kalmanOneStepLTV
Updated:
Tags: documentation, estimation, ltv, one-step
Sintax
[K,Ppred,Pfilt] = kalmanOneStepLTV(system,E,Pprev)
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 $k$.
Let matrix $\mathbf{E} \in\mathbb{R}^{n\times o}$ denote a sparsity pattern. The set of matrices which obey the sparsity constraint determined by $\mathbf{E}$ is defined as
\[\mathrm{Sparse}(\mathbf{E}) :=\left\{[\mathbf{K}]_{ij}\in\mathbb{R}^{n\times o}: [\mathbf{E}_{ij}] = 0 \implies [\mathbf{K}]_{ij}= 0;\: i= 1,...,n, \:j=1,...,o \right\}.\]The command
[K,Ppred,Pfilt] = kalmanOneStepLTV(system,E,Pprev)computes the one-step filter gain at time instant $k$, that aims at solving the optimization problem
\[\begin{aligned} & \underset{\begin{subarray}{c}\mathbf{K}(k)\in \mathbb{R}^{n\times o} \end{subarray}}{\text{minimize}} & & \mathrm{tr}(\mathbf{P}(k|k)) \\ & \text{subject to} & & \mathbf{K}(k) \in \mathrm{Sparse}(\mathbf{E})\:, \end{aligned}\]where $\mathbf{P}(k|k)$ is the estimation error covariance matrix, using the one-step method proposed in [Section 3, 1].
Computational complexity
The one-step optimization problem is solved using the efficient sparse equation solver proposed in [2]. See sparseEqSolver for the implementation of the solver.
Define the set $\chi$ of integer pairs of the form $(i,j)$ to index the nonzero entries of $\mathbf{E}$ as
\[\begin{cases} (i,j) \in \chi &\;,\;\left[\mathbf{E}\right]_{i,j} \neq 0\\ (i,j) \notin \chi &\;,\;\text{otherwise} \end{cases}, i = 1,...,n,\: j = 1,...,o\:.\]It is shown in [2] that each gain computation of the algorithm requires $\mathcal{O}(|\chi|^3)$ floating-point operations, where $|\chi|$ denotes the cardinality of set $\chi$. In the field of distributed estimation and control theory, $|\chi|$ is usually given by $|\chi| \approx cn$, where $c\in \mathbb{N}$ is a constant. It, thus, follows that each iteration requires $\mathcal{O}(n^3)$ floating-point operations, thus it has the same complexity as a centralized gain computation.
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}: \(\mathbf{R}(k)\)
E: sparsity pattern $\mathbf{E}$Pprev: predicted estimation error covariance matrix, i.e., $\mathbf{P}(k|k-1)$
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 $\mathbf{P}(k|k)$
Examples
See One-step Kalman filter tutorial for LTV systems for a tutorial.