Nonlinear Filters. Simon Haykin

Nonlinear Filters

Baseline bold upper C Subscript i Baseline bold upper Phi left-parenthesis i comma j plus 1 right-parenthesis period"/>

As before, the system (2.49) and (2.50) is observable, if and only if the observability Gramian matrix bold upper W Subscript o Baseline left-parenthesis j comma k right-parenthesis is full‐rank (nonsingular) [9].

2.5.3 Discretization of LTV Systems

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) bold x left-parenthesis t right-parenthesis equals bold upper Phi left-parenthesis t comma t 0 right-parenthesis bold x left-parenthesis t 0 right-parenthesis plus integral Subscript t 0 Superscript t Baseline bold upper Phi left-parenthesis t comma tau right-parenthesis bold upper B left-parenthesis tau right-parenthesis bold u left-parenthesis tau right-parenthesis normal d tau comma

where bold upper Phi left-parenthesis t comma tau right-parenthesis is the state‐transition matrix as described in (2.40) and (2.41). Using zero‐order‐hold sampling results in a piecewise‐constant input bold u left-parenthesis t right-parenthesis , which remains constant at bold u left-parenthesis t Subscript k Baseline right-parenthesis over the time interval t element-of left-bracket t Subscript k Baseline comma t Subscript k plus 1 Baseline right-parenthesis . Setting t 0 equals t Subscript k and t equals t Subscript k plus 1 , 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) bold upper Gamma left-parenthesis t Subscript k plus 1 Baseline comma t Subscript k Baseline right-parenthesis equals integral Subscript t Subscript k Baseline Superscript t Subscript k plus 1 Baseline Baseline bold upper Phi left-parenthesis t Subscript k plus 1 Baseline comma tau right-parenthesis bold upper B left-parenthesis tau right-parenthesis normal d tau period

2.6 Observability of Nonlinear Systems

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 bold x Subscript a Baseline left-parenthesis t 0 right-parenthesis and bold x Subscript b Baseline left-parenthesis t 0 right-parenthesis are indistinguishable, if their corresponding outputs are equal: bold y Subscript a Baseline left-parenthesis t right-parenthesis equals bold y Subscript b Baseline left-parenthesis t right-parenthesis for t 0 less-than t less-than upper T , where upper T is finite. If the set of states in the neighborhood of a particular initial state bold x left-parenthesis t 0 right-parenthesis 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 bold x 0 element-of double-struck upper R Superscript n Super Subscript x . 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].

There is another difference between linear and nonlinear systems regarding observability and that is the role of inputs in nonlinear observability. While inputs do not affect the observability of a linear system, in nonlinear systems, some initial states may be distinguishable for some inputs and indistinguishable for others. This leads to the concept of uniform observability, which is the property of a class of systems, for which initial states are distinguishable for all inputs [12, 20]. Furthermore, in nonlinear systems, distinction between time‐invariant and time‐varying systems is not critical because by adding time as an extra state such as bold x Subscript n plus 1 Baseline equals t minus t 0 , the latter can be converted to the former [21].

Nonlinear Filters. Simon Haykin