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DATE DUE DATE DUE DATE DUE Nuv :39.Q.-...] -« "an" I M,» A» i MSU Is An Affirmative ActiorVEqual Opponunlty Institution cmma-M A COMPARATIVE STUDY IN AUTOMOTIVE ACTIVE SUSPENSION SYSTEMS By thg- Chi LIN A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1992 {73- s/ 8% ABSTRACT A COMPARATIVE STUDY IN AUTOMOTIVE ACTIVE SUSPENSION SYSTEMS By Yung- Chi LIN The vehicle active suspension problem is investigated using a quarter car model which consists of one fourth of the body mass, suspension components and one wheel. A State space approach is used and three control methodologies including LQG/LTR, constant gain output feedback and singular perturbation are applied. The perfor- mance criteria of suspension systems are formulated into minimizing sprung mass ac- celeration, suspension deflection, and tire deflection for the concerns of ride comfort, working space constraint, and road holding ability, respectively. Various measurement schemes which contain position, velocity and acceleration are investigated. It is shown that sprung mass acceleration gain can be attenuated significantly over a wider fre- quency band, reaching into the wheel resonance via using acceleration feedback, then characterizing the dynamics into slow and fast models and designing compensators individually to meet desired specifications for low and high frequency ranges. Non- ideal integration, controller bandwidth, force level requirement and robustness are also studied. All designs demonstrate comparable robustness properties when system parameters are perturbed. To my Parents For their afiection and devotion iii ACKNOWLEDGMENTS I am grateful to my advisor, Dr. Hassan K. Khalil, for inspiring and instructing me all the way through this research. His patience and provocative insights have been major motivations for this thesis. With his help through the many stages where I got stuck with my progress, I came to realizing my potential in doing research. I want to thank Dr. G. L. Park and Dr. R. L. Tummala for their time and energy to be the members of my thesis committee. My deepest respect and gratitude go to my wife, Tsailing Liang, for her con- stant love, encouragement, and sacrifice. She has tried every means possible to help me concentrate on doing my research, even in her poor health after our first baby, Alexander, was born on Oct 1“, 1991. I really owe thousands of thanks to many people in the world — mentioned or not mentioned in this acknowledgement. iv Contents 1 Active Suspension 1 1.1 What Is Active Suspension? ....................... 1 1.2 Classification of Suspension Systems .................. 2 1.3 Mathematical Model ........................... 3 1.4 Performance Measure ........................... 4 1.5 Organization of Thesis .......................... 5 2 Review of Some Control Strategies 6 2.1 Linear Quadratic Regulator (LQR) ................... 6 2.2 Linear Quadratic Gaussian (LQG) .................... 8 2.3 LQG Applied to ASS ........................... 9 2.4 Constant Gain Output Feedback (CGOF) ............... 11 3 Acceleration Feedback 14 3.1 LQG with Acceleration Feedback (LQG-A) ............... 15 3.2 Performance of LQG-A .......................... 15 4 Singular Perturbation Approach 17 4.1 Singularly Perturbed Systems ...................... 18 4.2 Sequential Design Procedure ................. P ...... 2 0 4.3 Modeling in Singular Perturbation (SPT) ................ 4.4 Sequential Design Example I (SPT-2) .................. 4.4.1 Fast Design ............................ 4.4.2 Slow Design ............................ 4.4.3 Full Design ............................ 4.5 Sequential Design Example II (SPT) .................. 4.5.1 Fast Design ............................ 4.5.2 Slow Design ............................ 4.5.3 Full Design ............................ Comparative Study 5.] Nonideal Acceleration Integration .................... 5.2 Evaluation of Performance Criteria ................... 5.3 Controller Bandwidth and Force Level ................. 5.4 Robustness Analysis ........................... 5.5 Performance Variation Due to Parameter Perturbation ........ 5.5.1 Perturbation of 50% Damping (b,) Increase ........... 5.5.2 Perturbation of 25.7% Sprung Mass (m,) Decrease ...... 5.5.3 Perturbation of i50% Tire Stiffness (kt) ............ 6 Conclusions Figures Appendix A Bibliography vi 2O 23 23 23 24 25 26 27 29 29 30 31 31 33 34 34 34 36 38 72 75 List of Tables 1.1 2.1 2.2 2.3 3.1 3.2 4.1 4.2 4.3 Vehicle Parameters ............................ 3 LTR p of LQG-P a‘nd LQG-V ...................... 11 Designed p of LQG-P and LQG-V .................... 11 Designing Kc of CGOF systems ..................... 13 LTR a of LQG-A ............................. 15 Observer Gain K for Various Schemes ................. 16 Eigenvalue Approximation for the Open Loop System ......... 22 Closed Loop Eigenvalues of SPT-2 .................... 25 Closed Loop Eigenvalues of SPT ..................... 28 vii List of Figures [O 9 10 11 13 14 15 16 Structure of an Active Suspension System ............... 38 Frequency Response of a Passive System ................ 39 LTR of LQG-P System .......... I ................ 4O LTR of LQG-V System .......................... 41 LTR of LQG-A System .......................... 42 Performance of Passive, LQR, LQG-P & LQG-V Systems ....... 43 Performance of Passive, LQR, & LQG-A Systems ........... 44 Design of CGOF (Velocity Feedback) System .............. 45 Root. Locus of CGOF Systems (Position, Velocity & Acceleration Feed- back) .................................... 46 Transfer Functions of z'r —-> f, ...................... 47 Transfer Functions of 2', -> (z, — zu) ................... 48 Transfer Functions of 2', —> (2,, — z.) .................. 49 Transfer Functions of u —» 23', ....................... 50 Transfer Functions of u -+ (z, — zu) ................... 51 Transfer Functions of u -> (2,, — 2,) ................... 52 Root Locus of Slow and Fast Models (SPT-2) ............. 53 Performance of Slow, Fast, Full 3; Passive Systems (SPT-2) ...... 54 Root Locus of Slow and Fast Models (SPT) .............. 55 Performance of Slow, Fast, Full & Passive Systems (SPT) ....... 56 Performance and Controller Gain (2", —> u) of SPT & SPT-2 ..... 57 Performance and Force Gain (é, -* u.) of SPT & SPT-2 ........ 58 Performance and Controller Gain(é', -> u) of LQG-A, LQG-V, CGOF & SPT ................................... 59 viii 23 24 25 26 27 28 29 30 31 32 33 34 Performance & Force Gain (i, —> u) of LQG-A, LQG-P & SPT Performance 85 Force Gain (23, -—> u) of LQG-V, CGOF & SPT . . . . Singular Value of Various Designs .................... Performance of 50% Damping (b,) Perturbed LQG-P System ..... Performance of 50% Damping ((2,) Perturbed CGOF System ..... Performance of 50% Damping (b,) Perturbed SPT System ...... Performance of 25.7% Body Mass (m,) Reduced LQG-P System . . Performance of 25.7% Body Mass (rm) Reduced CGOF System Performance of 25.7% Body Mass (m,) Reduced SPT System ..... Performance of 50 -— 150% Tire Stiffness (kt) Perturbed LQG-P System Performance of 50 —— 150% Tire Stiffness (kt) Perturbed CGOF System Performance of 50 — 150% Tire Stiffness (kt) Perturbed SPT System . ix Chapter 1 Active Suspension The automotive industry is always pursuing more comfortable and safer cars. Im- provement in suspension systems plays an important role in achieving these goals. Naturally, the suspension system is then expected to have more intelligence to ac- commodate itself in various road conditions. The increasing capability and decreasing cost of electronic components motivate people’s strong interest in this topic. 