Backstepping control and revamped recurrent fuzzy neural network with mended ant colony optimization applied in SCRIM drive system

Abstract

A six-phase copper rotor induction motor (SCRIM) drive system causes a lot of nonlinear effects such as nonlinear uncertainties. To obtain better performance, the backstepping control system using switching function is firstly proposed for controlling the SCRIM drive system. To reduce chattering in control effort, the backstepping control system using revamped recurrent fuzzy neural network (RFNN) with mended ant colony optimization (ACO) is secondly proposed for controlling the SCRIM drive system to raise robustness of system. Furthermore, four variable learning rates of the weights in the revamped RFNN are adopted by using mended ACO to speed-up parameter’s convergence. Finally, comparative performances through some experimental results are verified that the proposed backstepping control system by means of revamped RFNN with mended ACO has better control performances than the other methods for the SCRIM drive system.

Keywords

Ant colony optimization backstepping control Lyapunov stability recurrent fuzzy neural network six-phase copper rotor induction motor

1. Introduction

A six-phase copper rotor induction motor (SCRIM) is chosen as an alternating current (AC) motor drive system because of providing lower torque ripple, better reliability, and higher efficiency for their size compared to a three-phase aluminum rotor induction motor [1 –5]. However, a SCRIM has been used in many commercial and industrial drive applications [6 –8]. Lin et al. [6] proposed multiobjective optimization design technique for a SCRIM, and then a SCRIM was applied in a scroll compressor to achieve minimum manufacturing cost and starting current, and maximum efficiency and power factor. Lin [7] proposed modellling and control of six-phase induction motor servo-driven continuously variable transmission (CVT) system using blend modified recurrent Gegenbauer orthogonal polynomial neural network control system and amended artificial bee colony optimization. Lin [8] proposed a blend modified recurrent Gegenbauer orthogonal polynomial neural network control system for controlling the SCRIM driving continuously variable transmission (CVT) system to achieve high control performance. However, position tracking control of the SCRIM drive system under uncertainties is yet not proposed in any literatures. To improve control performance by using backstepping technique combined with neural network is the prime motivation in this paper research.

Backstepping control [9 –12] is known as a construction approach in the sense that it has a systematic way of constructing the Lyapunov function along with the control input design. To ensure stability or the negativeness of the derivative of the every-step Lyapunov function, it usually requires the cancellation of the indefinite cross-coupling terms, which may be the most prominent difference between backstepping control and sliding mode based control. While this cancellation results in the perfect-looking of the derivative of the Lyapunov function, it does not necessarily mean that good performance is ensured. Some methods use a linear model of the machine, which may not be suitable for high-performance applications under the occurrence of the uncertainties. Therefore, some nonlinear system control methods such as the activation control [13], the linear separation control [14], the linear coupling control [15], the sliding mode control [16], the Chaos control [17], the networked synchronization control [18] and the hybrid recurrent wavelet neural network control [19] were thus proposed to improve control performance. In additionally, an adaptive control method for a class of nonlinear multiple input multiple output unknown time-varying delay systems with full state constraints is proposed by Li et al. [20]. An adaptive neural control strategy for an n-link rigid robotic manipulator with both state constraints and unknown time-varying delayed states is proposed by Li and Li [21]. An adaptive control scheme for nonlinear stochastic systems with unknown parameters is proposed by Liu et al. [22]. An adaptive neural network (NN) control approach for nonlinear pure-feedback systems with time-varying full state constraints is propose by Gao et al. [23]. The finite-time fault-tolerant control for a class of switched nonlinear systems in lower-triangular form under arbitrary switching signals is proposed by Liu et al. [24]. However, these methods were not proposed any error compensation mechanism. The backstepping technique using the neural network with an error compensation mechanism is thus the study motivation in this paper.

In recent years, the concept of incorporating fuzzy logic into a neural network has been grown into a popular research topic [25 –27]. In contrast to the pure neural network or fuzzy system, the fuzzy neural network (FNN) possesses both their advantages; it combines the capability of fuzzy reasoning in handling uncertain information [25 –27] and the capability of artificial neural networks in learning from processes. In [25], an integral-proportional position controller and an on-line trained FNN controller are proposed to control a permanent magnet synchronous servo motor drive. The major drawback of the existing FNN is that their application domain is limited to static problem due to their feedforward network structure. Therefore, the neural adaptive backstepping tracking control [28], the adaptive control combined with fuzzy neural network [29], the adaptive backstepping control using recurrent fuzzy neural network [30] and the adaptive backstepping sliding mode control using recurrent wavelet fuzzy neural network [31] are developed to improve these problems. However, this control strategy lacks a good adjustion method of the learning rates.

Dorigo, et al. [32, 33] proposed an ant colony op-timization (ACO), which is a probabilistic skill for solving computational problems. The initial applications of ACO were in the domain of non-deterministic polynomial-time hard (NP-hard) com-binatorial optimization problems. Dorigo and Stützle [34] proposed the application of routing in telecommunication networks by using ACO. Dorigo and Blum [35] proposed a survey on theoretical results on ant colony optimization. However, many concerning applications by using ACO are solved for dynamic, multiobjective, stochastic and mixed-variable optimization problems as well as the creation of parallel implementations capable of taking advantage of the new available parallel hardware. Recently year, improved ACO algorithms [36 –38] by using hybrid metaheuristic skills were proposed to apply in traveling salesman and job scheduling. Therefore, the mend ACO with revised pheromone method applied for adjusting four varied learning rates of four parameters in the re-vamped RFNN is proposed to prevent precocious convergence and obtain faster convergence. This paper presents the backstepping control system using revamped RFNN with mended ACO for controlling the SCRIM drive system to enhance the robustness of system under the parameter variations and the external load torque disturbances.

