− α
−
∀
j )
ke x(t1)
{
exp
(t2 t1)}, x ≠ 0 , x∈Ci
(34)
whereby
λmax { i
P }
k =
l
, i = 1,2,...,N
(35)
λmin { i
P }
236
New Approaches in Automation and Robotics
The trajectories in each cell C
V x
i approach the origin with a exponential decrease in
( )
along the trajectory. According with the philosophy of control strategy, the trajectories will
enter the smallest ellipsoid corresponding to ρN . The exponential stability is assured by (33).
Remark 2. In the paper (De Dona et al., 2004) is considered the case when in the model (12)
uncertainty matrix B
Δ (w) = 0 and the degree of prescribed stability α = 0 . Also, in that
reference the Riccati equation approach is used until in this paper the Lyapunov approach is
used.
Remark 3. When unmodeled dynamic is absent, i.e. in the Theorem 1
a = 0 , D(w) = E(w) = 0
the conditions A.5) and A.7) in the Theorem have the form
λ
{ }>
min Q
0
[
4λ
Q
β
=
+
max ]
{ }
min
min
1
jS
i=1,2,...,,N
⎛ m
⎞
2
j
⎜
l
j
⎟
S
∑ ir Ki
⎜
⎟
⎝l=p+1 ⎠
These assumption are identical with the assumptions in reference (De Dona et al., 2002).
4. Conclusion
In this paper the switching controller with low-and-high gain and alloved over-saturation
for uncertain system is considered. The unmodeled dynamics satisfies matching conditions.
Using picewise quadratic Lyapunov function it is proved the exponential stability of the
closed loop system.
It would be interesting to develop the theory for output case and for the descrete-time case.
Also, extremely is important application of hybrid systems in distributed coordination
problems (multiple robots, spacecraft and unmanned air vehicles).
8. References
Anderson, B.D.O. & Moore , J.B. (1989). Optimal Control. Linear Quadratic Methods, Prentice-
Hall, 0-13-638651-2, New Jersey
Barmish, B.R. & Leitmann, G. (1982). On ultimate boundedness control of uncertain system
in the absence of matching conditions. IEEE Trans. Automatic Control, Vol. 27, No. 2,
(Feb. 1982) 153-158
Blanchini, F. (1999). Set invariance in control a – survey. Automatica, Vol.35, No. 11, (Nov.
1999) 1747-1767
Blanchini, F. & Miani, S. (2008). Set –Theoretic Methods on Control, Birkhauser, 978-0-8176-
3255-7, Basel
Switching Control in the Presence of Constraints and Unmodeled Dynamics
237
Cassandras, C.G. & Lafortune, S. (2008). Introduction to Discrete – Event Systems, Springer
Verlag, , 978-0-387-33332-8, Berlin
De Dona. J.A., Goodwin, G.C. & Moheimani, S.O.R. (2002). Combining switching, over –
saturation and scaling to optimize control performance in the presence of model
uncertainty and input saturation. Automatica, Vol. 38, No.11, (Nov. 2002) 1153-1162
Crama, P. & Schoukens, J. (2001). Initial estimates of Wiener and Hammerstein Systems
using multisine excitation. IEEE Trans. Instrum Measure., vol. 50, No. 6, ( Dec. 2005)
1791-1795
Filipovic, V.Z. (2005). Performance guided hybrid LQ controller for time-delay systems.
