An interval-parameter fuzzy nonlinear optimization model

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基于NT降阶算法的区间二型模糊系统辨识王哲【摘要】Due to the defects in description system uncertainty of the traditional T-S fuzzy description system, type-2 T-S fuzzy system has received extensive attention. In according with the low efficiency of common type reduction algorithm for interval two type fuzzy set, the NT type reduction algorithm was used for interval type-2 fuzzy system identification. The NT type reduction algorithm avoid the complexity iterative search operation, directly using the upper and lower bounds of first membership function, and then directly get the results of the fuzzy system. The simulations result shows that NT type reduction algorithm can improve identification efficiently without reduce identification accuracy.%由于传统T-S模糊描述系统不确定性方面的缺陷,二型T-S模糊系统得到了广泛关注.针对常见区间二型模糊集合的降阶算法存在的效率低下的问题,本文利用NT降阶算法进行区间二型模糊系统的辨识.NT降阶算法避免了复杂的迭代搜索操作,直接利用首隶属度函数的上、下限进行降阶运算,然后直接得到解模糊化结果.仿真实例表明,利用NT降阶算法能够在不降低辨识精度的情况下,提高辨识效率.【期刊名称】《仪器仪表用户》【年(卷),期】2018(025)006【总页数】4页(P17-20)【关键词】区间二型模糊集合;降阶;T-S模糊系统;模糊辨识;NT降阶算法【作者】王哲【作者单位】天津现代职业技术学院,天津 300350【正文语种】中文【中图分类】TP273+.40 引言近些年，T-S模糊模型在非线性系统辨识方面取得了很好的效果。

模糊认知图的学习此篇随笔仅⽤于记录学习内容⽅便以后查阅，主要参考学习了 迟亚雄. 模糊认知图智能学习算法与应⽤研究[D].基本理论与⽅法基于Hebbian的学习⽅法 核⼼思想：神经元的激活顺序和⽅式会影响权重的变化，若两神经元异步激活则降低权值，同步激活则增⾼权值。

特点：总是依赖专家知识。

⾮线性Hebbian学习算法[1]（Nonlinear Hebbian Learning, NHL）的初始化需要专家对concept进⾏⼲预，⽐如建议各concept的模糊值，这些模糊值的取值范围以及各concept之间的因果关系等。

数据驱动型NHL[2]（Data-driven Nonlinear Hebbian Learning, DDNHL），与NHL类似，但DDNHL利⽤了可观测到的数据进⾏学习来提⾼ FCMs 的模型质量。

集成学习与NHL相结合的学习算法[3]，⽤NHL训练模型，再使⽤集成学习算法提⾼性能。

在学习模糊认知图的准确性⽅⾯由于DDNHL。

基于进化计算的学习⽅法 Hebbian的⽅法⽐较依赖专家知识，进化计算的⽬的是从数据中学习模糊认知图，搜索最优的模糊认知图。

进化算法是⼀种启发式算法，⽐较常见的启发式算法有：遗传算法，粒⼦群优化算法，模拟退⽕，蚁群优化等。

⾯临的问题 模糊认知图扩展到⼀定规模时，⾯临的问题是多维度优化问题，因为随着决策变量的增加，需要确定的权重关系的数量会呈指数增长，会造成维度灾难。

从优化的⾓度看，训练样本数量不变的情况下，决策变量的数量增加会导致过拟合。

当数据的维度到达⼀定⾼度，如果要找到最优解或者仅仅达到低维度的同等性能，则需要⼏何数量增长的数据。

真实模糊认知图的⽹络密度要⽐算法学习得到的⽹络模型的密度低的多。

基于神经⽹络和进化计算的模糊认知图学习Reference[1] Elpiniki Papageorgiou, Chrysostomos Stylios, Peter Groumpos. Fuzzy Cognitive Map Learning Based on Nonlinear Hebbian Rule[M]// AI 2003: Advances in Artificial Intelligence. Springer Berlin Heidelberg, 2003.[2] Stach W , Kurgan L A , Pedrycz W . Data-driven Nonlinear Hebbian Learning method for Fuzzy Cognitive Maps[C]// IEEE IEEE International Conference on Fuzzy Systems. IEEE, 2008.[3] Papageorgiou E I , Kannappan A . Fuzzy cognitive map ensemble learning paradigm to solve classification problems: Application to autism identification[J]. Applied Soft Computing Journal, 2012, 12(12):3798-3809.[] 迟亚雄. 模糊认知图智能学习算法与应⽤研究[D].。