2. Configuration of SCRIM drive system

Fig.1

The coordinate frames transformation from the six-phase a ₁ - b ₁ - c ₁, a ₂ - b ₂ - c ₂ axes to the q ₁ - d ₁, q ₂ - d ₂ axes in the SCRIM: (a) coordinate frames location of stator and rotor coils, and (b) vector diagram of coordinate frames with respect to location of stator and rotor coils.

For convenience, the coordinate frames transformation from the six-phase a ₁ - b ₁ - c ₁, a ₂ - b ₂ - c ₂ axes to the q ₁ - d ₁, q ₂ - d ₂ axes in the SCRIM is shown in Fig. 1. The q ₁ - d ₁, q ₂ - d ₂ axes voltage equations of the SCRIM are given as [1 –8]

$\begin{matrix} v_{q 1}^{e} & = & r_{s} i_{q 1}^{e} + ω_{e} (L_{ss} i_{d 1}^{e} + L_{M} i_{dr}^{e}) \\ + d (L_{ss} i_{q 1}^{e} + L_{M} i_{qr}^{e}) / dt \end{matrix}$ (1)

$\begin{matrix} v_{d 1}^{e} & = & r_{s} i_{d 1}^{e} + ω_{e} (L_{ss} i_{q 1}^{e} + L_{M} i_{qr}^{e}) \\ + d (L_{ss} i_{d 1}^{e} + L_{M} i_{dr}^{e}) / dt \end{matrix}$ (2) $v_{q 2}^{e} = r_{s} i_{q 2}^{e} + d (L_{ss} i_{q 2}^{e}) / dt$ (3) $v_{d 2}^{e} = r_{s} i_{d 2}^{e} + d (L_{ss} i_{d 2}^{e}) / dt$ (4)

$\begin{matrix} 0 & = & r_{r}^{'} i_{qr}^{e} + (ω_{e} - ω_{r}) (L_{ss} i_{dr}^{e} + L_{M} i_{d 1}^{e}) \\ + d (L_{rr} i_{qr}^{e} + L_{M} i_{q 1}^{e}) / dt \end{matrix}$ (5)

$\begin{matrix} 0 & = & r_{r}^{'} i_{dr}^{e} + (ω_{e} - ω_{r}) (L_{rr} i_{qr}^{e} + L_{M} i_{q 1}^{e}) \\ + d (L_{ss} i_{dr}^{e} + L_{M} i_{d 1}^{e}) / dt \end{matrix}$ (6) where $v_{q 1}^{e}$ , $v_{d 1}^{e}$ , and $v_{q 2}^{e}$ , $v_{d 2}^{e}$ , are the q ₁ - d ₁ axes and q ₂ - d ₂ axes voltages, respectively; $i_{q 1}^{e}$ , $i_{d 1}^{e}$ and $i_{q 2}^{e}$ , $i_{d 2}^{e}$ are the q ₁ - d ₁ axes and q ₂ - d ₂ axes currents, respectively; $i_{qr}^{e}$ , $i_{dr}^{e}$ are q _r - d _r axes currents; L _ss = Lls + 3L _ms and L _rr = L _lr + 3L _m s are the self inductances of stator and rotor, respectively; is the mutual inductance between stator and rotor; r _s and are the resistances of stator and rotor, respectively; ω _e and ω _r = P ₁ ω ₁/2 are the electrical angular speeds of synchronous flux and rotor, respectively. When an indirect field-oriented control is adopted for SCRIM, the electromagnetic torque T _e of SCRIM can be presented as $T_{e} = (3 P_{1} L_{M}^{2} i_{q 1}^{e} i_{d 1}^{e}) / (4 L_{rr}) = k_{t} i_{q 1}^{e}$ (7)

Where $k_{t} = (3 P_{1} L_{M}^{2} i_{d 1}^{e}) / 4 L_{rr})$ is the torque constant when d ₁ axis flux was kept in a fixed value for the indirect field-oriented control. P ₁ is the numbers of poles.

The SCRIM drive system is adopted by the indirect field-oriented control [1 –8]. The position and torque dynamic equation for the SCRIM are simplified as $J_{1} ω_{1} + B_{1} ω_{1} + T_{1} = T_{e}$ (8)

The block diagram shown in Fig. 2 indicates the SCRIM drive and control systems.

Fig.2

Block diagram of the SCRIM drive and control systems.

The SCRIM drive system constitutes of a speed control, an indirect field-oriented control and the voltage source inverter with six sets of insulated-gate bipolar transistor (IGBT) power modules driven by a sinusoidal pulse width modulation (PWM) control modulator, the interlock and isolated circuits. In order to attain good dynamic response, all gains for well-known PI current loop controller are listed as follows: k _pc = 14.5 and k _ic = k _pc/T _ic = 5.2 through some heuristic knowledge [39 –41]. The indirect field-oriented control is comprised of a sin θ _e/cos θ _e generation, a lookup table generation, a coordinate transformation and a PI current controller. The TMS320F28335 digital-signal-processor (DSP) control system with mix signal field-programmable-gate-array (FPGA) manufactured by Microcontroller Company was used to execute speed control and indirect field-oriented control. The SCRIM drive system was operated under bulk nonlinear extrinsic disturbances.

