Understanding stochastic process disturbances and the discrete time domain

Developing a successful control algorithm often requires proper identification of the disturbances. In Chapter 1 (PDF downloadable here), unautocorrelated process disturbances (white noise) and autocorrelated process disturbances were presented using the "large hotel water tank" example. In this chapter these terms and concepts will be revisited with a little more rigor using the autocorrelation, the line spectrum, the cumulative line spectrum, and the expectation operator. The ability of a PI controller to deal with different kinds of disturbances will be discussed. Chapter 9 will revisit the discrete time domain and introduce the Z-transform.

8-1 The discrete time domain
In the previous chapters, for the most part, the time domain was considered as continuous. Differential equations were derived based on this concept. Laplace transforms were used to solve these differential equations and also to provide a path to the frequency domain which was also considered continuous. In this chapter the time domain will be discrete in the sense that a data stream will now consist of a sequence of numbers usually sampled at a constant interval of time. For example, a data stream might consist of samples of a temperature T(t), as is:

T(t₁), T(t₂), . . . ,T(t_N)

or

T₁, T₂, . . . ,T_N

with the sample-instants in time being equally spaced, in the sense that: t_i = t_i-1 + h
i = 1,2, . . .

where h is the sampling interval, which will be assumed to be constant unless otherwise stated. The sampling frequency is 1/h.

Instead of differential equations where the independent variable is continuous time, there will be algebraic equations with the independent variable being an index, such as i, to an instant of time. A simple example of an indexed equation would be a running sum of a data stream consisting of sampled values of the variable x, as in:

S_i = S_i-1+x_i i = 2, 3, . . . ., N

The average of the x₁, x₂, , , . . . data after N samples would be:

The sample average:

8-2 White noise and sample estimates of population measures
Consider a data stream of infinite extent:

w₁, w₂, . . .

from which N contiguous samples have been taken. Figure 8-1 shows two views of the data stream. The infinite data stream represents a population having certain population characteristics and the subset of size N mentioned above is a sample of that population. The subset has certain sample characteristics, which can be used as estimates of the population characteristics. The data shown in Figure 8-1 will soon be shown to be samples of "white noise." For now, we simply refer to it as a stochastic sequence. The word "stochastic," means "nondeterministic" in that the value at time t_i does not completely determine the value at time t_i+1. In the white noise stochastic sequence shown in Figure 8-1, the value at t_i has no influence whatsoever on the value at t_i+1. In other non–white stochastic sequences to be covered later in the chapter, the value at t_i+1 still is nondeterministic but the value at t_i does have an influence. Note that the two streams shown in Figure 8-1 have different sample standard deviations.

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