Appendix A

Matrix Algebra and Control

Boldface lower case letters, e.g., a or b, denote vectors, boldface capital letters, e.g., A, M, denote matrices. A vector is a column . Containing m elements (entries) it is referred to as an m-vector. The number of rows and columns of a matrix A is nand m, respectively. Then, A is an (n, m)-matrix or n x m-matrix (dimension n x m). The matrix A is called positive or non-negative if A>, 0 or A :2:, 0 , respectively, i.e., if the elements are real, positive and non-negative, respectively.

A.1

Two matrices A and B may only be multiplied, C = AB , if they are conformable. A has size n x m, B m x r, C n x r. Two matrices are conformable for multiplication if the number m of columns of the first matrix A equals the number m of rows of the second matrix B. Kronecker matrix products do not require conformable multiplicands. The elements or entries of the matrices are related as follows Cij = 2::;;'=1 AivBvj 'Vi = 1 ... n, j = 1 ... r . The jth column vector C,j of the matrix C as denoted in Eq.(A.I3) can be calculated from the columns Av and the entries BVj by the following relation; the jth row C j ' from rows Bv. and Ajv:

column C j = L A,vBvj , row Cj ' = (CT),j = LAjvBv, (A. 1) /1=1 11=1

A matrix product, e.g., AB = (c: c:) (~b ~b) = 0 , may be zero although neither multipli• cand A nor multiplicator B is zero. Without A or B being the nullmatrix the product AB only vanishes if both A and B are singular. The matrix B = A 1. is the (right) annihilator of A, i.e., AA 1. = O.

A.2 Properties of Matrix Operations

Distributivity: A(B + C) = AB + AC . Associativity of addition (A + B) + C = A + (B + C) and multiplication (AB)C = A(BC). Commutativity of addition, non-commutativity of multiplication and raise to higher powers:

A+B=B+A, AB f- BA, (A.2)

Exceptions: Consider, first, a multi variable control with transfer matrix G( s) in the forward path and unity feedback and, second, H in the forward path and F in the feedback where G = FH. The overall transfer matrix is given by

(I + FH)-IFH = FH(I + FH)-I . (A.3)

The inverse of the return-difference matrix and G commute unexpectedly. Another exceptional case is the product A exp (At), i.e. the coefficient matrix and the state transition matrix. Finally, suppose A and B nonsingular. Then, A and B commute if their product is the : Both AB = I and BA = I yield A = B-1 . Generally, the matrices A and B commute with respect to multiplication if B is a function of A , e.g. as given by a matrix or by the decomposition in Eq.(A.45). 610 A Matrix Algebra and Control

Note that F(I + HF)-I = (I + FH)-I F and F(I + HF)-I H = (I + FH)-IFH , particularly observe the change of order within the parentheses. 0 Properties when transposing or inverting a matrix product:

(AA) Inverse and operations (symbols) may be permuted: (AT)-I = (A-If. If A-I = AT is true then A is referred to as an . An A has the property A 2 = A . This result can be observed in least squares, estimation and sliding mode theory, e.g.,

or A = I - B(C,B)-IC, . (A.5)

A matrix A is nilpotent if A k = 0 for some k. Such a matrix appears in the case of the state-space representation of a k-tuples integrator.

A.3 Diagonal Matrices

A A is a with non-zero entries A;; in the main diagonal, only, e.g.,

(A.6)

If these entries A;; are equal to each other A is a matrix. The identity matrix Inis a scalar matrix with elements 1 and dimension n x n: In ~ dia~(1,I,I, ... ,1), In E nnxn . Given a rectangular (n, r)-matrix B, premultiplying B by the identity matrix In or postmultiplying B by Ir yields InB = B or Blr = B . Premultiplying [postmultiplying] a matrix A by a diagonal matrix yields

...... ) ) , (A.7) i.e., a new matrix the rows [columns] of which are successively multiplied (scaled) by d; (i.e., i-th row [column] with di).

A.4 Triangular Matrices

A lower is a square matrix having all elements zero above the main diagonal, an upper triangular matrix only contains zero elements below the main diagonal. The product of two triangular matrices produces a triangular one again. If A is given as a diagonal matrix or an upper or lower triangular matrix, the eigenvalues A[A] are already given by the entries in the main diagonal A;; Vi = L.n.

A.S Column Matrices (Vectors) and Row Matrices

The unit m-vector with k-th component 1 is termed ek . Defining this m-vector ek and the n-vector ei , the or Kronecker matrix E;k is given by the dyadic product

ek = (0,0, ... I, ... ,0, of = e~mx I) = (Im).k E;k E nnxm (A.8) as an (n,m)-matrix with entry 1 only in the i,k-element and zero elsewhere. Thus, the (n,m)-matrix A can be established element by element: A = L7=1 L;'=I A;kEik where A E nnxm or dim A = n X m . The identity matrix can be achieved by the sum In = L7=1 e;er = L7=1 E;; . The sum vector with elements throughout unity 1 = (I, 1, ... ,If serves as a summation operator for an m-vector: ITa= L~ai. A.6 Reduced Matrix, , Cofactor, Adjoint 611

The inner product of two vectors a and b is a scalar aTb = bT a . Orthogonal vectors have zero inner product. Assume a vector output signal y given by the of a vector input signal x governed by the transfer equation y=Cx where yEnr , xEnn , CEnrxn . (A.9)

The entry Gij of C is considered as an operational factor from the input component Xj to the output component Yi. Note that the output subscript i (effect) is written first and j (source) is written second. A partitioned vector u is denoted by the "vec" symbol u- (:~ ) =vec(uI,u2, ... uN)=vecui=(uf,uI, ... u~f uiEnm " uEnm , m=Lmi. - u~ (A.lO) The or length of a is the distance to the nullvector (origin) and is defined by the Frobenius norm lIaliF = .,;;T; . The inner product is always smaller than the product of the norms of the multiplicators (Schwartz inequality): laTbl ::; lIallFllbliF . Triangle inequality (for any kind of norm): lIa + bll ::; Iiall + IIbll . The angle 0 between two vectors u and v is defined by cosO = uTv/(lluIlFllvIlF) . Mapping a matrix A

(A.ll) to a vector a is provided by the operator "col" (or by the operator "vec")

colA = vecA = a = (All ... AmI: AI2 ... Am2 ... Amnf (A.I2) The operator col lists the entries of A column-wise. Separating the ith column of A will be termed by (A).i. For abbreviation also A.i is used although, usually, only matrices are denoted by upper case boldface letters, irrespective of the subscript. A.i is defined as a column matrix (vector). The operator col (column string) can also be written as

col A = (A·f : A·I ... A.~f. (A.I3) The inner product of two real matrices A and B of equal dimension m x n is a scalar and coincides with the trace of the matrix product: (A, B) = (coIA)T colB = trATB = L:::I L:j=1 AijBij ::; IIAIIFIIBIIF . The Frobenius or Euclidian norm of a real matrix A is given by

(A.I4)

A.6 Reduced Matrix, Minor, Cofactor, Adjoint

Given the (n, n)-matrix A , the reduced matrix A"d ik of the size (n - 1) x (n - 1) is obtained by cancelling row i and column k. Repeating for NI ... n, k'v'I ... n yields n 2 different matrices. The minor on the i, k-component of A is defined as the det Ared ik . The cofactor of the i, k-element of A is obtained by permuting the sign of the minor, precisely by multiplying the minor with (-1 )iH, that is, COfikA = (_1)iH det Ared ik . Given the (n, n)-matrix A on the elements Aik, then A = (Aik) = matrix[AikJ and the adjoint is given as the transposed matrix of cofactors: adjA = [matrix( COfikA)JT . The determinant can be decomposed with respect to the row i , or with respect to the kth column, that is, n det A = L AikcofikA Vi = 1 ... n or det A = L AikcofikA 'v'k=l. .. n. (A.I5) k=1 i=l Interpretation of the equations above as a matrix multiplication yields

I det A = A adjA = (adjA) A A -I = adjA . (A.I6) detA 612 A Ma.trix Algebra. and Control

A.7 Similar Matrices

Matrices A and A are similar, i.e., A ~ A, if A = TAT-I. The preceding equation is named similarity transformation. Similar matrices A and A are characterized by the property det A = det A, by the same eigenvalues, eigenvalue multiplicities and eigenvalue indices (and by the same number of generalized eigenvectors), see Eq.(B.I07). Examples of similar matrices are A and A = diag ~i[A) = T-I AT in Eq.(B.I3).

A.S Some Properties of

Multiplying an (n, n)-matrix A by a scalar Jl yields a (n, n)-matrix r = JlA = (JlAik) = matrix!JlAik) . The determinants of A and r comply with the relation det r = det JlA = Jln det A . Determinant of the transpose: det AT = det A . Determinant of the product AB if both A and B are square: det(AB) = (det A)(det B). Determinant of the inverse: det(A -I) = (det A)-I = 1/ det A . In the case of an orthogonal matrix R the relation RT = R -1 holds and det R = ±I . The eigenvalues A[A) of an (n, n)-matrix A are related with the determinant and with the trace as follows n n det A = II ~i[A) trA = LAii = LAi[A). (A.I7)

Selecting two conformable matrices, the (m, n)-matrix A and the (n, m)-matrix B, the equivalence holds

det(Im + AB) = det(In + BA). (A. IS)

A.9 Singularity

The matrix U is singular and no inverse U-I exists if det U = 0 . The matrix U is nonsingular if its determinant does not vanish. A nonsquare matrix always is singular.

A.I0 System of Linear Equations

Solving a system of n linear algebraic equations Ax = b yields

x=A-Ib= adjA 1 b or Xk = -dA L(cofikA)bi (A.I9) detA et .. where the transposed matrix of cofactors replaces the adjoint.

A.ll Stable Matrices

Consider the set A(A) of eigenvalues AdA) of the (square) matrix A

A[A) = P;(A) E C det(A - ~iI) = 0 } . (A.20)

If and only if A is a subset of C- (complex numbers with negative real part), i.e., A[A) ~ C- , the matrix A is said to be asymptotically stable.

A.12 Range Space. . Null Space

The range (or image) of a linear operator (map) X -+ Y is the set of all linear combinations of the columns of the matrix A of this operator and is termed range space n[A), see Figs. A.I and A.2. The range space of A is the set of all vectors Ax where x ranges over the set nn .

n[A)~ {y:y EY, x E X, Y = Ax} C Y x E X Enn , A E nmxn . (A.2I) A.12 Range Space. Rank. Null Space 613

Example: Consider m > n, m = 3, n = 2, A column-like

(A.22)

Within the three-dimensional Euclidian space y the range space of A is a plane through the origin spanned by both columns of A . In a case m < n the columns of A are linearly dependent which yields redundancy but is not in contradiction to the definition Eq.(A.2I). 0 The dimension of the range space is the rank of A : dim R[A] = rank A , i.e., the number of linearly independent columns of A . In any square or rectangular matrix the maximum number oflinearly independent columns is equal to the maximum number of linearly independent rows. The rank can be checked either from rows or columns of a matrix. Given the (m, n)-matrix A , if m ~ n and rank A = m the matrix is of full row rank. If m :::: n and rank A = n the matrix is of full column rank. In the case m = n = rank A the matrix A is nonsingular. An (m, n)-matrix A is said to be of full rank if rank A is n or m whichever is less, thus

rank A = min {n,m}. (A.23) A square matrix of full rank is nonsingular, a nonsingular square matrix is of full rank. If a matrix is nonsingular, the inverse of the matrix exists. Since the rank of a matrix A is the maximum number of linear independent rows or columns of A the rank may be considered the order of its largest nonsingular square submatrix (non-vanishing minor). Two matrices A and B, B = rAn, are equivalent if rank A = rank B and both rand n are nonsingular. The rank of a square matrix A equals the number of eigenvalues A[A] that are nonzero,

e. g., rank A = 1 A[A] = 0 ; -8 . 0 (A.24)

Usually, the input matrix B E R nxm in state-space representation is full rank: rankB = m . If not, the deficiency of rankB corresponds to redundant input signals U; Additionally, rank (BBIL - In) = n - rank B .

1 if a i 0 and b i 0 Rank of the dyadic (outer) product: rank (abT) = { 0 (A.25) if a = 0 or b = 0 . Some more properties:

rank a = 1 if a i 0, rank (A+B) ~ rank A+rank B, rank (AB) ~ min (rank A, rank B) . (A.26) The null space (or ) of a matrix A is termed N[A] and is defined as a subspace

N[A] ~ {x: x E X where Ax = O} eX (A.27) In other words: If Ax = 0, then x belongs to the null space of A , i.e., x E N[A], see Figs. A.I and A.2. The range space R[AT] and the null space N[A] are orthogonal since Y1 = ATx and AY2 = 0 yields YI Y2 = xT AY2 = 0 . The dimension of the null space plus dimension of column space of the (n, m)-matrix B is dim N[B] + dim R[B] = n . (A.28) Example: Consider a multi variable system x(t) = Ax(t) + Bu(t) , (A.29) its Kalman controllability matrix SK

(A.30) and a subspace SK spanned by the columns of SK . Then SK is the controllable subspace of the system given in Eq.(A.29). Only intitial conditions x(O) E SK can be controlled (Kalman, R.E., 1963). Within a limited time interval any system state in SK can be reached. If any vector z satisfies z E R[SK] then also Az E R[SK]' Consider a special case (m = 2,n = 2)

= B=b= ( 1 ) ~ • = ( 0.707 ) (A.31) A (0-5 -1)4 1 a 1 0.707 614 A Matrix Algebra and Control

y eYe 'R'"

Figure A.l: Range space (or image or column space) and null space (or kernel)

-1 ) rank SK = 1 f. 2, hence (A, B) not completely controllable. (A.32) -1

SK =R[A] = {y y = a ( ~ ), a E (-oo,oo)} (A.33) i.e., only the line a(l 1) is the controllable subspace of the x-plane. In this example with distinct eigenvalues

0.707 ( -0.196 ) T- I = ( 1.178 0.236) and T-IB = T-Ib = C·~04) T = (aj : a;) = 0.707 0.981 ' -0.850 0.850 . (A.34) As far as controllability is concerned, Eq.(B.20) shows that xio cannot be influenced by u since the second entry of T-I b is zero and the system equations in x mo are decoupled. There is a transient motion in xio , only, starting from the initial conditions, and decaying to zero. The first variable xjO is generated from u(t) via PTI-delay, conventionally. Invoking x = Txmo and omitting, for brevity, the homogeneous part of the solution, yields

x = (XI) = T(XjO) = T(XjO) = (0.707)x mo (A.35) X2 xio 0 0.707 I which corresponds to the line a(l : 1) . IfB = b coincides with an eigenvector b = a~ then the controllable subspace is reduced to this eigenvector. 0

A.13 Trace

The trace of a square matrix is defined as the sum of the elements Aii in the main diagonal. Moreover, the trace is identical to the sum of the eigenvalues. For comparison: The product of the eigenvalues yields the determinant. The trace of the dyadic product of two vectors gives the inner product: tr abT = aTb . Taking the trace of a matrix product, the matrix factors are commutative tr AB = tr BA = L:~=I L:~=I AikBki although AB or BA and their dimensions are strongly different, A is an (n,p)• matrix and B a (p, n)-matrix. This cyclic property for a product matrix in a trace argument is very convenient when matrix derivative operations with respect to matrices are studied. For applications see e.g. sensitivity theory. Further properties of the trace are

tr(A + B) = trA + trB, tr In = n . (A.36) A.14 Matrix Functions 615

A.14 Matrix Functions

Consider a function I(A) and assume that its Taylor expansion

I(A) = f ;"(Qi I(A») (A - AiY (A.37) j=O J! aA] A=A, exists in the vicinity of Ai where Ai = Ai[A) is one of the p eigenvalues of A of multiplicity mi and degeneracy qi. Inverting the characteristic polynomial e(A) = det(AI - A) and expanding into partial fractions, using a numerator polynomial bi (A)

(A.38)

The definition of ai( A) is

a'(A) ~ bi(A)e(A) (A.39) • - (A - Ai)m,

Multiplying Eq.(A.38) with e(A) and invoking Eq.(A.39) yields

1 = t ai(A) = e(A) t bi(A) (A.40) i=1 i=1 (A - Ai)m,

The matrix function I(A) associated with the scalar function I(A) is obtained by substituting A by A . From Eqs.(A.39) ,(A.40) and using Cayley-Hamiltion theorem etA) = 0

(A - M)m'ai(A) = bi(A)e(A) = 0 (A.41) and 2:;=1 ai(A) = I . Multiplying I(A) with ai(A) yields

00 1(j)(A;) . I(A)ai(A) = L -'-1-(A - AiI)]ai(A) . (A.42) j=O J.

The upper bound in the sum above can be replaced by mi. This results since one has (A-AiI)j b>m,-I = 0 in Eq.(A.41). If the multiplicity mmi of Ai in the minimal polynomial is mmi < mi then the upper bound can be replaced by mmi . Thus, taking the sum with respect to i yields

p p m,-l/(j)(A') L I(A)ai(A) = I(A) = L L -'-1-' (A - AiI)j ai(A) . (A.43) i:::l i=l j=o J.

With the abbreviation Di,HI ~ (A - AiI)j ai(A) = j! Zij (A.44) where Zij denotes the interpolating matrix polynomial or component of A , one has

p m,-I/U)(A;) I(A) = L L -'-1-D i ,j+1 . (A.45) i=l j=O J.

Note that the matrices Di,HI or Zij do not depend on the function I but only on A . Hence, any f can be used to determine Di,HI or Zij for a given A. The derivatives are very simple if I is taken the rth power I(A) = A r A r = t 1:' (r) A~-j Di,HI . (A.46) i=1 j=O J The last equation can be used to determine Di,HI . Now, defining column-like partitioned matrices Tn Dc = (DII : DIz : ... DJm p f and Ac = (I : AT : ... A f, Eq.(A.46) is rewritten to V dmg Dc = Ac where V dmg is the generalized . In the case mmi < mi one has Di,mm,+2 = 0 through Di,m,+1 = 0 (Frame, J.S., 1964; Ganlmacher, F.R., 1986). 616 A Matrix Algebra and Control

A.15 Metzler Matrices

A A E nnxn is characterized iii, j = [1, n] by

<0 i=j main diagonal { (A.47) aij = 2: 0 i f- j elsewhere.

A Metzler matrix is stable if and only if all leading principal minors Vk = [1, n] premultiplied by {_I)k

all ... alk a2l ... a2k 1 {_I)k det ( : . . > O. (A.48)

akl ... akk

Remark: To check the stability of an ordinary matrix one has to test all the principal minors. A Metzler matrix is stable if and only if it is quasi dominant negativ diagonal. A matrix A is said quasi dominant negative dominant if all aii < 0 and Vi,j = [1, n] there exist positive numbers di such that

n n either dda .. I > L dj laij I or dj lajj I > L dilaijl (A.49) j=l, iti i=l, itj is satisfied (Metzler, L.A., 1950; !iiljak, D.D., 1978; Mansour, M., 1987; Xin, L.X., 1987). Example: 0.5) A-_ (-1 0.5 -3 Al,2[A] = -3.118; -0.882 (A.50)

leading principal minors x ( _1)k :

There exists another definition of M-matrices which is explained using the notation P . The matrix P is an M-matrix if Pij ::; 0 Vi f- j and ~e A[P] > 0 . Then, Eq.{A.48) has to be rewritten with aij := Pij and {_I)k must be omitted. For any non-negative matrix B the matrix P = vI - B is an M-matrix if and only if v> 1I'[B] where 11'[.] denotes the Perron root. Finally, if P and D are square and non-negative and 1I'[P] < 1 then all the matrices

I-P-D; I-D{I-P)-l; I-{I-P)-ID (A.52) are M-matrices and the Perron root of all the following matrices is less than unity

P + D; D{I - p)-l; (I _ P)-ID . (A.53)

A.16 Projectors

Consider the n-dimensional space decomposed into two subspaces namely the range space and null space. The matrix P r is a projector ifit projects nn on the range space n[Prl ofPr along the null space N[Pr ] of Pro A matrix acts as a projector if it is idempotent, i.e. P; = Pro The projection operator function and the range space details can be seen by premultiplying a vector x of nn by P r

Prx = X . (A.54)

The above projection is essentially augmented by another projection: If the same vector x is considered to be projected by the projector In - P r the projection result

(I - Pr)x = 0 (A.55) is the null space, in other words, In - P r projects x of nn on the null space N[Pr 1along the range space n[Pr ]. Summarizing, P r projects nn on n[Pr ] along N[Prl (A.56) A.17 Projectors and Rank 617

II .. - P,

- plan 4xI -f "l.:rl r·, = 0

Figure A.2: Range space and null space of a projector

1- P r projects nn on N[Prl along n[Prl . (A. 57) Hence, n[Prl = N[I -P r land N[Prl = n[I - Prl .

Example: Range space and null space: Consider n = 3 and

0 0 0) P r = ( 0 0 0 ; (A.58) 4 2 1

Hence, XI = 0, X2 = 0, X3 undetermined, i.e. , the range space n[Prl is given by the x3-axis in this example (see Fig. A.2). Check 1 0 0) I-Pr = ( 0 1 0 ; (A.59) -4 -2 0 The null space N[P r 1 is obtained from P rX' = 0 or from the range space of I - p .. i.e., n[I - Prl: (I - Pr)x' = x' (A .50)

(A.5 1) (I-Pr)x'=x' ~ (i4 52 ~) (~~) (~~)

Thus, x~ = x~; x~ = x~; -4x~ - 2x~ = x;, i.e. the null space N[Prl is the plane 4xI + 2X2 + X3 = 0 (see Fig. A.2). The matrix P r is a projector on the x3-axis along the plane. The matrix (I - Prj is a projector on the plane along the x3-axis and rank P r = 1, rank (I - Prj = 2, tr P r = 1. 0

A.17 Projectors and Rank

Note that rank P r = tr P r and rank (In - Prj = n - rank P r . If the matrices B, K, H, C have full rank, the following relations are true

n[BK] = n[Bl and N[HC] = N[Cl (A.52) which is often used when treating the matter of sliding mode (EI-Ghezawi, O.M.E., et al. 1983 ). Sum• marizing: The rank of a projector matrix equals the dimension of its range space.

If Pr=B(CB)-IC ~ rankPr=rankB=m and rank(In-Pr)=n-rankPr=n-m. (A.53) 618 A Matrix Algebra and Control

Premultiplying any square matrix by I,. - P r will cause the product to have the rank of n - m (or less):

rank (I,. - Pr)A = n - m . (A.64)

A.IS Projectors. Left-Inverse and Right-Inverse

Let C be any (m, n )-matrix then the following list of idempotent matrices, subdivided into left-inverse and right-inverse matrices, serve as projectors: CCLl , C RI C, I". - CCLl , I,. - C Rr C . For instance, the projector P r = B(CB)-IC := CRIC is considered as a product of C RI and C. Then B(CB)-I = C RI is true since CCRI = CB(CB)-I = 1m.

A.19 Trigonal Operator

The P is characterized by the property that each row and column has exactly one element equal to unity while the other entries are zero. The matrix P can be derived from the identity matrix I by permuting columns or rows. Postmultiplying and premultiplying a matrix by P causes permutation of columns and rows of this matrix, respectively (Qianhua, W., and Zhongjun, Z., 1987). Finally detP = 1. As a special case of the permutation matrix the matrix is defined as the unity matrix with unity entries in the secondary diagonal, only,

001) 1'" = ( 0 1 0 . For comparison, I = I" = diag{l, 1, ... , I} . (A.65) 1 0 0

Postmultiplication of a matrix A with 1/ yields a of A across the vertical symmetric axis

( :: :~ ::) 1/ = (:: :~ ::) (~ ~ ~) (:::~::) (A.66) a7 as a9 a7 as a9 1 0 0 a9 as a7 Premultiplication determines a reflection ;'cross horizontal symmetric axis

(A.67)

The reflection operation holds for every kind of vector or matrix. The secondary diagonal matrix possesses the following properties: (1/)-1 = 1/ , (1/)2 = I ,1/ = vi, (1/ C)T = CTI/T = CTI/ . With regard to the symmetry of 1/ and trigonal matrices

(1/ trig bf = (trig b)TI/T = (trig b)I/, trig a = trig(ai) = trig[I/ x (an+I_;)). (A.68)

For a given vector b the trigonal matrix is defined as trig b

4 bo ) . . ~ ... trIgonal matrix (A.69)

The secondary diagonal of the original trigonal matrix is filled with bn , i.e., the entry hi of highest index n. Reflecting the vector b

(A.70)

The elements of the secondary diagonal of the result are troughout hi. A.20 Transfer Function Zeros and Initial Step Transients 619

Further properties:

b3 b2 b. b3 b2 Upper triangular matrix: (trig b)I/ = ~ (A.71) 0 b. "63 ) (" 0 0 b4

t( transpose)

0 0 b. 0 Lower triangular matrix: 1/ (trig b) = :: (A.72) b3 b. C'b1 b2 b3 n b2 b3 b4 b b 6 l 2 3 (A.73) 0 bl b2 ) 0 0 bl !( transpose)

~ ). (A.74) b1

The trigonal matrices in Eqs.(A.71) through (A.74) are persymmetric and Toeplitz matrices because the elements along and parallel to the main diagonal are identical (Grenander, U., and SzegiJ, G., 1958, Makhoul, J., 1981; Brillinger, D.R., 1981). The remaining trigonal matrices are of Toeplitz type with respect to the secondary diagonal.

