A simple stable adaptive neuro-fuzzy speed controller for induction motors

1 Introduction

Induction motors (IMs) have high robustness, easy structure, and generally satisfactory efficiency, so they are used widely in industry [1]. The speed control of IMs is a challenging problem for high dynamic performance applications due to the non-linearity and parameter variation [2]. Lots of algorithms have been employed to improve the performance of the IM control; they have helped to make a better situation for the stability and robustness against the parameter variation and disturbance [3–6]. Generally, three main strategies can be distinguished in the literature for the speed control of IMs: FOC, DTC, and nonlinear Lyapauov-basedstrategies.

FOC is an attractive strategy for the speed control of IMs due to the independency of torque and flux dynamics [3]. As a result, IMs can be considered as separately excited DC motors and the problem of speed control of IMs is simplified significantly. However, the FOC strategy suffers from one major disadvantage. Motor parametric variations such as the rotor time constant will degrade the speed control system performance considerably [7–9].

DTC is one of the popular strategies in the speed control of IMs which has some advantages in its simple implementation like the absence of coordinate transformation, current regulator, and separate voltage modulation block [10, 11]. However, the conventional DTC suffers from some disadvantages such as high torque ripple and slow transient response to step changes in torque during its start-up. To improve the performance of conventional DTC, many researchers have applied artificial intelligence techniques such as neuro-fuzzy systems [12]. In these approaches, adaptive neuro-fuzzy systems are designed for the generation of the voltage space-vector. These controllers combine fuzzy logic and artificial neural networks to decouple flux and torque controls. However, they lack a mathematical and rigorous stability analysis to guarantee a satisfactory speed regulation and boundedness of currents and fluxes.

In contrast with FOC and DTC strategies, in nonlinear Lyapauov-based strategies the asymptotic convergence of the speed tracking error and boundedness of fluxes and currents can be guaranteed. For the position control of induction motor servo drives, many Lyapanov-based controllers have been proposed in the literature [13–16]. However, very few controllers based on stability analysis have been proposed for the speed control of larger induction motors. For example, in [17], an adaptive fuzzy controller has been presented for the speed control of a medium-sized induction motor which guarantees the control system stability. The control law is designed based on indirect adaptive fuzzy control [18]. However, the proposed controller requires feedbacks from all state variables such as speed, fluxes and currents. In addition, the acceleration signal is needed in this method. Usually, the acceleration feedback is contaminated with noise and will degrade the controller performance in practical implementations. Moreover, the high number of fuzzy rules imposes a considerable computational load on the controller. Thus, it seems that designing a simple stable nonlinear speed controller for IMs has been left as an open problem.

This paper proposes a simple and novel adaptive neuro-fuzzy speed control for IMs, in which asymptotic convergence of the speed tracking error is guaranteed. For online parameter adjustment of the neuro-fuzzy system, some adaptation laws are driven, using a stability analysis. Moreover, the reconstruction error of the neuro-fuzzy system is compensated in the control law. Since the uncertainty upper bound is required in the control law and it is unknown, an adaptation law is designed to estimate it. Then, to attenuate the chattering phenomenon, the proposed control law is modified using a PI control structure. Comparing to the previous works in the literature, the proposed controller presents excellent results since it has the following features.

The control system stability is guaranteed. It should be mentioned that most of available speed controllers presented in the literature are based on DTC and FOC without any mathematical and rigorous stability analysis.

This paper is organized as follows –Section 2 describes the IM model. Section 3 presents the proposed control law and stability analysis. Simulation results are given in Section 4 and finally, Section 5 concludes the paper.

