Hi, firends:
I am studying automatic control system. Seemsly, only in second system or
first system the theory offers the mathmatic equation for settling time and

overshot, for the high order system, we only have to get the diagram of time
response to check overshot and settling time, isn't it? Thank you very much!

More or less. You could, if you were persistent, come up with a
mathematical equation for each different type of system. Doing so,
however, would take more time out of your life than you'd ever save by
diagramming and checking each instance you run into.
Frankly, in my ever so humble opinion the primary value of the various
performance calculations is more to give you an idea of the effect of
system characteristics on the system performance, rather than to have a
firm guide for system design.

'...the theory...' to which you refer is actually the mathematics of
differential (or possibly difference) equations and their solutions. This
topic is not exclusive to automatic control.
Because automatic control is primarily concerned with the dynamic response of
feedback control systems, it uses that same mathematics that has been
developed over hundreds of years (since Newton) to model and understand the
dynamic response of any physical system, whether mechanical, electrical,
biological, etc. Simple and memorable analytical solutions are readily
available for low-order linear differential equations, and because so many
real-world systems can be approximated by 1st, 2nd-order dynamic responses,
then any engineer concerned with system dynamics in any discipline (not just
in control engineering) needs to be familiar with the essential
characteristics and the shortcuts and rules-of-thumb that can be used to
describe their behaviour. For higher-order equations the analytical solutions
can be onerious to resolve, and in any case they do not have a form that is
indicative of the time-response: there are no simple equivalents of things
like natural frequency and damping ratio that can be picked-out from the
analytical solution. So one, turns to numerical solutions and response plots.
Kelvin B. Hales
Kelvin Hales Associates Limited
Consulting Process Control Engineers
E-mail: snipped-for-privacy@khace.com
Web: www.khace.com

...
My hat is off to you, Sir!
That is a cogent summary and statement of purpose that belongs in the
introductions to texts on math and engineering. Unfortunately many
engineers reach retirement without ever figuring that out.
Jerry

--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

I spent many years in Systems Analysis which encompassed the working and
understanding of control systems (hydromechanical,analog & digital). Unlike
what typically occurs in a plant, system response was very fast; in the .020
to 3 sec time frame--control of the steady state and transient response of a
jet engine. I'm not into the conceptual mathematical modeling that defined
the dynamics of the complete system but once that was out of the way, got
very involved into the block (functional & logic) diagrams. These systems
are very complex usually with more than one parameter controlling the engine
at the same time (eg. speed, temperature). Add to that the effects of a
variable exhaust nozzle and an afterburner system and you end up with a
nightmare of interaction between several control loops transitioning (on the
fly) from one control mode to another. Then you have the fact that the
dynamic response of the engine does not remain the same as the operating
conditions change--You have to control this beast from sea level static up
to Mach 2.0; from sea level to 60,000 ft. and at inlet temperatures of minus
65 F to 130F. So the control system has to be flexible and robust enough to
provide a transient response that meets rigid requirements and at the same
time protects against overspeed, overtemperature, flameouts, stall etc. and
to transition into any one of a number of failure modes in the event there
is a system malfunction.
It is next to impossible to analytically and mathematically define
transient system behavior under all the operating conditions noted above. So
what's the alternative?? What has not been discussed in this thread and what
we relied very much on was the use of a transient model. The model allowed
us to simulate any operating condition and any transient and gave us a good
insight into predicting the behavior of the engine and to observe the time
response of any defined parameter, control or engine.. More importantly,
however, was the ability to input any operating condition and simulate any
transient (throttle push) while in the throes of investigating and
troubleshooting a field problem. There was a pretty big separation between
those that did the analytical modeling and those of us that had to work with
the hardware and diagnose/troubleshoot and fix operating problems.
Having said all this, I guess my purpose was to interject and mention that
one can only go so far with the mathematics, stability plots and analytical
methods. Do those of you involved in plant systems etc. stop with the block
diagrams or do you move on into developing the transient models?
MLD

Actually, the process guys build dynamic models for training purposes and
for doing gross tests of the controls for the more complex process (not
necessarily the more complex loops).
But in the end we build the plant, make all controllers adjustable and then
tune the controls live during early operations. If things get seriously out
of hand, the automatic shutdowns will cover our ass, save our necks, etc. I
understand this procedure is not optimal for aircraft :-) Or is that why
test pilots have parachutes?
Walter.

What do you mean by a transient model? Is this a linearized
mathematical model at one operating point, or do you measure the
response of an engine, or what?
I do a lot of work with small motion controllers for production
machines. On those systems I generally start from a mathematical model
which I use to inform the process of setting sampling rates, determining
required ADC and DAC accuracy, and coming up with the initial controller
structure. Once I have real hardware I do sine sweeps to measure
responses, then use that to feed back into the loop tuning. I have not
so far, however, needed to deal with the range of operating conditions
that you do.

