1. Introduction
Owing to its several advantages, 5-phase PMSM becomes over the years one of the best choices for the electric vehicle and other applications which require robustness and efficiency. In addition to the classical PMSM advantages such as long life, small size, simple structure, high torque to inertia ratio, easy control, etc., the high number of phases offer more robustness toward phase failures and reduces considerably the the torque ripples [
1,
2,
3]. Nevertheless, the stability and precision of the control system is highly affected by the system non-linearity, the load torque and parametric variations and external perturbations [
4]. Several strategies were introduced to improve the overall control performance by using model linearization of the multi-phase PMSM and decomposing it into virtual sub-machines mechanically coupled and magnetically decoupled. The developed control techniques can successfully stabilise the system under normal conditions but reached their limits in presence of unmodeled or unexpected disturbances. To overcome these limitations, the Active Disturbance Rejection Controller (ADRC) is investigated in this paper.
ADRC is a modern control strategy characterized by its real-time estimation and compensation for internal and external disturbances [
5,
6]. It is composed mainly by a Tracking Differentiator (TD), an Extended State Observer (ESO), and Non-Linear State Error Feedback (NLSEF) control law. This technique is known by its simple structure, fast response, high precision, and easy parameter setting, beside its ability to deal with disturbances and uncertainties without the needs of an accurate mathematical model of the controlled system [
7,
8,
9]. It also proved its worth in many applications, such Servo System of Self-Propelled Gun [
10], Energy Conservation Controller [
11], Wind Turbine [
12,
13], Humanoid robot [
14], and especially in the field of motor control [
15,
16,
17].
For electric vehicle application, the ADRC technique was used in the speed control loop [
18]. The objective of this approach was to cancel the impact of unknown disturbances and parameter uncertainties on cruise control by compensating nonlinear vehicle dynamics. The proposed strategy proved its ability to overcome the effect of road slope and its capacity to improve stability and tracking performance of the closed-loop system during load change. ADRC was also used to control the braking current by a real time estimation including the unmodeled perturbation of regenerative braking system of EV [
19]. This technique showed a high recovery efficiency against unknown disturbance. To improve the dynamic and steady state execution of the Brushless motor used in EV, authors in [
20] implemented ADRC approach to control the speed loop. This technique has an important capacity to reduce torque ripple and enhance the overall performance of the controlled system under different operation conditions.
Over and above the modeling uncertainties and unknown perturbation, a failure in electrical devices affects considerably the control performance of electric machines. A faulty mode is an irregular operating condition, where the physical parameters of the system deviate from nominal value or states [
21]. There are several kinds of failures in electrical devices, among them, those related to sensors. Because of the importance of the sensors to get feedback information, it is unavoidable to find out solutions that can ensure the continuity and the stability of the motor even in presence of sensor failures. The core of the resilient control strategies proposed in literature is to find a technique able to switch between two different control strategies: one for the normal operation and the second for the faulty mode. This switching mechanism is based on the use of an observer to detect, isolate, and accommodate the sensor failures [
22,
23]. There are several study results dealing with the PMSM resilient control, mostly focused on an open-phase failure [
24] and stator winding shorts [
25] while others consider speed sensor sensor failure resilient control. Many of the proposed resilient control strategies are based on a detection and isolation mechanism [
22,
23,
26,
27,
28].
This paper discusses the ADRC technique’S ability to control the 5-phase PMSM in case of speed and load torque variation while experiencing speed sensor failures. Thus, the ADRC approach is considered to be a resilient control algorithm without the need for the conventional switching mechanism or an observer, usually used to estimate the failure and the system internal state. A standard ADRC model has three main parts: TD, ESO, and NLSEF. This control approach is associated with a discretization process, which increase the nonlinear ADRC adjustment complexity. For simplification purposes, the TD module was removed and the NLSEF function was replaced by a proportional gain [
29]. This modification consists of transforming the traditional form of the nonlinear ADRC to a linear one (LADRC) [
12]. The LADRC mainly comprises the proportional controller and the ESO while keeping the same high performance of the nonlinear ADRC approach [
30,
31].
