Impulsive adaptive observer design for a class of hybrid ODE–PDE cascade systems with uncertain parameters
Introduction
In reality, many processes can be described as coupled systems formed by ordinary differential equations (ODEs) cascading with partial differential equations (PDEs), such as wind turbine tower [1], [2], overhead crane [3], [4], vehicular traffic flow [5], [6], etc. Due to the infinite-dimensional nature of the PDE subsystem and the existence of the coupling between the two subsystems, it may be difficult to directly measure all states of hybrid cascade ODE–PDE systems for the whole space variable. Therefore, using observers to achieve estimation of these coupled systems receives much attention.
In recent years, some researchers have paid attention to the observation problems of PDEs [7], [8], [9], [10], [11], [12]. A boundary observation method was firstly proposed by Smyshlyaev and Krstic in [13] for a class of parabolic PDEs via a backstepping approach. This method was applied to observer design for various systems, such as drilling systems [9] and microfluidic systems [10]. In [11], Krstic solved the observation problem addressed for linear ODE–PDE cascade systems when the connection state of the cascade systems between ODEs and diffusive PDEs cannot be measured. Furthermore, an adaptive observer approach was designed in [12] for the class of ODE–PDE cascade systems with unknown parameters.
On the other hand, the impulsive observer is usually used to obtain the online estimates of the certain state which cannot be directly accessible and has been paid on for the promising advantages on reducing the energy consumption. In recent years, the impulsive observation method for systems modeled by ODEs has attracted considerable interests [14], [15], [16], [17]. A robust hybrid observer for linear system was designed in [14], and this approach was applied to deal with the observation problem for linear systems with measurement noise in [15]. Although there have been some impulsive controllers designed for systems modeled by PDEs [18], [19], [20], researches about impulsive observation for PDEs are little.
Inspired by aforementioned works, an impulsive adaptive observer is firstly designed to deal with the observation problem of the hybrid ODE–PDE cascade system with uncertainties. Through establishing an estimation error system and using backstepping design technique, the error system can be transformed into the target system by the designed kernel functions. In addition, the impulsive adaptive law for system parameters is designed by utilizing the designed kernel functions. In order to get the convergence of the impulsive adaptive observer, the convergence of the target system is proved based on the Lyapunov stability theory.
The paper is organized as follows. In Section 2, the ODE–PDE cascade system under study is described and some basic definitions are introduced. In Section 3, the impulsive adaptive observer and the kernel functions are provided, and the boundedness of the kernel functions are also discussed. In Section 4, the process to transform the estimation error system into a target system is given by using the designed kernel functions. Then the well-posedness analysis of the target system is also presented. Finally, a numerical simulation is shown to illustrate the proposed results in Section 5. Conclusion is given in Section 6.
Notations In this paper, denotes the dimensional real space with Euclidean -norm defined as for vectors, while denotes the set of nonnegative real numbers. Accordingly, , for all . stands for the set of natural numbers. represents continuously differentiable function space. Then, for a function , . is the Hilbert space with Euclidean -norm defined as , that is , for all . is the Sobolev space of absolutely continuous functions: with . For any such that or , the following Wirtinger’s inequality holds: A function belongs to class- if it is continuous, zero at zero, and is strictly increasing. A function belongs to class- if it satisfies: (i) for each , the mapping is continuous and strictly decreasing with , and (ii) for each , the mapping belongs to class-.
Section snippets
Model formulation
In this section, we mainly discuss about the model of an ODE–PDE cascade system as shown in Fig. 1, where the ODE subsystem described by ODEs is coupled with the subsystem described by a parabolic PDE.
Therefore, the ODE–PDE cascade system can be modeled as follows: where and denote the state vector and the distributed state, respectively. For clarity, , ,
Adaptive impulsive observer statement
In this section, the statement of the impulsive adaptive observer for system (1) will be provided. Furthermore, the boundedness of the kernel functions which are used to transform the system (1) into the target system will be discussed. Some studies with regard to coupled systems formed by ODEs and PDEs indicate advantages and feasibility by using adaptive observers [22]. For the purpose of fewer measurements, we propose an impulsive adaptive observer as follows:
Stability analysis
In this section, the estimation error system is described. In addition, the error system is transformed into a target system by utilizing backstepping method. To prove the stability of the estimation errors, the Lyapunov stability analysis for the target system is provided.
Numerical simulations
In this section, the design steps of observation gain and are interpreted, and the numerical simulation results about the impulsive adaptive observation approach proposed in above sections are exhibited.
Example 1 Consider the ODE–PDE cascade system in [12]
In order to illustrate the feasibility and advantages of impulsive adaptive observer designed in above sections, comparison between common adaptive observer and observer
Conclusion
In this paper, we have designed an impulsive adaptive observer for an ODE–PDE cascade model to achieve online estimates of distributed state and state vector . The unknown parameters have been estimated and the convergence of the observation error has been achieved by the backstepping transformations regarding both state vector estimation and distributed state estimation. Finally, a numerical example has been provided to confirm the feasibility and effectiveness of the impulsive adaptive
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the editors and the anonymous referees for constructive feedback, which allowed us to improve the quality of our work and this article. This work was supported by Jiangsu Provincial Natural Science Foundation of China (BK20201340) and China Postdoctoral Science Foundation (2018M642160).
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