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## Stability and Robustness Conditions using Frequency Dependent Half Planes

### Citations

7687 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ... largest and smallest eigenvalue respectively. The field of values (also called the numerical range) of a matrix A ∈ C n×n is the set F (A) :={x ∗ Ax | x ∈ C n ,x ∗ x =1} . We will use the properties =-=[4]-=-: σ(A) ⊂ F (A), σ(AB −1 ) ⊂ F (A)/F (B) for 0 /∈ F (B), and Re (F (A)) = F ((A + A ∗ )/2) where A ∗ is the conjugate transpose of A. Negative feedback is always implicitly assumed if not stated otherw... |

804 | Consensus and cooperation in networked multi-agent systems
- Olfati-Saber, Fax, et al.
- 2007
(Show Context)
Citation Context ...hibits complex dynamics over a wide range of time scales, and has a bandwidth that is sensitive to parameter changes and is consequently not explicitly known a priori. For such scenarios we have to rely on methods that hide the “details”, at the price of somewhat conservative results. The dynamics of interconnections of even very simple dynamical systems tends to be very complex. They are inherently hard to analyze due to the decentralized structure and typically requires a fairly sophisticated machinery. Analysis and synthesis of such systems has lately attracted much interest; see [3], [8], [11], [13] and references therein. This line of work, however, does not explicitly address scalability. Here, scalability means that a class of interconnection structures can be shown to be stable, without requiring a new centralized analysis whenever an agent is added or removed, which subsequently alters the system equilibrium. Such scalability is required by many applications, such as data networks, flocking phenomena and power networks, but K. Jacobsson, krister@caltech.edu, is with the Division of Engineering and Applied Science, California Institute of Technology, Pasadena CA 91125, USA. L. ... |

369 | FAST TCP: motivation, architecture, algorithms, performance
- Wei, Jin, et al.
- 2006
(Show Context)
Citation Context ...rding to Wi(s) using the observed queuing delay as input signal, which is considered to reflect the amount of congestion in the network. The model is thus suitable for studying, for example, FAST TCP =-=[19]-=- and TCP Vegas [2]. The parameter c corresponds to the capacity of the link, xc models the amount of uncontrolled traffic (UDP traffic and short lived TCP flows), and xi and τi are the ith source’s eq... |

291 |
Nonlinear Systems, 3rd ed
- Khalil
- 2002
(Show Context)
Citation Context ...in ⊂ C ˜ ψmin ⊂ Cθ. Remark 6: It follows directly from the definition of the gain margin, that any scalar system satisfying Theorem 1 must have gain margin GM ≥ 1/α. Remark 7: By the circle criterion =-=[7]-=-, it is possible to derive sector conditions similar to (6) that guarantee stability when sector bounded static nonlinearities are present in the the loop. IV. EXAMPLES The results of Section III will... |

270 |
Dissipative dynamical systems part I: General theory
- Willems
- 1972
(Show Context)
Citation Context ... condition is transformed from finding infinitely many bounding half planes, into finding a suitable factorization of the loop gain. Our criterion is related to notions of dissipativity and passivity =-=[20]-=-, which have recently been applied to communication systems [21]. They have also long been applied to process control engineering, which also requires that closed loop stability be maintained in a mul... |

159 |
TCP Vegas: end-to-end congestion avoidance on a global Internet”,
- Brakmo, Peterson
- 1995
(Show Context)
Citation Context ...g the observed queuing delay as input signal, which is considered to reflect the amount of congestion in the network. The model is thus suitable for studying, for example, FAST TCP [19] and TCP Vegas =-=[2]-=-. The parameter c corresponds to the capacity of the link, xc models the amount of uncontrolled traffic (UDP traffic and short lived TCP flows), and xi and τi are the ith source’s equilibrium sending ... |

98 | On the stability of networks operating TCP-like congestion control,
- Vinnicombe
- 2002
(Show Context)
Citation Context ...et (−∞, −1]. Frequency dependent half plane conditions have previously been used in the context of interconnected systems [12], [15]; similarly, the convex hull conditions for stability in [9], [10], =-=[18]-=-, need only hold pointwise over frequency (cf. the conditions in [6]). Considering half planes per frequency reduces conservatism, at the price of greater complexity in the stability test. The approac... |

