An Improved Robust Fuzzy-PID Controller With Optimal Fuzzy ...
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 35, NO. 6, DECEMBER 2005
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An Improved Robust Fuzzy-PID Controller With Optimal Fuzzy Reasoning Han-Xiong Li, Senior Member, IEEE, Lei Zhang, Kai-Yuan Cai, and Guanrong Chen, Fellow, IEEE
Abstract—Many fuzzy control schemes used in industrial practice today are based on some simplified fuzzy reasoning methods, which are simple but at the expense of losing robustness, missing fuzzy characteristics, and having inconsistent inference. The concept of optimal fuzzy reasoning is introduced in this paper to overcome these shortcomings. The main advantage is that an integration of the optimal fuzzy reasoning with a PID control structure will generate a new type of fuzzy-PID control schemes with inherent optimal-tuning features for both local optimal performance and global tracking robustness. This new fuzzy-PID controller is then analyzed quantitatively and compared with other existing fuzzy-PID control methods. Both analytical and numerical studies clearly show the improved robustness of the new fuzzy-PID controller. Index Terms—Fuzzy control, fuzzy reasoning, optimal fuzzy reasoning, fuzzy-PID controller, robustness.
I. INTRODUCTION
T
HE MAIN idea of fuzzy logic control (FLC) was introduced by Zadeh [1], and first applied by Mamdani [2] in an attempt of controlling structurally ill-modeled systems. FLC has two main parts that need to be designed: one is the control structure composing of rules and gains and the other is a fuzzy reasoning method. Various inference for the fuzzy reasoning method have been proposed, including the compositional rule of inference (CRI) [3], evidence reasoning [4], approximate analogical reasoning approach based on similarity measures [5], triple implication method [6], and so on. The most popular inference method is perhaps the CRI method. However, [7] discusses the robustness problem of various inference methods in terms of -equalities of fuzzy sets [8], i.e., how errors in premises affect consequences in fuzzy reasoning, and arrives at the conclusion that the robustness of the CRI methods is not very satisfactory. It is well known that this traditional fuzzy reasoning has inconsistent inference outcomes. In contrast to most existing fuzzy reasoning methods of being processes of logical inference, the optimal fuzzy reasoning method, introduced in our previous work [9], [10], treats (fuzzy) reasoning as a process of optimization instead of (an
Manuscript received October 18, 2004. This work was supported in part by a grant from RGC of Hong Kong (CityU 1207/04E) and by a Grant from the National Science Foundation of China for a distinguished young scholar (50 429 501). This paper was recommended by Associate Editor T. H. Lee. H.-X. Li is with the Department of MEEM, City University of Hong Kong, Hong Kong, China (e-mail: [email protected]). L. Zhang and K.-Y. Cai are with the Department of Automation, Beihang University, Beijing, China. G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China. Digital Object Identifier 10.1109/TSMCB.2005.851538
Fig. 1.
Two-dimensional configuration for PID type of FLC.
extension of) the classical logical inference process. From the control perspective, the processes of the other methods are of open-loop but the optimal fuzzy reasoning method is of closed-loop with some objectives of performance optimization. Owing to the closed-loop characteristic, the optimal fuzzy reasoning method, which introduces the feedback optimization into traditional fuzzy reasoning, can significantly improve robustness and consistency. Although systematic analysis and design for FLC are still considered premature in general, significant progress has been gained recently in the pursuit of this technology. FLC structure can be classified into different types [11], and the most popular one for the low-level control purpose is PID type of FLC [12], [13]. Significant research efforts have been spent on controllers design of the fuzzy-PID type [12], [14]–[19], including its simplified versionsoffuzzy-PD[20],[21]andfuzzy-PI[22]controlfordifferent applications. Comparisons between fuzzy control and its conventional control have also been reported from different viewpoints [23]–[29]. As the rule base conveys a general control policy, it should be sustained throughout the control process, leaving most design and tuning work to the scaling gains of the controller [30]. However, many of these analyzes and comparisons are qualitative or descriptive due to the lack of a precise mathematical model of the inference logic. A quantitative model for the max-min inference logic was first introduced by Ying [31] for analysis of the Mamdani-type of FLC, which was lately modified by Li [32] for the case of a linear rule base. On the other hand, most FLC is based on the simplified fuzzy reasoning [33], which loses much of the original fuzzy characteristics and therefore usually affect the robustness. A proper integration of the fuzzy reasoning method and its outer control structure is obviously crucial for achieving optimal control performance. In this paper, an improved robust fuzzy-PID control scheme is proposed by incorporating the optimal fuzzy reasoning into the well-developed PID type of control framework [13]. The model of the optimal fuzzy-PID control is developed here by using the quantitative method [31], [32], which will then be compared with the existing fuzzy-PID control schemes based on other reasoning methods. Finally, simulation study is reported, which demonstrates the robustness of the proposed reasoning
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Fig. 2.
Traditional fuzzy reasoning model for control.
Fig. 3.
Optimal fuzzy reasoning model for control.
method and the effectiveness of its integration with the PID type of control structures. II. FUNDAMENTALS OF FLC A. Fuzzy-PID Control Structure There are different types of fuzzy-PID controllers [16]. One of the earliest and very effective structures use a two-dimensional linear rule base with standard triangular membership functions (MFs) [13], which actually combines fuzzy-PI and fuzzy-PD controllers, as shown in Fig. 1. And it is a very simple controller. There are four input/output scaling gains, , for control action adjustment, where the input gain ratio is defined as . B. Three Inference Methods In this paper, three kind of fuzzy reasoning methods (FRMs) will be studied. These are simplified, traditional and optimal FRM, respectively. Simplified FRM treats crisp input value as a singleton, which is a nonfuzzy variable. The traditional FRM treats both premises and consequences as “real” fuzzy variables and forms an inference space via the implication operation. This inference space is discretely represented by a fuzzy relation matrix , which can be obtained off-line from the rule base. The simplified FRM is usually used by the FLC due to its relatively small computational burden. The optimal FRM introduces the idea of optimization and feedback into the reasoning process of
the traditional FRM. Their key difference is clearly explained by comparing Fig. 2 with Fig. 3. The basic idea of the optimal FRM is explained as follows. •
Suppose that rule base which can be expressed as a fuzzy relation matrix is gained from the experience of an expert, i.e., heuristic rules or data. Then, one should trust and treat it as a reference for reasoning evaluation. • While in the process of inference, if a fuzzy premise is given, a corresponding consequence will be generated (through ) via the inference method. Then, this pair of and form a new relation matrix , which unfortunately is different from the original for most FRMs. • In the proposed optimal reasoning method, the feedback introduced in the reasoning will search for the best from the consequence domain, so as to minimize the “reasoning . For the existing error” defined by . FRMs, In the optimal reasoning process, may be analytically gained in some cases, but in most cases it must be gained from some existing optimization methods. The basic idea is to according to the change of continuously adjust and stop computing when satisfies a given criterion, because analytically evaluating is often complicated and impractical. It can be easily seen that the optimal FRM treats the fuzzy reasoning as a process of optimization instead of a process of
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TABLE I A TWO-DIMENSIONAL LINEAR RULE BASE WITH n = 3
Fig. 4. Decomposition of the rule base into inference cells.
logical inference. Since the inner loop is in the feedback loop, the effect of the disturbance in the fuzzy reasoning process, such as the perturbation of reasoning premises due to the noise of inputs, can be greatly reduced [10]. Since the inner loop will help the local performance more, and the outer loop will help the global performance more, the combination of these two loops will clearly yield a better performance. III. MODELS OF DIFFERENT FRMS A. FRM With Linear Rule Base and Standard Triangular MFs The fuzzy-PID shown in Fig. 1 has two inputs and one output. All fuzzy variables are normalized to the same domain, , number of triangular MFs ( is odd each of them has number) with equal spread of 2W as shown in Fig. 4. The grades . A linear rule base is of MFs are represented by used, for which an example with = 3 is shown in Table I. The rule base plane can be decomposed into many inference , and ) on its four cells (ICs) with output rules ( corners, as shown in Fig. 4 [32]. The inference can be operated on these ICs. Four different types of ICs are generated in terms , and , as follows: of different
The total number of ICs in a linear rule base is with from type 1, from type 2, from type 3, and from type 4. The percentages of these four types of ICs in a linear , and , rerule base are spectively. When is large enough, most ICs in the rule base
Fig. 5.
