# Neural Network Based Active Disturbance Rejection Control of a Novel Electrohydraulic Servo System for Simultaneously Balancing and Positioning by Isoactuation Configuration

Authors: Qiang Gao; Yuanlong Hou; Kang Li; Zhan Sun; Chao Wang; Runmin Hou
Journal: Shock and Vibration
 (2016)

---

## Abstract

To satisfy the lightweight requirements of large pipe weapons, a novel electrohydraulic servo (EHS) system where the hydraulic cylinder possesses three cavities is developed and investigated in the present study. In the EHS system, the balancing cavity of the EHS is especially designed for active compensation for the unbalancing force of the system, whereas the two driving cavities are employed for positioning and disturbance rejection of the large pipe. Aiming at simultaneously balancing and positioning of the EHS system, a novel neural network based active disturbance rejection control (NNADRC) strategy is developed. In the NNADRC, the radial basis function (RBF) neural network is employed for online updating of parameters of the extended state observer (ESO). Thereby, the nonlinear behavior and external disturbance of the system can be accurately estimated and compensated in real time. The efficiency and superiority of the system are critically investigated by conducting numerical simulations, showing that much higher steady accuracy as well as system robustness is achieved when comparing with conventional ADRC control system. It indicates that the NNADRC is a very promising technique for achieving fast, stable, smooth, and accurate control of the novel EHS system.


---

## Body

## 1. Introduction

Large pipe weapons, arms of heavy machineries, industrial robotics, and space-borne manipulators possess a sort of long and heavy linkages which have high ratio of length to diameter and time-varying unbalancing torque induced by the misalignment between its gravity center and the corresponding trunnion. The unbalancing torque serving as a sort of strong disturbance highly deteriorates control performances of the system [1, 2].Currently, there are two kinds of system balancing strategies, namely, the external force based system balancing and the internal driving based system balancing. The external methods mainly depended on adopting proper balance weights or adopting balance machinery, and the static unbalancing components can be well compensated. Practically, the unbalancing part including partial static and dynamic unbalancing components was treated as external disturbance and further compensated during active control of the system. Regarding the large diameter of the pipe, the unbalancing components were heavy, highly challenging the control system for positioning [3, 4]. Moreover, the extra machinery for balancing strongly increased weights and costs of mechanical systems, being difficult to meet the lightweight requirements. With the internal driving based system balancing methods, the most directly way was to use the driving forces for system balancing [5, 6]. Generally, the currently adopted method based on direct driving was essentially a feedforward inverse compensation strategy, and it cannot achieve real-time balancing. It was obvious that not enough driving forces can be provided by the motor servo system in terms of the heavy pipes. Another kind of adaptive balancing method combining the hydraulic accumulator and balancing machinery was recently developed in [7]. However, only partial unbalancing forces can be compensated during the process. Also, this method cannot be applied for active rejection of the unbalancing forces during the positioning.As discussed above, active compensation for the unbalancing components in pipe positioning is still an outstanding issue. In the present study, a novel electrohydraulic servo (EHS) system where the hydraulic cylinder possesses three cavities is developed. In the EHS system, the balancing cavity of the EHS is especially designed for active compensation for the unbalancing force of the system, whereas the driving cavity is employed for real-time positioning and disturbance rejection of the large pipe. By means of the hydraulic cylinder with three cavities, simultaneous control and disturbance rejection can be achieved by using only one driving source, namely, isoactuation, greatly reducing weights of weapons with heavy pipes. However, there are certain extremely complicated segments with strong nonlinearities in gun control systems, including time-varying parameters induced by varying working conditions, random external applied loads, and complex friction forces between the cannon and trunnion. All these nonlinear behaviors add difficulties in achieving high control performances (both static and dynamic), blocking improvements of working performances of the gun control system. In terms of nonlinear control, considerable work has been done based on certain types of adaptive control strategies, such as fuzzy control, adaptive sliding mode robust control, and adaptive equivalent disturbance compensation control [4–7]. All these nonlinear control methods can significantly improve the uncertainties of the control system in terms of its tolerance and robustness, but only at the expense of the positioning accuracy and response speed.The recently developed active disturbance rejection control (ADRC) is an efficient nonlinear digital control strategy which regards the unmodeled part and external disturbance as overall disturbance of the controlled system [8–10]. The ADRC mainly consists of a tracking differentiator (TD), an extended state observer (ESO), and a nonlinear state error feedback (NLSEF) control law. In this ADRC approach, the processes with higher orders, uncertainties, and unmodeled dynamics are viewed as lower-ordered systems with general disturbances; meanwhile, the general disturbances are estimated by ESO and are actively compensated [11–20]. There are a large number of adjustable parameters in the ADRC, and choosing proper parameters can contribute to excellent performances of the control system. To optimally get the system parameters, the global optimization strategy based off-line tune methods [11, 12] and the artificial intelligent based online tune methods [13–16] were developed. Generally, the off-line tune methods are highly dependent on identified physical model of the controlled system, and the obtained parameters cannot adapt to variable working conditions. The online updating method can adjust control parameters to get optimal control performance in real time. In [13], the fuzzy control scheme was introduced in the ADRC to estimate the states and accordingly update the compensation factors of the ESO. In [14], the diagonal recurrent neural network was introduced to realize online tuning of the NLSEF. In terms of parameters of the NLSEF, the compensation gains in ESO highly affect estimation performance of the control system, especially the estimation accuracy of overall disturbances. Thereby, the backpropagation neural network (BPNN) was adopted in [15] to tune the gains in ESO. However, the BPNN suffers easily trapping in local minimum and slow convergent speed.Motivated by this, the radial biases function (RBF) neural network is introduced in an improved ADRC to solve the practical issue in control of the newly developed EHS system with three hydraulic cavities for simultaneous balancing and positioning, constructing the novel neural network based active disturbance rejection control (NNADRC) strategy. Four adjustable compensation gains in the ESO are online updated by adopting the RBF neural network.


