On stability in fuzzy linear programming problems

admin[非线性薛定谔方程数值解的MATLAB仿真]——利用分步快速傅里叶变换对光纤中光信号的传输方程进行数值求解1、非线性薛定谔方程非线性薛定谔方程(nonlinear Schrodinger equation ，NLSE)是奥地利物理学家薛定谔于1926 年提出的，应用在量子力学系统中。

由于量子力学主要研究粒子的动力学运动状态，所以不能运用牛顿力学公式来表示。

通常在量子力学中，研究系统的状态一般通过波函数(x ，t)来表示。

而对波函数的研究主要是求解非线性薛定谔方程。

本文主要研究光脉冲在光纤中传输状态下的演变。

一般情况下，光脉冲信号在光纤中传输时，同时受到光纤的色散和非线性效应的影响。

通过Maxwell 方程，考虑到光纤的色散和非线性效应，可以推导出光信号在光纤中的传输方程，即非线性薛定谔方程。

NLSE 是非线性偏微分方程，一般很难直接求出解析解，于是通过数值方法进行求解。

具体分为两大类：(1)分布有限差分法(split-step finite differencemethod ，SSFD)；(2)分步傅里叶变换法(split-step Fourier transform method ，SSFT)。

一般情况，在达到相同精度，由于分步傅里叶变换法采用运算速度快的快速傅里叶变换，所以相比较有限差分法运算速度快一到两个数量级。

于是本文介绍分步傅里叶变换法来对光纤中光信号的传输方程，即非线性薛定谔方程进行数值求解。

并通过MATLAB 软件对结果数值仿真。

非线性薛定谔方程的基本形式为:22||t xx iu u u u =+其中u 是未知的复值函数.目前，采用分步傅立叶算法(Split step Fourier Method)求解非线性薛定谔方程的数值解应用比较多。

分步傅立叶方法最早是在1937年开始应用的，这种方法己经被证明是相同精度下数值求解非线性薛定愕方程最快的方法，部分原因是它采用了快速傅立叶变换算法(Fast Fourier Transform Algorithm)。

巴特沃斯滤波器求阶数n
【最新版】
目录
一、巴特沃斯滤波器概述
二、巴特沃斯滤波器的阶数选择
三、巴特沃斯滤波器的设计方法
四、应用实例与结论
正文
一、巴特沃斯滤波器概述
巴特沃斯滤波器（Butterworth filter）是一种常用的数字滤波器，以英国数学家巴特沃斯（Butterworth）的名字命名。

