# Design and Performance Assessment of Adaptive Harmonic Control for a Half-Car Active Suspension System

Authors: Maria-Geanina Unguritu; Teodor-Constantin Nichițelea; Dan Selișteanu
Journal: Complexity
 (2022)

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## Abstract

The vehicle suspension system is represented by a complex group of components which connect the wheels to the frame or body. Its primary function is to reduce or absorb various vehicle vibrations generated by road disturbances, providing comfort and safety for passengers. Most modern vehicles have independent, active, or semiactive front and rear suspensions which allow the use of electronic actuation. For this reason, automotive engineers conduct research on the active suspension model to determine the most suitable control algorithm. Three active suspension models are intensely used within simulations: the quarter-car, the half-car, and the full-car models. This paper proposes an adaptive harmonic control for a half-car active suspension system. The mathematical model of the suspension system is implemented in the MATLAB/Simulink environment. The control approach is tested via simulation, and several comparisons with the classical proportional-integral controller are provided. The simulation results show that the controller behaves quite similarly on the half-car model as it did on a quarter-car model. Additionally, an improvement to the harmonic control algorithm has been accomplished.


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## Body

## 1. Introduction

Automotive vehicles are comprised of numerous parts and components which are classified into systems based on the function they perform. There are many complex systems incorporated in a modern vehicle, but the major ones are the engine, the fuel system, the ignition system, the electrical system, the exhaust system, the drive train, the suspension system, the steering system, the brake system, and the frame and body. These systems have been consolidated and improved throughout the vehicle’s evolution and history.The primary function of the suspension system is to optimize the overall performance of the vehicle while it travels, absorbing the shock caused by the road disturbances and providing a safe and comfortable ride. Vehicle suspensions are categorized into dependent, independent, and semi-independent suspensions based on their construction and as passive, active, and semiactive suspensions based on their control method. Most modern vehicles have independent, active, or semiactive front and rear suspensions.The interest of automotive engineers in the active suspension control raised the need for the mathematical model of the suspension system used within simulations to design, compare, and optimize control solutions. For this reason, three models are actively used in this research field: the quarter-car, the half-car, and the full-car models. Based on a comparison study [1], the quarter-car model does not give results close to the full-car model in nearly all the cases. However, when the driving condition is changed to a road with a bump, the quarter-car model gives a better approximation to the full-car model for three out of the eight components of accelerations of different body parts. The half-car model provides results closer to the full-car model than the quarter-car model. Though this advantage of the half-car model is more noticeable while considering the car on a random irregular profiled road rather than on a road with sudden bumps.Many works use a half-car model of an active suspension system to design, compare, and optimize various control algorithms, e.g., an analysis of the half-car active suspension model using a proportional-integral-derivative (PID), linear quadratic regulator, fuzzy and adaptive neuro-fuzzy inference system [2], a linear quadratic regulator optimal control, and PID classic control on a half-car active suspension system [3], a bioinspired dynamics-based adaptive fuzzy method for half-car active suspension systems with input dead zones and saturations [4], an adaptive controller for the half-car active suspension systems with partial performance constraints [5], a H-infinity fault-tolerant control applied to an active half-car suspension systems with actuators failure accounts [6], a hybrid fault-tolerant controller that consists of a nominal state-feedback controller and a robust H-infinity observer applied to a half-vehicle active suspension under different running conditions [7], a black-box compatible simulation-based approach for solving nonlinear model predictive control (MPC) problem via a parameterized technique to control the vertical dynamics of a half-car vehicle equipped with the semiactive suspension system [8], an adaptive dynamic surface control strategy for a half-car active suspension systems [9], an analysis of adaptive finite-time fault-tolerant control with output constraints for a class of uncertain nonlinear half-car active suspension systems [10], and a direct adaptive neural network controller for a four degree of freedom nonlinear half-car suspension system [11].Some research directions focus on the improvement of the active suspension control using a half-car model when the operating conditions change. This topic was mostly studied considering a PID controller capable of dynamic tuning of its parameters using fuzzy logic [12], machine learning algorithms [13], or specific criteria [14, 15].In previous research [16], an adaptive harmonic controller was designed and applied on a quarter-car active suspension system, evaluating its response alongside a proportional-integral controller, a H-infinity controller, and an MPC. The active suspensions are complex systems which require distributed control [10, 17, 18]. This paper is based on the adaptive harmonic controller from [16] and continues its research by evaluating and comparing the responses on a half-car model of the active suspension system controlled by hydraulic actuators. The model reactions are compared with a proportional-integral (PI) controller and with the harmonic controller. The PI controller has been preferred because it is a well-known control solution which demonstrated good results in the previous research. One benefit of the harmonic solution is that it considers only one feedback signal represented by the body acceleration. Compared with the previous research [16], two main contributions of this paper are achieved. First, the proposed harmonic control solution is applied on a more complex half-car active suspension system. Second, an improvement of the control solution with a variable amplification factor is proposed over the previous research for the harmonic controller algorithm. Also, a specific disturbance delayed profile was used for the half-car suspension system.The advantages of the proposed harmonic control approach are the following: it is a simpler design procedure compared with other adaptive control systems since it does not require a high level of mathematical understanding; it uses only one feedback signal which leads to costs and weight reduction due to the small number of sensors needed in the control loop; it is not a hardware-dependent control method; and it can be easily implemented and adjusted on a real-time embedded system which is capable of harmonic signal generation.The structure of the paper is as follows. Section2 introduces the model of the half-car active suspension complex system and the hydraulic actuator model used within the design and the simulation. Section 3 evaluates the harmonic control solution by comparing it with a PI control algorithm. Section 4 presents the conclusions of the paper.


