Disturbance rejection and performance analysis for nonlinear systems based on nonlinear equivalent-input-disturbance approach

Abstract

This paper presents a nonlinear equivalent-input-disturbance (NEID) approach to rejecting an unknown exogenous disturbance in a nonlinear system. An NEID compensator has two parts: a conventional equivalent-input-disturbance estimator and a nonlinear state feedback term. This design ensures that only the exogenous disturbance is rejected and the useful nonlinearity of the system is retained. Unlike other active disturbance-rejection methods, a Lipschitz condition is not necessary to guarantee the convergence of the observation error. Analysis of control performance provides upper bounds for the evaluation of disturbance rejection and the degree of nonlinearity retention. Numerical examples show the validity and superiority of this method.

Appendix

Stability of the NEID-based disturbance-rejection system for (4) is analyzed below.

The state-space representation of the NEID-based disturbance-rejection system is

$$\begin{aligned} \frac{{\mathrm{d}}{\xi }(t)}{{\mathrm{d}}t}=\bar{A}{\xi }(t)+\bar{B}f(\hat{x}(t))+\bar{B}_b{f}(x(t))+\bar{B}_dd(t), \end{aligned}$$

(A.1)

where

$$\begin{aligned} \bar{A}= & {} { \left[ \begin{array}{ccc} A &{}0 &{} -BC_{F}\\ 0 &{} A-LC &{} BC_F \\ 0&{}-B_{F}B^{+}LC &{} A_F+B_FC_F \end{array} \right] },\\ \bar{B}= & {} { \left[ \begin{array}{c} B \\ 0\\ 0 \end{array} \right] },~ \bar{B}_b= \left[ \begin{array}{c} B \\ -B\\ 0 \end{array} \right] ,~ \bar{B}_d= \left[ \begin{array}{c} 0 \\ -B\\ 0 \end{array} \right] . \end{aligned}$$

If $\bar{A}$ is Hurwitz, then, for any positive-definite symmetric matrix K, the solution, $P_g$, of

$$\begin{aligned} P_g\bar{A}+\bar{A}^{\mathrm{T}}P_g=-K \end{aligned}$$

(A.2)

is positive definite. Let

$$\begin{aligned} V_g(\xi (t))=\xi ^{\mathrm{T}}(t)P_g\xi (t) \end{aligned}$$

(A.3)

be a Lyapunov function candidate of (A.1). The derivative of $V_g(\xi (t))$ along the trajectories of (A.1) is given by

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}{V_g(\xi (t))}}{{\mathrm{d}}t}&=-\xi ^{\mathrm{T}}(t)K\xi (t)+{{\Pi }}_1+{{\Pi }}_2+{{\Pi }}_3, \end{aligned} \end{aligned}$$

(A.4)

where

$$\begin{aligned} \begin{aligned}&{{\Pi }}_1{=}2\xi ^{\mathrm{T}}(t)P_g\bar{B}f(\hat{x}(t)){,}~{{\Pi }}_2{=}2\xi ^{\mathrm{T}}(t)P_g\bar{B}_bf({x}(t)){,}\\&{{\Pi }}_3{=}2\xi ^{\mathrm{T}}(t)P_g\bar{B}_dd(t){.}\\ \end{aligned} \end{aligned}$$

Since

$$\begin{aligned}&\begin{aligned} {{\Pi }}_1&\le \frac{1}{\theta _1}\xi ^{\mathrm{T}}(t){{\Gamma }}_1\xi (t)+\theta _1f^{\mathrm{T}}(\hat{x}(t))f(\hat{x}(t))\\&\le \frac{1}{\theta _1}\xi ^{\mathrm{T}}(t){{\Gamma }}_1\xi (t)+\theta _1\Vert f\Vert ^2_\infty \\&\le \frac{1}{\theta _1}\xi ^{\mathrm{T}}(t){{\Gamma }}_1\xi (t)+\theta _1\varphi ^2, \end{aligned} \end{aligned}$$

