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The concepts of linearization, stability analysis, modal analysis, PID and LQR control synthesis, controllability, and observability in the context of linear systems analysis and control. It includes examples of linearization for a pendulum system and a Rosetta stone problem, as well as an explanation of the Cayley-Hamilton theorem and its application to controllability.
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The
complete
set
of
coupled
translational and rotational equa-tions of motion for a rigid body:
ω
−
1
h
v
m
p
f
f
r
v
ω
g
g
r
v
ω
r
m
p
ω
×
r
p
ω
×
p
f
¯q
q
ω
h
ω
×
h
g
The
di
ff
erential
equations
are
coupled
through
ω
as
well
as
through the dependence of forceand moment on position, velocity,attitude, and angular velocityThe force and moment can alsoinclude
controls
for
example,
the torque on a submersible ve-hicle can include
environmental
torques due to viscosity,
buoy-
ancy, and gravity, as well as
con-
trol
torques due to actuators such
as momentum wheels and
fi
ns
The equations are also
nonlinear
The
function
f
is
sometimes
called the vector
fi
eld
Often
f
will not depend on
t
, and
in that case the system is called autonomous
, or
time-invariant
Furthermore,
y
usually does not
depend on
u
; if
y
does depend on
u
, then the system is said to have
a
feedforward
connection between
the input and the output
Generally, the equations describ-ing dynamics and control prob-lems
can
be
developed
in
the
form:
x
f
x
u
, t
y
g
x
u
, t
where
x
n
y
m
, and
u
p
The vector
x
is the state vector,
the vector
u
is the input vector,
and the vector
y
is the output
vector
De
fi
ne
small
perturbations away from the equilibrium state and
control:
x
x
∗
δ
x
u
u
∗
δ
u
Recall that for a scalar function
f
x
), the Taylor series is
f
x
f
x
∗
δ
x
f
x
∗
f
0
x
∗
δ
x
f
00
x
∗
δ
x
2
For the vector functions of vector arguments here, the Taylor seriesis expressed similarly:
f
x
f
x
∗
δ
x
f
x
∗
f
x
x
∗
δ
x
where the
represent the higher order terms in the Taylor series.
The vector
fi
eld depends on
x
and
u
x
f
x
u
, t
f
x
∗
δ
x
u
∗
δ
u
, t
Apply the Taylor series expansion to
f
x
u
, t
) as
f
x
u
, t
f
x
∗
u
∗
, t
f
x
x
∗
u
∗
, t
δ
x
f
u
x
∗
u
∗
, t
δ
u
By de
fi
nition, the term
f
x
∗
u
∗
, t
, so that the di
ff
erential
equation is approximated as
δ
x
f
x
x
∗
u
∗
, t
δ
x
f
u
x
∗
u
∗
, t
δ
u
The terms
∂
f
∂
x
x
∗
u
∗
, t
) and
∂
f
∂
u
x
∗
u
∗
, t
) are
n
n
and
n
p
ma-
trices, respectively
The second matrix in the linearized equation
δ
x
f
x
x
∗
u
∗
, t
δ
x
f
u
x
∗
u
∗
, t
δ
u
is
f
u
x
∗
u
∗
, t
∂
f
1
∂
u
1
∂
f
1
∂
u
p
∂
f
n
∂
u
1
∂
f
n
∂
u
p
(
x
∗
,
u
∗
)
t
This matrix,
t
) is the
input
matrix
Using these matrix de
fi
nitions, the linear equation can be written ˙
δ
x
t
δ
x
t
δ
u
The linearized equation
δ
x
f
x
x
∗
u
∗
, t
δ
x
f
u
x
∗
u
∗
, t
δ
u
is abbreviated as
δ
x
t
δ
x
t
δ
u
Frequently, the “
δ
” and the time-dependence are
understood
, and
the equation is written
x
Ax
Bu
This is
a
standard form for the linear state-space di
ff
erential equa-
tion One also sees
x
Fx
Gu
Consider the motion of a spinning rigid body with a constant“environmental” torque.
