# Random Modeling and Control of Nonlinear Active Suspension

Authors: Mingyue Cui; Liangchao Geng; Zhaojing Wu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

For a quarter car with nonlinear active suspension in rough road, the problem of random modeling and control is considered. According to the relative motion principle, the influence of rough road can be seen as that force is disturbed by the noise and a random model is constructed. By an appropriate transform, the model is transformed into a lower triangular system, which can be used as backstepping method. Then a controller is designed such that the mean square of the state converges to an arbitrarily small neighborhood of zero by tuning design parameters. The simulation results illustrate the effectiveness of the proposed scheme. Therefore, the active suspension system offers better riding comfort and vehicle handing to the passengers.


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## Body

## 1. Introduction

The active suspension is the key technology for vehicles to achieve both ride comfort and control performance. Compared with the passive and semiactive suspension system, the active suspension can supply energy from an external source and generate force to achieve the desired performance. Therefore, the performance of active suspension system is better. In recent decades, the control of active suspension systems has been enthusiastically studied by many researchers. Great efforts have been made in modeling and developing control techniques to obtain the ride comfort. Many control methods, for instance, adaptive control [1], PD control [2], robust control [3], fuzzy logic control [4, 5], intelligent control [6], and sliding mode control [7, 8], have been recently proposed. However, all of these studies are based on deterministic systems.In the actual life, the car often runs on rough roads and the influence of the rough road is not negligible. Therefore, it is expected that the passenger still feels comfortable in rough road. For increasing the passenger’s comfort, the vertical acceleration of the vehicle caused by road vibrations must be limited which means that the suspension system must absorb the road vibrations and prevent it from transferring to the vehicle body and passengers. It is very necessary to study the random model and control of nonlinear active suspension system in rough road.In [9–12], the design methods of the controller with different stochastic mechanical systems were studied. However, the stochastic disturbance was described as white noise in these literatures. Because of absorbers, it is more reasonable to describe the final effect of road irregularities as stationary processes. To overcome the conservation, [13] constructed a theoretical framework on stability of random differential equation systems (RDEs) where stochastic disturbance is stationary processes. For a class of Lagrange systems with colored noise, [14] designed a tracking controller such that the mean square of the tracking error converges to an arbitrarily small neighborhood of zero.Inspired by these, the model and control problem of nonlinear active suspension in rough road are considered in this paper. The main work consists of the following aspects.(
1
) Different from the linear deterministic system in [2–5], the active suspension with nonlinear damper in rough road is considered in this paper, which increases the difficulty of modeling and design.(
2
) The main difficulty for dynamics modeling is how to transform the effect of the road irregularities to the suspension. In this paper, regardless of the rough road, dynamics model of the system is firstly constructed. According to the dynamic-static method and the relative motion, the road irregularities are transformed to the force disturbed by the stationary processes. Then the random dynamic model is established.(
3
) Because the system is an underactuated system which is not the quasi lower triangular structure, one cannot use the design method of vector controller in [9–12]. In this paper, according to special form of the model, the system is transformed into a lower triangular system by a transform. Then using backstepping design method, a state feedback controller is designed such that the state can be made arbitrarily small by tuning design parameters. The simulation results illustrate the effectiveness of the proposed scheme.This paper is organized as follows. In Section2, the mathematical preparation is given and the problem is formulated. The random model is constructed in Section 3. In Section 4, the tracking controller design and stability analysis are addressed. A simulation result is given in Section 5. Section 6 concludes the paper.Notations. The following notations are used throughout the paper: For a vector x, xT denotes its transpose; · denotes the usual Euclidean norm of “·”; E denotes the mathematical expectation; R+ denotes the set of all nonnegative real numbers; Rn denotes the real n-dimensional space; Rn×m denotes the real n×m matrix space; Ci denotes the set of all functions with continuous ith partial derivative; K denotes the set of all functions: R+→R+, which are continuous, strictly increase, and vanish at zero; K∞ denotes the set of all functions which are of class K and unbounded. KL denotes the set of all functions β(s,t):R+×R+→R+, which is of class-K for each fixed t and decreases to zero as t→∞ for each fixed s. For simplicity, sometimes the arguments of functions are dropped.


