Interval Observers for Linear Impulsive Systems

Degue, Kwassi H.; Efimov, Denis; Richard, Jean‐Pierre

doi:10.1016/j.ifacol.2016.10.275

Cited by 18 publications

(24 citation statements)

References 20 publications

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“…Let us consider here the example from [26] to which we add disturbances as also done in [21]. The matrices of the system are given by…”

Section: Example 2 Minimum Dwell-timementioning

confidence: 99%

L1×ℓ1-to-L1
Briat¹
2019
Nonlinear Analysis: Hybrid Systems
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“…Let us consider here the example from [26] to which we add disturbances as also done in [21]. The matrices of the system are given by…”

Section: Example 2 Minimum Dwell-timementioning

confidence: 99%

L1×ℓ1-to-L1
Briat¹
2019
Nonlinear Analysis: Hybrid Systems
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“…Let us consider here the example from [37] to which we add disturbances as also done in [10,18]. We consider the system (49) with the matrices:…”

Section: Minimum Dwell-timementioning

confidence: 99%

“…Over the past recent years, this problem witnessed an increase in its popularity and various methodologies for their design in different setups have been proposed. To cite a few, those observers have been obtained for systems with inputs [3,4], linear systems [5][6][7], timevarying systems [8], delay systems [4,9], impulsive systems [10], uncertain/LPV systems [11][12][13], discrete-time systems [4,14], systems with samplings [15,16], impulsive systems [10,17,18], switched systems [18][19][20] and Markovian jump systems [21].…”

Section: Introductionmentioning

confidence: 99%

Stability andL₁× ℓ₁-to-L₁× ℓ₁performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

Briat¹

2019

International Journal of Control

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Solutions to the interval observation problem for delayed impulsive and switched systems with L1performance are provided. The approach is based on first obtaining stability and L1/ 1-to-L1/ 1 performance analysis conditions for uncertain linear positive impulsive systems in linear fractional form with norm-bounded uncertainties using a scaled small-gain argument involving time-varying D-scalings. Both range and minimum dwell-time conditions are formulated -the case of constant and maximum dwell-times can be directly obtained as corollaries. The conditions are stated as timer/clock-dependent conditions taking the form of infinite-dimensional linear programs that can be relaxed into finite-dimensional ones using polynomial optimization techniques. It is notably shown that under certain conditions, the scalings can be eliminated from the stability conditions to yield equivalent stability conditions on the so-called worst-case system, which is obtained by replacing the uncertainties by the identity matrix. These conditions are then applied to the special case of linear positive systems with delays, where the delays are considered as uncertainties, similarly to as in [1]. As before, under certain conditions, the scalings can be eliminated from the conditions to obtain conditions on the worst-case system, coinciding here with the zero-delay system -a result that is consistent with all the existing ones in the literature on linear positive systems with delays. Finally, the case of switched systems with delays is considered. The approach also encompasses standard continuous-time and discrete-time systems, possibly with delays and the results are flexible enough to be extended to cope with multiple delays, time-varying delays, distributed/neutral delays and any other types of uncertain systems that can be represented as a feedback interconnection of a known system with an uncertainty. the design of interval observers. In particular, the design of structured state-feedback controllers is convex in this setup [23,24], the L p -gains for p = 1, 2, ∞ of linear positive systems can be exactly computed by solving linear or semidefinite programs [24][25][26], optimal state-feedback controllers and observers gains enjoy an interesting invariance property with respect to the input and output matrices of a linear positive system, respectively [4,27]. Linear positive systems with discrete-delays are also stable provided that their zero-delay counterparts are also stable; see e.g. [24,[28][29][30][31][32][33][34]. In particular, all those exact results have been shown to be consequences of small-gain arguments using various L p -gains in [1] by exploiting the robust analysis results reported in [25,26,35]. Some extensions have also been provided, notably pertaining on the analysis of linear positive systems with time-varying delays.Before delving into the objective and the contributions of this paper, it seems important to motivate the consideration of uncertain linear impulsive systems. Impulsive systems are, in fact, a powerful cla...

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“…Interval observers are a particular type of observers that aim at estimating upper and lower bounds on the state value at all times. They have been successfully designed for a wide variety of systems including systems with inputs [6,17], linear systems [19], delay systems [6,13], LPV systems [8,14], discrete-time systems [6,18], impulsive systems [5,7,11] and switched systems [5,15,20]. To the best of the author's knowledge, no results have been obtained in the context of Markov jump linear systems albeit those systems are important for practical purposes.…”

Section: Introductionmentioning

confidence: 99%

A Class of ${L}_{1}$ -to-${L}_{1}$ and ${L}_{\infty}$ -to-${L}_{\infty}$ Interval Observers for (Delayed) Markov Jump Linear Systems

Briat¹

2019

IEEE Control Syst. Lett.

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We exploit recent results on the stability and performance analysis of positive Markov jump linear systems (MJLS) for the design of interval observers for MJLS with and without delays. While the conditions for the L 1 performance are necessary and sufficient, those for the L ∞ performance are only sufficient. All the conditions are stated as linear programs that can be solved very efficiently. Two examples are given for illustration.

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Interval Observers for Linear Impulsive Systems

Cited by 18 publications

References 20 publications

L1×ℓ1-to-L1
Briat¹
2019
Nonlinear Analysis: Hybrid Systems
25
0
17
0
View full text Add to dashboard Cite

L1×ℓ1-to-L1
Briat¹
2019
Nonlinear Analysis: Hybrid Systems
25
0
17
0
View full text Add to dashboard Cite

Stability andL₁× ℓ₁-to-L₁× ℓ₁performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

A Class of ${L}_{1}$ -to-${L}_{1}$ and ${L}_{\infty}$ -to-${L}_{\infty}$ Interval Observers for (Delayed) Markov Jump Linear Systems

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Interval Observers for Linear Impulsive Systems

Cited by 18 publications

References 20 publications

L1×ℓ1-to-L1Briat1 2019Nonlinear Analysis: Hybrid Systems250170View full textAdd to dashboardCite

L1×ℓ1-to-L1Briat1 2019Nonlinear Analysis: Hybrid Systems250170View full textAdd to dashboardCite

Stability andL1× ℓ1-to-L1× ℓ1performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

A Class of ${L}_{1}$ -to-${L}_{1}$ and ${L}_{\infty}$ -to-${L}_{\infty}$ Interval Observers for (Delayed) Markov Jump Linear Systems

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Product

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About

L1×ℓ1-to-L1
Briat¹
2019
Nonlinear Analysis: Hybrid Systems
25
0
17
0
View full text Add to dashboard Cite

L1×ℓ1-to-L1
Briat¹
2019
Nonlinear Analysis: Hybrid Systems
25
0
17
0
View full text Add to dashboard Cite

Stability andL₁× ℓ₁-to-L₁× ℓ₁performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays