Supervisory Multiple-Model Approach to Multivariable
Lambda and Torque Control of SI Engines
Pawel Majecki*, Hossein Javaherian** and Michael J. Grimble***
*Industrial Systems and Control Ltd, Glasgow, United Kingdom (e-mail: )
**General Motors R&D Center, Warren, MI, USA (e-mail: )
***Industrial Control Centre, University of Strathclyde, Glasgow, United Kingdom
This research was supported by General Motors Corporation
Abstract: The problem of simultaneous air-fuel ratio regulation and torque tracking in a spark ignition engine with electronic throttle control is considered. The proposed methodology involves the use of a set of piecewise-affine models to represent the nonlinear engine dynamics. These models are the basis of a supervisory multiple-model control scheme, which, in its simplest form, consists in switching among the predefined bank of controllers. In the following a monitoring signal generator is driven by a bank of observers, and a supervisor ensures the robustness of the switching scheme. An optimal linear-quadratic cost function enables the trade-off between emissions performance and drivability to be adjusted. Simulation results using the data obtained from a vehicle with a 5.3L V8 engine on Federal Test Procedure (FTP) driving cycles are presented, with a nonlinear regression model of the engine identified from the FTP data. The results indicate that both tight lambda regulation and fast torque tracking are possible using the proposed designs.
Keywords: powertrain systems, air-fuel ratio, multiple models, optimal control
1. INTRODUCTION
Air-fuel ratio and torque control in spark ignition engines has been the subject of investigations in numerous publications (Moskwa 1988, Hendricks and Sorenson 1990, Dutka 2005). These authors mostly used physical mean-value models to describe engine dynamics. In this paper, we employ a simpler multivariable nonlinear regression model that was identified from driving cycle data collected from a vehicle with a 5.3L V8 engine. The model is assumed to have two outputs: torque (TRQ) and air-fuel ratio or lambda (l) and three inputs: throttle position (TPS), fuel pulse width (FPW) and engine speed (RPM). Other measured engine variables are considered internal states of the system.
In a conventional engine control scheme, the throttle plate is directly linked to the acceleration pedal, and as a result the throttle position and hence the torque produced depends on the driver. The fuel flow is adjusted by a single-loop feedforward plus feedback controller, to maintain a steady stoichiometric ratio of the air-fuel mixture. The controller configuration utilized in this paper uses the electronic throttle control (ETC) which effectively decouples the throttle from the pedal (drive-by-wire). That is, the pedal position, together with other measurements and design specifications, are used to determine the optimal torque set-point via a nonlinear mapping, and the angle set-point for the throttle servo is manipulated by the electronic control unit. This arrangement enables multivariable control to be used.
The control methodology used in the following is presented in Section 2 and is based on the concept of multiple model switching control (Narendra and Balakrishnan 1994, Giovanini et al. 2006), and in its simplest form involves supervisory switching among one of a finite number of controllers. The multiple piecewise affine (PWA) models were determined from the nonlinear model, for a prespecified set of operating points. The scheduling variables were chosen as TPS and RPM, and for each (TPS, RPM) pair the FPW value was computed that would result in the stoichiometric l in the steady state, as well as the corresponding steady-state torque value. The determination of the models is presented in Section 3.
The supervisory switching scheme provides a general framework, which is independent of the actual control algorithm. In this work, the individual controllers were designed as linear-quadratic (LQ) regulators, allowing a trade-off between lambda regulation and torque tracking performance to be defined. Two switching schemes were implemented, one involving simple switching based on the “shortest distance” to the model, and the other involving a bank of observers and a monitoring signal generator. The details of the control design and implementation are presented in Section 4. Finally, simulation results are presented in Section 5 and conclusions are summarized in Section 6.
