- Email: [email protected]
MODEL-MATCHING CONTROL FOR STEER-BY-WIRE VEHICLES
MODEL-MATCHING CONTROL FOR STEER-BY-WIRE VEHICLES Julien Coudon ∗,1 Carlos Canudas-De-Wit ∗∗ Xavier Claeys ∗ ∗
RENAULT Research Department, Technocentre, 1 avenue du Golf, 78288 Guyancourt Cedex, FRANCE Email: [email protected], [email protected] ∗∗
Laboratoire d’Automatique de Grenoble, ENSIEG-INPG, BP 46, 38402 St Martin d’H`eres, FRANCE Email: [email protected]
Abstract: We propose a new architecture for Steer-by-Wire vehicle control based on Model-Matching strategy. The two-time scale nature of the considered systems leads to a ”multi-loop” control. An inner-loop is first designed to damp fast dynamics then an outer loop is computed on the slow dynamics to achieve ModelMatching. Finally, an extra loop is added to reproduce a virtual steering column and to synchronize the two systems. Keywords: Steer-by-Wire systems, vehicle control, Model-Matching.
1. INTRODUCTION In recent years, important efforts have been undertaken in drive-by-wire technologies. The Automotive industry is highly motivated by the introduction of decoupled actuators in order to improve driving comfort and security, reduce vehicle cost and accelerate proving ground testing. Steer-by-Wire is such a technology that replaces the mechanical interface between the driver steering wheel and the vehicle front wheels. Two actuators are used: the steering wheel actuator generates a force-feedback on the steering wheel and the front wheels actuator moves the front wheels. Expected benefits are improved safety (by the suppression of the mechanical link between driver and wheels), better vehicle maneuverability and handling characteristics. Steer-by-Wire can also be further exploited to enhance vehicle dynamic behavior. Different Steer-by-Wire strategies have been reported in the literature. Some of these approaches 1
only use local feedback (i.e. the steering wheel angle) to produce a reaction torque computed so as to give a comfortable driving feel (Segawa, 2000; Wook, 2003; Segawa, 2002). However, true road information is not transmitted to the driver. Without road force feedback, the driver can not be aware of the hazards of the road, such as variations of the road surface conditions, curbs or also ruts. These approaches are similar to the unilateral tele-operation strategy. It is thus well accepted that force-feedback needs to be transmitted to the driver in a Steer-by-Wire system. In this context, other works have developed control structures including force-feedback strategies. In works of (Bajcinca et al., 2003; Seltur et al., 2002; Fukao et al., 2001) control strategies informing the driver of the real status of his vehicle and especially the front wheel slip have been proposed. However, in most of these applications the impact of the force-feedback in the global vehicle dynamical behavior is often not considered. In (Coudon et al., 2006), an new reference model for Steer-by-Wire application has been developed to account for force reflection as
This work was supported by RENAULT
614
Steering Wheel
Gear
Torsional Bar
Motor 1 Motor 2 Gear Valve Rack
Hydraulic steering actuator
Fig. 1. SBW vehicle well as for vehicle dynamics interaction. It has been designed to reproduce a virtual stiff steering column with the driver torque as an input. The model also depends on lateral contact forces and a virtual feedback force. These properties make this model innovative with respect to existing contributions in the field. In this paper, we propose a control architecture which aims at reproducing the behavior of this particular reference model. This can be treated by a Model-Matching problem. We use classic feedback technics to perform Model-Matching on the Steer-by-Wire systems. However, such a technic needs the reference model and the systems to have same dimensions, which is not the case here. Then, an additional inner-feedback loop is necessary to reduce the dimension of Steer-by-Wire systems before designing Model-Matching control law. 2. REFERENCE MODEL AND SBW SYSTEM 2.1 SBW system The SBW vehicle is composed by two subsystems: the steering wheel system and the steering system. Steering wheel system: A state-space representation for the steering wheel system is the following 2 : · ¸ Td X˙ sw = Asw Xsw + Bsw1 usw + Bsw2 (1) Text £ ¤ ˙ δ, θ˙sw , θsw T and δ = θsw − θm1 . with Xsw = δ, Rm1 θ˙sw and θsw are the velocity angle and position of the steering wheel. θ˙m1 and θm1 are the velocity angle and position of the feedback motor. Matrices Asw , Bsw1 and Bsw2 depend on Jsw the steering wheel inertia, βsw the damping on the steering wheel axis, Km1 the stiffness of the steering wheel 2
See Appendix for expressions
axis, Rm1 the ratio of the gear on the feedback motor, Jm1 the inertia of the feedback motor axis, βm1 the damping on the feedback motor axis. usw , Td and Text are the steering wheel actuator output torque, the driver torque and the torque due to external forces on the front wheels respectively. Front wheels system: The front wheels system is defined by the following linearized equations 2 : · ¸ Td X˙ f w = Af w Xf w +Bf w1 uf w +Bf w2 (2) Text £ ¤T with Xf w = α, and α = Rdr θf w − ˙ α, θ˙f w , θf w 1 ˙ Rm2 θm2 . θf w and θf w are the velocity angle and position of front wheels. θ˙m1 and θm2 are the velocity angle and position of the steering motor. Matrices Af w , Bf w2 and Bf w1 depend on Mr the rack mass, βr the damping on the rack, Kt the stiffness of the hydraulic valve, Rm2 the ratio of the gear on the steering motor, Jm2 the inertia of the steering motor axis, βm2 the damping on the steering motor axis, Kh defined as the linearized hydraulic gain and S the piston area, d the length of the steering arm on the front axle. us , Td and Text are the steering motor output torque, the driver torque and the torque due to external forces on the front wheels respectively.
