Brief paperBoundary observer-based control for hyperbolic PDE–ODE cascade systems with stochastic jumps☆
Introduction
Much attention has been drawn to the systems of hyperbolic PDEs cascaded with ODEs in research communities during the past decades, because this kind of systems can be utilized to describe many physical systems and various processes such as vehicular traffic flow (Goatin, 2006), gas flow pipeline (Gugat & Dick, 2011), communication networks (Espitia, Girard, Marchand, & Prieur, 2017), and fluid flow in transmission lines, etc. Meanwhile, it is aware that some physical systems are subject to sudden changes in the parameters, due to the environmental factors, the component failures, or the variability in the system structure. For example, the traffic flow has two distinct dynamic modes, i.e. the free-flow mode and the congestion mode (Colombo, 2003); the random flux can cause changes for the boundary conditions in the gas flow pipeline (Gugat, Dick, & Leugering, 2011). Those systems can be described by the hyperbolic PDE–ODE cascade systems with switching modes. Meanwhile, in the physical systems and processes, the variables cannot all be directly measured accurately, and only limited number of measurements can be obtained at the boundary. Therefore, using a limited number of measurements to estimate the system variables is a typical problem. Those estimated variables are then utilized in control design to maintain the desired system state trajectories. Those observations motivate the work presented in this paper.
For the ODEs coupled with a first-order hyperbolic PDE, the backstepping control method was used, which resembles the finite-dimensional systems with actuator and sensor delays, and the Korteweg–de Vries equation (Krstic & Smyshlyaev, 2008). However, the backstepping can only be applicable to the systems where the PDE–ODE cascade is in the strict-feedback form. Then a forwarding-backstepping transformation was proposed to convert the coupled system into a stable form. The exponential stability of the original system was established by the explicit inverse transformations (Bekiaris-Liberis & Krstic, 2011). The state estimation and controller design are attractive to the ODEs coupled with a pair of counter-convecting transport PDEs with local couplings. The focus is emphasized on the boundary observer design with limited numbers of measurements at the boundaries, and on the controller design with actuation at one end of the domain. The work presented in Vazquez, Krstic, and Coron (2011) is concerned with the design of a collocated output feedback law for a 2 × 2 linear hyperbolic system, which combined the state feedback controller and the collocated boundary observer with measurements at the controlled end. The backstepping method was applied to obtain both control and observer kernels. Results from this study were extended to a class of coupled systems in which a 2 × 2 linear hyperbolic PDE describes the transport phenomenon, and an ODE describes the disturbance affecting the left boundary of the PDE. A disturbance attenuating control law combined with an observer were designed (Aamo, 2013). The same type of coupled systems with the 2 × 2 hyperbolic PDE being quasilinear was revisited in Hasan, Aamo, and Krstic (2016), and the boundary observer was designed in a collocated setup.
When the state equations are switched, the initial–boundary value problem and the stability analysis for the purely hyperbolic PDEs have been addressed (Amin et al., 2011, Hante et al., 2009, Michel et al., 2005, Prieur et al., 2014). For the linear hyperbolic systems of conservation laws, with the Lyapunov functional method, the stabilizing switching controller was synthesized in Lamare, Girard, and Prieur (2015), the stability analysis by considering the Markovian jumping parameters was addressed in Zhang and Prieur (2017), and the result was applied to verify the usefulness of a feedback controller of the boundary values in a traffic flow system.
This work is concerned with the observer-based control for the hyperbolic PDE–ODE cascade systems with stochastic jumps, named the Markovian jump hyperbolic PDE–ODE cascade systems (MJHPOCS). In the systems, the ODEs couple the PDEs through the boundary. A homogeneous Markov process models the jumping parameters of the systems, and its right continuous trajectory candidate values are taken from a finite set. A boundary observer is designed, which relies only on limited quantity of sensors at the boundary of the PDEs. Then, a control input is constructed with the estimated variables to attenuate the effect of the ODE state to the PDE boundary. Based on the Lyapunov functional approach, the observer-error system and the evolution profiles of the PDEs in the closed-loop are proved to be exponentially mean square stable, respectively. The existence conditions of the observer and the stability conditions of the PDEs are given via a set of compact space-varying linear matrix inequalities (SVLMIs). The SVLMIs are approximated by a finite number of LMIs, which guarantee the SVLMIs, and the computational cost is reduced greatly. Finally, the effectiveness of the proposed methods are verified in the application to a traffic flow system with simulations.
In Zhang and Prieur (2017), the plant is in the form of linear conservation laws. The main results are the stability conditions for the systems with switching modes, which depend on the boundary conditions of the plant. If the boundary conditions satisfy the stability conditions, then they are said to be dissipative [22]. The PDEs in the observer-error system in this paper are not in the form of linear conservation laws. We first convert the observer-error system into a stabilizable system, and then design the observer gain matrices to guarantee the stability of the observer-error system by the explicit inverse transformations. The method in Zhang and Prieur (2017) and that in this paper for deriving the main results are different.
