ISS output feedback synthesis of disturbed reaction–diffusion processes using non-collocated sampled-in-space sensing and actuation

Abstract

Linear proportional ISS synthesis of parabolic systems is developed within the practical framework of in-domain embedded sensing and actuation. The underlying system is affected by external disturbances and it is governed by a non-homogeneous reaction–diffusion PDE with a priori unknown spatially varying parameters. The present investigation focuses on practically motivated sampled-in-space sensing and actuation. A finite number of available sensing and actuating devices are assumed to be located along the one-dimensional spatial domain of interest. Tuning of the controller gains is then constructively developed by means of the Lyapunov approach to achieve a desired attenuation level for external distributed disturbances, affecting the system in question. Dual observer design is additionally developed within the present framework, and it is involved into the non-collocated output feedback synthesis. Theoretical results are supported by simulations made for the proposed synthesis over non-collocated sensing and actuation.

Introduction

The ISS (input-to-state stability) concept has been introduced to characterize the response of the closed-loop system against external disturbances (Sontag, 2008). ISS means that the state norm is upper bounded by a continuous disturbance-dependent function, escaping to zero when the disturbance magnitude is nullified, whereas the effect of an arbitrary initial condition is captured by an additional term, depending on both the magnitude of the initial condition and on time, which asymptotically decays as time goes to infinity.

The ISS of DPS (distributed parameter systems), basically with boundary and/or point-wise sensing and actuation, has been addressed in the literature in such works as Bribiesca Argomedo, Prieur, Witrant, and Bremond (2013), Dashkovskiy and Mironchenko (2013), Jacob, Mironchenko, Partington, and Wirth (2018), Karafyllis and Krstic (2016), Mazenc and Prieur (2011), Mironchenko and Ito (2015) and Pisano and Orlov (2017), (see also references therein). Within the linear ${\mathscr{H}}_{\infty }$ framework, a constructive output feedback design has been proposed in Morris (2001) in terms of Riccati operator equations and later (Fridman & Orlov, 2009) in terms of operator LMIs. In Fridman and Bar (2013) and Fridman and Blighovsky (2012), the LMI approach has been extended to sampled-data ${\mathscr{H}}_{\infty }$ synthesis whereas in Kasinathan and Morris (2013), a methodology has been given to optimally locate a pre-specified number of actuators in the spatial domain. A relevant problem of determining how many point-wise collocated actuator/sensor pairs would be needed for a one-dimensional reaction–diffusion process to achieve a desired decay rate with a pre-specified arbitrarily small level of attenuation of mismatched distributed disturbances has been solved in Pisano and Orlov (2017).

It is worth noticing that the desired decay rate of the stabilization of an approximately controllable unperturbed system can be achieved by means of one actuator only, e.g., through a spectral/modal approach (Curtain & Zwart, 1995) or via the backstepping method (Karafyllis & Krstic, 2018), However, it might be impossible to use just one actuator for managing both the decay rate and disturbance attenuation to their desired levels as it is the case of the backstepping boundary control proposed in Karafyllis and Krstic (2016) for a parabolic PDE. Such a drawback appears to be typical for scalar control of distributed parameter plants in the presence of mismatched disturbances. Thus motivated, the interest of choosing many different actuators merges to designing the closed-loop with desired both exponential decay rate and attenuation level in the presence of general distributed disturbances.

It is the primary concern of the present work to develop the ISS synthesis for one-dimensional reaction–diffusion processes, located within a finite scalar spatial domain and governed by parabolic PDEs (partial differential equations) with spatially-varying heat capacity, diffusivity, and reaction coefficients. A practically motivated framework of sampled-in-space actuators and state measurements is adopted throughout thereby making a step beyond the existing literature where the available sensing and actuation are located point-wise, e.g., at the plant boundary.

The ISS synthesis to be developed for the underlaying parabolic system deals with a finite number of non-collocated sampled-in-space actuators and sensors. The development is inspired from Pisano and Orlov (2017) where the investigation was confined to specific reaction–diffusion–advection processes withNeumann boundary conditions, constant plant parameters, and collocated point-wise sensing and actuation. In contrast to Pisano and Orlov (2017), the present work captures Robin boundary conditions and spatially varying plant parameters, and admits non-collocated sampled-in-space sensing and actuation which are not necessarily point-wise. The proposed synthesis relies on the well-known weighted quadratic Lyapunov functional, earlier used in such works as Bribiesca Argomedo et al., 2013, Karafyllis and Krstic, 2016, Mazenc and Prieur, 2011 and Orlov (1983) to name a few, and it follows the methodology, recently developed in Orlov, Autrique, and Perez (2019) for a class of parabolic systems with collocated sensor-actuator pairs. Linear proportional state feedback and dual observer design are developed side by side. The resulting non-collocated output feedback synthesis is accompanied with a guideline of ensuring a desired closed-loop decay rate and disturbance attenuation level by pre-specifying the required number and specific location of actuators and sensors and by properly tuning the actuator and sensor gains.