2.2 Reference Model Two reference model have been designed in (Coudon et al., 2006). One computes the desired dynamics of the steering system based on mesures of vehicle dynamic variables, external forces on the steering system and the driver torque on the steering wheel. The second model does not use mesures from vehicle variables. Both models could be used for this study. For simplicity, we will only use the second one in this paper. This model is described by the following equations 2 : · ¸ Td ˙ Xref = Aref (V ) Xref + Bref (3) Text £ ref ref ¤T θ ref ref with Xref = θ˙sw and θsw = sw θsw n . ˙θref and θref are the velocity angle and the sw sw position of the virtual steering wheel respectively. Matrices Aref and Bref depend on J the inertia of the virtual steering column and τ the steering assistance gain. Td is the driver torque on the steering wheel and Text accounts for the resulting torque on the virtual column due to external ref forces applied on the front wheels. θw is the desired front wheels angle and n is the steering ratio. Note that the model is vehicle velocity dependent. However, in this paper, we consider the vehicle
615
velocity V to be slowly time varying. As a consequence, matrix Aref is supposed, in what follows, to be constant and function of V . 3. CONTROL STRUCTURE As mentioned before, the vehicle to be controlled is composed of two decoupled systems, the steering wheel and the front wheels ones. The main goal here is to achieve Model-Matching for both systems (1) and (2), so that the global steering system of the Steer-by-Wire vehicle behaves like the reference model (3). However, both systems present higher dimensions than the reference model which is not compatible with the ModelMatching control strategy used. Additional feedback loops need to be added in view to approximate systems (1) and (2) by low-order models compatible with the Model-Matching control at hand.
a reduced-order model is used to design ModelMatching control strategy. Steering wheel system: Considering the steering wheel system and making the assumption that βsw ' 0 and βm1 ' 0, the singularly perturbed model is written as:
²2sw z¨sw
Jsw θ¨sw = −zsw + Td (4) 1 1 1 Td − usw (5) =− zsw + Jsw Jm1 Rm1 J sw
with zsw = Km1 δ, ²2sw
1 Km1 .
1 J sw
=
1 Jsw
+
1 2 Jm1 Rm1
and
= Now the main goal is to damp the fast dynamic variable zsw and to apply the ModelMatching control strategy on the slow one θsw . As a consequence, usw is defined as follows: usw = usw1 + ²sw usw2
(6)
The general structure used to control the two systems of the Steer-by-Wire vehicle is the following (Figure 2):
where usw1 accounts for the slow dynamic variable control and usw2 accounts for the fast dynamic variable control.
• an inner loop is computed to damp fast dynamics and to provide reduced-order models; • an outer loop is used to achieve ModelMatching; • a synchronization loop is used to reproduce a virtual column, synchronizing the steering wheel and the front wheels.
By setting ²sw = 0 in (5), the “steady state value” u ˆf of uf is obtained:
-
Outer Loop +
Synchronization -
Front Wheels
+ -
(7)
Substituting (6) and (7) in the fast dynamics equation (5) leads to: 1 1 ²sw usw2 (zsw − zˆsw ) − Jm1 Rm1 J sw (8) Now, defining ζsw = zsw − zˆsw the fast error dynamics can be written:
System Inner Loop
-
Jm1 Rm1 ˆ Jm1 Rm1 Td − zˆsw Jsw J sw
²2sw z¨sw = −
Steering Wheel
-
u ˆsw =
System Inner Loop Outer Loop
Fig. 2. Control law principle 3.1 Inner loop: damping loop Steering wheel and steering systems have the same structure, they are both constituted with two subsystems linked by a torsional bar. Then, a classic technic in system control, the singular perturbation approach, see for example (Canudas de Wit et al., 1996), is used to approximate the original system by a low order one. The idea is to exploit the two-time scale nature of the flexible part and the rigid part of the dynamic equations. Fast dynamics are damped and
1 1 ²2sw ζ¨sw = − ζsw − ²sw usw2 J J sw m1 Rm1
(9)
The fast control usw2 must stabilize this fast error dynamics so that the fast variable zsw quickly converges to its “steady state” value zˆsw . Then, usw2 is chosen as following: usw2 = λsw ζ˙sw = λsw z˙sw = λsw Km1 δ˙
(10)
λsw is computed in order to insure a good damping to the fast error dynamics (9). Finally, the control law of the steering wheel system can be summarized by: usw = usw1 + Lsw δ˙
(11)
with Lsw = ²sw Km1 λsw . Once the fast dynamics is damped, the slow dynamic variable equation is obtained by getting ²sw = 0 in (5) and replacing the solution zsw in (4): µ ¶ J sw J sw ¨ usw1 + 1 − Td (12) Jsw θsw = Jm1 Rm1 Jsw
616
A reduced-order model is obtained, describing the dynamics of the steering wheel. A state-space representation for(12) is 2 : · ¸ Td r r r r r ˙ Xsw = Asw Xsw +Bsw1 usw1 +Bsw2 (13) Text £ ¤T r with Xsw = θ˙sw θsw .
The reference model is supposed to be stable, all the eigenvalues of Aref have negative real parts. u must be found in order to achieve: (Aref − A) X + (Bref − Bv ) v − Bu u = 0. Defining u = −K X + N v, the Model-Matching Problem is equivalent to find K and N such that: −Bu K = (Aref − A) Bu N = (Bref − Bv )
(21)
Front wheels system: The front wheels system has the same structure as the steering wheel one, the control strategy used is consequently very similar.
Replacing (19) in (21) leads to:
Then, the control law of the steering system is the following: uf w = uf w1 + Lf w α˙ (14) with Lf w = ²wv Kt λf w .
If the two conditions in (18) are verified, then K and N are solutions to the Model-Matching Problem.
