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but the dummy variables y$ and ydot$ need not be declared as state variables 3-18 Dealing with Sampled-data Assignments, Limiters, and Switches A user-defined function involving sampled-data assignments, limiters and/or switches (Table 3-1) generates only one line of DYNAMIC-segment code, and can be thus programmed following an OUT, SAMPLE m, or step statement as
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References
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discussed in Sections 2-10 and 2-11 However, submodels can generate multiple lines that cannot be separated by OUT, SAMPLE m, or step statements in a submodel definition As a result, a submodel must generate only differentialequation-system ( analog ) code, only limiter/switch operations operating on analog variables, or only sampled-data operations Sampled-data assignments can safely include limiter/switch operations It is then, strictly speaking, incorrect to invoke the submodel defined by
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SUBMODEL signal(y$, p$, w$) d/dt y$ = w$ * p$ p$ = sgn(p$ - y$) end
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to produce triangle waves and square waves in the manner of Section 2-17 Serendipitously, the resulting code usually works anyway, presumably because we are only integrating a constant input equal to either a or a
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REFERENCES 1 D Hetrick, Dynamics of Nuclear Reactors, University of Chicago Press, Chicago, 1971 2 G A Korn, A Simulation-model compiler for all seasons, Simulation Practice and Theory, 9, 2001, pp 21 25 3 G A Korn, Interactive Dynamic System Simulation with Microsoft Windows, Taylor and Francis, London, 1998 4 G F Franklin, Digital Control of Dynamic Systems, Addison-Wesley, Reading, MA, 1990 5 G A Korn, A new software technique for submodel invocation, Simulation, March 1987, pp 93 97
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PARAMETER-INFLUENCE STUDIES AND VECTORIZATION 4-1 Exploring the Effects of Parameter Changes Parameter-influence studies explore effects of different combinations of model and experiment parameters Initial state-variable values are treated simply as extra model parameters For a system of differential equations or difference equations, for example,
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(d/dt) x = f(t; x, y; a, b, ) y = g(t; x; c, d, )
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(4-1)
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with suitably differentiable functions f and g, we can measure the sensitivity of x = x(t) and y = y(t) to small changes in the parameter a by computing time histories of the parameter-sensitivity coefficients u(t) x/ a and v(t) y/ a Differentiation of the system equations (4-1) with respect to the parameter a produces the differential-equation system
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(d/dt)u = ( f/ x)u + ( f/ y)v + f/ a y/ a = ( g/ x)u
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Advanced Dynamic-system Simulation: Model-replication Techniques and Monte Carlo Simulation By Granino A Korn Copyright 2007 by John Wiley & Sons, Inc
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In principle, the parameter-sensitivity equations (4-2) can be solved together with the given system equations (4-1) to produce time histories of u and v Parameter-influence coefficients are theoretically interesting But for a system with N equations [Eq (4-1)], in general, 2N equations (4-1) and (4-2) have to be solved even when only the sensitivity of one system variable to a single parameter is needed Even that reveals only effects of small parameter changes It is usually easier to just solve the given system equations for different parameter combinations (Sections 4-2 and 4-3) Monte Carlo simulation with randomly perturbed parameter values (Section 4-4) is also a form of parameter-influence study and permits, for instance, statistical regression of performance measures on parameter values [1] 4-2 Repeated Runs and Model Replication (Vectorization)
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(a) A Simple Repeated-run Study
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Repeated-run parameter-influence studies simply repeat simulation runs with different parameter values As an example, the response x(t) of a damped harmonic oscillator after an initial displacement x(0) = 1 is modeled by the DYNAMIC program segment
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DYNAMIC d/dt x = xdot d/dt xdot = - ww * x - r * xdot X = x - scale | -- offset the display dispt X | -display
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(4-3)
We let xdot(0) default to 0 A small repeated-run parameter-influence study explores the effects of different positive damping coefficients r with the experiment-protocol script
TMAX = 05 | DT = 00001 | NN = 1001 ww = 400 | -- fixed system parameter x = 1 | -- given initial displacement n = 5 | -- number of simulation runs for i = 1 to n | -- set parameter values r=5*i drunr next
(4-4)