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Stability analysis of a flatness-based controller driving a battery emulator with constant power load

Stabilitätsanalyse eines flachheitsbasierenden Reglers für einen Batterieemulator mit konstanter Leistungsaufnahme
Michael Zauner

Michael Zauner was born in Baden, Austria in 1990. He received the Dipl. Ing. degree from TU Wien, Vienna, Austria, in 2018. Since 2018, he has been a Project Assistant at the Institute of Mechanics and Mechatronics at TU Wien. His current research interests include optimization, nonlinear control, and digital systems.

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, Philipp Mandl

Philipp Mandl was born in Mistelbach, Austria, in 1996. He received the B.S. degree in mechanical engineering from the TU Wien, Vienna, Austria, in 2019, where he is currently working toward the M.S. degree in mechanical engineering. His current research interests include power system modeling, nonlinear control and digital systems.

, Oliver König

Oliver König received the M.S. degree from Graz University of Technology, Austria, in 2009. Starting from 2010, he was working at the Institute of Mechanics and Mechatronics at TU Wien, Austria, from where he received the Ph.D. degree in 2013. In 2013 he joined AVL List GmbH in Graz to work on digital control for automotive electrification test systems. In 2019, he became department manager responsible for control systems in the segment of Energy Storage Test and Emulation Products. His main area of research is embedded model predictive control with a focus on power electronics.

, Christoph Hametner

Christoph Hametner received the M.S. degree in mechanical engineering, the Ph.D. degree in technical sciences, and the Habilitation (professorial qualification) in control theory and system dynamics from TU Wien, Vienna, Austria, in 2005, 2007, and 2014, respectively. He is the Head of the Christian Doppler Laboratory for Innovative Control and Monitoring of Automotive Powertrain Systems, TU Wien. His research interests are nonlinear system identification, modeling, and control.

and Stefan Jakubek

Stefan Jakubek received the M.S. degree in mechanical engineering in 1997, the Ph.D. degree in technical sciences, in 2000, and the Habilitation (professorial qualification) in control theory and system dynamics in 2007, all from TU Wien, Vienna, Austria. From 2006 to 2009, he was the Head of Development for Hybrid Powertrain Calibration and Battery Testing Technology with AVL List GmbH, Graz, Austria. He is currently a Professor with the Institute of Mechanics and Mechatronics, TU Wien. His research interests include fault diagnosis, nonlinear system identification, and simulation technology.

Abstract

This contribution deals with the control of a battery emulator used in automotive testbeds for electric drivetrains. The battery emulator, which is realized as a DC-DC converter, is connected to a unit-under-test (UUT), e. g., an electric motor inverter. To accurately emulate the dynamic impedance of a battery, a highly dynamic output is required. Additionally, battery emulators should be applicable for a large variety of UUTs, hence robust performance in a large operating range is also required. This is especially challenging when the UUT behaves like a constant power load, as this can cause stability issues. To meet the requirements, a flatness-based control concept is presented that establishes feedback equivalence between a nonlinear and a linearized system representation. By examining the stability of the concept, an estimation of the region of attraction is found.

Zusammenfassung

Dieser Artikel beschreibt die Regelung eines Batterie-Emulators, welcher in Fahrzeugprüfständen für elektrische Antriebe verwendet wird. Der als DC-DC Konverter realisierte Emulator wird mit einem Prüfling verbunden, z. B. einem E-Motor-Inverter. Um das dynamische Verhalten einer Batterie wiedergeben zu können, muss der Ausgang hochdynamische Spannungsverläufe realisieren können. Zum Testen einer weiten Bandbreite von Prüflingen, sollte der Batterie-Emulator einen großen Arbeitsbereich vorweisen. Schwierig ist dies, wenn der Prüfling sich wie eine Last mit konstanter Leistungsaufnahme verhält, was zu Stabiltätsproblemen führen kann. Um dies dennoch zu bewältigen, stellt der präsentierte Flachheits-basierende Regler Rückführäquivalenz zwischen einer nichtlinearen und einer linearisierten Modellbeschreibung her. Des Weiteren wird mittels einer Stabilitätsanalyse eine Abschätzung des Einzugsbereiches angegeben.

Funding statement: The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development, and the Christian Doppler Research Association is gratefully acknowledged.

About the authors

Michael Zauner

Michael Zauner was born in Baden, Austria in 1990. He received the Dipl. Ing. degree from TU Wien, Vienna, Austria, in 2018. Since 2018, he has been a Project Assistant at the Institute of Mechanics and Mechatronics at TU Wien. His current research interests include optimization, nonlinear control, and digital systems.

Philipp Mandl

Philipp Mandl was born in Mistelbach, Austria, in 1996. He received the B.S. degree in mechanical engineering from the TU Wien, Vienna, Austria, in 2019, where he is currently working toward the M.S. degree in mechanical engineering. His current research interests include power system modeling, nonlinear control and digital systems.

