# A New Constrained State Estimation Method Based on Unscented Hoo Filtering
## Preliminaries
### The Principle of the Unscented H\(\infty\) Filter  
The UKF is a recursive MMSE estimator based on the optimal Gaussian approximation Kalman filter framework [17]. The UKF does not require model linearization, but performs a direct estimate of the state propagation using a set of deterministically selected sample points. A set of 2L+1 sigma points are chosen to capture the true mean and covariance of the prior random variable \(x_{k}\). The nonlinear filtering problem is depicted in a discrete-time state-space form as follows. \[x_{k+1}=f(x_{k},u_{k})+w_{k} \tag{14}\]  
\[y_{k}=h(x_{k})+v_{k} \tag{15}\]  
where \(f\) and \(h\) are the nonlinear functions that describe the process and measurement models; \(x_{k}\) is the system state vector, \(y_{k}\) is the measurement vector, \(u_{k}\) is the system input vector; \(w_{k}\) and \(v_{k}\) are the system process and measurement noise, respectively, which are assumed to be independent and identically distributed with zero mean and covariance matrices \(Q_{k}\) and \(R_{k}\), respectively. Since the statistical characteristics of the system noises are usually unknown or inaccurate in practical applications, the traditional UKF faces uncertain interference, which always performs poorly. Unscented H\(\infty\) filtering (UHF) is an improvement to the UKF. The UHF updates the estimation error covariance through auto adjustment of the measurement error covariance by selecting a proper scalar parameter \(\gamma\), as shown in Equation (32). The UHF includes two steps of time update and measurement update. The time update equation will get 2L+1 sigma points. The current state and the estimated value of the covariance are obtained by the unscented transform (UT), which constitutes the prior state of the next moment. The idea of a linear combination of the Kalman filter combines the measured value with the preceding value. The error covariance is corrected by H\(\infty\) filtering to obtain the best estimate of the state. The specific UKF algorithm steps are as follows:  
(1) Initialization:  
\[\hat{x}_{0}=E[x_{0}] \tag{16}\]\[P_{0}=E\big{[}(x_{0}-\hat{x}_{0})(x_{0}-\hat{x}_{0})^{T}\big{]} \tag{17}\] \[\hat{x}_{0}^{a}=E[x^{a}]=E[\hat{x}_{0}\quad 0\quad 0]^{T}\] (18) \[P_{0}^{a}=E[(x_{0}^{a}-\hat{x}_{0}^{a})(x_{0}^{a}-\hat{x}_{0}^{a})^{T}]= \left[ \begin{array}{ccc}P_{0}&0&0\\ 0&Q_{k}&0\\ 0&0&R_{k}\end{array} \right]\] (19) where the \(Q_{k}\) and \(R_{k}\) are the system process noise and measurement noise covariance. In the UKF, the effective vector of this augmented state is \(x_{k}^{a}=[x_{k}\,w_{k}\,v_{k}]^{T}\), the superscript \"a\" here means the new state is an \"augmented\" state. It consists of a target state vector, process noise and observation noise. To calculate the sigma points and weights: 1. Generate 2L+1 Sigma points: \[\left\{ \begin{array}{l}\chi_{k-1}^{0}=\hat{x}_{k-1},i=0\\ \chi_{k-1}^{i}=\hat{x}_{k-1}+(\sqrt{(L+\lambda)P_{k-1}})_{i},i=1,\ldots,L\\ \chi_{k-1}^{i}=\hat{x}_{k-1}-(\sqrt{(L+\lambda)P_{k-1}})_{i-L},i=L+1,\ldots,2L \end{array} \right.\] 2. The weights are determined as follows: \[\left\{ \begin{array}{l}\omega_{0}^{(m)}=\frac{\lambda}{L+\lambda}\\ \omega_{0}^{(c)}=\frac{\lambda}{L+\lambda}+\left(1-\alpha^{2}+\beta\right)\\ \omega_{i}^{(m)}=\omega_{i}^{(c)}=\frac{1}{2(L+\lambda)},i=1,\ldots,2L\end{array} \right.\] (21) where \((\sqrt{(L+\lambda)P_{k-1}})_{i}\) denotes the \(i\)-th column of the square root of the matrix [18]. \(\lambda\) is a compound scaling parameter,\(\lambda=\alpha^{2}(L+\kappa)-L;L\) is the dimension of the state variable; \(\alpha\) is a scale parameter, which is generally a small positive value and determines the distribution of the sigma points [19]; \(\beta\) is a non-negative weighting term that affects the weight of the sigma points used for covariance calculations and can be set to 2; \(\kappa\) is a scale parameter, which is usually greater than or equal to 0 to ensure the positive definiteness of the covariance matrix.