State space courses

2.14 Analysis and Design of Feedback Control Systems
State-Space Representation of LTI Systems
Derek Rowell
October 2002
1 Introduction
The classical control theory and methods (such as root locus) that we have been using in
class to date are based on a simple input-output description of the plant, usually expressed
as a transfer function. These methods do not use any knowledge of the interior structure of
the plant, and limit us to single-input single-output (SISO) systems, and as we have seen
allows only limited control of the closed-loop behavior when feedback control is used.
Modern control theory solves many of the limitations by using a much “richer” description
of the plant dynamics. The so-called state-space description provide the dynamics as a set
of coupled first-order differential equations in a set of internal variables known as state
variables, together with a set of algebraic equations that combine the state variables into
physical output variables.
1.1 Definition of System State
The concept of the state of a dynamic system refers to a minimum set of variables, known
as state variables, that fully describe the system and its response to any given set of inputs
[1-3]. In particular a state-determined system model has the characteristic that:
A mathematical description of the system in terms of a minimum set of variables
xi(t), i = 1, . . . , n, together with knowledge of those variables at an initial time t0
and the system inputs for time t ≥ t0, are sufficient to predict the future system
state and outputs for all time t > t0.
This definition asserts that the dynamic behavior of a state-determined system is completely
characterized by the response of the set of n variables xi(t), where the number n is defined
to be the order of the system.
The system shown in Fig. 1 has two inputs u1(t) and u2(t), and four output vari-
ables y1(t), . . . , y4(t). If the system is state-determined, knowledge of its state variables
(x1(t0), x2(t0), . . . , xn(t0)) at some initial time t0, and the inputs u1(t) and u2(t) for t ≥ t0 is
sufficient to determine all future behavior of the system. The state variables are an internal
description of the system which completely characterize the system state at any time t, and
from which any output variables yi(t) may be computed.
Large classes of engineering, biological, social and economic systems may be represented
by state-determined system models. System models constructed with the pure and ideal
(linear) one-port elements (such as mass, spring and damper elements) are state-determined
1

Figure 1: System inputs and outputs.
system models. For such systems the number of state variables, n, is equal to the number of
independent energy storage elements in the system. The values of the state variables at any
time t specify the energy of each energy storage element within the system and therefore
the total system energy, and the time derivatives of the state variables determine the rate
of change of the system energy. Furthermore, the values of the system state variables at
any time t provide sufficient information to determine the values of all other variables in the
system at that time.
There is no unique set of state variables that describe any given system; many different
sets of variables may be selected to yield a complete system description. However, for a
given system the order n is unique, and is independent of the particular set of state variables
chosen. State variable descriptions of systems may be formulated in terms of physical and
measurable variables, or in terms of variables that are not directly measurable. It is possible
to mathematically transform one set of state variables to another; the important point is
that any set of state variables must provide a complete description of the system. In this
note we concentrate on a particular set of state variables that are based on energy storage
variables in physical systems.
1.2 The State Equations
A standard form for the state equations is used throughout system dynamics. In the standard
form the mathematical description of the system is expressed as a set of n coupled first-order
ordinary differential equations, known as the state equations, in which the time derivative of
each state variable is expressed in terms of the state variables x1(t), . . . , xn(t) and the system
inputs u1(t), . . . , ur(t). In the general case the form of the n state equations is:
˙x1 = f1 (x, u, t)
˙x2 = f2 (x, u, t)
... =
... (1)
˙xn = fn (x, u, t)
2

