Quasi continuous sliding mode control with fuzzy switching gain for an induction motor

Abstract

This manuscript investigates speed and flux tracking control method based on quasi-continuous sliding mode control (SMC) for induction motor (IM). To overcome the problem of conventional SMC, a quasi-continuous second-order sliding mode controller (QC2SMC) is first designed. QC2SMC produces continuous control and less chattering. Chattering exists during the control action of QC2SMC when the system remains on the sliding manifold. However, it is difficult to select sliding mode (SM) controller gain that minimizes the reaching time on sliding manifold. So as to reduce the chattering, fuzzy-adapted QC2SMC (FQC2SMC) is designed, in which the fuzzy system is employed to adaptively tune the SM controller gain. Moreover, with the intention of reduce chattering and achieve high tracking accuracy, quasi-continuous third-order sliding mode controller (QC3SMC) is investigated. However, the transient response of QC3SMC is very slow. This is the main constraint of QC3SMC and overcome by applying a fuzzy-adapted QC3SMC (FQC3SMC) is intended. Control structure incorporates a state observer based on a super twisting algorithm to estimate the rotor flux and torque from stator currents and rotor speed. By using robust exact differentiator, estimation of the sliding surface derivative and designing of an observer for load torque are not needed. The convergence of controllers can be guaranteed and analyzed by using Lyapunov function. Here, simulation results are included to validate the effectiveness of the proposed strategy.

Keywords

quasi continuous sliding mode sliding mode observer (SMO)chattering phenomenon robustness induction motor (IM)

1 Introduction

An induction motor (IM) is widely used in industry, mainly due to its reliability and relatively low cost. The control of an IM has attracted much attention in the past few decades. However, IMs constitute a theoretically challenging control problem, since the dynamical system is nonlinear, the electric rotor variables are not measurable, and the physical parameters are most often imprecisely known. In the past years, various approaches have been developed to estimate rotor flux and speed including conventional schemes [1 –4], model reference adaptive schemes (MRAS) [5 –10], sliding mode control (SMC) theory [11 –18] and fuzzy logic control (FLC) [19 –28]. [29] is concerned with the problem of passivity-based asynchronous SMC for a class of uncertain singular Markovian jump systems (MJSs) with time-delay and nonlinear perturbations. [30] investigates the problem of the fault detection filter design for nonhomogeneous MJSs by a Takagi-Sugeno fuzzy approach. From the existing literature, SMC has received a great deal of attention due to its attractive features such as order reduction, disturbance rejection, robustness and simplicity of implementation. In most of application to IMs, the first order sliding mode controller (FOSMC) approach is impractical due to chattering phenomenon and offer lower accuracy. So that high order sliding modes (HOSMs) employed in real time applications to eliminate chattering effect with high tracking accuracy [31, 32].

The quasi continuous SMC existing in [33 –36] is the class of HOSM controller. Formulation of HOSM controllers involved higher time derivatives of the switching surface in the design of control law. Their implementation is more complicated and need further computation time. However, the benefits of HOSM are that elimination of chattering effect and offers higher accurateness in realization. In [33 –36], quasi-continuous HOSM controller with robust exact differentiators techniques have been developed and applied successfully for real applications. Besides, quasi continuous second-order sliding mode controller (QC2SMC) and quasi continuous third-order sliding mode controller (QC3SMC) and differentiators are applied to quaternion-based spacecraft-attitude-tracking maneuvers in [37]. [38, 39] proposed controller using the quasi- continuo control algorithm to IM. Quasi-continuous HOSM generates continuous control and less chattering than other HOSM controllers. However, quasi-continuous HOSM require large SM controller gain values to bring the system state on the sliding manifold $s = \dot{s} = \ddot{s} = . . . . = s^{r - 1} = 0$ . Significant chattering is produced owing to the presence of switching delays, measurement noises and singular perturbations. In addition, it is complicated to choose a suitable SM controller gain that minimize the reaching time on sliding manifold. Moreover, in order to reduce chattering, fast transient response and achieve high tracking accuracy, so that a fuzzy-adapted QC3SMC (FQC3SMC) based robust exact differentiators is proposed to IM. In order to these, the fuzzy system is employed to adaptively tune the SM controller gain. This technique has been developed and applied successfully for robot manipulatorsin [40].

In SMC, auxiliary control effort should be designed to eliminate the effect of the unpredictable perturbations. The auxiliary control effort is referred to as hitting control effort. The hitting control gain concerned with an upper bound of uncertainties and sign function. However, the upper bound of uncertainties, which is required in the control law, is difficult to obtain precisely in advance for practical applications. Several methods exist [41 –46] in literature, which has significant advantage that, convergence speed increased and reaching time is reduced. In path tracking systems, however, the system invariance properties are observed only during the sliding phase. In reaching phase, tracking may be hindered by disturbances or parameter variations. The straightforward way to reduce tracking error and reaching time by increasing hitting gain, which may cause chattering effect. The chattering can also be reduced by using small boundary layer thickness. The selection of hitting gain value is based on minimization of tracking error and reaching time, whenever the tracking error is negative then we have to choose a small gain value for desired performance of system and vice-versa. The sliding hyperplane highly depend upon the dynamics of error and change in error so that we have to consider this variable as input to the fuzzy logic module for updating hitting gain. By using above consideration, the general rule is composed as, if sliding surface in negative region, then select a small value of hitting gain and vice versa. In this way, hitting gain can be determined by using fuzzy logic tuning approach.

In this paper, quasi continuous SM approach includes state observers based on super twisting algorithm (STA) to estimate the rotor flux and torque from stator currents and rotor speed. QC2SMC with first order differentiator and QC3SMC with second order differentiator are developed to robust for the speed and the flux module in three phase IMs. QC2SMC provides continuous control and less chattering. Chattering exists during the control action of QC2SMC when the system remains on the sliding manifold. However, it is difficult to select SM controller gain that minimize the reaching time on sliding manifold. So as to reduce the chattering, fuzzy QC2SMC (FQC2SMC) is designed, in which the fuzzy system is employed to adaptively tune the SM controller gain. Moreover, with the intention of reduce chattering and achieve high tracking accuracy, a QC3SMC is investigated. However, tracking response of QC3SMC is very slow. This is main constraint of QC3SMC and to overcome this problem, a fuzzy-adapted QC3SMC (FQC3SMC) is proposed. The convergence of controllers can be guaranteed and analyzed by using Lyapunov function.

2 Induction Motor Model description

Equations for an IM can be modeled in the (α, β) orthogonal stationary reference frame as [47, 48] $\begin{matrix} \dot{x} = f (x) + g (x) u \\ y = h (x) \end{matrix}$ (1) $\begin{matrix} f (x) & = & [\begin{matrix} - c_{3} λ_{α} - ω λ_{β} + c_{4} i_{α} \\ - c_{3} λ_{β} + ω λ_{α} + c_{4} i_{β} \\ - c_{1} a_{1} i_{α} - c_{1} c_{2} (- c_{3} λ_{α} - ω λ_{β}) \\ - c_{1} a_{1} i_{β} - c_{1} c_{2} (- c_{3} λ_{β} + ω λ_{α}) \\ \frac{1}{J} {c_{5} (i_{β} λ_{α} - i_{α} λ_{β}) - T_{l}} \end{matrix}] \\ g (x) & = & [\begin{matrix} 0 & 0 \\ 0 & 0 \\ c_{1} & 0 \\ 0 & c_{1} \\ 0 & 0 \end{matrix}] \end{matrix}$

