Robust Discrete Time Disturbance Observer with Finite Control Set Model Predictive Control in UPS System

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IFAC PapersOnLine 52-4 (2019) 51–56 Robust Discrete Time Disturbance Observer with Finite Control Set Model Robust Discrete Time Disturbance Observer with Finite Control Set Model Control in UPS System Robust Discrete TimePredictive Disturbance Observer with Finite Control Set Model Control in UPS System Robust Discrete TimePredictive Disturbance Observer with Finite Control Set Model Predictive Control in UPS System Predictive Control in UPS System

Yahya Danayiyen* İsmail Hakkı Altaş ** Yahya Danayiyen* İsmail Hakkı Altaş ** Young IL Lee*** Yahya Danayiyen* İsmail Hakkı Altaş ** Young IL Lee*** Yahya Danayiyen* İsmail Hakkı Altaş ** Young IL Lee*** * Electrical and Electronics Engineering Department, Karadeniz Technical University, Young Department, IL Lee*** Karadeniz Technical University, * Electrical and Electronics Engineering Trabzon, Turkey, (Tel: +905324534150; e-mail: [email protected]) * Electrical and Electronics Engineering Department, Karadeniz Technical University, Trabzon, Turkey, (Tel: +905324534150; e-mail: [email protected]) ** Electrical and Electronics Engineering Department, Karadeniz Technical University, * Electrical and Electronics Engineering Department, Karadeniz Trabzon, Turkey, (Tel: +905324534150; e-mail: [email protected]) ** Electrical and Electronics Engineering Department, KaradenizTechnical TechnicalUniversity, University, Trabzon, Turkey, (e-mail: [email protected]) Trabzon, Turkey, (Tel: +905324534150; e-mail: [email protected]) ** Electrical and Electronics Engineering Department, Karadeniz Technical University, Trabzon, Turkey, (e-mail: [email protected]) *** Electrical and Information Engineering, Seoul National University Science and Technology, Seoul, Korea ** Electrical and Electronics Department, Karadeniz Technical University, Trabzon,Engineering Turkey, [email protected]) *** Electrical and Information Engineering, Seoul(e-mail: National University Science and Technology, Seoul, Korea (Tel: +82-02-970-6544; e-mail: [email protected]) Trabzon, Turkey, *** Electrical and Information Seoul(e-mail: National University Science and Technology, Seoul, Korea (Tel:Engineering, +82-02-970-6544; e-mail:[email protected]) [email protected]) *** Electrical and Information Seoul National University Science and Technology, Seoul, Korea (Tel:Engineering, +82-02-970-6544; e-mail: [email protected]) (Tel: +82-02-970-6544; e-mail: [email protected]) Abstract: In this paper, output voltage of a three phase uninterrupted power supply is controlled using finite Abstract: In this paper, output voltage a three phasetime uninterrupted power supplyisis used controlled using finite control set model predictive control.of A discrete disturbance observer to estimate the Abstract: In this paper, output voltage of A a three phasetime uninterrupted power supplyisis used controlled using finite control set model predictive control. discrete disturbance observer to estimate the disturbance load current. A robust method is proposed to determine gain matrix of the observer. This Abstract: In this paper, output voltage of A a three phasetime uninterrupted power controlled using finite control set model predictive control. discrete disturbance observer isis used estimate the disturbance load current. A robust method is proposed to determine gainsupply matrix of thetoobserver. This optimal gain matrix was computed using matrix inequality method. control is designed control set load model predictive control. Alinear discrete time disturbance observer is used estimate the disturbance current. A robust method is proposed to determine gainThe matrix of system thetoobserver. This optimal gain matrix was computed using linear matrix inequality method. The control system is designed in alpha-beta reference and implemented Matlab environment. The comparative simulation results disturbance current. A robust method is in proposed to determine gainThe matrix of system the observer. This optimal gainload matrix wasframe computed using linear matrix inequality method. control is designed in alpha-beta reference frame and implemented in Matlab environment. The comparative simulation results arealpha-beta givengain for reference three different observers inlinear orderinmatrix to see inequality performance of theThe proposed robust method. The optimal wasframe computed usingin method. controlrobust system is designed in and implemented environment. comparative simulation results are given for matrix threethe different observers order toMatlab see performance ofThe the optimal proposed method. The results show that proposed LMI matrix can be used to calculate the observer gain matrix. in frame observers andLMI implemented intoMatlab environment. comparative simulation results arealpha-beta given for reference threethe different in order performance ofThe the optimal proposedobserver robust method. The results show that proposed matrix can besee used to in calculate the gain matrix. Furthermore, an additional integration term can be included the disturbance observer to improve FCSare given for three different observers in order to see performance of the proposed robust method. The results show that the proposed LMI matrix can be used to calculate the optimal observer gain matrix. Furthermore, an additional integration term canperiod be included in the disturbance observer to improve FCSMPC performance. The effect ofLMI the sampling on FCS-MPC is evaluated under different sampling results show that the proposed matrix can be used to calculate the optimal observer gain matrix. Furthermore, an additional integration term can be included in the disturbance observer to improve FCSMPC performance. Theshow effect of athe sampling period on FCS-MPC is evaluated under different sampling frequency. The results that small sampling frequency can bedisturbance used with proposed observer in FCSFurthermore, anresults additional integration term canperiod befrequency included in thebe observer to improve MPC performance. Theshow effect of athe sampling on FCS-MPC is evaluated under different sampling frequency. The that small sampling can