ACTIVE SUSPENSION SYSTEM OF QUARTER CAR

Transcription

1 ACTIVE SUSPENSION SYSTEM OF QUARTER CAR By JIE FANG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2014

2 c 2014 Jie Fang 2

3 To my parents, Zheng Fang and Pin Zhang, and my friends, Moses and Darsan 3

4 ACKNOWLEDGMENTS I would like to express my special thanks of gratitude to my advisor, Dr. Crane as well as my PHD instructor, Olugbenga Moses Anubi and Darsan Petal who gave me the precious opportunity to work on this wonderful project on the topic, active suspension system of quarter car. I appreciate their patience to me and significant instruction about the project. Secondly, I would also like to thank my parents and friends who gave me lots of support and help in finishing this project within the limited time. THANKS AGAIN TO ALL WHO HELPED ME. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Passive Suspension System Active Suspension System Semi-active Suspension System SYSTEM DESCRIPTION Suspension System Model Performance of Suspension System Suspension System Combined with Nonlinear Energy Sink and Skyhook Active Suspension System CONTROL DEVELOPMENT BASED ON SINGULAR PERTURBATION Dynamic Analysis Control Design Stability Analysis Simulation MULTIPLE SLIDING MODE CONTROL Introduction of Sliding Mode Control Control Design Stability Analysis Simulation MODEL PREDICTIVE CONTROL Introduction of Model Predictive Control Control Design Simulation Model Predictive Control with Constraints CONCLUSION AND FUTURE WORK Conclusion

6 6.2 Future Work REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 2-1 Dynamic System Parameter Values Parameter Values for Combined Suspension System Hydraulic System Parameter Values

8 Figure LIST OF FIGURES page 1-1 Quart Car Model Simplified Quarter Car Model Passive Suspension System Skyhook Suspension System Nonlinear Energy Sink Suspension System Suspension System Combined with Nonlinear Energy Sink and Skyhook Car Body Acceleration Suspension Travel Wheel Deflection Quarter Car Model with Hydraulic System Simulation of Quarter Car Model Force Tracking Suspension Force Car Body Acceleration Suspension Deflection Force Tracking by Sliding Mode Control Car Body Position Car Body Velocity Unsprung Mass Position Unsprung Mass Velocity Control Input for Hydraulic System Car Body Position Car Body Velocity Unsprung Mass Position Unsprung Mass Velocity

9 5-1 Force Tracking Car Body Position Car Body Velocity Unsprung Mass Position Unsprung Mass Velocity Control Input Car Body Position Car Body Velocity Unsprung Mass Position Unsprung Mass Velocity Constraint Control Input

10 Chair: Carl D. Crane Major: Mechanical Engineering Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ACTIVE SUSPENSION SYSTEM OF QUARTER CAR By Jie Fang May 2014 This research is mainly about the control design, analysis, and simulation of a vehicle suspension system. In order to decrease the vibration of the vehicle, a passive suspension system, semi-active suspension system, and active suspension system are taken into consideration. The central idea is based on the combination of a nonlinear energy sink and a skyhook. The car performance such as car body acceleration, suspension deflection, and wheel deflection are measured to compare the proposed control with a passive suspension system, a skyhook suspension system (semi-active suspension system), and a nonlinear energy sink suspension system (active suspension system). According to the comparison, it shows that the proposed system comprised of a nonlinear energy sink and skyhook achieves better performance with regards to the performance objectives. Based on the result, a hydraulic actuator is applied to mimic the force obtained from the proposed suspension system. Three control methods are utilized to achieve the purpose. The first control input is designed by using a singular perturbation technique combined with adaptive control. The second control method is based on multiple surface sliding mode control. By designing two sliding surfaces, an ideal tracking performance can also be obtained. Lastly, model predictive control is utilized, which can be treated as a special optimal control. By optimizing the cost function involved in tracking error and control input at each sample time, an ideal control input can be obtained from it. The 10

11 first two methods are supported by Lyapunov based stability analysis to prove that the tracking error will approach zero asymptotically. Simulation results for each method are also given to show the vehicle performance and tracking performance. 11

12 CHAPTER 1 INTRODUCTION A vehicle suspension system is used to separate the car body physically from the wheels and allow relative motion between the two parts. It is typically rated by its ability to provide good roadholding, isolate passenger from road disturbance, and improve passenger comfort. The road disturbance may be caused by various reasons such as road unevenness, aerodynamics forces, non-uniformity of the tire/wheel assembly, and even braking force. The quality of roadholding especially during cornering and swerving determines the active safety of the vehicle. The ability of absorbing vibration from road disturbance is mainly discussed in this research. Typically, a suspension system consists of the system of springs, shock absorbers, and linkages that connect a vehicle to its wheels. The structure of a quarter car model is shown in Figure 1-1. Figure 1-1. Quarter Car Model 12

