Diesel Engine Identification and Predictive Control using Wiener and Hammerstein Models E. P´erez, X. Blasco, S. Garc´ıa-Nieto, J. Sanchis Abstract— Air management process in a turbocharged Diesel engine is a multivariable, highly coupled nonlinear system with fast dynamics. Because of this, control algorithms with reasonably low computation times (enabling real time application) must be used. Furthermore, testing of new algorithms on a real engine is expensive. Therefore, a detailed non-linear engine simulator based on a first principles model is developed. A brief description of this model is shown. Next, identification and control schemes based on model predictive control and Wiener and Hammerstein models are proposed. Finally, some results of these algorithms implemented on the engine simulator are offered, and compared with those obtained by applying standard Generalized Predictive control (GPC).
effective flow area and the angle of attack of the exhaust gases to the blades of the turbine. Turbine
Compressor
.
.
mt
ma Intercooler
.
%VGT
megr %EGR
.
Kegr
me
Intake Manifold
.
K2, p a
Exhaust Manifold
K3 , p3
mf
I. M OTIVATION A turbocharged Diesel engine is a very complex system which must fulfill the user’s requirements (high power, low fuel consumption, etc.) as well as accomplishing increasingly strict emission standards. In order to face this tradeoff, it is needed a better development of the whole system, specially the air management process. In particular, some improvements can be achieved by developing new control schemes. Moreover, the designed control algorithms must be implemented into an electronic control unit (ECU) with limited computation resources, in order to run on a real engine. Therefore, computationally expensive control techniques are not suitable. II. P ROCESS D ESCRIPTION AND C ONTROL O BJECTIVES In order to improve their power and driveability, most of Diesel engines use a turbocharging group (turbine and compressor), which uses part of the power of the exhaust gases to increase the pressure of fresh air coming into the engine, and therefore its flow (Fig. 1). This increase of air into the cylinders allows the combustion of a higher quantity of fuel, and therefore higher power and torque of the engine [2]. The turbocharging group is usually designed to offer a fast response at low engine speed and fuel mass flow rate. However, at higher speeds and fuel mass flow rates, the engine could be damaged. To avoid this, a variable geometry turbine (VGT) is used to change the turbogroup speed. This is achieved by adjusting, during engine operation, the All authors are members of Predictive Control and Heuristic Optimization Group (CPOH), Department of Systems Engineering and Control, Polytechnic University of Valencia, Camino de Vera S/N, 46022-Valencia (Spain),
[email protected]
http://ctl-predictivo.upv.es Research partially supported by MEC (Spain) and FEDER funds: projects DPI2004-8383-C03-02 and DPI2005-07835.
Engine
Fig. 1.
Ke ,Ve
Turbocharged Diesel engine.
On the other hand, in order to reduce N Ox emissions, a portion of the exhaust gases is recirculated from the exhaust manifold to the intake manifold by the exhaust gas recirculation (EGR) valve. The recirculated gases act as inert gases, decreasing the combustion temperature, and therefore the amount of N Ox . Furthermore, smoke emission depends mainly on the air-fuel ratio (AFR) in the combustion. For a fixed fuel mass flow rate, a minimum amount of fresh air is needed to have a combustion without an excessive emission of smoke. Therefore, a clear tradeoff between the different control objectives exists. Moreover, the variables that should be controlled , air-fuel ratio and burnt gas fraction in the intake manifold (λ2 ), are not measurable with the usual sensors configuration in a commercial engine. Instead of these, present algorithms consider pressure in the intake manifold (Pa ) and air mass flow rate through the compressor (m ˙ a ). These variables are related to the previous ones, so that setpoints for AFR and ˙ a [3]. λ2 can be translated into setpoints for Pa and m At last, it is needed to consider into the control scheme two additional variables that, although not manipulable by the air management control system, affect heavily the behaviour of the system, engine speed (N) and fuel mass flow rate (m ˙ f ). In this situation, an important part of pollutant emissions occurs during transitions until the engine variables achieve
their new setpoints. It is very interesting then to design controllers that reduce the duration of these transitions and therefore, this is the main control objective. III. F IRST PRINCIPLES ENGINE MODEL The Diesel engine is described by a mean-model which is based in first-principles knowledge [1], [2]. The relations between the different thermodynamic an mechanical variables of the engine are described in the block diagram in Fig. 2, where:
Te (N · m): T1 (N · m): uegr (V): uvgt (V): λ2 : λ3 :
torque produced by the engine. resistence torque. command signal to the EGR valve. command signal to the VGT valve. ratio of air and fuel flow into the cylinder. ratio of the recirculated mass flow to the total exhaust gas flow.
