Appendix D A Review of Complex Variables This appendix is a brief summary of some results on complex variables theory, with emphasis on the facts needed in control theory. For a comprehensive study of basic complex variables theory, see standard textbooks such as Brown and Churchill (1996) or Marsden and Ho/man (1998). D.1 Denition of a Complex Number The complex numbers are distinguished from purely real numbers in that they also contain the imaginary operator, which we shall denote j . By denition, j 2 = 1 or j = p 1: (D.1) A complex number may be dened as A = + j!; (D.2) where is the real part and ! is the imaginary part, denoted respectively as = Re(A); ! = Im(A): (D.3) Note that the imaginary part of A is itself a real number. Graphically, we may represent the complex number A in two ways. In the Cartesian coordinate system (Fig. D.1a), A is represented by a single point in the complex plane. In the polar coordinate system, A is represented by a vector with length r and an angle ; the angle is measured in radians counterclockwise from the positive real axis (Fig. D.1b). In polar form the complex number A is denoted by A = jAj \ arg A = r \= re j ; 0 2; (D.4) where r called the magnitude, modulus, or absolute value of A is the length of the vector representing A, namely, r = jAj = p 2 + ! 2 ; (D.5) and where is given by tan = ! (D.6) or = arg(A) = tan 1 ! : (D.7) Care must be taken to compute the correct value of the angle, depending on the sign of the real and imaginary parts (i.e., one must nd the quadrant in which the complex number lies). The conjugate of A is dened as A = j!(D.8) 35
Transcript
Appendix D
A Review of Complex Variables
This appendix is a brief summary of some results on complex variables theory, with emphasis onthe facts needed in control theory. For a comprehensive study of basic complex variables theory, seestandard textbooks such as Brown and Churchill (1996) or Marsden and Ho¤man (1998).
D.1 De�nition of a Complex Number
The complex numbers are distinguished from purely real numbers in that they also contain theimaginary operator, which we shall denote j. By de�nition,
j2 = �1 or j =p�1: (D.1)
A complex number may be de�ned as
A = � + j!; (D.2)
where � is the real part and ! is the imaginary part, denoted respectively as
� = Re(A); ! = Im(A): (D.3)
Note that the imaginary part of A is itself a real number.Graphically, we may represent the complex number A in two ways. In the Cartesian coordinate
system (Fig. D.1a), A is represented by a single point in the complex plane. In the polar coordinatesystem, A is represented by a vector with length r and an angle �; the angle is measured in radianscounterclockwise from the positive real axis (Fig. D.1b). In polar form the complex number A isdenoted by
A = jAj � \ argA = r � \� = rej�; 0 � � � 2�; (D.4)
where r� called the magnitude, modulus, or absolute value of A� is the length of the vectorrepresenting A, namely,
r = jAj =p�2 + !2; (D.5)
and where � is given by
tan � =!
�(D.6)
or� = arg(A) = tan�1
�!�
�: (D.7)
Care must be taken to compute the correct value of the angle, depending on the sign of the realand imaginary parts (i.e., one must �nd the quadrant in which the complex number lies).The conjugate of A is de�ned as
A� = � � j!� (D.8)
35
36 APPENDIX D. A REVIEW OF COMPLEX VARIABLES
Figure D.1: The complex number A represented in (a) Cartesian and (b) polar coordinates
Because each complex number is represented by a vector extending from the origin, we can add orsubtract complex numbers graphically. The sum is obtained by adding the two vectors. This we doby constructing a parallelogram and �nding its diagonal, as shown in Fig. D.2(a). Alternatively, wecould start at the tail of one vector, draw a vector parallel to the other vector, and then connect theorigin to the new arrowhead.Complex subtraction is very similar to complex addition.
D.2.2 Complex Multiplication
For two complex numbers de�ned according to Eq. (D.15),
A1A2 = (�1 + j!1)(�2 + j!2)
= (�1�2 � !1!2) + j(!1�2 + �1!2): (D.17)
The product of two complex numbers may be obtained graphically using polar representations, asshown in Fig. D.2(b).
D.2.3 Complex Division
The division of two complex numbers is carried out by rationalization. This means that both thenumerator and denominator in the ratio are multiplied by the conjugate of the denominator:
A1A2
=A1A
�2
A2A�2
=(�1�2 + !1!2) + j(!1�2 � �1!2)
�22 + !22
: (D.18)
From Eq. (D.4) it follows that
A�1 =1
re�j�; r 6= 0: (D.19)
Also, if A1 = r1ej�1 and A2 = r2ej�2 , then
A1A2 = r1r2ej(�1+�2); (D.20)
where jA1A2j = r1r2 and arg(A1A2) = �1 + �2, andA1A2
=r1r2ej(�1��2); r2 6= 0; (D.21)
where���A1A2 ��� = r1
r2 and arg�A1A2
�= �1 � �2: The division of complex numbers may be carried out
graphically in polar coordinates as shown in Fig. D.2(c).
Example D.1 Frequency Response of First-Order SystemFind the magnitude and phase of thetransfer function G(s) = 1
s+ 1 , where s = j!.SOLUTION Substituting s = j! and rationalizing, we obtain
G(j!) =1
� + 1 + j!
� + 1� j!� + 1� j!
=� + 1� j!(� + 1)2 + !2
:
Therefore, the magnitude and phase are
jG(j!)j =
p(� + 1)2 + !2
(� + 1)2 + !2=
1p(� + 1)2 + !2
;
arg(G(j!)) = tan�1�Im(G(j!))Re(G(j!))
