Original Differential-algebraic Equations Research Articles

Trajectory sensitivity analysis of hybrid systems with sliding motion

This paper concerns hybrid control systems exhibiting the sliding motion. It is assumed that the system’s motion on the switching surface is described by index-2 differential–algebraic equations (DAEs), which guarantee the accurate tracking of the sliding motion surface. For those systems the sensitivity analysis is performed with the help of solutions to system’s linearized equations. The paper states conditions under which the solutions to the linearized equations for original DAEs and the solutions to linearized equations for underlying ordinary differential equations (ODEs) exhibit similar properties. Due to the presence of sliding motion, we restrict the class of admissible control functions to piecewise differentiable functions. The presented sensitivity analysis might be useful in deriving the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding motion and in establishing the global convergence of algorithms for solving those problems.

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.

The differentiation index of nonlinear differential‐algebraic equations versus the relative degree of nonlinear control systems

AbstractIt is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differentiation index for nonlinear differential‐algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems.

Bicycle dynamics and its circular solution on a revolution surface

In this paper, we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution. We employ symbolic manipulations to derive the contact constraint equations from an ordered process, and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations (DAEs), leaving nine coupled differential equations, six contact equations, two holonomic constraint equations and four nonholonomic constraint equations. We then present a complete description of hands-free circular motions, in which the time-dependent variables are eliminated through a rotation transformation. We find that the circular motions, similar to those of the bicycle moving on a horizontal surface, nominally fall into four solution families, characterized by four curves varying with the angular speed of the front wheel. Then, we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface. Furthermore, we directly linearize the nonlinear DAEs, from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates. The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position. We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families. Finally, based on the numerical simulation of the original nonlinear DAEs system, we show that the stable circular motion is not asymptotically stable.

A METHODOLOGY TO OBTAIN ANALYTICAL MODELS THAT REDUCE THE COMPUTATIONAL COMPLEXITY FACED IN REAL TIME IMPLEMENTATION OF NMPC CONTROLLERS

Abstract Model Predictive Control, MPC and NMPC, and real-time optimization, RTO and D-RTO, are known to help plant operability through the mitigation of impacts caused by external disturbances. However, the usage of these tools in industry requires overcoming some challenges, for instance: accurate models of the process, particularly in regard to nonlinearities; suitable computational time for obtaining the solution of large-scale problems and model mismatch between the RTO or D-RTO and NMPC. In this paper, we present a methodology to obtain analytical model predictions based on a Hammerstein structure to represent the process nonlinearities, reducing the computational effort in real-time applications. Unlike most common approaches that transform NMPC internal models, described by differential-algebraic equations (DAE), into an approximate system of nonlinear algebraic (NLA) equations using, for instance, orthogonal collocation, in the proposed approach, the obtained NLA is an exact description of the original DAEs system. The proposed algorithm was applied to a non-isothermal CSTR (continuous stirred tank reactor) integrated with an optimization layer. The results show that the proposed structure presented a significant reduction in computational time without performance loss, when compared with the NMPC using a rigorous model. Moreover, the proposed strategy demonstrated good performance in tracking the targets sent by the optimization layer, without model mismatches between layers.

Parameterized model order reduction for linear DAE systems via ε-embedding technique

In this paper, we present a new method in the reduction of large-scale linear differential-algebraic equation (DAE) systems. The approach is to first change the DAE system into a parametric ordinary differential equation (ODE) system via the ε-embedding technique. Next, based on parametric moment matching, we give the parameterized model order reduction (MOR) method to reduce this parametric system, and a new Arnoldi parameterized method is proposed to construct the column-orthonormal matrix. From the reduced-order parametric system, we get the reduced-order DAE system, which can preserve the structure of the original DAE system. Besides, the parametric moment matching for the reduced-order parametric systems is analyzed. Finally, the effectiveness of our method is successfully illustrated via two numerical examples.

Index-analysis for a method of lines discretising multirate partial differential algebraic equations

In radio frequency applications, electric circuits generate signals, which are amplitude modulated and/or frequency modulated. A mathematical modelling typically yields systems of differential algebraic equations (DAEs). A multivariate signal model transforms the DAEs into multirate partial differential algebraic equations (MPDAEs). In the case of frequency modulation, an additional condition is required to identify an appropriate solution. We consider a necessary condition for an optimal solution and a phase condition. A method of lines, which discretises the MPDAEs as well as the additional condition, generates a larger system of DAEs. We analyse the differentiation index of this approximate DAE system, where the original DAEs are assumed to be semi-explicit systems. The index depends on the inclusion of either differential variables or algebraic variables in the additional condition. We present results of numerical simulations for an illustrative example, where the index is also verified by a numerical method.

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.

Observation of Nonlinear Differential-Algebraic Systems With Unknown Inputs

A method to carry out the state estimation is proposed for a class of nonlinear systems with unknown inputs whose dynamics is governed by differential-algebraic equations (DAE). We achieve, under suitable conditions, to replace the original DAE for a system with differential equations only by using a zeroing manifold algorithm inducing a state space dimension reduction. Observability conditions can be checked using the original system parameters. The state estimation is done using a sliding mode high order differentiator.

