On the fuel-optimal singular control of nonlinear second-order systems

Given a body subject to quadratic drag forces so that the positiony(t)and the applied control thrustu(t)are related byddot{y}(t)+adot{y}(t)|dot{y}(t)| = u(t), |u(t)| leq 1, the controlu(t)is found which forces the body to a desired position, and stops it there, and which minimizes the costJ=intliminf{0} limsup{T}{k + |u(t)|}dt. The response timeTis not fixed,k > 0, and|u(t)|is proportional to the rate of flow of fuel. Repeated use of the necessary conditions provided by the Maximum Principle results in the optimum feedback system. It is shown that ifkleq 1, then singular controls exist and they are optimal; ifk > 1, then singular controls are not optimal. Techniques for the construction of the various switch curves are given, and extensions of the results to other nonlinear systems are discussed.     

有中断时间代价的一致并行机抢先调度问题

提出了一种具有中断时间代价的抢先调度问题(P|ptmn(δ)|C_max):在抢先调度中,一个任务发生一次中断,其总的执行时间会增加一个δ.该问题在工程任务分配、分布式计算和网络通信等实际问题中有着广泛的应用背景.证明了这是一个NP-hard问题,给出了一个时间复杂度为O(nlogn+m)的脱线近似算法LPT-Wrap,其近似比小于等于1.40825,并分析了P|ptmn(δ)|C_max的在线特性,给出一个线性时间复杂度的在线近似算法,其竞争比为2.     

Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability

We consider nonlinear systems with input-to-output stable (IOS) unmodeled dynamics which are in the range of the input. Assuming the nominal system is globally asymptotically stabilizable and a nonlinear small-gain condition is satisfied, we propose a first control law such that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin. The design of this controller is based on a gain assignment result which allows us to prove our statement via a Small-Gain Theorem [JTP, Theorem 2.1]. However, this control law exhibits a high-gain feature for all values. Since this may be undesirable, in a second stage we propose another controller with different characteristics in this respect. This controller requires morea priori knowledge on the unmodeled dynamics, as it is dynamic and incorporates a signal bounding the unmodeled effects. However, this is only possible by restraining the IOS property into the exp-IOS property. Nevertheless, we show that, in the case of input-to-state stability (ISS)—the output is the state itself-ISS and exp—ISS are in fact equivalent properties.Yuan Wang was supported in part by NSF Grant DMS-9403924 and by a scholarship from Université Lyon I, France.     

Construction for a class of smooth wavelet tight frames

From the inequality |P(z)|2 + |P(-z)|2 ≤1, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows the design more freedom, and both the low-pass filters and high-pass filters have symmetries or anti-symmetries. We give the algorithm for filters with odd and even lengths separately, some concrete examples of wavelet tight frames with the length 4, 5, 6, 7, and at last we give the result of decomposing Lena image with them.     

Bounds for closed-loop transfer functions of multivariable systems

A multivariable feedback system y(s)=G(s)x(s), x(s) = u(s)- F(s)y(s) is treated where G(s) ≜ (gkl(s)) is the transfer function matrix of a plant and F(s) ≜ diag(f1(s),...,fn(s)) is that of a controller. A new bound for the transfer function hj(s) that relates yj(s) to uj(s) when fj(s)≡ 0 is given. The main result reads |hj(s)- gjj(s)| < aj(s) if |fk(s)^-1+ gkk(s)| > ak(s) for k = 1,... ,n; k≠j. Here, A ≜ diag(a1(s),...,an(s)) is a diagonal matrix which makes A-B a semi-M-matrix where B ≜ (bkl) is given by bkk=0, bkl= |gkl(s)| (k≠l). A similar result is also obtained for the inverse transfer function.     

Comments on the robust stability analysis of adaptive controllersusing normalizations

Some results on robustness to unmodeled dynamics of adaptive controllers using normalization factors are analyzed. It is shown that the proofs given by various authors fail to establish that the proposed adaptive controllers provide a nonvanishing robustness margin for the class of unmodeled dynamics considered     

含有一个6-圈的共轭圈问题

Yeo于1999年提出猜想:每一个至少有8个点的c-部(c≥4)正则竞赛图D包含一对不相交的长分别为t和｜V(D)｜-t的共轭圈,其中t∈｛3,4,…,｜V(D)｜-t｝。文章证明当c≥9时猜想对于t=6是成立的。     

