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Abstract

Synchronous motion is an essential function for multi-motor drive systems, which have been applied in high-precision space collaborative manipulators, satellite systems, and other aerospace industrial fields. To maintain the desired high precision of synchronization in the case of motor inverter fault, a coordinated fault-tolerant control (FTC) scheme is proposed in this paper for the drive system consisting of multiple brushless direct current motors with an adjacent cross-coupling structure. Firstly, the proposed scheme develops a robust coordinated FTC control law based on synergetic control theory, in which the expected manifold of the fault-tolerant synergetic controller is devised to guarantee the synchronization preferentially in the fault case. To enhance system robustness, the control framework incorporates a disturbance observer and an adaptive technique to effectively mitigate the impact of load disturbances. Secondly, the asymptotic stability of the FTC system is proven via Lyapunov theory. Finally, simulation and test experiments on the speed synchronous control of a triple-motor system verified the effectiveness of the developed FTC strategy.

Keywords

coordinated fault-tolerant control multi-motor drive system synergetic control synchronous precision inverter fault

Introduction

Multi-motor drive systems have been successfully applied in high-precision space station collaborative manipulators,^1,2 twin-gyros synchronization attitude control of spacecraft,³ satellite antenna servo systems,^4,5 distributed electric propulsion aircraft,⁶ and other aerospace industrial fields, where high synchronous precision among motors is vital to system reliability, safety, operating performance, and missions. Thus, numerous synchronous control strategies for multi-motor drive systems have been developed in recent years, such as coupling-based (including cross-coupling, deviation coupling, adjacent coupling, and relative coupling),^6,7 concurrent,⁴ and leader-following approaches.⁸ Zhang et al.⁹ analyzed the prevalent synchronization control structures employed in multi-motor systems, with applications in industrial drones, robotic arms, and electric vehicles. These structures included master command control (namely concurrent), master–slave control and coupling-based controls. The authors also introduced an improved deviation coupling strategy incorporating a virtual motor to achieve multi-motor speed synchronization by synchronization coefficient and tracking coefficient. The coefficients were optimized using the particle swarm optimization algorithm. Masroor et al.⁸ investigated the brushless direct current (BLDC) multi-motor speed synchronization problem, a critical challenge in applications such as aerospace, unmanned aerial vehicles, and robotic manipulation. It was addressed by using the leader-following multi-agent system consensus and variable-structure model reference adaptive control algorithm. It is indicated by Yang and Chang³ and Niu et al.¹⁰ that the concurrent and leader-following methods suffer from inherent deficiencies of the poor anti-interference and the low response speed. They are usually applied in the early aerospace systems or other industrial scenarios like paper-making and textile manufacturing with low-synchronization requirements. In modern satellites and space structures applications, the coupling-based synchronous strategies are often employed with other techniques to meet higher levels of synchronization.^7,11 For examples, Weng et al.⁶ proposed a thrust synchronous control strategy for multiple propulsion motors with relative coupling structure in distributed electric propulsion aircraft to keep the synchronous speed of each motor. To meet the synchronization control requirements of multi-servo motor drive systems in radar antennas and air defense artillery, Wang et al.⁷ developed an adaptive robust H∞ control method that integrates an average deviation coupling strategy with distributed synchronization controllers. Han et al.¹¹ developed an attitude synchronous control strategy by combining cross-coupling structure and non-singular fast terminal sliding mode control for satellite formation systems under disturbances. However, the synchronization of systems may be degraded or failed due to component faults, abnormal disturbances or system parameter variations. For instance, power inverters are the key actuators in BLDC motor-based servo systems which are often used in satellite antennas.^4,5 It has been reported that more than 82.5% faults were caused by inverters.¹² Such faults may not only lead to accidents in the faulty motor subsystem, but also propagate through the coupling structure to impair normal subsystems, causing the entire system to lose synchronization. Therefore, it is required to find a proper fault-tolerant control (FTC) strategy for the high synchronization applications.

At present, many FTC methods can tolerate the faults of power inverters, sensors and windings components in motor systems (see e.g. Lv and Li,¹³ Bhuiyan et al.¹⁴). Most of them rely on complex fault diagnostic modules to obtain in-depth failure information, and their FTC controllers are designed for failed motors without leveraging the latent useful information in multi-motor synchronous systems. To address such problems, an active FTC strategy with improved command filtered backstepping control scheme was proposed by Wang et al.¹⁵ for a four-motor synchronously driving servo system with symmetric mechanical structure. This approach can isolate the faulty motor while maintaining relatively high synchronization among the remaining fault-free motors under unexpected motor faults. Mao et al.¹⁶ integrated model-based and data-based methods to tolerate sensor faults in multi-motor synchronous systems with deviation coupling structures. Recently, sliding-mode control (SMC) has exhibited outstanding performance advantages in addressing complex fault-tolerant issues due to its superior disturbance rejection capability.¹⁷ It is also noted by Bhuiyan et al.¹⁴ that the generalized FTC schemes such as SMC and adaptive extended Kalman filter can ensure a moderate performance. However, conventional SMC inherently suffers from discontinuities, and the complex coupling structures in multi-motor systems exacerbate system chattering. Some improved SMC methods such as second-order SMC¹⁸ and super-twisting algorithm¹⁹ introduced significant complexity, although they demonstrated superior chattering suppression and control performance. Consequently, these methods are difficult to be applied directly in FTC of multi-motor synchronization systems due to complex coupling-based structure.¹⁶

In recent decades, synergetic control (SC) approach, similar to SMC,²⁰ has provided solutions for solving the high-precision control problem of nonlinear systems, and applied in the fields such as healthcare,²¹ aerospace,^22,23 and power apparatus.²⁴ The design of macro-variable in SC can reconcile multiple control indices and lead the system to a reference trajectory in finite time. It not only brings about the advantages such as robustness, fast response, high accuracy and order reduction, but also eliminates the chattering in SMC through the designed macro-variable or manifold.²¹ Huang et al.²⁵ proposed a fixed-time synergetic control method employing load torque observer to synchronize the speeds of wheels in a multi-interior permanent magnet synchronous motor traction system, leveraging the relative coupling structure. To the best of our knowledge, only a few studies have focused on the FTC issue for coupling-based synchronization of multi-motor systems subject to inverter faults.

