# Robust Observer-BasedH∞ Controller Design for Motorcycle Lateral Dynamics

Authors: A. Ferjani; I. Zaidi; M. Chaabane
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The present work deals with the design problem of a robust observer-based controller for a motorcycle system using LPV approach. The designed model is specifically uncertain and disturbed one, whose uncertainties are related to variations of both the cornering stiffness and the longitudinal velocity. The nonlinear motorcycle model is firstly transformed on an uncertain LPV model with two vertices; then an observer-basedH∞ robust controller is designed. Both the controller and observer gain matrices are computed by solving a unified convex optimization problem under LMI constraints using YALMIP solver. Numerical simulation results are given to illustrate the effectiveness of the designed method.


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## Body

## 1. Introduction

In recent years, more and more people are moving towards fast, light, less expensive means of transport and small jigs. Among these means of transport, motorized two-wheeled vehicles are increasingly popular because of the great freedom of driving and the possibility of avoiding traffic congestion and parking difficulties.At the present time, scooters and motorcycles have become necessary means of transport in our society and the last statics of sales of these vehicles reinforce this statement. The increase in the number of motorcycles and scooters is, unfortunately, followed by an increase in accidents on the road.In these circumstances, safety in two-way transport has become a central concern for public bodies and car manufacturers. Several preventive and repressive measures have been put in place to warn drivers of the risks associated with certain behavior on the steering wheel and the handlebars. On the other hand, several research projects have been initiated in order to provide solutions in the field of prevention and driver assistance systems.In the last years, many motorcycle equipment manufacturers focus on active and semiactive safety systems. The first work to stabilize a motorcycle has been developed in [1], by considering a simple linear model to describe the motorcycle behavior. The control of motorcycles has been investigated in [2, 3] and the problem of observation and estimation of unmeasurable state variables have been studied in [4–7]. Overall, motorcycle safety system design is still an open problem and few results exist in the literature. To the best of our knowledge, there are no studies that deal with both the motorcycle stability and estimation in presence of disturbance, except the work in [7], which has studied the two problems separately, without considering external disturbance.Overall, safety system design for motorcycle is still an open problem and few results exist in the literature. In the modelling aspect, we develop an uncertain polytope that can cover the time-varying longitudinal velocity range and has few vertices. Moreover, the nonlinear tyre model is reformulated into a Linear Parameter-Varying (LPV) model with norm-bounded uncertainties. LPV models have been studied in many works like [8].Motivated by these observations, a robustH∞ observer-based controller will be investigated. The observer will be used to estimate the unmeasured variables of the motorcycle like the yaw rate, the roll rate, and the steering rate and the controller ensures the robust stability despite the variation of the road conditions and external disturbances. The less conservative LMI-based design conditions of both the observer and the controller are proposed and can be solved in one step.The rest of the paper is structured as follows: the analysis of the motorcycle described by an LPV model is introduced in Section3. Hereafter, in Section 4, the analysis and the design problems of the robust observer-based controller are studied. To show the performance of the proposed observer in different cases, some simulations are given in Section 5. Finally, conclusions are given in Section 6.


## 2. Notations and Preliminaries

The following lemmas are used to prove our results.Lemma 1 (see [9]).
Given a positive scalarϵ and two matrices G and H, the following inequality holds: (1)GTH+HTG≤ϵGTG+1ϵHTHLemma 2 (see [10]).
Considering a negative definite matrixΞ<0, a given matrix Z, and a scalar μ>0, the following inequality holds: (2)ZTΞZ<-μZ+ZT-μ2Ξ-1


