Extended Kalman Filter — Control Systems/State Estimation

Extended 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 – [x1 + 0.05*x2; x2 + 0.05*((1 - x1^2)*x2 - x1 + u)], say. Each sample the filter linearizes them at the current estimate, corrects the estimate with the measurement, and predicts the next one.

It is MATLAB's own extended Kalman filter (extendedKalmanFilter, which Simulink's block runs) in its square-root form: the covariance is carried as a factor S with P = S·S', and both the correction and the prediction update S by a QR factorization rather than forming P. That form keeps P symmetric and positive semidefinite however long the filter runs.

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.
• Jacobians – Numerical (the default, as in MATLAB) linearizes f and h by forward differences with MATLAB's own step, max(√eps, √eps·|x_j|); Exact differentiates the expressions symbolically, which is what supplying the Jacobian functions gives MATLAB.
• 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 semidefinite, or a scalar.
• 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, h and their Jacobians written in the target's own syntax, the two QR factorizations and the triangular solves unrolled – so every target does the same arithmetic in the same order. The noise factors 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.

⚠ VHDL linearizes with the exact Jacobian even when Jacobians is Numerical (whenever f and h can be differentiated). VHDL's math_real functions – SIN, ARCTAN and the rest – are accurate to about 1e-8 in GHDL, and a forward difference over a step of 1.5e-8 turns that into an error of order one in the Jacobian. The exact Jacobian differs from the difference quotient by about 1e-8 relative.

Simulink bridge

None (Support::None). Simulink's block (Control System Toolbox and System Identification Toolbox, cstblocks/State Estimation/Extended 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.
• ⚠ Numerical Jacobians agree with MATLAB to about 1e-8, not bit for bit. A forward difference amplifies the last bit of f and h, and MATLAB evaluates x^2 as x*x where this block uses a power. With Exact Jacobians the two agree to 1e-13.
• Carries no state space: the filter is nonlinear, and f and h describe the PLANT, not this block.