�= tan�1
��!� + 1
��
38 APPENDIX D. A REVIEW OF COMPLEX VARIABLES
Figure D.3: Graphical determination of magnitude and phase
D.3 Graphical Evaluation of Magnitude and Phase
Consider the transfer function
G(s) =
Qmi=1(s+ zi)Qni=1(s+ pi)
: (D.22)
The value of the transfer function for sinusoidal inputs is found by replacing s with j!. The gain andphase are given by G(j!) and may be determined analytically or by a graphical procedure. Considerthe pole-zero con�guration for such a G(s) and a point s0 = j!0 on the imaginary axis, as shownin Fig. D.3. Also consider the vectors drawn from the poles and the zero to s0. The magnitude ofthe transfer function evaluated at s0 = j!0 is simply the ratio of the distance from the zero to theproduct of all the distances from the poles:
jG(j!0)j =r1
r2r3r4: (D.23)
The phase is given by the sum of the angles from the zero minus the sum of the angles from thepoles:
argG(j!0) = \G(j!0) = �1 � (�2 + �3 + �4): (D.24)
This may be explained as follows. The term s + z1 is a vector addition of its two components. Wemay determine this equivalently as s � (�z1), which amounts to translation of the vector s + z1starting at �z1, as shown in Fig. D.4. This means that a vector drawn from the zero location tos0 is equivalent to s+ z1. The same reasoning applies to the poles. We re�ect p1, p2, and p3 aboutthe origin to obtain the pole locations. Then the vectors drawn from �p1, �p2, and �p3 to s0 arethe same as the vectors in the denominator represented in polar coordinates. Note that this methodmay also be used to evaluate s0 at places in the complex plane besides the imaginary axis.
D.4 Di¤erentiation and Integration
The usual rules apply to complex di¤erentiation. Let G(s) be di¤erentiable with respect to s. Thenthe derivative at s0 is de�ned as
G0(s0) = lims!s0
G(s)�G(s0)s� s0
; (D.25)
provided that the limit exists. For conditions on the existence of the derivative, see Brown andChurchill (1996).
D.5. EULER�S RELATIONS 39
Figure D.4: Illustration of graphical computation of s+ z1
The standard rules also apply to integration, except that the constant of integration c is a complexconstant: Z
G(s)ds =
ZRe[G(s)]ds+ j
ZIm[G(s)]ds+ c: (D.26)
D.5 Euler�s Relations
Let us now derive an important relationship involving the complex exponential. If we de�ne
A = cos � + j sin �; (D.27)
where � is in radians, then
dA
d�= � sin � + j cos � = j2 sin � + j cos �
= j(cos � + j sin �) = jA: (D.28)
We collect the terms involving A to obtain
dA
A= jd�: (D.29)
Integrating both sides of Eq. (D.29) yields
lnA = j� + c; (D.30)
where c is a constant of integration. If we let � = 0 in Eq. (D.30), we �nd that c = 0 or
A = ej� = cos � + j sin �: (D.31)
Similarly,A� = e�j� = cos � � j sin �: (D.32)
From Eqs. (D.31) and (D.32) it follows that Euler�s relations
cos � =ej� + e�j�
2; (D.33)
sin � =ej� � e�j�
2j: (D.34)
40 APPENDIX D. A REVIEW OF COMPLEX VARIABLES
Figure D.5: Contours in the s-plane: (a) a closed contour; (b) two di¤erent paths between A1 andA2
D.6 Analytic Functions
Let us assume that G is a complex-valued function de�ned in the complex plane. Let s0 be in thedomain of G, which is assumed to be �nite within some disk centered at s0. Thus, G(s) is de�nednot only at s0 but also at all points in the disk centered at s0. The function G is said to be analyticif its derivative exists at s0 and at each point in the neighborhood of s0.
D.7 Cauchy�s Theorem
A contour is a piecewise-smooth arc that consists of a number of smooth arcs joined together. Asimple closed contour is a contour that does not intersect itself and ends on itself. Let C be aclosed contour as shown in Fig. D.5(a), and let G be analytic inside and on C. Cauchy�s theoremstates that I
C
G(s)ds = 0: (D.35)
There is a corollary to this theorem: Let C1 and C2 be two paths connecting the points A1 and A2as in Fig. D.5(b). Then Z
C1
G(s)ds =
ZC2
G(s)ds: (D.36)
D.8 Singularities and Residues
If a function G(s) is not analytic at s0 but is analytic at some point in every neighborhood of s0, it issaid to be a singularity. A singular point is said to be an isolated singularity if G(s) is analyticeverywhere else in the neighborhood of s0 except at s0. Let G(s) be a rational function (thatis, a ratio of polynomials). If the numerator and denominator are both analytic, then G(s) will beanalytic except at the locations of the poles (that is, at roots of the denominator). All singularitiesof rational algebraic functions are the pole locations.Let G(s) be analytic except at s0. Then we may write G(s) in its Laurent series expansion form:
G(s) =A�n
(s� s0)n+ : : :+
A�1(s� s0)
+B0 +B1(s� s0) + : : : : (D.37)
The coe¢ cient A�1 is called the residue of G(s) at s0 and may be evaluated as
A�1 = Res[G(s); s0] =1
2�j
IC
G(s) ds; (D.38)
D.9. RESIDUE THEOREM 41
Figure D.6: Contour around an isolated singularity
where C denotes a closed arc within an analytic region centered at s0 that contains no other singu-larity, as shown in Fig. D.6. When s0 is not repeated with n = 1, we have
A�1 = Res[G(s); s0] = (s� s0)G(s)js=s0 : (D.39)
This is the familiar cover-up method of computing residues.