Our contribution in this paper is to investigate the efficiency of ADRC technique in controlling the 5-phase PMSM for EV as a resilient control strategy, and under different operation conditions. For this purpose, this paper is structured as follows: the
Section 2 presents a general model of the EV and the 5-phase PMSM. The
Section 3 introduces the mathematical model of the Active Disturbance Rejection Controller and Linear ADRC used to simplify the motor control. The
Section 4 presents the speed and current control by the ADRC and LADRC, while in the
Section 5, the simulation results of the 5-phase PMSM operation are presented and compared to PI regulator. The
Section 6 is reserved for the conclusion.
3. Active Disturbance Rejection Controller
The PMSM physical model can be affected by several internal and external perturbations, parametric variations, and potential sensor failures. Those perturbations will decrease the motor control performance. Using ADRC, as a robust and resilient control strategy, those mixed uncertainties and failures are treated as a total disturbance, which is estimated and compensated in real time. The particularity of this control technique is its ability to estimate the whole disturbance as an extended state [
33,
34].
3.1. Resilient Control
Fault-tolerant or resilient control strategies should allow the controlled system, the EV propulsion in our case; to be able accommodating failures while preserving stability conditions and maintaining tracking performance as close as possible to the set-point, even in presence of unknown perturbations [
35]. Several studies focused on resilient control techniques that consider sensor failures producing wrong feedback signals therefore perturbing the system control. Most of the proposed resilient control approaches use two control strategies, one devoted to the healthy state while switching to another one when a sensor failure is detected [
22].
In contrast with traditionally adopted resilient control approaches, the ADRC technique do not use a separate observer or a switching mechanism, since it includes an extended observer, which is able to estimate the internal state and the total disturbances. By defining the sensor failure as an external unknown perturbation, the ADRC is able to estimate and compensate its effect as a part of the total disturbance.
3.2. Nonlinear ADRC Design
The ADRC technique comprises three principal parts: a TD responsible for tracking signals efficiently, the ESO which estimates the total disturbances and observe the internal state of the system and the NLSEF or control law, used to provide a stable and effective output signal [
36].
3.2.1. Nonlinear Function Design
The nonlinear function is the key of the entire design in the ADRC approach. It is used in TD, ESO and NLSEF. The fal function, which is generally used, is described as follows:
This equation can satisfy the conditions of continuity and derivability, where
and
are two adjustable parameters, representing the filter and the nonlinear factor, respectively, with:
3.2.2. TD Design
Through the nonlinear function in ADRC, TD algorithm is able to arrange the transition process according to the reference input and achieve a smooth approximation of the generalized derivative of the input signal. Considering the first order system in a canonical form:
In Equation (
11)
u is the known input, b is a constant, and the total disturbances
is treated as the external state variable to be estimated.
The first order system TD is described by [
37]:
where
v is the reference input,
is the trace output after TD arrangement and r is the tracking speed factor.
3.2.3. ESO Design
The ESO estimates in real time the known, unknown, internal and external perturbations applied to the system. The observed disturbances will be compensated, simultaneously as a feedback in order to achieve object reconstruction and best performance. In this part, the ESO is introduced to estimate the total disturbance
and update the state variable in real time. Since the system to be controlled is a first order, then the observer must be designed as a second-order extended state observer ESO. The algorithmic expression of ESO can be designed as:
where
and
are the output and the total disturbance estimations, respectively.
e is the state error and
is an estimated value of the constant
b.
and
are the correction gains of the output error factor.
3.2.4. NLSEF Design
The NLSEF part is responsible to arrange the compensation of the control quantity and improve the efficiency of the feedback control. It is described as the error between TD and ESO and expressed by the following equations:
where
and
are the objective control law and the final control signal after compensation, respectively. The different gains
,
and
depend mainly on the control system sampling time.