It is now possible to define a reduced-order model describing the front wheels dynamics and defined by 2 : · ¸ Td r r r r r ˙ Xf w = Af w Xf w + Bf w1 uf w1 + Bf w1 Text (15) £ ¤T r ˙ with Xf m = θf w θf w 3.2 Outer loop: matching loop Principle: Let us consider a reference model described by its state-space representation: X˙ ref = Aref Xref + Bref v (16) with Aref ∈ Rn×n , Bref ∈ Rn×nv Supposing that the system to be controlled is defined by: X˙ = A X + Bv v + Bu u (17) n×n
with A ∈ R
n×nv
, Bv ∈ R
n×nu
and Bu ∈ R
−Bu Bu+ (A − Aref ) = (Aref − A) Bu Bu+ (Bref − Bv ) = (Bref − Bv )
(22)
Moreover, if (BuT Bu )−1 exists, analytic expressions for K and N can be given: K = (BuT Bu )−1 BuT (A − Aref ) N = (BuT Bu )−1 BuT (Bref − Bv )
(23)
We can now apply this property to each system. Steering wheel system: A brief look to (3), (13) and to the expression of the matrices involved given in Appendix shows that the reducedorder model and the reference model meet the two necessary and sufficient conditions for an exact Model-Matching by state feedback, i.e.: ¡ r ¢ r + r ¡ Brsw1 (Brsw1 )+ − I¢ ¡(Aref − Arsw )¢= 0 Bsw1 (Bsw1 ) − I Bref − Bsw2 = 0 And, as: µ
.
r (Bsw )T 1
r Bsw 1
=
J sw Jsw Jm1 Rm1
¶2 6= 0
The MMP is defined as follows: find u so that the transfert v 7→ X of the system is equal to the transfert v 7→ Xref of the reference model.
it is then possible to compute the following ModelMatching control law:
Property 1. Perfect Model-Matching with bounded control efforts can be achieved if and only if: ¡ ¢ + ¡ Bu B+u − I¢ (Aref − A) = 0 (18) Bu Bu − I (Bref − Bv ) = 0
r + Nsw [Td , Text ] usw1 = −Ksw (V ) Xsw (24) © r T r ª−1 r T r Ksw (V ) = (Bsw1 ) Bsw1 (Bsw1 ) {Asw − Aref (V )} © r T r ª−1 © ª r r Nsw = (Bsw1 ) Bsw1 (Bsw )T Bref − Bsw 1 2
where Bu+ is the pseudo-inverse of Bu . This property has been demonstrated in (Erzberger, 1968). If the two previous conditions are fulfilled, a solution is given by: u = −K X + N v, where: K = Bu+ (A − Aref ) N = Bu+ (Bref − Bv )
(19)
Proof 1. Let us define the Model-Matching error ² = Xref − X: ²¨ = Aref Xref − A X + Bref v − Bv v − Bu u = Aref ² + (Aref − A) X + (Bref − Bv ) v − Bu u (20)
T
Applying (24) to the reduced-order model (13) yields to perfect Model-Matching on this reduced order model. Front wheels system: The reduced-order model defined in (15) and the reference model (3) meet the two necessary and sufficient conditions for an exact Model-Matching by feedback 2 . Moreover, ¶2 µ K2 J f w T −1 (Bf w1 Bf w1 ) = 6= 0 d Mr Kt Jm2 Rm2 the following control law can be computed: 2
See Appendix for expressions
617
Steering Wheel
T
) Xfrw
uf w1 = −Kf w (V + Nf w [Td , Text ] (25) n o−1 © ª Kf w (V ) = (Bfr w1 )T Bfr w1 (Bfr w1 )T Arf w − Aref (V ) ½ ¾ n o−1 1 r T r r T r Nf w = (Bf w1 ) Bf w1 (Bf w1 ) Bref − Bf w2 n 3.3 Synchronization
+ -
δ˙
Td
-
+
Nsw
Ksw (V ) +
PI
-
Front Wheels
Due to uncertainties on the initial conditions or other possible disturbances (measurement errors), the structure does not insure good synchronization between θsw and θf w . Therefor, an additional loop may be required to perform such synchronization.