## 2. The Isoactuation Electrohydraulic Servo System

### 2.1. System Configuration and Working Principle

Figure1 illustrates the schematic of the isoactuation EHS system for simultaneous balancing and positioning. As shown in Figure 1, the EHS system mainly consists of a positioning controller, a variable capacity pump, a proportional servo valve, a rotary transformer, a RDC modular, a balancing controller, a constant displacement pump, a proportional pressure-reducing valve, a pressure sensor, and an actuation hydraulic cylinder. The actuation hydraulic cylinder has three cavities, namely, the upper, lower, and balancing cavity. A combination of the upper and lower cavities is adopted for system actuation.Figure 1
Schematic of the isoactuation EHS system.In practice, the balancing controller compares and calculates the deviation between the measured and desired pressure in the balancing cavity. The control signal for the proportional pressure-reducing valve is obtained based on the deviation to accurately control the pressure in the balancing cavity, accordingly cancelling the load and weight torque on the system. As for the positioning, the deviation between the measured and desired positions of the pipe is similarly calculated, and accordingly a control signal is generated for the proportional servo valve to control the flow rate and direction of the hydraulic fluid for the upper and lower cavities. Thereby, position of the pipe can be accurately controlled by adopting the feedback control strategy.


### 2.2. Modelling the EHS System

Assume the state variables arex
=
x
1
x
2
x
3
T with x
1
=
θ, x
2
=
θ
˙, and x
3
=
θ
¨; the state space equation for the isoactuation EHS system can be determined by(1)
x
˙
1
=
x
2
,
x
˙
2
=
x
3
,
x
˙
3
=
f
x
+
g
u
1
+
d
t
,where(2)
f
x
=
-
β
e
C
t
G
J
V
0
x
1
-
β
e
J
V
0
D
e
2
+
C
t
B
m
+
V
0
β
e
G
x
2
,
-
β
e
J
V
0
J
C
t
+
V
0
β
e
B
m
x
3
,
g
=
β
e
J
V
0
D
e
K
a
K
1
,
d
t
=
β
e
C
t
J
V
0
T
L
+
1
J
T
˙
L
,
d
t
≤
Const
,where θ denotes the actual position of the pipe, u
1 denotes the output signal of the positioning controller, β
e is the effective bulk modulus of elasticity, C
t is the overall leakage coefficient, G represents the load elastic stiffness, J is the rotational inertia, V
0 is the volume of a cavity, D
e is the equivalent surface total displacement, B
m is the viscoelastic damping coefficient, K
1 represents the current-flow rate amplification ratio, K
a is the gain of the servo amplifier, and T
L represents the external loads. Practically, the overall leakage coefficient, the viscoelastic damping coefficient, the equivalent surface total displacement, and the external loads vary with respect to working conditions, showing strong nonlinear behavior of the working system.


### 2.3. Controlling Principle of the EHS System

#### 2.3.1. Principle of Balancing Control

In (1), the external load T
L denotes a combination of unbalancing torque, external disturbance torque, and launch impact torque. To compensate for these external disturbances, a novel active actuation configuration based on a hydraulic cylinder with three cavities is proposed for simultaneous positioning and balancing. The basic principle can be summarized as follows: the weight torque varies with respect to the rotation position θ, which can be expressed by T
G
(
θ
). With a specified arm of force l
(
θ
) and the acting area A, the required pressure P
G
(
s
) provided by the balancing cavity can be accordingly determined. To actively balance the weight torque, the pressure in the balancing cavity is controlled by means of the proportional pressure-reducing valve. Thereby, the required torque for balancing the weight torque can be achieved.Since the working frequency of the proportional pressure-reducing valve is much higher than the natural frequency of its actuation, it can be simply regarded as a proportional element. Thus, the relationship between the control signalU
2
(
s
) and output flow rate Q
1
(
s
) of the valve can be expressed by (3)
Q
1
s
=
K
2
K
q
1
U
2
s
,where K
2 is the voltage-displacement coefficient and K
q
1 is the flow rate gain for the valve.The theoretical pressure in the balancing cavity can be determined by(4)
P
p
s
=
1
V
1
/
β
e
s
+
C
L
1
Q
s
1
s
-
Q
L
1
s
-
Q
1
s
=
1
V
1
/
β
e
s
+
C
L
1
Q
s
1
s
-
1
A
l
θ
s
θ
s
-
K
2
K
q
1
U
2
s
,where Q
s
1
(
s
) and Q
L
1
(
s
) denote the flow rate of the constant displacement pump and the flow rate into the balancing cavity, V
1 is the volume between the output end of the pump, the input end of the valve, and the balancing cavity, and C
L
1 is the leakage coefficient of the balancing cavity.


#### 2.3.2. Principle of Positioning Control

The feedback control scheme is employed for the positioning. Since there are certain nonlinear components and unbalanced torque in the system, the ESO is further employed for the estimation of these unbalanced components; the control signal is then obtained by following the nonlinear state error feedback control law with consideration of the disturbance compensation.


## 2.1. System Configuration and Working Principle

Figure1 illustrates the schematic of the isoactuation EHS system for simultaneous balancing and positioning. As shown in Figure 1, the EHS system mainly consists of a positioning controller, a variable capacity pump, a proportional servo valve, a rotary transformer, a RDC modular, a balancing controller, a constant displacement pump, a proportional pressure-reducing valve, a pressure sensor, and an actuation hydraulic cylinder. The actuation hydraulic cylinder has three cavities, namely, the upper, lower, and balancing cavity. A combination of the upper and lower cavities is adopted for system actuation.Figure 1
Schematic of the isoactuation EHS system.In practice, the balancing controller compares and calculates the deviation between the measured and desired pressure in the balancing cavity. The control signal for the proportional pressure-reducing valve is obtained based on the deviation to accurately control the pressure in the balancing cavity, accordingly cancelling the load and weight torque on the system. As for the positioning, the deviation between the measured and desired positions of the pipe is similarly calculated, and accordingly a control signal is generated for the proportional servo valve to control the flow rate and direction of the hydraulic fluid for the upper and lower cavities. Thereby, position of the pipe can be accurately controlled by adopting the feedback control strategy.