## 2. Half-Car Active Suspension Mathematical Model

This paper uses the half-car active suspension model applied in several similar works such as [2]. The following equations of motion in form (1) represent the vertical motion at vehicle body, the pitch motion at vehicle body, the vertical motion at unsprung mass of vehicle front, and the vertical motion at unsprung mass of vehicle rear. These equations describe the model shown in Figure 1:(1)mbx¨b=ksfxwf−ksfxbf+bsfx˙wf−bsfx˙bf+103fsf+ksrxwr−ksrxbr+bsrx˙wr−bsrx˙br+103fsr,Ibθ¨b=Lfksfxwf−ksfxbf+bsfx˙wf−bsfx˙bf+103fsf−Lrksrxwr−ksrxbr+bsrx˙wr−bsrx˙br+103fsr,mwfx¨wf=ktfrf−ktfxwf+ksfxbf−ksfxwf+bsfx˙bf−bsfx˙wf−103fsf,mwrx¨wr=ktrrr−ktrxwr+ksrxbr−ksrxwr+bsrx˙br−bsrx˙wr−103fsr..Figure 1
Half-car model of the active suspension.The equations in form (1) are rewritten to obtain form as follows:(2)x¨b=−1mbksfxbf−xwf+bsfx˙bf−x˙wf−103fsf+ksrxbr−xwr+bsrx˙br−x˙wr−103fsr,θ¨b=−LfIbksfxbf−xwf+bsfx˙bf−x˙wf−103fsf+LrIbksrxbr−xwr+bsrx˙br−x˙wr−103fsr,x¨wf=1mwfksfxbf−xwf+bsfx˙bf−x˙wf−ktfxwf−rf−103fsf,x¨wr=1mwrksrxbr−xwr+bsrx˙br−x˙wr−ktrxwr−rr−103fsr..The constraints in relations (3) are introduced in form (2), resulting the equations seen in (4):(3)xb=LfXbr+LrxbfL,θb=xbf−xbrL,.(4)Lfx¨br+Lrx¨bf=−Lmbksfxbf−xwf+bsfx˙bf−x˙wf−103fsf+ksrxbr−xwr+bsrx˙br−x˙wr−103fsr,x¨bf−x¨br=−LfLIbksfxbf−xwf+bsfx˙bf−x˙wf−103fsf+LrLIbksrxbr−xwr+bsrx˙br−x˙wr−103fsr,x¨wf=1mwfksfxbf−xwf+bsfx˙bf−x˙wf−ktfxwf−rf−103fsf,x¨wr=1mwrksrxbr−xwr+bsrx˙br−x˙wr−ktrxwr−rr−103fsr..x¨bf is determined from the second relation and substituted in the first relation of form (4), obtaining x¨br. The resulting expression of x¨br is substituted in the second relation of form (4), obtaining x¨bf. The final form of the equations is given as follows:(5)x¨br=LfLrIb−1mbksfxbf−xwf+bsfx˙bf−x˙wf−103fsf−Lr2Ib+1mbksrxbr−xwr+bsrx˙br−x˙wr−103fsr,x¨bf=−Lf2Ib+1mbksfxbf−xwf+bsfx˙bf−x˙wf−103fsf+LfLrIb−1mbksrxbr−xwr+bsrx˙br−x˙wr−103fsr,x¨wf=1mwfksfxbf−xwf+bsfx˙bf−x˙wf−ktfxwf−rf−103fsf,x¨wr=1mwrksrxbr−xwr+bsrx˙br−x˙wr−ktrxwr−rr−103fsr..The following state-space variables notations are used, resulting the state-space equations given in form (7).(6)x1=xbf;x2=x˙bf;x3=xwf;x4=x˙wf;x5=xbr;x6=x˙br;x7=xwr;x8=x˙wr,(7)x˙1=x2,x˙2=−Lf2Ib+1mbksfx1−x3+bsfx2−x4−103fsf+LfLrIb−1mbksrx5−x7+bsrx6−x8−103fsr,x˙3=x4,x˙4=1mwfksfx1−x3+bsfx2−x4−ktfx3−rf−103fsf,x˙5=x6,x˙6=LfLrIb−1mbksfx1−x3+bsfx2−x4−103fsf−Lr2Ib+1mbksrx5−x7+bsrx6−x8−103fsr,x˙7=x8,x˙8=1mwrksrx5−x7+bsrx6−x8−ktrx7−rr−103fsr..Table1 provides a summary of the input-state-output variables correlated with the active suspension model. Within this table, sdfm=xbf−xwf=x1−x3 and sdrm=xbr−xwr=x5−x7 are the front and rear suspension deflection which describe the road handling, and abfm/s2=x¨bf=x˙2 and abrm/s2=x¨br=x˙6 are the front and rear body acceleration which indicate the passenger comfort.Table 1
