Linear quadratic regulator controller design for active ... - NOPR

Journal of Scientific & Industrial Research Vol. 65, March 2006, pp. 213-226

Linear quadratic regulator controller design for active suspension system subjected to random road surfaces M Senthil Kumar* and S Vijayarangan Department of Mechanical Engineering, PSG College of Technology, Coimbatore 641 004 Received 04 July 2005; revised 14 December 2005; accepted 02 January 2006 Present work aims at developing an active suspension for a passenger car by designing a controller using linear quadratic optimal control theory, based on two different control approaches [conventional method (CM), acceleration dependent method (ADM)]. Performance of the active suspension system of a quarter car model with 3 degrees-of-freedom with two control approaches has been compared with that of passive one. Active suspension system had a better potential to improve both the ride comfort and road holding, since the passenger acceleration has been reduced for active CM system (19.58%) and for active ADM system (34.08%) compared to passive one. Also suspension travel has reduced (37.5%). Keywords: Active suspension, Optimal control, Ride comfort, Road handling IPC Code: B60G; G09B17/00

Introduction Suspension system isolates a vehicle body from road irregularities in order to maximize passenger ride comfort and retain continuous road-wheel contact to provide road holding1,2. Traditionally, automotive suspension designs have been a compromise between the three conflicting criteria of passenger comfort road, suspension travel and road holding, which are also called as design goals3. Good ride comfort requires a soft suspension, whereas insensitivity to applied loads requires stiff suspension. Electronically controlled suspension systems can potentially improve the ride comfort as well as the road handling of vehicle4. Road handling relates contact forces between the tires and road surface and related to tire displacement and suspension travel. In passive system, parameters are fixed, being to achieve a certain level of compromise between road holding, load carrying and comfort. An active suspension system (ASS), on the other hand, has the capability to adjust itself continuously to changing road conditions resulting in a better set of design trade-offs compared to passive suspension. ASS employs pneumatic or hydraulic actuators for additional energy. The actuator is secured in parallel with a spring and shock absorber5. ASS requires sensors to be located at different points of the vehicle to measure motions of the body. This information is fed as input for the __________ *Author for correspondence E-mail: [email protected]

controller in order to provide exact amount of force required through the actuator. Many controller designs3,4,9 have been proposed to develop ASSs. In this paper, a linear quadratic regulator (LQR) control theory with two different approaches, namely CM and ADM, is used to develop ASS. Seat stiffness and damping are also included in the modelling. Mathematical Modeling In this work, a quarter car model with three degrees-of-freedom, which leads to simplified analysis and also represents most of the features of the full model, was used. The model consists of passenger seat, sprung mass, which refer to the part of the car that is supported on springs and unsprung mass, which refers to the mass of wheel and axle assembly. The tire has been replaced with its equivalent stiffness and tire damping is neglected. The suspension, tire and passenger seat are modelled by linear springs in parallel with dampers. Passive Suspension

For passive suspension (Fig. 1), using Newton’s Second Law of Motion and free-body diagram concept, the following equations of motion are derived: m p && x p + k p (x p - xs )+ c p (x& p - x&s )= 0

…(1)

ms && xs + k p ( xs − x p ) + c p ( x&s − x& p )

+k s (xs - xus )+ cs (x& s - x&us )= 0

…(2)

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mus && xus + ks (xus - xs ) + cs (x&us - x&s ) +kt (xus - r ) = 0

over the letters indicates differentiation with respect to time). Therefore,

…(3)

where, mp = Passenger seat mass (kg), ms = Quarter car sprung body mass (kg), mus = Unsprung mass (kg), kp = Seat stiffness (N/m), ks = Spring stiffness (N/m), kt = Tire stiffness (N/m), cp = Seat damping coefficient (N-s/m), cs = Suspension damping coefficient (N-s/m), xp = Passenger vertical displacement (m), xs = Sprung mass vertical displacement (m), xus = Unsprung mass vertical displacement (m), and r = Road profile height (m).

x& 2 = &x&p = −

x&4 = && x3 = −

0

0

0

0

0

0

ms

ks ms

cs ms

0

1

cp

kp

cp

mp

mp

0

0

0

1

kp

cp

ms

ms

0

0

0

0

0

0

ks mus

cs mus

−

mp

       where A =           

−

0 −

kp mp 0

−

(ks + k p ) ms

−

(ks + k p )

−

(ks + k p )

