A reduced-order extended state observer–based trajectory tracking control for one-degree-of-freedom pneumatic manipulator

Abstract

In this article, a reduced-order extended state observer is proposed to deal with uncertainties and disturbances in tracking control for a one-degree-of-freedom manipulator pneumatic system. A tracking differentiator is designed to generate a smooth tracking signal and a corresponding differential signal of the reference input. A non-linear state error feedback is adopted to ensure the response rapidity and control precision. Furthermore, stability analysis of the proposed reduced-order extended state observer and the closed-loop system is also presented. Finally, the experimental results show the efficiency of the proposed scheme for the one-degree-of-freedom manipulator pneumatic system.

Keywords

Reduced-order extended state observer tracking control stability analysis one-degree-of-freedom manipulator pneumatic artificial muscle

Introduction

Pneumatic systems have been increasingly used in automation industry in recent years. Since pneumatic systems provide a lower cost, more flexible, and safer alternative to electric motors and actuators, pneumatic field has drawn a lot of attention¹ and has been widely used in many fields.^2–5 Especially in pneumatic robots, the flexibility of dexterous robot hands plays an important role in space exploration, industrial manufacture, service robot industry, and other fields. Due to gripping characteristics of low cost, safety operation, cleanliness, and high power–weight ratio, pneumatic artificial muscles (PAMs) are popular in robotic applications.⁶ With the rapid development of networks and computer technologies, networked control systems have attracted a lot of attention.^7–9 PAMs are also considered as a pneumatic manipulator in networked control systems.¹⁰ Nevertheless, it is difficult to obtain good performance of control accuracy in PAM systems for the existence of highly non-linear and time-varying dynamics caused by elasticity of PAMs and compressibility of air. In order to deal with those problems, various control strategies are put forward to improve the performance of the PAMs, such as proportional–integral–derivative (PID) methods, adaptive control laws, and sliding mode approaches. An adaptive robust control strategy is proposed to achieve precise posture trajectory tracking control for a three-pneumatic-muscle-driven parallel manipulator.¹¹ A switching algorithm using learning vector quantization neural networks is proposed to optimize the control performance of a pneumatic muscle manipulator.¹² A sliding mode controller is designed to increase the robustness of robotic manipulator tracking systems.¹³ A novel non-linear PID controller enhanced by neural networks is introduced to improve the control performance of 2-axis PAM manipulator.¹⁴ An integrated intelligent non-linear control method is presented for a PAM to realize good control performance by effective compensation.¹⁵ An adaptive neural network (NN)–based decentralized control scheme with the prescribed performance is proposed for uncertain switched non-strict-feedback interconnected non-linear systems.¹⁶ However, the above studies do not deal with the dynamic uncertainties, transition processes, and stability analysis directly, which motivates us to study a much effective scheme.

Active disturbance rejection control (ADRC), which is effective for a wide range of non-linear systems that have large uncertainties, was proposed by Jingqing Han in the late 1990s. Those uncertainties, including internal model dynamics uncertainties and external disturbances, can be estimated by an extended state observer (ESO) in real time.¹⁷ Note that the ADRC has been successfully implemented on many uncertain systems, such as flywheel energy storage systems,¹⁸ pneumatic cylinders,¹⁹ motor systems,²⁰ micro-electro-mechanical systems (MEMS),²¹ metal composite actuators,²² chemical process control systems,²³ thermal plant control systems,²⁴ and ALSTOM gasifier system.²⁵ A typical ADRC consists of a tracking differentiator (TD), an ESO, and a feedback controller. The TD is designed to track an input signal and generate its differential signal in order to avoid overshoot.²⁶ Weak convergence has been shown for a high-gain non-linear TD based on finite-time stable systems.²⁷ The ESO is a key component in the ADRC approach and is used to observe state variables online and estimate total disturbances. Because of the good properties of estimation and capacity to compensate, ESOs have been widely used in many complicated uncertain systems, such as hypersonic vehicles,²⁸ permanent magnet synchronous motor (PMSM) servo systems,²⁹ multi-agent systems,³⁰ and spacecraft formation systems.³¹ Besides non-linear ESOs, there also exist other ESO structures. A kind of linear ESOs for non-linear time-varying plants is proposed, and a complete stability analysis of the proposed linear ESO is also presented in Zheng et al.³² Moreover, reduced-order ESOs have been proposed and discussed in Shao and Wang.³³ Although various kinds of ESOs have been developed for a wide range of applications, there are still some pending problems on tuning parameters and stability analysis.

