Modeling and speed control design of an ethanol engine for variable speed gensets

Control Engineering Practice 35 (2015) 54–66

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Modeling and speed control design of an ethanol engine for variable speed gensets Jonas Roberto Tibola a,n, Thompson Diórdinis Metzka Lanzanova b, Mario Eduardo Santos Martins b, Hilton Abílio Gründling a, Humberto Pinheiro a a b

Power Electronics and Control Research Group (GEPOC), Federal University of Santa Maria, Santa Maria, RS, Brazil Engines, Fuels and Emissions Research Group (GPMOT), Federal University of Santa Maria, Santa Maria, RS, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 20 February 2014 Accepted 12 November 2014

This paper presents the development of a discrete dynamic mean value engine model (MVEM) suitable for the design of speed controllers of ethanol fueled internal combustion engines (ICE), to be used in variable speed gensets. Two MVEMs are developed for the ICE: the Time Based model and the Crank Based model. The speed controller design is held through the discretization and linearization of the Crank Based MVEM. This model is used due to the advantages over the time based MVEM especially with respect to the transport delay which becomes constant. Two approaches for the ICE speed control are investigated: (i) a single loop gain-scheduled proportional integral (PI) controller and (ii) a dual loop control based on an internal gain-scheduled Manifold Absolute Pressure (MAP) feedback loop and an outer loop composed of a gain-scheduled PI controller. The control design is developed in the frequency domain and its stability is ensured by the phase and gain margins. In addition, an integral anti-windup and a feed forward action are also proposed to improve the behavior during control law saturation, improve transient responses and disturbance rejection capability. Experimental results on a 50 kW generator set are provided to validate the controllers and to demonstrate the performance of the system. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Ethanol engine modeling Ethanol engine speed control Variable speed gensets

1. Introduction The use of fossil fuels for electric energy production is considered as one of the main villains in the climate change scenario (Akpan & Akpan, 2012). The use of environmentally friendly fuels has grown in the last decades in an effort to mitigate green house gases (GHG) emissions from thermal machines. For instance, the ethanol life cycle has reduced GHG emissions due its positive energy balance, when compared to fossil fuels. This means that the ethanol life cycle releases more energy during the burn phase than in all the production phases (Foteinis, Kouloumpis, & Tsoutsos, 2011). In countries such as USA and Brazil ethanol has a full deliver system already implanted and its market is mature (Jun, Fubing, Gesheng, & Xiaohong, 2010). Therefore, the use of ethanol stands out as an excellent player as a fuel for SI (spark ignited) engines through the “green energy” concept, and can be considered as a fuel candidate for gensets as well. Diesel gensets have been widely used for backup power and for peak shaving to reduce energy costs for many decades, mainly due

n

Corresponding author. Tel.: þ 55 55 9146 4085; fax: þ55 55 3220 9497. E-mail addresses: [email protected], [email protected] (J.R. Tibola).

http://dx.doi.org/10.1016/j.conengprac.2014.11.002 0967-0661/& 2014 Elsevier Ltd. All rights reserved.

their ruggedness and simplicity. Although constant-speed diesel gensets have been widely used they have drawbacks such as (i) usage of non-renewable fossil fuel; (ii) high NOx and particulate emissions; (iii) high audible noise; (iv) poor energy quality. The basic and most used concept of gensets consists of a Diesel compression ignition (CI) internal combustion engine coupled to a wound rotor synchronous machine (WRSM). To generate AC voltage with constant frequency, gensets speed must be constant. This fact comes against an important characteristic of SI engines: higher engine efficiency does not occur at the same speed for different loads. This implies that a SI engine running at constant speed for a given load range should have less overall efficiency than a variable speed SI engine running at the same load range. Furthermore, the use of a permanent magnet synchronous generator (PMSG) and a back-to-back converter enables variable speed operation which can lead to higher efficiency than the conventional WRSM and constant speed SI engine (Koczara et al., 2008). The efficient use of a variable speed SI engine in gensets implies in finding the best operating points, and in the development of a closed loop speed control for the engine. Therefore, it is paramount to develop SI engine dynamic models to predict its behavior and to design its controllers. This paper describes a mean

J.R. Tibola et al. / Control Engineering Practice 35 (2015) 54–66

value engine model (MVEM) for an ethanol engine applied to grid connected variable speed gensets. To validate the models a genset composed of a 4-cylinder 3.9 l four-stroke ethanol fueled engine, a three phase PMSG and a 50 kVA three phase PWM back-to-back converter connected to the grid is considered. Then, three dynamic models are derived. The first two are a time based and an event based nonlinear models. The nonlinear functions of these models are derived experimentally. Then, for the purpose of a speed control design, a linear event based model is obtained. Experimental results are given to validate the developed models and to demonstrate the behavior of the proposed control strategies. For the sake of convenience a list of symbols and acronyms used in this paper are given in Appendix A.

2. Background mean value engine models (MVEMs) are an intermediate category between complex cyclic simulation models and simplistic phenomenological transfer function models (Asl, Saeedi, Fraser, Goossens, & McPhee, 2013; Hendricks & Sorenson, 1990). The MVEM derivation usually starts with a nonlinear mean value description of the engine variables and ignores the characteristics of individual cylinders. Such model is a mix of first-principle, physical laws, and identification techniques using dynamic and steady-state experiments. It usually takes into account the engine intake mass air flow (MAF) rate and torque as functions of engine speed, throttle opening area, manifold absolute pressure (MAP), air–fuel ratio and spark advance (Hazell & Flower, 1971; Powell, 1979; Powell & Cook, 1987). This type of model is also referred as control-oriented model (Chaumerliac, Bidan, & Boverie, 1994; Cook & Powell, 1988). Generally, control oriented engine models are developed for engine speed control design purpose. The reported engine speed control can be classified according to the engine speed operation range. Two main approaches for the control-oriented models are idle speed control (ISC) and speed control for all operating points, named here as wide range speed control (WRSC). One of the main goals of ISC is to keep speed as low as possible to improve fuel economy. The ISC also has to guarantee engine combustion quality without compromising vehicle accessories performance (e.g. alternator) and also prevent any possibility of engine stalling (Hrovat & Sun, 1997). Many different control strategies for ISC have been addressed in the literature. Among them, strategies are based on estimation techniques which can be used to find unknown engine parameters and then design a controller using for example linear quadratic regulator (LQR) (Nicolao, Rossi, Scattolini, & Suffritti, 1999) or to estimate the unmeasurable disturbances and then compensate them (Stotsky, Egardt, & Eriksson, 1999). More robust techniques like adaptive control (Hsieh, Chen, & Wu, 2007; Yildiz, Annaswamy, Yanakiev, & Kolmanovsky, 2011) and sliding mode control (Alt, Blath, Svaricek, & Schultalbers, 2009; Li & Yurkovich, 2001) have also been considered. Proportional-integral-derivative (PID) type controllers were also proposed which are designed offline via linearization of the non-linear MVEM (Wang, Stefanopoulou, & Levin, 1999) or tuned online (Howell & Best, 2000). In Klawonn, Gebhardt, and Kruse (1995) and Martinez and Jamshidi (1993) Fuzzy control were applied for ISC problem instead of relying on a mathematical formalization of the engine model. Further details on ISC can be found in Thornhill, Thompson, and Sindano (2000). All the previous cited MVEM and control strategies are developed in time based domain. However, there is another approach to describe the behavior of an ICE which uses crank angle instead of time. This approach is known as crank-angle based model or event based model. In Chin and Coats (1986) it is shown that most engine dynamics (with the exception of fuel) are less varying in the crankangle domain than in the time domain.

