Optimal Vibration Control for Tracked Vehicle Suspension Systems

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.

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 Section 2, 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 Figure 1 (see [15, 16]), in which 𝑚_𝑠 is the sprung mass representing the tracked car chassis; 𝑚_𝑢 is the unsprung mass representing the wheel assembly; 𝑘_𝑠 and 𝑏_𝑠 are stiffness and damping of the uncontrolled suspension, respectively; 𝑘_𝑡 and 𝑏_𝑡 stand for stiffness and damping of the wheel, respectively; 𝑥_𝑠(𝑡) and 𝑥_𝑢(𝑡) are the displacements of the sprung and unsprung masses, respectively; 𝑥_𝑟(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 Figure 1; the dynamic equations of 𝑚_𝑠 and 𝑚_𝑢 are given by 𝑢(𝑡)=𝑚_𝑠𝑥̈_𝑠(𝑡)+𝑏_𝑠𝑥̇_𝑠(𝑡)−𝑥̇_𝑢(𝑡)+𝑘_𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡,𝑢(𝑡))=−𝑚_𝑢𝑥̈_𝑢(𝑡)+𝑏_𝑠𝑥̇_𝑠(𝑡)−𝑥̇_𝑢(𝑡)+𝑘_𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)−𝑘_𝑡𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡)−𝑏_𝑡𝑥̇_𝑢(𝑡)−𝑥̇_𝑟(𝑡), (1) where 𝑥̈_𝑠(𝑡) and 𝑥̇_𝑠(𝑡), respectively, are acceleration and velocity of the sprung and unsprung masses; 𝑥̈_𝑢(𝑡) and 𝑥̇_𝑢(𝑡), respectively, are acceleration and velocity of the unsprung masses; 𝑥̇_𝑟(𝑡) is the road velocity.

Define the set of state variables for (1): 𝑥₁(𝑡)=𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡),𝑥₂(𝑡)=𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡),𝑥₃(𝑡)=𝑥̇_𝑠(𝑡),𝑥₄(𝑡)=𝑥̇_𝑢(𝑡), (2) 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: 𝑥(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)^𝑇. (3)

The principal variables for the active suspension design and evaluation are sprung mass acceleration 𝑥̈_𝑠(𝑡), which determines the ride comfort; the suspension deflection 𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡), which indicates the limit of the vehicle body motion; and the wheel deflection 𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡), which ensures the road holding ability. To satisfy the performance requirements, the controlled output vector is chosen as 𝑦_𝑐(𝑡)=𝑥̈_𝑠(𝑡)𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡)=𝑥̈_𝑠(𝑡)𝑥₁(𝑡)𝑥₂(𝑡). (4)

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: 𝑥̇(𝑡)(𝑡)(𝑡)(𝑡),𝑦=𝐴𝑥+𝐵𝑢+𝐷𝑝_𝑐(𝑡)=𝐶𝑥(𝑡)+(𝑡),𝑥(0)𝐵𝑢=𝑥₀, (5) where 𝑝(𝑡)=𝑥̇_𝑟(𝑡) is external input disturbance, and 𝐴=001−10001−𝑘_𝑠𝑚_𝑠0−𝑏_𝑠𝑚_𝑠𝑏_𝑠𝑚_𝑠𝑘_𝑠𝑚_𝑢−𝑘_𝑡𝑚_𝑢𝑏_𝑠𝑚_𝑢−𝑏_𝑡+𝑏_𝑠𝑚_𝑢,00𝐵=1𝑚_𝑠−1𝑚_𝑢00𝑏,𝐷=−1_𝑡𝑚_𝑢,1𝐵=𝑚_𝑠00,𝐶=−𝑘_𝑠𝑚_𝑠0−𝑏_𝑠𝑚_𝑠𝑏_𝑠𝑚_𝑠1000.0100 (6)

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 𝑆𝑊₁=𝐶_𝑠𝑊₁⁻²=4^𝑘×10⁻⁷𝑊₁⁻², (7) where 𝑊₁ is the spatial frequency, 𝐶_𝑠 is the road roughness constant, and 𝑘 denotes the sorts of road grade as shown in Table 1.

Table 1 

Road grades and PSDs.

Assume that the vehicle travels at a certain constant horizontal velocity 𝑣₀ 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 𝑥_𝑟(𝑡) can be approximately simulated by a finite series sum 𝑥_𝑟(𝑡)=^𝑖_𝑗=1𝜉_𝑗(𝑡)≜^𝑖_𝑗=1𝜙_𝑗sin𝑗𝜔₀𝑡+𝜃_𝑗 (8) with amplitudes 𝜙_𝑗 and random phases 𝜃_𝑗 which follows a uniform distribution in [0,2𝜋]. According to the random process theory, the 𝑗th average power is as follows: 𝑆𝑗Δ𝑊₁×Δ𝑊₁=𝜙²_𝑗2. (9) Hence, we get 𝜙_𝑗=2𝑆𝑗Δ𝑊₁Δ𝑊₁=2^𝑘10³𝑗𝑙.10𝜋 (10)

