Computationally advantageous and stable

48

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 50, NO. 1, FEBRUARY 2003

Computationally Advantageous and Stable Hierarchical Fuzzy Systems for Active Suspension Waratt Rattasiri and Saman K. Halgamuge, Member, IEEE

Abstract—A new type of hierarchical fuzzy system (HFS), namely, Hierarchical Classifying-Type Fuzzy System (HCTFS), is developed and proposed in the paper. While the HCTFS enjoys the full benefits of a traditional HFS, one of which is to suppress the effects of the unwanted phenomenon, “the curse of dimensionality,” it also offers one great advantage that all rule strengths are preserved when passing through subsystem layers. To demonstrate the potential of the HCTFS, computational complexity analysis will be conducted on the complete rule-base models of a conventional fuzzy system and the HCTFS. Furthermore, a methodology of stability analysis is proposed incorporating the use of the the HCTFS, providing the reader with another option of hierarchical fuzzy controller design upon stability concerns. To verify and conclude our proposal, a mathematical example and simulations are provided. In our simulated example, the the HCTFS controller incorporating the proposed stability analysis technique are applied to the active suspension system. The results obtained from the active suspension system are then discussed and compared with the results of the ideal and passive suspension systems. Index Terms—Alternative Hierarchical Fuzzy System (AHFS), Classifying-Type Fuzzy System (CTFS), computational complexity, stability.

I. INTRODUCTION

O

NE OF THE MAIN purposes of using a hierarchical fuzzy system (HFS) is to minimize the size of rule base by eliminating “the curse of dimensionality,” Furthermore, the computational complexity in the process can be reduced as a consequence of the rule-base size reduction, which has become one of the main concerns among system designers. It has been reported in [1] that the more mathematically complex circuits could notably increase the simulation and CPU time. This is further supported in [2]–[4] that circuits with more components would require more simulation and CPU time compared to simplified ones. Therefore, these suggest that the less computationally complex system would be technically preferable in order to be more responsive in producing an output, as well as more economical due to the less number components used, thus implying smaller production costs. To underline the need for the less computational complex system, it has been reported in [2] and [5]–[7] about the difference between the simManuscript received June 26, 2001; revised March 26, 2002. Abstract published on the Internet November 20, 2002. The authors are with the Mechatronics Research Group, Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Melbourne, Australia (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIE.2002.807676

ulation time and hardware-experiment time, among which the results obtained from hardware experiment conducted in [6] reflected faster response comparing to those obtained from simulation. This may suggest that the response obtained from conventional fuzzy system (CFS), whose rule base is greater in size and more complex, would be lagging that of an HFS, whose rule base is extremely smaller and less complex. Moreover, if testing on hardware, this performance difference between a CFS and an HFS could be quite obvious. The greater CPU-time difference could be expected if the circuit size increased, according to the suggestion in [6]. Many researchers have proposed different types of HFS to address these concerns. One of the pioneering works on HFS, having defuzzification between subsystem layers, was proposed in [8]. A type of HFS using Differential Evolution technique was reported in [9]. Another type of HFS, called Structured Takagi–Sugeno Type Fuzzy Logic Unit (SFLU) was proposed in [10]. In the proposed system, the outputs of the previous layer are not used in the IF parts, but used in the THEN parts of the fuzzy rules in the next layer to prevent the loss of the fuzzy variables’ meaning, which may have occurred during the repetitive defuzzification process. The use of TSK-based HFS was reported in [11]. An HFS using Hierarchical Prioritized Structure (HPS) technique in which the firing level of a rule is determined upon the certainty qualification is discussed in [12]. There has been much work involving HFS implementation techniques, but there have been very few studies contributing to computational complexity of HFS. However, there were a number of studies involving other aspects of HFS, such as sensitivity analysis [13], and stability analysis [14] based on the work of [15] and extended to application in generic HFS-type structure. To address this gap, this paper will discuss a type of HFS, called Hierarchical Classifying-Type Fuzzy System (HCTFS), which relies on Alternative Hierarchical Fuzzy System (AHFS) structure and Classifying-Type Fuzzy System instead of repetitive defuzzification process between subsystem layers to prevent the loss of fuzzy information, analyzes the computational complexity in terms of mathematical process and electronic components used, and proposes a way to use an HCTFS as a controller under stability concerns. In Section III, the structure of AHFS [16] will be discussed, providing information on its internal and rule base structures. Section IV will discuss the computational process of CFS and HCTFS, which includes computational complexity and computational circuits. Section V provides a discussion on stability analysis and criteria. In Section VI, simulations will be provided

0278-0046/03$17.00 © 2003 IEEE

RATTASIRI AND HALGAMUGE: HFSs FOR ACTIVE SUSPENSION

49

Fig. 1. CFS.

showing the practicality of the use of HCTFS with active suspension system. II. CFS In the CFS of Fig. 1, the rule base IF is AND THEN is where the inputs output

is

is of the form AND

is (1)

, and the are the fuzzy sets and in and

. and and , respectively [17]. Definition 1: The rule base is said to be complete [8] if there is an onto and into map between and where

Fig. 2. AHFS (n is even.)