## 3. Motorcycle Model Analysis

### 3.1. Nonlinear Model of the Motorcycle

The model used in this work describes the motorcycle’s lateral and roll dynamics, which are obtained by considering Sharp’s motorcycle model [11] (see Figure 1). The dynamics of a motorcycle can be represented by a model with four equations [12, 13] describing the lateral motion mainly caused by lateral forces (Fyf,Fyr) and the yaw and roll motions under rider’s steering actions.Figure 1
Illustration of motorcycle’s motion.The movements that correspond to lateral, roll, yaw, and steering motions, respectively, are expressed by the following equations:(3)Mv˙y+vxψ˙+Mfζrψ¨+d1ϕ¨+Mfeδ¨=Fyf+Fyrd1v˙y+b2ϕ¨+a2ψ¨+b1δ¨+b5vxψ˙+d3vxδ˙=∑MxMfζrv˙y+vxψ˙+a2ϕ¨+a3ψ¨+a1δ¨-a4vxϕ˙-d2vxδ˙=∑MzMfev˙y+b1ϕ¨+a1ψ¨+c1δ¨-d3vxϕ˙+c3vxψ˙+Kδ˙=∑MsUsing the moment principle, the following expressions can be obtained: (4)∑Mx=b4sin⁡ϕ-b3sin⁡δ∑Mz=lfFyf-lrFyr∑Ms=-b3sin⁡ϕ-c2sin⁡δ-ηfyf+τwhere coefficients ai, bi, ci, and di are given in the Appendix.Lateral and roll forces have the following expressions:(5)Fyf=-Cf1αf+Cf2γf,Fyr=-Cr1αr+Cr2γrwhere(6)αf=vy+lfψ˙-ηδ˙vx-δcos⁡εαr=vy-lrψ˙vxγf=ϕ+δsin⁡εγr=ϕSubstituting (4) and (5) in model (3), we obtain the following form: (7)Mv˙y+Mfζrψ¨+d1ϕ¨+Mfeδ¨=a11vxvy+a12vxψ˙+a14ϕ+a15vxδ˙+b16δd1v˙y+b2ϕ¨+a2ψ¨+b1δ¨=a22vxψ˙+a24ϕ+a25vxδ˙+b26δMfζrv˙y+a2ϕ¨+a3ψ¨+a1δ¨=a31vxvy+a32vxψ˙+a33vxϕ˙+a34ϕ+a35vxδ˙+b36δMfev˙y+b1ϕ¨+a1ψ¨+c1δ¨=a51vxvy+a52vxψ˙+a53vxϕ˙+a54ϕ+a55vxδ˙+b56δ+τwhere motorcycle parameters are defined in Nomenclature and parameters aij and bij are given in the Appendix.Let us consider state vectorxt=vyψ˙ϕ˙ϕδ˙T; system (7) can be written as follows:(8)Ex˙t=A11vxxt+B11δt+B21τtwhere(9)E=MMfζrd10MfeMfζra3a20a1d1a2b20b100010Mfea1b10c1A11vx=a11vxa12vx0a14vxa15vx0a22vx0a24a25vxa31vxa32vxa33vxa34a35vx00100a51vxa52vxa53vxa54a55vxB11=b16b26b360b56TB21=00001TMatrix E is a nonsingular matrix, so system (8) can be expressed as follows: (10)x˙t=Avxxt+B1δt+B2utwhere(11)Avx=E-1A11vx,B1=E-1B11,B2=E-1B21,ut=τtTo simplify the development, vx is written as v.In the following, we mention that the yaw control is only tripped at a nonzero of the longitudinal velocity. For this, we propose to accommodate the variation of this speed.


