A self-organizing fuzzy logic controller for the ... - Semantic Scholar
Applied Soft Computing 1 (2002) 271–283
A self-organizing fuzzy logic controller for the active control of flexible structures using piezoelectric actuators Gustavo Luiz C.M. de Abreu∗ , José F. Ribeiro School of Mechanical Engineering, Federal University of Uberlˆandia, Uberlˆandia-MG, Brazil Accepted 8 January 2002
Abstract This paper proposes an on-line self-organizing fuzzy logic controller (FLC) design applied to the control of vibrations in flexible structures containing distributed piezoelectric actuator patches. In this methodology, the fuzzy rules are generated using the history of input/output (I/O) pairs without using any plant model. The generated rules are stored in the fuzzy rule space and updated on-line by a self-organizing procedure. The validity of the proposed fuzzy logic control has been demonstrated experimentally in a steel cantilever test beam and a set of experimental tests are made in the system to verify the efficiency of the on-line self-organizing fuzzy controller. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Self-organizing fuzzy controller; Active vibration control; Piezoelectric actuators; Flexible structures
1. Introduction The great progress experimented by the theory of the fuzzy controllers in the last years has been opening new possibilities of practical application for these controllers. In the last two decades, the subject area of smart/ intelligent materials and structures has experienced tremendous growth in terms of research and development [1]. One reason for this activity is that it may be possible to create certain types of structures and systems capable of adapting to or correcting for changing operating conditions. The advantage of incorporating these special types of materials into the structure is that the sensing and actuating mechanism becomes part of the structure by sensing and actuating strains directly. This is more known like piezoelectric phenomena, i.e. ∗ Corresponding author. E-mail addresses: [email protected] (G.L.C.M. de Abreu), [email protected] (J.F. Ribeiro).
direct and converse piezoelectric effects. When a mechanical force is applied to a piezoelectric material, an electric voltage or change will be generated. On the other hand, when an electric field is applied to the material, a mechanical force will be induced because of the converse piezoelectric effect. With the recent advances in piezoelectric technology, it has been shown that the piezoelectric actuators based on the converse piezoelectric effect can offer excellent potential for active vibration control techniques, especially for vibration suppression or isolation. Recent research has focused on the applications of piezoelectric sensor and actuator in smart structures. Crawley and de Luis [1] were among the first to embed piezoelectric materials in composite laminated beams. Chang-qing et al. [2] and Chien-Chang and Huang-Nan [3] presented a formulation methodology to the control of vibrations in composite structures with bonded piezoelectric sensors and actuators. Vibration suppression of composite smart structures by using piezoelectric sensors and actuators were also
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analyzed numerically by Jyh-Horng et al. [4]. Therefore, it is difficult to implement classical controllers to these systems, especially to a system which is complex and non-linear. Effective applications in vibration control, however, require that the system dynamics can be adequately and/or accurately determined and that the controller design can be easily implemented. Vibration control of smart structures using fuzzy controllers has thus been receiving attention for their ability to deal with non-linearities, uncertainties in terms of vagueness, ignorance, and imprecision, and provide a feasible alternative since they can easily capture qualitative aspects of human knowledge. Fuzzy controllers are most suitable for systems that cannot be precisely described by mathematical formulations. In this case, a control designer captures operators knowledge and converts it into a set of fuzzy control rules. The idea of the fuzzy logic is useful for representing linguistic terms numerically and making reliable decisions with ambiguous and imprecise events or facts. The benefit of the simple design procedure of a fuzzy controller leads to the successful applications of a variety of engineering systems [5]. Zeinoun and Khorrami [6] proposed a fuzzy logic algorithm for vibration suppression of a clamp-free beam with piezoelectric sensor/actuator, and Ofri et al. [7] also used a control strategy based on fuzzy logic theory for vibration damping of a large flexible space structure controlled by bonded piezoceramic actuators. In general, the fuzzy logic controllers (FLCs) use fuzzy inference with rules preconstructed by an expert. Therefore, the most important task is to form the rule base which represents the experience and intuition of human experts. When the rule base of human experts is not available, an efficient control cannot be expected. The self-organizing fuzzy controller is a rule-based type of controller which learns how to control on-line while being applied to a system, and it has been used successfully for a wide variety of processes [8]. This controller combines system identification and control based on experience. Therefore, only a minimal amount of information about the environment needs to be provided. The main purpose of this paper is to minimize the role of human experts in designing a fuzzy logic controller. For this, the on-line self-organizing controller is used which uses the input and output history in its
rules [9]. This controller has no rule at its initial stage, but forms rules by defining membership functions using the plant input/output (I/O) data as singletons and stores them in a rule base. The rule base is updated as experience is accumulated using a self-organizing procedure. A new method for defuzzification is also developed by adding a predictive capability using a prediction model. The self-organizing controller is experimentally verified in a steel cantilever test beam. A set of experimental tests is made in the system to evaluate the performance of the self-organizing fuzzy controller and the conventional fuzzy controller. Experimental results are shown, to support the effectiveness of the active vibration control, and finally, a conclusion of this study is given.