Control Engineering and Applied Informatics. Vol. 7, No. 2, (Mar. 2005) 34-44
Goodwin, G.C., Seron, M.M. & De Dona, J.A. (2005). Constrained Control and Estimation. An
optimization Approach, Springer Verlag, 1-85233-548-3, Berlin
Johansson, M. (2003). Picewise Linear Control Systems, Springer-Verlag, 3-54044-124—7, Berlin
Hippe, P. (2006). Windup in Control, Springer Verlag, 1-84628-322-1, Berlin
Li, Z., Soh, Y. & Wen, C (2005). Switched and Impulsive Systems. Analysis, Design and
Applications, Springer Verlag, 3-540-23952-9, Berlin
Lin, Z., Pachter, M, Banda, S. & Shamash,Y. (1997). Stabilizing feedback design for linear
systems with rate limited actuators, In Control of Uncertain Systems with Bounded
Inputs, Tarbouriech, S. & Garcia, G. (Ed.), pp. 173-186, Springer Verlag, 3-540-
761837, Berlin
Lin, Z. (1999). Low Gain Feedback, Springer Verlag, 1-85233-081-3, Berlin
Michel, A., Hou, L. & Liu, D. (2008). Stability of Dynamical Systems: Continuous, Discontinuous
and Discrete Systems, Birkhauser, 978-0—8176-4486-4, Basel
Ren, W. & Atkins, E. (2007). Distributed multi-vehicle coordinated control via local
information exchangle. I nternational Journal of Robust and Nonlinear Control, Vol. 17,
No. 10-11 , (July, 2007) 1002-1033
Ren, W. & Beard, R.W. (2008). Distributed Consensus in Multi-vehicle Cooperative Control.
Theory and Applications, Springer Verlag, 978-1-84800-014-8, Berlin
Saberi, A., Stoorvogel, A.A. & Sannuti, P. (2000). Control of Linear Systems with Regulation and
Input Constraints. Springer Verlag, 1-85233-153-4, Berlin
Sun, Z. & Ge, S.S. (2005). Switched Linear Systems. Control and Design, Springer Verlag, 1-
85233-893-8, Berlin
Tarbouriech, S. & Da Silva, J.M.G. (2000). Sinthesis of controllers for continuous – time delay
systems with saturating control via LMI’s. IEEE Trans. Automatic Control, Vol. 45,
No. 1, (Jan. 2000) 105-111
Tsay, S.C. (1991). Robust control for linear uncertain systems via linear quadratic state
feedback, Systems and Control Letters, Vol.15, No. 2, (Feb. 1991) 199-205
Wredenhagen, G.F. & Belanger, P.R. (1994). Picewise linear L Q control for systems with
input constraints. Automatica, Vol.30, No. 3, (Mar. 1994) 403-416
Wu, F., Lin, Z. & Zheng, Q. (2007). Output feedback stabilization. IEEE Trans. Automatic
Control, Vol. 45, No. 1, (Jan. 2007) 122-128
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Zhao, W.X. & Chen, H.F. (2006). Recursive identification for Hammerstein system with ARX
subsystem. IEEE Trans. Automatic Control, vol. 51, No. 12, (Dec. 2006) 1966-1974
14
Advanced Torque Control
C. Fritzsche and H.-P. Dünow
Hochschule Wismar – University of Technology, Business and Design
Germany
1. Introduction
Torque is one of the fundamental state variables in powertrain systems. The quality of
motion control highly depends on accuracy and dynamics of torque generation. Modulation
of the air massflow by opening or closing a throttle is the classical way to control the torque
in gasoline combustion engines.
In addition to the throttle and the advance angle several other variables are available to
control the torque of modern engines - either directly by control of mixture or indirectly by
influence on energy efficiency. The coordination of the variables for torque control is one of
the major tasks of the electronic control unit (ECU).
In addition to the generation of driving torque other objectives (emission threshold, fuel
consumption) have to be taken into account. The large number of variables and pronounced
nonlinearities or interconnections between sub-processes makes it more and more difficult
to satisfy requirements in terms to quality of control by conventional map-based control
approaches. For that reason the investigation of new approaches for engine control is an
important research field (Bauer 2003).
In this approach substitute variables instead of the real physical control variables are used
for engine torque control. The substitute variables are used as setpoints for subsidiary
control systems. They can be seen as torque differences comparative to a torque maximum
(depending on fresh air mass). One advantage of this approach is that we can describe the
controlled subsystems by linear models. So we can use standard design methods to
construct the torque controller. Of course we have to consider variable constraints. Due to
the linearization the resulting control structure can be used in conventional controller
hardware.
