Observer-based Event-triggered Boundary Control of the One-phase Stefan Problem

In recent decades, the study of Stefan-type moving boundary problems driven by parabolic equations has found a new momentum due to the expansion of its interests into thriving areas of research such as additive manufacturing (Petrus \BOthers., \APACyear2017; Chen \BOthers., \APACyear2020), cyrosurgical operations (Rabin \BBA Shitzer, \APACyear1997), modeling of tumor growth (Friedman \BBA Reitich, \APACyear1999), and information diffusion in social media (Lei \BOthers., \APACyear2013). The mathematical formulation of the classical one-phase Stefan problem for a monocomponent two phase material involves a diffusion partial differential equation (PDE) in cascade with an ordinary differential equation (ODE). The PDE describes the thermal expansion of one phase along its dynamic spatial domain whereas the ODE captures the dynamics of the moving interface between the two phases (Rubinstein, \APACyear1979).

The control of the Stefan problem deals with the stabilization of the temperature profile and the moving interface to a desired setpoint. During the past decade, inspired by the seminal work (Dunbar \BOthers., \APACyear2003) on boundary control of the Stefan problem, numerous works have made contributions to tackle this challenging moving boundary PDE control problem (Petrus \BOthers., \APACyear2012; Chen \BOthers., \APACyear2020; Petrus \BOthers., \APACyear2017; Chen \BOthers., \APACyear2019; Maidi \BBA Corriou, \APACyear2014; Koga, Diagne\BCBL \BBA Krstic, \APACyear2018; Koga, Bresch-Pietri\BCBL \BBA Krstic, \APACyear2020; Koga, Karafyllis\BCBL \BBA Krstic, \APACyear2018; Ecklebe \BOthers., \APACyear2021). An enthalpy-based full-state feedback boundary controller is proposed in (Petrus \BOthers., \APACyear2012) to ensure the asymptotic convergence of the closed-loop system to the setpoint. Compensating for the effect of input hysteresis, the authors of (Chen \BOthers., \APACyear2019) and (Chen \BOthers., \APACyear2020) develop full-state feedback and output feedback designs for the control of the Stefan problem, respectively. Using a geometric control approach, Maidi \BBA Corriou (\APACyear2014) achieves exponential stability of the closed-loop system for the one-phase Stefan problem via Lyapunov analysis. In recent years, Koga and coauthors have addressed the control of the Stefan problem in both theoretical settings (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018; Koga \BBA Krstic, \APACyear2020) and application settings (Koga, Straub\BCBL \BOthers., \APACyear2020; Koga \BBA Krstic, \APACyear2020) using the infinite-dimensional backstepping control approach which has been instrumental in the control of a wide variety of PDEs (Krstic \BBA Smyshlyaev, \APACyear2008). For the one-phase Stefan problem, the pioneering contribution (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018) discusses a full-state feedback control design along with robustness guarantees to parameter uncertainties, an observer design, and the corresponding output feedback control design under both Dirichlet and Neumann boundary actuations via backstepping approach, ensuring the exponential stability of the closed-loop system in $H_{1}$-norm.

