Chapter 3. Fuzzy Logic Control
3.2 Fuzzifiers
Usually, the intersection and union operators are denoted by∧and , respectively. The complement
3.2 Fuzzifiers
The fuzzifier stage transforms crisp input from real values into fuzzy sets.
Here we introduce three fuzzifiers as following:
1. Singleton fuzzifier: the singleton fuzzifier maps a real valued point into the fuzzy singleton A in U, in which the membership value is 1 at and 0 at other
= 0 otherwise ) 1
(
x*
x x
µA (3.4)
2. Triangular fuzzifier: the triangular fuzzifier maps into the fuzzy set A in U, in which the membership function is written as:
U
where biare positive parameters and symbol ⊗ is often chosen as algebraic product
or minimum.
3. Gaussian fuzzifier: the Gaussian fuzifier maps x*∈U into the fuzzy set A in U, in
which the membership function is written as:
where δiare positive parameters and symbol ⊗ is often chosen as algebraic product
or minimum.
Finally, we summarize the above fuzzifiers. The singleton fuzzifier greatly simplifies the computation involved in the fuzzy inference engine for all membership functions. And the Gaussian and triangular fuzzifiers do, too. The Gaussian and triangular fuzzifiers can restrain noise in the input, but the singleton fuzzifier cannot.
3.3 Deffuzzifiers
The defuzzifier is defined as a mapping from a fuzzy set D in to a crisp point Hence, the task of the defuzzifier is to specify a point in V that represents the fuzzy set D. There are three types of defuzifiers introduced below.
R V ⊂
* . V y ∈
1. Center of gravity Defuzzifier
The center of gravity defuzzifier specifies as the center of the area covered by the membership function of D.
y*
where ∫V is the conventional integral.
2. Center Average Defuzzifier
Let ylbe the center of the lth fuzzy set and wlbe its height. The center
average defuzzifier presents y*as
3. Maximum Defuzzifier
The maximum defuzzifier chooses as the point in V, at which achieves its maximum value. Define
y* µD(y) maximum defuzzifier is defined as an arbitrary element in hgt(D), i.e.,
)
D( y µ
y*
y*=any point in hgt(D).The mean of maximum defuzzifier is defined as:
∫
where is an integration for the continuous part of hgt(D) and it is a summation
for the discrete part of hgt(D).
∫hgt(D)
3.4 Fuzzy Rule Bases
The fuzzy rule base consists of fuzzy IF-THEN rules. It is the core of the fuzzy system in a sense. And all other stages are used to implement these rules in a reasonable and efficient manner. Hence, the fuzzy rule base comprises the following fuzzy IF-THEN rules:
Rule i: IF xl is A1i and …and xn is Ani THEN y is Di (3.11)
The canonical fuzzy IF-THEN rules in the form of (3.11) includes the following ones:
(1) Partial rules:
IF x1 is A1i and …and xm is Ami THEN y is Di (3.12) (2) Or rules
IF x1 is A1i and …and xm is Ami or xm+1 is Ami+1 and …xn isAni THEN y is
Di (3.13)
(3) Singles fuzzy statement
y is Di (3.14) 3.5 Fuzzy Inference
The fuzzy inference is a reasoning method using the fuzzy theory, and whereby the expert knowledge is presented using linguistic rules. The fuzzy inference introduced as following.
Product Inference: (y)=max=1 [sup∈ ( A(x)
∏
A (xi) D(y))] (3.15)U x M
D l l
i
µ µ
µ µ
Minimum Inference: ( ) max[supmin( ( ), ( 1)..., ( ), ( ))] (3.16)
1 1
y x
x x
y l l
n
l A n D
A A U
x M
D l µ µ µ µ
µ
= ∈
=
The product inference and minimum inference are the most commonly used fuzzy inference in the fuzzy system and other fuzzy applications. In the Fig.3-2 shows the product inference and minimum inference [25-26].
µ µ µ
µ µ µ
µ
x x
x x
y y y
(a)
Max Minmin
y y y
Max
(b) x
x
x
x
Fig. 3-2 (a) Minimum inference (b) Product inference
3.6 Longitudinal Fuzzy Logic Control of a Platoon of Vehicles
In this section, we design a FLC by imitating a PD controller [6] for a platoon of vehicles. Design procedures are stated below.
