Navigation control for electric vehicles using nonlinear state feedback H∞ control

Nonlinear Analysis 71 (2009) e2920–e2933

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Navigation control for electric vehicles using nonlinear state feedback H∞ control Katsumi Moriwaki ∗ , Katsuyuki Tanaka Department of Mechanical Systems Engineering, The University of Shiga Prefecture, Hikone, Shiga 522-8533, Japan

article

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Keywords: Autonomous vehicles Motion control Nonlinear state feedback control H∞ optimal control Navigation

abstract The problem of modelling and control for autonomous vehicles is considered. Mutual interactions among vehicle motion dynamics are evaluated. The mathematical model suitable for describing and simulating the whole motion of autonomous passenger vehicles is proposed. The passenger vehicles are evaluated from many points of view, such as riding comfort, vehicle position, stability, manipulability and so on. The performance of vehicle control in technically is separated into several control items and considered for each item, independently. The mathematical model for steering control of an autonomous vehicle usually has two degrees of freedom, which consider the lateral motion and the yawing motion. The model for suspension dynamics, which is deeply related to riding comfort, also has two degrees of freedom, which consider the bouncing motion and the pitching motion. The above mentioned models are not enough to treat the problem of total motion control of autonomous vehicles. The specifications of tires must also be considered in the whole motion control of vehicles, and they have strong nonlinearity. There are, furthermore, mutual interactions among them, which are inevitably considered for the problem of the whole motion control of autonomous passenger vehicles. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The passenger vehicles are evaluated from many points of view, such as riding comfort, vehicle position, stability, manipulability and so on (Fig. 1). In the recent year, the electric control systems of chassis functions – suspension, steering, drivetrain and brake – have been developed rapidly. These systems have been developed for the purpose of obtaining the maximum performance, independently [1]. The technical performance of vehicle control is separated into several control items and also considered for each item, independently. The mathematical model for steering control of an autonomous vehicle usually has two degrees of freedom, which consider the lateral motion and the yawing motion [2,3]. The model for suspension dynamics, which is strongly related to riding comfort, also has two degrees of freedom, which consider the bouncing motion and the pitching motion [4,5]. The above mentioned models are not enough to treat the problem of total motion control for autonomous vehicles. The specifications of tires must also be considered in the whole motion control of vehicles, and they have strong nonlinearity [6]. There are, furthermore, mutual interactions among them, which are inevitably taken into account for the problem of the whole motion control for autonomous passenger vehicles (Fig. 2).

∗

Corresponding author. Tel.: +81 749 28 8593. E-mail address: [email protected] (K. Moriwaki).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.07.053

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

e2921

Fig. 1. Integrated vehicle control [1].

Fig. 2. Rigid-body model of four wheel vehicles.

2. Modelling of an autonomous vehicles 2.1. Steering model of an autonomous vehicle The features of car steering dynamics in a horizontal plane are described by Fig. 3 [3–7]. In Fig. 3, the angle δ is the front steering angle and the angles βf 1 , βf 2 , βr1 , βr2 are the sideslip angles of front tires and rear tires, respectively. The angle β between the vehicle center line and the velocity vector V is called ‘‘vehicle sideslip angle’’. The cornering forces Yf 1 , Yf 2 , Yr1 , Yr2 are the forces transmitted from the road surface via the wheels to the car chassis. The distance between the center of gravity (P) and the front axle (resp. rear axle) is lf (resp. lr ) and together l = lf + lr is the wheel base. In the horizontal plane of Fig. 3 an inertially fixed coordinate system (X , Y ) is shown together with a vehicle fixed ˙ will appear as a coordinate system (x, y) that is rotated by a‘‘yaw angle’’ ψ . In the dynamic equations the yaw rate r := ψ state variable. Assuming that βf = βf 1 = βf 2 = δ − β − lf r /V , βr = βr1 = βr2 = −β + lr r /V , and |βf |  1, |βr |  1, |δ|  1, then the‘‘two wheel model’’ [8] (Fig. 4) can be regarded as the equivalent model to the four wheel model (Fig. 3). The side forces ff := 2Yf , fr := 2Yr are projected through the steering angle into chassis coordinate (x, y), where they appear as forces fx , fy and the torque mz around a z-axis which is pointing upward from the center of gravity (P).

