary to expand their range of applications. In this paper, a modular approach is proposed for dynamic modeling of cable-driven multisegment continuum robots. The previous dynamic models are usually formulated as a set of ordinary differential equations. However, the derived dynamic model in this paper is formulated as a set of differential-algebraic equations, resulting in the coefficient matrices of adjacent modules being decoupled. Reformulation of the existing coefficient matrices is not required as the number of modules increases, which makes the presented approach suitable for modular modeling. The continuum deformations, including elongation or contraction, bending, and torsion, are taken into account. Hermite interpolation is adopted to describe the flexible shape of each backbone, avoiding the singularity and additional constraints of constant strains. Static and dynamic experiments were implemented on continuum robot prototypes. The normalized errors are less than 3.83% and 6.93% of the robot length in the static and dynamic experiments. The presented model is experimentally validated.

Data-driven reduced bond graph for nonlinear multiphysics dynamic systems

We propose a data-driven approach aiming at reducing Bond Graphs by using simulation data. The output of the approach is a hybrid model that contains a Reduced Bond Graph coupled to a simple artificial neural network. In this paper, the neural network is obtained by a linear calibration procedure. We propose a method that contains two training steps. First, the method selects the components of the original Bond Graph that are kept in the reduced Bond Graph. Secondly, the method builds an artificial neural network that supplements the reduced Bond Graph. Because the output of the proposed approach is a hybrid model, not solely data, it becomes difficult to use a usual Backpropagation Through Time to calibrate the weights of the neural network. So, in a first attempt, a very simple neural network is proposed by following a model reduction approach. As an industrial application, we consider the modeling of the automotive cabins thermal behavior. The data used for the training step are obtained via solutions of differential algebraic equations by means of a design of experiment. Simple cooling simulations are run during the training step. We show a simulation speed-up when the reduced bond graph is used to simulate the driving cycle of the WLTP vehicles homologation procedure, while preserving accuracy on output variables. The variables of the original bond graph are split into a set of primary variables, a set of secondary variables and a set of tertiary variables. The reduced bond graph contains all the primary variables, but none of the tertiary variables. While secondary variables are coupled to primary ones via an artificial neural network. Besides, the mathematical modeling used to describe the proposed method allows generalizing it to any set of differential algebraic equations, and not only to bond graphs. Finally, we discuss the extension of the coupling approach to more complex artificial neural networks.

Tracking control for underactuated non-minimum phase multibody systems

We consider tracking control for multibody systems which are modeled using holonomic and non-holonomic constraints. Furthermore, the systems may be underactuated and contain kinematic loops and are thus described by a set of differential-algebraic equations that cannot be reformulated as ordinary differential equations in general. We propose a control strategy which combines a feedforward controller based on the servo-constraints approach with a feedback controller based on a recent funnel control design. As an important tool for both approaches, we present a new procedure to derive the internal dynamics of a multibody system. Furthermore, we present a feasible set of coordinates for the internal dynamics avoiding the effort involved with the computation of the Byrnes–Isidori form. The control design is demonstrated by a simulation for a nonlinear non-minimum phase multi-input, multi-output robotic manipulator with kinematic loop.

Frequency Stability Constrained Optimal Power Flow Incorporating Differential Algebraic Equations of Governor Dynamics

Frequency dynamics during primary frequency control is closely related to pre-disturbance operating conditions, the disturbance, and equipment characteristics. How to rigorously express such a relation in a frequency stability constrained optimal dispatch problem is still an open question. This paper presents an optimal power flow (OPF) model for generation re-dispatch that considers dynamic frequency response constraints to ensure frequency stability during primary frequency regulation. The distinct feature of the model is that the dynamic frequency response is formulated as a set of differential algebraic equations (DAEs), which allows considering the transient behavior of frequency and mechanical/electromagnetic power for each generator during primary frequency control period. Based on the solution of the optimization problem, a definition of the primary reserve for each unit is proposed according to the maximum of incremental mechanical power. This definition helps meet the adequacy requirement of primary frequency response and achieve partial frequency restoration. Simulation results on WSCC 3-machine 9-bus system, New England 10-machine 39-bus system and the modified IEEE 54-machine 118-bus system validate the effectiveness of the proposed model and reveal the strong coupling between frequency dynamics and the pre-disturbance generation.

Lyapunov-based economic model predictive control for nonlinear descriptor systems

A large number of physical applications such as electrical circuits, chemical processes and multibody systems can be accurately described by a set of differential algebraic equations (DAEs). For instance, sustainable water desalination techniques such as membrane distillation (MD) is an example of a chemical process that is described by DAEs. Optimal control strategies such as economic model predictive control (EMPC) has many operational advantages (e.g., maximizing process economics and enhancing operational safety) for chemical applications. However, constructing EMPC schemes for a general class of nonlinear DAEs systems while ensuring closed-loop stability and recursive feasibility is an open research challenge that has not been considered. Motivated by the above considerations, this note introduces a Lyapunov-based economic model predictive control (LEMPC) design that can economically operate nonlinear descriptor systems while satisfying input and states constraints. To guarantee closed-loop stability and recursive feasibility of the proposed control design, a Lyapunov-based control law is introduced to characterize the stability region at which the LEMPC can maximize the process economics. Finally, a chemical batch process modeled by nonlinear differential algebraic equations (DAEs) is utilized to demonstrate the applicability of the proposed framework.

Adaptive Fuzzy Control for Attitude Stabilization of Spacecraft with Deployable Composite Laminated Solar Array

Modern spacecraft are often equipped with large-scale, complex, and lightweight solar arrays whose deployment involves a highly dynamic movement. This paper proposed a novel adaptive proportional-derivative typed fuzzy logic control scheme for the attitude stabilization of a flexible spacecraft during the deployment of a composite laminated solar array. First, a constrained rigid-flexible coupling spacecraft model consisting of a rigid main body and a flexible solar array was proposed. The solar array, which is composed of composite laminated shells, was described by the absolute nodal coordinate formulation. Then, the detailed derivation of the adaptive fuzzy PD controller for attitude stabilization of the spacecraft was discussed. In addition, the spacecraft dynamic model which integrated the adaptive fuzzy PD controller was derived as a set of differential-algebraic equations. Several simulations were developed to investigate the solar array deployment dynamics and to verify the effectiveness of the proposed adaptive fuzzy PD controller. The results suggested that the proposed dynamic model is able to exactly describe the deployment dynamics of the composite laminated solar array. The solar array deployment causes obvious translational and rotational motions of the spacecraft. The proposed adaptive fuzzy PD control scheme has better performance in terms of the control precision and time response in stabilizing spacecraft during the deployment of the composite laminated solar array, comparing with that of the conventional PD controller.