﻿ 一类区间二型模糊PI控制器设计算法
«上一篇
 文章快速检索 高级检索

 智能系统学报  2018, Vol. 13 Issue (5): 836-842  DOI: 10.11992/tis.201703039 0

### 引用本文

SHI Jianzhong, LI Rong, YANG Yong. An interval type 2 fuzzy PI controller design algorithm[J]. CAAI Transactions on Intelligent Systems, 2018, 13(5): 836-842. DOI: 10.11992/tis.201703039.

### 文章历史

An interval type 2 fuzzy PI controller design algorithm
SHI Jianzhong, LI Rong, YANG Yong
School of Energy and Power Engineering, Nanjing Institute of Technology, Nanjing 211167, China
Abstract: The interval type-2 fuzzy controllers outperform their type-1 counterparts in processing uncertainty; however, the type-2 fuzzy controller needs to be reduced, and the commonly used iterative reduction algorithms such as Karnik-Mendel (KM) algorithm are inefficient and difficult to use in real-time situations. In this paper, an interval type-2 fuzzy PI controller algorithm that combines the dynamic defuzzification method and direct reduction algorithm is proposed. The algorithm considers the effects of error and error variation on the controller output during reduction, thus avoiding iterative reduction such as in KM. Simulations of a second-order delay object and nonlinear object show that the proposed algorithm can effectively suppress the system overshoot and reduce the time of the system to reach steady state; furthermore, the controller outputs around the set value are smoother.
Key words: type-2 fuzzy sets    KM-type reduction    type-2 fuzzy control    type-2 fuzzy PI    uncertain domain    dynamic defuzzification    direct reduction    incremental PI

1 KM降阶算法

 ${c_l} =\frac{{\displaystyle\sum\limits_{i =1}^L {{x_i}\overline {f({x_i})} + \displaystyle\sum\limits_{i =L + 1}^N {{x_i}\underline {f ({x_i})}} } }}{{\displaystyle\sum\limits_{i =1}^L {\overline {f({x_i})} + \displaystyle\sum\limits_{i =L + 1}^N {\underline {f ({x_i})}} } }}$ (1)
 ${c_r} =\frac{{\displaystyle\sum\limits_{i =1}^R {{x_i}\underline {f ({x_i})} + \displaystyle\sum\limits_{i =R + 1}^N {{x_i}\overline {f({x_i})}} } }}{{\displaystyle\sum\limits_{i =1}^R {\underline {f ({x_i})} + \displaystyle\sum\limits_{i =R + 1}^N {\overline {f({x_i})}} } }}$ (2)

 ${c_l} =\frac{{\displaystyle\int_a^{{c_l}} {\overline {f(x)}x{\rm{d}}x + \displaystyle\int_{{c_l}}^b {\underline {{f} (x)}x{\rm{d}}x} } }}{{\displaystyle\int_a^{{c_l}} {\overline {f(x)}{\rm{d}}x + \int_{{c_l}}^b {\underline {{f} (x)}{\rm{d}}x} } }}$ (3)
 ${c_r} =\frac{{\displaystyle\int_a^{{c_r}} {\underline {{f} (x)}x{\rm{d}}x + \int_{{c_r}}^b {\overline {f(x)}x{\rm{d}}x} } }}{{\displaystyle\int_a^{{c_r}} {\underline {{f} (x)}{\rm{d}}x + \displaystyle\int_{{c_r}}^b {\overline {f(x)}{\rm{d}}x} } }}$ (4)

2 区间二型模糊控制系统

 Download: 图 1 区间二型模糊控制系统结构 Fig. 1 Interval type-2 fuzzy control system structure

 Download: 图 2 三角型区间二型模糊控制器的首隶属度函数 Fig. 2 Triangle interval type-2 fuzzy controller primary membership function

then ${y^1} ={K_{\rm P}} \times (d\dot e) + {K_{\rm I}} \times (de)$

then ${y^2} ={K_{\rm P}} \times (d\dot e) + {K_{\rm I}} \times ( - de)$

then ${y^3} ={K_{\rm P}} \times ( - d\dot e) + {K_{\rm I}} \times (de)$

then ${y^4} ={K_{\rm P}} \times ( - d\dot e) + {K_{\rm I}} \times ( - de)$

3 本文算法

 $\overline \mu _e^{\tilde P} =\left\{ \begin{gathered} 1,\quad{\rm{ }}e > de - {d_1} \\ \frac{{e + de + {d_1}}}{{2 \times de}},\quad {\text{其他}} \\ \end{gathered} \right.$
 $\underline {\mu } _e^{\tilde P} =\left\{ \begin{gathered} 0,\quad{\rm{ }}e < - de + {d_1} \\ \frac{{e + de - {d_1}}}{{2 \times de}},\quad{\text{其他}} \\ \end{gathered} \right.$

 $\overline \mu _e^{\tilde N} =\left\{ \begin{gathered} 1,\quad{\rm{ }}e < - de + {d_1} \\ \frac{{de + {d_1} - e}}{{2 \times de}},\quad{\text{其他}} \\ \end{gathered} \right.$
 $\underline {\mu } _e^{\tilde N} =\left\{ \begin{gathered} 0,\quad{\rm{ }}e > de - {d_1} \\ \frac{{de - {d_1} - e}}{{2 \times de}},\quad{\text{其他}} \\ \end{gathered} \right.$