Summary of the Input-State-Output variables for the half-car active suspension.
InputStateOutputu1=rfx1=xbfy1=xbfu2=fsfx2=x˙bfy2=sdfu3=rrx3=xwfy3=abfu4=fsrx4=x˙wfy4=xbrx5=xbry5=sdrx6=x˙bry6=abrx7=xwrx8=x˙wrThe state-space equations in form (7) together with the chosen input-state-output variables defined in Table 1 describe the state-space representation of the system given in (8) and (9).(8)A=A11A12A21A22,A11=0100−ksfmb−ksfLf2Ib−bsfmb−bsfLf2Ibksfmb+ksfLf2Ibbsfmb+bsfLf2Ib0001ksfmwfbsfmwf−ksfmwf−ktfmwf−bsfmwf,A12=0000−ksrmb+ksrLfLrIb−bsrmb+bsrLfLrIbksrmb−ksrLfLrIbbsrmb−bsrLfLrIb00000000,A21=0000−ksfmb+ksfLfLrIb−bsfmb+bsfLfLrIbksfmb−ksfLfLrIbbsfmb−bsfLfLrIb00000000,A22=0100−ksrmb−ksrLr2Ib−bsrmb−bsrLr2Ibksrmb+ksrLr2Ibbsrmb+bsrLr2Ib0001ksrmwrbsrmwr−ksrmwr−ktrmwr−bsrmwr,B=00000103mb+103Lf2Ib0103mb−103LfLrIb0000ktfmwf−103mwf0000000103mb−103LfLrIb0103mb+103Lr2Ib000000ktrmwr−103mwr.(9)C=C11C12C21C22,C11=100010−10−ksfmb−ksfLf2Ib−bsfmb−bsfLf2Ibksfmb+ksfLf2Ibbsfmb+bsfLf2Ib,C12=00000000−ksrmb+ksrLfLrIb−bsrmb+bsrLfLrIbksrmb−ksrLfLrIbbsrmb−bsrLfLrIb,C21=00000000−ksfmb+ksfLfLrIb−bsfmb+bsfLfLrIbksfmb−ksfLfLrIbbsfmb−bsfLfLrIb,C22=100010−10−ksrmb−ksrLr2Ib−bsrmb−bsrLr2Ibksrmb+ksrLr2Ibbsrmb+bsrLr2Ib,D=000000000103mb+103Lf2Ib0103mb−103LfLrIb000000000103mb−103LfLrIb0103mb+103Lr2Ib.The hydraulic control of the active suspensions is performed with actuators modelled through the first-order transfer function as can be seen in (10). The controller computes the input signals for the actuators which generate the front and rear output forces fsf and fsr.(10)Hacts=11/60s+1.The road disturbance influencing the active suspensions was modelled as a road bump of 5 cm height through relation (11). The front and rear suspension is affected by the road perturbation with a specific time delay computed using relation (12), where v0 is the initial velocity of the vehicle.(11)rft=rrt=0.025⋅1−cos8πt.(12)Time Delay=Lf+Lrv0.This paper evaluates the closed-loop response of the half-car active suspension model with the adaptive harmonic control solution and the conventional PID controller which have the measurement of the front and rear body accelerationabf and abr as feedback.