Based on the foregoing discussion, a coordinated FTC strategy with an adjacent cross-coupling structure is proposed in this paper to maintain the desired synchronous precision for a multi-BLDC-motor drive system against inverter fault. The main contributions are stated as follows: (1) The proposed FTC scheme achieves the desired synchronization through coordinated operation among all motor subsystems, rather than relying solely on the faulty motor subsystem as in conventional approaches. This method features simple decision logic and does not require a stringent fault detection and diagnosis unit or complex FTC mechanisms. (2) A numerical FTC controller is developed based on SC theory, in which the expected manifold is designed to give priority to synchronous precision while mitigating the negative effects of the coupling structure on FTC performance. The resulting control algorithm can be readily executed by numerical control programs without inducing the chattering phenomenon of conventional SMC. (3) An adaptive weighting coefficient and an exponential convergent disturbance observer are adopted to enhance the FTC performance. (4) The asymptotic convergence of the FTC system is proven via Lyapunov stability analysis.

The remainder of this paper is organized as follows: The motor model with inverter faults and the FTC problem are described in Section 2. The coordinated fault-tolerant controller is developed in Section 3. In Section 4, the simulative and experimental results are demonstrated to verify the proposed FTC method. Conclusions are given in Section 5.

Problem formulation

Adjacent cross-coupling synchronization structure

Figure 1 is a typical speed synchronous control system with n BLDC motors and adjacent cross-coupling structure.²⁶ The tracking error for the i-th motor control subsystem is

δ_{i} (t) = θ_{i} (t) - w_{i} (t), i = 1, 2, \dots, n,

(1)

where w_i(t) is the angular speed output; θ_i(t) is the reference speed input after compensation from the coupling structure.

θ_{i} (t) = x_{d} (t) - ξ_{i} (t),

(2)

where x_d(t) is the command speed; ξ_i(t) is the compensation of synchronous errors, denoted by

{\begin{matrix} ξ_{1} (t) = K_{α} ε_{1} (t) - K_{β} ε_{n} (t), \\ ξ_{i} (t) = K_{α} ε_{i} (t) - K_{β} ε_{i - 1} (t), i = 2, 3, \dots, n, \end{matrix}

(3)

where K_α > 0 and K_β > 0 are the gain coefficients, and ε_i(t) is the synchronous error of speed between the i-th and (i+1)-th motor control subsystems, expressed by

{\begin{matrix} ε_{i} (t) = w_{i} (t) - w_{i + 1} (t); i = 1, 2, \dots, n - 1 \\ ε_{n} (t) = w_{n} (t) - w_{1} (t) . \end{matrix}

(4)

Both tracking error δ_i(t)→0 and synchronous error ε_i(t)→0 are satisfied in normal case. Whereas, when the fault occurs in motor inverter, it can cause the entire system synchronization failure. To address this problem, the inverter fault model is firstly deduced in the sequel.

Figure 1.

Multi-motor speed synchronous control system with adjacent cross-coupling structure.

Inverter fault model of BLDC motor and problem statement

According to BLDC servo-motor operating principle, the following simplified state-space model

{\begin{matrix} {\overset{\cdot}{x}}_{i, 1} (t) = x_{i, 2} (t) \\ {\overset{\cdot}{x}}_{i, 2} (t) = a_{i, 1} x_{i, 2} (t) + a_{i, 2} x_{i, 1} (t) + b_{i} u_{i} (t) + f_{i} (t) \\ w_{i} (t) = x_{i, 1} (t), \end{matrix}

(5)

is derived by Kirchhoff Circuit Laws²⁷ for the i-th motor control subsystem, where x_i,₁(t) is the speed, u_i(t) is the output voltage of inverter, a_i,₁ = −(2D(L−M)+RJ)/(2J(L−M)), b_i = −(K_t)/ (2J(L−M)), f_i(t) = −RT_L(t)/(2J(L−M)) with df_i(t)/dt = 0, a_i,₂ = (−RD−K_eK_t)/(2J(L−M)), R is the resistance of winding, L and M are the coefficients of self-inductance and mutual-inductance, J is the moment of inertia, K_t is the torque coefficient, K_e is the coefficient of back electromotive force (EMF), D is the coefficient of viscous damping, T_L(t) is the load torque with gradual variation and the assumption dT_L(t)/dt = 0. According to the commutative principle of inverter with pulse width modulation (PWM), u_i(t) is also expressed by

u_{i} (t) = \frac{u_{c, i} (t) \cdot u_{M} (t) \cdot n_{sp}}{(u_{W} \cdot T_{PWM})},

(6)

where u_c,i(t) is the controller output, u_M(t) > 0 is the bus DC voltage of inverter, T_PWM is the period of PWM signal, u_W is the nominal value of bus DC voltage, n_sp is the value of PWM counter. Consequently, we have u_i(t) = u_c,i(t) in normal case when ignoring the influences of PWM precision and power switch transient durations in the motor drive system.

Common faults in inverters include bus DC voltage fluctuations, electrical component damage, and PWM pulse counting biases. They are usually manifested by the abnormal drop of inverter output voltage, and lead to the abnormal deceleration of motor.²⁸ Such kinds of inverter faults are equivalently regarded as the abnormal changes of output u_i(t) in (6), which are the gain type faults theoretically. Thus, the output voltage of the faulty inverter in system (5) can be denoted by

u'_{i} (t) = g (t) u_{i} (t),

(7)

where g(t)∈[GL, GH] is the fault factor depicting the severity, GL and GH are the lower and upper bounds of g(t) with 0 < GL < 1 and GH ≥ 1. The inverter is fault free when g(t) = 1, otherwise the anomaly or fault happens. Consequently, the faulty system model of (5) is

{\begin{matrix} {\overset{\cdot}{x}}_{i, 1} (t) = x_{i, 2} (t) \\ {\overset{\cdot}{x}}_{i, 2} (t) = a_{i, 1} x_{i, 2} (t) + a_{i, 2} x_{i, 1} (t) + b_{i} u_{i} (t) + f_{i} (t) + b_{i} {u_{i}}^{f} (t) \\ w_{i} (t) = x_{i, 1} (t), \end{matrix}

(8)

where $u_{i}^{f} (t) = u'_{i} (t) - u_{i} (t) = u'_{i} (t) - u_{c, i} (t)$ is the residual of inverter output. Inverter is in health condition if u_i^f(t) = 0. As for the inverter fault detection, many techniques may be applied (see e.g. Delpha et al.,²⁹ Li,³⁰ Jiang and Chowdhury³¹). To facilitate the FTC strategy development, the multi-sequential probability ratio test (M-SPRT) technique is adopted in this paper by using the control signal u_c,i(t) and the inverter output signal. The main procedure is listed in the Appendix 1 to save pages and interested readers can refer to references.^29,30

To this end, the FTC problem investigated in this paper is stated as follows: As for the multi-motor synchronous control system in Figure 1 subject to inverter fault in one of the BLDC motor subsystems, a coordinated FTC strategy is studied to preferentially guarantee the expected synchronous speed of the overall multi-motor system, thereby ensuring the satisfied speed synchronous errors among the motors.