Step 1) Define ∆i and ∆ from Eq.(2.1) as two input variables of ith FLC; and &i
u as an output variable of ith FLC; the membership functions for input i ,∆ and output are shown in Fig. 3-3.
∆i &i ui
-1.0 -0.4 0 0.4 1.0 PO
NG ZR
membership functions for inupt
-1.0 -0.4 0 0.4 1.0
PO
NG ZR
membership functions for inupt
-1.0 -0.4 0 0.4 1.0
ZR
membership functions for output u
NS PS PB
NB
1.0
1.0
1.0
∆
∆&
Fig. 3-3 Membership function for input and output variables
Step 2) The main idea is : “ if ∆i >0, input positive net force to decrease the space of two vehicles; if , input negative net force to increase the space of two vehicles; is used to amend the strategy above” Table. 3-1 is the rule base we obtain.
<0
∆i
∆&i
Step 3) Select one type of the fuzzifiers, defuzzifiers, and fuzzy inferences. The most frequently used triangular membership, the center-of-gravity defuzzification, and the
“max-min” reasoning method are adopted here to carry out the algorithm.
NG ZR PO NG NB NS ZR ZR NS ZR PS PO ZR PS PB
∆&
∆
Table. 3-1 Rule Base of the Fuzzy Logic Controller
3.7 Simulation Results
To examine the behavior of a platoon of vehicles under the above controller, we run simulations for a platoon consisting of 3 different types of vehicles. Three types of vehicles with their relevant parameters referring to [7][8] are shown in the Table. 3-2 and used in the simulations. In the simulation conducted, all the vehicles are assumed to be initially traveling at the steady-state velocity of v0 = 17.9 m/s (i.e., 40 mph). Beginning at time t = 0 s, the lead vehicle’s velocity is increased from its steady-state velocity value of 17.9 m/s until it reaches its final value of 21.9 m/s (i.e., 50 mph): the maximum jerk and the peak acceleration values corresponding to this velocity time profile are 0.5 m/s3 and 1m/s2 , respectively (see Fig. 3-4 and Fig. 3-5).
Take three following vehicles as a simulation example, the order of vehicles in the
platoon followed the lead vehicle is as follows: Daihatsu Charade CLS followed by Buick Regal Custom followed by BMW 750iL. Consider the vehicle loading, total mass of each following vehicle is conditioned by adding vehicle curb mass and passengers’ mass. Fig. 3-6 ~ Fig. 3-9 show the simulation results :
Type Daihatsu Charade
CLS
Buick Regal Custom
BMW 750iL
Vehicle i 1 2 3
Curb mass(kg) 916 1464 1925
Passengers’ mass(kg) 91, 91, 91 64, 64 75
Vehicle mass mi(kg) 1189 1592 2000
Kdi(kg/m) 0.44 0.49 0.51
)
i(s
τ 0.2 0.25 0.2
kmi(N) 352 392 408
Table. 3-2 Simulation Model Parameters of Fuzzy Logic Control
0 2 4 6 8 10 12 14 16 18 17.5
18 18.5 19 19.5 20 20.5 21 21.5 22
time(s)
lead vehicle velocity(m/s)
Fig. 3-4 Lead Vehicle’s velocity time profile: vl versus t
0 2 4 6 8 10 12 14 16 18
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(s) lead vehicle acceleration(m/s2 )
Fig. 3-5 Lead Vehicle’s acceleration time profile: al versus t
0 2 4 6 8 10 12 14 16 18 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
time(s) delta i(m)
delta 1 delta2 delta
3
Fig. 3-6 ∆1,∆2,and∆3versus t
0 2 4 6 8 10 12 14 16 18
17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5
time(s)
velocity(m/s)
lead follow
1 follow
2 follow3
Fig. 3-7 vl,v1,v2,andv3versus t
0 2 4 6 8 10 12 14 16 18 -0.4
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
time(s) acceleration(m/s2 )
lead follow
1 follow
2 follow3
Fig. 3-8 al,a1,a2,anda3versus t
0 2 4 6 8 10 12 14 16 18
-500 0 500 1000 1500 2000 2500 3000 3500
time(s)
control u(N)
follow 1 follow2 follow
3
Fig. 3-9 u ,u ,andu versus t
Chapter 4 H
∞Observer-based Sliding Mode Control with Adaptive Fuzzy Neural Approach
4.1 Fundamental Conceptions of Sliding Control
Model imprecision, which is a common trouble in system control, can have strong adverse effects on nonlinear control systems. Therefore, any practical design must address it explicitly. A simple approach to robust control is the so-called sliding mode control (SMC) methodology. It allows an nth-order problem to be replaced by an equivalent 1st-order problem, which is intuitively easier to be addressed. For the class of systems to which it applies, sliding controller design provides a systematic approach to the problem of maintaining stability and consistent performance in the face of modeling imprecision.
