12.2.1  Solution to the Finite-Horizon Discrete-Time Mixed H2/H Nonlinear Filtering Problem

We similarly consider the following class of estimators:

daf:{ x^k+1=f(xk)+L(x^k,k)[ ykh2(xk) ],x^(k0)=x^0z^=h1(x^k)

(12.46)

where x^kX is the estimated state, L(.,.)n×m(X×Z) is the error-gain matrix which is smooth and has to be determined, and z^s is the estimated output of the filter. We can now define the estimation error or penalty variable, z, which has to be controlled as:

zk:=zkz^=h1(x^k)h1(x^k).

Then, we combine the plant (12.42) and estimator (12.46) dynamics to obtain the following augmented system:

xk+1=f(xk)+g(xk)wk,x(k0)=(x0Tx^0T)Tzk=h1(xk)},

(12.47)

where

xk=(xkxk),f(x)=(

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