Tham khảo tài liệu 'adaptive control 2011 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 18 Adaptive Control stated in web08 Linear regression is probably the most widely used and useful statistical technique for solving environmental problems. Linear regression models are extremely powerful and have the power to empirically tease out very complicated relationships between variables. Due to the importance of model we list several simple examples for illustration Assume that a series of stationary data xk yk k 1 2 N are generated from the following model Y p P1X s where fi0 P1 are unknown parameters xk are i. i. d. taken from a certain probability distribution and Sk N 0 CT2 is random noise independent of X . For this model let 0 @0 P1 T pk 1 xk T then we have yk Q ộk Sk . This example is a classic topic in statistics to study the statistical properties of parameter estimates 0N as the data size N grows to infinity. The statistical properties of interests may include E Q Q Var Q and so on. Unlike the above example in this example we assume that xk and xk have close relationship modeled by xk 1 P0 Px xk Sk where p0 fi-1 are unknown parameters and Sk N 0 Ơ2 are i. i. d. random noise independent of x1 x2 xk . This model is an example of linear time series analysis which aims to study asymptotic statistical properties of parameter estimates V under certain assumptions on statistical properties of Sk . Note that for this example it is possible to deduce an explicit expression of xk in terms of sj j 0 1 L k 1 . In this example we consider a simple control system xk 1 p pi xk buk Sk where b 0 is the controller gain Sk is the noise disturbance at time step k. For this model in case where b is known a priori we can take Q P0 p ộk 1 xk 1 T zk xk buk 1 otherwise we can take Q P0 P1 b ệk 1 xk 1 T zk xk buk 1. In both cases the system can be rewritten as zk QT k Sk Adaptive Estimation and Control for Systems with Parametric and Nonparametric Uncertainties 19 which implies that intuitively 0 can be estimated by using the identification algorithm since both data .