used with proposed observer in FCSFCSMPC. MPC Theshow effect of athe sampling period on FCS-MPC evaluated under different frequency. The results that small sampling frequency can be is used with proposed observersampling in FCSMPC.performance. frequency. The results showFederation that a small can be used withdisturbance proposed observer in linear FCSMPC. © 2019, IFAC (International of sampling Automaticfrequency Control) by Elsevier Ltd. All rights reserved. Keywords: Model predictive control, uninterruptible powerHosting supply (UPS), observer, Keywords: Model (LMI). predictive control, uninterruptible power supply (UPS), disturbance observer, linear MPC. matrix inequalities Keywords: Model (LMI). predictive control, uninterruptible power supply (UPS), disturbance observer, linear matrix inequalities Keywords: Model (LMI). predictive control, uninterruptible power supply (UPS), disturbance observer, linear matrix inequalities  matrix inequalities (LMI).   disturbance observer based FCS-MPC was applied a three 1. INTRODUCTION disturbance observer FCS-MPC was applied J.a etthree  phase inverter in motorbased control application in (Wang, al., 1. INTRODUCTION disturbance observer based FCS-MPC was applied J.a etthree phase inverter in load motor control application in (Wang, al., 1. INTRODUCTION 2017). Flux and currents were estimated using proposed disturbance observer based FCS-MPC was applied phase inverter in load motor control application in (Wang, J.a etthree al., Flux and currents were estimated usingbased proposed FCS-MPC has gained more attention in the last decade in the 2017). 1. INTRODUCTION observer and reference states were computed on FCS-MPC has gained more attention inItthe last decade in the observer phase inreference motor control application in (Wang, J. et al., 2017).inverter Flux and load currents were estimated usingbased proposed and states were computed on control of power electronics devices. predicts the system estimated sate values. 16 kHz sampling frequency was used in FCS-MPC has gained more attention inItthe last decade in the 2017). control of power electronics devices. predicts the system Flux and load currents were estimated using proposed observer and reference states were computed based on estimated sate values. 16 kHz sampling frequency was used in state based on possible voltage vectors of the converter and all the simulation. The disturbance observers can be used to FCS-MPC has gained more attention in the last decade in the control of power electronics devices. It predicts the system state based on are possible voltage vectors of the converter and all observer and values. reference states were computed based estimated sate 16 kHz sampling frequency was usedon in the simulation. The disturbance observers can be used to these vectors evaluated in a predefined cost function in compensate the disturbances (Jun, Y., et al., 2010, 2011; control of power electronics devices. It predicts the system state based on possible voltage vectors of the converter and all these vectors arethe evaluated in astate. predefined cost function in estimated satethe values. 16 kHz sampling frequency was used in the simulation. The disturbance observers can 2010, be used to order to choose switching The selected switching compensate disturbances (Jun, Y., et al., 2011; A., et al., 2015; (Jun, Kim, K.S., et al., 2013). A state based on are possible voltage of the converter and all these vectors evaluated in vectors astate. predefined cost function in Aboudonia, order to choose the switching The selected switching the simulation. The disturbance observers can be used to compensate the disturbances Y., et al., 2010, 2011; Aboudonia,time A., disturbance et al., 2015; Kim, was K.S., et al., by 2013). A state isvectors applied directly to theinconverter without a function modulator. continuous observer proposed Jun and these are evaluated a predefined cost in order to choose the switching state. The selected switching state is applied directly to the converter without a modulator. compensatetime the disturbance disturbances (Jun, Y., et et al.,al., 2010, 2011; Aboudonia, A., et al., 2015; Kim, was K.S., 2013). A continuous observer proposed by Jun and The main drawback of FCS-MPC is offset error due to the Chen (2010) for bank to observer turnKim, system. The observer was order choose the switching state.is The selected switching state istoapplied directly to the converter without a modulator. Aboudonia, A., et al., 2015; K.S., et al., 2013). A continuous time disturbance was proposed by Jun and The main drawback of FCS-MPC offset error due to the Chen (2010) for bankspace to turn system. The observer was variable switching frequency. all oferror theacalculations designed based on state model of theproposed system considering state is applied directly theMoreover, converter without modulator. The main drawback of to FCS-MPC is offset due to the Chen variable switching frequency. Moreover, all of the calculations continuous time disturbance observer was by Jun and (2010) for bank to turn system. The observer was designed based on state space model of the system considering should beswitching done in afrequency. one sampling period. order to the parameter uncertainties. Although a good reference The main drawback of FCS-MPC is offset error dueobtain to theaa Chen variable Moreover, allIn the calculations (2010) bankspace to turn system. The observer was designed basedfor on state model of the system considering should be done in a one sampling period. Inof order to obtain the parameter uncertainties. Although a good reference good controller performance, the sampling period should be tracking performance was obtained, it was not clear how the variable Moreover, the calculations shouldcontroller beswitching done in afrequency. one sampling period.allInof order toshould obtainbea designed good performance, the sampling period based on statewas space model of the system considering the parameter uncertainties. Although anot good reference tracking performance obtained, it was clear how the chosen properly and the disturbances should be compensated gain matrix was obtained, optimally similar should be done in a one period. In period order toshould obtainbea observer