"

fx fy mz

#

" − sin δ = cos δ lf cos δ

0 1 −lr

# 

ff . fr

(1)

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K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

Y

V

O

X Fig. 3. Four wheel model for car steering.

Fig. 4. Two wheel model for car steering.

Via the dynamics model the forces cause state variables β, V , r. The equations of motions for three degrees of freedom in the horizontal plane are 1. longitudinal motion

− mV (β˙ + r ) sin β + mV˙ cos β = fx .

(2)

2. lateral motion mV (β˙ + r ) cos β + mV˙ sin β = fy .

(3)

3. yaw motion I r˙ = mz .

(4)

It is obtaind that from Eq. (2) to Eq. (4) mV (β˙ + r )  = mV˙ I r˙





" − sin β cos β 0

cos β sin β 0

0 0 1

fx fy mz

#"

# .

(5)

The side force fy is known to be the nonlinear function of the tire sideslip angles βf , βr , i.e., fy = ff (βf ) + fr (βr ). Two wheel model (5) and (6) is nonlinear and we will introduce the additional assumptions [3] as follows.

(6)

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

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(A1) The sideslip angle β is assumed to be small. Then, Eq. (5) becomes mV (β˙ + r )



mV˙





 −β = 1 0

I r˙



fx



1

0

β

0  fy  .

0

1

(7)

mz

(A2) The velocity is constant, V˙ = 0. Then, the second row of Eq. (7) yields fx = −β fy and with β 2  1, then we have



mV (β˙ + r )



 =

I r˙





1

0

fy

0

1

mz

.

(8)

The velocity V is treated as an uncertain constant parameter. (A3) The nonlinear characteristic of Eq. (6) is approximated by the nominal value of the tangent at βf = βr = 0 and small nonlinear functions, i.e., ff (βf ) = cf µ(βf + ∆1 (β, r ))

(9)

fr (βr ) = cr µ(βr + ∆2 (β, r ))

where ∆i (β, r ), (i = 1, 2) is C ∞ nonlinear function and k∆i (β, r )k  1 [8,9]. The constant coefficients cf , cr are called ‘‘cornering stiffness’’, and µ is the adhesion factor (treated here as a disturbance) between road surface and tire. Typical experimental values of µ [3] are

µ=1 µ = 0.5 µ = 0.15

dry road wet road ice.

The steering model follows from Eq. (7) to Eq. (9) and using Eq. (1) as



mV (β˙ + r ) I r˙



 =

1

1

lf

−lr

cf µ(δ − β − lf r /V + ∆1 (β, r ))



cr µ(−β + lr r /V + ∆2 (β, r ))



.

(10)

The uncertain parameters in this model are mass m, moment of inertia I, velocity V and road friction factor µ. Solving Eq. (10) for β˙ and r˙ and rearranging terms yields the nonlinear state space model

   a β˙ = 11 a21 r˙

a12 a22

  β r

b1 g (β, r ) δ+ 1 µ b2 g2 (β, r )

  +





(11)

where a11 = −

cf + cr

a21 =

˜I

cf

b1 =

˜ mV m

˜ := m

,

˜ mV cr lr − cf lf

µ

,

,

a12 = −1 +

,

b2 =

˜I := I µ

a22 = − cf lf

cr lr − cf lf

˜ 2 mV cf l2f + cr l2r ˜IV

˜I

and g1 (β, r ) = cf ∆1 (β, r ) + cr ∆2 (β, r ), g2 (β, r ) = lf cf ∆1 (β, r ) − lr cr ∆2 (β, r ).

2.2. Geometric model of steering motion for |V | < 1 The steering motion of autonomous vehicles with low velocity (|V | ≈ 0) cannot be described by Eq. (11), because some coefficients of Eq. (11) include V in their denominators. In this case, there can be assumed to be no sideslip angle (β ≡ 0) and the velocity vector of P is given by (V , 0) in a vehicle fixed coordinate system (x, y) with the yaw rate : r =

V l

tan δ.