2. Design of the self-organizing fuzzy controller Based on the steps in designing a conventional FLC, we proposed a self-organizing fuzzy controller design that consists of five steps: (1) the definition of input/output variables of the self-organizing fuzzy controller; (2) definition of the control rules; (3) fuzzification procedure; (4) inference logic procedure; (5) a new defuzzification procedure; (6) the self-organization of the rule base. 2.1. Definition of input/output variables of the self-organizing fuzzy controller In general, the output of a system can be described with a function or a mapping of the plant input/output history. For a single-input single-output (SISO) discrete time systems, the mapping can be written in the form of a non-linear function as follows: x(k + 1) = f (x(k), x(k − 1), . . . , u(k), u(k−1), . . . ) (1) where x(k) and u(k) are, respectively the output and input variables at the kth time step. The objective of the control problem is to find a control input sequence which will drive the system to an arbitrary reference set point xref . Rearranging Eq. (1) for control purposes, the value of the input u
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at the kth step that is required to yield the reference output xref can be written as follows: u(k) = g(xref , x(k), x(k − 1), . . . , u(k − 1), u(k − 2), . . . )
(2)
which is viewed as an inverse mapping of Eq. (1). While a typical conventional FLC uses the error and the error rate as the inputs, the proposed controller uses the input and output history as the input terms: xref , x(k), x(k−1), x(k−2), . . . , u(k), u(k−1), u(k− 2), . . . , where k is the sampling time. This implies that u(k) is the input to be applied when the desired output is xref as indicated explicitly in Eq. (2). 2.2. Definition of the control rules In this work, the key idea behind the self-organizing fuzzy controller is not to use rules preconstructed by experts, but forms rules with input and output history at every sampling step. Therefore, a new rule with the input and output history can be defined as follows: R (i) : if x(k) is A1i , x(k − 1) is A2i , . . . , x(k − n + 1) is Ani , u(k − m) is Bmi ,
and
u(k − 1) is B1i , . . . ,
then u(k) is Ci
(3)
where n, m are the number of output and input variables, Aij , Bij the antecedent linguistic values for the ith rule and Ci the consequent linguistic values for the ith rule. 2.3. Fuzzification procedure In a conventional FLC, an expert usually determines the linguistic values Aij , Bij , and Ci by partitioning each universe of discourse. In this paper, however, this linguistic values are determined
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from the crisp values of the input and output history at every sampling step and a fuzzification procedure for fuzzy values is developed to determine A1i , A2i , . . . , A(n+1)i , B1i , B2i , . . . , Bmi , and Ci from the crisp x(k), x(k − 1), x(k − 2), . . . , x(k − n + 1), u(k − 1), u(k − 2), . . . , u(k − m), and u(k), respectively. The fuzzification is done with its base on a reasonably assumed input or output ranges. When the assumed input or output range is [a, b], the membership function for crisp xi is determined in a triangular shape as follows: 1 + (x − x ) i , if a ≤ x < xi (b − a) µAi = 1 − (x − xi ) (4) , if xi ≤ x < b (b − a) 0 Note that all linguistic values overlap on the entire range [a, b], and furthermore, every crisp value uniquely defines the membership function with the unity center or vertex value and identical slopes: −1/(b − a) and 1/(b − a) for the right and left lines, respectively (see Fig. 1). Fig. 1 shows the fuzzification procedure for crisp variables x1 and x2 , where A1 and A2 are the corresponding linguistic values (fuzzy sets) with membership functions defined on the range [a, b]. Thus, this fuzzification procedure requires only the minimal information in forming the membership functions. 2.4. Inference logic procedure To attain the output fuzzy set, it is necessary to determine the membership degree (wi ) of the input fuzzy set with respect to each rule. If input fuzzy variables are considered as fuzzy singletons, the membership degree of the input fuzzy variables for each rule may be calculated by using a specific operator (AND).
Fig. 1. Fuzzification procedure for Aij , Bij or Ci .
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Fig. 2. Inference mechanism.
As the conventional FLC, the operator used in this work was the min operator described for the ith rule as follows: wi = min[(A1i ∧ x1 ), . . . , (A(n+1)i ∧ x(n+1) ), (B1i ∧ u1 ), . . . , (Bmi ∧ um )]
(5)
where ∧ is AND operation. This mechanism considers the minimum intersection degree between input fuzzy variables and the antecedent linguistic values, for example ith and jth rules, as shown in Fig. 2. The membership degrees wi and wj thus defined reflects the contribution of all input variables in ith and jth rule. The evaluation of the membership degree value w with three fuzzy input variables, x(k), x,(k−1), and u(k − 1), is shown in Fig. 2, where the ith rule is closer to the input variables than the jth rule and thus wi > wj . The consequent linguistic value or the net linguistic control action, Cn is calculated for taking the α-cut of the Cn , where α = max[µ(Cn )]. To find a control range for the example shown in Fig. 2, each operation forms the consequent fuzzy set, and the range with its membership degree one is deduced as a control range for each rule, i.e. [a, b] for the ith rule, and [c, d] for the jth rule as the respective ranges. As a result of this inference, the net linguistic range (i.e. the net control range, NCR), which is the intersection of all control ranges, is determined, i.e. [c, b] as shown in Fig. 3, where Ci and Cj are consequent fuzzy sets for ith and jth rules, respectively.
2.5. Defuzzification procedure Defuzzification is a procedure to determine a crisp value from a consequent fuzzy set. The often used methods are the center of area and the mean of maxima. In this paper, defuzzification is to determine a crisp value from the NCR resulting from the inference. Any control value within the NCR has a potential as a control, however, some controls may cause overshoot while others may be too slow.
Fig. 3. NCR with two rules.
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Fig. 4. The defuzzification procedure.
This problem can be avoided by adding a predictive capability in the defuzzification. A method is presented which modifies the NCR to compute a crisp value by using the prediction of the output response. The series of the last outputs is extrapolated in time domain to estimate x(k + 1) by the Newton backward-difference formula [10]. If the extrapolation order is n, using the binomial-coefficient notation: s s(s − 1) · · · (s − k + 1) = (6) k! k the estimate x(k ˆ + 1) is calculated as follows: n −1 i x(k ˆ + 1) = ∇ i x(k) (−1) i
(7)
To modify the control range, the sign of u(k) − u(k − 1) is assumed to be the same as the sign of ˆ + 1). Thus, for Case 1 the sign of xr (k + 1) − x(k xr (k +1)− x(k ˆ +1), hence the sign of u(k)−u(k −1), is positive implying that u(k) has to be increased from the previous input u(k − 1). The final crisp control value u(k) is then selected as one of the mid-points of the modified NCR (as shown in Fig. 4): (u(k − 1) + q) , for Case 1 2 u(k) = (9) (p + u(k − 1)) , for Case 2 2
where
where p and q are the respective lower and upper limits of the NCR resulting from the inference mechanism (Section 2.4).
