In-Domain Control of a Heat Equation: An Approach Combining Zero-Dynamics Inverse and Differential Flatness

Zheng, Jiqiang; Zhu, Guchuan

doi:10.1155/2015/187284

Cited by 6 publications

(8 citation statements)

References 27 publications

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“…Denote by h i ( t ), i = 1,…, n , the basic outputs. The solution to is given in a previous work

\begin{matrix} right {\tilde{ξ}}_{0}^{i} = & left (k_{1} k_{2} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{x^{2 k + 1} {(x_{i} - 1)}^{2 (n - k) + 1}}{(2 k + 1)! (2 (n - k) + 1)!} h_{i}^{(n)} - k_{1} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{x^{2 k + 1} {(x_{i} - 1)}^{2 (n - k)}}{(2 k - 1)! (2 (n - k))!} h_{i}^{n} & right \\ right & left + k_{2} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{x^{2 k} {(x_{i} - 1)}^{2 (n - k) + 1}}{(2 k)! (2 (n - k) + 1)<...>}) \end{matrix}

…”

Section: Differential Flatness‐based Set‐point Control and Trajectorymentioning

confidence: 99%

“…The parameter γ i , i = 1,…, n , can be determined by the Green's functions G i , which comply with the following differential equations:

\begin{matrix} A G_{i} & = δ_{i}, i = 1, …, n, \\ G_{i x} false(0 false) & = k_{1} G_{i} false(0 false), G_{i x} false(1 false) = - k_{2} G_{i} false(1 false) . \end{matrix}

The Green's functions G i ( x ), i = 1,…, n , can be explicitly expressed as

G_{i} (x) = \{\begin{matrix} \frac{false(1 - k_{2} x_{i} + k_{2} false) false(k_{1} x + 1 false)}{α false(k_{1} + k_{2} + k_{1} k_{2} false)}, & 0 \leq x \leq x_{i}, \\ \frac{false(1 - k_{2} x + k_{2} false) false(k_{1} x_{i} + 1 false)}{α false(k_{1} + k_{2} + k_{1} k_{2} false)}, & x_{i} < x < 1 . \end{matrix}

…”

Section: Differential Flatness‐based Set‐point Control and Trajectorymentioning

confidence: 99%

“…Denoting

G = [\begin{matrix} C_{1} G_{1} & C_{1} G_{2} & C_{1} G_{3} & … & C_{1} G_{n} \\ C_{2} G_{1} & C_{2} G_{2} & C_{2} G_{3} & … & C_{2} G_{n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ C_{n} G_{1} & C_{n} G_{2} & C_{n} G_{3} & … & C_{n} G_{n} \end{matrix}]

and noting that G is an invertible n × n matrix, we obtain

([], \begin{matrix} center \\ array γ_{1} \\ array γ_{2} \\ array ⋮ \end{matrix})

…”

Section: Differential Flatness‐based Set‐point Control and Trajectorymentioning

confidence: 99%

“…The aim of this paper is to extend the approach developed in the previous work to a class of semilinear parabolic PDEs with in‐domain control and a nonlinear term that can be expressed as a polynomial. It should be noted that certain boundary controlled nonlinear parabolic PDEs can be converted to a linear one by using Hopf‐Cole transformation .…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown that Adomian series exhibits a very fast convergence, and this technique becomes a powerful tool in numerical analysis of a wide range of both static and evolutional PDEs . In the context of implementation of dynamic in‐domain control, we use ADM to solve the zero dynamics by taking the solution to the corresponding linear parabolic equation as the initial approximation, which can be obtained by applying the technique development in the previous work . Note that the technique of flat systems can be straightforwardly applied to solve motion planning problems of nonlinear PDEs under boundary control .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Yang

Zheng

Zhu

2019

Intl J Robust & Nonlinear

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Summary This paper addresses the problem of asymptotic output tracking of a class of semilinear parabolic equations with pointwise in‐domain actuation. First, the assessment of the well‐posedness of the considered systems is performed, and then, the stability of boundary controlled systems is analyzed via Chaffee‐Infante equation and Fisher's equation. The application of the zero dynamics inverse design results in a dynamic control scheme that is implemented by using the technique of trajectory planning for flat systems and the Adomian decomposition method. The convergence of the solution of the original systems to that of the corresponding zero dynamics and the convergence of the solution expressed by an Adomian series are also analyzed. Numerical simulations are carried out to illustrate the effectiveness of the developed approach.

show abstract

“…Denote by h i ( t ), i = 1,…, n , the basic outputs. The solution to is given in a previous work

\begin{matrix} right {\tilde{ξ}}_{0}^{i} = & left (k_{1} k_{2} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{x^{2 k + 1} {(x_{i} - 1)}^{2 (n - k) + 1}}{(2 k + 1)! (2 (n - k) + 1)!} h_{i}^{(n)} - k_{1} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{x^{2 k + 1} {(x_{i} - 1)}^{2 (n - k)}}{(2 k - 1)! (2 (n - k))!} h_{i}^{n} & right \\ right & left + k_{2} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{x^{2 k} {(x_{i} - 1)}^{2 (n - k) + 1}}{(2 k)! (2 (n - k) + 1)<...>}) \end{matrix}

…”

Section: Differential Flatness‐based Set‐point Control and Trajectorymentioning

confidence: 99%

“…The parameter γ i , i = 1,…, n , can be determined by the Green's functions G i , which comply with the following differential equations:

\begin{matrix} A G_{i} & = δ_{i}, i = 1, …, n, \\ G_{i x} false(0 false) & = k_{1} G_{i} false(0 false), G_{i x} false(1 false) = - k_{2} G_{i} false(1 false) . \end{matrix}

The Green's functions G i ( x ), i = 1,…, n , can be explicitly expressed as

G_{i} (x) = \{\begin{matrix} \frac{false(1 - k_{2} x_{i} + k_{2} false) false(k_{1} x + 1 false)}{α false(k_{1} + k_{2} + k_{1} k_{2} false)}, & 0 \leq x \leq x_{i}, \\ \frac{false(1 - k_{2} x + k_{2} false) false(k_{1} x_{i} + 1 false)}{α false(k_{1} + k_{2} + k_{1} k_{2} false)}, & x_{i} < x < 1 . \end{matrix}

…”

Section: Differential Flatness‐based Set‐point Control and Trajectorymentioning

confidence: 99%

“…Denoting

G = [\begin{matrix} C_{1} G_{1} & C_{1} G_{2} & C_{1} G_{3} & … & C_{1} G_{n} \\ C_{2} G_{1} & C_{2} G_{2} & C_{2} G_{3} & … & C_{2} G_{n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ C_{n} G_{1} & C_{n} G_{2} & C_{n} G_{3} & … & C_{n} G_{n} \end{matrix}]

and noting that G is an invertible n × n matrix, we obtain

([], \begin{matrix} center \\ array γ_{1} \\ array γ_{2} \\ array ⋮ \end{matrix})

…”

Section: Differential Flatness‐based Set‐point Control and Trajectorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Yang

Zheng

Zhu

2019

Intl J Robust & Nonlinear

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show abstract

Modeling the growth dynamics of Spartina alterniflora and the effects of its control measures

Zheng

Shao

Asaeda

et al. 2016

Ecological Engineering

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The method of fundamental solutions for the scattering problem of an open cavity

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Guo³

2023

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In-Domain Control of a Heat Equation: An Approach Combining Zero-Dynamics Inverse and Differential Flatness

Cited by 6 publications

References 27 publications

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

Modeling the growth dynamics of Spartina alterniflora and the effects of its control measures

The method of fundamental solutions for the scattering problem of an open cavity

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