## 1 Introduction  
Parameter and state estimation has played a significant role in system identification, controller designs, signal filtering and fault diagnosis for decades [1]. Accurate estimation of states and parameters is essential for effective monitoring, control and identification in many fields, such as chemical, biological and mechanical [2; 3; 4; 5; 6; 7; 8]. However, the majority of the system parameters and state variables are difficult to measure and have to be inferred from the available online data and the models relating the measurements. Further, unmeasured model parameters may slowly drift from their nominal values over a period of time. Thus, it becomes necessary to track the changing parameters/unmeasured states and use them for improving performances of monitoring, control and real-time fault diagnosis [9]. The typical state-parameter estimation algorithms include the recursive methods and the iterative methods [1]. Estimation and identification problems are equivalent to the state and parameter estimation problems, respectively. The two problems have conventionally been viewed as two separate problems [10]. To address them, one conventional strategy is to employ two step process-first identify the process parameters in an offline settings, and then use the identified model for real-time state estimation. The second one is to employ dual filters [11], where one filter is used for state estimation, and another one for parameter estimation. In dual filters, the estimation and identification are achieved through iterations between the two filters. The third one is to design adaptive observer which augment the state space with the parameters for the joint estimation [12]. This formulates the estimation and identification problems simultaneously under a single observer framework. Usually the problem of joint state-parameter estimation is solved by recursive algorithms known as adaptive observers [10]. While the adaptive observers were originally developed in continuous system, the estimation and identification are guaranteed to asymptotically converge to the true values only if the process is run long enough. If the parameters change faster than the convergence rate, it is difficult to converge within a shorter duration time. So observation with time constraints (non-asymptotic observation)become a more practical problem for both control theory and practice [13]. In term of the time varying parameters estimation for nonlinear systems, it is usually very difficult to analyze the convergence and stability of the observer [14]. Similarly, it is difficult to adjust the gain matrix of the adaptive observer to obtain satisfactory convergence performance whether it is stable or time-varying [12]. Adaptive observer can be widely used to solve the state estimation problem of a nonlinear system. However, existing methods generally perform state estimation based on the assumption that the initial state is known and accurate. Nevertheless, in the actual industrial process, the initial state is generally unknown or biased. Improper initial value of the state can lead to inaccuracy state estimation result. In this paper, the problem of initial state uncertainty in state estimation of nonlinear systems is considered. Although a great deal of knowledge has been accumulated in the literature about adaptive observer design for nonlinear systems, a customary approach is to linearise the nonlinear model around the current states estimate, and then to apply linear systems techniques for states estimation of nonlinear system. However, this strategy is only effective if linearization does not result in a large mismatch between the linear model and the nonlinear behavior. In order to improve the estimation performance, it is necessary to constrain the nonlinear system when designing the observer. As a result, the analysis and design process is often very complex. To overcome this problem, [PERSON] and [PERSON] propose a genetic adaptive observer [15].