MPC for Active Torsional Vibration Reduction of Hybrid Electric Powertrains

MPC for Active Torsional Vibration Reduction of Hybrid Electric Powertrains

Preprints, Preprints, 8th 8th IFAC IFAC International International Symposium Symposium on on Preprints, IFAC International on Advances 8th in Automot...

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Preprints, Preprints, 8th 8th IFAC IFAC International International Symposium Symposium on on Preprints, IFAC International on Advances 8th in Automotive Automotive Control Symposium Preprints, 8th IFAC International Symposium on Advances in Control Advances in2016. Automotive Control Available online at www.sciencedirect.com June 19-23, Norrköping, Sweden Advances Automotive Control June 19-23,in2016. Norrköping, Sweden June 19-23, 2016. Norrköping, Sweden June 19-23, 2016. Norrköping, Sweden

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IFAC-PapersOnLine 49-11 (2016) 756–761

MPC for Active Torsional Vibration MPC for Active Torsional Vibration MPC for Active Torsional Vibration MPC for Active Torsional Vibration Reduction of Hybrid Electric Powertrains Reduction of Hybrid Electric Powertrains Reduction of Hybrid Electric Reduction of Hybrid Electric Powertrains Powertrains Institute Institute Institute Institute

 , Christian Beidl Raja Sangili Vadamalu Raja Sangili Vadamalu  , Christian Beidl Raja Sangili Vadamalu Raja Sangili Vadamalu  ,, Christian Christian Beidl Beidl for Internal Combustion Engines and Powertrain for Internal Combustion Engines and Powertrain for Internal and Technische Universitaet Darmstadt, Germany for Internal Combustion Combustion Engines and Powertrain Powertrain Technische UniversitaetEngines Darmstadt, Germany Technische Universitaet Darmstadt, Technische Universitaet Darmstadt, Germany Germany

