Linear Active Disturbance Rejection Control for Nonaffine Strict-Feedback Nonlinear Systems

This paper presents a linear active disturbance rejection controller (LADRC) for nonaffine nonlinear systems with strict-feedback form. The proposed LADRC results in a closed-loop system with a three-time-scale structure, in which a reduced-order linear extended state observer estimates unmeasured states and total disturbance in the fastest time scale and dynamic inversion based on sector conditions is used to deal with nonaffine inputs in the intermediate time scale while the system dynamics evolves in the slowest time scale. The singular perturbation method is used to analyze the stability and performance of the system. The effectiveness of the proposed LADRC is demonstrated through simulation studies and experimental validation on a linear motor servo system with nonaffine uncertainties.

SECTION I. Introduction

Control of uncertain nonlinear systems is one of the most important issues in control theory. Among many different methods developed in the past few decades, the active disturbance rejection control (ADRC), first proposed by Han [1], is an efficient method to deal with uncertain nonlinear systems. For its ability in dealing with a vast range of uncertainties, simplicity in engineering implementation, and superior performance in practice, ADRC is becoming an emerging technology in control engineering [2]. The last two decades have witnessed ADRC’s success in many industrial applications, including speed control of induction motor [3] and permanent magnet synchronous motor [4], fan control in server [5], air-fuel ratio control in gasoline engines [6], hydraulic servo systems [7]. In recent years, the theoretical research has been gradually developed. In [8], [9], the characteristics of the linear ADRC (LADRC), first proposed by Gao [10], are comprehensively studied both in time-domain and frequency-domain. In [11]–​[13], the convergence of nonlinear extended state observer (NLESO) and nonlinear ADRC (NLADRC) are proved, respectively. Several modified ADRCs for nonlinear uncertain systems with time delay are studied and compared with theoretical analysis in [14].

Most of the existing ADRC studies consider that the considered systems are affine-in-control, i.e., the control inputs appear linearly in the control systems, especially in the known nominal models. However, there are many practical control systems, such as the inverted pendulum [15], and flight control system [16], [17], which cannot be expressed in an affine-in-control form. Thus, it is necessary to study how to design ADRC for uncertain nonaffine nonlinear systems.

Various approaches for nonaffine uncertain nonlinear systems have been proposed recently [18]–​[25]. An output feedback controller is proposed for uncertain, single-input-single-output nonaffine input systems using neural network observers together with dynamic inversion in [18]. Using neural networks to approximate the nonaffine nominal model and uncertainties, the stability of the closed-loop system can be proved [19], [20]. All of these methods based on neural networks require heavy computations and good prior knowledge of the system. To deal with heavy computation burdens, an extended high-gain observer (EHGO) is used to estimate system states and uncertainties instead of neural networks. And dynamic inversion is used to deal with the nonaffine inputs and input uncertainties based on the estimates provided by EHGO in [17], [21]. Based on the control design method for feedback linearization systems in [21], an ADRC design method, combining NLESO and dynamic inversion, is proposed for uncertain nonaffine strict-feedback nonlinear systems in [22]. In [23], an indirect dynamic inversion approach is developed for uncertain nonaffine strict-feedback nonlinear systems. However, the model uncertainties estimation provided by EHGO is not used and information of initial states are required in the indirect dynamic inversion approach. In [24], an observer-based adaptive fuzzy method is proposed to control an uncertain nonaffine feedback linearization system. Similarly, an adaptive fuzzy controller is designed for a class of uncertain nonaffine feedback linearization systems based on the passivity theorem in [25].

In classical ADRC framework, the choice of nominal value $b_{0}$ of control gain $b$ has a great impact on the performance of the closed-loop systems. In [26], the stability of the closed-loop system can only be guaranteed with the condition $\frac {b}{b_{0}} \in \left ({{0,2} }\right)$ be satisfied. The range of $b_{0}$ is extended to $\frac {b}{b_{0}} \in \left ({{0,2 + \frac {2}{n}} }\right)$ in [8], [9]. Compared with these results in [26] and [8], [9], the advantage of dynamic inversion is that only the control direction, i.e., the sign of control gain $b$ , is needed. This idea is successfully applied to the stabilization of the inverted pendulum on a cart to solve the uncertainties of control gain in [27].

This paper presents a LADRC design method for a class of uncertain nonaffine strict-feedback nonlinear systems. Reduced-order linear extended state observer (RLESO) and dynamic inversion are combined together with a multi-time-scale structure to deal with model uncertainties and nonaffine inputs. Numerical simulation and experimental results demonstrate the effectiveness of the proposed method. The contributions of this paper can be summarized as follows.

1. We extend the applications of dynamic inversion from feedback linearization systems in [21], [24], [25] to a more general nonaffine systems with strict feedback form.

2. Compared with the indirect dynamic inversion approach for strict-feedback nonlinear systems in [23], a novel design method is proposed, where the uncertainties estimation of the system is adopted effectively and the requirements of the initial states can be removed. The stability and performance analysis for the multi-time-scale structure is rigorously studied via the singular perturbation method.

3. The proposed LADRC provides an extra ADRC design method to handle the difficulty of the choice of nominal value of control gain in the classical ADRC framework.

This paper is organized as follows. In Section II, the problem is formulated for a class of uncertain, nonaffine, nonlinear systems. In Section III, the LADRC design is presented. Section IV provides the stability and performance analysis. Comparative simulation and experimental results with the existing approaches are presented in Section V. Concluding remarks are provided in Section VI.

SECTION II. Problem Statements

Consider a class of uncertain nonaffine nonlinear systems with strict-feedback form:

$$\begin{align*} {\dot x_{i}}=&{x_{i + 1}} + {f_{i}}({x_{1}}, \cdots,{x_{i}}), \\&i = 1,2, \cdots,n - 1,\quad {x_{i}}(0) = {x_{i0}}, \\ {\dot x_{n}}=&{\bar f_{n}}(x,z,w,u),\quad {x_{n}}(0) = {x_{n0}}, \\ \dot z=&{f_{0}}(x,z,w),\quad z(0) = {z_{0}}, \\ y=&{x_{1}}\tag{1}\end{align*}$$ where $x = {[{x_{1}},{x_{2}}, \cdots,{x_{n}}]^{T}} \in {D_{x}} \subset {\mathbb {R}^{n}}$ denotes the system states, $z \in {D_{z}} \subset {\mathbb {R}^{m}}$ is the states of internal dynamics, $u \in {D_{u}} \subset \mathbb {R}$ denotes the control input, the domains ${D_{x}},{D_{z}},{D_{u}}$ contain their respective origins. $y$ is the measured output, $w \in \mathbb {R}$ denotes external disturbances. Functions ${f_{i}}(\cdot):{\mathbb {R}^{i}} \to \mathbb {R},i = 1,2, \cdots,n - 1$ are known, and ${f_{0}}(\cdot)$ is unknown, the function ${\bar f_{n}}(\cdot):{\mathbb {R}^{n + m + 2}} \to \mathbb {R}$ can be expressed as $$\begin{equation*}{\bar f_{n}}(x,z,w,u) = {f_{n}}(x,u) + {\sigma _{f_{n}}}(x,z,w,u),\end{equation*}$$ where ${f_{n}}:{\mathbb {R}^{n + 1}} \to \mathbb {R}$ is known, and ${\sigma _{f_{n}}}:{\mathbb {R}^{n + m + 2}} \to \mathbb {R}$ is unknown.

The goal of this paper is to design the control input $u$ that can make the system (1) follow a target system. The target system is defined by the following dynamics:

$$\begin{align*} {\dot r_{i}}=&{r_{i + 1}} + {f_{i}}({x_{1}}, \cdots,{x_{i}}), \\&i = 1,2, \cdots,n - 1,\quad {r_{i}}(0) = {r_{i0}}, \\ {\dot r_{n}}=&{f_{rn}}(t),\quad {r_{n}}(0) = {r_{n0}},\tag{2}\end{align*}$$ where ${f_{rn}}(t)$ belongs to a compact set ${D_{f}} \subset \mathbb {R}$ , and $r = {[{r_{1}},{r_{2}}, \cdots,{r_{n}}]^{T}}$ belongs to a compact set ${D_{r}} \subset {D_{x}}$ .

