VEHICLE WITH MULTIPLE TRAILERS
8.1 FUZZY MODELING OF A VEHICLE WITH TRIPLE TRAILERS
⭈ ⌬t
x0Žtq1.sx0Ž .t q tanŽu tŽ .., Ž8.1. l
x t1Ž .sx0Ž .t yx2Ž .t , Ž8.2.
⭈ ⌬t
x2Žtq1.sx2Ž .t q sinŽx t1Ž .., Ž8.3. L
x3Ž .t sx2Ž .t yx4Ž .t , Ž8.4.
⭈ ⌬t
x4Žtq1.sx4Ž .t q sinŽx3Ž .t ., Ž8.5. L
x5Ž .t sx4Ž .t yx6Ž .t , Ž8.6.
⭈ ⌬t
x6Žtq1.sx6Ž .t q sinŽx5Ž .t ., Ž8.7. L
Fig. 8.1 Vehicle model with triple trailers.
x6Žtq1.qx6Ž .t
x7Žtq1.sx7Ž .t q ⭈ ⌬tcosŽx5Ž .t .sinž 2 /, Ž8.8. x6Žtq1.qx6Ž .t
x8Žtq1.sx8Ž .t q ⭈ ⌬tcosŽx5Ž .t .cosž 2 /, Ž8.9.
where
x t0Ž .sangle of vehicle,
x t1Ž .sangle difference between vehicle and first trailer, x t2Ž .sangle of first trailer,
x t3Ž .sangle difference between first trailer and second trailer, x t4Ž .sangle of second trailer,
x t5Ž .sangle difference between second trailer and third trailer, x t6Ž .sangle of third trailer,
x t7Ž .svertical position of rear end of third trailer, x t8Ž .shorizontal position of rear end of third trailer,
u tŽ .ssteering 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
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:
ls0.087 m, Ls0.130 m, s y0.10 mrsec., ⌬ts0.5 sec., where l is the length of the vehicle, L is the length of the trailer, ⌬t is the sampling time, and is the constant speed of the backward movement. For
Ž . Ž . Ž .
x t1 , x t3 , and x t5 , 90⬚ and y90⬚ correspond to eight ‘‘jack-knife’’ posi- tions.
Ž .
The control objective is to back the vehicle into the straight line x7s0 without any forward movement, that is,
x t1Ž .™0, x3Ž .t ™0, x5Ž .t ™0, x6Ž .t ™0, x7Ž .t ™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
Ž . Ž .
represent the nonlinear equations 8.1᎐ 8.8 . To facilitate the control design,
Ž . Ž . Ž . Ž .
with the assumption that the values of u t , x t1 , x t3 , and x t5 are small, we further simplify the model to be of the following form:
⭈ ⌬t
x0Žtq1.sx0Ž .t q u tŽ ., Ž8.10. l
⭈ ⌬t ⭈ ⌬t
x t1Ž q1.sž1y L /x t1Ž .q l u tŽ ., Ž8.11.
⭈ ⌬t
x2Žtq1.sx2Ž .t q x t1Ž ., Ž8.12. L
⭈ ⌬t ⭈ ⌬t
x3Žtq1.sž1y L /x3Ž .t q L x t1Ž ., Ž8.13.
⭈ ⌬t
x4Žtq1.sx4Ž .t q x3Ž .t , Ž8.14. L
⭈ ⌬t ⭈ ⌬t
x5Žtq1.sž1y L /x5Ž .t q L x3Ž .t , Ž8.15.
⭈ ⌬t
x6Žtq1.sx6Ž .t q x5Ž .t , Ž8.16. L
⭈ ⌬t
x7Žtq1.sx7Ž .t q ⭈ ⌬t⭈sinžx6Ž .t q 2L x5Ž .t /. Ž8.17.
Ž .
In this simplified model, the only nonlinear term is in 8.17 ,
⭈ ⌬t
⭈ ⌬t⭈sinžx6Ž .t q 2L x5Ž .t /. Ž8.18.
This term can be represented by the following Takagi-Sugeno fuzzy model:
⭈ ⌬t
⭈ ⌬t⭈sinžx6Ž .t q 2L x5Ž .t /
⭈ ⌬t sw1Žp tŽ ..⭈ ⭈ ⌬t⭈žx6Ž .t q 2L x5Ž .t /
⭈ ⌬t
qw2Žp tŽ ..⭈ ⭈ ⌬t⭈g⭈žx6Ž .t q 2L x5Ž .t /, Ž8.19. where
⭈ ⌬t p tŽ .sx6Ž .t q x5Ž .t ,
2L gs10y2r,
°sinŽ p tŽ ..yg⭈p tŽ .