The nonlinear equations for each Diesel engine block are introduced in [1], [2] and [4]. Fig. 2 shows in bold face the variables used in the MIMO control scheme, where m˙ a and Pa are treated as outputs and uegr and uvgt as control signals. This non-linear model, based on equations of first physical principles (mass and energy) and on empirically adjusted equations, is programmed into Simulink and constitutes the core of an engine simulator, used to test the control algorithms [4]. The model is executed in real time in a stand-alone PC, by means of the xPC-target MATLAB toolbox. As regards the acquisition and control, they are made with an industrial equipment based on PXI standard, which allows deterministic real-time performance by an embedded controller working with a real-time operating system.
IV. PROPOSED CONTROL STRATEGIES
Fig. 2.
m ˙ a (Kg/h): m ˙ t (Kg/h): m ˙ f (Kg/h): m ˙ e (Kg/h): m ˙ egr (Kg/h): Pa (bar): P1 (bar): P3 (bar): P4 (bar): K1 (K): K2 (K): K3 (K):
Complete development platform.
air mass flow through the compressor. exhaust mass flow trough the turbine. fuel mass flow through the engine. exhaust gas mass flow. recirculated mass flow. pressure in the intake manifold. pressure before the compressor. pressure in the exhaust manifold. pressure after the turbine. air temperature after the compresor. air temperature in the intake manifold. exhaust gas temperature in front of the turbine. air temperature after the compressor. Kc (K): exhaust gas temperature in the EGR. Kegr (K): Exhaust gases temperature. Ke (K): wtc (r.p.m.): turbocharger rotational speed. we (r.p.m.): engine rotational speed. Tc (N · m): torque produced by the turbocharger.
As it has been shown in section II, from the control point of view the system has two manipulable inputs (EGR and ˙ a ). There are V GT ) and two controlled variables (Pa and m also two other inputs to the system, engine speed and fuel mass flow rate, which are measurable but not manipulable by the air management control system. Due to the high non-linear response of the system, it is very difficult to treat these variables as measurable disturbances in the control scheme. Instead of that, the solution proposed is to develop several models and controllers for a range of operating points and apply a gain-scheduling philosophy between them [5]. However, in the present work the system is controlled for a single N − m ˙ f point, and the whole control system will be developed in future work. Considering that, the problem is to control a 2 × 2 non-linear system, and several strategies are proposed to approach it. A model predictive control (MPC) approach is proposed because it allows a simple treatment of input and output constraints [6] and copes in a natural way with multivariable systems, such as the Diesel engine. With MPC it is possible to weight some of the outputs over the others. Specifically, this is useful in different working points where it is more important to get a fast response in m ˙ a than in Pa [7].
The particular Predictive controller chosen is GPC [9], adapting its formulation for MIMO systems. GPC calculates the control moves which optimize an index with the form:
J(Δu)
=
n
i=1
+
N2
αik [ˆ yi (t + k|t) − wi (t + k)]2
k=N1
N m u j=1
2
λjk [Δuj (t + k − 1)]
k=1
where: m, n: yˆi (t + k|t):
Number of inputs and outputs, respectively. Predicted outputs, obtained from the CARIMA model (1), being Ti a filtering polynomial and ξi white noise. Setpoints. wi : System inputs. uj : Nu , N1 , N2 : Control, lower prediction and higher prediction horizons. λjk and αik : Input and Output weightings. Aii (z
−1
)yi (t) =
m
Bij (z −1 )uj (t − 1) +
j=1
Ti ξi (t) Δ
(1)
In absence of constraints, the described optimization can be obtained analitically. When constraints are present, the usual solution is to formulate the problem as a quadratic programming (QP) problem [6]. However, QP can be very time-consuming and is not suitable for fast processes as the engine. Therefore, another option is needed. The approach taken is to perform a clipping on the control moves.
Hammerstein and Wiener models can represent reasonably well the engine simulator because many of the non-linearities present in the model have a static nature, as can be seen in [4]. Moreover, control of these non-linear structures is simple enough to be implemented in control units with limited computation resources. Hammerstein and Wiener MIMO models are represented by a linear dynamic part (transfer function matrix G) and a non-linear static function (f ). Therefore the main idea in the present work is to cancel the non-linearity and apply the well-known theory of predictive control to the linear part. To do so, static non-linear functions (g) that cancel the non-linear part of the system must be included in the control scheme. A. Wiener model Fig. 3 shows the control scheme when the system is treated as a Wiener model. As the non-linear part of the system is supposed to be after the linear part, the non-linear static part of the controller, g, must be included right after the engine simulator. Therefore, the MPC works with y1 and y2 instead of m ˙ a and Pa . Because of this, it is also necessary to modify the setpoints by passing them through the non-linear static function too. .
ma,sp Pa,sp
y1,sp
.
g(ma,Pa)
y2,sp
MPC y1
.
