非线性部分习题答案

1、精选优质文档-倾情为你奉上精选优质文档-倾情为你奉上专心-专注-专业专心-专注-专业精选优质文档-倾情为你奉上专心-专注-专业描述函数法：1.根据题意有：, 根据题意可得图形（略）。其交汇点在实轴上，即 ，得到此时频率为：幅值为即：由于极限环为系统从不稳定区域到稳定区域极限环，因此该极限环是稳定的。2.根据题意有：系统没有交汇点，并且曲线在的外面，故系统是稳定的。3.根据题意有：饱和特性为在实轴上的最大值为。如果存在交汇点，则交汇点在实轴上，即 ，得到此时频率为：临界值为：得到：。4.根据题意画出BODE图，然后得到低频带宽即可！5.如果存在交汇点，则交汇点在实轴上，即 ，得到此时频率为：临界

2、值为：得到：。系统拥有极限环。当，系统不稳定。当，系统稳定。相平面法：1.得到：因此有：得到：绘制相平面为：2.上课讲过（略）。3.比较复杂，略。4. 根据公式有：得到奇点为：分别以该点进行线性化可得：5.分段即可。6.以这道题为例，进行讲解。第一问为课本例题，系统存在临界极限环。第二问：有， 即， 当， 当绘制系统的相平面轨迹为：略。可得系统是稳定的。根据非线性方程得到非线性状态方程，并且基于状态方程绘制系统的相平面图形，进而判定系统的稳定性及其它特性。李雅普洛夫函数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)圆 是一个椭圆 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) 平衡点，线性化，asymptotic stability. 将平衡点移到原点位置，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.实对称矩阵的正交对角化。 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，进而。故，。令，4.7 The second law of thermodynamics states that the entropy of an i