**4.1. MPC control application for EM2**

In this research, we assume that the MPC outputs horizon length is set equal to the inputs control horizon, i.e., *Nu* = *Ny* = *Np* (equal to the predictive lengths). The quadratic objective junction *J*(*U* , *x*(*t*)) in equation (4.3) is minimized over a vector *Np* future prediction inputs

(31) can be rewritten as a function of the current state *x*(*t*) and the current setpoints *r*(*t*):

( ) 1 1 ' ' ( ), ( ) ( ) ( ) min ()() , 2 2

*U x t r t x t Yx t U HU x t r t FU* ì ü Y= + + í ý

subject to the hard combined constraints of *GU* ≤*W* + *Ex*(*t*), where the column vector

and *E* are proceeding matrices obtained from *Q*, *R* and *x*(*t*), *r*(*t*) in equation (4.1). Because that we use only the optimizer *U* , other terms involving *Y* are normally removed from equation (32). Therefore, the optimal problem in (4.4) will be a purely quadratic formula and depending on only the current state variables *x*(*t*), and the current set-points *r*(*t*). Implementation of MPC always requires the real-time solution of each quadratic program for each discrete time steps.

In fact, the system may have always both input and output constraints. Difficulties will be arised due to the inability to respond to all output constraints because of the already input constraints. Since MPC is applied for the real-time implementation, any infeasible solution of the optimal control problems cannot be tolerated. Basically if the input constraints are set from the system physical limits and usually considered as the hard/unchanged constraints. At the same time, if the system outputs constraints are the measured velocities and the unmeasured torques which are not so much strictly imposed and can be violated somewhat during the movement of the vehicles. In order to guarantee the system stability if some outputs may violate the constraints, equation (32) can be transformed to some other soften constraints.

is the MPC optimization vector; *H* =*H* '

*x*(*t*) + ∑ *k*=0 *Np*−1

î þ (32)

*<sup>A</sup>kBut*+*Np*−1−*<sup>k</sup>* , equation

>0, and then, *H* , *F* , *Y G*, *W*

e

(*t*) are represent the violation

(33)

starting from the state *x*(*t*).

46 New Applications of Electric Drives

*U* ≜ Δ*ut*, ..., Δ*ut*+*Np*−<sup>1</sup> '∈Δ

For the MPC with hard constraints, by substituting *xt*+*Np*|*<sup>t</sup>* <sup>=</sup> *<sup>A</sup> Np*

*t*

( ) ' ' '

*t*

 ee

ë û

e

The new weighting items, *ε<sup>i</sup>*

penalty terms (*ε<sup>i</sup>*

(ject to ) ; , and

(generally some small values) become the weighting factors, *ε<sup>i</sup>*

 e

min | max min | max sub

the MPC's subject to these soften state constraints can be read from reference [16].

*i i <sup>U</sup>*

e

+ +

(*t*), are added into the MPC soften objective function: Λ>0

 e

 e

*i yu y t kt y y t kt u*

e

î þ = -£ £ + -£ £ + é ù

(*t*)≥0) for the scheme objective function. These values will keep the output

*t y yy u uu*

violations at low levels until the constrained solution can be appeared. Further reference of

11 1 ( ), ( ) ( ) ( ) min ()() () () , 22 2

*x t r t x t Yx t U HU x t r t FU t t*

ì ü Y = + + +L í ý

U

The MPC control for EM2 is used for driving the HEVs at the slow speed (less 50 km/h). In this moment the friction clutch is open and both ICE and EM1 are turned off. The dynamic equation of the system in this case is formulated in equation (3.24). The MPC objective function applied in this system is developed in equations (4.3), (4.5), and (4.6). The discrete time interval of all simulations is set at 0.01 sec.

Parameters used for the EM2 are: torsional rigidity, *k<sup>θ</sup>* =1158; motor constants, *kE*<sup>2</sup> =*kT* <sup>2</sup> =10; motor inertia, *J*<sup>2</sup> =1; load inertia, *J*<sup>3</sup> =2; gear ratio, *i* =2.34; motor damping, *kβ*<sup>2</sup> =0.5; load damping, *kβ*<sup>3</sup> =12; armature resistance, *RI* <sup>2</sup> =5.