In this article, a feedback controller, which compensates for the total disturbances by a reduced-order ESO, is designed to stabilize pneumatic manipulator systems. The main contributions of this article are summarized as follows:

An application of a TD avoids the overshoot and smoothens the transition process for the one-degree-of-freedom (DoF) manipulator pneumatic system.

A reduced-order ESO is implemented to estimate and compensate for the total disturbances. The lower the model order is and the fewer the parameters are, the easier the stability analysis and parameter tuning are.

A non-linear feedback controller combined with compensation of disturbances is designed to guarantee a satisfying performance of the closed-loop manipulator pneumatic system.

Experimental setup and system model

Experimental setup

In this article, a 1-DoF manipulator system, which is illustrated in Figure 1, driven by McKibben-type PAMs is considered. Moreover, analog output and collection cards are installed in an industrial control computer, and air with a certain pressure is produced by an air compressor, which passes a pneumatic filter regulator lubricator (FRL) into pressure proportional valves. The accessary components are listed in Table 1. Meanwhile, the schematic diagram of the 1-DoF manipulator system is depicted in Figure 2.

Figure 1.

The experimental setup of the 1-DoF manipulator system driven by PAMs: (a) experimental setup and (b) 1-DoF manipulator system.

Table 1.

Experimental setup.

Name	Element type	Number
Industrial control computer	Advantech	1
Pressure proportional valve	SMC ITV0050	2
Analog output card	PCI-1727	1
Analog collection card	PCI-1784	1
Pneumatic FRL	SMC 3000	1
Angle encoder	OMRON E6B2-CWZ3E	1
PAM	Self-made	2

FRL: filter regulator lubricator; PAM: pneumatic artificial muscle.

Figure 2.

The schematic diagram of the 1-DoF manipulator system.

Model of PAMs

PAMs, despite the existence of many variants, have some common features, such as they have a closed volume and are dependent on contraction of the mesh grid fiber to produce huge pulling force and displacement. Many works have been done on the structure and the mathematical model of PAMs.^34,35 A McKibben PAM is considered for our study. The three-dimensional (3D) structure of its mesh grid fiber and the structure of its rubber tube are illustrated in Figure 3; the two-dimensional (2D) structure of its mesh grid fiber is shown in Figure 4.

Figure 3.

3D structure of the mesh grid fiber and rubber tube.

Figure 4.

Structure of the mesh grid fiber.

In Figure 4, L denotes the actual length of the PAM, D is its diameter, b is the total length of rayon, N is the winding number of rayon, and $α$ represents the actual angle of rayon to the axial direction. Let $L_{0}$ denote the static length of the PAM, $α_{0}$ the static angle of rayon to the axial direction, and $ε$ the shrinkage of the PAM; thus, the geometrical equations are given as follows

{\begin{matrix} L = b \cos α \\ N π D = b \sin α \\ \frac{L_{0} - L}{L_{0}} = \frac{\cos α_{0} - \cos α}{\cos α_{0}} = ε \end{matrix}

(1)

Based on the analysis in the literature,^32,33 the output force $f_{m}$ is a function of inflation pressure P and actual length L

f_{m} (L, P) = - P \frac{dV}{dL} = P \frac{3 L^{2} - b^{2}}{4 π N^{2}}

(2)

where V is the approximate operating volume of PAM.

By considering the non-linearities caused by uncertain parameters, elastic forces of the rubber tube, and the friction, equation (2) is further formulated as follows

f_{ml} (L, P) = {\begin{matrix} η f_{m} (L, P) + f_{r 1} - f_{r 2} \\ - f_{c}, P > P_{d} \\ f_{r 1} - f_{r 2}, P \leq P_{d} \end{matrix}

