模糊控制器設計外文資料翻譯--離散模糊雙線性系統(tǒng)的靜態(tài)輸出反饋控制

1、<p><b>　　中文2094字</b></p><p><b>　　外文資料翻譯</b></p><p>　　Static Output Feedback Control for Discrete-time Fuzzy Bilinear System</p><p>　　Abstract The paper

2、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 g

3、uarantee 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 to

4、olbox. In comparison with the existing re</p><p>　　Keywords discrete-time fuzzy bilinear system (DFBS); static output feedback control; fuzzy control; linear matrix inequality (LMI)</p><p>　　1 I

5、ntroduction</p><p>　　It 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

6、 rules. Based on T-S model, a great number of results have been obtained on concerning analysis and controller design[1]-[11]. Most of the above results are designed based on either state feedback control or observer-bas

7、ed control[1]-[7].Very few results deal with fuzzy output feedback[8]-[11]. The scheme of stat</p><p>　　Bilinear systems exist between nonlinear and linear systems, which provide much better approximation of

8、 the original nonlinear systems than the linear systems [12].The research of bilinear systems has been paid a lot of attention and a series of results have been obtained[12][13].Considering the advantages of bilinear sys

9、tems and fuzzy control, the fuzzy bilinear system (FBS) based on the T-S fuzzy model with bilinear rule consequence was attracted the interest of researchers[14]-[16]. The paper</p><p>　　In this paper, a new

10、 approach for designing a fuzzy static output feedback controller for the </p><p>　　DFBS is proposed. Some sufficient conditions for synthesis of fuzzy static output feedback controller are derived in terms

11、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 be

12、en eliminated. </p><p>　　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 st

13、ands for a block-diagonal matrix. The notionmeans.</p><p>　　2 Problem formulations</p><p>　　Consider a DFBS that is represented by T-S fuzzy bilinear model. The th rule of the DFBS is represent

14、ed by the following form</p><p><b>　　(1)</b></p><p>　　Wheredenotes the fuzzy inference rule, is the number of fuzzy rules. is fuzzy set andis premise variable.Is the state vector,is

15、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.</p><p>　　By us

16、ing singleton fuzzifier, product inference and center-average defuzzifier, the fuzzy model</p><p>　　(1) Can be expressed by the following global model</p><p>　　(2)Where.is the grade of Membershi

17、p 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 ru

18、le is written as</p><p>　　(3)Hence, the overall fuzzy control law can be represented as</p><p><b>　　(4)</b></p><p>　　Where.is a vector to be determined and is a scalar

19、to be assigned.</p><p>　　By substituting (4) into (2), the closed-loop fuzzy systems can be represented as </p><p><b>　　(5) </b></p><p><b>　　where.</b></

20、p><p>　　The objective of this paper is to design fuzzy controller (4) such that the DFBS (5) is asymptotically stable.</p><p>　　3 Main results</p><p>　　Now we introduce the following L

21、emma which will be used in our main results.</p><p>　　Lemma 1 Given any matricesandwith appropriate dimensions such that, the inequality holds.</p><p>　　Proof: Note that </p><p>　　A

22、pplying Lemma 1 in [1]: , the inequality can be obtained. Thus the proof is completed.</p><p>　　Theorem 1 For given scalarand, the DFBS (5) is asymptotically stable in the large, if there exist matricesand s

23、atisfying the inequality (6).</p><p><b>　　(6)</b></p><p><b>　　Where .</b></p><p>　　Proof: Consider the Lyapunov function candidate as</p><p><

24、;b>　　(7) </b></p><p>　　where is to be selected.</p><p>　　Define the difference, and then along the solution of (5), we have</p><p><b>　　(8)</b></p><

25、;p>　　Applying Lemma 1 again, it follows that</p><p><b>　　(9)</b></p><p>　　Substituting (9) into (8) leads to</p><p><b>　　(10)</b></p><p>　　Ap

26、plying the Schur complement, (6) is equivalent to </p><p><b>　　(11)</b></p><p>　　Pre- and post multiplying both side of (11) with, respectively, we have</p><p><b>