HCTFS and CFS, respectively. To further analyze the worthiness of HCTFS decision, let the number of rules obtained from a certain fuzzy system be , hence, the ratio of the number of rules in a certain CFS and the number of rules obtained from the full combination of adjectives used in the CFS in comparison will be (4)

or

In the case that (2)

is used in In other words, every fuzzy set . It is obvious that a complete rule every rule different rules where is the number of base has fuzzy sets defined on each fuzzy variable and is the number of fuzzy variables in the system [8]. III. ALTERNATIVE MODEL OF HFSS Prior to the HCTFS decision, it should be examined that it is worth opting for a HCTFS. Let the number of rules in the complete rule-base model of CFS be and be the number of rules in the HCTFS rule base when all combinations of fuzzy adjectives are applied. Therefore, the ratio of HCTFS and CFS may be defined as (3) where HCTFS and CFS represent the total number of rules obtained from the full combination of fuzzy adjectives used in an

the system designer should consider adding HCTFS into his selection while, on the other hand,

the system designer should remain with his present system configuration. In the HFS domain, the complete HFS rule base is at a minand each subsystem has only imum when two inputs [8], [17]. In situations where the design of an HFS has to deal with system inputs and their rate of change, or other inputs that are closely related to each other, then it is an advantage to have the two inputs closely bonded in pair. These paired parameters should always be placed in the same subsystem. In order to minimize the rule-base size and to keep the bonds between closely related inputs still preserved, it is advantageous to use AHFS [16], which incorporates a CTFS [18] in each subsystem, thus named HCTFS. The use of the HCTFS leads to the elimination of the need for defuzzification process between layers, thus greatly the reducing computational complexity and loss of meaningful information during repetitive defuzzification process. The AHFS is illustrated in Figs. 2 and 3 where takes

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.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

Fuzzy System 2:

.. .

.. .

Layer 2: Fuzzy System 3:

.. .

Fig. 3. Odd-n AHFS.

even and odd numbers, respectively. It will indeed be worth . trying if the original system has

.. .

.. .

.. .

Fuzzy System 4:

A. AHFS With Even “ ” layers with fuzzy systems. The There are structure of an even- AHFS is shown in Fig. 2. .. .

Layer 1: Fuzzy System 1: IF is Fuzzy System 2: IF

is

Layer 2: Fuzzy System 3: IF Fuzzy System 4: IF .. .

is

THEN

AND

is

THEN

AND is

.. .

.. .

.. .

.. .

.. .

.. .

.. .

(6)

is .. .

THEN .. .

are the subscripts indicating the rule strength where of the th rule in a particular subsystem

: IF

AND

THEN

is

(5)

.. .

, and

where and Layer 1: Fuzzy System 1:

.. .

Layer :

THEN

AND .. .. . .

.. .

Layer : Fuzzy System

AND

.. .

and

.. .

.. .


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base of the HCTFS will appear as follows:

B. AHFS With Odd “ ” layers with fuzzy systems. There are The structure of “odd- ” AHFS is shown in Fig. 3. The rule base will be similar to the “even- ” case however there will be .” Furthermore, the inputs to the 1 subsystem in Layer “ last subsystem in Layer “ ” will be the system input “ ” and ” from the previous layer. class attributes “

IF is THEN

AND

is

IF is THEN

AND

is

IF IV. COMPUTATIONAL PROCESS

THEN

AND is

(10)

A. Computational Cost for CFS We will establish an assumption on an example in which the original CFS has four inputs, each defined by three fuzzy sets, and one output defined by three fuzzy sets, and the consequent is normal with center . Moreover, the comparfuzzy set ison analysis will be made after the fuzzification process as these two systems rely upon the same fuzzification engine through which the computation steps will all be similar. The system employs fuzzy rule base, product inference engine, a fuzzification, and center average defuzzifier. The rule base is, thus, of the following form [17]: IF is AND is AND is THEN where

AND is

, and the complete where rule base is composed of 27 rules. Hence, the output of each subsystem can be acquired by the following:

is (7)

and . The resulting crisp output then is

where

(8)

and . where operator can be accomplished by algebraic If we assume product, consequently, we need 8 81 (numerator) 7 81 1215 multiplications, 80 (numerator) 80 (denominator) (denominator) 160 additions, and one division to complete the computation process. The computational complexity of CFSs can be described as where Multiplications Additions Division

