強健性與可靠度控制在電動載具上之研究

國

立

交

通

大

學

電控工程研究所

碩士論文

強健性與可靠度控制在電動載具上之研究

Study of Robust and Reliable Control

for Electric Vehicles

研 究 生：魏源廷

指導教授：梁耀文 博士

強健性與可靠度控制在電動載具上之研究

Study of Robust and Reliable Control

for Electric Vehicles

研 究 生：魏源廷

Student: Yuan-Tin Wei

指導教授：梁耀文 博士 Advisor: Dr. Yew-Wen Liang

國立交通大學電控工程研究所

碩士論文

A Thesis

Submitted to institute of Electrical and Control Engineering

College of Electrical Engineering and Computer Science

National Chiao Tung University

In Partial Fulfillment of the Requirements

For the Degree of Master

In

Electrical and Control Engineering

July 2010

Hsinchu, Taiwan, Republic of China

強健性與可靠度控制在電動載具上之研究

研究生：魏源廷

指導教授：梁耀文 博士

國立交通大學電控工程研究所

摘要

本論文首先針對電動載具系統具有不確定參數與外在干擾提出強健控

制方法，利用State Dependent Riccati Equation設計法可預測系統性能的特

性，積分型順滑模控制具有強健性及擁有額外設計自由度等優點來提升電

動載具煞車系統之強健性，並且應用在三輪載具，模擬結果顯示了所提之

結合方法可以提升系統性能。除此之外，由於電動載具對於行車安全之需

求，本論文也利用了順滑模可靠度控制與錯誤偵測與估測機制針對當部分

輪胎制動器完全損壞時，提升電動載具變換車道時的操控性。模擬結果顯

示了，當一個、兩個或三個輪胎制動器完全損壞時，電動載具仍能達到良

好的可靠度性能。

Study of Robust and Reliable Control for Electric Vehicles

Student: Yuan-Tin Wei

Advisor: Dr. Yew-Wen Liang

Institute of Electrical and Control Engineering

National Chiao Tung University

ABSTRACT

In this thesis, we first address the issues regarding the robust stabilization of a

class of nonlinear uncertain systems using a combined scheme. The scheme

consists of the state-dependent Riccati equation (SDRE) technique for the

control of nominal systems and the integral sliding mode control (ISMC)

strategy to compensate for the error when the system state deviates from the

nominal system trajectory. It is shown that the state of the uncertain system

using the combined scheme and that of the nominal system under the SDRE

scheme are identical when the uncertainties are of the matched type. These

analytic results are also applied to the brake control of a three-wheeled vehicle.

Simulation results show that the control under the combined scheme could be

intelligently adjusted so that the yaw rate and lateral velocity is as small as

possible. Second, we deal with the reliable issues of yaw rate tracking for

electric vehicles using yaw moment control strategy. The presented reliable

controllers are organized via the sliding mode control technique. As a result, this

reliable scheme is robust and is shown to be able to tolerate some of the

actuators' faults. Simulation results demonstrate the benefits of the approach.

誌 謝

感謝許多人的關心與協助，使本論文能夠順利完成。首先特別要感謝

指導教授梁耀文博士的用心指導，感謝老師兩年來細心與耐心的指導及對

我孜孜不倦的教誨，使我不僅在研究過程中受益良多，且在待人處世各方

面有許多成長。同時也要感謝口試委員鄧清政博士、廖德誠博士和徐勝均

博士給予建設性的建議與指導使得本論文更臻完備。此外，感謝當我在台

灣科技大學求學時，時常鼓勵我的黃世欽博士與黃安橋博士，由於你們的

鼓勵，當我在研究上遇到瓶頸時，依然能夠保持活力與自信面對與處理複

雜的問題。

接下來要感謝王士昕學長、丁立偉學長、吳家榮學長以及徐勝均學長

在我遇到困難時能給予適時的幫助與鼓勵。謝謝實驗室的同學宜展、立岡

和旭志，學弟智強、君豪、榮人以及偉庭使我兩年研究生涯多采多姿充滿

回憶。謝謝你們的關懷與照顧，沒有你們的幫忙，論文不可能順利完成。

最後要感謝我的家人的大力支持與鼓勵，讓我可以無後顧之憂的在學業

上勇往直前，更要感謝在交大求學生涯中陪伴著我的好友佳穗，謝謝你們，

謹將此論文獻給你們。

TABLE OF CONTENTS

i

ii

iii

iv

vi

LIST OF FIGURES...

vii

1. INTRODUCTION

1.1. Motivation ... 1

1.2. Outline ... 5

2. PRELIMINARIES

2.1. State Dependent Riccati Equation... 6

2.2. Integral Sliding Mode Control... 7

2.3. Sliding Mode Control Design... 9

2.4. Burckhardt Tire Friction Model………. 11

2.5 Magic Formula for Longitudinal and Lateral Force………... 12

3. AN INTELLIGENT SDRE AND ISMC COMBINED SCHEME WITH

APPLICATION TO VEHICLE BRAKE CONTROL

3.1. Problem Formulation ... 15

3.2. Controller Design ... 16

3.2.1. SDRE Design for Nominal System ... 16

3.2.2. ISMC Design for the Compensation of Uncertainties ... 17

3.3. Simulation Results ... 19

3.3.1. Vehicle dynamics..……….. 19

3.3.2. Employment of the Combined Scheme ... 22

3.3.3. Simulation Results... 23

APPENDIX 3.A... 29

4. STUDY OF RELIABLE YAW MOMENT CONTROL FOR ELECTRIC

VEHICLE

4.1. Problem Formulation ... 31

4.2. Reliable Controller Design…... 33

4.2.1. Output tracking formulation……..………. 33

4.2.2. SMC reliable design………..……….. 34

4.2.3. FDD mechanism….………..……….. 36

4.3. Simulation Results ... 37

4.3.1. One actuator fault……….…..………. 38

4.3.2. Two actuator faults……..………..……….. 39

4.3.3. Three actuator faults……..………..………...………. 40

5. CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

REFERENCES ...

LIST OF TABLES

Table 2.1. Parameters for friction coefficient characteristics (Burckhardt Tire Friction Model)……...12 Table 2.2. Magic Formula coefficient…... 13

LIST OF FIGURES

Figure 2.1. Typical friction coefficient... 12

Figure 2.2. Longitudinal force use Magic Formula... 13

Figure 2.3. Cornering force use Magic Formula…... 14

Figure 3.1. Three wheeled vehicle model………... 20

Figure 3.2. Time history of the first four system states... 25

Figure 3.3. Time history of the last three system states... 26

Figure 3.4. Time history of the four sliding variables for SDRE+ISMC scheme... 26

Figure 3.5. Time history of the four control inputs…... 27

Figure 3.6. Time history of the first four system states……….. 27

Figure 3.7. Time history of the last three system states…………...…... 28

Figure 3.8. Time history of the four sliding variables for SDRE+ISMC scheme……….. 28

Figure 3.9. Time history of the four control inputs ………..……... 29

Figure 4.1. Four wheeled vehicle model………... 31

Figure 4.2. Time histories of (a) longitudinal velocity, (b) body side slip angle, (c) yaw rate and (d) output tracking error for one actuator fault... 42

Figure 4.3. Time histories of (a) front-right, (b) front-left, (c) rear-right and (d) rear-left wheel angular speeds for one actuator fault…... 43

Figure 4.4. (a) Residual signals; (b) alarm signal; (c) controls of RSMC and (d) Controls of SMC for one actuator fault…... 43

Figure 4.5. Sliding variables of (a) RSMC and (b) SMC schemes for one actuator fault.. 44

Figure 4.6. Time histories of (a) longitudinal velocity, (b) body side slip angle, (c) yaw rate and (d) output tracking error for two actuator faults...………..44

Figure 4.7. Time histories of (a) front-right, (b) front-left, (c) rear-right and (d) rear-left wheel angular speeds for two actuator faults... 45

Figure 4.8. (a) Residual signals; (b) alarm signal; (c) controls of RSMC and (d) Controls of SMC for two actuator faults...45 Figure 4.9. Sliding variables of (a) RSMC and (b) SMC schemes for two actuator faults 46 Figure 4.10. Time histories of (a) longitudinal velocity, (b) body side slip angle, (c) yaw

rate and (d) output tracking error for three actuator faults...46 Figure 4.11. Time histories of (a) front-right, (b) front-left, (c) rear-right and (d) rear-left

wheel angular speeds for three actuator fauls... 47 Figure 4.12. (a) Residual signals; (b) alarm signal; (c) controls of RSMC and (d)

Controls of SMC for three actuator faults………47 Figure 4.13. Sliding variables of (a) RSMC and (b) SMC schemes for three actuator

faults... 48

CHAPTER ONE

INTRODUCTION

1.1

Motivation

Recently, electric vehicles have achieved remarkable driving performance and has strong incentives of energy efficiency. The advantages of Electric Vehicle (EV) can be summarized as follows [15]:

1. Torque generation of an electric motor is fast and precise. The electric motor’s torque response is several milliseconds, 10-100 times as fast as that of the internal combustion engine or hydraulic braking system. Because a motor can generate both acceleration or deceleration torque, therefore, the electric motor’s can be integrated high performance antilock braking system and traction control system with minor feedback control.

