# Quantized Feedback Control of Active Suspension Systems Based on Event Trigger

Authors: Jinwei Sun; Jingyu Cong; Weihua Zhao; Yonghui Zhang
Journal: Shock and Vibration
 (2021)

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## Abstract

As the deviation error will accumulate during the data acquisition and transcoding process of an active suspension system, this paper presents a sliding-mode-based quantized feedback control method. The aim of the controller is to improve the vertical performance of vehicles in the presence of external interferences. A 7-DOF suspension model with nonlinear springs and actuator dynamics is built for the control purpose. Firstly, a static quantizer on the uplink channel and a dynamic quantizer on the downlink channel are considered in the sliding mode controller to reduce the cumulative error and suppress the sprung mass motions. Secondly, an event trigger mechanism is introduced in the controller design process to reduce energy consumption and operation frequency of the actuator. The overall stability of the designed controller is proved by the Lyapunov functions. Finally, numerical simulations are carried out to evaluate the efficacy of the proposed controller. Different quantitative and trigger conditions are discussed, and the random road excitation is considered as the external disturbance input. The results of the control method indicate that the designed controller can improve the riding comfort with little loss of handling stability compared with the passive system. In addition, the trigger mechanism can improve the working efficiency of actuators effectively.


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## Body

## 1. Introduction

As the passive suspension system can hardly provide a better compromise between riding comfort and handling stability, the controllable suspension system has received extensive attention in recent years [1, 2], and the control technology can be classified into a semiactive and active suspension system [3]. Despite the high cost and power requirement, an active suspension system can provide desired control force and improve the vehicle’s dynamic performance significantly [2].Many active suspension control approaches such as optimal control [4], adaptive control [5], and linear quadratic regulator control have been proposed. An adaptive backstepping controller with hard constraints for the active suspension system is designed in [6] to stabilize the attitude of vehicle and improve ride comfort. A saturated adaptive robust controller to improve suspension performance is proposed in [7], while in [8], backstepping control to compensate actuator lag in the nonlinear full vehicle model is designed. As the accurate dynamic model of the suspension system is difficult to determine, many different control techniques have been proposed for active suspension systems. A fuzzy logic controller has been used to provide desired actuation force for the capacity of resisting external disturbances and parameter uncertainty [9]. An output feedback finite-time controller was designed in [10] for active suspension systems to compensate the unknown external disturbance and improve the suspension performance, and a bioinspired dynamics-based adaptive tracking active suspension controller was presented in [11] to cancel vibration energy transmitted by suspension inherent nonlinearity. Another well-known control method which can exert robustness on unmeasured disturbances and parameter uncertainties for nonlinear systems is the sliding mode control [12]. The traditional sliding mode scheme can hardly be implemented on complicated systems with unknown parameters and the system model must be known. Hence, many research studies focused on the combination of control algorithms for uncertain systems to achieve better performance [13]. An enhanced adaptive self-organizing fuzzy sliding mode controller for an active suspension system is proposed in [14], and a hybrid self-organizing fuzzy and radial basis-function neural-network controller is used to reduce the sprung acceleration in [15]. The model-free adaptive sliding controller proposed in [16] uses a functional approximation and online learning technique to improve the control performance.With respect to the existing results, most of conventional control algorithms tend to ignore the performance degradation caused by quantization errors, which may cause a certain deviation from the original data after decoding at the receiving end. Therefore, quantization error should be considered in the active suspension design process. In addition, for active suspension systems, a certain amount of energy will be generated during the working process of the controller. The time-triggered mechanism [17] can be utilized to save energy and reduce the actuator action frequency while extending the life of the actuator.This study proposes an event-trigger-based quantitative feedback active suspension system with quantization error, both the static quantizer and dynamic quantizer are utilized, and the sliding mode controller is also adopted. The objective of the proposed controller is to reduce the influence of accumulated sensor signal errors, enhance the actuator efficiency, and improve active suspension system properties. Compared with the recent active suspension control systems, the main contributions of this study are as follows:(1)
The connection channels between the suspension model and the controller are considered, and robust quantization control theory is designed for the active suspension system to reduce the signal transmission during data acquisition and transcoding(2)
The event trigger mechanism is introduced in the controller design process to reduce the operation frequency of the active suspension actuatorThe rest of the paper is organized as follows. The 7-DOF suspension model with actuator dynamics is presented in the upcoming section. The procedure for the design of the quantized feedback controller and event trigger mechanism is illustrated next. Simulation and discussion of the results are summarized later, and concluding remarks are given in the last section.