D.9 Residue Theorem
If the contour C contains l singularities, then Eq. (D.39) maybe generalized to yield Cauchy�sresidue theorem:
1
2�j
IG(s) ds =
lXi=1
Res[G(s); si]: (D.40)
D.10 The Argument Principle
Before stating the argument principle, we need a preliminary result from which the principle followsreadily.
Number of Poles and Zeros
Let G(s) be an analytic function inside and on a closed contour C, except for a �nite number ofpoles inside C. Then, for C described in the positive sense (clockwise direction),
1
2�j
IG0(s)
G(s)ds = N � P (D.41)
or1
2�j
Id(lnG) = N � P; (D.42)
where N and P are the total number of zeros and poles of G inside C, respectively. A pole or zeroof multiplicity k is counted k times.
Proof Let s0 be a zero of G with multiplicity k. Then, in some neighborhood of that point, wemay write G(s) as
G(s) = (s� s0)kf(s); (D.43)
42 APPENDIX D. A REVIEW OF COMPLEX VARIABLES
where f(s) is analytic and f(s0) 6= 0. If we di¤erentiate Eq. (D.43), we obtainG0(s) = k(s� s0)k�1f(s) + (s� s0)kf 0(s): (D.44)
Equation (D.41) may be rewritten as
G0(s)
G(s)=
k
s� s0+f 0(s)
f(s): (D.45)
Therefore, G0(s)=G(s) has a pole at s = s0 with residue K. This analysis may be repeated for everyzero. Hence, the sum of the residues of G0(s)=G(s) is the number of zeros of G(s) inside C. If s0 isa pole with multiplicity l, we may write
h(s) = (s� s0)lG(s); (D.46)
where h(s) is analytic and h(s0) 6= 0. Then Eq. (D.46) may be rewritten as
G(s) =h(s)
(s� s0)l: (D.47)
Di¤erentiating Eq. (D.47) we obtain
G0(s) =h0(s)
(s� s0)l� lh(s)
(s� s0)l+1; (D.48)
so thatG0(s)
G(s)=
�ls� s0
+h0(s)
h(s): (D.49)
This analysis may be repeated for every pole. The result is that the sum of the residues of G0(s)=G(s)at all the poles of G(s) is �P .
The Argument Principle
Using Eq. (D.38), we get1
2�j
IC
d[lnG(s)] = N � P; (D.50)
where d[lnG(s)] was substituted for G0(s)=G(s). If we write G(s) in polar form, thenI�
Because � is a closed contour, the �rst term is zero, but the second term is 2� times the netencirclements of the origin:
1
2�j
I�
d[lnG(s)] = N � P: (D.52)
Intuitively, the argument principle may be stated as follows: We let G(s) be a rational functionthat is analytic except at possibly a �nite number of points. We select an arbitrary contour in thes-plane so that G(s) is analytic at every point on the contour (the contour does not pass throughany of the singularities). The corresponding mapping into the G(s)-plane may encircle the origin.The number of times it does so is determined by the di¤erence between the number of zeros andthe number of poles of G(s) encircled by the s-plane contour. The direction of this encirclement isdetermined by which is greater, N (clockwise) or P (counterclockwise). For example, if the contourencircles a single zero, the mapping will encircle the origin once in the clockwise direction. Similarly,if the contour encloses only a single pole, the mapping will encircle the origin, this time in thecounterclockwise direction. If the contour encircles no singularities, or if the contour encloses anequal number of poles and zeros, there will be no encirclement of the origin. A contour evaluation ofG(s) will encircle the origin if there is a nonzero net di¤erence between the encircled singularities.The mapping is conformal as well, which means that the magnitude and sense of the angles betweensmooth arcs is preserved. Chapter 6 provides a more detailed intuitive treatment of the argumentprinciple and its application to feedback control in the form of the Nyquist stability theorem.
D.11. BILINEAR TRANSFORMATION 43
D.11 Bilinear Transformation
A bilinear transformation is of the form
w =as+ b
cs+ d; (D.53)
where a, b, c, d are complex constants and it is assumed that ad�bc 6= 0. The bilinear transformationalways transforms circles in the w-plane into circles in the s-plane. This can be shown in severalways. If we solve for s, we obtain
s =�dw + bcw � a : (D.54)
The equation for a circle in the w-plane is of the form
jw � �jjw � �j = R: (D.55)
If we substitute for w in terms of s in Eq. (D.53), we get
js� �0jjs� �0j = R
0; (D.56)
where
�0 =�d� ba� �c ; �0 =
�d� ba� �c ; R0 =
���� a� �ca� �c
����R; (D.57)
which is the equation for a circle in the s-plane. For alternative proofs the reader is referred to Brownand Churchill (1996) and Marsden and Ho¤man (1998).
44 APPENDIX D. A REVIEW OF COMPLEX VARIABLES
Appendix E
Summary of Matrix Theory
In the text, we assume you are already somewhat familiar with matrix theory and with the solutionof linear systems of equations. However, for the purposes of review we present here a brief summaryof matrix theory with an emphasis on the results needed in control theory. For further study, seeStrang (1988) and Gantmacher (1959).
E.1 Matrix De�nitions
An array of numbers arranged in rows and columns is referred to as a matrix. If A is a matrix withm rows and n columns, an m� n (read �m by n�) matrix, it is denoted by
A =
26664a11 a12 � � � a1na21 a22 � � � a2n...
......
am1 am2 � � � amn
37775 ; (E.1)
where the entries aij are its elements. If m = n, then the matrix is square; otherwise it is rectan-gular. Sometimes a matrix is simply denoted by A = [aij ]. If m = 1 or n = 1, then the matrixreduces to a row vector or a column vector, respectively. A submatrix of A is the matrix withcertain rows and columns removed.
