Half-Car Model Based LQR Control of Active …xingchenfan.weebly.com/uploads/2/4/9/9/24992009/xingchen...Fan 1 Half-Car Model Based LQR Control of Active Suspension Xingchen Fan ME

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Half-Car Model Based LQR Control of

Active Suspension

Xingchen Fan

ME 131 Final Project

Professor Hedrick

Spring 2013

1. INTRODUCTION

Suspension is one of the key connections between road and vehicle body. It not only

supports the static weight of the vehicle, but also influences the comfort of passengers, the

road holding abilities of tires and vehicle’s handling characters. Several criteria are usually

adopted to evaluate the performance of suspension, namely ride quality, suspension

deflection (also called rattle space) and tire deflection. For passive suspension, when damping

and stiffness are tuned to improve one performance, the other two are likely to be penalized.

The inevitable tradeoffs in passive suspension design motivate the implementation of active

suspensions, in which a controlled actuator is able to provide force in addition to that from

conventional spring/damper unit. Since the evaluation of suspension performance consists of

several criteria, it is reasonable to weight each criterion and put them in a performance index.

Linear quadratic regulator (LQR) is a good choice for such optimization based control. LQR

control has already been successfully implemented on quarter-car model (Rajamai). This

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paper is dedicated to see how LQR control works on a more complicated half-car model with

pitch movement. Therefore, in addition to vertical acceleration, angular acceleration about

center of gravity (CG) is also used to assess ride quality. In fact, the half-car model based

LQR controller for active suspension also works for roll control; however, this paper only

focuses on pitch control.

2. HALF-CAR MODEL

A two-degree-of-freedom quarter-car model is usually used for LQR control on active

suspension (Figure 1). Sprung mass (𝑚𝑠) represents the vehicle body (quarter weight in this

case) and unsprung mass (𝑚𝑢) represents components including drive axle, suspension,

wheel and brake. Damper has damping coefficient 𝑏𝑠. Suspension and tire has stiffness 𝑘2

and 𝑘1 respectively. Tires usually have negligible damping; otherwise, a large amount of

heat will be generated. 𝐹𝑎 is the output from the active actuator.

Figure 1 Quarter-car model

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In our analysis, we consider the most common conditions where wheels are always in

contact with ground. Thus, all displacements are measured from the static equilibrium

position, allowing gravity to be neglected.

The half-car model combines two quarter-car models with a shared sprung mass, where

a human model could be attached. The model has four degrees of freedom: sprung mass’s

vertical translation, its rotation about center of gravity, front and rear unsprung masses’

vertical translations. By adopting the same assumptions in quarter-car model, the

gravitational effect is neglected.

Figure 2 Half-car model

The equations of motion for half-car model are shown in Appendix I. Notice the positive

pitch is defined when the vehicle body rotates clockwise, or the front end of the vehicle

moves down.

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With four degrees of freedom and their first time-derivatives, an eight-state linear

time-invariant (LTI) system is developed to represent the plant. The state-space representation

of half-car model is shown below:

�̇� = 𝐴𝑥 + 𝐵𝑢 + 𝐿𝑟

𝑦 = 𝐶𝑥 + 𝐷𝑟 (1)

Matrices A, B, L, C and D are included in Appendix I. In this case, x is an 8 × 1 state vector;

u is a 2 × 1 input vector including front and rear actuator forces (𝐹𝑎𝑓 and 𝐹𝑎𝑟); r is a 2 ×

1 disturbance vector containing road inputs to front and rear tires (𝑧𝑟𝑓 and 𝑧𝑟𝑟); y is the

output vector with all criteria, which will be discussed in the next section. For road inputs, the

vehicle is assumed to travel at constant speed, so the rear tire experiences the same road input

as the front tire does, but with a constant time delay.

3. LQR CONTROL WITH VARIOUS CRITERIA

The design of the controller is associated with the criteria in which we are interested. For

suspension, we are interested in:

i. Ride quality

Passengers feel uncomfortable when they experience large vertical accelerations.

Hence the acceleration of center of gravity ( �̈�𝑐𝑔 ) is one of the criteria.

Furthermore, with only �̈�𝑐𝑔, it’s possible that CG is at constant height, while

front and rear ends of the vehicle are oscillating. Therefore, the pitch acceleration

about the center of gravity (�̈�𝑐𝑔) is also taken into account. Penalties on �̈�𝑐𝑔 and

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�̈�𝑐𝑔 are able to reduce the acceleration of every point on sprung mass, even

though the acceleration of specific passenger point is not considered in this paper.

Moreover, people feel uncomfortable when acceleration itself is changing fast; so

the first time-derivative of acceleration, jerk (𝑧𝑐𝑔 and 𝜃𝑐𝑔), can also be used as a

criterion. However, to simplify the system, it is not included in the performance

index. Jerk data is extracted from simulation results through numerical

differentiation of �̈�𝑐𝑔 and �̈�𝑐𝑔.

ii. Suspension deflection (rattle space)

Suspension deforms under load so it can absorb some energy from the road.

