## 4 Adaptive Observer Based on Chaotic De
### The Process of State Estimation
#### 4.4.2 Real Time Tracking Estimation of the Observer State  
In order to compensate the influence of external disturbances and of the system model changes on the state estimate, the state estimate needs to track the system state on-line in real time manner.  
After complete the state estimate of the last time window, a new time window with a same time length as the previous time window is recreated. The start time and the terminal time of the new time window are re-denoted by \(t_{0}\) and \(t_{l}\), and all the variables use the same notations in the previous time window. The DE based state estimate procedure which was executed in the previous time window is repeated.  
The state estimation value \(x(t)\) at the beginning time of the new time window calculated using the estimated initial state \(x_{0}\) in the previous time window is reserved as a pre-estimate initial state \(\hat{x}_{0}(i,g)\), that is,, an individual of the initial population \(P(\hat{x}_{0}(g))\) in the new time window state estimate calculation, so as to inherit the state estimate result of the previous time window to accelerate the search of optimal point. However, in order to track the changes in real time, the remaining \(m-1\) pre-estimate initial states are reset according to the rule of the chaos DE algorithm. If the state estimation of the previous time window is still globally optimal, it will be retained in the new time window, otherwise it will be replaced by the new optimal estimation.  
After the task of the current time window is finished, another new time window is recreated, and the procedure in the previous time window is repeated. It operates repeatedly in this way all the time. The new time window can partially overlap, not overlap or even not connect with the previous time window according to the necessary of the update. Generally, the number \(m\) of the individuals of the initial population in the normal time windows is smaller than the one of the first time window.  
The purpose of adaptive observer is to realize the joint estimate of state and parameter, and the detection of system parameter changes. The system parameters to be estimated are considered as the extended state variables with zero dynamics as (4), thus the adaptive observer is equivalent to the nonlinear state observer. When the system parameters are considered as the extended state variables, their values are hence the initial values of the extended state variables if they remain unchange.  
It is easy to understand that the dimension of state \(x_{0}\) directly affects the calculation load of the estimate analysis. Therefore, the dimension of state \(x_{0}\) to be estimated should be minimized, as mentioned previously, the measurable state variable should not be considered in the estimation. For the system parameters which are considered as the extended state variables, only unknown parameters and the parameters that maybe subject to change are considered, while other parameters are considered as the parts of the system model and are not included in the estimation. Even the unknown parameters if they do not subject to change, they should not be included in the estimation unless in the first time window. Because they have been estimated in the period of the first time window therefore their values become known.