The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze

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Polynomial Toolbox 3.0 Manual
© COPYRIGHT 2009 by PolyX, Ltd.
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Printing history: September 2009. First printing.

Contents
1 Quick Start ........................................................................................................ 8
Initialization .................................................................................................................... 8
Help ................................................................................................................................. 8
How to define a polynomial matrix .............................................................................. 9
Simple operations with polynomial matrices ............................................................. 10
Addition, subtraction and multiplication ........................................................ 10
Entrywise and matrix division ........................................................................ 11
Operations with fractions ................................................................................ 11
Concatenation and working with submatrices ................................................ 12
Coefficients and coefficient matrices .............................................................. 13
Conjugation and transposition ........................................................................ 13
Advanced operations and functions ........................................................................... 14
2 Polynomial matrices ....................................................................................... 16
Introduction .................................................................................................................. 16
Polynomials and polynomial matrices ........................................................................ 16
Entering polynomial matrices ..................................................................................... 16
The pol command ......................................................................................... 17
The Polynomial Matrix Editor ........................................................................ 17
The default indeterminate variable ................................................................. 17
Changing the default indeterminate variable .................................................. 18
Basic manipulations with polynomial matrices ......................................................... 19
Concatenation and working with submatrices ................................................ 19
Coefficients ..................................................................................................... 19
Degrees and leading coefficients .................................................................... 20
Constant, zero and empty matrices ................................................................. 21
Values ............................................................................................................. 21
Derivative and integral .................................................................................... 21
Arithmetic operations on polynomial matrices ......................................................... 22
Addition, subtraction and multiplication ........................................................ 22
Determinants, unimodularity and adjoints ...................................................... 23

Rank ................................................................................................................ 25
Bases and null spaces ...................................................................................... 26
Roots and stability ........................................................................................... 27
Special constant matrices related to polynomials ........................................... 29
Divisors and multiples.................................................................................................. 30
Scalar divisor and multiple ............................................................................. 30
Division ........................................................................................................... 31
Division with remainder ................................................................................. 31
Greatest common divisor ................................................................................ 31
Coprimeness .................................................................................................... 31
Least common multiple ................................................................................... 32
Matrix divisors and multiples ......................................................................... 33
Matrix division ................................................................................................ 33
Matrix division with remainder ...................................................................... 33
Greatest common left divisor .......................................................................... 34
Least common right multiple .......................................................................... 35
Dual concepts .................................................................................................. 36
Transposition and conjugation ................................................................................... 36
Transposition ................................................................................................... 36
Complex coefficients ...................................................................................... 36
Conjugated transposition ................................................................................ 37
Reduced and canonical forms ..................................................................................... 38
Row degrees .................................................................................................... 38
Row and column reduced matrices ................................................................. 39
Row reduced form ........................................................................................... 39
Triangular and staircase form ......................................................................... 39
Another triangular form .................................................................................. 40
Hermite form ................................................................................................... 40
Echelon form ................................................................................................... 41
Smith form ...................................................................................................... 41
Invariant polynomials ..................................................................................... 42
Polynomial matrix equations ...................................................................................... 42
Diophantine equations .................................................................................... 42
Bézout equations ............................................................................................. 44

Matrix polynomial equations .......................................................................... 44
One-sided equations ........................................................................................ 45
Two-sided equations ....................................................................................... 45
Factorizations ............................................................................................................... 46
Symmetric polynomial matrices ..................................................................... 47
Zeros of symmetric matrices ........................................................................... 47
Spectral factorization ...................................................................................... 50
Non-symmetric factorization .......................................................................... 52
Matrix pencil routines.................................................................................................. 53
Transformation to Kronecker canonical form ................................................. 54
Transformation to Clements form ................................................................... 55
Pencil Lyapunov equations ............................................................................. 55
3 Discrete-time and two-sided polynomial matrices ........................................ 57
Introduction .................................................................................................................. 57
Basic operations ............................................................................................................ 57
Two-sided polynomials ................................................................................... 57
Leading and trailing degrees and coefficients ................................................ 57
Derivatives and integrals ................................................................................. 58
Roots and stability ........................................................................................... 59
Norms .............................................................................................................. 59
Conjugations ................................................................................................................. 59
Conjugate transpose ........................................................................................ 60
Symmetric two-sided polynomial matrices................................................... 61
Zeros of symmetric two-sided polynomial matrices ....................................... 62
Matrix equations with discrete-time and two-sided polynomials ............................ 65
Symmetric bilateral matrix equations ............................................................. 65
Non-symmetric equation ................................................................................. 66
Discrete-time spectral factorization ................................................................ 66
Resampling ................................................................................................................... 68
Sampling period .............................................................................................. 68
Resampling of polynomials in z -1 .................................................................... 69
Resampling and phase ................................................................................... 69
Resampling of two-sided polynomials ............................................................ 70

Resampling of polynomials in z ................................................................... 70
Dilating ........................................................................................................... 70
4 The Polynomial Matrix Editor ...................................................................... 72
Introduction .................................................................................................................. 72
Quick start .................................................................................................................... 72
Main window ................................................................................................................ 73
Main window buttons ..................................................................................... 73
Main window menus ....................................................................................... 73
Matrix Pad window ...................................................................................................... 74
Matrix Pad buttons .......................................................................................... 74
5 Polynomial matrix fractions .......................................................................... 76
Introduction .................................................................................................................. 76
Scalar-denominator-fractions ..................................................................................... 76
The sdf command ......................................................................................... 78
Comparison of fractions .................................................................................. 78
Coprime ........................................................................................................... 79
Reduce ............................................................................................................. 79
General properties cop, red ...................................................................... 79
Reverse ............................................................................................................ 80
Matrix-denominator-fractions .................................................................................... 81
The mdf command ......................................................................................... 82
Left and right polynomial matrix fractions ............................................................... 82
Right-denominator-fraction ............................................................................ 83
The rdf command ......................................................................................... 83
Left-denominator-fraction ............................................................................... 84
The ldf command ......................................................................................... 84
Mutual conversions of polynomial matrix fraction objects ...................................... 84
Computing with polynomial matrix fractions ........................................................... 86
Addition, subtraction and multiplication ........................................................ 86
Entrywise and matrix division ........................................................................ 88
Concatenation and working with submatrices ................................................ 89

Coefficients and coefficient matrices .............................................................. 90
Conjugation and transposition ........................................................................ 90
Polynomial matrix fractions, transfer functions and transfer function matrices .. 91
Transfer functions ........................................................................................... 91
Transfer function matrices .............................................................................. 92
Transfer functions from state space model and vice versa ............................. 93
Conversions of polynomial matrix fraction objects to and from other Matlab
objects ............................................................................................................................ 95
Conversion to the Control System Toolbox Objects ...................................... 96
Conversion from the Control System Toolbox Objects ................................ 100
Conversion to the Symbolic Math Toolbox .................................................. 102
Conversion from the Symbolic Math Toolbox ............................................. 104
6 Conclusions .................................................................................................. 107

1 Quick Start
Initialization
Help
Every Polynomial Toolbox session starts with the initialization command pinit:
pinit
Polynomial Toolbox 3.0 initialized. To get started, type one of
these: helpwin or poldesk. For product information, visit
www.polyx.com or www.polyx.cz.
This function creates global polynomial properties and assigns them to their default
values. If you place this command in your startup.m file then the Polynomial
Toolbox is automatically initialized. If you include lines such as
path(path,'c:\Matlab\toolbox\polynomial')
pinit
in the startup.m file in the folder in which you start MATLAB then the toolbox is
automatically included in the search path and initialized each time you start
MATLAB.
To see a list of all the commands and functions that are available type
help polynomial
To get help on any of the toolbox commands, such as axxab, type
help axxab
Some commands are overloaded, that is, there exist various commands of the same
name for various objects. To get help on overloaded commands, you must also specify
the object class of interest. So
help pol/rank
returns help on rank for pol objects (polynomial matrices) while
help frac/rank
returns help on rank for frac objects (polynomial fractions) etc.

How to define a polynomial matrix
Basic objects of the Polynomial Toolbox are polynomial matrices. They may be looked
at in two different ways.
You may directly enter a polynomial matrix by typing its entries. Upon initialization
the Polynomial Toolbox defines several indeterminate variables for polynomials and
polynomial matrices. One of them is s. Thus, you can simply type
P = [ 1+s s^2
2*s^3 1+2*s+s^2 ]
to define the polynomial matrix
2
1
s s
Ps () 3 2
2s12ss MATLAB returns
P =
1 + s s^2
2s^3 1 + 2s + s^2
The Polynomial Toolbox displays polynomial matrices by default in this style.
We may also render the polynomial matrix P as
P() s
2s12ss 1
0 0 1
1 0 0 0
s
2 0 1
s
1 0
2 0
s
0
P
0
P
1
P
2
P
3
2
1s s 2 3
3 2
Polynomial matrices may be defined this way by typing
P0 = [ 1 0; 0 1 ];
P1 = [ 1 0; 0 2 ];
P2 = [ 0 1; 0 1 ];
P3 = [ 0 0; 2 0 ];
P = pol([P0 P1 P2 P3],3)
MATLAB again returns first
P =
1 + s s^2
2s^3 1 + 2s + s^2
The display format may be changed to ―coefficient matrix style‖ by the command
pformat coef

Typing the name of the matrix
P
now results in
Polynomial matrix in s: 2-by-2, degree: 3
P =
Matrix coefficient at s^0 :
1 0
0 1
Matrix coefficient at s^1 :
1 0
0 2
Matrix coefficient at s^2 :
0 1
0 1
Matrix coefficient at s^3 :
0 0
2 0
Simple operations with polynomial matrices
Addition,
subtraction and
multiplication
In the Polynomial Toolbox polynomial matrices are objects for which all standard
operations are defined.
Define the polynomial matrices
P = [ 1+s 2; 3 4], Q = [ s^2 1-s; s^3 1]
P =
Q =
1 + s 2
3 4
s^2 1-s
s^3 1
The sum and product of P and Q follow easily:
S = P+Q
S =
R = P*Q
1 + s + s^2 3 - s
3 + s^3 5

Entrywise and
matrix division
Operations with
fractions
R =
s^2 + 3s^3 3 - s^2
3s^2 + 4s^3 7 - 3s
Entrywise division of polynomial matrices P and Q yields a fraction
T = P./Q
results in
T =
1 + s 2
----- -----
s^2 1 - s
3 4
--- -
s^3 1
Right matrix division gets a fraction
U = P/Q
U =
1
PQ
1 + s 2 / s^2 1 - s
3 4 / s^3 1
Left matrix division gets a fraction
V = Q\P
U =
1
Q P
s^2 1 - s \ 1 + s 2
s^3 1 \ 3 4
Polynomial fractions may also enter arithmetic operations, possibly mixed with
numbers or polynomials such as
or
2*U
ans =
2 + 2s 4 / s^2 1 - s
6 8 / s^3 1
P+T
ans =

Concatenation
and working
with
submatrices
1 + s + s^2 + s^3 -4 + 2s
----------------- -------
s^2 -1 + s
3 + 3s^3 8
-------- -
s^3 1
In the realm of fractions, inverses are possible
T.^-1
ans =
s^2 0.5 - 0.5s
----- ----------
1 + s 1
0.33s^3 0.25
------- ----
1 1 U^-1
U^-1
ans =
s^2 1 - s / 1 + s 2
s^3 1 / 3 4
All standard MATLAB operations to concatenate matrices or selecting submatrices
also apply to polynomial matrices. Typing
W = [P Q]
results in
W =
1 + s 2 s^2 1 - s
3 4 s^3 1
The last row of W may be selected by typing
w = W(2,:)
w =
3 4 s^3 1
Submatrices may be assigned values by commands such as
W(:,1:2) = eye(2)
W =
1 0 s^2 1 - s

Coefficients and
coefficient
matrices
Conjugation
and
transposition
0 1 s^3 1
For fraction similarly
T(1,:)
ans =
1 + s -2
----- ------
s^2 -1 + s
It is easy to extract the coefficient matrices of a polynomial matrix. Given W, the
coefficient matrix of s 2 may be retrieved as
W{2}
ans =
0 0 1 0
0 0 0 0
The coefficients of the (1,3) entry of W follow as
Given
W{:}(1,3)
ans =
0 0 1 0
W
W =
1 0 s^2 1 - s
0 1 s^3 1
the standard MATLAB conjugation operation results in
W'
ans =
1 0
0 1
s^2 -s^3
1 + s 1
Transposition follows by typing
W.'

ans =
1 0
0 1
s^2 s^3
1 - s 1
The command transpose(W) is synonymous with W.'
Advanced operations and functions
The Polynomial Toolbox knows many advanced operations and functions. After
defining
P = [ 1+s 2; 3 4+s^2 ]
try a few by typing for instance det(P), roots(P), or smith(P).
The commands may be grouped in the categories listed in Table 1. A full list of all
commands is available in the companion volume Commands. The same list appears
after typing
help polynomial
in MATLAB.
Table 1. Command categories
Global structure Equation solvers
Polynomial, Two-sided polynomial and
Fraction objects
Linear matrix inequality functions
Special matrices Matrix pencil routines
Convertors Numerical routines
Overloaded operations Control routines
Overloaded functions Robustness functions
Basic functions (other than overloaded) 2-D functions
Random functions Simulink

Advanced functions Visualization
Canonical and reduced forms Graphic user interface
Sampling period functions Demonstrations and help

2 Polynomial matrices
Introduction
In this chapter we review in a tutorial style many of the functions and operations
defined for polynomials and polynomial matrices. This exposition continues in
Chapter 3 Discrete-time and two-sided polynomial matrices. Chapter 5 is devoted to
next objects, Polynomial matrix fractions. Functions and operations for linear timeinvariant
systems defined by polynomial matrix fractions are discussed in Chapter 6,
LTI systems. Chapter 6, Control system design covers the applications of polynomial
matrices in control system design.
More detailed information is available in Chapter 12, Objects, properties, formats and
in the manual pages included in the companion volume Commands.
Polynomials and polynomial matrices
Polynomials are mathematical expressions of the form
2
n
P s P0 Ps 1 P2 s Pn s
where P0 , P1 , P2 , , P n are real or complex numbers, coefficients, and s is the
indeterminate variable.
Matrix polynomials are of the same form but with P0 , P1 , P2 , , P n numerical matrices,
coefficients matrices. Alternatively and usually, these objects can be viewed as
polynomial matrices, i.e. matrices whose individual entries are polynomials. These
two concepts will be used interchangeably.
For system and control engineers, s may be treated as the derivative operator d dt
acting on continuous-time signals xt . It can be also considered as the Laplace
transform complex variable.
Entering polynomial matrices
Polynomials and polynomial matrices are most easily entered using one of the
indeterminate variables s, p that are recognized by the Polynomial Toolbox, combined
with the usual MATLAB conventions for entering matrices. Thus, typing
P = [ 1+s 2*s^2
2+s^3 4 ]

