and for all then it is possible to determine to within an additive constant vector which lies in the null space of defined by where is the state-transition matrix. It is possible to determine a unique if is nonsingular. In fact, it is not possible to distinguish the initial state for from that of if is in the null space of . Note that the matrix defined as above has the following properties: is symmetric is positive semidefinite for satisfies the linear matrix differential equation satisfies the equation Observability matrix generalization The system is observable in if and only if there exists an interval in such that the matrix is nonsingular. If are analytic, then the system is observable in the interval [,] if there exists and a positive integer k such that where and is defined recursively as Example Consider a system varying analytically in and matrices Then , and since this matrix has rank = 3, the system is observable on every nontrivial interval of . Nonlinear systems Given the system , . Where the state vector, the input vector and the output vector. are to be smooth vector fields. Define the observation space to be the space containing all repeated Lie derivatives, then the system is observable in if and only if , where Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar, Kou, Elliot and Tarn, and Singh. There also exist an observability criteria for nonlinear time-varying systems. Static systems and general topological spaces 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 . 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 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. See also Controllability Identifiability State observer State space (controls) References External links MATLAB function for checking observability of a system Mathematica function for checking observability of a system Classical control theory fr:Représentation d'état#Observabilité et détectabilité