Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
WeM08-2
MCS Adaptive Control of Vehicle Dynamics: an Application of Bifurcation Techniques to Control System Design Bruno Catino 1 , Stefania Santini1 , and Mario di Bernardo23 uation software tool MATCONT [4]. We then shall seek to make explicit use of the system bifurcation diagrams to synthesize an appropriate adaptive control law to achieve the desired dynamics. The effectiveness of such control law will be extensively tested through numerical simulation of the time-trajectories and the numerical continuation of the steady-state behaviour of the closed-loop vehicle.
Abstract In vehicle dynamics, active control is often necessary to extend the open-loop stability range in critic or hard road conditions. Active control of steering angle results in an extension of car stability and safety. In this paper, an Adaptive Minimal Control Synthesis algorithm is proposed for active steering. This method offers an efficient way of solving control problem strongly affected by parametric and modelling uncertainties, as in our case of study, where tire-force characteristics, modelled with the ”Magic Formula”, are not known a priori. The control action is synthesized starting from a bifurcation analysis of the open-loop nonlinear plant. The effectiveness of the proposed method is tested through numerical simulation of the time-trajectories and the numerical continuation of the steady-state behaviour of the closed-loop vehicle.
In order to achieve the desired control objective we use a novel Model Reference adaptive control strategy, namely the Minimal Control Synthesis Algorithm (MCS), see for example [3] [9]. We will see that the proposed control law is effective in extending the stability range of the vehicle dynamics model considered. In so doing, we will also see that continuation methods and bifurcation analysis can indeed be explicitly used in the control design. The rest of the paper is outlined as follows: in section 2 a model for vehicle dynamics is proposed; in section 3 we perform a bifurcation analysis of the previous model in low friction conditions; then, in 4.1, the essential features of Minimal Control Synthesis Algorithm are described while in 4.2 we apply MCS control to vehicle dynamics. Results are shown in section 5.
Keywords: Adaptive Control, Vehicle Dynamics, Bifurcation Analysis
1 Introduction Recently, the design of appropriate control strategies for vehicle dynamics has become increasingly relevant in automotive engineering. The aim is two-fold: (i) to improve the overall performance, driveability and comfort and (ii) to address safety issues concerning driving on low friction surfaces (e.g. icy roads etc). Recently it has been proposed [7] that the dynamical behaviour of a vehicle model for different values of the steering angle can be assessed by performing a bifurcation analysis. Namely, the steady-state behaviour of the model is plotted against different values of the steering angle via so-called continuation methods (see [5] for further details).
2 Model description The essential features of car steering dynamics in an horizontal plane are described by the ”single-track model” [6]. This is obtained by lumping the two front wheels into one wheel in the center line of the car; the same is done with the two rear wheels. Moreover pitch and roll motions are neglected. In this paper, we use a nonlinear version of the two degree of freedom model presented in [7] (see also Fig. 1) which takes the form:
In this paper, we carry out an extensive numerical analysis of the car steady-state behaviour using the contin-
mV (β˙ + r) = Ff + Fr J r˙ = (af Ff − ar Fr ) cos β
1 Dipartimento
di Informatica e Sistemistica, Universit` a degli Studi di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy 2 Dipartimento di Ingegneria, Universit` a del Sannio, Corso Garibaldi 107, 82100 Benevento, Italy, Tel. +39 0824 30 5832, Fax. +39 0824 30 5840. E-mail:
[email protected]. 3 Corresponding Author
0-7803-7924-1/03/$17.00 ©2003 IEEE
(1)
where m is the vehicle mass; af and ar are the distances between the front and rear axle, respectively; V is the vehicle velocity at the Center of Gravity, CoG,
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equilibria in the case δ 6= 0 and to assess their robustness to parameter variations. In fact they can strongly modify the dynamics of a nonlinear plant by destroying and/or creating equilibria points [10].