3. Design of control system

The Equations (8) of the actual SCRIM drive system including parameter variations and external load disturbances can be rewritten as

$\begin{matrix} {\dot{g}}_{1} & = & a_{1} g_{1} + b_{1} h_{1} + c_{1} T_{1} + Δ a_{1} g_{1} \\ + Δ b_{1} f_{1} = a_{1} g_{1} + b_{1} h_{1} + w_{l} \end{matrix}$ (9) where $g_{1} = ω_{1} = {\dot{θ}}_{1}$ is the rotor speed of the SCRIM. a ₁ = - B ₁/ - J ₁, b ₁ = k _t/ - J ₁ > 0 and c ₁ = - 1/ - J ₁. Δa ₁ and Δb ₁ denote the uncertainties introduced by system parameters J ₁ and B ₁; h ₁ is the control input of the SCRIM drive system, i.e., the torque current $i_{q 1}^{e *}$ . w _l ≡ Δa ₁ g ₁ + Δbh ₁ + c ₁ T ₁ is the lumped uncertainty. The design purpose of control system is to carry out the better position tracking performance under nonlinear uncertainty disturbance for the SCRIM drive system. The position tracking error is e _f = θ ^* - θ ₁, where θ ^* = g _d (t) is the reference trajectory. The derivative of e _f is ${\dot{e}}_{f} = {\dot{θ}}^{*} - {\dot{θ}}_{1} = {\dot{g}}_{d} - g_{1}$ . The virtual tracking error is defined as $e_{h} = g_{1} - x_{1}$ (10) where x ₁ is the virtual control law and is defined as $x_{1} = {\dot{g}}_{d} + d_{1} e_{f} + d_{2} e_{g}$ (11) where d ₁ and d ₂ are positive constants, and e _g = ∫e _f (τ) dτ is the integral function. By taking the derivative of (11), then by substituting the derivative of g ₁ and the derivative of (11) into the derivative of (10), then

$\begin{matrix} {\dot{e}}_{h} & = & {\dot{g}}_{1} - {\dot{x}}_{1} = (a_{1} g_{1} + b_{1} h_{1} + w_{l}) \\ - ({\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g}) \end{matrix}$ (12)

The Lyapunov function is first selected as $L_{y 1} = e_{f}^{2} / 2$ (13)

By substituting the derivative of e _f, (10) and (11) into the derivative of (13)

$\begin{matrix} {\dot{L}}_{y 1} & = & e_{f} {\dot{e}}_{f} = e_{f} ({\dot{g}}_{d} - g_{1}) = - d_{1} e_{f}^{2} \\ - d_{2} e_{f} e_{g} - e_{f} e_{h} \end{matrix}$ (14)

The Lyapunov function is then second selected as $L_{y 2} = L_{y 1} + d_{2} e_{g}^{2} / 2 + e_{h}^{2} / 2$ (15)

By substituting (12), (14), e _g = ∫e _f (τ) dτ and ${\dot{e}}_{g} = e_{f}$ into the derivative of (15) $\begin{matrix} {\dot{L}}_{y 2} = {\dot{L}}_{y 1} + d_{2} e_{g} {\dot{e}}_{g} + e_{h} {\dot{e}}_{h} \\ = - d_{1} e_{f}^{2} - d_{2} e_{f} e_{g} - e_{f} e_{h} + d_{2} e_{g} {\dot{e}}_{g} \\ + e_{h} [(a_{1} g_{1} + b_{1} h_{1} + w_{l}) - ({\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g})] \\ = - d_{1} e_{f}^{2} - e_{f} e_{h} + e_{h} [(a_{1} g_{1} + b_{1} h_{1} + w_{l}) \\ - ({\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g})] \end{matrix}$ (16)

Suppose that the lumped uncertainty w _l is bounded, i.e., $| w_{l} | \leq {\bar{w}}_{l}$ . The backstepping control using switching function is first designed as

$\begin{matrix} h_{1} = i_{q 1}^{e *} = b_{1}^{- 1} [{\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g} + e_{f} - a_{1} g_{1} \\ - {\bar{w}}_{l} sgn (e_{h}) - d_{3} e_{h}] \end{matrix}$ (17)

By use of the lumped uncertainty bound, substitute (17) into (16) to find $\begin{matrix} {\dot{L}}_{y 2} (e_{f}, e_{h}) = - d_{1} e_{f}^{2} - d_{3} e_{h}^{2} + e_{h} w_{l} - | e_{h} | {\bar{w}}_{l} \\ \leq - d_{1} e_{f}^{2} - d_{3} e_{h}^{2} \leq 0 \end{matrix}$ (18)

The function by using (18) is defined as $μ (t) = d_{1} e_{f}^{2} + d_{3} e_{h}^{2} \leq - {\dot{L}}_{y 2} (e_{f}, e_{h})$ (19)