(12)

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2.3. Suspension model of an autonomous vehicle The features of car vertical dynamics in (X –Z )-plane are described by Fig. 6 [4–6] with active suspension system in which the external control forces is used to suppress the uncomfortable bouncing motion and pitching motion. The equations of motions for two degrees of freedom in (X –Z )-plane with the constant velocity V are i. bouncing motion mz¨ = Ff + Fr .

(13)

ii. pitching motion Iy θ¨ = −lf Ff + lr Fr

(14)

where z is the vertical displacement of the center of gravity (CG) of a car (Fig. 6), θ is the pitching angle of the center of gravity (CG) of a car and the external forces acting on a front wheel and the rear wheel from the road surface are written by Ff and Fr , respectively. The state space model of the vertical dynamics is derived from Eqs. (13) and (14) as follows [5].

   z˙ 0  θ˙   0  = w ˙  a˜ 31

0

0

a˜ 32

a˜ 33

a˜ 41

a˜ 42

a˜ 43

q˙

0

0   0 1   θ  +  a˜ 34  w  b˜ 31 q a˜ 44 b˜ 41

1

0

  z

0



0  Uf 

 

b˜ 32  Ur b˜ 42

0 0 + h˜ 31 h˜ 41

0



0  z0f 



h˜ 32  z0r

 (15)

h˜ 42

where w := z˙ , q := θ˙ and Kf + Kr

a˜ 31 = −

m Cf + Cr

a˜ 33 = −

m

,

a˜ 32 =

,

a˜ 34 =

Kf lf − Kr lr

a˜ 41 =

Iy Cf lf − Cr lr

a˜ 43 =

Iy 1

b˜ 31 =

m

Iy

Kf

,

h˜ 41 = −

,

a˜ 44 = −

Iy

h˜ 32 =

,

m Kf l2f + Kr l2r Iy Cf l2f + Cr l2r Iy

1 m lr

b˜ 42 =

,

m Kf lf

m Cf lf − Cr lr

a˜ 42 = −

b˜ 32 =

lf

b˜ 41 = − h˜ 31 =

,

,

Kf lf − Kr lr

Iy

Kr m

b˜ 42 =

Kr lr Iy

Kf : stiffness coefficient for front suspension Kr : stiffness coefficient for rear suspension Cf : damping coefficient for front suspension Cr : damping coefficient for rear suspension. In Eq. (15), the external input zof , zor are assumed to be the random variations at the front tire and the rear tire, respectively, which are induced by the steering motion. According to Eq. (9) and (11), zof , zor are assumed to be induced as the form zof = σf ff (βf ) zor = σr fr (βr )

(16)

where σf , σr are unknown coefficients. Using sensors for the height of CG of the car and the pitching angle of CG of the car, it can be assumed that the output equation for Eq. (15) is given by

 

  w q



0 = 0

0 0

1 0

z 0 θ  . 1 w  q



(17)

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

e2925

P

z

y M M

reference path y S V

Y

X Fig. 5. Scheme of automatic vehicle steering.

Fig. 6. Active suspension control [5].

3. Extended model for automatic vehicle control 3.1. Extended state space model for vehicle steering In order to consider the problem of automatic car steering, the extended model of vehicle is introduced. The extended model must include not only velocities, but also the vehicle heading and the lateral position of the sensor with respect to the reference path. This extended model is derived using a nonlinear model that is valid for deviations from a stationary path. It is assumed that the reference path is given as an arc with radius Rref and center M (See Fig. 5) [3]. For a straight path segment the radius is Rref = ∞. It is more convenient to introduce the curvature ρref := 1/Rref as input that generates the reference path. The curvature is defined positive for left cornering and negative for right cornering. The radial line from the center M passing through the center of gravity (P) of the vehicle intersects a unique point zM on the desired path. It is assumed that there is a small deviation from the reference point zM to the center of gravity which is the deviation yp and that a vehicle fixed coordinate system (x, y) is rotated from the inertially fixed coordinate system (X , Y ) by the yaw angle ψ . The tangent to the reference path at zM is rotated by a reference yaw angle ψt with respect to X . Thus, the rate of change of yp is given by V sin(β + 4ψ) where β is the vehicle sideslip angle and 4ψ := ψ − ψt is the angle between the tangent to the reference path at zM and the center line of the vehicle. With the linearization sin(β + 4ψ) ≈ β + 4ψ the deviation yp changes according to y˙ p = V (β + 4ψ).

