外文翻译-具有积分滑模控制的内埋式永磁同步电动机基于线性矩阵不等式的模糊控制

1、外文翻译Linear Matrix Inequality-Based Fuzzy Control for Interior Permanent Magnet Synchronous Motor with integral sliding mode controlFaGuang Wang, Seung Kyu Park, Ho Kyun AhnDepartment of Electrical Engineering, Changwon National University, KoreaAbstract-Recently, interior permanent magnet synchronou

2、s motor (IPMSM) is widely used in various applications, such as electric vehicles and compressors. It has a high requirement in wide load variations, high speed condition, stability, providing a fast response and most important thing is that it can be applied easily and efficiently. However, the con

3、trol of IPMSM is more difficult than surface permanent magnet synchronous motor (SPMSM) because its nonlinearity due to the non-zero daxis current which can be zero in SPSM but not IPMSM. In this paper, the IPMSM is controlled very efficient algorithm by using the combination of linear control and f

4、uzzy control with linear models depending on certain operating points. The Hlinear matrix inequality (LMI) based integral sliding mode control is also used to ensure the robustness. The membership functions of this paper are easy to be determined and implemented easily. Index Terms-Fuzzy control, Hc

5、ontrol, integral sliding mode control, interior permanent magnet synchronous motor (IPMSM), linear matrix inequality.I. INTRODUCTIONFrom 1980s, with the development of semiconductor, IPMSM supplied by converter source has been widely studied 1 2. The development of microcomputer made the vector cont

6、rol system of IPMSM well controlled by single chip. IPMSM possesses special features for adjustable-speed drives which distinguish it from otherclasses of ac machines, especially surface permanent magnet synchronous motor. The main criteria of high performance drives are fast and accurate speed resp

7、onse, quick recovery of speed from any disturbances and insensitivity to parameter variations 3. In order to achieve high performances, the vector control of IPMSM drive is employed 4-6. Control techniques become complicated due to the nonlinearities of the developed torque for non-zero value of d-a

8、xis current. Many researchers have focused their attention on forcing the daxis current equals to zero in the vector control of IPMSM drive, which essentially makes the motor model linear 4,7. However, in real-time the electromagnetic torque is non-linear in nature. In order to incorporate the nonli

9、nearity in a practical IPMSM drive, a control technique known as maximum torque per ampere (MTPA) is devised which provides maximum torque with minimum stator current 3. This MTPA strategy is very important from the limitation of IPMSM and inverter rating points of view, which optimizes the drive ef

10、ficiency. The problem associated MTPA control technique is that its implementation in real time becomes complicated because there exists a complex relationship between d-axis and q-axis currents. Thus, one of the main objectives of this paper is to make a new efficient control method for IPMSM and i

11、ts calculation easy and efficient. The LMI fuzzy Hcontrol has been applied and solved the nonlinearity of the IPMSM model to a set of linear model. To increase the robustness for disturbances, an ISMC technique is added to the Hcontroller. By ISMC, the proposed controller gives performances of the H

12、control system without disturbances which satisfy the matching condition. It has a good compatible with linear controllers. T-S fuzzy control 8 is based on the mathematical model which is the combination of local linear models depending on the operating points. Linear controllers are designed for ea

13、ch linear model and they are combined as a controller and make it possible to use linear control theories for nonlinear systems. Linear controls via parallel distributed compensation (PDC) and linear matrix inequality (LMI) is a most popular method considering the stability of the system with PDC 9.

14、 HLMI T-S fuzzy controller is considered as a practical Hcontroller which eliminates the effects of external disturbance below a prescribed level, so that a desired Hcontrol performance can be guaranteed 10-12. In this paper, the robustness of SMC 13 is added to the HLMI T-S fuzzy controller for the

15、 control of IPMSM. We can divide the disturbances in the IPMSM into two parts. First part is that SMC can deal with and other part is dealt by HLMI fuzzy controller. By using ISMC, the robustness of SMC and Hperformance can be combined. Integral sliding mode control (ISMC) is a kind of SMC which has