, p tŽ ./0,
~ p t ⭈ 1yg
w1Ž p tŽ ..s Ž . Ž . Ž8.20.
¢1, p tŽ .s0,
°p tŽ .ysinŽp tŽ ..
, p tŽ ./0,
~ p t ⭈ 1yg
w2Ž p tŽ ..s Ž . Ž . Ž8.21.
¢0, p tŽ .s0.
Ž . Ž . Ž Ž .. Ž Ž ..
From 8.20 and 8.21 , it can be seen that w p t1 s1 and w2 p t s0
Ž . Ž Ž .. Ž Ž .. Ž .
when p t is about 0 rad. Similarly,w p t1 s0 andw2 p t s1 when p t is about ory rad.
Ž Ž .. Ž Ž .. Ž .
When w p t1 s1 and w2 p t s0, that is, p t is about 0 rad, substi-
Ž . Ž .
tuting 8.19 into 8.17 , we have
⭈ ⌬t 2
Ž .
x7Žtq1.sx7Ž .t q ⭈ ⌬t⭈x6Ž .t q ⭈x5Ž .t . 2L
As a result the simplified nonlinear model can be represented by
⭈ ⌬t
Ž .
x t1 q1 1y 0 0 0 0
L
⭈ ⌬t ⭈ ⌬t
Ž .
x t3 q1 1y 0 0 0
L L
⭈ ⌬t ⭈ ⌬t
Ž .
x t5 q1 s 0 1y 0 0
L L
⭈ ⌬t
Ž .
x t6 q1 0 0 1 0
L Ž ⭈ ⌬t.2
Ž .
x7 tq1 0 0 ⭈ ⌬t 1
2L
=
⭈ ⌬t x t1Ž .
l x t3Ž . 0
x t5Ž . q 0 u tŽ .. Ž8.22.
x t6Ž . 0 x7Ž .t 0
Ž Ž .. Ž Ž .. Ž .
Whenw p t1 s0 and w2 p t s1, that is, p t is about ory rad, Ž8.17 is represented as.
g⭈Ž ⭈ ⌬t.2
x7Žtq1.sx7Ž .t qg⭈ ⭈ ⌬t⭈x6Ž .t q ⭈x5Ž .t . 2L
The resulting simplified nonlinear model can be represented by
⭈ ⌬t
Ž .
x t1 q1 1y 0 0 0 0
L
⭈ ⌬t ⭈ ⌬t
Ž .
x t3 q1 1y 0 0 0
L L
⭈ ⌬t ⭈ ⌬t
Ž .
x t5 q1 s 0 1y 0 0
L L
⭈ ⌬t
Ž .
x t6 q1 0 0 1 0
L
Ž .2
g⭈ ⭈ ⌬t
Ž .
x7 tq1 0 0 g⭈ ⭈ ⌬t 1
2L
=
⭈ ⌬t x t1Ž .
l x t3Ž . 0
x t5Ž . q 0 u tŽ .. Ž8.23.
x t6Ž . 0 x7Ž .t 0
Ž .
In this representation, if gs0, system 8.23 becomes uncontrollable. To alleviate the problem, we select gs10y2r. With this choice of g, the
Ž . Ž .
nonlinear term of 8.18 is exactly represented by the expression of 8.19 under the condition
y179.4270⬚-p tŽ .-179.4270⬚.
To this end, in application to the vehicle with triple trailers, we arrive at the following Takagi-Sugeno fuzzy model:
Rule 1
IFp tŽ .is ‘‘about 0 rad,’’
Ž . Ž . Ž .
THEN x tq1 sA x1 t qB1u t , Ž8.24. Rule 2
IFp tŽ .is ‘‘about rad ory rad,’’
Ž . Ž . Ž .