## 3. Assessment of Control Algorithms and Discussion

Figure2 shows the Simulink block diagram designed to compare the adaptive harmonic controller to the conventional PID controller. The diagram employs the half-car suspension model (8) and (9) and the hydraulic actuator model (10). The parameter values (13) which were estimated based on the values used in the previous research [16] characterize the simulated suspension model. The state-space block of the suspension has the parameter initial conditions set to 0.(13)mb=800kg,Ib=1600kgm2,mwf=mwr=60kg,bsf=bsr=1000Ns/m,ksf=ksr=16000N/m,ktf=ktr=190000N/m,Lf=1m,Lr=1.5m,v0=10m/s.Figure 2
Simulink block diagram with the half-car active suspension model.In order to generate the road disturbance input, a signal builder source block is used, which abides by equation (11) and was implemented using the code lines (14). Additionally, the time delay (12) of the road disturbance for the rear suspension is implemented using a variable time delay block.(14)t=0:0.0025:1;r=zerossizet;r1:101=0.025∗1−cos8∗pi∗t1:101;.For each suspension model, the front and rear forcesfsf and fsr are applied by the actuators.(1)
For the first suspension model, the actuators are decoupled to simulate the open-loop system response; therefore,fsf=fsr=0.(2)
For the second suspension model, the actuators are controlled by the adaptive harmonic controller output given in (15) and implemented using a MATLAB function block. This function has three input variables: the simulation time t and the front and rear body acceleration abf and abr. The implementation of the algorithm is similar as designed in the previous research [16], and it is presented in (16). The amplification factor Ka is adaptive based on the interval where ab is found. Additionally, compared with the previous work, Ka is also variable when ab is within the interval −k,k, with the selected k=0.4. Through this change, a better performance is gained regarding the profile of the utilized force by moderating it around small acceleration values. Figure 3 presents the detailed design process of the proposed harmonic controller which can be afterwards implemented and validated on a real-time embedded system.(15)Ka=10∗ab0.4,ab∈−k,k5,ab∉−k,kA=Ka∗−aby=A∗sin2∗π∗f∗t,ab∈−k,kA∗sin22∗π∗f∗t,ab∉−k,k..(16)functionyf,yr=Harmonict,abf,abrff=100∗absabf;fr=100∗absabr;ifabf<0.4&&abf>−0.4Ka=10∗absabf/0.4;A=Ka∗−abf;yf=A∗sin2∗pi∗ff∗t;elseKa=5;A=Ka∗−abf;yf=A∗sin2∗pi∗ff∗t^2;endifabr<0.4&&abr>−0.4Ka=10∗absabr0.4;A=Ka∗−abr;yr=A∗sin2∗pi∗fr∗t;elseKa=5;A=Ka∗−abr;yr=A∗sin2∗pi∗fr∗t^2;end.(3)
For the third suspension model, the actuators are controlled by a conventional PI control law implemented using a continuous-time PID controller block. Both PID blocks were configured as in the previous work [16] with a parallel form (17), with the proportional gain Kp=−0.8, the integral gain Ki=−3, and the integrator initial conditions set to 0.(17)HPIs=Kp+Ki1s.The simulation was executed for 3.5 seconds using the variable-step solver ode45 (Dormand-Prince).Figure 3
Detailed design process of the adaptive harmonic controller.Figure4 shows the road disturbance simulated as a road bump of 5 cm height which influences the front suspension with the rf signal, and after a time delay computed according to (12), it influences the rear suspension with the rr signal.Figure 4
Road disturbance affecting the front and rear suspension.Figures5 and 6 show the simulation results for the front body travel xbf, the front body acceleration abf, the front suspension deflection sdf, and the front actuator force fsf.Figure 5
Front body travelxbf and front body acceleration abf results: blue dash line means open-loop, red continuous line means harmonic, and green dash line means classic PI.Figure 6
