Optimized Virtual Model Reference Control for Ride and Handling Performance-Oriented Semiactive...

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1

AbstractmdashThis paper proposes an optimized virtual model

reference (OVMR) control synthesis method for semi-active sus-pension control based on ride and vehicle handling characteristics First we present the semi-active Macpherson suspension system as an Hinfin robust output feedback-oriented control model Then by using the combination of a set of linear matrix inequalities (LMIs) and genetic algorithm (GA) the desired internal states for the tracking control problem of the semi-active suspension can be obtained via an OVMR To achieve the Hinfin performance of ride comfort and vehicle handling against the influence of parameter uncertainties and external disturbances of the system a robust adaptive controller is designed so that the controlled system can track the desired states generated from OVMR The tracking control can be converted into a stabilization problem with as-ymptotic convergence in the sense of Lyapunov stability theorem To validate the effectiveness of the proposed approach the co-simulation technique is employed to bridge the gap between the mathematically well-defined system model and the optimization quality of control It can be confirmed that the designed control system can achieve performance-effective suspension control through the confident software-in-the-loop (SITL) simulation

Index Termsmdashautomobile control semi-active suspension ride and handling optimization adaptive robust control

I INTRODUCTION

utomotive suspension systems are able to isolate the pas-sengers from road disturbances to improve ride comfort

and also force contact between the tire and the road at all times to enhance road handling During the past few decades vehicle suspension design has been extensively explored due to their considerable contribution in ride handling and safety for road vehicle performances Conventional passive suspension sys-tems include springs and dampers that mitigate harmful and uncomfortable vibrations so that they can offer certain ad-vantages in realizing a certain desired degree of compromise between ride and handling [1 2] However such systems easily achieve conflicting performance requirements among ride comfort road handling ability suspension travel and cost Alternatively great interests as well as emerging demands are being devoted to the controllable suspensions in both academia and the industry for a few decades Due to the rapid advances in mechatronics a large amount of active and semi-active sus-

Manuscript received hellip

pension control approaches have been the potential solutions to resolve the inherent tradeoffs between the performance re-quirements [3-5] Active suspension systems have been investigated and de-veloped for decades [6-8] Although such systems can effec-tively provide viable improvement for aforementioned vehicle performances the main obstacle is the significant power con-sumption On the other hand failures caused in active suspen-sion systems might result in handling problems due to road disturbances Compared with active suspension systems a semi-active suspension requires considerably less power and is less complex Therefore semi-active suspensions have raised considerable attention in recent decades since semi-active de-vices can provide the most favorable compromise between cost-effectiveness and control performance In addition semi-active devices are more stable and fail-safe thus they can act as pure passive dampers in case of damping control failure Since there still exists a trade-off between comfort-oriented design and road handling ability semi-active suspension tech-niques require to address the associated challenges to date In addition to reducing complexity and cost while improving ride comfort vehicle handling and control performance the ad-vanced development of semi-active suspensions with promis-ing results has emerged

To effectively handle nonlinear dynamics in the geometry and structure of vehicle suspensions considerable challenges exist in the development of semi-active control systems which attempt to achieve a compromise between ride control and vehicle handling performance [9 10] Conventional semi-active control strategies typically apply a switching nature However this type of design makes matters difficult for control systems and is highly dependent on an extensive trial-and-error process To facilitate a certain degree of robustness against external disturbances and parameter uncertainties in systems a number of robust control schemes have been proposed in-cluding H [7 8][11 12] sliding mode control (SMC) [13]-[15][30] and adaptive control [16 17][29] Improvement in body acceleration can be observed when investigating these approaches but usually with an increase in tire load So far intelligent control methodologies have been proposed to ad-dress the mechanical nonlinear dynamics and system com-plexity behind practical and industrial application domains [5][18 19] Since the major challenge arising is to design model-based controllers for the nonlinear and complex char-

Optimized Virtual Model Reference Control for Ride and Handling Performance-oriented

Semi-active Suspension Systems

Hsin-Han Chiang Member IEEE and Lian-Wang Lee Member IEEE

A




2

acteristics exhibited in vehicle suspension systems integrated computational intelligence and adaptive approaches including fuzzy inference systems [20 21] neural networks [22 23] and genetic algorithms [24 25] in vehicle suspension control sys-tems have been thoroughly proposed Although these intelli-gent control methodologies have been successfully applied to various practical control systems the design challenge still remains when it comes to determining suitable control structure and appropriate selection of parameters especially when such methodologies are adopted to manipulate complicated systems with nonlinearities and uncertainties The other unresolved problem brings about the bottleneck in the mathematical proof for stability which is still difficult to demonstrate at present

In many studies on vehicle semi-active suspension devel-opment the results have been obtained from active suspensions by using a simple projection to handle the dissipative constraint In other words the damping force applied by the semi-active suspension is chosen by the controller to be as close as possible to the active force This results in the so-called ldquosynthesize and try methodrdquo potentially leading to unpredictable behaviors which make it impossible to ensure either closed-loop internal stability or performance Moreover most existing methods of semi-active suspension control lack the systematic strategy to involve an optimization procedure while guaranteeing the dis-sipative constraint owing to parameter-dependent structures and multi-objective constraints handling To this end some studies (eg [26 27]) have also proposed hybrid intelligent control systems in order to pursue the effective balance be-tween ride comfort and vehicle handling quality However the problem that arises is satisfactory real-time computing effi-ciency of the control system to fulfill the adaptive mul-ti-objective optimization required in different road conditions

To address the difficulties of the aforementioned control schemes this paper presents a systematic and practical meth-odology for designing optimized vehicle semi-active suspen-sion systems A robust SMC system with an adaptive scheme is also proposed in our approach to identify the unknown upper bound of system uncertainties and external disturbances so that higher robustness and adaptability can be achieved Unlike most past studies of suspension control systems which only demonstrate the performance under a quarter-vehicle or half-vehicle model we focus on developing a semi-active suspension system that allows for a systematic design with multi-objective optimization considerations for a full-vehicle Moreover this study addresses the associated practical diffi-culties of using semi-active suspension controllers with avail-able sensors Up to now the co-simulation technique has be-come indispensable support in the design process of automotive industry Thus through the co-simulation analysis in support of suspension design the proposed approach can benefit from the evaluation analysis of a unit vehicle performance based on the high-level perspective of the physical environment

The rest of this paper is organized as follows Section II presents the model of the Macpherson semi-active suspension system and addresses the development of the optimized virtual model reference (OVMR) control synthesis In Section III the adaptive robust tracking control design is introduced to achieve

the state tracking objective in terms of comfort and vehicle handling Section IV shows the feasibility and effectiveness of our proposed control scheme by using co-simulation execution in a software-in-the-loop (SITL) environment Conclusions and recommendations for future work are given in Section V

zr

zs

zu Ks

Fsa

Z

Y0+

O

B

A

C

Fig 1 The schematic model for the Macpherson suspension system

II DYNAMIC MODELING

A System Description

Various vehicle suspension models have been proposed to study semi-active suspensions (eg [28]) In this study a comprehensive and realistic model of the Macpherson suspen-sion is presented for semi-active control systems A schematic diagram of control-oriented Macpherson suspension is shown in Fig 1 in which the rotational motion of the unsprung mass is admitted Such a suspension system is composed of the fol-lowing major elements the chassis the control arm the strut and the wheel assembly In this section a two-dimensional model with the two generalized coordinates zs and θ in Fig 1 is firstly investigated for the control-oriented problem Ac-cording to the equations of motion in [29 30] the nonlinear model with Macpherson suspension for the two generalized coordinates is given as

20 0( ) cos( ) sin( )s u s u c u cm m z m l m l

0 0 [ sin( ) sin ] 0t s c rK z l z (1-1) 2

0 0

0 0

cos( ) cos( )

1[sin( ) sin ] sin( )[

2

u c u c s t c s

c r s l

m l m l z K l z

l z K b

]cos( )

lb sa

l l

dl F

c d

(1-2)

with 2 2

l a ba l l 2l a bb l l 2 cos( )l l l lc a a b 2 cos( )l l l ld a b b and

0

where ms and mu are the sprung and unsprung masses respec-tively and Ks and Kt are the spring constants for the suspension and tire respectively zs represents the displacement of the sprung mass is the angular displacement of the control arm zr

is terrain height disturbance la lb and lc are the distances from point O to points A B and C respectively α is the angle be-




3

tween the Y-axis and line OA 0 is the initial angle of the control arm at the static equilibrium point resulting from structure design Fsa denotes the semiactive damping force and can be defined as

( - ) 0( )( - )

( - ) 00 s s usa s u

sas s u

if z z zC t z zF

if z z z

(2)

where Csa(t) represents the actual damping coefficients of the shock absorber Based on the accelerations of the generalized coordinates from (1) the overall system dynamics can be yielded as the followings state-space representation