Functional composition of the inference cell IC(i; j ).
are type 1, particularly around the equilibrium point or far away from the boarder. Therefore, only type 1 IC is considered in this study. Analysis of the rule base can be simplified to the analysis of . As shown in Fig. 5, two MFs from type 1 inference cell (ith and th) and (jth and th) divide IC into four with inference outputs generated by sub-regions four rules as follows: and and and and where is the th fuzzy variable of is the th fuzzy variable of , and is the th fuzzy variable of , with their , and , respectively. MF grades denoted by The grades of fuzzification can be defined as
As seen from Figs. 1 and 4, they can be further derived as and , or (1) with
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B. Model of the Simplified FRM
1) Fuzzy relation matrices for each rule are, respectively
Most FLC schemes use a simplified FRM with different operators, such as the max-min, sum-product, etc. The sum-product method is chosen for the simplified FRM in this paper with the inference output expressed as and where . Using (1), the same inference output will be obtained in four regions in each IC, as shown in (2), at the bottom of the page. With this simplified FRM, the fuzzy-PID scheme shown in Fig. 1 can be easily derived, as
2) Fuzzy relation matrix of IC(i, j) is
(4)
Finally, the fuzzy inference is defined as
(5-1) with inference output (3)
(5-2)
This actually is a conventional PID controller with proportional , integral time constant and derivative time constant gain , where
From Appendix A, the model of the traditional FRM is obtained as
(6) (3-2) with
.
Then, the model of fuzzy-PID in Fig. 1 with traditional FRM becomes
C. Traditional FRM The max-min method is chosen for the traditional FRM in this paper. With the linear rule base and standard triangular MFs, it gives the same inference output as the simplified FRM with the max-min method [32]. The fuzzification result (reasoning premise) is first given as follows. 1) for for . for 2) and . Then, the fuzzy relation matrix is formed as follows.
(7) with the nonlinear parameter
defined in (6).
(2)
LI et al.: AN IMPROVED ROBUST FUZZY-PID CONTROLLER WITH OPTIMAL FUZZY REASONING
This actually is a nonlinear PID controller plus a relay, , with proportional gain , integral time constant and derivative time constant , where
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From Appendix B, the reasoning output is found to be
with
(10)
D. Optimal FRM For the optimal FRM, the reasoning premise and the fuzzy relation matrix can be formed in the same way as the traditional FRM. The objective function of optimal FRM is chosen as
Similar to (7), the model of the fuzzy-PID control in Fig. 1 with the optimal FRM becomes
with as (4) and the first equation shown at the is the reasoning bottom of the page, where consequence. can be The optimization of the complex function further decomposed into three simple optimization functions, , in
and in (8), as shown at the bottom of the page. The optimization scheme divides IC into nine sub, where different outputs are genregions, that enclose erated. The lines can be figured out from Appendix B and they are expressed as
, and , respectively. The output of the fuzzy reasoning is expressed as
and
are chosen to minimize
defined same as in (10). with This is a nonlinear PID controller plus a relay, , with proportional gain , integral time constant and derivative time constant , defined as
When the control process is on the equilibrium state, only depends on . IV. THEORETICAL ANALYSIS Fundamental features of the three methods are analyzed and discussed, including their stability and robustness in process control.
(9) where in (8).
(11)
and
A. Mathematical Explanation From (2)–(11) and Figs. 5 and 6, one can arrive at the following conclusions.
(8)
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and is a local switching function. 3) Optimal FRM . Define a switching function Then
where
Fig. 6.
Partition in inference cell IC(i; j ).
1) There is an adjustable parameter for the traditional FRM for and a group of adjustable parameters the optimal FRM; however, there is no adjustable parameter for the simplified FRM. is on the boundary of , 2) When where the inference results of both traditional and optimal FRM are the same as that of the simplified FRM. are in in Fig. 6, both the tra3) When ditional and the optimal FRMs have the same inference output, which however is different from that of the simplified FRM. 4) The simplified FRM is actually a conventional PID controller; the traditional FRM plays the role as a nonlinear PID controller with a varying parameter plus a relay C; the optimal FRM behaves as a variable structure controller with a group of varying parameters , which make it become: ; • a PID plus a relay C in regions of ; • a PD plus a relay C in regions of ; • a PI plus a relay C in regions of • a relay control C in region . B. Feature of Sliding Mode Control The model of the three FRMs, (3), (7), and (11), can be expressed in terms of the sliding mode structure, as follows: 1) Simplified FRM: . Define a switching function . Then, 2) Traditional FRM . Define a switching function, Then
where
can be treated as an equivalent control, is a switching control term
is an equivalent control term and is a switching control term
In view of sliding-mode control [34], the equivalent control term is to push the system trajectory toward the sliding surface (it handles the major nonlinearity) and the switching control term is to maintain the trajectory on the sliding surface (it deals with the unmodeled dynamics). A fuzzy-PID controller on either the traditional or the optimal FRM usually displays the feature of sliding mode control. However, the one in the simplified FRM has very little feature of sliding mode because of the missing equivalent control term. Since both equivalent and switching control terms in the optimal FRM are more flexible than the one in the traditional FRM, the control in the optimal FRM can handle more complex processes than the one in the traditional FRM, and thus should have a better performance in more realistic situations. C. Robustness Analysis The issue of robustness in control systems is for the stability and performance that the control system can maintain under parameter perturbations and external disturbances. Previous work [9], [10] has shown that the optimal FRM can improve the reasoning robustness under perturbations of rule base and fuzzy premises. Here, the robustness of the optimal FRM will be further discussed. 1) Structure Robustness of the System: When the control law or the plant parameter varies, it can be considered as a perturbation of the rule base, from the correct one, to the current one, , as shown in Fig. 7, where and the line is a . perpendicular bisector of the line For the same rule base, different FRMs still get different inference results. So, the new fuzzy relation matrix , which is produced under the given reasoning premise, has a nonzero diswith tance to . Assume that the optimal FRM produces distance to and the other FRM produces with distance to . In Fig. 7, is the origin and the correct rule base in . the control, and is the nominal rule base, with . From the naDefinition of the optimal FRM gives ture of the perpendicular bisector, should be on the left side of appears inside the circle centered line . Since has a large probability to at with the radius . Therefore, . In other words, the opbe on the left side of line , i.e., timal FRM has better robustness to the perturbation of the rule base.
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•
Robustness of the optimal method is better than, equal to, or worse than the traditional one in probability of 0.161, 0.75, or 0.089, respectively. Thus, the optimal method has the best robustness in view of probability and the simplified one is the worst in this comparison. V. SIMULATION A fuzzy-PID controller shown in Fig. 1, with the linear rule base given in Table I and some standard triangular MFs, is chosen to control the plant
Fig. 7. Robustness analysis for perturbation of the rule base.
2) Statistics Based Robustness to External Disturbance: When the reference signal varies or noise appears in the control loop, they can be considered as disturbances to the fuzzy reasoning premises. Previous work [10] has shown better robustness of the optimal FRM against errors in the premises than the CRI method in Monte Carlo simulations. Since the results are measured in the statistical sense, the robustness should be measured in terms of probability . First, compare with the three methods when is fixed is changing from 0 to 1. A smaller means a and better robustness here. Then, the probability , that one method is better than, equal to and worse than another method on robustness, are obtained via integration over the whole IC area. The following results (12)–(14) are derived from (2) and Appendices A and B. 1) Simplified FRM
(15) The design of the scaling gains follows the well-known Ziegler-Nichols method [35]. Using the frequency method, the and are obtained ultimate gain and frequency for gain calculation. Since the plant is relatively simple, the derivative action is not necessary and can be chosen as zero for simplicity. Thus, the classical PID control parameters will be chosen as (16) Since the fuzzy-PID controller in the simplified FRM is actually a conventional PID controller shown in (3), its PID gains in (3-2) can be chosen to be the same as that of the well-tuned classical PID controller in (16). Using (3-2) and (16), one can easily derive and select the following gains:
(12) 2) Traditional FRM
(13)
where regions 3) Optimal FRM
are classified in Fig. 5.