For the superordinate controller a Model Predictive Controller (MPC) based on state space
models is used because with this control approach constraints, and also setpoint- and
disturbance progressions respectively, are considered.
In chapter 2 we briefly explain the most important engine processes and the main torque
control variables. The outcome of this explanations is the use of a linear multivariable model
as a bases for the design of the superordinate torque controller. The MPC- algorithm and its
implementation are specified in chapter 3. Chapter 4 contains several examples of use.
In this chapter it is also shown how the torque controller can change to a speed- or
acceleration controller simply by manipulation of some weighting parameters.
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New Approaches in Automation and Robotics
2. Process and torque generation
Combustion and control variables: In figure 1 the four cycles of a gasoline engine are
shown in a pressure/volume diagram. The mean engine torque results from the difference
energy explained by the two closed areas in the figure.
The large area describes the relation of pressure and volume during compression and
combustion. The charge cycle is represented by the smaller area.
For good efficiency the upper area should be as large as possible and the area below should
be minimal (for a given gas quantity). The best possible efficiency in theory is represented
by the constant-volume cycle (or Otto cycle) (Grohe 1990, Urlaub 1995, Basshuysen 2002).
Fig. 1. p-V-Diagram (gasoline engine)
The maximum torque of the engine basically depends on the fresh air mass which is
reaching the combustion chamber during the charge cycle. A desired air/fuel mixture can
be adjusted by injecting an appropriate fuel quantity. The air/fuel-ratio is called lambda (λ).
If the mass of fresh air corresponds to the mass of fuel (stoichiometric ratio) the lambda
parameter is one (λ=1). By several reasons the combustion during the combustion cycle is
practically never complete. So after the cycle there remain some fresh air and also unreacted
particles of fuel. A more complete reaction of the air can be reached by increasing fuel mass
(λ<1). This results in rising head supply and hence to an increase of the potential engine
torque. The maximum of torque is reached by approximately λ=0.9. The torque decreases
more and more if lambda increases. So one can realize fast torque changes for direct
injection engines by modulation of the fuel.
The maximum of lambda is bounded by combustibility of the mixture (depends on
operating conditions). For engine concepts with lean fuel-air ratio (λ>1) the fuel feed is the
main variable for torque control. Because of the exhaust after treatment λ=1 is requested for
most gasoline engines at present. In case of weighty demands however it is possible to
change lambda temporary.
Advanced Torque Control
241
In addition to air and fuel the torque can be controlled by a number of other variables like
the advance angle or the remaining exhaust gas which can be adjusted by the exhaust gas
recirculation system.
For turbocharged engines the air flow depends in addition to the throttle position on
turbocharger pressure which can be controlled by compressors or by exhaust gas turbines.
Pressure charging leads to an increase of the maximum air mass in the combustion chamber
and hence to an increasing maximum of torque even if the displaced volume of the engine is
small (downsizing concepts).
Because of the mass inertia of the air system it takes time until a desired boost pressure is
achieved. The energy for the turbocharging process is taken from the flue gas stream. This
induces a torque load which can result in nonminimumphase system behaviour.
In addition to the contemplated variables there are further variables imaginable which
influence the torque (maximum valve opening, charge-motion valve, variable compression
ratio, etc.).
Furthermore the total torque of the engine can be influenced by concerted load control. For
example one can generate a negative driving torque by means of the load of the electric
generator. The other way around one can generate a fast positive difference torque by
abrupt unloading. This torque change can be much faster than via throttle control.
In hybrid vehicles we have a number of extra control variables and usable parameters for
torque control.
More details on construction and functionality of internal combustion engines are described
by Schäfer and Basshuysen 2002, Guzella and Onder 2004 and Pulkrabeck 2004.