$u = {[\begin{matrix} u_{α} & u_{β} \end{matrix}]}^{T},: with, state,: x = {[\begin{matrix} λ_{α} & λ_{β} & i_{α} & i_{β} & ω \end{matrix}]}^{T}$ The parameters of the model c₁, c₂, c₃, c₄, c₅ and a₁ are defined as $\begin{matrix} c_{1} & = & \frac{1}{σ L_{s}}, c_{2} = \frac{L_{m}}{L_{r}}, c_{3} = η, \\ c_{4} & = & η L_{m}, c_{5} = \frac{3 N_{r}}{2} \frac{L_{m}}{L_{r}}, \end{matrix}$ and $a_{1} = R_{s} + \frac{L_{m}^{2}}{L_{r}^{2}} R_{r}$ . with η, σ, β, and γ are positive constants described as $η = (\frac{R_{r}}{L_{r}}), σ = (1 - \frac{L_{m}^{2}}{L_{s} L_{r}}), β = c_{1} c_{2} = (\frac{L_{m}}{σ L_{s} L_{r}})$ and $γ = c_{1} a_{1} = \frac{1}{σ L_{s}} (R_{s} + \frac{L_{m}^{2}}{L_{r}^{2}} R_{r})$ R_s and L_s are stator resistance and inductance; R_r and L_r are rotor resistance and inductance; and L_m is the mutual inductance. ω is the electrical rotor speed, two-dimensional vectors λ^T = (λ_α, λ_β), i^T = (i_α, i_β) and u^T = (u_α, u_β) are rotor fluxes, stator currents, and stator voltages in (α, β) coordinate, respectively; T and T_l are the torque developed by the motor and the load torque; J is the moment of inertia; N_r is the number of pole pairs. The control output vectors are chosen as

$y = [\begin{matrix} h_{1} (x) \\ h_{2} (x) \end{matrix}] = [\begin{matrix} ω \\ φ \end{matrix}] = [\begin{matrix} ω \\ λ_{α}^{2} + λ_{β}^{2} \end{matrix}]$ (2) where ω and φ are an electrical rotor speed and square of the rotor flux module respectively.

3 Problem statements

The control problems stated in the manuscript are mentioned as:

To design a controller scheme for an output tracking based on QC2SMC with first order differentiator, QC3SMC with second order differentiator.

To investigate a controller by employing fuzzy logic system to adaptively tune the SM controller gain in order to minimize chattering, high accuracy and fast response.

Lyapunov function employed to verify convergence of suggested controllers in presence of the parametric uncertainties and disturbances effect. It implies that the speed and rotor flux tracking error maintain towards zero.

Error variables are defined as

$ɛ_{1} = [\begin{matrix} ɛ_{1}^{1} \\ ɛ_{2}^{1} \end{matrix}] = [\begin{matrix} ω - ω_{ref} \\ φ - φ_{ref} \end{matrix}]$ (3) where ω_ref and φ_ref are the references value for the electrical rotor speed and square rotor flux module respectively. Deriving (3) and introducing the desired dynamics, we have ${\dot{ɛ}}_{1} = ɛ_{2} = [\begin{matrix} ɛ_{1}^{2} \\ ɛ_{2}^{2} \end{matrix}] = [\begin{matrix} {\dot{ɛ}}_{1}^{1} \\ {\dot{ɛ}}_{2}^{1} \end{matrix}] = [\begin{matrix} \dot{ω} - {\dot{ω}}_{ref} \\ \dot{ϕ} - {\dot{ϕ}}_{ref} \end{matrix}] = [\begin{matrix} \dot{ω} \\ \dot{ϕ} \end{matrix}]$ (4) In a state regulator problem, change in reference signals should be zero ${\dot{ω}}_{ref} = 0,: {\dot{φ}}_{ref} = 0$ and

${\dot{ɛ}}_{2} = [\begin{matrix} {\dot{ɛ}}_{1}^{2} \\ {\dot{ɛ}}_{2}^{2} \end{matrix}] = [\begin{matrix} {\ddot{ɛ}}_{1}^{1} \\ {\ddot{ɛ}}_{2}^{1} \end{matrix}] = [\begin{matrix} \ddot{ω} \\ \ddot{ϕ} \end{matrix}]$ (5) Finally, the new variable system is represented in the following way $\begin{matrix} {\dot{ε}}_{1} = \\ {\dot{ε}}_{2} = \end{matrix} \begin{matrix} ε_{2} \\ \bar{F} (ε_{1}, ε_{2}, η) + {\bar{B}}_{2} (ε_{1}, ε_{2}) u \end{matrix}$ (6)

$\bar{F} = [\begin{matrix} {\bar{f}}_{3} \\ {\bar{f}}_{4} \end{matrix}], {\bar{B}}_{2} = [\begin{matrix} - \frac{c_{1} c_{5}}{J} λ_{β} & \frac{c_{1} c_{5}}{J} λ_{α} \\ 2 c_{1} c_{4} λ_{α} & 2 c_{1} c_{4} λ_{β} \end{matrix}], u = [\begin{matrix} u_{α} \\ u_{β} \end{matrix}]$ $\begin{matrix} {\bar{f}}_{3} & = & \frac{c_{5}}{J} (λ_{α} f_{31} + i_{β} f_{32} - λ_{β} f_{33} - i_{α} f_{34}) \\ {\bar{f}}_{4} & = & - 2 c_{3} d ϕ + 2 c_{4} (λ_{α} f_{33} + i_{α} f_{32} + λ_{β} f_{31} + i_{β} f_{34}) \\ f_{31} & = & - c_{1} a_{1} i_{β} - c_{1} c_{2} (- c_{3} λ_{β} + ω λ_{α}) \\ f_{32} & = & - c_{3} λ_{α} - ω λ_{β} + c_{4} i_{α} \\ f_{33} & = & - c_{1} a_{i} i_{α} - c_{1} c_{2} (- c_{3} λ_{α} - ω λ_{β}) \\ f_{34} & = & - c_{3} λ_{β} + ω λ_{α} + c_{4} i_{β} \\ d ϕ & = & - 2 c_{3} ϕ + 2 c_{4} (λ_{α} i_{α} + λ_{β} i_{β}) \end{matrix}$ Here $\bar{F}$ and ${\bar{B}}_{2}$ are smooth and bounded functions [49].

4 Fuzzy-adapted quasi continuous second order sliding mode controller

In the controller design procedure the state variables of a system are known. The relative degree is determined for the respective control output. The relative degree [49] for electrical rotor speed and rotor flux is 2 for each one and the system relative degree is (r = 4). The order of system state equation (1) is 5. As a result, the new system has internal dynamics with lesser order. This means that the system can be partly linearized, controllable and observable. Using linearization technique, the system is converted into a new system where sliding surface is designed. A control law is employed to the sliding variables. An observer is used to estimate the unavailable variables and differentiators to determine the sliding surface derivative.

4.1. QC2SMC

4.1.1. Sliding Manifold

$s = ɛ_{1} = [\begin{matrix} ɛ_{1}^{1} \\ ɛ_{2}^{1} \end{matrix}] = [\begin{matrix} s_{α} \\ s_{β} \end{matrix}] [\begin{matrix} ω - ω_{ref} \\ ϕ - ϕ_{ref} \end{matrix}]$ (7) The time derivative of $s = {[\begin{matrix} s_{α} & s_{β} \end{matrix}]}^{⊤}$ expressed by

$\frac{d}{dt} s = \bar{F} + {\bar{B}}_{2} \dot{i}$ (8) where $\dot{i} = ({\dot{i}}_{α}, {\dot{i}}_{β})$ is the control input vector. The sliding variables in the subspace s^∗=0 are given as

$\frac{d}{dt} s^{*} = D \bar{F} + {\bar{B}}_{2} s^{*} \frac{d}{dt} D + \dot{i}$ (9) Where D= ${\bar{B}}_{2}^{- 1}$ is non singular matrix and it remains all over in the state space.

4.1.2 Control Law

A SMC algorithm is used to lead the sliding variable towards zero, within fixed time in presence of internal and external disturbances (6). The suggested control law is given by [33, 34]

$u = - k \frac{{\dot{s}}^{*} + | s^{*} |^{1 / 2} sign (s^{*})}{| {\dot{s}}^{*} | + | s^{*} |^{1 / 2}}$ (10) where $u = {[\begin{matrix} u_{α} & u_{β} \end{matrix}]}^{⊤}$ is a quasi-continuous signal of the controller and k is a control gain as adjustment parameter.