∆Th −Λ (28) n R Defining ∆Th = KP (V )²+KI (V ) ²dt, for a given vehicle velocity, we finally obtain in the Laplace domain: J ²¨ + kv (V ) ²˙ + kp (V ) ² = −
²= (29) ³ Js3
+ kv (V
)s2
sΛ
+ kp (V ) +
τ KP (V ) n
´ s+
τ KI (V ) n
Among all the possible values for KP (V ) and KI (V ), we can find a single adjustment KP and KI , which gives satisfactory results for all considered values of V . This PI controller in fact simulates a virtual column linking the front wheels system to the steering wheel system. This controller is added to the front wheels system control law with the purpose of simulating an additional torque that the driver should provide in order to synchronize both systems. Full control expressions: The final full control expressions for the two systems are given by: h iT T ˙ ˙ usw = Lsw δ−K (V ) θ , θ +Nsw [Td , Text ] sw sw sw (30)
θf w
α˙ Lf w
+
+
-
+
Nf w
Kf w (V )
Vehicle
Fig. 3. Control law bloc diagram and, uf w =
T
−Lf·w α˙ − Kf w (V ) Xfrw + NZf w [Td , Text ] ¸ θsw θsw +Nf w KP ( − θf w ) + KI ( − θf w ) dt n n (31) 4. SIMULATION AND RESULTS
Figure 3 represents the final control strategy for the front wheels system and the steering wheels system. In reality, external efforts Text are linked to the front wheels position θw by the the transfert function of the vehicle. For simulation purpose, Text is computed as follows: Text = −Tmax arctan(Ct θf w )
(32)
where Ct is the stiffness of tires and front axle seen from the rack and Tmax the maximum tyre/road torque. During simulations, Kh in (2) is replaced by the real non-linear hydraulic law of the valve defined by: k0 (α + k1 α3 ) (33) Moreover, measurement errors of 20 % are simulated on Td and Text . 0.05 Position (rad)
Th + ∆Th Text + + Λ (27) n n where Λ stands for torque disturbances and ∆Th is the amount of driver torque applied to the front wheels system necessary to get rid of Λ. From equations (26) and (27) it is possible to define the synchronization error ² = θsw n − θf w between steering wheel and front wheels:
Xs
System +
θfw
θsw /n
0 ref
θsw /n −0.05 0
1
2
0.05 Position (rad)
J θ¨f w + kv (V ) θ˙f w + kp (V ) θf w =
n
Lsw
+
J θ¨sw + kv (V ) θ˙sw + kp (V ) θsw = Th + Text (26)
θsw
System
Text
Supposing that the two systems reproduce the behavior of the reference model, they are described by the following equations:
Xf
3
Time (s)
4
5
6
7
θsw /n
θfw
0 ref
θsw /n −0.05 0
1
2
3
Time (s)
4
5
6
7
Fig. 4. Time response to a sinusoidal driver torque at 20 km/h
618
In Figures 4 and 5, time responses of the steering wheel, the front wheels and the reference model are shown. For each Figure, time responses without and with synchronization loop are reproduced. The benefits of th synchronization control is clearly visible: both systems, are well synchronized and reproduce the behavior of the reference model, even in presence of non-linearities (hydraulic valve law) and measurement errors. Position (rad)
0.06
θsw /n
0.04
ref
θsw /n
0.02
θfw
0 −0.02 0
1
2
3
Time (s)
4
5
6
7
5
6
7
Position (rad)
0.06 ref
θsw /n
0.04 0.02 0 −0.02 0
θsw /n θfw
1
2
3
Time (s)
4
based on model reference adaptive nonlinear control. In: Proceedings of the 2001 IEEE Intelligent transportation Systems Conference. Oakland. Segawa, M. (2000). A study of vehicle stability control by steer-by-wire system. In: Proceedings of the 5th International Symposium on Advanced Vehicle Control. Segawa, M. (2002). A study of reactive torque control for steer-by-wire system. In: Proceedings of the 7th International Symposium on Advanced Vehicle Control. Seltur, P., D.Dawson, J. Chen and J.Wagner (2002). A nonlinear tracking controller for a haptic interface steer by wire systems. In: Proceedings of the 41st IEEE, Conference on Decision and Control. Wook, S. (2003). The development of an advanced control method for the steer-by-wire system to improve the vehicle maneuvrability and stability. In: Proceedings of SAE International Congress and Exhibition.
Fig. 5. Time response to a driver step of torque at 20 km/h 5. CONCLUSION A new architecture for Model-Matching control on Steer-by-Wire vehicle has been proposed. The Steer-by-Wire vehicle is constituted with two uncoupled systems (steering wheel and front wheels). For both systems, an inner-loop feedback gain is defined to damp fast dynamics. On this basis, the systems are approximated by a reduced-order model respectively. Then Exact Model-Matching control laws are computed on the reduced-order models and applied on each system. Finally, a PI controller is added to the control law of the front wheels system. It reproduces a virtual column between the steering wheel and the front wheels and then insures a good synchronization between the two controlled systems under measurement errors and dynamics differences.
kv (V ) kp (V ) − − J J 1 0
Aref (V ) =
Asw =
−
β2 Jf m 1
0 Km1 − Jsw 0
0
Bsw1 =
Af w =
βm1 βsw − Jm1 Jsw 0 βsw − Jsw 1
2
−ωδ
0
−
βm2 Jm2 1 0
·
Bref =
ωδ = 0
r Km1 Km1 + 2 Jsw Jm1 Rm1
0 0
Bsw2 =
0 0
0 0 1
0 Jsw 0 0
βm2 d βr d − Jm2 Rm2 Mr Rr 0 βr − Mr 1
2
−ωα 0 K2 − d Mr 0
¸
τ 1 J J 0 0
0
−1 Jm1 Rm1 0 0 0
0
0
0
0 0
r K2 Kt + 2 Mr Rr Jm2 Rm2
ωα =
REFERENCES Bajcinca, N., R. Corteso, M. Hauschild, J. Bals and G. Hirzinger (2003). Haptic control for steer-by-wire systems. In: Proceedings of the 2003 IEEE/RSJ Conference on Intelligent Robots and Systems. Las Vegas. Canudas de Wit, C., B. Siciliano and G. Bastin (1996). Theory of Robot Control. Coudon, J., C. Canudas de wit and X. Claeys (2006). A new reference model for steer-bywire applications with embedded vehicle dynamics. In: Submited to ACC 2006. Erzberger, H. (1968). Analysis and design of model following control systems by state space techniques. In: JACC. Fukao, T., S. Miyasaka, K. Mori, N. Adachi and K. Osuka (2001). Active steering systems
6. APPENDIX ¸
·
Bf w 1
−1 Jm2 Rm2 0 0 0
h r Asw
=
0 0 1 0
1 Mr Rr2 0 0 = 1 0 M r Rr d 0 0 0
Bf w2
· r Bsw 1
=
=
µ 1 Jsw
1−
J sw Jsw Jm1 Rm1 0 J sw Jsw
0
h r
Af w =
=
·
i r
Bf w = 1
µ
" r Bf w 2
0 0 1 0
0 0
1 d Mr
K2 J f w 1 − Rr Kt Mr Rr2 0
Kt Rr
i "
r Bsw 2
K2 = Kh S +
¸
¶ # 0 0
K2 J f w d Mr Kt Jm2 Rm2 0
¸
¶# 1 Jfw
=
K2 1 + Mr Rr Kt Jm2 Rf2 m
619
RENAULT Research Department, Technocentre, 1 avenue du Golf, 78288 Guyancourt Cedex, FRANCE Email: [email protected], [email protected] ∗∗
Laboratoire d’Automatique de Grenoble, ENSIEG-INPG, BP 46, 38402 St Martin d’H`eres, FRANCE Email: [email protected]
Abstract: We propose a new architecture for Steer-by-Wire vehicle control based on Model-Matching strategy. The two-time scale nature of the considered systems leads to a ”multi-loop” control. An inner-loop is first designed to damp fast dynamics then an outer loop is computed on the slow dynamics to achieve ModelMatching. Finally, an extra loop is added to reproduce a virtual steering column and to synchronize the two systems. Keywords: Steer-by-Wire systems, vehicle control, Model-Matching.