1.1 What Is Active Suspension? An active suspension system is considered as one with the following two features [1]. 1. Energy is constantly supplied to the suspension system and the force generated by that energy is constantly controlled. 2. The suspension system incorporates various types of sensors and a unit for processing signals; it generates forces which are functions of the output signals. One possible structure of active suspension systems is depicted in Fig.1. The system employs sensors to measure signals, a compensator to process signal and an electro- hydraulic actuator to generate force. Shown in this figure is the so-called quarter 1 car model consisting of one fourth of the body mass, suspension components and one wheel. The model describes the y — axis motion under road disturbances when the vehicle moves in the :c — axis direction. It does not describe other motions of the body like roll or pitch. 1.2 Classification of Suspension Systems The design of traditional suspension systems requires a careful choice of spring and damper which are assumed to behave linearly. This system is referred to as passive suspension system (PSS) since no external energy is added to generate any control force. An active suspension system (ASS) usually employs an electrohydraulic sys- tem to achieve its performance criteria, requiring extra power supply and a signal processing unit. Another kind of suspension system between the previous two sys- tems is called semi-active suspension system (SAS). Based on different control laws, the damper of the SAS can be switched on/off or adjusted continuously to meet var- ious road conditions. It does not require power supply as large as the ASS. This will introduce nonlinear characteristics into the system. Some research [3, 4, 5, 6] has been dedicated to the study of SAS. 1 .3 Mathematical Model The quarter car model used in this thesis is shown in Fig.1. The state variables are chosen as follows [12], Z, '— zu is Zu—Zr 2}, L .1 where z, — 2“, z',, 2,, — z, and z}, are, respectively, called suspension deflection, sprung mass velocity, tire deflection and tire velocity. The state space equation is then given by :5: = Ax + Bu+ Ez', (1.2) where 0 1 0 —1 0 0 -—k, m, —b,, m, 0 b, m, ' 1 m, 0 A z / / / ,3: / ,E’: 0 0 0 1 0 —1 k,/m,, b,/m,, -—k¢/m,, —b,/m,, —1/mu 0 ' ‘ (1.3) The above vehicle parameters are defined in Fig.1. In our study, a typical set of vehicle parameters shown in Table 1.1 is taken from [2}. m, = 504.5kg mu = 62199 b, = 1328N.3/m k, = 13100N/m k, = 252000N/rl Table 1.1: Vehicle Parameters When active control is used, the damping constant, b,, is replaced by 400N .s / m. The damper is not completely removed due to a practical safety concern in case of the failure of the control system. The observed outputs of interest are position (suspension deflection, z, — zu), velocity (sprung mass velocity, 2,) and acceleration (sprung mass acceleration, 5,). The suspension deflection can be measured by acoustic or radar transmitter & receiver; while the velocity 23, is typically obtained by integrating the acceleration a", which is measured using accelerometer [l6]. 1 .4 Performance Measure Traditionally, a good suspension system is supposed to be able to provide passengers with ride comfort, while maintaining necessary road holding ability to satisfy maneu- vering safety concerns, subject to the design constraint of limited suspension working space. Ride comfort is related to vehicle body motion sensed by passengers and, generally, measured by vehicle body acceleration (sprung mass acceleration for the quarter car model). Road holding ability is affected by wheel load dynamic variations, i.e., fluctuation of the contact force between the tire and the road surface. Clearly, this fluctuation is directly related to tire deflection. When road disturbances come into the suspension system, the relative displacement between the body and wheel keeps varying in such a way that the disturbance transmission is kept as small as pos- sible, to provide best comfort. However, the allowable relative displacement is limited by the usable working space. Thus, the previous discussions lead us to formulate the performance criteria into minimizing sprung mass acceleration, suspension deflection, and tire deflection for ride comfort, working space constraint, and road holding ability [9, 10, 11, 12, 13, 14], respectively. It is well known that there is always a trade-off in minimizing these performance criteria [9, 10, 11, 12, 13, 14]. In this research, control effort, force actuator bandwidth, robustness and nonideal signal processing in using velocity feedback are also considered. Hrovat [7] has identified the advantages of tak- ing jerk (derivative of acceleration) into consideration for ride comfort, but his idea has not yet received much support. 1.5 Organization of Thesis This thesis is organized into six chapters. The first chapter is an introduction to the active suspension system, including the mathematical model description, and the performance measure of suspension systems. Chapter 2 reviews some control method- ologies of LQR, LQG, and constant gain output feedback. The simulation results with the applications of the above methodologies are also presented under velocity and position feedback. The value of acceleration feedback is first explored with LQG method in chapter 3. Chapter 4 introduces the singularly perturbed systems and the sequential design procedure. Two sequential design examples are presented in this chapter. A comprehensive study of the above schemes is performed in chapter 5. The issues discussed in this chapter include nonideal integration, performance evaluation, controller bandwidth, force level, and robustness. The robustness is investigated via evaluating both the singular values of the complementary sensitivity functions and the performance under system parameter perturbations. Chapter 6 presents conclu- sions. For the purpose of comparison, some performance results reported in other literature are also mentioned briefly. Chapter 2 Review of Some Control Strategies In this chapter, the theory of LQR, LQG [8] and constant gain output feedback are briefly described. The performance of ASS by the application of LQR is evaluated as a reference performance. After that, with the observed outputs of position and velocity, LQG designs and output feedback are done. In doing LQG designs, the concept of LQG/LTR (Loop Transfer Recovery) is introduced to achieve the desired frequency loop shaping. The work of LQG is similar to the work of [12]; the work of constant output feedback is similar to [2]. 2.1 Linear Quadratic Regulator (LQR) Consider the system :i:=Ax+Bu (2.1) Assuming measurements of all states available, the LQR problem can be formulated into seeking a linear control law it = —G9: (2.2) where G is a suitable stabilizing gain matrix, to minimize the performance index PI =/ (x’Qa: + u'Ru)dr (2.3) o R is a positive definite symmetric matrix and Q is a positive semidefinite symmetric matrix. It is well known that the optimal gain, G, is given by G=RAEM (@ [\D where M satisfies the algebraic Riccatic equation (ARE) 0 = MA + A'M -— MBR’IB’M + Q (2.5) When (A, B) is controllable and (A, (Q) is observable, the Riccatic equation (2.5) has a unique solution M = M’ > 0 such that (A — BR'IB’M) is Hurwitz. 