Mathematical Systems Theory I

Hinrichsen, D.; Pritchard, A. J.

doi:10.1007/b137541

Cited by 238 publications

(199 citation statements)

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“…1. Lur'e systems are a common and important class of nonlinear systems and are at the centre of the classical subject of absolute stability theory which includes the well-known real and complex Aizerman conjectures, circle and Popov criteria, see [13,14,18,20,23,24,45,47]. An absolute stability criterion for (1.1) is a sufficient condition for stability, usually formulated in terms of frequency-domain properties of the linear system given by (A, B, C) and sector or boundedness conditions for f , guaranteeing stability for all nonlinearities f satisfying these conditions.…”

Section: X(t) = Ax(t) + Bu(t) + V(t) Y(t) = C X(t) U(t) = F (Y(t))mentioning

confidence: 99%

“…Indeed, it is well known that, for given v ∞ , the state x and output y of (1.1) have respective limits The main contribution of this paper is the establishment of sufficient conditions for the CICS property which are reminiscent of the complex Aizerman conjecture [17,18,20,36], the circle criterion for ISS [19,20,36] and the "nonlinear" ISS smallgain condition for Lur'e systems [36] and involve the transfer function matrix of the linear system (A, B, C) and an incremental condition (in terms of norm or sector inequalities) on the nonlinearity f . Recent ISS results for Lur'e systems [36] play a key role in the development of the CICS theory in Sect.…”

Section: X(t) = Ax(t) + Bu(t) + V(t) Y(t) = C X(t) U(t) = F (Y(t))mentioning

confidence: 99%

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The converging-input converging-state property for Lur’e systems

Bill¹,

Guiver²,

Logemann³

et al. 2017

Math. Control Signals Syst.

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Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur'e systems, we derive necessary and sufficient conditions for a class of Lur'e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur'e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called "quasi CICS" property for Lur'e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples. Keywords Absolute stability · Circle criterion · Converging-input converging-state property · Input-to-state stability · Lur'e systems · Non-negative systems Mathematics Subject Classification

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Section: X(t) = Ax(t) + Bu(t) + V(t) Y(t) = C X(t) U(t) = F (Y(t))mentioning

confidence: 99%

Section: X(t) = Ax(t) + Bu(t) + V(t) Y(t) = C X(t) U(t) = F (Y(t))mentioning

confidence: 99%

The converging-input converging-state property for Lur’e systems

Bill¹,

Guiver²,

Logemann³

et al. 2017

Math. Control Signals Syst.

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“…all eigenvalues of the closed loop given by A+BF are contained in the interior of the unit circle, cf. Hinrichsen and Pritchard (2005). As a consequence, constants C ≥ 1 and σ ∈ (0, 1) exist such that, for each state x 0 ∈ R n , the closed loop solution (x F (k; x 0 )) k∈N 0 generated by x F (k + 1;…”

Section: Characterization Of the Viability Kernel For Linear Systemsmentioning

confidence: 99%

Stability and feasibility of state constrained MPC without stabilizing terminal constraints

Boccia

Grüne

Worthmann

2014

Systems & Control Letters

143

122

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In this paper we investigate stability and recursive feasibility of a nonlinear receding horizon control scheme without terminal constraints and costs but imposing state and control constraints. Under a local controllability assumption we show that every level set of the infinite horizon optimal value function is contained in the basin of attraction of the asymptotically stable equilibrium for sufficiently large optimization horizon N .For stabilizable linear systems we show the same for any compact subset of the interior of the viability kernel. Moreover, estimates for the necessary horizon length N are given via an analysis of the optimal value function at the boundary of the viability kernel.

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“…This is not for all structures the case, see [116] for more details. Depending on the point of view, the problem of finding the structured backward error can also be regarded as a structured inverse eigenvalue problem [36] or computing the smallest structured singular value [66,73].…”

Section: Backward Error and Backward Stabilitymentioning

confidence: 99%

Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew‐symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Mathematical Systems Theory I

Cited by 238 publications

References 1 publication

The converging-input converging-state property for Lur’e systems

The converging-input converging-state property for Lur’e systems

Stability and feasibility of state constrained MPC without stabilizing terminal constraints

Structured Eigenvalue Problems

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