带有饱和的电机伺服系统非奇异终端滑模funnel控制陈强;汤筱晴【摘要】本文提出一种非奇异终端滑模funnel控制(NTSMFC)方法,实现带有饱和输入电机伺服系统的指定性能跟踪控制.根据中值定理,非光滑饱和函数转化为放射形式,并且应用一个简单的神经网络进行逼近和补偿.为保证跟踪误差被限制在指定的界限内,同时为避免构建复杂的barrier李雅普诺夫函数或逆函数,本文采用一个新的限制变量.然后,构建非奇异终端滑模funnel控制器保证电机伺服系统的指定跟踪性能.该方法无需事先已知输入饱和函数的界限等先验知识,且基于李雅普诺夫函数设计可以保证位置跟踪误差的收敛性,最后给出仿真对比实例证明了该方法的有效性.【期刊名称】《控制理论与应用》【年(卷),期】2015(032)008【总页数】8页(P1064-1071)【关键词】funnel控制;非奇异终端滑模;神经网络;输入饱和【作者】陈强;汤筱晴【作者单位】浙江工业大学信息工程学院,浙江杭州310023;浙江工业大学信息工程学院,浙江杭州310023【正文语种】中文【中图分类】TP273Over the past decades，motor servo systems have been widely studied in motion control applications［1-3］.The mechanical connection between servo motors and mechanical devices produces non-smooth nonlinear constraints on their outputs and/or inputs in the form of the physical stoppage，saturation，hysteresis，and dead-zone.During operation，violation of the constraints leads to performance degradation，hazards or system damage.For the output constraint，there are some effective methods for position，velocity，and force constraints existing in motor servo systems.By using the logarithmic function in the Lyapunov function design，a barrier Lyapunov function（BLF）is constructed，in which a symmetric or asymmetric constraint is utilized to constrain the state variable of the control system，so that the tracking errors can be indirectly constrained［4-6］.However，the expression of BLF is complex， and extra efforts are needed to ensure the continuity and differentiability.In［7-9］，a prescribed performance control（PPC）scheme is proposed，and the tracking error of a nonlinear system is transformed into a new error by constructing the inverse of the transformation function.Therefore，the prescribed tracking performance of the transient property and the steady-state error can be guaranteed.But PPC scheme may cause a singularity problem since the inverse transformation function includes a partial differential terms.Being a non-model-based（memory less）constraint technique，the funnel control is proposed to guarantee the prescribed transient behavior and asymptotic tracking of the system［10-13］.This technique bypassesthe difficulties of identification and estimation of traditional high-gain adaptive control.Recently，a new error-constraint variable is designed as a virtual control variable in the backstep-ping design to ensure the prescribed transient and steadystate performance［14］，and thus the aforementioned complex transformation function is not needed any more.In［15］，a funnel dynamic surface control with prescribed performance is proposed to overcome the explosion of complexity problem in the backstepping technique，and the tracking performance of closed-loop system is guaranteed.Sliding mode control（SMC）is one of the most useful approaches to deal with system uncertainties and bounded disturbances，and has been widely applied in various fields［16-17］.The traditional linear sliding mode control scheme can guarantee the asymptotical convergence of tracking errors when time goes to the infinity.Recently，many research works have focused on the finite-time convergence of tracking errors.Man and Yu［18］proposed a terminal sliding mode control（TSMC）scheme by introducing a nonlinear term in the SMC design and the tracking error can be guaranteed to converge within a finite time.However，there are two disadvantages of TSMC，i.e.，the singularity problem and requirement of the uncertainty bound.To overcome the singularity problem，Feng，et al.［19］and Yu，et al.［20］proposed nonsingular terminal sliding mode control（NTSMC）methods.Besides，Chen，et al ［21］utilized a recurrent Hermite neural network（RHNN）to estimate the lumped uncertainty online when designing the nonsingular terminalsliding surface，and hence the lumped uncertainty bound is unnecessary. As one of the most important non-smooth nonlinearities，saturation on hardware dictates that the magnitude of the control signal is always constrained，which often severely limits system performance，giving rise to undesirable inaccuracy or leading instability［22］.So far，many significant results on the control design for the systems with input saturation have been obtained［23-27］.However，the lower and upper limits of the saturation constraints should be exactly known or estimated for controllers design.Recently，several research work has been investigated without using the prior knowledge of saturation bounds. Wen et al.［28］uses a smooth non-affine function of the control input signal to approximate the non-smooth saturation function，and a Nussbaum function is introduced to compensate for the nonlinear term arising from the input saturation.Based on the idea of［28］，Wang et al.［29］transforms the non-affine function of the system into an affine form，and an adaptive control scheme is derived without requiring the prior knowledge of input saturation bounds.In this paper，we propose a nonsingular terminal sliding mode funnel control to achieve a prescribed tracking performance for motor servo systems with unknown input saturation.A smooth and affine function is used to solve the input saturation problem.Meanwhile，to avoid using the complex barrier Lyapunov function or inverse transformed function，a funnel constraint variable is utilized in constructingthenonsingularterminalslidingmodescheme to force thetracking error fall within prescribe boundaries. No prior knowledge of the input saturation bounds is required in the proposed method，and the effectiveness is demonstrated by simulation results.2.1 System descriptionThe mechanical dynamics of the motor servo system can be described as follows:where x=［x，］T∈R2，u（t）∈R，y∈R are state variables，the control input voltage to the motor and the output from the motor，respectively；x is the position，m is the inertia，k0is a positive control gain（the force constant），f（x，t）is the friction force；d（x，t）represents a bounded disturbance modeling nonlinear elastic forces generated by coupling and protective covers，measurement noise，power electronics disturbances and other uncertainties.v（u）∈R is the plant input subject to saturation nonlinearity described bywhere vmaxis an unknown parameter of input saturation.For convenience of the controller design，defining x1=x，x2=，the mechanical dynamics of the motor servo system can be transformed into Without loss of generality，two technical assumptions are made to pose the problem in a tractable manner.1）The desired position trajectory yd，the time derivative˙ydand¨ydare both bounded and smooth signals.2）The angular position and velocity，x1and x2，are measurable.2.2 Nonlinear saturation modelClearly，the relationship between the applied control v（t）and thecontrol input u（t）has a sharp corner when |u（t）|=vmax.As shown in Fig.1，the saturation is approximated by a smooth non-affine function defined asThen，v=sat（u）in（2）can be expressed in the following formwhere d1（u）=sat（u）-g（u）is a bounded function and its bound can be obtained aswhere D is the upper bound of|d1（u）|.According to the mean-value theorem［29］，there exists a constant ξ with 0＜ξ＜1，such thatwhereand u0∈［0，u］.By choosing u0=0，（6）can be rewritten in the following affine formSubstituting（7）and（5）into（3），we can obtainwhere2.3 Neural network approximationDue to good capabilities in function approximation，neural networks （NNs）are usually used for the approximation of nonlinear functions.The following neural network with a simple structure and a fast convergence property will be used to approximate the continuous functionwhere W∗∈Rn1×n2is the ideal weight matrix，φ（X）∈Rn1×1is the basis function of the neural network，ε is the neural network approximation error satisfying|ε|≤εN，φ（X）can be chosen as the commonly used sigmoid function，which is in the following formwith a，b，c and d being appropriate parameters.Remark 1 The employed neural network with sigmoid function representsa class of linearly parameterized approximation methods，and can be replaced by any other approximation approaches such as spline functions，RBF functions or fuzzy systems.However，the structure of the employed neural network in the this paper is simpler than the other neural networks that are commonly used in other works.There is no hidden layer in the employed NN，in which five inputs and one output are included and the corresponding weight matrix is 5×1.3.1 Funnel error variableFunnel control is a strategy that employs a timevarying gain ρ（t）to control systems of class S with a relative degree r=1 or 2，stable zero dynamics，and known high-frequency gains.The system S is governed by the funnel controller with the control inputwhere e（t）=y1-ydis the tracking error，ρ（·）denotes the control gain. As shown in Fig.2，evaluate the vertical distance at the actual time between the funnel boundary Fφ（t）and the Eucli dian error norm‖e （t）‖asThe funnel boundary is given by the reciprocal of an arbitrarily chosen bounded，continuous and positive function φ（t）＞0 for all t≥0 with＜∞.The funnel is defined asTo ensure that the error e（t）evolves inside the funnel Fφ（t），theexp ression of ρ（·）can be chosen asFrom（12）we can see that，when the gain ρ（t）increases，the error e （t）approaches the boundary Fφ，and when the gain ρ（t）decreases conversely，the error e（t）becomes small.A proper funnel boundary toprescribe the performance is selected aswhere δ0≥δ∞＞0，inf Fφ（t），and|e（0）|＜Fφ（0）.According to（11）and（12），define a new funnel error variable s1（t）aswhere the funnel boundary Fφ（t）satisfies the condition given in （13）.This variable will be employed to ensure the prescribed output performance.The derivative of（14）iswhereandwhere3.2 Controller designConsidering（15）and（16），the sliding mode manifold is designed as where α＞0.Differentiating s2，we haveSubstituting（8）and（16）into（18）yieldswhere the nonlinear function κ isSince κ is not ea sy to be exactly known，the modelbased controllers cannot be applied directly.Hence，we adopt a neural network（9）to approximate the nonlinear function κ.Assume that there exists a constant ideal weight matrix W∗so that the nonlinear function κ can be expres sed aswhere the input vectorIn the following，a nonsingular terminal sliding mode neural funnel control approach is developed for tracking control of the motor servo system（8）.To force s2converge to zero within a finite time，thenonsingular terminal sliding mode manifold is employed aswhere β＞0，p and q are positive odd integers with p＜q.Substituting（19）into（21）and using（20），the controller is designed aswhere b0is the lower bound of b；p and q are positive odd integers with p ＜q；is the estimate of the ideal weight W∗andµis the upper bound of sum of the neural network approximation error ε andTφ（X），where=W-is the weight estimation error of the neural network.The adaptive law ofis given bywhere K is a positive definite and diagonal matrix，and ν is a positiv e constant.Substituting（22）into（19）yieldsIn this section，a lemma and a theorem is provided to show the boundedness of all signals and the stability of the system（8）inboththereachingphaseandtheslidingphase，respectively.Lemma 1 Assume that there exists a continuous positive definite function V（t）satisfying the following inequality:where n＞0，0＜γ＜1 are constants.Then，for any given t0，V（t）satisfies the following inequality:andwith tsgiven byTheorem 1 Consider the motor servo system（8）with unknown nonlinear saturation（2），nonsingular terminal sliding manifold（21），control law（22），and weight update law（23），then1）All signals of the closed-loop system are bounded.2）The nonsingular terminal sliding manifold s2can converge to zero in finite time by using controllers（22），if the design parameterµ＞εN+‖Tφ（X）‖F.3）The tracking error e will fall into proscribed boundaries.Proof 1）Choose the following Lyapunov function candidatewhere km=FφΦF＞0.Differentiating（26）with respect to time and using（24），we have Substituting（23）into（27）yieldsInequality（27）implies that both s2andare bounded.Meanwhile，considering（17）and the boundedness of W∗，we can conclude s1，˙s1，andare bounded，and thus from（22）and（14），we can obtain u，e andare all bounded.Furthermore，the boundedness of yd，˙ydand¨ydcan lead to the boundedness of s2according to（17）.As a result，and˙s2is bounded due to the boundedness of guξ. Therefore，all signals of the closed loop system are bounded.From（26）-（28），the stability of the system（8）with control laws（22）and weight update law（23）has been proved.However，it is not necessary for the terminal sliding manifold s2to converge to zero in finite time.Therefore，further proof should be given to guarantee that the terminalslidingmanifolds2convergetozeroin finitetime.2）From（29），we can see that the sigmoid function φ（X）is boundedby 0＜φi（X）＜n0，i=1，···，n1， with n0=maxTherefore，φ（X） is bounded bywhere‖·‖denotestheEuclideannormofavector，φ（X）=［φ1（X）φ2（X）···φn1（X）］T.From the property of Forensics norm，it can be obtained thatSelect another Lyapunov function candidateDifferentiating（29）with respect to time and using（24），we have Sinceµ＞εN+‖Tφ（X）‖F，（30）can be rewritten asThen，we can obtainAccording to Lemma 1，it can be concluded that the fast terminal sliding manifold s2can converge to the equilibrium point within a finite timet1given by3）Once the sliding surface s2=0 is achieved.the states of system（8）will remain on it and the system has the invariant properties.On the sliding surface s2=0，we can obtainConstructing the following Lyapunov candidateand differentiating V2along（34），we haveThen，we can conclude that the funnel error s2will converge to the equilibrium point.Thus，from（14），the tracking error e will fall into the prescribed boundaries.Remark 2 Since the discontinuous switching functionsgn（·）shown in （22）may result in the chattering phenomenon，the following continuousΔ（s）is employed instead in the simulation section:whereζis a positive constant defining the thickness of the boundary layer. When|s|≥ζ，the proof can be easily accomplished according to the proof of Theorem 1.When|s|＜ζ，following the proof steps in［30］，we can alsoobtainandActually，there exists a small positive constantϵsuch that ϵ≤（|s|+ζ）.With this modified controller，the chattering can be eliminated and the finite-time convergence is guaranteed in the whole tracking process.In this section，the following three other control approaches are presented for the performance comparison with the proposed NTSMFC scheme.1）PID controlwhere kp=20，ki=0.05，and kd=4.2）Neural-network sliding mode control（SMC）［31］where s1=e，b0=6，α=2，k1=10，andµ=0.1.3）Neural-network sliding mode control（NTSMC）［21］where s1=e，α=2，β=0.2，k1=10，p=5，q=7，b0=6 andµ=0.1.For fair comparison，all control parameters are fixed for various reference signals.The initial states of the system are x1（0）=0，x2（0）=0.The NN parameters are K=0.1，a=2，b=10，c=1，d=-10.The parameters of funnel boundary（13）are chosen as δ0=100，δ∞=0.3 and a0=3.And the control law（22），where α=2，β=0.2，k1=10，p=5，q=7，b0=6 and µ=0.1.The system is select as h=0.2x2sinx2and b=6.The saturation bound vmax=1.In the following，three different cases are performed to compare four different controllers.Case 1 Sinusoidal wave.yd=0.5sint is employed as thereference.Simulation has been conducted，and two comparative resultsare shown in Fig.3.The tracking performance and tracking errors are depicted in Fig.3（a）and Fig.3（b），respectively.AsshowninFig.3（a），whentracking the sinusoidal wave，the NTSMFC and the NTSMC have the comparatively lower overshoot，while the PID control scheme has the largest overshoot.From Fig.3（b），we can see that the NTSMFC has the smallest tracking error and fastest convergence speed；NTSMC has the largest overshoot at the beginning，and the PID scheme has the largest steady tracking error.Case 2 Sinusoidal wave with harmonic.The trackingperformanceofthereferencesignal0.5（sint+sin0.5t）is shown in Fig.4.Among the four schemes，the NTSMFC has the smallest overshoot and tracking error with the fastest convergence speed.The SMC and NTSMC have a large overshoot at the beginning，and PID control has largest tracking error when time goes to the infinity.Obviously，NTSMFC has the best performance when tracking the the reference signal 0.5（sint+sin0.5t）.Case 3 Step signal.To further justify the transient performance（e.g.，overshoot），a step signal with amplitude 1 rad is employed.Control parameters are set the same as those given before.As shown in Fig.5，we can see that PID control has comparatively larger overshoot.Besides，the proposed scheme can converge within 1 s，while PID control costs 1.5 s；SMC and NTSMC cost more than 4 s to achieve the same performance.Therefore，we can conclude that the proposed NTSMFC has the best transient performance.In order to show the comparison performance more convincingly，several indices are provided to evaluate the performance of the four controllers. 1），which is the integrated absolute value of the error to measure intermediate tracking result.2）which is the integral of the time multiplied by the absolute value of the error，and used to measure the tracking performance with time behaving as a factor to emphasize errors occurring late.3）which is the integrated square error and used to demonstrate the smoothness of the profile.The simulation results in terms of performance indices are provided by Tables 1-3.From Tables 1-2，we can see that when tracking sinusoidal waves，the proposed NTSMFC scheme has the smallest IAE，ITAE and ISDE，which means it performs best among four controllers.From Table 3，it can be concluded that when tracking the step signal，although PID controller has the smallest ISDE，it has the larger ITAE and IAE than NTSMFC，while SMC and NTSMC have relatively large IAE，ITAE and ISDE. Therefore，all the aforementioned simulation results clearly show that the proposed NTSMFC scheme can achieve the best tracking performance with respect to tracking errors and convergence speed.In this paper，a nonsingular terminal sliding mode funnel control （NTSMFC）scheme is proposed to achieve a prescribed tracking performance for motor servo systems with unknown input saturation.The non-smooth saturationistransformedintoanaffineformbydefiningasmoothnon-affine function and using the mean-value theorem.A new constraint variable is employed and the tracking error will be forced to fall into prescribe boundaries.By using a simple sigmoid neural network to approximate the unknown system nonlinearity，a nonsingular terminal sliding mode funnel control is developed for the prescribed tracking performance of the motor servo system.With the proposed scheme，no prior knowledge is required on the input saturation bound，and the convergence of the position tracking error is guaranteed via the Lyapunov synthesis.Our further work is to apply the proposed scheme to a practical motor servo system.References:［1］LI S H，LIU Z.Adaptive speed control for permanent-magnet synchronous motor system with variations of load inertia［J］.IEEE Transactions on Industrial Electronics，2009，56（8）:3050-3059.［2］MORAWIEC M.The adaptive backstepping control of permanent magnet synchronous motor supplied by current source inverter［J］. IEEE Transactions on Industrial Informatics，2013，9（2）:1047-1055.［3］LIU G，CHEN L，ZHAO W，et al.Internal model control of permanent magnet synchronous motor using support vector machine generalized inverse［J］.IEEE Transactions on Industrial Informatics，2013，9（2）:890-898.［4］TEE K P，REN B B，GE S S.Control of nonlinear systems with time-varying output constraints［J］.Automatica，2011，47（11）:2511 -2516. ［5］NIU B，ZHAO J.Barrier Lyapunov functions for the output trackingcontrol of constrained nonlinear switched systems［J］.Systems& Control Letters，2013，62（10）:963-971.［6］LI Y，LI T，JING X.Indirect adaptive fuzzy control for input and output constrained nonlinear systems using a barrier Lyapunov function ［J］.International Journal of Adaptive Control and Signal Processing，2014，28（2）:184-199.［7］BECHLIOULIS C P，ROVITHAKIS G A.Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities［J］.IEEE Transactions on Automatic Control，2011，56（9）:2224-2230.［8］NA J.Adaptive prescribed performance control of nonlinear systems with unknown dead zone［J］.International Journal of Adaptive Control and Signal Processing，2013，27（5）:426-446.［9］NA J，CHEN Q，REN X M，et al.Adaptive prescribed performance motion control of servo mechanisms with friction compensation［J］. 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求变系数椭圆形偏微分方程的非齐次柯西问题近似特解的一种正则化方法李明陈文 C. H. Tsai z2011年7月摘要用径向基函数(RBFs)表示函数公认很灵活。