From (3-2) we know that the solutions of is not unique, but the performance with different solutions on the steady state are same, which is displayed in simulations. And different solutions are equivalent to the same PID parameters . For the plant (15) without perturbation and noise, the performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 8, which clearly demonstrates that three fuzzy-PID methods all have good response. In the following simulations, reference signal is a unit step, simulation time is 150 s and sampling step is 0.01 s. A. Robustness Against System Perturbation
(14)
are classified in Fig. 6. where regions The following conclusions are drawn from Appendix C. • Robustness of the traditional method is better or worse than the simplified one in probability of 0.688 or 0.312, respectively. • Robustness of the optimal method is better or worse than the simplified one in probability of 0.698 or 0.302, respectively.
The simulation steps are summarized as follows. 1) Generate disturbance by changing either the plant parameter or one of the control parameters. 2) Compare the responses under disturbance. Case 1: Under Plant Perturbation: When the parameter “15” in the plant (15) adds, the responses of all the three methods become worse. On condition that control system has an acceptable performance, that is, the relative error is lower than 2%, the larger the range of the parameter is, the better the robustness of control method is. Assume that the plant (15) is perturbed to (17)
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Fig. 10. Performance comparison w.r.t. controller perturbation. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
Fig. 8. Performance comparison. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
Fig. 11. Performance comparison w.r.t. noise. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
B. Robustness Against Noise
Fig. 9. Performance comparison w.r.t. plant perturbation. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
The performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 9, which clearly demonstrates better robustness of the optimal FRM against system perturbation in the plant. Case 2: Under Controller Perturbation: When the parameter in the controllers greatly adds, the responses of the three methods all become worse. Similar to the above, the is, the better the robustness of control larger the range of is perturbed from method is. The controller input gain 1 to 8. The performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 10, which clearly demonstrates better robustness of the optimal FRM against system perturbation in the controller.
When noises are added to control system, the responses of of the three methods all become worse. Inputs the fuzzy-PID controller in Fig. 1 are added with uniformly , to become distributed noises . The performance criteria are defined as and Since the influence of the noise is more important while in the steady state, the integral action will start from the mid-time of the simulation, , to the end time, . For one simulation, the performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 11. And a large amount of simulations have been carried out with the mean performance shown in Table II. The optimal FRM clearly demonstrates better robustness than the other methods with respect to noise during the fuzzy-PID
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From (5-1), one can derive and then from (5-2) one has
TABLE II STATISTICAL PERFORMANCE TO NOISE
(a-1) control processes, although the outer control loop can filter out some influence of the noise.
Further derivation will result in
VI. CONCLUSION Both the concept and the technique of optimal fuzzy reasoning have been proposed and discussed in this paper, to integrate with fuzzy PID-type of control structures for better robust control. A mathematical model of the optimal fuzzy reasoning has been derived and then compared with other reasoning methods. Based on this quantitative model, both theoretical analysis and numerical simulations have been carried out to study the fuzzy-PID control with different reasoning methods. It is found that a fuzzy-PID control scheme with different reasoning methods becomes different types of nonlinear PID controllers plus a relay. The optimal fuzzy reasoning method brings in more flexibility in tuning their nonlinear gains, which work better for more complex problems. Both analytical and simulation studies have clearly demonstrated that the optimal fuzzy reasoning method can enhance the control system robustness thanks to its optimality and inherent feedback tuning. It therefore works better under complex and realistic environments. And by simulations, we can arrive at the conclusions: total operation time of optimal FRM is about six times that of simplified FRM and about four times that of traditional FRM, which display the computational cost of optimal FRM is not large; the output of FLC with optimal FRM is stable when the control process is stable. The integration of the proposed optimal fuzzy reasoning method and some good control structures seems to have great potential in achieving both local optimal performance and global tracking robustness. APPENDIX A. Mathematical Model for the Traditional FRM For region For
in Fig. 5, the derivation is shown as follows. , one has for for
and
(a-2) Let as
. Then, the inference out of region
is given
(a-3) For regions , using the same derivation for (a-1), one can easily obtain the results shown in (a-4). Then, following the same way as in (a-2) and (a-3), one has all the results of (6)
(a-4)
B. Mathematical Model for the Optimal FRM In this part, the IC can be studied under the following eight conditions: and and and and and and and and
(b-1)
For condition and in (b-1), the criteria shown at the bottom of the page can be obtained from (8).
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Optimization of : The results of under different conditions are as shown in the first equation at the bottom of the page. To minimize , one has the second equation at the bottom of the page. Further development results in the third equation at the bottom of the page. Optimization of and : Similar to the optimization of , : one can obtain the following result by optimizing
can easily derive the results (b-2) and thus the results in (10)
(b-2)
Similar to the optimization of , one can obtain the results shown in the fourth equation at the bottom of the page by optimizing . Reasoning Development: The results of optimization of – are summarized as follows. See the fifth equation shown at the bottom of the page. Substituting those results into (9) gives the first equation at the top of the next page. Following similar derivation in (a-2) and (a-3), one obtains
with the second equation shown at the top of the next page. For the remaining seven conditions in (b-1), in the same way, one
C. Probability for Robustness Because the probability when is the same as that when , one only needs to discuss the prob. First, fix , and then compare ability when of different methods for changing from 0 to 1. In the end, same results are combined and summarized in length means a better robustness. of . A smaller Comparison of Simplified and Traditional FRMs: Define the probability for robustness of the traditional (T) FRM being
and and
and and
and and and
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and and and
and and and
better than, equal to, and worse than that of the simplified (S) FRM, as and . From (12) and (13), one can easily compare of the traditional FRM and the simplified FRM. After summarizing the result, one arrives at the conclusion that the traditional one is smaller than, equal of to, or bigger than that of the simplified one in length of , and , is fixed in ; and in respectively, when , and , lengths of of is fixed in . These respectively, when results will be integrated over all the IC area, so to obtain the last equation shown at the top of the page. Comparison of Other Methods: Similarly, one can compare the simplified and the optimal (O) FRMs, and the traditional and the optimal FRMs. Following the same way as above, one obtains , and , respectively. ACKNOWLEDGMENT The authors greatly appreciate the useful comments provided by the anonymous reviewers. REFERENCES [1] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst., Man, Cybern., vol. SMC-3, no. 1, pp. 28–44, Jan. 1973. [2] E. H. Mamdani, “Applications of fuzzy algorithms for simple dynamic plant,” Proc. Inst. Elect. Eng., vol. 121, no. 12, pp. 1585–1588, Dec. 1974.
[3] L. A. Zadeh, “The concept of a linguistic variable and its applications to approximate reasoning, I, II, III,” Inform. Sci., vol. 8, no. 8, pp. 199–249, 1974, 1975. 8(8), 301–357; 9(9), 43–93. [4] J. W. Guan and D. A. Bell, “Approximate reasoning and evidence theory,” Inform. Sci., vol. 96, no. 96, pp. 207–235, 1997. [5] I. B. Turksen and Z. Zhong, “An approximate analogical reasoning approach based on similarity measures,” IEEE Trans. Syst., Man Cybern., vol. 18, no. 6, pp. 1049–1056, Dec. 1988. [6] G. J. Wang, “On the logic foundation of fuzzy reasoning,” Inform. Sci., vol. 117, no. 117, pp. 47–88, 1999. [7] K. Y. Cai, “Robustness of fuzzy reasoning and -equalities of fuzzy sets,” IEEE Trans. Fuzzy Syst., vol. 9, no. 5, pp. 738–750, Oct. 2001. [8] , “ -equalities of fuzzy sets,” Fuzzy Sets Syst., vol. 76, no. 76, pp. 97–112, 1995. [9] L. Zhang and K. Y. Cai, “A new fuzzy reasoning method,” Fuzzy Syst. Math., vol. 16, no. 3, pp. 1–5, 2002. , “Optimal fuzzy reasoning and its robustness analysis,” Int. J. In[10] tell. Syst., vol. 19, no. 11, pp. 1033–1049, 2004. [11] P. P. Wang et al., “Fuzzy dynamic systems and fuzzy linguistic controller classification,” Automatica, vol. 30, no. 11, pp. 1769–1774, 1994. [12] W. Z. Qiao and M. Mizumoto, “PID type fuzzy controller and parameters adaptive method,” Fuzzy Sets Syst., vol. 78, pp. 23–35, 1996. [13] H. X. Li and H. B. Gatland, “Conventional fuzzy control and its enhancement,” IEEE Trans. Syst., Man, Cybern. B, vol. 26, no. 5, pp. 791–797, Oct. 1996. [14] H. X. Li and G. Chen, “Feature-based integrated design of fuzzy control systems,” in Intelligent Systems: Techniques and Applications, C. Leondes VI, Ed. Boca Raton, FL: CRC, 2002, pp. VI253–VI283. [15] J. Carvajal, G. Chen, and H. Ogmen, “Fuzzy PID controller: Design, performance evaluation, and stability analysis,” Inf. Sci., vol. 123, pp. 249–270, 2000. [16] G. K. I. Mann, B. G. Hu, and R. G. Gosine, “Analysis of direct action fuzzy PID controllers structures,” IEEE Trans. Syst., Man, Cybern. B, vol. 29, no. 3, pp. 371–388, Jun. 1999. [17] B. G. Hu, G.K.I. Mann, and R. G. Gosine, “A systematic study of fuzzy PID controllers-function-based evaluation approach,” IEEE Trans. Fuzzy Syst., vol. 9, no. 5, pp. 699–712, Oct. 2001. [18] P. Sooraksa, T. Pattaradej, and G. Chen, “Design and implementation of fuzzy P2ID controller for handlebar of a bicycle robot,” Integr. Comput.Aided Eng., vol. 9, no. 4, pp. 319–331, 2002.
