## 7 Ground-Segment Optimization Aspects
### Applicable Optimization Methods for Ground Segment
#### 7.2.2 Long Short-Term Memory (LSTM)
Similarly, the 'Longest Period Method' focuses on the visibility of the satellites and chooses the one with longest time spans, which contains almost all of the negatives associated with the elevation method but with different metrics. Other methods, such as the 'Downdate Method' proposed in [31] or the method utilized in [29], combine some of the traditional ideas together, such as applying a greedy search metric to the ranking idea of these traditional methods. #### 8.3.1 Kalman Filtering (KF)

KF, first introduced in [59], is an estimation method that uses a series of measurements observed over time and estimates the unknown variables within the system by estimating a joint probability distribution over the known variables for each time frame. It requires a dynamical system model represented by two equations: a state Equation (39) and an observation Equation (40). Table 2.3

Table 2.4

In the above equations, \(F\) represents the state transition matrix, \(H\) represents the observation matrix, \(B\) is the control input model, \(x_{k}\) is the state vector \(z_{k}\) is the observation vector, \(w_{k}\) is the system noise vector, \(v_{k}\) is the observation noise vector, and \(u_{k}\) is the control vector. The subscript \(k\) denotes the iteration number (or alternatively, time or sample). We provide the steps of the linear KF process in Algorithm 5, where notation from the state and observation equations carry over, and \(\hat{x}\) is the predicted estimation vector, \(\hat{P}\) is the estimated covariance matrix, \(R\) is the covariance matrix of the measurement noise, and \(Q\) is the covariance matrix of the process noise. If the system represented by Equations (39) and (40) are not linear or the noises are not Gaussian, the process seen in Algorithm 5 requires modifications, which is where the modified KF methods, such as EKF and UKF, emerge. ```
Step 1: Initialization Initialize the parameters of the state and observation models given in Equations (39) and (40) (initial states, observations, matrices, noises, etc.). Step 2: Predict State and Covariance \(x_{k}=F\hat{x}_{k-1}\) \(P_{k}=F\hat{P}_{k-1}F^{T}+Q\) Step 3: Compute Kalman Gain (\(K\)) \(S=HP_{k}H^{T}+R\) \(K=P_{k}H^{T}+S^{-1}\) Step 4: Correction \(\hat{x}_{k}=x_{k}+K(z-Hx_{k})\) \(\hat{P}_{k}=P_{k}+KHP_{k}\) Step 5: Termination Repeat steps 2-4 until a termination criterion is met (i.e., total iterations, change in optimality metric, etc.). ```

**Algorithm 5** KF

Both EKF and UKF apply a linearization step to \(H\) and/or \(F\) matrices before the equations in steps 2-4 of Algorithm 5 are applied to address this issue. In the case of EKF, the non-linear matrix \(H\) is approximated via a Taylor series expansion about the estimated state vector, as in Equation (41), where the notation from Equations (39) and (40) carries over. Table 2.5

In the case of UKF, an unscented transformation is applied, as seen in Equation (42), where \(\chi^{v}\) are the sigma points describing the measurement noise and \(\chi^{x}\) are the sigma points describing the prior predicted states. \(n_{x}\) is the number of weighted samples, which is chosen as a sum of the number of process states and the dimensions of \(w_{k}\) and \(v_{k}\). Again, notation carries over from Equations (39) and (40). Table 2.6

While KF can be used as an offline or online algorithm depending on what the optimization objective is (i.e., what the system model from Equations (39) and (40) represents), in practical applications concerning autonomous vehicles, it is typically used in an online fashion. Although not always in the LEO context, the most common examples of KF implementations in the field of autonomous vehicles are sensor fusion [169] and/or trajectory tracking [170]. The method typically does not consume significant resources from the system and adds negligible delay to the operation; thus, it remains a highly utilized method, although not always for optimization but rather as a method for estimation, tracking, etc.