Page 186 - Modern Control Systems

160 Chapter 2 Mathematical Models of Systems

Homogeneity The property of a linear system in which Principle of superposition The law that states that if two
the system response, y(t), to an input «(0 leads to the inputs are scaled and summed and routed through a
response /3y(f) when the input is (3u(t). linear, time-invariant system, then the output will be
identical to the sum of outputs due to the individual
Inverse Laplace transform A transformation of a function scaled inputs when routed through the same system.
F(s) from the complex frequency domain into the
time domain yielding f(t). Reference input The input to a control system often
representing the desired output, denoted by R(s).
Laplace transform A transformation of a function /(t)
from the time domain into the complex frequency Residues The constants k\ associated with the partial
domain yielding F(s). fraction expansion of the output Y(s), when the out-
put is written in a residue-pole format.
Linear approximation An approximate model that re-
sults in a linear relationship between the output and Signal-flow graph A diagram that consists of nodes con-
the input of the device. nected by several directed branches and that is a
graphical representation of a set of linear relations.
Linear system A system that satisfies the properties of Simulation A model of a system that is used to investigate
superposition and homogeneity.
the behavior of a system by utilizing actual input signals.
Linearized Made linear or placed in a linear form. Taylor Steady state The value that the output achieves after all
series approximations are commonly employed to the transient constituents of the response have faded.
obtain linear models of physical systems.
Also referred to as the final value.
Loop A closed path that originates and terminates on the 5-plane The complex plane where, given the complex
same node of a signal-flow graph with no node being number s = s + jw, the x-axis (or horizontal axis) is
met twice along the path.
the s-axis, and the y-axis (or vertical axis) is the ;w-axis.
Mason loop rule A rule that enables the user to obtain
a transfer function by tracing paths and loops with- Taylor series A power series defined by g(x) =
in a system. m
^ :—(x — x 0) . For m < 00, the series is an
Mathematical models Descriptions of the behavior of a
system using mathematics. approximation which is used to linearize functions
Natural frequency The frequency of natural oscillation and system models.
that would occur for two complex poles if the damp- Through-variable A variable that has the same value at
ing were equal to zero. both ends of an element.
Necessary condition A condition or statement that must
be satisfied to achieve a desired effect or result. For ex- Time constant The time interval necessary for a system to
from
change
one
by a specified
per-
state
to another
ample, for a linear system it is necessary that the input centage. For a first order system, the time constant is
m(t) + M 2 (0 results in the response y^t) + frit),
where the input u^(t) results in the response y^t) and the time it takes the output to manifest a 63.2%
change due to a step input.
the input 112(() results in the response yi(t).
Transfer function The ratio of the Laplace transform of
Node The input and output points or junctions in a the output variable to the Laplace transform of the
signal-flow graph.
input variable.
Nontouching Two loops in a signal-flow graph that do not Underdamped The case where the damping ratio is f < 1.
have a common node.
Unity feedback A feedback control system wherein the
Overdamped The case where the damping ratio is £ > 1.
gain of the feedback loop is one.
Path A branch or a continuous sequence of branches Viscous damper A type of mechanical damper where the
that can be traversed from one signal (node) to model of the friction force is linearly proportional to
another signal (node) in a signal-flow graph.
the velocity of the mass.
Poles The roots of the denominator polynomial (i.e., Zeros The roots of the numerator polynomial of the
the roots of the characteristic equation) of the trans- transfer function.
fer function.
Positive feedback loop Feedback loop wherein the output
signal is fed back so that it adds to the input signal.