Engineering Aspects on Nonlinear H∞ Control for Longitudinal Vehicle Dynamics

Copyright © IFAC Control in Transportation Systems, Tokyo,Japan,2003

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ENGINEERING ASPECTS OF NONLINEAR H.. CONTROL FOR LONGITUDINAL VEHICLE DYNAMICS Lothar Ganzelmeier, Eckehard Schnieder Institute for Traffic Safety and Automation Engineering Technical University of Braunschweig Langer Kamp 8 38106 Braunschweig, Germany email: {L.GanzelmeierlE.Schnieder}@tu-bs.de Phone: +495313913317 Fax: +495313915197

Abstract: In this paper a nonlinear robust controller design task for longitudinal dynamics of autonomous road vehicles is considered. This work presents a longitudinal controller that covers the nonlinearities of the vehicle dynamics by a nonlinear model based controller design while the variation in vehicle parameters is covered by the unstructured robustness of the controller. How model nonlinearities in longitudinal vehicle dynamics are described and how they are taken into account during the controller design process is shortly presented. The used control concept for nonlinear plants in form of a nonlinear H~ control task is described. The approach involves the numerical solution, by power series approximation, of the time-varying Hamilton-Jacobi-Isaac equation and results in a nonlinear controller approximating the optimal tracking control law for a specified desired trajectory and cost function. Mathematical and computational aspects of nonlinear H~ control or of L2 gain synthesis are considered. The first practical results with a nonlinear robust control law for longitudinal dynamic of an autonomous vehicle are presented. Copyright © 2003 IFAC Keywords: nonlinear control, robust control, autonomous vehicles, combustion engine, vehicle, velocity control, practical result

changing. In addition large intluences exist from nonlinear vehicle dynamics like shown in figure I. For this reason it is a large challenge for every control task to operate appropriate under such a wide variety of vehicle dynamics. The basic idea is to take full advantage of recent robust and nonlinear controller design methodologies and using these advantages for longitudinal and lateral control of autonomous vehicles. The studied cars are equipped with actuators that are able to turn the steering wheel, to press the clutch. brake. throttle and to change the gears in cars with manual shift. Therefore the controllers have the full range to affect the vehicle in the way a human driver operates a car. In order to get sufficient information about the state of the vehicle an additional sensor is fixed to the car.

I. INTRODUCTION During the last decades the subject of design and analysis of various longitudinal and lateral control laws has been studied extensively. Throughout the literature numerous topics such as sliding mode control, parameter scheduling, look-ahead curvature processing and automated guideway transit systems have been reported (Byme, et al.. 1998; Chen, 1992; German, 1996; Hingwe. 1997; Manigel, 1993; Sohnitz. et aI., 1999). Even though much effort has been spent on various control laws for longitudinal and lateral control of autonomous vehicles this paper presents a mostly neglected aspect of autonomous vehicle control.

In previous publications the obtained results of a lateral controller that covers not only broad changes in longitudinal velocity but also various types of cars has been presented (Ganzelmeier, et aI., 200 I). Therefore the most recent results in controlling the

The main task of the presented project is to drive customary cars completely autonomously. With varying vehicle parameters like the velocity or mass the dynamic of one vehicle is continuously

407

longitudinal dynamic of autonomous cars are to be regarded. The main objective of a longitudinal controller is to ensure that the vehicle drives with the desired velocity. Hereby the largest problem is the nonlinear behaviour of a combustion engine. Even within one gear the combustion engine modifies its characteristic significantly by passing through all different engine speeds. Especially during the acceleration phase this effect is particularly relevant and has to be taken into account for stability and performance aspects in controller design. The typical approach is to linearize the system around some operating point and to analyze the resulting linear system. As the motion of the system is large, the linear model of the system becomes invalid. Therefore it is desirable to consider the full nonlinear model of the system. However, nonlinear systems are difficult to analyze mathematically.