2. A motor can be attached to each wheel. We can distribute motor location to en-hance the performance of Vehicle Stability Control (VSC) such as Direct Yaw Con-trol (DYC). It is not allowed for an Internal Combustion engine Vehicle (ICV) to use four engines, however, EV is permitted to insert four motors without increase significantly cost.

3. Motor torque can be measured easily. This advantage will contribute greatly to application of new control strategies based on road condition estimation.

To improve the safety of the driver and passengers without sacrificing the stability and steering ability of a vehicle, an antilock braking system (ABS) had been proposed to realize maximum negative acceleration, while preventing the wheels from locking [2], [14], [19], [20], [30]. Among the existing studies, there are mainly two methods of realizing the

ABS. One method uses the slip ratio to adjust the duration of brake signal pulses, i.e., to discretely “pump” the brakes [19], [30]. The other uses the fact that the friction between the road and the tire is a nonlinear function of wheel slip to regulate the slip ratio at its optimum value so that the vehicle has maximum deceleration [2], [14], [20]. In this thesis, we uses the fact that the friction between the road and the tire is a nonlinear function of wheel slip to regulate the slip ratio at its optimum value so that the vehicle has maximum deceleration to realize the ABS.

Moreover, the studies of electric vehicles’ (EVs’) yaw moment control and reliable (or fault tolerant) control have attracted considerable attention [4], [8], [10], [11], [18], [24]-[28], [34]-[36], [38]-[41]. EVs are known to have many advantages [11], [38], including 1) the motor torque can be measured easily and be controlled more precisely; 2) the individual motor at each wheel can generate differential distribution of braking/driving forces between the right and the left sides of the vehicle; 3) the motor enhances the diagnostic capability of the braking system. Thus, the lateral stability of EVs can be improved significantly if an appropriate controller is organized. Among the various control designs for EVs, it was reported that the yaw moment control is one of the most effective means for active chassis control, which may considerably improve the vehicle stability and controllability [10]. The yaw moment control is also an important technique behind the vehicle dynamic control systems, especially for controlling the lateral motion of a vehicle during a severe driving maneuver [10]. From the advantages mentioned above, EVs have the ability to generate more accurate yaw moment than the conventional vehicles.

Recently, the study of nonlinear control using the state-dependent Riccati equation (SDRE) approach and the integral sliding mode control (ISMC) design have attracted considerable attention [1], [5]-[7], [31], [32]. The SDRE scheme factorizes the nonlinear drift term into a (nonunique) linear structure with state-dependent coefficient (SDC) matrices and then employs the linear quadratic regulation (LQR) technique to organize an optimal controller at every nonzero state. This adoption of the LQR strategy is intuitive, and yet provides a systematic and effective controller design. The benefits of the SDRE design also include 1) the ability to predictably address system performance through the specification of the performance index by adjusting the state and the control

weightings, for instance, the engineer may tune up the weightings on the system state to speed up the response at the expense of more control effort, 2) an extra design degree of freedom arising from the non-uniqueness of the SDC representation of the nonlinear drift term, which can be utilized to enhance controller performance and 3) the preservation of the essential system nonlinearities, as it does not truncate any system nonlinear term. Many practical and meaningful applications that are successfully performed by the SDRE design, including vehicle control, have been reported [1], [7], [31], [32]. However, due to the direct adoption of the LQR strategy at every nonzero state, the SDRE design might not robustly compensate for model uncertainties and/or disturbances, which can also be observed from the simulations in this thesis. Hence, the SDRE scheme should incorporate another robust scheme for better performance.

On the other hand, the sliding mode control (SMC) design is known to have the benefits of rapid response, easy implementation and robustness to model uncertainties and/or external disturbances [2], [3], [5], [6], [21]-[23], [33]. However, it has been reported that the resulting closed-loop system might be sensitive to uncertainties and/or disturbances during the period in which the system state has not yet reached the sliding manifold [5]. To overcome the reaching phase problem, an integral sliding mode control design, which guarantees that the system trajectories will start in the manifold from the first time instant, has been studied recently [5], [6]. In addition to the absence of a reaching phase feature, the ISMC design also maintains the above-mentioned advantages of the SMC and has the following three characteristics: 1) the matched uncertainties and/or disturbances will be completely rejected whenever the system state remains on the sliding manifold, 2) the maximum control magnitude required for ISMC is usually smaller than those of SMC designs because the maximum control magnitude of SMC designs usually occurs at the beginning of reaching phase period and 3) the states of the nominal system and the matched-type uncertain system are exactly the same if the system state stays on the sliding manifold. The last feature provides an extra degree of freedom to organize a suitable controller for the nominal system, creating a desired system state trajectory for the state of the uncertain system to follow. In light of the benefits of the SDRE and the ISMC approaches mentioned above, in this thesis, investigates the ABS controller design

issues from the ISMC viewpoint, while adopting the SDRE design for nominal system. It is known that a control system is not to be able to operate in normal (non-faulty) situation all the time, and the repair and maintenance services are in general not able to be provided instantly. These make the reliable control issues of paramount importance. The objective of reliable control is to design an appropriate controller such that the closed-loop system can tolerate abnormal operation of specific control components and retain the overall system stability with acceptable system performance. Within the existing reliable control studies, several approaches have been presented. These approaches include the linear-matrix inequality-based approach [24], the algebraic Riccatti equation-based approach [35], the coprime factorization approach [36], the Hamilton-Jacobi (HJ)-based approach [27], and the sliding-mode control (SMC)-based approach [22], [28]. Among the aforementioned reliable control studies, only the HJ-based and the SMC-based approaches deal with reliability issues for nonlinear systems. However, because the HJ-based approach was designed under an optimal strategy, its reliable controller is inevitably dependent upon the solution of an associated HJ equation, which is, in general, difficult to solve. Although a power series method [13] may alleviate the difficulty through computer calculation, the solution obtained is only approximate, and the computational load grows quickly when the system is complicated. In contrast, the SMC reliable controllers [22] do not require the solution of any HJ equation, and they retain the advantages of conventional SMC designs. Those advantages include rapid response, easy implementation, and robustness to model uncertainties and/or external disturbances.

In EVs, a fault might happen in sensor or actuator (see e.g., [8], [34], [38], [41]) that results in hazard, such as loss of steering, loss of traction force for wheel by wheel motor, brake system and inverter failure [41]. However, due to the limit of space, a vehicle is in general not allowed to insert a backup of control component. Thus, it is important for a vehicle using its analytic redundancy to equip with a suitably fault detection and diagnosis (FDD) mechanism and an active fault tolerant controller for ensuring driving safety when fault happens. For instance, when wheel actuator(s) experiences fault, the vehicle is likely to spin and result in catastrophic situation. Thus, a yaw moment reliable controller is expected to activate for preventing possible driving instability. Two recent papers

have investigated this reliable issue from feedback linearization approach [8] and LQR scheme [41]. Though the proposed two schemes are able to enhance the driving safety, one of them only considers the linear case [41] (i.e., it considers the issue based on the linearized model at a free-rolling equilibrium point), and the mentioned two approaches require incorporating a robust scheme for better robustness performance when the EVs experience model uncertainties, measurement noises and/or external disturbances. In light of the many benefits of SMC reliable design as mentioned above, in this thesis we will study the reliable control for EVs’ nonlinear model from the SMC approach viewpoint.

1.2

Outline

This thesis is organized as follows. In Chapter 2, we recall some basic concepts of state dependent Riccati equation (SDRE), integral sliding mode control, sliding mode control, Burckhardt tire friction model and Magic Formula. Chapter 3 proposes an intelligent SDRE and ISMC combined scheme with application to vehicle bake control, and the as-sociated simulations. Chapter 4 gives detailed of reliable yaw moment control for electric vehicle. Finally, in Chapter 5, we give the conclusions and suggestions for the researches in the future.

CHAPTER TWO

PRELIMINARIES

2.1

State Dependent Riccati Equation

In this section, we recall the concepts of State Dependent Riccati Equation (SDRE) [1], [5]-[7], [31], [32].