其特点是通频带的频率响应曲线最平滑，能够有效地抑制噪声和杂波，广泛应用于信号处理、通信系统等领域。

二、巴特沃斯滤波器的阶数选择
在设计巴特沃斯滤波器时，一个重要的参数是滤波器的阶数 n。

阶数n 决定了滤波器的性能，如通带截止频率、阻带衰减等。

一般来说，阶数n 越大，滤波器的性能越理想，但同时计算复杂度和成本也会增加。

因此，需要在满足性能要求的前提下，选择合适的阶数 n。

三、巴特沃斯滤波器的设计方法
巴特沃斯滤波器的设计方法通常采用拉普拉斯变换或模拟滤波器原
型法。

拉普拉斯变换是一种数学工具，可以将数字滤波器设计问题转化为一个关于 s（复变量）的方程，然后通过求解该方程得到滤波器的传递函数。

而模拟滤波器原型法则是通过构建一个模拟滤波器，然后根据模拟滤波器的特性设计数字滤波器。

四、应用实例与结论
巴特沃斯滤波器在信号处理和通信系统中有广泛的应用。

例如，在音频处理中，可以使用巴特沃斯滤波器对音频信号进行降噪和音质改善；在通信系统中，可以使用巴特沃斯滤波器对信号进行预处理，以提高信号的可靠性和抗干扰性。

总之，巴特沃斯滤波器是一种优秀的数字滤波器，具有良好的性能和实用性。

模糊控制器设计外文资料翻译--离散模糊双线性系统的静态输出反馈控制中文2094字外文资料翻译Static Output Feedback Control for Discrete-time Fuzzy Bilinear System Abstract The paper addressed the problem of designing fuzzy static output feedback controller for T-S discrete-time fuzzy bilinear system (DFBS). Based on parallel distribute compensation method, some sufficient conditions are derived to guarantee the stability of the overall fuzzy system. The stabilization conditions are further formulated into linear matrix inequality (LMI) so that the desired controller can be easily obtained by using the Matlab LMI toolbox. In comparison with the existing results, the drawbacks such as coordinate transformation, same output matrices have been eliminated. Finally, a simulation example shows that the approach is effective.Keywords discrete-time fuzzy bilinear system (DFBS); static output feedback control; fuzzy control; linear matrix inequality (LMI)1 IntroductionIt is well known that T-S fuzzy model is an effective tool for control of nonlinear systems where the nonlinear model is approximated by a set of linear local models connected by IF-THEN rules. Based on T-S model, a great number of results have been obtained on concerning analysis and controllerdesign[1]-[11]. Most of the above results are designed based on either state feedback control or observer-based control[1]-[7].Very few results deal with fuzzy output feedback[8]-[11]. The scheme of static output feedback control is very important and must be used when the system states are not completely available for feedback. The static output feedback control for fuzzy systems with time-delay was addressed [9][10] and a robust H∞ controller via static output feedback was designed[11]. But the derived conditions are not solvable by the convex programming technique since they are bilinear matrix inequality problems. Moreover, it is noted that all of the aforementioned fuzzy systems were based on the T-S fuzzy model with linear rule consequence.Bilinear systems exist between nonlinear and linear systems, which provide much better approximation of the original nonlinear systems than the linear systems [12].The research of bilinear systems has been paid a lot of attention and a series of results have been obtained[12][13].Considering the advantages of bilinear systems and fuzzy control, the fuzzy bilinear system (FBS) based on the T-S fuzzy model with bilinear rule consequence was attracted the interest of researchers[14]-[16]. The paper [14] studied the robust stabilization for the FBS, then the result was extended to the FBS with time-delay[15]. The problem of robust stabilization for discrete-time FBS (DFBS) was considered[16]. But all the above results are obtained via state feedback controller.In this paper, a new approach for designing a fuzzy static output feedback controller for theDFBS is proposed. Some sufficient conditions for synthesis of fuzzy static output feedback controller are derived in terms of linear matrix inequality (LMI) and the controller can be obtained by solving a set of LMIs. In comparison with the existing literatures, the drawbacks such as coordinate transformation and same output matrices have been eliminated.Notation: In this paper, a real symmetric matrix 0P > denotes P being a positive definite matrix. In symmetric block matrices, an asterisk (*) is used to represent a symmetric term and {...}diag stands for a block-diagonal matrix. The notion ,,1si j l =∑means 111s s si j l ===∑∑∑. 2 Problem formulationsConsider a DFBS that is represented by T-S fuzzy bilinear model. The i th rule of the DFBS is represented by the following form11 ()...() (1)()()()()()()1,2,...,i i v vi i i i i R if t is M and and t is M then x t A x t B u t N x t u t y t C x t i sξξ+=++== (1)Where iR denotes the fuzzy inference rule, s is the number of fuzzy rules.,1,2...ji M j v=is fuzzy set and()j t ξis premise variable.()nx t R ∈Is the statevector,()u t R ∈is the control input and T 12()[(),(),..,()]q q y t y t y t y t R =∈ is the system output. The matrices ,,,i i i iA B N C are known matrices with appropriatedimensions. Since the static output feedback control is considered in this paper,we simply set v q =and11()(),...,()()v q t y t t y t ξξ==.By using singleton fuzzifier, product inference and center-averagedefuzzifier, the fuzzy model(1) Can be expressed by the following global model11(1)(())[()()()()]()(())()si i i i i si i i x t h y t A x t B u t N x t u t y t h y t C x t ==+=++=∑∑(2)Where 11(())(())/(()),(())(())qsi i i i ij i j h y t y t y t y t y t ωωωμ====∑∏.(())ij y t μisthe grade of Membership of ()j y t in jiM . We assumethat (())0iy t ω≥and 1(())0si i y t ω=>∑. Then we have the following conditions:1(())0,(())1si i i h y t h y t =≥=∑.Based on parallel distribute compensation,the fuzzy controller shares the same premise parts with (1); that is, the static output controller for fuzzy rule i is written as11T T ()... () ()()1i i v vi i i i R if y t is M and and y t is M then u t F y t y F F yρ=+(3)Hence, the overall fuzzy control law can be represented as111()sin cos ()1ss si i i i i i i i i i T T i i u t h h h F y t y F F y ρθρθ======+∑∑∑ (4)WhereT TTTsin [,],1,2,...,2211i i i i i i i i i s y F F yy F F yππθθθ==∈-=++.1qi F R ⨯∈is a vector to be determined and 0ρ>is a scalar to be assigned.By substituting (4) into (2), the closed-loop fuzzy systems can be represented as,,11(1)()()()si j l ijl i j l si i i x t h h h x t y t h C x t ==+=Λ=∑∑ (5)wherecos sin ijl i i j l j i jA B F C N ρθρθΛ=++.The objective of this paper is to design fuzzy controller (4) such that the DFBS (5) is asymptotically stable. 3 Main resultsNow we introduce the following Lemma which will be used in our main results.Lemma 1 Given any matrices ,M N and 0P >with appropriate dimensionssuch that 0ε>, the inequality T T T 1TM PN N PM M PM N PN εε-+≤+holds.Proof: Note that 11112222()()()()T TTTM PN N PM P M P N P N P M +=+Applying Lemma 1 in [1]: 1T T T TM N N M M M N N εε-+≤+, the inequalityT T T 1T M PN N PM M PM N PN εε-+≤+can be obtained. Thus the proof iscompleted.Theorem 1 For given scalar 0ρ>and 0,,1,2,...,ij i j sε>=, the DFBS (5) is asymptotically stable in the large, if there exist matrices 0Q >and ,1,2,...,i F i s=satisfying the inequality (6).T T T 111()0000 ,,1,2,...,i i i j l ijl Q QA QN B F C Q a Q b Q b Q i j l s---⎡⎤-⎢⎥*-⎢⎥Φ=<⎢⎥**-⎢⎥***-⎣⎦= (6)Where211,(1)ij ij a b ερε-=+=+.Proof: Consider the Lyapunov function candidate asT ()()()V t x t Px t =(7)where 1P Q -= is to be selected.Define the difference ()(1)()V t V t V t ∆=+-, and then along the solution of (5), we haveT TT ,,1,,1T T T T 12,,1,,1T T T ,,1()()()()()()()()()() ()()()ssi j l m n p ijl mnp i j l m n p ssi j l m n p ijl mnp ijl mnp i j l m n p s i j l ijl ijl i j l V t h h h h h h x t P x t x t Px t h h h h h h x t P P x t x t Px t h h h x t P x t x t P =====∆=ΛΛ-=ΛΛ+ΛΛ-≤ΛΛ-∑∑∑∑∑()x t (8)Applying Lemma 1 again, it follows thatT T T 21T T (cos sin )(cos sin ) (1)(1)[()()]ijl ijl i i j l j i j i i j l j i j ij i i ij i j l i j l i i P A B F C N P A B F C N A PA B F C P B F C N PN ρθρθρθρθερε-ΛΛ=++++≤++++ (9)Substituting (9) into (8) leads toT T T T ,,1()()[()()]()si j l i i i j l i j l i i i j l V t h h h x t aA PA b B F C P B F C bN PN P x t =∆≤++-∑ (10)Applying the Schur complement, (6) is equivalent toT 1T 1T 1()(Q)0i i i j l i j l i i Q aQA Q A Q b B F C Q Q B F C bQN Q N Q ----+++<(11)Pre- and post multiplying both side of (11) with P , respectively, we haveT T T ()()0i i i j l i j l i i P aA PA b B F C P B F C bN PN -+++< (12)Therefore, it is noted that ()0V t ∆<, then the DFBS (5) is asymptotically stable. Thus the proof is completed.The matrix inequality (6) leads to BMI optimization, a non-convexprogramming problem. In the following theorem, we will derive a sufficient condition such that the matrix inequality (6) can be transformed into an LMI problem.Theorem 2 For given scalar 0ρ>and 0,,1,2,...,ij i j sε>=, the DFBS (5) is asymptotically stable in the large, if there exist matrices 0Q >and ,1,2,...,i F i s=satisfying the inequality (13).T T T 1110()000000 ,,1,2,...,i i l ijl i j Q QA QN C Q a Q I b Q I b Q B F I i j l s ---⎡⎤-⎢⎥*-+⎢⎥⎢⎥ψ=<**-+⎢⎥***-⎢⎥⎢⎥****-⎣⎦= (13)Proof: It is trivial thatT T ()()00000()()l l ijl i j i j C Q C Q I I B F B F φ⎡⎤⎢⎥*⎢⎥=>⎢⎥**⎢⎥***⎢⎥⎣⎦(14)Then ifijl ijl φΦ+<, we can conclude that 0ijl Φ<.TT T T 11T 1TTT111()()000()000000()()()0000 00l l i i i j l ijl ijl i j i j l i i i j C Q C Q Q QA QN B F C Q I a Q I b Q B F B F b Q C Q Q QA QN a Q I b Q IB F b Q φ------⎡⎤⎡⎤-⎢⎥⎢⎥**-⎢⎥⎢⎥Φ+=+⎢⎥⎢⎥****-⎢⎥⎢⎥******-⎢⎥⎣⎦⎣⎦⎡⎡⎤-⎢⎢⎥*-+⎢⎢⎥=+⎢⎥**-+⎢⎥***-⎣⎦⎣T 00()l i j C Q B F ⎤⎥⎥⎡⎤⎣⎦⎢⎥⎢⎥⎢⎥⎦(15)By applying Schur complement, (13) is equivalent to 0ijl ijl φΦ+<. Then wegetijl Φ<. According to Theorem 1, the DFBS (5) is asymptotically stable. Thusthe proof is completed.4 Numerical examplesIn this section, an example is used for illustration. The considered DFBS is1: (1)()()()()()() 1,2i ii i i i R ify is M then x t A x t B u t N x t u t y t C x t i +=++==Where1212129.78-1010101,,,510510011A A N N B B ----⎡⎤⎡⎤⎡⎤⎡⎤======⎢⎥⎢⎥⎢⎥⎢⎥---⎣⎦⎣⎦⎣⎦⎣⎦ [][]1210,10.C C ==-The membership functions are defined as111()(1cos())/2,M y y μ=-2111()1()M M y y μμ=-.By letting111221220.72,0.2, 1.51,ρεεεε===== applying Theorem 2 andsolving the corresponding LMIs, we can obtain the following solutions:[][]12-1.5014,11.0790 2.2785,-3.0452.2.27850.5984F Q F =⎡⎤=⎢⎥=⎣⎦Simulation results with the initial conditions: []T1.4-1.6 respective, areshown in Fig.1 and Fig.2. One can find that all these state converge to the equilibrium state after 17 seconds.tx (t )x 1x 2tu (t )Fig.1. State responses of system Fig.2. Control trajectory of system 5 ConclusionsIn this paper, a new and simple approach for designing a fuzzy static output feedback controller for the discrete-time fuzzy bilinear system is presented. The result is formulated in terms of a set of LMI-based conditions. By the proposed approach, the local output matrices are not necessary to be the same. Thus, the constraints had been relaxed and applicability of the static output feedback is increased.References[1] Wang R J, Lin W W and Wang W J. Stabilizability of linear quadratic state feedback for uncertain fuzzy time-delay systems [J]. IEEE Trans. Syst., Man, and Cybe., 2004, 34(2):1288-1292.[2] Cao Y Y and Frank P M. Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach [J]. IEEE Trans. Fuzzy Syst., 2000, 18(2): 200-211.[3] Yoneyama J. Robust stability and stabilization for uncertain Takagi-Sugeno fuzzy time-delay systems [J]. Fuzzy Sets and Syst., 2007, 158(4): 115-134.[4] Shi X Y and Gao Z W. Stability analysis for fuzzy descriptor systems [J]. Systems Engineering and Electronics, 2005, 27(6):1087-1089. (In Chinese)[5] Jiang X F and Han Q L. On designing fuzzy controllers for a class of nonlinear networked control systems[J].IEEE Trans. Fuzzy Syst., 2008, 16(4): 1050-1060.[6] Lin C, Wang Q G, Lee T H, et al. Design of observer-based H∞ control for fuzzy time-delay systems[J]. IEEE Trans. Fuzzy Syst., 2008, 16(2): 534-543.[7] Kim S H and Park P G. Observer-based relaxed H∞ control for fuzzy systems using a multiple Lyapunov function[J]. IEEE Trans. Fuzzy Syst., 2009, 17(2): 476-484.[8] Zhang Y S, Xu S Y and Zhang B Y. Robust output feedback stabilization for uncertain discrete-time fuzzy markovian jump systems with time-varying delays[J]. IEEE Trans. Fuzzy Syst., 2009, 17(2): 411-420.[9] Chang Y C, Chen S S, Su S F, et al. Static output feedback stabilization for nonlinear interval time-delay systems via fuzzy control approach [J]. Fuzzy Sets and Syst., 2004, 148(3): 395-410.[10] Chen S S, Chang Y C, Su S F, et al. Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approach[J]. IEEE Trans. Fuzzy Syst., 2005, 13(2): 263-272.[11] Hua ng D and Nguang S K. Robust H∞ static output feedback control of fuzzy systems: a LMIs approach [J]. IEEE Trans. Syst., Man, and Cybe., 2006, 36: 216-222.[12] Mohler R R. Nonlinear systems: Vol.2 Application to Bilinear control [M]. Englewood Cliffs, NJ: Prentice-Hall, 1991[13] Dong M and Gao Z W. H∞ fault-tolerant control for singular bilinear systems related to output feedback[J]. Systems Engineering and Electronics, 2006, 28(12):1866-1869. (In Chinese)[14] Li T H S and Tsai S H. T-S fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems [J]. IEEE Trans. Fuzzy Syst., 2007, 3(15):494-505.[15] Tsai S H and Li T H S. Robust fuzzy control of a class of fuzzy bilinear systems with time-delay [J]. Chaos, Solitons and Fractals (2007), doi: 10.1016/j. chaos.2007.06.057.[16] Li T H S, Tsai S H, et al, Robust H∞ fuzzy control for a class of uncertain discrete fuzzy bilinear systems [J]. IEEE Trans. Syst., Man, and Cybe., 2008, 38(2) : 510-526.离散模糊双线性系统的静态输出反馈控制摘要：研究了一类离散模糊双线性系统(DFBS)的静态输出反馈控制问题。

