Two-end observability of elastic vibrations in distributed and lumped parameter systems

The boundary observation problems (initial state reconstruction) of vibrations in objects with distributed and lumped parameters are solved. The vibrations in an object with distributed parameters are described by boundary value problems with boundary conditions of various types. An object with lumped parameters is described by a second-order ordinary differential equation.     

Two-end controllability of elastic vibrations of systems with distributed and lumped parameters

Boundary control problems for the vibrations of a system with distributed and lumped parameters are solved. The vibrations of a distributed-parameter object are described by boundary value problems with boundary conditions of various types. A lumped-parameter object is described by a second-order ordinary differential equation.     

Boundary observability of elastic vibrations in a system of sequentially connected strings

The observability problem for the elastic vibrations of a system consisting of sequentially connected objects with distributed parameters is solved. An object with lumped parameters is connected to the system. The initial state of the system is recovered from observations on its boundary and at the connection point of the objects with distributed and lumped parameters.     

Controllability of vibrations of a net of coupled objects with distributed and lumped parameters

The problem of damping vibration of a net consisting of m distributed parameter objects is solved. A lumped parameter controlled object affects the distributed parameter objects through boundary conditions.     

分布参数与集中参数耦合系统的极点配置问题

设 H 是可分的 Hilbert 空间,A 是空间 H 中的线性算子,b∈H 是非零元.考察空间H 中的一阶发展方程描述的控制系统(dx)/(dt)=Ax+bu(t),x(0)=x_0,(1)这里 u(t) 是控制量,是一数值函数.考察反馈控制律u(t)=〈x(t),g〉,(2)这里 g∈H 是非零元,〈·,·〉是 H 上的内积.     

Reduction of an elastic system to a prescribed state by use of a boundary force

A solution is constructed for the problem of optimal control of the motion of a distributed elastic system, using a lumped boundary force. The system's state is described by a constant-coefficient hyperbolic equation. A general case of arbitrary initial and final distributions is examined. Questions of control by lumped and distributed forces are discussed.     

State observability of elastic string vibrations under the boundary conditions of the first kind

We solve state observation problems for string vibrations, i.e., problems in which the initial conditions generating the observed string vibrations should be reconstructed from a given string state at two distinct time instants. The observed vibrations are described by the boundary value problem for the wave equation with homogeneous boundary conditions of the first kind. The observation problem is considered for classical and L ₂-generalized solutions of this boundary value problem.     

A computational method for solving optimal control of a system of parallel beams using Legendre wavelets

The optimal control of transverse vibration of two Euler–Bernoulli beams coupled in parallel by discrete springs is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. By use of the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of a linear time-invariant lumped-parameter systems. A computationally attractive method based on Legendre wavelets in time domain for solving the optimal control of the lumped parameter systems for any finite interval is proposed. Legendre wavelet integral operational matrix and the properties of a Kronecker product are used to find the approximated optimal trajectory and optimal law of the linear systems with respect to a quadratic cost function by only solving a linear system of algebraic equations. This method provides a straightforward and convenient approach for digital computation. A numerical example is provided to demonstrate the applicability and effectiveness of the proposed method.     

Coupled Galerkin and parameterization methods for optimal control of discretely connected parallel beams

Galerkin and wavelet methods for optimal boundary control of a couple of discretely connected parallel beams are proposed. First, the problem with boundary controls is converted into a problem with distributed controls. The problem is, then, reduced by a Galerkin-based approach into determining the optimal control of a linear time-invariant lumped parameter system, which will be solved by a wavelet-based method using Legendre wavelets. The integration-operational matrix and Kronecker product are utilized to significantly simplify the optimization problem into a system of linear equations. A numerical example is presented to demonstrate the applicability and the efficiency of the proposed method.     

An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.     

一类非线性抛物型分布参数控制系统的研究

对一类有广泛应用的活动边界域上非线性抛物型分布参数系统进行建模。采用将系统转化为固定域系统或转化为集中参数系统的方法,从理论上分析了系统的基本特征,得出控制变量存在两个临界值,大者区分边界活动与否,小者区分边界活动后会自动停止与否。将平面、柱面和球面的一维系统表述成统一形式,通过计算机仿真研究了系统开环控制和闭环反馈控制的动态特性,数值结果与理论结果一致。计算机仿真表明系统是适定的、稳定的,而且是可测的和可控的。     

Observability of elastic vibrations of a beam

The observability problem for beam vibrations described by a fourth-order partial differential equation with various boundary conditions is considered. Dynamic observability problems are solved in terms of boundary conditions and observations of the beam state at certain fixed instants of time.     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

Control of distributed time delay systems with Dirichlet boundary conditions

Various optimization problems associated with the optimal controlof distributed parameter systems with time lags appearing inthe boundary conditions have been studied recently by Wang (1975),Knowles (1978), Wong (1987), Kowalewski (1987a,b, 1988a,b,c,1990a,b,c,d, 1991, 1993a,b,c,d, 1995) and Kowalewski & Duda(1992). In this paper optimal boundary control problems fordistributed systems described by linear partial differentialequations of parabolic and hyperbolic type in which constanttime delays appear in the state equations are considered. Sufficientconditions for the existence of a unique solutions of such equationswith the Dirichlet boundary conditions are proved. The performancefunctional has the quadratic form. The time horizon T is fixed.Finally, we impose some constraints on the control. Making useof Lions' scheme (Lions 1971) necessary and sufficient conditionsof optimality for the Dirichlet problem with the quadratic performancefunctional and constrained control are derived. The flow chartof the algorithm, which can be used in the numerical solvingof certain optimization problems for distributed parameter systems,is also presented.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

System dynamics and feedback control problem formulations for real time dynamic traffic routing

This paper formulates the Dynamic Traffic Routing (DTR) problem as a real-time feedback control problem. Three different forms of the formulation are presented:

• 1.(1) distributed parameter system form derived from the conservation law;
• 2.(2) space discretized continuous lumped parameter form;
• 3.(3) space and time discretized lumped parameter form.

These formulations can be considered as the starting points for development of feedback control laws for the different control problems stated in this paper. This paper presents the feedback control problems, and does not discuss in detail the methodology of solution techniques which could be used to solve these problems. However, for the sake of completeness a brief treatment of the three forms are included in this paper to show possible ways to design the controllers.     

Mixed Frank–Wolfe Penalty Method with Applications to Nonconvex Optimal Control Problems

We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.     

Parametric Vibrations of a Three-Layer Conical Piezoshell

Based on the Kirchhoff-Love hypotheses and adequate supplementary hypotheses for the distribution of electric field quantities, a model for parametric vibrations of composite shells of revolution made of a passive (without a piezoeffect) middle layer and two active (with a piezoeffect) surface layers under the action of harmonic mechanical and electric loads is developed. The dissipative material properties are taken into account by linear viscoelastic models. Since the vibrations on the boundary of the main domain of dynamic instability (MDDI) are harmonic, the investigation of this domain, in a first approximation, is reduced to generalized eigenvalue problems, which are solved by the finite-element method. The problem on parametric vibrations of a three-layer conical shell under harmonic mechanical loading is considered. The influence of the shell thickness, dissipation, and electric boundary conditions on the MDDI is investigated. Two limiting cases of electric boundary conditions are considered, where the electrodes are short-circuited or not. The curves presented show a considerable influence of the electric boundary conditions on the characteristics of the MDDI, namely on its width and position on the frequency axis and on the critical parameter of excitation.