Sensitivity-Based Hierarchical Controller Architectures for ... - CiteSeerX

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008

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Sensitivity-Based Hierarchical Controller Architectures for Active Suspension Waratt Rattasiri, Nalin Wickramarachchi, and Saman K. Halgamuge, Member, IEEE

Abstract—In this brief, a sensitivity analysis of hierarchical fuzzy system (HFS) is conducted, allowing a sensitivity order of controller inputs to be established and used in a hierarchical fuzzy system inputs placement process. The frequency response analysis method is used to analyze the sensitivity of the controller output with respect to perturbation. This sensitivity knowledge thus constitutes a platform on which three different designs of HFS are investigated. An automotive active suspension system is chosen as an application for the HFS models for performance verification purposes. The simulation results are provided and discussed in this brief. Index Terms—Active suspension, frequency response analysis, hierarchical fuzzy system (HFS), sensitivity analysis, stability.

I. INTRODUCTION INCE THE emergence of fuzzy control theory, it has become one of the areas that are increasingly and extensively researched by researchers around the world. One of its great advantages has been its applicability, which allows human experts to apply their knowledge through human’s natural language in designing fuzzy systems as well as operating them with ease. Consequently, most applications of fuzzy systems have a low number of inputs and, hence, low dimensions and complexity. A large number of inputs cause the effect, widely known as “the curse of dimensionality,” in which the number of rules inside the complete fuzzy rule base grows exponentially as the number of input increases as in

S

(1) represents the number of rules, the number of input where membership functions, and the number of inputs. Many researchers have addressed the problem involving system performance degradation resulting from greater computational complexity and expressed their concerns through a number of research works. In [1], Strollo reported that the more mathematically complex circuits could suffer from a remarkable increase in simulation and CPU time. This was further supported by [2]–[4] that the circuits with more components would require more simulation and CPU time compared to the more simplified ones. Practical comparisons were made between hardware and simulation time in [3] and [5]–[7]. Among these practical experiments, the results from real hardware obtained in [7] reflected a more responsive output response comparing to those obtained Manuscript received July 5, 2005. Manuscript received in final form January 12, 2007. Recommended by Associate Editor C.-Y. Su. W. Rattasiri and S. K. Halgamuge are with the Dynamic Systems Control, Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Victoria 3150, Australia (e-mail: [email protected]; saman@ unimelb.edu.au). N. Wickramarachchi is with Department of Electrical Engineering, University of Moratuwa, Moratuwa, Katubadda 10400, Sri Lanka (e-mail: wick@elect. mrt.ac.lk). Digital Object Identifier 10.1109/TCST.2007.899739

from simulation. It was further suggested in [7] that the greater CPU time difference could be expected, should the circuit size be increased. From the proposed research in [7], a fuzzy system with a large rule base will severely be affected since its rule base naturally utilizes a great number of numerical computation processes. According to the analytical work proposed in [8], the size of an extremely large rule base could slow down the overall performance of the fuzzy controller as well as complicate the fuzzy logic controller design. From these grounds, fuzzy system designers should be well informed of the possible difficulties in case they are coping with a large rule-based fuzzy system. If possible, a less computationally complex system would technically be more preferable to the more complicated one in order to make the system more responsive in producing an output as well as to provide the ease of controller design. To reduce the rule base size, the concept of forming a hierarchically structured set of subsystems having only a few inputs connected was introduced. These fuzzy subsystems form a “hierarchical fuzzy system (HFS)” and offer a capability to suppress the effects of “the curse of dimensionality” as they only allow the fuzzy rule base size to expand linearly as the number of inputs increases, as described in (2) From (2), the HFS rule base will have a less number of rules compared to that of a single-layer fuzzy system which has the same number of inputs, which results in less computational complexity as well as better system performance. According to [7], the difference in system performance between single-layer fuzzy system and HFS could be more obvious when testing on hardware. Many researchers have proposed different types of HFS to address these computational complexity concerns. One of the early works in HFS was proposed by Zhou et al. [9]. In their proposed HFS, the first two fuzzy inputs are connected to the first subsystem. The second subsystem takes the output from the first subsystem as well as the third fuzzy input. This procedure repeats until the last fuzzy input is connected to the last subsystem. This type of HFS, shown in Fig. 1, is as desirable as the rule base size is kept at its minimum and can be described by (2). A type of HFS using the differential evolution technique was reported in [10]. Another type of HFS, called structured Takagi–Sugeno-type fuzzy logic unit (SFLU), was proposed in [11], in which the outputs of the previous layer are not used in the IF-parts but in the THEN-parts of the fuzzy rules. This is 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 [12]. An HFS using a hierarchical prioritized structure (HPS) technique, in which the firing level of a rule is determined upon the certainty qualification is discussed in [13]. Evolutionary algorithms were also

