# Nonlinear Time-Delay Suspension Adaptive Neural Network Active Control

Authors: Yue Zhu; Sihong Zhu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Considering the time-delay in control input channel and the nonlinear spring stiffness characteristics of suspension, a quarter-vehicle magneto rheological active suspension nonlinear model with time-delay is established in this paper. Based on the time-delay nonlinear model, an adaptive neural network structure for magneto rheological active suspension is presented. By recognizing and training the adaptive neural network, the adaptive neural network active suspension controller is obtained. Simulation results show that the presented method can guarantee that the quarter-vehicle magneto rheological active suspension system has satisfying performance on the E_level very poor ground.


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## Body

## 1. Introduction

For engineering vehicles, farm tractors, military vehicles, and so on, the road conditions are usually very poor. Therefore, vibration problem is prominent particularly [1]. Seat suspension has been widely used as a simple and effective method to improve ride quality of vehicle which is directly related to driver fatigue, discomfort, and safety [2]. In recent years, the active suspension control research has received the widespread attention of scholars at home and abroad, in the aspect of theoretical analysis and physical test [3–8].Vehicle suspension system is a complex dynamic system; the road input is a random process and has a strong uncertainty, while the system itself has strong nonlinearity characteristic; therefore, it is difficult to obtain ideal control effect by using conventional control methods [5, 6].The neural network is a kind of powerful tool to deal with uncertainty and nonlinearity; it can parallel computing and distribute information storage, while it has strong fault tolerance and self-learning ability. Hence, neural network is suitable for the complex system modeling and control [9–11].In this paper, in order to reduce the vibration of magneto rheological active suspension system, adaptive neural network active suspension controller is designed. At first, a quarter-vehicle active suspension nonlinear model with time-delay is established by considering the time-delay in control input channel of magneto rheological active suspension system and the nonlinear spring stiffness characteristics of suspension. Then, an adaptive neural network structure for magneto rheological active suspension is presented according to the time-delay nonlinear model. Next, the adaptive neural network active suspension controller is obtained by recognizing and training the adaptive neural network. At last, E_level ground which is very poor road condition is considered for a quarter-vehicle magneto rheological active suspension system. Simulation shows that the presented method can guarantee that the system has satisfying performance.The remainder of this paper is organized as follows. In Section2, the considered quarter-vehicle magneto rheological active suspension time-delay nonlinear model is presented. Adaptive neural network structure, neural network identification, and neural network training are given in Section 3 and the adaptive neural network active suspension controller is obtained in Section 3 as well. Simulation on a quarter-vehicle magneto rheological active suspension system under E_level very poor ground is illustrated in Section 4. Section 5 draws the conclusions of this paper.


## 2. Quarter-Vehicle Magneto Rheological Active Suspension Time-Delay Nonlinear Model

In this paper, quarter-vehicle model for active control of magneto rheological seat suspension system is studied. At the same time, the nonlinear stiffness characteristics of the suspension spring and control time-delay characteristics are considered. Figure1 shows a quarter-vehicle model of an active suspension system.Figure 1
Quarter-vehicle model with a magneto rheological active suspension.Among them,m
2 is body quality, m
1 is wheel quality, k
2 is suspension stiffness, c
2 is damping coefficient, k
1 is tire equivalent stiffness coefficient, z
2 is the displacement of body, z
1 is displacement of wheels, z
0 is the ground displacement excitation, and f
a
(
t
-
τ
) is active control. Here, the variable stiffness spring of restoring force-displacement relationship is expressed as [12] f
s
=
k
2
Δ
z
+
ε
k
2
Δ
z
3, Δ
z
=
z
2
-
z
1, ε is nonlinear spring ratio. When k
2
=
27358, ε
=
10 restoring force f
s and displacement Δ
z relationship is shown in Figure 2.Figure 2
Force-displacement curve of nonlinear spring.Dynamic differential equation for the suspension system is(1)
m
2
z
¨
2
+
c
2
(
z
˙
2
-
z
˙
1
)
+
k
2
(
z
2
-
z
1
)
+
ε
k
2
(
z
2
-
z
1
)
3
=
f
a
(
t
-
τ
)
,
m
1
z
¨
1
+
c
2
(
z
˙
1
-
z
˙
2
)
+
k
1
(
z
1
-
z
0
)
-
k
2
(
z
2
-
z
1
)
-
ε
k
2
(
z
2
-
z
1
)
3
=
-
f
a
(
t
-
τ
)
as x
=
[
x
1
,
x
2
,
x
3
,
x
4
]
T
=
[
z
2
,
z
1
,
z
˙
2
,
z
˙
1
]
T
=
[
q
T
,
p
T
]
T.u
=
f
a; assume acceleration z
¨
2, velocity z
˙
2, z
˙
1, and displacement z
2, z
1 are measurable; the introduction of auxiliary control variable is written as
(2)
u
-
(
t
-
τ
)
=
u
(
t
-
τ
)
-
ε
k
2
(
z
2
-
z
1
)
3
and (1) can be rewritten into
(3)
x
˙
=
A
x
+
B
u
-
(
t
-
τ
)
+
B
w
z
0
,
A
=
[
0
0
1
0
0
0
0
1
-
k
2
m
2
k
2
m
2
-
c
2
m
2
c
2
m
2
k
2
m
1
-
k
2
-
k
1
m
1
c
2
m
1
-
c
2
m
1
]
,
B
=
[
0
0
1
m
2
-
1
m
1
]
T
,
B
w
=
[
0
0
0
k
1
m
1
]
.
The corresponding nominal system model of (3) is
(4)
x
˙
=
[
A
11
A
12
A
21
A
22
]
[
q
p
]
+
[
B
1
B
2
]
u
-
(
t
-
τ
)
,
where
(5)
A
11
=
[
0
0
0
0
]
,
A
12
=
[
1
0
0
1
]
,
B
1
=
[
0
0
]
,
A
21
=
[
-
k
2
m
2
k
2
m
2
k
2
m
1
-
k
2
-
k
1
m
1
]
,
B
2
=
[
1
m
2
-
1
m
1
]
,
A
22
=
[
-
c
2
m
2
c
2
m
2
c
2
m
1
-
c
2
m
1
]
.