E.2 Elementary Operations on Matrices
If A and B are matrices of the same dimension, then their sum is de�ned by
C = A+B; (E.2)
wherecij = aij + bij : (E.3)
That is, the addition is done element by element. It is easy to verify the following properties ofmatrices: Commutative
law for additionAssociative lawfor addition
A+B = B+A; (E.4)
(A+B) +C = A+ (B+C): (E.5)
Two matrices can be multiplied if they are compatible. Let A = m � n and B = n � p. Then them� p matrix
The trace of a square matrix is the sum of its diagonal elements:
trace A =nXi=1
aii: (E.10)
E.4 Transpose
The n�m matrix obtained by interchanging the rows and columns of A is called the transpose ofmatrix A:
AT =
26664a11 a21 : : : am1a12 a22 : : : am2...
......
a1n a2n : : : amn
37775 �A matrix is said to be symmetric if
AT = A: (E.11)
It is easy to show thatTransposition
(AB)T
= BTAT (E.12)
(ABC)T
= CTBTAT (E.13)
(A+B)T
= AT +BT : (E.14)
E.5 Determinant and Matrix Inverse
The determinant of a square matrix is de�ned by Laplace�s expansion
detA =nXj=1
aij ij for any i = 1; 2; : : : ; n; (E.15)
where ij is called the cofactor and
ij = (�1)i+j detMij ; (E.16)
where the scalar det Mij is called a minor. Mij is the same as the matrix A except that its ith rowand jth column have been removed. Note that Mij is always an (n� 1)� (n� 1) matrix, and thatthe minors and cofactors are identical except possibly for a sign.
E.6. PROPERTIES OF THE DETERMINANT 47
The adjugate of a matrix is the transpose of the matrix of its cofactors:
adj A = [ ij ]T : (E.17)
It can be shown thatA adj A = (detA)I; (E.18)
where I is called the identity matrix:Identity matrix
I =
266641 0 : : : : : : 00 1 0 : : : 0...
.... . .
...0 : : : : : : 0 1
37775 ;that is, I has ones along the diagonal and zeros elsewhere. If detA 6= 0, then the inverse of a matrixA is de�ned by
A�1 =adj AdetA
(E.19)
and has the propertyAA�1 = A�1A = I: (E.20)
Note that a matrix has an inverse� that is, it is nonsingular� if its determinant is nonzero.The inverse of the product of two matrices is the product of the inverse of the matrices in reverse
order:(AB)�1 = B�1A�1 (E.21)
and Inversion(ABC)�1 = C�1B�1A�1: (E.22)
E.6 Properties of the Determinant
When dealing with determinants of matrices, the following elementary (row or column) operationsare useful:
1. If any row (or column) of A is multiplied by a scalar �, the resulting matrix �A has thedeterminant
det �A = � detA: (E.23)
Hencedet(�A) = �n detA: (E.24)
2. If any two rows (or columns) of A are interchanged to obtain �A, then
det �A = �detA: (E.25)
3. If a multiple of a row (or column) of A is added to another to obtain �A, then
det �A = detA: (E.26)
4. It is also easy to show thatdetA = detAT (E.27)
anddetAB = detA detB: (E.28)
48 APPENDIX E. SUMMARY OF MATRIX THEORY
Applying Eq. (E.28) to Eq. (E.20), we have
detA detA�1 = 1: (E.29)
If A and B are square matrices, then the determinant of the block triangular matrix is theproduct of the determinants of the diagonal blocks:
det
�A C0 B
�= detA detB: (E.30)
If A is nonsingular, then
det
�A BC D
�= detA det(D�CA�1B): (E.31)
Using this identity, we can write the transfer function of a scalar system in a compact form:
G(s) = H(sI� F)�1G+ J =
det
�(sI� F) G�H J
�det(sI� F) : (E.32)
E.7 Inverse of Block Triangular Matrices
If A and B are square invertible matrices, then�A C0 B
��1=
�A�1 �A�1CB�1
0 B�1
�: (E.33)
E.8 Special Matrices
Some matrices have special structures and are given names. We have already de�ned the identitymatrix, which has a special form. A diagonal matrix has (possibly) nonzero elements along theDiagonal matrixmain diagonal and zeros elsewhere:
A =
2666664a11 0
a22a33
. . .0 ann
3777775 : (E.34)
A matrix is said to be (upper) triangular if all the elements below the main diagonal are zeros:Upper triangu-lar matrix
A =
26666664
a11 a12 � � � a1n0 a22... 0
...
0...
. . .. . .
0 0 � � � 0 ann
37777775 : (E.35)
The determinant of a diagonal or triangular matrix is simply the product of its diagonal elements.A matrix is said to be in the (upper) companion form if it has the structure
Note that all the information is contained in the �rst row. Variants of this form are the lower, left,or right companion matrices. A Vandermonde matrix has the following structure:
A =
266641 a1 a21 � � � an�11
1 a2 a22 � � � an�12...
......
