Hankel Norm Research Articles

Leveraging the Hankel norm approximation and data‐driven algorithms in reduced order modeling

SummaryLarge‐scale linear time‐invariant (LTI) dynamical systems are widely used to characterize complicated physical phenomena. Glover developed the Hankel norm approximation (HNA) algorithm for optimally reducing the system in the Hankel norm, and we study its numerical issues. We provide a remedy for the numerical instabilities of Glover's HNA algorithm caused by clustered singular values. We analyze the effect of our modification on the degree and the Hankel error of the reduced system. Moreover, we propose a two‐stage framework to reduce the order of a large‐scale LTI system given samples of its transfer function for a target degree of the reduced system. It combines the adaptive Antoulas–Anderson (AAA) algorithm, modified to produce an intermediate LTI system in a numerically stable way, and the modified HNA algorithm. A carefully computed rational approximation of an adaptively chosen degree gives us an algorithm for reducing an LTI system, which achieves a balance between speed and accuracy.

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

<abstract><p>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.</p></abstract>

Two-Dimensional Hankel Norm Performance of Roesser-Type Filters

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.

Transformation of Two-Dimensional Roesser Model Into Two-Dimensional Decoupled Model: Application to Model Reduction via Balanced Framework

Two transformations are presented in this research; the first approach takes a generic two-dimensional (2-D) framework and transforms it into a diagonalized 2-D framework without resorting to minimal rank-decomposition criteria. Afterward, it converts the diagonalized system into two subsystems (i.e., one-dimensional (1-D) cascaded systems). Moreover, the proposed transformation retains the symmetrical structure of the two decomposed subsystems. Likewise, the second transformation converts 1-D submodels to the balanced structure used for model reduction operations. The proposed model reduction approach efficiently performs the model reduction task on a given 2-D diagonalized model by utilizing optimal Hankel norm approximation. The numerical results demonstrate that the proposed transformations efficiently convert the standard 2-D framework to a diagonalized model form and develop a stable reduced-order model.

Hankel norm performance of 2-D digital filters described by Roesser model with overflow arithmetic

ABSTRACT This paper is concerned with the problem of asymptotic stability with Hankel norm performance (HNP) of two-dimensional (2-D) digital filter characterised by Roesser model, where the 2-D system includes overflow nonlinearities and external interference or excitation of finite duration. The underlying nonlinearities cover the usual types of overflow arithmetic utilised in practice such as saturation, two’s complement, zeroing and triangular. To solve this problem, new sufficient criterion is proposed by employing quadratic 2-D Lyapunov function. The proposed criterion guarantees that the addressed system has 2-D HNP bound. This criterion can examine the unwanted memory effect (ME) reduction of 2-D interfered digital filters for past excitations. In the absence of external inputs, the asymptotic convergence of 2-D digital filters is established under the proposed criterion. The developed HNP criterion is in linear matrix inequality framework and, therefore, computationally tractable. In addition, the optimisation problem is formulated to obtain the minimum HNP of the 2-D interfered digital filter. To illustrate the effectiveness of the proposed criterion, a numerical example is provided. The criteria addressed in current paper can give a whole analysis framework for the unwanted MEs of filters.

External Positivity of Sampled-data Systems and Their <i>L<sub>q</sub></i>/<i>L</i><sub>∞</sub> Hankel Norm Analysis

This paper extends the theory of positive systems to sampled-data systems. When considering positivity of LTI systems, where to take the initial time is unimportant. However, since sampled-data systems are periodically time-varying from the perspective of continuous-time, introducing an adequate definition for their positivity should be relevant to considering which time to take as the initial time. This paper begins by studying positivity of sampled-data systems first by considering the case where a sampling instant is regarded as the initial time, and secondly by further considering the case where an arbitrary intersample instant is regarded as the initial time. These arguments are combined to derive the necessary and sufficient conditions for a unified notion of (i.e., initial-time-independent) positivity of sampled-data systems. This paper then applies this concept to the analysis of the Lq/L∞ Hankel norm, known as a measure for evaluating the effects of disturbances in dynamical systems. It is shown that positive sampled-data systems lead to concise expressions for the (quasi) Lq/L∞ Hankel norms not only for q = ∞ but also for q = 1,2, where the latter are hard to handle for general sampled-data systems. Some relevant properties for these norms are also shown. The theoretical results derived in this paper are confirmed through numerical examples.

Parameterized Synthesis of Discrete-Time Positive Linear Systems: A Geometric Programming Perspective

This letter focuses on optimization problems of discrete-time positive linear systems. To this end, the synthesis problem is presented by introducing parameterized system coefficient matrices and optimizing system parameters directly. Based on results concerning positive linear systems and nonnegative matrix theory, we demonstrate that optimization problems of minimizing the parameter tuning cost while satisfying the H2 norm, H∞ norm, and ℓ1/ℓ∞ Hankel norm constraints can be reduced to corresponding geometric programming problems. In turn, by imposing reasonable assumptions on system matrices, these geometric programming problems can be further transformed into convex optimization problems owing to the convexity of the logarithm transformation on posynomials. Finally, simulation experiments on a numerical example and epidemic spreading process example are used to show the validity of the main results.

Hankel Norm Based Strict Passivity Performance of Digital Filter Using Saturation Nonlinearities

The work proposes a new technique to examine and reduce the unusual memory effects in digital filters with saturation nonlinearity when external interference is present via strict passivity approach. Novel criterion is given to show the performance of Hankel norm in interfered digital filters by using strict passivity approach. The unusual behaviour of digital filters about previous excitations can be checked by using this proposed criterion. Given criterion ensures the asymptotic stability nature when there is no external input. The dominance of the proposed method is represented by a mathematical example.

Robust Controller Design for AUV Subsystem Using µ Synthesis

The paper considers the problem of synthesizing the pitch control system of an autonomous underwater vehicle (AUV), considering the uncertainty of its parameters and underwater disturbances. The µ-synthesis using D-K iteration is applied to solve the problem. For the subsequent implementation of the controller in practice, a method for reducing its order based on the Hankel norm approximation is applied. The proposed approach is tested on the mathematical model of the MAYA AUV with the parameters deviating from the nominal values. The simulation result shows that the proposed robust control scheme effectively compensates for the parametric uncertainty and underwater disturbances.

Controller and sensor placement for a 3D irregular building based on Hankel norm

Placing controllers and sensors properly is important in structural health monitoring and control. Many optimization methods require much computation efforts. This paper used Hankel norms to develop the placement rules, because they involve the input and output gains and thus could be shaped by the locations. In modal form, their computations are relatively simple. The location and mode influences on norms were arranged in rows and columns, respectively, to form a matrix, and was normalized by the column (mode) root mean square. The optimization goal is to choose locations with higher index values and lower correlations to ensure higher controllability and observability, and with less effort to be compensated for by gains. Hankel norm is compatible with the LQR control objectives in that they are both 2-norm, so the methodology is appropriate to be applied to the base isolation benchmark building for structural control, which is an eight-story irregular building with ninety-two candidate locations for controllers and thirty-six locations for sensors. Following the method, ten controller locations and eighteen sensor locations were determined. Earthquake time history analysis using LQG technique validated the effectiveness of thus determined subset of locations by comparing with other subset of locations.

Hankel Norm Model Reduction of Discrete-Time Interval Type-2 T–S Fuzzy Systems With State Delay

This article focuses on the model reduction problem of discrete-time time-delay interval type-2 Takagi-Sugeno (T–S) fuzzy systems. Compared with the type-1 T–S fuzzy system, the interval type-2 T–S fuzzy system has more advantages in expressing nonlinearity and capturing uncertainties. In addition, in order to simplify the analysis process, complex high-order systems can be approximated as low-order systems, which is called model reduction. In previous studies, there are few researches on model reduction of the interval type-2 T–S fuzzy system with time delay. Hankel norm is adopted to limit the error after model reduction. Based on Jensen's inequality, a linear matrix inequality (LMI) condition for the Hankel norm performance of the error system is obtained. A membership-function-dependent method based on piecewise linear membership functions is utilized to deal with mismatched membership functions where information of membership functions will be used for relaxing analysis results. Next, by a convex linearization design, the model reduction problem is formulated as a convex LMI feasibility/optimization condition. Numerical examples are given to verify the validity of the analysis.

Efficient Technique for Model Order Reduction Retaining Non-Minimum Phase Characteristics using Clustering Dominant Pole-Zero Algorithm

<span style="font-family: 'Times New Roman',serif; font-size: 10pt; -ms-layout-grid-mode: line; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-GB; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-GB">Model Order Reduction (MOR) challenges a high dimensional problem and plays a key role in areas where dynamic simulation studies are necessary for modern simulation strategy. Many conventional reduction methods namely, reduced order models based on Least Square Method (LSM), Balanced Truncation, Hankel Norm reduction, Dominant Pole Algorithm (DPA) and CDPA method have been developed in the field of control theory. Among these, recently proposed Clustering Dominant Pole Algorithm (CDPA) is able to compute the full set of dominant poles and their cluster center efficiently. In this paper, a hybrid algorithm for model order reduction known as Clustering Dominant Pole-Zero Algorithm (CDPZA) is proposed to identify and preserve the dominant zeros of the processes exhibiting non-minimum phase behaviour. The CDPZA method combines the features of clustering method and DPA. Further, the cluster centers of the dominant zeros in the numerator polynomial are determined using factor division algorithm. The Benchmark HiMAT system of 6<sup>th</sup> order is considered for testing and validation of the proposed algorithm. The simulation studies are carried out to show the efficacy of the proposed algorithm over conventional MOR algorithms.</span>

Model reduction for discrete-time periodic systems with dissipativity

ABSTRACTThis paper is concerned with the model reduction problem for discrete-time periodic systems with time-varying delay. By utilising an adjusted reciprocally convex combination approach, some sufficient criteria are established to guarantee that the discrete-time periodic system with time-varying delay is asymptotically stable. Then, the reduced-order model is designed to approximate the original system. In the design, both the dissipativity and the Hankel norm performances are considered to test the model approximation, and correspondingly, the conditions are also established, which guarantee that the approximation error system is asymptotically stable with the dissipativity performance and the Hankel norm property, respectively. The convex linearisation and the projection are presented to design the reduced-order model, respectively. Finally, numerical examples are provided to verify the effectiveness of the design scheme.

Geometric Programming for Optimal Positive Linear Systems

This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a \emph{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 tunning 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 parametrization 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.

The ${L}_{{q}}/{L}_{{p}}$ Hankel Norms of Positive Systems

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.

Model reduction analysis using Hankel Norm Approximation on discrete-time linear system with the special shape of matrix A

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.

Optimal Hankel norm model reduction for discrete-time descriptor systems

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.

Hankel Norm Performance of Interfered Fixed-Point State-Space Digital Filters with Quantization/Overflow Nonlinearities

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.

Research on robust reduction control method of steering-by-wire based on MBSE

Based on the theory and method of MBSE, a model of Steering-By-Wire (SBW) system with mechanical and electrical integration is established. Based on the model, a robust control strategy with H∞ mixed sensitivity is used, and a robust controller is designed. The high order robust controller is poor in real-time and has high cost. Routh, Hankel norm and ISE index reduction method are used to reduce the controller. Through time domain and frequency domain analysis, the effects of various reduction methods on the controller are discussed. Simulation results show that the ISE index reduction method has better reduction effect and has good robust performance and robust stability.

Synthesis of robust controllers for the control systems of technological units at iron ore processing plants

In order to synthesize a robust system of control over technological units, an analysis of appropriate mathematical models was performed. The uncertainty of models of technological iron ore processing units was accounted for by connecting a diagonal block at the top using a fractional linear transformation. To study robust control systems of technological units, we applied the following types of robust controllers: suboptimal H ∞ -controller, a controller that was synthesized using the method of circuit formation, and µ-controller. We performed an analysis of results of the study into indicators of robust quality and stability of control, created on the basis of these types of controllers. The best results were obtained using μ-controller, which ensures a minimal overshoot value of 2 %. Reducing the order of the selected μ-controller to the fourth order was performed by approximation using the Hankel norm. Under such a condition, a root-mean-square error relative to the base controller is 0.027. Results of present research could be used in the synthesis of control over technological iron ore processing units under conditions of uncertainty in parameters.