# LQR-Based Power Train Control Method Design for Fuel Cell Hybrid Vehicle

Authors: Yun Haitao; Zhao Yulan; Liu Zunnian; Hao Kui
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Based on the mathematical model of fuel cell hybrid vehicle (FCHV) proposed in our previous study, a multistate feedback control strategy of the hybrid power train is designed based on the linear quadratic regulator (LQR) algorithm. A Kalman Filter (KF) observer is introduced to estimate state of charge (SOC) of the battery firstly, and then a linear quadratic regulator is constructed to compute the state feedback gain matrix of the closed-loop control system. At last, simulation and actual test are utilized to demonstrate this new approach.


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## Body

## 1. Introduction

Fuel cell vehicle (FCV) is one of the most important trends of new energy vehicles. Since the dynamic response of fuel cell has a property of time lag, which fails to meet the demands of the fast-changing vehicle load, a hybrid power system equipped with auxiliary energy storage device is indispensable. This is so-called fuel cell hybrid vehicle. For such a complex system which has multiple power sources, it is very important for energy management strategy to maintain the power balance of the power train.There have been many academic papers discussing the energy management of the FCHV in recent years. Hyun et al. [1] discussed the feature of hydrogen FCHV power train based on experimental study. Pede et al. [2] discussed hybrid degree of fuel cell vehicle power train. Mohammadian et al. [3] utilized neural network and genetic algorithm to solve the control problem of the fuel cell power system. Kim and Peng [4] designed a power allocation algorithm based on Markov decision programming (MDP). Eren et al. [5] proposed a control strategy for fuel cell hybrid power system based on fuzzy logic. Ryu et al. [6] designed different energy distribution strategies according to the driving modes of the motor. Torreglosa et al. [7] put forward a power control method based on optimal theory. Recently, there are several patents [8–10] with the focus on configuration and control of FCHV power train. In addition, several research teams pay attention to model and simulation of FCHV [11–14].


## 2. Simulation Model of the FCHV Power Train

The fuel cell hybrid vehicle (FCHV) studied in this research isSHANGHAI prototype vehicle from Shanghai Motor Company, whose power train configuration is shown in Figure 1. The mathematic model of SHANGHAI power train has been constructed based on test data and the equivalent circuit model in our previous research [15].Figure 1
Power train configuration ofSHANGHAI FCHV.The following equation is the state-space equation of the mathematic model:(1)x˙=Ax+Bu,y=Cx+Du,where the state vectorx=[IdcfImUEU1U2]T, the input vector u=[IdcfsetImset]T, the output vector y=Ubus, and A, B, C, and D are the state matrix, as defined in (2), respectively:(2)A=[-1Tf00000-1Tmc0001CE-1CE0001C1-1C10-1R1C101C21C200-1R2C2],B=[KfTf00KmcTmc000000],C=[R0-R0111],D=[00].Herein,Kf, Tf, Kmc, Tmc, CE, C1, C2, R0, R1, and R2 are constants of the model, which can be estimated by the least-squares parameter estimation method.Based on the mathematic model, a simulation model is constructed. Figure2 is the overall system model block diagram, which indicates the input and output signals of all component models and how they are interconnected.Figure 2
The block diagram of the simulation model.To verify the mathematic model and simulation model of the power train, the exampled comparison between simulation results with test data is presented in Figure3, confirming that the FCHV model can meet the demand of model-based FCHV control design.Comparison between simulation results and test data (part of J1015 cycle).
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## 3. Multistate Feedback Control Strategy Designing Based on LQR

