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\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, y_i 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, u_i = Kx_i is feedback control and Ω ⊂ ℝn is a polytope convex of vertices w₁, w₂, . . . , w_p. 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, . . . , α_p ≥ 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 y_i 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.