Category: CDS 101/110 Fall 2003

Identifiers: FN H0 H1 H2 H3 H4 H5 H6 H7 H8 L0.0 L1.1 L1.2 L10.1 L2.1 L2.2 L3.1 L3.2 L4.1 L4.2 L5.1 L5.2 L6.2 L8.1 L9.1 L9.2

Questions

• What is the advantage of having a model?

• Where/when do we get graded homeworks back?

• Should d_r be d_f in the predator-prey model?

• Can you define the variables in all of the equations from lecture 2.1?

• In a difference equation, how is the state continuous even though the time is discrete?

• Can we get more information about state space formulation?

• What is the advantage of having a model?
  Submitted by: waydo
  Submitted on: October 6, 2003
  Identifier: L2.1

  The exact question was, "Is the advantage of having a model that you can choose appropriate lengths/constants/gains before spending money and finding out that it doesn't work right? Why not just build it an watch the behavior (trial & error)?" This is exactly right. We use models to study systems when it is easier, cheaper, safer, or faster than working with a real system. Careful modeling allows us to design and build systems that work well with a minimum of trial and error. Safety is often key here - a test pilot will be much happier getting into a new airplane that has been thoroughly modeled prior to first flight than one that was built just to see if it works! A model can also allow us to study a system that is too large and complicated to build or that cannot be isolated. For example, the predator-prey model discussed in class lets us see the effects of things like a sudden increase in food supply, etc. very clearly, without having to account for the effects of other disturbances that might be present in nature. While we can observe this system in real life, the model may give us more information about specific input-output relationships than we can get from our observations. It should be noted, however, that modeling is not the correct choice 100% of the time (although it very often is) - occasionally modeling a system may be extremely difficult and time consuming while building the system is relatively easy. In such a case trial and error may be a good alternative. The word of caution here, though, is that modeling often gives us a good understanding of the system in general (i.e. we can see how performance changes as a function of particular parameters), allowing us to make inferences about many systems, while trial & error teaches us only about the systems we actually build.

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• Where/when do we get graded homeworks back?
  Submitted by: murray
  Submitted on: October 6, 2003
  Identifier: L2.1 , H0 , H1 , H2

  Graded homework will be handed back one week after they are turned in. When possible, we will put them out on the table outside 74 Jorgensen before lecture. After that, they will be available outside 109 Steele.

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• Should d_r be d_f in the predator-prey model?
  Submitted by: demetri
  Submitted on: October 6, 2003
  Identifier: L2.1

  Yes. This parameter gives the number of foxes which die in one time unit as a fraction of the overall population.

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• Can you define the variables in all of the equations from lecture 2.1?
  Submitted by: lars
  Submitted on: October 7, 2003
  Identifier: L2.1

  Many of the variables are defined in the slides, so I won't review them all. If there are specific ones you are unsure about, feel free to email me and I'll post another FAQ.

  Here's something that I think may not have been clear from reviewing the slides: In much of the material we will see in this course, x will denote a state variable or vector, u will denote the input and y will denote the output. You may want to refer back to slide 7 for reference when encountering these variables on the other slides.

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• In a difference equation, how is the state continuous even though the time is discrete?
  Submitted by: mreiser
  Submitted on: October 8, 2003
  Identifier: L2.1 , L2.2

  This type of system model arises naturally in many problems. Consider for example the predator-prey from Monday's lecture. In this model the state vector (consisting of the number of rabbits and the number of foxes) is treated as real-valued (though of course thinking about non-integer animal quantities seems silly), and the rate at which we update the population occurs at a discrete time interval, one year or one months or similar.

  Perhaps a better example is any sample-based system. Suppose our cruise controller uses some standard microprocessor system, that samples the instantaneous car velocity at some fixed rate. Here the sampling in our sensor generates the discreteness of the problem, but of course the quantity we sample can take on any value. We could try to write continuous time control laws for this system, but it seems foolish to do so, since we only get information about our system at discrete time steps. One simple thing we might want to do with these sampled values is smooth them so we are less likely to be effected by a sensor error. This can be done with a difference equation that implements a moving average, where the state vector keeps track of the last few measurements.

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• Can we get more information about state space formulation?
  Submitted by: waydo
  Submitted on: October 9, 2003
  Identifier: L2.1

  Absolutely.

  ``State-space'' refers to modeling a system using some number of quantities, called the states, that are sufficient to fully charactarize its behavior. For example, the spring-mass-damper system we see so frequently can be fully described by two states: the position and velocity of the mass. We know this is true because when we solve the ODE if we know the position and velocity at a given instant that is all we need to fully charactarize the behavior - no information about past behavior or other quantities is needed. When we model a system using state-space methods, we always end up with a system of first order differential equations in the states,

  dx/dt = f(x).

  It is important to note that our choice of states is not unique, but the number of states needed to describe a given system is fixed. For example, for a coupled spring mass system where we have two masses and two springs we can use the position and velocity of each mass as our states (4 states), but we can just as well use the sum and difference of the positions of the two masses and their derivatives (also 4 states) and we will get the same answers.

  I'm not 100% sure I gave you the answer you were looking for - please feel free to email me at waydo@cds.caltech.edu if you want more specific information.

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