【正文】 to the Lyapunov equation [53, 56, 74, 951]Interesting problems arise immediately if the matrix A is perturbed by another matrix, say A’. For example, it may be of interest to understand the relationship between eA and eA+A’ (where t = 1 for convenience here). If A and A’ mute, then clearly eA+A’= eAeA’= eAeA, whereas if A and A’ do not mute then eA+A’, eAeA, and eAeA are generally different [13]. Furthermore,as shown by examples in [85], eAeA’= eAeA does not imply eAeA’= eA+A’, eA+A’= eAeA’= eAeA does not imply AA’= A’A ,and eAeA’=eA+A’, and eAeA’= however, A and A’ have only algebraic entries then eAeA’= eAeA implies that A and A’ mute [85, 86]. If A and A’have algebraic entries and eAeA’= eA+A’, then it is reasonable to conjecture that A and A’ must mute, but this case is not discussed in [85] and remains open. Specializing to the case A’ = AT, a related open question is the following [14]: Does there exist a nonnormal matrix A satisfying either Some relevant results are given in [75]. In a somewhat different vein, the CampbellBakerHausdorff formula from Lie group theory [l0, 79, 83, 84, 87] states that if A and A’ have sufficiently small norm, then there exists a matrix D in the Lie algebra generated by { A, A’} that satisfies Specifically, D is given by the expansion where . Of course, at least one such matrix D satisfying () must always exist, and it need not be unique . The expansion (), however, is only locally convergent [83]. Thus () can only be used to determine the existence of D in the Lie algebra generated by { A, A’} when the norm of [A, A’] is sufficiently small.A remarkable result of a related, but slightly different, nature is given in [80]. If A and A’ have sufficiently small norm, then there exist invertible matrices S and T (depending upon A and A’) such that Furthermore, it is known that S and T are of the form ep and eQ where P and Q are elements of the Lie algebra generated by A and A’. An alternative, globally convergent expansion is given by [70] where, for k = 0, 1,…， Another class of related results involves inequalities for spectral functions of products of exponentials. Such bounds may be useful for robust stability of sampleddata control systems [16]. For example, if A and A’ are symmetric, then [24, 57]while for arbitrary A we also have [12, 23]A closely related result is An interesting open question that immediately arises is whether or not it is possible to derive ()()directly from any of the formulas (), () ,or (). In this regard () appears to be the most promising candidate. Finally, note that for implies Hence if A is stable, then the lefthand side of ()will converge to zero, whereas the righthand side may beunbounded, rendering the bound useless. A generalization of () in the spirit of () with A’= AT may be useful here. To resolve the possible conservatism in () for it is natural to conjecture generalizations of () to include terms of the form , where the positive definite matrix P is chosen as in Lyapunov stability theory to render ATP + PA negative definite. 