Systems, Systems, Systems, Systems,

Abstract: Abstract: Trends Trends such such as as downsizing downsizing and and downspeeding downspeeding of of internal internal combustion combustion engines engines (ICE) (ICE) Abstract: Trends such and downspeeding of internal combustion engines (ICE) shift the excitation of automotive by the ICE torque leading to demanding Abstract: Trends spectrum such as as downsizing downsizing and powertrains downspeeding of internal combustion engines (ICE) shift the excitation spectrum of automotive powertrains by the ICE torque leading to demanding demanding shift the excitation spectrum of automotive powertrains by the ICE torque leading to requirements for passive vibration reduction measures. Hybrid electric offer shift the excitation spectrum of automotive powertrains the ICE torquevehicles leading(HEV) to demanding requirements for passive passive vibration reduction measures.by Hybrid electric vehicles (HEV) offer aaa requirements for vibration reduction measures. Hybrid electric vehicles (HEV) offer possibility to implement active vibration reduction using the electric traction machine (ETM). requirements for passive vibration reduction measures. Hybrid electric vehicles (HEV) offer a possibility to implement active vibration reduction reduction using using the the electric electric traction traction machine machine (ETM). (ETM). possibility implement active vibration This paper paperto proposes a model model predictive controller using (MPC) forelectric activetraction vibration reduction of aa possibility toproposes implement activepredictive vibration reduction thefor machine (ETM). This a controller (MPC) active vibration reduction of of This paper proposes model for vibration reduction plug-in parallel HEV with downsized 2-cylinder combustion engine. Both time This paper proposes apowertrain model predictive predictive controller (MPC) for active active vibration reduction of aa plug-in parallel HEV a powertrain with aaa controller downsized(MPC) 2-cylinder combustion engine. Both time time plug-in parallel HEV powertrain with downsized 2-cylinder combustion engine. Both and frequency domain variants have been implemented. The frequency domain MPC-variant plug-in paralleldomain HEV powertrain withbeen a downsized 2-cylinder combustion engine. Both time and frequency frequency variants have have implemented. The frequency frequency domain MPC-variant and domain variants been implemented. The domain MPC-variant uses a cost formulation. Low bandwidth the the energy and variants have been frequencycontroller domain of MPC-variant uses frequency a spectral spectral domain cost based based formulation. Lowimplemented. bandwidth of of The the tracking tracking controller of the energy of energy uses a cost formulation. of tracking controller management (EM) has been been extendedLow by bandwidth the regulation regulation based MPC vibration controller. uses a spectral spectral(EM) cost based based formulation. Low bandwidth of the thebased tracking controller of the the energy management has extended by the MPC vibration controller. management (EM) been by regulation MPC controller. Further, degree freedom (DoF) controller has been adopted for the integration management (EM) of has been extended extended by the the structure regulation based MPC vibration vibration controller. Further, aa a 22 2 degree degree ofhas freedom (DoF) controller controller structure hasbased been adopted adopted for the the integration integration Further, of freedom (DoF) structure has been for of the MPC-based vibration controller with the EM controller of the HEV. Simulation Further, a 2 degree of freedom (DoF) controller structure has been adopted for the integration of the MPC-based vibration controller with the EM controller of the HEV. Simulation analysis analysis of vibration controller with EM controller of the analysis the proposed stationary and dynamic ICE conditions shall be of the the MPC-based vibrationunder controller with the the EM controller of operating the HEV. HEV. Simulation Simulation analysis the MPC-based proposed approach approach under stationary and dynamic ICE operating conditions shall be under of the and ICE operating conditions shall discussed focusing on time-variant of the proposed proposed approach under stationary stationary performance and dynamic dynamicand ICE operating constraint conditionshandling. shall be be discussed focusingapproach on the the stationary/dynamic stationary/dynamic performance and time-variant constraint handling. discussed discussed focusing focusing on on the the stationary/dynamic stationary/dynamic performance performance and and time-variant time-variant constraint constraint handling. handling. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Torsional Vibration Reduction, Model Predictive Control, Spectral Model Keywords: Torsional Torsional Vibration Vibration Reduction, Reduction, Model Model Predictive Predictive Control, Control, Spectral Spectral Model Model Keywords: Predictive Control, Control, Plug-In Hybrid ElectricModel Vehicle Keywords: TorsionalPlug-In Vibration Reduction, Predictive Control, Spectral Model Predictive Hybrid Electric Vehicle Predictive Control, Control, Plug-In Plug-In Hybrid Hybrid Electric Electric Vehicle Vehicle Predictive 1. INTRODUCTION tion of aa Hybrid Electric Vehicle (HEV) is aa combination 1. INTRODUCTION INTRODUCTION tion of Electric Vehicle (HEV) is 1. tion of aa Hybrid Hybrid Electric Vehicle (HEV) is aa combination combination of a ICE, an electric traction machine (ETM) and 1. INTRODUCTION tion of Hybrid Electric Vehicle (HEV) is combination of a ICE, an electric traction machine (ETM) and an an of a ICE, an electric traction machine (ETM) and electric energy storage device. Electrical components in Automotive powertrain development faces challenges to of a ICE, an electric traction machine (ETM) and an an electric energy storage device. Electrical components in Automotive powertrain development faces challenges to Automotive powertrain development faces challenges to electric energy storage device. Electrical components in a hybridized powertrain not only offer an opportunity to reduce fuel consumption and pollutant emissions besides energy storage device. Electrical components in Automotive powertrain development faces challenges to electric aa hybridized powertrain not only offer an opportunity to reduce fuel consumption and pollutant emissions besides hybridized powertrain not only offer an opportunity to reduce fuel consumption and pollutant emissions besides enhance the operational efficiency of the powertrain but meeting the customer demands for better performance and a hybridized powertrain not only offer an opportunity to reduce fuel consumption and pollutant emissions besides enhance the operational efficiency of the powertrain but meeting the customer demands for better performance and meeting the customer demands for better performance and enhance the efficiency the but also possibility dampen oscillations. comfort. The European Union has decided to reduce the the aaoperational operational efficiency oftorsional the powertrain powertrain but meeting demands performance also provide provide possibility to to dampenof torsional oscillations. comfort. the Thecustomer European Union for hasbetter decided to reduce reduce and the enhance comfort. The European Union has decided to the also provide a possibility to dampen torsional oscillations. emissions by 40% in 2030 compared to the emission CO also provide a possibility to dampen torsional oscillations. 2 comfort. The European Union has decided to reduce the CO2 emissions by 40% in 2030 compared to the emission This paper presents a model predictive controller (MPC) CO by 40% in to the This paper presents aa model predictive controller (MPC) 2 emissions levels in 1990. A legal framework and binding tar2 CO by 40% in 2030 2030 compared compared to emission the emission emission This paper predictive controller 2 emissions levels in 1990. A legal framework and binding emission tarfor reduction of torsional oscillation in hybridized powerThis paper presents presents a model model predictive controller (MPC) (MPC) levels in 1990. A legal framework and binding emission tarfor reduction of torsional oscillation in hybridized powergets have been formulated to achieve this goal. Emissions levelshave in 1990. Aformulated legal framework and binding emission tar- train gets been to achieve this goal. Emissions for reduction of torsional oscillation in hybridized powerand its integration with HEV/Plug-in HEV Energy for reduction of torsional oscillation in hybridized powergets have been formulated to achieve this goal. Emissions train and its integration with HEV/Plug-in HEV Energy of the vehicle fleet of automotive manufacturers shall not gets have been formulated to achieve this goal. Emissions train and its integration with HEV/Plug-in HEV Energy of the vehicle fleet of automotive manufacturers shall not Management (EM) controller. This paper is structured as train and its integration with HEV/Plug-in HEV Energy of the vehicle fleet of automotive manufacturers shall not Management (EM) controller. This paper is structured exceed 130 g g CO CO per kilometer by bymanufacturers 2015 and and 95 95 g g shall CO2 not per Management (EM) controller. This paper is structured as 2 per of the vehicle fleet of automotive as exceed 130 kilometer 2015 CO per 2 2 follows. Following the introduction, section 2 presents the Management (EM) controller. This paper is structured as exceed 130 