线性系统理论资料线控辅导第7章

7.1 Given . Find a three-dimensional controllable realization check its observalility 找 的一三维可控性突现，判断其可观性。 Solution : using (7.9) we can find a three-dimensional controllable realization as following this realization is not observable because (s-1)and are not coprime 7.2 find a three-dimensional observable realization for the transfer function in problem 7.1 check its controllability 找 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 7.1 中 的一三维可控性突现，判断其可观性。 Solution : using (7.14) ，we can find a 3-dimensional observable realization for the transfer function : and this tealization is not controllable because (s-1) and are not coprime 7.3 Find an uncontrollable and unobservable realization for the transfer function in problem 7.1 find also a minimal realization 求题7.1中传函的一个不可控且不可观突现以及一个最小突现。 Solution : a minimal realization is , an uncontrollable and unobservable realization is 7．4 use the Sylvester resultant to find the degree of the transfer function in problem 7.1 用Sylvester resultant求题7.1 中传函的阶 solution : rank(s)=5. because all three D-columns of s are linearly independent . we conclude that s has only two linearly independent N-columns . thus deg 7.5 use the Sylvester resultant to reduce to a coprime fraction 用Sylvester resultant 将 化简为既约分式。 Solution : null thus we have and 7.6 form the Sylvester resultant of by arranging the coefficients of and in descending powers of s and then search linearly independent columns in order from left to right . is it true that all D-columns are linearly independent of their LHS columns ?is it true that the degree of equals the number of linearly independent N-columns? 将 和 按s 的降幂排列形成Sylvester resultant ，然后从左至右找出线性无关列。所有的D-列是否都与其LHS线性无关？ 的阶数是否等于线性无关N-列的数回？ Solution ； from the Sylvester resultant : (the number of linearly independent N-columns 7.7 consider and its realization show that the state equation is observable if and only if the Sylvester resultant of is nonsingular .考虑 及其实现 证明该状态方程可观及其当且仅当 的Sylvester resultant 非奇异 proof: is a controllable canonical form realization of from theorem 7.1 , we know the state equation is observable if and only id are coprime from the formation of Sylvester resultant we can conclude that are coprime if and only if the Sylvester resultant is nonsingular thus the state equation is observable if and only if the Sylvester resultant of is nonsingular 7.8 repeat problem 7.7 for a transfer function of degree 3 and its controllable-form realization , 对一个3阶传函及其可控型实现重复题7.7 solution : a transfer function of degree 3 can be expressed as where are coprime writing as , we can obtain a controllable canonical form realization of EMBED Equation.3 the Sylvester resultant of is nonsingular because are coprime , thus the state equation is observable . 7.9 verify theorem 7.7 fir .对 验定定理7.7 verification : is a strictly proper rational function with degree 2, and it can be expanded into an infinite power series as EMBED Equation.3 EMBED Equation.3 7.10 use the Markov parameters of to find on irreducible companion-form realization 用马尔科夫参数求 的一个不可简约的有型实现 solution deg =2 the triplet is an irreducible com-panion-form realization 7.11 use the Markov parameters of to find an irreducible balanced-form realization 用 的马尔科夫参数求它的一个不可简约的均衡型实现. Solution : Using Matlab I type This yields the following balanced realization 7.12 Show that the two state equation and are realizations of , are they minimal realizations ? are they algebraically equivalent? 证明以上两状态方程都是 的突现,他们是否最小突现?是否代数等价? Proof : thus the two state equations are realizations of the degree of the transfer function is 1 , and the two state equations are both two-dimensional .so they are nor minimal realizations . they are not algebraically equivalent because there does not exist a nonsingular matrix p such that 7.13 find the characteristic polynomials and degrees of the following proper rational matrices note that each entry of has different poles from other entries 求正则有理矩阵 , 和 的特征多项式和阶数. Solution : the matrix has and det =0 as its minors thus the characteristic polynomial of is s(s+1)(s+3) and =3 the matrix has , , , . det as its minors , thus the characteristic polynomial of Every entry of has poles that differ from those of all other entries , so the characteristic polynomial of is and 7.14 use the left fraction to form a generalized resultant as in (7.83), and then search its linearly independent columns in order from left to right ,what is the number of linearly independent N-columns ?what is the degree of ? Find a right coprime fraction lf ,is the given left fraction coprime?用 的左分式构成(7.83)中所示的广义终结阵,然后从左到右找出其线性无关列.线性无关的N-列的数目是多少> 的阶是多少?求出 的一个既约右分式.题给的左分式是否既约? Solution : Thus we have And the generalized resultant rank s=5 , the number of linearly independent N-colums is l. that is u=1, So we have the given left fraction is not coprime . 7.15 are all D-columns in the generalized resultant in problem 7.14 linearly independent .pf their LHS columns ?Now in forming the generalized resultant ,the coefficient matrices of and are arranged in descending powers of s , instead of ascending powers of s as in problem 7.14 .is it true that all D-columns are linearly independent of their LHS columns? Doe’s he degree of equal the number of linearly independent N-columns ?does theorem 7.M4hold ?