systems, controllability, and observability are presented in this chapter. Basic'design
metl~odsbased on state-feedback control are given in Chapter 12.
Outline of the Chapter. Section 11-1 has presented an introduction to state-space
analysis of control systems. Section 11-2 deals with the state-space representation of
transfer-function systems. Here we present various canonical forms of state-space equations. Section 11-3 discusses the transformation of system models (such as from transfer function to state-space models, and vice versa) with MATLAB. Section 11-4 presents
the solution of time-invariant state equations. Section 11-5 gives some useful results in
vector-matrix analysis that are necessary in studying the state-space analysis of control
systems. Section 11-6 discusses the controllability of control systems and Section 11-7
treats the observability of control systems.
1 1-2 STATE-SPACE REPRESENTATIONS OF
TRANSFER-FUNCTION SYSTEMS
Many techniques are available for obtaining state-space representations of
transfer-function systems. In Chapter 3 we presented a few such methods. This section
presents state-space representations in the controllable, observable, diagonal, or Jordan
canonical form. (Methods for obtaining such state-space representations from transfer
functions are discussed in Problems A-11-1 through A-11-4.)
State-Space Representation in Canonical Forms. Consider a system defined
by
+ a l (fl-11
y + . . . + n,-,y + a,,y
( ~ ~ 1
y
(/I)
=
bULL
f
(n- 1 )
LL
b1
+ . * . + b , - , i ~+ b,,u
(11-1)
where LL is the input and y is the output. This equation can also be written as
In what follows we shall present state-space representations of the system defined by
Equation (11-1) or (11-2) in controllable canonical form, observable canonical form, and
diagonal (or Jordan) canonical form.
Controllable Canonical Form. The following state-space representation is called
a controllable canonical form:
Section 11-2
/ State-Space Representations of Transfer-Function Systems
albol
[il
xn
The controllable canonical form is important in discussing the pole-placement approach
to the control systems design.
Observable Canonical Form. The following state-space representation is called
an observable canonical form:
Note that the n X n state matrix of the state equation given by Equation (11-5) is the
transpose of that of the state equation defined by Equation (11-3).
Diagonal Canonical Form. Consider the transfer function system defined by Equation (11-2). Here we consider the case where the denominator polynomial involves
only distinct roots. For the distinct-roots case, Equation (11-2) can be written as
The diagonal canonical form of the state-space representation of this system is given by
Chapter 1 1 / Analysis of Control Systems in State Space
Jordan Canonical Form. Next we shall consider the case where the denominator
polynomial of Equation (11-2) involves multiple roots. For this case, the preceding
diagonal canonical form must be modified into the Jordan canonical form. Suppose, for
example, that the p,'s are different from one another, except that the first three p,'s are
equal, or p, = p2 = p,. Then the factored form of Y(s)/U(s) becomes
Y(s)
- -
busn+ blsn-'
+ ... + bn-,s + bn
(s + pJ3(s + p4)(s + p5) ... (s + p,)
The partial-fraction expansion of this last equation becomes
U(s)
A state-space representation of this system in the Jordan canonical form is given by
Section 1 1-2 / State-Space Representations of Transfer-Function Systems
EXAMPLE 1 1-1
Consider the system given by
s +3
Y(s)s2
+
3s + 2
U(s)
Obtain state-space representations in the controllable canonical form, observable canonical form,
and diagonal canonical form.
--
I
Controllable Canonical Form:
Observable Canonical Form:
Diagonal Canonical Form:
Eigenvalues of an n x n Matrix A. The eigenvalues of an n
roots of the characteristic equation
X
n matrix A are the
lhI - A/ = 0
The eigenvalues are also called the characteristic roots.
Consider, for example, the following matrix A:
The characteristic equation is
h.-1
\ A 1 - A1 = 0 A
6 11
O
-1
h+6
The eigenvalues of A are the roots of the characteristic equation, or -1, -2, and -3.
Diagonalization of n x n Matrix. Note that if an n
eigenvalues is given by
Chapter 1 1 / Analysis of Control Systems in State Space
X
n matrix A with distinct
the transformation x = Pz, where
A,, A 2 , . .. , A,, = n distinct eigenvalues of A
will transform P-' AP into the diagonal matrix, or
If the matrix A defined by Equation (11-12) involves multiple eigenvalues, then
diagonalization is impossible. For example, if the 3 X 3 matrix A, where
has the eigenvalues A,, A,, A,, then the transformation x = Sz, where
will yield
This is in the Jordan canonical form.
Section 11-2
/ State-Space Representations of Transfer-FunctionSystems
EXAMPLE 1 1-2
Consider the following state-space representation of a system.
Equations (11-13) and (11-14) can be put in a standard form as
where
A
=
[:
i
'1 .=[!I,
c=Li
0 01
-6 -11 -6
The eigenvalues of matrix A are
Thus, three eigenvalues are distinct. If we define a set of new state variables zl, zz, and z3 by the
transformation
or
where
then, by substituting Equation (11-17) into Equation (11-15), we obtain
By premultiplying both sides of this last equation by P-I,
we get
Chapter 1 1 / Analysis of Control Systems in State Space
Simplifying gives
Equation (11-20) is also a state equation that describes the same system as defined by Equation
(11-13).
The output equation, Equation (11-16), is modified to
Notice that the transformation matrix P, defined by Equation (11-18), modifies the coefficient
matrix of z into the diagonal matrix. As is clearly seen from Equation (11-20), the three scalar state
equations are uncoupled. Notice also that the diagonal elements of the matrix P-' AP in Equation
(11-19) are identical with the three eigenvalues of A. It is very important to note that the eigenvalues of A and those of P-I AP are identical. We shall prove this for a general case in what follows.
Invariance of Eigenvalues. To prove the invariance of the eigenvalues under a
linear transformation, we must show that the characteristic polynomials IAI - A / and
(A1 are identical.
Since the determinant of a product is the product of the determinants, we obtain
P-'API
Noting that the product of the determinants
we obtain
uct
IP-~PI,
IAI
-
IP-'~
P - ~ A P=~
and /PIis the determinant of the prod-
IP-~P~/AI
- A(
= /A1 - A1
Thus, we have proved that the eigenvalues of A are invariant under a linear
transformation.
Nonuniqueness of a Set of State Variables. It has been stated that a set of state
variables is not unique for a given system. Suppose that x , , x,, . .. , x, are a set of state
variables.Then we may take as another set of state variables any set of functions
Section 11-2 / State-Space Representations of Transfer-Function Systems
759