55

Event-based approach has also been addressed in the literature to design many different control strategies. In Butts, Sivashankar, and Sun (1999) an event-based MVEM is developed for ISC and a feedback linear quadratic Gaussian (LQG) and a feed forward controller are designed via ℓ1 optimization methodology. In Yurkovich and Simpson (1997) both a linear and a non-linear event-based MVEM were developed for ISC, and three control techniques were compared for these two models, which are linear quadratic regulator (LQR), fuzzy and sliding mode control. In Jimbo and Hayakawa (2011) a SI modeling concept was proposed based on an analytical state variable discrete-crank angle model which enables an optimal ISC design for a periodic engine model. The main limitation of the ISC models comes from the fact that it is derived considering: (i) a limited speed range, (ii) that the throttle body usually remains in choked flow, therefore the intake air mass flow rate is only a function of the throttle angle, and is not a function of the manifold pressure, (iii) that in idle speed the parametric uncertainties are smaller, (iv) and that the intake-topower stroke delay at idle speed is longer than that in nominal speed, but it can be considered constant due to the limited speed range. On the other hand, for variable speed gensets, it is important to track the engine speed behavior in a wider range than idle. Also, in the majority of the time, the intake air flow is not choked at the throttle. Hence, the intake air mass flow rate is now a non-linear function of the throttle angle and manifold absolute pressure. In Puleston, Spurgeon, and Monsees (2001) a sliding-mode controller is proposed for speed control based on the engine model presented in Crossley and Cook (1991). Since in WRSC the parametric uncertainties are larger than for ISC it is common to use speed and manifold absolute pressure (MAP) as feedback measurements. In Zhang and Shen (2009) an adaptive feedback controller is proposed also using speed and MAP measurements. In Pan, Wei, and Zhao (2008) an H 1 state feedback controller is proposed for a nonlinear Takagi–Sugeno fuzzy systems with timedelay allowing a good control design over all operation points. In Johansen, Hunt, Gawthrop, and Fritz (1998) a gain-schedule control is proposed and it is designed using off-equilibrium linearization technique. Engine speed control is also designed for SI engines with powertrain dynamics connected to continuously variable transmissions (CVT). In Guzzella and Schmid (1995) a feedback linearization is developed for a truck engine speed control in a wide range of operation. In Liu and Stefanopoulou (2002) an engine plus power dynamics and CVT were also used and a MIMO feedback controller was proposed improving the engine operation at best efficiency conditions. Despite many different techniques have been developed for engine speed control, not much attention had been given to simpler controllers like PI controllers and its discrete design. Besides that, gain-schedule dual loop PI controllers with feed forward action have not been reported for ICE. In the following sections the two main models used for engine speed control are described, which are the time based model (Cook & Powell, 1988) and the event based model (Butts et al., 1999). The last one is used in the development of a linear discrete model more suitable for speed controller design.

3. Engine modeling 3.1. Time based model The development of the time based model was first described in Cook and Powell (1988). In there, a dynamic mean value model in time domain was developed to describe throttle body behavior,

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where Cp is a fitted constant, in this case Cp ¼9, as proposed by Wagner et al. (2003), Pfeiffer (1997), and Cho and Hedrick (1989). For transient engine operation conditions, where mass flow rapidly changes, the air mass flow is assumed to be a quasi-static phenomena therefore Eq. (1) can be used. For consecutive transient cycles the air mass flow is calculated considering the hypothesis of consecutive changes of steady states. 3.1.2. Engine pumping The estimative of the output air mass flow rate from the intake _ ao ) can be achieved considering the engine as a manifold (m volumetric pump (Cook & Powell, 1988). Therefore the output air mass flow rate can be expressed as a separable polynomial function of the crankshaft speed (n) and intake manifold absolute pressure (pm), as shown below: _ ao ðpm ; nÞ ¼ f p ðpm Þn m

Fig. 1. Simplified representation of one cylinder of the engine.

engine pumping, manifold filling, engine brake torque and rotational dynamics. Fig. 1 illustrates one cylinder of the engine, its sensors and the main engine variables.

3.1.1. Throttle body The throttle body behavior can be modeled by using the principle of conservation of mass and energy and assuming the air as one-dimensional, steady, isentropic compressible flow of an _ ai ) through the throttle ideal gas, then the intake mass flow rate (m body can be written as a nonlinear separable function of the throttle body angle ðφÞ and the manifold absolute pressure (pm). This equation is known as equation of compressible flow through a restriction, which is well described in Guzzella and Onder (2010) and Heywood (1988). However, it has a few limitations to be named: (i) the discharge coefficient Cd requires additional flow bench tests which are not always available, (ii) the throttle area function Ath ðφÞ is a complicated function of the throttle body geometry and throttle angle (Moskwa & Hedrick, 1987), (iii) Ψ ðÞ has an inconvenient of being a conditional equation which is not an adequate option for model based control design. To overcome these drawbacks an alternative option is given in Wagner, Dawson, and Zeyu (2003) and Pfeiffer (1997) as
 pm _ ai ðuth ; pm ; pin Þ ¼ TCðuth ÞΨ~ m ð1Þ pin where pin is the atmospheric pressure which is considered constant (pin  101:3 kPa), the function TCðÞ is the throttle body characteristic function expressed by a polynomial function of the throttle body signal uth instead of φ. This can be performed since the throttle angle φ is controlled by the throttle driver through the throttle signal uth, thereby φ p uth , as shown in Fig. 1. The throttle driver has a dynamic response, which can be modeled by a first order transfer function with pole place at ωth ¼231 rad/s. This dynamic has been disregarded in the model but it has implications in the control design to be considered. The function TCðÞ is shown as follows: TCðuth Þ ¼ u0 þ u1 uth þ u2 u2th þ u3 u3th

