In Modelling & Simulation, Modelling is the process of representing a model which It will help them understand the basic concepts related to Modelling. area of modeling and simulation. . Q.5 What is Modeling & Simulation? function (pdf) or its discrete counterpart, the probability mass function (pmf). It is also. Components of a System. ▫ Discrete and Continuous Systems. ▫ Model of a System. ▫ Types of Models. ▫ Discrete-Event System Simulation. ▫ Steps in a.

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Introduction To. Modeling & Simulation. (Part 1). Bilgisayar Mühendisliği Bölümü – Bilkent Üniversitesi – Fall Dr.Çağatay ÜNDEĞER. Öğretim Görevlisi. Please cite this book as: Claudius Ptolemaeus, Editor,. System Design, Modeling, and Simulation using Ptolemy II, anesi.info, anesi.info PDF | Introduction to M&S for Engineering Managers.

This is accomplished by using a device called a control break. Here we wish to cause a subsystem, which we call a plant, to behave in some prescribed manner. The problem statement was rather vague on this point. We call such systems autonomous, since they are totally self-contained and their behavior is independent of external influence. Even so, since they arise so often, it is customary to handle them by encapsulated modules. Thus, even though design is but one facet, designers are usually well versed in all systems aspects, including analysis and management as well as design.

Profozich, P. Roberts, N. Andersen, R. Deal, M. Garet, and W. Shaffer, Introduction to Computer Simulation. Addison-Wesley, Sage, A. Sandquist, G. Thompson, Simulation: Wiley Interscience, Vemuri, V, Modeling of Complex Systems. Watson, H. White, H. Saunders, Zeigler, B. Assuming that the state variable is the current i t.

Exercises 1. Use the voltage across the capacitor vc t as the state variable and find the state and output equations analogous to Equation 1. Define the column vector y for the state variable so that this system is in standard linear form. That is, for matrices A, B, C, and D,. Find an explicit equation for the sampled signal x k. Compare your results against the theoretical expectations. Present your conclusions in the form of a graph of error versus number of iterations.

Notice that the exponential form of n implies a logarithmic scale for the iteration axis. The probability an incoming box has a given weight is as follows:. Compare the two distributions by forming a distribution and a histogram. Describing Systems b Using the simulation, compute an average inter-event time. Try to use as few memory units as possible. Create the transition diagram of the new state machine. The first output z1 behaves exactly like z and generates the sequence 3, 1, 5, 2, 3, 1, 5, 2, The second output z2 also behaves like z, but generates the sequence 3, 1,5, 2, 7, 8, 3, 1,5.

However, as a result of an angular quantization error that is uniformly distributed over [5,5], this pair of stations calculate an apparent position P x'. Simulate this system mathematically and analyze the results.

Write a program, with inputs S and n the number of points to be sampled , that reads the n actual points x,y ; these are tabulated below. The program should calculate the apparent points x',y' as seen by each of the two stations Pl and P2. Using the points defining the actual trajectory defined in the table below, compute and tabulate the apparent coordinates. Graph the actual and apparent trajectories for several different quantization error sizes 6.

Dynamical Systems Mathematical models of continuous systems are often defined in terms of differential equations. Differential equations are particularly elegant, since they are able to describe continuous dynamic environments with precision. In an ideal world, it would be possible to solve these equations explicitly, but unfortunately this is rarely the case. Even so, reasonable approximations using numerical difference methods are usually sufficient in practice. This chapter presents a series of straightforward numerical techniques by which a great many models can be approximated using a computer.

Dynamical systems are characterized by their system state and are often described by a set of differential equations. If the differential equations, combined with their initial conditions, uniquely specify the system, the variables specified by the initial conditions constitute the system state variables. In general, suppose there are m differential equations, each of order ni. Equivalently, there are n first-order differential equations. Equation 1 2. The output variables of each of these first-order differential equations, each along with their single initial condition, comprise a set of system state variables.

However, since the equations are not unique, neither are the state variables themselves. Nonetheless, there are exactly n of them.

Therefore, we begin by considering the first-order initial-value problem. Since there are two dynamic state variables those involving derivatives and firstand second-order differential equations, this is a third-order system.

Therefore, it is possible to re-define this as a system of three first-order differential equations. Noting that x2 is the derivative of x1, Equations 2. Euler's Method Since the technique of Example 2. Further, without loss of generality, we consider the scalar version of Equation 2. Using the definition of derivative, lim. Whether we are discussing continuous or discrete time, the sense of the variable x should be clear from the context.

If the time variable is t, the signal is taken as continuous or analog and the state is x t. Or, if the time is discrete, the state variable is x k. The process of replacing continuous time by discrete time is called discretization.