3.3. Linear ADRC Design
As described, the ADRC controller requires the adjustment of a significant number of parameters such as:
,
and
which complicate their tuning and may cause overshoot and hyper harmonic noise [
5]. To simplify the controller, some researchers propose a Linear ADRC (LADRC) to reduce perturbations and non-linearity in the control system, and to drive the PMSM with more robustness [
30,
31]. In LADRC approach, the linear TD (LTD) is expressed by:
and the Linear ESO (LESO) can be defined as follows:
where
is the estimated error,
,
and
are the estimated state variable, the estimation of total disturbance and the actual state variable, respectively. TD and LTD are used to smooth the input and reduce the overshoot, but it may extend the system’s adjustment time. For this reason, taking into account adjustment time and overshoot, requires to take a compromise solution. Moreover, to simplify the system structure, NLSEF, used in ADRC, can be replaced by a simple proportional controller and defined as a control law of LADRC [
12,
38].
This control law is presented by:
Then, the LADRC is composed mainly by a proportional controller and LESO. The LESO is used to estimate the total disturbances which include internal and external ones as well as the uncertainties and the system states. For the 5-phase PMSM, five controllers are required, where four are used to control the currents of primary and secondary dq frames and one for the speed loop.
3.4. Stability Analysis
Stability study of a closed-loop linear control system is usually carried out by the direct determination of its own values. However, for a nonlinear control system, the use of a Lyapunov function is the most popular technique for stability analysis. In [
39], the stability study of the ADRC regulator and ESO observer was demonstrated using a Lyapunov function. Similarly for the linear ADRC regulators presented in [
34,
40], the analysis of its stability based on the Lyapunov function justifies the existence of appropriate gains ensuring estimation errors convergence.
Firstly, based on the linear ADRC study presented in [
38], the nonlinear system defined by Equation (
11) can be rewritten as:
Then, the matrix form of Equation (
18) is:
with
,
,
, and
The LESO presented by Equation (
16) can then be deduced and described by the following expression:
with
;
L denote the LESO gains vector and
its bandwidth. For the stability analysis, let define the dynamic of the tracking error
of the observer by the following function:
with
and
.
Lemma 1. It can be noted that for any bounded h, the error e is bounded if the matrix is Hurwitz.
Proof. Assuming the matrix
is Hurwitz, let define respectively the Lyapunov function
V and the Lyapunov equation by:
and
where
X is the unique solution of the Lyapunov equation and
P is a definite positive matrix.
According to [
41], the derivative of Lyapunov function is expressed in the form:
which implies that
if:
or equivalently
For
, an identity matrix, the derivative of Lyapunov function
defined by Equation (
24) is negative if:
which implies for all error
e satisfying Equation (
27),
decreases. Then, it can be noted that
e is bounded. □
Lemma 1 can be generalized for any system described by:
with
and
. The corresponding lemma is:
Lemma 2. If the matrix N is Hurwitz and the function is bounded, then the state μ in Equation (28) is also bounded. Combining Lemmas 1 and 2, the boundedness of LADRC can be defined by [
41]:
Theorem 1. If the control law Equation (17) and the observer Equation (20) are stable, the linear ADRC design of Equations (20) and (17) presents a stable closed-loop system with bounded input and output. The above-presented stability analysis shows the ability of the ADRC controller to solve the uncertainties problem in a real system.
5. Simulation Results
The ADRC control performance is compared simultaneously with that of LADRC and the conventional PI regulator to show the properties of each regulator and their capabilities. The Simulation were carried out in Matlab/Simulink platform. The PMSM and EV numerical parameters are presented in
Table 1 and
Table 2.
For comparison purposes, the following fitness functions are defined to study the regulators performance [
43]:
Equations (
33)–(
36) present the Integral of the Square Error (ISE), the Integral of Absolute Error (IAE), the Integral of the Time-weighted Absolute Error (ITAE), and the Integral of the Time Square Error (ITSE), respectively [
44].
As already announced, the control of PMSM requires the use of five control block for both of currents and rotor speed, with an adjustment of a significant number of gains. Then, it is important to analysis the effect of variation of control parameters on PMSM operation. As presented in Equation (
9), nonlinear law used in ADRC depends mainly on
and
. For
, the time response increase and the signal track rapidly its reference to the increasing of gain
which improve the control efficiency as shown in
Figure 3a. Also, the control law depends on
variation, where the time response increase with decreasing of
as presented in
Figure 3b.