2.2 Linear Quadratic Gaussian (LQG) Consider the state equation :i: = Az+ Bu+Ev (2.6) and the observed output equation 31 = Ca: + w A (27) where v and w are Gaussian white noise processes with E [v(t)v(t + 7)] = 6(7), E [w(t)w(t + 7)] = p6(7) (2.8) The LQG problem seeks an optimal control u that minimizes the performance index PI = 7111.20 lE{/OT (x’Qa: + u'Ru) d7} (2.9) The optimal solution of the LQG problem is given by u = —G.i: (2.10) where G is the same optimal control gain defined by Equation (2.4), and i: is the optimal state estimate, defined by the optimal observer or Kalman filter a=Aa+Bu+K(y—ce) (2.11) The observer gain K is given by K = lPC' (2.12) p where P satisfies the ARE 0 = AP + PA’ — iPC’CP + EE’ (2.13) Although p indicates the value of the sensor noise, it is often treated as a design parameter to reflect the bandwidth of the observer. By increasing the bandwidth of the observer, i.e. p —» 0, the feedback loop transfer function of an LQR system can be recovered by an LQG system. This is the so—called LQG/LTR (Loop Transfer Recovery) methodology [23, 26]. 2.3 LQG Applied to ASS Looking at the system dynamic equation (1.2), there are two inputs coming into the picture. The design task is to select a control it to reject the effect of the road disturbance 7.} on the sprung mass acceleration 2",. Fig.2 shows the frequency response of the original passive system. It is easily noticed that the suspension system dynamics contain two distinct oscillatory modes: one mode corresponds to body resonance (z 1 Hz); the other corresponds to wheel resonance (z 10 Hz). The damping constant of the passive system is 1328N.s/m which will be replaced by a smaller value of 400N .s / m when control is applied. As stated previously, the reference performance is designed by an LQR procedure [8]. 10 The following Q and R Q=diagio 1225 o 156] (2.14) R = [0.000056] (2.15) yield the state feedback gain G, given by G=l0 4374 -14448 -1274] (2-16) which will be used in cascade with observer. In particular, both the states 2,, — z, and z}, are not easily measured. The LQG design is used, assuming only position (z, —z,,) or velocity (2],) are available. [The observer equation is then given by i=Air+Bu+If(y—C.i:) (2.17) with the measured output being yposition = 0px + w =l 1 0 0 0 l3 + w (2-18) 01' y..i...-..=C.:c+w=[o 1 0 0]:c+w (2.19) In order to apply LQG, the road velocity 2', and sensor noise w are modelled as white noise processes, i.e., E [z',.(t)z',(t + 7)] = 6(7), E [w(t)w(t + 7)] = p6(7) (2.20) As stated in Sec.2.2, p is used as a design parameter to determine the bandwidth of ”the observer. In Fig.3 and Fig.4 the LTR results are seen. The dotted lines indicate 11 the performance of LQR designs. Table 2.1 shows the values of it used in these Figures. p(Pos.) 10-3 10-4 5x10-5 10-6 p(VeI.) 10-3 10-4 5x10-5 10-6] Table 2.1: LTR p of LQG-P and LQG-V The arrows in these plots indicate the corresponding directions of change. It is easily noticed that larger suspension deflection 2,, — 2,, and high frequency tire deflection 2,, — 2r come along with the improvement of sprung mass acceleration 2', and low frequency tire deflection. Based on the principle of “equal working space 1” the design parameters are picked up as in Table 2.2. Once the design parameter p is picked up, the observer is determined. The frequency response of the LQG—P and LQG-V designs are shown in Fig.6. Note that the suspension deflections 2, — 2,, are very close to each other. 2.4 Constant Gain Output Feedback (CGOF) CGOF has been applied in [2] by the centralized/local optimization procedures to solve a full car ASS problem. Motivated by this idea, CGOF is used to investigate various measurements. The control is obtained by multiplying measured signals with a constant gain, denoted by Kc. In single-input-single—output systems, Kc is scalar. 1This principle 18 discussed 1n [11]. It 18 primarily to express a common usable space constraint which exists in designing vehicle suspension systems. Even when active control are applied, it is still considered as a fair comparison baseline for various designs. p(Position) 0.0001 a(Velocity) 0.000075 Table 2.2: Designed p of LQG-P and LQG-V 12 This fact simplifies our analysis. Consider a system i=Am+Bu y=Cx+Du where u and y are scalar variables. Suppose the control u is obtained by u = -Kcy = —KC(C:c + Du) = —KCG$ - KcDu if I + KCD is nonsingular, then u = —(I + KCD)“KCC:1: The closed loop system is represented by 5: = (A — B(I + KCD)“KCC)a: (2.21) (2.22) (2.23) (2.24) (2.25) Since K6 is scalar, the stabilizing range of K, can be determined by looking at the root locus of the closed loop system. Therefore, we study all possible values of K c to pinpoint a satisfactory K c for each of the different measurement schemes. The root locus analysis of CGOF for various measurements shows that only velocity feedback can provide with enough damping force. Thus, only the results of velocity feedback will be discussed. Fig.8 shows the performance of several CGOF designs. Table 2.3 summarizes the values of K0 used in this Figure. Likewise, the arrows in this Figure indicate the corresponding directions of change. Both of the open loop performance and LQR are also shown for comparison. For convenience, the root locus of position,velocity 13 and acceleration feedback schemes are reported in Fig.9. Note that, for the case of velocity feedback, the two fast poles are insensitive to Kc range of interested, as shown in Table 2.3. In other words, when K, is tuned not greater than 20000, their motions toward imaginary axis are negligible. This characterizes the CGOF loop shaping primarily in low frequency range. Note that the performance changes only in [[K, I 1000 5000 10000 200001] Table 2.3: Designing Kc of CGOF systems the low-frequency range when K c varies, i.e., the high frequency shape of open loop system is still kept in feedback system. This scheme increases damping force around body resonance to eliminate the two body resonance peaks of acceleration and tire deflection. Compared to the LQR performance, the price paid in larger suspension deflection and high frequency tire deflection is also observed. Chapter 3 Acceleration Feedback The output feedback controllers designed in the previous chapter use measurements of suspension deflection 2, — 2,, (LQG-P) or the sprung mass velocity 2, (LQG-V & CGOF). The suspension deflection can be measured by acoustic or radar transmitter & receiver; while the velocity 2', is typically obtained by integrating the acceleration 2’, which is measured using accelerometer [16]. This integration scheme shows that the actual observed output is acceleration, and requires an integrator part in cascade with the compensator designed using velocity feedback. The nonideal effects of such integration have been discussed in [17]. The restriction of the controller to have an integral component might be limiting the performance which can be achieved with a more general use of acceleration feedback. Hence, the design of LQG controller using acceleration feedback is explored in this chapter. 