Let

$$\begin{align*} {e_{i}}=&{x_{i}} - {r_{i}},\quad e = {\left [{ {\begin{array}{cccc} {e_{1}}&\quad {e_{2}}&\quad \cdots &\quad {e_{n}} \end{array}} }\right]^{T}},\\ A=&\left [{ {\begin{array}{cccc} 0&\quad 1&\quad {}&\quad {}\\ \vdots &\quad {}&\quad \ddots &\quad {}\\ 0&\quad {}&\quad {}&\quad 1\\ 0&\quad 0&\quad \cdots &\quad 0 \end{array}} }\right],\quad B = \left [{ {\begin{array}{c} 0\\ \vdots \\ 0\\ 1 \end{array}} }\right],\end{align*}$$ the error dynamics are $$\begin{align*} {\dot e_{i}}=&{e_{i + 1}},i = 1,2, \cdots,n - 1, \\ {\dot e_{n}}=&- Le + F(x,z,w,u,{f_{rn}}(t)),\tag{3}\end{align*}$$ where the error gain matrix $L = {[{\ell _{1}},{\ell _{2}}, \cdots,{\ell _{n}}]} \in {\mathbb {R}^{n}}$ can be chosen that $(A - BL)$ is Hurwitz, $F(x,z,w,u,{f_{rn}}(t))$ is $$\begin{equation*}F(x,z,w,u,{f_{rn}}(t)) = {\bar f_{n}}(x,z,w,u) - {f_{rn}}(t) + Le.\end{equation*}$$

The following mild assumptions are made for the system (1) and error dynamics (3), and they are common in the relevant literature.

Assumption 1:

$w(t)$ belongs to a known compact set $D_{w} \subset \mathbb {R}$ , and $\dot w(t)$ is bounded.

Remark 1:

Assumption 1 is a rational assumption, which is also presented in [26].

Assumption 2:

The functions ${f_{i}}(\cdot),i = 1, \cdots,n,{\sigma _{f_{n}}}(\cdot), {f_{0}}(\cdot)$ are continuously differentiable. Besides, for any given compact set in ${D_{x}} \times {D_{z}} \times {D_{u}}$ , there exist ${k_{i}} \in \mathbb {R},i = 1,2, \cdots,n,{k_{u}} \in \mathbb {R}$ satisfying:

$$\begin{align*}&\hspace {-2pc}\left |{ {f_{i}({x_{1}} + {\delta _{1}}, \cdots,{x_{i}} + {\delta _{i}}) - {f_{i}}({x_{1}}, \cdots,{x_{i}})} }\right | \\&\le {k_{i}}\sum \limits _{j = 1}^{i} {\left |{ {\delta _{j}} }\right |},\quad \forall {\delta _{j}} \in \mathbb {R},\\&\hspace {-2pc}\left |{ {f_{n}({x_{1}} + {\delta _{1}}, \cdots,{x_{n}} + {\delta _{n}},u) - {f_{n}}({x_{1}}, \cdots,{x_{n}},u)} }\right |\\&\le {k_{n}}\sum \limits _{j = 1}^{n} {\left |{ {\delta _{j}} }\right |},\quad \forall {\delta _{j}} \in \mathbb {R}, \\&\hspace {-2pc}\left |{ {{\sigma _{f_{n}}}(x,z,w,u + {\delta _{u}}) - {\sigma _{f_{n}}}(x,z,w,u)} }\right | \\&\le {k_{u}}\left |{ {\delta _{u}} }\right |,\quad \forall {\delta _{u}} \in \mathbb {R}.\end{align*}$$

Remark 2:

Assumption 2 is a Lipschitz-like condition, which guarantees the existence of solutions for (1).

Assumption 3:

There exists a positive definite function ${V_{z}}$ such that for all $x \in {D_{x}}, z \in {D_{z}}, w \in {D_{w}}$

$$\begin{align*} {\alpha _{1}}(\left \|{ z }\right \|)\le&{V_{z}}(z) \le {\alpha _{2}}(\left \|{ z }\right \|), \\ {\dot V_{z}}\le&0,\quad { {for}}\;\left \|{ z }\right \| \ge {\alpha _{3}}(\left \|{ {x,w} }\right \|),\end{align*}$$ where ${\alpha _{1}}(\cdot), {\alpha _{2}}(\cdot)$ and ${\alpha _{3}}(\cdot)$ are class $\kappa$ functions.

Remark 3:

Assumption 3 implies that the $z$ -subsystem in (1) is bounded-input-bounded-state stable and if ${\left \|{ x }\right \|}$ is bounded in a finite time, the states of the internal dynamics are also bounded. It leads our attention to the trajectory of $x$ .

Assumption 4:

There is a unique continuously differentiable function $\phi (x,z,w,{f_{rn}}(t))$ such that $u_{r} = \phi (x,z,w,{f_{rn}}(t))$ satisfies

$$\begin{equation*}F(x,z,w,{u_{r}},{f_{rn}}(t)) = 0,\end{equation*}$$ and its derivative ${\dot u_{r}} \stackrel { \Delta } = {\phi _{d}}(x,z,w,{f_{rn}}(t),{\dot f_{rn}}(t))$ is bounded on compact sets of ${D_{x}} \times {D_{z}} \times {D_{u}}$ . Furthermore, there’s a known continuously differentiable $K(x,{f_{rn}}(t))$ , for any $(x,z,u) \in {D_{x}} \times {D_{z}} \times {D_{u}}$ , satisfying $$\begin{equation*}sK(x,{f_{rn}}(t))F(x,z,w,s + {u_{r}},{f_{rn}}(t)) \ge \beta {s^{2}},\quad \beta > 0,\end{equation*}$$ where $s = u - {u_{r}}$ , $K(x,{f_{rn}}(t))$ satisfies $$\begin{equation*}\left |{ {K(x + {\delta _{x}},{f_{rn}}(t)) - K(x,{f_{rn}}(t))} }\right | \le {k_{b}}\left \|{ {\delta _{x}} }\right \|,\end{equation*}$$ where ${k_{b}} \ge 0$ .

Remark 4:

Assumption 4 guarantees the uniform controllability of the system (1), and the requirement of $K(x,{f_{rn}}(t))$ in Assumption 4 is a sector condition. An available method is selecting $K$ as the generalized direction of ${\bar f_{n}}(\cdot)$ with respect to $u$ , i.e., $K = sign\left ({{\frac {{\partial {\bar f_{n}}}}{\partial u}(x,z,w,u)} }\right)$ . In this case, according to the mean value theorem there exists a constant $k_{u}^ {*}$ such that

$$\begin{align*}&\hspace {-2pc}{sK\left [{ {\bar f_{n}(x,z,w,s + {u_{r}}) - {f_{rn}}(t) + Le} }\right]} \\=&sign\left ({{\frac {{\partial {\bar f_{n}}}}{\partial u}(x,z,w,u)} }\right)s{{[}}k_{u}^{*}s + {\bar f_{n}}(x,z,w,{u_{r}}) \\&-\, {f_{rn}}(t) + Le{{]}} \\\ge&\left |{ {k_{u}^{*}} }\right |{s^{2}},\end{align*}$$ Therefore, this selection of $K(x,{f_{rn}}(t))$ satisfies Assumption 4.

SECTION III. LADRC Design

A novel LADRC design method for the system (1) will be introduced, and we use dynamic inversion to deal with nonaffine inputs and input uncertainties.

Let

$$\begin{align*} F(x,z,w,u,{f_{rn}}(t))=&{\bar f_{n}}(x,z,w,u) - {f_{rn}}(t) + Le \\=&{f_{n}}(x,u) + {\sigma _{f_{n}}}(x,z,w,u)\\&- \,{f_{rn}}(t) + Le \\\stackrel { \Delta } =&{F_{c}}(x,{\sigma _{f_{n}}},u,{f_{rn}}(t)).\end{align*}$$ If the system states $x$ and ${\sigma _{f_{n}}}(x,z,w,u)$ are known, the dynamic inversion [18], [21], [28] can be expressed as follows: $$\begin{equation*} \mu \dot u = - K(x,{f_{rn}}(t)){F_{c}}(x,{\sigma _{f_{n}}},u,{f_{rn}}(t)),\quad u(0) = {u_{0}},\tag{4}\end{equation*}$$ where $\mu$ is a small positive number.