### 3.2. Motorcycle Uncertain LPV Model Using Polytopic Approach and Reducing Vertices

Let us consider that the longitudinal speedv is time-varying in interval vm,vM. Then, variable 1/v is time-varying in 1/vM,1/vm. For pair (v,1/v), we can use a rectangular polytope [14] to describe it.Since the choices for the parameter set(v,1/v) only occur on the solid line and most of the area inside the rectangle is not achievable, the description using a rectangular polytope could be conservative. Moreover, if the number of the polytope vertices increases, the computational load and the complexity also increase. In the following, we propose to use the idea proposed by [15] to reduce the number of the polytope vertices and also to simplify the modelling complexity.Based on [15] and considering matrix A(vx) defined in Section 3.1, for the set of the variables (v,1/v), we consider a rectangular polytope given in Figure 2.Figure 2
Illustration of transforming the rectangular polytope to the uncertain straight line segment.As is shown in Figure2, the rectangular polytope obtained by the four vertices Ω1Ω2Ω3Ω4 illustrates the variation of the set of variables (v,1/v). A straight line D1 crosses Ω1 and Ω3; then, shifting the straight line until the tangent to the hyperbola y=1/v, we obtain another straight line D2. D2 crosses the straight line Ω1Ω2 in point p2 and, on the other hand, Ω3Ω4 in point p4. Then, the line crosses the middle of lines Ω1p2 and Ω3p4 via p1 and p3. The sweep above and below D3 along the two segments Ω1p2 and Ω3p4 can achieve all possibilities of most of the area inside the polytope.The two newly obtained vertices are defined by the following coordinates:(12)p1=vm,vM-vm2vmvM+1vmvM+vm+vM2vmvM-1vmvMNtp3=vM,vm-vM2vmvM+1vmvM+vm+vM2vmvM-1vmvMNtSince Nt is less than one, the uncertain straight line segment crossing points p1 and p3 can cover the whole parallelogram Ω1p2p4Ω3. Then, based on the work of [16], each point inside the parallelogram Ω1p2p4Ω3 can be represented by a linear combination of the new uncertain vertices p1 and p3 as follows:(13)pn=μ1p1+μ2p3where μ1=(v-vm)/(vM-vm) and μ2=(vM-v)/(vM-vm) and μ1+μ2=1.The description of the two parametersv and 1/v, in expression (13), has only two vertices and it can overlay all the possibilities for the set of v and 1/v.To take into account the road condition variations, the cornering stiffness is given by the following equations:(14)C-f1=Cf1+ΔCf1,C-f2=Cf2+ΔCf2,C-r1=Cr1+ΔCr1,C-r2=Cr2+ΔCr2Considering the representation in (10) and (13), the obtained model is(15)x˙t=∑i=12μiAi+ΔAixt+B1+ΔB1δt+B2utWe define the state matrices by the first term containing nominal parameters. So, the state matrices of the multiple models are written as (16)A1=f1c11f2c11-Mvm0Cf2+Cr2f3ηc11f2c11f4c11-Mfζrvma4vmCf2lf-Cr2lrd2vm+lff3ηc110-b5vm0b4-d3vm00100f3ηc11lff3ηc11-c3vmd3vm-b3-ηCf2-K-η2f3c11A2=f1c12f2c12-MvM0Cf2+Cr2f3ηc12f2c12f4c12-MfζrvMa4vMCf2lf-Cr2lrd2vM+lff3ηc120-b5vM0b4-d3vM00100f3ηc12lff3ηc12-c3vMd3vM-b3-ηCf2-K-η2f3c12The uncertain ones are given by(17)ΔA1=G11N-tH11,ΔA2=G12N-tH12where(18)G11=f11f210ΔCf2+ΔCr2ηf31f21f410ΔCf2lf-ΔCr2lrηlff310000000000ηf31ηlff310-ηΔCf2-η2f31G12=f12f220ΔCf2+ΔCr2ηf32f22f420ΔCf2lf-ΔCr2lrηlff320000000000ηf32ηlff320-ηΔCf2-η2f32N-=N0000∗N000∗∗N00∗∗∗N0∗∗∗∗NH11=I5,H12=I5B1=Cf1cos⁡ε+Cf2sin⁡εlfCf1cos⁡ε+Cf2sin⁡ε-b30Cf1cos⁡ε+Cf2sin⁡εΔB1=G2N-tH2where(19)G2=ΔCf1cos⁡ε+ΔCf2sin⁡εlfΔCf1cos⁡ε+ΔCf2sin⁡ε00ΔCf1cos⁡ε+ΔCf2sin⁡εH2=I5,1with (20)c11=vM-vm2vmvM+1vmvM,d11=vm+vM2vmvM-1vmvMc12=vm-vM2vmvM+1vmvM,d12=vm+vM2vmvM-1vmvMf1=-Cf1-Cr1,f2=-Cf1lf-Cr1lr,f3=Cf1,f4=-lf2Cf1+lr2Cr1.For more details, see [16].


