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Ride Comfort-Road Holding Trade-off Improvement of Full Vehicle Active Suspension System by Interval Type-2 Fuzzy Control
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
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Ride Comfort-Road Holding Trade-off Improvement of Full Vehicle Active Suspension System by Interval Type-2 Fuzzy Control Meral Özarslan Yatak a, Fatih Sß ahin b,⇑ a b
Department of Electrical-Electronic Engineering, Faculty of Technology, Gazi University, Ankara 06500, Turkey Department of Automotive Engineering, Faculty of Technology, Gazi University, Ankara 06500, Turkey
a r t i c l e
i n f o
Article history: Received 6 July 2020 Revised 22 September 2020 Accepted 20 October 2020 Available online xxxx Keywords: Active suspension system Hybrid fuzzy control system Interval type-2 fuzzy logic controller Ride comfort Road holding
a b s t r a c t This paper presents a hybrid fuzzy controller structure to improve ride comfort-road holding trade-off characteristics of a full vehicle active suspension system. The controller includes two interval type-2 fuzzy logic controllers, optimized for ride comfort and road holding, for each tire. Type C random road profiles according to ISO 8608 are applied to each tire in the model and ride index, road holding, crest factor, and RMS acceleration values are evaluated. Stability of the overall closed loop system is presented by phase plane analysis. The simulation results reveal that simultaneous improvement of ride index and road holding can be made possible with the proposed controller. Ó 2020 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Vehicle suspension system plays an important role for ride comfort and road holding. Ride comfort is achieved by minimizing transmitted acceleration through tire to vehicle body. Suspension systems ensure establishing contact between the tires and the road for road holding while providing ride comfort by isolating roadinduced vibration. It is apparent that increasing number of studies on the improvement of suspension system performance will lead to enhancement of ride comfort and road holding by using computational intelligence methods [5,12,41]. Vehicle suspension systems can be classified according to the controlled component. Passive suspension system, which does not contain a controllable component, comprises a constant spring and a fixed damping coefficient shock absorber. Passive suspension system limits the forces exerted by the road with hydraulic shock absorber [38]. Semi-active suspension system including a variable damping coefficient damper in addition to fixed shock absorber provides a controllable damping ratio [35,37,47]. Besides, it has been extensively accepted that active suspension systems with fewer physical constraints and flexible structure are more efficient in getting better suspension performance than the conventional passive and semi-active ones by the help of linear force actuator ⇑ Corresponding author at: Gazi Üniversitesi Teknoloji Fakültesi Otomotiv Mühendislig˘i Bölümü, Emniyet Mahallesi, Bandırma Caddesi, No: 6/19 06560 Yenimahalle, Ankara, Türkiye E-mail address: [email protected] (F. Sßahin).
[4,17,19,36]. This actuator regulates the force between vehicle body and tire axle. The actuator force is determined by a control method to overcome road-induced vibrations. Various control algorithms have been used to control the actuator including H1 control [9], LQR control [27], adaptive robust control [40], fuzzy sliding-mode control [22,46], disturbance observer-based adaptive tracking control [33], artificial intelligence control techniques [7,39], and nonlinear output feedback finite-time control [34]. In addition, application of a bioinspired dynamics based adaptive control to suspension systems is available in the literature [32]. Active suspension system control can be evaluated as a specific control problem, and therefore the studies are extending into the different control methods. Type-1 fuzzy logic controller (T1 FLC) is one of the common control methods for complex nonlinear systems. Nonlinear dynamics of suspension systems caused by dynamic behavior of shock absorber, actuator, suspension spring, etc. are included in the control process with fuzzy rules [8,16,30]. Wen et al. [42] proposed a fuzzy controller for active suspension system via dynamic slidingmode method. Montazeri-Gh and Soleymani [25] improved ride comfort with fuzzy logic controller by tuning the parameters with genetic algorithm based on human sensitivity. A self-organizing fuzzy controller for determining suitable membership function (MF) and fuzzy rules was proposed by Lin and Lian [21]. Nagarkar et al. [28] used genetic algorithm-based optimization technique for tuning PID parameters and FLC MFs’ range and scaling factors. The mentioned studies have shown that T1 FLC has various parameters to tune. Consequently, determination of type, number, limits of
https://doi.org/10.1016/j.jestch.2020.10.006 2215-0986/Ó 2020 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Meral Özarslan Yatak and F. Sß ahin, Ride Comfort-Road Holding Trade-off Improvement of Full Vehicle Active Suspension System by Interval Type-2 Fuzzy Control, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.10.006
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
and four tires, and angular displacements of vehicle COG form 7DOF of the model. Full vehicle suspension system parameters are given in Table 1 [3]. Full vehicle suspension model is developed with motion equations by Newton’s Law. Equation of motion for each unsprung and sprung masses are given as:
MFs and formation of rule base cause controller sourced uncertainties in the system. Besides, T1 FLC cannot handle uncertainties caused by different meanings of the words, various comments from experts, noisy measurements and noisy data [49]. This problem has revealed new control algorithms to include these uncertainties in control process. Uncertainties affect the system control performance adversely. Systematic design approach of the controller, modeling errors, and the uncertain behaviors of the system components are the system sourced uncertainties and should be included in control process. High-level uncertainties cannot be handled completely with T1 FLC. To overcome these weaknesses, type-2 fuzzy set used in type-2 fuzzy control system was proposed by Zadeh [48]. Type-2 fuzzy logic controller (T2 FLC) has been preferred for applications with uncertainties and nonlinear characteristics such as active suspension systems by researchers in recent years because of the ability to handle data ambiguousness and imprecision [1,23,49]. Interval type-2 fuzzy set, a kind of type-2 fuzzy sets, can handle linguistic uncertainties [20] and interval type-2 fuzzy logic controller (IT2 FLC) has been mostly preferred for real time control problems due to the simplicity and low computational cost [2,4,20,26,29]. Uncertain properties of suspension system, uncertainties caused by road vibration effects, and control method can be incorporated with IT2 FLC. The main objective of suspension system control is to increase ride comfort and road holding concurrently. Ride comfort and road holding are conflicting goals for suspension system design and exhibit a trade-off behavior. Improving ride comfort inescapably declines road holding and vice versa [18]. Therefore, it is necessary to design different controller structures for ride comfort and road holding. Ride comfort can be improved by low frequency control while road holding needs high frequency control. These requirements can be achieved with a hybrid controller structure. In this study, a hybrid IT2 FLC for full vehicle active suspension system is proposed. The motivation of this study is to overcome the conflict between ride comfort and road holding. Improvement of one of them causes the deterioration of the other. The main contribution and novelty of this paper is the proposal of hybrid fuzzy control system based on IT2 FLC that improves both ride comfort and road holding simultaneously. The existing studies related to active suspension system control are realized by single controller. Unlike existing studies, this study presents a hybrid control structure for active suspension system. Evaluation of the proposed controller in terms of ride comfort and road holding is carried out by applying Type-C road profile according to the ISO 8608 [13]. Ride comfort is determined by using ride index which includes bounce, roll and pitch accelerations of the vehicle center of gravity (COG). On the other hand, road holding is assessed by root mean square of dynamic tire loads (RMS-DTL). Performance of the proposed IT2 FLC is compared to T1 FLC and optimized passive suspension system. Stability of the closed loop system including proposed controller structure is conducted within the phase plane analysis. Simulation results reveal that IT2 FLC outperforms in improvement of trade-off between ride comfort and road holding. Consequently, the proposed hybrid controller structure has the ability of better control of active suspension system.
_ _ mi zi ¼ kui ðzri zi Þ ki ðzi zbi Þ ci zi zbi F i 4 _ _ P mb zb ¼ ki ðzi zbi Þ þ ci zi zbi þ F i ; ði ¼ 1; 2; 3; 4Þ i¼1 2 3 _ _ 6 k1 ðz1 zb1 Þ þ c1 z1 zb1 þ F 1 þ k2 ðz2 zb2 Þ 7 6 7 Jh ¼ 6 7lf _ _ 4 5 þc2 z2 zb2 þ F 2 3 2 _ _ 6 k3 ðz3 zb3 Þ þ c3 z3 zb3 þ F 3 þ k4 ðz4 zb4 Þ 7 7 6 6 7 lr _ _ 5 4 þc4 z4 zb4 þ F 4 _ _ I / ¼ k1 ðz1 zb1 Þ þ c1 z1 zb1 þ F 1 t fl _ _ k2 ðz2 zb2 Þ þ c2 z2 zb2 þ F 2 t fr _ _ þ k3 ðz3 zb3 Þ þ c3 z3 zb3 þ F 3 trl _ _ k4 ðz4 zb4 Þ þ c4 z4 zb4 þ F 4 trr
ð1Þ
Vertical displacements of body connection points for each suspension are calculated by using body vertical displacement, roll and pitch angular displacements as:
zb1 ¼ zb þ lf h þ t fl / zb2 ¼ zb þ lf h t fr /
ð2Þ
zb3 ¼ zb lr h þ t rl / zb4 ¼ zb lr h trr /
Tire positions, tire velocities, body position, body velocity, roll angle, roll angular velocity, pitch angle, and pitch angular velocity are chosen as state variables of the system. Actuator forces are control inputs. Road inputs for each tire are disturbances while tire positions, body position, roll angle and pitch angle are the outputs. State space model parameters are given in Table 2. The statespace equations can be written as: _
x ¼ Ax þ Bu þ Ed y ¼ Cx þ Du
ð3Þ
where A; B; C; D and E are the coefficient matrices for the state variables, control inputs and disturbances. State vector x, input vector u, and external disturbance vector d are represented as follows: x ¼ ½x1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 ; x8 ; x9 ; x10 ; x11 ; x12 ; x13 ; x14 u ¼ ½F 1 ; F 2 ; F 3 ; F 4 T d ¼ ½d1 ; d2 ; d3 ; d4
T
ð4Þ
T
A; B; C; D and E coefficient matrices are given in Appendix. Ride comfort and road holding are crucial parameters for active suspension control problem and exhibit a trade-off characteristic. The main task in this study is to present an IT2 FLC based hybrid controller structure in order to improve ride comfort and road holding simultaneously. Ride comfort and road holding parameters can be examined with ride index and dynamic tire load respectively. The objective of the control is to minimize ride index and dynamic tire load parameters.
2. System Description and Problem Statement A seven-degrees of freedom (7-DOF) full vehicle active suspension system model is constructed as shown in Fig. 1. The model consists of sprung mass, four suspension assemblies, and tires. Each suspension assembly includes a linear damper, a linear spring and a linear force actuator. Each tire is modeled as an unsprung mass and a linear spring. Vertical displacements of vehicle COG 2
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 1. 7-DOF full vehicle active suspension system model.
Table 2 State space model parameters of active suspension system
Table 1 Active suspension system model parameters. Symbol
Unit
Value
Definition
lf lr tfl tfr trl trr mb m1;2 m3;4 kui k1;2 k3;4 c1;2 c3;4 I J Fi zi zb zri zbi / h
m m m m m m kg kg kg N/m N/m N/m Ns/m Ns/m kgm2 kgm2 N m m m m radian radian
1.011 1.803 0.761 0.761 0.755 0.755 1460 40 35.5 175500 19960 17500 1290 1620 460 2460
Distance between front axle and COG Distance between rear axle and COG Distance between front left tire and roll axis Distance between front right tire and roll axis Distance between rear left tire and roll axis Distance between rear right tire and roll axis Body mass Front tire masses Rear tire masses Tire spring constants Front suspension spring constants Rear suspension spring constants Front suspension damping coefficients Rear suspension damping coefficients Roll inertia Pitch inertia Actuator forces Tire vertical positions Body vertical position Road excitations Suspension and body link point positions Roll angle Pitch angle
State variables Parameter Definition
Parameter
Definition
x1 ¼ z1
x8 ¼ z_ 4
Rear right tire velocity
x9 ¼ zb
Body position
x10 ¼ z_ b
Body velocity
x11 ¼ h
Pitch angle
x12 ¼ h_ x13 ¼ / x14 ¼ /_
Pitch angular velocity
x2 ¼ z_ 1 x3 ¼ z2 x4 ¼ z_ 2 x5 ¼ z3 x6 ¼ z_ 3 x7 ¼ z4
Front left tire position Front left tire velocity Front right tire position Front right tire velocity Rear left tire position
Rear left tire velocity Rear right tire position Control Inputs (actuator forces) Parameter Definition u1 ¼ F 1 Front left actuator force u2 ¼ F 2 Front right actuator force u3 ¼ F 3 Rear left actuator force u4 ¼ F 4 Rear right actuator force Outputs Parameter Definition y1 ¼ z1 Front left tire position Front right tire y2 ¼ z2 position Rear left tire position y3 ¼ z3 y4 ¼ z4 Rear right tire position
3. Design of Proposed Controller Reduction of the effects of road condition on vehicle ride performance by overcoming road-induced vibrations is the main task of the controller. General structure of IT2 FLC, hybrid fuzzy control system, IT2 FLC for ride comfort and road holding are discussed in this section. 3.1. Interval Type-2 Fuzzy Logic Controller Fuzzy logic control is a problem-solving control method for especially complex nonlinear systems. The nonlinearities of the system are modeled with linguistic approach. Fuzzy logic control transforms linguistic variables obtained from expert knowledge to automatic control [39]. This process is one of the sources of
Roll angle Roll angular velocity
External disturbances Parameter Definition Front left tire road d1 ¼ zr1 disturbance Front right tire road d2 ¼ zr2 disturbance Rear left tire road d3 ¼ zr3 disturbance Rear right tire road d4 ¼ zr4 disturbance Parameter y5 ¼ zb
Definition Body position
y6 ¼ h
Pitch angle
y7 ¼ /
Roll angle
uncertainty. Furthermore, the meaning of the words and the consequent in the rules, the measurements and the data used to tune the parameters are the other sources of uncertainties. Type1 fuzzy logic controller is unable to directly handle such uncertainties [24]. However, the uncertainties are handled in type-2 fuzzy set in Footprint of Uncertainty (FOU). When type-1 fuzzy set is blurred by shifting the points on the set either to the left or to 3
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
the right and not necessarily by the same amounts, this type-2 set is called as generalized type-2 fuzzy set. If this blurring process is done by shifting the points by forming a definite interval, the formed type-2 set is called as interval type-2 fuzzy set [24]. These sets are shown in Fig. 2. IT2 fuzzy set is preferred primarily for real-time applications, due to handling higher order uncertainty factors considering simple structure with low computational cost [45]. Block diagram of type-2 FLC that is composed of fuzzifier, rule base, inference engine, type-reducer, and defuzzifier is shown in Fig. 3. Crisp inputs are mapped into type-2 fuzzy sets in Fuzzifier.
Fig. 3. Block diagram of T2 FLC.
h i h i l l F l ðxÞ ¼ f ðxÞ; f l ðxÞ ¼ f ; f l
A type-2 fuzzy set is denoted A and characterized by a MF l ðx0 Þ. In A
this study, Gaussian MFs with uncertain standard deviation are used as interval type-2 MF and a Gaussian primary MF is defined as follows:
This interval determines LMFs and UMFs of the antecedent FOUs and is defined as:
1 2
lA ðxÞ ¼ exp ððx mÞ=rÞ2 ; r 2 ½r1 ; r2
l
h
i
h
i
lA ðxÞ ¼ Nðx; m; r1 Þ exp 12 ððx mÞ=r1 Þ2
0
f ðx Þ ¼ l
ðxl1 Þ F ı1
F2
lF ı ðxl2 Þ
ð8Þ
2
that has two antecedents, in an IT2 Mamdani FLC for singleton fuzzification and minimum t-norm. Type-reduction maps a type-2 fuzzy set into a type-1 fuzzy set. Various iterative methods have been developed to compute switch points L and R, and subsequently two end points yl and yr .The first and the most widely used iterative method is Karnik-Mandel (KM) Method [15]. Enhanced KM (EKM) method [43] was developed because of the lack of KM method on optimization of initialization and stopping. The average number of iterations for the EKM algorithm is smaller than that for the original KM algorithm. To simplify computations for KM and EKM methods, an Iterative Algorithm Stopping Condition (IASC) was improved. The number of elements N is smaller than 100 in most practical typereduction computations and Enhanced Iterative Algorithm with Stopping Condition (EIASC) is about 50% more efficient in terms of computational cost than KM and EKM methods [44]. In this study, EIASC method has been used for road holding and ride comfort controllers. An interval set determined by its two end points yl and yr is formed as type-reduced set in regardless of typereduction method or MF. These points are defined as:
ð6Þ
Rules are formed as IF hAntecedenti THEN hConsequent i. The antecedent is a combination of fuzzy logic expression of fuzzy phrases and the consequent is an output fuzzy set [6]. The structure of the lth interval type-2 Zadeh rule is defined as:
F1
This MF has a fixed mean,m, and an uncertain standard deviation that takes on values in ½r1 ; r2 . The Upper Membership Function (UMF) and Lower Membership Function (LMF) can be expressed respectively as:
f ðx0 Þ ¼ lı ðxl1 Þ lı ðxl2 Þ
ð5Þ
l
lA ðxÞ ¼ Nðx; m; r2 Þ exp 12 ððx mÞ=r2 Þ2
ð7Þ
RlZ : IF x1 is F l1 and:::and xp is F lp
THEN y is Gl ðl ¼ 1; ::; M Þ where x ¼ ðx1 ; :::; xp Þ is the input vector, y are linguistic variables, l F p ðp ¼ 1; 2; :::; pÞ and G ¼ yll ; ylr are interval type-2 fuzzy sets [24]. An IT2 fuzzy system is called as IT2 Mamdani fuzzy system when the rules are IT2 Zadeh rules and a Mamdani implication operator is used [24]. In this study, IT2 Mamdani fuzzy system has been realized. Inference engine combines all the fired rules and maps from interval type-2 input fuzzy antecedent sets to interval type-2 output fuzzy consequent sets [24]. Various antecedents in rules are associated by the minimum or product t-norm. Minimum t-norm operator has been used for this study. Fuzzy inference process is described as visually in Fig. 4. This figure is for a two-antecedents–single consequent rule, singleton fuzzification, and minimum t-norm. The firing strength set of the lth rule is an interval and is defined as: l
PL i i PM i i f yl þ f yl i¼1 i¼Lþ1 yl ¼ P L PM i i i¼1
f þ
i¼Lþ1
i¼1
f þ
i¼Rþ1
f
PR i i PM i i f yr þ f yr i¼1 Pi¼Rþ1 yr ¼ P R M i i
ð9Þ
f
Defuzzification maps type-1 fuzzy set into a crisp number. Defuzzifier of this set is done by averaging yl and yr . Hence, the defuzzified crisp output is defined as
y¼
yl þ yr 2
Fig. 2. Fuzzy sets a) type-1fuzzy set b) general type-2 fuzzy set c) interval type-2 fuzzy set. 4
ð10Þ
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 4. Visualization of input, antecedent, and consequent operations for rule l
comfort and road holding controller outputs multiplied by a weighting coefficient. Input and output fuzzy sets have been classified into reasonable number of fuzzy sets to reflect on their impacts. The input fuzzy sets have been determined as seven MFs, while the output fuzzy sets have been determined as nine in order to achieve desired control. The MFs have been formed initially by examining the simulation results of passive suspension system. Afterwards, tuning of the controller parameters have been performed by analyzing of repetitive simulation results. The rule bases have been formed with expert knowledge. Ride comfort controller and road holding controller in hybrid fuzzy control system have been optimized individually. Optimization has been realized by setting the weighting coefficient to 1 for ride comfort and to 0 for road holding. For the comparison of designed IT2 FLC, T1 FLC with the same input-output sets and
3.2. Hybrid Fuzzy Control System for Active Suspension Active suspension system reduces effects of road excitation by controlling the relative motion of the vehicle body with an actuator to provide ride comfort and road holding. Ride comfort is related to ride index and requires low frequency control. On the contrary, road holding is related to tire deflection and requires high frequency control. Therefore, increasing ride comfort and road holding concurrently with single controller structure seems inappropriate. This trade-off characteristic between ride comfort and road holding constitutes the basis of the proposed controller in this study. Thus, a hybrid controller structure has been proposed to overcome this drawback. The proposed hybrid fuzzy control system is illustrated in Fig. 5. The hybrid fuzzy control system contains ride comfort and road holding controllers and the output is obtained by summing ride
Fig. 5. Structure and operating principle of hybrid fuzzy control system for active suspension. 5
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rules have been used. After designing of ride comfort and road holding controllers, the weighting coefficient has been determined empirically.