∇ i x(k) ∇(∇ i−1 x(k)) for i ≥ 2
2.6. Self-organization of the rule base
i=0
∇x(k) x(k) − x(k − 1) Defuzzification is performed by comparing the two values, the estimate x(k ˆ + 1) and the reference output xref or the temporary target xr (k + 1), generated by: xr (k + 1) = x(k) + α(xref − x(k))
(8)
where xr (k+1) is the reference output or the temporary target and α the target ratio constant (0 < α ≤ 1). The value α describes the rate with which the present output x(k) approaches the reference output value. The value α is chosen by the user to obtain a desirable response. When the estimate exceeds the reference output, the control has to slow down. On the other hand, when the estimate has not reached the reference, the control should speed up. Two possible cases will therefore be considered: Case (1), x(k ˆ + 1) < xr (k + 1) and Case (2), x(k ˆ + 1) > xr (k + 1).
The rules of the self-organizing fuzzy controller are generated at every sampling time. If every rule is stored in the rule base, two problems will occur: (1) the memory will be exhausted, and (2) the rules which are performed improperly during initial stages also affect the later inference. For this reason, the fuzzy rule space is partitioned into a finite number of domains of different sizes and only one rule, is stored in each domain. Fig. 5 shows an example of the division of a rule space for two output variables x(k) and x(k − 1). Fig. 6 shows the rule base updating procedure. If there are two rules in a same domain, the selection of a rule is based on comparison of x(k) in both rules. That is, if there is a new rule which has the output smaller to the output in a given domain (old rule), the old rule is replaced by the new one. This updating procedure of the rule base makes the proposed fuzzy
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Fig. 5. Division of a two-dimensional rule space.
logic controller capable of learning the object plant and self-organizing the rule base. The number of rules increases as new input/output data is experienced. It converges to a finite number in steady state, however, and never exceeds the maximum number of domains partitioned in the rule space. Fig. 7 shows the architecture of the proposed FLC system. Initially, since there is no control rule in the rule base, the control input u(k) for the first step is the medium value of the entire input range. As time increases, the defuzzification procedure begins to de-
Fig. 6. The self-organization of the rule base.
termine whether the input has to be increased or decreased depending on the trend of the output. The sign of ∇u(k) and the magnitude of u(k) are determined in the defuzzification procedure. The self-organization of the rule base, in other words “learning” of the system, is performed at each sampling time k. 3. Control problem formulation In this section, we apply the self-organizing fuzzy controller and a conventional fuzzy controller to control the vibrations in a flexible structure using piezoelectric actuators. An experimental apparatus was constructed and it is constituted by a flexible cantilever steel beam type structure with piezoelectric patches symmetrically bonded on both sides (see Fig. 8) to provide a bending movement in the structure and this system has been employed to illustrate the suggested self-organizing control algorithm. These piezoelectric elements are fed by a voltage amplifier (provide by company ACX [11]), that amplifies the entrance voltage in the order of 30 V/V. The flexible structure which is 400 mm long and has a cross-sectional dimensions of 34.5 mm × 1.2 mm, contains two piezoelectric patches having a cross-sectional dimensions of 20.574 mm × 0.254mm and its length is 46 mm. Each pair of piezoceramics (PZT) is bonded side by side to form two sets of actuators located 92.93 mm to rigid support. The schematic diagram for the vibration suppression experiment is shown in Fig. 9. One accelerometer (B & K 4375) attached at the tip through the
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Fig. 7. The self-organizing fuzzy logic control system architecture.
charge amplifiers is used to measure the system displacement with integration circuits. Since the output voltage of the interface board is limited to ±5 V, a voltage amplifier (ACX) is needed to drive the actuator in the range of ±150 V for effective control performance. According to the schematic diagram shown in Fig. 9, a PENTIUM-100 was used as the central processing unit (implementation of a processing digital system using the self-organizing fuzzy control algorithm and the conventional fuzzy controller algorithm) to handle all the I/O data for the flexible structure and to calculate the control actions at 20 ms.
3.1. Parameters of the self-organizing fuzzy controller In the experimental tests, x(k), x(k −1), x(k −2), and u(k − 1), u(k − 2) were used as input variables to the self-organizing fuzzy controller and the variables x(k), x(k − 1) and x(k − 2), were divided into five segments to partition the rule space. The third-order extrapolation (Eq. (7)) was performed to estimate x(k + 1) as follows: x(k + 1) = 3x(k) − 3x(k − 1) + x(k − 2)
(10)
The output range is −0.5 to 0.5 mm, input range is −5 to 5 V, the target ratio constant α was 0.01 (determined by trial and error), and xref is zero. 3.2. Parameters of the conventional fuzzy controller
Fig. 8. Experimental apparatus.
The structure of the conventional fuzzy controller consists of four steps: the definition of input/output fuzzy variables, the decision making of fuzzy control rules, fuzzy inference and defuzzification. In this paper, the error and error rate of system variables are the inputs, and the output voltage applied to voltage amplifier is the output. A triangle-type membership function, shown in Fig. 10, is employed to convert these input and output variables into linguistic control variables (N, Z and P), where [a, b] is the range
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Fig. 9. Schematic diagram of the vibration control experiment.
of the universe of discourse. The input ranges are: −0.5 to 0.5 mm for the error and −100 to 100 mm/s for the error rate, and the output range is −5, to 5 V. Fuzzy control rules of state evaluation, which are similar to the intuitional thinking of humans, are employed in this paper: R
(i)
: if e is A1i ,
e is A2i ,
then u is Ci
(11)
where e, e and u are the system variables (error, error rate and output voltage) and A1i , A2i and Ci are the linguistic values of the fuzzy variable to express the universe of discourse of the fuzzy sets (N, Z and P). In this paper, the fuzzy inference method employs the max–min product composition to operate the fuzzy control rules. In order to obtain the correct control output for this control system, it is necessary to defuzzify the fuzzy sets. The centroid of area method [12] was employed to defuzzify the output variable.