Take a single-input dynamic system for example:
u b f
x(n) = (x)+ (x) (4.1) where the scalar x is the output of interest, the scalar is the control input, and
is the state vector. In equation (4.1), (in general nonlinear) is not exactly known, but the extent of the imprecision on is upper bounded by a known continuous function of ; similarly, the control gain is not exactly known, but is of known sign and bounded by a known, continuous function of . The control problem is to get the state to track a specific
u x T
x
x ]
[ 1)
= & L
x (n− f(x)
f(x)
x b(x)
x x
time-varying state in the presence of model impression
be the tracking error vector. Furthermore, let us define a time-varying surface in the state-space
) linear differential equation whose unique solution is
>0 s≡0
~ ≡x 0
s
as . Thus, the problem of tracking the n-dimensional vector , i.e., the original n
∞
th-order tracking
problem in , can be replaced by a 1x st-order problem of keeping the scalar quantity at zero [11]. This simplified 1
s st-order problem of keeping at zero can be
achieved by choosing the control law such that outside of (
, (4.4) where is a strictly positive constant [11]. Eq. (4.4) is called sliding condition.
Essentially, (4.4) states that the squared “distance” to the surface, as measured by , decreases along all system trajectories.
s2
However, in order to account for the presence of modeling imprecision and disturbances, the control has to be discontinuous across . Since the implementation of the associated control switching is necessarily imperfect, this leads to chattering.
) (t s
In general, chattering must be eliminated for the controller to perform properly.
This can be achieved by smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface
, 0 }
| )
; (
|, { )
(t = x s x t ≤φ φ>
B (4.5)
where φ is a boundary layer thickness, and is the boundary layer width, as Fig. 4-1 illustrates for the case
/ −1
=φ λn ε
=2
n . In other words, outside B(t), we choose
ε
ε φ
Boundary
x&
x
=0 s
Fig. 4-1 Boundary layer
u
φ
− φ
s
Fig. 4-2 Control interpolation in boundary layer
control u satisfying sliding condition (4.4), which guarantees that the boundary layer is attractive - hence invariant: all trajectories starting inside B(t=0) remain inside B(t) for all t≥ . Then, we interpolate u inside B(t), as illustrated in Fig. 4-2. This leads to tracking within a guaranteed precision
0
ε (rather than “perfect” tracking), and in the
meantime eliminates the chattering.
4.2 Sliding Mode Control
Let’s consider the first two vehicles in the forefront of the platoon. We will discuss the first couple of vehicles for detail thereinafter since the behavior of other couples of vehicles are similar. By rearranging (2.7) the first follow vehicle dynamic is written as
And, the lead vehicle dynamic is stated as
vehicle for increasing its velocity from to in a period of time. Substituting Eq. (4.6) and (4.7) to Eq. (2.1), we have
1 1 1 1 1 1 1 control law such that the tracking error e
[
e1(1) e1(2) e1(3)T1 is zero. Furthermore, let the sliding
surface be expressed in the state-space by the scalar equation as
From (4.11), (4.12), and following similar derivations in [5], we can use the sliding mode control method and obtain a control law derived in Lemma 1.