## 2. Vehicle Suspension Model

A full vehicle suspension model with heave, pitch, and roll motions of the sprung mass and four unsprung mass vertical motions is utilized to describe the dynamics of the vehicle., as shown in Figure1. The terms ms and mui represent the sprung and unsprung masses, respectively. θ and φ are the pitch and roll angle of the vehicle sprung mass. The linear damping and spring coefficients are denoted as ci and ki, respectively, and kti represents the tire stiffness. The external road disturbance is denoted as zri. The suspension deflection can be expressed as(1)zsui=zsi−zui,i=1,2,3,4.Figure 1
Active suspension model.Then, the spring and damping forces can be given by(2)Fsi=kizsui+knizsui3,i=1,2,3,4,Fdi=ciz˙sui,i=1,2,3,4,where kni is the nonlinear spring coefficient. zsi and zui are the sprung and unsprung mass displacements of each corner, respectively. Assuming the pitch and roll angle of the sprung mass are small, the following approximation can be obtained:(3)zs1=zs+asinθ+csinφ,zs2=zs+asinθ−dsinφ,zs3=zs−bsinθ+csinφ,zs4=zs−bsinθ−dsinφ,where a,b,c, and d are the distances of the suspensions to the center of the vehicle. Then, the vehicle suspension model with sprung and unsprung mass dynamics can be given as(4)msz¨s+∑Fsi+Fdi−Fai+Fati1+Fati2+Fati3+Fati4=0,Iyθ¨+aFs1+Fd1+Fs2+Fd2+Fati1+Fati2−bFs3+Fd3+Fs4+Fd4+Fati3+Fati4−Fθ=0,Ixφ¨−dFs2+Fd2+Fati2+Fs4+Fd4+Fati4+cFs1+Fd1+Fati1+Fs3+Fd3+Fati3−Fφ=0,muiz¨ui−Fsi−Fdi+Fti−Fai=0,i=1,2,3,4,where Ix and Iy are the roll and pitch moment inertia of the sprung mass; Fti=ktizui−zri is the dynamic tire load. The antiroll bar forces are(5)Fati1=kafc+dφ−zu2−zu1c+d,Fati2=−kafc+dφ−zu2−zu1c+d,Fati3=−krfc+dφ−zu3−zu4c+d,Fati4=krfc+dφ−zu3−zu4c+d,where kaf and krf are the antiroll bar stiffness.Active forceFai is generated by the actuator located between the sprung and unsprung masses. The control input of the vertical, pitch, and roll motions can be expressed as(6)Fz=Fa1+Fa2+Fa3+Fa4,Fφ=cFa1+Fa3−dFa2+Fa4,Fθ=aFa1+Fa2−bFa3+Fa4.The electrohydraulic actuator model can be represented as(7)F˙ai=−CdApFai−αAp2z˙sui+λUiPs−signUiFaiAp,where i=1,2,3,4, Cd is the discharge coefficient, Ap is the actuator piston area, Ps is the hydraulic supply pressure, α is a function of the bulk modulus of the fluid, w is the spool valve area gradient, λ=ApCdwα/ρ, ρ is the hydraulic fluid density, and Ui is the spool valve input that can be the controlled voltage or current.


## 3. Controller Design

A quantize-event-trigger-based sliding mode controller is designed in this section for the active suspension system, and the control diagram of the proposed controller is shown in Figure2.Figure 2
Diagram block of the proposed controller.