For control engineering purposes the torque generation process of a gasoline engine is
multivariable and nonlinear. It has a large number of plant inputs, a main variable to be
controlled (the torque) and some other aims of control like the quality of combustion
concerning emission and efficiency.
Considerable differences in the dynamic behaviour of the subsystems generate further
problems particularly for the implementation of the control algorithms. For example we can
change the torque very fast by modulation of the advance angle (compared to the throttle).
Normally the value that causes the best efficiency is used for the advance angle. However to
obtain the fastest torque reduction it can be helpful to degrade the efficiency and so the
engine torque is well directed by the advance angle. Fast torque degradation is required e. g.
in case of a gear switching operation. For the idle speed control mode fast torque
interventions can be useful to enhance the stability and dynamics of the closed loop. Fast
load changes also necessitate to generate fast torque changes. In many cases the dynamic of
sub-processes depends on engine speed and load.
Whether the torque controller uses fast- or slow-acting variables depends on dynamic,
efficiencies and emission requirements. Mostly the range of the manipulated variables is
bounded.
In the normally used "best efficiency" mode one can only degrade the torque by the
advanced angle. For special cases (e. g. idle speed mode) the controller chooses a concerted
permanent offset to the optimal advance angle. So a "boost reserve" for fast torque
enhancement occurs. Because of the efficiency this torque difference is as small as possible
and we have a strong positive (and changeable) bound for the control variable. This should
be considered in the control approach. The main target variables and control signals are
pictured in figure 2.
242
New Approaches in Automation and Robotics
We can summarize: The torque control of combustion engine is a complex multivariable
problem. In addition to the torque we have to control other variables like lambda, the
efficiency, EGR , etc. The sub-processes are time variant, nonlinear and coupled. The control
variables are characterized by strong bounds.
Fig. 2. Actuating and control variables
Torque control: Below we describe the most important variables for torque control and the
effect chain.
a.) fresh air mass:
As aforementioned the engine torque mainly depends on the fresh air mass as the base for
the maximum fuel mass. Classical the air mass is controlled by a throttle within the intake.
In supercharged engines the pressure in front of the throttle can be controlled by a
turbocharger or compressor. The result of supercharging is a higher maximum of fresh air
mass in the combustion chamber. This leads to several benefits. In addition to the throttle or
charging pressure the air mass flow can also be controlled by a variable stroke of the inlet
valve.
It is not expedient to use all variables of the air system for torque control directly. For the
torque generation the air mass within the cylinder is mainly important (not the procedure of
the adjustion). Hence it is more clever to use the setpoint for the fresh air mass as the plant
input for the torque controller. This can be realized by the mentioned control variables with
a the aforementioned sub-control system. One advantage of the approach is, that the sub-
control system at large includes measures for the linearization of the air system.
Furthermore the handling of bounds is a lot easier in this way.
b.) exhaust gas recirculation (EGR):
By variable manipulating of the in- and outlet valves it is possible to arrange, that both
valves are simultaneously open during the charge cycle (valve lap). So in addition to the
fresh air some exhaust gas also remains the combustion chamber. This is advantageous for
the combustion and emission. In case of valve lap the throttle valve must be more open for
the same mass of fresh air. This is another benefit because the wastage caused by the throttle
decreases. The EGR can also be realized by an extern return circuit. Here we only consider
the case of internal EGR. The EGR induced a deviation from the desired set point of fresh
air. This deviation usually will be corrected by a controller via the throttle. As
aforementioned this correction is relatively slow. If the mechanism for the valve lap allows
fast shifting we can realize fast torque adjustments because the change of air mass by the
valve lap occurs immediately.