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the plant output sensitivity with respect to the plant inputs perturbation as well as the HFS controller output sensitivity with respect to the controller input perturbation. Three HFS designs will be tested and simulated, and their results will be provided and discussed in Section IV. Section II provides a discussion on the frequency response analysis based on Bode plots conducted on the active suspension system. Section III offers a discussion on the designs of hierarchical fuzzy system with minimized rule base (HFSwMRB) and alternative hierarchical fuzzy system (AHFS) [8] based on sensitivity information obtained in Section II. The three HFS controller models, being HFSwMRB with the most sensitive input at the bottom system; HFSwMRB with the least sensitive input at the bottom system; and AHFS, are tested with the automotive active suspension, for which the simulation results are provided in this section. Section IV provides the discussions on the simulation results as well as concluding comments on the HFS designs. II. FREQUENCY RESPONSE ANALYSIS

Fig. 1. Raju et al.’s hierarchical fuzzy system.

used in finding the optimized HFS pattern and parameters in automated generation of HFS [14], [15]. There has been much work involving HFS implementation techniques while there are very few studies contributed to the discussion on the manual design of HFS. This essentially involves the very first fundamental steps starting from the input placement process in subsystems, which could severely result in the degradation of HFS performance, if improperly performed. Wang [16] proposed a method in determining the ranking of fuzzy inputs based on their sensitivity quality with respect to the HFS output. The mean-value theorem was used as a means to analyze the change in HFS output , when each of the inputs experiences a perturbation. Based on the HFS structure in Fig. 1, Wang [16] further suggested that the most imporis most sensitive, be placed at , tant fuzzy input, to which whereas the least important one be placed at . However, the paper only emphasized the positions of fuzzy inputs but did not include an example to visualize the effects of different fuzzy system arrangements. Apart from HFS controller, it has not been taken into consideration that the plant, or the system to be controlled, itself can also exhibit sensitivity characteristics with respect to perturbation at the plant inputs. Although external to the HFS controller, this information should also be considered in the design process. To address this gap, this brief analyzes

The frequency response analysis methods widely used in classical control theory are Nyquist stability critereon [17] and Bode plots. The proposal on Bode plots was made in 1938, using the magnitude and phase frequency response plots of a complex function to investigate closed-loop stability using the notions of gain and phase margin. Reference [18] was later published in 1940. These frequency response analysis techniques are advantageous because they allow the transfer function to be computed and modeled using the physical data experimentally measured or the characteristics of the system to be analyzed using the plots if the system model is unavailable. However, in this brief, the Bode plot technique will be used to analyze the sensitivity of the system to be controlled, and therefore, will only be emphasized in the sequel. A. Sensitivity Analysis Using Frequency Response Technique This section provides the basis on which the sensitivity analysis using Bode plots is conducted. The transfer function of the system, which implies the system characteristics as a function of frequency, is presented in the following: (3) where is the transfer function, the magnitude of , and the phase angle of . The transfer function can be plotted as a function of frequency whose the magnitude and phase plots are graphed separately. To analyze its characteristics in frequency domain, the transfer function of the format in (4) is required in order to construct the magnitude Bode plot which is essentially the plot , where dB of magnitude curve in decibel (dB) versus . This magnitude plot further implies the amplification factor on the system input which will subsequently yield different system output characteristics if operated through different system transfer functions. In other words, the magnitude


