TP1 - 3 2 0 A Sliding Mode Control Approach to Automatic Car Steering

Jurgen Guldner Vadim I. Utkin Jurgen Ackermann

DLR - Institute for Robotics and System Dynamics D-82230 Wessling, Germany

Abstract

Highway automation is a promising approach to cope with increasing road traffic and congestions. An important control subtask of an Intel- ligent VehicIe/Ifaghwoy System is automatic steering for lateral vehicle control. Due to system uncertainties and the wide range of operating conditions, state-of-the-art robust control techniques are required. This paper introduces two automatic steering controllers for cars driving un- der highway conditions. The control design is based on sliding mode control and robust state observation. It is shown that good tracking of a reference path, delineated either continuously or discretely, can be achieved with a minimum effort in sensing and without preview of the road curvature.

1. Introduction

Road traffic has steadily increased during the last decades. Especially me- tropolitan areas suffer from growing traffic congestions not only during rush hours. Numerous strategies to improve automotive mass transit have been under consideration for several years. Some progress has been made by enhan- cing public transport systems and promoting car pooling (“high occupancy lanes”). However, such approaches cannot considerably reduce road traffic since they restrict the flexibility and mobility of the individual. Varaiya argues that measures assisting the driver, for example by advanced route and road informations systems, cannot significantly ameliorate road traffic and full automation of highway traffic is indispensable for increasing highway throughput and driving safety [I]. A recent comprehensive systems study of design, development and deployment of fully automated highway systems was presented by Bender [Z]. A suitable system architecture for an Intelligent Vehicle/Highway System (IVHS) is the subject of on-going research e.g. at Ohio State University [3] and in the Californian PATH project [4]. There are a number of well-defined control subtasks which can be addressed a priori. The two major control tasks for highway automation are longitudinal control (spacing between cars) and lateral vehicle control (lane keeping). There are several approaches to lateral vehicle control. Complete autonomy of the vehicle can be achieved through an intelligent vision system [5] with on- line generation of the desired path. A perception system replicating a human driver can sense the upcoming road geometry and, in addition, objects in front of the vehicle. This would open the possibility of a common sensing device for lateral and longitudinal vehicle control. Enabling the vehicle t o “see” e.g. the lane edge stripping under varying weather and road conditions, however, challenges present limits of real-time vision systems. The alternative is an indirect sensing approach with a reference for lateral control built into the road surface. A more detailed discussion on direct and indirect sensing can be found e.g. in [6].

Different realizations of reference systems supplying the vehicle with lateral information are possible. A continuous reference guideline using an electric wire was studied by Daimler-Benz and MAN in Germany for a city-bus [7,8]. A radar reflecting guard rail at the side of the lane was examined at OSU [3,9]. Alternatively, permanent magnets can be used as discrete markers in the road surface [4,6,10]. The goal of control is to track the delineated pathway.

A study of vehicle lateral control techniques [ll] suggested that an intelligent reference system for automatic steering should possess anticipatory capability rather than merely having compensatory behavior. In other words, a preview of the road geometry, conveyed to the car and used as feed-forward aides the control algorithm. In the discrete marker approach pursued in the Californian PATH project, preview of the roadway curvature is encoded bit-wise in the permanent magnets [4] and used to specify a desired steering angle [6,12,13]. However, the feed-forward controller depends on knowledge of the plant pa- rameters, which is not available in general and requires an estimation scheme e.g. for the cornering stiffness of the tires [4].

Due to the wide range of possible operating conditions (vehicle mass and inertia, road adhesion, etc.), automatic steering is a task for robust control. Steering actuator saturation and limited state information represent further challenges to control design. In addition, passenger comfort and safety consi- derations have to be taken into account. We present two controllers based on sliding mode control and robust state observation. The control design in Section 3 uses only measurements of the

lateral displacement from the reference path. The variable under control is the input of an integrating steering actuator, the actual steering angle also being unknown. Stability and robustness of the proposed control algorithm under parametric uncertainty is proved using Lyapunov stability theory.

