Dynamic Analysis and Optimized Design of Synergetic Control for a PMSM Drive System

Abstract

This chapter presents an optimum design of synergetic control for a permanent magnet synchronous motor (PMSM) drive system. New macro-variables are proposed to improve the performance of the standard controller. The controller’s performance is compared with that of the field-oriented control scheme. The chapter also investigates the regenerative braking mode of operation in PMSM. Regenerative braking is achieved by operating the motor in torque control mode. The different algorithms are validated through experiments using a 1-hp PMSM drive system. We also provide an extensive study of the controller parameters tuning for optimal performance. The experimental results show that the proposed macro-variables improve the performance of the synergetic controller significantly. The synergetic controller is able to overcome nonlinearities in the system, such as static friction, faster than the field-oriented controller. The system also experiences fewer harmonics under the synergetic controller. The synergetic controller shows also better performance under wide signal variations. As for regenerative braking, the torque control mode of operation is shown to be suitable for harvesting energy and both techniques showed similar performance levels. The proposed synergetic control strategy will be very useful in electric vehicle (EV) applications, as it allows to improve the dynamic response and efficiency of the drive system required by the EV dynamics.

1. Introduction

Electric vehicles are gaining increasing attention by automotive manufacturers in an attempt to reduce carbon footprints and produce more eco-friendly vehicles. Electric vehicles are powered by electric motor drives instead of the conventional combustion engine. Permanent Magnet Synchronous Motor (PMSM) drives have received a lot of interest in this field for their powerful torque profiles that are suitable for vehicle traction [1, 2, 3, 4, 5, 6].

Regenerative braking is crucial for electric vehicle applications since the mechanical energy of the motor during deceleration or braking periods can be harvested and used to charge the vehicle’s battery or contained within an energy storage unit designed for such purpose. In electrical braking, the kinetic energy of the motor can be harvested, with the proper switching sequence of the three-phase inverter, in order to redirect the current flowing in the motor windings back to the source or to an integrated hybrid energy storage system. It should be noted that not all energy can be recovered during the braking process as there exist some boundaries along the torque-speed plane where regenerative braking occurs [7]. In most applications, this regenerated current charges a supercapacitor module, or the vehicle’s battery, connected to the DC side of the inverter through a DC-DC converter. For AC motor drives, the power switches of the three-phase inverter can be utilized to harvest mechanical energy and feed it back to the DC source [8, 9]. The three-phase inverter works as a bidirectional DC-AC converter that controls energy flow in both directions between the power source and the motor in the case of both driving and regenerative braking modes of operation.

Several control techniques are available for controlling AC motor drives, such as vector control and direct torque control (DTC). Vector control, also known as field-oriented control (FOC), is the most commonly utilized control technique in AC motor drives. It is a reliable control scheme in most drive applications. The main role of field orientation is to decouple the machine’s generated flux from the torque in order to control both parameters separately as desired. This is done by replacing the three-phase stationary representation of the stator field with a dynamic two-phase rotating d-q frame, where the rotor magnetic field is aligned with the d-axis. In this process, the motor torque can be controlled separately by controlling the current in the quadrature (q) axis. Similarly, the flux is controlled independently by controlling the current in the direct (d) axis. Operating a three-phase AC machine under field orientation is similar to the operation of a separately excited DC machine where the field circuit controls the generated flux, while the armature circuit regulates the torque of the motor [3].

The FOC scheme consists of two control loops: an inner loop consisting of current regulators and an outer loop controlling the speed of the motor. In general, the torque or current regulators have to be tuned before tuning the speed controller, and the speed controller cannot be tuned without the assumption that the current regulators are tuned perfectly [1, 2].

Unlike FOC, DTC does not require any current regulator, coordinate transformation and PWM signals generator [3, 10]. In spite of its simplicity, DTC allows a good torque control in steady state and transient operating conditions. In addition, this controller is very little sensible to the parameters detuning in comparison with FOC. The main principle of DTC is to adjust the stator voltage vector based on the errors between the actual torque and stator flux linkage as well as their references. This method requires stator resistances be known and precludes both current controllers and motor parameters. Hence, it is insensitive to parameter variations of the machine. DTC also provides fast torque responses and simple implementation. However, DTC is known for its high torque and flux ripples that lead to vibrations and increase of losses [11, 12, 13].

On the other hand, it is acknowledged that DTC and FOC presents some disadvantages that can be summarized in the following points: 1) difficulty to control torque and flux at very low speed; 2) high current and torque ripple; 3) variable switching frequency behavior; 4) high noise level at low speed; 5) sensitivity to the stator resistance, rotor time constant, or mutual inductance parameters.

Various control strategies have been also proposed in the literature to solve these problems for PMSM drives and improve the robustness to modeling or parametric uncertainties, such as sliding mode control, adaptive control, predictive control, neural networks, as well as hybrid control methods [14, 15, 16, 17]. It is recognized that these different strategies have their own merits, and all meet the requirements of modern high-performance drives. However, most techniques are found to be computationally intensive, and show some performance limitations in terms of flexibility, implementation, and required bandwidth.

Therefore, new control techniques have to be developed to improve the performance of PMSM drives. It is essential to develop advanced control strategies for power conversion systems so that the robustness, flexibility, and dynamic performance of PMSM can be significantly improved.

