Adaptive sliding mode backstepping control - based maximum torque per ampere control of permanent magnet-assisted synchronous reluctance motor via nonlinear disturbance observer

Abstract

Combined with two approaches of sliding mode control and backstepping control, an adaptive sliding mode–based backstepping control scenario on the basis of nonlinear disturbance observer is proposed to complete maximum torque per ampere control for permanent magnet–assisted synchronous reluctance motor. The constraint relation of permanent magnet–assisted synchronous reluctance motor under maximum torque per ampere control is built, and the design of the controller is elaborated in detail. The uncertainties of modeling errors considering unmatched items are estimated through a presented nonlinear disturbance observer. The adaptive law reflecting the modeling error of the system is constructed. Globally asymptotic stability and convergence of the tracking error for the system are validated through Lyapunov stability criterion. Simulation and experimental results illustrate that external disturbances and uncertainties are observed correctly by nonlinear disturbance observer; the close-loop system controlled by the proposed controller can track the references rapidly and precisely, and the designed controller has a good robust ability.

Keywords

Backstepping control maximum torque per ampere control permanent magnet–assisted synchronous reluctance motor sliding mode control nonlinear disturbance observer

Introduction

In the past decades, direct current (DC) motor drive has been gradually replaced by alternating current (AC) motor drive. Nowadays, AC motor drive occupies a dominant position in a large amount of speed control applications. There into, squirrel cage induction motors have been extensively used in many AC drive systems due to its low cost and simple structure. However, its inherently low efficiency and difficult to control makes the utilization of induction motor in high-precision control field not so ideal. With the popularization of permanent magnet materials, permanent magnet synchronous motor (PMSM) catches the attention of the public with its high efficiency, simple structure, and high torque–inertia ratio. Nevertheless, high cost of magnetic materials and sensitivity of magnetic materials to temperature are hindering wide application of such motor to a certain extent. In addition, synchronous reluctance motor (SRM) developed in early times has almost been ignored by current researchers in AC motor drives. One of the great advantages for this type of motor is that through designing its d axis and q axis reactance ratio reasonably, which is also called as changing the saliency ratio, it can achieve speed control of high efficiency and acquire high torque and wide constant power. Therefore, an appealing point is to combine PMSM with SRM to develop a new type of motor, which is permanent magnet–assisted synchronous reluctance motor (PMa-SynRM).

As a combination of two kinds of motors, PMa-SynRM possesses many special advantages of broad speed range of flux-weakening control, good inverter utilization, high efficiency, and natural synchronization, which receive more and more attentions. Due to vital importance of saliency ratio for PMa-SynRM, current research primarily concentrates on optimizing motor body and designing new motor structure. Lohninger et al.¹ propose to use ferrite magnets to reduce manufacturing cost, and they also points out that the placement of magnets in stator is only to enhance the torque efficiency. Niazi et al.² designed an innovative motor structure with permanent magnets laid up in rotor core to acquire low-cost and high average electromagnetic torque. For achieving extensive constant-power operation and reducing torque ripple, d axis flux linkage is designed to approach to zero, and the size of permanent magnet is altered by Kim et al.³

Moreover, the control technology for PMa-SynRM is an issue of great significance but researched relatively inadequately. Due to strict synchronization requirement of speed and power frequency for synchronous motor, its speed adjustment is achieved through variable voltage and variable frequency. There are three frequently used control methods: U/f control, direct torque control (DTC), and vector control (VC). As a scalar control method, U/f control is easy to implement and has a low cost. However, poor dynamic performance, slow torque response, and low motor torque utilization restrict its promotion.⁴ On the basis of torque and flux errors between reference values and observed values, DTC manages switching states of inverter directly through real-time feedback control. Therefore, the merit of DTC is responding to the input commands rapidly. But large on-line computations lead to a certain delay, and impulse current and torque ripple worsen control performance during startup and load variation.⁵ VC is known as high-accuracy, high-dynamic performance, and strong speed stability.⁶ For VC, in comparison with i_d = 0 control, maximum torque per ampere (MTPA) enables PMa-SynRM to output a larger low-speed torque and reduces the capacity of inverter.

Actually, in contrast with conventional PMSM, one of the specific features for PMa-SynRM is that its reluctance torque is supplemented by a permanent magnet torque, which increases its output torque and improves power factor. For sufficient utilization of the two torques, MTPA control is favorable for motor operation, which is a hot spot indeed and state-of-the-art technology.^7,8 The key of MTPA control is to distribute d axis and q axis currents reasonably. Due to complicated nonlinear relationship between d axis and q axis currents, look-up table method is frequently introduced to improve the real-time ability.⁹ Acquiring reference currents through off-line look-up table method is convenient for engineering application, but it requires massive storage space, and partly off-line look-up table method is proposed.¹⁰ Anyways, look-up table method increases the workload of off-line test and is inevitably to occupy some storage space. Hence, some improved algorithms are applied in this area. Finite element method (FEM) as a popular method is used to optimize the currents¹¹ and considers the effect of saturation.¹² For various types of PMa-SynRM, optimization control through FEM needs considerably complex modeling and analysis in spite of simpleness and effectiveness.¹³ Furthermore, due to the parameters of PMa-SynRM such as stator resistance, flux linkage vary with temperature.¹⁴ Niazi et al.¹⁵ proposes a parameter estimation based robust MTPA control strategy via the technology of multiple reference frame, and sliding mode control (SMC) is utilized to construct robust MTPA controller by Chakali et al.¹⁶ and Foo and Rahman.¹⁷

