Suboptimality and stability of linear distributed-parameter systems with finite-dimensional controllers

In order to implement feedback control for practical distributed-parameter systems (DPS), the resulting controllers must be finite-dimensional. The most natural approach to obtain such controllers is to make a finite-dimensional approximation, i.e., a reduced-order model, of the DPS and design the controller from this. In past work using perturbation theory, we have analyzed the stability of controllers synthesized this way, but used in the actual DPS; however, such techniques do not yield suboptimal performance results easily. In this paper, we present a modification of the above controller which allows us to more properly imbed the controller as part of the DPS. Using these modified controllers, we are able to show a bound on the suboptimality for an optimal quadratic DPS regulator implemented with a finite-dimensional control, as well as stability bounds. The suboptimality result may be regarded as the distributed-parameter version of the 1968 results of Bongiorno and Youla.This research was supported by the National Science Foundation under Grant No. ECS-80-16173 and by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0124. The author would like to thank the reviewer for many helpful suggestions.     

Optimal control of linear stochastic systems described by functional differential equations

The optimal control is determined for a class of systems by assuming the configuration of the feedback loop. The feedback loop consists of an unbiased estimator and controller. The gain matrices of the estimator and the controller are so determined that the mean-squared estimation error and the average value of a quadratic cost functional, respectively, are minimized. This is accomplished by the application of the matrix maximum principle to a distributed parameter system. The results indicate that the optimal estimation and the optimal control can be computed independently (separation principle).This work was supported in part by the Air Force Office of Scientific Research, Grant No. 69-1776.     

Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

Performance modeling of heterogeneous data networks

A heterogeneous data network consists of Local Area Networks (LANs) interconnected with either leased lines, packet-switched networks, Metropolitan Area Networks (MANs), or combinations thereof. Heterogeneous networks are characterized by bursty traffic, nested segmentation and reassembly of packets, window flow control and round-robin channel access. We develop a performance methodology for estimating user perceived delay and buffer overflow in heterogeneous data networks. The methodology is based on well-known two-moment approximation schemes. Modifications of these schemes are proposed in order to model the bandwidth allocation policies of the MANs and to capture the burstiness of the traffic. The methodology is applied to several LAN-MAN and LAN-MAN-WAN network examples.Supported partially through AT&T Grant No. 5-20491 and partially through NSF Grant No. NCR-8914447.     

The empirical performance of a polynomial algorithm for constrained nonlinear optimization

In [6], a polynomial algorithm based on successive piecewise linear approximation was described. The algorithm is polynomial for constrained nonlinear (convex or concave) optimization, when the constraint matrix has a polynomial size subdeterminant. We propose here a practical adaptation of that algorithm with the idea of successive piecewise linear approximation of the objective on refined grids, and the testing of the gap between lower and upper bounds. The implementation uses the primal affine interior point method at each approximation step. We develop special features to speed up each step and to evaluate the gap. Empirical study of problems of size up to 198 variables and 99 constraints indicates that the procedure is very efficient and all problems tested were terminated after 171 interior point iterations. The procedure used in the implementation is proved to converge when the objective is strongly convex.Supported in part by the Office of Naval Research under Grant No. N00014-88-K-0377 and Grant No. ONR N00014-91-J-1241.     

Delay-Dependent Nonfragile <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Observer-Based Control for Neutral Systems with Time Delays in the State and Control Input

The problem of the delay-dependent nonfragile H _∞ observer-based control for a class of neutral systems with time delays is investigated. The additive gain variations under consideration are contained in both the controller gain and the observer gain. Novel delay-dependent criteria are derived to guarantee the stability of the nonfragile H _∞ observer-based control system using the Lyapunov function approach combined with linear matrix inequalities (LMI). The controller and observer gains are given from the LMI feasible solutions. Based on the result of this paper, the constraint of matrix equality is not necessary for designing a nonfragile H _∞ observer-based control. The computer software Matlab can be applied to solve the proposed problems. Finally, a numerical example is given illustrating the design of the nonfragile H _∞ observer-based control. The research was supported by the National Science Council of Taiwan, ROC under Grant NSC 96-2221-E-507-003.     

