Singularity structure analysis of fractal relaxation systems

The fractional slope frequency spectrum has been observed in many physical phenomena such as electrical noise, the relaxation of polarized dielectrics, viscous and magnetic materials as well as the interface between two dissimilar conducting materials. Such systems are recognized as fractal systems and several empirical mathematical models have been suggested to model their dynamical behaviour. The singularity structure model to characterize the steady-state dynamics of fractal systems in the linear range is proposed. The approach incorporates fractal concepts applied to classical system theory. The singularity structure model of fractal systems can be mathematically represented as a rational system function. Differing from the standard system function, the poles and zeros of fractal systems function are located very close to each other in the complex plane and are distributed in a special manner. The local singularity structure of fractal systems, i.e. the placement of poles and zeros, can be constructed by a recursive procedure following a simple rule which relates the global parameters to local ones, using the self-similarity property. The standard singularity structure is also introduced to generalize the singularity structure of fractal systems. A fractal circuit is constructed to test the model and the synthesized singularity structure for the impedance of the electrode-electrolyte interface is compared with experimental data.     

Dual switched positive systems – a less conservative condition for diagonal quadratic stability

ABSTRACT

The key result of the paper exploits the duality of arbitrary switching positive linear systems, in order to derive a sufficient condition for the existence and construction of diagonal quadratic copositive Lyapunov functions. This condition is less conservative than a result also relying on duality, recently reported in the literature; our condition operates with milder inequalities than those required by the existing result. We prove that the algebraic relaxation of these inequalities is related to a relaxation of the primal and dual systems’ properties, referring to the trajectory growth rates, or, equivalently, to the spectra of the associated column representatives. The existing result is incorporated into the new one as a particular case, and the advantages offered by the latter are illustrated by a relevant numerical example. Our developments focus on the continuous-time case, so as to permit direct comparisons with the existing result; some brief discussion suggests how the proposed approach can be applied, mutatis mutandis, to discrete-time dynamics.     

分形曲线生成的频域方法

目的 分形几何学的理论研究与应用实践方兴未艾,在分形的计算机生成领域,传统方法是在空间域中,通过对生成元的迭代操作而形成。为了扩展分形的生成方法,本文将频谱分析引入到分形几何中。方法 正交函数系是频谱分析的核心问题之一。考虑到分形曲线是一类连续而不光滑的折线型信号,通常的三角函数（Fourier变换）、连续小波变换仅适用于光滑的对象,否则会出现所谓“Gibbs现象”;另一方面,以V-系统为代表的正交分段多项式函数系适用于表达包含间断性的对象,否则会出现信息冗余。因此,通常的正交函数系均不适合分形的频谱表达与分析。针对分形曲线的特点,本文将其视为一次样条函数,通过引入一类正交样条函数系-Franklin函数系,实现了对分形曲线的有限项精确正交表达,得到Franklin频谱,从而完成分形的时频变换。然后,对Franklin频谱系数在不同尺度上进行修改。最后,通过正交重构得到新的分形。结果 对比实验验证了Franklin函数系在分形曲线频域表达方面的优越之处,它既能通过最小项数实现分形的正交表达,而且不会出现Gibbs现象。本文以von Koch曲线、Sierpinski square曲线和Hilbert曲线这3个经典分形为例,通过对Franklin谱在不同尺度上的自由调节,能够方便地生成大量形态各异的新的分形曲线。结论 Franklin谱不仅能够实现对分形曲线的有限精确重构,而且还能在不同尺度上刻画分形的形态特征。基于Franklin频谱调节实现的分形生成方法,只要修改频谱就可以得到大量的新型分形曲线,而且这些分形的样式千变万化,几乎不可预测,这种分形生成方式为分形设计带来了巨大的自由空间,为分形的生成提供了新的思路与方案。     

-like control for nonlinear stochastic systems

In this paper we develop a H_∞-type theory, from the dissipation point of view, for a large class of time-continuous stochastic nonlinear systems. In particular, we introduce the notion of stochastic dissipative systems analogously to the familiar notion of dissipation associated with deterministic systems and utilize it as a basis for the development of our theory. Having discussed certain properties of stochastic dissipative systems, we consider time-varying nonlinear systems for which we establish a connection between what is called the L₂-gain property and the solution to a certain Hamilton–Jacobi inequality (HJI), that may be viewed as a bounded real lemma for stochastic nonlinear systems. The time-invariant case with infinite horizon is also considered, where for this case we synthesize a worst case-based stabilizing controller. Stability in this case is taken to be in the mean-square sense. In the stationary case, the problem of robust state feedback control is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear matrix inequalities.     

