A New Intelligent Fuzzy Controller for Nonlinear
Hysteretic Electronic Throttle in Modern
Intelligent Automobiles
Chi-Hsu Wang, Fellow, IEEE, and De-Yu Huang, Member, IEEE
Abstract—In order to control the nonlinear hysteretic elec-tronic throttle adopted in modern automobiles, a new result for the tracking control of nonlinear hysteretic system is first proposed in this paper. Therefore we can realize the intelligent fuzzy logic controller in a well-behaved and systematic manner. A new closed-loop Back-propagation tuning is also proposed for the tuning of the fuzzy output membership functions to yield better tracking result. Finally, the controller synthesis is performed in a real-time environment using dSpace MicroAutobox and advanced microcontroller development board to yield excellent tracking results with cost-effective implementation.
Index Terms—Electronic throttle (ET), fuzzy logic, hysteretic systems, intelligent control.
I. INTRODUCTION
M
ANY OF THE vital functions of today’s cars are shift-ing from a purely mechanical to an electromechanical implementation. These so-called “X-by-wire” systems act as an interface between the driver and the targeted mechanical subsystem of the vehicle (e.g., brakes, throttle valve) [1]–[12]. Coupled with the use of advanced control strategies the “X-by-wire” systems can, in general, provide wider functionality and better research and development in vehicle control techniques, such as electric braking control, posture stabilization control, longitudinal control of road vehicles, and electronic throttle (ET) valve control [13]–[15]. ET can replace its mechanical counterpart if a control loop satisfies prescribed requirements, which are a fast transient response without overshoot, position-ing within the measurement resolution, and the control action that does not wear out the components [3]. These high-quality controldemands are difficult to accomplish since the throttle is burdened with strong nonlinear hysteretic effects of friction and return spring [3], [16], [17]. Moreover, the control strategy should be simple enough to be implemented on a low-costManuscript received November 1, 2011; revised January 20, 2012 and March 1, 2012; accepted March 28, 2012. Date of publication May 3, 2012; date of current version February 6, 2013. This work was supported by the National Science Council, Taiwan, under the title of “Design and Implementation of Intelligent Controller for Electronic Throttle of Advanced Vehicles,” 100-2221-E-009-027-, 2011∼ 2012.
C.-H. Wang is with the National Chiao-Tung University, Department of Electrical engineering, Hsihchu 300, Taiwan (e-mail: [email protected]). D.-Y. Huang is with the Department of Electrical engineering, Institute of Electrical control engineering, National Chiao-Tung University, Hsihchu 300, Taiwan (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2012.2193861
automotive microcontroller system, while it has to be robust to plant parameter variations.
Recently, several control strategies for ET have been pre-sented; differing in the underlying philosophy and complexity [3], [18]–[34]. However, these ET control (ETC) systems have their weak points in practical applications. In [17], the nonlinear hysteretic ET was also discussed and controlled by variable structure (VS) controller via the expensive dSpace DSP boards [35]. This kind of expensive laboratory implementation is still questionable in real industrial cost-effective implementations. Furthermore, its performance still has chattering effects due to the nature of VS controller under high gain.
The principal aim of this paper is to present a new intelligent controller for nonlinear hysteretic ET. Several new properties for the control of nonlinear hysteretic system (NHS) will be explored in this paper, which will lead to the design of a stable fuzzy logic controller for the tracking of drivers’ pedal signals [36], [37]. Under the proposed feedforward configuration for tracking control, a new theorem is developed to show the existence with special properties of a stable controller (of any kind) for the NHS. These special properties will allow us to realize the stable intelligent fuzzy controller in a well-behaved and systematic manner. To improve the tracking performance, a new closed-loop Back-propagation tuning is also proposed for the tuning of the fuzzy output membership functions to yield better tracking result. We adopt the ET of SAAB-91-88-186, which have been used in Saab 95 from 1998 [34]. The MicroAutobox (MABX) [35] from dSPACE will mainly be adopted for the identification of the hysteretic loop in ET. The expensive MABX can also be used for the real-time control of ET with good confidence. For comparison, the PID-tuned controller is also designed by using Matlab Toolbox [38] and realized using MABX. However, for real industrial applica-tions, we adopt the cost-effective advanced microcontroller to implement the real-time FLC by converting the floating point numbers to fix point integers. The final FLC is also realized in look-up table stored in the microcontroller for fast real-time operations. Excellent tracking results are obtained using the cost-effective microcontroller, which is even better than the PID-tuned controller in MABX under our new development.
II. NONLINEARHYSTERETICELECTRONIC THROTTLEUSINGBLDC (ET-BLDC)
The ET in modern automobiles is usually a brushless dc (BLDC) motor with 12 V/5 A rating. It is also a strong nonlinear 0278-0046/$31.00 © 2012 IEEE
Fig. 1. (a) The nonlinear electronic throttle. (b) The electronic throttle with controller.
device with hysteretic phenomenon, friction, and return spring nonlinearities, as shown in the following Fig. 1(a). The dotted block in Fig. 1(b) shows the schematic diagram for ET body. The H-bridge is the power amplifier to boost the power of the control output from the electronic controller.
The nonlinear ET has four sensors, including two pedal sensors and two throttle sensors. When the driver pushes or relaxes the pedal to vary the speed, ET’s pedal sensors will receive the signals in the same time. The throttle valve is controlled by a brushless throttle motor with a 600-Hz pulse width modulation (PWM) signal.
