# A New Constrained State Estimation Method Based on Unscented Hoo Filtering
## 1 Introduction  
The Kalman filter (KF) was proposed by [PERSON] in the 1960s to solve discrete-time linear filtering problems [1], which is a recursive filter based on the minimum mean-square-error (MMSE) criterion. The Kalman filter has been widely used in navigation systems [2], robot motion planning [3], target tracking [4], state of charge (SOC) estimation [5], etc. The Kalman filtering method is described by the state transition model and measurement model. The state transition model is responsible for projecting the current state estimate ahead in time [6]. The measurement model adjusts the projected estimate by an actual measurement at that time. The models are listed as follows [7]:  
\[\begin{array}{l}x_{k+1}=A_{k}x_{k}+B_{k}u_{k}+w_{k},\\ y_{k}=C_{k}x_{k}+v_{k},\end{array} \tag{1}\]  
where \(k\) is the index, \(x_{k}\) is the unobserved state of the system, \(y_{k}\) is the measurement, \(w_{k}\) is the process noise and \(v_{k}\) is the measurement noise. \(A_{k}\), \(B_{k}\), \(C_{k}\) are the state transition matrix, the control input matrix, and the measurement matrix, respectively.  
However, traditional KF can only deal with linear systems [8]. In practical applications, nonlinear systems often appear. In order to apply Kalman filtering to nonlinear systems, some improved methodshave been proposed. [PERSON] et al. [9] proposed the extended Kalman filter (EKF) to deal with nonlinear cases. [PERSON] and [PERSON] et al. [6] proposed the Unscented Kalman Filter (UKF). The EKF is uncomplicated and widely used in various fields. However, the EKF has limitations when it is applied to nonlinear systems. It first needs to linearize the system using a 1 st-order truncation of the multi-dimensional Taylor series expansion, which introduces significant errors. In contrast to the EKF, true nonlinear models are used in UKF, and they are similar to the distribution of the random state variables. The UKF approximates the parameter distribution with discrete sigma points by the Unscented Transform (UT) and propagates them through the original nonlinear system [10].  
The UKF method needs an accurate system model and perfect knowledge of the noise statistics [6]. Different from the UKF, Ho filtering (HF) requires no a priori knowledge of the noise statistics but instead requires knowledge of finite bounded energies [11]. Ho filtering aims to minimize the effect of the worst possible disturbances on the estimation errors [12]. However, Ho filtering is used for linear systems. It can be applied to nonlinear systems when it is combined with the EKF and UKF, which forms the Extended Ho filtering (EHF) and the Unscented Ho filtering (UHF). The authors of [13] propose an Ho extended Kalman filtering (HEKF) based on the robust control theory, and numerical results on the IEEE 39-bus system demonstrate the effectiveness of the HEK. [PERSON], [PERSON] et al. [14] combined Ho filtering with the UKF to process the state estimator of the power system by integrating robust statistics and robust control theory. It is more robust against model uncertainty. The performance of the new filter is better.  
In some applications, state variables are often accompanied with many constraints, such as the route limitation and the mass conservation law in electronic navigation. Adding these constraints to the Kalman filter can effectively improve the estimation accuracy. This paper focuses on the state estimation problems with constraints. A new constrained state estimation method based on unscented Ho filtering is proposed.