模糊控制器設(shè)計(jì)外文資料翻譯離散模糊雙線性系統(tǒng)的靜態(tài)輸出反饋控制

1、中文2094字外文資料翻譯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 condi

2、tions 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 drawba

3、cks 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 w

4、ell 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

5、. 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 f

6、eedback. 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. Mor

7、eover, 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 researc

8、h 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 researcher

9、s14-16. 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

10、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 compar

11、ison 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

12、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

13、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 d

14、efuzzifier, 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

15、 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

16、 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: ,

17、 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 th

18、e 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 tha

19、t, 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 proble

20、m.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

21、 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 fol

22、lowing 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 s

23、imple 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 constra

24、ints 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

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33、Trans. Syst., Man, and Cybe., 2008, 38(2) : 510-526. 離散模糊雙線性系統(tǒng)的靜態(tài)輸出反饋控制摘要：研究了一類離散模糊雙線性系統(tǒng)(DFBS)的靜態(tài)輸出反饋控制問題。使用并行分布補(bǔ)償算法(PDC)，得到了閉環(huán)系統(tǒng)漸近穩(wěn)定的充分條件，并把這些條件轉(zhuǎn)換成線性矩陣不等式(LMI)的形式，使得模糊控制器可以由一組線性矩陣不等式的解得到。和現(xiàn)有的文獻(xiàn)相比，這種方法不要求相同的輸出矩陣和相似轉(zhuǎn)換等條件。最后，通過仿真例子驗(yàn)證了方法的有效性。關(guān)鍵詞：離散模糊雙線性系統(tǒng)；靜態(tài)輸出反饋控制；模糊控制；線性矩陣不等式；0引言眾所周知，基于T-S模型的模糊控制是研究非線

34、性系統(tǒng)比較成功的方法之一，在穩(wěn)定性分析和控制器設(shè)計(jì)方面，已有很多成果面世1-10。然而大部分控制器是關(guān)于狀態(tài)反饋或基于觀測(cè)器的狀態(tài)反饋 1-3，關(guān)于輸出反饋的結(jié)果則很少4-10。輸出反饋控制直接利用系統(tǒng)的輸出量來設(shè)計(jì)控制器，不用考慮系統(tǒng)狀態(tài)是否可測(cè)可觀，而且靜態(tài)輸出反饋控制器結(jié)構(gòu)簡(jiǎn)單，因此具有良好的應(yīng)用價(jià)值。文56研究了模糊時(shí)滯系統(tǒng)的靜態(tài)輸出反饋控制問題，文8第一次提出了模糊靜態(tài)輸出反饋H控制的問題。但是上述結(jié)果所得到的條件常常是雙線性矩陣不等式(BML），為了化成線性矩陣不等式（LMI）求解，需引入了相似變換或是要求所有的輸出矩陣全部相同，這樣的結(jié)果往往具有很強(qiáng)的保守性。雙線性系統(tǒng)是一類比較

35、特殊的非線性系統(tǒng)，它的模型比一般的非線性系統(tǒng)模型結(jié)構(gòu)簡(jiǎn)單，描述對(duì)象的近似程度比線性系統(tǒng)模型要高的多11-13。對(duì)于很多實(shí)際系統(tǒng)，當(dāng)用線性系統(tǒng)模型不能描述時(shí)，往往可以用雙線性系統(tǒng)模型來描述。考慮T-S模型的有效性及雙線性系統(tǒng)的特點(diǎn)，對(duì)T-S模糊雙線性系統(tǒng)（FBS）的研究引起了很多學(xué)者的關(guān)注1415。和常用的T-S模糊模型不同的是，F(xiàn)BS的模糊規(guī)則的后件部分由一個(gè)雙線性函數(shù)表示，F(xiàn)BS的局部動(dòng)態(tài)可由雙線性狀態(tài)空間模型表示。文14研究了一類連續(xù)FBS系統(tǒng)的魯棒穩(wěn)定性問題，并把結(jié)果推廣到了帶有時(shí)滯的FBS中15。但是上述結(jié)果都是連續(xù)時(shí)間系統(tǒng)并且是基于狀態(tài)反饋控制器的，目前還沒有關(guān)于離散模糊雙線性系統(tǒng)（

36、DFBS）靜態(tài)輸出反饋控制的文獻(xiàn)。綜上分析，本文研究了一類用T-S模型表示的DFBS靜態(tài)輸出反饋控制問題。給出了系統(tǒng)漸近穩(wěn)定的充分條件，并把這種條件轉(zhuǎn)換成LMIs形式，使模糊控制器可以通過求解LMI而得到。這種方法不要求系統(tǒng)的輸出矩陣相同，也不需要相似轉(zhuǎn)換。最后，由數(shù)例仿真驗(yàn)證了結(jié)果的有效性。注1：在本文中，表示維空間，表示是一個(gè)正定（正半定）實(shí)對(duì)稱矩陣。在矩陣表達(dá)式中，用“”來表示對(duì)稱項(xiàng)，用來表示合適維數(shù)的單位矩陣。用來表示。如果不做特別說明，矩陣均表示合適維數(shù)的矩陣。1系統(tǒng)的模型描述由T-S模型描述的不確定模糊雙線性系統(tǒng),它的第條規(guī)則可描述如下:（1）其中：是模糊集合，是前提變量。分別是狀