(18)

If the sensor S is mounted in a distance ls in front of the center of gravity with ls  Rref , the measured deviation ys from the ˙ . Taking this into account, the rate of reference path changes both with y˙ p and under the influence of the yaw rate r = ψ change of the measured deviation is y˙ s = V (β + 4ψ) + ls r .

(19)

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K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

Determination of y˙ s requires knowledge of three variables β, r and 4ψ . The variables β and r are given by Eq. (11). The angle 4ψ will be obtained by

4ψ˙ = r − V ρref .

(20)

Combining Eqs. (11), (19) and (20), the extended state space model is obtained as [2,3,9,10].

 ˙   a11 β  r˙  a21  ˙= 0 4ψ V y˙ s

a12 a22 1 ls

0 g1 (β, r ) 0 β b1 0  r  b2   0  g2 (β, r ) δ +   ρref +  +  µ. −V 0 0 4ψ   0  0 0 0 0 ys



0 0 0 V

 











(21)

Using sensors for the yaw rate r (ex. a gyro) and the deviation ys (ex. a GPS), it can be assumed that the output equation for Eq. (21) is given by

   4ψ ys

 =

0 0

0 0



1 0

β



0  r  . 1 4ψ  ys

(22)

The system (21), (22) with the control input δ , the reference input ρref , the disturbance input µ and the output 4ψ, ys is shown to be controllable and observable w.r.t. δ . 3.2. State space model for the case : |V | < 1, |V˙ | < 1 In this case, the automatic vehicle steering can be described by Eqs. (19) and (20)

   0 4ψ˙ = V y˙ s

  4ψ

0 0

ys

  +





1 −V r+ ρref . ls 0

(23)

Substituting Eq. (12) into Eq. (23) with the approximation tan δ ≈ δ + w(δ), in which w(δ) is small nonlinear function (∈ C ∞ ), we have

   0 4ψ˙ = V y˙ s

  4ψ

0 0

ys

+

V

 

l





V 1 −V δ+ ρref + ls 0 l

 

1 w(δ). ls

(24)

Using sensors for the yaw rate r (ex. a gyro) and the deviation ys (ex. a GPS), it can be assumed that the state variables of Eq. (24) is directly obtained. The system (24) with the control input δ and the reference input ρref is shown to be controllable. 4. Optimized servo-controllers for automatic vehicle traveling 4.1. Nonlinear state feedback H∞ optimal controller It is considered to design the H∞ optimal controller so that the output [4ψ T , yTs ]T can be driven to zero as t → ∞ [2]. In this paper, the problem of H∞ optimal control is considered so that the output [4ψ T , yTs ]T can be driven to zero as t → ∞ with the output [w T , qT ]T can be driven to zero as t → ∞ , which are related with the riding comfort. From the steering system (21), (22) with the suspension system (15), (17) and the steering system (23), (24) with the suspension system (15), (17), the vehicle system is able to be rewritten as the same form x˙ i (t ) = Ai xi (t ) + Bi ui (t ) + Di ρref (t ) + Gi (∆)di (t ) y(t ) = Ci xi (t ), (i = 1, 2) where x1 := [β T r T 4ψ T yTs | z T θ T w T qT ]T , x2 := [4ψ ui := [ δ | T

T

θ w q ],

yTs z T T UfT UrT T

|

],

T

T T

(for the case : |V | > 1, |V˙ | < 1) (for the case : |V | < 1, |V˙ | < 1)

(i = 1, 2)

and d1 := µ

(for the case :, |V | > 1, |V˙ | < 1) d2 := w(δ) (for the case :, |V | < 1, |V˙ | < 1).

(25)

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

e2927

Each coefficient matrix is as follows.

