Direct Torque Control of PMSM with Modiﬁed Finite Set Model Predictive Control

: A direct torque control (DTC) with a modiﬁed ﬁnite set model predictive strategy is proposed in this paper. The eight voltage space vectors of two-level inverters are taken as the ﬁnite control set and applied to the model predictive direct torque control of a permanent magnet synchronous motor (PMSM). The duty cycle of each voltage vector in the ﬁnite set can be estimated by a cost function, which is designed based on factors including the torque error, maximum torque per ampere (MTPA), and stator current constraints. Lyapunov control theory is introduced in the determination of the weight coe ﬃ cients of the cost function to guarantee stability, and thus the optimal voltage vector reference value of the inverter is obtained. Compared with the conventional ﬁnite control set model predictive control (FCS-MPC) method, the torque ripple is reduced and the robustness of the system is clearly improved. Finally, the simulation and experimental results verify the e ﬀ ectiveness of the proposed control scheme.


Introduction
Permanent magnet synchronous motor (PMSM) direct torque control (DTC) has been widely used in industry due to its simple control structure, fast dynamic response, and high efficiency [1]. However, the traditional DTC control results in large torque ripple due to the insufficient switching frequency of the inverters and two nonlinear hysteresis comparators [2,3].
To solve the problem of torque ripple, many scholars have put forward many improved methods. An improved method is calculating the effective voltage vector action time in real time to ensure the minimum torque ripple by the current torque error [4][5][6]. This method reduces the torque ripple to a certain extent, but the calculation process is complicated. At present, voltage space vector modulation (SVM) is also introduced into DTC [7,8]. This method can effectively reduce the torque ripple, but it cannot eliminate the steady-state error of the torque. At the same time, the calculation process is highly dependent on the parameters of the motor and has poor robustness.
As a real-time online optimization control method, model predictive control (MPC) has received extensive attention in the field of electric drives and power converters due to its high dynamics and resistance to parameter disturbance [9,10]. The method of finite control set MPC (FCS-MPC) directly utilizes the discreteness of inverter output voltage and the finiteness of switching state; at the same

Discrete Mathematical Model of PMSM and Drive
The main circuit of the electric drive system under consideration is illustrated in Figure 1, which also corresponds to the experimental system layout. The PMSM has the characteristics of being multivariable and nonlinear and has strong coupling [20]. In order to simplify the mathematical model of PMSM, the following assumptions are made: (1) Y-shaped connection of stator windings, symmetrical distribution of three-phase windings, and space difference of each winding is 120 • .
(2) Eddy current loss, hysteresis loss, and the change of motor parameters are neglected. (3) It is assumed that permanent magnets on the rotor generate a main magnetic field in the stator-rotor airgap (the magnetic field is distributed sinusoidally along the circumference of the airgap) 22. So the continuous time model for PMSM in the d-q coordinate system can be described as: where u d , u q , i d and i q are the stator voltages and currents, R is the stator winding resistance, L d and L q are the d-and q-axes inductances, ψ f is the flux linkage of the rotor, p is the number of pairs of poles, ω is the mechanical angular speed, and T e is the electromagnetic torque. i q is proportional to the electromagnetic torque and i d is proportional to the reactive power. A predictive current control scheme is thus formed in which the reference current is generated by an external speed control loop. ( 1) ( ) .where s T is the sampling period. Then, the discrete current model of the PMSM is obtained: The commutation process of two-level voltage source inverters is accomplished by use of DC bus. The switching state can be represented by the switching signals a S and b S and switching on and off on different bridge arms. The upper and lower switches of each bridge arm of the inverter cannot be turned on at the same time to prevent short circuit c S , as shown in Table 1: The relationship between output voltage vector with the switch state can be defined as: where dc U is the DC source voltage and out U is the output voltage of the inverter. Discretize the mathematical model of PMSM to obtain the necessary conditions for model prediction. Since the sampling time T s is sufficiently small, the Euler approximation is used for the stator current derivative di/dt of the sampling time T s .
where T s is the sampling period. Then, the discrete current model of the PMSM is obtained: The commutation process of two-level voltage source inverters is accomplished by use of DC bus. The switching state can be represented by the switching signals S a and S b and switching on and off on different bridge arms. The upper and lower switches of each bridge arm of the inverter cannot be turned on at the same time to prevent short circuit S c , as shown in Table 1: where a = e j2π/3 . The relationship between output voltage vector with the switch state can be defined as: where U dc is the DC source voltage and U out is the output voltage of the inverter. Considering the different switching states of the inverters, eight different voltage vectors are obtained, as shown in Table 2. It is noted that V0 = V7, resulting in a finite set of seven different voltage vectors in the plane.