题7.14 中的广义终结阵中是否所有的D-列都与其LHS 列线性无关?现将 和 的系数矩阵按S得降幂.排列构成另一种形式的广义终结阵.在这样的终结阵中是否所有的D-列也与其LHS 列线性无关? 的阶数是否等于线性无关N-列的数目?定理7.M4是否成立? Solution : because ALL THE D-columns in the generalized resultant in problem7.14 are linearly independent of their LHS columns Now forming the generalized resultant by arranging the coefficient matrices of and in descending powers of s : we see the -column in the second colums block is linearly dependent of its LHS columns .so it is not true that all D-columns are linearly independent of their LHS columns . the number of linearly independent -columns is 2 and the degree of is I as known in problem 7.14 , so the degree of does not equal the number of linearly independent -columns , and the theorem 7.M4 does not hold . 7.16, use the right coprime fraction of obtained in problem 7.14 to form a generalized tesultant as in (7.89). search its linearly independent rows in order from top to bottom , and then find a left coprime fraction of 用题7.14中得到的 的既约右分式,如式(7.89)构造一个广义终结阵, 从上至下找出其线性无关行, 求出 的一个既约左分式. Solution : The generalized resultant thus a left coprime fraction of is 7.17 find a right coprime fraction of snd then a minimal realization 求 的一个既约右分式及一个最小突现] solution : where the generalized resultant is rank the monic null vectors of the submatrices that consist of the primary dependent -columns and -columns are , respectively thus a right coprime fraction of is we define then we have thus a minimal realization of is _1100023145.unknown _1100199021.unknown _1100199030.unknown _1100244027.unknown _1100245962.unknown _1100255584.unknown _1100256411.unknown _1100256421.unknown _1100256435.unknown _1131222162.unknown _1131389707.unknown _1131390187.unknown _1131389427.unknown _1100256443.unknown _1100256502.unknown _1100256436.unknown _1100256432.unknown _1100256433.unknown _1100256422.unknown _1100256416.unknown _1100256418.unknown _1100256419.unknown _1100256417.unknown _1100256413.unknown _1100256415.unknown _1100256412.unknown _1100256406.unknown _1100256409.unknown _1100256410.unknown _1100256407.unknown _1100255740.unknown _1100255774.unknown _1100255924.unknown _1100255593.unknown _1100254696.unknown _1100254760.unknown _1100255301.unknown _1100255304.unknown _1100254715.unknown _1100246322.unknown _1100254458.unknown _1100245990.unknown _1100246193.unknown _1100245590.unknown _1100245595.unknown _1100245667.unknown _1100245736.unknown _1100245597.unknown _1100245641.unknown _1100245593.unknown _1100245594.unknown _1100245591.unknown _1100245259.unknown _1100245587.unknown _1100245589.unknown _1100245586.unknown _1100244767.unknown _1100244863.unknown _1100245027.unknown _1100244239.unknown _1100244245.unknown _1100241571.unknown _1100241583.unknown _1100243443.unknown _1100243671.unknown _1100243715.unknown _1100244018.unknown _1100243659.unknown _1100243044.unknown _1100243023.unknown _1100243027.unknown _1100242258.unknown _1100241584.unknown _1100242241.unknown _1100241578.unknown _1100241580.unknown _1100241581.unknown _1100241579.unknown _1100241574.unknown _1100241576.unknown _1100199034.unknown _1100240409.unknown _1100241134.unknown _1100241393.unknown _1100240507.unknown _1100240917.unknown _1100238565.unknown _1100240394.unknown _1100238623.unknown _1100199036.unknown _1100238269.unknown _1100199032.unknown _1100199033.unknown _1100199031.unknown _1100199026.unknown _1100199028.unknown _1100199029.unknown _1100199027.unknown _1100199023.unknown _1100199024.unknown _1100199022.unknown _1100199012.unknown _1100199017.unknown _1100199020.unknown _1100199018.unknown _1100199019.unknown _1100199015.unknown _1100199016.unknown _1100199013.unknown _1100186831.unknown _1100186835.unknown _1100186840.unknown _1100186842.unknown _1100186844.unknown _1100186849.unknown _1100186845.unknown _1100186843.unknown _1100186841.unknown _1100186837.unknown _1100186838.unknown _1100186836.unknown _1100186833.unknown _1100186834.unknown _1100186832.unknown _1100185600.unknown _1100186160.unknown _1100186829.unknown _1100186026.unknown _1100186157.unknown _1100185711.unknown _1100185782.unknown _1100185705.unknown _1100023676.unknown _1100024239.unknown _1100023570.unknown _1097568072.unknown _1099853041.unknown _1100022235.unknown _1100022725.unknown _1100022905.unknown _1100023141.unknown _1100022879.unknown _1100022900.unknown _1100022750.unknown _1100022617.unknown _1100022076.unknown _1097741563.unknown _1097742510.unknown _1097755758.unknown _1099848107.unknown _1099848167.unknown _1099848846.unknown _1097756013.unknown _1097748227.unknown _1097741721.unknown _1097568468.unknown _1097581559.unknown _1097581463.unknown _1097581071.unknown _1097581238.unknown _1097568597.unknown _1097568173.unknown _1097568459.unknown _1097568136.unknown _1097562950.unknown _1097566688.unknown _1097566693.unknown _1097565399.unknown _1097565654.unknown _1097566173.unknown _1097566201.unknown _1097565510.unknown _1097565139.unknown _1097565296.unknown _1097565013.unknown _1097519627.unknown _1097519920.unknown _1097562654.unknown _1097519730.unknown _1097518540.unknown _1097518874.unknown _1097519205.unknown _1097518676.unknown _1097518306.unknown