good controller performance, the sampling chosen properly and the sampling disturbances should be compensated the parameter uncertainties. Although good reference tracking performance was it determined. wasanot clearA how the observer gain matrix was optimally determined. A similar with a proper observer. To decrease the offset error and disturbance observer was applied toit determined. awas missile system with good acontroller performance, the sampling period should be tracking chosen properly and the disturbances should be compensated with proper observer. To decrease the offset error and performance was obtained, not clear how the observer gain matrix was optimally A similar disturbance observerbywas applied a missile with increase the controller several nonlinear gain dynamics Chen and Lito (2011) and system aA quadrotor chosena properly and the performance, disturbances beobserves compensated with proper observer. To decreaseshould the offset error with and observer matrix was optimally determined. similar disturbance observer was applied to a missile system with increase the controller performance, several observes with nonlinear by and ChenRashad and Li(2015). (2011) A anddiscrete a quadrotor FCS-MPC in power converters have bydynamics Aboudonia time with a proper observer. To electronic decreaseseveral the offset error been and system increase themethod controller performance, observes with disturbance observerbywas applied a missile system with nonlinear dynamics Chen and Lito(2015). (2011) anddiscrete a quadrotor FCS-MPC method in et power electronic converters have been system by Aboudonia and Rashad A time published (Cortes, P. al., 2009; Wang, J. et al., 2017). A disturbance observer based on continuous counterpart used by increase the controller performance, several observes with FCS-MPC method in power electronic converters have been nonlinear dynamics by Chen and Li (2011) and a quadrotor system by Aboudonia and Rashad (2015). A discrete time published (Cortes, P. et al., 2009; Wang, J. et al., 2017). A disturbance observer based on continuous counterpart usedThe by simple load current estimation method with FCS-MPC Jun and Chen (2010) is proposed by Kim and Rew (2013). FCS-MPC method in power electronic converters have been published (Cortes, P. et al., 2009; Wang, J. et al., 2017). A simple load current estimation method with FCS-MPC system Aboudonia andon Rashad (2015). A discrete time disturbance observer based continuous counterpart used by Jun and by Chen (2010) was is proposed byusing Kim and Rew (2013). The ((Cortes, P. et al., 2009) is used in a three phase UPS to control disturbance observer designed reduced order system publishedload P. etestimation 2009; Wang, J.with etUPS al.,FCS-MPC 2017). A disturbance simple current method ((Cortes, P.(Cortes, et al., 2009) isal., used in a three phase to control observer based on continuous counterpart used by Jun and Chen (2010) was is proposed byusing Kim reduced and Reworder (2013). The disturbance observer designed system the output voltage. The load current dynamics discretized available states and the by gain matrix formulation was simple load current estimation method with FCS-MPC ((Cortes, P. et al., 2009) is used a three phase UPSdiscretized to control based the output voltage. The load in current dynamics Jun andon Chen (2010) is proposed Kim and Rew (2013). The disturbance observer was designed using reduced order system based on available states and the gain matrix formulation was based on Euler forward method and used as an estimator. The given for this reduced order observer. The given formulation ((Cortes, et al., 2009) is used a three phase to control the output voltage. Themethod load in current discretized based on P. Euler and useddynamics asunder an UPS estimator. The observer was designed using reduced order system based on available states and the gain matrix formulation was given for this reduced order observer. The given formulation performance offorward the controller was tested resistive and disturbance is not robust against parameter uncertainties and mismatches. the output voltage. The load current dynamics discretized based on Euler forward method and used as an estimator. The performance of the controller was tested under resistive and based on available states and the gain matrix formulation was given for this reduced order observer. The given formulation nonlinear loadofconditions. Thewas sampling time chosenThe 33 is not robust against parameter uncertainties and mismatches. based on Euler forward method andtested used as an was estimator. performance the controller under resistive and nonlinear load conditions. The sampling time was chosen 33 given for this reduced order observer. The given formulation is not robust against parameter uncertainties and mismatches. µs. Although a fast response was time obtained, the output performance thetransient controller was tested underwas resistive and nonlinear loadof conditions. The sampling chosen 33 The objective this paperuncertainties is to control the voltage µs. Although awas fast transient response was obtained, theload. output is notmain robust againstofparameter andoutput mismatches. voltage THD still high under linear and nonlinear A The maindevice objective ofthe thissimilar paper isobserver to control the output voltage nonlinear load conditions. The sampling time was chosen 33 µs. Although a fast transient response was obtained, the output of UPS using defined in (Jun, Y., voltage THD was still high under linear and nonlinear load. A The main objective of this paper is to control the output voltage of UPS device using the similar observer defined in (Jun, Y., µs. Although faststill transient response was obtained, theload. output voltage THD awas high under linear and nonlinear A The maindevice objective ofthe thissimilar paper isobserver to control the output voltage of UPS using defined in (Jun, Y., voltage THD wasIFAC still high under linear and nonlinear load. A 51 Copyright © 2019 of UPS device using the similar observer defined in (Jun, Y., 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 51 Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 51 Control. 10.1016/j.ifacol.2019.08.154 Copyright © 2019 IFAC 51

