Part A | 10.1
234 Part A Fundamentals
Commonly, the aforementioned mathematical law
is expressed as a set of n first-order differential equations with respect to time, dynamic equations. The
dynamic equations connect the input vector with the
state vector
P
x.t/ ,
d
dt
x.t/ D f .x.t/; u.t/; t/ :
(10.31)
The description of the system in state-space requires
also a set of p algebraic output equations connecting
the input and state vectors with the output one
y.t/ D g.x.t/; u.t// :
(10.32)
By employing (10.31), it is directly concluded that if the
initial state of a system is given at a time instant t 0 then
the state at any future time instant can be calculated as
follows
x.t/ D x.t 0 / C
t
Z
t0
f .x../; u../; ;/d :
(10.33)
Multivariable LTI Systems
In the case of LTI systems (10.31) and (10.32) assume
the following form
P
x D Ax C Bu ;
y D Cx C Du :
(10.34)
In the above A is an n n matrix, B an n m matrix, C
a p n matrix, and D a p m matrix.
An input–output relationship can be deduced for
an LTI system described in state-space by the state
and output equations of the form (10.34). This relationship comes as a natural extension of the transfer
function concept already introduced for single-input,
single-output (SISO) LTI systems to the case of MIMO
(multi-input, multioutput) LTI systems. To obtain it the
Laplace transform is applied to the time-domain expression (10.34) by employing property (10.4)
sX.s/ x.0/ D AX.s/ C BU.s/ ;
Y.s/ D CX.s/ C DU.s/ :
(10.35)
In the above, it has been assumed that t 0 D 0 for convenience and without any loss of generality. The complex
frequency domain algebraic (10.35) can be solved to
yield the following
X.s/ D .sI n A/
1 BU.s/
C .sI n A/
1 x.0/ ;
Y.s/ D
h
C .sI n A/
1 B C D
i
U.s/
C C .sI n A/
1 x.0/ :
(10.36)
In the above, I n stands for the n n identity matrix. Formulas (10.36) allow the analytic calculation of any LTI
system’s response and are referred to as full-response
equations in the literature [10.1–6]. The following special cases are investigated:
1. Zero initial state response, x.0/ D 0
Y.s/ D
h
C .sI n A/
1 B C D
i
U.s/ : (10.37)
2. Zero input response, u D 0
Y.s/ D C .sI n A/
1 x.0/ :
(10.38)
In order to obtain the response in the time domain,
the following formula of the Laplace transform theory
is needed
.sI n A/
1 D L
˚
e
At
« ; t > 0 :
(10.39)
The above is the matrix generalization of (10.16).
Expression e
At , also referred to by the term system
transient matrix, is defined as an n n matrix with the
following properties
e
At
D I n C At C
1
2Š
A
2 t
2
C
1
3Š
A
3 t
3
C C C C
,
d
dt
e
At
D A e
At
:
(10.40)
By using (10.40), the full-response equation in the time
domain is obtained
x.t/ D
t
Z
0
e
Av Bu.t v /dv C e
At x.0/ ;
t > 0 :
(10.41)
The following matrix version of the theorem of convolution has been employed in the above
.sI n A/
1 BU.s/ D L
˚
e
At
Bu.t/
«
D L
8
<
:
t
Z
0
e
Av Bu.t v /dv
9
=
;
:
(10.42)
Transfer Function Matrix
In the input–output relationship (10.37), the p m matrix H.s/ D C .sI n A/
1 B CD is the system’s transfer matrix.
It is straightforward to show that
.sI n A/
1 D jsI n Aj
1 adj.sI n A/ :
234 Part A Fundamentals
Commonly, the aforementioned mathematical law
is expressed as a set of n first-order differential equations with respect to time, dynamic equations. The
dynamic equations connect the input vector with the
state vector
P
x.t/ ,
d
dt
x.t/ D f .x.t/; u.t/; t/ :
(10.31)
The description of the system in state-space requires
also a set of p algebraic output equations connecting
the input and state vectors with the output one
y.t/ D g.x.t/; u.t// :
(10.32)
By employing (10.31), it is directly concluded that if the
initial state of a system is given at a time instant t 0 then
the state at any future time instant can be calculated as
follows
x.t/ D x.t 0 / C
t
Z
t0
f .x../; u../; ;/d :
(10.33)
Multivariable LTI Systems
In the case of LTI systems (10.31) and (10.32) assume
the following form
P
x D Ax C Bu ;
y D Cx C Du :
(10.34)
In the above A is an n n matrix, B an n m matrix, C
a p n matrix, and D a p m matrix.
An input–output relationship can be deduced for
an LTI system described in state-space by the state
and output equations of the form (10.34). This relationship comes as a natural extension of the transfer
function concept already introduced for single-input,
single-output (SISO) LTI systems to the case of MIMO
(multi-input, multioutput) LTI systems. To obtain it the
Laplace transform is applied to the time-domain expression (10.34) by employing property (10.4)
sX.s/ x.0/ D AX.s/ C BU.s/ ;
Y.s/ D CX.s/ C DU.s/ :
(10.35)
In the above, it has been assumed that t 0 D 0 for convenience and without any loss of generality. The complex
frequency domain algebraic (10.35) can be solved to
yield the following
X.s/ D .sI n A/
1 BU.s/
C .sI n A/
1 x.0/ ;
Y.s/ D
h
C .sI n A/
1 B C D
i
U.s/
C C .sI n A/
1 x.0/ :
(10.36)
In the above, I n stands for the n n identity matrix. Formulas (10.36) allow the analytic calculation of any LTI
system’s response and are referred to as full-response
equations in the literature [10.1–6]. The following special cases are investigated:
1. Zero initial state response, x.0/ D 0
Y.s/ D
h
C .sI n A/
1 B C D
i
U.s/ : (10.37)
2. Zero input response, u D 0
Y.s/ D C .sI n A/
1 x.0/ :
(10.38)
In order to obtain the response in the time domain,
the following formula of the Laplace transform theory
is needed
.sI n A/
1 D L
˚
e
At
« ; t > 0 :
(10.39)
The above is the matrix generalization of (10.16).
Expression e
At , also referred to by the term system
transient matrix, is defined as an n n matrix with the
following properties
e
At
D I n C At C
1
2Š
A
2 t
2
C
1
3Š
A
3 t
3
C C C C
,
d
dt
e
At
D A e
At
:
(10.40)
By using (10.40), the full-response equation in the time
domain is obtained
x.t/ D
t
Z
0
e
Av Bu.t v /dv C e
At x.0/ ;
t > 0 :
(10.41)
The following matrix version of the theorem of convolution has been employed in the above
.sI n A/
1 BU.s/ D L
˚
e
At
Bu.t/
«
D L
8
<
:
t
Z
0
e
Av Bu.t v /dv
9
=
;
:
(10.42)
Transfer Function Matrix
In the input–output relationship (10.37), the p m matrix H.s/ D C .sI n A/
1 B CD is the system’s transfer matrix.
It is straightforward to show that
.sI n A/
1 D jsI n Aj
1 adj.sI n A/ :