14 15 3.1 LQG with Acceleration Feedback (LQG-A) In order to use sprung mass acceleration 2', which is given by ‘58 =l -ks/ms “bs/ms 0 bs/ms l3 +l1/m8lu (3'1) Equations (2.7) and (2.11) are modified as follows. y=Car+Du+w - (3.2) i = .43 + Bu + K(y — g) (3.3) where g = C5: + Du (3.4) 3.2 Performance of LQG-A The LQG/LTR design performance for the acceleration feedback are shown in Fig.7 with the design parameters [1 shown in Table 3.1, where p = 1.0 is picked up for LQG-A design. p(Acc.) 10.0 2.0 1.0 0.1 || Table 3.1: LTR p of LQG-A The observer gain K and design parameter a of the three LQG designs are sum- marized in Table 3.2. Looking at Fig.6 and Fig.7, the LQG-A design used in this work does not provide with remarkable advantages over the LQG-P and LQG-V designs. The reasons will be discussed in chapter 5. Compared to the performance of LQG-V 82 LQG-P, the LQG-A design has a smaller suspension deflection 2, - 2,, along with 16 type K’ # LQG-P [ 103.80 -82.3 -420 5474.40] 0.0001 LQG-V [ 423.70 127.1 9.20 6262.10] 0.00075 LQG-A [ -072 0.0 -027 53.58] 1.0 Table 3.2: Observer Gain K for Various Schemes a sacrifice in sprung mass acceleration and tire deflection. Chapter 4 ' Singular Perturbation Approach In vehicle suspension systems, the presence of body (slow) and wheel (fast) resonance shows the existence of a two-time—scale structure [21]. This can be utilized by the application of singular perturbation methodology. In this chapter, a brief descrip- tion of singularly perturbed systems and the sequential design procedure of [18] is first presented. The suspension system is then cast into a singularly perturbed form which is composed of two submodels called slow and fast models. The value of using acceleration feedback is easily seen at this point because the transfer function from the control input to the acceleration output has nontrivial slow and fast components. The corresponding transfer functions for position and velocity outputs have zero fast models. This chapter concludes with two design examples where the sequential design procedure is employed to design the acceleration feedback controllers. 17 18 4.1 Singularly Perturbed Systems A linear time-invariant singularly perturbed system is represented by If? = A1113 + A122 + Blu (4.1) 62 = A212: + A222 + Bgu, det [A22(0)] # 0 (4.2) y = 013: + 022 + Du (4.3) where a: 6 R" denotes the slow state vector; 2 E R“ denotes the fast state vector; u E R” is the control input and y E R9 is the output. The separation between the slow and fast dynamics can be represented by the small positive constant e in the sense that i: is 0(1), whereas 2' is 0(%). In other words, as c -> 0, the singularly perturbed system of (4.1)—(4.3) has a two-time-scale structure and the eigenvalues cluster into a group of slow 0(1) eigenvalues and a group of fast 06) eigenvalues. The full system can be approximated by the slow and fast models. The slow model is given by :i:, = A02, + Bou (4.4) y: = 003: + DD” (4.5) Where A0 = All - A12A;21A21,Bo = B] - AHA-{2182,00 = Cl — 02142-211421 and D0 = D — 02142—2132. The fast model is given by 2, = 14fo + Bf" (4.6) 311 = 0121 + DJ“ (4-7) 19 where A, = 1422/6, B, = 32/130; = Cz, and D; = D. Various properties of the singularly perturbed system (4.1)-(4.3) can be approximated by the slow and fast models (4.4)—(4.7). Two approximations that are used in this paper are the eigenvalue and transfer function approximations. Eigenvalue Approximation [20] As 6 -+ 0 the slow eigenvalues of the full singularly perturbed system (4.1)—-(4.3) approach the eigenvalues of the slow model of (4.4)—(4.5); the fast eigenvalues of (4.1)-—(4.3) approach the eigenvalues of the fast model of (4.6)—(4.7). Transfer Function Approximation [19] The transfer function, denoted by G(s,c), of the full singularly perturbed system (4.1)-(4.3) can be approximated by G(s, e) = G,(s) + Gf(€3) — G,(oo) + 0(6) (4.8) on the imaginary axis 3 = jw, where G,(s) and G 1(cs) denote the transfer functions of the slow and fast models, respectively, i.e., G,(S) = Co(SI — Ao)-lBo + Do 01(63): Cf(8] — Af)-le + Df = 02(631 — 2422)-le + D Equation (4.8) is valid when G,(s) and G f(€3) have no poles on the imaginary axis, which is the case when they are stable. 20 4.2 Sequential Design Procedure A stabilizing output feedback compensator, with two-time—scale structure, can be obtained by the following sequential design procedure [18]. 1. Design a fast compensator G,(es) to stabilize the high frequency feedback loop [0 f(€3), G f(€3)] and to meet high-frequency design specifications. 2. Design a slow compensator G,(s) to stabilize the low-frequency feedback loop [C,(s),G,(s)] and to meet low-frequency design specifications, subject to the constraint C.(°°) = Cf(0) (4-9) 3. A composite compensator G(s,6), taken as the parallel connection of Cf(€3) and the strictly proper part of G,(s), will stabilize the closed-loop system [G(s, 6), G(s, 6)] for sufficiently small 6. Moreover, any point to point the trans- fer function of the closed loop system [C(s, 6), G(s, 6)] is 0(6) close to the one approximated by the corresponding slow and fast models, as stated in (4.8). 4.3 Modeling in Singular Perturbation (SPT) In vehicle suspension systems, the time scale characteristics are composed of two resonance modes: body resonance and wheel resonance [21]. To apply singular per- turbation theory, a singularly perturbed model of suspension systems is needed. First, the small positive constant, 6, which represents the separation of the slow and fast dynamics can be chosen as the ratio between sprung mass resonance and unsprung V Mm' (4.10) Wei/mu mass resonance, i.e., c: 21 For the typical data of Table 1.1, 6 z 0.1. It is shown in the Appendix that the suspension system is a singularly perturbed system with the first two state variables, 2, — 2,, and 2', as slow variables and the other two state variables, 2,, — 2, and 2,, as fast variables. Hence, assuming acceleration measurement £.=[—k,/m, —b,/m, 0 b,/m,]x+[1/m.lu=Cx+Du (4-11) and using the vehicle parameters given in Table 1.1, the full system can be approxi- mated by the slow and fast models given below. Slow model -1 0 l -1 0 A0 = A11 — 14121422 A21 = , 30 = Bl - A12/122 32 = —25.97 —0.79 0.002 00 = Cl -' 0214511421 = [ —2597 —O79 ] ,D0 = D — 02/123132 = [0.002] (4.12) Fast model 0 1 i 0 A! = , Bf = , —4064.5 —6.5 ] -0.016 (4.13) C, = [ o _o,793 ] , D, = [0.002] The value of using acceleration feedback is seen from the fact that the measurement matrices C; 82 D f in the fast model are nonzero. Consequently, the fast model G f( es) is not trivial. If position or velocity feedback is used, i.e., yposition=[10 0 0]x, yvelocity=[010 0].? we have Cfmclocitv = Cfmosition = 01 Dfmelocity = Dimosition = 0 22 Hence, the fast model G f(€3) will be identically zero, and feedback will have a little effect on the performance of the system in the high-frequency range. More precisely, the effect of feedback on the closed-100p transfer function in the high-frequency range will be 0(6). In the case of acceleration feedback, the fast model in not trivial and feedback could have a significant effect in the high—frequency range. Eigenvalue Approximation The eigenvalue approximation for the open loop system is demonstrated in Table 4.1. Eigenvalues fast slow Approximate —3.23 :l: 63.67j —0.40 :1: 5.08j Exact —3.26 :l: 65.28j -0.36 :l: 4861' Table 4.1: Eigenvalue Approximation for the Open Loop System Transfer Functions Approximation F ig.10—Fig.15 show the frequency response of the transfer functions of the system from control force, u, and road velocity, 2,, to the three controlled outputs. They are shown in the order of (1) slow, (2) fast, (3) composite and (4) full. In the case of Fig.11 & Fig.14, the fast transfer function is identically zero. The composite transfer function is 0(6) close to the full transfer function. It is easily noticed that there is a “sharp dip” at the point of wheel resonance in the transfer function of u to 2,. This implies that the control it has no (or very little) influence on the ride comfort at that frequency point. That. point is referred to as the Invariant Point, and the fact that it cannot be changed by feedback is proved in [12]. 