III. NONLINEARHYSTERETICSYSTEM(NHS) The throttle is a valve used in vehicles to regulate air in-flow into the engine combustion system. The air throughput is controlled by the opening angle of the valve plate in the air tube. Traditionally, the throttle valve plate was directly connected to the gas pedal by a mechanical cable. The reason is that the amount of air inflow is proportional to the desired engine speed, which is decided by the position of the gas pedal. Nowadays, however, the throttle cable has been substi-tuted with the throttle-by-wire system since around 1998. The throttle-by-wire system consists of a dc motor with computer controller to convert the position of the gas pedal to the an-gle of throttle valve position. The gas pedal sensor provides the driver command to a throttle control module (TCM) in Engine Management System which then specifies proper air-fuel mixture to be fed into the engine. In particular, the TCM finds a proper control output that positions the throttle valve in the desired opening angle, the ETC system. Fig. 2 shows a typical illustration of hysteretic phenomenon. Hysteretic phe-nomena occur in magnetic materials, ferromagnetic materials, and ferroelectric materials, as well as in the elastic, electric, and magnetic behavior of materials, in which a lag occurs between
Fig. 2. Nonlinear hysteretic system.
Fig. 3. Control strategy to control the NHS.
the application and the removal of a force. Electric hysteretic occurs when applying a varying electric field and elastic hys-teretic occurs in response to a varying force, such as the dc brushless motor in ET adopted in this paper. The hysteretic is often used specifically to represent rate-independent state. The magnetized iron or the thermostat has this property. In the paper, the ET is a NHS with rate dependency, and its property is shown in Fig. 2. The NHS shown in Fig. 2 is input/output bounded, i.e., x1≤ x(t) ≤ x2 and y1≤ y(t) ≤ y2. In Fig. 2,
when the input force x(t) increases, i.e., Δx(t) > 0, the ET will operate from close to open, which is a→ b → c in Fig. 2. When
x(t) decreases, i.e., Δx(t) < 0, the ET will operate from open
to close, which is c→ d → a in Fig. 2. Note that although we have two cases of NHS in Fig. 2, all the discussions in this paper will be applied equally. For simplicity, the NHS in Fig. 2(a) will be adopted for illustrations in the remaining of this paper.
IV. STABLECONTROLSTRATEGY FORNHS To properly control the nonlinear dc brushless motor with hysteretic phenomenon, we propose the following feedforward closed-loop configuration.
In Fig. 3, the R(t) is input signal to the system and the feed-forward controller will decide a nominal output command u(t) based on R(t) to steer the NHS. The feedforward controller is actually a transducer to convert the physical R(t) signal into the electrical signal u(t). For instance, the R(t) can be the mechanical pedal signal by the automobile driver, the u(t) will be the equivalent electrical signal generated by the transducer. The e(t) and Δe(t) between R(t) and Y (t) will be the inputs to the controller. The controller will then generate a Δu(t) to be added with the nominal control command u(t) to the NHS. It is hoped that the controller will generate a proper Δu(t) to compensate the nominal command u(t) to steer the NHS to the desired Y (t).
To better illustrate the control strategy for the controller to generate Δu(t), we have the following arrangements. In Fig. 4,
Fig. 4. Control strategy of the hysteretic loop when Δx(t) > 0 and Δx(t) < 0.
Fig. 5. Relationship between e and Δu(= f (e)).
when Δx(t) > 0, the output Y (t) will follow the direction pointed by the arrows if Δu = 0. However, the goal is to let
Y (t) to track R(t), so we need to generate a proper nonzero
Δu to let Y (t) follow R(t). When x(t) = u1(t), the error
is e1(t) > 0. By drawing a horizontal line across point 1 to
intersect the hysteretic loop at point a in Fig. 4, we have the desired Δu1 to let x(t) = u1(t) + Δu1(t), so that Y (t)
will be equal to R(t) at that moment. In this case, Δu1(t) is
positive. Similarly, when x(t) = u2(t), the error is e2(t) > 0.
By drawing a horizontal line across point 2 to intersect the hysteretic loop at point b in Fig. 4, we have the desired Δu2to
let x(t) = u2(t) + Δu2(t). In this case, Δu2(t) is also positive.
In fact, we can say that ef (e) = eΔu > 0 when Δx(t) > 0. Similar discussions can be made at points 3 to c and 4 to d in Fig. 4 for Δx(t) < 0. This will yield e < 0 and Δu < 0 to have
ef (e) = eΔu > 0. It is also obvious from Fig. 4 that if e = 0
will imply Δu = f (e) = 0.
Furthermore, Fig. 5 shows the relationship between e and Δu by
Δu = f (e) = e cot(θ), 0 < θ≤ cot−1 2m n < π 2. (1) To facilitate the proofs of the following theorems, the fol-lowing Fig. 6 provides a general presentation for the hysteretic loop, where all n1, n2, n3, and m are greater than zero.
It is desired for output y(t) to track R(t) whose equation is in the following form:
R(t) = m n1
x. (2)
The right-hand-side (RHS) loop (Δx > 0) in Fig. 6 is y = n2m 1+n2x + m(n2−n1) n1+n2 , if−n2≤ x ≤ n1 y =−m, if−n1≤ x < −n2 . (3)
Fig. 6. Hysteretic loop when Δx > 0 and Δx < 0.
