Active disturbance rejection adaptive control of uncertain nonlinear systems: theory and application

Abstract

This paper proposes an active disturbance rejection adaptive controller for tracking control of a class of uncertain nonlinear systems with consideration of both parametric uncertainties and uncertain nonlinearities by effectively integrating adaptive control with extended state observer via backstepping method. Parametric uncertainties are handled by the synthesized adaptive law and the remaining uncertainties are estimated by extended state observer and then compensated in a feedforward way. Moreover, both matched uncertainties and unmatched uncertainties can be estimated by constructing an extended state observer for each channel of the considered nonlinear plant. Since parametric uncertainties can be reduced by parameter adaptation, the learning burden of extended state observer is much reduced. Consequently, high-gain feedback is avoided and improved tracking performance can be expected. The proposed controller theoretically guarantees a prescribed transient tracking performance and final tracking accuracy in general while achieving asymptotic tracking when the uncertain nonlinearities are not time-variant. The motion control of a motor-driven robot manipulator is investigated as an application example with some suitable modifications and improvements, and comparative simulation results are obtained to verify the high tracking performance nature of the proposed control strategy.

Appendices

Appendix 1

Definition of matrix ${\varLambda }$

$$\begin{aligned} \Lambda =\left[ {{\begin{array}{ccc} {\Lambda _1 }&{}\quad \mathbf{0}&{}\quad {\Lambda _3 } \\ \mathbf{0}&{}\quad {\varpi _1 }&{}\quad \mathbf{0} \\ {\Lambda _3^T }&{}\quad \mathbf{0}&{}\quad {\Lambda _2 } \\ \end{array} }} \right] \end{aligned}$$

(42)

where 0 denotes zero vector with proper dimensions, and $\Lambda _{1}\in {\mathbb {R}}^{n\times n}, \Lambda _{2}\in {\mathbb {R}}^{(2n-1) \times (2n-1)}$ and $\Lambda _{3}\in {\mathbb {R}}^{(2n-1)\times (2n-1)}$ are defined as

$$\begin{aligned} \begin{array}{l} \Lambda _1 =\left[ {{\begin{array}{ccccc} {k_1 w_1 }&{} {-\frac{\rho _1 }{2}}&{} 0&{} \cdots &{} 0 \\ {-\frac{\rho _1 }{2}}&{} {k_2 w_2 }&{} {-\frac{\rho _2 }{2}}&{} \ddots &{} \vdots \\ 0&{} {-\frac{\rho _2 }{2}}&{} {k_3 w_3 }&{} \ddots &{} 0 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} {-\frac{\rho _{n-1} }{2}} \\ 0&{} \cdots &{} 0&{} {-\frac{\rho _{n-1} }{2}}&{} {k_n w_n } \\ \end{array} }} \right] ,\\ \Lambda _2 =\left[ {{\begin{array}{cccc} {\varpi _1 }&{} 0&{} \cdots &{} 0 \\ 0&{} {\varpi _2 }&{} \ddots &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{} 0 \\ 0&{} \cdots &{} 0&{} {\varpi _n } \\ \end{array} }} \right] \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \Lambda _3 =\left[ {{\begin{array}{cccccccccc} {-\frac{\beta _1 }{2}}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ {-\frac{\gamma _{2,1} }{2}}&{} 0&{} {-\frac{\beta _2 }{2}}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ {-\frac{\gamma _{3,1} }{2}}&{} 0&{} {-\frac{\gamma _{3,2} }{2}}&{} 0&{} {-\frac{\beta _3 }{2}}&{} 0&{} 0&{} 0&{} 0 \\ {-\frac{\gamma _{4,1} }{2}}&{} 0&{} {-\frac{\gamma _{4,2} }{2}}&{} 0&{} {-\frac{\gamma _{4,3} }{2}}&{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} {-\frac{\beta _{n-1} }{2}}&{} 0&{} 0 \\ {-\frac{\gamma _{n,1} }{2}}&{} 0&{} {-\frac{\gamma _{n,2} }{2}}&{} 0&{} {-\frac{\gamma _{n,3} }{2}}&{} \cdots &{} {-\frac{\gamma _{n,n-1} }{2}}&{} 0&{} {-\frac{\beta _n }{2}} \\ \end{array} }} \right] \\ \end{array} \end{aligned}$$

Appendix 2

Proof of Theorem 1

In this case, define $x_{ei} =\Delta _i (\bar{{x}}_i ,\;t)$ and consider the following Lyapunov function

$$\begin{aligned} V_a =\frac{1}{2}\sum \limits _{i=1}^n w_i z_i^2 +\frac{1}{2}\sum \limits _{i=1}^n \mu _i \varepsilon _i^T P\varepsilon _i +V_\theta (\tilde{\theta },\theta ) \end{aligned}$$

(43)