2. CONTROL METHODOLOGY
A general block diagram of the multiple-model adaptive switching scheme is presented in Fig. 1. The engine is modelled by a black-box nonlinear structure estimated from the actual vehicle data obtained on FTP driving cycles, as discussed in Section 3. The following components of the scheme can be listed:
§ A finite number of estimators designed for a grid or set of (TPSi , RPMi) pairs, corresponding to all the linear multivariable models for a given model partitioning
§ A monitoring signal generator, computing the weighted average of past errors
§ A supervisor, switching on the controller corresponding to the smallest monitoring signal
These components are described in more detail below.
Fig. 1. General block diagram of the supervisory control scheme
2.1 Banks of observers and controllers
The output of each estimator yj(t) (a vector containing torque and lambda output estimates) is compared with the measured output y(t) to form j estimation errors ej(t) = y(t) − yj(t), whose norms measure the difference in the behaviour between the j models and the system.
The estimators, monitoring signal generator and the switching logic block generate a vector of switching variables S(t) at every sample instant, independently of the controller design. Then, the control law K(t) is designed, which in this case is obtained by solving an LQ optimization problem. The controller design and synthesis are normally performed off-line for each region using linear models. The resulting controller gains are stored and switched in at appropriate times by examining the states of the switching variables. As the engine operating points change the control law will therefore change accordingly. As a result, since the controller is based on a set of known models, the adaptation should be more predictable and simpler to validate.
2.2 Monitoring signal generation, switching logic and dwell time
The monitoring signal generator is a dynamical system which generates monitoring signals mj(t). These are suitably defined integral norms of the estimation errors ej(t):
1)
where b, H and l Î [0, 1) are design parameters affecting the estimation sensitivity and hence the switching frequency. The size of these monitoring signals indicates which of the multi-estimators is “closest” to the true plant.
These signals are used to choose a controller to place in the feedback loop that is designed using the model from an estimator which has the smallest monitoring (model error) signal. The time history of the monitoring signal mj(t) can be viewed as a measure of the similarity of the jth nominal model to the actual system and drives the decision process of the supervisory control S. From time to time, S searches for the monitoring signal mj(t) with the smallest value, sets S(t) equal to the corresponding index, which switches in the corresponding controller, and maintains S(t) fixed at the value until a new search is completed and a new minimal value is found. A minimum time Tmin (“dwell time”) can be set to elapse between the subsequent switches, thus avoiding undesirable rapid switching of the controllers at regional boundaries.
3. MULTIPLE MODELS
The key component of the multiple-model control scheme is a set of linear (or piecewise affine – PWA) models covering the operating space. In this work, the modelling approach was to first identify a nonlinear regression model of the engine from the measured driving cycle data, and then linearise it around the pre-specified operating points.
3.1 NARX model identification
The NARX model of the engine was identified from the FTP driving cycle data collected for a vehicle with a 5.3L V8 engine. The data were used to identify a multi-input, multi-output (MIMO) NARX model. A related approach was to consider a diagonal model constructed by combining two already available multi-input, single output (MISO) models, however such a model might neglect important interactions between the outputs – hence, a fully multivariable model structure was adopted. In this section, we briefly present the identification methodology.
Fig. 2. Generic system model and NARX structure
Consider a model with a control input vector ut, a disturbance input vector dt, and an output vector yt, as shown in Fig. 2. The objective is to fit the assumed NARX model structure to the measurement data. The NARX model structure includes linear and quadratic inputs, as well as a constant input, followed by a Linear Time-Invariant (LTI) model.
The NARX model may therefore be represented in the state-space form as:
2)
where, for the engine model:
and .
The measurements of brake (load) torque, which approximates the produced (unmeasured) torque in the steady-state sense, were used to identify the above model.
The MIMO transfer-function model was first identified from the first half of the data (identification set), and since, in effect, we deal with a linear model (with “nonlinear” inputs), the simple least squares algorithm could be used for estimating the parameters. The resulting model was then converted to the state-space form.