The left-hand-side (LHS) loop (Δx < 0) in Fig. 6 is y = m, if n2< x≤ n1 y = 2m n1+n2x + m(n1−n2) n1+n2 , if−n1≤ x ≤ n2 . (4) As previously mentioned that the feedforward controller is only a transducer to convert the physical signal R(t) into the electrical signal u(t), we let
u(t) = αR(t).
Therefore,
e(t) = R(t)− NHS (u(t) + Δu(t)) . (5) Because the hysteretic loop is a nonlinear and multiple valued function with different outputs according to the sign of Δx, i.e., Δx > 0 and Δx < 0. When Δx > 0, the y(t) will follow the RHS hysteretic loop to produce e(t) and Δu(t). We also define cot θ3= n1+ n2 2m and cot θ33= (n2− n1) (|x| + n1) 2m (|x| − n1) , when Δx > 0 and Δx < 0. (6)
It is obvious that cot θ3> 0, cot θ33> 0 due to the fact that
{n1, n2, m > 0, and n1> n2}. The following Theorem 1 will
prove the above discussions in a formal way.
Theorem 1: For the closed-loop configuration in Fig. 3 with
its controller output Δu defined in Fig. 4 to control the NHS, if and only if the closed-loop system in Fig. 3 is asymptotically stable i.e., lim
t→∞e(t) = 0, then
ef (e) = eΔu≥ 0, with
Δu = f (e) = ⎧ ⎪ ⎨ ⎪ ⎩ n2−n1 2n1 [x(t)− n1] = e(t) cot θ ∗, if Δx≥ 0 and θ∗is θ3or θ33 n2−n1 2n1 [x(t) + n1] = e(t) cot θ ∗, if Δx≤ 0 and θ∗is θ3or θ33 .
Proof: Part A: We will show that if the system is asymptot-ically stable then
ef (e) = eΔu≥ 0, with
Δu = f (e) = ⎧ ⎪ ⎨ ⎪ ⎩ n2−n1 2n1 [x(t)− n1] = e(t) cot θ ∗, if Δx≥ 0 and θ∗is θ3or θ33 n2−n1 2n1 [x(t) + n1] = e(t) cot θ ∗, if Δx≤ 0 and θ∗is θ3or θ33 .
When Δx > 0, we can have two cases, i.e.,−n2≤ x ≤ n1
and−n1≤ x ≤ −n2.
1) When−n2≤ x ≤ n1, the error signal e(t) without
con-trol Δu(t) at time t is
e(t) = R(t)− Y (t) = m(n2− n1) n1(n2+ n1)
[x(t)− n1]≥ 0. (7)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4, so that the system will be asymptotically stable. Since the system is assumed to be asymptotically stable, we can have lim t→∞e(t) = limt→∞ m n1 x(t)− 2m n1+ n2 (x(t) + Δu(t)) +m(n2− n1) (n2+ n1) = 0. (8)
This will let us find the approximate Δu(t) in finite time t. Therefore, the proper Δu(t) can be found as
Δu(t) =n2− n1 2n1 [x(t)− n1] = m(n2− n1) n1(n2+ n1) [x(t)− n1]∗ (n2+ n1) 2m = e(t)∗ cot θ3≥ 0. (9)
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0 with
Δu(t) shown in (9). Further, if e(t) = 0, then x(t) = n1,
therefore Δu(t) = f (e(t)) = f (0) = 0.
2) When −n1≤ x ≤ −n2, the error signal e(t) without
control Δu(t) at time t is
e(t) = R(t)− Y (t) = m n1
[x(t) + n1]≥ 0. (10)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4, so that the system will be asymptotically stable. Since the system is assumed to be asymptotically stable, we can have lim t→∞e(t) = limt→∞ m n1 x(t)− 2m n1+ n2 (x(t) + Δu(t)) +m(n2− n1) (n2+ n1) = 0. (11)
This will let us find the approximate Δu(t) in finite time t. Therefore, the proper Δu(t) can be found as
Δu(t) =n2− n1 2n1 [x(t)− n1] =m n1 [x(t) + n1]∗ (n2− n1) (|x| + n1) 2m (|x| − n1) = e(t)∗ cot θ33≥ 0. (12)
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0 with
Δu(t) shown in (12). Further, if e(t) = 0, then
x(t) =−n1, therefore Δu(t) = f (e(t)) = f (0) = 0. It
is obvious that Δu(t) defined in (9) and (12) are identical for Δx > 0.
When Δx < 0, the y(t) will follow the LHS hysteretic loop, and it produces e(t) and Δu(t) in Fig. 6. When Δx < 0, we can also have two cases, i.e.,−n1≤ x ≤ n2and n2≤ x ≤ n1.
