Page 236 - Modern Control of DC-Based Power Systems

200 Modern Control of DC-Based Power Systems

extension for nonlinear plants were developed. We will only consider lin-
ear control loops, as these systems allow us to determine the essential
basics of sliding state control. In certain cases, the design of sliding state
controllers for nonlinear control loops is not more difficult than that for
linear control loops. This applies, e.g., to input linear controlled systems
according to the methodology described in Section 5.1.

5.8.1 Design Approach
The SM control can be divided into three stages:
1. The arrival phase, in which the trajectories converge to the switching
surface or switching plane and reach them in finite time.
2. The sliding state, in which the trajectory slides on the manifold into
the resting position.
3. The equilibrium point x e 5 0, in which the system remains stable.
For a globally stable equilibrium point x e 5 0, it must be ensured that
all trajectories of the system go to the switching plane in finite time and
then to the equilibrium point x e 5 0, in the sliding state. We consider lin-
ear control loops
_ x 5 Ax 1 bu (5.229)

with the switching surface
s xðÞ 5 0 (5.230)

and the control law

u 1 xðÞ; for s xðÞ . 0
uxðÞ 5 (5.231)
u 2 xðÞ; for s xðÞ . 0
For a nonlinear control loop the linearization methodology presented
in Section 5.1.1 can be used, and afterwards the SM control is applied on
this linearized system. The manipulated variable u is not defined here for
sxðÞ 5 0. In most cases, the following switching function is used:
T
s xðÞ 5 r x (5.232)
T
Which corresponds to a switching surface of r x 5 0. First of all, the
reachability of the switching surface for all trajectories of the state-space
has to be ensured. A necessary condition for this is that the trajectories of
the control loop run from both sides towards the switching surface, as it
must be