## 3. The Design of Adaptive Neural Network Active Suspension Controller

### 3.1. Adaptive Neural Network Structure

Adaptive neural network system structure is shown in Figure3. The system consists of three parts: the controlled active suspension, the neural network identifier AN1, and the neural network controller AN2. AN1 is used for online identification of active suspension systems; AN2 is used to realize the network parameters adjustment by tuning the network weights and the threshold online and then to get the best effect of vibration reduction.Figure 3
Adaptive neural network structure.In Figure3, output y is the body acceleration, control signal u is damping force, and its normalized range is 0~1. The common way to implement u is to regulate electrohydraulic proportional throttle valve flow area or valve core displacement. z
˙
2
(
z
˙
2
-
z
˙
1
) is as an input for the neural network controller AN2. The input for neural network identifier AN1 is the body acceleration and control signal u.


### 3.2. Neural Network Identification

The road excitation can be viewed as a white noise signal and it is suitable to identify active suspension system by using nonlinear autoregressive moving average (NARMA) model; namely,(6)
y
(
k
)
=
f
[
y
(
k
-
1
)
,
…
,
y
(
k
-
n
)
,
u
-
(
k
-
1
-
τ
)
,
…
,
u
-
(
k
-
m
-
τ
)
,
v
(
k
-
1
)
,
…
,
v
(
k
-
l
)
y
]
,
where y
(
k
) is for body acceleration at the kth step; u
-
(
k
-
1
-
τ
) is for the control signal at the (
k
-
1
-
τ
)th step; v
(
k
-
1
) is for road random input (white noise) at the (
k
-
1
)
th step.AN1 identifies the system model and multilayer feed-forward neural network was used by the network structure. Network is composed of two layers: the first layer is input layer and hasn
+
m inputs. Hyperbolic function is chosen as neurons transformation; each of the neurons has a threshold. The second layer is output layer and is with one neuron which represents the estimate of the system output. Proportion function is selected as neuron function and it also has neuron threshold.