Adding variables to quasi-Newton Hessian approximations

The paper examines methods for increasing the dimension of a quasi-Newton approximation to a Hessian matrix when the dimension of the problem is increased. A new method is proposed, and numerical results given to demonstrate the superiority of the new method to existing methods.The work of the first author was partially supported by the Air Force Office of Scientific Research under Grant No. AFOSR-86-0170.     

Approximation of nonlinear functional differential equation control systems

We develop a general approximation framework for use in optimal control problems governed by nonlinear functional differential equations. Our approach entails only the use of linear semigroup approximation results, while the nonlinearities are treated as perturbations of a linear system. Numerical results are presented for several simple nonlinear optimal control problem examples.This research was supported in part by the US Air Force under Contract No. AF-AFOSR-76-3092 and in part by the National Science Foundation under Grant No. NSF-GP-28931x3.     

Adaptive Modified Function Projective Lag Synchronization for Two Different Chaotic Systems with Stochastic Unknown Parameters

In this paper, we investigate the modified function projective lag synchronization for two different stochastic chaotic systems using adaptive control method. We design an adaptive controller to make the mean square of synchronization error convergence to an arbitrarily small bound around zero depending on the controller feedback gain according to the Lyapunov stability theory. One example is presented to demonstrate the effectiveness of the proposed controller.

     

Optimal quasi-static shape control for large aerospace antennae

In this paper, we propose an on-line control approach which will adjust the steady-state shape of a large antenna arbitrarily close to any achievable desired profile. The method makes use of distributed parameter system theory and allows refocusing using a limited number of control actuators and sensors.The controller gains are calculated by approximating the solution to an infinite-dimensional optimal quasi-static control problem. We prove a very general convergence result for such quasi-static controllers here and apply it to the antenna controller to show convergence using any Galerkin (finite-element) approximation method.This research was partially supported by the National Science Foundation, the National Aeronautics and Space Administration, and the Air Force Office of Scientific Research under Grants ECS-80-16173, NAG-1-171, and AFOSR-83-0124, respectively.     

Solution of a periodic optimal control problem by asymptotic series

In order to understand the numerical behavior of a certain class of periodic optimal control problems, a relatively simple problem is posed. The complexity of the extremal paths is uncovered by determining an analytic approximation to the solution by using the Lindstedt-Poincaré asymptotic series expansion. The key to obtaining this series is in the proper choice of the expansion parameter. The resulting expansion is essentially a harmonic series in which, for small values of the expansion parameter and a few terms of the series, excellent agreement with the numerical solution is obtained. A reasonable approximation of the solution is achieved for a relatively large value of the expansion parameter.This work was sponsored partially by the National Science Foundation, Grant No. ECS-84-13745.     

Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed methods. The work of S. Wu was partially supported by the National Science Council, Taiwan, ROC (Grant No. 19731001). S.-C. Fang’s research has been supported by the US Army Research Office (Grant No. W911NF-04-D-0003) and National Science Foundation (Grant No. DMI-0553310).     

Best Approximation and Perturbation Property in Hilbert Spaces

We study the perturbation property of best approximation to a set defined by an abstract nonlinear constraint system. We show that, at a normal point, the perturbation property of best approximation is equivalent to an equality expressed in terms of normal cones. This equality is related to the strong conical hull intersection property. Our results generalize many known results in the literature on perturbation property of best approximation established for a set defined by a finite system of linear/nonlinear inequalities. The connection to minimization problem is considered.The authors thank the referees for valuable suggestions.K.F. Ng - This author was partially supported by Grant A0324638 from the National Natural Science Foundation of China and Grants (2001) 01GY051-66 and SZD0406 from Sichuan Province. Y.R. He -This author was supported by a Direct Grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.     

