Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 989674, 13 pages

http://dx.doi.org/10.1155/2015/989674

## Robust Control of the Air to Fuel Ratio in Spark Ignition Engines with Delayed Measurements from a UEGO Sensor

^{1}Department of Electronic Engineering, CUCEI UDG, 44430 Guadalajara, JAL, Mexico^{2}CONACYT, CINVESTAV del IPN, 45019 Zapopan, JAL, Mexico^{3}Department of Electronic Engineering, CINVESTAV del IPN, 45019 Zapopan, JAL, Mexico

Received 17 July 2015; Revised 8 October 2015; Accepted 8 October 2015

Academic Editor: Zhike Peng

Copyright © 2015 Javier Espinoza-Jurado et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A precise control of the normalized air to fuel ratio in spark ignition engines is an essential task. To achieve this goal, in this work we take into consideration the time delay measurement presented by the universal exhaust gas oxygen sensor along with uncertainties in the volumetric efficiency. For that purpose, observers are designed by means of a super-twisting sliding mode estimation scheme. Also two control schemes based on a general nonlinear model and a similar nonlinear affine representation for the dynamics of the normalized air to fuel ratio were designed in this work by using the super-twisting sliding mode methodology. Such dynamics depends on the control input, that is, the injected fuel mass flow, its time derivative, and its reciprocal. The two latter terms are estimated by means of a robust sliding mode differentiator. The observers and controllers are designed based on an isothermal mean value engine model. Numeric and hardware in the loop simulations were carried out with such model, where parameters were taken from a real engine. The obtained results show a good output tracking and rejection of disturbances when the engine is closed loop with proposed control methods.

#### 1. Introduction

The air to fuel ratio (AFR) control [1, 2] is one of the most important control problems for conventional gasoline engines. The performance of the AFR control can strongly impact key aspects in the SI engine, such as emissions, fuel economy, and output torque [3, 4]. The objective of the AFR control is to keep a stoichiometric value ( for gasoline fuel) or the normalized factor () equal to in the air to fuel mixture in the presence of environmental perturbations and high nonlinear dynamics involved in the combustion process. One important aspect to ensure a good AFR control performance is the determination of how much air is aspirated by the cylinders in order to inject the exact amount of fuel in each stroke. However, because of the short cycle time available and the flow restrictions due to the purity of the air, intake manifold, and intake valves, less air enters into the cylinder than the circulating air from the throttle valve. The relationship between the cylinder geometric volume and the actual volume or air aspirated into the cylinder is defined as volumetric efficiency . Another issue to consider is the time delay in the factor measurement provided by the Universal Exhaust Gas Oxygen (UEGO) sensor [5, 6], which is basically the time between the fuel injection and the burned gases reaching the UEGO sensor. But, in particular, this time delay is due to the transient response of the sensor, to the delay between the fuel injection and the time valve opening, to combustion and to the intake and exhaust valves opening, and to the transportation of gases. Currently, there are several works related to the control of the AFR for SI engines, where a variety of well-known established control designs are applied. In [7], an -infinity control is designed based on a linearized model of the engine where the high order controller is only valid around an operating point. A classical sliding mode technique and a global linearized control strategy are compared in [8] without considering time delays measurements from sensor; although the classical sliding mode controller showed the best results with parameter uncertainties, it is well known that it introduces the undesirable chattering phenomenon to systems in closed loop [9]. An artificial neural network design is presented in [10] where the engine is initially in open loop for about 2 s while the neural network adapts its weights, thus generating large transient peaks in the rate of injected fuel each time the process is initiated. A fuzzy logic based PID controller is presented in [11] where measurements are realized with a binary sensor, but omitting its intrinsic time delays. In [12], a super-twisting algorithm is designed where again the delayed measurements are omitted and the complexity of the dynamics is not revealed. Nevertheless, the super-twisting algorithm reduces the chattering phenomenon; hence, it is gaining popularity over classical sliding mode designs [13].

It is clear from the revision of mentioned works that the delay of measurements is not taken into consideration. Also the estimation of the volumetric efficiency is not given its corresponding importance. Moreover, in such works, the dynamics is not explicitly revealed and, as it will be demonstrated in this work, this dynamics is complex since it depends on the time derivative of the control input. If a more accurate regulation of the normalized air to fuel ratio () is desired, then the delays, the right knowledge of , and the explicit dynamics for must be taken into consideration in the control design.