Table 1 The fuzzy control rules of the conventional fuzzy controller e
N Z P
e N
Z
P
1P
2P
3Z
4P
5Z
6N
7Z
8N
9N
The fuzzy control rules employed in controlling this vibration control system are listed in Table 1. 4. Experimental results In order to evaluate the performance of the self-organizing fuzzy controller and the conventional fuzzy controller, the following experiments were performed. 4.1. Vibration suppression under an impulse excitation
Fig. 10. The membership functions of the conventional fuzzy controller.
Fig. 11 shows the experimental open-loop response. It can be observed initially that the maximum amplitude was 5 mm and had a lower steady state error of about 20 s. Under an initial impulse excitation by the piezoelectric actuators of 0.5 s, the experimental results of using a conventional fuzzy controller and the proposed controller are shown in Fig. 12 for comparison.
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Fig. 11. Open-loop response.
Fig. 12. Close-loop response.
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Fig. 13. Output voltage, u.
Fig. 14. Number of generated rules.
G.L.C.M. de Abreu, J.F. Ribeiro / Applied Soft Computing 1 (2002) 271–283
Fig. 15. Closed-loop response.
Fig. 16. Output voltage, u.
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Fig. 17. Number of generated rules.
The continuous curve represents the response of the system with a conventional fuzzy controller and the dashed line depicts the response of the system using the self-organizing fuzzy controller. Note that the performance of the proposed controller is better than that of the conventional fuzzy controller. In Fig. 13, the continuous curve and the dashed line represent the output voltage applied to amplifier voltage with a conventional fuzzy controller and the proposed controller, respectively. In Fig. 14, the proposed control algorithm starts with no initial rule and the number of generated rules is increased monotonically to 33 rules (each rule can be represented by Eq. (3)).
amplifier with a conventional fuzzy controller compared with the self-organizing fuzzy controller are shown in Figs. 15 and 16, respectively. Note that the amplitude of vibration is reduced to about one-fifth of the original value (5 mm) instantly with the proposed controller (continuous curve), while the amplitude of vibration was suppressed at the same reduction of about 1 s later (see Fig. 15) with conventional fuzzy controller (dashed line). The number of newly-generated rules is larger in Section 4.1 (33 rules) than this experimental test (7 rules, Fig. 17). This is because more system conditions are experienced by the proposed controller, i.e. more rules are stored for larger system changes.
4.2. Vibration suppression under constant amplitude sinusoidal disturbance 5. Conclusions The conventional fuzzy controller and the proposed controller were employed to suppress actively the amplitude of vibration of this system. The external sinusoidal disturbance was introduced as the vibration source by B & K Vibration Exciter type 4808 located next to rigid support and the control output started at 3 s (see Fig. 16). The output response of the system and the output voltage applied to voltage
A self-organizing fuzzy controller which implemented in real-time was developed to control the vibrations of the flexible beam type structure using piezoelectric actuators without a modeling procedure or preconstructed rules of an expert. It mimics the human learning process with only a minimal information on the environment. A new defuzzification
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method was developed and an updating procedure of the rule was developed which makes the proposed fuzzy logic controller capable of learning the system and self-organizing the controller. This proposed controller has effectively reduced the effort of constructing a conventional fuzzy controller. This can facilitate the implementation of a self-organizing fuzzy control scheme for complex systems. The experimental results have shown that piezoceramic actuators bonded on a beam control efficiently the vibrations of this flexible structure and show that the present control methodology is effective and the control behaviors exhibit our predicted characteristics. Acknowledgements The authors are thankful to Capes Foundation (Brazil) for the financial support. References [1] E.F. Crawley, J. de Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA J. 10 (1987) 1373– 1385.
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[2] C. Chang-qing, W. Xiao-ming, S. Ya-peng, Finite element approach of vibration control using self-sensing piezoelectric actuators, Comput. Struct. 3 (1996) 505–512. [3] L. Chien-Chang, H. Huang-Nan, Vibration control of beamplates with bonded piezoelectric sensors and actuators, Comput. Struct. (1999) 239–248. [4] C. Jyh-Horng, C. Shinn-Horng, C. Min-Yung, P. An-Jia, Active robust vibration control of flexible composite beams with parameter pertubations, Int. J. Mech. Sci. 7 (1997) 751– 760. [5] M. Lee, Fuzzy logic in control systems: fuzzy logic controller. Parts I and II, IEEE Trans. Syst., Man, Cybernetics 2 (1990) 404–435. [6] I.J. Zeinoun, F. Khorrami, An adaptive control scheme based on fuzzy logic and its application to smart structures, Smart Mater. Struct. 3 (1994) 266–276. [7] A. Ofri, W. Tanchum, H. Guterman, Active Control for Large Space Structure by Fuzzy Logic Controllers, IEEE Press, Silver Spring, MD, 1996, pp. 515–518. [8] S. Shao, Fuzzy self-organizing controller and its application for dynamic processes, Fuzzy Set Syst. 26 (1988) 151–164. [9] Y.-M. Park, U.-C. Moon, Y.L. Kwang, A self-organizing fuzzy controller for dynamic systems using a fuzzy auto-regressive moving average (FARMA) model, IEEE Trans. Fuzzy Syst. 3 (1995) 75–82. [10] R.L. Burden, J.D. Faires, Numerical Analysis, PWS-KENT, 1989. [11] ACX, Active Control eXperts Inc., http://www.acx.com. [12] L.A. Zadeh, Informat. Control 8 (1965) 338–353.