 $\overline \mu _{\Delta e}^{\tilde P} =\left\{ \begin{gathered} 1,\quad{\rm{ }}\Delta e > d\dot e - {d_2} \\ \frac{{\Delta e + d\dot e + {d_2}}}{{2 \times d\dot e}},\quad{\text{其他}} \\ \end{gathered} \right.$
 $\underline {\mu } _{\Delta e}^{\tilde P} =\left\{ \begin{gathered} 0,\quad{\rm{ }}\Delta e < - d\dot e + {d_2} \\ \frac{{\Delta e + d\dot e - {d_2}}}{{2 \times d\dot e}},\quad{\text{其他}} \\ \end{gathered} \right.$

 $\overline \mu _{\Delta e}^{\tilde N} =\left\{ \begin{gathered} 1,\quad{\rm{ }}\Delta e < - d\dot e + {d_2} \\ \frac{{d\dot e + {d_2} - \Delta e}}{{2 \times d\dot e}},\quad{\text{其他}} \\ \end{gathered} \right.$
 $\underline {\mu } _{\Delta e}^{\tilde N} =\left\{ \begin{gathered} 0,\quad{\rm{ }}\Delta e > d\dot e - d_2 \\ \frac{{d\dot e - d_2 - \Delta e}}{{2 \times d\dot e}},\quad{\text{其他}} \\ \end{gathered} \right.$

 ${c_l} =\frac{{\displaystyle\sum\limits_{k =1}^M {{{\underline {f} }^k}} {y^k}}}{{\displaystyle\sum\limits_{k =1}^M {{{\underline {f} }^k}} }}$ (5)
 ${c_r} =\frac{{\displaystyle\sum\limits_{i =1}^M {{\overline{ f^k}}} {y^k}}}{{\displaystyle\sum\limits_{i =1}^M {{\overline {f^k}}} }}$ (6)

 $a(t) =\frac{{e(t) + \Delta e(t)}}{2} + 0.5$ (7)
 $\Delta u(t) =a(t) \times {c_l} + (1 - a(t)) \times {c_r}$ (8)

 $\Delta u(t) =a(t) \times \min ({c_l},{c_r}) + (1 - a(t)) \times \max ({c_l},{c_r})$ (9)

 Download: 图 3 本文算法的控制系统框图 Fig. 3 Control system diagram of the proposed method

4 仿真实例 4.1 仿真实例1

 $G(s) =\frac{K}{{s(Ts + 1)}}{{\rm e}^{ - \tau s}}$

PI控制器参数KP=0.044 9，KI=0.001 4，采样周期为0.1 s。

ISE、ITSE、ITAE为误差积分准则，在单位阶跃扰动下，系统的设定值与输出之间偏差的某个函数的积分数值，分别表示为

 ${\rm {ISE}} =\int_0^{{t_s}} {e{{(t)}^2}{\rm d}t}$
 ${\rm {ITSE}} =\int_0^{{t_s}} {t \times e{{(t)}^2}{\rm d}t}$
 ${\rm{ITAE }}=\int_0^{{t_s}} {t \times \left| {e(t)} \right|{\rm d}t}$

 Download: 图 5 PI控制器输出增量随着偏差和偏差变化量变化的三维曲线图 Fig. 5 PI controller output increment with respect to error and error variation
 Download: 图 6 本文算法控制器输出增量随着偏差和偏差变化量变化的三维曲线图 Fig. 6 The proposed method controller output increment with respect to error and error variation

 Download: 图 7 二阶迟延系统阶跃响应控制量曲线 Fig. 7 Second-order delay system control variable curve under step response
4.2 仿真实例2

 $\frac{{y\left( t \right)}}{{dt}} =- y\left( t \right) + 7 \times y{\left( t \right)^2} + u\left( t \right)$

PI控制器参数：KP=56.25，KI=669.375，采样周期0.01 s。

 Download: 图 9 PI控制器输出增量随着偏差和偏差变化量变化的三维曲线图 Fig. 9 PI controller output increment with respect to error and error variation
 Download: 图 10 本文算法控制器输出增量随着偏差和偏差变化量变化的三维曲线图 Fig. 10 The proposed method controller output increment with respect to error and error variation

 Download: 图 11 非线性系统阶跃响应控制量曲线 Fig. 11 Nonlinear system control variable curve under step response
5 结束语

