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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: p.majecki@strath.ac.uk)

**General Motors R&D Center, Warren, MI, USA (e-mail: hossein.javaherian@)

***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 (λ) 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 λ 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.

[pic]

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 (j(t). These are suitably defined integral norms of the estimation errors ej(t):

[pic] (1)

where (, H and ( ( [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 (j(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((j(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.

[pic]

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:

[pic] (2)

where, for the engine model:

[pic]and [pic].

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 kλ 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 [pic] and [pic] 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.

[pic]

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.

[pic]

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:

1. A set of m multi-controllers or dynamic compensators must be designed off-line.

2. A set of m Kalman filters or multi-estimators (observers) must be found.

3. 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.

[pic]

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:

[pic] (3)

with [pic].

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 λ = 1 under steady state conditions. With the NARX model structure (2), 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.

2. Linearize the NARX model around the operating points specified by the input triples (TPS0, FPW0, RPM0). This can be easily executed by exploiting the special structure of the NARX model:

[pic] (4)

The linearised model follows by taking the derivatives:

[pic] (5)

Note: the eigenvalues (poles) of the linearised system remain the same as for the LTI part of the NARX model. It is only the system zeros that change their positions.

3. Construct the piecewise-affine models. For the ith operating point, the linear model for the deviations around the operating point is obtained from (5) as:

[pic] (6)

However, since [pic], [pic], (6) can be rewritten as

[pic] (7)

which is the required piecewise-affine system description. The states of the PWA model are equivalent to those of the original NARX model, and in particular the initial conditions can be set to the same values.

4. CONTROL DESIGN

The approximation of the system with a set of linear/PWA models is the basis of the control method. For this purpose, it is required to select/estimate the closest model at any given moment and design the corresponding controllers and the switching mechanism. Thus the bank of observers and controllers is constructed offline.

4.1 Bank of observers

Each model has a Kalman filter associated with it, which will produce the optimal estimates of the outputs. These estimates are then compared with the output measurements and a set of model errors is produced. The scheme is shown in Fig. 6, where the output delays were also taken into account. In the actual implementation, an array of LTI models was used for the bank of observers, with common initial conditions. Note that the constant input for the NARX model has a different meaning to that of PWA models.

If each local estimated state is multiplied by the associated local control gain Ki, this results in the multiple model control laws. The global control applied to the plant may be defined using a probabilistic weighting Pi of the local control:

[pic] (8)

[pic]

Fig. 6. Bank of multi-observers

Under suitable assumptions (stationarity and ergodicity), this architecture is essentially identical to that of the sum of Gaussians estimators used in nonlinear filtering (Anderson and Moore 1979).

Note: a simple switching version of the multiple model control can be implemented if the local control with the largest posterior probability is used as the global control.

4.2 LQ control design

The design of LQ controllers for PWA models is presented in this section. Consider the ith PWA system, which can be represented in the state-space form as:

[pic] (9)

The controller is designed to minimize the following quadratic cost function:

[pic] (10)

where et = rt – yt is the control error and rt is the reference signal.

By defining rt, and dt as additional states, and dropping the superscript i, the following system model is obtained:

[pic] (11)

In order to achieve offset-free tracking, an integrator needs to be added to the system and the state vector augmented. The cost will therefore include the control increment Δut rather than ut. Additionally, to account for plant-model mismatch, a “robustness” signal needs to be modelled, which is defined as the output model error [pic]. This effectively introduces feedback to the controller and if the above output disturbance is modelled with an integrator, then the steady-state offset due to mismatch is rejected asymptotically.

With the above modifications the system equations become:

[pic]

The overall system model can then be represented in vector-matrix form as:

[pic] (12)

with [pic], which leads to the equivalent LQ criterion:

[pic] (13)

with [pic].

This optimization problem involves the solution of a Riccati equation and can be solved using standard tools such as the dlqr routine in Matlab. The general structure of the controller is shown in Fig. 7.

[pic]

Fig. 7. Switched LQ state-feedback controller

Two switching methods have been considered, which are described in the following subsections.