(assumed always positive); J is the moment of inertia with respect to a perpendicular axis through CoG; β is the angle between the velocity vector and the center line (the so called sideslip angle); r is the yaw rate; Ff and Fr represent tire-road contact forces along the directions reported in Figure 1.
In this work we perform a stability analysis of the vehicle dynamics as the steering angle δ varies. In so doing, we make explicit use of the continuation software package MATCONT [4] to perform such analysis. The results are summarized in the diagrams shown in Figure 2. Here the asymptotic states β and r of the system are plotted against δ. Starting from the stable equilibrium at the origin when δ = 0 we note the occurrence of two ”saddle-node” bifurcations [10], symmetric with respect to the origin at δ ≈ ±0.015 [rad]. In practice a branch of stable equilibria rooted at the origin collides with an unstable branch and is therefore annihilated. Thus, at low steering angles, the vehicle dynamics will be attracted onto the stable branch of equilibria. Because of the saddle-node bifurcations, any value of |δ| > 0.015 [rad] will induce the vehicle to exhibit unstable behaviour due to the lack of any stable attractor in that region. Hence, without any appropriate stabilizing control action, our analysis predicts that the vehicle will not achieve the desired trajectory (requested by the driver) but will spin instead in an unsafe manner.
δ α1 V1
β
Ff
V
β r af
CoG
α2 V2 ar
Fr
β
Figure 1: Vehicle Model
Ff Fr
=
Sideslip angle β [rad]
Tire forces are calculated with the well-known Magic Formula [2] as Df sin[Cf arctan{Bf (1 − Ef )αf + +Ef arctan(Bf αf )}] (2a) Dr sin[Cr arctan{Br (1 − Er )αr +
=
+Er arctan(Br αr )}]
(2b)
High Friction Road
0.1
SN SN
0 SN
−0.1
−0.2 −0.06
with
SN
Low Friction Road
−0.04
−0.02 δ [rad] 0
0.02
0.04
0.06
0.4
αf αr
= =
Yaw Velocity r [rad/s]
High Friction Road
¶ af r δ − arctan tan β + V cos β¶ µ ar r − arctan tan β − V cos β µ
(3)
SN
0.2
SN
0 Low Friction Road SN
−0.2 SN
−0.4 −0.08
All the coefficient values are reported in Table 1 and Table 2.
−0.06
−0.04
−0.02 δ [rad] 0
0.02
0.04
0.06
0.08
Figure 2: Bifurcation diagrams for a vehicle on low friction road and high friction road. Saddle-node points are labelled as SN
3 Stability Analisys Choosing δ = 0, it is easy to see that the origin is a stable equilibrium of the system (Eqs. (1), (2), (3)). It is relevant to explore the existence and stability of af m V
1.2 m 1500 Kg 20 m/s
ar J
LFR Front Rear HFR Front Rear
1.3 m 3000 Kgm2
Bf ,Br 11.275 18.631 Bf ,Br 6.7651 9.0051
Cf ,Cr 1.5600 1.5600 Cf ,Cr 1.3000 1.3000
Df ,Dr 2574.7 1749.7 Df ,Dr 6436.8 5430.0
Ef ,Er -1.9990 -1.7908 Ef ,Er -1.9990 -1.7908
Table 2: Coefficients for the Magic Formula for Low FricTable 1: Vehicle parameters
tion Road (LFR) and High Friction Road (HFR)
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This scenario suggests the use of an adaptive control method which offers an efficient way of solving control problems affected by parametric and modelling uncertainties. For this purpose, we make use of a novel adaptive algorithm Minimal Control Synthesis, first presented in [9], which is based on Landau’s original Model Reference adaptive scheme. MCS requires no a priori knowledge of the system parameters and no initial guess of the gain values which are adapted from homogeneous initial conditions. A proof of asymptotic stability is given in [3] together with the scheme robustness to nonlinear perturbations.