The input constraints in this example are set as follows: control constraints for the DC input voltage for the vehicle is |*V*<sup>2</sup> | ≤300*V* ; Δ*u*(*t*)≤inf, or in this case, there is no need to set the input limit increments on Δ*u*(*t*). The physical output constraints are set for the motor shaft with an allowable shear stress (carbon steel), *τ* =25 (MPa or N/mm²). The constraint for the output torque on shaft2 is |*T* | =*τπ d* 3 <sup>16</sup> , where *<sup>d</sup>* =0.05*m* is the output shaft diameter of the motor. Then, the output torque constraint will be |*T*<sup>2</sup> | ≤455*Nm*.

The MPC parameters will be selected as the horizon prediction length, *Nu* = *Ny* = *Np* =5, the weighting matrices, *<sup>Q</sup>* <sup>=</sup> <sup>10</sup> <sup>0</sup> <sup>0</sup> <sup>10</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> . Figure 11 shows the MPC performance with the input voltage, the output speed, and torque.

**Figure 11.** MPC performance with *Np* =5 and *Q* =10*R*.

Because the MPC performance is highly relying on the selection of the values in weighting matrices *Q* and *R*. If we choose that *Q* is much bigger than *R* (*Q*≫*R*), then, the voltage input control increment Δ*u*(*t*) will become much bigger than the output penalty in (*y*(*t*)−*r*) as indicated in the MPC objective function in (6.3). The controller will drive the vehicle to track the output set-points very fast, but in return, the vehicle will need very much great energy for input torques as illustrated in Figure 12.

In the simulation, the horizon prediction is chosen as *Nu* = *Ny* = *Np* =5 and the values for weighting matrices are selected as *<sup>Q</sup>* <sup>=</sup> <sup>100</sup> <sup>0</sup> <sup>0</sup> <sup>100</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> :

The MPC horizon length of prediction is highly influenced by its performance. The system will become looser if a longer prediction horizon is selected and the MPC will achieve better performance since the system becomes more flexible and then it is easier to find out better

**Figure 12.** MPC performance with *Np* =5 and *Q* =100*R*.

torque on shaft2 is |*T* | =*τπ*

48 New Applications of Electric Drives

weighting matrices, *<sup>Q</sup>* <sup>=</sup> <sup>10</sup> <sup>0</sup>

*d* 3

the output torque constraint will be |*T*<sup>2</sup> | ≤455*Nm*.

input voltage, the output speed, and torque.

**Figure 11.** MPC performance with *Np* =5 and *Q* =10*R*.

input torques as illustrated in Figure 12.

weighting matrices are selected as *<sup>Q</sup>* <sup>=</sup> <sup>100</sup> <sup>0</sup>

<sup>16</sup> , where *<sup>d</sup>* =0.05*m* is the output shaft diameter of the motor. Then,

<sup>0</sup> <sup>10</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> . Figure 11 shows the MPC performance with the

The MPC parameters will be selected as the horizon prediction length, *Nu* = *Ny* = *Np* =5, the

Because the MPC performance is highly relying on the selection of the values in weighting matrices *Q* and *R*. If we choose that *Q* is much bigger than *R* (*Q*≫*R*), then, the voltage input control increment Δ*u*(*t*) will become much bigger than the output penalty in (*y*(*t*)−*r*) as indicated in the MPC objective function in (6.3). The controller will drive the vehicle to track the output set-points very fast, but in return, the vehicle will need very much great energy for

In the simulation, the horizon prediction is chosen as *Nu* = *Ny* = *Np* =5 and the values for

The MPC horizon length of prediction is highly influenced by its performance. The system will become looser if a longer prediction horizon is selected and the MPC will achieve better performance since the system becomes more flexible and then it is easier to find out better

<sup>0</sup> <sup>100</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> :

solutions. However with the longer prediction, the burden of the computer calculation will exponentially rise up and the time for calculating the optimal actions will depend on the ability of the CPU and the speeds of the communication protocols. In the next simulation, we run the MPC controller with a very shorter horizon length of *Nu* = *Ny* = *Np* =2; and with medium values

of *<sup>Q</sup>* <sup>=</sup> <sup>50</sup> <sup>0</sup> <sup>0</sup> <sup>50</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> . Result of this simulation is shown in Figure 13.