### 3.1. Robust Quantization Feedback Controller

Robust quantization control theory contains control problems based on the quantizer, in which the dynamic quantizer expands the quantization range by adjusting the quantization level to achieve asymptotic stability of the system. This section mainly focuses on the connection channels between the suspension model and the controller. Both static quantizer and dynamic quantizer are utilized, and a sliding mode controller is also adopted. The main feature of the dynamic quantizer is that it can increase the quantization range of the quantizer by adjusting the quantization level. It can also reduce the size of the limit cycle when the system is in a steady state and achieve asymptotic stability. Compared with the static quantizer, its control effect is better, but the structure is more complicated. For practical consideration, we utilize a static quantizer on the uplink channel and a dynamic quantizer on the downlink channel. The quantization system is shown in Figure3.Figure 3
Flow chart of the quantization system.Assuming the suspension system considered in this paper in bounded-state stable, the road profile and its derivate are totally bounded. Selecting the state variables asx1x2x3x4x5x6T=zsz˙sθθ˙φφ˙T, the suspension system model including heave, pitch, and roll motions can be expressed as(8)x˙1=x2,x˙2=fz+GzFz,fz=−1ms∑Fsi+Fdi,Gz=1ms,x˙3=x4,x˙4=fθ+GθFθ,fθ=−1IyaFs1+Fd1+Fs2+Fd2−bFs3+Fd3+Fs4+Fd4,Gθ=1Iy,x˙5=x6,x˙6=fφ+GφFφ,fφ=−1IxcFs1+Fd1+Fs3+Fd3−dFs2+Fd2+Fs4+Fd4,Gφ=1Ix.Then, the abovementioned functions can be written as [18](9)xt=A0x˙t+BjUforce+djt,where xt=xit;xi+1t; i=1,3,5, j=1,2,3, and Uforce=Fz;Fθ;Fφ represent the vertical, pitch, and roll force generated by the active suspension, respectively. A0=01;00; B1=0;Gz, B2=0;Gθ, B3=0;Gφ; and d1t=0;fz/Gz, d2t=0;fθ/Gθ, and d3t=0;fφ/Gφ.Assumption 1.
djt satisfies the inequality djt≤d1+d2xt+d3Uforce≤d4, where d1≥0,d2≥0,d4>0,0≤d3≤1, and d1,d2,d3,d4 are constants.
As shown in Figure3, a static quantizer qμsv is used in the uplink channel, where v is the control law designed by the quantized state, μs is a fixed number; the down channel utilized a dynamic quantizer qμx to transfer system quantized states, and μ is the quantization parameter adjusted statically. Defining a function that takes the nearest integer, the uniform quantizer with the quantization level parameter μ is(10)qμx=defμjroundxμj,μj>0;j=1,2,3.
The uniform quantization error is(11)eμ=defqμx−x.
Then,(12)eμ=defqμx−x≤Γμj,where Γ=p/2, p is the dimension of the vector x.
For the dynamic quantizer, a special quantization level parameterμj=0 is given when the system runs to the sliding surface.(13)qμx=def0,μj=0.
The linear sliding surface is defined as follows:(14)sxt=Cxt=0,where C is a given vector to ensure that the system has stable eigenvalues and CBj≠0. The object of sliding mode control is to force the system state trajectory to reach the sliding surface and maintain the sliding mode in the presence of uncertainties and disturbances.Lemma 1.
(see [19]). There is a fixed constant τ>0, and we assume μ>0 which satisfies(15)μ≤Cxτ+1CΓ.Then, the following equation can be obtained:(16)Ceμ≤CΓμ≤1τCqμx.By considering the uplink and down channel quantize, the following control law can be designed [19]:(17)v=−τ+1τ−χCBj−1CAqμxt+CAΓμj+CBjΓ1μs+CBjd1+d2qμxt+d2Γμj+d3Γ1μssgnCqμxt−τρτ−χCBj−1sgnCqμxt,(18)Uforce=qμsv,where χ=1+τ+1d3,τ>1+d3/1−d3,ρ>0. The control law can keep the system states reach and maintain on the sliding surface sxt=0.A simple and effective design of the adjustment law is as follows [20]:(19)μj=Cxtτ+1CΓ,Cxt≥1,μj=αλτ+1CΓ,0<Cxt<1,μj=0,Cxt=0,where α∈0,1,αλ<Cx<αλ−1.