1) When−n1≤ x ≤ n2, the error signal e(t) without
con-trol Δu(t) at time t is
e(t) = R(t)− Y (t) = m(n2− n1) n1(n2+ n1)
[x(t) + n1]≤ 0. (13)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 3, so that the system will be asymptotically stable. Since the system is assumed to be asymptotically stable, we can have lim t→∞e(t) = limt→∞ m n1 x(t)− 2m n1+ n2 (x(t) + Δu(t)) +m(n2− n1) (n2+ n1) = 0. (14)
This will let us find the approximate Δu(t) in finite time t. Therefore, the proper Δu(t) can be found as
Δu(t) =(n2− n1) 2n1 (x(t) + n1) =m(n2− n1) n1(n2+ n1) [x(t) + n1]∗ n1+ n2 2m = e(t)∗ cot θ3≤ 0. (15)
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0 with
Δu(t) shown in (15). Further, if e(t) = 0, then
x(t) =−n1, therefore Δu(t) = f (e(t)) = f (0) = 0.
2) When n2≤ x ≤ n1, the error signal e(t) without control
Δu(t) at time t is
e(t) = R(t)− Y (t) = m n1
[x(t)− n1]≤ 0. (16)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4, so that the
system will be asymptotically stable. Since the system is assumed to be asymptotically stable, we can have lim t→∞e(t) = limt→∞ m n1 x(t)− 2m n1+ n2 (x(t) + Δu(t)) +m(n2− n1) (n2+ n1) = 0. (17)
This will let us find the approximate Δu(t) in finite time t. Therefore, the proper Δu(t) can be found as
Δu(t) =(n2− n1) 2n1 (x(t) + n1) = m n1 [x(t)− n1]∗ (n2− n1) (|x| + n1) 2m (|x| − n1) = e(t)∗ cot θ33≤ 0. (18)
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0 with
Δu(t) shown in (17). Further, if e(t) = 0, then
x(t) = n1, therefore Δu(t) = f (e(t)) = f (0) = 0. It is
obvious that Δu(t) defined in (15) and (17) are identical for Δx < 0.
Part B: We will show that if the system has error e(t) and
ef (e) = eΔu≥ 0, with
Δu = f (e) = ⎧ ⎪ ⎨ ⎪ ⎩ n2−n1 2n1 [x(t)− n1] = e(t) cot θ ∗, if Δx≥ 0 and θ∗is θ3or θ33 n2−n1 2n1 [x(t) + n1] = e(t) cot θ ∗, if Δx≤ 0 and θ∗is θ3or θ33
then it is asymptotically stable. When Δx > 0, we can have two cases, i.e.,−n2≤ x ≤ n1and−n1≤ x ≤ −n2.
1) When−n2≤ x ≤ n1, the error signal e(t) without
con-trol Δu(t) at time t is
e(t) = R(t)− Y (t) = m(n2− n1) n1(n2+ n1)
[x(t)− n1]≥ 0. (19)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4. We can have the approximate Δu(t) in finite time t
Δu(t) =n1+ n2 2m y(t)−m(n2− n1) (n2+ n1) −n1 my(t) =n2− n1 2n1 [x(t)− n1] . (20)
The Δu(t) can be rewritten as Δu(t) =n2− n1 2n1 [x(t)− n1] = m(n2− n1) n1(n2+ n1) [x(t)− n1] ∗(n2+ n1) 2m = e(t)∗ cot θ3≥ 0.
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0.
Further, if e(t) = 0, then x(t) = n1, therefore
Δu(t) = f (e(t)) = f (0) = 0.
Then, we consider the error e(t) with Δu(t), the error can be found as
lim
t→∞e(t) = limt→∞(R(t)− y(t))
= lim t→∞ m n1 x(t)− 2m n2+ n1 (x(t) + Δu(t)) +m(n2− n1) n2+ n1 = 0. (21)
The error will be decreased to zero as time goes to infinity, and the system will be asymptotically stable. 2) When −n1≤ x ≤ −n2, the error signal e(t) without
control Δu(t) at time t is
e(t) = m n1
x(t)− (−m) = m n1
[x(t) + n1]≥ 0. (22)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4. We can have the approximate Δu(t) in finite time t
Δu(t) =n1+ n2 2m y(t)−m(n2− n1) (n2+ n1) −n1 my(t) =n2− n1 2n1 [x(t)− n1] . (23)
The Δu(t) can be rewritten as Δu(t) =n2− n1 2n1 [x(t)− n1] = m n1 [x(t) + n1] ∗ (n2− n1)(x− n1) 2m(x + n1) = e(t)∗ cot θ33≥ 0.
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0. Further, if e(t) = 0, then x(t) =−n1, therefore Δu(t) =
f (e(t)) = f (0) = 0. Then, we consider the error e(t)
with Δu(t), the error can be found as lim
t→∞e(t) = limt→∞(R(t)− y(t))
= lim t→∞ m n1 x(t)− 2m n2+ n1 (x(t) + Δu(t)) +m(n2− n1) n2+ n1 = 0. (24)
The error will be decreased to zero as time goes to infinity, and the system will be asymptotically stable. When Δx < 0, the y(t) will follow the LHS hysteretic loop and it produces e(t) and Δu(t) in Fig. 6. When Δx < 0, we can also have two cases, i.e.,−n1≤ x ≤ n2and n2≤ x ≤ n1.
I. When−n1≤ x ≤ n2, the error signal e(t) without
con-trol Δu(t) at time t is
e(t) =m n1 x(t)− 2m n1+ n2 x(t) +m(n1− n2) n1+ n2 =m(n2− n1) n1(n2+ n1) [x(t) + n1]≤ 0. (25)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4. We can have the approximate Δu(t) in finite time t
Δu(t) =n2− n1 2m y(t)−m(n1− n2) n1+ n2 −n1 my(t) =(n2− n1) 2n1 (x(t) + n1) . (26)
The Δu(t) can be rewritten as Δu(t) =(n2− n1) 2n1 (x(t) + n1) = m(n2− n1) n1(n2+ n1) [x(t) + n1] ∗n1+ n2 2m = e(t)∗ cot θ3≤ 0.