Noting (7), the time derivative of $V_{a}$ can be given as

$$\begin{aligned} \dot{V}_a= & {} \sum \limits _{i=1}^n w_i z_i \dot{z}_i +\frac{1}{2}\sum \limits _{i=1}^n \mu _i \left( \dot{\varepsilon }_i^T P\varepsilon _i +\varepsilon _i^T P\dot{\varepsilon }_i \right) \nonumber \\&+\,\tilde{\theta }_\pi ^T \Gamma ^{-1}\dot{\hat{{\theta }}} \end{aligned}$$

(44)

Based on (11), (20), (25), and (28), and noting $h_{i}(t)=0$, then

$$\begin{aligned} \dot{V}_a= & {} -\sum \limits _{i=1}^n k_i w_i z_i^2 + \sum \limits _{i=1}^{n-1} w_i g_i (\bar{{x}}_i )z_i z_{i+1} +w_1 z_1 \tilde{x}_{e1} \nonumber \\&+\sum \limits _{i=2}^n \left( w_i z_i \tilde{x}_{ei} -w_i \mathop \Sigma \limits _{j=1}^{i-1} \frac{\partial \alpha _{i-1} }{\partial x_j }\tilde{x}_{ej} z_i \right) \nonumber \\&+\,\tilde{\theta }_\pi ^T \left( \Gamma ^{-1}\dot{\hat{{\theta }}}-\tau _n \right) \nonumber \\&+\left( \sum \limits _{k=2}^n w_k z_k \frac{\partial \alpha _{k-1} }{\partial \hat{{\theta }}}\right) \left( \Gamma \tau _n -\dot{\hat{{\theta }}}\right) \nonumber \\&-\frac{1}{2}\mathop \Sigma \limits _{i=1}^n \mu _i \omega _{ei} \left\| {\varepsilon _i } \right\| ^{2} \end{aligned}$$

(45)

Combing the adaptive law in (30), we can upper bound (45) as

$$\begin{aligned} \dot{V}_1\le & {} - \sum \limits _{i=1}^n k_i w_i z_i^2 + \sum \limits _{i=1}^{n-1} w_i g_i (\bar{{x}}_i )z_i z_{i+1} +w_1 z_1 \tilde{x}_{e1} \nonumber \\&+ \sum \limits _{i=2}^n (w_i z_i \tilde{x}_{ei} -w_i \sum \limits _{j=1}^{i-1} \frac{\partial \alpha _{i-1} }{\partial x_j }\tilde{x}_{ej} z_i ) \nonumber \\&-\frac{1}{2}\sum \limits _{i=1}^n \mu _i (\omega _{ei} -1)\left\| {\varepsilon _i } \right\| ^{2}-\frac{1}{2} \sum \limits _{i=1}^n \mu _i \left\| {\varepsilon _i } \right\| ^{2}\nonumber \\ \end{aligned}$$

(46)

Noting that the defined matrix $\Lambda$ is positive definite and the definition of $\varepsilon _{i}$, we have

$$\begin{aligned} \dot{V}_a\le & {} - \sum \limits _{i=1}^n k_i w_i z_i^2 + \sum \limits _{i=1}^{n-1} w_i g_i (\bar{{x}}_i )\left| {z_i } \right| \left| {z_{i+1} } \right| \nonumber \\&+\,w_1 \omega _{e1} \left| {z_1 } \right| \left| {\varepsilon _{12} } \right| \nonumber \\&+ \sum \limits _{i=2}^n (w_i \omega _{ei} \left| {z_i } \right| \left| {\varepsilon _{i2} } \right| \nonumber \\&+\,w_i \sum \limits _{j=1}^{i-1} \omega _{ej} \left| {\frac{\partial \alpha _{i-1} }{\partial x_j }} \right| \left| {\varepsilon _{j2} } \right| \left| {z_i } \right| ) \nonumber \\&-\frac{1}{2} \sum \limits _{i=1}^n \mu _i (\omega _{ei} -1)\left\| {\varepsilon _i } \right\| ^{2} \nonumber \\\le & {} -Z^{T}\Lambda Z\le -\lambda _{\min } (\Lambda )(z^{T}z+\varepsilon ^{T}\varepsilon )\buildrel \Delta \over = -W \end{aligned}$$

(47)

where $Z=[z^{\mathrm{T}}, \varepsilon ^{\mathrm{T}}]^{\mathrm{T}}\in {\mathbb {R}}^{3n}, z=[ {\vert }z_{1}{\vert }, {\ldots }, {\vert }z_{\mathrm{n}}{\vert }]^{\mathrm{T}}, \varepsilon =[{\vert }\varepsilon _{11} {\vert }, {\vert }\varepsilon _{12}{\vert }, {\ldots }, {\vert }\varepsilon _{\mathrm{n1}} {\vert }, {\vert }\varepsilon _{\mathrm{n2}}{\vert }]^{\mathrm{T}}, \lambda _{\min }(\Lambda )$ is the minimal eigenvalue of matrix $\Lambda$ and W is a positive function. Hence, $V_{a}\in L_{\infty }$ and $W\in L_{2}$ and the error signal Z is bounded. From Assumptions 2 and 3, it can be inferred that all the states $x_{1}, {\ldots }, x_{\mathrm{n}}$ and $x_{\mathrm{e1}}, {\ldots }, x_{\mathrm{en}}$, and their estimates are bounded. Noting (17), we know that all the virtual control laws are bounded. Based on the smooth projection mapping (3) and the expression of the derivative of virtual control laws, the boundness of the final control input u can be concluded. That is to say, all signals in the closed loop system are bounded. Based on the dynamics of $z_{1}$, ..., $z_{\mathrm{n}}$, it is easy to check that the time derivative of W is bounded, thus W is uniformly continuous. By applying Barbalat’s lemma, $W\rightarrow 0$ as $t\rightarrow \infty$, which leads to the results in Theorem 1. $\square$