As shown in Figure 13, for very short horizon length prediction of *Np* =2. The MPC controller performance becomes worse since the system cannot properly track the output set-points. And then, compared to the next simulation with a longer horizon length prediction, the MPC will achieve a much better performance.

The horizon length prediction is now chosen even longer for *Nu* = *Ny* = *Np* =10 and for the same values of weighting *<sup>Q</sup>* <sup>=</sup> <sup>50</sup> <sup>0</sup> <sup>0</sup> <sup>50</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> , the performance of this controller is shown in Figure 14.

The MPC scheme with longer prediction length in Figure 14 indicates better performance compared to that in Figure 13. However in this simulation, the much higher input energy is required for the input DC voltages.

A fact is that if we impose an input constraint for voltage of |*V*<sup>2</sup> | ≤300*V* and because that the DC motor basically runs in only positive voltage or 0≤*V*<sup>2</sup> ≤300*V* . The braking and the recharged system will be started if we want to slow down the vehicle speed. In the next example, we will

**Figure 13.** MPC performance with horizon length *Np* =2 and *Q* =50*R*.

**Figure 14.** MPC performance with horizon length *Np* =10 and *Q* =50*R*.

set the input voltage for the DC motor on only positive values of 0≤*V*<sup>2</sup> ≤300*V* . This simulation allows for the regenerative braking option when we start braking to slow down the speed. The performance of this simulation is shown in Figure 15 for the horizon length of *Nu* = *Ny* = *Np* =5

and high weighting values of *<sup>Q</sup>* <sup>=</sup> <sup>1000</sup> <sup>0</sup> <sup>0</sup> <sup>1000</sup> and small weighting value of *<sup>R</sup>* <sup>=</sup> <sup>1</sup> .

**Figure 15.** MPC performance with 0≤*V*<sup>2</sup> ≤300*V* , *Np* =5 and *Q* =1000*R*.

Simulations for the MPC performance with the motor EM2 have been conducted with the change of parameters in the weighting matrices, horizon prediction length, and with the input voltage constraints. For the next part, we will investigate the MPC controller for the ICE and the EM1 to track the desired speed set-points applied for the vehicle high speeds (above 50 km/h) and investigate how to synchronize the velocities of these two parts with the friction clutch engagement.

## **4.2. MPC performance for ICE and EM1**

**Figure 13.** MPC performance with horizon length *Np* =2 and *Q* =50*R*.

50 New Applications of Electric Drives

**Figure 14.** MPC performance with horizon length *Np* =10 and *Q* =50*R*.

If the speed of the vehicle exceeds 50 km/h, EM1 will start and activate the ICE to drive the vehicle. Depending on the needed requirements of the speeds and the torques, this hybrid vehicle can configure to operate only the ICE or all ICE, and/or EM1 and/or EM2. For the starting time, the friction clutch is still open and the dynamic formula for the vehicle in the ICE and EM1 is indicated as in equation (27).

The example parameters used for the EM1 are: motor constants, *kE*<sup>2</sup> =*kT* <sup>2</sup> =15; motor inertia, *J*<sup>1</sup> =1; motor damping, *kβ*<sup>1</sup> =0.5; armature resistance, *RI* <sup>1</sup> =7; compensation factor, *ς* =0.5. The system is discretized at a time interval of 0.01 sec. The air drag resistance torque at *ω*<sup>3</sup> =2000 rpm is chosen as *Mv*<sup>0</sup> =30 Nm.

The physical input constraints for this vehicle are set for the DC voltage in range of |*V*<sup>1</sup> | ≤48*V* ; The input increment constraint is Δ*u*(*t*)≤180*V* / sec. The output constraint of torque for the shaft1 is set for |*T*<sup>1</sup> | ≤628*Nm*.

