Fractional Interval Research Articles

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

SummaryThis paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust ‐stability. Firstly, general result is presented to check the robust ‐stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides ‐stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID ‐stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the ‐stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi‐linear interval uncertainty and mixed‐type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method. Copyright © 2015 John Wiley & Sons, Ltd.

Probabilistic robust stabilization of fractional order systems with interval uncertainty

This study investigates effects of fractional order perturbation on the robust stability of linear time invariant systems with interval uncertainty. For this propose, a probabilistic stability analysis method based on characteristic root region accommodation in the first Riemann sheet is developed for interval systems. Stability probability distribution is calculated with respect to value of fractional order. Thus, we can figure out the fractional order interval, which makes the system robust stable. Moreover, the dependence of robust stability on the fractional order perturbation is analyzed by calculating the order sensitivity of characteristic polynomials. This probabilistic approach is also used to develop a robust stabilization algorithm based on parametric perturbation strategy. We present numerical examples demonstrating utilization of stability probability distribution in robust stabilization problems of interval uncertain systems.

Cost‐effective sampling clock synchronisation scheme with its implementation in optical orthogonal frequency division multiplexing systems

A cost-effective scheme for sampling clock synchronisation based on the interpolation algorithm is proposed and implemented via field programmable gate array (FPGA). This scheme can provide a simpler and more economical achievement than the schemes that utilise expensive external clock recovery circuits. Low system complexity is another advantage of the proposed scheme due to the utilisation of the results of the frame synchronisation. Besides, a novel fractional sampling interval calculating algorithm which aims to calculate the fractional sampling time offset in a sampling interval is also proposed and this simpler operation algorithm based on least squares method is proved to fully satisfy the multi-subcarrier characteristic of orthogonal frequency division multiplexing (OFDM) symbol. The design and experimental results of a 2.5 Gb/s FPGA-based optical OFDM receiver by using the scheme that was proposed earlier are presented here. A major improvement in this scheme for sampling clock synchronisation is demonstrated to be suitable for practical application.

Robust stability region of fractional order PIλ controller for fractional order interval plant

In this article, by using the fractional order PIλ controller, we propose a simple and effective method to compute the robust stability region for the fractional order linear time-invariant plant with interval type uncertainties in both fractional orders and relevant coefficients. The presented method is based on decomposing the fractional order interval plant into several vertex plants using the lower and upper bounds of the fractional orders and relevant coefficients and then constructing the characteristic quasi-polynomial of each vertex plant, in which the value set of vertex characteristic quasi-polynomial in the complex plane is a polygon. The D-decomposition method is used to characterise the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters, which can obtain the stability region by varying λ orders in the range (0, 2). These regions of each vertex plant are computed by using three stability boundaries: real root boundary (RRB), complex root boundary (CRB) and infinite root boundary (IRB). The method gives the explicit formulae corresponding to these boundaries in terms of fractional order PIλ controller parameters. Thus, the robust stability region for fractional order interval plant can be obtained by intersecting stability region of each vertex plant. The robustness of stability region is tested by the value set approach and zero exclusion principle. Our presented technique does not require sweeping over the parameters and also does not need linear programming to solve a set of inequalities. It also offers several advantages over existing results obtained in this direction. The method in this article is useful for analysing and designing the fractional order PIλ controller for the fractional order interval plant. An example is given to illustrate this method.

Multi-controller approach to uncertain discrete-time-delay systems

In this paper, a switched robust control is designed for a class of discrete-time systems subject to linear fractional uncertainty and interval time-delays. The controller is based on state feedback and the LMIs-based stability conditions are derived using an improved Lyapunov-Krasovskii functional. Moreover, the switching rule as well as the state feedback gains are determined from the minimisation of a guaranteed cost function. The theoretical results are illustrated with numerical examples.

Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0

In this paper we consider a class of fractional order linear time invariant (FO-LTI) interval systems with linear coupling relationships among the fractional order, the system matrix and the input matrix. We present the sufficient conditions for the robust stability and stabilization of such coupling FO-LTI interval systems with the fractional order α satisfying 0<α<1. All the results are proposed in terms of linear matrix inequalities (LMI). Two numerical examples show that our results are effective for checking the robust asymptotical stability and designing the stabilizing controller for FO-LTI interval systems.

Algorithm of robust stability region for interval plant with time delay using fractional order PIλDμ controller

By using PI λ D μ controller, we investigate the problem of computing the robust stability region for interval plant with time delay. The fractional order interval quasi-polynomial is decomposed into several vertex characteristic quasi-polynomials by the lower and upper bounds, in which the value set of the characteristic quasi-polynomial for vertex quasi-polynomials in the complex plane is a polygon. The D-decomposition technique is used to characterize the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters. We investigate how the fractional integrator order λ and the derivative order μ in the range (0, 2) affect the stabilizability of each vertex characteristic quasi-polynomial. The stability region of interval characteristic quasi-polynomial is determined by intersecting the stability region of each quasi-polynomial. The parameters of PI λ D μ controller are obtained by selecting the control parameters from the stability region. Using the value set together with zero exclusion principle, the robust stability is tested and the algorithm of robust stability region is also proposed. The algorithm proposed here is useful in analyzing and designing the robust PI λ D μ controller for interval plant. An example is given to show how the presented algorithm can be used to compute all the parameters of a PI λ D μ controller which stabilize a interval plant family.