27、　　(12)</b></p><p>　　Therefore, it is noted that, then the DFBS (5) is asymptotically stable. Thus the proof is completed. </p><p>　　The matrix inequality (6) leads to BMI optimization, a n

28、on-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.</p><p>　　Theorem 2 For given scalara

29、nd, the DFBS (5) is asymptotically stable in the large, if there exist matricesand satisfying the inequality (13).</p><p><b>　　(13)</b></p><p>　　Proof: It is trivial that </p>

30、<p><b>　　(14)</b></p><p>　　Then if , we can conclude that.</p><p><b>　　(15)</b></p><p>　　By applying Schur complement, (13) is equivalent to. Then we g

31、et. According to Theorem 1, the DFBS (5) is asymptotically stable. Thus the proof is completed. </p><p>　　4 Numerical examples</p><p>　　In this section, an example is used for illustration. The

32、considered DFBS is</p><p><b>　　Where</b></p><p>　　The membership functions are defined as.</p><p>　　By letting applying Theorem 2 and solving the corresponding LMIs, we

33、can obtain the following solutions: </p><p>　　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 af

34、ter 17 seconds.</p><p>　　Fig.1. State responses of system Fig.2. Control trajectory of system</p><p>　　5 Conclusions</p><p>　　In this paper, a new and simple approach for d

35、esigning 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

36、matrices are not necessary to be the same. Thus, the constraints had been relaxed and applicability of the static output feedback is increased.</p><p>　　References</p><p>　　[1] Wang R J, Lin W W

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48、J]. Systems Engineering and Electronics, 2006, 28(12):1866-1869. (In Chinese)</p><p>　　[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]. I

49、EEE Trans. Fuzzy Syst., 2007, 3(15):494-505.</p><p>　　[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: 10.1

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51、-526. </p><p>　　離散模糊雙線性系統(tǒng)的靜態(tài)輸出反饋控制</p><p>　　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)有的文獻相比，這種方法不要求相同的輸出矩陣和相似轉換等條件。最

52、后，通過仿真例子驗證了方法的有效性。</p><p>　　關鍵詞：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；</p><p><b>　　0引言</b></p><p>　　眾所周知，基于T-S模型的模糊控制是研究非線性系統(tǒng)比較成功的方法之一，在穩(wěn)定性分析和控制器設計方面，已有很多成果面世[1]-[10]。然而大部分控制器

53、是關于狀態(tài)反饋或基于觀測器的狀態(tài)反饋 [1]-[3]，關于輸出反饋的結果則很少[4]-[10]。輸出反饋控制直接利用系統(tǒng)的輸出量來設計控制器，不用考慮系統(tǒng)狀態(tài)是否可測可觀，而且靜態(tài)輸出反饋控制器結構簡單，因此具有良好的應用價值。文[5][6]研究了模糊時滯系統(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

54、相似變換或是要求所有的輸出矩陣全部相同，這樣的結果往往具有很強的保守性。</p><p>　　雙線性系統(tǒng)是一類比較特殊的非線性系統(tǒng)，它的模型比一般的非線性系統(tǒng)模型結構簡單，描述對象的近似程度比線性系統(tǒng)模型要高的多[11]-[13]。對于很多實際系統(tǒng)，當用線性系統(tǒng)模型不能描述時，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，對T-S模糊雙線性系統(tǒng)（FBS）的研究引起了很多學者的關注[14

55、][15]。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雙線性函數(shù)表示，F(xiàn)BS的局部動態(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，并把結果推廣到了帶有時滯的FBS中[15]。但是上述結果都是連續(xù)時間系統(tǒng)并且是基于狀態(tài)反饋控制器的，目前還沒有關于離散模糊雙線性系統(tǒng)（DFBS）靜態(tài)輸出反饋控制的文獻。</p><p>　　綜上分析，本文研究了一類用T-S模型表

56、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，也不需要相似轉換。最后，由數(shù)例仿真驗證了結果的有效性。</p><p>　　注1：在本文中，表示維空間，表示是一個正定（正半定）實對稱矩陣。在矩陣表達式中，用“”來表示對稱項，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

