Observer-based Event-triggered Boundary Control of the One-phase Stefan Problem Bhathiya Rathnayakeaand Mamadou Diagneb

2025-05-02 0 0 929.44KB 22 页 10玖币

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

Stefan Problem

Bhathiya Rathnayakeaand Mamadou Diagneb

aDepartment of Electrical and Computer Engineering, University of California San Diego,

9500 Gilman Dr, La Jolla, CA 92093, USA;

bDepartment of Mechanical and Aerospace Engineering, University of California San Diego,

9500 Gilman Dr, La Jolla, CA 92093, USA

ABSTRACT

This paper provides an observer-based event-triggered boundary control strategy for

the one-phase Stefan problem using the position and velocity measurements of the

moving interface. The inﬁnite-dimensional backstepping approach is used to design

the underlying observer and controller. For the event-triggered implementation of

the continuous-time observer-based controller, a dynamic event triggering condition

is proposed. The triggering condition determines the times at which the control

input needs to be updated. In between events, the control input is applied in a

Zero-Order-Hold fashion. It is shown that the dwell-time between two triggering

instances is uniformly bounded below excluding Zeno behavior. Under the proposed

event-triggered boundary control approach, the well-posedness of the closed-loop

system along with certain model validity conditions is provided. Further, using Lya-

punov approach, the global exponential convergence of the closed-loop system to

the setpoint is proved. A simulation example is provided to illustrate the theoretical

results.

KEYWORDS

Backstepping control design, event-triggered control, moving boundaries,

output-feedback, Stefan problem.

1. Introduction

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 (Chen, Bentsman, &

Thomas, 2020; Petrus, Chen, Bentsman, & Thomas, 2017), cyrosurgical operations

(Rabin & Shitzer, 1997), modeling of tumor growth (Friedman & Reitich, 1999), and

information diﬀusion in social media (Lei, Lin, & Wang, 2013). 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, 1979).

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

CONTACT Bhathiya Rathnayake. Email: brm222@ucsd.edu

arXiv:2210.00412v2 [eess.SY] 11 Aug 2023

by the seminal work (Dunbar, Petit, Rouchon, & Martin, 2003) on boundary control of

the Stefan problem, numerous works have made contributions to tackle this challenging

moving boundary PDE control problem (Chen, Bentsman, & Thomas, 2019; Chen et

al., 2020; Ecklebe, Woittennek, Frank-Rotsch, Dropka, & Winkler, 2021; Koga, Bresch-

Pietri, & Krstic, 2020; Koga, Diagne, & Krstic, 2018; Koga, Karafyllis, & Krstic, 2018;

Maidi & Corriou, 2014; Petrus, Bentsman, & Thomas, 2012; Petrus et al., 2017). An

enthalpy-based full-state feedback boundary controller is proposed in (Petrus et al.,

2012) to ensure the asymptotic convergence of the closed-loop system to the setpoint.

Compensating for the eﬀect of input hysteresis, the authors of (Chen et al., 2019) and

(Chen et al., 2020) develop full-state feedback and output feedback designs for the

control of the Stefan problem, respectively. Using a geometric control approach, Maidi

and Corriou (2014) 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, & Krstic, 2018; Koga & Krstic, 2020) and application settings (Koga & Krstic,

2020; Koga, Straub, Diagne, & Krstic, 2020) using the inﬁnite-dimensional backstep-

ping control approach which has been instrumental in the control of a wide variety of

PDEs (Krstic & Smyshlyaev, 2008). For the one-phase Stefan problem, the pioneering

contribution (Koga, Diagne, & Krstic, 2018) discusses a full-state feedback control

design along with robustness guarantees to parameter uncertainties, an observer de-

sign, 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 H1-norm.

In (Koga, Karafyllis, & Krstic, 2021), the authors consider the Zero-Order-Hold

(ZOH) implementation of the full-state feedback continuous-time stabilizing controller

introduced in (Koga, Diagne, & Krstic, 2018), 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 et al., 2017), comprehensive and relevant

insights are provided into aperiodic sampled-data controller design, as well as limita-

tions 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 di-

ameter. 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 sam-

pling 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, Johansson, & Tabuada,

2012). 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 numer-

ous contributions during the past decade introducing event-triggered control strategies

to control for PDE systems (Diagne & Karafyllis, 2021; Espitia, 2020; Espitia, Karafyl-

lis, & Krstic, 2021; Katz, Fridman, & Selivanov, 2020; Rathnayake & Diagne, 2022;

Rathnayake, Diagne, Espitia, & Karafyllis, 2021; Rathnayake, Diagne, & Karafyllis,