4 Active Steering by Adaptive Control The primary purpose of vehicle dynamics active control is to provide extended stability range even in critic or hard road conditions (snow, icy road...,etc. ). Active Steering can be used to extend vehicle stability and safety with an appropriate compensation of the steer angle commanded by the driver. In particular, from the knowledge of desired steering angle δ, of yaw velocity r, and side slip angle β, the controller computes the appropriate compensation angle, thus providing the actual steering angle δf to the vehicle actuators (see Figure 3). In so doing, it is assumed that measurements or accurate estimations of β and r values are always available [8].
4.1 MCS Algorithm: an overview In the MCS algorithm the controlled system is assumed to have unknown parameters but a known phase canonical structure given by: x˙ = Ax + Bu + f (x), where A=
Figure 3: Schematics of Active Steering Control
F
f
... 0 .. .
0 0 .. .
0 −a1
0 −a2
0 ...
0 ...
1 −an
0 0 .. .
B= 0 b
Fr
u(t) = K(t)x(t) + KR (t)r(t)
Characteristics for High Friction Road
4000
Ff
f
r
F , F [N]
0 1 .. .
The control signal u is made up of a feed-forward and a feedback action with time-variant gains:
8000
2000
F
where r(t) is the reference to track. Gains are computed by the continuous-time equations Z t K(t) = α ye (τ )xT (τ )dτ + βye (t)xT (t)
r
Characteristics for Low Friction Road 0 0
1 0 .. .
and b > 0. The structure reference model reflects that of the plant, and it will be defined with Am .
The control synthesis has to take directly into account the highly nonlinear nature of the plant. Moreover,
6000
0 0 .. .
(4)
0.05
0.1
0.15
0.2
αf , αr [rad]
0.3
0.35
0.4
0.45
0.5
Figure 4: Tire Force Characteristics
0
K(0) = K0 , K ∈ Rn
saturation on tire forces makes it impossible to get larger steady yaw velocity r, without unstable vehicle behaviour. Looking at the tire force characteristics in Figure 4 it should be clear that, to get a better vehicle behavior, tire slip angles, αf and αr , should be controlled and moved to values in which tire characteristics give maximum contact force. Notice that this is not easy to be achieved with classical control methods because the parameters of the Magic Formula cannot be estimated online with sufficient accuracy, as road conditions, weight transfer load and velocity change rapidly.
KR (t) = α
Z
t
ye (τ )r(τ )dτ + βye (t)r(t) 0
KR (0) = KR0 , KR ∈ R
with α e β positive scalar adaptation weights. The output error is calculated as ye (t) = Ce xe (t), xe (t) = xm (t) − x(t) £ ¤ Ce = 0 . . . 0 1 P ,
(5a) (5b)
where P is the solution of Lyapunov problem
Another source of uncertainty is represented by the characteristic parameters of the vehicle conditions: mass distribution changes due to passengers induce also changes in moment of inertia and distance line between tires and CoG (see equations (1)).
P Am + ATm P = −Q , Q > 0. The action of the MCS controller forces the state of the controlled plant to evolve along the solution of the reference system; hence the origin of the closed-loop
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in Eq.(1), we then obtain
error dynamics becomes an asymptotically stable equilibrium point. Robustness of MCS, moreover, assure that even if systems contains a bounded, rapidly varying nonlinearity, f (x) = [0 . . . d(x, t)]T , the control goal is achieved up to a bounded tracking error. Formula to describe an upper bound of the tracking error can be found in [3]. 4.2 MCS Active Steering The use of the MCS algorithm for Active Steering can be attractive because of the hardness in estimating tire characteristics and, in our specific case, Magic Formula’s parameters.