The closed-loop system can be expressed as

$$\begin{align*} \dot e=&(A - BL)e + B{F_{c}}(x,{\sigma _{f_{n}}},s + u_{r},{f_{rn}}(t)), \\ \dot z=&{f_{0}}(x,z,w), \\ \mu \dot s=&- K{F_{c}}(x,{\sigma _{f_{n}}},s + u_{r},{f_{rn}}(t)) - \mu {\phi _{d}}.\tag{5}\end{align*}$$

Let

$$\begin{equation*}{V_{s}} = \frac {1}{2}{s^{2}},\quad {V_{e}} = {e^{T}}{P_{e}}e,\end{equation*}$$ where $P_{e}$ is the solution of the Lyapunov equation $P_{e}(A - BL) + {(A - BL)^{T}}P_{e} = - I$ and $I$ is the identity matrix. From Assumption 4, $$\begin{align*} {\dot V_{s}}\le&- \left ({{\frac {\beta }{\mu }} }\right){s^{2}} + \left |{ s }\right |\left |{ {\phi _{d}} }\right |, \\ {\dot V_{e}}=&- {e^{T}}e + 2{e^{T}}{P_{e}}B{F_{c}}(x,{\sigma _{f_{n}}},s + u_{r},{f_{rn}}(t)), \\\le&- {\left \|{ e }\right \|^{2}} + {k_{e}}\left |{ s }\right |\left \|{ e }\right \|,\end{align*}$$ on the compact sets, for sufficiently small $\mu$ . Therefore, there exist constants ${b_{1}} > 0,{b_{2}} > 0,{c_{0}} > 0$ such that $$\begin{equation*}{\Omega _{b}} = \left \{{ {V_{e} \le {b_{1}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {b_{2}}} }\right \}\end{equation*}$$ is a compact, positively invariant set with respect to (5).

In output feedback control and ${\sigma _{f_{n}}}(x,z,w,u)$ is unknown, the system states $x$ and ${\sigma _{f_{n}}}(x,z,w,u)$ are estimated by the following RLESO:

$$\begin{align*} {\dot z_{i}}=&- \frac {{{\alpha _{i - 1}}}}{{{\varepsilon ^{i - 1}}}}\left ({{z_{2} + {f_{1}}({x_{1}})} }\right) + {z_{i + 1}} \\&+\, {f_{i}}({x_{1}},{\hat x_{2}}, \cdots,{\hat x_{i}}) + \left ({{\frac {\alpha _{i}}{\varepsilon ^{i}} - \frac {{\alpha _{1}{\alpha _{i - 1}}}}{\varepsilon ^{i}}} }\right){x_{1}}, \\ {\hat x_{i}}=&{z_{i}} + \frac {{{\alpha _{i - 1}}}}{{{\varepsilon ^{i - 1}}}}{x_{1}}, \\&i = 2,3, \cdots,n,\quad {z_{i}}(0) = - {\beta _{i - 1}}{x_{1}}(0), \\ {\dot z_{n + 1}}=&- \frac {\alpha _{n}}{\varepsilon ^{n}}\left ({{z_{2} + {f_{1}}({x_{1}})} }\right) - \frac {{\alpha _{1}{\alpha _{n}}}}{{{\varepsilon ^{n + 1}}}}{x_{1}}, \\ {\hat \sigma _{f_{n}}}=&{z_{n + 1}} + \frac {\alpha _{n}}{\varepsilon ^{n}}{x_{1}},\quad {z_{n + 1}}(0) = - {\beta _{n}}{x_{1}}(0),\tag{6}\end{align*}$$ where the parameters ${\alpha _{1}},{\alpha _{2}}, \cdots,{\alpha _{n}}$ are chosen such that $$\begin{equation*}{s^{n}} + {\alpha _{1}}{s^{n-1}} + \cdots + {\alpha _{n-1}}s + {\alpha _{n}}\end{equation*}$$ is Hurwitz, and $\varepsilon$ is a small positive number. Especially, when $n=1$ , RLESO in (6) can be simplified as: $$\begin{align*} {\dot z_{2}}=&- \frac {\alpha _{1}}{\varepsilon }\left ({{z_{2} + {f_{1}}({x_{1}})} }\right) - \frac {\alpha _{1}^{2}}{\varepsilon ^{2}}{x_{1}}, \\ {\hat \sigma _{f_{1}}}=&{z_{2}} + \frac {\alpha _{1}}{\varepsilon }{x_{1}},\quad {z_{2}}(0) = - {\beta _{1}}{x_{1}}(0).\tag{7}\end{align*}$$

Since the dynamic inversion uses the estimates provided by the RLESO, $\varepsilon$ is smaller than $\mu$ to make the EHGO faster than the dynamic inversion. Therefore, the parameters $\varepsilon$ and $\mu$ are chosen such that $0 < \varepsilon \ll \mu \ll 1$ . Using the RLESO in (6) together with the dynamic inversion in (4), the output feedback control is designed as

$$\begin{align*}&\hspace {-1pc}\mu \dot u = - K({\hat x_{s}},{f_{rn}}(t))\left [{ {f_{n}({\hat x_{s}},u) + {\hat \sigma _{f_{n}s}} - {f_{rn}}(t) + L{\hat e_{s}}} }\right], \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad u(0) = {u_{0}},\tag{8}\end{align*}$$ where $$\begin{align*} {\hat x_{s}}=&{\left [{ {x_{1},{M_{2}}sat\left ({{\frac {\hat x_{2}}{M_{2}}} }\right), \cdots,{M_{n}}sat\left ({{\frac {\hat x_{n}}{M_{n}}} }\right)} }\right]^{T}}, \\ {\hat \sigma _{f_{n}s}}=&{M_\sigma }sat\left ({{\frac {{{\hat \sigma _{f_{n}}}}}{M_\sigma }} }\right),\\ \hat e_{s}=&\hat x_{s} - r,\end{align*}$$ the saturation function $sat(\cdot)$ is defined as $sat(\xi) = \min \left \{{ {1,\left |{ \xi }\right |} }\right \}sign(\xi)$ and is used to prevent peaking from degrading the system performance.

SECTION IV. Main Results

In this section, we will prove that in the presence of uncertainties, the output feedback control (6) and (8) can force the system (1) to follow the target system (2).

A. $n \gt 1$

Consider fast variables

$$\begin{equation*} {\eta _{i}} = \frac {{x_{i} - {\hat x_{i}}}}{{{\varepsilon ^{n + 1 - i}}}},i = 2, \cdots,n,\quad {\eta _{n + 1}} = {\sigma _{f_{n}}} - {\hat \sigma _{f_{n}}},\tag{9}\end{equation*}$$ the closed-loop system can be expressed as a standard singularly perturbed form $$\begin{align*} \dot e=&(A - BL)e + {\delta _{f}} \\&+\, B\left [{ {\bar f_{n}(x,z,w,s + {u_{r}}) - {f_{rn}}(t) + Le} }\right], \\ \dot z=&{f_{0}}(z,x,w), \\ \mu \dot s=&- K\left [{ {f_{n}({\hat x_{s}},s + {u_{r}}) + {\hat \sigma _{f_{n}s}} - {f_{rn}}(t) + L\hat e} }\right] - \mu {\phi _{d}}, \\ \varepsilon \dot \eta=&\Lambda \eta + {\delta _{f/\varepsilon }} + {B}(\varepsilon /\mu){\psi _{1}},\tag{10}\end{align*}$$ where $$\begin{align*} \Lambda=&\left [{ {\begin{array}{ccccc} { - {\alpha _{1}}}&\quad 1&\quad 0&\quad \cdots &\quad 0\\ { - {\alpha _{2}}}&\quad 0&\quad 1&\quad \ddots &\quad \vdots \\ \vdots &\quad \vdots &\quad \ddots &\quad \ddots &\quad 0\\ { - {\alpha _{n-1}}}&\quad 0&\quad \cdots &\quad 0&\quad 1\\ { - {\alpha _{n}}}&\quad 0&\quad 0&\quad \cdots &\quad 0 \end{array}} }\right], \\ {\delta _{f}}=&\left [{ {\begin{array}{c} {0}\\ {\left ({{f_{2}({x_{1}},{x_{2}}) - {f_{2}}({x_{1}},{\hat x_{2}})} }\right)}\\ \vdots \\ {\left ({{{f_{n - 1}}({x_{1}}, \cdots,{x_{n - 1}}) - {f_{n - 1}}({x_{1}}, \cdots,{\hat x_{n - 1}})} }\right)}\\ 0 \end{array}} }\right], \\ {\delta _{f/\varepsilon }}=&\left [{ {\begin{array}{c} {\dfrac {1}{{{\varepsilon ^{n - 2}}}}\left ({{f_{2}({x_{1}},{x_{2}}) - {f_{2}}({x_{1}},{\hat x_{2}})} }\right)}\\ {\dfrac {1}{{{\varepsilon ^{n - 3}}}}\left ({{f_{3}({x_{1}},{x_{2}},{x_{3}}) - {f_{3}}({x_{1}},{\hat x_{2}},{\hat x_{3}})} }\right)}\\ \vdots \\ {\left ({{f_{n}(x, s + u_{r}) - {f_{n}}({\hat x}, s + u_{r})} }\right)}\\ 0 \end{array}} }\right], \\ {\psi _{1}}=&- \frac {{\partial {\sigma _{f_{n}}}}}{\partial u}K{F_{cs}} + \mu \left \{{ {\frac {{\partial {\sigma _{f_{n}}}}}{\partial x}\left [{ {Ax + f(x,z,s + {u_{r}})} }\right]} }\right \} \\&+ \,\mu \left \{{ {\frac {{\partial {\sigma _{f_{n}}}}}{\partial z}{f_{0}}(x,z) + \frac {{\partial {\sigma _{f_{n}}}}}{\partial w}\dot w} }\right \}, \\ {F_{cs}}=&{f_{n}}({\hat x_{s}},s + {u_{r}}) + {\hat \sigma _{f_{n}s}} - {f_{rn}}(t) + L\hat e.\end{align*}$$

