Volume 13 Number 4
August 2016
Article Contents
Salim ZianiNew Time-varying Fuzzy Sets Based on a PSO Midpoint of the Universe of Discourse. International Journal of Automation and Computing, vol. 13, no. 4, pp. 392-400, 2016. doi: 10.1007/s11633-016-0988-0
Cite as: Salim ZianiNew Time-varying Fuzzy Sets Based on a PSO Midpoint of the Universe of Discourse. International Journal of Automation and Computing, vol. 13, no. 4, pp. 392-400, 2016. doi: 10.1007/s11633-016-0988-0

New Time-varying Fuzzy Sets Based on a PSO Midpoint of the Universe of Discourse

Author Biography:
  • Salim Ziani received the B.Sc., M.Sc.and Ph.D.degrees in control system from the University of Constantine, Algeria in 1996, 2004 and 2010 respectively.Currently, he is a professor in automatic control, Department of Electronics University of Constantine1-Constantine, Algeria.He is a member of Automatic and Robotics Laboratory (LARC) University of Constantine.Since 2011, he is responsible of the specialty in the same department.He is the founder of the International Conference on Electrical Engineering and Control Applications (ICEECA).His research interests include automatic control (optimization Problem, robust control, adaptive control, fuzzy sets and fuzzy systems, predictive control), embedded system (field programmable gate array (FPGA) & very high description language (VHDL) applications, microcontroller and arduino), the programming of the siemens automate and supervisory control and data acquisition (SCADA)(Step7 & WinCC).E-mail:ziani_salim@umc.edu.dz; zianide_s@yahoo.fr ORCID iD:0000-0001-9176-9533

  • Received: 2014-03-27
  • Accepted: 2015-03-19
  • Published Online: 2016-06-29
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New Time-varying Fuzzy Sets Based on a PSO Midpoint of the Universe of Discourse

Abstract: The paper presents a robust parallel distributed compensation (PDC) fuzzy controller for a nonlinear and certain system in continuous time described by the Takagi-Sugeno (T-S) fuzzy model. This controller is based on a new type of time-varying fuzzy sets (TVFS). These fuzzy sets are characterized by displacement of the kernels to the right or left of the universe of discourse, and they are directed by a well-defined criterion. In this work, we only focused on the movement of midpoint of the universe. The movements of this midpoint are optimized by particle swarm optimization (PSO) approach.

Salim ZianiNew Time-varying Fuzzy Sets Based on a PSO Midpoint of the Universe of Discourse. International Journal of Automation and Computing, vol. 13, no. 4, pp. 392-400, 2016. doi: 10.1007/s11633-016-0988-0
Citation: Salim ZianiNew Time-varying Fuzzy Sets Based on a PSO Midpoint of the Universe of Discourse. International Journal of Automation and Computing, vol. 13, no. 4, pp. 392-400, 2016. doi: 10.1007/s11633-016-0988-0
  • The interest in takagi-sugeno (T-S) model[1, 2] is due to the fact that the stability and performance characteristics of the system can be analyzed using a Lyapunov function approach. It achieved great success in its application to many systems in real world[3-5]. The so-called parallel distributed compensation (PDC)[2, 6, 7] utilizes a nonlinear state feedback controller which mirrors the structure of associated T-S model. The stability analysis is based on the quadratic Lyapunov function[7, 8], which is easy to implement and can be expressed as a convex optimization problem in linear matrix inequality (LMI) formalism[7-11].

    Fuzzy controller’s design depends mainly on the rules based of fuzzy sets and membership functions, which contain the linguistic elements who characterize the functioning of the industrial process. In reality, we cannot exactly evaluate the length of an element of fuzzy sets. For example, temperature’s linguistic variables are ``Low'', ``Medium'' and ``High''. These linguistics values of fuzzy sets do not have a well-defined numeric range at all the time and they also depend on the process. In general, we approximate the linguistics values of fuzzy sets by a proper numeric range, where the membership functions are fixed during the computation time called fixed fuzzy sets (FFS). Conventionally, this type of fuzzy sets is known as type-1 fuzzy sets. The type-2 fuzzy set is a set where we also have uncertainty on the membership function[12, 13].