ð2Þ

where u0–3 are experimental constants. Eq. (2) can be obtained through experimental tests using a MAF sensor as shown in Fig. 1. On the other hand, the function Ψ~ ðÞ is an approximation of the one proposed in Guzzella and Onder (2010), with no conditional action which facilitates its use in a model based controller design. This function is shown as follows:
 
 p p Ψ~ m ¼ 1  EXP C p m  1 ð3Þ pin pin

ð4Þ

where the function f p ðÞ can be defined as f p ðpm Þ ¼ p0 þ p1 pm þ p2 p2m þ p3 p3m

ð5Þ

where p0–3 are experimental constants. The constants p0–3 are obtained from experimental tests using the MAF and MAP sensors, as shown in Fig. 1, with the engine in steady state operation. The manifold absolute pressure pm can be determined employing the principles of conservation of mass and energy. Conservation of momentum is also applied by assuming that a uniform pressure exists in the intake manifold between the throttle body and the intake valves (Powell & Cook, 1987). Therefore the intake manifold pressure dynamic can be expressed as _ ai  m _ ao Þ p_ m ¼ ρðm

ð6Þ

where ρ ¼ RT m =V m , and R is the gas constant of air, Vm is the manifold volume and Tm is the manifold air temperature. It is also assumed that the density ρ remains constant in the entire operation range. 3.1.3. Induction to power stroke delay Considering the engine as a reciprocating pump then a mixture sample of fuel plus air is admitted by each cylinder and will produce torque through the combustion process after 3601 CA (crank angle). This lag is known as induction to power stroke delay and can be expressed in terms of the engine speed (n) as

τd ¼

60 n

ð7Þ

3.1.4. Air-to-fuel ratio and spark advance control There are two other variables that affect directly the performance of the engine regarding torque production, fuel economy and pollutant emissions, which are (i) the air-to-fuel ratio and (ii) the spark advance. The air-to-fuel ratio is defined as the relationship between air _ ao and the injected fuel mass flow rate m _ fi , also mass flow rate m expressed normalized by the stoichiometric ratio known as excess air ratio or λ, that is

λ¼

_ ao 1 m _ fi AFRs m

ð8Þ

where AFRs is the stoichiometric ratio, typically AFRs  9 for ethanol. The impact of the excess of air ratio on the engine torque can be expressed by a quadratic function f λ ðλÞ ¼ 1  αλ ðλ  λMBT Þ2

ð9Þ

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where αλ is an experimental constant and λMBT is the excess air ratio of maximum brake torque. A graphical representation of Eq. (9) is shown in Fig. 2. While a genset engine operating with lower than stoichiometric lambda values enhances the output power, it also increases the specific fuel consumption (g/kW h), which directly affects the output power specific monetary cost. On the other hand, the use of high air excess is proved to enhance the engine thermal efficiency, leading to lower output power cost, but higher in-cylinder temperature during combustion and higher cyclic variability must be considered. Thus, λ values around stoichiometric are often used (0:95 o λ o 1:05) in a compromise between stable operation, engine integrity and specific output power cost. Since the intake air mass flow rate is controlled by the throttle angle, then the variable λ is controlled through the commanded _ fc ). Not all the fuel commanded by the ECU fuel mass flow rate (m enters immediately into the cylinder as fuel vapor, but part of it deposits on the intake ports as liquid. This phenomena is known as Wall-Wetting and can be described by the Aquino (1981) model, shown in Fig. 3. Mathematically speaking the Aquino model is a dynamic equation that relates the commanded fuel mass flow rate _ fc ) and the injected fuel mass flow rate (m _ fi ). (m Fuel injection and spark timing are controlled by an ECU, capable of providing allocations of fuel injected at any given load/rpm. The ECU also has an internal throttle driver which controls the throttle angle through the throttle signal uth as seen in Fig. 1. The air-to-fuel ratio control is beyond the scope of this paper, for more information about air-to-fuel ratio control refer to Guzzella, Simons, and Geering (1997), Wagner et al. (2003), and Ebrahimi et al. (2012). The spark advance (uδ ) is the angular position, before top dead center (BTDC), in which an electrical discharge is applied in the spark plug to start combustion and produce engine torque. More details on each air-to-fuel ratio and spark advance control can be found in Guzzella and Onder (2010). The impact of the spark advance on the engine torque can be expressed by a quadratic function f δ ðuδ Þ ¼ 1  αδ ðuδ  δMBT Þ2

ð10Þ

where

αδ

is an experimental Fig.

57

constant and δMBT is the minimum spark advance which produces the maximum brake torque. A graphical representation of Eq. (10) is shown in Fig. 2. For this specific engine, in the operation speed range, experimental tests showed that the δMBT is located between 251 and 301 CA BTDC. Spark timing control is also performed by the ECU, through the use of a previous calibrated look-up table. One fact that can limit the operation in MBT is the knocking phenomenon. This phenomenon is the auto ignition and instantaneous burn of the entire in-cylinder charge which leads to higher in-cylinder peak pressure and temperature. Continuous operation under knocking conditions can cause severe damage to the internal parts of the engine. 3.1.5. Engine torque The torque produced by the engine can be expressed as a function of the main variables responsible for the combustion, that are air mass flow rate delayed by the power stroke delay, the fuel mass flow rate included in the air-to-fuel ratio, spark timing, and the crankshaft speed which modify how the combustion process happens. A similar idea has been reported in Xiaohong and Tielong (2011):

τe ðm_ dao ; n; uδ ; λd Þ ¼ f τ ðm_ dao ; nÞf δ ðuδ Þf λ ðλd Þ

ð11Þ

_ ao ðt  τd Þ and λ ¼ λðt  τd Þ and function f τ ðÞ is where ¼m shown as follows: d

