4.3 LQ based control
5.1.2 Design model
Design model is the nonlinear model that is linearized in operating point, and used for the feedback control design. Only two states ωf and ωrare considered in the design model. The vehicle model dynamics is then transformed to the following equation ˙ ωi = 1 I + ratioi m (1 ± λi) r2i · (τi− ratioi Fdr) , (5.4)
where I is inertia of the wheel, ratioi is ratio of the mass which is driven
by the i-th wheel, m is whole mass of the vehicle, lambdai is i-th slip ratio
demand, ri is diameter of the i-th wheel, τi is i-th moment applied to the i-th wheel (this is input to the system and comes from the engine) and Fd is
aerodynamic drag.
The part of this equation where the inertia of wheel is extended by the mass of the vehicle comes from well know equations
F = m a , (5.5) M = F r , (5.6) v = (1 ± λ) ω r =⇒ a = (1 ± λ) ˙ω r (5.7) F = m (1 ± λ) ˙ω r (5.8) M = m (1 ± λ) ˙ω r2 (5.9) M = I ˙ω =⇒ I = m (1 ± λ) r2 . (5.10)
5. Acceleration/Longitudinal Slip Ratio Control Using Wheel Angular Velocity Tracking with LQ Techniques
The equation (5.7) is valid under assumption that λ is constant. The equation for Fdwas already mentioned in equation (3.24), where the velocity is replaced using equation (5.7). So the whole equation is
ratioi Fd= ratioi
1
2ρair A cD ((1 ± λi) ωi ri)
2 . (5.11)
All other physical phenomena are assessed as disturbances. These are mainly:
.
Pacejka magic fromula coefficients - different types of tires, riding sur- faces,....
Disturbances coming from the nature of the operation:.
Normal force disturbances - wheel goes through hole.
Disturbances in control action (torque) - vehicle goes to the hill.
Lateral dynamics effect on longitudinal motion (mainly the effect of the friction ellipse as described in [EHH19] and in Section 3.2.3).5.1.3 ω controller
Reference coming to ω controller will be from it’s nature (see Section 5.1.1) hyperbolic. Provided this fact the controller contains two integrators in order to be able of the reference tracking with zero steady state error. The controller is designed to use only the τf and τr. This is because only the acceleration is
in the scope of this work.
Controller was developed using LQ techniques. The LQ gain was computed for augmented system as two integrators were added. It’s state space matrices
Aaug and Baug are
Aaug = A 0 0 C 0 0 0 I 0 , Baug = B D 0 , (5.12)
where the A, B and C are state space matrices of the linerized design model (described in Section 5.1.2).
ωi which are regulated (brought to zero) are in fact differences of the ωi
...
5.1. Control architecture5.1.4 Controller summary
Alternative (let us say "robust") approach compared to the controller proposed in the Chapter 4 is presented in this chapter. The "robust" means the ω tracking controller is able to follow ωRef coming form the ω reference generator. The behavior was explored experimentally in simulation environment (see the Chapter 6). Both of these controllers are solving the same problem. The biggest advantage of this controller is its invariance to vehicle velocity v. It is also much simpler which brings many advantages like better implementation, tuning, error proneness and so on.
Chapter
6
Results
Longitudinal slip ratio/acceleration controllers were introduced in previous chapters. In this chapter their verification and comparison of them will be shown. Also virtual riding tests made using vehicle dynamics blockset toolbox in Simulink will be shown here.
6.1
Gain scheduling controller
In this subsection results using the controller from Chapter 4 are presented. The designed controller was tested with two different scenarios. The testing of the controller was done using the adopted non-linear model (see Section 3.2.5). The first test case was to slightly modify the Pacejka model in order to test controller robustness. The time response of the controller can be seen on the figure Fig. 6.3 for the front wheel and on the figure Fig. 6.4 for the rear wheel. On these figures are also for better insight the Pacejka models of the non-linear model against which the controller was tested. Here, on the Pacejka curves, are also depicted the operating regions of the system. Three cases were considered. Pacejka model for which the system was designed, then the same one but with the D coefficient 1.3 times bigger and the last one was also the same as the for which the system was linearized but with 70% of the original D coefficient. The second test case was inputs’ disturbance rejection. The rejection of disturbance can be seen on the figure Fig. 6.1 for the front wheel and on the figure Fig. 6.2 for the rear
6. Results
...
0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.15 0.2 0.25 0.3 -100 -50 0 50 100Disturbance of input Torque
Reference
Linearized model Pacejka D coeff 70 % of Linearized model Pacejka D 130 % of Linearized model Pacejka D Front wheel torque disturbance
Figure 6.1: Response of the designed controller to disturbance. On this figure
is slip ratio of front wheel.