where ˙xi = dxi/dt and each of the functions fi (x, u, t), (i = 1, . . . , n) may be a general
nonlinear, time varying function of the state variables, the system inputs, and time. 1
It is common to express the state equations in a vector form, in which the set of n state
variables is written as a state vector x(t) = [x1(t), x2(t), . . . , xn(t)]T
, and the set of r inputs
is written as an input vector u(t) = [u1(t), u2(t), . . . , ur(t)]T
. Each state variable is a time
varying component of the column vector x(t).
This form of the state equations explicitly represents the basic elements contained in
the definition of a state determined system. Given a set of initial conditions (the values of
the xi at some time t0) and the inputs for t ≥ t0, the state equations explicitly specify the
derivatives of all state variables. The value of each state variable at some time ∆t later may
then be found by direct integration.
The system state at any instant may be interpreted as a point in an n-dimensional state
space, and the dynamic state response x(t) can be interpreted as a path or trajectory traced
out in the state space.
In vector notation the set of n equations in Eqs. (1) may be written:
˙x = f (x, u, t) . (2)
where f (x, u, t) is a vector function with n components fi (x, u, t).
In this note we restrict attention primarily to a description of systems that are linear and
time-invariant (LTI), that is systems described by linear differential equations with constant
coefficients. For an LTI system of order n, and with r inputs, Eqs. (1) become a set of n
coupled first-order linear differential equations with constant coefficients:
˙x1 = a11x1 + a12x2 + . . . + a1nxn + b11u1 + . . . + b1rur
˙x2 = a21x1 + a22x2 + . . . + a2nxn + b21u1 + . . . + b2rur
...
...
˙xn = an1x1 + an2x2 + . . . + annxn + bn1u1 + . . . + bnrur
(3)
where the coefficients aij and bij are constants that describe the system. This set of n
equations defines the derivatives of the state variables to be a weighted sum of the state
variables and the system inputs.
Equations (8) may be written compactly in a matrix form:
d
dt






x1
x2
...
xn






=






a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
an1 an2 . . . ann












x1
x2
...
xn






+






b11 . . . b1r
b21 b2r
...
...
bn1 . . . bnr










u1
...
ur



 (4)
which may be summarized as:
˙x = Ax + Bu (5)
1
In this note we use bold-faced type to denote vector quantities. Upper case letters are used to denote
general matrices while lower case letters denote column vectors. See Appendix A for an introduction to
matrix notation and operations.
3

where the state vector x is a column vector of length n, the input vector u is a column vector
of length r, A is an n × n square matrix of the constant coefficients aij, and B is an n × r
matrix of the coefficients bij that weight the inputs.
1.3 Output Equations
A system output is defined to be any system variable of interest. A description of a physical
system in terms of a set of state variables does not necessarily include all of the variables of
direct engineering interest. An important property of the linear state equation description is
that all system variables may be represented by a linear combination of the state variables
xi and the system inputs ui. An arbitrary output variable in a system of order n with r
inputs may be written:
y(t) = c1x1 + c2x2 + . . . + cnxn + d1u1 + . . . + drur (6)
where the ci and di are constants. If a total of m system variables are defined as outputs,
the m such equations may be written as:
y1 = c11x1 + c12x2 + . . . + c1nxn + d11u1 + . . . + d1rur
y2 = c21x1 + c22x2 + . . . + c2nxn + d21u1 + . . . + d2rur
...
...
ym = cm1x1 + cm2x2 + . . . + cmnxn + dm1u1 + . . . + dmrur
(7)
or in matrix form:






y1
y2
...
ym






=






c11 c12 . . . c1n
c21 c22 . . . c2n
...
...
cm1 cm2 . . . cmn












x1
x2
...
xn






+






d11 . . . d1r
d21 d2r
...
...
dm1 . . . dmr










u1
...
ur



 . (8)
The output equations, Eqs. (8), are commonly written in the compact form:
y = Cx + Du (9)
where y is a column vector of the output variables yi(t), C is an m×n matrix of the constant
coefficients cij that weight the state variables, and D is an m × r matrix of the constant
coefficients dij that weight the system inputs. For many physical systems the matrix D is
the null matrix, and the output equation reduces to a simple weighted combination of the
state variables:
y = Cx. (10)
4

