On the optimal design of the output transformation for discrete-time linear systems

The optimization of the output matrix for a discrete-time, single-output, linear stochastic system is approached from two different points of view. Firstly, we investigate the problem of minimizing the steady-state filter error variance with respect to a time-invariant output matrix subject to a norm constraint. Secondly, we propose a filter algorithm in which the output matrix at timek is chosen so as to maximize the difference at timek+1 between the variance of the prediction error and that of the a posteriori error. For this filter, boundedness of the covariance and asymptotic stability are investigated. Several numerical experiments are reported: they give information about the limiting behavior of the sequence of output matrices generated by the algorithm and the corresponding error covariance. They also enable us to make a comparison with the results obtained by solving the former problem.This work was supported by the Italian Ministry of Education (MPI 40%), Rome, Italy.     

Fast algorithms for designing variable FIR notch filters

The paper presents fast algorithms for designing finite impulse response (FIR) notch filters. The aim is to design a digital FIR notch filter so that the magnitude of the filter has a deep notch at a specified frequency, and as the notch frequency changes, the filter coefficients should be able to track the notch fast in real time. The filter design problem is first converted into a convex optimization problem in the autocorrelation domain. The frequency response of the autocorrelation of the filter impulse response is compared with the desired filter response and the integral square error is minimized with respect to the unknown autocorrelation coefficients. Spectral factorization is used to calculate the coefficients of the filter. In the optimization process, the computational advantage is obtained by exploiting the structure of the Hessian matrix which consists of a Toeplitz plus a Hankel matrix. Two methods have been used for solving the Toeplitz‐plus‐Hankel system of equations. In the first method, the computational time is reduced by using Block–Levinson's recursion for solving the Toeplitz‐plus‐Hankel system of matrices. In the second method, the conjugate gradient method with different preconditioners is used to solve the system. Comparative studies demonstrate the computational advantages of the latter. Both these algorithms have been used to obtain the autocorrelation coefficients of notch filters with different orders. The original filter coefficients are found by spectral factorization and each of these filters have been tested for filtering synthetic as well as real‐life signals. Copyright © 2007 John Wiley & Sons, Ltd.     

Markov跳变系统滚动时域有限记忆控制

针对离散时间Markov跳变系统,提出滚动时域有限记忆控制的方法.在一段有限滤波时域上,利用系统输入与输出变量的线性组合构造一段有限控制时域上的输出反馈控制器.首先,不考虑跳变系统均方可镇定,基于最优控制的方法,获得以迭代计算形式给出的控制器,并使其在无偏条件下能优化二次型性能指标.其次,进一步考虑在成本衰减条件下确定终端加权矩阵,并以它作为边界条件计算得到最优控制律,调节系统均方稳定.为便于求解,成本衰减条件以线性矩阵不等式的形式给出.仿真实例验证了所提方法的可行性和有效性.     

Algorithm Research Based on the PDE Sensitivity Filter北大核心CSCD

采用PDE灵敏度滤波器可以消除连续体结构拓扑优化结果存在的棋盘格现象、数值不稳定等问题,且PDE灵敏度滤波器的实质是具有Neumann边界条件的Helmholtz偏微分方程.针对大规模PDE灵敏度滤波器的求解问题,有限元分析得到其代数方程,分别采用共轭梯度算法、多重网格算法和多重网格预处理共轭梯度算法对代数方程进行求解,并且研究精度、过滤半径以及网格数量对拓扑优化效率的影响.结果表明：与共轭梯度算法和多重网格算法相比,多重网格预处理共轭梯度算法迭代次数最少,运行时间最短,极大地提高了拓扑优化效率.     

Stability Analysis of Controlled, Partially-Connected, Distributed-Parameter Systems Through Perturbation Methods

A stability result for locally-controlled, interconnected, distributed-parameter systems (DPS) is developed. Using a special perturbation operator, exponential stability is shown to be a function of both the value of the perturbation operator and the characteristics of the interconnected DPS. Proof of the bound is shown using the expansion of the matrix operator and the solution vector through a set of gauge functions. Each expansion term is power matched and individually bound using stationary phase methods. Special consideration is given to interconnected systems of a structural dynamic nature.     