(A.5)

$$\begin{aligned}&\begin{aligned} {{\Pi }}_2&\le \frac{1}{\theta _2}\xi ^{\mathrm{T}}(t){{\Gamma }}_2\xi (t)+\theta _2f^{\mathrm{T}}({x}(t))f({x}(t))\\&\le \frac{1}{\theta _2}\xi ^{\mathrm{T}}(t){{\Gamma }}_2\xi (t)+\theta _2\Vert f\Vert ^2_\infty \\&\le \frac{1}{\theta _2}\xi ^{\mathrm{T}}(t){{\Gamma }}_2\xi (t)+\theta _2\varphi ^2, \end{aligned} \end{aligned}$$

(A.6)

and

$$\begin{aligned} \begin{aligned} {{\Pi }}_3&\le \frac{1}{\theta _3}\xi ^{\mathrm{T}}(t){{\Gamma }}_3\xi (t)+\theta _3d^{\mathrm{T}}(t)d(t)\\&\le \frac{1}{\theta _3}\xi ^{\mathrm{T}}(t){{\Gamma }}_3\xi (t)+\theta _3d^2_M, \end{aligned} \end{aligned}$$

(A.7)

(A.4) becomes

$$\begin{aligned} \begin{aligned}&\frac{{\mathrm{d}}{V_g(\xi (t))}}{{\mathrm{d}}t}\\&\quad \le -\xi ^{\mathrm{T}}(t)K\xi (t) +\frac{1}{\theta _1}\xi ^{\mathrm{T}}(t){{\Gamma }}_1\xi (t) +\frac{1}{\theta _2}\xi ^{\mathrm{T}}(t){{\Gamma }}_2\xi (t)\\&\qquad \,+\frac{1}{\theta _3}\xi ^{\mathrm{T}}(t){{\Gamma }}_3\xi (t)+\theta _1\varphi ^2+\theta _2\varphi ^2+\theta _3d_M^2\\&\quad \, \le -[\lambda _{\mathrm{min}}(K)-\frac{1}{\theta _1}\lambda _{\mathrm{max}}({{\Gamma }}_1)-\frac{1}{\theta _2}\lambda _{\mathrm{max}}({{\Gamma }}_2) \\&\qquad -\frac{1}{\theta _3}\lambda _{\mathrm{max}}({{\Gamma }}_3)]\Vert \xi \Vert _2^2+\theta _1\varphi ^2+\theta _2\varphi ^2+\theta _3d^2_M, \end{aligned} \end{aligned}$$

where $\theta _1,~\theta _2,~\theta _3$ are positive numbers and

$$\begin{aligned} {{\Gamma }}_1{=}P_g\bar{B}\bar{B}^{\mathrm{T}}P_g{,}~{{\Gamma }}_2{=}P_g\bar{B}_b\bar{B}_b^{\mathrm{T}}P_g{,}~{{\Gamma }}_3{=}P_g\bar{B}_d\bar{B}_d^{\mathrm{T}}P_g{.} \end{aligned}$$

It is easy to check that there exist positive numbers $\theta _1,~\theta _2,~\theta _3$, such that

$$\begin{aligned} \begin{aligned} \lambda _{\mathrm{min}}(K)>\frac{\lambda _{\mathrm{max}}({{\Gamma }}_1)}{\theta _1} +\frac{\lambda _{\mathrm{max}}({{\Gamma }}_2)}{\theta _2} +\frac{\lambda _{\mathrm{max}}({{\Gamma }}_3)}{\theta _3} \end{aligned} \end{aligned}$$

(A.8)

is true. Thus,

$$\begin{aligned} \frac{{\mathrm{d}}{V_g(\xi (t))}}{{\mathrm{d}}t}<0, \end{aligned}$$

which means that the NEID-based disturbance-rejection system is global uniformly boundedness for (4).