...1 an a2n � � � an�1n
37775 : (E.37)
E.9 Rank
The rank of a matrix is the number of its linearly independent rows or columns. If the rank of A isr, then all (r+ 1)� (r+ 1) submatrices of A are singular, and there is at least one r� r submatrixthat is nonsingular. It is also true that
row rank of A = column rank of A: (E.38)
E.10 Characteristic Polynomial
The characteristic polynomial of a matrix A is de�ned by
a(s) , det(sI�A)= sn + a1s
n�1 + � � �+ an�1s+ an; (E.39)
where the roots of the polynomial are referred to as eigenvalues of A. We can write
a(s) = (s� �1)(s� �2) � � � (s� �n); (E.40)
where f�ig are the eigenvalues of A. The characteristic polynomial of a companion matrix (e.g.,Eq. (E.36)) is
a(s) = det(sI�Ac)
= sn + a1sn�1 + � � �+ an�1s+ an: (E.41)
E.11 Cayley�Hamilton Theorem
The Cayley�Hamilton theorem states that every square matrix A satis�es its characteristic polyno-mial. This means that if A is an n� n matrix with characteristic equation a(s), then
a(A) , An + a1An�1 + � � �+ an�1A+ anI = 0 (E.42)
E.12 Eigenvalues and Eigenvectors
Any scalar � and nonzero vector v that satisfy
Av = �v (E.43)
are referred to as the eigenvalue and the associated (right) eigenvector of the matrix A [becausev appears to the right of A in Eq. (E.43)]. By rearranging terms in Eq. (E.43), we get
(�I�A)v = 0: (E.44)
50 APPENDIX E. SUMMARY OF MATRIX THEORY
Because v is nonzero,det(�I�A) = 0; (E.45)
so � is an eigenvalue of the matrix A as de�ned in Eq. (E.43). The normalization of the eigenvectorsis arbitrary; that is, if v is an eigenvector, so is �v. The eigenvectors are usually normalized to haveunit length; that is, kvk2 = vTv = 1.If wT is a nonzero row vector such that
wTA = �wT ; (E.46)
then w is called a left eigenvector of A [because wT appears to the left of A in Eq. (E.46)]. Notethat we can write
ATw = �w (E.47)
so that w is simply a right eigenvector of AT .
E.13 Similarity Transformations
Consider the arbitrary nonsingular matrix T such that
�A = T�1AT: (E.48)
The matrix operation shown in Eq. (E.48) is referred to as a similarity transformation. If A hasa full set of eigenvectors, then we can choose T to be the set of eigenvectors and �A will be diagonal.Consider the set of equations in state-variable form:
From Eq. (E.55) we can see that �F and F both have the same characteristic polynomial, giving usthe important result that a similarity transformation does not change the eigenvalues of a matrix.From Eq. (E.50) a new state made up of a linear combination from the old state has the sameeigenvalues as the old set.
E.14. MATRIX EXPONENTIAL 51
E.14 Matrix Exponential
Let A be a square matrix. The matrix exponential of A is de�ned as the series
eAt = I+At+1
2!A2t2 +
A3t3
3!+ � � � � (E.56)
It can be shown that the series converges. If A is an n� n matrix, then eAt is also an n� n matrixand can be di¤erentiated:
d
dteAt = AeAt: (E.57)
Other properties of the matrix exponential are
eAt1eAt2 = eA(t1+t2) (E.58)
and, in general,eAeB 6= eBeA: (E.59)
(In the exceptional case where A and B commute� that is, AB = BA� then eAeB = eBeA.)
E.15 Fundamental Subspaces
The range space of A, denoted by R(A) and also called the column space of A, is de�ned bythe set of vectors
x = Ay (E.60)
for some vector y. The null space of A, denoted by N (A), is de�ned by the set of vectors x suchthat
Ax = 0: (E.61)
If x 2 N (A) and y 2 R(AT ), then yTx = 0; that is, every vector in the null space of A isorthogonal to every vector in the range space of AT .
E.16 Singular-Value Decomposition
The singular-value decomposition (SVD) is one of the most useful tools in linear algebra andhas been widely used in control theory during the last two decades. Let A be an m�n matrix. Thenthere always exist matrices U, S, and V such that
A = USVT : (E.62)
Here U and V are orthogonal matrices; that is
UUT = I;VVT = I: (E.63)
S is a quasidiagonal matrix with singular values as its diagonal elements; that is,
S =
�� 00 0
�; (E.64)
where � is a diagonal matrix of nonzero singular values in descending order:
�1 � �2 � � � � � �r > 0: (E.65)
52 APPENDIX E. SUMMARY OF MATRIX THEORY
The unique diagonal elements of S are called the singular values. The maximum singular value isdenoted by ��(A), and the minimum singular value is denoted by �(A). The rank of the matrix isthe same as the number of nonzero singular values. The columns of U and V,
U = [ u1 u2 : : : um ];
V = [ �1 �2 : : : �n ]; (E.66)
are called the left and right singular vectors, respectively. SVD provides complete informationabout the fundamental subspaces associated with a matrix:
N (A) = span[ �r+1 �r+2 : : : �n ];
R(A) = span[ u1 u2 : : : ur ];
R(AT ) = span[ �1 �2 : : : �r ];
N (AT ) = span[ ur+1 ur+2 : : : um ]: (E.67)
Here R denotes the null space, and N denotes the range space respectively.
The norm of the matrix A, denoted by kAk2, is given by
kAk2 = ��(A): (E.68)
If A is a function of !, then the in�nity norm of A, kAk1, is given by
kA(j!)k1 = max!��(A): (E.69)
E.17 Positive De�nite Matrices
A matrix A is said to be positive semide�nite if
xTAx � 0 for all x: (E.70)
The matrix is said to be positive de�nite if equality holds in Eq. (E.70) only for x = 0. A symmetricmatrix is positive de�nite if and only if all of its eigenvalues are positive. It is positive semide�niteif and only if all of its eigenvalues are nonnegative.
An alternate method for determining positive de�niteness is to test the minors of the matrix. Amatrix is positive de�nite if all the leading principal minors are positive, and positive semide�niteif they are all nonnegative.