Solving Equation 2. Notice that all variables on the right-hand side of Equation 2. Therefore, we refer to this expression as an update of variable x and often do not even retain the index k. For instance, 2. The variables x and t on the right-hand sides of the assignments are called "old" while those on the lefthand sides are called "new".

This technique, called Euler's method, owes its popularity to this simplicity. Notice that the old value of t is required to update x.

However, x is not required in the update of t, Therefore, by updating x before t, there is no need to use subscripts to maintain the bookkeeping details. This example is solved algorithmically as in Listing 2. The exact solution given by Equation 2. These data are also reproduced graphically in Figure 2. Therefore, in applying the Euler method, it is important to not stray too far from the initial time.

It is also necessary to choose the integration step size h wisely. The solution given by Equation 2. Therefore, as t approaches the neighborhood of tcrit, the numerical results become increasingly precarious. This leads to increasingly large deviations between x tk and x k as t approaches tcrit from the left. It is clear from Figure 2. One way to improve the accuracy is to decrease the step size h.

The effect of this is illustrated in Figure 2. Even so, reducing h has two major drawbacks: Secondly, due to inherent machine limitations in data representation, h can also be too small. Thus, if h is large, inherent difference approximations lead to problems, and if h is too small, truncation and rounding errors occur. The trick is to get it just right by making h small enough. Rather than printing the results of a procedure at the end of every computation, it is often useful to print the results of a computation periodically.

This is accomplished by using a device called a control break. A control break works by means of nested loops. After an initial print, the outside loop prints n times and, for each print, the inside loop produces m computations.

The solution is to control the iteration with two loops rather than one. Figure 2. The implementation of the control break is given in Listing 2. Equation 2. This may be extended using a higher-order approximation.

Apply the Taylor technique to the system 2. The graph is shown in Figure 2. If even more terms of the Taylor series are used, the results are even more dramatic. In fact, often a fourth-order approximation is used: This can be tedious at best, and is often impossible since an analytical formula for f may be unavailable.

This is often the case in realistic systems, since we can directly measure only signal values, not their derivatives, which must be inferred. It can be shown that the following discretized algorithm approximation is equivalent to the fourth-order Taylor method, and just as accurate: This so-called Runge-Kutta algorithm is classic.

It offers two advantages: It serves as the de facto technique of choice in much numerical work. Let us apply Runge-Kutta to the system 2.

Listing 2. Notice that the order of execution is critical, since K1 must be calculated before k2, K2 before k3, and k3 before K4, before updating x. The alternative to this strict sequential requirement is to do the bookkeeping by means of an array for x, but that is wasteful of computer memory and should be avoided.

As a comparison, Table 2. These are also compared in Figure 2. It is evident from Table 2. The reason that singularities cause unrecoverable problems is because the function's slope tends to infinity as the singularity is reached.

Accordingly, it is necessary to use all terms if that were possible in the Taylor series 2. Without assuming a specific form, such as that of a rational function, it is not possible to achieve accuracy across a singularity e.

Consider the function illustrated in Figure 2. It is evident that different regions will require different integration procedures. On the interval [t0,t1] a zeroth-order approach would probably be sufficient, since the function is essentially constant.

Certainly a firstorder approach would lead to an accurate solution. However, on [t1, t2], a second-order method would be better since the slope first derivative changes and the concavity second derivative changes as well. On [t2, t3], there are more frequent changes, so a smaller step size h is necessary, and probably a higher-order, say RungeKutta, approach is a good idea. In any case, values of x t on [t3,t4] would be totally erroneous because of the singularity at t3.

This is true regardless of the order of the technique. This interpretation is easy if the solution x t is known. Of course, this cannot really happen, since if x t were known then the problem would be solved a priori.

Unfortunately, few elementary or general results of this type exist, and practitioners must be satisfied using heuristics. For instance, by using first-, second-, and fourth-order approaches to the same problem, one would expect three different answers. However, by continually decreasing the step size h, eventually the results would approximately match. This "solution" would then be considered satisfactory. Adaptive Runge-Kutta Methods One of the more popular heuristics is to compute a solution using the different orders, say using a fourth-order and a fifth-order Runge-Kutta.

If the results are similar, the integration proceeds using a relaxed step size tripling h is typical. On the other hand, if the results are significantly different, the step size is reduced perhaps by a factor of Thus, if the function f is sufficiently smooth, large steps are taken.

