[Practical Exercise] Gaussian Belief Propagation Algorithm for DC State Estimation Model in MATLAB

# 2.1 Principles of Gaussian Belief Propagation Algorithm

Gaussian Belief Propagation (GBP) is a nonlinear state estimation algorithm based on the Bayesian filtering framework. It approximates the posterior probability density function with non-Gaussian distributions using Gaussian distributions, thus achieving state estimation in nonlinear systems.

### 2.1.1 State Prediction and Update Equations

The core concept of the GBP algorithm is to update the state distribution through message passing. In the state prediction step, the prior distribution of the system state is updated to the posterior distribution through the transition probability density function:

p(x_k | y_{1:k-1}) = \int p(x_k | x_{k-1}) p(x_{k-1} | y_{1:k-1}) dx_{k-1}

Here, x_k represents the state at time k, y_{1:k-1} represents the observed data from time 1 to k-1, p(x_k | x_{k-1}) is the transition probability density function, and p(x_{k-1} | y_{1:k-1}) is the state posterior distribution at time k-1.

In the state update step, the posterior distribution is updated to a new posterior distribution through the observation probability density function:

p(x_k | y_{1:k}) = \frac{p(y_k | x_k) p(x_k | y_{1:k-1})}{\int p(y_k | x_k) p(x_k | y_{1:k-1}) dx_k}

Here, p(y_k | x_k) is the observation probability density function.

### 2.1.2 Weight Update Equations

In the GBP algorithm, the weight of each state variable represents its importance in the posterior distribution. The weight update equations are used to update the weights to reflect the credibility of the state variables under the observed data:

w_k(x_k) = \frac{p(y_k | x_k) w_{k-1}(x_k)}{\sum_{x_k} p(y_k | x_k) w_{k-1}(x_k)}

Here, w_k(x_k) represents the weight of state x_k at time k, and w_{k-1}(x_k) represents the weight at time k-1.

# 2. Gaussian Belief Propagation Theory

### 2.1 Principles of Gaussian Belief Propagation

Gaussian Belief Propagation (GBP) is an approximate reasoning algorithm used for nonlinear, non-Gaussian state estimation. Its basic principle is to use Gaussian distributions to approximate non-Gaussian distributions and propagate these Gaussian distributions to approximate the calculation of the posterior distribution.

#### 2.1.1 State Prediction and Update Equations

In the GBP algorithm, the system state is represented by a Gaussian distribution, namely:

p(x) = N(x; μ, Σ)

Here, μ and Σ represent the mean and covariance matrix of the Gaussian distribution, respectively.

In the state prediction phase, the state prediction distribution can be obtained based on the system model and prior distribution:

p(x_t | x_{t-1}) = N(x_t; f(x_{t-1}), Q)

Here, f(x_{t-1}) represents the system model, and Q represents the process noise covariance matrix.

In the state update phase, the state posterior distribution can be obtained based on the measurement model and prediction distribution:

p(x_t | y_t, x_{t-1}) = N(x_t; μ_t, Σ_t)

Here, μ_t and Σ_t represent the mean and covariance matrix of the posterior distribution, respectively, and the calculation formulas are:

μ_t = μ_t^- + K_t(y_t - h(μ_t^-))
Σ_t = Σ_t^- - K_tH_tΣ_t^-

Here, μ_t^- and Σ_t^- represent the mean and covariance matrix of the prediction distribution, respectively, K_t is the Kalman gain, and H_t is the Jacobian matrix of the measurement model.

#### 2.1.2 Weight Update Equations

In the GBP algorithm, each Gaussian distribution is associated with a weight representing its importance in the posterior distribution. The weight update equations are used to update these weights, and the formula is:

w_t = w_{t-1}N(y_t; h(μ_t), H_tΣ_tH_t^T + R)

Here, w_t and w_{t-1} represent the updated and previous weights, respectively, and R represents the measurement noise covariance matrix.

### 2.2 Advantages and Limitations of Gaussian Belief Propagation

#### 2.2.1 Advantages: Nonlinear, Non-Gaussian State Estimation

The advantage of the GBP algorithm is that it can be used for nonlinear, non-Gaussian state estimation. Traditional Kalman filters can only be used for linear, Gaussian systems, while the GBP algorithm approximates non-Gaussian distributions as Gaussian distributions, allowing it to handle more complex systems.

#### 2.2.2 Limitations: High Computational Load, Sensitive to Initial Conditions

The limitation of the GBP algorithm is that its computational load is high, especially when the system dimension is high. I