## 3.1. Nonlinear Model of the Motorcycle

The model used in this work describes the motorcycle’s lateral and roll dynamics, which are obtained by considering Sharp’s motorcycle model [11] (see Figure 1). The dynamics of a motorcycle can be represented by a model with four equations [12, 13] describing the lateral motion mainly caused by lateral forces (Fyf,Fyr) and the yaw and roll motions under rider’s steering actions.Figure 1
Illustration of motorcycle’s motion.The movements that correspond to lateral, roll, yaw, and steering motions, respectively, are expressed by the following equations:(3)Mv˙y+vxψ˙+Mfζrψ¨+d1ϕ¨+Mfeδ¨=Fyf+Fyrd1v˙y+b2ϕ¨+a2ψ¨+b1δ¨+b5vxψ˙+d3vxδ˙=∑MxMfζrv˙y+vxψ˙+a2ϕ¨+a3ψ¨+a1δ¨-a4vxϕ˙-d2vxδ˙=∑MzMfev˙y+b1ϕ¨+a1ψ¨+c1δ¨-d3vxϕ˙+c3vxψ˙+Kδ˙=∑MsUsing the moment principle, the following expressions can be obtained: (4)∑Mx=b4sin⁡ϕ-b3sin⁡δ∑Mz=lfFyf-lrFyr∑Ms=-b3sin⁡ϕ-c2sin⁡δ-ηfyf+τwhere coefficients ai, bi, ci, and di are given in the Appendix.Lateral and roll forces have the following expressions:(5)Fyf=-Cf1αf+Cf2γf,Fyr=-Cr1αr+Cr2γrwhere(6)αf=vy+lfψ˙-ηδ˙vx-δcos⁡εαr=vy-lrψ˙vxγf=ϕ+δsin⁡εγr=ϕSubstituting (4) and (5) in model (3), we obtain the following form: (7)Mv˙y+Mfζrψ¨+d1ϕ¨+Mfeδ¨=a11vxvy+a12vxψ˙+a14ϕ+a15vxδ˙+b16δd1v˙y+b2ϕ¨+a2ψ¨+b1δ¨=a22vxψ˙+a24ϕ+a25vxδ˙+b26δMfζrv˙y+a2ϕ¨+a3ψ¨+a1δ¨=a31vxvy+a32vxψ˙+a33vxϕ˙+a34ϕ+a35vxδ˙+b36δMfev˙y+b1ϕ¨+a1ψ¨+c1δ¨=a51vxvy+a52vxψ˙+a53vxϕ˙+a54ϕ+a55vxδ˙+b56δ+τwhere motorcycle parameters are defined in Nomenclature and parameters aij and bij are given in the Appendix.Let us consider state vectorxt=vyψ˙ϕ˙ϕδ˙T; system (7) can be written as follows:(8)Ex˙t=A11vxxt+B11δt+B21τtwhere(9)E=MMfζrd10MfeMfζra3a20a1d1a2b20b100010Mfea1b10c1A11vx=a11vxa12vx0a14vxa15vx0a22vx0a24a25vxa31vxa32vxa33vxa34a35vx00100a51vxa52vxa53vxa54a55vxB11=b16b26b360b56TB21=00001TMatrix E is a nonsingular matrix, so system (8) can be expressed as follows: (10)x˙t=Avxxt+B1δt+B2utwhere(11)Avx=E-1A11vx,B1=E-1B11,B2=E-1B21,ut=τtTo simplify the development, vx is written as v.In the following, we mention that the yaw control is only tripped at a nonzero of the longitudinal velocity. For this, we propose to accommodate the variation of this speed.


