## 4 Adaptive Observer Based on Chaotic De
### Fitness Judement and Convergence Analysis  
In studying the genetic and evolutionary phenomena of biological organisms in nature, biologists use the term fitness to measure the extent to which a species adapt to its living environment. Species that have more adaptation to the environment will have more reproductive opportunities. The species that have a low degree of adaptation to the environment are relatively small and even extinct. The fitness of the individual (fitnessy) refers to a measure of the extent of the body's survival in the population, which is used to distinguish the individual's \"good and bad\". The probability that the fitness is higher is the probability of the next generation. And the relatively low fitness of the next generation is relatively small. In this case, the evolutionary algorithm also use the concept of fitness to measure the ability of individuals in the optimization of the optimization of or to achieve or help to find the optimal solution. The function of measuring the fitness of the individual is a fitness function system. The fitness function is also called the evaluation function, which is mainly through individual characteristics to determine the fitness of the individual. The design of fitness function needs to be proper, otherwise it is easy to make the algorithm premature and fall into the local optimal. The evolutionary search of DE algorithm is mainly based on fitness function and makes little use of external information. Therefore, the selection of fitness function directly affects the convergence performance. The fitness function is the main component of DE algorithm complexity, so the design of fitness function should be as simple as possible to minimize the time complexity of calculation. The fitness function can be directly transformed by the objective function \(f_{ob}(X)\), and its design requirements are as follows:  
1. Single value, continuous, non-negative and maximized;
2. Reasonable and consistent;
3. Small amount of calculation;
4. Strong versatility. According to the above requirements, we define the fitness function  
\[f_{t}(X)=-f_{ob}(X)\]  
and a searching vector  
\[X_{j}=[x_{1},x_{2},...,x_{k}],\]  
where elements \(X_{j},j=1,2,...,Np\) are in accordance with initial values of the unobtained state variables and unknown parameters. \(k\) is the dimension of the estimator system, Np is the number of individual of population. The fitness \(fit(X_{j})\) is greater, the observer output vector \(\hat{Y}(t)\) is closer to the real system output vector \(Y(t)\). **Theorem 1**: _Convergence of chaotic DE based State Estimate Chaotic DE based state estimate guarantees global convergence of pre-estimated value \(\hat{x}_{0}(i,g)\), \(i=1,2,3,\ldots,m\) to \(x_{0}\) with the iteration times \(g\)._  
(i) \(Y(\hat{x}_{0}(i,g))\) converging to \(Y(x_{0})\) implies \(\hat{x}_{0}(i,g)\) converging to \(x_{0}\);  
It is obvious that if the system is observable through the output \(y(t)\). According to the definition of system observability, \(x_{0}\) corresponds one-to-one to the output trajectory \(\mathcal{Y}_{t_{0},x_{0}}(t)\), \(\hat{x}_{0}\) corresponds one-to-one to the output trajectory \(\mathcal{Y}_{t_{0},\hat{x}_{0}(i,g)}(t)\), so if the sampling points of the output sample are enough, \(x_{0}\) will correspond to the sample vector \(Y(x_{0})\) one-to-one, and \(\hat{x}_{0}\) will correspond to the sample vector \(Y(\hat{x}_{0}(i,g))\) one-to-one. Therefore \(Y(\hat{x}_{0}(i,g))\) converging to \(Y(x_{0})\) implies \(\hat{x}_{0}(i,g)\) converging to \(x_{0}\). (ii) The greedy selection mechanism of the chaotic DE algorithm guarantees global convergence of sample vector \(Y(\hat{x}_{0}(i,g))\) to \(Y(x_{0})\) with the iteration times \(g\).