The desired motion of spinning body is
ω
∗
1
2
3
]. We want to linearize the equations of motion
about the desired motion.Expressed in a principal frame, the environmental torque is
g
e
The “control” torque,
g
∗
, required to maintain the desired motion
is easily computed using Euler’s equations:
ω
−
1
ω
×
ω
−
1
g
e
−
1
g
f
x
u
−
1
ω
∗
×
ω
∗
−
1
g
e
−
1
g
∗
f
x
∗
u
∗
g
∗
ω
∗
×
ω
∗
g
e
Given the desired motion,
ω
∗
and the environmental torque
g
e
, we
can easily compute
g
∗
, which is clearly constant
In this problem,
x
ω
u
g
x
∗
ω
∗
, and
u
∗
g
∗
Rosetta stone:
x
ω
u
g
x
∗
ω
∗
, and
u
∗
g
∗
Equations in scalar form:
ω
1
2
3
1
ω
2
ω
3
g
e
1
1
g
1
1
f
1
x
u
ω
2
3
1
2
ω
1
ω
3
g
e
2
2
g
2
2
f
2
x
u
ω
3
1
2
3
ω
1
ω
2
g
e
3
3
g
3
3
f
3
x
u
The matrix,
∂
f
∂
x
x
∗
u
∗
, t
), is
f
x
x
∗
u
∗
, t
I
2
−
I
3
I
1
3
I
2
−
I
3
I
1
2
I
3
−
I
1
I
2
3
I
3
−
I
1
I
2
1
I
1
−
I
2
I
3
2
I
1
−
I
2
I
3
1
Clearly
depends only on the inertias and the desired steady
motion
ω
∗
Thus, the linearized equations of motion are
δ
ω
δ
ω
δ
g
or
x
Ax
Bu
where A
I
2
−
I
3
I
1
3
I
2
−
I
3
I
1
2
I
3
−
I
1
I
2
3
I
3
−
I
1
I
2
1
I
1
−
I
2
I
3
2
I
1
−
I
2
I
3
1
and
1 I
1
1 I
2
1 I
3
Note that
δ
ω
ω
ω
∗
, and
δ
g
g
g
∗
. A typical control problem
is to determine the control
δ
g
that will maintain the desired motion
in the presence of initial condition errors and other disturbances.A recommended exercise is to choose values of
g
e
, and
ω
∗
, and
then compare the results of integrating the nonlinear di
ff
erential
equations with the results of integrating the linear equations.
The linearization process servesto
shift
the origin to the equilib-
rium of interestBefore
applying
a
control,
we
should
determine
whether
the
origin is
stable
in the absence of
any control (
i.e.
u
x
Ax
We further assume that
is con-
stant,
so that we have a time-
invariant systemStability analysis is based on theeigenvalues of
Consider the special case where A
is
diagonal
The di
ff
erential equation
decou-
ples
into the
n
fi
rst-order di
ff
er-
ential equations
x
1
11
x
1
x
j
jj
x
j
x
n
nn
x
n
and each solution is immediatelyintegrable as
x
j
t
e
A
jj
t
x
j
Exercise: Solve for
x
j
t
More generally,
is not diagonal,
and its eigenvalues are either
real
or
complex conjugate pairs
Denote the real and imaginaryparts of the eigenvalues by
Re
λ
j
and
Im
λ
j
The stability condition can thenbe written as
Re
λ
j
j
stability
Re
λ
j
0 for any
j
instabil-
ity
If any eigenvalue has
Re
λ
j
then the linear stability analysisis inconclusiveWe will develop the eigenvaluedecomposition next time
The general nonlinear system of equations is
x
f
x
u
, t
y
g
x
u
, t
where the state is
x
n
, the input is
u
p
, and the output is
y
m
Equilibrium motions, (
x
∗
u
∗
) satisfy
x
f
x
∗
u
∗
, t
Linearization about (
x
∗
u
∗
), and simplifying notation with
δ
x
x
gives the linear system
x
Ax
Bu
y
Cx
Du