基于模拟方程法的概念和径向基函数。

在本文，首次考虑使用近似特解方法(MAPS)求解变系数椭圆形偏微分方程（PDEs）的不适定柯西问题。

我们证明，使用Tikhonov 正则化，近似特解方法会给椭圆形偏微分方程及不规则解空间带来一个有效的、准确的数值算法。

有效且准确。

比较上面提出的近似特解方法和Kansa 方法，计算结果表明用上面提出的近似特解方法求解不适定柯西问题是有效的、准确的且稳定的。

1 引言根据给定的便接受或者内部的信息，椭圆偏微分方程(PDEs) 大致可以分为以下五个主要问题：1.边界值问题[2，6]。

这些问题已经发展和应用很长一段时间。

例如：Dirichlet, Neumann 或者 Robin 混合问题。

2.Cauchy问题[23]。

这些问题也被称为边值的确定问题。

对于椭圆形Cauchy问题，Dirichlet和Neumann边界条件只在解空间边界的一部分给定。

3.边界的确定问题[11]。

这些是工程上典型的非破坏性测试问题。

这些问题确定解空间的未知边界。

4.系数的重建问题[26]。

5.潜在源的确定问题。

上述问题中。

只有第一个问题是适定的。

适定问题的定义是由Hadamard【7】给出的。

他认为，为某物理现象建立的合适的数学模型应具有以下三个属性：存在，唯一且稳定。

如若其中之一不满足，则建立在物理现象之上的这个数学模型就是不适定的。

然而,越来越多的不适定问题正出现于科学cs,中国香港特别行政区，香港城市大学相关作者，中国南京河海大学工程力学系型的逆问题，从Hadamard角度上来说，他们是不适定的，因为测量数据任何小误差都可能导致解的大差距。

偏微分方程的系数通常对应于问题的材料参数。

在非均匀介质中，材料参数可能会随位置而变化。

因此在非均匀介质的物理问题的控制方程中很可能会涉及到变系数。

Contents:Editorial Board (i)Call for Papers (vi)< PART 1 >Solve Combinatorial Optimization Problem Using Improved Genetic Algorithm (1)Hanmin Liu, Qinghua Wu, Xuesong YanOptimization for the Row Heights in Medium-Length Hole Blasting Design by Genetic Algorithms (9)YANG Zhen, WANG Cheng-Jun, GUO LiAn Improved Differential Evolution with Adaptive Disturbance for Numerical Optimization .. 16 Dan MengA Radial Basis Function based Artificial Immune Recognition System for Classification (24)DENG Ze-lin, TAN Guan-zheng, HE PeiThe Forecasting Algorithm Based on User Access Intention (33)Jun Guo, Fang Liu, Yongming Yan, Bin ZhangPublicly Available Visualization System of Environmental Remote Sensing Information (40)Rustam Rakhimov Igorevich, Yang Dam Eo, Dugki Min, Mu Wook Pyeon, Ki Ho HongExamining Hospitality and Tourism Majors’ Intensions of Entering Hospitality and Tourism Professions Based on Theory of Planned Behavior (51)Ya-Ling Wu, Cheng-Wu Chen, Ya-Hui LiaoResearch Overview of Manifold Learning Algorithm (58)Wei Zhan, Guangming Dai, Hanmin LiuFormal Verification of Process Layer with Petri nets and Z (68)Yang Liu, Jinzhao Wu, Rong Zhao, Hao Yang, Zhiwei ZhangA Robust Text Zero-watermarking Algorithm based on Dependency Parsing (78)Yuling Liu, Ting JiangSimulation Modeling of Inter-firm Financial Contagion Process: a Network Perspective (86)WU BaoNew Cryptanalysis on 6-round Khazad (94)Yonglong TangA New Risk Assess Model for Urban Rail Transit Projects (104)Zhu Xiangdong, Xiao Xiang, Wu ChaoranAnt Colony Algorithm Optimized by Vaccination (111)He Haitao, Xin NingComputer Network Security and Precaution Evaluation based on Incremental Relevance Vector Machine Algorithm and ACO (120)Guangyuan SongRealization of an Embedded and Automated Performance Testing System for a MEMS Transducer (128)LIAO Hai-yang, XIONG Kui, WEN Zhi-yuMulti-level Cache Prediction and Partitioning Mechanism in CMP (135)Shuo Li, Gaochao Xu, Xiaozhong Geng, Xiaolin Qiao, Feng WuNew Classes of Sequences for Encryption Procedures in Symmetric Cryptography (145)Amparo Fuster-SabaterExplore and Analysis of Environmental Policy Based on Green Industry Development (152)Chunhong Zhu, Zhe Liu, Yue Zhou, Xuehua ZhangResearch on Service Encapsulation of Manufacturing Resources Based on SOOA (158)Lingjun Kong, Wensheng Xu, Nan Li, Jianzhong ChaA Twice Ant Colony Algorithm Based on Simulated Annealing for Solving Multi-constraints QoS Unicast Routing (167)Yongteng Lv, Yongshan Liu, Wei Chen, Xuehui Shang, Yuanyuan Han, Chang LiuA Two-phase Multi-Constraint Web Service Selection Approach for Web Service Composition (176)Zhongjun Liang, Hua Zou, Fangchun Yang, Rongheng LinAdvanced Coupled Map Lattice Model for the Cascading Failure on Urban Street Network . 186 ZHENG Li, SONG Rui, Xiao YunTransition Probability Matrix Based Tourists Flow Prediction (194)Yuting Hu, Rong Xie, Wenjun ZhangA Study on the Macro-Control Policy of China Real Estate Development (202)Lu ShiAn Effective Construction of a Class of Hyper-Bent Functions (212)Yu Lou, Feng Zhou, Chunming TangSpeaker-independent Recognition by Using Mel Frequency Cepstrum Coefficient and Multi-dimensional Space Bionic Pattern Recognition (221)Guanglin Xian, Guangming XianIntelligent Decision Support System (IDSS) for Cooling/Heating Sources Scheme Selection of City Buildings Based on AHP Method (228)Liu Ying, Jiang Kun, Jiang ShaModeling of Underwater Distributed Target Based on FDTD and Its Scale Characteristic Extraction (237)P AN Yu-Cheng, SHAO Jie, ZHAO Wei-Song, ZHONG Ya-QinMethod for Dynamic Multiple Attribute Decision Making under Interval Uncertainty and its Application to Supplier Selection (246)Xu JingStudy on Multi-Agent Information Retrieval Based on Concern Domain (254)Sun JianmingA New Method for Solving Numerical Solution of Fractional Differential Equations (263)Jianping Liu, Xia Li, HuiQuan Ma, XueZhi Mao Guoping ZhenAssessment and Analysis of Hierarchical and Progressive Bilingual English Education Based on Neuro-Fuzzy approach (269)Hao Xin< PART 2 >Computer-based Case Simulations in China: 2001-2011 (277)Tianming Zuo, Peng Wang, Baozhi Sun, Jin Shi, Yang Zhang, Hongran BiHybrid Monte Carlo Sampling Implementation of Bayesian Support Vector Machine (284)Zhou Yatong, Li Jin, Liu LongA Face Recognition Method based on PCA and GEP (291)WANG Xue-guang, CHEN Shu-hongResearch and Application of Higher Vocational College Library Personalized Information Service Based on Cloud Computing (298)Meiying Nie, XinJuan Zhou，Qingzhi WenA Calculation Model for the Rover’s Coverage Boundary on the Lunar Surface Based on Elevation (307)Hu Yasi, Meng Xin, Pan Zhongshi, Li Dalin, Liang Junmin, Yang YiAn Evaluation of firms’ Best Strategies with the ANP, AHP and Sensitivity Test Approaches 316 Catherine W. Kuo, Shun-Chiao ChangDesign of Embedded Vehicle Safety Monitoring System (326)Jing-Lian, Lin-Hui Li, Hu-Han, Ya-Fu Zhou, Feng-Hu, Ze-Quan ZengThe Approach to Obtain the Accurate TOA of VHF Lightning Signals Based on FastICA Algorithm (334)Xuquan Chen, Wenguang ZhaoIntegrating Augmented Reality into Consumer’ Tattoo Try-on Experience (341)Wen-Cheng Wang, Hao-Hsiang Ku, Yen-Wu TiResearch on Location for Emergency Logistics Center Based on Node Cost (348)Wang ShouqiangLogistics Terminal Facility Location Model Based on Customer Value in Competitive Environment (354)Han Shuang Wang XiaoxiaA Decision Support Model for Risk Analysis with Interval-valued Intuitionistic Fuzzy Information (362)Guoqing WuPeak to Average Power Ratio Reduction with Bacterial Foraging Algorithm for OFDM Systems (370)Jing Gao, Jinkuan Wang, Bin WangUsing Linear and Nonlinear Inversion Algorithm Combined with Simple Dislocation Model Inversion of Coal Mine Subsidence Mechanism (379)Yu-Feng ZHU，Qin-Wei WU，Tie-Ding Lu, Yan LuoApplication of Multi- media in Education of Schoolgirl’s Public P. E. in College and University—set Popular Aerobics as Example (388)Yuanchao zhouA Fuzzy Control System for Trailers Driven by Multiple Motors in Side Slipway to Launch and Pull Out Ships (395)Nyan Win Aung, Wei HaijunA Novel Image Encryption Algorithm based on Virtual Optical Imaging and Hyper-chaos .. 403 Wei Zhu , Geng Yang, Lei Chen, Zheng-Yu ChenThe Structure Character of Market Sale Price in the Coordinating Supply Chain (412)Jun Tian, Zhichao WangDesign Parameter Analysis for Inducers (419)Wei Li, Weidong Shi, Zhongyong Pan, Xiaoping Jiang, Ling ZhouAutomatic Recognition of Chinese Traffic Police Gesture Based on Max-Covering Scheme .. 428 Fan Guo, Jin Tang, Zixing CaiDetermination of Acrylamide Contents in Fried Potato Chips Based on Colour Measurement 437 Peng He, Xiao-Qing Wan, Zhen zhou, Cheng-Lin WangAn Efficient Method of Secure Startup and Recovery for Linux (446)Lili Wu, Jingchao Liu,Research on Life Signals Detection Based on Parallel Filter Bank and Higher Order Statistics (454)Jian-Jun LiResearch for Enterprise Logistics Dynamic Optimization Based on the Condition of Production Ability Limited (462)Guo QiangTechnological Progress in Macroeconomic Volatility and Employment Impact analysis - Based on Endogenous Labor RBC Model (469)Wang Qine, Hu honghaiResearch on Bottleneck Identification in Multiple Products Small lot Production Logistics of Manufacturing Enterprise Based on TOC (476)Jian Xu, Hongbo WangThe Spiral Driven and Control Method Research of the Pipe Cleaning Robot (484)Quanyu Yu, Jingyuan Yu, Jun Wang, Jie LiuHoisting Equipment of Coal mine Condition Monitoring and Early Warning Based on BP Neural Network (491)Shu-Fang Zhao, Li-Chao ChenSpatial Temporal Index-based Historic Closing Event Query or Moving Objects (497)Xianbiao Ji, Hong Mi, Zheping ShaoResearch on the Risk-sharing Mechanism of Energy Management Contract Project in Building Sector (510)CaiWeiguang, Ren Hong, Qin BeibeiFinancial Crisis and Financial Index Structure Break (516)Keng-Hsin Lo, Yen-Chang Chen, Yi -Wei ChuangThe Research of Cooling System for the High-Energy Storage Flywheel (522)Wang Wan, He LinNumerical Models and Seismic Design of Steel Frames Equipped with Supplemental Fluid Viscous Devices (528)Marco ValenteStudy on Positive and Dynamic Enterprise Crisis Management based on Sustainable Business Model Innovation (535)Shi-chang Fu, Hui-fen Wang, Dalen ChiangQuantitatively Study on the Mechanism of Cooperating Profit Distribution within Business Ecosystem (544)Bin HuStudy on the Landscape Design of Urban Commemorative Squares Based on Sustainable Development (552)Wenting Wu, Ying Li, Yi Ren< PART 3 >SRPMS: A web-based Project Management System for Scientific Research (559)Yanbao Ji, Xiaopeng Yun, Zhao Jun, Quanjiang Bai, Lingwang GaoEmpirical Study on Influence Factors of Carbon Dioxide Emissions in Liaoning Province based on PLS (567)Yu-xi Jiang, Su-yan He, Xiang-chao WeiHarmony Factor Considered Evaluation of Science Popularization Talents Based on Grey Relational Analysis Model (575)Li MingStudy on the Safety of High-Speed Trains under Crosswind (582)Xian-Liang Sun, Bin-Jie Wang, Ming Gong, San-San Ding, Ai-Qing TianControl Method of Giant Magnetostrictive Precise Actuator Based on the Preisach Hysteresis Theory (589)Yu Zhang, Huifang Liu, Feng SunDynamic Modeling and Characteristics Analysis of Rolls along Axial Direction for Four High Mill Based on Timoshenko Theory (602)Jian-Liang Sun, Yan PengConstruct on Maintenance Requirement Analysis Model of Pavement Management System 610 Xiu-shan Wang, Yun-fang YangAutomatically Generate Test Data Based on Intelligent Algorithms Method (617)Jian Ni, Ning-NingYangAnalysis of the Functions of a High-Speed Railway Station in China (623)Li-Juan Wang, Tian-Wei Zhang, Fan Wang, Qing-Dong ZhouEconometric Analysis of Expectation in Savings-to-Investment in Capital Market Converting Process (630)Wang Yantao , Yu Lihua ,Mao BeibeiAnalysis Model and Empirical Research on Product Innovation Process of Manufacturing Enterprises Based on Entropy-Topsis Method (637)Hang Yin, Bai-Zhou Li, Tao Guo, Jian-Xin ZhuThe Market Analysis and Prediction of Chinese Iron and Steel Industry (645)Li Xiaohan, Sun Qiubai , Li HuaSystem Dynamics Mode Construction and System Simulation in the Product Innovation Process (653)Jian-Xin Zhu, Jun DuResearch on Organization Innovation of Enterprise Based on Complexity Theory (664)Yu Zheng, Tao GuoThe Study on TFP of Iron and Steel Industry inChina Based on DEA-Malmquist Productivity Index Model (672)Xiaodong Dong, Yuzhi ShenResearch and Implementation of Energy Balance Control System in Metallurgy Industry (681)Qiu DongA Study of Opportunities and Threats in the Implementation of International Marketing for Production Design of Corporate Brand Licensing – A Case Study of POP 3D Co., Ltd. (689)Min-Wei Hsu，Tsai-Yun Lo，Liang,K.C.Research on Risk Forming Mechanism and Comprehensive Evaluation of the Enterprise Group (695)Dayong XUMode Construction of Dining Reform in Universities Based on Theory of Institutional Transformation (704)Li PingjinResearch on QR Decomposition and Algorithm of Linguistic Judgment Matrix (711)Lu YuanA Comparison of the Mahalanobis-Taguchi System to A Selective Naïve Bayesian Algorithm for Semiconductor Chemical Vapor Deposition Process (720)Jui-Chin Jiang, Tai-Ying LinStudy of Policy-making Model for Producer Service: Empirical Research in Harbin (730)Xin Xu, Yunlong DingKrein space H∞ filtering for initial alignment of SINS with large azimuth misalignment (738)Jin Feng, Fei Yu, 3Meikui Zou, Heming JiaInternet Word-Of-Mouth on Consumer Online Purchasing Behavior Analysis in China (747)Jie Gao, Weiling YeConvex Relaxation for Array Gain/Phase Calibration in ULAs and UCAs with Unknown Mutual Coupling (758)Shu CaiFinancial-Industrial Integration Risk Management Model of Listed Companies Base on Logistic (767)Ke WenBehavior Equilibrium Analysis for The Cross-Organizational Business Process Reengineering in Supply Chain (775)Jianfeng Li, Yan ChenA Secure Scheme with Precoding Approach in Wireless Sensor Networks (782)Bin Wang, Xiao Wang, Wangmei GuoA New Method on Fault Line Detection for Distribution Network (789)Bo LiManagers’ Power and Earnings Manipulating Preference (796)J. Sun, X. F. Ju, Y. M. Peng, Y. ChangStudy and Application of the Consistency of Distributed Heterogeneous Database Based on Mobile Agent (804)Zhongchun Fang, Hairong Li, Xuyan Tu。

