Optimization design of general Type-2 fuzzy logic controllers for an uncertain Power-line inspection robot

Abstract

In this paper, a general type-2 fuzzy logic controller (GT2FLC), which is optimized by the particle swarm optimization (PSO) algorithm, is applied to a power-line inspection (PLI) robot. The information fusion is used to design the GT2FLC to avoid the rule explosion. The proposed controller has the ability to deal with uncertainties when the PLI robot works on the insulated access cable. In order to compare the performance of the proposed controller with that of other controllers, the type-1 fuzzy logic controller (T1FLC) and the interval type-2 fuzzy logic controller (IT2FLC) are both optimized by the PSO to adjust the PLI robot. To show the ability of different controllers to deal with uncertainties, external disturbances and parameter perturbations are added to the PLI robot. According to simulations, the performance of the proposed controller is better than that of other controllers, and the proposed controller has better ability to deal with uncertainties.

Keywords

Power-line inspection robot particle swarm optimization algorithm general type-2 fuzzy logic controller

1 Introduction

For a long time, the inspection of power-line depends on the manpower. It is not only dangerous but also inefficient. To solve this problem, the PLI robot is presented to replace the manpower completely, which has the advantages of low energy consumption, light weight and is more efficient than the manpower. Because of its advantages, it has attracted a lot of attention from the academic circles [1 –6]. Whereas, the PLI robot is a typical underactuated nonlinear system, which has two degrees of freedom and only a single actuation, so the control of the PLI robot is more difficult than the fully actuated systems [7 –9]. Furthermore, the PLI robot can be disturbed by some uncertainties, such as the wind and vibration of the power line, so the designed controller needs to have the ability of dealing with uncertainties.

There are many methods which are presented to control the underactuated system [10 –14]. The adaptive gain-scheduled backstepping scheme was proposed in [1], which is dependent on the model of the PLI robot linearized at a nominal equilibrium point. The GT2FLC is a model-free controller which does not depend on the precise mathematical model of the controlled object. In addition, it has the strong ability of dealing with uncertainties. In recent years, the study of GT2FLC has received a lot of attention in applications and theory [15 –21]. The GT2FLC is proposed on the foundation of the T1FLC by Zadeh in 1975, which has potentials to outperform the T1FLC in a highly uncertain environment. However, The GT2FLC has high complexities because of its three-dimensional structure of the general type-2 fuzzy set (GT2FS). To avoid the huge amount of the computation of the GT2FLC, the interval type-2 fuzzy logic controller (IT2FLC) as a special type of GT2FLC has been presented in previous study. The IT2FLC sacrifices certain performance to get easier calculations [22]. The IT2FLC is applied in various fields because of its simple calculation [23 –29]. With the intensive study of the GT2FLC, Liu introduces the sets operation and type-reduction based on the α - plane representation theorem, which simplify the computation of GT2FLC greatly [18]. Based on the α - plane representation theorem, which converts the computation of GT2FSs into the computation of IT2FSs. For a GT2FLC, multiple inputs and only a single output will cause the problem of rules explosion. To avoid the problem of rules explosion of T1FLCs, the information fusion was proposed in [30]. In this paper, the information fusion method is extended to the GT2FLC design of the PLI robot.

The three dimensional membership function (MF) of GT2FSs is the significant part in the GT2FLC. The variations of parameters of the MF will have a great effect on the performance of the GT2FLC. Generally, the choice of the MF depends on the experience, practice and data of experiment. In order to obtain better performance, the parameters of the MF need to be tuned with some optimization algorithms [31 –36]. The PSO algorithm, as an evolutionary algorithm proposed by J. Kennedy and R.C. Eberhart in 1995, has been studied for decades by many scholars [37 –41]. The PSO is used to seek the global optimal values of parameters in the multiple parameters system. Because of its advantages of high precision and fast convergence, it is used in the optimization of fuzzy logic system. However, the numerous studies are based on the T1FLC and IT2FLC. The MF of the GT2FLC has extra mathematical dimension. How to optimize the GT2FLC and applying it to the PLI robot is an urgent problem to be solved.

In this paper, the GT2FLC is optimized by the PSO and is applied to the PLI robot. The main contributions of this paper are as follow: 1) For the balance adjusting of the PLI robot with uncertainties, we design a GT2FLC. To avoid the rule explosion, the information fusion technology is applied in the process of designing the GT2FLC. 2) Using PSO algorithm, parameters of MFs in the GT2FLC are optimized, which makes the GT2FLC get better performance. 3) Comparing the GT2FLC with the IT2FLC and T1FLC, the superiority of the GT2FLC to adjust the balance of the uncertain PLI robot is verified.