## 2. Mathematical Preliminaries and Problem Formulation

### 2.1. Mathematical Preliminaries

Consider the following random nonlinear affine system:(1)x˙=fx,t+gx,tξt,xt0=x0,where x∈Rn is the state of system, ξ(t)∈Rm is a stochastic process, and the underlying complete probability space is taken to be the quartet (Ω,F,Ft,P) with a filtration Ft satisfying the usual condition (i.e., it is increasing and right continuous while F0 contains all P-null sets). Both functions f:Rn×R+→Rn and g:Rn×R+→Rn×m are locally Lipschitz in x piecewise continuous in t; that is, for each R>0, there exists a constant LR>0 such that(2)fx1,t-fx2,t+gx1,t-gx2,t≤LRx2-x1for any t∈R+ and x1,x2∈UR={x:x≤R}, x1≠x2. Moreover, f(0,t) and g(0,t) are bounded a.s.In order to obtain the stability, processξ(t) satisfies the following assumption.Assumption 1.
Processξ is a Ft-adapted process and piecewise continuous, and there exist positive constants c0, d0 such that(3)supt0≤s≤tEξs2≤d0ec0t,∀t≥t0.The following definition, criterion, and inequality are represented now for the stability analysis.Definition 2 (see [13]).
System (1) with Assumption 1 is said to be noise-to-state stable in probability (NSS-P) if there exist a function β(·,·)∈KL and a function γ(·)∈K∞ such that, for any ϵ>0, x0∈Rn, t≥t0,(4)Pxt≤βx0,t-t0+γsupt0≤s≤t⁡Eξs2≥1-ϵ.Lemma 3 (see [13]).
Suppose that for system (1) with conditions (3), there exist parameters c>0 and d>0 and a function V∈C1 such that(5)α1x≤Vx≤α2x,V˙xt≤-cVxt+dξt2,where α1, α2 are functions of class K∞. Then system (1) has a unique solution, and the system is NSS-P.Definition 4 (see [15]).
A stochastic processx(t) is said to be bounded in probability if the random variables xt are bounded in probability uniformly in t; that is,(6)limR→∞⁡supt>0⁡Pxt>R=0.Lemma 5 (see [13]).
Under Assumption1, if there exist a positive-definite function V(x,t)∈C and a constant dc>0 such that(7)limr→∞⁡infx>r⁡Vx,t=∞,EVx,t≤dc,then system (1) has a unique solution, and the solution is bounded in probability.Lemma 6 (see [16]).
Consider the continuous functionsk(t), h(t), and they are integrable over every finite interval. If a continuous function y(t) satisfies the inequality(8)y˙t≤ktyt+ht,∀t≥t0,then(9)yt≤yt0e∫t0tksds+∫t0te∫stkuduhsds,∀t≥t0.


### 2.2. Problem Formulation

The simplified quarter-car active suspension model is shown in Figure1 (see [17]). The car runs at a constant speed on a rough road. The road irregularity is described as white noise ξ0, which is the acceleration of vertical oscillation. Mb is the mass of car body. The wheel is modeled as an unsprung mass Mus with a linear spring Kt. The actuator is connected in parallel with a linear spring Ka and a nonlinear damper Ca. ua is the control force from the actuator such as hydraulic actuator. When the components are stationary, the system is in equilibrium. s1 and s2 denote the barycenter displacements of wheel and vehicle body relative to the equilibrium position, respectively.Figure 1
The quarter-car model for active suspension design with parallel connection of actuator with spring and damper.The objective of this paper is to design a controller to get more comfortable riding. To this end, two efforts will be taken in the following sections.(
1
) Considering the road roughness and the nonlinear damper, construct an appropriate random model to describe the suspension motion of the car.(
2
) Design a controller ua such that the states s1 and s2 can be small as much as possible.


## 2.1. Mathematical Preliminaries

Consider the following random nonlinear affine system:(1)x˙=fx,t+gx,tξt,xt0=x0,where x∈Rn is the state of system, ξ(t)∈Rm is a stochastic process, and the underlying complete probability space is taken to be the quartet (Ω,F,Ft,P) with a filtration Ft satisfying the usual condition (i.e., it is increasing and right continuous while F0 contains all P-null sets). Both functions f:Rn×R+→Rn and g:Rn×R+→Rn×m are locally Lipschitz in x piecewise continuous in t; that is, for each R>0, there exists a constant LR>0 such that(2)fx1,t-fx2,t+gx1,t-gx2,t≤LRx2-x1for any t∈R+ and x1,x2∈UR={x:x≤R}, x1≠x2. Moreover, f(0,t) and g(0,t) are bounded a.s.In order to obtain the stability, processξ(t) satisfies the following assumption.Assumption 1.
Processξ is a Ft-adapted process and piecewise continuous, and there exist positive constants c0, d0 such that(3)supt0≤s≤tEξs2≤d0ec0t,∀t≥t0.The following definition, criterion, and inequality are represented now for the stability analysis.Definition 2 (see [13]).
System (1) with Assumption 1 is said to be noise-to-state stable in probability (NSS-P) if there exist a function β(·,·)∈KL and a function γ(·)∈K∞ such that, for any ϵ>0, x0∈Rn, t≥t0,(4)Pxt≤βx0,t-t0+γsupt0≤s≤t⁡Eξs2≥1-ϵ.Lemma 3 (see [13]).
Suppose that for system (1) with conditions (3), there exist parameters c>0 and d>0 and a function V∈C1 such that(5)α1x≤Vx≤α2x,V˙xt≤-cVxt+dξt2,where α1, α2 are functions of class K∞. Then system (1) has a unique solution, and the system is NSS-P.Definition 4 (see [15]).
A stochastic processx(t) is said to be bounded in probability if the random variables xt are bounded in probability uniformly in t; that is,(6)limR→∞⁡supt>0⁡Pxt>R=0.Lemma 5 (see [13]).
Under Assumption1, if there exist a positive-definite function V(x,t)∈C and a constant dc>0 such that(7)limr→∞⁡infx>r⁡Vx,t=∞,EVx,t≤dc,then system (1) has a unique solution, and the solution is bounded in probability.Lemma 6 (see [16]).
Consider the continuous functionsk(t), h(t), and they are integrable over every finite interval. If a continuous function y(t) satisfies the inequality(8)y˙t≤ktyt+ht,∀t≥t0,then(9)yt≤yt0e∫t0tksds+∫t0te∫stkuduhsds,∀t≥t0.