1( )s s s sa rz f z z F z

2( )s s sa rf z z F z (3)

The above equations of f1() and f2() present the acceleration of the sprung mass and the angular acceleration of the control arm respectively as nonlinear functions of the state variables con-trol input and road disturbance Their detailed expressions are described in Appendix A

The Macpherson suspension is widely used in the semi-active suspension systems due to the various merits such as light weight compact size and low-cost However its var-iable geometry including large and asymmetric variations in kinematic provokes a nonlinear behavior and causes quite a few of obstacles for control purpose The linearized model of the Macphseson suspension has been proposed by [31] where detailed comparison shows good dynamic performance when using the linearized model However the measurement of the angular displacement of the control arm is very difficult obtain using the practical sensors For minimizing the cost and ad-dressing the associated practical issues only the vertical vi-brations of the sprung mass and the unsprung mass are availa-ble in this study

ms

mu

Ku zr

zs

zu

Ks fsa

Accelerometer

Semi-active control strategy

Accelerometer

Fig 2 Quarter-car semi-suspension model

The linear dynamics of a quarter-car semi-active suspension

as shown in Fig 2 can be expressed as ( )s s s s u sam z K z z F (4-1)

( ) ( )u u s u s t u r sam z K z z K z z F (4-2)

where zu is the displacement of the unsprung mass It is not sufficient to represent the inherent nonlinearities and uncer-tainties existing in real system (1) with a linear model (4) In order to obtain the proper parameters for the linear quarter-car

model the least square method is applied for system identifi-cation For the sake of simplicity we assume the same spring coefficient for both directions and no loss of tire contact By setting Fsa = 0 the linear model (4) can be rewritten as follows

1( ) Ts s s s um z K z z (5-1)

2( ) Tu u s u s tirem z K z z F (5-2)

where ( )tire t u rF K z z presents the tire force

1 [ 0 ]Ts s uz z z and

2 [0 ]Tu u sz z z are regression

vectors [ ]Ts u sm m K is the parameter vector Define the

estimated errors as 1 1 ˆTe and

2 2 ˆTe and

ˆˆ ˆ ˆ[ ]Ts u sm m K is the estimated parameter vector The error

cost function for the least square algorithm can be defined as

2 21 2

1

[ ( ) ( )]N

k

E e k e k

(6)

where k denotes the k-th measurement and N is the total number of measurements The total error E can be minimized by using the following condition

1 1 2 2 21 1

ˆ[ ( ) ( ) ( ) ( )] ( ) ( ) 0ˆ

N NT T

tirek k

Ek k k k k F k

(7)

As a result the estimated parameter vector can be obtained as 1

1 1 2 2 21 1

ˆ [ ( ) ( ) ( ) ( )] ( ) ( )N N

T Ttire

k k

k k k k k F k

(8)

Now define the system state vector as

1 2 3 4[ ]Tx x x x =[ ]Ts u u r s uz z z z z z output as [ ]T

s uy z z

and thus the state-space model of a quarter-car suspension is

sax Ax BF G

say Cx DF v (9)

where

-

-

0 0 1 -1

0 0 0 1

0 0 0

0 0

s

s

s u

u u

K

m

K K

m m

A

-1

1

0

0

s

u

m

m

B

0

1

0

0

G

-

-

0 0 0

0 0

s

s

s u

u u

K

m

K K

m m

C

-1

1

s

u

m

m

D

rz and 1 2[ ]Tv v v is the sensor noise

B Virtual Reference Model Formulation

For the purpose of virtual reference model design the hybrid strategy using sky-hook and ground-hook controllers are ap-plied for a semi-active Macpherson suspension system by means of the robust output feedback control (OFC) technique The reference model should behave as a desired system with a suitable control force while the mathematical model must be as similar to the practical system as possible Accordingly the combination of the Hinfin OFC theory and the genetic algorithm (GA) is adopted in this study to find the optimized feedback gains for the reference model so that the stability and control objectives can be achieved The aim of the sky-hook control strategy is to minimize the




4

vertical motion of the sprung mass by connecting a virtual damper between the body and the sky This strategy hence certainly improves the ride comfort but may lead to poor vehi-cle handling In practice the adjustable damper is approxi-mated to mimic the virtual damper and the performed damping force can be represented as

( - ) 0

( - ) 00 s s usky s

skys s u

if z z zC zF

if z z z

(10)

Complementary to the comfort-oriented skyhook strategy the ground-hook control strategy is originated to connect a damper between the ground and the wheel This strategy can result in a reduction of vertical tire motion so that vehicle handling can be improved Similarly in practice the adjustable damping force can be expressed as

( - ) 0

( - ) 00 u s ugrd u

grdu s u

if z z zC zF

if z z z

(11)

With the combination of sky-hook and ground-hook strategy the hybrid control strategy for the semi-active damping force can be formulated as

(1 )

(1 )sa sky grd

sky s grd u

F F F

C z C z

(12)

where Csky Cgrd are the damping coefficients and β gt 0 deter-mines the degree of contribution of the sky-hook and ground-hook control actions inside the hybrid control strategy

To obtain the optimized control gain of (12) in the virtual reference model this study applies the Hinfin OFC theory on the control formulation in an attempt to minimize the transfer characteristics from road irregularities to the chassis while retaining the other requirements within a desired level Based on the state-space representation (9) the dynamics of the ref-erence system are formulated in the following form

r r sarx Ax BF G

r r ry C x

1 1obj r sarz C x D F

sar r r r r rF K y K C x (13)

where Kr = [βCsky (1-β)Cgrd] stands for the output feedback gain to calculate the desired damping force Fsar xr and yr de-note the reference state vector and output vector respectively

[ ]Tobj s s uz z z z denotes the objective vector which constitutes

the ones to be minimized A B and G are matrices as obtained from (9) and other matrices are defined as follows

0 0 1 0

0 0 0 1rC

- -1

1 1 1

2

0 0 0

0 0 ( ) 0 0

0 0 0 ( ) 0

ss s

Km m

C s D

s

(14)

where ( ) 1 2i s i are defined as the filter functions to es-

timate the vertical displacement from the velocity value re-spectively The utilized filters take the following form

1 2 21 1

( )2s s s

n n

sz s z z

s

(15)

2 2 22 2

( )2u u u

n n

sz s z z

s

(16)

Here the damping ratio is chosen as 0707 ωn1 and ωn2 are set as 01 Hz and 10 Hz respectively The filter ρ1(s) performs as differentiator below 01 Hz and as an integrator above 01 Hz for estimating the displacement of sprung mass Similarly the displacement of unsprung mass can be estimated by using the filter ρ2(s) which acts as an integrator above 10 Hz These filters provide a satisfactory displacement estimation of the sprung mass and the unsprung mass by excluding a DC offset [29] From (13) the objective vector zobj includes the vertical accel-eration of the sprung mass

sz vertical chassis displacement zs

and vertical tire displacement zu Accordingly the minimiza-tion of the Hinfin norm of the transfer characteristics of accelera-tion and the displacement of sprung mass and the displacement of unsprung mass from road disturbances yield better ride quality and road-contact ability

In the control formulation for the virtual reference model the uncertainty of sprung mass ms should be taken into account This uncertainty arises from changed under different driving conditions such as the mass distribution between the front and the rear axles during acceleration or deceleration and load variations of the vehicle In reality the uncertain mass ms is reasonably bounded between its minimum value msmin and its maximum value msmax Let ς(t) = 1 ms the following result can be represented

1 2

min max

( ( )) ( ( ))1

s s

t t

m m

(17)

where 0 1 2i i and 2

1

( ( )) 1ii

t

The function

( ( ))i t can be calculated as

min max

min max min max

1 1 1 1

1 21 1 1 1( ) ( )s s s s

s s s s

m m m m

s s

m m m m

m m

(18)

Next the following two-vertex combination under the as-sumption of limited sprung mass changes

min max[ ]s s sm m m

can be defined 2

1 1 1 11

( ( ) ( ) ( ) ( )) ( ( ))( )i i i i ii

A t B t C t D t t A B C D

(19)

where Ai Bi C1i and D1i are the matrices that are obtained by replacing ms with msmin and msmax for i = 1 and 2 respectively In addition it is noted that the desired damping force should be subject to asymmetric saturation due to the manufacturing limitations of practical actuators Therefore the passivi-ty-constraint of the actuator in the form of asymmetric satura-tion is defined as follows

max max

min max

min min

( )

sar

sar sar sar

sar

F if F F

sat F F if F F F

F if F F

(20)

Besides firm uninterrupted contact of tires with the road is essential for vehicle handling ability A constraint for dynamic tire load is considered as

max( ) ( )t u r s uK z z m m g (21)