(9)

the number of inputs, and the number of fuzzy where sets defined on the inputs. Note that this calculation process is based on the assumption that each input is defined by the same number of fuzzy sets. B. Computational Cost for AHFS Now, we will continue with HCTFS under the same restrictions. According to Section III-A, we will need a two-layer HCTFS being composed of three subsystems. The complete rule

(11)

Each subsystem in Layer 1 needs 27 multiplications to get the from every rule, and nine subtractions, nine rule strength 9 comparisons for the three MAX operators. In the last layer, the defuzzification process needs 18 (numerator) 9 (denominator) 27 multiplications, 8 (numerator) 8 (denominator) 16 additions, and 1 division. Therefore, the entire process may need up to 81 multiplications, 16 additions, 18 subtractions, 18 comparisons, and 1 division.

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Fig. 4. Comparisons between computational complexity of CFS and HCTFS.

Following are the formulas to calculate the computational complexity of HCTFSs: number of fuzzy sets defined on each inputs; number of class attributes; number of rules assigned to the class attributes ; computational dummy variable for subtractions; computational dummy variable for comparisons;

computational dummy variable for subtractions; computational dummy variable for comparisons; Subtractions

Comparisons

(13)

The rest remain unchanged. Number of subsystems

C. Computational Circuit

even , Number of layers

odd , Multiplications Additions Subtractions

Comparisons Division

(12)

Please note that the use of HCTFS will be suitable for the case , and there are only two inputs to each subsystem. when Furthermore, the calculation process for subtractions and comparisons is made on the assumption that each input is defined by the same number fuzzy sets, and the same number of rules is classified by each of the class attributes. In other . words, If the numbers of rules classified by each class attribute are not equal the following formulas should be used: ; ; number of rules assigned to each class attribute;

Let us further continue on the assumption that the word size for the data path is 4-bit thus we will need a 4-bit parallel full adder, a 4-bit parallel multiplier, a 4-bit divider, and a 4-bit magnitude comparator exclusively for the HCTFS computational circuit. Even though a single 4-bit parallel subtractor has already been included in the divider circuit one will also be added to the HCTFS to perform the comparison operation. This is because of our intention to exemplify the possible maximum number of electronic components needed in excess to those required in CFS. Then, in Fig. 4, a comparison between CFS and HCTFS is presented [19]–[26]. Fig. 4 shows the comparisons of the computational circuits, the computational complexity, the logic gates, as well as electronics parts, which are needed by our experimental CFS and HCTFS. It was found that CFS may need 1500% multiplications more and up to 1000% more additions than are needed by HCTFS, while HCTFS may need on average 30% more electronic parts than are needed by CFS. This excess contributes to the comparator circuit, which is not needed in the CFS computational circuit, and can be reduced if one can manage to use the subtractor circuit inside the divider circuit. V. STABILITY ANALYSIS In this section, the stability analysis will be conducted on the HCTFS using the structure of the AHFS [16] shown in Fig. 2, having even “ ” inputs with “ ” fuzzy sets defined on each , . Therefore, the hierarchical system will ” subsystems, each having “ ” class attributes contain “


Fig. 5.

Open-loop system.

Fig. 6.

Control system with state feedback.

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the state variables and is the control inputs. is a vector of uncertain nonlinear parameter, which is restricted to a prescribed bounding set, being a compact set. is assumed to is a continuous matrix of be Lebesgue measurable, and . The state model in (19) can be viewed as an uncertain linear dynamical model. We will mainly be focused on the analysis of stability boundaries of the state matrix , which represents the characteristics of the original plant. In this paper, we will consider the following structure of uncertainty:

[18] as the outputs, except the one in the last layer, which yields , the complete rule base the final output . By letting rules shown in (5) and in every subsystem will comprise of can be mathematically represented by (6).

in which , and are known real matrices that govern the . The characteristics the structure of uncertainty, and uncertainty is bounded as follows [28]: (20)

A. Open-Loop System We will then conduct stability analysis on the original plant in (14) of Fig. 5, which, for simplicity purposes, is assumed to be linear time invariant (LTI) and strictly proper

in which all where is some real value, and are Lebesgue measurable. elements in With a controller yielding , the system of (19) can be viewed in Fig. 6. To analyze stability, we let of (19) be 0 and consider

(14) , , , and , where respectively. Assumption: Throughout the following, it will be assumed that the system of (14) satisfies the following assumption: is both controllable and stabilizable. The control inputs may be defined as (15) represents the control input and . Therefore, where , which represents the state variables, will be the range of as follows:

(21) matrix is said to be Definition 2—[29]: An for all where stable or Hurwitz if . Definition 3—[29]: A matrix is said to be a Hicks matrix if all odd-order principal minors of are negative and all evenorder principal minors of are positive. matrix is a Metzler Definition 4—[29]: An matrix if (22)

(16) Then, the response from each of these internal state variables is fed back to the controller as inputs so that the controller can produce the controller signal to the plant. Therefore, the statefeedback controller whose inputs are of (16) may be regarded as a function whose output can be represented as follows:

. for all Theorem 1—[29]: A Metzler matrix is said to be stable if and only if it is Hicks. Example 1: Let us consider a 2 2 system matrix in which

(17) and, thus,

(23) (18)

where

Thus, (21) can be represented by

.

