Robust stability: perturbed systems with perturbed equilibria

Michel, Anthony N.; Wang, Kaining

doi:10.1016/0167-6911(93)90118-p

Cited by 11 publications

(11 citation statements)

References 5 publications

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“…where x is a real n-vector, f and h are continuously differentiable functions, and h(x) represents the uncertainties or perturbation terms. Given an exponentially stable equilibrium x e , [14] establishes sufficient conditions by using the linearization of the system to produce Lyapunov functions which prove existence and local exponential stability of an equilibrium x e for (17) with the property |x e −x e | < ε where ε is sufficiently small. Since the approach in [14] is essentially based on a fixed Lyapunov function, it is more limited than our approach using contraction theory and SOS programming, and can prove stability only under quite conservative ranges of allowable uncertainty.…”

Section: Uncertainty Analysis With Contraction Metrics and Sos Programentioning

confidence: 99%

Section: Scalar Additive Uncertaintymentioning

confidence: 99%

“…Using the techniques in [14] we computed the allowable uncertainty range for the system given in (19) as |δ| ≤ 5.1 × 10 −3 . In the notation of [14], we calculated the other parameters in Assumption 1 of [14] as: h = [δ, 0] T , |A −1 | ∞ = 1, |Dh(x e )| ∞ = 0, a = 1 30 , and |h(x e )| ∞ = |δ|, where δ is the perturbation term in (19). The allowable range |δ| ≤ 1.023 computed via contraction theory and SOS programming is much larger than the allowable uncertainty range |δ| ≤ 5.1 × 10 −3 computed with the techniques in [14].…”

Section: Scalar Additive Uncertaintymentioning

confidence: 99%

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Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

Aylward¹,

Parrilo²,

Slotine³

2008

Automatica

148

142

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Contraction analysis is a stability theory for nonlinear systems where stability is defined incrementally between two arbitrary trajectories. It provides an alternative framework in which to study uncertain interconnections or systems with external inputs, where it offers several significant advantages when compared with traditional Lyapunov analysis. Contraction-based methods are particularly useful for analyzing systems with uncertain parameters and for proving synchronization properties of nonlinear oscillators. Existence of a contraction metric for a given system is a necessary and sufficient condition for global exponential convergence of system trajectories. For systems with polynomial or rational dynamics, the search for contraction metrics can be made fully algorithmic through the use of convex optimization and sum of squares (SOS) programming. The search process is made computationally tractable by relaxing matrix definiteness constraints, whose feasibility indicate existence of a contraction metric, into SOS constraints on polynomial matrices. We illustrate the results through examples from the literature, emphasizing the advantages and contrasting the differences between the contraction approach and traditional Lyapunov techniques.

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Section: Uncertainty Analysis With Contraction Metrics and Sos Programentioning

confidence: 99%

Section: Scalar Additive Uncertaintymentioning

confidence: 99%

Section: Scalar Additive Uncertaintymentioning

confidence: 99%

See 1 more Smart Citation

Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

Aylward¹,

Parrilo²,

Slotine³

2008

Automatica

148

142

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show abstract

“…represents the 2" extreme systems for the system given by (12). Therefore, equation (11) in Lemma 1, when applied to system (1 2), becomes…”

Section: (W(k))=a:!jw(k) W ( K ) E R/j D V ( I~) ( W (~) )mentioning

confidence: 99%

“…Systems with saturation nonlinearities (and with no parameter uncertainties) have been investigated by many researchers (see, for example, References 2,[5][6][7]11,15,16). Systems with parameter uncertainties characterized by interval matrices have also been widely investigated (see, for example, References 1,3,4,8,12,14,[17][18][19][20][21]. The stability and stabilizability of systems with saturation nonlinearities and parameter uncertainties do not appear to have been addressed.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a class of systems with parameter uncertainties and with state saturation nonlinearities

Michel

Liu

Wang

1995

Intl J Robust & Nonlinear

Self Cite

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For linear systems with parameter uncertainties and subject to state saturation, we establish results concerning the global asymptotic stability of an equilibrium. In addition to providing a means for a qualitative analysis, these results also enable us to address the stabilizability of such systems by means of linear state feedback. Systems of the type considered herein capture two important phenomena commonly encountered in the modelling process: (i) system parameter uncertainties (which in the present case are modelled by means of interval matrices), and (ii) operation of systems over a wide range (which in the present case is accounted for by state saturation nonlinearities). We demonstrate the applicability of the present results by means of several specific examples.

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Introduction

Michel¹,

Hou²,

Liu³

Stability of Dynamical Systems

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Robust stability: perturbed systems with perturbed equilibria

Cited by 11 publications

References 5 publications

Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

Stability analysis of a class of systems with parameter uncertainties and with state saturation nonlinearities

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