Guide to Biometrics for Large-Scale Systems Technological, Operation
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The descriptor form of Lagrangian DAEs and the semiexplicit form of Hamiltonian DAEs are featured prominently in this book for two reasons. First, in these forms equations of motion are obtained from work, energy, and constraint expressions in a systematic manner. Second, recent advances in numerically integrating DAEs makes more practicable than ever their use in modeling and simulation. In this chapter the systematic aspects of the analytical approach are featured and issues relating to the numerical solution of DAEs are outlined. The Lagrangian DAE in descriptor form is used to illustrate basic concepts. An approach to automating the modeling procedure is outlined and numerical examples are given.
6.1. 6.1.1.
Analysis Schematic
A schematic of a physical system must be of sufficient detail to identify the necessary state variables and to illustrate the connections among system elements. No formal schematic-building algorithm is proposed. The central modeling problem of course is to keep the model as simple as possible yet include enough detail to capture significant dynamics. An example of a system schematic is shown in Fig. 6.1. In this electromechanical system, a power supply provides AC electrical power through a rectifier to a DC motor driving a slider-crank mechanism. To simplify this 115
R. A. Layton, Principles of Analytical System Dynamics © Springer-Verlag New York, Inc. 1998
116
6. Modeling and Simulation
Q2.h
power
supply
AC/DC
COllverter
DC ulOtor
slider
crank
FIGURE 6.1. Illustrative electromechanical system.
example, the rectifier is modeled in primitive form and speed control for the crank is omitted. This example is used throughout this section to illustrate the method of analysis.
6.1.2.
Coordinate Selection
The analyst has the freedom to choose the representation of motion and the DAE formulation-Lagrangian, Hamiltonian, or one of the complementary forms-best suited for the problem at hand. Coordinates need not be independent nor comprise a set of minimum dimension. It is often convenient to add coordinates to simplify the work and energy expressions. Each coordinate added above the minimum number of coordinates necessary to describe the motion of the system adds another algebraic constraint to thc DAEs. Selecting a particular representational variable pair implies the selection of a particular formulation. The pair (q,f) is Lagrangian, the pair (q,pT) is Hamiltonian, the pair (p,e) is co-Lagrangian, and the pair (qV,p) is coHamiltonian. Lagrangian and Hamiltonian formulations admit a broader class of constraints than the complementary forms since the complements hold only for constraints that are independent of displacement. Systems that do not meet this criterion must be modeled using either Lagrangian and Hamiltonian formulations. Choosing between Lagrangian and Hamiltonian formulations is largely a matter of personal preference, although Hamiltonian DAEs may have superior numerical properties. For the example at hand, the Lagrangian formulation is used and displacement an
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