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Event-triggered Backstepping Control of 2×2 Hyperbolic PDE-ODE Systems

Motivated by vibration control of a mining cable elevator avoiding frequent actions of the actuator which is a massive hydraulic cylinder at the head sheave, we present an event-triggered backstepping boundary controller for a 2×2 coupled hyperbolic PDE-ODE system. A two-step design is proposed including the design of a low-pass-filter-based backstepping boundary stabilization law and the sequent design of an event-trigger mechanism. The proof of the existence of a nonzero minimal dwell-time between two triggering times, and the exponential stability result of the event-based closedloop system are given in this paper.


INTRODUCTION
A mining cable elevator is a vital device used to transport the mines and miners between thousands of metres underground and ground. In order to suppress vibrations in the long compliant cable to reduce the fatigue damage, a vibration control force is applied at the head sheave, which is designed based on the PDE model of the mining cable elevator Wang et al. (2018b)-Wang et al. (2018d via backstepping. Two challenges existing in implementation of the designed PDE backstepping controller into practice are 1) the control input signal is changing rapidly so that it is hard for the actuator which is a heavy hydraulic cylinder shown in Fig. 1 to follow; 2) the high-frequency components in the control input may in turn become a vibration source for the cable. It is thus required to reduce the action frequency of the actuator, and meanwhile ensure the suppression of the vibrations in the cable.
Designing sampling schemes applied into the control input is a potential solution. Designs of sampled-data control laws of parabolic and hyperbolic PDEs were presented in Fridman et al. (2012); Karafyllis et al. (2018) and Davo et al. (2018); Karafyllis et al. (2017) respectively. Compared with the periodic sampled-data control where unnecessary movements of the massive actuator may exist, event-triggered control where the massive actuator is only animated at the necessary times which are determined by an event-triggered mechanism of evaluating the operation of the elevator, is more feasible for the mining cable elevator from the point of view of energy saving.
The event-triggered control system consists of two elements, namely, a continuous-time feedback control law, and an ETM that determines triggering times of updating the control law Heemels et al. (2012). The key task in the design of ETM is to make sure the minimal dwell time between two triggering times is nonzero meanwhile the exponential stability of the closed-loop system is ensured. Most of ETM implementations are based on feedback control laws of ODE systems. Tabuada (2007) introduced a static ETM based on the existence of an input to state stable control Lyapunov function (ISS-CLF) and a dynamic triggering mechanism which uses an internal dynamic variable was proposed in Girard (2015). The eventtriggered algorithms by the value of the derivative of the Lyapunov function were proposed in Marchand et al. (2013) and Seuret et al. (2014). Recently, Espitia et al. (2016a) and Espitia et al. (2016b) originally developed event-triggered strategies to boundary control of linear hyperbolic PDEs with dissipativity boundary conditions. Event-triggered boundary control of 2 × 2 coupled transport PDEs and reaction-diffusion PDEs are also addressed in Espitia et al. (2018) and Espitia et al. (2019), respectively.
Compared with Espitia et al. (2018) which designed an eventtriggered backstepping controller for a 2 × 2 hyperbolic PDEs, in addition to an ODE coupled at the uncontrolled boundary of the PDE in our paper, the main contribution here lies in considering the proximal reflection term in the 2 × 2 hyperbolic PDEs, where ETM design would become more difficult because a higher-order continuous-time boundary stabilization law is required. A two-step design is proposed to solve this problem, where a low-pass-filter-based backstepping boundary stabilization law is designed, based on which the ETM is designed sequently to determine the triggering times, shown in Fig.2.
The rest of the paper is organized as follows. The concerned model and a continuous-time state-feedback controller combining the backstepping method and low-pass filter design are presented in Section 2. The event-trigged mechanism is built in Section 3, and the exponential stability of the event-based closed-loop system is proved via Lyapunov analysis in Section 4. The conclusion and future work are proved in Section 5.