鞅差中心极限定理鞅差中心极限定理（Yoneda’s Lemma）是范畴论中经常被引用的定理之一。

它是由日本数学家Yoneda实性（Y.A. Uyehara）于1954年提出的，并由日本数学家Nobuo Yoneda在1955年推广和证明。

该定理在数学研究中提供了一种方法，通过研究与对象之间的关联来研究对象的性质。

这篇文章将介绍鞅差中心极限定理的概念和应用。

先介绍一下范畴论的基本概念。

范畴（category）是由对象（objects）和态射（morphisms）组成的mathcal{C}=(Obj({mathcal{C}),Hom({mathcal{C})}。

每个态射都有一个源对象（source object）和一个目标对象（target object）。

对于任意两个对象A, B于范畴mathcal{C}，Hom({mathcal{C})(A,B）表示从A到B的所有可能的态射的集合。

态射之间可以进行复合（composition），并且满足结合律。

鞅差中心极限定理是关于范畴Hom-Set的性质的一个结果。

给定范畴mathcal{C}和一个对象X于mathcal{C}，鞅差中心极限定理的陈述如下：对于任意对象A于mathcal{C}，Hom({mathcal{C})(A,X)与Hom({mathcal{C})(A,X+n)之间存在一个自然变换。

其中n表示任意对象。

这个定理的关键在于“中心极限”（central limit）。

直观的来说，这个定理说明了给定一个对象A于范畴mathcal{C}和一个差n，在Hom-Set Hom({mathcal{C})(A,X)和Hom({mathcal{C})(A,X+n)之间存在一个特殊的变换。

它将每个态射映射到一个稍微偏移一点的态射，表示了A到X的“鞅差”或“波动”。

在实际应用中，鞅差中心极限定理可以帮助数学家在给定了一个对象A时，研究从A到X的态射的性质。

这些态射在形式上可能非常复杂，但鞅差中心极限定理提供了一种方法，将这些态射转化为一种更简单的形式。

robust solutions of uncertain linear programs文献讲解在不确定线性规划（Uncertain Linear Programming，ULP）中，我们不仅要考虑经典的优化问题，还要考虑模型参数的不确定性。

Robust优化是一种处理不确定性的有效方法，旨在为决策者提供在各种可能的不确定性情况下都能保持最优的解决方案。

Robust solutions of uncertain linear programs这一研究领域，旨在为不确定环境下的线性规划问题找到稳健的解决方案。

线性规划是一种经典的优化方法，广泛应用于各种实际问题，如生产计划、资源分配和投资决策等。

然而，当模型参数存在不确定性时，传统的线性规划方法可能无法给出可靠的解决方案。

Robust优化的基本思想是，找到一个解决方案，该方案在所有可能的模型参数下都能保持最优。

这需要对不确定性进行建模，并使用适当的优化技术来找到稳健的解决方案。

Robust线性规划是Robust优化的一个分支，它专门处理线性规划问题中的不确定性。

在Robust线性规划中，通常采用的方法包括鲁棒对偶、鲁棒中心和鲁棒外包等。

这些方法在处理不确定性时采用了不同的策略，例如鲁棒对偶方法将原始问题转化为一个鲁棒对偶问题，而鲁棒中心方法则试图找到一个中心解决方案，该方案在所有可能的模型参数下都能保持最优。

Robust线性规划的应用非常广泛，包括供应链管理、风险管理、金融和能源等领域。

通过使用Robust线性规划，决策者可以在不确定环境下做出更可靠的决策，从而提高组织的效率和竞争力。

综上所述，Robust solutions of uncertain linear programs是一个重要的研究领域，它为不确定环境下的线性规划问题提供了稳健的解决方案。

通过使用适当的优化技术和不确定性建模方法，我们可以找到在各种可能的不确定性情况下都能保持最优的解决方案，从而提高组织的效率和竞争力。

模糊赋范线性空间的紧性与完备性模糊赋范线性空间（FuzzyNormedLinearSpace，简称FNLS）一个常见的数学概念，它涉及到紧性和完备性这两个概念。

本文将从定义、性质及存在性等方面对模糊赋范空间做进一步阐述，同时探讨其紧性和完备性的性质及其在小的范数空间中是否还存在。

一、模糊赋范空间的定义模糊赋范空间是指具有特定模糊赋范结构的线性空间，它定义为：设E是一个完备的具有欧几里得范数的线性空间，其上存在一个对应于E上所有子集的模糊赋范：F(X)= [f1(X),f2(X),…，fn(X)]其中，f1(X)，f2(X)，…，fn(X)X上某种模糊赋范。

则称E为模糊赋范空间。

二、模糊赋范空间的性质1、紧性性质模糊赋范空间的紧性性质是指除零外的任何元素都具有非负的模糊赋范值，且满足有限性。

若x,y∈E，则有：f(x+y)≤f(x)+f(y)其中，f(x)，f(y)，f(x+y)示x，y，x+y上的模糊赋范值；2、完备性性质模糊赋范空间具有完备性性质，即它是一个完备的线性空间。

它满足Cauchy序列定理：若在一个完备的线性空间E上存在一个序列（xn），且对任何ε >0有lim（n→∞） f(xn)=0则称序列（xn）在E上模糊赋范收敛。

三、模糊赋范空间在小范数空间中的存在性模糊赋范空间既可以是一个大的完备空间，也可以是一个小的范数空间。

一个小的范数空间的模糊赋范空间是满足以下条件的：1、有边界：范数空间E是一个具有模糊赋范结构的边界空间，即其上存在一组模糊赋范；2、不改变结构：范数空间E上的模糊赋范不改变空间E的结构和性质；3、紧性：模糊赋范空间满足紧性性质；4、完备性：模糊赋范空间满足完备性性质。