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From the schematic diagram in Fig. 2, four sensors are used to read the car body displacement and acceleration as well as the wheel displacement and acceleration. These signals are sent to the HFS controller which in turn yields the control signal to the hydraulic actuator, adjusting it accordingly to the rule base design in the HFS. To obtain thorough knowledge about the characteristics of the plant prior to HFS controller design, system outputs supplied to the HFS controller should be included in the analysis to determine their sensitivity order which is subsequently used in the HFS inputs placement process. To simplify the analysis, the hydraulic actuator functionality is represented by and we consider the following regulated variable: (5) represents the suspension travel to be controlled, where the car body displacement of , and being a filtered version of the wheel displacement of . The filtered version of wheel displacement can be calculated as follows: Fig. 2. HFS-controlled active suspension system.

(6)

plot can be used to determine the sensitivity levels of the system outputs with respect to the system input disturbance since (4) represents the system output, where the system input. transfer function, and

the system

B. Sensitivity Analysis of Active Suspension System To conduct the sensitivity analysis on the active suspension system [19] designated as the plant to be controlled by the HFS controller, the plant is analyzed for its output sensitivity with respect to the system perturbation. The quarter-car active suspension system of interest is shown in Fig. 2 [19]. An adjustable hydraulic actuator is attached to the suspension, enabling the suspension setting to vary from soft, to provide comfort when driving on a smooth road, to stiffened, to prevent the suspension from possible damages when driving on a rough road surface. Active suspension parameters and values used are listed in the following: 290 kg; • quarter car body mass, • wheel mass, 59 kg; 16812 N/m; • spring coefficient, 1000 N/(m/s); • damping coefficient, 190 000 N/m; • Tyre’s spring coefficient, • ; the natural frequency of unsprung mass • ; ; • car body displacement, • wheel displacement, ; • road disturbance, ; and • control force yielded from hydraulic actuator .

where takes a positive real value greater than 0. For small value of , the term multiplying serves as a low-pass filter. At very low frequencies and in steady state, the regulated variable becomes almost identical to the suspension travel , indicating the suspension system operating in a soft mode. For tends to be an input containing high-frequency components, to approach , resulting in a reapproaching zero, forcing jection of those high-frequency portions. As the value of increases, more high-frequency components of the road input are allowed to pass through to . Thus, is approximately equal with the high-frequency components appearing in the to response since the high bandwidth filter is rendering . For large value of , the suspension may be considered as operating in a stiffened mode which sacrifices a significant amount of passenger comfort with a reduction in the amount of suspension’s rattle-space use. In our simulations, the active suspension is given an input . To preserve the origiperturbation for the time nality of the simulation model and to demonstrate the difference in controller response, the input perturbation being a 5-cm speed bump is chosen. The speed bump can be mathematically modeled by (7) otherwise

(7)

which operates at the frequency of 4 Hz. The active suspension and are given in (8) as follows: transfer functions of

(8)

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Fig. 3. Frequency response of the active suspension at = 1:5.


Hence, the frequency response analysis is performed on the following four plant outputs: car body displacement ; car body acceleration ; wheel displacement ; and wheel acceleration . To preserve the originality of the active suspension model, the range of has been chosen and verified to be in the stable region [20]. The values indicate soft to stiffened settings, respectively. Therefore, based on linear system theory, our analysis will be focussed on the operating frequency of 4 Hz. The frequency response plots of the active suspension when 1.5, 5.75, and 10.0 are shown in Figs. 3–5, operating at respectively. ” (dashed line), “ ” Please also note that the lines “ (dashed-dotted line), “ ” (thick line), and “ ” (dotted line), represent the responses of car body displacement, car body acceleration, wheel displacement, and wheel acceleration, respectively. , it can be seen that, from the When operating at frequency of approximately 1 rad/s or 0.1592 Hz, the output response gains can be put in the following order from the highest to the lowest: wheel acceleration, car body acceleration, wheel


