Differential Algebraic Framework Research Articles

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as the Newmark methods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier. However, the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses, overcomes such barriers and allows a wide variety of arbitrary implicit-implicit, implicit-explicit, and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement, velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains. By selecting an appropriate spatial method and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body, the proposed work has the potential to better capture the physics of a given simulation. The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility. The accuracy and efficacy of the present work have not yet been seen in the current field, and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

Computational Differential Algebraic Equation Framework and Multi Spatial and Time Discretizations Preserving Consistent Second-Order Accuracy: Nonlinear Dynamics

Abstract We propose the novel design, development, and implementation of the well-known generalized single step single solve (GS4) family of algorithms into the differential algebraic equation framework which not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains but also additionally allows for the selection of different space discretized methods such as the finite element method and particle methods, and other spatial methods as well in a single analysis unlike existing state-of-the-art. For the first time, the user has the flexibility and robustness to embed different algorithms for time integration and different spatial methods for space discretization in a single analysis. In addition, the GS4 family enables a wide variety of choices of time integration methods in a single analysis and also ensures the second-order accuracy in time of all primary variables and Lagrange multipliers. This is not possible to date. However, the present framework provides the fusion of a wide variety of choices of time discretized methods and spatial methods and has the bandwidth and depth to engage in various types of research investigations as well and features for fine tuning of numerical simulations. It provides generality/versatility of the computational framework incorporating subdomains with different spatial and time integration algorithms with improved accuracy. The robustness and accuracy of the present work is not feasible in the current state of technology. Various numerical examples illustrate the significant capabilities and generality and effectiveness for general nonlinear dynamics.

A leader-following consensus problem of multi-agent systems in heterogeneous networks

Within a differential–algebraic framework, this paper studies the leader-following consensus problem for networks of a class of strictly different nonlinear systems. In this case, by allowing any type of interplay between followers and assuming a directed spanning tree in the network, a dynamic consensus protocol with diffusive coupling terms is designed to achieve the leader-following consensus. In this setting, it is revealed that the full closed loop network can be interpreted as an input-to-state convergent system. Moreover, with the premise that differential–algebraic techniques allow us to completely characterize its synchronization manifold, a stability analysis for this manifold is presented. Finally, the effectiveness of the approach is shown in two numerical examples that consider a network of chaotic systems.

Trajectory feasibility evaluation using path prescribed control of unmanned aerial vehicle in differential algebraic equations framework

ABSTRACTMission simulation is a critical activity in the development and operation of Unmanned Aerial Vehicles (UAVs). It is important to ascertain the feasibility of a trajectory in a mission. In this work, an algorithm has been developed for feasibility study of a trajectory of a UAV using prescribed path optimal control through an inverse simulation method. This has been done under a Differential Algebraic Equations (DAE)/Inequalities (DAI) framework. The UAV model together with constraints is represented as a high index DAE system. The trajectory that UAV shall take is prescribed as one of the constraint equations. The solution for the DAE system is obtained using a variation of the alpha method that is capable of handling both equality and inequality constraints on system dynamics. The algorithm involves direct numerical integration of a DAI formulation in a time-stepping manner using a Sequential Quadratic Programming (SQP) solver that detects and satisfy active path constraints at each time step (mesh point). In this unique approach, the model and the constraints are always solved together. The method ensures stable solution at each time step, local minimum at each iteration of simulation and provides a regularised basis to the solver. A typical UAV trajectory has been simulated and demonstrated in this paper. This new approach can be used for path planning of UAVs before the actual control law is designed for flight control computer. Compared to other existing computationally intensive techniques, this approach is computationally simple, ensures continuous constraint satisfaction and provides a viable option for model predictive control of UAVs.

Generalized multi-synchronization: A leader-following consensus problem of multi-agent systems

In this paper, the problem of Generalized Multi-Synchronization (GMS) in master-slave topology is addressed. Within a differential algebraic framework this problem is interpreted as a leader-following consensus problem of Multi-Agent Systems (MAS). Here, a multi-agent system is treated as a network of interconnected systems with strictly different dynamics of same dimension, fixed and not strongly connected topology. Multi-agent system is carried out to a Multi-output Generalized observability Canonical Form (MGOCF) with a family of transformations obtained from an adequate selection of the differential primitive element as a linear combination of state measurements and control inputs. This allow us to explicitly give the synchronization algebraic manifold and design a dynamic consensus protocol able to asymptotically achieve consensus for all agents in the network. Finally a worked out example is provided to illustrate the methodology proposed.

Simulation of nanoparticle coagulation under Brownian motion and turbulence in a differential–algebraic framework: Developments and applications

This paper is devoted to the modeling of the dynamics of nano-aerosols under Brownian motion and turbulence effects. An Eulerian point of view is considered through a population balance equation which is solved in a differential–algebraic framework (DAE-QMOM). This choice is supported by making a brief comparison with reference numerical methods. In order to evaluate the physical insights of the model, a comparison with five available experiments is carried out. The present results reveal the necessity to take into consideration the Brownian and turbulence effects together in the coagulation kernel and also highlight the need to introduce the fractal dimension of aggregates.