Advanced Torque Control
243
c.) advance angle:
An important parameter for the efficiency of the engine is the initiation of combustion. The
point of time mainly depends on the advance angle. The ignition angle which leads to the
best efficiency is called the optimized advance angle. It depends on engine speed and the
amount of mixture. A delay in the advance angle leads to less efficiency and to the decrease
of engine torque. The relation between advance angle and torque is nonlinear.
d.) lambda:
The amount of fuel determines (for a given fresh air mass) the torque significantly. As
aforementioned bounded we can manipulate the torque by lambda. For direct injection
concepts the torque changes immediately if lambda is modified. Torque control by means of
lambda is limited for engine concepts with λ=1.
e.) other variables
The quality of mixture has also an effect on the efficiency and so on the torque. For the
quality several construction parameters are important. Further the quality depends on
adjustable variables like fuel pressure, point of injection or the position of a charge-motion
valve. The setpoints for these parameters are primarily so chosen that we obtain best
mixture efficiency.
In figure 3 the main variables and the effects on the process are outlined.
Fig. 3. Main variables to influence the energy conversion and the torque generation
From the explanation up to now we can deduce appropriate variables for torque control:
− fresh air mass in the combustion chamber (adjustable via charge pressure, throttle,
valve aperture and valve lap),
− fuel, lambda (adjustable by injection valve),
− advance angle,
− internal exhaust gas recirculation (adjustable via valve lap)
All variables are bounded.
Subsidiary control systems: For simplification of the torque control problem it is expedient
to divide the process in to several sub-control circuits. The following explanations are aimed
244
New Approaches in Automation and Robotics
to gasoline engines with turbo charging, internal EGR, direct injection and homogeneous
engine operation (λ=1).
The sub-control circuits are outlined in figure 4. The most complicated problem is the design
of the subsystem for fresh air control. Because of the strong connection of fresh air and EGR
it makes sense to control both simultaneously in one sub-system. The control system has to
consider or compensate a number of effects and nonlinear dependences.
The main task is to control the fresh air mass and the desired amount of exhaust gas which
reaches the combustion chamber during the charge cycle. The control system should reject
disturbances and adjust new setpoint values well (fast, small overshoot). Another
requirement is that it should be possible to describe the complete control circuit by a linear
model.
For the control structure we propose a reference-model controller (figure 5). One of the aims
of this control structure is to achieve a desired dynamic behaviour for the circuit. The
mentioned requirement for simplification of the modelling is implicitly given in this way.
Nonlinear effects can be compensated by appropriate inverted models.
Fig. 4. Model Following Control (MFC)
In the explained approach the fresh air mass is used as the main control variable. From this
air mass results the maximum possible torque with approximately λ = 0.9.
In case of modification of the exhaust gas portion we only consider the influence on the
fresh air. The influence on combustion quality is disregarded. Increasing of exhaust gas
portion leads to decreasing of fresh air and so to reduction of torque. This influence is
compensated by a special controller so that the action of EGR on the engine torque is
comparable to the characteristic of a high-pass filter. An advantage is obtained if the acting
effect from EGR-setpoint to the fresh air is faster than via the throttle. In this case it is
possible to realize fast (but transient) changes of the torque.
For torque manipulation via the advance angle it is not necessary to use a feedback system.
The influence on the efficiency of the engine can be modelled by a characteristic curve. So
for this control path a feedforward control action is adequate. We can find the same
conclusion for lambda.
Second-level control system: By means of the described sub-control systems the actuating
variables of the torque control system can be defined. The main variables are the setpoints of
the fresh air mass, the exhaust gas, the advance angle and the setpoint for lambda. The
relation between the torque and the mentioned variables is nonlinear. Following the torque
controller also had to be nonlinear and the controller design could be difficult.
A more convenient way is to linearize the sub-control systems first by stationary functions
at the in and outputs. In the described control approach the control variables are substituted
by a number of partial torque setpoints (delta torques). The partial torques represent the
Advanced Torque Control
245
influence of the used control variables to the whole engine torque. The over-all behaviour of
the system to be controlled is linear (widely).
Base control variable is the theoretical maximum of the torque for a given fresh air mass.
The other variables can be seen as desired torque differences to a reference point (the
maximum of torque). That means if the delta variables all set to zero, the maxim