4. Stability Analysis Lyapunov stability gives information about stability of an equilibrium point. For an equilibrium point, if all possible trajectories start within a definite radius d (d > 0) and stay within a definite radius e (e > d) then the system is stable in the sense of Lyapunov. In addition to these requirements, if the trajectories converge to the equilibrium point then the system is asymptotically stable. Fig.9 shows phase planes for passive suspension system and active suspension system with the proposed controller. The trajectories have been obtained by applying step inputs to the tires in different configurations. Step inputs have been applied to all tires at the same time for Fig. 9a and Fig. 9b. Step inputs have been applied to only front tires and only left tires for Fig. 9c and Fig. 9d respectively. It can be clearly seen from Fig. 9, the trajectories converge to the equilibrium point. Besides, trajectories show that energy dissipation capability of active suspension system with the proposed controller outperforms passive system. The phase planes reveal that the active suspension system is asymptotically stable in the sense of Lyapunov.
3.3. IT2 FLC for Ride Comfort Ride comfort is related to the transmitted vibrations sensed by human and is achieved with ride index parameter. The transmitted acceleration components as amplitude, direction, and frequency are measured in different key points of the vehicle. Ride index is determined by using weighted accelerations according to effect to the occupant comfort and passengers’ body posture [10,25]. Inputs of T1 FLC and IT2 FLC for ride comfort controllers are velocity and acceleration at vehicle body points to which each suspension system connected (zbi ) as shown in Fig. 1. The velocity and accelerations of these points are calculated by using vertical and angular motions of vehicle COG. T1 and IT2 input fuzzy sets have been realized as Gaussian MFs and outputs of controllers are singleton as shown in Fig. 6. 3.4. IT2 FLC for Road Holding Road holding, which is comprised with braking, cornering, and traction abilities during maneuvers, is related to the contact forces between road surface and tires. The dynamic tire load should not exceed the static ones so as to ensure the contact of wheels with the road robustly [11]. Minimizing the relative displacement between tire and road improves road holding ability [18]. T1 FLC and IT2 FLC for road holding controllers have two inputs for each tire: the first input is road error, which is difference between tire position and road input, and the second one is change of road error which includes derivative of the first input. The formed T1 and IT2 fuzzy sets for the inputs are Gaussian MFs and the outputs are singleton for road holding controllers. MFs of road holding controllers are shown in Fig. 7. The control surfaces for ride comfort and road holding controllers are shown in Fig. 8.
5. Simulation Analysis Road input has been modeled to simulate random Class-C type road profile according to the ISO 8608 [13]. Total length of the road profile has been set to 100 m and vehicle speed is 20 m/s. Four different Class-C type random road profiles have been obtained to apply the tires. Road profiles are shown in Fig. 10. Determination of the ride sensation to human is ensured by measuring ride characteristic of a vehicle and relating it to vibration evaluation methods. In order to determine ride comfort, the relative importance of magnitude, frequency, axis and duration are taken into account to weight the motion, and vibration dose is evaluated [10]. RMS value of frequency-weighted acceleration
Fig. 6. Input and output MFs for T1 FLC and IT2 FLC for ride comfort controllers. 6
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Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 7. Input and output MFs for T1 FLC and IT2 FLC for road holding controllers.
Fig. 8. Control surfaces: (a) ride comfort controller, (b) road holding controller.
based on ISO 2631-1 quantifies ride comfort [14]. Ride index, which is used for ride comfort evaluation in this study, can be calculated as:
"Z ai ¼
1:12f c
armsw ¼
#1=2 ðW i ai Þ
2
ð12Þ
i
#1=2 Sy ðf Þdf
" X
h i1=2 2 2 2 RI ¼ ðkb armswb Þ þ kp armswp þ ðkr armswr Þ
ð11Þ
0:89f c
7
ð13Þ
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Fig. 9. Phase planes: (a) front left tire position and velocity, (b) vehicle COG vertical position and velocity, (c) vehicle COG pitch angle and pitch angular velocity, (d) vehicle COG roll angle and roll angular velocity.
RC-T1: Active suspension system with T1 FLC for ride comfort RC-IT2: Active suspension system with IT2 FLC for ride comfort RH-T1: Active suspension system with T1 FLC for road holding RH-IT2: Active suspension system with IT2 FLC for road holding H-T1: Active suspension system with hybrid T1 FLC H-IT2: Active suspension system with hybrid IT2 FLC (Proposed) Performance results of ride comfort controller, road holding controller, and hybrid controller are given in Table 3. The table gives comparisons among the aforementioned systems using ride index, crest factor (CF), RMS-DTL, and RMS acceleration values. The percentages of change compared to passive suspension system are also listed in parenthesis. Improvement or deterioration of the parameters is shown with up arrow or down arrow respectively. Smaller values for ride index, CF, RMS-DTL, and RMS acceleration mean an improvement in related parameter, while larger values state deterioration. According to the results of ride comfort controllers, it can be clearly seen that considerable improvement in ride comfort can be provided when road holding is not taken into account. According to Table 3, RC-T1 and RC-IT2 have improved ride index by 39.14% and 39.76% comparing to P. RC-IT2 has slightly better performance. Dynamic tire loads have been deteriorated by both controllers with an increase of 25-34% for all tires. RMS accelerations of vehicle COG vertical, pitch and roll motions have been reduced significantly by both controllers. On the other hand, vehicle COG vertical acceleration CF has increased slightly while CFs of pitch and roll accelerations have shown a big increase. These increases can be explained with the reduction of RMS accelerations. Results of road holding controllers have been given in terms of dynamic tire loads. RH-IT2 has better performance for tires 1 and 3 with 22.28% and 23.32% reduction respectively, while for tires 2 and 4 RH-T1 has shown better performance with 23.33% and
Fig. 10. Random road profiles for each tire.