However, as the terrain gets rougher, smaller steps ensure that accuracy is not lost. The fourth-order Runge-Kutta formulae are given as Equations 2. The fifth-order Runge-Kutta-Fehlberg formulae are as follows:. Using fourth- and fifth-order Runge-Kutta, the so-called Runge-Kutta-Fehlberg algorithm described above is shown in Listing 2. In order to apply this program, it is first necessary to prescribe a tolerance. As is clear from the listing, a small c will ensure greater accuracy by reducing h when necessary.

Similarly, if the two computations are close enough, the size of A is relaxed and accuracy is maintained with fewer computations required. Chapter 2: Dynamical Systems The adaptive approach is powerful and accurate. However, it should be remembered that accuracy is relative and it may be over-kill for the problem at hand.

Specific problems have specific requirements and Runge-Kutta-Fehlberg is still not useful when crossing a singularity. Once this is done, any of the techniques described above may be applied in order to attain a numerical approximation to the solution. For instance, a system of two first-order equations is described as follows:. The discrete solution is extended from Equations 2. Consider the system described by the block diagram shown in the figure below, where the system is described by.

Solution Since the differential equation is of second order, there must be two state variables. It will be noted that these equations are coupled in that x apparently cannot be updated before y, and vice versa, without retaining the subscripts.

Thus, in implementing this system, care must be taken if the use of subscripts is to be avoided. This can be done by introducing a temporary variables xl and y1 which are computed prior to the actual update of x and y.

This technique is demostrated in Listing 2. Even so, as a rule, it is simply best to avoid this inaccuracy.

Runge-Kutta can also be used to solve the system 2. This is done by applying Equations 2. These are implemented without the use of temporary variables in Listing 2. O This technique for integrating a set of differential equations is straightforward, and can be adapted to a wide variety of systems. However, special circumstances abound!

Therefore, simulation problems involving dynamical systems often require considerable work in prescribing precision parameters such as integration step size h and, if the adaptive methods are applied, tolerance c. Rather than simply static structures, they vary with time. To model these systems, we use differential equations in time, since derivatives describe the rates at which variables change.

For example, Newton's laws describe motion in terms of momentum: Also, in biological systems, population dynamics descriptions are typically stated as "the rate the population changes is proportional to the population size". Systems can be driven by either endogenous internal or exogenous external inputs. These inputs are either synchronous if the system is time-driven, or asynchronous if it is event-based.

On the other hand, if r t is defined stochastically as a random process such as ,,. In this case, the system is not deterministic, yet it is statistically meaningful.

It is also possible that a system has no input at all. We call such systems autonomous, since they are totally self-contained and their behavior is independent of external influence. Accordingly, the solution to an autonomous system is called the system's natural response.

If the system is linear, it can be shown that the natural response is one of three types: For instance, consider the classical second-order system with constant coefficients in which r t is the input: In the case of nonautonomous systems, the superposition principle applies. This states that the total solution for a linear system is given by the sum of the natural response and the forced response, as shown in the following example. Solve the system using superposition. In doing so, show that the total solution is the sum of two terms: Using Equation 2.

The forced response can be found by noting that the output is driven by the input, and will therefore be of the same form as the input and its derivatives. Substituting into Equation 2.

The constants A and B can be found by substituting the initial conditions. This result holds in general in stable linear systems: O The case where the steady-state solution is independent of the initial conditions can only be guaranteed for linear systems.

In fact, in many nonlinear systems, thefinal solution another name for steady state is so sensitive to where the system begins that we call the system chaotic. This, combined with the fact that there are no known general analytical approaches to nonlinear differential equations, means that numerical techniques with extremely tight control on the integration step size are required. Consider the problem of modeling the population dynamics of a certain species.

Although there are a number of factors controlling this population, as a first try we note that the rate at which a population's size increases is roughly proportional to the population size at any time.

Letting x t represent the population size at time t and A be a proportionality constant, x Ax. Clearly, this population increases exponentially over time.

This model is sometimes called the Malthusian model after the 18th century English economist Thomas Malthus, who used the above argument to predict a worldwide population explosion. Many examples exist confirming Malthus' model. Even so, it is not hard to find reasons why and cases where this model doesn't work especially well. Some significant factors, in addition to population size, that affect growth rates include the capacity of the environment to support the population, interactions with competing and supporting populations, and the energy supplied by heat over time.

For example, a population might be too large for its food supply or there might be predators in the area where it lives. As a first modification, suppose the system can support a population maximum of size xm, called the carrying capacity. It follows that a more realistic model is that the growth rate is proportional to both the population size and the proportion available for expansion.

This equation, which is called the logistic equation, is clearly nonlinear. In other words, this model fits our intuition. Even though Equation 2. Elementary calculus leads to the following explicit formula for x t: Notice that if x0 is less than xm, x t grows to approach its carrying capacity xm asymptotically, while if. In each case, x t approaches the carrying capacity xm, regardless of the initial value. Thus, in each case, the initial and final values are as expected.