Suspension deflection (𝑧𝑠𝑓 − 𝑧𝑢𝑓 and 𝑧𝑠𝑟 − 𝑧𝑢𝑟 ) refers to the change in distance

between sprung and unsprung mass. How much suspension is allowed to deflect

depends on the limited space between wheels and vehicle body. If suspension

bottoms out, there will be a rigid connection between wheels and vehicle body.

Passengers will feel really uncomfortable. Small vehicles usually have a low ride

height, so their suspensions must deflect within a smaller range. Therefore,

suspension deflection should be held under control.

iii. Tire deflection

Tires are generally stiffer than suspension but they still deflect under load. Tire

deflection (𝑧𝑢𝑓 − 𝑧𝑟𝑓 and 𝑧𝑢𝑟 − 𝑧𝑟𝑟 ) is defined as the change in distance

between unsprung mass and road. Tires’ road holding ability is sensitive to

change in normal force and the shape of the contact patch. Excessive tire

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deflection can cause tires to lose grip and result in slipping. Thus tire deflection

has to be minimized.

iv. Actuator force

Actuators only provide limited forces (𝐹𝑎𝑓 and 𝐹𝑎𝑟). Typical hydraulic actuators

in the market today are able to supply as much as 20,000 N. In addition,

compared to passive and semi-active suspensions, higher power consumption is

one of the drawbacks of active suspension. Therefore, it is necessary to constrain

the actuator output both to save power and to ensure that the desired force is

achievable.

The resultant performance index is defined as below:

𝐽 = ∫ [𝜌0�̈�𝑐𝑔2 + 𝜌1�̈�𝑐𝑔

2+ 𝜌2(𝑧𝑠𝑓 − 𝑧𝑢𝑓)

2+ 𝜌3(𝑧𝑠𝑟 − 𝑧𝑢𝑟)2

∞

0

+ 𝜌4�̇�𝑠𝑓2 + 𝜌5�̇�𝑠𝑟

2

+𝜌6(𝑧𝑢𝑓 − 𝑧𝑟𝑓)2

+ 𝜌7(𝑧𝑢𝑟 − 𝑧𝑟𝑟)2 + 𝜌8 �̇�𝑢𝑓2 + 𝜌8�̇�𝑢𝑟

2 + 𝜌10𝐹𝑎𝑓2 + 𝜌11𝐹𝑎𝑟

2]𝑑𝑡 (2)

It is the cost function for a quadratic programming problem, which should be minimized. For

weights in the performance index, only relative values matter. Therefore, the weight for z̈cg2

is assumed to be 1 whenever this term is penalized. All other weights are referenced to it. The

performance index is rearranged to the standard form as below:

𝐽 = ∫ [∞

0𝑥𝑇𝑄𝑥 + 2𝑥𝑇𝑁𝑢 + 𝑢𝑇𝑅𝑢]𝑑𝑡 (3)

where Q, N and R are shown in Appendix II. These three matrices are large and have

complicated entries. Hence instead of putting them down in state-space representation

explicitly, they are defined elementwise in MATLAB code.

It is shown that the solution to this optimization problem is feedback control: 𝑢 = −𝐺𝑥,

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where 𝐺 = 𝑅−1(𝐵𝑇𝑃 + 𝑁), and P is the solution to algebraic Riccati equation (Rajamani):

(𝐴 − 𝐵𝑅−1𝑁)𝑇𝑃 + 𝑃(𝐴 − 𝐵𝑅−1𝑁) + (𝑄 − 𝑁𝑇𝑅−1𝑁) − 𝑃𝐵𝑅−1𝐵𝑇𝑃 = 0 (4)

Accordingly, equation 1 becomes:

�̇� = (𝐴 − 𝐵𝐺)𝑥 + 𝐿𝑟

𝑦 = 𝐶𝑥 + 𝐷𝑟 (5)

In this case, r becomes the input to the system; y is a 6 × 1 vector containing the criteria:

y = [�̈�𝑐𝑔 �̈�𝑐𝑔 𝑧𝑠𝑓 − 𝑧𝑢𝑓𝑧𝑠𝑟 − 𝑧𝑢𝑟 𝑧𝑢𝑓 − 𝑧𝑟𝑓 𝑧𝑢𝑟 − 𝑧𝑟𝑟]𝑇 (6)

With corresponding matrices C and D, we are able to obtain transfer functions from road

inputs to each criterion. Firstly, the state-space representation is written in s-domain:

𝑠𝑋 = (𝐴 − 𝐵𝐺)𝑋 + 𝐿𝑅

𝑌 = 𝐶𝑋 + 𝐷𝑅 (7)

Then the equations are reorganized and X is cancelled out. We obtain the closed-loop transfer

function from R to Y:

𝑌

𝑅= 𝐶(𝑠𝐼 − 𝐴 + 𝐵𝐺)−1𝐿 + 𝐷 (8)

This will be used in frequency analysis of the LQR controller in the next section.