目录1、电力设计基本术语2、给水排水设计基本术语3、水泵专业英语词汇4、阀门种类英汉术语对照5、阀门专用英语词汇6、照明术语7、工程结构设计基本术语电力设计基本术语abrasion-Proof component of burner 燃烧器耐磨件arm-brace 撑脚ash conditoner 调灰器basket removal panel 元件盒检修护板BDV blow down valve 疏水阀，排污阀blind 堵板blind flange 法兰堵板/盲板法兰（盖calling 催交campell diagram 叶片埃贝尔曲线dado 墙裙daily service fuel tank level switch 日用油缸液位掣damage 损毁damper 挡板damper linkage 风闸联动装置damper motor 风闸马达damping mat 阻尼垫dangerous earth potential 危险性对地电势dashpot 减震器data transmission 数据传输DC/AC converter 直流电／交流电转换器dead 不带电dead weight 自重decanter 沉淀分取器declaration of conformity 符合标准声明decommissioning 解除运作；停止运作decompression chamber 减压室decorative lighting 装饰照明；灯饰deep bore well pump 深钻井泵defect liability period 故障修理责任期；保用期defectograph 钢缆探伤仪；故障检查仪defence in depth 纵深防御definite sequence 固定次序deflection 偏转；挠度deflector sheave 折向轮；导向轮defrost timer 防霜时间掣defrost unit 溶雪组合dehumidifier 抽湿机deleterious substance 有害物质delivery and return air temperature 送风及回风温度delivery connection 出油接头delivery pressure 输出压力demand side management 用电需求管理demand side management agreement 用电需求管理协议demand side management programme 用电需求管理计划dent 凹痕dental instrument 牙科仪器dental scaler 洗牙具Departmental Administration Division [Electrical and Mechanical Services Department] 行政部〔机电工程署〕Departmental Safety Unit [Electrical and Mechanical Services Department] 部门安全组〔机电工程署〕deposition 沉积物depth measuring facility 深度测量装置derating factor 额定值降低因子derust 除锈descale 清除氧化皮design current 设计电流design parameter 设计参数designated employee 指定雇员detachable grip 可拆除的夹扣Details of Branch Offices of Registered Electrical Contractors 注册电业承办商分行详情申报deterioration 变质；变坏Deutsche Industrie Normen [DIN] 德国工业标准device 器件；装置dewatering 脱水；排水diaphragm 膜片；隔板dielectric strength test 电介质强度测试diesel fuel tank 柴油燃料缸diesel oil 柴油differential gasket 差速器衬垫differential lock 差速器锁differential oil 差速器机油diffuser 透光罩；扩散器dilute 稀释dim sum trolley 点心手推车dim transformer 光暗变压器diminution of value 减值dimmer 调光器；光暗掣；光暗器dip tube 液位探测管Diploma in Electrical Engineering 电机工程学文凭dipstick 量油尺direct current [DC] 直流电direct current control 直流控制direct current electric drive 直流电电力驱动direct current reactor 直流电抗器direct drive 直接驱动direct purging 直接驱气direct-acting lift 直接驱动升降机direct-fired vaporizer 明火直热式汽化器direction arrow 方向箭头direction arrow plate 方向指示板direction indicator 方向指示器Director of Electrical and Mechanical Services 机电工程署署长Directory of Accredited Laboratories 认可实验所名册Directory of Quality System Registration Bodies 品质系统注册团体指南disassemble 拆散discharge 放电；卸载discharge lamp 放电灯；放电管discharge lighting 放电照明设施discharge of electricity 释电；放电discharge valve 排水阀disciplinary board 纪律审裁委员会disciplinary board panel 纪律审裁委员团disciplinary tribunal 纪律审裁小组disciplinary tribunal panel 纪律审裁委员团；纪律审裁委员会discolouring 变色disconnection 截断；截离steam hamerring analysis 汽锤分析steam packing unloading valve 汽封卸载阀steam purity 蒸汽纯度steam seal diverting valve 汽封分流阀steam seal feed valve 汽封给水阀steam water mixture 汽水混合物steel bar 扁钢steel supporting 钢支架steel wire brush 钢丝轮steel works 钢结构step load change 负荷阶跃still air 蒸馏气体stirrup 镫形夹stoikiometric ratio 化学当量比stopper 制动器、塞子storage vessell 贮水箱stppage alarm 停转报警stranded copper cable 铜绞线电缆strength 强度strong backs 支撑stud bolt 柱头螺栓、双头螺栓sub cooling line 欠热管submerged arc welding 埋弧焊substation 配电装置substation island 电气岛superficial corrosion 表面腐蚀superheat 过热度supersaturation 过饱和supervisory instrument 监测装置supply transformer 供电变压器support trunnion 支撑端轴surfactant 表面活性剂surge 喘振suspended diode 中断二极管suspended particles 悬浮颗粒switch board 开关柜switch gear 开关柜sychronization 并网sychroscope 同步指示器、同步示波器T square 丁字尺T/G transformer 发变组tackling system 起吊系统tamped/compacted backfill 夯实回填土tanks and accessories 箱罐和附件taper land thrust bearing 斜面式推力轴承tar epoxy paint 柏油环氧漆tarpaulin 防水布temperature digital display meter 温度数显表tensile test 拉伸试验tension test 拉伸试验，张力试验tensioning rod 拉杆terminal box 接线盒terminal poit 接口termination flange 接口法兰tertiary air 三次风test connection 试验接头test permition 试验合格the expansion coordinate system 热膨胀系统theodilite\transit instrument 经纬仪thermal insulatiion for tuebine casing 汽缸保温thermo resistor 热电阻thermostat 恒温器、恒温调节器thinner 稀释剂threaded flange 螺纹法兰throudh type 直通式、穿入式through bolt 贯穿螺栓、双头螺栓thrust plate 推力板tier tube 间隔管tilting pad 可倾瓦块tilting pad bearing 可倾瓦块轴承tip shroud 围带、环形叶栅外柱面tip speed 叶顶速度toe board/plate (kick plate) 踢脚板top crown plate seal 高冠板式密封装置top girder 顶板top penthouse 顶部雨棚top plan view 俯视图torquemeter 扭矩测量仪totalnumber of welding 焊口总数trajectory 轨道、轨迹transducer board 变送器屏transfer pipe 引出管transition piece 过渡连接件transtion piece 过渡段transverse strength 弯曲强度、抗挠强度transverse stress 横向应力、弯曲应力transverse test 抗弯试验trapezoid corrugated plate seperater 梯形波形板分离器、顶帽travelling crab 小车起重机travelling hoist 移动卷扬机tread width 踏步宽度trestle 组合支架trim and grind the welding 修磨焊点trisector air preheater 三分仓空预器trunk cable pair 主电缆对trunnion air seal assembly 端轴空气密封tube exchanger 管式热交换器tubing stress analysis 管系应力分析turbidity analyser 浊度分析仪turbine lube oil and conditioning system 汽机润滑油及净化系统turning oil 循环油twisted pair conveyer 双绞线传送器undercut 坡口underflow 地流、潜流、下溢union 活接头、管节unit control 单元控制unloadding spout vent fan 卸料口通风风机unloading valve 卸载阀urgent need equipment 急需设备urgtented need equipment 急需设备u-shape hanger chains u形曲链片吊挂装置UT ultrasonic testing 超声波探伤UTS ultimate tensible strength 极限抗拉强度vacuum belt filter 皮带真空吸滤器valve opening chart at load rejection 甩负荷阀门开启阀valve seat body seat 阀座valve spindle 阀轴、阀杆valve stem 阀杆vapor proof 防水灯variable inlet guide vane centrifugal fan 进口可调导叶离心式风机variable moning blade axial flow fan 动叶可调轴流式风机variable moving blade double stage axial fan 动叶可调双级轴流式风机variable speed driver 变速马达variables 变量vent capacity 排放量vent line 放气管ventilator valve 通风阀vernier caliper 游标卡尺vertical deflection 垂直挠度vertical movement 垂直位移vertical spindle coal pulveriser 立式磨煤机vibration isolation 隔振装置viewing lamp 观察指示灯viscosity 粘滞度、内摩擦viscous fluid 粘性液体visual examination of coating 外观质量vlve body 阀体void 无效volatily 挥发分voltage class 电压等级vortex gasket 涡流垫片wall type and retractable soot blower 墙式、伸缩式吹灰器warm air curtain 热风幕rwarming line 加热管water balance 水平衡water induction prevent control 防进水控制water level gauge 水位计water stop flange 止水法兰water supply facility island 水工岛wear hardness 可抗磨能力wear template 防磨板wearing bush 防磨套wearing plate 防磨板、护板weigh feeder 重量计量进料器weld bolt 焊接螺栓weld contamination 焊接杂质weld groove 焊缝坡口weld pass 焊道weld penetration 熔深weld preparation 焊缝坡口加工weld with shop beveled ends 工厂加工坡口焊接welder helment 面罩welding line 焊缝welding plate flange 焊接板式法兰welding rod 焊条welding rods dryer barrel 焊条保温筒welding run 焊道welding seam 对接焊缝welding technological properties 焊接工艺性能welding tool 电焊钳welding torch 焊枪welding wire 焊丝welds counting quantity 焊口统计数量wellington boot 防水长统靴whirl plate 折流板wide column 宽立柱winding resistance 绕组电阻wire feed speed 送丝速度wire netting/metal mesh 铁丝网wire wool 擦洗用的）钢丝绒，百洁丝withstand voltage test 耐压试验working medium 工质worm hole （焊缝）条虫状气孔yield strength 屈服强度yoke 磁轭、人孔压板、座架联板firproof paint 防火漆manifold valve 汇集阀saw trace 锯痕tapping point 取样点bushing current transformer 套管式电流互感器light gauge plate/sheet 薄钢板notch 槽口、凹口holding strip 压板straight edge 校正装置trailing edge 后缘lance 喷枪lighting off 点火gaseous fuel 气体燃料entrain 夹带、传输combustion air 助燃风hot stand by 热备用行波travelling wave模糊神经网络fuzzy-neural network神经网络neural network模糊控制fuzzy control研究方向research direction副教授associate professor电力系统the electrical power system大容量发电机组large capacity generating set输电距离electricity transmission超高压输电线supervltage transmission power line 投运commissioning行波保护Traveling wave protection自适应控制方法adaptive control process动作速度speed of action行波信号travelling wave signal测量信号measurement signal暂态分量transient state component非线性系统nonlinear system高精度high accuracy自学习功能selflearning function抗干扰能力antijamming capability自适应系统adaptive system行波继电器travelling wave relay输电线路故障transmission line malfunction仿真simulation算法algorithm电位electric potential短路故障short trouble子系统subsystem大小相等，方向相反equal and opposite in direction 电压源voltage source故障点trouble spot等效于equivalent暂态行波transient state travelling wave偏移量side-play mount电压electric voltage附加系统add-ons system波形waveform工频power frequency延迟变换delayed transformation延迟时间delay time减法运算subtraction相减运算additive operation求和器summator模糊规则fuzzy rule参数值parameter values可靠动作action message等值波阻抗equivalent value wave impedance附加网络additional network修改的modified反传算法backpropagation algorithm隶属函数membership function模糊规则fuzzy rule模糊推理fuzzy reasoning样本集合sample set给定的given模糊推理矩阵fuzzy reasoning matrix采样周期sampling period三角形隶属度函数Triangle-shape grade of membership function 负荷状态load conditions区内故障troubles inside the sample space门槛值threshold level采样频率sampling frequency全面地all sidedly样本空间sample space误动作malfunction保护特性protection feature仿真数据simulation data灵敏性sensitivity小波变换wavelet transformation神经元neuron谐波电流harmonic current电力系统自动化power system automation继电保护relaying protection中国电力China Power学报journal初探primary exploration标准的机组数据显示(Standard Measurement And Display Data) 负载电流百分比显示Percentage of Current load(%)单相/三相电压Voltage by One/Three Phase (Volt.)每相电流Current by Phase (AMP)千伏安Apparent Power (KVA)中线电流Neutral Current (N Amp)功率因数Power Factor (PF)频率Frequency(HZ)千瓦Active Power (KW)千阀Reactive Power (KVAr)最高/低电压及电流Max/Min. Current and Voltage输出千瓦/兆瓦小时Output kWh/MWh运行转速Running RPM机组运行正常Normal Running超速故障停机Overspeed Shutdowns低油压故障停机Low Oil Pressure Shutdowns高水温故障停机High Coolant Temperature Shutdowns起动失败停机Fail to Start Shutdowns冷却水温度表Coolant Temperature Gauge机油油压表Oil Pressure Gauge电瓶电压表Battery Voltage Meter机组运行小时表Genset Running Hour Meter怠速-快速运行选择键Idle Run – Normal Run Selector Switch运行-停机-摇控启动选择键Local Run-Stop-Remote Starting Selector Switch其它故障显示及输入Other Common Fault Alarm Display and input给水排水设计基本术语一、通用术语给水排水工程的通用术语及其涵义应符合下列规定：1、给水工程water supply engineering 原水的取集和处理以及成品水输配的工程。