where ai is RMS acceleration of ith one third octave band, Sy ðf Þ is the power spectral density of acceleration, armsw is weighted RMS acceleration, W i is frequency weighting coefficient of ith one third octave band, kb , kp and kr are multiplying factors. kb , kp and kr have been chosen as 1, 0.4 and 0.63 respectively. 6. Simulation Results Performance of the proposed hybrid fuzzy control system has been evaluated by comparing results of the following suspension systems. P: Passive suspension system. 8
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Table 3 Results of passive and active suspension systems. Parameter
P
RC-T1
RC-IT2
RH-T1
RH-IT2
H-T1
H-IT2
Ride Index
0.3981
0.2423 ("39.14%)
0.2398 ("39.76%)
0.4711 (;18.34%)
0.4683 (;17.63%)
0.3243 ("18.54%)
COG Vertical Acceleration CF COG Pitch Acceleration CF
3.0068 3.2506
3.0214 (;0.49%) 6.1921 (;90.49%)
3.1172 (;3.67%) 6.0715 (;86.78%)
3.0076 (;0.03%) 3.2109 ("1.22%)
3.016 (;0.31%) 3.1029 ("4.54%)
COG Roll Acceleration CF RMS-DTL1
3.3082 539.229
5.9333 (;79.35%) 681.038 (;26.3%)
2.992 ("9.56%) 420.445 ("22.03%)
RMS-DTL2
562.0445
RMS-DTL3
499.3457
RMS-DTL4
514.9311
707.3671 (;25.86%) 654.8275 (;31.14%) 688.9964 (;33.8%)
COG RMS Vertical Acceleration COG RMS Pitch Acceleration COG RMS Roll Acceleration
0.3337
0.2387 ("28.47%)
5.9061 (;78.53%) 684.4425 (;26.93%) 710.0954 (;26.34%) 654.8259 (;31.14%) 688.5267 (;33.71%) 0.2362 ("29.22%)
430.9118 ("23.33%) 383.9089 ("23.12%) 394.1369 ("23.46%) 0.4119 (;23.43%)
3.0012 ("9.28%) 419.0784 ("22.28%) 432.4891 ("23.05%) 382.9015 ("23.32%) 395.2678 ("23.24%) 0.4097 (;22.77%)
0.3263 ("18.04%) 3.2794 (;9.07%) 3.8413 (;18.17%) 3.1059 ("6.12%) 499.631 ("7.34%) 519.703 ("7.53%) 477.654 ("4.34%) 486.458 ("5.53%) 0.31("5%)
0.1488 0.3316
0.0334 ("77.55%) 0.0628 ("81.06%)
0.0334 ("77.55%) 0.0623 ("81.21%)
0.1446 ("2.82%) 0.3511 (;5.88%)
0.1477 ("0.74%) 0.3475 (;4.79%)
0.054("63.1%) 0.117("64.48%)
0.0555 ("62.7%) 0.1179 ("64.45%)
3.2446 (;7.91%) 3.8797 (;19.35%) 3.0738 ("7.09%) 499.1901 ("7.43%) 519.0671 ("7.65%) 475.5626 ("4.76%) 487.3152 ("5.36%) 0.3149 ("5.63%)
23.46% reduction respectively. On the other hand, road holding improvement has caused to decline of ride index with a rate of 18.34% and 17.63% for RH-T1 and RH-IT2 respectively. COG RMS vertical acceleration has been increased approximately 23% by both controllers. COG RMS pitch acceleration has been reduced with a rate of 2.82% and 0.74% while COG RMS roll acceleration has been increased with a rate of 5.88% and 4.79% by RH-T1 and RH-IT2. COG vertical acceleration CF has remained approximately same. COG pitch and roll CFs have been decreased by road holding controllers. According to the results of hybrid controllers, H-T1 and H-IT2 have improved ride index with a rate of 18.04% and 18.54% respectively. In addition, RMS-DTLs have been improved by 4.76%-7.65% with H-IT2. H-T1 has shown better performance than H-IT2 for Tire 4. COG vertical acceleration CF has been increased by both controllers with rates of 9.07% and 7.91%. COG roll acceleration CF has been reduced by 6.12% and 7.09% while COG pitch acceleration CF has been increased by 18.17% and 19.35%. This increase is caused by significant reduction of COG RMS pitch acceleration. Force outputs of RC-IT2, RH-IT2, and H-IT2 are given in Fig. 11. The force requirements for ride comfort and road holding can be seen from the figure. RC-IT2 output force has relatively low frequency and amplitude change while RH-IT2 output has high frequency and large amplitude change. H-IT2 merges these controller outputs and its’ output can be seen from the figure.
Fig. 12 shows ISO 2631-1 frequency weighted vehicle COG vertical accelerations and their power spectral densities (PSD) for P, HT1 and H-IT2. According to the graphs, H-T1 and H-IT2 have slightly better performance comparing to P in view of COG vertical acceleration especially for frequencies below 10 Hz. P has slightly better performance comparing to H-T1 and H-IT2 for higher frequencies. However, H-IT2 has improved COG RMS vertical acceleration by 5.63% while H-T1 has improved by 5%.
Fig. 11. Controller output forces of RC-IT2, RH-IT2, and H-IT2 for Tire 1.
Fig. 12. Vehicle COG vertical accelerations and PSD curves for P, H-T1, and H-IT2. 9
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 13 shows ISO 2631-1 frequency weighted vehicle COG pitch accelerations and their PSDs for P, H-T1 and H-IT2. Vehicle COG RMS pitch acceleration has been improved by 62.7% and 63.1% for H-IT2 and H-T1 respectively. H-T1 has provided slightly better performance than H-IT2. On the other hand, H-T1 and H-IT2 have shown lower performance comparing to passive system for higher frequencies. According to frequency weighted roll acceleration graphs and their PSD curves in Fig. 14, hybrid controllers have given the best results for roll acceleration performance. H-IT2 has improved COG RMS roll acceleration by 64.45% while H-T1 has improved by 64.48%. H-IT2 and H-T1 have provided worthwhile reduction of vehicle COG roll acceleration. When performance results of the proposed hybrid fuzzy control system are compared to the studies of Nagarkar et al. [28] and Nieto et al. [31], it can be seen that the proposed system gives better results. In addition to these, altough Cao et al. [4] achieves better ride index value, road holding was not taken into account. Thus, the simulation results confirm the advantages of the proposed control structure.
7. Conclusion This paper presents a hybrid controller structure based on IT2 FLC to improve ride comfort and road holding of an active suspension system simultaneously. A full vehicle model has been constructed to simulate performance of the proposed controller structure. The proposed IT2 FLC hybrid controller performance results have been compared to T1 FLC hybrid controller and passive system. The conflict between ride comfort and road holding,
Fig. 14. Vehicle COG roll accelerations and PSD curves for P, H-T1, and H-IT2.
improving one of them causes the deterioration of the other, is the inspiration point of this study. For the purpose of joining different controller requirements of road holding and ride comfort, a hybrid structure has been developed. The proposed hybrid controller consists of a ride comfort controller and a road holding controller. These controllers have different inputs and operate independently from each other. Output of the hybrid controller is obtained by summing ride comfort and road holding controller outputs weighted with a weighting coefficient which has a range of 0-1. Increasing the value of the weighting coefficient to 1 results in improvement of ride comfort while lowering to 0 results in improvement of road holding. The weighting coefficient has been chosen as 0.5 by evaluating the simulation results. Ride index results, RMS-DTLs, the frequency weighted acceleration curves and their PSD curves obtained from the simulation results have shown that the proposed controller provides more improvements in ride comfort and road holding performance in comparison with the T1 FLC hybrid controller (H-T1), and passive system (P). The proposed controller has reduced ride index by 18.54% and dynamic tire loads by 7.43%, 7.65%, 4.76% and 5.36% for each tire respectively. These improvements can be explained with the ability of modeling uncertainties by IT2 FLC. On the other hand, the overall performance scores of IT2 FLC and T1 FLC hybrid controllers is not too different from each other. It is considered that this difference will be much more in implementation of the proposed controller in a real system due to addition of more uncertainty sources caused by suspension system components. Another problem to work on, is the weighting coefficient. The weighting coefficient can be determined by system dynamics. Furthermore, it can
Fig. 13. Vehicle COG pitch accelerations and PSD curves for P, H-T1, and H-IT2. 10
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
D¼0
be determined by an additional controller. It can be concluded that the proposed IT2 FLC hybrid controller scheme provides an effective control concept for active suspension systems to improve ride comfort and road holding performances simultaneously.
2
0 6 60 6 E¼6 6 60 4 0
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
ku1 m1
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Appendix State-space matrices of full vehicle active suspension system model
2
0 6 6 61 6 6 60 6 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 6 6 6 6 6 6 60 6 6 A¼6 6 6 60 6 6 6 6 6 6 60 6 6 6 6 6 6 6 60 6 6 6 6 6 6 60 6 6 6 6 4 0 2
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k 1 lf m1
k2 lf m2
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km4 4lr
c 1 lf m1
c 2 lf m2
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7 7 7 7 7 7 5
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I
Meral Özarslan Yatak and F. Sßahin
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Engineering Science and Technology, an International Journal journal homepage: www.elsevier.com/locate/jestch
Full Length Article
Ride Comfort-Road Holding Trade-off Improvement of Full Vehicle Active Suspension System by Interval Type-2 Fuzzy Control Meral Özarslan Yatak a, Fatih Sß ahin b,⇑ a b
Department of Electrical-Electronic Engineering, Faculty of Technology, Gazi University, Ankara 06500, Turkey Department of Automotive Engineering, Faculty of Technology, Gazi University, Ankara 06500, Turkey
a r t i c l e
i n f o
Article history: Received 6 July 2020 Revised 22 September 2020 Accepted 20 October 2020 Available online xxxx Keywords: Active suspension system Hybrid fuzzy control system Interval type-2 fuzzy logic controller Ride comfort Road holding
a b s t r a c t This paper presents a hybrid fuzzy controller structure to improve ride comfort-road holding trade-off characteristics of a full vehicle active suspension system. The controller includes two interval type-2 fuzzy logic controllers, optimized for ride comfort and road holding, for each tire. Type C random road profiles according to ISO 8608 are applied to each tire in the model and ride index, road holding, crest factor, and RMS acceleration values are evaluated. Stability of the overall closed loop system is presented by phase plane analysis. The simulation results reveal that simultaneous improvement of ride index and road holding can be made possible with the proposed controller. Ó 2020 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Vehicle suspension system plays an important role for ride comfort and road holding. Ride comfort is achieved by minimizing transmitted acceleration through tire to vehicle body. Suspension systems ensure establishing contact between the tires and the road for road holding while providing ride comfort by isolating roadinduced vibration. It is apparent that increasing number of studies on the improvement of suspension system performance will lead to enhancement of ride comfort and road holding by using computational intelligence methods [5,12,41]. Vehicle suspension systems can be classified according to the controlled component. Passive suspension system, which does not contain a controllable component, comprises a constant spring and a fixed damping coefficient shock absorber. Passive suspension system limits the forces exerted by the road with hydraulic shock absorber [38]. Semi-active suspension system including a variable damping coefficient damper in addition to fixed shock absorber provides a controllable damping ratio [35,37,47]. Besides, it has been extensively accepted that active suspension systems with fewer physical constraints and flexible structure are more efficient in getting better suspension performance than the conventional passive and semi-active ones by the help of linear force actuator ⇑ Corresponding author at: Gazi Üniversitesi Teknoloji Fakültesi Otomotiv Mühendislig˘i Bölümü, Emniyet Mahallesi, Bandırma Caddesi, No: 6/19 06560 Yenimahalle, Ankara, Türkiye E-mail address: [email protected] (F. Sßahin).