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0.
Further, if e(t) = 0, then x(t) =−n1, therefore
Δu(t) = f (e(t)) = f (0) = 0. Then, we consider the error e(t) with Δu(t), the error can be found as lim
t→∞e(t) = limt→∞(R(t)− y(t))
= lim t→∞ m n1 x(t)− 2m n2+ n1 (x(t) + Δu(t)) +m(n1− n2) n1+ n2 = 0. (27)
The error will be decreased to zero as time goes to infinity and the system will be asymptotically stable. II. When n2≤ x ≤ n1, the error signal e(t) without control
Δu(t) at time t is e(t) = m n1 x(t)− m = m n1 [x(t)− n1]≤ 0. (28)
In Fig. 6, we can see the error e(t) and then find out the Δu(t). However, the proper Δu(t) will be added in the next time instant t≈ t + Δt, which follows closely from the control principle shown in Fig. 4. We can have the approximate Δu(t) in finite time t
Δu(t) =n2+ n1 2m y(t)−m(n1− n2) n1+ n2 −n1 my(t) =(n2− n1) 2n1 (x(t) + n1) . (29)
Fig. 7. Hysteretic loop when Δx(t) > 0 and Δx(t) < 0. The Δu(t) can be rewritten as
Δu(t) =(n2− n1) 2n1 (x(t) + n1) = m n1 [x(t)− n1] ∗(n2− n1) (|x| + n1) 2m (|x| − n1) = e(t)∗ cot θ33≤ 0.
Therefore, e(t)∗ f(e(t)) = e(t) ∗ Δu(t) ≥ 0. Further, if e(t) = 0, then x(t) = n1, therefore Δu(t) = f (e(t)) =
f (0) = 0. Then, we consider the error e(t) with Δu(t),
the error can be found as lim
t→∞e(t) = limt→∞(R(t)− y(t))
= lim t→∞ m n1 x(t)− 2m n2+ n1 (x(t) + Δu(t)) +m(n1− n2) n1+ n2 = 0. (30)
The error will be decreased to zero as time goes to infin-ity, and the system will be asymptotically stable. Q.E.D. When the NHS is not symmetrical, as shown in the following Fig. 7, we still have the following similar results, where n1, n2,
n3, and m are all assumed to be greater to the zero. The proof
of Theorem 2 is skipped for simplicity.
Theorem 2: For the closed-loop configuration in Fig. 2 with
its controller output Δu defined in Fig. 7 to control the NHS, if and only if the closed-loop system in Fig. 3 is asymptotically stable, i.e., lim
t→∞e(t) = 0, then
ef (e) = eΔu≥ 0, with
Δu = f (e) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n2−n1 2n1 [x(t)− n1] = e(t) cot θ ∗, if Δx≥ 0 and θ∗is θ2or θ22 −(n1+n3) 2n1 [x(t) + n1] = e(t) cot θ ∗, if Δx≤ 0 and θ∗is θ1or θ11.
It is noted that the above Theorem 1 and Theorem 2 are valid regardless of the type of controller. The controller can be the classical PID controller or the intelligent controller. The following Section V will further elaborate the advantages of designing an intelligent fuzzy controller for NHS.
Fig. 8. Fuzzy logic controller.
V. SYNTHESIS OFFUZZYLOGICCONTROLLER FOR NONLINEARHYSTERETICSYSTEMS
For the past two decades, the FLC has been widely and successfully utilized and implemented in numerous industrial applications [21]–[24], [39], [40]. A general FLC consists of four modules: A fuzzification module, a fuzzy rule base, a fuzzy inference engine, and a defuzzification module [40]–[44], as shown in the dotted area in Fig. 8.
The ith control rule Ri in fuzzy controller in Fig. 8 has the
following form:
Ri: IF x1is Ai1and . . . xnis AinTHEN y1is Wi1
and . . . ymis Wim
where i is a rule number, Aiq are membership functions of antecedent part, and Wip are membership functions of the
consequent part. In order to guarantee the stability of the closed-loop system in Fig. 3 to control the NHS, we consider the following 2N + 1 fuzzy IF-THEN rules:
IF e is Ai1and Δe is Ai2THEN Δu is Bi (31) where i = 1, 2, . . . , 2N + 1. The 2N + 1 triangular member-ship functions for e, Δe, and Δu, respectively, are chosen in the following way.
That is, we use the N membership functions A1, . . . , AN
to cover the negative intervals [α, 0), the other N membership functions AN+2, . . . ,A^{2{\rm N} + 1} to cover the positive intervals (0, β], and choose the center xN +1of AN+1equals to zero. Also, the centers yiof fuzzy sets Biare chosen such that
yi= ⎧ ⎨ ⎩ ≤ 0 for i = 1, . . . , N = 0 f or i = N + 1 ≥ 0 for i = N + 2, . . . , 2N + 1 (32) Lemma 1: [39]: If the fuzzy logic controller defined in
Fig. 8, with their membership functions defined using (31), Fig. 9 and (32), then the fuzzy logic controller f (e) using the center average defuzzifier (CAD) scheme will have the following properties:
(a) f (0) = 0 and ef (e)≥ 0 for all e ∈ R.
(b) |f(e1)− f(e2)| ≤ α|e1− e2|, ∀e1, e2∈ R, for some
constant α.
Although we have two input fuzzy variables, i.e., e and Δe, in the fuzzy rule base, the validity of the above Lemma 1 still holds. Since the fuzzy variable Δe is generated inside the fuzzy controller, only the physical signal e is shown in all the block diagrams in this paper.
Fig. 9. Proposed triangular membership functions.
Fig. 10. NHS control system in Example 1.
Fig. 11. NHS in Fig. 10.
Theorem 3: The closed-loop system shown in Fig. 3 using
the fuzzy controller defined in Lemma 1 with the property of
{Δu = f(e) = e cot θ, 0 < θ < π/2}, will yield a
asymptoti-cal stable system, i.e., lim
t→∞e(t) = 0, in Fig. 3.
Proof: The result is quite obvious from Theorem 1 and
Lemma 1. Q.E.D.
The condition of f (e) = e cot(θ), 0 < θ < π/2 will be the guide to design the fuzzy rules using the configuration defined in Lemma 1.
Example 1: For the closed-loop system shown in the
follow-ing Fig. 10.
It is desired to control the following NHS by using the above fuzzy controller (Fig. 11).
The membership functions for e(t), Δe(t), and Δu(t) are defined by the above (1), (2) and Figs. 9 and 12.
Table I shows the fuzzy rule table, which consists of 32= 9 rules for the two-input FLC. There are, however,
only three major output membership functions for the conse-quent part.
The following Fig. 13 shows the simulation of Example 1 by letting
R(t) =
3− 0.3 ∗ e5t, if t≤ 0.6 sin(5t) + 2 cos(20t) + 0.01 sin(100t), if t > 0.6.
Fig. 12. Triangular membership functions for (a) e(t), (b) Δe(t), and (c) Δu(t). TABLE I
FUZZYRULETABLE FOR THETWO-INPUTFUZZYLOGICCONTROLLER
Fig. 13. Response with (df 2, df 0, df 1) = (−1.5, 0, 1.5).
Fig. 14. Tracking performance in Example 1.
The above complicated R(t) is to mimic the pedal signals by automobile drivers. In the first time, the (df2, df0, df1) = (−1.5, 0, 1.5), and the response is shown in Fig. 13.
Then, we let (df2, df0, df1) = (−0.1, 0, 0.1) again, and the response is shown in Fig. 14. It is obvious that excellent track-ing performance after changtrack-ing the values of (df2, df0, df1) from (−1.5, 0, 1.5) to (−0.1, 0, 0.1). is obtained by using the above stable fuzzy controller. And the curves of e and f (e) in Example 1 are shown in Fig. 15.
The condition of {ef(e) ≥ 0, f(0) = 0} is obviously true in Fig. 15, where (df2, df0, df1) = (−0.1, 0, 0.1). Through comparing the responses in Figs. 13 and 14, the different linguistic hedge dfi have the different control effects. Because
Fig. 15. Curves of e and Δu = f (e) in Example 1.
the dfi in the Δu will decide the performance, we need to tune the dfi to realize the FLC controller. Next, we will need a new back-propagation algorithm to fine tune dfi to complete the controller design so as to track the input signal in good performance. This new control strategy will be adopted to control real world device, i.e., the nonlinear hysteretic ET for modern automobiles.
VI. FINETUNING THECENTERS OFFUZZYOUTPUTMFS In this paper, the CAD is adopted to determine the defuzzifier output Δu[39]. Since Theorem 1 shows the existence of Δu of the output of the FLC to eliminate the nonlinear hysteretic phonemenon in Fig. 3. To realize a better Δu to achieve better tracking performance, we choose to fine tune the center of the output Membership functions (MFs) to minimize the errors by a sinusoidal training signal. The reason to choose a sinusoidal training signal to fine tune the output MFs is that the pedal signals by the drivers are the up-and-down signals, which is actually the distorted sinusoidal signal. The new back-propagation algorithm to tune the output MFs of the FLC can be proceeded as follows: di(t + 1) = di(t)− η ∂e2(t) ∂di = di(t)− 2η ∗ e(t) ∗ ∂e(t) ∂di (33) where e(t) is defined in (1) in Fig. 3, di(t) is the center of the ith output MF, and η is the learning rate. And we can have
∂e(t) ∂di =∂(R− Y ) ∂di = ∂(−Y ) ∂di =−∂ [N HS(u + Δu)] ∂di . (34) Assuming that we have the following symmetric NHS like Fig. 16.
When Δx > 0, we have the following three cases.
Case 1:−n1≤ x ≤ −n2, then from (34), we have
∂e(t) ∂di
= wi
wi
Fig. 16. NHS for fine tuning the centers of output MFs.
Therefore, we have the following equation from (33)
di(t + 1) = di(t)− 2η ∗ m n1 [x(t) + n1] wi wi .
Case 2:−n2≤ x ≤ n1, then from (34), we have
∂e(t) ∂di
= wi
wi
.
Therefore, we have the following equation from (33)
di(t + 1) = di(t)− 2η ∗ m(n2− n1) n1(n2+ n1) [x(t)− n1] wi wi .
