Generalized ℋ︁₂ model approximation for differential linear repetitive processes

Abstract

This paper investigates the generalized H 2 model approximation for differential linear repetitive processes (LRPs). For a given LRP, which is assumed to be stable along the pass, we are aimed at constructing a reduced-order model of the LRP such that the generalized H 2 gain of the approximation error LRP between the original LRP and the reduced-order one is less than a prescribed scalar. A sufficient condition to characterize the bound of the generalized H 2 gain of the approximation error LRP is presented in terms of linear matrix inequalities (LMIs). Two different approaches are proposed to solve the considered generalized H 2 model approximation problem. One is the convex linearization approach, which casts the model approximation into a convex optimization problem, while the other is the projection approach, which casts the model approximation into a sequential minimization problem subject to LMI constraints by employing the cone complementary linearization algorithm. A numerical example is provided to demonstrate the proposed theories. 66 L. WU, W. X. ZHENG AND X. SU method [1][2][3][4][5], the optimal Hankel-norm approximation method [6,7], and the optimal H 2 model approximation [8,9], just to mention a few. As is well known, H ∞ and L 2 -L ∞ (called also generalized H 2 ) settings have been considerably used in optimal synthesis over the past decades [10][11][12]. They have been well recognized to be most appropriate for systems with noise input, whose stochastic information is not precisely known. H ∞ setting minimizes the worst-case energy gain from the noise inputs to the system output; whereas L 2 -L ∞ setting minimizes the worst-case energy to peak gain from the noise inputs to the system output. In this paper, we are interested in the generalized H 2 model approximation method, which was studied in [13] for time-varying delay systems. One of its main advantages is the fact that this method is insensitive to the exact knowledge of the statistics of the noise signals. The generalized H 2 model approximation procedure ensures that the generalized H 2 gain from the input signal to the approximation error will be less than a prescribed level, where the noise input is an arbitrary energy-bounded signal.Linear repetitive processes (LRPs) are a distinct class of 2-D systems (that is, information propagation in two independent directions) of both system-theoretic and application interests [14]. The essential unique characteristic of a repetitive, or multipass, process is a series of sweeps, termed passes, through a set of dynamics defined over a fixed finite duration known as the pass length. On each pass, an output, termed the pass profile, is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile [14]. To introduce a formal definition, let <+∞ denote the pass length (assumed constant). Then in an LRP the pass profile y k (t), 0 t , generated on pass k acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile y k+1...