B. Stabilization of Linear Systems With Norm-Bounded Time-Varying Uncertainty

(24)

Considering the control system with state feedback [27] of Fig. 6, we obtain the following equation:

whose candidate can Let us consider Lyapunov function be chosen as the Euclidean distance squared [29], [30] (25)

(19) and where and are of appropriate dimensions, and

, and matrices represents

for all which implies as inite function, and of (24), it is required that

or

being positive-def. To ensure stability for all trajectories of the

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having three membership functions {Lo, Med, Hi} defined on each of the inputs, and on the output. Thus, the rule base can be represented as follows. Layer 1: Fuzzy System 1: IF IF

Fig. 7.

HCTFS.

system. Thus, we havep

(26) To ensure stability of , let us further assume that is both a Metzler matrix, and a Hicks matrix. Hence, (26) can be obtained as follows:

(27) Given that

IF IF IF IF IF IF IF Fuzzy System 2: IF IF IF IF IF IF IF IF IF

everywhere except at

, and its consequence

as well as the assumption as follows:

, (27) can be manipulated

(28) which implies that algebraic conditions

Layer 3: Fuzzy System 2: IF IF IF IF IF IF IF IF IF

is Hi AND is Hi AND

is Hi THEN is Med THEN

is Hi AND is Med AND is Med AND is Med AND is Lo AND is Lo AND is Lo AND

is Lo THEN is Hi THEN is Med THEN is Lo THEN is Hi THEN is Med THEN is Lo THEN

is Hi AND is Hi AND is Hi AND is Med AND is Med AND is Med AND is Lo AND is Lo AND

is Hi THEN is Med THEN is Lo THEN is Hi THEN is Med THEN is Lo THEN is Hi THEN is Med THEN

is Lo AND

is Lo THEN

AND AND AND AND AND AND AND AND AND

THEN THEN THEN THEN THEN THEN THEN THEN THEN

is is is is is is is is is

is always negative provided that the which can, thus, be mathematically represented by (29)

are satisfied. From (25) and (28), we can conclude that , for all states except at the equilibrium which implies the convergence as

Layer 1: Fuzzy System 1: , (30)

C. Controller Analysis Let us take a further step from Section IV into controller analysis and consider a four-input HCTFS controller of Fig. 7,

(31)


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Fuzzy System 2

Layer 2:

(32)

, , where in Fuzzy System 1, , and in Fuzzy System 2, , , . From (32), we can see that the value of each class will range from attribute (33) which results in the product of them to also be in the same range. By analyzing the defuzzification process in Layer 2, we can is the weighted value of the centers of see that the output membership functions, which implies (34) and, thus, (35) is bounded by Hence, we know that the range of and for which the closed-loop system will be stable if Definition 2.1–2.3 and Theorem 2.1 are satisfied. VI. SIMULATION WITH ACTIVE SUSPENSION In this section, HCTFS with the stability analysis technique introduced in Section V will be applied to the active suspension system adopted from [31]. In the HCTFS controlled active suspension system, the controller employs four inputs, namely, Vertical Car Body Displacement, Vertical Car Body Acceleration, Vertical Wheel Displacement, and Vertical Wheel Acceleration. The major task of the controller is to prevent the active suspension displacement from hitting the physical limits, which are meter. Each of the inputs is defined by five membership functions, thus totaling up to 75 rules from threetwo-inputsubsystems in two layers instead of 625 rules in a single fuzzy system having four inputs. In our simulation, all parameters are taken from real-world values, and are widely used as the reference values [31] in the literature, except only the in Fig. 8, which is an adjustable component of the filter bandwidth representing the hydraulic actuator, allowing us to focus the scope of our work in the HFS domain in depth. The parameter can be adjusted from soft, providing comfort when driving on smooth roads, to stiff, preventing suspension damage when driving on rough roads.

Fig. 8. Active suspension.

