Design of Digital Controllers in the Presence of Random Disturbances
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Design of Digital Controllers in the Presence of
Figure 4.3. Recording of a controlled variable in regulation
By examining Figure 4.3, one observes that the evolution during one day may be described by a deterministic function f(t), but that this function will be different every day (f(t) is known as the “realization” of the stochastic process). If the time for carrying out the measurement of the observed variable is fixed (e.g. at 10 a.m.), each day (at each test) a new value will be measured (this it what is known as a random variable). However, for all values measured every day at the same time, statistics can be defined characterized by the mean value and the variance of the measurements. The probabilities of the occurrence of different values may be defined as well. The stochastic process (partially) represented in Figure 4.3, is dependent on the time (during a day) and on the experiment (first, second...fourth day). More formally, a stochastic process may be described as a function f(t, ξ) where t represents the time and belongs to the set T of real variables, and ξ Control in the Presence of Random Disturbances 171 represents the stochastic variable (the outcome of an experiment), which belongs to a probability space S1. For a given ξ = ξ0, the function f(t, ξ0) is a regular time function called a realization. For fixed t = t0 the function f(t0, ξ) is a random variable. The argument ξ is often omitted. If the stochastic (random) process is ergodic, the statistics related to an experiment (in our example over one day) are significant, i.e. the result obtained is identical to that obtained from measurements taken on several experiments when the time is maintained constant (at the same time of the day). If, in addition, the stochastic process is gaussian, the knowledge of the mean value and of the variance allows the probability of occurrence of a given value to be specified (Gauss's bell – see Appendix A). In practice, the majority of random disturbances occurring in automatic control systems may be accurately described as a discrete-time white noise passed through a filter. This discrete-time white noise is a random signal having an energy uniformly distributed at all frequencies between 0 and 0.5 fs. Note that the discretetime white noise has a physical realization, since it is a finite energy signal (the frequency band is finite), whereas the continuous-time white noise does not correspond to a physical reality since the energy is constant over an infinite frequency range (infinite energy signal). The filters that will constitute the random disturbance models will modify the frequency spectrum of the energy distribution of the white noise in order to obtain a distribution corresponding to the frequency distribution of energy of the various random disturbances encountered. The white noise has, in the random case, the same role as the Dirac pulse in the deterministic case. It constitutes the fundamental generator signal. The gaussian discrete-time white noise will henceforward be considered as the generator signal. This is a sequence of independent equally distributed gaussian random variables of zero mean value and variance σ2. This sequence will be noted {e(t)} and will be characterized by the parameters (0, σ), in which the first term indicates the mean value and σ is the standard deviation (square root of the variance). A part of such a sequence is represented in Figure 4.4. Download 64.18 Kb. Do'stlaringiz bilan baham: |
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