5

Since using more suspension stroke is unavoidable to reduce the vertical acceleration of the sprung mass the likelihood of hitting the suspension travel limits may increase [8][32] Thus the constraint of the suspension traveling within the maximum deflection range (denoted as zmax) is defined as

max| |s uz z z (22)

The time-domain constraints (20)-(22) which represent per-formance requirements for vehicle suspensions should be re-spected to the design of the optimized output feedback solution Based on the formulated closed-loop convex polytopic model in (19) the following LMI-based constrained Hinfin OFC can thus be applied to design the output feedback control law for a given number gt 0 the minimum Hinfin norm can be feasible using the following semi-definite programming

0min

TrP P K

subject to

2 0 0

T Ti i i iP P PB

I

I

(23)

where i = 1 and 2 i i i rA BK C and

1 1i i i rC D KC In the

above P is a real matrix with appropriate dimension The output feedback control Kr is optimum when is found to be minimum

The OFC design problem with respect to LMI solving methods is still a challenging issue both analytically and nu-merically due to its non-convex nature [33] To avoid an in-feasible analytical solution to the LMI optimization problem a hybrid procedure which combines the LMIs with the GA is proposed to solve the minimization problem in (23) with the accuracy and efficiency while respecting the realistic require-ments in vehicle suspensions including time-domain con-straints in (20)-(22) saturation nonlinearity and polytopic uncertainty Based on the LMIs and the GA the hybrid ap-proach begins to work with a population of candidate solutions and hence the optimized output feedback controller with the minimum Hinfin norm can be achieved via the genetic operators selection crossover and mutation [34] Due to the properties of the proposed approach the generic operators are only applied on each candidate gain matrices Kr so that the computation burden can be reduced Moreover additional performance constraints in (20)-(22) can also be incorporated into the opti-mization procedures The hybrid algorithm to the optimization loop is sketched as follows

Initially a larger value of is assigned into the LMI solver to generate the initial population as a formation of population pool Accordingly the GA Toolbox in MATLAB is then applied to implement the GA algorithm This study chooses the floating representation for the initial population In each generation the GA searches for a better solution for the offspring which needs to fulfill the LMI constraints in (23) and satisfy the following stability requirement of the closed-loop vertex subsystems and the closed-loop nominal system that is

max

max

( ) 0

( ) 0 1 2N N r r

i i r r

A B K C

A B K C i

(24)

where 2 2

1 1

( ) ( )N i s i N i s ii i

A m A B m B

represent the nominal system matrix and the nominal input vector respectively λmax() denotes the maximum eigenvalue of the matrix Then the minimization problem of (23) is carried out by using the LMI solver in MATLAB LMI Toolbox If the LMI solver cannot yield a feasible solution or the resulted close-loop vertex subsystems and the nominal system are not all strictly Hurwitz (the stability conditions in (24) cannot be guaranteed) then the generic operations including crossover and mutation will be activated so as to move toward a better solution The new yielded value of is indicated as the cost function for evaluating each individual Kr in the GA process going from the current population to the next generation In each generation better solutions are reproduced to give off-spring that replace the relatively bad or infeasible solutions Based on the evolutionary theory only the most suited indi-viduals in the population are likely to exist and generate off-spring that can achieve the smaller value of Note that the hybrid algorithm combining the LMI solver and the GA oper-ation continues until the minimum of cost function is found Consequently robust controller gain of OFC Kr can be obtained and the optimized control objectives of the semi-suspension system can be achieved The other potential advantage is that nearly all of the tasks of the OFC in the virtual reference model can be accomplished automatically

III CONTROL SYSTEM DESIGN

In this section the semi-active suspension controller design is presented based on the idea of making the plant track the motion of the optimized reference model and forcing the sys-tem to follow the virtual reference model and minimize the error dynamics between the plant output and the reference model in an asymptotically stable condition To confront the high nonlinearities and uncertainties raised from the practical applications of semi-active suspension this study applies the SMC design to track the desire reference states with high pre-cision The designed control system also possesses the salient advantage of stable tracking control performance and offers a straightforward and adaptive tuning of its control gains for further assuring robust control performance

Considering the semi-active system (9) with parameter un-certainties and external disturbances the state equations can be rewritten as

0( ) ( ) sa dx A A x B B F G f (25)

where ∆A and ∆B denote the uncertainties introduced by system parameters of the matrix A and B respectively fd0 represents the unknown nonlinearity of the system or external disturb-ances To ensure the closed-loop system achieves stable and robust performance in a wide range of shock and vibration environments the following assumptions are made

Assumption 1 There exist unknown matrix functions of ap-propriate dimensions Bda(t) Bdb(t) and Bdf(t) such that

0( ) ( ) ( ) ( ) ( ) ( )da db d dfA t BB t B t BB t f t BB t (26)




6

Optimized virtual reference model

Semi-activesuspension system

Virtual control calculation

Adaptive state tracking

calculation

Control calculation

c hF F

aF

Road disturbance

rx

xΣ

Fig 3 Block diagram of the proposed control scheme

Under this assumption system (21) can be arranged as follows

0( )

sa sa d

sa lm

x Ax BF G Ax BF f

Ax BF G BF

(27)

where Flm represents the lumped matched uncertainties and external disturbance of the suspension system and defined as

lm da db sa dfF B Ax B F B (28)

Assumption 2 There exists a bounded positive constant Q

which is large enough to suppress all the uncertainties Flm such that || Flm || le Q Now define the tracking error as

re x x (29)

and its derivative is

sa sar lme Ae BF BF BF (30)

Define the sliding surface as follows ( )rS e x x (31)

where λ is a constant sliding vector The derivative of the sliding surface can be expressed as

sa sar lmS e Ae BF BF BF (32)

To guarantee the sliding condition and asymptotic stability the following control should be applied

S KS (33) where the constant parameter K gt 0 By virtue of (32) and (33) the control law can be yielded as follows

sa c h aF F F F

1 1 1( ) ( ) ( )|| ||

T T

sarT T

B SK B S B Ae Q B BF

B S

(34)

where Fc represents the control input for tracking the reference state and the hitting control ( ) || ||T T T T

hF Q B S B S repre-

sents the nonlinear feedback control for eliminating the un-predictable perturbations the last term Fa represents the desired control effort calculated from OVMR control strategy The control Fsa in (34) is proposed to be the sum of three control efforts as illustrated in the control system block dia-gram in Fig 3 This control system can assure the asymptotic stability of the error regulation objective however the overall stability of the closed-loop system cannot be well guaranteed during the entire control process when facing unknown model uncertainties and external disturbances To overcome this drawback an adaptive tuning mechanism is utilized in this

study to estimate the optimal value of the hitting control gains to achieve the minimum control effort and stable reference model tracking performance In the standard adaptive control scheme the hitting control gain can be structured as a state-dependent function Q = 1 +2 ||x|| in which 1 and 2 are non-negative and adjustable parameters There exists the op-timal parameters 1

and 2 to achieve the sliding condition

Owning to the unknown lumped uncertainties the optimal gains 1

and 2 cannot be obtained exactly in advance for

practical applications To estimate the optimal parameter val-ues define the estimated errors as

1 1 1 2 2 2

ˆ ˆ (35)

where 1 and

2 are the estimated values of the hitting control

gains Now the control law can be represented via (34) as

ˆ|| ||

T T

sa eq T T

B SF F Q

B S

(36)

with 1 2

ˆ ˆ ˆ || ||Q x Choose a Lyapunov candidate as

2 21 2 1 1 2 2

1 1 1( )

2 2 2TV S S S (37)

where σ1 and σ2 are positive constants Differentiating

1 2( )V S with respect to time can yield

1 2 1 1 1 2 2 2

1ˆ ˆ( )

2TV S S S (38)

After substituting (32) and (34)-(36) into (38) the following results can be obtained

1 2( )V S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT

lm T T

B SS KS BF BQ

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT T T

lm T T

B SKS S S BF QS B

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||T T T TlmKS S S BF Q B S

1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || || || ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )

T T T T TlmKS S B S F x B S

1 2 1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || ( || ||) ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )

T T T T TKS S B S x x B S

1 1 1 2 2 2ˆ ˆ( || ||) ( || || || ||)T T T T TKS S B S B S x (39)

Now if the adaption laws for the hitting control gains are cho-sen as

11

|| ||ˆ =

T TB S

22

|| || || ||ˆ

T TB S x

(40)

then (35) becomes

1 2( ) 0TV S KS S (41)

From the above result 1 2( )V S is negative semi-definite

that is 1 2 1 2( ( ) ( ) ( )) ( (0) (0) (0))V S t t t V S and this im-

plies that 1( ) ( )S t t and

2 ( )t are all bounded Set a function

as W(t) = KSTS le 1 2( )V S and integrate this function




7

with respect to time

1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))

tW d V S V S t t t (42)