The contribution of the present work to the existing literature is thus as follows. The ISS state feedback synthesis, developed for the perturbed reaction–diffusion process with spatially-varying plant parameters, is coupled to a dual ISS observer, independently constructed for the underlying plant with an arbitrary non-collocated control input, to constitute the output feedback synthesis over non-collocated sampled-in-space sensors and actuators. The decay rate and disturbance attenuation level of the resulting closed-loop system are then established as explicit functions of both the plant parameter magnitudes and the number of available actuators and sensors to guide the designer on attaining the desired decay rate and disturbance attenuation level in the closed-loop.

The rest of the paper is outlined as follows. The present section is completed by the notational conventions and by instrumental lemmas to be used in the sequel. The ISS problem of interest is stated in Section 2. The collocated ISS synthesis is then developed in Section 3 whereas Sections 4 ISS observer design, 5 Non-collocated output feedback synthesis present the dual non-collocated observer design and non-collocated output feedback synthesis, respectively. In Section 6, the effectiveness of the proposed synthesis is supported by numerical simulations. Finally, Section 7 collects conclusions and discusses further research challenges.

The symbol $H^{\ell }(a,b)$ with $a<b$ and $\ell =0,1,2,\dots$ denotes the Sobolev space of absolutely continuous scalar functions $z\left(x\right)$ on $(a,b)$ with square integrable derivatives $z^{\left(i\right)}\left(x\right)$ up to the order $\ell$ and the $H^{\ell }$-norm

$${\Vert z\left(\cdot \right)\Vert }_{H^{\ell }(a,b)}=\sqrt{{\int }_a^b{\Sigma }_{i=0}^{\ell }{\left[z^{\left(i\right)}\left(x\right)\right]}^{2}dx}.$$

Throughout the paper, $C^0(a,b)$ and $C^1(a,b)$ are for the spaces of continuous and, respectively, continuously differentiable functions on $[a,b]$, and the standard notations $H^0(a,b)=L_2(a,b)$ and $L_{\infty }(a,b)$ are used as well. The symbol $L_{\infty }(0,T;L_2(a,b))$ is reserved for the set of functions $f(x,t)$ such that $f(\cdot ,t)\in L_2(a,b)$ for almost all $t\in (0,T)$, ${\int }_a^{b}f(x,t)\varphi \left(x\right)dx$ is Lebesgue measurable in $t$ for all $\varphi \left(\cdot \right)\in L_2(a,b)$, and $ess{sup}_{t\in (0,T)}{\int }_a^b{f}^2(x,t)dx<\infty$. It is said that $z\left(\cdot \right)\in L_{\infty }^{loc}$ iff $z\left(\cdot \right)\in L_{\infty }(0,T)$ for all $T>0$; respectively, $f\left(\cdot \right)\in L_{\infty }^{loc}\left(L_2(a,b)\right)$ iff $f\left(\cdot \right)\in L_{\infty }(0,T;L_2(a,b))$ for all $T>0$.

Given a function $\phi \left(x\right):\mathbf{R}\to \mathbf{R}$, the symbol $supp\phi \left(x\right)$ stands for the closure of the set $\{x\in \mathbf{R}:\phi \left(x\right)\ne 0\}$ where the function $\phi$ takes nonzero values.

The following well-known results are instrumental for the subsequent ISS analysis.

Lemma 1 Mean-value Lemma for Definite Integrals

Let $\psi \left(x\right):[a,b]\to \mathbf{R}$ be a continuous function and let $\phi \left(x\right)\in L_2(a,b)$ be sign definite on $[a,b]$. then there exists $\xi \in (a,b)$ such that

$${\int }_a^b\psi \left(x\right)\phi \left(x\right)dx=\psi \left(\xi \right){\int }_a^b\phi \left(x\right)dx.$$

Lemma 2 Poincare Inequality

Let $z\left(x\right)\in H^1(a,b)$ and $x_1,x_2\in \mathbf{R}$ be such that $a\le x_1\le x_2=x_1+h\le b$ for some $h\ge 0$. Then, the following inequality

$${\Vert z\left(\cdot \right)\Vert }_{L_2(x_1,x_2)}^2\le 2h\left[z^2\left(x_i\right)+h{\Vert z_x\left(\cdot \right)\Vert }_{L_2(x_1,x_2)}^2\right]$$

holds for $i=1,2$ and $z_x\left(\cdot \right)=dz∕dx$.