16、 sliding mode dynamics with the same order of the controlled system and can have the properties of the other control method.II. HT-S FUZZY CONTROL AND ISMCA. HT-S fuzzy controlConsider a nonlinear system as follows.x(t)f (x)g(x)u(t)w(t) (1)where |w(t)|Wb and Wb is the boundary of disturbance. Depend

17、ing on the operating points, the nonlinear system can be expressed as follows.The i-th model is that in the case z1(t) is Mi1 and and z p(t) is Mip ,(2)And HT-S fuzzy feedback controller is uikiX(t) (3)where i=1,2, ,r and Mij is the fuzzy set and r is thenumber of model rulesGiven a pair of (x(t),u(

18、t), the fuzzy systems areinferred as follows:where and i(z(t) is themembership for every fuzzy rule.From (1) we get (7)Take (6) into (7), we can get the closed loop systemequations.If we set Apresent the error boundary of every ruleand satisfy the following condition:In the same way we get: (9)Based

19、 on these, the approximation error can bebounded by matrix Ap and Bp . Hcontrol performance is:(10)where is the prescribed Hnorm. If we get theminimized for (10) we can make the effect of w(t) of (1) on x(t) is minimized.If consider the initial condition, the Hnorm (10) canbe modified as the followi

20、ng form:where P is some symmetric positive definite weighting matrix.The following result is given in 14:Theorem 1: If system (1) is controlled by T-S fuzzy controller (6), and there is a positive definite matrix P such that（12）then the closed loop system is uniformly ultimately bounded (UUB) and Hc

21、ontrol performance (11) is guaranteed.It is not easy to obtain P and, fortunately, after small change of (12), it can be solved by LMI toolbox. So we need to do some changes. Assume Utilize the Schur complements for (12), we can get:（13）where Now the problem changes to find the positive definite mat

22、rix L and F to satisfy the condition (13) and we can obtain k j at last. The (13) can be solved by LMI toolbox on computer easily.B. Sliding mode controlThe system (1) with input signal noise or disturbance d(t) is:（14）In the system (14), it can be considered that the disturbance is the summation of

23、 two different kinds of disturbancesw(t)=w1(t)+w2(t) (15)where w1(t) satisfies the following matching condition: （16）For the disturbance w1(t) , ISMC gives the desired response of the following system: （17）where x0 represents the state trajectory of the system with the disturbance w2(t) only under H

24、T-S fuzzy control uo . Assume that w(t) is bounded and that an upper bound can be found as（18）where wmax is a known positive scalar.For system (14), first redesign the control law to beu(t)=u0(t)+u1(t) （19）where is the ideal control defined in (6) and is designed to reject the perturbation term w1(t

25、) .A sliding manifold is defined ass=s0 (x)+z(x) （20）where s,s0 (x), , which consists of two parts: the first part s0(x)is designed as a linear combination of the system states; the second part z introduces the integral term and will be determined below. （21）where initial condition z(0) is determine

26、d based on the requirement s(0)=0. Different from the conventional design approach, the order of the motion equation in ISMC is equal to the order of the original system, rather than reduced by the dimension of the control input. As a result, robustness of the system can be guaranteed starting from

27、the initial time instance.III. COMBINATION HT-S FUZZY CONTROL ANDINTEGRAL SMCThe mathematic model of an IPMSM in the d-q synchronously rotating reference frame for assumed sinusoidal stator excitation is given as 3:（22）where p is the differential operator.The overall scheme of the HLMI T-S fuzzy con

28、trol system is as follows. HLMI T-S fuzzy based ISMC controller designed as following steps.Step.1. utilize the equilibrium point to calculate the error system.System (22) can be presented by state form as:（23）where x1(t) iq , x2(t) id , x3(t) wr ,u10(t) vq andu20(t) vd .Based on (23), a reference s

29、ystem can be given as:（24）where f means the required value.Then the following error dynamic system is derived. （25）where etxtxf tStep.2. determine for membership function.For x1 minimum case: For x1 maximum case: For x2 minimum case: For x2 maximum case: The fuzzy rules are as the follows:Rule.1 x1