Remark 1: In applications such as electric propulsion aircrafts, satellite antenna drive systems, and space station collaborative manipulators, the operating speeds of spaceborne systems usually vary along with the changing operational conditions. However, maintaining synchronization in multi-motor systems is critical to ensure the required reliability, safety and mission success (see e.g. Wang et al.,² Yang and Chang,³ Heera et al.,⁵ Weng et al.⁶). In other words, the synchronous precision requirement is prioritized over other control performance. Hence, this FTC problem is particularly significant for the aerospace applications.

Coordinated fault-tolerant control based on synergetic control theory

Coordinated synchronous FTC design

The proposed coordinated FTC scheme is illustrated in Figure 2. It embraces n BLDC motor-drive subsystems and an FTC decision-making module, where a fault detector and a fault-tolerant SC (FTSC) controller are included in every motor control subsystem. For the i-th subsystem, fault detector is used to get inverter fault information by the M-SPRT algorithm in Appendix 1. It provides a Boolean flag F_i(t) to the FTC decision-making module, where F_i(t) = 0 indicates fault-free state and F_i(t) = 1 otherwise. FTC decision-making module generates the Boolean signal s_i(t) based on the current values of all flags F_i(t). s_i(t) enables the fault-tolerant executer_i (see Figure 3) in the corresponding FTSC controller. The decision-making logic is:

When the multi-motor synchronous system in normal case, $\sum_{i = 1}^{n} F_{i} (t) = 0$ and s_i(t) = 0;

When inverter fault occurred in the j-th subsystem, F_j(t) = 1 and F_i(t) = 0 (i = 1,2,…, n, i≠j), the outputs s_j(t) = 1 and s_i(t) = 0 accordingly;

If inverter faults occurred in more subsystems, $\sum_{i = 1}^{n} F_{i} (t) > 1$ , then the FTC strategy cannot maintain the synchronization and the system suspends operation to avoid further serious accidents.

Figure 2.

Coordinated FTC scheme for multi-motor speed synchronous control system.

Figure 3.

FTSC controller scheme for i-th motor control subsystem.

Remark 2: The proposed FTC scheme is an active FTC method taking full advantage of adjacent cross-coupling mechanism. It achieves the desired FTC objective through the coordination among the motor control subsystems. Conventional FTC solutions usually focus on the faulty motor control subsystem, and need more complex fault detection and identification modules with rigorous requirements.^14,16 Comparatively, the proposed scheme only needs a simple fault detector tagging the occurrence of inverter fault, reducing the difficulty of design and implementation.

As for the FTSC controller design in Figure 2, we take the i-th subsystem as example to demonstrate its procedure. The proposed FTSC controller structure is illustrated in Figure 3. It has five inputs (x_d(t), x_i,₁(t), x_i_−1,1(t), x_i_+1,1(t) and s_i(t)), and the control signal is

u_{c, i} (t) = u_{b, i} (t) + u_{k, i} (t),

(9)

where u_c,i(t) is the robust coordinated FTC signal, u_k,i(t) is the FTSC control signal and u_b,i(t) is the compensation signal for load disturbances.

First of all, a disturbance observer with exponential convergence is adopted to attenuate the load influence. According to (5), the following model is got

{\begin{matrix} {\overset{\cdot}{x}}_{i, 1} (t) = x_{i, 2} (t) \\ {\overset{\cdot}{x}}_{i, 2} (t) = a_{i, 2} x_{i, 1} (t) + a_{i, 1} x_{i, 2} (t) + b_{i} u_{i} (t) + f_{i} (t) \\ w_{i} (t) = x_{i, 1} (t) \end{matrix}

(10)

for the i-th subsystem, where f_i(t) is the bounded load disturbance with df_i(t)/dt = 0. Complying with the principle in Fazal and Choudhry³² and Song and Zhang,³³ we develop the following exponential convergence observer

{\begin{matrix} {\overset{\cdot}{z}}_{i} (t) = β_{i} (- a_{i, 1} x_{i, 2} (t) - a_{i, 2} x_{i, 1} (t) - b_{i} u_{i} (t)) - β_{i} {\hat{f}}_{i} (t) \\ {\hat{f}}_{i} (t) = z_{i} (t) + β_{i} x_{i, 2} (t), \end{matrix}

(11)

to estimate f_i(t) in real-time and improve the robustness of FTSC controller, where z_i(t) is the observer state, ${\hat{f}}_{i} (t)$ is the real-time estimation of f_i(t), β_i > 0 is the observer gain. In view of df_i(t)/dt = 0, we derive ${\overset{\cdot}{\tilde{f}}}_{i} (t) = - {\overset{\cdot}{z}}_{i} (t) - β_{i} {\overset{\cdot}{x}}_{i, 2} (t)$ from the observing error ${\tilde{f}}_{i} (t) = f_{i} (t) - {\hat{f}}_{i} (t)$ with an upper-bound | ${\tilde{f}}_{i}$ (t)|<B_i. The differential equation of the observing error can be obtained by

{\overset{\cdot}{\tilde{f}}}_{i} (t) + β_{i} {\tilde{f}}_{i} (t) = 0 .