Classical controller design techniques for fractional order case

This paper presents some classical controller design techniques for the fractional order case. New robust lag, lag-lead, PI controller design methods for control systems with a fractional order interval transfer function (FOITF) are proposed using classical design methods with the Bode envelopes of the FOITF. These controllers satisfy the robust performance specifications of the fractional order interval plant. In order to design a classical PID controller, an optimization technique based on fractional order reference model is used. PID controller parameters are obtained using the least squares optimization method. Different PID controller parameters that satisfy stability have been obtained for the same plant.

Frequency response computation of fractional order interval transfer functions

The paper present extensions of some results developed in the parametric robust control to fractional order interval control systems (FOICS). Computation of the Bode and Nyquist envelopes of FOICS are studied. Using the geometric structure of the value set of fractional order interval polynomials (FOIP), a technique is proposed for computing the Bode and Nyquist envelopes of transfer functions whose numerator and denominator polynomials are fractional order polynomials with interval uncertainty structure. The results obtained are useful for the analysis and design of FOICS. Numerical examples are included to illustrate the benefit of the method presented.

Robust stability testing function and Kharitonov-like theorem for fractional order interval systems

This study deals with the subject of robust bounded-input bounded-output (BIBO)-stability of a family of fractional order interval systems. Employing the idea of ‘robust stability testing function’ and extending it to the case of intended systems, a simple graphical procedure for checking the robust BIBO-stability applicable to both commensurate and incommensurate orders is developed. Moreover, a Kharitonov-like theorem is provided that presents necessary and sufficient conditions for checking the mentioned stability of the fractional order interval systems with commensurate order α belonging to [1,2), but only sufficient conditions for commensurate order α in interval (0,1). Besides, lower and upper bounds applicable to both commensurate and incommensurate cases are provided which are useful for simulation purposes. Finally, three numerical examples are given to illustrate the results.

Author’s reply: A comment on the “Robust stability analysis of fractional order interval polynomials”, by Nusret Tan et al.

This author’s reply addresses the comment given in the note mentioned in the title. Theorem 3 given in Tan et al. (2009) [1] uses zero exclusion principle for the stability analysis of Fractional Order Interval Polynomial (FOIP). We show that the constant degree assumption is exist in the definition of zero exclusion principle. Although it has not been clearly stated in Tan et al. (2009) [1] that FOIP of Eq. (1) is a constant degree polynomial, this condition is implicit in the zero exclusion principle. Therefore, Theorem 3 is true under the constant degree assumption which is a requirement for zero exclusion principle.

Switched state feedback for uncertain continuous‐time systems with interval delays

AbstractThe objective of this paper is to design a switched robust control for a class of continuous‐time systems subject to linear fractional uncertainty and interval time delays. The controller is based on state feedback and the LMI‐based stability conditions are derived using an improved Lyapunov–Krasovskii functional. Moreover, the switching rule as well as the state feedback gains are determined from the minimization of a guaranteed cost function. The theoretical results are illustrated with a numerical example. Copyright © 2010 John Wiley &amp; Sons, Ltd.

A comment on the “Robust stability analysis of fractional order interval polynomials”, by Nusret Tan et al.

It is shown that Theorem 3 in the article mentioned in the title is not true and some modification is suggested to eliminate the mistake.

Robust stability analysis of fractional order interval polynomials

The paper deals with the robust stability analysis of a Fractional Order Interval Polynomial (FOIP) family. Some new results are presented for testing the Bounded Input Bounded Output (BIBO) stability of dynamical control systems whose characteristic polynomials are fractional order polynomials with interval uncertainty structure. It is shown that the Kharitonov theorem is not applicable for this type of polynomial. A procedure is given for computation of the value set of FOIP. Based on the value set, an algorithm is presented for testing the stability of FOIP. The results presented in the paper are useful for the analysis and design of Fractional Order Interval Control Systems (FOICS). Examples are given to show how the proposed method can be used to assess the effects of parametric variations on the stability in feedback loops with fractional order interval transfer functions.