### 3.2. Event Trigger Mechanism

In recent years, event triggers have been used to reduce computational costs and energy consumption, resulting in a wide range of applications in control systems [21, 22]. The event trigger mechanism is based on the physical quantity signal of the sensor; when the semaphore reaches a certain condition, sampling is performed. In this way, the average sampling period is longer than the fixed sampling period, so that the sampling frequency of the system can be effectively reduced. As the nest sample instant is dependent on the previous sampling information, the control law is held constant between sampling instants. Defining the event trigger instant as tk, the system maintains the control output at time tk in the time interval tk,tk+1. The discretization state error of the system is(20)et=xt−xtk.The sampling is not periodic; hence,tk+1−tk is not constant. For computational purposes, the deviation from the system state to the desired state is given by(21)eT=x−xd,where xd is the set reference. As the error variable is required to vanish or settle in close vicinity of zero, the reference state is set as xd=0.The triggering instanttk is completely characterized by a triggering rule, so a triggering rule is introduced [23].(22)κ=ζeT+ξe˙T2−ψm1+m2e−ςt,where ζ>0,ξ>0,ψ∈0,1,ς∈0,1,m1≥0,m2≥0,m1+m2>0. The term ζeT+ξe˙T2 represents the overall error; the system is triggered when the error is small, guaranteeing the control output. When κ≤0, the system is triggered at tk, and the control output is maintained until tk+1. The following relation determines the triggering instants in an iterative manner:(23)tk+1=inft∈tk,∞,κ>0.The event trigger time isTk=tk+1−tk. For Tk≤tk+1−tk, and lower bound on the event trigger time is ensured. Tk is bounded by [23](24)Tk≥1AlnA⋅e∞Aϖ+Bjqμsv+d4+1,where ϖ>0.Proof:.
Considering system (9),(25)x˙t=Axt+BjUforcet+Bjdjt.
At time intervaltk,tk+1, discretization state error rises from 0 to the final value at Tk instant, so(26)ddtet≤ddte¯t≤ddtxt≤Axt+BjUforcet+Bjdjt,by inserting the control law (17) into abovementioned equation.(27)ddtet≤Axt+Bjqμsv+Bjdjt≤Axt+Bjqμsv+Bjdjt≤Aet+xtk+Bjqμsv+Bjdjt≤Aet+Axtk+Bjqμsv+Bjdjt≤Aet+Aϖ+Bjqμsv+d4.
∀t∈tk,tk+1, with the initial condition et=0, and the solution of inequality (27) can be understood by utilizing the comparison lemma [24] and is given as(28)et≤Aϖ+Bjqμsv+d4AeAt−tk−1.
Similarly, for triggering time instanttk+1,(29)e∞=etk+1≤Aϖ+Bjqμsv+d4AeATk−1,and then, inequality (24) is satisfied.
Discretized state error after quantization is(30)eqt=qμxt−qμxtk.
Event is triggered at discrete moments∀t∈tk,tk+1, when st>0 or st<0, st=stk. So,the following equation is satisfied:(31)sgnCqμxt=sgnCqμxtk,t∈tk,tk+1.
Therefore, the triggered control law applied to the system is(32)Fai=Uforce=−τ+1τ−χCBj−1CAqμxtk+CAΓμj+CBjΓ1μs+CBjd1+d2qμxtk+d2Γμj+d3Γ1μssgnCqμxtk−τρτ−χCBj−1sgnCqμxtk.


### 3.3. Stability Analysis

In order to illustrate the reliability of the proposed algorithm, the stability of the quantize-event-trigger-based sliding mode controller needs to be considered. From (11) and (18), eμs=qμsv−v=Uforce−v; then, Uforce=qμsv=v+eμs. Combining with xt=qμxt−eμ, the derivative of the sliding surface is written as(33)s˙xt=Cx˙t=CAqμxt−eμ+Bjv+eμs+dj=CAqμxt−Aeμ+Bjv+Bjeμs+Bjdj.From inequalitieseμ=qμx−x|≤Γμj and Ceμ|≤CΓμj≤1/τCqμx|,(34)sxts˙xt=CqμxtCAqμxt−Aeμ+Bjv+Bjeμs+Bjdj−CeμCAqμxt−Aeμ+Bjv+Bjeμs+Bjdj≤τ+1τCqμxtCAqμxt+τ+1τCqμxtCAΓμj+τ+1τCqμxtCBjΓ1μs+τ+1τCqμxtCBjdj+1τCqμxtCBjv+CqμxtCBjv.If Assumption1 is satisfied, equation (34) can be written as(35)sxts˙xt≤τ+1τCqμxtCAqμxt+τ+1τCqμxtCAΔμj+τ+1τCqμxtCBjΓ1μs+τ+1τCqμxtCBj|d1+d2qμxt+d2Γμj+d3Uforce+1τCqμxtCBjv+CqμxtCBjv.AsUforce=qμ1v=v+eμs≤v+Γ1μs and χ=1+τ+1d3, equation (35) can be written as(36)sxs˙x≤τ+1τCqμxCAqμx+τ+1τCqμxCAΓμj+τ+1τCqμxCBjΓ1μs+τ+1τCqμxCBjd1+d2qμx+d2Γμj+d3Γ1μs+χ−1τCqμxCBjv+1τCqμxCBjv+CqμxCBjv,=τ+1τCqμxCAqμx+τ+1τCqμxCAΓμj+τ+1τCqμxCBjΓ1μs+τ+1τCqμxCBjd1+d2qμx+d2Γμj+d3Γ1μs+χτCqμxCBjv+τ−χτCqμxCBjv+χτCqμxCBjv.Forτ>1+d3/1−d3, d3<τ−1/τ+1, χ=1+τ+1d3<τ. By substituting the control law (17), the following inequality can be obtained:(37)sxts˙xt≤−ρCqμxt−τ+1τCqμxtCAeq+CBjd2eq.Definingσ=ρ+τ+1/τCAeq+CBjd2eq, (37) can be written as(38)sxts˙xt≤−σCqμxt.According toCeμ≤CΓμj≤1/τCqμx and triangle inequality, one can obtain(39)Cxt=Cqμxt−Ceμ≤Cqμxt+Ceμ≤τ+1τCqμxt⇒ττ+1Cxt≤Cqμxt.Then, (38) it can be observed as(40)sxts˙xt≤−ρCqμxt≤−ττ+1Cxt,=−τρτ+1sxt.Hence, the state trajectories of the plant can be driven to the sliding surface, and the overall stability of the proposed suspension controller holds.