EGR
SAT
VGT
SAT
ma
ENGINE SIMULATOR
Pa
y2
.
g(ma,Pa)
Standard GPC is intended to control linear systems, an it usually offers poor results when used to control non-linear systems in regions far from where the linear model used was identified. In this situation, models able to represent more accurately the system, could offer better controller performance. With this purpose, the engine is modelled as a Wiener or a Hammerstein system [8]. A Wiener model consists of a linear system followed by a static nonlinearity and can be described by the following equations: yj (t) x(t) xi (t)
= fj (x(t)) [x1 (t) x2 (t) . . . xI (t)] m Bij (z −1 ) uj (t − dij )) = ( Aij (z −1 ) j=1
Fig. 3.
B. Hammerstein model On the other hand, Fig. 4 shows the control scheme when the system is treated as a Hammerstein model. In an ideal case, the system is actually a Hammerstein system and its static non-linear part is cancelled exactly by the function g(x1 , x2 ). This way, the predictive controller only needs to cope with a linear system.
=
.
ma,sp Pa,sp
I Bji (z −1 ) xi (t − dji )) ( Aji (z −1 ) i=1
yj (t)
=
xi (t)
= fi (u(t)) =
[u1 (t)
EGR
x1
MPC
On the other hand, a Hammerstein system is represented by a static nonlinearity followed by a linear system:
u(t)
Model Predictive Control of Wiener model.
Fig. 4.
x2
g(x1,x2)
.
SAT
VGT SAT
ma
ENGINE SIMULATOR
Pa
Model Predictive Control of Hammerstein model.
V. MODELLING FOR CONTROL u2 (t)
...
um (t)]
The real-time engine model described in section III is suitable for real-time simulation but it is too complex to
be used in the predictive control scheme. Instead of that, Hammerstein and Wiener models as described in section IV are developed. In order to implement the proposed control schemes, mainly two different elements need to be identified: • The non-linear function g. • Linear models to be used by the MPC. Identification schemes of these models also require the non-linear function f , which needs to be identified too. A. Static non-linearities The first step to work with Wiener and Hammerstein models is to obtain the non-linear functions that represent the static relations between the inputs and the outputs (f (EGR, V GT )). This function will be obtained experimentally on the engine simulator, and the same function will be used for both Wiener and Hammerstein models. The experiments consist of sweeping all the range of inputs, waiting until the outputs stabilize, and saving those steady state values. This gives a set of values for different combinations of the inputs. When a value for a combination of inputs not measured in these experiments is needed, linear interpolation is applied. Fig. 5 and 6 show graphically these values.
10 6
8 6 EGR 4
4 VGT 2
2 0
1.05
2.4
100 0
Pa
m ˙a
2
2
4
6 VGT
4 6 8 1010 8 EGR
2
0
1.2 0
2
4
6 VGT
8
10
8
4 6 EGR
2
0
m ˙a
Fig. 7.
1.2 120
B. Identification of linear models As can be seen in Fig. 3 and 4, the control strategies proposed use a static cancelation part (g) and a standard
Fig. 8.
1.2
120
Steady-state VGT.
linear Model Predictive Control, specifically GPC. Therefore, linear models are needed to implement this controller. These linear models will be obtained in a different way depending on the model type. Firstly, for Wiener models, the identification scheme shown in Fig. 9 is used. Identification inputs are applied to the engine simulator and the outputs of this are passed trough function g(m ˙ a , Pa ). A linear identification procedure will be applied that uses EGR and V GT as inputs and EGRi∗ and V GTi∗ as outputs. EGR VGT
.
ENGINE SIMULATOR
ma Pa
EGRi*
.
g(ma,Pa)
VGTi*
Identification scheme of Wiener model.
On the other hand, Fig. 10 shows the identification scheme for a Hammerstein system. In this case function ∗ . Now, the f (EGR, V GT ) is used to obtain m ˙ ∗a,i and Pa,i inputs to the linear identification procedure will be m ˙ ∗a,i and ∗ Pa,i and the outputs m ˙ a and Pa .
Fig. 6. Steady-state intake manifold pressure.
