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 linear system are mathematical duals. The concept of observability was introduced by 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 any 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 systemsEdit

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 matrixEdit

If the row rank of the observability matrix, defined as


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

Related conceptsEdit

Observability indexEdit

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


Unobservable subspaceEdit

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]

Nonlinear observers

sliding mode and cubic observers[7] can be applied for state estimation of time invariant linear systems, if the system is observable and fulfills some additional conditions.

Linear time-varying systemsEdit

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 generalizationEdit

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[9]


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 systemsEdit

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  .

Note:  [10]

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,[11] Kou, Elliot and Tarn,[12] and Singh.[13]

Static systems and general topological spacesEdit

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 alsoEdit


  1. ^ Kalman R. E., "On the General Theory of Control Systems", Proc. 1st Int. Cong. of IFAC, Moscow 1960 1481, Butterworth, London 1961.
  2. ^ Kalman R. E., "Mathematical Description of Linear Dynamical Systems", SIAM J. Contr. 1963 1 152
  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
  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. ^ Pasand, Mohammad Mahdi Share (2020). "Luenberger-type cubic observers for state estimation of linear systems". International Journal of Adaptive Control and Signal Processing. n/a (n/a): 1148–1161. arXiv:1909.11978. doi:10.1002/acs.3125. ISSN 1099-1115. S2CID 202888832.
  8. ^ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
  9. ^ Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
  10. ^ 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.
  11. ^ Griffith E. W. and Kumar K. S. P., "On the Observability of Nonlinear Systems I, J. Math. Anal. Appl. 1971 35 135
  12. ^ Kou S. R., Elliott D. L. and Tarn T. J., Inf. Contr. 1973 22 89
  13. ^ Singh S.N., "Observability in Non-linear Systems with immeasurable Inputs, Int. J. Syst. Sci., 6 723, 1975
  14. ^ Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)
  15. ^ Stanley G.M., and Mah R.S.H., "Observability and Redundancy Classification in Process Networks", Chem. Engng. Sci. 36, 1941 (1981)

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