## 2.2. Problem Formulation

The simplified quarter-car active suspension model is shown in Figure1 (see [17]). The car runs at a constant speed on a rough road. The road irregularity is described as white noise ξ0, which is the acceleration of vertical oscillation. Mb is the mass of car body. The wheel is modeled as an unsprung mass Mus with a linear spring Kt. The actuator is connected in parallel with a linear spring Ka and a nonlinear damper Ca. ua is the control force from the actuator such as hydraulic actuator. When the components are stationary, the system is in equilibrium. s1 and s2 denote the barycenter displacements of wheel and vehicle body relative to the equilibrium position, respectively.Figure 1
The quarter-car model for active suspension design with parallel connection of actuator with spring and damper.The objective of this paper is to design a controller to get more comfortable riding. To this end, two efforts will be taken in the following sections.(
1
) Considering the road roughness and the nonlinear damper, construct an appropriate random model to describe the suspension motion of the car.(
2
) Design a controller ua such that the states s1 and s2 can be small as much as possible.


## 3. Modeling of Nonlinear Active Suspension System

Consider the system of particles consisting of the wheel and the car body which are regarded as the mass points, and select(s1,s2) as the generalized coordinate.


### 3.1. Modeling under the Assumption That the Road Is Smooth

The total kinetic energy of the system isK=(1/2)Muss˙12+(1/2)Mbs˙22. The total potential energy of the system equals P=(1/2)Ka(s2-s1)2+(1/2)Kts12. Then the Lagrange function(10)L=K-P=12Mbs˙22+12Muss˙12-12Kas2-s12-12Kts12. According to the Lagrangian mechanics [18], the system model is(11)Muss¨1-Kas2-s1+Kts1=τ1,Mbs¨2+Kas2-s1=τ2,where τ1 and τ2 are the generalized forces.Considering the damper nonlinearity, the Rayleigh dissipation function is(12)R=Cap+1s˙2-s˙1p+1,where p>0 is an index, which represent the nonlinear form of damping force. Then the dissipative force is(13)τd=τd1τd2=-∂R∂s˙=-∂R∂s˙1∂R∂s˙2=Casigns˙2-s˙1s˙2-s˙1p-Casigns˙2-s˙1s˙2-s˙1p,where sign(·) is the sign function. The control force u=[u1,u2]T, where u1 and u2 are the control force acting wheel and car body, respectively. Then by replacing τi with τdi+ui, the control system can be modeled as(14)Muss¨1-Kas2-s1+Kts1=Casigns˙2-s˙1s˙2-s˙1p+u1,Mbs¨2+Kas2-s1=-Casigns˙2-s˙1s˙2-s˙1p+u2.Remark 7.
In this paper, the suspension uses the nonlinear damper. In the dissipative force (13), p>0 is an index, which represents the nonlinear form of damping force (see [19]). When p=1, it is the common linear damping force.


### 3.2. Modeling under the Assumption That the Road Is Rough

Road irregularities are often described as white noises. Because of the spring, the final influence of rough road to the wheel and the car body is stationary processesξ1 and ξ2, respectively. Therefore, there exists positive constant K such that(15)supt0≤s≤t⁡Eξis2≤K,∀t≥t0,i=1,2.According to dynamic-static method and relative motion [20], the effect of rough road can be seen as the force is disturbed by the noise. Replacing u1 and u2 with u1-Musξ1 and u2-Mbξ2 in (14) results in(16)Muss¨1-Kas2-s1+Kts1=Casigns˙2-s˙1s˙2-s˙1p+u1-Musξ1,Mbs¨2+Kas2-s1=-Casigns˙2-s˙1s˙2-s˙1p+u2-Mbξ2.Considering the control force from the actuator, u1=-ua and u2=ua. Thus, (16) can be rewritten as(17)Muss¨1-Kas2-s1+Kts1=Casigns˙2-s˙1s˙2-s˙1p-ua-Musξ1,Mbs¨2+Kas2-s1=-Casigns˙2-s˙1s˙2-s˙1p+ua-Mbξ2.Remark 8.
In order to model, the control forceua provided by the actuator can be seen as two independent forces u1 and u2, which act on the car body and the wheel, respectively. Then according to the Lagrangian mechanics and the dynamic-static method, the random model (16) is constructed. Finally, considering the constraint u1=-ua and u2=ua, the final random model (17) is established.