g CO kilometer by and CO Following the introduction, section 22 presents the kilometer In order to meet the stringent exceed 130by g 2020. CO222 per per kilometer by 2015 2015 and 95 95 g gdemands, CO222 per per follows. follows. Following the introduction, section presents the kilometer by 2020. In order to meet the stringent demands, problem and state of the art solution techniques. Section follows. Following the introduction, section 2 presents the kilometer by 2020. In order to meet the stringent demands, problem and state of the art solution techniques. Section among other measures three technologies are being kilometer by 2020. In order tomajor meet the stringent demands, problem and state of art techniques. Section among other measures three major technologies are being 3problem theory to solve the vibration andthe state of the theemployed art solution solution techniques. Section among other measures three major technologies are being 33 describes describes the theory employed to solve the vibration employed : Downsizing, Downspeeding and Hybridization. among other measures three major technologies are being describes the theory employed to solve the vibration employed : Downsizing, Downspeeding and Hybridization. reduction problem using MPC. The results obtained from 3reduction describes the theory employed toresults solve obtained the vibration employed : Downsizing, Downspeeding and Hybridization. problem using MPC. The from employed : Downsizing, Downspeeding andengine Hybridization. reduction problem using MPC. The results obtained from the implementation are discussed in section 4. Downsizing refers to the reduction of the displacereduction problem using MPC. The results obtained from the implementation are discussed in section 4. Downsizing refers to the reduction of the engine displaceDownsizing refers to the reduction of the engine displacethe implementation are discussed in section 4. ment or the number of cylinders or a combination of both. the implementation are discussed in section 4. Downsizing refers to the reduction of the engine displacement or the number of cylinders or a combination of both. 2. STATE OF THE ART ment or number of or both. In order order to retain retain performance, downsized engines of operate 2. STATE OF THE ART ment or the the number of cylinders cylindersdownsized or a a combination combination of both. In to performance, engines operate 2. STATE OF THE ART In order to retain performance, downsized engines operate 2. STATE OF THE ART at higher mean effective pressure usually with charging deIn order to retain performance, downsized engines operate at higher mean effective pressure usually with charging deTorsional excitation resulting from cyclic at higher mean effective pressure usually with charging deTorsional excitation resulting from cyclic irregularity irregularity is is vices reducing thereby the pumping / gas exchange losses. at higher mean effective pressure usually with charging devices reducing thereby the pumping / gas exchange losses. Torsional excitation resulting from cyclic irregularity is an inherent problem of ICE powered automotive powvices reducing thereby the pumping / gas exchange losses. Torsional excitation resulting from cyclic irregularity is an inherent problem of ICE powered automotive powExhaust gas turbo-charging methods enable downsized vices reducing thereby the pumping / gas exchange losses. Exhaust gas turbo-charging methods enable downsized an inherent problem of ICE powered automotive powertrains. Displacement of eccentric masses and periodic an inherent problem of ICE powered automotive powExhaust gas turbo-charging methods enable downsized ertrains. Displacement of eccentric masses and periodic engines to deliver torque levels comparable to naturally Exhaust gas turbo-charging methods enabletodownsized engines to deliver torque levels comparable naturally ertrains. Displacement of eccentric masses and periodic energy conversion in the aa pulsating engines to deliver levels comparable to naturally ertrains. Displacement of cylinder eccentric result massesin and periodic energy conversion in the cylinder result in aspirated combustion (ICE) with larger engines tointernal deliver torque torque levelsengine comparable to naturally aspirated internal combustion engine (ICE) with larger energy conversion in the cylinder result in aa pulsating pulsating engine torque. This torque when left unchecked excites aspirated internal combustion engine (ICE) with larger energy conversion in the cylinder result in pulsating engine torque. This torque when left unchecked excites displacement and higher stationary torque over a larger aspirated internal combustion engine (ICE) with displacement and higher stationary torque over aa larger engine torque. This torque when left unchecked excites the torsionally non-stiff powertrain resulting in vibrations displacement and higher stationary torque over larger engine torque. This torque when left unchecked excites the torsionally non-stiff powertrain resulting in vibrations speed range. Downspeeding reduces operating speed hence displacement and higher stationary torque over a larger speed range. Downspeeding reduces operating speed hence the torsionally non-stiff powertrain resulting in vibrations that are detrimental to passenger comfort. There exists speed range. Downspeeding reduces operating speed hence the torsionally non-stiff powertrain resulting in vibrations that are detrimental to passenger comfort. There shifting the combustion engine operation to fuel efficient speed range. Downspeeding reduces operating speed hence that are detrimental to passenger comfort. There exists shifting the combustion engine operation to fuel efficient exists different approaches to reduce vibration in lightly damped shifting the combustion engine operation to fuel efficient are approaches detrimentaltotoreduce passenger comfort. There exists different vibration in lightly damped regions. As an undesired side-effect these trends shifting the combustion engine operation to fuel increase efficient that regions. As an undesired side-effect these trends increase different approaches to reduce vibration in lightly damped flexible structures such as stiffening, damping and isoregions. As an undesired side-effect these trends increase different approaches to reduce vibration in lightly damped flexible structures such as stiffening, damping and isothe cyclic irregularity (torque oscillations) due to either regions. As an undesired side-effect these trends increase the cyclic irregularity (torque oscillations) due to either flexible structures such as stiffening, damping and isolation (Preumont, 2011). Stiffening shifts the structural flexible structures such as stiffening, damping and isothe cyclic irregularity (torque oscillations) due to either lation (Preumont, 2011). Stiffening shifts the structural high amplitude and/or low frequency torque oscillations. the cyclic irregularity (torque oscillations) due to either high amplitude and/or low frequency torque oscillations. lation (Preumont, 2011). Stiffening shifts the structural resonance outside the excitation band. Through isolation high amplitude and/or low frequency torque oscillations. lation (Preumont, 2011). Stiffening shifts the structural resonance outside the excitation band. Through isolation Hybridized powertrains comprises of more than one energy high amplitude and/or comprises low frequency torque oscillations. Hybridized powertrains of more than one energy resonance outside excitation band. Through isolation measures propagation oscillations to vibration Hybridized powertrains comprises of more than energy outside the the of excitation band. Through sensitive isolation measures propagation of oscillations to vibration sensitive converter offering additional degree freedom in meeting Hybridized powertrains comprises of of more than one one energy resonance converter offering additional degree of freedom in meeting measures propagation of oscillations to vibration sensitive devices is prevented. Damping dissipates the vibration converter offering additional degree of freedom in meeting measures propagation of oscillations to vibration sensitive devices is prevented. Damping dissipates the vibration the motive power demand. Common powertrain configuraconverter offering additional degree of freedom in meeting the motive power demand. Common powertrain configura- devices is Damping dissipates the energy hence reducing the amplitude of resonant peaks. is prevented. prevented. Damping dissipates the vibration vibration the energy hence reducing the amplitude of resonant peaks. the motive motive power power demand. demand. Common Common powertrain powertrain configuraconfigura- devices  energy hence reducing the amplitude of resonant peaks. From the energy perspective, damping can be realised by Corresponding Author,(e-mail: [email protected] vkm.tu-darmstadt.de)  Corresponding Author,(e-mail: [email protected] vkm.tu-darmstadt.de) energy hence reducing the amplitude of resonant peaks. From the energy perspective, damping can be realised by  Corresponding Author,(e-mail: [email protected] vkm.tu-darmstadt.de) From the energy perspective, damping can be realised  Corresponding Author,(e-mail: [email protected] vkm.tu-darmstadt.de) From the energy perspective, damping can be realised by by Copyright © 2016, 2016 IFAC IFAC 770Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 770 Copyright 2016 responsibility IFAC 770Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC 770 10.1016/j.ifacol.2016.08.110