Front suspension deflectionsdf and front actuator force fsf results: blue dash line means open-loop, red continuous line means harmonic, and green dash line means classic PI.Figures7 and 8 illustrate the simulation results for the rear body travel xbr, the rear body acceleration abr, the rear suspension deflection sdr, and the rear actuator force fsr.Figure 7
Rear body travelxbr and rear body acceleration abr results: blue dash line means open-loop, red continuous line means harmonic, and green dash line means classic PI.Figure 8
Rear suspension deflectionsdr and rear actuator force fsr results: blue dash line means open-loop, red continuous line means harmonic, green dash line means classic PI.In order to compare the two controllers and to have a similar overview as in the previous research [16], Table 2 contains some performance criteria represented by the maximum value and the minimum value but also the integral time-weighted absolute error (ITAE) for each signal. The ITAE was calculated with (18), and it was implemented using function (19) where t is the time and ε is the error. In Table 2, bold text represents the favourable values, and italic text represents the unfavourable ones.(18)ITAE=∫tεdt.(19)ITAE=trapzt,absε.∗t;.Table 2
Performance criteria of body travel, body acceleration, suspension deflection, and actuator force.
Performance criteriaOpen-loopHarmonicClassic PIFrontRearFrontRearFrontRearBody travelMax. value0.027770.028160.006460.006610.012960.01283Min. value−0.02044−0.01299−0.00308−0.00361−0.00284−0.00215ITAE0.03070.01950.00880.00660.00730.0078Body accelerationMax. value2.3223.0150.42510.68871.1211.261Min. value−1.622−2.527−0.4686−0.4834−0.5544−0.6477ITAE0.91030.94140.25080.23280.10200.1581Suspension deflectionMax. value0.027690.030900.013500.013230.015770.01702Min. value−0.04688−0.04431−0.05951−0.05835−0.05512−0.05497ITAE0.02870.01880.00900.00840.00770.0094Actuator forceMax. value000.43770.48970.30270.3804Min. value00−1.464−1.371−0.9230−1.030ITAE000.06960.09440.09070.1238Bold texts represent the favourable values. Italic texts represent the unfavourable values.From the performance criteria, it can be observed that better simulation results are obtained using the harmonic control approach for the body acceleration, which represents the passenger comfort, since it is the only feedback signal in the closed-loop system. For the suspension deflection which represents the road handling, the body travel influenced by the body acceleration, and the actuator force, the results are quite similar with both control solutions.


## 4. Conclusions

This paper continues the research of the adaptive harmonic controller which was designed for a quarter-car active suspension in previous work by evaluating and comparing its responses on the half-car active suspension. The mathematical model extended to the half-car active suspension is presented and targeted for simulation in MATLAB/Simulink. Compared with the previous work, the proposed controller was improved with a variable amplification factor in a certain interval of the body acceleration. The simulation results show that the harmonic controller which generates a harmonic control signal with variable amplitude and frequency based on the body acceleration feedback behaves quite similarly on the complex system of the half-car active suspension as it did on the quarter-car active suspension, proving the applicability of the algorithm. For future research, the proposed control algorithm can be implemented on an embedded system with a real-time operating system to obtain some experimental results which can validate the proposed solution. The implementation difficulties can be related to the software configuration for real-time performance and to the capability of the hardware to generate a harmonic signal. The applicability of the adaptive harmonic controller can be extended and evaluated on a full-car active suspension system. However, for the full-car model, the roll angle and the roll angular velocity signals should be considered.



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