Suboptimal feedback control of linear periodic systems

A method is developed for the approximate design of an optimal state regulator for a linear periodically varying system with quadratic performance index. The periodic term is taken to be a perturbation to the system. By making use of a power-series expansion in a small parameter, associated with periodic terms, a set of matrix equations is derived for determining successively a feedback gain. Given periodic terms of a Fourier-series form, explicit solutions are obtained for those matrix equations. A sufficient condition for existence and periodicity of the solution is also shown. Further, the performance degradation resulting from a truncation of the power-series solution is investigated. The method may effectively be used in a computer-programmed computation.     

求解带箱式约束全局优化问题的滤子填充函数方法

提出一个基于滤子技术的填充函数算法, 用于求解带箱式约束的非凸全局优化问题. 填充函数算法是求解全局优化问题的有效方法之一, 而滤子技术以其良好的数值效果广泛应用于局部优化算法中. 为优化填充函数方法, 应用滤子来监控迭代过程. 首先给出一个新的填充函数并讨论了其特性, 在此基础上提出了理论算法及算法性质. 最后列出数值实验结果以说明算法的有效性.     

Writing representations over minimal fields

The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E:F|d³) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.     

Minimal polynomial systems for parametric matrices

In this article, we study the minimal polynomials of parametric matrices. Using the concept of (comprehensive) Gröbner systems for parametric ideals, we introduce the notion of a minimal polynomial system for a parametric matrix, i.e. we decompose the space of parameters into a finite set of cells and for each cell we give the corresponding minimal polynomial of the matrix. We also present an algorithm for computing a minimal polynomial system for a given parametric matrix.     

Linear programming approach to the control of discrete-time periodic systems with uncertain inputs

The problem is considered of finding a control strategy for a linear discrete-time periodic system with state and control bounds in the presence of unknown disturbances that are only known to belong to a given compact set. This kind of problem arises in practice in resource distribution systems where the demand has typically a periodic behavior, but cannot be estimated a priori without an uncertainty margin. An infinite-horizon keeping problem is formulated, which consists in confining the state within its constraint set using the allowable control, whatever the allowed disturbances may be. To face this problem, the concepts of periodically invariant set and sequence are introduced. They are used to formulate a solution strategy that solves the keeping problem. For the case of polyhedral state, control, and disturbance constraints, a computationally feasible procedure is proposed. In particular, it is shown that periodically invariant sequences may be computed off-line, and then they may be used to synthesize on-line a control strategy. Finally, an optimization criterion for the control law is discussed.     

Alternating convex projection methods for discrete-time covariance control design

The problem of designing a controller for a linear, discretetime system is formulated as a problem of designing an appropriate plant-state covariance matrix. Closed-loop stability and multiple-output performance constraints are expressed geometrically as requirements that the covariance matrix lies in the intersection of some specified closed, convex sets in the space of symmetric matrices. We solve a covariance feasibility problem to determine the existence and compute a covariance matrix to satisty assignability and output-norm performance constraints. In addition, we can treat a covariance optimization problem to construct an assignable covariance matrix which satisfies output performance constraints and is as close as possible to a given desired covariance. We can also treat inconsistent constraints, where we look for an assignable covariance which best approximates desired but unachievable output performance objectives; we call this the infeasible covariance optimization problem. All these problems are of a convex nature, and alternating convex projection methods are proposed to solve them, exploiting the geometric formulation of the problem. To this end, analytical expressions for the projections onto the covariance assignability and the output covariance inequality constraint sets are derived. Finally, the problem of designing low-order dynamic controllers using alternating projections is discussed, and a numerical technique using alternating projections is suggested for a solution, although convergence of the algorithm is not guaranteed in this case. A control design example for a fighter aircraft model illustrates the method.This research was completed while the first author was with the Space Systems Control Laboratory at Purdue University. Support from the Army Research Office Grant ARO-29029-EG is gratefully acknowledged.     

On the stability and spectral radius of a finite set of matrices

This paper studies some problems related to the stability and the spectral radius of a finite set of matrices. A seasonal epidemic model is given to illustrate the use of the obtained results. In this example, the relationship between the obtained results and the stability of a discrete time periodic linear system is obtained.     