The pol
command
The Polynomial
Matrix Editor
The default
indeterminate
variable
defines the matrix
2
1s 2s
Ps () 3
2s4
and returns
P =
1 + s 2s^2
2 + s^3 4
For other available indeterminate variables such as z, q, z^–1 or d see Chapter 3,
Discrete-time and two-sided polynomial matrices.
Polynomials and polynomial matrices may also be entered in terms of their
coefficients or coefficient matrices. For this purpose the pol command is available.
Typing
P0 = [1 2;3 4];
P1 = [3 4;5 1];
P2 = [1 0;0 1];
P = pol([P0 P1 P2],2,'s')
for instance, defines the polynomial matrix
P =
according to
1 + 3s + s^2 2 + 4s
3 + 5s 4 + s + s^2
P( s) P0 P1s P2 s
2
More complicated polynomial matrices may be entered and edited with the help of the
Polynomial Matrix Editor (see Chapter 4).
Note that if any of the default indeterminates is redefined as a variable then it is no
longer available as an indeterminate. Typing
s = 1;
P = 1+s^2
results in

Changing the
default
indeterminate
variable
P =
2
To free the variable simply type
clear s;
P = 1+s^2
P =
1 + s^2
After the Polynomial Toolbox has been started up the default indeterminate variable
is s. This implies, among other things, that the command
P = pol([P0 P1 P2],2)
returns a polynomial matrix
P =
1 + 3s + s^2 2 + 4s
3 + 5s 4 + s + s^2
in the indeterminate s. The indeterminate variable may be changed with the help of
the gprops command: typing
gprops p; pol([P0 P1 P2],2)
for instance, results in
ans =
1 + 3p + p^2 2 + 4p
3 + 5p 4 + p + p^2
To enter data independently on the current default indeterminate variable, one can
use a shortcut v: The indeterminate variable v is automatically replaced by the
current default indeterminate variable. Thus, after the default indeterminate
variable has been set to p by the command
then
gprops p
V = 1+v^2+3*v^3
returns
V =
1 + p^2 + 3p^3

Basic manipulations with polynomial matrices
Concatenation
and working
with
submatrices
Coefficients
Standard MATLAB conventions may be used to concatenate polynomial and standard
matrices:
P = 1+s^2;
Q = [P 1+s 3]
results in
Q =
1 + s^2 1 + s 3
Submatrices may be selected such as in
Q(1,2:3)
ans =
1 + s 3
Submatrices may also be changed; the command
yields
Q(1,2:3) = [1-s 4]
Q =
1 + s^2 1 - s 4
All usual MATLAB subscripting and colon notations are available. Also the commands
like sum, prod, repmat, resape, tril, triu, diag and trace work as
can be expected.
It is easy to extract the coefficients of a polynomial. Given
R=2+3*s+4*s^2+5*s^3;
the coefficient matrix of s 2 may be retrieved as
R{2}
ans =
4
The coefficient can also be changed
yields
R{2}=6

Degrees and
leading
coefficients
R =
2 + 3s + 6s^2 + 5s^3
In this way, new coefficients may be inserted. Thus
yields
R{5}=7
R =
2 + 3s + 6s^2 + 5s^3 + 7s^5
For polynomial matrices, it works similarly. Given
T = [ 1+s 2 s^2 s
3 4 s^3 0 ];
the coefficient matrix of s 2 may be retrieved as
T{2}
ans =
0 0 1 0
0 0 0 0
Taking coefficients and taking submatrices may be combined. So the coefficients of
the (1,3) entry of T follow as
T{:}(1,3)
ans =
0 0 1 0
The degree of a polynomial matrix is available by
deg(T)
ans =
3
The leading coefficient matrix, in our example that of
lcoef(T)
ans =
0 0 0 0
0 0 1 0
3
s may be obtained as

Constant, zero
and empty
matrices
Values
Derivative and
integral
A special case of polynomial matrix is that of degree 0; in effect, it is just a standard
MATLAB matrix. The Polynomial Toolbox treats both these forms of data
interchangeably: when a polynomial is expected, the standard matrix is also
accepted. For explicit conversions, pol and double commands are available:
P = pol([1 2;3 4]);
D = double(P);
Another special case is the zero polynomial; its degree is –Inf . Still another case:
the empty polynomial matrix, similar to empty standard matrix. Its degree is empty.
A polynomial may also be considered as a function of real or complex variable. For
evaluating the polynomial function, the value command is available:
F=2+3*s+4*s^2; X=2;
Y=value(F,X)
Y =
24
When F or X is a matrix then the values are evaluated entrywise:
X=[2 3];
Y=value(F,X)
Y =
24 47
A bit different command is mvalue. For scalar polynomial F and square matrix X, it
computes the matrix value Y according to the matrix algebra rules:
X=[1 2;0 1];
Y=mvalue(F,X)
Y =
9 22
0 9
For matrix polynomial function Fs , the derivative and the integral are computed
deriv(F)
ans =

3 + 8s
integral(F)
ans =
2s + 1.5s^2 + 1.3s^3
Arithmetic operations on polynomial matrices
Addition,
subtraction and
multiplication
In operations with two or more polynomial matrices, the indeterminates should be
the same. If they are not (usually by a mistake), a warning is issued and the
indeterminate is changed to the default.
Define the polynomial matrices
P = [1+s 2; 3 4], Q = [s^2 s; s^3 0]
P =
Q =
1 + s 2
3 4
s^2 s
s^3 0
The sum and product of P and Q follow easily:
S = P+Q
S =
R = P*Q
R =
The command
R+3
ans =
1 + s + s^2 2 + s
3 + s^3 4
s^2 + 3s^3 s + s^2
3s^2 + 4s^3 3s
3 + s^2 + 3s^3 3 + s + s^2

Determinants,
unimodularity
and adjoints
3 + 3s^2 + 4s^3 3 + 3s
obviously is interpreted as the instruction to add three times the unit matrix to R.
The command
3*R
ans =
3s^2 + 9s^3 3s + 3s^2
9s^2 + 12s^3 9s
yields the expected result. Also expected can be results of entrywise multiplication
R.*S and of powers R^N or R.^N with nonnegative integer N. For negative N, see
Chapter 4, Polynomial matrix fractions.
The determinant of a square polynomial matrix is defined exactly as its constant
matrix counterpart. In fact, its computation is not much more difficult:
P = [1 s s^2; 1+s s 1-s; 0 -1 -s]
P =
det(P)
ans =
1 s s^2
1 + s s 1 - s
0 -1 -s
1 - s - s^2
If its determinant happens to be constant then the polynomial matrix is called
unimodular:
U=[2-s-2*s^2, 2-2*s^2, 1+s; 1-s-s^2, 1-s^2, s;-1-s, -s, 1]
U =
det(U)
2 - s - 2s^2 2 - 2s^2 1 + s
1 - s - s^2 1 - s^2 s
-1 - s -s 1
Constant polynomial matrix: 1-by-1
ans =
1
If a matrix is suspected of unimodularity then one can make it sure by a special
tester

isunimod(U)
ans =
1
Also the adjoint matrix is defined as for constant matrices. The adjoint is a
polynomial matrix and may be computed by typing
adj(P)
ans =
1 - s - s^2 0 s - s^2 - s^3
s + s^2 -s -1 + s + s^2 + s^3
-1 - s 1 -s^2
Quite to the contrary, the inverse of a square polynomial matrix is usually not a
polynomial but a polynomial fraction
inv(P)
ans =
-1 + s + s^2 0 -s + s^2 + s^3
-s - s^2 s 1 - s - s^2 - s^3
1 + s -1 s^2
----------------------------------------
-1 + s + s^2
For more details, see Chapter 4, Polynomial matrix fractions.
The only cases do have polynomial inverse are unimodular matrices. In our example
inv(U)
ans =
Indeed,
U*inv(U)
ans =
1 -2 - s + s^2 -1 + s + s^2 - s^3
-1 3 + s - s^2 1 - 2s - s^2 + s^3
1 -2 + s^2 s - s^3
1.0000 0 0
0 1.0000 0
0 0 1.0000

Rank
If the matrix is nonsquare but has full rank then a usual partial replacement for the
inverse is provided by the generalized Moore-Penrose pseudoinverse, which is
computed by the pinv function.
Q = P(:,1:2)
Q =
1 s
1 + s s
0 -1
Qpinv = pinv(Q)
Qpinv =
1 - s^3 1 + s + s^3 2s + s^2
s^2 + s^3 -s^2 -2 - 2s - s^2
------------------------------------------
2 + 2s + s^2 + s^4
In general, the generalized Moore-Penrose pseudoinverse is a polynomial matrix
fraction. Once again,
Qpinv*Q
ans =
1.0000 0
0 1.0000
A polynomial matrix P( s)
has full column rank (or full normal column rank) if it has
full column rank everywhere in the complex plane except at a finite number of points.
Similar definitions hold for full row rank and full rank.
Recall that
P = [1 s s^2; 1+s s 1-s; 0 -1 -s]
P =
The rank test
1 s s^2
1 + s s 1 - s
0 -1 -s
isfullrank(P)

Bases and null
spaces
ans =
1
confirms that P has full rank.
The normal rank of a polynomial matrix P( s)
equals
max sC P( s)
rank
Similar definitions apply to the notions of normal column rank and normal row rank.
The rank is calculated by
rank(P)
ans =
3
As for constant matrices, rank evaluation may be quite sensitive and an ad hoc
change of tolerance (which may be included as an optional input parameter) may be
helpful for difficult examples.
A polynomial matrix is nonsingular if it has full normal rank.
issingular(P)
ans =
0
There are two important subspaces (more precisely, submodules) associated with a
polynomial matrix A( s):
its null space and its range (or span). The (right) null space
is defined as the set of all polynomial vectors x( s)
such that A( s) x( s)
0 . For matrix
A = P(1:2,:)
A =
1 s s^2
1 + s s 1 - s
the (right) nullspace is computed by
N = null(A)
N =
0.35s - 0.35s^2 - 0.35s^3
-0.35 + 0.35s + 0.35s^2 + 0.35s^3
-0.35s^2
Here the null space dimension is 1 and its basis has degree 3.

Roots and
stability
The range of A( s)
is the set of all polynomial vectors y( s)
such that y( s) A( s) x( s)
for
some polynomial vector x( s)
. In the Polynomial Toolbox, the minimal basis of the
range is returned by the command
minbasis(A)
>> minbasis(A)
ans =
0 1.0000
1.0000 0
The roots or zeros of a polynomial matrix P( s)
are those points si in the complex
plane where P( s)
loses rank.
roots(P)
ans =
-1.6180
0.6180
The roots can be both finite and infinite. The infinite roots are normally suppressed.
To reveal them, type
roots(P,'all')
ans =
-1.6180
0.6180
Inf
Unimodular matrices have no finite roots:
roots(U)
ans =
Empty matrix: 0-by-1
but typically have infinite roots:
roots(U,'all')
ans =
Inf
Inf
Inf

If P( s)
is square then its roots are the roots of its determinantdet P( s ) , including
multiplicity:
roots(det(P))
ans =
-1.6180
0.6180
The finite roots may be visualized as in Fig. 1 by typing
zpplot(P)
Fig. 1. Locations of the roots of a polynomial matrix
A polynomial matrix is stable if all its roots fall within a relevant stability region. For
polynomial matrices in s and p it is the open left half plane for polynomial matrices,
which means Hurwitz stability used for continuous-time systems
The macro isstable checks stability:
isstable(s-2)
ans =
0
isstable(s+2)

Special
constant
matrices related
to polynomials
ans =
0
For other stability regions considered in the Polynomial Toolbox, see Chapter 3,
Discrete-time and two-sided polynomial matrices.
There are several interesting constant matrices that are composed from the
coefficients of polynomials (or the matrix coefficients of polynomial matrices) and are
frequently encountered in mathematical and engineering textbooks. Given a
polynomial
p( v) p p v p v p v
0 1 2 2
d n
of degree n we may for instance define the corresponding n n Hurwitz matrix
H
p
L
M
NM
p p p
n 1 n 3 n 5
p p p
0
0
n n 2 n 4
p p p
n 1 n 3 n 5
p p p
0 0 0
n n 2 n 4
0 0 0 0
a k ( n k)
Sylvester matrix (for some k 1)
S
p
L
M
N
M
p
n
p p p 0 0
0 1
n
0 p p p 0 0
0 1
0 0 p p p
or an n n companion matrix
C( p)
L
M
NM
0 1 0
n
0 0 1 0
0 1
0 0 1
p0
p1
pn
1
pn
pn
pn
Using the Polynomial Toolbox, we take
p = pol([-1 1 2 3 4 5],5)
p =
-1 + s + 2s^2 + 3s^3 + 4s^4 + 5s^5
O
P
QP
.
p
0
n
O
P
QP
,
O
P
Q
P

and simply type
or
or
hurwitz(p)
ans =
sylv(p,3)
ans =
4 2 -1 0 0
5 3 1 0 0
0 4 2 -1 0
0 5 3 1 0
0 0 4 2 -1
-1 1 2 3 4 5 0 0 0
compan(p)
ans =
Divisors and multiples
Scalar divisor
and multiple
0 -1 1 2 3 4 5 0 0
0 0 -1 1 2 3 4 5 0
0 0 0 -1 1 2 3 4 5
0 1.0000 0 0 0
0 0 1.0000 0 0
0 0 0 1.0000 0
0 0 0 0 1.0000
0.2000 -0.2000 -0.4000 -0.6000 -0.8000
For polynomial matrices the block matrix versions are defined and computed in a
fairly obvious manner
To understand when division of polynomial matrices is possible, begin with scalar
polynomials. Consider three polynomialsa( s),
b( s)
and c( s)
such thata( s) b( s) c( s)
.