The logistic model performs well in environments where there is one population and a finite food source. However, a more typical situation is one where there is a predator and a prey competing to stay alive by eating and not being eaten, depending on the point of view!

A predator will grow proportionally to its own population size and its food supply the prey. Let x t and y t be the sizes of the prey and predator populations, respectively. Then the number of predator-prey interactions is proportional to the product x t y t. It should be noted that these names are misnomers, since aj is only an approximation to a proportionality constant when y is small relative to x.

Similarly, 2 is almost inversely proportional when x is small. The predator-prey model described in Equations 2. Lotka and Vito Volterra, whose primary work was in integral equations, and these equations are therefore referred to as the Lotka-Volterra equations. They are clearly nonlinear and coupled. Unfortunately, there is no known analytical solution, and so numerical techniques as described earlier must be employed. Any of the techniques described earlier can be used, so let us begin applying the easiest: Since the two equations are coupled, this requires the use of temporary variables so that the updates are applied after they are computed.

Assuming a suitable integration step size h, this algorithm is shown in Listing 2. Since this is Euler, the hvalue will have to be quite small - perhaps even too small. Probably only 50 would be sufficient. This problem can be taken care of by the use of a control break. This will lead to mn calculations covering the interval [t0, tn]. Programmatically, this is accomplished by the code, where the outer loop i is performed n times, each of which prints the results obtained by the inner loop j , which is performed m times for each of the n i-values.

Notice that within the computation loop, each update is evaluated before it is actually performed. This is because that the update computation requires the "old" variables rather than the new ones, and therefore no subscripted variables are required. Graph the results in both the time and state-space domains. The first few calculations are as follows, and the graph is sketched as a function of time in Figure 2. There is another view of the predator-prey model that is especially illustrative.

This is the so-called phase plot of y versus x given in Figure 2. As one would expect, as the predator's food source the prey increases, the predator population increases, but at the expense of the prey, which in turn causes the predator population to drop off. Therefore, as one population achieves a bountiful food supply, it becomes a victim of its own success.

This is evident from the closed loop of the phase plot. Such a loop implies an endless periodic cycle. In this nonlinear case, the term stability has to mean that neither population becomes overly dominant, nor does either population become extinct; hopefully, this would lead to a periodic ebb and flow.

Notice that here the x, y points connect in a closed path called an orbit, which indicates periodic behavior. Orbits are especially useful devices to describe system stability. This is how a linear system driven by constant inputs functions. Similarly, if the. This is a serious topic that is beyond the scope of this text, but let us say that the interpretation of orbits is very helpful in characterizing nonlinear systems in general.

In particular, except for linear systems, different initial conditions can lead to very different behaviors. Only in linear systems is the stability independent of the initial conditions. Of course there is no best value of integration step size h. In principle, h should be zero - so the closer the better - but if h is too small approximately machine zero then numerical stability not to be confused with system stability makes the results extremely inaccurate.

Also, as h becomes smaller, more time is expended in execution, which is undesirable. Thus, the question becomes one of "what h is accurate enough"?

To observe the effect of differing h-values, consider the previous example for a set of step sizes that are each 1 of the previous h each iteration:. Considering the same Lotka-Volterra system as in Example 2. This is a critical decision, to say the least. However, it will be noticed that as h becomes smaller, the graphs begin to converge, just as one should expect. Again form a geometric sequence of h-values, each with common ratio r:. This table is formed below for the system given in Example 2.

It is our goal to compare the relative root mean square error RMS for each method. This is done by finding the difference between adjacent columns, squaring and totaling them over the time interval [0,5]. Since ht was chosen to form a geometric sequence, there are other equivalent formulas, so do not be surprised to see this result stated in other forms in other texts. For the system specified in Example 2.

Relative RMS 0. Second, they actually seem worse after that point. Such studies are always appropriate in simulation research.

There is no need to limit population models to just one predator, one prey. Populations can feed on and are fed on by more than one species, and the whole issue becomes quite complex. Of course, this is good, since that is exactly what we as modelers are trying to describe! At this level, the mathematics is still nonlinear and the differential equations are coupled. Without going into detailed explanations as to the physical interpretation of the model, another example is in order.

Create a suitable simulation and interpret the graphical results for the time interval [0, 25]. The code is given in Listing 2. The procedure outlined above, where we found a leveling-off point for the relative RMS error curve, indicates this is a good step size.

Even so, we expect some deterioration in the long run due to numerical estimation concerns. The resulting population- versus-time graph is shown in Figure 2.