4. SIMULATION RESULTS

The half-car model based LQR controller is simulated in MATLAB. Instead of the

analytical methods introduced in the previous section, some built-in MATLAB functions are

used to significantly simplify simulation and analysis: given A, B, Q, R and N, “lqr” computes

the feedback gain matrix G, which is used in every iteration in transient analysis. One of the

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requirements for input argument to “lqr” function is that R matrix:

𝑅 = [

1

𝑚2 + 𝜌101

𝑚2

1

𝑚2

1

𝑚2 + 𝜌11

] (9)

has to be positive definite, where 𝜌10 and 𝜌11 are from equation 2. Therefore, for “lqr” to

work, there must be some penalty on actuator force, i.e. 𝜌10 = 𝜌11 ≠ 0. Small 𝜌10 and 𝜌11

(e.g. 10−10) represent fully operating actuators, and large 𝜌10 and 𝜌11 (e.g. 10−1) make the

system behave like a passive one. In addition, given A, B, C and D, built-in function “ss2tf”

computes the corresponding transfer function, which is used in frequency analysis.

The front and rear halves of the model are assumed to be symmetrical. The vehicle

parameters used in simulation are (SI units): 𝑘𝑠𝑓 = 𝑘𝑠𝑟 = 16000, 𝑏𝑠𝑓 = 𝑏𝑠𝑟 = 1000, 𝑘𝑡𝑓 =

𝑘𝑡𝑟 = 160000, 𝑚𝑢𝑓 = 𝑚𝑢𝑟 = 45, 𝑚 = 650, 𝐼 = 2200, 𝐿 = 2.8, 𝑙𝑓 = 𝑙𝑟 = 1.4 (Rajamani).

The speed of the vehicle is assumed to be 15 m/s (54 km/h).

Simulation consists of two parts: transient analysis and frequency analysis.

I. Transient analysis

In this part of simulation, the results are shown in time domain. The road input is a

typical speed hump with a height of 20 cm and a chord length of 4 m. The front tire hits the

hump at time t = 1 s. The results for passive half-car model (u = 0) are shown below:

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Figure 3 CG position and pitch movement for passive suspension

When the front tire hits the hump, the sprung mass lifts and rotates counterclockwise.

That’s why a positive vertical displacement and a negative pitch movement are observed in

Figure 3. As the vehicle passes the hump, the perturbations gradually die out and the vehicle

returns to its equilibrium position. This is further confirmed by Figure 4 and 5, and jerk data

in Figure 6.

Figure 4 CG acceleration for passive suspension

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Figure 5 Pitch acceleration for passive suspension

Figure 6 CG jerk and pitch jerk for passive suspension

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Figure 7 Front suspension deflection and tire deflection for passive suspension

Figure 8 Rear suspension deflection and tire deflection for passive suspension

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From Figure 7 and 8, it is shown that the front suspension is first compressed and then

stretched. The rear suspension behaves similarly but with a time delay of about 0.2 s. Also,

we can see that a large portion of the displacement of the speed hump (0.2 m) is transferred to

CG and rattle space.

When a fully operating LQR controller (𝜌10 = 𝜌11 = 10−10) with penalty only on CG

acceleration is implemented, some results are shown below:

Figure 9 CG position and pitch movement for 𝜌0 = 1

It is obvious from Figure 8 that CG of sprung mass barely moves up (less than 1 mm).

There is a significant improvement in CG acceleration and jerk in Figure 10 and 11 compared

to passive ones in Figure 4 and 6. However, comparing Figure 3 and 9 shows that nothing has

improved in terms of pitch movement, suggesting that when the front and rear actuators are

working to reduce �̈�𝑐𝑔, the vehicle is still rotating about CG.

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Figure 10 CG acceleration for 𝜌0 = 1

Figure 11 CG jerk for 𝜌0 = 1

Then we implement a fully operating LQR controller with penalty on both CG

acceleration and pitch acceleration. The results are shown below:

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Figure 12 CG position and pitch movement for 𝜌0 = 𝜌1 = 1

Compared to the passive system in Figure 3, both vertical translation and pitch

movement have decreased considerably. The vehicle is barely moving up and down or

rotating about the CG. The CG acceleration in Figure 13 and the pitch acceleration in Figure

14 both confirm this improvement. This also suggests that every point on sprung mass has

small acceleration.

Figure 13 CG acceleration for 𝜌0 = 𝜌1 = 1

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Figure 14 Pitch acceleration for 𝜌0 = 𝜌1 = 1

Figure 15 shows the front and rear actuators’ force outputs. Again a time delay of 0.2 s

between front and rear is obvious in the plots. The desired force is no more than 7000 N,

which is totally achievable with available actuators.

Figure 15 Front and rear actuator forces for 𝜌0 = 𝜌1 = 1

If power consumption is a concern, the weights on 𝐹𝑎𝑓 and 𝐹𝑎𝑟 can increase. The

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performance will deteriorate though since actuators will not work as hard as they can.