1. INTRODUCTION In recent years, important efforts have been undertaken in drive-by-wire technologies. The Automotive industry is highly motivated by the introduction of decoupled actuators in order to improve driving comfort and security, reduce vehicle cost and accelerate proving ground testing. Steer-by-Wire is such a technology that replaces the mechanical interface between the driver steering wheel and the vehicle front wheels. Two actuators are used: the steering wheel actuator generates a force-feedback on the steering wheel and the front wheels actuator moves the front wheels. Expected benefits are improved safety (by the suppression of the mechanical link between driver and wheels), better vehicle maneuverability and handling characteristics. Steer-by-Wire can also be further exploited to enhance vehicle dynamic behavior. Different Steer-by-Wire strategies have been reported in the literature. Some of these approaches 1
only use local feedback (i.e. the steering wheel angle) to produce a reaction torque computed so as to give a comfortable driving feel (Segawa, 2000; Wook, 2003; Segawa, 2002). However, true road information is not transmitted to the driver. Without road force feedback, the driver can not be aware of the hazards of the road, such as variations of the road surface conditions, curbs or also ruts. These approaches are similar to the unilateral tele-operation strategy. It is thus well accepted that force-feedback needs to be transmitted to the driver in a Steer-by-Wire system. In this context, other works have developed control structures including force-feedback strategies. In works of (Bajcinca et al., 2003; Seltur et al., 2002; Fukao et al., 2001) control strategies informing the driver of the real status of his vehicle and especially the front wheel slip have been proposed. However, in most of these applications the impact of the force-feedback in the global vehicle dynamical behavior is often not considered. In (Coudon et al., 2006), an new reference model for Steer-by-Wire application has been developed to account for force reflection as
This work was supported by RENAULT
614
Steering Wheel
Gear
Torsional Bar
Motor 1 Motor 2 Gear Valve Rack
Hydraulic steering actuator
Fig. 1. SBW vehicle well as for vehicle dynamics interaction. It has been designed to reproduce a virtual stiff steering column with the driver torque as an input. The model also depends on lateral contact forces and a virtual feedback force. These properties make this model innovative with respect to existing contributions in the field. In this paper, we propose a control architecture which aims at reproducing the behavior of this particular reference model. This can be treated by a Model-Matching problem. We use classic feedback technics to perform Model-Matching on the Steer-by-Wire systems. However, such a technic needs the reference model and the systems to have same dimensions, which is not the case here. Then, an additional inner-feedback loop is necessary to reduce the dimension of Steer-by-Wire systems before designing Model-Matching control law. 2. REFERENCE MODEL AND SBW SYSTEM 2.1 SBW system The SBW vehicle is composed by two subsystems: the steering wheel system and the steering system. Steering wheel system: A state-space representation for the steering wheel system is the following 2 : · ¸ Td X˙ sw = Asw Xsw + Bsw1 usw + Bsw2 (1) Text £ ¤ ˙ δ, θ˙sw , θsw T and δ = θsw − θm1 . with Xsw = δ, Rm1 θ˙sw and θsw are the velocity angle and position of the steering wheel. θ˙m1 and θm1 are the velocity angle and position of the feedback motor. Matrices Asw , Bsw1 and Bsw2 depend on Jsw the steering wheel inertia, βsw the damping on the steering wheel axis, Km1 the stiffness of the steering wheel 2
See Appendix for expressions
axis, Rm1 the ratio of the gear on the feedback motor, Jm1 the inertia of the feedback motor axis, βm1 the damping on the feedback motor axis. usw , Td and Text are the steering wheel actuator output torque, the driver torque and the torque due to external forces on the front wheels respectively. Front wheels system: The front wheels system is defined by the following linearized equations 2 : · ¸ Td X˙ f w = Af w Xf w +Bf w1 uf w +Bf w2 (2) Text £ ¤T with Xf w = α, and α = Rdr θf w − ˙ α, θ˙f w , θf w 1 ˙ Rm2 θm2 . θf w and θf w are the velocity angle and position of front wheels. θ˙m1 and θm2 are the velocity angle and position of the steering motor. Matrices Af w , Bf w2 and Bf w1 depend on Mr the rack mass, βr the damping on the rack, Kt the stiffness of the hydraulic valve, Rm2 the ratio of the gear on the steering motor, Jm2 the inertia of the steering motor axis, βm2 the damping on the steering motor axis, Kh defined as the linearized hydraulic gain and S the piston area, d the length of the steering arm on the front axle. us , Td and Text are the steering motor output torque, the driver torque and the torque due to external forces on the front wheels respectively.