A.20 Transfer Function Zeros and Initial Step Transients

Consider the single-input single-output system with the transfer function O( s)

Y(s) = O(s) = L~ /ksk = Z(s) (A.75) U(s) Lo aksk N(s) and with input signal u(t) (ic 0 t 2': 0;= 0 t < 0). For the sake of abbreviation, the following vectors of constants, signal derivatives and powers of the Laplace operator s are defined

fr = (to II··· In-If (A.76)

u(t) = (u(t) u(t) ... u(n-I)(t)f y(t) = (y(t) y(t) ... y(n-I)(t)f S = (1 S s2 ... sn-If. (A.77) If O(s) has zeros, i.e., coefficients Ii exist (i > 0), the output signal y changes suddenly between 0- and 0+ (FijI/inger, 0., 1961; Wunsch, G., 1971 ). The initial conditions of y obey the relation

(A.78)

is a matrix, relating the initial condition step transients in a general view of a and f (Weinmann, A., 1988 ). Note that the Laplace transform of the differential equation corresponding to Eq.(A.75) does not depend on y(O+) but only on y(O-) (Wunsch, G., 1971 )

Y(s)N(s) - yT(O-)(trig a) s = U(s)Z(s). (A.79)

The nth derivatives u(n)(o+) and y(n)(o+) comply with the following relation

y(n)(O+) = (t; - a;O)u(O+) - Inu(n)(o+) (A.80) an 620 A Matrix Algebra and Control

u(t) weighting yet) function get)

Figure A.3: System output analysis

Even in the case of non-existing zeros of O(s), i.e., t;li;tO = 0 and 0 = 0, the nth derivative y(n)(o+) has the non-zero amount (A.8I)

Using the state-space representation for the single-input single-output system of Eq.(A.75)

itt) = Ax(t) + butt) y(t) = eT x(t) + du(t) (A.82) and observing the definitions of boldface u(t) and y(t) in Eq.(A.77) which are different from the scalars u(t) and y(t), it results

y(O+)=(e ATe A 2,Te ... An-l,Tefx(O+)+L~u(O+) (A.83) where 0 = 1/ trig(I/ (d cTb cT Ab ... fJ is equivalent to Eq. (A.78).

A.21 Convolution Sum and Thigonal Operator

Definitions: u(!l.t) y(!l.t) g(o) u(2!l.t) y(2!l.t) g(!l.t)

u=(u;)~ , Y = (y;) ~ (A.84) u( i!l.t) y(i!l.t) g(i!l.t)

urN !l.t) y(N !l.t) g[(N - l)!l.tJ

Convolution integral and convolution sum (see Fig. A.3):

k-l k-l y(t) = l' g(r)u(t - r)dr , y(Mt) = L g(i!l.t)u(Mt - i!l.t)!l.t = L u(i!l.t)g(Mt - i!l.t)!l.t (A.85) i=Q System analysis: y = 1/ [trig (1/ u)) g!l.t or y = 1/ [trig (1/ g)) u !l.t (A.86) System identification: (A.87)

Input synthesis: (A.88)

Compensator design (see Fig. A.4):

e = !l.~2 (trig 1/ r)-l 1/ (trig 1/g)-l 1/ Y . (A.89)

Considering k -> 00 the equation above corresponds to infinite matrix equations in expanded form. For stability considerations and norm definitions of multivariable infinite matrices etc. see Makhlouf, M.A., 1972. A.21 Convolution Sum and Trigonal Operator 621

compensator plant reference r( t) v(t) output y(t) c(t) g(t) ~I I ~I I ~

Figure A.4: Compensator design Appendix B

Eigenvalues and Eigenvectors

Assume a stable (n, n)-matrix A with the following definitions: Characteristic matrix of A: AI - A . Characteristic function (polynomial) in A: C(A) = det (AI - A). Characteristic equation of A : C(A) = det (AI - A) = (A - AI)(A - A2)'" (A - An) = 0 . Roots of the characteristic equation yield the eigenvalues A[A). The coefficients of the characteristic polynomial C(A) = An + Cn_1 An- I + ... Co (B.1) can be written as K where tr(.) A = L det (principal minors of order r), K= (~). (B.2) j=1 Note that tr(n) A = det A and tr(1) A = tr A . The polynomial C(A) is a monic polynomial, i.e., a polynomial with coefficients one of the highest power in A . The coefficients Cn _. of the characteristic polynomial can be calculated from the trace of A up to the power n - i (starting with Cn = 1) according to the following formulas -tr A and 'Vi = [1,2 ... n) (B.3) 1 2'I . Cn-i -"7(Cn_.+1 tr A + Cn _.+2 tr A + ... + Cn_1 tr A'- + tr A') . (B.4) I The inverse matrix (AI - A)-I ~ iI'(A) is known as resolvent matrix for A and is identical to the fundamental or transition matrix iI'(A) . Cayley-Hamilton Theorem: In the characteristic equation A may be substituted by A : c(A) = o. The matrix A satisfies its own characteristic equation. Examples: A[In) = l(n times); A[dia&a;) = ai, a2, ... an; A[aA) = aA[A) . If any eigenvalue A[A) = 0 then det(A - AI) = detA = 0 and A -I does not exist. See Eq.(A.16). 0

B.l Right-Eigenvectors

The eigenvector a. is derived from Aa. = A.ai = AdA) a •. (B.5) These eigenvectors are denoted as right-eigenvectors. The transpose AT possesses the same eigenvalues A•. The right-eigenvectors Pi of AT are defined by (B.6)

B.2 Left-Eigenvectors

If eigenvectors a. are taken into consideration, they are interpreted as right-eigenvectors a. = a~ = aHA). Left-eigenvectors a~ are defined as a~T A = A.a~T . Transposing and comparing with Eq. B.6 shows the conformity with the right-eigenvector of AT. The left-eigenvector of A is identical to the right-eigenvector of AT (B.7) 624 B Eigenvalues and Eigenvectors

B.3 Complex-Conjugate Eigenvalues

If the eigenvalues Ai[A] of the real matrix A are complex-conjugate note the following properties

Aai Aj 8 i right-eigenvectors ai = ai of A, (B.8) aiTA ..\;a;T left-eigenvectors at of A, (B.g) ATat ..\;a; right-eigenvectors a?. of AT, (B.lO) arT ai == 1 and a;T a: = 0 or a;T a: == Dik . (B.ll) The asterix superscript denotes the complex conjugation. It does not matter if these properties are ap• plied in the case of real eigenvalues but not vice versa. The pair (Ai, ai) is denoted eigensolution of A. Then, the corresponding eigensolution of the transpose AT is (Ai, an . See also Eq.(6.70).

Example: 0 -1) (B.12) A= ( 2 -2

End of Example

Perturbation theory of eigenvalues see separate chapter. Numerical solutions see Wilkinson, J.H., 1965.

B.4 Modal Matrix of Eigenvectors

Assume all eigenvalues AdA] distinct. Then, the modal matrix T of a given square matrix A is built up by the right-eigenvectors T = (a"a2,'" ,an) . Multiplying T by A from the left and substituting Eqs.(B.8) and (A.7), right-hand side,

AT = A(al,a2, ... ,an) = T diagAi T-l AT = diagAi = A . (B.13)

In this expression A = A[A] = diag Ai[A] is the diagonal matrix of the eigenvalues Ai of A in the main diagonal. The eigenvector matrix associated with AT is named P, i.e., P = (Pl, P2,'" ,Pn) . Combining the modal matrices T and P and referring to Eq.(B.13),

(B.14)

Taking the transpose and comparing with the left-hand side of the equation above yields

(B.15)

In other words: The modal matrices T and P associated with A and AT are orthogonal. Hence, the corresponding eigenvectors are also orthogonal:

(B.16)

Table B.1 presents an overview of the matrices cited above where double lines denote identities. If a matrix A is symmetric the eigenvectors associated with A and AT are identical. The right and left-eigenvectors are identical, too. With regard to P = T and Eq.(B.15)

T= T[A) if (B.17)

Although the eigenvectors are normalized to unity the eigenvectors are only fixed with the exception of the sign. See also Eq.(6.70). B.5 Complex Matrices 625

Table B.1: Modal matrices T = T[A] and P = T[AT]

T Ii T T I Tl X p-ITX P p-I pT

In the case of real and complex conjugate eigenvalues and eigenvectors, the following relations are used. The definition of the modal matrix P!; (ai, a~ ... a~) associated with the transposed matrix AT corresponds to AT at = Ai at. Then, since at! at = 6ik

(B.lS)

B.5 Complex Matrices

If A E en xn the following notation is used: Aai D. A.a. and A H a1 D. Ai a1 where aJ at af: ai = Oki .

B.6 Modal Decomposition

Transformation of the original system

x(t) '" Ax(t) + Bu(t), x(O) '" Xo y(t) '" Cx(t) (B.19)

into the diagonalized modal system by defining the modal state variable x mo is achieved by substituting x '" Txmo. Thus,

y(t) '" CTxmo . (B.20)

If the sign of eigenvector i is altered, the sign of the i-th component of the modal vector xmo is merely changed.

B.7 Linear Differential Equations and Modal Transformations

The interrelations between original and modal domain are listed in Table B.2. An eigenvector a., in any case, is determined except a constant factor ai. To verify that the modal matrices T '" (al : a2 ... an) and T '" (alaI: a2a2 ... anan ) '" T diag a. possess the same modal decomposition, T is substituted by T in the expression A '" TAT-I

(B.21)

Example:

-3 -1) T '" (1/..;5 1/V2) I A", ( 2 0 ' -2/V2 -1/V2 ' T- '" (;:;: -:) , AI,2 '" -1; -2 (B.22)

- s~1 + s~2 - S~I + S~2) . 0 (B.23) 2 2 2 I s+1 - s+2 s+1 - s+2 626 B Eigenvalues and Eigenvectors

Table B.2: Transformation into the modal domain

Original domain Transformation Modal domain x=Ax x = Txmo x mo = Axmo A T lAT=AorA=TAT 1 A x(t) = eAt x(o+) xmo(t) = eAt xmo(o+) CI>(t) = eAt eAt = TeAt T 1 eAt = diag {exp( Ait)} 1_ CI>(s) = (sI - A) (sI - A) I=T(sI-A) IT 1 (sI - A) diag s~

B.B Eigenvalue Assignment

Defining a polynomial

PI(>') ;; Plo + PI 1>' + ...... + PI,n_l>.n-l + >.n = (pJ : 1)~ (B.24)

where PI ;; (Plo P/l" 'PI,n_dT and ~;; (1 >. ... >.n-l >.nf (B.25) consider an (n, n)-matrix A and an n-vector b and let (A, b) be an observable pair, i.e. det R;6 0 where R = (b, Ab, A 2b, ... An-1b). Then, the zeros of PI(>') are assigned to the eigenvalues of A + bkT if

(B.26)

The unique solution, given by Rissanen (Ackermann, J., 1980), is

(B.27)

the vector (R-1)n' being the last row of R-1 . It can be rewritten

(B.28) where en is the unit n-vector ek with k = n. Note that the matrix W has the property of transforming the system (A, b) to the control canonical form, i.e., W-l AW is the and W-1b = en .

B.9 Eigensystem Assignment

The closed-loop system given by the plant and the state feedback, respectively,

x(t) = Ax(t) + Bu(t), x(O) = x o , u(t) = Kx(t) (B.29) can be rewritten to x(t) = (A + BK)x(t) ;; Fx(t) . (B.30) Using the diagonal matrix of distinct eigenvalues or the Jordan canonical form J , and the modal matrix of eigenvectors T, one has

F = A+BK = TJT-1 AT-TJ= -BKT (B.31) and a Lyapunov-type of closed-loop system equation is obtained. Let rna denote a non-negative , rna ~ 0 and a > IIAIIF a positive constant. Then, it will be proved in what follows that choosing T according with the Lyapunov equation

- (A + rnaaIn)T + T[-(A + rnaaInW = -(rna + 1)BR-1BT where R> 0, T = TT > 0 (B.32) B.10 Complete Modal Synthesis 627 makes all the closed-loop eigenvalues ..\[F] lie to the left of -mall' in the complex s-plane, i.e., the controller matrix K is K == -(ma + I)R-lBTT-l ..\[F] < -mall' . (B.33) Proof: From Eq.(B.32) - AT - TAT - 2maO'T == -(ma + I)BR-lBT (B.34) AT + T(AT + 2maO'In) - maBR-lBTT-lT == BR-lBT (B.35) (A-maBR-lBTT-l)T+T(AT +2maO'!n)==BR-l BT . (B.36) , v '~ A, -J From Eq.(B.36) the following statements can be derived: First, Al == A - maBR-lBTT-l ~ A + BKl Kl == -maR-lBTT-l (B.37) where Kl is a feedback gain to stabilize A. Second, comparing Eq.(B.36) with Eq.(B.31), BKT == _BR-lBT K2 == _R-lBTT-l (B.38) where K2 is an additional feedback to shift ..\[F] to the left of -0'. Hence F == Al + BK2 == A + B(Kl + K 2) == A + BK (B.39) K == Kl + K2 == -(ma + I)R-lBTT-l . (B.40) Third, in order to obtain all the eigenvalues of J == _(AT + 2maO'In) in the left of -mall' in the complex plane, the decay factor 0' must satisfy 0' > IIAIIF since ..\[J] == "\[_AT - 2ma O'I] == "\[_AT]_ 2ma O' < -mall' (B.4I)

- "\[A] < mall' 0' > -"\[A]/ma ¢= 0' > IIAIIF . (B.42) End of the Proof Example: ..\i[A] == -1;-2 R==I (B.43)

ma == 1 IIAIIF == v'i4 == 3.7417 17m • x [A] == 3.7025 . (B.44) l 0 0' > 17m • x [A] "" 0' == 4 . From Lyapunov equation T == 301 (1 -4 -4)22 ' T- == (12100 25 ) (B.45) K == -(ma + 1)R-lBTT-l == -2(0 1) (110 == (-40 -10) (B.46) 20 20)5

F==A+BK==(~2 ~3)+(~)(-40 -10)==(_~2 -~3) ..\i[F]==-6;-7. (B.47) Check "\[-A]- 2ma O' < -4. End of Example

B.lO Complete Modal Synthesis

Consider the nth order open-loop system and its matrices A, B E nnxm, the closed-loop matrix with state feedback F == A + BK and the eigenvectors fi associated with F. Then,

(A + BK)fi == Ffi == ..\i[F] fi (A - ..\i[F] I)fi == -BKfi ~ - Bpi. (B.48)

With the aforementioned definition Pi ~ Kfi of n parameter vectors Pi E nm and a parameter matrix P E nmxn it results, using the modal matrix T[F] of the closed loop (Roppenecker, G., and Lohmann, B., 1989; Roppenecker, G., 1987 and 1988), fi == -(A - "\i[F]Inl-lBpi (B.49)

P == (Pl : P2 ... Pn) == (Kfl : Kf2 ... Kfn) == K(fl : f2 ... fn) == K T[F] . (B.50) 628 B Eigenvalues and Eigenvectors

B.ll Vandermonde Matrix

Consider a system with (n, n)-matrix A in companion form (regulator form, controllable canonical form) and distinct eigenvalues,

1 o o A= ( ; (B.51) o o -ao

Then, the eigenvectors (the eigenvector direction) and the modal matrix are given by the Vandermonde matrix Vdm

A2 T = Vdm = ( "~I A~ '. 1 (B.52) An-I An-I ,:~, 1 2 which simplifies computation to a high extent. Note the helpful relation that the determimant of the Vandermonde matrix is equal to rr~j=l;i

1 0 ... (-".-,-a n -2 0 ... A' ;; 1/ ATI/ = A = (B.53) -al 0 0 -ao 0 0 .. ;1 where A' is reflected from A with respect to the secondary axis and A is taken from the aforementioned A in companion form. Then, the corresponding modal matrix is denoted T' = T[A/J, the eigenvalues are the same AdA] = AdA/]. Combining TAT-I = A and T/A(T/)-I = A' = 1/ ATI/ in order to eliminate A , and referring to 1/ = (1/)-1

Example:

1 Ai[A] = -1, 5; (B.55) ai= CJG) T = Vdm = ( ~1 5 )

0 5 -1 A = T diagAi[A] T- 1 (B.56) (~1 ;) ( ~1 5 ) ( ) /6 . Eq.(B.54) ""

-1 I)T = (B.57) ( -11 5 (0 1 01) (-1 5 1) 1 T' = ( 5 ) /6 which is equivalent to T' or 1" , calculated directly from

1 A' = (: yielding, e.g., 1" ) . 0 (B.58) ~) ( -5 B.12 Decompostion into Eigenvectors 629

Consider the nth order system as given in Eq.(B.20) but single-input (u := u) and distinct eigen• values, only. The Vandermonde matrix V dm also plays an important role in decomposing the Kalman controllability matrix Sk as mentioned in Eq.(A.30). (B.59)

Then, rank Sk < n occurs if one or more elements (T-I b)i should vanish and controllability is no longer given. Furthermore, rankSk is the dimension of the controllable subspace (Ackermann, J., 1988).

B.12 Decompostion into Eigenvectors

In this section the solution of a system of linear algebraic equations is decomposed into a linear combi• nation of the eigenvectors of the system matrix. Consider the nonhomogenous equation Ax = b with the (n, n)-matrix A real and symmetric. The eigenvalues ~[Al are assumed to be distinct. The (right) eigen• vectors a; of A are orthogonal. Moreover, ai are normalized to unity. Expanding the unknown solution x into vector components according with ai n n n n x= LaiR;, Ax = A Laiai = LaiAa; = La; ~i[Al a; = b. (B.60) ;=1 ;=1 i=1 ;=1 Premultiplying this equation with aJ yields

n n arb aj~j[AlaJ aj+ L ai~i[AlaJ ai = aj~j[Alx1+0 = arb and x = L ,J[ laj . i=l;i;;!:j j=1 I\J A (B.61)

B.13 Properties of Eigenvalues B.13.1 Smallest and Largest Eigenvalue of Symmetric Matrices

Consider ~min and ~max , the smallest and largest eigenvalues of a symmetric (n, n)-matrix A , respec• tively. Then, the following relations hold for any vector x (B.62) If the matrix A is positive definite, the notation for the modulus can be omitted. All eigenvalues ~[Al are real and positive if A is positive definite. Assume all eigenvalues distinct. Thus, all eigenvectors a are distinct, too. With regard to the symmetry A = AT the eigenvectors are orthogonal, see Eq.(B.16), and form an orthogonal in n-dimensional space. Any vector variable x can be assumed of the form n X= Laia;. (B.63) ;=1 The quadratic forms xT Ax and xT x using Eq.(B.63) can be deduced as follows n n Ax = L aiAai = L a;~i[Al ai (B.64) ;=1 ;=1 n n xTAx = (LajaJ) Lai~i[Al a; = LLaiaj~iaJa; = La~~iarai . (B.65) j=1 i=1 j i i Using ~max leads to x T Ax = Li a~ ~iar a; $; Li a~ ~maxaT a; = ~maxxT x . Finally, an upper and a lower bound for the quadratic (Lyapunov) form is

~min[Al xT x$; xT Ax $; ~max[Al xT x . (B.66) From Eq.(B.64) one can derive

(B.67) 630 B Eigenvalues and Eigenvectors

If A and B are symmetric matrices and A > 0 then a nonsingular matrix S exists such that

ST(A + B)S = 1+ diagPi[A -IBn (B.68)

(Thrall, R.M., and Tornheim, L., 1957, p.188; Petkovski, D.B., 1985). For A, B, C E 1lnxn symmetric and A, B ;::: 0, C > 0 one has

Amax[AB] S Amax[AC] Amax[C-IB] . (B.69)

Proof: Setting C = TTT and using A;lM] = Ai[TMT- I ] and Eq.(22.25)

Amax[ATTT] Amax[T-IT-I,TB] = Amax[TATT] Amax[T-I,TBT-I] (B.70) IITATTII,IIT-1,TBT-1II, ;::: IITABT-1II, = Amax[TABT-1] = Am.x[AB]

(Corless, M., and Da, D., 1989). 0

B.13.2 Eigenvalues and Trace A similar property is given as follows. If G, H E 1lnxn where H ;::: 0 and G such that either the relations hold G ,H ;::: 0 (Kleinman, D.L., and Athans, M., 1968) or H = HT (Wang, S.D., et al. 1986), then,

Amin[G] tr H S tr [GH] S Am.x[G] tr H . (B.71)

Calculating bounds of the Riccati and Lyapunov matrix equation, these conditions are useful (Kwon, B.H., et al. 1985; Mori, T., et al. 1987).

B.13.3 Maximum Real Part of an Eigenvalue For A E cnxn consider Aai = A[A]ai, the expression af1Aa, = Ai[A]af1ai and its conjugate transpose af1 A H ai = Ai[A]af1 ai. Adding both expressions above,

(B.72)

Since the sum A + AH is Hermitian, from Rayleigh's theorem it is known that

af1 A\AH ai A + AH max H = Amax [--2--] . (B.73) a i ai 8i

Hence, (B.74)

If A E 1lnxn then the symmetric part A, is used to evaluate 3?e Ai[A]

(B.75)

Another property is

Amin[A,] + Amin[B,] S 3?e A[A + B] S Amax[A,] + Amax[B,] (B.76) where A, ~ (A + AT)/2, A, BE 1lnxn 0 (Jiang, C.L., 1987).

B.13.4 Definiteness of As and Stability of A

The symmetric part of a matrix A is A, ~ (A + AT)/2. If the matrix A, is negative definite (xT A,x < 0 for any x) since Eq.(B.73) and since ai E {x} "" Amax[A,] < O. Since Eq.(B.75) Amax[A,] > 3?e Ai[A]. Hence, A, < 0 3?e Ai[A] < 0 and A is stable. (B.77) Further properties concerning norms of a matrix see separate chapter and H eitzinger, W., et al. 1985. B.14 Rayleigh's Theorem 631

B.13.S Adding the Identity Matrix Adding /-II (/-I positive) to a matrix A causes increased eigenvalues

(B.78)

The eigenvalues are uniformly shifted to the right. The amount of shifting is /-I. Frequently A serves as a system coefficient matrix in state-space approach. Then, the displacement follows towards the imaginary axis, towards instability. Adding the /-I-scaled identity matrix to A -1 results in .\fJ.II+ A -1] = /-1+ 1/'\[A] . Example: Inverting (sIn - A) , the determinant det(sIn - A) is needed. Note that the s-dependent eigenvalues of the combined matrix (sIn - A) are related to '\;[A] as follows .\;[sIn - A] = -'\;[A] + s . For the sake of completeness,

n '\;[-A] = -'\;[A] det(-A) = (-1)" det A = (_I)n II '\;[A]. (B.79) ;=1 Characteristic equation and the sum and product of eigenvalues (Vieta's rule):

n det(sIn - A) = 0 sn + sn-1 I:{-.\i[A]} + + II{-.\;[A]} = O. (B.80) i=1 i=l

End of Example

B.13.6 Eigenvalues of Matrix Products Note the property '\[AB] = '\[BA] if A and B are square. (B.81)

B.13.7 Eigenvalue of a Matrix Polynomial For a matrix polynomial J(A) (or a function which can be represented by a matrix polynomial) there exists the relation between eigenvectors

.\[J(A)] = J('\[A]) . (B.82)

B.13.S Weyl Inequality Let A and E be Hermitian matrices and let a perturbed matrix be defined as Ap = A + E where the largest eigenvalue is denoted by the subscript n etc.

(B.83)

Then, .\i[A] + .\max[E] 2: '\;[Ap] 2: A;[A] + Amin[E] Vi = [1, n] . (B.84) If E > 0 then Ai[Ap] > Ai[A] Vi = [1, n] and I A;[Ap]- A;[A] I::; IIEII, Vi = [1, n] (B.85)

(Franklin, J.N., 1968, p.157). For comparison see Eq.(15.43).

B.14 Rayleigh's Theorem

Consider G = G H E c nxn and let G n_r E c(n-r)x(n-r) be a principal submatrix of G Vr = 1 ... n-l .

If the eigenvalues of G n_r are A1[Gn_r]::; A2[Gn_r] ::; ::; An-r [G n_r] and the eigenvalues of G are AdG] ::; A2[G] ::; ... ::; An[G] (B.86) then (Lancaster, P., and Tismenetsky, M., 1985, p. 294)

(B.87) 632 B Eigenvalues and Eigenvectors

B.15 Eigenvalues and Eigenvectors of the Inverse

Let an eigenvalue of A and A -I be A and J-l, respectively,

det(AI - A) = 0 (B.88) Multiplying by det A yields

det A det(J-lI - A -I) = det(J-lA - I) = 0 . (B.89)

Comparing with Eq.(B.88) yields J-l = 1/ A . If A['] is used as an operator for determining the eigenvalue, then generally, A[A -I] = A-I[A] . Denoting the eigenvectors of A -I as a· then A -Ia· = A-Ia· . Multi• plying by A from the left shows that a· = a is the solution. If a square matrix A has eigenvalues Ai and eigenvectors ai , then, the inverted matrix A -I has inverted eigenvalues 1/ Ai but identical eigenvectors a: = ai. Eigenvalues are reciprocals, the eigenvectors are shared.

Example: A= (~5 ~1) detA =-5 (B.90)

A-I = (-~i8 -~.2) det A -I = -0.2 (B.91)

J-lI,2 = -1; 0.2 • (0.7071) (-0.1961) (B.92) a l ,2 = 0.7071 ' . 0.9806 .

End of Example

B.16 Dyadic Decomposition (Spectral Representation)

Given the m-vector a and the n-vector b , the (m, n)-matrix abT is the dyadic product (or ). Using the modal matrices T[A] and P = T[AT] of A and AT, respectively, decomposed to columns (eigenvectors) T = T[A] = (al,a2' .. an) (B.93) one can decompose the (n, n)-matrix A

A = T(diag Ai)T- I = T(diag Ai)pT (B.94)

n n n n A = L Ai(aipT) = L(Aiai)pT = L(Aa;)pT = A L aiPT = AI = A . (B.95) ;=1 ;=1 ;=1 In this equation use is made of the following result

L aiPT = matrix[L aikPiI] = TpT = TT-I = I (B.96) ;=1 caused by the orthonormality of T and pT. Both expressions of A , mentioned in the right-hand side of Eqs. (B.94) and (B.95), are equal to a square matrix the (k, I)-element of which is 2::7=1 Ai aik Pil .

Example:

A = -1; 5 T _ (0.707 -0.196) p = (T-I)T = (1.178 -0.850) - 0.707 0.981 0.236 0.850 (B.97) A = (-1) ( ~:~~~ ) (1.178 0.236) + (5) ( ~~g~;6 ) (-0.850 0.850) (B.98)

A = (-1) (0.833 0.167) (5) (0.167 -0.167) (B.99) 0.833 0.167 + -0.834 0.834 . End of Example B.17 Spectral Representation of the Exponential Matrix 633

Starting once more from Eq.(B.94), the decomposition of A is A = T(diag A;)T-1 . Inverting yields

(B.100)

Comparison with Eq.(B.94) gives an expression corresponding with Eq.(B.95)

A-I = L Ail(aipTl . (B.I0l) ;=1

The dyadic product matrices (aiPTl are the same no matter if A or A -I is decomposed. The factors of these decomposition matrices are Ai and Ai I, respectively. With respect to the quadratic structure of the dyadic decomposition in the variables ai and Pi , the notation spectml representation is also of common use.