Then $\int_{0}^{t} μ (τ) d τ \leq L_{y 2} (e_{f} (0), e_{h} (0)) - L_{y 2} (e_{f} (t), e_{h} (t))$ (20)

Because L _y2 (e _f (0) , e _h (0)) is bounded, and L _y2 (e _f (t) , e _h (t)) is also nonincreasing and bounded, then $\lim_{t \to \infty} \int_{0}^{t} μ (τ) d τ < \infty$ . Moreover, $\dot{μ} (t)$ is bounded, and μ (t) is also uniformly continuous bounded [42, 43]. By using Barbalat’s lemma [42, 43], μ (t) will converge to zero as t→ ∞, i.e., e _f and e _h will converge to zero as t→ ∞. Moreover, ω ₁ will converge to ${\dot{g}}_{d}$ as t→ ∞, and θ ₁ will converge to g _d as t→ ∞. Therefore, the backstepping control system using switching function, which is shown in Fig. 3, can be guaranteed.

Fig.3

Block diagram of the backstepping control system using switching function.

Because the lumped uncertainty w _l is unknown in practical application, and the upper bound ${\bar{w}}_{l}$ is difficult to determine, and the estimated value ${\hat{w}}_{l}$ of the lumped uncertainty is not precisely to estimate the lumped uncertainty in real time. Therefore a revamped, which acted as adaptive uncertainty observer, is proposed to adapt the real value of the lumped uncertainty. The architecture of the proposed four-layer revamped RFNN, which is composed of the input layer, the membership function layer, the rule layer and the output layer. The signal actions of all nodes in the four-layer revamped RFNN can be described as follows: $v_{i}^{1} (N) = \prod_{o} u_{i}^{1} (N) q_{oi}^{1} v_{o}^{4} (N - 1), o = 1, i = 1, 2$ (21) $v_{j}^{2} (N) = {exp}^{- {(u_{i}^{2} (N) - m_{ij})}^{2} / {(σ_{ij})}^{2}}, i = 1, 2, j = 1, \dots, n$ (22) $v_{k}^{3} (N) = \prod_{j} q_{jk}^{3} u_{j}^{3} (N), j = 1, \dots, n, k = 1, \dots, l$ (23)

$\begin{matrix} v_{o}^{4} (N) & = & \sum_{k} q_{ko}^{4} u_{k}^{4} (N) + ɛ v_{o}^{4} (N - 1), \\ k = 1, \dots, l, o = 1 \end{matrix}$ (24) where ∏ and Σ are the multiplier and the summation. $u_{1}^{1} = e_{f}$ is the tracking error. $u_{2}^{1} = e_{f} (1 - z^{- 1}) = Δ e_{f}$ is the tracking error increment. N denotes the number of iterations. ɛ is the self-feedback gain of the output layer which is between 0 and 1. The Gaussian function is adopted as the membership function in the membership function layer. m _ij and σ _ij are the mean and the standard deviation of the Gaussian function in the jth term of the ith input linguistic variable $u_{i}^{2}$ to the node of the membership function layer, respectively. The connecting weights $q_{oi}^{1}$ are the recurrent weights between the output layer and the input layer. $q_{jk}^{3}$ is the connective weight between the membership function layer and the rule layer, are assumed to be unity. $q_{ko}^{4}$ is the connective weight between the rule layer and the output layer. $v_{i}^{1} (N) = u_{i}^{2} (N)$ is the output value from the input layer. $v_{j}^{2} (N) = u_{j}^{3} (N)$ is the output value from the membership function layer. $v_{k}^{3} (N) = u_{k}^{4} (N)$ is the output value from the rule layer. $v_{o}^{4} (N)$ is the output value from the output layer. n is the total number of the linguistic variables with respect to the input nodes. l = (n/i) ⁱ is the number of rules with complete rule connection if each input node has the same linguistic variables. The output value $v_{o}^{4} (N)$ of the revamped RFNN can be denoted by $v_{o}^{4} (N) = {\hat{w}}_{l} (Y) = Y^{T} Ψ$ (25) where $Y = {[q_{11}^{4} q_{21}^{4} \dots \dots q_{l, 1}^{4}]}^{T}$ is the collections of the adjustable parameters of revamped RFNN. $u_{o}^{4} (N) = v_{k}^{3} (N)$ represents the kth input to the node of output layer. $Ψ = {[u_{1}^{4} u_{2}^{4} \dots \dots u_{l}^{4}]}^{T}$ .

The minimum reconstructed error e _m is defined as $e_{m} = w_{l} - w_{l} (Y^{*}) = w_{l} - (Y^{*})^{T} Ψ$ (26) where Y ^* is an optimal weight vector that achieves the minimum reconstructed error. Suppose that the absolute value of e _m is less than a small positive constant ${\bar{e}}_{m}$ , i.e., $| e_{m} | ⩽ {\bar{e}}_{m}$ . To develop adaptive law of the revamped RFNN and error estimated law, the Lyapunov function is third selected as