In (Koga \BOthers., \APACyear2021), the authors consider the Zero-Order-Hold (ZOH) implementation of the full-state feedback continuous-time stabilizing controller introduced in (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018), leading to an aperiodic sampled-data control approach for the one-phase Stefan problem. Aperiodic sampled-data control strategies, which rely on nonuniform sampling schedules, are quite appealing as they point towards efficient use of limited hardware, software, and communication resources. In a relatively recent survey paper (Hetel \BOthers., \APACyear2017), comprehensive and relevant insights are provided into aperiodic sampled-data controller design, as well as limitations and challenges in their practical implementation. Although nonuniform sampling schedules in aperiodic sampled-data control offer increased flexibility, designers still have to manually select a schedule that adheres to the maximum allowable sampling diameter. This selection remains independent of the closed-loop system state, rendering the decision-making process open-loop. Event-triggered control strategies, on the other hand, provide a systematic solution to this drawback by bringing feedback to the sampling process. An event-triggered system transmits the system’s states/outputs to a controller/actuator when the freshness in the sample exceeds an appropriate threshold involving the current state of the closed-loop system (Heemels \BOthers., \APACyear2012). Only at the event times is the feedback loop closed, and between successive event times, the control is executed in an open-loop fashion. There have been numerous contributions during the past decade introducing event-triggered control strategies to control for PDE systems (Espitia, \APACyear2020; Katz \BOthers., \APACyear2020; Espitia \BOthers., \APACyear2021; Diagne \BBA Karafyllis, \APACyear2021; Rathnayake \BOthers., \APACyear2021, \APACyear2022; Wang \BBA Krstic, \APACyear2021; Rathnayake \BBA Diagne, \APACyear2022), to name a few. For 2$\times$2 linear hyperbolic systems, an output feedback event-triggered boundary control strategies relying on dynamic triggering conditions is proposed in (Espitia, \APACyear2020). The authors of (Espitia \BOthers., \APACyear2021) propose a full-state feedback event-triggered boundary control approach for reaction-diffusion PDEs with Dirichlet boundary conditions using ISS properties and small gain arguments. Using dynamic event-triggering conditions, the works (Rathnayake \BOthers., \APACyear2021) and (Rathnayake \BOthers., \APACyear2022) develop output feedback control strategies for a class of reaction-diffusion PDEs under anti-collocated and collocated boundary sensing and actuation, respectively. A full-state feedback event-triggered boundary control strategy for the one-phase Stefan problem is proposed in (Rathnayake \BBA Diagne, \APACyear2022) using a static triggering condition.

This paper considers the output feedback boundary control of the one-phase Stefan problem using the position and velocity measurements of the moving interface. We propose an observer-based event-triggered boundary control strategy using a dynamic triggering condition under which we show that the closed-loop system is free from Zeno phenomenon. To the best of our knowledge, this work is the first to present an observer-based event-triggered boundary control approach for moving boundary type problems. In (Koga, Makihata\BCBL \BOthers., \APACyear2020), the authors propose a sampled-data observer-based boundary control design for the one-phase Stefan problem, yet with no theoretical guarantees. At event-times dictated by the proposed triggering condition, the continuous-time observer-based boundary control law derived in (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018) is computed and applied to the plant in a ZOH fashion. The dynamic event-trigger makes use of a dynamic variable that depends on some information of the current states of the closed-loop system and the actuation deviation between the continuous-time boundary feedback and the event-triggered boundary control. We also prove that the closed-loop system is well-posed satisfying certain model validity conditions and globally exponentially converges to the setpoint subject to the proposed event-triggered control. The present work differs from (Rathnayake \BOthers., \APACyear2021, \APACyear2022) in that this paper involves a moving boundary making the Lyapunov analysis substantially different. Moreover, the Lyapunov candidate function involves the $H_{1}$-norm of the observer error target system unlike in (Rathnayake \BOthers., \APACyear2021, \APACyear2022) where the $L_{2}$-norm is sufficient. As opposed to (Rathnayake \BOthers., \APACyear2021, \APACyear2022), dwell-times between consecutive events in the Stefan problem has to be upper-bounded to maintain the positivity of the control input. Thus, careful design of the event-triggering mechanism is required to ensure that the minimal dwell-time is smaller than the largest dwell-time, otherwise, the well-posedness of the closed-loop system fails to exist.

The paper is organized as follows. Section 2 describes the one-phase Stefan problem and Section 3 presents the continuous-time observer-based backstepping boundary control and its emulation. In section 4, we introduce the event-triggered boundary control approach and present the main results of the paper. We conduct simulations in Section 5 and conclude the paper in Section 6.

Notation: $\mathbb{R}_{+}$ is the nonnegative real line whereas $\mathbb{N}$ is the set of natural numbers including zero. $t^{+}$ and $t^{-}$ respectively denote the right and left limit at time $t$. Let $u:[0,s(t)]\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ be given. $u[t]$ denotes the profile of $u$ at certain $t\geq 0$, i.e., $\big{(}u[t]\big{)}(x)$, for all $x\in[0,s(t)]$. By $\|u[t]\|=\Big{(}\int_{0}^{s(t)}u^{2}(x,t)dx\Big{)}^{1/2}$ we denote $L_{2}(0,s(t))$-norm. $I_{m}(\cdot),$ and $J_{m}(\cdot)$ with $m$ being an integer respectively denote modified Bessel and (nonmodified) Bessel functions of the first kind.

