Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

10.3  Aggregate Filters

If the coordinate transformation, φ, discussed in the previous section cannot be found, then an aggregate filter for the system (10.1) must be designed. Accordingly, consider the following class of filters:

${\text{F}}_{3ag}^a:\{\begin{array}{l}{\stackrel{‵}{x}}_1=f_1({\stackrel{‵}{x}}_1,{\stackrel{‵}{x}}_2)+g_{11}({\stackrel{‵}{x}}_1,{\stackrel{‵}{x}}_2)\stackrel{‵}{w^{\star }}+{\stackrel{‵}{L}}_1(\stackrel{‵}{x},y)(y-h_{21}({\stackrel{‵}{x}}_1)+h_{22}({\stackrel{‵}{x}}_2));\\ {\stackrel{‵}{x}}_1(t_0)=0\\ \epsilon {\stackrel{‵}{x}}_2=f_1({\stackrel{‵}{x}}_1,{\stackrel{‵}{x}}_2)+g_{12}({\stackrel{‵}{x}}_1,{\stackrel{‵}{x}}_2)\stackrel{‵}{w^{\star }}+{\stackrel{‵}{L}}_2(\stackrel{‵}{x},y)(y-h_{21}({\stackrel{‵}{x}}_1)+h_{22}({\stackrel{‵}{x}}_2));\\ {\stackrel{‵}{x}}_2(t_0)=0\\ \stackrel{}{\stackrel{‵}{z}=y-h_{21}({\stackrel{‵}{x}}_1)+h_{22}({\stackrel{‵}{x}}_2)}\end{array}$

where ${\stackrel{‵}{L}}_1\in {\Re }^{n_1\times m},{\stackrel{‵}{L}}_2\in {\Re }^{n_2\times m}$ are the filter gains, and $\stackrel{‵}{z}$ is the new penalty variable. Then the following result can be derived using similar steps as outlined in the previous section.

Theorem 10.3.1 Consider the nonlinear system (10.1) and the NLHIFP for it. Suppose the plant P^a_sp is locally asymptotically-stable about ...