Lemma 1: Consider the nonlinear system (4.11) with given nonlinear function . Suppose that control input is chosen as
)
and that P is positive definite symmetric matrix, P ∈ satisfies the Lyapunov matrix equation
R2x2
A1TP+PA1 =−Q (4.14)
where sign(s(e1;t))
= ,
AConsider a Lyapunov function candidate as following:
2 here P is positive definite symmetric matrix.
By using (4.11), (4.12), (4.14), and (4.15), the time derivative of (4.16) is 0 )
Apply (4.13) to (4.17), we have the following relationship:
0 Note that if we include the design of (4.5), (4.13) becomes
)]
4.3 Sliding Mode Control with Fuzzy Neural Network Approximator
In practical applications, however, is generally uncertain rather than given. The controller of (4.13) derived above is not always obtainable. Therefore, a new controller needs to be designed taking into account the unknown nonlinear function, which will be adequately approximated by a fuzzy-neural approximator.
)
The configuration of the fuzzy-neural network shown in Fig. 4-3 consists of a fuzzy system and neural network. The fuzzy system can be divided into two parts:
some fuzzy IF-THEN rules and a fuzzy inference engine. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input linguistic vector
to an output linguistic variable .
[
(1) (2) (3) R3e = e e e T ∈ o(e )∈R
The ith fuzzy IF-THEN rule is written as
Ri: If is and is and is than is (4.20)
center-average and singleton fuzzifier, the output of the fuzzy-neural network can be )
( i
Bi y
µ yi µBi(yi)=1
expressed as
Layer I Layer II Layer III Layer IV
L
L
L
Fig. 4-3 Configuration of a fuzzy-neural approximator
)
∑ ∏
When the inputs are given into the fuzzy-neural network shown in Fig. 4-3, the truth value (layer III) of the antecedent part of the ith implication is calculated by (4.22).
Among the commonly used defuzzification strategies, the output (layer IV) of the fuzzy-neural network is expressed as (4.21). Therefore, the fuzzy logic approximator based on the neural network can be established. Fig. 4-3 shows the configuration of the fuzzy-neural function approximator. The approximator has four layers. At layer I, input nodes stand for the input linguistic variables (1), , and . At layer II, nodes represent the values of the membership functions. At layer III, nodes are the values of the fuzzy basis vector
ϕi
e1 e1(2) e1(3)
ϕ. Each node of layer III performs a fuzzy rule. The
links between layer III and layer IV are full connected by the weighting factors , i.e., the adjusted parameters. At layer IV, the output stands for the value of . adaptive update laws to adjust the parameter vector in (4.21) of the fuzzy-neural approximator need to be developed. Let be the estimation function for the uncertain nonlinear function . That is,
)
In order to derive the adaptive update law, the following assumption is required.
Assumption 1 [27]:
Let e1 belongs to a compact setUe1 =
{
e1∈R3 : e1 ≤me1 <∞}
and isdesigned parameter. It is known a prior that the optimal parameter vector me
] )
| ˆ( sup
[ min
θ 1 1
e
*
1
θ e1
θ θ e1
F F
M U −
=
∈ ∈
arg
{
lies in some convex region
θ
}
θ θ R θ m
M = ∈ 3: ≤ , where the radius mθ is constant.
□ Thus, to approximate the uncertain nonlinear term F1, Eq. (4.10) becomes
∆&&1
& =θ*Tϕ(e1)+G1(v1)u1+w (4.24)
where w=F1−θ*Tϕ(e1) is the lumped uncertainty.
To facilitate the design process of the controller, the lumped uncertainty is generally assumed to have an upper bound,
w ≤wu, wuis a positive constant (4.25)
Based on above condition, a control law via fuzzy-neural approximator can be obtained from Lemma 2 below.