B.17 Spectral Representation of the Exponential Matrix

Suppose A E nnxn, Ai[A] distinct, the eigenvectors ai and ai linear independent and, finally, ai norma• lized: aiT ak = Cik. Decomposing A = 2:7=1 aiaiT AdA] and applying the well-known Taylor expansion for eA , n eA = L aia;T e·qAI (B.102) i=l n c}(t) = eAt = Laia;Te·>.;[Alt (B.103) i=l

B.18 Perron-Frobenius Theorem

A non-negative (n, n)-matrix A always has a non-negative eigenvalue .. [A] , named Perron root or Perron eigenvalue or Perron-Frobenius radius of A, such that IAdA] I :s: .. [A] Vi = [1, n]. If A is irreducible then .. [A] is positive and simple and the corresponding eigenvector can be chosen as a positive vector. A matrix is reducible if by identical row and column transpositions the matrix can be brought to an upper trigonal (A~I ~~~) where Au and A22 are square.

B.19 Multiple Eigenvalues. Generalized Eigenvectors

Consider A E cnxn . The n eigenvalues AdA] Vi = 1 ... n are the zeros of the characteristic polynomial det(Ai[A] In - A) with multiplicities counted. Multiple eigenvalues Ai[A] of (algebraic) multiplicity ffii correspond to an ffii-order zero. If there are p distinct eigenvalues 2:;=1 ffii = n . In the case ffii > 1 a full set of n linearly independent eigenvectors need not exist. Assume qi linearly independent eigenvectors associated with Ai. The number qi is the degeneracy of AdA] In - A

qi ~ n - rank (Ai [A] In - A) where 1:S: qi :s: ffii . (B.104)

In addition to the qi distinct eigenvectors, ffii - qi generalized eigenvectors (principal vectors) associated with AdA] are needed. There is the ambiguity which of the qi linearly independent eigenvectors the chain is associated with. Associated with an eigenvector means that the generalized eigenvector at equals a;, see Eq. (B.I07) and Fig. B.l. These ffii - q; generalized eigenvectors appear in qi chains (see e.g. Fig. B.l). The generalized eigen• vectors associated with Ai[A] are given as follows: Calculating (Ai [A] In - A)~ for I-' = 1,2 ... until for some I-' = k the relation

rank (Ai [A] In - A)k = rank (Ai[A] In - A)k+1 (B.105) is obtained. Then, it can be verified that the highest generalized eigenvector afj of rank k is given by

(B.106) 634 B Eigenvalues and Eigenvectors see ~l in Fig. B.l. The superscript k is used to distinguish between various generalized eigenvectors within a chain. This superscript, of course, cannot be mistaken with an exponent. If k f. mi there are two or more chains of generalized eigenvectors. The entire chain of generalized eigenvectors of rank IJ < k is given by VIJ= l. .. (k-l). (B.107) Totally, there are k generalized eigenvectors a;j VIJ = 1 ... k within this chain. This manifold k is denoted k = 8ij. The generalized eigenvectors are linearly independent, as can be easily verified. Each eigenvector or each chain of eigenvectors is labelled by an additional subscript j. The number of eigenvectors belonging to a chain is denoted 8ij. Following this procedure, j is affixed to every aij in the following Eqs. (B. lOS) up to (B.llI). Writing aij , the index j varies from 1 to qi. The overall sum of the manifold eigenvectors is E1~:; 8ij = m; . Note that Eq.(B.107) can be rewritten to

a~j-l = (Ai [A] In - A)afj or Aa~j = a~j-l + Ai[A]afj (B. lOS) af;2 = (Ai [A] In - A)2a~j = (Ai [A] I,. - A)a~j-l or Aa~j-l = a~j-2 + Ai[A]a~-l (8.109)

alj = (Ai[A] I,. - A)k-lafj = (Ai [A] In - A)a~j or A~j = alj + A;[A]a~j . (8.110)

PremuItiplication of the last row by A;[A] In - A and referring to Eq.(B.I06) yields

(B.lll) which reveals that alj (within the chain of generalized eigenvectors) corresponds to the ordinary eigen• vector I1;j. Synopsis of symbols: mi ... algebraic multiplicity of the eigenvalue A;[A] qi . . .. geometric multiplicity, degeneracy, number oflinear independent (distinct) eigenvectors associated with Ai [A] , dimension of the space spanned by the eigenvectors, number of chains m; - qi··· number of generalized eigenvectors associated with A;[A] (total number for all j associated with Ai) mmi.. index of the eigenvalue Ai in A given by the largest order of the Jordan blocks associated with Ai , Le., mmi = maxj 8lj 8ij . .. length of the chain containing the eigenvector aij and the generalized eigenvectors ~j through a:ji associated with Ai, Blj E [Sib sn ... Sill.,).

B.20 Jordan Canonical Form and Jordan Blocks

In order to obtain a canonical representation the following definitions are used: i) The Jordan canonical form J, i.e., a block diagonal matrix containing qi Jordan blocks for each eigenvalue Ai[A] (one Jordan block for each distinct eigenvector aij) . ii) The Jordan block Jij is given by

Ai 0 0 0 0 0 Ai 0 0 0 0 0 Ai 1 0 0 0 0 0 Ai 0 Jij ~ E C'ijX6ij (B.1I2)

0 0 0 0 0 Ai i.e., a matrix with the same Ai[A] for all main diagonal elements and ones on the diagonal just above the main diagonal. The size of the Jordan blocks within the Jordan canonical form is 8ij X 8ij and is not given merely by mi or qi. (In the example of Fig. B.I there are three Jordan blocks of dimension B.20 Jordan Canonical Form and Jordan Blocks 635

generalized eigenvectors 3 I \ "";!1.__ ", ~ j = 1 multiplicity mj 11""'"""-----_ ..,...... a;l first chain of the eigenvalue AdA] ••••• ___"'i'i'""~a/;'·2 .a.._- a;2 i = 1 .. ------~--~second cham ~ j = 2 qj distinct eigenvectors

Figure B.l: Brief sketch to obtain orientation about eigenvectors and generalized eigen• vectors (associated with Ai = 2, i = 1) and their multiplicity. Two chains (qi = 2), length of eigenvector chains is 3 and 2 vectors (Si1 = 3, Si2 = 2), mi = 5

3 x 3,2 x 2,1 x 1 associated with Ai') The order of the Jordan blocks within the Jordan canonical form is given by choosing Sil > Si2 > ... > Siq, > 1. Of course, ordering the eigenvectors within the modal matrix T, see Eq.(B.117), must correspond to the Jordan canonical form J. The Jordan canonical form J is given by

J ~ block diag (Jll, J 12 ,··., J lq" J 2l ... Jij ... J 2q" ... J pqp ) E cnxn . (B.113) Rewriting the right-hand equation block of Eqs.(B.108) to (B.110), one has Aialj (B.114)

aIj + Aia;j (B.115)

(B.116) which can be easily abbreviated to and identified as a special part of AT = TJ given in Eq.(B.117). The nonsingular modal matrix T E cnxn is composed by the eigenvectors and generalized eigenvectors. With the help of Jordan canonical form J and the modal matrix T the matrix A can be decomposed as follows A = TJT-l . (B.117)

Example (corresponding to Fig. B.1 if the single-input case is selected), n = 6,p = 2

T-1AT = J (B.118)

generalized eigenvectors associated with Al

T = (all all all a12 a12 a2) (B.119) '"'0 3 '" 2 /~ \ eigenvectors associated with Al eigenvector associated with A2 0 0 0.876 0 0 -1.142 0 0 0 0 1 0 0 0 -2 1.142 4 0 0 0 -0.438 1 4 0 0 0 1.~42 ~) T-l = 0.25 0 0.5 0.25 0 1.142 0 0 0 T{;5 0 0 0 -0.438 -1 ) ( 0 0 -1.142 0 -1~142 ~ 0 0.25 -0.876 0 0 0 0 0.5 0 -0.5 0 (B.120) 636 B Eigenvalues and Eigenvectors

ql = n - rank (A - All) = 6 - 4 = 2 (B.121)

i AdA] qi mi mmi 1 2 2 5 3 2 0 1 1 1

p C(A) = (A - 2)5 A = II (A - Ai)m; characteristic polynomial (B.122) ;=1

p m(A) = (A - 2)3 A = II(A - Ai)m~; minimal polynomial (B.123) ;=1

1 0 0 0 2 1 0 0 1 0 2 0 0 J= block diag [( 2 (B.124) 0 0 2 1 = ~ , ~ ) ,0]. 0 ~ ) ( ~ 0 0 0 2 (! 0 0 0 0 !) B.21 Special Cases

In the special case qi = mi (full degeneracy), there are mi separate eigenvectors and Sij = 1 'Vj = 1 ... mi' The Jordan blocks become 1 x 1 matrices, identical to the scalar Ai , the diagonalization of A by similarity transformation is still available if qi = mi 'Vi, see Eq. (B.117). Whenever qi '" mj for any i , diagonalization is substituted by the Jordan canonical form.

In the special case of qi = 1 (simple degeneracy), one has Sijlj=l = Sil = mi and J ij E cm,xm;, i.e. only one Jordan block for AdA]. If this should be true for each Ai then J = diag (J I , J 2 ... J p). This case arises, e.g., if the (n, n)-matrix A is given in companion form, irrespective of the multiplicity of the roots Ai since rank (AiIn - A) = n - 1 is always true. Various examples are given by Chen, C. T., 1970.

B.22 Fundamental Matrix

Calculating the fundamental matrix (state transition matrix) yields

(B.125)

The of the Jordan canonical form can be calculated block by block, thus

eJ , = block diag (811 ... 8ij ... 8 pqp ) eblock diag (J", .. J,j' J pqp ) (B.126) where the trigonal matrices as given below are used

2 3 t t /2! t /3' ,'•• -' Ii'" - 1)' 1 t t 2 /2! 1 t Sij = ( j 0 1 ... E CJI,X$IJ . (B.127) 0 0 1

For comparison, in the case of distinct eigenvalues there is Sij ;: 1 and the simple version

exp( diag Ait) = diag e~;' . (B.128) B.23 Eigenvector .;\ssignment 637

Table B.3: Multiple eigenvalues and degeneracy

Only distinct Multiple eigenvalues Ai with multiplicity mi eigenvalues simple degeneracy degeneracy, general case full degeneracy mi = 1 qi = 1 qi = 1 qi = n - rank(A - Ail,,) qi=mi Sij = Sil = mi(> 1) Sij E [Sib ... Siq.] Sij = 1 mmi = 1 mmi=mi mmi = maxj Sij mmi = 1 m(A) = C(A) m(A) = C(A) m(A) = niP - A;}mm, m(A) = ni(A - A;) diagonalization diagonalization diagonalization diagonalization feasible where impossible impossible feasible J = diag Ai only if qi = mi Vi one eigenvector one eigenvector qi eigenvectors qi eigenvectors per eigenvalue irrespective of mi(> 1) mi - 1 generalized mi - qi generalized no generalized eigenvectors eigenvectors eigenvectors only one Jordan block qi Jordan blocks mi Jordan blocks mixmi Sij x Sij 1 x 1 identical per eigenvalue per eigenvalue to the scalars Ai e.g., A in companion e.g., identity form regardless of mi matrix I"

B .23 Eigenvector Assignment B.23.1 Assignable Subspaces. Parametrization of Controllers Applying state feedback by (m, n)-matrix K in muItivariable systems, the degree of freedom is mn in order to assign n closed-loop eigenvalues. The remaining degree of freedom (m - I)n Can be utilized to assign eigenvectors. Assigning eigenvectors in addition to eigenvalues is an important matter because the eigenvectors are responsible for the relation between the state variables and the modal state variables (see Table B.2) and thus the eigenvectors determine the strength of eigenvalue dominance in the state variables. For a given control system in state space representation, i.e., (A, B) and state feedback K with the closed-loop behaviour (A + BK) = F, there exists a subspace within which the (right) eigenvectors fi of the closed-loop system are located and can be assigned arbitrarily.

B.23.2 Single Real or Complex-Conjugate Eigenvalues From the eigenvector definition (A + BK)f = Ff= ~[F) f (B.I29) BKf= (~[F) In -A) f. (B.I30) Referring to Eq.(C.55), by substitution of M := K, P := B, Q := f, L:= (A[F) In - A)f

BBI(~[F) In - A) f flf = (A[F) In - A)f . (B.I3I) Noting that the pseudoinverse BI of the column-like matrix B is BIL and that f flf = f a solution K exists if (Sinswat, V., and Fa/lside, F., 1977) (BBIL - In)(~[F) In - A)f = 0 , (B.132) i.e., the vector [(A[F) In - A)f) is in the null space of (BBIL - In). Since BBILB - B = 0 "" (BBIL -In)B = 0 (B.I33) 638 B Eigenvalues and Eigenvectors and the column space of B equals the null space of (BnIL - In). Hence, the eigenvector f to be assigned premultiplied by a modified characteristic matrix lies within the column space n[B) (A[F) In - A)f E n[B) . (B.134) Note that the modified characteristic matrix uses the eigenvalues of the closed loop and, thus, differs from the ordinary characteristic matrix (A[A) In - A).

B.23.3 Multiple Eigenvalues and Linearly Independent Eigenvectors Find a controller gain matrix K to a given system A, B where a number of k eigenvalues and k linearly independent eigenvectors should be assigned. The numbers At[F) ... AdF) are multiple real or conjugate complex closed-loop eigenvalues. The vectors fi must be linearly independent (qi = mi, Sij = 1). If A1 ... At and f1 ... ft belong to an assignable set

(A + BK)(f1 : f2 ... ft ) = (f1 : f2 ... ft) diag Ai[F) or (A +BK) TJ = TJ diag Ai[F) (B.135) BKTJ = TJ diag Ai[F)- ATJ . (B.136) The matrix diag Ai[F) is a (k, k)-diagonal matrix with A;[F) in the main diagonal. Referring to Eq.(C.55) and applying the following substitutions yields

M:= K, P:= B, Q:= TJ, L:= TJ diag Ai -ATJ (B.137)

BBIL(TJ diag Ai[F)- ATJ)T}TJ = TJ diag Ai[F)- ATJ . (B.138) Since T~TJ = It (BBIL - In)(TJ diag Ai[F)- ATJ) = 0 . (B.139) Resubstituting the definition T J ' one has

(B.140) Postmultiplication by a diagonal matrix yields column-wise multiplication with Ai

(B.141)

(B.142) From the above equation, a necessary and sufficient condition is obtained for fi being in the nullspace of (BBIL - In)(A;In - A) fi E N[(BBIL - In)(Ai[F) In - A») . (B.143) Referring to Eq.(C.56) and applying the same substitutions as mentioned above, the solution K of Eq.(B.136) is given by (B.144) where Z is any (m, n)-matrix. Eq.(B.144) points out the parametrization of the result. In the case of complex conjugate eigenvalues Ai[F) , the matrices TJ and diag Ai[F) can be modified in order to obtain a real-valued controller gain matrix1. Consider A1 and A2 complex-conjugate and the remaining k - 2 eigenvalues real, (B.145) f1 ~ f1re + jflim , f2 ~ f1re - jflim . (B.146) It can be easily verified that the result as given by Eq.(B.144) is obtained by the modified real matrices TJ and diag Ai (Korn, U., and Wilfert, H.H., 1982; Schwarz, H., 1971)

T J := (f1re : f1im : f3 ... f k ) (B.147)

diag Ai[F):= block diag [( A,lre ~lim) ,A3 ... Ak). (B.148) -"lim "lre 1The modification can be applied but need not be applied if complex controller gain K is overcome by any other programming facility. B.23 Eigenvector Assignment 639

B.23.4 Multiple Eigenvalues and Generalized Eigenvectors First, the conditions for the existence of K are derived. The desired set of eigenvalues Ai[F] of the closed• loop system should comprise p distinct eigenvalues, with multiplicity mi each. The design of K should provide qi linear independent eigenvectors fij with manifold Sij each (see Fig. B.I but a replaced by f). The overall number of linear independent eigenvectors is d. Thus,

qi LSij = mi, (B.149) j=1

If the set of eigenvalues A;(F] and eigenvectors fi is preassumed assignable, then, referring to Eq.(B.1l7), the closed-loop system matrix F can be decomposed using the nonsingular modal matrix T, and the Jordan canonical form J,

F = (A + BK) = TJ,T-1 (B.150)

where the Jordan canonical form is defined by Eq.(B.1l3)

J,=blockdiag(Jll , J I2 ... J 1q " J21 ... J2q" ... Jij ... ,Jpq.)ECnxn. (B.151)

The Jordan blocks are given by Eq.(B.1l2) with Ai = Ai[F]. The modal matrix Tis

T = (Tll : T I2 ... T 1q , T 21 ... T 2q, ... T rq.) E cnxn where Tij = (fi) : fi~ : ... f:t) E CnX$i; . (B.152) Note that the vectors f~ are linearly independent eigenvectors and generalized eigenvectors, to be assigned to F = A + BK satisfying

(A + BK)fi) Ai[F] Cij (B.153)

(A +BK)f~ Ai [F] ftj + fCI "IJl = 2 ... Sij . (B.154) Applying the condition Eq.(C.55) to Eqs.(B.153) and (B.154), there result d conditions

"Ij=I,2 ... qi; i=I,2 ... p (B.155)

associated with the linearly independent eigenvectors Cij == Ci~ , and n - d conditions associated with the generalized eigenvectors

"IA: = 2,3 . .. Sij (B.156)

(Sinswat, V., and Fallside, F., 1977). In the case of a given system (A, B) and A: predetermined eigen• values and eigenvectors (A: < n) (B.157) is used. Referring to Eq.(C.56),

K=BUL(T,JJ -ATJ)T~L+Z(I-T,T~L). (B.158)

In the case A: = n, K = BU(TJ,T-1 - A) + Z - Z = BUL(TJ,T-1 - A) (B.159) all degrees of freedom are required and the solution K is unique.

B.23.5 Assignable Subspace. Concluding Remarks In order to determine n eigenvalues and n eigenvectors, the degree offreedom must be n for the eigenvalues and n - I for each given eigenvector since the direction of the eigenvectors is sufficient. This leads to the (m, n)-matrix K of the controller. One has to satisfy the condition

n+(n-I)A: ~ nm. (B.160)

The designer is free to predetermine a complete set of eigenvalues and/or an incomplete set of additionally given eigenvectors. Determining the null space N[H] = N[(BB'L - In)(AiIn - A)] for the given A, B 640 B Eigenvalues and Eigenvectors and the desired Ai , a matrix can be established whose columns span a subspace (or whose column space is) identical to the assignable eigenvector space. It has to be checked if the desired set of eigenvalues and eigenvectors belongs to the assignable eigenvalues and the assignable eigenvector subspace. If the open-loop system has an uncontrollable eigenvalue, then the closed-loop system must comprise this eigenvalue. Regardless of the fact that this eigenvalue cannot be moved the eigenvector associated with this uncontrollable eigenvalue can be arbitrarily assigned. The only condition is that the eigenvalue to be assigned lies within the assignable subspace N[H] associated with the uncontrollable eigenvalue. The eigenvector assignment (or eigenstructure method) is carried out by finding the right and left• eigenvectors for the closed loop such that the eigenvalues meet the expected closed-loop values. The eigenvector assignment has the disadvantage that the eigenvalue sensitivity with respect to eigenvector errors may be very high, resulting in a non-robust approach. In the case of state feedback, the sensitivity can be reduced by choosing the eigenvectors as orthogonal to each other as possible. Prespecifying the closed-loop eigenvectors associated with unchanged open-loop eigenvalues is denoted partial eigenstructure assignment (Jin Lu et al., 1991). The output feedback case is investigated by Ho, w.e., and Fletcher, L.R., 1988. Norm bounds are given on the closed-loop eigenvalues, caused by the perturbation of the assigned closed-loop eigenvectors. Moreover, the influence of the plant and controller matrices A, B, C and K is estimated by using their spectral norm. Appendix C

Matrix Inversion

In addition to the simple inversion as given by Eq.(A.16), the control engineer is involved in many problems associated with matrix inversion such as the right and left-inverse, the pseudo-inverse and the inverse of partitioned matrices. Furthermore, in many applications the inversion of matrices is related to the operations of conditioning, and orthogonalizing.

C.I Matrix Inversion Using Cayley-Hamilton Theorem

The Cayley-Hamilton theorem gives a tool to calculate the matrix inverse via powers of the same ma• trix. In detail, a matrix polynomial up to the power An-I yields the result. Defining the characteristic polynomial C(A) = An + Cn_IAn- 1 + ... + Co where Co = det A the Cayley-Hamilton theorem is given by

c(A) = An + cn_IAn- 1 + ... + CoIn = O. (C.I) Postmultiplying by A -I (assuming A -I exists) yields

(C.2)

(C.3)

E.g. A = (~ !), n = 2, c(A) = A2 - 4A - 5 ; CI = -4, Co = -5 = det A (CA)

A-I = -[G!) -4G ~)1I(-5) = (~5 -;)/(-5) = (-0;8 ~.2) .0 (C.5)

C.2 Matrix Inversion Lemma

The matrix inversion lemma proves very helpful in inverting complicated matrices and economizing com• putational effort. Take into consideration: The quadratic (n, n)-matrix A, its inverse A -I and two rec• tangular matrices, the (n, r)-matrix B and the (r, n)-matrix C. These matrices are given. The dimensions are related as r < n or r < n . The inverse of A + BC is to be calculated, i.e.,

r ~ (A + BC)-I (BC) E nnxn, (CB) E nrxr (C.6)

r x I r-I A+BC (C.7) In rA+rBC I x A-I (C.8) A-I r+rBCA-I IxB (C.g) A-IB rB + rBCA -IB = rB(Ir + CA -IB) (C.1O) A-IB(Ir +CA-IB)-I rB. (C.ll)

PostmuItiplying by (-CA-I) yields -A-IB(Ir + CA-IB)-ICA-I = -rBCA-I . Adding A-Ion both sides, (C.12) 642 C Matrix Inversion

The last equating operation follows directly from Eq.(C.9). So one has the result

(C.13)

If the inverse A -I is already given, then the matrix inversion lemma Eq.C.13 requires the inversion of an (r x r)-matrix (Ir + CA -I B), only. Eq.(C.13) can easily be proved: Examining riA + BC) results in the identity matrix In.

EX8Dlple: If IIFII < IIAII, from Eq.(C.13) it results

(A + F)-I A-I _ A -IF[I + A -IFtl A-I = A-I _ A -IF[I _ IA -1(1+ FA -1)-IFjA-I A-I _ A -IFA -I + A -IFA -1(1 + FA -I)-IFA -I = etc. (C.14)

Using (I + B)-I = I - B + B2 - B3 + ... where IIBII < 1

[A(I+ A -IF)j-1 = [I - A -IF + (A-IF? - (A -IF)3 + ... jA-I (C.15) A-I _ A -IFA -I + A -IFA -IFA -I _ A -IFA -IFA -IFA -I + ... (C.16)

End of Example

C.3 Simplified Version of the Matrix Inversion Lemma

Given the (n, r)-matrix B and the (r, n)-matrix C where r < n the inverse (In + BC)-I is derived as follows B + BCB = (In + BC)B = B(Ir + CB) I x (Ir + CB)-I (C.17)

(In + BC)B(Ir + CB)-I = B I X C (C. IS)

(I" + BC)B(Ir + CB)-IC = BC I + In (C.19)

(In + BC)-I X I (C.20)

(C.21)

C.4 Matrices in Partitioned Form

CA.1 Algebraic Properties

The transpose of a partitioned matrix is given by

(C.22)

Partial matrices or partial vectors within matrices and vectors (in partitioned form) may be multiplied or added as though the submatrices were scalar elements (numerical or scalar functions) provided the submatrices or sub vectors are conformable. The matrices can be treated as follows

A B) (u) = (AU + BV) (C.23) ( C D v Cu+Dv

AC DB) (E F) _ (AE + BG AF+BH) (C.24) ( G H - CE+DG CF+DH . C.4 Matrices in Partitioned Form 643

C.4.2 Inversion of a Partitioned Matrix Preassaming the matrix M in partitioned form is a square (n + q, n + q)-matrix containing the (n, n)• submatrix A, the (q, q)-submatrix D and corresponding rectangular (n, q)-matrix Band (q, n)-matrix C, then, find M-1. On condition det A "# 0 multiply the first submatrix row of M with -CA -1 and add the result to the second row in order to get a block triangular matrix. These operations can be combined by premultiplying M with a certain matrix as follows

(C.25)

Inverting this equation,

(C.26)

With regard to the upper trigonal property in the right-hand side of Eq.(C.26), it is easy to postulate the structure of the inverse containing a specific submatrix S and the inverse of the submatrices

B (C.27) (~ D-CA-1B In order to obtain I(n+q) by transferring the left-hand side of Eq.(C.27) to the right side, i.e., to get zero matrices outside the main diagonal, the condition is

AS + B(D - CA -IB)-1 = 0 (C.28)

Combining this result with Eqs.(C.26) and (C.27),

-1_ (A-1 -A-1B(D-CA-IB)-I) ( I M - 0 (D-CA-1B)-1 _CA-1 ~ ) (C.29) -A -IB(D - CA -IB)-1 (D - CA-1B)-1 ) . (C.30) This result is the Frobenius formula for inverting a partitioned matrix. Eq.(C.30) can be rewritten to

1 M-1 _ (In _A- B) ( A-I Onxq (C.31) - O,xn In Oqxn (D - CA -IB)-1 ) . If det D "# 0 can be preassumed instead of det A "# 0

) . (C.32)

C.4.3 Inversion of a Partitioned Matrix. Nonsingular Submatrices If both det A "# 0 and det D "# 0 can be presupposed, then -1 ( (A - BD-IC)-I -A-1B(D - CA-1B)-1 ) M = -D-1C(A _ BD-IC)-I (D _ CA -IB)-1 . (C.33)

The result can be confirmed by applying the matrix inversion lemma to the expression (D - CA -IB)-I in the left column of the matrix in Eq.(C.30). Deriving the above result by constituting

u M ( Xy) __ (AC BD ) (Xy) -_ (v) or Ax + By = u and Cx + Dy = v , (C.34) it follows

x (A - BD-1C)-1 (u - BD-Iv) (C.35) y (D - CA -IB)-1 (v - CA -Iu) . (C.36) 644 C Matrix Inversion

Rearranging yields

(C.37)

The elements in the secondary diagonal of Eq.(C.33) can simply be checked as equal to those in Eq.(C.37), e.g., -A-IB(D - CA-IB)-I == -(A - BD-IC)-IBD-I . Applying the Frobenius formula in any version, the inversion of a (n + q, n + q)-matrix can be reduced to the inversion of (n, n)-matrices and (q,q)-matrices. To the former class A,A-1 and BD-1C belong, to the latter D,D-1 and CA-IB.