## 3.1. Modeling under the Assumption That the Road Is Smooth

The total kinetic energy of the system isK=(1/2)Muss˙12+(1/2)Mbs˙22. The total potential energy of the system equals P=(1/2)Ka(s2-s1)2+(1/2)Kts12. Then the Lagrange function(10)L=K-P=12Mbs˙22+12Muss˙12-12Kas2-s12-12Kts12. According to the Lagrangian mechanics [18], the system model is(11)Muss¨1-Kas2-s1+Kts1=τ1,Mbs¨2+Kas2-s1=τ2,where τ1 and τ2 are the generalized forces.Considering the damper nonlinearity, the Rayleigh dissipation function is(12)R=Cap+1s˙2-s˙1p+1,where p>0 is an index, which represent the nonlinear form of damping force. Then the dissipative force is(13)τd=τd1τd2=-∂R∂s˙=-∂R∂s˙1∂R∂s˙2=Casigns˙2-s˙1s˙2-s˙1p-Casigns˙2-s˙1s˙2-s˙1p,where sign(·) is the sign function. The control force u=[u1,u2]T, where u1 and u2 are the control force acting wheel and car body, respectively. Then by replacing τi with τdi+ui, the control system can be modeled as(14)Muss¨1-Kas2-s1+Kts1=Casigns˙2-s˙1s˙2-s˙1p+u1,Mbs¨2+Kas2-s1=-Casigns˙2-s˙1s˙2-s˙1p+u2.Remark 7.
In this paper, the suspension uses the nonlinear damper. In the dissipative force (13), p>0 is an index, which represents the nonlinear form of damping force (see [19]). When p=1, it is the common linear damping force.


## 3.2. Modeling under the Assumption That the Road Is Rough

Road irregularities are often described as white noises. Because of the spring, the final influence of rough road to the wheel and the car body is stationary processesξ1 and ξ2, respectively. Therefore, there exists positive constant K such that(15)supt0≤s≤t⁡Eξis2≤K,∀t≥t0,i=1,2.According to dynamic-static method and relative motion [20], the effect of rough road can be seen as the force is disturbed by the noise. Replacing u1 and u2 with u1-Musξ1 and u2-Mbξ2 in (14) results in(16)Muss¨1-Kas2-s1+Kts1=Casigns˙2-s˙1s˙2-s˙1p+u1-Musξ1,Mbs¨2+Kas2-s1=-Casigns˙2-s˙1s˙2-s˙1p+u2-Mbξ2.Considering the control force from the actuator, u1=-ua and u2=ua. Thus, (16) can be rewritten as(17)Muss¨1-Kas2-s1+Kts1=Casigns˙2-s˙1s˙2-s˙1p-ua-Musξ1,Mbs¨2+Kas2-s1=-Casigns˙2-s˙1s˙2-s˙1p+ua-Mbξ2.Remark 8.
In order to model, the control forceua provided by the actuator can be seen as two independent forces u1 and u2, which act on the car body and the wheel, respectively. Then according to the Lagrangian mechanics and the dynamic-static method, the random model (16) is constructed. Finally, considering the constraint u1=-ua and u2=ua, the final random model (17) is established.


