# Optimal Vibration Control for Tracked Vehicle Suspension Systems

Authors: Yan-Jun Liang; Shi-Liang Wu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Technique of optimal vibration control with exponential decay rate and simulation for vehicle active suspension systems is developed. Mechanical model and dynamic system for a class of tracked vehicle suspension vibration control is established and the corresponding system of state space form is described. In order to
prolong the working life of suspension system and improve ride comfort, based on the active suspension vibration control devices and using optimal control approach,
an optimal vibration controller with exponential decay rate is designed. Numerical simulations are carried out, and the control effects of the ordinary optimal controller
and the proposed controller are compared. Numerical simulation results illustrate the effectiveness of the proposed technique.


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## Body

## 1. Introduction

The vibration caused by severe pavement conditions not only affects the normal use of vehicle-mounted instruments but also damages the instruments, reduces ride ability, and even threatens the safety of passengers and vehicles. Therefore, the vibration control of the vehicle suspension is necessary.Since people put forward higher requirements on the ride comfort and the operational stability, the passive suspension can not meet the current needs. However, the active suspension has outstanding advantage in the riding comfort and operational stability. Therefore, the active suspension vibration control technique has attracted many scholars' research interests. To improve suspension performance on driver ride comfort, integrated seat and suspension models and driver body models were investigated [1, 2], in which the problems of vibration control were studied. A nonlinear mathematical model of the dynamic suspension system with two degrees of freedom was developed and PID controller was designed for a kind of air suspension system in [3]. An active suspension system utilizing a low-cost high-performance linear switched reluctance actuator with proportional-derivative control was presented in [4]. To study the vibration control problems of the vehicle suspension, H∞ control theory was applied [5–9], in which robust H∞ control and linear matrix inequality optimization method, nonfragile H∞ control method, and adaptive neurofuzzy inference system inverse magnetorheological damper model were used, respectively. Sliding mode control strategy such as self-organizing fuzzy sliding-mode control and adaptive sliding-mode control methods were utilized in [10–12] to obtain vibration controllers for vehicle suspension systems. Considering structured uncertainties and unstructured uncertainties, adaptive robust controllers were designed in [13, 14].The response time of the controller, namely, the real-time of the controller, is not considered in most of the research on vibration control of active suspension. In this paper, considered control energy and response time, used optimal control approach, optimal vibration control for tracked vehicle suspension systems is studied. The model of active suspension for tracked vehicle is established, optimal vibration controller with exponential decay rate is designed, and the control effect is demonstrated by numerical simulations.The remainder of this paper is structured as follows. In Section2, we establish active suspension systems and disturbance model. In Section 3, optimal vibration controller is designed for the suspension systems. In Section 4, numerical experiments are presented. Finally, in Section 5 some conclusions are drawn.


## 2. Suspension Systems and Disturbance Model

### 2.1. Mechanical Model of Suspension

Vehicle suspensions are complex dynamic systems, and the responses of suspensions are influenced by many factors. The suspension systems of four-wheel drive and four-wheel steering vehicles, strictly speaking, are eight degrees of freedom vibration systems. Based on practical problem and principle of being sufficient to analyze and study the vibration control for vehicle suspensions, the mechanical model of quarter-car suspension system with two degrees of freedom is investigated in this paper, as shown in Figure1 (see [15, 16]), in which ms is the sprung mass representing the tracked car chassis; mu is the unsprung mass representing the wheel assembly; ks and bs are stiffness and damping of the uncontrolled suspension, respectively; kt and bt stand for stiffness and damping of the wheel, respectively; xs(t) and xu(t) are the displacements of the sprung and unsprung masses, respectively; xr(t) is the road displacement input; u(t) denotes the actuator control force, which is normally generated by an actuator between the two masses.Figure 1
Quarter-car model with active suspension.


