Design of Digital Controllers in the Presence of Random Disturbances
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Design of Digital Controllers in the Presence of
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- Figure 4.5.
- Figure 4.6.
- Figure 4.7a,b.
Figure 4.4. Discrete-time white noise sequence
Control in the Presence of Random Disturbances 173 Since E{e(t)} = const. and E{e(t) e(t-i)} is only a function of i, the gaussian discrete-time white noise sequence has the properties of a weakly stationary stochastic process (in fact it is a real stationary stochastic process since the properties of the sequence {e(t)} are the same as those of the sequence {e(t+τ)}). The knowledge of the covariances R(i) for a weakly stationary stochastic process makes it possible to compute the energy distribution in the frequency domain, known as the spectral density function. Figure 4.5. Normalized autocorrelations of the white noise This is given by (discrete Fourier transform of the covariance function) Since in the case of the discrete-time white noise all the R(i) = 0 for i ≠ 0, it results that the spectral density of the discrete-time white noise is constant and equal to The spectral density of the white noise is represented in Figure 4.6. A uniform energy distribution between 0 and 0.5 fs is observed. Figure 4.6. Spectral density of the discrete-time white noise 4.1.2 Models of Random Disturbances As it has already been mentioned in Section 4.1, different types of random disturbances, whose spectral density can be approximated by a rational function of the frequency, can be considered as resulting from the filtering of a white noise through a shaping filter. Several types of processes thus obtained will be examined. “Moving Average” Process (MA) Consider, for example, the process which corresponds to the filtering of a white noise through a filter (1+c1 q-1), as shown in Figure 4.7a. Figure 4.7a,b. Generation of a “Moving Average” random process: a first order; b general case The mean value of y(t) is 3 The variance of the process y(t) is since the third term is zero (see properties of the white noise, Equations 4.1.3 and 4.1.6). Control in the Presence of Random Disturbances 175 From Equation 4.1.7, one obtains by shifting Download 64.18 Kb. Do'stlaringiz bilan baham: |
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