    In context of the self organizing fuzzy control (SOFC)[14-25], this work is an extension of [22, 23] which were interested in membership functions by proposing that the ranges of linguistics values of the fuzzy sets vary during the computation time, called the time-varying fuzzy sets (TVFS). In general, movements of all support points are performed in both directions, i.e., left or right and with variable distances. Our approach is to only adjust the midpoint of the universe of FFS by a function depending on the error. The displacement of the midpoint is defined using a temporal function and is directed to both right and left sides of the universe by the position of the premise variable. All other support points of the fuzzy sets are shifted by the same function and direction except that the two extreme support points are fixed for all time.

    Salim et al.[22, 23] deal with identifying the parameters of TVFS method. Rererence [22] discusses an offline identification based on recursive least squares (RLS), where the parameters are fixed during the execution time, reference [23] is an online identification based on an adaptive approach, where the parameters vary in real-time. In this paper, we propose a new online identification based on optimization using particle swarm optimization (PSO) approach.

    By applying this TVFS for designing a PDC fuzzy controller for nonlinear system, we use a decay rate controller and relaxed stability conditions[26-28]. A sample system (inverted pendulum) is given to show the robustness of the PDC fuzzy controller based on this approach.

    The paper is organized as follow. The T-S fuzzy model and stability using Lyapunov approach and PDC fuzzy controller are recalled in Section 2. Section 3 discusses the time-varying fuzzy sets. Section 4 presents the design of the algorithm. A simulation example is provided to show the effectiveness of this approach in Section 5. Finally, conclusions are given in Section 6.

  • Consider a nonlinear system described by the T-S fuzzy model[1, 2]:

    Plant rule $i$ : IF $z_1(t)$ is $M_{i1}$ , $\cdots$ , and $z_p(t)$ is M $_{ip}$ , THEN

    $\left\{ \begin{array}{l} \dot x(t)={A_i}x(t) + {B_i}u(t)\\ y(t)={C_i}x(t), i=1, \cdots, r \end{array} \right.$

    (1)

    where $M_{ij}$ is the fuzzy set, and $r$ is the number of IF-THEN rules, $z(t)=[z_{1}(t)$ , $z_{2}(t), \ \cdots, \ z_{p}(t)]$ are the premise variables, $A_i \in \boldsymbol{R}^{n\times n}$ , $B_i\in {\bf R}^{n\times m}$ and $C_i\in \boldsymbol{R}^{n\times 1}$ are system matrices where $m \leq n, x \in \boldsymbol{R}^n$ is the state, $u\in \boldsymbol{R}^m$ is the control constrained as

    $\parallel u(t){\parallel _2} < \varphi.$

    (2)

    The output $y(t)$ is constrained as

    $\parallel y(t){\parallel _2} < \rho.$

    (3)

    The considered fuzzy model can be written as

    $x(t)=\frac{{\sum\limits_{i=1}^n {{w_i}({A_i}x(t) + {B_i}u(t))} }}{{\sum\limits_{i=1}^n {{w_i}} (t)}}$

    (4)

    where $w_i$ is defined as

    ${w_i}(z(t))=\prod\limits_{j=1}^r {{M_{ij}}} ({z_j}(t))$

    (5)

    $M_{ij}$ is membership function of the $j$ -th fuzzy set in the $i$ -th rule. Let us define

    ${h_i}(z(t))=\frac{{{w_i}(z(t))}}{{\sum\limits_{i=1}^r {{w_i}} (z(t))}}$

    (6)

    $\left\{ \begin{array}{l} \sum\limits_{i=1}^r {(z(t))}=1\\ {h_i}(z(t)) \ge 0, i=1, \cdots, r \end{array} \right.$

    (7)

    for every input $x(t)$ and $u(t)$ , the global output is obtained by

    $\left\{ \begin{array}{l} \dot x(t)=\sum\limits_{i=1}^r {{h_i}} (z(t))\{ {A_i}x(t) + {B_i}u(t)\} \\ y(t)=\sum\limits_{i=1}^r {{h_i}} (z(t))\{ {C_i}x(t)\}, i=1, \cdots, r \end{array} \right.$

    (8)

    where matrices $A_{i} $ and $B_{i} $ are constants of appropriate size and satisfy the assumption that: each pair ( $A_{i} $ , $B_{i} $ ) is stabilizable.