## 3.1. Robust Quantization Feedback Controller

Robust quantization control theory contains control problems based on the quantizer, in which the dynamic quantizer expands the quantization range by adjusting the quantization level to achieve asymptotic stability of the system. This section mainly focuses on the connection channels between the suspension model and the controller. Both static quantizer and dynamic quantizer are utilized, and a sliding mode controller is also adopted. The main feature of the dynamic quantizer is that it can increase the quantization range of the quantizer by adjusting the quantization level. It can also reduce the size of the limit cycle when the system is in a steady state and achieve asymptotic stability. Compared with the static quantizer, its control effect is better, but the structure is more complicated. For practical consideration, we utilize a static quantizer on the uplink channel and a dynamic quantizer on the downlink channel. The quantization system is shown in Figure3.Figure 3
Flow chart of the quantization system.Assuming the suspension system considered in this paper in bounded-state stable, the road profile and its derivate are totally bounded. Selecting the state variables asx1x2x3x4x5x6T=zsz˙sθθ˙φφ˙T, the suspension system model including heave, pitch, and roll motions can be expressed as(8)x˙1=x2,x˙2=fz+GzFz,fz=−1ms∑Fsi+Fdi,Gz=1ms,x˙3=x4,x˙4=fθ+GθFθ,fθ=−1IyaFs1+Fd1+Fs2+Fd2−bFs3+Fd3+Fs4+Fd4,Gθ=1Iy,x˙5=x6,x˙6=fφ+GφFφ,fφ=−1IxcFs1+Fd1+Fs3+Fd3−dFs2+Fd2+Fs4+Fd4,Gφ=1Ix.Then, the abovementioned functions can be written as [18](9)xt=A0x˙t+BjUforce+djt,where xt=xit;xi+1t; i=1,3,5, j=1,2,3, and Uforce=Fz;Fθ;Fφ represent the vertical, pitch, and roll force generated by the active suspension, respectively. A0=01;00; B1=0;Gz, B2=0;Gθ, B3=0;Gφ; and d1t=0;fz/Gz, d2t=0;fθ/Gθ, and d3t=0;fφ/Gφ.Assumption 1.
djt satisfies the inequality djt≤d1+d2xt+d3Uforce≤d4, where d1≥0,d2≥0,d4>0,0≤d3≤1, and d1,d2,d3,d4 are constants.
As shown in Figure3, a static quantizer qμsv is used in the uplink channel, where v is the control law designed by the quantized state, μs is a fixed number; the down channel utilized a dynamic quantizer qμx to transfer system quantized states, and μ is the quantization parameter adjusted statically. Defining a function that takes the nearest integer, the uniform quantizer with the quantization level parameter μ is(10)qμx=defμjroundxμj,μj>0;j=1,2,3.
The uniform quantization error is(11)eμ=defqμx−x.
Then,(12)eμ=defqμx−x≤Γμj,where Γ=p/2, p is the dimension of the vector x.
For the dynamic quantizer, a special quantization level parameterμj=0 is given when the system runs to the sliding surface.(13)qμx=def0,μj=0.
The linear sliding surface is defined as follows:(14)sxt=Cxt=0,where C is a given vector to ensure that the system has stable eigenvalues and CBj≠0. The object of sliding mode control is to force the system state trajectory to reach the sliding surface and maintain the sliding mode in the presence of uncertainties and disturbances.Lemma 1.
(see [19]). There is a fixed constant τ>0, and we assume μ>0 which satisfies(15)μ≤Cxτ+1CΓ.Then, the following equation can be obtained:(16)Ceμ≤CΓμ≤1τCqμx.By considering the uplink and down channel quantize, the following control law can be designed [19]:(17)v=−τ+1τ−χCBj−1CAqμxt+CAΓμj+CBjΓ1μs+CBjd1+d2qμxt+d2Γμj+d3Γ1μssgnCqμxt−τρτ−χCBj−1sgnCqμxt,(18)Uforce=qμsv,where χ=1+τ+1d3,τ>1+d3/1−d3,ρ>0. The control law can keep the system states reach and maintain on the sliding surface sxt=0.A simple and effective design of the adjustment law is as follows [20]:(19)μj=Cxtτ+1CΓ,Cxt≥1,μj=αλτ+1CΓ,0<Cxt<1,μj=0,Cxt=0,where α∈0,1,αλ<Cx<αλ−1.