4.3 Simple switching solution

A simple possible switching solution utilizes the set of trim conditions for each model. That is, at each time t, the closest (in terms of Euclidian distance) trim conditions are determined for the current pair (TPSt-1, RPMt), and the appropriate controller gain is switched on, as well as the corresponding state estimator.

To prevent fast switching at region boundaries, two separate mechanisms were put in place. First, the TPS and RPM scheduling signals were passed through the small-signal deadzone operators, making the region selection relatively insensitive to small fluctuations in these signals. In addition, a dwell time block was implemented as described in Section 2.2. The implementation of the dwell time using triggered integrator blocks is shown in Fig. 8.

Note: the two above mechanisms perform a similar function and both need not be active at the same time. Either can be disabled by setting the deadzone width to 0 or the dwell time to 1 event.

[pic]

Fig. 8. Dwell time implementation

The gains Ki are pre-computed for each operating point and stored in an array. In the controller implementation, the closest operating point is first found (based on the previous TPS and the current RPM input), and the corresponding state feedback gain is switched on. The control scheme is shown in Fig. 9.

[pic]

Fig. 9. Control scheme with simple switching

4.4 Supervisory scheme

The supervisory scheme consists of the following subsystems:

▪ bank of observers

▪ monitoring signal generator

▪ switching logic

▪ bank of LQ controller gains

The supervisor selects one of the controllers as active, while the others remain in the stand-by mode. To facilitate bumpless transfer during switching, the controllers have been defined to have identical structure and order, and hence they share a common set of states, with different gains. In the basic case, the supervisor output is the model index corresponding to the smallest monitoring signal, which switches in the corresponding observer and controller gain (the dwell time can also be implemented as in the above simple switching scheme).

In this work, a smooth transition between regions was obtained by interpolating between p closest models, with the scaling factors in Equation (8) determined by the size of the monitoring signals. In the simulations, the parameter p was set to 3.

[pic]

Fig. 10. Control scheme with the supervisory control

The detailed block diagram of the control structure is shown in Fig. 10. The operator minp() returns the indices of the p smallest monitoring signals, which are then used to compute the interpolated state feedback gain, according to:

[pic] (14)

where the scaling factors are inversely proportional to the monitoring signal magnitudes. A similar interpolated scheduling is also performed by the observer block.

5. SIMULATION RESULTS

In this section we present the simulation results and an assessment of the multiple-model based controllers for the vehicle with a 5.3L V8 engine for the FTP driving cycle data. As mentioned before, the torque reference for the supervisory controller is determined by a nonlinear mapping from the accelerator pedal position. However, for simulation purposes, it is taken as the filtered torque measurement from the dataset, as is the RPM signal (which is normally measured in a vehicle and therefore available).

The multiple PWA models were extracted from the NARX model based on the trim conditions given in Table 2, and the LQ control weights in (10) were parameterized as:

[pic]

where qλ, rTPS and rFPW are the tuning parameters. Following some trial and error tuning, they were eventually selected with a slight emphasis on lambda regulatory performance as: [pic], [pic] and [pic].

The following cases have been considered:

▪ Fixed LQ controllers based on each individual model

▪ Simple switching (Section 4.3)

▪ Supervisory scheme (Section 4.4).

The parameters β used in the supervisory scheme to generate the scalar monitoring signal in Equation (1) are usually chosen to normalize the observation errors for all the outputs. In this case, they were selected as [pic] and [pic], with the horizon H = 0. As indicated in Section 4.4, the controller gain for this scheme was computed by interpolating between the three closest models (in terms of the monitoring signal size).

The input/output plots for the simple switching and supervisory scheme are shown in Figs. 11 and 12 (only a fragment is shown for better clarity). In addition, the performance measures in terms of the integral square error (computed for the whole dataset) for both lambda and torque output have been collected in Table 3.

Generally speaking, the torque tracking performance is acceptable and comparable for both the simple switching and supervisory control scheme, and even for the individual LQ controllers. More emphasis in the design was devoted to lambda performance, and here the supervisory scheme provides tighter regulation. This can be observed from reduced peaks on the graph, as well as from a smaller overall ISE benchmark figure. This is of course at the expense of a more complex and computationally demanding algorithm. The basic state-feedback structure of the controller is however the same in both cases.

