Duality and eventually periodic systems

Seiler

2017

Summary The paper investigates the robust stability and performance of uncertain linear time‐varying (LTV) systems using an integral quadratic constraint (IQC) based analysis approach. Specifically, previous theoretical work on IQC‐based robustness analysis of linear time‐invariant (LTI) systems is extended to discrete‐time LTV systems. In the case of a general LTV nominal system, the analysis solution is provided in terms of an infinite‐dimensional convex optimization problem. This optimization problem reduces into a finite‐dimensional semidefinite program when the nominal system in question is finite horizon, periodic, or, more generally, eventually periodic. Finally, the results are applied to an unmanned aircraft control system executing an aggressive maneuver, where the developed techniques are used to find the region in which the aircraft is guaranteed to reside at the end of its planned trajectory. Copyright © 2017 John Wiley & Sons, Ltd.

Section: Introductionmentioning

confidence: 99%

IQC‐based robustness analysis of discrete‐time linear time‐varying systems

Fry

Seiler

2017

“…Eventually time‐invariant LPV models are NSLPV models such that the explicit time variation in the state‐space realization is eventually time‐invariant, ie, aperiodic for an initial amount of time and then time‐invariant afterwards. () The definition of the isomorphic system can be further extended to cover all negative time instants by simply setting the state‐space matrix‐valued functions for t <−1 equal to zero. The resulting system is then defined for all integers

t \in false(- \infty, \infty false)

with A ( k ) ( t ,δ ( k ) ( t ))=0 for t <0 (in the isomorphic system, we have A ( k ) (−1, · )=0).…”

Section: Preliminariesmentioning

confidence: 99%

Distributed control of LPV systems over arbitrary graphs with communication latency

2017

Summary This paper focuses on spatially distributed control systems where the controller sensing and actuation topology is inherited from that of the plant. Specifically, this paper considers distributed systems composed of discrete‐time linear parameter‐varying subsystems interconnected over general graph structures. These distributed systems are subject to a communication latency of one sampling period, where the information sent by a subsystem at the current time step is received by the target subsystem at the next time step. Analysis and synthesis conditions are derived for control design in this setting using a parameter‐dependent Lyapunov approach with the ℓ2‐induced norm as the performance measure. Fast and easy‐to‐implement algorithms are also provided for constructing the controller in real time. The techniques developed are finally applied to a leader‐follower formation problem with intermittent communications.

“…We show that, for such systems, the synthesis conditions in Corollary reduce to a finite‐dimensional semidefinite programming problem. An eventually periodic system arises when linearizing the nonlinear system equations about an eventually periodic trajectory. Such a trajectory can be arbitrary for an initial amount of time but then settles into a periodic orbit.…”

Section: Finite‐dimensional Synthesis Conditionsmentioning

confidence: 99%

“…Clearly, a γ ‐admissible synthesis for G is a 1‐admissible synthesis for

\overset{G}{true}

, where

\overset{G}{true}

has the same realization as G except that

{\overset{C}{true}}_{1 i} (t MathClass-punc, k) MathClass-rel= \frac{1}{γ} C_{1 i} (t MathClass-punc, k)

{\overset{D}{true}}_{11} (t MathClass-punc, k) MathClass-rel= \frac{1}{γ} D_{11} (t MathClass-punc, k)

, and

{\overset{D}{true}}_{12} (t MathClass-punc, k) MathClass-rel= \frac{1}{γ} D_{12} (t MathClass-punc, k)

for all

t MathClass-rel\in {double-struckN}_{0}

, k = 1,2,3,4, and i = 0,1, … , s . Furthermore, by employing the Schur complement formula, we can reformulate the synthesis conditions to be linear in γ 2 as in , Problem (16). With this said, let γ min denote the minimum γ , up to a certain tolerance, that is achievable by a ( 45,120)‐eventually periodic synthesis.…”

Section: Example: Distributed Control Of Hovercraftsmentioning

confidence: 99%

“…Specifically, the results in this special case are provided in terms of finite‐dimensional semidefinite programs. The concept of an eventually periodic system is introduced for the purely time‐varying single system case in . These systems are aperiodic for an initial amount of time and then become periodic afterwards; they contain LTI, finite horizon, and periodic systems as special cases.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Distributed control of linear time‐varying systems interconnected over arbitrary graphs

Dullerud

2013

SUMMARYIn this paper, we focus on designing distributed controllers for interconnected systems in situations where the controller sensing and actuation topology is inherited from that of the plant. The distributed systems considered are composed of discrete-time linear time-varying subsystems interconnected over arbitrary graph structures. The main contribution of this paper is to provide results on general graph interconnection structures in which the graphs have potentially an infinite number of vertices. This is accomplished by first extending previous machinery developed for systems with spatial dynamics on the lattice Z n . We derive convex analysis and synthesis conditions for design in this setting. These conditions reduce to finite sequences of LMIs in the case of eventually periodic subsystems interconnected over finite graphs. The paper also provides results on distributed systems with communication latency and gives an illustrative example on the distributed control of hovercrafts along eventually periodic trajectories. The methodology developed here provides a unifying viewpoint for our previous and related work on distributed control.