In Section 4, the automatic steering system is improved by additional feedback of the vehicle yaw rate measured by a gyroscope. A cascaded controller is designed and its performance is illustrated in numerical studies of typical driving maneuvers in Section 5. Special emphasis is placed on eliminating chattering caused by unmodelled dynamics via observers.

Both controllers enable direct inclusion of specifications l i e actuator con- straints and ride quality requirements into the design procedure. The crucial issue of implementation with a digital computer is discussed in detail. The discrete control algorithms are implementable both for continuous and for discrete reference systems without modification. It is concluded that satisfac- tory performance under uncertainty conditions is achievable without preview information of the road curvature.

2. Dynamic Model

2.1. Model for Car Steering The classical single-track model of Riekert-Schunck [14] is used to model the steering dynamics. It 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. The variables in Figure 1 denote:

J /3 T yaw rate, Sf front wheel steering angle, e, (e,)

velocity vector with magnitude w > 0, sideslip angle between vehicle centerline and velocity vector at CG,

distance of CG from front (rear) axle

For simplicity, control design is based on a linearized model [15]. Extension to a more detailed nonlinear model is straightforward, since the design procedure uses nonlinear control theory. The lineaxized dynamics of the sideslip angle and the yaw rate T are

The front (rear) wheel cornering stifl’nesses C, (4) are empirically det:rmined tire parameters entering into the side force equations. a = m / p and J = J / f i are a ”viiual mass” and a “virtual moment of inertia” obtained through normalization by the road adheaion factor p. All car parameters are taken to be uncertain within known bounds e.g. m E [m-; m+] or fi E [cr-; cl+]. We assume that the vehicle velocity v is known, constant (or varying slowly), and lower bounded.

I I I I I r i I I I I I I I

1- +-- c 1- 4 - Fig. 1: Single-track model for car steering

The front wheel steering actuator is modeled as an integrator

8f =uj

with constraints l6fl 2 6fo, and Iufl 5 ufo

3. Control with Displacement Measurement

In this section, we design a nonlinear controller for automatic steering of a car under the assumption that the lateral displacement a, is the only measured state in model (6) and that no preview of the road curvature is available. The design procedure is based on sliding mode control. The basic idea is to force the dynamic system to restrict its motion to a manifold called the "sli- ding manifold" s(2) = 0. This is achieved by directing the system trajectories towards this manifold "from both sides" using two different controls U+ and U-, as shown in Figure 3.

The main benefits of sliding mode control are its invariance property and the ability to decouple high dimensional problems into sub-tasks of lower dimen- sionality. The interested reader is referred to [l6] for a tutorial introduction and to [17] for a more detailed discussion of sliding mode control.

2.2. Model of the Reference Path The model (1) is extended to include vehicle heading and lateral position with respect to the reference path, see Figure 2. The sensor S for measuring the displacement from the reference path is mounted in the center of the front bumper at a distance e, from the center of gravity (CG). A linearized model is derived for small deviations from a stationary curved path. The road curvature p7.f := l/+ef is the reference input defined positive for left cornering and negative for right cornering. The rate of change of YCG is obtained from Figure 2 as

~ C G = wsin(p + A$) = w(P + A+), (3) where A$ = ll, - is the angle between the tangent to the path at z,,, and the centerline of the car. The linearization sin(p+ A$) = p + A$ is valid due to small angles p and A$. Smce e. << R,,f, the lateral velocity at the sensor S with respect to the guiding wire, y, is

= v(p +A$) +tar, (4) where 0 and r are given by the basic car model (1). The angular displacement All, is obtained from

M

(5)