(3)

where $η$ denotes the structural parameter, $f_{c}$ is the friction, $f_{r 1}$ represents the restoring force of the rubber tube, and $f_{r 2}$ represents the binding force of the rubber tube. $P_{d}$ is the pressure of the dead zone, which is usually a small value, and is solved by $η f_{m} (L, P_{d}) - f_{r 2} (L) = 0$ . Furthermore, $f_{c}$ , $f_{r 1}$ , and $f_{r 2}$ are calculated as follows

\begin{matrix} f_{c} = (\frac{1}{n_{s}}) \frac{μ}{N} (\frac{b^{2} - L_{0}^{2}}{\sqrt{b^{2} - L^{2}}}) \frac{L_{0}^{2}}{L} P sign (\overset{\cdot}{Δ} L) \\ f_{r 1} = \frac{Et}{N} \sqrt{b^{2} - L^{2}} (\frac{L}{L_{min}} - 1) \\ f_{r 2} = \frac{Et}{N} L^{2} (\frac{1}{\sqrt{b^{2} - L_{max}^{2}}} - \frac{1}{\sqrt{b^{2} - L^{2}}}) \end{matrix}

where $L_{max}$ (or $L_{min}$ ) is the maximum (or minimum) length that the PAM can be extended (or compressed), $t_{0}$ is the thickness of the rubber tube, E is the rubber tube’s modulus of elasticity, $μ$ is the frictional coefficient, $n_{s}$ represents the modifying factor, and $Δ L$ is defined as $Δ L = 1 / 2 (L_{2} - L_{1})$ , where $L_{1}$ and $L_{2}$ are the actual lengths of PAM-1 and PAM-2, respectively.

System model

A dynamical model of the 1-DoF manipulator is described as follows

τ = J \overset{\cdot\cdot}{θ} + b_{v} \overset{\cdot}{θ} = [f_{ml 1} (L_{1}, P_{1}) - f_{ml 2} (L_{2}, P_{2})] r

(4)

where $τ$ is the torque driving the pulley wheel, $θ$ is the rotation angle of the pulley wheel, $b_{v}$ stands for the damping ratio, r represents the radius of the pulley wheel, J is the rotary inertia of the pulley wheel, $P_{1}$ and $P_{2}$ are the internal pressures of PAM-1 and PAM-2, respectively, and $f_{ml 1}$ and $f_{ml 2}$ are the output forces of PAM-1 and PAM-2, respectively. The relation between PAM length and its rotation angle is described as follows

{\begin{matrix} L_{1} = L_{0} - θ r \\ L_{2} = L_{0} + θ r \end{matrix}

(5)

System (4), which is with the double inputs $P_{1}$ and $P_{2}$ , is rewritten as a system with the single input $Δ P$ as follows

{\begin{matrix} P_{1} = P_{0} + Δ P \\ P_{2} = P_{0} - Δ P \end{matrix}

(6)

where $P_{0}$ is the initial given pressure of the PAMs.

Substituting equations (3), (5), and (6) into equation (4) and then ignoring the high-order terms of its Taylor series expanded at $θ = 0$ yield the following state space system

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f_{w} + b_{k} (x_{1}, x_{2}) Δ u \\ y = x_{1} \end{matrix}

(7)

where $x_{1} = θ$ , $x_{2} = \overset{\cdot}{θ}$ , and $Δ u = Δ p / K_{0}$ , where $K_{0} = 0.1 MPa / V$ is the gain of the pressure proportional valves. Notation $f_{w}$ is an unknown term and it represents the non-linearity of the system model. It is assumed that $f_{w}$ is continuously differentiable and bounded. $Δ u$ is the control input for PAM-1 and PAM-2 and $b_{k} (x_{1}, x_{2})$ is the gain of $Δ u$ . In the rest of this article, $b_{k} (x_{1}, x_{2})$ is replaced by $b_{k}$ for brevity.

The filter gain $b_{k}$ is further divided into two parts, a constant term $b_{0}$ and a time-varying term $(b_{k} - b_{0})$ . Considering the input $Δ u$ , let the unknown term $(b_{k} - b_{0}) Δ u$ be a part of the total disturbance to estimate. The rewritten state space representation of the system is shown as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f_{t} + b_{0} Δ u \\ y = x_{1} \end{matrix}

(8)

where $f_{t} = f_{w} + (b_{k} - b_{0}) Δ u$ .

Remark 1

The total disturbance is estimated and compensated by the proposed control method in this article. Therefore, the proposed control method in this article is not dependent on an accuracy model of the 1-DoF manipulator system. There is a little relationship between the range of the input signal and the accuracy of the model.