Since 1 2( (0) (0) (0))V S is bounded and

1 2( ( ) ( ) ( ))V S t t t is

non-increasing and bounded the following result is yielded

0lim ( )

t

tW d

(43)

and ( )W t is also bounded Thus on the basis of Barbalatrsquos

lemma [35] it can be concluded that lim ( ) 0t W t that is S(t)

rarr 0 as t rarr infin Consequently the designed control system is stable even when unknown uncertainties occur In addition the tracking error converges to zero owing to S(t) rarr 0 Remark 1 To remedy the undesired chattering phenomenon induced by the discontinuous control component in (36) a ldquosmoothedrdquo version of the sliding mode component can be used as

ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)

where ε gt 0 is a small constant This continuous approximation method produces a small boundary which is larger around the sliding surface so that the error state trajectory can be sustained in the close neighborhood of the switching manifold by a high-gain control [36] As stated in Appendix B the replace-ment of using (44) can also preserve the closed-loop system stability in the sense of Lyapunov theorem

Remark 2 Generally the sliding surface S will not be equal to zero all the time when the continuous approximation in the SMC is used Unfortunately this fact will deteriorate the tracking accuracy of the control system because the adaptive gains still increase boundlessly even in the steady-state condi-tion To this end the adaptation laws in (40) are modified as

1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)

where c1 and c2 are positive constants This method can elim-inate the phenomena of integral windup during the adaptation of the upper bound while suppressing unknown disturbances

Remark 3 In each semi-active suspension control system an acceleration-based observer is developed to provide a state estimation in the tracking of the optimized reference model For the case where the acceleration measurements may be cor-rupted with high-frequency noise an observer based on the dynamical system (9) can be constructed as

ˆ ˆ ˆ( )sa obx Ax BF L y y

ˆ ˆ say Cx DF (46)

where x is the estimated state vector y is the estimated output

and Lob is the observer parameter For the minimization of the covariance of the error between the actual state x and the ob-served state the Kalman filter algorithm [9][30][37] is applied to obtain the proper observer gain matrix Lob

Hence the following procedures summarize the designed controller for system (9) such that the closed-loop system achieves robust stability with the bounded disturbance attenu-

ation level while respecting the constraints as follows Step 1) Choose P and solve the LMI optimization problem (23) with the GA operation to obtain an optimal solution and the corresponding OFC Kr Step 2) Obtain the desired damping force Fsar from the virtual reference model in (13) Step 3) Select λ for the defined sliding surface in (31) and choose K and then calculate the equivalent control law Feq via (34) Step 4) Assign the initial hitting control gain Q and select the parameters σi and ci i = 1 2 to build the adaptive tuner Step 5) Tuning Q via (45) and obtain the control law for the closed-loop system via (36)

IV CO-SIMULATION SETUP AND ANALYSES

In this section the co-simulation using MATLABSimulink and CarSim software is conducted to evaluate the dynamic behavior of the designed semi-suspension control system in the full-car maneuver We synchronize CarSim and Simulink with the entire vehicle system via establishing a control interface In virtual test environments the developed control system and the vehicle dynamic system involves transferring and citing all the relevant parameters resulting in an organic link to achieve real-time interaction between two systems for ensuring confi-dence simulation

TABLE I

NUMERICAL PARAMETERS OF THE QUARTER-CAR SUSPENSION MODEL Parameter Front-wheel Rear-wheel Max ms (kg) 446 425 Min ms (kg) 343 320 mu (kg) 58 52 Ks (Nm) 83271 82755 Kt (Nm) 242958 242958

A Co-simulation Setup

A D-class sedan is chosen as the testbed vehicle to validate the performance of the proposed system In CarSim the dy-namic model of a vehicle encompasses the complete vehicle system and inputs from the driver the ground and aerody-namics [38] This heavy complexity results in the need for developing simple analytical models of vehicle suspension system In the proposed approach the modeling experiment must be made initially and the obtained model is then verified and fine-tuned experimentally Because an in-depth study was conducted based on the state-space model (9) a priori knowledge of vehicle suspension systems was used to build an appropriate model with a multi-output structure The experi-mental inputoutput data of a suspension system can be ac-cessed in the CarSim test environment The testbed vehicle is excited by sweeping the frequency base excitations The test road input exhibits its maximum power at frequencies ranging from 01 to 20 Hz Accordingly the state-space model is treated as a gray box and the corresponding parameters can be identi-fied by the MATLAB toolkit using the algorithm from (5)-(8) The obtained parameters for the front- and rear-wheel quar-ter-vehicle suspension of the tested vehicle model are listed in Table I These parameters of the model are modified to




8

Fig 4 Co-simulation with Simulink through s-function

precisely approximate the behavior of the tested vehicle In addition the first-order system is used to represent the actuator dynamics of the semi-active damper which behaves similarly to low-pass filters with relatively high bandwidths [10 11][17][30]

The co-simulation scheme as shown in Fig 4 opens the multi-body vehicle model in CarSim and also in the co-simulation system VehicleSim (VS) Solver The dynamic behavior of the multi-body vehicle model is calculated by VS Solver at various steps of the co-simulation This model can more accurately reflect the fine structures of the vehicle The pre-defined output state variables are solved and values of the state variables are provided from the CarSim Outputs to the Simulink model at each iteration of the co-simulation The control system in Simulink drives the input damping force of the semi-active suspension for the required performance levels of ride comfort and vehicle handling The co-simulation results of VS Solver can be obtained in the CarSim Postprocessor The CarSim Postprocessor can also provide charts and virtu-al-reality (VR) animations of the multi-body vehicle model behavior in regard to the damping control of the suspension system

According to the proposed approach the main parameters for GA optimization are chosen as follows a chromosome length of 16 bits a population size of 50 bits a crossover frac-tion of 08 a number of generations of 100 and a mutation fraction of 001 The output feedback gain Kr via the OVMR control strategy can be computed as the minimum Hinfin norm = 672 The other controller parameters are set as λ = [-200 50 150 15] K = 25 σ1 = σ2 = 20 and c1 = c2 = 10

B Experimental Results and Analyses

Two types of test-road disturbance were conducted in this study The time-domain responses of tire deflection suspension stroke and sprung mass acceleration are evaluated For the sake of simplicity instead of showing all performances of the four suspension controllers of the entire vehicle only the responses of the front-left wheel are plotted The robustness of the pro-posed approach against the changed sprung mass and deterio-

rated suspension is also investigated Moreover as a compar-ison the passive semi-active and the active suspension are employed along with the proposed controller The semi-active suspension controller adopted in this study mainly applies the skyhook-based control strategy [9] The active suspension controller is designed based on linear optimal control theory for the sake of better handling and riding comfort [5] and the cost function is presented as a weighted sum of mean-square output performance values of body acceleration suspension travel and tire deflection

Fig 5 The sinusoidal-shape speed bump for the 1st test-road The left-top figure shows the 3D road shape

The first type of test-road is a sinusoidal-shape with the

height of 30 cm and the length of 12 m as shown in Fig 5 The performance of the proposed controller is validated under the vehicle speed at 50 kmh The comparison results obtained for the time responses of tire deflection suspension stroke and sprung mass acceleration are shown in Fig 6 As shown in Fig 6(a) the proposed approach can suppress about 17 of tire deflection while traveling over Test Road 1 In Fig 6(b) the proposed controller can regulate the suspension stroke back to the equilibrium point faster than the others can Fig 6(c) compares the individual sprung mass acceleration and it can be seen that our approach shows better performance indicating the ride comfort is improved

Investigating the robustness of suspension systems is cri-




9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive

Our ApproachSemi-active (CarSim)

Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)

(c) Fig 6 Time responses for going over the sinusoidal-shape speed bump (a) Tire deflection (b) Suspension stroke (c) Sprung mass acceleration

2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)

(c) Fig 7 Time responses for going over the sinusoidal-shape speed bump subject to robustness study (a) Tire deflection (b) Suspension stroke (c) Sprung mass acceleration

tical because some vehicle parameters such as body weight and the spring and damping coefficients of suspensions may change depending on road and load conditions To validate the robustness of the proposed controller for frequency responses the weight of sprung mass was increased by 30 and the spring constant was reduced by 20 The time responses of tire de-flection suspension stroke and sprung mass acceleration for robustness study are shown in Fig 7 From this comparison we can see that the proposed control system can achieve better sprung mass acceleration with less displacement for both tire deflection and suspension stroke From Figs 6 and 7 it can be observed that the proposed approach can offer better ride comfort and vehicle handling and preserve the small values of suspension working space This means that the proposed sys-tem will not damage the suspension system structure

