Boundary observer-based control for hyperbolic PDE–ODE cascade systems with stochastic jumps

Abstract

This work is concerned with the boundary observer-based control for a category of cascade systems with stochastic jumps. The cascade system consists of ordinary differential equations (ODEs) entering the hyperbolic partial differential equations (PDEs) through the boundary. A continuous-time, discrete-state homogeneous Markov process models the jumping parameters of the system, 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. Based on the Lyapunov functional approach, the observer-error system is firstly proved to be exponentially mean square stable. Then, the controller is presented by utilizing the estimated states of the ODEs as feedback variables. The evolution profiles of the PDEs in the closed-loop are proved to be exponentially mean square stable at the desired steady-state. The existence conditions of the boundary observer and the stability conditions of the PDEs in the closed-loop form are presented via a set of compact space-varying linear matrix inequalities (SVLMIs). The SVLMIs are then approximated by a finite number of LMIs, which guarantee the SVLMIs, and the computational cost is reduced greatly. Finally, the obtained theoretical results are applied to a traffic flow control system, which confirms that the proposed estimation and control methods are effective.

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.