Discrete time

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

Figure C3.6.4(a) shows an experimental chaotic attractor reconstmcted from tire Br electrode potential, i.e. tire logaritlim of tire Br ion concentration, in tlie BZ reaction [F7]. Such reconstmction is defined, in principle, for continuous time t. However, in practice, data are recorded as a discrete time series of measurements (A (tj) / = 1,...

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

U. Schmitt and J. Brinkmann. Discrete time-reversible propagation scheme for mixed quantum classical dynamics. Chem. Phys., 208 45-56, 1996. 

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

Discrete-time solution of the state vector differential equation... 

The discrete-time solution of the state equation may be considered to be the vector equivalent of the scalar difference equation method developed from a z-transform approach in Chapter 7. 

The continuous-time solution of the state equation is given in equation (8.47). If the time interval t — to) in this equation is T, the sampling time of a discrete-time system, then the discrete-time solution of the state equation can be written as... 

Equation (8.76) is called the matrix vector difference equation and can be used for the recursive discrete-time simulation of multivariable systems. 

The discrete-time control matrix B T) from equations (8.75) and (8.76) is... 

If the sampling time is 0.1 seconds, the values of the discrete-time state transition and control matrices AfT) and BfT) calculated in Example 9.8 may be used in the recursive solution. 

Equation (10.57) is in the same form as the discrete-time solution of the state equation (8.76). 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

Discrete-time steady-state feedback gain... 

Cadzow, J.A. and Martens, H.R. (1970) Discrete-Time and Computer Control Systems, Prentice-Hall, Inc., Englewood Cliffs, N.J. 

Ogata, K. (1995) Discrete-Time Control Systems, 2nd ed., Prentice-Hall, Inc., Upper Saddle River, NJ. 

Payne, H.J. and Silverman, L.M. (1973) On the Discrete Time Algebraic Riccati Equation, IEEE Trans. Automatic Control, AC-18, pp. 226-234. 

If the distribution curve is only known at a number of discrete time values, tj, then the mean is expressed by... 

These concepts were implemented according to the following scheme the liquid element surrounding the bubble and the bulk are considered as two separate dynamic reactors that operate independent of each other and interact at discrete time intervals. In the beginning of the contact time, the interface is being detached from the bulk. When overcome by the bubble, it returns to the bulk and is mixed with it. Hostomsky and Jones (1995) first used such a framework for crystal precipitation in a flat interface stirred cell. To formulate it for a... 

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

In many ways, May s sentiment echoes the basic philosophy behind the study of CA, the elementary versions of which, as we have seen, are among the simplest conceivable dynamical systems. There are indeed many parallels and similarities between the behaviors of discrete-time dissipative dynamical systems and generic irreversible CA, not the least of which is the ability of both to give rise to enormously complicated behavior in an attractive fashion. In the subsections below, we introduce a variety of concepts and terminology in the context of two prototypical discrete-time mapping systems the one-dimensional Logistic map, and the two-dimensional Henon map. 

Since the absolute value of the Jacobian J = a qn+i,Pn+i)/d qn,Pn) = 1, we see that this discrete-time map is indeed area-preserving. 

Consider, once again, a one-dimensional discrete-time map... 

The time evolution of the discrete-valued CA rule, F —> F, is thus converted into a two-dimensional continuous-valued discrete-time map, 3 xt,yt) —> (a y+i, /y+i). This continuous form clearly facilitates comparisons between the long-time behaviors of CA and their two-dimensional discrete mapping counter-... 

The case of multidimensional discrete-time mappings of the form... 

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