  • To stabilize the system represented by (6), we use a PDC controller defined by[2, 6, 7]. Control rule $i$ : IF $ z_{1}(t) $ is $ M_{i1} $ and, $\cdots $ , and $ z_{p}(t)$ is $ M_{ip} $ THEN

    $u(t)=-{K_i}x(t)i=1, \cdots, r$

    (9)

    where $ K_{i} $ is the controller stabilizing the $i$ -th subsystem. The global control will be given by

    $\begin{array}{l} u(t)=-\frac{{\sum\limits_{i=1}^r {{w_i}} (z(t)){K_i}.x(t)}}{{\sum\limits_{i=1}^r {{w_i}} (z(t))}}=\\ \; \;\; \;\; \;\; -\sum\limits_{i=1}^r {{h_i}} (z(t)){K_i}x(t), i=1, \cdots, r. \end{array}$

    (10)
  • For guaranteeing the synthesizable fuzzy controller stability, we use the theorems giving the sufficient conditions of Lyapunov quadratic stability which exploits LMI formalism[29-32].

  • On the fuzzy sets form, we have no confusion to give the two extreme numeric ranges of their corresponding linguistics values. Around the midpoint of the universe, there is always a wide margin for intersection of linguistics values where we cannot determine their exact numeric ranges fixed throughout the computation time. We propose that the ranges of the fuzzy sets vary with time on the universe [ $\underline{Z}$ , $\overline{Z}$ ], called time-varying fuzzy sets.

    These linguistics ranges are inversely proportional to membership grades (Figs. 1-3) which are directly proportional to the control law, defined by the PDC fuzzy controller. Let us define the error $e(t)$ as

    Figure 1.  Membership functions with the time-varying midpoint (2D and 3D)( $\mu$ : membership grade, $Z$ : premise variable, $R_R(t)$ : right range, $R_L(t)$ : left range, $\alpha(t)$ : midpoint)

    Figure 2.  Area of membership functions covered by the time-varying midpoint: called Footprint of Shifting (FOS)

    Figure 3.  Variation of the membership grades generated by the time-varying midpoint}

    $e(t)=x(t)-{x_d}(t)$

    (11)

    where $x(t)$ is the current system state and $x_{d}(t)$ is the desired system state.

    If $e(t)$ is large, then one needs more control effort to decrease when approaching the desired state by an adequate acceleration. For example, consider the fuzzy set Low or High, if the range of one decreases by one step, the other range increases by the same step.

    To carry out this objective, we propose to adjust the midpoint of the universe defined by $\alpha$ in Fig. 1. The displacements of this midpoint are characterized by a continuous function depending on the error[23], where

    $\alpha (t)=\frac{f(e(t))}{\alpha (t)}\in \left[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha }, \bar{\alpha } \right]\subset \left[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Z}, \bar{Z} \right].$

    (12)

    Fig. 3 presents a case where $\alpha(t)$ is displaced to the left side, then the membership grade of the $z_2(t)$ premise variable is considerably increased, while those of the $z_1(t)$ premise variable is slightly decreased.

    Through Fig. 1, if the midpoint is shifted towards the left $\{\alpha(t_0), (\alpha(t_1), \cdots, (\alpha(t_i)\}$ , the decrease of the left range, it causes a higher membership grades on the left side $\{\mu_{z1}(t_0) < \mu_{z1}(t_1) < \cdots < \mu_{z1}(t_i) \}$ and the increase of these membership grades generates an increase of the control effort, see (10).

  • From the (5) that

    ${{h}_{i}}=\frac{{{w}_{i}}}{\sum\limits_{i=1}^{r}{w}i}=\frac{wi}{{{w}_{1}}+{{w}_{2}}+\cdots +{{w}_{i}}+\cdots +{{w}_{r}}}\ge $

    (13)

    $W={{w}_{1}}+{{w}_{2}}+\cdots +{{w}_{i-1}}+{{w}_{i+1}}\cdots +{{w}_{r}}>0.$

    (14)

    For $w_{i}{'}$ , we have

    ${{h}_{i}}'=\frac{{{w}_{i}}'}{W+{{w}_{i}}'}.$

    (15)

    Then,

    $\begin{align} & {{h}_{i}}'-{{h}_{i}}=\frac{{{w}_{i}}'}{W+{{w}_{i}}'}-\frac{{{w}_{i}}'}{W+{{w}_{i}}}=\\ & \frac{W}{\left(W+{{w}_{i}} \right)\times \left(W\text{+}{{w}_{i}}' \right)}\times \left({{w}_{i}}'-{{w}_{i}} \right). \\ \end{align}$

    (16)

    So,

    $\begin{array}{l} {\rm{if }}{w_i}{\rm{'}}-{w_i} > 0, {\rm{then}}\\ \; \;{h_i}{\rm{'}}-{h_i} > 0. \end{array}$

    (17)

    So increase of $w_{i} $ generates an increase in $h_{i} $ , and from (10), the increase of $h_{i} $ generates an increase in the control effort $u(t)$ .