Consider the following class of nonlinear control systems

˙x = f(x) + B(x)u (2.1)

where x ∈ IRnand u ∈ IRmdenote the system and control inputs, respectively, f(x) ∈ IRn,

B(x) ∈ IRn×m _{and f(0) = 0. In addition, we consider the following performance index}

J =

Z ∞

0

h

xTQ(x)x + uTR(x)uidt (2.2)

where QT_{(x) = Q(x) ≥ 0 and R}T_{(x) = R(x) > 0. The procedure of SDRE is summarized}

as follows:

• Symbolically factorize f(x) into the form of f(x) = A(x)x, where A(x) ∈ IRn×n. • Check the stabilizability of (A(x), B(x)) and the observability of (A(x), C(x))

sym-bolically to ensure the solvability of the following SDRE:

AT(x)P (x) + P (x)A(x) − P (x)B(x)R−1(x)BT(x)P (x) + Q(x) = 0. (2.3)

where C(x) ∈ IRp×n has full rank and satisfies Q(x) = CT_(x)C(x).

• Solve the SDRE for P (x) to produce the SDRE controller u = −R−1_(x)BT_{(x)P (x)x.}

Note that the SDRE scheme to the stabilization of nonlinear control systems need to symbolically factorize the drift term in the form of f(x) = A(x)x, and then using this A(x)

to check system’s stabilizability and observability symbolically at every state for ensuring the solvability of an associated state-dependent Riccati equations. In doing so, the SDRE algorithm fully captures the nonlinearities of the system, bringing the nonlinear system to a (non-unique) linear structure having state-dependent coefficient (SDC) matrices, and minimizing a nonlinear performance index having a quadratic-like structure. Moreover, the nonuniqueness of the factorization creates extra degrees of freedom, which can be used to enhance controller performance.

2.2

Sliding Mode Control

The implementation of the Sliding Mode Control (SMC) consists of two main phases. First, we should construct the sliding surface such that the system states restricted to the sliding surface will produce the desired behavior. Second, we construct switched feedback gain which derive the plant state trajectory to the sliding surface in finite time (i.e. σT _{˙σ ≤ −η ksk for some η ¿0) and restrict the state to sliding surface (i.e. s = 0}

and ˙s = 0). Suppose at t0, the state trajectory of the plant intercepts the sliding surface

and a sliding mode exists for all t > t0. The existence of a sliding mode implies (1) ˙s = 0,

and (2)s = 0 for all t > t0. The system’s motions on the sliding surface can be given

an interesting geometric interpretation, as an “averag” of the systems’ dynamics on both sides of the surface. The system while in sliding mode can be written as

˙s = 0. (2.4)

By solving the above equation formally for the control input, we obtain an expression for u called the equivalent control, ueq _{which can be interpreted as the continuous control}

law that would maintain ˙s = 0 if the dynamics were exactly known. For instance, for a system of the form

¨

x = f + u, x ∈ IR. (2.5)

In order to be converged to a desired trajectory x(t)≡xd(t), we define a sliding surface

s = 0.

s = (d

here, define the tracking error ˜x = x − xd. We then have

˙s = ¨x − ¨xd+ λ ˙˜x = f + u − ¨xd+ λ ˙˜x (2.7)

and the system dynamics while in sliding mode is, of course,

ueq = −f + ¨xd− λ ˙˜x. (2.8)

Controller design is the second phase of the SMC design procedure. Here the goal is to determine switched feedback gains which derive the plant state trajectory to the sliding surface and maintain a sliding mode condition. The presumption is that the sliding surface has been designed. Among several approach (e.g. the diagonalization method and hierarchical control method), augmenting the equivalent control is one popular approach. This structure of control of system (2.5) is

u = ueq+ ure (2.9)

where ure_{is the discontinuous or the switched part of Eq.(2.9). Consider the system (2.5),}

we have ueq _{= −f + ¨x}

d− λ ˙˜x. In order to satisfy sliding condition Eq.(2.12), we add to

ueq _{a term discontinuous across the surface s = 0, and let}

u = ueq+ ure

= ueq− ksgn(s) (2.10)

where sgn denotes the sign function.

sgn(s) =      1 if s > 0 0 if s = 0 −1 if s < 0 (2.11)

By choosing k to be a positive scalar, 1

2 d dts

2 _{= ˙s · s = −ksgn(s) · s = −k|s|. (2.12)}

Therefore, the sliding variable s will keep at zero. Practically, Eq. (2.12) states that the squared “distance” to the surface, as measured by s2_{, decrease along system trajectory.}

2.1. In particular, once on the surface, the system trajectories remain on the the surface. In other words, satisfying sliding condition Eq. (2.12), makes the surface an invariant set. Furthermore, as we shall see, Eq. (2.12) also implies that some disturbances or dynamics uncertainties can be tolerated while still keeping the surface an invariant set.

2.3

Integral Sliding Mode Control

In this section, we review the concepts of integral sliding mode control scheme. First, consider the following nonlinear matched uncertainties system :

˙x(t) = f(x) + G(x) {u + ∆dm} (2.13)

where x ∈ IRn is a vector of states, u ∈ IRm is a vector of control inputs. f(x) and

G(x) are known nonlinear functions. kdmk ≤ γm, γ > 0 which represent the matched

uncertainties. In the ISMC approach, a law of the form

u(x, t) = u0 + u1 (2.14)

is proposed. The nominal control u0(x) is responsible for the performance of the nominal

system; u1(x) is a discontinuous control action that rejects the perturbations by ensuring

the sliding motion, design as u1 =    0, if σ = 0 −ρ · [DG(x)]Ts k[DG(x)]T_sk, if σ 6= 0 (2.15) where ρ > γm. and sliding surface define as

σ(x, t) = Ds  x(t) − x(t₀) − t Z t0 (f(x) + G(x) · uo) · dτ   (2.16)

where t0 is the initial time, Ds∈ IRm×n, and DsG(x) has full rank. The term

x(t0) + t

Z

t0

(f((x)) + G(x) · uo) · dτ (2.17)

in (2.16) can be though as a trajectory of the system in the absence of perturbations and in the presence of the nominal control u0, that is, as a nominal trajectory for a given

initial condition x(t0). With this remark in mind, σ(x) can be considered a penalizing

G (hence, the name projection matrix, not to be confused with a projection operator). Notice that at t = t0, s(x, t) = 0, so the system always starts at the sliding manifold.

To determine the motion equations at the sliding manifold we use the equivalent control method. The derivative of s along time is

˙s = Ds· [ ˙x − (f (x) + G(x)u0)]

= Ds· [(f(x) + G(x)u + G(x)dm) − (f (x) + G(x)u0)]

= DsG(x) · (u + dm− u0). (2.18)

Therefore, the equivalent control ueq = u0 − dm by solving the equation ˙σ = 0. By

substituting ueq for u1 in (2.13), we obtain the sliding dynamics (motion eqations on the

sliding manifold) is

˙xeq= f(x) + G(x) · u0. (2.19)

It is found from (2.19) that the matched type uncertainties can be completely rejected, and the sliding dynamics and the nominal system dynamics are exactly the same. To

prove u1 can maintain the sliding mode, we choose a Lyapunov function as V = 1/2σTσ.

Differentiating V with respect time using with respect to time using (2.14), (2.15) and (2.18) yields ˙ V = σT ˙σ = σT_D sG(x) · (u + dm− u0) = σTDsG(x) ·  −ρ · [DsG (x)]T s [DsG (x)] T s + dm   ≤ −ρ (DsG(x)) T_σ + kdmk · (DsG(x)) T_σ ≤ (−ρ + γm) · (DsG(x)) T_σ < 0. (2.20)

since DG(x) is of full rank and σ(x(t0), t0) = 0, then the controller (2.14) guarantees that

the sliding mode σ = 0 can be maintained, ∀t ∈ [t0, ∞).

2.4

Burckhardt Tire Friction Model

The friction behavior of the wheel can be approximated with parametric characteristics, as shown in Fig. 2.2. The friction, or adhesion coefficient µ is defined as the ratio of the

frictional force in the wheel plane Ff ric and the wheel ground contact force Fz:

Ff ric

Fz

. (2.21)

The calculation of friction forces can be carried out using the method of Burckhard [16]: µ (λ) = c1·  1 − e−c2·λ_{− c} 3λ  · e−c4·λ·vcog_·_{1 − c} 5FZ2  (2.22) λ = V − Rwω max(V, Rwω) (2.23) where c1, c2, and c3 are given for various road surfaces in table 2.1. c4 lies in the rang

0.02s/m to 0.04s/m which influence of a higher drive velocity. c5 is the influence of a

higher wheel load. c4 and c5 have a maximum value of 1, i.e. they lead to a reduction of

the friction coefficient. In this thesis, we neglect the influence of a higher wheel load and adopt as tire friction model of vehicle brake control.