13 In order to simplify the analysis of the suspension system, a simplified quarter car model is shown in Figure 1-2. Figure 1-2. Simplified Quarter Car Model In the simplified quarter car model, m s represents sprung mass (vehicle body), m u represents unsprung mass (wheel body). According to the component used to generate the control force F to connect the sprung mass (m s ) and unsprung mass (m u ), the suspension system can be classified as a Passive Suspension System, Semi-active Suspension System, and Active Suspension System. 1.1 Passive Suspension System Based on the simplified quarter car model, a passive suspension system is one in which the coefficients of the components are constant. The structure of the passive suspension system is shown in Figure 1-3. Lots of researchers have worked on the passive suspension system [1-3] including its structure, performance, and safety. The main components, determined by the designer of the suspension system, are k s, b s, and k t. k s and b s represent the spring and damper coefficients between the wheel and the vehicle body while k t represents the compliance of the tire. r represents the road disturbance. Usually a sine function is taken into consideration. Vehicle deflection y s, car body velocity ẏ s, acceleration ÿ s, wheel deflection y u, wheel velocity ẏ u, and wheel acceleration ÿ u are usually measured 13

14 Figure 1-3. Passive Suspension System to determine the vehicle performance. A passive suspension system is able to store energy from the road disturbance via a spring and to dissipate it via a damper. A certain level of compromise among roadholding, load carrying, and ride comfort can be reached by choosing appropriate values for the coefficients of the spring and damper. The passive suspension system is an open loop control system. There is no way to adjust the characteristics of a passive suspension system at run time. It has been shown in [2], when the passive suspension system is heavily damped, the suspension system will have good vehicle handling, but lots of road input will be transferred or the system will throw the car due to the unevenness of the road. When the vehicle is run at low speed on an uneven road or at high speed along a straight line, it will be perceived as a harsh ride or it may damage cargo. When the passive suspension system is lightly damped or soft suspension, the stability of the car in turns or change lane maneuvers will be reduced or the suspension system will swing the car. Thus the performance of the passive suspension system depends on the characteristic of the suspension elements. 1.2 Active Suspension System Considering the shortcoming of the passive suspension system, the active suspension system is definitely a promising topic. The advantage of the active suspension system is obvious. Lots of researchers [4-11] have showed that the active 14

15 suspension system can improve safety and ride comfort of the vehicle significantly. In an active suspension system, a force actuator is used instead of a passive damper or both the passive damper and spring. A significant difference between a passive suspension system and an active suspension system is that active suspension system can add and dissipate energy from the system via a force actuator, unlike a passive suspension system which can only dissipate energy via a passive damper. But the introduction of active force actuator also introduces other problems. The first problem is the requirement of larger power which decreases the overall performance of the vehicle. Meanwhile, the force actuator also adds to the complexity of the whole system which leads to a larger range of control algorithms. Further, most control methods are based on the accurate knowledge of the suspension system such as the parameters of the suspension system, and the vehicle system states. Thus the requirement of sensors also increases the cost of the suspension system. Another problem for active suspension systems are unacceptable failure modes. It will be dangerous for both the vehicle and passengers because of the possible failure of an actuator. 1.3 Semi-active Suspension System The semi-active system [12-18] retains the conventional spring element of the passive suspension but uses a controllable damper. External power is required and the power is used to adjust the damping level, and operate controller and sensors. Thus the semi-active suspension system has less complexity, cost and more reliability compared to the active suspension system. The skyhook control strategy [17] is the most widely used control policy for semi-active suspension systems. It can reduce the resonant peak of the body mass and thus reach a good quality of performance. The structure of the skyhook suspension system is shown in Figure 1-4. It consists of a linear spring with k s, a damper with b s between the sprung mass and unsprung mass and a damper with coefficient b sky which is effectively attached to 15

16 Figure 1-4. Skyhook Suspension System an absolute reference. The special damper used in the skyhook suspension system ranges between hard and soft envelopes. In compression or extension, such a damper can provide a symmetric damping force. This kind of special semi-active damper is called HH/SS damper. Theoretically, the skyhook control strategy requires two sensors to measure the displacement of the sprung mass and acceleration. In the practical implementation, the velocity of the sprung mass is measured and then employed to obtain the ideal damping level. Lastly, the corresponding damping control signal will be sent to a controllable damper to reduce vibration. Recently, the development of the electro-rheological (ER) fluid damper and the magneto-rheological (MR) fluid damper which are also HH/SS dampers make the semi-active suspension system more applicable. Electro-rheological fluid is a kind of smart material whose yield strength, and apparent viscosity can be externally controlled by the application of an electric field. Because the Electro-rheological fluid works in the electric field, it has a very fast response characteristic to the electric field and wide control bandwidth. The power requirement to activate the phase change is very low. Magneto-rheological (MR) fluid consists of a synthetic hydrocarbon or silicone base coupled with a suspension of magnetically soft particles. When no magnetic field 16

17 is applied, the particles disperse randomly and the fluid exhibits Newtonian behavior. When a magnetic field is applied, the rheological behavior changes from Newtonian to Bingham plastic, which makes the fluid more viscous. Based on this, the amount of torque transmitted through the device can be controlled by changing the magnetic field. In the off state, both the electro-rheological (ER) and magneto-rheological (MR) fluids have similar viscosity, but once switched to the on state, MR fluids shows a much greater increase in yield strength, therefore viscosity. Meanwhile, it has been shown that the device based on an ER fluid will have roughly the same overall power requirement as similar device based on an MR fluid, though the ER device requires high voltage, low current power, while the MR device requires low voltage, high current power. The high requirement of voltage for the ER device makes it impractical for most commercial applications. Moreover, the MR is less sensitive to contaminants, and has a much broader useful temperature range than ER fluids. Thus according to different applications, different semi-active suspension systems can be applied to reach desired performance. 17

