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Software for Modeling and Simulating Control Systems
Published in William S. Levine, Control System Fundamentals, 2019
Martin Otter, François E. Cellier
Using the object-oriented modeling methodology, this approach to controller design can be implemented elegandy. The user would start out with the reference model and the plant dynamics model. The input of the reference model is then declared as external input, the output of the reference model is connected to the output of the plant dynamics model, and the input of the plant dynamics model is declared as external output. Object-oriented modeling systems, such as Dymola, are capable of generating either a DAE or an ODE model from such a description. However, the original set of equations resulting from connecting the submodels in such a fashion is invariably of higher index. The Pantelides algorithm is used to reduce the index down to one, leading to a DAE formulation containing algebraic loops but no dependent storage elements.
Modelica as a Platform for Real-Time Simulation
Published in Katalin Popovici, Pieter J. Mosterman, Real-Time Simulation Technologies, 2017
John J. Batteh, Michael M. Tiller, Dietmar Winkler
The Modelica language has evolved as a collaboration between computer scientists and engineers to provide a modeling language to describe the behavior of multi-domain physical systems in an open format not tied to a particular tool. The roots of the Modelica language can be traced back to Hilding Elmqvist’s PhD thesis [3] in which he designed and implemented the Dymola modeling language. The Dymola language used an object-oriented, equation-based approach to formulate models of physical systems. Rather than forcing modelers to pose the problem using ordinary differential equations (ODEs) for which many solvers exist, the Dymola language allowed the problem to be posed as differential-algebraic equations (DAEs), which is generally considered a more natural way to describe physical problems [4]. Sophisticated symbolic algorithms were used in the language implementation to transform the mathematical model into a form capable of solution by existing numerical solvers. As symbolic algorithms advanced, in particular, the Pantelides algorithm [5] for DAE index reduction, it became possible to solve an even larger class of problems using the new approach pioneered by the Dymola language. Within the context of the ESPRIT project “Simulation in Europe Basic Research Working Group (SiE-WG),” Hilding Elmqvist in 1996 sought to bring together a group of object-oriented modeling language developers and engineering simulation experts in an attempt to develop a new modeling language to describe physical systems over a wide range of engineering domains. After a series of 19 meetings over the course of the next 3 years, version 1.3 of the Modelica language specification was released in December 1999.
The structural index of sensitivity equation systems
Published in Mathematical and Computer Modelling of Dynamical Systems, 2018
Atiyah Elsheikh, Wolfgang Wiechert
Applying numerical solvers on the reduced resulting systems alone via Pantelides algorithm may not lead to accurate numerical results. Numerical results tend to drift away from the original algebraic constraints that have been differentiated by the process of index reduction [53]. The numerical integration of systems of differential equations at each time step resolves to problem of nonlinear system equation where hidden constraints are not explicitly included. In practice, the Pantelides algorithm is combined with the method of dummy derivatives [18]. This method rather augments a higher-index DAE with the differentiated equations selected by the Pantalides algorithm. However for each new generated equation, a new dependent algebraic variable is introduced replacing the derivative of a selected variable (i.e. a dummy derivative). In this way, the generated reduced system of equations is well-determined in terms of the number of unknowns. Overall, the index gets reduced while the original algebraic constraints are kept in the system of equations and hence they are still subject to satisfaction by the numerical solvers. Algorithm 2 lists a modification of Algorithm 1 that combines the method of dummy derivatives with Pantelides algorithm. Here, line 6 from Algorithm 1 is modified and lines 9–11 from Algorithm 2 are inserted in Algorithm 1. Section 4 provides two examples demonstrating the index reduction process.
An algorithm to modify consistent initialization of differential-algebraic equations obtained by pantelides algorithm using minimally singular subsets
Published in SICE Journal of Control, Measurement, and System Integration, 2021
Keisuke Shimako, Masanobu Koga
MATLAB [2] and Mathematica [3] can solve the DAE systems. These tools deal with high-index DAE systems by reducing the index to at most 1. The tools listed here use Pantelides algorithm [4] to reduce the index. Pantelides algorithm uses the condition to differentiate the equations of the DAE systems and reduces the index until they reach 1 at most. However, the index of some DAE system cannot be reduced correctly under the condition provided by the algorithm. Figure 1 shows the result in MATLAB when Pantelides algorithm is actually applied in MATLAB to a DAE system whose index cannot be reduced correctly with Pantelides algorithm.
Differential algebraic observer-based trajectory tracking for parallel robots via linear matrix inequalities
Published in International Journal of Systems Science, 2022
J. Álvarez, J. Servín, J. A. Díaz, M. Bernal
Due to the nature of the DAEs considered in this work, once the LMIs in Theorem 3.1 are feasible and the nonlinear controller/observer gains are calculated, the simulation of the closed-loop system is performed in the recently appeared DAE Toolbox (Nedialkov et al., 2013). Obviously, the consistent initial conditions and the dynamics of the solution are restricted to a subset of the state space where algebraic restrictions hold. Alternatively, the Pantelides algorithm can be employed to reduce the DAEs to ODEs, thus allowing ordinary simulation tools to be employed, though numerical robustness can be compromised in the latter case.