The rest of the paper can be divided into the following sections. Preliminaries are introduced in section 2. Section 3 presents the optimization design of the GT2FLC for the uncertain PLI robot. Section 4 provides numerical simulations to verify the effectiveness of the proposed GT2FLC. The last section draws the conclusion.

2 Preliminaries

2.1 General type-2 fuzzy set

The MF of GT2FS $\tilde{A}$ has the three-dimensional structure as shown in Fig. 1. The point-valued representation of $\tilde{A}$ can be expressed by: $\tilde{A} = {(x, u), μ_{\tilde{A}} (x, u) | \forall x \in X | \forall u \in [0, 1]}$ (1)

Fig.1

MF of GT2FS.

where x is the primary variable and u is the secondary variable; $μ_{\tilde{A}} (x, u)$ is the secondary grade; X is the universe for the primary variable. The two-dimensional support of $μ_{\tilde{A}} (x, u)$ named the footprint of uncertain (FOU) of $\tilde{A}$ can be expressed by: $FOU (\tilde{A}) = {(x, u) \in X \times [0, 1] | μ_{\tilde{A}} (x, u) > 0} .$ (2) The upper membership function (UMF) and lower membership function (LMF), denoting $UMF (\tilde{A})$ and $LMF (\tilde{A})$ , are boundary of FOU: $\begin{matrix} UMF (\tilde{A}) = \inf {u | u \in [0, 1], μ_{\tilde{A}} (x, u) > 0} \\ LMF (\tilde{A}) = \sup {u | u \in [0, 1], μ_{\tilde{A}} (x, u) > 0} . \end{matrix}$ (3)

The vertical-slice representation of a GT2FS $\tilde{A}$ is described by: $\tilde{A} = \int_{x \in X} \tilde{A} (x) / x$ (4) $\tilde{A} (x) = \int_{\forall u \in J_{x}} μ_{\tilde{A} (x)} (u) / u$ (5) where $\tilde{A} (x)$ is named as secondary MF which is a type-1 fuzzy set (T1FS) and $μ_{\tilde{A} (x)} (u)$ is named as secondary grade of $\tilde{A}$ ; J_x is the domain of the secondary MF, which is named primary membership. For a GT2FS, $μ_{\tilde{A} (x)} (u)$ is not equal to 1. When $μ_{\tilde{A} (x)} (u)$ are equal to 1, an interval type-2 fuzzy set (IT2FS) is obtained. Fig. 1 shows four vertical slices, and the shape of secondary MF is symmetrical trapezoid.

To simplify for calculation of GT2FS, Liu presents the α - plane representation theorem in [18]. The α - cut of the T1FS $μ_{\tilde{A} (x)} (u)$ denotes ${\tilde{A}}_{α} (x)$ , which can be expressed by [18]: ${\tilde{A}}_{α} (x) = {u | μ_{\tilde{A} (x)} (u) \geq α}, α \in [0, 1] .$ (6) An α - plane for a GT2FS $\tilde{A}$ , denoted ${\tilde{A}}_{α}$ , can be expressed by: ${\tilde{A}}_{α} = \int_{\forall x \in X} {\tilde{A}}_{α} (x) / x .$ (7) The lower and upper MFs of an IT2FS are shown in Fig. 2. In particular, when α = 0, the α - plane denoted by ${\tilde{A}}_{0}$ can be expressed by: ${\tilde{A}}_{0} = FOU (\tilde{A})$ (8) Finally, a GT2FS $\tilde{A}$ can be obtained by the union of ${\tilde{A}}_{α}$ : $\tilde{A} = \underset{α \in [1, 0]}{\cup} [α / {\tilde{A}}_{α}]$ (9)

Fig.2

MF of IT2FS.

2.2 General type-2 fuzzy logic controller

In general, a GT2FLC contains five components, named as the fuzzifier, interference engine, rulebase, type-reducer and defuzzifier, shown in Fig. 3. The inputs and outputs of GT2FLC are the crisp number. The fuzzifier, as an important link to realize fuzzy control, is the process of converting the crisp input into the corresponding fuzzy language variable value by the MF. Generally, the LMF and UMF can be selected as gaussian, triangle and trapezoid commonly; the secondary MF can be selected as triangle or trapezoid.

Fig.3

Schematic diagram of GT2FLC.