Hence, in this work, we propose the following enhancements to the regulation of :(i)To design two higher order sliding mode (HOSM) controllers, based on the super-twisting algorithm and on the explicit dynamics of .(ii)To deal with delays in the measurements provided by the UEGO sensor by designing a super-twisting based observer for .(iii)To design a super-twisting observer for the nonmeasurable parameter .

For these purposes, the controller and the observer designs are based on an isothermal mean value engine model (MVEM) developed in the works by [14, 15]. This model is well accepted as still it is reported in the recent literature [16, 17]. It is well known that this model is control oriented and so neglects discrete cycles of the engine and assumes that all processes and effects are spread out over the engine cycle. Moreover, the time delay model for the UEGO sensor is approximated through a third order Padé approximation [18] of the frequency domain function (where is the time delay and is the Laplace variable) and then expressed in a state space representation. Numeric and real-time hardware in the loop (HIL) simulations are carried out for the validation of proposed control methods.

The remainder of this work is organized as follows. Section 2 reviews the mean value engine model and the time delay of the UEGO sensor. Section 3 deals with the controller and observer designs for the unity tracking of the AFR ratio. A simulation study is carried out in Section 4, and finally some comments conclude the work in Section 5.

#### 2. Mean Value Engine Model for SI Engine

This section describes a MVEM for a SI engine with an electronic fuel injection system. The MVEM of the SI engine primarily consists of 3 subsystems as explained in [14, 19, 20]:(i)The intake manifold filling dynamics.(ii)The fuel mass flow rate.(iii)The crank shaft speed.

##### 2.1. The Intake Manifold Filling Dynamics

The intake manifold filling dynamics is based on an isothermal model, where the temperature exchange between the ambient temperature and the intake manifold temperature occurs slowly; therefore, both temperatures are assumed to be the same. The intake manifold filling dynamics is separated into three equations: () the intake manifold pressure, () the throttle air mass flow, and () the intake port air mass flow.

###### 2.1.1. The Intake Manifold Pressure

The intake manifold pressure is determined by two parts: the air mass flow to the throttle valve and the air mass flow to the intake valve . Then, the manifold intake pressure is expressed as follows [14]:where is the ideal gas constant, is the air temperature in the manifold, and is the intake manifold volume.

###### 2.1.2. Throttle Air Mass Flow

This part of the model is based on the isentropic flow equation for a converging-diverging nozzle [14]: where and are constant functions, is the ambient pressure, is the angle of the throttle plate, and is the ratio of the throttle throat diameter to the throttle plate shaft diameter where is the close angle throttle plate. Function is useful only when the throttle plate has a circular shape. In other cases, another equation that can appropriately describe it must be found. Expression is the isentropic flow where and is the critical pressure (turbulent flow).

###### 2.1.3. Intake Port Air Mass Flow

The air mass flow at the intake port can be obtained from a speed density equation [14]:where is the ambient temperature, is the engine displacement, and is the volumetric efficiency. The volumetric efficiency behavior can be modeled in polynomial form taken from [14]:

##### 2.2. The Fuel Mass Flow Rate

According to the experiments reported by [20], the equations that describe the fuel mass flow rate into the cylinder are as follows:where is the mass of the fuel film adhered to the manifold wall, is the fuel flow rate from the injector, is the fraction of injected fuel that remains as fuel film, is the fuel evaporation time constant, and is the portion of fuel that enters into the cylinder valve. The fraction in the fuel film is approximated by [20]: where is the maximum air mass flow for the engine.

##### 2.3. The Crankshaft Speed

The crankshaft speed is derived based on the conservation of the rotational energy on the crankshaft [20]: where is the crankshaft speed, is the inertial moment of the crankshaft, , , and are the power losses by friction, pumping, and load, respectively, is the fuel burn value, is the thermal efficiency, and is the mass fuel rate into the cylinder with a torque time delay . The friction and pumping losses in the engine can be expressed as polynomials of the crankshaft speed and the intake manifold pressure:and where is the load factor. The thermal efficiency can be expressed as the effect of four individual factors:where with as the maximum brake torque and , , , and as constants. Figure 1 shows a block diagram of the MVEM.