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• W3217113380 abstract "Abstract Consider the linear discrete-time fractional order systems with uncertainty on the initial state <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable columnalign=left> <m:mtr columnalign=left> <m:mtd columnalign=left> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=normal>Δ</m:mi> </m:mrow> <m:mi>α</m:mi> </m:msup> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>A</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> <m:mo>+</m:mo> <m:mi>B</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=left> <m:mrow> <m:mi>i</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mtd> </m:mtr> <m:mtr columnalign=left> <m:mtd columnalign=left> <m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mn>0</m:mn> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>τ</m:mi> </m:mrow> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mi>n</m:mi> </m:msup> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=left> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr columnalign=left> <m:mtd columnalign=left> <m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>C</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> </m:mrow> </m:msub> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi>i</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mtd> <m:mtd columnalign=left> <m:mrow /> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:math> 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{lower3pthbox{$scriptscriptstylefrown$}}over tau } }_0} in {mathbb{R}^n},} hfill & {{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}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, x 0 is the initial state, y i is the signal output, α the order of the derivative, τ 0 and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> {mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over tau } _0} are the known and unknown part of x 0 , respectively, u i = Kx i is feedback control and Ω ⊂ ℝn is a polytope convex of vertices w 1 , w 2 , . . . , w p . According to the Krein–Milman theorem, we suppose that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> <m:mo>=</m:mo> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>p</m:mi> </m:munderover> <m:mrow> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> <m:msub> <m:mrow> <m:mi>w</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> {mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over tau } _0} = sumlimits_{j = 1}^p {{alpha _j}{w_j}} for some unknown coefficients α 1 ≥ 0, . . . , α p ≥ 0 such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>p</m:mi> </m:munderover> <m:mrow> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> sumlimits_{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( <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> {mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over tau } _0} , ϵ) of all possible gain matrix K that makes the system insensitive to the unknown part <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> {mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over tau } _0} , which means <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:mi>χ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:mo>∈</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>K</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>/</m:mo> <m:mrow> <m:mo>‖</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> </m:mrow> </m:mfrac> </m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mo>∈</m:mo> <m:mo>,</m:mo> <m:mo>∀</m:mo> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi> </m:mi> <m:mo>∀</m:mo> <m:mi>i</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:math> chi left( {{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:mrow> <m:mo>‖</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mi>i</m:mi> </m:msub> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> </m:mrow> </m:mfrac> </m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mo>∈</m:mo> </m:mrow> </m:math> left| {{{partial {y_i}} over {partial {alpha _j}}}} right| le in showing the sensitivity of y i relatively to uncertainties <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msubsup> <m:mrow> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>p</m:mi> </m:msubsup> </m:mrow> </m:math> 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( <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msub> <m:mrow> <m:mover accent=true> <m:mi>τ</m:mi> <m:mo>⌢</m:mo> </m:mover> </m:mrow> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> {mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over tau } _0} , ϵ) and we propose an algorithmic approach to made explicit characterisation of such set." @default.
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• W3217113380 date "2021-11-29" @default.
• W3217113380 modified "2023-10-11" @default.
• W3217113380 title "An Output Sensitivity Problem for a Class of Fractional Order Discrete-Time Linear Systems" @default.
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