## 2.2. Modelling the EHS System

Assume the state variables arex
=
x
1
x
2
x
3
T with x
1
=
θ, x
2
=
θ
˙, and x
3
=
θ
¨; the state space equation for the isoactuation EHS system can be determined by(1)
x
˙
1
=
x
2
,
x
˙
2
=
x
3
,
x
˙
3
=
f
x
+
g
u
1
+
d
t
,where(2)
f
x
=
-
β
e
C
t
G
J
V
0
x
1
-
β
e
J
V
0
D
e
2
+
C
t
B
m
+
V
0
β
e
G
x
2
,
-
β
e
J
V
0
J
C
t
+
V
0
β
e
B
m
x
3
,
g
=
β
e
J
V
0
D
e
K
a
K
1
,
d
t
=
β
e
C
t
J
V
0
T
L
+
1
J
T
˙
L
,
d
t
≤
Const
,where θ denotes the actual position of the pipe, u
1 denotes the output signal of the positioning controller, β
e is the effective bulk modulus of elasticity, C
t is the overall leakage coefficient, G represents the load elastic stiffness, J is the rotational inertia, V
0 is the volume of a cavity, D
e is the equivalent surface total displacement, B
m is the viscoelastic damping coefficient, K
1 represents the current-flow rate amplification ratio, K
a is the gain of the servo amplifier, and T
L represents the external loads. Practically, the overall leakage coefficient, the viscoelastic damping coefficient, the equivalent surface total displacement, and the external loads vary with respect to working conditions, showing strong nonlinear behavior of the working system.


## 2.3. Controlling Principle of the EHS System

### 2.3.1. Principle of Balancing Control

In (1), the external load T
L denotes a combination of unbalancing torque, external disturbance torque, and launch impact torque. To compensate for these external disturbances, a novel active actuation configuration based on a hydraulic cylinder with three cavities is proposed for simultaneous positioning and balancing. The basic principle can be summarized as follows: the weight torque varies with respect to the rotation position θ, which can be expressed by T
G
(
θ
). With a specified arm of force l
(
θ
) and the acting area A, the required pressure P
G
(
s
) provided by the balancing cavity can be accordingly determined. To actively balance the weight torque, the pressure in the balancing cavity is controlled by means of the proportional pressure-reducing valve. Thereby, the required torque for balancing the weight torque can be achieved.Since the working frequency of the proportional pressure-reducing valve is much higher than the natural frequency of its actuation, it can be simply regarded as a proportional element. Thus, the relationship between the control signalU
2
(
s
) and output flow rate Q
1
(
s
) of the valve can be expressed by (3)
Q
1
s
=
K
2
K
q
1
U
2
s
,where K
2 is the voltage-displacement coefficient and K
q
1 is the flow rate gain for the valve.The theoretical pressure in the balancing cavity can be determined by(4)
P
p
s
=
1
V
1
/
β
e
s
+
C
L
1
Q
s
1
s
-
Q
L
1
s
-
Q
1
s
=
1
V
1
/
β
e
s
+
C
L
1
Q
s
1
s
-
1
A
l
θ
s
θ
s
-
K
2
K
q
1
U
2
s
,where Q
s
1
(
s
) and Q
L
1
(
s
) denote the flow rate of the constant displacement pump and the flow rate into the balancing cavity, V
1 is the volume between the output end of the pump, the input end of the valve, and the balancing cavity, and C
L
1 is the leakage coefficient of the balancing cavity.


### 2.3.2. Principle of Positioning Control

The feedback control scheme is employed for the positioning. Since there are certain nonlinear components and unbalanced torque in the system, the ESO is further employed for the estimation of these unbalanced components; the control signal is then obtained by following the nonlinear state error feedback control law with consideration of the disturbance compensation.


## 2.3.1. Principle of Balancing Control

In (1), the external load T
L denotes a combination of unbalancing torque, external disturbance torque, and launch impact torque. To compensate for these external disturbances, a novel active actuation configuration based on a hydraulic cylinder with three cavities is proposed for simultaneous positioning and balancing. The basic principle can be summarized as follows: the weight torque varies with respect to the rotation position θ, which can be expressed by T
G
(
θ
). With a specified arm of force l
(
θ
) and the acting area A, the required pressure P
G
(
s
) provided by the balancing cavity can be accordingly determined. To actively balance the weight torque, the pressure in the balancing cavity is controlled by means of the proportional pressure-reducing valve. Thereby, the required torque for balancing the weight torque can be achieved.Since the working frequency of the proportional pressure-reducing valve is much higher than the natural frequency of its actuation, it can be simply regarded as a proportional element. Thus, the relationship between the control signalU
2
(
s
) and output flow rate Q
1
(
s
) of the valve can be expressed by (3)
Q
1
s
=
K
2
K
q
1
U
2
s
,where K
2 is the voltage-displacement coefficient and K
q
1 is the flow rate gain for the valve.The theoretical pressure in the balancing cavity can be determined by(4)
P
p
s
=
1
V
1
/
β
e
s
+
C
L
1
Q
s
1
s
-
Q
L
1
s
-
Q
1
s
=
1
V
1
/
β
e
s
+
C
L
1
Q
s
1
s
-
1
A
l
θ
s
θ
s
-
K
2
K
q
1
U
2
s
,where Q
s
1
(
s
) and Q
L
1
(
s
) denote the flow rate of the constant displacement pump and the flow rate into the balancing cavity, V
1 is the volume between the output end of the pump, the input end of the valve, and the balancing cavity, and C
L
1 is the leakage coefficient of the balancing cavity.


## 2.3.2. Principle of Positioning Control

The feedback control scheme is employed for the positioning. Since there are certain nonlinear components and unbalanced torque in the system, the ESO is further employed for the estimation of these unbalanced components; the control signal is then obtained by following the nonlinear state error feedback control law with consideration of the disturbance compensation.


## 3. Design of the Controllers

### 3.1. Balancing Controller

The typical PID controller is adopted for the balancing of the gun control system. Assume that the desired and practical pressures in the balancing cavity areP
G
(
t
) and P
p
c
(
t
), respectively; the command signal for the balancing system can be determined by (5)
u
2
t
=
U
m
-
k
p
e
p
t
+
k
i
∫
0
t
e
p
t
d
t
+
k
d
d
e
p
t
d
t
,
e
p
t
=
P
G
t
-
P
p
c
t
,where U
m denotes the maximum output voltage for the control and k
p, k
i, and k
d are proportional, integral, and differential coefficients of the PID controller.


### 3.2. The Improved ADRC Controller

To better track the trajectory of the control system, an improved ADRC control strategy is developed. To suppress the inherent chatter phenomenon, a novel nonlinear functionn
f
a
l
(
·
) featuring smooth switching behavior is employed to replace the conventionally adopted nonlinear function f
a
l
(
·
) which is a key component of the ADRC controller. Besides, the radial biases function (RBF) neural network is adopted to online update the compensation gains of the ESO in the ADRC.


