•The robust stabilization criteria for singular switched FOS subject to actuator saturation under the designed switching law is proposed.•An observer to stabilize the closed-loop singular switched ...FOS is designed and the observer and controller gains are obtained.•The stability region by solving an optimization problem in terms of LMIs is estimated.•The presented methods are illustrated by the practical and numerical simulations.
The present paper proposes the robust control for singular switched fractional order systems (FOS). The main objective of this paper is to design an observer to stabilize the closed-loop singular switched FOS with actuator saturation under the designed switching law. By using generalized singular value decomposition (SVD) and linear matrix inequalities (LMIs), the observer and controller gains are obtained. Then, based on the proposed stabilization conditions, the stability region by solving an optimization problem is estimated. Finally, the presented methods are illustrated by the practical and numerical simulations.
•Multilevel information cryptosystem using GSVD, optical interference, and DVFL encoding is proposed.•The parameters of DVFL are exploited as highly sensitive decryption keys for the first time.•The ...individual key and two analytically produced POMs are utilized as highly robust decryption keys.•The proposed method avoids problems that result from misalignment.•Numerical simulation results show the practicability and effectiveness of the proposed system.
A novel multilevel information cryptosystem based on generalized singular value decomposition (GSVD), optical interference, and devil's vortex Fresnel lens (DVFL) encoding, is proposed. The proposed cryptosystem exploits a set of four fused LL sub-bands from four RGB images, which are converted into a CMYK image and split into C, M, Y, and K channels. The GSVD operation is used to produce five matrices from C and M channels, and five matrices from Y and K channels independently. A single-channel image of the first set formed by fusion of the two unitary matrices from each pair is gyrator transformed. In a similar fashion, the transformed images of a number of sets are combined into a complex image, which is then inverse gyrator transformed. The proposed system exploits optical interference of one phase-only mask (POM) as DVFL and two analytically produced POMs. The parameters of DVFL are used as remarkably sensitive decryption keys. Moreover, the individual keys and two POMs are used as decryption keys to advance the security against potential attacks. To avoid the strict alignment of the three POMs in different arms during the experiment, the summation of the three POMs is displayed on a single spatial light modulator (SLM). Further, gyrator transform does not require axial movements. Therefore, the proposed method avoids problems that result from misalignment. The proposed method can be implemented by using a hybrid optoelectronic system. Numerical simulation results demonstrate the practicability and effectiveness of the proposed system.
This paper proposes a new methodology for moving force identification (MFI) from the responses of bridge deck. Based on the existing time domain method (TDM), the MFI problem eventually becomes ...solving the linear algebraic equation in the form Ax=b. The vector b is usually contaminated by an unknown error e generating from measurement error, which often called the vector e as ‘‘noise’’. With the ill-posed problems that exist in the inverse problem, the identification force would be sensitive to the noise e. The proposed truncated generalized singular value decomposition method (TGSVD) aims at obtaining an acceptable solution and making the noise to be less sensitive to perturbations with the ill-posed problems. The illustrated results show that the TGSVD has many advantages such as higher precision, better adaptability and noise immunity compared with TDM. In addition, choosing a proper regularization matrix L and a truncation parameter k are very useful to improve the identification accuracy and to solve ill-posed problems when it is used to identify the moving force on bridge.
•A truncated generalized singular value decomposition method is proposed for identifying force.•The regularization matrix L is introduced which can improve solving the ill-posed problems.•The truncation parameter k is introduced which can avoid noise disturbance and ensure robustness.•The proposed method has high precision, good adaptability and immunity of ill-posed problems.
In recent years, the multiple-input multiple-output (MIMO) non-orthogonal multiple-access (NOMA) systems have attracted a significant interest in the relevant research communities. As a potential ...precoding scheme, the generalized singular value decomposition (GSVD) can be adopted in MIMO-NOMA systems and has been proved to have a good trade-off for complexity and performance. In this paper, the performance of the GSVD-based MIMO-NOMA communications with Rician fading is studied. In particular, the distribution characteristics of generalized singular values (GSVs) of channel matrices are analyzed. Two novel mathematical tools, the linearization trick and the deterministic equivalent method, which are based on operator-valued free probability theory, are exploited to derive the Cauchy transform of GSVs. An iterative process is proposed to obtain the numerical values of the Cauchy transform of GSVs, which can be exploited to derive the average data rates of the communication system. In addition, the special case when the channel is modeled as Rayleigh fading, i.e., the line-of-sight propagation is trivial, is analyzed. In this case, the closed-form expressions of average rates are derived from the proposed iterative process. Simulation results are provided to validate the derived analytical results.
