Abstract
The paper studies the problem of exponential stability and \(H_{\infty }\) performance of externally interfered discrete-time systems (DTS) with time-varying delays and finite wordlength nonlinearities. The presented approach uses Wirtinger-based inequality (WBI) and reciprocally convex inequality (RCI) that are suitable to approximate the sum terms in the forward difference of Lyapunov–Krasovskii functional (LKF). The results obtained can be used to determine the exponential stability and to diminish the effects of external interference. Examples are presented to illustrate the usefulness of the obtained results.
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Appendix
Appendix
Proof of Theorem 1
Choose the following LKF [50] as
where \({{\varvec{\Pi}}}^{\text{T}} (v) = [\begin{array}{*{20}c} {{\varvec{x}}^{\text{T}} (v)} & {\sum\limits_{{s = v - d_{l} }}^{v - 1} {{\varvec{x}}^{\text{T}} (s)} } & {\sum\limits_{{s = v - d_{u} }}^{{v - d_{l} - 1}} {{\varvec{x}}^{\text{T}} (s)} } \\ \end{array} ]\) and \({{\varvec{\upsigma}}}(v) = {\varvec{x}}(v + 1) - {\varvec{x}}(v) = {{\varvec{\Omega}}}({\varvec{y}}(v)){ + }{\varvec{e}}(v) - {\varvec{x}}(v).\)
Utilizing Lemmas 1, 2 and following [50], it can be shown that
where
In view of (3), the quantity ‘\(\delta\)’ given by (27b) is nonnegative [16, 17]. If the condition \({{\varvec{\Theta}}}_{1} (d(v)) < {\varvec{0}}\) along with (8) is fulfilled, then
By taking summation on both sides of (28) from 0 to \(\infty\), we obtain
As \(V({\varvec{x}}(\infty )) \ge {0}\) and\(V({\varvec{x}}(0)) = {0,}\) one can see that relation (4) always holds true in view of (29).
In the following, the system (1)–(3) with \({\varvec{e}}(v) = {\varvec{0}}\) is shown to be exponentially stable under the condition \({{\varvec{\Theta}}}_{1} (d(v)) < {\varvec{0}}\). Note that \(V({\varvec{x}}(v))\) satisfies Rayleigh inequality [47]
With \({\varvec{e}}(v) = {\varvec{0}}{,}\) (28) becomes \(\Delta V({\varvec{x}}(v)) < - {\varvec{x}}^{\text{T}} (v){\varvec{Sx}}(v) \le - \lambda_{\min } ({\varvec{S}})\left\| {{\varvec{x}}(v)} \right\|^{2} .\) According to Theorem 3.1 in [25], this condition together with (30) guarantees the exponential stability of the system (1)–(3). With respect to the time-varying delay \(d(v)\), it is obvious that \({{\varvec{\Theta}}}_{1} (d(v))\) is an affine matrix function. Therefore, \({{\varvec{\Theta}}}_{1} (d(v)) < {\varvec{0}}\) if (9) is met. This completes the proof of Theorem 1.
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Singh, K., Kandanvli, V.K.R. & Kar, H. Limit Cycle-Free Realization of Discrete-Time Delayed Systems with External Interference and Finite Wordlength Nonlinearities. Circuits Syst Signal Process 41, 4438–4454 (2022). https://doi.org/10.1007/s00034-022-02007-5
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DOI: https://doi.org/10.1007/s00034-022-02007-5