C.4.4 Inversion of a Block-Diagonal Matrix o 1 ) = block diag (A-1,D-1) . M = (~ ~) = block diag (A,D) M- = ( D-1 (C.38)

C.4.S Determinants of Matrices in Partitioned Form A determinant remains unchanged if any row (column), multiplied by a certain constant factor, is added to another row (column). Thus, presuming A square and det A # 0, premultiply the first sub matrix with -CA-I and add the result to the second submatrix row

D _ ~A -I B ) = [det A)[det(D - CA -IB)] (C.39) since submatrices in triangle determinants may be treated as scalar numbers. Similarly, if D is square and det D # 0

det ( AC DB) = det ( A - BD-1CC D0) = [det D)[det(A - BD-1C)] . (C.40)

When a partitioned matrix with submatrices A through D of equal dimensions is considered, the result detM = det(AD - ACA-1B) is obtained. If the matrices A and C commute (e.g., if A or C is the identity matrix) the result is det(AD - CB), regardless if det A = 0 . Considering Eq.(C.39) in the special case C = 0 and A, D square of any order, the result is [det A)[det D] .

C.4.6 Reducible Matrix

If a partitioned decomposition (~ ~) = PHP-I with C = 0, A and D square can be deduced from a matrix H where P is the permutation matrix then the matrix H is denoted reducible.

C.S Right-Inverse

Define a rectangular matrix WRI as the right-inverse of the nonsquare matrix W

WWRI =1. (C.4l)

The right-pseudo-inverse (C.42) satisfies Eq.(C.4l) WWRI = 1 and is, therefore, a possible solution WRI = WIR but not the only one. Given an np-vector p, an nm-vector y and an (nm, np )-matrix M in the relation np > nm the equation Mp = y has more unknowns than knowns. There exist solutions in a large variety p = MRly . The matrix MRI is an (np, nm)-matrix and the right-inverse (inverse to the right). The matrix MRI exists if rank M ~ nm and obeys MMRI = Inm . The right-pseudo-inverse MIR is the minimum right-inverse (Momri, M., 1983), i.e., (C.43) G.6 Left-Inverse 645

C.6 Left-Inverse

Given an op-vector p, an nm-vector y and an (n m , np)-matrix M the formula Mp = y is inadmissible if the relation Om > op holds, if more knowns exist than unknowns, if the number of scalar equations exceeds the number of unknown variables in p. The left-inverse (or inverse form the left) MLI does not supply the solution. Although a variety of MLI exists, none of the associated products p? = MLly may be declared as a solution that is compatible to the overdetermined system, completely.

Example: 0 ( 0.6 ) M=( -0.5 y= 1.3 (C.44) -1.1506 °t) 0.5 -2 - 2.3012 0 MLI = ( 4 + 4.60240 MLlM =12 '10,(3. (C.45) 2 + 4.6024 (3 -2.3012 (3 ~ ) Selecting, e.g., 0 = (3 = 0 then MLly = (-0.2 1.2JT = p? . Computing Mp? = (0.6 1.3 0.230)T, this result is not consistent with y in Eq.(C.44). 0

The formulation by addition (subtraction) of e, i.e., Mp = y - e , is permissible even in the case Om > op. Within the infinite variety of MLI the minimum left-inverse is defined

(C.46)

C.7 Pseudo-Inverse

C.7.1 General Pseudo-Inverse Let the (n, m)-matrix M with rankM = r contain only real elements. With regard to rank r, the product form can be attained

r m ... rank V = r M=VW= rank W = r (C.47) 0 V r W ( -) r < n, r< m.

The general pseudo-inverse MI is defined by the following four equations

(C.48)

(C.49) Explicitly, MI is given by (C.50) The pseudo-inverse M' is an (m, D)-matrix. Inserting Eq.(C.50) into Eq. (C.48), i.e., VWWT(WWT)-I(VTV)-IVTVW = VW makes evident that it is not permissible to omit M neither in the left nor in the right half of Eq.(C.48). Some more properties for the pseudoinverse:

(MI)I = M (M'l = (MT)I ((3M)1 = (lj(3)M' (if (3 =1= 0) (C.51)

(MTM)I = MI(MI)T (M'M)' = MIM (MMI)' = MM' (MB)' =1= BIMI (C.52)

rank M = rank MI = rank MTM = rank M'M tr M'M = rank M . (C.53) 646 C Matrix Inversion

C.7.2 General Pseudo-Inverse and a General Matrix Equation Given the matrix equation with known matrices P, Q, L and unknown M

PMQ=L. (C.54)

All matrices are assumed of adequate dimensions. If the condition

(C.55) is satisfied, the solution of Eq.(C.54) is (Rao, C.R., and Mitra, S./(., 1971)

(C.56) for any matrix Z. This result can easily be proved by substitution.

The condition Eq.(C.55) is necessary and sufficient for Eq.(C.54) to have a solution.

(i) Necessity: It is unimaginable that the equation to be solved, Eq.(C.54), and the condition Eq.(C.55) for the existence of the solution are incompatible. Hence, it is a necessary condition. Consistency of Eq.(C.54) and (C.55) requires that L of Eq.(C.54) substituted into the left-hand side of Eq.(C.55) gives a true statement

ppl(PMQ)QIQ = pplpMQQIQ = PMQ = L . (C.57)

(ii) Sufficiency: Substituting the solution Eq.(C.56) into Eq.(C.54) yields the expression Eq.(C.55) irre• spective of the value of Z. Thus, Eq.(C.55) is sufficient.

C.7.3 Right-Pseudo-Inverse

If n = r and V = I,. then M = Wand the right-pseudo-inverse is

(C.58)

Example: Specializing Eqs.(C.54) and (C.56), namely Mp = y, M E nn~xnp, np > nm . If MM'y = y exists (M' = MIR) then the solution p is

(C.59) where z is any np-vector. This can easily be proved by substituting the solution into Mp = y and using Eq.(C.48)

Mp = MMly + M(lnp - MIM)z = y + Mz - MMIMz = y . (C.60) Differentiating pTp with respect to z yields z = 0, i.e. the special case p = Mly has minimal norm and M' = MIR. The solution Eq.(C.59) can. be generalized to solve Eq.(C.54) by rewriting Eq.(C.54) into vector form using Eq.( 4.40) (QT 0 P)col M = col L (C.BI)

col M = (QT 0 p)IR col L + [I - (QT 0 p)IR (QT 0 P)Jcol Z (C.62)

col M = (QT 0 p)IR col L + col Z - (QTIR 0 pIR)(QT 0 P)col Z (C.63)

M = plLQI + Z - plpZQQI . (C.64)

End of Example C.8 General System Inverse 647

C.7.4 Left-Pseudo-Inverse Considering m = rand W = Ir then M = V, the left-pseudo-inverse is

(C.65) and M'LM = Ir . The left-pseudo-inverse is frequently applied in linear regression. Applying the left• pseudo-inverse, only, the superscript L is often omitted.

Example: Specializing the general pseudo-inverse to the left-pseudo-inverse,

Q = B (input matrix), P = 1m, BE cnxm , then L = 1m . (C.66)

Eq.(C.55) yields BIB = 1m or BU = BIL. From Eq.(C.56) one has

M = BI + Z(ln - BBI) = BUL + Z(ln - BBIL) . 0 (C.67)

Note that for M E nnxm , n > m ,

1 1 det(sln - MMU) sn det(ln - -MM') = sn det(lm - -MUM) (C.68) s s sn-m det(slm _ 1m) = sn-m(s _ l)m . (C.69)

C.7.5 Projector Properties of MMtt and MttRM The matrix product, in this order, possesses the property of a projector matrix, in both cases left-pseudo• inverse and right-pseudo-inverse.

C.S General System Inverse

Consider the system x(t) = Ax(t) + Bu(t) x(t) E nn, u(t) E nm (C.70) y(t) = Cx(t) + Du(t) y(t) E nr, rank D = min(r,m) . (C.7l)

C.S.l Case r < m Eq.(C.71) is solved with respect to the input vector using right-pseudo-inverse and an arbitrary m-vector z as shown in Eq.(C.59) (C.72) where (C.73) Deriving z from x by defining an arbitrary (m, n)-matrix N, i.e., z = Nx, from Eqs.(C.70) and (C.72), it results

Ax + B[DIR(y - Cx) + (1m - DIRD)Nx] (C.74) [A - BDURC + B(Im - DIRD)N]x + BD"Ry . (C.75)

Eqs.(C.75) and (C.72) as a system with left-hand vectors (x, u) is called general system inverse of the system with left-hand vectors (x, y).

C.S.2 Case r = m Equating DUR = D-1 yields a unique inverse

u D-1y _ D-1Cx (C.76) X (A - BD-1C)x + DD-1y . (C.77) 648 C Matrix Inversion

C.S.3 Case r > m With regard to r > m, Eq.(C.71) is overdetermined and incompatible in this version. Considering a dual system (Sinswat, V., 1976)

X' = AT x' + CT u' y' = BTx' +DTu' (C.78) with an r-vector y' as an input and an m-vector u' as an output, the aforementioned calculus can be applied again. Using the right-pseudo-inverse of D T , i.e.,

(C.79) the transpose of the left-pseudo-inverse comes into action, corresponding to the fact that the system matrices are defined as transposed matrices in the case of Eq.(C.7S).

C.9 Pseudo-Inverse and Singular-Value Decomposition

If the singular-value decomposition of a complex matrix G is given by G = U1:VH , the pseudo-inverse amounts to where 1: = (1:ra a) a ' r = rankG (C.SO) since the basic properties of the pseudo-inverse are satisfied by the decomposition cited above.

C.lO Pseudo-Inverse of a Matrix Partitioned into Submatrices

If a matrix M is partitioned into matrices A through D

M = ( AC DB) = (A)C A -1 (A:.' B) = (A)C (I:. A -1 B) = ( AC (C.SI) and the submatrix rows in M are linearly dependent, i.e., the second row results from the first one by multiplying with CA -1 (A square and nonsingular) then D = CA -IB , and rankM = rank A . In this case, the pseudo-inverse can be calculated in partitioned form

(C.82)

C.l1 Pseudo-Inverse of a Matrix Partitioned into Columns

Using Greville's method, the pseudo-inverse can be calculated in a recursive way. The matrix M is disassembled into columns M = (mcl.mc2, ...• mcn) . On the other hand, the matrix M is recursively composed by defining

(C.S3)

With the definitions

I {aT1 j(aTl al) if al 1: a M ~aA' - (C.S4) 1 = 1 - aT if al = a

(ak - Mk_1dk)' if ak - Mk-ldk 1: a = { (C.S5) hi (1 + df dk)-ldfML if ak - Mk_ldk = a . Taking these abbreviations, the pseudo-inverse associated with Mk V k = 2, ... ,n is obtained

(C.S6) C.12 Successive Application of Right and Left-Pseudo-Inverse Operator 649

C.12 Successive Application of Right and Left-Pseudo-Inverse Operator

According to the relation np and nrn , the following identities hold

(C.87)

Checking, e.g., Eq.(C.87) (C.88)

(MIL)IR = [(MTM)-IMTf ([(MTM)-IMTJ[(MTM)-IMTf)-1 (C.89)

(MIL)IR = M(MTM)-I(MTM)-IMTM(MTM)-Ifl = M (C.90)

(Barnett, S., 1971; Rao, C.R., and Mitra, S.K., 1971; Lancaster, P., and Tismenetsky, M., 1985) .

C.13 Conditioning and Scaling

C.l3.l Condition Number of a Matrix The condition number ",,[A] of a matrix A is defined for any u and v

IIAu[[F maxllullp=1 IIAuliF urn,x[A] ",,[A] max (C.91) '" lIullF=I, II v IlF=1 IIAvllF minllvlIF=1 IIAv[[F urnin[A] IIAII, ",,[A] IIAv[[F = = IIAII,IIA-III, ?: 1 . (C.92) minIlA-'AvIIF=1 111~1[:i;1

Using the relation between norm and maximum/minimum modulus of eigenvalues,

",,[A] = ~ > m~", [.AdA] [ (C.93) _1_ - mmi [.AdA] [ IIA-III,

Solving, e.g., Ay = b where A is nonsingular, b and A are only known to be in the limit of a tolerance 6b and 6A, an estimate of the error 6y of the solution y is (Franklin, J.N., 1968)

(C.94)

A matrix A is ill-conditioned if "" [A] is large. Then, the solution of the equation Ay = b or the computation of the inverse of A or of the eigenvalues .A[A] becomes inaccurate, especially as far as the smaller eigenvalues are concerned. Conditioning and scaling methods help to overcome this problem. A general approach transforming a given (n, n )-matrix A into a decomposed version D is considered as D = T[A] . Some transformations T are listed in Table C.l. With regard to their quadratic structure, they are referred to as spectral decompositions.

C.l3.2 General Spectral Decomposition

In the general case of spectral decomposition A = EDET, the matrices D and A are congruent matrices. Substituting x' = ET X to the quadratic form a(x)

(C.95)

If D is desired diagonal, D = diag di, the quadratic form turns out a sum a(x) = L~ di:ct 2 • 650 C Matrix Inversion

Table C.l: Various decompositions (matrix transformations)T

T I name, details, main properties A = EDEl general spectral decomposition (congruent matrix transformation) A = TDT-1 eigenvalue decomposition, D ... diagonal (A=AT --... T-l = TT --... E=T) (orthogonal matrix transformation) A=EDE scaled decomposition A=D2 square root decomposition A=DD1 Cholesky decomposition

C.13.3 Eigenvalue Decomposition Assume the matrix A symmetric, only. The modal matrix T = T[A] associated with A then becomes TT = T-1. The matrix T has orthogonal property. Using T for transformation E := T, ET = E-1 . Since AT = A Pi = ai and from Eq.(B.95)

n A = T( diag Ai[A])TT = L: Ai(aiai) (C.96) i=l

n A-I = (TT)-l(diag A;1[A])T-1 = T(diag A;l)TT = L: A; 1(aiai) . (C.97) i=l Since E = T, D = T-l AT and det D = det A, the matrices D and A are similar, in this special case.

C.13.4 Orthogonal Transformation Applying an orthogonal transformation T to second-order or quadratic forms, it can be arranged that the transformed variables Vi are non-interacting. Consider a quadratic polynomial in the vector-valued variable x

(C.9S)

By the linear transformation x = TT v, the scalar J(x) is transformed to

(C.99)

The T is defined as the modal matrix associated with Q . Invoking Eq.(B.16),

TT = T-1 and TQTT = TQT-1 = diag Ai[Q] = A (C.lOO)

(C.lOI)

The gradient of J(v) with respect to v is 8J(v)/8v = 2Av + 2Tb . Since A is diagonal, the gradient component [8J(v)/8v]i only depends on the component Vi. In order to find the optimum of J(v), the variables Vi may be adjusted independently of each other. Moreover, the optimum is given by (Roberts, P.D., 1967)

v* = -A-1Tb, (C.I02)

Important applications are given in the of parameter identification and optimal system design. 0.13 Conditioning and Scaling 651

C.13.S Scaled Decomposition Consider A symmetric, only, and E diagonal, E := diag ei, E = ET, then A =EDE. (C.103) The matrix D is the scaled version of A. The elements ei are assigned to

if Aii 0 e· - { .jjA;";T # (C.104) ,- 1 if Aii = 0 (initial data). In the following paragraphs, Aii has to be replaced by 1 if the initial data Aii are zero. SUbstituting Eq.(C.104) into Eq.(C.103) and rearranging yields

(C.105)

The matrix D is symmetric since A is symmetric. The modal matrix associated with D is T = T[D] . With regard to the symmetry of D it results TT = T-1 and D = T( diag Ai[D])TT . Note that T[D] is used, not T[A]. Finally, the scaled decomposition of A is A = EDE = ET(diag A;[D])TTE = ET(diag Ai[D])(ETf . (C.106) Inverting Eq.(C.106), the scaled decomposition of A-1 is achieved A-1 = E-1T(diag Ai1)(E-1Tf . (C.107) Partitioning the matrices ET and E-1T into columns (ET).i and (E-1T).i, respectively, and denoting the kth element with the additional subscript k, the relations can be proved

(C.10S) where di is the eigenvector of D . The scaled decomposition of A and A-1 can be written as the sum of Ai-weighted dyadic products

n n A = L Ai[D] (ET).i(ET)·T A -1 = L Ai1[D] (E-1T).i (E-1T).[ . (C.109) i=1 .=1

Example:

A = (1010000 \0) E = (1000 01 ) n,(A) = I A[A]lmax/l A[A]lmin = 10000.01 I 0.99 = 10101 (C.llO)

Scaled version of A: D = (O.gl ~) (1010000 \0) (O.gl ~) = (0\ Oil) (c.m) Eigenvalues A[D]: A1 = 1.10, A2 = 0,90; n,(D) =1.10 I 0.90 = 1.22 (C.1l2) M d I . T (0.7071 0.7071) (dO dO) o a matnx: = 0.7071 -0.7071 = 1 2 (C.113)

70,71 70,71) ( . ) ET = ( 0.71 -0.71 = (ET).l: (ETh (C.114)

. ( 70 71 70,71) (1.1 0) ( 70,71 0.71) Matnx decomposed: A = 0.'71 -0.71 0 0.9 70,71 -0.71 (C.115)

-1 (0.0071 0.0071) ( -1) : ( -1 ) ) E T = 0.7071 -0.7071 = E T.1 . E T.2 (C.116)

d -1 (0.0071 0.0071) ( 0.9091 0 ) (0.0071 0.7071) Inverse decompose: A = 0.7071 -0.7071 0 1.1111 0.0071 -0.7071 . (C.1l7) End of Example 652 C Matrix Inversion

C.13.6 Square Root Decomposition Considering nonsingular A = DDT and symmetric D, the square root decomposition is obtained. The matrix D is named the square root of A . That is, A = D2 and D = .,fA . In view of the eigenvalue decomposition Eq.(C.96),

A = T(diag ~;)TT = T Jdiag ~; TTT Jdiag ~; TT . (C.lIS)

The term TTT in Eq.(C.lIS) is 1 . Thus, it results D = T ~ TT .

C.13.7 Cholesky Decomposition Consider nonsingular A, only, and a lower triangular matrix L with elements L;j = 0 Vi < j . The decomposition n or A;j = LL;.L;. Vi,j = 1 ... n (C.1I9) v=l is chosen. Then, L;j can be found out by a simple recursion formula. If a new variable y is defined when solving Ax = b then LLTx = b and LT x = Y Ly=b. (C.120) Both equations for y and x can easily be solved on account of the triangular nature of L and its simple inverse.

C.14 Orthogonalizing

A given set of data Z;, i.e., a set of n-vectors, possibly contains linear dependent information. Composing these data Z; to a matrix Z = (z;, Z2, ...), the information of Z should be transferred into the column information of a new matrix X. The aim is to receive orthogonal columns x;, only. The matrix X;-l is the submatrix of X, containing columns Xj (j:$ i-I) already transformed by previous operations

X; = (Xl,X2 ... x;) X = (Xl,X2 ... Xq) X; E nnx;, X E nnx q , x; E nn, i < q:$ n. (C.121) Remember the orthogonality properties treated when deriving Eq.(D.37). Thus, the operator matrix (premultiplication matrix) to be applied to Z; , in order to obtain x; orthogonal to each column of Xi_l , is (I - X;-lX!:l)' i.e., x; = (I - Xi-1X!:1)Zi . (C.122) Multiplying xT with X i_l yields the null row OT

zT(1 - Xi-1X!:lfxi-l = zT (Xi_l - Xi_l(xLXi_l)-lxLxi_l) = zTo = OT . (C.123)

An alternative way to achieve Xi is to apply the pseudo-left-inverse X!: 1 to the vector Zi

Pi = (XLl Xi_d-lxL1Zi . (C.124) The parameter vector Pi fits Zi optimally into the Xi_l-space. The vector Xi-1Pi is the optimal least square regression of Zi. The residual Zi - Xi-1Pi is orthogonal to all previous vectors from Xl until Xi-l' This residual is declared as x;

(C.12S) and is chosen to complete Xi-l to X; . The algorithm is started with Xl = Zl , without any regression manipulation. After having regressed all input vectors Zi and having computed the residuals, all the vectors Xi are orthogonal.

Example:

(C.126) C.14 Orthogonalizing 653 1) (-0.02) 5) ( 12 -0.360.30 1. Xl . (C.127) End of Example

If a zero column Xi appears, the linear dependence is obvious. This datum Xi = 0 has to be cancelled to avoid singularity of xLlxi- 1 . Linear dependence occurs if at least one column of Xi-l is a linear combination of the other ones. Then, rank(XLIXi- l ) < n, detXLIXi- 1 = 0 and xLlxi- 1 is singular.

Example:

XrX2 = C~ ~). (C.128)

End of Example Appendix D

Linear Regression and Estimation

Linear regression is considered as the deterministic problem of minimizing the error between measurement sequences and a deterministic model. The aim is to find the optimum model parameters based on least squares. Linear models are taken into account, only. The model is defined as

Mp=y (0.1)

The matrix M is an (nrn,np)-matrix with fixed elements, p (usually) a vector with np elements of unknown parameters and y an observation vector with dimension nrn (the number of measurements or observations).

D.l Parameter Demarcation

When measurements are available to state nrn linear equations with np parameters and if nrn < np , the system is underdetermined and a solution does not exist. The shape of the matrix M is row-like. Although there is no solution the matrix M defines a demarcation which relation between the elements of p exists. Within the scope given by M and y, find p such that the least square pT p or the Frobenius norm !!p!!} is minimum. This investigation can be formulated with the help of a vector-Lagrange-multiplier .\

(0.2)

8 8p [pTp+.\T(Mp_y)]=O p = -0.5 MT.\ . (0.3)

In view of Eq.(O.l) and (0.3),

M( -0.5 MT.\) = y and .\ = -2(MMT)-ly . (0.4)

The matrix MMT is preassumed nonsingular. Combining Eq.(0.3) and (0.4), the result is

(0.5)

The starred parameter p* is the optimal (minimal least square) parameter vector, complying with Eq. (0.1). Eq.(0.5) shows the linear relationship between y and p*. The matrix operator to be applied to y is the right-pseudo-inverse MIH It is a right-inverse of M because M has to be multiplied by MIR from the right to yield the identity matrix In m.

Example 1: Suppose that nrn = 2 measurements are available and np = 3 parameters are to be determined, one has two linear equations with three variables PI, P2 and P3. Each equation corresponds to a plane in Fig. 0.1.

-0.5 -1.1506) o 0 (MMT)-I = (0.7279 0.2911) (0.6) M=( 0.5 0.2911 0.9165 656 D Linear Regression and Estimation

} equation

Figure D.l: Illustration of two linear equations with three variables

0.1456 0.4583) 0.6001 ) MIR = MT(MM1')-1 = ( -0.0729 0.7710 , y = ( 0.5 ) p* = MIRy = ( 0.8507 -0.8375 -0.3349 1.1506 -0.8041 (0.7) Vectors perpendicular to the two planes given by M:

-050) and (0.8) ( -1.1506

Vector g in the direction of the intersection of both planes:

-0.5 x 0 + 1.1506 Xl) 1.1506 ) g= ( -1.1506xO.5-0xO ( -0.5753 . (0.9) o x 1 + 0.5 x 0.5 0.25

Checking if g is perpendicular to p*:

gT p* = 0.6001 x 1.1506 + 0.8507( -0.5753) - 0.8041 x 0.25 = 4.23 x 10-5 =O. 0 (0.10)

Example 2: Optimal initial conditions for observers. The initial condition is usually chosen X(O) = 0 . This default choice does not reduce x(t) as much as possible and is not compatible with the initial output y(O) . The optimal choice is given by IIx(O)IIF -+ min, subject to Cx(O) = y(O) (Johnson, C.D., 1988) which is exactly the problem posed in this section. Substituting M := C, y := y(O), p:= X(O), the optimal initial condition is

(0.11)

End of Examples

D.2 Interpolation Consider the case nm = np' The number of measurements is supposed equal to the number of parameters to fit the model of Eq. (0.1). The measurements are preassumed linearly independent. Eq.(D.1) with square matrix M can be solved (0.12) The inverse exists with regard to the independence of the measurements. The unique solution in the parameter vector p* provides a unique interpolation of the observation y. D.3 Weighted Least Squares Approximation 657

D.3 Weighted Least Squares Approximation

In the case nm > np , more measurement information is given than one needs at minimum to solve Eq.(D.l), see Fig. D.3. The system of equations of this model is overdetermined. Hence, Eq.(D.l) must be rewritten to Mp=y-t: (D.13) in order to have a consistent system of equations. The solution of Eq.(D.13) yields an optimal approximation of the measurements without rejecting any information. Optimal approximation is gained by postulating p* in such a way that an index of performance is minimized. In most cases this index of performance e is defined as the sum of squares t:~ produced by a special p "- e= L t:r = t:Tt: min. (D.14) 1'=1 This case is shown in a subsequent section in detail. The function e can also be established in a more general way, using different weighting factors for t:; and o.Oj thus forming a weighting matrix W

(D.15)

where W denotes an arbitrary positive , normally symmetric. Following the necessary condition for the minimum of e with respect to p yields oe 0 0 - = -[(-Mp+yfW(-Mp+y)] = _[pTMTWMp - yTWMp - pTMTWy+yTWy] = 0 op op op (D.16) [MTWM + (MTWMf]p - (yTWMf - MTWy = 0 (D.17) p = p* = (MTWM)-IMTWy . (D.18) Calculating the second derivative of e with respect to p, this derivative matrix has to be positive definite. IfW=WT 02e opTop = MTWM > O. (D.19)

Both Eqs.(D.18) and (D.19) together are a necessary and sufficient condition for achieving a minimum of e with respect to p. The condition Eq.(D.19) is satisfied if W is a positive definite matrix. If the matrix H = MTWM is singular, linear independent information must be cancelled. If cancelling leads to nm < np then Eq.(D.5) has to be employed. After cancelling, the matrix H red plays the same role as M in Eq.(D.l). Using the right-pseudo-inverse H~~d yields the optimal solution as in Eq.(D.5). A necessary condition to uniquely estimate p in the model Mp = y when applying the least square method is the nonsingula.rity of MTM or det MTM '" O. It is a necessary condition even in the weighted least square case. This can easily be realized by inverting the assertion: If one has detMTM = 0 then also det MTWM = O. It is not a sufficient condition: If det MTM '" 0 then det MTWM '" 0 or = 0 is possible, depending on W. Thus, for the model Mp = y the statement det MTM '" 0 is denoted identifiability condition.