The rulebase, as the heart of the GT2FLC, has the N if-then rules, which are described by: $\begin{matrix} ℝ^{n} : if \begin{matrix} x_{1} \begin{matrix} is \end{matrix} \begin{matrix} {\tilde{X}}_{1}^{n} \begin{matrix} and \end{matrix} \begin{matrix} \dots \end{matrix} \end{matrix} \end{matrix} \begin{matrix} and \begin{matrix} x_{I} \begin{matrix} is \end{matrix} \end{matrix} \end{matrix} \begin{matrix} {\tilde{X}}_{I}^{n} \end{matrix} \\ then \begin{matrix} y \begin{matrix} is \begin{matrix} Y^{n} \end{matrix} \end{matrix} \end{matrix} n = 1, \dots, N \end{matrix}$ (10) where ${\tilde{X}}_{i}^{n} (i = 1, 2, \dots, I)$ are the GT2FSs, which are named as the antecedents in the system and x_i is the crisp input of the GT2FLC; Yⁿ is the consequents of the GT2FLC and y, as the final output, is the crisp number. The if-then rules are depicted by the nature language, which is easy to be accepted by human. By the inference engine, the GT2FSs of the outputs are calculated according to the baserule. The common operation of GT2FSs is minimum t-norm or product t-norm. For a input vector x = (x₁, x₂, …, x_I), the firing interval with level of α, named as $F_{α}^{n}$ , can be expressed as: $F_{α}^{n} = [\underline{f} α^{n}, {\bar{f}}_{α}^{n}]$ (11) $\underline{f} α^{n} = {\underline{μ}}_{α x_{1}^{n}} (x_{1}) \times \dots \times {\underline{μ}}_{α x_{I}^{n}} (x_{I})$ (12) $\bar{f} α^{n} = {\bar{μ}}_{α x_{1}^{n}} (x_{1}) \times \dots \times {\bar{μ}}_{α x_{I}^{n}} (x_{I})$ (13) where ${\bar{μ}}_{α x_{i}^{n}} (x_{i})$ and ${\underline{μ}}_{α x_{i}^{n}} (x_{i})$ are the upper and lower membership degree of x_i on each ${\tilde{X}}_{i}^{n}$ with level of α. The outputs of inference engine are the GT2FSs, which are not input into the defuzzifier directly. The GT2FSs are needed to be converted into the type-1 fuzzy set (T1FS) before defuzzification. The type-reducer is exclusive to the type-2 fuzzy logic controller. There are many methods of designing the type-reducers. The most common method is the center-of-sets type-reducer, which can be expressed by: $y_{l α} = \frac{\sum_{n = 1}^{L} {\underline{y}}_{α}^{n} {\bar{f}}_{α}^{n} + \sum_{n = L + 1}^{N} {\underline{y}}_{α}^{n} {\underline{f}}_{α}^{n}}{\sum_{n = 1}^{L} {\bar{f}}_{α}^{n} + \sum_{n = L + 1}^{N} {\underline{f}}_{α}^{n}}$ (14) $y_{l α} = \frac{\sum_{n = 1}^{R} {\bar{y}}_{α}^{n} {\underline{f}}_{α}^{n} + \sum_{n = R + 1}^{N} {\bar{y}}_{α}^{n} {\bar{f}}_{α}^{n}}{\sum_{n = 1}^{R} {\underline{f}}_{α}^{n} + \sum_{n = R + 1}^{N} {\bar{f}}_{α}^{n}}$ (15) $Y_{cos, α} = [y_{l α}, y_{r α}]$ (16) where ${\bar{y}}_{α}^{n}$ and ${\underline{y}}_{α}^{n}$ are the consequent of left-end and right-end point with level of α; L and R are the switch point, which can be determined by some iterative algorithms such as the KM (Karnik-Mendel) algorithm and EIASC (Enhanced Iterative Algorithm with Stopping Condition).

In the end, the final output can be obtained by the defuzzifier. The expression of centroid defuzzification is as follows: $y = \frac{\sum_{α} α (y_{l α} + y_{r α})}{2 \sum_{α} α}$ (17) where α = {0, 1/G, …, (G - 1)/G, 1}; G is the number of divisions for a GT2FS. There are G + 1 α - planes in a GT2FS. In general, the type-reducer and defuzzifier can be seen as a whole in the GT2FLC.