## 3.2. Event Trigger Mechanism

In recent years, event triggers have been used to reduce computational costs and energy consumption, resulting in a wide range of applications in control systems [21, 22]. The event trigger mechanism is based on the physical quantity signal of the sensor; when the semaphore reaches a certain condition, sampling is performed. In this way, the average sampling period is longer than the fixed sampling period, so that the sampling frequency of the system can be effectively reduced. As the nest sample instant is dependent on the previous sampling information, the control law is held constant between sampling instants. Defining the event trigger instant as tk, the system maintains the control output at time tk in the time interval tk,tk+1. The discretization state error of the system is(20)et=xt−xtk.The sampling is not periodic; hence,tk+1−tk is not constant. For computational purposes, the deviation from the system state to the desired state is given by(21)eT=x−xd,where xd is the set reference. As the error variable is required to vanish or settle in close vicinity of zero, the reference state is set as xd=0.The triggering instanttk is completely characterized by a triggering rule, so a triggering rule is introduced [23].(22)κ=ζeT+ξe˙T2−ψm1+m2e−ςt,where ζ>0,ξ>0,ψ∈0,1,ς∈0,1,m1≥0,m2≥0,m1+m2>0. The term ζeT+ξe˙T2 represents the overall error; the system is triggered when the error is small, guaranteeing the control output. When κ≤0, the system is triggered at tk, and the control output is maintained until tk+1. The following relation determines the triggering instants in an iterative manner:(23)tk+1=inft∈tk,∞,κ>0.The event trigger time isTk=tk+1−tk. For Tk≤tk+1−tk, and lower bound on the event trigger time is ensured. Tk is bounded by [23](24)Tk≥1AlnA⋅e∞Aϖ+Bjqμsv+d4+1,where ϖ>0.Proof:.
Considering system (9),(25)x˙t=Axt+BjUforcet+Bjdjt.
At time intervaltk,tk+1, discretization state error rises from 0 to the final value at Tk instant, so(26)ddtet≤ddte¯t≤ddtxt≤Axt+BjUforcet+Bjdjt,by inserting the control law (17) into abovementioned equation.(27)ddtet≤Axt+Bjqμsv+Bjdjt≤Axt+Bjqμsv+Bjdjt≤Aet+xtk+Bjqμsv+Bjdjt≤Aet+Axtk+Bjqμsv+Bjdjt≤Aet+Aϖ+Bjqμsv+d4.
∀t∈tk,tk+1, with the initial condition et=0, and the solution of inequality (27) can be understood by utilizing the comparison lemma [24] and is given as(28)et≤Aϖ+Bjqμsv+d4AeAt−tk−1.
Similarly, for triggering time instanttk+1,(29)e∞=etk+1≤Aϖ+Bjqμsv+d4AeATk−1,and then, inequality (24) is satisfied.
Discretized state error after quantization is(30)eqt=qμxt−qμxtk.
Event is triggered at discrete moments∀t∈tk,tk+1, when st>0 or st<0, st=stk. So,the following equation is satisfied:(31)sgnCqμxt=sgnCqμxtk,t∈tk,tk+1.
Therefore, the triggered control law applied to the system is(32)Fai=Uforce=−τ+1τ−χCBj−1CAqμxtk+CAΓμj+CBjΓ1μs+CBjd1+d2qμxtk+d2Γμj+d3Γ1μssgnCqμxtk−τρτ−χCBj−1sgnCqμxtk.


