The Design of LQG Controller for Active Suspension Based on Analytic Hierarchy Process

Abstract

A full vehicle model with seven degrees of freedom based on active suspension control is established, and linear quadratic gaussian (LQG) is designed by applying optimal control theory. Especially, the methodology of Analytic Hierarchy Process (AHP) is used to make the selection of weighted coefficients of performance indexes, which can reduce ineffective job in contrast with experience method. From the simulation results, it is shown that ride quality of the vehicle with active suspension has been effectively improved in comparison with the vehicle of passive suspension by the methodology of AHP applying to the selection of the weights.

1. Introduction

Active suspension plays a major role for improving ride comfort of ground vehicle, which depends on its control algorithm design to a great extent, due to its effective isolation from road surface unevenness. Therefore, many experts have done researching for suspension control algorithms, such as optimal control theory, fuzzy control and neural net technology, and robust control [1–3]. Among them, optimal control theory is widely used, in which the linear quadratic gaussian control can ascertain weighted matrix of state variable and control variable by giving the optimal performance indexes. So the designer is offered an approach associated with weights selection of overall consideration of various factors of suspension system, to improve ride quality of ground vehicle.

The approach of selecting weighted coefficients is conventionally designed by experience, and results in a time-consuming and inefficient job. In [4, 5], it does not provide a method with the selection of weighting coefficients. In [6], AHP applying to the selection of the weights is only used in quarter vehicle model with 2 degrees of freedom (DOF), which makes the limitations to design and analysis of vehicle performance. This paper makes use of AHP to optimize the weights of performance indexes in the design of active suspension LQG controller based on full vehicle model with 7 DOF. The simulation results illustrate that the body acceleration of active suspension has been effectively reduced, and ride quality of the vehicle has been improved by AHP applying to the selection of the weights.

2. Modeling of Active Suspension with 7 Degrees of Freedom

Building a dynamic model of vehicle suspension is the basis of system design and performance analysis. Quarter vehicle model with 2 DOF and half vehicle model with 4 DOF have limitations to design and analyze vehicle performance. In order to reflect vertical vibration of the vehicle, quarter vehicle model is often used to conceptual design of suspension and research of control theory. Half vehicle model can only reflect vertical and pitch vibration of the vehicle. Therefore, the completed vehicle model with 7 DOF in this paper is used to grasp comprehensive ride quality including body vertical jump, pitch and roll vibration of the vehicle. The model is shown in Figure 1.

Figure 1 

Model of completed vehicle with seven degrees freedom of suspension.

The completed vehicle suspension itself is shown to consist of four springs 𝑘₁,𝑘₂,𝑘₃,and𝑘₄, four dampers 𝑐₁,𝑐₂,𝑐₃,and𝑐₄ and four active forces actuator 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄. The unsprung mass 𝑚₁, 𝑚₂, 𝑚₃, and 𝑚₄ represent the equivalent mass due to the axle and tire. The vertical stiffness of the tires is represented by the springs 𝑘₅, 𝑘₆, 𝑘₇, and 𝑘₈. The variables 𝑧₁, 𝑧₂, 𝑧₃, and 𝑧₄ represent the vertical displacements from static equilibrium of the unsprung mass. The variables 𝑧₅, 𝑧₆, 𝑧₇, and 𝑧₈ represent the vertical displacements from static equilibrium of the sprung mass. The variables 𝑧₀₁, 𝑧₀₂, 𝑧₀₃, and 𝑧₀₄ represent the vertical displacements from static equilibrium of the road. 𝑧_𝑏 is vertical displacement of body mass. 𝑚_𝑏 is body mass. 𝜃 which encloses 𝑥-axis rotation is roll angle. 𝜙 which encloses 𝑦-axis rotation is pitch angle. The displacement from center of mass to front axle is 𝑎. The displacement from center of mass to rear axle is 𝑏. The displacement from center of mass to the center of left tire is 𝑙_𝑙. The displacement from center of mass to the center of right tire is 𝑙_𝑟.

