Impulsive adaptive observer design for a class of hybrid ODE–PDE cascade systems with uncertain parameters

Abstract

In this paper, an impulsive adaptive observer is proposed for online estimates of a class of hybrid cascade systems with uncertain parameters. The hybrid cascade system is formed by an ordinary differential equation (ODE) coupled with a partial differential equation (PDE) with uncertain parameters. In the hybrid cascade system, the state of the connection point between the two subsystems cannot be directly available, which makes it difficult to obtain online estimates of the inaccessible states. To deal with the complex observation problem, an impulsive adaptive observer is designed using the backstepping approach and the extended Kalman observer. In addition, the convergence of the proposed observer is examined. The backstepping transformation plays a crucial role in the convergence proof. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.

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, ${\mathbb{R}}^n$ denotes the $n\times 1$ dimensional real space with Euclidean $l_2$-norm defined as $\Vert \cdot \Vert$ for vectors, while ${\mathbb{R}}_{\ge 0}$ denotes the set of nonnegative real numbers. Accordingly, $\Vert X\Vert ≔{\left(X^{T}X\right)}^{\frac{1}{2}}$, for all $X\in {\mathbb{R}}^n$. $N=\{0,1,2,\dots \}$ stands for the set of natural numbers. ${\mathcal{C}}^1$ represents continuously differentiable function space. Then, for a function $s\in {\mathcal{C}}^1[0,+\infty )\to {\mathbb{R}}^n$, ${\Vert s\Vert }_{\tau }:={sup}_{\tau \in [0,t]}\Vert s\Vert$. ${\mathcal{L}}_2[0,L]$ is the Hilbert space with Euclidean ${\mathcal{L}}_2$-norm defined as $\left|\cdot \right|$, that is $\left|w\right|≔{\left({\int }_0^L{w}^2\left(x\right)dx\right)}^{\frac{1}{2}}$, for all $w\in {\mathcal{L}}_2[0,L]$. ${\mathcal{H}}^1[0,L]$ is the Sobolev space of absolutely continuous functions: $w:[0,L]\to \mathbb{R},x\to w(x,\cdot )$ with $dw∕dx\in {\mathcal{L}}_2[0,L]$. For any $w\in {\mathcal{H}}^1[0,L]$ such that $w(0,t)=0$ or $w(L,t)=0$, the following Wirtinger’s inequality holds:

$$\begin{array}{c}{\int }_0^L{w}^2(x,t)dx\le \frac{4L}{{\pi }^2}{\int }_0^L{w}_x^2(x,t)dx.\hfill \end{array}$$

A function ${\kappa }_1:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ belongs to class-$\mathcal{K}$ if it is continuous, zero at zero, and is strictly increasing. A function ${\kappa }_2:{\mathbb{R}}_{\ge 0}\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ belongs to class-$\mathcal{KL}$ if it satisfies: (i) for each $r\in {\mathbb{R}}_{\ge 0}$, the mapping ${\kappa }_2(\cdot ,r)$ is continuous and strictly decreasing with ${lim}_{t\to \infty }{\kappa }_2(t,r)=0$, and (ii) for each $t\in {\mathbb{R}}_{\ge 0}$, the mapping ${\kappa }_2(t,\cdot )$ belongs to class-$\mathcal{K}$.