Observability

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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.[1][2]

Definition[edit]

Formally, a system is said to be observable if, for any possible sequence of state and control vectors[clarification needed], the current state can be determined in finite time using only the outputs. (This definition is slanted towards 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 states cannot be determined through output sensors. This implies that their value is unknown to the controller (although they can be estimated by various means).

For time-invariant linear systems in the state space representation, there is a convenient test to check whether a system is observable. Consider a SISO system with states (see state space for details about MIMO systems). If the row rank of the following observability matrix

is equal to (where the notation is defined below), then the system is observable. The rationale for this test is that if rows are linearly independent, then each of the states is viewable through linear combinations of the output variables .

A module 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.

Observability index

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

Unobservable subspace

The unobservable subspace N of the linear system (A,,C) is the kernel of the linear map G given by[3]

,

where is the set of continuous functions and is the state-transition matrix associated to A.


If (A,,C) is an autonomous system, N can be written as [4]

Example: Consider A and C given by:

, .

If the observability matrix is defined by , it can be calculated as follows:

Let's now calculate the kernel of observability matrix.

the system is observable if Rank()=n where n is the number of independent columns in the observability matrix. In this example det()=0, then Rank()<n and the systems is unobservable.

Since the kernel of a linear application, the unobservable subspace is a subspace of . The following properties are valid: [5]

Detectability

A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.[6]

Continuous time-varying system[edit]

Consider the continuous linear time-variant system

Suppose that the matrices are given as well as inputs and outputs 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
[7]

Observability[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

Example[edit]

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 case[edit]

Given the system , . Where the state vector, the input vector and the output vector. are to be smooth vectorfields.

Define the observation space to be the space containing all repeated Lie derivatives, then the system is observable in if and only if .

Note: [9]

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,[10] Kou, Elliot and Tarn,[11] and Singh.[12]

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 ,.[13][14] 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]

References[edit]

  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. ^ Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
  4. ^ Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
  5. ^ Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998
  6. ^ http://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf
  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. and Kumar K. S. P., "On the Observability of Nonlinear Systems I, J. Math. Anal. Appl. 1971 35 135
  11. ^ Kou S. R., Elliott D. L. and Tarn T. J., Inf. Contr. 1973 22 89
  12. ^ Singh S.N., "Observability in Non-linear Systems with immeasurable Inputs, Int. J. Syst. Sci., 6 723, 1975
  13. ^ Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)
  14. ^ 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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