1.4 State Equation Based Modeling Procedure
The complete system model for a linear time-invariant system consists of (i) a set of n state
equations, deﬁned in terms of the matrices A and B, and (ii) a set of output equations that
relate any output variables of interest to the state variables and inputs, and expressed in
terms of the C and D matrices. The task of modeling the system is to derive the elements
of the matrices, and to write the system model in the form:
˙x = Ax + Bu (11)
y = Cx + Du. (12)
The matrices A and B are properties of the system and are determined by the system
structure and elements. The output equation matrices C and D are determined by the
particular choice of output variables.
The overall modeling procedure developed in this chapter is based on the following steps:
1. Determination of the system order n and selection of a set of state variables from the
linear graph system representation.
2. Generation of a set of state equations and the system A and B matrices using a well
deﬁned methodology. This step is also based on the linear graph system description.
3. Determination of a suitable set of output equations and derivation of the appropriate
C and D matrices.
2 Block Diagram Representation of Linear Systems
Described by State Equations
The matrix-based state equations express the derivatives of the state-variables explicitly in
terms of the states themselves and the inputs. In this form, the state vector is expressed as
the direct result of a vector integration. The block diagram representation is shown in Fig.
2. This general block diagram shows the matrix operations from input to output in terms
of the A, B, C, D matrices, but does not show the path of individual variables.
In state-determined systems, the state variables may always be taken as the outputs
of integrator blocks. A system of order n has n integrators in its block diagram. The
derivatives of the state variables are the inputs to the integrator blocks, and each state
equation expresses a derivative as a sum of weighted state variables and inputs. A detailed
block diagram representing a system of order n may be constructed directly from the state
and output equations as follows:
Step 1: Draw n integrator (S−1
) blocks, and assign a state variable to the output of each
block.
5

Figure 2: Vector block diagram for a linear system described by state-space system dynamics.
Step 2: At the input to each block (which represents the derivative of its state variable)
draw a summing element.
Step 3: Use the state equations to connect the state variables and inputs to the summing
elements through scaling operator blocks.
Step 4: Expand the output equations and sum the state variables and inputs through a set
of scaling operators to form the components of the output.
Example 1
Draw a block diagram for the general second-order, single-input single-output
system:
˙x1
˙x2
=
a11 a12
a21 a22
x1
x2
+
b1
b2
u(t)
y(t) = c1 c2
x1
x2
+ du(t). (i)
Solution: The block diagram shown in Fig. 3 was drawn using the four steps
described above.
3 Transformation From State-Space Equations to Clas-
sical Form
The transfer function and the classical input-output diﬀerential equation for any system vari-
able may be found directly from a state space representation through the Laplace transform.
The following example illustrates the general method for a ﬁrst-order system.
6

Figure 3: Block diagram for a state-equation based second-order system.
Example 2
Find the transfer function and a single first-order differential equation relating
the output y(t) to the input u(t) for a system described by the first-order linear
state and output equations:
dx
dt
= ax(t) + bu(t) (i)
y(t) = cx(t) + du(t) (ii)
Solution: The Laplace transform of the state equation is
sX(s) = aX(s) + bU(s), (iii)
which may be rewritten with the state variable X(s) on the left-hand side:
(s − a) X(s)) = bU(s). (iv)
Then dividing by (s − a), solve for the state variable:
X(s) =
b
s − a
U(s), (v)
7