Backstepping control (BC) is a novel and valid control approach suitable for nonlinear system.^18–20 Through introducing virtual controllers and back-step design, an initial high-dimension system is converted into multiple low-dimension subsystems. For MTPA, BC is successfully applied to regulate the speed of PMSM.^21,22 However, the main defect of conventional BC method is that the regression matrices increase the complexity of control under the assumption of linear parameters.²³ The accurate computation of derivatives in virtual variables may also induce “calculation expansion” or “calculation explosion,” and the singularity and over-parameterization are difficult to avoid. So some improved approaches combining of BC and other design methods are put forward.^24,25 SMC is an approach with the merits of simpleness, robustness, and reliability, and is particularly suitable for controlling the uncertain system. For an uncertain system with external disturbances, SMC can guarantee the robustness of the whole system.²⁶ But the presence of chattering limits its application. Combination of BC with SMC can simplify the design of controller, avoid the defect of computation explosion in conventional BC and inhibit the mismatching uncertainties. An algorithm composited with adaptive variable structure control and BC is proposed by Song et al.²⁷ The variable structure based BC has no sliding mode. So it does not solve the problem of calculation explosion, and its realization is relatively difficult. A careful design guidance laws integrated with SMC and fuzzy methods are illustrated by Chen et al.²⁸ In reality, the construction of a fuzzy logic system greatly depends on expert knowledge, and the updating of the weights requires spending a lot of learning time. The technology of nonlinear disturbance observer (NDO) enables to eliminate the uncertainties of external disturbances and unmodeled dynamics and is used in many actual systems.^29,30 An adaptive backstepping-based MTPA control of PMa-SynRM is proposed to deal with fully uncertain parameters in Yu et al.³¹ Adaptive laws are established to determine these uncertain parameters. As mentioned above, a certain amount of computations are a challenging issue in implementation. The main contribution of the article, as an extension of Niazi and Toliyat,³² is to relax the adaptive laws reflecting the uncertainties of fully parameters and reduce the computation. Meanwhile, the uncertainties of fully parameters by Yu et al.³¹ are extended to a composite unknown uncertainties including modeling errors, parameter variations and external disturbances in this article. As in addition to Yu et al.,³¹ employment of BC into PMa-SynRM is seldom found to the extent of our knowledge. In allusion to MTPA control of PMa-SynRM below base speed and considering model parameter uncertainties and external disturbances with mismatching items, a composite NDO-based adaptive SMC BC algorithm is proposed by integrating the theories of adaptive BC and SMC. Simultaneously, NDO is designed to compensate for the effects acted by parameter uncertainties and external disturbances. The adaptive law designed in controller construction process furthermore fullfils the estimation of uncertainties and enables the designed controller to rid of the uncertainties’ dependence of unknown upper bound.

The remaining of the article is organized as follows. The constraint relation of PMa-SynRM under MTPA control principle for PMa-SynRM is presented in section “Constraint relation of currents in PMa-SynRM under MTPA control principle.” In section “Adaptive SMC-based backstepping controller design,” an adaptive SMC backstepping controller design based on NDO is proposed, and the stability of the control scheme is testified through Laypunov stability criterion. Section “Experiment and analysis” conducts the simulations and hardware experiments to validate the effectiveness and accuracy of the proposed algorithm. Ultimately, the conclusions are summarized in section “Conclusion.”

Constraint relation of currents in PMa-SynRM under MTPA control principle

In d–q axis synchronous rotating coordinate system, the mathematical model of PMa-SynRM with permanent magnets in q axis can be described as³²

{\begin{matrix} {\overset{\cdot}{θ}}_{r} = ω_{r} \\ {\overset{\cdot}{ω}}_{r} = \frac{1}{J_{m}} (T_{e} - B_{m} ω_{r} - T_{L}) + F_{ω} \\ {\overset{\cdot}{i}}_{d} = - \frac{R_{s}}{L_{d}} i_{d} + \frac{n_{p} L_{q}}{L_{d}} ω_{r} i_{q} - \frac{n_{p} ψ_{f}}{L_{d}} ω_{r} + \frac{1}{L_{d}} u_{d} + F_{d} \\ {\overset{\cdot}{i}}_{q} = - \frac{R_{s}}{L_{q}} i_{q} - \frac{n_{p} L_{d}}{L_{q}} ω_{r} i_{d} + \frac{1}{L_{q}} u_{q} + F_{q} \end{matrix}

(1)

where θ_r and ω_r are the mechanical rotor angle and speed, respectively. J_m is the moment of inertia, B_m is the viscous damping coefficient, T_e is the electromagnetic torque, T_L is the load torque, i_d and i_q are the d axis and q axis stator currents, u_d and u_q are the d axis and q axis stator voltages, L_d and L_q are the d axis and q axis inductances, R_s is the stator resistance, n_p is the number of pole pairs, and ψ_f is the flux linkage produced by permanent magnets. F_ω, F_d, and F_q depict unknown uncertainties indicating modeling errors in ω_r, i_d and i_q, respectively.

The electromagnetic torque T_e can be written as

T_{e} = \frac{3}{2} \frac{n_{p}}{2} [ψ_{f} i_{d} + (L_{d} - L_{q}) i_{d} i_{q}]

(2)

Utilization of MTPA control is expected to obtain maximum electromagnetic torque through stator current as low as possible, which can be converted into a solution of extreme problem given by

{\begin{matrix} min : i_{s} = \sqrt{i_{d}^{2} + i_{q}^{2}} \\ condition : T_{e} = \frac{3}{2} \frac{n_{p}}{2} [ψ_{f} i_{d} + (L_{d} - L_{q}) i_{d} i_{q}] \end{matrix}

(3)

where i_s is the stator current.

Through introduction of a Lagrangian multiplier λ, the auxiliary function is constructed as

F = \sqrt{i_{d}^{2} + i_{q}^{2}} + λ {T_{e} - \frac{3}{2} \frac{n_{p}}{2} [ψ_{f} i_{d} + (L_{d} - L_{q}) i_{d} i_{q}]}

(4)

Using partial derivatives of i_d, i_q, and λ in equation (4), respectively, we can obtain

{\begin{matrix} \frac{\partial F}{\partial i_{d}} = \frac{i_{d}}{\sqrt{i_{d}^{2} + i_{q}^{2}}} - \frac{3}{4} λ n_{p} [ψ_{f} + (L_{d} - L_{q}) i_{q}] \\ \frac{\partial F}{\partial i_{q}} = \frac{i_{q}}{\sqrt{i_{d}^{2} + i_{q}^{2}}} - \frac{3}{4} λ n_{p} (L_{d} - L_{q}) i_{d} \\ \frac{\partial F}{\partial λ} = T_{e} - \frac{3}{2} \frac{n_{p}}{2} [ψ_{f} i_{d} + (L_{d} - L_{q}) i_{d} i_{q}] \end{matrix}

(5)

Let every function in equation (5) equal to zero, the relationship of d–q axis currents in MTPA control can be gained as

i_{q} = - \frac{ψ_{f}}{2 (L_{d} - L_{q})} + \sqrt{i_{d}^{2} + {(\frac{ψ_{f}}{2 (L_{d} - L_{q})})}^{2}}

(6)

Equation (6) indicates that if i_d is known, q axis reference current i_qref can be determined.

