Image watermarking using separable fractional moments of Charlier–Meixner

This paper proposes a new set of discrete orthogonal separable moments of fractional order, named Fractional Charlier–Meixner Moments (FrCMMs). The latter are constructed from fractional Charlier polynomials (FrCPs) and fractional Meixner polynomials (FrMPs) proposed in this paper. The proposed FrMPs are constructed algebraically using the spectral decomposition of classical Meixner polynomials and singular value decomposition (SVD). The proposed FrCMMs generalize the separable moments of Charlier–Meixner of integer order (CMMs). In addition, FrCMMs are characterized by the polynomial parameters and by the fractional orders of the two fractional kernel functions of Charlier and Meixner, which allows them to be used efficiently for different applications such as local and global image reconstruction and image watermarking. Based on the proposed FrCMMs, a new watermarking scheme for copyright protection of digital images in the transform domain is proposed where the watermark is embedded in the FrCMM coefficients leading to an efficient watermarking scheme in terms of imperceptibility, robustness and security. The performances of the proposed moments are evaluated and compared with discrete fractional moments existing in the literature and with classical separable moments of integer order.     

Polynomial Root Clustering

A sequence of tests on derived polynomials to be strictly Hurwitz polynomials is shown to be equivalent to a given (typically real) polynomial having all its zeros in an open sector, symmetric with respect to the real axis, in the left half-plane. The number of tests needed is at most 1 + ?(ln k)/(ln 3)?, where k is the integer associated with the central angle π/k of the sector. An extension of this result on the sector as a region of root clustering is given which shows that only a limited number of tests are needed to verify that the roots are clustered in a region composed as the intersection of a set of primative (sector-like) regions. The results reported evolve from application of a collection of mappings on the complex plane defined by a particular collection of Schwarz-Christoffel transformations.     

Robust FOPID stabilization of retarded type fractional order plants with interval uncertainties and interval time delay

This study investigates the robust stability of the retarded type of interval fractional order plants with an interval time delay. To this end, the characteristic quasi-polynomial is divided into two terms. The first term is simply the denominator interval polynomial of the open loop system and the second term is the multiplication of the interval delay term in the numerator of the open loop system which is an interval polynomial. Each of these two terms of the characteristic quasi-polynomial makes their own value sets in the complex plane for a given frequency. In this paper, based on these two value sets and by using the zero exclusion principle, the robust stability of the closed loop system by applying a FOPID controller is analyzed. Finally, two numerical examples and an experimental verification are provided to demonstrate the effectiveness of the proposed method in the robust stabilization of fractional order plants with interval uncertainties and interval time delay.     

Two dimensional quaternion valued singular spectrum analysis with application to image denoising

Compared to the traditional single color plane based image denoising methods, the quaternion valued singular value decomposition (QSVD) exploits the relationship among different color planes. Hence, it has been applied to the color image denoising. On the other hand, compared to the non-overlapping based image denoising methods, the two dimensional real valued singular spectrum analysis (2DSSA) constructs the trajectory matrix with many elements in the matrix being overlapped. Since the 2DSSA exploits the local information within each color plane, it has also been applied to the single color plane based image denoising. However, neither these two image denoising methods can exploit the relationship among the color planes and the local information within each color plane simultaneously. Therefore, this paper proposes a two dimensional quaternion valued singular spectrum analysis (2DQSSA) based method for performing the color image denoising. Our proposed method can enjoy the advantages of both methods. However, the most critical issue for the 2DQSSA is on the selection of these 2DQSSA components. This paper finds that the optimal total number of the 2DQSSA components used for performing the reconstruction is monotonic decreasing with respect to the power of the noise in the image. Therefore, the polynomial fitting approach is proposed to model this relationship. Since the 2DQSSA based denoising method exploits the relationship among the red color plane, the green color plane and the blue color plane, the 2DQSSA based denoising method outperforms the conventional single color plane based denoising methods. Moreover, since the 2DQSSA based denoising method also exploits the local relationship within each color plane, the 2DQSSA based denoising method outperforms the non-overlapping based methods.     

Mikhailov stability criterion for fractional commensurate order systems with delays

In this paper, we present the extension of the Mikhailov stability criterion to linear fractional commensurate order systems with delays of the retarded type. The extension is obtained by generalizing the Mikhailov stability criterion of fractional commensurate order and integer order delay systems. The validity of the results is illustrated by means of several examples.     