The MPC parameters are chosen for the horizon length prediction of *Nu* = *Ny* = *Np* =5, the weighting values for the matrices of *<sup>Q</sup>* <sup>=</sup> <sup>10</sup> <sup>0</sup> <sup>0</sup> <sup>10</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> <sup>0</sup> <sup>0</sup> <sup>1</sup> . Figure 16 shows the MPC performance at the starting time.

From Figure 16, after a certain delay time of 1 second, the ICE is fully started and after about 2.4 seconds the peed of ICE has reached the set-point of 2000 rpm and operate stably at a power of 6 kW, generating an output torque of 30 Nm (at this moment, the friction clutch is still open).

Figure 16. MPC performance at the starting time. **Figure 16.** MPC performance at the starting time.

60

5 10 15

Input ICE (Kw)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

time

Engine ICE (Kw)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

time

Next, we run the hybrid vehicle with this EM1 and ICE to track the speed set-points and test for the clutch engagement. It is assumed that if the motor EM2 operating at more than 1500 rpm, the motor EM1 will start and activate the ICE to engage to the system. Regarding the improvement of the driving comfort and the reduction of jerk, the clutch engagement will be taken place for only when1 2 or 1 2 1.05. or the EM1 and ICE must track the EM2 at 5% positive offset. The MPC objective function in (4.3) is now changed from setpoints*r t*( ) to track 2 ( )*t* with 5% positive offset. Results of the simulation are shown in Figure 17. The system reaches the setpoint and ready for the clutch engagement after 2.5 seconds. Next, we run the hybrid vehicle with this EM1 and ICE to track the speed set-points and test for the clutch engagement. It is assumed that if the motor EM2 operating at more than 1500 rpm, the motor EM1 will start and activate the ICE to engage to the system. Regarding the improvement of the driving comfort and the reduction of jerk, the clutch engagement will be taken place for only when *ω*<sup>1</sup> ≥*ω*2 or *ω*<sup>1</sup> =1.05.*ω*2 or the EM1 and ICE must track the EM2 at 5% positive offset. The MPC objective function in (4.3) is now changed from setpoints *r*(*t*) to track *ω*2(*t*) with 5% positive offset. Results of the simulation are shown in Figure 17. The system reaches the setpoint and ready for the clutch engagement after 2.5 seconds.

20 40 Input Voltage (V) Start Motor (V) 500 1000 1500 2000 Output Speed (RPM) Setpoints Speed In the simulation in Figure 17, we operate both motor EM1 and the ICE to track motor EM2 and we can observe that after around 2 sec, the speed of the left hand side clutch disk has exceeded more than 5% of the speed on the right hand side disk and ready for the clutch

2500

Output Torque (Nm)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

Time (Sec)

Output Torque = Start Motor + Engine ICE

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

Time (Sec)

Figure 17. MPC controller for tracking setpoint with both EM1 and ICE. In the simulation in Figure 17, we operate both motor EM1 and the ICE to track motor EM2 and we can observe that after around 2 sec, the speed of the left hand side clutch disk has exceeded more than 5% of the speed on the right hand side disk and ready for the clutch engaged. However, the motor EM1 functions only for as the ICE starter and once the ICE is fully run, the EM1 can be turned on to as an electrical generator to charge the batteries. In the next example, we will turn off the motor EM1 and use only the ICE to track the speed of the motor EM2. The MPC objective function becomes now similar to the example that we have simulated in Figure 17. Results of the performance are illustrated in Figure 18. The vehicle

will reach the speed set-point and ready for the clutch engaged after around 4.4 sec.