Adaptive SMC-based backstepping controller design

NDO design

System (1) is rewritten as the form of state space

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{F} = f (x) + bu + F \end{matrix}

(7)

where $x_{1} = θ_{r}$ , $x_{2} = ω_{r}$ , $x_{F} = {[x_{2}, x_{3}, x_{4}]}^{T}$ , $x = {[x_{1}, x_{2}, x_{3}, x_{4}]}^{T}$ , $f (x) = [\begin{matrix} \frac{1}{J_{m}} (T_{e} - B_{m} x_{2} - T_{L}) \\ - \frac{R_{s}}{L_{d}} x_{3} + \frac{n_{p} L_{q}}{L_{d}} x_{2} x_{4} - \frac{n_{p} ψ_{f}}{L_{d}} x_{2} \\ - \frac{R_{s}}{L_{q}} x_{4} - \frac{n_{p} L_{d}}{L_{q}} x_{2} x_{3} \end{matrix}]$ , $b = [\begin{matrix} 1 / L_{d} & 0 \\ 0 & 1 / L_{q} \end{matrix}]$ , $F = {[F_{ω}, F_{d}, F_{q}]}^{T}$ , and $u = {[u_{d}, u_{q}]}^{T}$ .

Let

\tilde{F} = \hat{F} - F

(8)

where $\tilde{F} = {[{\tilde{F}}_{ω}, {\tilde{F}}_{d}, {\tilde{F}}_{q}]}^{T}$ ; $\hat{F} = {[{\hat{F}}_{ω}, {\hat{F}}_{d}, {\hat{F}}_{q}]}^{T}$ ; ${\hat{F}}_{ω}$ , ${\hat{F}}_{d}$ , and ${\hat{F}}_{q}$ are the estimations of $F_{ω}$ , $F_{d}$ , and $F_{q}$ ; ${\tilde{F}}_{ω}$ ${\tilde{F}}_{d}$ , and ${\tilde{F}}_{q}$ are the estimation errors of $F_{ω}$ , $F_{d}$ , and $F_{q}$ . In general, priori knowledge with respect to the first-order differential of F is difficult to acquire. Variation of disturbance F is assumed to be very slowly relative to dynamic property of disturbance observer, that is $\overset{\cdot}{F} = 0$ . By derivative of equation (8), we can get

\overset{\cdot}{\tilde{F}} = \overset{\cdot}{\hat{F}} - \overset{\cdot}{F} = \overset{\cdot}{\hat{F}}

(9)

NDO is designed as

{\begin{matrix} \hat{F} = e_{F} + N (x) \\ {\overset{\cdot}{e}}_{F} = - L e_{F} + L (- N (x) - f (x) - bu) \end{matrix}

(10)

where $e_{F}$ is the internal variable in NDO, $\hat{F}$ is the output of NDO, $N (x)$ is the nonlinear function pending design, L(x) is the gain matrix. Let $L {\overset{\cdot}{x}}_{F} = d N (x) / dt$ , equation (9) can be rearranged as

\overset{\cdot}{\tilde{F}} = \overset{\cdot}{\hat{F}} = {\overset{\cdot}{e}}_{F} + \overset{\cdot}{N} (x)

(11)

By substituting equation (10) into equation (11), we can obtain

\overset{\cdot}{\tilde{F}} = - L \tilde{F}

(12)

Equation (12) implies that if a suitable $L (x) > 0$ is selected, the observation error will converge to zero exponentially. For convenience, the gain matrix L is selected as a constant matrix a ( a > 0) and the solution of equation (12) is

\tilde{F} (t) = \tilde{F} (0) e^{- at}

(13)

where t is the time, $\tilde{F} (0)$ is the initial values of $\tilde{F} (t)$ . Equation (13) displays that observation error of NDO will converge to zero exponentially.

Let $u = u_{s} - u_{F}$ and $u_{F} = \hat{F} / b$ , b is a constant vector, b > 0, we can obtain

\begin{matrix} {\overset{\cdot}{x}}_{F} = f (x) + bu + F = f (x) \\ + b (u_{s} - u_{F}) + F = f (x) + b u_{s} - \tilde{F} \end{matrix}

(14)

Equation (14) shows that the uncertainties are decreased by NDO. System (7) can be changed into

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{F} = f (x) + b u_{s} + Δ \end{matrix}

(15)

where $Δ = - \tilde{F} = [Δ_{ω}, Δ_{d}, Δ_{q}]$

Proposed adaptive control

In terms of BC principle, the error variables e_ω, e_d, and e_q are defined as

{\begin{matrix} e_{ω} = ω_{ref} - ω_{r} \\ e_{d} = i_{dref} - i_{d} \\ e_{q} = i_{qref} - i_{q} \end{matrix}

(16)

where $ω_{ref}$ , $i_{dref}$ , and $i_{qref}$ are the references of ω_r, i_d, and i_q, respectively.

Step 1: By derivative of e_ω in equation (16)

\begin{matrix} {\overset{\cdot}{e}}_{ω} = {\overset{\cdot}{ω}}_{ref} - {\overset{\cdot}{ω}}_{r} = {\overset{\cdot}{ω}}_{ref} - \frac{T_{e}}{J_{m}} + \frac{B_{m} ω_{r}}{J_{m}} \\ + \frac{T_{L}}{J_{m}} - Δ_{ω} = {\overset{\cdot}{ω}}_{ref} - \frac{3 n_{p} [ψ_{f} i_{d} + (L_{d} - L_{q}) i_{d} i_{q}]}{4 J_{m}} \\ + \frac{B_{m} ω_{r}}{J_{m}} + \frac{T_{L}}{J_{m}} - Δ_{ω} \\ = {\overset{\cdot}{ω}}_{ref} + \frac{B_{m} ω_{r}}{J_{m}} + \frac{T_{L}}{J_{m}} - Δ_{ω} - \frac{3 n_{p}}{4 J_{m}} \\ [ψ_{f} i_{dref} + (L_{d} - L_{q}) i_{dref} i_{qref}] - \frac{3 n_{p}}{4 J_{m}} \\ [- ψ_{f} - (L_{d} - L_{q}) i_{qref}] e_{d} \\ + \frac{3 n_{p}}{4 J_{m}} \cdot (L_{d} - L_{q}) i_{dref} e_{q} - \frac{3 n_{p}}{4 J_{m}} (L_{d} - L_{q}) e_{d} e_{q} \end{matrix}

(17)