Division
Division with
remainder
Greatest
common divisor
Coprimeness
We say that b( s)
is a divisor (or factor) of a( s)
or a( s)
is a multiple ofb( s),
and write
a( s) b( s).
This is sometimes also stated as b( s)
dividesa( s).
For example, take
b = 1-s; c = 1+s; a = b*c
a =
1 - s^2
As b( s)
is a divisor of a( s),
the division
a/b
can be done and results in
ans =
1 + s
If b( s)
is not a divisor of a( s)
then the result of division is not a polynomial but a
polynomial fraction1 , see Chapter 4, Polynomial matrix fractions.
On the other hand, any n( s)
can be divided by any nonzero d( s)
using another
operation called division with a remainder. This division results in a quotient q( s)
and
a remainder r( s):
n( s) q( s) d( s) r( s)
Typically, it is required that deg r( s) deg d( s)
, which makes the result unique. Thus,
dividing
by
yields
n = (1-s)^2;
d = 1+s;
[q,r] = ldiv(n,d)
q =
-3 + s
Constant polynomial matrix: 1-by-1
r =
4
Division with remainder is sometimes called Euclidean division.
If a polynomial g( s)
divides both a( s)
and b( s)
then g( s)
is called a common divisor of
a( s)
and b( s).
If, furthermore, g( s)
is a multiple of every common divisor of a( s)
and
b( s)
then g( s)
is a greatest common divisor of a( s)
and b( s).
1 Users familiar with older versions of the Polynomial Toolbox (2.5 and below) should
notice that the performance of slash and backward slash operators has been changed:
the operators are now primarily used to create fractions and not only to extract
factors as before. For deeper understanding, experiment or see Chapter 4, Polynomial
matrix fractions and Chapter 13, Compatibility with Version 2. To extract a factor
directly, one should better use equation solvers as axb and alike.
31

Least common
multiple
If the only common divisors of a( s)
and b( s)
are constants then the polynomials a( s)
and b( s)
are coprime (or relatively prime). To compute a greatest common divisor of
and
type
a = s+s^2;
b = s-s^2;
gld(a,b)
ans =
s
Similarly, the polynomials
c = 1+s; d = 1-s;
are coprime as
gld(c,d)
is a constant
ans =
1
Coprimeness may also be tested directly
isprime([c,d])
ans =
1
As the two polynomials c and d are coprime, there exist other two polynomials e and f
that satisfy the linear polynomial equation
ce df 1
The polynomials e and f may be computed according to
[e,f] = axbyc(c,d,1)
e =
f =
0.5000
0.5000
We check the result by typing
c*e+d*f
ans =
1
If a polynomial m ( s)
is a multiple of both a( s)
and b( s)
then m ( s)
is called a common
multiple of a( s)
and b( s).
If, furthermore, m ( s)
is a divisor of every common multiple
of a( s)
and b( s)
then it is a least common multiple of a( s)
and b( s):
m = llm(a,b)
m =
s - s^3
32

Matrix divisors
and multiples
Matrix division
Matrix division
with remainder
The concepts just mentioned are combined in the well-known fact that the product of
a greatest common divisor and a least common multiple equals the product of the two
original polynomials:
isequal(a*b,gld(a,b)*llm(a,b))
ans =
1
Next consider polynomial matrices A( s),
B( s)
and C( s)
of compatible sizes such that
A( s) B( s) C( s)
. We say that B( s)
is a left divisor of A( s),
or A( s)
is a right multiple of
B( s)
. Take for instance
B = [1 s; 1+s 1-s], C = [2*s 1; 0 1], A = B*C
B =
C =
A =
1 s
1 + s 1 - s
2s 1
0 1
2s 1 + s
2s + 2s^2 2
As B( s)
is a left divisor of A( s),
the matrix left division
B\A
can be done and results in a polynomial matrix
ans =
2s 1
0 1
If B( s)
is not a left divisor then the result of left division is not a polynomial matrix
but a polynomial matrix fraction, see Chapter 4, Polynomial matrix fractions. This is
also the case when B( s)
is a divisor but from the other ("wrong") side: A/B.
A/B
ans =
1 + 3s 2s / 1 + s 1
2 + 2s + 2s^2 2s + 2s^2 / 2 1 + s
On the other hand, a polynomial matrix N ( s)
can be divided by any compatible
nonsingular square matrix D( s)
with a remainder, resulting in a matrix quotient
Q( s)
and a matrix remainder R( s)
such that
N ( s) D( s) Q( s) R( s)
1
If it is required that the rational matrix D ( s) R( s)
is strictly proper then the
division is unique. Thus, dividing a random matrix
N = prand(4,2,3,'ent','int')
N =
33

Greatest
common left
divisor
0 -3 + 11s -1 + s
5 - 4s^3 1 - 7s + 4s^2 + 8s^3 - 3s^4 4 + 6s
from the left by a random square matrix
D = prand(3,2,'ent','int')
D =
results in
-7 3 - 2s + 3s^2
4 + 4s + 6s^2 0
[Q,R] = ldiv(N,D)
Q =
R =
Indeed,
while
0.44 - 0.67s -0.11 + 1.7s - 0.5s^2 0
0 -1.2 0
3.1 - 4.7s -0.28 + 20s -1 + s
3.2 + 0.89s 1.4 - 13s 4 + 6s
deg(det(D))
ans =
4
deg(adj(D)*R,'ent')
ans =
3 3 3
3 3 3
1
so that each entry of the rational matrix D ( s) R( s) (adj D) R / det D is strictly
proper.
If a polynomial matrix G( s)
is a left divisor of both A( s)
and B( s)
then G( s)
is called
a common left divisor of A( s)
and B( s)
. If, furthermore, G( s)
is a right multiple of
every common left divisor of A( s)
and B( s)
then it is a greatest common left divisor of
A( s)
and B( s)
. If the only common left divisors of A( s)
and B( s)
are unimodular
matrices then the polynomial matrices A( s)
and B( s)
are left coprime. To compute a
greatest common left divisor of
and
A = [1+s-s^2, 2*s + s^2; 1-s^2, 1+2*s+s^2]
A =
1 + s - s^2 2s + s^2
1 - s^2 1 + 2s + s^2
B = [2*s, 1+s^2; 1+s, s+s^2]
B =
2s 1 + s^2
34

Least common
right multiple
type
gld(A,B)
ans =
1 + s s + s^2
0 1
1 + s 0
Similarly, the polynomial matrices
C = [1+s, 1; 1-s 0], D = [1-s, 1; 1 1]
C =
D =
1 + s 1
1 - s 0
1 - s 1
1 1
are left coprime as
gld(D,C)
ans =
1 0
0 1
is obviously unimodular. You may also directly check
isprime([C,D])
ans =
1
As the two polynomial matrices C and D are left coprime there exist other two other
polynomial matrices E and F that satisfy
CE DF I
They may be computed according to
[E,F] = axbyc(C,D,eye(2))
E =
F =
0 0
1.0000 -1.0000
0 0
0 1.0000
If a polynomial matrix M ( s)
is a right multiple of both A( s)
and B( s)
then M ( s)
is
called a common right multiple of A( s)
and B( s)
. If, furthermore, M ( s)
is a left
divisor of every common right multiple of A( s)
and B( s)
then M ( s)
is a least common
right multiple of A( s)
and B( s)
.
35

Dual concepts
M = lrm(A,B)
M =
0.32 + 1.3s + 0.95s^2 -0.3 - 1.4s - 0.3s^2 + 0.15s^3
0.63 + 1.3s + 0.63s^2 -0.75 - 1.1s - 0.15s^2 + 0.15s^3
which is verified by
A\M, B\M
ans =
ans =
0.32 + 0.32s -0.3 - 0.15s
0.32 + 0.32s -0.45
0.63 + 0.32s -0.75
0.32 -0.3 + 0.15s-3
The dual concepts of right divisors, left multiples, common right divisors, greatest
common right divisors, common left multiples, and least common left multiples are
similarly defined and computed by dual functions grd and llm.
Transposition and conjugation
Transposition
Complex
coefficients
Given
T = [1 0 s^2 s; 0 1 s^3 0]
T =
1 0 s^2 s
0 1 s^3 0
the transposition follows by typing
T.'
ans =
1 0
0 1
s^2 s^3
s 0
The command transpose(T)is synonymous with T.'
The Polynomial Toolbox supports polynomial matrices with complex coefficients such
as
C = [1+s 1; 2 s]+i*[s 1; 1 s]
C =
1+0i + (1+1i)s 1+1i
2+1i (1+1i)s
36

Conjugated
transposition
Many Toolbox operations and fractions handle the accordingly. For example the
transposition
C.'
ans =
1+0i + (1+1i)s 2+1i
1+1i (1+1i)s
In addition, there are several special functions for complex coefficient polynomial
matrices such as
real(C)
ans =
1 + s 1
2 s
imag(C)
ans =
conj(C)
ans =
s 1
1 s
1+0i + (1-1i)s 1-1i
2-1i (1-1i)s)
The operation of conjugated transposition is a bit more complicated. It is connected
with the scalar products xt yt of continuous-time systems. Here the theory
requires s s.
For matrix polynomials, the operation consists of the conjugation,
the transposition and the change of the argument s to –s. So,
C'
ans =
1+0i + (-1+1i)s 2-1i
1-1i (-1+1i)s
The command ctransp(C) is synonymous with C' .
For real polynomial matrix
T
sign of s hence T s
T = [1 0 s^2 s; 0 1 s^3 0]
T =
T'
ans =
1 0 s^2 s
0 1 s^3 0
1 0
T s , this operation makes transposition with change of
37

0 1
s^2 -s^3
-s 0
Reduced and canonical forms
Row degrees
Suppose that we have a polynomial matrix
P = [1+s^2, -2; s-1 1]
P =
of degree
deg(P)
ans =
1 + s^2 -2
-1 + s 1
2
with leading coefficient matrix
Pl = P{2}
Pl =
1 0
0 0
Besides the (overall) degree of P we may also consider its row degrees
deg(P,'row')
ans =
2
1
and column degrees
deg(P,'col')
ans =
2 0
Associated with the row degrees is the leading row coefficient matrix
lcoef(P,'row')
ans =
1 0
1 0
and associated with the column degrees is the leading column coefficient matrix
lcoef(P,'col')
ans =
1 -2
38

Row and
column reduced
matrices
Row reduced
form
Triangular and
staircase form
0 1
A polynomial matrix is row reduced if its leading row coefficient matrix has full row
rank. Similarly, it is column reduced if its leading column coefficient matrix has full
column rank. The matrix P is definitely not row reduced
isfullrank(lcoef(P,'row'))
ans =
0
but it is column reduced
isfullrank(lcoef(P,'col'))
ans =
1
Any polynomial matrix with full row rank may be transformed into row reduced form
by pre-multiplying it by a suitable unimodular matrix. To compute a row reduced
form of P, call
P_row_reduced = rowred(P)
P_row_reduced =
1 + s -2 - s
-1 + s 1
Indeed, the row rank of
lcoef(P_row_reduced,'row')
ans =
is full.
1 -1
1 0
There are several special forms of a polynomial matrix that can be achieved by pre-
and/or post-multiplying it by suitable unimodular matrix. These operations preserve
many important properties and indeed serve to make these visible.
Thus, a lower-left triangular form T ( s)
of A( s)
resulting from column operations
T ( s) A( s) U( s)
can be computed by the macro tri:
A = [s^2 0 1; 0 s^2 1+s]
A =
T = tri(A)
T =
s^2 0 1
0 s^2 1 + s
-1 0 0
-1 - s -1.2s^2 0
The corresponding unimodular matrix is returned by
[T,U] = tri(A); U
U =
39

Another
triangular form
Hermite form
0 0.29 0.5
0 -0.87 + 0.29s 0.5 + 0.5s
-1 -0.29s^2 -0.5s^2
If A( s)
has not full row rank then T ( s)
is in staircase form. Similarly, an upper-right
triangular (row staircase) form is achieved by row (unimodular) operations. It results
from the call tri(A,'row').
If B( s)
is a square polynomial matrix with nonsingular constant term then another
upper-triangular form may be obtained by the overloaded macro lu:
B = [ 1 1 s; s+1 0 s; 1-s s 2+s]
B =
1 1 s
1 + s 0 s
1 - s s 2 + s
[V,T] = lu(B)
V =
T =
1 0.33s 0.11s
1 + s 1 + 1.3s + 0.33s^2 0.44s + 0.11s^2
1 - s 1 - 1.7s - 0.33s^2 1 - 0.56s - 0.11s^2
1 1 + 0.33s 0.78s + 0.11s^3
0 -1 -0.67s - s^2 + 0.33s^3
0 0 2 + 2s + 2s^2 - s^3
The triangular forms described above are by no means unique. A canonical triangular
form is called the Hermite form. An n m polynomial matrix A( s)
of rank r is in
column Hermite form if it has the following properties:
it is lower triangular
the diagonal entries are all monic
each diagonal entry has higher degree than any entry on its left
in particular, if the diagonal element is constant then all off-diagonal elements in
the same row are zero
if n r then the last n r columns are zero
The nomenclature in the literature is not consistent. Some authors (in particular
Kailath, 1980) refer to this as the row Hermite form. The polynomial matrix A is in
row Hermite form if it is the transpose of a matrix in column Hermite form. The
command
H = hermite(A)
returns the column Hermite form
H =
1 0 0
1 + s s^2 0
40

Echelon form
Smith form
while the call
[H,U] = hermite(A); U
U =
0 -0.25 0.5
0 0.75 - 0.25s 0.5 + 0.5s
1 0.25s^2 -0.5s^2
provides a unimodular reduction matrix U such that H ( s) A( s) U( s)
.
Yet another canonical form is called the (Popov) echelon form. A polynomial matrix E
is in column echelon form (or Popov form) if it has the following properties:
it is column reduced with its column degrees arranged in ascending order
for each column there is a so-called pivot index i such that the degree of the i -th
entry in this column equals the column degree, and the i-th entry is the lowest
entry in this column with this degree
the pivot indexes are arranged to be increasing
each pivot entry is monic and has the highest degree in its row
A square matrix in column echelon form is both column and row reduced
Given a square and column-reduced polynomial matrix D( s)
the command [E,U] =
echelon(D) computes the column echelon form E( s)
of D( s).
The unimodular
matrix U ( s)
satisfies E( s) D( s) U( s)
.
By way of example, consider the polynomial matrix
D = [ -3*s s+2; 1-s 1];
To find its column echelon form and the associated unimodular matrix, type
[E,U] = echelon(D)
MATLAB returns
E =
U =
2 + s -6
1 -4 + s
0 -1.0000
1.0000 -3.0000
The ultimate, most structured canonical form for a polynomial matrix is its Smith
form. A polynomial matrix A( s)
of rank r may be reduced to its Smith form
S ( s) U( s) A( s) V ( s)
by pre- and post- multiplication by unimodular matrices U ( s)
and
V ( s),
respectively. The Smith form looks like this:
S ( s)
L N M
S r ( s)
0
0 0
O
QP
with the diagonal submatrix
Sr ( s) diag( a1( s), a2 ( s), , ar ( s))
41