TABLE II

PERCENTAGE VARIATION OF THE RMS VALUE IN THE SPRUNG MASS

ACCELERATION AND THE TIRE DEFLECTION (TEST-ROAD 1) Side Passive Semi-active Active Our approachFL body 100 856 785 679 FL wheel 100 902 881 813 RL body 100 838 772 683 RL wheel 100 894 885 815 FR body 100 861 781 682 FR wheel 100 910 892 806 RR body 100 857 769 677 RR wheel 100 904 885 812

To clearly compare control performance between our ap-

proach and other controllers the percentage variation of the RMS values for each quarter car in this test-road is shown in

Table II The results show the worst performance for the pas-sive suspension Noticeably the variation of the RMS values is the minimum for sprung mass vibrations in our approach Be-sides the tire vibrations can be more suppressed so that our system realizes the better handling performance

Fig 8 The sinusoidal-sweep road disturbance for the 2nd test-road The right-top figure shows the 3D road shape

In the second testing the performance of the proposed

controller was validated under the sine sweep testing which sweeps from 05 Hz to 15 Hz with the amplitude 005 m as shown in Fig 8 At a traveling speed of 37 kmh time-domain responses of tire deflection suspension stroke and sprung mass acceleration are shown in Fig 9 In Fig 9(a) the tire deflection in the case of our proposed approach can be more suppressed This implies that the road handling capability of our system




10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)

(c) Fig 9 Time responses for going over the sinusoidal-sweep road (a) Tire deflection (b) Suspension stroke (c) Sprung mass acceleration

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)

(c) Fig 10 Time responses for going over the sinusoidal-sweep road subject to robustness study (a) Tire deflection (b) Suspension stroke (c) Sprung mass acceleration

outperforms that of other controllers In Fig 9(b) our system can perform the less value of suspension stroke after 8 seconds and return to equilibrium faster than the others can As in Fig 9(c) the amplitudes of sprung mass acceleration for the pro-posed controller show a significant reduction which results in the evident improvement in the riding comfort performance Again the time responses of tire deflection suspension stroke and sprung mass acceleration when evaluating robustness un-der the same system parameter variations are shown in Fig 10 Obviously it can be seen that our approach achieves better control performance in suppressing more sprung mass accel-eration and tire deflection and in reducing the suspension stroke under higher sweeping frequencies of road disturbance input

TABLE III



To clearly compare control performance Table III lists the

percentage variation of the RMS values for each quarter car under the second type of test-road The proposed approach can achieve better performance than the other methods performed

with significantly suppressed sprung mass acceleration and tire deflection Consequently the proposed controller can effec-tively improve ride quality without deteriorating the suspension stroke and tire deflection as well

From the experimental studies of the aforementioned two testing conditions the proposed control system clearly shows superior performance over the active suspension the conven-tional semi-active suspension as well as the passive suspension control in suppressing the sprung mass acceleration and re-ducing the tire deflection with smaller degenerations of sus-pension working space Moreover it can be observed that alt-hough there is a large amount of variation in vehicle parameters less effective changes in the sprung mass acceleration and tire deflection occur for the proposed approach Thus the robust-ness of our control system is not affected as expected because it possesses the online adaptive tuning capability for tracking the optimized reference model with guaranteed system operation stability Based on co-simulation technology the benchmark of the suspension control strategies is used to evaluate both comfort and vehicle handling performance and provide an evident comparison of the proposed approach with other methods Finally the proposed systematic strategy of the sus-pension control can then be easily extended to a wide range of other application domains such as cabins in trucks or tractors trains seats instruments or architectural structures

V CONCLUSION AND FUTURE WORK

This paper has presented an OVMR control strategy for ve-hicle semi-active suspension systems Based on the con-




11

trol-oriented model of the Macpherson suspension system the control gains in virtual reference model are optimized by means of combining Hinfin OFC theory with GA while respecting the suspension constraints as required in real situation Subse-quently the adaptive sliding mode controller is developed to steer the state variables toward the virtual desired signals such that the Hinfin performance of ride comfort and vehicle handling performance can be achieved Furthermore a simple and easily realized adaptive algorithm with exponentially stability is de-veloped The closed-loop system can achieve best possible ride comfort and handing performance against parameter variations and uncertainty Through co-simulations the effectiveness of the controller is evaluated and is compared to that of passive conventional semi-active and LQR-based active systems The performance of our approach is validated through SITL simu-lations

APPENDIX A

OVERVIEW OF THE DETAILED NONLINEAR SYSTEM MODEL

The overall dynamics of Macpherson suspension can refer to [28-30] and the detailed functions f1() and f2() in (3) are given as follows

1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s

s t c s s r s u b sa

f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2

2 0( ) sin ( )s u c u cg m m l m l

3( )cos( )

ll

l l

dg b

c d

0 0( ) [sin( ) sin ]s s r s c rz z z z l z

APPENDIX B

STABILITY ISSUE OF THE CLOSED-LOOP SYSTEM USING THE CONTINUOUS APPROXIMATION

Without loss of generality we are assuming that the optimal hitting control gains

1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a

bounded positive constant Q The Lyapunov candidate in (37) can thus be rewritten as V(S)=STS2 After using (44) in the control law the following results can be concluded

( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)

Since || ||lmF Q for all t notice that

|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)

Since K gt 0 and 0Q are bounded constants one can obtain

( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)

which from (31) leads to

22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)

Consider the designed control law which can achieve ( ) 2V S Q K we can find ( ) 0V S with the bounded V(S)

that ensures ||e|| is bounded Accordingly this guarantees that the closed-loop system is stable and V rarr 0 as t rarr 0 then e rarr 0

ACKNOWLEDGMENT

The authors would like to thank reviewers and the editor for their helpful and detailed comments which have assisted in improving the presentation of the paper

REFERENCES [1] D Hrovat ldquoSurvey of advanced suspension developments and related

optimal control applicationsrdquo Automatica vol 33 no 10 pp 1781-1817 Oct 1997

[2] D Cao X Song and M Ahmadian ldquoEditors perspectives road vehicle suspension design dynamics and controlrdquo Vehicle System Dynamics vol 49 no 1-2 pp 3-28 2011

[3] S M Savaresi and C Spelta ldquoA single-sensor control strategy for semi-active suspensionsrdquo IEEE Trans Control Systems Technology vol 17 no 1 pp 143-152 2009

[4] J Wang W Wang and K Atallah ldquoA linear permanent-magnet motor active vehicle suspensionrdquoIEEE Trans Vehicular Technology vol 60 no 1 pp 55-63 2011

[5] J Cao H Liu P Li and D J Brown ldquoState of the art in vehicle active suspension adaptive control systems based on intelligent methodologiesrdquo IEEE Trans Intelligent Transportation Systems vol 9 no 3 pp 392-405 2008

[6] J Lin K W E Cheng Z Zhang N C Cheung X Xue and T W Ng ldquoActive suspension system based on linear switched reluctance actuator and control schemesrdquo IEEE Trans Vehicular Technology vol 62 no 2 pp 562-572 2013

[7] W Sun H Gao and O Kaynak ldquoFinite frequency Hinfin control for vehicle active suspension systemsrdquo IEEE Trans Control Systems Technology vol 19 no 2 pp 416-422 2011

[8] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspension control with frequency band constraints and actuator input delayrdquo IEEE Trans Industrial Electronics vol 59 no 1 pp 530-537 2012

[9] V Sankaranarayanan M E Emekli B A Guvenc L Guvenc E S Ozturk S S Ersolmaz I E Eyol and M Sinal ldquoSemiactive suspension control of a light commercial vehiclerdquo IEEEASME Trans Mechtronics pp 598-604 vol 13 no 5 2008

[10] C P Vassal C Spelta O Sename S M Savaresi and L Dugard ldquoSurvey and performance evaluation on some automotive semi-active suspension control methods A comparative study on a single-corner modelrdquo Annual Reviews in Control vol 36 no 1 pp 148-160 2012

[11] M M Ma and H Chen ldquoDisturbance attenuation control of active suspension with non-linear actuator dynamicsrdquo IET Control Theory and Applications vol 5 no 1 pp 112-122 2011

[12] R Vepa ldquoNonlinear unscented H suspension and tracking control of mobile vehiclesrdquo IEEE Trans Vehicular Technology vol 61 no 4 pp




12

1543-1553 2012 [13] Y Sam J Osman and M Ghani A class of proportionalndashintegral

sliding mode control with application to active suspension system Systems Control Letters vol 51 no 34 pp217-223 2004

[14] N Yagiz and Y Hacioglu ldquoBackstepping control of a vehicle with active suspensionsrdquo Control Engineering Practice vol 16 no 12 pp 1457-1467 2008

[15] H Kim and H Lee ldquoHeight and leveling control of automotive air suspension system using sliding mode approachrdquo IEEE Trans Vehicular Technology vol 60 no 5 pp 2027-2041 2011

[16] S Huang and H Chen ldquoAdaptive sliding controller with self-tuning fuzzy compensation for vehicle suspension controlrdquo Mechatronics vol16 no 10 pp607-622 2006

