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 the equilibrium-point x = 0 and observable for all $\epsilon \in [0,{\epsilon }^{\star })$. Further, suppose for some γ > 0 and $\epsilon \in [0,{\epsilon }^{\star })$, there exists a C¹ positive-semidefinite function $\stackrel{‵}{V}:\stackrel{‵}{N\times }\stackrel{‵}{\Upsilon }\to {\Re }_{+}$, locally defined in a neighborhood $\stackrel{‵}{V}:\stackrel{‵}{N\times }\stackrel{‵}{\Upsilon }\subset \mathcal{X}\times \mathcal{Y}$ of the origin $({\stackrel{‵}{x}}_1,{\stackrel{‵}{x}}_2,y)=(0,0,0),$, and matrix functions ${\stackrel{‵}{L}}_i:\stackrel{‵}{N\times }\stackrel{‵}{\Upsilon }\to {\Re }^{n_i\times m}$, i = 1, 2, satisfying the HJIE:

together with the side-conditions

Then, the filter F^a_3ag with

$\stackrel{‵}{w^{\star }}=\frac{1}{{\gamma }^2}[g_{{}_{11}}^T(\stackrel{‵}{x}){\stackrel{‵}{V}}_{{\stackrel{‵}{x}}_1}(\stackrel{‵}{x},y)+\frac{1}{\epsilon }{g}_{21}(\stackrel{‵}{x}){\stackrel{‵}{V}}_{{\stackrel{‵}{x}}_2}(\stackrel{‵}{x},y)]$

solves the NLHIFP for the system locally in $\stackrel{‵}{N}$.

Proof: Proof follows along the same lines as Proposition 10.2.1. □

The above result, Theorem 10.3.1, can similarly be specialized to the LSPS P^l_sp.

To obtain the limiting filter (10.40) as ε ↓ 0, we use Assumption 10.2.1 to obtain a reduced-order model of the system (10.1). If we assume the equation

has k ≥ 1 isolated roots, we can denote any one of these roots by

for some smooth function p : X × W → X. The resulting reduced-order system is given by

and the corresponding reduced-order filter is then given by

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

where all the variables have their corresponding previous meanings and dimensions, while

$\begin{array}{l}\stackrel{‵}{w^{\star }}=\frac{1}{{\gamma }^2}{g}_{{}_{11}}^T(\stackrel{‵}{x}){\stackrel{‵}{V}}_{{\stackrel{‵}{x}}_1}(\stackrel{‵}{x},y)\\ {\stackrel{‵}{V}}_{{\stackrel{‵}{x}}_1}(\stackrel{‵}{x},y){\stackrel{‵}{L}}_1(\stackrel{‵}{x},y)=-\text{(}y-h_{21}({\stackrel{‵}{x}}_1)+h_{22}{\left(p\left({\stackrel{‵}{x}}_1,\stackrel{‵}{w^{\star }}\right)\right)}^T\end{array}$

and $\stackrel{‵}{V}$ satisfies the following HJIE:

with $\stackrel{\prime }{V}$ (0, 0) = 0. In the next section, we consider some examples.

10.4  Examples

Consider the following singularly-perturbed nonlinear system

$\begin{array}{l}{\dot{x}}_1=-x_1^3+x_2\\ \epsilon {\dot{x}}_2=-x_1-x_2+w\\ y=x_1+x_2+w,\end{array}$

where w ∈ L₂[0, ∞), ε ≥ 0. We construct the aggregate filter F^a_3ag presented in the previous section for the above system. It can be checked that the system is locally zero-input observable, and the function $\stackrel{\prime }{V}(\stackrel{‵}{x})=\frac{1}{2}\left({\stackrel{‵}{x}}_1^2+\epsilon {\stackrel{‵}{x}}_2^2\right),$, solves the inequality form of the HJIE (10.40) corresponding to the system. Subsequently, we calculate the gains of the filter as

where ${\stackrel{‵}{L}}_1(\stackrel{‵}{x},y),{\stackrel{‵}{L}}_2(\stackrel{‵}{x},y)$ are set equal to zero if $\Vert \stackrel{‵}{x}\Vert <\epsilon$ (small) to avoid the singularity at $\stackrel{‵}{x}$ = 0.

10.5  Notes and Bibliography

This chapter is mainly based on [26]. Similar results for the ${\mathcal{H}}_2$ filtering problem can be found in [25]. Results for fuzzy TS nonlinear models can be found in [40, 41, 283].