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Fig. 2. Structure of the nonlinear controller design task for robust stability If the nonlinear plant with its definition equation in accordance with (1) is called system P, then the nonlinear controller design task can be brought into the general form shown by figure 2. The controller to be designed is named K. The overall system with the plant P and the controller K in the closed loop is called M.

x=A(x)+ B, (x,w )+B 2 (x,u) z=c 1(x)+ DII (x,W)+D I2 (x,u) y=C 2 (x)+D 21 (X,W)

steering transmission ratio gear transmission ratio manual I automatic transmission petrol I diesel engine motor characteristics (power I torque

wheelbase

(1)

The approach generally considers nonlinear systems of the given form, where x E Rn is the state of the system, Z E Rpl is the controlled output, WE R ml is the exogenous input including all commands and unmeasured disturbances, U E R m2 represents the control input and y E RP2 is the measured output (Isidori, et al., 1992).

mass ai r drag coefflCie nt center of gravity moment of inertia

Fig. 1. Influence parameters on vehicle dynamics

The controller design task is defined in accordance with the standard problem shown in figure 2 as follows:

Most scientific works made so far deal with this problem and suggest a wide range of different solutions like feedback linearization and plant inversion (Chen, 1992; German, 1996). The basic approach of this work however goes one step beyond. With the aim to control autonomous vehicles with highly nonlinear dynamics and absolutely varying parameters, much harder boundary conditions are found by the controller. For this reason it is a large challenge for every control task to operate appropriate under such a wide variety of vehicle dynamics. Designing control laws that cause a non linear system to track a desired trajectory is a problem of fundamental importance in control theory.

It is looked for the controller K, which

1.

2.

minimizes the effects of the disturbances W on the output signals z concerning the L2 norm and stabilizes the closed loop asymptotically.

The controller has to guarantee that the closed-loop system has an ~-gain less than or equal to a prespecified constant level y from the disturbance input W to the output cost functions z. We write this ~-gain condition as T

T

o

0

fllz(t)IF dt::;)'2 fllw(t)W dt 2. NONLINEAR ROBUST CONTROL OBJECTIVES

(2)

The optimization has the task to minimize a control criterion for all possible continuous controller functions u with consideration of the system dynamics and additional boundary conditions. The nonlinear optimal controller design is based on a cost function J, which is needed during the design process, in order to evaluate the quality of the designed controller. The goal of the nonlinear controller design is to find a controller

Nonlinear H.. control is an extension of the linear H_ method, which has received much attention during the 1990's. In contrast to linear control theory nonlinear H.. control theory is formulated in the time domain and depends on ideas and methods of differential games and nonlinear partial differential equations or partial differential inequalities.

408

u=u(x), u(O)=O

model. However substantial changes and additions were made regarding the modelling of nonlinear influences.

(3)

which meets the cost function As represented in figure 3 the model of the longitudinal vehicle dynamics with the submodels engine, transmission, wheel and vehicle illustrates the longitudinal dynamic behaviour of the complete vehicle. The position of the throttle serves as input and the vehicle speed forms the output variable. The sub-models are coupled over feedbacks loops with one another. Due to the modular structure the model receives a universal character and can likewise be used for hardware in the loop simulation, by replacing parts like the engine model by the real engine.

J(x, w,u) 00

sup

in!

f L(x(t),w(t),u(t))dt

.·eLJO.oo)"eL,[O.oo) 0

(4)

with

L(x, w, u)=lIzW-

rllwlF

(5)

that minimizes the manipulated variable u and at the same time maximizes the exogenous influences w. Controller design methodologies, which use the concept of the ~-gain, are closely related to the linear H.. design method, as the infinite norm of the transfer function matrices of linear systems are equal to the ~ induced system norm.

a n M '---

For output-feedback H_ control a nonlinear state feedback controller is combined with a nonlinear observer based on the plant dynamics. For nonlinear systems, the optimal control problem reduces to the solution of the Hamilton-Jacobi-Isaac equation, which is a nonlinear partial differential equation. In the case where the system state is available for control, called the state feedback case, the solution of the problem reduces to the solution of a certain Hamilton Jacobi PDE. In general, however, solving this PDE is computationally a hard problem, with complexity increasing rapidly as the state dimension grows. As the equation is extremely diffcult to solve researchers have looked for methods of approximating its solution (Beard, et al., 1998; Cloutier, et al., 1996). The solution of the resulting Hamilton-Jacobi-Isaac differential inequalities is approximated by power series and implemented in Matlab according to (Christen, et al., 1997). In consideration of figure 2 block P includes the nominal plant model, the description of the model uncertainties and the specifications for robust control performance by weighting the penalty functions z. Guaranteed stability for closed loop systems for plants with model uncertainties is called robust stability. The nonlinear H.. controller design leads to a controller that guarantees the best possible stability for nonlinear models with unstructured uncertainties. The resulting controller was implemented on a Infineon 32-bit microcontroller TC 1775. Since the matrix inversion necessary for calculating the output injection gain usually cannot performed symbolically, it is carried out online in C Code function.

