Hankel Norm Research Articles

The $L_\infty/L_2$ Hankel Operator/ Norm of Sampled-Data Systems

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 impor...

Modified Hankel Interaction Index Array for Input-Output Pairing with Improved Characteristics

In this study, a modified version of Hankel Interaction Index Array (HIIA) for control configuration selection is presented which can overcome some of its shortcomings, like e.g. scaling dependency, or not relating to closed loop system properties. Inspired by the relative gain array approach, the HIIA is reformulated in the relative gain thinking by considering the effect of closing loops. The ratio of the Hankel norm of the subsystems in closed and open loop are used to state a modified version of HIIA, which has improved characteristics compared to the original HIIA. Properties of the modified HIIA are discussed and benchmarked with established methods on three example cases.

Analysis of Memory Effects in Digital Filters with Overflow Arithmetic

<p>This paper deals with the problem of undesired memory effects in nonlinear digital<br />filters 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.</p>

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.

A Method for the Hankel-Norm Approximation of Fractional-Order Systems

A model-reduction methodology for fractional-order systems based on the Hankel-norm is presented. The methodology involves the truncation of a Laurent series associated with the fractional-order system in a transformed domain. The truncated Laurent series coefficients are used to construct a finite-order transfer function to approximate the original system. Standard model-reduction techniques are then applied to obtain a final low-order approximation. The Hankel norm of the approximation error can be specified a priori. The approximation method is applied to several fractional-order and other infinite-order systems. It is shown to be more generally applicable than standard finite-order modeling techniques.

Hankel Norm Performance of Digital Filters Associated With Saturation

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.

A Hankel norm for quadrature rules solving random linear dynamical systems

In linear time-invariant dynamical systems, random variables are included to quantify uncertainties. The solution can be expanded into a series with predetermined orthogonal basis functions, which depend on the random variables. We define a norm of Hankel-type associated to a truncated series. A quadrature rule or a sampling method yields approximations of the unknown time-dependent coefficient functions in the truncated series. We arrange a Hankel norm for the quadrature technique in this context. Assuming a convergent sequence of quadrature rules, we show that the Hankel norms of the quadrature methods converge to the Hankel norm of the truncated series. Hence a numerical method is obtained to calculate the Hankel norm of the truncated series approximately. Results of numerical computations confirm the convergence property in a test example.

Identification-Based Robust Motion Control of an AUV: Optimized by Particle Swarm Optimization Algorithm

In this paper, the problem of identification-based robust motion control of an Autonomous Underwater Vehicle (AUV) is investigated. The unknown system parameters are estimated by using an adaptive parameter identifier, whose gains are optimized by Particle Swarm Optimization (PSO) algorithm. Removing the trial and error procedure and ensuring the convergence property together with fast response, are the benefits of such identification scheme. On the other hand, the system uncertainties, hydrodynamic parameter variations and external disturbances which affect the identified dynamical model, are also taken into account. The cross-coupling effects between subsystems are also considered as model uncertainties. Such uncertain model is then adopted in control synthesis procedure, in the steering and diving modes. In order to achieve the robust stability and performance, two robust control strategies are presented here to solve the motion control problem. First, an Hź$H_{\infty }$ mixed sensitivity problem is formulated in which the weighting functions are selected based on an optimization criterion, by using PSO algorithm. Controller order reduction is also applied to the resulting diving and steering controllers, using the Hankel norm approximation. Then, an Adaptive Sliding Mode Control (ASMC), whose sliding surface coefficients are optimized by PSO algorithm, is developed for the identified AUV model. Possessing the robustness properties with respect to system perturbations, the developed Sliding Mode Control (SMC) removes the complexity of uncertain model representation and the limitations on choosing the weighting functions in the Hź$H_{\infty }$ control problem. The upper bounds of perturbations are not required to be known in the proposed control schemes. The simulation results are also presented to demonstrate the performance of the proposed identification-based control methods.

Multivariable System with Level Control

The paper presents the mathematical modeling procedure of a multivariable process with two coupled tanks and it is determined the multivariable control algorithms, imposing a desired behavior of the closed-loop and open-loop system. They ensure the imposed performances on direct connection channels, compensate the effect of the inter-influences between channels, and the system operates decoupled. The control algorithms resulted by the design are high-order, so, based on the Hankel norm, there are determined the truncated (reduced) forms of the controllers. The computing, as well as the behavior of the system, are performed using the Matlab environment.

Assessment of the Design Criteria Effect on Performance of Multiple Tuned Mass Dampers

In this research, the effect of different design criteria on performance of multiple tuned mass dampers (MTMDs) under earthquake excitation has been studied. For optimal design of MTMDs, the parameters of TMDs are determined based on minimizing an objective function. Two different objective functions including minimizing (a) the Hankel's norm; and (b) the maximum displacement of structure have been selected using Distributed Genetic Algorithms (DGAs) for solving the optimization problem. For numerical simulation, the method has been utilized on an eight-storey shear frame and optimal MTMDs have been designed. Results show that using the Hankel's norm as the objective function has led to design MTMDs which are effective in reducing the root-mean-square (RMS) of structural response under different earthquakes while in reducing the maximum response both design criteria have had the same results. Through numerical analysis the effect of design record, MTMDs mass ratio, TMD's number and stroke length on MTMDs effectiveness has been studied.