## 4. Control of Nonlinear Active Suspension System

### 4.1. Control Design

In order to design controller, choose(18)x1=Muss1+Mbs2,x2=x˙1,x3=s2-s1,x4=x˙3.(19)x˙1=x2,x˙2=-KtMus+Mbx1+KtMbMus+Mbx3-Musξ1-Mbξ2,x˙3=x4,x˙4=-Ka1Mus+1Mbx3+KtMusMb+Musx1-KtMbMusMb+Musx3-Ca1Mus+1Mbsignx4x4p+1Mus+1Mbua-ξ2+ξ1.Remark 9.
Because system (17) is underactuated system which is not transformed into the quasi lower triangular structure, the design methods of vector controller in [9–12] are not applicable to this system. In order to design the controller with backstepping design method [17], the system is transformed into a lower triangular system by transform (18).Then the backstepping controller can be given step by step.Step 1.
Introduce the first two error variables(20)z1=x1,z2=x2-α1,where α1 is a function to be designed. For the Lyapunov function V1=(1/2)z12, by choosing α1=-c1z1 with design parameter c1>0, the derivative of V1 is(21)V˙1=-c1z12+z1z2.Step 2.
Introducing(22)z3=x3-α2,then(23)z˙2=-KtMus+Mbx1+KtMbMus+Mbz3+α2+c1x2-Musξ1-Mbξ2. For the Lyapunov function V2=V1+(1/2)z22, the derivative of V2 is(24)V˙2=-c1z12+KtMbMus+Mbz2z3+z2z1-KtMus+Mbx1+KtMbMus+Mbα2+c1x2-z2Musξ1-z2Mbξ2.Applying Young’s inequality (for any vectors x,y∈Rn and any scalars ϵ>0, p>1, there holds xTy≤(ϵp/p)|x|p+(1/qϵq)yq, where q=p/(p-1)) to the last two terms in (24), one has(25)-z2Musξ1≤d1Mus2z22+14d1ξ12,-z2Mbξ2≤d2Mb2z22+14d2ξ22,where d1>0 and d2>0 are design parameters. Substituting (25) into (24), the resulting V˙2 is(26)V˙2≤-c1z12+KtMbMus+Mbz2z3+z2z1-KtMus+Mbx1+KtMbMus+Mbα2+c1x2+d1Mus2z2+d2Mb2z2+14d1ξ12+14d2ξ22.Choose(27)α2=Mus+MbKtMb-z1-c2z2+KtMus+Mbx1-c1x2-d1Mus2z2-d2Mb2z2,where c2>0 is a design parameter. Then (26) can be rewritten as(28)V˙2≤-c1z12-c2z22+KtMbMus+Mbz2z3+14d1ξ12+14d2ξ22.Step 3.
Introducing(29)z4=x4-α3,then(30)z˙3=z4+α3-∂α2∂x1x2-∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3-Musξ1-Mbξ2. For the Lyapunov function V3=V2+(1/2)z32, the derivative of V3 is(31)V˙3≤-c1z12-c2z22+z3z4+14d1ξ12+14d2ξ22+z3KtMbMus+Mbz2+α3-∂α2∂x1x2-∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3-z3∂α2∂x2Musξ1-z3∂α2∂x2Mbξ2.Applying Young’s inequality to the last two terms in (31), one has(32)-z3∂α2∂x2Musξ1≤d3∂α2∂x22Mus2z32+14d3ξ12,-z3∂α2∂x2Mbξ2≤d4∂α2∂x22Mb2z32+14d4ξ22,where d3>0 and d4>0 are design parameters. Then (31) can be rewritten as(33)V˙3≤-c1z12-c2z22+z3z4+z3KtMbMus+Mbz2+α3-∂α2∂x1x2-∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3+d3∂α2∂x22Mus2z3+d4∂α2∂x22Mb2z3+14d1+14d3ξ12+14d2+14d4ξ22.Choose(34)α3=-KtMbMus+Mbz2-c3z3+∂α2∂x1x2+∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3-d3∂α2∂x22Mus2z3-d4∂α2∂x22Mb2z3,where c3>0 is a design parameter. Then(35)V˙3≤-c1z12-c2z22-c3z32+z3z4+14d1+14d3ξ12+14d2+14d4ξ22.Step 4.
From (29), one has(36)z˙4=ψx1,x2,x3,x4+1Mus+1Mbua+∂α3∂x2Mus+1ξ1+∂α3∂x2Mb-1ξ2,where ψ(x1,x2,x3,x4) = -Ka(1/Mus+1/Mb)x3+(Kt/MusMb+Mus)x1 − (KtMb/MusMb+Mus)x3-Ca(1/Mus+1/Mb)sign(x4)x4p − (∂α3/∂x1)x2-(∂α3/∂x2)(-(Kt/Mus+Mb)x1 + (KtMb/Mus+Mb)x3)-(∂α3/∂x3)x4. For the Lyapunov function V4=V3+(1/2)z42, the derivative of V4 is(37)V˙4≤-c1z12-c2z22-c3z32+14d1+14d3ξ12+14d2+14d4ξ22+z4z3+ψ+1Mus+1Mbua+z4∂α3∂x2Mus-1ξ1+z4∂α3∂x2Mb-1ξ2.Applying Young’s inequality, one has(38)z4∂α3∂x2Mus+1ξ1≤d5∂α3∂x2Mus+12z42+14d5ξ12,z4∂α3∂x2Mb-1ξ2≤d6∂α3∂x2Mb-12z42+14d6ξ22,where d5>0 and d6>0 are design parameters. Substituting (38) into (37) leads to(39)V˙4≤-c1z12-c2z22-c3z32+z4z3+ψ+1Mus+1Mbua+d5∂α3∂x2Mus+12z4+d6∂α3∂x2Mb-12z4+14d1+14d3+14d5ξ12+14d2+14d4+14d6ξ22.Choose(40)ua=MusMbMus+Mb-z3-c4z4-ψ-d5∂α3∂x2Mus+12z4-d6∂α3∂x2Mb-12z4,where c4>0 is a design parameter. Then(41)V˙4≤-∑i=14cizi2+14d1+14d3+14d5ξ12+14d2+14d4+14d6ξ22.
Up to now, the closed-loop system is obtained(42)z˙1=-c1z1+z2,z˙2=-z1-c2z2+KtMbMus+Mbz3-Mus+MbKtMbd1Mus2+d2Mb2z2-Musξ1-Mbξ2,z˙3=-KtMbMus+Mbz2-c3z3+z4-d3∂α2∂x22Mus2z3-d4∂α2∂x22Mb2z3+∂α2∂x2Musξ1+Mbξ2,z˙4=-z3-c4z4-MusMbMus+Mbd5∂α3∂x2Mus+12+d6∂α3∂x2Mb-12z4+∂α3∂x2Mus+1ξ1+∂α3∂x2Mb-1ξ2.