Invariant
polynomials
The entries a1 ( s), a2 ( s), , ar ( s)
are monic polynomials such that a1 ( s)
divides
ai+1 ( s)
for i 1, 2, , r 1.
The Smith form is particularly useful for theoretical
considerations as it reveals many important properties of the matrix. Its practical
use, however is limited because it is quite sensitive to small parameter perturbations.
The computation of the Smith form becomes numerically troublesome as soon as the
matrix size and degree become larger. The Polynomial Toolbox offers a choice of three
different algorithms to achieve the Smith form, all programmed in macro smith.
For larger examples, a manual change of tolerance may be necessary. To compute the
Smith form of a simple matrix
A=[1+s, 0, s+s^2; 0, s+2, 2*s+s^2]
A =
simply call
smith(A)
ans =
1 + s 0 s + s^2
0 2 + s 2s + s^2
1 0 0
0 2 + 3s + s^2 0
The polynomials a1 ( s), a2 ( s), , ar ( s)
that appear in the Smith form are uniquely
determined and are called the invariant polynomials of A( s).
They may be retrieved
by typing
diag(smith(A))
ans =
1
2 + 3s + s^2
Polynomial matrix equations
Diophantine
equations
The simplest type of linear scalar polynomial equation — called Diophantine
equation after the Alexandrian mathematician Diophantos (A.D. 275) — is
a( s) x( s) b( s) y( s) c( s)
The polynomials a( s),
b( s)
and c( s)
are given while the polynomials x( s)
and y( s)
are
unknown. The equation is solvable if and only if the greatest common divisor of a( s)
and b( s)
divides c( s)
. This implies that with a( s)
and b( s)
coprime the equation is
solvable for any right hand side polynomial, including c( s)
1.
The Diophantine equation possesses infinitely many solutions whenever it is
solvable. If x( s),
y( s)
is any (particular) solution then the general solution of the
Diophantine equation is
42

x( s) x( s) b ( s) t( s)
y( s) y( s) a ( s) t( s)
Here t( s)
is an arbitrary polynomial (the parameter) and a ( s)
, b ( s)
are coprime
polynomials such that
b ( s)
b( s)
a ( s)
a( s)
If the polynomials a( s)
and b( s)
themselves are coprime then one can naturally take
a( s) a( s)
and b ( s) b( s)
.
Among all the solutions of Diophantine equation there exists a unique solution pair
x( s)
, y( s)
characterized by
deg x( s) deg b ( s)
.
There is another (generally different) solution pair characterized by
deg y( s) deg a( s)
.
The two special solution pairs coincide only if
deg a( s) deg b( s) deg c( s)
.
The Polynomial Toolbox offers two basic solvers that may be used for scalar and
matrix Diophantine equations. They are suitably named axbyc and xaybc. For
example, consider the simple polynomials
a = 1+s+s^2; b = 1-s; c = 3+3*s;
When typing
[x,y] = axbyc(a,b,c)
x =
y =
2
1 + 2s
b g b g
MATLAB returns the solution pair x( s), y( s) The alternative call
2, 1 2 s with minimal overall degree.
[x,y,f,g] = axbyc(a,b,c)
x =
y =
f =
g =
2
1 + 2s
-0.45 + 0.45s
0.45 + 0.45s + 0.45s^2
retrieves the complete general solution in the form x( s) f ( s) t( s), y( s) g( s) t( s)
an arbitrary polynomial parameter t( s)
.
43
b g with

Bézout
equations
Matrix
polynomial
equations
To investigate the case of different minimal degree solutions, consider a right hand
side of higher degree
c = 15+15*s^4;
As before, the call
[x1,y1] = axbyc(a,b,c)
x1 =
y1 =
8 - 13s + 15s^2
7 + 12s + 2s^2
results in the solution of minimal overall degree (in this case deg x1 deg y1
2 ).
A slightly different command
[x2,y2] = axbyc(a,b,c,'minx')
x2 =
y2 =
10.0000
5 - 5s - 15s^2 - 15s^3
returns another solution with the minimal degree of the first unknown. Finally,
typing
[x2,y2] = axbyc(a,b,c,'miny')
x2 =
y2 =
10 - 15s + 15s^2
5 + 10s
produces the solution of minimal degree in the second unknown.
Should the equation be unsolvable, the function returns NaNs..
[x,y] = axbyc(s,s,1)
x =
y =
NaN
NaN
A Diophantine equation with 1 on its right hand side is called a Bézout equation. It
may look like
a( s) x( s) b( s) y( s)
1
with a( s),
b( s)
given and x( s)
, y( s)
unknown.
In the matrix case, the polynomial equation becomes a polynomial matrix equation.
The basic matrix polynomial (or polynomial matrix) equations are
and
A( s) X ( s) B( s)
X ( s) A( s) B( s)
44

One-sided
equations
Two-sided
equations
or even
A( s) X ( s) B( s) C( s)
A( s),
B( s)
and, if applicable, C( s)
are given while X ( s)
is unknown. The Polynomial
Toolbox functions to solve these equations are conveniently named axb, xab, and
axbc. Hence, given the polynomial matrices
A= [1 s 1+s; s-1 1 0]; B = [s 0; 0 1];
the call
X0 = axb(A,B)
X0 =
1 -1
1 - s s
-1 + s 1 – s
solves the first equation and returns its solution of minimal overall degree. Typing
[X0,K] = axb(A,B); K
K =
1.7 + 1.7s
1.7 - 1.7s^2
-1.7 - 1.7s + 1.7s^2
also computes the right null-space of A so that all the solutions to A( s) X ( s) B( s)
may easily be parameterized as
X ( s) X0 ( s) K ( s) T ( s)
T ( s)
is an arbitrary polynomial matrix of compatible size. The other equations are
handled similarly.
In systems and control several special forms of polynomial matrix equations are
frequently encountered, in particular the one-sided equations
and
A( s) X ( s) B( s) Y ( s) C( s)
X ( s) A( s) Y ( s) B( s) C( s)
Also the two-sided equations
and
A( s) X ( s) Y ( s) B( s) C( s)
X ( s) A( s) B( s) Y ( s) C( s)
are common. A( s), B( s)
and C( s)
are always given while X ( s)
and Y ( s)
are to be
computed. A( s)
is typically square invertible.
The solutions of the one- and two-sided equations may be found with the help of the
Polynomial Toolbox macros axbyc, xaybc, and axybc . Thus, for the matrices
A= [1 s; 1+s 0]; B = [s 1; 1 s]; C = [1 0; 0 1];
the call
[X,Y] = axbyc(A,B,C)
45

Factorizations
returns
X =
Y =
0.25 - 0.5s 0.5
0.25 + 0.5s -0.5
-0.25 - 0.5s 0.5
0.75 + 0.5s -0.5
Various other scalar and matrix polynomial equations may be solved by directly
applying appropriate solvers programmed in the Polynomial Toolbox, such as the
equation
A( s) X ( s) X ( s) A ( s) B( s)
Table 2 lists all available polynomial matrix equation solvers.
Table 2. Equation solvers
Equation Name of the routine
AX B
axb
AXB C
axbc
AX BY C
axbyc
A X X A B
axxab
A X Y A B
axyab
AX YB C
axybc
XA AX B
xaaxb
XA B
xab
XA YB C
xaybc
To indicate the polynomial matrix equation is unsolvable, the solver returns matrices
of correct sizes but full of NaNs.
X0 = axb(A,B)
X0 =
NaN NaN
NaN NaN
46

Symmetric
polynomial
matrices
Zeros of
symmetric
matrices
Besides linear equations special quadratic equations in scalar and matrix
polynomials are encountered in various engineering fields. The equations typically
contain a symmetric polynomial matrix on the right hand side.
A square polynomial matrix M which is equal to its conjugated transpose M' is called
(para-Hermitian) symmetric. In the scalar case with real coefficients,
2 4
2 3 4
M s s s M s
only even coefficients are nonzero. In the scalar case with complex coefficients,
2 3 4
2 3 3 4
M s is s is s M s
even coefficients are real while odd ones are imaginary. In the matrix case
or
2 1 0 1 3203 M s s s s M s
1 2
10
2 4
3 0
2 3
2 2i 1 i i 1
i
M s s M s
1 i 2 2i
1 i 7i
even coefficients are (Hermitian) symmetric, odd coefficients are (Hermitian) antisymmetric.
M s as a function of complex variable, we can set s to various points s i of
the complex plane. Thus we obtain numerical matrices Ms i that are (Hermitian)
symmetric and hence they have only real eigenvalues. If all eigenvalues of Ms i are
positive for all s i on the imaginary axis then Ms is called positive definite. Such a
polynomial matrix, of course, has no zeros on the imaginary axis. If all the
eigenvalues are nonnegative then Ms is called nonnegative definite. Yet another
important class consists of matrices with constant number of positive and negative
eigenvalues of Ms i over the whole all imaginary axis. Such Ms is called
indefinite (in matrix sense). All the above cases can be factorized. Should the number
of positive and negative eigenvalues change as s i ranges the imaginary axis,
however, the matrix Ms is indefinite in the scalar sense and cannot be factorized in
any symmetric sense.
Treating
Nonnegative definite symmetric polynomial matrix Ms with real coefficients has
its zeros distributed symmetrically with respect to the imaginary axis. Thus if it has
a zero s i , it generally has a quadruple of zeros si , si, si , si
, all of the same
multiplicity. If s i is real, then it is only a couple si, si.
If s i is imaginary, then it is
also a couple si, s i but it must be of even multiplicity. Finally, if si 0 then it is a
singlet of even multiplicity.
zpplot(4+s^4), grid
47

zpplot(1-s^2), grid
zpplot(1+2*s^2+s^4), grid
48

zpplot(-s^2), grid
For polynomial matrices with complex coefficients, the picture is more simple:
generally a couple si, si,
especially in imaginary case (including the zero case) a
singlet with even multiplicity.
zpplot(2+2*i*s-s^2),grid
49

Spectral
factorization
zpplot(1+2*i*s-s^2),grid
One of quadratic equations in scalar and matrix polynomials frequently encountered
is the polynomial spectral factorization
A( s) X (
s) JX ( s)
and the spectral co-factorization
A( s) X ( s) JX (
s)
.
In either case, the given polynomial matrix A( s)
satisfies A( s) A ( s)
(we say it is
para-Hermitian symmetric) and the unknown X ( s)
is to be stable. The case of A( s)
positive definite on the stability boundary results in J I .
Spectral factorization with J = I is the main tool to design LQ and LQG controllers as
well as Kalman filters. On the other hand, if A( s)
is indefinite in matrix sense then
J diag l+1, + 1, , + 1, 1, 1, 1q
. This is the famous J -spectral factorization
problem, which is an important tool for robust control and filter design based on H
50

norms. The Polynomial Toolbox provides two macros called spf and spcof to
handle spectral factorization and co-factorization, respectively.
By way of illustration consider the para-Hermitian matrix
A =
34 - 56s^2 -13 - 22s + 60s^2 36 + 67s
-13 + 22s + 60s^2 46 - 1e+002s^2 -42 - 26s + 38s^2
36 - 67s -42 + 26s + 38s^2 59 - 42s^2
Its spectral factorization follows by typing
[X,J] = spf(A)
X =
J =
2.1 + 0.42s 5.2 + 0.39s -2 + 0.35s
-5.5 + 4s 4.3 + 0.64s -7.4 - 5.5s
0.16 + 6.3s -0.31 - 10s -0.86 + 3.5s
1 0 0
0 1 0
0 0 1
while the spectral co-factorization of A is computed via
[Xcof,J] = spcof(A)
Xcof =
J =
2.7 + 0.42s 4.8 + 4s 2 + 6.3s
-1.6 + 0.39s 0.93 + 0.64s -6.5 - 10s
4.3 + 0.35s 2.7 - 5.5s 5.8 + 3.5s
1 0 0
0 1 0
0 0 1
The resulting J reveals that the given matrix A is positive-definite on the imaginary
axis. On the other hand, the following matrix is indefinite
B =
5 -6 - 18s -8
-6 + 18s -41 + 81s^2 -22 - 18s
-8 -22 + 18s -13
Its spectral factorization follows as
[Xf,J] = spf(B)
Xf =
J =
3.3 2.9 - 0.92s -1.6
1.8 6.6 + 0.44s 3.4
1.6 2.5 + 9s -2
51

Non-symmetric
factorization
or
1 0 0
0 -1 0
0 0 -1
[Xcof,J] = spcof(B)
Xcof =
J =
3.3 -1.8 1.6
-1.9 + 0.92s -4.4 + 0.44s -5 - 9s
-1.6 -3.4 -2
1 0 0
0 -1 0
0 0 -1
General non-symmetric factorization can be computed by function fact which
returns a factor with zeros arbitrarily selected from the zeros of the original matrix.
For example, it is sometimes useful to express a given non-symmetric polynomial
matrix As as the product of its anti-stable and stable factors A Aantistab Astab
. So for
factorize polynomial matrix
A=prand(3,3,'int')
A =
-1 + s + 2s^2 + 7s^3 -2 + 3s + 4s^2 + 5s^3 -5 + s + s^2 - 5s^3
-4 + 5s - s^2 + 2s^3 -3s - 3s^2 + 2s^3 -5 + 4s + 3s^2 - 4s^3
-1 - 6s - 11s^2 + 5s^3 -3 + 2s + s^2 -5 - 5s - 2s^2 - 6s^3
having stable as well as unstable zeros
r=roots(A)
r =
4.2755
-1.6848
-0.8988 + 0.5838i
-0.8988 - 0.5838i
0.5927 + 0.7446i
0.5927 - 0.7446i
0.3336
-0.1425 + 0.2465i
-0.1425 - 0.2465i
the stable factor should have all the stable zeros
r_stab=r(find(real(r)

-0.8988 + 0.5838i
-0.8988 - 0.5838i
-0.1425 + 0.2465i
-0.1425 - 0.2465i
while the anti-stable should have all the remaining (unstable) ones. The (right) stable
factor is directly returned by typing
A_stab=fact(A,r_stab)
A_stab =
-5.9 - 4.1s + s^2 4.3 + 3.3s -2
-7.2 - 6.5s 5.6 + 5.4s + s^2 -2
5.2 + 4.2s -3.3 - 2.8s 2.5 + s
The anti-stable is then computed indirectly as
A_antistab=xab(A_stab,A)
A_antistab =
53
A AA
-1
antistab stab
26 + 28s -18 - 9.2s 4.3 + 14s - 5s^2
-19 + 19s 8.5 - 9.3s -10 + 13s - 4s^2
-22 + 30s 9.3 - 17s -12 + 13s - 6s^2
The result is correct as
and
isequal(A_antistab*A_stab,A)
ans =
1
roots(A_stab),roots(A_antistab)
ans =
-1.6848
-0.8988 + 0.5838i
-0.8988 - 0.5838i
-0.1425 + 0.2465i
-0.1425 - 0.2465i
ans =
Matrix pencil routines
4.2755
0.5927 + 0.7446i
0.5927 - 0.7446i
0.3336
Matrix pencils are polynomial matrices of degree 1. They arise in the study of
continuous- and discrete-time linear time-invariant state space systems given by