Synergetic control (SC) is a nonlinear control technique that is well-suited for nonlinear systems. It overcomes the disadvantages of the previous controllers for PMSM, and linear model-based controllers who do not perform well with nonlinear plants under large signal variations [18, 19, 20]. The theory of synergetic control, however, overcomes these disadvantages via the use of all of its nonlinearity in the controller design process. This control theory is close to that of sliding mode controllers; both theories share similar properties and advantages, such as order reduction of the system and decoupling design procedure. Despite this, SC is still more advantageous compared to the sliding mode control primarily because it does not require high bandwidth to operate, as is the case with the sliding mode control. This makes it suitable for digital implementation. Furthermore, contrary to sliding mode controllers, SC maintains constant switching frequencies which lead to less noise due to switching in the system. Finally, synergetic control operates continuously as opposed to the sliding mode controller, which is known for its discontinuity. As previously mentioned, the system model is used for the controller synthesis process. Although this leads to better control as it provides more information about the system dynamics, it also makes the controller sensitive to errors in system parameters considered in the model itself. However, this can be overcome by proper selection of the macro-variables which is further discussed in Section 3.

Synergetic control has been previously used in motor drive applications [14, 15, 21, 22, 23, 24]. It has been implemented as a nonlinear speed control of multiple induction motor (IM) and PMSM drives. In [21], the authors derive a control law that reduces the order of the system and devise a simple method to select the closed loop poles of the system. However, they do not take into account any load variations applied on the system, which will affect the steady state speed of the drive. On the other hand, researchers in [22] introduce an adaptive controller based on the synergetic approach to control a PMSM drive. Although this controller takes into consideration any application of load torque on the motor, there remains some steady state error in the motor speed. In [23], the authors implement a synergetic controller to control the speed of an IM drive. They show the effect of changing some controller parameters on the response speed, transient behavior, steady state error and response to load applications. However, they propose excessive numbers of manifolds for a two control input system. In [24], the authors use both synergetic control and vector control to control an IM drive. The proposed controller uses the synergetic control theory to control the speed and generate reference d-q currents. The motor currents are regulated using vector control and hysteresis current control. In [14], the authors compare between the synergetic and the sliding mode controllers in controlling the speed of an IM. The results show that the synergetic controller was able to reduce the effect of load torque application as well as vibrational torques caused by the discontinuities presented by the sliding mode controller. In [15], three adaptation based nonlinear controllers namely adaptive terminal sliding mode, adaptive terminal synergetic and adaptive synergetic controllers have been used for the unified model of fuel cell, battery and ultracapacitor based hybrid electric vehicle to regulate the output voltage of the DC bus and to cater for the parametric variations of the system.

In this paper an optimized synergetic controller has been applied for the speed control of a PMSM drive. The particularity of the method is the incremental design and optimization of the macro-variables to improve the performance of the SC. The new macro-variables enhance the behavior of the d-axis current and eliminate any steady-state errors. The performance of this controller is improved compared to the standard controller used in the literature. A special macro-variable is also designed and implemented to control the torque and operate the motor under regenerative braking.

The synergetic controller is designed and implemented on a 1-hp prototype setup. FOC is also implemented and used as a reference to test the performance of the proposed synergetic controller under different operating conditions, such as response time, response to load torque variations, and regenerative braking. Regenerative braking under FOC and SC is also investigated and implemented. The synergetic controller showed very good robustness against disturbances with a reduced level of harmonics in the system. The proposed control scheme has also the characteristics of finite time convergence and chattering free phenomena.

2. PMSM dq model

In this paper, we consider a three-phase surface mounted PMSM with sinusoidal stator windings distribution. In order to simplify the analysis of the PMSM, the mathematical model is transformed into a two-phase reference frame using the Clarke-Park transformation [1]. This transforms the voltages and currents from a three-phase reference frame (uvw) to a fixed two-phase reference frame (αβ) as illustrated in Figure 1. The PMSM stator consists of three-phase windings (abc). The (αβ) reference frame is fixed to the stator and the (dq) frame is rotating with an angular speed ω equal to the synchronous speed. The permanent magnets are surface mounted on the rotor and thus create a rotating magnetic field. Under field orientation, the PMSM model in the rotor reference frame is given by

Figure 1.

PMSM reference frames.

$$v_d=R{i}_d+L_d\frac{d{i}_d}{\mathit{dt}}-L_q{i}_q{\omega }_e$$E1

$$v_q=R{i}_q+L_q\frac{d{i}_q}{\mathit{dt}}+L_d{i}_d{\omega }_e+{\omega }_e{\lambda }_{\mathit{PM}},$$E2

$$J\frac{\mathit{d\omega }}{\mathit{dt}}=T_{\mathit{em}}-\mathit{B\omega }-T_L$$E3

$$T_e=\frac{3}{2}\frac{P}{2}\left[{\lambda }_{\mathit{PM}}{i}_q+\left(L_d-L_q\right)i_d{i}_q\right]$$E4

where $\left(v_d,v_q,i_d,i_q\right)$ are the dq components of voltages and currents in the rotating dq frame, R is the stator resistance per phase, $\left(L_d,L_q\right)$ are the direct and quadrature self-inductances, ${\mathrm{\lambda }}_{\mathit{PM}}$ is the flux produced by the permanent magnets on the rotor, $T_{\mathit{em}}$ is the generated electromagnetic torque, and P is the number of poles. For the case of surface mounted or non-salient PMSM, the inductances in both d and q axes are equal.