23 4.4 Sequential Design Example I (SPT-2) 4.4.1 Fast Design Under sprung mass acceleration feedback, with a lighter damper, b, = 400N.s/m, the ' fast subsystem is not a strictly proper plant as given below: 0.00232 + 8.0565 Grads) = 32 + 5.5., + 4064.5 (4.14) The fast transfer function is written in terms of 3 rather than 63 since the parameter 6 is substituted in the fast model by its numerical value. Under the consideration of controller order not exceeding plant order, a constant gain controller is first tried and causes instability. Therefore, in order to improve stability and satisfy the well- posedness requirement [22], a controller with one zero and two poles is proposed for the fast subsystem. The transfer function of this controller is in the form 8+2 Chm“) : K32 + 2Cwns + w}. (4.15) Using root-locus (Fig.16) and Bode plot techniques, the following compensator pa- rameters are chosen, with emphasis on ride comfort improvement: K = 180000,2 = 36,( = 0.5,w,, = 100 (4.16) 4.4.2 Slow Design The transfer function of the slow subsystem is 0.00232 32 + 0.79298 + 25.9663 G,(ow(3) = (4.17) 24 In the low frequency range, the design task is to choose a slow controller to meet the slow design damping force requirements and eliminate the resonance peak, subject to the following constraint: Calow(m) = Cfast(0) = 648 (4.18) In order to satisfy the above constraint, the proposed controller is taken in the form 3 2 0,10,,(3) = 648( + p +1) (4.19) For the concern of the separation of the slow and fast dynamics, the choice of the zero and the pole should not exceed the mid-point between the sprung and unsprung mass resonance frequencies, which is about 5 Hz. Using root locus (Fig.16) and Bode plot, the slow compensator is taken as 12 0.1015(3) — 648 (8 + 0.8 +1) (4.20) 4.4.3 Full Design The two-timescale stabilizing controller is taken as the parallel connection of C 15.1(3) and the strictly proper part of 0,10,,(3), i.e. 0(3) = Calow(3) ‘l' Cfast(3) - Celow(°0) 12 + 180000 x 3 + 36 (4.21) = 4 6 8" s+0.8 32+1003+1002 The full design is applied to the system. The closed-loop performance of the full system (under the composite compensator (4.21)) is shown in Fig.17. The closed- loop eigenvalues are summarized in Table 4.2. It is noticed that a very wide band reduction and a very sharp wheel resonance 25 fast -426.16 -46.85 4.77 21:64.242' slow -1055 -0.57:l:0.8i Table 4.2: Closed Loop Eigenvalues of SPT-2 reshaping in sprung mass acceleration have been achieved. This significant improve- ment, however, introduces other lightly damped peaks in suspension and tire defec- tion, which implies larger working space and worse road holding ability. Actually, the necessity of high peak force level at wheel resonance is also noticed, which will be mentioned later. Due to the concern of actuator saturation, a force roll-off might be preferred in high frequency range. Thus, the performance of SPT-2 might not be satisfactory enough. 4.5 Sequential Design Example II (SPT) In the previous section, the restriction of controller order not exceeding plant is im- posed on the fast design, and results in a pair of lightly damped poles which are responsible for several undesirable peaks. This restriction will be relaxed a little bit but still imposed on the full system in this section, i.e., a 3" order fast controller and a 1" order slow controller will be considered. 4.5.1 Fast Design Under sprung mass acceleration feedback, with a lighter damper, b, = 400N .s / m, the fast subsystem is repeated below: 0.00232 + 8.0565 Gfau(3) = 32 + 6.53 + 4064.5 (4.22) 26 which has two zeros at :l:63.46j and two poles at —3.23:l:63.67j. A 3” controller with two zero and three poles is considered for the fast subsystem. The transfer function of this controller is in the form of s2 + 2(,w,,,s + w2 C as = K nz f 43) (32 'l" 2prnps + 103...)“ + P) (4.23) Under stability and well-posedness concerns as stated previously, the design idea is to choose a pair of complex zeros around the neighborhood of the fast poles to keep the open loop high frequency damping characteristics; while the choice of the poles is done without causing another significant resonance. Moreover, due to the actuator behaviors, the high frequency ride comfort improvement is also desired while not increasing force level in that range. As shown in the lower half of F ig.18, the open loop fast poles move slightly left before meeting with the controller poles. If the controller gain is appropriately chosen, the open loop high frequency damping is then expected to be preserved. This leads us to choose the following compensator: 32 + 7.5.4 + (65.8)2 as = 4 4.24 cf 48) 5 000 (s2 -l- 60.53 + (72.0)2.)(8 + 30) ( ) 4.5.2 Slow Design The transfer function of the slow subsystem is repeated again 0.00232 Gslow(3) — 32 + 0.79293 + 25.9663 (425) In the low frequency range, the design task is to choose a slow controller to meet the slow design damping force requirement and eliminate the resonance peak, subject to 27 the following constraint: 0,10,,(00) = C;,,,g(0) = 1504 (4.26) In order to satisfy the above constraint, the proposed controller is taken in the form of Z Cslow(8) = 1504(8—‘5 + 1) (4.27) Likewise, for the separation concern of the slow and fast dynamics, the choice of the zeros and the pole should not exceed the mid-point between the sprung and unsprung mass resonance frequencies which is about 5 Hz. The slow compensator is taken as 7 s+0.8 C,,..,,(s) = 1504( +1) (4.28) 4.5.3 Full Design The two-timescale stabilizing controller can be taken as the parallel connection of Cj,,,(s) and the strictly proper part of 0,10,,(3), i.e. C"(3) = 031011.7(3) + Cfast(3) ‘- Cslow(°°) + 54000 82 + 7.53 + (65.8)2 = 15048 + 0.8 (32 + 60,5, + (72.0)?)(:: + 30) (4.29) The full design is applied to the system. The closed-loop eigenvalues are summarized in Table 4.3. Note that the open loop fast poles, —3.26:l: 65.28j (shown in Table 4.1), have been moved to -3.59:l: 65.87j. Thus, the fast open loop damping is preserved in this design. The closed-loop performance of the slow, fast, and full systems (under the composite compensator (4.29)) is shown in Fig.19. For the purpose of comparison, the same transfer functions are shown under the passive system. The performance of SPT 28 fast -192.18 —3.59 :1: 65.87j -10.84 :1: 60.68j slow -4.35 —0.55 :l: 0.84j Table 4.3: Closed Loop Eigenvalues of SPT and SPT-2 are shown in Fig.20 & Fig.21 again. Compared to SPT-2, the undesirable peaks including high frequency tire deflection, suspension deflection, and force level are reduced. Although a sharp reduction in high frequency sprung mass acceleration is lost, we still have some more ride comfort and road holding improvement at the low frequency range (below 5 Hz). Generally speaking, SPT performs more satisfactorily than SPT-2. Chapter 5 Comparative Study Five controllers have been designed. They are LQG—P, LQG-V, LQG-A, CGOF and SPT. A comparative study in those various control schemes will be done in this chap- ter based on the perspectives of nonideal integration, controller bandwidth, actuator force level requirement and system robustness. 