1

0

0

cp

kp

cp

mp

mp

mp

0

1

0

( ks + k p )

1 [ k3 ( x5 − x3 ) mus

mus

−

cs mus

0

0

0

0

ms

ks ms

cs ms 1

( cs + c p )

cp ms

0

0

0

0

0

0

0

ks mus

cs mus

k +k  −  s t   mus 

−

 0   x1    x   0  2    0   x3      +  0 r x  4  0   x5       kt   x6     mus 

0

kp

ms

                   

0

ms

−

…(6)

Putting Eqs (4-6) in state space representation form7 ( x&6 = Ax + Gr ), it gives

0

mp

−

kp

0

…(5)

+ c3 ( x6 − x4 ) + kt ( x5 − r ) ]

x5 = xus, x6 = &x&us ), as in Eq. (6), Eqs (1-3) can be written in state variable form where x1 , x2 , x3 , x4 , x5 , x6 are taken as the states (the dot

1

]

+ k3 ( x3 − x5 ) + c p ( x4 − x2 ) 

Using notations (x1 = xp, x2 = x& p , x3 = xs, x4 = x&s ,

0

[

1 k p  ( x3 − x1 ) + c p ( x4 − x2 ) ms 

x&6 = && xus = −

      x&1   &    x2    x&3    =  x&4    x&    5   x&6        

1 k p ( x1 − x3 ) + c p ( x 2 − x4 ) …(4) mp

−

cs mus

…(7)

     0    0       0  and G =      0    0    kt       m us      

SENTHIL KUMAR & VIJAYARANGAN: LQR CONTROLLER FOR ACTIVE SUSPENSION SYSTEM OF CAR

m p && x p + k p ( x p − xs ) + c p ( x& p − x&s ) = 0

215 …(8)

ms && xs + k p ( xs − x p ) + c p ( x&s − x& p ) + k s ( xs − xus ) + cs ( x&s − x&us ) − f a = 0

…(9)

mus && xus + ks ( xus − xs ) + cs ( x&us − x&s ) + kt ( xus − r ) + f a = 0 …(10) where, fa denotes the actuator force (N). Using notations (x1 = xp, x2 = x& p , x3 = xs, x4 = x&s , x5 = xus, x6 = &x&us ) as in Eq. (6), it gives x&2 = && xp = −

1  k p ( x1 − x3 ) + c p ( x2 − x4 )  mp 

x&4 = && x3 = −

1 k p ( x3 − x1 ) + c p ( x4 − x2 ) + k3 ( x3 − x5 ) ms 

+ k3 ( x3 − x5 ) + c p ( x4 − x2 ) − f a 

Fig. 1 — Quarter car model of passive suspension

x&6 = && xus = −

Active Suspension System (ASS)

ASS has hydraulic actuator in addition to passive elements (Fig. 2). Hydraulic actuator is located parallel to suspension spring and shock absorber. Using Newton’s Second Law of Motion and freebody diagram concept, following equations are derived:       x&1   &    x2    x&3    =  x&4    x&    5   x&6        

0

1

0

0

cp

kp

cp

mp

mp

mp

0

0

0

1

kp

cp

ms

ms

0

0

0

0

kp

−

−

mp

where, B =

 0   0  0   1   0  − 1

         

and

−

+ c3 ( x6 − x4 ) + kt ( x5 − r ) + f a ]

0

0

0

0

ms

ks ms

cs ms

0

0

0

1

ks mus

cs mus

ms

u = [ fa].

…(13)

Putting Eqs (11-13) in state space representation form ( x& = Ax + Bu + Gr ) , represented as Eq. (14), gives Eq. 15 as shown below: 0

−

…(12)

1 [ k3 ( x5 − x3 ) mus

0

(ks + k p )

…(11)

(ks + k p )

−

(ks + k p ) mus

−

cs mus

                   

 0   x1   0   0  x      0 2      0   x3   0      +   fa +  0  r x 1  4    0   x5   0         kt   x6   −1 m   us 

…(15)

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∞

J = ∫ ( x T Q x + u T R u )dt

…(17)

0

where u = [f a ] and Q and R are positive definite weighting matrices. Here the passenger acceleration, which is indicator of ride comfort, is not included. Linear optimal control theory provides the solution of Eq. (17) in the form of Eq. (16). The gain matrix K is computed from, K = R −1 B T P