Furthermore, it is clear from the plots above that it takes a long time for the system to settle

down to its equilibrium position. In Figure 15, the actuators still have small oscillating

outputs after the speed hump. This problem can be solved by penalizing the actuator outputs

more heavily. An example is shown in Figure 16 and 17:

Figure 16 CG position and pitch movement for 𝜌0 = 𝜌1 = 1 and 𝜌10 = 𝜌11 = 10−6

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Figure 17 Front and rear actuator forces for 𝜌0 = 𝜌1 = 1 and 𝜌10 = 𝜌11 = 10−6

Overall, a half-car model based LQR controller is able to significantly improve ride

quality. However, just like passive suspension tuning, active suspension tuning has tradeoffs.

The inverse relationship between actuator force and ride quality improvement is already

discussed above. In addition, there are suspension deflection and tire deflection criteria.

When ride quality improves, those two criteria are usually sacrificed. This will be studied

through frequency analysis in the next section.

II. Frequency analysis

In this part of simulation, the results are shown in frequency domain, and mainly based

on Bode magnitude plots of transfer functions from road input to each criterion. For a typical

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commercial vehicle, the natural frequencies for sprung mass and unsprung mass are about 1

Hz and 10 Hz. Also, humans are most sensitive to frequencies in this range. Therefore, the

frequency band at which the analysis is done is from 0.1 Hz and 100 Hz. In the following

plots, for comparison, the dashed lines illustrate the frequency response of the passive system,

while the solid lines represent the system with LQR control. Since the front and rear halves of

the vehicle has identical parameters, only the frequency response of the front suspension is

plotted. Moreover, because the CG acceleration and the pitch acceleration are often weighted

equally, they have very similar frequency responses. Only one of them will be plotted.

When the LQR controller is tuned for ride quality (𝜌0 = 𝜌1 = 1) without any penalty on

other criteria, the frequency responses of CG acceleration and pitch acceleration are shown in

Figure 18 and 19 respectively:

Figure 18 CG acceleration for 𝜌0 = 𝜌1 = 1

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It is shown that ride quality has improved over the entire frequency band except the peak

at about 10 Hz. This is consistent with the results shown by Rajamani: acceleration transfer

function has an invariant point at ω = √𝑘𝑡

𝑚𝑢, which is approximately the natural frequency of

unsprung mass.

As ride quality improves, suspension deflection and tire deflection become worse

(Figure 20), especially at unsprung mass resonant frequency. Moreover, the suspension

deflection transfer function has an asymptote at low frequencies, which does not exist in

passive suspension. The results are consistent with those obtained from quarter-car model

(Rajamani).

Figure 19 Front suspension deflection for 𝜌0 = 𝜌1 = 1

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Figure 20 Front tire deflection for 𝜌0 = 𝜌1 = 1

Then, if the LQR controller is tuned for suspension deflection and tire deflection

(𝜌2 = 𝜌6 = 104) with some penalty on CG acceleration and pitch acceleration (𝜌0 = 𝜌1 = 1),

ride quality becomes worse as a tradeoff (Figure 21). Moreover, it is shown that the CG

acceleration rolls off more slowly at frequencies higher than 10 Hz, which is called high

frequency harshness. It is clear from Figure 22 and 23 that both suspension deflection and tire

deflection have improved considerably at unsprung mass resonant frequency. However, an

asymptote still exists in suspension deflection transfer function at low frequency; tire

deflection does not improve that much either at low and medium frequencies.

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Figure 21 CG acceleration for 𝜌0 = 𝜌1 = 1, 𝜌2 = 𝜌6 = 104

Figure 22 Front suspension deflection for 𝜌0 = 𝜌1 = 1, 𝜌2 = 𝜌6 = 104

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Figure 23 Front tire deflection for 𝜌0 = 𝜌1 = 1, 𝜌2 = 𝜌6 = 104

To sum it up, like passive suspensions, tradeoffs are inevitable in active suspension

design. A considerable amount of improvement in ride quality requires a significant sacrifice

in suspension deflection and tire deflection, and vice versa; improvement at sprung mass

frequency results in deterioration at unsprung mass frequency, and vice versa.

5. CONCLUSION

A half-car model based LQR controller gives vehicle designers more freedom in

suspension tuning. The suspension is allowed to have different setups with fixed geometry

and fixed-rate spring/damper unit. However, a balance between ride quality, suspension

deflection, tire deflection and power consumption is important.

Another thing to notice is that most results from the half-car model based LQR

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controller are consistent with those obtained from the quarter-car model based LQR controller.

This suggests that the decoupled quarter-car model may be sufficient for all desired

improvements from active suspension. An intuition behind this is as follows. For the half-car

model, two degrees of freedom define the position of sprung mass. In this paper, the vertical

translation of center of gravity and the rotation about center of gravity are used. However,

another choice will be the vertical translations of front and rear suspension attachment points

(�̈�𝑠𝑓 and �̈�𝑠𝑟). In that case, the model is basically decoupled into two quarter-car models. In

addition, the quarter-car model is much simpler mathematically.