Let us consider a physical model that describes the melting or solidification process in a pure one-component material of length $L$ in one dimension. The position $s(t)$ at which the phase transition occurs divides the domain $[0,L]$ into two time-varying sub-domains; the interval $[0,s(t)]$ containing the liquid phase, and the interval $[s(t),L]$ containing the solid phase. The dynamics of the position of the liquid-solid interface is driven by a heat flux entering through the boundary at $x=0$ (the fixed boundary of the liquid phase). The heat equation coupled with the dynamics that describes the moving boundary is used to characterize the heat propagation in the liquid phase and the phase transition. Fig. 1 illustrates this configuration.

Under the assumption that the temperature in the liquid phase is not lower than the melting temperature $T_{m}$ of the material, the conservation of energy and heat conduction laws can be used to derive the following PDE-ODE cascade system known as the one-phase Stefan Problem.

with the boundary conditions

and the initial values

where $T(x,t),q(t),\rho,C_{p},$ and $k$ are the liquid phase distributed temperature, applied heat flux, the liquid density, the liquid heat capacity, and the liquid heat conductivity, respectively. By considering the local energy balance at the liquid-solid interface $x=s(t)$, the following ODE associated with the time-evolution of the spatial domain can be obtained:

where $\Delta H^{*}$ is the latent heat of fusion.

The validity of the physical model (1)-(5) relies on two physical conditions (Koga \BOthers., \APACyear2019):

The first condition implies that the liquid phase should not be frozen to the solid phase from the boundary $x=0$. The second condition implies that the material should not be completely melted or frozen to single phase through the disappearance of the other phase.

To be consistent with the conditions (6) and (7), we make the following assumptions on the initial data:

The well-posedness of the solution of the one-phase Stefan problem (1)-(5) has been presented in (Cannon \BBA Primicerio, \APACyear1971) and Lemma 1 in (Koga \BOthers., \APACyear2019) which we state as follows:

The steady-state solution $(T_{eq}(x),s_{eq})$ of the system (1)-(5) with zero input $q(t)=0$ delivers a uniform temperature distribution $T_{eq}(x)=T_{m}$ and a constant interface position determined by initial data. In (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018), the authors proposed a continuous-time observer-based backstepping boundary controller using $s(t)$ and $T_{x}(s(t),t)$ (or equivalently $\dot{s}(t)$ due to (5)) as the available measurements to exponentially stabilize the interface position $s(t)$ at a desired reference setpoint $s_{r}$ through the design of $q(t)$ as

where $c>0$ is the control gain and $\hat{u}(x,t)$ and $X(t)$ are reference error variables defined as

Here, $\hat{T}(x,t)$ is the observer state which satisfies

for $0<x<s(t)$ and

with $p(x,s(t))$ being the observer gain given by

for $0<x<s(t)$.

We aim to stabilize the closed-loop system containing the plant (1)-(5) and the observer (12)-(15) while sampling the continuous-time controller $q(t)$ given by (9) at a certain sequence of time instants $(t_{j})_{j\in\mathbb{N}}$. These time instants will be fully characterized later via a dynamic event-trigger. The control input is held constant between two consecutive time instants. Therefore, we define the control input for all $t\in[t_{j},t_{j+1}),j\in\mathbb{N}$ as

Accordingly, the boundary conditions (3) and (14) are modified as follows:

for $t\in[t_{j},t_{j+1}),j\in\mathbb{N}$. Let the observer error state be defined as

Therefore, considering (1),(2),(12),(13),(17),(18), we can obtain that

for all $t>0$. In (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018), the authors show that, subject to the following invertible backstepping transformation

where

for $0\leq x\leq y\leq s(t)$, the observer error system (20)-(22) with the gain $p(x,s(t))$ chosen as in (15) gets transformed into the following globally $H_{1}$-exponentially stable observer error target system

The inverse transformation of (23) is given by

where

for $0\leq x\leq y\leq s(t)$. Considering (10),(12),(13), and (18), we can obtain that $\hat{u}$-system satisfies

whereas considering (5),(10),(11),(19), we can show that the dynamics of $X(t)$ satisfies

Let us consider the following backstepping transformation on $t\in[t_{j},t_{j+1}),j\in\mathbb{N}$ as in (Koga \BOthers., \APACyear2019),

where

It can be shown that the backstepping transformation (34),(35) transforms system (30)-(33) into the following system valid for $t\in[t_{j},t_{j+1})$:

where

and

for $t\in[t_{j},t_{j+1}),j\in\mathbb{N}$.