displacement, and car body displacement, respectively. Particularly at the input frequency of 4 Hz which is approximately 25.1327 rad/s, the response gains of the signals are as follows: wheel acceleration at 57.9208 dB, car body acceleration at 33.4224 dB, wheel displacement at 1.9113 dB, and car body , the order of sensidisplacement at 22.5871 dB. At tivity is as follows: wheel acceleration at 58.1186 dB, car body acceleration at 45.0856 dB, wheel displacement at 2.1090 dB, and car body displacement at 10.9240 dB. When the active , from the frequency of suspension is operating at about 1 rad/s, the car body and wheel acceleration responses are very close to each other. This also applies to the car body and wheel displacement responses. This exhibits the characteristics of stiffened suspension setting that, at low frequencies, the magnitudes of the wheel and car body displacement, as well as the wheel and car body acceleration are very close to each other. Their magnitudes begin to differ when the frequency is approximately 4 rad/s or 0.6366 Hz and above. However, at the input frequency of 4 Hz or 25.1327 rad/s, the response gains order as follows: wheel acceleration at 58.5170 dB, car body acceleration at 49.8740 dB, wheel displacement at 2.5075 dB, and car body displacement at 6.1356 dB. From the results obtained, they correspond to our expectation because, by nature, accelerations should exhibit their inherent characteristics of high sensitivity due to movement, thus resulting in higher response gains in the Bode plot than the displacement responses. Furthermore, in the lower value range which implies a soft setting, when , wheel acceleration has the highest gain magnitude because the wheel is allowed to make more of movement at faster rates, absorbing all the forces exerted to the car, resulting in the fast wheel movements, hence higher wheel acceleration response gain. The second highest gain is that of the car body acceleration. This is because the car body can be expected to move at a slower rate than that of the wheel because all the forces to exert on the car body have already been absorbed by the wheel moments. The gains of displacement responses, by nature, are lower than those of accelerations. Moreover, the displacement response results obtained correspond to the acceleration response results that the wheel


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Fig. 6. AHFS controller.

displacement reflects a higher gain than the car body displacement because of larger magnitudes of wheel displacement resulted from the soft suspension setting. , which is the most When the suspension is operating at stiffened setting, similar characteristics of responses still generally apply that the order of gains from the highest to the lowest is wheel acceleration, car body acceleration, wheel displacement, and car body displacement, respectively. However, the increase in response gains on the four outputs when operating at and at are as follows: 0.5962 dB on wheel acceleration, 16.4516 dB on car body acceleration, 0.5962 dB on wheel displacement, and 16.4515 dB on car body displacement. These increments in response gains may suggest that when the active suspension has changed its operation from the softest to the most stiffened mode, the responses that seem to have been affected the most are car body acceleration and car body displacement. , the suspension is said to be This is because, when at the most stiffened, hence less wheel movement is allowed, and, therefore, more of the forces acting on the car are passed onto the car body, resulting in faster, larger car body movement, hence larger magnitudes of its response gains. III. HIERARCHICAL FUZZY SYSTEM DESIGNS AND SIMULATIONS Since we have obtained the information on the sensitivity order of the controller inputs and the membership functions, we will investigate further on the system performance on three different designs of HFS controllers. These three controller designs are the AHFS controller [8], [20] of Fig. 6 which supports the duality feature for closely related inputs; the HFSwMRB with the most sensitive input at the bottom subsystem of Fig. 7 which utilizes the HFS structure of Fig. 1 with the most sensitive input in place of , the second most sensitive in place of , until the least sensitive one in place of ; and the HFSwMRB with the least sensitive one at the bottom subsystem, in which the input placement order is reversed, shown in Fig. 8. In our controller designs, each of the three controllers will have four inputs and each input is defined by five fuzzy membership functions which were designed using expert knowledge and fine-tuned using trial-and-error method, shown in Fig. 9(a)–(d).

