Optimal control problem for a class of bilinear systems via block pulse functions
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Cited by 7 publications
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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.
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.
Tracking control design for nonlinear polynomial systems via augmented error system approach and block pulse functions technique
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The present study tackles the tracking control problem for unstructured uncertain bilinear systems with multiple time-delayed states subject to control input constraints. First, a new method is introduced to design memory state feedback controllers with compensator gain based on the use of operational properties of block-pulse functions basis. The proposed technique permits transformation of the posed control problem into a constrained and robust optimization problem. The constrained robust least squares approach is then used for determination of the control gains. Second, new sufficient conditions are proposed for the practical stability analysis of the closed-loop system, where a domain of attraction is estimated. A real-world example, the headbox control of a paper machine, demonstrates the efficiency of the proposed method.
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