Fractional weak discrepancy and interval orders

The fractional weak discrepancy w d F ( P ) of a poset P = ( V , ≺ ) was introduced in [A. Shuchat, R. Shull, A. Trenk, The fractional weak discrepancy of a partially ordered set, Discrete Applied Mathematics 155 (2007) 2227–2235] as the minimum nonnegative k for which there exists a function f : V → R satisfying (i) if a ≺ b then f ( a ) + 1 ≤ f ( b ) and (ii) if a ∥ b then | f ( a ) − f ( b ) | ≤ k . In this paper we generalize results in [A. Shuchat, R. Shull, A. Trenk, Range of the fractional weak discrepancy function, ORDER 23 (2006) 51–63; A. Shuchat, R. Shull, A. Trenk, Fractional weak discrepancy of posets and certain forbidden configurations, in: S.J. Brams, W.V. Gehrlein, F.S. Roberts (Eds.), The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishburn, Springer, New York, 2009, pp. 291–302] on the range of the w d F function for semiorders (interval orders with no induced 3 + 1 ) to interval orders with no n + 1 , where n ≥ 3 . In particular, we prove that the range for such posets P is the set of rationals that can be written as r / s , where 0 ≤ s − 1 ≤ r < ( n − 2 ) s . If w d F ( P ) = r / s and P has an optimal forcing cycle C with up ( C ) = r and side ( C ) = s , then r ≤ ( n − 2 ) ( s − 1 ) . Moreover when s ≥ 2 , for each r satisfying s − 1 ≤ r ≤ ( n − 2 ) ( s − 1 ) there is an interval order having such an optimal forcing cycle and containing no n + 1 .

Toward using the network as a switch: on the use of TDM in linear optical networks

A common problem in optical networking is that the large quantity of raw bandwidth available in such networks is often difficult to access. We show that time-division multiplexing (TDM) can be used to operate bus and ring architectures in a manner akin to a switch. Doing so substantially reduces the amount of hardware [particularly, add-drop multiplexers (ADMs)] needed to utilize fully the available bandwidth in a range of optical networks. We show that a significant fraction (and in some cases all) of the bandwidth available to the system can be utilized even if each node in the system has only a single ADM. Our approach is probabilistic in nature, using generalizations of the Birkhoff-von Neumann statistical multiplexing approaches that have been successful in switching theory. Our techniques rely on decompositions of fractional matchings (for architectures without erasures) and fractional interval graph colorings (for architectures with erasures) into integral matchings and colorings.

Frequency and character of alternative somatic recombination fates of paralogous genes during T-DNA integration

A synthetic RBCSB gene cluster was transformed into Arabidopsis in order to simultaneously evaluate the frequency and character of somatic illegitimate recombination, homologous recombination, and targeted gene replacement events associated with T-DNA-mediated transformation. The most frequent type of recombination event observed was illegitimate integration of the T-DNA without activation of the silent DeltaRBCS1B: LUC transgene. Sixteen luc(+) (firefly luciferase positive) T1 plants were isolated. Six of these were due to illegitimate recombination events resulting in a gene trapping effect. Nine resulted from homologous recombination between paralogous RBCSB sequences associated with T-DNA integration. The frequency of somatic homologous recombination associated with T-DNA integration was almost 200 times higher than previously reported rates of meiotic homologous recombination with the same genes. The distribution of (somatic homologous) recombination resolution sites generally fits a fractional interval length model. However, a small region adjacent to an indel showed a significant over-representation of resolution sites, suggesting that DNA mismatch recognition may also play an important role in the positioning of somatic resolution sites. The frequency of somatic resolution within exon-2 was significantly different from that previously observed during meiotic recombination.

Left ventricular dysfunction and impaired exercise tolerance in a chronic canine model

Twelve months of right ventricular tachypacing was used in a group of five male dogs to produce chronic stable left ventricular dysfunction (LVD). Dogs were decrementally tachypaced (200 bpm for 7 months, then 160 bpm for 3 months, then 120 bpm for 2 months) to achieve and maintain LVD. The six minute walk test (6MWT) is commonly used to evaluate functional impairment in human heart failure, and we hypothesized that the 6MWT could be used to evaluate LVD in our chronic canine model. Dogs were conditioned prior to data acquisition. The 6MWT was done in a standardized fashion in a 60 M hallway. Functional assessments included serial measurements of LV fractional shortening, QRS interval and 6MWT. A group of 6 normal dogs were used as controls. In the LVD dogs, we observed a significant reduction in LV fractional shortening (p<0.05). There was a prolongation of the QRS interval, which was significant after 7 months of tachypacing, and was further prolonged after 12 months of tachypacing (p<0.05). There was a significant inverse correlation between LV fractional shortening and the QRS interval (r=−0.81, p<0.0001). In the tachypaced dogs, there was a significant time-dependent reduction in the 6MWT (p<0.05), which correlated with LV fractional shortening (r=0.63, p<0025). In our canine LVD model, the 6MWT provides a functional assessment which correlates with the severity of LVD. This canine model of chronic LVD results in reduced left ventricular systolic function, delayed intra-ventricular conduction, and reduced exercise tolerance.

Fractional integration and interval prediction

The motivation for fractional integration in terms of low-frequency spectral behavior and long-lag auto-correlation behavior is well known. Using results on the rate of growth of variances of sums of integrated random variables, we provide additional and complementary time-domain motivation for fractional integration in terms of the long-horizon behavior of (1) the variance-time function, and (2) confidence intervals for predictions. The results are illustrated with an empirical application to real interest rate forecasting.

A note on integer solutions to linear fractional interval programming problems by a branch &amp; bound technique

AbstractThis paper provides a method for solving linear fractional interval programming problems in integers with the help of a branch and bound technique.