18 CHAPTER 2 SYSTEM DESCRIPTION A detailed description of the suspension system of the quarter car model used in this thesis will be given in this chapter. Meanwhile, the comparison of the passive suspension system, skyhook suspension system, nonlinear energy sink suspension system, and suspension system combined with a nonlinear energy sink and skyhook based on car body deflection, velocity and wheel deflection will also be presented. 2.1 Suspension System Model The suspension analysis is based on the simplified quarter car model which has been shown in Figure 1-2. The simplified model consists of the sprung mass m s and unsprung mass m u including the mass of the tire and axles. The tire is modeled as a linear spring with stiffness k t. The suspension system is controlled by force F which is designed by the engineer. Typically, F is generated by a linear spring, a damper and another force actuator which may be an active actuator or a semi-active actuator. r represents the road disturbance which is treated as a sine function and can be showed as follows: r = A 1 sin(ω t) (2 1) where A 1 represents the amplitude of the road disturbance signal, for the purpose of frequency response generation. The values of the parameters used have been given in Table 2-1. And L 0s represents the original length of the spring K s. Parameter Value m s 315 Kg m u 37.5 Kg K t N/m K s N/m b s 1500 N/m/s A m L 0s 0.6 m Table 2-1. Dynamic System Parameter Values 18

19 A certain level of performance and safety will be reached by the passive elements by choosing appropriate value of k t, k s and b s, while the active element will be used to further improve the roadholding, passenger comfort, responsiveness and safety. Usually one focuses on the car body acceleration Ÿ s, suspension deflection Y s Y u, and wheel deflection Y u r which determine the vehicle performance. The main idea of the active suspension system is to design an active element to generate force F a which can adjust itself continuously to changing road conditions. 2.2 Performance of Suspension System The items used to judge the performance of the suspension system in this work are mainly passenger comfort and road holding. The passenger s comfort is a combination of different factors such as the safety of the driver, driving environment which can t be controlled, and vibration. In this work, vibration is the only factor taken into consideration. It can be judged by the isolation between the road disturbance and primary system. The better the isolation is, the better the passenger s comfort is. The sprung mass acceleration Ÿ s is utilized to represent the passenger s comfort. The lower the sprung mass acceleration is, the better the passenger s comfort is. The second factor, which determine the performance of the suspension system, is road holding. It is the ability of the vehicle to keep contact with the road and maximize wheel tracking to road unevenness and to guarantee road contact whatever the road profile and load transfer situations. In this work, the road holding is determined by the suspension deflection which can be presented as Y s -Y u. It has been shown that there is a trade-off between the passenger s comfort and road holding. It is impossible to avoid this for active suspension systems or semi-active suspension systems. Thus it is necessary for the engineer to design an appropriate suspension system to minimize it and obtain the desired result. 19

20 2.3 Suspension System Combined with Nonlinear Energy Sink and Skyhook A suspension system combined with a nonlinear energy sink and skyhook is a combined control strategy based on both nonlinear energy sink and skyhook technology. As shown in Chapter 1, the skyhook control strategy is a semi-active suspension system which is based on a linear spring. The nonlinear energy sink control strategy [19-30] is an active suspension system with a highly nonlinear spring. Recently, it has been shown that a suspension system combined of linear substructures and strongly nonlinear parts has the abilities of localization and irreversible transient transferring of energy to prescribed fragments of the structure dependent on initial conditions and external force. This new active suspension system can react on the amplitude characteristics of the external force in a wide range for frequencies. Unlike a passive tuned absorber which can only work in a narrow band of frequencies and can t absorb multi-frequency transient disturbances, the nonlinear energy sink suspension system can transfer vibrational energy from a primary system to the nonlinear energy sink part where the vibrational energy localizes and diminishes in time due to dissipation. As shown in [28-30], a transient resonance capture on a 1:1 resonance manifold of the system is at the origin of an irreversible and almost all energy is transferred from the primary system to the nonlinear energy sink part. However, the transfer is very selective as the two oscillators must be well tuned, and the primary system must have a specific amount of energy(the nonlinear energy sink is at rest initially). The nonlinear energy sink suspension system has be shown in Figure 2-1. The nonlinear energy sink consists of a linear spring with coefficient K 1 and a nonlinear spring with coefficient K 2 Y 2 s. Obviously, the stiffness of the nonlinear spring will change with the car body deflection Y s which, to some extent, determines the passenger comfort. Thus if Y s is large which means vibration is heavy, then the nonlinear spring will become hard by increasing the stiffness to decrease the vibration. According to 20

21 Figure 2-1. Nonlinear Energy Sink Suspension System Hooke s Law, the force generated from the nonlinear energy sink part can be presented as: F NES = K 1 (L 01 Y s ) K 2 (L 02 Y s ) 3 (2 2) where L 01,L 02 are the free lengths of the linear and nonlinear springs. The model of the suspension system with the nonlinear energy sink and skyhook is shown in Figure 2-2. Figure 2-2. Suspension System Combined with Nonlinear Energy Sink and Skyhook Obviously, the combined suspension system has two parts: the nonlinear energy sink part and skyhook part. Based on the forces generated by the nonlinear energy sink 21

