Page 295 - Distributed model predictive control for plant-wide systems

High-Speed Train Control with Distributed Predictive Control 269

To represent the performance of a local subsystem of a coach, we define state variables of
a local subsystem by the velocity of the coach and the relative displacement of the adjacent
spring, as shown in (12.12):
[ ]
v
Z = i , i = 1, … , n − 1 (12.12)
i x
i
With the definition of the state variables, we can rewrite the longitudinal dynamic model by n
subsystems as follows:
[ ] [ ( ) ][ ]
̇ v 1 = − c + c v −k ∕m 1 v 1
1
21 r
11
̇ x 1 1 0 x 1
[ ][ ] [ ]
0 0 v 1∕m
+ 2 + 1 u 1 (12.13)
−1 0 x 0
2
[ ] [ ( ) ][ ] [ ][ ]
̇ v i = − c + c v −k ∕m i v i + 0 0 v i+1
2i r
i
1i
̇ x i 1 0 x i −1 0 x i+1
[ ][ ] [ ]
0 k ∕m i v i−1 1∕m i
i
+ + u i
0 0 x 0
i−1
i = 2, … , n − 1 (12.14)
̇ v =−(c + c v )× v + k ∕m × x + 1∕m × u (12.15)
n 1n 2n r n n−1 n n−1 n n
In the real train system, there are n coaches with n − 1 couplers. As the last coach, there is
only the velocity v of the last coach without the relative displacement x . As a result, there is
n
n
only one state variable in the local subsystem of the last coach shown in (12.15). To unite the
representing forms, we assume an augmented variable x in the last coach. All the parameters
n
related to the x are set up with zero as shown in (12.16).
n
[ ] [ ( ) ][ ]
̇ v − c + c v 0 v
n = 1n 2n r n
̇ x 0 0 x
n n
[ ][ ] [ ]
0 k n−1 ∕m n−1 v n−1 1∕m n
+ + u n (12.16)
0 0 x n−1 0
The subsystem can be rewritten as follows:
̇ z = A z + B u + A z
1 11 1 1 1 12 2
̇ z = A z + B u + A z + A z i = 2, … , n − 1 (12.17)
i ii i i i ii+1 i+1 ii−1 i−1
̇ z = A z + B u + A z
n nn n n n nn−1 n−1
where A is the matrix of the ith subsystem, which is not influenced by other subsystems,
ii
A ii + 1 is the matrix of the ith subsystem, which is influenced by the i+1th coach, and A ii − 1 is
the matrix of the ith subsystem, which is influenced by the i − 1th coach.
In Figure 12.5, we can see how subsystems communicate and work. To represent the system
better, we have to propose some definitions.