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[19] M. Mizumoto, “Realization of PID controls by fuzzy control methods,” in Proc. IEEE Conf. Fuzzy Systems, 1992, pp. 709–715. [20] H. A. Malki, H. Li, and G. Chen, “New design and stability analysis of fuzzy proportional-derivative control systems,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 345–354, Apr. 1994. [21] B. M. Mohan and A. V. Patel, “Analytical structures and analysis of the simplest fuzzy PD controllers,” IEEE Trans. Syst., Man, Cybern. B, vol. 32, no. 2, pp. 239–248, Apr. 2002. [22] H. Ying, “Practical design of nonlinear fuzzy controllers with stability analysis for regulating processes with unknown mathematical models,” Automatica, vol. 30, no. 7, pp. 1185–1195, 1994. [23] K. E. Arzen, M. Johansson, and R. Babuska, “Fuzzy control versus conventional control,” in Fuzzy Algorithm for Control, H. B. Verbruggen, H.-J. Zimmermann, and R. Babuska, Eds., 1999, pp. 59–82. [24] L. Foulloy and S. Galichet, Fuzzy and Linear Controllers, Fuzzy Systems, H. T. Nguyen and M. Sugeno, Eds. Norwell, MA: Kluwer, 1998, pp. 197–225. [25] S. Galichet and L. Foulloy, “Fuzzy controllers: Synthesis and equivalencies,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 140–148, Jun. 1995. [26] F. Matia, A. Jimenez, and R. Galan, “Fuzzy controllers: Lifting the linear/nonlinear frontiers,” Fuzzy Sets Syst., vol. 52, no. 2, pp. 113–128, 1992. [27] R. Palm, “Sliding mode fuzzy control,” in Proc. 1st IEEE Int. Conf. Fuzzy Systems, 1992, pp. 519–526. [28] B. S. Moon, “Equivalence between fuzzy logic controllers and pi controllers for single input systems,” Fuzzy Sets and Syst., vol. 69, no. 2, pp. 105–113, 1995. [29] R. Ketata, D. De Geest, and A. Titli, “Fuzzy controllers: Design, evaluation, parallel, and hierarchial combination with a PID controllers,” Fuzzy Sets Syst., vol. 71, no. 1, pp. 113–129, 1995. [30] W. Pedrycz, “Fuzzy control engineering: Reality and challenges,” in Proc. 4th Int. Conf. Fuzzy Systems, 1995, pp. 437–446. [31] H. Ying et al., “Fuzzy control theory: A nonlinear case,” Automatica, vol. 26, pp. 513–520, 1990. [32] H. X. Li, H. B. Gatland, and A. W. Green, “Fuzzy variable structure control,” IEEE Trans. Syst., Man, Cybern. B, vol. 27, no. 2, pp. 306–312, Apr. 1997. [33] Y. F. Li and C. C. Lau, “Development of fuzzy algorithms for servo systems,” IEEE Control Syst. Mag., vol. 9, no. 4, pp. 65–71, Apr.. 1989. [34] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [35] K. J. Astrom and T. Hagglund, Automatic Tuning of PID Controllers: Instrum. Soc. America, 1988.
Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree from the National University of Defence Technology, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1997. Currently, he is an Associate Professor in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. He is holder of the “Lotus Scholar” at Central South University—an honorary professor endowed by the Ministry of Hunan Province, China. Over the last 20 years, he has had opportunities to work in different fields, including, military service, industry, and academia. His research interests include fuzzy and intelligent control, industrial process control with special interest to electronic packaging. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B. He was awarded the (overseas) Distinguished Young Scholar award in 2004 by the China National Science Foundation.
Lei Zhang was born in Tianjin, China, in 1977. He received the B.S. degree in automatic control from NanKai University, Tianjin, in 1999. He is currently pursuing the Ph.D degree from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. His research interests include fuzzy systems and fuzzy control.
Kai-Yuan Cai was born in April 1965. He received the B.S. degree in 1984, the M.S. degree in 1987, and the Ph.D. degree in 1991, all from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. He is is a Cheung Kong Scholar (Chair Professor), jointly appointed by the Ministry of Education of China and the Li Ka Shing Foundation of Hong Kong in 1999. He has been a Full Professor at Beihang University since 1995. He was a Research Fellow at the Centre for Software Reliability, City University, London, U.K., and a Visiting Scholar at City University of Hong Kong, Swinburge University of Technology, Australia, University of Technology, Sydney, Australia, and Purdue University, West Lafayette, IN. He has published over 70 research papers and is the author of three books: Software Defect and Operational Profile Modeling (Norwell, MA: Kluwer, 1998); Introduction to Fuzzy Reliability (Kluwer, 1996); Elements of Software Reliability Engineering (Beijing, China: Tsinghua University Press, 1995, in Chinese). He serves on the editorial board of the international journal Fuzzy Sets and Systems and is the editor of the Kluwer International Series on Asian Studies in Computer and Information Science (http://www.wkap.nl/prod/s/ASIS). His main research interests include software reliability and testing, intelligent systems and control, and software cybernetics. Dr. Cai served as program committee co-chair for the Fifth International Conference on Quality Software (Melbourne, Australia, September 2005), the First International Workshop on Software Cybernetics (Hong Kong, September 2004), and the Second International Workshop on Software Cybernetics (Edinburgh, U.K., July 2005). He also served (will serve) as guest editor for Fuzzy Sets and Systems (1996), the International Journal of Software Engineering and Knowledge Engineering (2006), and the Journal of Systems and Software (2006).
Guanrong (Ron) Chen (M’89–SM’92–F’97) received the M.Sc. degree in computer science from Zhongshan University, Guangzhou, China, in 1981 and the Ph.D. degree in applied mathematics from Texas A&M University, College Station, in 1987. He was previously with Rice University, Houston, TX, and the University of Houston for 15 years. Currently, he is a Chair Professor in the Department of Electronic Engineering, City University of Hong Kong. He has (co)authored 16 research monographs and advanced textbooks, more than 300 journal papers, and about 200 conference papers, published since 1981 in the field of nonlinear systems dynamics and controls, with applications to Internet technology as well as intelligent control systems. Dr. Chen is Chief, Advisory, Feature, and Associate Editors for eight IEEE TRANSACTIONS and International Journals. He is also an Honorary Professor of the Central Queensland University of Australia. He received the Harden-Simons Outstanding Prize for the Best Journal Paper Award from the American Society of Engineering Education in 1998, the M. Barry Carlton Best Transactions Paper Award from the IEEE Aerospace and Electronic Systems Society in 2001, the Best Journal Paper Award from the Czech Academy of Sciences in 2002, and the Guillemin-Cauer Best Transactions Paper Award from the IEEE Circuits and Systems Society in 2005.