[4,17,19,36]. This actuator regulates the force between vehicle body and tire axle. The actuator force is determined by a control method to overcome road-induced vibrations. Various control algorithms have been used to control the actuator including H1 control [9], LQR control [27], adaptive robust control [40], fuzzy sliding-mode control [22,46], disturbance observer-based adaptive tracking control [33], artificial intelligence control techniques [7,39], and nonlinear output feedback finite-time control [34]. In addition, application of a bioinspired dynamics based adaptive control to suspension systems is available in the literature [32]. Active suspension system control can be evaluated as a specific control problem, and therefore the studies are extending into the different control methods. Type-1 fuzzy logic controller (T1 FLC) is one of the common control methods for complex nonlinear systems. Nonlinear dynamics of suspension systems caused by dynamic behavior of shock absorber, actuator, suspension spring, etc. are included in the control process with fuzzy rules [8,16,30]. Wen et al. [42] proposed a fuzzy controller for active suspension system via dynamic slidingmode method. Montazeri-Gh and Soleymani [25] improved ride comfort with fuzzy logic controller by tuning the parameters with genetic algorithm based on human sensitivity. A self-organizing fuzzy controller for determining suitable membership function (MF) and fuzzy rules was proposed by Lin and Lian [21]. Nagarkar et al. [28] used genetic algorithm-based optimization technique for tuning PID parameters and FLC MFs’ range and scaling factors. The mentioned studies have shown that T1 FLC has various parameters to tune. Consequently, determination of type, number, limits of
https://doi.org/10.1016/j.jestch.2020.10.006 2215-0986/Ó 2020 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Meral Özarslan Yatak and F. Sß ahin, Ride Comfort-Road Holding Trade-off Improvement of Full Vehicle Active Suspension System by Interval Type-2 Fuzzy Control, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2020.10.006
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
and four tires, and angular displacements of vehicle COG form 7DOF of the model. Full vehicle suspension system parameters are given in Table 1 [3]. Full vehicle suspension model is developed with motion equations by Newton’s Law. Equation of motion for each unsprung and sprung masses are given as:
MFs and formation of rule base cause controller sourced uncertainties in the system. Besides, T1 FLC cannot handle uncertainties caused by different meanings of the words, various comments from experts, noisy measurements and noisy data [49]. This problem has revealed new control algorithms to include these uncertainties in control process. Uncertainties affect the system control performance adversely. Systematic design approach of the controller, modeling errors, and the uncertain behaviors of the system components are the system sourced uncertainties and should be included in control process. High-level uncertainties cannot be handled completely with T1 FLC. To overcome these weaknesses, type-2 fuzzy set used in type-2 fuzzy control system was proposed by Zadeh [48]. Type-2 fuzzy logic controller (T2 FLC) has been preferred for applications with uncertainties and nonlinear characteristics such as active suspension systems by researchers in recent years because of the ability to handle data ambiguousness and imprecision [1,23,49]. Interval type-2 fuzzy set, a kind of type-2 fuzzy sets, can handle linguistic uncertainties [20] and interval type-2 fuzzy logic controller (IT2 FLC) has been mostly preferred for real time control problems due to the simplicity and low computational cost [2,4,20,26,29]. Uncertain properties of suspension system, uncertainties caused by road vibration effects, and control method can be incorporated with IT2 FLC. The main objective of suspension system control is to increase ride comfort and road holding concurrently. Ride comfort and road holding are conflicting goals for suspension system design and exhibit a trade-off behavior. Improving ride comfort inescapably declines road holding and vice versa [18]. Therefore, it is necessary to design different controller structures for ride comfort and road holding. Ride comfort can be improved by low frequency control while road holding needs high frequency control. These requirements can be achieved with a hybrid controller structure. In this study, a hybrid IT2 FLC for full vehicle active suspension system is proposed. The motivation of this study is to overcome the conflict between ride comfort and road holding. Improvement of one of them causes the deterioration of the other. The main contribution and novelty of this paper is the proposal of hybrid fuzzy control system based on IT2 FLC that improves both ride comfort and road holding simultaneously. The existing studies related to active suspension system control are realized by single controller. Unlike existing studies, this study presents a hybrid control structure for active suspension system. Evaluation of the proposed controller in terms of ride comfort and road holding is carried out by applying Type-C road profile according to the ISO 8608 [13]. Ride comfort is determined by using ride index which includes bounce, roll and pitch accelerations of the vehicle center of gravity (COG). On the other hand, road holding is assessed by root mean square of dynamic tire loads (RMS-DTL). Performance of the proposed IT2 FLC is compared to T1 FLC and optimized passive suspension system. Stability of the closed loop system including proposed controller structure is conducted within the phase plane analysis. Simulation results reveal that IT2 FLC outperforms in improvement of trade-off between ride comfort and road holding. Consequently, the proposed hybrid controller structure has the ability of better control of active suspension system.
_ _ mi zi ¼ kui ðzri zi Þ ki ðzi zbi Þ ci zi zbi F i 4 _ _ P mb zb ¼ ki ðzi zbi Þ þ ci zi zbi þ F i ; ði ¼ 1; 2; 3; 4Þ i¼1 2 3 _ _ 6 k1 ðz1 zb1 Þ þ c1 z1 zb1 þ F 1 þ k2 ðz2 zb2 Þ 7 6 7 Jh ¼ 6 7lf _ _ 4 5 þc2 z2 zb2 þ F 2 3 2 _ _ 6 k3 ðz3 zb3 Þ þ c3 z3 zb3 þ F 3 þ k4 ðz4 zb4 Þ 7 7 6 6 7 lr _ _ 5 4 þc4 z4 zb4 þ F 4 _ _ I / ¼ k1 ðz1 zb1 Þ þ c1 z1 zb1 þ F 1 t fl _ _ k2 ðz2 zb2 Þ þ c2 z2 zb2 þ F 2 t fr _ _ þ k3 ðz3 zb3 Þ þ c3 z3 zb3 þ F 3 trl _ _ k4 ðz4 zb4 Þ þ c4 z4 zb4 þ F 4 trr
ð1Þ
Vertical displacements of body connection points for each suspension are calculated by using body vertical displacement, roll and pitch angular displacements as:
zb1 ¼ zb þ lf h þ t fl / zb2 ¼ zb þ lf h t fr /
ð2Þ
zb3 ¼ zb lr h þ t rl / zb4 ¼ zb lr h trr /
Tire positions, tire velocities, body position, body velocity, roll angle, roll angular velocity, pitch angle, and pitch angular velocity are chosen as state variables of the system. Actuator forces are control inputs. Road inputs for each tire are disturbances while tire positions, body position, roll angle and pitch angle are the outputs. State space model parameters are given in Table 2. The statespace equations can be written as: _
x ¼ Ax þ Bu þ Ed y ¼ Cx þ Du
ð3Þ
where A; B; C; D and E are the coefficient matrices for the state variables, control inputs and disturbances. State vector x, input vector u, and external disturbance vector d are represented as follows: x ¼ ½x1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 ; x8 ; x9 ; x10 ; x11 ; x12 ; x13 ; x14 u ¼ ½F 1 ; F 2 ; F 3 ; F 4 T d ¼ ½d1 ; d2 ; d3 ; d4
T
ð4Þ
T
A; B; C; D and E coefficient matrices are given in Appendix. Ride comfort and road holding are crucial parameters for active suspension control problem and exhibit a trade-off characteristic. The main task in this study is to present an IT2 FLC based hybrid controller structure in order to improve ride comfort and road holding simultaneously. Ride comfort and road holding parameters can be examined with ride index and dynamic tire load respectively. The objective of the control is to minimize ride index and dynamic tire load parameters.
2. System Description and Problem Statement A seven-degrees of freedom (7-DOF) full vehicle active suspension system model is constructed as shown in Fig. 1. The model consists of sprung mass, four suspension assemblies, and tires. Each suspension assembly includes a linear damper, a linear spring and a linear force actuator. Each tire is modeled as an unsprung mass and a linear spring. Vertical displacements of vehicle COG 2
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 1. 7-DOF full vehicle active suspension system model.
