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This letter is concerned with the $L_{q}/L_{p}$ Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued $L_{p}$ -past inputs to vector valued $L_{q}$ -future outputs. The $L_{q}/L_{p}$ Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the $L_{q}/L_{p}$ Hankel norms of positive systems, which are useful in the case where we analyze the $L_{q}/L_{p}$ Hankel norms of positive systems affected by parametric uncertainties.

In this paper, we present frequency‐weighted optimal Hankel‐norm model reduction algorithms for linear time‐invariant continuous‐time systems by representing an original higher‐order syste...

This paper presents a new linear matrix inequality (LMI)-based approach for the Hankel norm performance of fixed-point state-space digital filters subjected to external interference or excitation of finite duration and operating under various concatenations of quantization and overflow nonlinearities. The approach can be used to check the attenuation of undesired memory effects of past excitations on future outputs in the digital filters and to test the asymptotic stability in the absence of external input. The convex optimization problem with LMI constraints is formulated to determine the minimum Hankel norm performance of the digital filter. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed approach.

We propose and develop a new Hankel-norm approach to the robust receding horizon (RH) finite impulse response (FIR) filter design in discrete-time state space under intensive external disturbances. A new condition is developed for the RH Hankel-norm FIR filter (HNFF) design based on the linear matrix inequality and an equality constraint. The proposed RH HNFF ensures unwanted memory reduction and reduces the effect of memory on errors caused by past disturbances. Another condition is also examined to avoid using the equality constraint. The approach is tested and compared with existing filters based on a numerical example to verify its high robustness against unpredictable model changes for an F-404 turbofan engine system model. An experimental study on the one-degree-of-freedom torsion system is also provided to demonstrate its validity.

This paper is concerned with the Hankel operator and the Hankel norm of sampled-data systems. Even though these systems are intrinsically periodically time-varying, no studies have taken this important feature into account in the treatment of the Hankel operator/ norm. We characterize the Hankel operator/ norm of sampled-data systems adequately under the treatment with the $L_2$ norm for the past input and the $L_infty$ norm for the future output. Such treatment still captures the essential issue on the periodicity but contributes to neat arguments compared with the case when the $L_2$ norm is also considered on the future output.

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.

This brief proposes a new approach to the examination and reduction of the undesired memory effect of digital filters joined with saturation, which is known as system ringing. A new criterion is established to ascertain the Hankel norm performance of digital filters joined with saturation. Using this criterion, it is possible to check the reduction of undesired responses of the filters to past excitations. The asymptotic stability for the digital filters is ensured without external input under the proposed criterion. Three improved criteria for the Hankel norm performance of digital filters are also presented based on the diagonally dominant matrix approach. A numerical simulation shows the usefulness of the developed method.

We use a novel extension of the Adamyan–Arov–Krein theorem to develop an algorithm for designing banks of digital filters by optimal in the Hankel norm reduction of a reference filter bank's order. Summary of the relevant mathematical background is followed by outline of the numerical implementation. We conclude by presenting an example of designing a bank of bandpass filters for splitting a signal into frequency bands.

Optimal Hankel norm model reduction for dynamical systems is of great significance in model-based simulation and design. For the class of linear time-invariant systems, it is among the few optimal reduction methods for which a prior error bound between the original system and its approximation is known. However, for descriptor systems, this optimal approximation technique no longer applies. In this paper, we propose several definitions of the Hankel operator for dynamical discrete-time descriptor systems. We investigate the implications of these definitions for the problem of optimal model approximation of descriptor systems in the sense of the Hankel norm. Novel reduction algorithms are derived for this class of systems with and without preservation of the DAE-index. The performance of the proposed methods is illustrated by numerical examples.

This article studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parameterization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a geometric program , which, in turn, can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the $H^2$ norm, $H^\infty$ norm, Hankel norm, and Schatten $p$ -norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tuning problem jointly constrained by the exponential decay rate, the $\mathcal L^1$ -gain, and the $\mathcal L^\infty$ -gain can be solved by convex optimization. The assumption on the system parameterization is stated in terms of posynomial functions, which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from, for example, dynamical buffer networks and epidemic spreading processes. We present numerical examples to illustrate the effectiveness of the proposed optimization framework.

This article introduces a novel approach to examining the unfavorable memory effects of two-dimensional (2-D) Roesser-type filters based on the 2-D Hankel norm performance. Novel analysis criteria are obtained such that the 2-D Roessertype filters associated with saturation have the guaranteed 2-D Hankel norm performance bound. These criteria can analyze the undesired effect reduction of Roesser-type filters for past horizontal and vertical excitations. Without external horizontal and vertical input signals, the 2-D asymptotic convergence of Roesser-type filters is ascertained under the proposed novel criteria. Finally, a numerical demonstration is given to verify the validity of the obtained criteria. The criteria presented in this paper for 2-D filters and those reported in our recent paper for one-dimensional (1-D) filters can provide a whole analysis framework for the undesired memory effects of filters.

An Illustration of reduction technique of higher order model into its lower order subsystems for highly unstable NASA HIGHMAT FIGHTER using optimal Hankel Norm Approximation is explained in this paper. For higher order system a decomposition approach has been done to resolve the highly unstable system into its unstable and stable combination. Hankel Norm Approximation has been used for obtaining the stable part of reduced order model and hence the overall reduction model system has been developed by combining the both parts. One is reduced model of stable part and second one is unstable part of decomposition. A mathematical strategy of the plant had been done to demonstrate the adequacy of the technique.