Nonlinear estimation in a class of gene transcription process

In this work the Goodwin model applied to gene transcription is employed as a benchmark system for estimation purposes, considering two dynamic behaviors, monotone decreasing and sustained oscillations, each one under a specific parameter’s set. The preceding observability analysis of the Goodwin model was done via linear observability and the differential–algebraic framework, where is proved that the system is fully observable from mRNA concentration measurements. Therefore a class of nonlinear observer which considers a class of sigmoid and linear functions of the output feedback, considering model uncertainties, is proposed and a sketch of proof of the observer’s convergence is provided under the background of the Lyapunov theory, in order to demonstrate asymptotic convergence. Numerical experiments are carrying out in order to show the performance of the proposed methodology which is compared with a standard Luenberger (Proportional) observer and a proportional sliding-mode observer (PSMO).

On finite element method–flux corrected transport stabilization for advection–diffusion problems in a partial differential–algebraic framework

An extension of the finite element method–flux corrected transport stabilization for hyperbolic problems in the context of partial differential–algebraic equations is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ-scheme when applied to the time integration of the resulting differential–algebraic equation is shown, under a mild restriction on the time step size. As a crucial tool in the analysis, the Drazin inverse and the corresponding Drazin ordinary differential equation are explicitly derived. Numerical results are presented for non-constant and time-dependent boundary conditions in one space dimension and for a two-dimensional advection problem with a sinusoidal inflow boundary condition and the advection proceeding skew to the mesh.

Adaptive fault-tolerant control of heavy lift launch vehicle via differential algebraic observer

A novel adaptive fault-tolerant control scheme in the differential algebraic framework was proposed for attitude control of a heavy lift launch vehicle (HLLV). By using purely mathematical transformations, the decoupled input-output representations of HLLV were derived, rendering three decoupled second-order systems, i.e., pitch, yaw and roll channels. Based on a new type of numerical differentiator, a differential algebraic observer (DAO) was proposed for estimating the system states and the generalized disturbances, including various disturbances and additive fault torques. Driven by DAOs, three improved proportional-integral-differential (PID) controllers with disturbance compensation were designed for pitch, yaw and roll control. All signals in the closed-loop system were guaranteed to be ultimately uniformly bounded by utilization of Lyapunov’s indirect method. The convincing numerical simulations indicate that the proposed control scheme is successful in achieving high performance in the presence of parametric perturbations, external disturbances, noisy corruptions, and actuator faults.

Nonlinear filtering methods for spacecraft navigation based on differential algebra

The paper investigates the problem of nonlinear filtering applied to spacecraft navigation. Differential algebraic (DA) techniques are proposed as a valuable tool to implement the higher-order numerical and analytic extended Kalman filters. Working in the DA framework allows us to consistently reduce the required computational effort without losing accuracy. The performance of the proposed filters is assessed on different orbit determination problems with realistic orbit uncertainties. The case of nonlinear measurements is also considered. Numerical simulations show the good performance of the filter in case of both complex dynamics and highly nonlinear measurement problems.

Robust Fault‐Tolerant Control of Launch Vehicle Via GPI Observer and Integral Sliding Mode Control

AbstractA robust fault‐tolerant attitude control scheme is proposed for a launch vehicle (LV) in the presence of unknown external disturbances, mismodeling dynamics, actuator faults, and actuator's constraints. The input‐output representation is employed to describe the rotational dynamics of LV rendering three independently decoupled second order single‐input‐single‐output (SISO) systems. In the differential algebraic framework, general proportional integral (GPI) observers are used for the estimations of the states and of the generalized disturbances, which include internal perturbations, external disturbances, and unknown actuator failures. In order to avoid the defects of the conventional sliding surface, a new nonlinear integral sliding manifold is introduced for the robust fault‐tolerant sliding mode controller design. The stability of the GPI observer and that of the closed‐loop system are guaranteed by Lyapunov's indirect and direct methods, respectively. The convincing numerical simulation results demonstrate the proposed control scheme is with high attitude tracking performance in the presence of various disturbances, actuator faults, and actuator constraints.

Difference algebra and system identification

The framework of differential algebra, especially Ritt’s algorithm, has turned out to be a useful tool when analyzing the identifiability of certain nonlinear continuous-time model structures. This framework provides conceptually interesting means to analyze complex nonlinear model structures via the much simpler linear regression models. One difficulty when working with continuous-time signals is dealing with white noise in nonlinear systems. In this paper, difference algebraic techniques, which mimic the differential-algebraic techniques, are presented. Besides making it possible to analyze discrete-time model structures, this opens up the possibility of dealing with noise. Unfortunately, the corresponding discrete-time identifiability results are not as conclusive as in continuous time. In addition, an alternative elimination scheme to Ritt’s algorithm will be formalized and the resulting algorithm is analyzed when applied to a special form of the nfir model structure.