3. Synergetic control strategy

3.1 Synergetic control law formulation

The synergetic control design procedure is based on the Analytical Design of Aggregated Regulators (ADAR) method [18, 19, 25]. We consider a non-linear system, described by its state space equation in the form:

$$\dot{x}=f\left(x,u,t\right)$$E5

where, x is the state vector defining all the state variables of the system, u is the control input vector, and t denotes time. First, a macro-variable is defined as a function of the system state variables

$$\psi =\psi \left(x\right)$$E6

The objective of the control effort is to force the system dynamics to operate on the manifold ψ = 0. Defining a manifold adds a new constraint on the system and hence reduces the order of the system. This will also force the system towards stability, preventing it from diverging. The macro-variable should be carefully selected to achieve the control specifications and limitations, such as avoiding saturation. A macro-variable is defined for each desired control input and its dynamics are subjected to the following constraint

$$T\dot{\psi }+\psi =0,T>0$$E7

where T is a time constant that defines the convergence speed to the manifold specified by the macro-variable.

T can be tuned and selected by the designer. Defining the appropriate macro-variable will force the system towards global stability. As discussed earlier, the controller synthesis depends greatly on the system model. Therefore, any uncertainties in the model parameters may affect the controller’s performance. To avoid the use of costly complex observers in estimating system parameters, studies show this drawback can be overcome by appropriate selection of the macro-variables [25].

By using the chain rule of differentiation, the expression of and the constraint equation are computed as

$$\frac{\mathit{d\psi }}{\mathit{dt}}=\frac{\mathit{d\psi }}{\mathit{dx}}\times \frac{\mathit{dx}}{\mathit{dt}}$$E8

$$T\frac{\mathit{d\psi }}{\mathit{dt}}f\left(x,u,t\right)+\psi =0$$E9

3.2 Conventional synergetic controller design for the PMSM

The PMSM model is first written in state space form by selecting the dq current components (id, iq) and the speed ω as state variables. Eqs. (1)–(2) are next as follows:

$$\frac{d{i}_d}{\mathit{dt}}=\frac{1}{L_d}\left[v_d-R{i}_d+L_q{i}_q{\omega }_e\right]$$E10

$$\frac{d{i}_q}{\mathit{dt}}=\frac{1}{L_q}\left[v_q-R{i}_q-L_d{i}_d{\omega }_e-{\omega }_e{\lambda }_{\mathit{PM}}\right]$$E11

It can be seen from this representation that the PMSM has two control input channels: vd and vq. Thus, two macro-variables are needed. A macro-variable is defined for each axis, which also means that torque and flux will be decoupled similar to FOC. In the literature, the following set of macro-variables are used [21].

$${\psi }_1=i_d$$E12

$${\psi }_2=K_1\left(\omega -{\omega }_{\mathit{ref}}\right)+K_2{i}_q+K_3\int \left(\omega -{\omega }_{\mathit{ref}}\right)\mathit{dt}$$E13

where K1, K2, K3 are the SC gains. The control effort forces the macro-variable to operate on the manifold $\mathrm{\psi }=0$, following the dynamics given by Eq. (7). Therefore, the first macro-variable ${\mathrm{\psi }}_1$ is selected to be equal to the direct-axis current $i_d$ since it is preferable to keep $i_d=0$ in PMSM systems to avoid any flux weakening or demagnetization of the permanent magnets. As for the second macro-variable ${\mathrm{\psi }}_2$, the first and last terms act as a speed controller and, together, they generate a command reference for the current in the q-axis $i_q$. ${\omega }_{\mathit{ref}}$ is the reference speed set by the designer. By selecting $K_2=-1$, the resultant will be the error in the q-axis current $i_q^{\ast }-i_q$, which converges to zero automatically following the dynamics of the macro-variable.

To derive the control effort, the constraint given by Eq. (7) is applied for each macro-variable. The system Eq. (10)–(13) are next combined and solved to find the control input voltages as follows:

$$v_d^{\ast }=\left(R-\frac{L_d}{T_d}\right)i_d-L_q{\omega }_e{i}_q,$$E14

$$\begin{array}{l}{v}_q^{\ast }=-\frac{K_1}{K_2}\frac{L_q}{T_q}\left(\omega -{\omega }_{\mathit{ref}}\right)-\frac{L_q}{T_q}{i}_q-\frac{K_3}{K_2}\frac{L_q}{T_q}\int \left(\omega -{\omega }_{\mathit{ref}}\right)\mathit{dt}-\frac{K_1}{K_2}{L}_q\frac{\mathit{d\omega }}{\mathit{dt}}\\ -\frac{K_3}{K_2}{L}_q\left(\omega -{\omega }_{\mathit{ref}}\right)+R{i}_q+L_d{\omega }_e{i}_d+{\omega }_e{\lambda }_{\mathit{PM}}.\end{array}$$E15

As seen from Eqs. (14) and (15), the expression for vd and vq are a function of the system parameters and the variables Td and Tq, which are selected by the designer. The controller gains K1, K2, K3 are also tuned and selected by the designer. In conclusion, the synergetic controller requires four inputs; desired motor speed ${\omega }_{\mathit{ref}}$, dq currents (id, iq), and actual motor speed. The expressions for each of the two control inputs are computed and the commands voltages (vd, vq) are computed. These voltages are then converted into three phase voltages using the inverse Clarke-Parke transformation and the corresponding SPWM signals are generated to control the switching of the VSI as desired. Figure 2 shows a block diagram representation of the synergetic controller.

Figure 2.

Synergetic controller block diagram.