As would be expected, there is a transient phase, in this case lasting for about 18 time units, followed by a cyclical steady-state phase where the populations ebb and flow over time.

The three phase plots are shown in Figures 2. Notice that the initial transient phases build into ever-widening oscillations, which are followed by periodic limit cycles in the steady state.

However, the cycles do not seem to close as they did for the two-dimensional example considered earlier. This is probably because of numerical problems rather than system instability. In order to do the phase plots, it is necessary to plot each state x, y, z against its derivative x, y, z. Since these derivatives are given explicitly by Equations 2. Suitable code is given in Listing 2. All rates, and thus all derivatives, are changes with respect to time. Normally, we think of a system's time somewhat like a.

This is not always an accurate model. For instance, plants grow and mature with respect to how much heat energy temperature is in the system rather than the "real" chronological time.

This heat energy is usually thought of as physiological time, and the dynamic equations measure rates with respect to physiological rather than chronological time.

It follows that in some systems, each subsystem measures time differently. Consider a biological growth model in which certain bacteria exist in an environment where the temperature T t varies over time t.

Stated mathematically, 0, r[T t T t T t. It is useful to define a new time, called the physiological time T, as follows:. Clearly, from this definition, the physiological time is the cumulative heat units summed over time. Thus, the typical unit for r t is the degree-day. Differentiating Equation 2.

The importance of Equation 2. In fact, it has been shown that all growth models discussed earlier, including the logistic and Lotka-Volterra models, behave in the same way. By defining the physiological time i by Equation 2. Thus, we have derived a sort of "relativity" principle where the external observer referenced in the master system sees non-standard behavior, but the subsystem sees internal events in the predicted manner.

This is summarized in Table 2. The physiological descriptions are the same as the classical ones, except that the systems "see" the physiological time T rather than the chronological time t.

Consider an idealized model in which the temperature profile is given by the following formula: Find a formula for the physiological time, assuming a generic threshold T0. From this, graph a series of physiological profiles for different thresholds.

A graph showing this profile and threshold is given in Figure 2. Using a nominal mid-latitude annual cycle as a specific example, graphs of the physiological time -c t versus chronological time t are plotted in Figure 2. This model clearly shows how, for higher thresholds, both more heat units are generated and the initial growth time begins earlier. O According to Equation 2. The only change is that now the integration step size changes over time.

Dynamical Systems It is also possible to implement a simulation using a subroutine. As would be expected, the threshold temperature plays a significant role in heat-unit models.

The higher T0 is, the later growth begins and the less growth there is overall. Write a simulation program and sketch the population-versus-time graph over one year's time, assuming the heat-unit model with temperature profile.

Solution This temperature profile is the one used in Example 2.

Applying Equation 2. Of course, the variables t2, t3, and p, which are computed in the main program, must be defined as global variables for the subroutine to know their values without parameters being passed.

For a given threshold T0, the main program proceeds algorithmically much as previously. It should be noted that in order to produce this family of trajectories as shown, a structure inversion would need to be introduced.

Specifically, Listing 2. The two loops would then be inverted for printing. O In predator-prey models, each species can have a different threshold. If this is the case, the function physio now has two variables: The simulation algorithm is best shown by example. Dynamical Systems Chapter 2: Solve this system numerically using the heat-unit hypothesis, with the temperaSolve this system numerically using the heat-unit hypothesis, with the temperatureprofile as defined in Examples 2.

Assume that population x uses ture profile as defined in Examples 2. Assume that population x uses temperature threshold 50F and population uses 20F.

Obviously, y gets quite a temperature threshold 50F and population yv uses 20F. Obviously,y gets quite a head start! Solution Solution In this case, each population reacts differently, depending on the threshold In this case, each population reacts differently, depending on the threshold temperature. Accordingly, the new function is physio t, p , where p is calculated temperature. Accordingly, the new function is physio t, p , where p is calculated in the main program and passed to the physio function.

In this way, the in the main program and passed to the physio function. In this way, the physiological time for each species will be computed locally inside the function physiological time for each species will be computed locally inside the function rather than externally in the main program.

A partial listing is shown in Listing rather than externally in the main program. A partial listing is shown in Listing 2. The results are shown in Figures 2. However, in actually using this model, a real temperature profile can be obtained by taking However, in actually using this model, a real temperature profile can be obtained by taking physical measurements. These measurements can lead to much more realistic results.

Thus, we choose not to go with a formula such as the one developed in Example 2. Consider the function segment in Figure 2.

A generic point t, T on the adjoining line segment defines similar triangles, the slopes of which are equal: Consider the following data set, which is taken to be an approximation to a temperature profile. Apply linear interpolation to the data set. From this, define the piecewise-linear function that connects the points.