Transformation
to Kronecker
canonical form
x( t) Ax( t) Bu( t)
y( t) Cx( t) Du( t)
x( t 1)
Ax( t) Bu( t)
y( t) Cx( t) Du( t)
and continuous- and discrete-time descriptor systems given by
Ex( t) Ax( t) Bu( t)
Ex( t 1)
Ax( t) Bu( t)
y( t) Cx( t) Du( t)
y( t) Cx( t) Du( t)
The transfer matrix of the descriptor system is
1
H ( s) C( sE A) B D
in the continuous-time case, and
1
H ( s) C( zE A) B D
in the discrete-time case. The polynomial matrices
sE A , zE A
that occur in these expressions are matrix pencils. In the state space case they reduce
to the simpler forms sI A and zI A .
A nonsingular square real matrix pencil P( s)
may be transformed to its Kronecker
canonical form
C( s) QP( s) Z
L
NM
a sI
0
0
O
QP
I se
Q and Z are constant orthogonal matrices, a is a constant matrix whose eigenvalues
are the negatives of the roots of the pencil, and e is a nilpotent constant matrix. (That
is, there exists a nonnegative integer k such that e i 0 for i k . The integer k is
called the nilpotency of e.)
This transformation is very useful for the analysis of descriptor systems because it
separates the finite and infinite roots of the pencil and, hence, the corresponding
modes of the system.
By way of example we consider the descriptor system
L
M
O
P
L
M
O
P
L
M
O
P
1 0 0 1 0 0 1
M0
0 1 0 1 0 0
NM
P
0 0 0QP
0 0 1 1
1 1 0 2
M
NM
P
QP
x x M
NM
Pu
QP
2
E A B
y x u
C
D
The system is defined by the matrices
A = [ -1 0 0; 0 1 0; 0 0 1 ];
B = [ 1; 0; 1 ];
C = [ 1 -1 0 ];
D = 2;
E = [ 1 0 0; 0 0 1; 0 0 0 ];
We compute the canonical form of the pencil sE A by typing
c = pencan(s*E-A)
54

Transformation
to Clements
form
Pencil
Lyapunov
equations
c =
1 + s 0 0
0 1 -s
0 0 1
Inspection shows that a = 1 has dimensions 1 1 and that e is the 2 2 nilpotent
matrix
e
L
NM
O
QP
0 1
0 0
The descriptor system has a finite pole at 1 and two infinite poles.
A para-Hermitian real pencil P( s) sE A , with E skew-symmetric and A
symmetric, may — under the assumption that it has no roots on the imaginary axis
— be transformed into its ―Clements‖ form (Clements, 1993) according to
T
C( s) UP( s) U
L
M
NM
0 0
0
55
sE A
1 1
A sE A
2 3 3
T T T T
1 1 3 3 4 3
sE A sE A sE A
The constant matrix U is orthogonal and the finite roots of the pencil sE1 A1
all
have negative real parts. This transformation is needed for the solution of various
spectral factorization problems that arise in the solution of H 2 and H optimization
problems for descriptor systems.
Let the para-Hermitian pencil P be defined as
P( s)
L
M
N
M
100 0.
01 s 0
0. 01 0.
01 0 1
s 0 1
0
0 1 0 0
It Clements form follows by typing
O
P
Q
P
P = [100 -0.01 s 0; -0.01 -0.01 0 1; -s 0 -1 0; 0 1 0 0];
C = pzer(clements(P))
C =
0 0 0 10 - s
0 -1 0 -3.5e-005 -
0.0007s
0 0 1 -0.014 + 0.00071s
10 + s -3.5e-005 + 0.0007s -0.014 - 0.00071s 99
The pzer function is included in the command to clear small coefficients in the
result.
When working with matrix pencils the two-sided matrix pencil equation
A( s) X YB( s) C( s)
is sometimes encountered. A and B are square matrix pencils and C is a rectangular
pencil with as many rows as A and as many columns as B. If A and B have no
O
P
QP

common roots (including roots at infinity) then the equation has a unique solution
pair X, Y with both X and Y a constant matrix.
By way of example, let
Then
A = s+1;
B = [ s 0
1 2];
C = [ 3 4 ];
[X,Y] = plyap(A,B,C)
gives the solution
X =
Y =
1 0
-1 2
56

3 Discrete-time and two-sided
polynomial matrices
Introduction
Basic operations
Two-sided
polynomials
Leading and
trailing degrees
and coefficients
Polynomials in s and p introduced in Chapter 2, Polynomial matrices are usually
associated with continuous-time differential operators, acting on signals xt . Here
we introduce polynomials in z and 1
z , typically associated with discrete-time shift
operators acting on signals x t with integer t.
The advance shift z acts zxt xt1, the delay shift
1
inverses each of others zz 1.
57
1
z 1
acts z xt xt1
. They are
Many operations and functions work for discrete-time polynomials exactly as
described before in Chapter 2, Polynomial matrices. Here we present what is
different.
1
Both z and z can be used as an indeterminate variable for polynomials. Here z^-1
can be shortened to zi. In arithmetic formulae when only one of them is used, the
usual polynomial algebra is followed. However, when both of them are used, the
1
relation zz 1 is respected.
(3*z^3+4*z^4+5*z^5)*zi^2
ans =
3z + 4z^2 + 5z^3
It may happen that both powers of z and those of
(1+z)*(1+zi)
ans =
z^-1 + 2 + z
1
z remain in the result
In such a way, new objects, two-sided polynomials (tsp) are created. Their algebra is
naturally similar to that of polynomials.
Besides degree deg and leading coefficient lcoef, two-sided polynomials have
trailing degree tdeg and trailing coefficient tcoef:
T=[2*zi+1+3*z^3, 2*zi; zi^2, 1+z*2]
T =
2z^-1 + 1 + 3z^3 2z^-1
z^-2 1 + 2z
deg(T),lcoef(T)

Derivatives and
integrals
ans =
ans =
3
3 0
0 0
tdeg(T),tcoef(T)
ans =
ans =
-2
0 0
1 0
In deriv command, the derivation is taken with respect to the indeterminate
variable:
F=1+z+z^2;
deriv(F)
ans =
1 + 2z
G=1+z^-1+z^-2;
deriv(G)
ans =
1 + 2z^-1
For two-sided polynomials, unless stated otherwise, the derivative is taken with
respect to z:
T=z^-2+z^-1+1+z+z^2;
deriv(T)
ans =
-2z^-3 - z^-2 + 1 + 2z
We can also explicitly state the independent variable for the derivative
deriv(T,z)
ans =
-2z^-3 - z^-2 + 1 + 2z
deriv(T,zi)
ans =
2z^-1 + 1 - z^2 - 2z^3
The same holds for integrals. As the integral of 1 z is log z , we obtain the logarithmic
term separately:
[intT,logterm]=integral(T)
intT =
58

Roots and
stability
Norms
Conjugations
-z^-1 + z + 0.5z^2 + 0.33z^3
logterm =
z
The roots of polynomials in z and 1
z , unless stated otherwise, are understood in the
complex variable equal to the indeterminate:
roots(z-0.5)
ans =
0.5000
roots(zi-0.5)
ans =
0.5000
However, we can state the variable for roots explicitely:
>> roots(zi-0.5,z)
ans =
2.0000
The stability region for z is the interior of the unit disc
isstable(z-0.5)
ans =
while for
1
1
z it is the exterior of the unit disc
>> isstable(zi-0.5)
ans =
0
>> isstable(zi+2)
ans =
1
Polynomials in z and 1
z may be considered (by means of z-transform), as discretetime
signals of finite duration. The H2-norm of such a signal may be found
h2norm(1+z^-1+z^-2)
ans =
1.7321
while its H∞-norm is returned by typing
hinfnorm(1+z^-1+z^-2)
ans =
3
59

Conjugate
transpose
The operation of conjugate transposition is connected with scalar product of discrete-
1
time signals. Here the theory requires z z
1 or z z and, indeed, the
Polynomial Toolbox returns
or
z'
ans =
zi'
ans =
z^-1
z
For more complex polynomial matrices, the operation consists of the complex
conjugation, the transposition and the change of argument. So
C=[1+z 1;2 z]+i*[z 1;1 z]
C =
D=C'
D =
1+0i + (1+1i)z 1+1i
2+1i (1+1i)z
1+0i + (1-1i)z^-1 2-1i
1-1i (1-1i)z^-1
Hence, conjugate of a polynomial in z is a polynomial in
D'
ans =
1+0i + (1+1i)z 1+1i
2+1i (1+1i)z
For two-sided polynomials, the operation works alike
T=[z 2-z; 1+zi 1+z-zi]
T =
T'
ans =
z 2 - z
z^-1 + 1 -z^-1 + 1 + z
z^-1 1 + z
-z^-1 + 2 z^-1 + 1 – z
60
1
z vice versa2 2 Clearly, this operation with discrete-time polynomials is in Version 3 of the
Polynomial Toolbox implemented more naturally than before. On the other hand,
users familiar with older versions should beware of this incompatibility.

Symmetric
two-sided
polynomial
matrices
In particular, a two-sided polynomial matrix that equals it conjugate is called (para
Hermitian) symmetric. So is, for example,
M=[zi+2+z 3*zi+4+5*z;3*z+4+5*zi zi+3+z]
M =
M'
ans =
z^-1 + 2 + z 3z^-1 + 4 + 5z
5z^-1 + 4 + 3z z^-1 + 3 + z
z^-1 + 2 + z 3z^-1 + 4 + 5z
5z^-1 + 4 + 3z z^-1 + 3 + z
Expressed by matrix coefficients as
M z M z M M z ,
1
1
0 1
the matrix 0 M must be symmetric and the matrices M 1, M1
must be transposed each
to other. Indeed,
and
M{0}
ans =
M{-1}
ans =
M{1}
ans =
or, together,
2 4
4 3
1 3
5 1
1 5
3 1
pformat coef
M
Two-sided polynomial matrix in z: 2-by-2,degree: 1, tdegree: -1
M =
Matrix coefficient at z^-1 :
1 3
5 1
Matrix coefficient at z^0 :
2 4
4 3
Matrix coefficient at z^1 :
61

Zeros of
symmetric twosided
polynomial
matrices
1 5
3 1
In case of complex coefficients, the complex conjugacy steps also into the play: in the
above, "symmetric" is replaced by "conjugate symmetric" (Hermitian symmetric) and
"transposed" by "conjugate transpose" (Hermitian transpose).
pformat
M=[(1-i)*zi+2+(1+i)*z (4+5*i)+(6+7*i)*z;(4-5*i)+(6-7*i)*zi ...
(1-i)*zi+2+(1+i)*z]
M =
(1-1i)z^-1 + 2+0i + (1+1i)z 4+5i + (6+7i)z
(6-7i)z^-1 + 4-5i (1-1i)z^-1 + 2+0i + (1+1i)z
pformat coef
M
Two-sided polynomial matrix in z: 2-by-2,degree: 1, tdegree: -1
M =
Matrix coefficient at z^-1 :
1.0000 - 1.0000i 0
6.0000 - 7.0000i 1.0000 - 1.0000i
Matrix coefficient at z^0 :
2.0000 4.0000 + 5.0000i
4.0000 - 5.0000i 2.0000
Matrix coefficient at z^1 :
1.0000 + 1.0000i 6.0000 + 7.0000i
0 1.0000 + 1.0000i
A nonnegative definite symmetric two-sided polynomial matrix Mz with real
coefficients has its zeros distributed symmetrically with respect to the unit circle.
1 1
Thus if it has a zero z i , it generally has a quadruple of zeros zi , zi , zi , zi
, all of the
1
same multiplicity. If z i is real, then it is only a couple zi, zi . If z i is on the unit circle,
then it is also a couple zi, z i but it must be of even multiplicity. Finally, if zi 1 then
it is a singlet of even multiplicity.
zpplot(sdf(2*z^-2+6*z^-1+9+6*z+2*z^2)), grid
62

zpplot(sdf(2*z^-2+6*z^-1+9+6*z+2*z^2)), grid
zpplot(sdf(z^-2+2*sqrt(2)*z^-1+4+2*sqrt(2)*z+z^2)), grid)
zpplot(sdf(z^-2+4*z^-1+6+4*z+z^2)), grid
63

For polynomial matrices with complex coefficients, the picture is simpler: generally a
z z ,
couple ,
i i
zpplot(sdf((1-i)*z^-1+3+(1+i)*z)), grid
specially in the unit circle case a singlet with even multiplicity
a=sqrt(2)/2; zpplot(sdf(a*(1-i)*z^-1+2+a*(1+i)*z)), grid
64

Matrix equations with discrete-time and two-sided polynomials
Symmetric
bilateral matrix
equations
All the various matrix equations introduced in Chapter 2, Polynomial matrices are of
the same use for discrete-time and two-sided polynomials as well. Moreover, most of
them look identically for standard and discrete-time polynomials. Only equations
that include conjugation differ. Due to a different meaning of discrete-time
conjugation, discrete-time polynomials in different operators as well as two-sided
polynomials may encounter in such an equation.
A frequently encountered polynomial matrix equation with conjugation is
AXXA B
where AXare , polynomial matrices in z while A, Xare
1
polynomial matrices in z
1
and B is a symmetric two-sided polynomial matrix in z and z . As usually, A and
hence A is given, B is also given while X is to be computed along with its conjugate
X .
The Polynomial Toolbox function to solve this equation is conveniently named axxab.
Hence, given the polynomial matrix
A=[1+5*z z;0 1-2*z]
A =
1 + 5z z
0 1 - 2z
and the symmetric two-sided polynomial matrix
the call
B=[7*z^-1+24+7*z,0;0 4*z^-1-10+4*z]
B =
7z^-1 + 24 + 7z 0
0 4z^-1 - 10 + 4z
X=axxab(A,B)
X =
solves the equation
0.96 + 2.2z -0.49z
0.23 -1.2 + 1.7z
1 1
5z 1 0 15z z 7z 24 7z 0
X X
1 1
1
z 2z 1 0 1 2z
0 4z 10 4z
It is easy to check that
A'*X+X'*A-B
ans =
0 0
0 0
Another symmetric equation with conjugation is
65