Fig. 2: Measured displacement fl from the reference path

Combining ( l ) , (2), (4), and (5) yields the extended state space model. As outlined in the introduction, there exist continuous and discrete reference systems. For a continuous reference guidelime, it is appropriate to adjust the sampling interval AT of a discrete controller depending on the vehicle velocity. Obviously, fast motion requires a higher sampling rate than slow motion. Sampling data is collected with regard to covered distance rather than with regard to elapsed time, i.e. Ad = VAT. This method coincides with the discrete reference method using uniformly spaced markers. The following control development thus is valid both for a continuous guidelime and for a discrete reference system. To facilitate the design procedure, the motion equations are rewritten in terms of d = wt instead of time 1. For convenience, we will denote "distance" derivatives with "primes". However, differentiation is understood with respect to d instead o f t in the sequel. The overall model is

x = f ( x , .+) s(x) = 0

x = f(z, U-) 7- Fig. 3: State vectors in the vicinity of the sliding surface

3.1. Continuous-Time Controller Design The sliding manifold [17] is defined to be

s i ( d ) = Y" + ~ I Y ' +by, (7)

where d = wt. The goal of control is to restrict the motion of (6) to the manifold sl(d) = 0 in (7). In this case, the tracking behavior of the vehicle is entirely determined by the gains ko and kl. To establish stability of s1 = 0, we examine the Lyapunov function candidate Vl(d) = $5:. Differentiation of Vl(d) along the system trajectories yields

V;(d) = SI ( f i (P ,~ ,A$,6f ,p+. f ,p~=f) + c ~ u f ) t (8)

where = bl + $&. Under normal operating conditions, all states, contribu- ting linearly to fl(.), are bounded. In the sequel, j 2 ( . ) denote linear functions of the states. Furthermore, we assume that the rate of change of road curva- ture p:., is piecewise bounded with isolated points of discontinuities, where pVef changes jumpwise. Using the given uncertainty bounds on the car para- meters, an upper bound (f1)" > Ifl(.)l can be determined, which is valid almost everywhere. Hence there exists a finite scalar ~0 such substitution of the control law

uf = -ugsign(sl)

into (8) results in

for some positive scalar [I. Substitution of VI (d) = $3: into (10) yields

Vl(4 I -FiK(d)~. D e h e a positive definite function Vo(d) with

(9)

vanishes after a finite driving distance:

6) &(IT) is an upper bound for &(d) for d >_ 4, i.e.

h ( d ) I Wd) Vd E do. (15)

Consequently, Vz(d) = 0 after finite distance &, which implies si = 0 in (7) Vd 1 do, and convergence of y to zero as specified by ko and kl .

1870

At locations where ptef changes jumpwise, the above boundedness conditions are violated and the manifold s1 = 0 is left. Such steps in the reference curvature occur only at isolated points, which means that convergence to si = 0 is guaranteed after each step-change of pref within finite time. Direct implementation of the control law (9) requires measurement of y, yf and y“. It is assumed that only y is available from the displacement sensor mounted at the front end of the car. Numerical differentiation is undesirable due to noise considerations and the introduction of parasitic dynamics pre- venting ideal sliding mode to occur along si = 0 (7). To obtain estimates of y’ and y“, a robust observer is designed for the system

r o 1 0 0 1 r o i , . . . .

0 0 1 0 t’ = lo 0 0 + + I ‘ ’

0 0 0 0 d5 1 - -

where zT = [ y yf y” fz(.) 1, cs is given in (8) and ds = (all + $ ~ ~ z l ) b l +

(012 + 1 + +22)b2. Under the assumption that f2(.) is uncertain, but slowly varying, i.e. f; zz 0, an observer is defined as

where “hats” denote estimates (constant for parameters like cs and ds ) and “bars” denote estimation errors, e.g. g = y - 8. The gains 0 << el1 <<

< e13 << e14 determine the desired observer poles, and should be chosen at least one order of magnitude faster than the dynamics of fz(.) for “good” observation. The observation error decays according to

Provided the observer dynamics are sufficiently fast in comparison to the plant dynamics, small errors (18) are achieved even for j i +. 0. According to singu- lar perturbation theory and the motion separation principle, the observation errors are of vi =O(&) order for a = 1, ..., 4.