  • We propose five steps

    1) The design (inclusion) of $\alpha(t)$ midpoint into the algorithm of fuzzy system

    2) The functions defining the $\alpha(t)$ displacements

    3) The direction criterion of the displacement of the $\alpha(t)$ midpoint

    4) The effect of the $\alpha(t)$ function on the stability

    5) Identification of the parameters of $\alpha(t)$ .

    We explain this algorithm using triangular membership functions and PDC fuzzy controller.

  • Let us consider triangular membership functions in Fig. 4. The premise variable $z_i(t)$ is given by

    Figure 4.  Membership functions with $\alpha$ fixed

    ${{z}_{i}}(t)={{M}_{1i}}({{z}_{i}}(t))\times {{\bar{z}}_{i}}+{{M}_{2i}}({{z}_{i}}(t))\times {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z}}_{i}}$

    (18)

    [ $\overline{z}_{i}$, $\underline{z}_{i}$ ] represent minimum and maximum of $z_{i} $ for $x(t)\in$[ $\overline{x}$, $\underline{x}$].

    $M_{1i}$ and $M_{2i}$ are the membership grades of $z_{i} (t)$ with $\alpha$ fixed, as shown in Fig. 4.

    $\begin{align} & {{M}_{1i}}({{z}_{i}}(t))=\frac{{{z}_{i}}(t)-{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z}}}_{i}}}{{{{\bar{z}}}_{i}}-{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z}}}_{i}}} \\ & {{M}_{2i}}({{z}_{i}}(t))=\frac{{{{\bar{z}}}_{i}}-{{z}_{i}}(t)}{{{{\bar{z}}}_{i}}-{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z}}}_{i}}}. \\ \end{align}$

    (19)

    The value of $\alpha$ is fixed as the midpoint of the universe.

    And we propose to calculate the premise variable $z_{i} $ (t) by

    $\begin{align} & z_{i}(t)=M_{1i}(z_{i}(t))\alpha(t)+M_{2i}(z_{i}(t))\underline{z}_{i}+\notag\\ & \qquad \quad M_{3i}(z_{i}(t))\overline{z}_{i}+M_{4i}(z_{i}(t))\alpha(t) \end{align} $

    (20)

    where

    $\alpha (t)\in \left[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha }, \bar{\alpha } \right]\subset \left[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Z}, \bar{Z} \right]$

    (21)

    $\begin{align} & M_{1i}(z_{i}(t))=\frac{1}{2}\times \frac{z_{i}(t)-\underline{z}_{i}}{\alpha(t)-\underline{z}_{i}}\notag\\ & M_{2i}(z_{i}(t))=\frac{1}{2}\times \frac{\alpha(t)-z_{i}(t)}{\alpha(t)-\underline{z}_{i}}\notag\\ & M_{3i}(z_{i}(t))=\frac{1}{2}\times \frac{z_{i}(t)-\alpha(t)}{\overline{z}_{i}\alpha(t)}\notag\\ & M_{4i}(z_{i}(t))=\frac{1}{2}\times \frac{\overline{z}_{i}-z_{i}(t)}{\overline{z}_{i}\alpha(t)} \end{align}$

    (22)

    $M_{1i}$ , $M_{2i}$ , $M_{3i}$ and $M_{4i}$ are the membership grades of $z_{i}(t)$ with $\alpha$ time-varying.

  • Let $\alpha_{R}(t)$ be the right displacement of the midpoint on the universe, ensured by a function depending on the error, as shown in Fig. 5.

    Figure 5.  Direction of the displacement of the time-varying midpoint

    ${{\alpha }_{R}}(t)={{f}_{R}}(e(t))={{\theta }_{1R}}\times \left(1-{{\text{e}}^{-{{\theta }_{2R}}\times |e(t)|}} \right)$

    (23)

    where $\theta_{1R} $ and $\theta_{2R} $ are the maximum and the growth rate of $\alpha_{R}(t)$ , respectively.