Table 2.1: Parameters for friction coefficient characteristics (Burckhardt Tire Friction Model)

c1 c2 c3

Asphalt, dry 1.2801 23.99 0.52

Asphalt, wet 0.857 33.822 0.347

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 λ friction coefficient µ asphalt, dry snow asphalt, wet

Fig. 2.1. Friction coefficient (Burckhardt Tire Friction Model)

2.5

Magic Formula for Longitudinal and Lateral Force

In general, there are many descriptions of the model for longitudinal and lateral force. In recent years, an well-known empirical method , the so-called Magic Formula, is devel-oped for characterizing tire behavior and used in vehicle handling simulation. The Magic Formula can be used to fit experimental tire data for characterizing the relationships between the cornering face and tire slip angle or braking effort and skid. Thus, in this thesis, we adopt the Magic Formula form [37] to represent the nonlinear cornering forces. It is expressed by

y(x) = D sin n C tan−1 h Bx − EBx − tan−1Bx i o (2.24)

where B is called the stiffness factor, C the shape factor, D the peak factor, and E the curvature factor, respectively. Where y(x) represent longitudinal force fx and lateral force

fy. When y(x) represent longitudinal force fx, then x denotes slip ratio at wheel.When

y(x) represent lateral force, then x denotes tire slip angle at wheel.One example of the parameter values for Magic Formula type longitudinal and cornering force are adopted from [37] as given in Table 2.2. Fig. 2.3 depicts the relationship between longitudinal force and skid. Fig. 2.4 depicts the relationship between cornering force and tire side slip

angle. In this thesis, we adopt Magic Formula as tire model of yaw moment control.

Table 2.2: Magic Formula coefficient (Fz = 3118.3N)

B C D E longitudinal force 0.1664 1.65 3579.4 0.6645 lateral force 0.23 1.3 3152.9 -0.4216 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 2500 3000 3500 4000

slip ratio using Magic Formula.

longitudinal force

F x

(N)

0 5 10 15 20 0 500 1000 1500 2000 2500 3000 3500

tire side slip angle α (deg.)

cornering force

F y

(N)

CHAPTER THREE

AN INTELLIGENT SDRE AND ISMC COMBINED

SCHEME WITH APPLICATION TO VEHICLE BRAKE

CONTROL

In this chapter, combination of the SDRE and ISMC scheme is studied and applied to vehicle brake system when actuators degradation. The organized of this chapter as follow. Section 3.2 states the problem and the main goal of this paper. Section 3.3 then describes the design of a combination of the SDRE and ISMC schemes. Section 3.4 discusses the application of the analytic results to the antilock brake control of a three-wheel vehicle system.

3.1

Problem Formulation

Consider a class of nonlinear systems described by the following equation: ˙

x = f(x) + G(x) [(I + ∆G)u + d] (3.1)

where x ∈ IRn

and u ∈ IRm

denote the system states and the control inputs respectively,

∆G and d describe possible matched type uncertainties, f(x) ∈ IRn

and G(x) ∈ IRn×m

. In this chapter, we assume that G(x) 6= 0 for all x 6= 0, and make the following three assumptions:

Assumption 3.1 : The origin of the nominal system ˙x = f(x) + G(x)u of System (3.1)

is asymptotically stabilizable.

Assumption 3.2 : I + ∆G ≥ κI for some κ > 0 in the sense of ξT(I + ∆G)ξ ≥ κ · kξk2

for all ξ 6= 0.

Assumption 3.3 : There exist a positive constant ρg and a nonnegative function ρd(x)

When ∆G is a diagonal matrix and a controller has been organized, Assumption 3.2 implies that each actuator experiences no change in the output direction and only experiences degradation and/or amplification. In electrical vehicles, this degradation and/or amplification are associated with the loss of control effectiveness of steering and/or wheel torque controllers [41]. The objective of this chapter is to organize an appropriate controller to realize the regulation performance x → xd, where xd is a nonzero constant

vector.

3.2

Controller Design

Due to the many advantages of the SDRE and the ISMC approaches mentioned in the Introduction section, we combine the two schemes for the controller design in this chapter. First, the SDRE design is used for the nominal system. Then, we adopt the ISMC strategy to compensate for the error when the system state deviates from the nominal system’s trajectory.

3.2.1 SDRE Design for Nominal System

To employ the SDRE approach, the regulation problem needs to be converted into the stabilization problem. For this purpose, we introduce the error state e1 = x − xd. System

(3.1) then becomes

˙e1 = f1(e1) + G1(e1) [(I + ∆G)u + d] (3.2)

where f1(e1) := f(e1+ xd) and G1(e1) := G(e1+ xd). Using this error state dynamics, the

regulation problem becomes the stabilization problem, i.e., x → xd if and only if e1 → 0.

To implement the SDRE design, we assume that the quadratic performance index is given by (3.3) below: J = Z ∞ 0 h eT₁Q(e1)e1+ uTR(e1)u i dt (3.3) where QT_(e

1) = Q(e1) ≥ 0 and RT(e1) = R(e1) > 0. The SDRE scheme requires the

condition f1(0) = 0 to factorize the drift term f1(e1) in Eq. (3.2) into the form of A1(e1)e1.

i.e.,

f1(e1) = A1(e1)e1+ b(e1) (3.4)

with b(0) 6= 0. To solve this problem, we adopt a strategy from [7] to express b(e1) as

b(e1) =

hb_(e1)

z

i

z and augment the system with a stable auxiliary state z as (3.5) below:

˙z = −k1z (3.5)

where k1 is a small positive constant that makes z change slowly, and z(0) 6= 0 so that

b_(e1)

z is smooth during the control period. With these settings, the augmented system

formed by (3.5) and the nominal system of (3.2) has following form:

˙e = A(e)e + B(e)u (3.6)

where A(e) = A1(e1) b_(e1) z 0 −k1 ! , B(e) = G1(e1) 0 ! and e = [eT 1, z]T. To successfully

implement the SDRE scheme, the following assumption for the existence of a unique positive definite matrix solution of an associated algebraic Riccati equation (ARE) is needed [17]:

Assumption 3.4 : The pair nA(e), Q1/2_(e)o _{is pointwise observable and {A(e), B(e)}}

is pointwise stabilizable.

According to the SDRE design procedure, the SDRE controller then has the following form [7]

u₀ = −R−1(e)BT(e)P (e)e (3.7)

where P (e) is the unique positive definite matrix solution of the following ARE:

AT(e)P (e) + P (e)A(e) − P (e)B(e)R−1(e)BT(e)P (e) + Q(e) = 0. (3.8)

3.2.2 ISMC Design for the Compensation of Uncertainties

After selecting the nominal controller, the system might experience uncertainties, in-cluding external disturbances and actuators’ output degradation and/or amplification. To compensate for these effects, we adopt the ISMC design because of its advantages, including robustness, rapid response and easy implementation.

According to the ISMC design procedure [5], [6], the sliding manifold for System (3.2) is given by the following equation:

σ = Ds  e1(t) − e1(t0) − Z t t0 f1(e1) + G1(e1)u0 dτ  (3.9) where Ds∈ IR m×n

is choose such that DsG1(e1) has full rank and u0 is given by (3.7). It

follows from Eqs. (3.2) and (3.9) that

˙σ = Ds[ ˙e1− f1(e1) − G1(e1)u0]

= DsG1(e1) [(I + ∆G)u + d − u0] . (3.10)

To keep the system state on the sliding manifold, we chose the overall control law to be

u= u0+ u1 (3.11)

where u0 is given by (3.7) and

u₁ =    0 if σ = 0; −ρ [DsG1(e1)]Tσ k[DsG1(e1)]Tσk otherwise (3.12) with ρ chosen to satisfy ρ > ρgku0k + ρd(e1). Although the coefficient ρ requires the

information about u0, it can be easily obtained after the calculation of u0 because the

upper bounds ρg and ρd(e1) can be estimated offline. We then have the next result:

Theorem 3.1 : Suppose that Assumptions 3.1-3.4 hold. Then the error states e1 given

in the nominal system (3.6) with the SDRE controller and the uncertain system described by (3.2) with the control law given by (3.11)-(3.12) are identical.

proof: From Eqs. (3.10)-(3.12) and Assumptions 3.2-3.3 we have σT ˙σ = σTDsG1(e1) [(I + ∆G)u1− ∆G · u0+ d] ≤ (−ρκ + k∆Gk · ku0k + kdk) · [DsG1(e1)] T_σ ≤ κ (−ρ + ρgku0k + ρd(e1)) · [DsG1(e1)] T_σ < 0 (3.13)

whenever [DsG1(e1)]Tσ 6= 0 or σ 6= 0 because DsG1(e1) is a nonsingular matrix. From

i.e., the system state remains on the sliding manifold for all t ≥ t0. To determine the

sliding dynamics (the equations of motion on the sliding manifold), the equivalent control method is used [6]. The equivalent control is obtained by solving the equation ˙σ = 0 from Eq. (3.10) as follows:

u_eq = (I + ∆G)−1(u0 − d). (3.14)

By substituting ueq into (3.2), we have the sliding dynamics ˙e1 = f1(e1)+G1(e1)u0, which

is the nominal system under the SDRE controller and the proof is completed.