Adaptive Dynamic Surface Control for Uncertain Nonlinear Systems With Interval Type-2FuzzyNeural NetworksYeong-Hwa Chang,Senior Member,IEEE,and Wei-Shou ChanAbstract—This paper presents a new robust adaptive control method for a class of nonlinear systems subject to uncertainties. The proposed approach is based on an adaptive dynamic sur-face control,where the system uncertainties are approximately modeled by interval type-2fuzzy neural networks.In this paper, the robust stability of the closed-loop system is guaranteed by the Lyapunov theorem,and all error signals are shown to be uniformly ultimately bounded.In addition to simulations,the proposed method is applied to a real ball-and-beam system for performance evaluations.To highlight the system robustness, different initial settings of ball-and-beam parameters are con-sidered.Simulation and experimental results indicate that the proposed control scheme has superior responses,compared to conventional dynamic surface control.Index Terms—Ball-and-beam system,dynamic surface control, interval type-2fuzzy neural network.I.IntroductionN ONLINEAR phenomena,popularly existing in physical systems,have attracted a lot of attention in both industry and academia.The design of stabilizing controller for a class of nonlinear systems is usually full of challenges,especially for the systems with uncertainties.To achieve desired control goals,several techniques have been developed for uncertain nonlinear systems.In[1],a fuzzy hyperbolic model,consisting of fuzzy,neural network,and linear models,was proposed for a class of complex systems.In[2],the guaranteed cost control problem was presented for uncertain stochastic T-S fuzzy systems with multiple time delays.Recently,type-2fuzzy logic has been developed to deal with uncertain information in intelligent control systems[3]–[5].The concept of type-2fuzzy sets(T2FSs)is an extension of type-1fuzzy sets (T1FSs).Type-2fuzzy techniques with uncertainties in the antecedent and/or consequent membership functions have at-tracted much interest.The membership functions of T2FSs are Manuscript received June23,2012;revised December5,2012;accepted March4,2013.Date of publication April19,2013;date of current version January13,2014.This work was supported in part by the National Science Council,Taiwan,under Grant NSC101-2221-E-182-041-MY2,and by the High Speed Intelligent Communication Research Center,Chang Gung Uni-versity,Taiwan.This paper was recommended by Associate Editor H.Zhang. The authors are with the Department of Electrical Engineering, Chang Gung University,Kwei-Shan,Tao-Yuan333,Taiwan(e-mail: yhchang@.tw;d9721002@.tw).Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TCYB.2013.22535483-D,where a footprint of uncertainty is particularly addressed. The primary membership function grade in type-2can be a subset in the universe[0,1].Moreover,corresponding to each primary membership,there is a secondary membership,which is also in[0,1].T2FSs provide additional degrees of freedom such that modelling and minimizing the effects of uncertainties become more feasible.However,the same works usually cannot be executed by type-1fuzzy logic systems(T1FLSs) [6],[7].Although the types of membership functions are different,the structures of type-1fuzzy logic control(T1FLC) and type-2fuzzy logic control(T2FLC)are similar,in which a fuzzifier,a rule base,a fuzzy inference engine,and an output processor are included.In particular,the output processor of a T2FLC has a type reducer.The outcome of a type reducer is a type-1fuzzy set,and then a crisp output can be obtained through a defuzzifier.However,heavy computational loads would be the main concern with regard to T2FLSs[8].To simplify the computation complexity,the grades of secondary membership functions can be simply set to1,which becomes the so-called interval T2FLSs(IT2FLCs).IT2FLCs have been successfully applied in many control applications due to the capability of modeling uncertainties.In[9],an ant optimized fuzzy controller is applied to a wheeled-robot for wall-following under a reinforcement learning,where the interval type-2fuzzy sets are adopted in the antecedent part of each fuzzy rule to improve robustness.Recently,interval type-2 fuzzy neural networks(IT2FNN),combining the capability of interval type-2fuzzy reasoning to handle uncertain information and the capability of artificial neural networks to learn from processes,have been popularly addressed.In[10],an IT2FNN control system was designed for the motion control of a linear ultrasonic motor based moving stage.The proposed controller combines the merits of FLS and neural networks,and the update laws of the network parameters are obtained based on the backpropagation algorithm.However,the stability of the IT2FNN control system cannot be guaranteed.As a breakthrough in the area of nonlinear control,a system-atic and recursive design procedure,named the backstepping control,has been provided to design a stabilizing controller for a class of strict-feedback nonlinear systems.The backstepping control is basically attributed to the stabilization consideration of subsystems using the Lyapunov stability theory.A number of applications have been addressed in the utilization of backstepping techniques.In[11],the problem of a one degree-2168-2267c 2013IEEEFig.1.Structure of IT2FNN.of-freedom master-slave teleoperated device was discussed by the adaptive backstepping haptic control approach.The adaptive backstepping control was also applied to a plane-type 3-DOF precision positioning table,in which the unmodeled nonlinear effects and parameter uncertainties were considered [12].In[13],a backstepping controller design was investigated for the maneuvering of helicopters to autonomously track predefined trajectories.In[14],a voltage-controlled active magnetic bearing system was addressed,where a robust con-troller was designed to overcome unmodeled dynamics and parametric uncertainties by backstepping techniques.In[15], a backstepping oriented nonlinear controller was proposed to improve the shift quality of vehicles with clutch-to-clutch gearshifts.In[16],theflocking of multiple nonholonomic wheeled robots was studied,where distributed controllers were proposed with the backstepping techniques,graph theory and singular perturbation theory.The position control of a highly-nonlinear electrohydraulic system was considered,where an indirect adaptive backstepping was utilized to deal with param-eter variations[17].In[18],a neural network and backstepping output feedback controller was proposed for leader-follower-based multirobot formation control.In[19],an adaptive back-stepping approach was presented for the trajectory tracking problem of a closed-chain robot,where radial basis function neural networks were used to compensate complicated non-linear terms.In[20],an adaptive neural network backstepping output-feedback control was presented for a class of uncertain stochastic nonlinear strict-feedback systems with time-vary delays.Although backstepping control can be applied to a wide range of systems,however,there is a substantial drawback of the explosion of terms with conventional backstepping techniques.The dynamic surface control(DSC)has been proposed to overcome the“explosion of terms"problem, caused by repeated differentiations.Similar to the backstep-ping approach,DSC is also an iteratively systematic design procedure.The complexity arisen from the explosion terms in conventional backstepping control methods can be avoided by introducingfirst-order low passfilters[21]–[23].In[24], an integrator-backstepping-based DSC method for a two-axis piezoelectric micropositioning stage was proposed,where the robustness of tracking performance can be improved.In[25],a feedback linearization strategy including the dynamic surface control was developed to improve paper handling performance subject to noises.In[26]and[27],an adaptive DSC was adopted for mobile robots,where the exact parameters of robot kinematics and dynamics including actuator dynamics were assumed to be unknown a priori.In[28],a T-S fuzzy model based adaptive DSC was proposed for the balance control of a ball-and-beam system.Based on the T-S fuzzy modelling,the dynamic model of the ball-and-beam system was formulated as a strict feedback form with modelling errors.An observer-based adaptive robust controller was developed via DSC tech-niques for the purpose of achieving high-performance for servo mechanisms with unmeasurable states[29].A robust adaptive DSC approach was presented for a class of strict-feedback single-input-single-output nonlinear systems,where radial-basis-function neural networks were used to approximate those unknown system functions[30],[31].In[32],a novel-function approximator was constructed by combining a fuzzy-logic system with a Fourier series expansion to approximate the un-known system function depending nonlinearly on the periodic disturbances.Based on this approximator,an adaptive DSC was proposed for strict-feedback and periodically time-varyingCHANG AND CHAN:ADAPTIVE DYNAMIC SURFACE CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS295systems with unknown control-gain functions.In[33]and [34],an adaptive fuzzy backstepping dynamic surface control approach was developed for a class of multiple-input multiple-output nonlinear systems and uncertain stochastic nonlinear strict-feedback systems,respectively.In[35],an adaptive DSC was proposed for a class of stochastic nonlinear systems with the