    $\alpha _{L} (t)$ is the left displacement of the midpoint on the universe, ensured by

    ${{\alpha }_{L}}(t)={{f}_{L}}(e(t))=-{{\theta }_{1L}}\times \left(1-{{\text{e}}^{-{{\theta }_{2L}}\times |e(t)|}} \right)$

    (24)

    where $-$ $\theta_{1L} $ and $\theta_{2L} $ are the minimum and the decay rate of the $\alpha_{L}(t)$ , respectively.

  • The direction criterion depends on the relation between the membership grades and the control law based on the error distance $e(t)$ , see Section 3.1. In this note, we use a PDC controller and we based it on the relationship (10). The displacement of the $\alpha(t)$ midpoint to both left or right is directed by the position of the premise variable $Z(t)$ , i.e., if the premise variable is set to the left of the midpoint, then $\alpha(t)$ must approach to the minimum $\underline{\alpha}$ by the function defined in relationship (24). If the premise variable is set to the right of the midpoint, then $\alpha(t)$ must approach to the maximum $\overline{\alpha}$ by the function defined in relationship (24). The relationship (21) is always checked.

    So, we propose that the displacements $\alpha(t)$ will follow the $Z(t)$ premise variable by an acceleration determined by the output reference of the Mamdani fuzzy model, shown in Figs. 6 and 7. The switch function between the right and the left displacements is illustrated in Fig. 5, and is given by this sub-program:

    Figure 6.  Closed loop system with time-varying fuzzy sets

    Figure 7.  Representation of the reference model

    $\begin{array}{l} \alpha (t) & z(t + 1){\rm{then}}\alpha (t + 1)={\alpha _R}(t + 1)\\ \; \;\; \;\; \;\; \;\; \;{\rm{else}}\quad \alpha (t + 1)={\alpha _L}(t + 1). \end{array}$

    (25)
  • It is simple to demonstrate that (18) and (20) are exactly equal for any value of $\alpha(t)$ , by substituting respectively equation (19) in (18) and (22) in (20). This implies that the subsystems [ $A_i$ $B_i$ ; $C_i$ 0] of T-S[2, 7] do not change as the value of $\alpha(t)$ , and also the criteria of the stability (Stability theorems[29-32]) do not change.

  • In this part, we use an online identification of $\alpha(t)$ function shown by Fig. 6 based on the traditional model reference adaptive control (MRAC).

    where

    $Y_{m} $ (t): The output of the Mamdani model reference

    $e(t)$ : Error

    $\triangle e(t)$ : Change in error

    $e _{a} (t)$ : Adaptive error

    $r(t)$ : Reference

    $u(t)$ : Control law

    $\alpha_{L}(t)$ : The left displacement function of the midpoint

    $\alpha_{R}(t)$ : The right displacement function of the midpoint

    $\alpha(t)$ : The general displacement function of the midpoint

    [ $ z_{1} (t)$ , $\cdots$ , $ z_{n} (t)$ ]: Premise variable vector

    [ $\alpha_{1}(t)$ , $\cdots$ , $\alpha_{i}(t)$ ]: The vector of the midpoint function

    $y(t)$ : Output of the system.

  • The reference model is defined by the output $Y_m(t)$ of Mamdani fuzzy system, which represents the acceleration of the midpoint displacements presented in Fig. 7.

  • As an example we give in Table 1 the rule-base of the system.

    Table 1.  Rule base

  • The PSO method is a member of wide category of swarm intelligence methods for solving the optimization problems, which is a population based search algorithm where each individual is referred to as particle and represents a candidate solution. Each particle in PSO flies through the search space with an adaptable velocity that is dynamically modified according to its own flying experience and also to the flying experience of the other particles. In PSO, all particles strive to improve themselves by imitating traits of their successful peers. Further, each particle has a memory and hence it is capable of remembering the best position in the search space ever visited by it. The position corresponding to the best fitness of one particle is known as $pbest$ and the overall best out of all the particles in the population is called gbest[33, 34].

