A Backstepping Control Framework for m-Triangular Systems

TitleA Backstepping Control Framework for m-Triangular Systems
Publication TypeReport
Year of Publication2011
AuthorsKottenstette, N., H. LeBlanc, E. Eyisi, and J. Porter
Refereed DesignationDoes Not Apply
Date Published04/28/2011
InstitutionInstitute for Software Integrated Systems
Report NumberISIS-11-104

$m$-Triangular Systems are dynamical physical systems which can be
described by $m$ triangular subsystem models. Many physical system
models such as those which describe fixed-wing and quadrotor aircraft
can be realized as $m$-Triangular Systems. However, many control engineers
try to fit their dynamical model into a $1$-Triangular System model.
This is commonly seen in the backstepping control community in
which they have developed pioneering adaptive control laws which can
explicitly account for operating state constraints. We shall
demonstrate that such control laws can even be implemented in a
non-adaptive form while still addressing actuator limitations such as
saturation. However, most importantly, by removing the adaptation
component, a {\em strictly output passive} input-output mapping can be
realized. This important property is most applicable to the
networked control community. For the networked control community,
this {\em key property} allows us to integrate an aircraft into our framework such that a {\em discrete-time
lag compensator} can be used by a ground control station for remote
navigation in a {\em safe and stable manner in spite of time-varying delays and random data loss}. The applicability of our result shall
be made clear as we demonstrate how an inertial navigation system for
a quadrotor aircraft can be constructed. Specifically: i) the desired inertial
position ($\zeta_s=[\zeta_{Ns},\zeta_{Es},\zeta_{Ds}]\tr$) and yaw ($\psi_s$)
setpoints can be concatenated to consist of the {\em virtual} desired
setpoint ($\bar{u}=[\zeta_s \tr, \psi_s]\tr$); ii) the {\em virtual}
desired setpoint corresponds to the $m=3$-concatenated state outputs
$\bar{x}=[x_{(1,1)}\tr,x_{(2,1)}\tr,x_{(3,1)}\tr]\tr =
[[\zeta_{N},\zeta_{E}],\zeta_{D},\psi]\tr$; which iii) are augmented
such that the output $\bar{v}$ equals $\bar{x}$ at
steady-state operation; iv) using Lemma~\ref{L:sop_bstep} we can show
that the backstepping framework renders the quadrotor aircraft to be
strictly output passive (sop) ($\dot{V}(v) \leq -\epsilon_b \bar{v}\tr \bar{v} + \bar{v}\tr
\bar{u}$) such that $V(v)=\frac{1}{2}v\tr v$ is a Lyapunov function in
terms of all concatenated system states $v$ associated with the
$m$-Triangular System. Lemma~\ref{L:PassiveClosedLoop} then shows how the resulting
continuous-time strictly output passive system involving the quadrotor
aircraft can be integrated into an advanced digital control framework
such that a strictly output passive {\em discrete-time lag}
compensator can be used to control the inertial position from a
ground-station in an $L^m_2$-stable manner such that time-delays and
data loss will not cause instabilities.

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