Hybrid Synthesis for Almost Asymptotic Regulation of Linear Impulsive Systems with Average Dwell Time Chengzhi Yuan and Fen Wu Abstract— This paper deals with the hybrid output regulation problem for a class of linear impulsive systems with average dwell time. The hybrid regulator is constructed as a linear impulsive system that undergoes synchronous impulses with the controlled plant, and the hybrid synthesis conditions are formulated in terms of two matrix equations plus a set of linear matrix inequalities (LMIs). With the proposed hybrid synthesis scheme, both continuous-time and discrete-time dynamics of the hybrid regulator can be jointly synthesized by solving a convex optimization problem of minimizing the weighted L2 gain from the perturbation signal to the error output. A numerical example is used to demonstrate the proposed approach. I. I NTRODUCTION The problem of output regulation, which concerns con- trolling a given plant such that its output tracks references or rejects disturbances generated by an exosystem, have been extensively studied over the past decades for various dynamical systems (see, [1], [2], [3] and the references there- in). In particular, necessary and sufficient conditions for the solvability of the output regulation problem for linear time- invariant (LTI) systems were nicely established as a set of linear algebraic equations in [1], which paves the fundamen- tals for the subsequent researches on, for instance, optimal output regulation [4]; output regulation with saturations [5]; output regulator syntheses with multi performance objectives [6], and also involves in many practical applications [7]. It is not until the nineties that the output regulation problem for general nonlinear systems was first pursued by the pioneering works [8] and [2]. Recent research on the output regulation problem has shifted to hybrid dynamical systems that combine continuous-time and discrete-time behaviors, such as [9], [10], [3], [11]. In this paper, we study the hybrid output regulation prob- lem for a class of linear systems subject to impulse effects. This type of systems are classified as linear impulsive dy- namical systems that are typically modeled by the combina- tion of ordinary differential equations and instantaneous state jumps or resets (also referred to as impulse) [12], [13]. As a special case of hybrid systems, linear impulsive systems have been widely involved in the research in control community, due mainly to their presence in practical systems and poten- tials in overcoming limitations of traditional controllers (see, e.g., [12], [14]). Stability properties of such systems have been extensively investigated, see, for instance, [12], [15], [13]. However, few results on the output regulation problem This work is supported in part by the NSF grant CMMI-1200242. Both authors are with the Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA cyuan2@ncsu.edu; fwu@eos.ncsu.edu of linear impulsive systems can be found in the literature. In particular, it is worth to mention the latest contribution [3] in which the classical output regulation concepts were extended to single-input single-output (SISO) linear systems with periodic state jumps and a series of fundamental theories for hybrid regulator design were proposed. This work was subsequently applied in [11] to design hybrid regulators with different structures. These results, however, rely heavily on the periodicity of the impulse sequences. In this paper, the impulse instants of the linear im- pulsive system under consideration are allowed to occur non-periodically, but with a frequency constraint, which is formalized in terms of an average dwell time (ADT) condition [16], [13]. On the other hand, as opposed to the typical treatments in classical output regulation problem of considering an exosystem with autonomous dynamics and the reference/disturbance signals that are precisely known, we will also generalize the hybrid output regulation problem for linear impulsive systems with unknown perturbations on both controlled plant and the exosystem, which leads to the notion of almost asymptotic regulation. From this respect, the hybrid regulator design problem is treated under a multiobjective framework, such that both objectives of output regulation and weighted H ∞ controlled performance (from [17]) can be achieved. The proposed hybrid output regulator is constructed as a linear impulsive system, and the resulting hybrid multiobjective synthesis conditions are formulated in terms of tractable conditions, consisting of linear algebraic equations and linear matrix inequalities (LMIs). Another novelty of the proposed hybrid synthesis scheme is that the hybrid regulator gain matrices with respect to both continuous-time and discrete-time dynamics can be jointly (simultaneously) synthesized in a systematic and unified framework, i.e., by solving a convex LMI-based optimization. Notation. R stands for the set of real numbers and R + for the positive real numbers. R m×n is the set of real m × n matrices. The transpose of a real matrix M is denoted by M T . The hermitian operator He {·} is defined as He {M } = M + M T for real matrices. The identity matrix of any dimension is denoted by I . S n and S n + are used to denote the set of real symmetric n × n matrices and positive definite matrices, respectively. A block diagonal matrix with matrices X 1 ,X 2 , ··· ,X p on its main diagonal is denoted by diag {X 1 ,X 2 , ··· ,X p }. Furthermore, we use the symbol ⋆ in LMIs to denote entries that follow from symmetry. For x ∈ R n , its norm is defined as ∥x∥ := (x T x) 1/2 . The space of square integrable functions is denoted by L 2 , that is, for 53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA 978-1-4673-6088-3/14/$31.00 ©2014 IEEE 4691