The inverse transform of (34) is given by

for $t\in[t_{j},t_{j+1}),j\in\mathbb{N}$ where

with

and $0<\varepsilon<2\frac{\sqrt{\alpha c}}{\beta}$ to be chosen later.

Next we present the observer-based event-triggered boundary control approach for the one-phase Stefan problem. The closed-loop system consisting of the plant, the observer-based controller, and the event-trigger is shown in Fig. 2.

Proof: For $t\in(t_{j},t_{j+1})$ where $j<j^{*}$, differentiating (9) along the solution of (30)-(33), we can obtain that

Subject to the conditions (49) and (50), following Lemma 3 and 4 in (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018) which make use of Maximum principle and Hopf’s lemma, one can show that $\tilde{u}_{x}(s(t),t)>0$. Note from (15) that $p(x,s(t))<0$ for $0<x<s(t)$. Thus, we can obtain that $\dot{q}(t)>-cq_{j},$ for $t\in(t_{j},t_{j+1})$ which we integrate in $t\in[t_{j},t_{j+1})$ to obtain

We use this recursively to derive that

One can easily show that $q_{0}>0$ when the conditions (49) and (51) are met. Furthermore, under the event-triggered boundary control approach in Definition 1, it is ensured that $1-c(t_{i+1}-t_{i})\geq 0$. Therefore, we have that $q_{j}>0$ for all $j\in\mathbb{N}$ such that $j<j^{*}$. Thus, recalling Lemma 1, we can conclude that the plant (1),(2),(4),(5),(16),(17) has a unique solution satisfying the conditions (6) and (8) in the interval $[0,\sup(I))$. The observer error system (20)-(22) has a unique solution due to the transformation (23) and as the observer error target system (25)-(27) admits a unique solution. Thus, the observer PDE (12),(13),(15),(16),(18) also admits a unique solution in the interval $[0,\sup(I))$ due to (19). Again, subject to the conditions (49) and (50), one can show that $\tilde{u}(x,t)<0$ as in Lemma 3 and 4 in (Koga, Diagne\BCBL \BBA Krstic, \APACyear2018). Thus, considering (6),(10),(19), we can verify that $\hat{u}(x,t)>0$. Further, note from (53) that $q(t)>0$ for $t\in[0,\sup(I))$. Thus, considering (9), we can show that $X(t)<0$ for $t\in[0,\sup(I))$ which in combination with (8) and (11) leads to the property (7) for $t\in[0,\sup(I))$. ∎

Proof: According to Definition 1, the triggering of events guarantee that $d^{2}(t)\leq\gamma m(t),t\in[0,\sup(I))$. This inequality in combination with (47) yields:

for $t\in(t_{j},t_{j+1})$ and $j\in\mathbb{N}$ such that $j<j^{*}$ where $j^{*}$ is given by (52). Thus, considering the time-continuity of $m(t)$, we can obtain the following estimate:

for all $t\in[t_{j},t_{j+1}]$. From Definition 1, we have that $m(t_{0})=m(0)>0$. Therefore, it follows from (56) that $m(t)>0$ for all $t\in[0,t_{1}]$. Again using (56) on $[t_{1},t_{2}]$, we can show that $m(t)>0$ for all $t\in[t_{1},t_{2}]$. Applying the same reasoning successively to the future intervals, it can be shown that $m(t)>0$ for $t\in[0,\sup(I))$. ∎

Proof: For $t\in(t_{j},t_{j+1}),j\in\mathbb{N}$, differentiating $d(t)$ given by (41) along the solution of (30)-(33) and using (41), we can obtain that

Using Cauchy-Schwarz inequality and Young’s inequality along with the fact that $0<x<s(t)<L$, we can obtain from (59) that

for all $t\in(t_{j},t_{j+1})$ and $j\in\mathbb{N}$ such that $j<j^{*}$ where $j^{*}$ is given by (52). ∎