2.3 PSO algorithm

In this paper, the PSO is used to optimize the parameters of MFs in the GT2FLC. The PSO is a simplified model, which is inspired by the predation of the bird swarm. During the running time of the PSO algorithm, each particle can be seen as a bird and all the particles constitute a swarm. The optimal solution can be seen as the food what the bird swarm want to seek. Each particle has two information, named as weights and velocities. The weight and velocity of the individual particle can be changed based on the information of all the swarm. For the each iteration, the variables of weights and velocities for the ith particle can be computed by [38]: $\begin{matrix} v_{i} = ω v_{i} + c_{1} r_{1} (pbes t_{i} - w_{i}) + c_{2} r_{2} (gbest - w_{i}) \\ w_{i} = v_{i} + w_{i} \end{matrix}$ (18) where v_i and w_i are the velocity and weight of the ith particle respectively; pbest_i is the optimal solution of all iterations for ith particle and gbest is the best optimal solution of all iteration for entire swarm; ω is the inertia weight; c₁ and c₂ are the learning factors which are the positive constants; r₁ and r₂ are random numbers between 0 and 1.

The flow diagram of the PSO algorithm is shown in Fig. 4. Before the beginning of the PSO algorithm, all the parameters need to be selected. The updating of weights and velocities is depended on Eq.(18). The fitness values, as the quality measure, are obtained by the cost function. The updating of gbest and pbest_i is depended on the fitness value. In generally, the stopping criterion is the number of iteration or the fitness value meeting the expectations.

Fig.4

The flow diagram of the PSO algorithm.

3 Optimization of the GT2FLC for the PLI robot

3.1 Mathematical model of the PLI robot

This section is referenced from [1]. The model of the PLI robot is shown in Fig. 5. When the PLI robot works on the insulated access cable, it should adjust the self-balance mechanism to maintain the tilt angle balance. The PLI robot as a typical underactuated system, has two freedom degrees, which are adjusted by rotating the counter-weight box.

Fig.5

The model of the PLI robot.

The motion equations for the motion balance adjusting of the PLI robot are obtained by the Euler-Lagrange [42]: $\begin{matrix} u_{i} = \frac{d}{dt} [\frac{\partial L}{\partial \dot{θ_{i}}}] - \frac{\partial L}{\partial θ_{i}} i = 1, \dots, m \end{matrix}$ (19) where u_i is the external torque acting on the ith generalized coordinate; L can be expressed by: $\begin{matrix} L = K - P \end{matrix}$ (20) where K and P denote the kinetic and potential energy of the motion balance adjusting of the PLI robot, respectively, which can be computed by:

$\begin{matrix} K = \frac{m_{1} h_{1}^{2} \dot{θ_{1}^{2}}}{2} + \frac{m_{2} l^{2} \dot{θ_{2}^{2}}}{2} + \frac{m_{2} [h^{2} + (- h_{20} + lsin θ_{2})^{2}] \dot{θ_{1}^{2}}}{2} \end{matrix}$ (21) $\begin{matrix} P = - m_{1} g h_{1} \sin θ_{1} + m_{2} g [(h_{20} + lsin θ_{2}) \sin θ_{1} - hcos θ_{1}] \end{matrix}$ (22) where θ₁ is the angle between the cable and the PLI robot; θ₂ is the angel of rotation of the actuator; m₁ and m₂ are the mass of the robot body and the counter-weight box respectively; l is the length of actuator bar and h is the height of the T-shaped base. h₂₀ is the distance from the center of mass of the counter-weight box to the cable when the PLI robot is in the horizontal position; g is the gravitational acceleration. The table I shows the values of parameters for the PLI robot. According to the table I, we have: $m_{1} h_{1} = m_{2} h_{20} .$ (23) Substituting Eq.(23) into Eq.(22), we get: $P = m_{2} g (- hcos θ_{1} + lsin θ_{2} \sin θ_{1})$ (24)

Table 1

value of parameters for the PLI robot

m₁ (kg)	m₂ (kg)	h₁ (m)	h₂0 (m)	l (m)	h (m)
63	27	0.18	0.42	0.5	0.5

In the end, substituting Eq. (21) and Eq. (24) into Eq. (19), the dynamic equations of motion balance adjusting for the PLI robot are obtained, which can be expressed by: $\begin{matrix} u_{1} = {m_{1} h_{1}^{2} + m_{2} [h^{2} + (- h_{20} + l sin θ_{2})^{2}]} \ddot{θ_{1}} \\ + 2 m_{2} l (- h_{20} + l sin θ_{2}) (cos θ_{2}) \dot{θ_{1}} \dot{θ_{2}} \\ + m_{2} gh sin θ_{1} + m_{2} gl sin θ_{2} cos θ_{1} \\ u_{2} = m_{2} l^{2} \ddot{θ_{2}} - m_{2} l (- h_{20} + lsin θ_{2}) (\cos θ_{2}) \dot{θ_{1}^{2}} \\ + m_{2} glcos θ_{2} \sin θ_{1} \end{matrix}$ (25)

