模糊控制器设计外文资料翻译.doc

外文资料翻译Static Output Feedback Control for Discrete-time Fuzzy Bilinear SystemAbstract 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 controller design1-11. Most of the above results are designed based on either state feedback control or observer-based control1-7.Very few results deal with fuzzy output feedback8-11. 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 910 and a robust H controller via static output feedback was designed11. 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 obtained1213.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 researchers14-16. The paper 14 studied the robust stabilization for the FBS, then the result was extended to the FBS with time-delay15. The problem of robust stabilization for discrete-time FBS (DFBS) was considered16. 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 the DFBS 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 denotes being a positive definite matrix. In symmetric block matrices, an asterisk (*) is used to represent a symmetric term and stands for a block-diagonal matrix. The notionmeans.2 Problem formulationsConsider a DFBS that is represented by T-S fuzzy bilinear model. The th rule of the DFBS is represented by the following form (1)Wheredenotes the fuzzy inference rule, is the number of fuzzy rules. is fuzzy set andis premise variable.Is the state vector,is the control input and is the system output. The matrices are known matrices with appropriate dimensions. Since the static output feedback control is considered in this paper, we simply setand.By using singleton fuzzifier, product inference and center-average defuzzifier, the fuzzy model(1) Can be expressed by the following global model (2)Where.is the grade of Membership ofin. We assume thatand. Then we have the following conditions:.Based on parallel distribute compensation, the fuzzy controller shares the same premise parts with (1); that is, the static output controller for fuzzy rule is written as (3)Hence, the overall fuzzy control law can be represented as (4)Where.is a vector to be determined and is a scalar to be assigned.By substituting (4) into (2), the closed-loop fuzzy systems can be represented as (5) where.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 matricesandwith appropriate dimensions such that, the inequality holds.Proof: Note that Applying Lemma 1 in 1: , the inequality can be obtained. Thus the proof is completed.Theorem 1 For given scalarand, the DFBS (5) is asymptotically stable in the large, if there exist matricesand satisfying the inequality (6). (6) Where .Proof: Consider the Lyapunov function candidate as(7) where is to be selected.Define the difference, and then along the solution of (5), we have (8)Applying Lemma 1 again, it follows that (9)Substituting (9) into (8) leads to(10)Applying the Schur complement, (6) is equivalent to (11)Pre- and post multiplying both side of (11) with, respectively, we have (12)Therefore, it is noted that, then the DFBS (5) is asymptotically stable. Thus the proof is completed. The matrix inequality (6) leads to BMI optimization, a non-convex programming 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 scalarand, the DFBS (5) is asymptotically stable in the large, if there exist matricesand satisfying the inequality (13). (13)Proof: It is trivial that (14)Then if , we can conclude that. (15)By applying Schur complement, (13) is equivalent to. Then we get. According to Theorem 1, the DFBS (5) is asymptotically stable. Thus the proof is completed. 