3.3 Proposed synergetic controller

The selected macro-variable ${\mathrm{\psi }}_1$, given by Eq. (12), was found to lead to steady-state errors in the d-axis variables. Therefore, to eliminate any steady-state errors, the following modified macro-variable is proposed to improve the performance and maintain a constant value for the d-axis current $i_d=0$.

$${\psi }_{1m}=K_1{i}_d+K_2\int i_d\mathit{dt}.$$E16

$${\psi }_2=K_3\left(\omega -{\omega }_{\mathit{ref}}\right)+K_4{i}_q+K_5\int \left(\omega -{\omega }_{\mathit{ref}}\right)\mathit{dt}$$E17

The introduction of the gain K1 in the macro-variable ${\psi }_{1m}$ will control the peak magnitude of id during the transient periods. The addition of an integral term removes any steady state error and the gain K2 controls the speed of convergence during the transient periods. The macro-variable ${\psi }_2$ controlling the q-axis command voltage vq remains unchanged similar to Eq. (13). Only the gain subscripts have been updated.

Next, the same steps from Section 3-B are followed to compute the new control law based on the proposed macro-variable in Eq. (16). The new control voltage is computed as follows:

$$v_d^{\ast }=-\frac{L_d}{T_d}{i}_d-\frac{K_2}{K_1}\frac{L_d}{T_d}\int i_d\mathit{dt}-\frac{K_2}{K_1}{L}_d{i}_d+R{i}_d-L_q{i}_q{\omega }_e$$E18

This command effectively maintains the d-axis current $i_d$ at zero despite any discrepancies between the actual and system model parameters.

4. Regenerative braking strategy

4.1 Regenerative braking using FOC

For the PMSM model, regenerative braking (RB) is achieved by operating the motor in torque control mode, under FOC. RB analysis is next performed to determine the lower and upper boundaries of the applied torques that identify the RB region on the speed-torque plane. Through balancing the input and output power, the following expression for the input power can be formulated to account for power losses in the windings.

$$P_{\mathit{\text{in}}}=\frac{3P}{4}{\lambda }_{\mathit{PM}}{i}_q\omega +R\left(i_d^2+i_q^2\right)$$E19

To find the regenerative braking boundaries, Pin is set to 0 indicating that no power is drawn from or supplied to the DC source. Next, the equivalent electromagnetic torque is found and minimized with respect to id.

$$T_{\mathit{em}}=0\mathit{\text{or}}{T}_{\mathit{em}}=-\frac{\omega }{R}{\left(\frac{3P}{4}{\lambda }_{\mathit{PM}}\right)}^2$$E20

Then, the local minimum points to minimize the input power to the motor are found, and the maximum current absorbed by the DC source during regenerative braking can be calculated as follows.

$$\begin{array}{l}{i}_d=0,\\ i_q=-\frac{3P}{8R}{\lambda }_{\mathit{PM}}\omega .\end{array}$$E21

To achieve RB under FOC, the current controllers are given the command signals $\left(i_d,i_q\right)$ equal to the results found in Eq. (20). This will force the motor to operate in the RB region and guarantee maximum current harvested from the motor’s rotation. For that reason, the speed control loop is removed from the FOC controller and instead, the current commands $\left(i_d,i_q\right)$are fed directly to the current regulators. Figure 3 shows the block diagram of the FOC scheme in RB mode.

Figure 3.

FOC scheme in regenerative braking mode.

4.2 Proposed regenerative braking using SC

Two macro-variables are defined for the synergetic controller to operate the PMSM in regenerative braking mode. The design of these macro-variables adopts the same approach to control the currents $\left(i_d,i_q\right)$ and force them to follow their references described in Eq. (20). Figure 4 shows the block diagram of the synergetic controller in regenerative braking mode. The proposed SC d-axis macro-variable ψ1sc is described by the same Eq. (16) used previously for the speed control scheme (ψ1sc = ψ1m). The reference d-axis current $i_d^{\ast }$ is set to 0. The gain K4 is used to force id to zero and reduce the error between id and its reference. The integral term is added to eliminate any steady-state error.

Figure 4.

Synergetic controller in regenerative braking mode.

The dynamic response of this macro-variable is governed by Eq. (7), where Td is the convergence time of ψ1. Following the design procedure explained in section III, the control law vd is derived as illustrated in Eq. (17). Applying this voltage vd will ensure that id remains at 0.

The proposed SC q-axis macro-variable ψ2sc to control iq is defined as

$${\psi }_{2\mathit{sc}}=K_6\left(i_q-{i_q}^{\ast }\right)+K_7\int \left(i_q-{i_q}^{\ast }\right)\mathit{dt}$$E22

where, $i_q^{\ast }$ is the reference q-axis current that allows maximum current absorption by the DC supply as given by Eq. (20). The dynamic response of this new macro-variable will also be governed by Eq. (7), where $T_q$ is the convergence time.

To evaluate the voltage command $v_q^{\ast }$, the design procedure explained in Section 3 is followed to get:

$$v_q^{\ast }=\begin{array}{l}{L}_q\frac{d{i}_q^{\ast }}{\mathit{dt}}-\frac{K_7}{K_6}{L}_q\left(i_q-i_q^{\ast }\right)-\frac{L_q}{T_q}\left(i_q-i_q^{\ast }\right)-\frac{K_7}{K_6}\frac{L_q}{T_q}\int \left(i_q-i_q^{\ast }\right)\mathit{dt}+R{i}_q+L_d{i}_d{\omega }_e\\ +{\omega }_e{\lambda }_{\mathit{PM}}.\end{array}$$E23

Applying this voltage will generate a reverse torque that forces the motor to operate under regenerative braking mode.