where u₁ is an external disturbance; u₂ is the torque acting on the active joint which produces the θ₂. To study the balance adjustment progress of the PLI robot, the u₁ is assumed to be zero. Define the state variable vector $[\begin{matrix} q_{1} & q_{2} & q_{3} & q_{4} \end{matrix}] = [\begin{matrix} θ_{1} & {\dot{θ}}_{1} & θ_{2} & {\dot{θ}}_{2} \end{matrix}]$ . According to the dynamic equations, the state space model of the motion balance adjusting of the PLI robot can be expressed by [1]: $\begin{array}{l} {\dot{q}}_{1} = q_{2} \\ {\dot{q}}_{2} = \frac{- 2 m_{2} l (- h_{20} + l \sin q_{3}) (\cos q_{3}) q_{2} q_{4}}{m_{1} h_{1}^{2} + m_{2} [h^{2} + {(- h_{20} + l \sin q_{3})}^{2}]} \\ \begin{matrix} \begin{matrix} - \frac{m_{2} g h \sin q_{1} + m_{2} g l \sin q_{3} \cos q_{1}}{m_{1} h_{1}^{2} + m_{2} [h^{2} + {(- h_{20} + l \sin q_{3})}^{2}]} \end{matrix} \end{matrix} \\ {\dot{q}}_{3} = q_{4} \\ {\dot{q}}_{4} = \frac{u_{2} + m_{2} l (- h_{20} + l \sin q_{3}) (\cos q_{3}) q_{2}^{2}}{m_{2} l^{2}} \\ \begin{matrix} \begin{matrix} - \frac{m_{2} g l \cos q_{3} \sin q_{1}}{m_{2} l^{2}} \end{matrix} \end{matrix} \end{array}$ (26)

3.2 Design of GT2FLC with PSO algorithm for the PLI robot

The PLI robot is controlled by a GT2FLC which is optimized by the PSO algorithm. The overall control diagram for the PLI robot is shown in Fig. 6. The whole control flow can be divided into two parts. The first part is the closed-loop control for the PLI robot. The outputs of the PLI robot are four states, which can be selected as the feedback inputs to the GT2FLC. The increase of inputs of the GT2FLC will cause an increase in the rule number of GT2FLC exponentially, which make the design of the GT2FLC very complicated. In order to avoid the rule explosion of fuzzy system, the four states are fused to two states by the fusion function [30]: $[\begin{matrix} {\tilde{θ}}_{1} \\ \dot{{\tilde{θ}}_{1}} \end{matrix}] = [\begin{matrix} 1 & \frac{k (2)}{k (1)} & 0 & 0 \\ 0 & 0 & \frac{k (3)}{k (2)} & \frac{k (4)}{k (2)} \end{matrix}] \times [\begin{matrix} θ_{1} \\ \dot{θ_{1}} \\ θ_{2} \\ \dot{θ_{2}} \end{matrix}]$ (27) where k is the state feedback gain matrix, which can be obtained by the method of the linear quadratic regulator; ${[\begin{matrix} {\tilde{θ}}_{1} & {\dot{\tilde{θ}}}_{1} \end{matrix}]}^{T}$ are the new states infused all the states, whose physical meaning is the same as ${[\begin{matrix} θ_{1} & {\dot{θ}}_{1} \end{matrix}]}^{T}$ . The output of the GT2FLC is the control force of the PLI robot. K_e and K_ec are the quantification factors for two inputs, which make the input proportional change to meet the requirements of the fuzzy domain. K_u, named as the scale factor, is the coefficient of variation from the fuzzy domain to the physical domain.

Fig.6

Overall control diagram for the PLI robot.