   0 a a12 0 0   11 a21 a22 0 0  0 A1 = diag  , 1 0 0 a˜ 31   0 a˜ 41 V ls V 0   0 0 1    0 0 0 0 0 A2 = diag , ˜ a31 a˜ 32 a˜ 33   V 0 a˜ 41 a˜ 42 a˜ 43     0 0     b1  0  b   0  B1 = diag  2  ,  b˜ 31 b˜ 32  0      0 b˜ 41 b˜ 42    0 0     V    0 0  1   , B2 = diag  l ls b˜ 31 b˜ 32      b˜ 41 b˜ 42      0  0     0  0 D1 = diag  ,   0     −V 0 0    0      −V 0 D2 = diag ,  0    0 

0 0 a˜ 32 a˜ 42

1 0 a˜ 33 a˜ 43



0   1   a˜ 34   a˜ 44



0   1   a˜ 34   a˜ 44

0

and

    0 g (β, r )    1  g2 (β, r )  0  G1 (∆) = diag  ,   h1 (β, r )  0    h2 (β, r ) 0    0   V    1  0  G2 (∆) = diag ,  h1 (β, r )    l ls  h2 (β, r ) where h1 (β, r ) = h˜ 31 σf cf (β + ∆1 (β, r )) + h˜ 32 σr cr (β + ∆2 (β, r )), h2 (β, r ) = h˜ 41 σf cf (β + ∆1 (β, r )) + h˜ 42 σr cr (β + ∆2 (β, r )). Now consider the problem of nonlinear state feedback H∞ optimal control for Eq. (25), in which it is to find, if existing, the smallest value γ ∗ ≥ 0 such that for any γ > γ ∗ there exists a state feedback u = l(x)

(26)

such that L2 -gain from d to [ y , u ] is less than or equal to γ , where the system (25) is said to have L2 - gain less than or equal to γ if T

T

Z

k y k2 dt ≤ γ 2 0

T T

T

Z

k u k2 dt

(27)

0

for all T ≥ 0 and all u ∈ L2 (0, T ). For this problem formulation, we can apply the theorem proposed by van der Schaft [10–12]. Then, we obtain the following theorem. Theorem 1. Consider the nonlinear system with disturbances Eq. (25). Let γ > 0. Suppose there exists a C ∞ solution V ≥ 0 to the Hamilton–Jacobi equation

    ∂ V (x) 1 ∂ V (x) 1 ∂ V (x) T 1 T T T T Ax + G ( ∆ ) G ( ∆ ) − BB × + x C Cx = 0, ∂x 2 ∂x γ2 ∂x 2

V ( x0 ) = 0

(28)

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K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

or to the Hamilton–Jacobi inequality [13]

    ∂ V (x) 1 ∂ V (x) 1 ∂ V (x) T 1 T T T T Ax + G(∆)G (∆) − BB × + x C Cx ≤ 0, ∂x 2 ∂x γ2 ∂x 2

V ( x0 ) = 0

(29)

then the closed-loop system for the feedback u = −BT



∂ V (x) ∂x

T (30)

has L2 - gain (from d to [ yT , uT ]T ) less than or equal to γ . Proof. Abbreviated.



The subscript i (= 1, 2) in Eq. (25) is abbreviated in the Theorem 1. 4.2. Suboptimal solution of Hamilton–Jacobi equation (inequality) We consider the following linearized system derived from Eq. (25). x˙ i (t ) = Ai xi (t ) + Bi u(t ) + Di ρref (t ) + Ei di (t ) y(t ) = Ci xi (t ), (i = 1, 2)

(31)

where Ei , (i = 1, 2) is the first-order approximation of Gi (∆), (i = 1, 2). Then, we obtain the following corollary from Theorem 1. Corollary 2. Consider the linearized system (31) derived from the nonlinear steering system (25) and assume Ai is asymptotically stable. The L2 - gain of the linearized system (31) is less than or equal to γ if and only if there exists a solution Pi ≥ 0 of the algebraic Riccati equation ATi Pi + Pi Ai + CiT Ci + Pi



1

γ

E E T − Bi BTi 2 i i



Pi = 0,

(i = 1, 2).

(32)

The H∞ optimal controller for Eq. (31), is given by [10,11]. u(t ) = −BTi Pi xi (t ), Proof. Abbreviated.