Predictive Control Based on Duty Cycle
Within one sampling interval, the cost function values that correspond to the seven voltage vectors are calculated by the FCS. The switching state that minimizes the cost function is chosen as the switching state of the inverter. The process of model prediction optimization is shown in Figure 2, where x, x p , and x * are the torque of the real response, the predicted value, and the reference value, respectively. If the corresponding x(k) is the best value at time k, the cost function value x(k + 1) corresponding to the seven voltage vectors is calculated at time (k + 1), and the seven calculated cost functions values are compared with reference values, where the closest is to be the optimal solution at the moment (k + 1). V 4 is the control signal of the voltage vector corresponding to the optimal value of the time (k + 1). In the same way, V 3 should be selected as the optimal control signal at time (k + 2) [12].

Predictive Control Based on Duty Cycle
Within one sampling interval, the cost function values that correspond to the seven voltage vectors are calculated by the FCS. The switching state that minimizes the cost function is chosen as the switching state of the inverter. The process of model prediction optimization is shown in Figure   2, where x , p x , and * x are the torque of the real response, the predicted value, and the reference value, respectively. If the corresponding x(k) is the best value at time k, the cost function value x(k + 1) corresponding to the seven voltage vectors is calculated at time (k + 1), and the seven calculated cost functions values are compared with reference values, where the closest is to be the optimal solution at the moment (k + 1). V4 is the control signal of the voltage vector corresponding to the optimal value of the time (k + 1). In the same way, V3 should be selected as the optimal control signal at time (k + 2) [12].

Design of Cost Function
The maximum torque/current control (MTPA) consumes the minimum stator current at the same electromagnetic torque output, which is usually used in interior PMSM systems. The space vector analysis of PMSM in the d-q coordinate system is shown in Figure 3. Assuming that the stator current vector s i leads q axis β , the current component in the d-q coordinate system is as follows:

Design of Cost Function
The maximum torque/current control (MTPA) consumes the minimum stator current at the same electromagnetic torque output, which is usually used in interior PMSM systems. The space vector analysis of PMSM in the d-q coordinate system is shown in Figure 3. Assuming that the stator current vector i s leads q axis β, the current component in the d-q coordinate system is as follows: If the amplitude of the stator current of the PMSM is a constant value, it can be concluded that the electromagnetic torque of the motor is only related to β. In order to find the maximum value of Energies 2020, 13, 234 5 of 16 the torque, that is, to satisfy the MTPA condition, and calculate Equation (11) with β as a variable, the following relationship is obtained: Because i s and β satisfy the Pythagorean theorem, the relationship of i d and i q under the MTPA condition can thus be derived as follows: If the amplitude of the stator current of the PMSM is a constant value, it can be concluded that the electromagnetic torque of the motor is only related to β . In order to find the maximum value of the torque, that is, to satisfy the MTPA condition, and calculate Equation (11) with β as a variable, the following relationship is obtained: Because s i and β satisfy the Pythagorean theorem, the relationship of d i and q i under the MTPA condition can thus be derived as follows: The most important part of MPC is the cost function design. The cost function is expressed in orthogonal coordinates and considering the torque tracking, MTPA condition, and current constraint: where T K , M K , and C K are the weighting factors of the cost function and are positive real numbers. The main aim of the MPC is to minimize the torque error. Thus, the cost for the error Since high currents lead to large losses, ( ) M J k is to have a low absolute current: The most important part of MPC is the cost function design. The cost function is expressed in orthogonal coordinates and considering the torque tracking, MTPA condition, and current constraint: where K T , K M , and K C are the weighting factors of the cost function and are positive real numbers. The main aim of the MPC is to minimize the torque error. Thus, the cost for the error J T (k) is: Since high currents lead to large losses, J M (k) is to have a low absolute current: Tertiary control goals can be added in order to ensure the operational safety of the system. J C1 (k) denotes the current limit, and the current amplitude of the system must be smaller than the maximum allowable value I max .
To make the point of the cost function converge to the parabola on the left side of the i q axis, as shown in Figure 3. Adding current constraints J C2 (k) to the cost function: Energies 2020, 13, 234 6 of 16 To ensure current constraints and avoid control variables converging to the wrong MTPA curve, K C K T , K M should be satisfied. For the choice of K T and K M , K T + K M = 1 can be made. A larger value of K T leads to faster convergence of the torque, while a larger value of K M indicates that the MTPA state converges quickly. K T > K M can be selected [21] to achieve a fast torque response.