REFERENCES

Rajamani, Rajesh. Vehicle Dynamics and Control. NewYork: Springer, 2012. Chapter 10, 11

and 12.

APPENDIX I

Equations of motion:

Acceleration of center of gravity:

𝑚�̈�𝑐𝑔 = −𝑘𝑠𝑓(𝑧𝑠𝑓 − 𝑧𝑢𝑓) − 𝑏𝑠𝑓(�̇�𝑠𝑓 − �̇�𝑢𝑓) − 𝑘𝑠𝑟(𝑧𝑠𝑟 − 𝑧𝑢𝑟) − 𝑏𝑠𝑟(�̇�𝑠𝑟 − �̇�𝑢𝑟) + 𝐹𝑎𝑓 + 𝐹𝑎𝑟

Pitch acceleration:

𝐼�̈� = −𝑙𝑓[−𝑘𝑠𝑓(𝑧𝑠𝑓 − 𝑧𝑢𝑓) − 𝑏𝑠𝑓(�̇�𝑠𝑓 − �̇�𝑢𝑓) + 𝐹𝑎𝑓] + 𝑙𝑟[−𝑘𝑠𝑟(𝑧𝑠𝑟 − 𝑧𝑢𝑟) − 𝑏𝑠𝑟(�̇�𝑠𝑟 − �̇�𝑢𝑟) + 𝐹𝑎𝑟]

Acceleration of front unsprung mass:

𝑚𝑢𝑓�̈�𝑢𝑓 = 𝑘𝑠𝑓(𝑧𝑠𝑓 − 𝑧𝑢𝑓) + 𝑏𝑠𝑓(�̇�𝑠𝑓 − �̇�𝑢𝑓) − 𝑘𝑡𝑓(𝑧𝑢𝑓 − 𝑧𝑟𝑓) − 𝐹𝑎𝑓

Acceleration of rear unsprung mass:

𝑚𝑢𝑟�̈�𝑢𝑟 = 𝑘𝑠𝑟(𝑧𝑠𝑟 − 𝑧𝑢𝑟) + 𝑏𝑠𝑟(�̇�𝑠𝑟 − �̇�𝑢𝑟) − 𝑘𝑡𝑟(𝑧𝑢𝑟 − 𝑧𝑟𝑟) − 𝐹𝑎𝑟

Displacement of front and rear sprung mass:

𝑧𝑠𝑓 = 𝑧𝑐𝑔 − 𝑙𝑓𝜃

𝑧𝑠𝑟 = 𝑧𝑐𝑔 + 𝑙𝑟𝜃

𝐴 =

[

0 1

−𝑘𝑠𝑓 + 𝑘𝑠𝑟

𝑚−

𝑏𝑠𝑓 + 𝑏𝑠𝑟

𝑚

0 0𝑙𝑓𝑘𝑠𝑓 − 𝑙𝑟𝑘𝑠𝑟

𝑚

𝑙𝑓𝑏𝑠𝑓 − 𝑙𝑟𝑏𝑠𝑟

𝑚0 0

𝑙𝑓𝑘𝑠𝑓 − 𝑙𝑟𝑘𝑠𝑟

𝐼

𝑙𝑓𝑏𝑠𝑓 − 𝑙𝑟𝑏𝑠𝑟

𝐼

0 1

−𝑙𝑓

2𝑘𝑠𝑓 + 𝑙𝑟2𝑘𝑠𝑟

𝐼−

𝑙𝑓2𝑏𝑠𝑓 + 𝑙𝑟

2𝑏𝑠𝑟

𝐼

0 0𝑘𝑠𝑓

𝑚

𝑏𝑠𝑓

𝑚

0 0𝑘𝑠𝑟

𝑚

𝑏𝑠𝑟

𝑚0 0

−𝑙𝑓𝑘𝑠𝑓

𝐼 −

𝑙𝑓𝑏𝑠𝑓

𝐼

0 0

𝑙𝑟𝑘𝑠𝑟

𝐼

𝑙𝑟𝑏𝑠𝑟

𝐼0 0 𝑘𝑠𝑓

𝑚𝑢𝑓

𝑏𝑠𝑓

𝑚𝑢𝑓

0 0

−𝑙𝑓𝑘𝑠𝑓

𝑚𝑢𝑓 −

𝑙𝑓𝑏𝑠𝑓

𝑚𝑢𝑓

0 0 𝑘𝑠𝑟

𝑚𝑢𝑟

𝑏𝑠𝑟

𝑚𝑢𝑟

0 0𝑙𝑟𝑘𝑠𝑟

𝑚𝑢𝑟

𝑙𝑟𝑏𝑠𝑟

𝑚𝑢𝑟

0 1

−𝑘𝑠𝑓 + 𝑘𝑡𝑓

𝑚𝑢𝑓 −

𝑏𝑠𝑓

𝑚𝑢𝑓

0

0

0

0

0

0

0

0

0 1

−𝑘𝑠𝑟 + 𝑘𝑡𝑟

𝑚𝑢𝑟−

𝑏𝑠𝑟

𝑚𝑢𝑟]

