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Tham khảo tài liệu 'robotics process control book part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 154 Dynamical Behaviour of Processes When replacing the arguments ti t2 in Equations 4.4.58 4.4.60 by t and T then Re t T E e t C T 4.4.63 and Cove t t E t - MƠ t - m t 4.4.64 If t T then Cove t t E t - M t 2 4.4.65 where Cove t t is equal to the variance of the random variable . The abbreviated form Cove t Cove t t is also often used. Consider now mutually dependent stochastic processes 1 t 2 t . n t that are elements of stochastic process vector t . In this case the mean values and auto-covariance function are often sufficient characteristics of the process. The vector mean value of the vector t is given as p t E t 4.4.66 The expression Cove ti t2 E ti - M ti t2 - R t2 T 4.4.67 or Cove t T E t - M t T - m t T 4.4.68 is the corresponding auto-covariance matrix of the stochastic process vector t . The auto-covariance matrix is symmetric thus Cove T t CovT t T 4.4.69 If a stochastic process is normally distributed then the knowledge about its mean value and covariance is sufficient for obtaining any other process characteristics. For the investigation of stochastic processes the following expression is often used 1 f T M lim -ặ- t dt 4.4.70 T 1 2T J T M is not time dependent and follows from observations of the stochastic process in a sufficiently large time interval and t is any realisation of the stochastic process. In general the following expression is used Mm 2T T t mdt 4.4.71 1 1 2T _T For m 2 this expression gives M2. Stochastic processes are divided into stationary and non-stationary. In the case of a stationary stochastic process all probability densities fl f2 . fn do not depend on the start of observations and onedimensional probability density is not a function of time t. Hence the mean value 4.4.55 and the variance 4.4.56 are not time dependent as well. Many stationary processes are ergodic i.e. the following holds with probability equal to one M i xf1 x dx M lim i t dt 4.4.72 1 T 1 2T T M2 M2 Mm Mm 4.4.73 The usual assumption in practice is that .