Enforcing stability constraints in set-membership identification of linear dynamic systems

In this paper, we consider the identification of linear systems, a priori known to be stable, from input–output data corrupted by bounded noise. By taking explicitly into account a priori information on system stability, a formal definition of the feasible parameter set for a stable linear system is provided. On the basis of a detailed analysis of the geometrical structure of the feasible set, convex relaxation techniques are presented to solve nonconvex optimization problems arising in the computation of parameter uncertainty intervals. Properties of the computed relaxed bounds are discussed. A simulated example is presented to show the effectiveness of the proposed technique.     

Robust H ∞ filtering for uncertain discrete piecewise time-delay systems

This paper investigates the problem of robust H _∞ filter design for uncertain discrete piecewise time-delay systems based on a piecewise Lyapunov functional. The parametric uncertainties are assumed to be time-varying but norm bounded. The purpose is the design of a piecewise filter such that, for all admissible uncertainties, the resulting filtering error system is asymptotically stable and satisfies a prescribed H _∞ performance level. By introducing some different extra matrix variables, a sufficient condition for the solvability of this problem is obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, the explicit expression of a desired piecewise filter is given. Two numerical examples are provided to demonstrate the effectiveness of the proposed design method.     

A drainage system analysis evaluation of,and comparison between,Landsat-3 RBV,Landsat-5 TM and SPOT PA imageries covering the Central Macedonia district,Greece

Abstract

Landsat-3 RBV, Landsat-5 TM imageries and SPOT PA stereopair diapositives were visually interpreted for the purpose of finding the accuracy of certain morphometric variables of three drainage basin sample areas in Central Macedonia, North Greece, drawn separately from each of the above three types of satellite imageries and comparisons were made between the efficiency of drainage systems drawn from each of the above imageries and the drainage systems extracted from the available topographic maps of 1:50000 scale.

The main findings were the following: (1) SPOT PA stereopair diapositives of 1:200000 scale can be used to map drainage systems to an order of magnitude slightly more than TM imagery of 1:125000 scale, but significantly more than RBV imagery of 1:125000 scale. This slight superiority of SPOT imagery over TM imagery implies that the greater spectral range of TM, compared with the shorter range of SPOT imageries, vastly outweighs the advantage of SPOT'S superior resolution, but not the superiority of stereoscopic view; (2) TM imagery can be used to map drainage systems to an order of magnitude significantly more than RBV imagery; and (3) RBV imagery can be used to map drainage systems to an order of magnitude less than topographic maps of 1:50000 scale but better than topographic maps of 1:100000 scale.     

A hybrid algorithm for queueing systems

Abstract In this paper, we propose a hybrid algorithm based on [12] for solving linear systems of equations. The hybrid algorithm combines the evolutionary algorithm and the successive over-relaxation (SOR) method. The evolutionary algorithm allows the relaxation parameter w to be adaptive in the SOR method. We prove the convergence of the hybrid algorithm for strictly diagonal dominant linear systems. We then apply it to solve the steady-state probability distributions of Markovian queueing systems. Numerical examples are given to demonstrate the fast convergence rate of the method.     

Generic properties and control of linear structured systems: a survey

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A,B,C and D, is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems.     

The need for a new experimental environment for HCI research into multi-agent,real-time systems

Abstract

Much of the current research in HCI is carried out using experimental environments based on word processors, database search, or other conventional office automation. While this approach meets many needs it lacks the power required for investigating many unconventional situations. Complex multi-agent real-time systems are not typically found in offices and cannot easily be investigated in typical word processing or office automation contexts. The paper refers to four environments where multi-agency exists in a real-time environment: flight systems, plant control, telephone networks, and complex office systems. Consideration is given to the requirements of an alternative experimental environment which could allow HCI research to explore a wider range of issues.     

Output feedback regulation of a class of triangular structural nonlinear systems with unknown measurement sensitivity

ABSTRACT

This paper investigates the problem of global regulation via output feedback for a class of triangular structural nonlinear systems with unknown measurement sensitivity. Two kinds of triangular structure nonlinear systems, namely upper triangular systems and lower triangular systems, are considered here, and the key features of our considered systems are that there are uncertain linear growth condition in the nonlinear terms. Firstly, for a class of upper triangular nonlinear systems with unknown measurement sensitivity, an output feedback controller is designed such that global regulation of the system is achieved. Then, for a class of lower triangular nonlinear systems with unknown measurement sensitivity, global regulation is realised in a unifying framework. Finally, two simulation examples are respectively given to demonstrate the effectiveness of the theoretical results.     

Achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters

The notion of interactor matrix or equivalently the Hermite normal form, is a generalization of relative degree to multivariable systems, and is crucial in problems such as decoupling, inverse dynamics, and adaptive control. In order for a system to be input-output decoupled using static state feedback, the existence of a diagonal interactor matrix must first be established. For a multivariable linear system which does not have a diagonal interactor matrix, dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system. Such precompensation often depends on the parameters of system, and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty. In this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters. More precisely, we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix. These conditions are associated with the so-called (non)generic singularity of certain matrix related to the system structure but independent of the system parameters. The result of this paper is expected to be useful in robust and adaptive designs.     