Example: Four simple cases are chosen to illustrate various assumptions on nm , np and rank M. To obtain definite solutions rank M 2: np must be satisfied. Case a) nm = 3, np = 2 :

M=(~ !) (D.20)

Third equation (row) linear dependent on the first and second one: rank M = 2

MTM _ (14 16) - 16 21 ' p = (1 2f. (D.21) 658 D Linear Regression and Estimation

Case b) Rm = 2, Rp = 2 : M=(: :) (0.22) Second equation linear dependent on the first: rank M = 1

T (4560) MM= 6080' (0.23)

Case c) Rm = 3, Rp = 2 :

M= (: :) (0.24) -3 -4 Second and third row linear dependent on the first one: rank M = 1

(0.25)

Case d): Case c) continued: The equations

H"p = MT y = (198)264 (0.26) are linear dependent just as Eq.(0.24). The system contains the same information as the first equation in Eq. (0.24) or any other one: (3 4)p = 11 . After cancelling useless information in Eq.(0.26),

H •• d = (54 72), p = H;.d(H.ed H;ed)-'(MT y) •• d = H~~d (MT y)•• d (0.27) " = (54) [(54 72) (54)]_, 198 = (0.0067) 198 = (1.32) (0.28) p 72 72 x 0.0089 1.76 . End of Example The error vector e at the minimum p*, e(p*) = e* = e is named residual e . In statistic estimation theory the expectation E[P*] turns out to be the optimal estimation p . Anticipating this fact, the superscript * is frequently omitted and the optimal parameter p* is equated with p, i.e. p* = p . The quantity Mp = j• is the regression value of y . Fig. 0.2 gives an impression in three dimensional sample space. The weighted squared error index of performance C = eTWe = C(p) depends on p. After having mi• nimized C, there remains the residual sum of squares C* = e*TWe* = C(p). In view of this, substituting e = e* = -Mp + y and pinto C yields GtvLS = (-Mp + y?W( -Mp + y) = pTMTWMp - yTWMp - pTMTWy + yTWy GtvLS = yTWTM(MTWM)-"TMTWM(MTWM)-'MTWy _ yTWM(MTWM)-'MTWy _ yTWTM(MTWM)-"TMTWy + yTWy . (0.29) The first and second term above can be cancelled. Supposing W symmetric, simplifying yields

(0.30) By substituting y = Mp + e, the expression Eq.(0.30) is rewritten by elementary algebraic operations Gtv LS (Mp + e)TW[I - M(MTWM)-'MTW](Mp + e) = (pTMT + eT)[W - WM(MTWM)-'MTW](Mp + e) = [pTMTW + eTW _ pTMTWM(MTWM)-'MTW -eTWM(MTWM)-'MTW](Mp + e) eTWMp - eTWM(MTWM)-'MTWMp +eTWe _ eTWM(MTWM)-'MTWe GtvLS = eTW[I - M(MTWM)-'MTW]e . (0.31) The matrixW[I-M(MTWM)-'MTW] weights the squares ofy and e in such a way that equal results GtvLS are obtained (Rosen, J.B., 1960). DA Ordinary Least Squares Approximation 659

(residual vector in error space) Figure D.2: Sample space, estimation space and error space in the case of nm = 3 measurements and np = 2 parameters

D.4 Ordinary Least Squares Approximation Setting W = I, the weighted least squares solution is specialized to the ordinary least squares one. The ordinary least squares solution is

(0.32)

The matrix MIL is the left-pseudo-inverse of the matrix M and is of dimension np x nm (vice versa to the dimension nm x np of the model matrix M). For the sake of comparison, previous results are repeated: If solutions of Mp = y exist then either M-1y is the unique solution or p = MIRy is the solution of minimum norm (length) IIpIiF. If solutions of Mp = y do not exist then the sum of squares £T £ of deviations £ = -Mp + y is minimized by p = MILy. For automatic control purposes the left-pseudo-inverse MIL plays a dominant role. If only the left• pseudo-inverse MIL is used within a section the superscript L is omitted in MI , for simplicity. In the ordinary least squares case the resu lting residual least squares becomes more simple

C1s = yT(I - MMI)y = eT(I - MMI)e . (0.33)

Corresponding with

Mp y - £ (arbitrary p) (0.34) and with y = Mp y - e (optimal parameter P) (0.35) 660 D Linear Regression and Estimation

Table D.l: Vectors and sum of squares

I vector I squared amount of the vector (distance) C(p) = cl·c (sum of squares in sample space representing the squared distance from any point Mp in estimation space to point y) M~p=Mp C(p) - C(p) Mp=y regression sum of squares e(= y) residual sum of squares Crnin = ~ S = C(p) = e T e

the squared amount of the distances Ilyll}, lIyll} and lIell} is investigated. With regard to the fact that e is minimum in quadratic sense, y = Mp and e = c(f» are vertical. Applying Eq. (0.32),

(0.36)

Hence, the above-mentioned distances y,y and e satisfy Ilyll} = lIyll} + lIell} (see Fig. 0.2). The regression sum of squares is defined by the Frobenius norm lIyll} = yT y . In terms of the observation vector y the regression sum can be expressed as

(0.37)

Referring to the between y and e, the same result is obtained from (MPVy or (Mf»Ty. The projection of the vector y on to the plane spanned by ml and mz in Fig. 0.2 is identical to y. The squared length of the vector y is given by yT y, the optimal C by G!s in Eq.(0.33). Combining Eqs. (0.37) and (0.33), (0.38) This expression G!s is named the residual sum of squares. Thus, the residual sum G!s plus the regression sum lIyll} yields the squared sum of the observation vector lIyll}

G!s + lIyll} = Ilyll} . (0.39) The vector quantity y = y + e = y + Y is used to define y = e. The residual e frequently is declared as the minimum estimation error y (deterministic equation error vector), see Table 0.1. Combining Eq.(0.18) and y = Mp* ,

(0.40) where TM is an idempotent transformation matrix. Applying TM twice, three times etc., the same result is to be expected. This can easily be verified in Fig. 0.2, by inspection. In Fig. 0.2 one has to distinguish between three spaces (Draper, N.R., and Smith, H., 1966). An additional space is sketched in Fig. 0.4. (i) The urn-dimensional sample space containing the observation vector y and all columns of M. Each of the up columns of M and the vector y is represented by a single point in the sample space. (ii) The up-dimensional subspace named estimation space defined by the Dp column vectors of M. (iii) The error space assembled by the error vectors" and the residual vector e. The dimension of the error space is Urn - up. In Fig. 0.2 it is a cone with the apex pointed out by y. (iv) The parameter space assembled by the components of p.

Example: Linear regression of two parameters on the basis of three observations Drn = 3, up = 2. a) Non-faulty observations:

0 0.5 ) M = ( -0.5 y = ( 1.1506 (0.41) -1.1506 0.3466 D.5 Left Inverse and Right Inverse. Mnemonic Aid 661

MTM = (0.7279 0.2911) MIL = (MTM)-IMT = (0.1456 -0.0728 -0.8375) (D.42) 0.2911 0.9165 0.4582 0.7709 -0.3350

* = M'L = (-0.3012) (D.43) p y 1.0000· b) Faulty observations:

0.6 ) 0.5548 ) • _ MIL _ (-0.426) ( y = ( 1.3 , p - y - 1.1096 ' Y = Mj) = 1.3226 , e = y - Mj) = ( ~O~~;;6 ) 0.5 0.4902 0.0098 (D.44) Check if e and sample space are orthogonal: e™ == (0 0). Residual sum of squares: eT e = Cis = C{j») = 0.0026, lIeliF = 0.0515 . Regression sum of squares: yTy = 2.2974, IIYIIF = v'2.2974 = 1.5157 . Norm of observation: yT y = 2.3000 , lIyllF = 1.5166 . Check: eT e + yT Y =? = yT y 0.0026 + 2.2974 = 2.3000 . Parameter space (see subsequent section):

!:!..G = 0.7279 P~ + 0.5822 P1P2 + 0.9165 P~ . (D.45)

End of Example

D.S Left Inverse and Right Inverse. Mnemonic Aid

Considering Mp = y, M E nnm xnp and comparing (i) the least squares approximation nm > np with (ii) the parameter demarcation nm < np , note the following facts when solving Mp = y, i.e. when "separating" p as a result. (i) Least squares approximation: j) = MILy can be achieved by "multiplying" Mp = y by MIL from the left although none of the scalar equations Mp = y is true (t: omitted) but the premultiplication by MIL is a good mnemonic aid. (ii) Parameter demarcation: p = MIRy cannot be obtained by a multiplication operation from the statement Mp = y although the result can easily be proved by multiplying the result p = MIRy by M from the left. Note that each scalar equation ofMp = y is true but multiplication ofMp = y by M,L from the left is inadmissible although one is induced to do this: MTM in the case nm < np has dimension np x n p , rankM™ = nm . Hence, the inverse (MTM)-l and MIL do not exist.

D.G Complex Matrix M

If the matrices involved are complex but only a real parameter vector p is admissible the result

p* = argminGwLs subject to !;)

w>o. (D.47)

D.7 Sum of Errors and Residual Sum in Parameter Space

The parameter space is defined by the np-dimensional parameter vector p. The sum of squared errors eT e is

(D.4B)

The smallest value Cis = G(I» = yT y - yTMMly is subtracted from G(p), thus obtaining

(D.49) 662 D Linear Regression and Estimation

Least squares approximation. Left-pseudo-inverse. System Parameter demarcation. Right-pseudo-inverse. overdetermined if E is neglected: System underdeterminecl:

n, Given d.t.:~ x ~P = ~ nm min

Left- ~ pseudo-inverse, u-'

Result:B

Figure D.3: Ordinary least squares approximation and parameter demarcation. Graphic interpretation: left-pseudo-inverse and right-pseudo-inverse

It can easily be checked that the formulations ~c = (p - p)T(MTM)(p - p) = [M(p - pW[M(p - p)l = (M~pf(M~p) = (Mpf(Mp) (D.50)

are identical to the previous version. The vector ~p = p is the parameter difference. The error sum difference to the residual sum expressed by ~C is equal to the squared sum of ~p weighted by the matrix MTM, as known from identifiability condition. The gradient of ~C in the parameter space reads as follows (D.51)

If the columns mei of the matrix M = (mel, m e2 .. . m en.) are orthogonal 'tip, /I = 1 ... np then

if and if 1'=/1. (D.52)

The matrix MTM becomes diagonal, MTM = diag a • . The increment ~C and its gradient turn out as n. ~C = ~pT(MTM)~p = ~pT(diag a.)~p = E a.(~p.)2 ~~~ = 2(diag a.) ~p. (D.53) 11=1

The shape ~C does not contain cross products. Hence, the shape is symmetrical to the axes ~P •. The component /I of the gradient only depends on ~P •. If MTM is diagonal recognize the following fact: Fixing P2 at an arbitrary value and minimizing ~C with regard to PI , the minimum PI min = PI is always the same, no matter which arbitrary P2 was initially chosen (see Fig. DA).

D.S Successive Estimation in Large-Scale Systems

In the field of large-scale systems, the parameter estimation is based on a decomposition technique. For small-scale systems, the model y = Mp+e is applied as given in Eq.(D.13). For a large-scale system, with regard to computational difficulties, it is troublesome to apply Eq.(D.13). Decomposing into L submodels yields

L L L mi npi Xn Yi = EMi;p;+ei Yi E'R,n , pi,ei E 1l , Mij E 1l"mi ,i, E npj = n p , E nmj =nm - ;=1 ;=1 ;=1 (D.54) D.S Successive Estimation in Large-Scale Systems 663

P2 ------

np = 2 parameter space

ih = plmin

Figure D.4: Shape C(p) and contour lines of constant amount b.C in parameter space

The parameter vector P in decomposed form is pT = (pT pI ... pI) . The least squares estimation is obtained by minimizing

(D.55) i=l with respect to Pi. Furthermore, try to find Pi in such a way that the interaction between the L sub models is minimized. Thus, an interaction variable

L Wi= L (D.56) i=l,i¢i is defined, providing the influence from all subsystems j -I i to the subsystem i. Following Eq.(D.55),

L C= L(Yi - MiiPi - Wi? (Yi - MiiPi - Wi) (D.57) i=l has to be minimized with respect to Pi and Wi within the constraint of Eq.(D.56). By application of the method of the Lagrange multiplier, the new criterion function I is

L L 1= L(Yi - MiiPi - Wi?(Yi - MiiPi - w;) + >o.T(Wi - L Mijpj). (D.58) i=1 j=l.j;!i Differentiating with respect to Pi, >o.i, Wi yields

L Pi = (MEMii)-I[M[;(Yi - Wi) + L MJ,>o.j] (D.59) j=I,j;!i

L Wi = L MijPj (D.60) i=l,j¢i respectively, Vi = 1 ... L. The previous equations can only be solved by iteration using appropriate starting values for w;O) and >0.;0) in Eq.(D.59). Eq.(D.59) gives p;O). Taking this value, Eqs. (D.60) yield w;l) and >0.;1), respectively. This interaction-prediction algorithm is carried out until some preset stopping condition in >o.;k) and w;k) is satisfied, e.g., in the norm of the difference between consecutive quantities. 664 D Linear Regression and Estimation

The above algorithm can be modified by substituting Eq.(D.60) into Eq.(D.59). In this way, a succes• sive algorithm in pIt) is formulated. In order to process on-line measurements, a recursive version of this algorithm also was developed (see Sultan, M.A., et al. 1988), corresponding with the recursive algorithm given in the next section.

D.9 Recursive Least-Squares Estimation

In practice, measurements are recorded successively with time. An estimator version updating the latest parameter state is of interest. By processing recent data this recursive estimator provides an incremental complement to the hitherto existing parameter result. Up to the kth sample assume to process the measurement matrix M(k) and the measurement vector y(k). Advancing from sample instant k to k + 1, the measurement matrix M( k) is supplemented by a new row m T (k + 1) and M( k) turns out as the new measurement matrix M(k + 1). Assume the model specified by the difference equation of nth order

y(k) + aly(k - 1) + ... + any(k - n) = blu(k - 1) + ... + bnu(k - n) (D.61) and define the parameter vector p ~ (al ... an bl ... bn)T . Let L + 1 measurements be available from sampling instant k - L through k. Without measurement errors this equation is rewritten, applying the definitions m(k), M(k) and y(k)

y(k - L) - mT(k - L)p = 0, y(k-L+ 1)-mT (k-L+ l)p = 0, ..... y(k)-mT(k)p = o. (D.62)

Taking into account some measurement error e,

y(k) - M(k)p = 0 y(k) - M(k)p = e (D.63)

-y(k -1)

-y(k - n) Y(k-L») m(k) = ----- y(k) = ( : E "R(L+l) (D.64) u(k - 1) y(k)

u(k - n)

-y(k-L-l) -y(k - L - 2).. -y(k - L - n) u(k - L - 1).. u(k - L - n) ) -y(k - L) -y(k - L -1) .. ( M(k)= -Y(k~L+l) (D.65)

-y(k - 1) u(k - 1) u(k - n)

M(k) = [m(k - L) m(k-L+l) M(k) E "R(L+I)x(2n) . (D.66) Starting the recursive algorithm, care must be taken of negative arguments and k 2:: L + n must be provided. Advancing from sample instant k to k + 1 and assuming enlarged measurement data from L + 1 to L + 2, the dimension of y(k) to y(k + 1) is increased by one

y(k-L») y(k - L) 1 : y(k) ( E "R(L+2) . (D.67) y(k) = ( : E "R(L+l), y(H 1) = y(·k) = ( y(k + 1) ) y(k) y(k + 1)

Within the vectors m(k) and m(k+ 1) only a shift operation is carried out. Thus, the dimension ofm(k) D.9 Recursive Least-Squares Estimation 665 does not increase when proceeding to m(k + 1).

-y(k-I) -y(k)

-y(k - n) -y(k - n + 1) m(k) E 2n , get = ------n From m(k) = - - - - - m(k+I) m(k + 1) E n2n. (0.68) u(k - 1) u(k)

u(k - n) u(k-n+I)

From M(k) to M(k + 1) proceed by enclosing a row mT(k + 1). Thus,

M(k + 1) = ( M(k) ) M(k + 1) E n(L+2)x2n . (0.69) mT(k + 1)

The least-squares estimator P(k) and p(k+ 1) resulting from (L+ 1) or (L+2) measurements between samples k and k + L or k + L + 1 is obtained by applying Eq.(0.32)

p(k + 1) = [MT(k + I)M(k + I)tlMT(k + I)y(k + 1), (0.70) respectively. Making use of Eq.(C.23) and substituting

A := MT(k), B:= m(k + 1), C = D := 0, u:= y(k), v:= y(k + 1), (0.71) one has

MT(k + l)y(k + 1) =,( m(k - L) m(k.- L + 1) ... m(k), m(k+ 1)) ( Y(~~)I) ) (0.72)

MT(k)

MT(k + l)y(k + 1) = MT(k)y(k) + m(k + l)y(k + 1). (0.73) Invoking Eq.(C.24) and substituting C = D = F = H := 0

A := MT(k), B := m(k + 1), E := M(k), G := mT(k + 1), (0.74)

T _ ( M(k) )T ( M(k) ) _ (T: ) ( M(k) ) M (k + I)M(k + 1) - mT(k + 1) mT(k + 1) - M (k). m(k + 1) mT(k + 1) (0.75)

MT(k + I)M(k + 1) = MT(k)M(k) + m(k + I)mT(k + 1) (0.76) is achieved. Taking Eqs.(0.73), (0.70) and (0.76) into consideration,

p(k + 1) [MT(k + I)M(k + I)]-I[MT(k)y(k) + m(k + I)y(k + 1)] (0.77) p(k + 1) [MT(k + I)M(k + I)]-I[MT(k)M(k)p(k) + m(k + I)y(k + 1)] (0.78) p(k + 1) [MT(k + I)M(k + 1)]-I[MT(k + I)M(k + I)p(k) -m(k + l)mT (k + I)p(k) + m(k + l)y(k + 1)] (0.79) p(k + 1) P(k) + [MT(k + I)M(k + l)tlm(k + l)[y(k + 1) - mT(k + 1)P(k)] . (0.80) / -1--- old estimate I "-- one sample step / . new meas,urement ahead prediction new estimate correcting gain vector measurement y- (k + 1 Ik) including I (Kalman gain factor) ,(k) ~ sample k + 1 prediction error Applying the matrix inversion lemma Eq.(C.13) to the inverse of Eq.(0.76), substituting and defining

B = C T := m(k + 1), (0.81) 666 D Linear Regression and Estimation the result may be considered as a

P(k + 1) = P(k) - !,(k)m(k + 1)[1 + mT(k + I)P(k)m(k + 1)tI,mT(k + I)P(k). (D.82) "I,

Note that the bracket expression in the above equation is a scalar. Hence, it can be shifted to any other position within the matrix product

mT P( k + 1) __P-,-( k-,-,Hc-I...c+_ _ ....:.(k_+-,--1 );-P....:.( k--".)"m;7( k;-+-:,-;-;1)~] "'-;"7P"":'( k-;);-m....:.( k-;-;+-,1 ),---m_T....:.( k_+_l ),---P....:.( k---,-) (D.83) - 1 + mT(k + I)P(k)m(k + 1)

Postmultiplying the above numerator by m(k + 1) yields

P(k)m(k+ 1) + P(k)m(k + l)mT(k+ I)P(k)m(k + 1) - P(k)m(k + l)mT(k + I)P(k)m(k+ 1) . (D.84)

The second and third term are cancelled. Hence, by Eq.(D.83),

P(k)m(k + I) P(k + l)m(k + 1) = 1 + mT(k + I)P(k)m(k + 1) . (D.85)

The left side above corresponds to the correcting gain vector "I(k) in Eq. (D.80), the right side above equals "II (k), as defined in Eq.(D.82): "II (k) = "I(k). Finally, the resulting recursive algorithm is expressed by the following three equations P(k)m(k + 1) (D.86) "I(k) = 1 + mT(k + I)P(k)m(k + 1)

P(k + 1) = P(k) - "I(k)mT(k + I)P(k) (0.87) i>(k + 1) = i>(k) + "I(kHy(k + 1) - mT(k + 1)i>(k)] . (0.88) Note that the matrix M increases its dimension from sample to sample but the product MTM does not. The matrix P as its inverse also keeps the dimension (2n) x (2n) unchanged. For the sake of comparison with block-pulse function approximation, see Eq.(37.58).

D.lO Recursive Instrumental Variable Method

The recursive (and one-shot) least-squares method only yields an unbiased parameter vector if the error signal e is uncorrelated both with the input and output signal of the process u and y, respectively. If uncorrelated error signals cannot be preassumed the method of instrumental variables is used since it also works with correlated e. The instrumental variable vector Yaux is synthesized as the output of the instrumental model, see Fig. D.5

-Yaux(k - 1)

-Yaux(k - n) _ ( Yaux(~-I) ) IDl1tu:(k) = ------Yaux - : . (0.89) u(k - I) Yaux(k - n)

u(k - n)

The instrumental variable Yaux is generated by exciting the instrumental model with u and by declaration Yaux = mIux i>aux using an auxiliary parameter vector i>aux. To avoid stochastic dependence between Yaux and e, the auxiliary parameter i>aux is taken from the real i> after having passed PTI-delay. A memory is employed to generate the vector Yaux from the past scalars Yaux( k - i) . The recursive least• squares algorithm runs as given by Eqs.(D.87), (0.88) and, additionally, by

(D.90) D.ll Linear Estimation 667

~ •" instrumental model >: =' I Your ~'== ~ maux maux(k) m;uxPauz memory II r------. I ~ instrumental variable "I;'• t a. II Paux '/ Y.- ~ ( u= (*-1) ) maux u(k- n) f>aux(O)~ PTl ~ P(O) ~ ,(0) first-order low pass filter u recursive least squares algorithm ,(k) In F== P(k 1) m=(-:) + f>(k + 1) Y p(k + 1)

Figure D.5: Recursive instrumental variable method

The matrix P(k) is a modified covariance P(k) = [M~ux(k)M(k»)-l and Maux(k) is the instrumental variable matrix

(0.91)

(Jakoby, W., 1985). The aim is to establish a signal Yaux strongly correlated with the undisturbed process output but weakly correlated with measurement noise (Isermann, R., 1988).

D.ll Linear Estimation

For x( k) scalar, the set {x( k)} with k = l...N is referred to as a scalar random process. The vector-valued variable {x( k)} denotes a vector process. Linear expectation (mean): x ~ E{x(k)} ~ limN_oo k L~=l x(k) .

Variance: IT; ~ E ([xi k) - xF} ~ limN _00 k L~=l [xi k) - xF· Autocorrelation function: Rxx(-r) ~ E{x(k)x(k + r)} ~ limN_oo k L~=l x(k)x(k + r) . Autocovariance function:

cov [x(k)x(k+r») ~ cov (x,x,r) ~ E{[x(k)-x)[x(k+r)-x)} ~ Qxx(r) ~ X(r). (0.92)

If the signal is Gauss distributed (= normally distributed), then the process is completely determined by x and cov [x(k )x(k+r»). The process is denoted stationary if x and cov [x(k)x(k+r») are time-invariant. Cross covariance function: cov [x, y, r) ~ E{[x(k) - x)[y(k) - jj]} = Rxy( r) - xjj ~ Qxy( r) . A random discrete-time process is named white if the signal x(k) has no stochastic interrelation to 668 D Linear Regression and Estimation earlier signals x(k - p). The covariance function of a white process (white noise) is I if r = 0 OT ( r) = { 0 if r t 0 (0.93) where oT(r) is the Kronecker delta. The power density function is constant versus frequency V Iwl ::; 7r/T where T is the sampling interval. For comparison: Continuous-time white random processes are characterized by a vanishing stochastic interrelation, even between instants of infinitesimally small time-shift. The power density function is constant for all frequencies up to infinity and signals may grow to infinity. Hence, a white continuous• time process does not exist in reality and is only imaginable as a limes. Kronecker delta in Eq.(0.93) is replaced by the dirac function o( r) . If the stochastic signal contains several components x(k) = [XI (k), x2(k) ... xn(k)Y a vector process {x(k)} is established and a covariance function matrix is given by

cov [x, r] l>. cov x ~ E{[x(k) - x][x(k + r) - xV} (0.94) cov [x1o XI, r] cov [XI, X2, r] ( coy [X2J Xl, r] coy [X2' X2, r] ::: ) g Q•• (d g X(,). (0.95) · . · . Note that Qxx is positive semidefinite symmetric for r = 0 and that the main diagonal positions of Qrr( r) are given by the autocovariance functions.

D.I1.1 Parametric Models. Markov Processes Markov signal processes (of first order) are defined by the conditioned probability distribution function p[x(k)lx(k - 1), x(k - 2) ... x(O)] = p[x(k)lx(k - 1)] depending only on the preceding x(k - 1). A signal process obeys the probability distribution above if it is generated by the first order difference equation x(k + 1) = aMx(k) + bMV(k + 1) . Higher-order time-invariant Markov signal processes are given by (0.96) If AM(k) and/or BM(k) are time-dependent the Markov process is time-varying. The process {x(k)} is a white Markov process if v( k) is white. Oefining cov [x(k + 1), x(k + 1), r = 0] = cov [x(k + 1)] ~ E{[x(k + 1) - x][x(k + 1) - xf} ~ X(k + 1) (0.91) and substituting Eq.(0.91) into (0.96) yields (calculations omitted) in the case of white noise v(k) X(k + 1) = AMX(k)A1 + BMV(k)B1 . (0.98)

If AM is constant and stable and BM is constant for k --+ 00 then X(k + 1) = X(k). Using Eq.(4.40) col X = (I - AM ® AM )-I(BM ® BM)col V . (0.99) Assuming a positive semidefinite weighting function matrix R, the Markov signal process x( k) can be assessed globally by 1= E{xT(k)Rx(k)} "" 1= xTRx + tr [RX] .

