IFAC PapersOnLine 52-2 (2019) 138–143 ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.08.024 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION The speed gradient method is long recognized (Fradkov, 2007) as a powerful tool of the energy control of Lagrangian systems. It is therefore of interest to develop a consistent extension of the method in the infinite-dimensional setting. The first applications to the sine-Gordon PDE (Dolgopolik et al., 2016; Orlov et al., 2017) have corroborated the utility of the method for controlling the energy of the sine-Gordon model, using the manipulable parameter (intensity) of the external electrical field or the boundary actuation. The present work makes a step beyond the aforementioned results by admitting the use of non-collocated in-domain sen- sors and actuators, which are placed within small spatial plant subdomains. First, the speed gradient method is revisited for the state feedback energy control of the string PDE model with several in-domain actuators. For certainty, the Dirichlet bound- ary conditions are chosen for the treatment. Being feasible in practice, the combination of the in-domain actuators is shown to yield more opportunities for reducing the invariant manifold with parasitic dynamics of undesirable energy levels. The proposed state feedback synthesis forms one of the main contributions of the paper. The Luenberger-type observer de- sign for the linear PDE model in question is another contribu- tion which is made over available in-domain position measure- ments. ⋆ This work was partially supported by CONACYT under Grant No.285279. Numerical study was supported by the Government of Russian Federation (Grant 08-08) and RFBR (Grant 17-08-01728). Coupled together, the proposed state feedback and Luenberger- type observer result in the output feedback synthesis of the underlying string PDE model over in-domain position measure- ments, thereby finalizing the present contribution to the PDE- flavored speed gradient method of the nonlinear energy control. Capabilities of the method are additionally supported by nu- merical simulations whereas its rigorous validation is expected to be published elsewhere (Orlov et al., 2019). 1.1 Notation Standard notation is used throughout. Also recall that the Sobolev space H l (a, b) with a natural index l consists of l times weakly differentiable functions x(r) : R → R, which are defined on the domain (a, b) ⊂ R and whose norm is given by ‖x(·)‖ H l (a,b) =     l ∑ j=0  b a  ∂ j x ∂ r j  2 dr. By default, H 0 (a, b)= L 2 (a, b) and H l (0, 1)= H l . 2. PROBLEM STATEMENT Consider the dimensionless string model x tt = kx rr + u(r , t ), t ≥ 0, 0 ≤ r ≤ 1 (1) where t is the time instant, r ∈ [0, 1] is the scalar spatial variable, x = x(·, t ) is the instant state of the system, the parameter k is the elasticity of the string, u(r , t ) is for the manipulable input. Throughout, the available in-domain actuation u(r , t )= M ∑ i=1 u i (t )I i (r) (2) Keywords: energy control, wave equation, speed-gradient Abstract: The output energy control is developed for 1-D string model with fixed endpoints. It is considered a practical situation where a finite number of spatially-sampled sensing and actuation are available. First, the speed-gradient method is generalized in the present framework to pump/dissipate the energy of underlying model to a desired level provided the state feedback is available. Next, Luenberger- type observers are additionally developed over collocated position and/or velocity measurements to be involved into the output feedback synthesis. Capabilities of the proposed synthesis and its robustness features are illustrated in numerical simulations. ∗ Department of Electronics and Telecomunications Mexican Scientific Research and Advanced Studies Center of Ensenada, Carretera Tijuana-Ensenada, B.C. 22860, M´ exico yorlov@cicese.mx ∗∗ Institute for Problems of Mechanical Engineering of the RAS, 61 Bolshoy prospekt, V.O., 199178, St.Petersburg, Russia fradkov@mail.ru ∗∗∗ St.Petersburg State University, St.Petersburg, Russia ∗∗∗∗ ITMO University, 49 Kronverksky Pr., 19710, St.Petersburg, Russia boris.andrievsky@gmail.com Yury Orlov ∗ Alexander L. Fradkov ∗∗,∗∗∗,∗∗∗∗ Boris Andrievsky ∗∗∗∗ Output Feedback Energy Control of String PDE Model ⋆