$\begin{matrix} L_{y 3} & = & L_{y 1} + d_{2} e_{g}^{2} / 2 + e_{h}^{2} / 2 + ({\hat{e}}_{m} - e_{m})^{2} / (2 γ) \\ + (Y - Y^{*})^{T} (Y - Y^{*}) / (2 ρ_{1}) \end{matrix}$ (27) where γ and ρ ₁ are positive constants. ${\hat{e}}_{m}$ is the estimated value of the minimum reconstructed error e _m. Take the derivative of (27) and using (16), then the derivative of (27) can be prented by ${\dot{L}}_{y 3} = - d_{1} e_{f}^{2} - d_{2} e_{f} e_{g} - e_{f} e_{h} + d_{2} e_{g} {\dot{e}}_{g} + e_{h} (a_{1} g_{1} + b_{1} h_{1} + w_{l}) - e_{h} ({\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g}) + ({\hat{e}}_{m} - e_{m}) {\dot{\hat{e}}}_{m} / γ + {(Y - Y^{*})}^{T} \dot{Y} / ρ_{1} = - d_{1} e_{f}^{2} - e_{f} e_{h} + e_{h} [(a_{1} g_{1} + b_{1} h_{1} + w_{l}) - ({\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g})] + ({\hat{e}}_{m} - e_{m}) {\dot{\hat{e}}}_{m} / γ + {(Y - Y^{*})}^{T} \dot{Y} / ρ_{1}$ (28)

Then, the backstepping control system using revamped recurrent RFNN with mended ACO is designed as

$\begin{matrix} h_{1} & = & h_{3} = i_{q 1}^{e *} = b_{1}^{- 1} [{\ddot{g}}_{d} + d_{1} {\dot{e}}_{f} + d_{2} {\dot{e}}_{g} \\ + e_{f} - a_{1} g_{1} - {\hat{e}}_{m} - {\hat{w}}_{l} (Y) - d_{3} e_{h}] \end{matrix}$ (29)

An adaptive law $\dot{Y}$ and an error estimated law from (28) are designed as $\dot{Y} = ρ_{1} e_{h} Ψ$ (30) $\dot{Y} = ρ_{1} e_{h} Ψ {\dot{\hat{e}}}_{m} = γ e_{h}$ (31)

Substitute (25), (26) and (29) into (28), then (28) can be rewritten as ${\dot{L}}_{y 3} = - d_{1} e_{f}^{2} - d_{3} e_{h}^{2} - e_{h} {\hat{e}}_{m} + e_{h} w_{l} - e_{h} {\hat{w}}_{l} (Y) + ({\hat{e}}_{m} - e_{m}) {\dot{\hat{e}}}_{m} / γ + {(Y - Y^{*})}^{T} \dot{Y} / ρ_{1} = - d_{1} e_{f}^{2} - d_{3} e_{h}^{2} - e_{h} {\hat{e}}_{m} + e_{h} (w_{l} - w_{l} (Y^{*})) - e_{h} ({\hat{w}}_{l} (Y) - w_{l} (Y^{*})) + ({\hat{e}}_{m} - e_{m}) {\dot{\hat{e}}}_{m} / γ Y - Y^{*})^{T} \dot{Y} / ρ_{1} = - d_{1} e_{f}^{2} - d_{3} e_{h}^{2} - e_{h} ({\hat{e}}_{m} - e_{m}) - e_{h} {(Y - Y^{*})}^{T} Ψ + ({\hat{e}}_{m} - e_{m}) {\dot{\hat{e}}}_{m} / γ + {(Y - Y^{*})}^{T} \dot{Y} / ρ_{1}$ (32)

Substitute (30) and (31) into (32), then (32) can be rewritten as ${\dot{L}}_{y 3} = - d_{1} e_{f}^{2} - d_{3} e_{h}^{2} = μ (t) ⩽ 0$ (33)

By using Barbalat’s lemma [42, 43], μ (t) will converge to zero as t→ ∞, i.e., e _f and e _h will converge to zero as t→ ∞. Moreover, ω ₁ will converge to ${\dot{g}}_{d}$ as t→ ∞, and θ ₁ will converge to g _d as t→ ∞. As a result, the stability of the backstepping control system using revamped RFNN with mended ACO, which is shown in Fig. 4, can be guaranteed. On the other hand, the guaranteed convergence of tracking error to be zero does not imply convergence of the estimated value of the lumped uncertainty to it real values. The persistent excitation condition [42, 43] should be satisfied to converge to its theoretic value.

Fig.4

Block diagram of the backstepping control system using revamped recurrent fuzzy neural network (RFNN) with mended ant colony optimization (ACO).

In order to describe the on-line training algorithm of the revamped RFNN, a cost function is defined as [44, 45] $Z_{1} = e_{h}^{2} / 2$ (34)

The adaptive law of the connective weight by using the gradient descent method can be represented as ${\dot{q}}_{ko}^{4} = ρ_{1} e_{h} Ψ \underline{\underline{Δ}} - ρ_{1} \frac{\partial Z_{1} \partial v_{o}^{4}}{\partial v_{o}^{4} \partial q_{ko}^{4}} = - ρ_{1} \frac{\partial Z_{1}}{\partial v_{o}^{4}} u_{k}^{4}$ (35)

The above Jacobian term of controlled system can be rewritten as $\partial Z_{1} / - \partial v_{o}^{4} = - e_{h}$ . The adaptive law of the mean of the Gaussian function from Jacobian term of controlled system can be updated as

