206 CHAPTER 10.FUNDAMENTALS OF MIMO FEEDBACK CONTROL and substituting (10.1-4)into (10.1-5)we find s=(C(sI-A)-B+Du(s） where (10.1-6 Gs)三C(sI-A-1B+D (10.1-7 is referred to as thesystem transfer matrin.The elements (gij(s)}of G(s)are transfer functions expressing the relationship between specific inputs uj(s)and outputs yas).In ths book.except for proofs and derivations we will use the transfer matrix rathe::han the state space description. The ma.trix Gls)w:e assuned to be of full normal ra2达一ie..rank iG(s三 min [m.n}for everysin te set of complex aumbers C,except for a finite number of clements of C. In general,the elss}will be allowed to include delays.The time domain realization ofransfer matrices with delays is complex and will not be addressed here.In orro be physically realizable the transfer matrices have to be proper and causal. Defimitien 1011imGls is proper ifallits elements ure proper and siricily proper f clements are stmcily proper.All systems which are not proper ure iniprop Definition 10.1-2.AstcmGls:is causal if ull its elements gs)are causnl All systems which a uure noncausul. 10.1.2 Poles Definition 10.1-3. 特纪值 .0e3!= t说肥2osu时hesi以10- ”etce 、T0 nomini5.secd2s fs=Π-， 10.1-S Thus the poles are ri roo:s of the characteristic equation (s=0 The poles determiue the ssrem'stability.) (10.1-9) Theorem10.1-1.Te'ystem (10,1-01-2以i这e2 and only乐akUi poles n are in the opeu ieft half plane. The following theore alls us to determine the system poles directly from the transfer matrix Gs without performing a realization and constructing the matrix A first
rn:>竹元？3分 10.1.DEFINITIONS AND BASIC PRINCIPLES 最小、么 西 207 Theorem 10.1-2. The pole polynomial (s)is the least common denominator of all non-identically-zero.minors of all orders of G(s). Example 10.1-1.Consider the matrix 1 (s-1)(s+2) 0 (s-12 G()=9+19+2)-1 -(s+1)(s+2) (3-18+1)(s-1)(3+1)/ The minors of order 1 are the elements themselves.The minors of order 2 are G- 2 G3- (s+1)川s+2) 行刘式道 s+1)(s+2) G- -(s-1) (5+1(s+22 上行 升大 (Here superscripts dente the rows and subseripts the columns used for compu- tation of the minors).Considering the minors of all orders (i.e..orders 1 and 2) we find the least comon denominator 5=s+1)s+2}2{s-11 0 10.1.3 Zeros Recall that if is a zo of the SISO system g(s)then g(s)=0.Furthermore, we know that s is a zer,of gts)if and only if s is a pole of gs).The following definition consisteutly"xtends this concept of a zero to MIMO systerus Definition-1014.〈aero of Gls)if the ra吵时i less than the norme rank of G(s) In other words.sines G(s)is assumed to be of full normal rank.the transfer matrix G(s)becomes rank deficient at the zero s=.The zero polynomial (s) is defined as 缺·不王 n: 5(s)=IΠ(s-S) (10.1-10) =l where n:is the number of finite zeros of G(s).Thus the zeros are the roots of (s)=0 (10.1-11) The following theorern provides a method for calculating the zeros. 犬么 约r Theorem 10.1-3.The zero polynomial ((s)is the greatest common divisor af the numerators of all order-r minors of G(s).where r is the normal rank of G(s) 公6
208 CHAPTER 10.FUNDAMENTALS OF MIMO FEEDBACK CONTROL provided that these minors have all been adjusted in such a way as to have the pole polynomial (s)as their denominator. Example 10.1-2.Consider the system from Ex.10.1-1.and adjust the denom- inators of all the minors of order 2 to be (s) G号=8-1s+2 T(s) G8- 2(s-1(s+2) π(s) G好=-2 and so (s) 5(s）=5-1) 7tc不线生 5(只 As Exs.10.1-1.and 10.1-2.show.MIMO systems ean have zeros and poles at the same location Therefore it is generally not possible to find all the zeros of a square system from the conditiondeGis)=0 because determinant,zeros and poles at the same location cancel. when forming the Definition1015.A system G(s2s九0u2n2ump边a,se (NMP头of its transfer matriz contains zeros in the RHP or ther czists a common time delay term that can be factored out of every matrin eleme Note that the zero locations of a MIMO system are in no way related to the 一上及不 zero location of the individual SISO transter functions constituting the MIMO system.Thus.it is possible for a AIMO ssem to be NMP even when all the SISO transier functions are MP and vice vers Example 10.1-3.The system G(s=1 43 5+13 has one finite zero at s=+3 though all the SISO transfer functions are MP. 10.1.4 Vector and Matrix Norms Let E be a linear space over the field (ivpically K is the field of real R or complex mumbers C)We say that a real valued functionI is a norm on E if and only if al>0a∈E、x≠0 (10.i-12a)
10.1.DEFINITIONS AND BASIC PRINCIPLES 209 1x1=0E三0一 (10.1-12b) lazll三al VaEK、xEE一 (10.1-13) x+l≤xi+yi,a,y∈E (10.1-14) A norm is a single number measuring the "size"of an element of E.Given a linear space E there may be many possible norms on E.Given the linear space E and a norm on E,the pair (E,i)is called a normed space. In this section let the linear space E be Cr.More precsely a Cn means that =(r1.22....)with ziEC.Vi.Three commonly used norms on C"are given by zp(P+z2”++rP)p=1.2.x (10.1-15) where is interpreted as max.The norm is the usual Euclidean length of the vector r. Let E=Cnxn,the set of all n x n matrices with clenients in C.E is a linear space.The following are norms on cnxn A1-max∑ay 10.1-16 Hlx=max∑al 10.1-1) 1 Frobenius or Euclidean norm 10.1-1S) 诸 IlAll2 max Af(4441 Spectral norn 10.1-19) 帆 像 where the superscript H is used to denote complex cdnjugate transpose. The eigenvalues A4)are (necessarily)real and nontegative. Some useful rela- tionships involving the spectral and Frobenius norms are l2≤H:≤VnA2 (10.1-20) where d s Cnxn These inequalities follow from the fact that is positive semidefinite and maxA(,AHA)≤Al=trace[]≤nmax.AHH) (10.1-21) ∠v A州：A复芝沉转置
210 CIIAPTER 10.FUNDAMENTALS OF MIMO FEEDBACK CONTROL Here we considered matrices as elements of a linear space.Next we shall cousider matrices as representation of linear maps and shall relate the matrix nonas to the vector norms of the domain and range spaces. First let us adopt the following notation.From now on we shall use for norms on R"or c"and for norms on function spaces (see Sec.10.1.6 or for induced norms of linear operators. Let be a norm on E and let be a linear map from E into E.Define the function AlA SuD 14x ≠0 (10.1-22 or equivalently ‖三sup |Az s=1 10.1-231 The guantity is called the indueed orm of the linear map tor the opefor norm induced by the aorm To interpret (:0.1-23)geometrically,consider the set of ail vectors oi nir length-i.e.,the uuit sphere.Then is the least upper bound on the magni- fication of the elerents of this set by the operator .4. It is easy to sw :om the derinition (10.1-22)that any induced norm satises 1≤A 0.i-2 ai=…. i0.1- i+B≤i4÷B 10.1-i ABI≤A·HB 你理绳10：- Any matrix nori V()which in addition to thelaxioms (10.1-12)-(10.1 satisfies V(AB)≤VH.Y(B) f10.1-2s is called compatible.(It is "compatible with itself.")An induced norm is an example of a compatible norm.It can be shown-that for every compatible max norm N()there exisis a vector norm such that Az≤V(Aa (10.1-29