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either passive or active means. Passive methods usually make use of inertial and/or damping properties of the components to absorb the vibration while active measures try to attenuate the undesired vibration by introducing external energy to actively reduce vibration. Dual Mass Flywheel (DMF), as a passive measure strikes a balance between effectiveness of vibration reduction and device complexity. Isolation in a DMF is achieved by tuning the spring rate and inertia of the flywheels such that the DMF eigenfrequency is below the relevant range of ICE excitation. As mentioned earlier, downsizing and downspeeding shift the ICE excitation to lower frequencies and higher amplitudes which necessitate measures beyond DMF such as Centrifugal Pendulum Adsorber (CPA). CPA has an adaptive tuning feature due to centrifugal force variation over rotational speed offering better isolation over a range of frequencies. Added complexity, cost and tuning effort to meet increasing customer demand has shifted the focus from passive measures to active vibration reduction (VR) techniques. There are numerous realizations of active VR for automotive powertrains based on classical, adaptive and modern LMI based approaches. (Gusev et al., 1997) proposes a virtual flywheel for torsional VR reducing the mass of the physical flywheel using a reversible alternator. (Nakajima et al., 2000) demonstrated reduction of engine speed oscillations using open loop control. Simulation based analysis of an adaptive method using online identification of ICE torque pulsation relying on Harmonic Activated Neural Network was presented in (Beuschel et al., 2000). In (Zaremba and Davis, 2000) different control techniques for periodic disturbance attenuation such as lead-lag compensator, H∞ , PID and notch filter were investigated to mitigate torsional oscillations along with analysis of realtime capability. (Philippe and Coirault, 2005) reported a harmonic controller synchronized with engine speed during stationary operation along with parameter adaption for transient conditions.