According to state equation of the FCHV power train, the power train is marginally stable and little random interference may cause system instability. Therefore, it is very important for energy management system to maintain the power balance of the power train.In this study, a multistate feedback control strategy is designed based on linear quadratic regulator (LQR) algorithm. A multistate feedback controller is designed using LQR technique and a Kalman filter (KF) observer is designed for battery open circuit voltage (OCV) estimation. Figure4 shows the schematic diagram of this new control strategy, where uf is feedforward control input, ub is feedback control input, u is actual input, y is actual output, xtarget is target setting of the controller, and x^ is the state observed by the KF observer.Figure 4
The schematic diagram of the multistate feedback control strategy.To make the control framework more suitable for the actual control input, the control framework shown in Figure4 is reformed, as shown in Figure 5. This control strategy can be described by
(3)u=uf+ubuf=[00Imsetff]Tub=[PfsetfbIdcfsetfbImsetfb]=K(xtarget-x^)ub=[K′11K′12K′13K′14K′15K21K22K23K24K25K31K32K33K34K35]ub=×([00OCVobj00]-[I^dcfI^mU^EU^1U^2])ub×[K11′K12′K13′K14′K15′]=Ubus[K11K12K13K14K15].Figure 5
The schematic diagram of the multistate feedback strategy (reformed).Herein,OCVtarget represents the target value of the battery OCV, and K is the state feedback matrix. The variables shown in (3) have been explained in Figure 2. In the actual control system, uf is computed by power demand of the power train and limited by some boundary condition ub is computed by control deviation and the state feedback matrix K.


### 3.1. Observer Designing Based on KF

Although battery OCV is not a linear function of battery SOC, according to [16], when battery SOC is in 20% ~ 90% (its work range), battery OCV has a corresponding relationship with battery SOC. Therefore, the fluctuated ranges of the SOC can be regulated by changing the target value of the battery OCV OCVtarget, namely, UE of the state-space model. However, the state UE is not a measurable variable in the power train. To estimate the state UE, a KF observer is designed. The discrete state-space model of the battery can be obtained from the state-space model of the battery equation as follows:
(4)x(k)=Gx(k-1)+Hu(k)+Γw(k),z(k)=Mx(k)+Nu(k)+v(k),where state vectorx(k)=[UE(k)U1(k)U2(k)]T, input vector u(k)=[Ib(k)], output vector y(k)=[Ubus(k)], Γ is the noise transfer matrix, w(k) and  v(k) are zero-mean white Gaussian stochastic process, and G, H, M, and N are discrete state matrices defined as
(5)G=[1000R1C1R1C1+T000R2C2R2C2+T],H=[-TCE-TR1C1R1C1+T-TR2C2R2C2+T],M=[111],N=[-R0].Herein,T is the sampling period. The filter is initialized with the best information available on the state and error covariance, and the recursive equation of the KF is summarized as follows:
(6)Initialization:X^(0)=E[X(0)],P(0)=E[X(0)X(0)T]Q=E[w×wT],R=E[v×vT]RecursiveComputation:P(k∣k-1)=GP(k-1)GT+ΓQΓT,Kk=P(k∣k-1)MT[MP(k∣k-1)MT+R]-1,X^(k)=[I-KkH][GX^(k-1)+Hu(k)]X^(k)=+Kkz(k)-Nu(k),P(k)=[I-KkM]P(k∣k-1).Although the hypothesis about the noise process cannot conform exactly to the fact, the result obtained from the test shows that the KF observer still performs well.To validate the KF observer designed here, a J1015 dynamic test was performed (sample period was 50 ms) As the input data of the KF observer, the current and voltage of the battery were collected in the test, as shown in Figure6. The estimated results of the battery OCV by the KF observer and current integral calculation method shown in Figure 7 indicate that, if the initial deviation is excluded from consideration, the maximum error is not more than 1% in the cycle test. Though the current test data has its inevitable error, in a relatively short time OCV estimated by current integral calculation approaches the actual value. Therefore, the test result indicates that the KF designed for battery OCV estimation is valid.Current and voltage of the battery (J1015 cycle).
(a)
(b)Figure 7
Battery OCV estimated by the KF observer and current integral method.


