Elsevier

Automatica

Volume 138, April 2022, 110137
Automatica

Delay-compensated event-triggered boundary control of hyperbolic PDEs for deep-sea construction

https://doi.org/10.1016/j.automatica.2021.110137Get rights and content

Abstract

In this paper, we present delay-compensated event-triggered boundary control of heterodirectional coupled hyperbolic PDE–ODE systems, with the purpose of employing a piecewise-constant control input which makes the control law more user-friendly in deep-sea construction vessels where the actuator, i.e., the ship-mounted crane, is massive, and compensating delays that exist in transmission of sensing signals from the seabed to the vessel on the ocean surface through a set of acoustics devices. After treating the time delay of arbitrary length as a transport PDE whose boundary connects with the plant, a state observer is built to estimate the states of the overall system of heterodirectional coupled hyperbolic PDEs–ODE–transport PDE, only using the measurements at the right boundary of the last transport PDE. An observer-based output-feedback continuous-in-time controller is designed to stabilize the overall system using the backstepping method, followed by an observer-based event-triggering mechanism, which is designed based on the evolution of the overall PDE–ODE–PDE system, to determine the updating times of the actuation signal. The absence of a Zeno behavior and exponential convergence in the event-based closed-loop system are proved. In the simulation, the obtained theoretical result is verified in control of a deep-sea construction vessel to place the equipment in the target area on the seabed. Performance deterioration under extreme and unmodeled disturbances is also illustrated.

Introduction

The motivation of this work arises from the control problem for a deep-sea construction vessel (DCV), which is used to place underwater parts of an off-shore oil drilling platform at the designated locations on the seabed (Stensgaard, White, & Schiffer, 2010), and is also used in the construction of artificial reefs for enabling the development and growth of marine life, in laying communication cables on the seabed, which is often uneven and rife with obstacles, and in other undersea applications. A diagram of DCV is shown in Fig. 1, where the top of a cable is attached to a crane on a vessel at the ocean surface and the cable’s bottom is attached to an object to be installed/moved on the seabed (How et al., 2010, How et al., 2011, Wang and Krstic, 2021d). Large oscillations often appear in the cable system, especially under the external ocean disturbances (How et al., 2010). The large oscillations would cause large deviation of the payload from the designated location on the seabed.

Several boundary control schemes have been proposed for the PDE-modeled cable systems with the purpose of suppressing the cable oscillations (He and Ge, 2016, He et al., 2018, Wang, Koga et al., 2018). In addition to the infinite-dimension property introduced by the cable structure, there are two challenges in the control design of DCV. The first one is the existence of sensor delays. As shown in Fig. 1, the sensor signal measured at the payload is transmitted over a large distance to the vessel on the ocean surface through a set of acoustics devices, because acoustic communication is the most versatile and widely used technique in underwater engineering applications considering low attenuation in water (Jiang, 2008, Manjula and Manvi, 2011). The transmission speed of acoustic signals in salty water is around 1500 m/s, which is a difference of five orders of magnitude lower than the speed of electromagnetic wave in free space (Jiang, 2008). As a result, signal propagation delay in underwater becomes significant (Jiang, 2008). Such sensor signal delay may result in information distortion or even make the control system lose stability, which motivates us to study the delay compensation in the control design of DCV. The second one is that the massive actuator, i.e., the ship-mounted crane shown in Fig. 1, is incapable of supporting the fast-changing control signal, due to its low natural frequency, which motivates us to consider applying sampling schemes into the continuous-in-time control signals to reduce the changes in the actuation. Compared with the periodic sampled-data control where unnecessary movements of the massive actuator may appear, event-triggered control is more feasible from the point of view of energy saving, because the massive actuator is only animated at the necessary times which are determined by an event-triggering mechanism of evaluating the operation of the DCV.