2 Induction motor model

The fifth-order model of an IM under the assumptions of equal mutual inductances and linear magnetic circuit is given by:

di sa dt = - ( M 2 R r + L r 2 R s σ L s L r 2 ) i sa + MR r σ L s L r 2 ψ ra + n p M σ L s L r ω ψ rb + 1 σ L s u sa di sb dt = - ( M 2 R r + L r 2 R s σ L s L r 2 ) i sb + MR r σ L s L r 2 ψ rb - n p M σ L s L r ω ψ ra + 1 σ L s u sb d ψ ra dt = - R r L r ψ ra - n p ω ψ rb + R r L r Mi sa d ψ rb dt = - R r L r ψ rb + n p ω ψ ra + R r L r Mi sb d ω dt = n p M JL r ( ψ ra i sb - ψ rb i sa ) - T L J(1) where u _s , i, T _L , and ψ indicate stator voltage input to the machine, current, external load torque, and flux linkage respectively; the subscripts r and s stand for rotor and stator; (a, b) denote the components of a vector with respect to a fixed stator reference frame and σ = 1 - (M ²/L _s L _r ) [4–6, 19]. According to (1), we can express this nonlinear system using the state space representation as X ̇ = F ( X , T L , u s ) in which X = [i _sa  i _sb  ψ _ra  ψ _rb  ω]  ^T is the state vector.

Consider the system dynamics as ω ̇ = f ( X , T L )(2) in which, f (X, T _L ) is given by f ( X , T L ) = n p M JL r ( ψ ra i sb - ψ rb i sa ) - T L J(3)

In this paper it is assumed that f (X, T _L ) is uncertain and a neuro-fuzzy system will be designed to estimate it. According to [20], the system relative degree is two because the time derivative of Equation (2) leads to ω ̈ = h 1 ( X ) + h 2 ( X ) u(4) which u is the amplitude of stator input voltage. Using feedback linearization, the control law is given by u = ω ̈ d + k d e ̇ + k p e - h 1 ( X ) h 2 ( X )(5)

In which e = ω _d  - ω is the speed tracking error and k _p , k _d are design parameters.

Therefore, the acceleration signal is required to implement the control law Equation (5). However, the acceleration feedback is contaminated with noise and will degrade the controller performance. According to [21], the uncertain function which should be estimated can include the control signal u. Thus in this paper in order to guarantee the asymptotic convergence of the speed tracking error and eliminate the acceleration feedback the system dynamics Equation (2) is rewritten as ω ̇ = f 1 ( X , u , T L ) + u .(6)

The neuro-fuzzy system objective is the estimation of f ₁ (X, u, T _L ). Using the feedback linearization technique, the ideal control law is u * = g + ω ̇ d + λ e(7) in which g = - f ₁ (X, u, T _L ) is uncertain, e is the speed tracking error and λ is a design parameter. Thus, in order to implement the control law Equation (7) g should be estimated using a neuro-fuzzy system. In this paper, we use a neuro- fuzzy system which is a mapping from U ⊂ R to V ⊂ R that follows fuzzy IF–THEN rules: R ( l ) : IF e is F l THEN y = θ l , l = 1 , … , N(8) where e ∈ U and y ∈ V are the input and output of the neuro-fuzzy system, N is the number of rules, and F ^l is the fuzzy set in U. There exists the fuzzy inference engine here that does the mapping of U (fuzzy set) to V (constant).

With the use of the singleton fuzzifier and product inference engine, the neuro-fuzzy system has this output: y = ∑ i = 1 N θ i ξ i ( e , m i , σ i ) = θ T ξ ( e , m , σ )(9) where θ = [θ ₁, θ ₂, …, θ _N ]  ^T is the consequent adjustable parameter vector. ξ ( e , m , σ ) = [ ξ 1 ( e , m 1 , σ 1 ) , ξ 2 ( e , m 2 , σ 2 ) , … , ξ N ( e , m N , σ N ) ] T is a set of Gaussian membership functions with the following definition. ξ i ( e , m i , σ i ) = exp ( - ( e - m i ) 2 / σ i 2 ) ) ∑ i = 1 N ξ i ( e , m i , σ i )(10) where m _i and σ _i are the center and the standard deviation of the Gaussian membership function Equation (10). According to the universal approximation theorem [22–24], there exist m, σ and θ such that g = g * ( m , σ , θ ) + Δ(11) where g ^* (m, σ, θ) is a neuro-fuzzy system given by

g * ( m , σ , θ ) = ∑ i = 1 N θ i ξ i ( e , m i , σ i ) = θ T ξ ( e , m , σ )(12) m = [m ₁, m ₂, …, m _N ]  ^T and σ = [σ ₁, σ ₂, …, σ _N ]  ^T are adjustable vectors; and Δ is the approximation or reconstruction error [25].