Fractional order sliding mode control of a pneumatic position servo system

Gas flow has fractional order dynamics; therefore, it is reasonable to assume that the pneumatic systems with a proportional valve to regulate gas flow have fractional order dynamics as well. There is a hypothesis that the fractional order control has better control performance for this inherent fractional order system, although the model used for fractional controller design is integer order. To test this hypothesis, a fractional order sliding mode controller is proposed to control the pneumatic position servo system, which is based on the exponential reaching law. In this method, the fractional order derivative is introduced into the sliding mode surface. The stability of the controller is proven using Lyapunov theorem. Since the pressure sensor is not required, the control system configuration is simple and inexpensive. The experimental results presented indicate the proposed method has better control performance than the fractional order proportional integral derivative (FPID) controller and some conventional integral order control methods. Points to be noticed here are that the fractional order sliding mode control is superior to the integral order sliding mode counterpart, and the FPID is superior to the corresponding integral order PID, both with optimal parameters. Among all the methods compared, the proposed method achieves the highest tracking accuracy. Moreover, the proposed controller has less chattering in the manipulated variable, the energy consumption of the controller is therefore substantially reduced.     

A test for robust Hurwitz stability of convex combinations of complex polynomials

In this paper we present a method for testing the Hurwitz property of a segment of polynomials (1−λ)p₀(s)+λp₁(s), where λ∈[0,1] and p₀(s) and p₁(s) are nth-degree polynomials with complex coefficients. The method consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real λ-polynomials of degrees 2 and 2n in the interval (0,1). The presented method is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure.     

On the robust stability of commensurate fractional-order systems

Based on a recent generalised version of the Mikhailov stability criterion, this paper presents a Kharitonov–like test for a class of linear fractional–order systems described by transfer functions whose coefficients are subject to interval uncertainties. To this purpose, first the transfer function is associated with an integer-order complex polynomial function of the generalised frequency (i.e. the current coordinate along the boundary radii of the instability sector) whose coefficients are uncertain. Then the geometrical form of the value set of this characteristic polynomial is determined from the direct examination of its monomial terms. To show how the test operates, it is finally applied to two fractional–order transfer functions whose coefficients belong to given intervals.     

A new approach based on the discriminant system of polynomial for robust stability and stabilization of two-dimensional systems

In this paper, we present necessary and sufficient stability and robust stability conditions for two-dimensional (2D) systems described by the Fornasini-Marchesini (FM) second model in terms of the discriminant systems of polynomial. This paper simplifies the traditional method of stability into a tractable method by the fractional linear transformation (FLT). More specifically, we reduce the stability analysis to a easy issue whether some polynomials are positive definite. Then we use the same idea to consider the stabilization and robust stabilization issues. Finally, the effectiveness of the proposed results is demonstrated by a practical example and two numerical examples.     

Stabilizing region of PDμ controller for fractional order system with general interval uncertainties and an interval delay

This paper concentrates on computing the stabilizing region of PD^μ controller for fractional order system with general interval uncertainties and an interval delay. The stabilizing region means the complete/approximate set of PD^μ controllers that stabilize the given closed-loop control system. General interval uncertainties refer to both coefficients and orders of the fractional system suffer from interval uncertainties. Interval delay indicates that the delay also vary in a specified interval.Firstly, a method is presented to calculate the stabilizing region for general interval fractional system with an interval time-constant delay. Based on a novel mapping function and the concept of critical controller parameters, the stabilizing region can be determined numerically. Secondly, the stabilizing region computation problem for general interval fractional system with an interval time-varyingdelay is considered. By applying a revised small-gain theorem, the stabilizing region can be calculated like the time-constant delay case. Thirdly, two alternative methods are proposed to improve the computational efficiency of stabilizing region calculation. Both methods can reduce the number of polynomials which are used to determine the stabilizing region. Examples are followed to illustrate the proposed results.     

Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state

This paper is concerned with the stabilization problem of singular fractional order systems with order α?∈?(0, 2). In addition to the sufficient and necessary condition for observer based control, a sufficient and necessary condition for output feedback control is proposed by adopting matrix variable decoupling technique. The developed results are more general and efficient than the existing works, especially for the output feedback case. Finally, two illustrative examples are given to verify the effectiveness and potential of the proposed approaches.     