It is found from the proof of Theorem 3.1 that the matched type uncertainties can be completely rejected, and that the sliding dynamics and the nominal system dynamics are identical. Because the system state under the ISMC scheme starts from the sliding manifold, it follows that the state trajectories of the uncertain system using the combined scheme and the nominal system under SDRE scheme are identical. Therefore, with this ISMC design the engineer can organize another optimal controller (other than the SDRE design) according to system requirements creating a desired system state trajectory for the state of the uncertain system to follow.

3.3

Simulation Results

3.3.1 Vehicle dynamics

A three wheeled vehicle (TWV) model has been described in [12]. For simplicity, we consider only the yaw plane motion, in which the simplified yaw plane vehicle model can be described as follows: the body frame, which is fixed to vehicle’s center of gravity (CG), is denoted by x and y. The positive x- and y-axes represent the forward direction and right-hand side respectively, as seen by the driver. The vehicle dynamics then have the following form:

Fig. 3.1. Three wheeled vehicle model. ˙ Vx ˙ Vy ! = 1 m Fx Fy ! + γ Vy −Vx ! (3.15) Jv˙γ = l1(Lfcδsat− Rfsδsat) − l2(Lrr+ Lrl) +l3(Rrr− Rrl) (3.16) Fx = −Rfcδsat− Rrr− Rrl− Lfsδsat (3.17) Fy = −Rfsδsat+ Lfcδsat+ Lrr+ Lrl (3.18) Lf = Cf δsat− Vy + l1γ Vx ! (3.19) Lrr = Crr l2γ − Vy Vx− l3γ ! (3.20) Lrl = Crl l2γ − Vy Vx+ l3γ ! . (3.21)

Here, Vx and Vy are the components of the velocity of the CG in the x- and y-directions

respectively, γ denotes the angular speed about the vertical axis (z-axis), m denotes the total mass of the TWV, Jv denotes the moment of inertia in the vertical axis of the body

frame, l1 and l2 are the longitudinal distances from the CG to the front axle and the

rear axle respectively, l3 denotes the lateral distance between CG and the left or right

wheel, Fx and Fy represent the external forces acting on the body along the x- and

y-axes respectively, Rα and Lα are the longitudinal and the lateral forces at each wheel

f , rl and rr denote the front, the rear-left and the rear-right wheels respectively, cα,

α = f, rl, rr, denotes the cornering stiffness for the three wheels, δsat is the actual output

value of the steering wheel angle δ, defined by δsat = ₁₆π if δ > ₁₆π ; δsat = −₁₆π if δ < −₁₆π;

δsat = ₁₆π · sin(δ) if −₁₆π ≤ δ ≤ ₁₆π, and c and s denote the cosine and the sine functions

respectively. In addition to the vehicle dynamics, we adopt the tire friction model from

Burckhardt [14] in this chapter to simulate the antilock brake system, where Rα at each

wheel can be expressed as

Rα = µαNα, α = f, rr, rl. (3.22)

Here, Nf = mgl_L1, Nrr = mgl_2L2, Nrl = mgl_2L2, L = l1+ l2, g is the acceleration of gravity and

µα, α = f, rr, rl, are the tire-road coefficients of friction defined by

µα = h c1  1 − e−c2λα_{− c} 3λα i · e−c4λαVx _(3.23)

where c1, c2, c3 and c4 are four parameters introduced in [14],

λα = ( _Vx−Rwωα Vx during braking; Vx−Rwωα rwωα during acceleration (3.24) for α = f, rr, rl are the slip ratios of the three wheels, and Rw and ωα denote the radius

and the angular velocity of the wheel respectively. To study the braking performance, we consider only the slip ratio in braking mode. To achieve an optimal antilock braking performance, the wheel slip ratio is guided to track its peak value for producing maximum negative acceleration [14]. For this purpose, we differentiate λα and using the fact that

˙ωα= (µαrwNα− Tα)/Jw [14], yield ˙λα= ˙ Vx(1 − λα) Vx + RwTα− µαR 2 wNα VxJw , α = f, rr, rl (3.25)

where Jw denotes the inertia moment of wheel and Tα, α = f, rl, rr, are the three brake

torques. Finally, the vehicle model is augmented with the following steering dynamics [1]:

τ ˙δ = −δ + δS (3.26)

where τ is time constant and δS is the steering wheel angle generated by the SDRE

3.3.2 Employment of the Combined Scheme

In [1], Acarman has demonstrated the efficiency of SDRE scheme in vehicle control. In this chapter, we further improve the SDRE robust performance by incorporating it with the ISMC scheme when there are model uncertainties and/or external disturbances. To employ the combined scheme, we define e1 = [e1, e2, e3, e4, e5, e6, e7]T = [Vx, Vy, γ, δ, λf −

λ∗

f, λrr− λ∗rr, λrl− λ∗rl]T and u = [δS, Tf, Trr, Trl]T, where λ∗α, α = f, rl, rr, are the peak

values, to be tracked, of the wheel slip ratio curves. The governing equations then have the form of (3.2), and the control objective becomes to organize an appropriate controller that effectively brings the error state e1 to the origin. Next, we factorize the nonlinear

drift term f1(e1) into a linear structure with SDC matrices. Because we require regulation

performance for λf, λrr and λrl, the factorization of f1(e1) exhibit a bias term b(e1), as

described by Eq. (3.4). Details of an expression for A1(e1), b(e1) and G1(e1) given in

Eq. (3.6) are presented in Appendix.

In this example we assume that the disturbance d = [0, 0, 0, 0, 0, 0, 0.5 sin 20t]T _and

the output of the brake torque at the rear-left wheel experiences a 40% degradation in magnitude, i.e., [∆G]ij = 0 for all i, j except for [∆G]44 = −0.4, where [·]ij denotes

the (i, j)-entry of a matrix. The degradation might result from the abnormal operation of the inverter, braking system and/or wheel motor [41]. The vehicle parameters are

assumed from [12] to be m = 403.87kg, Jv = 178.54kgm2, l1 = 1.39m, l2 = 0.61m,

l3 = 0.575m, Cf = 3885N/rad, Crr = 4050N/rad, Crl = 4050N/rad, Rw = 0.21m and

Jw = 0.567kgm2. The time constant is set to be τ = 30. The road is assumed to be dry

with c1 = 1.2801, c2 = 23.99, c3 = 0.52, c4 = 0.02 and λ∗α = 0.15 for all of the three wheels

[14]. The other parameters and initial state are κ = 0.6, ρg = 2₃, ρd(e1) = 0.5, k1 = 10−3,

Ds = [04×3 I4×4], R = diag[10−1, 1, 1, 1], Q = diag[10−5, 10−2, 10−2, 10−2, 106, 106, 106]

and e1(0) = [30, 0, 0, 10−4, −10−1, −10−1, −10−1]T, where the unit of velocity is meters

per second. Note that, we have promoted the weightings on the three slip ratios to

make λα → λ∗α as soon as possible and to maximize the antilock braking torque. To

alleviate chatter, the control u1 given by (3.12) is replaced with −ρ(DsG1(e1))

T

σ

k(DsG1(e1))T σk ≤ ǫ, and ǫ is selected to be 5 × 10−3.

Verification of Assumption 3.4:

Finally, we need to verify Assumption 3.4 so that the SDRE scheme can be success-fully implemented. Because Q is selected to be a nonsingular matrix, (A(e), Q12(e)) is

observable. Due to the special structure of A(e) and B(e) given by (3.6), (A(e), B(e)) is stabilizable if (A1(e1), G1(e1)) is controllable. To investigate the controllability of

(A1(e1), G1(e1)), we introduce a matrix M ∈ IR7×7 as follows: the first four columns

of M are G1(e1), while the last three columns of M are taken from the last three columns

of A1(e1)G1(e1). It is found that M =

03×4 M12

M21 ∗

!

and M21= diag[1_τ,_JR_ww_e1,_JR_ww_e1,_JR_ww_e1].

Clearly, M21 is nonsingular if e1 6= 0 (i.e., before the vehicle is fully stopped). Since M is

a block triangular matrix, we have that (A1(e1), G1(e1)) is controllable if det(M12) 6= 0.

By direct calculation, we have det(M12) =

−2E1E2E3R3wl3sδsat

J3

wJvm2e31 , where E1, E2 and E3 are

three nonzero scalars given in Appendix A. It follows that (A1(e1), G1(e1)) is controllable

if sin(δ) 6= 0 (or δ 6= 0). For δ = 0, we replace the fifth column of M by the first column of A2

1(e1)G1(e1). With this new M and the fact that e5 = λf− λ∗f, e6 = λrr− λ∗rr and e7 =

λrl−λ∗rl, it is found that det(M12) = 2E2E3l3R

3 w m3_J ve41J 3 w · h (1 − λf)(1 + R 2 w Jw)E 2 1 + (1 − λrr)E1E2+ (1 − λrl)E1E

From (3.24), we observe that 0 ≤ λα ≤ 1 during braking and λα = 1 only when

the wheel is locked. Thus, det(M12) 6= 0 unless all the three wheels are locked (i.e.,

λf = λrr = λrl= 1). These results verify Assumption 3.4 for the period before the vehicle

is fully stopped.

