A general theory of decoupling a system of lossless coupled lines using linear transformations

This paper presents a general theory of decoupling a system of nonuniform lossless coupled lines with the aid of linear transformations. The lines have a common return and support only TEM waves. It is shown that the solution of the decoupling problem is equivalent to that of the simultaneous diagonalization of a set of real constant matrices through specially related pairs of congruent and similarity transformations. The solution of the equivalent problem yields the necessary and sufficient conditions for decoupling the system of coupled lines. A systematic procedure for decoupling is then presented. A special case is studied for which the decoupling problem becomes simpler. Finally, we have derived a set of equivalent but considerably simpler necessary and sufficient conditions for decoupling the frequently used system of a pair of lossless coupled lines.     

On the Input—Output Decoupling of 2-D Systems by State Feedback

The problem of designing a linear state feedback controller for single-input single-output decoupling of linear multivariable 2-D (two-dimensional) systems is discussed. A method is presented for the determination of the decoupling controller matrices which, when applied to the open-loop system, yield a closed-loop system whose transfer function matrix is diagonal and nonsingular. Necessary and sufficient conditions are established for the state feedback decoupling problem to have a solution. Two examples are included to illustrate the proposed decoupling method.     

二阶系统数值解耦方法的研究

数值代数领域通过保持Lancaster结构来研究二阶系统的解耦问题,但寻找解耦变换涉及到了非线性方程组求解问题,难以实现. 提出了一种二阶系统数值解耦的新方法. 根据系统解耦前后的同谱信息确定解耦后的系统,将寻找解耦变换的非线性问题转化为齐次Sylvester方程求解问题; 并利用矩阵的Kronecker积理论求解二阶系统的解耦变换. 数值试验证明了该方法的可行性,为二阶系统的数值解耦找到了更便易的实现途径.     

Optimal stabilization for differential systems with delays - Malkin’s approach

The paper considers a process controlled by a system of delayed differential equations. Under certain assumptions, a control function is determined such that the zero solution of the system is asymptotically stable and, for an arbitrary solution, the integral quality criterion with infinite upper limit exists and attains its minimum value in a given sense. To solve this problem, Malkin’s approach to ordinary differential systems is extended to delayed functional differential equations, and Lyapunov’s second method is applied. The results are illustrated by examples, and applied to some classes of delayed linear differential equations.     

线性方程组求解的新方法

利用向量组的线性组合来讨论线性方程组的相容性,给出一种新的解法,即将方程组所确定的矩阵进行初等行变换以后,可以直接写出齐次线性方程组的基础解系和非齐次线性方程组的通解.它比通常所用的消元法简单明了,使用方便,容易掌握.     

The graph-theoretic field model—I. Modelling and formulations

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure.This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.     

Decentralized optimal controller design for multimachine power systems using successive approximation approach

In this paper, we propose to develop algorithmically and implement a nonlinear decentralized optimal control for multimachine power systems, based on a successive approximation approach for designing the optimal controller with respect to quadratic performance index. The advantage of this approach is to transform the high order coupling nonlinear two-point boundary value (TPBV) problem into a sequence of linear decoupling TPBV problem, which uniformly converges to the optimal control for nonlinear interconnected large scale systems. We apply this approach to a 3-machine power system which generators are strongly nonlinear interconnected, and containing possible uncertainties on the parameters. We demonstrate clearly via advanced simulations that this approach brings better performances than other decentralized controller, improving effectively transient stability of these power systems in few iterative sequences for different cases of perturbations.     