standard output-feedback form,where neural networks were used to approximate the unknown system functions. In[36],a neural-network-based decentralized adaptive DSC control design method was proposed for a class of large-scale stochastic nonlinear systems,where neural network method was used to approximate the unknown nonlinear functions. In this paper,a robust adaptive control method is presented for a class of uncertain nonlinear systems.The proposed con-trol scheme is based on adaptive DSC techniques combining with IT2FNN learning(IT2FNNADSC).To achieve a desired tracking goal,a stabilizing control action can be systematically derived.Also,the uncertainty modelling can be attained using IT2FNNs.The IT2FNN has the ability to accurately approxi-mate nonlinear systems or functions with uncertainties based on the merits of both the type-2fuzzy logic and the neural network.From the Lyapunov stability theorem,it is shown that the closed-loop system is stable,and all error signals are uniformly ultimately bounded.In addition,a ball-and-beam system is utilized for performance validations,including both simulation and experimental results.Moreover,the form of strict-feedback systems is a special case of addressed general nonlinear systems.Thus,the proposed IT2FNNADSC control method can be easily applied to many applications of interest. The organization of this paper is as follows.In Section II, the dynamic characteristics of uncertain nonlinear systems are discussed.The structure and design philosophy of IT2FNNs and the design procedures of the adaptive DSC were intro-duced in Section III.The stability analysis is investigated in Section IV.In Section V,a ball-and-beam system is applied for performance validations,where both simulation and ex-perimental results are included.The concluding remarks are given in Section VI.II.PreliminariesA.Problem FormulationConsider a class of nonlinear systems as follows.˙x1=f1(x)+g1(x1)x2˙x2=f2(x)+g2(x1,x2)x3...˙x i=f i(x)+g i(x1,x2,...,x i)x i+1...˙x n=f n(x)+g n(x)u(t)(1)where i=1,2,...,n−1,x=[x1x2···x n]T∈R n is the state vector,u(t)∈R is the control input.The f i,g i, i=1,2,...,n are smooth system and virtual control-gain functions,respectively.In this paper,the state vector x is assumed to be in a compact set in R n.Definition1:System(1)is uniformly ultimately bounded with ultimate bound b if there exist positive constants band Fig.2.Interval type-2membership function with an uncertain mean. c,independent of t0≥0,and for every a∈(0,c),there is T=T(a,b)≥0,independent of t0such that[37]x(t0) ≤a⇒ x(t) ≤b,∀t≥t0+T.(2) With the consideration of system uncertainties,the perturbed model of(1)can be represented in˙x1=f1(x)+g1(x1)x2+ 1(x,t)˙x2=f2(x)+g2(x1,x2)x3+ 2(x,t)...˙x i=f i(x)+g i(x1,x2,...,x i)x i+1+ i(x,t)...˙x n=f n(x)+g n(x)u(t)+ n(x,t)(3)where i=1,2,...,n−1.The i(x,t),i=1,2,...,n are bounded uncertainties,consisting of unknown modelling errors,parameter uncertainties,and/or external disturbances. Denote x1d(t)as a control goal of x1(t).The control objective is to develop an adaptive controller such that the corresponding closed-loop system of(3)is stable,all errors are uniformly ultimately bounded,and the magnitude of the tracking error x1(t)−x1d(t)can be arbitrary small as t→∞. Assumption1:There exist some constantsγi>0,ψi>0, i=1,2,...,n such thatψi≤ g i(x1,x2,...,x i) ≤γi.(4) Assumption2:The reference command x1d(t)is a continu-ous and bounded function of degree2such that x1d,˙x1d,and ¨x1d are well defined.Remark1:In particular casesf i(x)=f i(x1,x2,...,x i),i=1,2,...,n(5) the representation of(1)turns out to be a class of nonlinear systems with a general strict-feedback form.B.Interval Type-2Fuzzy Neural NetworksRecently,interval type-2fuzzy neural networks handling uncertainty problems have been popularly addressed.The design procedures of IT2FNNs will be introduced in this section.First,the IF-THEN rules for an IT2FNN can be expressed asR j:IF x(1)1is˜F1j AND···AND x(1)n is˜F njTHEN y(5)is[w(4)jL w(4)jR],j=1,2,...,m(6)296IEEE TRANSACTIONS ON CYBERNETICS,VOL.44,NO.2,FEBRUARY2014 where˜F ij is an interval type-2fuzzy set of the antecedentpart,[w(4)jL w(4)jR]is a weighting interval set of the consequentpart,and x(1)i is the input of IT2FNN,i=1,2,...,n.Thenetwork structure of IT2FNN is shown in Fig.1,in whichthe superscript is used to identify the layer number.Thefunctionality of each layer of the IT2FNN will be introducedin sequence.1)Input Layer:In this layer,x(1)1,...,x(1)n,denoted as thestate variables of(3),are the inputs of IT2FNN.2)Membership Layer:In this layer,each node performsits work as an interval type-2fuzzy membership function.Asshown in Fig.2,the associated Gaussian membership functionregarding the i th input and j th nodes is represented asμ(2)˜F ij (x(1)i)=exp⎛⎝−12x(1)i−m(2)ijσ(2)ij2⎞⎠≡N(m(2)ij,σ(2)ij,x(1)i)(7)where m(2)ij∈[m(2)ijL m(2)ijR]is an uncertain mean,andσ(2)ij is a variance,i=1,...,n,j=1,...,m.In Fig.2,the upper mem-bership function(UMF)and the lower membership function(LMF)are denoted as¯μ(2)˜F ij (x(1)i)andμ(2)˜F ij(x(1)i),respectively.The upper and lower bounds of each type-2Gaussian membership function can be respectively represented as¯μ(2)˜F ij (x(1)i)=⎧⎪⎨⎪⎩N(m(2)ijL,σ(2)ij,x(1)i),x(1)i<m(2)ijL1,m(2)ijL≤x(1)i≤m(2)ijRN(m(2)ijR,σ(2)ij,x(1)i),x(1)i>m(2)ijR,(8)μ(2)˜F ij (x(1)i)=⎧⎨⎩N(m(2)ijL,σ(2)ij,x(1)i),x(1)i>m(2)ijL+m(2)ijR2N(m(2)ijR,σ(2)ij,x(1)i),x(1)i≤m(2)ijL+m(2)ijR2.(9)Therefore,the outputs of membership layer can be describedas a collection of intervals[μ(2)˜F ij (x(1)i)¯μ(2)˜F ij(x(1)i)],i=1,...,n,j=1,...,m.3)Rule Layer:The operation of this layer is to multiply the input signals and output the result of product.For example, the output of the j th node in this layer is calculated as[f(3)j ¯f(3)j]=n i=1μ(2)˜F ij(x(1)i) n i=1¯μ(2)˜F ij(x(1)i).(10)4)Type-Reduction Layer:Type-reducer has been applied to reduce a type-2fuzzy set to a type-reduced set[6],[38]. This type-reduced set is then defuzzified to derive a crisp output.To simplify the computation complexity,singleton output fuzzy sets with the center-of-sets type reduction method are considered such that the output y(4)=[y(4)L y(4)R]is an interval type-1set.Then,the output of the type-reduction layercan be expressed asy(4)= mj=1f(3)j w(4)jmj=1f(3)j(11)where w(4)j=[w(4)jL w(4)jR]is consequent interval weighting andf(3)j=[f(3)j ¯f(3)j]is the interval degree offiring.Withoutloss of generality,w(4)jR and w(4)jL are assumed to be assigned in ascending order,i.e.,w(4)1L≤w(4)2L≤···≤w(4)mL and w(4)1R≤w(4)2R≤···≤w(4)mR.With reference to[39]and[40],the calculation of y(4)L and y(4)R can be derived asy(4)L=lj=1¯f(3)jw(4)jL+mj=l+1f(3)jw(4)jLj=1f j+j=l+1f(3)j,(12) y(4)R=rj=1f(3)jw(4)jR+mj=r+1¯f(3)jw(4)jRrj=1f(3)j+mj=r+1¯f(3)j.(13)Obviously,the numbers l and r in(12)and(13)are essential in the switching between the lowerfiring strength f(3)jand the upperfiring strength¯f(3)j.In general,the values of l and r can be obtained from a sorting process.In[9],a simplified type of reduction was proposed as follows:ˆy(4)L=mj=1f(3)jw(4)jLmj=1f(3)j≡W T L F(14)ˆy(4)R=mj=1¯f(3)jw(4)jRmj=1¯f(3)j≡W T R¯F(15)where W L and W R are the weighting vectors connected between the rule layer and type-reduction layer,W T L=w(4)1L w(4)2L···w(4)mL,W T R=w(4)1R w(4)2R···w(4)mR;F and¯F are the weightedfiring strength vectorF=f(3)1mj=1f(3)jf(3)2mj=1f(3)j···f(3)mmj=1f(3)jT,¯F=¯f(3)1mj=1¯f(3)j¯f(3)2mj=1¯f(3)j···¯f(3)mmj=1¯f(3)jT.It is noted that only the lower and upper extremefiring-strengths are used in(14)and(15),respectively,such that the computation burden can be reduced.5)Output Layer:The output y(5)is determined as the average ofˆy(4)L andˆy(4)Ry(5)=ˆy(4)L+ˆy(4)R2.(16) Remark2:The results obtained so far are mainly on the structure description of IT2FNNs.It is more interesting to investigate how to model the uncertain terms in(3)with IT2FNNs.Moreover,the closed-loop stability affected by the approximation errors will be addressed later on.III.Design of IT2FNNADSCIn this paper,the dynamic surface control method is utilized to design a stabilizing controller for a class of uncertain nonlinear systems of(3).With IT2FNNs,uncertainty terms of(3)can be approximately modeled asi(x,t)=12W T i F i+εi(17) where W T i=W T iL W T iRis an ideal constant weights vector, F i=F T i¯F T iTis thefiring strength vector,andεi is the approximation error,i=1,2,...,n.In(17),there exist some unknown bounded constants W M andεM such that W i ≤CHANG AND CHAN:ADAPTIVE DYNAMIC SURFACE CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS297 W M and εi ≤εM,respectively.Since there is no a prioriknowledge to determine exact W i s,an adaptive mechanismis applied such that the estimatedˆW i can asymptoticallyconverge to W i.Let the estimation error be defined as follows:˜Wi(t)=ˆW i(t)−W i,i=1,2,...,n.(18)For the aforementioned uncertain nonlinear systems,an adap-tive dynamic surface control(ADSC)will be proposed topreserve the closed-loop stability.The design procedures re-garding the ADSC are discussed in the following.To stayconcise,the notations i,i=1,2,...,n,will be used torepresent the uncertainties of(3).Step3.1:Defines1=x1−x1d.(19)It is desired to track the reference command x1d with giveninitial conditions.From(3)and(17),the derivative of(19)canbe obtained as˙s1=f1+g1x2+12W T1F1+ε1−˙x1d.(20)Let¯x2be a stabilizing function to be determined for(3)¯x2=g−11(−f1−12ˆW T1F1−k1s1+˙x1d)(21)where k1is a positive constant.An adaptive law is designated as˙ˆW 1=121F1s1−η 1ˆW1(22)with a symmetric positive definite matrix 1andη>0.Let x2d be the regulated output of¯x2through a low-passfilter τ2˙x2d(t)+x2d(t)=¯x2,x2d(0)=¯x2(0).