因此，在满足上述性质的情况下，模糊赋范空间可以存在于小的范数空间中。

本文论述了模糊赋范空间的定义和性质，及其是否可以存在于小的范数空间中。

紧性性质和完备性性质是模糊赋范空间的关键，两者缺一不可。

紧性性质可以保证空间上的结构不变，完备性性质可以保证空间上的元素收敛。

LOGIC AND FUZZY SYSTEMLESSON 14:SYSTEMS OF FUZZY LINEAR EQUATIONSFuzzy Analogy of Linear SystemsAbstract: In this start up study fuzzy logic theory is incorporated into modelling of linearsystems whose parameters and variables subject to uncertainty.The major intention is to try to understand the complicated mathematical background and therefore the principles and formulations can be properly interpreted into transparent engineering implementation. Two examples are selected to illustrate the fuzzy analogy of linear systems to accommodate uncertainties due to imprecise measurements or lack of complete information.Key words: Fuzzy numbers, convex fuzzy sets, fuzzy analogy,and implementation algorithms.The main advantage of fuzzy models is their ability to describe expert knowledge in a descriptive, human like way, in the form of simple rules using linguistic variables. The theory of fuzzy sets (Zimmermann 2000) allows the existence of uncertainty to vagueness (or fuzziness) rather than due to randomness. When using fuzzy sets, accuracy is traded for complexity – fuzzy logic models do not need an accurate definition for many systems (in terms of the parameters). This results in a natural reduction in number of variables and states for an ad hoc system structure.Simultaneous linear equations play a major role in representing various systems in natural science, engineering, and social domain. Since in many applications at least some of the system’s parameters and measurements are represented byexpert experience in terms of fuzzy rather than crisp numbers, it is immensely important to develop mathematical models and numerical procedures that would appropriately deal with those general fuzzy terms.One of the major applications using fuzzy number arithmetics is treating those linear systems, which their parameters are entirely or partially represented by fuzzy numbers. A general model for solving a n fuzzy linear system whosecoefficients matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector, which use the embedding method and replace the original linear fuzzy systemcrisp linear system with a matrix Swhich may be singular if A be nonsingular.The following contents are organised with five major sections.In section 2 some fundamental concepts and definitions are selected from classical fuzzy logic theory for treating fuzzy linear systems. In Section 3 fuzzy analogy method is presented to lay a basis for translating linear equations into their fuzzycounterparts. In Section 4 fuzzy solutions are described. InSection 5 two examples are used to demonstrate the algorithms.In section 6 the conclusions are drawn.2. PreliminariesAn arbitrary fuzzy number can be described with an ordered pair of functions where1. is a bounded left continuous non-decreasing functionover [0,1].2. is a bounded left continuous non-increasing function over [0,1].3.A crisp number a is simply representedbyDefinition 1 : Consider a nxnlinear system of equations= + · · · + +(2.1) may be expressed in a vector form(2.2)Expression (2.2) can be used as a Fuzzy Linear Equation (FLE),while . n x n matrix, and, ( are fuzzy vector, so called system transfermatrix, input, and output respectively in classical system notations.Let P denote the product and g be a mapping, which is The Extension Principle (Dubois 1980) states that g can be extended to five tuples (A, B, C, D, E)which are all fuzzy subsets ofas follows:Where sup is taken over allDefinition 2: A fuzzy set A of X is called convex if thefollowing relation holds:for any . A is said to be normalised if there exists an x such that A (x ) = 1. Theany. A is said to be normalised if thereexists an x such that A (x ) = 1. The a-level of a fuzzy subset A denoted by, if defined byA fuzzy set A can be shown to be convex if and only if its a cuts are intervals for all . Fuzzy set A can be defined by its acuts,wheredenotes the characteristic function of the setLOGIC AND FUZZY SYSTEMDefinition 3 Fuzzy number A is called a number of the R L .type if its membership functionhas the following form:where L and R are continuous non-increasing functions, defined on ), strictly decreasing to zero in those subintervals of the interval ) , in which they are positive, and fulfilling the condition The parameters a and â are non-negative real numbers.3. Fuzzy AnalogyIf X is the set of real numbers, by a fuzzy number x N , this means a fuzzy subset of X where •, if and only if y = x.•is continuous.• is convex.•vanishes at infinity..denotes the degree of belief that the value of x is infact y. Consider a class of functional equations of the formwhere f is unknown, may be either addition or multiplication of real numbers. (3.1) may be put in a generalformwhereis a function relating the unknownquantities stands for andstands forLet denote the degree of belief that x equals to y . Using extension principle (Dubois 1980), (3.2) can be expressed in thefuzzy formwhere the sup is taken over all y 1and y2 for whichAccording to Nguyen (1978), (3.3) and (3.1) may be the implies as following For the alevels are equalis a closed and bounded interval. Hence,,where are the left endpointand the right endpoint respectively associated with the a level of the fuzzy number w . If F is non-decreasing in its argument and Fis continuous, then it givesMoreover solutions to (3.5) and (3.6), say functions H 1and H2, respectively, are continuous and no-decreasing. ThusSo, the solution of (3.3) is given bywhere H 1 and H 2 are the solutions of (3.5) and (3.6),respectively. According to (3.3), the solution of (3.8) may alsobe represented bywhere the sup is taken over allis thecharacteristic function over the interval4. Fuzzy Solutions A solution tto (2.1) one should recall that forarbitrary fuzzy numbers and realnumber k ,•x = yif and only if•Definition 4: A fuzzy number vector givenby is called a solution ofthe FSLE ifConsider the ith equation of the system (2.1)it hasFrom (2.4) two crisp linear systems for all i that there can be extended to. crisp linear system as follows:LOGIC AND FUZZY SYSTEMThus FLE (2.1) is extended to a crisp (2.5) where A= S 2 + S 1.(4.5) can be write as follows:where the matrix S is nonsingular if and only if the matrices A=S1+S2 and S1-S2 are both non-singular.Definition 5: Let denote the unique solution of SX = Y . The fuzzy number vectoris defined byis called the fuzzy solution of SX = Y . Ifare all triangular fuzzy numbers thenand U is called a strongfuzzy solution. Otherwise, U is a weak fuzzy solution.In the general, the structure of S implies thatand thatWhere B contains the positive entries of A , C the absolute values of the negative entries of A , and A=B-C If linear system are fuzzy variables, for the inputsdenotes the degree of,inputThesolution of linear system are givenIf linear system are fuzzy coefficients and fuzzy variables, whichis in system (2.1)are fuzzy numbers, denotes the degree ofcoefficients and variable respectively.where so the solution of systemsAX = Yis5. Examples and SimulationsTwo examples were selected to demonstrate the understanding of the fuzzy of analogy of linear systems.Example 1:Consider a simple linear system B AX Y + = withfuzzy output and input variablesFig.1 shows a fuzzy line, which denotes what a line looks like with a fuzzy input.Example 2 :Consider a two inputs & single output fuzzy system with fuzzy outputsY = AX + B(5.3)Where .The extended 4 x 4 . matrix isThe fuzzy solution isFig.2 shows that variables x 1and x 2 are determined by the output y which is the ácut, when y = 1, an accurate value for thesolution can be obtained.LOGIC AND FUZZY SYSTEM6. ConclusionsIn this paper a general model structure is presented for solving of linear equations (systems) with fuzzy variables andparameters. With this structure, a fuzzy system with a matrix Ais transformed into a crisp linear system S . The system is then solved with crisp variables and parameters and thesolution vector is either a strong fuzzy solution or a weak fuzzy solution. Solutions of linear fuzzy equations have been well addressed in mathematics. However it is still a long distance to arrive in engineering field for applications. This study has attempted to bridge the two domains. Additionally fuzzy analogy of nonlinear systems is a new area and will be studied as the expansion of linear systems.References1.Zimmermann, H.J., Fuzzy Set Theory-and Its Applications,3rd Edition, Kluwer Academic Publishers, Dordrecht, 2000.2.Wang, L.X., Adaptive Fuzzy Systems and Control, New Jersey 07632, 1999.3.Klir, G.J., Folger, T.A., Fuzzy Sets, Uncerainty andInformation, Prentice-Hall, Englewood, Cli.s, NJ, 1988.4.Dubois, D., and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.5.Deeba, E., On a fuzzy difference equation. IEEE Trans.Fuzzy Systems 3(3):469-472, 1995.6.Deeba, E., On a fuzzy logistic difference equation.Differential Equations Dynam. Systems 4(2):149-156, 19967.Deschrijver, G., On the relationship between someextensions of fuzzy set theory, Fuzzy Sets and Systems 133:227-235, 2003.8.Friedman, M., Fuzzy Linear Systems, Fuzzy Sets and Systems 96(2): 201-209, 1998.9.Soliman, S.A., Fuzzy linear parameter estimation algorithms:a new formulation, International Journal of Electrical Power & Energy Systems 24(5): 415-420, 200210.Nguyen, H.T., A note on the extension principle for fuzzy sets, J. Math. Anl. 64(2): 369-380, 1978Notes。