and substitute into the Laplace transform of the output equation Y (s) = cX(s)+
dU(s):
Y (s) =
bc
s − a
+ d U(s)
=
ds + (bc − ad)
(s − a)
U(s) (vi)
The transfer function is:
H(s) =
Y (s)
U(s)
=
ds + (bc − ad)
(s − a)
. (vii)
The differential equation is found directly:
(s − a) Y (s) = (ds + (bc − ad)) U(s), (viii)
and rewriting as a differential equation:
dy
dt
− ay = d
du
dt
+ (bc − ad) u(t). (ix)
Classical representations of higher-order systems may be derived in an analogous set of
steps by using the Laplace transform and matrix algebra. A set of linear state and output
equations written in standard form
˙x = Ax + Bu (13)
y = Cx + Du (14)
may be rewritten in the Laplace domain. The system equations are then
sX(s) = AX(s) + BU(s)
Y(s) = CX(s) + DU(s) (15)
and the state equations may be rewritten:
sx(s) − Ax(s) = [sI − A] x(s) = Bu(s). (16)
where the term sI creates an n×n matrix with s on the leading diagonal and zeros elsewhere.
(This step is necessary because matrix addition and subtraction is only defined for matrices
of the same dimension.) The matrix [sI − A] appears frequently throughout linear system
theory; it is a square n × n matrix with elements directly related to the A matrix:
[sI − A] =






(s − a11) −a12 · · · −a1n
−a21 (s − a22) · · · −a2n
...
...
...
...
−an1 −an2 · · · (s − ann)






. (17)
8

The state equations, written in the form of Eq. (16), are a set of n simultaneous opera-
tional expressions. The common methods of solving linear algebraic equations, for example
Gaussian elimination, Cramer’s rule, the matrix inverse, elimination and substitution, may
be directly applied to linear operational equations such as Eq. (16).
For low-order single-input single-output systems the transformation to a classical formu-
lation may be performed in the following steps:
1. Take the Laplace transform of the state equations.
2. Reorganize each state equation so that all terms in the state variables are on the
left-hand side.
3. Treat the state equations as a set of simultaneous algebraic equations and solve for
those state variables required to generate the output variable.
4. Substitute for the state variables in the output equation.
5. Write the output equation in operational form and identify the transfer function.
6. Use the transfer function to write a single diﬀerential equation between the output
variable and the system input.
This method is illustrated in the following two examples.
Example 3
Use the Laplace transform method to derive a single diﬀerential equation for the
capacitor voltage vC in the series R-L-C electric circuit shown in Fig. 4
Solution: The linear graph method of state equation generation selects the
Figure 4: A series RLC circuit.
9

capacitor voltage vC(t) and the inductor current iL(t) as state variables, and
generates the following pair of state equations:
˙vc
˙iL
=
0 1/C
−1/L −R/L
vc
iL
+
0
1/L
Vin. (i)
The required output equation is:
y(t) = 1 0
vc
iL
+ 0 Vin (ii)
Step 1: In Laplace transform form the state equations are:
sVC(s) = 0VC(s) + 1/CIL(s) + 0Vs(s)
sIL(s) = −1/LVC(s) − R/LIL(s) + 1/LVs(s) (iii)
Step 2: Reorganize the state equations:
sVC(s) − 1/CIL(s) = 0Vs(s) (iv)
1/LVC(s) + [s + R/L] IL(s) = 1/LVs(s) (v)
Step 3: In this case we have two simultaneous operational equations in the state
variables vC and iL. The output equation requires only vC. If Eq. (iv) is
multiplied by [s + R/L], and Eq. (v) is multiplied by 1/C, and the equations
added, IL(s) is eliminated:
[s (s + R/L) + 1/LC] VC(s) = 1/LCVs(s) (vi)
Step 4: The output equation is y = vC. Operate on both sides of Eq. (vi) by
[s2
+ (R/L)s + 1/LC]
−1
and write in quotient form:
VC(s) =
1/LC
s2 + (R/L)s + 1/LC
Vs(s) (vii)
Step 5: The transfer function H(s) = Vc(s)/Vs(s) is:
H(s) =
1/LC
s2 + (R/L)s + 1/LC
(viii)
Step 6: The diﬀerential equation relating vC to Vs is:
d2
vC
dt2
+
R
L
dvC
dt
+
1
LC
vC =
1
LC
Vs(t) (ix)
10