Structural Control of Piezoelectric Laminated Beams under Thermal Load

The dynamic analysis and the active vibration control of piezoelectric laminated beams under thermal load are presented. The beam is modeled using two noded finite elements with four mechanical and a variable number of electric potential degrees of freedom at each node. In the thickness direction, the thermal and the electric fields are approximated as piecewise linear across an arbitrary number of sublayers in a layer. Cubic Hermite interpolation is used for the deflection and electric potentials at the sublayers and linear interpolation is used for the axial displacement and the shear rotation. The thermal field is computed using a consistent six-noded thermal finite element with a quadratic interpolation along longitudinal direction and a linear interpolation along thickness direction. The temperature distribution and undamped natural frequencies are obtained for composite and sandwich beam under cantilever and clamped-clamped boundary conditions and compared with 2D-FE Abaqus results. The finite element equations derived are converted into modal model to represent them in the state space form. This model is then reduced using Hankel norm for designing the LQG controller. Optimal control technique is used to control the vibration of the beam. The designed LQG controller controls the tip deflection of composite and sandwich cantilever beams and midpoint deflection of clamped-clamped beams.

Minimizing memory effects of a system

Given a stable linear time-invariant system with tunable parameters, we present a method to tune these parameters in such a way that undesirable responses of the system to past excitations, known as system ringing, are avoided or reduced. This problem is addressed by minimizing the Hankel norm of the system, which quantifies the influence of past inputs on future outputs. We indicate by way of examples that minimizing the Hankel norm has a wide scope for possible applications. We show that the Hankel norm minimization program may be cast as an eigenvalue optimization problem, which we solve by a nonsmooth bundle algorithm with a local convergence certificate. Numerical experiments are used to demonstrate the efficiency of our approach.

Experimental Identification and Control of a Cantilever Beam Using ERA/OKID with a LQR Controller

This paper presents an experimental study of the system identification and vibration control of a cantilever beam. For system identification, a white noise was applied, and the response signal was measured. These signals were used to feed the eigensystem realization algorithm–Observer/Kalman Filter identification method. The identified system was reduced using the Hankel norm model. An linear quadratic regulator controller was projected to operate just on the first two natural frequencies of the structure. The damping ratio of the first mode was effectively increased from 0.009 to 0.046.

T-S Fuzzy Model-Based Approximation and Filter Design for Stochastic Time-Delay Systems with Hankel Norm Criterion

This paper investigates the Hankel norm filter design problem for stochastic time-delay systems, which are represented by Takagi-Sugeno (T-S) fuzzy model. Motivated by the parallel distributed compensation (PDC) technique, a novel filtering error system is established. The objective is to design a suitable filter that guarantees the corresponding filtering error system to be mean-square asymptotically stable and to have a specified Hankel norm performance levelγ. Based on the Lyapunov stability theory and the Itô differential rule, the Hankel norm criterion is first established by adopting the integral inequality method, which can make some useful efforts in reducing conservativeness. The Hankel norm filtering problem is casted into a convex optimization problem with a convex linearization approach, which expresses all the conditions for the existence of admissible Hankel norm filter as standard linear matrix inequalities (LMIs). The effectiveness of the proposed method is demonstrated via a numerical example.

Model Reduction by Balanced Truncation for Systems with Nuclear Hankel Operators

We prove the H-infinity error bounds for Lyapunov balanced truncation and for optimal Hankel norm approximation under the assumption that the Hankel operator is nuclear. This is an improvement of the result from Glover, Curtain, and Partington [SIAM J. Control Optim., 26 (1998), pp. 863--898], where additional assumptions were made. The proof is based on convergence of the Schmidt pairs of the Hankel operator in a Sobolev space. We also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation.

Model reduction by extended minimal degree optimal Hankel norm approximation

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 Study of Non-Uniform SPST RF-MEMS Switch under the Effect of Squeeze Film Damping

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.

Interpolatory [formula omitted] model reduction

We introduce an approach to H∞ model reduction that is founded on ideas originating in realization theory, interpolatory H2-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.

Hankel norm model approximation of Takagi–Sugeno fuzzy time-delay systems

This paper is concerned with the Hankel norm model approximation for discrete-time Takagi–Sugeno (T–S) fuzzy systems with time delay. For a given stable T–S fuzzy system, our attention is focused on the construction of a reduced-order model which not only approximates the original system well with a Hankel norm performance but is also translated into a lower-dimensional system. By employing the modified delay partitioning method, a delay-dependent sufficient condition is developed for the asymptotic stability with a Hankel norm error performance for the error system. Then, the Hankel norm approximation problem is solved by employing the convex linearization approach, which casts the reduced-order model construction into a convex optimization problem subject to linear matrix inequality constraints. Finally, a numerical example is provided to show the effectiveness of the proposed methods.

Pade and Laguerre approximations applied to the active queue management model of internet protocol

This paper considers the problem of pure time delay in the Active Queue Management model, the delay element is approximated by using Pade and Laguerre presentations. Hankel norm is used to determine the order of truncation. Quantitative error bounds on delay approximations are given and calculated. The approximated models are compared with the exact model with delay to determine the efficiency of the presented methods. The example presented in illustration is the model of Congestion control in Internet Protocol.