Appunti del corso di Modeling and Control of ICE and HPS

SPEED-DENSITY SYSTEM

We consider the density of the fuel to calculate (indirectly) the air mass flow:

Speed n

→

Density →

This is why we named this way "Sped-Density"

42

∙

)

̇ = = ∙ ∙ → = (

2

= ∙ = ∙

,

where:

= ∙ → =

∙

To measure the manifold density we need to install 2 sensors to measure the temperature

and the pressure.

The efficiency depends by the operating conditions (atmospheric or manifold), and if we use the

intake manifold condition (so the relative conditions, not the absolute), we have:

,

=

,

∙

∙

It's not an energy conversion efficiency: it may be even greater than one.

But the relative volumetric efficiency is typically smaller than one (due to exhaust temperature,

wall losses, ecc) but there are also dynamic (inertia 12:36 - 12:37 --> inertial effect) effects and

compressibility effects, in fact: ≪

≫

≠

In the test cell we can measure the manifold pressure, the manifold temperature and the air

mass with a direct measure (ultra sonic sensors) or with indirect measurements like an



interpolation between and the fuel mass.

For each combination of speed and load (P ) we measure the relative volumetric efficiency

MAN

and we generate a "map": calibration to be used in real time via interpolation.

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Conclusion and critical comparison:

System (λ, n) MAF Speed-Density

Based on a direct

No need for additional Good dynamic

Advantages measurement of air

sensors behaviour

mass flow

Fully MAP-Based "Poor" dynamic 2 sensors needed

(Black Box Model) behaviour and and Physical model

Disadvantages and "Poor" dynamic Expensive (and partly based on

behaviour "delicate") sensor experimental data

The injection time in an open-loop system should ideally be as close as possible to the correct

value. However, in reality, this is often not accurate enough when compared to the precision

required for controlling the air-fuel ratio (λ). This means that open-loop systems have inherent

inaccuracies when determining the exact injection time.

Main Error Sources:

1. : The mass of air entering each cylinder. Inaccuracies in measuring or estimating

,

this value can lead to errors in the calculation of the necessary fuel quantity.

2. Injector Characteristics: Variations in the performance or behaviour of fuel injectors,

such as response time, spray pattern, and flow rate, can introduce errors.

3. : The target stoichiometric air-fuel ratio. Even small deviations from the ideal air-

( )

fuel mixture can affect the combustion process and emissions.

Additional Factors Not Considered in the Preliminary Analysis:

1. Fuel Dynamics: This refers to the physical behaviour of fuel within the system, including

factors like pressure, temperature, and delivery time. These dynamics can introduce

delays or variations in fuel delivery.

2. Need to Anticipate Injection: Especially for Port Fuel Injection (PFI) engines, there's a

need to anticipate the correct timing of fuel injection due to the delay between the fuel

injection event and the actual combustion process.

Open-Loop System Accuracy:

Ideally, the open-loop part of the system should only introduce errors less than 5% to

• ensure that the fuel delivery is within acceptable limits for efficient combustion.

44

PID SYSTEM

It is a Proportional-Integral-Derivative correction. The idea is that once the error is calculated,

the correction to bring the error to 0 is immediately available. The PID is made up of 3 terms

that summed together gives the final correction to apply:

Δ = ∙ + ∙ ∫ + ∙

: This term is proportional to the error. The larger the error, the larger the

• ∙

correction that is applied immediately. It’s the fast part of the control, but it never allows

the error to reach exactly 0.

This term is proportional to the integral of the error. It increases the accuracy

• ∙ :

∫

of the controller but is slow. Even if the controller starts with a large error, the first

correction will be small because that error is integrated over a small time interval. The

positive aspect is that this correction will always be applied until the error reaches 0.

Once the error reaches 0, this integral component remains constant at its final value.

: This term is proportional to the derivative of the error. It tries to anticipate the

• ∙

future trend of the error and compensates for it before it actually happens. If the error is

growing significantly, a high correction will be needed to avoid the error reaching a large

value. Normally, this term in ICE (Internal Combustion Engines) is never used ( = 0);

for example, in λ control, it is set to 0 due to the large delays between the measured λ

value and the actual λ value.