4.1.3 First order robust differentiator

To employ the QC2SMC, time derivative of the sliding variable ( $\dot{s}$ ) is required. But, it is extremely difficult to find out ( $\dot{s}$ ) for an IM. Owing to this cause, we employ the first order differentiator [50, 51] for evaluation of $\dot{s}$ . A first order robust differentiator using the STA to estimate the sliding variable has the form $\begin{matrix} {\dot{z}}_{00} & = & v_{0} \\ v_{0} & = & - λ_{0} {| z_{00} - s^{*} |}^{\frac{1}{2}} s i g n (z_{00} - s^{*}) + z_{11} \\ {\dot{z}}_{11} & = & - λ_{1} s i g n (z_{00} - s^{*}) \end{matrix}$ (11) where z₀₀, z₁₁ are the actual time estimation of s^∗ and $\dot{s^{*}}$ . λ₀ and λ₁, are the fine tune constraints of the differentiator and they are determined by an equation [51].

$| \dot{f (t)} | \leq l_{c}, λ_{1} = 1.1 * l_{c}, λ_{0} = 1.5 * l_{c}^{\frac{1}{2}}$ (12) where l_c is the Lipschitz constant.

The sliding surface of speed and rotor flux module φ formulate vector s and both are the functions of differentiation. With the use of differentiator, the requirement of load torque has been eliminated for estimating the sliding surface.

QC2SMC (10) can be revised as

$u = - k \frac{z_{11} + | z_{00} |^{1 / 2} sign (z_{00})}{| z_{11} | + | z_{00} |^{1 / 2}}$ (13)k > 0 is selected suitably large to guarantee convergence in fixed time and disturbance rejection in the system [34]. Second order dynamic systems (6), its stability and convergence are evaluated through Lyapunov functions.

Theorem 1. Consider an IM represented in (6) with the control law (10) leads the rotor speed ω and flux ϕ so that tracking error as (ω - ω_ref) and (ϕ - ϕ_ref) tends to zero as the time tends to infinity.

Proof. The Lyapunov function for quasi continuous SOSM flux observer is chosen as

$V_{s} = \frac{1}{2} (s_{α}^{* 2} + s_{β}^{* 2})$ (14) and V_s = 0 only when s = 0. Taking the first derivative of V_s, we have

${\dot{V}}_{s} = s_{α}^{*} {\dot{s}}_{α}^{*} + s_{β}^{*} {\dot{s}}_{β}^{*}$ (15) using (6), (8) and (9), then (15) becomes $\begin{matrix} {\dot{V}}_{s} = s_{α}^{*} (D_{α} {\bar{f}}_{3} + {\bar{B}}_{2 α} s_{α}^{*} \frac{d}{dt} D_{α} + u_{α}) \\ + s_{β}^{*} (D_{β} {\bar{f}}_{4} + {\bar{B}}_{2 β} s_{β}^{*} \frac{d}{dt} D_{β} + u_{β}) \end{matrix}$ where ${\bar{f}}_{3}$ , ${\bar{f}}_{4}$ and ${\bar{B}}_{2 α}$ , ${\bar{B}}_{2 β}$ are smooth and bounded functions and these are known. Let L_r = L_0r + Δ L_r, where L_0r and ΔL_r indicate the nominal and uncertain part of rotor inductance. J = J₀ + Δ J₀, where J₀ and ΔJ₀ indicate the nominal and uncertain part of inertia. Suppose that the external disturbances δ and uncertain parameters ΔJ, ΔL_r are all bounded and that these bounds are included in bounded functions. The bounded function becomes ${\bar{B}}_{2} = {\bar{B}}_{02} + Δ {\bar{B}}_{2}$ . Consider $δ_{α} = D_{α} {\bar{f}}_{3}$ , $δ_{β} = D_{β} {\bar{f}}_{4}$ , $ϒ_{α} = {\bar{B}}_{2 α} s_{α}^{*} \frac{d}{dt} D_{α}$ and $ϒ_{β} = {\bar{B}}_{2 β} s_{β}^{*} \frac{d}{dt} D_{β}$

Then above equation becomes

$\begin{matrix} {\dot{V}}_{s} & = s_{α}^{*} (δ_{α} + ϒ_{α} + u_{α}) + s_{β}^{*} (δ_{β} + ϒ_{β} + u_{β}) \\ = \sum_{i = α}^{β} s_{i}^{*} (δ_{i} + ϒ_{i} + u_{i}) \end{matrix}$ (16) suppose Ψ_i = δ_i + ϒ_i

${\dot{V}}_{s} = \sum_{i = α}^{β} s_{i}^{*} (Ψ_{i} + u_{i})$ (17) by setting the controller as (10) and we have

$\begin{matrix} {\dot{v}}_{s} = \sum_{i = α}^{β} s^{*}_{i} [ψ_{i} - k_{i} (\frac{{\dot{s}}_{i}^{*} + {| s_{i}^{*} |}^{\frac{1}{2}} s i g n (s_{i}^{*})}{| {\dot{s}}_{i}^{*} | + {| s_{i}^{*} |}^{\frac{1}{2}}})] \\ = \sum_{i = α}^{β} s_{i}^{*} ψ_{i} - \sum_{i = α}^{β} k_{i} s_{i}^{*} s i g n (s_{i}^{*}) (\frac{{\dot{s}}_{i}^{*} s i g n (s_{i}^{*}) + {| s_{i}^{*} |}^{\frac{1}{2}}}{| {\dot{s}}_{i}^{*} | + {| s_{i}^{*} |}^{\frac{1}{2}}}) \\ = \sum_{α}^{β} | s_{i}^{*} | k_{i} [\frac{ψ_{i} s i g n (s_{i}^{*})}{k_{i}} - \frac{{\dot{s}}_{i}^{*} s i g n {(s_{i}^{*})}^{\frac{1}{2}}}{| {\dot{s}}_{i}^{*} | + {| s_{i}^{*} |}^{\frac{1}{2}}}] \end{matrix}$ To assure the attaining and sliding on the manifold, we need

$\frac{{\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{1 / 2}}{| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{1 / 2}} \geq \frac{Ψ_{i} sign (s_{i}^{*})}{k_{i}}$ (18) Since $(({\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{1 / 2}) / (| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{1 / 2})) \leq 1$ , (18) can be written as

$k_{i} \geq Ψ_{i} sign (s_{i}^{*})$ (19) If we select the gain k_i so that $k_{i} \geq Ψ_{i} sign (s_{i}^{*})$ , then ${\dot{V}}_{s} < 0$ . It imply that tracking error as (ω - ω_ref) and (ϕ - ϕ_ref) tends to zero as the time tends to infinity. This assurances the attainment and sliding on the surface.

4.2 FQC2SMC

In previous section (i.e. design of QC2SMC), SM controller gain k in the control law of equations (10) and (13) should be selected to be larger than the upper bound of the uncertainty. Due to large values of SM controller gain, significant chattering is generated even if sliding manifold converges to zero, $(s = \dot{s} = 0)$ . In extra case, k should be chosen as time-varying due to the uncertainties are frequently unknown and time varying. In this section, we designed compensated control (CC), whose gain is to adaptively tune k and the time-varying.

