Image Processing Reference
In-Depth Information
3 Mathematical
Foundations
3.1 INTRODUCTION
The motivation for this chapter is to present mathematical tools for analysis and design
of open- and closed-loop continuous- and discrete-time control systems. A broad class
of linear time-invariant (LTI) systems can be represented by linear differential equa-
tions (DEs) with constant coef
cients in case of continuous-time and linear difference
equations with constant coef
cients in case of discrete-time systems. The Fourier and
Laplace transforms play important roles in analyzing and designing such systems. The
z-transform is a tool for analyzing and designing discrete-time systems and is the
counterpart of Laplace transform that is used for continuous-time systems. In this
chapter, we introduce LTI continuous-time systems, Laplace transform, discrete-time
systems, and z-transform. Matrices and linear algebra are also important tools in
analyzing control systems in the state-space form. Eigenvalues and eigenvectors,
singular value decomposition (SVD), and functions of matrices are other mathe-
matical tools used to design control systems with state-space approach. The rest of
this chapter is devoted to this important subject.
3.2 GENERAL CONTINUOUS-TIME SYSTEM DESCRIPTION
Consider a single-input single-output (SISO) continuous-time system with input
u ( t )
and output y ( t )
that can be represented by constant-coef
cients linear DE of
the form
N y ( t )
d t N
N 1
M u ( t )
d t M
M 1
d
þ a N 1 d
d t N 1 þþ a 0 y ( t ) ¼ b M d
y ( t )
þ b M 1 d
u ( t )
d t M 1 þþ b 0 u ( t )
(
3
:
1
)
where
N is the order of the DE
M is typically less than or equal to N
Given the input signal u ( t )
and N initial conditions y (
)
, y 0 (
)
...
, y ( N 1 ) (
)
0
0
,
0
, the
output signal y ( t )
can be determined uniquely by solving the DE given by Equation
3.1 either in time domain or in the transform domain using Fourier or Laplace
transform. We will
first solve the equation in time domain, and then we will use
Laplace transform to solve the equation in s-domain.
99
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