Output regulation for a first-order hyperbolic PIDE with state and sensor delays

Abstract

Considering disturbances within domain and at the boundary, a backstepping-based output boundary regulator design is developed for a class of first-order linear hyperbolic partial integro-differential equation (PIDE) in the presence of state and sensor delays. The delays are represented by two transport PDEs, which results in an extended spatial domain where the hyperbolic PIDE, transport PDEs and ordinary differential equation (ODE) are in cascade. The ODE is a finite-dimensional signal model describing exogenous signals. First, a state feedback regulator is realized to achieve a finite time stability by applying an affine Volterra integral transformation. Then, an output regulator is developed on the basis of the nominal plant transfer behavior, which results in an exponential stability. Numerical examples illustrate the performance of the proposed regulators.

Introduction

The output regulation design for boundary controlled first-order hyperbolic PIDE systems with state and measurement delays is developed. The first-order hyperbolic PIDE system due to the dominant role of transport by convection frequently appears in engineering process applications, such as plug-flow tubular reactor [23], [25], traffic flows [5], heat exchangers [13] and oil drilling applications [20], [27]. Therefore, a stabilization control problem for hyperbolic system has been considered by many researchers [4], [11], [28]. Recently, the infinite-dimensional backstepping technique [19] emerged and was further developed to provide a systematic design methodology for the control of PDE systems. An early paper [18] which addresses the first-order hyperbolic PDEs develops a backstepping-based controller for stabilization of an open-loop unstable hyperbolic PIDE. Along the same line, the contributions in [26] and [7] considered a boundary output-feedback controller design for a linear $2\times 2$ first-order PDE and for a quasilinear $2\times 2$ first-order hyperbolic PDE, respectively. As well as in [12], a backstepping-based output-feedback design approach is introduced for a $n+1$ coupled first-order hyperbolic PDEs. Further, the control problem of general $m+n$ heterodirectional hyperbolic systems are addressed in [15], [16]. A new integral transform is introduced in [6] for a first-order hyperbolic PIDE with Fredholm integrals. An adaptive controller and an observer are developed based on swapping for coupled first-order hyperbolic system with unknown parameters in [1], [2]. Recently, the research in [3] discusses the robustness of the output feedback for $2\times 2$ hyperbolic PDE with respect to small delays in the actuation and in the measurements. Besides the efforts to address stabilization of these systems, an important control design problem is to consider the asymptotic tracking of reference input signals in the presence of disturbances.

A promising extension of the backsteping method to the regulation problem design, that achieves asymptotic tracking of reference trajectories in the presence of disturbance, is developed in [8], [10], [22]. Based on the stabilization result of a $2\times 2$ hyperbolic PDE system proposed in [26], a finite-time output feedback regulator is designed [9] and achieves the asymptotic tracking of reference signals in the presence of disturbances. In addition, [29] extends the result for a first-order hyperbolic PIDE with Fredholm integrals developed in [6] to an output regulation problem. The key idea of these works is transforming the output regulation problem into a class of PDE-ODE problem, which is considered in an abstract framework in [21].

However, the aforementioned works for hyperbolic PDE systems have not considered the impact of delays in system setting. The existence of delays brings the meaningful modelling framework since sensor delays and/or state delays are frequently present and in addition induce a new challenge to the output regulator design. Along this theme, there are only few references considering the output regulation which account for the impact of delays. In particular, only in [14] the output regulation of an unstable reaction-diffusion PDE with input delay is considered.

Motivated by the above considerations, we extend our previous works [29] and [24] to develop backstepping-based regulator for a first-order hyperbolic PIDE with delays to ensure the output tracking of a reference in the presence of disturbances. An Ordinary Differential Equation (ODE) exo-system is introduced to describe the reference and disturbance signals. We employ the PDE backstepping boundary control to stabilize the system and the exterior state feedforward for output tracking and disturbance rejection. Slightly different from [29], in this paper the rejection of a disturbance at the controlled output is explored. In addition, both state feedback regulation and output feedback regulation are considered in the paper. Due to the state delay and measurement delay, the regulator equation is more complex than one given in [29], which brings greater difficulties for the regulator equation analysis of both state feedback and output feedback. We prove the existence of a unique solution of the regulator equations under some mild assumption on the exo-system properties. The state feedback output regulator achieves a finite time tracking and furthermore, a novelty of the output feedback regulator is that it combines the state observer, which achieves the output tracking of reference input in finite-time. Last but not least, based on the observer design which assumes that the observer used output and the tracking output do not collocate, we prove and demonstrate the output exponentially convergence to the reference signal.

This paper is organized as follows. Section 2 presents the system. In Section 3, the state feedback regulator is designed and the finite-time tracking is proved. Section 4 designs the observers for reference, disturbance and the state, further proves the exponential stability in $L^2$ norm of the observers. The output feedback regulator based on observer is presented in Section 5 and the supportive simulation results are provided in Section 6. The paper ends with concluding remarks and a discussion of the future work in Section 7.

Notation: Throughout this paper, the partial derivative is denoted as:

$${\partial }_{\zeta }g(\zeta ,t)=\frac{\partial g(\zeta ,t)}{\partial \zeta },{\partial }_{t}g(\zeta ,t)=\frac{\partial g(\zeta ,t)}{\partial t}.$$

The $L^2$-norm for $g(\zeta ,t)\in L^2[0,1]$ is defined as:

$${\parallel g(·,t)\parallel }^2={\int }_0^1{\left|g(\zeta ,t)\right|}^2\mathrm{d}\zeta .$$

The euclidean norm for $X\in {\mathbb{R}}^n$ is defined as:

$${\left|X\right|}^2=X^{\mathrm{T}}X.$$

Domains are defined as:

$${\mathcal{T}}_1=\{(\zeta ,\eta )\in {\mathbb{R}}^2:0\le \zeta \le \eta \le 1\},$$

$${\mathcal{T}}_2=\{(\zeta ,\eta )\in {\mathbb{R}}^2:0\le \zeta ,\eta \le 1\},$$

$${\mathcal{T}}_3=\{(\zeta ,\eta )\in {\mathbb{R}}^2:0\le \eta \le \zeta \le 1\}.$$