To minimize the complexity, we shall consider the case in which the regulated variable (36) being the car body displacement with is realized, where being a filtered version of the wheel displacement (37) Now, let us expand on the effects of positive-valued variable “ ” [31]. For small value of , (37) serves as a low-pass filter for which, at very low frequencies and in steady state, the regulated becomes almost identical to the suspension travel variable . For input containing high-frequency components, (37) tends to approach zero, making equal to , resulting in rejection of those high-frequency portions. Hence, for small value of , the suspension may be considered as operating in a soft setting. As the value of increases, more high-frequency components of the road input are allowed to pass through the filter (37). as the high bandwidth Thus, is approximately equal to . For large value of , the suspension may filter renders be considered as operating in a stiffened setting, which trades off a significant amount of passenger comfort with a reduction in the amount of its rattle-space use. The transfer functions of and are as follows:

(38) Thus, from these transfer functions, we first determine the state models in the Second Canonical form, which ensure controllability, and then convert them to controllable Jordan statespace model using MATLAB, which are found to be as follows:

(39)

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Fig. 9.


Comparison of suspension displacement from ideal, passive, and active suspensions.

The state matrices , , , and have to be presented in such a form containing functions of as their elements because of the extensively long content in each function of , which results from the conversion to Jordan model with the existence of variable when attempting to find the state matrices using MATLAB. However, this problem can then be solved after substituting a real value of into each element of the state matrices, thus returning the numerical value of the state model. In our simulation, we apply a road input, which is in a form of a bump of 5-cm height otherwise

(40)

to show the functioning of the HCTFS controller as well as the stability condition of the matrix . Our value ranges from 1.5 to 10, indicating soft to stiffened setting, respectively. Fig. 9 demonstrates the suspension travel results obtained from ideal (solid line), which is physically impossible to obtain, passive (dotted line), and active (with varying , represented by -line) suspensions. The resulting suspension travel is shown in Fig. 10, whereas the varying values are shown in Fig. 11. The simulation results of the ideal, passive, and active (using Filter Bandwidth to represent hydraulic actuator) suspensions (Fig. 9) demonstrate that none of the systems hit the hard physical limits, and the response obtained from the active suspension comes to rest much faster than the passive one, but it does not approach the ideal suspension result, which in anyway can never be obtained by any kind of suspensions [31].

Fig. 10.

Suspension displacement with varying .

Nevertheless, there are some benefits and tradeoffs using the proposed active suspension as follows. Please note that the authors employed the same range of in our simulation experiment as that used in [31] in order to preserve the originality and to prevent the loss of generality. Benefit: With appropriate choice of “ ,” the active suspension design is superior not only to the passive one, but also to the ideal one in some frequency ranges [31].


Fig. 11.

Values of .

Fig. 12.

Suspension displacement at

57

Fig. 13.

= 1:5.

Tradeoff: With small , the active suspension design reduces both car body displacement and acceleration compared with the passive suspension, but increases the suspension travel. On the other hand, if is increased the suspension travel can be reduced but then the car body displacement and acceleration are increased [31]. Consequently, according to the benefit and tradeoff mentioned, it can be seen in the time-domain representation that the active suspension response is inferior to the other two at some time intervals, and, more obviously, outperforms the passive suspension with faster settling time than the passive suspension, and approximately 1 s after the ideal suspension. From our results, the HCTFS controller design proposed in the paper has satisfactorily shown its potential to be practically operational. However, in the case that HCTFS is to be put into a hardware application, it could be further researched and then implemented in that field with the superior performance set as an achievable goal. Figs. 12–14 illustrate the suspension displacement of the active suspension when is set at a value of, 1.5, 5.0, and 10.0, respectively.

Suspension displacement at = 5:0.

Fig. 14. Suspension displacement at = 10:0.

Fig. 15.

Value of varying pole () (real).

Resulting from the varying values of , the three system poles , , and of are also varied. The varying pole , which can be viewed in Fig. 15, is found to be a negative real value whereas the other two poles are found to be complex

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Fig. 16.



Fig. 19.

Value of varying pole () (imaginary).

VII. CONCLUSIONS

Fig. 17.

Value of varying pole () (imaginary).

Fig. 18.


conjugate numbers with negative real parts. The varying real and are shown in Figs. 16 and 17, respecimaginary parts of can be viewed in Figs. 18 and 19, tively, while those of respectively.