The mathematical model is set up as follows:𝑚_𝑏𝑧̈_𝑏=𝑐₁(𝑧̇₁−𝑧̇₅)+𝑘₁(𝑧₁−𝑧₅)+𝑢₁+𝑐₂(𝑧̇₂−𝑧̇₆)+𝑘₂(𝑧₂−𝑧₆)+𝑢₂+𝑐₃(𝑧̇₃−𝑧̇₇)+𝑘₃(𝑧₃−𝑧₇)+𝑢₃+𝑐₄(𝑧̇₄−𝑧̇₈)+𝑘₄(𝑧₄−𝑧₈)+𝑢₄,𝐼_𝑃𝜙̈=[𝑐₃(𝑧̇₃−𝑧̇₇)+𝑘₃(𝑧₃−𝑧₇)+𝑢₃+𝑐₄(𝑧̇₄−𝑧̇₈)+𝑘₄(𝑧₄−𝑧₈)+𝑢₄]𝑏−[𝑐₁(𝑧̇₁−𝑧̇₅)+𝑘₁(𝑧₁−𝑧₅)+𝑢₁+𝑐₂(𝑧̇₂−𝑧̇₆)+𝑘₂(𝑧₂−𝑧₆)+𝑢₂𝐼]𝑎,_𝑟𝜃̈=[𝑐₂(𝑧̇₂−𝑧̇₆)+𝑘₂(𝑧₂−𝑧₆)+𝑢₂+𝑐₄(𝑧̇₄−𝑧̇₈)+𝑘₄(𝑧₄−𝑧₈)+𝑢₄]𝑙_𝑟−[𝑐₁(𝑧̇₁−𝑧̇₅)+𝑘₁(𝑧₁−𝑧₅)+𝑢₁+𝑐₃(𝑧̇₃−𝑧̇₇)+𝑘₃(𝑧₃−𝑧₇)+𝑢₃]𝑙_𝑙,𝑚₁𝑧̈₁=𝑘₅(𝑧₀₁−𝑧₁)+𝑘₁(𝑧₅−𝑧₁)+𝑐₁(𝑧̇₅−𝑧̇₁)−𝑢₁,𝑚₂𝑧̈₂=𝑘₆(𝑧₀₂−𝑧₂)+𝑘₂(𝑧₆−𝑧₂)+𝑐₂(𝑧̇₆−𝑧̇₂)−𝑢₂,𝑚₃𝑧̈₃=𝑘₇(𝑧₀₃−𝑧₃)+𝑘₃(𝑧₇−𝑧₃)+𝑐₃(𝑧̇₇−𝑧̇₃)−𝑢₃,𝑚₄𝑧̈₄=𝑘₈(𝑧₀₄−𝑧₄)+𝑘₄(𝑧₈−𝑧₄)+𝑐₄(𝑧̇₈−𝑧̇₄)−𝑢₄. (2.1)

When 𝜃, 𝜙 is very small, the displacement equations of four end point with sprung mass are𝑧₅=𝑧_𝑏+𝑙_𝑙𝑧𝜃−𝑎𝜙,₆=𝑧_𝑏−𝑙_𝑟𝑧𝜃−𝑎𝜙,₇=𝑧_𝑏+𝑙_𝑙𝑧𝜃+𝑏𝜙,₈=𝑧_𝑏−𝑙_𝑟𝜃+𝑏𝜙. (2.2)

The road input model is the basis of automobile dynamic response and control. In general, the mathematical models of the road disturbances in time domain include random road disturbances model and discrete event road disturbances model. This paper will use them as road excitation input. There are four road inputs corresponding to left front tire, right front tire, left rear tire and right rear tire in this paper. So tread correlated of left and right tires and wheelbase lag of front and rear tires are considered. Based on these, random road input model [7] is built.