𝐵 =

[

01

𝑚0

−𝑙𝑓𝐼

01

𝑚0𝑙𝑟𝐼

0

−1

𝑚𝑢𝑓

0

0

0

0

0

−1

𝑚𝑢𝑟]

, 𝐿 =

[

0000

0000

0𝑘𝑡𝑓

𝑚𝑢𝑓

0

0

0

0

0𝑘𝑡𝑟

𝑚𝑢𝑟]

1

function [Q, R, N] = generate_QRN(P)

Import parametersl_r = P.l_r;l_f = P.l - P.l_r;m = P.m; % Half car sprung mass [kg]I = P.I;m_uf = P.m_uf; % Quarter car unsprung mass [kg]m_ur = P.m_ur; % Quarter car unsprung mass [kg]k_sf = P.k_sf; % Spring constant of sprung mass [N/m]b_sf = P.b_sf; % Damping constant of sprung mass [N*s/m]k_tf = P.k_tf; % Spring constant of tire [N/m]k_sr = P.k_sr; % Spring constant of sprung mass [N/m]b_sr = P.b_sr; % Damping constant of sprung mass [N*s/m]k_tr = P.k_tr; % Spring constant of tire [N/m]

rho_0 = P.rho_0; % weight of z_cg_ddotrho_1 = P.rho_1; % weight of theta_cg_ddotrho_2 = P.rho_2; % weight of z_s_dotrho_3 = P.rho_3; % weight of z_u-z_rrho_4 = P.rho_4; % weight of z_u_dotrho_5 = P.rho_5; % weight of F_arho_6 = P.rho_6; % weight of z_s-z_urho_7 = P.rho_7; % weight of z_s_dotrho_8 = P.rho_8; % weight of z_u-z_rrho_9 = P.rho_9; % weight of z_s_dotrho_10 = P.rho_10; % weight of z_u-z_rrho_11 = P.rho_11; % weight of z_u-z_r

Input argument "P" is undefined.

Error in ==> generate_QRN at 3l_r = P.l_r;

Generate Q R N% R for z_cg_ddotR0 = [1/m^2, 1/m^2;... 1/m^2, 1/m^2];

% N for z_cg_ddotN0 = zeros(8,2);

N0(1,1) = (-k_sf - k_sr)/m^2;N0(2,1) = (-b_sf - b_sr)/m^2;N0(3,1) = (k_sf*l_f - k_sr*l_r)/m^2;N0(4,1) = (b_sf*l_f - b_sr*l_r)/m^2;N0(5,1) = k_sf/m^2;N0(6,1) = b_sf/m^2;N0(7,1) = k_sr/m^2;

2

N0(8,1) = b_sr/m^2;N0(:,2) = N0(:,1);

% Q for z_cg_ddotQ0 = zeros(8,8);

Q0(1,2) = (b_sf*k_sf + b_sf*k_sr + b_sr*k_sf + b_sr*k_sr)/m^2;Q0(1,3) = (-k_sf^2*l_f + k_sr^2*l_r - k_sf*k_sr*l_f + k_sf*k_sr*l_r)/m^2;Q0(1,4) = (-b_sf*k_sf*l_f - b_sf*k_sr*l_f + b_sr*k_sf*l_r +... b_sr*k_sr*l_r)/m^2;Q0(1,5) = (-k_sf^2 - k_sf*k_sr)/m^2;Q0(1,6) = (-b_sf*k_sf - b_sf*k_sr)/m^2;Q0(1,7) = (-k_sr^2 - k_sf*k_sr)/m^2;Q0(1,8) = (-b_sr*k_sf - b_sr*k_sr)/m^2;

Q0(2,3) = (-b_sf*k_sf*l_f - b_sr*k_sf*l_f + b_sf*k_sr*l_r +... b_sr*k_sr*l_r)/m^2;Q0(2,4) = (-b_sf^2*l_f + b_sr^2*l_r - b_sf*b_sr*l_f + b_sf*b_sr*l_r)/m^2;Q0(2,5) = (-b_sf*k_sf - b_sr*k_sf)/m^2;Q0(2,6) = (-b_sf^2 - b_sf*b_sr)/m^2;Q0(2,7) = (-b_sf*k_sr - b_sr*k_sr)/m^2;Q0(2,8) = (-b_sr^2 - b_sf*b_sr)/m^2;

