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Study on discrete-force suspension system for a car on randomly profiled road Yang, Dajiang 1991

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STUDY ON DISCRETE-FORCE SUSPENSION S Y S T E M FOR A C A R ON R A N D O M L Y PROFILED ROAD By Dajiang Yang B.A.Sc. (Engineering) Shanghai Institute of Railway Technology M.A.Sc. (Engineering) Shanghai Institute of Railway Technology  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTER OF APPLIED SCIENCE  in T H E FACULTY OF G R A D U A T E STUDIES MECHANICAL ENGINEERING  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA  June 1991 © Dajiang Yang, 1991  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2788)  Abstract  An active suspension system possesses the ability to generate arbitrarily controllable forces in an automotive suspension. It has widely been accepted as the most promising car suspension scheme. Prototype active suspension systems have shown impressively smooth ride, firm attitude control ability and good handling quality. Present implementations are, however, not perfect and still leaves some problems which represent the limitations of an active suspension system. The most conspicuous problem of performance is ride harshness over small, sharp bumps. Active suspension is worse than the passive variety at isolating high frequency vibration because the force actuator bandwidth is not big enough to cover the frequency range for both sprung and unsprung masses. The energy consumption is also a problem. The power required by system's hydraulic pump is more than three horsepower. A new suspension scheme called discrete force suspension (DFS) is introduced to solve these problems. In order to improve the high frequency characteristics of the active suspension and significantly reduce the extra external power required by the hydraulic system , a new type of hydraulic actuator -r- digital force actuator (DFA) is developed to replace the conventional one. A DFA only outputs a limited number of force levels but does not restrict the flow in the system. Therefore, it transmits little high frequency disturbance through it and has a very little energy loss. It has been studied through theoretical analysis and numerical simulation. Results show satisfyingly small equivalent stiffness and damping coefficient, and a quite linear relation between output and input. Unfortunately, no sensible outcome was obtained for the physical experiment due to mistakes made during the manufacturing and design.  11  Then the idea to control the low frequency body mode and the high frequency wheel mode separately by a D F A and a conventional damper (or a dynamic absorber) respectively is developed to avoid the necessity of wide bandwidth actuators and to achieve a better high frequency performance. Due to introducing D F A into the control system, a new control strategy — step-wise reference control, was also developed, analyzed and adopted to cope with the problem of discrete control force coming up with D F A . Numerical simulations have been performed for a quarter car model to demonstrate the feasibility and effectiveness of a DFS in terms of the independence of the D F A output to the suspension motion, the significant reduction of the extra power requirement of the DFS in comparison with an active suspension, and the vibration isolation characteristics.  in  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  viii  Acknowledgement  x  1 Introduction  1  1.1  1.2  1.3  Introduction to Car Suspension Problem  .  1  1.1.1  Suspension performance requirements and criteria  1  1.1.2  Limitation of currently used passive suspension  2  Literature Review and Existing Unsolved Problems .  3  1.2.1  Active suspension  3  1.2.2  Semi-active suspension  6  Discrete-force Suspension  .  2 Discrete Force Actuator  6 10  2.1  Basic theory  10  2.2  Overall structure  12  2.3  Principle of operation  13  2.4  Actuator Modelling  15  2.5  Effects of Actuator Motion on Force .  21  2.6  Transient Output During Switching iv  .  25  3  2.7  Hydraulic Power Losses  2.8  DFA Test  .  32  2.8.1  Test setup  32  2.8.2  Test procedure  33  2.8.3  Failure analysis  35  C o n t r o l System  37  3.1  Model Establishment  37  3.1.1  Car model  37  3.1.2  Road surface model  41  3.2  3.3  3.4  4  28  Linear Stochastic Quadratic Optimal Control  42  3.2.1  Procedure of control law design  42  3.2.2  Unavailable state information  46  3.2.3  Limitation of LQ method  '  48  Conditional Minimum M.S. Control  49  3.3.1  Control law construction  49  3.3.2  Advantages and limitation  52  Step-wise Reference Control  53  3.4.1  Quantization of the control force and construction of the control law 54  3.4.2  Error analysis  55  3.4.3  Influence of this control strategy on system dynamics  57  N u m e r i c a l Simulation and Analysis  4.1  4.2  59  Simulation. .  •  59  4.1.1  System parameters  59  4.1.2  Simulation algorithm  59  Results  • v  61  5  4.2.1  Ideal active vs. passive  62  4.2.2  Limited active vs. ideal active  4.2.3  Active with dynamic absorber vs. limited active  66  4.2.4  DFS vs. limited active  67  4.2.5  DFS vs. ideal active and passive  4.2.6  DFS with limited measurements  72  4.2.7  Control force power  72  4.2.8  Summary  75  62  x  .  Conclusions  67  76  Appendix  78  A Discretization of Coefficient Matrices of State-Space Model  78  Appendix  79  B D F A Transient Behaviour Modelling  79  Bibliography  83  vi  List of Tables  2.1  Hydraulic and structural parameters for simulation  23  4.2  System parameters . .  60  4.3  Comparison of RMS responses  75  vii  List of Figures  1.1  Schematic configuration of an active suspension  3  1.2  Schematic suspension with skyhook dampers  1.3  Schematic configuration of DFS  2.4  Schematic 1 DOF mass-spring system with a DFA  11  2.5  Cross section of a rotary DFA .  13  2.6  Schematic fluid flow of hydraulic system  16  2.7  Schematic equivalent system of DFA  22  2.8  Sinusoidal output of the DFA  23  2.9  DFA equivalent stiffness  24  5 7  2.10 DFA equivalent damping coefficient  25  2.11 Input-output relationship of DFA . .  26  2.12 DFA transient pressures during switching  28  2.13 DFA transient outputs with different rotor velocities  29  2.14 DFA transient outputs with different valve orifice areas  29  2.15 DFA transient outputs with different accumulator volumes  30  2.16 DFA transient outputs with different force level settings  30  2.17 Comparison of the actual and the ideal output of the DFA  31  2.18 Setup for DFA test . .  32  2.19 DFA assembly  34  3.20 Quarter car model with a DFS  .  3.21 DFS scheme with a dynamic absorber viii  38 40  3.22 Block diagram of the system with a Kalman filter  47  3.23 Non-linear model of the DFA  54  3.24 Quantization characteristics, (a) truncating, (b) rounding-off  55  3.25 Comparison of u (fc), u (k) and u(t)  56  3.26 Statistic model of the DFA  56  4.27 PSD of passive suspension versus ideal active suspensions  63  0  q  4.28 PSD of two active systems of / = 10 Hz: one with damper, another without 64 fe  4.29 PSD of limited active system vs. ideal active system  65  4.30 PSD of limited active system vs. the active system with dynamic absorber  68  4.31 PSD of limited active suspension vs. DFS with f = 10 Hz  69  4.32 RMS responses of DFS vs. the force level width  70  c  4.33 Comparison of time histories of the control forces for limited active suspension and DFS  70  4.34 PSD of DFS vs. ideal active and passive suspensions 4.35 PSD of DFS with Kalman filter and DFS with full state feedback  71 . . . .  73  4.36 PSD of DFS with limited state feedback and DFS with full state feedback i  ix  74  Acknowledgement  I will always be indebted to my advisor, Dr. Bruce Dunwoody, for his support and encouragement through this research. Thanks are also due to the personnel of the Mechanical Engineering workshop who assisted in the construction of a prototype actuator. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council. Finally, I would like to express the special thanks to my wife, Yaojin, who in many ways has helped this thesis become what it is now.  x  Chapter 1  Introduction  1.1  Introduction to Car Suspension Problem  1.1.1  Suspension performance requirements and criteria  The suspension system of a passenger car must have three basic functions: First of all, it must support the car body yet isolate it from external disturbing inputs, mainly road irregularities, in order to obtain a satisfying ride quality. Secondly, it should maintain a firm contact between the road and the tires and decrease the tire loadfluctuationin the presence of external disturbances as far as possible, thus providing a good handling and steering behaviour. Finally, it should offer good attitude control under load variation due to cornering, acceleration and braking maneuvers. It is widely accepted that vertical acceleration of car body is a suitable measure for ride quality and that tire spring deflection is a good indication of how well the tire remains in contact with the road, while suspension stroke indirectly shows the attitude control ability. There is certainly a maximum value for suspension stroke which forms two "stops". That is another reason to choose it as a performance criterion. Due to the fact that any passenger car runs on a randomly profiled road, the perfor-  mance criteria can be expressed as the root mean square (RMS) of vertical acceleratio of body — reflecting the passenger discomfort and RMS suspension deflection and RMS tire deflection — indicating the ability of attitude control and manoeuvring. The whole performance goal, therefore, can be compactly and briefly described as  1  Chapter 1. Introduction  follows:  2  minimizing body's RMS vertical acceleration under the constraints of RM  suspension deflection and RMS tire deflection smaller than some given values. 1.1.2  Limitation of currently used passive suspension .  Unfortunately, these requirements conflict with each other. Current suspensions consist of only passive components — springs and dampers, which have fixed characteristics. It is impossible to meet all varying requirements by choosing any values for stiffness and damping rate because the optimal stiffness and damping values for these requirements are quite different. The softer the spring, the more capable of isolating the sprung mass from road irregularities and therefore subjecting the passengers to low levels of vertical acceleration and hence good ride. However, inertia forces due to cornering, braking and etc., will result in large attitude changes of a softly-suspended car plus a requirement for large wheel movements and associated suspension linkage deflection. The designer is not only faced with the problem of choosing soft spring for good body isolation, versus stiffer spring for better control of both body and wheel motions, but also with the conflict in choosing damping rate which has different requirements for sprung mass and unsprung mass.  Acceptable damping of body modes Avill lead  to excessive damping of the unsprung mass and hence shock loads being transmitted through to the body. Increased damping is also used to control body motion during transient manoeuvres. The real problem for passive suspensions is that they can only response to the relative motion between the car body and wheel mass and generate a resistance to this relative motion no matter how this motion is induced. Based on the above limitations, conventional passive suspensions are always some kind of compromise.  