22 part and skyhook part, the force generated by the combined suspension system can be presented as F a = F NES + F sky = K 1 (L 01 Y s ) K 2 (L 02 Y s ) 3 b sky Ẏ s (2 3) The values used in this study for the parameters mentioned above are given in Table 2-2. Parameter Value K N/m K N/m L 01 50mm L 02 50mm b sky 2000N/m/s Table 2-2. Parameter Values for Combined Suspension System In order to show the advantages of the suspension system consist of the nonlinear energy sink and skyhook, it is necessary to make a comparison of the passive suspension, skyhook suspension system, nonlinear energy sink suspension system, and the suspension system consist of the nonlinear energy sink and skyhook. An approximate frequency response from the road disturbance r to the sprung mass acceleration Ÿ s, suspension travel Y s Y u, and wheel deflection Y u r are computed by variance gains[31-32]. Usually the bode plot is utilized to analyze the frequency response for linear system. However, here the system studied here is a nonlinear suspension system. Thus a new technique (Variance Gain) is used to analyze the frequency response. The approximate variance gain is given by 2πN ω G(jω) = Z 0 2 dt 2πN ω A 0 2 sin 2 (ωt)dt (2 4) where Z is the performance measurement (sprung mass acceleration, suspension travel, and wheel deflection in this work). The road disturbance is r = A 1 sin(ωt), t [0,2πN/ω], where N is an integer large enough to ensure that the system reaches a steady state. 22

23 The variance gains corresponding to different Z from are shown in Figure 2-3 through 2-5: 10log10(Gain) Traditional 10 Skyhook NES NES Skyhook Frequency (HZ) Figure 2-3. Car Body Acceleration log10(Gain) Traditional Skyhook 40 NES 50 NES Skyhook Frequency (HZ) Figure 2-4. Suspension Travel Based on Figure 2-3, in low frequency (<8 HZ), the suspension system with the nonlinear energy sink has better vibration isolation compared to the passive suspension system and skyhook suspension system. In the high frequency (> 8 HZ), the skyhook suspension system shows better performance compared to the other two suspension systems. Obviously, the suspension system with nonlinear energy sink and skyhook combines the two advantages which shows the better vibration isolation. Figure

24 log10(Gain) Traditional Skyhook 50 NES Skyhook N ES Frequency (HZ) Figure 2-5. Wheel Deflection refers to the variance gain for the suspension travel. The improvement of the skyhook suspension system and nonlinear energy sink suspension system achieve in vibration isolation compared to the passive suspension system results in the degradation in suspension travel. The combined suspension system doesn t show a big advantage compared to the passive suspension system. This result agrees with the trade-off in suspension design. Figure 2-5 shows the similar result for wheel deflection which determines the vehicle road holding ability, and the degree of wear of the tire. Although the combined suspension system doesn t show a huge advantage compared to the passive suspension system in suspension travel and wheel deflection (Figure 2-4 and 2-5), the improvement of vibration isolation (Figure 2-4) for the combined suspension system is obvious. Thus based on the three plots showed above, the suspension system combined with nonlinear energy sink and skyhook shows better vehicle performance. If the controller can control the actuator to track the force given by the suspension system combined with nonlinear energy sink and skyhook, then the whole suspension system can obtain good road holding, passenger comfort, and safety. 24

25 2.4 Active Suspension System The simplified combined suspension system with the nonlinear energy sink and skyhook model has been shown in Figure 2-2. The corresponding quarter car model together with the schematics for the hydraulic system is showed as Figure 2-6. Quarter Car Model with Hydraulic System Typically, hydraulic servomechanisms are used to control hydraulically actuated suspensions. The hydraulic pressure to the servos is provided by a high pressure radial piston hydraulic pump. Sensors are used to monitor body movement and vehicle ride level continuously and transfer the new data to the computer. As the computer receives and processes the data, it operates the hydraulic servos, mounted beside each wheel. The servo-regulated suspensions generates counter forces to body lean, dive, and squat during driving maneuvers. Hydraulic actuators [33] are one of the most viable choices due to their high power-to-weight ratio, low cost, and the fact that force can be generated over a prolonged period of time without overheating. The hydraulic system consists of a source of hydraulic pressure, a spool valve, and a hydraulic cylinder which are shown in Figure 2-6. A hydraulic pump which is typically augmented with accumulators to reduce pressure fluctuations and supply additional fluid for peak demands is used to supply 25

26 hydraulic pressure. The hydraulic cylinder is a double acting cylinder. The position of the piston can be changed by modulating the oil flow into and out of the cylinder chambers, which are connected to the spool valve through cylindrical ports. The modulation is provided by the spool valve. The dynamic function of the hydraulic actuator and the spool valve are as follows: P L = αav p βp L + γx v Ps sgn(x v )P L (2 5) ẋ v = 1 τ x v + K τ u (2 6) F = AP L (2 7) where A is the pressure area in the actuator,p L is the load pressure, v p = ḋ is the actuator piston velocity, and F is the output force generated by the hydraulic actuator. The parameters α, β, γ are determined by actuator pressure area, effective system oil volume, effective oil bulk modulus, oil density, hydraulic load flow, total leakage coefficient of the cylinder, discharge coefficient of the cylinder, and servo valve area gradient. x v is the spool valve position, τ is the actuator electrical time constant, K = 1 is the DC gain of the four-way spool valve, and u is the input current to the servo valve. The values of the hydraulic parameters used in this study are shownin Table 2-3. Parameter Value α N/m 5 β 1sec 1 γ N/m 5/2 Kg 1/2 τ 1/30 sec P s Pa A m 2 Table 2-3. Hydraulic System Parameter Values Based on the knowledge of hydraulic system, different control methods are utilized to design the control input u to let the hydraulic actuator track the force generated by the suspension system combined with the nonlinear energy sink and skyhook which has been shown in Figure

