Simon Haykin - Nonlinear Filters

From its structure, it is obvious that the observability Gramian matrix is symmetric and nonnegative. If we apply a transformation, , to the state vector, , such that , the output energy:

(2.47)

can be rewritten as:

(2.48)

If the transformation is chosen in a way that the transformed observability Gramian matrix, , is diagonal, then, its diagonal elements can be viewed as the contribution of different state variables in the initial state vector to the energy of the output. The continuous‐time LTV system of ( 2.36) and ( 2.37) is observable, if and only if the observability Gramian matrix is nonsingular [9, 14].

The state‐space model of a discrete‐time LTV system is represented by the following algebraic and difference equations:

(2.49)

(2.50)

Before proceeding with a discussion on the observability condition, we need to define the discrete‐time state‐transition matrix , , as the solution of the following difference equation:

(2.51)

with the initial condition:

(2.52)

The reason that is called the state‐transition matrix is that it describes the dynamic behavior of the following autonomous system (a system with no input):

(2.53)

with being obtained from

(2.54)

Following a discussion on energy of the system output similar to the continuous‐time case, we reach the following definition for the discrete‐time observability Gramian matrix:

(2.55)

As before, the system ( 2.49) and ( 2.50) is observable, if and only if the observability Gramian matrix is full‐rank (nonsingular) [9].

This section generalizes the method presented before for discretization of continuous‐time LTI systems and describes how the continuous‐time LTV system ( 2.36) and ( 2.37) can be discretized. Solving the differential equation in ( 2.36), we obtain:

(2.56)

where is the state‐transition matrix as described in ( 2.40) and ( 2.41). Using zero‐order‐hold sampling results in a piecewise‐constant input , which remains constant at over the time interval . Setting and , from ( 2.56), we will have:

(2.57)

Therefore, dynamics of the discrete‐time equivalent of the continuous‐time system in ( 2.36) and ( 2.37) will be governed by the following state‐space model [19]:

(2.58)

(2.59)

where

(2.60)

As mentioned before, observability is a global property for linear systems. However, for nonlinear systems, a weaker form of observability is defined, in which an initial state must be distinguishable only from its neighboring points. Two states and are indistinguishable , if their corresponding outputs are equal: for , where is finite. If the set of states in the neighborhood of a particular initial state that are indistinguishable from it includes only , then, the nonlinear system is said to be weakly observable at that initial state. A nonlinear system is called to be weakly observable if it is weakly observable at all . If the state and the output trajectories of a weakly observable nonlinear system remain close to the corresponding initial conditions, then the system that satisfies this additional constraint is called locally weakly observable [13, 20].