Particle Filter — Control Systems/State Estimation

Particle Filter

Control Systems / State Estimation

Estimates the states of a nonlinear discrete-time plant from a noisy measurement by carrying a cloud of weighted particles, each a guess of the state. The plant is

x[k+1] = f(x[k], u[k]) + w[k],   y[k] = h(x[k], u[k]) + v[k],   with w ~ N(0, Q) and v ~ N(0, R),

and f and h are expressions you type over the state and input variables you name. Each sample the filter weighs every particle by how well h(particle) explains the measurement, resamples the cloud when too few particles carry the weight, outputs the estimate, and moves every particle through f with fresh process noise. Unlike a Kalman filter it assumes nothing Gaussian about the estimate, so it follows multimodal and strongly nonlinear problems.

It is MATLAB's own particle filter – the code Simulink's Particle Filter block runs – with the state transition and the measurement likelihood written as expressions instead of MATLAB functions. The block outputs the estimate after the correction, as Simulink's does with Use the current measurements to improve state estimates on (its default).

The factory setting tracks a forced Van der Pol oscillator, stepped by forward Euler at 0.05 s, from its noisy position x1 with 1000 particles.

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 8 of them.
• Input Variables – the p input names, 0 to 8 of them; leave it empty for a plant with no input. The state and input names together are at most 12.
• State Transition f – the next state f(x, u) before noise: a column of n expressions.
• Process Noise Q – the covariance of w, n×n symmetric positive definite, or a scalar meaning that times the identity. Zero means no process noise.
• Measurement h – the measurement h(x, u): a column of 1 to 8 expressions.
• Measurement Noise R – the covariance of v, m×m symmetric positive definite, or a positive scalar. The likelihood of a particle is the Gaussian density of y − h(particle) with this covariance.
• Number of Particles – 1 to 5000.
• Initial Distribution – Gaussian (from Initial Mean and Initial Covariance) or Uniform (inside Initial State Bounds). The cloud is drawn on the first sample.
• Initial Mean – a column of n values.
• Initial Covariance – n×n symmetric positive definite, or a positive scalar.
• Initial State Bounds – an n×2 matrix, one [lower upper] row per state.
• Resampling Method – Multinomial, Systematic or Stratified.
• Resampling Trigger – Ratio resamples when the effective number of particles, 1/Σw², falls below Minimum Effective Particle Ratio times the count; Interval resamples every Sampling Interval corrections.
• Minimum Effective Particle Ratio – 0 to 1.
• Sampling Interval – a whole number of corrections, 1 or more, or Inf for never.
• State Estimation Method – Mean (the weighted mean of the cloud) or Max Weight (the first particle of largest weight).
• Seed – the random generator's seed, a whole number: the same seed gives the same run, in simulation and in every export.
• Sampling Time (s) – the filter's sample time; zero or less inherits the solver's rate.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Each carries the cloud, the weights and its own copy of the random generator, and runs the same step in the same order, so an export reproduces the simulation's particles draw for draw.

⚠ The three HDL targets run in simulation-only real arithmetic, quantizing only at the port boundaries.

Simulink bridge

None (Support::None). Simulink's Particle Filter names its state transition and measurement likelihood as MATLAB functions on the path, and this block's are typed expressions, so nothing a model file carries maps onto it; the bridge reports this block rather than dropping it silently.

Notes

• Discrete only, and stateful: the cloud, the weights and the generator persist from sample to sample.
• ⚠ The random numbers are not MATLAB's. Simulink's block draws from MATLAB's global stream; this one uses the generator of Simulink's Random Number block, so the same seed does not give the same particles as MATLAB. The filter's arithmetic is MATLAB's to the last digits: fed the same draws, it agrees with MATLAB's own code to 4e-15.
• ⚠ Multinomial positions are drawn as normalized sums of exponentials – the same distribution as MATLAB's sorted uniforms, with no sort to carry into ten languages.
• The process noise is additive Gaussian and the likelihood Gaussian. Residual resampling, circular state variables, several measurement likelihoods, enable ports, and particles or weights on an output port are not offered.
• Carries no state space: the filter is nonlinear, and f and h describe the PLANT, not this block.