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An Improved Robust Fuzzy-PID Controller With Optimal Fuzzy Reasoning Han-Xiong Li, Senior Member, IEEE, Lei Zhang, Kai-Yuan Cai, and Guanrong Chen, Fellow, IEEE
Abstract—Many fuzzy control schemes used in industrial practice today are based on some simplified fuzzy reasoning methods, which are simple but at the expense of losing robustness, missing fuzzy characteristics, and having inconsistent inference. The concept of optimal fuzzy reasoning is introduced in this paper to overcome these shortcomings. The main advantage is that an integration of the optimal fuzzy reasoning with a PID control structure will generate a new type of fuzzy-PID control schemes with inherent optimal-tuning features for both local optimal performance and global tracking robustness. This new fuzzy-PID controller is then analyzed quantitatively and compared with other existing fuzzy-PID control methods. Both analytical and numerical studies clearly show the improved robustness of the new fuzzy-PID controller. Index Terms—Fuzzy control, fuzzy reasoning, optimal fuzzy reasoning, fuzzy-PID controller, robustness.
I. INTRODUCTION
T
HE MAIN idea of fuzzy logic control (FLC) was introduced by Zadeh [1], and first applied by Mamdani [2] in an attempt of controlling structurally ill-modeled systems. FLC has two main parts that need to be designed: one is the control structure composing of rules and gains and the other is a fuzzy reasoning method. Various inference for the fuzzy reasoning method have been proposed, including the compositional rule of inference (CRI) [3], evidence reasoning [4], approximate analogical reasoning approach based on similarity measures [5], triple implication method [6], and so on. The most popular inference method is perhaps the CRI method. However, [7] discusses the robustness problem of various inference methods in terms of -equalities of fuzzy sets [8], i.e., how errors in premises affect consequences in fuzzy reasoning, and arrives at the conclusion that the robustness of the CRI methods is not very satisfactory. It is well known that this traditional fuzzy reasoning has inconsistent inference outcomes. In contrast to most existing fuzzy reasoning methods of being processes of logical inference, the optimal fuzzy reasoning method, introduced in our previous work [9], [10], treats (fuzzy) reasoning as a process of optimization instead of (an
Manuscript received October 18, 2004. This work was supported in part by a grant from RGC of Hong Kong (CityU 1207/04E) and by a Grant from the National Science Foundation of China for a distinguished young scholar (50 429 501). This paper was recommended by Associate Editor T. H. Lee. H.-X. Li is with the Department of MEEM, City University of Hong Kong, Hong Kong, China (e-mail: [email protected]). L. Zhang and K.-Y. Cai are with the Department of Automation, Beihang University, Beijing, China. G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China. Digital Object Identifier 10.1109/TSMCB.2005.851538
Fig. 1.
Two-dimensional configuration for PID type of FLC.
extension of) the classical logical inference process. From the control perspective, the processes of the other methods are of open-loop but the optimal fuzzy reasoning method is of closed-loop with some objectives of performance optimization. Owing to the closed-loop characteristic, the optimal fuzzy reasoning method, which introduces the feedback optimization into traditional fuzzy reasoning, can significantly improve robustness and consistency. Although systematic analysis and design for FLC are still considered premature in general, significant progress has been gained recently in the pursuit of this technology. FLC structure can be classified into different types [11], and the most popular one for the low-level control purpose is PID type of FLC [12], [13]. Significant research efforts have been spent on controllers design of the fuzzy-PID type [12], [14]–[19], including its simplified versionsoffuzzy-PD[20],[21]andfuzzy-PI[22]controlfordifferent applications. Comparisons between fuzzy control and its conventional control have also been reported from different viewpoints [23]–[29]. As the rule base conveys a general control policy, it should be sustained throughout the control process, leaving most design and tuning work to the scaling gains of the controller [30]. However, many of these analyzes and comparisons are qualitative or descriptive due to the lack of a precise mathematical model of the inference logic. A quantitative model for the max-min inference logic was first introduced by Ying [31] for analysis of the Mamdani-type of FLC, which was lately modified by Li [32] for the case of a linear rule base. On the other hand, most FLC is based on the simplified fuzzy reasoning [33], which loses much of the original fuzzy characteristics and therefore usually affect the robustness. A proper integration of the fuzzy reasoning method and its outer control structure is obviously crucial for achieving optimal control performance. In this paper, an improved robust fuzzy-PID control scheme is proposed by incorporating the optimal fuzzy reasoning into the well-developed PID type of control framework [13]. The model of the optimal fuzzy-PID control is developed here by using the quantitative method [31], [32], which will then be compared with the existing fuzzy-PID control schemes based on other reasoning methods. Finally, simulation study is reported, which demonstrates the robustness of the proposed reasoning
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Fig. 2.
Traditional fuzzy reasoning model for control.
Fig. 3.
Optimal fuzzy reasoning model for control.
method and the effectiveness of its integration with the PID type of control structures. II. FUNDAMENTALS OF FLC A. Fuzzy-PID Control Structure There are different types of fuzzy-PID controllers [16]. One of the earliest and very effective structures use a two-dimensional linear rule base with standard triangular membership functions (MFs) [13], which actually combines fuzzy-PI and fuzzy-PD controllers, as shown in Fig. 1. And it is a very simple controller. There are four input/output scaling gains, , for control action adjustment, where the input gain ratio is defined as . B. Three Inference Methods In this paper, three kind of fuzzy reasoning methods (FRMs) will be studied. These are simplified, traditional and optimal FRM, respectively. Simplified FRM treats crisp input value as a singleton, which is a nonfuzzy variable. The traditional FRM treats both premises and consequences as “real” fuzzy variables and forms an inference space via the implication operation. This inference space is discretely represented by a fuzzy relation matrix , which can be obtained off-line from the rule base. The simplified FRM is usually used by the FLC due to its relatively small computational burden. The optimal FRM introduces the idea of optimization and feedback into the reasoning process of
the traditional FRM. Their key difference is clearly explained by comparing Fig. 2 with Fig. 3. The basic idea of the optimal FRM is explained as follows. •
Suppose that rule base which can be expressed as a fuzzy relation matrix is gained from the experience of an expert, i.e., heuristic rules or data. Then, one should trust and treat it as a reference for reasoning evaluation. • While in the process of inference, if a fuzzy premise is given, a corresponding consequence will be generated (through ) via the inference method. Then, this pair of and form a new relation matrix , which unfortunately is different from the original for most FRMs. • In the proposed optimal reasoning method, the feedback introduced in the reasoning will search for the best from the consequence domain, so as to minimize the “reasoning . For the existing error” defined by . FRMs, In the optimal reasoning process, may be analytically gained in some cases, but in most cases it must be gained from some existing optimization methods. The basic idea is to according to the change of continuously adjust and stop computing when satisfies a given criterion, because analytically evaluating is often complicated and impractical. It can be easily seen that the optimal FRM treats the fuzzy reasoning as a process of optimization instead of a process of
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TABLE I A TWO-DIMENSIONAL LINEAR RULE BASE WITH n = 3
Fig. 4. Decomposition of the rule base into inference cells.
logical inference. Since the inner loop is in the feedback loop, the effect of the disturbance in the fuzzy reasoning process, such as the perturbation of reasoning premises due to the noise of inputs, can be greatly reduced [10]. Since the inner loop will help the local performance more, and the outer loop will help the global performance more, the combination of these two loops will clearly yield a better performance. III. MODELS OF DIFFERENT FRMS A. FRM With Linear Rule Base and Standard Triangular MFs The fuzzy-PID shown in Fig. 1 has two inputs and one output. All fuzzy variables are normalized to the same domain, , number of triangular MFs ( is odd each of them has number) with equal spread of 2W as shown in Fig. 4. The grades . A linear rule base is of MFs are represented by used, for which an example with = 3 is shown in Table I. The rule base plane can be decomposed into many inference , and ) on its four cells (ICs) with output rules ( corners, as shown in Fig. 4 [32]. The inference can be operated on these ICs. Four different types of ICs are generated in terms , and , as follows: of different
The total number of ICs in a linear rule base is with from type 1, from type 2, from type 3, and from type 4. The percentages of these four types of ICs in a linear , and , rerule base are spectively. When is large enough, most ICs in the rule base
Fig. 5.
