Latest / Mechanical Engineering Made Simple / (#49) Modeling Control Measurement The Engineering Cycle

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(#49) Modeling Control Measurement The Engineering Cycle · Mechanical Engineering Made Simple

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  Welcome back design junkies, to Mechanical Engineering Made
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  Simple. Today we are doing a deep dive
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  into the the foundational principles that engineers use to
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  understand, predict, and ultimately control dynamic
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  systems. Right.
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  It's really a classic three-part journey for anyone building
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  complex machinery. Yeah, first you have to create a
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  model, then you design the control logic and 3rd and this
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  is key, you have to make sure your measurements are accurate
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  enough to validate the whole process.
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  So our mission today is to really pull out the key insights
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  from these sources that make that whole cycle work.
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  Exactly. And it all starts with modeling.
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  I mean, why do we spend so much time creating these mathematical
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  representations of reality? Well, the answer is pretty
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  simple actually. Models are the ultimate
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  engineering shortcut. A shortcut how?
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  They're cheaper for one. Yeah, and infinitely faster to
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  build than a physical system. You can test things you know,
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  failure modes, component swaps, things would be way too
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  dangerous or just plain impossible to try on real
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  hardware. So let's you ask those what if
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  questions. Precisely for systems that might
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  only exist on paper. OK, but that brings up this
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  fundamental paradox of engineering, doesn't it?
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  We assume the best model is the most accurate one, the one that
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  captures every tiny detail. And that sounds like good
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  science, but it could be really poor engineering, right?
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  It's the trade off. The key insight here is that the
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  best engineering model isn't the most accurate one.
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  It's the simplest one that still gives you the information you
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  need. Because if it's too complex.
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  If it's so complex you can't solve the equations
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  analytically, then you're stuck running experiments on a
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  computer simulation anyway. So simplicity is king.
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  You want the lowest order, most linear model you can get away
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  with, which I guess brings us to lumped parameter modeling.
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  Yes, and that phrase is so important.
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  The real world is full of distributed parameters.
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  Think of a long steel beam where mass and stiffness are spread
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  out everywhere infinitely. A lumped parameter model just
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  says, let's pretend all that mass is in one spot and all that
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  stiffness is in another. We replace the continuous world
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  with a set of discrete, idealized springs, masses and
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  dampers. Which makes the math possible.
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  And the really beautiful thing is that this approach is
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  universal. Whether you're designing a
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  robot, a fluid circuit, an electrical filter, you're using
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  the same basic alphabet of elements.
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  And that alphabet is built around energy.
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  You have these idealized 2 terminal elements, and they're
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  defined by two variables. Right.
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  First is the through variable. That's the quantity that flows
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  or gets transmitted through the elements like force in a spring
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  or current in a wire. And the other one is the cross
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  variable. Exactly.
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  That's what you measure across the elements terminals.
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  So the velocity difference across a damper, or a voltage
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  difference, or a pressure drop. And with just those two
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  variables, you can classify every passive element based on
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  how it handles energy. We put them into three groups.
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  It's all about storage or dissipation.
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  So let's start with storage, OK? So we have the T type or
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  inductive storage elements. These store energy because of
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  the through variable. The classic electrical example
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  is an inductor which stores energy based on current.
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  And in the mechanical world? That's a spring, a translational
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  or rotational spring. Hang on, I thought springs
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  stored potential energy based on displacement, which feels like a
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  state across its ends. Why is it AT type?
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  That's a great question. It's because displacement is the
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  integral of velocity, the across variable.
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  If you look at the fundamental math of the element, the energy
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  storage really depends on the accumulation of the through
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  variable, which is force in this case.
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  OK, that's a subtle distinction. It is, but let's look at the
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  other storage type which is maybe a bit clearer, the A type
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  or capacitive storage elements. Right, the A type elements store
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  energy based on the across variable.
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  Correct. And the perfect mechanical
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  example is a mass or an inertia. It stores kinetic energy based
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  on its velocity, which is an across variable.
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  The electrical equivalent is of course, a capacitor.
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  So T type for springs and inductors, a type for masses and
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  capacitors that covers storage. What's the last?
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  One, that's the D type dissipative.
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  These are your resistors. They just get rid of energy,
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  usually as heat. So friction dampers, electrical
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  resistors, they all fit here. And with just those three TA and
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  D, we can build a model for basically any dynamic system.
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  It's. The universal alphabet.