27 CHAPTER 3 CONTROL DEVELOPMENT BASED ON SINGULAR PERTURBATION In this chapter, the control method based on singular perturbation technology [36] is utilized to let the hydraulic system track the ideal force generated from the suspension system combined with nonlinear energy sink and skyhook. Typically, if the system has the form shown in Equation 3-1, then singular perturbation technology can be applied to solve the system. ẋ = f (t, x, z, ϵ), x(t 0 ) = ξ(ϵ) ϵż = g(t, x, z, ϵ), z(t 0 ) = ζ(ϵ) (3 1) The idea of applying singular perturbation theory is to obtain the knowledge about the solution of the whole system when ϵ is small by using limiting behavior of the system. 3.1 Dynamic Analysis According to the simplified quarter car model in Figure 1-2, the states and dynamic system of the simplified quarter car model are given as follows: x 1 = y s x 2 = ẏ s x 3 = y u x 4 = ẏ u x 5 = P L x 6 = x v actuator load pressure spool valve position The dynamic equation is Ẋ 1 = X 2 Ẋ 2 = K s m s X 1 b s m s X 2 + K s m s X 3 + b s m s X 4 + A m s X 5 Ẋ 3 = X 4 Ẋ 4 = Ks m u X 1 + bs m u X 2 ( Kt m u + Ks m u )X 3 bs m u X 4 A m u X 5 + Kt m u r (3 2) Ẋ 5 = βx 5 A(X 2 X 4 ) + γx 6 Ps sgn(x 6 )X 5 Ẋ 6 = 1 τ ( X 6 + K u) 27

28 The values of parameter used in this study have been shown in Tables 2-1, 2-2, and Control Design According to the dynamic system from Equation 3-2 and hydraulic system equation from Equations 2-5 to 2-7, the force generated by the hydraulic system F can be obtained by taking a derivative of Equation 2-7 and substituting Equation 2-5 into it which can be written as Ḟ = βf αa 2 ḋ + γaū (3 3) where ū = x v P s sgn(x v ) F A. (3 4) The actuator force tracking error can be defined as e = F F d (3 5) where F d represents a desired force which can be obtained by analyzing the combined suspension system. F d is written as F d = F NES + F sky + F passive (3 6) where F NES represents the force generated by the nonlinear energy sink part, F sky represents the force generated by the skyhook part, and the F passive represents the force generated by the passive elements which include a linear spring K s and a damper b s. Thus F NES = K 1 (L 01 Y s ) K 2 (L 02 Y s ) 3 (3 7) F sky = b sky Ẏ s (3 8) F passive = Ks(L 0s d) b s ḋ (3 9) Thus F d can be written as F d = K 1 (L 01 Y s ) K 2 (L 02 Y s ) 3 b sky Ẏ s + Ks(L 0s d) b s ḋ. (3 10) 28

29 where d = Y s Y u (3 11) ḋ = Ẏ s Ẏ u (3 12) The desired force F d is tracked by the F which is exerted by the hydraulic actuator in Equation 2-7. Taking the derivative of the tracking error e, ė can be written as ė = Ḟ Ḟ d = βf αa 2 ḋ + raū Ḟ d (3 13) Then add and subtract both βf d and ˆF d, where ˆF d represents the estimation parameter of Ḟ d. Thus, (3-13) can be written as ė = βf αa 2 ḋ + raū Ḟ d = βf + βf d αa 2 ḋ + raū Ḟ d + ˆF d βf d ˆF d. (3 14) Let F d be the error between the actual value and estimate value of F d. Thus F d = Ḟ d ˆF d. (3 15) Substituting equation 3-15 into equation 3-14 gives ė = βe αa 2 ḋ + raū F d βf d ˆF d = βe Fd + ra(ū Y T θ) (3 16) where [ ] Y = ḋ F d ˆF d [ θ = αa r β ra 1 ra ]. The estimate value of F d can be obtained by using the high gain observer [20] ϵ 2 Ṗ = A hg P b hg F d (3 17) ˆF d = sat( 1 ϵ 2 c T hgp, a, b) (3 18) 29

30 where sat represents the saturation function a, if χ < a sat(χ, a, b) = χ, if a χ b b, if χ > b (3 19) and A hg = , b hg = 1 1, c hg = 0 1, ϵ 2 1. (3 20) The saturation function can be used to overcome the peaking phenomenon associated with high gain observers. The stability analysis about the hydraulic suspension system is based on the Lyapunov Function. Thus, the fictitious control Ū is designed as ū = Y T ˆθ k 0 e c 0 sgn(e) (3 21) where ˆθ represents the estimated value of parameter θ and k 0, c 0 are control gains. Then the closed-loop error can be obtained by substituting (3-21) into (3-16), ė = (β + k 0 γa)e Fd c 0 γasgn(e) γay T θ (3 22) where θ represents the error between the actual value and estimate value of θ θ = θ ˆθ. (3 23) The control input can be designed by using the singular perturbation technique [21] to simplify the controller design for the actuators. It is given by u = K f x v KK f K u s (3 24) Then by substituting (3-24) into (3-2), the valve psuedo-closed loop dynamics is given by ϵẋ v + x v = u s (3 25) 30

