State-space models discrete-time

Several identification methods result in a state space model, eithejp by direct identification in the state space structure or by identjLfication in a structure that can be transformed into a state space model. In system identification, discrete-time models are used. The discrete-time state-space model is given by... 

I. is the identity matrix and z is defined by z (k-v)AT), is determined. In the second step, this model is transformed into a discrete-time state space model. This is achieved by making an approximate realization of the markov parameters (the impulse responses) of the ARX model ( ). The order of the state space model is determined by an evaluation of the singular values of the Hankel matrix (12.). 

Since real measurements are taken at discretetime instants and in order to fit model Eq. 14 to the measurements, this model needs to be converted into discrete time. Hence, assuming a constant sampling period At and that the input is piecewise constant over the sampling period, the continuous-time equations (Eq. 14) are discretized and solved at all discrete-time instants fjt = kAt, obtaining the discrete-time state-space model ... 

In this chapter, discrete linear-state space models will be discussed and their similarity to ARX models will be shown. In addition Wiener models are introduced. They are suitable for non-linear process modeling and consist of a linear time variant model and a non-linear static model. Several examples show how to develop both types of models. 

To determine the state space model with system Identification, responses of the nonlinear model to positive and negative steps on the Inputs as depicted in Figure 4 were used. Amplitudes were 20 kW for P,, . 4 1/s for and. 035 1/s for Q. The sample interval for the discrete-time model was chosen to be 18 minutes. The software described In ( 2 ) was used for the estimation of the ARX model, the singular value analysis and the estimation of the approximate... 

In this section, classical state-space models are discussed first. They provide a versatile modeling framework that can be linear or nonlinear, continuous- or discrete-time, to describe a wide variety of processes. State variables can be defined based on physical variables, mathematical solution convenience or ordered importance of describing the process. Subspace models are discussed in the second part of this section. They order state variables according to the magnitude of their contributions in explaining the variation in data. State-space models also provide the structure for... 

State-space models can also be developed for discrete-time systems. Let the current time be denoted as k and the next time instant where input values become available as A - - 1. The equivalents of Eqs. 4.44-4.45 in discrete time are... 

These kinds of equations have to be specified to obtain continuous time discrete state space models of chemical reactions. 

The proposed procedure is shown in Figure 1 and consists of five basic steps. First a standard ODE model is derived from first engineering principles and the constitutive equations containing unknown functional relations are identified. The ODE model is then translated into a stochastic state space model consisting of a set of SDE s describing the dynamics of the system in continuous time and a set of discrete time measurement equations signifying how the available experimental data was obtained. 

In the second step of the procedure the ODE model is translated into a stochastic state space model with r as an additional state variable. This is straightforward, as it can simply be done by replacing the ODE s with SDE s and adding a set of discrete time measurement equations, which yields a model of the following type ... 

Higher-order linear differential equations can be converted to a discrete-time, difference equation model using a state-space analysis (Astrom and Wittenmark, 1997). 

For stable models, the predicted unforced response, Y k + 1) in Eq. 20-38, can be calculated from a recursive relation (Lundstrom et al., 1995) that is in the form of a discrete-time version of a state-space model ... 

Formulated on a lattice and with discrete time, the WR model becomes a variant of a cellular automaton (it was later used in the construction of more complex cellular automata for excitable media, see [30-32]). However, Wiener and Rosenblueth actually assumed in [6] that both time and space were continuous. In their model smooth excitation fronts propagate into the regions where the medium is in the state of rest. The duration Tg of excitation is taken... 

It is very important to remark that an AVT provides information only on the vibration responses of a structure excited by unmeasured inputs. Consequently, it is impossible to distinguish the input term /k from the noise terms iPk and Vk in Eq. 17. This results in the following discrete-time stochastic state-space model ... 

A discrete-time Anite-dimensional linear system can be represented by the following state space model ... 

In LGCA models, time and space are discrete this means that the model system is defined on a lattice and the state of the automaton is only defined at regular points in time with separation St. The distance between nearest-neighbor sites in the lattice is denoted by 5/. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ch where i e 1, 2,. .., b. The set c can be chosen in many different ways, although they are restricted by the constraint that... 

The first one is to decompose the dynamical system into the control and the state spaces. In the next step, only the control variables are discretized and remain as degrees of freedom for the NLP solver [5]. The method is called the sequential approach. The DAE system has to be solved at each NLP iteration. The disadvantages of the approach are problems of handling path constraints on the state variables, since these variables are not included directly in the NLP solver [5] the time needed to reach a solution can be very high in case the model of the dynamic system is too complex difficulties may arise while handling unstable systems [4]. 

Each of the optional dynamical models mentioned above involves a homogeneous Markov process Xt = Xt teT in either continuous or discrete time on some state space X. The motion of Xt is given in terms of the stochastic... 

Markov chains or processes are named after the Russian mathematician A.A.Markov (1852-1922) who introduced the concept of chain dependence and did basic pioneering work on this class of processes [1]. A Markov process is a mathematical probabilistic model that is very useful in the study of complex systems. The essence of the model is that if the initial state of a system is known, i.e. its present state, and the probabilities to move forward to other states are also given, then it is possible to predict the future state of the system ignoring its past history. In other words, past history is immaterial for predicting the future this is the key-element in Markov chains. Distinction is made between Markov processes discrete in time and space, processes discrete in space and continuous in time and processes continuous in space and time. This book is mainly concerned with processes discrete in time and space. 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

In this subsection, we develop a very simple queueing model. This model is a Markov chain IXf) representing the number of jobs present in a queueing system observed at regular discrete times t = 0, 1, 2,. . . The state space is 0, 1, 2,. . . . There are two types of transitions possible arrivals and departures. We write p for the probability that a job arrives in the next time step. We write q for the probability that a job will complete service in the next time step, assuming that there is at least one job present (L(t) > 0). If we write r for 1 — p — q, which is the probability of no state change when there is at least one job present, then the transition matrix of the chain is... 

Crew Station/equipment characteristics The crew station design module and library is a critical component in the MIDAS operation. Descriptions of discrete and continuous control operation of the equipment simulations are provided at several levels of functiontil deteiil. The system can provide discrete equipment operation in a stimulus-response (blackbox) format, a time-scripted/ event driven format, or a full discrete-space model of the transition among equipment states. Similarly, the simulated operator s knowledge of the system can be at the same varied levels of representation or can be systematically modified to simulate various states of misunderstanding the equipment function. 

One of the most widely used approaches for the simulation of sprays is the stochastic discrete droplet model introduced by Williams [30]. In this approach, the droplets are described by a probability density fxmction (PDF),/(t,X), which represents the probable number of droplets per unit volume at time t and in state X. The state of a droplet is described by its parameters that are the coordinates in the particle state space. Typically, the state parameters include the location x, the velocity v, the radius r, the temperature Td, the deformation parameter y, and the rate of deformation y. As discussed in more detail in Chapter 16, this spray PDF is the solution of a spray transport equation, which in component form is given by... 

At least eight different kinetic models can be defined, depending on the specification of time (X), state-space (T) and nature of determination (Z). As was explained earlier, time can be discrete (D) or continuous (C), the state-space can be also discrete (D) or continuous (C), and the nature of determination is deterministic (D) or stochastic (S). 

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