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

Abstract

Consider the linear discrete-time fractional order systems with uncertainty on the initial state { Δ α x i + 1 = A x i + B u i , i ≥ 0 x 0 = τ 0 + τ ⌢ 0 ∈ ℝ n , τ ⌢ 0 ∈ Ω , y i = C x i , 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 = 1 p α j w j {\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 = 1 p α j = 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 ∈ ℝ m × n / ‖ ∂ y i ∂ α j ‖ ≤ ∈ , ∀ 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 ‖ ∂ y i ∂ α j ‖ ≤ ∈ \left\| {{{\partial {y_i}} \over {\partial {\alpha _j}}}} \right\| \le \in showing the sensitivity of yi relatively to uncertainties { α j } j = 1 p \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.