On exact filters for continuous signals with discrete observations

Many filtering applications have continuous state dynamics X_t =∫₀^tm(X_s)ds+σW_t +ρ, discrete observations Y_j=Y(t_j), and nonadditive or non-Gaussian observation noise. One wants to calculate the conditional probability Pr{Xt∈dz|Y_j, 0⩽t_j ⩽t} economically. In this paper we show that a combination of convolution, scaling, and substitutions efficiently solves this problem under certain conditions. Our method is easy to use and assumes nothing about the observations other than the ability to construct p(Y_j )|X(t_j), the conditional density of the jth observation given the current state     

Identification in the presence of classes of unmodeled dynamics andnoise

Identification involves obtaining a model from an a priori chosen model class(es) using finite corrupted data. The corruption may be due to several reasons ranging from noise to unmodeled dynamics, since the real system may not belong to the model class. Two popular approaches-probabilistic and set-membership identification-deal with this problem by imposing temporal constraints on the noise sample paths. We differentiate between the two sources of error by imposing different types of constraints on the corruption. If the source of corruption is noise, we model it by imposing temporal constraints on the possible realizations of noise. On the other hand, if it results from unmodeled dynamics informational constraints are imposed. Contrary to probabilistic identification where the parameters of the identified model converge to the true parameters in the presence of noise, current results in set-membership identification do not have this convergence property. Our approach leads to bridging this gap between probabilistic and set-membership identification when the source of corruption is noise. For the case when both unmodeled dynamics and noise are present, we derive consistency results for the case when the unmodeled dynamics can be described either by a linear time-invariant system or by a static nonlinearity     

数据与未建模动态驱动的非线性PID切换控制

本文充分利用系统的数据信息和知识,把数据驱动控制、PID控制与一步超前最优控制策略相结合,提出了数据与未建模动态驱动的非线性PID切换控制方法.该方法首先利用被控对象往往运行在工作点附近的特点及系统丰富可测的数据信息,把被控对象表示成低阶控制器设计模型与高阶非线性项(未建模动态)和的形式.与以往方法的本质区别在于,所提的方法直接将未建模动态分解为前一拍数据与未知增量的和,并充分利用未建模动态可测数据信息补偿系统未知的非线性动态特性,设计非线性PID控制器,对未建模动态的未知增量采用自适应神经模糊推理系统(ANFIS)进行估计,从而设计带有未建模动态增量估计的非线性PID控制器.将控制器的跟踪误差引入切换指标,两个控制器通过切换机制协调控制系统,既保证系统的稳定,同时提高系统的性能.为解决PID控制器参数难以选择的问题,采用一步超前最优控制策略进行参数设计,从理论上给出了PID控制器参数选择的一般原则和方法,推导了保证闭环系统输入输出稳定性的条件;最后,通过数值仿真实验以及在水箱液位控制系统的物理对比实验,实验结果验证了所提算法的有效性和实用性.     

Robust continuous-time adaptive control by parameter projection

The problem of adaptive control of a continuous-time plant of arbitrary relative degree, in the presence of bounded disturbances as well as unmodeled dynamics, is addressed. The adaptation considered is the usual gradient update law with parameter projection, the latter being the only robustness enhancement modification employed. It is shown that if the unmodeled dynamics, which consists of multiplicative as well as additive system uncertainty, is small enough, then all the signals in the closed-loop system are bounded. This shows that extra modifications are not necessary for robustness with respect to bounded disturbances and small unmodeled dynamics. In the nominal case, where unmodeled dynamics and disturbances are absent, the asymptotic error in tracking a given reference signal is zero. Moreover, the performance of the adaptive controller is also robust     

A robust direct adaptive controller

This paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics. The algorithm is designed based on the reduced-order plant, which is assumed to be minimum phase and of known order and relative degree, but is analyzed with respect to the overall plant which, due to the unmodeled dynamics, may be nonminimum phase and of unknown order and relative degree. It is shown that if the unmodeled dynamics are sufficiently small in the low-frequency range, then the algorithm guarantees boundedness of all signals in the adaptive loop and "small" residual tracking errors for any bounded initial conditions. In the absence of unmodeled dynamics, the residual tracking error is shown to be zero.     