Controller design

In the rest of this work, an ADRC with a reduced-order ESO is designed to deal with the tracking problem for the PAM-driven 1-DoF manipulator system introduced in the previous section. The diagram of the designed control system is shown in Figure 5.

Figure 5.

An ADRC system diagram of the 1-DoF manipulator system.

Design of TD

TD is designed to generate a smooth track of any input signal and its differential signal of reference input to arrange the transient process. The TD is usually implemented to preprocess the reference signal of control systems, aiming to avoid overshoot and optimize the system response. The TD is constructed as follows

{\begin{matrix} f_{h} = f_{han} (v_{1} - v, v_{2}, r_{0}, h_{0}) \\ v_{1} = v_{1} + h v_{2} \\ v_{2} = v_{2} + h f_{h} \end{matrix}

(9)

where v is the input reference signal, $v_{1}$ is the output which tracks v, $v_{2}$ is the differential signal of $v_{1}$ , h is the step length, $r_{0}$ is a velocity factor, and $h_{0}$ is a filter factor. $f_{han} (\cdot)$ is a non-linear function, which is defined as follows

{\begin{matrix} d = r_{0} h_{0}^{2}, a_{0} = h v_{2}, a_{1} = λ + a_{0} \\ a_{2} = \sqrt{d (d + 8 | a_{1} |)} \\ a_{3} = a_{0} + sign (a_{1}) (a_{2} - d) / 2 \\ s_{y} = (sign (a_{1} + d) - sign (a_{1} - d)) / 2 \\ a = (a_{0} + a_{1} - a_{3}) s_{y} + a_{3} \\ s_{a} = (sign (a + d) - sign (a - d)) / 2 \\ f_{han} (λ, v_{2}, r_{0}, h_{0}) = - r_{0} (\frac{a}{d} - sign (a)) s_{a} - r_{0} sign (a) \end{matrix}

where $λ = v_{1} - v$ denotes the tracking error.

Design of a reduced-order ESO

The non-linear dynamics and uncertainties of the PAMs are regarded as a total disturbance. This disturbance is estimated by a reduced-order ESO. Considering equation (8), the term $f_{t}$ is regarded as an unknown and extended variable. Defining $f_{t} = x_{3}$ reformulates equation (8) as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} + b_{0} Δ u \\ {\overset{\cdot}{x}}_{3} = w \\ y = x_{1} \end{matrix}

(10)

where $x_{3}$ is continuous and differentiable. Its derivative, w, is assumed to be bounded.

In this section, a reduced-order ESO is designed to estimate the total disturbance which is produced by the non-linear and uncertain dynamics of the system. The reduced-order ESO for the 1-DoF manipulator system is given as follows

{\begin{matrix} {\overset{\cdot}{z}}_{2} = - β_{1} z_{2} + z_{3} + (β_{2} - β_{1}^{2}) x_{1} + b_{0} Δ u \\ {\overset{\cdot}{z}}_{3} = - β_{2} z_{2} - β_{1} β_{2} x_{1} \\ {\hat{x}}_{2} = z_{2} + β_{1} x_{1} \\ {\hat{x}}_{3} = z_{3} + β_{2} x_{1} \end{matrix}

(11)

where ${\hat{x}}_{2}$ and ${\hat{x}}_{3}$ are the estimates of $x_{2}$ and $x_{3}$ , respectively, $z_{2}$ and $z_{3}$ are two intermediate variables of the reduced-order ESO, both $β_{1}$ and $β_{2}$ are two positive gains of the observer (equation 11). Since the output $x_{1}$ is directly measured by an angle encoder, there is no need to estimate $x_{1}$ . In addition, the traditional non-linear ESO is given as follows

{\begin{matrix} e_{t} = z_{1 t} - y \\ {\overset{\cdot}{z}}_{1 t} = - β_{1 t} e_{t} + z_{2 t} \\ {\overset{\cdot}{z}}_{2 t} = - β_{2 t} fal (e_{t}, c_{t}, δ_{t}) + b_{0 t} Δ u_{t} + z_{3 t} \\ {\overset{\cdot}{z}}_{3 t} = - β_{3 t} fal (e_{t}, c_{t}, δ_{t}) \end{matrix}