In our computational complexity experiments, we conducted the analysis on the assumption that all computation be accomplished serially, indicating a minimal estimate on the amount of electronic components needed to construct the computational circuit. From the results obtained, it can be seen that the CFS obviously requires significantly larger number of computational steps than that required in the HCTFS while the HCTFS needs some small number of extra computational steps, which are not needed in the CFS. This may as well be an indication that the HCTFS would take less processing time in computing final output as it has been reported [1] that the more mathematically complex systems could eminently increase the simulation and CPU time. This proposal was further supported by [2]–[4] that the circuit with more components would require more simulation and CPU time comparing to the more simplified one. The need for the less computational complex system was further emphasized that the difference between the simulation time and hardware-experiment time was observed in [2] and [5]–[7], among which the results obtained from hardware experiment conducted in [6] yielded faster response compared to those obtained from simulation. This may suggest that the performance difference between the CFS and HCTFS in the hardware experiment could also be anticipated. Furthermore, if testing on hardware, this performance difference between the CFS and HCTFS could be quite obvious, according to the proposal in [6]. Nonetheless, it has been reported in [3] and [32] that different software used in simulating the same system could result in different outcomes and simulation times. Leonardi and Raciti [7] agreed that there were some factors not available to be included in simulation as well as the software that caused some difference in the simulation results. The difference in CPU times required for simulating the same system on different techniques was observed in [33] and [34], which Bataineh and Özgüner [33] further suggested that the difference would even be larger if the circuit size increased. Lee et al. [35] reported that the CPU speeds of the controller would definitely effect to the controller performance, thus, the overall system performance. In general, since most of the simulations are performed on either personal


Fig. 20.

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Computational complexity of HCTFS and CFS.

computer or workstation, in which internal resources are shared and so is the CPU power, the results obtained from the controller, which is a more dedicated resource to the system, could be better. Therefore, it could be summarized from the work of [3], [7], and [32]–[35] that the HCTFS results obtained from software simulation could differ as they are also dependent upon which simulation software was used, whether or not all factors were included in the simulation in order to produce the most accurate outcome, and which technique was used to implement the system in the simulation. It has also been found that CFS may require more memory space to store those intermediate results obtained from each of the rules before the last process to compute the final output. On the other hand, the HCTFS may require more electronic parts to construct a comparator circuit, which is not needed in the CFS. However, the total amount of electronic parts needed in the HCTFS can be reduced if one could manage to share the subtractor circuit in the divider circuit. To summarize the computational complexity of the HCTFS and CFS, the table of Fig. 20 is provided. It is to note that “Case 1” refers to the occurrence whereas “Case 2” indicates the ocin which currence in which the numbers of rules classified by each class attribute are not equal. In our active suspension simulation, the HCTFS controller design was aimed and focused on providing a smooth ride with the shortest settling time while eliminating the unwanted ripple occurring in passive suspension response, shown in Fig. 9. The results have satisfactorily shown that none of the systems hit meter. As mentioned in Benefit and the physical limits of Tradeoff in the previous section, there were some time intervals in which active suspension performance was inferior to the other two, and some other time intervals in which active suspension excelled the passive suspension. It was reported in [31] that the active suspension using Filter Bandwidth as a representation of hydraulic actuator could outperform the ideal and passive sus-

pensions at some frequency ranges, when analyzing in the frequency domain. From the simulation results of Figs. 15–19, we can see that all the system poles are lying well into the negative real-part region, implying the system being stable, as wanders in its norm-bounded range. This agrees with our proposal that the controlled system be stable if the output range of the stabilizing controller in (35) satisfies Definition 2–4 and Theorem 1. Practically speaking, in this paper, the acquisition of stability condition depends upon the matrix being Metzler or not. To conclude this paper, it should be reemphasized that the stabilizing HCTFS controller will be of great advantage if the plant to be controlled has some adjustable component, and the plant’s state model and operating range of the adjustable components be known. Hence, our stability analysis proposal may be readdressed as follows. 1) Obtain the state model of the system or plant to be controlled including the adjustable component. 2) If the state model is not in any of the Canonical forms, then convert it into the Second Canonical form, which guarantees controllability, allowing all poles to be movable. 3) Convert it into Jordan Canonical form. After the Jordan Canonical form state model is obtained the designer will be able to determine the value range of the adjustable parts that forces all components on the diagonal of matrix A’s to remain negative at all times. 4) By doing so, this will agree with Definition 2–4 and Theorem 1 in the paper, which automatically guarantees the system stability, provided that the adjustable values all remain in the range pre-identified in Step 3), which makes all of the components on the matrix diagonal negative at all times. 5) Having obtained the range, the designer may simply move on with the HCTFS controller design.