Non-symmetric
equation
Discrete-time
spectral
factorization
XAAX B
where again AXare , polynomial matrices in z , A, Xare
polynomial matrices in
and B is a symmetric two-sided polynomial matrix. As before, A and hence A is
given and B is also given. The unknown X along with its conjugate X can be
computed using the Polynomial Toolbox function xaaxb
X=xaaxb(A,B)
X =
1 + 2.2z -0.2
0.2 - 0.4z -1 + 2z
Equations with conjugations may also be non-symmetric such as
axya b
1
where a is a given polynomial in z , a is a polynomial in z (also given as it is the
conjugate of a ) and b is a given two-sided polynomial, not necessarily symmetric.
The equation has two unknowns, xy, , both polynomials in z . Notice that the second
unknown appears in the equation in its conjugate, y which is actually a polynomial
1
in z . This equation is of a special use only and hence only a scalar version of its
solver is programmed in the Polynomial Toolbox function axyab . Given a polynomial
a=(1+2*z)*(1-3*z)
a =
1 - z - 6z^2
and a non-symmetric two-sided polynomial
b=z^-1 +2*z
b =
z^-1 + 2z
the solution is computed by calling
[x,y] = axyab(a,b)
x =
y =
0.006 - 0.24z + 0.071z^2
0.012 - 0.39z + 0.036z^2
Also special quadratic equations called spectral factorizations contain conjugations as
explained in Chapter 2, Polynomial matrices. In discrete-time spectral factorizations,
1
therefore, polynomials in z meet polynomials in z as well as two-sided
polynomials.
This happens in the spectral factorization
A z z X z X z
1 ( , ) (
) ( )
66
1
z

as well as in the spectral co-factorization
A z z X z X z .
1 ( , ) ( ) (
)
By way of illustration consider the symmetric polynomial
b =
roots(b)
-6z^-2 + 5z^-1 + 38 + 5z - 6z^2
3.0000
-2.0000
-0.5000
0.3333
Its spectral factorization follows by typing
x=spf(b)
x =
Indeed,
-1 + z + 6z^2
roots(x)
ans =
x*x'
ans =
-0.5000
0.3333
-6z^-2 + 5z^-1 + 38 + 5z - 6z^2
When working with polynomials in
typing
xi=pol(x*zi^deg(x))
xi =
Then again
xi*xi'
ans =
but, naturally,
6 + z^-1 - z^-2
-6z^-2 + 5z^-1 + 38 + 5z - 6z^2
roots(xi)
ans =
3.0000
-2.0000
1
z , you may convert the result in those simply by
67

Resampling
Sampling
period
Discrete-time and two-sided polynomials often describe, discrete-time signals of finite
duration. Such a way, a discrete-time signal (sequence)
n
f fttm can be expressed as
f f z f z f z f f z f z f z
m m1 1 2 n
m m1 1 0 1 2
n
1
where the powers of z actually serve as position makers. In particular, polynomials
1
in z describe causal signals that are zero for t 0 while polynomials in z
represent anti-causal signals zero for t 0 .
1
Every polynomial f in z or z bears a sampling period h. Unless specified
otherwise, default sampling period is used which equals 1. The sampling period can
be accessed and changed by function props
f=1+0.5*zi
f =
props(f)
1 + 0.5z^-1
POLYNOMIAL MATRIX
size 1-by-1
degree 1
PROPERTY NAME: CURRENT VALUE: AVAILABLE VALUES:
variable symbol z^-1 's','p','z^-1','d','z','q'
sampling period 1 0 for c-time, nonneg or [] for d-time
user data [] arbitrary
props(f,2)
POLYNOMIAL MATRIX
size 1-by-1
degree 1
PROPERTY NAME: CURRENT VALUE: AVAILABLE VALUES:
variable symbol z^-1 's','p','z^-1','d','z','q'
sampling period 2 0 for c-time, nonneg or [] for d-time
user data [] arbitrary
or, better by a structure-like notation using simply f.h
f=1+0.5*zi
68

Resampling of
polynomials in
z -1
Resampling
and phase
f =
f.h
ans =
f.h=2
f =
f.h
ans =
1 + 0.5z^-1
1
1 + 0.5z^-1
2
In arithmetic operations and polynomial equations solvers, all input polynomias must
be of the same sampling period; otherwise a warning message is issued.
1
Polynomials in z represent, by means of z-transform, causal discrete-time signals
of finite duration. They may, or may not, be considered as a result of sampling of
continuous-time signals. Despite of that, they can be "resampled" . For example,
resampling by 3 means taking only every third sample
f=1+0.9*zi+0.8*zi^2+0.7*zi^3+0.6*zi^4+0.5*zi^5+0.4*zi^6+0.3*
zi^7+0.2*zi^8
f =
1 + 0.9z^-1 + 0.8z^-2 + 0.7z^-3 + 0.6z^-4 + 0.5z^-5 +
0.4z^-6 + 0.3z^-7 + 0.2z^-8
g=resamp(f,3)
g =
g.h
ans =
1 + 0.7z^-1 + 0.4z^-2
3
1
In the resulting g, the indeterminate z still means the delay operator but the
delay, when measured in continuous-time units, is three times longer than that for f.
It is captured by the fact that g.h is three times f.h .
Resampling with ratio 3 can be performed with "phase" 0,1 or 2, the default being 0:
g0=resamp(f,3,0)
g0 =
1 + 0.7z^-1 + 0.4z^-2
g1=resamp(f,3,1)
g1 =
0.9 + 0.6z^-1 + 0.3z^-2
g2=resamp(f,3,2)
g2 =
0.8 + 0.5z^-1 + 0.2z^-2
69

Resampling of
two-sided
polynomials
Resampling of
polynomials
in z
Dilating
Two-sided polynomials are considered as signals in discrete time t which may be
nonzero both for t 0 and t 0 . According to the usual habit of z-transform, the
1
ordering is according to powers of z . This is followed in resampling of two-sided
polynomials with a ratio and a phase
T=1.3*z^3+1.2*z^2+1.1*z+1+0.9*zi+0.8*zi^2+0.7*zi^3
T =
0.7z^-3 + 0.8z^-2 + 0.9z^-1 + 1 + 1.1z + 1.2z^2 +
1.3z^3
resamp(T,3)
ans =
0.7z^-1 + 1 + 1.3z
resamp(T,3,1)
ans =
0.9 + 1.2z
resamp(T,3,2)
ans =
0.8 + 1.1z
Polynomials in z are considered as discrete-time signals for t 0 (anticausal). To be
compatible with the above described cases, their resampling follows
W=1.5*z^5+1.4*z^4+1.3*z^3+1.2*z^2+1.1*z+1
W =
1 + 1.1z + 1.2z^2 + 1.3z^3 + 1.4z^4 + 1.5z^5
resamp(W,3)
ans =
1 + 1.3z
resamp(W,3,1)
ans =
1.2z + 1.5z^2
resamp(W,3,2)
ans =
1.1z + 1.4z^2
Dilating is a process in some sense inverse to resampling. With
g0,g1,g2
g0 =
g1 =
g2 =
1 + 0.7z^-1 + 0.4z^-2
0.9 + 0.6z^-1 + 0.3z^-2
0.8 + 0.5z^-1 + 0.2z^-2
70

computed before we have
and
f0=dilate(g0,3,0)
f0 =
1 + 0.7z^-3 + 0.4z^-6
f1=dilate(g0,3,1)
f1 =
z^-1 + 0.7z^-4 + 0.4z^-7
f2=dilate(g0,3,2)
f2 =
f=f0+f1+f2
f =
z^-2 + 0.7z^-5 + 0.4z^-8
1 + z^-1 + z^-2 + 0.7z^-3 + 0.7z^-4 + 0.7z^-5 + 0.4z^-6
+ 0.4z^-7 + 0.4z^-8
restores the original f . Note that
g0.h,f.h
ans =
ans =
3
1
71

4 The Polynomial Matrix Editor
Introduction
Quick start
The Polynomial Matrix Editor (PME) is recommended for creating and editing
polynomial and standard MATLAB matrices of medium to large size, say from about
4 4 to30 35.
Matrices of smaller size can easily be handled in the MATLAB
command window with the help of monomial functions, overloaded concatenation,
and various applications of subscripting and subassigning. On the other hand,
opening a matrix larger than 30 35 in the PME results in a window that is difficult
to read.
Type pme to open the main window called Polynomial Matrix Editor. This window
displays all polynomial matrices (POL objects) and all standard MATLAB matrices (2dimensional
DOUBLE arrays) that exist in the main MATLAB workspace. It also
allows you to create a new polynomial or standard MATLAB matrix. In the Polynomial
Matrix Editor window you can
create a new polynomial matrix, by typing its name and size in the first (editable)
line and then clicking the Open button
modify an existing polynomial matrix while retaining its size and other
properties: To do this just find the matrix name in the list and then double click
the particular row
modify an existing polynomial matrix to a large extent (for instance by changing
also its name, size, variable symbol, etc.): To do this, first find the matrix name in
the list and then click on the corresponding row to move it up to the editable row.
Next type in the new required properties and finally click Open
Each of these actions opens another window called Matrix Pad that serves for editing
the particular matrix.
In the Matrix Pad window the matrix entries are displayed as boxes. If an entry is too
long so that it cannot be completely displayed then the corresponding box takes a
slightly different color (usually more pinkish.)
To edit an entry just click on its box. The box becomes editable and large enough to
display its entire content. You can type into the box anything that complies with the
MATLAB syntax and that results in a scalar polynomial or constant. Of course you can
use existing MATLAB variables, functions, etc. The program is even more intelligent
and handles some notation going beyond the MATLAB syntax. For example, you can
drop the * (times) operator between a coefficient and the related polynomial symbol
(provided that the coefficient comes first). Thus, you can type 2s as well as 2*s.
72

Main window
Main window
buttons
Main window
menus
To complete editing the entry push the Enter key or close the box by using the mouse.
If you have entered an expression that cannot be processed then an error message is
reported and the original box content is recovered.
To bring the newly created or modified matrix into the MATLAB workspace finally
click Save or Save As.
The main Polynomial Matrix Editor window is shown in Fig. 2.
The main window contains the following buttons:
Open Clicking the Open button opens a new Matrix Pad for the matrix specified
by the first (editable) line. At most four matrix pads can be open at the same
time.
Refresh Clicking the Refresh button updates the list of matrices to reflect
changes in the workspace since opening the PME or since it was last refreshed.
Close Clicking the Close button terminates the current PME session.
The main PME window offers the following menus:
Menu List Using the menu List you can set the type of matrices listed in the main
window. They may be polynomial matrices (POL objects) – default – or standard
MATLAB matrices or both at the same time.
Menu Symbol Using the menu Symbol you can choose the symbol (such as s,z,...)
that is by default offered for the 'New' matrix item of the list.
List menu to
control what
matrices are
displayed
Editable row to
create or modify
properties
List of existing
matrices
Button to open
new matrix pad
Symbol menu to
change default
symbol
Fig. 2. Main window of the Polynomial Matrix Editor
73
Properties of
edited matrix
Button to close
PME

Matrix Pad window
Matrix Pad
buttons
To edit a matrix use one of the ways described above to open a Matrix Pad window for
it. This window is show in Fig. 3.
The Matrix Pad window consists of boxes for the entries of the polynomial matrix. If
some entry is too long and cannot be completely displayed in the particular box then
its box takes a slightly different color.
To edit an entry you can click on its box. The box pops up, becomes editable and large
enough to display its whole content. You can type into the box anything that complies
with the MATLAB syntax and results in a scalar polynomial or constant. Of course,
you can use all existing MATLAB variables, functions, etc. The editor is even a little
more intelligent: it handles some notation going beyond the MATLAB syntax. For
example, you can drop the * (times) operator between a coefficient and the related
polynomial symbol (provided that the coefficient comes first). Thus, you can type 2s
as well as 2*s. Experiment a little to learn more.
When editing is completed push the Enter key on your keyboard or close the box by
mouse. If you have entered an expression that cannot be processed then an error
message is appears and the original content of the box is retrieved.
When the matrix is ready bring it into the MATLAB workspace by clicking either the
Save or the Save As button.
The following buttons control the Matrix Pad window:
Save button: Clicking Save copies the Matrix Pad contents into the MATLAB
workspace.
Save As button: Clicking Save As copies the matrix from Matrix Pad into the
MATLAB workspace under another name.
Browse button: Clicking Browse moves the cursor to the main window (the same
as directly clicking there).
Close Button: Clicking Close closes the Matrix Pad window.
The Matrix Pad window consists of boxes for the entries of the polynomial matrix. If
some entry is too long and cannot be completely displayed then its box takes a
slightly different color.
To edit an entry you can click on its box. The box pops up, becomes editable and large
enough to display its whole content. You can type into the box anything that complies
with the MATLAB syntax and results in a scalar polynomial or constant. Of course,
you can use all existing MATLAB variables, functions, etc. The editor is even a little
more intelligent: it handles some notation going beyond the MATLAB syntax. For
example, you can drop the * (times) operator between a coefficient and the related
polynomial symbol (provided that the coefficient comes first). Thus, you can type 2s
as well as 2*s. Experiment a little to learn more.
When editing is completed push the Enter key on your keyboard or close the box by
mouse. If you have entered an expression that cannot be processed then an error
message is appears and the original content of the box is retrieved.
74

When the matrix is ready bring it into the MATLAB workspace by clicking either the
Save or the Save As button.
Box with
completely
displayed
contents
Box with
incompletely
displayed
contents
Buttons to save
the matrix
Name of edited
matrix
Button to
activate the
main window
Fig. 3. Matrix Pad with an opened editable box
75
Properties of
edited matrix
Opened editable
box
Button to close
Matrix Pad

5 Polynomial matrix fractions
Introduction
In some point of view, polynomials behave like integer numbers: they can be added
and multiplied but generally not divided. Instead of that, the concepts of divisibility,
greatest common divisors and last common multiples are constructed.
To be able to perform division of integer numbers, we create new objects: fractions
like1 2,2 3 , etc. Similarly, polynomial fractions and polynomial matrix fractions are
created.
In this chapter, matrix polynomial fractions are introduced and operations on them
are described. To satisfy as many needs of control engineers and theorists as possible,
four kinds of fractions are implemented in the Polynomial Toolbox: scalardenominator-fraction
(sdf), matrix-denominator-fraction (mdf), left-denominatorfraction
(ldf) and right-denominator-fraction (rdf). All of them are nothing more
than various forms of the same mathematical entities (fractions); each of them can be
converted to each other.
In systems, signals ad control, the polynomial matrix fractions are typically used to
describe special rational transfer matrices. For more details, see Chapter 5, LTI
systems.
The following general rule is adopted for all kinds of fractions. The standard MATLAB
matrices, polynomials and two-sided polynomials may contain Inf's or NaN's in their
entries. The fractions, however, may not. Moreover, the denominators must not be
zero or singular, as will be explained later for individual kinds of matrix polynomial
fractions.
Scalar-denominator-fractions
These objects have a matrix numerator and a scalar denominator. They usually arise
as inverses of (square nonsingular) polynomial matrices:
P=[1 s s^2; 1+s s 1-s;0 -1 -s]
P =
F=inv(P)
F =
1 s s^2
1 + s s 1 - s
0 -1 -s
-1 + s + s^2 0 -1s + s^2 + s^3
-1s - s^2 s 1 - s - s^2 - s^3
1 + s -1 s^2
----------------------------------------
-1 + s + s^2
76

The numerator and the denominator can be accessed or changed by F.num and F.den
F.num
ans =
F.den
ans =
-1 + s + s^2 0 -1s + s^2 + s^3
-1s - s^2 s 1 - s - s^2 - s^3
1 + s -1 s^2
-1 + s + s^2
They are also returned by special functions NUM and DEN:
num(F),den(F)
ans =
ans =
-1 + s + s^2 0 -s + s^2 + s^3
-s - s^2 s 1 - s - s^2 - s^3
1 + s -1 s^2
-1 + s + s^2
Recall that F.n and F.d are, up to a constant multiple, equal to adj(P) and det(P)
adj(P)
ans =
det(P)
ans =
1 - s - s^2 0 s - s^2 - s^3
s + s^2 -s -1 + s + s^2 + s^3
-1 - s 1 -1s^2
1 - s - s^2
Sdf objects also arise as negative powers of polynomial matrices
Q=[1 1;1-s s]
Q =
Q^-1
ans =
Q^-2
1 1
1 - s s
0.5s -0.5
-0.5 + 0.5s 0.5
-------------------
-0.5 + s
77