自组织结构的变论域模糊控制

针对变论域模糊控制,提出一种新的自组织结构的变论域模糊控制方法。自组织结构算法可以调整变论域模糊系统结构以及动态获得模糊规则,进一步减小变论域模糊控制项的稳态逼近误差。通过进一步理论分析可知,自组织结构算法仅仅保证了系统瞬时的切换是平稳的,但不能保证系统的闭环稳定性。给出了所提出控制方法的适用条件。通过与固定模糊系统结构的变论域模糊控制比较,仿真结果表明,所提出控制方法不仅使得系统的稳态跟踪误差更平稳,而且使得输入控制信号更加平滑。     

Differential games,dynamic optimization,and generalized control theory

In this paper, we examine the problem of control in a generalized framework in which there are more than one criterion, more than one intelligent controller, each of which has access to different information. It is argued that optimal control, differential games, dynamic team theory, etc., can all be viewed as special cases of this generalized control theory. Useful concepts and potential difficulties are discussed.The work reported in this paper is a result of research support extended over a period of years by Joint Services Electronic Contract No. N00014-67-A-0298-0006, NASA Grant No. NGR-22-007-068, and AFOSR Grant No. 69-1768 administered through Harvard University. The author has benefited from discussions with numerous colleagues, in particular, K. C. Chu, A. Starr, W. Willman, H. Witsenhausen, and S. Mitter.     

A semiparametric Wald statistic for testing logistic regression models based on case-control data

We propose a semiparametric Wald statistic to test the validity of logistic regression models based on case-control data. The test statistic is constructed using a semiparametric ROC curve estimator and a nonparametric ROC curve estimator. The statistic has an asymptotic chisquared distribution and is an alternative to the Kolmogorov-Smirnov-type statistic proposed by Qin and Zhang in 1997, the chi-squared-type statistic proposed by Zhang in 1999 and the information matrix test statistic proposed by Zhang in 2001. The statistic is easy to compute in the sense that it requires none of the following methods: using a bootstrap method to find its critical values, partitioning the sample data or inverting a high-dimensional matrix. We present some results on simulation and on analysis of two real examples. Moreover, we discuss how to extend our statistic to a family of statistics and how to construct its Kolmogorov-Smirnov counterpart. This work was supported by the 11.5 Natural Scientific Plan (Grant No. 2006BAD09A04) and Nanjing University Start Fund (Grant No. 020822410110)     

Stochastic independent modal-space control of distributed-parameter systems

A method capable in theory of estimating and controlling all the modes of a distributed-parameter system is presented. A linear distributed estimator using a distributed Kalman function is defined. It is shown that a particular choice of the Kalman function, in conjunction with the independent modal-space control method, leads to an infinite set of independent second-order modal-space Kalman filters cascaded with spatial modal filters. Independent modal-space Kalman filters experience no computational difficulties, regardless of the order of the system, and closed-form solutions for the modal Kalman gain matrices can be obtained with relative ease, making real-time implementation feasible. It is also shown that the independent modal-space Kalman filters are theoretically free of observation spillover.This work was supported in part by NSF Grant No. PFR-80-20623.     

Approximation of infinite delay and Volterra type equations

Summary We consider a class of infinite delay equations of neutral type which includes Volterrra type integral und integrodifferential equations. Using abstract approximation results (cf. Trotter-Kato-type) for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by Laguerre (and Legendre) polynomials. Numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data.Supported in part by the Air Force Office of Scientific Research under Contracts AFORS-84-0398 and AFORS-85-0303 and the National Aeronautics and Space Administration under NASA Grant NAG-1-517 and by NSF under Grant UINT-8521208Supported in part by the Air Force Office of Scientific Research under Contract AFORS-84-0398 and in part by the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under project No. S3206     

A general approach to nonlinear multiple control problems with perturbation consideration

In this paper, we consider a general nonlinear optimal control problem involving multiple criteria. We show that the problem can be transformed into a standard optimal control problem, and hence, is solvable by conventional techniques. However, the optimal control so obtained is of open loop nature and is rather sensitive to perturbations. Based on the first-order approximation, neighboring extremal approach is used to obtain a local linear feedback correction control law, leading to a combined controller. Two numerical examples are solved using the proposed method to demonstrate the effectiveness of the combined control.