Cramer’s Rule, for the solution of a set of linear algebraic equations, is a useful method
to apply to the solution of these equations. In solving for the variable xi in a set of n linear
algebraic equations, such as Ax = b the rule states:
xi =
det A(i)
det [A]
(18)
where A(i)
is another n × n matrix formed by replacing the ith column of A with the vector
b.
If
[sI − A] X(s) = BU(s) (19)
then the relationship between the ith state variable and the input is
Xi(s) =
det [sI − A](i)
det [sI − A]
U(s) (20)
where (sI − A)(i)
is deﬁned to be the matrix formed by replacing the ith column of (sI − A)
with the column vector B. The diﬀerential equation is
det [sI − A] xi = det (sI − A)(i)
uk(t). (21)
Example 4
Use Cramer’s Rule to solve for vL(t) in the electrical system of Example 3.
Solution: From Example 3 the state equations are:
˙vc
˙iL
=
0 1/C
−1/L −R/L
vc
iL
+
0
1/L
Vin(t) (i)
and the output equation is:
vL = −vC − RiL + Vs(t). (ii)
In the Laplace domain the state equations are:
s −1/C
1/L s + R/L
Vc(s)
IL(s)
=
0
1/L
Vin(s). (iii)
11

The voltage VC(s) is given by:
VC(s) =
det (sI − A)(1)
det [(sI − A)]
Vin(s) =
det
0 −1/C
1/L (s + R/L)
det
s −1/C
1/L (s + R/L)
Vin(s)
=
1/LC
s2 + (R/L)s + (1/LC)
Vin(s). (iv)
The current IL(t) is:
IL(s) =
det (sI − A)(2)
det [(sI − A)]
Vin(s) =
det
s 0
1/L 1/L
det
s −1/C
1/L (s + R/L)
Vin(s)
=
s/L
s2 + (R/L)s + (1/LC)
Vin(s). (v)
The output equation may be written directly from the Laplace transform of Eq.
(ii):
VL(s) = −VC(s) − RIL(s) + Vs(s)
=
−1/LC
s2 + (R/L)s + (1/LC)
+
−(R/L)s
s2 + (R/L)s + (1/LC)
+ 1 Vs(s)
=
−1/LC − (R/L)s + (s2
+ (R/L)s + (1/LC))
s2 + (R/L)s + (1/LC)
Vs(s)
=
s2
s2 + (R/L)s + (1/LC)
Vs(s), (vi)
giving the diﬀerential equation
d2
vL
dt2
+
R
L
dvL
dt
+
1
LC
vL(t) =
d2
Vs
dt2
. (vii)
For a single-input single-output (SISO) system the transfer function may be found directly
by evaluating the inverse matrix
X(s) = (sI − A)−1
BU(s). (22)
Using the deﬁnition of the matrix inverse:
[sI − A]−1
=
adj [sI − A]
det [sI − A]
, (23)
12

X(s) =
adj [sI − A] B
det [sI − A]
U(s). (24)
and substituting into the output equations gives:
Y (s) = C [sI − A]−1
BU(s) + DU(s)
= C [sI − A]−1
B + D U(s). (25)
Expanding the inverse in terms of the determinant and the adjoint matrix yields:
Y (S) =
C adj (sI − A) B + det [sI − A] D
det [sI − A]
U(s)
= H(s)U(s) (26)
so that the required differential equation may be found by expanding:
det [sI − A] Y (s) = [C adj (sI − A) B + det [sI − A] D] U(s). (27)
and taking the inverse Laplace transform of both sides.
Example 5
Use the matrix inverse method to find a differential equation relating vL(t) to
Vs(t) in the system described in Example 3.
Solution: The state vector, written in the Laplace domain,
X(s) = [sI − A]−1
BU(s) (i)
from the previous example is:
Vc(s)
IL(s)
=
s −1/C
1/L s + R/L
−1
0
1/L
Vin(s). (ii)
The determinant of [sI − A] is
det [sI − A] = s2
+ (R/L)s + (1/LC) , (iii)
and the adjoint of [sI − A] is
adj
s −1/C
1/L s + R/L
=
s + R/L 1/C
−1/L s
. (iv)
From Example .5 and the previous example, the output equation vL(t) = −vC −
RiL + Vs(t) specifies that C = [−1 − R] and D = [1]. The transfer function, Eq.
(26) is:
H(s) =
det [sI − A]−1 . (v)
13