Table 2 State space model parameters of active suspension system
Table 1 Active suspension system model parameters. Symbol
Unit
Value
Definition
lf lr tfl tfr trl trr mb m1;2 m3;4 kui k1;2 k3;4 c1;2 c3;4 I J Fi zi zb zri zbi / h
m m m m m m kg kg kg N/m N/m N/m Ns/m Ns/m kgm2 kgm2 N m m m m radian radian
1.011 1.803 0.761 0.761 0.755 0.755 1460 40 35.5 175500 19960 17500 1290 1620 460 2460
Distance between front axle and COG Distance between rear axle and COG Distance between front left tire and roll axis Distance between front right tire and roll axis Distance between rear left tire and roll axis Distance between rear right tire and roll axis Body mass Front tire masses Rear tire masses Tire spring constants Front suspension spring constants Rear suspension spring constants Front suspension damping coefficients Rear suspension damping coefficients Roll inertia Pitch inertia Actuator forces Tire vertical positions Body vertical position Road excitations Suspension and body link point positions Roll angle Pitch angle
State variables Parameter Definition
Parameter
Definition
x1 ¼ z1
x8 ¼ z_ 4
Rear right tire velocity
x9 ¼ zb
Body position
x10 ¼ z_ b
Body velocity
x11 ¼ h
Pitch angle
x12 ¼ h_ x13 ¼ / x14 ¼ /_
Pitch angular velocity
x2 ¼ z_ 1 x3 ¼ z2 x4 ¼ z_ 2 x5 ¼ z3 x6 ¼ z_ 3 x7 ¼ z4
Front left tire position Front left tire velocity Front right tire position Front right tire velocity Rear left tire position
Rear left tire velocity Rear right tire position Control Inputs (actuator forces) Parameter Definition u1 ¼ F 1 Front left actuator force u2 ¼ F 2 Front right actuator force u3 ¼ F 3 Rear left actuator force u4 ¼ F 4 Rear right actuator force Outputs Parameter Definition y1 ¼ z1 Front left tire position Front right tire y2 ¼ z2 position Rear left tire position y3 ¼ z3 y4 ¼ z4 Rear right tire position
3. Design of Proposed Controller Reduction of the effects of road condition on vehicle ride performance by overcoming road-induced vibrations is the main task of the controller. General structure of IT2 FLC, hybrid fuzzy control system, IT2 FLC for ride comfort and road holding are discussed in this section. 3.1. Interval Type-2 Fuzzy Logic Controller Fuzzy logic control is a problem-solving control method for especially complex nonlinear systems. The nonlinearities of the system are modeled with linguistic approach. Fuzzy logic control transforms linguistic variables obtained from expert knowledge to automatic control [39]. This process is one of the sources of
Roll angle Roll angular velocity
External disturbances Parameter Definition Front left tire road d1 ¼ zr1 disturbance Front right tire road d2 ¼ zr2 disturbance Rear left tire road d3 ¼ zr3 disturbance Rear right tire road d4 ¼ zr4 disturbance Parameter y5 ¼ zb
Definition Body position
y6 ¼ h
Pitch angle
y7 ¼ /
Roll angle
uncertainty. Furthermore, the meaning of the words and the consequent in the rules, the measurements and the data used to tune the parameters are the other sources of uncertainties. Type1 fuzzy logic controller is unable to directly handle such uncertainties [24]. However, the uncertainties are handled in type-2 fuzzy set in Footprint of Uncertainty (FOU). When type-1 fuzzy set is blurred by shifting the points on the set either to the left or to 3
Meral Özarslan Yatak and F. Sßahin
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the right and not necessarily by the same amounts, this type-2 set is called as generalized type-2 fuzzy set. If this blurring process is done by shifting the points by forming a definite interval, the formed type-2 set is called as interval type-2 fuzzy set [24]. These sets are shown in Fig. 2. IT2 fuzzy set is preferred primarily for real-time applications, due to handling higher order uncertainty factors considering simple structure with low computational cost [45]. Block diagram of type-2 FLC that is composed of fuzzifier, rule base, inference engine, type-reducer, and defuzzifier is shown in Fig. 3. Crisp inputs are mapped into type-2 fuzzy sets in Fuzzifier.
Fig. 3. Block diagram of T2 FLC.
h i h i l l F l ðxÞ ¼ f ðxÞ; f l ðxÞ ¼ f ; f l
A type-2 fuzzy set is denoted A and characterized by a MF l ðx0 Þ. In A
this study, Gaussian MFs with uncertain standard deviation are used as interval type-2 MF and a Gaussian primary MF is defined as follows:
This interval determines LMFs and UMFs of the antecedent FOUs and is defined as:
1 2
lA ðxÞ ¼ exp ððx mÞ=rÞ2 ; r 2 ½r1 ; r2
l
h
i
h
i
lA ðxÞ ¼ Nðx; m; r1 Þ exp 12 ððx mÞ=r1 Þ2
0
f ðx Þ ¼ l
ðxl1 Þ F ı1
F2
lF ı ðxl2 Þ
ð8Þ
2
that has two antecedents, in an IT2 Mamdani FLC for singleton fuzzification and minimum t-norm. Type-reduction maps a type-2 fuzzy set into a type-1 fuzzy set. Various iterative methods have been developed to compute switch points L and R, and subsequently two end points yl and yr .The first and the most widely used iterative method is Karnik-Mandel (KM) Method [15]. Enhanced KM (EKM) method [43] was developed because of the lack of KM method on optimization of initialization and stopping. The average number of iterations for the EKM algorithm is smaller than that for the original KM algorithm. To simplify computations for KM and EKM methods, an Iterative Algorithm Stopping Condition (IASC) was improved. The number of elements N is smaller than 100 in most practical typereduction computations and Enhanced Iterative Algorithm with Stopping Condition (EIASC) is about 50% more efficient in terms of computational cost than KM and EKM methods [44]. In this study, EIASC method has been used for road holding and ride comfort controllers. An interval set determined by its two end points yl and yr is formed as type-reduced set in regardless of typereduction method or MF. These points are defined as:
ð6Þ
Rules are formed as IF hAntecedenti THEN hConsequent i. The antecedent is a combination of fuzzy logic expression of fuzzy phrases and the consequent is an output fuzzy set [6]. The structure of the lth interval type-2 Zadeh rule is defined as:
F1
This MF has a fixed mean,m, and an uncertain standard deviation that takes on values in ½r1 ; r2 . The Upper Membership Function (UMF) and Lower Membership Function (LMF) can be expressed respectively as:
f ðx0 Þ ¼ lı ðxl1 Þ lı ðxl2 Þ
ð5Þ
l
lA ðxÞ ¼ Nðx; m; r2 Þ exp 12 ððx mÞ=r2 Þ2
ð7Þ
RlZ : IF x1 is F l1 and:::and xp is F lp
THEN y is Gl ðl ¼ 1; ::; M Þ where x ¼ ðx1 ; :::; xp Þ is the input vector, y are linguistic variables, l F p ðp ¼ 1; 2; :::; pÞ and G ¼ yll ; ylr are interval type-2 fuzzy sets [24]. An IT2 fuzzy system is called as IT2 Mamdani fuzzy system when the rules are IT2 Zadeh rules and a Mamdani implication operator is used [24]. In this study, IT2 Mamdani fuzzy system has been realized. Inference engine combines all the fired rules and maps from interval type-2 input fuzzy antecedent sets to interval type-2 output fuzzy consequent sets [24]. Various antecedents in rules are associated by the minimum or product t-norm. Minimum t-norm operator has been used for this study. Fuzzy inference process is described as visually in Fig. 4. This figure is for a two-antecedents–single consequent rule, singleton fuzzification, and minimum t-norm. The firing strength set of the lth rule is an interval and is defined as: l
PL i i PM i i f yl þ f yl i¼1 i¼Lþ1 yl ¼ P L PM i i i¼1
f þ
i¼Lþ1
i¼1
f þ
i¼Rþ1
f
PR i i PM i i f yr þ f yr i¼1 Pi¼Rþ1 yr ¼ P R M i i
ð9Þ
f
Defuzzification maps type-1 fuzzy set into a crisp number. Defuzzifier of this set is done by averaging yl and yr . Hence, the defuzzified crisp output is defined as
y¼
yl þ yr 2
Fig. 2. Fuzzy sets a) type-1fuzzy set b) general type-2 fuzzy set c) interval type-2 fuzzy set. 4
ð10Þ
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Fig. 4. Visualization of input, antecedent, and consequent operations for rule l
comfort and road holding controller outputs multiplied by a weighting coefficient. Input and output fuzzy sets have been classified into reasonable number of fuzzy sets to reflect on their impacts. The input fuzzy sets have been determined as seven MFs, while the output fuzzy sets have been determined as nine in order to achieve desired control. The MFs have been formed initially by examining the simulation results of passive suspension system. Afterwards, tuning of the controller parameters have been performed by analyzing of repetitive simulation results. The rule bases have been formed with expert knowledge. Ride comfort controller and road holding controller in hybrid fuzzy control system have been optimized individually. Optimization has been realized by setting the weighting coefficient to 1 for ride comfort and to 0 for road holding. For the comparison of designed IT2 FLC, T1 FLC with the same input-output sets and
3.2. Hybrid Fuzzy Control System for Active Suspension Active suspension system reduces effects of road excitation by controlling the relative motion of the vehicle body with an actuator to provide ride comfort and road holding. Ride comfort is related to ride index and requires low frequency control. On the contrary, road holding is related to tire deflection and requires high frequency control. Therefore, increasing ride comfort and road holding concurrently with single controller structure seems inappropriate. This trade-off characteristic between ride comfort and road holding constitutes the basis of the proposed controller in this study. Thus, a hybrid controller structure has been proposed to overcome this drawback. The proposed hybrid fuzzy control system is illustrated in Fig. 5. The hybrid fuzzy control system contains ride comfort and road holding controllers and the output is obtained by summing ride
Fig. 5. Structure and operating principle of hybrid fuzzy control system for active suspension. 5
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rules have been used. After designing of ride comfort and road holding controllers, the weighting coefficient has been determined empirically.