### 4.2. Stability Analysis

Theorem 10.
Consider the random model (17) of quarter-car active suspension. Under assumption (15), choose controller (40).
(i) The closed-loop system (42) is NSS-p and all the signals of the closed-loop system are bounded in probability.
(ii) The statez=[z1,z2,z3,z4]T of the closed-loop system satisfies(43)limt→∞⁡Ez2≤2dKc,where the right-hand can be made small enough by tuning parameters.Proof.
Obviously, the functions of the closed-loop system satisfy the local Lipschitz condition. The Lyapunov function for the whole system is(44)V=V4=∑i=1412zi2.From (41), one has(45)V˙≤-cV+dξ2,where c=2min{c1,c2,c3,c4}, d=max⁡{1/4d1+1/4d3+1/4d5,1/4d2+1/4d4+1/4d6}, and ξ=[ξ1,ξ2]2. From Lemma 3, the closed-loop system is NSS-P.
Furthermore, by definingv(t)=EV(z(t)), from (45), one has(46)v˙t=EV˙xt≤-cvt+dEξt2. From Lemma 6, one has(47)vt≤v0e-ct-t0+dsupt0≤s≤t⁡Eξs2,which together with (15) implies(48)EVt≤Vx0e-ct-t0+dKc≤Vx0+dKc.According to Lemma 5, (48) and the definition of V, z is bounded in probability. By z1=x1, x1 is bounded in probability. Since z2=x2-c1z1, x2 is bounded in probability, too. Similarly, x3 and x4 are bounded in probability. Then the control u is also bounded in probability by (40).
From (15), (44), and (47), one has(49)Ez2≤2EVt≤2Vx0e-ct-t0+2dKc,which leads to (43). Noting c=2min⁡{c1,c2,c3,c4}, d=max{1/4d1+1/4d3+1/4d5,1/4d2+1/4d4+1/4d6}, it is clear that the right-hand sides of (43) can be made small enough by choosing ci(i=1,…,4) and dj(j=1,…,6) large enough.Remark 11.
Sincez1=Muss1+Mbs2, z3=s2-s1-α2=s2-s1-((Mus + Mb)/KtMb)(-z1-c2z2+(Kt/(Mus+Mb))z1-c1(z2-c1z1)-d1Mus2z2-d2Mb2z2), then s1, s2 can be small enough by choosing parameters appropriately. As a result, the passenger feels comfortable. From Figure 2, the simulation results also demonstrate the comfortableness.Figure 2
The system simulation results.