Since g ^* is unknown we propose the control law u = g ˆ + g c + ω ̇ d + λ e(13) in which g ˆ = θ ˆ T ξ ˆ ( m ˆ , σ ˆ , e , e ̇ )(14) is the neuro-fuzzy estimation of g ^* and g _c is added to this control law in order to compensate the reconstruction error Δ. The neuro-fuzzy control strategy is summarized in Fig. 1.

Substituting the control law Equation (13) in the system dynamic Equation (6) and using g = - f ₁ (X, u, T _L ) yields ω ̇ = - g + g ˆ + g c + ω ̇ d + λ e .(15)

According to Equations (11), (15) can be rewritten as:

ω ̇ = - g * - Δ + g ˆ + g c + ω ̇ d + λ e .(16)

In the other word, we have g * - g ˆ + Δ - g c = e ̇ + λ e .(17)

Using Equations (12) and (14), the closed loop Equation (17) is expressed by e ̇ + λ e = θ * T ξ * - θ ˆ T ξ ˆ + Δ - g c .(18)

Suppose θ ˜ = θ * - θ ˆ and ξ ˜ = ξ * - ξ ˆ, then e ̇ + λ e = ( θ ˜ + θ ˆ ) T ( ξ ˜ + ξ ˆ ) - θ ˆ T ξ ˆ - θ ˆ T ξ ˆ + Δ - g c(19) which can be simplified as: e ̇ + λ e = θ ˜ T ξ ˜ + θ ˜ T ξ ˆ + θ ˆ T ξ ˜ + Δ - g c(20)

By using Taylor series expansion, ξ ˜ can be rewritten as [25]: ξ ˜ = A m ˜ + B σ ˜ + h ot(21) in which m ˜ = m * - m ˆ, σ ˜ = σ * - σ ˆ, h _ot shows the terms of higher order and A = [ ∂ ξ 1 ∂ m ∂ ξ 2 ∂ m … ∂ ξ N ∂ m ] | m = m ˆ(22) B = [ ∂ ξ 1 ∂ σ ∂ ξ 2 ∂ σ … ∂ ξ N ∂ σ ] | σ = σ ˆ(23) where

∂ ξ i ∂ m = [ ∂ ξ i ∂ m 1 ∂ ξ i ∂ m 2 … ∂ ξ i ∂ m N ] T(24)

∂ ξ i ∂ σ = [ ∂ ξ i ∂ σ 1 ∂ ξ i ∂ σ 2 … ∂ ξ i ∂ σ N ] T .(25)

Inserting Equation (21) into Equation (20):

e ̇ + λ e = θ ˆ T ( A σ ˜ + B m ˜ + h ot ) + θ ˜ T ξ ˆ + θ ˜ T ξ ˜ + Δ - g c = θ ˆ T A σ ˜ + θ ˆ T B m ˜ + θ ˜ T ξ ˆ + Δ 2 - g c(26)

In the above equation Δ 2 = θ ˆ T h ot + θ ˜ T ξ ˜ + Δ is an uncertain term limited by an unknown constant bound; |Δ ₂| ≤ E, and E ˆ is an estimation of E.