[17] M Zapateiro F Pozo H R Karimi N Luo ldquoSemiactive control methodologies for suspension control with magnetorheological dampersrdquo IEEEASME Trans Mechatronics vol 17 no 2 pp 370-380 2012

[18] H H Chiang S J Wu J W Perng B F Wu and T T Lee ldquoThe human-in-the-loop design approach to the longitudinal automation system for an intelligent vehiclerdquo IEEE Trans Syst Man Cybern A Syst Humans vol 40 no 4 pp 708ndash720 Jul 2010

[19] J G Juang L H Chien and F Lin ldquoAutomatic landing control system design using adaptive neural network and its hardware realizationrdquo IEEE Systems Journal vol 5 no 2 pp 266-277 June 2011

[20] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-mode control of active suspensionsrdquo IEEE Trans Industrial Electronics vol 55 no 11 pp 3883-3890 2008

[21] J Lin R J Lian C N Huang W T Sie ldquoEnhanced fuzzy sliding mode controller for active suspension systemsrdquo Mechatronics vol19 pp 1178-1190 2009

[22] N A Holou T Lahdhiri D S Joo J Weaver and F A Abbas ldquoSliding mode neural network inference fuzzy logic control for active suspension systemsrdquo IEEE Trans Fuzzy Systems vol 10 no 2 pp 234-246 2002

[23] D L Guo H Y Hu and J Q Yi ldquoNeural network control for a semiactive vehicle suspension with a magnetorheological damperrdquo J Vib Control vol 10 no 3 pp 461ndash471 2004

[24] Y P Kuo and T H S Li ldquoGA-based fuzzy PIPD controller for automotive active suspension systemrdquo IEEE Transactions on Industrial Electronics vol 46 no 6 pp 1051-1056 Dec 1999

[25] L Sun X Cai and J Yang ldquoGenetic algorithm-based optimum vehicle suspension design using minimum pavement load as designrdquo Journal of Sound and Vibration vol 301 no 20 pp 18-27 March 2007

[26] J Cao P Li and H Liu ldquoAn interval fuzzy controller for vehicle active suspension systemsrdquo IEEE Trans Intelligent Transportation systems vol 11 no 4 pp 885ndash895 Dec 2010

[27] J Lin and R J Lian ldquoIntelligent control of active suspensions systemsrdquo IEEE Trans Industrial Electronics vol 58 no 2 pp 618-628 2011

[28] J Hurel A Mandow and A Garcacuteıa-Cerezo ldquoNonlinear two-dimensional modeling of a McPherson suspension for kinematics and dynamics simulationrdquo in Proc 12th IEEE Workshop on Advanced Motion Control 2012 March Sarajevo Bosnia and Herzegovina

[29] K S Hong H C Sohn and J K Hedrick ldquoModified skyhook control of semi-active suspensions A new model gain scheduling and hardware-in-the-loop tuningrdquo Journal of Dynamic Systems Measurement and Control vol 124 pp 158-167 2002

[30] B C Chen Y H Shiu and F C Hsieh ldquoSliding-mode control for semi-active suspension with actuator dynamicsrdquo Vehicle Dynamic Systems vol 49 no 1-2 pp 277-290 2011

[31] M S Fallah R Bhat and W F Xie ldquoNew model and simulation of Macpherson suspension system for ride control applicationsrdquo Journal of Vehicle Syst Dyn vol 47 no 2 pp 195ndash220 2009

[32] H Du W Li and N Zhang ldquoIntegrated seat and suspension control for a quarter car with driver modelrdquo IEEE Trans Vehicular Technology vol 61 no 9 pp 3893-3908 2012

[33] C Lee and S M Salapaka ldquoFast robust nanopositioningminusA linear-matrix-inequalities-based optimal control approachrdquo IEEEASME Trans Mechatronics vol 14 no 4 pp 414ndash422 Aug 2009

[34] R L Haupt and S E Haupt Practical Genetic Algorithms 2nd ed Hoboken NJ Wiley 2004

[35] J J E Slotine and W P Li Applied Nonlinear Control Englewood Cliffs NJ Prentice-Hall 1991

[36] X Yu and O Kaynak ldquoSliding-mode control with soft computing A surveyrdquo IEEE Trans Industrial Electronics vol 56 no 9 pp 3275-3285 2009

[37] W Y Hsiao and H H Chiang ldquoSliding-mode-control with filtered

state-feedback for disturbance attenuation of active suspension systemsrdquo International Journal of Engineering and Industries vol 3 no 1 pp 67-78 2012

[38] CarSim CarSim Reference Manual Ver 803 Ann Arbor MI Mechanical Simulation Corporation 2010

Hsin-Han Chiang (Srsquo02ndashMrsquo04) received the BS and PhD degree in electrical and control engineering from National Chiao-Tung University Hsinchu Taiwan in 2001 and 2007 respectively He was a Postdoctoral Researcher in electrical engineering at National Taipei University of Technology in 2008 and 2009 Since 2009 he has been an Assistant Professor with the Departement of Electrical Engineering Fu Jen Catholic University New Taipei City Taiwan His research interests include the areas of intelligent systems and control

vehicle and robot control and automation systems

Lian-Wang Lee was born in Taipei Taiwan ROC in 1974 He received MS degree in automation and control form National Taiwan University of science and Technology Taipei Taiwan in 2000 And he received PhD degree at National Taiwan University of Science and Technology Taipei Taiwan in 2009

Currently he is an Assistant Professor with the Department of Mechanical Engineering in Lunghwa University of Science and Technology Taoyuan His research interests include fluid power control mechatronics nonlinear control

intelligent control adaptive control and sliding mode control




2

acteristics exhibited in vehicle suspension systems integrated computational intelligence and adaptive approaches including fuzzy inference systems [20 21] neural networks [22 23] and genetic algorithms [24 25] in vehicle suspension control sys-tems have been thoroughly proposed Although these intelli-gent control methodologies have been successfully applied to various practical control systems the design challenge still remains when it comes to determining suitable control structure and appropriate selection of parameters especially when such methodologies are adopted to manipulate complicated systems with nonlinearities and uncertainties The other unresolved problem brings about the bottleneck in the mathematical proof for stability which is still difficult to demonstrate at present

In many studies on vehicle semi-active suspension devel-opment the results have been obtained from active suspensions by using a simple projection to handle the dissipative constraint In other words the damping force applied by the semi-active suspension is chosen by the controller to be as close as possible to the active force This results in the so-called ldquosynthesize and try methodrdquo potentially leading to unpredictable behaviors which make it impossible to ensure either closed-loop internal stability or performance Moreover most existing methods of semi-active suspension control lack the systematic strategy to involve an optimization procedure while guaranteeing the dis-sipative constraint owing to parameter-dependent structures and multi-objective constraints handling To this end some studies (eg [26 27]) have also proposed hybrid intelligent control systems in order to pursue the effective balance be-tween ride comfort and vehicle handling quality However the problem that arises is satisfactory real-time computing effi-ciency of the control system to fulfill the adaptive mul-ti-objective optimization required in different road conditions

To address the difficulties of the aforementioned control schemes this paper presents a systematic and practical meth-odology for designing optimized vehicle semi-active suspen-sion systems A robust SMC system with an adaptive scheme is also proposed in our approach to identify the unknown upper bound of system uncertainties and external disturbances so that higher robustness and adaptability can be achieved Unlike most past studies of suspension control systems which only demonstrate the performance under a quarter-vehicle or half-vehicle model we focus on developing a semi-active suspension system that allows for a systematic design with multi-objective optimization considerations for a full-vehicle Moreover this study addresses the associated practical diffi-culties of using semi-active suspension controllers with avail-able sensors Up to now the co-simulation technique has be-come indispensable support in the design process of automotive industry Thus through the co-simulation analysis in support of suspension design the proposed approach can benefit from the evaluation analysis of a unit vehicle performance based on the high-level perspective of the physical environment

The rest of this paper is organized as follows Section II presents the model of the Macpherson semi-active suspension system and addresses the development of the optimized virtual model reference (OVMR) control synthesis In Section III the adaptive robust tracking control design is introduced to achieve

the state tracking objective in terms of comfort and vehicle handling Section IV shows the feasibility and effectiveness of our proposed control scheme by using co-simulation execution in a software-in-the-loop (SITL) environment Conclusions and recommendations for future work are given in Section V

zr

zs

zu Ks

Fsa

Z

Y0+

O

B

A

C

Fig 1 The schematic model for the Macpherson suspension system

II DYNAMIC MODELING

A System Description

Various vehicle suspension models have been proposed to study semi-active suspensions (eg [28]) In this study a comprehensive and realistic model of the Macpherson suspen-sion is presented for semi-active control systems A schematic diagram of control-oriented Macpherson suspension is shown in Fig 1 in which the rotational motion of the unsprung mass is admitted Such a suspension system is composed of the fol-lowing major elements the chassis the control arm the strut and the wheel assembly In this section a two-dimensional model with the two generalized coordinates zs and θ in Fig 1 is firstly investigated for the control-oriented problem Ac-cording to the equations of motion in [29 30] the nonlinear model with Macpherson suspension for the two generalized coordinates is given as