E.18 Matrix Identity
If A is an n�m matrix and B is an m� n matrix, then
det[In �AB] = det[Im �BA];
where In and Im are identity matrices of size n and m respectively.
Appendix F
Controllability and Observability
Controllability and observability are important structural properties of dynamic systems. First iden-ti�ed and studied by Kalman (1960) and later by Kalman et al. (1961), these properties have con-tinued to be examined during the last four decades. We will discuss only a few of the known resultsfor linear constant systems with one input and one output. In the text we discuss these concepts inconnection with control law and estimator designs. For example, in Section 7.4 we suggest that ifthe square matrix given by
C = [G FG F2 G : : : Fn�1G] (F.1)
is nonsingular, then by transformation of the state we can convert the given description into controlcanonical form. We can then construct a control law that will give the closed-loop system an arbitrarycharacteristic equation.
F.1 Controllability
We begin our formal discussion of controllability with the �rst of four de�nitions:
De�nition 1 De�nition I The system (F, G) is controllable if, for any given nth-order polynomial�c(s), there exists a (unique) control law u = �Kx such that the characteristic polynomial of F�GKis �c(s).
From the results of Ackermann�s formula (see Appendix G), we have the following mathematicaltest for controllability: (F, G) is a controllable pair if and only if the rank of C is n. De�nition Ibased on pole placement is a frequency-domain concept. Controllability can be equivalently de�nedin the time domain.
De�nition 2 De�nition II The system (F, G) is controllable if there exists a (piecewise continu-ous) control signal u(t) that will take the state of the system from any initial state x0 to any desired�nal state xf in a �nite time interval.
We will now show that the system is controllable by this de�nition if and only if C is full rank. We�rst assume that the system is controllable but that
rank[G FG F2G : : : Fn�1G] < n: (F.2)
We can then �nd a vector v such that
v[G FG F2G : : : Fn�1G] = 0; (F.3)
orvG = vFG = vF2G = : : : = vFn�1G = 0: (F.4)
53
54 APPENDIX F. CONTROLLABILITY AND OBSERVABILITY
The Cayley�Hamilton theorem states that F satis�es its own characteristic equation, namely,
�Fn = a1Fn�1 + a2Fn�2 + : : :+ anI: (F.5)
Therefore,�vFnG = a1vF
n�1G+ a2vFn�2G+ : : :+ anvG = 0: (F.6)
By induction, vFn+kG = 0 for k = 0; 1; 2; : : :, or vFmG = 0 for m = 0; 1; 2; : : :, and thus
veFtG = v
�I+ Ft+
1
2!F2t2 + : : :
�G = 0 (F.7)
for all t. However, the zero initial-condition response (x0 = 0) is
x(t) =
Z t
0
veF(t��)Gu(�) d�
= eFtZ t
0
e�FtGu(�) d� : (F.8)
Using Eq. (F.7), Eq. (F.8) becomes
vx(t) =
Z t
0
veF(t��)Gu(�) d� = 0 (F.9)
for all u(t) and t > 0. This implies that all points reachable from the origin are orthogonal to v.This restricts the reachable space and therefore contradicts the second de�nition of controllability.Thus if C is singular, (F, G) is not controllable by De�nition II.Next we assume that C is full rank but (F, G) is uncontrollable by De�nition II. This means
that there exists a nonzero vector v such that
v
Z tf
0
eF(tf��)Gu(�) d� = 0; (F.10)
because the whole state space is not reachable. But Eq. (F.10) implies that
veF(tf��)
G = 0; 0 � � � tf : (F.11)
If we set � = tf , we see that vG = 0. Also, di¤erentiating Eq. (F.11) and letting � = tf givesvFG = 0. Continuing this process, we �nd that
vG = vFG = vF2G = : : : = vFn�1G = 0; (F.12)
which contradicts the assumption that C is full rank.We have now shown that the system is controllable by De�nition II if and only if the rank of C
is n, exactly the same condition we found for pole assignment.Our �nal de�nition comes closest to the structural character of controllability:
De�nition 3 De�nition III The system (F, G) is controllable if every mode of F is connected tothe control input.
Because of the generality of the modal structure of systems, we will treat only the case of systemsfor which F can be transformed to diagonal form. (The double-integration plant does not qualify.)Suppose we have a diagonal matrix Fd and its corresponding input matrix Gd with elements gi. Thestructure of such a system is shown in Fig. (F.1). By de�nition, for a controllable system the inputmust be connected to each mode so that thegi are all nonzero. However, this is not enough if the poles (�i) are not distinct. Suppose, for
instance, that �1 = �2. The �rst two state equations are then
_x1d = �1x1d + g1u;
_x2d = �1x2d + g2u: (F.13)
F.1. CONTROLLABILITY 55
Figure F.1: Block diagram of a system with a diagonal matrix
Figure F.2: Examples of uncontrollable systems
If we de�ne a new state, � = g2x1d � g1x2d, the equation for � is
which does not include the control u; hence, � is not controllable. The point is that if any two polesare equal in a diagonal Fd system with only one input, we e¤ectively have a hidden mode that isnot connected to the control, and the system is not controllable (Fig. F.2a). This is because the twostate variables move together exactly, so we cannot independently control x1d and x2d. Therefore,even in such a simple case, we have two conditions for controllability:
1. All eigenvalues of Fd are distinct.
2. No element of Gd is zero.
Now let us consider the controllability matrix of this diagonal system. By direct computation,
C =
266664g1 g1�1 : : : g1�
n�11
g2 g2�2 : : :...
...... � � �
...gn gn�n : : : gn�
n�1n
377775
=
26664g1 0
g2. . .
0 gn
377752666641 �1 �21 : : : �n�11
1 �2 �22 : : :...