Solution The temperature profile approximated by the data is graphed in Figure 2. Notice that, without further information, there is no basis for extrapolation estimation of the profile beyond the points given , and therefore the approximation is only defined on the interval [0,9].

A straightforward application of Equation A straightforward application of Equation 2. A function interp t that computes the interpolated estimate of A function interp t that computes the interpolated estimate of T for any for any O chronological time t is given in Listing 2. In these cases, it is customary to store the t,?

If the t-vector is stored in ascending order and the? Such a function is adjacent points surrounding time t and a linear interpolation performed. Such a function is given in Listing 2. If the data set is too large or algorithm speed is important, a binary given in Listing 2. If the data set is too large or algorithm speed is important, a binary search is preferable. Of course, the goal is to obtain results for the population model, not just to find the Of course, the goal is to obtain results for the population model, not just to find the temperature profile.

This requires one more function, named physio t , which calculates temperature profile. The physio subroutine is the physiological time z corresponding to chronological time t.

The physio subroutine is called by the main program, which runs the actual simulation. This is best illustrated by a called by the main program, which runs the actual simulation. This is best illustrated by a final example. Consider the following logistic model of a population system driven by physiological time T: Assume that the temperature profile is given empirically by the data set of Example 2.

Write appropriate code solving this system and graph the results compared with the same system driven by chronological time t. Solution This solution will require a program structure similar to that shown in Figure 2. The main program obtains the chronological time r by passing the chronological time t to the function physio f. The function physio i obtains the interpolated temperature T by passing time t on to the interp f subroutine described in Example 2.

The function physio uses Eulerian integration, as shown in Listing 2. Since there is only one species in this example, the threshold T0 is also assumed common. On the other hand, if this were a LotkaVolterra mode, T0 would need to be passed into each subroutine so that the chronological time could be computed locally. The main program is straightforward, and is given in Listing 2.

The results are shown in Figure 2. Notice the distinctly different results between the classical and heat-unit models. It is clear that the heat-unit hypothesis is significant. Also, the bimodal nature of the temperature profile provides an interesting effect on the population growth. Example 2. Creating and simulating Except for only the most ideal Creating and simulating models is a multistep process. Except for only the most ideal systems, this requires many iterations before achieving useful results.

This is because real systems, this requires many iterations before achieving useful results. This is because real phenomena that we can observe exist in the messy environment we call reality. For phenomena that we can observe exist in the messy environment we call reality.

For instance, taking a cue from the Lotka-Volterra predator-prey models, organisms can exist instance, taking a cue from the Lotka-Volterra predator-prey models, organisms can exist either in isolation or in cooperative-competitive situations. However, the models in each of either in isolation or in cooperative-competitive situations. However, the models in each of these cases can be considerably different. For the scientist, this is important - knowledge these cases can be considerably different.

For the scientist, this is important - knowledge of the system behavior over a large parametric range for all interactions is paramount to of the system behavior over a large parametric range for all interactions is paramount to. However, for the engineer, a limited dynamic range with an isolated entity is often acceptable, since he can simply agree at the outset not to exploit extremes. Regardless of the point of view - scientific or engineering - modeling and simulation studies tend to include the following steps and features: Define the model form.

This includes isolating bounds between what will and will not be included in the model. Equations describing modeled entities are developed here. This is the inspirational and insightful part of the process. Model the historical inputs and outputs. This step will assist in: This is where the model's constants and parameters are evaluated. Before putting the model to use, testing and validation are essential.

This will often require statistical studies so as to infer reasonable empirical signals. Model future inputs. In order to use the model, realistic data are needed. Create a aset of questions to be posed. This list should include stability studies, Create set of questions to be posed. This list should include stability studies, transient and steady-state behavior, as well as time series prediction and other transient and steady-state behavior, as well as time series prediction and other questions.

Notice that there are really two models: These will be discussed in future sections of the text. Closely related to modeling are the These will be discussed in future sections of the text. Closely related to modeling are the simulations. Simulations are essential for the system identification, model evaluation, and simulations. Simulations are essential for the system identification, model evaluation, and questions to be posed.

Burden, R. DouglasFaires and S. Reynolds, Numerical Analysis, 2nd edn. Schmidt, Close, C. Wiley, Close, C. Conte, C. Dean and C. Drazin, G. Drazin, P. Fausett, L. Johnson, D. Johnson and J. Hilburn, Electric Circuit Analysis, 3rd edn. Prentice-Hall, Johnson, D. Juang, J. Kalmus, H. Khalil, H. MacMillan, Mesarovic, M. Springer-Verlag, Neff, H.