5. Experimental analysis of synergetic control

5.1 Test bench description

Figure 5 shows a picture of the PMSM test bench. The system consists of a PMSM supplied with a voltage source inverter Myway MWINV-9R144 [26]. The inverter switches are controlled using the dSPACE 1103 board. The PMSM is coupled with a Bühler DC Motor through flexible couplings and additional disc inertia mounted on the same shaft. The DC motor acts as a mechanical load and is controlled using a DC-DC converter, which in turn is controlled using a dSPACE 1104 board. Two encoders are used: one incremental encoder is directly coupled to the DC motor side, while the other sine/cosine encoder is connected to the PMSM side. The analog encoder provides high resolution and accurate measurements necessary for field orientation in FOC. An interface board is designed to interface the analog encoder with dSPACE 1103. The PMSM is a 1.65 hp. Unimotor UM series from Emerson. Table 1 summarizes the technical specifications of the motor [27].

Figure 5.

PMSM experimental setup.

Variable	Value	Unit
Torque Constant $\left(K_t\right)$	1.6	Nm/A
EMF Coefficient $\left(K_e\right)$	98	V/krpm
Rated Torque $\left({\mathrm{\tau }}_{\mathit{\text{rated}}}\right)$	3.9	Nm
Stall Current $\left(I\right)$	2.7	A
Rated Speed $\left({\mathrm{\omega }}_{\mathit{\text{rated}}}\right)$	3000	rpm
Rated Power $\left(P_{\mathit{\text{rated}}}\right)$	1.23	kW
Stator Resistance $\left(R_{\mathit{ph}-\mathit{ph}}\right)$	6.8	Ω
Stator Inductance $\left(L_{\mathit{ph}-\mathit{ph}}\right)$	24.3	mH
Inertia $\left(J\right)$	2.5	${\text{kgcm}}^2$
Number of Poles $\left(P\right)$	6	—

Table 1.

PMSM parameters.

5.2 Initial motor alignment

For proper field orientation, an absolute encoder or a resolver is usually used with the PMSM. Therefore, an initial motor alignment is needed to operate the PMSM with the Sin/Cos encoder [28]. The main idea is to align the stator current vector is with the rotor flux vector ${\mathrm{\Psi }}_r$ along the stator phase a-axis. Hence, the angle between both vectors will be 0. This can be done by connecting phase a of the PMSM to the positive terminal of the DC link and connecting phases b and c to the negative terminal of the DC link. The application of a constant DC voltage will create a constant current flowing in all phases, and will place the stator current vector along the a-axis. This will in turn force the rotor flux vector to align itself with the stator current vector. This means that the d-axis is aligned with the a-axis, and the initial electrical angle of the rotor is 0. By this, the readings of an incremental encoder become sufficient to apply field orientation.

5.3 Synergetic controller testing

In this section, the performance of the SC with the modified macro-variable ${\psi }_{1m}$, proposed in Section 4-B, is compared to that of the conventional macro-variable. The performance of both macro-variables is tested by analyzing the behavior of id(t) andψ 1(t). Since the q-axis macro-variable ψ2(t) is not altered, the speed response and iq(t) remain unchanged. In this experiment, the PMSM is subjected to a speed reference step of 500 RPM followed by another step to 1000 RPM. The DC motor load is controlled to apply a load torque of 0:6 N.m. The load torque is then removed, and the speed reference is brought back to 0.

Figure 6 shows the speed reference profile and the speed response of both controllers. Figure 7 shows the response of the macro-variable ${\mathrm{\psi }}_1\left(\mathrm{t}\right)$resulting from both controllers. The conventional macro-variable is given the subscript A and the proposed modified macro-variable is given the subscript B. The macro-variables and d-axis current signals have been filtered using a low pass filter with a cutoff frequency at 20 Hz to reduce noise and signal ripples.

Figure 6.

Speed reference profile.

Figure 7.

Macro-variable ψ1(t) response: (a) refers to the conventional macro-variable and (b) refers to the proposed macro-variable.

Figure 8 shows a zoomed view of each transient response. It can be seen that the proposed controller improves the response of ${\mathrm{\psi }}_1\left(\mathrm{t}\right)$significantly. First, the macro-variable always converges back to zero after any transient event. The original controller, on the other hand, maintains a steady-state error and does not converge to zero as designed. Second, with the proposed controller, the magnitudes of the peak values reached during the transients are clearly much smaller than those reached by the original controller.

Figure 8.

Zoomed view at each transient event of ψ1(t): (a) 1st speed step (b) 2nd speed step (c) applying and removing load (d) braking.

Table 2 summarizes the recorded peaks in Figure 8. It is shown that the proposed controller B improves the peak response of ψ1(t) by approximately 90%. The proposed controller shows good robustness to load changes. However, there still remains a very small steady state error in the response ofψ 1 m(t). The reason is that the internal friction torque and external load torque were not accounted for in the controller synthesis. Hence, all resistive load torques act as a disturbance to the system. It is worth noting that the response of controller B can be further improved by tuning the gains $\left({\mathrm{K}}_4,{\mathrm{K}}_5\right)$ in Eq. (16). The same previous analysis is done for the d-axis current $i_d\left(t\right)$. Figure 9 shows the response of $i_d\left(t\right)$for the two controllers (conventional vs. proposed) and Figure 10 shows a zoomed view at each transient event. Studying the results, it can be seen that $i_d\left(t\right)$exhibits lower peak magnitudes in the case of the proposed controller in all scenarios. Additionally, the proposed controller displays a very good disturbance rejection response. The current $i_d\left(t\right)$maintains a zero- steady state error upon adding or removing a load torque.