......
... � � �...
1 �n �2n : : : �n�1n
377775 : (F.15)
Note that the controllability matrix C is the product of two matrices and is nonsingular if and onlyif both of these matrices are invertible. The �rst matrix has a determinant that is the product of
56 APPENDIX F. CONTROLLABILITY AND OBSERVABILITY
the gi, and the second matrix (called a Vandermonde matrix) is nonsingular if and only if the �i aredistinct. Thus De�nition III is equivalent to having a nonsingular C also.Important to the subject of controllability is the Popov�Hautus�Rosenbrock (PHR) test
(see Rosenbrock, 1970, and Kailath, 1980), which is an alternate way to test the rank (or determinant)of C. The system (F, G) is controllable if the system of equations
vT [sI� F G] = 0T (F.16)
has only the trivial solution vT = 0T� that is, if the matrix pencil
rank[sI� F G] = n (F.17)
is full rank for all s, or if there is no nonzero vT such that1
vTF = svT ; (F.18)
vTG = 0: (F.19)
This test is equivalent to the rank-of-C test. It is easy to show that if such a vector v exists, then Cis singular. For, if a nonzero v exists such that vTG = 0 then by Eqs. (F.18) and (F.19),
vTFG = svTG = 0: (F.20)
Then, multiplying by FG, we �nd that
vTF2G = svTFG = 0; (F.21)
and so on. Thus we determine that vTC = 0T has a nontrivial solution, that C is singular, and thatthe system is not controllable. To show that a nontrivial vT exists if C is singular requires moredevelopment, which we will not give here (see Kailath, 1980).We have given two pictures of uncontrollability. Either a mode is physically disconnected from
the input (Fig. F.2b), or else two parallel subsystems have identical characteristic roots (Fig. F.2a).The control engineer should be aware of the existence of a third simple situation, illustrated inFig. F.2c, namely, a pole-zero cancellation. Here the problem is that the mode at s = 1 appearsto be connected to the input but is masked by the zero at s = 1 in the preceding subsystem; theresult is an uncontrollable system. This can be con�rmed in several ways. First let us look at thecontrollability matrix. The system matrices are
F =
��1 01 1
�; G =
��21
�;
so the controllability matrix is
C = [ G FG ] =
��2 21 �1
�; (F.22)
which is clearly singular. The controllability matrix may be computed using the ctrb command inMATLAB: [cc]=ctrb(F,G). If we compute the transfer function from u to x2, we �nd that
H(s) =s� 1s+ 1
�1
s� 1
�=
1
s+ 1: (F.23)
Because the natural mode at s = 1 disappears from the input�output description, it is not connectedto the input. Finally, if we consider the PHR test,
[sI� F G] =
�s+ 1 0 �2�1 s� 1 1
�; (F.24)
and let s = 1, then we must test the rank of�2 0 �2�1 0 1
�;
which is clearly less than 2. This result means, again, that the system is uncontrollable.1vT is a left eigenvector of F.
F.2. OBSERVABILITY 57
De�nition 4 De�nition IV The asymptotically stable system (F, G) is controllable if the control-lability Gramian, the square symmetric matrix Cg; given by the solution to the Lyapunov equation
FCg + CgFT +GGT = 0; (F.25)
is nonsingular. The controllability Gramian is also the solution to the following integral equation:
Cg =Z 1
0
e�FGGT e�FT
d� : (F.26)
One physical interpretation of the controllability Gramian is that if the input to the system is whiteGaussian noise, then Cg is the covariance of the state. The controllability Gramian (for an asymp-totically stable system) can be computed with the following command in MATLAB: [cg]=gram(F,G).
In conclusion, the four de�nitions for controllability� pole assignment (De�nition I), state reach-ability (De�nition II), mode coupling to the input (De�nition III), and controllability Gramian(De�nition IV)� are equivalent. The tests for any of these four properties are found in terms of therank of the controllability or controllability Gramian matrices or the rank of the matrix pencil [sI- F G]. If C is nonsingular, then we can assign the closed-loop poles arbitrarily by state feedback,we can move the state to any point in the state space in a �nite time, and every mode is connectedto the control input.2
F.2 Observability
So far we have discussed only controllability. The concept of observability is parallel to that of con-trollability, and all of the results we have discussed thus far may be transformed to statements aboutobservability by invoking the property of duality, as discussed in Section 7.7.2. The observabilityde�nitions analogous to those for controllability are as follows:
1. De�nition I : The system (F, H) is observable if, for any nth-order polynomial �e(s), thereexists an estimator gain L such that the characteristic equation of the state estimator error is�e(s).
2. De�nition II : The system (F, H) is observable if, for any x(0), there is a �nite time � suchthat x(0) can be determined (uniquely) from u(t) and y(t) for 0 � t � � .
3. De�nition III : The system (F, H) is observable if every dynamic mode in F is connected tothe output through H.
4. De�nition IV : The asymptotically stable system (F, H) is observable if the observabilityGramian is nonsingular.
As we saw in the discussion for controllability, mathematical tests can be developed for observ-ability. The system is observable if the observability matrix
O =
26664HHF...
HFn�1
37775 (F.27)
is nonsingular. If we take the transpose of O and let HT = G and FT = F, then we �nd thecontrollability matrix of (F, G), another manifestation of duality. The observability matrix O may
2We have shown the latter for diagonal F only, but the result is true in general.