The function g(x1 , x2 ) can’t be obtained by inverting f , because a mathematical formulation of it is not available. Therefore, it is obtained by making closed loop experiments. Two independent PI controllers are designed, one of them to control m ˙ a with EGR and the other one to control Pa with V GT . These controllers are designed with a high settling time, to avoid problems due to the non linear nature of the system. In the experiments, setpoints to the controllers are ˙a chosen to cover the whole feasible range of Pa and m and once the outputs have stabilized, a table is built for the values of EGR and VGT needed to achieve those setpoints. As it is done with f , if intermediate values are needed, linear interpolation is applied. The values of those tables are shown graphically in Fig. 7 and 8.
m ˙a ˙
Pa
Steady-state EGR.
EGR
Fig. 5. Steady-state air mass flow rate .
1.05
˙
Pa
Fig. 9. 220
60
0
60
VGT
.
ENGINE SIMULATOR
ma Pa
.
ma i*
f(EGR,VGT)
Fig. 10.
Pa i*
Identification scheme of Hammerstein model.
For comparison purposes, standard GPC will also be applied to the system. In this case, identification inputs are directly applied to the system, there is no need of any special scheme. Once the manipulated identification Hammerstein
inputs and outputs have been correctly depending on the model type, the three procedures (for standard GPC, Wiener and models) are essentially the same.
The identification procedure used is a Prediction Error Method (as implemented in the MATLAB System Identification Toolbox), which consists of choosing the model parameters that minimize an index with the difference between
Engine Sim.
the real measures and a prediction with horizon k = 1: N 1 θˆ = argmin εF (t) N t=1
Linear
1.2
Hammerstein Wiener
1.15
Where: Pa
εF (t) = L(z)(y(t) − yˆ(t, θ|t − 1))
1.1
L(z) is a filter that affects the weighting applied to the fit between the model and the data and can be used to assure that the model approximates the true system well over certain frequency intervals [10]. The chosen structure for the lineal models is autoregressive with exogenous input (ARX):
1.05
1
5
10
15
20 t(s)
Fig. 12.
Aii (z −1 )yi (t) =
m
Bij (z −1 )uj (t − 1) + ξi (t)
In order to compare how the different models reproduce the system behaviour, a simultaneous simulation of the three models is carried out for a changing range in EGR and V GT . Fig. 11 shows real and simulated m ˙ a . It can be seen that both the Wiener and the Hammerstein models work better than the linear one, and between them, Hammerstein slightly better. This can be due to the number of static non-linearities at the input of the non-linear model of the engine simulator [4], which is an structure more similar to the Hammerstein model. Similarly, Fig. 12 shows how the different models simulate Pa . It can be seen again how the linear model is not able of reproducing right the whole range of the output, while Hammerstein and Wiener models offer a better response.
35
Pa simulation.
Conserv.
Aggr.
Ts
Wiener
Hammerstein
0.1s x1 45-118.5 x2 1.02-1.32
EGR 0-10 VGT 0-10
Sats
C. Model validation
30
TABLE I P REDICTIVE C ONTROLLERS PARAMETERS .
j=1
The identification inputs to the system are pseudorandom binary signals (PRBS) varying in the effective range of operation (EGR between 0 − 9 and V GT between 0 − 4).
25
Nu (samples)
4
8
8
8
N2 (samples)
20
100
100
100
α1
0.2
0.5
0.5
0.5
α2
0.8
0.5
0.5
0.5
λ1
15
100
100
100
λ2
7
100
100
100
T
(1 − 0.9z −1 )5
VI. CONTROL RESULTS The proposed identification and control algorithms were tested in the described platform. The fine tuning of the controllers applied to the engine simulator was made in order to get a fast response with little overshoot, avoiding unstable behaviour because of the non-linear nature of the system.
100 Engine Simulator Linear
Table I shows the parameters of the controllers finally chosen for the different schemes, as defined in section IV. It is important to note that all models are scaled, and therefore weightings have a relative meaning.
Hammerstein Wiener
95
90
m ˙ a (Kg/h) 85
As regards the saturations, linear and Wiener models use the physical limits of the engine actuators (EGR and V GT ). However for the Hammerstein model the GPC outputs are not the actuators levels, but the variables x1 and x2 . Therefore, the saturation values must be different, and the maximum feasible values of m ˙ a and Pa are used.
80
75
70
5
10
15
Fig. 11.
20 t(s)
25
m ˙ a simulation.
