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Improved Stability and Passivity Results for Discrete Time-Delayed Systems with Saturation Nonlinearities and External Disturbances

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Abstract

This study considers the stability and passivity analysis of discrete-time systems with variable time-lags, saturation overflow nonlinear effects and external disturbances. Summation inequality based on free-weighting matrices is employed to develop an enhanced stability criterion that is in the form of linear matrix inequalities. A sufficient passivity condition for discrete-delayed systems involving overflow nonlinearities and subjected to external interferences is also proposed. Illustrative examples are presented to demonstrate the effectiveness of the obtained results.

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Acknowledgements

The authors wish to thank the anonymous reviewers and the Associate Editor for their constructive comments and suggestions for improving the quality of the manuscript.

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The author(s) received no financial support for the research, authorship, and/or publication of this article from any funding agency in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Electronics and Telecommunication Engineering, Bhilai Institute of Technology, Durg, Bhilai House, Durg, CG, 491001, India

    Suchitra Pandey

  2. Department of Electronics Engineering, National Institute of Technology Uttarakhand, NH 58, Srinagar, Uttarakhand, 246174, India

    Siva Kumar Tadepalli & Rishi Nigam

  3. Department of Electrical and Electronics Engineering, Bhilai Institute of Technology, Durg, Bhilai House, Durg, CG, 491001, India

    Surekha Bhusnur

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Appendix. Proof of Theorem 1

Appendix. Proof of Theorem 1

Proof

Let us consider the LKF [6, 41, 55] as

$$\begin{aligned} V(\varvec{\chi }(\vartheta ))&= V_1(\varvec{\chi }(\vartheta )) + V_2(\varvec{\chi }(\vartheta )) + V_3(\varvec{\chi }(\vartheta )) + V_4(\varvec{\chi }(\vartheta )), \end{aligned}$$
(A.1)

where

$$\begin{aligned} V_1(\varvec{\chi }(\vartheta ))&= \varvec{\Gamma }^T(\vartheta ) {\varvec{P}} \varvec{\Gamma }(\vartheta ), \\ V_2(\varvec{\chi }(\vartheta ))&= \sum _{\beta =\vartheta -d_l}^{\vartheta -1} \varvec{\chi }^T(\beta ) \varvec{Q_1}\varvec{\chi }(\beta ) + \sum _{\beta =\vartheta -d_h}^{\vartheta -d_l-1} \varvec{\chi }^T(\beta ) \varvec{Q_2}\varvec{\chi }(\beta ), \\ V_3(\varvec{\chi }(\vartheta ))&= \sum _{\beta =-d_l}^{-1} \sum _{\gamma =\vartheta +\beta }^{\vartheta -1} \Delta \varvec{\chi }^T(\gamma ) \varvec{R_1} \Delta \varvec{\chi }(\gamma ) + \sum _{\beta =-d_h}^{-1-d_l} \sum _{\gamma =\vartheta +\beta }^{\vartheta -1} \Delta \varvec{\chi }^T(\gamma ) \varvec{R_2} \Delta \varvec{\chi }(\gamma ), \\ V_4(\varvec{\chi }(\vartheta ))&= \sum _{\beta =-d_h}^{-1-d_l} \sum _{\gamma =\vartheta +\beta }^{\vartheta -1} \varvec{\chi }^T(\gamma ) \varvec{R_3} \varvec{\chi }(\gamma ). \end{aligned}$$

Next, on calculating the finite difference of the above LKF along the trajectories of system (1) we have