The sdf
command
Comparison of
fractions
ans =
0.25 - 0.25s + 0.25s^2 -0.25 - 0.25s
-0.25 + 0.25s^2 0.5 - 0.25s
----------------------------------------
and alike. It can be verified that
Q*Q^-1
ans =
Q^2*Q^-2
ans =
and so on.
1 0
0 1
1 0
0 1
0.25 - s + s^2
Scalar-denominator-fractions may also be entered in terms of their numerator
matrices and denominator polynomials. For this purpose the sdf command is
available. Typing
N = [-1+s+s^2 0 -s+s^2+s^3;-s-s^2 s 1-s-s^2-s^3;1+s -1 s^2];
d = -1+s+s^2;
sdf(N,d)
for instance, defines the scalar-denominator-fraction
ans =
-1 + s + s^2 0 -s + s^2 + s^3
-s - s^2 s 1 - s - s^2 - s^3
1 + s -1 s^2
----------------------------------------
-1 + s + s^2
Note that for two fractions to be equal it is not necessary to have the same numerator
and the same denominator. Common polynomial factors may be present:
G=(s-1)*inv(s^2-1)
G =
-1 + s
--------
-1 + s^2
H=inv(1+s)
H =
1
-----
1 + s
78

Coprime
Reduce
General
properties
cop, red
G==H
ans =
1
Every fraction can be converted to the coprime case
K=coprime(G)
K =
1
-----
1 + s
If one does not care about common factors and only needs putting the leading (or
trailing) denominator coefficient to unity, the fraction may be reduced
Nct=sdf(1,2+3*s)
Nct =
1
------
2 + 3s
reduce(Nct)
ans =
0.33
--------
0.67 + s
Ndt=sdf(1,2+3*zi)
Ndt =
1
---------
2 + 3z^-1
reduce(Ndt)
ans =
0.5
-----------
1 + 1.5z^-1
Of course, an equal fraction must results so that only one of the denominator
coefficients can be made unity depending on the indeterminate.
Results of operations with fractions are generally not guaranteed to be coprime and
reduced, even if the operands are. It depends on algorithm used. Sometimes it is
convenient to require such cancelling and reducing after every operation; it can be
79

Reverse
managed by switching the Polynomial Toolbox into a special mode of operation that is
achieved by setting the general properties
gprops('cop','red')
The default state is restored by
gprops('ncop','nred')
To display the status (together with all other general properties), type
gprops
Global polynomial properties:
PROPERTY NAME: CURRENT VALUE: AVAILABLE VALUES:
variable symbol s 's','p','z^-1','d','z','q'
zeroing tolerance 1e-008 any real number from 0 to 1
verbose level no 'no', 'yes'
display format nice 'symb', 'symbs', 'nice', 'symbr',
'block'
80
'coef', 'rcoef', 'rootr', 'rootc'
display order normal 'normal', 'reverse'
coprime flag ncop 'cop', 'ncop'
reduce flag nred 'red', 'nred'
defract flag ndefr 'defr', 'ndefr'
discrete variable disz 'disz', 'disz^-1'
continuous variable conts 'conts', 'contp'
This option should be used carefully as it increases the computation time. The test for
coprimeness is influenced by the zeroing tolerance globally
or locally
tol=1.e-5;
gprops(tol)
coprime(G,tol);
The same fraction may be expressed both by z and by
other, reverse command is used:
Q=sdf(2-zi,1-.9*zi)
Q =
2 - z^-1
-----------
1 - 0.9z^-1
R=reverse(Q)
R =
Q==R
-1 + 2z
--------
-0.9 + z
1
z . To convert each form to

ans =
1
1
Be careful with using reverse with fractions in other operators than z or z . So for
1
a fraction in s , as no polynomials in s exist, the result is again in s , and Q==R no
more holds:
Q=(2+s)*inv(3+s)
Q =
2 + s
-----
3 + s
R=reverse(Q)
R =
Q==R
ans =
1 + 2s
------
1 + 3s
Matrix-denominator-fractions
0
These objects have both matrix numerator and matrix denominator and are
especially convenient for rational matrices as one can see separately numerators of
all rational entries in the matrix numerator and similarly all particular
denominators in the denominator matrix.
They usually arise from array (entry-by-entry) division of polynomial matrices:
N=[1,1;s,s-1],D=[s s-1; 2 s+1]
N =
D =
F=N./D
F =
1 1
s -1 + s
s -1 + s
2 1 + s
1 1
- ------
s -1 + s
81

The mdf
command
-1 + s
0.5s ------
1 + s
The numerator and the denominator can be accessed or changed by F.num and F.den
F.num,F.den
ans =
ans =
1 1
0.5s -1 + s
s -1 + s
1 1 + s
Scalar-denominator-fractions may also be entered in terms of their numerator
matrices
F = mdf([s+2 s+3], [s+3,s+1])
F =
2 + s 3 + s
----- -----
3 + s 1 + s
Needless to say that all entries of this denominator matrix must be nonzero. All
functions described above for sdf work for mdf as well.
Left and right polynomial matrix fractions
Mathematicians sometimes like to describe matrices of fractions (rational matrices)
as fractions of two polynomial matrices. So the 2x3 rational matrix
2
s s
1
Rs ()
1s 1s 1s
1 1 1
1s 1s 1s
can either be described by the polynomial matrices
as the so called right matrix fraction
or by other two polynomial matrices
1s00 2
s s
1
Ar( s)
0 1 s 0
, Br( s)
1 1 1
0 0 1
s
R( s) B ( s) A ( s)
82
r r
1

Rightdenominatorfraction
The rdf
command
as the so called left matrix fraction
1 s s 0 1
Al( s) , Bl( s)
0 1 s
1 1 1
R s A s B s .
1
( ) l( ) l(
)
Such fractions polynomial matrix fractions are now often used in automatic control
and other engineering fields. To handle the right and the left polynomial matrix
fractions, Polynomial Toolbox is equipped with two objects, rdf and ldf. These
objects have both matrix numerator and matrix denominator (the right one or the left
one).
These objects are usually entered using standard Matlab division operator (slash)
Ar=[1+s,0,0;0,1+s,0;0,0,1+s],Br=[s^2,-s,1;1,1,1]
Ar =
Br =
R=Br/Ar
R =
1 + s 0 0
0 1 + s 0
0 0 1 + s
s^2 -s 1
1 1 1
s^2 -s 1 / 1 + s 0 0
1 1 1 / 0 1 + s 0
/ 0 0 1 + s
Right-denominator-fractions may also be entered in terms of their numerator and
right denominator matrices
F = rdf(Br,Ar)
F =
s^2 -s 1 / 1 + s 0 0
1 1 1 / 0 1 + s 0
/ 0 0 1 + s
Note the order of input arguments that come as they appear in the right fraction.
Needless to say that here the denominator matrix must be here nonzero but maz
eventually possess zero entries.
83

Leftdenominatorfraction
The ldf
command
These objects are usually entered using standard Matlab left division operator
(backslash)
Al=[1 s;0 s+1],Bl=[s 0 1; 1 1 1],G=Al\Bl
Al =
Bl =
G =
1 s
0 1 + s
s 0 1
1 1 1
1 s \ s 0 1
0 1 + s \ 1 1 1
Left-denominator-fractions may also be entered in terms of their numerator and left
denominator matrices
>> G = ldf(Al,Bl)
G =
1 s \ s 0 1
0 1 + s \ 1 1 1
Note the order of input arguments, which come as they appear in the left fraction.
Needless to say that here the denominator matrix must be here nonzero but may
eventually possess zero entries.
Mutual conversions of polynomial matrix fraction objects
Polynomial matrix fraction objects can be mutually converted by means of their basic
constructors sdf, mdf, rdf, and ldf. For example,
N = [-1+s 0 -s+s^3;-s-s^2 s 1-s]; d = -1+s+s^2;
Gsdf=sdf(N,d)
Gsdf =
-1 + s 0 -s + s^3
-s - s^2 s 1 - s
--------------------------
>> class(Gsdf)
ans =
sdf
-1 + s + s^2
can be converted as follows
84

Gmdf=mdf(Gsdf)
Gmdf =
-1 + s 0 -s + s^3
------------ ------------ ------------
-1 + s + s^2 -1 + s + s^2 -1 + s + s^2
-s - s^2 s 1 - s
------------ ------------ ------------
-1 + s + s^2 -1 + s + s^2 -1 + s + s^2
class(Gmdf)
ans =
mdf
Grdf=rdf(Gsdf)
Grdf.numerator =
-1 + s 0 -s + s^3
-s - s^2 s 1 - s
Grdf.denominator =
-1 + s + s^2 0 0
0 -1 + s + s^2 0
0 0 -1 + s + s^2
class(Grdf)
ans =
rdf
Gldf=ldf(Gsdf)
Gldf =
-1 + s + s^2 0 \ -1 + s 0 -s + s^3
0 -1 + s + s^2 \ -s - s^2 s 1 - s
class(Gldf)
ans =
ldf
Of course, the real entity behind remains the same so that the "contents" of all the
various expressions equal
Gsdf==Gmdf
ans =
1 1 1
1 1 1
85

Gmdf==Grdf
ans =
Grdf==Gldf
ans =
1 1 1
1 1 1
1 1 1
1 1 1
The user should keep in mind, however, that the conversions above may sometimes
require heavy commutations and can therefore be both time-consuming and subject
numerical inaccuracies. Definitely, they should be avoided if possible.
Operations with polynomial matrix fractions
Addition,
subtraction and
multiplication
In the Polynomial Toolbox polynomial matrix fractions are objects for which all
standard operations are defined. Different object types may usually enter an
operation.
Define the polynomial fractions
F=s^-1,G=s/(1+s)
F =
G =
1
-
s
s / 1 + s
The sum and product follow easily:
F+G
ans =
G+F
ans =
F*G
1 + s + s^2
-----------
s + s^2
1 + s + s^2 / s + s^2
86

ans =
G*F
ans =
s
-------
s + s^2
s / s + s^2
For polynomial matrix fractions
a=1./s;b=1./(s+1);F=[1 a; a+b b],G=[b a;a b]
F =
G =
1
1 -
s
1 + 2s 1
------- -----
s + s^2 1 + s
1 1
----- -
1 + s s
1 1
- -----
s 1 + s
the operations work as expected
F+G
ans =
F*G
2 + s 2
----- -
1 + s s
2 + 3s 2
------- -----
s + s^2 1 + s
87

Entrywise and
matrix division
ans =
1 + s + s^2 2 + s
----------- -------
s^2 + s^3 s + s^2
2 + 3s 1 + 3s + 3s^2
-------------- ----------------
s + 2s^2 + s^3 s^2 + 2s^3 + s^4
Entrywise division of polynomial matrix fractions
T = F./G
results in another fraction
ans =
s
1 + s -
s
s + 2s^2 1 + s
-------- -----
s + s^2 1 + s
while matrix division yields
F/G
ans =
F\G
ans =
0.5s + s^2 - 0.5s^4 0.5s^3 + 0.5s^4
------------------- ---------------
0.5s + s^2 0.5s + s^2
-0.5s^3 - 0.5s^4 0.5s + 2s^2 + 2s^3 + 0.5s^4
------------------- ---------------------------
0.5s + 1.5s^2 + s^3 0.5s + 1.5s^2 + s^3
-1s - 2s^2 0
--------------------- ---------------------
-s - 3s^2 - s^3 + s^4 -s - 3s^2 - s^3 + s^4
88

Concatenation
and working
with
submatrices
s^4 -1s - 3s^2 - s^3 + s^4
--------------------- ----------------------
-s - 3s^2 - s^3 + s^4 -1s - 3s^2 - s^3 + s^4
All standard MATLAB operations to concatenate matrices or selecting submatrices
also apply to polynomial matrix fractions. Typing
W = [F G]
results in
W =
1 1 1
1 - ----- -
s 1 + s s
1 + 2s 1 1 1
------- ----- - -----
s + s^2 1 + s s 1 + s
The last row of W may be selected by typing
w = W(2,:)
w =
3 4 s^3 1
Submatrices may be assigned values by commands such as
w =
1 + 2s 1 1 1
------- ----- - -----
s + s^2 1 + s s 1 + s
The user should be aware that, depending on fraction type used, even concatenation
and submatrix selection may require extensive computing when working with
fraction
L=[s 1; 1 s+1]\[1;1]
L =
L(1)
ans =
s 1 \ 1
1 1 + s \ 1
-1 + s + s^2 \ s
89

Coefficients and
coefficient
matrices
Conjugation
and
transposition
Coefficient of a fraction is defined as the coefficient in its Laurent series
>> l=(z-1)^-1
l =
1
------
-1 + z
>> l{-1}
ans =
1
>> l{-2}
ans =
1
>> l{-2:0}
ans =
1 1 0
Conjugation and transposition work as expected for fractions
T=[a b;a 0]
T =
T'
1 1
- -----
s 1 + s
1
- 0
s
ans =
T.'
ans =
1 1
-- --
-s -s
1
----- 0
1 - s
90

Signals and transforms
1 1
- -
s s
1
----- 0
1 + s
Sampling, unsampling and resampling
Polynomial matrix fractions, transfer functions and transfer function matrices
Transfer
functions
To describe transfer function of a single-input single-output linear (SISO) time
invariant system or transfer function matrix of a multi-input and/or multi-output
linear time invariant system, any of the four polynomial matrix fraction objects, sdf,
mdf, ldf or rdf, can be used. The particular object choice may depend of the form of
input data, on a task to be solved or operations to be performed or just on a personal
preference in displaying the polynomial fractions. As there are numerous possibilities
to create a proper object for a given transfer function, we will just show several
examples.
SISO system transfer function is usually given in the form of two polynomials, one for
the numerator and one for the denominator. Typically, these polynomials are entered
first and then used to create a fraction. So all one must do to form the transfer
function
is to type
a=1+s+2*s^2,b=s-s^2,f1=sdf(b,a)
a =
b =
f1 =
or
1 + s + 2s^2
s - s^2
s - s^2
------------
1 + s + 2s^2
f2=b/a
2
ss f () s
1s2s 91
2