_ dao m

_ dao ; nÞ ¼ τ0 þ τ1 m _ dao Þ2 þ τ4 m _ dao þ τ2 n þ ðm _ dao n þ τ5 n2 f τ ðm

ð12Þ

where τ0–5 are experimental constants. 3.1.6. Rotational dynamic Rotational dynamics are described by Newton's second law, and two parameters are needed to described it properly: (i) friction coefficient b which is in fact not constant and depends on the crankshaft speed and (ii) moment of inertia J ¼ J gen þ J en ðθÞ, where the first Jgen is the moment of inertia of the electrical generator which is constant and the second one J en ðθÞ is the moment of inertia of the ICE which is a function of the crank shaft position ðθÞ (Brusa, Delprete, & Genta, 1997), due to the variable distribution of mass around the crankshaft axis. However, for low frequency analysis only constant lumped moment of inertia is enough to describe rotational dynamics in the range of speed considered in this paper. And the friction coefficient is considered as a linear function of speed b(n) as shown below: bðnÞ ¼ b0 þ b1 n

ð13Þ

where b0–1 are experimental constants. Engine rotational dynamics follow

2. Lambda function (left) and spark function (right). The dots correspond to experimental data. Both functions are dimensionless.

π π τe  τl  bðnÞ n ¼ J n_ 30

Fig. 3. Block diagram of the event based model.

30

ð14Þ

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where τl is the load torque and the factor π =30 is due to the unit conversion of engine speed (from rpm to rad/s).

Hence Eq. (17d) can be rewritten as _ daoθ ; nθ ; uδ ; λd Þ  f τlθ ðτlθ ; nθ Þ n_ θ ¼  f frθ ðnθ Þ þ f τθ ðm

ð20Þ

3.2. Event based model 4. Model validation Another model used for control and analysis purposes is the model in the crankshaft angle domain or Event Based model. This _ ai , m _ ao , pm, λ as a function of model treats the engine variables n, m crankshaft angle instead of the time. Fig. 3 shows the entire block diagram of the event based model. The main advantages of the event based model over the time based model are:

 The time delay between the air–fuel mixture induction and the torque production becomes constant.

 The air mass flow rate (g/rad) admitted by the engine is practically a linear function of the manifold absolute pressure.

 The engine control actions are synchronized with the crankshaft angle. The representation of the variables in the crankshaft angle domain (θ) can be achieved through a change of variables. Noting that dθ π n ¼ dt 30

ð15Þ

then using the chain rule, any variable at the form dx=dt can be represented in the crankshaft angle domain dx=dθ through the transformation dx dx dt dx 30 ¼ ¼ dθ dt dθ dt π n

ð16Þ

Using the transformation above the following equations are obtained: _ aiθ ¼ m

dmai 30 _ m ¼ π nθ ai dθ

ð17aÞ

_ aoθ ¼ m

dmao 30 ¼ f p ðpmθ Þ π dθ

ð17bÞ

p_ mθ ¼ n_ θ ¼

  dpm _ aoθ _ aiθ  m ¼ρ m dθ

The event based model stated above is implemented in simulation and compared with experimental tests to validate the model. A comparison of the experimental engine speed and manifold absolute pressure against the proposed model is shown in Fig. 4. The experimental result is taken when the engine operates with a load torque of τlθ ¼ 100 N m and with a random input signal uth. The errors between the simulation and experimental signals are also shown. As can be seen in Fig. 4 the absolute errors for engine speed and MAP are smaller than 75 rpm and 7.5 kPa, respectively, which means a relative error smaller than 10%. The same accuracy is found for all load and speed range, which makes the proposed model suitable for model based control design. Usually the engine speed controller is implemented in a digital signal processor (DSP), then a discrete representation is desired. It is also desired to obtain a linear model, which is useful for a controller design using classical methods, e.g. frequency domain approaches. The next section describes the development of a linear and discrete model in the crankshaft angle domain.

5. Linear discrete event based model The nonlinearities of the system are concentrated in Eq. (19), which can be linearized over an equilibrium point as shown in Eqs. (21)–(24), where the (~) correspond to the deviation around the equilibrium point.

ð17cÞ

dn 30 900 τ  τlθ ¼  bðnθ Þ þ 2 eθ Jπ nθ Jπ dθ

and for the engine torque:  π n d θ _d m ;n ;u ;λ τeθ m_ daoθ ; nθ ; uδ ; λd ¼ τe 30 aoθ θ δ θ

ð17dÞ

ð18Þ

_ daoθ ¼ m _ aoθ ðθ  2π Þ and λθ ¼ λθ ðθ  2π Þ. where m Some auxiliary functions which group the system nonlinearities are also defined:  30TCðuth ÞΨ~ ppmθ inθ ð19aÞ f thθ ðuth ; pmθ ; pinθ ; nθ Þ ¼ π nθ d

f pθ ðpmθ Þ ¼

30 f p ðpmθ Þ

_ daoθ ; nθ ; uδ ; λ Þ ¼ f τθ ðm d

f τ lθ ðτ l ; nθ Þ ¼ f frθ ðnθ Þ ¼

ð19bÞ

π

900 τ J π 2 nθ lθ

30 bðnθ Þ Jπ

900 τ ðm_ d ; n ; u ; λd Þ J π 2 nθ eθ aoθ θ δ

ð19cÞ ð19dÞ ð19eÞ

Fig. 4. Comparison of simulation and experimental results for the engine operating with τlθ ¼ 100 N m with a random input signal uth results for engine speed and manifold absolute pressure.