-+-=

---J

Fig. 3. Simplified block diagram of the controlled system The entire engine can be modelled only by very extensive considerations on the basis of the individual combustion process. In this context the mass flows, the intake tube dynamics, fuel-wall interactions, bearing and piston friction, efficiency of the thermodynamic cyclic processes as well as thermo-chemical influences play a substantial role. For analysis of the vehicle dynamics such a complex modelling of the individual combustion process is not of advantage, why in the following an empirical approach is proposed. The torque delivered by the petrol or diesel engine is essentially a function of the number of revolutions and the air mass stream. This can be proven on the basis of detailed measurements on an engine test stand. If the throttle valve and the intake tube are also integrated into the engine model, then a transfer characteristic with the throttle valve angle and the engine speed results as inputs and the engine torque as output value. This behaviour can be described by a surface in three-dimensional space. In order to use the measured behaviour of a combustion engine for the controller design, in the context of this work a new approach is suggested. The behaviour of a combustion engine identified in form of a characteristic diagram is reproduced by a mathematical function with two input variables. The effective torque at the output side of the crankshaft is represented in dependence of the engine speed nM and throttle valve position a as a polynomial with two variables with reciprocal effect and the order r.

3. LONGITUDINAL MODELLING For modelling the longitudinal vehicle dynamics combined model structures are used. A physical basic structure is supplemented by the integration of phenomenological descriptive models. The synthesized model is based in parts on the one wheel

409

The model of the engine characteristic diagram corresponds to the surface polynomial of the form r

relevant operating areas also the limitations in the boundary regions are considered.

r- j

!(a,n M )= L Lcs"ai.n~=M M

4. LONGITUDINAL CONTROL

(6)

;-0;=0

with the degree of r and s as control variable of s = 0

fk

to t., . Due to the given engine characteristic diagram the data points (av, nM,v, MM,v) with v = I,Zoom are wellknown. The interpolation procedure has the task to design a function in such a manner that the received surface in the place (a.", nM.v) takes the given values MM.v as function values. In order to determine the coefficients of the surface polynomial Cs, the method of the smallest error squares is extended to several input variables.

The given task is concerned with the design of a velocity controller for motor vehicles. A higher-level control, like a driver assistance system or trajectory generator, creates the required velocity inputs. While in conventional systems, e.g. cruise control, a fixed point control is used, with which a once reached speed can only be held, the designed controller should be able to accelerate or decelerate the vehicle to the given required velocity.

12 , - - - - - - - - - - - - - - - , 100 10

80

~

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60 .!!

To show the efficiency of the suggested approach, exemplarily the delivery characteristic diagram of a d~rect-injecting diesel engine was approximated by dIfferent surface polynomials. Already starting from the 3rd order of the approximation polynomial acceptable accuracies in engine torque output are reached. With increasing order likewise the deviations decrease. Beyond that no disturbing oscillations in the boundary regions are to be recognized, so that during increased requirements concerning the approximation accuracy it can easily be changed over to higher order approximation.

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Fig. 5. Velocity control VW Passat, 1st gear

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Fig. 6. Velocity control VW Passat, 4th gear

!

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100 4000

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engine speed [rpml

2000

50 1000

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During the designing process of the nonlinear H.. controllers for the I SI to 4th gear of the vehicle the closed loop system was analyzed and optimized under most diverse criteria by multiple simulations. Now these results are to be confirmed by practical tests. For the experimental test a commercial Volkswagen Passat with a direct-injecting 1,9l-TDI engine was used.