The CC is now considered as [40]

$u = - \hat{k} \frac{z_{11} + | z_{00} |^{1 / 2} sign (z_{00})}{| z_{11} | + | z_{00} |^{1 / 2}}$ (20) where $\hat{k}$ is the estimated value of k. Using fuzzy logic system with adaptive tune, the estimation gain $\hat{k}$ can be obtained. The general structure of the proposed FQC2SMC and FQC3SMC are given in Figure 1. The proposed control scheme describes using fuzzy system to adaptively tune the quasi continuous SM controller gain $\hat{k}$ . We take into consideration operating condition (initial condition, magnitude of the uncertainties, etc.) while constructing fuzzy IF-THEN rules. In the presence of the uncertainties, the operating conditions of the system can be illustrated as a set of points OP_i in the sliding surface s and the derivative of the sliding surface $\dot{s}$ state space as shown in figure 2. Each OP_i, . . . , OP_n represents an operating point corresponding to the magnitude of the uncertainties of the system at each time. Based on the knowledge of the controlled system, we decide the operation of the SM controller gain $\hat{k}$ corresponding to the output of the fuzzy system as follows: when the operating point OP is far from the sliding surface $(s, \dot{s} = 0)$ such as at point OP₁, the control signal of QC2SMC is continuous. The SM controller gain should be correspondingly large to force the state trajectories to reach the sliding surface rapidly. In contrast, when the system remains around the sliding surface such as at point OP_n, the control signal is now discontinuous [40]. The SM controller gain should be very small to reduce chattering. The FQC2SMC scheme to adaptively tune sliding gain is executed as follow:

Fig.1

The recommended control scheme of the fuzzy quasi continuous second / third order sliding mode control (FQC2SMC / FQC3SMC) for induction motor

Fig.2

Operating point corresponding to the magnitude of the uncertainties of the system at each time

Table 1

Fuzzy rule base

$\dot{s}$
s	P	Z	N
P	PB	P	Z
Z	P	Z	N
N	Z	N	NB

We considered two inputs s, $\dot{s}$ and one output u_fqc for fuzzy control with adaptively tune SM controller gain system as demonstrated in figure 1

Switching surface s represented as input variable, which is defined by using linguistic variables P (positive), Z (zero), N (negative). Another input variable $\dot{s}$ is also defined with the help of above mention linguistic variables. Then the output of fuzzy inference system as u_fqc described by using linguistic variables NB (negative big), N (negative), Z(zero), P (positive), PB (positive big). Table 1 shows the fuzzy tuning rule base, which are constructed based on the above operating condition. A fuzzy system with two non-interactive inputs x₁ and x₂ (antecedents) are expressed in the form of s and $\dot{s}$ respectively. Then, a single output u_fqc (consequent) is described by a collection of 5 linguistic IF-THEN proposition in the Mamdani form: [52]

IF x₁ is $A_{1}^{l}$ and x₂ is $A_{2}^{l}$ THEN $u_{fqc}^{l}$ is B^l for l = 1, 2, . . . .5

where $A_{1}^{l}$ and $A_{2}^{l}$ are the fuzzy sets representing the l^th antecedent pairs, and B^l is the fuzzy set representing the l^th consequent. i^th = 1, 2 are number of antecedents. By using the center of gravity defuzzification, the output value of fuzzy system is[40, 52]

$u_{fqc} = \frac{\int_{i} μ_{F_{i}^{l}} (x_{i}) \cdot x_{i} dx}{\int_{i} μ_{F_{i}^{l}} (x_{i}) dx}$ (21) where x_i is a running point in the continuous universe, $μ_{F_{i}^{l}} (x_{i})$ is the membership functions of the input variable x_i. $μ_{F_{1}^{l}} (x_{1})$ and $μ_{F_{2}^{l}} (x_{2})$ are the membership functions of the fuzzy sets s and $\dot{s}$ and the membership functions of the fuzzy output u_fqc are shown in figure 3a, 3b and 3c respectively. With two inputs and one output the input-output mapping is a surface. Figure 4 is a mesh plot of the control surface of the fuzzy inference engine not a flat surface. The plot results from a rule base with nine rules and the surface is more or less bumpy. The horizontal plateaus are due to flat peaks on the input sets. The plateau around the origin implies a low sensitivity towards changes in either error or derivatives of error near the reference. Flat input sets produce flat plateaus and large gains far away from thereference.

Fig.3

The membership functions of the fuzzy subsets (a): s, (b): $\dot{s}$ and (c): fuzzy output u_fqc

Fig.4

Mesh plot of the control surface of the fuzzy inference engine

Next, SM controller gain with the adaptively tune is designed as

$\hat{k} = u_{fqc}$ (22) from the equation (10), (13) and (20), the FQC2SMC is designed as

$u = u_{fqc} \frac{z_{11} + | z_{00} |^{1 / 2} sign (z_{00})}{| z_{11} | + | z_{00} |^{1 / 2}}$ (23)

5 Fuzzy-adapted quasi continuous third order sliding mode controller

In case of QC2SMC, some chattering effect occurs, thus relative degree is increased upto 3, and third order SMC may be used. QC3SMC normally provides very accurate output.

5.1 QC3SMC

5.1.1 Control Law

Consider the QC3SMC [34]

$\begin{matrix} u = - k [{\ddot{s}}^{*} + β_{1} {(| {\dot{s}}^{*} | + | s^{*} |^{2 / 3})}^{- 1 / 2} \\ ({\dot{s}}^{*} + | s^{*} |^{2 / 3} sign (s^{*}))] \\ / [| {\ddot{s}}^{*} | + β_{2} {(| {\dot{s}}^{*} | + | s^{*} |^{2 / 3})}^{1 / 2}] \end{matrix}$ (24)

The positive gains k, β₁ and β₂ are the adjustment parameters of the controller and it is calculated by simulations. It is extremely difficult to find out ${\dot{s}}^{*}$ and ${\ddot{s}}^{*}$ from this system, we employ the second order differentiator [50] for evaluation of ${\dot{s}}^{*}$ and ${\ddot{s}}^{*}$ .

5.1.2 Second order robust differentiator

A second order Levant differentiator [50] is $\begin{matrix} {\dot{z}}_{00} & = & v_{0} \\ v_{0} & = & - λ_{0} {| z_{00} - s^{*} |}^{\frac{2}{3}} s i g n (z_{00} - s^{*}) + z_{11} \\ {\dot{z}}_{11} & = & v_{1} \\ v_{1} & = & - λ_{1} {| z_{11} - v_{0} |}^{\frac{1}{2}} s i g n (z_{11} - v_{0}) + z_{22} \\ {\dot{z}}_{22} & = & - λ_{2} s i g n (z_{22} - v_{1}) \end{matrix}$ (25)

where z₀₀, z₁₁ and z₂₂ are estimates of s^∗, $\dot{s^{*}}$ and $\ddot{s^{*}}$ respectively.

QC3SMC (24) can be rewritten as $\begin{matrix} u = - k [z_{22} + β_{1} {(| z_{11} | + | z_{00} |^{2 / 3})}^{- 1 / 2} \\ (z_{11} + | z_{00} |^{2 / 3} sign (z_{00}))] \\ / [| z_{22} | + β_{2} {(| z_{11} | + | z_{00} |^{2 / 3})}^{1 / 2}] \end{matrix}$ (26)

The adjustment parameters of the differentiator are λ₀, λ₁ and λ₂ [50].

5.2 FQC3SMC

The compensated control u is now designed by replacing k by $\hat{k}$ in (26). where the adjustable gain constant $\hat{k}$ is the estimated value of k. To obtain the estimation gain $\hat{k}$ , fuzzy control with an adaptive tune is employed. Also consider $\hat{k} = u_{fqc}$ is the output of an adaptive fuzzy control which is defined same as in FQC2SMC section. The general structure of proposed FQC3SMC scheme is demonstrated in figure 1.

Theorem 2. Consider an IM represented in (6) with the control law (24)and the second order exact differentiator (25) leads the rotor speed ω and flux ϕ so that tracking error as (ω - ω_ref) and (ϕ - ϕ_ref) tends to zero as the time tends to infinity.