(12)

where $Δ u_{t}$ is the control input for PAM-1 and PAM-2; $b_{0 t}$ is an adjustable parameter; $e_{t}$ is the estimation error; $z_{1 t}$ , $z_{2 t}$ , and $z_{3 t}$ are the estimates of $x_{1}$ , $x_{2}$ , and $x_{3}$ , respectively; $β_{1 t}$ , $β_{2 t}$ , and $β_{3 t}$ are the positive gains of a traditional non-linear ESO; and $fal (\cdot)$ is a non-smooth function, which is defined as follows

fal (e, c, δ) = {\begin{matrix} \frac{e}{δ^{1 - c}}, & | e | \leq δ \\ | e |^{c} sign (e), & | e | > δ \end{matrix}

(13)

where $δ$ and c are two given parameters.

Compared to the traditional non-linear ESO (equation 12), the structure of the reduced-order ESO only contains the estimations of $x_{2}$ and $x_{3}$ .

Based on ESO (equation 11) and system description (equation 10), define $e_{2} = {\hat{x}}_{2} - x_{2}$ and $e_{3} = {\hat{x}}_{3} - x_{3}$ ; the error system is obtained as follows

{\begin{matrix} {\overset{\cdot}{e}}_{2} = e_{3} - β_{1} e_{2} \\ {\overset{\cdot}{e}}_{3} = - w - β_{2} e_{2} \end{matrix}

(14)

In the following, convergence analysis of system (14) is presented.

Theorem 1

Considering error system (14) and choosing suitable parameters for $β_{1}$ and $β_{2}$ which satisfy

(β_{1} - β_{2} + 1)^{2} - β_{1} + 3 < 0

(15)

and $β_{1}$ as a large enough number, the system (14) is asymptotically uniformly stable. Meanwhile, the estimation errors $e_{2}$ and $e_{3}$ converge to zero, respectively.

Proof

Consider the following Lyapunov function

V_{e} = (e_{2} - e_{3})^{2} + e_{2}^{2}

(16)

It is obvious that equation (16) is positive definite. The derivation of $V_{e}$ is as follows

\begin{matrix} {\overset{\cdot}{V}}_{e} = 2 e_{2} e_{3} - 2 (β_{1} - β_{2}) e_{2}^{2} + 2 e_{2} w - 2 e_{3}^{2} \\ + 2 (β_{1} - β_{2}) e_{2} e_{3} - 2 e_{3} w + 2 e_{2} e_{3} - 2 β_{1} e_{2}^{2} \end{matrix}

Letting $β_{1} - β_{2} = λ$ and $λ$ be an arbitrary real number, the above equation can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{e} = - (2 β_{1} + 2 λ) e_{2}^{2} + 2 e_{2} w - 2 e_{3}^{2} - 2 e_{3} w \\ + (2 λ + 4) e_{2} e_{3} \\ = - (β_{1} + 2 λ) e_{2}^{2} - e_{3}^{2} + (2 λ + 4) e_{2} e_{3} - (β_{1} - 1) e_{2}^{2} \\ + 2 w^{2} - e_{3}^{2} - 2 e_{3} w - w^{2} - e_{2}^{2} + 2 e_{2} w - w^{2} \end{matrix}

Let $Φ = - (β_{1} + 2 λ) e_{2}^{2} - e_{3}^{2} + (2 λ + 4) e_{2} e_{3}$ . When the following inequality holds

(2 λ + 4)^{2} < 4 (β_{1} + 2 λ)

(17)

then $Φ$ is negative. Rewriting equation (17) as

(λ + 1)^{2} < β_{1} - 3

(18)

there always exists a suitable parameter $β_{1}$ to satisfy with the inequality (equation 18).

Moreover, it is obvious that $- e_{3}^{2} - 2 e_{3} w - w^{2} \leq 0$ and $- e_{2}^{2} + 2 e_{2} w - w^{2} \leq 0$ hold. Let $- e_{3}^{2} - e_{3} w - w^{2} = - Γ_{1}^{2}$ and $- e_{2}^{2} + e_{2} w - w^{2} = - Γ_{2}^{2}$ . Then $\overset{\cdot}{V}$ is rewritten as follows

{\overset{\cdot}{V}}_{e} = - (β_{1} - 1) e_{2}^{2} + 2 w^{2} + Φ - Γ_{1}^{2} - Γ_{2}^{2}

Wrapping up the aforementioned analysis yields a conclusion that $\overset{\cdot}{V}$ is negative definite if there exists a large enough $β_{1}$ , or in other words the system (14) is asymptotically uniformly stable if there exists a large enough $β_{1}$ .