Back to TopInformation flows from left to right, from two inputs to a single output. The parallel nature of the rules is one of the moreimportant aspects of fuzzy logic systems. Instead of sharp switching between modes based on breakpoints, logic flows smoothly from regions where the system's behavior is dominated by either one rule or another.Fuzzy inference process comprises of five parts: fuzzification of the input variables, application of the fuzzy operator (AND or OR) in the antecedent, implication from the antecedent to the consequent, aggregation of the consequents across the rules, and defuzzification. These sometimes cryptic and odd names have very specific meaning that are defined in the following steps.Step 1. Fuzzify InputsThe first step is to take the inputs and determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions. In Fuzzy Logic Toolbox software, the input is always a crisp numerical value limited to the universe of discourse of the input variable (in this case the interval between 0 and 10) and the output is a fuzzy degree of membership in the qualifying linguistic set (always the interval between 0 and 1). Fuzzification of the input amounts to either a table lookup or a function evaluation.This example is built on three rules, and each of the rules depends on resolving the inputs into a number of different fuzzy linguistic sets: service is poor, service is good, food is rancid, food is delicious, and so on. Before the rules can be evaluated, the inputs must be fuzzified according to each of these linguistic sets. For example, to what extent is the food really delicious? The following figure shows how well the food at the hypothetical restaurant (rated on a scale of 0 to 10) qualifies, (via its membership function), as the linguistic variable delicious. In this case, we rated the food as an 8, which, given your graphical definition of delicious, corresponds to µ = 0.7 for the delicious membership function.In this manner, each input is fuzzified over all the qualifying membership functions required by the rules.Step 2. Apply Fuzzy OperatorAfter the inputs are fuzzified, you know the degree to which each part of the antecedent is satisfied for each rule. If the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one number that represents the result of the antecedent for that rule. This number is then applied to the output function. The input to the fuzzy operator is two or more membership values from fuzzified input variables. The output is a single truth value.As is described in Logical Operations section, any number of well-defined methods can fill in for the AND operation or the OR operation. In the toolbox, two built-in AND methods are supported: min (minimum) and prod (product). Two built-in OR methods are also supported: max (maximum), and the probabilistic OR method probor. The probabilistic OR method (also known as the algebraic sum) is calculated according to the equationprobor(a,b) = a + b - abIn addition to these built-in methods, you can create your own methods for AND and OR by writing any function and setting that to be your method of choice.The following figure shows the OR operator max at work, evaluating the antecedent of the rule 3 for the tipping calculation. The two different pieces of the antecedent (service is excellent and food is delicious) yielded the fuzzy membership values 0.0 and 0.7 respectively. The fuzzy OR operator simply selects the maximum of the two values, 0.7, and the fuzzy operation for rule 3 is complete. The probabilistic OR method would still result in 0.7.Step 3. Apply Implication Method Step 4. Aggregate All OutputsStep 5. DefuzzifyThe input for the defuzzification process is a fuzzy set (the aggregate output fuzzy set) and the output is a single number. As much as fuzziness helps the rule evaluation during the intermediate steps, the final desired output for each variable is generally a single number. However, the aggregate of a fuzzy set encompasses a range of output values, and so must be defuzzified in order to resolve a single output value from the set.Perhaps the most popular defuzzification method is the centroid calculation, which returns the center of area under the curve. There are five built-in methods supported: centroid, bisector, middle of maximum (the average of the maximum value of theoutput set), largest of maximum, and smallest of maximum.Back to TopThe Fuzzy Inference DiagramThe fuzzy inference diagram is the composite of all the smaller diagrams presented so far in this section. It simultaneouslydisplays all parts of the fuzzy inference process you have examined. Information flows through the fuzzy inference diagramas shown in the following figure.In this figure, the flow proceeds up from the inputs in the lower left, then across each row, or rule, and then down the rule outputs to finish in the lower right. This compact flow shows everything at once, from linguistic variable fuzzification all the way through defuzzification of the aggregate output.The following figure shows the actual full-size fuzzy inference diagram. There is a lot to see in a fuzzy inference diagram, but after you become accustomed to it, you can learn a lot about a system very quickly. For instance, from this diagram with these particular inputs, you can easily see that the implication method is truncation with the min function. The max function is being used for the fuzzy OR operation. Rule 3 (the bottom-most row in the diagram shown previously) is having the strongest influence on the output. and so on. The Rule Viewer described in The Rule Viewer is a MATLAB implementation of the fuzzy inference diagram.Back to TopCustomizationOne of the primary goals of Fuzzy Logic Toolbox software is to have an open and easily modified fuzzy inference system structure. The toolbox is designed to give you as much freedom as possible, within the basic constraints of the process described, to customize the fuzzy inference process for your application.Building Systems with Fuzzy Logic Toolbox Software describes exactly how to build and implement a fuzzy inference system using the tools provided. To learn how to customize a fuzzy inference system, see Building Fuzzy Inference Systems UsingCustom Functions.Back to TopWas this topic helpful?Yes NoFoundations of Fuzzy Logic Building Systems with Fuzzy Logic Toolbox Software © 1984-2010 The MathWorks, Inc. •Terms of Use•Patents•Trademarks•Acknowledgments。

fuzzy数学在传感器设计中的应用
Fuzzy数学在传感器设计中的应用非常广泛。

传感器设计涉及到
诸多不确定性因素，比如噪声、干扰、温度变化等。

这些不确定性因
素会影响传感器的准确度和精度。

为了解决这些问题，传感器设计师
可以基于Fuzzy数学的理论来设计模糊控制器，从而提高传感器的性能。

Fuzzy数学的理论基础是模糊逻辑，它可以用来描述不确定性因素。

在传感器设计中，模糊逻辑可以用来设计模糊控制器，从而实现
自适应控制。

模糊控制器的设计需要考虑多个因素，包括输入变量、
输出变量、隶属度函数等。

在实践中，传感器设计师可以根据具体的
应用场景来选择合适的隶属度函数，以及确定输入和输出变量之间的
关系。

通过模糊控制器的设计，传感器的响应速度和准确度可以得到
显著提升。

另外，Fuzzy数学还可以用来优化传感器的数据处理和信息提取。

它可以处理非线性和模糊的信息，从而实现更加精准和可靠的信息提取。

比如，基于Fuzzy数学的算法可以应用于运动传感器的数据处理，从而提高动作检测的准确度。

此外，Fuzzy数学还可以用于传感器网络的优化设计，从而提高网络的稳定性和可靠性。

综上所述，Fuzzy数学在传感器设计中具有广泛的应用前景。

它
可以帮助传感器设计师解决因不确定性因素带来的问题，提高传感器
的性能和精度，进而满足不同应用场景的需求。

数字信号内蒙古科技⼤学2009/2010学年第⼀学期《数字信号处理》A卷考试试题使⽤专业、年级：通信、电信07 任课教师：杨⽴东⼀、填空题（共5空，每空2分，共10分）1. 当N=1024时，基2 时间抽取FFT的信号流图由级构成，如采⽤原位运算，最少需要个存储单元。

