Quasi L₂/L₂ Hankel Norms and L₂/L₂ Hankel Norm/Operator of Sampled-Data Systems

This article is relevant to appropriately defining the L₂/L₂ Hankel norm of sampled-data systems through setting a general time instant Θ at which past and future are to be separated and introducing the associated quasi L₂/L₂ Hankel operator/norm at Θ . We first provide a method for computing the quasi L₂/L₂ Hankel norm for each Θ , which is carried out by introducing a shifted variant of the standard lifting technique for sampled-data systems. In particular, it is shown that the quasi L₂/L₂ Hankel norm can be represented as the l₂/l₂ Hankel norm of a Θ -dependent discrete-time system. It is further shown that an equivalent discretization of the generalized plant exists, which means that the aforementioned discrete-time system can be represented as the feedback connection of the discretized plant and the same discrete-time controller as the one in the sampled-data system. It is also shown that the supremum of the quasi L₂/L₂ Hankel norms at Θ belonging to a sampling interval is actually attained as the maximum, which means that what is called a critical instant always exists and the L₂/L₂ Hankel operator is always definable (as the quasi L₂/L₂ Hankel operator at the critical instant). Finally, we illustrate those theoretical developments through a numerical example

Kernel Approximation Approach to the L1 Optimal Sampled-Data Controller Synthesis Problem

This paper is concerned with a new framework called the kernel approximation approach to the L1 optimal controller synthesis problem of sampled-data systems. On the basis of the lifted representation of sampled-data systems, which contains an input operator and an output operator, this paper introduces a method for approximating the kernel function of the input operator and the hold function of the output operator by piecewise constant functions. Through such a method, the L1 optimal sampled-data controller synthesis problem could be (almost) equivalently converted into the discrete-time l1 optimal controller synthesis problem. This paper further establishes an important inequality that forms the theoretical validity of the kernel approximation approach for tackling the L1 optimal sampled-data controller synthesis problem. © 201711Ysciescopu

Causality of general input–output systems and extended small-gain theorem for their feedback connection

For the small-gain theorem derived by Zames in 1966, the later studies after a few decades elaborated on its derivation through defining system causality, which was not assumed by Zames. In connection with the treatment of causality, however, these studies made some unnecessary assumptions on the subsystems in feedback connection and failed to handle general systems described by an input–output relation rather than mapping (which we call input-intolerant/-output-unsolitary systems). On the other hand, although the treatment by Zames can handle such subsystems, it instead turns out to lead to larger values for the induced norms of subsystems compared with the later treatment. This paper is concerned with developing an extended form of the small-gain theorem through the same induced norms as in the later studies while dealing with general input–output causal subsystems. Since causality of subsystems plays a key role in such development, our research direction strongly motivates us to study how causality should be defined for general input–output systems. Thus, much of the arguments in this paper is devoted to such a study, which provides us with profound and thorough understandings on causality of different restricted classes of general input–output systems. Mutual relationships among adequate causality definitions for different classes are also clarified, which should be important in its own right. After deriving an extended form of the small-gain theorem, an example illustrates the importance of dealing with such general subsystems, as well as usefulness of the extension

Numerical Calculation for Discretization of Continuous Quadratic Performance Index

A new procedure of numerical calculation to discretize a quadratic performance index denned for a linear time-invariant continuous system is proposed. The procedure is based on the Padé approximation with scaling and repeated squaring. Theoretical bounds of truncation errors involved in the resulting discretized weighting matrices are provided in terms of the maximum singular value norm for the proposed procedure. It is also shown that the paper by Van Loan which proposed another procedure of numerical calculation for the same problem contains some errors and a certain modification is required for his procedure. Numerical examples show that the new procedure is superior to the old one (of the modified version) from the viewpoint of accuracy, efficiency, and reliability

Characterization of Quasi L∞/L2 Hankel Norms of Sampled-Data Systems

This paper is concerned with the Hankel operator of sampled-data systems. The Hankel operator is usually defined as a mapping from the past input to the future output and its norm plays an important role in evaluating the performance of systems. Since the continuous-time mapping between the input and output is periodically time-varying (h -periodic, where h denotes the sampling period) in sampled-data systems, it matters when to take the time instant separating the past and the future when we define the Hankel operator for sampled-data systems. This paper takes an arbitrary Θ ϵ [0,h) as such an instant, and considers the quasi L∞/L2 Hankel operator defined as the mapping from the past input in L2(-∞ Θ) to the future output in L∞Θ ∞). The norm of this operator, which we call the quasi L∞/L2 Hankel norm at Θ is then characterized in such a way that its numerical computation becomes possible. Then, regarding the computation of the L∞L2 Hankel norm defined as the supremum of the quasi L∞L2 Hankel norms over Θ ϵ [0,h), some relationship is discussed between the arguments through such characterization and an alternative method developed in an earlier paper that is free from the computations of quasi L∞/L2 Hankel norms. A numerical example is studied to confirm the validity of the arguments in this paper. © 201711Ysciescopu

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

Induced Norm Analysis of Linear Systems for Nonnegative Input Signals

This paper is concerned with the analysis of the $L_p\ (p\in[1,\infty), p=\infty)$ induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the $L_{p+}$ induced norm in this paper. It has been shown recently that the $L_{2+}$ induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the $L_{2+}$ induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. Then, in the second part of the paper, we derive uniform lower bounds of the $L_{p+}\ (p\in[1,\infty), p=\infty)$ induced norms with respect to the standard $L_{p}$ induced norms that are valid for all linear systems including infinite-dimensional ones. For each linear system, we finally derive a computation method of the lower bounds of the $L_{2+}$ induced norm that are larger than (or equal to) the uniform one. The effectiveness of the upper/lower bound computation methods are fully illustrated by numerical examples.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at ECC 2022 (arXiv:2401.03242) and IFAC WC 202

Separator-type robust stability theorem of sampled-data systems allowing noncausal LPTV scaling

This paper derives a necessary and sufficient condition for robust stability of sampled-data systems, which is stated by using the notion of separators that are dealt with in an operator-theoretic framework. Such operator-theoretic treatment of separators provides a new perspective, which we call noncausal linear periodically time varying scaling and leads to reducing conservativeness in robust stability analysis. A numerical example is given to demonstrate the results

Properties of discrete-time noncausal linear periodically time-varying scaling and their relationship with shift-invariance in lifting-timing

This article is concerned with the technique called discrete-time noncausal linear periodically time-varying (LPTV) scaling for robust stability analysis. Noncausal LPTV scaling has already been shown to be effective for reducing the conservativeness of robustness analysis in theoretical and numerical ways. However, there still remain some issues to be resolved for further understanding and exploiting noncausal LPTV scaling, e.g. its relationship with the conventional analysis approach of causal linear time-invariant scaling. In this article, by introducing the key idea of shift-invariance in lifting-timing, we discuss the difference and corresponding relationship between the conventional approach and noncausal LPTV scaling