Parameter estimation of an electrochemistry-based lithium-ion battery model

Parameters for an electrochemistry-based Lithium-ion battery model are estimated using the homotopy optimization approach. A high-fidelity model of the battery is presented based on chemical and electrical phenomena. Equations expressing the conservation of species and charge for the solid and electrolyte phases are combined with the kinetics of the electrodes to obtain a system of differential-algebraic equations (DAEs) governing the dynamic behavior of the battery. The presence of algebraic constraints in the governing dynamic equations makes the optimization problem challenging: a simulation is performed in each iteration of the optimization procedure to evaluate the objective function, and the initial conditions must be updated to satisfy the constraints as the parameter values change. The ε-embedding method is employed to convert the original DAEs into a singularly perturbed system of ordinary differential equations, which are then used to simulate the system efficiently. The proposed numerical procedure demonstrates excellent performance in the estimation of parameters for the Lithium-ion battery model, compared to direct methods that are either unstable or incapable of converging. The obtained results and estimated parameters demonstrate the efficacy of the proposed simulation approach and homotopy optimization procedure.

ORDER-REDUCED MODELS BASED ON TWO SIDES TECHNIQUES FOR INPUT-OUTPUT SYSTEMS GOVERNED BY DIFFERENTIAL- ALGEBRAIC EQUATIONS

In this paper, a new model order reduction method is presented for solving large-scale differential-algebraic equation (DAE) systems. By nonsingular matrix transforms, the large-scale DAE system is decomposed into an ordinary differential equation (ODE) subsystem and a DAE subsystem with the same index as the original system. A dual weighted H2 model order reduction method is used to reduce the ODE subsystem, which can avoid the problem of large calculation caused by solving the Lyapunov equations. In order to keep the stability of the original DAE subsystem, we present a modified Lanczos model reduction (MLMR) method, which can produce a reduced-order model with better performances. Numerical experiments illustrate the effectiveness of our method.

A Symbolic–Numerical Method for Integration of DAEs Based on Geometric Control Theory

This work describes a symbolic–numerical integration method for a class of differential algebraic equations (DAEs) known as semi-explicit systems. Our method relies on geometric theory of decoupling for nonlinear systems combined with efficient numerical analysis techniques. It uses an algorithm that applies symbolic and numerical calculations to build an explicit vector field $$\tau $$ , whose integral curves with compatible initial conditions are the same solutions of the original DAE. Here, compatible initial conditions are the ones that respect the algebraic restrictions and their derivatives up to their relative degree. This extended set of restrictions defines a submanifold $$\varGamma $$ of the whole space, which is formed by all the variables of the system, and all solutions of the DAE lie on this submanifold. Furthermore, even for nonexactly compatible initial conditions, the solutions of this explicit system defined by $$\tau $$ converge exponentially to $$\varGamma $$ . Under mild assumptions, an approximation result shows that the precision of the method is essentially controlled by the distance of the initial condition from $$\varGamma $$ . A scheme to compute compatible initial conditions with the DAE is also provided. Finally, simulations with benchmarks and comparisons with other available methods show that this is a suitable alternative for these problems, specially for nonexactly compatible initial conditions or high-index problems.

Converting DAE models to ODE models: application to reactive Rayleigh distillation

This paper illustrates the application of an index reduction method to some differential algebraic equations (DAE) modelling the reactive Rayleigh distillation. After two deflation steps, this DAE is converted to an equivalent first-order explicit ordinary differential equation (ODE). This ODE involves a reduced number of dependent variables, and some evaluations of implicit functions defined, either from the original algebraic constraints, or from the hidden ones. Consistent initial conditions are no longer to be computed; at the opposite of some other index reduction methods, which generate a drift-off effect, the algebraic constraints remain satisfied at any time; and, finally, the computational effort to solve the ODE may be less than the one associated to the original DAE.

Extended space method for parameter identifiability of DAE systems

Mathematical models of physical systems often have parameters that must be identified from physical data. This makes the analysis of the parameter identifiability of the given model system an essential prerequisite. Thus far, several methods have been proposed for analyzing the parameter identifiability of ordinary differential equation (ODE) systems. But, to the best of our knowledge, the parameter identifiability of differential algebraic equation (DAE) systems has scarcely been analyzed as a specific topic. Traditional differential algebraic (DA) methods developed for ODE systems are often applied directly on DAE systems. These methods, however, are not always applicable, e.g., when the prime ideal condition is not satisfied by a DAE system. In this paper, we propose a novel method to analyze the identifiability of DAE systems, based on the concept of space extension, through which the algebraic and differential variables can be decoupled. Furthermore, an inherent, low-dimensional, regular ODE system can be obtained, which is the external equivalent of the original DAE system. Subsequently, the differential algebraic (DA) method can then be used to analyze the identifiability of the low-dimension ODE system. Theoretical analysis is also presented for the proposed method. Two examples, including a simplified interaction model and an isothermal reactor system, are presented to illustrate the detailed steps and effectiveness of the proposed method.