Q0(3,4) = (b_sf*k_sf*l_f^2 + b_sr*k_sr*l_r^2 - b_sf*k_sr*l_f*l_r -... b_sr*k_sf*l_f*l_r)/m^2;Q0(3,5) = (k_sf^2*l_f - k_sf*k_sr*l_r)/m^2;Q0(3,6) = (b_sf*k_sf*l_f - b_sf*k_sr*l_r)/m^2;Q0(3,7) = (-k_sr^2*l_r + k_sf*k_sr*l_f)/m^2;Q0(3,8) = (b_sr*k_sf*l_f - b_sr*k_sr*l_r)/m^2;

Q0(4,5) = (b_sf*k_sf*l_f - b_sr*k_sf*l_r)/m^2;Q0(4,6) = (b_sf^2*l_f - b_sf*b_sr*l_r)/m^2;Q0(4,7) = (b_sf*k_sr*l_f - b_sr*k_sr*l_r)/m^2;Q0(4,8) = (-b_sr^2*l_r + b_sf*b_sr*l_f)/m^2;

Q0(5,6) = b_sf*k_sf/m^2;Q0(5,7) = k_sf*k_sr/m^2;Q0(5,8) = b_sr*k_sf/m^2;

Q0(6,7) = b_sf*k_sr/m^2;Q0(6,8) = b_sf*b_sr/m^2;

Q0(7,8) = b_sr*k_sr/m^2;

Q0(1,1) = (k_sf^2 + k_sr^2 + 2*k_sf*k_sr)/m^2;Q0(2,2) = (b_sf^2 + b_sr^2 + 2*b_sf*b_sr)/m^2;Q0(3,3) = (k_sf^2*l_f^2 + k_sr^2*l_r^2 - 2*k_sf*k_sr*l_f*l_r)/m^2;Q0(4,4) = (b_sf^2*l_f^2 + b_sr^2*l_r^2 - 2*b_sf*b_sr*l_f*l_r)/m^2;Q0(5,5) = k_sf^2/m^2;Q0(6,6) = b_sf^2/m^2;Q0(7,7) = k_sr^2/m^2;Q0(8,8) = b_sr^2/m^2;

% R for theta_cg_ddot

3

R1 = [l_f^2/I^2, -l_f*l_r/I^2;... -l_f*l_r/I^2, l_r^2/I^2];

% N for theta_cg_ddotN1 = zeros(8,2);

N1(1,1) = (-k_sf*l_f^2 + k_sr*l_f*l_r)/I^2;N1(2,1) = (-b_sf*l_f^2 + b_sr*l_f*l_r)/I^2;N1(3,1) = (k_sf*l_f^3 + k_sr*l_f*l_r^2)/I^2;N1(4,1) = (b_sf*l_f^3 + b_sr*l_f*l_r^2)/I^2;N1(5,1) = k_sf*l_f^2/I^2;N1(6,1) = b_sf*l_f^2/I^2;N1(7,1) = -k_sr*l_f*l_r/I^2;N1(8,1) = -b_sr*l_f*l_r/I^2;N1(1,2) = (-k_sr*l_r^2 + k_sf*l_f*l_r)/I^2;N1(2,2) = (-b_sr*l_r^2 + b_sf*l_f*l_r)/I^2;N1(3,2) = (-k_sr*l_r^3 - k_sf*l_f^2*l_r)/I^2;N1(4,2) = (-b_sr*l_r^3 - b_sf*l_f^2*l_r)/I^2;N1(5,2) = -k_sf*l_f*l_r/I^2;N1(6,2) = -b_sf*l_f*l_r/I^2;N1(7,2) = k_sr*l_r^2/I^2;N1(8,2) = b_sr*l_r^2/I^2;

% Q for theta_cg_ddotQ1 = zeros(8,8);

Q1(1,2) = (b_sf*k_sf*l_f^2 + b_sr*k_sr*l_r^2 - b_sr*k_sf*l_f*l_r -... b_sf*k_sr*l_f*l_r)/I^2;Q1(1,3) = (-k_sf^2*l_f^3 + k_sr^2*l_r^3 - k_sf*k_sr*l_f*l_r^2 +... k_sf*k_sr*l_f^2*l_r)/I^2;Q1(1,4) = (-b_sf*k_sf*l_f^3 + b_sf*k_sr*l_f^2*l_r + b_sr*k_sr*l_r^3 -... b_sr*k_sf*l_f*l_r^2)/I^2;Q1(1,5) = (-k_sf^2*l_f^2 + k_sf*k_sr*l_f*l_r)/I^2;Q1(1,6) = (-b_sf*k_sf*l_f^2 + b_sf*k_sr*l_f*l_r)/I^2;Q1(1,7) = (-k_sr^2*l_r^2 + k_sf*k_sr*l_f*l_r)/I^2;Q1(1,8) = (b_sr*k_sf*l_f*l_r - b_sr*k_sr*l_r^2)/I^2;

