Observability

In control theory, observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. The observability and controllability of a system are mathematical duals. The concept of observability was introduced by Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems.Formally, a system is said to be observable if, for any possible sequence of state and control vectors (the latter being variables whose values one can choose), the current state (the values of the underlying dynamically evolving variables) can be determined in finite time using only the outputs. (This definition uses the state space representation.) Less formally, this means that one can determine the behavior of the entire system from the system's outputs. If a system is not observable, this means that the current values of some of its state variables cannot be determined through output sensors. This implies that their value is unknown to the controller (although they can be estimated by various means).Consider the continuous linear time-variant systemGiven the system x ˙ = f ( x ) + ∑ j = 1 m g j ( x ) u j {displaystyle {dot {x}}=f(x)+sum _{j=1}^{m}g_{j}(x)u_{j}}  , y i = h i ( x ) , i ∈ p {displaystyle y_{i}=h_{i}(x),iin p}  . Where x ∈ R n {displaystyle xin mathbb {R} ^{n}}   the state vector, u ∈ R m {displaystyle uin mathbb {R} ^{m}}   the input vector and y ∈ R p {displaystyle yin mathbb {R} ^{p}}   the output vector. f , g , h {displaystyle f,g,h}   are to be smooth vectorfields.Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in R n {displaystyle mathbb {R} ^{n}}  . Just as observability criteria are used to predict the behavior of Kalman filters or other observers in the dynamic system case, observability criteria for sets in R n {displaystyle mathbb {R} ^{n}}   are used to predict the behavior of data reconciliation and other static estimators. In the nonlinear case, observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.