### 2.2. Dynamic Model of Suspension

Consider the quarter-car suspension system with two degrees of freedom as shown in Figure1; the dynamic equations of ms and mu are given by
(1)u(t)=msx¨s(t)+bs[x˙s(t)-x˙u(t)]+ks[xs(t)-xu(t)],u(t)=-mux¨u(t)+bs[x˙s(t)-x˙u(t)]+ks[xs(t)-xu(t)]u(t)=-kt[xu(t)-xr(t)]-bt[x˙u(t)-x˙r(t)],
where x¨s(t) and x˙s(t), respectively, are acceleration and velocity of the sprung and unsprung masses; x¨u(t) and x˙u(t), respectively, are acceleration and velocity of the unsprung masses; x˙r(t) is the road velocity.Define the set of state variables for (1):
(2)x1(t)=xs(t)-xu(t),x2(t)=xu(t)-xr(t),x3(t)=x˙s(t),x4(t)=x˙u(t),
where x1(t) is the suspension deflection, x2(t) is the wheel deflection, x3(t) is the sprung mass velocity, and x4(t) is the unsprung mass velocity. Then, the state vector is in the following form:
(3)x(t)=[x1(t),x2(t),x3(t),x4(t)]T.The principal variables for the active suspension design and evaluation are sprung mass accelerationx¨s(t), which determines the ride comfort; the suspension deflection xs(t)-xu(t), which indicates the limit of the vehicle body motion; and the wheel deflection xu(t)-xr(t), which ensures the road holding ability. To satisfy the performance requirements, the controlled output vector is chosen as
(4)yc(t)=[x¨s(t)xs(t)-xu(t)xu(t)-xr(t)]=[x¨s(t)x1(t)x2(t)].Then, from motion equation (1), state variables (2), state vector (3), and output vector (4), the dynamic system model of vehicle suspension is rewritten in the state-space representation:
(5)x˙(t)=Ax(t)+Bu(t)+Dp(t),ffffffyc(t)=C-x(t)+B-u(t),ffffffffffffx(0)=x0,
where p(t)=x˙r(t) is external input disturbance, and
(6)fffffffffffffA=[001-10001-ksms0-bsmsbsmsksmu-ktmubsmu-(bt+bs)mu],B=[001ms-1mu],D=[0-10btmu],B-=[1ms00],fffffffffffffC-=[-ksms0-bsmsbsms10000100].


### 2.3. Disturbance Model

To evaluate the suspension characteristics, the road profiles variability is taken into account. According to the ISO 2631 standards, the road displacement power spectral density (PSD) is approximately represented in the formulation of(7)S(W1)=CsW1-2=4k×10-7W1-2,
where W1 is the spatial frequency, Cs is the road roughness constant, and k denotes the sorts of road grade as shown in Table 1.Table 1
Road grades and PSDs.
Road grade
A
B
C
D
E
C
s (10−7 m3/rad)
1
4
16
64
256
Road sortk
0
1
2
3
4Assume that the vehicle travels at a certain constant horizontal velocityv0 and the given road segment length is l. The road disturbances are approximately considered as periodic vibrations. Since vehicle wheels and the active suspension system have the low pass filter characteristic, the road displacement xr(t) can be approximately simulated by a finite series sum
(8)xr(t)=∑j=1i‍ξj(t)≜∑j=1i‍ϕjsin(jω0t+θj)
with amplitudes ϕj and random phases θj which follows a uniform distribution in [0,2π]. According to the random process theory, the jth average power is as follows:
(9)S(jΔW1)×ΔW1=ϕj22.
Hence, we get
(10)ϕj=2S(jΔW1)ΔW1=2k103jl10π.Choose the spatial frequency intervalΔW1=2π/l and then the time frequency internal ω0=2πv0/l. Positive integer i limits the considered frequency band, and in this simplified model, it is generally lower than 20 Hz.Defining the disturbance state vector,(11)w(t)=[w1(t),w2(t),…,w2i(t)]T=[ξ1(t),…,ξi(t),ξ˙1(t),…,ξ˙i(t)]T.
The road velocity disturbance p(t)=x˙r(t) is described by the following exosystem:
(12)w˙(t)=Gw(t),p(t)=Fw(t),
where
(13)fffffffG=[0Ii-Ω-20]∈ℝ2i×2i,F=[1,…,1,︸i0,…,0︸i]∈ℝ1×2i,
in which Ω-=diag{ω0,2ω0,…,iω0}∈ℝi×i. Evidently, the pair (F,G) is observable completely.


## 2.1. Mechanical Model of Suspension

Vehicle suspensions are complex dynamic systems, and the responses of suspensions are influenced by many factors. The suspension systems of four-wheel drive and four-wheel steering vehicles, strictly speaking, are eight degrees of freedom vibration systems. Based on practical problem and principle of being sufficient to analyze and study the vibration control for vehicle suspensions, the mechanical model of quarter-car suspension system with two degrees of freedom is investigated in this paper, as shown in Figure1 (see [15, 16]), in which ms is the sprung mass representing the tracked car chassis; mu is the unsprung mass representing the wheel assembly; ks and bs are stiffness and damping of the uncontrolled suspension, respectively; kt and bt stand for stiffness and damping of the wheel, respectively; xs(t) and xu(t) are the displacements of the sprung and unsprung masses, respectively; xr(t) is the road displacement input; u(t) denotes the actuator control force, which is normally generated by an actuator between the two masses.Figure 1
Quarter-car model with active suspension.