2.2 Reference Model Two reference model have been designed in (Coudon et al., 2006). One computes the desired dynamics of the steering system based on mesures of vehicle dynamic variables, external forces on the steering system and the driver torque on the steering wheel. The second model does not use mesures from vehicle variables. Both models could be used for this study. For simplicity, we will only use the second one in this paper. This model is described by the following equations 2 : · ¸ Td ˙ Xref = Aref (V ) Xref + Bref (3) Text £ ref ref ¤T θ ref ref with Xref = θ˙sw and θsw = sw θsw n . ˙θref and θref are the velocity angle and the sw sw position of the virtual steering wheel respectively. Matrices Aref and Bref depend on J the inertia of the virtual steering column and τ the steering assistance gain. Td is the driver torque on the steering wheel and Text accounts for the resulting torque on the virtual column due to external ref forces applied on the front wheels. θw is the desired front wheels angle and n is the steering ratio. Note that the model is vehicle velocity dependent. However, in this paper, we consider the vehicle
615
velocity V to be slowly time varying. As a consequence, matrix Aref is supposed, in what follows, to be constant and function of V . 3. CONTROL STRUCTURE As mentioned before, the vehicle to be controlled is composed of two decoupled systems, the steering wheel and the front wheels ones. The main goal here is to achieve Model-Matching for both systems (1) and (2), so that the global steering system of the Steer-by-Wire vehicle behaves like the reference model (3). However, both systems present higher dimensions than the reference model which is not compatible with the ModelMatching control strategy used. Additional feedback loops need to be added in view to approximate systems (1) and (2) by low-order models compatible with the Model-Matching control at hand.
a reduced-order model is used to design ModelMatching control strategy. Steering wheel system: Considering the steering wheel system and making the assumption that βsw ' 0 and βm1 ' 0, the singularly perturbed model is written as:
²2sw z¨sw
Jsw θ¨sw = −zsw + Td (4) 1 1 1 Td − usw (5) =− zsw + Jsw Jm1 Rm1 J sw
with zsw = Km1 δ, ²2sw
1 Km1 .
1 J sw
=
1 Jsw
+
1 2 Jm1 Rm1
and
= Now the main goal is to damp the fast dynamic variable zsw and to apply the ModelMatching control strategy on the slow one θsw . As a consequence, usw is defined as follows: usw = usw1 + ²sw usw2
(6)
The general structure used to control the two systems of the Steer-by-Wire vehicle is the following (Figure 2):
where usw1 accounts for the slow dynamic variable control and usw2 accounts for the fast dynamic variable control.
• an inner loop is computed to damp fast dynamics and to provide reduced-order models; • an outer loop is used to achieve ModelMatching; • a synchronization loop is used to reproduce a virtual column, synchronizing the steering wheel and the front wheels.
By setting ²sw = 0 in (5), the “steady state value” u ˆf of uf is obtained:
-
Outer Loop +
Synchronization -
Front Wheels
+ -
(7)
Substituting (6) and (7) in the fast dynamics equation (5) leads to: 1 1 ²sw usw2 (zsw − zˆsw ) − Jm1 Rm1 J sw (8) Now, defining ζsw = zsw − zˆsw the fast error dynamics can be written:
System Inner Loop
-
Jm1 Rm1 ˆ Jm1 Rm1 Td − zˆsw Jsw J sw
²2sw z¨sw = −
Steering Wheel
-
u ˆsw =
System Inner Loop Outer Loop
Fig. 2. Control law principle 3.1 Inner loop: damping loop Steering wheel and steering systems have the same structure, they are both constituted with two subsystems linked by a torsional bar. Then, a classic technic in system control, the singular perturbation approach, see for example (Canudas de Wit et al., 1996), is used to approximate the original system by a low order one. The idea is to exploit the two-time scale nature of the flexible part and the rigid part of the dynamic equations. Fast dynamics are damped and
1 1 ²2sw ζ¨sw = − ζsw − ²sw usw2 J J sw m1 Rm1
(9)
The fast control usw2 must stabilize this fast error dynamics so that the fast variable zsw quickly converges to its “steady state” value zˆsw . Then, usw2 is chosen as following: usw2 = λsw ζ˙sw = λsw z˙sw = λsw Km1 δ˙
(10)
λsw is computed in order to insure a good damping to the fast error dynamics (9). Finally, the control law of the steering wheel system can be summarized by: usw = usw1 + Lsw δ˙
(11)
with Lsw = ²sw Km1 λsw . Once the fast dynamics is damped, the slow dynamic variable equation is obtained by getting ²sw = 0 in (5) and replacing the solution zsw in (4): µ ¶ J sw J sw ¨ usw1 + 1 − Td (12) Jsw θsw = Jm1 Rm1 Jsw
616
A reduced-order model is obtained, describing the dynamics of the steering wheel. A state-space representation for(12) is 2 : · ¸ Td r r r r r ˙ Xsw = Asw Xsw +Bsw1 usw1 +Bsw2 (13) Text £ ¤T r with Xsw = θ˙sw θsw .
The reference model is supposed to be stable, all the eigenvalues of Aref have negative real parts. u must be found in order to achieve: (Aref − A) X + (Bref − Bv ) v − Bu u = 0. Defining u = −K X + N v, the Model-Matching Problem is equivalent to find K and N such that: −Bu K = (Aref − A) Bu N = (Bref − Bv )
(21)
Front wheels system: The front wheels system has the same structure as the steering wheel one, the control strategy used is consequently very similar.
Replacing (19) in (21) leads to:
Then, the control law of the steering system is the following: uf w = uf w1 + Lf w α˙ (14) with Lf w = ²wv Kt λf w .
If the two conditions in (18) are verified, then K and N are solutions to the Model-Matching Problem.