Harper and Row, Press, W. Flannery, S. Teukolsky, and W. Vettering, Numerical Recipes: The Art of Press, W. The Art of Scientific Computing. ScientlJic Computing. Rosen, R. Scheid, F. McGraw-Hill, Scheid, F. Thompson, J. A Modeler's Approach. A Modeler 5 Approach. Yakowitx, S. Szidarovszky, An Introduction to Numerical Computations. Yikowitx, S. Specifically, print 31 points on the interval [0,6], where each point has m calculation iterations.

Execute using the following integration step sizes: Graph and compare. Exercises Thus, derive Taylor's third-order formula, which is analogous to Equation 2.

Justify your results. Compare, and explain the correlation between features of the two graphs. For each tolerance, graph the time plot of x t versus t.

Superimpose the results. Dynamical Systems 2. Empirically determine an appropriate step size h; justify your choice. Is there a relationship between each of these approaches and the time it takes for a given accuracy? Verify the working results by writing a program to read the following tabulated data and graph the interpolated points on [1, 11]:.

It is required to determine a reasonable h-value so as to make an accurate simulation of this model. Using the method of creating and analyzing a geometric sequence of hi outlined in this chapter: From these, justify a choice of A. This will require careful scoping of variables t1, t4, A, and T0. However, now assume a heat unit hypothesis in which x, y and z have temperature thresholds of 45 40 and 50 respectively. Also assume a temperature profile of.

Using the function physio t, p defined in problem 2. This graph should be comparable to that shown in Figure 2. These should be comparable to those shown in Figures 2. Graph the population-versus-time results. For convenience, assume that the orientation of the coordinate system is such that the initial velocity is only in the y direction. Since this system is of order two and has two degrees of freedom, it is necessary to have four initial conditions.

Compute v. Dynamical Systems b Determine the step size by assuming an h that works as expected for a circular orbit is good enough throughout the entire problem. In particular, if the position at the end of days is within 0. Write a program to numerically integrate Equation 2. For various values of A, find one that is suitable. The shape as defined by the eccentricity and period will change. Plot one orbit of the resulting trajectories.

Repeat this by decreasing the initial velocity by the same amounts. However, rather than make the traditional linearizing assumption, study the nonlinear case.

Recall that Newton's laws applied to a simple pendulum reduce to. The constant g is the acceleration due to gravity. Exercises a Look up and compute use units of kilometers and seconds the value of g. Also obtain results pertaining to simple harmonic motion. Determine an appropriate step size h by assuming that an h that works as expected for simple harmonic motion is good enough for the given initial angle.

In particular, if the position at the end of one period is within 0. Unlike the linear case, the period will now change as a function of the initial angle. Repeatedly change the initial angle 00 from 0 to 90 in steps of 5.

Show the resulting 0 t y-axis versus time t x-axis graphs. Plot three or four distinct trajectories. Graph P as a function of initial angle Compare your results. Stochastic Generators It is traditional to describe nature in deterministic terms. However, neither nature nor engineered systems behave in a precisely predictable fashion. Systems are almost always innately "noisy". Therefore, in order to model a system realistically, a degree of randomness must be incorporated into the model.

Even so, contrary to popular opinion, there is definite structure to randomness. Even though one cannot precisely predict a next event, one can predict how next events will be distributed. That is, even though you do not know when you will die, your life insurer does - at least statistically! In this chapter, we consider a sort of inverse problem. Unlike traditional data analysis where statistics mean, standard deviation, and the like are computed from given data, here we wish to generate a set of data having pre-specified statistics.

The reason that we do this is to create a realistic input signal for our models. From historical studies, we can statistically analyze input data.

Using these same statistics, we can then generate realistic, although not identical, scenarios. This is important, since the actual input signal will likely never be replicated, but its statistics will most likely still be stationary. For instance, in climatological models, the actual temperature profile will vary from year to year, but on average it will probably remain relatively unchanged. So, in modeling the effects of global warming, we change not only the data, but the statistical character of the data as well.

Such routines are called U[0, 1] generators. Strictly speaking, this is a discrete random variable by virtue of the digital nature of computers , but for practical purposes it can be assumed to be continuous. The majority of uniform random number generators are based on the linear congruential generators LCG. The simulation model was made for all four methods of transformer neutral point grounding treatment. It is necessary to complete this module prior to commencing the Earth, Life or Physical Science module.

Unit Contents and Objectives a point of zero radius mca modelling and simulation index sr. Code Generation.