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  So once we've built the model, we need a standard way to write
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  down the math. Classically that was the input
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  output form the IO form. Yeah, that's what most people
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  learn first. It's kind of the black box view.
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  You don't care what's inside, you just want to relate a single
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  input to a single output with one big differential equation.
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  And that was powerful because it led to things like the Laplace
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  transform. Exactly.
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  You could turn these horrible differential equations into
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  simple algebra, which made them much easier to solve, but.
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  Getting that one equation sounds.
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  Awful. Oh, it's brutal.
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  Think about modeling even a simple milling machine.
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  To get the relationship between the input torque and the final
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  table position, you have to algebraically eliminate every
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  single internal variable, every velocity, every internal force.
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  And if you change just one little damper value in your
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  model. You have to do the entire
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  derivation all over again. It was incredibly tedious, and
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  that's exactly why the state variable form took over in
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  modern control theory. Because the IO form just
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  couldn't handle complex systems. It becomes unwieldy, especially
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  with multiple inputs and outputs, and more importantly,
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  it hides what's actually going on with the energy inside the
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  system. So the state variable form makes
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  us look inside the box. How does it work?
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  You define a set of state variables.
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  They're the minimum number of variables you need to completely
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  describe the energy state of the system at any instant.
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  If you know them, and you know the current input, you can
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  predict the entire future of the system.
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  And what's the, you know, the natural choice for those
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  variables? This is the really elegant part.
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  The natural state variables are the through variables on your T
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  type elements and the across variables on your a type
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  elements. Yeah, the variables that
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  literally describe the stored energy.
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  Precisely. It makes perfect physical sense.
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  You need to know how much energy is in all your storage tanks to
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  know what's going to happen next.
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  And this lets us write the whole system down in a really compact
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  matrix form. Right, we use a state vector X
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  of T which depends on an input vector U of T and two matrices
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  A&B. It might look abstract, but the
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  A matrix just describes how the energy states interact with each
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  other, and the B matrix describes how your external
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  inputs affect them. It scales beautifully.
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  OK, so once we have that solid model, we can finally move on to
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  actually controlling the system. What's the standard structure
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  for a control system? It's always a loop.
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  You start with sensors to measure the output.
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  That measurement goes to an error detector, which compares
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  it to what you want it to be. The error then goes to the
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  controller, the brains, which sends a command to the actuators
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  which physically changed the system.
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  Let's talk hardware for a second.
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  The sensors today are just incredible, like optical
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  encoders. Oh absolutely.
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  I think about a modern 16 bit absolute encoder.
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  It gives you something like 10,435 counts per Radian.
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  You're resolving the position down to a tiny tiny fraction of
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  a degree. Your model is useless without
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  that kind of fidelity. And then on the output side you
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  have the actuator. The classic armature controlled
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  DC motor is a good example. The torque is just proportional
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  to the current you put in. Yes, but there's a really cool
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  piece of physics in there. The back EMF.
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  As the motor spins it generates its own voltage that pushes back
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  against the voltage you're applying.
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  It's a natural self breaking mechanism.
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  And if you don't account for that in your model.
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  Your controller will never work correctly.
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  OK, so you've got good sensors, powerful actuators.
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  Now for the brains of the operation, the PID controller.
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  PID, the bedrock of control. Proportional, integral and
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  derivative. Let's break that down.
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  Proportional or P is the simple one.
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  Yeah, it just responds instantly.
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  The bigger the error, the bigger the control action.
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  It's fast, but it has one major flaw.
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  The offset error. The offset error exactly.
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  If there's any damping in the system, P control by itself
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  can't quite push the error all the way to 0.
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  It gets stuck in some small steady state value.
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  Which is unacceptable in most cases, so that's why we need the
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  I term integral. Right, it's sometimes called
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  reset action. It basically keeps adding up the
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  error over time. As long as there's any error,
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  the control signal keeps growing, which eventually forces
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  that offset to 0. It's the essential tool for
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  accuracy. It is, but here's that classic
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  trade off again. Integral action is great for
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  steady state, but it often hurts your transient performance.
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  It can make the system sluggish or even unstable if you turn the
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  gain up too high. Which brings us to D for
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  derivative. This is the one that looks into
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  the future. Sort of, yeah.
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  It responds to the rate of change of the air, not the air
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  itself. It's all about providing
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  damping. If the system is oscillating
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  wildly, the rate of change is high, so the D term pushes back
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  hard to calm things down. It helps the system settle
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  faster, so why not just use decontrol by itself if it's so
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  good at damping? Because it amplifies noise.