Design of a non-linear error feedback controller

In order to stabilize the closed-loop pneumatic system, a non-linear feedback controller with the ability to cancel the total disturbances is introduced in this section. The controller is designed as follows

Δ u = \frac{k_{1} fal (s_{1}, c_{1}, δ) + k_{2} fal (s_{2}, c_{2}, δ) - {\hat{x}}_{3}}{b_{0}}

(19)

where $k_{1}$ and $k_{2}$ are two gains, $c_{1}$ and $c_{2}$ are the tuning parameters satisfying $0 < c_{1} < 1 < c_{2}$ , and $s_{1}$ and $s_{2}$ are defined as the tracking errors of the TD

{\begin{matrix} s_{1} = v_{1} - x_{1} \\ s_{2} = v_{2} - {\hat{x}}_{2} \end{matrix}

(20)

For this closed-loop system, errors between the given signal and output signal are written as

{\begin{matrix} r_{1} = v_{1} - x_{1} \\ r_{2} = v_{2} - x_{2} \end{matrix}

(21)

Therefore, $fal (s_{1}, c_{1}, δ)$ and $fal (s_{2}, c_{2}, δ)$ are denoted by $fa l_{1}$ and $fa l_{2}$ for brevity, respectively. Combining equations (20), (21), and (19) yields an error differential system as follows

{\begin{matrix} {\overset{\cdot}{s}}_{1} = r_{2} \\ {\overset{\cdot}{r}}_{2} = v_{3} + e_{3} - (k_{1} fa l_{1} + k_{2} fa l_{2}) \end{matrix}

(22)

where $v_{3}$ is the derivative of $v_{2}$ and is bounded.

Theorem 2

The closed-loop system (22) is asymptotically uniformly stable by the reduced-order ESO (equation 11) and the error feedback controller (equation 19) with two properly tuned controller gains $k_{1}$ and $k_{2}$ .

Proof

A Lyapunov function for the closed-loop system (22) is given as follows

V = (s_{1} + r_{2} - e_{2})^{2} + (s_{1} - e_{2})^{2}

(23)

Then the derivative of V is as follows

\begin{matrix} \overset{\cdot}{V} = 2 (s_{1} + r_{2} - e_{2}) (r_{2} + v_{3} - k_{1} fa l_{1} - k_{2} fa l_{2} + β_{1} e_{2}) \\ + 2 (s_{1} - e_{2}) (r_{2} - e_{3} + β_{1} e_{2}) \\ = - 2 (s_{1} + s_{2}) (k_{1} fa l_{1} + k_{2} fa l_{2}) + 2 r_{2}^{2} + 2 s_{1} r_{2} \\ - 2 r_{2} e_{2} + 2 (s_{1} + r_{2} - e_{2}) (v_{3} + β_{1} e_{2}) \\ + 2 (s_{1} - e_{2}) (r_{2} - e_{3} + β_{1} e_{2}) \end{matrix}

Letting $k_{1} = k_{2} = k$ for the sake of simplicity, it is obvious that $(s_{1} + s_{2}) (k_{1} fa l_{1} + k_{2} fa l_{2}) = k (s_{1} + s_{2}) (fa l_{1} + fa l_{2})$ . Since the function $fal (\cdot)$ is both monotonous and odd, the relation between $fa l_{1}$ with $s_{1}$ and $fa l_{2}$ with $s_{2}$ is described as follows:

If $| s_{1} | \leq 1$ , there exists $| fa l_{1} | \geq | s_{1} |$ ; if $| s_{1} | > 1$ , then $| fa l_{1} | < | s_{1} |$ holds.

If $| s_{2} | \leq 1$ , there exists $| fa l_{2} | \leq | s_{2} |$ ; if $| s_{2} | > 1$ , then $| fa l_{2} | > | s_{2} |$ holds.