## 4.1. Control Design

In order to design controller, choose(18)x1=Muss1+Mbs2,x2=x˙1,x3=s2-s1,x4=x˙3.(19)x˙1=x2,x˙2=-KtMus+Mbx1+KtMbMus+Mbx3-Musξ1-Mbξ2,x˙3=x4,x˙4=-Ka1Mus+1Mbx3+KtMusMb+Musx1-KtMbMusMb+Musx3-Ca1Mus+1Mbsignx4x4p+1Mus+1Mbua-ξ2+ξ1.Remark 9.
Because system (17) is underactuated system which is not transformed into the quasi lower triangular structure, the design methods of vector controller in [9–12] are not applicable to this system. In order to design the controller with backstepping design method [17], the system is transformed into a lower triangular system by transform (18).Then the backstepping controller can be given step by step.Step 1.
Introduce the first two error variables(20)z1=x1,z2=x2-α1,where α1 is a function to be designed. For the Lyapunov function V1=(1/2)z12, by choosing α1=-c1z1 with design parameter c1>0, the derivative of V1 is(21)V˙1=-c1z12+z1z2.Step 2.
Introducing(22)z3=x3-α2,then(23)z˙2=-KtMus+Mbx1+KtMbMus+Mbz3+α2+c1x2-Musξ1-Mbξ2. For the Lyapunov function V2=V1+(1/2)z22, the derivative of V2 is(24)V˙2=-c1z12+KtMbMus+Mbz2z3+z2z1-KtMus+Mbx1+KtMbMus+Mbα2+c1x2-z2Musξ1-z2Mbξ2.Applying Young’s inequality (for any vectors x,y∈Rn and any scalars ϵ>0, p>1, there holds xTy≤(ϵp/p)|x|p+(1/qϵq)yq, where q=p/(p-1)) to the last two terms in (24), one has(25)-z2Musξ1≤d1Mus2z22+14d1ξ12,-z2Mbξ2≤d2Mb2z22+14d2ξ22,where d1>0 and d2>0 are design parameters. Substituting (25) into (24), the resulting V˙2 is(26)V˙2≤-c1z12+KtMbMus+Mbz2z3+z2z1-KtMus+Mbx1+KtMbMus+Mbα2+c1x2+d1Mus2z2+d2Mb2z2+14d1ξ12+14d2ξ22.Choose(27)α2=Mus+MbKtMb-z1-c2z2+KtMus+Mbx1-c1x2-d1Mus2z2-d2Mb2z2,where c2>0 is a design parameter. Then (26) can be rewritten as(28)V˙2≤-c1z12-c2z22+KtMbMus+Mbz2z3+14d1ξ12+14d2ξ22.Step 3.
Introducing(29)z4=x4-α3,then(30)z˙3=z4+α3-∂α2∂x1x2-∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3-Musξ1-Mbξ2. For the Lyapunov function V3=V2+(1/2)z32, the derivative of V3 is(31)V˙3≤-c1z12-c2z22+z3z4+14d1ξ12+14d2ξ22+z3KtMbMus+Mbz2+α3-∂α2∂x1x2-∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3-z3∂α2∂x2Musξ1-z3∂α2∂x2Mbξ2.Applying Young’s inequality to the last two terms in (31), one has(32)-z3∂α2∂x2Musξ1≤d3∂α2∂x22Mus2z32+14d3ξ12,-z3∂α2∂x2Mbξ2≤d4∂α2∂x22Mb2z32+14d4ξ22,where d3>0 and d4>0 are design parameters. Then (31) can be rewritten as(33)V˙3≤-c1z12-c2z22+z3z4+z3KtMbMus+Mbz2+α3-∂α2∂x1x2-∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3+d3∂α2∂x22Mus2z3+d4∂α2∂x22Mb2z3+14d1+14d3ξ12+14d2+14d4ξ22.Choose(34)α3=-KtMbMus+Mbz2-c3z3+∂α2∂x1x2+∂α2∂x2-KtMus+Mbx1+KtMbMus+Mbx3-d3∂α2∂x22Mus2z3-d4∂α2∂x22Mb2z3,where c3>0 is a design parameter. Then(35)V˙3≤-c1z12-c2z22-c3z32+z3z4+14d1+14d3ξ12+14d2+14d4ξ22.Step 4.
From (29), one has(36)z˙4=ψx1,x2,x3,x4+1Mus+1Mbua+∂α3∂x2Mus+1ξ1+∂α3∂x2Mb-1ξ2,where ψ(x1,x2,x3,x4) = -Ka(1/Mus+1/Mb)x3+(Kt/MusMb+Mus)x1 − (KtMb/MusMb+Mus)x3-Ca(1/Mus+1/Mb)sign(x4)x4p − (∂α3/∂x1)x2-(∂α3/∂x2)(-(Kt/Mus+Mb)x1 + (KtMb/Mus+Mb)x3)-(∂α3/∂x3)x4. For the Lyapunov function V4=V3+(1/2)z42, the derivative of V4 is(37)V˙4≤-c1z12-c2z22-c3z32+14d1+14d3ξ12+14d2+14d4ξ22+z4z3+ψ+1Mus+1Mbua+z4∂α3∂x2Mus-1ξ1+z4∂α3∂x2Mb-1ξ2.Applying Young’s inequality, one has(38)z4∂α3∂x2Mus+1ξ1≤d5∂α3∂x2Mus+12z42+14d5ξ12,z4∂α3∂x2Mb-1ξ2≤d6∂α3∂x2Mb-12z42+14d6ξ22,where d5>0 and d6>0 are design parameters. Substituting (38) into (37) leads to(39)V˙4≤-c1z12-c2z22-c3z32+z4z3+ψ+1Mus+1Mbua+d5∂α3∂x2Mus+12z4+d6∂α3∂x2Mb-12z4+14d1+14d3+14d5ξ12+14d2+14d4+14d6ξ22.Choose(40)ua=MusMbMus+Mb-z3-c4z4-ψ-d5∂α3∂x2Mus+12z4-d6∂α3∂x2Mb-12z4,where c4>0 is a design parameter. Then(41)V˙4≤-∑i=14cizi2+14d1+14d3+14d5ξ12+14d2+14d4+14d6ξ22.
Up to now, the closed-loop system is obtained(42)z˙1=-c1z1+z2,z˙2=-z1-c2z2+KtMbMus+Mbz3-Mus+MbKtMbd1Mus2+d2Mb2z2-Musξ1-Mbξ2,z˙3=-KtMbMus+Mbz2-c3z3+z4-d3∂α2∂x22Mus2z3-d4∂α2∂x22Mb2z3+∂α2∂x2Musξ1+Mbξ2,z˙4=-z3-c4z4-MusMbMus+Mbd5∂α3∂x2Mus+12+d6∂α3∂x2Mb-12z4+∂α3∂x2Mus+1ξ1+∂α3∂x2Mb-1ξ2.


