An Output Sensitivity Problem for a Class of Fractional Order Discrete-Time Linear Systems
Consider the linear discrete-time fractional order systems with uncertainty on the initial state {Δαxi+1=Axi+Bui,i≥0x0=τ0+τ⌢0∈ℝn,τ⌢0∈Ω,yi=Cxi, i≥0\left\{ {\matrix{{{\Delta ^\alpha }{x_{i + 1}} = A{x_i} + B{u_i},} \hfill & {i \ge 0} \hfill \cr {{x_0} = {\tau _0} + {{\mathord{\buildrel{\lower3pt...
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| Autores principales: | , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
Sciendo
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/a89b459a812146c88a5d6f9c041340f0 |
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| Sumario: | Consider the linear discrete-time fractional order systems with uncertainty on the initial state
{Δαxi+1=Axi+Bui,i≥0x0=τ0+τ⌢0∈ℝn,τ⌢0∈Ω,yi=Cxi, i≥0\left\{ {\matrix{{{\Delta ^\alpha }{x_{i + 1}} = A{x_i} + B{u_i},} \hfill & {i \ge 0} \hfill \cr {{x_0} = {\tau _0} + {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0} \in {\mathbb{R}^n},} \hfill & {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0} \in \Omega ,} \hfill \cr {{y_i} = C{x_{i,}}\,\,\,i \ge 0} \hfill & {} \hfill \cr } } \right. where A, B and C are appropriate matrices, x0 is the initial state, yi is the signal output, α the order of the derivative, τ0 and τ⌢0{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0} are the known and unknown part of x0, respectively, ui = Kxi is feedback control and Ω ⊂ ℝn is a polytope convex of vertices w1, w2, . . . , wp. According to the Krein–Milman theorem, we suppose that τ⌢0=∑j=1pαjwj{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0} = \sum\limits_{j = 1}^p {{\alpha _j}{w_j}} for some unknown coefficients α1 ≥ 0, . . . , αp ≥ 0 such that ∑j=1pαj=1\sum\limits_{j = 1}^p {{\alpha _j} = 1}. In this paper, the fractional derivative is defined in the Grünwald–Letnikov sense. We investigate the characterisation of the set χ(τ⌢0{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}, ϵ) of all possible gain matrix K that makes the system insensitive to the unknown part τ⌢0{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}, which means χ(τ⌢0,∈)={K∈ℝm×n/‖∂yi∂αj‖≤∈,∀j=1,…,p, ∀i≥0}\chi \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0}, \in } \right) = \left\{ {K \in {\mathbb{R}^{m \times n}}/\left\| {{{\partial {y_i}} \over {\partial {\alpha _j}}}} \right\| \le \in ,\forall j = 1, \ldots ,p,\,\forall i \ge 0} \right\}, where the inequality ‖∂yi∂αj‖≤∈\left\| {{{\partial {y_i}} \over {\partial {\alpha _j}}}} \right\| \le \in showing the sensitivity of yi relatively to uncertainties {αj}j=1p\left\{ {{\alpha _j}} \right\}_{j = 1}^p will not achieve the specified threshold ϵ > 0. We establish, under certain hypothesis, the finite determination of χ(τ⌢0{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}, ϵ) and we propose an algorithmic approach to made explicit characterisation of such set. |
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