#### 3.2.1. Configuration of the Control System

Schematic of the ADRC is illustrated in Figure2; it mainly consists of a TD, an ESO, and a NLSEF control law. In this ADRC approach, the processes with higher orders, uncertainties, and unmodeled dynamics are viewed as lower-ordered systems with general disturbances which are further estimated by ESO and are actively compensated.Figure 2
Schematics of ADRC.With the TD process, it is employed for transient process and command signal generation. Fast tracking without overshoots as well as suspension of rapid fluctuations of the control signal when the presetting parameters are suddenly changed can be achieved by employing the TD. The definition of the TD process can be expressed by(6)
v
1
k
+
1
=
v
1
k
+
h
v
2
k
,
v
2
k
+
1
=
v
2
k
+
h
v
3
k
,
v
3
k
+
1
=
v
3
k
+
h
f
s
t
,
f
s
t
=
-
r
r
r
v
1
-
θ
d
+
3
v
2
+
3
v
3
,where θ
d
(
t
) denotes the desired trajectory of the control system, r and h determine the speed and the step length, respectively, and v
1
(
t
), v
2
(
t
), and v
3
(
t
) track θ
d
(
t
), θ
˙
d
(
t
), and θ
¨
d
(
t
), respectively.With the ESO, the change rate and total disturbance of the error can be real-time estimated using ESO from the control variables and the position errors of the input signal. Thereby, dynamic compensation for the disturbances can be conducted properly using estimated total disturbances. By introducing the novel nonlinear functionn
f
a
l
(
·
), the improved third-order discrete ESO can be configured as follows:(7)
e
k
=
z
1
k
-
θ
k
,
z
1
k
+
1
=
z
1
k
+
h
z
2
k
-
β
01
e
k
,
z
2
k
+
1
=
z
2
k
+
h
z
3
k
-
β
02
n
f
a
l
e
k
,
c
1
,
b
1
,
γ
1
,
z
3
k
+
1
=
z
3
k
+
h
z
4
k
-
β
03
n
f
a
l
e
k
,
c
1
,
b
2
,
γ
1
+
b
0
u
k
,
z
4
k
+
1
=
z
4
k
-
h
β
04
n
f
a
l
e
k
,
c
1
,
b
3
,
γ
1
,where β
01, β
02, β
03, and β
04 are adjustable compensation gains, and a
1, a
2, a
3, δ
1, and b
0 are design parameters. z
1
(
t
), z
2
(
t
), z
3
(
t
), and z
4
(
t
) are the estimate outputs of the ESO process, respectively, tacking θ
(
t
), θ
˙
(
t
), θ
¨
(
t
), and the estimated total disturbance.The nonlinear functionn
f
a
l
(
·
) can be expressed by(8)
n
f
a
l
e
k
,
c
,
b
,
γ
=
b
a
r
c
tan
⁡
c
e
k
-
γ
-
μ
0
π
/
2
-
μ
0
,
μ
0
=
a
r
c
tan
⁡
c
0
-
γ
,where v determines the shape of the operator, b determines the range of the operator, and γ determines the center of the operator.Essentially, the NLSEF controller is a nonlinear PD controller; nonlinear combination of the error components deriving from the outputs of the TD and the state estimation from the ESO is properly designed to construct the command signalu
o
(
t
). The third-order discrete governing law of the NLSEF can be determined by(9)
e
1
k
+
1
=
v
1
k
+
1
-
z
1
k
+
1
,
e
2
k
+
1
=
v
2
k
+
1
-
z
2
k
+
1
,
e
3
k
+
1
=
v
3
k
+
1
-
z
3
k
+
1
,
u
0
k
+
1
=
β
1
n
f
a
l
e
1
k
+
1
,
c
0
,
b
4
,
γ
0
+
β
2
n
f
a
l
e
2
k
+
1
,
c
0
,
b
5
,
γ
0
+
β
3
n
f
a
l
e
3
k
+
1
,
c
0
,
b
6
,
γ
0
,where β
1, β
2, and β
3 represent the control gains, respectively, and a
4, a
5, a
6, and δ
0 are the parameters for the design process.By combining the estimated disturbance through the ESO, the actual control variables applied to the actuator of the control system can be obtained as follows:(10)
u
1
k
+
1
=
u
0
k
+
1
-
z
4
k
+
1
b
0
,where b
0 is the compensation factor.


#### 3.2.2. Adaptive Updating of the ESO

Overall, there are four important parameters in the ESO that govern the control accuracy and system robustness of the whole control system, namely,β
01, β
02, β
03, and β
04. To achieve optimal control during the whole working process, an online updating of the four parameters based on RBFNN is developed in the present study. The configuration of the adaptive control system is illustrated in Figure 3. As shown in Figure 3, a three-layer RBFNN is adopted where e
1
(
t
), e
2
(
t
), e
3
(
t
), and θ
(
t
) are serving as the input nodes, while β
01, β
02, β
03, and β
04 are serving as the output nodes. The number of node in hidden layer is chosen as 6.Figure 3
The diagram of ADRC based on RBF neural network.The goal of adaptive adjustment of the control parameters is to seek for a control signal that can minimize the difference between the process output and the desired output. The performance criterion employed in this paper for the parameter updating is defined by(11)
E
k
=
1
2
θ
d
k
-
θ
k
2
=
1
2
e
2
k
.The output of each hidden node in the RBFNN can be obtained as follows:(12)
φ
i
k
=
exp
⁡
-
X
-
C
i
2
2
b
i
2
,where X
=
[
e
1
,
e
2
,
e
3
,
θ
]
T denotes the input vector of the NN, C
i
=
[
c
i
1
,
c
i
2
,
c
i
3
,
c
i
4
]
T denotes the center vector of the ith hidden node, and b
i is the width of the radial basis function of this node.Thus, output of the RBFNN can be determined by(13)
β
0
l
k
=
∑
i
=
1
6
ω
l
i
φ
i
k
,where ω
l
i represents the connection weight between the ith (i
=
1,2
,
3
,
4) hidden node and the lth output node.The update law for the weights of the RBFNN yields the following:(14)
ω
l
i
k
=
ω
l
i
k
-
1
+
Δ
ω
l
i
k
+
ρ
Δ
ω
l
i
k
-
1
,
Δ
ω
l
i
k
=
-
η
∂
E
k
∂
ω
l
i
k
=
-
η
∂
E
k
∂
θ
k
∂
θ
k
∂
β
0
l
k
∂
β
0
l
k
∂
ω
l
i
k
=
η
e
k
∂
θ
k
∂
β
0
l
k
φ
i
k
,where ρ and η are the momentum and learning factors of this RBFNN, respectively. It is to be noticed that the term ∂
θ
(
k
)
/
∂
β
0
l
(
k
) is unknown. Thereby, it will be replaced by s
g
n
l
(
k
) in practice.