D.I1.2 Observation as a Random Process Techniques of linear regression yield deterministic linear models with weighted least squares 11£11 2 where £ is a vector of distinct errors. If there is a high number of data the error space is of high dimension. With regard to the high amount of data it is suitable to consider e: as a stochastic process. The observation vector y now is regarded as the sum of the model function Mp plus an additive zero mean noise £ where the matrix M is given by y = Mp + £ . The parameter p should be estimated optimally with respect to a risk function cov p . Supposing a linear relationship p* = ry , in order to derive the optimal estimation p* from the observation y, and using the model e = -Mp + y p = E{p*} = E{r(Mp + e)} = E{rMp} + E{re}. (0.100) The second term in Eq.(O.lOO) is named bias b = E {re} . The bias can be separated into E {r} and E{e:} if both processes are uncorrelated. Regardless of E{r}, the bias b vanishes if e has zero mean as preassumed. The condition of being uncorrelated is met if e is a white noise (see Bard, Y., 1974; isermann, R., 1988). If b = 0 then only the first term remains in Eq. (O.lOO) P = E{rMp} = E{rM}p . In order to obtain p = E{p*} = p, the condition rM = 1 must be satisfied. Hence, r is a left inverse of M. D.ll Linear Estimation 669

D.11.3 Minimum Variance Estimator. Gauss-Markov Theorem Which linear estimator p = ry or which r matches the minimum of some scalar valuation of the covariance cov p ? The variance of the estimation error vector

cov p = E{[P - E{P}][P - E{pW} (0.101)

could be reduced since E{p} = 0 (unbiased estimator). The estimation error p = p - p, the optimum estimate p = ry and the model y = Mp +" are substituted in Eq.(0.101). Thus, cov p = E{(p-p)(p_p)T} = E{[P-r(Mp+,,)][P-r(Mp+.Y} = E{(r,,)(r,,)T} = r(cov ,,)rT = rvrT (0.102) where V is assumed a given function. As scalar valuation of cov p, the determinant is chosen. Hence, the objective is to minimize det cov p subject to the matrix condition p = ry or rM = I . Introducing a matrix of Lagrange multipliers A and selecting the trace as a scalar measure of the constraint, the resultant form det (rVrT) + tr A(rM - I) is achieved. The minimum with respect to r is obtained by differentiation

:r {det (rVrT) + tr A(rM - I)} = 0 (0.103)

Comparing Eq.(0.103) with the optimum weighted least squares estimation in Eq.(0.18) yields the fol• lowing result. If the weighting matrix W is defined as W = V-I (the inverse of the covariance matrix V = cov ,,), the optimal linear estimator p = ry in accordance with Eq.(0.103) yields an unbiased estimate, additionally a minimal weighted risk function C = "TW" = "TV-I" and, finally, a mini• mum determinant of the variance of the parameter cov p = r(cov ,,)rT . Using biased and nonlinear estimators better results are available. The linear estimator discussed above is the best linear one and is unbiased but does not supply the least squares estimate at all (Hoerl, A.E., 1962; Hoerl, A.E., and Kennard, R. W., 1970 ). The minimum variance estimator can only be realized if the covariance matrix of the equation error is known a priori. Otherwise it is known after a recursive process. Substituting Eq.(0.103) into (0.102), the variance in the minimum variance estimation case is (cov p)MV = (MTV-IM)-I . Then, substituting the identity matrix V = I in Eqs.(0.103) and (0.102) yields the variance in the least squares estimation case (cov P)LS = (MTM)-IMT(cov ,,)M(MTM)-I which cannot be simplified any further for a general matrix M. The covariance (cov P)MV is smaller than (cov p)LS. The expectation of the ordinary least squares risk function C = "T" is now investigated, provided the standard assumptions are valid (additive zero mean non correlated measurement errors with constant variance cov" = ".21 ; nonrandom parameters; independent variables; no other prior information) (Beck, J. V., and Arnold, K.J., 1977). In the ordinary least squares case r is the left pseudo-inverse MIL. Thus, the expectation of C turns out

D.H.4 Estimation Sensitivity Consider the case that the exact value of the covariance matrix V = cov " is not known but an erroneous (perturbed) V p = V + .:l V is available as a basis for the linear regression formula Eq.(0.103). Then, rp is obtained and the erroneous estimation is Pp = rpy . Using Eq.(0.102) with the true value V but with the erroneous rp yields cov Pp = rp vr;1 . A degree of inefficiency f/e can be defined. This efficiency factor is always greater than or equal to unity because p is the minimum variance estimation. An upper bound for f/e can be formulated

f:>. det cov Pp (1 + a)2 1

The condition number a is given by the ratio between largest and smallest eigenvalue of H with Hermite property, see Hoerl, A.E., and Kennard, R. W., 1970; Wilkinson, J.H., 1965. Appendix E

Notations

E.1 General Conventions

• For detailed definition see boldface page in subject index. • Boldface lower and capital letters denote vectors and matrices, respectively. Calligraphic letters denote sets.

• Closed set ~ or [) , e.g., 1 ~ a ~ 5 or a E [1,5) for a . • Open set < or ( ), e.g., 2 < b < 4 or bE (2,4) for b real number. • Set of { }, e.g., i = 1,2, ... 8 or i = {I, 2, ... 8} for i integer.

• 0 end of the proof, end of the example, end of discussion. • Dots stand for undesignated variables.

E.2 Abbreviations and General Symbols =, -I, == equal, is different from, approximately equals, respectively t:>. equality by definition equality by simple substitution identity LO angle corresponding to the argument of the > positive definite (matrix) (P > Q means P - Q positive definite) >.0 ~. 0 positive and non-negative (matrix), respectively, element-by-element relation, ratio of size comparison II· lip p-norm or boundary of the region r 0[·) degree II modulus (absolute value) of a complex number II modulus (matrix), e.g., IMI =matrix[ IMijl) 1. perpendicular to leads to ~e , ~m real part and imaginary part of a complex number, respectively e field of complex numbers en n-dimensional linear over e, en == enXI e- set of complex numbers with negative real part (open complex left-half plane) e+ closed right-half complex plane o the empty space {ail set (with elements ai) V for all E is an element of, belongs to ¢ does not belong to ~ is a subset of (is contained in) or identical to u union (of sets) 672 E Notations

intersection (of sets) A=> B means: If the statement A (e.g. equation, inequality) is true then B is also true or, equivalently, "A is sufficient for B" or "A implies B", e.g., lRe A;[A) < -4 => A is stable. N ~ T means "N is implied by T' or, equivalently expressed as a necessary condition, "only if N then T', e.g., the system with state space coefficient matrix A is stable only if trA < 0 . implies and is implied, if and only if. E.g., B <;;; A ~ "Ix : (x E B) => (x E A) . (- .. ) inner product of two functions or vectors F , F-I Fourier transformation and its inverse, respectively. Fy(t) = y(jw) or, if necessary, y(jw) £ £-1 Laplace transformation and its inverse, respectively. £y(t) = y(s) or, if necessary, y(s) Z', Z-I z-transformation and its inverse, respectively H2 Hardy space satisfying H 2-norm, (i) for continuous-time systems: Hardy space of functions analytic in the open right-haif plane and square-integrable in the closed right-half plane

f(s)EH2 ~ suPjOO f*(u+jw)f(u+jw)dw < 00, 0'>0 -00 (ii) for discrete-time systems: Hardy space of functions analytic in the open unit disc (Izl < 1) and square-integrable in the closed unit disc

g(z) E H2 ~ sup f2~ g*(r ej9 ) g(r ej9 )dlJ < 00 . r0 or functions on Loo with bounded analytic continuation in the right-half plane, (ii) for discrete-time systems: Hardy space of functions analytic in the open unit disc and bounded in the closed unit disc g(z) E Hoo ~ suplg(r ej9)1 < 00 , r<1 set of asymptotically stable transfer functions with IIGlloo < 00 . subspace of functions in Hardy space H 00 that are real rational space of functions Hilbert space of functions square-integrable on jR f(s) E L2 ~ fr Coo f*(jw)f(jw)dw ~ IIf(s)ll~ < 00 L~ space of vector-valued functions L~. extended space L~ Loo Banach space essentially bounded on jR N null space (kernel) R.['] range space (image) R field of real numbers Rn n-dimensional linear vector space over R, R n == R nx1 R+ field of non-negative real numbers R used as a prefix denotes real rational V'p Nabla operator (gradient) with respect to p, 8()/8p bX exponential of base b logbx logarithm of a real number x to base b o generalized product corresponding to a polynomial product o $ Kronecker sum convolution * e.g., M(s) 1.=1 means function M(s) selected for s = 1 {a; EX: f(a;) = O} the set of elements of X having the property f(ai) = 0 adj adjoint of a matrix arg argminx(.), that is, value of x that minimizes () circ clockwise encirclements E.3 Superscripts 673

cof cofactor (of a matrix) col column string cony convex hull, convex combination coy covariance matrix det determinant (of a matrix)

diagn {ai} diagonal matrix of dimension n x n with entries ai dim dimension of a vector or a matrix (number of rows x number of columns) E{x} (expected) mean value of a stochstic process {x}, expectation operator expAt matrix exponential function of the matrix At gradp gradient operator, 80/8p h.o.t. higher-order-terms infu infimum over u, largest lower bound over u log logarithm to base 10 matrix[aij j arranging a matrix with entries aij min{a, b} selecting the minimum of a and b rank rank of a matrix row row string sign signum function (sign x = ±1 for x > 0 and x < 0, respectively, and = 0 for x = 0) supu supremum over u, least upper bound over u tr trace of a matrix tr(i) generalized trace trig trigonal matrix vec stacking operator

E.3 Superscripts

-1 inverse (of a matrix) * (star) optimal * (asterix) complex conjugate + e.g., 0+, i.e., 0 + € where € > 0, € --.. 0, e.g., t = 0+ means instant immediately following zero + a+(s) is a polynomial free of zeros in the closed right-half s-plane a-(s) is a polyomial free of zeros in the open left-half s-plane .L all-pass extension .L (right) annihilator (of a matrix) o corner matrix (n x m) matrix M of dimension n x m (overline dot) first derivative with respect to time (k) k-th derivative with respect to time Uj jth Kronecker power 8 specified bounds according to Eq.(12.80) ~ specified bounds according to Eq.(12.66) (overline bar) mean value, linear expectation (overline bar) signals in the case of unperturbed plant " (written as a superscript) piecewise linear (function expansion coefficient) (overline hat) (if necessary) referring to the frequency domain or the Laplace domain (overline hat) estimate (overline tilde) estimation error (casually) characterizing a modified signal n block-pulse (function expansion coefficient) general orthogonal (coefficient vector) # general pseudo-inverse #L, #R left-pseudo-inverse and right-pseudo-inverse, respectively. BUL left-pseudo-inverse where B E cnxm and rankB = m < n. KIR right-pseudo-inverse where K E cmxn and rankK = m < n . right (-eigenvalue) left (-eigenvalue) 674 E Notations

D Hadamard product h homogeneous solution H conjugate transpose (AH = A·T ) LI (L), RI left-inverse, right-inverse, respectively mo modal variable, modal coordinate op operator p particular solution R reciprocated (polynomial) R para-Hermitian transpose (of a matrix) T transpose w Walsh function (expansion coefficient)

E.4 Subscripts

er e.g., Fer means transfer matrix from the input r to the output e sum vector norm, largest absolute column sum 00 vector infinity norm, largest absolute row sum (n x m) matrix M of dimension n x m OJ jth column of a matrix ();. ith row of a matrix c observer based compensator c servo-compensator cp combined servo-compensator and plant CL closed-loop d dead-time delay D diagonally weighted or D-weighted Holder norm e element by element eq equivalent sliding mode condition f final F Frobenius norm H Hankel (singular value or norm) inner, see Eq.(30.166) ij (i, j)-partition of a matrix I interval k index denoting sampling instant L L-step ahead prediction L least favourable L8 least squares I, r left-coprime and right-coprime, respectively L, R left and right polar decomposition, respectively m measurement M Markov process MV minimum variance nom nominal (value) (casually 0) o initial o outer p plant p perturbed system p p-norm 1I·lIp , p-measure I'p p 1I·lIp function norm p e.g., Ap abbreviates 8A/8p pr index denoting a system with combined plant state and dynamic controller state r dynamic feedback controller red ij reduced (matrix), obtained by cancelling row i and column j ref reference, setpoint, particularly YreJ 8 symmetric part of a matrix, e.g., A. E.5 Glossary of Symbols in Alphabetic Order 675

spectral norm, e.g., IIAII, slow model, neglecting high-frequency dynamics (Sobolev) S2-bounded truncation of a function variable multiplied by e"'

E.5 Glossary of Symbols in Alphabetic Order a(5) unperturbed polynomial in s a,(5) extreme polynomial aik(s) edge polynomial ap (5) perturbed polynomial a( s) (right) eigenvector of A associated with Ak[A], abbreviation for ilk left-eigenvector of A associated with Ak[A] Ilnom nominal coefficient vector associated with the nominal polynomial a( s) A matrix of coefficients of a system in state-space representation, having elements aij in row i and column j A average matrix Ab bias matrix Ap perturbed matrix A AI real interval matrix dA perturbation matrix B input 'matrix in state-space representation C performance, index of performance C, C (matrix) transfer function of the controller C output matrix C, matrix expressing sliding mode condition cmxn set of complex matrices with m rows and n columns db decibels, e.g., a db means a gain of 10(;'0) d path in the s-plane, see Eq.(21.113) d[SI,52] chordal metric between the points 51 and 52 dG(G1 , G 2 ) graph metric distance d,i stochastic parameter D Nyquistcont~r enclosing the r.ight half of the s-plane DR Nyquist contour enclosing the right half of the s-plane with indentations of radius 1/ R along the imaginary axis Do(w) performance deterioration, see Eq.(24.43) D matrix in the dynamic-free part of the observer D diagonal scaling matrix Del, Dcr common left-divisor and common right-divisor, respectively Dd operational matrix for dead-time delay D/, Dr left-divisor and right-divisor, respectively D M operational matrix for differentiation optimal p-norm weight set of structured and unstructured perturbations, respectively base of natural logarithm (e = 2.71828) perturbation factor (dA = eE) e max maximum perturbation parameter tracking error, see Eq.(1.5) e residual error e error vector (e = Yr,! - y) e; ith vector (unit i-vector) with 1 only in the ith position E matrix in the dynamic-free part of the observer E error matrix E unidirectional perturbation matrix upper bound for structured perturbation of continuous-time systems 676 E Notations

and discrete-time systems, respectively Ei constant perturbation matrix Ep perturbation bound Eij Kronecker matrix J(z) polynomial in z fi eigenvector of F associated with Ai(F] F matrix of closed-loop coefficients in state-space description F closed-loop system transfer matrix F aggregated matrix F power series expansion matrix of orthogonal functions, see Eq.(36.41) Fo return-ratio matrix, open-loop transfer matrix Fuw transfer matrix of a system with output u and input w Fz system matrix of the observer g(t) continuous impulse response function, weighting function g perturbation vector g(x, t) nonlinear time-varying perturbation vector G,G (matrix) transfer function, plant (matrix) transfer function G aggregated input matrix Gil G n invertible and non-invertible part of the matrix G GN nearest normal approximation GOp p operator associated with the perturbed plant h high-frequency signal, dither h degree of stability hi (" high") upper limit of a coefficient ai hi, h general shifted orthogonal polynomial and polynomial function vector, respectively H Hamiltonian function H Hurwitz testing matrix H observer input matrix associated with the control variable H upper bound of an interval matrix H measurement transfer matrix i, 'l integer numbers I index of performance I identity matrix of appropriate dimension I,. identity matrix of dimension n x n 1° 1/ 0 null matrix of appropriate dimension On null matrix of dimension n x n j, J integer numbers j =H J matrix in its Jordan canonical form, Jordan form, Jordan canonical form J ij Jordan block k integer number k number of terms of generalized shifted orthogonal polynomials in t which are taken into consideration ki dimension of ~i, see Eq.(26.15) km stability margin k, stability margin with respect to block-structured uncertainty ku attenuation factor k output feedback controller K, K (matrix) transfer function of the controller, matrix gain factor KK steady-state Kalman gain matrix Kv equivalent sliding mode state feedback controller Kg output feedback controller matrix I integer number I dimension of the aggregated state vector z Ii lower limit of (a coefficient ai) lo(w) bound for ~Lo E.5 Glossary of Symbols in Alphabetic Order 677

lq dimension of the parameter vector q L number of submodels L observer input matrix associated with the output variable of the plant L lower bound of an interval matrix L scaling matrix Le, La controllability and observability gramian, respectively AL scalar multiplicative uncertainty ALo , Lo , ALi, Li output and input associated uncertainty, respectively LJ proportional-integral observer input matrix associated with the output of the plant LA, Lv matrix-Lagrange-multipliers m dimension of the input variable u m scalar function replacing the norm of the perturbation matrix Mi mi number of partitions ai, see Eq.(26.15) mi mUltiplicity of the eigenvalue Ai mm order of the minimal polynomial mmi index of the eigenvalue Ai, equivalent to the multiplicity of the eigenvalue in the minimal polynomial mM measure of stabilty robustness, see Eq.(23.2) maux auxiliary measurement vector M maximal bias matrix, deviation matrix M measurement matrix M matrix transfer function of the model M matrix partition of the unperturbed control system interacting with the uncertainty which is pulled out and is considered as an external part of the control system instrumental variable matrix feedback controller or sensor transfer matrix Mi perturbation matrix associated with the sensor MR partitioned Riccati coefficient matrix M u , My component connection model input matrices n dimension of state variable, order of a dynamical system, state dimension, order of the characteristic polynomial nE number of fixed perturbation matrices nm dimension of the observation vector y np dimension of the parameter vector P n measurement noise N number of zeros in the closed right half s-plane N number of corners N number of local subsystems N. smoothed nonlinearity No resolvent matrix of L N°P operator associated with a nonlinear system N u , Ny component connection model output matrices p scalar parameter varying the interconnection p, P parameter, parameter vector p[x(k)lx(k - 1)] conditioned probability distribution function p( s) polynomial in s Po(w) low-frequency performance PM(jW) modified polynomial for Mikhailow hodograph Pi right-eigenvector of AT P number of unstable poles of the open-loop system P permutation matrix P operational premultiplication matrix P solution matrix of the Lyapunov equation P solution matrix of the Riccati equation P matrix weighting the state vector x(t) P Pick matrix P covariance matrix PK matrix solution of the Riccati equation determining a Kalman-Bucy filter 678 E Notations

Pr projector matrix PI solution of the Lyapunov equation when Q = I P M & operational matrix of integration using block-pulse functions PMI operational matrix of integration using piecewise linear polynomial functions PM(, PM I, PM general operational matrices of integration PMp operational matrix of integration associated with the power series expansion vector, see Eq.(36.102) stretched operational matrix of integration operational matrix of integration using Walsh functions operator substituting the multiplication by eal truncation operator family of perturbed polynomials family of perturbation polynomials degeneracy modified spectral radius, see Eq.(28.17) q real uncertain parameter vector which allows for linear dependent coefficient perturbations Q positive definite matrix Q , weighting the state vector x(t) Qa weighting matrix, weighting the differential sensitivity vector IT; r dimension of the parameter vector p r dimension of the output vector y re['], rR['] complex and real stability radius, respectively rL, r 0 uncertainty radius bounding I~L(jw) I R region R[x] Rayleigh quotient R scaling matrix R weighting matrix, weighting the control vector u( t) ROP retardation operator n,(mxn) set of real matrices with m rows and n columns 8 complex Laplace variable Sij entry of the corner matrix 8ij length of the chain containing the eigenvector and the generalized eigenvectors 8 w sliding mode condition So output noise of a plant (measurement noise) Su input noise of a plant S intersection of hyperplanes Sj Sj hyperplane 8(8) sensitivity matrix of the nominal system S; normalized sensitivity function of the closed-loop transfer function F( 8) with respect to the characteristic polynomial a( s) of the plant sf; differential sensitivity t, to real time variable, initial time, respectively tm discrete sampling instants t, sliding mode initial time T sampling interval T time constant Td time delay TI time constant of integration 11 mapping given by the matrix sum weighted with powers of time T matrix relating the plant state variable and the observer state variable T transformation matrix (modal matrix) T(8) complementary sensitivity matrix function TF modal matrix associated with F = A + BK T transformation u, u control variable, controlling variable (vector), input vector il input signal in the case of unperturbed plant singular vector of GGH E.5 Glossary of Symbols in Alphabetic Order 679

U U matrix of singular vectors U;, see Eq.(22.96) U e normalized perturbation bound Ukl permutation matrix U kl self-derivative matrix U~ orthogonal spectrum of U v vector of measurement noise v plant state vector applying the orthogonal transformation T, see Eq.(32.54) v state vector of the dynamic feedback controller v; singular vector of G H G V gain constant V, Vk Lyapunov function, Lyapunov function at the sampling instant k, respectively VN norm-like Lyapunov function V(cPp , d) normalized value set of perturbation polynomials, see Eq.(21.113) V matrix of singular vectors v; , see Eq.(22.97) V prefilter matrix V dm, V dmg Vandermonde matrix, generalized Vandermonde matrix, respectively w weighting function w distortion, disturbance vector w vector of process noise W weighting matrix W A Kalman controllability matrix X,Xo state vector, initial state vector, respectively x~ general orthogonal coefficient vector x(t) signal vector after having multiplied x(t) by the operator polynomial <1>(8) Xin, Xout output and input ( Note the order !) of the block-diagonal uncertainty, see Fig. 26.3 Xr state of the dynamic feedback controller X covariance matrix X(t) state variable matrix X" piecewise linear spectrum X~ orthogonal spectrum of x Xn block-pulse spectrum X set of matrices X Xoo set of block-diagonal matrices with no restriction on the norm Y measurement vector, output vector y output signal in the case of unperturbed plant Y residual sum of squares YouX' instrumental variable vector Yre! reference, set point y set of matrices Y z complex z-transform variable z variable characterizing the E-contour z observer state vector z (fast) state vector of parasitic dynamics z aggregated model state vector z, Z observer state and observer estimation error, respectively Z (preferably) any matrix Z;j interpolating matrix polynomial, component of a matrix Z set of matrices Z

1(t) unit step function at t = 0 (= 1 for t 2: 0, and = 0 for t < 0) 1 sum vector a degree of stability a positive constant determining the radius of a disc a, spectral abscissa Po robustness measure PG constant, bounding the plant operator, see Eq.(2.26) 680 E Notations

constant, bounding the nonlinearity, see Eq.(2.26) I sector bound I maximum gain IG gain term, bounding the plant operator, see Eq.(2.26) IN gain term, bounding the nonlinearity, see Eq.(2.26) -y(k) correcting gain vector (Kalman gain vector) r stability region in the s-plane r c complement of the region r in the s-plane r system matrix of a singularly perturbed system including dynamic feedback controller r aggregation matrix r generalized system transfer matrix r A , rv matrix constraints {j e.g., ox, first-order infinitesimal difference (first variation) of x o(t) unit Dirac delta function, unit impulse occuring at t = 0 Oij Kronecker delta (= 1 for i = j, and = 0 for i -# j) oT(r) Kronecker delta (= 1 for r = 0, and = 0 for r -# 0) 6( G 1 , G 2) directed gap between two systems G 1 and G 2 ~ e.g., ~t, (small) increment of t Abd block-diagonal uncertainty matrix Ai partition of the block-diagonal uncertainty matrix Abd € small quantity € small scalar parameter representing high-frequency parasitic dynamics € quantity preferably moving from 0 to 1 € parameter determining the size of the polytope £ error vector 8 inverse matrix of power series expansion of orthogonal functions, see Eq.(36.46) 8(s) norm-bound uncertainty associated with an additive perturbance, see Eq.(26.1) 1<, [] spectral condition number A, .x scalar and vector-Lagrange-multiplier, respectively A coefficient of stretched time scale Ai [A] ith eigenvalue of the matrix A Ap eigenvalue of the perturbed plant transfer matrix .x costate variable, adjoint variable A set of eigenvalues A diagonal matrix of the eigenvalues Ai A matrix-Lagrange-multiplier, costate matrix P integer number PeN upper bound for nonlinear time-varying perturbation, see Eq.(13.75) PD[] structured singular value, see Eq.(26.18) pp[] matrix measure pp factor Amin[QJ/Amax[P] , see Eq.(13.41) p,[] spectral matrix measure v integer number ~lik , 6ik multiplicative noise 1r 1r = 3.14159265 1r[] Perron-Frobenius radius, Perron eigenvalue (root) P scalar parameter characterizing the p-system, see Eq.(25.173) PH distance from Hurwitz stability, see Eq.(21.100), radius of largest hypersphere Popt maximum radius of robust stabilizability, see Eq.(30.1l6) PH numerical radius P, spectral radius Pi modified differential sensitivity variable ot] , (Ti['] singular value (THi['] Hankel singular value (T" standard deviation, square root of the constant variance of observation errors (Tyv, (Tuv maximum plant sensitivity and maximum controller sensitivity, respectively (Ti differential sensitivity vector Epj region in k-space where r-stability is satisfied for a fixed plant parameter vector E.5 Glossary of Symbols in Alphabetic Order 681

(n, m)-matrix of singular values, see Eq.(22.88) T time shift T fast time scale, treating singular perturbation behaviour q,( s) least common multiple of the minimal polynomials of Aw and Ar where Aw and Ar are system matrices associated with the disturbance and reference, respectively 'P;{() orthogonal polynomial 'Pm phase margin 'P; right-eigenvector of 41(T) 41(t, T) transition matrix for a time-varying system 41cl(t) transition matrix of the closed-loop system 41cL(T) closed-loop coefficient matrix in state-space description of a discrete-time system ~ input matrix of a discrete-time system w (real) frequency in radians per second crossover frequency ellipsoidal set transfer matrix from the input Zj to the internal signal V; Appendix F