In this paper, we pursue a delay-compensated event-triggered control scheme, which compensates the sensor delay of arbitrary length, and has a reduction of the changes in the actuator signal, i.e., the control input employs piecewise-constant values. The delay-compensated event-triggered control scheme is designed for a 2 × 2 hyperbolic PDE–ODE system, which not only covers the DCV dynamics, and can be suitable in some other applications, such as road traffic (Goatin, 2006, Yu and Krstic, 2019), water level dynamics (Diagne et al., 2017, Prieur and Winkin, 2018), and flow of fluids in transmission lines (Hasan, Aama, & Krstic, 2016). The classical control results of the class of coupled hyperbolic PDEs are presented in Coron et al., 2013, Hu et al., 2016 and Vazquez, Krstic, and Coron (2011), which are further developed for the case including couplings with ODEs in Deutscher et al., 2018, Meglio et al., 2020, Saba et al., 2019 and Wang, Krstic and Pi (2018).

Most of the current designs on event-triggering mechanisms are for ODE systems, such as Girard, 2015, Heemels et al., 2012, Marchand et al., 2013, Seuret et al., 2014 and Tabuada (2007). There exist some results on distributed (in-domain) control of PDEs, such as Selivanov and Fridman (2016) and Yao and El-Farra (2013). For boundary control, event-triggered control designs for reaction–diffusion PDEs were proposed in Espitia, Karafyllis, and Krstic (2021) and Karafyllis, Espitia, and Krstic (2021), and the results for first-order hyperbolic PDEs were presented in Diagne and Karafyllis (2021) and Espitia et al., 2016a, Espitia et al., 2016b. Regarding the 2 × 2 first-order hyperbolic PDE, the first attempt of event-based boundary control was Espitia, Girard, Marchand, and Prieur (2018), which is in the state-feedback form. Based on the observer design, the output-feedback event-triggered boundary controllers of the 2 × 2 first-order hyperbolic PDEs were proposed in Espitia (2020) and Wang and Krstic (2022), and an adaptive event-triggered boundary controller is further developed in Wang and Krstic (2021a). However, a large delay is not allowed in the above event-triggered control designs.

Recently, regarding taking into consideration time delay, boundary control designs for hyperbolic PDEs have been proposed. For example, delay-robust stabilizing feedback control designs for coupled first-order hyperbolic PDEs were introduced in Auriol, Aarsnes, Martin and Di Meglio (2018) and Auriol, Bribiesca-Argomedo, Saba, Di Loreto and Di Meglio (2018), achieving robustness to small delays in actuation. In order to compensate arbitrarily long delays, a delay compensation technique was developed in Krstic (2009b) and Krstic and Smyshlyaev (2008), where the delay was captured as a transport PDE and the original ODE plant with sensor delay was treated as an ODE-transport PDE cascade in the controller and observer designs. Therein, the observer was built of a “full-order” type, which estimates both the plant states and the sensor states, compared with some classical results about delay-compensated observer designs (Ahmed-Ali et al., 2013, Cacace et al., 2010, Germani et al., 2002), which only estimate plant states, namely, observers of the “reduced-order” type. While compensation of arbitrarily long delays by this technique is commonly available for finite-dimensional systems, only very few examples exist for PDEs, where the result for coupled first-order hyperbolic PDEs was presented in Wang and Krstic (2020a), and the results for parabolic (reaction–diffusion) PDEs were proposed in Koga, Bresch-Pietri, and Krstic (2020) and Krstic (2009a). Delay compensation for the wave PDEs with arbitrarily long delays is more complex than the one for reaction–diffusion PDEs, for which the primary reason is the second-order-in-time character of the wave equation. In Krstic (2011), by treating the delay as a transport PDE and applying the backstepping method, a boundary controller was designed for a wave PDE with a compensation of an arbitrarily long input delay and with a guarantee of exponential stability in the closed-loop system. Through truncation of the original infinite dimensional system and applying predictor feedback, control designs for PDEs with long input delays were also presented in Guzman et al., 2019, Lhachemi and Prieur, 2020, Lhachemi et al., 2019 and Prieur and Trelat (2018). All the above delay compensation/robustness results employ the continuous-in-time control input, instead of a piecewise-constant fashion.

  • Compared with the results in Espitia, 2020, Espitia et al., 2018, Espitia et al., 2021 and Wang and Krstic (2022) about event-triggered control of PDEs without delay compensation, we achieve compensation of arbitrarily long sensor delays in event-triggered control of hyperbolic PDE–ODE systems.

