摘要:針對(duì)廣義時(shí)滯系統(tǒng),基于李雅普諾夫第二方法和廣義系統(tǒng)的受限等價(jià)變換,結(jié)合積分不等式技術(shù),給出一個(gè)線性矩陣不等式(linear matrix inequality,LMI)形式的容許性條件。首先,利用廣義系統(tǒng)的受限等價(jià)變換得出廣義時(shí)滯系統(tǒng)是正則且無(wú)脈沖的;然后,通過(guò)選取增廣型LyapunovKrasovskii泛函(LK泛函)和多重積分型LK泛函,引入松弛型LK泛函構(gòu)建新的LK泛函,利用Jensen積分不等式和Wirtinger積分不等式對(duì)LK泛函求導(dǎo)后產(chǎn)生的積分項(xiàng)進(jìn)行處理,得出廣義時(shí)滯系統(tǒng)的穩(wěn)定性條件,進(jìn)而得到廣義時(shí)滯系統(tǒng)的容許性條件;最后,利用MATLAB中的LMI工具箱,通過(guò)數(shù)值算例驗(yàn)證所用方法的可行性和有效性。
關(guān)鍵詞:廣義時(shí)滯系統(tǒng); 容許性條件; LyapunovKrasovskii泛函; Jensen積分不等式; Wirtinger積分不等式
中圖分類號(hào):O231文獻(xiàn)標(biāo)志碼:A
doi:10.3969/j.issn.16735862.2024.02.011
CUI Song LYU Yan CHEN Lanfeng SUN Xin, WANG Hanxuan
(1. College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China)
(College of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)
Abstract:Based on Lyapunov′s second method and limited equivalent transformations of descriptor systems, combined with the integral inequality technique, an admissibility condition for descriptor delay systems is given in the form of linear matrix inequality (LMI). Firstly, it is concluded that the descriptor delay system is regular and impulse free by using limited equivalent transformations of descriptor systems. Secondly, a new LyapunovKrasovskii functional (LK functional) is constructed by selecting the augmented LK functional, multiple integral LK functional and introducting the relaxed LK functional, and then, the integral terms producted by derivation of LK functional are dealt with by Jensen integral inequality and Wirtinger integral inequality, respectively. Thus, a stability condition for the descriptor delay system is obtained, correspondingly, an admissibility condition for the descriptor delay system is obtained. Finally, a numerical example is provided to demonstrate feasibility and validity of the proposed method by virtue of LMI toolbox of MATLAB.
Key words:descriptor delay systems; admissibility condition; LyapunovKrasovskii functional; Jensen integral inequality; Wirtinger integral inequality
廣義系統(tǒng),又稱奇異系統(tǒng)、微分代數(shù)系統(tǒng)、廣義狀態(tài)空間系統(tǒng)等[1]。與正常系統(tǒng)相比,廣義系統(tǒng)的形式更加一般,可以更精準(zhǔn)地描述一些物理現(xiàn)象,其在許多實(shí)際系統(tǒng)中,如電力系統(tǒng)、航空航天系統(tǒng)、社會(huì)經(jīng)濟(jì)系統(tǒng)中得到了廣泛應(yīng)用。近年來(lái),學(xué)者們對(duì)于廣義系統(tǒng)的研究主要集中在穩(wěn)定性分析方面[23]。時(shí)滯是工程實(shí)際中普遍存在的現(xiàn)象,信息或物質(zhì)的傳遞都會(huì)產(chǎn)生時(shí)滯,如流體的傳輸、電力的輸送等,但時(shí)滯的存在往往會(huì)降低系統(tǒng)的性能,甚至使系統(tǒng)變得不穩(wěn)定。因此,研究廣義時(shí)滯系統(tǒng)具有重要的理論意義和實(shí)用價(jià)值。目前,對(duì)廣義時(shí)滯系統(tǒng)的研究已經(jīng)非常深入,如指數(shù)穩(wěn)定性[4]、魯棒穩(wěn)定性[5]、H∞濾波[6]、滑??刂疲?]等。
對(duì)廣義時(shí)滯系統(tǒng)的研究既要考慮廣義系統(tǒng)的正則性和無(wú)脈沖性,又要考慮時(shí)滯對(duì)系統(tǒng)穩(wěn)定性的影響,難度更大,更具有挑戰(zhàn)性。針對(duì)廣義時(shí)滯系統(tǒng)容許性問(wèn)題,為了降低容許性條件的保守性,常采用以下2種方式:1)構(gòu)造合適的LK泛函,充分利用系統(tǒng)本身的狀態(tài)信息和時(shí)滯信息;2)選取合適的積分不等式處理LK泛函求導(dǎo)后產(chǎn)生的積分項(xiàng),將容許性條件寫成線性矩陣不等式(linear matrix inequality,LMI)形式,便于利用MATLAB求解。構(gòu)造LK泛函時(shí)通常采用簡(jiǎn)單型[8]、多重積分型[910]、增廣型[11]、松弛型[1214]等形式。常用的積分不等式主要有Jensen積分不等式、Wirtinger積分不等式、輔助函數(shù)積分不等式等。
1問(wèn)題描述
2主要結(jié)果
3數(shù)值算例
4結(jié)語(yǔ)
本文通過(guò)引入增維的松弛型LK泛函、增廣型LK泛函和多重積分型LK泛函構(gòu)造了一個(gè)新的LK泛函,結(jié)合積分不等式技術(shù),得到了一個(gè)新的廣義時(shí)滯系統(tǒng)容許性條件。與同類文獻(xiàn)相比,結(jié)論具有較小的保守性,數(shù)值算例說(shuō)明了結(jié)論的可行性和優(yōu)越性。以后可以考慮設(shè)計(jì)狀態(tài)反饋控制器或輸出反饋控制器對(duì)廣義時(shí)滯系統(tǒng)進(jìn)行容許性控制。
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