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Fig. 1. Schematic of Combustion Engine Assist powertrain such as ETM energy consumption minimization. In this work, the Combustion Engine Assist (CEA) powertrain (Beidl et al., 2012) as depicted in Fig. 1 is employed as a use-case. The energy converters ICE and ETM are separated by a clutch C1. Clutch C2 is positioned betweent the power pack and the gearbox. s, i, T and ω denote the state, transmission ratio torque and speed respectively. 3. MPC VIBRATION REDUCTION CONTROLLER This section presents the theory behind the developed MPC scheme. MPC is an optimization based control scheme which computes feedback based control action by iterative solution of a finite horizon optimal control problem conditioned on the initial state. Initially the MPC problem is formulated, followed by a discussion on the prediction model, actuator compensation techniques and applied solution method. In the subsequent discussion, a differentiation between the signals into low frequency part (mean value) known as DC signal and high frequency part referred to as AC signal shall be made. The low frequency part accounts for the moving average of one ICE working cycle and the high frequency part accounts for the cyclic irregularity due to combustion. 3.1 Problem formulation

The aim here is to devise an MPC scheme which computes optimal control action as a function of known quantities A disturbance input decoupling combining feedforward (measured/observed), such that the constraints are satisand feedback techniques with an integrated starter/alternator fied and the cost is minimal over the prediction horizon. (SA) is proposed for a hybrid vehicle with in (Davis and Lorenz, 2003). An adaptive notch filter based active VR Model The torsional dynamics of the powertrain is controller employing FxLMS for adaption has been re- modelled as linear dynamical system with 3 DoF as ported in (Tareilus et al., 2011). In (Njeh et al., 2011) LMI in (1) to (3). DoF associated with ICE and ETM are based control approach for control of ICE torque ripple in retained as first and second torsional DoF, reducing the hybrid vehicle modelling it as LPV system is presented rest powertrain to a third DoF. . (Beidl et al., 2012) proposes VR method termed Harx(k) ˙ = Ac x(k) + Bc u(k) + Dc w(k) monically Oriented Control (HOC) which employs a time y(k) = Cc x(k) (1) varying kalman filter for harmonic separation with PI con b1  k1 b1 trollers tuned for the different harmonics. directE (Beidl 0 0 − J1 − J1 J1 et al., 2013) alternatively presents an approach of fre 1 0 −1 0 0   b  k1 b1 +b2 k2 b2  quency doubling instead of attenuation using virtual cylin1  Ac =  J2 J2 − J2 − J2 J2  (2) ders emulated by ETM, which enables 2 cylinder engine  0 0 1 0 −1  to emulate the acoustic behaviour of a 4 cylinder engine. b2 k2 b2 0 0 J J3 − J3 (Buch, 2016) discusses further development, calibration of    1 3 0 the directE concept and its implementation in an engine J1 testbed. None of the above mentioned approaches provide 0 0    1  a direct way to handle torque limits of the actuator (ETM) Bc =  J2  , Dc =  0  , Cc = [0 0 1 0 0] (3)    which might result from thermal derating in closed loop 0 0 operation. This forms the motivation for this contribution 0 0 investigating MPC approach for VR, further the use of T MPC provides opportunity to define economic objectives where x = [ω1 φ1 ω2 φ2 ω3 ] , y = ω2 . ETM torque u is the input signal and the periodic ICE torque excitation is 771

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modelled as disturbance input w. Further model reduction can be applied to yield a 2 DoF system as shown in (4). This reduced model retains ICE and ETM DoF. Spectral cost MPC (Gondhalekar et al., 2011) involves augmentation of the model with a filter to suppress specific frequency bands. In order to keep the online computation tractable, fewer number of states are desired which is the reason behind model.   b1the k21 DoF   1 − J1 − J1 Jb11 0 J1 0 −1  Bc =  0  Dc =  0  Ac =  1 (4) 1 b1 b1 k1 0 J2 J2 J2 − J2 T

where x = [ω1 φ1 ω2 ] , y = ω2 . The augmented model contains a bandpass filter as in (5) along with the discretized version of the system model. The filter serves to filter the frequency band which shall be attenuated (frequency binning). The discretized version of the augmented model (6) is used for spectral cost MPC design discussed below. ξ(k + 1) = Af ξ(k) + Bf z(k) (5) ψ(k) = Cf ξ(k) + Df z(k) ˆx(k) + Bu(k) ˆ x ˆ(k + 1) = Aˆ

(6)

T

x ˆ(k) = [x(k) ξ(k)]     B A 0 ˆ B= Aˆ = 0 Bf C Af

(7)

Cost function In a nominal MPC, the quality of control is quantified by a deterministic cost function. Since the goal of VR is to attenuate torsional oscillations, the cost function shall be defined in terms of oscillation at the various DoF. Hence, the usage of oscillation / deviation from the mean value of speeds and torsional deflection is imminent. Further from computational viewpoint and heavier penalty for large deviations from the DC value, a quadratic formulation has been adopted. The resulting cost function as given in (8). N −1  (x(k)T Qx(k) + u(k)T Ru(k)) + x(N )T P x(N ) (8) k=1

where Q weighs the state deviation of the torsional system from their mean value, R weighs the control action and is a scalar in case of VR regulator. Q is positive semidefinite and R is positive definite matrix. P is the solution of the algebraic Ricatti equation (9), which results from the optimal cost of the equivalent infinite horizon linear quadratic regulation problem. P = AT P A + Q − AT P B(R + B T P B)−1 B T P A (9)   T T T T ˆ = Q + CT Df Df C C DTf Cf (10) Q Cf Df C Cf C ˆ=R R (11) The spectral cost MPC which acts on the filtered system states can be formulated using the same cost function as in (8) but replacing the states with x ˆ and the corresponding ˆ and R ˆ respectively. An involved weight matrices with Q discussion of placing weights on harmonic content of system trajectories can be found in (Gondhalekar et al., 2011) and references therein. Constraints Constraints encode the physical limitations of the torsional system in the problem formulation. In 772

case of torsional VR, the limits of ETM torque form the constraints. In addition the maximum torsional deflection of the shafts or clutches could be formulated as box constraints. N −1  (x(k)T Qx(k) + u(k)T Ru(k)) + x(N )T P x(N ) min. x,u

s. t.