The estimates for I, y’, y” replace the true values in (7) and (9)

(19) $1 = K l i , K1 = [ ko ki 1 0 1, ut = -uOsign(&),

and provide sliding mode within the observer system (17) with the actual states z in (16) being close to the estimate i in (17) as determined by the dynamics of the observer error (18). It is worth noting two direct consequences of (17)-(19):

Direct implementation of the control (9) with (7) would cause high &e- quency oscillations in the system even for pure feedback of the deriva- tives y’ and y”. This problem is referred to as “chattering” in sliding mode literature and is attributed to unmodeled dynamics like parasitics in sensors and actuators neglected in the linearized model (6). The ob- server loop (17) with control (19) is free of unmodeled dynamics, which allows ideal sliding mode to occur, leading to chattering-free motion in the real system (16), see e.g. [18]. The observer (17) has the structure of a Kalman filter. An appropriate choice of gains e,,, i = 1, ..., 4 results in noise filtering. A trade-off is required between the need for high gains el, for good uncertainty suppression and low gains el, for good filtering.

3.2. Discrete-Time Controller Design Micro-computer realization requires redesign of the continuous control algo- rithm in a discrete form of similar structure. The linear observer equations (17) are transformed using the standard formulas

where Ad is the sampliig interval, e.g. the distance between discrete markers. Immediate implementation of the control law (19) inevitably leads to chat- tering at the sampling rate, even in the absence of unmodeled dynamics and

measurement noise. This undesired effect is caused by the control being con- stant during the sampling interval and eual to either -U, or +U,,, which results in overshoot, see Figure 4.

s(n) A u(n)

Fig. 4: Chattering in discrete time Systems

A remedy is utilization of the equivalent control method for discrete sliding mode systems [19]. The key idea is t o exactly nullify the sliding variable s, i.e. to design the control ~ ( n ) such that s ( n + l ) 3 0. If the control constraint prevents reaching s = 0 in one step, maximum control action is applied to reduce the distance to the sliding manifold. Convergence is guaranteed within a finite number of steps j and s(n) 5 0 V n 2 j , see Figure 5.

. . . _ . _ . , . , . . n . . . -U0

Fig. 5: Chattering-free motion using equivalent control method

Given the observer dynamics

i(n + 1) = A:ba12(n) -k Bibs,Uf(n) + L;g(n), (21) the control uf(n) is designed to reach &(n + 1) = Kli(n + 1) = 0 identically in equation (19). Solving (21) for the control yields

.eq(n) = -[KI B~a,,l-’[Ki(A~b,,i(n) + LIB(n)l,

4. Control Design with Yaw Rate Feedback

In this section we discuss incorporation of yaw rate measurement into the control design. Yaw rate feedback enables design of a cascaded controller, reducing the order of the highest derivative in (7), thus increasing robustness and decreasing noise susceptibility of the automatic steering system.

4.1. Continuous-Time Cascaded Control Design It is important to note that control design in this section does not require measurement of the car velocity if the controller is designed in time domain rather than with regard to the driven distance d. Obviously, this is only possible for a continuous guideline. However, for the sake of a uniform pre- sentation, the controller is designed in d-domain for a known vehicle speed v > 0. Transformation to time-domain is straight forward. The design procedure follows the ideas of control using regular form, see chap. 6 in [17]. The inner loop of the cascaded control system comprises the “output” dynamics (4)

y’ = @ + A @ + $ . (23)

The yaw rate T is considered as a “fictitious” control input to (23) and a “desired” yaw rate Td is defined assunling r could be controlled directly. The control input uf is used to nullify the error e, = r - Td between the actual, measured yaw rate and the desired yaw rate defined as

r d = -E (h(’) + by)) 1 (24)

with state-bounded f3 = @+A$. Observer design assumes slow varying fs(.),

1971

where 0 << e21 <( e22 are the observer gains and @ = y - Q is the exponentially stable observation error. The observer poles should be at least one order of magnitude faster than the dynamics of f3. As outlined previously, observer (25) introduces advantageous ( K h a n - ) noise filtering into the system. Large deviations 8, lead to high Td, undesirable from a point of view of pas- senger comfort. Utilization of a saturation function,

steering Controller

where X limits the feedback of 6 and the ratio 4 determines the gain at the origin p = 0, allows direct inclusion of ride quality considerations. To obtain satisfactory behavior of the overall system, the actual input U! has to be designed to provide the desired yaw rate. The control objective is to drive the error e, = 7 - Td to zero. The sliding manifold is defined as