## 3.2. Motorcycle Uncertain LPV Model Using Polytopic Approach and Reducing Vertices

Let us consider that the longitudinal speedv is time-varying in interval vm,vM. Then, variable 1/v is time-varying in 1/vM,1/vm. For pair (v,1/v), we can use a rectangular polytope [14] to describe it.Since the choices for the parameter set(v,1/v) only occur on the solid line and most of the area inside the rectangle is not achievable, the description using a rectangular polytope could be conservative. Moreover, if the number of the polytope vertices increases, the computational load and the complexity also increase. In the following, we propose to use the idea proposed by [15] to reduce the number of the polytope vertices and also to simplify the modelling complexity.Based on [15] and considering matrix A(vx) defined in Section 3.1, for the set of the variables (v,1/v), we consider a rectangular polytope given in Figure 2.Figure 2
Illustration of transforming the rectangular polytope to the uncertain straight line segment.As is shown in Figure2, the rectangular polytope obtained by the four vertices Ω1Ω2Ω3Ω4 illustrates the variation of the set of variables (v,1/v). A straight line D1 crosses Ω1 and Ω3; then, shifting the straight line until the tangent to the hyperbola y=1/v, we obtain another straight line D2. D2 crosses the straight line Ω1Ω2 in point p2 and, on the other hand, Ω3Ω4 in point p4. Then, the line crosses the middle of lines Ω1p2 and Ω3p4 via p1 and p3. The sweep above and below D3 along the two segments Ω1p2 and Ω3p4 can achieve all possibilities of most of the area inside the polytope.The two newly obtained vertices are defined by the following coordinates:(12)p1=vm,vM-vm2vmvM+1vmvM+vm+vM2vmvM-1vmvMNtp3=vM,vm-vM2vmvM+1vmvM+vm+vM2vmvM-1vmvMNtSince Nt is less than one, the uncertain straight line segment crossing points p1 and p3 can cover the whole parallelogram Ω1p2p4Ω3. Then, based on the work of [16], each point inside the parallelogram Ω1p2p4Ω3 can be represented by a linear combination of the new uncertain vertices p1 and p3 as follows:(13)pn=μ1p1+μ2p3where μ1=(v-vm)/(vM-vm) and μ2=(vM-v)/(vM-vm) and μ1+μ2=1.The description of the two parametersv and 1/v, in expression (13), has only two vertices and it can overlay all the possibilities for the set of v and 1/v.To take into account the road condition variations, the cornering stiffness is given by the following equations:(14)C-f1=Cf1+ΔCf1,C-f2=Cf2+ΔCf2,C-r1=Cr1+ΔCr1,C-r2=Cr2+ΔCr2Considering the representation in (10) and (13), the obtained model is(15)x˙t=∑i=12μiAi+ΔAixt+B1+ΔB1δt+B2utWe define the state matrices by the first term containing nominal parameters. So, the state matrices of the multiple models are written as (16)A1=f1c11f2c11-Mvm0Cf2+Cr2f3ηc11f2c11f4c11-Mfζrvma4vmCf2lf-Cr2lrd2vm+lff3ηc110-b5vm0b4-d3vm00100f3ηc11lff3ηc11-c3vmd3vm-b3-ηCf2-K-η2f3c11A2=f1c12f2c12-MvM0Cf2+Cr2f3ηc12f2c12f4c12-MfζrvMa4vMCf2lf-Cr2lrd2vM+lff3ηc120-b5vM0b4-d3vM00100f3ηc12lff3ηc12-c3vMd3vM-b3-ηCf2-K-η2f3c12The uncertain ones are given by(17)ΔA1=G11N-tH11,ΔA2=G12N-tH12where(18)G11=f11f210ΔCf2+ΔCr2ηf31f21f410ΔCf2lf-ΔCr2lrηlff310000000000ηf31ηlff310-ηΔCf2-η2f31G12=f12f220ΔCf2+ΔCr2ηf32f22f420ΔCf2lf-ΔCr2lrηlff320000000000ηf32ηlff320-ηΔCf2-η2f32N-=N0000∗N000∗∗N00∗∗∗N0∗∗∗∗NH11=I5,H12=I5B1=Cf1cos⁡ε+Cf2sin⁡εlfCf1cos⁡ε+Cf2sin⁡ε-b30Cf1cos⁡ε+Cf2sin⁡εΔB1=G2N-tH2where(19)G2=ΔCf1cos⁡ε+ΔCf2sin⁡εlfΔCf1cos⁡ε+ΔCf2sin⁡ε00ΔCf1cos⁡ε+ΔCf2sin⁡εH2=I5,1with (20)c11=vM-vm2vmvM+1vmvM,d11=vm+vM2vmvM-1vmvMc12=vm-vM2vmvM+1vmvM,d12=vm+vM2vmvM-1vmvMf1=-Cf1-Cr1,f2=-Cf1lf-Cr1lr,f3=Cf1,f4=-lf2Cf1+lr2Cr1.For more details, see [16].