Firstly, the tread correlated equation of left and right tires is set up as follows:𝜂̇_𝑡1𝜂̇_𝑡2=𝐴_𝜂1𝜂_𝑡1𝜂_𝑡2+𝐵_𝜂1𝑊₁,𝑊₂=𝐶_𝜂1𝜂_𝑡1𝜂_𝑡2+𝐷_𝜂1𝑊₁. (2.3)

Secondly, the wheelbase lag equation of front and rear tires is set up and second-order Pade approximation algorithm is used to express time delay. So the model is𝜂̇_𝑡3𝜂̇_𝑡4=−6𝑡−12𝑡²𝜂10𝜂^𝑡3_𝑡4+12𝑡²0𝑊₁,𝑊₃=𝜂−𝑡0^𝑡3𝜂_𝑡4+[0]𝑊₁. (2.4)

Lastly, a filtered white noise model of the four wheels random road inputs is𝑧̇₀=𝐹_𝑤𝑧₀+𝐵_0𝜂𝜂_𝑡+𝐵₀𝜂̇𝑊,_𝑡=𝐴_𝜂𝜂_𝑡+𝐵_𝜂𝑊, (2.5)𝑧=[𝑧₀₁𝑧₀₂𝑧₀₃𝑧₀₄]^T, 𝑧₀₁, 𝑧₀₂, 𝑧₀₃, 𝑧₀₄ represent road inputs of left front, right front, left rear, right rear tire, respectively. 𝜂_𝑡 is the state transition vector, 𝜂_𝑡=[𝜂_𝑡1𝜂_𝑡2𝜂_𝑡3𝜂_𝑡4]^T. 𝑊 is the white noise, 𝑊=[𝑊₁𝑊₂𝑊₃𝑊₄]^T. 𝑡 is time delay.

Besides random road profile, the model [8] of discrete event road disturbances is also set up as follows: −ℎ𝑧=2cos[2𝜋𝑣𝑡𝑥_𝑙ℎ]+2𝑥,0≤𝑡≤^𝑙𝑣,𝑥0,𝑡>^𝑙𝑣. (2.6) The length of arc road is 𝑥_𝑙. The height from road surface to convex closure is ℎ. 𝑣 is velocity of vehicle running.

3. The Optimal Controller Design of Active Suspension

3.1. The State Equation of the System

For the completed vehicle model, the roll, pitch, jump vibration, suspension and tire deflection are very important performance items and should be chosen as output variables. So the state variables 𝑋^T, and output variables 𝑌^T are shown as below: 𝑋^T=𝑧_𝑏𝑧̇_𝑏𝜃𝜃̇𝜙𝜙̇𝑧₁𝑧̇₁𝑧₂𝑧̇₂𝑧₃𝑧̇₃𝑧₄𝑧̇₄,𝑌^T=𝑧̈_𝑏𝜃̈𝜙̈𝑧₅−𝑧₁𝑧₆−𝑧₂𝑧₇−𝑧₃𝑧₈−𝑧₄𝑧₁−𝑧₀₁𝑧₂−𝑧₀₂𝑧₃−𝑧₀₃𝑧₄−𝑧₀₄. (3.1)

The state space model of the completed vehicle system can be written as𝑋̇=𝐴𝑋+𝐵𝑢+𝐹z,𝑌=𝐶𝑋+𝐷𝑢. (3.2)

In the above description, the state vector is denoted as 𝑋. 𝑢 is the control vector. 𝑢=[𝑢₁𝑢₂𝑢₃𝑢₄]^T. A is the system matrix. 𝐵 is control matrix. 𝐹 is disturbance matrix. 𝐶 is the output matrix. 𝐷 is transfer matrix.