According to Assumption 4 and (9), we have:

$$\begin{align*} \left \|{ {\delta _{f}} }\right \|\le&{\varepsilon ^{2}}\sum \limits _{j = 2}^{n - 1} {j{k_{j}}} \left \|{ \eta }\right \|, \\ \left \|{ {{\delta _{f/\varepsilon }}} }\right \|\le&\varepsilon \sum \limits _{j = 2}^{n} {j{k_{j}}} \left \|{ \eta }\right \|.\tag{11}\end{align*}$$

We are going to analyze the stability of the boundary layer and reduced systems in the closed-loop system (10). Since the $z$ -subsystem with input $x$ is bounded-input-bounded-state stable, we are focusing on the $e/s/\eta$ subsystems. The stability analysis of each subsystem will be shown one by one using the time-scale structure of the closed-loop system (10).

Firstly, by considering $\eta$ -subsystem as the fast system and the $e/s$ -subsystems as the slow system, the boundary layer system can be expressed as

$$\begin{equation*} \varepsilon \dot \eta = \Lambda \eta,\quad \eta (0) = {\eta _{0}}.\tag{12}\end{equation*}$$ Since $\Lambda$ is Hurwitz, the boundary layer system (12) is exponentially stable at $\eta = 0$ .

After the fast variable $\eta$ reaches its quasi-steady state, $\eta = 0$ , the reduced system is obtained by setting $\eta = 0$ for $e/s$ -subsystems. In the reduced system, $s$ -subsystem is the fast system and the $e$ -subsystem is slow, the boundary layer system can be obtained as

$$\begin{equation*} \mu \dot s = - KF(x,z,w,s + {u_{r}},{f_{rn}}(t)),\quad s(0) = {s_{0}},\tag{13}\end{equation*}$$ according to Assumption 4, we have $$\begin{equation*}\mu {\dot V_{s}} = - sKF \le - \beta {s^{2}}.\end{equation*}$$ Therefore, the boundary layer system in (13) will be exponentially stable at $s = 0$ . Meanwhile, the reduced system is $$\begin{equation*} \dot e = (A - BL)e,\quad e(0) = {e_{0}}.\tag{14}\end{equation*}$$ Since $(A - BL)$ is Hurwitz, $e$ -subsystem will be exponentially stable at $e = 0$ .

Let

$$\begin{equation*}{\Omega _{a}} = \left \{{ {V_{e} \le {a_{1}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {a_{2}}} }\right \},\end{equation*}$$ where $$\begin{equation*}0 < {a_{1}} < {b_{1}},\quad 0 < {a_{2}} < {b_{2}},\end{equation*}$$ and ${Q_{x}} \subset {\mathbb {R}^{n}}$ and ${Q_{s}} \subset \mathbb {R}$ be any compact subsets containing their respective origins.

Theorem 1:

Under Assumptions 1–​4, suppose the trajectory $(e,z,s,\hat x,\hat \sigma)$ of the closed-loop system (10) starts from $({e_{0}},{z_{0}},{s_{0}},{\hat x_{0}},{\hat \sigma _{0}}) \in {\Omega _{a}} \times {Q_{x}} \times {Q_{s}}$ , then there exists a constant $\varsigma \in \left]{ {0{{,}}1} }\right [$ such that for $\max \left \{{ {\mu,(\varepsilon /\mu)} }\right \} < \varsigma$ ,

1. all trajectories of (10) are bounded.

2. for any given positive number $a$ ,

   $$\begin{equation*}\lim \limits _{\mu,(\varepsilon /\mu) \to 0} \left \|{ {x - \hat x} }\right \| = 0,\quad \lim \limits _{\mu,(\varepsilon /\mu) \to 0} \left |{ {u - {u_{r}}} }\right | = 0,\end{equation*}$$ hold uniformly on $\left [{ {a,\infty } }\right [$ , and $$\begin{equation*}\lim \limits _{\mu,(\varepsilon /\mu) \to 0,t \to \infty } \left \|{ {x - {r}} }\right \| = 0.\end{equation*}$$

Proof:

The proof of Theorem 1 is given in Appendix A.

B. $n = 1$

Define

$$\begin{equation*}{e_{1}} = {x_{1}} - {r_{1}},\quad \eta = {\sigma _{f_{1}}} - {\hat \sigma _{f_{1}}},\end{equation*}$$ the closed-loop system can be expressed as a standard singularly perturbed form $$\begin{align*} {\dot e_{1}}=&- \ell {e_{1}} + \left [{ {\bar f_{1}({x_{1}},z,w,s + {u_{r}}) - {f_{r1}}(t) + \ell {e_{1}}} }\right], \\ \dot z=&{f_{0}}(z,x,w), \\ \mu \dot s=&- K\left [{ {f_{1}({x_{1}},s + {u_{r}}) + {\hat \sigma _{f_{1}s}} - {f_{r1}}(t) + \ell {e_{1}}} }\right] - \mu {\phi _{d}}, \\ \varepsilon \dot \eta=&- {\alpha _{1}}\eta + (\varepsilon /\mu){\psi _{1}},\tag{15}\end{align*}$$ where $$\begin{align*} {\psi _{1}}=&- \frac {{\partial {\sigma _{f_{1}}}}}{\partial u}K{F_{cs}} + \mu \left \{{ {\frac {{\partial {\sigma _{f_{1}}}}}{{\partial {x_{1}}}}{\bar f_{1}}({x_{1}},z,w,s + {u_{r}})} }\right \} \\&+ \,\mu \left \{{ {\frac {{\partial {\sigma _{f_{1}}}}}{\partial z}{f_{0}}(x,z,w) + \frac {{\partial {\sigma _{f_{1}}}}}{\partial w}\dot w} }\right \}, \\ {F_{cs}}=&{f_{1}}({x_{1}},s + {u_{r}}) + {\hat \sigma _{f_{1}s}} - {f_{r1}}(t) + \ell {e_{1}}.\end{align*}$$

The closed-loop system (15) can be viewed as a special case of (10) in Subsection IV-A. Therefore, the stability and performance analysis results in Subsection IV-A is also applicable to (15).

SECTION V. Simulation and Experiment

A. Numerical Simulations

In order to verify the effectiveness of the proposed LADRC, consider the modified Van der Pol oscillator [23]

$$\begin{align*} {\dot x_{1}}=&{x_{2}} + {f_{1}}({x_{1}}),\quad {x_{1}}(0) = 0, \\ {\dot x_{2}}=&{f_{2}}(x,u) + {\sigma _{f_{2}}}(x,z,w,u),\quad {x_{2}}(0) = 0, \\ \dot z=&{f_{0}}(z,x),\quad z(0) = 0.5, \\ y=&{x_{1}} + n(t),\tag{16}\end{align*}$$ where $n(t)$ is measurement noises, ${f_{1}}({x_{1}}) = 0.2\sin ({x_{1}})$ and ${f_{2}}(x,u) = - {x_{1}} + 0.01u$ are known, ${f_{0}}(z,x) = - z + x_{1}^{2} + x_{2}^{2}$ is unknown, and ${\sigma _{f_{2}}}(x,z,w,u)$ is $$\begin{align*}&\hspace{-0.5pc}{\sigma _{f_{2}}}(x,z,w,u) = (1 - x_{1}^{2}){x_{2}} + \tanh ({x_{1}} + z + u + 3) \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \,\,+\, \tanh (u - 3) + w,\end{align*}$$ where $w = 0.1\sin (t)$ denotes external disturbances.