$\begin{matrix} {\dot{m}}_{ij} & = & - ρ_{2} \frac{\partial Z_{1}}{\partial m_{ij}} = [- ρ_{2} \frac{\partial Z_{1}}{\partial v_{j}^{2}} \frac{\partial v_{j}^{2}}{\partial m_{ij}}] \\ = & - ρ_{2} \frac{\partial Z_{1}}{\partial v_{j}^{2}} \frac{2 (u_{i}^{2} - m_{ij})}{{(σ_{ij})}^{2}} \end{matrix}$ (36)

The adaptive law of the standard deviation of the Gaussian function from Jacobian term of controlled system can be updated as

$\begin{matrix} {\dot{σ}}_{ij} & = & - ρ_{3} \frac{\partial Z_{1}}{\partial σ_{ij}} = [- ρ_{3} \frac{\partial Z_{1}}{\partial v_{j}^{2}} \frac{\partial v_{j}^{2}}{\partial σ_{ij}}] \\ = & - ρ_{3} \frac{\partial Z_{1}}{\partial v_{j}^{2}} \frac{2 {(u_{i}^{2} - m_{ij})}^{2}}{{(σ_{ij})}^{3}} \end{matrix}$ (37)

The recurrent weight $q_{oi}^{1}$ from Jacobian term of controlled system can be updated as

$\begin{matrix} {\dot{q}}_{oi}^{1} & \equiv & - ρ_{4} \frac{\partial Z_{1}}{\partial q_{oi}^{1}} = - ρ_{4} \frac{\partial Z_{1}}{\partial v_{j}^{2}} \frac{\partial v_{j}^{2}}{\partial q_{oi}^{1}} \\ = & - ρ_{4} \frac{\partial Z_{1}}{\partial v_{i}^{1}} u_{i}^{1} (N) v_{o}^{4} (N - 1) \frac{2 (m_{ij} - u_{i}^{2} (N))}{(σ_{ij})^{2}} \end{matrix}$ (38)

To improve convergence and obtain four optimal learning rates of the weights in the revamped RFNN, the mend ACO is presented as follows as below.

The basic ACO algorithm [32 –35] has a significant impact on performance of the algorithm with respect to four parameters as the probabilistic choice of solution and the pheromone updated values. The pheromone updated values are affected by two factors as the evaporation rate and the length of the best tour. The known rule of the probabilistic choice of solution [37, 38] is typically defined as: $p (c_{ij} | p^{s}) = \frac{(ρ_{ij})^{α} (η_{ij})^{β}}{[\sum_{c_{ij} \in M (p^{s})} (ρ_{il})^{α} (η_{il})^{β}]} \forall c_{ij} \in M (p^{s})$ (39)

where M (p ^s) is the feasible neighborhood given the current partial solution p ^s. ρ _ij and η _ij are the pheromone value and the heuristic value associated with the component c _ij, respectively. α and β are positive real parameters whose values determine the relative importance of pheromone and heuristic information. By moving from vertex i to vertex j, the ants add the associated solution component c _ij to their partial solution p ^s until they reach their terminal vertex and complete their candidate solutions. In order to improve these methods, the mend ACO algorithm works as follows. In each state, the ants α have to determine which action to choose from. At each trial, the ants are initialized randomly over the set of states. No heuristic values are associated with the vertices, as there is no a priori information available about the quality of solution components. This is implemented by setting all heuristic values to one. As the values of α and β tune the relative importance of the pheromones ρ _m,ij and the heuristics η _ij, these are also eliminated. The probability of an ant k being in a state i taking an action j is now:

$\begin{matrix} p_{m, k} (c_{ij} | p^{s}) & = & \frac{(ρ_{m, ij})}{\sum_{l \in U_{i} (p^{s})} (ρ_{m, il})}, m = 1, 2, 3, 4 \\ \forall c_{ij} \in U_{i} (p^{s}) \end{matrix}$ (40) with U _i (p ^s) the action set the ant has to its disposal in state i. The choice of the next action thus only depends on the value of the pheromone ρ _m,ij, m = 1, ⋯ , 4 associated with this vertex. Note that (40) contains no constants that need tuning, making this algorithm much more straightforward to implement than the original ACO algorithm based on (40). The pheromones are initialized equally for all vertices and set to a small value, different from zero. In every trial, all ants construct their solutions until they either have reached the goal state, or the trial exceeds a certain pre-specified limit. The pheromones are then updated according to the following rule:

$\begin{matrix} ρ_{m, ij} (N + 1) \\ = (1 - φ_{m}) ρ_{m, ij} (N) + φ_{m} \sum_{k = 1}^{M} Δ ρ_{m, k, best, ij}, \\ m = 1, \dots, 4 \forall c_{ij} \in M (p^{s}) \in S_{iter} \end{matrix}$ (41) where φ _m ∈ (0, 1] , m = 1, ⋯ , 4 is the evaporation rate for learning rate ρ _m,ij, m = 1, ⋯ , 4. S _iter is the set of all candidate solutions found in the trial. The value of Δρ _{m, k, best, ij}, m = 1, ⋯ , 4 reflects the amount of pheromone for an ant k deposits on the vertices it has visited. This value is defined as follows:

$\begin{matrix} Δ ρ_{m, k, best, ij} & = & \frac{1}{\sqrt{[t_{m, k} - (1 - s_{m})] s_{m}}} \\ - \frac{1}{\sqrt{[t_{m, max} - (1 - s_{m})] s_{m}}}, \\ m = 1, \dots, 4 \end{matrix}$ (42)

where t _m,k, m = 1, ⋯ , 4 is the number of steps at the ant k needed to reach the goal state, s _m, m = 1, ⋯ , 4 is the sample time, used to express the time in seconds, and t _m,max, m = 1, ⋯ , 4 is the maximum number of steps allowed in a trial. The value of (1 - s _m) , m = 1, ⋯ , 4 is used to make the amount of pheromone deposit approximately equal to 1/s _m, m = 1, ⋯ , 4 when the ant reaches the goal in just one step.