20 0( ) cos( ) sin( )s u s u c u cm m z m l m l

0 0 [ sin( ) sin ] 0t s c rK z l z (1-1) 2

0 0

0 0

cos( ) cos( )

1[sin( ) sin ] sin( )[

2

u c u c s t c s

c r s l

m l m l z K l z

l z K b

]cos( )

lb sa

l l

dl F

c d

(1-2)

with 2 2

l a ba l l 2l a bb l l 2 cos( )l l l lc a a b 2 cos( )l l l ld a b b and

0

where ms and mu are the sprung and unsprung masses respec-tively and Ks and Kt are the spring constants for the suspension and tire respectively zs represents the displacement of the sprung mass is the angular displacement of the control arm zr

is terrain height disturbance la lb and lc are the distances from point O to points A B and C respectively α is the angle be-




3


( - ) 0( )( - )

( - ) 00 s s usa s u

sas s u

if z z zC t z zF

if z z z

(2)






ms

mu

Ku zr

zs

zu

Ks fsa

Accelerometer


Accelerometer







1( ) Ts s s s um z K z z (5-1)







2 2 ˆTe and



2 21 2

1

[ ( ) ( )]N

k

E e k e k

(6)


1 1 2 2 21 1

ˆ[ ( ) ( ) ( ) ( )] ( ) ( ) 0ˆ

N NT T

tirek k

Ek k k k k F k

(7)


1 1 2 2 21 1

ˆ [ ( ) ( ) ( ) ( )] ( ) ( )N N

T Ttire

k k

k k k k k F k

(8)



s uy z z


sax Ax BF G

say Cx DF v (9)

where

-

-

0 0 1 -1

0 0 0 1

0 0 0

0 0

s

s

s u

u u

K

m

K K

m m

A

-1

1

0

0

s

u

m

m

B

0

1

0

0

G

-

-

0 0 0

0 0

s

s

s u

u u

K

m

K K

m m

C

-1

1

s

u

m

m

D







4


( - ) 0

( - ) 00 s s usky s

skys s u

if z z zC zF

if z z z

(10)


( - ) 0

( - ) 00 u s ugrd u

grdu s u

if z z zC zF

if z z z

(11)


(1 )

(1 )sa sky grd

sky s grd u

F F F

C z C z

(12)



r r sarx Ax BF G

r r ry C x






0 0 1 0

0 0 0 1rC

- -1

1 1 1

2

0 0 0

0 0 ( ) 0 0

0 0 0 ( ) 0

ss s

Km m

C s D

s

(14)



1 2 21 1

( )2s s s

n n

sz s z z

s

(15)

2 2 22 2

( )2u u u

n n

sz s z z

s

(16)





1 2

min max

( ( )) ( ( ))1

s s

t t

m m

(17)


1

( ( )) 1ii

t

The function


min max

min max min max

1 1 1 1

1 21 1 1 1( ) ( )s s s s

s s s s

m m m m

s s

m m m m

m m

(18)



can be defined 2

1 1 1 11

( ( ) ( ) ( ) ( )) ( ( ))( )i i i i ii


(19)


max max

min max

min min

( )

sar

sar sar sar

sar

F if F F

sat F F if F F F

F if F F

(20)






5


max| |s uz z z (22)


0min

TrP P K

subject to

2 0 0

T Ti i i iP P PB

I

I

(23)






max

max

( ) 0

( ) 0 1 2N N r r

i i r r

A B K C

A B K C i

(24)

where 2 2

1 1


A m A B m B





0( ) ( ) sa dx A A x B B F G f (25)







6





calculation

Control calculation

c hF F

aF

Road disturbance

rx

xΣ



0( )

sa sa d

sa lm

x Ax BF G Ax BF f

Ax BF G BF

(27)





re x x (29)








sa c h aF F F F

1 1 1( ) ( ) ( )|| ||

T T

sarT T


B S

(34)


hF Q B S B S repre-







1 1 1 2 2 2

ˆ ˆ (35)

where 1 and



ˆ|| ||

T T

sa eq T T

B SF F Q

B S

(36)

with 1 2


2 21 2 1 1 2 2

1 1 1( )

2 2 2TV S S S (37)



1 2 1 1 1 2 2 2

1ˆ ˆ( )

2TV S S S (38)


1 2( )V S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT

lm T T

B SS KS BF BQ

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT T T

lm T T

B SKS S S BF QS B

B S

1 1 1 2 2 2


1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || || || ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )


1 2 1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || ( || ||) ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )




11

|| ||ˆ =

T TB S

22

|| || || ||ˆ

T TB S x

(40)

then (35) becomes

1 2( ) 0TV S KS S (41)









7


1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))



1 2( ( ) ( ) ( ))V S t t t is


0lim ( )

t

tW d

(43)




ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)



1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)











TABLE I







8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































3


( - ) 0( )( - )

( - ) 00 s s usa s u

sas s u

if z z zC t z zF

if z z z

(2)






ms

mu

Ku zr

zs

zu

Ks fsa

Accelerometer


Accelerometer







1( ) Ts s s s um z K z z (5-1)







2 2 ˆTe and



2 21 2

1

[ ( ) ( )]N

k

E e k e k

(6)


1 1 2 2 21 1

ˆ[ ( ) ( ) ( ) ( )] ( ) ( ) 0ˆ

N NT T

tirek k

Ek k k k k F k

(7)


1 1 2 2 21 1

ˆ [ ( ) ( ) ( ) ( )] ( ) ( )N N

T Ttire

k k

k k k k k F k

(8)



s uy z z


sax Ax BF G

say Cx DF v (9)

where

-

-

0 0 1 -1

0 0 0 1

0 0 0

0 0

s

s

s u

u u

K

m

K K

m m

A

-1

1

0

0

s

u

m

m

B

0

1

0

0

G

-

-

0 0 0

0 0

s

s

s u

u u

K

m

K K

m m

C

-1

1

s

u

m

m

D







4


( - ) 0

( - ) 00 s s usky s

skys s u

if z z zC zF

if z z z

(10)


( - ) 0

( - ) 00 u s ugrd u

grdu s u

if z z zC zF

if z z z

(11)


(1 )

(1 )sa sky grd

sky s grd u

F F F

C z C z

(12)



r r sarx Ax BF G

r r ry C x






0 0 1 0

0 0 0 1rC

- -1

1 1 1

2

0 0 0

0 0 ( ) 0 0

0 0 0 ( ) 0

ss s

Km m

C s D

s

(14)



1 2 21 1

( )2s s s

n n

sz s z z

s

(15)

2 2 22 2

( )2u u u

n n

sz s z z

s

(16)





1 2

min max

( ( )) ( ( ))1

s s

t t

m m

(17)


1

( ( )) 1ii

t

The function


min max

min max min max

1 1 1 1

1 21 1 1 1( ) ( )s s s s

s s s s

m m m m

s s

m m m m

m m

(18)



can be defined 2

1 1 1 11

( ( ) ( ) ( ) ( )) ( ( ))( )i i i i ii


(19)


max max

min max

min min

( )

sar

sar sar sar

sar

F if F F

sat F F if F F F

F if F F

(20)






5


max| |s uz z z (22)


0min

TrP P K

subject to

2 0 0

T Ti i i iP P PB

I

I

(23)






max

max

( ) 0

( ) 0 1 2N N r r

i i r r

A B K C

A B K C i

(24)

where 2 2

1 1


A m A B m B





0( ) ( ) sa dx A A x B B F G f (25)







6





calculation

Control calculation

c hF F

aF

Road disturbance

rx

xΣ



0( )

sa sa d

sa lm

x Ax BF G Ax BF f

Ax BF G BF

(27)





re x x (29)








sa c h aF F F F

1 1 1( ) ( ) ( )|| ||

T T

sarT T


B S

(34)


hF Q B S B S repre-







1 1 1 2 2 2

ˆ ˆ (35)

where 1 and



ˆ|| ||

T T

sa eq T T

B SF F Q

B S

(36)

with 1 2


2 21 2 1 1 2 2

1 1 1( )

2 2 2TV S S S (37)



1 2 1 1 1 2 2 2

1ˆ ˆ( )

2TV S S S (38)


1 2( )V S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT

lm T T

B SS KS BF BQ

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT T T

lm T T

B SKS S S BF QS B

B S

1 1 1 2 2 2


1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || || || ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )


1 2 1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || ( || ||) ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )




11

|| ||ˆ =

T TB S

22

|| || || ||ˆ

T TB S x

(40)

then (35) becomes

1 2( ) 0TV S KS S (41)









7


1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))



1 2( ( ) ( ) ( ))V S t t t is


0lim ( )

t

tW d

(43)




ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)



1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)











TABLE I







8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































4


( - ) 0

( - ) 00 s s usky s

skys s u

if z z zC zF

if z z z

(10)


( - ) 0

( - ) 00 u s ugrd u

grdu s u

if z z zC zF

if z z z

(11)


(1 )

(1 )sa sky grd

sky s grd u

F F F

C z C z

(12)



r r sarx Ax BF G

r r ry C x






0 0 1 0

0 0 0 1rC

- -1

1 1 1

2

0 0 0

0 0 ( ) 0 0

0 0 0 ( ) 0

ss s

Km m

C s D

s

(14)



1 2 21 1

( )2s s s

n n

sz s z z

s

(15)

2 2 22 2

( )2u u u

n n

sz s z z

s

(16)





1 2

min max

( ( )) ( ( ))1

s s

t t

m m

(17)


1

( ( )) 1ii

t

The function


min max

min max min max

1 1 1 1

1 21 1 1 1( ) ( )s s s s

s s s s

m m m m

s s

m m m m

m m

(18)



can be defined 2

1 1 1 11

( ( ) ( ) ( ) ( )) ( ( ))( )i i i i ii


(19)


max max

min max

min min

( )

sar

sar sar sar

sar

F if F F

sat F F if F F F

F if F F

(20)






5


max| |s uz z z (22)


0min

TrP P K

subject to

2 0 0

T Ti i i iP P PB

I

I

(23)






max

max

( ) 0

( ) 0 1 2N N r r

i i r r

A B K C

A B K C i

(24)

where 2 2

1 1


A m A B m B





0( ) ( ) sa dx A A x B B F G f (25)







6





calculation

Control calculation

c hF F

aF

Road disturbance

rx

xΣ



0( )

sa sa d

sa lm

x Ax BF G Ax BF f

Ax BF G BF

(27)





re x x (29)








sa c h aF F F F

1 1 1( ) ( ) ( )|| ||

T T

sarT T


B S

(34)


hF Q B S B S repre-







1 1 1 2 2 2

ˆ ˆ (35)

where 1 and



ˆ|| ||

T T

sa eq T T

B SF F Q

B S

(36)

with 1 2


2 21 2 1 1 2 2

1 1 1( )

2 2 2TV S S S (37)



1 2 1 1 1 2 2 2

1ˆ ˆ( )

2TV S S S (38)


1 2( )V S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT

lm T T

B SS KS BF BQ

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT T T

lm T T

B SKS S S BF QS B

B S

1 1 1 2 2 2


1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || || || ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )


1 2 1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || ( || ||) ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )




11

|| ||ˆ =

T TB S

22

|| || || ||ˆ

T TB S x

(40)

then (35) becomes

1 2( ) 0TV S KS S (41)









7


1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))



1 2( ( ) ( ) ( ))V S t t t is


0lim ( )

t

tW d

(43)




ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)



1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)











TABLE I







8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































5


max| |s uz z z (22)


0min

TrP P K

subject to

2 0 0

T Ti i i iP P PB

I

I

(23)






max

max

( ) 0

( ) 0 1 2N N r r

i i r r

A B K C

A B K C i

(24)

where 2 2

1 1


A m A B m B





0( ) ( ) sa dx A A x B B F G f (25)







6





calculation

Control calculation

c hF F

aF

Road disturbance

rx

xΣ



0( )

sa sa d

sa lm

x Ax BF G Ax BF f

Ax BF G BF

(27)





re x x (29)








sa c h aF F F F

1 1 1( ) ( ) ( )|| ||

T T

sarT T


B S

(34)


hF Q B S B S repre-







1 1 1 2 2 2

ˆ ˆ (35)

where 1 and



ˆ|| ||

T T

sa eq T T

B SF F Q

B S

(36)

with 1 2


2 21 2 1 1 2 2

1 1 1( )

2 2 2TV S S S (37)



1 2 1 1 1 2 2 2

1ˆ ˆ( )

2TV S S S (38)


1 2( )V S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT

lm T T

B SS KS BF BQ

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT T T

lm T T

B SKS S S BF QS B

B S

1 1 1 2 2 2


1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || || || ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )


1 2 1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || ( || ||) ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )




11

|| ||ˆ =

T TB S

22

|| || || ||ˆ

T TB S x

(40)

then (35) becomes

1 2( ) 0TV S KS S (41)









7


1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))



1 2( ( ) ( ) ( ))V S t t t is


0lim ( )

t

tW d

(43)




ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)



1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)











TABLE I







8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































6





calculation

Control calculation

c hF F

aF

Road disturbance

rx

xΣ



0( )

sa sa d

sa lm

x Ax BF G Ax BF f

Ax BF G BF

(27)





re x x (29)








sa c h aF F F F

1 1 1( ) ( ) ( )|| ||

T T

sarT T


B S

(34)


hF Q B S B S repre-







1 1 1 2 2 2

ˆ ˆ (35)

where 1 and



ˆ|| ||

T T

sa eq T T

B SF F Q

B S

(36)

with 1 2


2 21 2 1 1 2 2

1 1 1( )

2 2 2TV S S S (37)



1 2 1 1 1 2 2 2

1ˆ ˆ( )

2TV S S S (38)


1 2( )V S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT

lm T T

B SS KS BF BQ

B S

1 1 1 2 2 2

ˆ ˆ ˆ|| ||

T TT T T

lm T T

B SKS S S BF QS B

B S

1 1 1 2 2 2


1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || || || ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )


1 2 1 2

1 1 1 1 2 2 2 2

ˆ ˆ|| || ( || ||) ( || ||) || ||

ˆ ˆ ˆ ˆ ( ) ( )




11

|| ||ˆ =

T TB S

22

|| || || ||ˆ

T TB S x

(40)

then (35) becomes

1 2( ) 0TV S KS S (41)









7


1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))



1 2( ( ) ( ) ( ))V S t t t is


0lim ( )

t

tW d

(43)




ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)



1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)











TABLE I







8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































7


1 2 1 20( ) ( (0) (0) (0)) ( ( ) ( ) ( ))



1 2( ( ) ( ) ( ))V S t t t is


0lim ( )

t

tW d

(43)




ˆˆ|| ||

T T

h T T

B SF Q

B S

(44)



1 11

1

ˆ( || ||)ˆ

T Tc B S

2 22

2

ˆ( || || || ||)ˆ

T Tc B S x

(45)











TABLE I







8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































8















9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































9

2 25 3 35 4 45 5 55 6

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8

-60

-40

-20

0

20

40

60

80

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

p g

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


2 25 3 35 4 45 5 55 6-5

0

5

10

15

20

25

30

35

40

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

100p

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

2 25 3 35 4 45 5 55 6 65

-10

-5

0

5

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE II












10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































10

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

15

20

25

30

Time (sec)

Tire

Def

lect

ion

(mm

)

Passive


Active (LQR)

(a)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60

80

100

Time (sec)

Sus

pens

ion

Str

oke

(mm

)

Passive


Active (LQR)

(b)

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

30

Time (sec)

Spr

ung

Mas

s A

ccel

erat

ion

(ms

2 )

Passive


Active (LQR)



TABLE III












11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































11


APPENDIX A



1

2 20 0 3

1

20 0

( )

1 1 sin( ) sin( )cos( ) ( )

( ) 2

sin ( ) ( ) cos( )

s s sa r

u c s

t c s s r b sa

f z z F z

m l K gg

K l z z z l F

2

2 2 20 0

2

3

0

( )

1 sin( )cos( )

( )

1( ) sin( ) ( )

2cos( ) ( ) ( )

s s sa r

u c

s u s


f z z F z

m lg

m m K g

m K l z z z m m l F

where 2

1 0( ) sin ( )s c u cg m l m l 2 2 2 2


3( )cos( )

ll

l l

dg b

c d


APPENDIX B



1 2ˆ ( )Q while

1 1 2 2ˆ ˆ is a


( )

|| ||

T

T TT

lm T T

V S S S

B SS KS BF BQ

B S

|| || || ||

T T T TT T

lm T T T T

B S B SKS S S B F Q

B S B S

(B1)


|| ||( )

|| ||

T TT

T T

B SV S KS S Q

B S

2 ( )KV S Q (B2)


( ( )) ( (0)) KtQ QV S t V S e

K K

(B3)


22 2

2 2|| || ( (0))

|| || || ||KtQ Q

e V S eK K

(B4)



ACKNOWLEDGMENT


















12





































12


































Date post:	27-Jan-2017
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