Let $B_{m} / J_{m} = θ_{1}$ , $T_{L} / J_{m} = θ_{2}$ , $ψ_{f} / J_{m} = θ_{3}$ , and $(L_{d} - L_{q}) / J_{m} = θ_{4}$ , equation (17) can be rewritten as

\begin{matrix} {\overset{\cdot}{e}}_{ω} = {\overset{\cdot}{ω}}_{ref} + θ_{1} ω_{r} + θ_{2} - Δ_{ω} - \frac{3 n_{p}}{4} \\ [θ_{3} i_{dref} + θ_{4} i_{dref} i_{qref}] + \frac{3 n_{p}}{4} [θ_{3} + θ_{4} i_{qref}] e_{d} \\ + \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{q} - \frac{3 n_{p}}{4} θ_{4} e_{d} e_{q} \end{matrix}

(18)

Then, i_dref can be chosen as

\begin{matrix} i_{dref} = \frac{4}{3 n_{p}} \\ \frac{({\overset{\cdot}{ω}}_{ref} + θ_{1} ω_{r} + θ_{2} + k_{ω} e_{ω} - {\hat{Δ}}_{ω} + (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})))}{θ_{3} + θ_{4} i_{qref}} \end{matrix}

(19)

where ${\hat{Δ}}_{ω}$ is the estimation of $Δ_{ω}$ , β_ω and ε_ω are positive constants, s_ω is the sliding mode surface for ω_r, $s_{ω} = α_{ω} e_{ω}$ , α_ω is a positive constant.

By substituting equation (19) into equation (18)

\begin{matrix} {\overset{\cdot}{e}}_{ω} = {\overset{\cdot}{ω}}_{ref} + θ_{1} ω_{r} + θ_{2} - Δ_{ω} - \frac{3 n_{p}}{4} [θ_{3} + θ_{4} i_{qref}] (\frac{4}{3 n_{p}} \frac{({\overset{\cdot}{ω}}_{ref} + θ_{1} ω_{r} + θ_{2} + k_{ω} e_{ω} - {\hat{Δ}}_{ω} + (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})))}{θ_{3} + θ_{4} i_{qref}}) \\ + \frac{3 n_{p}}{4} [θ_{3} + θ_{4} i_{qref}] e_{d} + \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{q} - \frac{3 n_{p}}{4} θ_{4} e_{d} e_{q} \\ = {\overset{\cdot}{ω}}_{ref} + θ_{1} ω_{r} + θ_{2} - Δ_{ω} - ({\overset{\cdot}{ω}}_{ref} + θ_{2} ω_{r} + θ_{3} + k_{ω} e_{ω} - {\hat{Δ}}_{ω} + (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω}))) \\ + \frac{3 n_{p}}{4} [θ_{3} + θ_{4} i_{qref}] e_{d} + \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{q} - \frac{3 n_{p}}{4} θ_{4} e_{d} e_{q} \\ = - k_{ω} e_{ω} + {\tilde{Δ}}_{ω} - (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) + \frac{3 n_{p}}{4} [θ_{3} + θ_{4} i_{qref}] e_{d} + \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{q} - \frac{3 n_{p}}{4} θ_{4} e_{d} e_{q} \end{matrix}

(20)

where ${\tilde{Δ}}_{ω}$ is the estimation error of $Δ_{ω}$ , ${\tilde{Δ}}_{ω} = {\hat{Δ}}_{ω} - Δ_{ω}$ .

Step 2: By derivative of i_d in equation (16)

\begin{matrix} {\overset{\cdot}{e}}_{d} = {\overset{\cdot}{i}}_{dref} - {\overset{\cdot}{i}}_{d} = {\overset{\cdot}{i}}_{dref} + \frac{R_{s}}{L_{d}} i_{d} - \frac{n_{p} L_{q}}{L_{d}} ω_{r} i_{q} \\ + \frac{n_{p} ψ_{f}}{L_{d}} ω_{r} - \frac{1}{L_{d}} u_{sd} - Δ_{d} \end{matrix}

(21)

By derivative of equation (19)

\begin{matrix} {\overset{\cdot}{i}}_{dref} = \frac{4}{3 n_{p}} \frac{({\overset{\cdot\cdot}{ω}}_{ref} + θ_{1} {\overset{\cdot}{ω}}_{r} + k_{ω} {\overset{\cdot}{e}}_{ω} - {\overset{\cdot}{\hat{Δ}}}_{ω} + β_{ω} {\overset{\cdot}{s}}_{ω})}{(θ_{3} + θ_{4} i_{qref})} - \frac{4}{3 n_{p}} \\ \frac{({\overset{\cdot}{ω}}_{ref} + θ_{1} ω_{r} + θ_{2} + k_{ω} e_{ω} - {\hat{Δ}}_{ω} + (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω}))) (θ_{5} {\overset{\cdot}{i}}_{qref})}{{(θ_{3} + θ_{4} i_{qref})}^{2}} \end{matrix}

(22)

Let $R_{s} / L_{d} = θ_{5}$ , $L_{q} / L_{d} = θ_{6}$ , $ψ_{f} / L_{d} = θ_{7}$ , and $1 / L_{d} = θ_{8}$ , equation (21) can be rearranged as

{\overset{\cdot}{e}}_{d} = {\overset{\cdot}{i}}_{dref} + θ_{5} i_{d} - n_{p} θ_{6} ω_{r} i_{q} + n_{p} θ_{7} ω_{r} - θ_{8} u_{sd} - Δ_{d}

(23)

The control input u_sd can be chosen as

\begin{matrix} u_{sd} = \frac{1}{θ_{8}} ({\overset{\cdot}{i}}_{dref} + θ_{5} i_{d} - θ_{6} n_{p} ω_{r} i_{q} + n_{p} θ_{7} ω_{r} + k_{d} e_{d} - {\hat{Δ}}_{d}) \\ + \frac{e_{ω}}{θ_{8}} (\frac{3 n_{p}}{4} (θ_{3} + θ_{4} i_{qref})) + \frac{1}{θ_{8}} (β_{d} s_{d} + ε_{d} sgn (s_{d})) \end{matrix}

(24)

where ${\hat{Δ}}_{d}$ is the estimation of $Δ_{d}$ , β_d and ε_d are positive constants, s_d is the sliding mode surface for i_d, $s_{d} = α_{d} e_{d}$ , α_d is a positive constant.