4. Experimental results

A block diagram of the SCRIM drive system is depicted in Fig. 1. A photo of the experimental set-up with related circuit diagrams is shown in Fig. 5.

The proposed controllers are implemented by DSP control system. A DSP control board includes 4-channels of D/A converter and 2-channels encoder interface circuits. The coordinate translation in the field-oriented mechanism is implemented by DSP control system. For the position control system, the magnet-force brake machine is operated to provide constant disturbance torque. A mechanism with adjustable inertia is also coupled to the rotor of the SCRIM. The specification of the SCRIM is a six-phase 48 V, 1.5 kW, 3000 rpm. The parameters of the SCRIM are given as follows: ${\bar{J}}_{1} = 45.15 \times 10^{- 3} Nms$ , ${\bar{B}}_{1} = 2.12 \times 10^{- 3} Nms / rad$ , r _s = 2.2Ω, $r_{r}^{'} = 1.8 Ω$ , L _ss = 4.52mH, L _rr = 4.32mH, L _M = 2.64mH.

Fig.5

Photo of the experimental set-up with the circuit diagrams.

Fig.6

Experimental results of the renowned PI controller at Case 1: (a) position response of the rotor; (b) response of control effort.

Fig.7

Experimental results of the renowned PI controller at Case 2: (a) position response of the rotor; (b) response of control effort.

For comparison control performance with the renowned PI controller, the backstepping control system using switching function and the backstepping control system using revamped RFNN with mended ACO, four cases are provided in the experimentation. Case 1 is the nominal case due to periodic step command from 0 rad to 6.28 rad. Case 2 is the parameter disturbance case with 4 times the nominal value as increasing of the rotor inertia and viscous friction due to periodic step command from 0 rad to 6.28 rad. Case 3 is the nominal case due to periodic sinusoidal command from – 6.28 rad to 6.28 rad. Case 4 is the parameter disturbance case with 4 times the nominal value as increasing of the rotor inertia and viscous friction due to periodic sinusoidal command from -6.28 rad to 6.28 rad. To achieve good transient and steady-state control performance, two gains of the renowned PI controller are k _ip = k _pp/T _ip = 2.8 and k _pp = 5.5 by using the Kronecker method to construct a stability boundary in the k _pp and k _ip plane [39 –41] on the tuning of the PI controller at Case 1. The experimental results of the renowned PI controller for controlling the SCRIM drive system at Case 1, Case 2, Case 3 and Case 4 are shown in Figs. 6–9, respectively. The position responses at Case 1 and Case 2 are shown in Figs. 6(a) and 7(a), the associated control efforts are shown in Figs. 6(b) and 7(b). The position responses of the rotor at Case 3 and Case 4 are shown in Figs. 8(a) and 9(a), the associated control efforts are shown in Figs. 8(b) and 9(b). The favorable tracking responses of position can be obtained by using the renowned PI controller at Case 1 and Case 3 shown in Figs. 6(a) and 8(a). Moreover, worsen tracking responses of position shown in Figs. 7(a) and 9(a) are obvious due to bigger nonlinear disturbance. From the experimental results, sluggish tracking response of position is obtained for controlling the SCRIM drive system using the renowned PI controller. The linear controller has the weak robustness under bigger nonlinear disturbance because of no appropriately gains tuning.

Fig.8

Experimental results of the renowned PI controller at Case 3: (a) position response of the rotor; (b) response of control effort.

Fig.9

Experimental results of the renowned PI controller at Case 4: (a) position response of the rotor; (b) response of control effort.

The parameters of the backstepping control system using switching function are given as d ₁ = 2.2, d ₂ = 1.7, d ₃ = 2.3, ${\bar{w}}_{l} = 7.5$ according to heuristic knowledge [9, 10] due to periodic step command from 0 rad to 6.28 rad at Case 1 to achieve good transient and steady-state control performance. The experimental results of the backstepping control system using switching function at Case 1, Case 2, Case 3 and Case 4 are shown in Figs. 10–13, respectively.

Fig.10

Experimental results of the backstepping control system using switching function at Case 1: (a) position response of the rotor; (b) response of control effort.

Fig.11

Experimental results of the backstepping control system using switching function at Case 2: (a) position response of the rotor; (b) response of control effort.

Fig.12

Experimental results of the backstepping control system using switching function at Case 3: (a) position response of the rotor; (b) response of control effort.

Fig.13

Experimental results of the backstepping control system using switching function at Case 4: (a) position response of the rotor; (b) response of control effort.