A self-organizing fuzzy logic controller for the active control of flexible structures using piezoelectric actuators Gustavo Luiz C.M. de Abreu∗ , José F. Ribeiro School of Mechanical Engineering, Federal University of Uberlˆandia, Uberlˆandia-MG, Brazil Accepted 8 January 2002
Abstract This paper proposes an on-line self-organizing fuzzy logic controller (FLC) design applied to the control of vibrations in flexible structures containing distributed piezoelectric actuator patches. In this methodology, the fuzzy rules are generated using the history of input/output (I/O) pairs without using any plant model. The generated rules are stored in the fuzzy rule space and updated on-line by a self-organizing procedure. The validity of the proposed fuzzy logic control has been demonstrated experimentally in a steel cantilever test beam and a set of experimental tests are made in the system to verify the efficiency of the on-line self-organizing fuzzy controller. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Self-organizing fuzzy controller; Active vibration control; Piezoelectric actuators; Flexible structures
1. Introduction The great progress experimented by the theory of the fuzzy controllers in the last years has been opening new possibilities of practical application for these controllers. In the last two decades, the subject area of smart/ intelligent materials and structures has experienced tremendous growth in terms of research and development [1]. One reason for this activity is that it may be possible to create certain types of structures and systems capable of adapting to or correcting for changing operating conditions. The advantage of incorporating these special types of materials into the structure is that the sensing and actuating mechanism becomes part of the structure by sensing and actuating strains directly. This is more known like piezoelectric phenomena, i.e. ∗ Corresponding author. E-mail addresses: [email protected] (G.L.C.M. de Abreu), [email protected] (J.F. Ribeiro).
direct and converse piezoelectric effects. When a mechanical force is applied to a piezoelectric material, an electric voltage or change will be generated. On the other hand, when an electric field is applied to the material, a mechanical force will be induced because of the converse piezoelectric effect. With the recent advances in piezoelectric technology, it has been shown that the piezoelectric actuators based on the converse piezoelectric effect can offer excellent potential for active vibration control techniques, especially for vibration suppression or isolation. Recent research has focused on the applications of piezoelectric sensor and actuator in smart structures. Crawley and de Luis [1] were among the first to embed piezoelectric materials in composite laminated beams. Chang-qing et al. [2] and Chien-Chang and Huang-Nan [3] presented a formulation methodology to the control of vibrations in composite structures with bonded piezoelectric sensors and actuators. Vibration suppression of composite smart structures by using piezoelectric sensors and actuators were also
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analyzed numerically by Jyh-Horng et al. [4]. Therefore, it is difficult to implement classical controllers to these systems, especially to a system which is complex and non-linear. Effective applications in vibration control, however, require that the system dynamics can be adequately and/or accurately determined and that the controller design can be easily implemented. Vibration control of smart structures using fuzzy controllers has thus been receiving attention for their ability to deal with non-linearities, uncertainties in terms of vagueness, ignorance, and imprecision, and provide a feasible alternative since they can easily capture qualitative aspects of human knowledge. Fuzzy controllers are most suitable for systems that cannot be precisely described by mathematical formulations. In this case, a control designer captures operators knowledge and converts it into a set of fuzzy control rules. The idea of the fuzzy logic is useful for representing linguistic terms numerically and making reliable decisions with ambiguous and imprecise events or facts. The benefit of the simple design procedure of a fuzzy controller leads to the successful applications of a variety of engineering systems [5]. Zeinoun and Khorrami [6] proposed a fuzzy logic algorithm for vibration suppression of a clamp-free beam with piezoelectric sensor/actuator, and Ofri et al. [7] also used a control strategy based on fuzzy logic theory for vibration damping of a large flexible space structure controlled by bonded piezoceramic actuators. In general, the fuzzy logic controllers (FLCs) use fuzzy inference with rules preconstructed by an expert. Therefore, the most important task is to form the rule base which represents the experience and intuition of human experts. When the rule base of human experts is not available, an efficient control cannot be expected. The self-organizing fuzzy controller is a rule-based type of controller which learns how to control on-line while being applied to a system, and it has been used successfully for a wide variety of processes [8]. This controller combines system identification and control based on experience. Therefore, only a minimal amount of information about the environment needs to be provided. The main purpose of this paper is to minimize the role of human experts in designing a fuzzy logic controller. For this, the on-line self-organizing controller is used which uses the input and output history in its
rules [9]. This controller has no rule at its initial stage, but forms rules by defining membership functions using the plant input/output (I/O) data as singletons and stores them in a rule base. The rule base is updated as experience is accumulated using a self-organizing procedure. A new method for defuzzification is also developed by adding a predictive capability using a prediction model. The self-organizing controller is experimentally verified in a steel cantilever test beam. A set of experimental tests is made in the system to evaluate the performance of the self-organizing fuzzy controller and the conventional fuzzy controller. Experimental results are shown, to support the effectiveness of the active vibration control, and finally, a conclusion of this study is given.