### 3.3. The Adaptive Neural Network Controller

The structure of the neural network controller AN2 is shown in Figure4. The network has two inputs and one output. The first input z
1
(
0
)
=
ψ
z
2 is the root mean square value of the body acceleration; the second input z
2
(
0
)
=
z
˙
2
(
z
˙
2
-
z
˙
1
) is the product of the speed of the suspension. Network output value μ
=
r
/
r
max
⁡ is the variable damping. In order to facilitate network training, suppose r
max
⁡
=
1
×
1
0
5
(
N
·
s
2
/
m
2
). Network structure applies multilayer feed-forward neural network. The network has two layers (not including the input layer); the first layer is composed of 10 neurons with thresholds; the hyperbolic function is used as neurons transformation function S. The second layer has a neuron with threshold; the sigmoid function is utilized as neurons transformation function S, so that the output of the network can be always in the range of 0~1.Figure 4
Adaptive neural network controller.In order to obtain good performance, Marquardt back-propagation algorithm (MBP algorithm) which is of high training efficiency is used in training neural network identifier ANl. During training, the error between desired body acceleration and the output of neural network identifier ANl is as the input for MBP algorithm. And the weights and thresholds of AN1 keep unchanging.Because the desired control signal is unknown at the very beginning, neural network controller AN2 is cascaded with neural network identifier AN1, as shown in Figure3. During training, MBP algorithm is employed as well, but the weights and thresholds of AN2 are changed, in order to obtain and optimize adaptive neural network controller.The input-output relationship of the first layer is(7)
S
i
(
1
)
=
∑
j
=
0
2
ω
i
j
(
1
)
x
j
(
0
)
(
x
0
(
0
)
=
1
,
ω
r
0
(
1
)
=
θ
r
(
1
)
)
,
(8)
x
i
(
1
)
=
f
(
S
i
(
1
)
)
=
1
-
e
-
2
S
i
(
1
)
1
+
e
-
2
S
i
(
1
)
,
i
=
1,2
,
…
,
10
,
j
=
0,1
,
2
.The input-output relationship of the first layer is(9)
S
i
(
2
)
=
∑
j
=
0
10
ω
i
j
(
2
)
x
j
(
1
)
(
x
0
(
0
)
=
1
,
ω
r
0
(
2
)
=
θ
r
(
2
)
)
,
(10)
u
=
f
(
s
i
(
2
)
)
=
1
1
+
e
-
2
s
j
(
2
)
(
j
=
0,1
,
2
,
…
,
10
)
.To improve the training algorithm, the weights and thresholds are adjusted according to the following laws:(11)
ω
i
j
(
q
)
(
k
+
1
)
=
ω
i
j
(
q
)
(
k
)
+
α
[
(
1
-
η
)
D
i
j
q
(
k
)
+
η
D
i
j
q
(
k
-
1
)
]
,
D
i
j
q
=
-
∑
p
=
1
p
∂
E
p
∂
ω
i
j
,
E
p
=
1
2
(
y
d
p
-
y
^
p
)
2
;
q
=
1,2
.


## 3.1. Adaptive Neural Network Structure

Adaptive neural network system structure is shown in Figure3. The system consists of three parts: the controlled active suspension, the neural network identifier AN1, and the neural network controller AN2. AN1 is used for online identification of active suspension systems; AN2 is used to realize the network parameters adjustment by tuning the network weights and the threshold online and then to get the best effect of vibration reduction.Figure 3
Adaptive neural network structure.In Figure3, output y is the body acceleration, control signal u is damping force, and its normalized range is 0~1. The common way to implement u is to regulate electrohydraulic proportional throttle valve flow area or valve core displacement. z
˙
2
(
z
˙
2
-
z
˙
1
) is as an input for the neural network controller AN2. The input for neural network identifier AN1 is the body acceleration and control signal u.


## 3.2. Neural Network Identification

The road excitation can be viewed as a white noise signal and it is suitable to identify active suspension system by using nonlinear autoregressive moving average (NARMA) model; namely,(6)
y
(
k
)
=
f
[
y
(
k
-
1
)
,
…
,
y
(
k
-
n
)
,
u
-
(
k
-
1
-
τ
)
,
…
,
u
-
(
k
-
m
-
τ
)
,
v
(
k
-
1
)
,
…
,
v
(
k
-
l
)
y
]
,
where y
(
k
) is for body acceleration at the kth step; u
-
(
k
-
1
-
τ
) is for the control signal at the (
k
-
1
-
τ
)th step; v
(
k
-
1
) is for road random input (white noise) at the (
k
-
1
)
th step.AN1 identifies the system model and multilayer feed-forward neural network was used by the network structure. Network is composed of two layers: the first layer is input layer and hasn
+
m inputs. Hyperbolic function is chosen as neurons transformation; each of the neurons has a threshold. The second layer is output layer and is with one neuron which represents the estimate of the system output. Proportion function is selected as neuron function and it also has neuron threshold.


