kalmanOneStepLTI
Updated:
Tags: documentation, estimation, lti, one-step
Sintax
[K_inf,P_inf] = kalmanOneStepLTI(A,C,Q,R,E)
[K_inf,P_inf] = kalmanOneStepLTI(A,C,Q,R,E,opts)
Description
Consider a generic LTI system of the form
\[\begin{cases} \mathbf{x}(k+1)=\mathbf{A}\mathbf{x}(k)+\mathbf{B}\mathbf{u}(k)+\mathbf{w}(k)\\ \mathbf{y}(k)=\mathbf{C}\mathbf{x}(k)+\mathbf{v}(k) \end{cases}\:,\]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 $\mathbf{Q} \succeq \mathbf{0}\in\mathbb{R}^{n\times n}$ and $\mathbf{R} \succ \mathbf{0}\in\mathbb{R}^{o\times o}$, respectively.
Consider a standard Kalman filter. The prediction step follows
\[\hat{\mathbf{x}}(k|k-1) = \mathbf{A}\hat{\mathbf{x}}(k-1|k-1)+\mathbf{B}\mathbf{u}(k-1)\:,\]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
\[\hat{\mathbf{x}}(k|k) = \hat{\mathbf{x}}(k|k-1)+\mathbf{K}(k)\left(\mathbf{y}(k) - \mathbf{C}\hat{\mathbf{x}}(k|k-1)\right)\:,\]where $\mathbf{K}(k)\in\mathbb{R}^{n\times o}$ is the filter gain. It is often very useful to find a steady-state constant gain $\mathbf{K}_{\infty}$ instead.
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_inf,P_inf] = kalmanOneStepLTI(A,C,Q,R,E)
[K_inf,P_inf] = kalmanOneStepLTI(A,C,Q,R,E,opts)compute the steady-state distributed gain that aims at solving the optimization problem
\[\begin{aligned} & \underset{\begin{subarray}{c}\mathbf{K}_{\infty}\in \mathbb{R}^{n\times o} \end{subarray}}{\text{minimize}} & & \mathrm{tr}(\mathbf{P}_{\infty}) \\ & \text{subject to} & & \mathbf{K}_{\infty} \in \mathrm{Sparse}(\mathbf{E})\:, \end{aligned}\]where $\mathbf{P}_{\infty}$ is the steady-state estimation error covariance matrix, using the one-step method proposed in [Section 4, 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
A: matrix $\mathbf{A}$ of the dynamics of the LTI systemC: matrix $\mathbf{C}$ of the dynamics of the LTI systemQ: covariance matrix of the process noise, $\mathbf{Q}$R: covariance matrix of the observation noise, $\mathbf{R}$E: sparsity pattern $\mathbf{E}$
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)
maxIt: maximum number of iterations until convergence (default:opts.maxIt = 1000)verbose: display algorithm status messages (default:opts.verbose = false)P0: initialization estimation error covariance matrix (default:opts.P0 = zeros(n,n))
Output Arguments
K_inf: steady-state filter gain matrix $\mathbf{K}_{\infty}$P_inf: steady-state estimation error covariance matrix $\mathbf{P}_{\infty}$
Examples
See One-step Kalman filter tutorial for LTI systems for a tutorial.