3.2. The LQG Controller Design

The aim of optimal control of active suspension is to improve ride comfort and handling stability. The body acceleration, roll angle acceleration, pitch angle acceleration and tire deflection, therefore, should be depressed as much as possible. To avoid suspension bottoming, suspension working space is also limited. Considering the completed model is complex and has many trade-off performance items, the performance function is defined as follows:1𝐽=2^∞₀[𝑞₁𝑧̈²_𝑏+𝑞₂𝜃̈²+𝑞₃𝜙̈²+𝑞₄(𝑧₅−𝑧₁)²+𝑞₅(𝑧₆−𝑧₂)²+𝑞₆(𝑧₇−𝑧₃)²+𝑞₇(𝑧₈−𝑧₄)²+𝑞₈(𝑧₁−𝑧₀₁)²+𝑞₉(𝑧₂−𝑧₀₂)²+𝑞₁₀(𝑧₃−𝑧₀₃)²+𝑞₁₁(𝑧₄−𝑧₀₄)²+𝑟₁𝑢²₁+𝑟₂𝑢²₂+𝑟₃𝑢²₃+𝑟₄𝑢²₄=1]𝑑𝑡2^∞₀(𝑌^T𝑄𝑌+𝑢^T𝑅𝑢)𝑑𝑡, (3.3)𝑞₁ is the weighted coefficient of body vertical acceleration. 𝑞₂ is the weighted coefficient of roll angle acceleration. 𝑞₃ is the weighted coefficient of pitch angle acceleration. 𝑞₄, 𝑞₅, 𝑞₆, and 𝑞₇ are the weighted coefficients of suspension deflection. 𝑞₈, 𝑞₉, 𝑞₁₀, and 𝑞₁₁ are the weighted coefficients of tire deflection. 𝑟₁, 𝑟₂, 𝑟₃, and 𝑟₄ are the weighted coefficients of control vector. 𝑄 is the weight matrix of the state. 𝑅 is the weight matrix of the control.

The formula (3.2) is substituted into formula (3.3), and the performance function is then written as1𝐽=2^∞₀[𝑋^T𝑄𝑋+2𝑋^T𝑁𝑢+𝑢^T𝑅𝑢]𝑑𝑡. (3.4)

There are 𝑄=𝐶^T𝑄𝐶, 𝑁=𝐶^T𝑄𝐷, and 𝑅=𝑅+𝐷^T𝑄𝐷 in formula (3.4).

The solution to the optimal control problem that minimizes this given performance index is a state feedback law 𝑢=−𝐾𝑋 where the feedback gain 𝐾 is determined by solving the following Riccati equation:𝐴𝑃+𝑃𝐴^T+𝑄−PBR⁻¹𝐵^T𝑃=0,𝐾=𝑅⁻¹(𝑁^T+𝐵^T𝑃). (3.5) So the state equation of closed-loop is𝑋̇=(𝐴−𝐵𝐾)𝑋+𝐹z. (3.6)

3.3. The Selection of the Weights Based on AHP

AHP is a multicriteria decision aiding method based on a solid axiomatic foundation. It is a systematic procedure for dealing with complex decision making problems in which many competing alternatives exist. The procedure can be described as the following three steps [9].

(1) Making Pairwise Comparison
The evaluation matrices 𝐻=(ℎ_𝑖𝑗)_𝑛×𝑛 are built up through pairwise comparing each decision factor under the topmost goal 𝐻=(ℎ_𝑖𝑗)_𝑛×𝑛=1ℎ₁₂⋯⋯ℎ_1𝑛1ℎ₁₂1ℎ₂₃⋯ℎ_2𝑛⋯1ℎ₂₃1⋯⋯⋯⋮⋮⋮ℎ_1𝑛1ℎ_2𝑛⋯⋯1. (3.7)𝑛 is the number of factors.
The fundamental 1 to 9 scale can be used to rank the judgments as shown in Table 1.
According to 𝐻 matrices, the subject weights [6] are calculated as follows(1)calculating 𝐻 multiplying vector of every row 𝑀=𝑀₁𝑀₂⋯𝑀_𝑛^T,𝑀_𝑖=^𝑛_𝑗=1ℎ_𝑖𝑗,(𝑖,𝑗=1,2,…,𝑛), (3.8)(2)calculating √^𝑛𝑀𝐸=𝐸₁𝐸₂⋯𝐸_𝑛^T,𝐸_𝑖=^𝑛𝑀_𝑖,(𝑖=1,2,…,𝑛), (3.9)(3)calculating positive vector of 𝐸𝐸=𝐸∑^𝑛_𝑖=1𝐸_𝑖(𝑖=1,2,…,𝑛), (3.10)𝐸 is the subjective weighted coefficient of each evaluating index.

Table 1 

A fundamental scale of 1 to 9.