Duty Cycle Calculation
The duty cycle of each voltage vector in a finite set is calculated by Lyapunov function to eliminate torque and current ripple. Without considering the current limitation, that is, ignoring J C1 (k) and J C2 (k), the Lyapunov function is expressed as follows: In order to optimize the operation of the PMSM, the condition V(k + 1) = 0 must be fulfilled to ensure T k+1 e = T * e and its current operating point is on the MTPA curve. In addition, the condition dV(k + 1)/dt = 0 indicates that the PMSM is operating at the optimum state.

Lemma 1.
Defining the function of the current: If the torque of the PMSM is kept at a nonzero constant value, the function f (i d , i q ) is a strict monotonic function along the constant torque curve. The correlative poof is introduced in the Appendix A.
With the help of the function f (i d , i q ), the stator current trajectory is shown in Figure 4. Point A is the initial point; the trajectory first reaches the point B along the constant torque curve and then reaches the point C along the MTPA curve with the condition V(k) = 0. When the current trajectory is along → AB to → BC, the Lyapunov function is strictly decreasing.
Lemma 3. For a given current dynamics:

Lemma 2.
Under the initial condition of the PMSM, if the back EMF of the PMSM is within the voltage limit range of the inverter, there exists a feasible voltage vector u * dq that fulfills: If the back EMF of the PMSM is within the voltage limit range of the inverter, then at least one voltage vector is satisfied: The proofs of Lemma 2 and Lemma 3 are introduced in the Appendix A.
In order to minimize the ripple of the PMSM current and torque, the Lyapunov function is expected to keep V = 0 and dV(k + 1)/dt = 0 in a steady state. The duty cycle of each voltage vector satisfying the desired requirement is calculated as follows: Here, T σ T s is a time constant. The voltage vector with dV(k + 1)/dt > 0 has to be taken into account in the FCS-MPC, and then the current reaches the limit, T σ is introduced, and the current constraint plays a major role in the cost function. When there is disturbance in the PMSM system, such as machine parameter drift and voltage error in the inverter, we also need to select a smaller T σ to compromise the disturbance at this point. When the PMSM model and the prediction are accurate, T σ = 0 is the ideal value for the ripple reduction. All voltage vectors u j dq,k+1 with dV(k + 1)/dt < 0 are candidates to minimize the cost function.

Finite Control Set Model Prediction
Finally, the optimum output voltage which can be realized by the SVPWM is obtained by Equation (23). Only when voltage u j dq,k+1 acts on the PMSM will it affect the torque, that is, to ensure the implementation of MTPA and dV(k + 1)/dt = 0. It is noteworthy that setting the optimal duty cycle T k+1 duty equal to T s in the transient state of the proposed control strategy can minimize the cost function. Therefore, under the same sampling frequency, the torque response of the new strategy is as fast as that of the traditional FCS-MPC. In steady state, dV(k + 1)/dt = 0 can be maintained. Therefore, current ripple and torque ripple can be minimized. Figure 5 is a block diagram of the predictive direct torque control algorithm of the FCS-MPC proposed in this paper.
Energies 2020, 13, 234 8 of 16 predictive direct torque control algorithm of the FCS-MPC proposed in this paper.

Simulation Results
In order to investigate the effectiveness and feasibility of the proposed method, the FCS-MPC of the surface-mounted PMSM based on MATLAB/Simulink software was built. Then, it was compared with that of the classical FCS-MPC method. The parameter settings are shown in Table 3. Table 3. Surface-mounted permanent magnet synchronous motor (SPMSM) parameters for simulation.