5.1 Nonideal Acceleration Integration Velocity signal is typically obtained by integrating acceleration. The practical factors of using a nonideal integrator has been studied in [17]. The frequency response of an integrator should reject DC bias and roll off quickly before reaching the system frequency range of interest to behave like an ideal integrator. Thus, the integrator is considered in the following form: 6,, 1'1 8 Z, = (1,. +1)(72.s +1) (5.1) 29 30 where e, and 6,, denote acceleration and velocity signals, respectively. In this case, the following condition is used. Tl : T2 = 10 ms/ks (5.2) Equation (5.2) indicates the integrator roll-off frequency point is one decade away from the body resonance point. Under the assumption of (5.2), this nonideal scheme usually affects the performance around 0.1 Hz. This fact can be seen from the performance of LQG-V and CGOF systems. 5.2 Evaluation of Performance Criteria As LQG designs are done by the loop transfer recovery of LQR system, no matter what kind of feedback signal is used, the role of the observer is just to provide actuator with state estimate which is determined by the sensor’s noise intensity p. This explains why no significant difference is observed among the LQG performance results, as shown in Fig.6 86 F ig.7. The average ride comfort improvement compared to passive system is about 5—8 dB, primarily in the range of [1 112-5 Hz]. This improvement substantially eliminates the body resonance in acceleration and tire deflection. The price paid with the above improvement is the elevated low frequency suspension deflection, depending on how far the ride comfort is achieved. At high frequency range, ride comfort improvement is achieved at the expanse of higher suspension and tire deflection. In the cases of CGOF and SPT, the ride comfort improvement show larger and wider band reduction than the ones of LQG systems, especially reaching further into wheel resonance region (see Fig.22 82 Fig.23). The SPT [1 Hz-5 Hz] acceleration average reduction compared to passive is about 18 dB (see Fig.19). 31 In designing SPT controller, the closed loop high frequency damping is intentionally maintained the same as open loop; while in CGOF case, the controller gain K c is tuned without affecting high frequency mode (see Fig.8 8L Fig.9). Therefore, both CGOF and SPT will keep high frequency road holding ability and suspension deflection at least the same as the open loop system. 5.3 Controller Bandwidth and Force Level The ride comfort improvement can be used, qualitatively, to reflect the bandwidth of the controller. The L.R.H. side of Fig.22 summarizes the controller frequency response from acceleration to force (Note that velocity is obtained by integrating acceleration). LQG-P is not shown in this plot because it uses a different measurement scheme from the other four systems. However, a fair comparison baseline is the force level requirement. Fig.23 81. F ig.24 show the force level of all designs. Generally speaking, in low frequency range, SPT requires 10 dB more force level than LQG systems in [0.05 Hz—0.5 Hz], and has 10 dB less sprung mass acceleration in [0.5 Hz-5 Hz]; while at high frequency range, the LQG systems require higher force to maintain better damping; instead, SPT uses less amount of force to achieve better ride comfort. This fact is also true in the CGOF system. 5.4 Robustness Analysis For the concern of robustness, the stability and performance of the feedback loop system (C(s),G(s)) should be maintained in the presence of model uncertainties including load, damping and tire stiffness variations, etc. In MIMO systems, assuming 32 that the system model error can be characterized by multiplicative uncertainties, i.e., G(s) + 6G(s) = [I + L(s)]G(s) (5.3) where L(s) is an arbitrary stable transfer function matrix with 5[L(J'w)l S m(w) (5.4) the system’s robustness can be measured by the widely used “complementary sensi- tivity function” [26, 27], as defined below: T(s) = G(s)C(s)[I + G(s)C(s)]’l (5.5) With m(w) denoting the upper bound of normalized magnitude that the model error can tolerate, it is shown that stability is maintained in the presence of all possible uncertainties described by (5.3)-(5.4) if and only if 61'1”an s 517,—) (5.6) The m(w) is typically small at low frequency but goes up to unity and above as frequency increases [2, 26]. In active suspension systems, T( j w) is a scalar term. The singular value bode plot of various designs is shown in Fig.25 including SPT-2 design. It is easily seen that all of the acceleration measurement schemes, except LQG-P, have similar behavior. Without taking LQG-P into consideration, CGOF shows better robustness property than the others because its largest magnitude is kept at unity (which implies allowing 100% model error) and starts to roll off at 2 Hz. The unity level is also observed 33 in SPT and SPT-2 designs, but their roll-off frequencies are at about 20 Hz and 30 Hz, respectively. This is reasonable because SPT (and SPT-2) the fast compensators G,(s) are intentionally introduced in high frequency range to achieve high frequency design goals. This high frequency dynamics does not exist in the CGOF design due to the velocity 2, (slow variable) feedback and the limited stabilizing range of the gain K,. As for LQG-A and LQG—V, the singular value rises up to 8 dB around wheel resonance. Therefore, the allowable model error has to be limited to 40%. As stated previously, m( j w) is typically small in low frequency region. This might make LQG-P still acceptable in spite of its elevated DC value. At the frequency range above 1 Hz, it is always below unity and rolls off at 10 Hz. However, the relative advantage over the other systems can not be determined unless we can understand more characteristics about acceleration and position measurements schemes. 5.5 Performance Variation Due to Parameter Per- turbation Another approach of robustness analysis is to investigate the performance variations when parameters perturbation occur. The perturbation of 50% increase in damping coefficient (b,) and 25.7% decrease in sprung mass (m,) relative to nominal value have been investigated in [24]. For a real vehicle, the opposite perturbation also happens, but it is claimed that the change in this direction as stated above represents the worst case. We thus consider the above two cases in our analysis. Besides, the perturbation of tire stiffness, which plays an important role in high frequency dynamics, is also of interest. Due to modeling error or environmental influence, it is assumed that the tire stiffness could be varied from 50% decrease to 50% increase relative to its 34 nominal value. This is equivalent to a 50%-150% wide variation range. For the sake of simplicity, the analysis is performed only on LQG-P, CGOF and SPT, representing different measurement schemes of position, velocity and acceleration, respectively. 5.5.1 Perturbation of 50% Damping ((7,) Increase Fig.26, Fig.27, 82 F ig.28 show the damping perturbed performance, where the solid lines represent the nominal performance and the dashed lines represent the perturbed performance. This perturbation causes the three systems about 2—4 dB ride comfort degradation in [1 Hz—10 Hz]. A little bit loss in road holding ability is also noticed. As for suspension deflection, the influence is not significant. In general, the three systems have comparable robustness property to damping perturbation. 5.5.2 Perturbation of 25.7% Sprung Mass (m,) Decrease The vehicle load fluctuates quite often, and is usually characterized by sprung mass variation. The assumption of 25.7% decrease results in 1—3 dB ride comfort loss in the three systems, while requiring less suspension and tire deflection, as shown in F ig.29, Fig.30, 86 F ig.3l. Likewise, the solid lines represent the nominal performance and the dashed lines represent the perturbed performance. In fact, SPT sprung mass acceleration is more insensitive to mass variation except at the wheel resonance point. This property is not considered as relatively important because the other two systems are still quite comparable. 