## 3.3. The Adaptive Neural Network Controller

The structure of the neural network controller AN2 is shown in Figure4. The network has two inputs and one output. The first input z
1
(
0
)
=
ψ
z
2 is the root mean square value of the body acceleration; the second input z
2
(
0
)
=
z
˙
2
(
z
˙
2
-
z
˙
1
) is the product of the speed of the suspension. Network output value μ
=
r
/
r
max
⁡ is the variable damping. In order to facilitate network training, suppose r
max
⁡
=
1
×
1
0
5
(
N
·
s
2
/
m
2
). Network structure applies multilayer feed-forward neural network. The network has two layers (not including the input layer); the first layer is composed of 10 neurons with thresholds; the hyperbolic function is used as neurons transformation function S. The second layer has a neuron with threshold; the sigmoid function is utilized as neurons transformation function S, so that the output of the network can be always in the range of 0~1.Figure 4
Adaptive neural network controller.In order to obtain good performance, Marquardt back-propagation algorithm (MBP algorithm) which is of high training efficiency is used in training neural network identifier ANl. During training, the error between desired body acceleration and the output of neural network identifier ANl is as the input for MBP algorithm. And the weights and thresholds of AN1 keep unchanging.Because the desired control signal is unknown at the very beginning, neural network controller AN2 is cascaded with neural network identifier AN1, as shown in Figure3. During training, MBP algorithm is employed as well, but the weights and thresholds of AN2 are changed, in order to obtain and optimize adaptive neural network controller.The input-output relationship of the first layer is(7)
S
i
(
1
)
=
∑
j
=
0
2
ω
i
j
(
1
)
x
j
(
0
)
(
x
0
(
0
)
=
1
,
ω
r
0
(
1
)
=
θ
r
(
1
)
)
,
(8)
x
i
(
1
)
=
f
(
S
i
(
1
)
)
=
1
-
e
-
2
S
i
(
1
)
1
+
e
-
2
S
i
(
1
)
,
i
=
1,2
,
…
,
10
,
j
=
0,1
,
2
.The input-output relationship of the first layer is(9)
S
i
(
2
)
=
∑
j
=
0
10
ω
i
j
(
2
)
x
j
(
1
)
(
x
0
(
0
)
=
1
,
ω
r
0
(
2
)
=
θ
r
(
2
)
)
,
(10)
u
=
f
(
s
i
(
2
)
)
=
1
1
+
e
-
2
s
j
(
2
)
(
j
=
0,1
,
2
,
…
,
10
)
.To improve the training algorithm, the weights and thresholds are adjusted according to the following laws:(11)
ω
i
j
(
q
)
(
k
+
1
)
=
ω
i
j
(
q
)
(
k
)
+
α
[
(
1
-
η
)
D
i
j
q
(
k
)
+
η
D
i
j
q
(
k
-
1
)
]
,
D
i
j
q
=
-
∑
p
=
1
p
∂
E
p
∂
ω
i
j
,
E
p
=
1
2
(
y
d
p
-
y
^
p
)
2
;
q
=
1,2
.


## 4. Case Study

Consider the active suspension system [13]: m
1
=
70 kg, k
1
=
309.511 kg, m
2
=
310 kg, k
2
=
27.358 KN/m, and c
2
=
0.984 KN s/m.For E_level ground, the road roughness power spectrum density isG
q
(
n
0
)
=
1024
×
1
0
-
6, n
0
=
0.1. Suppose forward velocity v
0
=
7 km/h. The ground excitation displacement is shown in Figure 5.Figure 5
Road excitation.Figure6 shows the training results, which illustrates the comparison between the test data of actual output and the network training output data. One can see that the network training output is consistent with the system test output very well. Figure 7 shows the test results; one can see that the network test output is consistent with the system test output very well.Figure 6
Training result.Figure 7
Test result.In order to make comparison, LQR controller is designed as well in this paper, andQ
LQR
=
diag
⁡
{
1,1
,
1,1
}; R
LQR
=
0.000001 is chosen for simulation. The simulation results are shown in Figures 8, 9, and 10, where the solid black lines represent the control results under the presented adaptive neural network (ANN) controller, while the dash blue lines represent the control results under LQR controller.Figure 8
Tire displacement.Figure 9
Body acceleration.Figure 10
Input force.Figures8~10 show that, on the terrible E_level ground, compared with the LQR controller, the presented adaptive neural network controller in this paper can significantly reduce the peak values of tire displacement, body acceleration, and control signal, and the body acceleration and control signal are smoother. Therefore, the proposed adaptive neural network controller for magneto rheological active suspension system is effective.


## 5. Conclusions

In this paper, an adaptive neural network control strategy is presented for a magneto rheological active suspension system with time-delay in control input channel and the nonlinear spring stiffness characteristics. On the basis of the time-delay nonlinear model and adaptive neural network structure, and by recognizing and training the adaptive neural network, the adaptive neural network active suspension controller is obtained. Simulation on a quarter-vehicle magneto rheological active suspension system under E_level ground shows that the proposed method can significantly reduce the peak values of tire displacement, body acceleration, and control signal, and the body acceleration and control signal are smoother.



---

*Source: 765871-2014-08-27.xml*