## 3.1. Balancing Controller

The typical PID controller is adopted for the balancing of the gun control system. Assume that the desired and practical pressures in the balancing cavity areP
G
(
t
) and P
p
c
(
t
), respectively; the command signal for the balancing system can be determined by (5)
u
2
t
=
U
m
-
k
p
e
p
t
+
k
i
∫
0
t
e
p
t
d
t
+
k
d
d
e
p
t
d
t
,
e
p
t
=
P
G
t
-
P
p
c
t
,where U
m denotes the maximum output voltage for the control and k
p, k
i, and k
d are proportional, integral, and differential coefficients of the PID controller.


## 3.2. The Improved ADRC Controller

To better track the trajectory of the control system, an improved ADRC control strategy is developed. To suppress the inherent chatter phenomenon, a novel nonlinear functionn
f
a
l
(
·
) featuring smooth switching behavior is employed to replace the conventionally adopted nonlinear function f
a
l
(
·
) which is a key component of the ADRC controller. Besides, the radial biases function (RBF) neural network is adopted to online update the compensation gains of the ESO in the ADRC.


### 3.2.1. Configuration of the Control System

Schematic of the ADRC is illustrated in Figure2; it mainly consists of a TD, an ESO, and a NLSEF control law. In this ADRC approach, the processes with higher orders, uncertainties, and unmodeled dynamics are viewed as lower-ordered systems with general disturbances which are further estimated by ESO and are actively compensated.Figure 2
Schematics of ADRC.With the TD process, it is employed for transient process and command signal generation. Fast tracking without overshoots as well as suspension of rapid fluctuations of the control signal when the presetting parameters are suddenly changed can be achieved by employing the TD. The definition of the TD process can be expressed by(6)
v
1
k
+
1
=
v
1
k
+
h
v
2
k
,
v
2
k
+
1
=
v
2
k
+
h
v
3
k
,
v
3
k
+
1
=
v
3
k
+
h
f
s
t
,
f
s
t
=
-
r
r
r
v
1
-
θ
d
+
3
v
2
+
3
v
3
,where θ
d
(
t
) denotes the desired trajectory of the control system, r and h determine the speed and the step length, respectively, and v
1
(
t
), v
2
(
t
), and v
3
(
t
) track θ
d
(
t
), θ
˙
d
(
t
), and θ
¨
d
(
t
), respectively.With the ESO, the change rate and total disturbance of the error can be real-time estimated using ESO from the control variables and the position errors of the input signal. Thereby, dynamic compensation for the disturbances can be conducted properly using estimated total disturbances. By introducing the novel nonlinear functionn
f
a
l
(
·
), the improved third-order discrete ESO can be configured as follows:(7)
e
k
=
z
1
k
-
θ
k
,
z
1
k
+
1
=
z
1
k
+
h
z
2
k
-
β
01
e
k
,
z
2
k
+
1
=
z
2
k
+
h
z
3
k
-
β
02
n
f
a
l
e
k
,
c
1
,
b
1
,
γ
1
,
z
3
k
+
1
=
z
3
k
+
h
z
4
k
-
β
03
n
f
a
l
e
k
,
c
1
,
b
2
,
γ
1
+
b
0
u
k
,
z
4
k
+
1
=
z
4
k
-
h
β
04
n
f
a
l
e
k
,
c
1
,
b
3
,
γ
1
,where β
01, β
02, β
03, and β
04 are adjustable compensation gains, and a
1, a
2, a
3, δ
1, and b
0 are design parameters. z
1
(
t
), z
2
(
t
), z
3
(
t
), and z
4
(
t
) are the estimate outputs of the ESO process, respectively, tacking θ
(
t
), θ
˙
(
t
), θ
¨
(
t
), and the estimated total disturbance.The nonlinear functionn
f
a
l
(
·
) can be expressed by(8)
n
f
a
l
e
k
,
c
,
b
,
γ
=
b
a
r
c
tan
⁡
c
e
k
-
γ
-
μ
0
π
/
2
-
μ
0
,
μ
0
=
a
r
c
tan
⁡
c
0
-
γ
,where v determines the shape of the operator, b determines the range of the operator, and γ determines the center of the operator.Essentially, the NLSEF controller is a nonlinear PD controller; nonlinear combination of the error components deriving from the outputs of the TD and the state estimation from the ESO is properly designed to construct the command signalu
o
(
t
). The third-order discrete governing law of the NLSEF can be determined by(9)
e
1
k
+
1
=
v
1
k
+
1
-
z
1
k
+
1
,
e
2
k
+
1
=
v
2
k
+
1
-
z
2
k
+
1
,
e
3
k
+
1
=
v
3
k
+
1
-
z
3
k
+
1
,
u
0
k
+
1
=
β
1
n
f
a
l
e
1
k
+
1
,
c
0
,
b
4
,
γ
0
+
β
2
n
f
a
l
e
2
k
+
1
,
c
0
,
b
5
,
γ
0
+
β
3
n
f
a
l
e
3
k
+
1
,
c
0
,
b
6
,
γ
0
,where β
1, β
2, and β
3 represent the control gains, respectively, and a
4, a
5, a
6, and δ
0 are the parameters for the design process.By combining the estimated disturbance through the ESO, the actual control variables applied to the actuator of the control system can be obtained as follows:(10)
u
1
k
+
1
=
u
0
k
+
1
-
z
4
k
+
1
b
0
,where b
0 is the compensation factor.


