Event-Triggered Safe Stabilizing Boundary Control for the Stefan PDE System with Actuator Dynamics Shumon Koga1 Cenk Demir2 and Miroslav Krstic2

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Event-Triggered Safe Stabilizing Boundary Control for the Stefan PDE

System with Actuator Dynamics

Shumon Koga1, Cenk Demir2, and Miroslav Krstic2

Abstract— This paper proposes an event-triggered boundary

control for the safe stabilization of the Stefan PDE system with

actuator dynamics. The control law is designed by applying

Zero-Order Hold (ZOH) to the continuous-time safe stabilizing

controller developed in our previous work. The event-triggering

mechanism is then derived so that the imposed safety conditions

associated with high order Control Barrier Function (CBF)

are maintained and the stability of the closed-loop system is

ensured. We prove that under the proposed event-triggering

mechanism, the so-called “Zeno” behavior is always avoided,

by showing the existence of the minimum dwell-time between

two triggering times. The stability of the closed-loop system is

proven by employing PDE backstepping method and Lyapunov

analysis. The efﬁcacy of the proposed method is demonstrated

in numerical simulation.

I. INTRODUCTION

Safety is an emerging notion in control systems, ensur-

ing required constraints in the state, which is a signiﬁ-

cant property in numerous industrial applications including

autonomous driving and robotics. Classically, such a con-

strained control design has been treated by model predictive

control [1], reference governor [2], reachability analysis [3],

etc. Following the pioneering work by Ames, et. al [4],

the concept of Control Barrier Function (CBF) and the safe

control design by CBF using Quadratic Programming (QP)

have been widely spread to the control community, such as

robust CBF [5], adaptive CBF [6], ﬁxed-time CBF [7]. The

system is said to be safe if the required safe set is forward

invariant, guaranteeing the positivity of CBF.

While most of the research in safe/constrained control has

focused on the systems described by Ordinary Differential

Equations (ODEs), a few recent works have been developing

and analyzing the safety or state constraints in the systems

described by Partial Differential Equations (PDEs), where the

safety in the inﬁnite-dimensional state needs to be satisﬁed,

such as for distributed concentration [8], gas density [9], or

the liquid level [10]. Our recent work [11] has incorporated

the concept of CBF into the boundary control of a PDE

system, the so-called ”Stefan system” [12], [13], which is

a representative model for the thermal melting process [14]

and biological growth process [15]. We have designed the

nonovershooting control [16] to achieve both safety and

stabilization of the system around the setpoint, and also

developed a CBF-QP safety ﬁlter for a given nominal control

1Department of Electrical and Computer Engineering, UC San Diego,

9500 Gilman Drive, La Jolla, CA, 92093-0411, skoga@ucsd.edu

2Department of Mechanical and Aerospace Engineering, UC San Diego,

9500 Gilman Drive, La Jolla, CA, 92093-0411, cdemir@ucsd.edu,

krstic@ucsd.edu

input. The proposed control law is given in continuous

time; however, many practical control systems do not afford

sufﬁciently high frequency in the actuator to justify the

continuous-time control input.

Some practical technologies own constraints in sensors

and systems with respect to energy, communication, and

computation, which require the execution of control tasks

when they are necessary [17]. To deal with this problem, the

digital control design should be aimed to reduce the number

of closing the loop based on the system state, developed as

event-triggered control. Pioneering work on event-triggered

control is designed for PID controllers in [18]. Following the

literature, authors of [19] have demonstrated the advantages

of event-driven control over time-driven control for stochas-

tic systems. The authors in [20] proposed event-triggered

scheduling to stabilize systems by a feedback mechanism.

Subsequent works [21], [22] proposed novel state feedback

and output feedback controllers for linear and nonlinear

time-invariant systems. Unlike the former literature, [23] and

[24] have regarded the closed-loop system under the event-

triggered control as a hybrid system and guaranteed asymp-

totical stability with relaxed conditions for the triggering

mechanism. To utilize the effect of this relaxation, dynamic

triggering mechanisms are presented in [25].

While all the studies of the event-triggered control men-

tioned above are for ODE systems, authors of [26] and [27]

proposed event-triggered in-domain control for hyperbolic

PDE systems, and authors of [28] proposed event-triggered

boundary control for hyperbolic PDE systems. Following

these studies, [29] developed an event-triggered boundary

controller for a reaction-diffusion PDE, and [30] derived an

adaptive event-triggered control for a hyperbolic PDE with

time-varying moving boundary with bounded velocity. As a

state-dependent moving boundary PDE, an event-triggered

control has been developed to stabilize the one-phase Stefan

PDE in [31]. However, the event-triggering mechanism in

[31] owns a limitation that the upper bound of the dwell-

time is identical to the sole condition for sampling-time

scheduling developed in [32]. Indeed, the condition of the

sampling schedule in [32] serves as the necessary condition

for ensuring the required constraint in the Stefan system.

While, such a necessary condition has not been clariﬁed in

the Stefan system with actuator dynamics considered in [11],

which requires the methodology from high-relative-degree

CBF for satisfying the state constraint.