2022; Wang & Krstic, 2021), to name a few. For 2×2 linear hyperbolic systems, an

output feedback event-triggered boundary control strategies relying on dynamic trig-

gering conditions is proposed in (Espitia, 2020). The authors of (Espitia et al., 2021)

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 et al.,

2021) and (Rathnayake et al., 2022) develop output feedback control strategies for a

class of reaction-diﬀusion PDEs under anti-collocated and collocated boundary sens-

ing and actuation, respectively. A full-state feedback event-triggered boundary control

strategy for the one-phase Stefan problem is proposed in (Rathnayake & Diagne, 2022)

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, Chen, Krstic, & Pisano, 2020), 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, & Krstic, 2018) 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 et

al., 2021, 2022) in that this paper involves a moving boundary making the Lyapunov

analysis substantially diﬀerent. Moreover, the Lyapunov candidate function involves

the H1-norm of the observer error target system unlike in (Rathnayake et al., 2021,

2022) where the L2-norm is suﬃcient. As opposed to (Rathnayake et al., 2021, 2022),

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: R+is the nonnegative real line whereas Nis 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)] ×R+→Rbe given. u[t] denotes the proﬁle of uat certain t≥0, i.e.,

u[t](x), for all x∈[0, s(t)]. By ∥u[t]∥=Rs(t)

0u2(x, t)dx1/2we denote L2(0, s(t))-

norm. Im(·),and Jm(·) with mbeing an integer respectively denote modiﬁed Bessel

and (nonmodiﬁed) Bessel functions of the ﬁrst kind.

2. Description of the One-phase Stefan Problem

Let us consider a physical model that describes the melting or solidiﬁcation process

in a pure one-component material of length Lin 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 Tmof 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.

Tt(x, t) = αTxx(x, t), α := k

ρCp

,0< x < s(t),(1)

with the boundary conditions

T(s(t), t) = Tm,(2)

−kTx(0, t) = q(t),(3)

and the initial values

T(x, 0) = T0(x), s(0) = s0,(4)

where T(x, t), q(t), ρ, Cp,and kare 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:

˙s(t) = −βTx(s(t), t), β := k

ρ∆H∗,(5)

where ∆H∗is the latent heat of fusion.

The validity of the physical model (1)-(5) relies on two physical conditions (Koga,

Karafyllis, & Krstic, 2019):

T(x, t)≥Tm,∀x∈[0, s(t)],∀t > 0,(6)

0< s(t)< L, ∀t > 0.(7)

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.

Figure 1. Description of the one-phase Stefan problem.

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

on the initial data:

Assumption 1. s0∈(0, L), T0(x)≥Tmfor all x∈[0, s0],and T0(x)is continuously

diﬀerentiable in x∈[0, s0].

The well-posedness of the solution of the one-phase Stefan problem (1)-(5) has been

presented in (Cannon & Primicerio, 1971) and Lemma 1 in (Koga et al., 2019) which

we state as follows:

Lemma 1. Subject to Assumption 1, if q(t)is a bounded piece-wise continuous func-

tion producing nonnegative heat for a time interval, i.e., q(t)≥0, for all t∈[0,¯

t],

then there exists a unique solution for the Stefan problem (1)-(5) for all t∈[0,¯

t],and

the condition (6) is satisﬁed for all t∈[0,¯

t].Furthermore, it holds that

˙s(t)≥0,∀t∈[0,¯

t].(8)

3. Observer-based Backstepping Boundary Control and Emulation

The steady-state solution (Teq(x), seq) of the system (1)-(5) with zero input q(t)=0

delivers a uniform temperature distribution Teq(x) = Tmand a constant interface

position determined by initial data. In (Koga, Diagne, & Krstic, 2018), the authors

proposed a continuous-time observer-based backstepping boundary controller using

s(t) and Tx(s(t), t) (or equivalently ˙s(t) due to (5)) as the available measurements

to exponentially stabilize the interface position s(t) at a desired reference setpoint sr

through the design of q(t) as

q(t) = −ck

αZs(t)

ˆu(x, t)dx +k

βX(t),(9)

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Observer-basedEvent-triggeredBoundaryControloftheOne-phaseStefanProblemBhathiyaRathnayakeaandMamadouDiagnebaDepartmentofElectricalandComputerEngineering,UniversityofCaliforniaSanDiego,9500GilmanDr,LaJolla,CA92093,USA;bDepartmentofMechanicalandAerospaceEngineering,UniversityofCaliforniaSanDiego,9500G...

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Observer-based Event-triggered Boundary Control of the One-phase Stefan Problem Bhathiya Rathnayakeaand Mamadou Diagneb

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