Transient Event	ψ1 Max. Amplitude	ψ1m Max. Amplitude	Improvement (%)
1st step reference	−0.3077	−0.02526	91.79
2nd step reference	−0.8425	−0.08622	89.77
Apply/Remove load	−0.2829	−0.02407	91.40
Braking	1.526	0.142	90.69

Table 2.

Summary results.

Figure 9.

Direct axis current id response.

Figure 10.

Zoomed views at each transient period of i_d (t): (a) 1st speed step (b) 2nd speed step (c) applying load (d) removing load (e) braking.

5.4 Synergetic controller parameters tuning

In this section, the effect of all gains used in the proposed SC synthesis is studied. The controller tuning process is employed to find the best possible values for optimal performance. The same experimental procedure explained in Section 5.C is used to test the performance of the SC. Each gain is tuned while keeping the remaining gains constant to see the effect of this inspected gain separately. This procedure is done for each of the five gains $\left(K_1-K_5\right)$ used in the controller design Eqs. (16)–(17) as well as the time constants $T_d$ and $T_q$.

5.4.1 D-axis macro-variable ${\psi }_1$

The designed macro-variable ${\psi }_1$ uses two gains, $K_1$ and $K_2$, as shown in Eq. (16) and its dynamics depend on the time constant $T_d$. The effect of each parameter on the behavior of the macro-variable ${\psi }_1$ and the response of the d-axis current $i_{\mathrm{d}}$ will be analyzed. Table 3 shows the selected gains to test the effect of $K_1$.

$K_1$	0.01	0.05	0.1	0.15	0.2	0.25
$K_2$	0.3
$T_d$	0.001 sec

Table 3.

Summary of the selected values for $K_1$ inspection.

Figure 11 shows the response of the macro-variable ${\psi }_1$ upon changing the value of $K_1$ using the same speed reference profile as shown in Figure 6. Looking at the transient periods displayed in Figure 12, it can be observed that increasing the value of $K_1$ increases the magnitude of the peak value that ${\psi }_1$ reaches before it starts settling back to its reference. It is also noticed from Figure 12c that the steady state error when adding a load torque decreases for lower values of $K_1$. In the same line, it is noticed that increasing the value of $K_1$ increases the ripple of the macro-variable ${\psi }_1$.

Figure 11.

Macro-variable ${\psi }_1$ response to different values of $K_1$.

Figure 12.

Macro-variable ${\psi }_1$ response to different values of $K_1$. Zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying and removing load, (d) braking.

Figure 13 shows the response of $i_{\mathrm{d}}$ for different values of $K_1$. It can be realized that $i_{\mathrm{d}}$ always converges to its reference value ($i_{\mathrm{d}}=0$) in steady state.

Figure 13.

D-axis current i_dresponse to different values of K_1. Zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying load, (d) removing load, (e) braking.

The same procedure is repeated to study the effect of $K_2$ on ${\psi }_1$ and $i_{\mathrm{d}}$. Table 4 summarizes the selected values of $K_2$ in this experiment.

$K_1$	0.1
$K_2$	0.05	0.1	0.15	0.2	0.25	0.3
$T_d$	0.001 sec

Table 4.

Summary of the selected values for $K_2$ inspection.

It was observed that the gain $K_2$ does not affect the behavior of ${\psi }_1$ significantly. However, this is not the case for $i_{\mathrm{d}}$. Changing the value of $K_2$ clearly affects the settling time of $i_{\mathrm{d}}$ especially when applying or removing a load as shown in Figure 14. Increasing $K_2$ results in a faster response.

Figure 14.

D-axis current i_dresponse to different values of K_2. Zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying load, (d) removing load, (e) braking.

It should be noted that all signals for ${\psi }_1$ and $i_{\mathrm{d}}$ are filtered by a low pass filter with a cut-off frequency of 20 Hz. It is also worth mentioning that these gains affect only the behavior of ${\psi }_1$ and $i_{\mathrm{d}}$. The speed of the motor is not affected by changing these gains, since the speed and torque of the motor are controlled only by the second macro-variable ${\psi }_2$, which was not altered in this experiment.

Lastly, the effect of changing $T_d$ on the behavior of ${\psi }_1$ and $i_{\mathrm{d}}$ is studied. Table 5 shows the selected gains in this experiment. $T_d$ is responsible for the time taken by ${\psi }_1$ to reach the manifold $\psi =0$.

$K_1$	0.1
$K_2$	0.3
$T_d$	0.003	0.001	0.0007

Table 5.

Summary of the selected values for $T_d$ inspection.

Figure 15 shows the response of ${\psi }_1$ upon changing the constant $T_d$. Changing $T_d$ affects both the magnitude of the peaks as well as the steady state errors.

Figure 15.

Macro-variable ψ_1response to different values of T_d. zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying and removing load, (d) braking.

5.4.2 Q-axis macro-variable ${\psi }_2$

The designed macro-variable ${\psi }_2$ uses three gains, $K_3,K_4,$and $K_5$, as shown in Eq. (17) and its dynamics depend on the time constant $T_q$. The effect of each parameter on the behavior of the macro-variable ${\psi }_2$ and the response of the q-axis current $i_{\mathrm{q}}$ was analyzed. $K_3$ and $K_5$ directly affect the speed, while $K_4$ affects the current $i_{\mathrm{q}}$, which adjusts the machine torque. Table 6 shows the selected gains to test the effect of $K_3$.

$K_3$	0.01	0.05	0.1	0.15	0.2
$K_4$	1
$K_5$	0.15
$T_q$	0.001 sec

Table 6.

Summary of the selected values for $K_3$ inspection.