Choose the spatial frequency interval Δ𝑊₁=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, 𝑤(𝑡)=𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_2𝑖(𝑡)^𝑇=𝜉₁(𝑡),…,𝜉_𝑖(𝑡),𝜉̇₁(𝑡),…,𝜉̇_𝑖(𝑡)^𝑇. (11) The road velocity disturbance 𝑝(𝑡)=𝑥̇_𝑟(𝑡) is described by the following exosystem: 𝑤̇(𝑡)(𝑡),𝑝(𝑡)(𝑡),=𝐺𝑤=𝐹𝑤 (12) where 𝐺=0𝐼_𝑖−Ω²0∈ℝ^2𝑖×2𝑖,𝐹=1,…,1,_𝑖0,…,0_𝑖∈ℝ^1×2𝑖, (13) in which Ω=diag{𝜔₀,2𝜔₀,…,𝑖𝜔₀}∈ℝ^𝑖×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: 𝐽=lim_𝑇→∞1𝑇^𝑇₀𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+𝑢^𝑇(𝑡)𝑅𝑢(𝑡)𝑑𝑡, (14) where 𝑄=𝐶^𝑇𝐶∈ℝ^4×4 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: 𝑢^∗(𝑡)=−𝑅⁻¹𝐵^𝑇𝑃₁𝑥(𝑡)+𝑃₂𝑝(𝑡)𝑒^−𝛼𝑡+𝑃₃𝑝_𝜔(𝑡)𝑒^−𝛼𝑡, (15) where 𝑃₁ is the unique positive definite solution of the following Riccati matrix equation: (𝐴+𝛼𝐼)^𝑇𝑃₁+𝑃₁(𝐴+𝛼𝐼)−𝑃₁𝑆𝑃₁+𝑄=0, (16) where 𝑃₂ and 𝑃₃ are the unique solutions of the following Sylvester matrix equations: (𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆²𝑃₂+𝑃₂Ω²=−(𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆𝑃₁)𝐷,(𝐴+𝛼𝐼^𝑇−𝑃₁𝑆²𝑃₃+𝑃₃Ω²=𝑃₁𝐷, (17) in which 𝑆=𝑅⁻¹𝐵^𝑇, Ω=diag{Ω,Ω}, 𝑝_𝜔(𝑡)=𝑝(̇𝑡)=𝐹𝐺𝑤(𝑡).