Theorem 1. Considering the nonlinear system Equation (6) and the control law Equation (13), if the following conditions are met, the signals in the control system are bounded and the tracking error approaches to zero asymptotically: E ^ . = η 1 e(27) θ ^ . = η 2 e ξ ^(28) m ^ . = η 3 e B T θ ^(29) σ ^ . = η 4 e A T θ ^(30) g c = E ˆ sgn ( e )(31) Proof. Taking the time derivative of the following positive definite Lyapunov function V = 1 2 e 2 + 1 2 ( E ˜ 2 η 1 + θ ˜ T θ ˜ η 2 + m ˜ T m ˜ η 3 + σ ˜ T σ ˜ η 4 )(32) where E ˜ = E - E ˆ, with the use of Equation (26) and Equation (27)–Equation (31) we have:

V ˙ = e e ˙ + E ˜ E ∑ . η 1 + θ ˜ T θ ∑ . η 2 + m ˜ T m ∑ . η 3 + σ ˜ T σ ∑ . η 4 = − λ e 2 + m ˜ T e A T θ ^ − m ∑ . η 3 + σ ˜ T e B T θ ^ − σ ∑ . η 4 + θ ˜ T e ξ ^ − θ ∑ . η 2 + e Δ 2 − e g c − E ˜ E ∑ . η 1 = − λ e 2 + e Δ 2 − e g c − E ˜ E ∑ . η 1 = − λ e 2 + e ( − E ˜ − E ^ ) + e Δ 2(33) therefore V ̇ ≤ - λ e 2 - | e | ( E - | Δ 2 | )(34) that means V ̇ is negative semidefinite, i.e. V ( e ( t ) , E ˜ ( t ) , θ ˜ , σ ˜ , m ˜ ) ≤ V ( e ( 0 ) , E ˜ ( 0 ) , θ ˜ , σ ˜ , m ˜ ). Thus, it is concludedthat e, E ˜ ( t ), θ ˜, σ ˜ and m ˜ are bounded. Since V ( e ( 0 ) , E ˜ ( 0 ) , θ ˜ , m ˜ , σ ˜ ) is bounded and V ( e ( t ) , E ˜ ( t ) , θ ˜ , m ˜ , σ ˜ ) is a non-increasing bounded function, by using Barbalat’s lemma it can be shown that lim t → ∞ | e ( t ) | = 0.

Assume Ω ( t ) = λ e 2 ≤ - V ̇, thus: ∫ 0 t Ω ( τ ) d τ ≤ V ( e ( 0 ) , E ˜ ( 0 ) , θ ˜ , m ˜ , σ ˜ ) - V ( e ( t ) , E ˜ ( t ) , θ ˜ , m ˜ , σ ˜ ) < ∞ .(35)

Meanwhile Equation (26) shows e ̇ and Ω ̇ ( t ) = 2 λ e e ̇ are bounded, according to the Barbalat’s lemma [20], lim t → ∞ Ω ( t ) = 0 and lim t → ∞ e ( t ) = 0.

For eliminating the chattering phenomenon, two states are considered:

In within the boundary layer, |e| < φ, the following PI-type structure is used to approximate the control Esgn (e):

Since α = [K _p , K _I ]  ^T and γ are not known, we use g c ( e | α ˆ ) with γ = E ˆ and α ˆ is derived from Lyapunov Theorem for the approximation of Esgn (e). Finally, the following theorem modifies the first theorem.

Theorem 2. Considering the nonlinear system Equation (6) and the control law Equation (13), if γ = E ˆ and the following conditions are met, the signals in the control system are bounded and the tracking error approaches to zero asymptotically. E ^ . = η 1 e(37) θ ^ . = η 2 e ξ ^(38) m ^ . = η 3 e A T θ ^(39) σ ^ . = η 4 e B T θ ^(40) α ^ . = η 5 e ψ ( e )(41) g c = α ˆ T ψ ( e )(42) Proof. Assume the following positive definite Lyapunov function V = 1 2 e 2 + 1 2 ( E ˜ 2 η 1 + θ ˜ T θ ˜ η 2 + m ˜ T m ˜ η 3 + σ ˜ T σ ˜ η 4 + α ˜ T α ˜ η 5 )(43) where α ˜ = α * - α ˆ and α ^* is defined as the optimal value for α: α * = arg min α ˆ ∈ R 2 [ sup e ∈ R | g c ( e | α ˆ ) - E sgn ( e ) | ] .(44)