Robust resilient controllers synthesis for uncertain fractional-order large-scale interconnected system

The problem of the decentralized stabilization for fractional order large-scale interconnected uncertain system with norm-bounded parametric uncertainties and controller gain perturbations is studied. It is solved under two circumstances: one is under the additive controller gain perturbations; the other is under the multiplicative ones. Sufficient conditions on the decentralized stabilization of fractional order large-scale interconnected system with a commensurate order 0<α<1$0<\alpha <1$ are established by applying a complex Lyapunov inequality method. The state feedback non-fragile controller designs for fractional order large-scale interconnected uncertain system under the two classes of gain perturbations are obtained in terms of solutions to LMIs. Numerical examples are used to illustrate the effectiveness of the proposed method.     

Directions of Root-Loci in the Stability-Equation Method

In this paper, the characteristics of the root-loci which are of importance in applying the stability-equation method are presented. Characteristic equations with real and complex coefficients are analyzed, and a method of finding the directions of root-loci near the singular points is proposed.     

A direct method based on the Chebyshev polynomials for a new class of nonlinear variable-order fractional 2D optimal control problems

The main goal of this study is to develop an efficient matrix approach for a new class of nonlinear 2D optimal control problems (OCPs) affected by variable-order fractional dynamical systems. The offered approach is established upon the shifted Chebyshev polynomials (SCPs) and their operational matrices. Through the way, a new operational matrix (OM) of variable-order fractional derivative is derived for the mentioned polynomials.The necessary optimality conditions are reduced to algebraic systems of equations by using the SCPs expansions of the state and control variables, and applying the method of constrained extrema. More precisely, the state and control variables are expanded in components of the SCPs with undetermined coefficients. Then these expansions are substituted in the cost functional and the 2D Gauss-Legendre quadrature rule is utilized to compute the double integral and consequently achieve a nonlinear algebraic equation.After that, the generated OM is employed to extract some algebraic equations from the approximated fractional dynamical system. Finally, the procedure of the constrained extremum is used by coupling the algebraic constraints yielded from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the cost functional by a set of unknown Lagrange multipliers. The method is established for three various types of boundary conditions.The precision of the proposed approach is examined through various types of test examples.Numerical simulations confirm the suggested approach is very accurate to provide satisfactory results.     

On orthogonal polynomials

A study is made of the orthogonal polynomials on certain curves in the complex plane. Necessary and sufficient conditions for a set of polynomials to be orthogonal on the curves are obtained in terms of symmetric matrices. The relations of the symmetric matrices to Toeplitz matrices and innerwise matrices are shown.     

Moment matching model reduction for negative imaginary systems

In this paper, moment matching model reduction problem for negative imaginary systems is considered. For a given negative imaginary system with poles at the origin, our goal is to find a reduced-order negative imaginary system such that a prescribed number of the moments and the poles at the origin are preserved. Firstly, the original negative imaginary system is split into an asymptotically stable subsystem, a lossless negative imaginary subsystem and an average subsystem. Then, moment matching model reduction is implemented on the asymptotically stable subsystem and the lossless negative imaginary subsystem. The resulting reduced-order system preserves the negative imaginary structure and the poles at the origin. Also, the proposed model reduction method is extended to the positive real systems. Numerical examples demonstrate the effectiveness of the proposed model reduction method.     

A combined time and frequency domain method for model reduction of discrete systems

A new combined time and frequency domain method for the model reduction of discrete systems in z-transfer function is presented. First, the z-transfer functions are transformed into the w-domain by the bilinear transformation, z = (1+w)/(1?w). Then, four model reduction methods—Routh approximation, Hurwitz polynomial approxima- tion, stability equation, and retaining dominant poles—are used respectively to reduce the order of the denominator polynomials in the w-domain. Least squares estimate is then used to find the optimal coefficients in the numerator polynomials of the reduced models so that the unit step response errors are reduced to a minimum. The advantages of the proposed method are that both frequency domain and time domain characteristics of the original systems can be preserved in the reduced models, and the reduced models are always stable provided the original models are stable.     

Sum and product separabilities of multivariable functions and applications

Necessary and sufficient conditions for reducibility of certain classes of multivariable polynomials are considered. The special case when a polynomial in that class can be factored into a finite product of single variable polynomials is investigated. A set of necessary and sufficient conditions for decomposing a multivariable p.r.f. into a sum of single variable p.r.f.'s is given. Examples to illustrate applications of sum and product separabilities of functions are given.