## 3.3. Stability Analysis

In order to illustrate the reliability of the proposed algorithm, the stability of the quantize-event-trigger-based sliding mode controller needs to be considered. From (11) and (18), eμs=qμsv−v=Uforce−v; then, Uforce=qμsv=v+eμs. Combining with xt=qμxt−eμ, the derivative of the sliding surface is written as(33)s˙xt=Cx˙t=CAqμxt−eμ+Bjv+eμs+dj=CAqμxt−Aeμ+Bjv+Bjeμs+Bjdj.From inequalitieseμ=qμx−x|≤Γμj and Ceμ|≤CΓμj≤1/τCqμx|,(34)sxts˙xt=CqμxtCAqμxt−Aeμ+Bjv+Bjeμs+Bjdj−CeμCAqμxt−Aeμ+Bjv+Bjeμs+Bjdj≤τ+1τCqμxtCAqμxt+τ+1τCqμxtCAΓμj+τ+1τCqμxtCBjΓ1μs+τ+1τCqμxtCBjdj+1τCqμxtCBjv+CqμxtCBjv.If Assumption1 is satisfied, equation (34) can be written as(35)sxts˙xt≤τ+1τCqμxtCAqμxt+τ+1τCqμxtCAΔμj+τ+1τCqμxtCBjΓ1μs+τ+1τCqμxtCBj|d1+d2qμxt+d2Γμj+d3Uforce+1τCqμxtCBjv+CqμxtCBjv.AsUforce=qμ1v=v+eμs≤v+Γ1μs and χ=1+τ+1d3, equation (35) can be written as(36)sxs˙x≤τ+1τCqμxCAqμx+τ+1τCqμxCAΓμj+τ+1τCqμxCBjΓ1μs+τ+1τCqμxCBjd1+d2qμx+d2Γμj+d3Γ1μs+χ−1τCqμxCBjv+1τCqμxCBjv+CqμxCBjv,=τ+1τCqμxCAqμx+τ+1τCqμxCAΓμj+τ+1τCqμxCBjΓ1μs+τ+1τCqμxCBjd1+d2qμx+d2Γμj+d3Γ1μs+χτCqμxCBjv+τ−χτCqμxCBjv+χτCqμxCBjv.Forτ>1+d3/1−d3, d3<τ−1/τ+1, χ=1+τ+1d3<τ. By substituting the control law (17), the following inequality can be obtained:(37)sxts˙xt≤−ρCqμxt−τ+1τCqμxtCAeq+CBjd2eq.Definingσ=ρ+τ+1/τCAeq+CBjd2eq, (37) can be written as(38)sxts˙xt≤−σCqμxt.According toCeμ≤CΓμj≤1/τCqμx and triangle inequality, one can obtain(39)Cxt=Cqμxt−Ceμ≤Cqμxt+Ceμ≤τ+1τCqμxt⇒ττ+1Cxt≤Cqμxt.Then, (38) it can be observed as(40)sxts˙xt≤−ρCqμxt≤−ττ+1Cxt,=−τρτ+1sxt.Hence, the state trajectories of the plant can be driven to the sliding surface, and the overall stability of the proposed suspension controller holds.