(i = 1, 2).

(33)



We suppose an approximate solution of the Hamilton–Jacobi equation (28), ((29)) as V (x) =

1 2

xT Px + Vh (x)

(34)

where P is a solution of the Riccati Eq. (32) and Vh (x) is satisfied the following higher order equation

−

      1 ∂ Vh ( x ) 1 ∂ Vh (x) T 1 ∂ V (x) ∂ Vh (x) ∂ V (x) T T T A∗ x = EE ( ∆ ) − BB × + G ( x ) h ∂x 2 ∂x γ2 ∂x 2 ∂x ∂x

(35)

where A∗ := A − BBT P + γ12 EE T P and Gh (x) is the higher term of γ12 G(∆)GT (∆) − BBT in Eq. (28), (Eq. (29)). The m-th order terms V (m) (x) of V (x), m = 3, 4, . . . , can be computed inductively starting from V (2) (x) = 12 xT Px by using the approximation scheme [12]. 5. Numerical simulations In this section, we consider the numerical examples for the linearized system (31). The specification data of a typical passenger car is given in Table 1. (1) The case 1: V = 15[m/s], |V˙ | < 1 The extended state space model for steering is obtained as follows.

 ˙   β −1.99  r˙   6.69  ˙= 0 4ψ 15 y˙ s

−0.96 2.53 1 1.83

0 0 0 15

0 β 0.997 0 g1 (β, r ) 0  r   10.8   0  g2 (β, r ) + δ+ ρ + µ 0 4ψ   0  −15 ref  0  0 ys 0 0 0

















(36)

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

e2929

Phase (deg) Gain (dB)

Bode Diagrams 100 0 –100 –180 –360 –540 –2 10

10

–1

0

10 Frequency (rad/sec)

10

1

10

2

10

2

Phase (deg) Gain (dB)

Bode Diagrams 100 50 0 –50 –180 – 360 – 540 –1 10

10

0

10

1

Frequency (rad/sec) Fig. 7. Bode diagrams (upper: δ to ∆ψ , lower: δ to ys ) of steering system without control (Case 1).

Impulse Response 400 Amplitude

200 0 –200 – 400

0

1

2

4

5 6 Time (sec)

7

8

9

10

Impulse Response

6000 Amplitude

3

4000 2000 0

0

2

4

6

8

10

Time (sec) Fig. 8. Impulse responses (upper: δ to ∆ψ , lower: δ to ys ) of steering system without control (Case 1). Table 1 Data for a passenger car. l = 2.54 [m] lf = 0.97 [m] lr = 1.57 [m] ls = 1.83 [m] cf = 25,000 [N/rad] cr = 25,000 [N/rad] Kf = 20,000 [N/m] Kr = 18,000 [N/m] Cf = 1,500 [N s/m] Cr = 2,000 [N s/m] V = 15, (or 0.05) [m/s] m = 1, 170 [kg] µ = 0.7 i2 = 1.341 [m2 ]

according to Eq. (21). The steering system (36) is unstable under no control. The Bode diagrams and impulse responses of the steering system without any controller are shown by Figs. 7 and 8, respectively.

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K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

Phase (deg) Gain (dB)

Bode Diagrams 50 0 –50 –360 –180 0 –1 10

10

0

10

1

10

2

10

1

10

2

Frequency (rad/sec)

Phase (deg) Gain (dB)

Bode Diagrams 50 0 –50 180 0 –180 –1 10

10

0

Frequency (rad/sec) Fig. 9. Bode diagrams of steering system with the proposed controller (Case 1).

Impulse response

15 Amplitude

10 5 0 –5 –10 –15

0

1

2

3 Time (sec)

4

5

6

Impulse response

10

Amplitude

0 –10 –20 –30 –40

0

1

2

3 4 Time (sec)

5

6

7

Fig. 10. Impulse responses of steering system with the proposed controller (Case 1).