Q1(2,3) = (-b_sf*k_sf*l_f^3 + b_sr*k_sf*l_f^2*l_r - b_sf*k_sr*l_f*l_r^2 +... b_sr*k_sr*l_r^3)/I^2;Q1(2,4) = (-b_sf^2*l_f^3 + b_sr^2*l_r^3 - b_sf*b_sr*l_f*l_r^2 +... b_sf*b_sr*l_f^2*l_r)/I^2;Q1(2,5) = (-b_sf*k_sf*l_f^2 + b_sr*k_sf*l_f*l_r)/I^2;Q1(2,6) = (-b_sf^2*l_f^2 + b_sf*b_sr*l_f*l_r)/I^2;Q1(2,7) = (b_sf*k_sr*l_f*l_r - b_sr*k_sr*l_r^2)/I^2;Q1(2,8) = (-b_sr^2*l_r^2 + b_sf*b_sr*l_f*l_r)/I^2;

Q1(3,4) = (b_sf*k_sf*l_f^4 + b_sr*k_sr*l_r^4 + b_sf*k_sr*l_f^2*l_r^2 +... b_sr*k_sf*l_f^2*l_r^2)/I^2;Q1(3,5) = (k_sf^2*l_f^3 + k_sf*k_sr*l_f*l_r^2)/I^2;Q1(3,6) = (b_sf*k_sf*l_f^3 + b_sf*k_sr*l_f*l_r^2)/I^2;Q1(3,7) = (-k_sr^2*l_r^3 - k_sf*k_sr*l_f^2*l_r)/I^2;Q1(3,8) = (-b_sr*k_sf*l_f^2*l_r - b_sr*k_sr*l_r^3)/I^2;

Q1(4,5) = (b_sf*k_sf*l_f^3 + b_sr*k_sf*l_f*l_r^2)/I^2;

4

Q1(4,6) = (b_sf^2*l_f^3 + b_sf*b_sr*l_f*l_r^2)/I^2;Q1(4,7) = (-b_sf*k_sr*l_f^2*l_r - b_sr*k_sr*l_r^3)/I^2;Q1(4,8) = (-b_sr^2*l_r^3 - b_sf*b_sr*l_f^2*l_r)/I^2;

Q1(5,6) = b_sf*k_sf*l_f^2/I^2;Q1(5,7) = -k_sf*k_sr*l_f*l_r/I^2;Q1(5,8) = -b_sr*k_sf*l_f*l_r/I^2;

Q1(6,7) = -b_sf*k_sr*l_f*l_r/I^2;Q1(6,8) = -b_sf*b_sr*l_f*l_r/I^2;

Q1(7,8) = b_sr*k_sr*l_r^2/I^2;

Q1(1,1) = (k_sf^2*l_f^2 + k_sr^2*l_r^2 - 2*k_sf*k_sr*l_f*l_r)/I^2;Q1(2,2) = (b_sf^2*l_f^2 + b_sr^2*l_r^2 - 2*b_sf*b_sr*l_f*l_r)/I^2;Q1(3,3) = (k_sf^2*l_f^4 + k_sr^2*l_r^4 + 2*k_sf*k_sr*l_f^2*l_r^2)/I^2;Q1(4,4) = (b_sf^2*l_f^4 + b_sr^2*l_r^4 + 2*b_sf*b_sr*l_f^2*l_r^2)/I^2;Q1(5,5) = k_sf^2*l_f^2/I^2;Q1(6,6) = b_sf^2*l_f^2/I^2;Q1(7,7) = k_sr^2*l_r^2/I^2;Q1(8,8) = b_sr^2*l_r^2/I^2;

% Q R N for front suspension deflectionQ2 = zeros(8,8);Q4 = zeros(8,8);

Q2(1,1) = 1;Q2(1,3) = -l_f;Q2(1,5) = -1;Q2(3,3) = l_f^2;Q2(3,5) = l_f;Q2(5,5) = 1;

Q4(4,4) = l_f^2;Q4(2,4) = -l_f;Q4(2,2) = 1;

% Q R N for rear suspension deflectionQ3 = zeros(8,8);Q5 = zeros(8,8);

Q3(1,1) = 1;Q3(1,3) = l_r;Q3(1,7) = -1;Q3(3,3) = l_r^2;Q3(3,7) = -l_r;Q3(7,7) = 1;

Q5(4,4) = l_r^2;Q5(2,4) = l_r;Q5(2,2) = 1;

% Q R N for all costsR = rho_0*R0 + rho_1*R1;

5

N = rho_0*N0 + rho_1*N1;Q = rho_0*Q0 + rho_1*Q1 +... rho_2*Q2 + rho_3*Q3 + rho_4*Q4 + rho_5*Q5;

for i = 2:8 for j = 1:7 if i > j Q(i,j) = Q(j,i); end endend

% Additonal terms from rhoR(1,1) = R(1,1) + rho_10;R(2,2) = R(2,2) + rho_11;Q(5,5) = Q(5,5) + rho_6;Q(6,6) = Q(6,6) + rho_8;Q(7,7) = Q(7,7) + rho_7;Q(8,8) = Q(8,8) + rho_9;

end

Published with MATLAB® 7.12