Variable
Parameter Value

Simulation Results
In order to investigate the effectiveness and feasibility of the proposed method, the FCS-MPC of the surface-mounted PMSM based on MATLAB/Simulink software was built. Then, it was compared with that of the classical FCS-MPC method. The parameter settings are shown in Table 3.  Figure 6 shows the simulation results of traditional FCS-MPC and improved FCS-MPC under steady-state conditions. The reference value of motor speed is set to 500 rpm, and the load torque is 20 N·m. As can be seen from the figure, compared with conventional FCS-MPC, the torque ripple of the improved method is significantly reduced. In Figure 6, the average values of the torque ripples of proposed (0.9 N·m) are smaller compared to those of conventional FCS-MPC (3 N·m) under the same conditions. It is noted that, in this paper, torque ripples are calculated by the following equation [22]:

Steady-State Operation
where N is the number of samples and T ave is the average value of the torque.
Energies 2020, 13, 234 9 of 16 same conditions. It is noted that, in this paper, torque ripples are calculated by the following equation [22]: where N is the number of samples and ave T is the average value of the torque.

Dynamic Response
In order to compare the torque dynamic, a magnified view of the torque response is shown in Figure 7. From Figure 7, we can see that the torque suddenly increases from 10 to 20 N⋅m in only 0.003 s, and in the process of torque mutation, the conventional FCS-MPC method has obvious overshoot. Compared with the conventional FCS-MPC, the improved method has faster dynamic response because the motor adopts optimized duty cycle modulation in the whole control cycle.

Dynamic Response
In order to compare the torque dynamic, a magnified view of the torque response is shown in Figure 7. From Figure 7, we can see that the torque suddenly increases from 10 to 20 N·m in only 0.003 s, and in the process of torque mutation, the conventional FCS-MPC method has obvious overshoot. Compared with the conventional FCS-MPC, the improved method has faster dynamic response because the motor adopts optimized duty cycle modulation in the whole control cycle.

Motor Parameter Robustness
As we all know, a temperature rise of the motor will cause the stator internal resistance to increase. The influence of internal resistance on the torque is discussed in this paper. The torque waveforms of the two control methods in the condition of rated stator resistance (the blue solid line) and 1.5-times the rated value (the red dashed line) are given in Figure 8a,b, respectively. It is noticeable that slight torque ripples occur under the conventional FCS-MPC method when the resistance increases, while the proposed FCS-MPC control has better robustness in which the torque is insensitive to the resistance change. However, when entering a steady state, the two control systems are not affected by internal resistance changes.

Motor Parameter Robustness
As we all know, a temperature rise of the motor will cause the stator internal resistance to increase. The influence of internal resistance on the torque is discussed in this paper. The torque waveforms of the two control methods in the condition of rated stator resistance (the blue solid line) and 1.5-times the rated value (the red dashed line) are given in Figure 8a,b, respectively. It is noticeable that slight torque ripples occur under the conventional FCS-MPC method when the resistance increases, while the proposed FCS-MPC control has better robustness in which the torque is insensitive to the resistance change. However, when entering a steady state, the two control systems are not affected by internal resistance changes.

Motor Parameter Robustness
As we all know, a temperature rise of the motor will cause the stator internal resistance to increase. The influence of internal resistance on the torque is discussed in this paper. The torque waveforms of the two control methods in the condition of rated stator resistance (the blue solid line) and 1.5-times the rated value (the red dashed line) are given in Figure 8a,b, respectively. It is noticeable that slight torque ripples occur under the conventional FCS-MPC method when the resistance increases, while the proposed FCS-MPC control has better robustness in which the torque is insensitive to the resistance change. However, when entering a steady state, the two control systems are not affected by internal resistance changes.  Almost all permanent magnet devices, including permanent magnet motors, will exhibit the demagnetization phenomenon in varying degrees due to factors such as the working environment and usage time. The effect of changes in the motor flux on torque is also discussed. Figure 9a,b shows the torque waveforms of the two control methods when the rotor flux is 0.9-times the rated value, equal to the rated value, and 1.1-times the rated value, to simulate permanent magnet motors that have been used for a long time, used normally, and newly manufactured. As shown in Figure 9, the blue point line is 0.9-times the rated flux, the red solid line is the rated flux, and the green dotted line is 1.5-times the rated flux. Evidently, the conventional FCS-MPC for the change of flux is the change of the maximum torque. However, the improved predictive control is insensitive to the flux change.
Energies 2020, 13, x FOR PEER REVIEW 11 of 17 Almost all permanent magnet devices, including permanent magnet motors, will exhibit the demagnetization phenomenon in varying degrees due to factors such as the working environment and usage time. The effect of changes in the motor flux on torque is also discussed. Figure 9a,b shows the torque waveforms of the two control methods when the rotor flux is 0.9-times the rated value, equal to the rated value, and 1.1-times the rated value, to simulate permanent magnet motors that have been used for a long time, used normally, and newly manufactured. As shown in Figure 9, the blue point line is 0.9-times the rated flux, the red solid line is the rated flux, and the green dotted line is 1.5-times the rated flux. Evidently, the conventional FCS-MPC for the change of flux is the change of the maximum torque. However, the improved predictive control is insensitive to the flux change. For a mechanical load, the oscillation of torque is very bad for the shaft and load of the motor. Therefore, it can be seen from the simulation waveforms of the above two sections that the robustness of predictive control to parameter changes is better than that of conventional FCS-MPC. In theory, the switch meter of direct torque control is fixed in advance and cannot be adjusted according to the running state of the motor. Therefore, the robustness of direct predictive control to parameter changes For a mechanical load, the oscillation of torque is very bad for the shaft and load of the motor. Therefore, it can be seen from the simulation waveforms of the above two sections that the robustness of predictive control to parameter changes is better than that of conventional FCS-MPC. In theory, the switch meter of direct torque control is fixed in advance and cannot be adjusted according to the running state of the motor. Therefore, the robustness of direct predictive control to parameter changes is better than that of direct torque control, which is also consistent with the previous prediction results.