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6) The tips of the design are as follows. a) The controller will only yield the values which are the centers of the membership functions defined on the fuzzy output. b) A greater number of fuzzy output adjectives will cause the controller output value to be finer. c) It is also dependent on the designer’s experience, knowledge, and performance in both the system and rule-base building which definitely results in how well the controller and the controlled system will be performing. Due to the simplistic design concepts of the HCTFS incorporating the noble advantage of the CTFS, which eliminates the need for unnecessary and repetitive defuzzification between layers, thus preventing the valuable fuzzy information from being lost during the process, and stability analysis methodology proposed, the system designer only requires some minimal amount of effort to implement an HCTFS controller that yields an output which perfectly matches the system requirements, allowing the controlled system to operate in optimal operating condition. ACKNOWLEDGMENT The authors would like to express their appreciation and gratitude to E. D. Lindsay for his effort in proofreading the manuscript, and the Reviewers for their useful comments. REFERENCES [1] A. G. M. Strollo, “SPICE modeling of power PiN diode using asymptotic waveform evaluation,” in Proc. IEEE PESC’96, Baveno, Italy, June 1996, pp. 44–49. [2] G. Masetti, S. Graffi, and D. Golzio, “New macromodels and measurements for the analysis of EMI effects in 741 op-amp circuits,” IEEE Trans. Electromagn. Compat., vol. 33, pp. 25–34, Feb. 1991. [3] N. A. Crestani, D. Deschacht, M. Robert, and D. Auvergne, “Post-layout timing simulation of CMOS circuits,” IEEE Trans. Computer-Aided Design, vol. 12, pp. 1170–1177, Aug. 1993. [4] J.-F Huang and C.-W. Kuo, “A technique reducing simulation time in method of moment computation,” in Proc. Int. Conf. Microwave and Millimeter-Wave Technology, Beijing, China, Aug. 1998, pp. 929–932. [5] H.-Ch. Skudelny, O. Apeldoorn, and L. Schülting, “Modeling a GTO with PSPICE,” in Conf. Rec. IEEE-IAS Annu. Meeting, Houston, TX, Oct. 1992, pp. 1074–1081. [6] A. G. M. Strollo, “A new SPICE model of power P-I-N diode based on asymptotic waveform evaluation,” IEEE Trans. Power Electron., vol. 12, pp. 12–20, Jan. 1996. [7] F. Frisina, R. Letor, S. Musumeci, C. Leonardi, and A. Raciti, “A new PSPICE power MOSFET model with temperature dependent parameters: Evaluation of performances and comparison with available models,” in Conf. Rec. IEEE-IAS Annu. Meeting, New Orleans, LA, Oct. 1997, pp. 1174–1181. [8] J. Zhou, G. V. S. Raju, and R. A. Kisner, “Hierarchical fuzzy control,” Int. J. Control, vol. 54, no. 5, pp. 1201–1216, 1991. [9] F. Cheong and R. Lai, “Designing a hierarchical fuzzy logic controller using differential evolution,” in Proc. 1999 IEEE Int. Fuzzy Systems Conf., Seoul, Korea, Aug. 1999, pp. I277–I282. [10] M. G. Joo and J. S. Lee, “Hierarchical fuzzy control scheme using structured Takagi–Sugeno type fuzzy inference,” in Proc. 1999 IEEE Int. Fuzzy Systems Conf., Seoul, Korea, Aug. 1999, pp. I78–I83. [11] J. Stufflebeam and R. Prasad, “Hierarchical fuzzy control,” in Proc. 1999 IEEE Int. Fuzzy Systems Conf., Seoul, Korea, Aug. 1999, pp. I78–I83. [12] R. R. Yager, “On hierarchical structure for fuzzy modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. 23, pp. 1189–1197, July/August 1993. [13] L. X. Wang, “Analysis and design of hierarchical fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 617–624, Oct. 1999.