(2) Calculating Maximum Eigenvalue of Matrix 𝐻
𝜆_max=^𝑛_𝑖=1(𝐻𝐸)_𝑖𝑛𝐸_𝑖(𝑖=1,2,…,𝑛). (3.11)

(3) Checking for Consistency
If every element in matrix 𝐻 satisfies the equation ℎ_𝑖𝑗=1/ℎ_𝑗𝑖 and ℎ_𝑖𝑗=ℎ_𝑖𝑘⋅ℎ_𝑘𝑗, the matrix 𝐻 is the consistency matrix. The evaluation matrices are often not perfectly consistent due to people's random judgments. These judgment errors can be detected by a consistency ratio CR𝜆CR=^max−𝑛RI(𝑛−1) (3.12)RI is random index. When 𝑛 is 11, RI equals 1.51. The value CR calculated is less than 1, the result is right. If CR is more than 1, the evaluation matrix needs to be revised on the basis of [10].

The formula fourteen has many weights which are calculated by AHP.

Firstly, according to the performance data of a passive antitype suspension under the given working condition, objective weights of evaluating indexes can be determined. Passive suspension to the specific demand for ride comfort under a given operation condition has 𝜎. The objective weight of body acceleration is supposed to 1. According to [6], the equation of calculating other objective weights is𝜎²_𝑎×1=𝜎²×𝛽. (3.13)

Secondly, it needs to calculate subjective weighted proportion coefficient of evaluating indexes. The subjective weighted proportion coefficient of body vertical acceleration is supposed to be 1. The subjective weighted proportion coefficients of suspension and tire deflection are𝐸₁=𝐸_𝑖𝑟_𝑖(𝑖=1,2,…,𝑛). (3.14)

Lastly, the general weights of evaluating indexes of related to ride comfort can be obtained based on the objective weights and subjective weighted proportion coefficients𝑞=𝛽×𝛾. (3.15)

The weights of the active suspension are derived as follows:𝑞₁=1,𝑞₂≈1,𝑞₃≈1,𝑞₄=1567,𝑞₅𝑞=1538₆=764,𝑞₇=305,𝑞₈=4279,𝑞₉=22821,𝑞₁₀=2867,𝑞₁₁=2423. (3.16)

4. Simulation Results and Analysis

The parameters [11] of a car are used in this paper as shown in Table 2. The velocity of the car is 20 m/s. The simulation time is 20 seconds in all. The road profile is a mixed random road excitation (smooth and rough) with the road roughness 𝐺₀=5×10⁻⁶ at the first 10 seconds, and 16×10⁻⁶ at the rest. The system simulation model of active suspension and passive suspension is built and simulated as in Table 2.

Table 2 

Parameter of a car.

4.1. Root Mean Square (RMS) of the Active Suspension and Passive Suspension

From Table 3, it is shown that the active suspension with LQG controller reduces body acceleration and roll angle acceleration remarkably on two grade roads in contrast with the passive one. The RMS value of body acceleration of the active suspension on both grade roads reduces 13% and 8%, respectively. The RMS value of roll angle acceleration in two grade roads reduces 14% and 15%, respectively. But the pitch angle acceleration is not noticeable to depress. The simulation results also show that the RMS value of right front axle of the active suspension working space reduces 23% and 12%, respectively, on the two grade roads. But the RMS value of left rear axle working space reduces 10% and 2%, respectively. It is obviously that control effect for left rear axle working space is even worse than right front axle working space. It can be seen that left rear wheel dynamic deflection of the passive suspension is better than that of the active suspension in Table 3, which shows that handling stability and ride comfort are incompatible to some extent.

Table 3 

Comparison of RMS for the active and passive suspensions.

4.2. Time Response of the Two Different Systems in Mixed Road Profile

From Figures 2 and 3, it can be seen that the active suspension with LQG controller remarkably reduces body acceleration and roll angle acceleration on both grade roads in contrast with the passive one. For Figures 4 and 5, it can be seen that the axles working space are controlled at the scope of the design requirements (±100 mm). According to Gaussian characteristics of road profile distribution, the RMS of the largest suspension deflection is 33 mm. It is less than theoretical value and the design is reasonable. Compared to right front axle working space, the control effect of left rear axle working space is relatively worse. In addition, wheel dynamic deflection related to handling stability impacts the effect of tire and road adhesion. For Figures 6 and 7, the control has no effect on the left rear wheel dynamic deflection on two grade roads with comparison to the passive suspension, which results from the interaction and coupling of the indexes.