0 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.15 0.2 0.25 0.3 -100 -50 0 50 100
Disturbance of input Torque
Reference
Linearized model Pacejka D coeff 70 % of Linearized model Pacejka D 130 % of Linearized model Pacejka D Rear wheel torque disturbance
Figure 6.2: Response of the designed controller to disturbance. On this figure
is slip ratio of rear wheel.
wheel. The disturbance is realized as addition of disturbance signal to the applied torque from controller. The same three Pacejka model configuration which were used in the first test scenario were used.
...
6.2. ω tracking based control system 0 2000 4000 6000 8000 10000 12000 14000Linearized model Pacejka D coeff 70 % of Linearized model Pacejka D 130 % of Linearized model Pacejka D Simulation data 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Reference
Linearized model Pacejka D coeff 70 % of Linearized model Pacejka D 130 % of Linearized model Pacejka D
Figure 6.3: Comparison of the designed regulator for different Pacejka model
of tires/road. On this figure is slip ratio of front wheel. On the left part of the figure are different Pacejka models. The simulation data points shows the operating range of the slip ratio respectively longitudinal force. On the right part of the figure the corresponding time response is shown.
0 0.5 1 1.5 2 2.5 104
Linearized model Pacejka D coeff 70 % of Linearized model Pacejka D 130 % of Linearized model Pacejka D Simulation data 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Reference
Linearized model Pacejka D coeff 70 % of Linearized model Pacejka D 130 % of Linearized model Pacejka D
Figure 6.4: Comparison of the designed regulator for different Pacejka model of
tires/road. On this figure is slip ratio of rear wheel. On the left part of the figure are different Pacejka models. The simulation data points shows the operating range of the slip ratio respectively longitudinal force. On the right part of the figure the corresponding time response is shown.
6.2
ω tracking based control system
Results made with the control system proposed in chapter Chapter 5 are presented in this section.
6. Results
...
The proposed control system architecture was verified using the modified single-track model described in Chapter 3. The front wheel control system performance and robustness properties can be seen in the figure Fig. 6.7a. The same situation but for the rear wheel can be seen in the figure Fig. 6.7b.
λ tracking details for front and rear wheel respectively can be seen in the
figures Fig. 6.5 and Fig. 6.6. These figures of details are comparable to figures Fig. 6.3 and Fig. 6.4 made using the gain scheduling based controller.
With the proposed controller the following was achieved
.
The traction control that has function of nowadays widely used ASR (anti slip regulation) is designed..
The controller is invariant to significant changes of Fz (Change from 10% to 120% of the nominal Fz0 is shown in the figures Fig. 6.7a and Fig.6.7b). That means the controller is invariant to:
.
change of vehicle mass.
different Fzon front and rear wheel which means invariant to center of gravity location..
cornering maneuvers with acceleration and generally it is usable with nonzero value of α.Results are more in detail inspected in the following subsections.
6.2.1 Front wheel
The most important variables regarding the front wheel during the vehicle acceleration can be seen in the figure Fig. 6.7a. If it is not said else this section will describe this figure. The vertical red dashed line, that is present in all graphs, is the moment where engine reaches its power limit and is no more capable of λ tracking.
The acceleration reference aRef and its real value a can be seen in the
second graph. The velocity v is also shown in this graph. Ideally the ramp of v wouldn’t be dependent on the Fzfratio, but in the time where Fzfratio
goes to 10% (this can be e.g. as the vehicle goes on ice) the maximal possible acceleration is not sufficient and the controller isn’t capable of aRef tracking. This is caused by the saturation in the λRef. λRef is limited to linear region
...
6.2. ω tracking based control systemof the Pacejka’s slip curve (more about this in [VHH19]). λRef is saturated
on ±0.17 in this thesis, where the maximal Fx is generated.
Fzfratio and ω tracking is shown in the first graph. There are three
remarkable parts. At the beginning the reference has hyperbolic character, after that it goes to ramp. After the red dashed line the power of the powertrain is on its limits and so the controller isn’t capable to track the
ωRef.
λ tracking is shown in the third graph. As was written above, when the
Fzfratio goes to 10% the λRef is saturated on its maximal value. Comparing
the figure Fig. 6.7a to the figure Fig. 6.7b the power limits are reached earlier in the latter one, approximately at 4.3s. This can be also seen in the third chart of Fig. 6.7a. The slip ratio demand λRef ramps up because the rear wheel isn’t anymore capable to generate sufficient acceleration force from that time. Till that moment the λRef is piece-wise constant.
In the last graph torque applied to the front wheel is shown. That is the actual control action used as input to the plant - vehicle.
6.2.2 Rear wheel
The most important variables regarding the rear wheel during the vehicle acceleration can be seen in the figure Fig. 6.7b.
Most of this figure content is nearly the same as for the front wheel (described in section Section 6.2.1). Taking this fact into account only the
differences will be described here.