Case 3: x≥ n1, or x≤ −n2then from (33) and (34), we have
di(t + 1) = di(t) due to
∂e(t) ∂di
= 0. When Δx < 0, we have the following three cases.
Case 4: n2≤ x ≤ n1, then from (34), we have
∂e(t) ∂di
= wi
wi
.
Therefore, we have the following equation from (33).
di(t + 1) = di(t)− 2η ∗ m n1 [x(t)− n1] wi wi .
Case 5:−n1≤ x ≤ n2, then from (34), we have
∂e(t) ∂di
= wi
wi
.
Therefore, we have the following equation from (33):
di(t + 1) = di(t)− 2η ∗ m(n2− n1) n1(n2+ n1) [x(t) + n1] wi wi ◦ .
Case 6: x≤ −n1, or x≥ n2, then from (33) and (34), we have
di(t + 1) = dit) due to
∂e(t) ∂di
= 0.
It is quite obvious that the update of the center of ith output MFs, i.e., di, for the above cases is quite obvious from the above
cases. The following Example 2 is illustrated to go through the sinusoidal training process and tested by the same pedal signal
R(t) adopted in Example 1.
Example 2: The closed-loop system is shown in Fig. 10.
It is desired to find the fuzzy controller to cancel the NHS effects as shown in Fig. 17, so that we can have a good tracking performance.
Fig. 17. NHS in Fig. 10.
The membership functions for e(t), Δe(t), and Δu(t) are defined the same as in Example 1, which is shown in Fig. 18.
In this example, we let (df 2, df 0, df 1) = (−1, 0, 1) initially. The control effect of this initial FLC is shown in Fig. 19, which shows bad tracking performance. By using the sinusoidal signal 3sin(7t) as the training signal to go through the above back-propagation tuning process with learning rate η = 0.00112, the final updated values of (df2, df0, df1) becomes (−0.13928, 0.011291, 0.15057). The converged processes of fine tuning using 3sin(7t) is shown in Figs. 20 and 21.
Fig. 21 shows excellent tracking performance using (df 2, df 0, df 1) = (−0.13928, 0.011291, 0.15057) with pedal signal as input signal. It is quite obvious that the tracking performance has been dramatically enhanced from Figs. 19–21.
VII. SYSTEMIDENTIFICATION OF NHS ELECTRONICTHROTTLE
In order to achieve the control of the nonlinear ET, we need to perform the system identification of the ET in open-loop mode. This can be done by using the dSpace MABX [31] under Simulink in Matlab. Fig. 22 shows the connection of PC (under Windows), MABX, and the nonlinear ET.
The basic theme is to generate a PWM signal with varying duty cycles as the input signal for the nonlinear ET. The turning angle of the nonlinear ET will be read back from its internal sensor and recorded in a log file. As shown in Fig. 23, we can gather the nonlinear hysteretic loop from the ET in Fig. 24.
One of the major factors of the hysteretic of ET is the return spring. The controller will need more energy to resist the force from the return spring during acceleration. However, the controller will spend less energy to release valve of ET during deceleration, since the return spring will automatically pull the valve back to its closed position. The other reason is due to the copper coils inside the BLDC. This open-loop control can be described by the following Fig. 24. When the nonlinear ET is from close to open, the controller receives the pedal signals and throttle signals. Then, the controller sends the consistent duty signal, which is PWM to the H-bridge circuit and drives the throttle to operate in the correct position. When the nonlinear ET is from open to close, the controller also does the same action as above but using different settings due to the hysteretic loop in ET as described above.
It is noted that although the hysteretic loop shown in Fig. 24 is different from that in Fig. 2, all the properties discussed above, such as Theorem 1 and Theorem 2, are still valid.
Fig. 18. Triangular membership functions for (a) e(t), (b) Δe(t), and (c) Δu(t).
Fig. 19. Bad tracking performance using initial (df2, df0, df1) = (−1, 0, +1).
Fig. 20. Convergence process of (df2, df0, df1) using training signal 3sin(7t).
Fig. 21. Excellent tracking performance using (df 2, df 0, df 1) = (−0.13928, 0.011291, 0.15057) with pedal signal as input signal.
Fig. 22. System identification of ET using MABX.
VIII. CLOSED-LOOPCONTROLUSING FUZZYLOGICCONTROLLER FORET
The following Fig. 25 shows the closed-loop control of the nonlinear hysteretic ET using FLC.
Fig. 23. Hysteretic loop of ET.
Fig. 24. Open-loop control for nonlinear ET.
Fig. 25. Closed-loop control nonlinear ET using FLC.
A controller compares the throttle signal (Y(t)) with the pedal signal (R(t)) and produces the control signal to minimize the error. The equations for e(k), Δe(k) are
e(k) = T (k)− P (k); Δe(k) = e(k) − e(k − 1)
where k is the sampling instant of the process. The variables
e(k), Δe(k) are the conditions monitored by the FLC. These
conditions are expressed in terms of linguistic variables as negative large (NL), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), positive large (PL). The five membership functions {NL, NM, ZE, PM, PL} of e and Δe are shown in Fig. 26(a) and (b). The seven membership functions {NL, NM, NS ZE, PS, PM, PL} of output Δu are shown in Fig. 26(c).