(12)

It is obvious from (12) that the error is exponentially convergent. Thus, the compensation law for the load disturbance is designed by $u_{b, i} (t) = - {\hat{f}}_{i} (t) / b_{i}$ . So far, the i-th subsystem with the compensation law is expressed by

{\begin{matrix} {\overset{\cdot}{x}}_{i, 1} (t) = x_{i, 2} (t) \\ {\overset{\cdot}{x}}_{i, 2} (t) = a_{i, 2} x_{i, 1} (t) + a_{i, 1} x_{i, 2} (t) + b_{i} u_{k, i} (t) + {\tilde{f}}_{i} (t) \\ w_{i} (t) = x_{i, 1} (t) . \end{matrix}

(13)

Next, the FTSC control law u_k,i(t) is designed. Based on the adjacent cross-coupling structure, the compensation of synchronous error is produced by

\begin{matrix} ξ_{i} (t) = K_{α} ε_{i} (t) - K_{β} ε_{i - 1} (t) \\ = K_{α} [x_{i, 1} (t) - x_{i + 1, 1} (t)] + K_{β} [x_{i, 1} (t) - x_{i - 1, 1} (t)] \end{matrix}

(14)

from the errors in (3) and (4). Based on SC theory, the most important step of designing SC controller is to find a macro-variable and a control law based on the manifold equation and the system state-space model.³² For the i-th subsystem we devise the following manifold

ψ_{i} (t) = k_{i, 1} δ_{i} (t) + k_{i, 2} ξ_{i} (t) + x_{i, 2} (t) = 0

(15)

by combining the system variables x_i,₂(t), ξ_i(t), and δ_i(t), where ψ_i(t) is the macro-variable, k_i,₁ > 0 and k_i,₂ > 0 are the designed weighting coefficients. When an inverter fault is detected, the proposed FTC strategy will prioritize the synchronization. The operating logic of the fault-tolerant executer_i in Figure 3 is thus designed as follows to enable real-time computation of δ_i(t) using the flag s_i(t) and timely adjust the manifold for achieving FTC goal.

When inverter fault is not detected, flag s_i(t) = 0. All motor subsystems are running synchronously and tracking their command x_d(t), the tracking error δ_i(t):=δ_i′(t)=x_i,1(t)−x_d(t);

When inverter fault is detected, flag s_i(t) = 1. The equilibrium point of the entire synchronous system will alter due to the fault, and the new equilibrium point (representing the updated synchronous speed of the multi-motor system under fault condition) cannot be determined a priori. To realize the FTC goal in this case, we use a high-pass filter in Zhao³⁴ to approximate the variation of tracking error in the manifold, namely,

\begin{matrix} δ_{i} (t) : = Δ δ_{i} (t) = \frac{T_{h} s}{T_{h} s + 1} \\ δ_{i}' (t) = \frac{T_{h} \cdot s}{T_{h} \cdot s + 1} (x_{i, 1} (t) - x_{d} (t)), \end{matrix}

(16)

where T_h > 0 is the time constant of the high-pass filter, s is the Laplace operator.

By substituting the compensation ξ_i(t) in (14) and the above tracking error δ_i(t) into the manifold (15), the following equation with first-order inertia feature

T_{i} \cdot {\overset{\cdot}{ψ}}_{i} (t) + ψ_{i} (t) = 0

(17)

is yielded, where T_i > 0 is the design parameter. We have ${\overset{\cdot}{ψ}}_{i} (t) = - ψ_{i} (t) / T_{i}$ from (17). Meanwhile, differentiating the left-side of (15) yields ${\overset{\cdot}{ψ}}_{i} (t) = k_{i, 1} {\overset{\cdot}{δ}}_{i} (t) + k_{i, 2} {\overset{\cdot}{ξ}}_{i} (t) + {\overset{\cdot}{x}}_{i, 2} (t)$ . Then substituting (15) and the second equation in (13) into it can produce

\begin{matrix} k_{i, 1} \cdot {\overset{\cdot}{δ}}_{i} (t) + k_{i, 2} \cdot {\overset{\cdot}{ξ}}_{i} (t) + {\overset{\cdot}{x}}_{i, 2} (t) = - ψ_{i} (t) / T_{i} \\ \Rightarrow k_{i, 1} \cdot {\overset{\cdot}{δ}}_{i} (t) + k_{i, 2} \cdot {\overset{\cdot}{ξ}}_{i} (t) + a_{i, 2} x_{i, 1} (t) + a_{i, 1} x_{i, 2} (t) + b_{i} u_{k, i} (t) \\ + {\tilde{f}}_{i} (t) = - ψ_{i} (t) / T_{i} \\ \Rightarrow u_{k, i} (t) = \frac{- 1}{b_{i}} [\begin{matrix} \frac{k_{i, 1}}{T_{i}} δ_{i} (t) + \frac{k_{i, 2}}{T_{i}} ξ_{i} (t) + \frac{x_{i, 2} (t)}{T_{i}} + k_{i, 1} \cdot {\overset{\cdot}{δ}}_{i} (t) \\ + k_{i, 2} \cdot {\overset{\cdot}{ξ}}_{i} (t) + a_{i, 2} x_{i, 1} (t) + a_{i, 1} x_{i, 2} (t) + {\tilde{f}}_{i} (t) \end{matrix}] . \end{matrix}

(18)

Remark 3: Essentially, the developed FTSC law achieves the FTC goal of prioritizing high-precision synchronization, to some extent, through regulating the tracking error δ_i(t) in the manifold with the fault detector flag F_i(t) serving as the control prompt. Additionally, the adaptive design of k_i,₂ presented in Section 3.2 can further improve the synchronization in the fault case.

To further enhance the robustness of the FTC law against load disturbances, the upper bound B_i > 0 with | ${\tilde{f}}_{i} (t)$ |<B_i is estimated by the following adaptive law

{\overset{\cdot}{\hat{B}}}_{i} (t) = γ_{i} \cdot | ψ_{i} (t) |,

(19)

where γ_i > 0 is the given adaptive coefficient. Then the dynamics of the estimation error ${\tilde{B}}_{i} (t) = B_{i} - {\hat{B}}_{i} (t)$ is

{\overset{\cdot}{\tilde{B}}}_{i} (t) = - {\overset{\cdot}{\hat{B}}}_{i} (t) = - γ_{i} | ψ_{i} (t) | .