### 3.2. Multistate Feedback Controller

The state feedback matrixK is very important for the multistate feedback controller. In actual control strategy, elements of the matrix K need to be restrained by some limitation and boundary condition, as shown in the following:
(7)K11<m,K13<n,K14<o,K15<p,K21=K22=K23=K24=K25=0,wherem, n, o, and p are the limitations determined by configuration parameters of the power train and practical experience. K11 is selected according to the test, K13, K14, and K15 can be selected properly by analyzing the step responding of the model. To compute state feedback gain in an optimal way, LQR is constructed based on the linear mathematic model of the power train, and the quadratic characteristic index function of the regulator is as follows:
(8)J=12∫t0∞[a(Idcf-Im)2+bUE2+cU12+dU22ppppppp+eIdcfset2+fImset2(Idcf-Im)2]dt=12∫t0∞[xT(t)Qx(t)+uT(t)Ru(t)]dt.In (8), a, b, c, d, e, and f are the weight of each item, Q is the state weight matrix, and R is the control weight matrix. Using Hamilton minimum principle, the state feedback matrix can be obtained by solving (8) for analytic solution. Note that the selection of proper weight matrix is very important for solving the equation. There are several helpful rules on the selection of Q and R: (1)  Q and R are symmetrical matrixes, in which Q is a half positive definite matrix and R is a positive definite matrix; (2) because Q and R are interrelated, once R is determined firstly, Q can be determined by simulation and test. In addition, state feedback matrix is limited by the boundary conditions, namely, (7) in this paper.In state weight matrixQ,  b is very important, which affects the stability of the close-loop control system directly. To make control system stable, b must be properly selected to make sure that all poles of the close-loop control system are in left quadrant. In addition, it can be seen that the larger the b, the more stable the control system.Figure8 presents the exampled system step responding curves. It can be seen that the larger b leads to the faster converging behavior of UE. However, if b is over large, Ubus will become vibration. When b equals 2, UE and Ubus tend to balance position with moderate speed.Step responding of the model.
(a)
(b)


## 3.1. Observer Designing Based on KF

Although battery OCV is not a linear function of battery SOC, according to [16], when battery SOC is in 20% ~ 90% (its work range), battery OCV has a corresponding relationship with battery SOC. Therefore, the fluctuated ranges of the SOC can be regulated by changing the target value of the battery OCV OCVtarget, namely, UE of the state-space model. However, the state UE is not a measurable variable in the power train. To estimate the state UE, a KF observer is designed. The discrete state-space model of the battery can be obtained from the state-space model of the battery equation as follows:
(4)x(k)=Gx(k-1)+Hu(k)+Γw(k),z(k)=Mx(k)+Nu(k)+v(k),where state vectorx(k)=[UE(k)U1(k)U2(k)]T, input vector u(k)=[Ib(k)], output vector y(k)=[Ubus(k)], Γ is the noise transfer matrix, w(k) and  v(k) are zero-mean white Gaussian stochastic process, and G, H, M, and N are discrete state matrices defined as
(5)G=[1000R1C1R1C1+T000R2C2R2C2+T],H=[-TCE-TR1C1R1C1+T-TR2C2R2C2+T],M=[111],N=[-R0].Herein,T is the sampling period. The filter is initialized with the best information available on the state and error covariance, and the recursive equation of the KF is summarized as follows:
(6)Initialization:X^(0)=E[X(0)],P(0)=E[X(0)X(0)T]Q=E[w×wT],R=E[v×vT]RecursiveComputation:P(k∣k-1)=GP(k-1)GT+ΓQΓT,Kk=P(k∣k-1)MT[MP(k∣k-1)MT+R]-1,X^(k)=[I-KkH][GX^(k-1)+Hu(k)]X^(k)=+Kkz(k)-Nu(k),P(k)=[I-KkM]P(k∣k-1).Although the hypothesis about the noise process cannot conform exactly to the fact, the result obtained from the test shows that the KF observer still performs well.To validate the KF observer designed here, a J1015 dynamic test was performed (sample period was 50 ms) As the input data of the KF observer, the current and voltage of the battery were collected in the test, as shown in Figure6. The estimated results of the battery OCV by the KF observer and current integral calculation method shown in Figure 7 indicate that, if the initial deviation is excluded from consideration, the maximum error is not more than 1% in the cycle test. Though the current test data has its inevitable error, in a relatively short time OCV estimated by current integral calculation approaches the actual value. Therefore, the test result indicates that the KF designed for battery OCV estimation is valid.Current and voltage of the battery (J1015 cycle).
(a)
(b)Figure 7
Battery OCV estimated by the KF observer and current integral method.