5.5.3 Perturbation of 21:50% Tire Stiffness (kt) Next, a wide range, from 50% decrease to 50% increase, of tire stiffness is assumed. The results are demonstrated in Fig.32, Fig.33, 81. Fig.34. Clearly, all of their wheel 35 resonances are shifted lower or higher, and their performance band variations appear without any significant difference. No relative advantage is offered by any system. This fact shows that tire stiffness change has similar influence on each system. On the other hand, if we investigate the perturbational influence on the three performance criteria, it might be concluded that the increase of tire stiffness causes (1) deprivation of ride comfort, (2) requirement of larger working space, (3) loss of road holding ability. For the case of ride comfort, it can be considered as better if only the range of [1 Hz-10 Hz] is concerned, which is claimed in [24] as the main sensitivity region of human beingl. But for road holding ability, it is not sufficient to look at only tire deflection when tire stiffness is perturbed, because the road holding is measured by the contact force fluctuation between the tire and road surface as stated in Section 1.4. Thus, a much worse road holding comes out. For more references about this issue, please refer to [11]. 1Actually, the region is claimed as [3 Hz—8 Hz] in [24]. Chapter 6 Conclusions The potential of using acceleration feedback has been explored in this thesis. It is shown that ride comfort can be improved over a wider frequency band, reaching into the wheel resonance. The improved design is achieved via characterizing the dynamics into slow and fast models and designing compensators individually to meet desired specifications for low and high frequency ranges. The design takes advantage of a more general use of acceleration feedback, compared with the more typical limited use when acceleration is integrated to produce velocity. The ride comfort improvement reported in [24] using “frequency weighted output feedback” is 8.3 dB at 20 rad/ sec. (z 3 Hz); and in [12] using LQG-P is the elimination of the sprung mass resonance. Here, the average band reduction on the frequency range of [1 Hz—5 Hz] is about 18 dB compared with the passive system, while still satisfying other performance concerns. A simpler scheme like CGOF might be attractive if acceleration integration is done properly. However, compared with the original passive system, the price paid for these achievements is the requirement of larger working space and some loss of road holding ability at the wheel resonance. Another advantage of applying the two-timescale technique to ASS is that the 36 37 complexity of design task will be reduced a lot when we move up to solve a full- car problem. By the assumption of full car symmetry (see [2] for a full car model description), the original 14“ order full model can be decomposed into a 6“ order slow model (body) and four identical 2"“ order fast models (4 wheels). 38 : sprung mass : unsprung mass : suspension damping consrant : suspension spring consrant : tyre spring constant ., : road displacement u : control force 4 ”7'? “R‘- 91°- :5 as from sensorsl positiortvclocity or acceleration) 2.: ms(body) & acruator Figure 1: Structure of an Active Suspension System 39 823m almana— a he vacancy. 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To. 4‘41 1 d d ddqdq d d 1 ‘ Adda-44 - 1 d r 3v.- l l - om.- ...p?. . P ....... . . .......L . .25.: an. No. 3:. 3:03.58". 1: so. 2: qu nun-Db - 1 P iddd q 1" uqqdd-qd nun-Db. - D b-Dthh h d 2: 3.5:. 2.59: 00 _o. 2: av.- cNT . § av mo :90 M1 2: «:14: d a d uddqd 14 d d 111111 1 1 h bb-bbbh h D bbbbb- b n h l l .256 355 ow.- om.- ap umo 390 9141 § Figure 15: Transfer Functions of u —) (zu — 2,) [mag Axis lmag Axis 15 10 -5 ~10 ~15 150 100 50 -100 -150 53 Rom Locus of Slow Model f I V ' l ‘4 i - L :4 - 15 .10 -5 O 5 10 15 Real Axis Roo: Locus of Fast Model T. I- 4 l. . J I. i- .1 -150 -100 -50 0 50 100 150 Real Axis Figure 16: Root Locus of Slow and Fast Models (SPT-2) 54 No. Amara—m. ".8295 3.3:: a. :5. Sun. .255 .0 005.58.?— K. 2sz as”. a 32m l.- =£ ...... D>mmmml 3:. x5252”. 2: so. 1: We. qu-ddq d u a pPPp-n - - co.- 1 u) of SPT & SPT-2 58 No. ”Em a. #5 .o A: T .3 :80 3.9.. a. 8.56.2.9. .5 2:3". Q2. 5:33... .o. as. To. ----q u Z—uqfid d 4 «qqqqa H d a dq—qqqd‘J - 2: o pppb—_ - =:—- h p ::b__ P P ::—.b p b GI. 35:35 «o. .o. cc. To— Go— :qqdq4 J duqqqq+fi — :qdqdd 1 - 43d«« 4 d 8~| r 1 on- ... I 1 o =FPP- - _ =:- p p p =:——_ n h b-ppP- _ p - Hp HIRE) 90.105 01 9903 HP “W9 'J9CI 'sns No. 3:. 5.5232...— .0. so. To. we. =«____ d pp-hb p p J—«u—qd 1 _ «dc-dud q — quad—«1 « u GI. 55:58... .o. b:---— n _ cc. n-npbb - p — To. :pp-pn - - ow- HP men use an. HP 11199 eoov Figure 21: Performance and Force Gain (:3, —+ u) of SPT & SPT-2 59 Elm 0% WOOD 5-004 {-004 .0 A: T an... 500 3:05:00 0% 00::E.0.t0& ”mm 0.3:,”— ANIV 5:015... we. 5. co. 1: law mxx £000 2 - 0: 3:. 5:02.00... No. 5. 0o. 1: 1: HP 0129 IGHOIIUOD HP “F99 'JQCI 'SPS GI. 3:03.00... Na. .0. so. 1: woo—MW- cc- ow. l 1 O GI. 35:00.... No. 2: 0o. 1: «A: av mo :90 915.1. HP “F99 '933V ) of LQG-A, LQG-V, CGOF & (z',-—+u Figure 22: Performance and Controller Gain SPT 60 rim 0% :60; {-001— :0 A: T i :30 00.0... 0% 00:55.88: ”mm 0.3:”— ANI;0:0:G0.£ ANIV 03:030.: m No: 2: 0o. 1: «A: No. z: 0o. 1: NA: & Z... . ........ 1 33C ,3... . ........ . Om! p... .11! .. H G A. 00‘. m. 1 m. _ m m A, r G m. m 0. 3 9 L 9 d m. m. w, - p r t. m . a .( CE»... . =2...— . =..... . . ht... . . EEF—ILEF—Il ON .m . G :02: 5:03.00..— Anzv 5:03.05 m N... a. a: 1: l: 2: a. a: 1: 1: m w m v m a m m 9 .9 .ne .4. D P m m 0 . e 0 0 m 61 Em 0. n. 000 5.00:. .o 0. T .0 5.5 85,. 0 8.5.525: em 050.: ANIV 3:02.02”. 3:. 5:03.00.”— NA: .9 0.: _-o_ 2% No. z: 2: l: «A: an: :08 mo ’ . 8 >0qu .. m - ._ on ma \.\\\ M 9 9 m. I Em 1 9: P Er—FELEF—IL 8 GI: 3:02:02”. 3: 5:030.”— NA: z: a: l: 1: N: z: 0.: l: «A: «11‘ q udddqq+d d :qqudq 1 dl «duqdud u q gfil , S m . 8. m m p o H HP mo :90 9151. HP 0:99 "mv Figure 24: Performance & Force Gain (2", -+ u) of LQG-V, CGOF & SPT 62 05:00A— m:0_.:> :0 02.3 .fizwfim ”mm 053: NA: 2: co— u—J—«-q. a 1 l: d-_.uq ‘ u NA: :3. 1-44411 a p d - uq_d.qd_ _ _ b pp-pr— _ — —_—--bl_ l ONT .1: ov —~P—— -——__h 0:383 m:0_.:> :0 m0:_:> .:_:w:_m lgns lar Value of Various Des' ingu S Figure 25 63 E053 LAVOA Boston— AB. mcfiEan— Qoom .0 00:...:.0.t0n_ ”cm 0.3:". No. 2: 0.: l: «Aw—w- 8| ow. on- I c 3:. 5:02.00..— QI. 3:02.00.”— NS 2: 0c. 1: «A: no. 2: :2: 1: No— HP WD 390 “‘8 I 1 C HP mo “:90 91M. an men 'mv Figure 26: Performance of 50% Damping (b,) Perturbed LQG-P System 64 605% ”.000 Bates: 38 wiaEmfl 0:8 :0 00:08.88: Km 0.33.. No. z: a: l: «A: 8. e- m - ”are... :8 L a? m "J I . 0.- o m. I l o @ AN. : 0.2.0230...— Q... 5:030.”— NA: .2 0.: l: 1: «A: z: 0.: 1: «A: “sns HP 11:20 use an mo °aoov Figure 27: Performance of 50% Damping (b,) Perturbed CGOF System 65 E053 ,Em .000.=..0n_ 23V @5955 Quom .0 00508.83: “mm 053,: 20:: 20:02.00.”— 2: 2: .2: .2: .2: 2N2. 20:01.0."— ANIV 20:00:22 2: 2: 0.: l: .2: N2: 2: , 02: .2: .2: HP “199 390 'SPS =-——-— - --—---b P - E—D-pn - =-—-——- - up mo ”1°C! 9151 HP “F99 '999V Figure 28: Performance of 50% Damping (b,) Perturbed SPT System 66 529$ L604 3263— 225 $22 boom $52 ho 8:55.88“: ”om 95w."— 2: 2: c2: .2: N2: ,3qq4q ‘ a 1:: q q . R O”! 1 - l on; 8 ’m . 91 m .l —GEEOZ 1 ON! m u .. a Q Q— = 22523..”