 [1] ZADEH L A. The concept of a linguistic variable and its application to approximate reasoning-I[J]. Information sciences, 1975, 8(3): 199-249. DOI:10.1016/0020-0255(75)90036-5 (0) [2] MENDEL J M. Uncertain rule-based fuzzy logic systems: introduction and new directions[M]. Upper-Saddle River, NJ: Prentice Hall, 2001. (0) [3] KARNIK N N, MENDEL J M. Operations on type-2 fuzzy sets[J]. Fuzzy sets and systems, 2001, 122(2): 327-348. DOI:10.1016/S0165-0114(00)00079-8 (0) [4] WU Dongrui, MENDEL J M. Enhanced Karnik-Mendel algorithms[J]. IEEE transactions on fuzzy systems, 2009, 17(4): 923-934. DOI:10.1109/TFUZZ.2008.924329 (0) [5] KARNIK N N, MENDEL J M, LIANG Qilian. Type-2 fuzzy logic systems[J]. IEEE transactions on fuzzy systems, 1999, 7(6): 643-658. DOI:10.1109/91.811231 (0) [6] LIANG Qilian, MENDEL J M. Interval type-2 fuzzy logic systems: theory and design[J]. IEEE transactions on fuzzy systems, 2000, 8(5): 535-550. DOI:10.1109/91.873577 (0) [7] MENDEL J M, JOHN R I, LIU Feilong. Interval type-2 fuzzy logic systems made simple[J]. IEEE transactions on fuzzy systems, 2006, 14(6): 808-821. DOI:10.1109/TFUZZ.2006.879986 (0) [8] CASTILLO O, MELIN P. A review on the design and optimization of interval type-2 fuzzy controllers[J]. Applied soft computing, 2012, 12(4): 1267-1278. DOI:10.1016/j.asoc.2011.12.010 (0) [9] DERELI T, BAYKASOGLU A, ALTUN K, et al. Industrial applications of type-2 fuzzy sets and systems: a concise review[J]. Computers in industry, 2011, 62(2): 125-137. DOI:10.1016/j.compind.2010.10.006 (0) [10] MELGAREJO M. A fast recursive method to compute the generalized centroid of an interval type-2 fuzzy set[C]//Annual Meeting of the North American Fuzzy Information Processing Society. California, USA, 2007: 190–194. (0) [11] WU Dongrui, NIE Maowen. Comparison and practical implementation of type-reduction algorithms for type-2 fuzzy sets and systems[C]//Proceedings of 2011 IEEE International Conference on Fuzzy Systems. Taipei, China, 2011: 2131–2138. (0) [12] 胡怀中, 赵戈, 杨华南. 一种区间型二型模糊集重心的快速解法[J]. 控制与决策, 2010, 25(4): 637-640. HU Huaizhong, ZHAO Ge, YANG Hua’nan. Fast algorithm to calculate generalized centroid of interval type-2 fuzzy set[J]. Control and decision, 2010, 25(4): 637-640. (0) [13] WU Dongrui. Approaches for reducing the computational cost of interval type-2 fuzzy logic systems: overview and comparisons[J]. IEEE transactions on fuzzy systems, 2013, 21(1): 80-99. DOI:10.1109/TFUZZ.2012.2201728 (0) [14] WU Dongrui. On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers[J]. IEEE transactions on fuzzy systems, 2012, 20(5): 832-848. DOI:10.1109/TFUZZ.2012.2186818 (0) [15] NIE Maowen, TAN W W. Analytical structure and characteristics of symmetric Karnik-Mendel type-reduced interval type-2 fuzzy PI and PD controllers[J]. IEEE transactions on fuzzy systems, 2012, 20(3): 416-430. DOI:10.1109/TFUZZ.2011.2174061 (0) [16] ZHOU Haibo, YING Hao. A method for deriving the analytical structure of a broad class of typical interval type-2 mamdani fuzzy controllers[J]. IEEE transactions on fuzzy systems, 2013, 21(3): 447-458. DOI:10.1109/TFUZZ.2012.2226891 (0) [17] 雷宾宾, 保宏, 许谦. 区间二型模糊PI/PD控制器设计与结构分析[J]. 电机与控制学报, 2016, 20(6): 50-62. LEI Binbin, BAO Hong, XU Qian. Design and structural analysis of interval Type-2 fuzzy PI/PD controller[J]. Electric machines and control, 2016, 20(6): 50-62. (0) [18] 龙祖强, 许岳兵, 李龙. 一类乘积型区间二型模糊控制器的解析结构[J]. 控制理论与应用, 2016, 33(7): 929-935. LONG Zuqiang, XU Yuebing, LI Long. Analytical structure of a class of product and interval-type-2 fuzzy controllers[J]. Control theory and applications, 2016, 33(7): 929-935. (0) [19] ULU C, GÜZELKAYA M, EKSIN İ. A dynamic defuzzification method for interval type-2 fuzzy logic controllers[C]//Proceedings of 2011 IEEE International Conference on Mechatronics. Istanbul, Turkey, 2011: 318–323. (0) [20] MIZUMOTO M. Realization of PID controls by fuzzy control methods[J]. Fuzzy sets and systems, 1995, 70(2/3): 171-182. (0) [21] WU Dongrui, TAN W W. Interval type-2 fuzzy PI controllers: why they are more robust[C]//Proceedings of 2010 IEEE International Conference on Granular Computing. San Jose, California, USA, 2010: 802–807. (0) [22] NIE Maowen, TAN W W. Towards an efficient type-reduction method for interval type-2 fuzzy logic systems[C]//Proceedings of 2008 IEEE International Conference on Fuzzy Systems. Hong Kong, China, 2008: 1425–1432. (0)