The nominal normal force Fz0R that is almost twice as big as Fz0F makes the biggest difference. This is the reason of the red dashed line position. It is approximately at 4.3s for the rear wheel while for the front one approx. at time 5.3s.
6. Results
...
0 2000 4000 6000 8000 10000 12000 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 FzF = 100% FzF = 10% FzF = 120% Simulation data 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Reference FFigure 6.5: λf tracking results using the proposed control architecture. The
inability to track the reference in higher speed (later in the time) is caused by the power limitations on the wheels.
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 F zR = 100% F zR = 10% FzR = 120% Simulation data 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Reference R
Figure 6.6: λr tracking results using the proposed control architecture. The
inability to track the reference in higher speed (later in the time) is caused by the power limitations on the wheels.
6.3
Control systems comparison
Both proposed control systems were verified against the single-track model that was described in Chapter 3. Both of the control systems are capable of
...
6.4. Riding testsλ tracking and disturbance rejection. Compared to each other the ω tracking
based control system exhibits better performance results. It has also major advantage compared to the gain scheduling controller - It is vehicle velocity invariant.
Regarding these facts only the ω tracking based control system was chosen for virtual riding tests that are described in the following section.
6.4
Riding tests
The virtual Riding tests, which are presented in this section, are made using the Vehicle dynamics blockset (Toolbox in MATLAB/Simulink environment). As was mentioned in Section 6.3 the tests are made testing only the ω tracking based control system. There is a comparison between the variants with traction control and in open loop. The riding tests are made using university simulator that is shown in the figure Fig. 6.8.
Here is no comparison to the currently implemented ESP/ABS systems that also have sort of slip ratio control. This is because no Simulink based implementation was available.
The virtual riding test input is the driver computer interface (can be seen in the figure Fig. 6.8). The simulation of the vehicle is done using single-track model and visualization is made using the vehicle dynamics blockset.
6.4.1 Comparison of open loop and ω tracking based control system
To make also some simple nice comparable results the following scenario was set up. The driver interface was disconnected and the situation with full accelerator pedal command was simulated. That means highest aRef for ω tracking control system and full torques τf and τr for the open loop system.
The inputs (τf and τr) and wheel angular speed can be seen for both variants
in the figure Fig. 6.9. In the figures Fig. 6.11 and Fig. 6.12 can be seen slip ratios and Pacejka slip curve, that was slightly modified for this test (it corresponds to some slippery surface like ice). The resulting difference in vehicle velocity is depicted in the figure Fig. 6.10. Here it can be seen that
6. Results
...
the final velocity is approximately 33 m/s for the control system against the 13 m/s for the open loop system.
A video from this virtual ride is in attachment (details in Appendix C).
6.5
Conclusion
Vehicle traction control using LQ based controllers (control architectures) was proposed with two alternative approaches in this work. As the single-track model is nonlinear global stability results should be provided if possible. In this work the global stability was tested only experimentally, no rigorous proof was presented. The approach of this work brings many benefits that were described in Section 6.2. The single-track model is in some ways simplified model (contains only 2 wheels) and for implementation in real vehicle it should be (at least according to my opinion) tested against the twin-track model (derived in [Cib19]), but overall results looks promising.
Also virtual riding tests interface was implemented. This brings opportunity to test the proposed control solutions with real driver operating the platform (see picture Fig. 6.8).
...
6.5. Conclusion 0 1 2 3 4 5 6 0 500 1000 1500 2000 F 0 1 2 3 4 5 6 0 20 40 60 80 0 0.5 1 1.5 Front wheel Front wheel reference FzF ratio 0 1 2 3 4 5 6 -5 0 5 10 15 20 0 5 10 15 20 25 a a reference velocity 0 1 2 3 4 5 6 -0.05 0 0.05 0.1 0.15 0.2 F F reference(a) : This figures presents the most im-
portant variables for the front wheel dur-
ing the Fz force disturbance. Detailed
description can be found in section Sec- tion 6.2.1. 0 1 2 3 4 5 6 0 20 40 60 80 0 0.5 1 1.5 Rear wheel Rear wheel reference FzR ratio 0 1 2 3 4 5 6 -5 0 5 10 15 20 0 5 10 15 20 25 a a reference velocity 0 1 2 3 4 5 6 -0.05 0 0.05 0.1 0.15 0.2 R R reference 0 1 2 3 4 5 6 0 500 1000 1500 2000 R
(b) : This figures presents the most im-
portant variables for the rear wheel dur-
ing the Fz force disturbance. Detailed
description can be found in section Sec- tion 6.2.2.
6. Results
...
Figure 6.8: Platform used for virtual riding tests.