  • The previous results (Koga et al., 2020, Krstic, 2009a, Krstic, 2011, Wang and Krstic, 2020a) on delay-compensated control of PDEs use continuous-in-time control input signals, whereas the control input in this paper is piecewise constant.

  • In comparison to the previous continuous-in-time controllers for DCVs without delay compensation (How et al., 2010, How et al., 2011, Wang and Krstic, 2021d), this paper develops a DCV control scheme, which compensates the sensor delays arising from the long-distance transmission of the sensing signal via acoustics devices, and employs a more user-friendly piecewise-constant actuation for the massive ship-mounted crane.

The DCV model, and the general model used in control design are presented in Section 2. Observer design is proposed in Section 3, where the observer gains are determined in two transformations which convert the observer error system to a target observer error system whose exponential stability is straightforward to obtain. An observer-based output-feedback continuous-in-time controller with delay compensation is designed in Section 4 by using the backstepping method. In Section 5, a dynamic event-triggering mechanism is designed, and the existence of a minimal dwell-time between two successive triggering times is proved. The absence of a Zeno behavior and exponential convergence in the event-based closed-loop system are proved in Section 6. The obtained theoretical result is verified in the application of DCV control for seabed installation by simulation in Section 7. The conclusions and future work are provided in Section 8.

We adopt the following notation.

  • The symbol R denotes the set of negative real numbers, whose complement of the real axis is R+[0,+). The symbol Z+ denotes the set of all non-negative integers.

  • Let URn be a set with non-empty interior and let ΩR be a set. By C0(U;Ω), we denote the class of continuous mappings on U, which take values in Ω. By Ck(U;Ω), where k1, we denote the class of continuous functions on U, which have continuous derivatives of order k on U and take values in Ω.

  • We use the notation L2(0,1) for the standard space of the equivalence class of square-integrable, measurable functions defined on (0,1) and f=01f(x)2dx12<+ for fL2(0,1).

  • For an IR+, the space C0(I;L2(0,1)) is the space of continuous mappings Itu[t]L2(0,1).

  • Let u:R+×[0,1]R be given. We use the notation u[t] to denote the profile of u at certain t0, i.e., (u[t])(x)=u(x,t), for all x[0,1].

Section snippets

Model of DCV

We follow the modeling process in Section 2 in Wang and Krstic (2021d) but make the following three simplifications: (1) We take the cable length as constant, i.e., we set l(t) as L; (2) The spatially-dependent static tension of the cable, which considers the effect of the cable mass on the tension in the cable, is assumed as a constant static tension resulting from the payload mass and buoyancy, i.e., we set T(x) as T0; (3) We only focus on lateral vibrations instead of longitudinal–lateral

Observer design

In order to build the observer-based event-triggered output-feedback controller of the plant (14)–(19), in this section, we design a state-observer to track the overall system (14)–(19) using only the delayed measurement yout(t). Through the reformulation in Section 2, the estimation task is equivalent to designing a state-observer to recover the overall system (14)–(18), (20)–(22) only using the measurements at the right boundary x=2 of the v-PDE. The observer is built as a copy of the plant

Output-feedback control design

In the last section, we have obtained the observer which compensates the time-delay in the output measurements of the distal ODE, to track the states of the overall system (14)–(19). In this section, we design an output-feedback control law U(t) based on the observer (23)–(29) with output estimation error injection terms assumed absent for reducing notation burden, in accordance to the result on their convergence to zero in Theorem 1. The separation principle is then verified in the resulting

Event-triggering mechanism

In this section we introduce an event-triggered control scheme for stabilization of the overall PDE–ODE–PDE system (14)–(18), (20)–(22). This scheme relies on both the delay-compensated observer-based output-feedback continuous-in-time controller designed in the last section, and a dynamic event-triggering mechanism (ETM) which is realized using the states from the delay-compensated observer. The event-triggered control signal Ud(t) is the value of the continuous-in-time control signal U(t) at

Stability analysis

In this section we state the main result of this paper, after we first establish an intermediate result.

Lemma 4

With arbitrary initial data (α(x,0),β(x,0))TC0([0,1];R2), vˆ(x,0)C1([1,2];R), ζˆ(0)R, m(0)R, for (101)(106), (111), (117), the exponential convergence is achieved in the sense of the norm |ζˆ(t)|+α(,t)+β(,t)+vˆ(,t)+|m(t)|.