Functional composition of the inference cell IC(i; j ).
are type 1, particularly around the equilibrium point or far away from the boarder. Therefore, only type 1 IC is considered in this study. Analysis of the rule base can be simplified to the analysis of . As shown in Fig. 5, two MFs from type 1 inference cell (ith and th) and (jth and th) divide IC into four with inference outputs generated by sub-regions four rules as follows: and and and and where is the th fuzzy variable of is the th fuzzy variable of , and is the th fuzzy variable of , with their , and , respectively. MF grades denoted by The grades of fuzzification can be defined as
As seen from Figs. 1 and 4, they can be further derived as and , or (1) with
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B. Model of the Simplified FRM
1) Fuzzy relation matrices for each rule are, respectively
Most FLC schemes use a simplified FRM with different operators, such as the max-min, sum-product, etc. The sum-product method is chosen for the simplified FRM in this paper with the inference output expressed as and where . Using (1), the same inference output will be obtained in four regions in each IC, as shown in (2), at the bottom of the page. With this simplified FRM, the fuzzy-PID scheme shown in Fig. 1 can be easily derived, as
2) Fuzzy relation matrix of IC(i, j) is
(4)
Finally, the fuzzy inference is defined as
(5-1) with inference output (3)
(5-2)
This actually is a conventional PID controller with proportional , integral time constant and derivative time constant gain , where
From Appendix A, the model of the traditional FRM is obtained as
(6) (3-2) with
.
Then, the model of fuzzy-PID in Fig. 1 with traditional FRM becomes
C. Traditional FRM The max-min method is chosen for the traditional FRM in this paper. With the linear rule base and standard triangular MFs, it gives the same inference output as the simplified FRM with the max-min method [32]. The fuzzification result (reasoning premise) is first given as follows. 1) for for . for 2) and . Then, the fuzzy relation matrix is formed as follows.
(7) with the nonlinear parameter
defined in (6).
(2)
LI et al.: AN IMPROVED ROBUST FUZZY-PID CONTROLLER WITH OPTIMAL FUZZY REASONING
This actually is a nonlinear PID controller plus a relay, , with proportional gain , integral time constant and derivative time constant , where
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From Appendix B, the reasoning output is found to be
with
(10)
D. Optimal FRM For the optimal FRM, the reasoning premise and the fuzzy relation matrix can be formed in the same way as the traditional FRM. The objective function of optimal FRM is chosen as
Similar to (7), the model of the fuzzy-PID control in Fig. 1 with the optimal FRM becomes
with as (4) and the first equation shown at the is the reasoning bottom of the page, where consequence. can be The optimization of the complex function further decomposed into three simple optimization functions, , in
and in (8), as shown at the bottom of the page. The optimization scheme divides IC into nine sub, where different outputs are genregions, that enclose erated. The lines can be figured out from Appendix B and they are expressed as
, and , respectively. The output of the fuzzy reasoning is expressed as
and
are chosen to minimize
defined same as in (10). with This is a nonlinear PID controller plus a relay, , with proportional gain , integral time constant and derivative time constant , defined as
When the control process is on the equilibrium state, only depends on . IV. THEORETICAL ANALYSIS Fundamental features of the three methods are analyzed and discussed, including their stability and robustness in process control.
(9) where in (8).
(11)
and
A. Mathematical Explanation From (2)–(11) and Figs. 5 and 6, one can arrive at the following conclusions.
(8)
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and is a local switching function. 3) Optimal FRM . Define a switching function Then
where
Fig. 6.
Partition in inference cell IC(i; j ).
1) There is an adjustable parameter for the traditional FRM for and a group of adjustable parameters the optimal FRM; however, there is no adjustable parameter for the simplified FRM. is on the boundary of , 2) When where the inference results of both traditional and optimal FRM are the same as that of the simplified FRM. are in in Fig. 6, both the tra3) When ditional and the optimal FRMs have the same inference output, which however is different from that of the simplified FRM. 4) The simplified FRM is actually a conventional PID controller; the traditional FRM plays the role as a nonlinear PID controller with a varying parameter plus a relay C; the optimal FRM behaves as a variable structure controller with a group of varying parameters , which make it become: ; • a PID plus a relay C in regions of ; • a PD plus a relay C in regions of ; • a PI plus a relay C in regions of • a relay control C in region . B. Feature of Sliding Mode Control The model of the three FRMs, (3), (7), and (11), can be expressed in terms of the sliding mode structure, as follows: 1) Simplified FRM: . Define a switching function . Then, 2) Traditional FRM . Define a switching function, Then
where
can be treated as an equivalent control, is a switching control term
is an equivalent control term and is a switching control term
In view of sliding-mode control [34], the equivalent control term is to push the system trajectory toward the sliding surface (it handles the major nonlinearity) and the switching control term is to maintain the trajectory on the sliding surface (it deals with the unmodeled dynamics). A fuzzy-PID controller on either the traditional or the optimal FRM usually displays the feature of sliding mode control. However, the one in the simplified FRM has very little feature of sliding mode because of the missing equivalent control term. Since both equivalent and switching control terms in the optimal FRM are more flexible than the one in the traditional FRM, the control in the optimal FRM can handle more complex processes than the one in the traditional FRM, and thus should have a better performance in more realistic situations. C. Robustness Analysis The issue of robustness in control systems is for the stability and performance that the control system can maintain under parameter perturbations and external disturbances. Previous work [9], [10] has shown that the optimal FRM can improve the reasoning robustness under perturbations of rule base and fuzzy premises. Here, the robustness of the optimal FRM will be further discussed. 1) Structure Robustness of the System: When the control law or the plant parameter varies, it can be considered as a perturbation of the rule base, from the correct one, to the current one, , as shown in Fig. 7, where and the line is a . perpendicular bisector of the line For the same rule base, different FRMs still get different inference results. So, the new fuzzy relation matrix , which is produced under the given reasoning premise, has a nonzero diswith tance to . Assume that the optimal FRM produces distance to and the other FRM produces with distance to . In Fig. 7, is the origin and the correct rule base in . the control, and is the nominal rule base, with . From the naDefinition of the optimal FRM gives ture of the perpendicular bisector, should be on the left side of appears inside the circle centered line . Since has a large probability to at with the radius . Therefore, . In other words, the opbe on the left side of line , i.e., timal FRM has better robustness to the perturbation of the rule base.
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•
Robustness of the optimal method is better than, equal to, or worse than the traditional one in probability of 0.161, 0.75, or 0.089, respectively. Thus, the optimal method has the best robustness in view of probability and the simplified one is the worst in this comparison. V. SIMULATION A fuzzy-PID controller shown in Fig. 1, with the linear rule base given in Table I and some standard triangular MFs, is chosen to control the plant
Fig. 7. Robustness analysis for perturbation of the rule base.
2) Statistics Based Robustness to External Disturbance: When the reference signal varies or noise appears in the control loop, they can be considered as disturbances to the fuzzy reasoning premises. Previous work [10] has shown better robustness of the optimal FRM against errors in the premises than the CRI method in Monte Carlo simulations. Since the results are measured in the statistical sense, the robustness should be measured in terms of probability . First, compare with the three methods when is fixed is changing from 0 to 1. A smaller means a and better robustness here. Then, the probability , that one method is better than, equal to and worse than another method on robustness, are obtained via integration over the whole IC area. The following results (12)–(14) are derived from (2) and Appendices A and B. 1) Simplified FRM
(15) The design of the scaling gains follows the well-known Ziegler-Nichols method [35]. Using the frequency method, the and are obtained ultimate gain and frequency for gain calculation. Since the plant is relatively simple, the derivative action is not necessary and can be chosen as zero for simplicity. Thus, the classical PID control parameters will be chosen as (16) Since the fuzzy-PID controller in the simplified FRM is actually a conventional PID controller shown in (3), its PID gains in (3-2) can be chosen to be the same as that of the well-tuned classical PID controller in (16). Using (3-2) and (16), one can easily derive and select the following gains:
(12) 2) Traditional FRM
(13)
where regions 3) Optimal FRM
are classified in Fig. 5.
