Hierarchical Structural Analysis Method for Complex Equation-Oriented Models

Structural analysis is a method for verifying equation-oriented models in the design of industrial systems. Existing structural analysis methods need flattening of the hierarchical models into an equation system for analysis. However, the large-scale equations in complex models make structural analysis difficult. Aimed to address the issue, this study proposes a hierarchical structural analysis method by exploring the relationship between the singularities of the hierarchical equation-oriented model and its components. This method obtains the singularity of a hierarchical equation-oriented model by analyzing a dummy model constructed with the parts from the decomposing results of its components. Based on this, the structural singularity of a complex model can be obtained by layer-by-layer analysis according to their natural hierarchy. The hierarchical structural analysis method can reduce the equation scale in each analysis and achieve efficient structural analysis of very complex models. This method can be adaptively applied to nonlinear-algebraic and differential-algebraic equation models. The main algorithms, application cases and comparison with the existing methods are present in this paper. The complexity analysis results show the enhanced efficiency of the proposed method in the structural analysis of complex equation-oriented models. Compared with the existing methods, the time complexity of the proposed method is improved significantly

Approximate Solution of a Reduced-Type Index- k Hessenberg Differential-Algebraic Control System

This study focuses on developing an efficient and easily implemented novel technique to solve the index- k Hessenberg differential-algebraic equation (DAE) with input control. The implicit function theorem is first applied to solve the algebraic constraints of having unknown state differential variables to form a reduced state-space representation of an ordinary differential (control) system defined on smooth manifold with consistent initial conditions. The variational formulation is then developed for the reduced problem. A solution of the reduced problem is proven to be the critical point of the variational formulation, and the critical points of the variational formulation are the solutions of the reduced problem on the manifold. The approximate analytical solution of the equivalent variational formulation is represented as a finite number of basis functions with unknown parameters on a suitable separable Hilbert setting solution space. The unknown coefficients of the solution are obtained by solving a linear algebraic system. The different index problems of linear Hessenberg differential-algebraic control systems are approximately solved using this approach with comparisons. The numerical results reveal the good efficiency and accuracy of the proposed method. This technique is applicable for a large number of applications like linear quadratic optima, control problems, and constrained mechanical systems.

Probabilistic solutions to DAEs learning from physical data

The nonlinear chaotic differential/algebraic equation (DAE) has been established to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. The DAE is obtained only by learning the experimental data with sparse optimization method. However, the deterministic solution will become increasingly inaccurate for long time approximation of the continuous system. In this paper, we introduce two probabilistic solutions to compute the totally DAE, the Random branch selection iteration (RBSI) and Random switching iteration (RSI). The samples are also taken as the reference trajectory to learn random parameter. The proposed probabilistic solutions can be regarded as the discrete analogues of differential inclusion and switching DAEs, respectively. They have been also compared with the deterministic method, i.e. backward differentiation formula (BDF). The deterministic methods only give limited candidates of all the probability solutions, while the RSI can include all the possible trajectories. The numerical results and statistical information criterion show that RSI can successfully reveal the sustaining instabilities of the motion itself and long time chaotic behavior.

The structured modeling framework of anaerobic digestion in Modelica: development of ADMLib package

This paper presents the development of ADMLib, a new high-productivity and efficient Modelica package to model and simulate anaerobic digestion systems inside the structured modeling framework. Library components were organized into subpackages to encompass growth kinetics, non-biochemical reaction kinetics, acid-base, heat transfer, and inhibition processes, as well as the characteristics of substances and gas phase. A validation of the dynamic behavior response was performed where the implemented functions were used to simulate different bibliographic models. A brief performance analysis was carried out, in order to evaluate the component-based approach of ADMLib against the traditional differential algebraic equation (DAE) systems. The implementation testing demonstrated that the developed package was reliable, usable, and performant.

The numerical implementation of land models: Problem formulation and laugh tests

The intent of this paper is to encourage improved numerical implementation of land models. Our contributions in this paper are two-fold. First, we present a unified framework to formulate and implement land model equations. We separate the representation of physical processes from their numerical solution, enabling the use of established robust numerical methods to solve the model equations. Second, we introduce a set of synthetic test cases (the laugh tests) to evaluate the numerical implementation of land models. The test cases include storage and transmission of water in soils, lateral sub-surface flow, coupled hydrological and thermodynamic processes in snow, and cryosuction processes in soil. We consider synthetic test cases as “laugh tests” for land models because they provide the most rudimentary test of model capabilities. The laugh tests presented in this paper are all solved with the Structure for Unifying Multiple Modeling Alternatives model (SUMMA) implemented using the SUite of Nonlinear and DIfferential/Algebraic equation Solvers (SUNDIALS). The numerical simulations from SUMMA/SUNDIALS are compared against (1) solutions to the synthetic test cases from other models documented in the peer-reviewed literature; (2) analytical solutions; and (3) observations made in laboratory experiments. In all cases, the numerical simulations are similar to the benchmarks, building confidence in the numerical model implementation. We posit that some land models may have difficulty in solving these benchmark problems. Dedicating more effort to solving synthetic test cases is critical in order to build confidence in the numerical implementation of land models.

TIME DOMAIN METHOD FOR THE DETERMINATION OF THE STEADY STATE BEHAVIOUR OF NONLINEAR CIRCUITS DRIVEN BY MULTI-TONE SIGNALS

Time domain methods, while well suited to compute the steady state behaviour of strongly nonlinear non-autonomous electrical circuits, are inefficient if the periods of the forcing signals have a very large minimum common multiple. The solution of the periodicity constraint requires to integrate the differential algebraic equation (DAE) describing the circuit along the T period and this can be a CPU time consuming task. Literature reports several attempts to extend the SH method to simulate circuits driven by multi-tone signals [2] [4] [5]. However, as far as we know, all they suffer of limitations and it is our opinion that an efficient and general extension has not been found, yet. In this paper we present a possible extension that takes its origin from the previous approach reported in [2]. In this paper a modification of the conventional shooting method is presented that tries to overcome the above drawback.

A class of optimal control problems governed by singular systems via Balakrishnan’s method

The purpose of this paper is to discuss, by the use of the Balakrishnan’s epsilon method, a class of optimal control problems governed by continuous linear time invariant singular systems which have only a finite dynamic mode. The linear differential algebraic equation is handled using the epsilon technique to obtain a sequence of the calculus of variations problems. A convergence theorem is given to obtain approximate and, in the limit, an optimal solution of this class of optimal control problem by the use of the necessary optimality conditions of Euler–Lagrange type. A correspondence has been also shown between this penalty function and duality for this class of optimal control problems considered. As an application, an example of optimal linear quadratic problem is also given.