$$\begin{aligned} \Delta V_1(\varvec{\chi }(\vartheta ))&= \begin{bmatrix} \varvec{\zeta }(\varvec{\psi }(\vartheta )) \\ \\ \sum _{\beta =\vartheta -d_l}^\vartheta \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_l) \\ \\ \sum _{\beta =\vartheta -d_h}^{\vartheta -d_l} \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_h) \end{bmatrix}^T {\varvec{P}} \begin{bmatrix} \varvec{\zeta }(\varvec{\psi }(\vartheta )) \\ \\ \sum _{\beta =\vartheta -d_l}^\vartheta \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_l) \\ \\ \sum _{\beta =\vartheta -d_h}^{\vartheta -d_l} \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_h) \end{bmatrix} \nonumber \\&\quad - \begin{bmatrix} \varvec{\chi }(\vartheta ) \\ \\ \sum _{\beta =\vartheta -d_l}^\vartheta \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_l) \\ \\ \sum _{\beta =\vartheta -d_h}^{\vartheta -d_l} \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_h) \end{bmatrix}^T {\varvec{P}} \begin{bmatrix} \varvec{\chi }(\vartheta ) \\ \\ \sum _{\beta =\vartheta -d_l}^\vartheta \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_l) \\ \\ \sum _{\beta =\vartheta -d_h}^{\vartheta -d_l} \varvec{\chi }(\beta ) - \varvec{\chi }(\vartheta -d_h) \end{bmatrix}, \end{aligned}$$
(A.2)
$$\begin{aligned} \Delta V_2(\varvec{\chi }(\vartheta ))&=\varvec{\chi }^T(\vartheta ) \varvec{Q_1} \varvec{\chi }(\vartheta ) - \varvec{\chi }^T(\vartheta -d_l) \varvec{Q_1} \varvec{\chi }(\vartheta -d_l) + \varvec{\chi }^T(\vartheta -d_l) \varvec{Q_2} \varvec{\chi }(\vartheta -d_l) \nonumber \\&\quad - \varvec{\chi }^T(\vartheta -d_h) \varvec{Q_2} \varvec{\chi }(\vartheta -d_h), \end{aligned}$$
(A.3)
$$\begin{aligned} \Delta V_3(\varvec{\chi }(\vartheta ))&= d_l \Delta \varvec{\chi }^T(\vartheta ) \varvec{R_1} \Delta \varvec{\chi }(\vartheta ) - \sum _{\beta =\vartheta -d_l}^{\vartheta -1} \Delta \varvec{\chi }^T(\beta ) \varvec{R_1} \Delta \varvec{\chi }(\beta ) \nonumber \\&\quad + d_{lh} \Delta \varvec{\chi }^T(\vartheta ) \varvec{R_2} \Delta \varvec{\chi }(\vartheta ) - \sum _{\beta =\vartheta -d(\vartheta )}^{\vartheta -1-d_l} \Delta \varvec{\chi }^T(\beta ) \varvec{R_2} \Delta \varvec{\chi }(\beta ) \nonumber \\&\quad - \sum _{\beta =\vartheta -d_h}^{\vartheta -1-d(\vartheta )} \Delta \varvec{\chi }^T(\beta ) \varvec{R_2} \Delta \varvec{\chi }(\beta ), \end{aligned}$$
(A.4)
$$\begin{aligned} \Delta V_4(\varvec{\chi }(\vartheta ))&= d_{lh}\varvec{\chi }^T(\vartheta ) \varvec{R_3} \varvec{\chi }(\vartheta ) - \sum _{\beta =\vartheta -d(\vartheta )}^{\vartheta -1-d_l} \varvec{\chi }^T(\beta ) \varvec{R_3} \varvec{\chi }(\beta ) \nonumber \\&\quad - \sum _{\beta =\vartheta -d_h}^{\vartheta -1-d(\vartheta )} \varvec{\chi }^T(\beta ) \varvec{R_3} \varvec{\chi }(\beta ). \end{aligned}$$
(A.5)

Now, by employing Lemma 1 the summation terms in (A.4) can be expressed as

$$\begin{aligned} - \sum _{\beta =\vartheta -d_l}^{\vartheta -1} \Delta \varvec{\chi }^T(\beta ) \varvec{R_1} \Delta \varvec{\chi }(\beta )&\le \varvec{\xi }^T(\vartheta )[\varvec{\Lambda _1}\varvec{N_1}^T + \varvec{N_1} \varvec{\Lambda _1}^T + d_1 \varvec{N_1}\varvec{{\hat{R}}_1}^{-1} \varvec{N_1}^T] \varvec{\xi }(\vartheta ), \end{aligned}$$
(A.6)
$$\begin{aligned} - \sum _{\beta =\vartheta -d(\vartheta )}^{\vartheta -1-d_l} \Delta \varvec{\chi }^T(\beta ) \varvec{R_2} \Delta \varvec{\chi }(\beta )&\le \varvec{\xi }^T(\vartheta )[\varvec{\Lambda _2}\varvec{N_2}^T + \varvec{N_2} \varvec{\Lambda _2}^T \nonumber \\&\quad + (d(\vartheta )-d_l) \varvec{N_2}\varvec{{\hat{R}}_2}^{-1} \varvec{N_2}^T] \varvec{\xi }(\vartheta ), \end{aligned}$$
(A.7)

