Continuous time modeling

Markov model A mathematical model used in reliabihty analysis. For many safety apphcations, a discrete-state (e.g., working or failed), continuous-time model is used. The failed state may or may not be repairable. 

The main challenge in short-term scheduling emanates from time domain representation, which eventually influences the number of binary variables and accuracy of the model. Contrary to continuous-time formulations, discrete-time formulations tend to be inaccurate and result in an explosive binary dimension. This justifies recent efforts in developing continuous-time models that are amenable to industrial size problems. 

The models developed to take the PIS operational philosophy into account are detailed in this chapter. The models are based on the SSN and continuous time model developed by Majozi and Zhu (2001), as such their model is presented in full. Following this the additional constraints required to take the PIS operational philosophy into account are presented, after which, the necessary changes to constraints developed by Majozi and Zhu (2001) are presented. In order to test the scheduling implications of the developed model, two solution algorithms are developed and applied to an illustrative example. The final subsection of the chapter details the use of the PIS operational philosophy as the basis of operation to design batch facilities. This model is then applied to an illustrative example. All models were solved on an Intel Core 2 CPU, T7200 2 GHz processor with 1 GB of RAM, unless specifically stated. 

It is the objective of this paper to provide a comprehensive review of the state-of-the art of short-term batch scheduling. Our aim is to provide answers to the questions posed in the above paragraph. The paper is organized as follows. We first present a classification for scheduling problems of batch processes, as well as of the features that characterize the optimization models for scheduling. We then discuss representative MILP optimization approaches for general network and sequential batch plants, focusing on discrete and continuous-time models. Computational... 

Perhaps the biggest gap in terms of effective models is the capability of simultaneously handling changeovers, inventories and resource constraints. Sequential methods can handle well the first, while discrete time models (e.g., STN, RTN), can handle well the last two. While continuous-time models with global time intervals can theoretically handle all of the three issues, they are at this point still much less efficient than discrete time models, and therefore require further research. 

Other types of machines that are not based on neural networks were also suggested as continuous-time models. Pour-El [157] constructed a general-purpose analog computer using a finite number of the following units ... 

Identification of Continuous-time Models from Sampled Data... 

Where Xk are state variables, Wk and Vk are the model and process disturbances, respectively, Lk is a stage cost function and Zt-n (z) is the arrival cost function. A discrete-time model is adopted in the above formulation for illustrative purposes. A continuous-time model can also be used. 

A discussion of the continuous-time model, the time-nonhomogenous model, and the semi-Markov chain is beyond the scope of this chapter (e.g., see Norris (13),... 

But how can we convert a continuous-time model to an equivalent discrete-time one This is the question that we resolve in this section. 

Merton, R., 1971. Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373 13. 

Honore, P., 1998. Five essays on financial econometrics in continuous-time models. PhD Thesis, Aarhus School of Business. 

To estimate the Vasicek continuous stochastic time model, the model must be discretized. We discretized and estimated the continuous time model as follows ... 

Continuous-Time Model with Long-Range Dispersal then the Hamiltonian takes the form... 

The continuous state problem is rarely seen in the field of reliability engineering. Both continuous time and discrete time problems are used. Continuous time models typically have analytical solutions. Discrete time models typically have numerical solutions. The model t)rpes are listed in Table D-1. 

Beyond model development for the purpose of analysis and simulation of the dynamic or the frequency domain behaviour, bond graphs have also proven useful as a tool for model-based FDI in engineering systems represented by continuous time models. Only recently, bond graph model-based FDI has also been used for systems described by a hybrid model. Following common terminology a system will be called a hybrid system for short if its dynamic behaviour is appropriately described by a hybrid model. In Chap. 2.1 of [1] Kowalewski rightly remarks ... 

If fast switching devices are modelled as switches then parts of an overall system model are instantaneously disconnected or reconnected for a while. That is, an overall model is of variable structure. The result of such a structural change is a continuous time model that holds for the resulting system mode. 

Instead of discretising the time and using a BDF-based method for the numerical computation of a continuous-time model one may think of quantising the state variables. That is, instead of using a multistep method to compute an approximation of the value x(f +i) of a state variable x at time tjc+i, the question then is at what time the state x will deviate from its current value x (tk) by more than a given quantum A Q. In other words, the task is to find the smallest time step h so that... 

This approach replaces the traditional discrete time based computation of continuous time models by a discrete event simulation that advances the time from the time point of an event to the time of the next event which is attractive for the computation of hybrid models because discrete events, i.e. discontinuous mode changes, and the continuous time behaviour during system modes can be uniformly processed in the framework of the well-known Discrete EVent System (DEVS) specification introduced by Ziegler [47, 48]. Moreover, the DEVS formalism is supported by software libraries such as adevs [49, 50] or simulation environments such as PowerDEVS [51, 52],... 

An atomic DEVS model is defined as a tuple of sets and functions (cf. Appendix A.3). If the output events of an atomic DEVS model are converted into input events of another atomic DEVS model, i.e. if atomic DEVS models are coupled, then the result defines a new DEVS model. That is, complex systems can be modelled in the DEVS framework in a hierarchical manner. The DEVS formalism is widely used in computer science. Its application to the numerical solution of continuous-time models, however, is much less common. 

Another approach that also allows to approximate each continuous time element of a bond graph by a DEVS model so that a DEVS simulation can be performed has been reported in [57, 58]. The task is to transform piecewise continuous input and output trajectories of a bond graph element into discrete event trajectories. To translate, e.g. the continuous time model of a C element in integral causality into a discret event model, the input trajectory of the flow f t) between two time instances U and tj is approximated by a linear function f(t) = a t + ao- The output trajectory of e(t) is a second order polynomial e(t) = b2t + bit = aQlC)t. This... 

Given a causal bond graph of a hybrid model, the question is how ARRs containing only known system inputs, outputs, system parameters, and information about the system mode can be derived. This section considers three methods that have been reported in the literature. One approach is the so-called causality inversion method [2-4], It has been introduced for causal bond graphs of continuous time models but can also be applied to bond graphs of hybrid models [5, Chap. 7]. 

In some way, this coupling of two models by residual sinks may be compared to the temporary coupling of two bodies such as the plates of a clutch by a residual sink (cf. Fig. 2.15). As long as the clutch is disengaged, there is no force acting between the two plates. If, however, the clutch engages, a torque acts on both of them such that the difference of their angular velocities is zero. The load side is forced to adapt to drive side. This approach has been applied for the numerical computation of ARR residuals from continuous time models [14, 17] and recently also to systems described by a hybrid model [18]. 

If ARRs can be obtained in closed symbolic form, parameter sensitivities can be determined by symbolic differentiation with respect to parameters. If this is not possible, parameter sensitivities of ARRs can be computed numerically by using either a sensitivity bond graph [1 ] or an incremental bond graph [5, 6]. Incremental bond graphs were initially introduced for the purpose of frequency domain sensitivity analysis of LTI models. Furthermore, they have also proven useful for the determination of parameter sensitivities of state variables and output variables, transfer functions of the direct model as well as of the inverse model, and for the determination of ARR residuals from continuous time models [7, Chap. 4]. In this chapter, the incremental bond graph approach is applied to systems described by switched LTI systems. 

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