Experimental Test
The experiment aimed to verify the torque performance of this method with surface-mounted permanent magnet synchronous motor (SPMSM) and interior permanent magnet synchronous motor (IPMSM). The experimental platform as shown in Figure 10 was built by using dSPACE1102. The load of the PMSM is provided by the load machine. For a mechanical load, the oscillation of torque is very bad for the shaft and load of the motor. Therefore, it can be seen from the simulation waveforms of the above two sections that the robustness of predictive control to parameter changes is better than that of conventional FCS-MPC. In theory, the switch meter of direct torque control is fixed in advance and cannot be adjusted according to the running state of the motor. Therefore, the robustness of direct predictive control to parameter changes is better than that of direct torque control, which is also consistent with the previous prediction results.

Experimental Test
The experiment aimed to verify the torque performance of this method with surface-mounted permanent magnet synchronous motor (SPMSM) and interior permanent magnet synchronous motor (IPMSM). The experimental platform as shown in Figure 10 was built by using dSPACE1102. The load of the PMSM is provided by the load machine.

Experiment on SPMSM
The parameters of the SPMSM are shown in Table 3. The steady-state performance test results of conventional FCS-MPC and proposed FCS-MPC are shown in Figure 11. In the experiment, the reference load torque was 20 N·m, and the results show that the proposed FCS-MPC significantly reduced the torque ripple under the same average switching frequency, while the conventional FCS-MPC had a lot of torque ripples (6 N·m) compared to proposed FCS-MPC (2.8 N·m). This is consistent with the simulation results.

Experiment on SPMSM
The parameters of the SPMSM are shown in Table 3. The steady-state performance test results of conventional FCS-MPC and proposed FCS-MPC are shown in Figure 11. In the experiment, the reference load torque was 20 N⋅m, and the results show that the proposed FCS-MPC significantly reduced the torque ripple under the same average switching frequency, while the conventional FCS-MPC had a lot of torque ripples (6 N⋅m) compared to proposed FCS-MPC (2.8 N⋅m). This is consistent with the simulation results. In order to test the dynamic performance of the proposed FCS-MPC method, a load torque of 25 N⋅m was suddenly added when the steady speed of the motor was 500 rpm. Figure 12 shows that the response times of proposed FCS-MPC and conventional FCS-MPC are approximately the same; however, the proposed FCS-MPC strategy torque ripple is smaller than that of the conventional FCS-MPC. Meanwhile, the proposed FCS-MPC method enables the motor to quickly recover to the reference speed value compared to conventional FCS-MPC. In order to test the dynamic performance of the proposed FCS-MPC method, a load torque of 25 N·m was suddenly added when the steady speed of the motor was 500 rpm. Figure 12 shows that the response times of proposed FCS-MPC and conventional FCS-MPC are approximately the same; however, the proposed FCS-MPC strategy torque ripple is smaller than that of the conventional FCS-MPC. Meanwhile, the proposed FCS-MPC method enables the motor to quickly recover to the reference speed value compared to conventional FCS-MPC.
In order to test the dynamic performance of the proposed FCS-MPC method, a load torque of 25 N⋅m was suddenly added when the steady speed of the motor was 500 rpm. Figure 12 shows that the response times of proposed FCS-MPC and conventional FCS-MPC are approximately the same; however, the proposed FCS-MPC strategy torque ripple is smaller than that of the conventional FCS-MPC. Meanwhile, the proposed FCS-MPC method enables the motor to quickly recover to the reference speed value compared to conventional FCS-MPC.