### 3.2.2. Adaptive Updating of the ESO

Overall, there are four important parameters in the ESO that govern the control accuracy and system robustness of the whole control system, namely,β
01, β
02, β
03, and β
04. To achieve optimal control during the whole working process, an online updating of the four parameters based on RBFNN is developed in the present study. The configuration of the adaptive control system is illustrated in Figure 3. As shown in Figure 3, a three-layer RBFNN is adopted where e
1
(
t
), e
2
(
t
), e
3
(
t
), and θ
(
t
) are serving as the input nodes, while β
01, β
02, β
03, and β
04 are serving as the output nodes. The number of node in hidden layer is chosen as 6.Figure 3
The diagram of ADRC based on RBF neural network.The goal of adaptive adjustment of the control parameters is to seek for a control signal that can minimize the difference between the process output and the desired output. The performance criterion employed in this paper for the parameter updating is defined by(11)
E
k
=
1
2
θ
d
k
-
θ
k
2
=
1
2
e
2
k
.The output of each hidden node in the RBFNN can be obtained as follows:(12)
φ
i
k
=
exp
⁡
-
X
-
C
i
2
2
b
i
2
,where X
=
[
e
1
,
e
2
,
e
3
,
θ
]
T denotes the input vector of the NN, C
i
=
[
c
i
1
,
c
i
2
,
c
i
3
,
c
i
4
]
T denotes the center vector of the ith hidden node, and b
i is the width of the radial basis function of this node.Thus, output of the RBFNN can be determined by(13)
β
0
l
k
=
∑
i
=
1
6
ω
l
i
φ
i
k
,where ω
l
i represents the connection weight between the ith (i
=
1,2
,
3
,
4) hidden node and the lth output node.The update law for the weights of the RBFNN yields the following:(14)
ω
l
i
k
=
ω
l
i
k
-
1
+
Δ
ω
l
i
k
+
ρ
Δ
ω
l
i
k
-
1
,
Δ
ω
l
i
k
=
-
η
∂
E
k
∂
ω
l
i
k
=
-
η
∂
E
k
∂
θ
k
∂
θ
k
∂
β
0
l
k
∂
β
0
l
k
∂
ω
l
i
k
=
η
e
k
∂
θ
k
∂
β
0
l
k
φ
i
k
,where ρ and η are the momentum and learning factors of this RBFNN, respectively. It is to be noticed that the term ∂
θ
(
k
)
/
∂
β
0
l
(
k
) is unknown. Thereby, it will be replaced by s
g
n
l
(
k
) in practice.


## 3.2.1. Configuration of the Control System

Schematic of the ADRC is illustrated in Figure2; it mainly consists of a TD, an ESO, and a NLSEF control law. In this ADRC approach, the processes with higher orders, uncertainties, and unmodeled dynamics are viewed as lower-ordered systems with general disturbances which are further estimated by ESO and are actively compensated.Figure 2
Schematics of ADRC.With the TD process, it is employed for transient process and command signal generation. Fast tracking without overshoots as well as suspension of rapid fluctuations of the control signal when the presetting parameters are suddenly changed can be achieved by employing the TD. The definition of the TD process can be expressed by(6)
v
1
k
+
1
=
v
1
k
+
h
v
2
k
,
v
2
k
+
1
=
v
2
k
+
h
v
3
k
,
v
3
k
+
1
=
v
3
k
+
h
f
s
t
,
f
s
t
=
-
r
r
r
v
1
-
θ
d
+
3
v
2
+
3
v
3
,where θ
d
(
t
) denotes the desired trajectory of the control system, r and h determine the speed and the step length, respectively, and v
1
(
t
), v
2
(
t
), and v
3
(
t
) track θ
d
(
t
), θ
˙
d
(
t
), and θ
¨
d
(
t
), respectively.With the ESO, the change rate and total disturbance of the error can be real-time estimated using ESO from the control variables and the position errors of the input signal. Thereby, dynamic compensation for the disturbances can be conducted properly using estimated total disturbances. By introducing the novel nonlinear functionn
f
a
l
(
·
), the improved third-order discrete ESO can be configured as follows:(7)
e
k
=
z
1
k
-
θ
k
,
z
1
k
+
1
=
z
1
k
+
h
z
2
k
-
β
01
e
k
,
z
2
k
+
1
=
z
2
k
+
h
z
3
k
-
β
02
n
f
a
l
e
k
,
c
1
,
b
1
,
γ
1
,
z
3
k
+
1
=
z
3
k
+
h
z
4
k
-
β
03
n
f
a
l
e
k
,
c
1
,
b
2
,
γ
1
+
b
0
u
k
,
z
4
k
+
1
=
z
4
k
-
h
β
04
n
f
a
l
e
k
,
c
1
,
b
3
,
γ
1
,where β
01, β
02, β
03, and β
04 are adjustable compensation gains, and a
1, a
2, a
3, δ
1, and b
0 are design parameters. z
1
(
t
), z
2
(
t
), z
3
(
t
), and z
4
(
t
) are the estimate outputs of the ESO process, respectively, tacking θ
(
t
), θ
˙
(
t
), θ
¨
(
t
), and the estimated total disturbance.The nonlinear functionn
f
a
l
(
·
) can be expressed by(8)
n
f
a
l
e
k
,
c
,
b
,
γ
=
b
a
r
c
tan
⁡
c
e
k
-
γ
-
μ
0
π
/
2
-
μ
0
,
μ
0
=
a
r
c
tan
⁡
c
0
-
γ
,where v determines the shape of the operator, b determines the range of the operator, and γ determines the center of the operator.Essentially, the NLSEF controller is a nonlinear PD controller; nonlinear combination of the error components deriving from the outputs of the TD and the state estimation from the ESO is properly designed to construct the command signalu
o
(
t
). The third-order discrete governing law of the NLSEF can be determined by(9)
e
1
k
+
1
=
v
1
k
+
1
-
z
1
k
+
1
,
e
2
k
+
1
=
v
2
k
+
1
-
z
2
k
+
1
,
e
3
k
+
1
=
v
3
k
+
1
-
z
3
k
+
1
,
u
0
k
+
1
=
β
1
n
f
a
l
e
1
k
+
1
,
c
0
,
b
4
,
γ
0
+
β
2
n
f
a
l
e
2
k
+
1
,
c
0
,
b
5
,
γ
0
+
β
3
n
f
a
l
e
3
k
+
1
,
c
0
,
b
6
,
γ
0
,where β
1, β
2, and β
3 represent the control gains, respectively, and a
4, a
5, a
6, and δ
0 are the parameters for the design process.By combining the estimated disturbance through the ESO, the actual control variables applied to the actuator of the control system can be obtained as follows:(10)
u
1
k
+
1
=
u
0
k
+
1
-
z
4
k
+
1
b
0
,where b
0 is the compensation factor.