From (3-2) we know that the solutions of is not unique, but the performance with different solutions on the steady state are same, which is displayed in simulations. And different solutions are equivalent to the same PID parameters . For the plant (15) without perturbation and noise, the performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 8, which clearly demonstrates that three fuzzy-PID methods all have good response. In the following simulations, reference signal is a unit step, simulation time is 150 s and sampling step is 0.01 s. A. Robustness Against System Perturbation
(14)
are classified in Fig. 6. where regions The following conclusions are drawn from Appendix C. • Robustness of the traditional method is better or worse than the simplified one in probability of 0.688 or 0.312, respectively. • Robustness of the optimal method is better or worse than the simplified one in probability of 0.698 or 0.302, respectively.
The simulation steps are summarized as follows. 1) Generate disturbance by changing either the plant parameter or one of the control parameters. 2) Compare the responses under disturbance. Case 1: Under Plant Perturbation: When the parameter “15” in the plant (15) adds, the responses of all the three methods become worse. On condition that control system has an acceptable performance, that is, the relative error is lower than 2%, the larger the range of the parameter is, the better the robustness of control method is. Assume that the plant (15) is perturbed to (17)
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Fig. 10. Performance comparison w.r.t. controller perturbation. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
Fig. 8. Performance comparison. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
Fig. 11. Performance comparison w.r.t. noise. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
B. Robustness Against Noise
Fig. 9. Performance comparison w.r.t. plant perturbation. (a) Simplified FRM. (b) Traditional FRM. (c) Optimal FRM.
The performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 9, which clearly demonstrates better robustness of the optimal FRM against system perturbation in the plant. Case 2: Under Controller Perturbation: When the parameter in the controllers greatly adds, the responses of the three methods all become worse. Similar to the above, the is, the better the robustness of control larger the range of is perturbed from method is. The controller input gain 1 to 8. The performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 10, which clearly demonstrates better robustness of the optimal FRM against system perturbation in the controller.
When noises are added to control system, the responses of of the three methods all become worse. Inputs the fuzzy-PID controller in Fig. 1 are added with uniformly , to become distributed noises . The performance criteria are defined as and Since the influence of the noise is more important while in the steady state, the integral action will start from the mid-time of the simulation, , to the end time, . For one simulation, the performances of the fuzzy-PID controller with three different FRMs are compared in Fig. 11. And a large amount of simulations have been carried out with the mean performance shown in Table II. The optimal FRM clearly demonstrates better robustness than the other methods with respect to noise during the fuzzy-PID
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From (5-1), one can derive and then from (5-2) one has
TABLE II STATISTICAL PERFORMANCE TO NOISE
(a-1) control processes, although the outer control loop can filter out some influence of the noise.
Further derivation will result in
VI. CONCLUSION Both the concept and the technique of optimal fuzzy reasoning have been proposed and discussed in this paper, to integrate with fuzzy PID-type of control structures for better robust control. A mathematical model of the optimal fuzzy reasoning has been derived and then compared with other reasoning methods. Based on this quantitative model, both theoretical analysis and numerical simulations have been carried out to study the fuzzy-PID control with different reasoning methods. It is found that a fuzzy-PID control scheme with different reasoning methods becomes different types of nonlinear PID controllers plus a relay. The optimal fuzzy reasoning method brings in more flexibility in tuning their nonlinear gains, which work better for more complex problems. Both analytical and simulation studies have clearly demonstrated that the optimal fuzzy reasoning method can enhance the control system robustness thanks to its optimality and inherent feedback tuning. It therefore works better under complex and realistic environments. And by simulations, we can arrive at the conclusions: total operation time of optimal FRM is about six times that of simplified FRM and about four times that of traditional FRM, which display the computational cost of optimal FRM is not large; the output of FLC with optimal FRM is stable when the control process is stable. The integration of the proposed optimal fuzzy reasoning method and some good control structures seems to have great potential in achieving both local optimal performance and global tracking robustness. APPENDIX A. Mathematical Model for the Traditional FRM For region For
in Fig. 5, the derivation is shown as follows. , one has for for
and
(a-2) Let as
. Then, the inference out of region
is given
(a-3) For regions , using the same derivation for (a-1), one can easily obtain the results shown in (a-4). Then, following the same way as in (a-2) and (a-3), one has all the results of (6)
(a-4)
B. Mathematical Model for the Optimal FRM In this part, the IC can be studied under the following eight conditions: and and and and and and and and
(b-1)
For condition and in (b-1), the criteria shown at the bottom of the page can be obtained from (8).
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Optimization of : The results of under different conditions are as shown in the first equation at the bottom of the page. To minimize , one has the second equation at the bottom of the page. Further development results in the third equation at the bottom of the page. Optimization of and : Similar to the optimization of , : one can obtain the following result by optimizing
can easily derive the results (b-2) and thus the results in (10)
(b-2)
Similar to the optimization of , one can obtain the results shown in the fourth equation at the bottom of the page by optimizing . Reasoning Development: The results of optimization of – are summarized as follows. See the fifth equation shown at the bottom of the page. Substituting those results into (9) gives the first equation at the top of the next page. Following similar derivation in (a-2) and (a-3), one obtains
with the second equation shown at the top of the next page. For the remaining seven conditions in (b-1), in the same way, one
C. Probability for Robustness Because the probability when is the same as that when , one only needs to discuss the prob. First, fix , and then compare ability when of different methods for changing from 0 to 1. In the end, same results are combined and summarized in length means a better robustness. of . A smaller Comparison of Simplified and Traditional FRMs: Define the probability for robustness of the traditional (T) FRM being
and and
and and
and and and
LI et al.: AN IMPROVED ROBUST FUZZY-PID CONTROLLER WITH OPTIMAL FUZZY REASONING
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and and and
and and and
better than, equal to, and worse than that of the simplified (S) FRM, as and . From (12) and (13), one can easily compare of the traditional FRM and the simplified FRM. After summarizing the result, one arrives at the conclusion that the traditional one is smaller than, equal of to, or bigger than that of the simplified one in length of , and , is fixed in ; and in respectively, when , and , lengths of of is fixed in . These respectively, when results will be integrated over all the IC area, so to obtain the last equation shown at the top of the page. Comparison of Other Methods: Similarly, one can compare the simplified and the optimal (O) FRMs, and the traditional and the optimal FRMs. Following the same way as above, one obtains , and , respectively. ACKNOWLEDGMENT The authors greatly appreciate the useful comments provided by the anonymous reviewers. REFERENCES [1] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst., Man, Cybern., vol. SMC-3, no. 1, pp. 28–44, Jan. 1973. [2] E. H. Mamdani, “Applications of fuzzy algorithms for simple dynamic plant,” Proc. Inst. Elect. Eng., vol. 121, no. 12, pp. 1585–1588, Dec. 1974.
[3] L. A. Zadeh, “The concept of a linguistic variable and its applications to approximate reasoning, I, II, III,” Inform. Sci., vol. 8, no. 8, pp. 199–249, 1974, 1975. 8(8), 301–357; 9(9), 43–93. [4] J. W. Guan and D. A. Bell, “Approximate reasoning and evidence theory,” Inform. Sci., vol. 96, no. 96, pp. 207–235, 1997. [5] I. B. Turksen and Z. Zhong, “An approximate analogical reasoning approach based on similarity measures,” IEEE Trans. Syst., Man Cybern., vol. 18, no. 6, pp. 1049–1056, Dec. 1988. [6] G. J. Wang, “On the logic foundation of fuzzy reasoning,” Inform. Sci., vol. 117, no. 117, pp. 47–88, 1999. [7] K. Y. Cai, “Robustness of fuzzy reasoning and -equalities of fuzzy sets,” IEEE Trans. Fuzzy Syst., vol. 9, no. 5, pp. 738–750, Oct. 2001. [8] , “ -equalities of fuzzy sets,” Fuzzy Sets Syst., vol. 76, no. 76, pp. 97–112, 1995. [9] L. Zhang and K. Y. Cai, “A new fuzzy reasoning method,” Fuzzy Syst. Math., vol. 16, no. 3, pp. 1–5, 2002. , “Optimal fuzzy reasoning and its robustness analysis,” Int. J. In[10] tell. Syst., vol. 19, no. 11, pp. 1033–1049, 2004. [11] P. P. Wang et al., “Fuzzy dynamic systems and fuzzy linguistic controller classification,” Automatica, vol. 30, no. 11, pp. 1769–1774, 1994. [12] W. Z. Qiao and M. Mizumoto, “PID type fuzzy controller and parameters adaptive method,” Fuzzy Sets Syst., vol. 78, pp. 23–35, 1996. [13] H. X. Li and H. B. Gatland, “Conventional fuzzy control and its enhancement,” IEEE Trans. Syst., Man, Cybern. B, vol. 26, no. 5, pp. 791–797, Oct. 1996. [14] H. X. Li and G. Chen, “Feature-based integrated design of fuzzy control systems,” in Intelligent Systems: Techniques and Applications, C. Leondes VI, Ed. Boca Raton, FL: CRC, 2002, pp. VI253–VI283. [15] J. Carvajal, G. Chen, and H. Ogmen, “Fuzzy PID controller: Design, performance evaluation, and stability analysis,” Inf. Sci., vol. 123, pp. 249–270, 2000. [16] G. K. I. Mann, B. G. Hu, and R. G. Gosine, “Analysis of direct action fuzzy PID controllers structures,” IEEE Trans. Syst., Man, Cybern. B, vol. 29, no. 3, pp. 371–388, Jun. 1999. [17] B. G. Hu, G.K.I. Mann, and R. G. Gosine, “A systematic study of fuzzy PID controllers-function-based evaluation approach,” IEEE Trans. Fuzzy Syst., vol. 9, no. 5, pp. 699–712, Oct. 2001. [18] P. Sooraksa, T. Pattaradej, and G. Chen, “Design and implementation of fuzzy P2ID controller for handlebar of a bicycle robot,” Integr. Comput.Aided Eng., vol. 9, no. 4, pp. 319–331, 2002.