The target system of (16) is

$$\begin{align*} {\dot r_{1}}=&{r_{2}} + {f_{1}}({x_{1}}),\quad {r_{1}}(0) = 0.1, \\[3pt] {\dot r_{2}}=&{f_{r2}}(t),\quad {r_{2}}(0) = 0,\tag{17}\end{align*}$$ where $$\begin{align*} {f_{r2}}(t)=&- 25{r_{1}} - 10{r_{2}} + {u_{r}}(t), \\[3pt] {u_{r}}(t)=&{e^{ - 0.2t}}\sin (t + 2) + {e^{ - 0.2t}}\cos (2t + 10) \\[3pt]&+\, {e^{ - 0.2t}}\sin (3t + 5).\end{align*}$$

To make the system (16) follow the target system (17), the proposed LADRC can be designed as

$$\begin{align*} {\dot z_{2}}=&- \frac {\alpha _{1}}{\varepsilon }\left ({{z_{2} + 0.2\sin ({x_{1}})} }\right) + {z_{3}} - {x_{1}} + 0.01u_{s} \\[3pt]&+\, \left ({{\frac {\alpha _{2}}{\varepsilon ^{2}} - \frac {\alpha _{1}^{2}}{\varepsilon ^{2}}} }\right){x_{1}}, \\ {\hat x_{2}}=&{z_{2}} + \frac {\alpha _{1}}{\varepsilon }{x_{1}},\quad {z_{2}}(0) = - \frac {\alpha _{1}}{\varepsilon }{x_{1}}(0), \\[3pt] {\dot z_{3}}=&- \frac {\alpha _{2}}{\varepsilon ^{2}}\left ({{z_{2} + 0.2\sin ({x_{1}})} }\right) - \frac {{\alpha _{1}{\alpha _{2}}}}{\varepsilon ^{3}}{x_{1}}, \\[3pt] {\hat \sigma _{f_{2}}}=&{z_{3}} + \frac {\alpha _{2}}{\varepsilon ^{2}}{x_{1}},\quad {z_{3}}(0) = - \frac {\alpha _{2}}{\varepsilon ^{2}}{x_{1}}(0), \\[3pt] \mu \dot u=&- K\left [{ { - {x_{1}} + 0.01u_{s} + {\hat \sigma _{f_{2}s}} - {f_{r2}}(t) + L{\hat e_{s}}} }\right], \\&u(0) = 0,\tag{18}\end{align*}$$ where $$\begin{align*} L{\hat e_{s}}=&{\ell _{1}}({x_{1}} - {r_{1}}) + {\ell _{2}}({\hat x_{2s}} - {r_{2}}), \\ {\hat x_{2s}}=&{M_{2}}sat\left ({{\frac {\hat x_{2}}{M_{2}}} }\right),\quad {\hat \sigma _{f_{2}s}} = {M_\sigma }sat\left ({{\frac {{{\hat \sigma _{f_{2}}}}}{M_\sigma }} }\right), \\ u_{s}=&{M_{u}}sat\left ({\frac {u}{M_{u}}}\right).\end{align*}$$

The parameters of the proposed LADRC in (18) are given in Table 1.

TABLE 1 Parameters of the Proposed LADRC in (18)

The indirect dynamic inversion in [23] is introduced to be compared with the proposed LADRC in (18) and can be designed as

$$\begin{align*} {{\dot {\hat x}}_{1}}=&{\hat x_{2}} + 0.2\sin ({\hat x_{1}}) + \frac {\alpha _{1}}{\varepsilon }({x_{1}} - {\hat x_{1}}),\quad {\hat x_{1}}(0) = 0, \\ {{\dot {\hat x}}_{2}}=&{\hat \sigma _{f_{2}}}(t) + \frac {\alpha _{2}}{\varepsilon ^{2}}({x_{1}} - {\hat x_{1}}),\quad {\hat x_{2}}(0) = 0, \\ {{\dot {\hat \sigma } }_{f_{2}}}=&\frac {\alpha _{3}}{\varepsilon ^{3}}({x_{1}} - {\hat x_{1}}),\quad {\hat \sigma _{f_{2}}}(0) = 0, \\ \dot \chi=&- \frac {1}{\mu }K\left [{ { - {f_{r2}}(t) + {\ell _{1}}({\hat x_{1s}} - {r_{1}}) + {\ell _{2}}({\hat x_{2s}} - {r_{2}})} }\right], \\&\chi (0) = 0, \\ u=&\chi - \frac {1}{\mu }K\left ({{\hat x_{2s}(t)} - {x}_{2}(0) }\right),\tag{19}\end{align*}$$ where the parameters are given in Table 2. One thing to be noticed is that the initial state $x_{2}(0)$ is needed.

TABLE 2 Parameters of the Indirect Dynamic Inversion in (19)

The nonlinear active disturbance rejection control (NLADRC) in [22] is also introduced to be compared with the proposed LADRC in (18) and can be designed as

$$\begin{align*} {{\dot {\hat x}}_{1}}=&{\hat x_{2}} + 0.2\sin ({\hat x_{1}}) + \frac {\alpha _{1}}{\varepsilon }({x_{1}} - {\hat x_{1}}) + \varepsilon \varphi \left ({{\frac {{x_{1} - {\hat x_{1}}}}{\varepsilon ^{2}}} }\right), \\&{\hat x_{1}}(0) = 0{{,}} \\ {{\dot {\hat x}}_{2}}=&{\hat \sigma _{f_{2}}}(t) + \frac {\alpha _{2}}{\varepsilon ^{2}}({x_{1}} - {\hat x_{1}}),\quad {\hat x_{2}}(0) = 0, \\ {{\dot {\hat \sigma }}_{f_{2}}}=&\frac {\alpha _{3}}{\varepsilon ^{3}}({x_{1}} - {\hat x_{1}}),\quad {\hat \sigma _{f_{2}}}(0) = 0, \\ \mu \dot u=&- K\left [{ { - {\hat x_{1s}} + 0.01{u_{s}} + {\hat \sigma _{f_{2}s}} - {f_{r2}}(t) + L{\hat e_{s}}} }\right], \\&u(0) = 0,\tag{20}\end{align*}$$ where $$\begin{align*} \varphi (v)=&\begin{cases} { - \dfrac {1}{4},}&{v < - \dfrac {\pi }{2},}\\ {\dfrac {1}{4}\sin (v),}&{ - \dfrac {\pi }{2} \le v \le \dfrac {\pi }{2},}\\ {\dfrac {1}{4},}&{v > \dfrac {\pi }{2},} \end{cases}\\ L{\hat e_{s}}=&{\ell _{1}}({\hat x_{1s}} - {r_{1}}) + {\ell _{2}}({\hat x_{2s}} - {r_{2}}),\\ {\hat x_{1s}}=&{M_{1}}sat\left ({{\frac {\hat x_{1}}{M_{1}}} }\right),\quad {\hat x_{2s}} = {M_{2}}sat\left ({{\frac {\hat x_{2}}{M_{2}}} }\right),\\ {\hat \sigma _{f_{2}s}}=&{M_\sigma }sat\left ({{\frac {{{\hat \sigma _{f_{2}}}}}{M_\sigma }} }\right),\quad {u_{s}} = {M_{u}}sat\left ({{\frac {u}{M_{u}}} }\right),\end{align*}$$ and the parameters take the same value as those of the indirect dynamic inversion in Table 2.

In the absence of measurement noises, i.e., $n(t)=0$ , simulation results of the proposed LADRC in (18) are almost indistinguishable from those of the indirect dynamic inversion in (19) and the NLADRC in (20), so they are neglected here. In the presence of measurement noises, $n(t) = 2 \times {10^{ - 6}} \times N(t)$ , where $N(t)$ is the standard Gaussian noise generated by the MATLAB program command randn.

Figs. 1 and 2 show the numerical simulation results of the proposed LADRC, the indirect dynamic inversion, and the NLADRC, respectively. Three performance indices, maximum tracking error ${e_{\max }}$ , root-mean-square error ${{RMSE}}$ , and generalized energy consumption ${E}$

$$\begin{align*} {e_{\max }}=&\max \left \{{ {\left |{ {x_{1}(t) - {r_{1}}(t)} }\right |} }\right \}, \\ {{RMSE}}=&\sqrt {\frac {1}{t}\int _{0}^{t} {{{\left ({{x_{1}(\tau) - {r_{1}}(\tau)} }\right)}^{2}}d\tau } },\\ E=&\int _{0}^{t} {\left |{ {u(\tau)} }\right |d\tau },\end{align*}$$ are introduced and tabulated in Table 3. From Fig. 1 and Table 3, it is shown that these three approaches can achieve good estimation and regulation of $x_{1}$ . According to Fig. 2, it is obvious that the oscillation of the control law of the proposed LADRC in (18) is smallest among the three approaches. This feature makes the proposed LADRC more advantageous in engineering applications, which will be shown in the next experimental results.

TABLE 3 Performance Indices of the Proposed LADRC in (18), the Indirect Dynamic Inversion in (19), and the NLADRC in (20) in Simulations

FIGURE 1.