## 4. Analysis and Synthesis of the RobustH∞ Observer-Based Controller

The objective is to design a robust observer-basedH∞ controller such that the closed-loop system is globally stable and the H∞ performances are guaranteed.Based on LPV model (15), the observer and the controller are defined as (21)x^˙t=∑i=12μiAix^t+Liyt-y^t+B2ut+B1δty^t=Cx^tut=∑i=12μiKix^twhere x^(t) is the estimated state and Ki(i=1:2) are the controller gains to be determined, Li are the observer gains to be determined, as the yaw rate, the roll rate, and the steering rate can be generally measured, and C is an adequate matrix which is given by(22)C=010000010000001Define the estimation error as (23)e0t=xt-x^tThe derivative of the error estimation is given by(24)e˙0t=∑i=12μiΔAixt+Ai-LiCe0t+ΔB1δtBy using (15), (21), and (24), we can express the augmented system in the following form: (25)X˙t=∑i=12μiA-iXt+B-δtwhere(26)Xt=xte0tA-i=Ai+B2Ki+ΔAi-B2KiΔAiAi+LiC,(27)B-=B1+ΔB1ΔB1To improve motorcycle’s lateral dynamics stability, the objective in H∞ controller design is to possess robustness against external disturbance (steering angle) and uncertainties (cornering stiffness and the longitudinal velocity), which guarantees a given attenuation level of disturbance rejection attenuation.The objective of this work is to design an observer-basedH∞ controller such that the following requirements are satisfied:(i)
Ensure the stability for closed-loop system (25) with the controller when δ(t)=0(ii)
Ensure disturbance rejection for system (25) such that (28)Xt∞≤γ2δt∞The stability conditions of augmented system (25) are given in terms of LMIs in the following theorem and can be solved in one step.Theorem 3.
For given scalarsμ and ϵ2, system (25) is asymptotically stable via the robust observer-based H∞ controller (21), if there exist positive definite matrices Z and P2 and matrices Mi, Ji, and S1, linear variables ϵ1, ϵ3, and ϵ4, and a positive scalar γ, which guarantees the H∞ performance, such that the following LMIs are verified:(29)θi1ZH1TZH1T-B2Mi0000B10∗-ε1I00000000∗∗-ε2I0000000∗∗∗-2μZ0μI0000∗∗∗∗-2μI000μI0∗∗∗∗∗θi2ε2P2G1iP2G20μI∗∗∗∗∗∗-ε2I000∗∗∗∗∗∗∗-ε4I00∗∗∗∗∗∗∗∗θ3H2T∗∗∗∗∗∗∗∗∗-ε3I<0where (30)θi1=ZAiT+AiZ+MiTB2T+B2Mi+S1+ε1G1iG1iT+ε3G2G2Tθi2=AiTP2+P2Ai-JiTP2-JiC+Iθi3=-γ2I+ε4H2TH2And the gains of the controller and the observer are, respectively, (31)Ki=MiZ-1(32)Li=P2-1JiProof.