and

$$\begin{aligned} - \sum _{\beta =\vartheta -d_h}^{\vartheta -1-d(\vartheta )} \Delta \varvec{\chi }^T(\beta ) \varvec{R_2} \Delta \varvec{\chi }(\beta )&\le \varvec{\xi }^T(\vartheta )[\varvec{\Lambda _3}\varvec{N_3}^T + \varvec{N_3} \varvec{\Lambda _3}^T \nonumber \\&\quad + (d_h-d(\vartheta )) \varvec{N_3}\varvec{{\hat{R}}_2}^{-1} \varvec{N_3}^T] \varvec{\xi }(\vartheta ), \end{aligned}$$
(A.8)

On the same lines as above, the second and third sum terms of (A.5) can be estimated by making use of the last inequality in Remark 3 of [40] as follows:

$$\begin{aligned} - \sum _{\beta =\vartheta -d(\vartheta )}^{\vartheta -1-d_l} \varvec{\chi }^T(\beta ) \varvec{R_3} \varvec{\chi }(\beta )&\le \varvec{\xi }^T(\vartheta )[\varvec{\Lambda _4}\varvec{N_4}^T + \varvec{N_4} \varvec{\Lambda _4}^T \nonumber \\&\quad + (d(\vartheta )-d_l) \varvec{N_4}\varvec{{\hat{R}}_3}^{-1} \varvec{N_4}^T] \varvec{\xi }(\vartheta ), \end{aligned}$$
(A.9)

and

$$\begin{aligned} - \sum _{\beta =\vartheta -d_h}^{\vartheta -1-d(\vartheta )} \varvec{\chi }^T(\beta ) \varvec{R_3} \varvec{\chi }(\beta )&\le \varvec{\xi }^T(\vartheta )[\varvec{\Lambda _5}\varvec{N_5}^T + \varvec{N_5} \varvec{\Lambda _5}^T \nonumber \\&\quad + (d_h-d(\vartheta )) \varvec{N_5}\varvec{{\hat{R}}_3}^{-1} \varvec{N_5}^T] \varvec{\xi }(\vartheta ), \end{aligned}$$
(A.10)

Employing (A.2) to (A.10), we obtain the following inequality

$$\begin{aligned} \Delta V(\varvec{\chi }(\vartheta ))&\le \varvec{\xi }^T(\vartheta ) ( \varvec{\Omega _1} + \varvec{\Omega _2} + \varvec{\Omega _3}) \varvec{\xi }(\vartheta ) - \delta \end{aligned}$$
(A.11)

where

$$\begin{aligned} \delta&= 2[\varvec{\psi }^T(\vartheta ) {\varvec{M}} + \varvec{\chi }^T(\vartheta ) {\varvec{Q}} + \varvec{\zeta }^T(\varvec{\psi }(\vartheta )) {\varvec{N}}] [\varvec{\psi }(\vartheta ) - \varvec{\zeta }(\varvec{\psi }(\vartheta ))] \end{aligned}$$
(A.12)

In view of (4), the quantity ‘\(\delta \)’ given by (A.12) is non-negative [42]. Because of the non-negativeness of ‘\(\delta \)’, one can conclude from (A.11) that \(\Delta V(\varvec{\chi }(\vartheta )) < {\varvec{0}}\) if \((\varvec{\Omega _1} + \varvec{\Omega _2} + \varvec{\Omega _3})< {\varvec{0}}\). By Schur’s complement, \((\varvec{\Omega _1} + \varvec{\Omega _2} + \varvec{\Omega _3}) < {\varvec{0}}\) if and only if the inequalities (8) and (9) hold. Therefore, the conditions (8), (9) and (10) are the asymptotical stability conditions for the system characterised by (1)–(2). Hence, we reach the end of an improved stability condition with the above proof. \(\square \)

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Pandey, S., Tadepalli, S.K., Bhusnur, S. et al. Improved Stability and Passivity Results for Discrete Time-Delayed Systems with Saturation Nonlinearities and External Disturbances. Circuits Syst Signal Process 43, 103–123 (2024). https://doi.org/10.1007/s00034-023-02465-5

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