Proof. Similar to the proof of stability of QC2SMC in theorem 1 in design of FQC2SMC section. The Lyapunov function for quasi continuous SOSM flux observer is chosen as

$V_{s} = \frac{1}{2} (s_{α}^{* 2} + s_{β}^{* 2})$ (27) and V_s = 0 only when s = 0. Taking the first derivative of V_s, we have

${\dot{V}}_{s} = s_{α}^{*} {\dot{s}}_{α}^{*} + s_{β}^{*} {\dot{s}}_{β}^{*}$ (28) We choose the control law (24). By setting this controller into (17), and letting Ψ_i = δ_i + ϒ_i, we have

$\begin{matrix} {\dot{V}}_{s} & = \sum_{i = α}^{β} s_{i}^{*} [Ψ_{i} - k_{i} \\ [\frac{{\ddot{s}}_{i}^{*} + β_{1} {(| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{2 / 3})}^{- 1 / 2} ({\dot{s}}_{i}^{*} + | s_{i}^{*} |^{2 / 3} sign (s_{i}^{*}))}{| {\ddot{s}}_{i}^{*} | + β_{2} {(| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{2 / 3})}^{1 / 2}}] \end{matrix}$

$\begin{matrix} = & \sum_{i = α}^{β} s_{i}^{*} Ψ_{i} - \sum_{i = α}^{β} k_{i} s_{i}^{*} sign (s_{i}^{*}) \\ {[{\ddot{s}}_{i}^{*} sign (s_{i}^{*}) + β_{1} {({\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}^{- 1 / 2} * \\ ({\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{2 / 3})] \\ / [| {\ddot{s}}_{i}^{*} | + β_{2} {(| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{2 / 3})}^{1 / 2}]} \end{matrix}$

$\begin{matrix} = & \sum_{i = α}^{β} | s_{i}^{*} | k_{i} \\ {\frac{Ψ_{i} sign (s_{i}^{*})}{k_{i}} - \frac{{\ddot{s}}_{i}^{*} sign (s_{i}^{*}) + β_{1} \frac{({\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}{{({\dot{s}}_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}^{1 / 2}}}{| {\ddot{s}}_{i}^{*} | + β_{2} {(| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{2 / 3})}^{1 / 2}}} \end{matrix}$ To assurance the attaining and sliding on the manifold, we need

$\frac{{\ddot{s}}_{i}^{*} sign (s_{i}^{*}) + β_{1} \frac{({\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}{{({\dot{s}}_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}^{1 / 2}}}{| {\ddot{s}}_{i}^{*} | + β_{2} {(| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{2 / 3})}^{1 / 2}} \geq \frac{Ψ_{i} sign (s_{i}^{*})}{k_{i}}$ (29) Since $({\ddot{s}}_{i}^{*} sign (s_{i}^{*}) + β_{1} \frac{({\dot{s}}_{i}^{*} sign (s_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}{{({\dot{s}}_{i}^{*}) + | s_{i}^{*} |^{2 / 3})}^{1 / 2}}) / (| {\ddot{s}}_{i}^{*} | + β_{2} {(| {\dot{s}}_{i}^{*} | + | s_{i}^{*} |^{2 / 3})}^{1 / 2}) \leq 1$ , (29) can be written as

$k_{i} \geq Ψ_{i} sign (s_{i}^{*})$ (30) If we select the gain k_i so that $k_{i} \geq Ψ_{i} sign (s_{i}^{*})$ , then ${\dot{V}}_{s} < 0$ . It imply that tracking error as (ω - ω_ref) and (ϕ - ϕ_ref) tends to zero as the time tends to infinity. This assurances the attainment and sliding on the surface.

6 Observer design

Flux measurements are the challenging task in IM. Alternatively, by measuring stator currents and rotor speed, flux estimation using observer is possible under various circumstances like parametric uncertainties and disturbance. Due to this, robustness, elimination of chattering and cheapness have the main advantages offered by observer. For the elimination of chattering effect, an asymptotic observer is included in the control loop. Due to the load or machine parametric inaccuracy, chattering is occurred in control signal. In IM, the chattering occurrence is caused by the load or machine parameter inaccuracy. Another possible reason is the unmodelled dynamics in the system, e.g. the lag or transport delay in the inverter or the sensors [14].

6.1 Flux observer

The observer is designed to estimate rotor fluxes by measuring ω and (i_α, i_β) based on the switching control input with STA [14 , 54]. The recommended observer is given as $\frac{d {\hat{λ}}_{α}}{dt} = - η {\hat{λ}}_{α} - ω {\hat{λ}}_{β} + η L_{m} i_{α} + l_{1} V_{1 α}$ (31) $\frac{d {\hat{λ}}_{β}}{dt} = - η {\hat{λ}}_{β} + ω {\hat{λ}}_{α} + η L_{m} i_{β} + l_{2} V_{2 β}$ (32) $\begin{matrix} \frac{d {\hat{i}}_{α}}{dt} & = & β η {\hat{λ}}_{α} + β ω {\hat{λ}}_{β} - γ {\hat{i}}_{α} + \frac{1}{σ L_{s}} u_{α} - V_{1 α} \end{matrix}$ (33) $\begin{matrix} \frac{d {\hat{i}}_{β}}{dt} & = & β η {\hat{λ}}_{β} - β ω {\hat{λ}}_{α} - γ {\hat{i}}_{β} + \frac{1}{σ L_{s}} u_{β} + V_{2 β} \end{matrix}$ (34) where ${\hat{λ}}_{α}$ , ${\hat{λ}}_{β}$ , are observed fluxes of the rotor, ${\hat{i}}_{α}$ , ${\hat{i}}_{β}$ , are the observed stator currents and l₁, l₂ are the observer gains. The inputs V_1α, V_2β are derived from the STA [38, 54]. $\begin{matrix} V_{1 α} & = & - K_{λ} | i_{α} - {\hat{i}}_{α} |^{1 / 2} sign (i_{α} - {\hat{i}}_{α}) + w_{β} \end{matrix}$ (35) $V_{2 β} = - K_{λ} | i_{β} - {\hat{i}}_{β} |^{1 / 2} sign (i_{β} - {\hat{i}}_{β}) + w_{α}$ (36) $\frac{{dw}_{α}}{dt} = - K_{α} sign (i_{α} - {\hat{i}}_{α})$ (37) $\frac{{dw}_{β}}{dt} = - K_{α} sign (i_{β} - {\hat{i}}_{β})$ (38) l₁, l₂, K_λ and K_α are positive parameters suitably high to guarantee the convergence of observer.

6.2 Load Torque Observer

In [38] the observer design, the rotor fluxes and speed ω have been used as an input to the system. According to that, motor model is given as $\hat{ω} = \frac{1}{J} {c_{5} (i_{β} λ_{α} - i_{α} λ_{β}) - T_{l}}$ (39) $\dot{T_{l}} = 0$ (40) The suggested observer is given by

$\begin{matrix} \dot{\hat{ω}} = \frac{1}{J} {c_{5} (i_{β} {\hat{λ}}_{α} - i_{α} {\hat{λ}}_{β}) - \hat{T_{l}}} + l_{1} (ω - \hat{ω}) \end{matrix}$ (41) $\dot{T_{l}} = l_{2} (ω - \hat{ω})$ (42) By selecting observer gains l₁ and l₂ in such manner that closed loop system become stable.

7 Simulation results

In this section, performance of First-order SMC [55, 56], QC2SMC [39], QC3SMC [38], FQC2SMC and FQC3SMC (proposed) schemes have been investigated on the three phase IM. The investigations are recognized using simulation in MATLAB- Simulink through numerical integration technique of Euler with the sampling frequency of 10 KHz (a fixed step of 10^-4 sec.). An IM and controller parameters are shown in Table 3 and table4 respectively.