## 2.2. Dynamic Model of Suspension

Consider the quarter-car suspension system with two degrees of freedom as shown in Figure1; the dynamic equations of ms and mu are given by
(1)u(t)=msx¨s(t)+bs[x˙s(t)-x˙u(t)]+ks[xs(t)-xu(t)],u(t)=-mux¨u(t)+bs[x˙s(t)-x˙u(t)]+ks[xs(t)-xu(t)]u(t)=-kt[xu(t)-xr(t)]-bt[x˙u(t)-x˙r(t)],
where x¨s(t) and x˙s(t), respectively, are acceleration and velocity of the sprung and unsprung masses; x¨u(t) and x˙u(t), respectively, are acceleration and velocity of the unsprung masses; x˙r(t) is the road velocity.Define the set of state variables for (1):
(2)x1(t)=xs(t)-xu(t),x2(t)=xu(t)-xr(t),x3(t)=x˙s(t),x4(t)=x˙u(t),
where x1(t) is the suspension deflection, x2(t) is the wheel deflection, x3(t) is the sprung mass velocity, and x4(t) is the unsprung mass velocity. Then, the state vector is in the following form:
(3)x(t)=[x1(t),x2(t),x3(t),x4(t)]T.The principal variables for the active suspension design and evaluation are sprung mass accelerationx¨s(t), which determines the ride comfort; the suspension deflection xs(t)-xu(t), which indicates the limit of the vehicle body motion; and the wheel deflection xu(t)-xr(t), which ensures the road holding ability. To satisfy the performance requirements, the controlled output vector is chosen as
(4)yc(t)=[x¨s(t)xs(t)-xu(t)xu(t)-xr(t)]=[x¨s(t)x1(t)x2(t)].Then, from motion equation (1), state variables (2), state vector (3), and output vector (4), the dynamic system model of vehicle suspension is rewritten in the state-space representation:
(5)x˙(t)=Ax(t)+Bu(t)+Dp(t),ffffffyc(t)=C-x(t)+B-u(t),ffffffffffffx(0)=x0,
where p(t)=x˙r(t) is external input disturbance, and
(6)fffffffffffffA=[001-10001-ksms0-bsmsbsmsksmu-ktmubsmu-(bt+bs)mu],B=[001ms-1mu],D=[0-10btmu],B-=[1ms00],fffffffffffffC-=[-ksms0-bsmsbsms10000100].


## 2.3. Disturbance Model

To evaluate the suspension characteristics, the road profiles variability is taken into account. According to the ISO 2631 standards, the road displacement power spectral density (PSD) is approximately represented in the formulation of(7)S(W1)=CsW1-2=4k×10-7W1-2,
where W1 is the spatial frequency, Cs is the road roughness constant, and k denotes the sorts of road grade as shown in Table 1.Table 1
Road grades and PSDs.
Road grade
A
B
C
D
E
C
s (10−7 m3/rad)
1
4
16
64
256
Road sortk
0
1
2
3
4Assume that the vehicle travels at a certain constant horizontal velocityv0 and the given road segment length is l. The road disturbances are approximately considered as periodic vibrations. Since vehicle wheels and the active suspension system have the low pass filter characteristic, the road displacement xr(t) can be approximately simulated by a finite series sum
(8)xr(t)=∑j=1i‍ξj(t)≜∑j=1i‍ϕjsin(jω0t+θj)
with amplitudes ϕj and random phases θj which follows a uniform distribution in [0,2π]. According to the random process theory, the jth average power is as follows:
(9)S(jΔW1)×ΔW1=ϕj22.
Hence, we get
(10)ϕj=2S(jΔW1)ΔW1=2k103jl10π.Choose the spatial frequency intervalΔW1=2π/l and then the time frequency internal ω0=2πv0/l. Positive integer i limits the considered frequency band, and in this simplified model, it is generally lower than 20 Hz.Defining the disturbance state vector,(11)w(t)=[w1(t),w2(t),…,w2i(t)]T=[ξ1(t),…,ξi(t),ξ˙1(t),…,ξ˙i(t)]T.
The road velocity disturbance p(t)=x˙r(t) is described by the following exosystem:
(12)w˙(t)=Gw(t),p(t)=Fw(t),
where
(13)fffffffG=[0Ii-Ω-20]∈ℝ2i×2i,F=[1,…,1,︸i0,…,0︸i]∈ℝ1×2i,
in which Ω-=diag{ω0,2ω0,…,iω0}∈ℝi×i. Evidently, the pair (F,G) is observable completely.


