## 1 Introduction  
As a typical type of electrochemical power source, lithium-ion batteries (LIBs) undergo degradation in both energy capacity and internal resistance during their irreversible aging process [1]. Hence, a reliable and efficient operation of LIBs requires monitoring, control, and management [2, 3]. A battery management system (BMS) is of great importance for the batteries, and state estimation is one of its key roles [4]. Accurately assessing the battery state of charge (SOC) and state of health (SOH) can not only make batteries work closer to their physical limits, but also guarantee their safety and lifetime [5]. However, due to the unmeasurable characteristic of these indicators, battery monitoring is challenging. With a mathematical model that characterizes the dynamics of the battery, indicatorssuch as SOC, SOH, etc., need to be inferred from the limited available measurements such as current, terminal voltage, and surface temperature [6]. Up to now, the most effective approach for battery monitoring is to apply advanced estimation algorithms based on equivalent circuit models (ECMs) [7]. The ECM composed of an open circuit voltage (OCV) source connected in series with a resistor and one or more RC networks is widely used to reproduce the battery's electrical behavior and to further accomplish the objective of battery state estimation [8]. However, the model parameters are actually affected by many factors such as temperature, SOC, charge/discharge rates, and battery aging [4]. Hence, the state-of-the-art estimation methods usually focus on an on-line identified battery model, namely the adaption of model parameters is indispensable. In [9], the parameters of a Thevenin model were constantly updated by a forgetting factor recursive least squares algorithm, while the nonlinear Kalman filter was used to perform the SOC estimation. The series combination of the commonly used recursive least squares (RLS) method and nonlinear state estimator provides a feasible way to enhance the estimation accuracy. In addition, a multi-timescale estimator with a vector-type RLS parameter adaption mechanism was proposed in [4]. The vector-type RLS can address the different variation rates of battery parameters, and then, Kalman filter-based SOC estimator and RLS-based capacity estimator are formulated and integrated in the form of dual estimation. Furthermore, considering the the current and voltage sensors are usually corrupted by noises, a novel technique that integrates the Frisch scheme-based bias compensating RLS with a SOC observer was proposed in [10]. The proposed method estimates the noise statistics and compensates the noise effect so that the model parameters can be extracted without bias. Similarly, the on-line model parameterization process considering the measurement noises was discussed in [11,12] recently. In fact, the aforementioned battery monitoring process can be eventually converted to a classical state and parameter estimation problem. A usual method for estimating slowly varying unmeasurable parameters is to include them in the state vector with the zero-time derivative condition [13]. We can obtain the so-called battery extended ECM. Then, joint or dual estimation methods, mainly based on single or double observers, are preferred [14] and can be applied to the battery extended ECM. Apart from enhancing the accuracy of state estimation, the estimated parameters can also be used to build a battery SOH indicator. SOH, usually evaluated by some related battery parameters correlated with their aging degrees, can be effectively estimated with the zero-time derivative condition. SOC, as the battery energy state indicator, usually appears as a state variable derived from the common ampere-hour integral method [15]. On the one hand, the joint estimation method based on a single observer can cope with the battery monitoring problem in high-dimensional state space directly.