(20)

By virtue of the estimated value ${\hat{B}}_{i} (t)$ , we further use the item ${\hat{B}}_{i} (t) sgn (ψ_{i})$ instead of ${\tilde{f}}_{i} (t)$ in (18). Thus, the FTSC control law is

\begin{matrix} u_{k, i} (t) = \\ \frac{- 1}{b_{i}} {\begin{matrix} \frac{k_{i, 1}}{T_{i}} δ_{i} (t) + \frac{k_{i, 2}}{T_{i}} ξ_{i} (t) + \frac{x_{i, 2} (t)}{T_{i}} + k_{i, 1} \cdot {\overset{\cdot}{δ}}_{i} (t) + k_{i, 2} \cdot {\overset{\cdot}{ξ}}_{i} (t) \\ + a_{i, 2} x_{i, 1} (t) + a_{i, 1} x_{i, 2} (t) + sgn [ψ_{i} (t)] {\hat{B}}_{i} (t) \end{matrix}} . \end{matrix}

To this end, the robust coordinated FTC law is figured out by

\begin{matrix} u_{c, i} (t) = u_{k, i} (t) + u_{b, i} (t) \\ = \frac{- 1}{b_{i}} {\begin{matrix} \frac{k_{i, 1}}{T_{i}} δ_{i} (t) + \frac{k_{i, 2}}{T_{i}} ξ_{i} (t) + (\frac{1}{T_{i}} + a_{i, 1}) x_{i, 2} (t) + k_{i, 1} \cdot {\overset{\cdot}{δ}}_{i} (t) \\ + k_{i, 2} \cdot {\overset{\cdot}{ξ}}_{i} (t) + a_{i, 2} x_{i, 1} (t) + sgn [ψ_{i} (t)] {\hat{B}}_{i} (t) + {\hat{f}}_{i} (t) \end{matrix}} . \end{matrix}

(21)

Remark 4: As can be observed from (21), the developed FTC control law essentially operates as a state-feedback controller, with the signal u_c,i(t) is computed numerically. Thus, the control parameters can be determined off-line, while the FTC action is implemented on-line. Such an approach is preferable in aerospace engineering practice.

Control parameters of the FTC law

From the previous design and analysis, we can find that the design parameters in control law (21) may vary only if they are greater than zero. It is preferable to tune these parameters for better synchronization. In practice, they are mainly empirical parameters and determined individually either through prior knowledge or a trial-and-error process. Among the design parameters, only k_i,₁, k_i,₂, and T_i are closely related to the synchronization and are discussed here.

For the manifold (15), it is known based on SC theory that increasing k_i,₁ can improve the tracking accuracy, but the regulating process becomes slower for the synchronous error. It is designed trade-off by the technique in Bastos et al.³⁵

k_{i, 1} = | \frac{λ_{i, m}}{δ_{i, m}} |,

(22)

and then fine-tuned to meet the required tracking accuracy of the i-th subsystem, where λ_i,_m is the nominal acceleration of motor and δ_i,_m is the permitted maximal tracking error.

As for the time constant T_i, the settling time τ_i of the manifold convergence is about 3T_i since its dynamics is described by (17) with the first-order inertial feature. Thus, we preset the value τ_i in comply with the transient performance of motor system, and the time constant T_i = τ_i/3 can be obtained.

In terms of the weighting coefficient k_i,₂, it is noted from Remark 3 that the tracking error $δ_{i} (t)$ relates the system synchronization in the adjacent cross-coupling structure. To reduce the negative effect of the coupling structure and enhance the synchronization in the FTC process, inspired by the adaptive parameter in Zhao,³⁴ we use the following adaptive technique

k_{i, 2} (t) = k_{\min} + l \cdot \frac{| δ_{i} (t) |}{(T_{l} s + 1)}

(23)

to determine the coefficient k_i,₂, where k_min > 0 is the preset minimum value, l > 0 is gain coefficient, T_l = 0.1 is the time constant of first-order inertia.

Remark 5: For the multi-motor synchronous system, the manifold is fixed when the weighting coefficients k_i,₁ and k_i,₂ in macro-variable (15) are held constant. This indicates that the system lacks the flexibility to tolerate potential inverter faults. The above adaptive adjustment of k_i,₂ allows the dynamic behavior of each motor subsystem to be adjusted by timely varying the manifold in response to inverter faults. Hence, the synchronous errors are reduced, and the FTC performance of the overall system is enhanced.

Remark 6: Through the above discussion, parameters k_i,₁, k_i,₂, and T_i can be determined according to the corresponding criterions. The gain coefficients K_α and K_β are the ratios of the inertia of adjacent motors, derived from the motor parameter values.⁹ The observer gain β_i and high-pass filtering coefficient T_h can be selected by C-parameter method.³⁶ Other parameters are mainly obtained by prior knowledge and trial-and-error. All the fixed parameters can thus be determined and fine-tuned in applications. Additionally, these parameters are selected off-line. Future studies can optimize them by using techniques such as extremum-seeking algorithm,³⁷ particle swarm optimization,⁹ and artificial bee colony algorithm³⁸ to reduce trial-and-error burden and get optimal parameters.

Stability analysis of fault-tolerant control system

The stability of the multi-motor synchronous system is analyzed based on Lyapunov stability theory. The Lyapunov candidate function is constructed by

V_{i} (t) = ψ_{i}^{2} (t) / 2 + {\tilde{f}}_{i}^{2} (t) / 2 + {\tilde{B}}_{i}^{2} (t) / (2 γ_{i}) \geq 0

(24)

for the i-th subsystem. Differentiating the macro-variable ψ_i(t) in (15) yields

{\overset{\cdot}{ψ}}_{i} (t) = k_{i, 1} \cdot {\overset{\cdot}{δ}}_{i} (t) + k_{i, 2} \cdot {\overset{\cdot}{ξ}}_{i} (t) + {\overset{\cdot}{x}}_{i, 2} (t) .

(25)

Putting the control law (21) and the state equation (10) into (25) can obtain

{\overset{\cdot}{ψ}}_{i} (t) = \frac{- ψ_{i} (t)}{T_{i}} + {\tilde{f}}_{i} (t) - sgn [ψ_{i} (t)] {\hat{B}}_{i} (t) .