…(18)

where the matrix P is evaluated being the solution of the Algebraic Riccati Equation, Fig. 2 — Quarter car model of active suspension LQR Controller Design

Among design goals, passenger ride comfort is related to passenger acceleration, suspension travel is related to relative distance between the unsprung mass and sprung mass and road handling is related to the tire displacement. The controller should minimize all these quantities and hence suitable 6 states are selected to configure the controller. The various states considered are passenger displacement and velocity, sprung mass displacement and velocity, finally tire displacement and velocity. Hence, the state vector matrix is x = [ x1 x 2 x 3 x 4 x 5 x6 ] . Eq. (14) is a linear time invariant system (LTI). State feedback control for ASS is a powerful tool for designing a controller. For controller design, it is assumed that all the states are available and also could be measured exactly. This study considered following state variable feedback regulator8:

u = − kx

…(16)

where K is the state feedback gain matrix. Optimization of control system consists of determining the control input u, which minimizes the performance index (J), which represents the performance characteristic requirement as well as controller input limitations using CM and ADM.

AT P + PA − PBR−1 B T P + Q = 0

Substituting gain matrix K in Eq. (14), gives x& = ( A − BK ) x + Gr

In CM, J penalizes state variables and inputs. Thus, it has the standard form as,

…(20)

Acceleration Dependent Method (ADM)

ADM is new approach giving more importance to passenger comfort by including passenger acceleration in J. Suppose that the vector z represents the passenger acceleration, z = [x&&p ] Then, J could be written as, ∞

J = ∫ ( x T Q x + u T R u + z T S z )dt

…(21)

0

Here, S is weighting matrix, which can be suitably assumed. Therefore, the Eq. (21) becomes ∞

J = ∫ ( x T Q x + u T R u + &x&Tp S &x&p )dt

…(22)

0

This equation is further modified, since the passenger acceleration is linearly dependent on the state variables. Eq. (11) is rewritten as,

&& xp = v x

…(23)

where row vector v is evaluated using Eq. (11), as,

v= Conventional Method (CM)

…(19)

1 [ − k p − c p k p c p 0 0] mp

Eq. (22) can be written as,

…(24)

SENTHIL KUMAR & VIJAYARANGAN: LQR CONTROLLER FOR ACTIVE SUSPENSION SYSTEM OF CAR

217

∞

J = ∫ ( x T ( Q + v T S v )x + u T R u )dt

…(25)

0

Thus, Eq. (22) could be written from Eq. (25) as,

Table 1 — Road roughness values classified by ISO Classification S (Ω)

Road roughness K [m2/(cycles/m)] Range

Average

A (Very good)

2 × 10-6 ∼8×10-6

4 × 10-6

B (Good)

8 × 10-6 ∼32×10-6

16 × 10-6

C (Average)

32 × 10-6 ∼128×10-6

64 × 10-6

D (Poor)

128 × 10-6 ∼512×10-6

256 × 10-6

E (Very poor)

512 × 10-6 ∼2048×10-6

1024 × 10-6

∞

J = ∫ ( x Qn x + u R u )dt T

T

…(26)

0

where, Qn = Q + v T S v . Optimal solution for Eq. (26) could be found similarly to that of Eq. (17). Random Road Input A real road surface taken as a random exciting function is used as input to the vehicle road model. The main characteristic of a random function is uncertainty as there is no method to predict an exact value at a future time. The function should be described in terms of probability statements as statistical averages, rather than explicit equations. In road model, power spectral density (PSD) has been used to describe the basic properties of random data. PSD function for stationary record represents the rate of change of mean values with frequency. It is estimated by computing the mean square value in a narrow frequency band at various frequencies, and then dividing it by the frequency band. Since the terrain surface roughness is a spatial disturbance, rather than a disturbance in time, it is desirable to define PSD in terms of spatial frequency. In random vibrations, mean square value of amplitude, and not the value of amplitude, is of prime interest since it is associated with the average energy. For a harmonic component, zr(x) with amplitude, Zr and wavelength λ, function is expressed as,

 2πx  r = z r ( x ) = Z r sin  = Z r sin f r x  λ 

…(27)

 2πx  where fr =   is the spatial circular frequency of  λ  the harmonic component expressed in rad/m. The mean-square value of the component

z r2 =

1

λ

∫

λ

0

Z r2   2πx   Z sin dx =    r  2  λ  

z r2

is given by, …(28)