[pic]

Fig. 11. Torque and TPS: Simple switching (dashed) and Supervisory Scheme (solid)

Table 3 Simulation results in terms of Integral Square Error for different control schemes

|Control scheme |ISE (torque) |ISE (lambda) |

|LQ / PWA model OP_1 |2.39×107 |0.397 |

|LQ / PWA model OP_2 |2.60×107 |0.242 |

|LQ / PWA model OP_3 |2.29×107 |0.263 |

|LQ / PWA model OP_4 |1.45×107 |1.052 |

|Simple switching |2.30×107 |0.163 |

|Supervisory scheme |2.32×107 |0.138 |

[pic]

Fig. 12. Lambda and FPW: Simple switching (dashed) and Supervisory Scheme (solid)

6. CONCLUSIONS

The simulation results indicate that the methods considered have been effective in the control of a nonlinear MIMO model of the engine. Excellent control performance, as measured by the lambda regulation and torque tracking, was achieved for the powertrain. The results for both simple and supervisory switching schemes are quite satisfactory and indicate that a significant improvement over the conventional control is possible in principle. This can be ascertained by comparing Figures 4 and 12. In practice, a more definitive conclusion requires further validation of NARX models and controls, through the implementation of the algorithms in a vehicle. It is worth noting, however, that the off-line model identification stage can be decoupled from the overall supervisory control framework, and the model can easily be replaced with alternative ones, e.g. physically-based models, for practical realization of the control algorithm.

Future work will involve engine testing and further simplification of the algorithms to aid in the real-time implementation of the algorithms in the vehicle. A potential reduction in the execution time is also desirable. It is likely that the approach provides a natural robustness to modelling errors because of its inherent model matching/switching characteristics but it would be valuable to quantify and demonstrate the robustness achieved.

ACKNOWLEDGEMENTS

We would like to thank Dr. Leonardo Giovanini for many useful discussions during his time at the University of Strathclyde, and Dr. Man-Feng Chang of General Motors R&D for his support on the project.

REFERENCES

Anderson B., Brinsmead T., de Bruyne F, Hespanha J., Liberzon D. and A. Morse. (2000). “Multiple model adaptive control I: finite controller covering”, International Journal of Robust and Nonlinear Control, vol. 10(12), pp. 909—929

Anderson B. and J. Moore. (1979). “Optimal Filtering”, Information and System Sciences Series, Prentice-Hall

Baram Y. and N. Sandell. (1978). “Consistent estimation on finite parameter sets with application to linear systems identification”, IEEE Trans. on Automatic Control, vol. 23(3), pp. 451-454

Dutka, A. (2005). “Nonlinear Identification, Estimation and Control of Automotive Powertrains”, University of Strathclyde, PhD thesis

Giovanini, L., Ordys, A.W. and Grimble, M.J. (2006). “Adaptive Predictive Control using Multiple Models, Switching and Tuning”, Int. Journal of Control, Automation and Systems, vol. 4, no. 6, pp. 669-681

Hendricks, E. and Sorenson, S.C. (1990). “Mean value modeling of spark ignition engines”, SAE paper 9006161

Moskwa, J.J. (1988). “Automotive Engine Modeling for Real Time Control”, MIT, PhD thesis

Narendra, K.S. and Balakrishnan, J. (1994), “Improving Transient Response of Adaptive Control Systems using Multiple Models and Switching”, IEEE Transactions on Automatic Control, Vol. 39, No. 9, pp. 1861-1866

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RPM

Engine

TRQ

TPS

FPW

TRQref

lð

Monitoring

Signal

Generator

Bank of

Estimators

ej(t)

mðj(t)

Supervisory Control

lðref =1

LTI Model

yt

Sysλ

Monitoring

Signal

Generator

Bank of

Estimators

ej(t)

μj(t)

Supervisory Control

λref =1

LTI Model

yt

System

ut

dt

ut2

dt2

1

[pic]

NARX model

( )2

( )2

ut

dt

λ

RPM

TRQ

TPS

FPW

NARX

model

[pic]

[pic]

1

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