2. Design of the self-organizing fuzzy controller Based on the steps in designing a conventional FLC, we proposed a self-organizing fuzzy controller design that consists of five steps: (1) the definition of input/output variables of the self-organizing fuzzy controller; (2) definition of the control rules; (3) fuzzification procedure; (4) inference logic procedure; (5) a new defuzzification procedure; (6) the self-organization of the rule base. 2.1. Definition of input/output variables of the self-organizing fuzzy controller In general, the output of a system can be described with a function or a mapping of the plant input/output history. For a single-input single-output (SISO) discrete time systems, the mapping can be written in the form of a non-linear function as follows: x(k + 1) = f (x(k), x(k − 1), . . . , u(k), u(k−1), . . . ) (1) where x(k) and u(k) are, respectively the output and input variables at the kth time step. The objective of the control problem is to find a control input sequence which will drive the system to an arbitrary reference set point xref . Rearranging Eq. (1) for control purposes, the value of the input u
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at the kth step that is required to yield the reference output xref can be written as follows: u(k) = g(xref , x(k), x(k − 1), . . . , u(k − 1), u(k − 2), . . . )
(2)
which is viewed as an inverse mapping of Eq. (1). While a typical conventional FLC uses the error and the error rate as the inputs, the proposed controller uses the input and output history as the input terms: xref , x(k), x(k−1), x(k−2), . . . , u(k), u(k−1), u(k− 2), . . . , where k is the sampling time. This implies that u(k) is the input to be applied when the desired output is xref as indicated explicitly in Eq. (2). 2.2. Definition of the control rules In this work, the key idea behind the self-organizing fuzzy controller is not to use rules preconstructed by experts, but forms rules with input and output history at every sampling step. Therefore, a new rule with the input and output history can be defined as follows: R (i) : if x(k) is A1i , x(k − 1) is A2i , . . . , x(k − n + 1) is Ani , u(k − m) is Bmi ,
and
u(k − 1) is B1i , . . . ,
then u(k) is Ci
(3)
where n, m are the number of output and input variables, Aij , Bij the antecedent linguistic values for the ith rule and Ci the consequent linguistic values for the ith rule. 2.3. Fuzzification procedure In a conventional FLC, an expert usually determines the linguistic values Aij , Bij , and Ci by partitioning each universe of discourse. In this paper, however, this linguistic values are determined
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from the crisp values of the input and output history at every sampling step and a fuzzification procedure for fuzzy values is developed to determine A1i , A2i , . . . , A(n+1)i , B1i , B2i , . . . , Bmi , and Ci from the crisp x(k), x(k − 1), x(k − 2), . . . , x(k − n + 1), u(k − 1), u(k − 2), . . . , u(k − m), and u(k), respectively. The fuzzification is done with its base on a reasonably assumed input or output ranges. When the assumed input or output range is [a, b], the membership function for crisp xi is determined in a triangular shape as follows: 1 + (x − x ) i , if a ≤ x < xi (b − a) µAi = 1 − (x − xi ) (4) , if xi ≤ x < b (b − a) 0 Note that all linguistic values overlap on the entire range [a, b], and furthermore, every crisp value uniquely defines the membership function with the unity center or vertex value and identical slopes: −1/(b − a) and 1/(b − a) for the right and left lines, respectively (see Fig. 1). Fig. 1 shows the fuzzification procedure for crisp variables x1 and x2 , where A1 and A2 are the corresponding linguistic values (fuzzy sets) with membership functions defined on the range [a, b]. Thus, this fuzzification procedure requires only the minimal information in forming the membership functions. 2.4. Inference logic procedure To attain the output fuzzy set, it is necessary to determine the membership degree (wi ) of the input fuzzy set with respect to each rule. If input fuzzy variables are considered as fuzzy singletons, the membership degree of the input fuzzy variables for each rule may be calculated by using a specific operator (AND).
Fig. 1. Fuzzification procedure for Aij , Bij or Ci .
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Fig. 2. Inference mechanism.
As the conventional FLC, the operator used in this work was the min operator described for the ith rule as follows: wi = min[(A1i ∧ x1 ), . . . , (A(n+1)i ∧ x(n+1) ), (B1i ∧ u1 ), . . . , (Bmi ∧ um )]
(5)
where ∧ is AND operation. This mechanism considers the minimum intersection degree between input fuzzy variables and the antecedent linguistic values, for example ith and jth rules, as shown in Fig. 2. The membership degrees wi and wj thus defined reflects the contribution of all input variables in ith and jth rule. The evaluation of the membership degree value w with three fuzzy input variables, x(k), x,(k−1), and u(k − 1), is shown in Fig. 2, where the ith rule is closer to the input variables than the jth rule and thus wi > wj . The consequent linguistic value or the net linguistic control action, Cn is calculated for taking the α-cut of the Cn , where α = max[µ(Cn )]. To find a control range for the example shown in Fig. 2, each operation forms the consequent fuzzy set, and the range with its membership degree one is deduced as a control range for each rule, i.e. [a, b] for the ith rule, and [c, d] for the jth rule as the respective ranges. As a result of this inference, the net linguistic range (i.e. the net control range, NCR), which is the intersection of all control ranges, is determined, i.e. [c, b] as shown in Fig. 3, where Ci and Cj are consequent fuzzy sets for ith and jth rules, respectively.
2.5. Defuzzification procedure Defuzzification is a procedure to determine a crisp value from a consequent fuzzy set. The often used methods are the center of area and the mean of maxima. In this paper, defuzzification is to determine a crisp value from the NCR resulting from the inference. Any control value within the NCR has a potential as a control, however, some controls may cause overshoot while others may be too slow.
Fig. 3. NCR with two rules.
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Fig. 4. The defuzzification procedure.
This problem can be avoided by adding a predictive capability in the defuzzification. A method is presented which modifies the NCR to compute a crisp value by using the prediction of the output response. The series of the last outputs is extrapolated in time domain to estimate x(k + 1) by the Newton backward-difference formula [10]. If the extrapolation order is n, using the binomial-coefficient notation: s s(s − 1) · · · (s − k + 1) = (6) k! k the estimate x(k ˆ + 1) is calculated as follows: n −1 i x(k ˆ + 1) = ∇ i x(k) (−1) i
(7)
To modify the control range, the sign of u(k) − u(k − 1) is assumed to be the same as the sign of ˆ + 1). Thus, for Case 1 the sign of xr (k + 1) − x(k xr (k +1)− x(k ˆ +1), hence the sign of u(k)−u(k −1), is positive implying that u(k) has to be increased from the previous input u(k − 1). The final crisp control value u(k) is then selected as one of the mid-points of the modified NCR (as shown in Fig. 4): (u(k − 1) + q) , for Case 1 2 u(k) = (9) (p + u(k − 1)) , for Case 2 2
where
where p and q are the respective lower and upper limits of the NCR resulting from the inference mechanism (Section 2.4).