Preliminaries are given in Section 2. The problem is formulated in Section 3. The design procedure of the observer and the stability analysis of the closed-loop system are given in Section 4. The application example is presented in Section 5. Finally, the conclusions are offered in Section 6.
Section snippets
Preliminaries
Given a complete probability space , where is a non-empty sample set, is a -algebra of subsets of , and is a probability measure. With a slight abuse, we use to represent the jumping process. It is a continuous-time, discrete-state homogeneous Markov process, and its right-continuous trajectory candidate values are taken from the set , where is an integer denoting the number of all modes of the system. The mode transition probabilities of
Problem formulation
The MJHPOCS is presented firstly in this section. Then the form of an observer with only a limited number of sensors at is given, followed by the feedback controller of the estimated state variables. The considered problem is formulated mathematically at the end of the section.
Main results
In this section, we first give the existence conditions of the boundary observer in (6)–(10) which guarantee the observer-error system to be exponentially mean square stable. Then we present the exponential mean square stability conditions of the closed-loop system in (12)–(17).
Applications to the freeway traffic control
The traffic dynamic in a section of freeway is modeled by the following Aw–Rascle traffic flow equation (Aw & Rascle, 2000): Here denotes the vehicle density, and means the average speed of vehicles; is the traffic pressure, and define by the Green-shields function diagram, where and are the maximal speed and the maximal density, respectively. Then the
Concluding remarks
In this work, we have addressed the boundary observer design and stability analysis problem for the system of PDEs coupled with ODEs with Markovian jumping parameters. Only limited quantity of measurements at the boundary are used in the observer design, and the estimated variables at the boundary are utilized in the control input with respect to the boundary conditions. It is shown that the observer-error system is exponentially stable, and the evolution profiles of the PDEs in the closed-loop
Yan Zhao received the Ph.D. degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2010. She was a visiting scholar with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, from 2009 to 2010, and was a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, from 2016 to 2017. She is an associate professor with the School of Automation at the
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Yan Zhao received the Ph.D. degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2010. She was a visiting scholar with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, from 2009 to 2010, and was a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, from 2016 to 2017. She is an associate professor with the School of Automation at the Nanjing University of Science and Technology, Nanjing, China. Her current research interests include the control and estimation of PDEs with the applications.
Jianbin Qiu received the B.Eng. and Ph.D. degrees in Mechanical and Electrical Engineering from the University of Science and Technology of China, Hefei, China, in 2004 and 2009, respectively. He also received the Ph.D. degree in Mechatronics Engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2009.
He is currently a Full Professor at the School of Astronautics, Harbin Institute of Technology, Harbin, China. He was an Alexander von Humboldt Research Fellow at the Institute for Automatic Control and Complex Systems, University of DuisburgEssen, Duisburg, Germany. His current research interests include intelligent and hybrid control systems, signal processing, and robotics.
Prof. Qiu is a Senior Member of IEEE and serves as the chairman of the IEEE Industrial Electronics Society Harbin Chapter, China. He is an Associate Editor of IEEE Transactions on Cybernetics.
Shengyuan Xu received the M.Sc. degree from the Qufu Normal University, China, in 1996, and the Ph.D. degree from the Nanjing University of Science and Technology, China, in 1999. From 1999 to 2000, he was a Research Associate in the Department of Mechanical Engineering at the University of Hong Kong, Hong Kong. From December 2000 to November 2001, and December 2001 to September 2002, he was a Postdoctoral Researcher in CESAME at the Université catholique de Louvain, Belgium, and in the Department of Electrical and Computer Engineering at the University of Alberta, Canada, respectively. Since November 2002, he has joined the School of Automation at the Nanjing University of Science and Technology as a Full Professor.
Prof. Xu was a recipient of the National Excellent Doctoral Dissertation Award in 2002 from the Ministry of Education of China. He obtained a grant from the National Science Foundation for Distinguished Young Scholars of P. R. China in 2006. He was awarded a Cheung Kong Professorship in 2008 from the Ministry of Education of China.
Prof. Xu is a member of the Editorial Boards of the Transactions of the Institute of Measurement and Control, and the Journal of the Franklin Institute. His current research interests include robust filtering and control, singular systems, time-delay systems, neural networks, multidimensional systems and nonlinear systems.
Wenguo Li received the B.Sc. degree in application mathematics from the Hebei University of Technology, Tianjin, China, in 2001, and the M.Sc. Degree in Management Science and Engineering from the Hebei University of Engineering, Handan, China, in 2009. His research interests include uncertain information processing and distributed parameter systems.
Junli Wu received the Ph.D. degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2011. She is now a Professor in Jiamusi University, China. Her research interests include robust control and filtering, sampled data control, and neural network control.
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This work was partially supported by National Natural Science Foundation of China (NSFC: 61573056, 61673215, 61873311), the 333 Project, China (BRA2017380), and Doctor Research Foundation of Jiamusi University, China (22ZB201515). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Joachim Deutscher under the direction of Editor Miroslav Krstic.