37、態(tài)變量、控制輸入和測(cè)量輸出。是已知合適維數(shù)的系統(tǒng)矩陣?？紤]靜態(tài)輸出反饋控制，這里假設(shè)及。通過單點(diǎn)模糊化，乘積推理和中心平均反模糊化方法，模糊控制系統(tǒng)的總體模型為：（2）其中：。是在中隸屬度函數(shù)。在本文中假設(shè)：。由的定義可知：和，。以下在不引起混淆的情況下記為。根據(jù)并行分布補(bǔ)償算法，考慮靜態(tài)輸出反饋控制器：（3）則整個(gè)系統(tǒng)的狀態(tài)反饋控制律可表示為：（4）這里：。是待定的控制器增益，是待求的標(biāo)量。在控制律（4）的作用下，整個(gè)閉環(huán)系統(tǒng)的方程可表示為： （5）其中：以下給出在證明中要用到的引理：引理1：設(shè)是維數(shù)適合的實(shí)矩陣,是正定對(duì)稱矩陣，則對(duì)于標(biāo)量，有如下不等式成立：。（6證明：考慮：由文1中引理1

38、：，可得到：。2 主要結(jié)果定理1：對(duì)于給定的和常數(shù)，如果存在著矩陣滿足下面矩陣不等式（7），則DFBS（5）是漸近穩(wěn)定的。（7）其中：。證明：選取如下Lyapunov函數(shù)：（8）其中：是待求的正定對(duì)稱矩陣。沿著系統(tǒng)（5）的軌線，對(duì)求差分，可得到：（9）由引理1可知：（10）把（10）帶入（9）中，可以得到：（11）考慮：（12）由引理1可知：（13）把（13）帶入（11）式，可得：（14）根據(jù)Schur補(bǔ)定理，（7）等價(jià)于：（15）對(duì)（15）分別左、右乘且，則可知：（16）從而可知，所以可知系統(tǒng)（5）是漸近穩(wěn)定的。考慮（7）是雙線性矩陣不等式，為求解控制器，下面提出一個(gè)新的方法把BMI轉(zhuǎn)換成L

39、MI形式。定理2：對(duì)于給定的和常數(shù)，如果存在著矩陣滿足下面線性矩陣不等式（17），則DFBS（5）是漸近穩(wěn)定的。 （17）證明：考慮： （18）假設(shè)有，則可以得出：。（19） 由Schur補(bǔ)定理可知：（17）式等價(jià)于，進(jìn)一步可以得。根據(jù)定理1，則可知在靜態(tài)輸出反饋器下，系統(tǒng)（5）是漸近穩(wěn)定的。3 算例分析為了進(jìn)一步闡述前面的方法和結(jié)論，考慮如下雙線性模糊系統(tǒng)：其中：選取隸屬度函數(shù)：并選取，根據(jù)定理2，通過Matlab求解相應(yīng)的LMIs，可以得到：分別選取初始值為，利用MATLAB仿真，圖1是系統(tǒng)變量和的狀態(tài)響應(yīng)，圖2是控制律變化過程。由仿真結(jié)果可以看出，在所設(shè)計(jì)的控制器下，系統(tǒng)狀態(tài)變量在17秒

40、后趨于平衡點(diǎn)。圖1：系統(tǒng)分別在初始狀態(tài)：1.4 -1.6（實(shí)線）、-1.1 0.9（長(zhǎng)劃線）下的狀態(tài)響應(yīng)曲線圖2：系統(tǒng)分別在初始狀態(tài)：1.4 -1.6（實(shí)線）、-1.1 0.9（長(zhǎng)劃線）下的控制曲線4 結(jié)論本文對(duì)一類用T-S模型表示的DFBS研究了靜態(tài)輸出反饋控制問題。給出了系統(tǒng)漸近穩(wěn)定的充分條件，并把這種條件轉(zhuǎn)換成LMIs形式，使模糊控制器可以通過求解LMI而得到。這種方法不要求系統(tǒng)的輸出矩陣相同，也不需要相似轉(zhuǎn)換。最后，由數(shù)例仿真驗(yàn)證了結(jié)果的有效性。參考文獻(xiàn)：1 X.Li and C.E.de Souza, Delayed-dependent robust stability and st

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