Estimation and regulation of $x_{1}$ by the proposed LADRC in (18), the indirect dynamic inversion in (19) and the NLADRC in (20) in simulations.

FIGURE 2.

Control inputs by the proposed LADRC in (18), the indirect dynamic inversion in (19) and the NLADRC in (20) in simulations.

B. Experiments

To show the practical effectiveness of the proposed LADRC, we perform experiments in the linear motor servo system shown in Fig. 3. The linear motor servo system with nonaffine input and uncertainties can be expressed as

$$\begin{align*} {\dot x_{1}}=&{x_{2}}, \\ {\dot x_{2}}=&{\sigma _{f_{2}}}(x,w,u), \\ y=&{x_{1}},\tag{21}\end{align*}$$ where $x_{1}$ is the position of the mover, $y$ is the measured output, $u$ is the control voltage, and unknown ${\sigma _{f_{2}}}(x,w,u)$ is $$\begin{equation*}{\sigma _{f_{2}}}(x,w,u) = bu + \arctan (u) + d(x),\end{equation*}$$ where the parameter $b > 0$ is relative with the load. Since it is difficult to obtain the precise value of $b$ as the working conditions vary, we assume that absolute value of $b$ is unknown here. $\arctan (u)$ is a nonaffine uncertainty term added through software, $d(x)$ denotes the external disturbances, e.g., the friction between the mover and the racks.

FIGURE 3.

Experimental platform of the linear motor servo system.

The target system of (21) is

$$\begin{align*} {\dot r_{1}}=&{r_{2}},\quad {r_{1}}(0) = {l_{0}}, \\ {\dot r_{2}}=&{f_{r2}}(t),\quad {r_{2}}(0) = {v_{0}},\tag{22}\end{align*}$$ where ${f_{r2}}(t)$ is a bounded command signal generated by the LSPB (Linear Segment with Parabolic Blend) method (refer to [29] for specific principles of LSPB).

The proposed LADRC for (21) can be designed as

$$\begin{align*} {\dot z_{2}}=&- \frac {\alpha _{1}}{\varepsilon }{z_{2}} + {z_{3}} + \left ({{\frac {\alpha _{2}}{\varepsilon ^{2}} - \frac {\alpha _{1}^{2}}{\varepsilon ^{2}}} }\right){x_{1}}, \\ {\hat x_{2}}=&{z_{2}} + \frac {\alpha _{1}}{\varepsilon }{x_{1}},\quad {z_{2}}(0) = - \frac {\alpha _{1}}{\varepsilon }{x_{1}}(0), \\ {\dot z_{3}}=&- \frac {\alpha _{2}}{\varepsilon ^{2}}{z_{2}} - \frac {{\alpha _{1}{\alpha _{2}}}}{\varepsilon ^{3}}{x_{1}}, \\ {\hat \sigma _{f_{2}}}=&{z_{3}} + \frac {\alpha _{2}}{\varepsilon ^{2}}{x_{1}},\quad {z_{3}}(0) = - \frac {\alpha _{2}}{\varepsilon ^{2}}{x_{1}}(0), \\ \mu \dot u=&- K\left [{ {{\hat \sigma _{f_{2}s}} - {f_{r2}}(t) + {\ell _{1}}({x_{1}} - {r_{1}}) + {\ell _{2}}({\hat x_{2s}} - {r_{2}})} }\right], \\&u(0) = 0,\tag{23}\end{align*}$$ where $$\begin{align*} {\hat x_{2s}}=&{M_{2}}sat\left ({{\frac {\hat x_{2}}{M_{2}}} }\right),\quad {\hat \sigma _{f_{2}s}} = {M_\sigma }sat\left ({{\frac {{{\hat \sigma _{f_{2}}}}}{M_\sigma }} }\right), \\ {u_{s}}=&{M_{u}}sat\left ({{\frac {u}{M_{u}}} }\right).\end{align*}$$

The indirect dynamic inversion in [23] for (21) can be expressed as

$$\begin{align*} {{\dot {\hat x}}_{1}}=&{\hat x_{2}} + \frac {\alpha _{1}}{\varepsilon }({x_{1}} - {\hat x_{1}}),\quad {\hat x_{1}}(0) = 0, \\ {{\dot {\hat x}}_{2}}=&{\hat \sigma _{f_{2}}}(t) + \frac {\alpha _{2}}{\varepsilon ^{2}}({x_{1}} - {\hat x_{1}}),\quad {\hat x_{2}}(0) = 0, \\ {{\dot {\hat \sigma } }_{f_{2}}}=&\frac {\alpha _{3}}{\varepsilon ^{3}}({x_{1}} - {\hat x_{1}}),\quad {\hat \sigma _{f_{2}}}(0) = 0, \\ \dot \chi=&- \frac {1}{\mu }K\left [{ { - {f_{r2}}(t) + {\ell _{1}}({\hat x_{1s}} - {r_{1}}) + {\ell _{2}}({\hat x_{2s}} - {r_{2}})} }\right], \\&\chi (0) = 0, \\ u=&\chi - \frac {1}{\mu }K\left ({{{\hat x_{2s}}(t) - {x_{2}}(0)} }\right),\tag{24}\end{align*}$$ where the initial state $x_{2}(0) = 0$ is needed.

In addition, the conventional PID controller is introduced and can be expressed as

$$\begin{equation*} u = {k_{p}}{e_{1}} + {k_{d}}{\dot e_{1}} + {k_{i}}\int _{0}^{t} {e_{1}(\tau)d\tau },\tag{25}\end{equation*}$$ where ${e_{1}} = {r_{1}} - {x_{1}}$ and since numerical differentiation usually introduces significant high-frequency noise in velocity measurements, we estimate $\dot e_{1}$ from $e_{1}$ by utilizing a derivative filter given by $$\begin{equation*} {{DF}}(s) = \frac {\omega _{c}^{2}s}{{s^{2} + 2\zeta {\omega _{c}}s + \omega _{c}^{2}}},\end{equation*}$$ where $\zeta$ and $\omega _{c}$ are the damping ratio and the cutoff frequency of the filter, respectively. In our experiments, $\zeta = 0.707,{\omega _{c}} = 80$ .

In order to compare these three approaches fairly, their parameters are tuned to produce similar tracking performance without the nonaffine term as shown in Fig. 4. The designed control parameters of the LADRC in (23) are chosen in Table 4, and the parameters of the indirect dynamic inversion in (24) are given in Table 5. The parameters of the PID in (25) are selected in Table 6.

TABLE 4 Parameters of the Proposed LADRC in (23)

TABLE 5 Parameters of the Indirect Dynamic Inversion in (24)

TABLE 6 Parameters of the PID in (25)

FIGURE 4.

Experimental results with the proposed LADRC in (23), the indirect dynamic inversion in (24), and the PID in (25) that are tuned to achieve similar performance without the nonaffine uncertainty term.

Fig. 5 shows experimental results of the proposed LADRC, the indirect dynamic inversion, and the PID, respectively. Three performance indices, maximum tracking error ${e_{\max }}$ , root-mean-square error ${{RMSE}}$ , and generalized energy consumption ${E}$ tabulated in Table 7. From Fig. 5 and Table 7, the maximum tracking error ${e_{\max }}$ and the generalized energy consumption ${E}$ of the proposed LADRC is the smallest among these three controllers, but its ${{RMSE}}$ is a little larger than that of the indirect dynamic inversion and much less than that of the PID. Therefore, it can be concluded that the proposed LADRC in (23) shows overall a better performance than other two controllers, i.e., the indirect dynamic inversion in (24), and the PID in (25).

TABLE 7 Performance Indices of the Proposed LADRC in (23), the Indirect Dynamic Inversion in (24), and the PID in (25)

FIGURE 5.

Performance of the proposed LADRC in (23), the indirect dynamic inversion in (24), and the PID in (25) with the nonaffine uncertainty term in experiments.

SECTION VI. Conclusion

In this paper, we propose a LADRC design method for the output feedback control of a class of uncertain, nonaffine, strict-feedback nonlinear systems. A RLESO is designed to estimate the unmeasured states and uncertainties, and dynamic inversion is used to deal with the nonaffine inputs with sector conditions for the inputs. The singular perturbation method is used to analyze the stability and performance of the closed-loop system, and the comparison simulation and experimental results with the existing methods demonstrates the effectivpeness of the proposed LADRC.

Although the considered system in this paper is nonaffine, for affine nonlinear systems, the proposed LADRC provides an extra ADRC design method when the choice of nominal value of control gain is difficult in the classical ADRC framework.