    The velocity and the position of each particle can be calculated using the current velocity and the distances from the $pbest_{j, g}$ to $gbest_{g}$ as shown in the following formulas:

    $\begin{array}{l} v_{j, g}^{(ite + 1)}={I_w} \times v_{j, g}^{(ite)} + {c_1} \times {r_1} \times (pbes{t_{j, g}}-x_{j, g}^{(ite)}) + \\ \qquad \qquad {c_2} \times {r_2} \times (gbes{t_g}-x_{j, g}^{(ite)}) \end{array}$

    (26)

    $x_{j, g}^{(ite + 1)}=x_{j, g}^{(ite)} + v_{j, g}^{(ite + 1)}$

    (27)

    where $j=1, 2, \cdots, n$ , $g=1, 2, \cdots, m$ , $n$ is the number of particles in the swarm, $m$ is the number of components for the vector $v_j$ and $x_j$ , $ite$ is the number of iterations (generations), $v_{j, g}^{(ite)}$ is the { $g$ -{th}} component of the velocity of the particle $j$ at iteration $t$ , $Iw$ is the inertia weight factor, $c_1$ and $c_2$ are the cognitive and social acceleration factors, $r_1$ and $r_2$ are the random numbers uniformly distributed in the range [0, 1], $x_{j, g}^{(ite)}$ is the $g$ -{th} component of the position of particle at iteration $t$ , $pbest_j$ is the $pbest$ of particle $j$ , $gbest_g$ is the $gbest$ of group.

    The $j$ -{th} particle in the swarm is represented by a $d$ -dimensional vector { $x_j$ =( $x_{j, 1}$ , $x_{j, 2}$ , $\cdots$ , $x_{j, d}$ )} and its rate of position change (velocity) is denoted by another $d$ -dimensional vector { $v_j$ =( $v_{j, 1}$ , $v_{j, 2}$ , $\cdots$ , $v_{j, d}$ )}. The best previous position of the $j$ -th particle is represented as $pbest_j$ =( $pbest_{j, 1}$ , $pbest_{j, 2}, \cdots$ , $pbest_{j, d}$ ). The index of best particle among all of the particles in the swarm is represented by the $gbest_g$ . In PSO, each particle moves in the search space with a velocity according to its own previous best solution and its group’s previous best solution. The velocity update in a PSO consists of three parts, namely momentum, cognitive and social parts (27). The balance among these parts determines the performance of a PSO algorithm. The parameters { $c_1$ } and { $c_2$ } determine the relative pull of $pbest$ and $gbest$ and the parameters { $r_1$ } and { $r_2$ } help in stochastically varying these pulls. In the above equations, superscripts denote the iteration number. Fig. 8 shows the velocity and the position in two-dimensional parameter space.

    Figure 8.  Description of velocity and position updates in PSO for 2-dimensional parameter  

    Based on the closed loop system with TVFS illustrated in Fig. 6, we have

    ${e_a}(t)=\alpha (t)-{Y_m}(t).$

    (28)

    The general form of $\alpha(t)$ midpoint function is given by

    $\alpha (t)={\theta _1}(1-{{\rm{e}}^{-{\theta _2}.e(t)}}).$

    (29)

    So, the objective of using the PSO approach is to find the parameters $ \theta_{1}(t)$ and $\theta_{2} (t)$ such that the objective function $J$ is optimized which is defined by

    $\begin{array}{l} J=\min \left({e_a^2(t) + {e^2}(t)} \right)=\\ \quad \min \left[{{{(\alpha (t)-{Y_m}(t))}^2} + {e^2}(t)} \right]=\\ \quad \min \left[{{{\left({{\theta _1}(1-{{\rm{e}}^{-{\theta _2}e(t)}})-{Y_m}(t)} \right)}^2}} \right. + \\ \quad \left. {{{\left({x(t)-{x_d}(t)} \right)}^2}} \right]. \end{array}$

    (30)
  • To illustrate the idea of this note, we consider the problem of an inverted pendulum on a cart[2]:

    $\left\{ {\begin{array}{*{20}{l}} {{x_1}(t)={x_2}(t)}\\ {{x_2}(t)=\frac{{g{\rm{sin}}({x_1}(t))-a \times m \times x_2^2(t) \times \frac{{{\rm{sin}}(2{x_1}(t))}}{2}-h(t)}}{{\frac{{4l}}{3}-a \times m \times l{\rm{co}}{{\rm{s}}^2}({x_1}(t))}}}\\ {h(t)=a \times {\rm{cos}}({x_1}(t)) \times u(t).} \end{array}} \right.$

    (31)

    We approximate the system by the following two-rule fuzzy model:

    $\begin{array}{l} {A_1}=\left[{\begin{array}{*{20}{c}} 0 & 1\\ {\frac{g}{{4lm-aml}}} & 0 \end{array}} \right]\\ {B_1}=\left[{\begin{array}{*{20}{c}} 0\\ {\frac{a}{{\frac{{4lm}}{3}-a \times m \times l}}} \end{array}} \right]\\ {C_1}=\left[{\begin{array}{*{20}{c}} 1 & 0 \end{array}} \right]\\ {A_2}=\left[{\begin{array}{*{20}{c}} 0 & 1\\ {\frac{{2 \times g}}{{\pi (4 \times \frac{l}{3}-a \times m \times l{\gamma ^2})}}} & 0 \end{array}} \right]\\ {B_2}=\left[{\begin{array}{*{20}{c}} 0\\ {\frac{{-a\gamma }}{{4 \times \frac{l}{3}-a \times m \times l \times {\gamma ^2}}}} \end{array}} \right]\\ {C_2}=\left[{\begin{array}{*{20}{c}} 1 & 0 \end{array}} \right]\\ x(0)=\left[{\begin{array}{*{20}{c}} {\frac{\pi }{4}} & {}\\ {} & 0 \end{array}} \right]. \end{array}$

    The membership functions are shown in Fig. 4, where $\frac{\pi}{2} \leq x_{1}(t) \leq \frac{\pi}{2} $ .

    The rule-base of the reference Mamdani model is in Table 1.

    We can define the parameters of the membership function of the error and the change in error as

    $\begin{array}{l} {S_e} = 1\% \times \left| {{x_1}\left( 0 \right)} \right| = {0.5^ \circ }\\ {M_e} = 3\% \times \left| {{x_1}\left( 0 \right)} \right| = {1.5^ \circ }\\ {B_e} = 6\% \times \left| {{x_1}\left( 0 \right)} \right| = {3.0^ \circ }\\ {S_{\Delta e}} = 1\% \times {S_e}\\ {M_{\Delta e}} = 3\% \times {M_e}\\ {B_{\Delta e}} = 6\% \times {B_e}. \end{array}$

    Acceleration of the displacement is dependedent on the universe, we can take the maximum of the range: $li=0$ , $mo=0.785$ , $gr=1.57$ .

    For PSO simulation parameters, we can use n=49 as the number of particles in swarm, ite=100 as the number of iterations, $c_{1}=c_{2}=2$ , ${r}_{1}={r}_{2}=random [0, 1] $ , ${I}_w=w_{\rm max} \times $ $\frac {(w_{\rm max}-w_{\rm min})}{iter_{\rm max}}$ , and iter is the inertia weight factor[33], $\theta_1=1, \theta_2=0, v_1=1, v_2=0$ , initial parameters. The responses obtained are as

    Figure 9.  $x_1$ (t) response (---TVFS, $-$ $-$ FFS)

    Figure 10.  $x_2 (t)$ response (---TVFS, $-$ $-$ FFS)

  • The fuzzy system with the VFS approach gives a good stability and the dynamic specifications are better than the system with the FFS. Our results are improved compared to the results found in [2, 20], particularly the settling time in [14, 20]. We can see also that TVFS completely reduces oscillations but the control effort is relatively high compared to the FFS. We can observe that the effect of $\alpha(t)$ function is in the transient but later is got stabilized towards the final stability midpoint, which depends on the reference system, as shown in Fig. 7.

    In the end, we define the footprint of shifting (FOS), which represents the area covered by the variation of fuzzy sets, which itself is an interval-valued fuzzy set (Fig. 13). The initial $\alpha$ (0) is set to the midpoint of the universe, through the response of the Fig. 12, we can compute the universe of $\alpha(t)$ as

    Figure 11.  Control effort response (---TVFS, $-$ $-$ FFS)

    Figure 12.  $\alpha (t)$ response ( $-$ $-$ ) and $Y_m$ reference (---)

    Figure 13.  FOS of VFS membership function

    $FOS=\bar{\alpha }+\mid \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha }\mid=1.32+1.17=2.49\in [-\frac{\pi }{2}, \frac{\pi }{2}].$

    (32)

    By the PSO approach, we get a large FOS compared with the adaptive approach[23], provided that control effort and time of application is enough. This is explained by the good results, especially the settling time.

    The FOS of the PSO approach is illustrated by Fig. 13 which is generated by the program.