Table 2
Parameters of an induction motor used in simulation

Rotor Type: Squirrel Cage Nominal Power = 0.186 KW

Voltage(Line to line)= 220 V Frequency = 60 Hz

R_s= 14 Ω L_s= 0.4 H

R_r= 10.1 Ω L_r= 0.4128 H

L_m= 0.377 H J = 0.01N . m . s²

N_r= 2 1660 rpm

Rotor Type: Squirrel Cage	Nominal Power = 0.186 KW
Voltage(Line to line)= 220 V	Frequency = 60 Hz
R_s= 14 Ω	L_s= 0.4 H
R_r= 10.1 Ω	L_r= 0.4128 H
L_m= 0.377 H	J = 0.01N . m . s²
N_r= 2	1660 rpm

Table 3

Controller parameters for an induction motor used in simulation

Controller type	Flux observer	Speed observer	Differentiator	Control law
QC2SMC [39]	k_α =-0.1	l₁ =-0.0231	λ₁ =1.1068e+03	k =8.576217e+03
	k_λ =-0.1	l₂ =-0.0231	λ₂ =1.5e+03
FQC2SMC (proposed)	k_α =-0.1	l₁ =-0.0231	λ₁ =1.1068e+03	k =751
	k_λ =-0.1	l₂ =-0.0231	λ₂ =1.5e+03
QC3SMC [39]	k_α =0.9	l₁ =0.9	λ₀ =1.1068e+03	k =1.3790e+06
	k_λ =0.9	l₂ =1.5	λ₂ =1.1068e+03	β₁ =β₂ =200
			λ₁ =2.1e+05
			λ₃=2.1e+03
FQC3SMC (proposed)	k_α =0.9	l₁ =0.9	λ₀ =1.1068e+03	k =3e+04
	k_λ =0.9	l₂ =1.5	λ₂ =1.1068e+03	β₁ =β₂ =200
			λ₁ =2.1e+05
			λ₃=2.1e+03

The constants in the mathematical model of an IM are obtained from the table 2: η = 24.4671, σ = 0.1392, β = 16.3977, γ = 402.6218, c₁ = 17.9549, c₂ = 0.9133, c₃ =24.4671, c₄ =9.2241, c₅ = 273.9826, a₁ = 22.4241.

A. Rotor speed tracking The performance of these controllers have been investigated under various circumstances. The first case involves electrical rotor speed controlling under no load, while second and third are related to parametric uncertainties and disturbance. The each case of simulation is divided into two parts. In first part, we validate the potential of FQC2SMC compared with QC2SMC. In second part of simulation, the proposed FQC3SMC is compared with QC3SMC. Also the performance of controllers were assessed based on its robustness, exactness, fast tracking and chattering effect of the plant output.

Under no load condition : In the rotor speed tracking, behavior analysis of the controllers are carried at low speed so that reference value chosen as 100 rad/sec. for the first part of simulation, rotor speed tracking performance of QC2SMC and FQC2SMC are studied. From figure 5(a.ii) and 5(b.i), we see that FQC2SMC converges faster and with less error than QC2SMC. QC2SMC required large SM controller gain values (k = 8.576217 * 10³) to bring system states on sliding manifold $s = \dot{s} = 0$ . However, it is very difficult to select SM controller gain that minimize the reaching time on sliding manifold. Also significant chattering is generated by QC2SMC when the system remains in the sliding manifold. The control efforts of QC2SMC on u_α and u_β axis is shown in figures 6(a.ii) and 7(a.ii) respectively. By applying fuzzy system to adaptively tune the SM controller gain (k=751), FQC2SMC almost eliminates the chattering as compared to QC2SMC as shown in figure 6(b.i) and 7(b.i).

Fig.5

The response of actual rotor speed and the estimated rotor speed at no load

Fig.6

Control signal as stator current on alpha axis (u_α)

Fig.7

Control signal as stator current on beta axis (u_β)

In second part of simulation, the proposed FQC3SMC is compared with QC3SMC. A phase portrait is the graphical representation of speed tracking error variable moving towards the origin as shown in figure 8. In case of QC2SMC, we require to know ( $\dot{s}$ ), but, it is very difficult to determine. We employed the first order Levant differentiator to find ( $\dot{s}$ ). Similarly for QC3SMC, we must know ( ${\dot{s}}^{*}$ and ${\ddot{s}}^{*}$ ), which is very difficult to determine. Owing to this reason, we employ the second order differentiator for evaluation of ${\dot{s}}^{*}$ and ${\ddot{s}}^{*}$ . Therefore, sliding variable of QC3SMC required more time to reach the origin in state space as presented in phase portrait. Hence transient response of QC3SMC is very slow. Also QC3SMC require large SM controller gain values (k = 1.379 * 10⁶) as shown in figures 6(a.iii) and 7(a.iii). This is main constraint of QC3SMC as appraise to QC2SMC. By applying fuzzy system to adaptively tune the sliding gain (k = 3 *10⁴), FQC3SMC provides fast tracking output with less error as compared to QC3SMC, FQC2SMC and QC2SMC techniques as shown in figure 5(b.ii) (i.e. minimum settling time of FQC3SMC is around t = 0.1msec, while QC2SMC having around t = 2.5msec). Figure 5(a.iii) and figure 5(b.ii) are illustrated as rotor speed tracking performance of QC3SMC and FQC3SMC. The control efforts of FQC3SMC on u_α and u_β axis is shown in figures 6(b.ii) and 7(b.ii) respectively.

Fig.8

Phase Portrait of the sliding manifold

Discussion: It is very difficult to choose SM controller gain and require large values in case of QC3SMC. QC3SMC system becomes too slow and significant chattering is generated. This difficulty can be overcome by applying fuzzy system to adaptively tune the SM controller gain (which is also low value). So that FQC2SMC provides less tracking error, smooth control action and fastresponse.

Remark 1. As compared with quasi continuous SM approach described in [37 –39], the proposed controller required less hitting gain value as k = 751 under the influence of the lumped uncertainties in system model.

Under parametric uncertainties : In the actual situation, the parameter variation in the value of L_r i.e. ΔL_r =+3 % as the mass properties in an IM. Here robustness of suggested controllers are examined under this condition. Due to uncertain condition, the nominal part ${\bar{B}}_{2}$ varies to ${\bar{B}}_{2} + Δ {\bar{B}}_{2}$ , which is represented by the smooth and bounded function matrix as

$\begin{matrix} {\bar{B}}_{2} = 10^{5} * [\begin{matrix} - 2.6156 & 2.6156 \\ 0.0018 & 0.0018 \end{matrix}], \\ Δ {\bar{B}}_{2} = 10^{4} * [\begin{matrix} 4.6370 & - 4.6370 \\ - 0.0040 & - 0.0040 \end{matrix}] \end{matrix}$

The robustness of the suggested control strategy has been verified by including $Δ {\bar{B}}_{2}$ . QC2SMC and QC3SMC produced significant chattering due to large SM controller gain k values the uncertainties exists in the system. From the table 4, table 5, and figure 9(b.ii), FQC3SMC produce less tracking error than QC2SMC & FQC2SMC under parametric uncertainties. Due to the uncertainties, which are frequently unknown and time varying, error in the system becomes time varying. Figure 10 represents selection of k as time-varying with respect to error function.

Table 4

Mean Squared Error (MSE): Performance comparison of QC2SMC, FQC2SMC, QC3SMC and FQC3SMC

Controller	at noload	certain parameter variation	disturbance	load torque
QC2SMC [39]	7.3729	8.5033	9.2770	19.2674
FQC2SMC (proposed)	4.1197	4.4737	4.1444	16.2102
QC3SMC [39]	0.3604	0.7354	5.7687	16.2001
FQC3SMC (proposed)	0.2263	0.3808	0.5334	9.8498

Table 5

Mean absolute Error (MAE): Performance comparison of QC2SMC, FQC2SMC, QC3SMC and FQC3SMC

Controller	at noload	certain parameter variation	disturbance	load torque
QC2SMC [39]	1.4625	1.6551	1.4625	4.1465
FQC2SMC (proposed)	0.1250	0.3351	0.1712	3.7250
QC3SMC [39]	0.2049	0.5836	1.2085	3.3001
FQC3SMC (proposed)	0.0827	0.2813	0.5460	3.2115

Fig.9

The response of actual rotor speed and the estimated rotor speed with uncertain parameter variation

Fig.10

Behavior of error in speed and sliding mode controller gain k

Discussion: Robustness property i.e. insensitivity to uncertain variation in parameter of the HOSM is preserved and the chattering is reduced.

Remark 2. Fuzzy logic tool employed to generate hitting signal as stated in (22) of quasi continuous SMC algorithm, which has less magnitude value and also provide smooth control action in speed and flux tracking control problem. Fuzzy adaptation technique applied to minimize the effect of time-varying uncertainties which maintain the transient behavior in complex condition.

