## A Backstepping Control Framework for m-Triangular Systems

 Title A Backstepping Control Framework for m-Triangular Systems Publication Type Report Year of Publication 2011 Authors Kottenstette, N., H. LeBlanc, E. Eyisi, and J. Porter Refereed Designation Does Not Apply Pagination 1-13 Date Published 04/28/2011 Institution Institute for Software Integrated Systems City Nashville Report Number ISIS-11-104 Abstract $m$-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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