k=1

 x(k + 1) = Ax(k) + Bu(k) + Dw(k),   u (k) <= u(k) <= u (k), max

  

(12)

min

x(0) = x0 , k ∈ {1, ..., N − 1}

3.2 Predictor MPC works with non-causal estimates. These estimates can be explicitly specified either deterministically or stochastically, like the predicted velocity vector over the prediction horizon (Vadamalu et al., 2015) or estimated using data present assuming a model. In case of VR regulator there is no explicit information on the states. Hence a model based approach is employed to perform prediction. From the problem description of VR regulator in section (3.1) as state regulator with a disturbance observer, it is evident that the ICE torque shall be estimated for control. There are alternatives for ICE torque estimation using classical luenberger observers, disturbance observers based on Unkown Input Observer (UIO) approach. The proposed VR method does not place restrictions on the torque estimator apart from its real-time capability and accuracy. Without loss of generality, for initial simulation studies an ICE model developed and parameterized using testbed measurements (Buch, 2016) has been used. This model precisely reproduces the generated ICE torque and forms part of the validation enviroment. The dynamic torque exciting the torsional system is modelled as combination of gas torque, inertia torque and friction torque. For the control perspective, the combustion engine acts as a source of disturbance hence the aforementioned model acts a disturbance observer. Further the torque value is fed to single DoF torsional model of the powertrain. As discussed, the VR regulator acts on the AC states. Approaches such as discrete time varying kalman filter used for filtering harmonics of speeds can be employed. Here an observer based on an internal model is used to obtain the AC states for state regulation of the VR controller. This model is fed with the DC torque values computed by the EM controller. From the measurements and the model states, the value of current AC state for MPC VR is computed. 3.3 Compensation of Actuator dynamics ETM is used to reduce torsional oscillations in the drivetrain. Apart from limits on torque and torque rate, dead time and time lag behaviour of the actuator have to be considered. This section describes the extension of the MPC problem considering the actuator dynamics. Let us consider a linear dynamical system as in equation (13) with a time delayed input with delay value of d which is ideally a multiple of the controller sampling time T. x(k + 1) = Ax(k) + Bu(k − d) + Dw(k), y(k) = Cx(k) (13)

IFAC AAC 2016 June 19-23, 2016. Norrköping, SwedenRaja Sangili Vadamalu et al. / IFAC-PapersOnLine 49-11 (2016) 756–761

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Implicit dead time compensation scheme involves an augmented representation interpreting the dead-time as deadtime dynamics resulting in a delay free representation as in (14). ξ(k + 1) = Aξ ξ(k) + Bξ u(k), y(k) = Cξ ξ(k)

(14)

ξ(k) = [x(k)T u(k − d)T u(k − d + 1)T ...u(k − 1)T ]T  T T A B 0 0 ... 0 0 C  0   0 0 I 0 ... 0 0    0 0 0 I ... 0 0  0           Aξ =  (15) .   . . . . ... .  ; Bξ =  .  ; Cξ =     . . . . ... .  . .        0  0 0 0 0 ... I 0 0 0 0 0 ... 0 I 0

This representation uses the augmented system matrices Aξ , Bξ and Cξ as shown in (15). Disturbance vector is neglected in the representation due to usage of disturbance observer. Further only dead time for the EM actuation shall be considered and hence the emphasis is on the input vector u(k). Nevertheless it can be extended by appending the delayed or non-delayed disturbance vector in the ξ(k) and updating correspondingly the matrices Aξ , Bξ and Cξ . The implicit scheme results in increase of computational demand as the order of the system increases in the same order as the dead-time length. Using the prediction model, the value of states in the future which will be affected due to the present input can be obtained. This results in a modified state described as in equation (16). In this approach the disturbances are assumed to be stationary over the delay horizon. x ˜(k) := x(k + d|k) x ˜(k + 1) = A˜ x(k) + Bu(k) + Dw(k) y(k + d|k) = C x ˜(k)

(16) (17)

The resulting system is shown in (17). As it can be seen there is no system augmentation. Equation (18) provides the modified prediction equation including the effect of disturbances. This explicit compensation scheme for linear constrained system guarantees robust stability and robust constraint satisfaction (Santos et al., 2012). x ˜(k + d|k) = Ad x(k − d) + +

d  j=1

d  j=1

[Aj−1 Bu(k − j − d)] (18)

[Aj−1 Dw(k − j − d)]

As it can be observed, the prediction needs past values of the inputs which can be buffered during execution. The torque dynamics of the ETM used for active torsional vibration reduction shows a lag behaviour which can be modelled as a PT1 system as in equation (19) and in discretized version (20). TET M,actual =

TET M,set τs + 1

bz −1 TET M,actual (k) = G(z −1 ) = TET M,set (k) 1 − az −1

(19) (20)

The MPC VR scheme as in (12) shall be extended by an additional state xs which accounts for ETM torque dynamics resulting in (21). 773

Fig. 2. Controller structure for EM and VR integration min. x,u

s. t.