(23) With a properly chosenτ2,the smoothed x2d can be equiva-lently considered the required¯x2.Step3.2:Analogous to the discussion in Step3.1,an error variable is defined ass2=x2−x2d.(24) From(3)and(17),the derivative of(24)can be obtained as ˙s2=f2+g2x3+12W T2F2+ε2−˙x2d.(25) Let¯x3be a stabilizing function to be determined for(3)¯x3=g−12(−f2−12ˆW T2F2−k2s2+˙x2d)(26)where k2is a positive constant.An adaptation law is designed as˙ˆW 2=122F2s2−η 2ˆW2(27)with a symmetric positive definite matrix 2.Let x3d be the smoothed variable of¯x3through a low-passfilterτ3˙x3d(t)+x3d(t)=¯x3,x3d(0)=¯x3(0).(28) With a properly chosenτ3,the smoothed x3d can be equiva-lently considered the required¯x3.Step3.i:The state tracking error of x i is defined ass i=x i−x id.(29) From(3)and(17),the derivative of(29)can be obtained as˙s i=f i+g i x i+1+12W T i F i+εi−˙x id.(30) Let¯x i+1be a stabilizing function to be determined for(3)¯x i+1=g−1i(−f i−12ˆW T i F i−k i s i+˙x id)(31) where k i is a positive constant.Similarly,an adaptive law is given as˙ˆWi=12i F i s i−η iˆW i(32) with a positive definite matrix i= T i.Similarly,let x i+1,d be the smoothed counterpart of¯x i+1through a low-passfilter τi+1˙x i+1,d(t)+x i+1,d(t)=¯x i+1,x i+1,d(0)=¯x i+1(0).(33) With a properly chosenτi+1,d,the smoothed x i+1,d can be equivalently considered the required¯x i+1.Step3.n:The state tracking error of x n is defined ass n=x n−x nd.(34) From(3)and(17),the derivative of(34)can be obtained as˙s n=f n+g n u(t)+12W T n F n+εn−˙x nd.(35) Let u(t)be a stabilizing function to be determined for(3) u(t)=g−1n(−f n−12ˆW T n F n−k n s n+˙x nd)(36) where k n is a positive constant.Analogous to the aforemen-tioned discussion,an adaptive law is given as˙ˆWn=12n F n s n−η nˆW n(37) where n is a symmetric matrix, n>0.Remark3:So far,only the design procedures,combining DSC approach and IT2FNN,are identified step by step.The derivation of control actions and learning rules in each step will be investigated later,where the closed-loop stability will also be discussed in detail.IV.Stability AnalysisIn this section,the overall stability of a class of uncertain nonlinear systems will be investigated,where a DSC controller with an IT2FNN adaption is adopted.In this paper,let 0and n be feasible sets of IT2FNNADSC such that0={(x1d,˙x1d,¨x1d):x1d+˙x1d+¨x1d≤p0}(38)n=(s i,˜W i,h i):12ni=1s2i+˜W T i −1i˜W i+12ni=2h2i≤p(39)where p0and p are positive constants.It is noticed that 0 and n are compact sets in R3and R(2m+2)n−1.298IEEE TRANSACTIONS ON CYBERNETICS,VOL.44,NO.2,FEBRUARY 2014Theorem 1:Consider a class of uncertain nonlinear systems of (3).With the DSC controller and IT2FNN adaptive laws designed as (19),(21)–(24),(26)–(29),(31)–(34),(36)–(37),the corresponding s i and ˜Wi are uniformly ultimately bounded,i =1,2,...,n ,where (38)and (39)are satisfied.Proo f :First,the errors between the stabilizing variables and the smoothed variables are defined ash i +1=x i +1,d −¯x i +1=x i +1,d −g −1i (−f i −12ˆW Ti F i −k i s i +˙x id )(40)where i =1,2,...,n −1.From (30),(31),and (40),the derivative of s i can be derived as˙s i=g i s i +1+g i h i +1−12˜W Ti F i +εi −k i s i(41)in which i =1,2,...,n −1.From (35)and (36),the derivativeof the tracking error s n can be obtained as˙s n =−k n s n −12˜W Tn F n +εn .(42)From (33)and (40),the dynamics of the smoothed variablex id can be formulated as˙x id (t )=¯x i −x id τi =−h iτi,i =2,3,...,n.(43)Therefore,from (40)and (43),the dynamic changes of h i +1can be derived in the following:˙h i +1=˙x i +1,d −˙¯x i +1=−h i +1τi +1+B i +1,i =1,2,...,n −1(44)in whichB i +1(s 1,...,s n ,h 2,...,h n ,ˆW T 1,...,ˆW T i ,x 1d ,˙x 1d ,¨x 1d )=g −2i [(˙f i +12˙ˆW T i F i +12ˆW T i ˙F i +k i ˙s i+˙h i τi )g i −˙g i (f i +12ˆW TiF i −˙x id +k i s i )].Choose a Lyapunov function candidate asV =12 n i =1 s 2i +˜W T i −1i ˜W i +12 n i =2h 2i .(45)The derivative of (45)can be obtained as˙V = n i =1 s i ˙s i +˜W T i −1i ˙ˆW i + n i =2h i ˙h i .(46)From (41),(42),(44),and (46),it leads to ˙V≤s 1(g 1s 2+g 1h 2−12˜W T1F 1+ε1−k 1s 1)+s 2(g 2s 3+g 2h 3−12˜W T2F 2+ε2−k 2s 2)+···+s i (g i s i +1−12˜W Ti F i +g i h i +1+εi −k i s i )+···+s n (−k n s n −12˜W Tn F n +εn )+ n i =1˜W T i −1i ˙ˆW i + n i =2 −h 2i τi + h i B i.(47)It is true thats i s i +1≤s 2i +s 2i +1,s i h i +1≤s 2i +h 2i +1,i =1,2,...,n −1(48)ands i εi ≤s 2i +ε2i ,i =1,2,...,n.(49)Therefore,the following inequality can be obtained by substi-tuting (4),(22),(27),(32),(37),and (48),(49)into (47)˙V ≤ n −1i =1(2γi s 2i +γi s 2i +1+γi h 2i +1)+ n i =1(−k i s 2i +ε2i +s 2i )+ n i =1(−η˜W T i ˆW i )+ n i =2 −h 2i τi+ h i B i .(50)Let α0>0be a constant such that k 1=2γ1+1+α0,k n =γn −1+1+α0,and k i =2γi +γi −1+1+α0,i =2,3,...,n −1,are satisfied.With reference to [30],using ˜Wi 2− W i 2≤2˜WT i ˆW i ,it gives ˙V ≤ ni =1(−α0s 2i +ε2i )+ n −1i =1(γi h 2i +1)+ n i =2 −h 2iτi + h i B i− n i =112η˜W i 2− W i 2 ≤ n i =1 −α0s 2i −η2λmax ( −1i )˜W T i −1i ˜W i+ n −1i =1(γi h 2i +1)+ n i =2 −h 2i τi + h i B i+ n i =1 ε2i +η2 W i 2.(51)Denoted ε2i +η2 W i 2=c i ,it leads to c i ≤ε2M +η2W 2M =c M ,i =1,2,...,n .From the positive definiteness of i ,a positive ηcan be determined such that η2λmax ( −1i)≥α0.From (51),itresults in˙V ≤ n i =1 −α0 s 2i +˜W T i −1i ˜W i +nc M + n −1i =1(γi h 2i +1)+ n i =2 −h 2iτi+ h i B i .(52)Therefore,from the assumptions of boundedness,smoothness,and compactness,B i is bounded on 0× n ,i.e.,there exists an M i >0such that B i ≤M i .Let1τi =γi −1+M 2i 2β+α0,i =2,3,...,n (53)where βis a positive number.Substituting (53)into (52),the following inequality can be obtained:˙V ≤ n i =1 −α0 s 2i +˜W T i −1i ˜W i +nc M + n i =2 − M 2i2β+α0 h 2i+ h i B i .(54)Notice that h i B i ≤h 2i B 2i 2β+ β2,i =2,3,...,n .Then,one can have˙V ≤ n i =1 −α0 s 2i +˜W T i −1i ˜W i +nc M +(n −1)β2+ n i =2 −α0h 2i −(M 2i −B 2i )h 2i 2β ≤ n i =1−α0 s 2i +˜W T i −1i ˜W i − n i =2(α0h 2i )+nc M +(n −1)β2=−2α0V +nc M +(n −1)β2.(55)Solving (55)leads toV (t )≤12α0 nc M +(n −1)β2+ V (0)−12α0nc M +(n −1)β2e (−2α0t ),∀t >0.(56)CHANG AND CHAN:ADAPTIVE DYNAMIC SURFACE CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS299Fig.3.Experiment setup of the ball-and-beam system.From (56),it can be observed that V (t )is uniformly ultimately bounded,and the proof is completed.ٗRemark 4:By increasing the value of α0,the quantity of12α0(nc M +(n −1)β2)can be made arbitrarily small.Also,the convergence of V (t )implies that s i ,˜Wi ,and h i are uniformly ultimately bounded.Thus a large enough α0can make s 1=x 1−x 1d arbitrarily small.Equivalently,the stability of the proposed IT2FNNADSC control system can be guaranteed,and the purpose of tracking can be achieved in the boundness of steady-state errors.V .Example:Ball-and-Beam SystemTo verify the feasibility of the proposed IT2FNNADSC,anend-point driven ball-and-beam system is considered,where the scheme diagram is shown in Fig.3,and the associated dynamic equations are described as follows [41]:˙x 1=x 2+ 1(x ,t )˙x 2=Ax 1x 24−Agx 3+ 2(x ,t )˙x 3=x 4+ 3(x ,t )˙x 4=B cos x 3[C cos l b x 3d u −Dx 4cos l b x 3d−E −Hx 1]−BGx 1x 2x 4+ 4(x ,t )(57)where x (t )=[x 1(t )x 2(t )x 3(t )x 4(t )]T is the state vector,x 1is the ball position (m),x 2is the ball velocity (m/sec),x 3is the beam angle (rad),x 4is the angular velocity of the beam(rad/sec),A = 1+m −1B J B R −2 −1,B = J B +J b +m B x 21 −1,C =n g K b l b (R a d )−1,D = n g K b l b 2 R a d 2 −1,E =0.5l b m b g ,H =m B g ,G =2m B ,and u (t )is the input voltage of a DC motor.The physical parameters of the ball-and-beam system are listed in Table I.Remark 5:The representation of (57)can be fitted into the general form of (3).In consequence,the proposed IT2FNNADSC will be applied for the position control of a ball-and-beam system,where the ball is desired to be asymptotically balanced at a designated position.TABLE IParameters of the Ball-and-Beam System Symbol DefinitionValue m B Mass of the ball 0.0217kg m b Mass of the beam 0.334kg R Radius of the ball 0.00873m l b Beam length0.4m d Radius of the large gear 0.04mJ B Ball inertia 6.803×10−7kgm 2J b Beam inertia0.017813kgm 2K b Back-EMF constant 0.1491Nm/A R a Armature resistance 18.91 g Acceleration of gravity 9.8m /s 2n gGear ratio4.2TABLE IIParameters of ADSCi 1234k i 2840.887434.943727τi N/A0.0250.0250.025 i 100I 32α025η0.5A.Simulation ResultsFrom (21),(26),(31),and (57),stabilizing functions can be obtained as follows:¯x 2=−12ˆW T1F 1−k 1s 1+˙x 1d(58)¯x 3=−1Ag −12ˆW T 2F 2−k 2s 2−Ax 1x 24+˙x 2d (59)and¯x 4=−12ˆWT3F 3−k 3s 3+˙x 3d (60)where the employing neural networks are adaptively obtained from (32).From (36)and (57),the IT2FNNADSC based control action can be drived asu (t )=1BC cos x 3cos l b x 3d·[B cos x 3(Dx 4cos l b x 3d +E +Hx 1)+BGx 1x 2x 4+˙x4d −k 4s 4−12ˆW T4F 4].(61)The parameters for the conventional DSC and the IT2FNNADSC are initially given in Tables II and III,in which F 1=F 2=F 3=F 4=⎡⎣f (3)1 16j =1f (3)j···f (3)16 16j =1f (3)j¯f (3)1 16j =1¯f (3)j ···¯f (3)16 16j =1¯f(3)j ⎤⎦T ,[f (3)j¯f (3)j ]=[ 4i =1μ(2)˜F ij 4i =1¯μ(2)˜F ij ],j =1,2,...,16,and ˆW 1(0)=ˆW 2(0)=ˆW 3(0)=ˆW 4(0)=[ˆw(4)1L (0)···ˆw (4)16L (0)ˆw (4)1R (0)···ˆw (4)16R (0)]T .The rule table of an IT2FNN is shown in Table IV,where PS (positive small),PB (positive big),Neg (negative),Pos (positive),and Zero are interval type-2fuzzy sets.The input and output membership functions for the corresponding interval type-2fuzzy sets are shown in Fig.4.。