Transfer
function
matrices
or
f2 =
f3=a\b
f3 =
0.5s - 0.5s^2 / 0.5 + 0.5s + s^2
0.5 + 0.5s + s^2 \ 0.5s - 0.5s^2
etc. For simple transfer functions, one can directly type
g=1/s,h=(1-s)\s
g =
h =
1 / s
-1 + s \ -s
Transfer functions of discrete-time systems are created similarly using
polynomials in "discrete-time" indeterminates:
f=sdf(z,1+z)
f =
z
-----
1 + z
g=1/(2-z)
g =
-1 / -2 + z
and alike. For simple transfer functions, one can directly type
Transfer function matrices of MIMO transfer function matrices of MIMO systems
can either be created from individual entries or via two polynomial matrices. So to
create an object for the transfer function matrix
one types 3 entry-by-entry
F=[1./s s./(s-1);1 s./(s+1) ]
F =
1 s
1s s s1
Fs ()
1
1
s 1
3 Note how the final matrix object type is defined by the particular choice of the
scalar object types.
92

Transfer
functions from
state space
model and vice
versa
or, equivalently
- ------
s -1 + s
s
1 -----
1 + s
F=[1 s; 1 1]./[s s-1;1 s+1]
F =
1 s
- ------
s -1 + s
1
1 -----
1 + s
Transfer functions of systems given in the state space model can be created
directly by either of the polynomial matrix fraction constructors. So for
1 x
1
0 1 x u
1
2
1 y
0 0 0 x u
1
0
one first creates the state-space model matrices
A =[-1 0;1 1], B = [1 ;2], C = [1 0;0 1], D =[0; 0]
A =
B =
C =
D =
-1 0
1 1
1
2
1 0
0 1
0
93

0
and then just types
F=mdf(A,B,C,D)
F =
-1 + s
--------
-1 + s^2
3 + 2s
--------
-1 + s^2
When the D matrix is zero, it need not be created and typed in
F=mdf(A,B,C)
F =
-1 + s
--------
-1 + s^2
3 + 2s
--------
-1 + s^2
The reversed process of computing state-space matrices from a given transfer
function matrix is accomplished by function abcd:
[a,b,c,d]=abcd(F)
a =
b =
c =
d =
0 1.0000
1.0000 0
1
0
1.0000 -1.0000
2.0000 3.0000
0
0
94

Note that the resulting matrices differ from those above. This process of so called
"state space realization" is not unique at all. The function actually returns a
particular realization in so called "observer form."
Another indeterminate, different form s, can be specified when creating transfer
function form the state space matrices. So to for a discrete-time system
x
1
1
0 1
x
1 u
2
1 yk
0 0
xk
1
k 1
k k
one creates transfer function matrix in z by
G=mdf(A,B,C,'z')
G =
-1 + z
--------
-1 + z^2
3 + 2z
--------
-1 + z^2
or transfer function matrix in
H=mdf(A,B,C,'zi')
H =
z^-1 - z^-2
-----------
1 - z^-2
2z^-1 + 3z^-2
-------------
1 - z^-2
1
z by
Conversions of polynomial matrix fraction objects to and from other Matlab objects
Polynomial Toolbox can effectively cooperate with other Matlab toolboxes. In
particular, its objects can directly be converted 4 to the object of Control System
Toolbox and Symbolic Math Toolbox.
4 Of course, this functionality only works if the other toolbox is installed.
95

Conversion to
the Control
System Toolbox
Objects
Any polynomial matrix fraction object is directly converted into any Control System
Toolbox LTI object via overloaded constructors5 of the Control System Toolbox. This
functionality will become clear from the following examples.
So the polynomial fraction
F=s/(s+1)
F =
s / 1 + s
is converted into the state-space object of the Control System Toolbox
Fss=ss(F)
a =
b =
c =
d =
x1
x1 -1
u1
x1 1
x1
y1 -1
u1
y1 1
Continuous-time model.
or into the transfer function object of the Control System Toolbox
Ftf=tf(F)
Transfer function:
s
-----
s + 1
5 To learn more about LTI objects, consult the Control Systems Toolbox manual or
use Matlab help window.
96

or even into the zero-pole-gain object of the Control System Toolbox
Fzpk=zpk(F)
Zero/pole/gain:
s
-----
(s+1)
For MIMO systems it works as well
G=mdf(A,B,C)
G =
Gss=ss(G)
a =
b =
c =
d =
-1 + s
--------
-1 + s^2
3 + 2s
--------
-1 + s^2
x1 x2
x1 0 1
x2 1 0
u1
x1 1
x2 0
x1 x2
y1 1 -1
y2 2 3
u1
97

y1 0
y2 0
Continuous-time model.
Gtf=tf(G)
Transfer function from input to output...
s - 1
#1: -------
s^2 - 1
2 s + 3
#2: -------
s^2 - 1
Gzpk=zpk(G)
Zero/pole/gain from input to output...
(s-1)
#1: -----------
(s-1) (s+1)
2 (s+1.5)
#2: -----------
(s-1) (s+1)
During the conversion, one need not care about the discrete/continuous-time nature
as well as other data are handled accordingly
Hss=ss(H)
a =
b =
x1
x1 1
u1
x1 1
98

c =
d =
x1
y1 1
u1
y1 1
Sampling time: 1
Discrete-time model.
Htf=tf(H)
Transfer function:
z
-----
z - 1
Sampling time: 1
Hzpk=zpk(H)
Zero/pole/gain:
z
-----
(z-1)
Sampling time: 1
This unlimited possibility to convert objects allows the Polynomial Toolbox users
employing almost directly all the numerous functions of the Control Systems Toolbox.
Sometimes, one even need not type the convertor at all as the object is converted
automatically. So for
F=1/s
F =
1 / s
H=s./(s-1)
H =
s
99

Conversion
from the
Control System
Toolbox
Objects
typing
------
-1 + s
bode(F),nyquist(H),step(F),impulse(H)
returns expected results but
bode(F,H)
??? Error using ==> lti.bode at 95
In frequency response commands, the frequency grid must be a
vector of
positive real numbers.
Error in ==> frac.bode at 16
bode(ss(F),varargin{:});
fails. To be on the safe side, one should always type the convertor name in such as
bode(ss(F),tf(H))
which work well.
Quite naturally, any LTI object of the Control System Toolbox can be converted
into a matrix fraction object whenever a convenient object is available6 in the
Polynomial Toolbox. The conversion is accomplished by typing any of the
polynomial matrix fraction constructor's names. For example,
F=1./(s-1);Fss=ss(F)
a =
b =
c =
x1
x1 1
u1
x1 1
x1
y1 1
6 For example, the conversion of a continuous-time system having nonzero time delay
is not possible for obvious reasons.
100

d =
u1
y1 0
Continuous-time model.
sdf(Fss)
ans =
1
------
-1 + s
ldf(Fss)
ans =
or
-1 + s \ 1
H = zpk( {[];[2 3]} , {1;[0 -1]} , [-5;1] )
Zero/pole/gain from input to output...
-5
#1: -----
(s-1)
(s-2) (s-3)
#2: -----------
Hmdf=mdf(H)
Hmdf =
-5
------
-1 + s
s (s+1)
6 - 5s + s^2
------------
s + s^2
101

Conversion to
the Symbolic
Math Toolbox
Hrdf=rdf(H)
Hrdf =
-5s - 5s^2 / -1s + s^3
-6 + 11s - 6s^2 + s^3 /
or, similarly, for discrete-time systems
G=z/(z+1);Gtf=tf(G)
Transfer function:
z
-----
z + 1
Sampling time: 1
sdf(Gtf)
ans =
z
-----
1 + z
ldf(Gtf)
ans =
1 + z \ z
Polynomials and polynomial fractions can also converted into the symbolic objects
of the Symbolic Math Toolbox. A polynomial such as
p=1+s
p =
1 + s
can be converted by
psym=sym(p)
psym =
s + 1
class(psym)
ans =
sym
To avoid eventual confusion when working with 'polynomial' and 'symbolic'
indeterminates during one Matlab session, one may wish to produce the symbolic
polynomial having a pre-specified indeterminate symbol, usually different form
102

that used in polynomials. For example, if you want to produce a symbolic
polynomial in capital S, just type
pSym=sym(p,'S')
pSym =
S + 1
Polynomial fractions can be converted similarly
f=1./p
f =
1
-----
1 + s
fsym=sym(f)
fsym =
1/(s + 1)
class(fsym)
ans =
sym
fSym=sym(f,'S')
fSym =
1/(S + 1)
Everything works as expected even for matrices of polynomials
P=[s 1; s+1 0]
P =
s 1
1 + s 0
Psym=sym(P)
Psym =
[ s, 1]
[ s + 1, 0]
Psym=sym(P,'S')
Psym =
[ S, 1]
[ S + 1, 0]
and fractions
F=mdf(1/P)
F =
103

Conversion
from the
Symbolic Math
Toolbox
0 1
----- -----
1 + s 1 + s
1 + s -1s
----- -----
1 + s 1 + s
Fsym=sym(F)
Fsym =
[ 0, 1/(s + 1)]
[ 1, -s/(s + 1)]
>> FSym=sym(F,'S')
FSym =
[ 0, 1/(S + 1)]
[ 1, -S/(S + 1)]
Backward conversion of symbolic polynomials, polynomial matrices,
polynomial fractions and polynomial matrix fractions into the corresponding
Polynomial Toolbox object is also possible. However, the source symbolic object
must possess as its indeterminate one of the indeterminates supported by the
Polynomial Toolbox, that is 's','p','z','q','d','zi','z^-1'. So for
symbolic objects defined as
syms s z
psym=1+s,qsym=1-z,fsym=1/psym,gsym=1/qsym
psym =
s + 1
qsym =
1 - z
fsym =
1/(s + 1)
gsym =
-1/(z - 1)
are their Polynomial Toolbox counterparts
pol(psym)
ans =
1 + s
104

pol(qsym)
ans =
1 - z
sdf(fsym)
ans =
1
-----
1 + s
ldf(gsym)
ans =
-1 + z \ -1
Symbolic matrix objects are converted accordingly
class(s)
ans =
sym
pol([s 1])
ans =
s 1
class(ans)
ans =
pol
sdf([1/s s/(1+s); 1 s])
ans =
1 + s s^2
s + s^2 s^2 + s^3
---------------------
s + s^2
mdf([1/s s/(1+s); 1 s])
ans =
1 s
- -----
s 1 + s
1 s
105

class(ans)
ans =
mdf
106

6 Conclusions
This is just a first version of the Manual. Future corrections, modifications and
additions are expected.
107

A
abcd · 65
addition · 10, 22, 57
adj · 24
adjoint · 24
advanced operations and functions · 14
axb · 16
axbyc · 3, 6, 14, 16
B
backslash · 55
basic operations on polynomial matrices · 22
Bézout equations · 15
C
canonical forms · 9
clements · 26
Clements form · 26
coefficient · 61
coefficient matrices · 13, 19, 20
coefficients · 13, 19, 20
column degrees · 9
column reduced · 10
common
divisor · 2
common left multiple · 7
common right divisor · 7
common right multiple · 6
compan · 30
companion matrix · 29
concatenation · 12, 19, 60
Conjugate transpose · 31
conjugation · 13, 62
Conjugation · 61
constant
matrices · 29
Control System Toolbox · 66, 71
conversion
polynomial matrix fraction objects · 55
Conversions · 66
cop · 50
coprime · 3
left · 5
Coprime · 50
108
D
default indeterminate variable · 18
deg · 9
degree · 9
deriv · 29
descriptor system · 25
det · 23
determinant · 23
Diophantine equation · 13
Diophantine equations · 13
Discrete-time spectral factorization · 37
display format · 9
division · 2
entrywise · 59
matrix · 59
with remainder · 2
divisor · 30
right · 7
E
echelon · 12
echelon form · 12
entering polynomial matrices · 16
equation solvers · 17
Euclidean division · 2
F
factor · 2
freeing variables · 18
G
General properties · 50
gensym · 18
gld · 3, 6
greatest common
divisor · 2
left divisor · 5
right divisor · 7
H
help · 8

hermite · 11
Hermite form · 11
how to define a polynomial matrix · 9
hurwitz · 30
Hurwitz matrix · 29
Hurwitz stability · 28
I
indeterminate variables · 16
infinite roots · 27
initialization · 8
integral · 29
invariant polynomials · 13
inverse · 23
isequal · 4
isfullrank · 25, 10
isprime · 3, 6
issingular · 26
isstable · 28
K
Kronecker canonical form · 25
L
Laurent series · 61
lcoef · 9
ldf · 55
ldiv · 5
leading column coefficient matrix · 9
leading row coefficient matrix · 9
least common multiple · 3
least common right multiple · 6
Left-denominator-fraction · 55
llm · 3
lrm · 7
LTI · 66, 71
lu · 11
M
matrix denominator · 52
matrix division · 4, 59
matrix division with remainder · 4
matrix divisors · 4
matrix multiples · 4
matrix numerator · 52
matrix pencil · 25
109
matrix pencil Lyapunov equations · 26
matrix pencil routines · 24
matrix polynomial equations · 15
mdf · 53
minbasis · 27
minimal basis · 27
Moore-Penrose pseudoinverse · 25
multiple · 30
left · 7
multiplication · 10, 22, 57
N
null · 26
null space · 25, 26
O
object
LTI · 66
one-sided equations · 16
P
para-Hermitian · 21
pencan · 25
pinit · 8
pinv · 25
plyap · 27
pol · 17, 49
polynomial
basis · 27
polynomial matrix editor · 43
main window · 43, 44
matrix pad · 43
polynomial matrix Editor
matrix pad · 45
polynomial matrix equations · 13, 36
polynomial matrix fraction
computing with · 57
Polynomial matrix fractions · 47
Popov form · 12
prand · 4
prime
relatively · 3
product · 22
pseudoinverse · 25

Q
quick start · 8
quotient · 2
R
range · 26
rank · 25, 26
normal · 26
rdf · 54
red · 50
Reduce · 50
reduced forms · 9
Resampling · 39
reverse · 51
right matrix fraction · 53
Right-denominator-fraction · 54
roots · 27
roots · 27
row degrees · 9
row reduced
form · 10
rowred · 10
S
Sampling period · 39
scalar denominator · 47
Scalar-denominator-fraction · 47
sdf · 49
simple operations with polynomial matrices · 10,
57
slash · 54
smith · 13
Smith form · 12
span · 26
spcof · 22
spectral co-factorization · 21, 38
110
spectral factorization · 21, 37
spf · 22
stability · 27
staircase form · 10
startup · 8
state space model · 64
state space system · 24
submatrices · 12, 19, 60
subtraction · 10, 22, 57
sum · 22
sylv · 30
Sylvester matrix · 29
symbolic
polynomial · 75
Symbolic Math Toolbox · 73, 75
T
time delay · 71
Transfer function · 62
transformation
to Clements form · 26
to Kronecker canonical form · 25
transpose · 14
transposition · 13, 61, 62
tri . · 10
triangular form · 10, 11
tutorial · 16, 28, 47
two-sided equations · 16
U
unimodular · 23
Z
zeros · 27, 33
zpplot · 28