Page 238 - Modern Control of DC-Based Power Systems

202 Modern Control of DC-Based Power Systems

5.8.2 Dynamics in the Sliding State
When the trajectories x(t) reach the switching line and the sliding state
begins, the following question arises: Which dynamics does the control
loop have during the sliding state? The problem with this question is the
discontinuity of the differential equation

u 1 xðÞ; for s xðÞ . 0
_ x 5 Ax 1 bu; uxðÞ 5 (5.241)
u 2 xðÞ; for s xðÞ . 0
of the closed loop on the switching surface.
T
s xðÞ 5 r x 5 0 (5.242)
The differential equation is obviously not defined on the sliding sur-
face, i.e., the existence and uniqueness of its solution is not guaranteed
there. There are several methods to solve this problem [75], such as
Filippov’s method [76].
We determine the dynamics of the system in the sliding state accord-
ing to the following frequently used method. The controlled system is
transformed into the control standard form

_ x 1 5 x 2
^
(5.243)
_ x n21 5 x n
_ x n 52 a 0 x 1 2 a 1 x 2 2 ? 2 r n21 x n21
Furthermore, if solving Eq. (5.244) for x n one obtains Eq. (5.245)

T
r x 5 r 1 x 1 1 ? 1 r n21 x n21 1 r n x n 5 0 (5.244)
(5.245)
x n 52 r 1 x 1 2 ? 2 r n21 x n21
Without further restriction to the generality the coefficient r n was
selected such that r n 5 1. Now we substitute in the controllable canonical
form of the plant (5.243) the state variable x n with Eq. (5.245). We obtain
now the differential equation system (5.246).

_ x 1 5 x 2
^
(5.246)
_ x n22 5 x n21

_ x n21 5 r 1 x 1 2 ? 2 r n21 x n21