非線性部分習(xí)題答案

1、精選優(yōu)質(zhì)文檔-傾情為你奉上精選優(yōu)質(zhì)文檔-傾情為你奉上專心-專注-專業(yè)專心-專注-專業(yè)精選優(yōu)質(zhì)文檔-傾情為你奉上專心-專注-專業(yè)描述函數(shù)法：1.根據(jù)題意有：, 根據(jù)題意可得圖形（略）。其交匯點在實軸上，即 ，得到此時頻率為：幅值為即：由于極限環(huán)為系統(tǒng)從不穩(wěn)定區(qū)域到穩(wěn)定區(qū)域極限環(huán)，因此該極限環(huán)是穩(wěn)定的。2.根據(jù)題意有：系統(tǒng)沒有交匯點，并且曲線在的外面，故系統(tǒng)是穩(wěn)定的。3.根據(jù)題意有：飽和特性為在實軸上的最大值為。如果存在交匯點，則交匯點在實軸上，即 ，得到此時頻率為：臨界值為：得到：。4.根據(jù)題意畫出BODE圖，然后得到低頻帶寬即可！5.如果存在交匯點，則交匯點在實軸上，即 ，得到此時頻率為：臨界

2、值為：得到：。系統(tǒng)擁有極限環(huán)。當(dāng)，系統(tǒng)不穩(wěn)定。當(dāng)，系統(tǒng)穩(wěn)定。相平面法：1.得到：因此有：得到：繪制相平面為：2.上課講過（略）。3.比較復(fù)雜，略。4. 根據(jù)公式有：得到奇點為：分別以該點進(jìn)行線性化可得：5.分段即可。6.以這道題為例，進(jìn)行講解。第一問為課本例題，系統(tǒng)存在臨界極限環(huán)。第二問：有， 即， 當(dāng)， 當(dāng)繪制系統(tǒng)的相平面軌跡為：略?？傻孟到y(tǒng)是穩(wěn)定的。根據(jù)非線性方程得到非線性狀態(tài)方程，并且基于狀態(tài)方程繪制系統(tǒng)的相平面圖形，進(jìn)而判定系統(tǒng)的穩(wěn)定性及其它特性。李雅普洛夫函數(shù)The norm used in the definitions of stability need not be the u

3、sual Euclidian norm. If the state-pace is of finite dimension n (i.e., the state vector has n components), stability and its type are independent of the choice of norm (all norms are “equivalent”), although a particular choice of norm may make analysis easier. For n=2, draw the unit balls correspond

4、ing to the following norms: (Euclidian norm)圓 是一個橢圓 菱形 正方形Recall that a ball B(, R), of center and radius R, is the set of X such that , and that the unit ball is B(0, 1).For the following systems, find the equilibrium points and determine their stability. Indicate whether the stability is asymptoti

5、c, and whether it is global. (a) (b) 平衡點，線性化，asymptotic stability. 將平衡點移到原點位置，令，得：令，得：因此global stability.(c) (d) (e) Consider an matrix M of the form , where N is a matrix. Show that M is positive definite. if, and only if, and N has full rank.4.4 Show that if M is a symmetric matrix such that then

6、M=0.實對稱矩陣的正交對角化。 Show that if symmetric positive definite matrices P and Q exist such that then all the eigenvalues of A have a real part strictly less that.令即可。Consider the system where the vector is the state, and the matrices are all symmetric positive definite. Show that the system is globally asymptotically stable, with 0 as a unique equilibrium point.令，可得，進(jìn)而。故，。令，4.7 The second law of thermodynamics states that the entropy of an i