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
Examples
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A Kalman filter
(definition)
Undergraduate
A Kalman filter is an observer, because it uses a time-varying gain chosen to minimize the mean-square error.
-
A Luenberger observer
(definition)
Undergraduate
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
Things that may be mistaken for examples.
-
A least-squares estimate of a constant parameter from data
High school
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
High school
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)
Undergraduate
An extended Kalman filter is not an observer in general, because its error is not guaranteed to converge.
-
An unscented Kalman filter
(definition)
Graduate
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.
Distinctions involving this term
- Types of estimators: 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.