Lyapunov Theory for 2-D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability

Yeganefar, Nima; Yeganefar, Nader; Ghamgui, Mariem; Moulay, Emmanuel

doi:10.1109/tac.2012.2220012

Cited by 124 publications

(53 citation statements)

References 23 publications

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“…Remark 2. It follows from the results above that Theorem 3.1 in (Yeganefar et al, 2013) gives sufficient conditions for exponential stability but (Yeganefar et al, 2013) claim that their result gives sufficient conditions for asymptotic stability only.…”

Section: Exponential Stability Analysismentioning

confidence: 99%

“…For example, the stability of nonlinear deterministic systems described by a Fornasini-Marchesini model was analyzed in (Kurek, 2014). Discrete nonlinear systems described by the Roesser model were considered in Yeganefar et al (2013) where Lyapunov theorems to check for asymptotic and exponential stability were established and a converse Lyapunov theorem developed for exponential stability.…”

Section: Introductionmentioning

confidence: 99%

“…Using results in (Pakshin et al, 2015a) it follows that the conditions for asymptotic stability obtained in (Yeganefar et al, 2013) actually guarantee exponential stability and the conditions for asymptotic stability were derived as a consequence of exponential stability. Exponential stability is a natural property of linear dynamics and it is well known that for linear standard, termed 1D in this paper, systems asymptotic stability and exponential stability are equivalent.…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis of 2D Roesser systems via vector Lyapunov functions

et al. 2017

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The paper gives new results that contribute to the development of a stability theory for 2D nonlinear discrete and differential systems described by a state-space model of the Roesser form using an extension of Lyapunov's method. One of the main difficulties in using such an approach is that the full derivative or its discrete counterpart along the trajectories cannot be obtained without explicitly finding the solution of the system under consideration. This has led to the use of a vector Lyapunov function and its divergence or its discrete counterpart along the system trajectories. Using this approach, new conditions for asymptotic stability are derived in terms of the properties of two vector Lyapunov functions. The properties of asymptotic stability in the horizontal and vertical dynamics, respectively, are introduced and analyzed. This new properties arise naturally for repetitive processes where one of the two independent variables is defined over a finite interval. Sufficient conditions for exponential stability in terms of the properties of one vector Lyapunov function are also given as a natural follow on from the asymptotic stability analysis.

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Section: Exponential Stability Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stability analysis of 2D Roesser systems via vector Lyapunov functions

et al. 2017

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“…The development of a stability theory for nonlinear 2D systems is underway, e.g., (Kurek, 2012;Yeganefar et al, 2013). Also sufficient conditions guaranteeing Lyapunov stability, asymptotic stability and exponential stability of nonlinear 2D differential-discrete systems were developed in (Knorn and Middleton, 2016), where the conditions for Lyapunov stability and asymptotic stability require that the corresponding 2D Lyapunov function has the negative semi-definite property.…”

Section: Introductionmentioning

confidence: 99%

Pass profile exponential and asymptotic stability of nonlinear repetitive processes

et al. 2017

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This paper considers discrete and differential nonlinear repetitive processes using the state-space model setting. These processes are a particular class of 2D systems that have their origins in the modeling of physical processes. Their distinguishing characteristic is that one of the two independent variables needed to describe the dynamics evolves over a finite interval and therefore they are defined over a subset of the upper-right quadrant of the 2D plane. The current stability theory for nonlinear dynamics assumes that they operate over the complete upper-right quadrant and this property may be too strong for physical applications, particulary in terms of control law design. With applications in mind, the contribution of this paper is the use of vector Lyapunov functions to characterize a new property termed pass profile exponential stability.

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“…For instance, the stability of nonlinear Fornasini-Marchesini systems was analyzed in [10] and the publications [11,12] considered different types of stability in nonlinear discrete-time Roesser systems. In [13,14], the stability of discrete and differential nonlinear repetitive processes was considered and there is a need to extend this work to allow control law design.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of differential repetitive processes

Emelianov¹,

Pakshin²,

Gałkowski

et al. 2015

Autom Remote Control

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Abstract-Differential repetitive processes are a subclass of 2D systems that arise in modeling physical processes with identical repetitions of the same task and in the analysis of other control problems such as the design of iterative learning control laws. These models have proved to be efficient within the framework of linear dynamics, where control laws designed in this setting have been verified experimentally, but there are few results for nonlinear dynamics. This paper develops new results on the stability, stabilization and disturbance attenuation, using an H ∞ norm measure, for nonlinear differential repetitive processes. These results are then applied to design iterative learning control algorithms under model uncertainty and sensor failures described by a homogeneous Markov chain with a finite set of states. The resulting design algorithms can be computed using linear matrix inequalities.

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Lyapunov Theory for 2-D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability

Cited by 124 publications

References 23 publications

Stability analysis of 2D Roesser systems via vector Lyapunov functions

Stability analysis of 2D Roesser systems via vector Lyapunov functions

Pass profile exponential and asymptotic stability of nonlinear repetitive processes

Stabilization of differential repetitive processes

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