2019 IFAC CSGRES Jeju, Korea, June 10-12, 2019 52

Yahya Danayiyen et al. / IFAC PapersOnLine 52-4 (2019) 51–56

et al., 2010, 2011; Aboudonia, A., et al., 2015; Kim, K.S., et al., 2013) with FCS-MPC.

𝑑𝑑𝑖𝑖𝑓𝑓𝑓𝑓 (𝑡𝑡) 𝑉𝑉𝑖𝑖𝑖𝑖 (𝑡𝑡) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑡𝑡) = 𝑑𝑑𝑑𝑑 𝐿𝐿𝑓𝑓

To overcome the offset error and high output ripple in low sampling frequency, we added an integration term into the observer given in (Kim, K.S., et al., 2013). Furthermore we defined a LMI to compute the optimal observer gain matrix. The comparative results are given with and without integrated term. The simple observer defined (Cortes, P. et al., 2009) is also used for the comparison purpose. Furthermore, the performance of the observers were evaluated under different sampling period.

𝑑𝑑𝑖𝑖𝑓𝑓𝑓𝑓 (𝑡𝑡) 𝑉𝑉𝑖𝑖𝑖𝑖 (𝑡𝑡) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑡𝑡) = 𝑑𝑑𝑑𝑑 𝐿𝐿𝑓𝑓 𝑑𝑑𝑉𝑉𝑜𝑜𝑜𝑜 (𝑡𝑡) 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑡𝑡) 𝑖𝑖𝑜𝑜𝑜𝑜 (𝑡𝑡) = − 𝐶𝐶𝑓𝑓 𝑑𝑑𝑑𝑑 𝐶𝐶𝑓𝑓 𝑑𝑑𝑉𝑉𝑜𝑜𝑜𝑜 (𝑡𝑡) 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑡𝑡) 𝑖𝑖𝑜𝑜𝑜𝑜 (𝑡𝑡) = − 𝑑𝑑𝑑𝑑 𝐶𝐶𝑓𝑓 𝐶𝐶𝑓𝑓

The remainder of the paper proceeds as follows. We describe a three phase two level uninterruptible power supply system in alpha-beta reference frame in section 2. Dynamics of the three different disturbance observers are given in section 3. In section 4, the disturbance observer based finite control setmodel predictive controller is designed. Finally, the simulation results are given in section 5 followed by the conclusion in section 6.

(3)

These dynamics can be discretised using Euler forward method (J.S. Lim, et al., 2014) as follow:

𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘 + 1) = 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘) + 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘 + 1) = 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘) +

2. SYSTEM DYNAMICS 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘 + 1) =

A three phase two level inverter with output LC filter represented in Fig.1 is used as a system model in this paper (J.S. Lim, et al., 2014).

𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘 + 1) =

𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑉𝑉 (𝑘𝑘) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘) 𝐿𝐿𝑓𝑓 𝑖𝑖𝑖𝑖 𝐿𝐿𝑓𝑓

𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑉𝑉𝑖𝑖𝑖𝑖 (𝑘𝑘) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘) 𝐿𝐿𝑓𝑓 𝐿𝐿𝑓𝑓

𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑖𝑖 (𝑘𝑘) − 𝑖𝑖𝑜𝑜𝑜𝑜 + 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘) 𝐶𝐶𝑓𝑓 𝑓𝑓𝑓𝑓 𝐶𝐶𝑓𝑓

𝑇𝑇𝑠𝑠 𝑇𝑇𝑠𝑠 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘) − 𝑖𝑖𝑜𝑜𝑜𝑜 + 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘) 𝐶𝐶𝑓𝑓 𝐶𝐶𝑓𝑓

(4)

where 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘), 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘), 𝑉𝑉𝑖𝑖𝑖𝑖 (𝑘𝑘), 𝑉𝑉𝑖𝑖𝑖𝑖 (𝑘𝑘) are the filter currents and input voltages, 𝑖𝑖𝑜𝑜𝑜𝑜 , 𝑖𝑖𝑜𝑜𝑜𝑜 are the load currents and 𝑉𝑉𝑖𝑖𝑖𝑖 (𝑘𝑘), 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘) are the output voltages in 𝛼𝛼 − 𝛽𝛽 stationary reference frame. 𝑇𝑇𝑠𝑠 , 𝐿𝐿𝑓𝑓 , 𝐶𝐶𝑓𝑓 are the sampling time, filter inductance and filter capacitance respectively. The state space representation of (4) can be written as follow: Fig. 1. Three phase UPS model.

𝑥𝑥(𝑘𝑘 + 1) = 𝐴𝐴𝐴𝐴(𝑘𝑘) + 𝐵𝐵𝑣𝑣𝑖𝑖 (𝑘𝑘) + 𝑊𝑊𝑖𝑖𝑜𝑜 (𝑘𝑘)

Applying Kirchhoff’s laws, the three phase UPS dynamics can be obtained as in (1) and (2). 𝑑𝑑𝑖𝑖𝑓𝑓 (𝑡𝑡) 𝑉𝑉𝑖𝑖 (𝑡𝑡) − 𝑉𝑉𝑜𝑜 (𝑡𝑡) = 𝑑𝑑𝑑𝑑 𝐿𝐿𝑓𝑓

(1)

