Robust Fault Tolerant Control Framework Using Uncertain Takagi-Sugeno Fuzzy Models - Fuzzy Modeling and Control: Theory and Applications - page 120

m

= F L + F drag , R

¨ z F R − F drag , L −

θ = F R −

F drag , R r

(6.23)

1

2 mr 2

F L + F drag , L −

where:

F drag , L = k drag v L | v L | F drag , R = k drag v R | v R |

(6.24)

r θ r θ v L =˙ z − v R =˙ z +

with k drag = 11 kg/m being the drag coefficient in the normal operating conditions,

z the total covered distance,

θ

the angular
velocity. The system can be controlled using the available control inputs F L and F R ,
that are the forces acting on the left and the right wheel, which have velocities v L
and v r , respectively.
By considering the state vector x

z the linear velocity, ˙ θ the yaw angle and

= F L F R T ,
and embedding the nonlinearities in the parameters, ( 6.23 ) can be put in the following
form:
⎛
⎝

= z θ θ T , the input vector u

˙

z

⎞
⎠ =

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠ +

x 1 ( ˙ t ) 0 1 0 0 x 1 ( t )

B u 1 (

x 2 ( ˙ t ) 0 a 22 ( x 2 , x 4 ) 0 a 24 ( x 2 , x 4 ) x 2 ( t ) t )

(6.25)

x 3 ( ˙ t ) 0 0 0 1 x 3 ( t ) u 2 ( t )

x 4 ( ˙ t ) 0 a 42 ( x 2 , x 4 ) 0 a 44 ( x 2 , x 4 ) x 4 ( t )

with:

01 T

/ m 0 − 2 /( mr )

B = (6.26)

01 / m 02 /( mr )

⎧
⎨

− 2 k drag x 2 / m 2 ≥ | rx 4 |

− 2 rk drag x 4 / m − rx 4 < x 2 < rx 4

a 22 ( x 2 , x 4 ) = (6.27)

2 rk drag x 4 / mrx 4 < x 2 < − rx 4

⎩

2 k drag x 2 / mx 2 ≤ | rx 4 |

⎧
⎨

2 k drag r 2 x 4 /

− m 2 ≥ | rx 4 |

− 2 rk drag x 2 / m − rx 4 < x 2 < rx 4

a 24 ( x 2 , x 4 ) = (6.28)

2 rk drag x 2 / mrx 4 < x 2 < − rx 4

⎩

2 k drag r 2 x 4 / m 2 ≤ | rx 4 |

⎧
⎨

− 4 k drag x 4 / mx 2 ≥ | rx 4 |

− 4 k drag x 2 /( mr ) − rx 4 < x 2 < rx 4

a 42 ( x 2 , x 4 ) = (6.29)

4 k drag x 2 /( mr ) rx 4 < x 2 < − rx 4

⎩

4 k drag x 4 / mx 2 ≤ | rx 4 |

⎧
⎨

− 4 k drag x 2 / mx 2 ≥ | rx 4 |

− 4 rk drag x 4 / m − rx 4 < x 2 < rx 4

a 44 ( x 2 , x 4 ) = (6.30)

⎩ 4 rk drag x 4 / mrx 4 < x 2 < − rx 4

4 k drag x 2 / mx 2 ≤ | rx 4 |