Student : YI-AN,CHEN 4992C085 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 22, NO. 2, MARCH 2014

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.

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. VII. CONCLUSION