This paper presents an optimal Hankel Norm Approximation technique applied to a stable non-minimal system which may be arbitrarily close to being uncontrollable and/or unobservable. The final result of the approximation of original system contains a reduced number of states and therefore must be minimal order. Numerical example has been carried out to demonstrate the procedure.

This paper presents an extended minimal degree optimal Hankel norm approximation (MDOHNA) based order reduction algorithm using a basis-free descriptor which circumvents the requirement of computing balanced realized model for order reduction. Conjunction of system decomposition algorithm (Singh and Nagar, 2004) [21], (Kumar et al., 2012) [30,31] with MDOHNA is used as extension for order reduction of unstable systems. The developed algorithm is applicable for stable/unstable, linear time invariant, minimal/non-minimal, continuous/discrete-time systems as well. Further, effectiveness of the algorithm over existing techniques is validated with the help of a numerical example.

A system can describe real problems that occur in the environment. The system is the result of modeling of real problems. Based on the results of real problem modeling, it often produces a linear system with a special shape of matrix A. In general, to represent the actual conditions of the real problem accurately, a system with a large order is required. While on the other hand, in terms of analysis and computation of the system, it is not desired to have a system with a large order. Therefore, we propose to use a model reduction that produces a system with small orders but without significant errors. One of the commonly used model reduction methods is Hankel Norm Approximation (HNA). In this research, we analyze the process of model reduction with the HNA method in the discrete-time linear system which has special forms of the matrix A. The process of reduction of this model begins with the formulation of the initial system with a special shaped matrix A. Next, we analyze the nature of stability, controllability and observability of the initial system. Then a balanced system is formed and followed by model reduction using HNA. Based on the results of the model reduction simulation using HNA, it was found that the reduced system with HNA has the same performance and properties as the initial system, which is stable, controllable, and observable. Besides that, HNA model reduction is suitable for use at high frequencies and has a fast computation time.

We introduce an approach to H ∞ model reduction that is founded on ideas originating in realization theory, interpolatory H 2 -optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. By employing Loewner “data-driven” partial realizations within each optimization cycle, computationally demanding H ∞ norm calculations can be completely avoided. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better H ∞ performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal Hankel norm approximation.

This work presents a practical technique that can be used to construct the dynamic model of any RF MEMS switch regardless of its shape. The presented technique also allows for inclusion of squeeze film effect in the model without resorting to complex mathematical development of the latter. The technique utilizes Finite element methods to determine mode shapes and natural frequencies of the switch. A modal-model is then constructed from the FEA results. The model can be reduced using by retaining modes with highest Hankel norm modes to reduce calculations effort associated with large models. Simulation results have shown that the proposed model has merit and agrees with published experimental data.

This paper deals with the problem of undesired memory effects in nonlinear digitalfilters owing to the influence of past excitations on future outputs. The nonlinearities under consideration cover the usual types of overflow arithmetic employed in practice. Based on the Hankel norm performance, a new criterion is proposed to ensure the reduction of undesired memory effects in digital filters with overflow arithmetic. In absence of external input, the nonexistence of overflow oscillations is also confirmed by the proposed criterion. A numerical example together with simulation result showing the effectiveness of the criterion is given.

Popular model reduction methods can easily be adapted to retain the asymptotic response to inputs with rational transform. To this purpose, the forced response of the high-order system is decomposed into a transient and a steady-state component. Then, the reduced-order model is obtained by combining the unaltered steady-state component with an approximation of the transient component. Examples show that forcing the reduced-order model to retain the steady-state component does not compromise the transient accuracy.

This paper is concerned with the Hankel operator and the Hankel norm of sampled-data systems. Even though these systems are intrinsically periodically time-varying, no studies have taken this important feature into account in the treatment of the Hankel operator / norm. We characterize the Hankel operator / norm of sampled-data systems adequately under the treatment with the L 2 -type norm for the past input and the L ∞ -type norm for the future output. Such treatment still captures the essential issue on the periodicity but contributes to neat arguments compared with the case when the L 2 -type norm were also considered on the future output.

The aim of this paper is to show how Chebycheff approximation may be used in frequency weighted ℋ∞ system identification and model reduction. We show how these techniques may be combined with known results from Hankel norm model reduction theory to reduce the error between the plant and the low-order approximation. In identification we also apply an ℒ∞-norm variant of Levy's method (Levy, 1959). © 1997 Elsevier Science Ltd.

Hankel norm and Nehari-type approximation problems arise in model reduction and /spl Hscr//sub /spl infin//-control theory. Existing solutions to the discrete-time version of these problems may be derived using a standard state-space framework, but the resulting solution formulas require an invertible A-matrix. As a further complication, the D-matrix in the representation formula for all solutions becomes unbounded in the optimal case. The aim of this paper is to show that both these complications may be removed by analyzing these problems in a descriptor framework.

This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.

This article is concerned with Hankel-norm model approximation (HNMA) for linear parameter-varying systems with parameter-varying time delays. For a given stable system, our attention is focussed on the construction of reduced-order models, which approximate the original system well in a Hankel-norm sense. By applying the slack matrix approach, a delay-dependent sufficient condition is proposed for the robustly asymptotic stability with a Hankel-norm error performance for the error system. Then, the HNMA problem is solved by using the projection approach, which casts the model approximation into a sequential minimisation problem subject to linear matrix inequality constraints by employing the cone complementary linearisation algorithm. Finally, a numerical example is provided to illustrate the effectiveness of the proposed methods.

This paper addresses the Hankel norm of polytopic systems with time-varying delay. Delay-dependent sufficient conditions are presented such that the polytopic system is stable and the Hankel norm of the polytopic system is less than a prescribed scalar in terms of linear matrix inequalities. A numerical example is presented.