J.R. Tibola et al. / Control Engineering Practice 35 (2015) 54–66

First, for the intake air mass flow rate: _~ aiθ ¼ k0 u~ th þ k1 p~ mθ þ ki p~ inθ þ k2 n~ θ m

6. Engine speed control ð21Þ

where k0 ¼ ∂f thθ =∂uth ; k1 ¼ ∂f thθ =∂pmθ ; ki ¼ ∂f thθ =∂pinθ ; k2 ¼ ∂f thθ =∂nθ . It is assumed that the atmospheric pressure does not change so, ki p~ inθ ¼ 0. Then, for the output air mass flow rate: _~ daoθ ¼ k3 p~ mθ ðt  τd Þ m

ð22Þ

where k3 ¼ ∂f pθ =∂pmθ ; For the engine torque:

τ~ eθ ¼ k4 m_~ daoθ þ k5 n~ θ

ð23Þ

_ aoθ ; k5 ¼ ∂f τθ =∂nθ . where k4 ¼ ∂f τθ =∂m For the load torque: f~ τlθ ¼ k6 τ~ lθ þ k7 n~ θ

ð24Þ

k6 ¼ ∂f τlθ =∂τl ; k7 ¼ ∂f τlθ =∂nθ . And for the friction coefficient: b~ ¼ k8 n~ θ

ð25Þ

where k8 ¼ ∂f frθ =∂nθ . All the partial derivatives above are evaluated over the equilibrium points. The equilibrium points are founded making all the derivatives of Eq. (17) equal to zero and then solving the variables of interest in Eqs. (18) and (19) as a function of the engine speed and load torque, therefore the parameters k0–8 will also be in function of the engine speed and load torque operation points. The next step is the system discretization. As described by Chang, Fekete, Amstutz, and Powell (1995), the largest sampling interval which allows correctly the representation of the engine events is given by 4π Δθs ¼ ncyl

ð26Þ

where ncyl is the number of cylinders. As a four cylinder engine is used in this work the sampling interval is π rad/s, or one-half of a crankshaft revolution. For the discretization, it is first considered as a continuous integral with respect to the crankshaft angle θ, as shown below: Z ð27Þ y¼ uðθÞ dθ Then using the ZOH approximation with the sampling interval

Δθs as shown below:

y½k ¼ y½k  1 þ Δθs u½k

59

ð28Þ

In this section it will be presented the design of two speed controllers. The first one is a single loop gain-scheduled proportional plus integral (PI) controller. And the second one is a cascade controller, or as named here dual loop controller. This controller is an improvement over the single loop controller (Jelali, 2012), and it is used here due to the availability of a manifold absolute pressure (MAP) measurement, which is an important intermediate state variable related to the torque production. This controller is achieved with an inner gain-scheduled feedback loop which uses the MAP measurement, and an outer loop composed of a gainscheduled PI controller to track the speed reference with zero steady state error. A dual loop control is particularly useful when the disturbances are associated with the inner loop variable or the output variable exhibits a non-linear behavior (Shinskey, 1996). Beyond that, a dual loop control design usually leads to a smaller minimum-achievable variance than that with single loop control and can provide performance enhancement in comparison with single loop control (Ko & Edgar, 2000). An integral anti-windup scheme and a feed forward action are also proposed to improve transient performance for both strategies. 6.1. Single control loop design The first proposed strategy is a gain-scheduled PI controller. The control design is based on the linearized model shown in Fig. 6. The discrete transfer function of the linearized model is shown in Eq. (32). And the block diagram of the single control loop is shown in Fig. 8: Gsn; ~ u~ ðzÞ ¼

k0 kp kn z4  ðωn þ ωp Þz3 þ ωn ωp z2  k2 kp kn

ð32Þ

The basic idea used here for PI controller design is that since the parameters k0–7 of the transfer function are in function of the speed reference and load torque, then a PI controller can be designed for each operational point. After the design had been performed, the PI gains are fitted to a polynomial equation dependent on the engine speed reference and load torque to be used online in a DSP. The single loop PI control design is performed as follows. First consider a transfer function in Z domain as shown below: Gs ðzÞ ¼

z  ωs z1

ð33Þ

where ωs is the frequency of the zero of the PI. And the transfer function of the PI itself is given by PI s ðzÞ ¼

uPI ðzÞ ¼ ks Gs ðzÞ es ðzÞ

ð34Þ

It is also common for design of discrete controllers the usage of the Z transform, where the complex variable z is defined as z ¼ eT s ðj2π f Þ

ð29Þ

where Ts is the sampling time, f is the frequency ¼1/t, and t is the time. The complex variable z is redefined by the replacement of the time t by the crankshaft angle θ, as shown below: z ¼ eΔθs ðj2π =θÞ

ð30Þ

Fig. 5. Block diagram of the discrete linearized event based model.

Finally applying the Z transform in Eq. (28) one can obtain yðzÞ ¼

Δθs z1

uðzÞ

ð31Þ

The block diagram of the discrete linearized event based model is shown in Fig. 5, and can be simplified as shown in Fig. 6, where kp ¼ ρΔθs ; ωp ¼ 1 þ ρΔθs ðk1  k3 Þ; kn ¼ k3 k4 Δθs and ωn ¼ 1 þ Δθs ðk5  k7  k8 Þ.

Fig. 6. Block diagram of the simplified discrete linearized event based model used for control design.

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where uPI is the PI control action, es is the speed tracking error for single loop design, and ks is the discrete proportional gain. Since the PI controller is a lag compensator, the desired phase n lag ϕs that the PI inserts in a certain frequency, e.g. the desired crossover frequency ωnc , is a function of ωs, which can be found replacing (30) in (33), then taking its phase and isolating ωs :

ωs ¼

sin ðΔθs ωnc Þ þ 2 sin sin ðΔθs ωnc Þ  2 sin

 

Δθs ωnc 2



n

Δθs ωc 2

n

tan ðϕs Þ tan ðϕs Þ n

ð35Þ

where ϕs is a function of the desired phase margin PMn and the phase margin of the plant at ωnc : n

ϕns ¼ PMn ðπ þ ∠Gsn;~ u~ ðejΔθs ωc ÞÞ n

ð36Þ

The chosen desired phase margin and crossover frequency are PMn ¼901 and ωnc ¼ 0:01 rad=rad, respectively. The controller is designed for the entire operation speed range 1000 rpm r nref r 2000 rpm and load torque range 0 N m r τl r 200 N m. For these design criteria the frequency of the PI remains constant ωs  0:98. The PI gains are obtained considering that in ωnc the gain of the open loop transfer function is unitary, therefore the PI gain is expressed as ks ¼

1 jΔθs ωnc ÞjjG ðejΔθ s ωnc Þj jGsn; s ~ u~ ðe

ð37Þ

The analytical solution for the problem above is quite complex to be solved online, then a numerical solution is applied and a polynomial function is fitted as shown below: ref ref ref 2 ks ðnref θ ; τ lθ Þ ¼ κ s0 þ κ s1 nθ þ κ s2 τ lθ þ κ s3 nθ τ lθ þ κ s4 ðnθ Þ

ð38Þ

where κ s0–s4 are constants fitted with a coefficient of determination R2 ¼0.9847. The graphical representation of Eq. (38) is shown in Fig. 7.