Chapter 1. Introduction  3  accumulator pump  actuator  o M  w  servovalve  controller  C h reservoir  Ul  feedback information  Figure 1.1: Schematic configuration of an active suspension 1.2 1.2.1  Literature Review and Existing Unsolved Problems Active suspension  Limitations of passive suspensions have been well known for a long time. Since the late 1960's, many people have been doing research on this field, trying to find some new suspension with arbitrarily controllable suspension force to replace the passive components. Among many interesting ideas, the active suspension is the most promising solution. Active suspension has a general structure as shows in Figure 1.1 The damper in a conventional passive suspension has been replaced by an active force actuator which is driven by a hydraulic pump through a servovalve. The servovalve operates according to the given commands from the controller. The controller relies on a suite of sensors and a control algorithm to determine the required force at each suspension unit. Usually the feedback information includes suspension travel, and the absolute velocities of body and wheel masses. In such a system, a soft spring is still used in order to support the static car body weight and to offer a temporary measure in case of the active suspension's  Chapter 1. Introduction  4  failure. The biggest advantage of an active suspension is that suspension control force is a function of many variables and can be ideally independent of the body-wheel relative motion. When a tire hits a bump, for instance, the actuator acts as a soft damper which passes only a little disturbance through to the body and can even pull the wheel up to maintain a nearly constant load in the actuator. On the other hand, if the body pitches down during sudden braking, the actuator can be commanded to output a force big enough to alleviate or even eliminate this effect. Since active suspensions allow designers to modulate suspension behaviour arbitrarily, various control strategies have been suggested and studied during the last two decades. Full state variable feedback scheme was first proposed by A.G. Thompson in 1976 [12], and has been widely adopted in active suspension research. As we will see later, we have included this control strategy in our project and a detailed discuss on this issue will be given in chapter 3. With the purpose of overcoming the difficulty in implementing such a control law due to the unavailability of some state information such as the relative displacement of sprung/unsprung mass and road surface, some kind of state estimator (eg. Kalman Filter) can be introduced into the system. As another option, the concept oi absolute or skyhook damping has been suggested. Figure 1.2 is helpful to understand the idea. In this situation, the desired control force should be u  =  cx 2  w  u =  —C\X\,  for body mass and  for the wheel mass at the same time. Unfortunately, because of the very  fundamental limitation in structure (actuator position), this ideal control scheme can not be accurately implemented even by an active force actuator. Finally, it could only come up with a trade-off:  u = — C\X\,  +  cx 2  w  In 1987, Lotus Cars Ltd. of Great Britain brought forth thefirstpassenger car prototype with a successful active suspension [4]. Its smooth ride, firm attitude control  Chapter 1. Introduction  5 /////  Kb /////  C  x.w  2  Ul  Figure 1.2: Schematic suspension with skyhook dampers ability and good handling quality fully displayed all features and potential advantages of the active suspension. It was, of course, not perfect and still left some problems which represented the limitations of an active suspension system. Among the problems, the most conspicuous one of performance is ride harshness over small, sharp bumps. The active suspension is worse than the passive variety at isolating high frequency vibration because the force actuator bandwidth is not big enough to cover the frequency range for both sprung and unsprung masses. Typically sprung mass frequencies are in the range of 0.5 ~ 2 Hz and unsprung mass frequencies 8 ~ 15 Hz [14]. Road roughness can extend to above 100 Hz. Achieving a reasonable bandwidth is not easy and usually, the higher the bandwidth is made, the more power will the system tend to consume [11]. The energy consumption is also a problem. The power required by system's hydraulic pump is more than three horsepower [4], reducing performance and fuel economy.  Chapter 1. Introduction  1.2.2  6  Semi-active suspension  In order to get rid of the energy burden from active suspension, D.C. Karnopp suggested a semi-active suspension scheme in the late 1970's. In a semi-active suspension, instead of an actuator, a rapidly adjustable damper is used. Adopting the same control strategy as for active systems, it is turned on when actual damping force is consistent with the desired force in direction and modulates the damper force as close to the desired control force as possible. Otherwise, it remains inactive or in the off state and gives a zero output. Semi-active suspensions have reduced hardware requirements with clear savings in terms of hydraulic pump, accumulators, pipework and oil reservoirs. Simple configuration means low cost. Furthermore, it needs little extra energy in comparison with a passive suspension except a small amount for damper on-off control and output adjustment. While semi-active suspensions overcome the weakness of an active system, it also brings itself, at the same time, back into the built-in limitation of the passive suspension: suspension force can only oppose the relative motion between the wheel and the body. Therefore, a semi-active suspension is ineffective for controlling vehicle attitude during cornering and braking. Also, the overall performance of a semi-active suspension is not as good as that of an active one [2] even though its performance index in heave mode is quite close to the one of an active system.  1.3  Discrete-force Suspension  From the discussion in previous sections, it is reasonable to say that the active suspension is the most promising suspension system if a better high frequency isolation can be obtained, and the extra external power requirement by its hydraulic system can be greatly  Chapter 1. Introduction  7  accumulator  pump  reservoir  1.  accelerometer  2. L V D T  Figure 1.3: Schematic configuration of D F S reduced. In this section, we will give a brief introduction to a new suspension scheme— discrete-force suspension(DFS) which overcomes these two limitations of existing active suspension implementations. This section will end with a brief description of the purposes of our research. A D F S is basically an active suspension in the sense of arbitrariness of suspension force generation and has a similar overall configuration as shown in Figure 1.3. But nearly all major components in the control system have totally different functions. D F A is a discrete-force actuator, consisting of a number of cylinders or working chambers of different dimensions, and can generate a range of discrete force outputs. The valve block consists of as many simple on-off flow control valves as the number of cylinders in the actuator, each of them switching the corresponding cylinder to the low pressure or high pressure state according to the digital control signal received directly from the controller. While the low pressure state is always connected to the reservoir, the high pressure, unlike the active suspension, is supplied by the accumulator. Here, the accumulator  Chapter 1. Introduction  8  plays a more important role than it does in an active system. On the other hand, the pump has lost some "weight" and is only used to compensate for the leakage loss. Since a DFA works with on-off valves instead of the conventional servovalve, there is little restriction for the flow. Therefore, little high frequency excitation from the ground is transmitted through it to the car body. That means it is able to alleviate the high frequency harshness problem of the active system with existing implementations. An even better high frequency characteristics of a suspension can be achieved through the idea of controlling the low and high resonance frequencies independently. Generally, the unsprung mass has a natural frequency around 12 Hz, ten times higher than the one of > sprung mass. Due to the barriers in the way of building an actuator with a very high bandwidth, it is impractical to try to control both sprung and unsprung mass modes over the whole frequency range. While the DFA is mainly used to actively command low frequency motion, a conventional damper is introduced into the DFS to bring the resonance of the wheel mode under control. Note that the high frequency resonance control by the damper in a DFS should be superior to that in a passive system even though they look similar. The reason is that in the former, there is no limitation on choosing damping rate in order to achieve optimum wheel motion control. That freedom does not exist in a passive system because the damper must also control body motion. Instead of placing a conventional damper between the body and wheel to control the high frequency wheel mode, another option is available through using a dynamic absorber on the wheel mass. Such a system is believed to have better body isolation characteristics because there is no disturbance transmitted to the body through the damper, since a damper is not needed to control wheel motion. The system configuration, however, will obviously be more complex. For the same reason of no flow restriction, the power consumed by a DFS hydraulic  Chapter 1. Introduction  9  system will be very little, the purpose of energy saving can also be achieved by a DFS. The basic theory, modelling, equivalent stiffness and damping, and transient behaviours of a DFA will be explored in Chapter 2 and the control scheme analysis for the DFS will be done in Chapter 3. Chapter 4 will show simulation method and results. The general conclusion will be given in Chapter 5. We have more things to do than just improving high frequency performance and saving energy. In brief, the purposes of our research can be summarized as follows: • Determine how closely the DFA comes to the ideal of providing a force which is independent of the motion of the actuator. • Determine whether the discrete force suspension can achieve the vibration isolation characteristics of an ideal active suspension. • Determine the average energy consumption of a discrete force suspension.  X  Chapter 2  Discrete Force Actuator  2.1  Basic theory  The basic theory of a DFS can be explained with reference to Figure 2.4. It shows a schematic 1 DOF mass-spring system with a DFA of only one cylinder. When the system is subjected to a ground excitation, the mass vibrates because the structures in between pass some excitation energy to it. If a conventional hydraulic actuator with a servovalve is adopted here besides a soft spring, the low frequency motion of the mass will be under the control of the actuator within its bandwidth. So the actuator acts as a vibration isolator. When a high frequency excitation is presented however, the servovalve cannot adjust sufficiently quickly and the actuator will behave like a rigid rod. Instead of isolating the mass from the ground disturbance, the conventional actuator transmits it directly to the mass. The only way for the actuator to control the high frequency excitation is to have a higher response bandwidth, which is impractical or even impossible. On the other hand, the high frequency harshness is not a problem if a DFA is used to replace the conventional actuator. Because the DFA works only with on-off control valves, it does not at all restrict the flow between the actuator and the accumulator or the reservoir. Therefore, little or no harshness is passed through it to the mass. Also the fast switching speed of on-off valves, which can be as high as 100 Hz, offers the big potential for the DFA to enhance its force bandwidth. The DFA is therefore able to make a significant contribution to improvement of the high frequency  10  Chapter 2. Discrete Force Actuator  11  I  Figure 2.4: Schematic 1 DOF mass-spring system with a DFA isolation of an active suspension. The DFA can also ameliorate the energy consumption problem of the active suspension. There are basically two possible operation conditions for the DFA: 1.  Resisting the relative motion between the mass and the ground.  For instance,  when the mass tends to move away from the ground, the valves may operate such that chamber B is connected to the accumulator and in high pressure state and chamber A to the low pressure reservoir. The fluid in chamber B will be forced by the piston to flow into the accumulator without any restraint and part of the kinetic energy of the mass will be converted to potential energy and stored in the accumulator. In this situations, the DFA is working just like a pump. Note that the DFA is taking away some energy from the mass by storing it in some other place in the system instead of dissipating it through some restriction mechanism like the throttle in an active suspension. 2.  