86 | Congestion control for high performance, stability and fairness in general networks
- Paganini, Wang, et al.
- 2005
(Show Context)
Citation Context ...y, Australia. A. Tang, atang@ece.cornell.edu, is with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA. has received relatively little attention [9], [10], =-=[12]-=-. The results in the present paper sprung from an analysis [16] of Internet congestion control: a distributed, highly heterogeneous and continually changing system and a domain where scalability is es... |

80 | A Unifying Passivity Framework for Network Flow Control.
- Wen, Arcak
- 2004
(Show Context)
Citation Context ...half planes, into finding a suitable factorization of the loop gain. Our criterion is related to notions of dissipativity and passivity [20], which have recently been applied to communication systems =-=[21]-=-. They have also long been applied to process control engineering, which also requires that closed loop stability be maintained in a multiloop control system when any subset of the control loops is de... |

55 | Distributed control design for systems interconnected over an arbitrary graph
- Langbort, Chandra, et al.
- 2004
(Show Context)
Citation Context ...erently hard to analyze due to the decentralized structure and typically requires a fairly sophisticated machinery. Analysis and synthesis of such systems has lately attracted much interest; see [3], =-=[8]-=-, [11], [13] and references therein. This line of work, however, does not explicitly address scalability. Here, scalability means that a class of interconnection structures can be shown to be stable, ... |

8 | Queue dynamics with window flow control
- Tang, Andrew, et al.
- 2010
(Show Context)
Citation Context ...,...,N along the imaginary axis, and with a factorization Let L(s) =Q(s)P −1 (s). Cθ = {z | Im ( ze −jθ) < 0} (i.e., an open complex half plane with angle θ and which touches the origin), and let α ∈ =-=[0, 1]-=-. A system with open loop transfer function L(s) is then closed loop stable if for all ω ≥ 0,ω ̸= ωi,i =1,...,N, there exists a θ ∈ R such that both F (P (jω)) ⊆ Cθ, (1) F (Q(jω)) + αF (P (jω)) ⊆ Cθ (... |

8 | ACK-clocking dynamics: Modeling the interaction between windows and the network. In
- Jacobsson, Andrew, et al.
- 2008
(Show Context)
Citation Context .../c xc/c + ∑ xi i c 1 1−exp(−sτi) ∑ i e−sτi Wi(s), −sτi 1 − e (14) where c > 0, xc ∈ [0,c], xi ∈ [0,c], τi > 0, and the conservation law ∑ xi + xc = c (15) i holds. This transfer function derives from =-=[5]-=- (see also [1]) where the general version of the model is shown to accurately model the equilibrium dynamics of a communication network where sources apply window based congestion control (such as TCP... |

8 |
Scalable Decentralized Robust Stability Certificates for Networks of Interconnected Heterogeneous Dynamical Systems.
- Lestas, Vinnicombe
- 2006
(Show Context)
Citation Context ...f Technology, Australia. A. Tang, atang@ece.cornell.edu, is with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA. has received relatively little attention =-=[9]-=-, [10], [12]. The results in the present paper sprung from an analysis [16] of Internet congestion control: a distributed, highly heterogeneous and continually changing system and a domain where scala... |

7 |
A Popov Criterion for Networked Systems.
- Jonsson, Kao, et al.
- 2007
(Show Context)
Citation Context ...y been used in the context of interconnected systems [12], [15]; similarly, the convex hull conditions for stability in [9], [10], [18], need only hold pointwise over frequency (cf. the conditions in =-=[6]-=-). Considering half planes per frequency reduces conservatism, at the price of greater complexity in the stability test. The approach taken here circumvents the frequency dependence in the half planes... |

7 |
Multivariable Feedback Control, 2nd edition.
- Skogestad, Postlethwaite
- 2005
(Show Context)
Citation Context ... (B), and Re (F (A)) = F ((A + A ∗ )/2) where A ∗ is the conjugate transpose of A. Negative feedback is always implicitly assumed if not stated otherwise. The gain and phase margins are defined as in =-=[14]-=-: The gain margin (GM) of a stable feedback system with return ratio L(s) is defined as GM = 1/|L(jω180)|, where the phase cross-over frequency ω 180 is where the Nyquist curve of L(jω) crosses the ne... |

7 | An Accurate Link Model and Its Application to Stability Analysis of FAST TCP.
- Tang, Jacobsson, et al.
- 2007
(Show Context)
Citation Context ...l of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA. has received relatively little attention [9], [10], [12]. The results in the present paper sprung from an analysis =-=[16]-=- of Internet congestion control: a distributed, highly heterogeneous and continually changing system and a domain where scalability is essential. A centralized analysis, which would require distributi... |