30
35
Results of the experiments with the different controllers are shown in Fig. 13 and 14. It can be seen how, when actuators don’t saturate (t = 0s to t = 70s), the
Hammerstein scheme is the one which offers the fastest response, followed by the Wiener one. Linear models offer a more oscillatory (’Aggressive’) or slower response (’Conservative’), because they are more difficult to adjust. This is because the prediction model is not good enough, and the controller offers a worse performance with high prediction horizons. This can be seen in Fig. 13. Controller ’Aggressive’ is adjusted with the same parameters as the ones of Wiener and Hammerstein models, but it offers a much more oscillatory response, specially in Pa . Therefore, the linear controller must be adjusted in a more conservative way (’Conservative’). Although the Hammerstein model seems to be the one than can get the best results, this scheme shows problems in its behaviour when one of the inputs saturates. This is related with the non-linear nature of the process and the treatment given to constraints. Fig. 15 shows the feasible steady-state values of m ˙ a vs. Pa for all the range in EGR and VGT, as well as the saturation constraints chosen for the Hammerstein controller.
Hammerstein Constraints
FEASIBLE ZONE
Fig. 15.
Steady-state m ˙ a vs Pa .
Some x1 − x2 pairs supplied by the linear MPC controller on the Hammerstein model, although satisfying the imposed constraints, may in fact lay in a non-feasible zone and produce saturations in EGR and V GT that the controller is not able to take into account. Therefore, there exists wrong information about the past inputs actually applied to the system (no clipping), causing higher prediction errors. This fact affects the controller performance, as can be seen in Fig. 13 and 14 (from t = 60s to t = 130s). VII. CONCLUSIONS AND FUTURE WORKS A. Conclusions In this work, a new approach to control the air management process of a Diesel engine has been proposed. • Predictive Control and model identification schemes for Wiener and Hammerstein models have been shown. Proposed algorithms are easily implemented in a real engine, and don’t cause significant increases in computation times.
• •
A test platform, including complex non-linear behaviour and real hardware data acquisition, has been developed. Results of applying the proposed control schemes, offering an improvement in the system behaviour, have been shown.
B. Future Works •
•
Hammerstein models have shown several problems due to saturation of inputs. Some work in order to cope somehow with physical saturation of the actuators is currently being carried out. Proposed control algorithms offer a solution to control the engine for a single m ˙ f − N pair. This work needs to be extended to cover the whole engine range. VIII. ACKNOWLEDGMENTS
The authors gratefully acknowledge the contribution of Emanuele Pieroni, who actively took part in some phases of this work development. R EFERENCES [1] L. Guzzella and A. Amstutz, Control of Diesel Engines, IEEE Control Systems, 1998. [2] L. Guzzella and C.H. Onder, Introduction to Modeling and Control of Internal Combustion Engine Systems, Springer, 2004. [3] M. van Nieuwstadt, I.V. Kolmanovsky, P.E. Moraal, A. Stefanopoulou, and M. Jankovic. “EGR-VGT control schemes: Experimental comparison for a high-speed diesel engine” in IEEE Control Systems, 2000. [4] J.Salcedo, E. Pieroni, E. P´erez, X. Blasco, M. Mart´ınez and J. Garc´ıa, “Real-time control and simulation of a non-linear model of air management in a turbocharged Diesel engine” in FISITA World Automotive Congress, May 2004. [5] Danielle Dougherty and Doug Cooper, “A practical multiple model adaptive strategy for multivariable predictive control”,Control Engineering Practice,vol. 11, pp. 649-664, 2003. [6] F. Camacho and C. Bordons. Model Predictive Control. Springer, 1999. [7] M. van Nieuwstadt, P.E. Moraal, I.V. Kolmanovsky, A. Stefanopoulou, P. Wood and M. Criddle. “Decentralized and multivariable designs for EGR-VGT control of a Diesel engine.” in IFAC Workshop on Advances in Automotive Control, 1998. [8] Enso Ikonen and Kaddour Najim, Advanced process identification and control. Marcel Dekker, 2002. [9] D.W. Clarke, C. Mohtadi and P.S Tuffs, “Generalized Predictive Control – Part I”, Automatica, vol. 23(2), pp. 137-148, 1987. [10] Lennart Ljung, System Identification Toolbox User’s Guide, The MathWorks, Inc., 2002.
Outputs 120 110
m ˙ a (Kg/h)
100 90 80 70 60 50 40
0
20
40
60
80
100
120
t(s) 1.25
Setpoint 140 Conservative GPC Wiener GPC Hammerstein GPC Aggressive GPC
1.15 1.1 1.05 1
0
20
40
60
80
100
120
140
t(s)
Fig. 13.
System outputs. Inputs
10
8
EGR
6
4
2
0
0
20
40
60
80
100
t(s) 3
Conservative GPC 120 140 Wiener GPC Hammerstein GPC Aggressive GPC
2.5 2 V GT
Pa (bar)
1.2
1.5 1 0.5 0
0
20
40
60
80 t(s)
Fig. 14.
System inputs.
100
120
140