(26)

Now differentiating the Lyapunov function (24), and substituting (26) as well as (12) into (24) gets

\begin{matrix} {\overset{\cdot}{V}}_{i} (t) = \frac{- ψ_{i}^{2} (t)}{T_{i}} + ψ_{i} (t) {\tilde{f}}_{i} (t) - | ψ_{i} (t) | {\hat{B}}_{i} (t) \\ - β_{i} {\tilde{f}}_{i}^{2} (t) + \frac{1}{γ_{i}} {\tilde{B}}_{i} (t) \cdot {\overset{\cdot}{\tilde{B}}}_{i} (t) . \end{matrix}

(27)

Due to the upper bound $B_{i} > | {\tilde{f}}_{i} (t) |$ , we have $B_{i} | ψ_{i} (t) | > | {\tilde{f}}_{i} (t) | \cdot | ψ_{i} (t) | \geq {\tilde{f}}_{i} (t) \cdot ψ_{i} (t)$ . Then substituting it and (20) into (27) produces

\begin{matrix} {\overset{\cdot}{V}}_{i} (t) < \frac{- ψ_{i}^{2} (t)}{T_{i}} + B_{i} | ψ_{i} (t) | - {\hat{B}}_{i} (t) | ψ_{i} (t) | \\ - β_{i} {\tilde{f}}_{i}^{2} (t) + \frac{1}{γ_{i}} {\tilde{B}}_{i} (t) \cdot {\overset{\cdot}{\tilde{B}}}_{i} (t) \\ = \frac{- ψ_{i}^{2} (t)}{T_{i}} - β_{i} {\tilde{f}}_{i}^{2} (t) < 0 . \end{matrix}

(28)

According to Lyapunov stability principle, the system is asymptotically stable, namely, the states of all subsystems will converge to the desired equilibrium points and manifolds.

Simulative and experimental verifications

A multi-motor synchronous system consisting of three BLDC servo motors is used to verify the proposed FTC strategy. The system was constructed by adjacent cross-coupling structure as shown in Figure 1 with n = 3. One motor is EC45 made by MAXON Corporation of Swiss, the other two are SMP6224B motors made by Shanghai MINDONG Corporation of China. Their nominal values are presented in Table 1. The proposed FTC strategy was compared with the sliding mode fault-tolerant control (SMFTC) method,¹⁷ and the adaptive coordination coefficient fault-tolerant (ACC) method³⁹ to demonstrate its superiority.

Table 1.

Nominal values of BLDC motor parameters.

Parameters	EC45 (MotorI)	SMP6224B (MotorII and MotorIII)
Rated power/W	250	188
Rated voltage/V	24	24
Rated current/A	7.51	11.4
Rated speed/rpm	4300	3000
Rated torque/Nm	0.331	0.6
Torque coefficient	0.0455	0.058
Back EMF coefficient	0.0455	0.058
Rotational inertia/kgcm²	0.209	1.98
Winding resistance/Ω	0.43	0.31
Winding inductance/μH	170	35
Number of pole-pairs	1	4
Damping coefficient/Nms	0.051	0.029

SMC is promising for FTC of multi-motor synchronous systems.^10,40 The control law for SMFTC method is designed based on the conventional SMC algorithm¹⁷ with the following sliding mode function

q_{i} (t) : = {\overset{\cdot}{e}}_{i} (t) + c_{1} \cdot e_{i} (t) + c_{2} \cdot ε_{i} (t),

(29)

and the control law

\begin{matrix} u_{c, i} (t) = \\ \frac{- 1}{b_{i}} {\begin{matrix} a_{i, 2} x_{i, 1} (t) + (c_{1} + c_{2} + a_{i, 1}) x_{i, 2} (t) - c_{1} k_{α} \cdot x_{i - 1, 2} (t) - {\overset{\cdot\cdot}{x}}_{d} (t) \\ - c_{1} {\overset{\cdot}{x}}_{d} (t) + ({\hat{B}}_{i} (t) - h_{i} \cdot ρ_{i}) sgn [q_{i} (t)] - h_{i} \cdot q_{i} (t) + {\hat{f}}_{i} (t) \end{matrix}} \end{matrix}

(30)

for the i-th subsystem, where error signal e_i(t) = x_d(t)-x_i,₁(t); h_i > 0, c₁>0, c₂ > 0, and ρ_i > 0 are the gain coefficients. The verifications were carried out respectively by MATLAB/Simulink simulations and experimental test in our laboratory.

Simulative validation

In the simulations, the command speed was set by step signal from 0 to 30 rad/s in the beginning and further to 50 rad/s at t = 0.06 s, that is, x_d(t) = 50 rad/s. The inverter fault in MotorI occurred at t = 0.1 s and resulted in the bus DC voltage dropped from the nominal 24 V to 14 V. At the same time, the load disturbance was imposed on MotorII with the rate 0.016 Nm/s. It can be omitted by comparing with the nominal torque 0.6 Nm, and so the parameters dT_L/dt = 0 and df_i/dt = 0. The overall system was operating with FTC function in the fault case.

By using the M-SPRT technique presented in Appendix 1,³⁰ the inverter fault was detected at t = 0.18 s. According to the proposed FTSC strategy, the fault detection information F₁ = 1 activated the FTC decision-making module to export the flag s₁ = 1, and further regulated the output u_c,1(t) to perform the FTC function. While by ACC method the coordinated coefficients of the coupling synchronization control were manipulated to achieve the FTC objective. For SMFTC scheme, the motor control subsystems were manipulated by the control law (30) to tolerate the inverter fault.

With respect to the control parameters determination of three FTC schemes, they can be firstly designed off-line following their techniques and then be readily adjusted in applications. Specifically, the regular PID control was adopted for ACC method. The resulted PID parameters are listed in Table 2 by the traditional poles-assignment technique, and the synchronous compensation coefficients are K_α = K_β = 0.5. For the proposed FTSC method, the designed parameters are shown in Table 3, and T_l = 0.1, k_min = 1500, k_max = 4788, l = 900 in (23) for the adaptive coefficients k_i,₂(t) in the simulation. Parameters in control law (30) for SMFTC method were as follows: h_i = 0.004, ρ_i = 17, c₁ = 56 and c₂ = 238 for MotorI; h_i = 0.004, ρ_i = 10, c₁ = 5, and c₂ = 42 for MotorII and MotorIII. To enhance the real-time performance of the three-motor synchronous system in simulations, we implemented the control algorithms for the above three FTC strategies by employing difference and cumulative sum operations in place of calculus operations. Furthermore, differential equations were solved using the fourth-order Runge-Kutta method and the improved Euler formula. Thereby, complex calculus-based computations were reduced to conventional algebraic operations. Consequently, the computational efforts were primarily manifested as multiplication/division operations, along with some addition/subtraction operations on the control parameter signals. Statistical analysis of the simulations indicated that within each synchronous control cycle (set to 4e–5s), the main computational cost lay in multiplication/division operations, while addition/subtraction imposed a minor burden. Among the FTC strategies, the PID algorithm in the ACC method required the fewest multiplication/division operations (only 6); the SMC algorithm in the SMFTC method required 33; and the control variable computations in the FTSC method entailed 36.

