Brief paperISS output feedback synthesis of disturbed reaction–diffusion processes using non-collocated sampled-in-space 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 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 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 with and denotes the Sobolev space of absolutely continuous scalar functions on with square integrable derivatives up to the order and the -norm
Throughout the paper, and are for the spaces of continuous and, respectively, continuously differentiable functions on , and the standard notations and are used as well. The symbol is reserved for the set of functions such that for almost all , is Lebesgue measurable in for all , and . It is said that iff for all ; respectively, iff for all .
Given a function , the symbol stands for the closure of the set where the function takes nonzero values.
The following well-known results are instrumental for the subsequent ISS analysis.
Lemma 1 Mean-value Lemma for Definite Integrals Let be a continuous function and let be sign definite on . then there exists such that
Lemma 2 Poincare Inequality Let and be such that for some . Then, the following inequality holds for and .
Section snippets
Problem statement
Consider the space- and time-varying scalar field , evolving in the space , with the spatial variable and time variable . Let it be governed by the perturbed parabolic Robin-type boundary-value problem (BVP) with the associated initial condition (IC) In the BVP (3), (4), is an uncertain distributed disturbance of class , and ,
Collocated output feedback synthesis
The collocated output feedback (6) is formed by the linear control signals which are involved with proportional gains to be tuned for achieving the control objective, stated above.
It is clear that the closed-loop system is linear, and due to the assumptions imposed, it possesses a unique solution in (Curtain & Zwart, 1995). With this in mind, the ISS analysis of the closed-loop system (3)–(8), (13) is developed next.
ISS observer design
The Luenberger-flavored observer design to be developed over the built-in-domain measurements (8) is dual to the proposed controller synthesis. It should be pointed out that it is developed for an arbitrary control input rather than for that specified with (6), and hence, it remains in force regardless of whether collocated or non-collocated actuation and sensing are in play. Such an observer design over a sampled-in-space, generally speaking non-collocated sensing (9) forms a basis for
Non-collocated output feedback synthesis
In order to synthesize an output feedback the state feedback law (6), (13) is modified according to where is the output of the state observer (44), (46). Under the conditions of Theorem 1, Theorem 3, the resulting closed-loop system proves to be exponentially ISS so that the state vector satisfies the following inequality where
Simulation results
Theoretical results are illustrated with a realistic thermal configuration, evolving in time in the operational domain, formed by a thin homogeneous plate whose thickness, width and length are respectively , , and , and whose volumic heat , thermal conductivity , and heat exchange coefficient are constants (the interested reader may refer (Ashby, 2016) for details). The spatial variables of such a plate are . Suppose that the domain is insulated on the boundaries
Conclusion
The ISS problem is addressed for one-dimensional reaction–diffusion processes with Neumann boundary conditions, with unknown spatially-varying uncertain parameters and mismatched external disturbances, and with a finite number of non-collocated built-in-domain actuators and sensors. The proportional ISS state feedback synthesis is proposed, and the decay rate and disturbance attenuation level are established as certain functions of both the plant parameter magnitudes and the number of available
Yury Orlov received his M.S. degree from the Mechanical-Mathematical Faculty of Moscow State University, in 1979, the Ph.D. and Dr. Sc. degrees in Physics and Mathematics from the Institute of Control Science, Moscow, in 1984, and from Moscow Aviation Institute, in 1990, respectively. He is a Full Professor of the Electronics and Telecommunication Department, Scientific Research and Advanced Studies Center of Ensenada, Mexico, since 1993. During the scientific career he shared visiting/temporal
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Cited by (5)
Output feedback stabilization of reaction–diffusion PDEs with a non-collocated boundary condition
2022, Systems and Control LettersCitation Excerpt :Beyond the sole output feedback stabilization of the plant, we show that the procedure developed in this paper also allows the establishment of an input-to-state stability (ISS) estimate with respect to an additive boundary perturbation in the application of the boundary control. Note that ISS estimates with respect to unmatched disturbances can also be obtained in our framework for additive perturbations applied in the domain of the PDE (see, e.g., [25] for the study of such a case in the context of collocated reaction–diffusion PDE with bounded input and output operators while using an infinite-dimensional observer) or in the measurement. Note however that such input perturbations apply to the closed-loop system dynamics as inputs of bounded operators.
Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs
2021, AutomaticaCitation Excerpt :In the past few years, the input-to-state stability (ISS) theoryfor infinite dimensional systems governed by partial differential equations (PDEs) has drawn much attention in the literature ofPDE control. The property of ISS was successfully established first for systems with disturbances distributed over the domain (see, e.g. Argomedo, Prieur, Witrant, & Bremond, 2013; Dashkovskiy & Mironchenko, 2013a, 2013b; Mazenc & Prieur, 2011; Mironchenko & Ito, 2015, 2016; Orlov, 2020; Orlov, Perez, Gomez, & Autrique, 2020; Pisano & Orlov, 2017), and then important achievements have been obtained for systems with disturbances located on the boundary (see, e.g., Jacob, Mironchenko, Partington, & Wirth, 2019; Jacob, Nabiullin, Partington and Schwenninger, 2018; Jacob, Schwenninger and Zwart, 2018; Karafyllis & Krstic, 2016, 2017, 2018a, 2018b, 2020; Lhachemi, Saussié, Zhu, & Shorten, 2020; Lhachemi & Shorten, 2019; Mironchenko, Karafyllis, & Krstic, 2019; Schwenninger, 2019; Zheng & Zhu, 2018a, 2018b, 2020). A comprehensive survey on this topic is presented in Mironchenko and Prieur (2019).
Input-to-state stability and integral input-to-state stability in various norms for 1-D nonlinear parabolic PDEs
2023, Journal of Control and DecisionFinite-time non-fragile boundary feedback control for a class of nonlinear parabolic systems
2021, Nonlinear DynamicsInput-to-state Stability in Different Norms for 1-D Parabolic PDEs
2021, Proceedings of the IEEE Conference on Decision and Control
Yury Orlov received his M.S. degree from the Mechanical-Mathematical Faculty of Moscow State University, in 1979, the Ph.D. and Dr. Sc. degrees in Physics and Mathematics from the Institute of Control Science, Moscow, in 1984, and from Moscow Aviation Institute, in 1990, respectively. He is a Full Professor of the Electronics and Telecommunication Department, Scientific Research and Advanced Studies Center of Ensenada, Mexico, since 1993. During the scientific career he shared visiting/temporal professor positions in Moscow Aviation Institute, CESAME (Catholic University in Louvain, Belgium), Ecole Centrale de Lille (France), Robotics Laboratory of Versalle University (France), INRIA (Grenoble, France), IRCCYN (University of Nantes, France), University of Angers (France), University of Cagliari (Italy), University of Kent (UK), and University of Lund (Sweden). The research interests include analysis and synthesis of nonlinear, nonsmooth, discontinuous, systems of finite and infinite dimensions, and their applications to electromechanical systems.
Laetitia Perez received the Postgraduate Degree (DEA) in Process Engineering from the University of Perpignan (France) in 2000 and the Ph.D. degree from the Ecole Nationale Supérieure des Arts et Métiers (ENSAM - France) in 2003. She has from 2004 to 2006 a temporary research and teaching position in the E.H.F department (Expertise Hauts Flux), D.G.A. (weaponry department of French Ministry of Defense) Font Romeu, (France). In 2006, she joined the Thermocinetic Laboratory of Nantes. She is currently associate professor in LARIS (Polytech, University of Angers, France). Her research interests are in the modeling of thermal process, the experimental benches development and the resolution of inverse problems.
Oscar Gómez received his B.S. degree in mathematics from the University of Sonora, in 1984, the M.S. degree in Industrial Engineering from the Technological Institute of Sonora, in 1992, and the Ph.D. in Electronics and Telecommunications from the Scientific Research Center CICESE in Ensenada, Mexico, in 2007. He is currently a full time Professor at the University of Sonora. His research of interest is the identification of parameters of distributed parameter systems.
Laurent Autrique received the postgraduate degree (DEA) in process control from the Ecole Centrale (Nantes, France) in 1992, and the Ph.D. degree from the same institute in 1995. He was a research scientist for the PROMES Research Institute (C.N.R.S. Perpignan, France) until 2002. Then he was from 2002 to 2007 with the E.H.F. Department (Expertise Hauts Flux), D.G.A. (Weaponry Department of French Ministry of Defense) Font Romeu, (France). He is currently professor in LARIS (Polytech, University of Angers, France). His research works are devoted to parametric identification, inverse problems, process analysis and systems control theory. Main publications are focused on non linear partial differential equations system describing state evolution of complex thermal processes. He is with the Department of Control Engineering and Computer Engineering, Polytech Angers, University of Angers, 62 avenue notre dame du lac, 49000 Angers, France (Email: [email protected]).
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This work was supported by the regional programme Atlanstic 2020, Atlanstic, funded by the French Region Pays de la Loire and the European Regional Development Fund, and it was partially supported via research grant A1-S-9270, funded by CONACYT, Mexico . The material in this paper was partially presented at the 58th IEEE Conference on Decision and Control, December 11–13, 2019, Nice, France. This paper was recommended for publication in revised form by Associate Editor Rafael Vazquez under the direction of Editor Miroslav Krstic.