57、數(shù)的矩陣。</p><p><b>　　1系統(tǒng)的模型描述</b></p><p>　　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:</p><p>　　（1）其中：是模糊集合，，是前提變量。分別是狀態(tài)變量、控制輸入和測量輸出。是已知合適維數(shù)的系統(tǒng)矩陣?？紤]靜態(tài)輸出反饋控制，這里假設及。</p><p>

58、;　　通過單點模糊化，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：</p><p>　　（2）其中：。是在中隸屬度函數(shù)。在本文中假設：。由的定義可知：和，。以下在不引起混淆的情況下記為。</p><p>　　根據(jù)并行分布補償算法，考慮靜態(tài)輸出反饋控制器： （3）</p><p>　　則整個系統(tǒng)的狀態(tài)反饋控制律可表示為：</p

59、><p>　?。?）這里：。是待定的控制器增益，是待求的標量。 </p><p>　　在控制律（4）的作用下，整個閉環(huán)系統(tǒng)的方程可表示為： （5）</p><p><b>　　其中： </b&g

60、t;</p><p>　　以下給出在證明中要用到的引理：</p><p>　　引理1：設是維數(shù)適合的實矩陣,是正定對稱矩陣，則對于標量，有如下不等式成立： </p><p>　　。 （6</p><p><b>　　證明：考慮：</b></p><p>　　由文[

61、1]中引理1：，可得到：</p><p><b>　　。</b></p><p><b>　　2 主要結果</b></p><p>　　定理1：對于給定的和常數(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。</p><p><b>　?。?）</b>&

62、lt;/p><p><b>　　其中：。</b></p><p>　　證明：選取如下Lyapunov函數(shù)： （8）其中：是待求的正定對稱矩陣。</p><p>　　沿著系統(tǒng)（5）的軌線，對求差分，可得到：</p><p>　　（9）由引理1可知：</p>

63、<p><b>　?。?0）</b></p><p>　　把（10）帶入（9）中，可以得到： （11）</p><p><b>　　考慮：</b></p><p><b>　?。?2）</b></p><p><b>　

64、　由引理1可知：</b></p><p><b>　?。?3）</b></p><p>　　把（13）帶入（11）式，可得： </p><p><b>　　（14）</b></p><p>　　根據(jù)Schur補定理，（7）等價于：</p><p><b

65、>　?。?5）</b></p><p>　　對（15）分別左、右乘且，則可知：</p><p><b>　　（16）</b></p><p>　　從而可知，所以可知系統(tǒng)（5）是漸近穩(wěn)定的。</p><p>　　考慮（7）是雙線性矩陣不等式，為求解控制器，下面提出一個新的方法把BMI轉換成LMI形式。&l

66、t;/p><p>　　定理2：對于給定的和常數(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。</p><p><b>　?。?7）</b></p><p><b>　　證明：考慮：</b></p><p><b>　　（18）</b></p&g

67、t;<p>　　假設有，則可以得出：。</p><p><b>　　（19） </b></p><p>　　由Schur補定理可知：（17）式等價于，進一步可以得。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。</p><p><b>　　3 算例分析</b></p><p

68、>　　為了進一步闡述前面的方法和結論，考慮如下雙線性模糊系統(tǒng)：</p><p><b>　　其中：</b></p><p>　　選取隸屬度函數(shù)：并選取，根據(jù)定理2，通過Matlab求解相應的LMIs，可以得到：</p><p>　　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，圖2是控制律變化過程。由仿真結果可以看出

69、，在所設計的控制器下，系統(tǒng)狀態(tài)變量在17秒后趨于平衡點。</p><p>　　圖1：系統(tǒng)分別在初始狀態(tài)：[1.4 -1.6]（實線）、[-1.1 0.9]（長劃線）下的狀態(tài)響應曲線 </p><p>　　圖2：系統(tǒng)分別在初始狀態(tài)：[1.4 -1.6]（實線）、[-1.1 0.9]（長劃線）下的控制曲線</p><p><b>　　4 結論&

70、lt;/b></p><p>　　本文對一類用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，也不需要相似轉換。最后，由數(shù)例仿真驗證了結果的有效性。</p><p><b>　　參考文獻：</b></p><

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