IEEE TRANSACTIONS ON CONTROL SYSTEMS
TECHNOLOGY, VOL. 22, NO. 2, MARCH 2014
Student : YI-AN,CHEN
4992C085
I. INTRODUCTION
PNEUMATIC actuators are commonly used in high power
positioning applications in automobile systems, such as
turbocharger control (waste-gate and VGT), exhaust gas
recirculation(EGR), and variable intake manifolds [1].
Although not as precise as electric motors [2], [3], they have the
advantages of high power-to-size ratio, low maintenance cost,
light weight, and the availability of pneumatic sources in the
engine[4], [5]. These advantages come at the price of more
friction and nonlinearity due to air compressibility and
aerodynamic forces etc. In presence of these traditional
drawbacks, precise and robust control strategies are required to
use pneumatic actuators effectively.
II. SYSTEM MODELING
The pneumatic actuator regulates the quantity of exhaust
gas entering into the turbine by adjusting its vanes.
This system (Fig. 1) consists of two parts, an EPC and a pneumatic
actuator. The EPC regulates the air mass-flow in the pneumatic
actuator, which varies the pressure in the actuator and produces a
linear motion in the diaphragm. This is converted into rotation via a
unison ring that actuates the VGT vanes. Detailed working and
modeling of this system have been published in [19] and [20]. In this
section, we will briefly review the results of these articles and
present complete model.
A. Actuator Mechanics
III. SLIDING MODE ESTIMATION
AND OBSERVATION
While commercial actuators are equipped with position
sensors only, velocity can be estimated by using robust differentiators
or finite time observers. In this paper, we have
estimated velocity from measured position using the robust
fixed time convergent differentiator. In this way, the velocity
is assumed to be known after a certain interval, and is used
for the pressure and friction state observer design.
A. Robust Fixed Time Differentiator
Real-time differentiation is an obstacle in control implementation.
Angulo et al. [30] have developed a uniform fixed time
differentiator that converges in two phases. The first phase
guarantees uniform convergence of the derivative estimate
from any initial condition, to a neighborhood of the real value
in a fixed maximum time duration tu. In the second phase,
the differentiator takes the form of the robust differentiator
[31], which then guarantees finite time convergence from the
neighborhood, exactly to zero, in finite time t f. After the
period tτ = tu + t f , the derivative estimate can be considered
as the real derivative for all t > tτ . For a given signal x1
whose derivative x2 is bounded by the Lipschitz constant L,
the first-order differentiator is given as for 0 ≤ t ≤ tu
B. State Observers
To design the pressure observer, we first define the error of
function f (u, x4) for observed and actual pressure, given as
follows:
f. = a1 + a2 x4 + xˆ4 ε4 = f˜ε4 (8)
where f. = f (u, x4)− f (u, xˆ4). From (3), f˜ < 0 for all values
of input u. The observer is designed using sliding mode,
IV. BACKSTEPPING CONTROL
DESIGN
The controller has been developed using backstepping
method, which is a recursive procedure based on Lyapunov’s
stability theory [34]. In this method, system states
are chosen as virtual inputs to stabilize the corresponding
subsystems.
Step 1: According to the control objective given in
Section II, the following error function is chosen:
s1 = λe1 + ˙e1 (19)
where e1 = x1 − xref is the tracking error. The term λ is a
positive constant and xref is the reference signal to be tracked.
The error function dynamics are
˙s1 = ˙x2 − [ ¨ xref − λ (x2 − ˙xref )] (20)
V. LYAPUNOV ANALYSIS
In this section, we
will demonstrate
the exponential
convergence
of the closed-loop
controller–
observer system.
Let us
recall VO, the
Lyaponov function
associated with
the observer
as given in (17),
and its derivative
from (18)
Fig. 8. Positioning at Ptrb = 0, 1500, and 3000 mbar.
Remark 2: When implemented on a discrete-time control
system, the velocity estimation error, due to sampling, will
be proportional to the sampling period T [31]. Therefore,
according to Levant, the term x2 in (16) will be replaced by
x2 + O(T ). In this case, the observation error (ε3, ε4) will
not converge to zero exactly, but to a neighborhood of zero
that is proportional to the sampling time. This is because the
derivative of the Lyapunov function (18) will take the form
VI. EXPERIMENTAL RESULTS
The experiments were performed on an industrial turbocharger
(GTB1244VZ) mounted on the test bench of a
commercial diesel engine DV6TED4 (shown in Fig. 5). The
maximum displacement of the VGT actuator is 16.4 mm,
which allows movement of the vanes in the range of 48°.
Its integrated potentiometric position sensor provides measurements
with an accuracy of 0.1 mm. The test bench
is equipped with a pressure sensor for measuring actuator
pressure and a force sensor on the actuator shaft for measuring
the total resistive force. These measurements were used for
validation of the observer. The low-pass bandwidth of the
latter does not permit us to capture the pulsating effects of
exhaust gases, therefore the mean effect of the aerodynamic
force was obtained. The controller output was calculated
using the observer values, whereas the measured values of
pressure and force were used for validating the observers later
on. National Instruments CompactRIO was used to implement
the controller and to record data and measurements.
The controller parameters were tuned to obtain the best tradeoff
between response time and saturation limits, such that for a
step reference of maximum possible displacement (16.4 mm)
at 3000 mbar turbine pressure, the controller output would
not exceed the voltage limit of 12 V (i.e., 100% PWM). The
sampling frequency was fixed at 1 kHz, which is the nominal
frequency used in commercial automobile computers (ECUs).
The exhaust air pressures (turbine inlet pressures) at which
the controller was evaluated were 0, 1500, and 3000 mbar,
resulting in three different conditions of the aerodynamics
force.
The estimated resistive force was validated, as shown in
Fig. 6. The force was generated by running the engine at
140 Nm load torque and varying the turbine inlet pressure,
VII. CONCLUSION
In this paper, an adaptive output feedback controller was
presented for controlling the electropneumatic actuator of a
diesel engine VGT. The controller was developed using backstepping
method. Friction and aerodynamic force were considered
and modeled as a composite resistive force by parameterizing
the aerodynamic force as a function of turbine inlet pressure.
Both forces were then combined through a modification
of the LuGre model. The states that are unavailable for measurement
were observed using sliding mode observers. Lyapunov
analysis showed that the complete closed-loop system
i.e., the system states, controller, and observers converge exponentially.
Experimental results showed the effectiveness of this
controller.
VII. CONCLUSION
quantizing errors introduced by the bus limitations of the
microprocessors may induce observation and estimation errors.
In future works, we aim to integrate dynamic adaptive laws in
the controller to make it robust against modeling uncertainty
and parametric drifts. Implementation issues in relation with
commercial ECUs, such as memory and calculation power
requirements will also be addressed in detail.