## 3. Optimal Vibration Controller Design

In order to study optimal vibration control problem for tracked vehicles, we choose an average performance index for system (5) as follows:
(14)J=limT→∞1T∫0T‍e2αt[xT(t)Qx(t)+uT(t)Ru(t)]dt,
where Q=CTC∈ℝ4×4 are positive semidefinite matrices, R∈ℝ is a positive definite matrix, and α≥0 is exponential decay rate, and if α=0, it is an ordinary optimal control performance index. We can modify the values of α, Q, and R to satisfy the control purpose, such as the response time of the control system and the balance between the control energy and control effect.The objective of this paper is to find a control law for system (5) and make the value of the performance index (14) minimum.Then, we design vibration controller for system (5). The optimal control law with exponential decay rate can be presented in the following theorem.Theorem 1.
Consider the optimal control problem described by system (5) with performance index (14); the optimal control law exists and is unique. Its form is as follows:
(15)u*(t)=-R-1BT[P1x(t)+P2p(t)e-αt+P3pω(t)e-αt],
where P1 is the unique positive definite solution of the following Riccati matrix equation:
(16)(A+αI)TP1+P1(A+αI)-P1SP1+Q=0,
where P2 and P3 are the unique solutions of the following Sylvester matrix equations:
(17)[(A+αI)T-P1S]2P2+P2Ω2=-[(A+αI)T-P1S]P1D,[(A+αI)T-P1S]2P3+P3Ω2=P1D,
in which S=R-1BT, Ω=diag{Ω-,Ω-}, pω(t)=p˙(t)=FGw(t).Proof.
Introduce model transform for the system (5) with performance index (14):
(18)x~(t)=x(t)eαt,u~(t)=u(t)eαt,p~(t)=p(t),(19)A~=A+αI,B~=B,D~=D.
Then we have
(20)x~˙(t)=x˙(t)eαt+αx(t)eαt=Ax(t)eαt+αx(t)eαt+Bu(t)eαt+Dp(t)eαt=A~x~(t)+B~u~(t)+D~p~(t),(21)J=limT→∞1T∫0T‍e2αt[xT(t)Qx(t)+uT(t)Ru(t)]dt=limT→∞1T∫0T‍[(x(t)eαt)TQ(x(t)eαt)gggggggggkk+(u(t)eαt)TR(u(t)eαt)]dt=limT→∞1T∫0T‍[x~T(t)Qx~(t)+u~T(t)Ru~(t)]dt.
Applying the maximum principle to the optimal control problem in (20) and (21), the optimal control law can be written as
(22)u~*(t)=-R-1B~Tλ~(t),
where λ~(t) is the solution to the following two-point boundary value problem:
(23)fffff-λ~˙(t)=Qx~˙(t)+ATλ~(t),x~˙(t)=Ax~(t)-R-1B~Tλ~(t)+Dp~(t),fffffffffx~(0)=x0eαt,fffffhhhhhλ~(∞)=0.
To solve (23), let
(24)λ~(t)=P1x~(t)+P2p~(t)+P3p~ω(t).
Substituting the equations of (20) and (22) into the first derivatives of (24), we get
(25)λ~˙(t)=P1x~˙(t)+P2p~˙(t)+P3p~¨(t)=(P1A~-P1SP1)x~(t)+(P1D~-P1SP2)p~(t)+(P2-P1SP3)p~˙(t)+P3p~¨(t).
Note that
(26)p~ω(t)=pω(t)=p˙(t),p~¨(t)=p¨(t)=G2p(t)p~¨(t)=-[Ω-200Ω-2]p(t)p~¨(t)=-Ω2p(t).
From (23) and (24), we obtain
(27)λ~˙(t)=-(Q+A~TP1)x~(t)-A~TP2p~(t)-A~TP3p~ω(t).
Adding (26) into (25), then comparing the coefficients of (25) and (27), we obtain matrix equations:
(28)ffA~TP1+P1A~-P1SP1+Q=0,A~TP2+P1D~-P3Ω2-P1SP2=0,fffffA~TP3+P2-P1SP3=0.
Then, adding the model transform (18), we can obtain
(29)(A+αI)TP1+P1(A+αI)-P1SP1+Q=0,[(A+αI)T-P1S]2P2+P2Ω2=-[(A+αI)T-P1S]P1D,[(A+αI)T-P1S]2P3+P3Ω2=P1D.
In the following, we prove the existence and uniqueness of the optimal control law. In fact, it is equivalent to prove the existence and uniqueness of the matricesP1, P2, and P3. It was well known that the matrix P1 is existent and unique.
According to the linear regulator theory, it follows that(30)Reλi[(A+αI)T-P1S]<-α<0,i=1,2,…,n.
Suppose that
(31)λi[(A+αI)T-P1S]=-αi+jβi,
where αi>0,  βi∈R, j=-1. Therefore, we have
(32)λi[(A+αI)T-P1S]2={λi[(A+αI)T-P1S]}2=αi2-βi2-j2αiβi.
Due to Ω=diag(Ω-,Ω-), we obtain
(33)λj(-Ω2)≤0,j=1,2,…,2m.
Assume that there is a certain λi(A-SP1)2=λj(-Ω2); from (32) and (33), we have a certain αi=0 at least. It is contrary to αi>0. Namely, λi(A-SP1)2≠λj(-Ω2).
Hence,P2 and P3, the solution of Sylvester matrix (17), are existent and unique [17].
When the matricesP1, P2, and P3 are derived, λ~(t) and the optimal control law u~*(t) can be obtained from (24) and (22), respectively. According to model transform (19), the optimal control law (15) is obtained.