Another part is about that the antecedent of GT2FLC is optimized by the PSO algorithm. In this paper, the fuzzy domain can be divided into seven GT2FSs, named as negative big (NB), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive medium (PM), positive big (PB), which mean the degree of deviation from the desired position. For example, the NB means that the deviation is big in the opposite direction. The LMFs and UMFs are triangle and the secondary MFs are symmetrical trapezoid. the UMF and LMF of GT2FS can be determined by a vector xMF and the secondary MF is determined by an element γ. Let ${\underline{μ}}_{0 x_{i}^{n}} (x_{i})$ and ${\bar{μ}}_{0 x_{i}^{n}} (x_{i})$ be the membership degree of LMF and UMF. For a input x_i, ${\underline{μ}}_{0 x_{i}^{n}} (x_{i})$ and ${\bar{μ}}_{0 x_{i}^{n}} (x_{i})$ can be computed by: ${\underline{μ}}_{0 x_{i}^{n}} (x_{i}) = {\begin{matrix} 0 & x_{i} \leq xMF (2) \\ \frac{x_{i} - xMF (2)}{xMF (3) - xMF (2)} & xMF (2) < x_{i} < xMF (3) \\ 1 & x_{i} = xMF (3) \\ \frac{xMF (4) - x_{i}}{xMF (4) - xMF (3)} & xMF (3) < x_{i} < xMF (4) \\ 0 & x_{i} \geq xMF (4) ∥ \end{matrix}$ (28) ${\bar{μ}}_{0 x_{i}^{n}} (x_{i}) = {\begin{matrix} 0 & x_{i} \leq xMF (1) \\ \frac{x_{i} - xMF (1)}{xMF (3) - xMF (1)} & xMF (1) < x_{i} < xMF (3) \\ 1 & x_{i} = xMF (3) \\ \frac{xMF (5) - x_{i}}{xMF (5) - xMF (3)} & xMF (3) < x_{i} < xMF (5) \\ 0 & x_{i} \geq xMF (5) \end{matrix}$ (29) where xMF (1), xMF (3) and xMF (5) represent the left vertex, upper vertex and right vertex of UMF, respectively; xMF (2), xMF (3) and xMF (4) represent the left vertex, upper vertex and right vertex of LMF, respectively. The secondary MF can be determined by xMF (6). The expressions are given as follows: $\begin{matrix} {\underline{μ}}_{α x_{i}^{n}} (x_{i}) = {\underline{μ}}_{0 x_{i}^{n}} (x_{i}) + \frac{1}{2} γ [{\bar{μ}}_{0 x_{i}^{n}} (x_{i}) - {\underline{μ}}_{0 x_{i}^{n}} (x_{i})] α \\ {\bar{μ}}_{α x_{i}^{n}} (x_{i}) = {\bar{μ}}_{0 x_{i}^{n}} (x_{i}) - \frac{1}{2} γ [{\bar{μ}}_{0 x_{i}^{n}} (x_{i}) - {\underline{μ}}_{0 x_{i}^{n}} (x_{i})] α \end{matrix}$ (30) where γ determines the shape of the secondary MF; when γ = 1, the secondary MF is a symmetrical triangle; when γ = 0, the secondary MF is a rectangle. The GT2FLC has two inputs which can be represented by fourteen GT2FSs. The fourteen GT2FSs are the object of optimization. In PSO algorithm, a particle is a vector which has 63 elements.

The entire process of optimization is off-line calculation. The parameters c1, c2 and ω of the PSO are selected as 2.03, 2.03 and 0.9, respectively. The range of velocity v_i is [-0.1, 0.1]. The number of iterations and the size of population are selected as 200 and 70, respectively. The cost function is expressed by: $MSE = \frac{1}{N} \sum_{i = 1}^{N} {(θ_{1} - θ_{d})}^{2}$ (31) where θ_d is the desired angle between the cable and the PLI robot. It is notable that the space of particle should be limited to a reasonable range, which can ensure the membership degrees of LUM and UMF are not equal to 0. By the iterative calculation of the PSO algorithm, the optimal parameters of the antecedents are obtained. The core code of the PSO is shown in Algorithm 1.

Algorithm 1

Core code of the PSO

1: Initialize the population size sizepop, the maximum number of iterations A, velocities v_i and weights w_i, ranges of velocities and weights, the inertia weight ω and learning factors c₁ and c₂.

2: fori = 1 : sizepop

3: apply the initially GT2FLC to the PLI robot and get θ₁

4: Compute the fitness fit (w_i) by Eq.(31)

5: pbest_i = fit (w_i)

6: end for

7: fori = 1 : sizepop - 1

8: iffit (w_i) < fit (w_i+1)

9: gbest = fit (w_i)

10: end if

11: end for

12: forj = 1 : A

13: fori = 1 : sizepop

14: Update v_i and w_i by Eq.(18) to get the new GT2FLC

15: apply the new GT2FLC to the PLI robot and get θ₁

16: Compute the fitness fit (p_i) by Eq.(31)

17: iffit (w_i) < pbest_ithen

18: pbest_i = fit (w_i)