Model-order reduction for differential-algebraic equation systems with higher index

In this paper, a new model-order reduction (MOR) approach is presented for reducing large-scale differential-algebraic equation (DAE) systems with higher index. This approach is based upon the balanced truncation, single-point, and multi-point MOR methods. We decompose the DAE system into an ordinary differential equation (ODE) subsystem and a DAE subsystem. The DAE subsystem has the same index as the original DAE system. Then, the balanced truncation method is applied to the ODE subsystem. Both single-point and multi-point methods are used to reduce the DAE subsystem. In generally, the multi-point method can perform better than the single-point method across a wide-range of frequencies. Some numerical examples demonstrate the effectiveness of our approach.

QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems

We present a projection-based nonlinear model order reduction method, named model order reduction via quadratic-linear systems (QLMOR). QLMOR employs two novel ideas: 1) we show that nonlinear ordinary differential equations, and more generally differential-algebraic equations (DAEs) with many commonly encountered nonlinear kernels can be rewritten equivalently in a special representation, quadratic-linear differential algebraic equations (QLDAEs), and 2) we perform a Volterra analysis to derive the Volterra kernels, and we adapt the moment-matching reduction technique of nonlinear model order reduction method (NORM) [1] to reduce these QLDAEs into QLDAEs of much smaller size. Because of the generality of the QLDAE representation, QLMOR has significantly broader applicability than Taylor-expansion-based methods [1]-[3] since there is no approximation involved in the transformation from original DAEs to QLDAEs. Because the reduced model has only quadratic nonlinearities, its computational complexity is less than that of similar prior methods. In addition, QLMOR, unlike NORM, totally avoids explicit moment calculations, hence it has improved numerical stability properties as well. We compare QLMOR against prior methods [1]-[3] on a circuit and a biochemical reaction-like system, and demonstrate that QLMOR-reduced models retain accuracy over a significantly wider range of excitation than Taylor-expansion-based methods [1]-[3]. QLMOR, therefore, demonstrates that Volterra-kernel based nonlinear MOR techniques can in fact have far broader applicability than previously suspected, possibly being competitive with trajectory-based methods (e.g., trajectory piece-wise linear reduced order modeling [4]) and nonlinear-projection based methods (e.g., maniMOR [5]).

The quasi-Weierstraß form for regular matrix pencils

Regular linear matrix pencils A-E∂∈Kn×n[∂], where K=Q, R or C, and the associated differential algebraic equation (DAE) Ex˙=Ax are studied. The Wong sequences of subspaces are investigate and invoked to decompose the Kn into V∗⊕W∗, where any bases of the linear spaces V∗ and W∗ transform the matrix pencil into the quasi-Weierstraß form. The quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V∗ and “pure” inconsistent initial values W∗⧹{0}. Furthermore, V∗ and W∗ are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A-E∂ lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of Ex˙=Ax.

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Lyapunov and exponential dichotomy spectral theory is extended from ordinary differential equations (ODEs) to nonautonomous differential-algebraic equations (DAEs). By using orthogonal changes of variables, the original DAE system is transformed into appropriate condensed forms, for which concepts such as Lyapunov exponents, Bohl exponents, exponential dichotomy and spectral intervals of various kinds can be analyzed via the resulting underlying ODE. Some essential differences between the spectral theory for ODEs and that for DAEs are pointed out. It is also discussed how numerical methods for computing the spectral intervals associated with Lyapunov and Sacker-Sell (exponential dichotomy) can be extended from those methods proposed for ODEs. Some numerical examples are presented to illustrate the theoretical results.

Index reduction for differential–algebraic equations by substitution method

Differential–algebraic equations (DAEs) naturally arise in many applications, but present numerical and analytical difficulties. The index of a DAE is a measure of the degree of numerical difficulty. In general, the higher the index is, the more difficult it is to solve the DAE. Therefore, it is desirable to transform the original DAE into an equivalent DAE with lower index.In this paper, we propose an index reduction method for linear DAEs with constant coefficients. The method is applicable to any DAE having at most one derivative per equality. In contrast to the other existing methods, it does not introduce any additional variables. Exploiting a combinatorial property of degrees of minors in polynomial matrices, we show that the method always reduces the index exactly by one. Thus the paper exhibits an application of combinatorial matrix theory to numerical analysis of DAEs.

The additional dynamics of least squares completions for linear differential algebraic equations

Several approaches have been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations (DAEs) for which more classical numerical methods such as backward differentiation or implicit Runge–Kutta may not be appropriate. One of these approaches is called explicit integration (EI). This approach is based on solving nonlinear DAE derivative arrays using nonlinear singular least squares methods. This results in a computed ODE, called the least squares completion, whose solutions contain those of the original DAE. This ODE is then integrated by a classical numerical method. The additional dynamics of the least squares completion can affect the numerical solution of the DAE. This paper begins the study of determining these extra dynamics.