Robust Observer Design and Fuzzy Optimization for Uncertain Dynamic Systems

Abstract

To achieve the state estimation of uncertain nonlinear systems, this paper proposes a novel optimal robust observer via the fuzzy bound information of the uncertainty. A robust observer scheme is first proposed to guarantee the uniform boundedness and uniform ultimate boundedness of the state estimation system regardless of the actual value of the uncertainty. To optimize the observer gain, a fuzzy set-theoretic-based optimal approach is then proposed by utilizing the fuzzy information of the uncertainty bound. The optimization problem is completely solved in analytic form. The resulting observer is able to guarantee the robustness of the state estimation system to the uncertainty, while minimizing an optimization index related to the state estimation performance and the observer cost. The novelty of this research is a creative optimal robust observer design by blending state estimation theory, fuzzy set theory, and optimization theory into an integrated framework.

Appendix: Mathematical Preliminary About Differential Inequality

Definition 2

(see [32]) Let $\omega (\psi ,t)$ be a scalar function of the scalars $\psi$, t in some open connected set ${\mathcal {D}}$. The function $\psi (t)$, $t\in [t_0,{\bar{t}}]$, is called the solution of the differential inequality

$$\begin{aligned} {{\dot{\psi }}}(t)\le \omega (\psi (t),t), \end{aligned}$$

(78)

on $[t_0,{\bar{t}}]$ when $\psi (t)$ is a continuous function on $[t_0,{\bar{t}}]$ and ${\dot{\psi }}(t)$ should meet the inequality (78).

Theorem 3

(see [32]) Let$\omega (\phi ,t)$be continuous on an open connected set${\mathcal {D}}\in \mathbf{R }^2$and such that the initial value problem for the scalar equation

$$\begin{aligned} {\dot{\phi }}(t)=\omega (\phi (t),t),\ \ \ \ \phi (t_0)=\phi _0 \end{aligned}$$

(79)

has a unique solution. If$\phi (t)$and$\psi (t)$are the solution of (79) and (78) on$t_0\le t\le {\bar{t}}$, respectively, and$\psi (t_0)\le \phi (t_0)$, then for each$t\in [t_0, {\bar{t}}]$, $\psi (t)\le \phi (t)$.

Remark

Due to the uniqueness of the solution to (79), this theorem implies the feasibility of exploring the upper bound of the solution to the differential inequality (78).

Theorem 4

(see [36]) Consider the differential inequality (78) and the differential equation (79). Let$\omega (\cdot )$denote a continuous function satisfying theLipschitz condition

$$\begin{aligned} \left| \omega (\nu _1,t)-\omega (\nu _2,t)\right| \le L\left| \nu _1-\nu _2\right| , \end{aligned}$$

(80)

for all points$(\nu _1,t),(\nu _2,t)\in {\mathcal {D}}$and the constant$L>0$. For each$t\in [t_0, {\bar{t}}]$, the function$\psi (t)$meet the following inequality

$$\begin{aligned} \psi (t)\le \phi (t), \end{aligned}$$

(81)

if$\psi (t)$meets the inequality (78).