Table 2.

PID parameters for ACC method.

Parameters	MotorI	MotorII and MotorIII
K _P	0.52	0.83
K _I	23.41	17.39
K _D	6.53	4.01

Table 3.

Parameters for FTSC method.

Parameters	MotorI	MotorII and MotorIII
β_i	40	30
Kα	0.5	0.5
Kβ	0.5	0.5
T _h	0.004	0.004
k _i,₁	2300	1600
T _i	0.008	0.008
γ_i	120	90

Figures 4 and 5 show the speed response and synchronous error curves of three motor control subsystems. In contrast, the superior synchronization was achieved by FTSC method in the FTC stage. It is obvious from Figure 5 that the maximum value 0.46 rad/s by FTSC method was better than both the max 1.59 rad/s by ACC method and 0.54 rad/s by SMFTC method. The errors by SMFTC generally fluctuated more than those by FTSC, although they were significantly better than those by the ACC method. The magnitudes of tracking errors are also illustrated in Figure 6. It is found that all the tracking errors were increasing notably during the FTC process, while FTSC and SMFTC methods responded faster than ACC method. They exhibited the comparable tracking performance overall. It is noted by observing the curves in Figures 5 and 6 that compared with ACC method, the other two methods enforced the more control efforts on the normal motors through the coupling structure to maintain preferentially the synchronization in the fault case, while the synchronous errors by FTSC method were generally smaller than those by SMFTC scheme, implying the superiority of our proposed method. Figure 7 shows the adaptive weighting coefficient k_i,₂(t) of FTSC method. It is reasonable from the previous analysis that k_1,2(t) increased with priority to suppress the worsening of synchronous error in MotorI when the fault happened. Subsequently, k_2,2(t) and k_3,2(t) increased simultaneously for MotorII and MotorIII through the coupling structure. By such a way the FTSC control law harmonized the operation of three motors to improve the synchronous precision and the stability during the fault tolerance process.

Figure 4.

Speed response curves in the inverter fault. (x_d(t): command speed; x_i,1(t): the i-th motor speed (i = 1,2,3).): (a) by ACC method, (b) by SMFTC method, and (c) by FTSC method.

Figure 5.

Magnitudes of synchronous errors in the inverter fault. (|ε_1,2(t)|: between MotorI and MotorII; |ε_2,3(t)|: between MotorII and MotorIII; |ε_3,1(t)|: between MotorIII and MotorI.).

Figure 6.

Magnitudes of tracking errors in the inverter fault. (|δ₁(t)|: MotorI; |δ₂(t)|: MotorII; |δ₃(t)|: MotorIII.).

Figure 7.

Adaptive weighting coefficient k_i,2(t). (i = 1,2,3).

Experimental test

The experimental system was established abiding by adjacent cross-coupling structure. Its physical experimental platform and the schematic diagram of each motor control subsystem are shown in Figures 8 and 9, respectively. The controller of each motor subsystem was realized by DSP chip TMS320F28335. The drive module DRV8301 operated with inverter. The sampler was designed by high-precision resistor RX70 to get the inverter voltage for fault detection. The motor speed was measured by incremental encoder TS6014N135. CAN Bus and communication transceiver SN65HVD230CAN were used for data transmission among the motor control subsystems. Hall transducer was installed for detecting the armature current. Slider PL60 with nominal torque 11 Nm was used as the gradual load disturbance during the experiments. FTC control algorithms were realized on the μcos-II real-time OS by C language programing. The desired FTC indices were the maximum synchronous error less than 0.8 rad/s in the fault case and 0.5 rad/s in the steady state, the maximum synchronous error in the transient stage less than 3 rad/s, and the time duration no more than 0.5 s for fault detection.

Figure 8.

Experimental platform of triple BLDC motor synchronous control system: (a) control and drive modules and (b) experimental platform.

Figure 9.

Schematic of motor control subsystem (i = 1,2,3).

The system was running with the command speed 50 rad/s and the fixed load torque 4 Nm exerted on the reducer plus gradual variation along the sliderail motion for each motor subsystem. In comparison, four control methods, the synchronous method without FTC, the ACC method, the SMFTC method and the proposed FTSC method, were applied in the test. For the first two methods, control parameters K_P = 0.31 and K_I = 15.11 for MotorI, K_P = 0.19 and K_I = 12.15 for MotorII and MotorIII were designed for the optimal PI control, and K_α = 0.5, K_β = 0.5 for synchronous compensation among motors. For SMFTC method, parameters in control law (30) were h_i = 0.03, ρ_i = 19, c₁ = 62 and c₂ = 212 for MotorI; h_i = 0.021, ρ_i = 10, c₁ = 5 and c₂ = 27 for MotorII and MotorIII. The designed parameters are listed in Table 4 for FTSC method. The real-time performance of the above FTC strategies was constrained by the hardware performance of the experimental system, the CAN communication speed, the multi-tasking management of the μcos-II system, and the C programing techniques. To improve the computational efficiency of the control algorithms, we employed the DSP chip (TMS320F28335) running at a high clock frequency of 150 MHz and adopted the incremental PID algorithm. In programing the control algorithms, all calculus-based operations were also converted into algebraic operations, with multiplication and division being the most time-consuming (each requiring 48 clock cycles). Considering the time required for CAN communication and for parameter conversion/estimation of motion signals, the synchronous control period was set to 0.004 s in the experiments to ensure reliable synchronization of the three-motor system. Based on the experimental measurements, the execution time within each synchronous control period were as follows: less than 0.0012 s for the PID algorithm, less than 0.00304 s for the SMFTC algorithm, and less than 0.00348 s for the proposed FTSC algorithm. The inverter fault was injected into MotorI at t = 6.16 s, which led the bus DC voltage decreased from 24 to 15.2 V (see Figure 10). It was perceived at 6.31 s by the M-SPRT technique and spent 0.15s to get the fault information. Figure 11 shows the speed response curves by four methods. It is evident that the synchronous errors were greater than 4 rad/s by the synchronous method without FTC and beyond the desired synchronization in the fault case. It can also be found that three FTC methods cannot eliminate the tracking errors, but compensate coordinately the speed variations to maintain the high synchronous precision in priority. It is very important in the actual processes such as the satellite antenna pointing operation and the delivery by space manipulators.