The Lyapunov function is chosen as(33)VXt=XTtP-Xtwhere the Lyapunov matrix P is chosen as (34)P-=P100P2The derivation of the Lyapunov function gives(35)V˙Xt=∑i=12μiA-iXt+B-δtTP-Xt+XTtP-A-iXt+B-δtThen, objective (28) is guaranteed by ensuring the following inequality: (36)V˙X+XTX-γ2δTδ≤0Therefore, we have(37)∑i=12μiXδTA-iTP-+P-A-i+IP-B-B-TP--γ2IXδThen, inequality (36) holds if the following condition is satisfied: (38)A-iTP-+P-A-i+IP-B-B-TP--γ2I≤0By substituting the expressions of A-i and B- defined in (26) and (27), inequality (38) is equivalent to(39)ϖi-P1B2Ki+ΔAiTP2P1B1+ΔB1∗ϑiP2ΔB1∗∗-γ2I≤0where(40)ϖi=AiTP1+P1Ai+KiTB2TP1+P1B2Ki+ΔAiTP1+P1ΔAi+Iϑi=AiTP2+P2Ai-CTLiTP2-P2LiC+IIn the following step, we expand matrix (39) in three matrices; that is, (41)Θi=Ti+Λi+ΛiTwith (42)Ti=ψi-P1B2KiP1B1∗ϕi0∗∗-γ2IΛi=P1ΔAiΔAiTP2P1ΔB100P2ΔB1000and (43)ψi=AiTP1+P1Ai+KiTB2TP1+P1B2Ki+Iϕi=AiTP2+P2Ai-CTLiTP2-P2LiC+IAccording to expressions of ΔAi and ΔB1 and using Lemma 1, inequality (39) holds if there exist real scalars ϵ1, ϵ2, ϵ3, and ϵ4 satisfying (44)Θi=ΠiΥi∗Ξi≤0with (45)Πi=AiTP1+P1Ai+KiTB2TP1+P1B2Ki+I+ε1-1H1TH1+ε1P1G1iG1iTP1+ε2-1H1TH1+ε3P1G2G2TP1Υi=-P1B2KiP1B1Ξi=χi00-γ2I+ε3-1H2TH2+ε4H2TH2χi=AiTP2+P2Ai-CTLiTP2-P2LiC+I+ε2P2G1iG1iTP2+ε4-1P2G2G2TP2Prepost multiplying (44) by the matrix diag(Z,Y) with Z=P1-1 and Y=diag(Z,I), we obtain (46)diagZ,YΘidiagZ,Y=Π-iΥ-i∗Ξ-iwith (47)Π-i=ZΠiZ=ZAiT+AiZ+MiTB2T+B2Mi+S1+ε1-1ZH1TH1Z+ε1G1iG1iT+ε2-1ZH1TH1Z+ε3G2G2TΥ-i=ZΥiY=-B2MiB1Ξ-i=YΞiY=ZχiZ00-γ2I+ε3-1H2TH2+ε4H2TH2Matrices Mi, Ji, and S1 are given by(48)Mi=KiZ,Ji=P2Li,S1=ZZThen, χi is rewritten as follows:(49)χi=AiTP2+P2Ai-CTJiT-JiC+I+ε2P2G1iG1iTP2+ε4-1P2G2G2TP2Using Lemma 2, there exists a positive scalar μ such that matrices Ξ-i can be rewritten: (50)YΞiY≤-2μY-μ2Ξi-1Using Schur’s complement, (50) can be written as (51)Ξ-i=-2μYμIμIΞiwhich is equivalent to (52)Ξ-i=-2μZ0μI0∗-2μI0μI∗∗χi0∗∗∗-γ2I+ε3-1H2TH2+ε4H2TH2Using Schur’s complement, sufficient conditions in Theorem 3 are established.