Moreover, considering the expression $(s_{1} + s_{2}) (fa l_{1} + fa l_{2})$ , four solutions are discussed in the following:

If $fa l_{1} \geq s_{1}$ and $fa l_{2} \geq s_{2}$ , then $(s_{1} + s_{2}) (fa l_{1} + fa l_{2}) \geq (s_{1} + s_{2})^{2}$ holds.

If $fa l_{1} \geq s_{1}$ and $fa l_{2} < s_{2}$ , then $(s_{1} + s_{2}) (fa l_{1} + fa l_{2}) > (s_{1} + fa l_{2})^{2}$ holds.

If $fa l_{1} < s_{1}$ and $fa l_{2} \geq s_{2}$ , then $(s_{1} + s_{2}) (fa l_{1} + fa l_{2}) > (fa l_{1} + s_{2})^{2}$ holds.

If $fa l_{1} < s_{1}$ and $fa l_{2} < s_{2}$ , then $(s_{1} + s_{2}) (fa l_{1} + fa l_{2}) > (fa l_{1} + fa l_{2})^{2}$ holds.

Define $ϒ$ as the smallest term among $(s_{1} + s_{2})^{2}$ , $(s_{1} + fa l_{2})^{2}$ , $(fa l_{1} + s_{2})^{2}$ , and $(fa l_{1} + fa l_{2})^{2}$ , and thus

k (s_{1} + s_{2}) (fa l_{1} + fa l_{2}) > k ϒ \geq 0

Therefore, it is obtained that

\begin{matrix} \overset{\cdot}{V} < - 2 k ϒ + 2 r_{2}^{2} + 2 s_{1} r_{2} - 2 r_{2} e_{2} \\ + 2 (s_{1} + r_{2} - e_{2}) (v_{3} + β_{1} e_{2}) \\ + 2 (s_{1} - e_{2}) (r_{2} - e_{3} + β_{1} e_{2}) \end{matrix}

Due to the guaranteed convergence of the reduced-order ESO, both $e_{2}$ and $e_{3}$ converge to zero. Meanwhile, $v_{3}$ , $s_{1}$ , and $r_{2}$ are bounded practically. Let

\begin{matrix} M = | 2 s_{1} r_{2} - 2 r_{2} e_{2} + 2 (s_{1} + r_{2} - e_{2}) (v_{3} + β_{1} e_{2}) \\ + 2 (s_{1} - e_{2}) (r_{2} - e_{3} + β_{1} e_{2}) | \end{matrix}

Then M is also bounded. Therefore, the following expression holds with a large enough k

\overset{\cdot}{V} < - 2 k ϒ + 2 r_{2}^{2} + M < 0

Because equation (23) is positive definite, and its differential is negative definite, the closed-loop system (22) is asymptotically uniformly stable.

Experimental results

An ADRC system combining the aforementioned TD (equation 9), the reduced-order ESO (equation 11), and the non-linear controller (equation 19) is written as follows

{\begin{matrix} v_{1} (k + 1) = v_{1} (k) + 0.01 v_{2} (k) \\ v_{2} (k + 1) = v_{2} (k) + 0.01 f_{han} (λ (k), v_{2} (k), 20, 0.02) \\ z_{2} (k + 1) = z_{2} (k) + 0.01 (- β_{1} z_{2} (k) + z_{3} (k) \\ + (β_{2} - β_{1}^{2}) x_{1} (k) + b_{0} Δ u (k)) \\ z_{3} (k + 1) = z_{3} (k) + 0.01 (- β_{2} z_{2} (k) - β_{1} β_{2} x_{1} (k)) \\ {\hat{x}}_{2} (k + 1) = z_{2} (k + 1) + β_{1} x_{1} (k) \\ {\hat{x}}_{3} (k + 1) = z_{3} (k + 1) + β_{2} x_{1} (k) \\ Δ u (k) = (k_{1} fal (s_{1} (k), 0.75, 0.1) + k_{2} fal (s_{2} (k), 1.5, 0.1) \\ - {\hat{x}}_{3} (k)) / b_{0} \end{matrix}

Based on the designed controller, comparisons of experimental results between the reduced-order ESO and a traditional non-linear ESO are shown in Figures 6 and 7. Note that more information on the traditional non-linear ESO has been described in Ahn and Nguyen.¹²

Figure 6.