Table 4.

Parameters for FTSC method in the test.

Parameters	MotorI	MotorII and MotorIII
β_i	180	150
Kα	0.5	0.5
Kβ	0.5	0.5
T _h	0.004	0.004
k _i,₁	2992	3282
T _i	0.008	0.008
γ_i	120	90
l	700	700
k _min	500	600
k _max	4788	5252

Figure 10.

Bus DC voltage curve in the inverter of MotorI.

Figure 11.

Speed curves of three motors in the inverter fault case. (dashed line: command x_d(t).).

For the sake of comparing the performance among three FTC methods, the synchronous errors are shown in Figure 12 and Table 5. It is obvious that the synchronous errors by FTSC method were generally the smallest among three methods, while SMFTC method yielded the errors higher than FTSC method. The maximum errors were far less than the required value 0.8 rad/s by the proposed FTSC method, which kept the superior synchronization. Additionally, the maximum phase current of windings in the faulty MotorI by FTSC method was 4.25 A, far less than the nominal value 7.51 A during the FTC operation. It implied that the proposed FTC method also ensured the safety of the faulty system.

Figure 12.

Synchronous errors in the inverter fault case. (|ε_1,2|: between MotorI and MotorII; |ε_2,3|: between MotorII and MotorIII; |ε_3,1|: between MotorIII and MotorI.).

Table 5.

Maximum synchronous errors during FTC operation.

Synchronous errors	ACC method	SMFTC method	FTSC method
Error between MotorI and MotorII	0.55 rad/s	0.45 rad/s	0.22 rad/s
Error between MotorII and MotorIII	0.08 rad/s	0.19 rad/s	0.09 rad/s
Error between MotorIII and MotorI	0.56 rad/s	0.42 rad/s	0.22 rad/s

Conclusions

A fault-tolerant speed synchronous control strategy based on SC theory has been developed to tolerate the possible power inverter faults for the multi-motor systems with adjacent cross-coupling structure. By virtue of the coupling structure, only the fault detector and the SC-based coordinated FTC unit are attached to each motor subsystem. In the proposed SC-based coordinated FTC design, the manifold is devised by combining the synchronous error, the tracking error, and the angular acceleration. The corresponding weighting coefficients and control parameters are also determined by giving priority to the synchronization. The FTC system stability is further proven based on Lyapunov theory. Consequently, the resulted FTC method can timely adjust the manifold to maintain the desired synchronization in the inverter fault case. It can be readily extended to the position synchronization control and other coupled-based structures, which is one of the future topics. In addition, some control parameters in the strategy need to be tuned by prior knowledge. The process is cumbersome and techniques such as extremum-seeking algorithm³⁷ are worth investigating in future study to reduce the difficulty and improve the performance.

Footnotes

Appendix 1

The main procedures of fault detection by M-SPRT technique algorithm in Li³⁰ is presented here, which was used in this study.

(31)

{\begin{matrix} Null hypothesis H_{0} : u_{i}^{f} (t) ~ N (μ_{0}, σ_{0}^{2}) \Rightarrow fault free; \\ Alternative hypothesis H_{1} : u_{i}^{f} (t) ~ N (μ_{1}, σ_{0}^{2}) \Rightarrow fault occurred . \end{matrix}

(32)

L_{j} (k) = L_{j} (k - 1) + \frac{[(μ_{0} - μ_{j}) \cdot (μ_{0} + μ_{j} - 2 u_{i}^{f} (k))]}{2 σ_{0}^{2}},

where μ_j is the mean value of the j-th alternative hypothesis, i = 1, 2, …, n_A, and n_A is the number of alternative hypothesis. μ_j is usually preset according to the failure severity of inverter. In the experimental study, we set n_A = 3 corresponding to the minor, middle and severe levels of the inverter failures, accordingly by 3%, 5%, and 10% of the nominal inverter voltage 15 V. Hence, the value of μ_j is set by: μ₁ = SPRT1 = 0.45, μ₂ = SPRT2 = 0.75, μ₃ = SPRT3 = 1.5. Then L_j(k) can be obtained by (A2) and the residual signal u_i^f(k).

(33)

t h_{0} = \ln (\frac{ρ}{(1 - ϕ)}), t h_{1} = \ln (\frac{(1 - ρ)}{ϕ})

can be determined by Neyman-Pearson criterion. In our experimental study, the FAR and FNR are set by 2%, and th₀ = −3.9, th₁ = 3.9.

(34)

L_{j} (k) \to {\begin{matrix} If any L_{j} (k) \geq t h_{1} \Rightarrow fault occurred; \\ If all L_{j} (k) \leq t h_{0} \Rightarrow no fault; \\ otherwise \Rightarrow keep the previous result until next k . \end{matrix}

ORCID iD

Dengfeng Zhang

Ethical consideration

There is no ethical consideration because this work did not involve humans and animals. Ethics approval was not required for this research.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number 62333010, 61972398, 61374133].

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Robust coordinated fault-tolerant control for aerospace multi-motor synchronous drive systems against inverter fault

Abstract

Keywords

Introduction

Problem formulation

Adjacent cross-coupling synchronization structure

Inverter fault model of BLDC motor and problem statement

Coordinated fault-tolerant control based on synergetic control theory

Coordinated synchronous FTC design

Control parameters of the FTC law

Stability analysis of fault-tolerant control system

Simulative and experimental verifications

Simulative validation

Experimental test

Conclusions

Footnotes

Appendix 1

ORCID iD

Ethical consideration

Consent to participate

Consent for publication

Funding

Declaration of conflicting interests

Data availability statement

References