By substituting equation (24) into equation (23)

\begin{matrix} {\overset{\cdot}{e}}_{d} = - k_{d} e_{d} - \frac{3 n_{p} e_{ω}}{4} (θ_{3} + θ_{4} i_{qref}) + {\hat{Δ}}_{d} - Δ_{d} \\ = - k_{d} e_{d} - \frac{3 n_{p} e_{ω}}{4} (θ_{3} + θ_{4} i_{qref}) \\ + {\tilde{Δ}}_{d} - (β_{d} s_{d} + ε_{d} sgn (s_{d})) \end{matrix}

(25)

where ${\tilde{Δ}}_{d}$ is the estimation error of $Δ_{d}$ , ${\tilde{Δ}}_{d} = {\hat{Δ}}_{d} - Δ_{d}$ .

Step 3: By derivative of i_q in equation (16)

\begin{matrix} {\overset{\cdot}{e}}_{q} = {\overset{\cdot}{i}}_{qref} - {\overset{\cdot}{i}}_{q} = {\overset{\cdot}{i}}_{qref} \\ - (- \frac{R_{s}}{L_{q}} i_{q} - \frac{n_{p} L_{d}}{L_{q}} ω_{r} i_{d} + \frac{1}{L_{q}} u_{sq} + Δ_{q}) \end{matrix}

(26)

By derivative of equation (6)

{\overset{\cdot}{i}}_{q r e f} = i_{d} {\overset{\cdot}{i}}_{d} {i_{d}^{2} + {[\frac{ψ_{f}}{2 (L_{d} - L_{q})}]}^{2}}^{- \frac{1}{2}}

(27)

Let $R_{s} / L_{q} = θ_{9}$ , $L_{d} / L_{q} = θ_{10}$ , and $1 / L_{q} = θ_{11}$ , equation (26) can be rewritten as

{\overset{\cdot}{e}}_{q} = {\overset{\cdot}{i}}_{qref} + (θ_{9} i_{q} + n_{p} θ_{10} ω_{r} i_{d} - θ_{11} u_{sq}) - Δ_{q}

(28)

The control input u_sq can be selected as

\begin{matrix} u_{sq} = \frac{1}{θ_{11}} ({\overset{\cdot}{i}}_{qref} + θ_{9} i_{q} + n_{p} ω_{r} i_{d} θ_{10} + k_{q} e_{q} - {\hat{Δ}}_{q}) \\ + \frac{e_{ω}}{θ_{1 q}} \frac{3 n_{p}}{4} θ_{4} i_{dref} - \frac{1}{θ_{1 q}} \frac{3 n_{p}}{4} θ_{4} e_{d} e_{ω} \\ + \frac{1}{θ_{11}} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \end{matrix}

(29)

where ${\hat{Δ}}_{q}$ is the estimation of $Δ_{q}$ , β_q and ε_q are positive constants, s_q is the sliding mode surface for i_q, $s_{q} = α_{q} e_{q}$ , α_q is a positive constant.

By substituting equation (29) into equation (28)

\begin{matrix} {\overset{\cdot}{e}}_{q} = - k_{q} e_{q} - \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{ω} + \frac{3 n_{p}}{4} θ_{4} e_{d} e_{ω} + {\hat{Δ}}_{q} - Δ_{q} \\ = - k_{q} e_{q} - \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{ω} + \frac{3 n_{p}}{4} θ_{4} e_{d} e_{ω} \\ + {\tilde{Δ}}_{q} - (β_{q} s_{q} + ε_{q} sgn (s_{q})) \end{matrix}

(30)

where ${\tilde{Δ}}_{q}$ is the estimation error of $Δ_{q}$ , ${\tilde{Δ}}_{q} = {\hat{Δ}}_{q} - Δ_{q}$ .

To utilize the controller u_sd and u_sq described in equations (24) and (29), respectively, the upper bound of $Δ$ should be known, which is unlikely to accomplish in reality. Hence, an adaptive item $\hat{Δ}$ is added to estimate $Δ$ on-line. Assuming $Δ$ varies very slowly in comparison to its dynamic property, that is $\overset{\cdot}{\tilde{Δ}} = \overset{\cdot}{\hat{Δ}} - \overset{\cdot}{Δ} = \overset{\cdot}{\hat{Δ}}$ .

Step 4: Lyapunov function is chosen as

\begin{matrix} V_{1} = \frac{1}{2} e_{ω}^{2} + \frac{1}{2} e_{d}^{2} + \frac{1}{2} e_{q}^{2} + \frac{1}{2} s_{ω}^{2} + \frac{1}{2} s_{d}^{2} + \frac{1}{2} s_{q}^{2} \\ + \frac{1}{2 γ_{ω}} {\tilde{Δ}}_{ω}^{2} + \frac{1}{2 γ_{d}} {\tilde{Δ}}_{d}^{2} + \frac{1}{2 γ_{q}} {\tilde{Δ}}_{q}^{2} \end{matrix}

(31)