6 |
A Frequency Domain Condition for Stability of Interconnected MIMO Systems.
- Gattami, Murray
- 2004
(Show Context)
Citation Context ...e inherently hard to analyze due to the decentralized structure and typically requires a fairly sophisticated machinery. Analysis and synthesis of such systems has lately attracted much interest; see =-=[3]-=-, [8], [11], [13] and references therein. This line of work, however, does not explicitly address scalability. Here, scalability means that a class of interconnection structures can be shown to be sta... |

6 | Window Flow Control: Macroscopic Properties from Microscopic Factors.
- Tang, Andrew, et al.
- 2008
(Show Context)
Citation Context ... considered to reflect the amount of congestion in the network. The model is thus suitable for studying, for example, FAST TCP [19] and TCP Vegas [2]. The parameter c corresponds to the capacity of the link, xc models the amount of uncontrolled traffic (UDP traffic and short lived TCP flows), and xi and τi are the ith source’s equilibrium sending rate and experienced loop (round trip) delay respectively. The conservation law (15) corresponds to a fully-utilized link: at the equilibrium point, the traffic flowing into the link equals the capacity. We remark that results from analyzing FAST TCP [17] suggests that applying standard low order Pade approximations of the exponential functions is not sufficient to study scalable stability of the closed loop system. This is due to the periodic narrow resonance peaks introduced by the multiple 1 − e−sτi terms in the denominator that are challenging to capture over a sufficiently wide frequency interval for arbitrary parameters. Apparently L(s) is quite a non-standard transfer function, and even assuming the simplest possible window control, a constant gain Wi(s) = ki, stability analysis seems quite cumbersome at a first glance. However, it can... |

5 |
Scalable robust stability for nonsymmetric heterogeneous networks.
- Lestas, Vinnicombe
- 2007
(Show Context)
Citation Context ...hnology, Australia. A. Tang, atang@ece.cornell.edu, is with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA. has received relatively little attention [9], =-=[10]-=-, [12]. The results in the present paper sprung from an analysis [16] of Internet congestion control: a distributed, highly heterogeneous and continually changing system and a domain where scalability... |

5 | Affine Controller Parameterization for Decentralized Control over Banach Spaces.
- Rotkowitz, Lall
- 2006
(Show Context)
Citation Context ...d to analyze due to the decentralized structure and typically requires a fairly sophisticated machinery. Analysis and synthesis of such systems has lately attracted much interest; see [3], [8], [11], =-=[13]-=- and references therein. This line of work, however, does not explicitly address scalability. Here, scalability means that a class of interconnection structures can be shown to be stable, without requ... |

2 |
Decentralized Unconditional Stability Conditions Based on the Passivity Theorem for Multi-loop Control Systems.
- Zhang, Bao, et al.
- 2002
(Show Context)
Citation Context ...lied to process control engineering, which also requires that closed loop stability be maintained in a multiloop control system when any subset of the control loops is detuned or even turned off; see =-=[22]-=- and references therein. In that context, the notion of scalable stability is referred to as decentralized unconditional stability. The paper is organized as follows. Preliminary concepts and notation... |

2 | Implementation of Provably Stable MaxNet.
- Suchara, Andrew, et al.
- 2008
(Show Context)
Citation Context ...hull of some aspect of the Nyquist curve [12], [9], potentially introducing a significant degree of conservativeness. This paper proposes a less conservative stability criterion, in which the loop gain is decomposed into two factors, which must satisfy two half-plane inclusion conditions. This criterion, which can involve a frequency-independent half plane, implicitly bounds the Nyquist curve by a set of frequencydependent half planes whose union does not include the set (−∞,−1]. Frequency dependent half plane conditions have previously been used in the context of interconnected systems [12], [15]; similarly, the convex hull conditions for stability in [9], [10], [18], need only hold pointwise over frequency (cf. the conditions in [6]). Considering half planes per frequency reduces conservatism, at the price of greater complexity in the stability test. The approach taken here circumvents the frequency dependence in the half planes. The complexity in the stability condition is transformed from finding infinitely many bounding half planes, into finding a suitable factorization of the loop gain. Our criterion is related to notions of dissipativity and passivity [20], which have recently b... |