Fig. 7. PI gain ks for single loop design as a function of the engine speed reference and load torque.

6.1.1. Feed forward action The electromagnetic torque produced by the generator is treated in the engine side as a disturbance. In Yang, Zheng, Shen, Kako, and Yoshida (2007) a feed forward action is proposed to compensate the magnetic torque. The same idea is used here since the magnetic torque is known and it is available to the speed control loop in a genset application through a communication network. The feed forward action is designed considering the engine in steady state operation. This is performed considering all the derivatives of Eq. (17) equal to zero and solving the control action uth in Eq. (18) and (19) as a function of the engine speed and load torque. The analytical solution for the problem above is also complex due to the nonlinearities involved, then a numerical solution is applied and a polynomial function is fitted as shown below: ref ref 2 uff ðnref θ ; τ lθ Þ ¼ f 0 þf 1 nθ þ f 2 τ lθ þf 3 ðnθ Þ ref 2 ref 2 2 3 þ f 4 nref θ τ lθ þ f 5 τ lθ þ f 6 ðnθ Þ τ lθ þ f 7 nθ τ lθ þ f 8 τ lθ

ð39Þ

where f 0–8 are constants fitted with a coefficient of determination R2 ¼0.9945. The graphical representation of Eq. (39) is shown in Fig. 9. 6.1.2. Anti-windup for single loop design Due to the use of an integral action in the presence of actuator limitation (uMIN o uth o uMAX ) an anti-windup action is used, which can improve transient behavior (Åström & Hägglund, 2006). The anti-windup algorithm implementation is shown in Eq. (40) together with the state space representation of the single PI controller and the feed forward action: vth ½k ¼ KI s xs ½k þ KP s es ½k þ uff ½k xs ½k þ 1 ¼ xs ½k þ es ½k  KW s ðvth ½k  uth ½kÞ

ð40Þ

where KP s ¼ ks and KI s ¼ ks ð1  ωs Þ are the state space PI gains; xs is the PI state; es is the tracking error; KWs is the anti-windup gain; and vth is the non-saturated control action and uth is the saturated

Fig. 9. Feed forward action as a function of the engine speed reference and load torque.

Fig. 8. Proposed dual and single loop (kf ¼ 0) controller with feed forward and anti-windup action applied to the linearized event based engine model.

J.R. Tibola et al. / Control Engineering Practice 35 (2015) 54–66

control action shown below: 8 > < uMIN if vth ½k o uMIN uth ½k ¼ uMAX if vth ½k 4 uMAX > : v ½k otherwise th

61

ð41Þ

The anti-windup gain is KW s ¼ ð1  ωsat Þ=KI s , where ωsat is the pole position of the PI at the saturation (0 r ωsat o 1). After many simulations it was chosen as ωsat ¼ 0:8. 6.2. Dual control loop design The second proposed strategy is a dual loop controller, in which the inner MAP loop is composed of a gain-scheduled feedback gain, and the outer speed loop is composed of a gain-scheduled PI controller. The inner loop is chosen as a simple feedback gain because it is the simplest controller that can perform the required task without an increase in the system order. The dual loop design procedure is performed as follows: first the inner loop is tuned using pole placement, with a crossover frequency about one-tenth of throttle driver crossover frequency. This condition makes the inner loop as fast as possible without interaction with the throttle driver control loop (Laughton & Warne, 2003). Then, the inner loop dynamics are included in a whole transfer function and then outer loop is designed as in single loop design. The crossover frequency of the inner and outer loop also must be at least one decade apart from each other to avoid interaction. 6.2.1. Inner control loop design To simplify the design of the MAP loop the constant k2 is disregarded since its value is small k2 A ð  1  10  4 ;  4  10  5 Þ and it does not change significantly the phase of the system at the crossover frequency. Therefore the transfer function from the input control signal to the absolute manifold pressure becomes Gp; ~ u~ ðzÞ ¼

k0p z  ωp

ð42Þ

where k0p ¼ k0 kp . Then applying the feedback gain kf as shown in Fig. 8, the closed loop form of the transfer function (42) becomes Gp; ðzÞ ¼ ~ u;CL ~

k0p z  ðωp  kf k0p Þ

ð43Þ

ωp  ωnp k0p

ð44Þ

where ωnp is the desired discrete pole place chosen approximately one decade below the crossover frequency of the throttle driver control loop in the crankshaft domain ωthθ :

ωnp ¼ e  Δθ ωthθ =10  0:5

ð45Þ

where ωthθ ¼ 30ωth =π nmin and nmin ¼1000 rpm. Therefore the gain kf is expressed as a function of the speed reference end load torque as shown below: ref ref ref 2 kf ðnref θ ; τ lθ Þ ¼ κ f 0 þ κ f 1 nθ þ κ f 2 τ lθ þ κ f 3 nθ τ lθ þ κ f 4 ðnθ Þ

feedback gain is shown as follows: Gdn; ~ u~ ðzÞ ¼

k0 kp kn z2 ðz  ωn Þ½ðz  ωp Þ þ k0 kp kf  k2 kp kn

ð47Þ

Again, considering a transfer function as shown in Eq. (48) in the Z domain Gd ðzÞ ¼

z  ωd z1

ð48Þ

where ωd is the frequency of the zero of the PI. And the transfer function of the PI dual loop is given by PI d ðzÞ ¼

uPI ðzÞ ¼ kd Gd ðzÞ ed ðzÞ

ð49Þ

where ed is the speed tracking error for dual loop design, and kd is the discrete proportional gain. The outer loop control is designed using the same procedure used in the single loop design, with the same desired crossover frequency ωnc and phase margin PMn. The frequency of the zero of the PI is shown as follows:  n Δθs ωnc tan ðϕd Þ sin ðΔθs ωnc Þ þ 2 sin 2  ð50Þ ωd ¼ n n Δθs ωc tan ðϕd Þ sin ðΔθs ωnc Þ  2 sin 2 and the desired phase lag ϕd is given by n

And the feedback gain kf can be calculated using pole placement: kf ¼

Fig. 10. Gain kf for dual loop design as a function of the engine speed reference and load torque.

