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This study investigates both static and dynamic anti-windup problems for rational systems subject to actuator saturation. New approaches for the synthesis of anti-windup compensators are proposed based on the linear-fractional representation of rational non-linear systems. By using quadratic Lyapunov function and with a modified sector condition, convex optimisation algorithms based on linear-matrix inequalities are devised to maximise the region of attraction and the ℒ2 bound on disturbances and minimise the ℒ2-gain from disturbance to performance output. Numerical examples illustrate the effectiveness of the proposed approach.

In this paper, we investigate the behavior of the solutions of the following rational nonautonomous three dimensional system of difference equations of fourth-order defined by xn+1=pn+ynpn+yn-3,yn+1=qn+znqn+zn-3,zn+1=rn+xnrn+xn-3,n=0,1,2,…, where pn,qn,rn are 3-periodic sequences of positive numbers, and the initial values xi , yi , zi ∈[0,∞) , for i=-3,-2,-1,0 .

This work presents a modeling and controller tuning method for non-rational systems. First, a graphical tool is proposed where transfer functions are represented in a four-dimensional space. The magnitude is represented in decibels as the third dimension and a color code is applied to represent the phase in a fourth dimension. This tool, which is called Phase Magnitude (PM) diagram, allows the user to visually obtain the phase and the magnitude that have to be added to a system to meet some control design specifications. The application of the PM diagram to systems with non-rational transfer functions is discussed in this paper. A fractional order Proportional Integral Derivative (PID) controller is computed to control different non-rational systems. The tuning method, based on evolutionary computation concepts, relies on a cost function that defines the behavior in the frequency domain. The cost value is read in the PM diagram to estimate the optimum controller. To validate the contribution of this research, four different non-rational reference systems have been considered. The method proposed here contributes first to a simpler and graphical modeling of these complex systems, and second to provide an effective tool to face the unsolved control problem of these systems.

We study the -adic dynamical system generated by the (2-2) rational function , with , which has three different fixed points. The behavior of these points depends on some conditions over and , to be an attractor, indifferent or repellent point. We find these conditions, and also the basin of attraction and the Siegel disk, in each case.

This study proposes a computationally efficient algorithm that characterises the set of allowable real parametric uncertainties while ensuring the desired output specifications are satisfied for rational systems. Along with the scaled main loop theorem, the proposed approach reformulates this NP-hard problem by using the skewed structured singular value ν, whose upper and lower bounds can be efficiently computed by existing algorithms and software. A short discussion on the extension to a multi-agent system is also included. Two numerical examples of pharmaceutical crystallisation and nasal spray demonstrate the effectiveness of the proposed algorithm.

When a feedback system has components described by non-rational transfer functions, a standard practice in designing such a system is to replace the non-rational functions with rational approximants and then carry out the design with the approximants by means of a method that copes with rational systems. In order to ensure that the design carried out with the approximants still provides satisfactory results for the original system, a criterion of approximation should be explicitly taken into account in the design formulation. This paper derives such a criterion for multi-input multi-output (MIMO) feedback systems whose design objective is to ensure that the absolute values of every error and every controller output components always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. The obtained criterion generalizes a known result which was derived for single-input single-output (SISO) systems; furthermore, for a given rational approximant matrix, it is expressed as a set of inequalities that can be solved in practice. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities. The numerical results clearly demonstrate that the usefulness of the criterion in obtaining a design solution for the original system.

In this paper we solve the problem of the existence of rational realizations of response maps. Sufficient and necessary conditions for a response map to be realizable by a rational system are presented. We provide also the characterization of the existence of rationally observable and canonical rational realizations for a given response map. [PUBLICATION ABSTRACT]

We consider a family of (2, 2)-rational functions given on the set of complex p -adic field C p . Each such function f has two distinct fixed points x 1 = x 1 ( f ) , x 2 = x 2 ( f ) . We study p -adic dynamical systems generated by the (2, 2)-rational functions. We prove that x 1 is always an indifferent fixed point for f , i.e., x 1 is a center of some Siegel disk S I ( x 1 ) . Depending on the parameters of the function f , the type of the fixed point x 2 may be any possibility: indifferent, attractive, repelling. We find the Siegel disk or the basin of attraction of the fixed point x 2 , when x 2 is indifferent or attractive, respectively. When x 2 is repelling we find an open ball any point of which is repelled from x 2 . Moreover, we study relations between the sets S I ( x 1 ) and S I ( x 2 ) when x 2 is indifferent. For each (2, 2)-rational function on C p there are two points x ^ 1 = x ^ 1 ( f ) , x ^ 2 = x ^ 2 ( f ) ∈ C p which are zeros of its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point x 1 ) containing some points such that the trajectories (under actions of f ) of the points after a finite step come to x ^ 1 or x ^ 2 . We study periodic orbits of the dynamical system and find an invariant set, which contains all periodic orbits. Moreover, we study ergodicity properties of the dynamical system on each invariant sphere. Under some conditions we show that the system is ergodic if and only if p = 2 .

α- and β-carotenes belong to the most essential carotenoids in the human body and display remarkable pharmacological value for health due to their beneficial antioxidant activities. Distinct high α-/β-carotene stoichiometries have gained increasing attention for their effective preventions of Alzheimer’s disease, cardiovascular disease, and cancer. However, it is extremely difficult to obtain α-carotene in nature, impeding the accumulations of high α-/β-carotene stoichiometries and excavation of their antioxidant activities. Herein, we developed a dynamically operable strategy based on lycopene cyclases (LCYB and LCYE) for concurrently enriching α- and β-carotenes along with high stoichiometries in E. coli. Membrane-targeted and promoter-centered approaches were firstly implemented to spatially enhance catalytic efficiency and temporally boost expression of TeLCYE to address its low competitivity at the starting stage. Dynamically temperature-dependent regulation of TeLCYE and TeLCYB was then performed to finally achieve α-/β-carotene stoichiometries of 4.71 at 37 °C, 1.65 at 30 °C, and 1.06 at 25 °C, respectively. In the meantime, these α-/β-carotene ratios were confirmed to result in diverse antioxidative activities. According to our knowledge, this is the first time that both the widest range and antioxidant activities of high α/β-carotene stoichiometries were reported in any organism. Our work provides attractive potentials for obtaining natural products with competitivity and a new insight on the protective potentials of α-/β-carotenes with high ratios for health supply.

The so-called “pinched disk” model of the Mandelbrot set is due to A. Douady, J. H. Hubbard and W. P. Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, “pinches” the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the “pinched disk” model of the Mandelbrot set.