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Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs.

In control theory, the observability and controllability of a linear system are mathematical duals.

The concept of observability was introduced by the Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems.[1][2] A dynamical system designed to estimate the state of a system from measurements of the outputs is called a state observer or simply an observer for that system.


Consider a physical system modeled in state-space representation. A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.

Linear time-invariant systems[edit]

For time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable. Consider a SISO system with state variables (see state space for details about MIMO systems) given by

Observability matrix[edit]

If and only if the column rank of the observability matrix, defined as

is equal to , then the system is observable. The rationale for this test is that if columns are linearly independent, then each of the state variables is viewable through linear combinations of the output variables .

Related concepts[edit]

Observability index[edit]

The observability index of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied: , where

Unobservable subspace[edit]

The unobservable subspace of the linear system is the kernel of the linear map given by[3]

where is the set of continuous functions from to . can also be written as [3]

Since the system is observable if and only if , the system is observable if and only if is the zero subspace.

The following properties for the unobservable subspace are valid:[3]


A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.[4]

Detectability conditions are important in the context of sensor networks.[5][6]

Linear time-varying systems[edit]

Consider the continuous linear time-variant system

Suppose that the matrices , and are given as well as inputs and outputs 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:

  • satisfies the equation

Observability matrix generalization[edit]

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[8]

where and is defined recursively as


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[edit]

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,[10] Kou, Elliot and Tarn,[11] and Singh.[12]

There also exist an observability criteria for nonlinear time-varying systems.[13]

Static systems and general topological spaces[edit]

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 .[14][15] 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[edit]


  1. ^ Kalman, R.E. (1960). "On the general theory of control systems". IFAC Proceedings Volumes. 1: 491–502. doi:10.1016/S1474-6670(17)70094-8.
  2. ^ Kalman, R. E. (1963). "Mathematical Description of Linear Dynamical Systems". Journal of the Society for Industrial and Applied Mathematics, Series A: Control. 1 (2): 152–192. doi:10.1137/0301010.
  3. ^ a b c Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
  4. ^ http://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf[bare URL PDF]
  5. ^ Li, W.; Wei, G.; Ho, D. W. C.; Ding, D. (November 2018). "A Weightedly Uniform Detectability for Sensor Networks". IEEE Transactions on Neural Networks and Learning Systems. 29 (11): 5790–5796. doi:10.1109/TNNLS.2018.2817244. PMID 29993845. S2CID 51615852.
  6. ^ Li, W.; Wang, Z.; Ho, D. W. C.; Wei, G. (2019). "On Boundedness of Error Covariances for Kalman Consensus Filtering Problems". IEEE Transactions on Automatic Control. 65 (6): 2654–2661. doi:10.1109/TAC.2019.2942826. S2CID 204196474.
  7. ^ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
  8. ^ Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
  9. ^ Lecture notes for Nonlinear Systems Theory by prof. dr. D.Jeltsema, prof dr. J.M.A.Scherpen and prof dr. A.J.van der Schaft.
  10. ^ Griffith, E. W.; Kumar, K. S. P. (1971). "On the observability of nonlinear systems: I". Journal of Mathematical Analysis and Applications. 35: 135–147. doi:10.1016/0022-247X(71)90241-1.
  11. ^ Kou, Shauying R.; Elliott, David L.; Tarn, Tzyh Jong (1973). "Observability of nonlinear systems". Information and Control. 22: 89–99. doi:10.1016/S0019-9958(73)90508-1.
  12. ^ Singh, Sahjendra N. (1975). "Observability in non-linear systems with immeasurable inputs". International Journal of Systems Science. 6 (8): 723–732. doi:10.1080/00207727508941856.
  13. ^ Martinelli, Agostino (2022). "Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems". IEEE Transactions on Automatic Control. 67 (9): 5002–5008. doi:10.1109/TAC.2022.3180771. ISSN 0018-9286. S2CID 251957578.
  14. ^ Stanley, G. M.; Mah, R. S. H. (1981). "Observability and redundancy in process data estimation" (PDF). Chemical Engineering Science. 36 (2): 259–272. doi:10.1016/0009-2509(81)85004-X.
  15. ^ Stanley, G.M.; Mah, R.S.H. (1981). "Observability and redundancy classification in process networks" (PDF). Chemical Engineering Science. 36 (12): 1941–1954. doi:10.1016/0009-2509(81)80034-6.

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