## 3.2. Multistate Feedback Controller

The state feedback matrixK is very important for the multistate feedback controller. In actual control strategy, elements of the matrix K need to be restrained by some limitation and boundary condition, as shown in the following:
(7)K11<m,K13<n,K14<o,K15<p,K21=K22=K23=K24=K25=0,wherem, n, o, and p are the limitations determined by configuration parameters of the power train and practical experience. K11 is selected according to the test, K13, K14, and K15 can be selected properly by analyzing the step responding of the model. To compute state feedback gain in an optimal way, LQR is constructed based on the linear mathematic model of the power train, and the quadratic characteristic index function of the regulator is as follows:
(8)J=12∫t0∞[a(Idcf-Im)2+bUE2+cU12+dU22ppppppp+eIdcfset2+fImset2(Idcf-Im)2]dt=12∫t0∞[xT(t)Qx(t)+uT(t)Ru(t)]dt.In (8), a, b, c, d, e, and f are the weight of each item, Q is the state weight matrix, and R is the control weight matrix. Using Hamilton minimum principle, the state feedback matrix can be obtained by solving (8) for analytic solution. Note that the selection of proper weight matrix is very important for solving the equation. There are several helpful rules on the selection of Q and R: (1)  Q and R are symmetrical matrixes, in which Q is a half positive definite matrix and R is a positive definite matrix; (2) because Q and R are interrelated, once R is determined firstly, Q can be determined by simulation and test. In addition, state feedback matrix is limited by the boundary conditions, namely, (7) in this paper.In state weight matrixQ,  b is very important, which affects the stability of the close-loop control system directly. To make control system stable, b must be properly selected to make sure that all poles of the close-loop control system are in left quadrant. In addition, it can be seen that the larger the b, the more stable the control system.Figure8 presents the exampled system step responding curves. It can be seen that the larger b leads to the faster converging behavior of UE. However, if b is over large, Ubus will become vibration. When b equals 2, UE and Ubus tend to balance position with moderate speed.Step responding of the model.
(a)
(b)


## 4. Verification of the Control Strategy

Simulation and practical driving test on the designed control strategy with three driving cycles are conducted. The driving cycle includes J1015, UDDS, and NEDC. Exampled simulation and test data from NEDC driving cycle are presented in Figures9–12. Figure 9 shows the changing process of vehicle velocity in the simulation and practical test during the NEDC cycle, indicating that the vehicle’s power performance can be ensured using the designed strategy. Figure 10 shows the changing process of the subsystem current. Figure 11 shows the changing processes of the battery voltage and SOC during the NEDC driving cycle, where the battery voltage and SOC fluctuate within the prescriptive range, indicating that the proposed control strategy can guarantee the stable control of the FCHV power train.Figure 9
Changing process of vehicle speed (ECE_EUDC cycle).Changing process of subsystem current (part of the ECE_EUDC cycle).
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(b)Figure 11
Changing process of battery voltage (ECE_EUDC cycle).Figure 12
Changing process of battery SOC (ECE_EUDC cycle).


## 5. Conclusions

In summary, a LQR-based control strategy designing method for FCHV power train is proposed in this study. A multistate feedback control strategy is designed and applied in the power train ofSHANGHAI FCHV. In this control strategy, a KF observer is introduced to estimate battery OCV and the state feedback gain matrix of the multistate feedback control system is computed by the LQR algorithm. To verify the designed control strategy, simulation and practical driving test with three types of driving cycles are conducted. The results of the simulation and practical testing show that the designed LQR-based power control strategy can ensure power performance of the vehicle and guarantee the stable control of the hybrid power train, which proves that the designed control strategy works excellently.The LQR-based control design method presented in this research well suits to control development for FCHV power train. And it provides valuable reference to the model-based control strategy development for other types of hybrid power train.



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*Source: 968203-2013-11-27.xml*