The H∞ optimal control input for Eq. (36) with Eq. (22) is given by

 β(t )  r (t )  δ(t ) = −[ 4.35 1.29 7.64 1.0]  . 4ψ(t ) ys (t ) 

(37)

The Bode diagrams, impulse responses and step responses of steering system with the proposed controller (37) are shown by Figs. 9, 10 and 11, respectively. Figs. 10 and 11 show that the proposed controller for steering system needs about 3 s to converge to the reference path, in which remain unsolved problems for the vehicle to obtain quick response. The state space model for the suspension system of a vehicle is obtained as follows.

   z˙ 0  θ˙   0 = w ˙  −32.48 −5.65 ˙q

0 0 −7.57 −40.27

1 0 −2.99 −1.07

 



0 0 z 0 1  θ   + −1.44 w   0.000855 −4.04 q −0.000618







0 0   0  Uf  0  + µ 0.000855 Ur h1 (β, r ) 0.001 h2 (β, r )

(38)

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

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Step response 8 Amplitude

6 4 2 0 –2 –4

0

1

2

3 Time (sec)

4

5

6

4

5

6

Step response Amplitude

0 –10 –20 –30 – 40

0

1

2

3 Time (sec)

Fig. 11. Step responses responses of steering system with the proposed controller (Case 1).

Bode Diagrams

Bode Diagrams Gain (dB)

–100 –150 90

Phase (deg)

Phase (deg)

Gain (dB)

–50

0 –90 –1 10

0

1

10 10 Frequency (rad/sec)

10

–50 –100 –150 90 0 –90 –1 10

2

Bode Diagrams Gain (dB)

Gain (dB)

10

2

10

2

Bode Diagrams

–100

Phase (deg)

–150 360 Phase (deg)

1

–50

–50

180 0 –1 10

0

10 10 Frequency (rad/sec)

0

1

10 10 Frequency (rad/sec)

10

–100 –150 90

2

0 –90 –1 10

0

1

10 10 Frequency (rad/sec)

Fig. 12. Bode diagrams (upper left: Uf to w , upper right: Ur to w , lower left: Uf to q, lower right: Ur to q) of suspension system with the proposed controller (Case 1).

according to Eq. (15). The suspension system (38) is intrinsically stable. The H∞ optimal control input for Eq. (38) with Eq. (17) is given by z (t ) 0.148  θ (t )  . −0.096 w(t ) q(t )





Uf (t ) −0.003 = Ur (t ) −0.025





0.030 0.003

−0.206 −0.102





(39)

The Bode diagrams, impulse responses and step responses of suspension system with the controller (39) are shown by Figs. 12, 13 and 14, respectively. Fig. 14 shows the suspension system with proposed controller extinguishes the influence of bouncing rate and pitching rate of the vehicle. The numerical simulations with several kinds of active suspension systems are shown in Fig. 15 [5].

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K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

1

Impulse response

x 10–3

–4

10

x 10

Impulse response

Amplitude

Amplitude

0.5 0

5

0

– 0.5 –5

–1 0

2

4

6

0

1

Time (sec) 4

x 10

–4

Impulse response

x 10

10

–4

2 Time (sec)

3

Impulse response

0

5 Amplitude

Amplitude

2

–2 –4

0

–6 –8

–5 0

1

2

3

4

5

0

1

Time (sec)

2

3

Time (sec)

Fig. 13. Impulse responses (upper left: Uf to w , upper right: Ur to w , lower left: Uf to q, lower right: Ur to q) of suspension system with the proposed controller (Case 1).

Step response

–4

2

x 10

Step response

–5

10

x 10

1

Amplitude

Amplitude

1.5

0.5 0

5

0

–0.5 –1

0

2

4

–5

6

0

1

2 Time (sec)

Time (sec) –5

10

0

–5

–10

0

1

2 3 Time (sec)

4

5

4

Step response

–5

Step response

x 10

Amplitude

Amplitude

5

3

x 10

5

0

–5

0

1

2 Time (sec)

3

Fig. 14. Step responses (upper left: Uf to w , upper right: Ur to w , lower left: Uf to q, lower right: Ur to q) of suspension system with the proposed controller (Case 1).