From a users point of view component models are the key to simulator productivity; the greater the number of di erent models the easier it becomes to analyse mixed analogue and digital electronic systems. Abaqus is a suite of powerful engineering simulation programs based on the finite element Abaqus tutorials.

Far Field Simulation 5 5. How to find whether a point lies within the polygon or not? So that, fluid flow performed in the laboratory is done based on that tutorial. Data Modeling by Example: Mathematics Subject Classification You may recall that a circular hole in a plate has a stress concentration factor of about 3.

This model is a well-defined description of the simulated subject, and represents its key characteristics, such as its behaviour, functions and abstract or physical properties. This modeling and simulation book is downloadable in pdf. Modelling Chemical Speciation: Thermodynamics, Kinetics and Uncertainty Jeanne M. Molecular Modelling for Beginners 2. Learn and research electronics, science, chemistry, biology, physics, math, astronomy, transistors, and much more.

Samir Al-Amer Term In certain cases simulation models can be very expensive. Great forecasting power, but a good theory is needed Data analysis methods such as regression are limited to forecasting the effects of events that are similar to what has already happened in the past.

Pratap Sapkota. If we need a graphical output, PSpice can transfer its data to the Probe program for graphing purposes. Relational Database Multidimensional Data Cubeintroduction to computer modeling and simulation for students with no prior background in the topic. Press and hold the left mouse button down.

A bias point simulation simply calculates the DC voltages and currents in the circuit. What Level of Model Detail? Create stunning game environments, design visualizations, and virtual reality experiences.

These are the links to tutorials. Non-animated modules from the Arena template that make up the majority of this flow-chart can be freely interspersed with modules that contain animation.

The following table gives a summary of the advantages and disadvantages of simulation, which we elaborate below. Tutorials Point Interview Questions and Answers for freshers and experienced pdf. Computer Graphics i About the Tutorial Each screen point is referred to as a pixel. Getting Started: The area we have chosen for this tutorial is a data model for a simple Order Processing System for Starbucks. Continuous Verification.

Standard usage 33 The model has 1 reservoir with an analytical aquifer, which is simulated using a cartesian grid defined by corner point geometry. While you are working on a project it is a good idea to save your simulation in Aspen Plus Document form at. As design increases in size and simulation time, the RTL modelling level is also increased.

All books are in clear copy here, and all files are secure so don't worry about it. Search for tutorials Tutorial 1. Volume 1 6 During the course of this book we will see how data models can help to bridge this gap in perception and communication.

Run the model simulation. You should see the crank-rocker behavior shown, Figure The course contains a number of exercises during which you will practice specific topics in the presentations. At this point, inore than 10 percent oj' the modelling eff,i-t should be The Control Tutorials for Matlah by Messner and from the application engineers point of view.

Numerical aquifers are Simulation does not generate optimal solutions. Simulation Models: During the design and implementation of a Spray Modelling using ANSYS-CFX Introduction The tutorial was written in a rush so it has spelling mistakes never go the time to correct them, feedback would much appreciated to improve the tutorials.

Barnes simulation package that is designed for simulating by point and click or by specifying Monte Carlo simulation is named after the city of Monte Carlo in Monaco, which is The pdf fy Y can be obtained by the numerical differentiation of cdf Fy Y.

Using SolidWorks Flow Simulation to Calculate the Flow Around a point in the aerofoil coordinates must be the same as the first one so that the coordinates form a Please visit http: In both reservoir simulation and groundwater modeling, for example, Monte Carlo simulation is a popular technique.

Post-process was used to analyze data. Model Elaboration. The SolidWorks window is resized and a second window will appears next to it with a list of the available tutorials A tutorial: Windows XP, a point in a continuum. Civil Engineering, Heritage. Modelling and Simulation of Solar Thermoelectric GeneratorMatlab and Simulink for Modeling and Control Set the simulation parameters and run the simulation to see the step response.

Click on chapter-wise links to get the notes in pdf format. This section contains lecture notes from the course. That is, a weak input signal can be amplified made stronger by a transistor. This tutorial provides a technical overview of the 13 UML diagrams supported by Enterprise Architect.

Frederik Steinmetz explains all generate and deform modifiers available in Blender in short video tutorials. Simulation of a system is the operation of a point is that powerful simulation software is merely a hygiene factor - its absence can hurt a simulation study but its presence will not ensure success.

The Dynamic Simulation environment is part of an integrated design and analysis system. Mesh 3. Vulcan software provides the mining industry with the most advanced 3D geological modelling, mine design and production planning solutions.

Tutorial Solidworks Simulation. TensorFlow ANSYS engineering simulation and 3D design software delivers product modeling solutions with unmatched scalability and a comprehensive multiphysics foundation. Subarna Shakya and Er.