Appendix Proof of Theorem 1

The main steps of the proof are outlined in three steps:

1. Firstly, all trajectories $(e,z,s,\eta)$ starting from ${\Omega _{a}} \times {Q_\eta }$ , i.e., $({e_{0}},{z_{0}},{s_{0}},{\eta _{0}}) \in {\Omega _{a}} \times {Q_\eta }$ , will enter the set ${\Omega _{b}} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ in finite time.

2. Secondly, all trajectories $(e,z,s,\eta)$ starting from ${\Omega _{b}} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ will enter the set $\left \{{ {V_{e} \le {b_{1}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ in finite time.

3. Lastly, all trajectories $(e,z,s,\eta)$ starting from $\left \{{ {V_{e} \le {b_{1}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ will enter the set $\left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ .

In the first step, since $(e,z,s,\hat x,\hat \sigma)$ start from $({e_{0}}, {z_{0}},{s_{0}},{\hat x_{0}},{\hat \sigma _{0}}) \in {\Omega _{a}} \times {Q_{x}} \times {Q_{s}}$ , there exist a compact set ${Q_\eta }$ such that ${\eta _{0}} \in {Q_\eta }$ . From the definition (9), $\eta (0) = o(1/{\varepsilon ^{n}})$ . According to Assumption 1, Assumption 2, and (11), there exist positive constants ${l_{1}},{l_{2}} > 0$ such that

$$\begin{equation*} \left \|{ {{\delta _{f/\varepsilon }} + {B}(\varepsilon /\mu){\psi _{1}}} }\right \| \le \varepsilon {l_{1}}\left \|{ \eta }\right \| + \frac {\varepsilon }{\mu }{l_{2}},\tag{26}\end{equation*}$$ where ${l_{1}} = \sum \limits _{j = 1}^{n} {j k_{j}}$ .

Let

$$\begin{equation*}{V_\eta } = {\eta ^{T}}{P_\eta }\eta,\end{equation*}$$ where ${P_\eta }$ is the solution of the Lyapunov function ${P_\eta }\Lambda + {\Lambda ^{T}}{P_\eta } = - I$ . Based on (10), we have $$\begin{align*} \varepsilon {\dot V_\eta }=&- {\left \|{ \eta }\right \|^{2}} + 2{\eta ^{T}}{P_\eta }\left ({{{\delta _{f/\varepsilon }} + {B_{1}}(\varepsilon /\mu){\psi _{1}}} }\right) \\\le&- {\left \|{ \eta }\right \|^{2}} + 2{l_{3}}\left \|{ \eta }\right \|\left ({{\varepsilon {l_{1}}\left \|{ \eta }\right \| + \frac {\varepsilon }{\mu }{l_{2}}} }\right) \\=&- {\left \|{ \eta }\right \|^{2}} + \varepsilon {l_{4}}{\left \|{ \eta }\right \|^{2}} + \frac {\varepsilon }{\mu }{l_{5}}\left \|{ \eta }\right \|,\tag{27}\end{align*}$$ where ${l_{3}} \stackrel { \wedge } = {\lambda _{\max }}({P_\eta }),{l_{4}} \stackrel { \wedge } = 2{l_{1}}{l_{3}},{l_{5}} \stackrel { \wedge } = 2{l_{2}}{l_{3}}$ .

When $\varepsilon \le \frac {1}{{2{l_{4}}}}$ ,

$$\begin{equation*}\varepsilon {\dot V_\eta } \le - \frac {1}{2}{\left \|{ \eta }\right \|^{2}} + \frac {\varepsilon }{\mu }{l_{5}}\left \|{ \eta }\right \|,\end{equation*}$$ then, there exist positive constants ${l_{6}},{l_{7}} > 0$ such that $$\begin{equation*} \varepsilon {\dot V_\eta } \le - {l_{6}}{V_\eta },\quad for~{V_\eta } \ge {\left ({{\frac {\varepsilon }{\mu }} }\right)^{2}}{l_{7}},\tag{28}\end{equation*}$$ where $$\begin{equation*}{l_{6}} < \frac {1}{{2{l_{3}}}},\quad {l_{7}} = {\left ({{\frac {{2\sqrt {l_{3}} {l_{5}}}}{{1 - 2{l_{6}}{l_{3}}}}} }\right)^{2}}.\end{equation*}$$

Taking into consideration that there is ${l_{8}} \in {\mathbb {R}^ {+} }$ such that $\left \|{ {\eta (0)} }\right \| \le {l_{8}}/{\varepsilon ^{n}}$ , so

1. if $\eta (0) \in \left \{{ {V_\eta \le {{\left ({{\frac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \}$ , then

   $$\begin{equation*}\eta (t) \in \left \{{ {V_\eta \le {{\left ({{\frac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \},\quad t \in \left [{ {0,\infty } }\right [,\end{equation*}$$

2. if $\eta (0) \notin \left \{{ {V_\eta \le {{\left ({{\frac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \}$ , then according to (28), we have:

   $$\begin{equation*} {V_\eta }(t) \le {V_\eta }(0){e^{ - \frac {l_{6}}{\varepsilon }t}} \le \frac {l_{9}}{{{\varepsilon ^{2n}}}}{e^{ - \frac {l_{6}}{\varepsilon }t}},\tag{29}\end{equation*}$$ where ${l_{9}} = {l_{3}}l_{8}^{2}$ .

To estimate the time ${T_{1}}(\varepsilon)$ that $\eta$ enters the set $\left \{{ {V_\eta \le {{\left ({{\frac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \}$ , let ${V_\eta } = {\left ({{\frac {\varepsilon }{\mu }} }\right)^{2}}{l_{7}}$ , then according to (29), we have

$$\begin{equation*}\frac {l_{9}}{{{\varepsilon ^{2n}}}}{e^{ - \frac {l_{6}}{\varepsilon }{T_{1}}}} = {\left ({{\frac {\varepsilon }{\mu }} }\right)^{2}}{l_{7}} \Leftrightarrow {T_{1}} = \frac {\varepsilon }{l_{6}}\ln \left ({{\frac {{l_{9}{\mu ^{2}}}}{{l_{7}{\varepsilon ^{2(n + 1)}}}}} }\right).\end{equation*}$$ By L’hopital’s rule, it can be shown $$\begin{align*} \lim \limits _{\varepsilon \to 0} {T_{1}}(\varepsilon)=&\lim \limits _{\varepsilon \to 0} \frac {\varepsilon }{l_{6}}\ln \left ({{\frac {{l_{9}{\mu ^{2}}}}{{l_{7}{\varepsilon ^{2(n + 1)}}}}} }\right) \\\le&\frac {{\ln ({l_{9}}/{l_{7}})}}{l_{6}}\lim \limits _{\varepsilon \to 0} \varepsilon - \frac {2(n + 1)}{l_{6}}\lim \limits _{\varepsilon \to 0} \varepsilon \ln \varepsilon \\=&0.\tag{30}\end{align*}$$

To summarize, we have

$$\begin{equation*} {T_{1}}(\varepsilon) = \begin{cases} {0,}&{\eta (0)\; \in \left \{{ {V_\eta \le {{\left ({{\dfrac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \},}\\ {0,}&{\varepsilon \to 0,\eta (0)\; \notin \left \{{ {V_\eta \le {{\left ({{\dfrac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \}.} \end{cases}\end{equation*}$$

Because ${a_{1}} < {b_{1}},{a_{2}} < {b_{2}}$ , $(e,z,s)$ -subsystems are bounded uniformly in $\varepsilon$ , there is time ${T_{0}}$ such that $(e,z,s) \in {\Omega _{b}},\;{{for}}\;t \in \left [{ {0,{T_{0}}} }\right]$ . By choosing $\varepsilon$ small enough, we can have ${T_{1}}(\varepsilon) = \frac {1}{2}{T_{0}}$ .