Comparison of experimental results for tracking a step signal at $10^{\circ}$ between the reduced-order ESO and the traditional non-linear ESO: (a) angle, (b) angle velocity, (c) control input, and (d) position error.

Figure 7.

Comparison of experimental results for tracking a sinusoidal signal at $5^{\circ}$ amplitude and 0.4 Hz between the reduced-order ESO and the traditional non-linear ESO: (a) angle, (b) angle velocity, (c) control input, and (d) position error.

Figure 6 shows the comparison of results between the reduced-order ESO and a traditional non-linear ESO for tracking a step signal at $10^{\circ}$ . In Figure 6(a), v is the given step signal at $10^{\circ}$ , $v_{1}$ tracks v, and $x_{1 r}$ and $x_{1 t}$ are two system outputs via using the reduced-order ESO and the traditional non-linear ESO, respectively. As we can see, $x_{1 r}$ has a faster response and a higher accuracy than $x_{1 t}$ to track the smooth signal $v_{1}$ . In Figure 6(b), $v_{2}$ is the differential signal of $v_{1}$ and $z_{2 r}$ and $z_{2 t}$ are the observations of $x_{2}$ using the reduced-order ESO and the traditional non-linear ESO, respectively. The control inputs $Δ u_{r}$ and $Δ u_{t}$ of two different ADRC methods are plotted in Figure 6(c). When the system output levels are off, $Δ u_{r}$ has a smaller stable solution error than $Δ u_{t}$ . In Figure 6(d), $e_{r}$ and $e_{t}$ denote two position errors of the given signal with the system output by adopting two different ADRC methods. Figure 6(d) also indicates that the reduced-order ESO has a faster response and a smaller position error than the traditional non-linear ESO.

Figure 7 shows the comparison of results between the reduced-order ESO and the traditional non-linear ESO for tracking a sinusoidal signal at $5^{\circ}$ amplitude and 0.4 Hz. In Figure 7(a), v is the given sinusoidal signal and $x_{1 r}$ and $x_{1 t}$ are two system outputs by adopting the reduced-order ESO and the traditional non-linear ESO, respectively. For tracking a dynamic response, it is clear to see that $x_{1 r}$ gets a faster response and a higher accuracy than $x_{1 t}$ . Figure 7(b) shows the differential signal $v_{2}$ , and $z_{2 r}$ and $z_{2 t}$ are the observations of $x_{2}$ using the reduced-order ESO and the traditional non-linear ESO, respectively. Figure 7(c) shows two control inputs $Δ u_{r}$ and $Δ u_{t}$ of two different ADRC methods. It is also seen that $Δ u_{r}$ has a smaller vibration than $Δ u_{t}$ . In Figure 7(d), it is much clear to observe that the reduced-order ESO has a faster response and a smaller tracking error than the traditional non-linear ESO.

In conclusion, based on the same TD and a feedback controller structure, it is easily known that the reduced-order ESO has an advantage on response speed and tracking steady-state error compared with the traditional non-linear ESO. That is, the reduced-order ESO has a better performance than the traditional non-linear ESO based on ADRC for a 1-DoF manipulator system.

Remark 2

In Figure 6(b) and (c), there are some spikes in the initial response position because $z_{2 r}$ has some spikes in the initial position and it also leads to spikes in the control signal. However, $z_{2 r}$ can track $v_{2}$ well within $0.5 s$ as shown in Figure 6(b) and (c).

Conclusion

In this article, a reduced-order ESO based on ADRC is proposed to deal with uncertainties in tracking control for a 1-DoF manipulator pneumatic system. A TD has been designed to get a corresponding smooth signal of a given input signal to avoid overshoot and a differential signal as well. Moreover, a non-linear error feedback controller has been adopted to guarantee a satisfying performance. Stability analysis of the reduced-order ESO and the closed-loop system has been proven by Lyapunov theory. Finally, the experimental results exhibit the efficiency and advantages of the reduced-order ESO with an ADRC method for the 1-DoF manipulator driven by PAMs.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of the paper.

Handling Editor: Xianzhi Zhang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the National Natural Science Foundation of China under grant nos 51375045 and 51505413 and the Hebei Provincial Natural Science Fund under grant no. E2016203264.

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