On the Maximal Output Admissible Set for a Class of Nonlinear Discrete Systems
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Cited by 5 publications
References 11 publications
Given a discrete-time controlled bilinear systems with initial state x 0 and output function y i , we investigate the maximal output set Θ(Ω)where Ω is a given constraint set and is a subset of R p . Using some stability hypothesis, we show that Θ(Ω) can be determined via a finite number of inequations. Also, we give an algorithmic process to generate the set Θ(Ω). To illustrate our theoretical approach, we present some examples and numerical simulations. Moreover, to demonstrate the effectiveness of our approach in real-life problems, we provide an application to the SI epidemic model and the SIR model.
Given a discrete-time controlled bilinear systems with initial state x 0 and output function y i , we investigate the maximal output set Θ(Ω)where Ω is a given constraint set and is a subset of R p . Using some stability hypothesis, we show that Θ(Ω) can be determined via a finite number of inequations. Also, we give an algorithmic process to generate the set Θ(Ω). To illustrate our theoretical approach, we present some examples and numerical simulations. Moreover, to demonstrate the effectiveness of our approach in real-life problems, we provide an application to the SI epidemic model and the SIR model.
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.
Identifying the set of all admissible disturbances: discrete-time systems with perturbed gain matrix
This paper focuses on linear controlled discrete-time systems which subject to the control input disturbances. A disturbance is said to be admissible if the associated output function verifies the output constraints. In this paper, we address the following problem: determine the set of all admissible disturbances from all disturbances susceptible to the deformation of control input. An algorithm for computing the maximum admissible disturbances set is described and the sufficient conditions for finite termination of this algorithm are given. Numerical examples are given. The case of discrete-time delayed systems is also considered.
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