Normally, the gain scheduling , and are not scalar but vector-based, proportional to the

error. For example, if the error is very large, the proportional gain has a strong contribution,

while the integral one is less significant. As the error decreases, the proportional gain is

reduced, and the integral gain is increased.

For a closed loop part of the controller, we need to control the error of lambda to calculate the

right time of the injection and we use a controller typically based on a PID controller:

We can use the proportional system where:

Δ = ∙

but the proportional term K is not able to reduce to 0 the error by definition.

P 45



We can correct our (open loop) by:

INJ 

1. Positive errors we need to decrease the ;

→ INJ



2. Negative errors we need to increase the

→ INJ.



The error in will serve as the input for the controller. A PID controller introduces three terms

which, when summed, produce the output. The first term is proportional, where the injection

time delta is equal to Kp multiplied by the lambda error. Here, we assume the PID controller

acts purely as a proportional controller.

The lambda error is less than 5%. If everything is functioning correctly, at t = t*, we activate

the pure closed-loop controller. When there’s a positive error, it indicates that our lambda value

is higher than the target lambda. Consequently, the injection time delta should be negative,

requiring us to reduce it, thereby applying a shorter injection time.

This correction reduces the error, leading to a subsequent, slower adjustment. Each successive

error will be slightly higher than the last, but still lower than the initial error, as we continue

applying corrections. This iterative process aims to achieve minimal error, although a small

residual error will remain.

Despite this, the method proves effective, allowing us to significantly reduce the error in a short

time.

We need to combine the proportional coefficient with the integral one:

∙ ∫

46

Typically we use both contributions:

Δ = ∙ + ∙ ∫ + ∙



but K = 0 for control because the system has significant delays and the sensor is slow!

D

At t = t*, we activate the closed-loop control. Initially, the integral term is zero because time

has passed, and it then remains constant. When the system performs the integral calculation

for the first time, it integrates over the interval from t* to t*+Δt. The correction is relatively

small at the beginning, as the error gradually decreases. The controller accounts not only for

the previous corrections but for the entire error history, resulting in a progressive reduction of

error and an increasing correction until the lambda value is accurate enough to achieve zero

error.

To achieve both speed and precision, we use a combination of proportional and integral control.

We avoid the derivative component, as it introduces delays and slows the response. The lambda

value we measure is delayed, as it reflects a previous combustion and, in particular, a blend of

exhaust gases from different cylinders, resulting in significant lag. The derivative action could

cause instability, so the key challenge is calibrating K and K . We can treat both parameters

p I

as variable depending on the error: when the error is large, we increase K , while for a smaller

p

error, we increase K , which is achieved through a one-dimensional lookup table.

I

Over time, the engine’s intake capacity may decline. In a new engine, the open-loop error is

initially below 5%, but after several years, the error can increase. To adapt, we can adjust the

open-loop control based on feedback from the closed-loop corrections, ensuring that the open-

loop control is better prepared for the next use. Today’s adaptive control systems periodically

modify the open-loop correction based on closed-loop feedback, but do so gradually to identify

any other potential issues.

Our ultimate goal is to maintain lambda equal to one, with a minor oscillation around this value

generated by adding a sinusoidal correction. We calculate the injection time needed to reach

lambda equal to one and superimpose a sinusoidal oscillation of a specific amplitude and

frequency, which defines the amount of extra O stored by the catalyst to reduce NOx and

2

enhance reaction times.

Typically, we phase the injection timing before the valve opens, accounting for vaporization

time. We calculate the injection timing near the bottom dead center of the previous cycle,

introducing a delay that could be problematic during transients. In Gasoline Direct Injection

(GDI) engines, we need to ensure a homogeneous mixture before spark timing. Additionally, we

must avoid wall impingement by timing the injection after top dead center during the intake

phase.

If a large fuel quantity is needed, such as during low load conditions, we may use multi-injection

(multiject) to prevent an exte