Boundary control and estimation of reaction–diffusion equations on the sphere under revolution symmetry conditions

ABSTRACT

Recently, the problem of designing boundary controllers and observers for unstable linear constant-coefficient reaction–diffusion equation on N-balls has been solved by means of the backstepping method. However, the extension of these results to spatially varying coefficients is far from trivial. This work deals with radially varying reaction coefficients under revolution symmetry conditions on a sphere (the three-dimensional case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, a similar type of singularity appears in the backstepping control and observer kernel equations. However, with a simple scaling transformation, we are able to reduce the singular equation to a regular equation, which turns out to be the same kernel equations appearing when using the one-dimensional backstepping method. In addition, the scaling transformation allows us to prove stability in the H ¹ space.     

New finite-time stability conditions of linear switched singular systems with finite-time unstable subsystems

ABSTRACT

This paper addresses the finite-time stability problem of linear switched singular systems with finite-time unstable subsystems. Dynamic decomposition techniques are used to transform such systems into equivalent one that is a reduced-order switched normal systems. Based on the mode-dependent average dwell time (MDADT) switching signal, new sufficient conditions are presented to guarantee the linear switched singular systems with finite-time unstable subsystems being finite-time stability, finite-time bounded and finite-time stabilization. Finally, a numerical example is employed to verify the efficiency of the preceding method.     

Robust predictive extended state observer for a class of nonlinear systems with time-varying input delay

ABSTRACT

This paper deals with asymptotic stabilisation of a class of nonlinear input-delayed systems via dynamic output feedback in the presence of disturbances. The proposed strategy has the structure of an observer-based control law, in which the observer estimates and predicts both the plant state and the external disturbance. A nominal delay value is assumed to be known and stability conditions in terms of linear matrix inequalities are derived for fast-varying delay uncertainties. Asymptotic stability is achieved if the disturbance or the time delay is constant. The controller design problem is also addressed and a numerical example with an unstable system is provided to illustrate the usefulness of the proposed strategy.     

Fractal evaluations of fish school movements in simulations and real observations

Fish schools behave like a single organism, and this offers considerable survival advantages. In our simulations, a fish school is well organized, without a leader, and behaves like a single creature depending solely on the interactions among individuals. This kind of system can be said to be typical of “complex systems.” In this article, it is shown that fractal evaluation is useful to understand the features of fish school movements. We make clear the validity of fractal analyses to quantify fish school movements through evaluations of simulated fish school movements andsardine movements. These fractal analyses show that we need two different fractal dimensions (D ₁,D ₂) to understand the features of fish school movements:D ₁ corresponds to thesmaller coarsening levels, andD ₂ corresponds to thelarge coarsening levels. The linear analyses in log-log space give an excellent fit with both the simulated movements and the sardine school movements. In approaching complex systems or complex behaviors, fractal analyses have attracted wide attention in mathematics, physical sciences, and information science. The fractal evaluations here convince us that we are coming close to understanding the structure of complex movements of animals. This work was presented, in part, at the Sixth International Symposium on Artificial Life and Robotics, Tokyo, Japan, January 15–17, 2001     

Identification of switched linear systems via sparse optimization

The work presented in this paper is concerned with the identification of switched linear systems from input-output data. The main challenge with this problem is that the data are available only as a mixture of observations generated by a finite set of different interacting linear subsystems so that one does not know a priori which subsystem has generated which data. To overcome this difficulty, we present here a sparse optimization approach inspired by very recent developments from the community of compressed sensing. We formally pose the problem of identifying each submodel as a combinatorial ?₀ optimization problem. This is indeed an NP-hard problem which can interestingly, as shown by the recent literature, be relaxed into a (convex) ?₁-norm minimization problem. We present sufficient conditions for this relaxation to be exact. The whole identification procedure allows us to extract the parameter vectors (associated with the different subsystems) one after another without any prior clustering of the data according to their respective generating-submodels. Some simulation results are included to support the potentialities of the proposed method.     

Approximate causal output tracking for linear perturbed systems via sliding mode control

ABSTRACT

A new approach to the solution of what is termed causal output tracking problem for linear time-invariant systems in the presence of both matched and unmatched unmodelled disturbances is presented. The proposed solution is addressed through the design of a dynamic steady state estimator based on the desired system structure and an observer, considering the reference as the system output. The unmatched disturbance is estimated and used in the steady state estimator to achieve invariance, while the matched disturbance robustness is provided via sliding mode control using the super-twisting algorithm. A useful application of the presented techniques is output tracking for non-minimum phase systems; thus, the performed simulation results confirm the effectiveness of the proposed control scheme for this kind of systems.