19: end if

20: end for

21: fori = 1 : sizepop - 1 do

22: ifpbest_i < pbest_i+1then

23: gbest = fit (w_i)

24: end if

25: end for

26: end for

27: Output gbest

4 Simulation

In this section, the GT2FLC for the uncertain PLI robot is designed, whose antecedents are optimized by the PSO algorithm. Fig. 7 and Fig. 8 show the initial LMF and UMF for the two inputs of GT2FLC, and γ is selected as 1. For convenience of calculation, we choose the product t-norm as the operation of the general type-2 fuzzy interference engine. The rulebase are zero-order Takagi-Sugeno fuzzy model as shown in Table II. With the optimization of the PSO algorithm, the optimal LMF and UMF are obtained, which are shown in Fig. 9 and Fig. 10. And γ is 0.55.

Fig.7

The UMF and LMF for ${\tilde{θ}}_{1}$ without optimization.

Fig.8

The UMF and LMF for $\dot{{\tilde{θ}}_{1}}$ without optimization.

Fig.9

The UMF and LMF for ${\tilde{θ}}_{1}$ with optimization.

Fig.10

The UMF and LMF for $\dot{{\tilde{θ}}_{1}}$ with optimization.

Table 2

rulebase of the GT2FLC

Rulebase		$\dot{{\tilde{θ}}_{1}}$
		NB	NM	NS	ZO	PS	PM	PB
$\dot{{\tilde{θ}}_{1}}$	NB	[12.5, 13.5]	[11.5, 12.5]	[9.5, 10.5]	[6.5, 7.5]	[5.5, 6.5]	[3.5, 4.5]	[-0.5, 0.5]
	NM	[11.5, 12.5]	[9.5, 10.5]	[6.5, 7.5]	[5.5, 6.5]	[3.5, 4.5]	[-0.5, 0.5]	[-4.5, - 3.5]
	NS	[9.5, 10.5]	[6.5, 7.5]	[5.5, 6.5]	[3.5, 4.5]	[-0.5, 0.5]	[-4.5, - 3.5]	[-6.5, - 7.5]
	ZO	[6.5, 7.5]	[5.5, 6.5]	[3.5, 4.5]	[-0.5, 0.5]	[-4.5, - 3.5]	[-6.5, - 5.5]	[-7.5, - 6.5]
	PS	[5.5, 6.5]	[3.5, 4.5]	[-0.5, 0.5]	[-4.5, - 3.5]	[-6.5, - 5.5]	[-7.5, - 6.5]	[-10.5, - 9.5]
	PM	[3.5, 4.5]	[-0.5, 0.5]	[-4.5, - 3.5]	[-6.5, - 5.5]	[-7.5, - 6.5]	[=10.5, - 9.5]	[-12.5, - 11.5]
	PB	[-0.5, 0.5]	[-4.5, - 3.5]	[-6.5, - 7.5]	[-7.5, - 6.5]	[-10.5, - 9.5]	[-12.5, - 11.5]	[-13.5, - 12.5]

4.1 The PLI robot without uncertainties

First, we consider the performance of GT2FLC for the PLI robot without uncertainties. Assume that the initial state is θ₁ = 0.3 rad and other states are equal to 0 rad. The desired balance position is selected as θ_d = 0 rad. The responses of the PLI robot controlled by GT2FLC with the optimization and without the optimization are shown in Fig. 11 and Fig. 12. In order to demonstrate the effectiveness of GT2FLC, the corresponding responses of T1FLC and IT2FLC are also depicted in Figs.11 and 12. All controllers can make the θ₁ keep at the desired balanced position without the steady state error. The proposed GT2FLC and the IT2FLC with the PSO algorithm have a smaller overshoot than other controllers.

Fig.11

responses of θ₁ and ${\dot{θ}}_{1}$ without uncertainties.

Fig.12

responses of θ₂ and ${\dot{θ}}_{2}$ without uncertainties.

4.2 The PLI robot with uncertainties

To further show the ability the GT2FLC with optimization in dealing with uncertainties, the external disturbance on the u₂ and parameter perturbation of m₂ are added to the PLI robot, meanwhile, the responses of the PLI robot with the GT2FLC are compared with those of the IT2FLC and T1FLC. Fig. 13 and Fig. 14 show the responses of three controllers when the external disturbance is d = 4 × sin(5 × t) N. After the external disturbance of sinusoidal signal is added, all responses of controllers swing on the 0 rad. However, the swing amplitude of the proposed GT2FLC is minimal.