2. Chebyshev I型模拟低通滤波器在带等波纹波动。

3．在利⽤双线性变换法设计数字滤波器的过程中，数字滤波器的⾓频率和模拟滤波器的⾓频率的关系式为。

4．利⽤DFT对连续⾮周期信号频谱进⾏近似分析过程中⼀般将会出现混叠现象、泄露现象和现象。

⼆、计算题（共6题，1-4每题10分，5-6每题15分，共70分）1. 设是⼀8点实序列的DFT，前5个点的值为，，，，，不计算IDFT，试确定下列表达式的值（1），（2），（3）2.已知2个有限长序列分别为，，求：（1）与的线性卷积；（2）与的4点循环卷积。

3．已知⼀稳定的LTI系统的H(z)为，试确定该系统的。

4．已知⼀实信号 x(t), 该信号的最⾼频率为=200 rad/s, ⽤=600 rad/s 对 x(t)进⾏抽样。

如对抽样信号做1024 点的DFT，试确定X[m]中 m=128 和m=768 点所分别对应的原连续信号的连续频谱点和。

5.已知某离散时间平稳⽩噪声X[K]通过⼀阶FIR滤波器，求输出序列的⾃相关函数和功率谱，以及输⼊输出序列的互相关函数和互功率谱。

6.已知8阶III型线性相位FIR滤波器的部分零点为：，，（1）写出该滤波器的其他零点；（2）设，求出该滤波器的系统函数。

（3）画出该滤波器的线性相位直接型结构框图。

三、问答题（共2题，每题10分，共20分）1.简述脉冲响应不变法设计数字滤波器的基本原理。

如果某因果模拟滤波器的系统函数为，写出直接利⽤脉冲响应不变法映射出来的数字滤波器的系统函数H(z)。

2.判断下列系统是否为线性、因果、⾮时变、稳定。

（1）（2）（3）内蒙古科技⼤学2009/2010学年第⼀学期《数字信号处理》B卷考试试题使⽤专业、年级：通信、电信07 任课教师：杨⽴东⼀、填空题（共5空，每空2分，共10分）1. 旋转因⼦具有周期性、性和性。

