MIMO Toolbox
For Use with M
ATLAB
Oskar Vivero
About the toolbox
The MIMO Toolbox is a collection of M
ATLAB
functions and a GUI. Its purpose is to
complement the Control Toolbox for M
ATLAB
with functions capable of handling the
multivariable input-output scheme. All the results and examples except for example
1.1.2.1 were obtained with the MIMO Toolbox and were corroborated with the
bibliography.
April, 2006
Installation
The installation is straightforward just copy the directory “Mimotools” and add the path
to the M
ATLAB
search path.
See path, in the M
ATLAB
documentation for more information.
Requirements
The MIMO Toolbox was created in Matlab 7.1 (R14) SP3, and requires the Symbolic
and Control Toolboxes.
Contact
Oskar Vivero
oskar.vivero@gmail.com
Contents
1. Theory behind the functions
1.1. SISO Systems
1.1.1. Feedback Basic Concepts
1.1.2. Nyquist’s Stability Criterion
1.2. MIMO Systems
1.2.1. Poles and Zeros of a MIMO System
1.2.1.1. Smith-McMillan Transformation
1.2.2. Stability of MIMO Systems
1.2.2.1. Generalized Nyquist’s Stability Criterion
1.2.3. Treating a MIMO System with SISO techniques
1.2.3.1. Coupling degree and pairings of inputs and outputs
1.2.3.2. Nyquist’s Arrays and Gershgorin Bands
1.2.3.3. Relative Gain Array (RGA)
1.2.3.4. Individual Channel Design (ICD)
2. Function Reference
2.1. The Symbolic Transfer Function
2.2. Function Description
2.2.1. tf2sym
2.2.2. sym2tf
2.2.3. ss2sym
2.2.4. smform
2.2.5. rga
2.2.6. nyqmimo
2.2.7. m_circles
2.2.8. icdtool
2.2.9. gershband
2.2.10. arrowh
3. References
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1
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2
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4
4
7
7
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MIMO Toolbox
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1. Theory behind the functions
The aim of this chapter is to introduce the MIMO control theories so that one can
understand both, the algorithm behind each function and its proper use. This chapter is
only intended to provide a brief description of such theories and it’s recommended that
the user refers to the bibliography listed at the end of this document.
1.1 SISO Systems
1.1.1 Feedback Basic Concepts
Assuming a lineal process that is time-invariant whose behavior is defined by lineal
differential equations with constant coefficients:
1 2 1 2
y a y a y b u b u
•• • •
+ + = +
where
(
)
y t
is the output signal and
(
)
u t
is the input signal, its possible to obtain a
transfer function by applying the Laplace’s Transform
(
)
( )
( )
(
)
( )
1 2
2
1 2
Y s N s
b s b
G s
U s D s
s a s a
+
= = =
+ +
(
)
( )
If 0 then is defined as a zero.
If then is defined as a pole.
Z Z Z
P P P
s s G s s
s s G s s
= =
= = ∞
If
(
)
G s
is rational, usually
(
)
D s
determines the dynamic characteristics of the system,
unless there exist cancellations between
(
)
N s
and
(
)
D s
.
Let
(
)
H s
and
(
)
G s
be two transfer functions
The stability of the system in a closed-loop configuration is given by
(
)
(
)
1
G s H s
+
if
and only if there is no cancellation of instabilities. For any design, it’s possible to
verify its stability by finding the singularities of
(
)
CL
G s
. If any of the singularities is
located in
2
+
or near the imaginary axis, it’s almost impossible to determine the
modifications needed on either
(
)
G s
or
(
)
H s
to avoid the location of the singularities.
The Nyquist’s stability criterion provides a tool for solving the problem
( )
(
)
( ) ( )
1
CL
G s
G s
G s H s
=
+
MIMO Toolbox
- 2 -
1.1.2 Nyquist’s Stability Criterion
The system in a closed-loop configuration is stable if and only if the trajectory of the
Nyquist diagram of
(
)
(
)
G j H j
ω ω
from
ω
−∞ < < ∞
surrounds the point
(
)
1,0
−
in a
counter-clockwise direction as much times as
(
)
(
)
G s H s
has unstable poles.
Theorem – suppose that a function
(
)
f z
is meromorphic in a simply connected domain
D, and that C is a simple closed positively oriented contour in
D
such that
(
)
f z
does
not contain any singularities. Then
(
)
( )
'
1
2
f f
f z
N dz Z P
i f z
π
= = −
where
N
is the winding number,
f
Z
is the number of zeros inside the contour and
f
P
is the number of poles inside the contour.
Example 1.1.2.1
The image of the circle of radius 2 centered at the origin under
(
)
2
f z z z
= +
is the
curve
(
)
(
)
(
)
(
)
(
)
(
)
, 4cos 2 2cos ,4sin 2 2sin
g x y t t t t
= + +
. Note that the curve
(
)
,
g x y
winds up twice around the origin. We check this by computing
(
)
( )
0 1
0 1
2 2
'
1
; Singulatiries at 0 and 1
2
2 1 2 1
Res Res 2
C
z z
f z
N z z
i f z
z z
N
z z z z
π
= = = −
+ +
= + =
+ +
Having defined
N
, it’s important to define a useful contour for the stability analysis.
It’s possible to know from the root locus analysis and the time response that the
unstable poles are at the right side of the S-plane. Since the zeros of the open loop
system are the poles of the closed loop system, we’ll focus on finding the unstable zeros
through Nyquist frequency analysis. The contours that we’ll consider are:
