10.1.DEFINITIONS AND BASIC PRINCIPLES 211 We say that N()is an (operator)norm compatibie with the (vector)norm It is left as an exercise for the reader to show that the norms,and are operator norms induced by the vector norms,2 ando respectively. The Frobenius norm is not an induced norm but it is compatible with Also,if (is an eigenvale of d and r is a corresponding eigenvector,then for compatible matrix and vector norms x=·z≤1Az (10.1-30) or (10.1-31) 、 (H≤想. Let p(A)be the spectral radius of4-i.e.. p4)=axA:(1 (10.1-32: Because /10.1-31)hoids for any eigenvalues of 4 p.A)≤ (10.1-33) Thus the spectral radius forms a lower bcund on any compatible matrix norm. 10.1.5 Singular Values and the Singular Value Decomposition The singular valses oia oulexnxm matrix 4.denoted )are the k largest nonnegative square ro心2 e eigeavalues of3」A wherek三infn以.that 04=.i）=1.2,…k (10.1-34 where we assume that-thare ordered such that.In the last section we asked the reader to show that the maximum singular value is the matrix norm induced by the vector norin2-i.e..the spectral norm. We can define the maximuin (and miningum (singular values alternatively bv Al2 .)=mx =12 (10.1-35) 0 g4)= x1-1 max =精4-2fA1 exists (10.1-36) r≠0 x x27-i |x2 可 min 84-1x2 =-m叫n =E1I1 0a2 费X车正月作网下它最大粮粒天家灯 最小效术到二位充人于月是奇并。一 过<入4<了
212 CHAPTER 10.FUNDAMENTALS OF MIMO FEEDBACK CONTROL Thus and g can be interpreted geometrically as the least upper bound and the greatest lower bound on the magnification of a vector by the operator The smallest singular valuc d)measures how near the matrix A is to being singular or rank deficient (a matrix is rauk deficient if both its rows and columns- are linearly dependent)To see this.consider finding a matrix L of minimum spectral norm that mnakes 4+L rank dieficient deficient there exists a nonzero vector such that 2 1 and (d+L)x =0. Thus by(10.1-35)and(10.1-36) Since +L must be rank A1:奇昂白es=Bs= Therefore,L must have spectral norm of at least g).Otherwise A+L cannot (10.1-3:) be rank deficient.The property that 八之) implies.that ALis.nonsingular (assuming square matrices)and will be a basic 10.1-38) inequality used in the formulation of various robustness tests. Definition 10.1-6.A compler matrin A is Hermitian if4=. Definition 10.1-7 A.compies matrirnry 里-a A convenient way of representing a matrix that exposes its iuternal structure is known as the Siuguiar Value Decomposition iSVD).For an n x m matrix A. the SVD of A is given'by .A=S= where U and V are unitary matrices with column vectors denoted by {10.1-39) U=(u,42,4n} (10.1-40a1 对捅线奶=，3… and S contains a diagonal nonnegative definite matrix Si of singular values ar- (10.1-406) ranged in descending order as in f= n≥m 9=(SD10; n≤m (10.1-41) 年摸裤行5营灭荐关心单疾一
10.1.DEFINITIONS AND BASIC PRINCIPLES 213 and EI=diag {o1,02,...,; k=minfm,n) (10.1-42) where 名网n 10=01202≥.≥0k=2 It can be shown easilyithat the columns of v and U are unit eigenvectors of A#A and respectively.They are known as the righi and left singular vectors of the matrix A.Trivially all unitary matrices have a spectral norm of unity.Thus by SVD an arbitrary matrix can be decomposed into a"rotation"(VH)followed by scaling (followed by a "rotation"(U). Example 10.1-4.The SVD of the matrix 0.8712 -1.31951 1= 1.5783 -0.0947 is 1 It is interpreted geometrically in Fig.10.1-1. 口 Let A()be the eigenvalue of minimum magnitude of d and the associated eigenvector.Then from (10.1-36)we find a()=min A2 H 0z2 =这4引 10.1-43) Combining (10.1-33)and (10.1-43)we conclude that z and o bound the magni- tude of the eigenvalues: 2(1)≤1入，(4川≤6(1) 致五视1) If is Hermitian then the singular values and the eigenvalues coincide. Define u=u.un =u.vI =6.Um=2.Then if follow's that 10=0u (10.1-45) 12=互业 (10.1-46) From a systems point of view the vector (u)corresponds to the input direction with ihe largest (smallest)amplification.Furthermore alu)is the output direction in which the inputs are most (least)effective
- 214 CHAPTER 10.FUNDAMENTALS OF MIMO FEEDBACK CONTROL [ U S Raration Sealing Rotation 】 :0.1-/.Geometric interpretation of the SVD for Ex.10.1-4. ：A5苑车空匿 吉 :疾2R7=47
10.1.DEFINITIONS AND.BASIC PRINCIPLES 215 两灯 If A is square and aonsingular then A--VS-IUH 0绿}(10.1-47 is the SVD of4-I but with the order-ef the singular values reversed. Let nj+1.Then it foilows from (10.1-47)that 01=1/o(A) (10.1-48) u,(.A)=v(A) (10.1-9a) .1-i）=u) 10.1-496 and in particular G.1)=1/A) (10.1-50: a(A-=iA) 10.1-51a1 A1}=() (10.1-516 When Giis er matrix we can plotthe singnlar values ,(G= 1.....k)as a functis:.of requency.These curves generalize the SISO ampl:ude- ratio Bode plot to 111O systems.In the MIIMO case the amplification of the inpur vector sinusoid depends on the direction of the complex yector the ampiification s hG)and at mostG) 弦地修 ，4 10.1.6 Norms ou Function Spaces 「6)昌受店星 Ia this section we w:ll illustrate the exteusion of the concept of a norm to iinear spaces whose elements are functions.Let us Arst consider the vector valued function y(s)of dimension n.WVe define the set La to be the set of all vector functions withlimension n.which are square-integrable on the imaginary axis -i.for which thewng quantits finite.多5乏 (10.1-52) Note that (10.1-52)defines the 2-norm of a function y(s)through an inner prod. uct.For the specia!case when y(s)has no poles in the closed RHP.Parseval's theorem yields an euivalent time domain expression for the 2-norm ofy(t)():