αr = −β +
ar r V
(6)
Then, it is possible to eliminate the derivative of the ˙ from Eq.(9) by using the hypothesteering angle, δ, sis of ideal mass distribution, J = maf ar , that generally holds for most passenger cars (see [1] and reference therein). By taking into account all the above considerations, finally we obtain the following vehicle dynamics model #· ¸ " · ¸ 0 1 x˙ 1 x1 ∗ ∗ = + Cf lC x˙ 2 x2 − marfV mar # · ¸ " 0 0 + + δf (10) Cf∗ f (x) ma
0.06
β [rad]
real plant Eqs. (1),(2),(3)
linearized tire side slip angle Eqs. (1),(2),(6)
0.03
LP
linearized tire side slip angle and front tire characteristic Eqs. (1),(2b),(6),(8)
LP
0.02
0.01
0 −0.02
−0.018
−0.016
−0.014
−0.012
−0.01 δ [rad]
−0.008
−0.006
−0.004
−0.002
0
r
0.12
linearized tire side slip angle and front tire characteristic Eqs. (1),(2b),(6),(8)
0.1
r [rad/s]
0.08
where x1 = αr , x2 = α˙ r and
LP
LP
linearized tire side slip angle Eqs. (1),(2),(6)
0.06
f (x) = −
real plant Eqs. (1),(2),(3)
0.04
0.02
0
0
0.002
0.004
0.006
0.008
0.01 δ [rad]
0.012
0.014
0.016
(8)
Differentiating Eq.(7b), after few algebraic manipulations, we get1 : ¶ µ ma2r + J dFr V V · − α˙ r + δ˙ + α ¨r = l JmV 2 dαr l · µ ¶ ¶ ¸ ½µ ma a − J V ma a f r f r −J Cf∗ − Fr + JmV 2 l JmV 2 ! # ¾ " Ã a2f V 1 V ∗ ˙ (δ + αr ) + δ − Cf + + · l mV JV l ¾ ¸ ½ · 2 1 V ar + (9) Fr − (δ + αr ) α˙ r + JV mV l
The effectiveness of this approximation can be assessed by the bifurcation diagrams in Fig. 5, in which no significant difference between the steady-state behaviour of the two models can be detected.
0.04
=
Bifurcation diagrams in figure 5 again show the accuracy of this simplification.
where we replaced δ with the control steering angle δf (see Figure 3).
0.05
α˙ r
af Ff + F r − (af Ff − ar Fr ) + mV JV V + (δ + αr − αf ) + δ˙ (7a) l Ff + F r ar − + (af Ff − ar Fr ) + mV JV V (δ + αr − αf ) (7b) l
−
Ff = Cf∗ αf = Bf Cf Df αf
A first approximation can be made by using Eqs. (1),(2) together with a linearized version of Eq. (3) as af r , V
=
Further, we consider a linearized version of front tire force characteristics
Unfortunately equations (1) are not in the form (4) requested by the MCS algorithm because the input variable δ is nonlinearly weighted by the front axle force characteristic. Hence, few transformations must be applied to system equations in order to design the controller.
αf = δ f − β −
α˙ f
0.018
l2 Cf∗ ar + V 2 mar lx2 dFr Fr (x1 ) − (x1 ) 2 JV m maf V dx1 (11)
With this reformulation we can now adopt Minimal Control Synthesis Algorithm.
0.02
1 With Eq. (8) it is possible to obtain α = α (α , α r ˙ r , δ) which f f shall be used in the differentiation of (7b). This is a necessary choice because, following the same derivation to obtain αf , α˙ f as state variables, bigger saturation on rear tire force would make it impossible to linearize relationship (2b) without generating intolerable errors.
Figure 5: Bifurcation diagrams for different plant models
Inverting the linearized Eqs.(6), and substituting them
2255
0.15
5 Results
LP
0.1 0.05 r [rad/s]
Simulation results refers to MCS control strategies based on two different reference models. First, we use a linearized version of Eq.(10) for a low friction road, stable for all steer angle; then the nonlinear reference model, (1), (2), (3), has been adopted describing the vehicle dynamics on a high friction road. Its stability is investigated in Fig. 2 and Magic Formula’s coefficients for high friction road can be found in Table 2.
