Types of estimators

Control Theory · Family

When the state of a system is not measured directly, it can be estimated in several ways. An estimator is any rule for estimating an unknown quantity. An observer is a dynamical system whose state estimate converges to the true state. A filter estimates the current state from noisy measurements, and a predictor estimates a future state.

Definitions

estimator

High school

Consider an unknown quantity and some data related to it. An estimator is a rule for guessing the unknown quantity from the data.

Undergraduate

Consider an unknown quantity, such as a state or a parameter, and the data available about it. An estimator is any rule for computing an estimate of the quantity from the data.

Examples

• A least-squares estimate of a constant parameter from data
  A least-squares estimate is an estimator because it estimates a quantity from data.
• A Kalman filter (definition)
  A Kalman filter is an estimator, because it estimates the state from noisy measurements.
• A Luenberger observer (definition)
  A Luenberger observer is an observer, and so an estimator, because it estimates the current state of a deterministic system from its inputs and outputs.
• A particle filter (definition)
  A particle filter is a filter, and so an estimator, because it estimates the current state from noisy measurements, by approximating the state’s distribution with weighted samples.
• An extended Kalman filter (definition)
  An extended Kalman filter is a filter, and so an estimator, because it estimates the current state of a nonlinear system from noisy measurements, by applying the Kalman filter to a linearization of the system.
• An unscented Kalman filter (definition)
  An unscented Kalman filter is a filter, and so an estimator, because it estimates the current state of a nonlinear system from noisy measurements, by passing sample points through the system’s model.

Properties

• Observers, filters, and predictors are all estimators of a state.

predictor

High school

Consider a quantity that changes over time, and the data about it available now. A predictor estimates a future value of the quantity from that data.

Undergraduate

Consider a system with state $x(t)$, measured up to time $t$, and a time $\tau > 0$ into the future. A predictor estimates the future state $x(t + \tau)$ from the measurements up to time $t$.

Examples

• A weather forecast for tomorrow
  A weather forecast is a predictor because it estimates a future state.

Non-examples

• A least-squares estimate of a constant parameter from data
  A least-squares estimate of a constant parameter from data is not predictor, because it estimates a constant parameter, not a changing state.
• A low-pass filter on a sensor signal
  A low-pass filter on a sensor signal is not a predictor, because it does not estimate the future state of a system.
• A Kalman filter (definition)
  A Kalman filter is not a predictor, because it estimates the present state, not a future one.
• A Luenberger observer (definition)
  A Luenberger observer is not a predictor, because it estimates the present, not the future.
• A particle filter (definition)
  A particle filter is not a predictor, because it estimates the present.
• An extended Kalman filter (definition)
  An extended Kalman filter is not a predictor, because it estimates the present state, not a future one.
• An unscented Kalman filter (definition)
  An unscented Kalman filter is not a predictor, because it estimates the present state, not a future one.

filter (estimation)

High school

Consider a quantity that is measured over time, with noisy measurements. In estimation, a filter estimates the current value of the quantity from the measurements made up to now.

Undergraduate

Consider a system with state $x(t)$ and noisy measurements of it. In estimation, a filter computes an estimate of the current state $x(t)$ from the measurements up to time $t$.

Examples

• A Kalman filter (definition)
  A Kalman filter is a filter, because it estimates the current state from the measurements up to the current time.
• A particle filter (definition)
  A particle filter is a filter, because it estimates the current state from noisy measurements up to now, by approximating the state’s distribution with weighted samples.
• An extended Kalman filter (definition)
  An extended Kalman filter is a filter, because it estimates the current state of a nonlinear system from noisy measurements up to now.
• An unscented Kalman filter (definition)
  An unscented Kalman filter is a filter, because it estimates the current state of a nonlinear system from noisy measurements up to now.

Non-examples

• A least-squares estimate of a constant parameter from data
  A least-squares estimate of a constant parameter from data is not filter, because it estimates a constant parameter, not a changing state.
• A low-pass filter on a sensor signal
  A low-pass filter is a filter in the signal-processing sense: it smooths a signal. It is not a filter estimator because it does not estimate an unknown value.

Properties

• The Kalman filter is the optimal (minimum mean-square error) filter for linear systems with Gaussian noise.

observer

High school

Consider a system with some quantities that you can measure and others that you cannot. An observer is a calculation that estimates the quantities you cannot measure directly from the ones you can.

Undergraduate

Consider a plant with input $u$, output $y$, and an unknown state $x(t)$. An observer is a dynamical system that takes $u$ and $y$ and produces a state estimate $\hat{x}(t)$ such that $\hat{x}(t) - x(t) \to 0$.

Examples

• A Kalman filter (definition)
  A Kalman filter is an observer, because it uses a time-varying gain chosen to minimize the mean-square error.
• A Luenberger observer (definition)
  A Luenberger observer $\dot{\hat{x}} = A \hat{x} + B u + L (y - C \hat{x})$, for $\dot{x} = A x + B u$ and $y = C x$, is an observer when $A - L C$ is Hurwitz, because then its estimation error converges to zero.

Non-examples

• A least-squares estimate of a constant parameter from data
  A least-squares estimate of a constant parameter from data is not an observer, because it estimates a constant parameter, not a changing state.
• A low-pass filter on a sensor signal
  A low-pass filter on a sensor signal is not an observer, because it does not estimate the state of a system.
• An extended Kalman filter (definition)
  An extended Kalman filter is not an observer in general, because its error is not guaranteed to converge.
• An unscented Kalman filter (definition)
  An unscented Kalman filter is not an observer in general, because its error is not guaranteed to converge.

Properties

• A Luenberger observer whose error converges to zero (that is, with $A - L C$ Hurwitz) exists exactly when $(A, C)$ is detectable.

Examples

Example	estimator	predictor	filter (estimation)	observer	Why
A Luenberger observer (definition)	✓	✗	–	✓
A Kalman filter (definition)	✓	✗	✓	✓
An extended Kalman filter (definition)	✓	✗	✓	✗
An unscented Kalman filter (definition)	✓	✗	✓	✗
A particle filter (definition)	✓	✗	✓	–
Missing example: a_moving_horizon_estimator
Missing example: a_high_gain_observer
A weather forecast for tomorrow	–	✓	–	–
A least-squares estimate of a constant parameter from data	✓	✗	✗	✗
A low-pass filter on a sensor signal	–	✗	✗	✗

Notes

• The Kalman filter alternates a prediction step (propagating the estimate forward with the model) and an update step (correcting it with a new measurement).

See also

All distinctions · All definitions