By derivative of V₁

\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{ω} {\overset{\cdot}{e}}_{ω} + e_{d} {\overset{\cdot}{e}}_{d} + e_{q} {\overset{\cdot}{e}}_{q} + s_{ω} {\overset{\cdot}{s}}_{ω} + s_{d} {\overset{\cdot}{s}}_{d} + s_{q} {\overset{\cdot}{s}}_{q} + \frac{1}{γ_{ω}} {\tilde{Δ}}_{ω} {\overset{\cdot}{\tilde{Δ}}}_{ω} + \frac{1}{γ_{d}} {\tilde{Δ}}_{d} {\overset{\cdot}{\tilde{Δ}}}_{d} + \frac{1}{γ_{q}} {\tilde{Δ}}_{q} {\overset{\cdot}{\tilde{Δ}}}_{q} \\ = e_{ω} {- k_{ω} e_{ω} + {\tilde{Δ}}_{ω} - (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) + \frac{3 n_{p}}{4} (θ_{3} + θ_{4} i_{qref}) e_{d} + \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{q} - \frac{3 n_{p}}{4} θ_{4} e_{d} e_{q}} \\ + e_{d} [- k_{d} e_{d} + {\tilde{Δ}}_{d} - (β_{d} s_{d} + ε_{d} sgn (s_{d})) - \frac{3 n_{p}}{4} (θ_{3} + θ_{4} i_{qref}) e_{ω}] \\ + e_{q} [- k_{q} e_{q} + {\tilde{Δ}}_{q} - (β_{q} s_{q} + ε_{q} sgn (s_{q})) - \frac{3 n_{p}}{4} θ_{4} i_{dref} e_{ω} + \frac{3 n_{p}}{4} θ_{4} e_{d} e_{ω}] \\ + s_{ω} {\overset{\cdot}{s}}_{ω} + s_{d} {\overset{\cdot}{s}}_{d} + s_{q} {\overset{\cdot}{s}}_{q} + \frac{1}{γ_{ω}} {\tilde{Δ}}_{ω} {\overset{\cdot}{\tilde{Δ}}}_{ω} + \frac{1}{γ_{d}} {\tilde{Δ}}_{d} {\overset{\cdot}{\tilde{Δ}}}_{d} + \frac{1}{γ_{q}} {\tilde{Δ}}_{q} {\overset{\cdot}{\tilde{Δ}}}_{q} \\ = - k_{ω} e_{ω}^{2} - k_{d} e_{d}^{2} - k_{q} e_{q}^{2} + e_{ω} {\tilde{Δ}}_{ω} + e_{d} {\tilde{Δ}}_{d} + e_{q} {\tilde{Δ}}_{q} \\ + s_{ω} {\overset{\cdot}{s}}_{ω} + s_{d} {\overset{\cdot}{s}}_{d} + s_{q} {\overset{\cdot}{s}}_{q} + \frac{1}{γ_{ω}} {\tilde{Δ}}_{ω} {\overset{\cdot}{\tilde{Δ}}}_{ω} + \frac{1}{γ_{d}} {\tilde{Δ}}_{d} {\overset{\cdot}{\tilde{Δ}}}_{d} + \frac{1}{γ_{q}} {\tilde{Δ}}_{q} {\overset{\cdot}{\tilde{Δ}}}_{q} \\ + e_{ω} (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) - e_{d} (β_{d} s_{d} + ε_{d} sgn (s_{d})) - e_{q} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \\ = - k_{ω} e_{ω}^{2} + e_{ω} {\tilde{Δ}}_{ω} + s_{ω} {\overset{\cdot}{s}}_{ω} + \frac{1}{γ_{ω}} {\tilde{Δ}}_{ω} {\overset{\cdot}{\tilde{Δ}}}_{ω} - e_{ω} (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) \\ - k_{d} e_{d}^{2} + e_{d} {\tilde{Δ}}_{d} + s_{d} {\overset{\cdot}{s}}_{d} + \frac{1}{γ_{d}} {\tilde{Δ}}_{d} {\overset{\cdot}{\tilde{Δ}}}_{d} - e_{d} (β_{d} s_{d} + ε_{d} sgn (s_{d})) \\ - k_{q} e_{q}^{2} + e_{q} {\tilde{Δ}}_{q} + s_{q} {\overset{\cdot}{s}}_{q} + \frac{1}{γ_{q}} {\tilde{Δ}}_{q} {\overset{\cdot}{\tilde{Δ}}}_{q} - e_{q} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \end{matrix}

(32)

The adaptive law of SMC is chosen as

{\begin{matrix} {\overset{\cdot}{s}}_{ω} = - β_{ω} s_{ω} - ε_{ω} sgn (s_{ω}) \\ {\overset{\cdot}{s}}_{d} = - β_{d} s_{d} - ε_{d} sgn (s_{d}) \\ {\overset{\cdot}{s}}_{q} = - β_{q} s_{q} - ε_{q} sgn (s_{q}) \end{matrix}

(33)

The adaptive law of $Δ$ is selected as

{\begin{matrix} {\overset{\cdot}{\hat{Δ}}}_{ω} = - e_{ω} γ_{ω} \\ {\overset{\cdot}{\hat{Δ}}}_{d} = - e_{d} γ_{d} \\ {\overset{\cdot}{\hat{Δ}}}_{q} = - e_{q} γ_{q} \end{matrix}

(34)

By substituting equations (33) and (34) into equation (32)

\begin{matrix} \overset{\cdot}{V} = - k_{ω} e_{ω}^{2} + e_{ω} {\tilde{Δ}}_{ω} + s_{ω} (- β_{ω} s_{ω} - ε_{ω} sgn (s_{ω})) \\ + \frac{1}{γ_{ω}} {\tilde{Δ}}_{ω} (- e_{ω} γ_{ω}) - e_{ω} (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) \\ - k_{d} e_{d}^{2} + e_{d} {\tilde{Δ}}_{d} + s_{d} (- β_{d} s_{d} - ε_{d} sgn (s_{d})) \\ + \frac{1}{γ_{d}} {\tilde{Δ}}_{d} (- e_{d} γ_{d}) - e_{d} (β_{d} s_{d} + ε_{d} sgn (s_{d})) \\ - k_{q} e_{q}^{2} + e_{q} {\tilde{Δ}}_{q} + s_{q} (- β_{q} s_{q} - ε_{q} sgn (s_{q})) \\ + \frac{1}{γ_{q}} {\tilde{Δ}}_{q} (- e_{q} γ_{q}) - e_{q} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \\ = - k_{ω} e_{ω}^{2} - k_{d} e_{d}^{2} - k_{q} e_{q}^{2} \\ + (s_{ω}) (- β_{ω} s_{ω} - ε_{ω} sgn (s_{ω})) - \frac{α_{ω} e_{ω}}{α_{ω}} (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) \\ + (s_{d}) (- β_{d} s_{d} - ε_{d} sgn (s_{d})) - \frac{α_{d} e_{d}}{α_{d}} (β_{d} s_{d} + ε_{d} sgn (s_{d})) \\ + (s_{q}) (- β_{q} s_{q} - ε_{q} sgn (s_{q})) - \frac{α_{q} e_{q}}{α_{q}} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \\ = - k_{ω} e_{ω}^{2} - k_{d} e_{d}^{2} - k_{q} e_{q}^{2} \\ + (s_{ω}) (- β_{ω} s_{ω} - ε_{ω} sgn (s_{ω})) - \frac{s_{ω}}{α_{ω}} (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) \\ + (s_{d}) (- β_{d} s_{d} - ε_{d} sgn (s_{d})) - \frac{s_{d}}{α_{d}} (β_{d} s_{d} + ε_{d} sgn (s_{d})) \\ + (s_{q}) (- β_{q} s_{q} - ε_{q} sgn (s_{q})) - \frac{s_{q}}{α_{q}} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \\ = - k_{ω} e_{ω}^{2} - k_{d} e_{d}^{2} - k_{q} e_{q}^{2} \\ + (s_{ω}) (- β_{ω} s_{ω} - ε_{ω} sgn (s_{ω})) - \frac{s_{ω}}{α_{ω}} (β_{ω} s_{ω} + ε_{ω} sgn (s_{ω})) \\ + (s_{d}) (- β_{d} s_{d} - ε_{d} sgn (s_{d})) - \frac{s_{d}}{α_{d}} (β_{d} s_{d} + ε_{d} sgn (s_{d})) \\ + (s_{q}) (- β_{q} s_{q} - ε_{q} sgn (s_{q})) - \frac{s_{q}}{α_{q}} (β_{q} s_{q} + ε_{q} sgn (s_{q})) \\ = - k_{ω} e_{ω}^{2} - k_{d} e_{d}^{2} - k_{q} e_{q}^{2} - (β_{ω} + \frac{β_{ω}}{α_{ω}}) s_{ω}^{2} \\ - (ε_{ω} + \frac{ε_{ω}}{α_{ω}}) | s_{ω} | \\ - (β_{d} + \frac{β_{d}}{α_{d}}) s_{d}^{2} - (ε_{d} + \frac{ε_{d}}{α_{d}}) | s_{ω} | \\ - (β_{q} + \frac{β_{q}}{α_{q}}) s_{q}^{2} - (ε_{q} + \frac{ε_{q}}{α_{q}}) | s_{q} | \\ < 0 \end{matrix}