Lemma 2: Consider the nonlinear system (4.11) with uncertain nonlinear function , which is approximated as (4.24). Suppose Assumption 1 and (4.25) are satisfied and control input is chosen as
) , (v a F1 1 1
))]
Consider the Lyapunov function θ
θ 4.4 Sliding Mode Control with Fuzzy Neural Network Approximator and performance advance to construct the control input , In practical applications, however, the
exact upper bound cannot be chosen so as to attenuate the uncertainties, large control chattering nevertheless occurs. Simulation with a sliding mode controller illustrate this effects for different k selected. To relax the impractical constraint, a new control law is designed by using the tracking design technique based on a much relaxed assumption [5], The lumped uncertainty is assumed such that
wu
which is approximated as (4.24). Suppose Assumption 1 ,Eq. (4.25), and Eq. (4.31)
are satisfied; the control input is chosen as attenuation level, is the sliding surface. Then the tracking performance [9], [10] for the overall system satisfies the following relationship:
s H∞
Consider the Lyapunov function
θ The time derivative of (4.34) is (see (4.29))
θ
By (4.31), we integrate (4.36) from t=0 to t=T, and obtain
∫
Substituting (4.34) to (4.37), we have the tracking performance, satisfying (4.33)
This completes the proof.
□ Remark:
If a set of initial condition e1(0)=0,s(0)=0, can be obtained, and Q=I, then control performance of the overall system satisfies
)
4.5 Observer-based Sliding Mode Control with Fuzzy Neural Network Approximator and H∞ performance
Utilizing sensors to obtain the measurements of the parameters in the vehicle system, such as the velocity, the acceleration,… etc., is difficult or expensive.
Considering the technical difficulties and the economic benefits, we adopt an observer to estimate the plant output state vector while we assume that only the headway information of two vehicles is measurable. Under this constraint, a sliding mode
observer is proposed for state estimation [3][37]. The sliding observers offer advantages similar to those of sliding controllers such as robustness to parameter uncertainty and easy application to important classes of nonlinear systems.
In this section, our task is to combine the plant and controller with an sliding observer which estimates the state vector ; the estimated vector is denoted as and used as the input both of the sliding mode controller and of the adaptive fuzzy-neural network approximator instead of state feedback used before.
e1 eˆ1 Rewrite (4.11) as
(4.39)
Consider the following observer [3][37][32]:
(4.40)
o is positive definite
symmetric matrix, Po satisfies the Lyapunov matrix equation
AoTPo +PoAo =−I (4.41)
is Hurwitz. Thus, the applied control to (4.39) and (4.40) is
solution of (4.32) replacing e by . Such an observer leads to the error
Then, we assume that:
Assumption 2:
~ 0 1(⋅)|1=
∆H e is globally Lipschitz continuous in its first argument
e1
Assumption 3:
o Theorem 1: Consider the nonlinear system (4.11) with uncertain nonlinear function , which is approximated as (4.24). And with control defined as in Lamma 3, to which observer (4.40) is associated. Suppose Assumption 1-3, (4.25), and (4.31) are satisfied. Then for any initial conditions, the state e of the observer converges toward the state of the system.
Proof:
Consider the partial Lyapunov function ~1(1)2 2 remaining estimation error can then be shown to decay exponentially using Filippov’s work. Taking a convex combination of the dynamics on each side, we have
and thus ensures that e′1 decreases to zero
in order to remedy the control chattering [24].
Naturally, to prove asymptotic stabilization of the controlled plant plus the observer, we use the following Lyapunov function :
θ
to (4.48), and let , we have following relationship:
))]
θ
This completes the proof.
□ 4.6 Simulation Results
To examine the behavior of a platoon of vehicles under the above controller, we run simulations for a platoon consisting of 3 different types of vehicles again.
Three types of vehicles with their relevant parameters are the same to the ones shown in the Table. 3-2 and are used in the simulations. In the simulation conducted, all the vehicles are assumed to be initially traveling at the steady-state velocity of v0 = 17.9 m/s (i.e., 40 mph). Beginning at time t = 0 s, the lead vehicle’s velocity is increased from its steady-state velocity value of 17.9 m/s until it reaches its final value of 21.9 m/s (i.e., 50 mph): the maximum jerk and the peak acceleration values corresponding to this velocity time profile are 0.5 m/s3 and 1m/s2 , respectively (also see Fig. 3-4 and Fig. 3-5). Take three following vehicles as an simulation example, the order of vehicles in the platoon followed the lead vehicle is as follows: Daihatsu Charade CLS followed by Buick Regal Custom followed by BMW 750iL. Consider the vehicle
loading, total mass of each following vehicle is conditioned by adding vehicle curb mass and passengers’ mass. From Fig. 4-5 to Fig. 4-9 are the simulations of the application using a sliding mode controller with the boundary layer. From Fig. 4-10 to Fig. 4-14, we can see that the fuzzy neural approximator can approximate the uncertain nonlinear term of the system. From Fig. 4-15 to Fig. 4-19 , we apply the
performance with the control law obtained in section 4.3. It shows that the performance is better and the chattering is attenuated, too. From Fig. 4-20 to Fig. 4-24, the observer is associated to estimate the system states, it shows that performance is still acceptable.