4 Numerical examplesIn this section, an example is used for illustration. The considered DFBS isWhereThe membership functions are defined as.By letting applying Theorem 2 and solving the corresponding LMIs, we can obtain the following solutions: Simulation results with the initial conditions: respective, are shown in Fig.1 and Fig.2. One can find that all these state converge to the equilibrium state after 17 seconds. Fig.1. State responses of system Fig.2. Control trajectory of system5 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.References1 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 systemsJ.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 systemsJ. 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 functionJ. 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 delaysJ. 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 approachJ. IEEE Trans. Fuzzy Syst., 2005, 13(2): 263-272.11 Huang 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 feedbackJ. 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。使用并行分布补偿算法(PDC)，得到了闭环系统渐近稳定的充分条件，并把这些条件转换成线性矩阵不等式(LMI)的形式，使得模糊控制器可以由一组线性矩阵不等式的解得到。和现有的文献相比，这种方法不要求相同的输出矩阵和相似转换等条件。最后，通过仿真例子验证了方法的有效性。关键词：离散模糊双线性系统；静态输出反馈控制；模糊控制；线性矩阵不等式；0引言众所周知，基于T-S模型的模糊控制是研究非线性系统比较成功的方法之一，在稳定性分析和控制器设计方面，已有很多成果面世1-10。然而大部分控制器是关于状态反馈或基于观测器的状态反馈 1-3，关于输出反馈的结果则很少4-10。输出反馈控制直接利用系统的输出量来设计控制器，不用考虑系统状态是否可测可观，而且静态输出反馈控制器结构简单，因此具有良好的应用价值。文56研究了模糊时滞系统的静态输出反馈控制问题，文8第一次提出了模糊静态输出反馈H控制的问题。但是上述结果所得到的条件常常是双线性矩阵不等式(BML），为了化成线性矩阵不等式（LMI）求解，需引入了相似变换或是要求所有的输出矩阵全部相同，这样的结果往往具有很强的保守性。双线性系统是一类比较特殊的非线性系统，它的模型比一般的非线性系统模型结构简单，描述对象的近似程度比线性系统模型要高的多11-13。对于很多实际系统，当用线性系统模型不能描述时，往往可以用双线性系统模型来描述。考虑T-S模型的有效性及双线性系统的特点，对T-S模糊双线性系统（FBS）的研究引起了很多学者的关注1415。和常用的T-S模糊模型不同的是，FBS的模糊规则的后件部分由一个双线性函数表示，FBS的局部动态可由双线性状态空间模型表示。文14研究了一类连续FBS系统的鲁棒稳定性问题，并把结果推广到了带有时滞的FBS中15。但是上述结果都是连续时间系统并且是基于状态反馈控制器的，目前还没有关于离散模糊双线性系统（DFBS）静态输出反馈控制的文献。综上分析，本文研究了一类用T-S模型表示的DFBS静态输出反馈控制问题。给出了系统渐近稳定的充分条件，并把这种条件转换成LMIs形式，使模糊控制器可以通过求解LMI而得到。这种方法不要求系统的输出矩阵相同，也不需要相似转换。最后，由数例仿真验证了结果的有效性。注1：在本文中，表示维空间，表示是一个正定（正半定）实对称矩阵。在矩阵表达式中，用“”来表示对称项，用来表示合适维数的单位矩阵。用来表示。如果不做特别说明，矩阵均表示合适维数的矩阵。1系统的模型描述由T-S模型描述的不确定模糊双线性系统,它的第条规则可描述如下: （1）其中：是模糊集合，是前提变量。分别是状态变量、控制输入和测量输出。是已知合适维数的系统矩阵。考虑静态输出反馈控制，这里假设及。通过单点模糊化，乘积推理和中心平均反模糊化方法，模糊控制系统的总体模型为： （2）其中：。是在中隶属度函数。在本文中假设：。由的定义可知：和，。以下在不引起混淆的情况下记为。根据并行分布补偿算法，考虑静态输出反馈控制器： （3）则整个系统的状态反馈控制律可表示为： （4）这里：。是待定的控制器增益，是待求的标量。 在控制律（4）的作用下，整个闭环系统的方程可表示为： （5）其中： 以下给出在证明中要用到的引理：引理1：设是维数适合的实矩阵,是正定对称矩阵，则对于标量，有如下不等式成立： 。 （6证明：考虑： 由文1中引理1：，可得到：。2 主要结果定理1：对于给定的和常数，如果存在着矩阵满足下面矩阵不等式（7），则DFBS（5）是渐近稳定的。 （7）其中：。证明：选取如下Lyapunov函数： （8）其中：是待求的正定对称矩阵。沿着系统（5）的轨线，对求差分，可得到：（9）由引理1可知： （10）把（10）带入（9）中，可以得到： （11）考虑： （12）由引理1可知： （13）把（13）带入（11）式，可得： （14）根据Schur补定理，（7）等价于： （15）对（15）分别左、右乘且，则可知： （16）从而可知，所以可知系统（5）是渐近稳定的。 考虑（7）是双线性矩阵不等式，为求解控制器，下面提出一个新的方法把BMI转换成LMI形式。定理2：对于给定的和常数，如果存在着矩阵满足下面线性矩阵不等式（17），则DFBS（5）是渐近稳定的。 （17）证明：考虑： （18）假设有，则可以得出：。（19） 由Schur补定理可知：（17）式等价于，进一步可以得。根据定理1，则可知在静态输出反馈器下，系统（5）是渐近稳定的。3 算例分析为了进一步阐述前面的方法和结论，考虑如下双线性模糊系统：其中：选取隶属度函数：并选取，根据定理2，通过Matlab求解相应的LMIs，可以得到：分别选取初始值为，利用MATLAB仿真，图1是系统变量和的状态响应，图2是控制律变化过程。由仿真结果可以看出，在所设计的控制器下，系统状态变量在17秒后趋于平衡点。图1：系统分别在初始状态：1.4 -1.6（实线）、-1.1 0.9（长划线）下的状态响应曲线 图2：系统分别在初始状态：1.4 -1.6（实线）、-1.1 0.9（长划线）下的控制曲线4 结论本文对一类用T-S模型表示的DFBS研究了静态输出反馈控制问题。给出了系统渐近稳定的充分条件，并把这种条件转换成LMIs形式，使模糊控制器可以通过求解LMI而得到。这种方法不要求系统的输出矩阵相同，也不需要相似转换。最后，由数例仿真验证了结果的有效性。参考文献：1 X.Li and C.E.de Souza, Delayed-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach J.IEEE Trans. Auto. Cont., 1997, 42(9):1144-1148 2 Y.Y.Cao and P.M.Frank, Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach J. IEEE Trans. Fuzzy Syst., 2000, 18(2): 200-211.3 J.Yoneyama, Robust stability and stabilization for uncertain Takagi-Sugeno fuzzy time-delay systems J. Fuzzy Sets and Syst., 2007, 158(4): 115-1344 S.W.Kau, H.J.Lee et al, Robust H fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties J. Fuzzy Sets and Syst., 2007, 158:135-146.5 Y.C.Chang, S.S.Chen, S.F.Su, et al. Static output feedback stabilization for nonlinear interval time-delay systems via fuzzy control approach J. Fuzzy Sets and Syst2004, 148(3): 395-410.6 S.S.Chen, Y.C.Chang, S.F. Su, et al. Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approachJ.