Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

Koga, Shumon; Krstić, Miroslav

doi:10.1109/tac.2018.2836018

Cited by 44 publications

(52 citation statements)

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“…The definition of the classical solution of the Stefan problem can be seen in the literature, for instance see appendix A in Reference . The proof of Lemma is provided by maximum principle as shown in Reference .…”

Section: Description Of the Physical Processmentioning

confidence: 99%

“…The control objective is to drive the moving interface s ( t ) to a desired setpoint s r by manipulating the heat flux q c ( t ). As derived in Reference , the steady‐state solution of the temperature profile T eq ( x ) governed by the Stefan problems to with setting the equilibrium interface position as the setpoint s r is uniquely given by the uniform melting temperature, namely, T eq ( x )= T m for all x ∈[0, s r ]. Hence, the primary objective is driving s ( t ) to s r , and consequently the convergence of T ( x , t ) to T m for all x ∈[0, s ( t )] is required: we aim to achieve

\begin{align} s false(t false) \to s_{normalr}, T false(x, t false) \to T_{normalm} for all x \in false[0, s false(t false) false], as t \to \infty, \end{align}

for given ( T 0 ( x ), s 0 ) which satisfies Assumption .…”

Section: Control Problem Statementmentioning

confidence: 99%

“…The transformation is the same transformation as the one proposed in Reference for delay‐free Stefan problem. The formulation of is motivated by a design in fixed domain introduced in Reference .…”

Section: Backstepping Transformationmentioning

confidence: 99%

“…In addition, by substituting x =0 in and , w x (0, t )=− z (0, t ) holds. On the other hand, because w transformation does not depend on v , w system is not changed from the delay‐free target system given in Reference . Thus, the target ( z , w , X )‐system is obtained by

\begin{align} z_{t} false(x, t false) & = prefix- z_{x} false(x, t false), prefix- D < x < 0, \end{align}

\begin{align} z false(prefix- D, t false) & = 0, \end{align}

\begin{align} w_{x} false(0, t false) & = prefix- z false(0, t false), \end{align}

\begin{align} w_{t} false(x, t false) & = α w_{x x} false(x, t false) + \frac{c}{β} \overset{s}{true} false(t false) X false(t false), 0 < x < s false(t false) . \end{align}

\begin{align} w false(s false(t false), t false) & = 0, \end{align}

\begin{align} \overset{X}{true} false(t false) & = prefix- c X false(t false... \end{align}

…”

Section: Backstepping Transformationmentioning

confidence: 99%

“…First, combining our previous result in Reference with the result in Reference , two nonlinear backstepping transformations for moving boundary PDE are employed. One is for the delay‐free control design of Stefan problem based on Reference , and the other is for the compensation of actuator delay formulated with Volterra‐ and Fredholm‐type transformations based on Reference . The associated boundary feedback controller remains positive under a setpoint restriction due to the energy conservation, which guarantees a condition of the model to be valid.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Delay compensated control of the Stefan problem and robustness to delay mismatch

Koga

Bresch‐Pietri

Krstić

2020

Intl J Robust & Nonlinear

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Summary This paper presents a control design for the one‐phase Stefan problem under actuator delay via a backstepping method. The Stefan problem represents a liquid‐solid phase change phenomenon which describes the time evolution of a material's temperature profile and the interface position. The actuator delay is modeled by a first‐order hyperbolic partial differential equation (PDE), resulting in a cascaded transport‐diffusion PDE system defined on a time‐varying spatial domain described by an ordinary differential equation (ODE). Two nonlinear backstepping transformations are utilized for the control design. The setpoint restriction is given to guarantee a physical constraint on the proposed controller for the melting process. This constraint ensures the exponential convergence of the moving interface to a setpoint and the exponential stability of the temperature equilibrium profile and the delayed controller in the ℋ1 norm. Furthermore, robustness analysis with respect to the delay mismatch between the plant and the controller is studied, which provides analogous results to the exact compensation by restricting the control gain.

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Section: Description Of the Physical Processmentioning

confidence: 99%