Proof. Introduce model transform for the system (5) with performance index (14): 𝑥(𝑡)=𝑥(𝑡)𝑒^𝛼𝑡,𝑢(𝑡)=𝑢(𝑡)𝑒^𝛼𝑡,𝑝(𝑡)=𝑝(𝑡),𝐴=𝐴+𝛼𝐼,𝐵=𝐵,𝐷=𝐷. (18)(19) Then we have ̇𝑥(𝑡)=𝑥̇(𝑡)𝑒^𝛼𝑡+𝛼𝑥(𝑡)𝑒^𝛼𝑡=𝐴𝑥(𝑡)𝑒^𝛼𝑡+𝛼𝑥(𝑡)𝑒^𝛼𝑡+𝐵𝑢(𝑡)𝑒^𝛼𝑡+𝐷𝑝(𝑡)𝑒^𝛼𝑡=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐷𝑝(𝑡),𝐽=lim_𝑇→∞1𝑇^𝑇₀𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+𝑢^𝑇(𝑡)𝑅𝑢(𝑡)𝑑𝑡=lim_𝑇→∞1𝑇^𝑇₀𝑥(𝑡)𝑒^𝛼𝑡𝑇𝑄𝑥(𝑡)𝑒^𝛼𝑡+𝑢(𝑡)𝑒^𝛼𝑡𝑇𝑅𝑢(𝑡)𝑒^𝛼𝑡𝑑𝑡=lim_𝑇→∞1𝑇^𝑇₀𝑥^𝑇(𝑡)𝑄𝑥(𝑡)+𝑢^𝑇(𝑡)𝑅𝑢(𝑡)𝑑𝑡. (20)(21)
Applying the maximum principle to the optimal control problem in (20) and (21), the optimal control law can be written as 𝑢^∗(𝑡)=−𝑅⁻¹𝐵^𝑇𝜆(𝑡), (22) where 𝜆(𝑡) is the solution to the following two-point boundary value problem: −̇𝜆(𝑡)̇=𝑄𝑥(𝑡)+𝐴^𝑇𝜆(𝑡),̇𝑥(𝑡)=𝐴𝑥(𝑡)−𝑅⁻¹𝐵^𝑇𝜆(𝑡)𝑝(𝑡),𝑥(0)+𝐷=𝑥₀𝑒^𝛼𝑡,𝜆(∞)=0. (23) To solve (23), let 𝜆(𝑡)=𝑃₁𝑥(𝑡)+𝑃₂𝑝(𝑡)+𝑃₃𝑝_𝜔(𝑡). (24) Substituting the equations of (20) and (22) into the first derivatives of (24), we get ̇𝜆(𝑡)=𝑃₁̇𝑥(𝑡)+𝑃₂̇𝑝(𝑡)+𝑃₃𝑝̈(𝑡)=𝑃₁𝐴−𝑃₁𝑆𝑃₁𝑥(𝑡)+𝑃₁𝐷−𝑃₁𝑆𝑃₂𝑝(𝑡)+𝑃₂−𝑃₁𝑆𝑃₃̇𝑝(𝑡)+𝑃₃𝑝̈(𝑡). (25) Note that 𝑝_𝜔(𝑡)=𝑝_𝜔(𝑡)=𝑝̇(𝑡),𝑝̈(𝑡)=𝑝̈(𝑡)=𝐺²𝑝(𝑡)=−Ω²00Ω²𝑝(𝑡)=−Ω²𝑝(𝑡). (26) From (23) and (24), we obtain ̇𝜆(𝑡)=−𝑄+𝐴^𝑇𝑃₁𝑥(𝑡)−𝐴^𝑇𝑃₂𝑝(𝑡)−𝐴^𝑇𝑃₃𝑝_𝜔(𝑡). (27)
Adding (26) into (25), then comparing the coefficients of (25) and (27), we obtain matrix equations: 𝐴^𝑇𝑃₁+𝑃₁𝐴−𝑃₁𝑆𝑃₁𝐴+𝑄=0,_𝑇𝑃₂+𝑃₁𝐷−𝑃₃Ω²−𝑃₁𝑆𝑃₂𝐴=0,_𝑇𝑃₃+𝑃₂−𝑃₁𝑆𝑃₃=0. (28) Then, adding the model transform (18), we can obtain (𝐴+𝛼𝐼)^𝑇𝑃₁+𝑃₁(𝐴+𝛼𝐼)−𝑃₁𝑆𝑃₁)+𝑄=0,(𝐴+𝛼𝐼^𝑇−𝑃₁𝑆²𝑃₂+𝑃₂Ω²=−(𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆𝑃₁)𝐷,(𝐴+𝛼𝐼^𝑇−𝑃₁𝑆²𝑃₃+𝑃₃Ω²=𝑃₁𝐷. (29)
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 matrices 𝑃₁, 𝑃₂, and 𝑃₃. It was well known that the matrix 𝑃₁ is existent and unique.
According to the linear regulator theory, it follows that Re𝜆_𝑖(𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆<−𝛼<0,𝑖=1,2,…,𝑛. (30) Suppose that 𝜆_𝑖(𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆=−𝛼_𝑖+𝑗𝛽_𝑖, (31) where 𝛼_𝑖>0,  𝛽_𝑖∈𝑅, 𝑗=√−1. Therefore, we have 𝜆_𝑖(𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆²=𝜆_𝑖(𝐴+𝛼𝐼)^𝑇−𝑃₁𝑆²=𝛼²_𝑖−𝛽²_𝑖−𝑗2𝛼_𝑖𝛽_𝑖. (32) Due to Ω=diag(Ω,Ω), we obtain 𝜆_𝑗−Ω²≤0,𝑗=1,2,…,2𝑚. (33) Assume that there is a certain 𝜆_𝑖(𝐴−𝑆𝑃₁)²=𝜆_𝑗(−Ω²); from (32) and (33), we have a certain 𝛼_𝑖=0 at least. It is contrary to 𝛼_𝑖>0. Namely, 𝜆_𝑖(𝐴−𝑆𝑃₁)²≠𝜆_𝑗(−Ω²).
Hence, 𝑃₂ and 𝑃₃, the solution of Sylvester matrix (17), are existent and unique [17].
When the matrices 𝑃₁, 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 Table 2 which have been used in some references [15, 16]. The associated matrices of system (5) are as follows: ,00000,𝐴=001−10001−43.94120−1.12631.1263376.0528−890.09689.6391−9.7676𝐵=0.001−0.0088,𝐷=0.1285𝐶=,−43.94120−1.12631.12631000010000𝐵=.0.001 (34)

Table 2 

Parameters of quarter-car model.

To generate D grade road profile, we select 𝐶_𝑠=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. Figure 2 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 acceleration 𝑥̈_𝑠(𝑡) are shown in Figure 3, suspension deflections 𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡) are shown in Figure 4, and wheel deflections 𝑥_𝑢(𝑡)−𝑥_𝑟(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.

Acknowledgments

The authors would like to thank the anonymous referees for their helpful suggestions, which greatly improve the paper. This research was supported by NSFC Tianyuan Mathematics Youth Fund (no. 11026040), the Nature Science Foundation of China (no. 11301009 and no. U1204402), Science & Technology Development Plan of Henan Province (no. 122300410316), and Natural Science Foundations of Henan Province Education Department (no. 12A120001, no. 13A520018, and no. 13A110022).