The time derivative of Equation (43) is

V ˙ = e e ˙ + E ˜ E ˜ . η 1 + θ ˜ T θ ∼ . η 2 + m ˜ T m ∼ . η 3 + σ ˜ T σ ∼ . η 4 + α ˜ T α ∼ . η 5 = e ( θ ^ T A m ˜ + θ ^ T B σ ˜ + θ ˜ T ξ ^ + ε − g c ( e α ) + g c ( e α * ) − g c ( e α * ) ) − E ˜ E ∼ . η 1 + θ ˜ T θ ∼ . η 2 + m ˜ T m ∼ . η 3 + σ ˜ T σ ∼ . η 4 + α ˜ T α ∼ . η 5 = e ( m ˜ T A T θ ^ + σ ˜ T B T θ ^ + θ ^ T ξ ^ + ε + α ˜ T ψ ( e ) − g c ( e α * ) ) − E ˜ E ∼ . η 1 + θ ˜ T θ ∼ . η 2 + m ˜ T m ∼ . η 3 + σ ˜ T σ ∼ . η 4 + α ˜ T α ∼ . η 5 = θ ˜ T e ξ ^ − θ ∼ . η 2 + m ˜ T e A T θ ^ − m ∼ . η 3 + σ ˜ T e B T θ ^ − σ ∼ . η 4 + α ˜ T e ψ ( e ) − α ∼ . η 5 + e ( ε − u c ( e α * ) ) − E ˜ E ∼ . η 1(45)

Considering Equations (37)–(42), note that g _c  (e|α ^*) stands in the first and third quadrants, so g _c  (e|α ^*) =0 for e = 0, and e g _c  (e|α ^*) ≥0 for all e. Therefore, e g _c  (e|α ^*) = |e|| g _c  (e|α ^*) | and V ̇ ≤ - λ e 2 - | e | ( E - | Δ 2 | ) .(46)

So V ̇ is negative semidefinite and lim t → ∞ | e ( t ) | = 0 (see the Theorem 1 proof).

To ensure the boundedness of internal dynamics of IM including i _ra , i _rb , ψ _sa and ψ _sb , Equation (1) can be rewritten as X ̇ = AX + v ( t )(47) in which

X = [ i sa i sb ψ ra ψ rb ] T(48)

A = [ - ( M 2 R r + L r 2 R s σ L s L r 2 ) 0 MR r σ L s L r 2 0 0 - ( M 2 R r + L r 2 R s σ L s L r 2 ) 0 MR r σ L s L r 2 R r M L r 0 - R r L r 0 0 R r M L r 0 - R r L r ](49) and v ( t ) = [ n p M σ L s L r ω ψ rb + 1 σ L s u sa - n p M σ L s L r ω ψ ra + 1 σ L s u sb - n p ω ψ rb n p ω ψ ra ] .(50)

Since the eigenvalues of A are negative, the state vector X in X ̇ = AX is exponentially stable. Moreover, the control law Equation (13) and the IM speed ω are bounded. Thus, the vector v (t) is bounded. Consequently, the state Equation (47) can be considered as a stable linear system with bounded inputs. Therefore, the system described in Equation (47) has the BIBO stability.

To make the superiority of the proposed method more obvious, its performance is compared with the controller designed in [17]. In Simulation 1 the proposed neuro-fuzzy control algorithm has been tested and Simulation 2 presents the performance of the adaptive fuzzy MIMO controller [17].

Simulation 1: Consider a three-phase standard IM with parameters given in the Table 1 [28]. To test the control system robustness against the thermal variation of motor parameters and external load disturbance, it is assumed that:

R s = R s 0 ( 1 + 0.2 sin ( t ) ) R r = R r 0 ( 1 + 0.2 cos ( t ) ) L s = L s 0 ( 1 + 0.1 sin ( t ) ) L r = L r 0 ( 1 + 0.1 cos ( t ) )(51) and T L = { 1 t < 12 7 12 ≤ t ≤ 2 t > 17 17 .(52)