In the second step, the $s$ -subsystem in (10) is a perturbation of the $s$ -subsystem in (13) with the perturbation term bounded by ${l_{10}}\left \|{ \eta }\right \| + {l_{11}}\mu$ for ${l_{10}},{l_{11}} > 0$ . Because $\eta (t)\; \in \left \{{ {V_\eta \le {{\left ({{\frac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \},t \in \left [{ {T_{1}(\varepsilon),\infty } }\right [$ ,

$$\begin{equation*} {l_{10}}\left \|{ \eta }\right \| + {l_{11}}\mu \le {l_{12}}\lambda,\tag{31}\end{equation*}$$ where $$\begin{equation*}{l_{12}} = {l_{10}}\sqrt {\frac {l_{7}}{{{\lambda _{\min }}({P_\eta })}}} + {l_{11}},\quad \lambda = \max \left \{{ {\mu,\frac {\varepsilon }{\mu }} }\right \}.\end{equation*}$$

Based on (31), we have

$$\begin{align*}&\hspace {-1pc} \mu {\dot V_{s}} = - s(KF - \mu {\phi _{d}}) \le - \beta {s^{2}} + \lambda {l_{12}}s \le - {l_{13}}{V_{s}}, \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad for\;{V_{s}} \ge {l_{14}}{\lambda ^{2}}.\tag{32}\end{align*}$$ where $$\begin{equation*}{l_{13}} \in \left]{ {0,2\beta } }\right [,{l_{14}} = \frac {1}{2}{\left ({{\frac {{{l_{12}}}}{{\beta - {l_{13}}/2}}} }\right)^{2}}.\end{equation*}$$

Next we estimate the time $T_{2}$ that $s(t)$ enters $\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \}$ .

1. If $s(0)\; \in \;\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \}$ , then

   $$\begin{equation*}s(t)\; \in \;\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \},\quad t \in \left [{ {T_{1}(\varepsilon),\infty } }\right [.\end{equation*}$$

2. If $s(0)\; \notin \;\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \}$ , based on (32), we have

   $$\begin{equation*}{V_{s}}(t) \le {V_{s}}(0){e^{ - \frac {{{l_{13}}}}{\mu }t}}.\end{equation*}$$

   • If $\lambda = \mu \ge \frac {\varepsilon }{\mu }$ , let ${V_{s}} = {l_{14}}{\mu ^{2}}$ , then

     $$\begin{align*} \displaystyle {V_{s}}(0){e^{ - \frac {{{l_{13}}}}{\mu }{T_{2}}}}=&{\mu ^{2}}{l_{14}} \Leftrightarrow \\ \displaystyle \quad {T_{2}}(\mu)=&\frac {\mu }{{{l_{13}}}}\ln \left ({{\frac {V_{s}(0)}{{{l_{14}}{\mu ^{2}}}}} }\right),\end{align*}$$ by L’hopital’s rule, it can be shown $$\begin{equation*}\lim \limits _{\mu \to 0} {T_{2}}(\mu) = \lim \limits _{\mu \to 0} \frac {\mu }{{{l_{13}}}}\ln \left ({{\frac {V_{s}(0)}{{{l_{14}}{\mu ^{2}}}}} }\right) = 0.\end{equation*}$$

   • If $\lambda = \frac {\varepsilon }{\mu } \ge \mu$ , let ${V_{s}} = {l_{14}}{\left ({{\frac {\varepsilon }{\mu }} }\right)^{2}}$ , then

     $$\begin{align*} {V_{s}}(0){e^{ - \frac {{{l_{13}}}}{\mu }{T_{2}}}}=&{l_{14}}{\left ({{\frac {\varepsilon }{\mu }} }\right)^{2}} \Leftrightarrow \\ {T_{2}}(\mu)=&\frac {\mu }{{{l_{13}}}}\ln \left ({{\frac {{V_{s}(0){\mu ^{2}}}}{{{l_{14}}{\varepsilon ^{2}}}}} }\right),\end{align*}$$ by L’hopital’s rule, $$\begin{align*} \lim \limits _{\mu \to 0} {T_{2}}(\mu)=&\lim \limits _{\mu \to 0} \frac {\mu }{{{l_{13}}}}\ln \left ({{\frac {{V_{s}(0){\mu ^{2}}}}{{{l_{14}}{\varepsilon ^{2}}}}} }\right) \\\le&\lim \limits _{\mu \to 0} \frac {\mu }{{{l_{13}}}}\ln \left ({{\frac {V_{s}(0)}{{{l_{14}}{\mu ^{2}}}}} }\right) = 0.\end{align*}$$

To summarize, we have

$$\begin{equation*}{T_{2}}(\mu) = \begin{cases} {0,}&{s(0)\; \in \;\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \},}\\ {0,}&{\mu \to 0,s(0)\; \notin \;\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \}.} \end{cases}\end{equation*}$$

By choosing $\mu$ small enough, we can ensure that $(e,z)$ still stays in $\left \{{ {V_{e} \le {b_{1}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \}$ . Therefore, after the time interval ${T_{1}}(\varepsilon) + {T_{2}}(\mu)$ , $(e,z,s,\eta)$ will enter $\left \{{ {V_{e} \le {b_{1}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ which is positively invariant for sufficiently small $\varepsilon$ and $\mu$ .

Lastly, the $x$ -subsystem in (10) is a perturbation of the $x$ -subsystem in (14) with the perturbation term bounded by ${\varepsilon ^{2}}\sum \limits _{j = 1}^{n - 1} {j k_{j}} \left \|{ \eta }\right \| + {k_{u}}\left |{ s }\right |$ . Because

$$\begin{align*}&\hspace{-0.8pc}\eta (t)\; \in \left \{{ {V_\eta \le {{\left ({{\frac {\varepsilon }{\mu }} }\right)}^{2}}{l_{7}}} }\right \},\quad s(t)\; \in \;\left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \}, \\&\qquad \qquad \qquad \qquad \qquad \qquad \quad ~~ t \in \left [{ {T_{1}(\varepsilon) + {T_{2}}(\mu),\infty } }\right [\end{align*}$$ we can obtain $$\begin{equation*}{\varepsilon ^{2}}\sum \limits _{j = 1}^{n - 1} {j k_{j}} \left \|{ \eta }\right \| + {k_{u}}\left |{ s }\right | < \lambda {l_{15}},\end{equation*}$$ where $$\begin{equation*}{l_{15}} = \left ({{\sum \limits _{j = 1}^{n - 1} {j k_{j}} \sqrt {\frac {l_{7}}{{{\lambda _{\min }}({P_\eta })}}} + {k_{u}}\sqrt {2{l_{14}}} } }\right).\end{equation*}$$

The derivative of ${V_{e}}$ can be expressed as

$$\begin{align*}&\hspace{-1.2pc} {\dot V_{e}} \le - {\left \|{ e }\right \|^{2}} + 2{\lambda _{\max }}({P_{e}})\lambda {l_{15}}\left \|{ e }\right \| \le - {l_{16}}{V_{e}}, \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad for\;{V_{e}} \ge {l_{17}}{\lambda ^{2}},\tag{33}\end{align*}$$ where $$\begin{equation*}{l_{16}} \in \left]{ {0,\frac {1}{{{\lambda _{\max }}({P_{e}})}}} }\right [,{l_{17}} = \frac {{4\lambda _{\max }^{3}({P_{e}})l_{15}^{2}}}{{{{(1 - {l_{16}}{\lambda _{\max }}({P_{e}}))}^{2}}}}.\end{equation*}$$

Next we estimate the time $T_{3}$ that $e(t)$ enters the set $\left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \}$ .

1. If $e(0) \in \left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \}$ , then

   $$\begin{equation*}e(t) \in \left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \},\quad t \in \left [{ {T_{1}(\varepsilon) + {T_{2}}(\mu),\infty } }\right [.\end{equation*}$$

2. If $e(0) \notin \left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \}$ , based on (33), we have

   $$\begin{equation*}{V_{e}}(t) \le {V_{e}}(0){e^{ - {l_{16}}t}}.\end{equation*}$$ Let ${V_{e}} = {l_{17}}{\lambda ^{2}}$ , we obtain $$\begin{equation*}{V_{e}}(0){e^{ - {l_{16}}{T_{3}}}} = {l_{17}}{\lambda ^{2}} \Leftrightarrow {T_{3}}(\lambda) = \frac {1}{{{l_{16}}}}\ln \left ({{\frac {V_{e}(0)}{{{l_{17}}{\lambda ^{2}}}}} }\right).\end{equation*}$$

To summarize, we have

$$\begin{equation*}{T_{3}}(\lambda) = \begin{cases} {0,}&{e(0) \in \left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \},}\\ {\infty,}&{\lambda \to 0,e(0) \notin \left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \},} \end{cases}\end{equation*}$$

Therefore, $(e,z,s,\eta)$ will enter the positively invariant set $\left \{{ {V_{e} \le {l_{17}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_{z} \le {c_{0}}} }\right \} \times \left \{{ {V_{s} \le {l_{14}}{\lambda ^{2}}} }\right \} \times \left \{{ {V_\eta \le {l_{7}}{(\varepsilon /\mu)^{2}}} }\right \}$ in the time ${T_{1}}(\varepsilon) + {T_{2}}(\mu) + {T_{3}}(\lambda)$ for sufficiently small $\varepsilon$ , $\mu$ , and $\lambda$ . Q.E.D.