Appendix 3

Proof of Theorem 2

If the uncertain nonlinearities $\Delta _{i}, i=1, {\ldots }, n$, are time-variant, we first use the definition of $x_{{ei}} =\Delta _i (\bar{{x}}_i ,t)-\tilde{\theta }_\pi ^T \varphi _i (\bar{{x}}_i)$. Based on (14), (20), (25), and (28), the time derivative of $V_{\mathrm{b}}$ defined in (31) is

$$\begin{aligned} \dot{V}_b= & {} -\sum \limits _{i=1}^n k_i w_i z_i^2 + \sum \limits _{i=1}^{n-1} w_i g_i (\bar{{x}}_i )z_i z_{i+1} +w_1 z_1 \tilde{x}_{e1} \nonumber \\&+\sum \limits _{i=2}^{n-1} \left( w_i z_i \tilde{x}_{ei} -w_i \sum \limits _{j=1}^{i-1} \frac{\partial \alpha _{i-1} }{\partial x_j }\tilde{x}_{ej} z_i \right) \nonumber \\&+\left( \sum \limits _{k=2}^n w_k z_k \frac{\partial \alpha _{k-1} }{\partial \hat{{\theta }}}\right) \left( \Gamma \tau _n -\dot{\hat{{\theta }}}\right) \nonumber \\&-\frac{1}{2}\sum \limits _{i=1}^n \mu _i \omega _{ei} \left\| {\varepsilon _i } \right\| ^{2} \nonumber \\&+\sum \limits _{i=1}^n \mu _i \varepsilon _i^T PB_2 \frac{h_i (t)}{\omega _{ei}} \end{aligned}$$

(48)

Combining the adaptive law in (30), and noting the definitions in (29), we can upper bound (48) as

$$\begin{aligned} \dot{V}_b\le & {} -\sum \limits _{i=1}^n k_i w_i z_i^2 +\mathop \sum \limits _{i=1}^{n-1} w_i g_i (\bar{{x}}_i )\left| {z_i } \right| \left| {z_{i+1} } \right| \nonumber \\&+\,w_1 \omega _{e1} \left| {z_1 } \right| \left| {\varepsilon _{12} } \right| \nonumber \\&+\sum \limits _{i=2}^n (w_i \omega _{ei} \left| {z_i } \right| \left| {\varepsilon _{i2} } \right| \nonumber \\&+\,w_i \mathop \sum \limits _{j=1}^{i-1} \omega _{ej} \left| {\frac{\partial \alpha _{i-1} }{\partial x_j }} \right| \left| {\varepsilon _{j2} } \right| \left| {z_i } \right| ) \nonumber \\&-\frac{1}{2} \sum \limits _{i=1}^n \mu _i (\omega _{ei} -1)\left\| {\varepsilon _i } \right\| ^{2}-\frac{1}{2} \sum \limits _{i=1}^n \mu _i \left\| {\varepsilon _i } \right\| ^{2} \nonumber \\&+ \sum \limits _{i=1}^n \mu _i \left\| {\varepsilon _i } \right\| \frac{\left\| {PB_2 } \right\| \left| {h_i } \right| _{\max } }{\omega _{ei} } \nonumber \\\le & {} -Z^{T}\Lambda Z+\zeta \end{aligned}$$

(49)

Noting that the matrix $\Lambda$ is positive definite, thus

$$\begin{aligned} \dot{V}_b\le & {} -\lambda _{\min } (\Lambda )(\left\| z \right\| ^{2}+\left\| \varepsilon \right\| ^{2})+\zeta \nonumber \\\le & {} -\lambda _{\min } (\Lambda )\left[ \lambda _1 \sum _{i=1}^n {w_i z_i^2 } +\sum _{i=1}^n {\frac{1}{\mu _i \lambda _{\max } (P)}\mu _i \varepsilon _i^T P\varepsilon _i } \right] +\zeta \nonumber \\\le & {} -\lambda V_b +\zeta \end{aligned}$$

(50)

which leads to (32) by using the Comparison Lemma [2]. Then the error signal Z is bounded. Similar to the proof of Theorem 1, the boundness of all closed loop system signals can also be ensured. $\square$