## 3.2.2. Adaptive Updating of the ESO

Overall, there are four important parameters in the ESO that govern the control accuracy and system robustness of the whole control system, namely,β
01, β
02, β
03, and β
04. To achieve optimal control during the whole working process, an online updating of the four parameters based on RBFNN is developed in the present study. The configuration of the adaptive control system is illustrated in Figure 3. As shown in Figure 3, a three-layer RBFNN is adopted where e
1
(
t
), e
2
(
t
), e
3
(
t
), and θ
(
t
) are serving as the input nodes, while β
01, β
02, β
03, and β
04 are serving as the output nodes. The number of node in hidden layer is chosen as 6.Figure 3
The diagram of ADRC based on RBF neural network.The goal of adaptive adjustment of the control parameters is to seek for a control signal that can minimize the difference between the process output and the desired output. The performance criterion employed in this paper for the parameter updating is defined by(11)
E
k
=
1
2
θ
d
k
-
θ
k
2
=
1
2
e
2
k
.The output of each hidden node in the RBFNN can be obtained as follows:(12)
φ
i
k
=
exp
⁡
-
X
-
C
i
2
2
b
i
2
,where X
=
[
e
1
,
e
2
,
e
3
,
θ
]
T denotes the input vector of the NN, C
i
=
[
c
i
1
,
c
i
2
,
c
i
3
,
c
i
4
]
T denotes the center vector of the ith hidden node, and b
i is the width of the radial basis function of this node.Thus, output of the RBFNN can be determined by(13)
β
0
l
k
=
∑
i
=
1
6
ω
l
i
φ
i
k
,where ω
l
i represents the connection weight between the ith (i
=
1,2
,
3
,
4) hidden node and the lth output node.The update law for the weights of the RBFNN yields the following:(14)
ω
l
i
k
=
ω
l
i
k
-
1
+
Δ
ω
l
i
k
+
ρ
Δ
ω
l
i
k
-
1
,
Δ
ω
l
i
k
=
-
η
∂
E
k
∂
ω
l
i
k
=
-
η
∂
E
k
∂
θ
k
∂
θ
k
∂
β
0
l
k
∂
β
0
l
k
∂
ω
l
i
k
=
η
e
k
∂
θ
k
∂
β
0
l
k
φ
i
k
,where ρ and η are the momentum and learning factors of this RBFNN, respectively. It is to be noticed that the term ∂
θ
(
k
)
/
∂
β
0
l
(
k
) is unknown. Thereby, it will be replaced by s
g
n
l
(
k
) in practice.


## 4. Simulation Results and Discussion

To demonstrate the effectiveness and superiority of the improved ADRC controller for the EHS system with isoactuation configuration, numerical simulation on step responses and constant velocity tracking of the system was conducted.In the simulation, parameters for the electrohydraulic servo system wereJ
=
1.55
×
1
0
5
K
g
·
m
2, C
t
=
1.5
×
1
0
-
13
(
m
3
·
s
-
1
)
/
P
a, B
m
=
1.35
×
1
0
5
N
·
m
/
(
r
a
d
·
s
-
1
), V
0
=
0.011781
m
3, β
e
=
700
M
P
a, and T
Z
=
1.28
×
1
0
5
N
m. With the balancing controller, three coefficients for the PID controller were set as k
p
=
4
×
1
0
-
5, k
i
=
2.3
×
1
0
-
6, and k
d
=
3.12
×
1
0
-
7. Parameters for the TD were h
=
0.01 and r
=
0.12, and those for the NLSEF are set as β
1
=
0.95, β
2
=
10.6, β
3
=
17.3, c
0
=
0.25, b
4
=
2.5, b
5
=
2.0, b
6
=
1.5, and γ
0
=
0.01. As for the ESO, the design parameters were set as c
1
=
0.5, b
1
=
2.5, b
2
=
2.0, b
3
=
1.5, and γ
1
=
0.01, while the adjustable compensation gains were online updated, and the initial values were set as β
01
=
150, β
02
=
40, β
03
=
40, and β
04
=
500.


### 4.1. Step Response

Positioning errors of the EHS system in step response obtained by both ADRC and the improved NNADRC controllers are illustrated in Figure4(a). No overshoots are observed for the two control systems, and the response time for the two control systems at which the positioning errors are within ±
0.0005236
r
a
d (≈
±
0.5
m
i
l) is about 2.545 s. The steady state error is about 2.7 × 10−4 rad. To further evaluate system robustness, external disturbances featuring square wave with an amplitude of 100
K
N
·
m were added in the control system from t
=
5 s to t
=
6 s. From the positioning errors in Figure 4(a), the maximum deviations for ADRC and NNADRC induced by the disturbances are 0.055 rad and 0.005 rad, respectively. This indicates that the NNADRC possesses much higher robustness than the ADRC.Figure 4
Step response of the control system: (a) errors of position signal with square wave disturbance, (b) position tracking control voltage, (c) square wave disturbance and the estimatedz
4, and (d) torque errors of position tracking.
(a)
(b)
(c)
(d)The command signal and the estimated total disturbance (z
4) during the step responses are further illustrated in Figures 4(b) and 4(c). As shown in Figure 4(b), much smooth estimation is obtained by the NNADRC controller, showing strong suspension capacity of the chatter phenomenon by adopting the improved nonlinear function n
f
a
l
(
·
). In addition, much more accurate estimation of the total disturbances is achieved in the NNADRC system which may attribute to the real-time updating capacity of the RBFNN. The weighting and balancing torques in the NNADRC system are illustrated in Figure 4(d); a deviation of the torque is about 50 Nm. It suggests that the output torque of the balancing cavity finely agrees with the weighting torque by adopting the balancing controller, being capable for active balance of the system weighting torque.To further investigate system nonlinearity induced performance variation of the control system, parameter perturbation was adopted to mimic the system state deviations caused by the nonlinear effects. Step responses of the ADRC and NNADRC systems are illustrated in Figure5 with respect to a variation of the rotational inertiaJ up to 30%. As shown in Figure 5, the overshoot of the ADRC system is about 0.4147 rad (3.67%), and the time for entering the positioning error tolerance is about 5.65 s. In contrast, the overshoot of the NNADRC system is only about 0.1152 rad (1.06%) which is only about 28.8% of that of the ADRC system, and the entering time is about 3.11 s. The parameter perturbation simulation indicates that the NNADRC system has high robustness against parameter perturbation of the control system.Figure 5
ARDC and NNARDC positioning results with 30% change ofJ.