Exerting a drive force in the direction of the relative motion between the mass  and the ground. Wtih the mass moving in the same direction, but with the positions of  Chapter 2. Discrete Force Actuator  12  the valves reversed, pressurized fluid is forced into chamber A from the accumulator in order to supply such a drive force. Obviously the system is absorbing more energy which does not come from any extra external power resource, instead, from where it has been stored. These two operation situations are actually opposite processes in terms of energy flow and complete an energy cycle 2.2  Overall structure  For different applications, the DFA may have different motion forms and structures such as a linear DFA and a rotary DFA. But all of them have the same principle of operation. In the rest of the chapter, we will only discuss a particular kind of DFA — the rotary DFA with the structure shown in Fig 2.5, which has been designed to find application in a DFS. The DFA consists of a stator and a rotor. All arcs within the stator and rotor have their centres at the centre of the shaft of the rotor. There are eight chambers formed by the teeth on the rotor and the stator. The moments of working areas of the chambers with respect to the centre are so designed that they form a sequence in the order of 1, 2, 4, 8, 16, 32 and 64, with an extra unused, or dummy, chamber. The pressured oil in the working chambers can cause the rotor to move in either direction: clockwise or counterclockwise, where counterclockwise is defined as positive. Therefore chambers 2, 16, 32 are positive and chambers 1, 4, 8, 64 negative. (See Figure 2.5) Each working chamber is connected to a two-position, three-way valve. The valve connects its chamber to either a high pressure accumulator or a low pressure reservoir. All valves can be put together physically to form a compact valve block. This valve block will work under the digital command signal which has as many bits as the number of  Chapter 2. Discrete Force Actuator  13  Figure 2.5: Cross section of a rotary DFA valves.  2.3  Principle of operation  As a link between controller and plant, an actuator receives the control signal as its input and outputs a control force. If the coupling between actuator dynamics and plant dynamics is quite weak and therefore negligible, the output of the actuator will only depend upon the control command — the only input, and be unrelated with the plant. The DFA is basically such an actuator. In a DFS, control signal comes to the DFA in a digital form, each bit controlling the corresponding valve by switching between two states: 0 (low potential) and 1 (high potential). This series of binary bits forms an integer number /,„ which is the real input of the DFA. After receiving a control signal, by switching mechanism valve block will make some chambers linked to the accumulator(high pressure) and others to the reservoir(low pressure). The high pressure fluid in positive chambers produces a counterclockwise moment  14  Chapter 2. Discrete Force Actuator  while an opposite moment is caused by the pressurized fluid in negative chambers. The difference of these two moments constitutes the output of DFA. The moment output is then converted to the linear suspension control force through a moment arm which connects the rotor of the DFA and the car body. If the area moment of-the ith chamber is notated by J,-, the high pressure by Ph and the low pressure by Pi, then the output of DFA can be expressed as u = jr( J P Ci  i  + dJP)  h  i  i  (2.1)  l  where c,- and d{ are constants with four possible combinations: c,- = l,di = 0 — for positive chambers in high pressure state. Ci = —l,di=0  — for negative chambers in high pressure state.  Ci = 0, di = 1 — for positive chambers in low pressure state. Ci = 0,di — — 1 — for negative chambers in low pressure state. We rewrite it as u = J P J2c 2  + J PiJ2d 2  i  1  h  i  i  1  i  t=i  »=i  where J\ is the moment of area of the smallest working chamber. Obviously the summations in the above equation are integers. If the first one is notated as I , the second out  one will equal —27 — I t with the given particular structure and code fashion.^Thus we ou  have u = J {P -P )I -21J P 1  h  l  out  l  l  (2.2)  In order to make the concept of digital force easier to understand, the output expression may be simplified by ignoring the constant term 2U\Pi which is generally quite small compared with the first term when a very high working pressure is used. u = MPh ~ P,)Iout  (2.3)  Chapter 2. Discrete Force Actuator  15  Divided by J\(Ph — Pi), the output will be normalized to u  nor  =I  out  — a series of integers.  Since the working chambers are coded in the same fashion as the multi-bit digital signal, the output of DFA I  oui  will be identical to the input digital signal J . For instance, if m  the signal has a value of I{ = 33 which can be decomposed to a series of 2's powers: 32, n  2,-1, then the valves will switch in such a way that only these three chambers are highpressure-connected. That obviously will result a normalized counterclockwise moment of hut = 33. So we have /,„ = I  out  =I  (2.4)  or u  n0T  =I  (2,5)  Now it is easy to see that a DFA only outputs a limited number of force levels which are corresponding to a series of discrete input integers. The number of force levels only depends on the number of working chambers n through the relation 2". For our particular DFA, the normalized maximum output is 50, which is achieved by switching only the positive chambers to the high pressure. Similarly, -77 is the lower-limit. So the DFA can generate any integer force moment output between these two limits.  2.4  Actuator Modelling  In the previous section, an ideal I/O relationship has been set up (See equation 2.5), which is based on the assumptions that there is no pressure drop through the valves and that the pressure in the working chambers is independent of the rotor motion. In reality, however, they are not true. Not only do the chamber pressures differ from the accumulator or the reservoir pressure, but also the pressures in different chambers (positive and negative) are not equal to each other at all. Therefore, we need to study a more accurate model of the DFA.  Chapter 2. Discrete Force Actuator  16  Figure 2.6: Schematic fluid flow of hydraulic system When the DFA is further modeled mathematically, we have made some reasonable assumptions: • No leakage occurs at valve'ports, between different working chambers and at any other concerned places. • The liquid in the pipeline between actuator and valve block and between valve block and accumulator is ideal and has no mass. In other words, the inertia effect and friction are ignored. • The liquid is incompressible. Under these assumptions, the steady state equations for the hydraulic system can be derived as follows. (Refer to Figure 2.6) 1.  Actuator.  There are two working conditions for both positive and negative chambers: connected to the high pressure or to the low pressure, under each of which the pressure in the  17  Chapter 2. Discrete Force Actuator  chamber will have different expressions: • positive chambers connected to the high pressure.  Qt where Q% = j£  =  = -  tff  (2-6)  *  flow rate of positive working chambers of high pressure total moment of the radial cross section area of the positive working chambers of high pressure  • positive chambers connected to the low pressure.  Qt = -Jt {  (2.7)  d  t  • negative chambers connected to the high pressure. Qh = Jn^t  <-) 2  8  • negative chambers connected to the low pressure.  (2.9)  QT = Jr^ 2.  Valves.  The valves used in a DFS are modeled as an open-closed orifice. Thus the flow equations for the valve between the actuator and the accumulator and the one between the actuator and the reservoir are -  P )A C \r  P  Qi = SIGN{P -  P )A C \r  P  Q  h  = SIGN(P  a  h  a  P  a  |  (2.10)  and  where  T  l  d  P r |  (2.11)  Chapter 2. Discrete Force Actuator  18  Ah, Ai = valve areas for high or low pressure passages Cd = discharge coefficient at each valve port P ,P a  = pressures in accumulator and tank  r  p = fluid density More particularly, equations 2.10 and 2.11 can be rewritten for the valves connecting the accumulator and positive working chambers, the accumulator and negative chambers, and so on. 2|^ +  Qt  Ql  A  Qt  -AtC  Ql  AjCd^  h  Cd^  Pa  2 | / V - Pa  (2.13)  2|P,+ - P | r  d  2|Pr-PJ  (2.12)  (2.14)  (2.15) (2.16)  3.  Accumulator.  For the type of spring-loaded accumulators, we have Qa =  dV . a  dt  (2.17)  and [9] dVa  dPa  =  dt ~  Ca  dt  (2.18)  Chapter 2. Discrete Force Actuator  where V  a  19  = accumulator volume  c„  =  A  = cross section area of accumulator  a  A\jk  a  k = spring stiffness of accumulator Combining equations 2.17 and 2.18 and considering the continuity offlowyield a  or ^  = f(^  +  +  OT  '  (2.20)  Substitute 2.6 and 2.8 into the above equation:  Doing integration gives P* = -{Jn-Jt)<l> + Paa c  (2-22)  a  Now we can get the equations for all concerned pressures which make contribution to the torque output, by comparing the valve equations and the actuator equations. They are four non-linear algebraic equations: Pi P t  =  P a ~  J  h  <f>  2  (2.23)  <f>  7  (2.24)  2  (2.25)  d  2 AtC dt> K  d  PH  Pi =  Pa  +  2  K  =  P r ~  2  y  Pi =  Pr  +  2  h  d  A -C h  pi PI"  J  K  t  J  A f C  dt  d  }  <f>  d  d  dt  }  r d<f>.  J  A j C  d  (2.26)  2  dt'  (2.27) Generally speaking, the pressures in the above equations are chamber-related because the ratios of the moments of area of the chambers to valve areas, i.e - ^ r , ^r, 4r and ^= A  h  A  h  A  l  A  l  Chapter 2. Discrete Force Actuator  20  may have different values for different chambers. This makes the analysis of DFA response much more difficult. Under the condition that the area of valve orifice is proportional to the moment of area of the corresponding chamber, however, these ratios will have the same value and be independent of the particular chambers. In other words, all chambers will have the same pressure responses as long as they are under the same working condition. Therefore, the number of the pressures which make contributions to the DFA output is only four, a combination of two kinds of chambers (positive and negative) and two working conditions. The equations 2.23 to 2.26 become Pit  =  P -A  (2.28)  P"  =  P +A  (2.29)  P,  =  P -A  (2.30)  =  Pr + A  (2.31)  fc  +  Pr  a  a  r  (2.33) where A is the pressure drop through the valve and J\ and A\ are the smallest area moment and valve passage area respectively. From the general DFA output expression « =  + P f t f - PHJH -  PfJr  (2-34)  the more compact form can be obtained by substituting equations 2.28 to 2.31 into it  «= Pain -  Jr) +  wt - Jr) - W£  + Jt + Jr  + Jr)  (2.35)  From equations 2.22 to 2.26, it can be seen that the dynamics of the DFA is coupled with the dynamics of the car through the rotor's position <f> and velocity  Chapter 2. Discrete Force Actuator  2.5  21  Effects of Actuator Motion on Force  From the analysis results we got in the previous section, it has been seen that no completely ideal DFA exists in the sense of the independence of the output to the rotor motion. The output of the DFA is always related, more or less, to the relative position and velocity of the rotor to the stator. In other words, the DFA does have some damping and stiffness effects. Since it is the main feature of the DFS for the DFA force generation to be independent of actuator motion, we have to do some quantitative analysis to find out how big the actuator's equivalent damping and stiffness are and therefore to examine the feasibility of using a DFA in the car suspension. To obtain the equivalent damping and stiffness, we convert the actual DFA into an equivalent system with a spring, a damper, and an ideal force generator whose output is entirely independent of the system motion (See Figure 2.7 and note that the DFA output is directly used as suspension control force for simplicity). For such a system, the resultant of the forces u will contain the information about the stiffness K , the damping e  coefficient C and the constant force u if a sinusoidal motion is input. e  c  u =  u - K <f> - C (f> c  </> =  e  e  A sin Lit  or u = u — K Asmut c  e  — C Aw cosut e  (2.36)  On the other hand, the output of DFA with sinusoidal input may be decomposed into a string of sinusoidal waves (Fourier series) because of its periodicity. Neglecting the higher frequency terms, we have u = ao + ai cosut -f bi sinut  (2.37)  Chapter 2. Discrete Force Actuator  22  Figure 2.7: Schematic equivalent system of D F A where  ZTT  Jo  udt  a,\ — — [ " u cos cotdt IT Jo  — f11 sin utdt .ZZ-  6i =  7T  Jo  Now the equivalent damping and stiffness come out from comparing equations 2.36 and 2.37: u K  c  e  C  e  =  a  0  ~ ~~A a i = Au  (2.38) (2.39) (2.40)  Using the model described by equations from 2.22 to 2.26 and the parameters given by Table 2.1, we did simulation for the D F A , with <p = 0.3142sinu;£ and u> = 20'7r rad/s. Figure 2.8 shows an example with force setting equal to 50. The result is shown in  23  Chapter 2. Discrete Force Actuator  Values  Parameters reservoir pressure P initial accumulator pressure P discharge coefficient Cd fluid density p bulk modulus 0 accumulator section area A accumulator length L accumulator stiffness K structure constant Oo smallest moment of J, smallest valve passage area valve switching time r  a  a  a  a  Units MPa MPa  0.1013 10 0.625 974 1.5 x 10 0.01 0.15 1.0577 x 10 0.7854 3.0 x IO" 1.5625/rimesl00.01  kg/m  3  MPa m  3  2  m N/m  6  rad m m 3  6  2  6  sec  Table 2.1: Hydraulic and structural parameters for simulation  -oa H — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r ~ i — i o  oo2  OJO*  ooe  OJM  at  0.13  ai«  aie  i  ate  I  time (sec) Figure 2.8: Sinusoidal output of the  DFA  aa  Chapter 2. Discrete Force Actuator  24  TO aoo aoo •  e t.  