4. Stability Analysis Lyapunov stability gives information about stability of an equilibrium point. For an equilibrium point, if all possible trajectories start within a definite radius d (d > 0) and stay within a definite radius e (e > d) then the system is stable in the sense of Lyapunov. In addition to these requirements, if the trajectories converge to the equilibrium point then the system is asymptotically stable. Fig.9 shows phase planes for passive suspension system and active suspension system with the proposed controller. The trajectories have been obtained by applying step inputs to the tires in different configurations. Step inputs have been applied to all tires at the same time for Fig. 9a and Fig. 9b. Step inputs have been applied to only front tires and only left tires for Fig. 9c and Fig. 9d respectively. It can be clearly seen from Fig. 9, the trajectories converge to the equilibrium point. Besides, trajectories show that energy dissipation capability of active suspension system with the proposed controller outperforms passive system. The phase planes reveal that the active suspension system is asymptotically stable in the sense of Lyapunov.
3.3. IT2 FLC for Ride Comfort Ride comfort is related to the transmitted vibrations sensed by human and is achieved with ride index parameter. The transmitted acceleration components as amplitude, direction, and frequency are measured in different key points of the vehicle. Ride index is determined by using weighted accelerations according to effect to the occupant comfort and passengers’ body posture [10,25]. Inputs of T1 FLC and IT2 FLC for ride comfort controllers are velocity and acceleration at vehicle body points to which each suspension system connected (zbi ) as shown in Fig. 1. The velocity and accelerations of these points are calculated by using vertical and angular motions of vehicle COG. T1 and IT2 input fuzzy sets have been realized as Gaussian MFs and outputs of controllers are singleton as shown in Fig. 6. 3.4. IT2 FLC for Road Holding Road holding, which is comprised with braking, cornering, and traction abilities during maneuvers, is related to the contact forces between road surface and tires. The dynamic tire load should not exceed the static ones so as to ensure the contact of wheels with the road robustly [11]. Minimizing the relative displacement between tire and road improves road holding ability [18]. T1 FLC and IT2 FLC for road holding controllers have two inputs for each tire: the first input is road error, which is difference between tire position and road input, and the second one is change of road error which includes derivative of the first input. The formed T1 and IT2 fuzzy sets for the inputs are Gaussian MFs and the outputs are singleton for road holding controllers. MFs of road holding controllers are shown in Fig. 7. The control surfaces for ride comfort and road holding controllers are shown in Fig. 8.
5. Simulation Analysis Road input has been modeled to simulate random Class-C type road profile according to the ISO 8608 [13]. Total length of the road profile has been set to 100 m and vehicle speed is 20 m/s. Four different Class-C type random road profiles have been obtained to apply the tires. Road profiles are shown in Fig. 10. Determination of the ride sensation to human is ensured by measuring ride characteristic of a vehicle and relating it to vibration evaluation methods. In order to determine ride comfort, the relative importance of magnitude, frequency, axis and duration are taken into account to weight the motion, and vibration dose is evaluated [10]. RMS value of frequency-weighted acceleration
Fig. 6. Input and output MFs for T1 FLC and IT2 FLC for ride comfort controllers. 6
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Fig. 7. Input and output MFs for T1 FLC and IT2 FLC for road holding controllers.
Fig. 8. Control surfaces: (a) ride comfort controller, (b) road holding controller.
based on ISO 2631-1 quantifies ride comfort [14]. Ride index, which is used for ride comfort evaluation in this study, can be calculated as:
"Z ai ¼
1:12f c
armsw ¼
#1=2 ðW i ai Þ
2
ð12Þ
i
#1=2 Sy ðf Þdf
" X
h i1=2 2 2 2 RI ¼ ðkb armswb Þ þ kp armswp þ ðkr armswr Þ
ð11Þ
0:89f c
7
ð13Þ
Meral Özarslan Yatak and F. Sßahin
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Fig. 9. Phase planes: (a) front left tire position and velocity, (b) vehicle COG vertical position and velocity, (c) vehicle COG pitch angle and pitch angular velocity, (d) vehicle COG roll angle and roll angular velocity.
RC-T1: Active suspension system with T1 FLC for ride comfort RC-IT2: Active suspension system with IT2 FLC for ride comfort RH-T1: Active suspension system with T1 FLC for road holding RH-IT2: Active suspension system with IT2 FLC for road holding H-T1: Active suspension system with hybrid T1 FLC H-IT2: Active suspension system with hybrid IT2 FLC (Proposed) Performance results of ride comfort controller, road holding controller, and hybrid controller are given in Table 3. The table gives comparisons among the aforementioned systems using ride index, crest factor (CF), RMS-DTL, and RMS acceleration values. The percentages of change compared to passive suspension system are also listed in parenthesis. Improvement or deterioration of the parameters is shown with up arrow or down arrow respectively. Smaller values for ride index, CF, RMS-DTL, and RMS acceleration mean an improvement in related parameter, while larger values state deterioration. According to the results of ride comfort controllers, it can be clearly seen that considerable improvement in ride comfort can be provided when road holding is not taken into account. According to Table 3, RC-T1 and RC-IT2 have improved ride index by 39.14% and 39.76% comparing to P. RC-IT2 has slightly better performance. Dynamic tire loads have been deteriorated by both controllers with an increase of 25-34% for all tires. RMS accelerations of vehicle COG vertical, pitch and roll motions have been reduced significantly by both controllers. On the other hand, vehicle COG vertical acceleration CF has increased slightly while CFs of pitch and roll accelerations have shown a big increase. These increases can be explained with the reduction of RMS accelerations. Results of road holding controllers have been given in terms of dynamic tire loads. RH-IT2 has better performance for tires 1 and 3 with 22.28% and 23.32% reduction respectively, while for tires 2 and 4 RH-T1 has shown better performance with 23.33% and
Fig. 10. Random road profiles for each tire.
where ai is RMS acceleration of ith one third octave band, Sy ðf Þ is the power spectral density of acceleration, armsw is weighted RMS acceleration, W i is frequency weighting coefficient of ith one third octave band, kb , kp and kr are multiplying factors. kb , kp and kr have been chosen as 1, 0.4 and 0.63 respectively. 6. Simulation Results Performance of the proposed hybrid fuzzy control system has been evaluated by comparing results of the following suspension systems. P: Passive suspension system. 8
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Table 3 Results of passive and active suspension systems. Parameter
P
RC-T1
RC-IT2
RH-T1
RH-IT2
H-T1
H-IT2
Ride Index
0.3981
0.2423 ("39.14%)
0.2398 ("39.76%)
0.4711 (;18.34%)
0.4683 (;17.63%)
0.3243 ("18.54%)
COG Vertical Acceleration CF COG Pitch Acceleration CF
3.0068 3.2506
3.0214 (;0.49%) 6.1921 (;90.49%)
3.1172 (;3.67%) 6.0715 (;86.78%)
3.0076 (;0.03%) 3.2109 ("1.22%)
3.016 (;0.31%) 3.1029 ("4.54%)
COG Roll Acceleration CF RMS-DTL1
3.3082 539.229
5.9333 (;79.35%) 681.038 (;26.3%)
2.992 ("9.56%) 420.445 ("22.03%)
RMS-DTL2
562.0445
RMS-DTL3
499.3457
RMS-DTL4
514.9311
707.3671 (;25.86%) 654.8275 (;31.14%) 688.9964 (;33.8%)
COG RMS Vertical Acceleration COG RMS Pitch Acceleration COG RMS Roll Acceleration
0.3337
0.2387 ("28.47%)
5.9061 (;78.53%) 684.4425 (;26.93%) 710.0954 (;26.34%) 654.8259 (;31.14%) 688.5267 (;33.71%) 0.2362 ("29.22%)
430.9118 ("23.33%) 383.9089 ("23.12%) 394.1369 ("23.46%) 0.4119 (;23.43%)
3.0012 ("9.28%) 419.0784 ("22.28%) 432.4891 ("23.05%) 382.9015 ("23.32%) 395.2678 ("23.24%) 0.4097 (;22.77%)
0.3263 ("18.04%) 3.2794 (;9.07%) 3.8413 (;18.17%) 3.1059 ("6.12%) 499.631 ("7.34%) 519.703 ("7.53%) 477.654 ("4.34%) 486.458 ("5.53%) 0.31("5%)
0.1488 0.3316
0.0334 ("77.55%) 0.0628 ("81.06%)
0.0334 ("77.55%) 0.0623 ("81.21%)
0.1446 ("2.82%) 0.3511 (;5.88%)
0.1477 ("0.74%) 0.3475 (;4.79%)
0.054("63.1%) 0.117("64.48%)
0.0555 ("62.7%) 0.1179 ("64.45%)
3.2446 (;7.91%) 3.8797 (;19.35%) 3.0738 ("7.09%) 499.1901 ("7.43%) 519.0671 ("7.65%) 475.5626 ("4.76%) 487.3152 ("5.36%) 0.3149 ("5.63%)
23.46% reduction respectively. On the other hand, road holding improvement has caused to decline of ride index with a rate of 18.34% and 17.63% for RH-T1 and RH-IT2 respectively. COG RMS vertical acceleration has been increased approximately 23% by both controllers. COG RMS pitch acceleration has been reduced with a rate of 2.82% and 0.74% while COG RMS roll acceleration has been increased with a rate of 5.88% and 4.79% by RH-T1 and RH-IT2. COG vertical acceleration CF has remained approximately same. COG pitch and roll CFs have been decreased by road holding controllers. According to the results of hybrid controllers, H-T1 and H-IT2 have improved ride index with a rate of 18.04% and 18.54% respectively. In addition, RMS-DTLs have been improved by 4.76%-7.65% with H-IT2. H-T1 has shown better performance than H-IT2 for Tire 4. COG vertical acceleration CF has been increased by both controllers with rates of 9.07% and 7.91%. COG roll acceleration CF has been reduced by 6.12% and 7.09% while COG pitch acceleration CF has been increased by 18.17% and 19.35%. This increase is caused by significant reduction of COG RMS pitch acceleration. Force outputs of RC-IT2, RH-IT2, and H-IT2 are given in Fig. 11. The force requirements for ride comfort and road holding can be seen from the figure. RC-IT2 output force has relatively low frequency and amplitude change while RH-IT2 output has high frequency and large amplitude change. H-IT2 merges these controller outputs and its’ output can be seen from the figure.