 or 1 2 1.05.or

1 2 

Figure 16. MPC performance at the starting time. Next, we run the hybrid vehicle with this EM1 and ICE to track the speed set-points and test for the clutch engagement. It is assumed that if the motor EM2 operating at more than 1500 rpm, the motor EM1 will start and activate the ICE to engage to the system. Regarding the improvement of the driving comfort and

the EM1 and ICE must track the EM2 at 5% positive offset. The MPC objective function in (4.3) is now

the reduction of jerk, the clutch engagement will be taken place for only when

In the simulation in Figure 17, we operate both motor EM1 and the ICE to track motor EM2 and we can **Figure 17.** MPC controller for tracking setpoint with both EM1 and ICE.

changed from setpoints*r t*( ) to track 2

The example parameters used for the EM1 are: motor constants, *kE*<sup>2</sup> =*kT* <sup>2</sup> =15; motor inertia, *J*<sup>1</sup> =1; motor damping, *kβ*<sup>1</sup> =0.5; armature resistance, *RI* <sup>1</sup> =7; compensation factor, *ς* =0.5. The system is discretized at a time interval of 0.01 sec. The air drag resistance torque at *ω*<sup>3</sup> =2000

The physical input constraints for this vehicle are set for the DC voltage in range of |*V*<sup>1</sup> | ≤48*V* ; The input increment constraint is Δ*u*(*t*)≤180*V* / sec. The output constraint of torque

The MPC parameters are chosen for the horizon length prediction of *Nu* = *Ny* = *Np* =5, the

From Figure 16, after a certain delay time of 1 second, the ICE is fully started and after about 2.4 seconds the peed of ICE has reached the set-point of 2000 rpm and operate stably at a power of 6 kW, generating an output torque of 30 Nm (at this moment, the friction clutch is still open).

Figure 16. MPC performance at the starting time. Next, we run the hybrid vehicle with this EM1 and ICE to track the speed set-points and test for the clutch engagement. It is assumed that if the motor EM2 operating at more than 1500 rpm, the motor EM1 will start and activate the ICE to engage to the system. Regarding the improvement of the driving comfort and

Next, we run the hybrid vehicle with this EM1 and ICE to track the speed set-points and test for the clutch engagement. It is assumed that if the motor EM2 operating at more than 1500 rpm, the motor EM1 will start and activate the ICE to engage to the system. Regarding the improvement of the driving comfort and the reduction of jerk, the clutch engagement will be taken place for only when *ω*<sup>1</sup> ≥*ω*2 or *ω*<sup>1</sup> =1.05.*ω*2 or the EM1 and ICE must track the EM2 at 5% positive offset. The MPC objective function in (4.3) is now changed from setpoints *r*(*t*) to track *ω*2(*t*) with 5% positive offset. Results of the simulation are shown in Figure 17. The system

the EM1 and ICE must track the EM2 at 5% positive offset. The MPC objective function in (4.3) is now

Output Torque (Nm)

Output Speed (RPM)

In the simulation in Figure 17, we operate both motor EM1 and the ICE to track motor EM2 and we can observe that after around 2 sec, the speed of the left hand side clutch disk has exceeded more than 5% of the speed on the right hand side disk and ready for the clutch

Figure 17. The system reaches the setpoint and ready for the clutch engagement after 2.5 seconds.

Figure 17. MPC controller for tracking setpoint with both EM1 and ICE. In the simulation in Figure 17, we operate both motor EM1 and the ICE to track motor EM2 and we can observe that after around 2 sec, the speed of the left hand side clutch disk has exceeded more than 5% of the speed on the right hand side disk and ready for the clutch engaged. However, the motor EM1 functions only for as the ICE starter and once the ICE is fully run, the EM1 can be turned on to as an electrical generator to charge the batteries. In the next example, we will turn off the motor EM1 and use only the ICE to track the speed of the motor EM2. The MPC objective function becomes now similar to the example that we have simulated in Figure 17. Results of the performance are illustrated in Figure 18. The vehicle

will reach the speed set-point and ready for the clutch engaged after around 4.4 sec.

the reduction of jerk, the clutch engagement will be taken place for only when

Start Motor (V)

Engine ICE (Kw)

reaches the setpoint and ready for the clutch engagement after 2.5 seconds.