3.3.3 Simulation Results

Numerical results are summarized in Figs. 3.1-3.4. Among these, we consider the following three cases: the first use the SDRE scheme for the nominal system (labeled SDRE0), while the other two adopt the SDRE scheme (labeled SDRE1) and the com-bined scheme (labeled SDRE+ISMC) for the uncertain system (experiencing actuator’s degradation and external disturbance in the actuator). It is observed from Fig. 1(a) that the longitudinal velocity converges to zero for all of the three cases. However, during the control period, the lateral velocity, the angular speed and the steering angle for SDRE1 given in Figs. 3.1(b)-3.1(d) are much larger than the other two cases, which might result

in undesirable instability. Due to the use of a saturation-type function instead of the sign-type function given in (3.12), the state trajectory of the SDRE+ISMC system deviates slightly from that of SDRE0, and it is found from simulation that the smaller the boundary layer width ǫ is, the closer the two trajectories are. From Fig. 3.3, all the sliding variables are found to be within the boundary layer. These results agree with the theoretical results. It is also observed from Fig. 3.2(c) that the performances of λrl → λ∗rl are not achieved

for SDRE1, which results in a worse braking performance, and the oscillation of e7 (i.e.,

λrl) for SDRE1 comes from the persistent oscillation of d. These results imply that the

SDRE+ISMC scheme is more robust than the SDRE1 scheme. Finally, Fig. 3.4 shows the

control efforts. Among these, the control curves of Trl for the SDRE1 and SDRE+ISMC

schemes have taken the degradation effect into account, i.e., those two curves describe the actual degradation torque output which is 60% of the magnitude of the designed control value. It is seen from Fig. 3.4(d) that the braking torque Trlfor the SDRE+ISMC systems

is automatically adjusted to a level such that its degradation curve approximates that of the SDRE0 scheme, to ensure that its state responses are close to those of the SDRE0

system. The oscillations of Trl for the SDRE1 and SDRE+ISMC systems resulted from

the compensation for the persistent disturbance excitation. The braking torque Trl for the

SDRE+ISMC system is found to be a little larger than that of the SDRE1 system, since

the combined scheme provides an additional control u1 for regulation when the system

state deviates from the sliding manifold. In contrast, the required steering wheel angle

δS for the SDRE+ISMC system is much smaller than that of the SDRE1 system. By

direct calculation, the quadratic performance of the three cases has the following relation: JSDRE1 = 9.2539 × 105 < JSDRE0 = 9.2265 × 105 < JSDRE+ISMC= 9.0261 × 105.

Though the above simulations demonstrate the effectiveness of the combined scheme, the controls are decoupled from each other since the matrix Ds is selected to be Ds = I.

To demonstrate the interactive relation of these actions, in the following we consider Ds

has the following form:

Ds=      0 0 0 1 0 0 0 0 0 0 0 1 0.7 0.7 0 0 0 0 0.7 1 0.7 0 0 0 0 0.7 0.7 1      . (3.27)

Under the choice of Ds, the sliding variables σ2, σ3, and σ4 will couple to each other. As a

result, the required control effort of Trl also be provided by Tf and Trr to compensate the

degradation effect. A same scenario for state response can also be observed form Figs. 3.6(a)-(d). However, from Figs. 3.7(a)-(c), the slip ratios of these three wheel are slight variation because the couple effect of sliding variable. Fig. 3.8 shows the couple effect for DsBσ2, DsBσ3, and DsBσ4. Fig 3.9 shows the control efforts. Among these, the control

curves of Tf, Trr, Trl are slight different to pervious simulation result.

Although the combined scheme requires a little more control effort than SDRE1, it enables a much smaller yaw rate and lateral velocity than the SDRE1 scheme. Thus, it is concluded from this example that the combined scheme is more robust and safer in braking control than the SDRE scheme alone.

0 1 2 3 0 10 20 30 (a) time e 1 0 1 2 3 −0.2 −0.1 0 0.1 0.2 (b) time e 2 0 1 2 3 −0.05 0 0.05 (c) time e 3 0 1 2 3 −0.03 −0.02 −0.01 0 0.01 (d) time δ sat SDRE+ISMC SDRE0 SDRE1 SDRE0 SDRE+ISMC SDRE1 SDRE1 SDRE+ISMC SDRE0 SDRE+ISMC SDRE0 SDRE1

0 0.5 1 1.5 2 2.5 3 −0.1 0 0.1 (a) time e 5 0 0.5 1 1.5 2 2.5 3 −0.1 0 0.1 (b) time e 6 0 0.5 1 1.5 2 2.5 3 −0.1 0 0.1 (c) time e 7 SDRE0 SDRE1 SDRE1 SDRE0 SDRE+ISMC SDRE+ISMC SDRE+ISMC SDRE0 SDRE1

Fig. 3.3. Time history of the last three system states.

0 0.5 1 1.5 2 2.5 3 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5x 10 −4 time DB σ DBσ 1 DBσ2 DBσ3 DBσ 4

0 1 2 3 −20 0 20 40 (a) time δ S 0 1 2 3 200 300 400 500 600 (b) time T f 0 1 2 3 300 400 500 600 (c) time T rr 0 1 2 3 200 300 400 500 600 (d) time T rl SDRE+ISMC SDRE+ISMC SDRE0 SDRE1 SDRE0 SDRE1 SDRE1 SDRE1 SDRE0 SDRE+ISMC SDRE0 SDRE+ISMC

Fig. 3.5. Time history of the four control inputs.

0 1 2 3 0 10 20 30 (a) time e 1 0 1 2 3 −0.2 −0.1 0 0.1 0.2 (b) time e 2 0 1 2 3 −0.05 0 0.05 (c) time e 3 0 1 2 3 −0.03 −0.02 −0.01 0 0.01 (d) time δ sat SDRE+ISMC SDRE0 SDRE1 SDRE+ISMC SDRE1 SDRE+ISMC SDRE0 SDRE0 SDRE1 SDRE+ISMC SDRE0 SDRE1

0 0.5 1 1.5 2 2.5 3 −0.1 0 0.1 (a) time e 5 0 0.5 1 1.5 2 2.5 3 −0.1 0 0.1 (b) time e 6 0 0.5 1 1.5 2 2.5 3 −0.1 0 0.1 (c) time e 7 SDRE0 SDRE1 SDRE+ISMC SDRE+ISMC SDRE1 SDRE0 SDRE0 SDRE+ISMC SDRE1

Fig. 3.7. Time history of the last three system states.

0 0.5 1 1.5 2 2.5 3 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5x 10 −4 time DB σ DBσ 1 DBσ 2 DBσ₃ DBσ 4

0 1 2 3 −20 0 20 40 (a) time δ S 0 1 2 3 200 400 600 800 (b) time T f 0 1 2 3 300 400 500 600 700 (c) time T rr 0 1 2 3 200 300 400 500 600 (d) time T rl SDRE0 SDRE1 SDRE+ISMC SDRE1 SDRE+ISMC SDRE0 SDRE1 SDRE+ISMC SDRE0 SDRE1 SDRE0 SDRE+ISMC

Fig. 3.9. Time history of the four control inputs.

Appendix 3.A

Here, we present A1(e1), b(e1) and G1(e1) for the TWV, which appear inside Eq.

(3.6):

(i) Let A1(e1) = [aij]. Then ai1 = a4j = a2k = 0 for i = 1, · · · , 7, j = 1, 2, 3, 5, 6, 7,

k = 6, 7 and a12= Cfsδsat me1 ; a13 = Cfl1sδsat me1 + e2; a14= −δsatCfsδsat me4 ; a15 = −E1cδsat m ; a16 = −E2 m ; a17= −E3 m ; a22= −1 m Cfcδsat e1 + Crr e1− l3e3 + Crl e1+ l3e3 ! ; a23= 1 m −Cfl1cδsat e1 + Crrl2 e1− l3e3 + Crll2 e1 + l3e3 ! − e1; a24 = δsat· Cfcδsat me4 ; a25= −E1sδsat m ; a32= 1 Jv −Cfl1cδsat e1 + Crrl2 e1− l3e3 + Crll2 e1+ l3e3 ! ; a33= −1 Izz Cfl12cδsat e1 + Crrl 2 2 e1− l3e3 + Crll 2 2 e1+ l3e3 ! ; a34= δsat· l1Cfcδsat Jve4 ; a35= −l1E1sδsat Jv ; a36= l3E2 Izz ; a37= −l3E3 Jv ; a44= −1 τ ; ai2= 1 − ei− λ∗α me1 · Cfsδsat e1 for i = 5, 6, 7, and

ai3= 1 − ei− λ∗α me1 Cfl1sδsat e1 + e2 ! for i = 5, 6, 7, and α = f, rr and rl if i = 5, 6 and 7, respectively; ai4= − 1 − ei− λ∗α me1 ·δsat· Cfsδsat e4 for i = 5, 6, 7, and α = f, rr and rl if i = 5, 6 and 7, respectively; a55= − 1 − e5− λ∗f me1 E1cδsat+ R2 w Jw E1 ! ; ai5= − 1 − ei− λ∗α me1