THEN x tq1 sA x2 t qB2u t ,
Here,
⭈ ⌬t p tŽ .sx6Ž .t q x5Ž .t ,
2L
xŽ .t s x t1Ž . x3Ž .t x5Ž .t x6Ž .t x7Ž .t T,
⭈ ⌬t
1y 0 0 0 0
L
⭈ ⌬t ⭈ ⌬t
1y 0 0 0
L L
⭈ ⌬t ⭈ ⌬t
0 1y 0 0
A1s ,
L L
⭈ ⌬t
0 0 1 0
L Ž ⭈ ⌬t.2
0 0 ⭈ ⌬t 1
2L
⭈ ⌬t l
0 B1s 0 ,
0 0
⭈ ⌬t
1y 0 0 0 0
L
⭈ ⌬t ⭈ ⌬t
1y 0 0 0
L L
⭈ ⌬t ⭈ ⌬t
0 1y 0 0
A2s L L ,
⭈ ⌬t
0 0 1 0
L
Ž .2
g⭈ ⭈ ⌬t
0 0 g⭈ ⭈ ⌬t 1
2L
⭈ ⌬t l
0 B2s 0 .
0 0
The overall fuzzy model is inferred as
2
xŽtq1.s ÝhiŽp tŽ .. A xi Ž .t qBiu tŽ .4. Ž8.25.
is1
Figure 8.2 shows the membership functions ‘‘about 0 rad’’ and ‘‘about rad ory rad.’’
Remark 21 As pointed out in Chapters 2᎐7, the stability conditions for the
Ž .
case of the common B matrix B1s ⭈⭈⭈ sBr can be simplified. In this
Fig. 8.2 Membership functions.
chapter we employ the general design conditions, that is, not the common B matrix case, although the fuzzy model of the vehicle shares common B among the rules.
Remark 22 As pointed out in Chapter 2, we construct the fuzzy model for a simplified nonlinear model. The fuzzy model has two rules. If we try to derive
Ž . Ž . 6
a fuzzy model for the original nonlinear system 8.1 ᎐8.9 , 2 rules are required to exactly represent the nonlinear dynamics. The rule reduction leads to significant reduction of the effort for the analysis and design of control systems. This approach is useful in practice.
8.1.1 Avoidance of Jack-Knife Utilizing Constraint on Output
Ž .
Let us recall the LMI constraint on the output shown in Chapter 3 to avoid the jack-knife phenomenon. The following theorem deals with this aspect of the control design.
THEOREM 30 Assume that the initial condition xŽ .0 is known. The con- 5 Ž .5 5 Ž .5 5 Ž .5
straints x t1 F1, x t3 F2, and x t5 F3 are enforced at all times tG0 if the LMIs
1 xTŽ .0
G0, Ž8.26. xŽ .0 X
X Xd1T
G0, Ž8.27. d X1 21I
X Xd2T
G0, Ž8.28. d X2 22I
X Xd3T
G0 Ž8.29. d X3 23I
y1 Ž . Ž .
hold, where XsP . In the triple-trailer case, we can select x t1 , x t3 , and x t as outputs:5Ž .
x t1Ž . x3Ž .t x Ž .t x t1Ž .sd x1 Ž .t s 1 0 0 0 0 5 ,
x6Ž .t x7Ž .t
x t1Ž . x3Ž .t x Ž .t x3Ž .t sd x2 Ž .t s 0 1 0 0 0 5 ,
x6Ž .t x7Ž .t x t1Ž . x3Ž .t x Ž .t x5Ž .t sd x3 Ž .t s 0 0 1 0 0 5 .
x6Ž .t x7Ž .t
Ž .
Proof. The proof of 8.27 is as follows. From x t1Ž . F1, x1TŽ .t x t1Ž .sxTŽ .t d1Td x1 Ž .t F21. Therefore,
12xTŽ .t d d x1T 1 Ž .t F1.
1
In the same way as in the proof of Theorem 12, we have 12xTŽ .t d d x1T 1 Ž .t FxTŽ .t Xy1xŽ .t .
1
The above inequality is
T 1 T y1
x Ž .t ž21d d1 1yX /xŽ .t F0.
Therefore, we have
1 T
Xy 2Xd d X1 1 G0.
1
Ž .
Inequality 8.27 can then be obtained from the above inequality. We obtain
Ž . Ž . Ž .
the LMI conditions 8.28 and 8.29 in the same fashion. Q.E.D.
As mentioned in Chapter 3, the above LMI design conditions for output constraints depend on the initial states of the system. To alleviate this problem, the initial-state-independent condition given in Theorem 13 may be utilized in the control design.