Fig. 7. HFSwMRB controller with the most sensitive input at the bottom subsystem.

Fig. 8. HFSwMRB controller with the least sensitive input at the bottom subsystem.

Therefore, since each input is defined by five membership functions which are to be used to make up all rules in full combination, there are 25 rules in each of the subsystems and hence 75 rules in total. The hierarchical fuzzy subsystems will utilize classifier-type fuzzy system (CTFS) [21]–[23], which are to produce the intermediate outputs classified into five classes, output class 1–5. At the top subsystem in hierarchy, the final controller yielding , will be discretized into five levels, susoutput pension setting 1–5, at which the value is equal to 1.5, 3.625, 5.75, 7.875, and 10.0, respectively. The three controller designs share the common design objectives which are to minimize the suspension travel, to minimize the suspension settling time, and to maintain the car body displacement level to provide for passenger comfort. Their computational engines are based on fuzzy

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Fig. 9. Membership functions of the four inputs. (a) The membership function plots of car body displacement, x . (b) The membership functions of the four inputs. The membership function plots of car body acceleration x . (c) The membership functions of the four inputs. The membership function plots of wheel . displacement x . (d) The membership functions of the four inputs. The membership function plots of wheel acceleration x TABLE I NUMERICAL DATA OF THE PERFORMANCE OF ACTIVE SUSPENSION USING HFS CONTROLLER

rule base, product inference engine, and center average defuzzifier. The complete rule bases of the three controller designs are presented in Table II. The HFS-controlled active suspension system schematic diagram can be viewed in Fig. 2. The value, the car body displacement, and the suspension travel plots of the active suspension based on each type of the controllers are presented in Figs. 10–18. The numerical results from the experiment on the suspension system equipped with three different HFS controller designs are also provided in Table I.

From the value plots of Figs. 10, 13, and 16, the characteristics of the generated from the AHFS controller seems to be more idle than that of the other two. The produced from the HFSwMRB with the most sensitive input at the bottom subsystem controller seems to be more responsive with respect to the change in the road conditions and the suspension travel, while the characteristics of are less flexible in the HFSwMRB with the least sensitive input at the bottom subsystem. It should be noted that in the current designs of HFS controllers, the initial and the steady-state values of are


Fig. 10. value generated from the AHFS controller.

Fig. 11. Car body displacement obtained from the active suspension using the AHFS controller.

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Fig. 13. value generated from the HFSwMRB controller with the most sensitive input at the bottom subsystem.

Fig. 14. Car body displacement obtained from the active suspension using the HFSwMRB controller with the most sensitive input at the bottom subsystem.

Fig. 12. Suspension travel obtained from the active suspension using the AHFS controller. Fig. 15. Suspension travel obtained from the active suspension using the HFSwMRB controller with the most sensitive input at the bottom subsystem.

largely governed by the rule base which computes and outputs the values accordingly. From the car body displacement plots of Figs. 11, 14, and 17, it can be seen that the controller that offered the least car

body displacement was the HFSwMRB with the least sensitive input at the bottom subsystem at 2.23 cm, the HFSwMRB with

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Fig. 16. value generated from the HFSwMRB controller with the least sensitive input at the bottom subsystem.