H∞
G u − F ˆ + (...)
=
d Gu
n =F+ +
∆( )
Controller
ϕ s Γ θ & =
ϕ F ˆ = θ
Tu
Observer
∆
∆
∆
&
&
&
ˆ ˆ ˆ
∆
∆ˆ
∆~ + _
Fig. 4-4 Block Diagram of Overall System
symbols parameters(values)
λ 1
λ 1
λ 2
λ3 1
1.2
ρ 0.3
P21 0.5
P22 1.5
φ 1.0
0 1|
ˆ =
∆ t -0.1
0 2 |
ˆ =
∆ t 0.2
0 3 |
ˆ =
∆ t 0.1
1 50I
2 40I
3 40I
1
2
k
Γ Γ Γ
Table. 4-1 Simulation Model Parameters of Modified Sliding Mode Control
Fig. 4-5 ∆1 ,∆2 ,and∆3versus t
Fig. 4-6 pl ,p1 ,p2 ,and p3versus t
Fig. 4-7 vl ,v1 ,v2 ,andv3versus t
Fig. 4-8 al ,a1 ,a2 ,and a3versus t
Fig. 4-9 u1 ,u2 ,andu3versus t
Fig. 4-10 ∆1 ,∆2 ,and∆3versus t
Fig. 4-11 pl ,p1 ,p2 ,and p3versus t
Fig. 4-12 vl ,v1 ,v2 ,andv3versus t
Fig. 4-13 al ,a1 ,a2 ,anda3versus t
Fig. 4-14 u1 ,u2 ,andu3versus t
Fig. 4-15 ∆1 ,∆2 ,and∆3versus t
Fig. 4-16 pl ,p1 ,p2 ,and p3versus t
Fig. 4-17 vl ,v1 ,v2 ,andv3versus t
Fig. 4-18 al ,a1 ,a2 ,anda3versus t
Fig. 4-19 u1 ,u2 ,andu3versus t
Fig. 4-20 ∆1 ,∆2 ,and∆3versus t
Fig. 4-21 pl ,p1 ,p2 ,and p3versus t
Fig. 4-22 vl ,v1 ,v2 ,andv3versus t
Fig. 4-23 al ,a1 ,a2 ,anda3versus t
Fig. 4-24 u1 ,u2 ,andu3versus t
Chapter 5 Conclusions
In this thesis, we modeled mathematically both vehicle and car-following systems. In order to gain better performance and ensure the robustness and global stability, a sliding mode controller with the fuzzy-neural network approximator and performance was proposed. Moreover, considering the technical difficulties and the economic benefits, we assumed that only the relative distance of two vehicles was measurable. Thus, an observer-based modified sliding mode controller was developed.
H∞
The control performance of the proposed system were simulated. Simulation results demonstrated the validity and effectiveness of the controlled systems. The system with the modified sliding mode controller showed a better performance than the one controlled by a sliding mode controller did. With these two controllers, the robustness and the global stability were both guaranteed during the vehicles-following process in the presence of the uncertainties and disturbances.
In designing the output feedback control law of an observer-based modified sliding mode controller, no differentiation of system outputs was performed in order to avoid the noise amplification associated with numerical differentiation, and no knowledge on nonlinearities of the nonlinear parts of the system was required. This
controller is subject to on-line tuning for a nonlinear system. Although the performance of the car-following system with an observer is not as good as the one with a modified sliding mode controller which were combined with an aprroximator and performance conception, it is still satisfying.
H∞
References
[1] Shahab Sheikholeslam and Charles A. Desoer, “Longitudinal control of a
[1] Shahab Sheikholeslam and Charles A. Desoer, “Longitudinal control of a