N −1 

(x(k)T Qx(k) + u(k)T Ru(k)) + x(N )T P x(N )

k=1

 x(k + 1) = Ax(k) + Bud (k) + Dw(k),     xs (k + 1) = Ad xs (k) + Bd u(k),    u (k) = C x (k), d

(21)

d s

umax (k) <= u(k) <= umin (k),      x(0) = x0   k ∈ {1, ..., N − 1} where Ad , Bd and Cd are obtained from the discrete time state space representation of the time lag of the ETM torque dynamics. 3.4 Solution Approach The formulation of VR regulator along with compensation of actuator dynamics result in a finite time horizon convex quadratic optimization problem with linear constraints. This problem is solved every sampling instant to obtain the AC torque set value for the ETM based on initial state values. There are various solution approaches to obtain online solution to such a problem. Choice of interior-point method is motivated by its ability to reliably and accurately solve with few iterations the optimization problem without warm start. Here CVXGEN, the real-time capable C implementation of CVX software for disciplined convex programming specific for QP transformable problems is used. CVXGEN uses a standard primal-dual interior point method with the Mehrotra predictor-corrector. For details refer (Mattingley and Boyd, 2012) The generated custom solver C code is integrated using s-function builder in simulink which is also ported to real-time hardware. 3.5 Integration with EM EM controller of a HEV/PHEV powertrain is a supervisory controller that coordinates the power flow by defining set values for the component level controllers of the different energy converters. The discussed MPC scheme for torsional VR is integrated with the EM controller of the HEV using 2 DoF controller structure (Alt et al., 2012). In the 2 DoF controller framework, the VR Regulator operates in tandem with the torque/speed controller of the ETM as shown in Fig. 2. EM performs tracking of battery State of Charge (SoC) using inputs udc where as the VR regulator actively dampens the torsional oscillations with uac . EM tracking controller acts on estimates of DC states x ˆdc and reference value xdc,ref , whereas the VR regulator uses the

IFAC AAC 2016 760 June 19-23, 2016. Norrköping, SwedenRaja Sangili Vadamalu et al. / IFAC-PapersOnLine 49-11 (2016) 756–761

uEM = [TICE , TET M ]T

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estimated AC states x ˆac . As there is a clear demarcation of the control objective and the set value signals on which the individual controllers operate (in DC and AC frequencies), the 2 DoF controller structure avoids interference between the VR and EM controllers. The EM controller considered is formulated as a predictive constrained economic MPC problem (Rawlings and Mayne, 2009) minimizing the energy consumption and solved using direct optimal implementing active set based algorithm. For a detailed discussion refer (Vadamalu et al., 2015). The DC values of ICE and ETM torques obtained are augumented with the AC torque values resulting in the set point torque values for the component controllers. (22) TICE,dc (k) = TGB (k) − TET M,dc (k) (23) udc = TICE,dc + TET M,dc uac = TICE,ac + TET M,ac (24)

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4. RESULTS AND DISCUSSION

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This section discusses the results of the developed MPC scheme using simulations. Performance of the controller is analysed in stationary and dynamic conditions. Further ability to handle time-varying constraints and integration with EM regulation is also studied. Fig. 3 illustrates the results of VR during stationary powertrain operation with normal MPC. As depicted MPC VR controller is able to reduce oscillations around the mean speed of 1500 rpm after it has been switched on at 1 second. It can be observed that initially the VR controller provides maximum torque (torque limit constraints become active) before settling to lower values as vibration reduces with time. The advantage of MPC based VR controller to handle Fig. 5. Spectrogram of ETM speed during transient without controller

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Fig. 3. Vibration reduction performance at stationary speed time varying constraints is shown in Fig. 4 exemplarily for spectral MPC with 3 DoF. It can be observed that the controller is able to adapt to time varying input constraints. Such a variation of torque limits can occur in practice during operation due to thermal derating of the battery or ETM. As a result of tightened input limits, the increase in amplitude of torsional oscillations can be noted. The constraint satisfaction is a consequence of the convergence of the optimization algorithm, in this case cvxgen used for recursive optimization of the MPC. The dynamic performance of VR controller based on spectral 2 DoF MPC is depicted in Fig. 5 and 6. The 774

Fig. 6. Spectrogram of ETM speed during transient with spectral 2 DoF MPC transient scenario starts with an ICE speed of 900 rpm and ramps up to 1300 rpm, followed by step change in speed to 1600 rpm and thereafter the speed ramps up to 2200 rpm. The set speed change can be conceived due to gear shift event. VR controller is switched on after 1 second. In

