Stability Theory of Dynamical Systems

Willems, J. L.; Pandit, Μ.

doi:10.1109/tsmc.1971.4308335

Cited by 147 publications

(39 citation statements)

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“…Example 3: Chua's circuit [11] can be described by the following state equations: (23) Clearly this is of the form of (19). The nonlinearity in (23) is given by for some positive and This nonlinearity is not smooth, but this does not affect the discussion.…”

Section: A Output Only Nonlinearitiesmentioning

confidence: 99%

An observer looks at synchronization

Nijmeijer

Mareels

1997

IEEE Trans. Circuits Syst. I

629

318

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Section: A Output Only Nonlinearitiesmentioning

confidence: 99%

An observer looks at synchronization

Nijmeijer

Mareels

1997

IEEE Trans. Circuits Syst. I

629

318

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“…2 with the dashed feedback block. Its stability follows from the same arguments as in Proposition 1, because a monotone first-third quadrant nonlinearity (9) cascaded with inverse-Zames-Falb multiplier (17) is passive (Willems, 1970;Zames & Falb, 1968): Fig. 2, where the mobile power control law is given by (12) and the base station price update (9) is replaced by…”

Section: Definition 2 the Class Of Proper Transfer Functions Z(s) Ismentioning

confidence: 93%

“…To this end, we exploit the monotone increasing property of the feedback nonlinearity in Fig. 1 and augment it with the following class of multipliers introduced by Zames and Falb (1968) and extended by Willems (1970), which preserves passivity of the feedback block:…”

Section: Extended Base Station Price Update Algorithmsmentioning

confidence: 99%

“…where x Z is the internal state of Z(s),x z = x z − x * z in which x * z is the equilibrium for the constant input (y * ), and (·) is a nonnegative function (see Willems, 1970, Zames & Falb, 1968 for proofs of this property). We now prove the closedloop stability by using the Lyapunov function V in Proposition 1.…”

Section: Theorem 3 Consider the Feedback Interconnection Shown Inmentioning

confidence: 99%

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A passivity approach to game-theoretic CDMA power control

et al. 2006

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This paper follows a game-theoretical formulation of the CDMA power control problem and develops new decentralized control algorithms that globally stabilize the desired Nash equilibrium. The novel approach is to exploit the passivity properties of the feedback loop comprising the mobiles and the base station. We first reveal an inherent passivity property in an existing gradient-type algorithm, and prove stability from the Passivity Theorem. We then exploit this passivity property to develop two new designs. In the first design, we extend the base station algorithm with Zames-Falb multipliers which preserve its passivity properties. In the second design, we broaden the mobile power update laws with more general, dynamic, passive controllers. These new designs may be exploited to enhance robustness and performance, as illustrated with a realistic simulation study. We then proceed to show robustness of these algorithms against time-varying channel gains. ᭧

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“…Ortega and Spong (1989) demonstrated passivity to hold for the robot dynamics from input τ to velocityq [21], [22], [23], [24].…”

Section: Passivitymentioning

confidence: 99%

Stability of haptic obstacle avoidance and force interaction

Johansson

Annerstedt

Robertsson

2009

2009 IEEE/RSJ International Conference on Intelligent Robots and Systems

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Abstract-Stability problems associated with haptics and robot control with obstacle avoidance are analyzed. Obstacle avoidance algorithms are revised to accomplish stable redesign using absolute stability and passivity theory. A modification of potential functions for haptic rendering and obstacle avoidance allowing stable operation for high stiffness is proposed. The modification leads to velocity-dependent potential-like repulsive stable haptic force interaction with obstacles. Using strictly positive real re-design, stable force interaction can be provided also for high stiffness of manipulated objects or obstacles.

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Stability Theory of Dynamical Systems

Cited by 147 publications

References 0 publications

An observer looks at synchronization

An observer looks at synchronization

A passivity approach to game-theoretic CDMA power control

Stability of haptic obstacle avoidance and force interaction

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