31 where ϵ = τ 1 + KK f. (3 26) The term ϵ is the perturbation constant. Let ϵ = 0, then the quasi-steady state solution (x vi (ϵ = 0)) x v is given by x v = u s. (3 27) The valve dynamics can be decomposed into fast and slow time scales by using the fast time scale ν = t ϵ and Tichonov s Theorem x v = x v + η + O(ϵ) (3 28) dη dν = η (3 29) where η(ν) is a boundary layer correction term. According to the (3-29), it is obvious that η(v) decays exponentially in the fast time scale. Usually, the time constant τ is designed to satisfy 0 < ϵ 1[22]. Thus, the perturbation constant ϵ can be as small as possible, if large control gain K f can be chosen. As a consequence, η + O(ϵ) will be negligibly small. Thus (3-27) exists. And (3-4) can be written as ū = u s P s sgn(u s ) F A. (3 30) Assuming sufficient pressure for the hydraulic pump, the term inside the square root operator is taken as positive. Thus sgn(ū) = sgn(u s ) (3 31) Then, (3-30) will be ū = u s P s sgn(ū) F A. (3 32) Solving for u s gives u s = ū(p s sgn(ū) F 1 A ) 2. (3 33) 31

32 At this point, the control design based on singular perturbation for the input to hydraulic system is completed. In this control design, the tracking error between the ideal force generated from the suspension system combined with nonlinear energy sink and skyhook and the force generated from hydraulic system is minimized. 3.3 Stability Analysis In this section, the Lyapunov based stability analysis is described. The corresponding adaptation law is also designed in this section to make the actuator force tracking error approach to zero asymptotically, if the control gains are chosen to meet certain sufficient conditions. Theorem 3.1: Given the adaptive update law ˆθ = ΓYe (3 34) where Γ is a positive definite adaptation gain matrix. If the control gain c 0 is chosen to meet the sufficient conditions as follows, c 0 F d γa. (3 35) Then the tracking error e in Equation 3-5 will approach to zero as time goes to infinity. i.e e(t) 0, as t 0. Proof: Let the Lyapunov function be V = 1 2 e2 + γa 2 θ T Γ 1 θ. (3 36) Taking the first derivative of Equation 3-36 yields V = eė + γa θ T Γ 1 θ. (3 37) 32

33 Because θ= θ- ˆθ and θ = 0. Thus θ = ˆθ. Then substituting 3-22, 3-34 into 3-37 yields V = eė γa θ T 1 Γ ˆθ = e[ (β + k 0 γa)e Fd c 0 γasgn(e) γay T θ] + γa θt Ye = (β + k 0 γa)e 2 + Fd e c 0 γaesgn(e). (3 38) Because of the math properties: F d e Fd e and esgn(e) e Equation 3-38 yields V (β + k 0 γa)e 2 + Fd e c 0 γa e. (3 39) Using the sufficient condition in 3-35, the inequality 3-39 becomes V (β + k 0 γa)e 2 0. (3 40) According to 3-36 and 3-40, it follows that V(t) is bounded, which means e(t) and θ are also bounded. Meanwhile, from 3-18, it follows that ˆF d is bounded. Then according to 3-16, ė is bounded, which implies that e(t) is uniformly continuous. Then integrating both sides of 3-40 yields that e(t) L 2. Based on the conditions metioned above, using Barbalat s Lemma [21] yields e(t) 0, as t. Thus, based on the stability analysis, the hydraulic system can track the ideal force generated by suspension system combined with nonlinear energy sink and skyhook perfectly as time goes to infinity. 33

34 3.4 Simulation In this section, the simulation of the behavior of the quarter car suspension system is done by using Matlab Simmechanics. The Quarter-car Model was modeled in Solidworks, and then translated to a Simmechanic model (In Figure 3-1). The vertical strut and tire damping and stiffness used are the ones given in the Renault Megane Coupe model [39]. Figure 3-1. Simulation of Quarter Car Model In the simulation, the vehicle travels at a steady horizontal speed of 40mph. The road disturbance is treated as a bump with amplitude 0.1 m Time Series Plot: 3500 Force (N) Time (s) F Fd Figure 3-2. Force Tracking 34

35 Force (N) NES NES Skyhook Passive Skyhook Figure 3-3. Suspension Force Time (s) The Figure 3-2 shows the force tracking between the ideal force generated from combined suspension system and the actual force generated by hydraulic actuator. Based on the figure, it is obvious that the hydraulic actuator tracks perfectly which also proves the control analysis presented in the previous section. Based on the perfect force tracking, the forces for difference suspension systems used to reduce vibration of the car are measured, as shown in Figure 3-3. From Figure 3-3, the nonlinear energy sink suspension system and the combined suspension system need larger forces than the other two suspension systems to let the system settle down. But in return, these two suspension systems can settle down more quickly which is also preferred. The Figure 3-4 shows the car body acceleration. The car body acceleration is the acceleration of the sprung mass which can be presented as Ÿ s. And the lower the car body acceleration, the better the passenger s comfort. According to Figure 3-4, it is obvious that the combined suspension system shows the lowest car body acceleration which means the best performance in passenger comfort. However, in order to achieve better comfort performance, the force used to reduce vibration is also large for the combined suspension system which has been shown from Figure 3-3. But the difference of force among these suspension systems is not very large, thus if enough control input can be applied, the combine suspension system with nonlinear energy sink and skyhook 35