The position responses of the rotor at Case 1 and Case 2 are shown in Figs. 10(a) and 11(a), the associated control efforts are shown in Figs. 10(b) and 11(b). The position responses of the rotor at Case 3 and Case 4 are shown in Figs. 12(a) and 13(a), the associated control efforts are shown in Figs. 12(b) and 13(b). The favorable tracking responses of position can be obtained by means of the backstepping control system using switching function at Case 1 and Case 3 shown in Figs. 10(a) and 12(a). Meanwhile, the fine tracking responses of position shown in Figs. 11(a) and 13(a) are obvious under bigger nonlinear disturbance.

Fig.14

Experimental results of the backstepping control system using revamped RFNN with mended ACO at Case 1: (a) position response of the rotor; (b) response of control effort.

Fig.15

Experimental results of the backstepping control system using revamped RFNN with mended ACO at Case 2: (a) position response of the rotor; (b) response of control effort.

The parameters of the backstepping control system using revamped RFNN with mended ACO are d ₁ = 2.2, d ₂ = 1.7, d ₃ = 2.3, γ = 0.1, ɛ = 0.5 through some heuristic knowledge [9 –12] at Case 1 for the position tracking in order to achieve good transient and steady-state control performance. Furthermore, to show the effectiveness of the control system with a small number of neurons, the revamped RFNN has 2, 6, 9 and 1 neurons in the input layer, the membership function layer, the rule layer and the output layer, respectively. The parameters adjustment process remains continually active for the duration of the experimentation. The parameter’s initialization of the revamped RFNN in [25] is adopted to initialize the parameters in this paper. The experimental results of the backstepping control system using revamped revamped RFNN with mended ACO for controlling the SCRIM drive system at Case 1, Case 2, Case 3 and Case 4 are shown in Figs. 14–17, respectively. The position responses of the rotor at Case 1 and Case 2 are shown in Figs. 14(a) and 15(a), the associated control efforts are shown in Figs. 14(b) and 15(b). The position responses of the rotor at Case 3 and Case 4 are shown in Figs. 16(a) and 17(a), the associated control efforts are shown in Figs. 16(b) and 17(b).

Fig.16

Experimental results of the backstepping control system using revamped RFNN with mended ACO at Case 3: (a) position response of the rotor; (b) response of control effort.

Fig.17

Experimental results of the backstepping control system using revamped RFNN with mended ACO at Case 4: (a) position response of the rotor; (b) response of control effort.

The best tracking responses of position can be obtained by means of the backstepping control system using revamped RFNN with mended ACO at Case 1 and Case 3 shown in Figs. 14(a) and 16(a). Moreover, the excellent tracking responses of position shown in Figs 15(a) and 17(a) are very conspicuous under bigger nonlinear disturbance. From these experimental results, better tracking responses of position are obtained by means of the backstepping control system and revamped RFNN with mended ACO for controlling the SCRIM drive system.

Finally, experimental result the measured rotor position response under step disturbance torque T _L = 2 Nm with adding load at 6.28 rad as Case 5 is shown in Fig. 18 by means of the renowned PI controller, the backstepping control system using switching function and the backstepping control system using revamped RFNN with mended ACO with mended ACO. Experimental results of measured rotor position response by means of the renowned PI controller, the backstepping control system using switching function and the backstepping control system using revamped RFNN with mended ACO at Case 5 are shown in Fig. 18(a-c), respectively.

Fig.18

Experimental results of measured rotor position response under load torque disturbance with adding load at 6.28 rad case: (a) by means of the renowned PI controller; (b) by means of the backstepping control system and switching function; (c) by means of the backstepping control system and revamped RFNN with mended ACO.

From these experimental results, transient response of the backstepping control system using revamped RFNN with mended ACO with mended ACO is better than the renowned PI controller and the backstepping control system using switching function at load regulation. However, the robust control performance of the backstepping control system using revamped RFNN with mended ACO was outstanding for controlling the SCRIM drive system in the tracking of periodic step and sinusoidal commands under the occurrence of parameter disturbance, and the load regulation owing to the on-line adaptive adjustment of the revamped RFNN with mended ACO.

5. Conclusion

The backstepping control system using revamped RFNN with mended ACO is proposed to control the SCRIM drive system under the occurrence of parameter disturbance, and the load regulation for the position tracking of periodic reference inputs.

The main contributions of this paper are (1) the field-oriented mechanism has been successfully applied for controlling the SCRIM drive system and formulate the dynamic equation; (2). the backstepping control system using switching function has been successfully derived according to the Lyapunov function to conquer influence under the lumped uncertainty disturbances; (3) the backstepping control system using revamped RFNN with mended ACO to estimate the lumped uncertainty and compensate estimated error has been successfully derived according to the Lyapunov function for diminishing the lumped uncertainty effect; (4) four optimal learning rates of the revamped RFNN have been successfully calculated by using mended ACO algorithm to speed-up parameter’s convergence. Furthermore, as indicated by the experimental results, the backstepping control system using revamped RFNN with mended ACO has smaller tracking error and better disturbance rejection in comparison with the renowned PI controller and the backstepping control system using switching function. In the future, the backstepping control system using revamped RFNN with sliding mode control law will further construct for adapting the unknown upper bound of the uncertainty and for enhancing robustness.

Footnotes

Acknowledgments

The author would like to acknowledge the financial support of the Ministry of Science and Technology in Taiwan, R.O.C. under its grant MOST 106-2221-E-239 -015.

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