If ^  \ \  400-  aoo • 100 •  0-4D  1  1  1  -«D  1 -40  1  1  -30  T O  90  40  input (force level setting) Figure 2.9: DFA equivalent stiffness Figure 2.8. By doing integration, we finally obtain the equivalent stiffness and the equivalent damping coefficients: K  e  C  e  =  0 ~ 564.4 Nm/rad  (2.41)  =  28.5 ~ 32.4 Nms/rad  (2.42)  With the estimation of 0.5m for the length of the moment arm which converts the DFA moment output to suspension force, the variations of suspension stiffness and damping rate due to the DFA are quite small and only present respectively 0 ~ 13.7 per cent of the suspension stiffness and 7.6 ~ 8.6 per cent of the suspension damping. In the load range of —40 ~ 40, where the DFA is supposed to work, the average stiffness is less than 76 Nm/rad — around 2% of the suspension stiffness. Figure 2.9 and 2.10 are plots of K and C versus the force level settings. e  e  Due to the small damping and stiffness of the DFA, the coupling effect of the DFA dynamics with car dynamics can therefore be ignored. In other, words, the control force  Chapter 2. Discrete Force Actuator  25  e  s  input (force level setting) Figure 2.10: DFA equivalent damping coefficient generated by the DFA only depends on the input signal from the controller. This simplification is reasonable and necessary. We do not have to combine the car dynamics with the DFA to form a system too complicated to analyze and design. The simulation results also show us a satisfactory relationship between the constant component of the output u and the input level(see Fig 2.11). It is quite linear and the c  slope (gain of the DFA) varies in the range of 29.9 ~ 30 Nm, quite close to the gain P Jj = o 0  30^771  of the ideal linear system u = P «7i/. Because of the stiffness and dampa0  ing effect, the actual DFA output u will fluctuate around u . The solid lines determine c  the variation range with maximum angular displacement and velocity of 0.3142 rad and 19.74 rad/s.  2.6  Transient Output During Switching  Since the discrete force levels are generated through the switching mechanism, it is necessary to have a look at the transient behaviour of the DFA between any two force levels  26  Chapter 2. Discrete Force Actuator  fe •4£  input (force level setting) Figure 2.11: Input-output relationship of DFA during the switching. A more accurate model which includes the compressibility of the fluid was developed. For any working chamber in DFA, there are totally four working conditions: connected to the high pressure or the low pressure and being switched from the low pressure up to the high pressure or from the high pressure down to the low pressure. With the two kinds of working chambers, positive and negative, eight pressures will be involved to form the output. The equations of them plus the equation for the pressure of the accumulator can be derived in the approach as shown in Appendix B. dP/t  s  dt  e  dP£_ dt dP? dt dP[_ dt  de  (  dt  {  P dO  AJC  A  Aid  (  e dt d$  e  {  dt  AfCg  AjC  e dt  j f  d  2\P+-P \ r  2\P -Pr\ r  (2.43) (2.44)  )  j+ \  3 d6 [  2\Pr-Pa\  \  K  3  )  j£ \  )  (2.45) (2.46)  Chapter 2.  27  Discrete Force Actuator  dt dP~ dt dfl dt dPl dt dPa dt  0,  d9  0  dt  {  pdO  Aj C h  _  AZtfh  o dt (  K  V  J+  de dt  V h  e dt  JJ \  dh  d  0C 0c + V a  a  \  _  AZfii  r  P  )  2 \ P - - P \  V  P  -  Jt P  AC dl  a  +  d  ]  )  \ 2\P ~-Pr\ d  2|P" -  Pa  (2.47) (2.48)  r  Ju  2|^ -Pa|  d  V  J+  P  2 P7  A~ C  J2\P+-P \  d  0  \,  d  t  2|P/-P | A % d  d  j+  Aj C  P  P  A$ c  3 de K  a  2 \ P - - P  J-  K  P e  l2\P+-P \  d  )  )  (2.49)  (2.50)  (2.51)  a  where subscript u indicates the case of switching from low to high or up and d from high to low or down. If the pressure difference under the square root is smaller than zero, the sign of the square root should be opposite. The transient process therefore can be described by equation 2.43 to 2.51. Similarly, the output has the same form as equation 2.25 but contains more terms. u  =  Zi(P?J?-P -Jr) t  i = h,l,u,d  (2.52)  With the parameter values in Table 2.1 and a valve switching time of 10 ms, the transient responses of the DFA were obtained by using the fourth order Runge-Kutta method. In the simulation, the angular velocity of the DFA was taken as a constant and the initial position of the rotor was just in the right between of the stroke. The dynamics of the suspension was not included because of its quite limited influence on the actuator dynamics. Figure 2.12 illustrates the transient pressures of the DFA. It has been observed that the pressures in the switching chambers go up or come down smoothly and arrive at the new steady state quickly. There is no evident pressure peak. The variations of the pressures in the accumulator and the chambers not undergoing the switching are quite small and only present less than 0.53% of the value of the high pressure state. Figure 2.13 to 2.16 show the transient outputs of the DFA influenced by different factors. For the  Chapter 2. Discrete Force Actuator  28  time (sec) Figure 2.12: DFA transient pressures during switching various rotor velocities, the DFA again displays a small damping. As we see, the resulted output differences when the steady state is reached are trifling in spite of bigger values occurring during the switching. Also, the output seems not sensitive to the areas of valve orifices. The accumulator volume, if not smaller than the certain value (about 600 - 700 cm ), does not affect the output very much either. 3  Since the DFA has a quick response to the force level control signal and its output can reach the steady state as soon as the switching ends, which takes only 10 ms or 10% of the sampling period, the time history of the DFA output is quite close to the ideal one (see Figure 2.17).  2.7  Hydraulic Power Losses  A major consideration for a DFA is power losses which have many different forms such as internal leakage losses, internal friction losses and throttle losses at valves.  29  Chapter 2. Discrete Force Actuator  6 fe;  9  740 730  0 = brad/s —i  0X08  r—i  O00A  r  T  1 OuOOB  1  -1—  001  time (sec) Figure 2.13: DFA transient outputs with different rotor velocities  time (sec) Figure 2.14: DFA transient outputs with different valve orifice areas  Chapter 2. Discrete Force Actuator  30  V = 960cm1 3  a  V = 1500cm  3  a  I OJOOS  1  1  1  OJOM  r — — i MOB  1 OOOB  1  1 OOI  r— O012  time (sec)  Figure 2.15: DFA transient outputs with different accumulator volumes  Figure 2.16: DFA transient outputs with different force level settings  Chapter 2. Discrete Force Actuator  31  11  n ideal  — actual  Figure 2.17: Comparison of the actual and the ideal output of the D F A Internal leakage losses, which were believed to be the largest contribution to the overall system losses, occur as the result of fluid leakage between adjacent chambers. They are mainly dependent on the clearance between rotor and stator. With a dimension of 0.05 mm for this gap and under the assumption that the inertial effects due to centripetal acceleration and gravitational effects would be negligible, a maximum loss of approximately 70 W was resulted in [5]. In the load range from -30 to 40, where the D F A is expected to operate, the average power loss is in order of 40% of the maximum, that is, around 30 W [5]. A pressure loss estimation has been done for the frictional flow losses through each passage between valve block and each actuator chamber [5]. According to the results of the pressure drops, a corresponding power loss around 10 W was obtained by multiplying pressure drops by volume flowrates. Also, some losses which cause a pressure drop are expected to happen at valve orifices. The product of the pressure drop and the volume flowrate will give the estimation of the  Chapter 2. Discrete Force Actuator  Accumulator  Compressed Air  32  Solenoids & Air Valves  o  Terminal |— Board  Laptop Pump oocroooooao  Electric Motor  Figure 2.18: Setup for DFA test power loss, which has been found at the level of 16 W with the biggest valve orifice area equal to 10 m . -4  2  The total hydraulic power losses can therefore be estimated as the summation of the above individual contributions. It lies in the range of 56 W to 96 W.  2.8  D F A Test  In order to confirm the results of the theoretical analysis and numerical simulation, and to make the idea of DFA closer to the final practical application, efforts were also given to the DFA physical model and related tests. Unfortunately, however, we could not obtained what we expected because of mistakes made in manufacturing and design.  2.8.1  Test setup  Chapter 2. Discrete Force Actuator  33  A diagram of the test setup is shown in Figure 2.18. It basically consists of four parts: DFA assembly, hydraulic power, data acquisition and control, and rotor excitation system. The hydraulic power offers a high pressure source by the accumulator and the pump with a capacity of 4.5 Litre/min, and a low one by the reservoir. The introduction of high pressure oil into the working chambers is controlled by seven two-position, threeway hydraulic valves whose opening and closing are in turn under the control of the same number of the air valves (Refer Fig 2.19). These air valves with working voltage of 12 volts can be actuated by digital signals coming from the terminal board. One strain gage bridge torque sensor is mounted on the rotor shaft and sends one channel of analog signal to the board. One card of ACJr — an interface card of data acquisition and control from Strawberry Tree Inc., is used as a communicating connection between the terminal board and a laptop computer. This card reads the analog input signal from the torque sensor into the RAM and logs the data to the disk if needed. It is also capable of taking the command of force level setting from the keyboard and sending it to the terminal board in a digital form. The variable electric motor is able to drive the rotor and to give it an approximately sinusoidal swing motion with any frequency up to 25 Hz. The detailed DFA assembly drawing is illustrated in Figure 2.19, which is helpful to understand what we described above and what we are going to discuss next.  2.8.2  Test procedure  With the test setup described above, the test procedure was planned as follows: • Static Response.  Let the rotor held still, change the input force level settings  within the limits and measure the corresponding steady state torque outputs. The measured points on the output-input diagram, would show a static DFA outputinput relationship. By this step, we could obtain the information about the linearity  Chapter 2. Discrete Force Actuator  Chapter 2. Discrete Force Actuator  35  of torque vs. force level settings. • Transient Response.  