Fig.13

responses of θ₁ and ${\dot{θ}}_{1}$ with the external disturbance d = 4 × sin(5 × t) N.

Fig.14

responses of θ₂ and ${\dot{θ}}_{2}$ with the external disturbance d = 4 × sin(5 × t) N.

Fig. 15 and Fig. 16 show the responses of the controllers when the external disturbance is d = 30 N where this disturbance is added from 5s to 7s. The external disturbance makes the responses shake significantly. The response of the proposed controller returning the balance is the fastest in three controllers.

Fig.15

responses of θ₁ and ${\dot{θ}}_{1}$ with the external disturbance d = 30 N.

Fig.16

responses of θ₂ and ${\dot{θ}}_{2}$ with the external disturbance d = 30 N.

The responses are shown in Fig. 17 and Fig. 18 when the perturbation occurs to the mass of the counter-weight box m₂ after 5s, where the uncertainty in m₂ is Δm₂ = 9. The responses of the PLI robot with the T1FLC has obvious static error. The IT2FLC and GT2FLC have the ability of dealing with mass perturbation. To study the robustness of proposed controllers, the value of Δm₂ is increased. When Δm₂ = 10, the responses of the PLI robot with the IT2FLC has static error. When Δm₂ = 15, the responses of the PLI robot with the GT2FLC has static error, which are shown in Fig. 19 and Fig. 20. Overall, the GT2FLC has more robustness than the T1FLC and IT2FLC.

Fig.17

responses of θ₁ and ${\dot{θ}}_{1}$ with the mass perturbation Δm₂ = 9.

Fig.18

responses of θ₂ and ${\dot{θ}}_{2}$ with the mass perturbation Δm₂ = 9.

Fig.19

responses of θ₁ and ${\dot{θ}}_{1}$ with the mass perturbation Δm₂ = 15.

Fig.20

responses of θ₂ and ${\dot{θ}}_{2}$ with the mass perturbation Δm₂ = 15.

To evaluate the performance of all controllers clearly, we use the following performance indexes: the integral of square error (ISE), the integral of the absolute value of the error (IAE) and the integral of the time multiplied by the absolute value of the error (ITAE), which can be expressed as: $\begin{matrix} ISE = \int_{0}^{\infty} {[e (t)]}^{2} dt \\ IAE = \int_{0}^{\infty} | e (t) | dt \\ ITAE = \int_{0}^{\infty} t | e (t) | dt \end{matrix}$ (32)

Table III, IV and V list the ISE, IAE and ITAE for all controllers. It can be seen that the GT2FLC has better performance of than T1FLC and IT2FLC in dealing with uncertainties.

Table 3

Evaluation index with external disturbance d = 4 × sin(5 × t) N

	T1FLC - PSO	IT2FLC - PSO	GT2FLC - PSO
ISE	0.0300	0.0284	0.0281
IAE	0.2714	0.2311	0.2201
ITAE	2.7140	2.3108	2.2010

Table 4

Evaluation index with external disturbance d = 30 N

	T1FLC - PSO	IT2FLC - PSO	GT2FLC - PSO
ISE	0.2003	0.1881	0.1513
IAE	0.8327	0.7684	0.6770
ITAE	8.3270	7.6844	6.7698

Table 5

Evaluation index with parameter perturbation Δm₂ = 9

	T1FLC - PSO	IT2FLC - PSO	GT2FLC - PSO
ISE	0.2252	0.0288	0.0269
IAE	3.1587	0.4925	0.2816
ITAE	252.6951	39.3982	22.5267

5 Conclusions

In this paper, a GT2FLC is designed for the balance adjusting of the PLI robot which is a typical underactuated nonlinear system. The information fusion is used in the design of the GT2FLC to solve the problem of ¡¯rule explosion¡¯. The antecedents of the proposed GT2FLC are optimized by the PSO algorithm. To show the ability of the proposed GT2FLC in dealing with uncertainties, the external disturbance and parameter perturbation are added to the PLI robot. The simulation results illustrate that the proposed GT2FLC has better performance than T1FLC and IT2FLC in dealing with uncertainties. The MF is the key factor to determine performances of the GT2FLC. The MF of the GT2FLC has three dimensions structure, which makes the GT2FLC has more potentials to obtain better results compared to the T1FLC and IT2FLC.

Footnotes

Acknowledgments

This work is supported by the National Key R&D Program of China (2018YFB1307401), the National Natural Science Foundation of China (61703291) and the Natural Science Foundation of Zhejiang Province (LQ19F030003)

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