0 −0.05
LP
−0.1
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.15
0.1
0.05
β [rad]
LP 0
LP −0.05
25
−0.1
−0.15 −0.06
20
−0.05
−0.04
−0.03
−0.02
−0.01
0 δ, δ
f
0.01
0.02
0.03
0.04
0.05
0.06
[rad]
HFR without control 15
10
Figure 7: Bifurcation diagrams for uncontrolled vehicle
LFR without control
(solid) and controlled vehicle (dashed)
5 LFR with control 0
30
40
50
60
70
LFR with control (Nonlinear Reference Model)
HFR without control
80
β [rad]
0 20
−0.01 −0.02 −0.03 0
Figure 6: Extension of stability by active steering at δ =
LFR with control (Linear Reference Model) 0.5
1
time [s]
1.5
2.5
3
3.5
4
0.2
0.03 [rad] r [rad/s]
0.15
Figure 6 shows, as an example, the evolution of the plant model in Eq.(10) controlled via the MCS algorithm using as reference model its linearized version for δ = 0.03 [rad]. The car without the control spins in an unsafe manner, while the behaviour of the controlled vehicle is always stable. Moreover, thanks to the adaptive MCS controller, the vehicle running on a low friction road is forced to follow quite well the trajectory taken by a safe vehicle on a high friction road. The bifurcation diagram of the closed-loop plant is shown in Figure 7, where there are no more saddle-node bifurcations. The efficacy of the MCS algorithm is clearly demonstrated in guaranteeing stability for all steer angles with the plant exhibiting a larger yaw velocity than the open-loop vehicle. Obviously, the yaw velocity is upper bounded due to the finite maximum value of tire forces. These maximum values can be reached by MCS through the efficient control of rear tire side slip angle.
HFR without control 0.1 LFR with control (Nonlinear Reference Model)
0.05 0 0
0.5
1
2.5
3
3.5
4
3
3.5
4
time [s]
0.03
LFR with control (Linear Reference Model)
0.02
δ
f
[rad]
LFR with control (Linear Reference Model)
1.5
0.01 0 0
LFR with control (Nonlinear Reference Model) 0.5
1
time [s]
1.5
2.5
Figure 8: Steer responses at δ = 0.02 [rad] with different reference models 0.01 0 HFR without control
β [rad]
−0.01
LFR with control (Nonlinear Reference Model)
−0.02 −0.03 −0.04 −0.05 LFR with control (Linear Reference Model) −0.06 0 0.5 1
1.5
time 2 [s]
2.5
3
3.5
4
0.25 0.2 r [rad/s]
HFR without control
We compare the vehicle dynamics achieved with MCS algorithm by using linear and nonlinear reference models. As an example, Figures 8 and 9 show results obtained for two different angle commanded by the driver. At δ = 0.02 [rad] the controller working with a nonlinear reference model ensures better tracking for the β state with respect to the linear reference model choice; the opposite happens for the r state tracking. No significant difference can be detected at δ = 0.04 [rad], but the control signal induced by the strategy with a linear reference model has a smaller spike.
0.15 LFR with control (Nonlinear Reference Model)
0.1
LFR with control (Linear Reference Model)
0.05 0
0
0.5
1
1.5
time 2 [s]
2.5
3
3.5
4
0.06
δf [rad]
0.04 LFR with control (Nonlinear Reference Model)
0.02 0 −0.02 0
LFR with control (Linear Reference Model) 0.5 1 1.5
time [s]
2.5
3
3.5
4
Figure 9: Steer responses at δ = 0.04[rad] with different reference models
Removing the assumption of ideal distribution mass and linear front tire characteristic Minimal Control
2256
Synthesis algorithm still ensures good results, as it can be seen in Figure 10.