To perceive the switching behaviour of the DFA, still keep  the rotor stationary and give any force setting to switch the valves. Keep track of the transient torque response by setting the data reading period as small as 5 ~ 10 ms. • Equivalent Stiffness and Damping Coefficient. Employ the electric motor to make the rotor oscillate sinusoidally with a constant frequency. Measurements of the DFA response to such a sinusoidal excitation offer the necessary information to find out the equivalent stiffness and damping coefficient of the DFA if higher frequency components are ignored. Also by setting different motor speeds, we would be able to discover the frequency characteristics of the DFA. 2.8.3  Failure analysis  During the process of setting up the test rig, it was observed that the high pressure could not be established even when all hydraulic valves were closed. By analysis and check, it was found that oil leakage occurred at the valve ports. Some measures including putting more seals between the spool and the cylinder were taken to solve the problem effectively. When we switched the valves, however, we faced the same problem again: as long as any one valve was kept open, the pressure would drop from 400 psi before switching to 250 psi and stayed there. The pressure difference between working chambers could not be set up, which is crucial to the response measurement. Measurement revealed that the clearance at rotor tooth ends and between rotor and face plate, rotor and valve block (Refer Fig 2.19) was 4 ~ 5 times bigger than the drawing requirement. The face plate deflected excessively when pressurized oil was switched into the working chamber. In spite of the various measures we tried many times, we could not eliminate the internal  Chapter 2. Discrete Force Actuator  36  leakage or reduce it to a satisfactory level. In brief, the internal leakage between different chambers prevented the possibility of establishing a pressure difference and continuing the test. We owe the failure to the following reasons: • The incapability and mistakes of manufacturing. • The complicated structure design which increases the degree of the manufacturing difficulty. From the information got during the test setup, we are convinced that the failure has nothing to do with the DFA idea itself and that with a simplified structure design and the ensured manufacturing quality, satisfactory physical test results woull be obtained.  Chapter 3  Control System  3.1  Model Establishment  3.1.1  Car model  In our research, we will use a single wheel station or quarter car model with some necessary and reasonable assumptions. First of all, the sprung mass is treated as a rigid body undergoing only heave motion. Secondly, in the frequency range of interest, it will be considered adequate to think of the tire making contact with the ground at a point [11]. Finally, the tire can be modeled as a single spring with a constant stiffness. This is because the tire damping is so located in the system that it does not make much difference to suspension performance whether or not it is included [11]. Under such assumptions, a quarter car model is set as shown in Figure 3.20 A quarter car model reflects the most important characteristics of the car suspension design. It not only includes a proper representation of the problem of controlling wheel load variations but also contains suspension system forces which are properly applied between wheel mass and body mass. By such a simple model, most performance criteria such as discomfort(body's vertical acceleration), working space, wheel load variation, static attitude variations and actuator force can be calculated. Furthermore, because a quarter car model is described by few design parameters and has few performance parameters, it is much easier to compute performance, to apply optimal control theory to derive control laws and to map and understand the relationships between design and  37  Chapter 3. Control System  38  Figure 3.20: Quarter car model with a DFS performance than a full car model. In establishment of a state-space description of the quarter car model, we include the following factors in our considerations: easy expression of performance index, easy application of optimal control theory and easy state measurement. With these considerations, the states are chosen as follows: X  X\  —  Xb  xi  =  ib  (3.53)  Then the state-space description of the system is derived by using Newton's equation of motion: —  £4  X\  —  X2  x  =  -KbXx/Mb - C(x - x )/M  2  2  4  b  + u/Mb  (3.54)  Chapter 3. Control System  X3  x  X4  =  4  39  Xg  K xi/M b  w  + C(x - x )/M 2  4  b  - K x /M w  3  w  - u/M  u  Its compact matrix form is x = A x + Bu + D i „  (3.55)  where 0  Xi  x =  A =  x  -K /M b  b  2  0 K /M b  X4  B =  0 l/M  b  0 -l/M  w  1  0  -1  -C/M  b  0  C/M  0  0  1  C/Mw  w  -K /Mw w  b  -C/M  u  0 D =  0 -1 0  For the system with a dynamic absorber (see Figure 3.21), there are two more states which have been defined as (3.56) —  X  a  Chapter 3.  40  Control System  Figure 3.21: D F S scheme with a dynamic absorber where x is the absolute displacement of the absorber mass. The system can still be dea  scribed by the standard equation 3.55 with some changes only in the coefficient matrices, 0 1  0  -1  0  0  0  0  0  0  0  0 0  0  1  0  0  -K /M b  A  b  =  0 -K /M ^  K /M b  w  w  =  a  0 0  0  0 0  0  0  0  1/M B  -C /M  w  0  b  0 -1/M  D=  -1 0  W  0  0  0  0  K /M  w  a  -1 CJM  a  C /M  w  a  w  0 -KJM  a  1 -CjM  a  Chapter 3. Control System  3.1.2  41  Road surface model  The vertical displacement of road surface is taken as a stationary process as the vehicle traverses the road surface. Although it can be expressed in the time domain, a spectral description in the frequency domain is commonly used. Since there is only one input in a quarter car model, the road excitation description requires only the specification of a single-profile spectral density. The single-slope spectrum of the form (3.57)  S(u) = ku~  m  where m is a constant around 2.5 and v the wave number, provides an adequate model for many purposes [10]. Roughness coefficient k varies widely according to the type of road. Spectrum of 3.57 can be expressed in terms of frequency instead of wave number. If the car has a constant speed [/(m/s), the excitation frequency /(Hz) and the wave number v are related by f = Uu By comparing the mean square value expressions of / °° S(v)dv and / °° S(f)df, and 0  H o S(u)du  0  and / °° S(f)df, we get 0  S(f) = S{yy% = kU ~ fm  l  (3.58)  m  df  or S(UJ) = ^ ( 2 7 r ) - i f c C / - a ; m  1  m  1  m  (3.59)  If we take m in 3.59 as 2, then we come up with the most popularly used road surface model: an integrated Gaussian white noise process with a zero mean. S(u) = ^  (3.60)  42  Chapter 3. Control System  Its corresponding time domain description is of the form av  (3.61)  where v is a zero-mean, unit spectral intensity Gaussian white noise and a has the value  This model will be adopted in our research.  3.2  Linear Stochastic Quadratic Optimal Control  It is convenient for active suspension design if there is a direct method for finding good control laws. Linear stochastic optimal control theory has been widely used for this purpose [11]. In the discussion of this section, perfect and complete state measurements are assumed and the dynamics of the control sensor and the actuator is neglected for the sake of simplicity. The second assumption is understood as reasonable from the discussion of the previous chapter.  3.2.1  Procedure of control law design  Generally, the dynamics of the system can be described by the matrix equation x(t) = Ax(i) + Bu(t) + Dv(t)  (3.62)  where v(t) is the disturbance vector which is assumed to be a Gaussian white noise process independent of the state x(<) and the control signal u(£), with zero mean and covariance matrix P. In the steady state, the performance index is  (3.63)  43  Chapter 3. Control System  where Q is a symmetric non-negative definite matrix and R a symmetric positive definite matrix. The optimal control law which minimizes the performance index 3.63 for a system governed by equation 3.62 is given by u(<) = -Gx(<)  (3.64)  where G is the constant gain matrix. So far, system model, performance index and control law are all expressed in continuous time. Due to the obvious reason that the control system will be implemented by a digital computer, all these expressions have to be sampled to get a corresponding discrete time version. W i t h the assumption that the control signal keeps unchanged during the sampling period T , system model 3.62 is discretized as (see Appendix A) x[(Jb + l ) r ] = $x(fcT) + r u ( j f c r ) + / e Dv[(fc + 1)T — \]d\ Jo AA  (3.65)  where $  =  e  A  (3.66) T  =  ge^dXB  Coefficient matrix D can not, however, be discretized in the same way used in discretizing B because assuming constant disturbance between sampling instances, necessary for such discretization, does not make any sense. The approach we used to solve this problem is shown as follows. Assume the desired discrete-time model has the form x[(Jb + 1)T] = *x(ifcT) + Tu(kT) + Slw(kT)  (3.67)  where w(fcT) is a discrete-time white noise, independent of x(fcT),u(A;T) and with zero mean and diagonal covariance matrix P j . Since both equations 3.65 and 3.67 describe  Chapter 3.  Control  44  System  the same process, they should possess the same stochastic properties, say, the second order moment, i.e E[(*x(kT) + Tu{kT) + Qw{kT))($x{kT) + Tu(kT) + flw(fcT)) ] = r  E[($x(kT) + Tu{kT) + [  e v[(ifc + l)T -  T  AA  Jo  (Qx(kT) + Tu(kT) + f  \]d\)  e v[(fc + 1)7/ - \)d\) \ AA  Jo  T  Doing simple manipulation with noting the independence between w and x, u, and between v and x, u leads to  nP n = I j T  d  Jo Jo  e  A A l  D(e  A A 2  D ) £ { v [ ( f c + 1)7/ — Ajjv-^fc + 1)7/ — A ]}dAidA r  2  2  where the correlation of the disturbances E{v[(k + 1)7/ — Ai]v [(A: + 1)T — A ]} equals T  2  P(5(Ai — A ) for a diagonal P. Then we have 2  nP n  T  d  = =  /  e  Jo  / e T  D [ / (e  A A l  A A  Jo  AA2  Jo  D(e  A A  D) P^(A! - A WA ]rfA T  2  2  1  D) PdA  (3.68)  T  w(fc) can be chosen so that its covariance matrix P is of the form P d  d  = l a , i.e, all 2  elements of w(fcT) are independent of each other and have the same variances cr . Then 2  fi can be found by ntf = -(  e DD (e ) <fAP  T  <7  2  AA  r  AA  T  (3.69)  Jo  Similarly, performance index 3.63 has its discrete-time form as follows (see Appendix A): J =  E^2[x {kT)u (kT)] T  T  k=0  Qi  Ox  ' x(*r)'  Of  R  _ n(kT)  where  Qi = /V(r)Q£(r)<fT Jo  t  (3.70)  Chapter 3.  Control System  Oi  =  Ri  =  45  [ f(T)(Qr,(T)  +  T  Jo  0)dT  / 0? (r)Qr/-(r) + r/(r)0 + R)dr Jo r  *(r) =  e ' A  f e ^- U\B  77(7) =  T  A  x  Jo  Noting that the performance index without cross terms is easier to deal with, we will eliminate them by defining ' Q2 = R2  Qi - O x R ^ O f  (3.71)  Ri  =  and u {kT) = R^O x{kT) T  x  + u(kT)  (3.72)  Substituting 3.71 and 3.72 back into 3.70, we finally get 00  J=  Ej:ix (kT)uT(kT)} k-o T  Q  2  0  0  x(fcT) "  R  (kT)  2  Ul  (3.73)  Solving the corresponding discrete-time Riccati equation s = Q + * s * - $ sr(R + r sr))- r s* T  T  T  2  2  1  T  (3.74)  where <*> = «!> — T R ^ O ^ , the feedback gain matrix which constitutes the control law Ui(fcT) = —  Gix(fcr)  is given as G  a  = (R + r ^ r ) " 2  1  ^ *  (3.75)  In order to find the optimal feedback gain G in u(kT) = — Gx(fcT), we just need to compare the following two equations: u(kT)  = -Gx(fcT)  u(fcT) =  Ui^-R^Ofx^T)  =  -(G +Rj 0[)x(fcT) 1  1  Chapter 3. Control System  46  that gives G = G + Rj Of  (3.76)  1  1  For the quarter car model, in order to represent all performance criteria such as discomfort, working space, wheel loadfluctuationand actuator force, the performance index should contain at least the following items: body vertical acceleration x , suspension deb  flection x — x , tire deflection x — x and control force u. Thus we have b  w  w  J = E  g  roo  [q x\ + q (x - x^) + q (x - x ) 2  Jo  a  b  b  w  w  2  g  + q u ]dt 2  u  or  J = E f°[x u ] r  r  Q O  t  O  X  R  u  dt  (3.77)  where q (K /M ) a  Q =  b  b  2  +q  b  a  b  a  b  b  0 -q K C/Mi a  b  0 -q K CjMl  2  q (C/M )  q K C/MZ a  q K C/M  a  0  2  -q K /M?  b  a  -q (C/M y a  b  0 q  0  w  -q (C/M ) a  b  0  2  o=  q (C/M y a  b  b  -q C/M a  2  0 qaC/M?  R = q + qa/Mt a  Once cost function 3.77 and system model 3.55 have been determined, an optimal control law is not difficult to obtain by the approach presented before.  3.2.2  Unavailable state information  In the previous section, availability of full state information was assumed. In practice, however, some state variables may be too difficult to be measured. The relative displacement of the wheel mass with respect to the ground is an example. We have two options to deal with this problem in the car suspension.  Chapter 3.  Control System  47  W  0  1-1  $  ——'  ,-i  c  -y  —-(s.  -G Figure 3.22: Block diagram of the system with a Kalman filter Using Kalman filter as a state estimator A Kalman filter is a state observer which is driven by measurements of the system and outputs the state estimation. The estimated states are then used as inputs to the optimal feedback gain. Figure 3.22 shows this process in a block diagram for an active suspension. The feedback gain of a Kalmanfiltercan be found as follows. The system to be estimated is described by equation 3.67 and the output equation  y{kT) =  Cx(fcr) + w {kT)  (3.78)  0  where w (fcT) is the sensor noise which is assumed to be white and with a covariance e  matrix V . Then the estimator has the form [1] x[(Jb + 1)7/] = $x(ifc7/) + Tu(kT) + L(kT)[y(kT) - Cx(kT)}  (3.79)  Solving the steady state Riccati equation  s = *S9  T  -  #sc '(v + c s c ) - c s $ + n p n j  T  1  i  d  (3.80)  48  Chapter 3. Control System  and L = *SC (V + CSC )T  T  (3.81)  1  gives the steady state Kalman filter gain L. The control law will be u(fcT) = — GSt(kT) where estimate x(fcT) is obtained from equation 3.79.  Ignoring some unknown state information We will see from the simulation results that high frequency performance of the active system is worse than the passive system. It has been argued that this high frequency harshness is due to the term  in full state feedback control  —53X3  u  = —g\X\  —  gx 2  2  —  g3%3 — ^4X4 and that #3 can be set to zero without hurting any performance index [3]. That means the state £ 3 is no longer needed to construct the control signal and by coincidence it is the one unavailable. Based on this argument, we just simply ignored the tire displacement in the feedback. The resulted control is of the form u(kT) = -gix {kT) - g x (kT) - g x (kT) x  3.2.3  2  2  4  4  (3.82)  Limitation of LQ method  From the procedure using the LQ method, we have noticed that once the system model is given, the control law u(fc), and therefore the system performance will only depend upon the weighting matrices, more specifically speaking in the case of quarter car model, the four weighting coefficients. Different combinations of weighting coefficients lead to different performances and all of them are optimal only in the mathematical sense. So in some cases where the physical optimal performance criterion can not be easily expressed by some particular combination of weighting coefficients, the arbitrariness in choosing weighting coefficients will result in the deviation from the real optimum in a physical sense. For instance, we only know the increase of the body acceleration weight will cause  Chapter 3. Control System  49  the decrease in corresponding RMS body acceleration, but we don't know exactly how large the weight should be in order to achieve a desired value. A series of tests in choosing weighting coefficients and calculating the performance, therefore, is common practice and that makes LQ actually a test-try method. Another limitation of LQ lies in its requiring full state information in feedback for the optimum. In many situations, it is very difficult or even impossible to measure all state variables. The relative tire displacement in quarter car model is an example. Such dilemma will obviously lead to the loss of the optimum. Aiming at weakening the influence of these two limitations of LQ, another control scheme is introduced, which will be discussed in next section.  3.3  Conditional Minimum M . S . Control  One major difference between conditional minimum mean-square control (CMM) and LQ is in the construction of the performance index. In CMM, the performance index is no longer constituted by a linear combination of quadratic form of each performance criterion, instead, by MS values of a goal and some constraints. Among all performance criteria, usually the most concerned one is chosen as the goal and the others Jorm the constraints. Any suitable analytical or numerical optimization approach can be applied to obtain the minimum goal under the constraints as long as the analytical expressions for goal and constraints can be derived.  3.3.1  Control law construction  Except for sports car, the performance emphasis should be put on the ride quality. Thus in the study of a quarter model, the control purpose is to keep the body acceleration as small as possible under the condition that suspension and tire displacements do not  50  Chapter 3. Control System  exceed the given limitations, i.e. to minimize E[x ] with constraints E[(xb — x ) } < ri 2  2  w  and E[(x - x ) ] < r . 2  w  g  2  The procedure of determining optimal gain matrix using CMM consists of two steps: • derive the analytical expressions for Efx^.Efi ,]), E[(xf, — x ) ](E[x ]) and E[(x — 2  2  w  2  w  x ) ](E[x ]) as functions of feedback gain along the way of impulse transfer function 2  2  g  —• spectrum —* M.S. value. System model:  x(f.) = Ax(t) + Bu(t) + Dv(*)  (3.83)  u{t) = -Gx(i)  (3.84)  Control law:  where  G = [ 9\ 92 93 94 Then we have  x(<) = (A - BG)x(t) + Dv(t) In order to find the transfer function between x and each Vj—the jth component of w, just let the other components be zero: x(<) = (A - BG)x(i) + DjVj(t) where Dj is the jth column of D. Taking Laplace transform: sX(s) = (A - BG)X(s) + DjVj(s) gives the transfer function:  ^[^-(A-BG)]" ^ 1  (3.85)  Chapter 3. Control System  whose each element Xi/Vj(i  51  = 1,... ,4) is the individual transfer function between the  ith state and the j'th disturbance. Because of the independence of each disturbance, component, the spectrum has the form [8] (3.86) From linear system theory [8], we know  J—oo +oo  ui S {u)dw 2  /  Si  -oo  In our case, Vj is unitary intensity white noise, i.e. S (u>) = 1- Therefore the M.S. values Vj  for goal and other two constraints have the forms as follows. E[xl]  =E / "  E[xl\  =  £/"  E[xl) =  £/_  w dw  (3.87)  dw  (3.88)  du  (3.89)  2  Wj  Simple formula are available [8] for doing the integration. These equations can be rewritten in another more general form: E[xl]  f3(9i,92,93,94)  (3.90)  E[xl)  f\(9i,92,93,9i)  (3.91)  E[xl]  h(g\, 92, g3, 9A)  (3.92)  where all / , are known algebraic functions. • minimize fz under the constraints f\ < r i , /  2  < r by using Lagrangian Multiplier 2  Method [6] to convert the problem into one with no constraint.  52  Chapter 3. Control System  First convert the constraints from inequality form to equality form by redefining the constraints as Fx = F  2  =  fi-n  + c  / -r 2  2  + c*  2  where c\ and c are variables in the real number range. Then introduce a new function 2  = /a + X1F1 +  F(gi,g2,g3,g4,ci,c ,\i,\ ) 2  2  XF 2  2  (3.93)  where A and A are Lagrangian multipliers. x  2  Now we only have to minimize F without any constraint. Solving following equations  ff = o ; = i,...,4 m dF . dci  SE  =  0 ,'=1,2  = 0 z=  or  l,2  g = 0 i-l,--.,4 fi c,  gives out the optimal gain G = [ g  1  (3.94)  (3.95)  = n . = 1,2 =0 g g 2  z=  3  l,2  g ]. In the case of too much manipulation 4  work involved, some numerical methods may be used.  3.3.2  Advantages and limitation  CMM has some obvious advantages over LQ method in deriving the feedback gain matrix for a car suspension control system. First of all, a more meaningful and clearer performance index can be given in terms of goal and some constraints. Such a performance index reflects exactly what we expect to get with the control system. Thus the optimum  Chapter 3. Control System  53  under it exists not only in mathematical sense but in a physical sense as well. In the case where handling ability is the top concern, just choose the tire loadfluctuationas the goal. Furthermore, for the given constraint values which reflect the performance requirements for a particular type of cars, only one calculation is needed. Another advantage of CMM is that it can guarantee the optimum in all situations where only limited state information is given. For instance, in a quarter car model, if the tire relative displacement is not available, CMM can still give out the minimum body acceleration in the condition that only the other three states are fedback. The major problem in using CMM is the complicated and tedious process of deriving analytical expressions for the goal and constraints. Since the inverse of a system matrix having an order doubling the degree of freedom of the system is involved and such inversion can only be done analytically, it seems impossible to avoid the cumbersome manipulation and calculation, especially for systems with high degree of freedom. Also CMM is generally a single-stage optimal control scheme which only ensures the optimum at each sampling point instead of through the whole process. The optimum of the whole process can be achieved by CMM only for stationary processes. Fortunately, in car suspension problem, any concerned process can be approximately treated as a stationary one. 3.4  Step-wise Reference Control  Due to introducing DFA into the control system, a new control concept is needed to cope with the problem of limited control force levels coming up with the DFA. In other words, any control law implemented by DFA consists of only a limited number of values. Such a control law based on some other control strategy but implemented by limited number of levels is called step-wise reference control(SRC).  Chapter 3. Control System  54  quantization  »„(*r)  DFA u{t)  u (kT) q  Figure 3.23: Non-linear model of the DFA  3.4.1  Quantization of the control force and construction of the control law  If u (k) is the control force derived under some control laws, say, LQ or CMM, the 0  real input of the DFA is actually u (k), the quantization of u (k) which makes the q  0  DFA output a step-wise reference control u(t). See Figure 3.23. In quantization, a constant quantization level is assumed. Suppose that the DFA has n output levels and its resolution is u . Then u (k) has to be quantized with n bits before entering the DFA. s  0  So u (k) has the form (if the DFA has a unit gain) q  ««(*) = E " -#« 2i lfl  »=i  where a,(fc) is either 1 or 0. By changing the combination of a;, u (k) can cover all the q  following values:  u , 2u ,3u ,..., (2 - l)u n  s  5  s  s  Apparently the quantization level is A = u . If u (k) is not exactly the multiple of u , B  0  a  there will be an error between u (k) and its quantized version u (k), which is smaller 0  q  than A . There are two ways to quantize u (k): one is to simply truncate it and the other to 0  round it off. Since rounding-off gives the closest approximation to u (k) or the smallest 0  Chapter 3. Control System  55  u (kT)  u (kT)  q  q  A u (kT)  • u (kT)  0  0  (a)  (b)  Figure 3.24: Quantization characteristics, (a) truncating, (b) rounding-off quantization error(see Figure 3.24), it is used in constructing u (k). Therefore the stepq  wise reference control law in discrete time domain is u (k) = INT[u {k)/A + SIGN(u (k))/2]A q  0  0  (3.96)  or (its recursive form) u (k) q  =  u (k-l)  +  q  INT[(u (k)-u (k-l))/A 0  q  +SIGN(u (k)-u (k-l))/2]A 0  [ u,(0)  q  (3.97)  = 0  where INT is an operator which takes the integer part of its argument. The rounding-off error at the sampling instant is 0 < |e(jfc)| < A/2  (3.98)  Figure 3.25 shows the differences of u (k), u (k) and the output of the DFA u(t). 0  3.4.2  q  Error analysis  We have noticed that on one hand, with the quantization error, the control system will essentially present some non-linear characteristics which will make analysis and design  Chapter 3. Control System  56  a  e  o  u  i  o u (kT) q  10  — u(t)  i  14 time (T)  ia u (fcT) 0  Figure 3.25: Comparison of u (k), u (k) and u(t) 0  u (kT)  q  u{t)  u (kT)  0  q  DFA  c(fcT)  Figure 3.26: Statistic model of the DFA much more difficult and on the other hand, such error can be reduced by increase the number of force levels n under a fixed range of u (k). Therefore, in order to make use 0  of mature linear control theory, it is reasonable to think of the error as a random noise and still to treat the control problem as a linear one plus some noise as long as we have enough force levels. This idea can be expressed as (see Figure 3.26) u (k) = u (k) + e(k) q  0  It is impossible and unnecessary to know all values of e(k). Under the assumption that  Chapter  3.  