· E1cδsat for i = 6, 7, and

α = rr and rl if i = 6 and 7, respectively; a66= − 1 − e6− λ∗rr me1 E2+ R2 w Jw E2 ! ; ai6= − (1 − ei− λ∗α)E2 me1 for i = 5, 7, and

α = f and rl if i = 5 and 7, respectively; a77= − 1 − e7− λ∗rl me1 E3 + R2 w Jw E3 ! ; ai7= − (1 − ei− λ∗α)E3 me1 for i = 5, 6, and

α = f and rr if i = 5 and 6, respectively; Ei= −Nαc3e−c4(λ

∗

α+ei+4)e1 _{for i = 1, 2, 3, and}

α = f, rr and rl if i = 1, 2 and 3, respectively. (ii) Let b(e1) = [b1, · · · , b7]T. Then

b1= − E5cδsat+ E6+ E7 m ; b2 = −E5sδsat m ; b3 = 1 Jv (−l1E5sδsat+ l3E6− l3E7) ; b4 = 0; bi= ei+ λ∗α− 1 me1 E5cδsat+ E6+ E7+ R2w Jw Ei ! for i = 5, 6, 7; Ei= Nαc1e−c4(λ ∗ α+ei)e1_{1 − e}−c2(λ∗α+ei)_{− λ}∗ αNαc3e−c4(λ ∗ α+ei)e1 _{for i = 5, 6, 7, and}

α = f, rr and rl if i = 5, 6 and 7, respectively.

(iii) Let G1(e1) = [gij]. Then gij = 0 for all i, j except for g41 = 1_τ and g52 = g63 =

CHAPTER FOUR

A STUDY OF RELIABLE YAW MOMENT

CON-TROL FOR ELECTRIC VEHICLE

In this chapter, the SMC reliable control scheme is applied to retain steerability and lead the vehicle to a neutral steer behavior. This chapter is organized as follows. Section 4.1 states the vehicle dynamical model and the problem. It is followed by the design of SMC reliable controllers for ensuring EVs’ safety when actuators’ fault happens. Section 4.4 demonstrates the simulation results.

4.1

Problem Formulation

Consider the following 7 DOF nonlinear vehicle model [16]:

Fig. 4.1. Four wheeled vehicle model.

   m ˙V mV ˙β Jv˙γ    =    0 −mV γ 0   +    cβ sβ 0 −sβ cβ 0 0 0 1    · 4 X i=1    fxicδi− fyisδi fxisδi+ fyicδi Mzi    (4.1)

˙ωi = Rwfxi − Ti Jw (4.2) and Mzi = lxi(fxicδi− fyisδi) +lyi(fxisδi+ fyicδi) (4.3)

for i = 1, · · · , 4. Here, c and s denote respectively the cosine and the sine functions (e.g., cβ = cos β and sδi = sin δi), m, Jv, Jw and Rw denote respectively the vehicle

mass, vehicle inertia, wheel inertia and wheel radius, V , β and γ are respectively the vehicle velocity, body side slip angle and yaw rate, ωi, δi and Ti for i = 1, · · · , 4 denote

respectively the wheel angular speed, steering wheel angle and brake/drive torque at the ith wheel, fxi and fyi denote the longitudinal and the lateral forces at the ith wheel

between the tire patch and the road, respectively, and lxi and lyi denote respectively the

lateral and longitudinal distances between the center of gravity (CG) to the ith wheel. In this chapter, the subscripts i = 1, · · · , 4 are employed to represent respectively the front-right, the front-left, the rear-right and the rear-left wheels, as seen from the driver. Besides, we assume that δ1 = δ2 = δ, δ3 = δ4 = 0, lx1 = lx3 = −lt, lx2 = lx4 = lt,

ly1 = ly2 = lf and ly3 = ly4 = −lr [16]. The longitudinal and the lateral forces are

expressed according to the Magic Formula [37] as follows: fxi(λi) = D sin n C tan−1[Bλi− E (Bλi − tan−1(Bλi) io (4.4) and fyi(αi) = D sin n C tan−1[Bαi− E (Bαi − tan−1(Bαi) io (4.5) for i = 1, · · · , 4, where the coefficients B, C, D and E denote respectively the stiffness, the shape, the peak, and the curvature factors, λi and αi are respectively the wheel slip

ratio and the tire slip angle defined by [16] as λi = (V − Rwωi) max{V, Rwωi} , i = 1, · · · , 4 (4.6) and αi =    − tan−1lfγ+V sβ V cβ  + δ, i = 1, 2 tan−1lrγ−V sβ V cβ  , i = 3, 4. (4.7)

Note that, λi < 0 if the ith wheel is in acceleration, and λi > 0 if it is in deceleration.

In order to have a satisfactory steerability and guide the vehicle to a neutral steer

speed, while negotiating a steady state cornering maneuver has been proposed as (4.8)-(4.9) below [4]: γd = γss 1 + τ pδ (4.8) and γss = 2V CfCr(lf + lr) 2CfCr(lf + lr)2− mV2(lfCf − lrCr) (4.9) where τ , p, Cf and Cr denote the time constant, the Laplace variable, the front and the

rear wheel cornering stiffness, respectively. When the vehicle is operated under normal (non-faulty) condition, the desired yaw rate can be easily tracked by creating a yaw moment from the difference of tire driving torques and/or braking forces between the right and left sides. However, if one of the wheel torque controllers fails to operate (which might result from the inverter failure, the brake system failure, or the wheel motor failure [41]), the vehicle will start to spin and an appropriate reliable controller is needed to assure driving safety and provide better performances. Thus, the objective of this chapter is to organize a suitably control to realize the performance γ → γd when a vehicle experiences

wheel actuator failure.

4.2

Reliable Controller Design

4.2.1 Output tracking formulation

Let x = [V, β, γ, ω1, ω2, ω3, ω4], u = [T1, T2, T3, T4] and y = γ. Then Eqs. (4.1)-(4.2)

together with the selected output γ constitute a multi-input and single-output (MISO) nonlinear affine system. It is found that the MISO system has relative degree 2, i.e., one has to differentiate y(t) twice to have u(t) explicitly appearing. By direct calculation we have

¨

γ = φ (x) + G (x) u + d (4.10)

where φ(x) ∈ IR, G(x) = [g1(x), · · · , g4(x)] ∈ IR4, d denotes possible uncertainties and/or

measurement noises, and the detailed expressions of φ(x) and gi(x) are given as below:

Define

̟1(x) = Bx − E

h

Bx − tan−1(Bx)i (4.11)

̟3(x) =

BCD cos(̟2(x))[(1 − E)(1 + B2x2) + E]

(1 + ̟2

1(x))(1 + B2x2)

. (4.13)

Then fxi(λi) = D sin(̟2(λi)), fyi(αi) = D sin(̟2(αi)),

d

dtfxi(λi) = ̟3(λi) ˙λiand

d

dtfyi(αi) =

̟3(αi) ˙αi. These together with Eqs. (4.1)-(4.3) and (4.6)-(4.7) yield

φ(x) = 1 Jv 4 X i=1 n − lxi(fxisδi+ fyicδi) ˙δi + lyi(fxicδi − fyisδi) ˙δi +(lxicδi+ lyisδi)̟3(λi) ˙λxi+(lyicδi− lxisδi)̟3(αi) ˙αi} (4.14) and gi(x) = 1 Jv (lxicδi+ lyisδi) ̟3(λi) ˙λui (4.15) for i = 1, · · · , 4, where ˙λxi=    JwRwωiV −R˙ 2wV fxi JwV2 if λi > 0 ˙ V Rwωi − V f_xi Jwω2_i if λi < 0 (4.16) ˙λui= ( _Rw JwV if λi > 0 V JwRwω_i2 if λi < 0 (4.17) ˙αi=    ˙δ − (V2_−l fγV sβ) ˙β+lfV cβ ˙γ−lfγcβ ˙V (V cβ)2_+(l fγ+V sβ)2 if i = 1, 2 (lrγV sβ−V2) ˙β+lrV cβ ˙γ−lrγcβ ˙V (V cβ)2_+(l rγ−V sβ)2 if i = 3, 4. (4.18) 4.2.2 SMC reliable design

In this chapter we consider the active reliable output tracking issues for System (4.1)-(4.2), that is, we assume that the actuators’ fault has been successfully detected and diagnosed by an FDD mechanism. The fault may be time varying and include degradation, amplification and outage [28]. Before the occurrence of faults, the engineers may take any kind of control strategy to fulfill their desired system performance. When the fault is detected and diagnosed, the control law is guided to switch to an active reliable law for ensuring system performance. Thus, after the fault is detected, we may divide the actuators into two groups H and F , within which we assume that all of the actuators in H are healthy, while those in F experience faults. This implies that Eq. (4.10) can be rewritten as

¨

γ = φ (x) + GH(x) uH+ GF(x) uF + d (4.19)

where u = [uTH uTF]T and G(x) = [GH(x) GF(x)]. In the rest of the this paper, we assume

necessary for the existence of the equivalent control in SMC design [9]. In addition, we assume that the control inputs in the set of F are diagnosed as

uF = ˆuF+ ∆uF (4.20)

where ˆuF and ∆uF denote the estimated control value and estimated error, respectively.