Fig. 26 shows the symmetrical triangular membership tions with 50% overlap with neighboring membership func-tions for FLC. It is obvious from Theorem 2 that this is the stable fuzzy controller which satisfies the constraint of
Fig. 26. Symmetrical triangular membership functions for: (a) e(k), (b) Δe(k), (c) Δu(k) of FLC.
TABLE II
RULEBASE FORFLCTOCONTROL THENONLINEARET
{ef(e) ≥ 0, f(0) = 0}. The control signal is in the fuzzified
form. To convert this into a crisp form, the center averaging defuzzification scheme is used. For this FLC, we have 72=
49 rules, which are shown in Table II. In order to regulate the output of FLC, i.e., Δu, to follow the trend of {Δu =
f (e) = e cot(θ), 0 < θ < π/2}, the membership functions of
Δu shown in Fig. 26(c) are updated several times to yield a satisfactory tracking performance.
For comparison purpose, the linearized transfer function of the ET is obtained from Matlab system identification toolbox as
G(s) = −6.112s
2− 21.75s − 3.168
s3+ 4.2392+ 9.412s− 2.322. (35)
Therefore, a linear PID controller can also be obtained as
C(s) = Kp+ KI/s + Kds
=− 0.8833 − 0.0506/s − 1.0235s. (36) The tracking performances will be compared in the following section.
Fig. 27 (a) ETC in laboratory. (b) Real-time ETC in Saab 9000.
Fig. 28. Tracking performance of PID controller.
IX. BENCHMARKEXPERIMENTS
Fig. 27(a) shows the ET and experiment plant adopted in our experiment. The real-time signal processing is carried out by an advanced microcontroller development board, in which the fuzzy controller can be easily coded in the form of fix point integer. Fig. 27(b) shows our ET and control circuits, which are set up on our experiment car. However, for the tracking performance of PID controller as shown in (36), we still need the expensive MABX to act as the real-time controller.
The real-time tracking is performed in this experiment using random pedal signals to simulate the real actions of automobile drivers. The tracking of the PID controller is shown in Fig. 28 with the red line representing the pedal signal and the blue
line representing the throttle signal.
In the processes of designing the fuzzy controllers, the following plots show three different parameter variations. The 3-D plots show different controller actions due to differ-ent fuzzy membership functions. The associated real-time re-sponses are also displayed.
Fig. 29. Three variations of fuzzy controller with their associated real-time responses.
It is obvious that the third fuzzy controller in Fig. 29 shows the best tracking result, although all the three fuzzy controllers show stable tracking results, as predicted from Theorem 1. The best tracking performance of FLC is shown in Fig. 30 with the red line representing the pedal signal and the blue line representing the throttle signal.
From the above real-time experimental results of PID and fuzzy controllers, we can get the comparative table between them in Table III.
It is obvious from Figs. 28 and 30 that the FLC can have a better tracking performance with very less cost than that of PID controller, which is designed from the expensive MABX.
TABLE III
COMPARATIVETABLEBETWEEN THEEXPERIMENT
RESULTS OFPIDANDFUZZYCONTROLLERS
Fig. 30. Excellent tracking performance of ETC using FLC. X. CONCLUSION
The intelligent control of industrial nonlinear ETs is designed and implemented in this paper. The experimental result shows that a stable fuzzy logic control can be realized to control the nonlinear hysteretic BLDC motor adopted in industrial ETs. The control strategy of a NHS is first investigated which can be used to systematically realize a stable FLC. The simulation of the stable FLC to control a NHS is first validated using computer simulation. The input signal is specially fabricated to mimic the pedal signals by automobile drivers. The successful results then carried over to the real-time implementation of FLC to control a real ET in Saab 95. The stable FLC is implemented using an advanced microcontroller development board at the minimum cost for real commercial applications. In order to compare the tracking results, a PID controller is also obtained using Matlab System Identification Toolbox and executed using dSpace MABX. It is obvious that the tracking results using FLC is much better than that using PID controller.
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Chi-Hsu Wang (M’92–SM’93–F’08) was born in Tainan, Taiwan, in 1954. He received the B.S. de-gree in control engineering from National Chiao Tung University, Hsinchu, Taiwan, the M.S. degree in computer science from the National Tsing Hua University, Hsinchu, Taiwan, and the Ph.D. degree in electrical and computer engineering from the Uni-versity of Wisconsin, Madison, in 1976, 1978, and 1986, respectively.
He was appointed Associate Professor in 1986, and Professor in 1990, in the Department of Elec-trical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. He is currently a Professor in the Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan. His current research interests and publications are in the areas of digital control, fuzzy neural network, intelligent control, adaptive control, and robotics.
Dr. Wang is currently serving as Associate Editor of IEEE TRANSACTIONS ONSYSTEMS, MAN,ANDCYBERNETICS, PARTB: CYBERNETICS/ and as a Webmaster of IEEE Systems, Man, and Cybernetics Society.
De-Yu Huang (M’10) was born in Taiwan, in 1976. He received the B.S. degree in electrical engineering from Tatung University, Taipei, Taiwan, in 1998, and received the M.S. degree in electrical control engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2004. He is currently work-ing toward the Ph.D. degree in electrical control engineering from National Chiao Tung University, Hsinchu, Taiwan.
His current research interests include intelligent control, fuzzy logic systems, and automobile elec-tronic technologies.
