Unscented Kalman Filter — Control Systems/State Estimation

Unscented Kalman Filter

Control Systems / State Estimation

Estimates the states of a nonlinear discrete-time plant from a noisy measurement. The plant is

x[k+1] = f(x[k], u[k]) + w[k],   y[k] = h(x[k], u[k]) + v[k],   with cov(w) = Q and cov(v) = R,

and f and h are expressions you type over the state and input variables you name. Instead of linearizing them, the filter pushes 2n + 1 sigma points – the estimate and a symmetric spread around it, sized by the covariance – through f and h, and takes the mean and covariance of what comes out. No derivative is ever formed, so f and h may be anything the expression language can write.

It is MATLAB's own unscented Kalman filter (unscentedKalmanFilter, which Simulink's block runs) in its square-root form: the covariance is carried as a factor S with P = S·S', updated by QR factorizations and rank-one Cholesky updates.

The factory setting is a forced Van der Pol oscillator, stepped by forward Euler at 0.05 s, whose position x1 is measured: feed the oscillator's input on u and its noisy position on y, and xhat recovers the velocity nobody measured.

Ports

• u – the known plant input, a column [p,1] in the order of Input Variables. With no input variables it must be a scalar, and it is ignored.
• y – the measurement, a column [m,1] (m = the entries of Measurement h).
• xhat – the state estimate, a column [n,1] in the order of State Variables.

Parameters

• State Variables – the n state names, separated by spaces: 1 to 6 of them.
• Input Variables – the p input names, 0 to 6 of them; leave it empty for a plant with no input.
• State Transition f – the next state f(x, u): a column of n expressions over the state and input variables.
• Measurement h – the measurement h(x, u): a column of 1 to 6 expressions.
• Alpha – the spread of the sigma points around the estimate, greater than 0 and at most 1; MATLAB's default 1e-3.
• Beta – prior knowledge of the distribution, 0 or greater; 2 is optimal for a Gaussian, MATLAB's default.
• Kappa – a secondary spread parameter, from 0 to 3; MATLAB's default 0.
• Process Noise Q – [n,n], symmetric positive semidefinite; a scalar is taken as that value times the identity.
• Measurement Noise R – [m,m], symmetric positive definite (the gain divides by it); a scalar expands the same way.
• Initial State – x[0], n entries.
• Initial State Covariance – P[0], [n,n] symmetric positive definite, or a positive scalar: the first correction updates its triangular Cholesky factor.
• Estimator Type – Current publishes x̂[k|k], corrected with the measurement of the same sample (Simulink's default); Delayed publishes x̂[k|k−1], the prediction made before it.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Each prints the same description of the filter that the simulation itself runs – f and h written in the target's own syntax at every sigma point, the QR factorizations, the Cholesky updates and their fallback unrolled – so every target does the same arithmetic in the same order. The noise factors, the weights and the initial state are baked in at export time.

⚠ The three HDL targets run the filter in simulation-only real arithmetic, quantizing only at the port boundaries: a covariance factor and a triangular solve span many decades, which a Q16.16 datapath does not carry.

Simulink bridge

None (Support::None). Simulink's block (Control System Toolbox and System Identification Toolbox, cstblocks/State Estimation/Unscented Kalman Filter) names its state transition and measurement as MATLAB functions on the path, not as expressions, so there is nothing a model could carry across; the bridge reports this block rather than dropping it silently. Code export verification still covers it across all ten languages.

Notes

• Discrete only, and stateful: the estimate and the covariance factor persist from sample to sample.
• ⚠ At MATLAB's default Alpha of 1e-3 the filter agrees with MATLAB to about 1e-10, not bit for bit: the sigma-point weights are about a million and cancel, and MATLAB run against itself with the initial state moved by one ulp differs by the same amount. From Alpha 0.05 upward the two agree to about 1e-15.
• When a rank-one Cholesky downdate would leave the factor indefinite – often, at small Alpha – the filter does what MATLAB does: it re-factors the covariance from an eigen-decomposition, taking the square root of each eigenvalue's magnitude.
• Carries no state space: the filter is nonlinear, and f and h describe the PLANT, not this block.