振动力学专业英语及词汇振动方面的专业英语及词汇振动方面的专业英语及词汇参见《工程振动名词术语》1 振动信号的时域、频域描述振动过程 (Vibration Process)简谐振动 (Harmonic Vibration)周期振动 (Periodic Vibration)准周期振动 (Ouasi-periodic Vibration)瞬态过程 (Transient Process)随机振动过程 (Random Vibration Process)各态历经过程 (Ergodic Process)确定性过程 (Deterministic Process)振幅 (Amplitude)相位 (Phase)初相位 (Initial Phase)频率 (Frequency)角频率 (Angular Frequency)周期 (Period)复数振动 (Complex Vibration)复数振幅 (Complex Amplitude)峰值 (Peak-value)平均绝对值 (Average Absolute Value)有效值 (Effective Value，RMS Value)均值 (Mean Value，Average Value)傅里叶级数 (FS，Fourier Series)傅里叶变换 (FT，Fourier Transform)傅里叶逆变换 (IFT，Inverse Fourier Transform)离散谱 (Discrete Spectrum)连续谱 (Continuous Spectrum)傅里叶谱 (Fourier Spectrum)线性谱 (Linear Spectrum)幅值谱 (Amplitude Spectrum)相位谱 (Phase Spectrum)均方值 (Mean Square Value)方差 (Variance)协方差 (Covariance)自协方差函数 (Auto-covariance Function)互协方差函数 (Cross-covariance Function)自相关函数 (Auto-correlation Function)互相关函数 (Cross-correlation Function)标准偏差 (Standard Deviation)相对标准偏差 (Relative Standard Deviation)概率 (Probability)概率分布 (Probability Distribution)高斯概率分布(Gaussian Probability Distribution) 概率密度(Probability Density)集合平均 (Ensemble Average)时间平均 (Time Average)功率谱密度 (PSD，Power Spectrum Density)自功率谱密度 (Auto-spectral Density)互功率谱密度 (Cross-spectral Density)均方根谱密度 (RMS Spectral Density)能量谱密度 (ESD，Energy Spectrum Density)相干函数 (Coherence Function)帕斯瓦尔定理 (Parseval''''s Theorem)维纳，辛钦公式 (Wiener-Khinchin Formula2 振动系统的固有特性、激励与响应振动系统 (Vibration System)激励 (Excitation)响应 (Response)单自由度系统 (Single Degree-Of-Freedom System) 多自由度系统(Multi-Degree-Of- Freedom System) 离散化系统(Discrete System)连续体系统 (Continuous System)刚度系数 (Stiffness Coefficient)自由振动 (Free Vibration)自由响应 (Free Response)强迫振动 (Forced Vibration)强迫响应 (Forced Response)初始条件 (Initial Condition)固有频率 (Natural Frequency)阻尼比 (Damping Ratio)衰减指数 (Damping Exponent)阻尼固有频率 (Damped Natural Frequency)对数减幅系数 (Logarithmic Decrement)主频率 (Principal Frequency)无阻尼模态频率 (Undamped Modal Frequency)模态 (Mode)主振动 (Principal Vibration)振型 (Mode Shape)振型矢量 (Vector Of Mode Shape)模态矢量 (Modal Vector)正交性 (Orthogonality)展开定理 (Expansion Theorem)主质量 (Principal Mass)模态质量 (Modal Mass)主刚度 (Principal Stiffness)模态刚度 (Modal Stiffness)正则化 (Normalization)振型矩阵 (Matrix Of Modal Shape)模态矩阵 (Modal Matrix)主坐标 (Principal Coordinates)模态坐标 (Modal Coordinates)模态分析 (Modal Analysis)模态阻尼比 (Modal Damping Ratio)频响函数 (Frequency Response Function)幅频特性 (Amplitude-frequency Characteristics)相频特性 (Phase frequency Characteristics)共振 (Resonance)半功率点 (Half power Points)波德图(Bodé Plot)动力放大系数 (Dynamical Magnification Factor)单位脉冲 (Unit Impulse)冲激响应函数 (Impulse Response Function)杜哈美积分(Duhamel’s Integral)卷积积分 (Convolution Integral)卷积定理 (Convolution Theorem)特征矩阵 (Characteristic Matrix)阻抗矩阵 (Impedance Matrix)频响函数矩阵 (Matrix Of Frequency Response Function) 导纳矩阵 (Mobility Matrix)冲击响应谱 (Shock Response Spectrum)冲击激励 (Shock Excitation)冲击响应 (Shock Response)冲击初始响应谱 (Initial Shock Response Spectrum) 冲击剩余响应谱(Residual Shock Response Spectrum) 冲击最大响应谱(Maximum Shock Response Spectrum) 冲击响应谱分析(Shock Response Spectrum Analysis 3 模态试验分析模态试验 (Modal Testing)机械阻抗 (Mechanical Impedance)位移阻抗 (Displacement Impedance)速度阻抗 (Velocity Impedance)加速度阻抗 (Acceleration Impedance)机械导纳 (Mechanical Mobility)位移导纳 (Displacement Mobility)速度导纳 (Velocity Mobility)加速度导纳 (Acceleration Mobility)驱动点导纳 (Driving Point Mobility)跨点导纳 (Cross Mobility)传递函数 (Transfer Function)拉普拉斯变换 (Laplace Transform)传递函数矩阵 (Matrix Of Transfer Function)频响函数 (FRF，Frequency Response Function)频响函数矩阵 (Matrix Of FRF)实模态 (Normal Mode)复模态 (Complex Mode)模态参数 (Modal Parameter)模态频率 (Modal Frequency)模态阻尼比 (Modal Damping Ratio)模态振型 (Modal Shape)模态质量 (Modal Mass)模态刚度 (Modal Stiffness)模态阻力系数 (Modal Damping Coefficient)模态阻抗 (Modal Impedance)模态导纳 (Modal Mobility)模态损耗因子 (Modal Loss Factor)比例粘性阻尼 (Proportional Viscous Damping)非比例粘性阻尼 (Non-proportional Viscous Damping) 结构阻尼(Structural Damping，Hysteretic Damping) 复频率(ComplexFrequency)复振型 (Complex Modal Shape)留数 (Residue)极点 (Pole)零点 (Zero)复留数 (Complex Residue)随机激励 (Random Excitation)伪随机激励 (Pseudo Random Excitation)猝发随机激励 (Burst Random Excitation)稳态正弦激励 (Steady State Sine Excitation)正弦扫描激励 (Sweeping Sine Excitation)锤击激励 (Impact Excitation)频响函数的H1 估计 (FRF Estimate by H1)频响函数的H2 估计 (FRF Estimate by H2)频响函数的H3 估计 (FRF Estimate by H3)单模态曲线拟合法 (Single-mode Curve Fitting Method)多模态曲线拟合法 (Multi-mode Curve Fitting Method)模态圆 (Mode Circle)剩余模态 (Residual Mode)幅频峰值法 (Peak Value Method)实频-虚频峰值法 (Peak Real／Imaginary Method)圆拟合法 (Circle Fitting Method)加权最小二乘拟合法 (Weighting Least Squares Fitting method) 复指数拟合法 (Complex Exponential Fitting method)1.2 振动测试的名词术语1 传感器测量系统传感器测量系统 (Transducer Measuring System)传感器 (Transducer)振动传感器 (Vibration Transducer)机械接收 (Mechanical Reception)机电变换 (Electro-mechanical Conversion)测量电路 (Measuring Circuit)惯性式传感器 (Inertial Transducer，Seismic Transducer) 相对式传感器 (Relative Transducer)电感式传感器 (Inductive Transducer)应变式传感器 (Strain Gauge Transducer)电动力传感器 (Electro-dynamic Transducer)压电式传感器 (Piezoelectric Transducer)压阻式传感器 (Piezoresistive Transducer)电涡流式传感器 (Eddy Current Transducer)伺服式传感器 (Servo Transducer)灵敏度 (Sensitivity)复数灵敏度 (Complex Sensitivity)分辨率 (Resolution)频率范围 (Frequency Range)线性范围 (Linear Range)频率上限 (Upper Limit Frequency)频率下限 (Lower Limit Frequency)静态响应 (Static Response)零频率响应 (Zero Frequency Response)动态范围 (Dynamic Range)幅值上限 Upper Limit Amplitude)幅值下限 (Lower Limit Amplitude)最大可测振级 (Max．Detectable Vibration Level)最小可测振级 (Min．Detectable Vibration Level)信噪比 (S/N Ratio)振动诺模图 (Vibration Nomogram)相移 (Phase Shift)波形畸变 (Wave-shape Distortion)比例相移 (Proportional Phase Shift)惯性传感器的稳态响应(Steady Response Of Inertial Transducer)惯性传感器的稳击响应 (Shock Response Of Inertial Transducer) 位移计型的频响特性(Frequency Response Characteristics Vibrometer)加速度计型的频响特性(Frequency Response Characteristics Accelerometer) 幅频特性曲线 (Amplitude-frequency Curve) 相频特性曲线 (Phase-frequency Curve)固定安装共振频率 (Mounted Resonance Frequency)安装刚度 (Mounted Stiffness)有限高频效应 (Effect Of Limited High Frequency)有限低频效应 (Effect Of Limited Low Frequency)电动式变换 (Electro-dynamic Conversion)磁感应强度 (Magnetic Induction， Magnetic Flux Density)磁通 (Magnetic Flux)磁隙 (Magnetic Gap)电磁力 (Electro-magnetic Force)相对式速度传 (Relative Velocity Transducer)惯性式速度传感器 (Inertial Velocity Transducer)速度灵敏度 (Velocity Sensitivity)电涡流阻尼 (Eddy-current Damping)无源微(积)分电路 (Passive Differential (Integrate) Circuit) 有源微(积)分电路(Active Differential (Integrate) Circuit) 运算放大器(Operational Amplifier)时间常数 (Time Constant)比例运算 (Scaling)积分运算 (Integration)微分运算 (Differentiation)高通滤波电路 (High-pass Filter Circuit)低通滤波电路 (Low-pass Filter Circuit)截止频率 (Cut-off Frequency)压电效应 (Piezoelectric Effect)压电陶瓷 (Piezoelectric Ceramic)压电常数 (Piezoelectric Constant)极化 (Polarization)压电式加速度传感器 (Piezoelectric Acceleration Transducer) 中心压缩式 (Center Compression Accelerometer)三角剪切式 (Delta Shear Accelerometer)压电方程 (Piezoelectric Equation)压电石英 (Piezoelectric Quartz)电荷等效电路 (Charge Equivalent Circuit)电压等效电路 (Voltage Equivalent Circuit)电荷灵敏度 (Charge Sensitivity)电压灵敏度 (Voltage Sensitivity)电荷放大器 (Charge Amplifier)适调放大环节 (Conditional Amplifier Section)归一化 (Uniformization)电荷放大器增益 (Gain Of Charge Amplifier)测量系统灵敏度 (Sensitivity Of Measuring System)底部应变灵敏度 (Base Strain Sensitivity)横向灵敏度 (Transverse Sensitivity)地回路 (Ground Loop)力传感器 (Force Transducer)力传感器灵敏度 (Sensitivity Of Force Transducer)电涡流 (Eddy Current)前置器 (Proximitor)间隙-电压曲线 (Voltage vs Gap Curve)间隙-电压灵敏度 (Voltage vs Gap Sensitivity)压阻效应 (Piezoresistive Effect)轴向压阻系数 (Axial Piezoresistive Coefficient)横向压阻系数 (Transverse Piezoresistive Coefficient)压阻常数 (Piezoresistive Constant)单晶硅 (Monocrystalline Silicon)应变灵敏度 (Strain Sensitivity)固态压阻式加速度传感器(Solid State Piezoresistive Accelerometer) 体型压阻式加速度传感器 (Bulk Type Piezoresistive Accelerometer) 力平衡式传感器 (Force Balance Transducer) 电动力常数 (Electro-dynamic Constant)机电耦合系统 (Electro-mechanical Coupling System)2 检测仪表、激励设备及校准装置时间基准信号 (Time Base Signal)李萨茹图 (Lissojous Curve)数字频率计 (Digital Frequency Meter)便携式测振表 (Portable Vibrometer)有效值电压表 (RMS Value Voltmeter)峰值电压表 (Peak-value Voltmeter)平均绝对值检波电路 (Average Absolute Value Detector)峰值检波电路 (Peak-value Detector)准有效值检波电路 (Quasi RMS Value Detector)真有效值检波电路 (True RMS Value Detector)直流数字电压表 (DVM，DC Digital Voltmeter)数字式测振表 (Digital Vibrometer)A/D 转换器 (A/D Converter)D/A 转换器 (D/A Converter)相位计 (Phase Meter)电子记录仪 (Lever Recorder)光线示波器 (Oscillograph)振子 (Galvonometer)磁带记录仪 (Magnetic Tape Recorder)DR 方式(直接记录式) (Direct Recorder)FM 方式(频率调制式) (Frequency Modulation)失真度 (Distortion)机械式激振器 (Mechanical Exciter)机械式振动台 (Mechanical Shaker)离心式激振器 (Centrifugal Exciter)电动力式振动台 (Electro-dynamic Shaker)电动力式激振器 (Electro-dynamic Exciter)液压式振动台 (Hydraulic Shaker)液压式激振器 (Hydraulic Exciter)电液放大器 (Electro-hydraulic Amplifier)磁吸式激振器 (Magnetic Pulling Exciter)涡流式激振器 (Eddy Current Exciter)压电激振片 (Piezoelectric Exciting Elements)冲击力锤 (Impact Hammer)冲击试验台 (Shock Testing Machine)激振控制技术 (Excitation Control Technique)波形再现 (Wave Reproduction)压缩技术 (Compression Technique)均衡技术 (Equalization Technique)交越频率 (Crossover Frequency)综合技术 (Synthesis Technique)校准 (Calibration)分部校准 (Calibration for Components in system) 系统校准 (Calibration for Over-all System)模拟传感器 (Simulated Transducer)静态校准 (Static Calibration)简谐激励校准 (Harmonic Excitation Calibration) 绝对校准 (Absolute Calibration)相对校准 (Relative Calibration)比较校准 (Comparison Calibration)标准振动台 (Standard Vibration Exciter)读数显微镜法 (Microscope-streak Method)光栅板法 (Ronchi Ruling Method)光学干涉条纹计数法 (Optical Interferometer Fringe Counting Method)光学干涉条纹消失法(Optical Interferometer Fringe Disappearance Method) 背靠背安装 (Back-to-back Mounting) 互易校准法 (Reciprocity Calibration)共振梁 (Resonant Bar)冲击校准 (Impact Exciting Calibration)摆锤冲击校准 (Ballistic Pendulum Calibration)落锤冲击校准 (Drop Test Calibration)振动和冲击标准 (Vibration and Shock Standard)迈克尔逊干涉仪 (Michelson Interferometer)摩尔干涉图象 (Moire Fringe)参考传感器 (Reference Transducer)3 频率分析及数字信号处理带通滤波器 (Band-pass Filter)半功率带宽 (Half-power Bandwidth)3 dB 带宽 (3 dB Bandwidth)等效噪声带宽 (Effective Noise Bandwidth)恒带宽 (Constant Bandwidth)恒百分比带宽 (Constant Percentage Bandwidth)1/N 倍频程滤波器 (1/N Octave Filter)形状因子 (Shape Factor)截止频率 (Cut-off Frequency)中心频率 (Centre Frequency)模拟滤波器 (Analog Filter)数字滤波器 (Digital Filter)跟踪滤波器 (Tracking Filter)外差式频率分析仪 (Heterodyne Frequency Analyzer) 逐级式频率分析仪 (Stepped Frequency Analyzer)扫描式频率分析仪 (Sweeping Filter Analyzer)混频器 (Mixer)RC 平均 (RC Averaging)平均时间 (Averaging Time)扫描速度 (Sweeping Speed)滤波器响应时间 (Filter Response Time)离散傅里叶变换 (DFT，Discrete Fourier Transform) 快速傅里叶变换 (FFT，Fast Fourier Transform)抽样频率 (Sampling Frequency)抽样间隔 (Sampling Interval)抽样定理 (Sampling Theorem)抗混滤波 (Anti-aliasing Filter)泄漏 (Leakage)加窗 (Windowing)窗函数 (Window Function)截断 (Truncation)频率混淆 (Frequency Aliasing)乃奎斯特频率 (Nyquist Frequency)矩形窗 (Rectangular Window)汉宁窗 (Hanning Window)凯塞-贝塞尔窗 (Kaiser-Bessel Window)平顶窗 (Flat-top Window)平均 (Averaging)线性平均 (Linear Averaging)指数平均 (Exponential Averaging)峰值保持平均 (Peak-hold Averaging)时域平均 (Time-domain Averaging)谱平均 (Spectrum Averaging)重叠平均 (Overlap Averaging)栅栏效应 (Picket Fence Effect)吉卜斯效应 (Gibbs Effect)基带频谱分析 (Base-band Spectral Analysis)选带频谱分析 (Band Selectable Sp4ctralAnalysis)细化 (Zoom)数字移频 (Digital Frequency Shift)抽样率缩减 (Sampling Rate Reduction)功率谱估计 (Power Spectrum Estimate)相关函数估计 (Correlation Estimate)频响函数估计 (Frequency Response Function Estimate) 相干函数估计 (Coherence Function Estimate)冲激响应函数估计 (Impulse Response Function Estimate) 倒频谱 (Cepstrum)功率倒频谱 (Power Cepstrum)幅值倒频谱 (Amplitude Cepstrum)倒频率 (Quefrency)4 旋转机械的振动测试及状态监测状态监测 (Condition Monitoring)故障诊断 (Fault Diagnosis)转子 (Rotor)转手支承系统 (Rotor-Support System)振动故障 (Vibration Fault)轴振动 (Shaft Vibration)径向振动 (Radial Vibration)基频振动 (Fundamental Frequency Vibration)基频检测 (Fundamental Frequency Component Detecting) 键相信号 (Key-phase Signal)正峰相位 (+Peak Phase)高点 (High Spot)光电传感器 (Optical Transducer)同相分量 (In-phase Component)正交分量 (Quadrature Component)跟踪滤波 (Tracking Filter)波德图 (Bode Plot)极坐标图 (Polar Plot)临界转速 (Critical Speed)不平衡响应 (Unbalance Response)残余振幅 (Residual Amplitude)方位角 (Attitude Angle)轴心轨迹 (Shaft Centerline Orbit)正进动 (Forward Precession)同步正进动 (Synchronous Forward Precession)反进动 (Backward Precession)正向涡动 (Forward Whirl)反向涡动 (Backward Whirl)油膜涡动 (Oil Whirl)油膜振荡 (Oil Whip)轴心平均位置(Average Shaft Centerline Position) 复合探头(Dual Probe)振摆信号 (Runout Signal)电学振摆 (Electrical Runout)机械振摆 (Mechanical Runout)慢滚动向量 (Slow Roll Vector)振摆补偿 (Runout Compensation)故障频率特征(Frequency Characteristics Of Fault) 重力临界(Gravity Critical)对中 (Alignment)双刚度转子 (Dual Stiffness Rotor)啮合频率 (Gear-mesh Frequency)间入简谐分量 (Interharmonic Component)边带振动 (Side-band Vibration)三维频谱图 (Three Dimensional Spectral Plot)瀑布图 (Waterfall Plot)级联图 (Cascade Plot)阶次跟踪 (Order Tracking)阶次跟踪倍乘器 (Order Tracking Multiplier)监测系统 (Monitoring System)适调放大器 (Conditional Amplifier)趋势分析 (Trend Analysis)倒频谱分析 (Cepstrum Analysis) 直方图 (Histogram)确认矩阵 (Confirmation Matrix) 通频幅值 (Over-all Amplitude) 幅值谱 (Amplitude Spectrum)相位谱 (Phase Spectrum)报警限 (Alarm Level)。