ð46Þ

where κ f 0–f 4 are constants fitted with a coefficient of determination R2 ¼0.9865. The graphical representation of Eq. (46) is shown in Fig. 10. 6.2.2. Outer control loop design For the outer loop design the plant model is modified by the pressure feedback loop. The new plant transfer function with the

ϕnd ¼ PMn  ðπ þ∠Gdn;~ u~ ðejΔθs ωc Þ Þ n

ð51Þ

The frequency of the PI also remains constant ωd  0:98 for dual control loop design. And the PI gain is given by kd ¼

1 jΔθ s ωc ÞjjG ðejΔθs ωc Þj jGdn; ~ u~ ðe d n

n

ð52Þ

As before, a numerical solution is applied in Eq. (52) and a polynomial function is fitted as shown below: ref ref ref 2 kd ðnref θ ; τ lθ Þ ¼ κ d0 þ κ d1 nθ þ κ d2 τ lθ þ κ d3 nθ τ lθ þ κ d4 ðnθ Þ

ð53Þ

where κ d0–d4 are constants fitted with a coefficient of determination R2 ¼0.9715. The graphical representation of Eq. (53) is shown in Fig. 11. 6.2.3. Anti-windup for dual loop control The same anti-windup action used for single loop control is used here, but with an addition of the MAP loop, as shown below: vth ½k ¼ KI d xd ½k þ KP d ed ½k  kf pm ½k þ uff ½k xd ½k þ 1 ¼ xd ½k þed ½k  KW d ðvth ½k  uth ½kÞ

ð54Þ

where KP d ¼ kd and KI d ¼ kd ð1  ωd Þ are the state space PI gains;

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KWd is the anti-windup gain; xd is the PI state; ed is the tracking error and uth is the saturated control action calculated in the same way as in Eq. (41). The anti-windup gain is KW d ¼ ð1  ωsat Þ=KI d , where ωsat is the pole position of the PI at the saturation which is the same for the single loop control. The same feed forward action described in Section 6.1.1 is used here for the dual loop controller.

A Bode diagram of the open loop transfer function (plant þ controller) is shown in Fig. 12, when nref θ ¼ 1500 rpm and τlθ ¼ 100 N m, for both single and dual loop control designs. It is clear to see that both designs have the same crossover frequency, as designed, however the gain at high frequency is higher for dual loop design, this characteristic is important for transient response. The entire block diagram of the dual control loop is shown in Fig. 8.

7. Experimental results

Fig. 11. Gain kd for dual loop design as a function of the engine speed reference and load torque.

Both controllers proposed in this paper have been implemented in the DSP TMS320F28335 from Texas Instruments on a heavy duty Cummins engine, originally running on diesel but then converted to ethanol fuel. The nominal parameters of the ICE are shown in Table 1. To measure the Manifold Absolute Pressure, a piezoresistive pressure transducer MPX4250A was installed at the intake runner. This sensor was chosen due to its linearity and response time smaller than 1 ms. For engine speed calculation an incremental encoder was used. This sensor also allows the generation of the sampling events synchronized with the crank angle. The load torque applied to the ICE is emulated by a setup, also developed in the research group, composed of a permanent magnet synchronous generator (PMSG) and a PWM grid-connected back-toback converter. The back-to-back converter is composed of two parts: (i) bidirectional PWM rectifier, responsible for the control of the PMSG currents and consequently controls the electromagnetic torque applied to the ICE, (ii) bidirectional PWM inverter connected to the grid, responsible to deliver the energy produced from the PMSG to the grid, and from the grid to the PMSG when necessary. A schematic of the full setup is shown in Fig. 13. The ICE and the PMSG of the developed test bench is shown in Fig. 14. This setup works as a regenerative dynamometer which can impose various load torque profiles (positive or negative torque), e.g. motored tests for friction model development and other studies, all are very useful for an engine test bench.

Fig. 12. Bode diagram of the open loop transfer function (OLTF) for single loop (red/ continuous) and dual loop (blue/dashed) design, when nref θ ¼ 1500 rpm and τlθ ¼ 100 N m.

Table 1 Engine specification. Model Cylinder number Power Displacement (cm3) Compression ratio Cooling

Cummins 4BT 3.9G4 4-cylinder, in-line 50 kW (  67 HP) 3920 11:1 Water-cooled Fig. 14. Engine test bench.

Fig. 13. Full setup, showing the ethanol ICE, three phase PMSG and PWM back-to-back converter.

J.R. Tibola et al. / Control Engineering Practice 35 (2015) 54–66

63

Fig. 17. Torque disturbance rejection test with constant speed reference of nref θ ¼ 1750 rpm for dual loop controller, with the presence of feed forward action and without it (1 Sample¼ 1/2 Revolution).

Fig. 15. Speed reference tracking test with constant load torque of τlθ ¼ 100 N m (1 Sample ¼1/2 Revolution).

the speed reference were applied for the single and dual loop controllers, in the crankshaft angle domain. It can be seen that as the speed reference evolves, the gains of the controller also evolve as expected. The steady state errors are zero for both strategies due to the presence of the integrator. However, the transient responses are different, the dual loop controller has a smaller under/overshoot than the single loop controller due to the presence of the MAP loop. And in Fig. 16 a result for the back-to-back holding of load torque at τlθ ¼ 200 N m is shown with the same speed reference profile used in Fig. 15, in the crankshaft angle domain. Again the dual loop controller has a smaller under/overshoot than the single loop controller. For the single loop controller when the speed reference goes from 1500 rpm to 1250 rpm the control action decays too much and so does the engine speed until the point that the engine stall. With a load torque of 200 N m the importance of the MAP loop becomes evident which was capable of preventing the engine to stop running. To also show the importance of the feed forward action, steps on load torque were applied to the ICE with a speed reference of 1750 rpm. The response is shown in Fig. 17 in the crankshaft angle domain. It is clear that the feed forward action avoids the excessive speed deviation over the load torque disturbance. Since the feed forward is an immediate action, this prevents the PI to integrate an excessive error which leads to a smaller under/overshoot.

8. Conclusion

Fig. 16. Speed reference tracking test with constant load torque of τlθ ¼ 200 N m (1 Sample ¼1/2 Revolution).

The first experiment was the speed tracking with constant load torque. In Fig. 15 an experimental result for the back-to-back holding of load torque at τlθ ¼ 100 N m is shown. Then steps on

This paper presented in detail the development of a discrete dynamic MVEM suitable for the design of speed controllers for an ethanol fueled ICE. The control design methodology presented is based on the discretization and linearization of the Crank Based MVEM as a function of the speed reference and load torque, thereby the controller gains were also in function of those variables, providing thus a very simple methodology for model based gain-schedule controller design. Two speed control approaches were investigated: a single loop controller and a dual loop controller. For the same design

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specifications the dual loop controller has shown a better transitory response than the single loop controller. That is possible since with two engine state measurements more information is available from the plant. And as the manifold absolute pressure changes faster than the engine speed, an improvement in the transitory response has been observed. In addition, the use of the feed forward action has played an important rule for load torque disturbance rejection mainly due to its immediate action when compared to the PI controller without feed forward.

