obsv

Observability matrix

Syntax

obsv(A,C)
Ob = obsv(sys)


Description

obsv computes the observability matrix for state-space systems. For an n-by-n matrix A and a p-by-n matrix C, obsv(A,C) returns the observability matrix

$$Ob=\left[\begin{array}{c}C\\ CA\\ \begin{array}{l}C{A}^2\\ :\end{array}\\ C{A}^{n-1}\end{array}\right]$$

with n columns and np rows.

Ob = obsv(sys) calculates the observability matrix of the state-space model sys. This syntax is equivalent to executing

Ob = obsv(sys.A,sys.C)

The model is observable if Ob has full rank n.

Examples

Determine if the pair

A =
     1     1
     4    -2

C =
     1     0
     0     1

is observable. Type

Ob = obsv(A,C);

% Number of unobservable states
unob = length(A)-rank(Ob)

These commands produce the following result.

unob =
     0

Tips

obsv is here for educational purposes and is not recommended for serious control design. Computing the rank of the observability matrix is not recommended for observability testing. Ob will be numerically singular for most systems with more than a handful of states. This fact is well-documented in the control literature. For example, see Section III in [1].

References

[1] Paige, C. C. "Properties of Numerical Algorithms Related to Computing Controllability." IEEE Transactions on Automatic Control. Vol. 26, Number 1, 1981, pp. 130-138.