一种新的模糊时间序列预测模型

利用模糊系数实变量的线性方程组建立了一种新的模糊时间序列预测模型。模型的求解被转化为一种二次规划问题,它可广泛地应用于不确定环境中的各种预测。     

Nonlinear thermo-inertial stability of thin circular FGM plates

In this research, linear and non-linear stability behaviour of a thin circular FGM plate subjected to the uniform temperature rise and the constant angular velocity loadings is analyzed. Properties of the FGM media are distributed across the thickness based on a power law form. Each property of the metal or ceramic constituents is considered to be the function of temperature based on the Touloukian model. General equilibrium equations for such conditions are obtained based on the classical plate theory. At first, the non-linear governing equations are established in a complete asymmetrical form. After that, two different analytical methods are presented to study the bifurcation behaviour. Existence of bifurcation phenomenon is examined. Pre-buckling analysis is performed for a plate with the immovable clamped edge. Stability equations are obtained based on the adjacent equilibrium criterion. The resulted equations are solved via the two distinct methodologies, i.e. the exact solution in terms of Coulomb wave functions and the power series method. A non-linear solution is also presented to detect the equilibrium path of the heated rotating FGM plate. It is found that the angular speed may stabilize the homogeneous circular plate which buckles during uniform heating. Furthermore, snapping may occur for FGM plates under the simultaneous action of heating and uniform rotation.     

Output regulation for a class of positive switched systems

In this paper, a complete procedure for the study of the output regulation problem is established for a class of positive switched systems utilizing a multiple linear copositive Lyapunov functions scheme. The feature of the developed approach is that each subsystem is not required to has a solution to the problem. Moreover, two types of controllers and switching laws are devised. The first one depends on the state together with the external input and the other depends only the error. The conditions ensuring the solvability of the problem for positive switched systems are presented in the form of linear matrix equations plus linear inequalities under some mild constraints. Two examples are finally given to show the performance of the proposed control strategy.     

Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations

A Chebyshev collocation method, an expansion method, has been proposed in order to solve the systems of higher-order linear integro-differential equations. This method transforms the IDE system and the given conditions into the matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the systems of differential and integral equations.     

Interflections in Peaked Ceilings

A solution to the interflection problem in rooms lit by peaked ceilings is investigated. The integral equation for interflections in the peaked ceiling is formulated. A method of solving the integral equation by employing linear approximations is developed. This shows that uniform incident light on the peaked ceiling will result in nonuniformity. An equation is derived which gives the degree of nonuniformity in the incident light required for a uniform ceiling. The actual incident distribution of light from fluorescent lamps is expressed simply in prolate spheroidal coordinates.     

Exact Model-Matching of 2-D Systems via State Feedback

This paper considers the problem of matching the transfer function matrix of a given two-dimensional (2-D) system to that of a desired 2-D model using state feedback. The approach followed refers to systems having square transfer function matrices and reduces the problem to that of solving a linear system of equations. Furthermore, necessary and sufficient conditions are established for exact matching. An example is included to illustrate the proposed method.     

Cooperative payload transportation with real-time formation control of multi-quadrotors in the presence of uncertainty

This paper considers the trajectory tracking for a cable-suspended rigid body carried by multiple quadrotors while maintaining the desired formation. A three-loop control structure is proposed based on the Super Twisting Sliding Mode (STSM) controller with high robustness to compensate for uncertainties and disturbances. In the proposed structure, a proper formation based on a centralized approach creates advantages such as collision avoidance and non-entanglement of the cables, which leads to uniform load distribution on all quadrotors to optimize energy consumption. The corresponding equations are formulated as a convex constrained optimization problem with an analytical solution for calculating the cables' tension. This approach is a straightforward and computationally efficient method for real-time applications. In addition to proving the system's stability by Lyapunov theory, the simulation results show good performance in the presence of disturbance and uncertainty.     

On the continuous time-varying JLQG

In this paper, the optimal control law for the continuous infinite time-varying stochastic control system with jumps and quadratic cost is found under the assumption that the coefficient have limits as time tends to infinity and the boundary system is absolutely observable and stabilizable. In addition, the asymptotic properties of the solution of the differential Riccati equations for continuous time Markovian jump linear quadratic control problem with time-varying coefficient are established.     