Ψ~ ðÞ exponential approximation of Ψ ðÞ (–) fitted constant of Ψ~ ðÞ (–)

CP

TCðÞ throttle body characteristic function (g/s) uth saturated throttle body control signal (–) ωth pole frequency of the throttle driver transfer function (rad/s) u0–3 coefficients of the fitted TCðÞ function n crankshaft speed (rpm) nmin minimum operable crankshaft speed (rpm) nref reference crankshaft speed (rpm) f p ðÞ pressure function (g/s/rpm)

Acknowledgments The authors would like to thanks FAPERGS and CNPq for the support. Appendix A. Notation

τd

_ fi m _ fc m

A.1. Acronyms and abbreviations MVEM ICE PI FF PID MAP

MAF GHG SI CI WRSM AC PMSG PWM ISC WRSC LQR LQG CVT MIMO CA BTDC ECU MBT DSP ZOH OLTF

mean value engine model internal combustion engine proportional integral feed forward proportional integral derivative manifold absolute pressure

mass air flow green house gases spark ignited compression ignited wound rotor synchronous machine alternating current permanent magnet synchronous generator pulse width modulation idle speed control wide range speed control linear quadratic regulator linear quadratic Gaussian continuously variable transmissions multi-input multi-output crank angle (deg) before top dead center engine control unit maximum brake torque digital signal processor zero order holder open loop transfer function

A.2. Mathematical notation _ ai m _ ao m

p0–3 R Vm Tm

intake air mass flow rate (g/s) in-cylinder air mass flow rate (g/s) φ throttle body angle (deg) pm manifold absolute pressure (kPa) pin atmospheric pressure (kPa) Cd discharge coefficient (–) Ath throttle area (m2) Ψ ðÞ normalized pressure influence function (–)

λ λd

AFRs f λ ðÞ

αλ λMBT

uδ f δ ðÞ

αδ δMBT

τe τl _ dao m f τ ðÞ

τ0–5

bðÞ b0–1

θ

Jgen Jen J f thθ ðÞ f pθ ðÞ f τθ ðÞ f τlθ ðÞ f frθ ðÞ ki

coefficients of the fitted f p ðÞ function (–) air gas constant (J/mol/K) manifold volume (m3) manifold air temperature (K) induction to power stroke delay (s) injected fuel mass flow rate (g/s) commanded fuel mass flow rate (g/s) excess air ratio (–) excess air ratio delayed by τd (–) stoichiometric air-to-fuel ratio (–) impact of the λ on the engine torque (–) coefficient of the f λ ðÞ function (–)

coefficient of the f λ ðÞ function, λ of maximum break torque (–) spark advance (deg) impact of the uδ on the engine torque (–) coefficient of f δ ðÞ function (deg  2) coefficient of f δ ðÞ function, minimum uδ for the maximum brake torque (deg) engine torque (N m) load torque (N m) in-cylinder air mass flow rate delayed by τd (g/s) torque function, proportional to τe (N m) coefficients of the fitted f τ ðÞ function (–) friction function (N m s/rad) coefficients of the fitted bðÞ function (–) crank shaft position (rad) moment of inertia of the electrical generator (kg m2) moment of inertia of the ICE (kg m2) total constant lumped moment of inertia (kg m2) _ aiθ (g/rad) auxiliary function, related to m _ aoθ (g/rad) auxiliary function, related to m

τe (rpm/rad) auxiliary function, related to τl (rpm/rad)

auxiliary function, related to

auxiliary function, related to b (rpm/rad) constants of the model linearized around an equilibrium point (g/rad/kPa) constants of the model linearized around an k0–8 equilibrium point (–) sampling interval (rad) Δθs ncyl number of cylinders (–) z complex variable of z transform kp ; kn gains of the simplified discrete (–) event based transfer function (–) ωp ; ωn pole frequency of the simplified discrete event based transfer function (rad/rad) Gsn; ~ u~ ðzÞ small signal transfer function from uth to n Gs(z) transfer function of the single loop PI without the proportional gain

J.R. Tibola et al. / Control Engineering Practice 35 (2015) 54–66

PIs(z) uPI es

transfer function of the single loop PI control action of PIs and PId (–) speed tracking error of PIs (–)

xs

ωs ; ks ϕns ωnc PMn

κ s0–s4 uff f 0–8 uMIN ; uMAX vth KP s ; KI s KWs

ωsat

Gp; ~ u~ ðzÞ k0p kf

κ f 0–f 4

ðzÞ Gp; ~ u;CL ~

ωnp Gdn; ~ u~ ðzÞ Gd(z) PId(z) ed xd ωd ; k d

ϕnd κ d0–d4

KP d ; KI d KWd

state of the PIs controller (–) zero and proportional gain of PIs (rad/rad), (–) desired phase lag inserted by PIs (rad) desired crossover frequency (rad/rad) desired phase margin (rad) coefficients of the fitted ks function (–) feed forward action (–) coefficients of the fitted uff function (–) minimum and maximum value of uth (–) non-saturated throttle control signal (–) state space gains of PIs (–) anti-windup gain of PIs (–) pole position of the PI at the saturation (rad/rad) small signal transfer function from uth to pm gain of Gp; ~ u~ ðzÞ feedback gain of the inner pressure loop (–) coefficients of the fitted kf function (–) closed loop transfer function of Gp; ~ u~ ðzÞ with the feedback gain kf desired pole place of the inner loop (rad/rad) small signal transfer function from uth to n, with the feedback gain kf transfer function of the dual loop PI without the proportional gain transfer function of the dual loop PI speed tracking error of PId (–) state of the PId controller (–) zero and proportional gain of PId (rad/rad), (–) desired phase lag inserted by PId (rad) coefficients of the fitted kd function (–) state space gains of PId (–) anti-windup gain of PId (–)

A.3. Subscripts

k s d

θ

ref ff

with respect to the sampling event k with respect to the single loop design with respect to the dual loop design with respect to crank angle reference or set point feed forward

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