SZ = ke, +e:, (27) where gain k > 0 determines the rate of decay of e, once the motion has been restricted to s2 = 0. Approach to the manifold s2 = 0 and subsequent stability is established using the Lyapunow function candidate h ( d ) = $3;. Differentiation along (27) yields

%e: - -

where j 4 is an unknown bounded function of the states, i.e. (.fd),,,= 2 If4(.)l. Hence there exists a finite value uo such that

uf = -uosign(sz) (29)

implies convergence of s2 to zero after finite time in the framework of (IO). The sliding variable s2 in (27) depends on r, r', rd , and rb. While T; can be calculated, T' has to be estimated by a third-order observer of the same structure as in (17):

0 1 0

i' = [ 0 0 0 0 0 , I S + [ k l U f + [ ; ; I F = (30)

= &bast + B o b * r U f + L3P1

where Ci = T, C2 = T', and gs = (aiiazi + aizazz)h + (oizazi + a&)b. 6 = fs(.) is assumed to be slow compared to the observer dynamics determined by 0 << &s << e32 << t a l . Similarly to before, ideal sliding mode occurs in the observer-loop system and guarantees chattering-free motion.

4.2. Discrete-Time Cascaded Control Design Using the equivalent control method for discrete systems, the continuous- time cascaded controller designed in the previous section takes the following discrete-time form:

P(n+1) =

Cd(n) =

S ( n + l ) =

&(n) =

f3(n+1) =

=

A block diagram of the controller structure is shown in Figure 6. -- Controller to determine

desired yaw rate

Remarks The functions f;(.), i = 1,2 ... 5 are not needed explicitly in the control development. Thus the above controllers are independent of the model as long as the f; are bounded, allowing all car parameters in (6) to be uncertain. However, bounds on uncertain parameters are vital for the stability analysis. Design based on a nonlinear model instead of (6) would lead to similar results since the controller only depends on the triple-integrator structure of the plant (6).

It is important to maintain the hierarchy of time scales resulting from the tripleintegrator structure when determining the gains of the control loops: The outermost loop in g has to be kept "slower" than the middle loop in p, r, and A$, whereas the innermost loop in 61 can be made arbitrarily fast, being constrained only by actuator limitations. The asymptotic observers, on the other hand, should have time constants at least one order of magnitude higher than the respective closed loop. The strict mathematical derivation in the previous sections neglected physical limitations of the actuator. In reality, steering angle and stee- ring angle rate are bounded. Such constraints require to determine the control parameters in inverse order, starting from the innermost loop and then following the hierarchy described above.

It was stated in [20] that high-gain feedback is not suitable for lateral motion control of a vehicle. At first sight, sliding mode control imple- ments infinitely high gains, a t least in the continuous-time versions of the control algorithms. However, sliding mode control is only used to restrict the motion of the plant to the respective sliding manifold despite system uncertainty. Thereafter the plant behavior is entirely determi- ned by the parameters K1 in (19), and K2 and X/E in (31), allowing incorporation of ride quality considerations into the design procedure.

5. Numerical Studies

In this section, we simulate the car 280 SEL of Daimler-Benz [15,21] in various typical driving maneuvers. A discrete marker reference system is used, but similar results can be obtained for a continuous reference.

5.1. Reference Driving Maneuvers All simulations were performed with the linear vehicle model (6) taking into account actuator saturation. Four typical driving maneuvers are used to eva- luate the performance of the automatic steering system, see Figure 7

llansition from manual to automatic steering. The car is assumed to drive in a distance of y = 0.30m parallel to the guideline when the control is activated.