### 4.2. Constant Velocity Tracking

Constant velocity tracking withv
=
0.8727 rad·s−1 was simulated to further investigate performances of the control system. A harmonic disturbance with amplitude and frequency of A
=
20
K
N
·
m and f
=
0.5 Hz is added to simulate external disturbance. The resulting tracking errors obtained by both ADRC and the NNADRC are illustrated in Figure 6(a). As shown in Figure 6(a), the maximum tracking error in the steady state for the NNADRC is about 0.0011 rad which is only about 1/9 that obtained by ADRC. Similarly, the command voltages and estimated total disturbances for the two control systems are, respectively, illustrated in Figures 6(b) and 6(c). Strong fluctuations of the command and estimated signals especially at the initial stage are observed in the ADRC control system, while those of the NNADRC are much smoother. This indicates that the NNADRC well outperform the ADRC control strategy. The weighting and balancing torques in the NNADRC system are illustrated in Figure 6(d). Slight harmonic fluctuation of the torques is observed, and the maximum unbalanced torque is only about 90 Nm. The result suggests that the developed EHS system is effective and very promising for simultaneous balancing and positioning of large pipes.Figure 6
Response of the control system with constant speed tracking. (a) Errors of speed signal with sinusoidal disturbance, (b) speed tracking control voltage, (c) sinusoidal disturbance and the estimatedz
4, and (d) torque errors of speed tracking.
(a)
(b)
(c)
(d)


## 4.1. Step Response

Positioning errors of the EHS system in step response obtained by both ADRC and the improved NNADRC controllers are illustrated in Figure4(a). No overshoots are observed for the two control systems, and the response time for the two control systems at which the positioning errors are within ±
0.0005236
r
a
d (≈
±
0.5
m
i
l) is about 2.545 s. The steady state error is about 2.7 × 10−4 rad. To further evaluate system robustness, external disturbances featuring square wave with an amplitude of 100
K
N
·
m were added in the control system from t
=
5 s to t
=
6 s. From the positioning errors in Figure 4(a), the maximum deviations for ADRC and NNADRC induced by the disturbances are 0.055 rad and 0.005 rad, respectively. This indicates that the NNADRC possesses much higher robustness than the ADRC.Figure 4
Step response of the control system: (a) errors of position signal with square wave disturbance, (b) position tracking control voltage, (c) square wave disturbance and the estimatedz
4, and (d) torque errors of position tracking.
(a)
(b)
(c)
(d)The command signal and the estimated total disturbance (z
4) during the step responses are further illustrated in Figures 4(b) and 4(c). As shown in Figure 4(b), much smooth estimation is obtained by the NNADRC controller, showing strong suspension capacity of the chatter phenomenon by adopting the improved nonlinear function n
f
a
l
(
·
). In addition, much more accurate estimation of the total disturbances is achieved in the NNADRC system which may attribute to the real-time updating capacity of the RBFNN. The weighting and balancing torques in the NNADRC system are illustrated in Figure 4(d); a deviation of the torque is about 50 Nm. It suggests that the output torque of the balancing cavity finely agrees with the weighting torque by adopting the balancing controller, being capable for active balance of the system weighting torque.To further investigate system nonlinearity induced performance variation of the control system, parameter perturbation was adopted to mimic the system state deviations caused by the nonlinear effects. Step responses of the ADRC and NNADRC systems are illustrated in Figure5 with respect to a variation of the rotational inertiaJ up to 30%. As shown in Figure 5, the overshoot of the ADRC system is about 0.4147 rad (3.67%), and the time for entering the positioning error tolerance is about 5.65 s. In contrast, the overshoot of the NNADRC system is only about 0.1152 rad (1.06%) which is only about 28.8% of that of the ADRC system, and the entering time is about 3.11 s. The parameter perturbation simulation indicates that the NNADRC system has high robustness against parameter perturbation of the control system.Figure 5
ARDC and NNARDC positioning results with 30% change ofJ.


## 4.2. Constant Velocity Tracking

Constant velocity tracking withv
=
0.8727 rad·s−1 was simulated to further investigate performances of the control system. A harmonic disturbance with amplitude and frequency of A
=
20
K
N
·
m and f
=
0.5 Hz is added to simulate external disturbance. The resulting tracking errors obtained by both ADRC and the NNADRC are illustrated in Figure 6(a). As shown in Figure 6(a), the maximum tracking error in the steady state for the NNADRC is about 0.0011 rad which is only about 1/9 that obtained by ADRC. Similarly, the command voltages and estimated total disturbances for the two control systems are, respectively, illustrated in Figures 6(b) and 6(c). Strong fluctuations of the command and estimated signals especially at the initial stage are observed in the ADRC control system, while those of the NNADRC are much smoother. This indicates that the NNADRC well outperform the ADRC control strategy. The weighting and balancing torques in the NNADRC system are illustrated in Figure 6(d). Slight harmonic fluctuation of the torques is observed, and the maximum unbalanced torque is only about 90 Nm. The result suggests that the developed EHS system is effective and very promising for simultaneous balancing and positioning of large pipes.Figure 6
Response of the control system with constant speed tracking. (a) Errors of speed signal with sinusoidal disturbance, (b) speed tracking control voltage, (c) sinusoidal disturbance and the estimatedz
4, and (d) torque errors of speed tracking.
(a)
(b)
(c)
(d)


## 5. Conclusions

To satisfy the lightweight requirements of large pipe weapons, a novel electrohydraulic servo (EHS) system where the hydraulic cylinder possesses three cavities is developed and investigated in the present study. In the EHS system, the balancing cavity of the EHS is especially designed for active compensation for the unbalancing force of the system, whereas the driving cavity is employed for the positioning and disturbance rejection of the large pipe. By adopting the isoactuation configuration, a more compact and lightweight gun control system can be achieved.Aiming at simultaneous balancing and positioning of the EHS system, a novel neural network based active disturbance rejection control (NNADRC) strategy is developed. In the NNADRC, the radial basis function (RBF) neural network is employed for online updating parameters of the extended state observer (ESO). By online updating of the estimator gains of the ESO, the nonlinear behavior and external disturbance of the system can be accurately estimated and compensated in real time.The efficiency and superiority of the control system are critically investigated by conducting numerical simulations. When suffering square disturbance in step response, the positioning error obtained by NNADRC is only 10% of that obtained by ADRC, and the overshoot and entering time of the NNADRC are about 28.8% and 55% of them in ADRC system when encountering parameter perturbation, respectively. Meanwhile, when suffering harmonic disturbance in constant speed tracking, the steady accuracy is enhanced by 9 times. In addition, the inherent chatter phenomenon in ADRC can also be well suspended by adopting the improved nonlinear function scheme in the NNADRC.



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*Source: 4921095-2016-01-11.xml*