(35)

Equation (35) indicates that Laypunov function is negative definition and the system is globally asymptotic stable. Hence, the system errors shown in equation (16) can converge to zero in a finite time. For elimination of the chattering problem in designed controller, signum function sgn in equations (24) and (29) are replaced by saturation function sat. The ultimate MTPA controller based on adaptive SMC backstepping is shown in equation (36) with NDO in equation (10) and adaptive law in equation (34)

{\begin{matrix} u_{d} = u_{sd} - L_{d} {\hat{F}}_{d} \\ u_{q} = u_{sq} - L_{q} {\hat{F}}_{q} \end{matrix}

(36)

Moreover, it is necessary to consider real limitation on domain of control signal in design phase. For PMa-SynRM, the outputs of speed loop and current loop cannot be arbitraryly large. Due to the output of the speed is the reference of d axis current i_dref, the current cannot exceed the limit of the motor. In other words, there are upper and lower limits for the output of speed loop. It supposes that the upper and lower limits for the motor are i_lim₊ and i_lim_– individually. The anti-windup approach is used after the output of speed loop, which is described as

i_{d} = {\begin{matrix} i_{lim +}, i_{d} \geq i_{lim +} \\ i_{d}, i_{lim -} < i_{d} < i_{lim +} \\ i_{lim -}, i_{d} \leq i_{lim -} \end{matrix}

(37)

The control schematic diagram is shown in Figure 1. The control problem in the article can be described as: for a PMa-SynRM with comprehensive interferences produced by the uncertainties of modeling errors, NDO in equation (10) is designed to estimate the comprehensive interferences; speed backstepping controller in equation (19), and current backstepping controller in equation (36) are proposed based on MTPA control in equation (6) and adaptive law in equation (34) to regulate the speed and d–q axis currents to track respective references rapidly.

Figure 1.

The schematic diagram of the proposed control scheme: (a) the overall configuration diagram and (b) the concrete nonlinear backstepping controller.

Selection of controller parameters through differential evolution algorithm

According to equations (24), (29), (33), and (34), there are 13 parameters that should be chosen in the controller, which are k_ω, k_d, k_q, β_ω, β_d, β_q, ε_ω, ε_d, ε_q, γ_ω, γ_d, γ_q, and L . The trial and error–based method frequently used is not adequate. Hence, differential evolution (DE) algorithm is introduced to optimize the control parameters.

The design procedure of the parameter optimization is displayed as follows:

Step 1: Initialization of the population. For a population in the study, the individual is defined as x =[k_ω, k_d, k_q, β_ω, β_d, β_q, ε_ω, ε_d, ε_q, γ_ω, γ_d, γ_q, L ]. A population is supposed to include NP individuals and x_ji,G is on behalf of the jth parameter of the ith individual in the Gth generation, i = 1, 2,…, NP, j = 1, 2,…, ND, where G, NP, and D are chosen as 1000, 50, and 13 individually.

Step 2: Determination of fitness function. The target of fitness function is to guarantee the control variables to track the references and reduce the estimation errors. The fitness function is shown as

\begin{matrix} f_{i} = \int_{0}^{t} t \cdot (k_{1} | e_{ω} | + k_{2} | e_{d} | + k_{3} | e_{q} | + k_{4} | {\tilde{Δ}}_{ω} | + k_{5} | {\tilde{Δ}}_{d} | + k_{6} | {\tilde{Δ}}_{q} |) dt \end{matrix}

(38)

where k₁, k₂, k₃, k₄, k₅, and k₆ are the adaptive constants and selected as 0.5, 0.1, 0.1, 0.1, 0.1, and 0.1, respectively.

Step 3: Updating the vectors of mutation and trial. The mutation vector v _i,G+1 and trial vector u _i,G+1 are obtained through the operations of mutation and crossover

\begin{matrix} v_{i, G + 1} = x_{r_{1}, G} + F_{i, G + 1} (x_{r_{2}, G} - x_{r_{3}, G}), \\ r_{1} \neq r_{2} \neq r_{3} \neq i \end{matrix}

(39)

u_{i, G + 1} = (u_{1 i, G + 1}, u_{2 i, G + 1}, \dots, u_{Di, G + 1})

(40)

u_{ji, G + 1} = {\begin{matrix} v_{ji, G + 1}, if r (j) \leq C R_{i, G} or j = rn (i) \\ x_{ji, G + 1}, if r (j) > C R_{i, G} and j \neq rn (i) \end{matrix}

(41)

where r₁, r₂, and r₃ are the random numbers and are not the same with each other, and the running index i, F_i,_G, and CR_i,_G are the scale factor and cross-over factor, F_i,₁ and CR_i,₁ are randomly determined from [0.1, 0.9] and [0, 1] at the start, r(j) is a random number between 0 and 1, and rn(i) is a random integer between 1 and D.

Step 4: Updating the factors F_i,G and CR_i,G

F_{i, G + 1} = {\begin{matrix} 0.1 + ran d_{1} \times 0.9, ran d_{2} < 0.1 \\ F_{i, G}, otherwise \end{matrix}

(42)

C R_{i, G + 1} = {\begin{matrix} ran d_{3}, ran d_{4} < 0.1 \\ C R_{i, G}, otherwise \end{matrix}

(43)

where rand_i indicates a random number.

