Electrical Engineering Dictionary - part 103
Nash equilibrium in such games is guaran-
teed under assumptions on the existence of
a unique solution for respectively defined
coupled Riccati equations. The equilibrium
strategies appear to be linear functions of the
associated state trajectories. The existence
and linear form for the feedback Nash strate-
gies could be guaranteed by the existence of
positive semidefinite solution to the relevant
coupled Riccati equations but it does not at-
tribute an uniqueness features to the solu-
tion set. For the zero-sum linear-quadratic
games, the situation becomes simpler both
in discrete-time and continuous-time cases.
The saddle point strategies could be found
by solving standard Riccati equations, and
whenever both open-loop and closed-loop
solutions do exist they generate the same state
trajectories. For example, in the continuous-
time zero-sum linear-quadratic game defined
by the state equation
˙x = Ax + B
1
u
1
+ B
2
u
2
and the quadratic cost functional
J =
1
2
Z
T
0
(x
0
Qx + u
0
1
u
1
+ u
0
2
u
2
)dt
the saddle point strategies are given by
u
i
= (−1)
i
B
0
i
K(t)x; i = 1, 2
where
K(t) is a unique symmetric bounded
solution to the matrix differential Riccati
equation
˙
K +A
0
K +KA+Q−K(B
1
B
0
1
−B
2
B
0
2
) = 0
with
K(T ) = 0.
linearity
a property of a system if that
system obeys the principle of superposition.
In other words, if the output
y(t) is a function
of the input
x(t), i.e., y(t) = f (x(t)), and
if
x(t) = αx
1
(t) + βx
2
(t), then for a linear
system,
y(t) will be = αy
1
(t)+βy
2
(t) where
y
1
(t) = f (x
1
(t)) and y
2
(t) = f (x
2
(t)).
linearization
approximation of a nonlin-
ear evolution equation in a small neighbor-
hood of a point by a linear equation. This is
obtained by keeping only the first order terms
in the Taylor series expansion of the nonlin-
earities about this point.
linearized machine equations
state equa-
tions obtained by linearizing the nonlinear
voltage and electromagnetic torque equa-
tions of induction or synchronous machines.
The state variables can be either currents, or
flux linkages. The linearization can be ac-
complished using a Taylor series expansion
of the machine variables about an operating
point, i.e.,
g(f ) = g(f
0
) + g
0
(f
0
)1f
+ higher order terms
where
f = f
0
+ 1f (and f
0
is the value of
f at a given operating point).
The small displacement characteristics
are then approximated as
g(f ) − g(f
0
) = 1g = g
0
(f
0
)1f
The linearized equations are typically ma-
nipulated into a standard state model form
dx
dt
= Ax + Bu
y = Cx + Du
and are used for Eigensystem, stability, and
control analysis and design.
linearizer
an equipment or circuit that is
used to reduce distorted components gener-
ated in nonlinear amplifiers such as traveling-
wave tube amplifier (TWTA) or solid state
power amplifier (SSPA). There are various
kinds of linearizers, such as predistortion,
feedback, and feedforward types.
lineman
utility employee working on pri-
mary facilities, distribution class equipment,
as opposed to customer service level facili-
ties.
linguistic hedge
See
linguistic variable
variable for which val-
ues are not numbers, but words or sentences
c
2000 by CRC Press LLC