36 6 Car Body Acceleration (m/s^2) Skyhook Time (s) NES NES Skyhook Passive Figure 3-4. Car Body Acceleration Suspension Deflection (m) NES NES Skyhook Passive Skyhook Time (s) Figure 3-5. Suspension Deflection is the better choice. Meanwhile, the comparison shown in Figure 3-3 and 3-4 agrees with the feature of the nonlinear energy sink which can transfer the energy between the primary system and nonlinear energy sink irreversibly and completely. Lastly, the suspension deflection is measured and shown in Figure 3-5. Suspension deflection is the vertical distance between the mass centers of the sprung mass and unsprung mass which can be expressed as Y s Y u. Theoretically, the vehicle suspension system with smaller car body acceleration will have larger suspension deflection which is not preferred. And as shown in Figure 3-5, the suspension deflection for the suspension 36

37 system with nonlinear energy sink part is larger than that for the other two suspension systems without the nonlinear energy sink part. However, the degradation in suspension deflection is not as much as the improvement achieved in the passenger comfort. Thus if the passenger comfort is the most significant requirement, then the suspension system with nonlinear energy sink part will be better choice. From Figure 3-3 to 3-5 the advantage for the combined suspension system is obvious. The peak values of car body acceleration and suspension deflection for the combined suspension system are smaller than that for the pure nonlinear energy sink part. Thus, the combined suspension system with nonlinear energy sink and skyhook combined the advantage of the nonlinear energy sink which can transfer the energy from road disturbance and dissipate it completely and irreversibly, and the skyhook which can decrease the peak value of resonance. 37

38 CHAPTER 4 MULTIPLE SLIDING MODE CONTROL 4.1 Introduction of Sliding Mode Control In this chapter, multiple sliding mode control strategy [37-40] will be used to design a control input to the hydraulic system so that the force generated by the hydraulic actuator can track the ideal force designed from the nonlinear energy sink and skyhook suspension system perfectly. Sliding mode control is a class of nonlinear control. The goal of the switching control law is to drive the nonlinear plant s state trajectory onto a designed surface in the state space and maintain the plant s state trajectory on the same surface all the time. Meanwhile, sliding mode control is also a variable structure control which can switch from a continuous structure to another. The surface defined here is designed by engineer which is also called a sliding surface (sliding manifold). A control input is designed so that the plant state can slide to the surface and stay on the surface. Moreover, a Lyapunov approach is also utilized to prove the stability of the whole system. 4.2 Control Design The dynamic system has been shown in Chapter 3 from Equation 3-2. As discussed in Chapter 2, the desired dynamics of the active system consisting of a passive suspension system with a nonlinear energy sink and skyhook has been shown in Figure 2-2. The ideal force is shown in Equation In order to simplify the problem, the following first order system is taken into consideration first. Assume the nonlinear system has the form, ẋ = f (x) + g(x)u (4 1) 38

39 where x is the state of the system and u is the control input to the system. If the control input is designed based on feedback linearization, then the control input should be u = 1 ( f (x) + v) (4 2) g(x) Substitute Equation 4-2 into Equation 4-1, then ẋ = v. If we design v = -k*x. Then asymptotically result can be obtained. The Lyapunov function V is chosen as V = 1 2 x T x. (4 3) Then taking the first derivative of the Lyapunov function, asymptotically stable or even exponentially stable can be achieved by designing an appropriate v. But the feedback linearization control is based on exact knowledge of the whole system, which means we need to know f(x) and g(x). In order to design a robust controller, the sliding mode control method is utilized. Firstly, the first sliding surface which is the error between the actual state and ideal state can be presented as follows s 1 (x, t) = x actual x desired. (4 4) The Lyapunov function is chosen as V = 1 2 s 1 T s 1. (4 5) If the first derivative of the Lyapunov function satisfies the following unequal equation, then the system can satisfy the robustness or sliding condition: V = d dt (1 2 s2 ) = sṡ ks 2 (4 6) where k is some positive constant which is designed by the engineer. According to the Lyapunov analysis, exponentially stability can be proved, which means the error between the actual state and ideal state will exponentially converge to zero as time goes to infinity. The value of k, to some extent, affects the convergence rate of the error. The 39

40 larger the k is, the faster the convergence is. But k is also constrained by the control input which is determined by the physical system. In the suspension system system, we expect that the force generated by the hydraulic actuator can track the ideal force generated from the combined suspension system. And assume the road disturbance of the suspension system is unknown and bounded by 0.1 m. Thus based on the state space from Equation 3-2, the first sliding surface can be defined as s 1 (x, t) = F actual F desired (4 7) where the F actual can be calculated as: F actual = AP L = Ax 5 (4 8) where A represents the area of the valve, and P L represents the pressure. Thus the force generated by the hydraulic actuator can be obtained by multiplying the area and pressure of the valve. Meanwhile, the desired force can also be presented as F desired = K 1 (L 01 Y s ) K 2 (L 02 Y s ) 3 b sky (Ẏ s Ẏ u ) = A x 5desired (4 9) Then the derivative of F desired can be written as Ḟ desired = K 1 (L 01 Ẏ s ) K 2 (L 02 Ẏ s ) 3 b sky (Ÿ s Ÿ u ) (4 10) substitute the states which have been shown in dynamic equation, then Ḟ desired = h(x) Hx r (4 11) where x r represents the unknown road disturbance and x r is bounded by 0.1 m and H is some positive constant determined by parameters b sky, k t, m u. Then according to the dynamic system, ẋ 5 is Ẋ 5 = βx 5 A(X 2 X 4 ) + γx 6 Ps sgn(x 6 )X 5 (4 12) 40

