Global robust regulation control for a class of cascade nonlinear systems subject to external disturbance

Abstract

The global robust regulation problem is studied for a class of cascade nonlinear systems subject to the external disturbance. The considered system represents more general classes of nonlinear uncertain systems, for example, the much weaker integral input-to-state stable (iISS) cascaded subsystem, the unknown control coefficients, the unmeasured states appearing in the nonlinear uncertainties and the external disturbance additively in the input channel. Combined the ideas of the Nussbaum-type gain and the disturbance as a generalized state, a dynamic extended state observer (ESO) based on a Riccati differential equation is constructed to overcome these difficulties. It is shown that the global robust regulation problem is well addressed by the proposed method. In the simulation part, the fan speed control system is used as a practical example to demonstrate its efficacy.

Appendix: Proof of Proposition 1

From $\dot{\overbrace{P^{-1}(t)}}=-P^{-1}(t)\dot{P}(t)P^{-1}(t)$, by virtue of (18) and (22), we can show that

$$\begin{aligned} \dot{V}_{e}= & {} -{e}^TP^{-2}(t){e}-{e}_1^2+\frac{2{e}_1x_1}{{\delta ^*}b}\nonumber \\&+\frac{2{e}^TP^{-1}(t)C_{n+1}h(t)}{\delta ^*} +\frac{2{e}^TP^{-1}(t)B{\cdot }G(\zeta ,y)}{{\delta ^*}b}.\nonumber \\ \end{aligned}$$

(103)

Given the choice of $\delta ^*$, by completing the squares, we have

$$\begin{aligned}&\frac{2{e}_1x_1}{{\delta ^*}b}\le {e}_1^2+y^2,\end{aligned}$$

(104)

$$\begin{aligned}&\frac{2{e}^TP^{-1}(t)B{\cdot }G(\zeta ,y)}{{\delta ^*}b}\le \frac{1}{4}{e}^TP^{-2}(t){e}\nonumber \\&\quad +\,8\sum \limits _{i=1}^{n}\Big (\phi _{i1}^2(|\zeta |)+\phi _{i2}^2(|y|)\Big ), \end{aligned}$$

(105)

$$\begin{aligned}&\frac{2{e}^TP^{-1}(t)C_{n+1}h(t)}{\delta ^*}\le \frac{1}{4}{e}^TP^{-2}(t){e} +4h^2(t).\nonumber \\ \end{aligned}$$

(106)

A direct substitution leads to (24).