Driving through a 60" curve with step changes in R,.f. The curve radius is chosen as Driving through a 60" curve with linear curvature transition to and f" the curve radius Ref = = y. Lane change with nominal lateral acceleration of 0.29.

= 9 = 9.

The simulations are performed with a velocity of IJ = 30m/s = 67.5mph and maximal parameter values, e.g. maximal virtual mass fi+. Due to the design in distance domain d rather than time domain t , similar results are obtained for different velocities. For control design, average parameter values were assumed, e.g. a virtual mass of %,,vevage = v. The distance between the markers of the discrete reference trajectory is chosen to be 0.3m R 1 ft, which is equivalent to a sampling time of T = 10ms.

We require the displacement from the guideline not to exceed 0.2m during transient and 0.05 m in steady state, and the lateral acceleration not to exceed the nominal value a =

5.2. Simulation Results For the sake of brevity, we confine the simulation to the discrete version of the cascaded controller designed in Section 4. Compared to the version in Section 3, the effects of measurement noise are decreased and incorporation of ride quality considerations into the control design is further facilitated, since a desired vehicle yaw rate can be specified. For the simulations in Figure 8, the following parameters were chosen: X = and E = 0.15 in (26), k = 0.0286 in (27), and poles pobsl = -2.5 for the observer (25) and Phi = -4 for (30).

during curve riding by more than fO . lg 1201.

The available robustness can be used e.g. to increase the spacing of the reference markers. Figure 9 depicts the performance of the controller (31) with A = & and c = 0.2 in (26), k = 0.02 in (27), P,,bal = -1 (25) and po).3 = -2.5 (30) for the transition from manual to automatic steering. The velocity and the marker spacing were doubled, i.e. v = 60 [m s-l] = 135 [mph]

and Ad = 0.6 [m] = 2 [ft]. Gaussian noise contents of U, = 0.002 [rads-’) for the yaw rate measurement and uv = (0.002 [m] +O.Ollpl) for the displacement measurement were assumed. Satisfactory performance is achieved, including small lateral acceleration. Additional studies, not shown here, for guste of sidewind, missing markers, and sudden change of road conditions (change of p ) resulted in no significant tracking errors. The critical effect turned out to be measurement noise.

6. Conclusions

This paper introduced two nonlinear control algorithms for automatic steering of cars, an important control subtask of Intelligent Vehicle/Highway Systems. Computer implementation and discretization of the control algorithms was discussed in detail. Direct inclusion of ride quality considerations into the control design was possible. Despite a high degree of system uncertainty, mi- nimal efforts in state measurement and no road curvature preview, satisfactory performance was achieved by utilizing sliding mode control methodology and robust state observation.

References [I] P. Varaiya, “Smart cam on smart roads: Problems of control,” IEEE “buns. on Automatic Control, vol. 38, pp. 195-207, 1993. 121 J. G. Bender, “An overview of systems studies of automated highway sy- stems,” IEEE Rum. on Vehicular Technology, vol. 40, no. 1, pp. 82-99, 1991. [3] R. E. Fenton and R. J. Mayhan, UAutomated highway studies at the Ohio State University - An overview,” IEEE Trans. on Vehicular Technology, vol. 40, no. 1, pp. 100-113, 1991. [4] S. E. Shladover et al., “Automatic vehicle control developments in the PATH Program,“ IEEE 7hms. on Vehicular Technology, vol. 40, no. 1, pp. 114-130, 1991. [5] E. Dickmanns and T. Christians, “Relative 3D state estimation for auton- mous visual guidance of road vehicles,” Intelligent Autonomous Systems, vol. 2, 1989. [6] W. Zhang and R. E. Parsons, “An intelligent roadway reference system for vehicle lateral guidance/control,” in Proc. American Control Con/., (San Diego, CA,

