kalmanCausalFiniteHorizonLTV
Updated:
Tags: causal-finite-horizon, documentation, estimation, ltv
Sintax
[K,P] = kalmanCausalFiniteHorizonLTV(system,E,T,P0)
[K,P] = kalmanCausalFiniteHorizonLTV(system,E,T,P0,opts)
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 commands
[K,P] = kalmanCausalFiniteHorizonLTV(system,E,T,P0)
[K,P] = kalmanCausalFiniteHorizonLTV(system,E,T,P0,opts)compute a window of causal finite-horizon filter gains, that aims at solving the optimization problem
\[\begin{aligned} & \underset{\begin{subarray}{c}\mathbf{K}(i) \in \mathbb{R}^{n\times o}\\i = 1,...,T \end{subarray}}{\text{minimize}} & & \mathrm{tr}(\mathbf{P}(T|T)) \\ & \text{subject to} & & \mathbf{K}(i) \in \mathrm{Sparse}(\mathbf{E})\:, i = 1,...,T \end{aligned}\]where $\mathbf{P}(k|k)$ is the estimation error covariance matrix at time instant $k$, using the causal finite-horizon method proposed in [Section III, 1].
Computational complexity
The finite-horizon 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 iteration 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.
Input arguments
Required
system: \(T\times 4\) cell array of the time-varying dynamics matrices of the LTV system, i.e.,system{i,1}: \(\mathbf{A}(i)\), $i = 1,…,T$system{i,2}: \(\mathbf{C}(i)\), $i = 1,…,T$system{i,3}: \(\mathbf{Q}(i)\), $i = 1,…,T$system{i,4}: \(\mathbf{R}(i)\), $i = 1,…,T$
E: sparsity pattern $\mathbf{E}$T: finite window lengthPprev: predicted estimation error covariance matrix, i.e., $\mathbf{P}(1|0)$
Optional
opts: struct of optional arguments (assumes default value for each parameter which is not assigned by the user)epsl: minimum relative improvement on the objective function of the optimization problem (default:opts.epsl = 1e-5)
alpha: ratio of the weighting geometric progression (default:opts.alpha = 0.1)maxOLIt: maximum number of outer loop iterations to run until convergence (default:opts.maxOLIt = 100)
verbose: display algorithm status messages (default:opts.verbose = false)
Output Arguments
K: \(T\times 1\) cell array of filter gain matrices $\mathbf{K}(i)$, $i = 1,…,T$P: \(T\times 1\) cell array of filtered estimation error covariance matrices $\mathbf{P}(i|i)$, $i = 1,…,T$
Examples
See Causal finite-horizon Kalman filter tutorial for LTV systems for a tutorial.
References
[1] Not published yet