Under the influence of disturbances : The internal disturbance rejection response of rotor speed tracking is presented in figure 11a and b. The external disturbance rejection response of rotor speed tracking is presented in figure 12a and b. Table 4 and 5 shows signified performance function such as the mean squared error (MSE) and mean absolute error (MAE), which exhibits a very low value of FQC3SMC in comparison with other techniques.

Fig.11

The response of actual rotor speed and the estimated rotor speed with internal disturbances

Fig.12

The response of actual rotor speed and the estimated rotor speed at load Torque

Discussion: The FQC3SMC has more accurate motion tracking with the speed remains significant unchanged in disturbance and load torqueconditions.

B. Observers The validity of recommended flux and torque observers are evaluated via simulation with initial conditions close to zero. In case of flux observer design, rotor flux has been estimated by considering rotor speed ω and stator currents (i_α, i_β) are as input of the system. Similarly, in torque observer design, the rotor speed has been estimated and the magnetic flux of rotor in company with stator currents are considered as input of the system. Due to external disturbances as vibration and measurement noise causes variation in measured rotor speed, which can be eliminated by using observer based solution. Parameters of observers, controller gain and differentiators parameters are selected suitably as shown in table 3. After selecting these parameters, estimated flux will converge to the real ones, when SM will happen as demonstrated in figure 13a and b.

Fig.13

The response of rotor flux module and its estimation

Figures 5, 9, 11 and 12 depicts responses of the estimated rotor speed and a real speed under no load, parametric uncertainties, influence of internal disturbances and load torque. It is examined that the estimated rotor speed converge quickly to the real values in FQC3SMC with less error. The performance index i.e. integral square error (ISE) for the square rotor flux module of QC2SMC, FQC2SMC, QC3SMC and FQC3SMC are presented in figure 14(a.ii), (b.i), (a.iii) and (b.ii) respectively. The performance index i.e. integral square error (ISE) for the rotor speed of QC2SMC, FQC2SMC, QC3SMC and FQC3SMC are presented in figure 15(a.ii), (b.i), (a.iii) and (b.ii)respectively.

Fig.14

Integral square error (ISE) of rotor flux

Fig.15

Integral square error (ISE) of rotor speed

Discussion: SMO can work properly in adverse condition. The performance of observers exhibit more precise, fast tracking and robustness in existence of uncertain variation in parameter, influence of internal disturbances and load torque.

Remark 3. In view of above cases, QC3SMC provides more accurate motion tracking and smooth control action than QC2SMC. Unfortunately, the time response is much slower compared with that of QC2SMC. This difficulty can be solved by employing fuzzy system to adaptively tune SM controller gain. The FQC3SMC has less tracking error, smooth control action and a fast transient response. From the results, FQC3SMC could be a good selection for IM controller design.

8 Conclusion

In this manuscript, speed and rotor flux tracking control schemes based on quasi continuous second and third-order SMC along with robust differentiators and fuzzy system for IM have been investigated. These investigations have been carried out under the influence of load disturbances and uncertain parameters variations. These quasi-continuous approach provides less chattering and higher accuracy compared with conventional SMC. In addition, a fuzzy logic control scheme was applied to adaptively tune the SM controller gain so as to reduce chattering, high accuracy and fast response. Usefulness of the recommended approach is investigated by comparing computer simulation results of QC2SMC, FQC2SMC, QC3SMC and FQC3SMC.

Footnotes

Acknowledgements

The authors would like to thank BCUD (Board of College and University Development), S. P. Pune University for funding the research project (Sanction No. BCUD/OSD/ 390).

References

P.L.

Jansen and

R.D.

Lorenz , Accuracy limitations of velocity and flux estimation in direct field oriented induction machines, In Power Electronics and Appli- cations, Fifth European Conference on, vol. 4, 1993, pp. 312–318.

P.L.

Jansen ,

R.D.

Lorenz and

D.W.

Novotny , Observer-based direct field orientation: Analysis and comparison of alternative methods, Industry Applications, IEEE Transactions on 30(4) (1994), 945–953.

F.-Z.

Peng and

Fukao , Robust speed identification for speed-sensorless vector control of induction motors, Indus Try Applications, IEEE Transactions on 30(5) (1994), 1234–1240.

D.A.

Dominic and

T.R.

Chelliah , Analysis of fieldoriented controlled induction motor drives under sensor faults and an overview of sensorless schemes, {ISA} Transactions 53(5) (2014), 1680–1694. {ICCA} 2013.

Tursini ,

Petrella and

Parasiliti , Adaptive sliding-mode observer for speed-sensorless control of induction motors, Industry Applications, IEEE Transactions on 36(5) (2000), 1380–1387.

Li ,

Xu and

Zhang , An adaptive sliding-mode observer for induction motor sensorless speed control, Industry Applications, IEEE Transactions on 41(4) (2005), 1039–1046.

Trabelsi ,

Khedher ,

M.M.

Faouzi and

Faouzi , Backstepping control for an induction motor using an adaptive sliding rotor-flux observer, Electric Power Systems Research 93 (2012), 1–15.

Barambones and

Alkorta , Position control of the induction motor using an adaptive sliding-mode controller and observers, Industrial Electronics, IEEE Transactions on 61(12) (2014), 6556–6565.

Kumar ,

T.R.

Chelliah and

S.P.

Srivastava , Adaptive control schemes for improving dynamic performance of efficiency-optimized induction motor drives, {ISA} Transactions 57 (2015), 301–310.

10.

Zaafouri ,

C.B.

Regaya ,

H.B.

Azza and

ChAcari , Dsp-based adaptive backstep ping using the tracking errors for high-performance sensorless speed control of induction motor drive, {ISA} Transactions 60 (2016), 333–347.

11.

V.I.

Utkin , Sliding mode control design principles and applications to electric drives, Industrial Electronics, IEEE Transactions on 40(1) (1993), 23–36.

12.

Benchaib ,

Rachid ,

Audrezet and

Tadjine , Realtime sliding-mode observer and control of an induction motor, Industrial Electronics, IEEE Transactions on 46(1) (1999), 128–138.

13.

Benchaib ,

Rachid and

Audrezet , Sliding mode inputoutput linearization and field orientation for real-time control of induction motors, Power Electronics, IEEE Transactions on 14(1) (1999), 3–13.

14.

Yan and

Utkin , Sliding mode observers for electric machines-an overview, In IECON02 [Industrial Electronics Society, IEEE 2002 28th Annual Conference of the], vol. 3, 2002, pp. 1842–1847.

15.

T.-J.

Fu and

W.-F.

Xie , A novel sliding-mode control of induction motor using space vector modulation technique, {ISA} Transactions 44(4) (2005), 481–490.

16.

Comanescu , Single and double compound manifold sliding mode observers for flux and speed estimation of the induction motor drive, Electric Power Applications, IET 8(1) (2014), 29–38.

17.

Hammou ,

Kebbati and

Mansouri , New algorithms of control and observation of the induction motor based on the sliding mode theory, Electric Power Components and Systems 43(5) (2015), 520–532.

18.

J.R.

Dominguez , Discrete-time modeling and control of induction motors by means of variational integrators and sliding modes-part ii: Control design, Industrial Electronics, IEEE Transactions on (PP99) (2015), 1–1.

19.

M.K.

Masood ,

W.P.

Hew and

N.A.

Rahim , Review of anfis-based control of induction motors, Journal of Intelligent & Fuzzy Systems 23(4) (2012), 143–158.

20.

Lokriti ,

Salhi ,

Doubabi and

Zidani , Induction motor speed drive improvement using fuzzy ip-self-tuning controller. A real time implementation, { ISA} Transactions 52(3) (2013), 406–417.

21.

Rafa ,

Larabi ,

Barazane ,

Manceur ,

Essoun-bouli and

Hamzaoui , Implementation of a new fuzzy vector control of induction motor, {ISA} Transactions 53(3) (2014), 744–754.

22.

Daryabeigi ,

N.R.

Abjadi and

G.R.

Arab Markadeh , Automatic speed control of an asymmetrical six-phase induction motor using emotional controller (belbic), Journal of Intelligent and Fuzzy Systems 26(4) (2014), 1879–1892.