实序列的傅里叶变换必是什么函数
实序列的傅里叶变换在频率域中必定是一对共轭对称的复指数
函数。

具体来说，如果我们有一个实序列x(n)，其傅里叶变换X(ω)满足以下关系：
X(ω) = X(-ω)。

其中X(-ω)表示X(ω)的共轭。

这意味着实序列的傅里叶变换
在频率域中是关于零频率的对称函数。

这是因为实序列的傅里叶变
换中包含了实部和虚部，而实部和虚部之间是共轭对称的关系。

从数学角度来看，这意味着实序列的傅里叶变换中不会出现奇
异的频率成分，而是以对称的方式分布在频率轴上。

这对于分析实
序列的频谱特性非常有用，因为它可以帮助我们理解实序列在频率
域中的性质。

另外，从信号处理的角度来看，实序列的傅里叶变换的共轭对
称性也意味着我们可以通过分析频率域中的一半数据来完全描述整
个频谱。

这样可以节省计算资源，并简化信号处理的过程。

总之，实序列的傅里叶变换必定是一对共轭对称的复指数函数，在频率域中呈现出对称的特性，这对于理解信号的频谱特性和进行
信号处理都具有重要意义。

boost的PID和FUZZY调节电路目录第一章绪论 ............................................................................................................ - 1 - 第二章PID对BOOST电路的控制及仿真...................................................................... - 2 -2.1 设计要求 ............................................................................................................. - 2 -2.2 设计思路 ............................................................................................................. - 2 -2.3 设计过程 ............................................................................................................. - 3 -2.4调制过程 ................................................................................................................ - 6 -2.5仿真结果及分析 .................................................................................................... - 8 - 第三章FUZZY对BOOST电路的控制及仿真 ............................................................. - 12 -3.1 设计要求 ........................................................................................................... - 12 -3.2 设计思路 ........................................................................................................... - 12 -3.3 设计过程 ........................................................................................................... - 12 -3.3 调试及仿真结果 ............................................................................................... - 14 - 附录参考文献第一章绪论本文采用的boost电路是一种开关直流升压电路，即可以使输出电压比输入电压高它。