Nonlinear Tools: Control and untrained · 2 © Nonlinear Tools: Control and untrained...

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2 ©

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

sigillo@sigicontrol.com

Livia Esposito

Fabio Esposito Legal Advisor

esposito@sigicontrol.com

Website: www.sigicontrol.com Email: info@sigicontrol.com

Alberta Graziani Financial Advisor

graziani@sigicontrol.com

Jarmila Muzykova Business Development

muzykova@sigicontrol.com

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“The ones who follow never come in first.” Michelangelo Buonarotti