## 4. Numerical Simulation and Discussion

The effectiveness of the proposed controller is discussed in this section through the simulation results of the MATLAB/Simulink. The model and controller parameters are listed in Tables1 and 2, respectively.Table 1
Suspension model parameters.
ParameterValueSprung massms1592kgUnsprung massmui120/150kgGravity to front axle distancea1.18mGravity to rear axle distanceb1.77mCenter of gravity to the left sidec0.7875mCenter of gravity to the right sided0.7875mPitch moment of inertiaIy2488kg•m2Roll moment of inertiaIx614kg•m2Spring stiffnesski146000N/mDamping coefficientbi3000Nm/sTire stiffnesskti230000N/mHydraulic supply pressurePs1.034×107N/m2Piston areaAp3.35×10−4Discharge coefficientCd0.61Spool valve area gradientw1.436×10−2mOil densityρ858kg/m3α1.19143×1013Table 2
Controller parameters.
ParameterValued10.1d20.1d30.01α0.5ρ4μs0.05p10.8p20.8p30.05p41e−5p50p60.97When the vehicle is driven in a straight line, road roughness is an important factor that affects the riding comfort and handling stability of the vehicle. Pavement roughness is often expressed in terms of statistical characteristic parameters in the frequency domain, such as power spectral density or variance. However, the power spectral density expression can hardly meet the research needs of vehicle dynamics control, and it is necessary to simulate the road roughness in the time domain. The power spectral density of signal wheel road disturbance can be expressed as [25](41)Gqn0=Gqn0nn0−W.Road roughnessqt can be seen as the first-order response of unite white noise in the filtered white noise method, and through introducing a low cutoff frequency, the system frequency response is(42)Hq∼wω=2πn0Gqn0vjω+2πn00v.The time-domain model of signal wheel excitation can be obtained by writing (42) into a differential form.(43)q˙t=2πn0Gqn0vwt−2πn00vqt.The definition of specific symbols can be seen in [25]. The four-wheel excitation model can be obtained by the time delay between front and rear axles and the coherence between left and right tracks [26]. Assuming that the vehicle is traveling on the B-level road with constant speed of 60 km/h, the time domain road profiles of four wheels are plotted in Figure 4.Figure 4
Four-wheel excitation input.The purpose of the active suspension controller is to improve the riding comfort while keeping the wheels in constant contact with the road surface. The riding comfort is evaluated by the movement of the sprung mass motion, and the handling stability is evaluated through the dynamic displacement of the tire. To verify the effectivness of the quantized feedback controller, different quantitative and trigger conditions are considered, and the road disturbance input is the B-level road.Figure5–Figure 7 reflect the control effect of the sprung mass motions with the quantized feedback controller. Sprung mass vertical acceleration, pitch acceleration, and roll acceleration with the combined quantizer, static quantizer, and uncontrolled condition are plotted. As can be seen from the figures, the passive system reported significantly more amplitude than the other two controlled systems. Partial enlarged detail of the figures indicates that compared with the static quantizer, a combined quantizer composed of the static and dynamic quantizer can achieve better performance.Figure 5
Sprung mass vertical acceleration.Figure 6
Sprung mass pitch acceleration.Figure 7
Sprung mass roll acceleration.Figure8 represents the simulation contrast curves of the suspension deflection of the vehicle. It can be seen from Figure 8 that the proposed combined quantizer in this study outperforms the static quantizer and passive system in time domain. The time responses of the vehicle’s tire deflection of four corners are set out in Figure 9. As shown in the figure, although the peak value of the controlled system is slightly greater than that of the passive system, the root mean square values are nearly consistent with the uncontrolled model. The partial enlarged details also show that the tire deflections with the static and combined quantizer are little changed. These results suggest that the quantized feedback controller is able to improve riding comfort effectively while maintaining the wheels in contact with the road surface.Figure 8
Suspension deflection.
(a)(b)(c)(d)Figure 9
Tire deflection.
(a)(b)(c)(d)The results of the trigger signals are shown in Figure10, and the quantization parameters are summarized in Figure 11. From the data in Figures 10 and 11, it is apparent that the trigger moments are corresponding to the change of quantization parameters. Actuator force with the event trigger mechanism is presented in Figure 12. The figure illustrates that compared with the control law without the trigger mechanism, the output force frequency with the trigger mechanism is significantly reduced, which has a certain effect on reducing power consumption and reducing fatigue damage of the actuator. Therefore, it is necessary to introduce the trigger mechanism to the controller.Figure 10
Trigger signals.
(a)(b)(c)Figure 11
Quantization parameters.
(a)(b)(c)Figure 12
Actuator forces with the event trigger mechanism.
(a)(b)(c)(d)Table3 compares the summary statistics for sprung mass accelerations, suspension deflection, and tire deflection with and without the trigger mechanism. Besides, the nonsingular fast terminal sliding mode (NFTSM) control method [27] is also added to verify the effectiveness of the proposed method. As Table 3 shows, the introduction of the trigger mechanism has very little effect on the vehicle’s riding comfort, and the designed controller has performance similar to that of the NFTSM controller in improving the sprung mass characteristics.Table 3
Comparison of root mean square values.
Vertical accelerationPitch accelerationRoll accelerationFL suspension deflectionFL tire deflectionWith trigger0.24370.35760.45650.00270.0036Without trigger0.24220.35590.45480.00270.0036NFTSM0.23300.35060.45090.00260.0037Passive0.47640.64270.81870.00320.0031


## 5. Conclusions

The aim of the present research was to deal with the quantization error and improve the actuator efficiency of the control system. A sliding-mode-based quantized feedback controller was designed for the active suspension system to improve the riding comfort with external disturbances. Then, an event trigger mechanism was introduced in the design process of the quantization controller to reduce the operation frequency of the actuator. Some conclusions can be drawn as follows:(1)
Compared with the static quantizer, the combined quantizer can achieve better performance(2)
The introduction of the trigger mechanism can effectively reduce the operating frequency of the actuator while maintaining the control effect of the active system(3)
The overall controller of the active suspension can improve riding comfort while maintaining handling stability without deterioration in the presence of external disturbancesThe performance of the proposed controller has been validated under random road excitation in the MATLAB/Simulink environment.In the future works, the influence of an in-wheel motor on vehicle vertical vibration will be considered, and critical test conditions will be applied. Besides, the longitude and lateral influence will also be considered.



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*Source: 8886069-2021-05-12.xml*