0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2716107, IEEE Transactions on Automatic Control 1 Robust Hybrid Output Regulation for Linear Systems with Periodic Jumps: Semi-classical Internal Model Design Daniele Carnevale, Sergio Galeani, Laura Menini, Mario Sassano Abstract—A complete procedure for the design of a robust out- put feedback regulator is proposed for a class of uncertain linear hybrid systems with periodic jumps, using a hybrid extension of the classical internal model principle. Simple conditions, testable on the plant nominal data, for the problem solvability are given. The plant is not restricted to be minimum phase, square or single- input-single-output. The proposed regulator has a key feature of containing an internal model composed by two main units, a flow internal model, in charge of providing the correct input to achieve regulation during flows, and a jump internal model, in charge of suitably resetting the state of the regulator at each period. The proposed procedure is illustrated by its application to a physically motivated example, for which the output regulation problem is not solvable by methods appeared thus far in the literature. I. I NTRODUCTION Control of hybrid systems, characterized by the interaction between a continuous-time (flow) dynamics and a discrete-time (jump) dynamics, has been widely studied in the last years, as shown, e.g., in [1], [2] and references therein. The problem of output regulation, well studied for linear [3], [4] and nonlinear [5], [6], [7], [8] systems, is also a very relevant problem in many applications involving hybrid systems. Due to several difficulties arising when trying to extend the classic output regulation theory to the hybrid setting, a specific scenario was identified in [9] in which hybrid phenomena arise without destroying the underlying linearity of the considered flow and jump dynamics; the study of the output regulation problem in the same setting was later addressed in several papers [10], [11], [12], [13], [14]. The assumption of [9] consists in considering that jumps only occur according to a periodic sequence of time instants, and that both the plant and the exosystem jump simultaneously. Despite its deceptively simple appearance, this problem formulation gives rise to a plethora of intriguing and unexpected phenomena that have no paral- lel in the classic regulation theory. The flow zero-dynamics internal model principle introduced in [13] is particularly relevant for the present discussion. The principle shows that, in general, the steady-state input achieving regulation contains a suitable copy of the modes of the flow zero dynamics of the plant. Such modes should be expected to be affected by uncertainties and then unknown. However, under some easily checkable and physically motivated structural conditions, it is Daniele Carnevale, Sergio Galeani, Laura Menini and Mario Sas- sano are with Dipartimento di Ingegneria Civile e Ingegneria Informat- ica, Universit` a di Roma “Tor Vergata”, Via del Politecnico, 1 - 00133 Roma, Italy Email: [daniele.carnevale@, sergio.galeani@, menini@disp., mario.sassano@]uniroma2.it possible to isolate a class of hybrid systems, namely semi- classical systems [15], [16], characterized by output-zeroing steady-state inputs that include only the natural modes of the exosystem. Such a class of systems is considered here. The robust version of the problem of hybrid output regulation proposed by [9] has been addressed in [11], [17]. While the contributions in [11], [17] focus on square, minimum-phase and (generally) relative degree one plants and Poisson stable exosystems, here no such assumptions are made, and the availability of more inputs than outputs is exploited to achieve solvability of the problem for a larger class of plants. The key difference in the two approaches consists in the allowed uncertainty affecting the plant, with the conditions in [11], [17] requiring a more structured uncertainty. The goal of this paper is to solve the problem of robust hybrid output regulation in the presence of additive uncertainties on the matrices of the state-space description of the plant. A preliminary version of the results was reported in [18]. The main novelty here is that sharper formal results are provided. Moreover, the compensator architecture has been revisited and simplified, thus yielding a clearer and more intuitive interpretation of its structure and inner working, as well as a simpler design procedure. Finally, the efficacy of the proposed compensator is illustrated by means of a physically motivated example, namely an RC electric circuit subject to periodic switchings whose physical parameters are allowed to vary arbitrarily in a neighborhood of their nominal values (the same system, but without parameter variations, was considered in [16]); it is worth stressing that the regulation problem for such a system cannot be solved by any other result currently available in the literature. Notation. The acronym GES is used for Global Exponential Stability, OR for Output Regulation whereas LTI/LTV stand for Linear Time Invariant/Varying. C g := {s ∈ C : |s| < 1}. The Kronecker product is denoted by ⊗, and the spectrum of matrix M by Λ(M ). For r ∈ R, the shortcuts N ≥r := {m ∈ N : m ≥ r} and R ≥r := {m ∈ R : m ≥ r} are used. The degree of polynomial p(s) is denoted deg(p(s)).A pair of matrices (A, B) is reachable if [B AB ··· A n−1 B] is full row rank, and a pair of matrices (A, C ) is observable if [C ′ A ′ C ′ ··· (A ′ ) n−1 C ′ ] ′ is full column rank. II. PRELIMINARIES AND PROBLEM DEFINITION This paper focuses on a class of hybrid systems, introduced in [9], which experience periodic jumps separated by a flow