30、is minimal and x2 is minimal:M1(t) E1(t)G1(t) （26）Rule.2 x1 is minimal and x2 is maximal:M2(t) E1(t)G2(t) (27)Rule.3 x1 is maximal and x2 is minimal:M3(t) E2(t)G1(t) (28)Rule.4 x1 is maximal and x2 is maximal:M4(t) E2(t)G2(t) (29)Step.3. obtain the matrixes A and B.Equation (25) can be of the follow

31、ing form: and the value of ( x1lim , x2lim ) is based on the rule1 to rule 4, it gets to be x1min,x1max,x2min and x2max .Step.4. calculate controller parameters K using LMI toolbox based on Theorem 1.By LMI, the error systemcontrol input is defined by (6) as （31）where k j is a 1by 3 matrix. Use ineq

32、uality (13) and Matlab LMI toolbox to calculate out the parameters k j . So that, HT-S fuzzy controller of the system is where u1 f and u2 f are reference inputs.Step.5. Design ISMC for system.Based on the SMC matching condition the system with disturbance is as follows: （32）where d(t) is the noise

33、or disturbance.The sliding surface is defined as: （33） x1r and x2r are required output values, x1n and x2n are states of nominal system: （34）Assume u1(t)u10(t)u1s(t) and u2(t)u20(t)u2s(t) .Derivate of sliding surfaces are: （35）where e1n(t)x1(t)x1n(t) , e2n(t)x2(t)x2n(t) , un(t) is the nominal contro

34、l input and us1 and us2 are sliding control inputs.The sliding controller finally is given out as: （36）where d1max and d1max are the maximal absolute values of disturbance.IV. SIMULATION RESULTSUse the controller design process in above sections with the parameters of Tab.1. Simulation results are:T

35、AB.1. IPMSM PARAMETERS. Fig.2. result of iq with parameter uncertainty and disturbance.Fig.3.result of id with parameter uncertainty and disturbance. Required output values are From the result of Fig.2 and Fig.3, we can see that some kind of disturbance can not be solved only by HLMI T-S fuzzy. Comb

36、ination with ISMC solves this perfectively.V. CONCLUSIONSThe Fuzzy LMI controller is used for IPMSM. It uses the linear models for each operating points. It is shown that only four operating points are enough for the proposed control method. The controller of this paper gives good control performanc

37、e with only four membership functions which are determined easily. Hfuzzy LMI solved the initial big input for IPMSM from ISMC, while ISMC solved the problem of Hfuzzy which is so dependent on fuzzy rules. The final results show that the combination control is efficient and perfect.具有积分滑模控制的内埋式永磁同步电

38、动机基于线性矩阵不等式的模糊控制王发光, Seung Kyu Park, Ho Kyun Ahn韩国昌原国立大学电机工程学系近期摘要，内埋式永磁同步电动机被广泛的用于各种各样的应用中，例如电动汽车和压缩机。它对宽负荷变化、高速度条件、稳定性、快速的反应有较高的要求，最重要的是它使用简便，效率高。但是对内埋式永磁同步电动机的控制要比表面式永磁同步电动机困难，这是由于它的非零d轴电流产生的非线性而造成的，而d轴电流在表面式永磁同步电动机中是可以为零的。在本论文中，内埋式永磁同步电动机通过运用有以确定工作点为基础的线性模型的线性控制和模糊控制的结合而变得非常有效率。线性矩阵不等式（LMI）的积分滑模

39、控制也能够用于确保鲁棒性，论文中的隶属函数可以很容易的确定和实施。索引条目模糊控制、控制、积分滑模控制、内埋式永磁同步电动机(IPMSM)、线性矩阵不等式。I. 导论从20世纪八十年代开始，伴随着半导体技术的发展，由转换器电源供应的内埋式永磁同步电动机已经被广泛的研究。微机技术的发展使得单芯片控制的内埋式永磁同步电动机的矢量控制系统相当的熟练，内埋式永磁同步电动机对于调速驱动器所具有的特别的特征使它区别于别的种类的交流电机，特别是表面式永磁同步电动机。高性能驱动器的主要标准是快速、精确的速度响应、从任何干扰速度的快速恢复和对参数变化的不敏感性。为了达到高性能，采用了内埋式永磁同步电动机的矢量控