## 4. Numerical Simulations

In this section, we apply the proposed optimal vibration controller to a tracked vehicle. The model parameters have the values listed in Table2 which have been used in some references [15, 16]. The associated matrices of system (5) are as follows:
(34)fffA=[001-10001-43.94120-1.12631.1263376.0528-890.09689.6391-9.7676],fffffB=[000.001-0.0088],D=[0000.1285],C-=[-43.94120-1.12631.126310000100],B-=[0.00100].Table 2
Parameters of quarter-car model.
Parameter
Variable
Value
Unit
Sprung mass
m
s
972.2
kg
Unsprung mass
m
u
113.6
kg
Suspension stiffness
k
s
42,719.6
N/m
Wheel stiffness
k
t
101,115
N/m
Suspension damping
b
s
1,095
N·s/m
Wheel damping
b
t
14.6
N·s/mTo generate D grade road profile, we selectCs=64×10-7m3/rad and k=3 in Table 1. Setting v0=20m/s, l=400m, and i=200 in (8) and (10) takes the frequency band from 0.05 Hz to 10 Hz.Using MATLAB software, numerical experiment is carried out for the proposed optimal vibration controller. Figure2 shows the road displacement curve.Figure 2
Road displacement curve.The main purpose of vibration control of vehicle suspensions is to reduce sprung mass acceleration to enhance ride comfort, to reduce the suspension deflection which indicates the limit of the vehicle body motion, and to reduce the wheel deflection to ensure road holding ability. So, to evaluate effectiveness of the proposed control strategy, sprung mass acceleration, suspension deflection, and wheel defection are considered. In order to demonstrate the effect of the proposed optimal vibration controller (α=1), we compare it with other two control strategies: open loop and traditional optimal control (α=1), see Figures 3–6.Figure 3
Sprung mass acceleration curves.Figure 4
Suspension deflection curves.Figure 5
Wheel deflection curves.Figure 6
Control force curves.The curves of sprung mass accelerationx¨s(t) are shown in Figure 3, suspension deflections xs(t)-xu(t) are shown in Figure 4, and wheel deflections xu(t)-xr(t) are shown in Figure 5, in which dash lines represent the open loop results of the suspension systems, dotted lines represent the traditional optimal vibration control results, and solid lines describe the results of the suspension systems controlled by the proposed control strategy. It can be seen from these numerical results that the proposed optimal controller with exponential decay rate is efficient, real-time, and robust in reducing sprung mass acceleration and suspension deflection, thereby enhancing ride comfort and ensuring safety of passengers and vehicles. Moreover, in Figure 6, it demonstrates that it needs less control forces than traditional optimal vibration control; therefore, it needs less energy.


## 5. Conclusions

Suspension control plays an important role in the modern vehicle and is one of the very important components to provide the ride comfort, in particular, to reduce driver fatigue due to long hours of driving. In order to enhance the ride comfort and ensure the safety of passengers and vehicles, dynamic system for a class of tracked vehicle suspension vibration control is established and a kind of optimal controller with exponential decay rate is designed. Numerical simulations demonstrated that the proposed strategy is efficient, real-time, and robust.



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