For a function containing discrete frequencies, its frequency content can be expressed in terms of the

mean-square values of the components, and the result is a discrete spectrum. Generally, mean-square value contribution in each frequency interval ∆f is of interest. By letting S(nfo) be the power spectral density of the mean-square value in the interval ∆f at frequency n ∆ fo, where fo is initial frequency step and the following relation can be obtained:

S ( nf 0 ) =

Z r2 = z r2 2

…(29)

and the discrete power spectral density becomes,

S (nf 0 ) =

Z r2 z2 = r 2∆ f ∆ f

…(30)

Classification of Road Surfaces

In this work, classifications of road profiles are based on the International Organization for Standardization (ISO), which proposed road roughness classification using PSD values9 (Table 1). To make use of the classification, a normal random input is generated with variable amplitude. Using fast Fourier transform (FFT), a trial and error approach has been used in order to obtain the desired PSD characteristics of the random input. Stochastical characteristics of the random input (Table 2) are used for analyzing the system, which corresponds to the poor road condition as being classified by ISO from Table 1. The random road surface (Fig. 3) has been used to analyze the developed ASS with two approaches.

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Table 2 — Characteristics of random road input under poor class Random road input Spatial, mm

PSD [m3/(cycles/m]

Min

-56

0

Max

49

2.6 × 10-3

Mean

0

3.07 × 10-4

STD

17.7

3.24 × 10-4

Since, all the state variables, acceleration and control force have been constrained, the problem of controller design is then a challenge for finding suitable weightings that satisfies the design performances. This is done by choosing arbitrary weighting matrices Q and R. Vaughan’s algorithm9 is used for solving the Ricatti equation. MATLAB software is used for the simulation. In comparison with passive system, the body motion and passenger accelerations in ASS (Figs 5-16) have reduced significantly, which guarantee better ride comfort. Moreover, the tire deflection is relatively same. Therefore, ASS improves the ride comfort while retaining the road handling characteristics, compared to passive system. For the purpose of quantitative comparison, since the road input to the system is in the form of normal random distribution, it is expected to have normal distributed outputs. Therefore, using the concept of variance for the output signals, useful probability values can be calculated. For a Gaussian normal distribution, the probability function of the random signal x (t) can be written as10: prob [ λσ ≤ x(t ) ≤ λσ ] = 1

Fig. 3 — Random road surface

Results and Discussion Suspension parameters9 taken for analysis of the system are: mp = 60 kg, ms = 290 kg, mus = 59 kg, kp = 10507.1 N/m, ks = 16812 N/m, kt = 190000 N/m, cp = 875.6 N-s/m, and cs = 1000 N-s/m. According to ISO 26318 standard for whole body vibration exposure limitations, root mean square (RMS) value of acceleration denotes the total energy across the entire frequency range, which is referred as a measure of comfort. Comfort levels to vibration response in vertical direction are defined by the vibration tolerance limits, which are illustrated through the graph plotted between RMS acceleration and the frequency (Fig. 4). In this analysis of system response with random road input, some limits are assigned for all the states and also to passenger acceleration and actuator force in order to satisfy the required ride comfort, road handling and the design restriction10. The limits are: xs, 0.01m; xp, 0.01 m; &x&s , 0.2% g m/s2; xs – xus, 0.08 m; xus, 0.05 m; and fa, 500 N.

σ 2π

x2

λσ

−

∫λσ e

2σ 2

 λ  dx = (erf )    2

…(31)

prob  x(t ) > λσ  =  λ  1 prob [ λσ ≤ x(t ) ≤ λσ ] = (erfc)    2

…(32)

where σ is the standard deviation (STD), λ is a real number, (erf) denotes error function. For a quantitative comparison between the two controllers, for each state variable the bounding limit (90% probability) is calculated. This quantity can be evaluated by using standard deviation of the signal together with Eqs (31) and (32). Using inverse error function, (erf)-1, Eq. (31) is transformed to,

λ 90 = 2 ( erf )

−1

( 0.9 )

…(33)

and x90 = λ90 X σ+mean x(t)

…(34)

SENTHIL KUMAR & VIJAYARANGAN: LQR CONTROLLER FOR ACTIVE SUSPENSION SYSTEM OF CAR

Fig. 4 — ISO 2631: RMS acceleration vs frequency

Fig. 5 — Passenger displacement for passive and active CM vs time

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Fig. 6 — Passenger displacement for active CM and active ADM vs time

