kalmanCentralizedLTV

Sintax

[K,Ppred,Pfilt] = kalmanCentralizedLTV(system,Pprev)

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Description

Consider a generic LTV system of the form

\[\begin{cases} \mathbf{x}(k+1)=\mathbf{A}(k)\mathbf{x}(k)+\mathbf{B}(k)\mathbf{u}(k)+\mathbf{w}(k)\\ \mathbf{y}(k)=\mathbf{C}(k)\mathbf{x}(k)+\mathbf{v}(k) \end{cases}\:,\]

where $\mathbf{x}(k)\in\mathbb{R}^{n}$ is the state vector, $\mathbf{u}(k)\in \mathbb{R}^{m}$ is the input vector, which is assumed to be known, and $\mathbf{y}(k)\in\mathbb{R}^{o}$ is the output of the system. The vectors $\mathbf{w}(k)$ and $\mathbf{v}(k)$ are the process and observation noise, modelled as zero-mean uncorrelated white Gaussian processes with associated covariance matrices $\mathbf{Q}(k) \succeq \mathbf{0}\in\mathbb{R}^{n\times n}$ and $\mathbf{R}(k) \succ \mathbf{0}\in\mathbb{R}^{o\times o}$, respectively.

Consider a standard Kalman filter. The prediction step follows

\[\hat{\mathbf{x}}(k|k-1) = \mathbf{A}(k-1)\hat{\mathbf{x}}(k-1|k-1)+\mathbf{B}(k-1)\mathbf{u}(k-1)\:,\]

where $\hat{\mathbf{x}}(k|k-1)$ denotes the predicted state estimate at time instant $k$ and $\hat{\mathbf{x}}(k|k)$ the filtered state estimate at time instant $k$. The filtering step follows

\[\hat{\mathbf{x}}(k|k) = \hat{\mathbf{x}}(k|k-1)+\mathbf{K}(k)\left(\mathbf{y}(k) - \mathbf{C}(k)\hat{\mathbf{x}}(k|k-1)\right)\:,\]

where $\mathbf{K}(k)\in\mathbb{R}^{n\times o}$ is the filter gain at time instant $k$.

The command

[K,Ppred,Pfilt] = kalmanCentralizedLTV(system,Pprev)

computes the centralized kalman filter gain at time instant $k$.

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Input arguments

Required

• system : \(1\times 4\) cell array of the time-varying dynamics matrices of the LTV system, i.e.,
  • system{i,1}: \(\mathbf{A}(k)\)
  • system{i,2}: \(\mathbf{C}(k)\)
  • system{i,3}: \(\mathbf{Q}(k)\)
  • system{i,4}: \(\mathbf{R}(k)\)
• Pprev : predicted estimation error covariance matrix, i.e., $\mathbf{P}(k|k-1)$

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Output Arguments

• K: filter gain matrix $\mathbf{K}(k)$
• Ppred: predicted estimation error covariance matrix $\mathbf{P}(k+1|k)$
• Pfilt: filtered estimation error covariance matrix $\mathbf{P}(k|k)$