what
allowed
good
time
any
between
with
that
analytical
block
To answer your question--Transient model is a mathematical simulation of the
engine and control system. The engine is defined via the thermodynamics of
the compressor, combustor, turbine, the mechanical components etc. It is not
a linear model as there many functional curves that define the thermodynamic
behavior and a whole slew of time constants as a function of the inlet
conditions (altitude, Mn, pressure recovery, inlet duct losses etc.). The
engine operating requirements change dramatically as a function of the
flight conditions and these changes have to accounted for in the Model.
There are also restraints that must be included in the Model-- such
as --stall characteristics, combustor blowout (flameout) boundaries,
compressor stability maps to mention a few. The thermodynamics of the
afterburner system and so on. Then there is the control system which is
designed to provide the steady state and transient operation as well as to
protect the engine against itself. Again, the control system dynamics are
not linear and are biased by the operating conditions--fuel flow, pressure,
servo flow to the various control components, engine (pump) speed and so on.
These all have to be built into the Model. And as previously noted, the
Model allows us to look at any parameter, engine or control as a function of
time or any number of parameters plotted against one another. When making a
design change to fix a problem at sea level or altitude for example, the
Model is used to evaluate the change over a range of flight conditions
through out the flight envelope, altitude and Mn, to insure no surprises
when it gets into the air.
When I said that things happen very quickly I guess quickly is a relative
term based on what you, or others, might define as fast. Let me give an
analogy of the time domain that we often deal with. You have a tachometer
in your car--when you step on the gas the tach moves, seemingly, almost
simultaneously with the change in the gas pedal position. Well, most of my
work took place in the time frame between stepping on the gas and the
initial movement of the tach. Think about it--a lot of things happen in
that time frame. Step on the gas pedal--linkages/cables have to
move--throttle position changes increasing the air flow into the engine and
increases the demand for fuel --the fuel pump responds increasing fuel
flow--the increased fuel mixes with the air and is drawn into the
cylinder--at the right time the spark plug fires--the fuel/air mixture
ignites powering the piston resulting in a change in engine speed.--All of
this takes maybe 20+ millisec or so. When you get into looking at something
like a compressor stall, you're dealing with .005-.010 secs in trying to
figure out what happened to cause the stall.
Don't know if I actually answered your question or not--just seemed to have
been carried away <g>.
MLD

So
what
allowed
good
with
analytical
block
I assumed 'transient model' was synonymous with 'dynamic model'; as in, for
example, dynamic simulation.
Kelvin B. Hales
Kelvin Hales Associates Limited
Consulting Process Control Engineers
E-mail: snipped-for-privacy@khace.com
Web: www.khace.com

Although here we do actually maintain a very clear distinction between the
activities of 'modelling' and 'simulation' and the results of those
activities; i.e. 'models' and 'simulations. Especially since there is more
than one way of finding the solution, in simulation, to any given model;
e.g. sequential vs. equation-based solvers, choices of numerical integration
methods, different simulation software packages & simulation environments,
etc.
Kelvin B. Hales
Kelvin Hales Associates Limited
Consulting Process Control Engineers
E-mail: snipped-for-privacy@khace.com
Web: www.khace.com

It is apparent that your definition of "simulation(s)/modeling" as compared
to mine most likely comes from where we sit. Obviously, I am much looser in
my definitions. I was not into the conceptional design of the
analytical/mathematical methods used in describing the dynamics of a system.
I have a great respect for the talent it takes to put together a complex
model. We have a unit whose sole function is to provide that kind of
modeling support (steady state or transient). Basically, I was an end
user--- for me, the dynamic model was a tool to be used for many varied
purposes----to evaluate/tune-up transient system behavior with respect to
meeting system requirements; investigate field/production problems; to
define/evaluate fixes; simulate off-design conditions and failure modes;
evaluate the introduction of proposed design changes and so on. In many
cases, key parameters taken from the infamous aircraft "black box" were used
as inputs to the model and used to help understand what was going on at the
time of an incident under investigation. Certainly not unlike the things
that are done with and used with the systems that you're involved in--just a
difference in the end product.
MLD

We typically work with Chemical and Process Engineers to develop rigourous
dynamic models of plant (based on physics, chemistry, thermodynamics, etc.)
and its controls; then we use both dynamic simulation studies and dynamic
analysis of the model to investigate the potential transient behaviour of the
plant under various operating and disturbance scenarios. We use various
feedback-controller design techniques from Control System Science to
determine critical controller tuning settings in advance of plant
commissioning - especially for those controls which are not amenable to
trial-and-error tuning during commissioning, or for which the closed-loop
performance is important in achieving/evaluating good overall system
behaviour. Case Studies have included gas compression plant for air and
natural gas; liquid-phase and fluidized-bed chemical reactors; boilers, HRSGs
and site-wide steam systems; distillation columns; water-treatment plant;
calciners; oil-field 1st-stage treatment, especially for slugging production
flow, etc. Currently completing simulation on a site-wide steam-system with
7x170t/h-boilers + 510t/h from HRSGs, with around 150 steam-pipes,
20-consumers. The blocks in our block diagrams are therefore rigourous models
of system elements, typically with many hundreds of blocks per system. If we
need linear models; e.g. for frequency-response analysis, then our
simulation software allows us to linearise our rigourous models between any
two points. I'll be talking next on what we do in Water Treatment Plant in
'Session3D: Modelling and Data Structures' at this event
<http://www.iee.org/events/water.cfm#prog1 .
Kelvin B. Hales
Kelvin Hales Associates Limited
Consulting Process Control Engineers
Web: www.khace.com

snip
It never ceases to amaze me how much more performance there can be in a
loop after you've tuned it b'guess & b'gosh, and how often that kind of
performance isn't necessary.
Kevin hit the nub here: There are times when you have to tune the loop
before you ever turn the system on, and there are times when you have to
formally tune the loop just to optimize performance.
I would add that there are times when you need to go from models when
you are implementing a loop that is going to exhibit a rapidly changing
structure due to nonlinearities (e.g. a motor with a current limit,
being driven into a stop in which it will jam if it's going too fast
when it hits).
But there are also 1000's of loops that don't need much tuning at all;
as long as they're stable and there's an integrator in there somewhere
keeping the setpoint correct they'll make money all day long.

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