58 APPENDIX F. CONTROLLABILITY AND OBSERVABILITY
be computed using the obsv command in MATLAB: [oo]=obsv(F,H). The system (F,H) is observableif the following matrix pencil is full rank for all s:
rank�sI� FH
�= n: (F.28)
The observability GramianOg, which is a symmetric matrix, and the solution to the integral equation
Og =Z 1
0
e�FT
HTHe�F d�; (F.29)
as well as the Lyapunov equation
FTOg +OgF+HTH = 0; (F.30)
can also be computed (for an asymptotically stable system) using the gram command in MATLAB:[og]=gram(F�,H�). The observability Gramian has an interpretation as the �information matrix� inthe context of estimation.
Appendix G
Ackermann�s Formula for PolePlacement
Given the plant and state-variable equation
_x = Fx+Gu; (G.1)
our objective is to �nd a state-feedback control law
u = �Kx (G.2)
such that the closed-loop characteristic polynomial is
�c(s) = det(sI� F+GK): (G.3)
First we have to select �c(s), which determines where the poles are to be shifted; then we haveto �nd K such that Eq. (G.3) will be satis�ed. Our technique is based on transforming the plantequation into control canonical form.We begin by considering the e¤ect of an arbitrary nonsingular transformation of the state,
x = T�x; (G.4)
where �x is the new transformed state. The equations of motion in the new coordinates, fromEq. (G.4), are
_x = T_�x = Fx+Gu = FT�x+Gu; (G.5)
_�x = T�1FT�x+T�1Gu = �F�x+ �Gu: (G.6)
Now the controllability matrix for the original state,
Cx = [ G FG F2G � � � Fn�1G ]; (G.7)
provides a useful transformation matrix. We can also de�ne the controllability matrix for the trans-formed state:
C�x = [ �G �F�G �F2 �G � � � �Fn�1 �G ]: (G.8)
The two controllability matrices are related by
C�x = [ �T�1G T�1FTT�1G � � � ] = T�1Cx (G.9)
and the transformation matrixT = CxC�1�x : (G.10)
59
60 APPENDIX G. ACKERMANN�S FORMULA FOR POLE PLACEMENT
From Eqs. (G.9) and (G.10) we can draw some important conclusions. From Eq. (G.9), we seethat if Cx is nonsingular, then for any nonsingular T, C�x is also nonsingular. This means that asimilarity transformation on the state does not change the controllability properties of a system. Wecan look at this in another way. Suppose we would like to �nd a transformation to take the system(F;G) into control canonical form. As we shall shortly see, C�x in that case is always nonsingular.From Eq. (G.9) we see that a nonsingular T will always exist if and only if Cx is nonsingular. Weconclude that
Theorem G.1 We can always transform (F;G) into control canonical form if and only if the sys-tem is controllable.
Let us take a closer look at control canonical form and treat the third-order case, although theresults are true for any nth-order case:
�F = Fc =
24 �a1 �a2 �a31 0 00 1 0
35 ; �G = Gc =
24 100
35 : (G.11)
The controllability matrix, by direct computation, is
C�x = Cc =
24 1 �a1 a21 � a20 1 �a10 0 1
35 : (G.12)
Because this matrix is upper triangular with ones along the diagonal, it is always invertible. Alsonote that the last row of C�x is the unit vector with all zeros, except for the last element, which isunity. We shall use this fact in what we do next.As we pointed out in Section 7.5, the design of a control law for the state �x is trivial if the
equations of motion happen to be in control canonical form. The characteristic equation is
s3 + a1s2 + a2s+ a3 = 0; (G.13)
and the characteristic equation for the closed-loop system comes from
where a and � are row vectors containing the coe¢ cients of the characteristic polynomials of theopen-loop and closed-loop systems, respectively.We now need to �nd a relationship between these polynomial coe¢ cients and the matrix F. The
requirement is achieved by the Cayley�Hamilton theorem, which states that a matrix satis�es itsown characteristic polynomial. For Fc this means that
Fnc + a1Fn�1c + a2F
n�2c + � � �+ anI = 0: (G.19)
61
Now suppose we form the polynomial �c(F), which is the closed-loop characteristic polynomial withthe matrix F substituted for the complex variable s:
�c(Fc) = Fnc + �1F
n�1c + �2F
n�2c + � � �+ �nI: (G.20)
If we solve Eq. (G.19) for Fnc and substitute into Eq. (G.20), we �nd that
where we use Eq. (G.18), which relates Kc to a and �.We now have a compact expression for the gains of the system in control canonical form as
represented in Eq. (G.25). However, we still need the expression for K, the gain on the originalstate. If u = �Kc�x, then u = �KcT
�1x, so that
K = KcT�1 = eTn�c(Fc)T
�1
= eTn�c(T�1FT)T�1
= eTnT�1�c(F): (G.26)
In the last step of Eq. (G.26) we used the fact that (T�1FT)k = T�1FkT and that �c is a polyno-mial, that is, a sum of the powers of Fc. From Eq. (G.9) we see that
T�1 = CcC�1x : (G.27)
With this substitution, Eq. (G.26) becomes
K = eTnCcC�1x �c(F): (G.28)
Now, we use the observation made earlier for Eq. (G.12) that the last row of Cc, which is eTnCc, isagain eTn . We �nally obtain Ackermann�s formula: Ackermann�s
formulaK = eTnC�1x �c(F): (G.29)
We note again that forming the explicit inverse of Cx is not advisable for numerical accuracy.Thus we need to solve bT such that
eTnC�1x = bT : (G.30)
We solve the linear set of equationsbTCx = eTn (G.31)
and then computeK = bT�c(F): (G.32)
Ackermann�s formula, Eq. (G.29), even though elegant, is not recommended for systems with a largenumber of state variables. Even if it is used, Eqs. (G.31) and (G.32) are recommended for betternumerical accuracy.