Oliver König

Oliver König received the M.S. degree from Graz University of Technology, Austria, in 2009. Starting from 2010, he was working at the Institute of Mechanics and Mechatronics at TU Wien, Austria, from where he received the Ph.D. degree in 2013. In 2013 he joined AVL List GmbH in Graz to work on digital control for automotive electrification test systems. In 2019, he became department manager responsible for control systems in the segment of Energy Storage Test and Emulation Products. His main area of research is embedded model predictive control with a focus on power electronics.

Christoph Hametner

Christoph Hametner received the M.S. degree in mechanical engineering, the Ph.D. degree in technical sciences, and the Habilitation (professorial qualification) in control theory and system dynamics from TU Wien, Vienna, Austria, in 2005, 2007, and 2014, respectively. He is the Head of the Christian Doppler Laboratory for Innovative Control and Monitoring of Automotive Powertrain Systems, TU Wien. His research interests are nonlinear system identification, modeling, and control.

Stefan Jakubek

Stefan Jakubek received the M.S. degree in mechanical engineering in 1997, the Ph.D. degree in technical sciences, in 2000, and the Habilitation (professorial qualification) in control theory and system dynamics in 2007, all from TU Wien, Vienna, Austria. From 2006 to 2009, he was the Head of Development for Hybrid Powertrain Calibration and Battery Testing Technology with AVL List GmbH, Graz, Austria. He is currently a Professor with the Institute of Mechanics and Mechatronics, TU Wien. His research interests include fault diagnosis, nonlinear system identification, and simulation technology.

Appendix A Flatness property of Σnl

The output y and its time-derivatives of the nonlinear system Σnl are given by (32)–(36). Note that, the input u only appears in y(4). Therefore, the system is said to have a relative degree of 4.

(32)y=v2
(33)y˙=i2C2PC2v2
(34)y¨=Pi2C2PC2v2C2v22v2L2vcL2+R2i2L2C2
(35)y(3)=R2C2L2PC22v22v2L2vcL2+R2i2L2i2C2PC2v21C2L2P2C22v24+2Pi2C2PC2v2C2v23+i1C1i2C1C2L2
(36)y(4)=v2L2vcL2+R2i2L21C2L2P2C22v24+2Pi2C2PC2v2C2v23C2+1C1C2L2R2R2C2L2PC22v22L2+2Pi2C2PC2v2C22v23i2C2PC2v26P2C22v256Pi2C2PC2v2C2v24i2C2PC2v2R2C2L2PC22v22L22Pv2L2vcL2+R2i2L2C22v23P1C2L2P2C22v24+2Pi2C2PC2v2C2v23C2v22+P1C2L2P2C22v24+2Pi2C2PC2v2C2v23C2v22i1C1i2C1R2C2L2PC22v22L2+1C1C2L2LgLf3h(x)u

From (32)–(36) one can see that when v20, the transformation z=t(x,P) is continuously differentiable. Note that, y=y(v2) is linear in v2, that y˙=y˙(v2,i2) is linear in i2, that y¨=y¨(v2,i2,vc) is linear in vc, and so on. Therefore, the states x and the input u can be uniquely represented as functions of the output and the first 4 time-derivatives. As the system is also square (it has the same number of inputs as outputs), all the conditions for flatness are met [17].

Remark.

The singularity atv2=0is not relevant for the physical system as the UUT would not be turned on below a threshold voltage and either draws no power at all or behaves like a constant current load at 0 V. Therefore, the resulting system dynamics below the threshold voltage would be linear.

Remark.

The flatness property ofΣlcan be shown much easier: SinceΣlis linear, it is flat if and only if it is controllable [11]. For real positive values ofC1,L2,C2the controllability-matrix has full rank, therefore the system is controllable.

Appendix B Nominal Lyapunov function change

The matrix S0 is the solution of the discrete-time algebraic Riccati equation, meaning it solves the following equation:

(37)AdTSAdSAdTSBd(BdTSBd+R)1BdTSAd+Q=0.

The state-feedback gain Kx of the LQR controller can then be expressed as

(38)Kx=(BdTSBd+R)1BdTSAd.

Inserting (20) into Vˆ(xˆ[k+1]) yields

(39)Vˆ(xˆ[k+1])=12Adxˆ[k]+Bduˆ[k]+EdPˆ[k]xw[k]S2,

where αS2 is short-hand for αTSα. Without loss of generality, one can introduce the shifted state-vector x˜=xˆxw, resulting in a new target state x˜w=0. For simplicity, a power draw Pˆ=0 is assumed, resulting in a uˆ=Kxx˜. The nominal Lyapunov function change is then given by

(40)ΔVˆ[k]=Vˆ(x˜[k+1])Vˆ(x˜[k])
(41)=12(AdBdKx)x˜[k]S2x˜[k]S2
(42)=12x˜[k]TAdTSAdSAdTSBdKxKxTBdTSAd+KxTBdTSBdKxx˜[k].
With (38) inserted into (37) one can see that the first 3 terms can be simplified to Q. The last 2 terms can also be rewritten with (38) into KxTRKx. Finally, one can write

(43)ΔVˆ[k]=12x˜[k]TQ+KxTRKxx˜[k],

which is negative definite.

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Received: 2020-06-26
Accepted: 2020-11-26
Published Online: 2021-01-30
Published in Print: 2021-02-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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