Author Index

Abdul- Wahab, A.A., 1989 Robustness measure bounds for generalized dynamic output feedback control• lers, Int.J.Systems Sci. 20, pp. 2095-2105 Abdul- Wahab, A.A., 1990, Lyapunov-type equations for matrix root-clustering in subregions of the complex plane, Int.J.Systems Sci. 21, pp. 1819-1830 Abdul- Wahab, A.A., 1990a, Lyapunov bounds for root clustering in the presence of system uncertainty, Int. J. Systems Sci. 21, pp. 2603-2611 Abdul- Wahab, A.A., 1990b, Lyapunov stability robustness measures for multivariable, continuous, time• invariant, linear systems, Int. J. Systems Sci. 21, pp.2577-2587 Abdul- Wahab, A.A., 1991, Perturbation bounds for root-clustering of linear continuous-time systems, Int. J. Systems Sci. 22, pp. 921-930 Abdul- Wahab, A.A., and Zohdy, M.A., 1988, Generalized linear transformations on the design of robust dynamic output feedback controllers, Int.J.Control 48, pp. 1241-1266 Abdul- Wahab, A.A., and Zohdy, M.A., 1989, Eigensystem assignment by feedback control, Int.J.Control 50, pp. 1619-1634 Ackermann, J.,1972, Der Entwurf linearer Regelungssysteme im Zustandsraum, Rege/ungstechnik 20, pp. 297-300 Ackermann, J., 1980, Parameter space design of robust control systems, IEEE-Trans. AC-25, pp. 1058- 1072 Ackermann, J., 1984, Robustness against sensor failures, Automatica 20, pp. 211-215 Ackermann, J.,(Ed.) 1985, Uncertainty and Control (Springer, Berlin New York) Ackermann, J.,1985a, Multi-model approaches to robust control system design, In: Ackermann, J.,{Ed.) 1985, Uncertainty and Control (Springer, Berlin New York) Ackermann, J., 1988, Abtastregelung, 3. Auflage (Springer, Berlin) Ackermann, J., and Barmish, B.R., 1988, Robust Schur stability of a polytope of polynomials, IEEE• Trans. AC-33, pp. 984-986 Ackermann, J., Hu, H.Z., and Kaesbauer, D., 1990, Robustness analysis: a case study, IEEE-Trans. AC-35, pp. 352-356 Ackermann, J., and Hu, H.Z., 1990a, Robustness of sampled-data control systems with uncertain phy• sical parameters, 11th IFAC-Congress Tallinn, Vol. 5, pp. 194-199 Ackermann, J., Kaesbauer, D., and Muench, R., 1991, Robust Gamma-stability analysis in a plant parameter space, Automatica 27, pp. 75-85 Adamjan, V.M., Arov, D.Z., and Krein, M.C., 1978, Infinite Hankel block matrices and related exten• sion problems, Am. Math. Soc., Transl. Series 2, 111, pp. 133-156 Aida, K., and Kitamori, T., 1990, Design of a PI-type state feedback optimal servo system, Int.J.Control 52, pp. 613-625 Aly, C.M., and Ali, W.C., 1990, Digital design of variable structure control systems, Int.J. Systems Sci. 21, pp. 1709-1720 Anagnost, J.I., Desoer, C.A., and Minichelli, R.I., 1989, Generalized Nyquist tests for robust stabi• lity: Frequency domain generalizations of Kharitonov's theorem In: Milanese, M., Tempo, R., and Vicino, A., (Eds.) 1989 Robustness in Identification and Control (Plenum Press, New York London) pp. 79-96 Anderson, B.D.O., 1969, Stability results for optimal systems, Electronics Letters 5, pp. 545. Anderson, B.D.O., 1973, Exponential data weighting in the Kalman-Buey filter, Information Sci- ences 5, pp. 217-230 684 F Author Index

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Page references in boldface type indicate definitions

A ~bounds, 308 ~ characteristic, 313 abscissa see spectral abscissa ~ part of a polynomial, 329 absolute bound, 301 ~ polynomial, 330 absolute value, 364, 394 asymptotically stable, 244 ~ of an interval matrix, 178 asynchronous model, 577 absolute-value-integral theorem, 121 attenuation factor, 318 actuator, 196, 520 attraction, 212 adaptation, 41, 42, 490 augmentation, 354 adaptive and robust, 222 autocorrelation, 667 additional feedback, 138 autocovariance, 667 additive uncertainty, 38, 181, 301, 385, 492, 497, autonomous system, 200 515 average adjoint, 79, 611 ~ degree, 338 affine coefficient function, 335, 340 ~ gain, 308 affine linear dependence, 340 ~ matrix, 169, 193, 209 aggregable, 554 axis-parallel box, 330 aggregation ~ algorithm, 552 B ~ conditions, 554 ~ model, 547 backward time direction, 268 algebra, 609 Banach space, 63, 502 algebraic Lyapunov equation, 202 Bauer-Fike theorem, 164, 207, 256 algebraic multiplicity, 633 Bellman-Gronwall lemma all-pass, 365, 501 ~ for continuous-time systems, 148, 155, ~ extension, 490, 496 162,206,293,296,298 ~ matrix, 516 ~ for discrete-time systems, 232 ~ transfer function, 496 Bezout identity, 249, 476, 485, 491, 511 amplitude distribution, 520 bialternate product, 170 amplitude margin, 371 bias, 668 analysis using matrices, 79 ~ matrix, 193 angle between two vectors, 611 BIBO stability, 150 annihilating polynomial, 466 BIBO system, 504 annihilator, 530, 609 bilinear form, 199 antimirror-image, 330 bilinear system, 212, 558, 580 antimirror polynomial, 330 bilinear transformation, 328 antisymmetric part, 378 block-diagonal, 218, 264, 272, 274, 401, 446 antisymmetric property, Hurwitz ~, 354 ~ identity matrix, 441 aperiodicity condition, 354 ~ matrix, 74, 438, 551, 553, 634 approximate norm of a ~ matrix, 54 ~ aggregation, 549 ~ structure, 98, 267 ~ model, 36, 523 ~ unitary matrix, 441 ~ system, 557 block problem, 517 approximation, optimal ~, 657 block-pulse function, 582 arbitrary connected region, 334 block-structured uncertainty, 433, 437 assignable subspace, 637, 639 block-triangle matrix, 140 assignment, 98, 117 block-triangular matrix, 643 associativity, 73, 609 Bacher formula, 110 asymmetric Bode plot, 38, 395 708 Index

bound, 153, 157, 160, 183, 195, 200, 203, 204, ~ system, 314 212, 216, 223, 231, 245, 255, 277, 292, coefficient parametrization, 323 301, 325, 327, 351, 353, 378, 382, 387, cofactor, 80, 611 391, 394, 398, 412, 422, 438, 441, 452, col operator, 573, 611, 646 493, 510, 519, 550, 559, 569, 629 collocation point approximation, 587 asymmetric ~, 308 column compression, 368 ellipsoid ~, 186 column-like partitioned matrix, 518 exponentially decaying ~, 150 column ~ for Lyapunov derivative, 221 ~ operator, 76 hyperellipsoid ~, 187 ~ string, 76 boundary, 189, 323 ~ sum, 49,176 ~ condition, 592, 605 ...... sum norm, 51 ~ layer, 541 combination of two polynomials, 326 ~ representation theorem, 342 combining plant and controller, 154 ~ theorem, 343 common left-divisor, 475 ~ value problem, 592 common right-divisor, 460, 476,479 bounded,37 commutativity, 609 bounded-input bounded-output system, 504 companion form, 111, 117, 128, 169, 185, 189, bounded-real, 504 190,381, 628 bounding differential equation solution, 57 companion matrix, 626, 628, 637 box of coefficients, 326 comparison theorem, 152 Box theorem, 337 compatible, 50 break point, 393, 482, 588 compensator, 463 bypass, 482 complementary sensitivity, 397, 499, 508 complete modal synthesis, 627 C complex-conjugate, 357 complex-conjugate eigenvalues, 624 cactus approach, 196 complex matrix, 357 calculus of variations, 592 complex matrix and real parameter regression, canonical equation, 260 660 canonical form, 626, 628 complex polynomial coefficients, 324 canonical form and sliding mode, 531 complex root boundary, 339, 340, 345 Cauchy's index, 331 complex stability radius, 376 Cauchy's residue theorem, 513 component Cauchy sequence, 63 ~ connection framework, 97, 274, 553 causal operator, 304, 494 ~ detection (failures), 607 causal system, 67 ~ of a matrix, 615 Cayley-Hamilton theorem, 615, 623, 641 composite system, 553 center of perturbed motion, 277 composite matrices, 378 centralized control, 97 compression, 368 chain rule, 88 computation delay, 236, 240 characteristic conditioning, 405, 649 ~ equation, 92, 168, 249, 623 condition number, 216, 256, 405, 649, 669 ~ function, 323, 623 cone-bounded transfer function, 433 ~ locus, 394,407,419,434 conformable, 609 ~ ~ inclusion region, 417 congruent matrices, 649 ~ matrix, 110, 373, 623 conjugate transpose, 358 ~ polynomial, 92, 109, 120, 186, 189, 241, conjugation, 624 252,323,347,408,623,633,636 conservatism, 43, 164, 180, 189, 218, 308, 329, chattering, 535 391,399,410,416,427,438,440,443 Chebychev polynomial, 332, 565, 567, 570 consistent norm, 48, 511 Cholesky decomposition, 650, 652 constraint, 131, 278, 300, 470 chordal metric, 430 control canonical form, 626, 628 see a/so com• circle criterion, 307, 313, 500 panion form circle region, pole assignment in a ~~, 254 control energy, 61 closed-loop control signal, 482 ~ eigenvalue sensitivity, 95 ~ ~ magnitude, 398 ~ singular values, 393 control uncertainty, 318 Index 709

control variable constraint, 404 degree, 476 controllability, 544 ~ of inefficiency, 669 ~ gramian, 60 ~ of interpolation, 496 ~ matrix, 548, 613, 629 ~ of polynomials, 565 controllable subspace, 614, 629 ~ ofrobust stability, 175 controlled signal, 482 ~ of robustness, 41, 400 controller, 96, 123, 181 ~ of stability, 181, 244, 273 ~ gain constraint, 470 degree-dropping boundary, 339, 345 ~ matrix, 116 delay, 574 ~ norm, 135 computation ~, 236, 241 ~ parameter, 250 delayed perturbation, 210 maximum ~ sensitivity, 316 demarcation, 562 convergence condition, 64 denominator, least common ~, 407 convex, 416,441 denominator matrix, 476 ~ combination, 353 derivative, 68, 79, 204, 363 ~~ of polynomials, 327 ~ of a singular value, 366 ~ hull, 182, 196, 324, 333 ~s overview, 79 ~ performance, 252 ~ with respect to a vector or a matrix, 79 ~ polygon, 343 ~ with respect to time, 88 convolution, 61, 67-70, 149, 157, 297, 312, 313, desensitized controller, 132, 137 563, 620 . design of a time-varying system, 576 ~ infinity norm, 68 detection observer, 607 coprime, 491, 515 determinant, 391, 408, 612 ~ factorization, 457 ~ derivative, 83 ~ polynomials, 476 ~ of a partitioned matrix, 644 core vector, 354 deviation matrix, 193 corner, 328 diagonal ~ matrix, 182, 184, 202 ~ dominance theorem, 168 ~~ norm, 182 ~ elements, 183 ~ points, 253, 328, 332 ~ feedback, 420 reduced number of ~~, 183 ~ matrix, 54, 610 ~ polynomial, 324 ~ perturbation, 59 corona, pole assignment in a~, 255 ~ scaling, 411 cost function, 300 ~ weighting, 164 cost matrix, 269 diagonalization, 192, 637 ~~ difference, 270 diagonally dominant, 371, 532 costate matrix, 260 diagonally perturbed system, 387 costate variable, 268 diagonally weighted norm, 59, 387 covariance, 266, 292, 667, 668 diagonizable, 206, 207, 212 perturbed ~, 294 ~ transition matrix, 231 cross-condition number, 423, 424 diamond, 325 crossover frequency, 35, 36, 38, 392 differential equation, 91, 107 crossover slope, 38 time-varying ~~, 145, 148 cyclic property of the trace, 614 differential operator, 475, 463 differential recurrence, 569 D differential sensitivity, 33, 71, 79, 95, 137, 395, 535 damping factor constraint, 470 ~~ feedback, 138 D-contour, 391, 419 minimize ~~, 256 dead time, 35, 432, 574 ~~ of characteristic polynomial, 115 decay rate, 158 ~~ of state variables, 111 decentralized control, 98, 172, 264, 272 performance ~~, 123 decomposition, 39, 210, 650 transfer function ~~, 119 ~ into orthogonal functions, 567 differentiation, 574 ~ into piecewise linear functions, 589 dimension, 196, 613 decoupling property, 568 Diophantine see Bezout identity definiteness, 202, 630 Dirac function, 668 degeneracy, 634, 636, 637 Dirac input, 297 710 Index direct matrix norm, 48 eigenvalue, 55, 82, 167, 170, 173, 180, 184, 190, directed gap metric, 520 200,214,227, 256, 273, 298, 394, 417, direction, most sensitive ~, 369 615, 623 disc, 174,188,254,310,325,371,417 ~ assignment, 117, 532,626 ~ bounding eigenvalues, 175 ~ bound, 371 open unit ~, 502 ~~ of an interval matrix, 174 ~ polynomial, 335 ~ decomposition, 650 discontinuous see discrete-time ~ differential sensitivity, 95 discrete lossless positive real function, 331 ~ distribution, 190 discrete-time interval system, 184, 352 dominant ~, 151 discrete-time system, 69, 69,112, 165, 177, 179, ~ exclusion circle, 417 188, 190, 195, 225, 249, 351, 374 ~ exclusion lemma, 383 distance from Hurwitz stability, 339, 342 generalized ~, 192 distance, graph metric ~, 519 ~, Hankel norm, 61 distance problem, 517 ~ inclusion region, 418 distinct eigenvalues, 354, 624, 628, 633, 636 ~ increment, 104, 164 distributed control, 323 largest. modulus of an ~, 49 distributed time-varying system, 604 multiple ~s, 102 distribution, 520 non-distinct ~s, 359 distributivity, 73, 609 ~ of a complex matrix, 357 disturbance, 34,42,397,482 ~ of the Kronecker product, 75 ~ of the Kronecker sum, 76 ~ differential equation, 459 ~ of the Lyapunov matrix, 204, 230 ~ rejection, 31,390,403,459,464,483,507, 527 ~ of the symmetric part, 210 dither, 520 ~ of the transition matrix, 225 ~ properties, 629 divergence, 81 ~ sensitivity, 640 divisive feedback structure, 435 eigenvector, 78, 90, 127, 216, 257, 275, 410, 533, domain of attraction, 212 623 dominant, 466 ~ assignment, 98, 637, 640 ~ eigenvalue, 151,547,552 ~ chain, 623 ~ pole location problem, 335 ~ derivative, 105 ~ state, 537, 539, 544, 550 ~ differential sensitivity, 95, 103, 107 double Laguerre expansion, 604 ~ increment, 104, 106 doubly-coprime, 486 normalized ~, 104,360 D-weighted norm, 59, 164,414 ~ of Hermite matrices, 359 dyadic ~ of the Kronecker product, 75 ~ decomposition, 632 ~ of the Kronecker sum, 76 ~product, 199,259,572,610,613,614,632 ~ of the transition matrix, 227 ~ representation, 78 element by element bound, 162, 165, 167, 181, dynamic 188 ~ control factor, 121 elementary matrix, 610 ~ controller, 154, 156, 218, 400, 540 ellipsoid sets, 277 ~ interval system, 167 ellipsoidal constraints, 278 ~ modelling uncertainty, 491 elliptic norm, 47 dynamically bounded, 149 encirclement, 310, 391, 408, 419, 435 enclave, 335 E energy ~ bounded, 510 E-contour, 417 ~ gain factor, 504 edge, 196,328,344 ~ of the error, 509 ~ perturbation, 443 ~ of the signal, 64 ~ polynomial, 333 entropy function, 509 ~ theorem, 334, 338,443 envelope, optimum ~, 300 eigenlocus, 380 equivalent controller, 526 eigensolution, 624 equivalent perturbed system, 428 eigenstructure assignment, 640 error, 511, 658 eigensystem assignment, 626 ~ energy, 509 Index 711

~ estimate, 649 first variation, 132 generalized ~, 483 fixed mode, 98 ~ matrix see perturbation matrix forcing phasor, 452 ~ polynomial, 323, 351, forgetting factor, 586 tracking ~, 36 Fourier transformation, 69, 307 estimate, 281, 291, 294 fractional representation, 477, 484, 515 ~ of a function, 66 fractional uncertainty, 39, 493 estimation, 607, 655, 658, 669 free of zeros in the closed right-half s-plane, 501 ~ error, 285, 292, 669 frequency-dependent uncertainty, 439 ~ in large-scale systems, 662 frequency domain, 52, 69, 407 linear~, 667 frequency richness, 42, 293 ~ sensitivity, 669 Frobenius formula for inversion, 643 Euclidian norm see Frobenius norm Frobenius norm, 47, 81, 95, 101, 149, 162, 175, even part, 325 186, 202, 215, 256, 311, 365, 412, 423, even-power term, 345 559,611,655,660 excess stability margin, 515 ~~ derivative, 90 exclusion circle, 417 expected function of the ~~, 292 exogenous system, 463 full-order observer, 284 expansion full rank, 613 ~ integral of the product, 572 full-state loop transfer recovery, 287, 289 power series ~ of orthogonal functions, 571 function norm, 53, 63 ~ product of two functions, 572 function of a matrix, 108 Tay lor series ~, 557 , 63, 301 expectation, 102, 667 functional analysis, 63 expected function of Frobenius norm, 292 functional differential equations, 576 exponential matrix, 107,633, fundamental matrix (see transition matrix) 623, exponential L~-stability, 307, 310 636 exponentially decaying bound, 150 future outputs, 61 exponentially stable, 244, 247, 477 exposed edges, faces, 334 G extended controllability matrix, 548 extended state controller, 139 g~n, 36, 227, 231,316, 399 external signal, 482 average ~, 308 externally skew-prime matrix, 461, 476 ~ and singular value, 52 extrapolation, 129 ~ factor, 226, 504, 665 extreme coefficients, 187 ~ margin, 289, 435, 491 extreme polynomial, 324 ~ matrix, 284, 291, 292 ~ range, 308 F ~ scheduling, 42 gap metric, 520 face, 196 Gastinel-Kahan theorem, 195, 373 factorization, 457, 475, 491 Gauss distributed, 294, 667 fractional ~, 515 Gauss-Markov theorem, 669 inner-outer ~, 497 Gaussian noise, 291 normalized ~, 488 general spectral ~, 504, 514 ~ distance problem, 517 Faddeev method, 110 ~ matrix equation, 646 failure, 34, 607 ~ orthogonal polynomial, 565 family of polynomials, 328, 333 ~ pseudo-inverse, 646 fast mode, 525, 534, 537, 539, 541, 544 ~ stability bounds, 146 fast time scale, 539 ~ system inverse, 647 feedforward, 465 generalized fictitious uncertainty, 447 ~ eigenvalue, 192 field of values, 358 ~ eigenvector, 633 final boundary condition, 270 ~ Nyquist criterion, 419 finite energy, 504, 506 ~ observer, 288 finite horizon case, 263 ~ Parseval theorem, 70 first-order expansion, 265 ~ plant, 508 712 Index

- polynomial, 331, 334 - determinant, 252 - product, 557 - polynomial, 325, 516 - resolvent matrix, 385 - stability, 176, 183, 194, 324, 339, 385 - signal, system, 482 - testing matrix, 327, 349 - stability, 170 hyperellipsoid, 346 geometric multiplicity, 634 hyperplane design, 525 Gershgorin's theorem, 176, 188, 313, 371 hyperplane motion, 530 globally stable, 201 hyperplane, supporting -, 196 globally weighted modulus matrix, 208 hyper-rectangle, 343 gradient,81,84,126,133,221,251,260,416,568, hypersphere, 219, 339, 345, 346 592,650 hysteresis, 31 gramian, 59, 60 graph metric perturbation, 519 greatest common divisor, 476 Greville's method, 648 idempotent matrix, 526, 610, 616, 618, 660 grey matrix, 193 identifiability condition, 657 guardian map, 170 identification, 42, 491, 537, 581, 602, 605, 607, 620 H identity matrix, 610, 618, 631, 637 eigenvalues of the --, 359 Hadamard product, 412 ill-conditioned matrix, 649 Hamiltonian, 279 imaginary axis, 167 - function, 260 imaginary eigenvalues, 191 - matrix, 379 imaginary part of the eigenvalue, sensitivity of - property, 125 the --,100 Hammerstein model, 563 impulse input, 297 Hankel impulse response, 312 - approximation, 517, 519 inaccurate calculation, 43 - norm, 59, 490 inclusion principle, 359 - operator, 60 inclusion region, 417 - singular value, 491 increment, 133, 135, 263 Hardy space, 64, 499, 502, 672 eigenvalue -, 164 harmonic function, 570 incremental Hermite - bound, 301 - form, 363 - notation, 90 - matrix, 46, 53, 358, 630 - sensitivity, 270 - polynomial, 565, 567 - updating, 125 , 128 incrementally conic nonlinearity, 305 hierarchical controller, 274 indentation, 419, 427 high-frequency independent information, 652 neglected - dynamics, 436 index of an eigenvalue, 634 - oscillations, 473 index of performance see performance - range, 392 index, robust stability -, 66 - signal, 520 individually weighted modulus matrix, 208 Hilbert norm, 50 (see also spectral norm) induced Hilbert space, 503 - gain, 399 H2-norm, 64, 65 - norm, 48, 49, 53, 374 minimum weighted -, 514 inefficiency, 669 Hoo-norm, 64, 299, 339, 380, 435, 443, 480, 490, inertia, 190 496,500,505,509,516,518,521,672 infimum, 164, 412 -, invariance re inner matrix, 497 infinite horizon case, 263 hodograph, 349 infinitely sensitive, 256 Holder infinity function norm, 351 - inequality, 65, 307, 311 infinity norm, 46, 51, 62, 63, 68, 174, 193, 313, - norm, 47, 49, 256 416, 511, 516 homegeneous solution, 145, 478 initial condition, 117, 279, 461, 479, 595, 599, Hurwitz 605, 619 - antisymmetric, 354 optimal -- for observers, 656 Index 713 initial function, 151 Jordan initial state, 60, 129, 140, 153,221,266,400,577, ~ block, 109, 634, 637 581,586,619 ~ canonical form, 370, 626, 634, 639 random ~~, 134 initial value, 91 K inner, 365 ~ matrix, 496, 497 Kalman-Bucy filter, 291 inner loop stability, 461 Kalman controllability matrix, 548 inner-outer factorization, 497 Kalman inequality, 425 inner product, 46, 81, 104, 233, 357, 363, 369, kernel see null space 502,611 Kharitonov input noise, 283, 291 ~ polynomial, 252, 324, 337, 349 input-output mapping, 66 ~ segment, 338, 339 input-output relations, 505 ~ theorem for continuous-time systems, input space, 505 184,324,339 insensitive, 256, 535 ~ theorem for discrete-time systems, 328 instrumental variable, 666 Kleinman lemma, 84, 263 integral of squared error, 582 Kronecker integration by parts, 262, 598 ~ algebra, 73, 79 integrity, 470 ~ delta, 668 interconnected system, 259, 553 ~ matrix, 610 interconnection, 247, 264, 274 ~ power, 89, 601 interdependent perturbation, 336 ~~ model, 559 interelement dependency, 80 ~product, 73, 78, 96,601,609 interlacing property, 325, 332 ~~ derivative, 87 internal ~~ of unitary matrices, 360 ~ model principle, 459, 464 spectral norm of ~~, 51 ~ parallel model, 402 ~ sum, 75, 78, 170, 195,378 ~ stability, 461, 481, 499 interpolating matrix polynomial, 615 L interpolation, 110, 495, 656 intersection, 525 Lachmann model, 563 ~ of parameter regions, 252 Lagrange multiplier, 131, 133, 141,252,267,339, ~ theorem, 344 interval 592,663,669 Laguerre polynomial, 565, 567, 604 ~ boundaries of orthogonal polynomials, 566 Laplace transformation, 70, 108, 127, 545, 573, 619 ~ matrix, 38, 167, 171, 174, 176, 187, 188, 193,196,229 large-scale system, 244, 264, 554, 662 largest ~ plant, 336 ~ absolute column sum, 154 ~ polynomial, 167, 185, 252, 323 ~ hypersphere, 339, 340, 345 ~ scalar, 178 ~ vector, 185, 250 ~ modulus of an eigenvalue, 49 inverse, 77,365,610,611,632,641 ~ stability box, 355 ~ complementary sensitivity function, 508 leading coefficient, 325 ~ matrix derivative, 88 leading principal minor, 349 ~ of a partitioned matrix, 643 least-favourable noise, 295 ~ return difference, 432, 482 least squares, 658 , 486 least upper bound, 49 , 362 Lebesque space, 503 irreducible matrix, 416 left iteration, 129, 251, 263, 271, 276,401,550,663 ~ convergent, 178 ~-coprime matrix, 477, 479 J ~-eigenvector, 623 ~-fractional, 480 Jacobi formula, 100 ~-inverse, 479, 529, 618, 645 Jacobi polynomial, 565, 566 ~-prime factorization, 461 Jacobian matrix, 85, 111 ~-prime matrix, 475 714 Index

-pseudo-inverse, 274, 548, 586, 599, 647, matricial gradient, 82, 86, 221, 260 652, 659, 673, 669 matrix ~ sector, 325 aggregation~, 547 ~ singular vector, 367 ~ algebra, 609 Legendre polynomial, 565, 567, 570 ~ analysis, 79 Leverrier's algorithm, 110, 186 ~ component, 108 L2-gain factor, 60 ~ decomposition, 650 limited-time exciting case, 470 ~ differential equation, 91, 202, 278 L~-induced norm, 67 ~ exponential, 75, 76, 107,633 line segment, 337 ~ function, 615 linear ~ infinity norm, 49, 51 ~ algebraic equations, error estimate, 649 ~ inversion lemma, 641 ~ causal stable operator, 494 ~ measure, 45, 56, 201, 212, 217 ~ dependent information, 652 ~ norm, 45 ~ optimal control via orthogonal functions, ~ I-norm, 49 591 ~ polynomial, 420, 631 ~ quadratic Gaussian theory, 504 ~ product approximation, 594 ~ quadratic regulator, 213, 288 ~ product rule, 86 ~ regression, 655 ~ set, 557 ~ system solution estimate, 151 symmetric ~, 57 L2-norm, 69, 307, 311, 316 ~ transpose, 642 localization of component failures, 607 ~-valued function, 86 loop max norm, 46 ~ gain, 36, 315, 508 maximally robust pole placement, 251 ~ shaping, 393, 442 maximizing robustness region, 221 ~ transfer function, 287, 393 maximum ~ transfer recovery, 287 ~ absolute column/row sum, 193 lossless positive real function, 329, 331 ~ bias matrix, 193 low-frequency ~ eigenvalue, 227 ~ parameter errors, 436 ~ gain, 227 ~ performance, 393 ~ modulus, 210 lower bound, sensitivity norm, 102 ~-modulus theorem, 64, 502 LQ regulator see Riccati controller ~ perturbation, 41, 185, 348, 349 LQG control, 43, 294 ~ positive real eigenvalue, 171 LR-Perron scaling, 424 ~ real part of an eigenvalue, 630 L2-S2-gain, 68 ~ singular value, 210, 312, 376, 392, 398, L;-stability, 63, 301,310, 313, 399 409,439 Lyapunov ~ stability bound, 342 ~ approach for unidirectional perturbation, mean value, 667 171 measure, matrix~, 45, 56 ~ equation, 78,135,139, 158, 160, 161, 189, measurement noise, 283, 291, 390 194, 202, 214, 225, 235, 244, 262, 266, measurement signal, 482 299,381,626 Metzler matrix, 229, 248, 616 ~ function, 173, 225 Mikhailow robustness criterion, 349 ~~ dynamics, 200 minimal ~ stability, 199, 201 ~ dimension of stable faces, 196 ~ distance to Hurwitz stability, 339 M ~ Frobenius norm, 412 ~ least square, 655, 659 main diagonal element, 176 ~ polynomial, 108,465, 469, 615, 636 manifold, 525 minimax mapping, 66 ~ frequency optimization, 457 ~ theorem, 443 Lyapunov ~ controller, 242 Markov ~ optimization of the spectral radius, 370 ~ parameter, 355 ~ problem, 278, 295 ~ process, 668 minimum ~ theorem on determinants, 354 ~ control energy, 61 Martin's theorem, 374 ~ degree of stability, 273 Index 715