Optimization of periodic systems

The problem of designing a regulator, optimal by a quadratic performance criterion, on an infinite time interval is examined for a linear periodic system. It is assumed that the control plant's motion is described by a system of linear periodic finite-difference equations. Controllable plants whose motion is described by differential and by finite-difference equations on different parts of the period are analyzed as well. The optimal regulator design problem is reduced to the determination of a periodic solution of an appropriate Riccati equation. An algorithm for constructing such a solution is derived. It is noted that this result can be used in periodic optimization problems /1/ and in the design of a stabilization system for a pacing apparatus.     

Stochastic design optimization of asynchronous flexible assembly systems

This paper presents the application of the stochastic quasigradient method (SQG) of Ermoliev and Gaivaronski to the performance optimization of asynchronous flexible assembly systems (AFAS). These systems are subject to blocking and starvation effects that make complete analytic performance modeling difficult. A hybrid algorithm is presented in this paper which uses a queueing network model to set the number of pallets in the system and then an SQG algorithm is used to set the buffer spacings to obtain optimal system throughput. Different forms of the SQG algorithm are examined and the specification of optimal buffer sizes and pallet numbers for a variety of assembly systems with ten stations are discussed. The combined Network-SQG method appears to perform well in obtaining a near optimal solution in this discrete optimization example, even though the SQG method was primarily designed for application to differentiable performance functionals. While a number of both theoretical and practical problems remain to be resolved, a heuristic version of the SQG method appears to be a reasonable technique for analyzing optimization problems for certain complex manufacturing systems.     

Switching control and time-delay identification

The unknown time delay makes the control design a difficult task. When the lower and upper bounds of an unknown time delay of dynamical systems are specified, one can design a supervisory control that switches among a set of controls designed for the sampled time delays in the given range so that the closed-loop system is stable and the control performance is maintained at a desirable level. In this paper, we propose to design a supervisory control to stabilize the system first. After the supervisory control converges, we start an algorithm to identify the unknown time delay, either on-line or off-line, with the known control being implemented. Examples are shown to demonstrate the stabilization and identification for linear time invariant and periodic systems with a single control time delay.     

Incremental learning optimization on knowledge discovery in dynamic business intelligent systems

As business information quickly varies with time, the extraction of knowledge from the related dynamically changing database is vital for business decision making. For an incremental learning optimization on knowledge discovery, a new incremental matrix describes the changes of the system. An optimization incremental algorithm induces interesting knowledge when the object set varies over time. Experimental results validate the feasibility of the incremental learning optimization.     

A new sequential systems of linear equations algorithm of feasible descent for inequality constrained optimization

Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations （SSLE） algorithm generating feasible iterates is proposed for solving nonlinear optimization problems with inequality constraints. In this paper, we introduce a new technique for constructing the system of linear equations, which recurs to a perturbation for the gradients of the constraint functions. At each iteration of the new algorithm, a feasible descent direction is obtained by solving only one system of linear equations without doing convex combination. To ensure the global convergence and avoid the Maratos effect, the algorithm needs to solve two additional reduced systems of linear equations with the same coefficient matrix after finite iterations. The proposed algorithm is proved to be globally and superlinearly convergent under some mild conditions. What distinguishes this algorithm from the previous feasible SSLE algorithms is that an improving direction is obtained easily and the computation cost of generating a new iterate is reduced. Finally, a preliminary implementation has been tested.     

A philosophy for the modelling of realistic nonlinear systems

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

     

Topology optimization of periodic Mindlin plates

Periodic structures exhibit unique dynamic characteristics that make them act as tunable mechanical filters for wave propagation. As a result, waves can propagate along the periodic structures only within specific frequency bands called the ‘pass bands’ and wave propagation is completely blocked within other frequency bands called the ‘stop bands’ or ‘band gaps’. The spectral width of these bands can be optimized using topology optimization. In this paper, topology optimization is used to maximize the fundamental natural frequency of Mindlin plates while enforcing periodicity. A finite element model for Mindlin plates is presented and used along with an optimization algorithm that accounts for the periodicity constraint in order to determine the optimal topologies of plates with various periodic configurations. The obtained results demonstrate the effectiveness of the proposed design optimization approach in generating periodic plates with optimal natural frequency and wide stop bands. The presented approach can be invaluable design tool for many structures in order to control the wave propagation in an attempt to stop/confine the propagation of undesirable disturbances.