Fig. 17. Car body displacement obtained from the active suspension using the HFSwMRB controller with the least sensitive input at the bottom subsystem.

the most sensitive input at the bottom subsystem at 2.48 cm, and the AHFS at 2.81 cm. However, in terms of the suspension travel and the suspension settling time, the best performing controller was the HFSwMRB with the most sensitive input at the bottom subsystem at 6.34 cm and settling in 1.1 s, followed by HFSwMRB with the least sensitive input at the bottom subsystem at 6.46 cm in 1.82 s, and the AHFS at 6.86 cm in 2.41 s. The suspension travel plots are shown in Figs. 12, 15, and 18, and the numerical results are presented in Table I. Please note ” (dashed line), that “ ” (dotted line), “ ” (thick line), and “ represent the result from the ideal, the active, and the passive suspension systems, respectively. The results obtained from the three controllers may suggest that the performance of the HFSwMRB with the most sensitive input at the bottom subsystem seems to be the most satisfactory with respect to the three design goals. It offered the smallest total suspension travel as well as the shortest suspension settling time. However, in terms of the maximum car body displacement,

Fig. 18. Suspension travel obtained from the active suspension using the HFSwMRB controller with the least sensitive input at the bottom subsystem.

it produced a slightly more than that of the HFSwMRB with the least sensitive input at the bottom subsystem controller. When analyzing the generated from the controller, it can be seen in Fig. 13 that the controller adjusted its output vigorously to the change in road conditions as well as to satisfy the design goals: when the suspension travel becomes negative, which implies , the is increased to prevent the wheel from hitting the car body and its travel limits; when the suspension travel , the is reduced to prois positive, which indicates vide the passenger comfort and smooth car body movement. As a result, the controller produced the smoothest car body movement when compared to that of the other two controllers, as can be seen in Fig. 14. An explanation behind its satisfactory performance would be that the most sensitive controller input is placed into the furthest subsystem from the HFS controller output and the signal has to pass three rule bases which were designed to achieve the same common goals: to minimize the suspension travel, and to minimize suspension settling time, and to provide passenger comfort. The information flow from the most to the least sensitive controller input is vital because it fundamentally describes the characteristics of the plant and is effectively used when building the rule bases. The next controller which has met one of the three design goals is the HFSwMRB with the least sensitive input at the bottom subsystem. Although it offered the minimum car body displacement, the produced seems to have lost its flexibility in dealing with the suspension travel, which may imply a more stiffened setting than the other two controllers, hence, the least car body displacement. This could have been resulted from the information flow being reversed and the most sensitive input being placed in the highest subsystem in hierarchy that makes the final decision on the value. The last controller, the AHFS, seems to produce the least responsive controller output, hence, was outperformed by the previous two controllers. While it was trying to catch up with the change in suspension travel, due to the inert reaction, it was not able to meet the design goals. However, the AHFS controller may offer a design advantage that the output from each of the rules as well as the outputs of the subsystems are more intuitive


TABLE II COMPLETE RULE BASES OF THE AHFS, THE HFSWMRB WITH THE MOST SENSITIVE INPUT AT THE BOTTOM SUBSYSTEM, AND THE HFSWMRB WITH THE LEAST SENSITIVE INPUT AT THE BOTTOM SUBSYSTEM CONTROLLERS

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plemented throughout the rule base system, and that the output from the subsystem having more sensitive input pair is given the same level of priority as the output from the subsystem having less sensitive input pair, which may explain why the suspension travel is less satisfactory than the other two. It should note that, in Figs. 12, 15, and 18, the plots may virtually look similar. This was because all three controllers were sharing the common design goals, hence, attempting to achieve their assigned tasks. However, the difference in characteristics lies in their control output , whose plots are shown in Figs. 10, 13, and 16, and the numerical data presented in Table II. IV. CONCLUSION From the previous discussion, it could be concluded that in HFS design, the order of HFS inputs plays a quite important role in achieving the HFS design objectives at the output. However, a rule base design that takes into account the structure of HFS is also very important that, if poorly designed, the essential information from the plant may not be fully utilized through the rule base system, which eventually results in poor HFS controller performance as well as an under performing system. Although none of the three structures could achieve all the three design goals at the same time, the HFSwMRB with the most sensitive input at the bottom subsystem controller seems to yield the best compromise among the three hierarchical structures tested. APPENDIX In Table II, the following denote: • CBD car body displacement, CBA car body acceleration, WD wheel displacement, and WA wheel acceleration; . • “MF ” membership function , where REFERENCES

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