Since
C adj (sI − A) B = −1 −R
s + R/L 1/C
−1/L s
0
1/L
= −
R
L
s −
1
LC
, (vi)
the transfer function is
H(s) =
−(R/L)s − 1/(LC) + (s2
+ (R/L)s + (1/LC)) 1
(s2 + (R/L)s + (1/LC))
=
s2
(S2 + (R/L)S + (1/LC))
, (vii)
which is the same result found by using Cramer’s rule in Example 4.
4 Transformation from Classical Form to State-Space
Representation
The block diagram provides a convenient method for deriving a set of state equations for
a system that is specified in terms of a single input/output differential equation. A set of
n state variables can be identified as the outputs of integrators in the diagram, and state
equations can be written from the conditions at the inputs to the integrator blocks (the
derivative of the state variables). There are many methods for doing this; we present here
one convenient state equation formulation that is widely used in control system theory.
Let the differential equation representing the system be of order n, and without loss of
generality assume that the order of the polynomial operators on both sides is the same:
ansn
+ an−1sn−1
+ · · · + a0 Y (s) = bnsn
+ bn−1sn−1
+ · · · + b0 U(s). (28)
We may multiply both sides of the equation by s−n
to ensure that all differential operators
have been eliminated:
an + an−1s−1
+ · · · + a1s−(n−1)
+ a0s−n
Y (s) =
bn + bn−1s−1
+ · · · + b1s−(n−1)
+ · · · + b0s−n
U(s), (29)
from which the output may be specified in terms of a transfer function. If we define a dummy
variable Z(s), and split Eq. (29) into two parts
Z(s) =
1
an + an−1s−1 + · · · + a1s−(n−1) + a0s−n
U(s) (30)
Y (s) = bn + bn−1s−1
+ · · · + b1s−(n−1)
+ b0s−n
Z(s), (31)
14

Figure 5: Block diagram of a system represented by a classical differential equation.
Eq. (30) may be solved for U(s)),
U(s) = an + an−1s−1
+ · · · + a1s−(n−1)
+ a0s−n
X(s) (32)
and rearranged to generate a feedback structure that can be used as the basis for a block
diagram:
Z(s) =
1
an
U(s) −
an−1
an
1
s
+ · · · +
a1
an
1
sn−1
+
a0
an
1
s−n
Z(s). (33)
The dummy variable Z(s) is specified in terms of the system input u(t) and a weighted sum
of successive integrations of itself. Figure 5 shows the overall structure of this direct-form
block diagram. A string of n cascaded integrator (1/s) blocks, with Z(s) defined at the input
to the first block, is used to generate the feedback terms, Z(s)/s−i
, i = 1, . . . n, in Eq. (33).
Equation (31) serves to combine the outputs from the integrators into the output y (t).
A set of state equations may be found from the block diagram by assigning the state
variables xi(t) to the outputs of the n integrators. Because of the direct cascade connection
of the integrators, the state equations take a very simple form. By inspection:
˙x1 = x2
˙x2 = x3
...
...
˙xn−1 = xn
˙xn = −
a0
an
x1 −
a1
an
x2 − · · · −
an−1
an
xn +
1
an
u(t). (34)
15