0 2 4 6 8 Time [s] 0 500 1000 1500 2000 f [Nm] 0 2 4 6 8 Time [s] 0 500 1000 1500 2000 r [Nm] 0 2 4 6 8 Time [s] 0 50 100 150 200 250 300 f [rad/s] 0 2 4 6 8 Time [s] 0 50 100 150 200 250 r [rad/s]
Figure 6.9: Comparison of inputs (τf and τr) and wheel angular velocities during
the test with open loop system and using ω tracking based control system. Left figures are for front wheel and right for the rear one.
...
6.5. Conclusion 0 1 2 3 4 5 6 7 8 Time [s] 0 5 10 15 20 25 30 35 velocity [m/s]Figure 6.10: Comparison of vehicle velocity during the test with open loop
system and using ω tracking based control system.
-500 0 500 1000 1500 2000 2500 30000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Slip curve Simulation data - ctrl system Simulation data - open loop
0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 F reference F - ctrl system F - open loop
Figure 6.11: Comparison of slip ratios of front wheel during the test with open
6. Results
...
-1000 0 1000 2000 3000 4000 50000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Slip curve Simulation data - ctrl system Simulation data - open loop0 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R reference R - ctrl system R - open loop
Figure 6.12: Comparison of slip ratios of rear wheel during the test with open
Appendix
A
Bibliography
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and yaw stabilization of 4ws cars, Automatica 30 (1994), no. 11,
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[Ada] Adams/Tire, Using the pac2002tire model.
[Cib19] Vit Cibulka, Mpc based control algorithms for vehicle control, 2019. [Efr18] Denis Efremov, Unstable ground vehicles and artificial stability
systems, 2018.
[EHH19] Denis Efremov, Tomas Hanis, and Martin Hromcik, Introduction
of driving envelope and full-time-full-authority control for vehi- cle stabilization systems, 2019 22nd International Conference on
Process Control (PC19), IEEE, jun 2019.
[GC03] Ryszard Gessing and Adam Czornik, About relation between lq and
h2 optimal control with output feedback, vol. 4, 07 2003, pp. 3508 –
3512 vol.4.
[GGK08] Bilin Aksun Guvenc, Levent Guvenc, and Sertaç Karaman, Robust
yaw stability controller design and hardware-in-the-loop testing for a road vehicle, IEEE Transactions on Vehicular Technology 58
(2008), no. 2, 555–571.
[Haf08] Lukas Haffner, Real-time tire models for lateral vehicle state esti-
mation, na, 2008.
[Pac02] Hans Pacejka, Tyre and vehicle dynamics, Elsevier LTD, Oxford, 2002.
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[SHB14] Dieter Schramm, Manfred Hiller, and Roberto Bardini, Vehicle
dynamics, Springer Berlin Heidelberg, 2014.
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namics control and controller allocation for rollover prevention, 11
2006, pp. 149 – 154.
[SS05] Ian Postlethwaite Sigurd Skogestad, Multivariable feedback control, John Wiley & Sons, 2005.
[VHH19] David Vosahlik, Tomas Hanis, and Martin Hromcik, Vehicle lon-
gitudinal dynamics control based on lq, 2019 22nd International
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Appendix
B
Parameters of used vehicle models
Vehicle body parameters
Name Value Unit Description
m 1190 kg mass of vehicle body
g 9.81 m/s−2 gravitational constant
Iz 1141 kg · m2 moment of inertia in z-axis
Rr 0.33 m rear wheel radius
Rf 0.33 m front wheel radius
Jr 1 kg · m2 rear wheel inertia
Jf 1 kg · m2 front wheel inertia
cD 0.33 aerodynamic drag coefficient
A 2 m2 aerodynamic reference area
Pmax 69 kW maximum engine power (for each engine)
τmax 2000 N m maximum engine torque (for each engine)
lf 1.1092 m front wheel distance from CG
lf 1.8908 m rear wheel distance from CG
Fz0f 4316.4 N front wheel nominal normal force
Fz0r 7357.5 N rear wheel nominal normal force
B. Parameters of used vehicle models
...
Tire parameters (Pacejka coefficients). Aggressive set used in Section 6.4.1, conservative everywhere else.
Conservative set Aggressive set B 0.15 0.4 C 2 2 D 1 1 E 0.95 0.1 Bx 4 3.5 Cx 1.4 3.1 Dx 2.5 2.5 Ex 0.1 0.95
Appendix
C
Attached files
Attached files on CD are described in the following table.
Path Description
Virtual riding test.mp4 Video from the virtual riding test (see Section 6.4.1). LPV Folder with MATLAB/Simulink implementation of
control system described in Chapter 4. LPV/functions These folders contains m-functions needed for the
LPV based model execution. LPV/linearization and trimming
LPV/LPV_init.mat Stored values that need to be loaded to the MATLAB workspace before LPV based control
simulation execution.
omegaTracking Folder with MATLAB/Simulink implementation of control system described in Chapter 5. omegaTracking/omegaTracking.mat Stored values that need to be loaded to the
MATLAB workspace before ω tracking based control simulation execution.