To overcome the complexity of chosing the adequate gains, the LADRC was proposed to minimize the number of parameters to be adjusted and to simplify the control approach.
5.1. Speed and Load Torque Variation
Figure 4 and
Figure 5 present the respectively the speed, electromagnetic torque and the q-axes currents responses during the motor operation. Firstly, the speed was maintained to 1500 RPM with a variable torque, which progressively increases until it reaches its nominal value, as is shown in
Figure 4. In the second case, the load torque was set to be constant with a speed variation, as presented in
Figure 5.
The three regulators, ADRC, LADRC and the PI show a high tracking performance where the physical parameters track perfectly its references, especially in the steady state. The difference between the three regulators appears in the transient state. In the case of Pi regulator, a reduction in response time is usually accompanied by a significant overshoot, whereas the resolution of this overshoot produces a longer response time or even a delay in the system, which is not the case for the ADRC and LADRC regulators as shown in the following simulation. The ADRC regulator presents a response time that is very close to that of PI, but with a reduced overshoot. By simplifying the structure of the ADRC and minimizing regulation parameters, the LADRC combine between the both characteristics: the stability and the faster response. It was totally remarkable that LADRC presents a better performance than the standard ADRC and the traditional PI regulator, with a faster speed response as presented in both of
Figure 4 and
Figure 5.
Also, we notice in
Figure 4c,d and
Figure 5c,d, that both of the ADRC and LADRC regulate the q axis currents to track their references so they can adapt perfectly to those variations. The performance of each regulator is given
Table 3,
Table 4 and
Table 5.
Performance indices of speed, , and are compared in case of starting stage, load torque and speed variations. The lowest performance indices for IAE, ISE, ITAE, and ITSE values prove the efficiency of proposed control strategy to achieve the highest tracking performance. These simulation results clearly show that the best control performance is achieved with ADRC and LADRC.
5.2. Speed Sensor Failure
In this section, the proposed control strategies are tested under speed sensor failure conditions. In this context,
Figure 6 illustrates the additive signals that describe the considered sensor failures. The first signal, inserted to the rotor speed sensor output, represents a simple step with an amplitude equal to 150 RPM (
Figure 6a), and the second one constitute an unmodeled noise with an amplitude below
of the nominal speed.
Figure 7 and
Figure 8 illustrate the PMSP rotor speed, torque, and currents when considering speed sensor failures.
In the obtained simulation results, it can be noted that the 5-phase PMSM achieves very interesting tracking performance in terms of speed and torque despite the speed sensor failure. Indeed, the 5-phase PMSM keeps its speed and produced electromagnetic torque too smooth, as in the healthy mode, and rapidly compensate the difference between the reference signal and the faulty one. Using the ESO, included in the ADRC controller, the sensor failure can be estimated in real time as a part of the total disturbance affecting the motor, and compensated by the control law to ensure the system prescribed performance. This mechanism justifies the choice of the ADRC controller as a resilient control strategy, in addition to its robustness ability. These simulations results clearly highlight the robustness and resilience control performance of the 5-phase PMSM thanks to the use of the ADRC and LADRC.
5.3. Simulation Results Using Driving Cycles
To test the validity of this control strategy, the proposed system operation will be tested with the Worldwide Harmonized Light Vehicle Test Procedure (WLTP) developed by the European Union [
45]. This developed driving cycle is based on real-driving data, gathered from around the world [
45,
46]. As shown in
Figure 9, WLTP is composed of four different parts with various average speeds: low, medium, high and extra high. Each one of those parts contains a variety of driving phases, stops, acceleration and braking phases.
It can be noted from the simulation results that because of the TD arrangement and the NLSEF, the PMSM can start continuously without overshoot but with the slow speed response.
In case of LADRC with elimination of TD, the system presents a neglected overshoot in a transient state, but it reaches quickly their reference value and have great stability in the steady state. The simulation results demonstrated the efficiency of suggested control strategy and the characteristics of each one in terms of fast tracking and robustness for internal and external disturbances.