## 4.2. Stability Analysis

Theorem 10.
Consider the random model (17) of quarter-car active suspension. Under assumption (15), choose controller (40).
(i) The closed-loop system (42) is NSS-p and all the signals of the closed-loop system are bounded in probability.
(ii) The statez=[z1,z2,z3,z4]T of the closed-loop system satisfies(43)limt→∞⁡Ez2≤2dKc,where the right-hand can be made small enough by tuning parameters.Proof.
Obviously, the functions of the closed-loop system satisfy the local Lipschitz condition. The Lyapunov function for the whole system is(44)V=V4=∑i=1412zi2.From (41), one has(45)V˙≤-cV+dξ2,where c=2min{c1,c2,c3,c4}, d=max⁡{1/4d1+1/4d3+1/4d5,1/4d2+1/4d4+1/4d6}, and ξ=[ξ1,ξ2]2. From Lemma 3, the closed-loop system is NSS-P.
Furthermore, by definingv(t)=EV(z(t)), from (45), one has(46)v˙t=EV˙xt≤-cvt+dEξt2. From Lemma 6, one has(47)vt≤v0e-ct-t0+dsupt0≤s≤t⁡Eξs2,which together with (15) implies(48)EVt≤Vx0e-ct-t0+dKc≤Vx0+dKc.According to Lemma 5, (48) and the definition of V, z is bounded in probability. By z1=x1, x1 is bounded in probability. Since z2=x2-c1z1, x2 is bounded in probability, too. Similarly, x3 and x4 are bounded in probability. Then the control u is also bounded in probability by (40).
From (15), (44), and (47), one has(49)Ez2≤2EVt≤2Vx0e-ct-t0+2dKc,which leads to (43). Noting c=2min⁡{c1,c2,c3,c4}, d=max{1/4d1+1/4d3+1/4d5,1/4d2+1/4d4+1/4d6}, it is clear that the right-hand sides of (43) can be made small enough by choosing ci(i=1,…,4) and dj(j=1,…,6) large enough.Remark 11.
Sincez1=Muss1+Mbs2, z3=s2-s1-α2=s2-s1-((Mus + Mb)/KtMb)(-z1-c2z2+(Kt/(Mus+Mb))z1-c1(z2-c1z1)-d1Mus2z2-d2Mb2z2), then s1, s2 can be small enough by choosing parameters appropriately. As a result, the passenger feels comfortable. From Figure 2, the simulation results also demonstrate the comfortableness.Figure 2
The system simulation results.


## 5. Simulation Result

In the simulation, the disturbancesξ1, ξ2 are produced by(50)ξ˙t=-ξt+wt,ξ0=0,where w(t)∈R is a zero-mean white noise. ξ=(ξ1,ξ2) is a zero-mean widely stationary process, and Eξ12=Eξ22=0.05; thus, K=0.05. The initial value is x10=0.5, x20=0, x30=-0.5, and x40=0, the parameters of the system are Mus=0.5, Mb=1, Kt=0.8, Ka=0.5, Ca=0.1, and p=3, and design parameters are c1=0.5, c2=1, c3=0.8, c4=1.5, d1=0.2, d2=0.1, d3=0.2, d4=0.2, d5=0.2, and d6=0.1.The simulation result demonstrates the effectiveness of the control scheme.


## 6. Conclusion

The stochastic modeling and control of a quarter car with active suspension in rough road are considered in the paper. According to the relative motion principle, the influence of rough road is regarded as the force, which is disturbed by the noise. Then a stochastic model is constructed. Based on the model, using backstepping method, a controller is designed such that the mean square of the state converges to an arbitrarily small neighborhood of zero by tuning design parameters. The simulation results illustrate the effectiveness of the proposed scheme.



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*Source: 4045796-2017-05-30.xml*