Start Motor (V)

Engine ICE (Kw)

changed from setpoints*r t*( ) to track 2

**Figure 16.** MPC performance at the starting time.

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

time

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

time

<sup>0</sup> <sup>10</sup> and *<sup>R</sup>* <sup>=</sup> <sup>1</sup> <sup>0</sup>

Output Torque (Nm)

Output Speed (RPM)

<sup>0</sup> <sup>1</sup> . Figure 16 shows the MPC

Setpoints Speed

1 2 

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

Time (Sec)

Output Torque = Start Motor + Engine ICE

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

Time (Sec)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>0</sup>

Time (Sec)

Output Torque = Start Motor + Engine ICE

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>0</sup>

Time (Sec)

( )*t* with 5% positive offset. Results of the simulation are shown in

 or 1 2 1.05.or

> Setpoints Speed

rpm is chosen as *Mv*<sup>0</sup> =30 Nm.

52 New Applications of Electric Drives

for the shaft1 is set for |*T*<sup>1</sup> | ≤628*Nm*.

performance at the starting time.

20 40 60

20 40 60

5 10 15

Input ICE (Kw)

Input Voltage (V)

Input ICE (Kw)

Input Voltage (V)

weighting values for the matrices of *<sup>Q</sup>* <sup>=</sup> <sup>10</sup> <sup>0</sup>

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>0</sup>

Time (Sec)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>0</sup>

Time (Sec)

engaged. However, the motor EM1 functions only for as the ICE starter and once the ICE is fully run, the EM1 can be turned on to as an electrical generator to charge the batteries. In the next example, we will turn off the motor EM1 and use only the ICE to track the speed of the motor EM2. The MPC objective function becomes now similar to the example that we have simulated in Figure 17. Results of the performance are illustrated in Figure 18. The vehicle will reach the speed set-point and ready for the clutch engaged after around 4.4 sec. Figure 17. MPC controller for tracking setpoint with both EM1 and ICE. In the simulation in Figure 17, we operate both motor EM1 and the ICE to track motor EM2 and we can observe that after around 2 sec, the speed of the left hand side clutch disk has exceeded more than 5% of the speed on the right hand side disk and ready for the clutch engaged. However, the motor EM1 functions only for as the ICE starter and once the ICE is fully run, the EM1 can be turned on to as an electrical generator to charge the batteries. In the next example, we will turn off the motor EM1 and use only the ICE to track the speed of the motor EM2. The MPC objective function becomes now similar to the example that we have simulated in Figure 17. Results of the performance are illustrated in Figure 18. The vehicle the speed on the right hand side disk and ready for the clutch engaged. However, the motor EM1 functions only for as the ICE starter and once the ICE is fully run, the EM1 can be turned on to as an electrical generator to charge the batteries. In the next example, we will turn off the motor EM1 and use only the ICE to track the speed of the motor EM2. The MPC objective function becomes now similar to the example that we have simulated in Figure 17. Results of the performance are illustrated in Figure 18. The vehicle will reach the speed set-point and ready for the clutch engaged after around 4.4 sec.

Figure 17. MPC controller for tracking setpoint with both EM1 and ICE.

observe that after around 2 sec, the speed of the left hand side clutch disk has exceeded more than 5% of

will reach the speed set-point and ready for the clutch engaged after around 4.4 sec.

How to control the 1 rapidly tracking 2 5% by MPC controller is still a big challenge. Next, we test a **Figure 18.** MPC controller for tracking setpoint with only ICE.

new MPC controller using the soften output constraints when we consider the output tracking setpoints as in (4.5) with some additional penalty terms added into the MPC objective function, <sup>2</sup> ( ) *Soften Hard i JJ t* . Results of the simulation are shown in Figure 19. How to control the *ω*<sup>1</sup> rapidly tracking *ω*<sup>2</sup> + 5*%* by MPC controller is still a big challenge. Next, we test a new MPC controller using the soften output constraints when we consider the output tracking setpoints as in (4.5) with some additional penalty terms added into the MPC objective function, *JSoften* = *JHard* + Λ*ε<sup>i</sup>* 2 (*t*). Results of the simulation are shown in Figure 19.