Figures 16 and 17 show the response of the motor speed and the micro-variable ${\psi }_2$ to different values of $K_3$.

Figure 16.

Macro-variable ψ_2response to different values of K_3. Zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying and removing load, (d) braking.

Figure 17.

Motor speed response to different values of K_3. Zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying load, (d) removing load, (e) braking.

It is observed that the amplitude of ${\psi }_2$ increases when $K_3$ is increased. The motor speed shows an underdamped response for transient for low values of $K_3$. And an overdamped behavior for large values of $K_3$. Similar analysis is made to study the effect of the gain $K_4$ and $K_5$. it is noticed that increasing $K_5$ increases the speed of convergence and eliminates steady state error faster. On the other hand, decreasing $K_4$ increases the speed of convergence of the motor speed and ${\psi }_2$.

Finally, the value of the constant $T_q$ is changed to investigate its effect on the convergence speed of ${\psi }_2$. Table 7 shows the selected gains in this experiment. Figure 18 shows the response of ${\psi }_1$ upon changing the constant $T_d$. Changing $T_d$ affects both the magnitude of the peaks as well as the steady state errors.

$K_3$	0.1
$K_4$	1
$K_5$	0.15
$T_q$	0.003	0.001	0.0007

Table 7.

Summary of the selected values for $T_q$ inspection.

Figure 18.

Macro-variable ψ_2response to different values of T_q. zoomed views of each transient period: (a) 1st speed step, (b) 2nd speed step, (c) applying and removing load, (d) braking.

6. Comparison between synergetic control and field oriented control

6.1 Speed reference tracking and disturbance rejection

In this section, the performance of the proposed synergetic controller (SC) is compared to that of the field-oriented controller (FOC), which is used as a benchmark.

Both controllers are tuned so that their responses have approximately similar settling times, around 0.05 s, and are tested under the same operating conditions. The following characteristics are used for comparison:

• Settling time and steady-state error in response to a step reference speed and to a load step disturbance.

• Transient response of the d-axis current.

• Current harmonics.

• Response to wide speed variation.

Figure 19 shows the speed profile used, which consists of a sequence of speed step references following the application and removal of a load step disturbance.

Figure 19.

Speed response of FOC and SC.

The observed results indicate that both controllers track the speed reference as desired without any steady-state errors. Both controllers are able to correct the speed after adding or removing the load. However, it was observed that when starting from rest, the SC shows a slightly faster response with a settling time of 0.076 s as opposed to 0.117 s for FOC. This shows that the speed of convergence, in the case of the SC, is better than that of the FOC by 35.042%, indicating that the SC was able to overcome the static friction that resists the motion of the motor when at rest faster than the FOC.

When both controllers are running at a fixed speed and the reference is increased, both exhibit the same settling time. Static friction acts as a disturbance to the system and the SC is able to overcome this disturbance more efficiently. This observation was confirmed by repeatedly conducting the step response experiment starting from rest for various speed step reference amplitudes. Figure 20 illustrates sample results obtained with step reference values of 50 rpm and 200 rpm. We can therefore conclude from this analysis that the SC has better performance during motor startup under low-speed references.

Figure 20.

Zoomed views of the motor speed response starting from rest for a reference speed step of (a) 50 rpm, and (b) 200 rpm.

Looking next at the d-axis current corresponding to the speed profile given in Figure 19, the following observations are made. Both controllers are able to maintain ${\mathrm{i}}_{\mathrm{d}}\left(\mathrm{t}\right)$ at 0 without any steady state errors. However, ${\mathrm{i}}_{\mathrm{d}}\left(\mathrm{t}\right)\text{with}\mathrm{SC}$ shows a faster response, less current ripple in steady state, as well as lower peaks values (seen in Figure 21 for the 1st step reference and the 1st load application). The d-axis $i_d$ current deviation for FOC could be due to friction and sensitivity to system parameters. Overall, it can be concluded that FOC has a good dynamic and steady state characteristics however, SC shows better performance.

Figure 21.

Zoomed views at the transient response of id(t)(a) 1st speed step (b) applying load.

When looking at the three phase stator currents, the waveforms from both controllers are sinusoidal. However, the stator current waveforms by the FOC controller are slightly distorted compared to the smoother waveforms obtained by the SC. This means that SC results in less harmonics in the output waveforms. Figure 22 shows the steady state currents waveforms at 300 rpm (15 Hz). The shown waveforms are filtered by means of a 16-sample moving averaging filter. The Fourier transform is next used to analyze the harmonics present in the current waveforms for each controller.

Figure 22.

Zoomed view of the three phase currents response at 300 rpm.

Figure 23 shows the single-sided amplitude spectrum of phase-a current for both controllers. Both controllers have the maximum magnitude at 15 Hz which is the fundamental frequency. However, the FOC amplitude spectrum shows higher magnitudes at higher frequencies as opposed to the SC.

Figure 23.

Magnitude response of the FFT.

The last comparative analysis is for the performance of the controllers to wide speed variation. This is tested by applying a square wave speed reference ranging between −1000 rpm to 1000 rpm. The results show that both controllers are able to track the speed reference and reach zero steady state error. However, the SC reaches the reference faster. In addition, the SC shows less current amplitude during each speed transient (phase current amplitude is 26.4% lower for the second speed step) as shown in Figure 24. This makes the SC more robust in wide signal variations, as it requires less control effort to overcome the large speed variation.

Figure 24.

Response of iq(t) to large speed variations (a) speed step from −1000 rpm to 1000 rpm (b) speed step from 1000 rpm to 0 rpm.