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[19] M. Mizumoto, “Realization of PID controls by fuzzy control methods,” in Proc. IEEE Conf. Fuzzy Systems, 1992, pp. 709–715. [20] H. A. Malki, H. Li, and G. Chen, “New design and stability analysis of fuzzy proportional-derivative control systems,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 345–354, Apr. 1994. [21] B. M. Mohan and A. V. Patel, “Analytical structures and analysis of the simplest fuzzy PD controllers,” IEEE Trans. Syst., Man, Cybern. B, vol. 32, no. 2, pp. 239–248, Apr. 2002. [22] H. Ying, “Practical design of nonlinear fuzzy controllers with stability analysis for regulating processes with unknown mathematical models,” Automatica, vol. 30, no. 7, pp. 1185–1195, 1994. [23] K. E. Arzen, M. Johansson, and R. Babuska, “Fuzzy control versus conventional control,” in Fuzzy Algorithm for Control, H. B. Verbruggen, H.-J. Zimmermann, and R. Babuska, Eds., 1999, pp. 59–82. [24] L. Foulloy and S. Galichet, Fuzzy and Linear Controllers, Fuzzy Systems, H. T. Nguyen and M. Sugeno, Eds. Norwell, MA: Kluwer, 1998, pp. 197–225. [25] S. Galichet and L. Foulloy, “Fuzzy controllers: Synthesis and equivalencies,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 140–148, Jun. 1995. [26] F. Matia, A. Jimenez, and R. Galan, “Fuzzy controllers: Lifting the linear/nonlinear frontiers,” Fuzzy Sets Syst., vol. 52, no. 2, pp. 113–128, 1992. [27] R. Palm, “Sliding mode fuzzy control,” in Proc. 1st IEEE Int. Conf. Fuzzy Systems, 1992, pp. 519–526. [28] B. S. Moon, “Equivalence between fuzzy logic controllers and pi controllers for single input systems,” Fuzzy Sets and Syst., vol. 69, no. 2, pp. 105–113, 1995. [29] R. Ketata, D. De Geest, and A. Titli, “Fuzzy controllers: Design, evaluation, parallel, and hierarchial combination with a PID controllers,” Fuzzy Sets Syst., vol. 71, no. 1, pp. 113–129, 1995. [30] W. Pedrycz, “Fuzzy control engineering: Reality and challenges,” in Proc. 4th Int. Conf. Fuzzy Systems, 1995, pp. 437–446. [31] H. Ying et al., “Fuzzy control theory: A nonlinear case,” Automatica, vol. 26, pp. 513–520, 1990. [32] H. X. Li, H. B. Gatland, and A. W. Green, “Fuzzy variable structure control,” IEEE Trans. Syst., Man, Cybern. B, vol. 27, no. 2, pp. 306–312, Apr. 1997. [33] Y. F. Li and C. C. Lau, “Development of fuzzy algorithms for servo systems,” IEEE Control Syst. Mag., vol. 9, no. 4, pp. 65–71, Apr.. 1989. [34] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [35] K. J. Astrom and T. Hagglund, Automatic Tuning of PID Controllers: Instrum. Soc. America, 1988.
Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree from the National University of Defence Technology, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1997. Currently, he is an Associate Professor in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. He is holder of the “Lotus Scholar” at Central South University—an honorary professor endowed by the Ministry of Hunan Province, China. Over the last 20 years, he has had opportunities to work in different fields, including, military service, industry, and academia. His research interests include fuzzy and intelligent control, industrial process control with special interest to electronic packaging. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B. He was awarded the (overseas) Distinguished Young Scholar award in 2004 by the China National Science Foundation.
Lei Zhang was born in Tianjin, China, in 1977. He received the B.S. degree in automatic control from NanKai University, Tianjin, in 1999. He is currently pursuing the Ph.D degree from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. His research interests include fuzzy systems and fuzzy control.
Kai-Yuan Cai was born in April 1965. He received the B.S. degree in 1984, the M.S. degree in 1987, and the Ph.D. degree in 1991, all from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. He is is a Cheung Kong Scholar (Chair Professor), jointly appointed by the Ministry of Education of China and the Li Ka Shing Foundation of Hong Kong in 1999. He has been a Full Professor at Beihang University since 1995. He was a Research Fellow at the Centre for Software Reliability, City University, London, U.K., and a Visiting Scholar at City University of Hong Kong, Swinburge University of Technology, Australia, University of Technology, Sydney, Australia, and Purdue University, West Lafayette, IN. He has published over 70 research papers and is the author of three books: Software Defect and Operational Profile Modeling (Norwell, MA: Kluwer, 1998); Introduction to Fuzzy Reliability (Kluwer, 1996); Elements of Software Reliability Engineering (Beijing, China: Tsinghua University Press, 1995, in Chinese). He serves on the editorial board of the international journal Fuzzy Sets and Systems and is the editor of the Kluwer International Series on Asian Studies in Computer and Information Science (http://www.wkap.nl/prod/s/ASIS). His main research interests include software reliability and testing, intelligent systems and control, and software cybernetics. Dr. Cai served as program committee co-chair for the Fifth International Conference on Quality Software (Melbourne, Australia, September 2005), the First International Workshop on Software Cybernetics (Hong Kong, September 2004), and the Second International Workshop on Software Cybernetics (Edinburgh, U.K., July 2005). He also served (will serve) as guest editor for Fuzzy Sets and Systems (1996), the International Journal of Software Engineering and Knowledge Engineering (2006), and the Journal of Systems and Software (2006).
Guanrong (Ron) Chen (M’89–SM’92–F’97) received the M.Sc. degree in computer science from Zhongshan University, Guangzhou, China, in 1981 and the Ph.D. degree in applied mathematics from Texas A&M University, College Station, in 1987. He was previously with Rice University, Houston, TX, and the University of Houston for 15 years. Currently, he is a Chair Professor in the Department of Electronic Engineering, City University of Hong Kong. He has (co)authored 16 research monographs and advanced textbooks, more than 300 journal papers, and about 200 conference papers, published since 1981 in the field of nonlinear systems dynamics and controls, with applications to Internet technology as well as intelligent control systems. Dr. Chen is Chief, Advisory, Feature, and Associate Editors for eight IEEE TRANSACTIONS and International Journals. He is also an Honorary Professor of the Central Queensland University of Australia. He received the Harden-Simons Outstanding Prize for the Best Journal Paper Award from the American Society of Engineering Education in 1998, the M. Barry Carlton Best Transactions Paper Award from the IEEE Aerospace and Electronic Systems Society in 2001, the Best Journal Paper Award from the Czech Academy of Sciences in 2002, and the Guillemin-Cauer Best Transactions Paper Award from the IEEE Circuits and Systems Society in 2005.