Experiment on IPMSM
Some parameters of the experiment motor are shown in Table 4. The following experiments were carried out in three aspects: torque steady-state responses, speed dynamics, and torque dynamics responses.  Figure 13 shows electromagnetic torque waveforms of two methods, with the load torque reference value set to 30 N⋅m. As can be seen from Figure 13, the method of proposed FCS-MPC can significantly reduce torque ripple by comparing torque waveforms; the average torque ripple of proposed method is only 3 N⋅m while that of conventional FCS-MPC is 7 N⋅m.

Experiment on IPMSM
Some parameters of the experiment motor are shown in Table 4. The following experiments were carried out in three aspects: torque steady-state responses, speed dynamics, and torque dynamics responses.  Figure 13 shows electromagnetic torque waveforms of two methods, with the load torque reference value set to 30 N·m. As can be seen from Figure 13, the method of proposed FCS-MPC can significantly reduce torque ripple by comparing torque waveforms; the average torque ripple of proposed method is only 3 N·m while that of conventional FCS-MPC is 7 N·m.  Figure 13 shows electromagnetic torque waveforms of two methods, with the load torque reference value set to 30 N⋅m. As can be seen from Figure 13, the method of proposed FCS-MPC can significantly reduce torque ripple by comparing torque waveforms; the average torque ripple of proposed method is only 3 N⋅m while that of conventional FCS-MPC is 7 N⋅m.   Figure 14 shows the waveforms of speed and electromagnetic torque of two methods when the starting moment of the motor and step change of load torque with load torque increasing from 10 to 20 N·m. It can be seen from Figure 14 that at the moment of starting the motor, the speed and electromagnetic torque of the conventional and proposed FCS-MPC methods increase sharply, reaching the given reference value quickly; however, compared with the conventional FCS-MPC, the strategy of proposed FCS-MPC has smaller torque ripple. When the load torque changes abruptly, the proposed FCS-MPC can also track the change of torque quickly, and the corresponding speed is faster than conventional FCS-MPC.
Energies 2020, 13, x FOR PEER REVIEW 14 of 17 Figure 14 shows the waveforms of speed and electromagnetic torque of two methods when the starting moment of the motor and step change of load torque with load torque increasing from 10 to 20 N⋅m. It can be seen from Figure 14 that at the moment of starting the motor, the speed and electromagnetic torque of the conventional and proposed FCS-MPC methods increase sharply, reaching the given reference value quickly; however, compared with the conventional FCS-MPC, the strategy of proposed FCS-MPC has smaller torque ripple. When the load torque changes abruptly, the proposed FCS-MPC can also track the change of torque quickly, and the corresponding speed is faster than conventional FCS-MPC. Through the analysis of the above experiments, compared with the conventional FCS-MPC control system, the proposed FCS-MPC in this paper effectively reduces the torque ripple and improves the following performance of the motor torque.

Summary
In this paper, a new scheme of direct torque control for PMSM based on a finite control set (FCS) Therefore, there exists a current derivative: so that: Here, µ can be a very small positive constant.
Proof of Lemma 3. Any reference vector u * dq (x) within the region of feasibility [having a magnitude of less than (2/3)U dc ] is contained within one of the six nonzero switching sectors of width (π/3) with vertices . . , 6} are the nonzero switching states to the left and right of the reference vector and (v 0 d , v 0 q ) is one of the two zero vectors. Containment within a switching sector ensures the existence of coefficients γ and η satisfying γ, η ≥ 0 and γ + η ≤ 1 such that the reference vector is expressible as a convex combination of the realizable inputs, given by: Plugging Equation (A10) into Equation (21) and noting that the system is control affine, we see that [23]: ∂V(k+1) ∂i k+1 dq (Ai k+1 dq + Bv 0 dq ) + ∂V(k+1) ∂i k+1 dq E (A10) Because γ, η, and (1 − γ − η) are all nonnegative, the following inequalities hold [12]: which completes the proof in Lemma 3. The theorem also guarantees the stability of the proposed control scheme.