40、制。由于d轴电流的非零值而造成的超前转矩的非线性使得对内埋式永磁同步电动机的控制技术变得复杂。在内埋式永磁同步电动机的矢量控制方面很多研究者把重点放在强迫d轴电流接近于零，这本质上是使点击模型线性化。但是，本质上在实时中它的电磁转矩是非线性的。为了在一个现实的内埋式永磁同步电动机中纳入非线性，一种叫做每安培最大转矩(MTPA)的控制技术被设计出来，它能够用最小的定子电流提供最大的转矩。这种策略从内埋式永磁同步电动机和逆变器等分级的限制的观点来看是非常的重要，它能够优化驱动器的效率。相关MTPA控制技术的困难是由于d轴和q轴电流存在复杂的关系使得在实时执行时变得复杂。因此，本论文的一个主要目的是

41、为内埋式永磁同步电动机作出一种高效的控制方法，而且它的计算需简单、高效。线性矩阵不等式模糊H控制已经被应用，而且已经完成了内埋式永磁同步电动机模型的非线性向一系列线性模型的转变。为了增加对干扰的鲁棒性，ISMC技术被增加到H控制器中。通过ISMC技术，先前的控制器能够给出H控制系统的性能而且没有干扰，这也满足了匹配条件。它和线性控制器有很好的兼容性。T-S模糊控制是建立在数学模型的基础之上，这种模型是建立在工作点上的局部线性模型的结合体。线性控制器是为了每一个线性模型而设计的，它们被合并为一个控制器使得对非线性系统运用线性控制理论而称为可能。线性控制凭借并行分布补偿，考虑到具有PDC系统的稳定

42、性线性矩阵不等式成为一种最流行方式。基于LMI的HT - S模糊控制器被视为实用的H控制器，它消除了低于规定水平的外部干扰影响，所以一个想得到的H控制器的性能可以被保证。在本论文中，为控制内埋式永磁同步电动机SMC的鲁棒性被添加到基于LMI的HT - S模糊控制器中。我们可以把内埋式永磁同步电动机中的干扰分成两个部分，第一部分是SMC能够处理的，另一部分是H线性矩阵不等式模糊控制器可以处理的。通过运用ISMC，SMC和H性能的鲁棒性能够结合起来。积分滑模控制是一种SMC，它具有与控制系统相同顺序的滑模动力学，还具有其他控制方式的特征。II. H T-S模糊控制与积分滑模控制A. HT-S模糊控

43、制x(t)f (x)g(x)u(t)w(t) (1)这里和是干扰的边界。以工作点为基础，非线性系统可以有以下表达方式。i-th模式是在情况下的在情况下的，（2）HT-S模糊反馈控制器是 uikiX(t) 这里i=1,2, , r、Mij是模糊方式、r是模型规则的数字。给出一对(x(t),u(t)的数据，模糊系统推断如下： （4） （5） （6）这里和i(z(t)是每个模糊规则的元素。从式（1）中可以得到：（7）把（6）式代入（7）式中，我们可以得到闭环系统方程。如果我们设置代表每个规则的边界错误且满足下列情况：（8）用同样的方法我们可以得到：（9）基于此，近似误差可以通过矩阵Ap和Bp得到限定

44、。H控制性能是（10）这里表示的是规定的H范数。如果我们在式10中对取最小值，我们可以使w对x的影响达到最小。 如果考虑到最初的条件，式10可以修改为下式：（11）这里P指的是一些对称正定加权矩阵。下面的结果在14式中给出：定理1：如果系统被T-S模糊控制器控制，而且有一个如下的正定矩阵（12）这时闭环系统是一致并且有界的，H控制性能可以得到保证。想得到矩阵P并不简单，幸运的是通过对式12作稍微的修改，这可以通过LMI工具箱来解决。所以我们要做一些改变，假设，利用舒尔补充式12，可以得到：（13）这里，现在的问题改为找到正定矩阵L和F满足式13的条件，我们最后可以获得k j，式13可以通过LMI在电脑上很容易的解决。B.滑模控制 具有输入噪音或干扰的系统一中的d(t)是：（14） 在系统（14）中，可以认为