In this screencast we will take a look at second-order underdamped transfer functions,
and most importantly how they respond to a step change in an input variable, or the manipulated
variable. The reason why this is of interest is the fact that some control schemes or some
control tuning parameters are dictated by particular landmarks on this response. So
to start lets write down the transfer function for a generic second-order transfer function,
which is what we see here, K over tau squared s squared plus two zeta tau s plus one. And
the process is considered underdamped when zeta is in between zero and one. What this
range represents is the fact that the real part of the two roots will both be negative,
and the imaginary part of the root will exist. So if you recall what that means, if the root
is real, and that means the fact that the process is stable, and the existence of an
imaginary root indicates the fact that the process will oscillate. So therefore we're
going to have a stable oscillatory response. We can do the Laplace transforms for this
but the response to a step input will be the equation provided below. So as we take a look
at this we can see how this response will be stable and oscillatory. If we look at the
exponential we see the fact that the exponential part has a negative sign present, indicating
the fact that since zeta, tau, and t must all be positive numbers, based on our definition
of an underdamped system, we know that this will head to zero. Additionally the term being
multiplied by it is a sign wave and the sign waves are known to have oscillations. So therefore
when we put these together what this means is the fact that the response to the step
input will be decaying oscillations, because as time goes to infinity this whole exponential
term will go away, and the system will converge to KM, so KM will be the steady state value,
where K is the steady state gain coming from the transfer function and M is the magnitude
of the step input. So if we where to graph the relationship for the output response provided
in the equation the graph would look like this. So as you can see in this graph we have
our decaying oscillations, which will converge at KM, but we can focus on a few interesting
landmarks here to gain more understanding about how quickly or to what magnitude the
oscillations will be. So I'll now highlight some of the landmarks on this graph that we
see for a second-order underdamped system. So we'll start by looking at t(r). t(r) represents
the rise time. The rise time represents the time it takes to reach the new steady state,
which will be KM for a step increase. So this rise time is approximated as one quarter of
the period. The period here, represented by t, is the typical definition of the period,
which is the time it takes to go from peak to peak, can be defined as a function of both
the time constant tau, and the damping coefficient zeta in the relationship two pi tau over the
square root of one minus zeta squared. Another value of interest here is the decay ratio,
so the decay ratio represents how much the peaks are decreasing each time we've reached
that point. So these peaks are related to the decay ratio, and the decay ratio can
be defined here as C over B, and in terms of tau and zeta, it can be represented as the
exponential of negative two pi zeta divided by the square root of one minus zeta squared.
So for the decay ratio the time constant doesn't influence how much this happens, its dictated
only by zeta. Another value which is dictated solely by zeta is something that is known
as the overshoot, and the overshoot represents how much does the first peak overshoot the
known steady stated value of KM? So the overshoot is defined here as B over A, where B is the
distance between the peak and KM, and A is the difference between where you start and
KM, which typically is the value KM itself, because typically we start in deviation variables
and this is zero. So the overshoot can be defined, very similar to the decay ratio,
as the exponential of negative pi zeta divided by the square root of one minus zeta squared.
So the only difference here is the fact that the decay ratio has a two in the exponential
which the overshoot does not. Finally the last value of interest is this t(s) value,
which is the settling time. So t(s) represents the settling time, and this is important because
its trying to give an idea of how quickly does it take the process to consistently stay
within some certain percentage of the steady state value of KM. So depending on what settling
time your're looking for they will have different values, but if we're looking at about one
percent settling time this can represented approximately as five tau over zeta. So therefore
if I am designing a process I am trying to tune a process. Often times I may be more
interested in making sure my system reaches a particular steady state. Other times I might
be more interested in keeping my overshoot low, or having my peaks decay at a fast rate
going from peak to peak. The important point to note here is the fact that you do have
a give and take between these variables because of where tau and zeta are located, and this
again can help dictate how you tune. For example, Cohen-Coon and Ziegler-Nichols tuning parameters
base their parameters on an assumption of the fact that the desired response to a change
is the quarter decay ratio. So in this screencast we have shown the importance of the second-order
underdamped system with a step input response, and some of the important landmarks of that
response.