𝑑𝑑𝑉𝑉𝑜𝑜 (𝑡𝑡) 𝑖𝑖𝑓𝑓 (𝑡𝑡) 𝑖𝑖𝑜𝑜 (𝑡𝑡) = − 𝑑𝑑𝑑𝑑 𝐶𝐶𝑓𝑓 𝐶𝐶𝑓𝑓

(2)

(5)

where 𝑥𝑥(𝑘𝑘)ℝ4𝑥𝑥1 is the system state vector, 𝑣𝑣𝑖𝑖 (𝑘𝑘)ℝ2𝑥𝑥1 is the input vector, 𝑖𝑖𝑜𝑜 (𝑘𝑘)ℝ2𝑥𝑥1 is the load current vector, 𝐴𝐴ℝ4𝑥𝑥4 , 𝐵𝐵ℝ4𝑥𝑥2 , 𝑊𝑊ℝ4𝑥𝑥2 are the system matrix, input matrix and disturbance matrix respectively and, 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘) 𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘) 𝑉𝑉𝑖𝑖𝑖𝑖 𝑖𝑖𝑜𝑜𝑜𝑜 , 𝑉𝑉 = [𝑉𝑉 ], 𝑖𝑖𝑜𝑜 = [𝑖𝑖 ], 𝑥𝑥(𝑘𝑘) = 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘) 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖 [𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘)]

(1) and (2) represent a single phase dynamics. The other phase dynamics are the same as (1) and (2). These three phase dynamics can be rewritten in alpha-beta stationary frame as in (3).

52

2019 IFAC CSGRES Jeju, Korea, June 10-12, 2019

𝐴𝐴 =

0

0

1

0

0

1

0

0

1

𝑇𝑇𝑠𝑠

𝐶𝐶𝑓𝑓

[

[

𝑇𝑇𝑠𝑠

0

−

𝑊𝑊 =

−

𝑇𝑇𝑠𝑠

1

𝐶𝐶𝑓𝑓

0 0 0

0 −

0 0 0

𝑇𝑇𝑠𝑠

𝐶𝐶𝑓𝑓

𝐿𝐿𝑓𝑓

−

𝑇𝑇𝑠𝑠

𝐶𝐶𝑓𝑓

Yahya Danayiyen et al. / IFAC PapersOnLine 52-4 (2019) 51–56

𝑇𝑇𝑠𝑠

𝐿𝐿𝑓𝑓

𝑇𝑇𝑠𝑠

, 𝐵𝐵 = ]

𝐿𝐿𝑓𝑓

0

0 [ 0

53

𝑉𝑉(𝑘𝑘) = 𝑒𝑒(𝑘𝑘)𝑇𝑇 𝑄𝑄𝑄𝑄(𝑘𝑘), 𝑉𝑉(𝑘𝑘 + 1) = 𝑒𝑒(𝑘𝑘 + 1)𝑇𝑇 𝑄𝑄𝑄𝑄(𝑘𝑘 + 1),

0

𝑇𝑇𝑠𝑠

(9)

𝐿𝐿𝑓𝑓

0 0 ]

and 𝑒𝑒(𝑘𝑘)𝑇𝑇 (𝐼𝐼

𝑉𝑉(𝑘𝑘 + 1) < 𝑉𝑉(𝑘𝑘) − 𝐿𝐿𝐿𝐿) − 𝐿𝐿𝐿𝐿) 𝑒𝑒(𝑘𝑘) − 𝑒𝑒(𝑘𝑘)𝑇𝑇 𝑄𝑄𝑒𝑒(𝑘𝑘) < 0 𝑇𝑇 𝑄𝑄(𝐼𝐼

𝑒𝑒(𝑘𝑘)𝑇𝑇 (𝐼𝐼 − 𝐿𝐿𝐿𝐿)𝑇𝑇 𝑄𝑄(𝐼𝐼 − 𝐿𝐿𝐿𝐿)𝑒𝑒(𝑘𝑘) − 𝑒𝑒(𝑘𝑘)𝑇𝑇 𝑄𝑄𝑒𝑒(𝑘𝑘) < 0 𝑒𝑒(𝑘𝑘)𝑇𝑇 ((𝐼𝐼 − 𝐿𝐿𝐿𝐿)𝑇𝑇 𝑄𝑄(𝐼𝐼 − 𝐿𝐿𝐿𝐿) − 𝑄𝑄)𝑒𝑒(𝑘𝑘) < 0

]

If the matrix (𝐼𝐼 − 𝐿𝐿𝐿𝐿) is stable, then a positive definite symmetric matrix 𝑄𝑄 can be find from (11). 𝑄𝑄 − (𝐼𝐼 − 𝐿𝐿𝐿𝐿) 𝑇𝑇 𝑄𝑄(𝐼𝐼 − 𝐿𝐿𝐿𝐿) > 0

3. LOAD CURRENT ESTIMATION

[

3.1. Observer-1

(−𝑌𝑌𝑌𝑌 + 𝑄𝑄)𝑇𝑇 ]>0 𝑄𝑄

(12)

3.2. Observer-2 An integration term (15) is included in observer-1 and state error vector is used as a state vector. In order to compute the state errors, the reference state should be defined. The reference voltages 𝑉𝑉𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) and 𝑉𝑉𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) are given and the current references can be computed using steady state condition in (4) as follow:

