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Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O. Wang
Copyright Q 2001 John Wiley & Sons, Inc. CHAPTER 8 ISBNs: 0-471-32324-1 ŽHardback.; 0-471-22459-6 ŽElectronic.
TRAJECTORY CONTROL OF A VEHICLE WITH MULTIPLE TRAILERS
This chapter contains an in-depth application study of the fuzzy control methodologies introduced in this book. The system under study is a vehicle with multiple trailers. The control objective is to back the vehicle into a straight-line configuration without forward motion. This is often referred as the problem of backing up control of a truck-trailer. A truck with a single trailer is often used as a testbed to study different control strategies. In this chapter, we consider the more challenging problem of backing up control of a vehicle with multiple trailers. Both simulation and experimental results w1]4x are presented. The results demonstrate that the designed fuzzy controller can effectively achieve the backing-up control of the vehicle with multiple trailers while avoiding the saturation of the actuator and ‘‘jack-knife’’ phenomenon. Moreover, the controller guarantees the stability and performance even in the presence of disturbance.
As mentioned above, the backing-up control of ‘‘trailer-truck,’’ that is, a vehicle with a trailer, has been used as a testbed for a variety of control design methods w1]11x. In particular, in order to successfully back up the trailer-truck, the so-called jack-knife phenomenon needs to be avoided throughout the operation. In the field of automatic control, a number of control methodologies including nonlinear control, fuzzy control, neural control, and hybrid neural-fuzzy control w5]8x have been applied to this testbed problem. Most of these are simulation-based studies; the important issue of the stability of the control systems was often left out. In our work, stabilizing fuzzy control was applied to the case of a truck with one trailer case in w9x and experimental demonstrations were reported in w1,10x.
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134 TRAJECTORY CONTROL OF A VEHICLE WITH MULTIPLE TRAILERS
This chapter mainly deals with the triple-trailer case w3,4x. The triple-trailer case, that is, backing-up control of a vehicle with triple trailers, is much more challenging than that of the one-trailer case. To the best of our knowledge, experimental results of the triple-trailer case had not been reported in the literature prior to our work. Part of the difficulties associated with multiple-trailer cases, the triple-trailer case included, lie in the exponentially increas-ing number of jack-knife configurations as the number of trailers increases. In the one-trailer case, only two jack-knife configurations exist. For the triple-trailer case, the number of jack-knife configurations increases to eight. Moreover, we need to address a number of practical constraints, for example, saturation of the steering angle and disturbance rejection, for such difficult control objects. In the control design for the vehicle with triple trailers, we utilize the LMI conditions described in Chapter 3 to explicitly handle the saturation of the steering angle and the jack-knife phenomenon. Both simula-tion and experimental results demonstrate that the fuzzy controller effec-tively achieves the backing-up control of the vehicle with triple trailers while avoiding the saturation of the actuator and jack-knife phenomenon. More-over, the feedback controller guarantees the stability and performance even in the presence of disturbance.
8.1 FUZZY MODELING OF A VEHICLE WITH TRIPLE TRAILERS
Figure 8.1 shows the vehicle model with triple trailers and its coordinate system. We use the following control-oriented model to design a fuzzy controller:
x0Žtq 1. s x0Žt. q n?lDt tanŽuŽt.., Ž8.1.
x1Žt. s x0Žt. y x2Žt., Ž8.2.
x2Žtq 1. s x2Žt. q n?Dt sinŽx1Žt.., Ž8.3.
x3Žt. s x2Žt. y x4Žt., Ž8.4.
x4Žtq 1. s x4Žt. q n?Dt sinŽx3Žt.., Ž8.5.
x5Žt. s x4Žt. y x6Žt., Ž8.6.
x6Žtq 1. s x6Žt. q n?Dt sinŽx5Žt.., Ž8.7.
