<abstract xmlns="http://www.w3.org/1999/xhtml">

Consider the linear discrete-time fractional order systems with uncertainty on the initial state
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mi>α</mi></msup><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub><mo>+</mo><mi>B</mi><msub><mrow><mi>u</mi></mrow><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mrow><mi>x</mi></mrow><mn>0</mn></msub><mo>=</mo><msub><mrow><mi>τ</mi></mrow><mn>0</mn></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub><mo>∈</mo><msup><mrow><mi>ℝ</mi></mrow><mi>n</mi></msup><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub><mo>∈</mo><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mrow><mi>y</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>C</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo></mrow></msub><mi> </mi><mi> </mi><mi> </mi><mi>i</mi><mo>≥</mo><mn>0</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{{{\Delta ^\alpha }{x_{i + 1}} = A{x_i} + B{u_i},} \hfill &amp; {i \ge 0} \hfill \cr {{x_0} = {\tau _0} + {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0} \in {\mathbb{R}^n},} \hfill &amp; {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0} \in \Omega ,} \hfill \cr {{y_i} = C{x_{i,}}\,\,\,i \ge 0} \hfill &amp; {} \hfill \cr } } \right.</tex-math>
</alternatives>
</inline-formula> 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 <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub></mrow></math>
<tex-math>{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}</tex-math>
</alternatives>
</inline-formula> 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 <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><mrow><msub><mrow><mi>α</mi></mrow><mi>j</mi></msub><msub><mrow><mi>w</mi></mrow><mi>j</mi></msub></mrow></mrow></math>
<tex-math>{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0} = \sum\limits_{j = 1}^p {{\alpha _j}{w_j}}</tex-math>
</alternatives>
</inline-formula> for some unknown coefficients α1 ≥ 0, . . . , αp ≥ 0 such that <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_004.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><mrow><msub><mrow><mi>α</mi></mrow><mi>j</mi></msub><mo>=</mo><mn>1</mn></mrow></mrow></math>
<tex-math>\sum\limits_{j = 1}^p {{\alpha _j} = 1}</tex-math>
</alternatives>
</inline-formula>. In this paper, the fractional derivative is defined in the Grünwald–Letnikov sense. We investigate the characterisation of the set χ(<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_005.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub></mrow></math>
<tex-math>{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}</tex-math>
</alternatives>
</inline-formula>, ϵ) of all possible gain matrix K that makes the system insensitive to the unknown part <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_006.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub></mrow></math>
<tex-math>{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}</tex-math>
</alternatives>
</inline-formula>, which means <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_007.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>χ</mi><mrow><mo>(</mo><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub><mo>,</mo><mo>∈</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo> <mrow><mi>K</mi><mo>∈</mo><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup><mo>/</mo><mrow><mo>‖</mo> <mrow><mfrac><mrow><mo>∂</mo><msub><mrow><mi>y</mi></mrow><mi>i</mi></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>α</mi></mrow><mi>j</mi></msub></mrow></mfrac></mrow> <mo>‖</mo></mrow><mo>≤</mo><mo>∈</mo><mo>,</mo><mo>∀</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi><mo>,</mo><mi> </mi><mo>∀</mo><mi>i</mi><mo>≥</mo><mn>0</mn></mrow> <mo>}</mo></mrow></mrow></math>
<tex-math>\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\}</tex-math>
</alternatives>
</inline-formula>, where the inequality <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_008.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mo>‖</mo> <mrow><mfrac><mrow><mo>∂</mo><msub><mrow><mi>y</mi></mrow><mi>i</mi></msub></mrow><mrow><mo>∂</mo><msub><mrow><mi>α</mi></mrow><mi>j</mi></msub></mrow></mfrac></mrow> <mo>‖</mo></mrow><mo>≤</mo><mo>∈</mo></mrow></math>
<tex-math>\left\| {{{\partial {y_i}} \over {\partial {\alpha _j}}}} \right\| \le \in</tex-math>
</alternatives>
</inline-formula> showing the sensitivity of yi relatively to uncertainties <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_009.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>{</mo> <mrow><msub><mrow><mi>α</mi></mrow><mi>j</mi></msub></mrow> <mo>}</mo></mrow></mrow></mrow> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup></mrow></math>
<tex-math>\left\{ {{\alpha _j}} \right\}_{j = 1}^p</tex-math>
</alternatives>
</inline-formula> will not achieve the specified threshold ϵ > 0. We establish, under certain hypothesis, the finite determination of χ(<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ama-2021-0029_eq_010.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mover accent="true"><mi>τ</mi><mo>⌢</mo></mover></mrow><mn>0</mn></msub></mrow></math>
<tex-math>{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}</tex-math>
</alternatives>
</inline-formula>, ϵ) and we propose an algorithmic approach to made explicit characterisation of such set.
</abstract>

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.

An Output Sensitivity Problem for a Class of Fractional Order Discrete-Time Linear Systems

Faculty of Sciences Ben M’Sik, Department of Mathematics and Computer Science

Acta Mechanica et Automatica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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...

An Output Sensitivity Problem for a Class of Fractional Order Discrete-Time Linear Systems

Online veröffentlicht: 29. Nov. 2021

Seitenbereich: 227 - 235

Eingereicht: 07. Feb. 2021

Akzeptiert: 20. Sept. 2021

DOI: https://doi.org/10.2478/ama-2021-0029

Schlüsselwörter
fractional order systems, output sensitivity, discrete-time systems, maximal output set admissible uncertainty

© 2021 Youssef Benfatah et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

An Output Sensitivity Problem for a Class of Fractional Order Discrete-Time Linear Systems

Online veröffentlicht: 29. Nov. 2021

Seitenbereich: 227 - 235

Eingereicht: 07. Feb. 2021

Akzeptiert: 20. Sept. 2021

DOI: https://doi.org/10.2478/ama-2021-0029

Schlüsselwörterfractional order systems, output sensitivity, discrete-time systems, maximal output set admissible uncertainty

© 2021 Youssef Benfatah et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Schlüsselwörter
fractional order systems, output sensitivity, discrete-time systems, maximal output set admissible uncertainty