(6)

where 𝑧𝑧(𝑘𝑘)ℝ2𝑥𝑥1 is auxiliary state variable, 𝑖𝑖̂𝑜𝑜 (𝑘𝑘)ℝ2𝑥𝑥1 is the estimated value of 𝑖𝑖𝑜𝑜 (𝑘𝑘), 𝐿𝐿 is the observer gain matrix to be calculated and 𝐼𝐼ℝ4𝑥𝑥4 identity matrix and

𝑖𝑖𝑓𝑓𝑓𝑓 ∗ (𝑘𝑘) =

(7)

The error dynamics can be obtained as follow:

𝑖𝑖𝑓𝑓𝑓𝑓 ∗ (𝑘𝑘) =

𝑒𝑒(𝑘𝑘) = 𝑖𝑖̂𝑜𝑜 (𝑘𝑘) − 𝑖𝑖𝑜𝑜 (𝑘𝑘)

𝑒𝑒(𝑘𝑘 + 1) = 𝑖𝑖̂𝑜𝑜 (𝑘𝑘 + 1) − 𝑖𝑖𝑜𝑜 (𝑘𝑘 + 1)

𝐶𝐶𝑓𝑓 (𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘 + 1) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘)) 𝑇𝑇𝑠𝑠 + 𝑖𝑖̂𝑜𝑜𝑜𝑜 (𝑘𝑘) 𝑇𝑇𝑠𝑠 𝐶𝐶𝑓𝑓 𝐶𝐶𝑓𝑓 (𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘 + 1) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘)) 𝑇𝑇𝑠𝑠

+

𝑇𝑇𝑠𝑠 𝑖𝑖̂ (𝑘𝑘) 𝐶𝐶𝑓𝑓 𝑜𝑜𝑜𝑜

(13)

(14)

The reference state vector 𝑥𝑥 ∗ (𝑘𝑘) and the error state vector 𝑥𝑥𝑒𝑒 (𝑘𝑘) can be defined as

𝑒𝑒(𝑘𝑘 + 1) = 𝐿𝐿𝐿𝐿(𝑘𝑘 + 1) − 𝑧𝑧(𝑘𝑘 + 1) − 𝑖𝑖𝑜𝑜 (𝑘𝑘 + 1)

𝑒𝑒(𝑘𝑘 + 1) = (𝐼𝐼 − 𝐿𝐿𝐿𝐿)𝑒𝑒(𝑘𝑘)

𝑄𝑄 (−𝑌𝑌𝑌𝑌 + 𝑄𝑄)

where 𝐿𝐿 = 𝑄𝑄 −1 Y.

A discrete time disturbance observer which proposed in (Kim, K.S., et al., 2013) is used to estimate the load current. The dynamics of the observer is given as follow:

𝑖𝑖̂𝑜𝑜𝑜𝑜 (𝑘𝑘) ] = 𝐿𝐿𝐿𝐿(𝑘𝑘) − 𝑧𝑧(𝑘𝑘). 𝑖𝑖̂𝑜𝑜 (𝑘𝑘) = [ 𝑖𝑖̂𝑜𝑜𝑜𝑜 (𝑘𝑘)

(11)

where 𝑄𝑄ℝ4𝑥𝑥4 positive definite symmetric matrix, 𝐿𝐿ℝ2𝑥𝑥4 matrix. Applying Schur Complement (Stephen Boyd et al., 2013) to (9) an LMI matrix can be obtained as follow:

In order to obtain a good satisfactory control performance, the disturbance effect of the load current in UPS system should be compensated. In this section, three different disturbance observers (observer-1, observer-2 and observer-3) are used to estimate the load current.

𝑧𝑧(𝑘𝑘 + 1) = 𝑧𝑧(𝑘𝑘) + 𝐿𝐿((𝐴𝐴 − 𝐼𝐼)𝑥𝑥(𝑘𝑘) + 𝐵𝐵𝑣𝑣𝑖𝑖 (𝑘𝑘) + W𝑖𝑖̂𝑜𝑜 (𝑘𝑘))

(10)

∗ 𝑥𝑥 ∗ (𝑘𝑘) = [𝑖𝑖𝑓𝑓𝑓𝑓 (𝑘𝑘)

(8)

𝑖𝑖𝑓𝑓𝑓𝑓 ∗ (𝑘𝑘)

𝑉𝑉𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘)

𝑉𝑉𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘)]𝑇𝑇 ,

(T denotes transpose of a matrix), 𝑥𝑥𝑒𝑒 (𝑘𝑘) = 𝑥𝑥 ∗ (𝑘𝑘) − 𝑥𝑥(𝑘𝑘). An integration term is defined as follow:

where 𝑒𝑒(𝑘𝑘) is the error term. The stability of the observer is determined by the term (𝐼𝐼 − 𝐿𝐿𝐿𝐿) in (8). To obtain a Hurwitz matrix to satisfy the asymptotic stability, 𝐿𝐿 should be chosen properly. Here we define an LMI matrix (Stephen Boyd et al., 2013) to find 𝐿𝐿. A monotonically decreasing Lyapunov function (𝑉𝑉(𝑘𝑘 + 1) < 𝑉𝑉(𝑘𝑘)) can be defined from the observer error dynamics (8) as follow:

𝑥𝑥𝑚𝑚 (𝑘𝑘) = 𝑥𝑥𝑚𝑚 (𝑘𝑘 − 1) + 𝑘𝑘𝑥𝑥𝑒𝑒 (𝑘𝑘)