  • The presented TVFS approach is based only on the distance of the error. The role of the $\alpha(t)$ midpoint function is to ensure an adequate effort and time required to apply this effort. Generally, it requires a high effort which accurately reflects that the time-varying fuzzy sets have given the necessary power and enough time to the controller for stabilizing the system.

    In addition, the TVFS approach is built on the rule-base that defines the displacement functions of the midpoint, where it is very important to give a high or low control amplitude, and on the criterion that defines the direction of shifting, that is to increase (decrease) the membership grades that generate the increasing (decreasing respectively) the control effort. The displacement functions of the midpoint can cause oscillations of the control (cattering). For reducing these oscillations, we focus on a well defined rule-base (that varies quickly or slowly) and on the choice of the direction criterion. Through the TVFS approach, the ranges of the linguistic values of the fuzzy sets change with time according to the variation of the linguistic values of the error.

    Finally, we can summarize this conclusion as Table 2.

    Table 2.  Comparison between three fuzzy sets

  • Decay rate controller design using relaxed stability conditions: The condition that $\dot{V}(x)+2\beta V(x(t))\leq 0$ for all trajectories is equivalent to[28-31]

    $\exists P>0, \exists Q>0\ \ G_{ii}^{\text{T}}P+P{{G}_{ii}}+(s-1)Q+2\beta P<0, \beta >0$

    (A1)

    $\begin{array}{l} {\left({\frac{{{G_{ij}} + {G_{ji}}}}{2}} \right)^{\rm{T}}}P + P\frac{{{G_{ij}} + {G_{ji}}}}{2}-Q + 2\beta P \le 0\\ i & j, \;\; \;{\rm{s}}.{\rm{t}}.\; \;{h_i} \cap {h_j} \ne \phi, \;\; {\rm{where}}1 & s & r. \end{array}$

    (A2)

    We can find the $K_i$ controller by this optimization problem:

    $\begin{align} & {{\max }_{(X, Z, {{Y}_{1}}, \cdots, {{Y}_{r}})}}\beta \\ &~~~~\left\{ \begin{array}{*{35}{l}}-XA_{i}^{\text{T}}-{{A}_{i}}X+{{B}_{i}}{{Y}_{i}}+Y_{i}^{\text{T}}B_{i}^{\text{T}}-(s-1)Z-\\ \qquad 2\beta X>0, \quad i=1, 2, \cdots, r \\-XA_{i}^{\text{T}}-{{A}_{i}}X-XA_{j}^{\text{T}}-{{A}_{j}}X+{{B}_{i}}{{Y}_{j}}+ \\ \quad \quad Y_{j}^{\text{T}}B_{i}^{\text{T}}+{{B}_{j}}{{Y}_{i}}+Y_{i}^{\text{T}}B_{j}^{\text{T}}+2Z-4\beta X>0 \\ X>0, \ \ Y\ge 0, \ \ \ i < j, \ \ \ \text{s}.\text{t}.\ \ {{h}_{i}}\cap {{h}_{j}}\ne \phi \\ \text{where}\ \ {{K}_{i}}={{Y}_{i}}{{X}^{-1}}, \ X={{P}^{-1}}, \ Z=XQX. \\ \end{array} \right. \\ \end{align}$

    (A3)
  • Assume that the initial condition $x(o)$ is known, then[28-31]:

    1) The constraint on control input $\parallel \text{u}(\text{t}){{\parallel }_{2}} < \varphi $ for t≥0 can be represented by

    $\left[\begin{matrix} 1 & x{{(0)}^{\text{T}}} \\ x(0) & X \\ \end{matrix} \right]\ge 0$

    (A4)

    $\left[\begin{matrix} X & Y_{i}^{\text{T}} \\ Y_{i}^{\text{T}} & {{\varphi }^{2}}I \\ \end{matrix} \right]\ge 0.$

    (A5)

    2) The constraint on output $\parallel y(t)\parallel_{2} < \rho$ for t≥0 can be represented by

    $\left[\begin{matrix} 1 & x{{(0)}^{\text{T}}} \\ x(0) & X \\ \end{matrix} \right]\ge 0$

    (A6)

    $\left[\begin{matrix} X & XC_{i}^{\text{T}} \\ XC_{i}^{\text{T}} & {{\rho }^{2}}.I \\ \end{matrix} \right]\ge 0$

    (A7)

    where

    ${{K}_{i}}={{Y}_{i}}{{X}^{-1}}, \ \ X={{P}^{-1}}.$

    (A8)
Reference (34)

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