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IFAC AAC 2016 June 19-23, 2016. Norrköping, SwedenRaja Sangili Vadamalu et al. / IFAC-PapersOnLine 49-11 (2016) 756–761

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Fig. 7. Integration of MPC based EM and VR comparison to the normal case in Fig 5, it can be noted that there is a considerable reduction in the vibration amplitude during the transient operation. In order to investigate the real-time capability of the MPC based VR controller, it was implemented in dSPACE real time board (DS1005). The observed maximum task turnaround time for a prediction horizon of 3 was found to be 8 ms. Fig. 7 presents the results of integration of MPC based EM and VR controllers. Due to the 2 DoF controller structure, the VR controller can be activated on demand by the supervisory EM controller. This enables the EM controller to decide when to enable the VR controller to balance between the energy consumption and vibration attenuation requirements. As it can be seen in the figure, during the phase with enabled VR, the speed oscillations at the gearbox input is reduced due to the compensation. 5. CONCLUSION This paper presented an MPC based VR controller, as a disturbance observer based state regulator for torsional vibration attenuation of downsized hybridized powertrains. Two variants, namely time domain (normal MPC) and frequency domain (spectral MPC) formulation based control algorithms were implemented. Computational demand was reduced in case of spectral MPC, which results due to the augmented filter states using 2 DoF formulation. An explicit dead time compensation was used to compensate for ETM torque dynamics. The developed controller was validated in Simulink in stationary and dynamic powertrain operation. Furthermore, MPCs ability to handle constraints is also validated by given time variant constraint to the controller. VR controller was integrated with EM controller using 2 DoF controller architecture and results of the integration has been presnted. Inital investigation of real-time performance was performed. Future research shall focus on experimental validation, for which explicit MPC implementation offer interesting prospects from the perspective of computational complexity along with robustness analysis of the MPC based VR controller. REFERENCES Alt, B., Antritter, F., and Svaricek, F. (2012). Flatness based Control for Electric and Hybrid Electric Vehicle 775

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Drivetrains. In Mediterranean Conference on Control & Automation (MED). Beidl, C., Buch, D., Hohenberg, G., Bacher, C., C, H., and Kufferath, A. (2012). Effizienter E-Fahrzeugantrieb mit dem kompakten CEA-Konzept Combustion Engine Assist. In MTZ-Fachtagung Der Antrieb von morgen. Beidl, C., Hohenberg, G., and Hoefler, D. (2013). Drehschwingungsberuhigung von Hybridantrieben mit niedrigen Zylinderzahlen. In Internationales Wiener Motorensymposium. Beuschel, M., Rau, M., and Schroder, D. (2000). Adaptive damping of torque pulsation using a starter generatoropportunities and boundaries. In IEEE Industry Applications Conference. Buch, D. (2016). Aktive Beruhigung verbrennungsmotorisch erregter Drehschwingungen im hybriden Fahrzeugantriebsstrang. TU Darmstadt. Davis, R.I. and Lorenz, R.D. (2003). Engine Torque Ripple Cancellation With an Integrated Starter, Alternator in a Hybrid Electric Vehicle. IEEE Transactions on Industry Applications, 39, NO. 6, 1765–1774. Gondhalekar, R., Jones, C., and T. Besselmann, J. Hours, M.M. (2011). Constrained spectrum control using MPC. In IEEE Conference on Decision and Control and European Control Conference. Gusev, S., Johnson, W., and Miller, J. (1997). Active flywheel control based on the method of moment restrictions. In American Control Conference(ACC). Mattingley, J. and Boyd, S. (2012). CVXGEN: A Code Generator for Embedded Convex Optimization. Optimization and Engineering, 13(1):127. Nakajima, Y., Uchida, M., Ogane, H., and Kitajima (2000). A study on the reduction of crankshaft rotational vibration velocity by using a motor-generator. JSAE Review 21, 3071–3076. Njeh, M., Cauet, S., and Coirault, P. (2011). LPV control of ICE torque ripple in hybrid electric vehicles. In IFAC World Congress. Philippe, M. and Coirault, P. (2005). A harmonic controller of engine speed oscillations for hybrid vehicles. In IFAC World Congress. Preumont, A. (2011). Vibration Control of Active Structures : An Introduction. Springer. Rawlings, J.B. and Mayne, D.Q. (2009). Model Predictive Control: Theory and Design. Nob Hill Publishing. Santos, T., Limon, D., Normey-Rico, J., and Alamo, T. (2012). On the explicit dead-time compensation for robust model predictive control. Journal of Process Control, Volume 22, Issue 1, Pages 236-246. Tareilus, A., Endres, H.D., Oflaz, A., and Schneider, H.J. (2011). Aktive schwingungsdaempfung im antriebsstrang von hybridfahrzeugen. In Fachtagung Schwingungen in Antrieben. Vadamalu, R.S., Buch, D., Xiao, H., and Beidl, C. (2015). Energy management of hybrid electric powertrain using predictive trajectory planning based on direct optimal control. In IFAC Workshop on Control Applications of Optimization. Zaremba, A. and Davis, R.I. (2000). Control design for active engine damping using a starter/alternator. In American Control Conference(ACC).