[14] R. B. Saad and S. K. Halgamuge, “Dynamic hierarchical fuzzy systems: Analysis and design using a state-space model,” in Proc. 19th Int. Conf. NAFIPS, Atlanta, GA, July 2000, pp. 373–377. [15] G. Feng, S. G. Cao, and N. W. Reese, “Analysis and design of fuzzy control systems using dynamic fuzzy-state space models,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 192–199, Apr. 1999. [16] W. Rattasiri and S. K. Halgamuge, “Computational complexity of hierarchical fuzzy systems,” in Proc. 19th Int. Conf. NAFIPS, Atlanta, GA, July 2000, pp. 383–387. [17] L. X. Wang, A Course in Fuzzy Systems and Control. Englewood Cliffs, NJ: Prentice-Hall, 1996. [18] S. K. Halgamuge, “Self-evolving neural networks for rule-based data processing,” IEEE Trans. Signal Processing, vol. 45, pp. 2766–2773, Nov. 1997. [19] J. R. Nowicki and L. J. Adam, Digital Circuits. London, U.K.: E. Arnold, 1990. [20] P. M. Chirlian, Analysis and Design of Electronic Circuits. New York: McGraw-Hill, 1965. [21] D. J. Comer, Introduction to Semiconductor Circuit Design. Reading, MA: Addison-Wesley, 1968. [22] M. H. Jones, A Practical Introduction to Electronic Circuits. Cambridge, U.K.: Cambridge Univ. Press, 1995. [23] A. R. Hambley, Electronics: A Top-Down Approach to Computer-Aided Circuit Design. Englewood Cliffs, NJ: Prentice-Hall, 1994. [24] H. Taub and D. Schilling, Digital Integrated Electronics. New York, NJ: McGraw-Hill, 1977. [25] T. L. M. Bartelt, Digital Electronics. Englewood Cliffs, NJ: PrenticeHall, 1991. [26] B. Holdsworth, Digital Logic Design. Oxford, U.K.: Butterworth-Heinemann, 1993. [27] B. C. Kuo, Automatic Control Systems. Englewood Cliffs, NJ: Prentice-Hall, 1991. [28] P. P. Khargonekar, I. R. Petersen, and K. Zhou, “Robust stabilization control of uncertain linear subsystems: Quadratic stabilization and theory,” IEEE Trans. Automat. Contr., vol. 35, pp. 356–361, Mar. 1990. ˇ [29] D. D. Siljak, Large-Scale Dynamics Systems. Amsterdam, The Netherlands: Elsevier, 1978. [30] H. Chapellat, S. P. Bhattacharyya, and L. H. Keel, Robust Control: The Parametric Approach. Englewood Cliffs, NJ: Prentice-Hall, 1995. [31] J. S. Lin and I. Kanellakopoulos, “Nonlinear design of active suspensions,” IEEE Contr. Syst. Mag., vol. 17, pp. 45–49, June 1997. [32] K.-P. Dyck and H. Grabinski, “LISIM—A simulator for time domain simulation of lossy transmission line systems in a nonlinear circuit environment,” in Proc. CompEuro’89, Hamburg, Germany, May 1989, pp. 5/82–5/85. [33] A. Bataineh and F. Özgüner, “Parallel-and-vector implementation of the event-driven logic simulation algorithm on the Cray Y-MP supercomputer,” in Proc. Supercomputing’92, Minneapolis, MN, Nov. 1992, pp. 444–452. [34] M. Bariani, I. Barbieri, and M. Raggio, “A VLIW architecture simulator innovative approach for HW-SW co-design,” in Proc. Int. Multimedia Expo, New York, Aug. 2000, pp. 1375–1378. [35] H.-K. Sung, J.-B. Lee, T.-B. Im, and Y.-O. Kim, “A low cost speed control system of brushless DC motor using fuzzy logic,” in Proc. Information, Decision, and Control Conf., Adelaide, Australia, Feb. 1999, pp. 433–437.

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Waratt Rattasiri was born in Chiangmai, Thailand, in 1973. He received the B.Eng. degree in electronics with a specialization in telecommunications from Assumption University, Bangkok, Thailand, in 1995, and the M.S. degree in electrical engineering with a specialization in control systems from Wichita State University, Wichita, KS, in 1997. He is currently working toward the Ph.D. degree in the Mechatronics Research Group, Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Melbourne, Australia After his graduation in 1995, he was with Seagate Technology (Thailand), Co. Ltd. as a Production Engineer. From June 1998 to March 1999, he was a full-time Lecturer in the Department of Electrical Engineering, Dhurakijpundit University, Bangkok. His research fields of interests are state-space control theory, hierarchical fuzzy systems, neuro-fuzzy systems, and genetic algorithms. Mr. Rattasiri is a recipient of the Melbourne International Research Scholarship Award. He was the organizer and chair of the Special Session on Soft Computing Applications in Mechatronic Systems at the 2002 International Conference on Fuzzy Systems and Knowledge Discovery (FSKD’02).


Saman K. Halgamuge (M’85) received the B.Sc. degree in electronic and telecommunication engineering from the University of Moratuwa, Moratuwa, Sri Lanka, in 1985, and the Dipl.-Ing. and Dr.-Ing. degrees in computer engineering from Darmstadt University (TUD), Darmstadt, Germany, in 1990 and 1995, respectively. In 1985, he was an Engineer with the Ceylon Electricity Board, Sri Lanka. From 1990 to 1995, he was a Research Associate at TUD. After lecturing in computer systems engineering and being associated with the Institute for Telecommunications Research of the University of South Australia, from 1996 to July 1997, he joined the Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Melbourne, Australia, as a Senior Lecturer in mechatronics. He is presently an Associate Professor and Reader in the same department. He has coauthored over 120 research papers and contributed to books in the areas of data mining and analysis, mechatronics, soft computing, and bioinformatics. His research interests also includes vehicle engineering, communication networks, and bio-inspired computing. His research papers have been cited by over 115 journal papers during the last ten years. Dr.-Ing. Halgamuge is an Executive Committee Member of the Asia Pacific Neural Network Assembly and co-chaired and chaired the Programme Committees of ICONIP’99 and FSKD’02, respectively.

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