Induced norms for sampled-data systems

In this paper, we consider a general linear interconnection of a continuous-time plant and a discrete-time controller via sample and hold devices. When the closed loop sampled-data feedback system is internally stable, bounded inputs produce bounded outputs. We present some explicit formulae for the induced norm of the closed loop system with L[infinity] (i.e. peak value) and 1 (i.e. integral absolute) norms on the input and output signals.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29776/1/0000115.pd

Frequency-Domain Analysis of Linear Time-Periodic Systems

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

Sampling from a system-theoretic viewpoint: Part I - Concepts and tools

This paper is first in a series of papers studying a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. In this paper we present the paradigm and revise underlying technical tools, such as the lifting technique and some topics of the operator theory. This material facilitates a systematic and unified treatment of a wide range of sampling and reconstruction problems, recovering many hitherto considered different solutions and leading to new results. Some of these applications are discussed in the second part

L2/L1 L_2/L_1 induced norm and Hankel norm analysis in sampled-data systems

This paper is concerned with the $L_2/L_1$ induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the $h$-periodicity of the input-output relation of sampled-data systems is taken into account, where $h$ denotes the sampling period; past and future are separated by the instant $\Theta\in[0, h)$, and the norm of the operator describing the mapping from the past input in $L_1$ to the future output in $L_2$ is called the quasi $L_2/L_1$ Hankel norm at $\Theta$. The $L_2/L_1$ Hankel norm is defined as the supremum over $\Theta\in[0, h)$ of this norm, and if it is actually attained as the maximum, then a maximum-attaining $\Theta$ is called a critical instant. This paper gives characterization for the $L_2/L_1$ induced norm, the quasi $L_2/L_1$ Hankel norm at $\Theta$ and the $L_2/L_1$ Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function $H(\varphi)$ on $[0, h)$ plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through $H(\varphi)$, even though $\varphi$ is a variable that is totally irrelevant to $\Theta$. The relevance of the induced/Hankel norm to the $H_2$ norm of sampled-data systems is also discussed

Sampling from a system-theoretic viewpoint

This paper studies a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. \ud \ud The paper is split into three parts. In Part I we present the paradigm and revise the lifting technique, which is our main technical tool. In Part II optimal samplers and holds are designed for various analog signal reconstruction problems. In some cases one component is fixed while the remaining are designed, in other cases all three components are designed simultaneously. No causality requirements are imposed in Part II, which allows to use frequency domain arguments, in particular the lifted frequency response as introduced in Part I. In Part III the main emphasis is placed on a systematic incorporation of causality constraints into the optimal design of reconstructors. We consider reconstruction problems, in which the sampling (acquisition) device is given and the performance is measured by the $L^2$-norm of the reconstruction error. The problem is solved under the constraint that the optimal reconstructor is $l$-causal for a given $l\geq 0,$ i.e., that its impulse response is zero in the time interval $(-\infty,-l h),$ where $h$ is the sampling period. We derive a closed-form state-space solution of the problem, which is based on the spectral factorization of a rational transfer function