## 5. Simulation Results

To show the performance and the effectiveness of the proposed observer-based controller law, we have considered both the variation of the longitudinal velocity as given in Figure3 and the cornering stiffness. The steering angle is given by Figure 4.Figure 3
Longitudinal velocity.Figure 4
Steering angle.Longitudinal velocity is varying betweenvm=11 m·s−1 and vM=18 m·s−1. The reduced model using two-vertex approach is firstly parameterized. Taking into account the uncertainties of the cornering stiffness which are 12% of the nominal values given in Table 1, the observer-based robust state feedback controller is then designed. Figures 5–9 show the performance of the designed observer-based controller by considering fixed parameters μ=1 and ε2=0.8 and the following initial conditions x0=0.2000.20T and x^(0)=0.020.50.5110T.Table 1
Main parameter values for the motorcycle.
Symbol Value M 248.1 Kg C f 1, Cf2 11173.9327 N⋅rad−1, 9385 N⋅rad−1 C r 1,Cr2 15831.2207 N⋅rad−1, 1325.5700 N⋅rad−1 lf 0.829 m lr 0.585 mFigure 5
Lateral velocity.Figure 6
Yaw rate.Figure 7
Roll rate.Figure 8
Roll angle.Figure 9
Control input.K i and Li controller gains are given by(53)K1=1.0e+03∗1.0375-1.28010.9633-8.8939-0.7356K2=1.0e+03∗0.3112-0.49610.3322-2.9184-0.3437L1=1.0e+03∗-0.01160.0448-0.0214-0.03990.01990.18540.0188-0.08520.3045-0.0000-0.00000.00000.26790.2131-9.8091L2=1.0e+03∗-0.01190.0447-0.0214-0.03960.02010.18550.0192-0.08510.3045-0.0000-0.00000.00000.26620.2106-9.8095MatrixZ is given by(54)Z=19.99574.1994-15.66860.27073.62784.199411.54282.8784-0.4382-23.0038-15.66862.87842.8784-0.4442-6.51090.2707-0.4382-0.44420.0438-0.04013.6278-23.0038-6.5109-0.04019.8105Parametersϵ1, ϵ3, and ϵ4 are obtained as(55)ϵ1=0.8910ϵ3=0.5028,ϵ4=0.5117Figures 5–9 show the results of a driving test with varying longitudinal velocity, Figures 5–8 present the comparison between one of the states of the model and its estimate from the robust H∞ observer-based controller. These figures show the good estimation, stabilization, and the robustness with respect to uncertainties. It can be seen that there is a close approximation of the estimated and measured lateral velocity, yaw rate, roll rate, and roll angle of the motorcycle. The yaw rate, however, shows a deviation that is related to steering angle of the motorcycle. We can also see that the designed robust H∞ observer-based controller improves the stability and safety under some unmeasured states and cornering manoeuver.


## 6. Conclusion

In this paper, the problem of the design of the observer-based robust control of the lateral dynamics of the motorcycle is studied. Using LPV approach and some geometrical transformations, the nonlinear motorcycle model is rewritten in uncertain LPV model form. Then, a robust observer-based controller has been designed using LMI approach. To prove the performance of the controller, simulations using Matlab-SIMULINK toolbox have been presented. The comparisons of the estimated and measured states evolution and then the stabilization and the robustness under some unmeasured states and cornering manoeuver with respect to parameters uncertainties of the LPV representation of lateral dynamic motorcycle model are only treated in this manuscript.



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*Source: 2158129-2018-06-10.xml*