The event-triggering mechanism has been developed for

the purpose of safety with CBF-based methods in [33], which

proves the nonexistence of the so-called “Zeno behavior”

arXiv:2210.01454v1 [math.OC] 4 Oct 2022

Fig. 1: Schematic of the one-phase Stefan problem with actuator dynamics

under event-triggered control.

by showing the existence of a lower bound of the dwell-

time. Following this work, [34] achieves safety under the

event-triggering mechanism by ensuring the existence of the

minimum bounded interevent time using input-to-state safe

barrier functions. The recent study in [35] has proposed the

safety-critical event-triggered controller design for general

nonlinear ODE systems. In addition, the event-triggered con-

trol with high-order CBFs under unknown system dynamics

has been handled in [36] by adaptive CBF approach and

in [37] by Gaussian Process learning approach. Such digital

safe control methods have been applied to spacecraft orbit

stabilization [38], network systems [39], and so on. However,

the safe event-triggered control for PDE systems has not been

established yet.

The contributions of the paper include (i) designing the

event-triggered mechanism for the Stefan PDE system with

actuator dynamics, so that both the safety and stability are

maintained, (ii) and proving the nonexistence of Zeno be-

havior by showing the existence of the minimum dwell-time

between two triggering times. Indeed, this paper provides the

ﬁrst study of the event-triggered boundary control for a PDE

system to achieve safety and stability. The safety is shown by

employing the high order CBF, and the stability is proven by

utilizing PDE backstepping method and Lyapunov analysis.

II. STEFAN MODEL AND CONSTRAINTS

Consider the melting or solidiﬁcation in a material of

length Lin one dimension (see Fig. 1). Divide the domain

[0, L]into two time-varying sub-intervals: [0, s(t)], which

contains the liquid phase, and [s(t), L], that contains the solid

phase. Let the heat ﬂux be entering at the boundary of the

liquid phase to promote the melting process

The energy conservation and heat conduction laws yield

the heat equation of the liquid phase, the boundary condi-

tions, the dynamics of the moving boundary, and the initial

values as follows:

Tt(x, t) = αTxx(x, t),for t > 0,0< x < s(t),(1)

−kTx(0, t) = qc(t),for t > 0,(2)

T(s(t), t) = Tm,for t > 0,(3)

˙s(t) = −βTx(s(t), t),(4)

s(0) = s0,and T(x, 0) = T0(x),for x∈(0, s0].(5)

The heat ﬂux qc(t)is manipulated by the voltage input U(t)

modeled by a ﬁrst-order actuator dynamics:

˙qc(t) = U(t).(6)

There are two requirements for the validity of the model

T(x, t)≥Tm,∀x∈(0, s(t)),∀t > 0,(7)

0< s(t)<L, ∀t > 0.(8)

First, the trivial: the liquid phase is not frozen, i.e.,

the liquid temperature T(x, t)is greater than the melting

temperature Tm. Second, equally trivially, the material is

not entirely in one phase, i.e., the interface remains inside

the material’s domain. These physical conditions are also

required for the existence and uniqueness of solutions [40].

Hence, we assume the following for the initial data.

Assumption 1. 0< s0< L,T0(x)∈C1([0, s0]; [Tm,+∞))

with T0(s0) = Tm.

We remark the following lemma.

Lemma 1. With Assumption 1, if qc(t)is a bounded piece-

wise continuous non-negative heat function, i.e.,

qc(t)≥0,∀t≥0,(9)

then there exists a unique classical solution for the Stefan

problem (1)–(4), which satisﬁes (7), and

˙s(t)≥0,∀t≥0.(10)

The deﬁnition of the classical solution of the Stefan

problem is given in Appendix A of [12]. The proof of Lemma

1is by maximum principle for parabolic PDEs and Hopf’s

lemma, as shown in [40].

Control Barrier Functions (CBFs) are introduced to render

the equivalency between the forward invariance of a safe set

satisfying the imposed constraints with the positivity of the

CBFs. Followng [11], let h1(t),h2(t),h3(t), and h(x, t)be

CBFs deﬁned by

h1(t) : = σ(t)

=− k

αZs(t)

(T(x, t)−Tm)dx +k

β(s(t)−sr)!,

(11)

h2(t) = qc(t),(12)

h3(t) = −qc(t) + c1σ(t),(13)

h(x, t) = T(x, t)−Tm.(14)

The functions h1(t)and h2(t)deﬁned above can be seen

as valid CBFs to satisfy the state constraints in (7) and (8)

as shown in the following lemma.

Lemma 2. With Assumption 1, suppose that the following

conditions hold:

h1(t)≥0,(15)

h2(t)≥0,(16)

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Event-TriggeredSafeStabilizingBoundaryControlfortheStefanPDESystemwithActuatorDynamicsShumonKoga1,CenkDemir2,andMiroslavKrstic2AbstractThispaperproposesanevent-triggeredboundarycontrolforthesafestabilizationoftheStefanPDEsystemwithactuatordynamics.ThecontrollawisdesignedbyapplyingZero-OrderHold(ZOH...

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Event-Triggered Safe Stabilizing Boundary Control for the Stefan PDE System with Actuator Dynamics Shumon Koga1 Cenk Demir2 and Miroslav Krstic2

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