In matrix form these equations are











˙x1
˙x2
...
˙xn−2
˙xn−1
˙xn











=











0 1 · · · 0 0
0 0 · · · 0 0
...
...
...
...
...
0 0 · · · 1 0
0 0 · · · 0 1
−a0/an −a1/an · · · −an−2/an −an−1/an






















x1
x2
...
xn−2
xn−1
xn











+











0
0
...
0
0
1/an











u (t) .
(35)
The A matrix has a very distinctive form. Each row, except the bottom one, is filled of
zeroes except for a one in the position just above the leading diagonal. Equation (35) is
a common form of the state equations, used in control system theory and known as the
phase variable or companion form. This form leads to a set of state variables which may not
correspond to any physical variables within the system.
The corresponding output relationship is specified by Eq. (31) by noting that Xi(s) =
Z(s)/s(n+1−i)
.
y (t) = b0x1 + b1x2 + b2x3 + · · · + bn−1xn + bnz (t) . (36)
But z (t) = dxn/dt , which is found from the nth state equation in Eq. (34). When substituted
into Eq. (36) the output equation is:
Y (s) = b0 −
bna0
an
b1 −
bna1
an
· · · bn−1 −
bnan−1
an






x1
x2
...
xn






+
bn
an
u(t). (37)
Example 6
Draw a direct form realization of a block diagram, and write the state equations
in phase variable form, for a system with the differential equation
d3
y
dt3
+ 7
d2
y
dt2
+ 19
dy
dt
+ 13y = 13
du
dt
+ 26u (i)
Solution: The system order is 3, and using the structure shown in Fig. 5 the
block diagram is as shown in Fig. 6.
The state and output equations are found directly from Eqs. (35) and (37):



˙x1
˙x2
˙x3


 =



0 1 0
0 0 1
−13 −19 −7






x1
x2
x3


 +



0
0
1


 u(t), (ii)
16

Figure 6: Block diagram of the transfer operator of a third-order system found by a direct
realization.
y(t) = 26 13 0



x1
x2
x3


 + [0] u (t) . (iii)
5 The Matrix Transfer Function
For a multiple-input multiple-output system Eq. 22 is written in terms of the r component
input vector U(s)
X(s) = [sI − A]−1
BU(s) (38)
generating a set of n simultaneous linear equations, where the matrix is B is n × r. The m
component system output vector Y(s) may be found by substituting this solution for X(s)
into the output equation as in Eq. 25:
Y(s) = C [sI − A]−1
B {U(s)} + D {U(s)}
= C [sI − A]−1
B + D {U(s)} (39)
and expanding the inverse in terms of the determinant and the adjoint matrix
Y(s) =
det [sI − A]
U(s)
= H(s)U(s), (40)
where H(s) is deﬁned to be the matrix transfer function relating the output vector Y(s) to
the input vector U(s):
H(s) =
(C adj (sI − A) B + det [sI − A] D)
det [sI − A]
(41)
17

For a system with r inputs U1(s), . . . , Ur(s) and m outputs Y1(s), . . . , Ym(s), H(s) is a m×r
matrix whose elements are individual scalar transfer functions relating a given component
of the output Y(s) to a component of the input U(s). Expansion of Eq. 41 generates a set
of equations:






Y1(s)
Y2(s)
...
Ym(s)






=






H11(s) H12(s) · · · H1r(s)
H21(s) H22(s) · · · H2r(s)
...
...
...
...
Hm1(s) Hm2(s) · · · Hmr(s)












U1(s)
U2(s)
...
Ur(s)






(42)
where the ith component of the output vector Y(s) is:
Yi(s) = Hi1(s)U1(s) + Hi2(s)U2(s) + · · · + Hir(s)Us(s). (43)
The elemental transfer function Hij(s) is the scalar transfer function between the ith output
component and the jth input component. Equation 41 shows that all of the Hij(s) trans-
fer functions in H(s) have the same denominator factor det [sI − A], giving the important
result that all input-output diﬀerential equations for a system have the same characteristic
polynomial, or alternatively have the same coeﬃcients on the left-hand side.
If the system has a single-input and a single-output, H(s) is a scalar, and the procedure
generates the input/output transfer operator directly.
18

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