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Nonlinear Tools: Control and untrained
Neural-Network
Case Study
Tuesday, October 31, 2017
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System Structure
The general dynamic system equations describing the vehicle motion are given by:
𝑣 𝑐𝑜𝑔
𝛽
𝜓
𝑞
=
𝑓1𝐴 𝑞, 𝛿,𝑚𝑐𝑜𝑔, 𝐹𝑍𝑖 , 𝑐𝑎𝑒𝑟𝑜
𝑓2𝐴 𝑞, 𝛿,𝑚𝑐𝑜𝑔, 𝐹𝑍𝑖 , 𝑐𝑎𝑒𝑟𝑜
𝑓3𝐴 𝑞, 𝛿, 𝐽𝑍, 𝑙𝐹/𝑅, 𝑏𝐹/𝑅, 𝐹𝑍𝑖𝐴(𝑞)
𝜇𝑆𝐹𝐿𝜇𝑆𝐹𝑅𝜇𝑆𝑅𝐿𝜇𝑆𝑅𝑅𝜇𝑆
+
𝑓1𝐵 𝑞, 𝛿,𝑚𝑐𝑜𝑔
𝑓2𝐵 𝑞, 𝛿,𝑚𝑐𝑜𝑔
𝑓3𝐵 𝑞, 𝛿, 𝐽𝑍, 𝑙𝐹/𝑅, 𝑏𝐹/𝑅𝐵(𝑞)
𝑢1𝑢2𝑢3𝑢4 𝑢
Where
𝑞 represents the vehicle state vector
𝐴(𝑞) represents the system matrix
𝜇𝑆 represents the lateral adherence coefficient vector
𝐵 𝑞 represents the control input matrix
𝑢 represents the control input vector
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Control Design: EMK and NN based
For the system to asymptotically reduce the control error
𝑒1 = 𝑞𝑑 − 𝑞
between the desired and actual state, the following control designs 𝑢 were adopted:
With exact model knowledge (EMK) assumption:
𝑢 = 𝐵 𝑞 −1 𝑞𝑑 − 𝐴 𝑞 𝜇𝑆 + 𝐾1𝑒1
Without any knowledge about the lateral adherence coefficient 𝜇𝑆 and replacing the
unknown coefficient dynamics by an untrained neural network approximation term 𝜇 𝑆:
𝑢 = 𝐵 𝑞 −1 𝑞𝑑 − 𝐴 𝑞 𝜇 𝑆 + 𝐾1𝑒1
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Simulation Results: Control Task
The controller was assigned with the simple task to follow a desired increasing vehicle
speed while a sinusoidal steering angle on the steering wheel was imposed.
0 5 10 15 20 250
2
4
6
8
10
12
14
time[s]
Vehic
le s
peed [m
/s]
Desired vehicle speed
Desired vehicle speed
0 5 10 15 20 25-60
-40
-20
0
20
40
60
80
time[s]
Ste
ering w
heel angle
[°]
Steering wheel angle
Steering wheel angle
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Simulation Results: Error and NN approx.
Despite the usage of an untrained neural network to approximate the lateral adhesion
coefficient dynamics, the error converges…
0 1 2 3 4 5 6 7 8 9 10-2
0
2
4
6
8
10
12
14x 10
-4
time[s]
Vehic
le s
peed e
rror
[m/s
]
Comparison velocity error with exact model knowledge vs. error using NeuralNetwork
error EMK
error_NN
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Simulation Results: Error and NN approx.
….while the coefficient behavior is estimated online:
0 5 10 15 20 25-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time[s]
La
tera
l ad
hesio
n c
oeff
icie
nt
Comparison acutal side grip vs. side grip represnetation NeuralNetwork
muS FL
muS_NN FL
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Simulation Results: Discussion
Even without the full knowledge of the system dynamics, it has been shown that it is
possible to reach a similar error reduction as with a controller that is able to
theoretically guarantee the best control performance possible (EMK).
In this specific case the a priori untrained neural network doesn’t just simulate the
behavior of the unknown lateral tire dynamics, but delivers also a good approximation
of it. Its precision increases with higher available control authority.
This simple simulation and proof of concept shows that the problematic of nonlinear
system behavior and signal estimation, can indeed be attacked using nonlinear control
tools.
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Team SIGICONTROL
Michele Sigilló Founder, Technical Solutions
Livia Esposito
Fabio Esposito Legal Advisor
Website: www.sigicontrol.com Email: [email protected]
Alberta Graziani Financial Advisor
Jarmila Muzykova Business Development
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“The ones who follow never come in first.” Michelangelo Buonarotti