53

(15)

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where 𝑥𝑥𝑚𝑚 (𝑘𝑘) is the new state variable and 𝑘𝑘 is the gain matrix. The dynamic of observer-2 can be written using 𝑥𝑥𝑚𝑚 (𝑘𝑘) as follow: 𝑧𝑧(𝑘𝑘 + 1) = 𝑧𝑧(𝑘𝑘) + 𝐿𝐿((𝐴𝐴 − 𝐼𝐼)𝑥𝑥𝑚𝑚 (𝑘𝑘) + 𝐵𝐵𝑣𝑣𝑖𝑖 (𝑘𝑘) + W𝑖𝑖̂𝑜𝑜 (𝑘𝑘))

are obtained with low frequency. This is the main drawback of FCS-MPC. In order to decrease the ripples, the sampling frequency should be increased. Table 1. Three phase inverter voltage vectors 𝑢𝑢𝑖𝑖 (𝑘𝑘) 𝑢𝑢𝑖𝑖 (𝑘𝑘) = 𝑢𝑢 ⃗ 𝛼𝛼𝛼𝛼 + 𝑗𝑗𝑢𝑢 ⃗ 𝛽𝛽𝛽𝛽

𝑢𝑢𝑖𝑖 (𝑘𝑘) 𝑢𝑢𝑖𝑖 (𝑘𝑘) = 𝑢𝑢 ⃗ 𝛼𝛼𝛼𝛼 + 𝑗𝑗𝑢𝑢 ⃗ 𝛽𝛽𝛽𝛽

3.3. Observer-3

𝑢𝑢 ⃗0

0

𝑢𝑢 ⃗4

A simple discrete time observer used in (Cortes, P., et al., 2009) with FCS-MPC is used as observer-3 for the comparison purpose. The dynamic of observer-3 is given as follow:

𝑢𝑢 ⃗1

2 𝑉𝑉 3 𝑑𝑑𝑑𝑑

⃗5 𝑢𝑢

𝑢𝑢 ⃗2

1 √3 𝑉𝑉𝑑𝑑𝑑𝑑 + 𝑗𝑗 𝑉𝑉 3 3 𝑑𝑑𝑑𝑑

𝑢𝑢 ⃗6

1 √3 𝑉𝑉𝑑𝑑𝑑𝑑 − 𝑗𝑗 𝑉𝑉 3 3 𝑑𝑑𝑑𝑑

𝑢𝑢 ⃗7

0

𝑖𝑖̂𝑜𝑜 (𝑘𝑘 − 1) = 𝑖𝑖𝑓𝑓 (𝑘𝑘 − 1) −

𝐶𝐶𝑓𝑓 (𝑉𝑉 (𝑘𝑘) − 𝑉𝑉𝑜𝑜 (𝑘𝑘 − 1)) 𝑇𝑇𝑠𝑠 𝑜𝑜

(16)

(17)

where 𝑖𝑖̂𝑜𝑜 (𝑘𝑘 − 1) is the estimated load current.

𝑢𝑢 ⃗3

4. FINITE CONTROL SET MODELPREDICTIVE

In this section, an FCS-MPC controller is designed for reference voltage tracking purpose. The discrete model of the three phase UPS inverter is used in the controller design. This model consist of 8 possible switching state. All of these states are evaluated in a prediction model to predict the system states. These predicted states are used in the cost function to compute 8 cost function values in one sampling period. The minimum one of these 8values is selected and the optimal switching state which makes the cost function minimum is determined. The selected switching state is send to the inverter gates in the next sampling time. The prediction model is defined as follow: (18)

where 𝑖𝑖 = 1,2, … ,8, 𝑖𝑖̂𝑜𝑜 (𝑘𝑘) is the estimated load current which estimated using disturbance observer, 𝑥𝑥𝑖𝑖 (𝑘𝑘 + 1|𝑘𝑘) is the prediction of 𝑥𝑥𝑖𝑖 (𝑘𝑘) at time step 𝑘𝑘, 𝑢𝑢𝑖𝑖 (𝑘𝑘) is the inverter input voltage vectors given in Table-1.

2. CONCLUSIONS A three phase UPS output voltage is controlled with FCS-MPC based on disturbance observer. The performance of the observers with FCS-MPC are evaluated under different sampling frequency. A discrete time disturbance observer called observer-1 is used with FCS-MPC. An LMI matrix is defined to calculate optimal gain matrix of the observer. An integration term and state error are included in observer-1 and the new observer is called observer-2. In order to see the effect of the effect of the sampling frequency, the simulations are carried out with different sampling frequency in order to see the effect of the frequency in FCS-MPC. In this study, the parameter uncertainties are not considered. They can be included for robust solution in the future works. The integration gain 𝑘𝑘 is chosen using trial and error method. It can be calculated using some robust methods such as linear matrix inequalities. The discrete observer which is used in (Kim, K.S., et al., 2013) called observer-1 in this paper can be further developed using its continuous counterpart with different discretization method in order to use in FCS-MPC to remove the offset error in the steady state.