Step 5: Updating the population x_i,G+1

\begin{matrix} x_{i, G + 1} = \\ {\begin{matrix} u_{i, G + 1}, if f (u_{i, G + 1}) < f (x_{i, G}), for minimization problems \\ x_{i, G}, otherwise \end{matrix} \end{matrix}

(44)

Experiment and analysis

Digital simulation

In this section, digital simulations are conducted in MATLAB to demonstrate the correctness and effectiveness of the proposed algorithm. The parameters of the system are shown as: stator resistance R_s = 3.16 Ω, d axis inductance L_d = 0.139 H, q axis inductance L_q = 0.062 H, flux linkage of permanent magnet ψ_f = 0.47 Wb, number of pole pairs n_p = 10, viscous damping coefficient B_m = 0.0005 N/rad/s, and moment of inertia J_m = 0.06 kg m².

The simulation time is 60 s and the sampling time is 0.001 s. For testing the performance of the proposed algorithm, the reference of speed is set as dynamic variation: startup to a low speed of 10 r/min with a increasing step to 300 r/min at t = 10 s; then continuous operation to t = 50 s with a decreasing step to 10 r/min. Furthermore, load torque is also changing with the time: from startup to 20 s, the load torque is 1.5 N m; at t = 20 s a 3 N m increased instruction is injected and lasting 20 s; then load torque returns to 1.5 N m at t = 40 s. For displaying the superiority of the approach, NDO-based BC control with no SMC and NDO-based PI control are also completed, which are denoted as NDO-BC and NDO-PI, respectively. The control parameters determined through DE optimization algorithm method are shown as: k_ω = 50.82, k_d = 500.12, k_q = 1500.04, β_ω = 100.52, β_d = 300.23, β_q = 400.01, ε_ω = 20.47, ε_d = 11.05, ε_q = 25.68, γ_ω = 0.60, γ_d = 1.82, γ_q = 2.58, and L = diag [38.10, 14.25, 14.25]. The uncertainties of modeling errors are described as: F_ω = 0.5 sin(t) + 0.4 T_L + 0.05 rand, F_d = i_q cos(t) + 0.2 sgn(i_d) + 0.05 rand, F_q = i_d sin(t) + 0.2 sgn(i_q) + 0.05 rand, where rand is an item of normal distribution random.

The digital simulation results are displayed in Figures 2 and 3, where the simulation results of the three algorithms are compared in Figure 2. Figure 2(a) illustrates the speed result, for the proposed algorithm in the article, they have no apparent fluctuations in the whole time, especially in speed reference variations at t = 10 and 50 s. However, the other algorithms of NDO-BC and NDO-PI have larger fluctuations when changing the speed references. Moreover, the changing of load torque at t = 20 and 40 s leads to larger overshoots for NDO-BC and NDO-PI, and the proposed algorithm experiences a smooth transition more or less. Figure 2(b) and (c) shows the results of i_d and i_q, respectively. There have almost no oscillations in current waveforms for the proposed algorithm at startup and at the time of changing reference in contrast with the other two approaches, which indicates that the proposed scheme ensures the state variables of the system to follow the references more quickly. Also, the integral of absolute error (IAE) is provided as an index for numerical metric. The values of IAE in Figure 2(a) for the three algorithms are 918.6 (NDO-BC + SMC), 1260.4 (NDO-BC), and 2425.8 (NDO-PI), respectively. For i_d in Figure 2(b), the values of IAE are 548.9 (NDO-BC + SMC), 886.3 (NDO-BC), and 1225.5 (NDO-PI) individually; and the values of IAE for i_q in Figure 2(c) are 410.0 (NDO-BC + SMC), 741.1 (NDO-BC), and 1212.0 (NDO-PI), respectively. Figure 3(a)–(c) demonstrates the observation results of F_ω, F_d, and F_q, respectively, which displays that the designed NDO can basically observe the disturbances and uncertainties rapidly and accurately.

Figure 2.

Comparative simulation results of ω_r, i_d and i_q with three control approaches: (a) ω_r, (b) i_d, and (c) i_q.

Figure 3.

Observation results of F_ω, F_d, and F_q: (a) F_ω, (b) F_d, and (c) F_q.

Hardware test

For validating the effectiveness of the proposed algorithm better, two hardware experiments with different dynamic reference speeds are implemented. The parameters in hardware test are the same as the digital simulation. First, the reference speed is set to 200 × sin(t) r/min. The experimental results are shown in Figure 4. Figure 4(a) plots the tracking of the reference speed. The speed regulated by the algorithm follows with the speed signal very well. Figure 4(b) and (c) shows the plots of d axis and q axis currents individually. Along with the changing of sinusoidal speed waveform, d axis and q axis currents demonstrate the sinusoidal waveforms similarly. Furthermore, the output d axis and q axis currents have basically completed the tracking of their actual commands, respectively. Second, the reference speed is set to the triangular waveform. The detail of the triangular waveform is shown in Figure 5(a). Controlled by the proposed algorithm, the speed of the motor shown in Figure 5(a) tracks the triangular reference speed very well. Followed with the triangular speed command, the output currents of d axis and q axis are shown in Figure 5(b) and (c). The currents also represent the characteristic of ramp waveforms. Similarly, the currents track their references well.

Figure 4.

Experimental results of speed, d axis current and q axis current with sinusoidal speed reference: (a) speed, (b) d axis current, and (c) q axis current.

Figure 5.

Experimental results of speed, d axis current and q axis current with triangular speed reference: (a) speed, (b) d axis current, and (c) q axis current.

Conclusion

An adaptive sliding mode–based BC algorithm is proposed for MTPA control of PMa-SynRM. An NDO is designed to estimate the uncertainties of modeling errors. In addition, an adaptive law is devised to further compensate for the disturbances and uncertainties. The theoretical analysis and experimental results show the advantages of the algorithm as follows: (1) the proposed approach guarantees globally asymptotic stability of the close-loop system and enables the system to track the reference signals correctly and quickly. (2) The designed NDO can estimate the disturbances and uncertainties precisely, and it is beneficial to reduce the chattering and improve the control performance. (3) Due to the existence of adaptive law, SMC-based BC is independent of uncertainties upper bound of disturbances.

Footnotes

Handling Editor: Jose Ramon Serrano

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Natural Science Foundation of China (grant no.: 51407077), the Fundamental Research Funds for the Central Universities of China (grants no.: 2016XS96 and 2017MS095), and the Science and Technology Project of State Grid Corporation of China Headquarter (grant no.: 5204BB16000 F).

ORCID iD

Yang Yu

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