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  Any real sensor has some high frequency noise, some little
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  jitter. The derivative operation just
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  mathematically magnifies those rapid changes.
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  So you'd be sending all that jitter straight to your motor.
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  And you could get this jerky destructive motion.
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  In practice, you almost never use a pure derivative.
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  You filter it or apply it in clever ways to get the damping
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  without the noise amplification. It's a constant juggling act.
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  So that brings us to the final piece.
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  We've got the model, we've got the controller, but none of it
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  matters if our measurements are wrong.
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  It all starts with the foundational standards from
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  NIST, right? Absolutely, yeah.
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  We have to remember all accuracy traces back to just four
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  independent standards. Length from the speed of light,
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  time from the cesium atom, that's 9,192,631,770 periods,
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  Mass from the International Prototype kilogram and
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  temperature from the triple point of water.
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  Every precise measurement is linked back to those.
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  And we have to get our terms right.
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  People mix up accuracy, precision and resolution all the
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  time. All the time.
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  Accuracy is how close you are to the true traceable value.
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  Precision is just how repeatable your measurements are.
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  You can be very precise, but completely inaccurate.
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  Like a broken clock that's stuck.
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  It's precisely at 3.15, but it's not accurate most of the day.
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  Good analogy. And resolution is just the
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  smallest change your instrument can even detect.
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  But the real pitfall in all of this is what's called the
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  loading problem. Yes, this is the inescapable
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  fact that the act of measuring something changes the thing
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  you're trying to measure. Give me a simple example.
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  Checking the air pressure in a tire.
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  The moment you connect the gauge, you hear that little
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  severe escaping. You've just lowered the pressure
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  by the very act of trying to measure it.
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  That's loading. So how do engineers deal with
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  that? We use a concept called
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  generalized input impedance, which we'll call Zgi.
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  It's the ratio of an effort variable to a flow variable, and
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  the rule is simple. When you're measuring an effort
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  variable like voltage or force, you want the input impedance of
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  your measuring device to be huge, as large as possible
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  compared to the system you're measuring.
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  So if I'm using a lewd cell to measure force in a structure, I
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  want that load cell to be incredibly stiff.
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  Exactly. You wanted stiffness to be way,
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  way higher than the stiffness of the thing you're pushing on.
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  If your load cell is too soft, it deforms.
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  It changes the system's dynamics and your reading is wrong.
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  This all comes together in error analysis.
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  Let's take that classic pendulum experiment for measuring gravity
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  G. The equation is g = 4π ^2 L / t
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  ^2. Right.
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  And if you run the numbers on the most probable error, you
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  find something amazing. What is it?
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  The error in measuring the period T contributes 4 times as
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  much to the final error in G is the error in measuring the
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  length though. Four times.
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  Wow, that's a huge insight it. Completely changes how you do
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  the experiment. It means you can be a little
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  casual with your measuring tape for the length, but you have to
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  be absolutely meticulous about timing the period.
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  Maybe a time 100 swings to average out the error.
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  It tells you exactly where to focus your effort.
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  And your money. But measurement fidelity isn't
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  just about buying expensive gear.
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  It's about understanding where the errors come from and how
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  they compound. So we've really covered the
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  whole cycle, building simple, robust models with T A&D
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  elements, designing effective PID controllers and ensuring our
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  measurements are actually trustworthy.
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  And it really shows the power of these analytical linear tools.
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  We spent this whole time talking about linear models because we
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  can solve them. But here's the final provocative
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  thought to leave everyone with Pretty much every real system is
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  nonlinear. That's right, springs don't
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  behave perfectly. Resistance changes with heat.
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  All physical systems are non linear to some degree.
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  And for those. For high order non linear
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  systems there are rarely any general analytical solutions, so
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  computer simulation is the only tool that works universally.
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  Which is why we're seeing this push into advanced digital
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  control. I'm thinking of modern robotics.
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  Exactly. To get that smooth precise
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  motion they use things like the computed torque method.
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  The controller is running so fast that it's actively
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  calculating all the complex non linear gravity and centrifugal
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  forces in real time. And just mathematically
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  cancelling them out. It's the ultimate evolution.
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  You use a complex non linear model, run it with incredibly
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  fast computation. Also that a simple linear
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  control system can work perfectly in a messy non linear
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  world. And you can't even begin to
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  design those computational masterpieces without mastering
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  the fundamentals we covered today.