Control  System  57  e(k) is a stationary white noise process, independent of u (k) and distributed uniformly Q  between —A/2 and A/2, it can be described by its statistical characteristics, such as mean and variance: (3.99) (3.100) The quantization error will also pass through the system and cause the resultant error e in states or output. The variance of e can be estimated by the following formula [13] x  x  based on the system's linearity: (3.101) where H(z) is the transfer function (zl — <*>)r. So the state noise induced by quanti_ 1  zation also depends on the position of poles or eigenvalues of the system. As we mentioned in the previous chapter, a DFA can be considered as a zero-orderhold A / D converter with limited word length. Generally speaking, non-linear roundingoff error due to the limitation of the word length will bring out the phenomenon of limit cycle and cause the system unstable. Fortunately, no such phenomenon will be resulted by the DFA because it is at the end of the digital part of the system and its dynamics can be ignored. 3.4.3  Influence of this control strategy on system dynamics  Due to-the quantization of the optimal control signal, the system performance will deviate from the optimum. Making such deviation as small as possible is an important task. In general, the closer the step-wise control u (k) is to the optimal reference u (k), or the q  0  finer the resolution of a DFA is, the better the system performance will be. The resolution of a DFA depends on two factors: required force range and the number of force levels.  Chapter 3. Control System  58  From the viewpoint of control accuracy, a DFA with more force levels is expected. But such a DFA will have more complicated structure and cost more money to build. In order to find its application in industry, a DFA should have as few force levels as possible.  Chapter 4 Numerical Simulation and Analysis  4.1  Simulation  4.1.1  System parameters  We have done a series of numerical simulations using a quarter car model with an integrated white noise modeled road surface. The system parameter values used in simulation are given in Table 4.2, which represent a typical sub-compact car [12] [2] traversing a fair-highway [2].  4.1.2  Simulation algorithm  The simulation using LQ control law for active systems and SRC based on LQ for discreteforce suspensions, was done in the discrete-time domain and frequency domain. The system time domain response was observed by simply iterating the state equations. In the frequency domain, since the all involved processes are stochastic, power spectral density (PSD) and root mean square (RMS) were chosen to reflect the characteristics of a process. While RMS presents the general criterion to evaluate the performance, PSD can offer more information about the energy distribution along the interested frequency range. There are many ready-made subroutines for PSD calculation. We used the one in the package of MATLAB. Once PSD response was obtained, RMS values simply came out from the square root of the integration of the PSD along the frequency axis. Since the randomly profiled road was modeled as an integrated white noise and the  59  Chapter 4. Numerical Simulation and Analysis  60  Description  Notation  Value  Unit  Body mass Wheel mass Suspension stiffness  M M Kb  Tire stiffness Suspension damping  K  288.90 28.58 (p)19960 (a)16500 155900 (p)1861 (a)1500 1.2* 10" 23.9  Kg Kg N/m N/m N/m Ns/m Ns/m m m/s  Road Roughness Car speed  b  w  w  C k u  5  Table 4.2: System parameters simulation was done in the discrete-time domain, a random number generator which is able to generate a sequence of discontinuous Gaussian random numbers, was adopted to imitate the road disturbances at the each sampling instant. In the simulation, four uncorrelated Gaussian random number sequences were needed, each of them coming from the same random number generator. They are uncorrelated because of the fact that the correlation between any two of the four discrete-time white noise sequences is less than 0.01. Although in the physical system, a sampling frequency of 10 Hz will be used, 100 Hz was assumed in our simulation. The reason is that we wanted to look at and compare the different controls around the resonance frequency of wheel mode, which is about 12 Hz. 10 Hz sampling frequency can only give a response spectrum up to the range of half of it — 5 Hz, which reflects little about the wheel mode control. In order to eliminate the inconsistency of numerical simulation and physical system in control frequency, we still designed the optimal gain under the sampling frequency of 10 Hz and just made the control active after each ten sampling periods in the simulation of 100 Hz.  Chapter 4. Numerical Simulation and Analysis  4.2  61  Results  Once the system parameters and simulation algorithm had been determined, the simulation could start. Since the body acceleration is the most important factor to evaluate a suspension system, we took it as the comparison index for all different suspension systems which had been given nearly the same suspension and tire displacements. In other words, we tried in simulation to get the suspension and tire displacements of the same level and compared the different systems only in terms of body acceleration. Also it is worth mentioning that exactly the same random number sequences were used for all different systems. Although the simulation was done independently for each system, the results will be presented in such a manner that the performance differences of the compared systems can be seen easily and clearly. The systems for which we did simulation are listed as follows. • passive system • ideal active system or active system with 100 Hz control frequency and no damper. • limited active system or active system with 10 Hz control frequency and a conventional damper. • active system with dynamic absorber. • DFS with 10 Hz control frequency and full state feedback. • DFS with Kalman filter. • DFS with limited state feedback.  Chapter 4. Numerical Simulation and Analysis  4.2.1  62  Ideal active vs. passive  Figure 4.27 illustrates the performance comparison between a passive suspension and two active suspensions. As we see, with the active control, the body acceleration can be significantly reduced in the whole frequency range without hurting suspension stroke and tire force variation. If much more improvement in body acceleration is required, however, a trade-off with ride comfort on one side, and suspension working space and wheel load control on the other side, will be inevitable. Better ride quality can only be achieved at the expense of the other two performances.  4.2.2  Limited active vs. ideal active  The control frequency for the active systems in Figure 4.27 is 100 Hz. If the control frequency is reduced to 10 Hz, the performance will seriously deteriorate in both low and high frequency ranges because the wheel motion mode will completely lose control and in turn pass more energy to the body motion mode. This is shown by Figure 4.28. As expected, the introducing of a conventional damper into the system is quite effective to solve this problem and gives out a nearly as good response as in the situation of 100 Hz control frequency. Figure 4.29 shows the PSD responses of two active systems: ideal vs. limited. While the latter shows a little deterioration in frequency range of 2 ~ 11 Hz, that is believed to be caused by the damper, it does make the same body isolation and wheel fluctuation around both resonance frequencies. In the high frequency range, the wheel fluctuation is even smaller. This indicates that the passive damper is better at controlling the high frequency wheel mode than the active actuator.  63  Chapter 4. Numerical Simulation and Analysis PSD of body acceleration  19  PSD of suspension  i »  i 2  o  12  U  16  IS  20 H z  displacement  i *  6  5  10  passive  +. active 1  Hz  12  14  16  ie  20  12  14  16  19  20 H i  PSD of tire displacement  —  active 2  Figure 4.27: PSD of passive suspension versus ideal active suspensions  Chapter 4. Numerical Simulation and Analysis  64  PSD of body acceleration  Figure 4.28: PSD of two active systems of f = 10 Hz: one with damper, another without c  Chapter 4. Numerical Simulation and Analysis PSD of body acceleration  Figure 4.29: PSD of limited active system vs. ideal active system  65  66  Chapter 4. Numerical Simulation and Analysis  4.2.3  Active with dynamic absorber vs. limited active  It has been noted in the previous section that although the conventional damper can effectively isolate the high frequency motion, it does pass some vibration energy to the car body and causes a higher PSD in the frequency range 2 ~ 11 Hz. It is anticipated that isolation would significantly improve if the damper is replaced by a dynamic absorber. In simulation, the mass of the absorber M  a  was taken as 10 per cent of the wheel mass  and the optimal natural frequency and damping ratio were obtained using the following equations [7], 1 + fi and  where \i is the ratio of absorber mass over wheel mass and u is the natural frequency of w  wheel motion mode, equal to 12.36 Hz. Substituting a and us into the above equations w  gives u  0  to  =  11.24  =  0.213  Simulation results are consistent with our expectation. Figure 4.30 shows that body acceleration has been significantly reduced to a quite low level although the tire fluctuation increases in the high frequency range. Of course, some gain of the body acceleration was obtained at expense of the tire fluctuation. If we could reduce the tirefluctuationto the same level as in the case of limited active system, we would have been able to clearly show the improvement of body acceleration caused only by introducing of dynamic absorber. Unfortunately, the tire displacement can only come down by using a bigger mass in the absorber, which is not acceptable for a practical system. By comparing RMS values of these two systems however, we are able to see that a dynamic absorber is not  Chapter 4. Numerical Simulation and Analysis  67  as good as a damper in the sense of controlling the high frequency wheel motion but it makes much better body isolation characteristics by not transmitting any disturbance energy to the body. The system with a dynamic absorber gives a gain of 0.8325 m/s  2  or 64.3% in RMS body acceleration and only a loss of 1.3048 mm or 34r5% in RMS tire displacement compared with the system with a damper.  4.2.4  D F S vs. limited active  When the continuous-output actuator is replaced by a DFA, the simulation results are quite surprising and exciting: no obvious change at all as shown in Figure 4.31. Even though the quantized force level is increased to a high value — around the RMS control force, the resulted difference is still too small to be considered (see Figure 4.32). The comparison of the time histories of the two control forces (Figure 4.33) may help us to understand this phenomenon: they are nearly the same except the amplitude during some sampling intervals. The results imply that in a car suspension system, the DFA is able to do the control work as well as the continuous-output actuator does. The sensitivity of the system response to the amplitude of control force seems quite small. Thus we do not have to worry about the error in the DFA output due to a series of assumptions which actually have some sort of deviations from the real situations.  4.2.5  D F S vs. ideal active and passive  Compared with an active suspension having ideal control frequency as high as 100 Hz, DFS does have some losses in the overall performance due to the passive damper, the much lower control frequency as well as the discretized control force. Nevertheless, it presents a clear superiority to its passive partner (see Figure 4.34).  