翻譯：在線性系統(tǒng)和控制中出現(xiàn)的矩陣?yán)碚撻_(kāi)放性問(wèn)題摘要控制理論一直以來(lái)都提供了矩陣?yán)碚摪l(fā)展的強(qiáng)大動(dòng)力。我發(fā)現(xiàn)許多相關(guān)的知識(shí)在書(shū)上都能找到，因此只要根據(jù)課本設(shè)計(jì)出來(lái)是沒(méi)多大困難的。與平時(shí)所做的實(shí)驗(yàn)都是按照實(shí)驗(yàn)指導(dǎo)書(shū)的說(shuō)明很機(jī)械的完成相比，這次畢業(yè)設(shè)計(jì)給了我很大的思考空間，在設(shè)計(jì)過(guò)程訓(xùn)練了我的自學(xué)能力，并也開(kāi)始學(xué)著在給定任務(wù)情況下該何如查找資料，何如在設(shè)計(jì)過(guò)程的時(shí)間內(nèi)能更好地分配所要學(xué)習(xí)的內(nèi)容服務(wù)于設(shè)計(jì)的需要，而不會(huì)沒(méi)有主次之分。6. 2心得剛開(kāi)始接到這題目時(shí)幾乎無(wú)從下手，這主要是因?yàn)樽约哼x的設(shè)計(jì)課題和其他同學(xué)的一點(diǎn)很不相似，沒(méi)有共同的要求，很難和其他同學(xué)一起溝通。根據(jù)設(shè)計(jì)任務(wù)和設(shè)計(jì)要求本人從多方面查找資料和學(xué)習(xí)相關(guān)的知識(shí)，在查找資料，學(xué)習(xí)相關(guān)知識(shí)和設(shè)計(jì)過(guò)程可分以下幾點(diǎn)：（1）根據(jù)設(shè)計(jì)任務(wù)和要求并學(xué)習(xí)教科書(shū)中第五章《線性系統(tǒng)的頻域分析》和第六章《線性系統(tǒng)的校正方法》的內(nèi)容；（2）在課本的理論知識(shí)的基礎(chǔ)上，學(xué)習(xí)Matlab軟件，主要是自動(dòng)控制在　Matlab軟件中的運(yùn)用部分，以及學(xué)習(xí)電路仿真軟件（Multisim軟件）。第6章設(shè)計(jì)總結(jié)本章主要是講一下本次設(shè)計(jì)的小結(jié)。5．2Multisim電路設(shè)計(jì)仿真方式使用matlab軟件中的Simulink仿真：其單位階躍相應(yīng)如下對(duì)應(yīng)的階躍響應(yīng)圖如下：校正后閉環(huán)傳遞函數(shù)可以看成一個(gè)積分環(huán)節(jié)與四個(gè)慣性環(huán)節(jié)及兩個(gè)微分環(huán)節(jié)。 10dB，500。grid。den=[ 165 1 0]。（4）繪出系統(tǒng)開(kāi)環(huán)傳函的bode圖，利用頻域分析方法分析系統(tǒng)的頻域性能指標(biāo)（相角裕度和幅值裕度，開(kāi)環(huán)振幅）。end %計(jì)算調(diào)節(jié)時(shí)間 stime=t(t1)stime = plot(t,y)grid on由圖形可以看到系統(tǒng)是穩(wěn)定的。y(t1)*yss。m=m+1。end %計(jì)算上升時(shí)間 m=1。while y(n)*yss。[ym,tm]=max(y)。t1=length(t)。 den0=[ 165 10]。 rlocus(num0,den0),grid（3）作出單位階躍輸入下的系統(tǒng)響應(yīng)，分析系統(tǒng)單位階躍響應(yīng)的性能指標(biāo)。程序如下： num0=[143 10]。 [z,p,k]=tf2zp(num0,den0)z = p = + k = 20 pzmap(sys1)可以得到校正后的閉環(huán)零點(diǎn)有兩個(gè)，分別為z =，z=；閉環(huán)極點(diǎn)有5個(gè)，分別為p =，p= + ，p= ，p=，p= 。2)為校正前系統(tǒng)的閉環(huán)零極點(diǎn)：閉環(huán)傳遞函數(shù)為：143 s^2 + s + 10 s^5 + s^4 + 165 s^3 + s^2 + s + 10程序如下： num=conv(conv([10],[ 1]),[10 1])；den=conv(conv(conv(conv([1 0],[1 1]),[ 1]),[ 1]),[100 1])；sys=tf(num,den) Transfer function: 143 s^2 + s + 10 s^5 + s^4 + 165 s^3 + s^2 + s sys1=feedback(sys,1) Transfer function: 143 s^2 + s + 10 s^5 + s^4 + 165 s^3 + s^2 + s + 10 num0=[143 10]。den=conv([ 1],[100 1])。的指標(biāo)要求。如需確保，可以通過(guò)減弱滯后校正部分對(duì)相角遲后的不利影響來(lái)達(dá)到。故校正裝置處的相角，即為所求相角裕量，從圖中測(cè)量（或計(jì)算）得幅值裕量等于16dB；基本上滿足指標(biāo)要求。因此，超前校正部分的傳遞函數(shù)為（5）將滯后超前校正部分的傳遞函數(shù)組合在一起，得滯后－超前校正網(wǎng)絡(luò)的傳遞函數(shù)（6）驗(yàn)算校正后系統(tǒng)的相角裕量。（3）當(dāng)已校正系統(tǒng)的截止頻率確定后，便可以初步確定滯后校正部分的第二個(gè)交接頻率，選取，于是，根據(jù)sinφm=（a1）/（a+1）選擇，則遲后部分第一個(gè)交接頻率即為。采用滯后－超前網(wǎng)絡(luò)是完全可以達(dá)到的。在未校正系統(tǒng)的相頻特性曲線中可以看出。本例這方面并沒(méi)有提出明確的要求，因而前述超前滯后校正設(shè)計(jì)的步驟失效。說(shuō)明未校正系統(tǒng)是不穩(wěn)定的。5. 1. 2校正裝置的設(shè)計(jì)過(guò)程及其性能指標(biāo)的仿真1應(yīng)用頻率法進(jìn)行串聯(lián)超前滯后校正的步驟如下：（1）根據(jù)的要求，繪制未校正系統(tǒng)的開(kāi)環(huán)對(duì)數(shù)頻率特性曲線。[gm,pm,pf,gf]=margin(num,den)Warning: The closedloop system is unstable. In warning at 26 In at 66 In margin at 98gm =