— G—: 22.2.72"— 2: . 2: 2: .2: «-2: 2: 2: .2: .2: «-2: SIP “199 3°C! '5115 SIP “199 '939V Figure 29: Performance of 25.7% Body Mass (m,) Reduced LQG-P System 67 8053 "500 3260: 225 awe—2 boom 922mg ho 8:56.85; ”or SEE 2: 2: c2: .2: N2: 5; 8a.: 2 . a... h a .I U... \ O... 1 G ‘7' 2N. : 29:259....— 2N:V 2o:o=..o.£ 2: 2: a2: .2: N2: 2: 2: c2: .2: «2: m. ON- G u. 9 o m. m cm \ c': T 9G SID men '933V Figure 30: Performance of 25.7% Body Mass (m,) Reduced CGOF System 68 Eo:w>m Em 600293. 3:: 35—2 35¢ §b.mm EC cutaEcctta . _ v $2.5". 23: 20:03:22 2N; 20:259.": REV 20:252.: 2: 2: .2: .2: N2: 2: 2: c2: .2: HP “FED 'J9CI ‘sns 1 Q «2: up no 390 9.15.1. HP “1'99 '939V Figure 31: Performance of 25.7% Body Mass (m,) Reduced SPT System 69 62% “.63 .6335“. .2. saga 2: $878 as 8255.3; an 2%.”. 2: 3...: 225379.... 2: oo_ To— «2: #44. d :ppn - p b 1:: 4+4 q =:—Pn - - «d—«qdd q 1 _:qq-« - d :..-_. . 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HP “339 '939V Figure 33: Performance of 50 - 150% Tire Stiffness (kt) Perturbed CGOF System 71 Eo.m>m ,Em 2.3.2.9: 2:: 305.25 2:. skew—-3 .c 8.5.5885. 6w 8.5.": 22: 228:3."— 2: 2: .2: .2: N2: d— 1 :uchd « d qua-qua q A .u-u—qd « H.\ cm. .2... o? m I .ta/..\.\.\\\\\ ”01.; 1 owl m / 5:82 1. 1 J... eon... . - cm. W I - o W 2N=v 2ocozco£ 2N2: 202.252,: 2: - 2: .2: .2: «be—2:- 2: 2: .2: .2: «2: HP “W9 '993V HP “199 3°C “sns :..... . :...... (Eu. . :...... . Figure 34: Performance of 50 — 150% Tire Stiffness (la) Perturbed SPT System Appendix A Through physical understanding of the state variables, it might be easy to conjecture that the state variables related to sprung mass dynamics, i.e., z, — 2,, & 72,, are slow variables; while the other two state variables related to unsprung mass dynamics, i.e., zu — z, & 2,, are fast variables. This conjecture can be confirmed by modeling the system in the singularly perturbed form. Consider the vehicle state equation as defined before, it = Ax+Bu+ Ez', (A.l) where F , 0 1 0 —1 l o 0 l —k, m, —b, m, 0 b, m, l m, 0 A = / / / ,B = / , E = O 0 0 l 0 -l k,/mu b,/mu —k¢/mu -b,/muj —1/mu l 0 ' ‘ (A.2) 72 73 k m . . . Let c = 3%? and choose a new state vector .7: and a new input u as l mu _ . ' l $1 2, — Zn :2 fim, hi, 5: = 2 = / ,a = i (A.3) $3 (zu "" Zr)/‘E 534 vms/kséu If we define a new time scale by A k, t = t — (AA) m, then, with ( ) denoting the derivative w.r.t. f, the state equation is expressed by £1 £1 £3 . o . = All + A12 + Blu + E12,- (A.5) i2 532 5:4 6.223 571 i3 ,, . = A21 + A22 + 321‘ + E2Zr (A5) 6174 532 .1 (54 where 0 1 0 —1 0 0 All— aAl2- aBl— 9E1- a 0 0 ' 0 1 A21 — ,Azz - , em,/mu eff; 1‘: 2% 6;?“- m,/k. 0 -l 82 = ,E2 = , (A.7) —cl"-t 0 74 For the typical numerical parameters given in Table 1.1, c is about 0.1 and all the following quantities are 0(1). 8 8 8k b8 8 —b./\/m.k.,b./\/m.k.,em./mu,6r-:- $.625—kt,ém \/m./k.,-€: u This shows that the model (A.5)—(A.6) is in the standard singularly perturbed form, and confirms the conjecture that z, — 2:u & 2", are the slow variables while zu - z, & é,, are fast variables. To put the system into the standard singularly‘perturbed form, we needed to scale some of the state and input variables. Scaling of state variables does not affect the input-output tansfer functions since it is an internal similarity transformation. Scaling of the input only multiplies the transfer function by a constant. Since the sequential design procedure uses transfer function models in designing the controllers, it is not necessary to model the system in the standard singularly perturbed form. It is sufficient to recognize the slow and fast variables and order the components of state vector so that the slow variables come first. Then, the slow and fast models can be defined as in Section 4.1. It can be verified that scaling of state variables does not affect the slow and fast transfer functions. Bibliography [1] Hedrick, J.K. and Wormley, D.N.,ASME' AMD, Vol.15,1975, p.21. [2] Majeed, K.N., “Centralized/Local Output Feedback Control and Robustness with Application to Vehicle Active Suspension System,” Ph.D. Thesis, U. of Dayton, 1989. [3] Alanoly, J ., “New Concept in Semi-active Vibration Isolation,” J. Mech. Trans. Auto. 065., Vol. 109, June 1987, pp.242-247. [4] Alanoly, J ., “Semi-active Force Generators for Shock Isolation,” J. Sou. Vib, Vol. 126, Oct. 1988, pp.145—156. [5] Karnopp, D., “Force Generation in Semi-active Suspensions Using Modulated ‘ Dissipative Elements,” Veh. Sys. Dyn., Vol. 16, 1987, pp.333-343. [6] Margolis, D. L., “Chatter of Semi-active On/Off Suspensions and Its Cure,”, Veh. Sys. Dyn., Vol. 13, Nov. 1984, pp.129—144. [7] Hrovat, D., “A Class of Active LQG Optimal Actuators,” Automatica, Vol. 18, No. 1, 1982, pp.117-119. [8] Friedland, B., Control System Design, McGraw Hill,1986. [9] Hall, B. B.& Gill, K. F., “Performance Evaluation of Motor Vehicle Active Sus- pension Systems,” Proc. Instn. Mech. Engrs.,Vol. 201, No. D2, 1987, pp.135—148. [10] Karnopp, D., “Two Contrasting Versions of the Optimal Active Vehicle Suspen- sion,” J. Dyn. Sys. Meas. Cont, Sept. 1986, pp.264—268. [11] Sharp, R.S., & Crolla, D.A., “Road Vehicle Suspension System Design — a Re- view,” Veh. Sys. Dyn., 16,1987, pp.167-192. [12] Yue, C., Butsuen, T. & Hedrick, .ll.K., “Alternative Control Laws for Automotive Active Suspensions,” J. Dyn. Sys. Meas. 5 Cont., Vol. 111, June 2989, pp.286— 291. 75 76 [13] Miller, L.R., “Tuning Passive, Semi-active and Fully Active Suspension Sys- tems,” Proc. of the 27‘“ CDC, 1988, pp. 2047—2053. [14] Hrovat, D., “Optimal Active Suspension Structures for Quarter-Car Vehicle Models,” Automatica, Vol. 26, 1990, pp.845-860. [15] Elemandy, M.M., “Optimal Linear Active Suspension with Multivariable Integral Control,” Veh. Sys. Dyn., 19, 1990, pp.313—329. [16] Thompson, A.G., “Optimal and Suboptimal Linear Active Suspensions,” Veh. Sys. Dyn., 13, 1984, pp.61—72. [17] Margolis, D.L. “The Response of Active and Semi-active Suspensions to Realistic Feedback Signals,” Veh. Sys. Dyn., 11,1982, pp.286-291. [18] Khalil, H.K., “Output Feedback of Linear Two-Time-Scale Systems,” IEEE Transactions on Automatic Control, Vol. AC-32, Sept. 1987, pp.784-792. [19] Luse, D.W. and Khalil, H.K.,“Frequency Domain Results for Systems with Slow and Fast Dynamics,” IEEE' Transactions on Automatic Control, AC-30, Dec. 1985, pp.1171—ll79. [20] Kokotovic, P.V., Khalil, H.K. and O’Reilly, J .,“Singular Perturbation in Control : Analysis and Design,”Academic, New York, 1986. [21] Salman, M.A., Fujimori, K., Uhlik, C., Kawatani, R. and Kimura, H., “Reduced Order Design of Active Suspension Control,” Proceedings of the 27‘“ CDC, Dec. 1988, pp.1038—1043. [22] Chen, C.T.,Linear System Theory and Design, Holt, Reinhart and Winston, 1984 [23] Tahk, M. and Speyer, J .L., “Modelling of Parameter Variations and Asymptotic LQG Synthesis,” IEEE Transactions on Automatic Control, AC-32, Sept. 1987, pp.793—801. [24] Yamashita, M., Fujimori, K., Uhlik, C., Kawatani, R. and Kimura, H., “H.,, Control of an Automotive Active Suspension System,” Proceedings of the 29‘“ CDC, Dec. 1990, pp.2244-2250. [25] Alexandridis, A.A. and Weber, T.R., “Active Vibration Isolation of Truck Cabs”, Proceedings of ACC, 1984, pp.1199—1208. [26] Stein, G. and Athans, M., “The LQG/LTR Procedure for Multivariable Feedback Control Design”, IEEE Transactions on Automatic Control, AC-32, Feb. 1987, pp.105-114. ' [27] Kwakernaak, H., “Robustness Optimization of Linear Feedback System”, Pro- ceedings of CDC, Dec. 1983. nrcurcnu STATE UNIV. Lrsnnn lllll[ll]ll]llll]ll]Ill]lllllllllllll]llllll 31293008770103