Fig. 7 — Passenger acceleration for passive and active CM vs time

SENTHIL KUMAR & VIJAYARANGAN: LQR CONTROLLER FOR ACTIVE SUSPENSION SYSTEM OF CAR

Fig. 8 — Passenger acceleration for active CM and active ADM vs time

Fig. 9 — Sprung mass displacement for passive and active CM vs time

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Fig. 10 — Sprung mass displacement for active CM and active ADM vs time

Fig. 11 — Suspension travel for passive and active CM vs time

SENTHIL KUMAR & VIJAYARANGAN: LQR CONTROLLER FOR ACTIVE SUSPENSION SYSTEM OF CAR

Fig. 12 — Suspension travel for active CM and active ADM vs time

Fig. 13 — Tire displacement for passive and active CM vs time

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Fig. 14 — Tire displacement for active CM and active ADM vs time

Fig. 15 — Control force for active CM vs time

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225

Fig. 16 — Control force for active ADM vs time

Table 3 — Comparison of 90% probability bounds for passive and active systems 90 % Probability bound States

Passive

CM

ADM

1

xp m

2.20 × 10-2

7.40 × 10-3

3.90 × 10-3

2

2 && m/s

2.5745

1.1767 × 10-1

1.572 × 10-2

3

xs m

1.56 × 10-2

5.4 × 10-3

1.6 × 10-3

4

xus m

5.48 × 10-2

4.85 × 10-2

5.18 × 10-2

5

xs-xus m

3.95 × 10-2

2.93 × 10-2

3.82 × 10-2

6

fa N

-

35.33

41.25

p

x

S No

where, x90 represents the bounding limit (90% probability) of the random signal x. This has been used to compare different systems quantitatively (Table 3). In this statistical comparison, the body bounce and passenger acceleration in ASS are reduced to more than half of their values in passive system. Tire displacement has been slightly reduced for ASS and suspension travel is also reduced (15%).

This confirms the efficiency of ASS in both ride comfort and road handling performance (Table 3). Body motions are lower in CM (Figs 5 & 6). However, passenger displacement and acceleration are significantly lower in ADM approach. Passenger acceleration in ADM approach has reduced (13.5%) than the CM approach (Figs 7 & 8). This is also evident through comparing passenger RMS accelerations for different systems. The passenger RMS acceleration values are: passive system, 1.0977; active CM system, 0.5518; and active ADM system, 0.0436 m/s2. This shows how effectively the passenger comfort has been improved with ADM approach. However, suspension travel values for ADM approach are slightly higher compared to CM approach but still very much less than that of passive system (Figs 11 & 12). This is due to more weightage has been given to ride comfort in ADM approach. Since ride comfort and road handling are mutually contradicting parameters, there is slight increase in suspension travel. However, the tire deflection values are almost same for both the active systems (Figs 13 & 14). This implies that active ADM suspension could improve the ride comfort compared to active

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CM suspension, retaining the road handling performance. The actuator forces are well below the applied limits and practically implementable. In ADM approach, gaining better ride comfort is possible by the cost of higher actuator force (Figs 15 & 16). However, optimal controller design could limit the actuator forces in some realistic bounds. The quantitative values (Table 3) could be an effective tool for the designer to satisfy the required performance or to compare different designs.

(90% probability) is calculated. Body bounce and passenger acceleration in active case has been found reduced to more than half of their values in passive system and suspension travel is also reduced (25.8%). Passenger acceleration in ADM approach has reduced by more than half to that of CM approach. The passenger acceleration has to be included in the performance index to improve ride comfort through active suspension system.

Conclusions A methodology was developed to design an active suspension for a passenger car by designing a controller, which improves performance of the system with respect to design goals compared to passive suspension system. Mathematical modelling has been performed using a three degrees-of-freedom model of a quarter car for passive and active suspension system considering only bounce motion to evaluate the performance of the suspension with respect to various contradicting design goals. Two controller design approaches (CM & ADM) have been examined for the active system. Realistic random road surface is modeled for simulation, which according to ISO road surface classification, falls under poor category of road. Also, some limits are assigned for all state variables and passenger acceleration and also actuator forces in order to satisfy the design goals. Almost all variables satisfy the constraints. A stochastical technique was used for quantitative comparison between the two controller design approaches of active system with respect to passive system, in which for each variable the amount of bounding limit

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