6.2 Regenerative braking

In this section, regenerative braking under FOC and SC is implemented, and its performance is analyzed. Torque control is used to run the motor under RB mode.

The reference q-axis currents evaluated by Eq. (20) is used to generate the required electromagnetic torque and guarantee maximum recovered power. The maximum torque is given by

$$T_{\mathit{reg}-\mathit{max}}=-\frac{9P^2}{32R}{\lambda }_{\mathit{PM}}^2\omega$$E24

Figure 25 shows the topology of the voltage source inverter used to supply the motor. The DC bus voltage is rectified from a 380 V-rms 3-phase line. Additional filter capacitors are used to maintain the DC bus voltage at 575 V. In order to observe the voltage variation during regenerative braking, the three-phase line is disconnected from the inverter. As a result, the DC link voltage begins to drop while the motor is operating under a constant speed control mode.

Figure 25.

Voltage source inverter supplying the PMSM.

Once the DC link voltage reaches 300 V, the regenerative braking command signal is triggered, and the motor operates under RB mode.

The regenerated power is calculated by multiplying the DC link voltage by the DC link current during RB. The regenerated energy is next calculated by integrating the power by means of numerical integration following the trapezoidal rule.

$$P=V_{\mathit{DC}}{I}_{\mathit{DC}}$$E25

$$E_{\mathit{reg}}\left(k\right)=E_{\mathit{reg}}\left(k-1\right)+\frac{T_s}{2}\left[P\left(k\right)+P\left(k-1\right)\right]$$E26

The mechanical energy of the motor is given by

$$E_{\mathit{\text{mech}}}=\frac{1}{2}J{\omega }^2$$E27

The efficiency of the regenerative braking process in terms of harvested energy is computed by

$$\eta =\frac{E_{\mathit{reg}}}{E_{\mathit{\text{mech}}}}\times 100.$$E28

Figure 26 shows the motor speed, DC link voltage, DC link current, q-axis current, regenerated power and energy on the DC link during the braking period under FOC. The motor comes to rest in a shorter period as well with a relatively high q-axis current value. This indicates that the generated electromagnetic torque is high resulting in a fast braking.

Figure 26.

Regenerative braking under torque control using FOC (a) motor speed (b) DC link voltage (c) DC link current (d) motor iq (e) DC link power (f) DC link energy.

The same experimental procedure is repeated under the SC. The macro-variables designed in section IV are used to control the motor in the braking period. Figure 27 shows the inverter and motor variables during braking under SC. Comparing the waveforms resulting from both controllers, it can be concluded that the motor behaves more or less the same way under both controllers and the waveforms are highly similar. Table 8 summarizes the results of the regenerative braking process under torque control using both controllers.

Figure 27.

Regenerative braking under torque control using SC (a) motor speed (b) DC link voltage (c) DC link current (d) motor iq (e) DC link power (f) DC link energy.

	$i_{\mathit{dc}-\mathit{max}}$ (A)	${\mathrm{\Delta }V}_{\mathit{dc}}$ (V)	$E_{\mathit{reg}}$ (J)	$P_{\mathit{max}}$ (W)	$i_{q-\mathit{max}}$ (A)	$E_{\mathit{\text{mech}}}$ (J)
FOC	−5.485	5.6	1.450	1626	−13.97	17.272
SC	−4.893	6.0	1.801	1452	−12.44	17.272

Table 8.

Summary of the regenerative braking results under torque control.

The oscillations in the motor speed caused by the flexible coupling hinder the controllers’ performances, since the generated q-axis current ${\mathrm{i}}_{\mathrm{q}}\left(\mathrm{t}\right)$ is a function of the motor speed. Therefore, the performance of the controllers in regenerative braking using this technique cannot be assessed properly unless a rigid coupling is used.

7. Conclusions

This chapter presented an optimum design of synergetic controllers for a permanent magnet synchronous motor (PMSM) drive system. We also provided an extensive study of the controller parameters tuning for optimal performance. The proposed macro-variables of the SC improve the performance of the synergetic controller significantly compared to the conventional controllers proposed in the literature by eliminating the steady-state errors in the direct axis current $i_{\mathrm{d}}$. The results also show that the error in the macro-variables ${\psi }_1$ was improved by 90%. The synergetic controller showed also more robustness against disturbances, such as static friction, than the field-oriented controller. Upon starting from rest, conducted experiments show that the SC performs better than FOC in terms of convergence speed at low-speed reference values. The synergetic controller also reduces the harmonics of the system compared to the FOC using both the same SPWM modulator and switching frequency. Since one of the advantages of the SC is operating under low bandwidths, this was validated using the Fourier transform of the phase currents, where SC waveforms show fewer distortions. The results show that FOC adds more harmonics at multiples of the fundamental frequency of the motor. The voltage waveforms resulting from the SC are different from those resulting from FOC due to the nonlinear output command law. However, the fundamental component remains sinusoidal as desired. Both controllers have been tested in case of wide signal variations, the results show that the SC shows better performance against wide variations, where it is able to converge in shorter time while using lesser amounts of energy compared to FOC. This confirms the benefit of using a nonlinear controller in nonlinear systems, where the linear PI controllers used in FOC respond poorly to wide signal variations as opposed to the SC. Finally, the results indicate that regenerative braking using the torque control mode allows for energy harvesting. The SC displays a similar energy harvesting efficiency compared to the FOC. However, the oscillations in the motor speed caused by the flexible coupling hinder the controllers’ performances, since the generated q-axis current $i_{\mathrm{q}}$ is a function of the motor speed.