PLANT AND CONTINUOUS-TIME CONTROL LAW
The plant considered in this paper is z(0,t) = pw(0,t) +U(t), (1) z t (x,t) = −q 1 z x (x,t) + c 1 w(x,t) + c 1 z(x,t), (2) w t (x,t) = q 2 w x (x,t) + c 2 w(x,t) + c 2 z(x,t), (3) w(1,t) = qz(1,t) +C 1 X(t), (4) X(t) = AX(t) + Bz(1,t) w(x,t) ∈ R are states of the 2 × 2 coupled hyperbolic PDEs. U(t) is the control input to be designed. q 1 and q 2 are positive transport velocities and p, q = 0 are arbitrary constants. Damping coefficients c 1 , c 2 are negative constants. The plant parameters satisfy e c 2 /q 2 +c 1 /q 1 |pq| < 1, which holds in the elevator application and also meets the condition of existence of delayrobust stabilization proposed in Auriol et al. (2018). (A, B) is assumed as controllable. Remark 1. That axial vibration dynamics of the mining cable elevator consisting of the mining cable and cage is described by a wave PDE -ODE system, which can be transformed to the 2 × 2 coupled transport PDE-ODE system considered in this paper via Riemann transformation Wang et al. (2018d).
For the concern plant, a feedback controller would be designed via traditional PDE backstepping as follows. Introduce a PDE backstepping transformation in the following form ; Wang et al. (2018a): the inverse transformation of which is Through the above backstepping transformation, (1)-(5) is converted to whereÂ is Hurwitz recalling (A, B) is controllable, and are obtained by matching (10)-(14) and (1) Taking Laplace transformation into (10)-(14), we have (21) We obtain the following relationship (26) (10) can thus be written in the frequency domain as Choosing the controller as where Ω(s) is a first-order low-passing filter as follows which can be realized by RC circuits. The constant a 0 is a design parameter making sure for s ∈ C, R(s) ≥ 0. Note thatψ(s) in (28) is a proper transfer function. Inserting (28) into (27), one obtain (30). Therefore, applying the backtepping transformation and the control input U(t) which is realized from U(s) in (28), then (10) can be regarded as (32) Therefore, the target system is obtained (11)-(14) and (32) and an extend dynamic of the dynamic controller (28), of which the realization is shown as following.
Realization of U(s): U(t) is the output signal of the low-passing filter Ω of which the input signal is U b (t) derived from previous backstepping design , Recalling the structure of the low-pass filter Ω (64), U(t) is the solution of the following ODE driven by U b (t)

EVENT-TRIGGER MECHANISM
In this section, we introduce an event-triggered control scheme for stabilization of the 2 × 2 coupled hyperbolic PDE-ODE system (1)-(5). It relies on both the continuous-time control U(t) and a dynamic event-triggered mechanism (ETM) which determines triggering times t k (k ≥ 0 and t 0 = 0) when the actuator signal is updated. In other words, the event-triggered form U d (t) is the value of the continuous-time U(t) at the time instants A deviation d(t) between a continuous-time controller and the event-based one is given as (37) Then (36) can be written as z(0,t) = pw(0,t) +U(t) − d(t).
(38) Recalling the backstepping transformations and designs of U(t) in Section 2, the target system becomes (11)-(14) with and an extend dynamics of the dynamic controller (34). The ETM to determine the triggering times of U d is designed as where m(t) satisfies the ordinary differential equation, with an initial condition m(0) < 0. Positive constants θ , η, λ d , σ , κ 1 , κ 2 are to be determined later. V (t) is given as where a positive definite matrix P = P T is the solution to the Lyapunov equationÂ T P + PÂ = −Q, for someQ =Q T > 0. The positive constants r a , r b , r c , δ 1 , δ 2 are to be determined in the next section. Lemma 1. Considering d(t) defined in (37), there exists a positive constant λ a such thaṫ for t ∈ (t k ,t k+1 ), where λ a only depends on the parameters of the plant, the continuous-time control law and the low-pass filter.
The following lemma proves the existence of a nonzero minimal dwell time independent of initial conditions. It contributes to reduction of the actuation frequency and the well-posedness of the resulting closed-loop system. Lemma 3. There exists a minimal dwell-time τ > 0 between two triggering times, i.e., t k+1 − t k ≥ τ > 0 for all k ≥ 0.
Proof. We know from (46), the events are triggered to guarantee d(t) 2 ≤ θV (t) − m(t) for all t ≥ 0. Define a function ψ as Note that ψ(t k+1 ) = 1 because the event is triggered and ψ(t k ) ≤ 0 because of m(t) ≤ 0 and d(t k ) = 0. Note that ψ(t) is a continuous function on [t k ,t k+1 ] due to continuity and wellposedness of this class of 2 × 2 hyperbolic PDE-ODE system according to . By the intermediate value theorem, there exists t * ∈ [t k ,t k+1 ] to make ψ(t * ) = 0. The minimal τ depends on the time it takes for ψ(t) from 0 to 1.
The proof of this lemma is completed.

CONCLUSION
In this paper, an event-triggered backstepping boundary controller for a 2 × 2 hyperbolic PDE-ODE system is proposed via a two-step design, including the design of a low-pass-filterbased backstepping boundary stabilization law and the design of an ETM which determines triggering times of updating the continuous-time control law. The nonzero minimal dwell time between two triggering times and the exponential stability of the event-based closed-loop system are proved. This design can be applied into vibration control of a mining cable elevator driven by hydraulic cylinders at the head sheaves, where the frequent actions of the massive cylinder can be avoided. The design of an output-feedback form of the event-triggered backstepping boundary controller and the simulation test would be conducted in the future work.