It is now possible to define a reduced-order model describing the front wheels dynamics and defined by 2 : · ¸ Td r r r r r ˙ Xf w = Af w Xf w + Bf w1 uf w1 + Bf w1 Text (15) £ ¤T r ˙ with Xf m = θf w θf w 3.2 Outer loop: matching loop Principle: Let us consider a reference model described by its state-space representation: X˙ ref = Aref Xref + Bref v (16) with Aref ∈ Rn×n , Bref ∈ Rn×nv Supposing that the system to be controlled is defined by: X˙ = A X + Bv v + Bu u (17) n×n
with A ∈ R
n×nv
, Bv ∈ R
n×nu
and Bu ∈ R
−Bu Bu+ (A − Aref ) = (Aref − A) Bu Bu+ (Bref − Bv ) = (Bref − Bv )
(22)
Moreover, if (BuT Bu )−1 exists, analytic expressions for K and N can be given: K = (BuT Bu )−1 BuT (A − Aref ) N = (BuT Bu )−1 BuT (Bref − Bv )
(23)
We can now apply this property to each system. Steering wheel system: A brief look to (3), (13) and to the expression of the matrices involved given in Appendix shows that the reducedorder model and the reference model meet the two necessary and sufficient conditions for an exact Model-Matching by state feedback, i.e.: ¡ r ¢ r + r ¡ Brsw1 (Brsw1 )+ − I¢ ¡(Aref − Arsw )¢= 0 Bsw1 (Bsw1 ) − I Bref − Bsw2 = 0 And, as: µ
.
r (Bsw )T 1
r Bsw 1
=
J sw Jsw Jm1 Rm1
¶2 6= 0
The MMP is defined as follows: find u so that the transfert v 7→ X of the system is equal to the transfert v 7→ Xref of the reference model.
it is then possible to compute the following ModelMatching control law:
Property 1. Perfect Model-Matching with bounded control efforts can be achieved if and only if: ¡ ¢ + ¡ Bu B+u − I¢ (Aref − A) = 0 (18) Bu Bu − I (Bref − Bv ) = 0
r + Nsw [Td , Text ] usw1 = −Ksw (V ) Xsw (24) © r T r ª−1 r T r Ksw (V ) = (Bsw1 ) Bsw1 (Bsw1 ) {Asw − Aref (V )} © r T r ª−1 © ª r r Nsw = (Bsw1 ) Bsw1 (Bsw )T Bref − Bsw 1 2
where Bu+ is the pseudo-inverse of Bu . This property has been demonstrated in (Erzberger, 1968). If the two previous conditions are fulfilled, a solution is given by: u = −K X + N v, where: K = Bu+ (A − Aref ) N = Bu+ (Bref − Bv )
(19)
Proof 1. Let us define the Model-Matching error ² = Xref − X: ²¨ = Aref Xref − A X + Bref v − Bv v − Bu u = Aref ² + (Aref − A) X + (Bref − Bv ) v − Bu u (20)
T
Applying (24) to the reduced-order model (13) yields to perfect Model-Matching on this reduced order model. Front wheels system: The reduced-order model defined in (15) and the reference model (3) meet the two necessary and sufficient conditions for an exact Model-Matching by feedback 2 . Moreover, ¶2 µ K2 J f w T −1 (Bf w1 Bf w1 ) = 6= 0 d Mr Kt Jm2 Rm2 the following control law can be computed: 2
See Appendix for expressions
617
Steering Wheel
T
) Xfrw
uf w1 = −Kf w (V + Nf w [Td , Text ] (25) n o−1 © ª Kf w (V ) = (Bfr w1 )T Bfr w1 (Bfr w1 )T Arf w − Aref (V ) ½ ¾ n o−1 1 r T r r T r Nf w = (Bf w1 ) Bf w1 (Bf w1 ) Bref − Bf w2 n 3.3 Synchronization
+ -
δ˙
Td
-
+
Nsw
Ksw (V ) +
PI
-
Front Wheels
Due to uncertainties on the initial conditions or other possible disturbances (measurement errors), the structure does not insure good synchronization between θsw and θf w . Therefor, an additional loop may be required to perform such synchronization.
∆Th −Λ (28) n R Defining ∆Th = KP (V )²+KI (V ) ²dt, for a given vehicle velocity, we finally obtain in the Laplace domain: J ²¨ + kv (V ) ²˙ + kp (V ) ² = −
²= (29) ³ Js3
+ kv (V
)s2
sΛ
+ kp (V ) +
τ KP (V ) n
´ s+
τ KI (V ) n
Among all the possible values for KP (V ) and KI (V ), we can find a single adjustment KP and KI , which gives satisfactory results for all considered values of V . This PI controller in fact simulates a virtual column linking the front wheels system to the steering wheel system. This controller is added to the front wheels system control law with the purpose of simulating an additional torque that the driver should provide in order to synchronize both systems. Full control expressions: The final full control expressions for the two systems are given by: h iT T ˙ ˙ usw = Lsw δ−K (V ) θ , θ +Nsw [Td , Text ] sw sw sw (30)
θf w
α˙ Lf w
+
+
-
+
Nf w
Kf w (V )
Vehicle
Fig. 3. Control law bloc diagram and, uf w =
T
−Lf·w α˙ − Kf w (V ) Xfrw + NZf w [Td , Text ] ¸ θsw θsw +Nf w KP ( − θf w ) + KI ( − θf w ) dt n n (31) 4. SIMULATION AND RESULTS
Figure 3 represents the final control strategy for the front wheels system and the steering wheels system. In reality, external efforts Text are linked to the front wheels position θw by the the transfert function of the vehicle. For simulation purpose, Text is computed as follows: Text = −Tmax arctan(Ct θf w )
(32)