The estimated error ∆uF is treated as an additional uncertainty that should be

compen-sated. Define the output error

e = γ − γd. (4.21)

The control objective is then to force e → 0 through SMC reliable design. Since Eq. (4.19) is a second-order system, we may assume the sliding surface in the form of

σ = ˙e + k2e (4.22)

where k2 is a positive constant. Clearly, if the system state remains on the sliding surface,

then the desired performance of e → 0 can be exponentially achieved with a convergence rate depending on the choice of k2 [9]. From Eqs. (4.19)-(4.22) we have

˙σ = φ(x) + GHuH+ GF(ˆuF + ∆uF) + d − ¨γd+ k2˙e. (4.23)

To guarantee the reaching performance, we impose the next assumption:

Assumption 4.1 : There exists a nonnegative functionρ(x, t) such that |GF∆uF + d| ≤

ρ (x, t).

Following the SMC design procedure [9], we choose

uH = −G+H(x) · [φ(x) + GF(x) ˆuF − ¨γd+ k2˙e + (ρ(x, t) + η) sgn (σ)] (4.24) where G+H(x) = GTH(x) h GH(x) GTH(x) i−1

and η is a positive constant which affects the

convergence speed of the system state to the sliding surface. Note that, uH, given by

(4.24), involves the information of diagnosis. Form (4.23), (4.24) and Assumption 1 we have

σ ˙σ = σ · [GF∆uF + d − (ρ(x, t) + η) sgn (σ)]

Thus, the system state will reach the sliding surface in a finite amount of time with reaching speed depending on the magnitude of η, and remain there hereafter [9]. After reaching to the sliding surface σ = 0, the tracking error e, from Eq. (4.22), satisfies ˙e + ke = 0. It implies that the tracking error is exponentially convergent to zero and the tracking performance is achieved. In addition to the yaw rate tracking, another factor that affects the vehicle driving stability and cannot be ignored is the magnitude of the body side slip angle β. It is in general expected to have β as small as possible for ensuring vehicle stability; however, when γ ≈ γd, | ˙V | ≈ 0, |β| ≪ 1, ltγd/V ≪ 1 and the tire forces

at the two sides are the same, the side slip angle is related to the yaw rate and the steering angle in the following relation [40]:

˙ β ≈ − 2C f + 2Cr mV  β + 2Crlr− 2Cflf mV2 − 1 ! γd + 2C f + 2Cr mV  δ (4.26)

which is clearly a stable system for β with inputs γdand δ. As a result, the side slip angle

is determined from the yaw rate and the steering angle, and it can be made small if γdand

δ are appropriately chosen. An analysis of lateral stability and bifurcation phenomena with respect to the variations in the front wheel steering angle has been presented in [29]. Besides, one may also deal with the tracking of yaw rate and side slip angle simultaneously by choosing the weightings on the error of yaw rate and on the error of side slip angle [11]. In this chapter, we only consider the yaw rate tracking matter, i.e., the weighting on the error of side slip angle is chosen to be zero.

4.2.3 FDD mechanism

To detect the actuators’ fault for active reliable task, in this chapter we assume that all the state variables are available for measurement or estimation. In fact, this assumption is feasible for EVs [4]. We adopt the observer and the associated residual signals ri from

[22] as (4.27 and (4.28) below: ˙ζi = Rwfxi− Ti Jw + ai(ωi− ζi) (4.27) and ri = ωi− ζi (4.28)

where i = 1, · · · , 4 and ai > 0 for all i. It was shown in [22] that the actuators’ fault can

be detected and diagnosed with diagnosed error converging to zero at an exponential rate ai.

4.3

Simulation Results

In this chapter, we adopt the vehicle’s parameters from [38] as follows: m = 1300kg, Jv = 2000kgm2, Jw = 0.6kgm2, lf = 1.25m, lr = 1.25m, lt= 0.8m and Rw = 0.3m. Under

these settings, the normal loads for the four wheels are calculated to be Fzi = mglr/2l =

3188N for i = 1, 2 and Fzi = mglf/2l = 3188N for i = 3, 4. Let the cornering stiffness

Cf = Cr = 8741N/rad. The coefficients of the Magic Formula can then be obtained as

follows: B = 0.1664, C = 1.65, D = 3579.4 and E = 0.6645 for longitudinal forces; and B = 0.2302, C = 1.3, D = 3152.9 and E = −0.0412 for lateral forces [37]. The control and the observer parameters are selected as τ = 10−3_{, k}

2 = 1, η = 1 and ai = 1 for i = 1, · · · , 4.

To alleviate chatter, sgn(σ) is replaced with the saturation function sat(σ/ǫ) and ǫ = 10−1_.

The steering command for lane change is chosen to be δ = 0.05 sin(1.5708(t − 1)) when 1 ≤ t ≤ 5, and δ = 0 elsewhere [4]. Finally, the road is assumed to be dry, and the initial state and the disturbance are taken as x(0) = [30, 0, 0.5, 102_{, 10}2_{, 10}2_{, 10}2_{] and}

d = sin(20t) + 0.5 sin(30t) + 0.1 sin(50t).

To demonstrate the reliable performances, in the following, we will consider three faulty cases: the first concerns only one actuator fault, two actuator faults and three actuators fault. The alarm will be fired if any one of the residual signals exceeds 1 (i.e., |ri| > 1). Numerical results are summarized in Figs. 4.2-4.13. Among these, we adopt the

following two schemes: the first uses the proposed reliable SMC scheme (labeled RSMC), while the other adopts the conventional (non-reliable) SMC design (labeled SMC). Before alarm, both the two schemes adopt their conventional non-reliable SMC design, i.e., con-trol in the form of (4.24) with all actuators being healthy. Whenever there is an alarm, the associate active reliable controllers are activated according to the FDD information.

4.3.1 One actuator fault

Here, we assume that the rear-left brake actuator fails at t = 2.5, i.e., T4 = 0 after

t = 2.5, H = {T1, T2, T3} and F = {T4} after the fault happens. It is observed from

Figs. 1(c) and 1(d) that the output tracking error for RSMC is much smaller than that of SMC after fault happens. The longitudinal velocity, as seen from Fig. 4.2(a), of RSMC decreases from 30m/sec to 24.48m/sec, while that of SMC decreases from 30m/sec to 16.3m/sec. This means that the non-reliable design leads to a larger deceleration than the RSMC; however, a large deceleration usually results in a loss of ride comfort [4], which is usually undesired. From Fig. 4.2(b), the magnitude of the side slip angle of RSMC is also significantly reduced after fault happens, compared with that of SMC. With the selected parameters and steering command, it is found that max{ltγd/V }|V =30 = 0.016

and Eq. (4.26) under V ≡ 30 becomes ˙β = −0.90β−γd+0.90δ, which yields the magnitude

of β has an upper bound maxt≥0|β| = 0.3678 at around t = 2.60. It is observed from

Fig. 4.2(b) that the magnitude of β for RSMC is actually within this bound. Besides, β of SMC is noticed from Fig. 4.2(b) to be convergent to zero slower than that of RSMC, because the output tracking error of SMC in Fig. 4.2(d) converges to zero slower than that of RSMC. Figures 4.3(a)-(d) show the angular speed of the four wheels. All of the four wheels are seen to be decelerated during the change lane period, and those of SMC having angular speed much decreased than those of RSMC. In addition, it is observed from Fig. 4.4(a) that the actuator fault is successfully detected by the observer at around t = 2.506. When the fault is detected, the RSMC scheme switches its controller to the reliable one and the residual signal r4 is seen to quickly decrease to zero. This can also

be seen from the alarm signals shown in Fig. 4.4(b), where the alarm value 1 denotes the fault of the fourth actuator. Figures 4.4(c) and 4.4(d) give the history of the four control torques for RSMC and SMC, respectively. It is noted that after the detection of the

outage of actuator T4, the other three actuators of RSMC contribute much more control

effort than those of SMC to realize the required yaw moment for yaw rate tracking, which are also reflected in Figs. 4.2(b) and 4.2(d) where the output tracking performances of RSMC are much better than those of SMC. After the change lane mission (i.e., t ≥ 5),