Decentralized robust adaptive control of nonlinear systems withunmodeled dynamics

The authors present a decentralized robust adaptive output feedback control scheme for a class of large-scale nonlinear systems of the output feedback canonical form with unmodeled dynamics. A modified dynamic signal is introduced for each subsystem to dominate the unmodeled dynamics and an adaptive nonlinear damping is used to counter the effects of the interconnections. It is shown that under certain assumptions, the proposed decentralized adaptive control scheme guarantees that all the signals in the closed-loop system are bounded in the presence of unmodeled dynamics, high-order interconnections and bounded disturbances. Furthermore, by choosing the design constants appropriately, the tracking error can be made arbitrarily small regardless of the interconnections, disturbances, and unmodeled dynamics in the system. An illustration example demonstrates the effectiveness of the proposed scheme     

随机系统最小二乘辨识的稳健性收敛性及适应控制

基于估计随机系统已建模部分未知参数的推广最小二乘算法,递推地定义了跟踪随机参考信号的适应控制器。我们证明了闭环系统稳定;当未建模动态特性在平均意义下有界时,估计误差随未建模 动态特性的衰减而减少且跟踪误差以一个微小的量偏离其最小值;当未建模动态特性在平均意义下趋于零时,可同时获得估计的强一致性及适应跟踪的渐近最优性。     

基于未建模动态补偿的非线性自适应切换控制方法

针对一类不确定的离散时间零动态不稳定的单输入-单输出(Single-input single-output, SISO)非线性系统,提出了一种基于未建模动态补偿的非线性控制器. 采用自适应神经模糊推理系统(Adaptive-network-based fuzzy inference system, ANFIS)和一一映射相结合的方法估计未建模动态.在此基础上,提出了由线性自 适应控制器、非线性自适应控制器以及切换机制组成的自适应切换控制方法.该方法通过对上述两种控制器的切换, 保证闭环系统输入输出信号有界的同时,改善系统性能.本文将要求未建模动态全局有界的条件放宽为线性增长, 建立了所提自适应控制方法的稳定性和收敛性分析.通过仿真比较和水箱的液位控制实验,验证了所提方法的有效性.     

Robust adaptive control: Conditions for global stability

An input-output approach is presented for analyzing the global stability and robustness properties of adaptive controllers to unmodeled dynamics. The concept of a tuned system is introduced, i.e., the control system that could be obtained if the plant were known. Comparing the adaptive system to the tuned system results in the development of a generic adaptive error system. Passivity theory is used to derive conditions which guarantee global stability of the error system associated with the adaptive controller, and ensure boundedness of the adaptive gains. Specific bounds are presented for certain significant signals in the control systems. Limitations of these global results are discussed, particularly the requirement that a certain operator be strictly positive real (SPR)-a condition that is unlikely to hold due to unmodeled dynamics.     

闭环随机系统精选—优化辅助变量辨识算法的稳健性

在闭环自适应控制系统设计下对含有有色噪声和未建模动态的随机系统,采用精选一优化辅助变量辨识方法和极点配置控制,可使参数估计收敛;参数估计误差随未建模动态的衰减而减少。当未建模动态在平均意义下趋于零时,参数一致估计。     

Determination of the roots and of the global extremum of a lipschitz function

Conclusion The obvious deficiency of the method (1.3), (1.9) is the possible difficulty of the operation . In connection with this one can note that all the above given statements remain valid if the number is replaced by some positive lower bound of |f(t _k ,x)| on .In computational methods, the presence of the Lipschitz constant is considered as a deficiency. In connection with this we can note that the Lipschitz constant L can be replaced by any of its upper estimates. For example, for a differentiable function f(z) one can take .Translated from Kibernetika, No. 2, pp. 71–74, March–April, 1987.     

Robustness of adaptive control of robots

In this paper we discuss the robustness of adaptive control of rigid robots and methods for improving robustness in the face of unmodeled dynamics and external disturbances. Robustness to unmodeled dynamics is achieved using a so-called composite control strategy based on a singular perturbation formulation of the manipulator dynamics together with -modificiation. Rigorous stability proofs are given using a composite Lyapunov function approach.Research partially supported by the National Science Foundation under Grants INT-8902476 and MSM-9100618 and by the University of Illinois Manufacturing Research Center.     

具有未建模动态系统的自适应动态面输出反馈控制

针对一类具有未建模动态和动态扰动且状态不可量测的非线性系统,利用神经网络逼近未知函数设计K-滤波器重构系统状态,提出一种自适应输出反馈控制策略。通过对未建模动态的新刻画,避免动态信号的引入。采用动态面设计方法,取消理论分析中产生的未知连续函数的估计,降低设计的复杂性。利用Lyapunov方法证明了闭环系统的所有信号是半全局一致终结有界的,并通过仿真结果验证了所提出方案的有效性。