Chapter 4. Numerical Simulation and Analysis  68  PSD of body acceleration O.J -  0.28 0.26 0.24 0.22 -  03. -  •V,  o.te 0.16 0.14 0.12 0.1 0.08 0.06 0.04 ' 0.02 6  0  10 '  PSD of s u s p e n s i o n  8  PSD of t i r e  1 0  1 2  1 4  1 6  I S  2 0 HZ  displacement  1 2  1 4  1 6  1 8  2 9 HZ  displacement  20 Hz  °  active with damper  active with absorber  Figure 4.30: PSD of limited active system vs. the active system with dynamic absorber  69  Chapter 4. Numerical Simulation and Analysis  PSD of body acceleration  i  i — i — i — i — i — i — i — i — r  PSD of suspension displacement  i  o  2 1.6 -  4  6  PSD of tire displacement  \S -  1A \A -  Te  1.2 1.1 1 0.9 ox -  0.7 0.6.OS OA 02 03. -  0.1 0 1  2  i—i—i—i—r-  44  o DFS  6  8  -I  10 *  1 1 1 1 1 1 1 1 112 14 16 10 20 HZ a c t i v e T = 0.1C» 1S00  Figure 4.31: PSD of limited active suspension vs. DFS with f = 10 Hz c  70  Chapter 4. Numerical Simulation and Analysis  A  RMSij  RMSxi  RMSzi  (m/100)  (m/s ) 1  (ro/600)  RMSu  (100JV)  Figure 4.32: RMS responses of DFS vs. the force level width  200  -A  100 N  o -100 -200  '"Ik 100  200 300 400 500 600 700  800  time (MC/1000) — limited active  — DPS  Figure 4.33: Comparison of time histories of the control forces for limited active suspension and DFS  Chapter 4. Numerical Simulation and Analysis PSD of body acceleration  Figure 4.34: PSD of DFS vs. ideal active and passive suspensions  71  Chapter 4. Numerical Simulation and Analysis  4.2.6  72  D F S with limited measurements  Figure 4.35 illustrates the PSD responses of the system containing a Kalmanfilteras the state estimator. The measurements are the suspension travel, and the absolute velocities of the two masses. According to the assumption of the independent white noise at each measurement, the covariance matrix of the measurement noises is of diagonal form and was chosen as: V = 10"  1 0  0  0 49  0  4  0  0 625  It can be seen that even though the tire displacement is unavailable, with the help of the state estimator, essentially the same response results except for a little loss in the range of 0.5 - 4 Hz. The results of the second option to the problem of unavailable state are shown in Figure 4.36. In this situation, where we just let the third term of the optimal feedback gain matrix be zero, no better high frequency performance appears as we expected. Fortunately, however, a PSD quite close to that of full state feedback DFS is obtained. 4.2.7  Control force power  In Chapter 2, we have calculated the internal leakage loss, fluid friction loss and valve throttle loss for a DFA. In order to make a complete estimation for the energy consumption of a DFS, another power term should be included. That is the power needed by the DFA to do the work towards the body and the wheel masses. This term can be estimated by the product of force output of DFA and relative velocity between those two masses. We did the calculation in the case of DFS with full state feedback control. The result of around 14 W for a quarter car model is quite satisfying.  Chapter 4. Numerical Simulation and Analysis  PSD of body acceleration  PSD of suspension displacement  2  4  10  12  1*  PSD of tire displacement  16  18  20  HZ  1.6 IA \A  1J 12 1.1 -  T e  1 -  09 0.8 0.7 0.6  oa o.* 0.3 -  02 0.1 -  —r a DFS futt-sUte feedback  12  -  20  Hz  DFS with kalman filter  Figure 4.35: PSD of DFS with Kalman filter and DFS with full state feedback  74  Chapter 4. Numerical Simulation and Analysis  PSD of body acceleration 0.3 -1 0.28 0.26 0.24 0.22 -  :  «C - • c  ,b  "5" 0.16 -  PSD of suspension displacement 8 -  0  2  4  6  8  10  12  14  PSD of tire displacement  16  20 Hz  IS  1.6 1JS 1/4 1J  1.2 1.1  TO  1  0.9  0.8 £ 0.7 0.6 0.S 0.4 0.3 0.2 0.1  0  T—i  9  2  1—i—i—i  4  6  ° DPS full-state feedback  i  i—i—i—r—i—i—I—r~~I  8  IO  12  14  16  1  I  18  I  20 H>  — - DFS limited-state feedback  Figure 4.36: PSD of DFS with limited state feedback and DFS with full state feedback  Chapter 4. Numerical Simulation and Analysis  1 2 3 4 5 *6 7 **g ***g  type  damper  fc(Hz)  P A A A A A D D D  Y N N N Y N Y Y Y  -  BA(m/s ) 2  BD(mm)  WD(mm)  Ctrl(N)  11.186 10.235 16.678 21.917 11.452 11.182 11.378 11.404 11.589  3.897 3.793 8.799 11.730 3.780 5.084 3.779 3.795 3.802  -  1.5644 1.1620 0.3747 2.1556 1.2948 0.4623 1.2937 1.3282 1.3406  100 100 10 10 10 10 10 10  75  331.04 253.79 •482.03 95.83 132.67 95.26 79.78 78.44  Note: step size of control force is 40 N. * system with dynamic absorber. ** system with Kalman filter. * * * system with ignorance of x — x in the feedback. w  g  Table 4.3: Comparison of RMS responses 4.2.8  Summary  Table 4.3 shows RMS responses for all situations, where the influences of different factors on the system performances can clearly be seen. From the work we have done so far, it becomes clear • The actuator in an active suspension can be replaced by a DFA and no big difference in overall performance will result. • The idea to control the body motion mode and wheel motion mode separately is feasible and effective.  Chapter 5  Conclusions  Our research work has included the study of DFA properties and of vibration isolation characteristics of a DFS. A rotary DFA has been studied to establish an understanding of the relationships between the moment output and the rotor motion in terms of equivalent stiffness and equivalent damping. While the damping rate keeps a nearly constant value, the stiffness shows a decreased trend with the reduced absolute value of the force level setting. The maximum output variation due to the dominant stiffness effect can be 6 times as large as the unit output (output resolution). But the average of the stiffness in the most operating range of the DFA is only about 14% of the maximum. How big the damping effect is depends on the rotor (excitation) velocity. The stiffness and damping effects are not negligible in the cases when the system response is quite sensitive to the DFA outputs or when the parallel stiffness and damping of the system containing a DFA is quite small. When the DFA is used in a car suspension system, however, the suspension stiffness and damping vary little with the inclusion of the DFA and therefore the DFA can be considered as being able to offer a control force independent of the suspension motion. Also the power loss calculations have been done for a rotary DFA with 7 working chambers and for the whole DFS system. A DFS needs only 20% of the power needed by an active suspension. Decreased DFA's hydraulic losses make the largest contribution to this reduction.  76  Chapter 5. Conclusions  77  As far as the vibration isolation characteristics are concerned, the DFS with a conventional damper can achieve the same low frequency body mode control and the even better high frequency wheel mode control. But in the frequency range between these two modes, the damper transmits more disturbance energy to the car body. Another promising option for the DFS configuration lies in the possibility of replacing the damper with a dynamic absorber. A dynamic absorber is able to control the wheel mode effectively and passes no energy to the body although a more complex structure is involved. In brief, the idea to control the body mode and the wheel mode separately is feasible and effective and a quite close or even better isolation can be obtained without the necessity to use an actuator with a large bandwidth. When fully making use of the simple quarter car model, we have noticed the one major shortcoming of such a heave-only model, which lies in its inability of reflecting the effect of the control strategy on the tire normal load distribution between the four corners of the vehicle, as it traverses a rough road. Therefore the study on using "DFS on a more elaborate full car model is necessary if both comprehensive ride and handling characteristics are desired.  Appendix A Discretization of Coefficient Matrices of State-Space Model  For a given system X(«) = AX(<) + Bu(«)  (A.102)  the solution is of the general form: X(t) = e ( ° ' X ( t ) + / ' e ( - » B u ( r ) ( / r A  M  A  0  (  T  Jto  (A.103)  Let t = kT and t = (fc + 1)T: 0  X[{k + 1)T) = e X(kT) AT  + /  ( f c + 1 ) T  e K A  fc+1  > - lBu(rWr r  T  JkT  Assuming u is unchanged between kT and (k + 1)T, B u can be moved out of the integral: X[(k + 1)T].= e X(kT)  p(*+l)T  +  AT  /  IkT JkT  e ^ - UrBu(kT)  (fc+1)T  A  T  T  or (with transformation (k + 1)T — r = A) X[(Jfc + 1)T] = e X(kT) AT  + f  T  e d\Bu{kT) Ax  (A.104)  Jo  Comparing the above equation with the desired form,  x[(fc + i)r] = *x(fcr) + ru(fcr)  (A.105)  we get *  =  r  =  e  (A.106)  AT  /  Jo  78  e d\ Ax  (A.107)  Appendix B  D F A Transient Behaviour Modelling  1.  Actuator. Theflowrateexpression for positive chambers is  where P  +  J  +  = pressure in positive working chambers =  total moment of the radial cross section area of the positive working chambers  6 = bulk modulus of the fluid The continuity equation is Q  +  = Qt + Qt  (B.109)  where Qt and Qt are theflowratesto the accumulator and to the reservoir respectively. Comparing equations B.108 and B.109 gives a differential equation of pressure P : +  Similarly, the equation for the pressure P~ in negative chambers is  T-fS- **) 2  <"»»>  where 6 = So — 6. 6 is the angular stroke of the DFA. 0  There are four possible working conditions for any chamber, under each of which the pressure in the chamber will have different expressions: 79  Appendix B. DFA Transient Behaviour Modelling  80  • being switched from the low pressure to the high pressure.  ~dT  e ~Tt 3 de QZH + QZI,  =  {  dPu ~dT  =  l Tt {  ~  J-  )  (B  -  112)  • ( B  -  1 1 3 )  • being switched from the high pressure to the low pressure.  ^  -  f(S  (B.115)  • being connected to the high pressure.  ditf h  3, de Qt  x  ~  dPr A  "  (B.H6) , (B.117)  £ ( - 1 7 - ^ )  3,de Or, §e( dt 1 7j~ -^) h  • being connected to the low pressure.  where subscript u indicates the case of switching from low to high or up, d from high to low or down and h represents high and / low. 2.  Valves.  Once the valves are modeled as an orifice, the flow equations for the valve between the actuator and the accumulator and the one between the actuator and the reservoir are Q  H  = SIGN(P -  Pa)A C \r h  d  P  Pa  (B.120)  Appendix B. DFA Transient Behaviour Modelling  81  and 2 P - P  Q^SIGNiP-P^AiC^ where A ,Ai  =  valve areas for high or low pressure passages  Cd  =  discharge coefficient at each valve port  P ,Pr  =  pressures in accumulator and tank  n  a  3.  r  (B.121)  p = fluid density Accumulator.  Spring-loaded accumulator has itsflowrateexpression of n V  _ Va.,V^dP d  ±  o  "  dt  0 dt  +  (B.122)  and [9] dP*  dVa  dt where V  =  c  a  =  A  =  a  a  1  (B.123)  dt  accumulator volume A\jh  a  cross section area of accumulator  k = spring stiffness of accumulator Combining equations B.122 and B.123 and considering the continuity offlowyield a  dP dt  0 0c + V  a  a  a  •EG*  (B.124)  The equations for all concerned pressures in the whole hydraulic system are nine non-linear differential equations obtained by substituting the valve equations into the actuator and the accumulator equations.  dt dfl dt  0 de e^ dt 0 de e dt (  K  Ajc 2\P£-P*\ j£ \ d  A~ c h  d  2\Pr-Pa\  )  (B.125) (B.126)  82  Appendix B. DFA Transient Behaviour Modelling  3  dt dPf dt dP+ dt  dP~ dt  dPt dt dPl_ dt dPa dt  A f C q  dO  e  dt  {  3 de ATC 2 \ P - P \ ) e dt jr \ &, dO At C /2|P+ - P.| e dt j + V P r  d  (B.127)  )  j+ \  (B.128)  r  (  h  d  Ai,c  {  l  M _ K £  (  e dt {  2\P--P  L  V  J-  a  (  dh  e  K  8 dS e dt  ul  dh  8C 8c + V d  a  a  d  a  V  r  P r  p  ]  Wt-PA ) Jt \] d  d  2\P -~Pr\  2|Pi -P +  a  2\Pf-Pa\  (B.130) • (B.131) (B.132)  d  JI \  P  (B.129)  }  Aj,C  dl  a  p\  2\P--P \  d  A~ C  2\PJ-P \  Y '\  I2\P+-  Jz V  p  d  A~ C Jr \  K  J+  A~ C  \  de A+ C 2 1 P / - P | dt jj \ P  3  d  )  (B.133)  Bibliography  Karl J. Astrom and Bjorn Wittenmark. Computer Controlled, Systems — theory and design. Prentice-Hall, Inc., Englewood Cliffs, NJ 07632, first edition, 1984. Pinhas Barak and Herbert Sachs. On the optimal ride control of a dynamic model for an automotive vehicle system. Vehicle System Dynamics, 15 suppl:15-29,1986. T. Bustsuen C. Yue and J.K. Hedrick. Alternative control laws for automotive active suspensions. J. of Dynamic Systems, Measurement, and Control, 111:286-291, 1989.  Csaba Csere. Lotus active suspension. Car and Driver, 51-57, June 1988. David Mumford & Mark Dabell. Design of A Digital Hydraulic Actuator. Technical Report, Dept. of Mech. Eng., U . B . C , 1990. Mathematics Handbook editing Group, editor. Mathematics Handbook. High Education Publishing House, Beijing, China, first edition, 1979. C M . Harris and C.E. Crede, editors. Shork and Vibration Handbook. McGRAWHILL Book Company, second edition, 1976. D. E . Newland. An Introduction to Random Vibrations and Spectral Analysis. Long-  man, London, first edition, 1975. Josef Prokes. Hydraulic Mechanisms in Automation. Elsevier Scientific Publishing Company, Amsterdam, first edition, 1977. [10 J.D. Robson. Road surface description and vehicle response. Int. J. of Vehicle Design, 1:25-35,1979. [11 R.S. Sharp and D.A. Crolla. Road vehicle suspension system design - a review. Vehicle System Dynamics, 16:167-192, 1987.  [12] A.G. Thompson. An active suspension with optimal linear state feedback. Vehicle System Dynamics, 5:187-203,1976.  [13] Shi-Yi Wang. Digital Signal Processing. Press of Beijing Industrial Institute, Beijing, China, first edition, 1987. [14] P. G. Wright. The application of active suspension to high performance road vehicles. C239/84 IMechE?, 23-27, 1984.  83  

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