Figure 18. MPC controller for tracking setpoint with only ICE.

In Figure 19, additional penalty terms *ε<sup>i</sup>* and a new weighting matrix Λ can be regulated independently together with values of matrices *Q* and *R* to obtain some good soften constraints

In Figure 19, additional penalty terms *<sup>i</sup>* and a new weighting matrix can be regulated independently **Figure 19.** Soften output constraints for tracking set-point with only ICE.

controller performance. The MPC controller becomes looser, more flexible with more regulated parameters. The new system reaches the set-point faster and ready for the clutch engagement after only about 3.4 sec. together with values of matrices *Q* and *R* to obtain some good soften constraints controller performance. The MPC controller becomes looser, more flexible with more regulated parameters. The new system reaches the set-point faster and ready for the clutch engagement after only about 3.4 sec. **5. Conclusions** 

As a main type of hybrid vehicle, HEVs have achieved better fuel economy and performances. Modern

sources from GRID, wind and solar. Due to rapid development in battery technology, the normal electric

#### **5. Conclusions** HEV can also improve the efficiency by using the energy from braking and bring other potential environmental benefits. The electric vehicle charging stations can use the low cost and green energy

As a main type of hybrid vehicle, HEVs have achieved better fuel economy and performances. Modern HEV can also improve the efficiency by using the energy from braking and bring other potential environmental benefits. The electric vehicle charging stations can use the low cost and green energy sources from GRID, wind and solar. Due to rapid development in battery technology, the normal electric recharging time has been reducing significantly from 8 h to less than 2 h. The fast recharging time for a modern electric vehicle is now reduced to less than 10 min. Hybrid electric vehicle technology has been applied now not only for the passenger cars but also for all heavy buses and trucks. recharging time has been reducing significantly from 8 h to less than 2 h. The fast recharging time for a modern electric vehicle is now reduced to less than 10 min. Hybrid electric vehicle technology has been applied now not only for the passenger cars but also for all heavy buses and trucks. The new modeling and control strategy for HEV using MPC has been developed. Reason for using this new control strategy is that, firstly, MPC can solve the optimization problems online with both linear and nonlinear systems, and secondly, MPC can deal with the constraints in the open-loop optimal control problems. MPC can find real-time solution for general constrained nonlinear models over a finite predictive horizon length. Therefore, the performances of the hybrid vehicle can be significantly improved. The new HEV dynamic modeling equations allow having better studied views for the acceleration of HEVs and the jerk reduction during the transitional engagement period. Examples show that MPC controllers can

The new modeling and control strategy for HEV using MPC has been developed. Reason for using this new control strategy is that, firstly, MPC can solve the optimization problems online with both linear and nonlinear systems, and secondly, MPC can deal with the constraints in the open-loop optimal control problems. MPC can find real-time solution for general con‐ strained nonlinear models over a finite predictive horizon length. Therefore, the performances of the hybrid vehicle can be significantly improved. control the speeds very well to track to any desired speeds. Examples also indicate that the MPC controller can be able to achieve fast and smooth engagement of clutch. The MPC performance can also be considerably improved when we select some appropriate prediction lengths and the values of the weighting values. MPC controller can provide online the optimal control actions subject to the input voltages and output torque constraints. The MPC modified schemes can improve the system performance robustness if some output torque constraints can be softened or turned into the constrained regions.

The new HEV dynamic modeling equations allow having better studied views for the acceleration of HEVs and the jerk reduction during the transitional engagement period. Examples show that MPC controllers can control the speeds very well to track to any desired speeds. Examples also indicate that the MPC controller can be able to achieve fast and smooth engagement of clutch. The MPC performance can also be considerably improved when we select some appropriate prediction lengths and the values of the weighting values. MPC controller can provide online the optimal control actions subject to the input voltages and **Acknowledgment**  The authors would like to thank the Tallinn University of Technology (TTU) for supporting the preparation of this research article.

output torque constraints. The MPC modified schemes can improve the system performance robustness if some output torque constraints can be softened or turned into the constrained regions.