FUZZY MODELING OF A VEHICLE WITH TRIPLE TRAILERS 135
Fig. 8.1 Vehicle model with triple trailers.
x7 Žtq 1. s x7 Žt. q n?DtcosŽx5Žt..sinžx6Žtq 1. q x6Žt. /, Ž8.8. x8Žtq 1. s x8Žt. q n?DtcosŽx5Žt..cosžx6Žtq 1. q x6Žt. /, Ž8.9.
where
x Žt.s angle of vehicle,
x Žt.s angle difference between vehicle and first trailer, x Žt.s angle of first trailer,
x Žt.s angle difference between first trailer and second trailer, x Žt.s angle of second trailer,
x Žt.s angle difference between second trailer and third trailer, x Žt.s angle of third trailer,
x Žt.s vertical position of rear end of third trailer,
x Žt.s horizontal position of rear end of third trailer, uŽt.s steering angle.
The model presented above is a discretized model with several simplifica-tions. It is not intended to be a model to study the detailed dynamics of the
136 TRAJECTORY CONTROL OF A VEHICLE WITH MULTIPLE TRAILERS
trailer-truck system. Because of the simplicity, its main usage is for control design. This is the same idea as the so-called control-oriented modeling in which some reduced-order type of models are sought instead of the full-fledged dynamic models. The trailer-truck model herein has proven to be effective in designing controllers for the experimental setup which is dis-cussed later in this chapter.
In the simulation and experimental studies the following parameter values are used:
ls 0.087 m, Ls 0.130 m, ns y0.10 mrsec., Dts 0.5 sec.,
where l is the length of the vehicle, L is the length of the trailer, Dt is the sampling time, and n is the constant speed of the backward movement. For x1Žt., x3Žt., and x5Žt., 908 and y908 correspond to eight ‘‘jack-knife’’ posi-tions.
The control objective is to back the vehicle into the straight line Žx7 s 0. without any forward movement, that is,
x1Žt. ™ 0, x3Žt. ™ 0, x5Žt. ™ 0, x6Žt. ™ 0, x7 Žt. ™ 0.
To employ the model-based fuzzy control design methodology described in this book, we start with the construction of a Takagi-Sugeno fuzzy model to represent the nonlinear equations Ž8.1.]Ž8.8.. To facilitate the control design,
with the assumption that the values of uŽt., x1Žt., x3Žt., and x5Žt. are small, we further simplify the model to be of the following form:
x0Žtq 1. s x0Žt. q n?lDt uŽt., Ž8.10.
x1Žtq 1. s ž1 y n?Dt /x1Žt. q n?lDt uŽt., Ž8.11. x2Žtq 1. s x2Žt. q n?Dt x1Žt., Ž8.12.
x3Žtq 1. s ž1 y n?Dt /x3Žt. q n?Dt x1Žt., Ž8.13. x4Žtq 1. s x4Žt. q n?Dt x3Žt., Ž8.14.
FUZZY MODELING OF A VEHICLE WITH TRIPLE TRAILERS 137
x5Žtq 1. s ž1 y n?Dt /x5Žt. q n?Dt x3Žt., Ž8.15. x6Žtq 1. s x6Žt. q n?Dt x5Žt., Ž8.16.
x7 Žtq 1. s x7 Žt. q n?Dt?sinžx6Žt. q n?Dt x5Žt./. Ž8.17. In this simplified model, the only nonlinear term is in Ž8.17.,
n?Dt?sinžx6Žt. q n?Dt x5Žt./. Ž8.18.
This term can be represented by the following Takagi-Sugeno fuzzy model:
n?Dt?sinžx6Žt. q n?Dt x5Žt./
s w1Ž pŽt.. ?n?Dt? žx6Žt. q n?Dt x5Žt./
q w2Ž pŽt.. ?n?Dt?g? žx6Žt. q n?Dt x5Žt./, Ž8.19.
where
pŽt. s x6Žt. q n?Dt x5Žt.,
gs 10y2rp ,
°sinŽ pŽt.. y g?pŽt. w1Ž pŽt.. s pŽt. ? Ž1 y g.
1,
pŽt. /0, Ž8.20.
pŽt. s 0,
°pŽt. y sinŽ pŽt.. w2Ž pŽt.. s pŽt. ? Ž1 y g.
0,
pŽt. /0, Ž8.21.
pŽt. s 0.
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