∇ i x(k) ∇(∇ i−1 x(k)) for i ≥ 2
2.6. Self-organization of the rule base
i=0
∇x(k) x(k) − x(k − 1) Defuzzification is performed by comparing the two values, the estimate x(k ˆ + 1) and the reference output xref or the temporary target xr (k + 1), generated by: xr (k + 1) = x(k) + α(xref − x(k))
(8)
where xr (k+1) is the reference output or the temporary target and α the target ratio constant (0 < α ≤ 1). The value α describes the rate with which the present output x(k) approaches the reference output value. The value α is chosen by the user to obtain a desirable response. When the estimate exceeds the reference output, the control has to slow down. On the other hand, when the estimate has not reached the reference, the control should speed up. Two possible cases will therefore be considered: Case (1), x(k ˆ + 1) < xr (k + 1) and Case (2), x(k ˆ + 1) > xr (k + 1).
The rules of the self-organizing fuzzy controller are generated at every sampling time. If every rule is stored in the rule base, two problems will occur: (1) the memory will be exhausted, and (2) the rules which are performed improperly during initial stages also affect the later inference. For this reason, the fuzzy rule space is partitioned into a finite number of domains of different sizes and only one rule, is stored in each domain. Fig. 5 shows an example of the division of a rule space for two output variables x(k) and x(k − 1). Fig. 6 shows the rule base updating procedure. If there are two rules in a same domain, the selection of a rule is based on comparison of x(k) in both rules. That is, if there is a new rule which has the output smaller to the output in a given domain (old rule), the old rule is replaced by the new one. This updating procedure of the rule base makes the proposed fuzzy
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Fig. 5. Division of a two-dimensional rule space.
logic controller capable of learning the object plant and self-organizing the rule base. The number of rules increases as new input/output data is experienced. It converges to a finite number in steady state, however, and never exceeds the maximum number of domains partitioned in the rule space. Fig. 7 shows the architecture of the proposed FLC system. Initially, since there is no control rule in the rule base, the control input u(k) for the first step is the medium value of the entire input range. As time increases, the defuzzification procedure begins to de-
Fig. 6. The self-organization of the rule base.
termine whether the input has to be increased or decreased depending on the trend of the output. The sign of ∇u(k) and the magnitude of u(k) are determined in the defuzzification procedure. The self-organization of the rule base, in other words “learning” of the system, is performed at each sampling time k. 3. Control problem formulation In this section, we apply the self-organizing fuzzy controller and a conventional fuzzy controller to control the vibrations in a flexible structure using piezoelectric actuators. An experimental apparatus was constructed and it is constituted by a flexible cantilever steel beam type structure with piezoelectric patches symmetrically bonded on both sides (see Fig. 8) to provide a bending movement in the structure and this system has been employed to illustrate the suggested self-organizing control algorithm. These piezoelectric elements are fed by a voltage amplifier (provide by company ACX [11]), that amplifies the entrance voltage in the order of 30 V/V. The flexible structure which is 400 mm long and has a cross-sectional dimensions of 34.5 mm × 1.2 mm, contains two piezoelectric patches having a cross-sectional dimensions of 20.574 mm × 0.254mm and its length is 46 mm. Each pair of piezoceramics (PZT) is bonded side by side to form two sets of actuators located 92.93 mm to rigid support. The schematic diagram for the vibration suppression experiment is shown in Fig. 9. One accelerometer (B & K 4375) attached at the tip through the
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Fig. 7. The self-organizing fuzzy logic control system architecture.
charge amplifiers is used to measure the system displacement with integration circuits. Since the output voltage of the interface board is limited to ±5 V, a voltage amplifier (ACX) is needed to drive the actuator in the range of ±150 V for effective control performance. According to the schematic diagram shown in Fig. 9, a PENTIUM-100 was used as the central processing unit (implementation of a processing digital system using the self-organizing fuzzy control algorithm and the conventional fuzzy controller algorithm) to handle all the I/O data for the flexible structure and to calculate the control actions at 20 ms.
3.1. Parameters of the self-organizing fuzzy controller In the experimental tests, x(k), x(k −1), x(k −2), and u(k − 1), u(k − 2) were used as input variables to the self-organizing fuzzy controller and the variables x(k), x(k − 1) and x(k − 2), were divided into five segments to partition the rule space. The third-order extrapolation (Eq. (7)) was performed to estimate x(k + 1) as follows: x(k + 1) = 3x(k) − 3x(k − 1) + x(k − 2)
(10)
The output range is −0.5 to 0.5 mm, input range is −5 to 5 V, the target ratio constant α was 0.01 (determined by trial and error), and xref is zero. 3.2. Parameters of the conventional fuzzy controller
Fig. 8. Experimental apparatus.
The structure of the conventional fuzzy controller consists of four steps: the definition of input/output fuzzy variables, the decision making of fuzzy control rules, fuzzy inference and defuzzification. In this paper, the error and error rate of system variables are the inputs, and the output voltage applied to voltage amplifier is the output. A triangle-type membership function, shown in Fig. 10, is employed to convert these input and output variables into linguistic control variables (N, Z and P), where [a, b] is the range
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Fig. 9. Schematic diagram of the vibration control experiment.
of the universe of discourse. The input ranges are: −0.5 to 0.5 mm for the error and −100 to 100 mm/s for the error rate, and the output range is −5, to 5 V. Fuzzy control rules of state evaluation, which are similar to the intuitional thinking of humans, are employed in this paper: R
(i)
: if e is A1i ,
e is A2i ,
then u is Ci
(11)
where e, e and u are the system variables (error, error rate and output voltage) and A1i , A2i and Ci are the linguistic values of the fuzzy variable to express the universe of discourse of the fuzzy sets (N, Z and P). In this paper, the fuzzy inference method employs the max–min product composition to operate the fuzzy control rules. In order to obtain the correct control output for this control system, it is necessary to defuzzify the fuzzy sets. The centroid of area method [12] was employed to defuzzify the output variable.
