Page 202 - Modern Control of DC-Based Power Systems

166 Modern Control of DC-Based Power Systems

An approach was proposed in [60] where u is chosen for minimizing
the control effort necessary to satisfy:
(5.145)
_ V #2 WðxÞ
Which leads to:

_
V (5.146)
u50 5 V x xðÞfxðÞ ,2 WðxÞ
It was mentioned in the beginning of Section 5.5.1.1, that it can
sometimes be very difficult to find a Lyapunov function for a given sys-
tem since by now there is no general approach to this problem.

5.5.2 Procedure of Backstepping
As already mentioned Backstepping or more specifically the integrator
Backstepping technique provides a schematic way of finding Lyapunov
control functions for systems that can be expressed in strict feedback
form, which corresponds to a lower triangular system matrix. In the fol-
lowing, the integrator Backstepping is explained for an example which
was taken from [6] and [61].
A general structure for a system that fulfills these requirements is given
in eq. (5.147). For strict feedback form it is important that nonlinearities
f i and h i of the derivative of state x i are only dependent on the previous
states which are fed back.

ðÞUx 2
_ x 1 5 f 1 x 1 1 h 1 x 1
ðÞ
_ x 2 5 f 2 x 1 ; x 2 Þ 1 h 2 x 1 ; x 2 ÞUx 3 (5.147)
ð
ð
^
_ x k 5 f k x 1 ; x 2 ; .. . ; x 3 Þ 1 h k x 1 ; x 2 ; .. . ; x k ÞUu
ð
ð
Compared to the LSF (Section 5.1) which is suitable for a similar sub-
class of nonlinear systems, Backstepping is advantageous because nonli-
nearities, which might be useful to keep for the system dynamics, are not
canceled out. Therefore, one could avoid the use of big control inputs
which lead to a high energy consumption or might be even impossible to
generate. Furthermore, the LSF requires a detailed knowledge of the sys-
tem to be controlled, whereas Backstepping allows attaining robustness
against parameter variation to some extent.
The Backstepping approach will be now explained for a general sys-
tem including two integrators. This two integrator system has similarities