[7] W. Darenberg, “Automatische Spurfiihrung von Kraftfahrzeugen (in Ger- man),” Automobil-Industrie, pp. 155-159, 1987. [8] J . Ackermann, W. Sienel, and R. Steinhauser, “Robust automatic steering of a bus,” in Proc. Second European Control Conference, vol. 3, (Groningen, The Netherlands), pp. 1534-1539, 1993. [9] R. J. Mayhan and R. A. Bishsl, “A two-frequency radar for vehicle automatic control,” IEEE %anS. on Vehicular Technology, vol. 31, no. 1, 1982. [lo] A. R. Johnston, T. Assefi, and J. Y. Lai, “Automated vehicle guidance using discrete markers,” IEEE Runs. on Vehicular Technology, vol. 28, no. 1, pp. 95-105, 1979. Ill] R. Parsons and W.-B. Zhang, “Program on advanced technology for the hig- hway - lateral guidance system requirements definition,” in Pruc. ASCE Int. Conf. on Appl. of Advanced Tech. in 7hnsportation Engineering, (San Diego, CA, USA), pp. 257-280, 1989. 1121 H. Peng and M. Tomizuka, “Preview control for vehicle lateral guidance in hig- hway automation,” in Proc. American Contml Conf., (Boston, MA, USA), pp. 2621- 2627, 1991. [13] M. Tomizuka and J. K. Hedrick, ”Automated vehicle control for IVHS sy- stems,” in Pmc. IFAC World Congress, (Sydney, Australia), 1993. (141 P. Riekert and T. Schunck, “Zur Fahrmechanik des gummibereiften Kraft- fahrzeugs (in German),” Ingenieur Archiu, vol. 11, pp. 210-224, 1940. [l5] J. Ackermann, A. Bartlett, D. Kaesbauer, W. Sienel, and R. Steinhawer, Robust contml: Analysis and design of linear control systems with uncertain physical pammeters. London: Springer, 1993. [IS] R. A. DeCarlo, S. H. Zak, and G. P. Matthews, ‘Variable structure control of nonlinear multivariable systems: A tutorial,” Proceedings of the IEEE, vol. 76, no.

[17] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992. [l8] A. G. Bondarev, S. A. Bondarev, N. E. Kostyleva, and V. I. Utkin, “Sli- ding modes in systems with asymptotic state observers,” Automation and Remote Control, vol. 46, no. 6 (P.l), pp. 679-684, 1985. [19] V. I. Utkin, “Sliding mode control in discrete-time and difference systems,” in Variable structure and Lyapunou Control (A. S. I. Zinober, ed.), pp. 83-103, London, UK: Springer-Verlag, 1993. [20] H. Peng and M. Tomizuka, ”Vehicle lateral control for highway automation,” in Pruc. American Control Con/.., (San Diego, CA, USA), pp. 788-794, 1990. [21] W. Darenberg, “Automatisehe Spnfihrung von Kraftfahrzeugen (in Ger-

.man),’ Presentation at Arbeitskis “Rohuste Regelung”, Interlaken, Switzerland, 1985.

USA), pp. 281-286, 1990.

. 3, pp. 212-232, 1988.

RefLaxnc4p.th m r .

xlO-3 R e f m c e Curvature xlO-3 Reference Curvature

Fig. 7: Reference paths for curve driving and lane change

o.3 Transition to Automatic Stcain

0.2

-0.01

-0.02 o so 100 150 200 [ml 0 100 200 MoIml

0.1

0

o.3 Transition to Automatic Stcain

0.2

-0.01

-0.02 o so 100 150 200 [ml 0 100 200 MoIml

0.1

0

xlO-3 Smooth Curvature Chan e

-0.02

Fig. 8: Performance of cascaded controller

La- Displacement 0.4 I I

I -0.1‘ o 50 100 I50 200 250 300 350 4OO[ml

Lateral Acceleration in P 0.2 I

Fig. 9: Controller performance under noisy measurements

1 973 c