Orthogonal functions approach to optimal control of delay systems with reverse time terms

Using block-pulse functions (BPFs)/shifted Legendre polynomials (SLPs) a unified approach for computing optimal control law of linear time-varying time-delay systems with reverse time terms and quadratic performance index is discussed in this paper. The governing delay-differential equations of dynamical systems are converted into linear algebraic equations by using operational matrices of orthogonal functions (BPFs and SLPs). The problem of finding optimal control law is thus reduced to the problem of solving algebraic equations. One example is included to demonstrate the applicability of the proposed approach.     

Finding exact periodic solutions in Lure systems through periodic signal mapping

A solution of the periodic problem in a nonlinear system comprising a single-valued symmetric nonlinearity (Lure system) and linear dynamics is presented. The solution is designed as an iterative application of Periodic Signal Mapping to refine the approximate solution obtained through the describing function method. The algorithm is based upon the transformation of the original nonlinear system into an equivalent nonlinear system for which the filtering hypothesis is satisfied exactly, and for that reason the latter being suitable for application of the developed algorithm. The solution sought for is a fixed point of the periodic signal mapping. Conditions of asymptotic convergence of the proposed algorithm are given. The proposed approach is illustrated by two examples of analysis of periodic motions in a relay feedback system and chattering in a terminal sliding mode.     

An initial value method for solving Fredholm integral equation of the first kind

The difficulties in solving Fredholm integral equations of the first kind are well known. A classical method has been to convert the equation into a set of m linear algebraic equations in n unknowns (m?n). For computational convenience, it is customary to force m = n and solve the resulting ill-conditioned system using one technique or other. In the general case, a feasible solution, if it exists, can be found by determining the generalized inverse of the coefficient matrix. One method of finding the generalized inverse is to reformulate the problem and observe the steady state response of a system of ordinary differential equations with prescribed initial conditions. Results obtained from this reformulation are found to be comparable in quality to those obtained earlier by other methods. Analog and digital computer implementations are discussed.     

Decoupling and robust control of velocity-varying four-wheel steering vehicles with uncertainties via solving Attenuating Diagonal Decoupling problem

This paper is devoted to solve the combined problem of input–output decoupling and robust control of the four-wheel steering vehicles. A more practical three-degree-of-freedom systems covering longitudinal, lateral and yaw motions are used to improve the safety and steerability while uncertainties and external disturbances are considered. A novel decoupling conception Attenuating Diagonal Decoupling and a new index Coupling Attenuation Index are introduced and the system is divided up into two systems with a special structure. The first system is caused by uncertainties and disturbances and the second system is a certain system coupling with the first one. A control scheme composed of a coupling attention controller and a decoupling controller are explored. The influences of the uncertainties and disturbances on the output are attenuated under the coupling index by the coupling attention controller designed for the first system while the input–output decoupling is achieved by employing the decoupling controller designed for the second system. Furthermore, we prove in theory that the input–output decoupling and robust control are both established for the closed-loop system of the control scheme and the primordial vehicle system. Besides these works, a switching law is introduced such that the above excellent performances are realizable in four-wheel steering vehicles with conventional steering interfaces. Simulations show that even with a large velocity varying range, the decoupling and robust performances are guaranteed simultaneously, i.e. the handling stability and steerability are improved.     

Optimization of multi-criteria facility-based systems via vector potential approach

The optimization of multi-criteria facility-based systems(MFBSs) is investigated through vector-potential function approach. To begin with, an MFBS with multiple pre-assigned system performance criteria is transformed into a multi-criteria potential game with given system performance criteria as its vector-potential function by designing proper facility-cost functions(FCFs). A linear system is presented, which turns the problem of designing suitable FCFs into checking whether a solution of the linear system exists. Then, a sufficient and necessary condition is presented for the verification of Pareto equilibria according to set relations. Next, dynamic behaviors of an MFBS is analyzed. The existence and convergence of Pareto equilibria are proved, showing that desirable global optimization can be achieved when each agent optimizes his(her) vector-utility functions. Finally, the approach is extended to constrained MFBSs, in which facilities are partly designable or restricted.