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Fig. 4. Three-dimensional surface polynomial with identified engine data To show the good convergence already starting from the 3'd order of the approximation polynomial the calculated surface polynomial is comparatively represented with the measured data in figure 4. The largest torque in the middle number of revolutions as well as the dropping to the boundary regions is a very good approximation of the original engine characteristic represented as circles in figure 4.

According to the simulation a required velocity was given. By step variation of the reference signal, the obtained performance of the robust controller can be evaluated by opposing the measured velocity signal of the car with the reference input. This procedure was repeated for all four gears. The obtained results for the I st and the 4th gear are shown in figure 5 and 6. The plots show the experimental results of a longitudinal controlled test drive achieved for a customary VW Passat with the above presented

The advantages of this approach are in the fact that the individual nonlinear engine characteristic can be used for controller design process. Apart from the exact modelling of the torque distribution in the

410

diesel engine. The vehicle was stimulated with a step input for the desired velocity. The input signal was chosen in view of the right velocity range for each gear. The closed loop system response is plotted over its input signal. Additional to these signals the percentage of the resulting throttle pedal is shown, too. In order to analyze the performance of the controller, large step inputs were given over the entire speed range of the combustion engine. With positive steps it was judged how good the reference signal is followed. By negative step inputs one additionally can measure, how good the assigned controller manages the different system dynamics in contrast to positive accelerations. In both kinds of regard one can observe very good behaviour. The controller shows robust behaviour in relation to the different dynamics with accelerations and deceleratic,ns. In order to judge apart from the transient responses the stationary accuracy of the controllers, between the reference jumps defined times with unchanged desired values were kept. Compared with commercial speed controllers, which must hold only the current speed, a substantial improvement was obtained. Thus the nonlinear controller performs better regarding stationary accuracy and variable disturbance sensitivity. The examined optimal controllers could use their nonlinearity fully in favour of an evenly good behaviour over the entire work area of a gear. Linear longitudinal controllers usually show to conservative behaviour in the lower and upper speed range and to aggressive behaviour in the middle speed range of the combustion engine. This behaviour can be avoided due to the nonlinear modelling of the combustion engine and the model based controller design methodology. Despite of possible deviations of the controlled system from the model, stable controllers can be guaranteed with the help of the consideration of unstructured uncertainties during the design process. By optimizing the quality function and minimization of the ~-gain highest accuracies can be obtained also under influence of variable disturbances, which also were present during the test procedures but cannot be specified more exactly. It can be outlined that the very strongly varying parameters are covered by the robustness of the designed controller. Therefore the presented controller not only handles the changing engine torque over different engine speeds but also various characteristics for different engine concepts.

Fig. 7. Picture series of a completely autonomous driven vehicle

5. CONCLUSION In this paper we presented a longitudinal control law for customary autonomous road vehicles. For the longitudinal control the largest problem is the nonlinear behaviour of a combustion engine. The main scope was to extend previous approaches for autonomous vehicles. In previous research works the robust design objective was mainly to cover nonlinearities in engine torque. This work goes a step beyond and presents a longitudinal controller that covers the nonlinearities of the vehicle dynamics by a nonlinear model based controller design while the variation in vehicle parameters is covered by the unstructured robustness of the controller. With the aim to control autonomous vehicles with highly . nonlinear dynamics and absolutely varying parameters, much harder boundary conditions are found by the controller. The nonlinear controller design used for all longitudinal specifications is described. The first practical results for longitudinal robust control autonomous vehicles are presented.

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Byme, R. H., Abdallah, C. T., Dorato, P. (1998). Experimental Results in Robust Lateral Control of Highway Vehicles. IEEE Control Systems, Vol. 18(2), p.70-76

Figure 7 shows the respective picture series of one of the autonomous rides. To additionally underline the achieved high accuracy of the autonomous vehicle control the track was marked with guidance cones standing only a few centimeters on each side of the car.

Chen, Z. (1992). Menschliche und automatische Regelung der Uingsbewegung von Personenkraftwagen. VDI-Verlag, Reihe 12, Nr. 168, Dtisseldorf

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Christen, U. and Cirillo, R. (1997). Nonlinear H_ Control - Derivation and Implementation. IMRT Repon,No. 31, Ztirich

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