Wireless Ad Hoc and Sensor Networks

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Background 592.3.2 Lyapunov StabilityAn equilibrium point x e is stable in the sense of Lyapunov (SISL) at k 0 ,if for every ε > 0, there exists δε ( , x 0 ), such that | x0 − xe|
60 Wireless Ad Hoc and Sensor NetworksThe term ultimate indicates that the boundedness property holds after antime lapse N. If S =R , the system is said to be globally UUB (GUUB).2.3.4 A Note on Autonomous Systems and Linear SystemsIf the system is autonomous so thatxk ( + 1) = f( xk ( ))(2.28)where f( x( k))is not an explicit function of time, the state trajectory isindependent of the initial time. This means that if an equilibrium pointis stable by any of the three definitions, the stability is automaticallyuniform. Nonuniformity is only a problem with nonautonomous systems.If the system is linear so thatxk ( + 1) = Akxk ( ) ( )(2.29)with A(k) being an n×n matrix, then the only possible equilibrium pointis the origin.For linear time-invariant (LTI) systems, matrix A is time-invariant. Thenthe system poles are given by the roots of the characteristic equation∆( z) = | zI− A| =0,(2.30)where||⋅ is the matrix determinant, and z is the Z transform variable. ForLTI systems, AS corresponds to the requirement that all the system polesstay within the unit disc (i.e., none of them are allowed on the unit disc).SISL corresponds to marginal stability, that is, all the poles are within theunit disc, and those on the unit disc are not repeated.2.4 Nonlinear Stability Analysis and Controls DesignFor LTI systems, it is straightforward to investigate stability by examiningthe locations of the poles in the s-plane. However, for nonlinear or nonautonomous(e.g., time-varying) systems there are no direct techniques.The (direct) Lyapunov approach provides methods for studying thestability of nonlinear systems and shows how to design control systemsfor such complex nonlinear systems. For more information see Lewis,Abdallah, and Dawson (1993), which deals with robot manipulator control,
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18. Chaos in Automatic Control, edi
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CRC PressTaylor & Francis Group6000
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Table of Contents1 Background on Ne
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3 Congestion Control in ATM Network
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5.4.2 Distributed Power Controller
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7.3 Adaptive and Distributed Fair S
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9.2.2 Overview.....................
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the number of connections in each l
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y Maheswaran Thiagarajan, and tirel
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2 Wireless Ad Hoc and Sensor Networ
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Page 72 and 73: 2BackgroundIn this chapter, we prov
Page 74 and 75: Background 51u(k)x n (k + 1)x n (k)
Page 76 and 77: Background 53with x( 0)being the in
Page 78 and 79: Background 55with tr(.) the matrix
Page 80 and 81: Background 57nGiven a function ft (
Page 84 and 85: Background 61as well as Jagannathan
Page 86 and 87: Background 63The subsequent example
Page 88 and 89: Background 65Evaluating this along
Page 90 and 91: Background 67Now, select the feedba
Page 92 and 93: Background 69is SISL if and only if
Page 94 and 95: Background 71L 1 (x)L(x, t)L 0 (x)0
Page 96 and 97: Background 73for some R > 0 such th
Page 98 and 99: Background 75Evaluating the first d
Page 100: Background 77Section 2.4Problem 2.4
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4Admission Controller Design for Hi
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7Distributed Fair Scheduling in Wir
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Distributed Fair Scheduling in Wire
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8Optimized Energy and Delay-Based R
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9Predictive Congestion Control for
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Predictive Congestion Control for W
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10Adaptive and Probabilistic Power
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Adaptive and Probabilistic Power Co
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Wireless Ad Hoc and Sensor Networks

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