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1、PROBLEMS OF CHAPTER 44.1 An oscillation can be generated by 一个振荡器可由下式描述： 试证其解为：Show that its solution is Proof: ,the eigenvalues of A are j,-j;Let .If ,then on the spectrum of A,then then so 4.2 Use two different methods to find the unit-step response of用两种方法求下面系统的单位阶跃响应：Answer: assuming the initial

2、 state is zero state. method1:we use (3.20) to computethen andthen for t>=0method2:for t>=04.3 Discretize the state equation in Problem 4.2 for T=1 and T=.离散化习题4.3中的状态方程，T分别取1和Answer:For T=1,use matlab:ab,bd=c2d(a,b,1)ab =0.5083 0.3096 -0.6191 -0.1108bd =1.0471 -0.1821for T=,use matlab:ab,bd=c

3、2d(a,b,3.1415926)ab =-0.0432 0.0000 -0.0000 -0.0432bd =1.5648 -1.04324.4 Find the companion-form and modal-form equivalent equations of求系统的等价友形和模式规范形。Answer: use ab ,bb,cb,db,p=canon(a,b,c,d,companion)We get the companion formab = 0 0 -4 1 0 -6 0 1 -4bb = 1 0 0cb = 1 -4 8db =0p =1.0000 1.0000 0 0.50

4、00 0.5000 -0.50000.2500 0 -0.2500use use ab ,bb,cb,db,p=canon(a,b,c,d) we get the modal formab = -1 1 0 -1 -1 0 0 0 -2bb = -3.4641 0 1.4142cb = 0 -0.5774 0.7071db = 0p = -1.7321 -1.7321 -1.7321 0 1.7321 0 1.4142 0 04.5 Find an equivalent state equation of the equation in Problem 4.4 so that all stat

5、e variables have their larest magnitudes roughly equal to the largest magnitude of the output.If all signals are required to lie inside volts and if the input is a step function with magnitude a,what is the permissible largest a?找出习题4.4中方程的等价状态方程使所有状态变量的最大量几乎等于输出的最大值。如果所有的信号需要在伏以内，输入为大小为a的的阶跃函数。求所允许

6、的最大的a值。Answer: first we use matlab to find its unit-step response .we type a=-2 0 0;1 0 1;0 -2 -2; b=1 0 1'c=1 -1 0; d=0;y,x,t=step(a,b,c,d)plot(t,y,'.',t,x(:,1),'*',t,x(:,2),'-.',t,x(:,3),'-')so from the fig above.we know max(|y|)=0.55, max(|x1|=0.5;max(|x2|)=1.0

7、5,and max(|x3|)=0.52for unit step input.Define thenthe largest permissible a is 10/0.55=18.24.6 Consider where the overbar denotes complex conjugate.Verify that the equation can be transformed intowith ,by using the transformation with 。试验证以上变换成立。Answer: let with ,we get so 4.7 Verify that the Jorda

8、n-form equationcan be transformed intowhere are defined in Problem 4.6 and is the unit matrix of order 2.试验证变换成立，其中是习题4.6中定义的形式。为二阶单位阵。PROOF: Change the order of the state variables from x1 x2 x3 x4 x5 x6 to x1 x4 x2 x5 x3 x6And then we get4.8 Are the two sets of state equationsandequivalent?Are the

9、y zero-state equivalent?上面两组状态方程等价吗？它们的零状态等价吗？Answer:obviously,so they are zero-state equivalent but not equivalent.4.9 Verify that the transfer matrix in (4.33)has the following realization:This is called the observable canonical form realization and has dimension rq.It is dual to (4.34).验证式(4.33)具

10、有如上的实现。这叫做能观标准型实现，维数为rq。它与式(4.34)对偶。Answer:this satisfies (4.33).4.10 Consider the proper rational matrix考虑如下有理正则矩阵Show that its observable canonical form realization can be reduced from Problem 4.9 as 试证习题4.9种系统的能观标准形实现能降低维数如下表示：Answer: In this case,r=4,q=1,soits observable canonical form realizati

11、on can be reduced from 4.9 to 4.104.11 Find a realization for the proper rational matrix求下面正则有理传递函数矩阵的一种实现。so the realization is4.12 Find a realization for each column of in Problem 4.11,and then connect them.as shown in fig4.4(a).to obtain a realization of .What is the dimension of this realization

12、 ?Compare this dimension with the one in Problem 4.11.求习题4.11中每一列的实现，在如图4.4(a)所示把它们连结得到的一种实现。比较其与习题4.11的维数。Answer:These two realizations can be combined asthe dimension of the realization of 4.12 is 3,and of 4.11 is 4.4.13 Find a realization for each row of in Problem 4.11 and then connect them,as s

13、hown in Fig.4.4(b),to obtain a realization of .What is the dimension of this realization of this realization?Compare this dimension with the ones in Problems 4.11 and 4.12. 求习题4.11中每一行的实现，在如图4.4(b)所示把它们连结得到的一种实现。比较其与习题4.11、4.12的维数。Answer:These two realizations can be combined asthe dimension of this

14、 realization is 4,equal to that of Problem 4.11.so the smallest dimension is of Problem 4.12.4.14 Find a realization for 求的一种实现Answer:4.16 Find fundamental matrices and state transition matrices for 求下列状态方程的基本矩阵和状态转移矩阵：Answer:for the first case: we have for the second case: we have the two initial s

15、tates are linearly independent;thus4.17 Show 试证Proof: first we know 4.18 Given ,show 已知A(t),试证Proof: 4.20 Find the state transition matrix of 求状态转移矩阵Answer:then we havethe two initial states are linearly independent;thus4.21 Verify that is the solution of 试验证为下面方程的解：Verify:first we know ,then4.23 Fi

16、nd an equivalent time-invariant state equation of the equation in Problem 4.20.求习题4.20中方程的等价时不变状态方程。Answer: let Then 4.24 Transform a time-invariant into is by a time-varying equation transformation.求把(A,B,C)转换成的时变状态转移方程。Answer: since (A,B,C)is time-invariant,then 4.25 Find a time-varying realization and a time-invariant realization of the inpulse response .求冲击响应的时变实现及时不变实现。Answer: the time-varying realization is4.26 Find a realization of . Is it possible to find a time-invar