Table 1 The fuzzy control rules of the conventional fuzzy controller e
N Z P
e N
Z
P
1P
2P
3Z
4P
5Z
6N
7Z
8N
9N
The fuzzy control rules employed in controlling this vibration control system are listed in Table 1. 4. Experimental results In order to evaluate the performance of the self-organizing fuzzy controller and the conventional fuzzy controller, the following experiments were performed. 4.1. Vibration suppression under an impulse excitation
Fig. 10. The membership functions of the conventional fuzzy controller.
Fig. 11 shows the experimental open-loop response. It can be observed initially that the maximum amplitude was 5 mm and had a lower steady state error of about 20 s. Under an initial impulse excitation by the piezoelectric actuators of 0.5 s, the experimental results of using a conventional fuzzy controller and the proposed controller are shown in Fig. 12 for comparison.
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Fig. 11. Open-loop response.
Fig. 12. Close-loop response.
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Fig. 13. Output voltage, u.
Fig. 14. Number of generated rules.
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Fig. 15. Closed-loop response.
Fig. 16. Output voltage, u.
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Fig. 17. Number of generated rules.
The continuous curve represents the response of the system with a conventional fuzzy controller and the dashed line depicts the response of the system using the self-organizing fuzzy controller. Note that the performance of the proposed controller is better than that of the conventional fuzzy controller. In Fig. 13, the continuous curve and the dashed line represent the output voltage applied to amplifier voltage with a conventional fuzzy controller and the proposed controller, respectively. In Fig. 14, the proposed control algorithm starts with no initial rule and the number of generated rules is increased monotonically to 33 rules (each rule can be represented by Eq. (3)).
amplifier with a conventional fuzzy controller compared with the self-organizing fuzzy controller are shown in Figs. 15 and 16, respectively. Note that the amplitude of vibration is reduced to about one-fifth of the original value (5 mm) instantly with the proposed controller (continuous curve), while the amplitude of vibration was suppressed at the same reduction of about 1 s later (see Fig. 15) with conventional fuzzy controller (dashed line). The number of newly-generated rules is larger in Section 4.1 (33 rules) than this experimental test (7 rules, Fig. 17). This is because more system conditions are experienced by the proposed controller, i.e. more rules are stored for larger system changes.
4.2. Vibration suppression under constant amplitude sinusoidal disturbance 5. Conclusions The conventional fuzzy controller and the proposed controller were employed to suppress actively the amplitude of vibration of this system. The external sinusoidal disturbance was introduced as the vibration source by B & K Vibration Exciter type 4808 located next to rigid support and the control output started at 3 s (see Fig. 16). The output response of the system and the output voltage applied to voltage
A self-organizing fuzzy controller which implemented in real-time was developed to control the vibrations of the flexible beam type structure using piezoelectric actuators without a modeling procedure or preconstructed rules of an expert. It mimics the human learning process with only a minimal information on the environment. A new defuzzification
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method was developed and an updating procedure of the rule was developed which makes the proposed fuzzy logic controller capable of learning the system and self-organizing the controller. This proposed controller has effectively reduced the effort of constructing a conventional fuzzy controller. This can facilitate the implementation of a self-organizing fuzzy control scheme for complex systems. The experimental results have shown that piezoceramic actuators bonded on a beam control efficiently the vibrations of this flexible structure and show that the present control methodology is effective and the control behaviors exhibit our predicted characteristics. Acknowledgements The authors are thankful to Capes Foundation (Brazil) for the financial support. References [1] E.F. Crawley, J. de Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA J. 10 (1987) 1373– 1385.
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[2] C. Chang-qing, W. Xiao-ming, S. Ya-peng, Finite element approach of vibration control using self-sensing piezoelectric actuators, Comput. Struct. 3 (1996) 505–512. [3] L. Chien-Chang, H. Huang-Nan, Vibration control of beamplates with bonded piezoelectric sensors and actuators, Comput. Struct. (1999) 239–248. [4] C. Jyh-Horng, C. Shinn-Horng, C. Min-Yung, P. An-Jia, Active robust vibration control of flexible composite beams with parameter pertubations, Int. J. Mech. Sci. 7 (1997) 751– 760. [5] M. Lee, Fuzzy logic in control systems: fuzzy logic controller. Parts I and II, IEEE Trans. Syst., Man, Cybernetics 2 (1990) 404–435. [6] I.J. Zeinoun, F. Khorrami, An adaptive control scheme based on fuzzy logic and its application to smart structures, Smart Mater. Struct. 3 (1994) 266–276. [7] A. Ofri, W. Tanchum, H. Guterman, Active Control for Large Space Structure by Fuzzy Logic Controllers, IEEE Press, Silver Spring, MD, 1996, pp. 515–518. [8] S. Shao, Fuzzy self-organizing controller and its application for dynamic processes, Fuzzy Set Syst. 26 (1988) 151–164. [9] Y.-M. Park, U.-C. Moon, Y.L. Kwang, A self-organizing fuzzy controller for dynamic systems using a fuzzy auto-regressive moving average (FARMA) model, IEEE Trans. Fuzzy Syst. 3 (1995) 75–82. [10] R.L. Burden, J.D. Faires, Numerical Analysis, PWS-KENT, 1989. [11] ACX, Active Control eXperts Inc., http://www.acx.com. [12] L.A. Zadeh, Informat. Control 8 (1965) 338–353.