23.

Senthilkumar and

Vijayan , High performance emotional intelligent controller for induction motor speed control, Journal of Intelligent & Fuzzy Systems 27(2) (2014), 891–900.

24.

Ramesh ,

A.K.

Panda and

S.S.

Kumar , Type-2 fuzzy logic control based MRAS speed estimator for speed sensorless direct torque and flux control of an induction motor drive, {ISA} Transactions 57 (2015), 262–275.

25.

Vahedi ,

M.H.

Zarif and

A.A.

Kalat , Speed control of induction motors using neuro-fuzzy dynamic sliding mode control, Journal of Intelligent & Fuzzy Systems 29(1) (2015), 365–376.

26.

Vahedi ,

M.H.

Zarif and

A.A.

Kalat , A simple stable adaptive neuro-fuzzy speed controller for induction motors, Journal of Intelligent and Fuzzy Systems 29(2) (2015), 571–581.

27.

Saghafinia ,

H.W.

Ping ,

M.N.

Uddin and

K.S.

Gaeid , Adaptive fuzzy sliding-mode control into chatteringfree im drive, Industry Applications, IEEE Transactions on 51(1) (2015), 692–701.

28.

Tir ,

O.P.

Malik and

A.M.

Eltamaly , Fuzzy logic based speed control of indirect field oriented controlled double star induction motors connected in parallel to a single six-phase inverter supply, Electric Power Systems Research 134 (2016), 126–133.

29.

Li ,

Du ,

Yang and

Gui , Passivity-based asynchronous sliding mode control for delayed singular markovian jump systems, IEEE Transactions on Automatic Control, 2017, pp. 1–1.

30.

Li ,

Shi ,

C.C.

Lim and

Wu , Fault detection filtering for nonhomogeneous markovian jump systems via a fuzzy approach, IEEE Transactions on Fuzzy Systems 26(1) (2018), 131–141.

31.

M.T.

Angulo and

R.V.

Carrillo-Serrano , Estimating rotor parameters in induction motors using high-order sliding mode algorithms, Control Theory Applications, IET 9(4) (2015), 573–578.

32.

Han and

Liu , Continuous higher-order sliding mode control with time-varying gain for a class of uncertain nonlinear systems, {ISA} Transactions 62 (2016), 193–201. SI: Control of Renewable Energy Systems.

33.

Fridman and

Levant , Sliding modes control in engineering. Control Engineering series, Ed.W. Perruquetti, J.P. Barbot, Marcel Dekker, New York, 2002.

34.

Levant , Quasi-continuous high-order sliding-mode controllers, Automatic Control, IEEE Transactions on 50(11) (2005), 1812–1816.

35.

Levant , Homogeneous quasi continuous sliding mode control. ser Lecture Notes in Control and Information Sciences, Berlin, Germany: Springer-Verlag, 2006, pp. 143–168.

36.

Levant and

Pavlov , Generalized homogeneous quasi continuous controllers, Int J Robust Nonlinear Control 18(4/5) (2007), 385–398.

37.

Pukdeboon ,

A.S.I.

Zinober and

M.W.L.

Thein , Quasi-continuous higher order sliding-mode controllers for spacecraft-attitude-tracking maneuvers, IEEE Transactions on Industrial Electronics 57(4) (2010), 1436–1444.

38.

J.M.

Canedo ,

Fridman ,

A.G.

Loukianov and

F.A.

Valen-zuela , Third-order quasi-continuo control of induction motor, In Electrical Engineering, Computing Science and Automatic Control, CCE, 2009 6th International Conference on, 2009, pp. 1–6.

39.

F.A.

Valenzuela ,

O.A.

Morfin ,

J.C.

Rodriguez ,

R.J.

Ramirez and

M.I.

Castellanos , High performance controller design applied to the induction motor via quasicontinuous sosm, 22nd International Conference on Electrical Communications and Computers (CONIELECOMP), 2012, pp. 124–129.

40.

Van ,

H.J.

Kang and

K.-S.

Shin , Adaptive fuzzy quasi-continuous high-order sliding mode controller for output feedback tracking control of robot manipulators, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228(1) (2014), 90–107.

41.

S.-H.

Ryu and

J.-H.

Park , Auto-tuning of sliding mode control parameters using fuzzy logic, In Proceedings of the 2001 American Control Conference (CatNo.01CH37148), vol. 1, 2001, pp. 618–623.

42.

C.-Y.

Liang and

Su , A new approach to the design of a fuzzy sliding mode controller, 139 (2003), 111–124.

43.

R.-J.

Wai and

K.-H.

Su , Adaptive enhanced fuzzy sliding-mode control for electrical servo drive, IEEE Transactions on Industrial Electronics 53(2) (2006), 569–580.

44.

A.F.

Amer ,

E.A.

Sallam and

W.M.

Elawady , Adaptive fuzzy sliding mode control using supervisory fuzzy control for 3 dof planar robot manipulators, Appl Soft Comput 11(8) (2011), 4943–4953.

45.

G.V.

Lakhekar , Tuning and analysis of sliding mode controller based on fuzzy logic, International Journal of Control and Automation 5(3) (2012), 93–110.

46.

G.V.

Lakhekar and

L.M.

Waghmare , Dynamic Fuzzy Sliding Mode Control of Underwater Vehicles, Springer International Publishing, Cham, 2015, pp. 279–304.

47.

Leonard , Control of Electrical Drives, Springer Verlag, Berlin, Germany, 1985.

48.

Peter , Sensorless Vector and Direct Torque Control, Oxford University Press, 1998.

49.

Fridman ,

Shtessel ,

Edwards and

Yan , Higher order sliding mode observer for state estimation and input reconstruction in nonlinear system, International Journal of Robust and Nonlinear Control (2007), 399–412.

50.

Levant , Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control 76(9–10) (2003), 924–941.

51.

Levant , Robust exact differentiation via sliding mode technique*, Automatica 34(3) (1998), 379–384.

52.

T.J.

Ross , Fuzzy Logic With Engineering Applications, 2ND ED. Wiley India Pvt Limited, 2007.

53.

Yan ,

Jin and

V.I.

Utkin , Sensorless slidingmode control of induction motors, Industrial Electronics, IEEE Transactions on 47(6) (2000), 1286–1297.

54.

Davila ,

Fridman and

Levant , Second-order sliding-mode observer for mechanical systems, Automatic Control, IEEE Transactions on 50(11) (2005), 1785–1789.

55.

Derdiyok ,

M.K.

Guven ,

Rehman ,

Inanc and

Xu , Design and implementation of a new sliding-mode observer for speed-sensorless control of induction machine, Industrial Electronics, IEEE Transactions on 49(5) (2002), 1177–1182.

56.

Rehman ,

Derdiyok ,

M.K.

Guven and

Xu , A new current model flux observer for wide speed range sensorless control of an induction machine, Power Electronics, IEEE ransactions on 17(6) (2002), 1041–1048.

Quasi continuous sliding mode control with fuzzy switching gain for an induction motor

Abstract

Keywords

1 Introduction

2 Induction Motor Model description

4.1. QC2SMC

4.1.1. Sliding Manifold

5.1 QC3SMC

5.1.1 Control Law

6.1 Flux observer

Table 2 Parameters of an induction motor used in simulation Rotor Type: Squirrel Cage Nominal Power = 0.186 KW Voltage(Line to line)= 220 V Frequency = 60 Hz R s = 14 Ω L s = 0.4 H R r = 10.1 Ω L r = 0.4128 H L m = 0.377 H J = 0.01N . m . s2 N r = 2 1660 rpm

Footnotes

Acknowledgements

References

Table 2
Parameters of an induction motor used in simulation

Rotor Type: Squirrel Cage Nominal Power = 0.186 KW

Voltage(Line to line)= 220 V Frequency = 60 Hz

R_s= 14 Ω L_s= 0.4 H

R_r= 10.1 Ω L_r= 0.4128 H

L_m= 0.377 H J = 0.01N . m . s²

N_r= 2 1660 rpm