Figure 2 

Body acceleration.

Figure 3 

Roll angle acceleration.

Figure 4 

Right front axle working space.

Figure 5 

Left rear axle working space.

Figure 6 

Right front wheel dynamic deflection.

Figure 7 

Left rear wheel dynamic deflection.

4.3. Time Response of the Two Different Systems in Random Road Excitation with Variations of Suspension Stiffness and Damping Parameters

Figures 8 and 9 are right front axle working space with suspension stiffness parameters decreasing 10 percent and increasing 10 percent, respectively. Figures 10 and 11 are right front axle working space with suspension damping coefficients decreasing 10 percent and increasing 10 percent, respectively. From Figures 8–11, it can be seen that active suspension control has shown a good effect at mixed road excitation in comparison with the passive suspension. It illustrates that LQG controller suspension system based on AHP has a good stability of performance when suspension stiffness and damping coefficients vary.

Figure 8 

Right front axle working space (stiffness-18000).

Figure 9 

Right front axle working space (stiffness-22000).

Figure 10 

Right front axle working space (damping-900).

Figure 11 

Right front axle working space (damping-1100).

4.4. Active Suspension Control Force at Mixed Random Road Excitation

It can be seen that, from Figures 12 and 13, suspension control force is within the control of actuator(±1000 N). Under the effect of suspension control force, active suspension can effectively reduce to vibration transmission from uneven roads to the body of the vehicle. Therefore, active suspension can availably depress values of body acceleration, roll angle acceleration and so on in contrast with passive suspension. So the vehicle of active suspension can effectively improve ride comfort.

Figure 12 

Right front suspension control force.

Figure 13 

Left rear suspension control force.

4.5. Power Spectral Density of the Two Different Systems in Conditions of the Same Road Roughness Coefficient (16×10⁻⁶)

From Figures 14, 15, and 16, it can be illustrated that vibration-isolation effect of active suspension is satisfactory at low frequency related to ride comfort of ground vehicle to a large extent, including body bounce vibration, roll and pitch mode and human sensitivity band. Meanwhile, the amplitudes near the tire resonance frequency associated with handling and stability of ground vehicle have no noticeable changes.

Figure 14 

Power spectral density of BA.

Figure 15 

Power spectral density of RA.

Figure 16 

Power spectral density of PA.

4.6. Time Response of the Two Different Systems in Discrete Event Road Excitation

It can be seen that, from Figures 17, 18, 19, and 20, the body acceleration, pitch angle acceleration, right front axle working space and left rear wheel dynamic deflection of the active suspension have improved noticeably, the amplitudes of performance indexes become small and the oscillation time are very short in contrast with the passive suspension.

Figure 17 

Body acceleration.

Figure 18 

Pitch angle acceleration.

Figure 19 

Right front axle working space.

Figure 20 

Left rear wheel dynamic deflection.

5. Conclusion

This paper applies the optimal control theory to design a Linear Quadratic Gaussian controller for the active suspension. Especially, the methodology of AHP is used to make the selection of weighted coefficients of performance indexes, which can reduce ineffective job in contrast with experience method. From the simulation results, it can be seen that ride quality of the vehicle with active suspension has been effectively improved in comparison with the vehicle of passive suspension. At the same time, it can also illustrate that active suspension control system based on AHP to select weighted coefficients of the LQG controller has a good stability when suspension damping coefficients and stiffness parameters vary. Furthermore, the future work will apply AHP to investigate stability of the integrated control system that consists of an active control of vehicle suspension and a steering system in cornering on curved by means of the optimal control theory.

Acknowledgments

The authors are grateful for the support by the Shanghai Municipal Natural Science Foundation (Grant no. 10ZR1421000) and Innovation Program of Shanghai Municipal Education Commission under Grant no. 08YZ99. This paper is also supported by Shanghai Leading Academic Discipline Project of Shanghai Municipal Education Commission, project number: J50503.