- differential sensitivity, 100 noise, 31, 43, 234, 274, 283, 284, 291, 389, 436, - distance from the critical point, 421 668 - distance problem, 490 --free output, 283 - intersection theorem, 344 least-favourable - uncertainty, 292, 295 - negative real eigenvalue, 171 measurement -, 479, 482 - norm controller, 252, 402 - reduction, 318 - number of edges, 333 - rejection/suppression, 397, 508 --phase, 478, 496 - uncertainty, 294, 295 - sensitivity, 516 nominal, 36, 137, 159, 243, 249, 302, 324, 345, - singular value, 365, 389,404,409,417 392,399,408,418,449 - variance estimator, 669 - parameter, 124 - variance scaling, 413 - polynomial, 324 Minkowski inequality, 66 non-anticipative, 67 minor, 349,408,611 non-autonomous system, 200 mirror-image/polynomial, 330 non-defective, 78 mixed product rule, 74, 601 non-diagonal scaling, 412 M-matrix, 616 nonlinear modal - control, 222, 301 - decomposition, 625 - estimator, 669 - matrix, 165, 188, 206, 207, 257, 360, 363, - perturbation, 159, 204, 211, 214, 225 370, 624, 650 - plant, 155, 520 - state, 614 - regression, 129 - transformation, 162 - singularly perturbed system, 541 model - system, 599, 601, 557 --, block pulse expansion approach, 585 --following approach, 142 - uncertainty, 301 --matching, 509, 516, 517 non penalizing control input, 471 - plant mismatch, 404 non-similarity Perron scaling, 415 - uncertainty gain, 317 nonsingular, 56, 169,202,374, 613 modified spectral radius, 451 nonsingular Kronecker sum of perturbed matri- modulus, 394 ces, 195 - matrix, 153, 167,207,229,381 nonsingularity approach, 169 - sensitivity, 101 nontruncated function, 67 monic polynomial, 185, 333, 381, 623 norm, 45, 178, 180, 184 most sensitive direction, 369,409 - bounds, 153 m-tuple of polynomials, 337 column sum -, 51 multi-model approach, 252 consistent ""', 48 multilinear coefficient function, 335 diagonally weighted -, 59 multiple eigenvalues, 256, 633, 637, 638 direct matrix -,47 multiplicative uncertainty, 38, 301, 366, 386, 390, Euclidian -, 48 392,493,508 Frobenius -, 48 multiplicity, 76, 92, 109, 364, 513, 615, 633, 639 function -, 63 multivariable circle criterion, 313 function p--, 502 multivariable system, gain of the --, 52 Hankel-,59 Holder -, 47 N H 2-, Hoo--, 503, 672 induced matrix -, 49 nabla operator, 81 infinity matrix -m 51 nearest normal approximation, 418 infinity -, 63 necessary condition, 199, 329 L~-induced -, 67 negative definite, 173, 182, 199, 210, 630 matrix 1-~, 49 negativity, 183 - of a matrix-valued function, 65 Nehari extension, 490 - of a vector-valued function, 63 Nevanlinna-Pick theory, 496, 497 - of mth power of a function, 65 Newton-Raphson method, 129 - of the controller, 135 nice stability, 252 operator -, 67 , 610 row sum~, 51 node, 484 Schur ~, 48 716 Index

sensitivity matrix -, 101 operator, 43, 66, 149, 301, 315, 399, 443, 463, Sobolev -, 68 469,494 spectral -, 50, 234 column -, 76 supremum -, 63 Hankel-,60 - uncertain plant, 315 - norm, 67, 304, 313, 505 vector -, 46 trigonal -, 618 weighted -, 59 optimal control, 123, 138, 160,238,259,262,265, norm-bounded matrix, 438 294,581 norm-bounded perturbation, 164 optimization, 62, 267, 279, 480 norm-like Lyapunov function, 244, 246 optimum gain, 228 , 205, 206, 228, 273, 300, 359, 360, order reduction, 526 364,384,418 orthogonal, 73, 257, 339, 624 normalized, 75, 104 - basis, 568 - factorization, 488 - coefficients, 567 normally distributed, 667 - component coefficients, 591 null space, 257, 525, 613, 616, 637 - expansion, 40, 599 number of - matrix, 551 - corners, 332 - polynomial, 565 - edges, 334 - projection, 518 - negative eigenvalues, 201 - transformation, 531, 650 - segments, 337 orthogonality, 359, 566 numerator matrix, 116 orthonormal basis, 368 numerical radius, 45 orthonormality, 568, 632 numerical range, 45, 418 outer matrix, 496 Nyquist criterion, 64, 122, 242, 308, 335, 350, outer product (see also dyadic product) 351,391,407,419,452 increment of the --, 105 outermost boundary, 178 o output, 390 - energy, 61 observability, 470, 544 - feedback, 97 - gramian, 60 - feedback controller, 130, 160, 164, 168, observation vector, 668 252,259,538 observer, 281,454, 607 - noise, 283, 291 - pole allocation, 288 - norm estimate, 52 observer-based - power, 505 - compensator, 472, 491 - regulation see disturbance rejection - control, singularly pert. syst., 546 - space, 505 - controller, 454 - spectral density, 511 odd part, 325, 326 - uncertainty, 452 odd-power term, 345 overall closed-loop transfer matrix, 398 on-off-control, 534 overdetermined, 657 one-parameter family of matrices, 196 overshoot, 297 one-sided Laplace transformation, 70 open-loop P - property, 285 - singular values, 393 para-Hermitian transpose, 511 - system, 313 parallel model, 402 - transfer function, 121 parallelotope, 343, 344 - uncertainty gain, 317 parameter, 37,250,657 open right-half s-plane, 503 - demarcation, 655 operational matrix - estimation, 576, 585, 595, 599 bidiagonal -, 605 - region, 252 - for delay, 574 - sensitivity, 111 - for differentiation, 574, 597, 605 - space, 203, 660 - for integration, 569, 576, 580, 582, 588, parametrization, 257, 274, 323, 475, 515, 637, 592,594,606 parasitic dynamics, 537 stretched -, 577 Parseval theorem, 69, 70, 298, 307, 312 operational product vector, 572 particular solution, 145, 478, Index 717

partitioned matrix, 433, 487, 518, 642, 648 piecewise linear functions, 579, 587 partitioned vector, 611 plant sensitivity, maximum ...... , 316 partitioning into columns, 648 plant uncertainty principle, 490 pathwise connected, 335 p-norm weight, 414 penalizing closed-loop poles, 254 point matrix, 178 penalty near a root, 471 polar decomposition, 2lO, 369 performance, 34,39, 123, 142, 172, 181,236,252, pole assignment/allocation/placement, 98, 118, n4,200,2~,~8,~4,~,W7,~, 154, 182, 249, 284, 289 389,392,395,400, 4lO, 436, 447, 505, pole excess, 121 510,532,549,558,560,592,657, pole zero cancellation, 476 ~ deterioration, 394, 397 polygon, 328 ~ gradient, 126 polynomial, 37, 74, 249, 323, 407, 420, 623 ~ kernel difference, 270 ~ coefficient, 339 ~ index, 213 ~~ function, 335 minimizing~, 141 corner ~, 324 ~ of qI-transformed signals, 471 extreme ~, 324 ~ robustness, 445, 517 generalized ~, 331 ~ robustness theorem, 416 ~ matrix, 408, 459, 475, 484 ~ sensitivity, 123, 130 minimal~, 108 permutation matrix, 73, 77, 126,416,618,644 orthogonal ~, 565 Perron eigenvector, 387, 414, 416 ~ product, 557 Perron (-Frobenius) eigenvalue, 164, 168, 169, vertex ~, 324 172, 385, 412, 416, 422, 424, 442, 616, polytope, 196, 203, 324, 333, 342, 353 633 Popov criterion, 3lO Perron-Frobenius theorem, 633 positive definite, 55,159,189,190,197,199,202, , 619 206, 216, 225, 359 perturbation, 34, 71, 95,101,161,164,171,179, positive matrix, 180 185, 188, 191, 195, 199, 209, 215, 232, positive real function, 329 234, 238, 277, 324, 336, 365, 369, 373, positive realness and Schur stability, 331 377,399,408,417,488 positivity, 183, 252 ~ bound, 204, 206 ~ test, 195 ~ by weighted sum of matrices, 218 postmultiplication, 610 ~ coefficients, 219 power delay ~, 211 ~ density, 668 diagonal ~, 59 ~ expansion, 121 ~ factor, 205,211 ~ of a matrix, 641 graph metric ~, 519 ~ series domain, 599 initial condition ~, 297 ~ series expansion, 557, 571, 576, 592 linear ~, 159 ~ spectral density, 505 ~ matrix, 219, 256 prediction, 235, 665 maximum~, 348 prefil ter, 283 nonlinear ~, 159,211,225,234,247 premultiplication, 6lO ~ parameter, 170 prescribed eigenvalue, 258 plant~, 293 prespecified controller, 134 ~ polynomial, 324, 343 primeness, 461 random ~~, 234 principal singular ~, 537 ~ direction, 367 structured ~, 238, 422 ~ gain see singular value time-varying~, 238 ~ minor, 349, 616, 623 unidirectional ~, 168, 239 ~ phase, 2lO unstructured ~, 238, 519 ~ vector, 633 ~ via constant matrices, 216 principle of argument, 419 phase margin, 35, 289, 425, 491 probability distribution, 668 phase variable form see companion form process noise, 294 phasor, 452 product Pick matrix, 496, 497 ~ of interval matrices, 178 PI-controller, 253, 454, 455 ~ of singular values, 365 piecewise-constant suboptimal controller, 271 ~ of two functions, 589 718 Index

~ rule, 74 - polytope, 324 projection, 430, 518 - representation, 358 ~ of a matrix, 369 recurrence ~ operator, 251 - coefficients, 565, 567 projector, 526, 529, 616, 647 - equation, 584 proper, 64, 244, 477, 493, 503, 545 recurSIve proportional-integral - algorithm, 558, 578 ~ controller, 455 - estimation, 664 ~ observer, 288 reduced matrix, 611 proportional observer, 284 reduced-order pseudo-commutativity property, 461 - model, 31, 60, 540 pseudo-inverse, 74,99, 135,637,645,648 - observer, 281 pulse functions, 579 reduced sensitivity, 429 purely fast variable, 540 reducible matrix, 644 reference, 117,284,398,465,469,479,482,511 Q - differential equation, 459 - energy, 510 quadratic form, 199, 629 ~ input, 389 quality see performance - norm, 318 quasi-diagonal form, 239 reflected argument, 496, 511, 519 quasi-dominant, 616 regression, 274, 549, 561,595, 647, 655 nonlinear ~, 129 R - sum, 660 regulator form, 628 radius relatively left-prime/right-prime, 476 numerical ~, 45 relatively prime, 461 ~ of robust stabilizability, 489 rescaling operation, 442 spectral -, 49 residual, 658 random, 102, 667 residue, 117,513 - initial state, 134, 266 resilience, 35, 403 - sampling, 225 resolvent matrix, 167, 180,373, 623 range, numerical -, 45 retardation operator, 66 range space, 257, 526, 529, 612, 616 return difference, 237, 240,287,508, 609 -- of stability, 328 return gain, 315 rank, 47, 531, 613, 617 return ratio matrix, 449 - reduction, 369 RH2 , RHoo, 64,486 rational transfer matrix, 479 Riccati Rayleigh's principle, 45, 204, 205, 228, 255 - coefficient matrix, 125 Rayleigh quotient, 45, 47, 216 - controller, 123, 134, 173, 214, 238, 294, Rayleigh's theorem, 364, 630, 631 426, 532, 592 reachable, 469 - differential equation, 264, 269, 472, real part of an eigenvalue, 630 - equation, 160, 191,214,272,549,561,595 sensitivity of the -~, 100 - matrix sensitivity, 125 real polynomial, 334 Riemann sphere, 335,429 real-rational functions, 64, 503 right real root boundary, 339, 340, 345 - convergent, 178 real stabilty radius, 377 --coprime factorization, 519 real-time process, 577 ~-coprime matrix, 477, 479 realizability, 403 --divisor, 460 reciprocated polynomial, 330 --eigenvector, 623 recovery --fractional, 479 loop transfer -,287, 289 --half-plane zero, 506 - matrix, 288 --inverse, 618, 644 robustness -, 287 --prime matrix, 475 sensitivity ~, 287, 291 --pseudo-inverse, 562, 646, 655, 673 rectangular - singular vector, 367 ~ box, 329 robust ~ coefficient space, 326 ~ dynamic controller, 154 Index 719

- observation, 607 Schiiflian form, 170 - pole placement, 250 Schur - rout locus, 348 - matrix norm, 48 - stability, 153, 175, 195 - polynomial, 351 - stability index, 66 - stability (see also discrete-time systems) robustification, 210, 218, 348 178,182,196,328,331,336,353,385 robustness, 31, 33, 40,42, 145, 167, 172, 181, Schwartz inequality, 611 195, 203, 206, 208, 210, 211, 213, 219, second-derivative sensitivity, 128 222, 248, 253, 272, 277, 297, 301, 307, secondary diagonal matrix, 618 352, 373, 377, 383, 389, 400, 409, 416, sector, 521 429, 436, 445, 459, 469, 483, 490, 495, - condition, 235 499,506,517,523,527,538,558 - gain, 307, 313, 499 - bounds, 244 --type inequality condition, 197 - in the frequency domain, 321 segment, 337 - in the steady state, 289 self-derivative matrix, 73 - in the time domain, 145 semidefinite, 58, 199 - measure, 344 sensitive to a single component, 607 Mikhailov - criterion, 349 sensitivity, 34, 41, 121, 390, 395, 397,427, 429, Nyquist - criterion, 350 480, 482, 499, 504, 506, 515, 516, 535, - of Kalman-Bucy filters, 291 640 - of proportional integral observers, 288 differential -, 33, 71, 79 - of proportional observers, 281 estimation differential -, 669 - recovery, 287 - function, 242 small-scale -, 33 - matrix, 101,302 stability -, 167 maximum plant -, 316 - using gap metric, 520 - minimization, 483 - via orthogonal decomposition, 607 - minimizing in H2 ,511 root clustering, 254 - norm, 102 root distribution, 189 - of eigenvalues, 256 root locus, 167, 334, 348 performance -, 123 Rosenbrock matrix, 118 - recovery, 287 rotation matrix, 618 - vector, 139 Routh's criterion, 328 sensor, 196 row - noise, 31, 293, 436 - compression, 368 - output, 482 --like, 517 - perturbation, 172 - rank, 496 separation principle, 288 - string, 77 servo-compensator, 459, 464 - sum, 49, 59,176 servo-controller, 464 set-theoretic approach, elliptic -, 277 setpoint of the performance, 134 S shape, 187, 278 shaping, 393, 442 saddle-point problem, 295 - signal, 470 sampling interval, 225 shifted orthogonal polynomial, 566 saturating actuator, 520 shifting eigenvalues, 631 saturation-like structure, 224 sign, 324, 325 saw-tooth signal, 520 - function, 531 scalar product, 46, 81, 104, 233, 357, 369, 502, signal energy, 64, 69, 506 611 similar matrices, 612 scalar product rule, 88 similarity scaling, 411, 422 scaled similarity transformation, 542 - argument, 577 simple - decomposition, 651 - connectedness, 335 - perturbation matrix, 172 -L2-stability, 306 scaling, 59, 187,217,225,410,422,442,443,515, - perturbation, 340 649 single-input single-output system, 392 - matrix, 172, 180, 379, 386,433, 438 single parameter perturbation, 192 720 Index singleton, 196 - condition, 195,301,399 singular - degree, 308 - matrix, 195 generalized -, 170 - perturbation, 38, 537 - hyperellipsoid/sphere, 345, 346 singular value, 163, 187, 205, 214, 217, 233, 238, - improvement, 293 239, 245, 287, 357, 364, 389, 396, 445, - index, 66 505 - margin, 188, 247, 284, 338, 339, 346, 387, -- decomposition, 163, 361, 367, 494 ~9,~2,400,4n,4~,«3,4~,OO8, --- and pseudo-inverse, 648 515 largest --, 50 - preserving perturbation, 191 maximum/minimum --, 90 stability radius, 172, 226, 366, 376 -- properties, 365 - of parametrized system, 380 structured --, 433, 438, 439 - of polynomials, 381 singular vector, 163,220, 366,423 stability singularity, 612 - region, 213 skew-Hermite, 358 - robustness, 158, 167, 181,445 - matrix, 190 -- measure, 298 skew-symmetric, 358 transformation of - to nonsingularity, 169 sliding mode, 40, 525 - with time delay, 151 slow mode, 537, 539 stabilizability slow sliding mode, 534 quadratical -, 222 slow state, 541 radius of robust -, 489 Small gain theorem, 304, 315, 339, 399, 490 robust -, 492 Smallil-theorem, 439 stabilization, 487 small-scale perturbation, 71 strong -, 495 small-scale robustness, 33, 95 stable face, 196 smallest but one singular value, 410 stable matrix, 55, 64, 168, 170, 171, 202, 612, smallest singular value and rank reduction, 369 630 Smith predictor, 241 stacking operator, 76 smoothed nonlinearity, 521 standard form of the actual plant, 538 Sobolev-boundedness, 521 state-dependent noise, 293 Sobolev space, 68 state feedback, 117, 134, 138, 160, 163, 213, 252, solution error estimate, 649 256, 283, 325, 525, 573, 626 spectral abscissa, 49, 162, 164, 169,205,206,298, state-space uncertainty, 161 374,402 state variable spectral condition number, 164, 206, 214, 231, -- differential sensitivity, 111 239, 240, 535, 649 -- matrix, 259 derivative of the --, 90 stationary stochastic process, 667 spectral statistical perturbation, 102 - cross-condition number, 423 step input, 297 - decomposition, 112, 649 stereographic projection, 430 - density, 511 stiff dynamic system, 55 - factorization, 504, 514 stochastic spectral norm, 38, 48, 50, 55, 65, 173, 178, 208, - parameter, 234 210, 230, 234, 255, 256, 297, 302, 311, - perturbation, 34, 234 364,375,435,559,640 - process, 668 -- bound, 207 - signal, 505 spectral radius, 49, 168, 172, 174, 178, 180, 184, stretched 227, 229, 244, 245, 370, 374, 387, 422, - operational matrix, 578 439, 449 - time scale, 577 spectral strict aperiodicity, 354 - representation, 78, 632 strictly proper, 64,503, 545 - richness, 398, 468 strong Kharitonov theorem, 324, 332 spectrum, 358, 360,379, 583 strong stabilization, 481, 495 speed requirements, 297 structural square root decomposition, 652 - constraint, 264 stability, 153,226,391,405,408,430,435 - information, 416 - bounds, 146, 149 - model error, 157 Index 721

structured tight bounds, 442 ~ E-contour, 425 time delay, 151 ~ Lyapunov function, 201 time-optimal,534 ~ perturbation, 179, 180 time scale, 539 ~ singular value, 433, 438, 439 time-variant system, 263 ~ stability radius, 379 time-varying ~ uncertainty, 153, 161, 229, 255, 410, 422, ~ differential equation, 145 447 ~ exponent, 162 sub determinant, 199 ~ matrix differential equation, 93 submatrix, 643 ~ perturbation, 39, 155,203,207,211,301 submodel, 663 ~ state-feedback gain, 268 suboptimal controller, 271 ~ system, 145,221,315,558,575,585,589 subpolytope, 196 , 619 subspace, 526 tolerance (see also perturbation, uncertainty) successive estimation, 662 391 successive matrices, 189 ~ of desired poles, 250 sufficient condition, 169, 183, 193, 199,202,207, ~ intervals, 268 208,215,329,395 trace, 82, 90, 105, 126, 260, 266, 295, 363, 513, sufficient stability condition, 153, 158, 175, 217, 549,612,614,623,630 230,233,235,399,409,435,452 tracking, 31, 36, 291, 389, 452, 459, 464, 509 sum trajectory differential sensitivity, 111, 137 ~ norm, 46, 154 transfer function ~ of matrices, 202 ~~ and chordal metric, 432 ~ of matrix polynomials, 420 ~~ and L2-norm, 307 ~ vector, 610 ~~ sensitivity, 119 summation junction, 461 transfer matrices, overview of ~~, 485 supporting hyperplane, 334 transfer zero, 115 supremum, 64, 67, 149, 303,312,397, 503 transformation, "'-~, 465 switching transition matrix, 93, 149, 151, 190, 206, 225, ~ hyperplane, 525 268,293,297,558,623,636 ~ mode, 525, 527 ~~ differential sensitivity, 107 Sylvester inequality, 183 transmission zero, 470 symmetric transpose, 623 ~ bounds, 308 conjugate ~, 358 ~ interval matrix, 168 transversality condition, 260 ~ matrix, 57, 183, 185 triangle symmetric part, 55, 183, 210, 359, 375, 630 ~ determinant, 644 ~~ of a polynomial, 329 ~ inequality, 48, 397 ~~ of corner matrices, 182 ~ signal, 520 symmetric polynomial, 330 triangular matrix, 610, 619, 652 ~~ coefficients, 332 triangular structure, 551 symmetrizing nonlinear charact.eristic, 309 trigonal matrix, 636 symplectic matrix, 480 trigonal operator, 618 system inverse, 647 truncation, 67, 304 two-frequency-scale decomposition, 544 two-level computational structure, 266 T two-matrix combination, 176 two-parameter family of matrices, 196 Taylor expansion, 89, 98, 123, 128, 135,222,615, two-polynomial combination, 326, 327, 331 633 two-sided Laplace transformation, 70 Taylor series two smallest singular values, 369 ~~ approximation, 596 two-time-scale decomposition, 539, 542 ~~ expansion, 557 ~~ model, 557 terminal condition, 219, 260, 279 u testing matrix Hurwitz ~~, 327 ultraspherical polynomial, 565, 567 Kalman controllability ~~, 548 unbiased estimator, 669 722 Index

uncertainty, 31, 37, 250, 292, 404, 446, 499 ~~ mode, 562 additive~, 301, 399, 409, 433, 450, 492, 497, variance, 102,235,667, 669 515 variation box, 329 block-structured ~, 433, 437 variation of parameters, method of ~~, 145 dynamic ~, 490 variation of the performance index, 131 fractional ~, 493 variational calculus, 512 identifying ~ via fuzzy logic, 294 varying uncertainty bounds, 345 ~ in norm, 315 vee operator, 611 induced norm of ~, 384 vector input associated ~, 392, 396,400, 409 ~ infinity norm, 416 ~ margin, 336 ~ norm, 45, 152 multiplicative~, 301,366,390,409,493,508 ~ process, stochastic, 668 noise ~, 293, 294 ~ product, 81 nonlinear ~, 301 ~-valued function, 84 ~ of polynomials, 323 vertex, 196, 344, 353 output associated ~, 392, 396, 400, 404, 452 ~ polynomial, 324 ~ parameter, 339 Vieta's rule, 631 parametric ~, 490 Volterra series, 563 plant ~, 293, 336 plant ~ isolating, 434 ~ plot, 410 W ~ radius, 395, 405 ~ reducibility, 456 Walsh functions, 579 spectral-noise-bounded ~, 255 weak structured ~, 153, 161, 255, 410, 422, 447 ~ control of parasitics, 537 unstructured ~, 153, 410 ~ Kharitonov theorem, 324, 328, 332 various structures, 436 ~ L; -stability, 303 varying~, 345 ~ observation of parasitics, 537 ~ with common factor n, 211 ~ robustness, 469 underdetermined, 655 weighted undesired region, 323 ~ linear regression, 586 unexcited case, 469 ~ matrix norm, 416 unidirectional perturbation, 37, 168, 169, 179 "'" norm, 59 unidirectional uncertainty margin, 336 ~ sensitivity, 515 unimodular, 475 ~ square deviation, 568 unit circle, 331 weighting unit vector control, 534 ~ function, 565 unitary ~ matrix, 142, 158, 164, 199, 209, 287, 297, ~ eigenvectors, 409 400,483,505,511,657,669 ~ matrix, 163,206,228,360,367,369,422, Weyl inequality, 631 439, 494 white noise/stochastic process, 294, 505, 667 ~ transformation, 364 whiting matrix, 169, 193 ~ vector, 360 unity feedback, 424 un modelled high-frequency dynamics, 537 Y unobservable disturbance model, 462 unstructured Youla parametrization, 479, 490 ~ stability margin, 339 ~ uncertainty, 153, 164, 179, 410 updating parameters, 664 z upper left corner, 580 U-stable, 323 zero, 341,408,495,496,619 ~ of a polytope, 334 v ~ placement, 117 ~~ robustness, 118 Vandermonde matrix, 615, 628 transfer ~, 115 variable structure z-transfer function and L2-norm, 307 ~~ control, 525 z-transform, 69 by the same author: Alexander Weinmann Regelungen - Analyse und technischer Entwurf

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