In order to examine the speed regulation capability in response to sudden variations of the speed command, ω _c is defined the summation of the constant value 150 and the square wave with altitude 20 and frequency 0.1 Hz. Also, the reference model ω d ( s ) ω c ( s ) = 125 s 3 + 15 s 2 + 75 s + 125(53) is chosen to regulate the transient response of the speed control system. Finally, in order to verify the ability of the proposed control law in rejecting input voltage disturbances, the following voltage disturbance has been inserted to u _sb . v dist = { 15 8 ≤ t ≤ 9 0 otherwise(54)

The structure of the neuro-fuzzy system with three neurons in the hidden layer is shown in Fig. 2. The initial values θ in neuro-fuzzy system is selected randomly in the interval [-50, 50], and the initial values of σ and m are set 140 and 250. Moreover, the learning rates of adaptation laws Equations (37)–(42) and the proportional gain λ in Equation (13) are: λ = 0.5, η ₁ = 1, η ₂ = 50, η ₃ = 50, η ₄ = 50 and η ₅ = 1.The thickness of the boundary layer |e| > φ is set φ = 20.

The tracking performance of the proposed control scheme and the speed tracking error are illustrated in Figs. 3 and 4, respectively. As shown in these figures, the asymptotic convergence of the motor speed to the command signal is satisfactory in terms of fast external load disturbance rejection and robustness against motor parameter variations and input voltage disturbances. The control effort (the amplitude of stator voltage) is presented in Fig. 5 which verifies its smoothness. Using Theorem 2, the chattering phenomenon is prevented. Moreover, as illustrated in this figure, the motor voltage is under the maximum permitted voltage.

In Fig. 6, the response of the control system without considering the reconstruction error compensator is illustrated which shows the essential role of this part in improving the performance of the control system. Finally the variation/adaptation of the neuro-fuzzy system parameters and the error compensator gains are shown in parts of Fig. 7.

Simulation 2: In this simulation, the adaptive fuzzy MIMO control presented in [17] is used for the speed control of the IM model described in Simulation 1. The reference model, external load torque, and input voltage disturbances are the same as Simulation 1. The tracking performance of this control scheme is illustrated in Fig. 8. As shown in the zoomed Figure, this control approach fails in rejecting the external load torque and input voltage disturbances. However, as shown in Fig. 3, the proposed method completely eliminates the effect of these disturbances. It should be noted that the adaptive fuzzy MIMO method [17], requires feedbacks from all state variables and also the acceleration signal is used, while the proposed controller needs just the speed feedback. Moreover, there are 6 uncertain functions which should be estimated in the adaptive fuzzy MIMO method.

In order to estimate each function, 243 fuzzy rules are needed. However, the neuro-fuzzy approach presented in this paper is much simpler and less computational. In addition, the non-singularity of the estimated input gain matrix in [17] is a critical condition which can be violated easily and make the control system unstable. Another superiority of the proposed controller is compensating the reconstruction error of the neuro-fuzzy estimator which has improved the performance of the controller.

In this paper, a simple stable speed controller for IMs is presented. In Section 2, the IM nonlinear model is described. In Section 3, a novel modeling of the system is proposed with the aim of making the control design simpler. Then, based on this new model, the control law is designed using feedback linearization. To estimate the uncertainty including un-modeled dynamics, motor parameter variations, load torque, and input voltage disturbances a simple neuro-fuzzy system has been designed. Moreover, the reconstruction error of the neuro-fuzzy system is compensated in the control law. The adaptation laws for online tuningof the parameters in the neuro-fuzzy system and approximation error compensator are derived based on the stability analysis. To guarantee the asymptotic convergence of the speed tracking error, Barbalat’s lemma has been applied. In addition, the boundedness of all state variables is ensured in the stability analysis, which makes the proposed method superior than previous related works. In Section 4, satisfactory performance of the proposed method is verified using computer simulations. Also, a comparison with an adaptive fuzzy method has been performed which obviously indicates that the proposed controller is much simpler, less computational and more efficient in rejecting external disturbances.