How conservative is the circle criterion?

Megretski, Alexandre

doi:10.1007/978-1-4471-0807-8_30

Cited by 6 publications

(5 citation statements)

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“…Furthermore we show that in the gain scheduling case, nonlinear compensator cannot outperform linear ones. This is in contrast with the robust stabilization case in which nonlinear controller can outperform linear ones [9] (see also [21] for further implications). Thus limiting the attention to linear compensator is not a restriction for gain scheduling state-feedback stabilization.…”

Section: Introductionmentioning

confidence: 91%

Stabilization of LPV Systems: State Feedback, State Estimation, and Duality

Blanchini

Miani

2003

SIAM J. Control Optim.

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In this paper, the problem of the stabilization of Linear Parameter Varying systems by means of Gain Scheduling Control, is considered This technique consists on designing a controller which is able to update its parameters on-line according to the variations of the plant parameters. We first consider the state feedback case and we show a design procedure based on the construction of a Lyapunov function for discrete-time LPV systems in which the parameter variations are affine and occur in the state matrix only. This procedure produces a nonlinear static controller. We show that, differently from the robust stabilization case, we can always derive a linear controller, that is nonlinear controllers cannot outperform linear ones for the gain scheduling problem. Then we show that this procedure has a dual version which leads to the construction of a linear gain scheduling observer. The two procedures may be combined to derive an observer-based gain scheduling compensator.

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Section: Introductionmentioning

confidence: 91%

Stabilization of LPV Systems: State Feedback, State Estimation, and Duality

Blanchini

Miani

2003

SIAM J. Control Optim.

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“…Indeed, the classical circle criterion is typically stated with a memoryless or static 9 nonlinearity Φ and a pointwise inner product characterizing the conic constraint. Finding a necessary and sufficient condition for robust stability under these assumptions remains an open problem [19].…”

Section: Memoryless Nonlinearitiesmentioning

confidence: 99%

“…Standard Lyapunov arguments may be applied to further refine Lemma 21. For example, If N 11 0 and G is stable, then we have P 0 in(18) or(19). Sufficient conditions for stability drawn from the literature.…”

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confidence: 99%

Generalized Necessary and Sufficient Robust Boundedness Results for Feedback Systems

Cyrus¹,

Lessard²

2021

Preprint

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Classical conditions for ensuring the robust stability of a system in feedback with a nonlinearity include passivity, small gain, circle, and conicity theorems. We present a generalized and unified version of these results in an arbitrary semi-inner product space, which avoids many of the technicalities that arise when working in traditional extended spaces. Our general formulation clarifies when the sufficient conditions for robust stability are also necessary, and we show how to construct worst-case scenarios when the sufficient conditions fail to hold. Finally, we show how our general result can be specialized to recover a wide variety of existing results, and explain how properties such as boundedness, causality, linearity, and time-invariance emerge as a natural consequence.

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“…with orthonormal eigenvector matrix Q as in (5). The matrices H = H (n) as defined in (8) are skew-symmetric for every n. Thus eigenvalues are purely imaginary and…”

Section: Theoremmentioning

confidence: 99%

“…In [8] Alexandre Megretski posed a problem 1 (Problem 30) with certain implications: in harmonic analysis a connection between the time domain and frequency domain multiplications, in control theory the conservatism of the circle criterion, the possiblility of robust stabilization of a second-order uncertain system using a linear and time-invariant controller, and the conjectured finiteness of the gap between the minimum in some specially structured non-convex quadratic optimization problem and its natural relaxation (cf. [1,Problem 30]).…”

mentioning

confidence: 99%