Stability of uncertain systems using Lyapunov functions with non-monotonic terms

Lacerda, Márcio J.; Seiler, Peter

doi:10.1016/j.automatica.2017.04.042

Cited by 28 publications

(20 citation statements)

References 25 publications

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“…Proof Different from the existing result, in this paper, according to the partition regions of state space mentioned above, a new piecewise LF consisting of multiple piecewise sub‐LFs is designed, in which each piecewise sub‐LF depends on the polyhedral regions of the system states at different instants

V^{k} false(ξ_{k} false) = true \sum_{t = 1}^{N} true \sum_{q = t}^{N} V_{q}^{k + t - 1},

where

\begin{matrix} alignleft \\ align-1 V_{q}^{k} (ξ_{k}) & align-2 = ξ_{k}^{T} P_{l s}^{q} ξ_{k}, x_{k} \in Φ_{l}, {\hat{x}}_{k} \in Φ_{s}, l, s \in Θ \\ align-1 V_{q}^{k + t} (ξ_{k + t}) & align-2 = ξ_{k + t}^{T} P_{i_{t} j_{t}}^{q} ξ_{k + t}, x_{k + t} \in Φ_{i_{t}}, {\hat{x}}_{k + t} \in Φ_{j_{t}}, i_{t}, j_{t} \in Θ \end{matrix}

in which

P_{l s}^{q}

P ...

…”

Section: Resultsmentioning

confidence: 99%

“…Remark In the work of Lacerda and Seiler, an LF consisting of multiple quadratic LFs is investigated, where each quadratic LF does not need to be positive definite. However, it should be pointed out that the stability of the system is studied in the work of Lacerda and Seiler, and the method designed in the aforementioned work will fail when the system is subject to external disturbances and fault signals. In order to handle the challenges incurred by external disturbances and fault signals, asynchronous piecewise FD observers are designed at different instants.…”

Section: Resultsmentioning

confidence: 99%

“…In order to overcome the difficulties caused by disturbances and fault signals, the asynchronous piecewise FD observers are designed at different instants, and a new method of dealing with different piecewise quadratic LFs in different regions of the states is proposed. Moreover, the proposed method solves the FD problem which cannot be solved by the existing method …”

Section: Resultsmentioning

confidence: 99%

“…In order to overcome the limitations of the existing method, Lacerda and Seiler reduce the conservatism of the existing result and achieve better FD performance, the asynchronous piecewise FD observers at different instants are established. Based on the new piecewise LF that depends on the polyhedral regions of the system states at different instants, the main design results of the asynchronous piecewise FD observers are given.…”

Section: Examplementioning

confidence: 99%

See 3 more Smart Citations

Asynchronous fault detection observer design for discrete‐time piecewise linear systems

Wang

Yang

2019

Intl J Robust & Nonlinear

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Summary This paper studies the problem of asynchronous fault detection (FD) observer design for piecewise linear systems. Considering that the states of the FD observer and the system may stay at different regions of the state space, asynchronous FD observers are designed at different instants to cope with the challenges incurred by exogenous disturbances and fault signals. By employing new piecewise Lyapunov functions that depend on the different regions where the states are located, it is proved that the proposed asynchronous FD observers ensure the stability and H∞ performance of the error systems. Three examples are given to show that the new design scheme provides better FD results than the existing design methods.

show abstract

V^{k} false(ξ_{k} false) = true \sum_{t = 1}^{N} true \sum_{q = t}^{N} V_{q}^{k + t - 1},

where

\begin{matrix} alignleft \\ align-1 V_{q}^{k} (ξ_{k}) & align-2 = ξ_{k}^{T} P_{l s}^{q} ξ_{k}, x_{k} \in Φ_{l}, {\hat{x}}_{k} \in Φ_{s}, l, s \in Θ \\ align-1 V_{q}^{k + t} (ξ_{k + t}) & align-2 = ξ_{k + t}^{T} P_{i_{t} j_{t}}^{q} ξ_{k + t}, x_{k + t} \in Φ_{i_{t}}, {\hat{x}}_{k + t} \in Φ_{j_{t}}, i_{t}, j_{t} \in Θ \end{matrix}

in which

P_{l s}^{q}

P ...

…”

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

Asynchronous fault detection observer design for discrete‐time piecewise linear systems

Wang

Yang

2019

Intl J Robust & Nonlinear

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“…Remark The Lyapunov function consisting of multiple quadratic Lyapunov functions is proposed in the work of Lacerda and Seiler in which the stability of the uncertain systems is studied, and note that the bias injection attack r k designed in this paper cannot be handled under the theoretical framework of To overcome this difficulty, similar to the work of Wang and Yang, a new method is proposed to copy with the Lyapunov function.…”

Section: Resultsmentioning

confidence: 99%

Cooperative attack strategy design via H₋/H_∞ scheme for linear cyber‐physical systems

Wang

Yang

2019

Intl J Robust & Nonlinear

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Summary This paper studies the problem of cooperative attack strategy design for cyber‐physical systems modeled by discrete‐time linear systems. A function that reflects the attacker's target is established, and a linear attack strategy is designed based on the objective function, in which the attack signal is unknown but energy bounded, while the statistical information of the existing ones is required to be known. Unlike the existing analysis methods for attack strategy, new Lyapunov functions that depend on the system states at different instants are constructed in the attack strategy analysis, and the sensitivity to the state estimator and the robustness against the detector are both enhanced on attack signals through using the H−/H∞ scheme. Finally, two simulation examples are given to show that the designed attack strategy achieves larger estimation errors and smaller detection probability simultaneously than the existing design methods.

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On discrete‐time LPV control using delayed Lyapunov functions

Peixoto

Lacerda

Palhares

2020

Asian Journal of Control

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This paper presents new stabilization conditions for discrete-time linear parameter-varying systems in the form of linear matrix inequalities. The use of Lyapunov functions with dependence on delayed scheduling parameters is introduced. In addition, a lifted condition based on a Lyapunov function with dependence on delayed scheduling parameters, constructed in terms of an augmented state vector that takes into account a generic number of higher-order shifted states, is presented. Numerical examples are provided to illustrate the advantages of the proposed approach when compared to other techniques from the literature.

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Stability of uncertain systems using Lyapunov functions with non-monotonic terms

Cited by 28 publications

References 25 publications

Asynchronous fault detection observer design for discrete‐time piecewise linear systems

Asynchronous fault detection observer design for discrete‐time piecewise linear systems

Cooperative attack strategy design via H₋/H_∞ scheme for linear cyber‐physical systems

On discrete‐time LPV control using delayed Lyapunov functions

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Stability of uncertain systems using Lyapunov functions with non-monotonic terms

Cited by 28 publications

References 25 publications

Asynchronous fault detection observer design for discrete‐time piecewise linear systems

Asynchronous fault detection observer design for discrete‐time piecewise linear systems

Cooperative attack strategy design via H−/H∞ scheme for linear cyber‐physical systems

On discrete‐time LPV control using delayed Lyapunov functions

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Product

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Cooperative attack strategy design via H₋/H_∞ scheme for linear cyber‐physical systems