Let J=0In−In0∈R2n×2n. A matrix H∈R2n×2n is called Hamiltonian if (HJ)⊤=HJ. In this paper, the inverse eigenvalue problem for Hamiltonian matrices is considered. The solvability condition for the ...inverse problem is derived and the representation of the general solution is presented by the generalized singular value decomposition of a matrix pair. Furthermore, the associated optimal approximation problem for this inverse eigenvalue problem is discussed and the expression of the solution for the optimal approximation problem is presented.
For a given matrix
G
∈
R
k
×
n
, let the set
Ω
G
=
Null
(
G
)
,
and let
A
S
Ω
G
(
R
n
×
n
)
=
{
A
∈
R
n
×
n
|
(
A
x
,
y
)
=
(
x
,
-
A
y
)
,
∀
x
,
y
∈
Ω
G
}
, that is,
AS
Ω
G
(
R
n
×
n
)
is the set of ...all anti-symmetric matrices on
Ω
G
. In this paper, we first consider the following problem (Problem 1): Given
A
∈
R
m
×
n
,
B
∈
R
m
×
m
, find
X
∈
AS
Ω
G
(
R
n
×
n
)
such that
A
X
A
⊤
=
B
. Then, we consider the associated optimal approximation problem: Given
X
~
∈
R
n
×
n
, find
X
^
∈
S
Ω
G
(
A
,
B
)
such that
X
^
=
arg
min
X
∈
S
Ω
G
(
A
,
B
)
‖
X
-
X
~
‖
, where
‖
·
‖
is the Frobenius norm and
S
Ω
G
(
A
,
B
)
is the solution set of Problem 1. By using the generalized singular value decomposition of a matrix pair, we deduce the solvability condition and the representation of the general solution of Problem 1. Moreover, we obtain the unique approximation solution
X
^
of the optimal approximation problem. Finally, a numerical example is presented to show the correctness of our result.
We present local biplots, an extension of the classic principal component biplot to multidimensional scaling. Noticing that principal component biplots have an interpretation as the Jacobian of a map ...from data space to the principal subspace, we define local biplots as the Jacobian of the analogous map for multidimensional scaling. In the process, we show a close relationship between our local biplot axes, generalized Euclidean distances, and generalized principal components. In simulations and real data we show how local biplots can shed light on what variables or combinations of variables are important for the low-dimensional embedding provided by multidimensional scaling. They give particular insight into a class of phylogenetically-informed distances commonly used in the analysis of microbiome data, showing that different variants of these distances can be interpreted as implicitly smoothing the data along the phylogenetic tree and that the extent of this smoothing is variable.
Multiple factor analysis (MFA, also called multiple factorial analysis) is an extension of principal component analysis (PCA) tailored to handle multiple data tables that measure sets of variables ...collected on the same observations, or, alternatively, (in dual‐MFA) multiple data tables where the same variables are measured on different sets of observations. MFA proceeds in two steps: First it computes a PCA of each data table and ‘normalizes’ each data table by dividing all its elements by the first singular value obtained from its PCA. Second, all the normalized data tables are aggregated into a grand data table that is analyzed via a (non‐normalized) PCA that gives a set of factor scores for the observations and loadings for the variables. In addition, MFA provides for each data table a set of partial factor scores for the observations that reflects the specific ‘view‐point’ of this data table. Interestingly, the common factor scores could be obtained by replacing the original normalized data tables by the normalized factor scores obtained from the PCA of each of these tables. In this article, we present MFA, review recent extensions, and illustrate it with a detailed example. WIREs Comput Stat 2013, 5:149–179. doi: 10.1002/wics.1246
This article is categorized under:
Data: Types and Structure > Categorical Data
Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
The generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large‐scale problems is challenging. ...Motivated by applications in hyper‐differential sensitivity analysis (HDSA), we propose new randomized algorithms for computing the GSVD which use randomized subspace iteration and weighted QR factorization. Detailed error analysis is given which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters. We demonstrate the performance of our algorithms on test matrices and a large‐scale model problem where HDSA is used to study subsurface flow.
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