where Ct is the stiffness of tires and front axle seen from the rack and Tmax the maximum tyre/road torque. During simulations, Kh in (2) is replaced by the real non-linear hydraulic law of the valve defined by: k0 (α + k1 α3 ) (33) Moreover, measurement errors of 20 % are simulated on Td and Text . 0.05 Position (rad)
Th + ∆Th Text + + Λ (27) n n where Λ stands for torque disturbances and ∆Th is the amount of driver torque applied to the front wheels system necessary to get rid of Λ. From equations (26) and (27) it is possible to define the synchronization error ² = θsw n − θf w between steering wheel and front wheels:
Xs
System +
θfw
θsw /n
0 ref
θsw /n −0.05 0
1
2
0.05 Position (rad)
J θ¨f w + kv (V ) θ˙f w + kp (V ) θf w =
n
Lsw
+
J θ¨sw + kv (V ) θ˙sw + kp (V ) θsw = Th + Text (26)
θsw
System
Text
Supposing that the two systems reproduce the behavior of the reference model, they are described by the following equations:
Xf
3
Time (s)
4
5
6
7
θsw /n
θfw
0 ref
θsw /n −0.05 0
1
2
3
Time (s)
4
5
6
7
Fig. 4. Time response to a sinusoidal driver torque at 20 km/h
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In Figures 4 and 5, time responses of the steering wheel, the front wheels and the reference model are shown. For each Figure, time responses without and with synchronization loop are reproduced. The benefits of th synchronization control is clearly visible: both systems, are well synchronized and reproduce the behavior of the reference model, even in presence of non-linearities (hydraulic valve law) and measurement errors. Position (rad)
0.06
θsw /n
0.04
ref
θsw /n
0.02
θfw
0 −0.02 0
1
2
3
Time (s)
4
5
6
7
5
6
7
Position (rad)
0.06 ref
θsw /n
0.04 0.02 0 −0.02 0
θsw /n θfw
1
2
3
Time (s)
4
based on model reference adaptive nonlinear control. In: Proceedings of the 2001 IEEE Intelligent transportation Systems Conference. Oakland. Segawa, M. (2000). A study of vehicle stability control by steer-by-wire system. In: Proceedings of the 5th International Symposium on Advanced Vehicle Control. Segawa, M. (2002). A study of reactive torque control for steer-by-wire system. In: Proceedings of the 7th International Symposium on Advanced Vehicle Control. Seltur, P., D.Dawson, J. Chen and J.Wagner (2002). A nonlinear tracking controller for a haptic interface steer by wire systems. In: Proceedings of the 41st IEEE, Conference on Decision and Control. Wook, S. (2003). The development of an advanced control method for the steer-by-wire system to improve the vehicle maneuvrability and stability. In: Proceedings of SAE International Congress and Exhibition.
Fig. 5. Time response to a driver step of torque at 20 km/h 5. CONCLUSION A new architecture for Model-Matching control on Steer-by-Wire vehicle has been proposed. The Steer-by-Wire vehicle is constituted with two uncoupled systems (steering wheel and front wheels). For both systems, an inner-loop feedback gain is defined to damp fast dynamics. On this basis, the systems are approximated by a reduced-order model respectively. Then Exact Model-Matching control laws are computed on the reduced-order models and applied on each system. Finally, a PI controller is added to the control law of the front wheels system. It reproduces a virtual column between the steering wheel and the front wheels and then insures a good synchronization between the two controlled systems under measurement errors and dynamics differences.
kv (V ) kp (V ) − − J J 1 0
Aref (V ) =
Asw =
−
β2 Jf m 1
0 Km1 − Jsw 0
0
Bsw1 =
Af w =
βm1 βsw − Jm1 Jsw 0 βsw − Jsw 1
2
−ωδ
0
−
βm2 Jm2 1 0
·
Bref =
ωδ = 0
r Km1 Km1 + 2 Jsw Jm1 Rm1
0 0
Bsw2 =
0 0
0 0 1
0 Jsw 0 0
βm2 d βr d − Jm2 Rm2 Mr Rr 0 βr − Mr 1
2
−ωα 0 K2 − d Mr 0
¸
τ 1 J J 0 0
0
−1 Jm1 Rm1 0 0 0
0
0
0
0 0
r K2 Kt + 2 Mr Rr Jm2 Rm2
ωα =
REFERENCES Bajcinca, N., R. Corteso, M. Hauschild, J. Bals and G. Hirzinger (2003). Haptic control for steer-by-wire systems. In: Proceedings of the 2003 IEEE/RSJ Conference on Intelligent Robots and Systems. Las Vegas. Canudas de Wit, C., B. Siciliano and G. Bastin (1996). Theory of Robot Control. Coudon, J., C. Canudas de wit and X. Claeys (2006). A new reference model for steer-bywire applications with embedded vehicle dynamics. In: Submited to ACC 2006. Erzberger, H. (1968). Analysis and design of model following control systems by state space techniques. In: JACC. Fukao, T., S. Miyasaka, K. Mori, N. Adachi and K. Osuka (2001). Active steering systems
6. APPENDIX ¸
·
Bf w 1
−1 Jm2 Rm2 0 0 0
h r Asw
=
0 0 1 0
1 Mr Rr2 0 0 = 1 0 M r Rr d 0 0 0
Bf w2
· r Bsw 1
=
=
µ 1 Jsw
1−
J sw Jsw Jm1 Rm1 0 J sw Jsw
0
h r
Af w =
=
·
i r
Bf w = 1
µ
" r Bf w 2
0 0 1 0
0 0
1 d Mr
K2 J f w 1 − Rr Kt Mr Rr2 0
Kt Rr
i "
r Bsw 2
K2 = Kh S +
¸
¶ # 0 0
K2 J f w d Mr Kt Jm2 Rm2 0
¸
¶# 1 Jfw
=
K2 1 + Mr Rr Kt Jm2 Rf2 m
619