SIM11 summary and conclusions (DRAFT)

This page is a draft copy of a (final?) SIM related post (SIM11?), intended to summarize all the previous SIM related posts in a coherent way. SIM5 also filled that role, but was posted at a mid-point in the process.

Outline of what I'd like to cover:

1. Introduce canonical state space representations of linear systems
1. Continuous time
1. Time-varying vs time-invariant (we'll stick with LTI here)
2. Over a field (e.g. ℝ or ℂ) (we'll stick with ℝ here)
3. Show a block diagram
4. Apply to simple circuit (use the simple RL circuit based on SIM)
2. Discrete time
1. Also LTI over ℝ here
2. Show a block diagram comparing it to the continuous time block diagram
3. Discretize the RL circuit
1. Show it can be done in different ways:
1. Impulse invariant (corrected or uncorrected)
2. Matched Z-transform method
3. Bilinear (Tustin) (w/ and w/o pre-warping)
2. Argue we are interested in impulse invariance (uncorrected) because we want our time samples to match (we're not as interested in the frequency response)
1. Provided our input is held constant over each sample period (refer to SIM4 for cases when that's not the case) 
3. Show how transfer functions (Laplace and Z-transform varieties) relate to the state space models, and briefly discuss poles and zeros and stability.
2. Introduce SIM
1. Use Jason's extracted tables from G&L's book, and provide a link to G&L's book
2. Show that it can be arranged to look like a discrete time state space model
1. Show that it's non-canonical in it's indices, and the definitions that need to be made to have a canonical version
2. Describe their period indexing, and that we'll use an alternative index starting at 0 corresponding to t=0 to avoid confusion (except where we tie our results back to theirs)
3. Describe the implicit dependence of the SIM model on the default sample period Ts0
1. Their flow variables go to zero as the sample period Ts goes to 0 and why that is.
2. This doesn't happen with stock variables (H) and why
3. How that's unlike the continuous time and discrete time circuit example you already covered and why (engineers are usually interested in what amounts to "instantaneous flows" themselves, rather than what G&L call a flow over a period)
4. Show that if the sample period remains fixed at its original value (Ts = Ts0) that the RL circuit can produce G&L's original discrete time SIM model (SIMd) (canonical version) outputs at the sample times, but that it won't necessarily obey G&L's SIM accounting identities at other sample times or with other sample periods (though it should be close)
3. Discuss the problem of changing SIMd's sample period or sample times
1. Show that a purely discrete time model in which the activity is confined only to sample times is unaffected by a change in sample times, but that the "adjustment time" (time constant = Tc) of the model then becomes completely dependent on the sample times
2. In all other cases (where we want the model to be independent of the sample times) we must assume an underlying continuous time model that we're sampling. This (as Nick Edmonds has pointed out) is a "fiction" and there are an infinite number of such models which will satisfy G&L's SIM accounting identities at the original sample times.
1. Introduce the alpha2 scaling method of Ramanan and "A H" and discuss it's pros (preserves the steady state (SS), satisfies accounting, is simple) and cons: it only approximately keeps Tc fixed, and there are two sets of limitations on the parameters (alpha1, alpha2 and theta): G&L's and those derived from other analysis (e.g. that it might start to oscillate and/or become unstable), which in turn limit the choices for sample times (Ts).
2. Cover the method already described by discretizing the RL circuit (SIM2 method): pros: preserves SS, and Tc, no restriction on choice of Ts, just slightly more complex than simple; cons: only approximately satisfies accounting when Ts ≠ Ts0, non-canonical offset must be accounted for separately. Point out this is the SIMm (m for "mean") version of a continuous time version of SIMd presented in SIM8.
3. Cover the method described in SIM6: adjusting both alpha1 and alpha2 for a new Ts. Explain why (using SIM7) theta is not changed. Pros: preserves SS, Tc, only moderately restricts choice of Ts (less restrictions that method 1. but more than method 2), preserves accounting (show how integration of flows (via summing in this case) must be used to verify this). Cons: slightly more complex than 2 and there's still a restriction on choice of Ts.
4. Show how SIM9 and especially SIM10 expand on the idea presented in SIM3 and partially tested with "alt" variables in SIM2 to construct a continuous time version of SIMd called SIMc which does not alter SIMd parameters to satisfy accounting, but rather inherently satisfies the accounting over any arbitrary period provided the flow variables are integrated.
1. Show the equations for SIMc and how they're derived
2. Argue that this version of an underlying continuous time version of SIMd is in some sense the "simplest" and smoothest and "most natural" (not that this is the simplest way to change sample time though).
3. Pros: preserves SS, Tc, works for any Ts, preserves accounting over any arbitrary period.
4. Cons: complexity: must integrate instantaneous flows to recover SIMd flow values (and you must select the same sample periods of course).
5. Show a procedure for changing to any arbitrary set of sample times with this procedure and for the special case of an arbitrary fixed sample period (with arbitrary phase) and for an arbitrary fixed sample period with the phase such that one sample time is at t=0. (I have yet to work this out yet actually).
3. Discuss how given the original sample period (Ts0) and given a set of SS values and a desired Tc, values for alpha1, alpha2 and theta (i.e. SIMd parameters) can be selected (this is the body of SIM7).
3. Briefly discuss analog computers and their implementation with op amps.
1. Refer to digital equivalents such as MatLab's SIMULINK or SciLab or Octave equivalents to SIMULINK.
2. Introduce op amp circuit implementation shown in SIM10. Also show how this op amps circuit can easily be modified to recreate the results of the RL circuit or the full SIMm model (change resistor values and remove ping-pong "accountant" integrators on output).
3. Discuss modeling non-linearities with analog computers or SIMULINK-like tools, or regular Matlab or Octave differential equation solving scripts.
1. Show the canonical 1st order non-linear differential equation x' = f(x,u)
2. How it can be solved numerically or by linearizing it around an operating point
4. Briefly discuss feedback control systems, estimation (Kalman filters), optimal control and robust control and how this relates to models such as SIM:
1. Discuss Laplace and Z-transforms and transfer functions, and show the Laplace and Z-transform transfer functions for SIM (perhaps this has already discussed in the 1st part which introduces general LTI state space models?)
2. Discuss frequency response, root locus, pole placement, Nyquist and Bode charts
3. Discuss noise and disturbances
4. Show an example with SIM (circuit model?)
5. Discuss LQC and optimal control and L2
6. Discuss model uncertainties, robust control, loop shaping, H∞, and μ-synthesis

Whew! That's a lot. Lol... I doubt that will be suitable for one post. Maybe I need to make another pass through and cut out some stuff. I'm caught between wanting to completely rework all my old posts to make their diagrams and nomenclature and symbols consistent and wanting to just leave all that as reference and summarize all of it in a consistent coherent fashion (plus introduce some things I haven't yet covered).

Finally a few other miscellaneous ideas (I'm not sure where (or if) to put them yet):

1. Show closed form expressions for SIMd, SIMm and SIMc outputs
2. Provide another spreadsheet which uses these closed form expressions to demonstrate that G&L's accounting identities for SIM are satisfied (or not satisfied as the case may be).
3. Provide a diagram like in SIM9 but simplified to be consistent with the "integrate over each sample period" function produced by the op amp "ping-pong" integrator "accountants" in SIM10.
4. Discuss SIM as a low pass filter with zeros
1. Show it's the frequency response of its transfer functions is discrete and continuous time
2. Show some pole-zero plots
3. Show how these kinds of systems can be easily manipulated using Matlab/Octave's control package.

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And at the conclusion of all this, I think I can say I've done about all I can do with SIM... except read G&L and figure out what it's really supposed to be about... Lol.
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