Fig. 12 shows ISO 2631-1 frequency weighted vehicle COG vertical accelerations and their power spectral densities (PSD) for P, HT1 and H-IT2. According to the graphs, H-T1 and H-IT2 have slightly better performance comparing to P in view of COG vertical acceleration especially for frequencies below 10 Hz. P has slightly better performance comparing to H-T1 and H-IT2 for higher frequencies. However, H-IT2 has improved COG RMS vertical acceleration by 5.63% while H-T1 has improved by 5%.
Fig. 11. Controller output forces of RC-IT2, RH-IT2, and H-IT2 for Tire 1.
Fig. 12. Vehicle COG vertical accelerations and PSD curves for P, H-T1, and H-IT2. 9
Meral Özarslan Yatak and F. Sßahin
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Fig. 13 shows ISO 2631-1 frequency weighted vehicle COG pitch accelerations and their PSDs for P, H-T1 and H-IT2. Vehicle COG RMS pitch acceleration has been improved by 62.7% and 63.1% for H-IT2 and H-T1 respectively. H-T1 has provided slightly better performance than H-IT2. On the other hand, H-T1 and H-IT2 have shown lower performance comparing to passive system for higher frequencies. According to frequency weighted roll acceleration graphs and their PSD curves in Fig. 14, hybrid controllers have given the best results for roll acceleration performance. H-IT2 has improved COG RMS roll acceleration by 64.45% while H-T1 has improved by 64.48%. H-IT2 and H-T1 have provided worthwhile reduction of vehicle COG roll acceleration. When performance results of the proposed hybrid fuzzy control system are compared to the studies of Nagarkar et al. [28] and Nieto et al. [31], it can be seen that the proposed system gives better results. In addition to these, altough Cao et al. [4] achieves better ride index value, road holding was not taken into account. Thus, the simulation results confirm the advantages of the proposed control structure.
7. Conclusion This paper presents a hybrid controller structure based on IT2 FLC to improve ride comfort and road holding of an active suspension system simultaneously. A full vehicle model has been constructed to simulate performance of the proposed controller structure. The proposed IT2 FLC hybrid controller performance results have been compared to T1 FLC hybrid controller and passive system. The conflict between ride comfort and road holding,
Fig. 14. Vehicle COG roll accelerations and PSD curves for P, H-T1, and H-IT2.
improving one of them causes the deterioration of the other, is the inspiration point of this study. For the purpose of joining different controller requirements of road holding and ride comfort, a hybrid structure has been developed. The proposed hybrid controller consists of a ride comfort controller and a road holding controller. These controllers have different inputs and operate independently from each other. Output of the hybrid controller is obtained by summing ride comfort and road holding controller outputs weighted with a weighting coefficient which has a range of 0-1. Increasing the value of the weighting coefficient to 1 results in improvement of ride comfort while lowering to 0 results in improvement of road holding. The weighting coefficient has been chosen as 0.5 by evaluating the simulation results. Ride index results, RMS-DTLs, the frequency weighted acceleration curves and their PSD curves obtained from the simulation results have shown that the proposed controller provides more improvements in ride comfort and road holding performance in comparison with the T1 FLC hybrid controller (H-T1), and passive system (P). The proposed controller has reduced ride index by 18.54% and dynamic tire loads by 7.43%, 7.65%, 4.76% and 5.36% for each tire respectively. These improvements can be explained with the ability of modeling uncertainties by IT2 FLC. On the other hand, the overall performance scores of IT2 FLC and T1 FLC hybrid controllers is not too different from each other. It is considered that this difference will be much more in implementation of the proposed controller in a real system due to addition of more uncertainty sources caused by suspension system components. Another problem to work on, is the weighting coefficient. The weighting coefficient can be determined by system dynamics. Furthermore, it can
Fig. 13. Vehicle COG pitch accelerations and PSD curves for P, H-T1, and H-IT2. 10
Meral Özarslan Yatak and F. Sßahin
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
D¼0
be determined by an additional controller. It can be concluded that the proposed IT2 FLC hybrid controller scheme provides an effective control concept for active suspension systems to improve ride comfort and road holding performances simultaneously.
2
0 6 60 6 E¼6 6 60 4 0
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
ku1 m1
0
0
0
0
0
0
0
0
ku2 m2
0
0
0
0
0
0
0
0
ku3 m3
0
0
0
0
0
0
0
0
ku4 m4
0 0 0 0 0 0
7 0 0 0 0 0 07 7 7 7 0 0 0 0 0 07 5 0 0 0 0 0 0
Appendix State-space matrices of full vehicle active suspension system model
2
0 6 6 61 6 6 60 6 6 6 60 6 6 60 6 6 60 6 6 60 6 6 60 6 6 6 6 6 6 60 6 6 A¼6 6 6 60 6 6 6 6 6 6 60 6 6 6 6 6 6 6 60 6 6 6 6 6 6 60 6 6 6 6 4 0 2
0
0
0
0
0
0
0
k1 mb
0
lf k 1 J
0
mc11
0
0
0
0
0
0
0
c1 mb
0
lf c 1 J
0
0
0
ku2 k2 m2
0
0
0
0
0
k2 mb
0
lf k 2 J
0 0
0
1
mc22
0
0
0
0
0
c2 mb
0
lf c 2 J
0
0
0
0
ku3 k3 m3
0
0
0
k3 mb
0
lr kJ 3
0
1
mc33
0
c3 mb
0
lr Jc3
0
0
k4 mb
0
lr kJ 4
0
0
c4 mb
0
lr Jc4
0
k1 þk2mþk3 þk4 b
1
c1 þc2mþc3 þc4 b
0
0
0
0
k1 m1
c1 m1
0
0
0
0
k2 m2
0
c2 m2
0
0
0
0
0
k3 m3
0
c3 m3
0
0
0
ku4 k4 m4
1
mc44
0
k4 m4
0
c4 m4
k 1 lf m1
k2 lf m2
0
0
km3 3lr
0
km4 4lr
c 1 lf m1
c 2 lf m2
0
k1 t fl m1
0
c1 t fl m1
0
0
0
k2 tfr m2
c t m2 2fr
0
0
0
cm3 l3r
k3 t rl m3
c3 t rl m3
0
0
0
cm4 l4r
km4 t4rr
cm4 t4rr
mb
k2 t fr þ k4 trr k1 t fl k3 t rl
0
mb
c2 t fr þ c4 t rr
0
3T
0
0
0
1 mb
0
lf J
0
0
0
0
0
1 mb
0
lf J
0
0
0
1 mb
0 lJr
0
0
1 mb
0 lJr
0 trrI
0 m12
0
0
0
0
0 m13
0
0
0
0
0
0 m14
0
0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 1 0 0 0 0
0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 1
0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0
t fl I
t fr I
trl I
7 7 7 7 7 7 5
3
0 0 0 0 07 7 7 0 0 0 0 07 7 7 0 0 0 0 07 7 0 0 0 0 07 7 7 0 1 0 0 05 0 0 0 1 0 11
0
2 lf ðk1
þ k2 Þ
2 lr ðk3 J
þ k4 Þ
2 lf ðc1
þ c2 Þ
2 lr ðc3 J
þ c4 Þ
!
lr t rl k3 lr t rr k4
0 0
J
lr trl c3 lr t rr c4 þlf t fr c2 lf tfl c1 J
0
0
!
þlf tfr k2 lf tfl k1
mb
0
0
c1 t fl c3 trl
0
0
1
lr c 3 þ lr c 4
0
J
0
lf c1 lf c2
0
J
lf c1 lf c2
lr c 3 þ lr c 4
lr k3 þ lr k4
mb
0
lf k1 lf k2
0
lf k1 lf k2
0
0
lr k 3 þ lr k 4
1 0 0
60 6 6 60 6 6 C ¼ 60 6 60 6 6 40
0
0
0 m11
6 6 60 B¼6 6 60 4 0 2
ku1 k1 m1
1
3T
t fl k1 I
3T
7 7 7 7 7 t fr k2 7 I 7 7 t fr c2 7 I 7 7 7 t rl k3 7 I 7 7 t rl c3 7 I 7 7 trr k4 7 I 7 7 t rr c4 7 I 7 7 7 t fr k2 t fl k1 7 7 7 þt rr k4 trl k3 7 I 7 7 t fr c2 t fl c1 7 7 7 þtrr c4 trl c3 7 I 7 7 7 7 lf t fr k2 lf t fl k1 7 þlr t rl k3 lr t rr k4 7 7 7 I 7 7 7 lf t fr c2 lf t fl c1 7 7 þlr trl c3 lr t rr c4 7 7 I ! 7 7 7 t 2fl k1 t 2fr k2 7 7 2 2 t rl k3 trr k4 7 7 I ! 7 7 2 2 7 t fl c1 t fr c2 7 5 2 2 trl c3 trr c4 t fl c1 I
I
Meral Özarslan Yatak and F. Sßahin
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