Page 170 - Modern Control of DC-Based Power Systems

134 Modern Control of DC-Based Power Systems

5.3 IMMERSION AND INVARIANCE CONTROL

The chapter introduces the Immersion and Invariance (I&I) control
method for the design of nonlinear controllers for nonlinear systems. The
method has been introduced by A. Astolfi and R. Ortega with the pur-
pose of combining the notions of system immersion and manifold invari-
ance [26] into a new method for the asymptotic stabilization of nonlinear
systems [26].
The I&I control realizes the immersion of the systems trajectories into
a low-order target-system with a desired dynamic. In its nonadaptive for-
mulation, the control theory does not require the definition of a
Lyapunov function, unlike many other methodologies, and, for its geo-
metrical interpretation as a manifold-based control, can be considered a
more general theory that includes the sliding mode [27] and synergetic
controllers [17].
The control method is based on the theorem, introduced in [28], that
contains the definition of the theory and describes the steps for defining
the stabilizing control output.
The adaptive formulation of the I&I, which is not part of this chapter,
has been described in [28] as a novel approach for the adaptive stabiliza-
tion of nonlinear system.
The chapter introduces the general theory of the nonlinear stabilizing
controller, describing the theorem and giving some interpretation to it.
Therefore, the application of the theory to a simple two-dimensional sys-
tem and to a buck converter in the presence of nonlinear CPL load are
performed.

5.3.1 The Immersion and Invariance Stabilization
The section defines the theory of the Immersion and Invariance control
for the stabilization of nonlinear systems towards an equilibrium point. In
the basic formulation, the control uses the state feedback for generating
the control output of a system in the following form:

_ x 5 fxðÞ 1 gðxÞu (5.65)
n
m
where xAR , uAR and fxðÞ; gxðÞ are two nonlinear functions. The role
of the controller is to define a control output u 5 cðxÞ that makes the
closed-loop system asymptotically stable.