Simultaneous Quadrature Method of Moments for the Solution of Population Balance Equations, Using a Differential Algebraic Equation Framework

The quadrature method of moments (QMOM) is a recent technique of solving population balance equations for particle dynamics simulation. In this paper, an alternative solution for the QMOM is described and thoroughly tested, which is based on the formulation and simultaneous solution of a semi-explicit differential algebraic equation (DAE) system. The DAE system consists of the ordinary differential equations resulting from the application of the method of moments, as well as a system of nonlinear algebraic equations derived by applying the quadrature theory for the approximation of the moments. It is shown that the proposed approach provides an efficient procedure for evolving the quadrature abscissas and weights from the QMOM. The Jacobian matrix of the DAE system is provided analytically to make the solution more robust. The DAE-QMOM method is compared to the well-established method for solving QMOM based on the product difference (PD) algorithm. The numerical results are compared to the analytical solutions in the case of breakage, aggregation, growth, and nucleation mechanisms. Excellent agreements are found on the moment evolution predicted by both methods. However, the DAE-QMOM method is found to be more accurate and robust than the PD-QMOM in some cases. Additionally, the DAE-QMOM is also capable of providing the solution significantly faster than the PD-QMOM method.

Classical electrical engineering questions in the light of Fliess's differential algebraic framework of non-linear control systems

This paper addresses two important questions of electrical engineering in the differential algebraic framework introduced by Fliess. This lead to an intrinsic formulation of the well known Park transformation and to a generalized state representation of (stand-alone) electrical generators. À mon maître en hommage respecteux pour son jubilé

Non-linear estimation is easy

Non-linear state estimation and some related topics like parametric estimation, fault diagnosis and perturbation attenuation are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with online estimations, illustrate our viewpoint.

Qualitative features of matrix pencils and DAEs arising in circuit dynamics¶

The dynamical behaviour of nonlinear electrical circuits is usually modelled in the time domain by differential–algebraic equations (DAEs). The differential–algebraic formalism drives qualitative analyses based on linearization to a matrix pencil setting. In this context, the present paper performs a spectral analysis of matrix pencils and DAEs arising in nonlinear circuit theory. Specifically, the non-singularity, hyperbolicity and asymptotic stability of equilibria are addressed in terms of circuit topology. The differential–algebraic framework puts the results beyond those already known for state-space models, unfeasible in many actual problems. The topological conditions arising in this qualitative study are proved independent of those supporting the index, and therefore they apply to both index-1 and index-2 configurations. The approach illustrates how graph theory, matrix analysis and DAE theory interact in the dynamical study of nonlinear circuits. ¶Preliminary results of this research were presented at the SCEE'04 Workshop in Capo D'Orlando, Sicilia, Italy 1.

Multipole expansion solution of the Laplace equation using surface data

This paper provides a computational method to model a three-dimensional static electromagnetic field within a finite source free volume starting from discrete field information on its surface. The method uses the Helmholtz vector decomposition theorem and the differential algebraic framework of COSY INFINITY to determine a solution to the Laplace equation. The solution is locally expressed as a Taylor expansion of the field which can be computed to arbitrary order. It provides a natural multipole decomposition of the field which is required for the computation of transfer maps, and also allows to obtain very accurate finite element representations with very small numbers of cells.

The fault detection problem in nonlinear systems using residual generators

In this chapter we study the fault detection problem using residual generators based upon high gain nonlinear observers in a differential algebraic framework. We analyze the stability of the residual generator when a fault occurs. We also consider two faults types: constant and time-varying faults. It is shown that under some mild conditions over the aforementioned faults the residual is different from zero.

Reconstructive correction of aberrations in nuclear particle spectrographs.

A method is presented that allows the reconstruction of trajectories in particle spectrographs and the reconstructive correction of residual aberrations that otherwise limit the resolution. Using a computed or fitted high order transfer map that describes the uncorrected aberrations of the spectrograph, it is possible to calculate a map via an analytic recursion relation that allows the computation of the corrected data of interest such as reaction energy and scattering angle as well as the reconstructed trajectories in terms of position measurements in two planes near the focal plane. The technique is only limited by the accuracy of the position measurements, the incoherent spot sizes, and the accuracy of the transfer map. In practice the method can be expressed as an inversion of a nonlinear map and implemented in the differential algebraic framework. The method is applied to correct residual aberrations in the S800 spectrograph which is under construction at the National Superconducting Cyclotron Laboratory at Michigan State University and to two other high resolution spectrographs.

Lie pseudogroups and differential sequences: New perspectives in two-dimensional conformal geometry

By introducing the general notion of Lie pseudogroups and related geometrical concepts, it is shown that all affine and projective structures on a Riemann surface emerge from the computation of certain differential invariants for a suitable differential sequence, namely the non-linear Janet sequence for the 2-d conformal pseudogroup. Similarly the corresponding gauge theory (for the same 2-d conformal pseudogroup) is related to the first non-linear Spencer sequence. It is proved that the latter incorporates exactly the vanishing curvature conditions obtained in a BRS differential algebraic framework by L. Baulieu, M. Bellon and R. Grimm (Phys. Lett. B 260 (1991) 63) for the “gauging of the Virasoro algebra”. A proper geometrical meaning of this reference is thus provided according to our jet theoretical formulation. At the same time the cohomological link between the non-linear Janet sequence and the first non-linear Spencer sequence is explained.