0 β [rad]
MCS Reference Model
−0.01
Numerical optimum weights −0.02
0
1
2
3 time [s]
Simplified Plant (with control)
0.15
r [rad/s]
β [rad]
0.2
Simplified Plant (with control) 0
1
2
3
4
5
6
7
time [s]
−0.02
−0.04
0
0.1
Linear Reference Model
−0.03
4
5
6
Numerical optimum weights
0.05
Reference Model
MCS
Linear Reference Model
0
0
1
2
3
4
5
6
7
3 time [s] 4
5
6
7
time [s] 0.1
0
1
2
0.08
Real Plant (with control)
0.04
4
5
6
0.04
MCS 0.02
0
Simplified Plant (with control)
Numerical optimum weights
0
1
2
δ
f
[rad]
0.06
3 time [s]
f
0
[rad]
Real Plant (with control)
0.05
δ
r [rad/s]
−0.03
Real Plant (with control)
−0.01
0.02 0
0
1
2
time [s]
3
4
5
Figure 11: Steer responses at δ = 0.02 [rad] of the con-
6
trolled vehicles with canonic MCS weights and numeric optimum weights
Figure 10: Steer responses at δ = 0.03 [rad] of the controlled plant and simplified plant
References [1] J. Ackermann and T. B¨ unte. Actuator rate limits in robust car steering control. Proc. of the 36th Conference on Decision & control, pages 4726–4731, 1997.
In order to assess MCS efficiency and robustness, here we explore the possibility of using the algorithm without transforming the plant equations, i.e. directly applying the nonlinear equations (1),(2),(3). In this case, however, the coefficients Ce (i) of the MCS algorithm (see Eq. 5b) cannot be set as in the original formulation but they must be found by a numerical optimization procedure. In this work we compare the performances of the MCS designed on the plant in canonical form (10) and the MCS algorithm with weights computed by numerical optimization. Both controllers use, as reference model, a linearized plant: the first one uses a linearized version of system (10) while the second works with a linearized version of the single track model (1),(2),(3). Obviously the two reference models are different in their state space form, but identical in behaviour. Applying the two algorithms on the single track plant, the MCS controller with the numeric optimum weights still produces an enlargement of stability regions and similar tracking performance are achieved, see Figure 11.
[2] E. Bakker, H.B. Pacjeka, and L. Lidner. A new tire model with an application in vehicle dynamics studies. Proc. Int. Congress and Exposition Detroit, 890087, 1989. [3] M. Di Bernardo and D.P. Stoten. An application of the minimal control synthesis algorithm to the control and synchronisation of chaotic system. International Journal of Control, 65:925–938, 1996. [4] A. Dhooge, W. Govaerts, Y. Kuznetsov, W. Mestrom, and A.M. Riet. Matcont: a continuation toolbox in matlab. 2002. [5] E.J. Doedel and X.J. Wang. Auto94: Software for continuation and bifurcation problems in ordinary differential equations. Technical Report CRPC-95-2, Center for Research on Parallel Computing, California Institute of Technology, Pasadena, CA, 91125, 1995. [6] T.D. Gillespie. Fundamental of Vehicle Dynamics. SAE Print, Warrendale, 1992. [7] E. Ono, S. Hosoe, H.D. Tuan, and S. Doi. Bifurcation in vehicle dynamics and robust front wheel steering control. IEEE Transactions on control system technology, 6(3):412–420, May 1998.
6 Conclusions
[8] L.R. Ray. Nonlinear state and tire force estimation for advanced vehicle control. IEEE Transactions on control system technology, 3 No.1:4750–4755, 1995.
MCS has been adopted to control vehicle dynamics by active steering. The proposed method ensures stability and safety even when driving on low friction surfaces. Bifurcation techniques have been used both for the analysis and control synthesis. The effectiveness of the proposed method has been successfully tested through numerical simulation of the time-trajectories and the numerical continuation of the steady-state behaviour of the closed-loop vehicle.
[9] D.P. Stoten and H. Benchoubane. Robustness of a minimal controller synthesis algorithm. International Journal of Control, 51(4):851–861, 1990. [10] S.H. Strogatz. Nonlinear Dynamics and Chaos. Perseus Publishing, 2000.
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