摘要模糊推理是以模糊集合论为基础描述工具，对以一般集合论为基础描述工具的数理逻辑进行扩展，从而建立了模糊推理理论，是不确定推理的一种。

在人工智能技术开发中有重要意义。

模糊建模是指利用模糊系统逼近未知的非线性动态，从而逼近于整个系统。

本文全面回顾了模糊推理的产生背景、研究现状和发展方向，并介绍了模糊系统、模糊集合以及模糊建模等的基础理论知识。

详细阐述了Sugeno模型的建模过程，利用模糊推理系统对非线性函数进行逼近，通过matlab仿真实例说明该建模方法的有效性。

最后，对全文进行总结，概括本篇文章的主旨，并提出今后的研究方向。

关键词：模糊推理，模糊建模，仿真AbstractFuzzy reasoning based on fuzzy sets theory to describe tool,it is based on general set theory of mathematical logic described tools, so as to establish the extended fuzzy reasoning theory,it is an uncertainty reasoning. It is important to the development of artificial intelligence technology. Fuzzy model is refered to the use of fuzzy system to approach unknown nonlinear dynamic,then approach the whole system.This paper reviews the background of fuzzy reasoning,research status and development direction,and introduces fuzzy system，the fuzzy set and the fuzzy model and basic theoretical knowledge. It also expounds the Sugeno modeling process,and use fuzzy inference system to approximate nonlinear function. Through matlab simulation example shows the effectiveness of the modeling methodFinally,the full text is summarized to express the purpose of this article, and puts forward the direction of future research.Keywords: fuzzy reasoning，fuzzy modeling，simulation目录1.绪论 (1)1.1 模糊思想的起源 (1)1.1.1 精确思维的缺陷 (1)1.1.2 逻辑推理与模糊性 (1)1.2 模糊推理理论研究的进展 (3)1.3 模糊推理的研究领域和成果 (4)1.3.1 模糊推理在模糊控制中的研究与应用 (4)1.3.2 模糊推理在人工智能中的研究与应用 (4)2．模糊系统基础 (6)2.1 模糊集 (6)2.2 模糊集的表示一隶属度函数 (6)2.3 If．．．then规则 (7)2.4 模糊推理 (8)2.5 模糊聚类 (10)3.模糊建模 (11)3.1 模糊模型建模过程 (11)3.2 非线性系统的T-S模糊模型 (12)3.3 T-S模型的参数辨识 (13)4.仿真实例 (16)4.1 仿真软件简介 (16)4.2 设计原理 (16)4.3 仿真实例 (18)4.4 结论 (22)结束语 (24)参考文献 (25)致谢 (26)1.绪论1.1 模糊思想的起源1.1.1 精确思维的缺陷迄今，经典逻辑和精确数学的成功推动了精确科学的迅速发展; 精确科学的巨大成就也造成了人类对“精确”的顶礼膜拜。