Distributed control of nonlinear diffusion systems by input–output linearization

Maidi, Ahmed; Corriou, Jean‐Pierre

doi:10.1002/rnc.2892

Cited by 36 publications

(22 citation statements)

References 39 publications

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“…Remark In this example, the constant parameter

c_{1}

is determined following the Kirchhoff transformation, a particular case of the Cole–Hopf tangent transformation, which yields

c_{1} = k_{r}

. More details can be found in Vadasz and Maidi and Corriou…”

Section: Illustrative Examplementioning

confidence: 99%

“…In summary, using transformation , with the operator

h false(. false)

satisfying and considering the expression of diffusivity

α

, the nonlinear diffusion equation will be converted to the following linear one:

\frac{\partial w false(z, t false)}{\partial t} = α \frac{\partial^{2} w false(z, t false)}{\partial z^{2}} + \frac{α}{c_{1}} b false(z false) u false(t false)

with the following boundary and initial conditions

w false(0, t false) = h^{- 1} false(x false(0, t false) false) = h^{- 1} false(x_{0} false) = w_{0},

w false(l, t false) = h^{- 1} false(x false(l, t false) false) = h^{- 1} false(x_{l} false) = w_{l},

w false(z, 0 false) = h^{- 1} false(ϕ false(z false) false) = ψ false(z false) .

Remark The tangent transformations represent an interesting tool that can be exploited for dynamical system analysis and design. For instance, the Cole–Hopf tangent transformation has been used by Maidi and Couriou to design a distributed state‐feedback controller to the nonlinear diffusion equation .…”

Section: Nonlinear Diffusion Systemmentioning

confidence: 99%

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Controllability of nonlinear diffusion system

Maidi

Corriou

2014

Can J Chem Eng

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International audienceIn this article, the semi-group theory is exploited to study the controllability of a particular but important class of distributed parameter systems with a uniform distributed control variable. This class is described by a partial differential equation that characterizes the nonlinear diffusion phenomenon. The main idea consists of using the Cole-Hopf tangent transformation, under some assumptions, in order to derive an equivalent linear model of the nonlinear diffusion system. Then, the study of the controllability of the linear model, based on the semi-group approach, allows us to conclude about the controllability of the nonlinear diffusion system. The study is illustrated by an application example

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“…Remark In this example, the constant parameter

c_{1}

is determined following the Kirchhoff transformation, a particular case of the Cole–Hopf tangent transformation, which yields

c_{1} = k_{r}

. More details can be found in Vadasz and Maidi and Corriou…”

Section: Illustrative Examplementioning

confidence: 99%

“…In summary, using transformation , with the operator

h false(. false)

satisfying and considering the expression of diffusivity

α

, the nonlinear diffusion equation will be converted to the following linear one:

\frac{\partial w false(z, t false)}{\partial t} = α \frac{\partial^{2} w false(z, t false)}{\partial z^{2}} + \frac{α}{c_{1}} b false(z false) u false(t false)

with the following boundary and initial conditions

w false(0, t false) = h^{- 1} false(x false(0, t false) false) = h^{- 1} false(x_{0} false) = w_{0},

w false(l, t false) = h^{- 1} false(x false(l, t false) false) = h^{- 1} false(x_{l} false) = w_{l},

w false(z, 0 false) = h^{- 1} false(ϕ false(z false) false) = ψ false(z false) .

Section: Nonlinear Diffusion Systemmentioning

confidence: 99%

Controllability of nonlinear diffusion system

Maidi

Corriou

2014

Can J Chem Eng

Self Cite

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“…In particular, feedback control of diffusion-type (parabolic) PDEs has been a subject of extensive research and several remarkable results have been produced [21][22][23][24]. For the control of the heat diffusion PDE boundary and distributed control methods have been developed [25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Differential Flatness Properties and Feedback Control of the Fokker–Planck PDE

2016

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The Fokker-Planck PDE describes the dynamics of several biochemical and biological processes, as well as of phenomena at molecular scale. In this paper, the problem of feedback control of the Fokker-Planck PDE is studied. It is shown that the procedure for numerical solution of Fokker-Planck PDE results into a set of nonlinear ordinary differential equations (ODEs) and an associated state equations model. For the local subsystems, into which a Fokker-Planck PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the Fokker-Planck PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the Fokker-Planck PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the Fokker-Planck PDE system so as to assure that all its state variables will converge to the desirable setpoints.

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“…The problem becomes more difficult in case that the distributed parameter systems are characterized by nonlinearities [4][5][6][7][8][9]. The paper treats the problem of piecewise control of nonlinear wave-type partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Control of the Nonlinear Wave-Type Dynamics Using the Derivative-Free Nonlinear Kalman Filter

Rigatos

Siano

2016

Intell Ind Syst

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The objective of the paper is to develop a pointwise control method for a 1D nonlinear wave equation and a filtering approach for estimating the dynamics of such a system from measurements provided from a small number of sensors. It is shown that the numerical solution of the associated partial differential equation results into a set of nonlinear ordinary differential equations. With the application of a diffeomorphism that is based on differential flatness theory it is shown that an equivalent description of the system in the linear canonical (Brunovsky) form is obtained. This transformation enables to obtain estimates about the state vector of the system through the application of the standard Kalman Filter recursion. For the local subsystems, into which the nonlinear wave equation is decomposed, it becomes possible to apply pointwise state estimation-based feedback control. The efficiency of the proposed filtering and control approach for nonlinear systems described by 1D partial differential equations of the wave type (e.g. sine-Gordon PDE) is confirmed through simulation experiments. It is shown that, by applying feedback control, the nonlinear wave-type dynamics can be made to track any reference setpoint.

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Distributed control of nonlinear diffusion systems by input–output linearization

Cited by 36 publications

References 39 publications

Controllability of nonlinear diffusion system

Controllability of nonlinear diffusion system

Differential Flatness Properties and Feedback Control of the Fokker–Planck PDE

Control of the Nonlinear Wave-Type Dynamics Using the Derivative-Free Nonlinear Kalman Filter

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