The system output time delays for torque and lambda were obtained using the correlation techniques. The time delays kTRQ and kl were chosen to minimize the difference between the data and model output, in the mean square error sense, and their optimal values were found as and events. The input delay on FPW was on the other hand assumed to be incorporated into the model. The resulting model structure is shown in Fig. 3.
Fig. 3. NARX model structure for identification (explicit time delays shown)
Table 1 Assessment of NARX models of increasing order
Model order / ISE (torque) / ISE (lambda)2 / 10.65×106 / 21.96
5 / 8.75×106 / 12.80
8 / 8.40×106 / 9.73
15 / 8.53×106 / 7.02
The model validation, in terms of the integral square of the prediction error for increasing model orders, is shown in Table 1. Based on the observed error, an 8th order model was finally selected as giving the best trade-off between model accuracy and complexity. The corresponding model validation plots for the second half of the dataset (validation set) are shown in Fig. 4.
Fig. 4. NARX model validation (8th order model): measurements (dashed) and model outputs (solid thick)
3.2 Region partitioning – general considerations
The multiple-model algorithms have the following common characteristics:
- A set of m multi-controllers or dynamic compensators must be designed off-line.
- A set of m Kalman filters or multi-estimators (observers) must be found.
- A switching/blending process by which the actual (global) control is generated must be implemented.
Clearly, the complexity of the switched system will depend on the number m of models that must be implemented. This set of models is called a “cover set” and was introduced in (Anderson et al. 2000). The number and distribution of these linear models are important contributors to the performance of the control scheme. Ideally, m should be as small as possible. However, if m is too small the performance of the system may be inadequate. On the other hand, if m is very large, one may reach the point of diminishing returns as far as the performance improvement is concerned. The increased complexity and switching frequency are also limiting factors.
Standard switching control schemes are usually based on the certainty equivalence philosophy. At each switching time, the supervisor selects the candidate controller that is best tuned to the current estimated system model. The compromise between robustness and performance is made off-line when the cover set is selected and the controllers designed.
3.3 Selection of PWA models
There are a number of possible ways of defining the operating points and hence the PWA model distribution. We have chosen to use the (TPS×RPM) grid and compute the FPW values, based on the model, such that at the steady state the lambda value equals unity. The motivation is to use the smallest possible number of scheduling variables, in order to reduce the complexity. The conditions for which lambda is close to unity are also likely to occur since lambda regulation is a control objective.
Some insight about the signal distribution can be obtained from the dataset by drawing histogram plots. The (TPS×RPM) histogram is shown in Fig. 5 and can be used as a basis for the choice of the regions. That is, the peaks on the histogram plot, corresponding to the most likely combinations of TPS and RPM (according to the dataset), were selected to define the operating points. The selected four trim conditions, including the corresponding FPW values, are given in Table 2.
Fig. 5. (TPS×RPM) histogram plot for the dataset
Table 2 Operating points
Oper. point / TPS [deg] / FPW [ms] / N [rpm]OP_1 / 4 / 3.46 / 500
OP_2 / 7 / 3.26 / 1000
OP_3 / 12 / 4.47 / 1250
OP_4 / 20 / 6.68 / 1700
3.4 PWA model determination
The PWA models were obtained from the NARX model and were assessed based on the dataset, using the difference between the output measurement and the multiple model PWA output as the optimization criterion. The approach involved linearising the NARX model around a number of points, defined by (TPS,RPM) pairs. The ith PWA model has the form:
3)
with .
Having specified the operating points (the previous section), it is necessary to compute the piecewise-affine models corresponding to those points/regions. This can be done in the following steps:
1. For the given values of TPS and RPM, compute the FPW input such that l = 1 under steady state conditions. With the NARX model structure , the solution to this problem is equivalent to solving a quadratic equation. The decision must be made as to which of the two roots is to be selected, and the natural choice is the smaller root so as to minimize the fuel consumption. Of course, the solution must also be feasible. This is normally the case if the trim conditions are chosen as described in Section 3.3.