Proof

Define e(t)=Υdeλet,for some positive Υd, λe such that |Ỹ(t)|e(t),recalling Theorem 1.

Let us consider the Lyapunov function Va(t)=V(t)m(t)+12Ree(t)2where m(

Application in deep-sea construction

In the simulation, we consider the application of DCV to place equipment to be installed on the seabed for off-shore oil drilling. The equipment has to be installed accurately at the predetermined location with a tight tolerance, such as the permissible maximum tolerance of 2.5 m for a typical subsea installation, according to How et al. (2011). Applying the design presented above, an output-feedback control force employing piecewise-constant values at the crane is obtained, to reduce

Conclusion

We present the design of delay-compensated event-triggered boundary control for heterodirectional coupled hyperbolic PDE–ODE systems, where arbitrarily long delays existing in sensing are compensated and the control input employs piecewise-constant values. To this end, we treat the time delay as a transport PDE, followed by designing a “full-order” state observer for the overall system of heterodirectional coupled hyperbolic PDEs–ODE–transport PDE, only using the measurements at the right

Ji Wang received the Ph.D. degree in Mechanical Engineering in 2018 from Chongqing University, Chongqing, China. From 2019 to 2021, He was a postdoctoral scholar in the Department of Mechanical and Aerospace Engineering at University of California, San Diego, La Jolla, CA, USA. He is currently an associate professor in the Department of Automation at Xiamen University, Xiamen, China. Since 2021, he serves as Associate Editor of Systems & Control Letters. His research interests include modeling

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    Ji Wang received the Ph.D. degree in Mechanical Engineering in 2018 from Chongqing University, Chongqing, China. From 2019 to 2021, He was a postdoctoral scholar in the Department of Mechanical and Aerospace Engineering at University of California, San Diego, La Jolla, CA, USA. He is currently an associate professor in the Department of Automation at Xiamen University, Xiamen, China. Since 2021, he serves as Associate Editor of Systems & Control Letters. His research interests include modeling and control of distributed parameter systems, active disturbance rejection control, event-triggered control, and adaptive control, with applications in cable-driven mechanisms.

    Miroslav Krstic is Distinguished Professor of Mechanical and Aerospace Engineering, holds the Alspach endowed chair, and is the founding director of the Cymer Center for Control Systems and Dynamics at UC San Diego. He also serves as Senior Associate Vice Chancellor for Research at UCSD. As a graduate student, Krstic won the UC Santa Barbara best dissertation award and student best paper awards at CDC and ACC. Krstic has been elected Fellow of seven scientific societies – IEEE, IFAC, ASME, SIAM, AAAS, IET (UK), and AIAA (Assoc. Fellow) – and as a foreign member of the Serbian Academy of Sciences and Arts and of the Academy of Engineering of Serbia. He has received the Richard E. Bellman Control Heritage Award, SIAM Reid Prize, ASME Oldenburger Medal, Nyquist Lecture Prize, Paynter Outstanding Investigator Award, Ragazzini Education Award, IFAC Nonlinear Control Systems Award, Chestnut textbook prize, Control Systems Society Distinguished Member Award, the PECASE, NSF Career, and ONR Young Investigator awards, the Schuck (’96 and ’19) and Axelby paper prizes, and the first UCSD Research Award given to an engineer. Krstic has also been awarded the Springer Visiting Professorship at UC Berkeley, the Distinguished Visiting Fellowship of the Royal Academy of Engineering, the Invitation Fellowship of the Japan Society for the Promotion of Science, and four honorary professorships outside of the United States. He serves as Editor-in-Chief of Systems & Control Letters and has been serving as Senior Editor in Automatica and IEEE Transactions on Automatic Control, as editor of two Springer book series, and has served as Vice President for Technical Activities of the IEEE Control Systems Society and as chair of the IEEE CSS Fellow Committee. Krstic has coauthored fifteen books on adaptive, nonlinear, and stochastic control, extremum seeking, control of PDE systems including turbulent flows, and control of delay systems.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Angelo Alessandri under the direction of Editor Thomas Parisini.

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