The eight predicted voltage values are evaluated in a cost function as follow: 𝑔𝑔𝑖𝑖 = (𝑉𝑉𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘 + 1))2 + (𝑉𝑉𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) − 𝑉𝑉𝑜𝑜𝑜𝑜 (𝑘𝑘 + 1))

2

1 √3 𝑉𝑉 − 𝑉𝑉𝑑𝑑𝑑𝑑 − 𝑗𝑗 3 3 𝑑𝑑𝑑𝑑

The reasonable frequency is obtained in 100 kHz, 75 kHz, 50 kHz and 30 kHz. Fig. 3 shows the comparison of observer-1 and observer-2 in 10 kHz. Although the voltage ripples are decreased with observer-2 in low frequencies, there is an offset error in high frequencies as it can be seen in Fig. 4. This offset error is another drawback of FCS-MPC. The three phase instantaneous voltage THD is given in Fig. 5 with observer-3 and observer-2 under linear load. It is clear that low THD (1%) can be obtained using observer-2. Fig. 6 shows the reference UPS output voltages and measured output voltages using observer-2 in alpha-beta frame with a small steady state offset error. The sampling frequency is chosen 30 𝑘𝑘𝑘𝑘𝑘𝑘 for this simulation. The peak values of the alpha-beta reference voltages, 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) and 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) are chosen156𝑉𝑉. It is worth nothing that the low THD can be obtained with high frequencies.

CONTROL

𝑥𝑥𝑖𝑖 (𝑘𝑘 + 1|𝑘𝑘) = 𝐴𝐴𝑥𝑥𝑖𝑖 (𝑘𝑘) + 𝐵𝐵𝑢𝑢𝑖𝑖 (𝑘𝑘) + 𝑊𝑊𝑖𝑖̂𝑜𝑜 (𝑘𝑘)

1 √3 𝑉𝑉 − 𝑉𝑉𝑑𝑑𝑑𝑑 + 𝑗𝑗 3 3 𝑑𝑑𝑑𝑑

2 − 𝑉𝑉𝑑𝑑𝑑𝑑 3

(19)

where 𝑔𝑔𝑖𝑖 is the cost function and 𝑔𝑔ℝ1𝑥𝑥1 . Then the minimum value of 𝑔𝑔𝑖𝑖 is selected and the corresponding switching state is send to the inverter. This switching state is determined by its voltage vector given in Table I. 1. SIMULATION RESULTS In this section simulation results are given under different scenarios. The results are obtained in MATLAB under linear load. The discrete model of the system is obtained in MATLAB and the simulation results are given based on this model. Fig. 2 shows the output d-axis voltage with different sampling frequency 𝑓𝑓𝑠𝑠 using observer-1. High voltage ripples 54

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55

Fig. 2. d-axis voltage response under different sampling periods with observer-1

Fig. 5. Simultaneous response of three phase output voltage THD with observer-3 and observer-2, 𝑓𝑓𝑠𝑠 = 30 𝑘𝑘𝑘𝑘𝑘𝑘.

Fig. 3. Behaviour of the d-axis voltage with observer-1 and observer-2 , 𝑓𝑓𝑠𝑠 = 10 𝑘𝑘𝑘𝑘𝑘𝑘.

Fig. 6. Reference and measured UPS output voltage in alphabeta frame with observer-2, 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) = 156𝑉𝑉, 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 ∗ (𝑘𝑘) = 156𝑉𝑉 𝑓𝑓𝑠𝑠 = 30 𝑘𝑘𝑘𝑘𝑘𝑘. REFERENCES Aboudonia, A., Rashad, R. & El-Badawy, A., 2015. Time domain disturbance observer based control of a quadrotor unmanned aerial vehicle. 2015 25th International Conference on Information, Communication and Automation Technologies, ICAT 2015 - Proceedings, pp.1–6. Chen, W.-H., Li, S. & Yang, J., 2011. Non-linear disturbance observer-based robust control for systems with mismatched disturbances/uncertainties. IET Control Theory & Applications, 5(18), pp.2053–2062. Cortes, P. et al., 2009. Model Predictive Control of an Inverter With Output LC Filter for UPS Applications. Industrial Electronics, IEEE Transactions on, 56(6), pp.1875– 1883.

Fig. 4. d-axis voltage and its reference with observer-3 and observer-2, 𝑓𝑓𝑠𝑠 = 30 𝑘𝑘𝑘𝑘𝑘𝑘.

J. S. Lim, C. Park, J.H. and Y.I.L., 2014. Robust Tracking Control of a Three-Phase DC–AC Inverter for UPS 55

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Applications. IEEE Transactions Electronics, 61(8), pp.4142–4151.

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Jun Yang, Shihua Li & Wen-Hua Chen, 2010. Autopilot Design of Bank-to-Turn Missiles Using State-Space Disturbance Observers. UKACC International Conference on CONTROL 2010, (20080769006), pp.1218–1223. Kim, K.S. & Rew, K.H., 2013. Reduced order disturbance observer for discrete-time linear systems. Automatica, 49(4), pp.968–975. Stephen Boyd, Laurent El Ghaoui, Eric Feron, V.B., 1994. Linear Matrix Inequalities in System and Control Theory illustrate., SIAM. Wang, J. et al., 2017. Design and Implementation of Disturbance Compensation-Based Enhanced Robust Finite Control Set Predictive Torque Control for Induction Motor Systems. IEEE Transactions on Industrial Informatics, 13(5), pp.2645–2656.

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