41 Then let Taking the first derivative of s 1 gives, f (x) = βx 5 A(X 2 X 4 ) (4 13) g(x) = γ P s sgn(x 6 )X 5 (4 14) ṡ 1 = Ḟ actual Ḟ desired = Ax 5actual Ax 5desired = A(f (x) + g(x)x 6desired x 5desired ) (4 15) According to the design principle mentioned above, x 6desired can be designed as x 6desired = 1 g(x) ( f (x) + ẋ 5desired k 3 s 1 k5 sgn(s1)) (4 16) where f(x), g(x) have been defined in Equation 4-13 and 4-14, and k 3 and k 5 are some positive constants. The value of k 3 which is designed by the engineer will affect the convergence rate of the error between the actual force and ideal force. At last x 5desired can be obtained as follows ẋ 5desired = d(f desired dt A ) Based on the state x 6, the second sliding surface can be defined as. (4 17) s 2 = x 6actual x 6desired (4 18) where x 6actual has been defined from Equation 3-2, and x 6desired has been defined from the first sliding surface from Equation Then taking the first derivative of the second sliding surface, gives ṡ 2 = ẋ 6actual ẋ 6desired = 1 τ ( x 6actual + u) ẋ 6desired (4 19) 41

42 In order to let the system satisfy robustness and for the nonlinear plant s states to slide along the two surfaces, the control input can be designed as u = x 6actual + τ(ẋ 6desired k 4 s 2 ) (4 20) where ẋ 6desired can be obtained by taking the derivative of Equation 4-15, and k 4 is some positive constant which is designed by the engineer. Up to now, the multiple sliding mode control based on defining two sliding surfaces has been defined which is aimed at letting the actual force generated by the hydraulic actuator track the ideal force generated by the combined suspension system. 4.3 Stability Analysis In this section, a stability analysis based on Lyapunov analysis is made. According to the previous section, two sliding surfaces are defined. Thus let the Lyapunov function be V = 1 2 s s2 2. (4 21) Then take the first derivative of the Lyapunov function. According to Equation 4-14, 4-15, 4-18, and 4-19, then V = s 1 ṡ 1 + s 2 ṡ 2 = k 3 s 2 1 k 4 s 2 2 Hs 1 x r + k 5 s 1 (4 22) If the value of k 5 satisfies Equation 4-20 can also be presented as 0.1H k 5 (4 23) V CV (4 24) where C is some positive constant determined by the values of k 3 and k 4. After solving the first derivative equation, V can be presented as V (t) V (0)e Ct. (4 25) 42

43 Obviously, from Equation 4-25, the V will be exponentially convergent to zero as time goes to infinity. And because V is defined from Equation 4-21, thus the s 1 and s 2 should also converge to zero exponentially as time goes to infinity. If s 1 goes to zero as time goes to infinite, it means the force generated by the hydraulic actuator can track the ideal force defined by the suspension system combined with the nonlinear energy sink and skyhook perfectly as time goes to infinity. 4.4 Simulation In the simulation, assume the vehicle travels at a steady horizontal speed of 40mph which is same as before. Also there is a road bump with amplitude 0.1 m. Based on the simulation, several aspects are taken into consideration. The first one is to verify that the force generated by the hydraulic actuator can track the desired force designed using sliding mode control. The desired force and actual force are shown in Figure Force (N) Actual Force Desired Force Time (s) Figure 4-1. Force Tracking by Sliding Mode Control It is seen in Figure 4-1 that the actual force can track the ideal force perfectly. Thus, the plant s states can slide along the defined sliding surface by designing the appropriate sliding mode control input. In addition to force tracking, it is also necessary to take the car deflection, car body velocity, and other states into consideration and analyze the vehicle performance based on these states. 43

44 Sprung Mass Position (m) Hydraulic System Passive System Time (s) Figure 4-2. Car Body Position Sprung Mass Velocity (m/s) Hydraulic System Passive System Figure 4-3. Car Body Velocity Time (s) Unsprung Mass Position (m) Hydraulic System Passive System Time (s) Figure 4-4. Unsprung Mass Position 44

45 Unsprung Mass Velocity (m/s) Hydraulic System Passive System Time (s) Figure 4-5. Unsprung Mass Velocity Figures 4-2 through 4-5 show a comparison between the passive suspension system and the suspension system with the hydraulic actuator. According to Figure 4-2, although the peak value of the car body position for the suspension suspension with the hydraulic system is larger than that for the passive suspension system, the car body settles down very quickly which is preferred. The same occurs to other states. As for the same road disturbance, the suspension system with the hydraulic actuator which is used to generate the force to track the nonlinear energy sink and skyhook force can settle down much faster than the passive suspension system. The control input of the suspension system with the hydraulic actuator is also shown in Figure 4-6. And the control input in the system is the input current to the servo valve. Figures 4-7 through 4-10 show the tracking performance between the original nonlinear energy sink and skyhook suspension system and the suspension system with hydraulic actuator by using sliding mode control. Figure 4-7 through Figure 4-10 show the tracking performance according to four states. There are some small differences between the original suspension system with the nonlinear energy sink and skyhook and the suspension system with the hydraulic actuator. However the trends are almost same. Based on these figures, the tracking performance by designing the sliding mode control is very good. 45