Problem formulation
A nonlinear state space system given by where is the state vector, is the known input vector and is the measurement vector. The functions and are known as the state dynamics and measurement models, respectively. One definition of observability states that given sufficient measurements and known inputs , it is possible to uniquely determine the initial state . In order to arrive at the derivation for the nonlinear system observability condition, let us track back to obtain the observability condition for linear systems.
Observability of a linear state space system
If the state dynamics and measurement model functions in () are linear, then the system in () can be expressed as where and are the state transition, input and measurement matrices, respectively. They are also assumed to be time invariant, i.e., their derivatives with respect to time are .
Applying the first derivative in time to the measurement equation in (), we get Substituting for in () from () Applying the second order derivative on both sides of (), and again substituting from (), we get As u and its derivatives with respect to time are assumed known, absorbing the terms related to and its derivates into the left hand side and simplifying the notations we get the following system where denotes the order derivative or .
Expressing () in matrix form, we have From () it is clear that, in order for a unique solution to exist for , the observability matrix needs to be full rank, i.e., rank needs to be .
Observability of a nonlinear state space system
From the previous section, it is clear that in order to derive the observability condition of the nonlinear state space system in (), we have to obtain the derivatives of . Consider the nonlinear measurement model Applying chain rule we get where the representation of has been substituted from (). The second representation in () can also be described using the concept of Lie derivatives from differential geometry as follows
where is the first order Lie derivative of the function with respect to another function . As a result the first order derivative of the measurement in () can now be expressed as
Taking the second order derivative of the measurement vector in (), we get Stacking up the derivatives of up to the order we get where and . Expressing () in matrix form, we get
It is straightforward to show that, if then (), reduces to (). From (), applying a partial derivative with respective to to both sides, we get Applying the same operation to both sides of () and using the definition of the observability matrix arrived at in (), we get Expressing () in vector form where . As has been shown in the section on linear observability, in a similar manner, for a nonlinear system () to be observable, the observability matrix defined in () has to be full rank.