6. Conclusions The problem of automatic driving control of autonomous vehicles which includes the automatic steering control and the induced active suspension control, has been considered. The extended model, including small nonlinear factors of the steering motion, the induced bouncing motion and the induced pitching motion, is derived. The H∞ optimal controller for the extended driving system is introduced, so that the autonomous vehicle can be driven along the reference path with riding stability. The proposed controller is supposed to be considered for autonomous vehicles in which no passengers ride. For the autonomous vehicles with passengers, the proposed controller does not seem to give them a comfortable ride because there are many problems (oscillation and yawing, etc.) which remain unsolved, for a comfortable ride. Recently, the integrated

K. Moriwaki, K. Tanaka / Nonlinear Analysis 71 (2009) e2920–e2933

0.2

0.05 pitching, rad

bouncing, m

0.1

LQR

0 preview 0

2

bouncing accel. m/s2

5

4

LQR 0

–0.05

0

2

4

0

2

4

2 time, sec

4

5

passive pitching accel. rad/s2

–0.1

passive

preview

passive

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LQR 0

0

preview –5

–5 0

2

4

1000

1000 LQR rear control Ur , N

front control Uf , N

preview 0

0

preview –1000 0

2 time, sec

4

–1000 0

Fig. 15. State responses and control sequence with active suspension control [5].

control of each chassis control system such as active 4WS system, active 4WD system, anti-lock brake system and traction control system [1,14,15] is confirmed to have possessed a higher potential of vehicle dynamic performance, and these control systems are applicable to the control of autonomous vehicles. References [1] Y. Matsuo, H. Harada, M. Yamamoto, Y. Kubota, Development of experimental vehicle with integrated chassis control, Toyota Engineering 40 (1990) 120–126. [2] K. Moriwaki, A method of automatic motion control with optimization, in: H. Asama, et al. (Eds.), Proceedings of the Fourth IFAC Symposium on Intelligent Autonomous Vehicles, IFAC, Elsevier, Sapporo, Hokkaido, Japan, 2001, pp. 393–398. [3] J. Ackermann, A. Bartlett, D. Kaesbauer, W. Sienel, R. Steinhauser, Robust Control Systems with Uncertain Physical Parameters, Springer-Verlag, London, 1993. [4] M. Abe, Vehicle Dynamics and Control, Saikai-do Press, Tokyo, 1992. [5] M. Nagai, A. Okada, K. Komoridani, Y. Suda, K. Tani, H. Amijima, S. Nakajiro, H. Harada, M. Miyamoto, H. Yoshioka, Dynamics and Control of Vehicles, Yoken-do Press, Tokyo, 1999. [6] R. Andrezejewski, J. Awrejcewicz, Nonlinear Dynamics of a Wheeled Vehicle, Springer, New York, 2005. [7] K. Kanai, Y. Ochi, T. Kawabe, Vehicle Control — Airplanes and Automotives, Maki Shoten, Tokyo, 2004. [8] J.R. Ellis, Vehicle Dynamics, Business Book Ltd, London, 1969. [9] K. Moriwaki, K. Tanaka, Motion control with optimization for autonomous vehicles, in: Proceedings of 2004 Japan — USA Symposium on Flexible Automation, pages No. JS–27, Denver, Colorado, USA, June 2004. [10] K. Moriwaki, Autonomous steering control for electric vehicles using nonlinear state feedback h∞ control, Nonlinear Analysis 63 (2005) e2257–e2268. [11] A.J. van der Schaft, l2 -gain analysis of nonlinear systems and nonlinear state feedback h∞ control, IEEE Transactions on Automatic Control AC-37 (1992) 770–784. [12] A.J. van der Schaft, L2 -gain and Passivity Techniques in Nonlinear Control, Springer-Verlag, London, 1996. [13] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequality in System and Control Theory, SIAM, Philadelphia, 1994. [14] H. Minegishi, K. Ise, S. Tanaka, S. Katayama, Y. Inoue, H. Miyazaki, Traction control system, Journal of the Society of Automotive Engineers of Japan 42 (1988) 1037–1044. [15] K. Nasukawa, Y. Miyashita, M. Shiokawa, Efficiency Tests for Running of Automotive Vehicles, Sankai-do Press, Tokyo, 1993.