Prelab 6

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State of State-Space.

We took a continuous-time path through state-space in lectures, staying with continuous time state-space for modeling, measured-state feedback, optimization-based gain selection. In lab, we used continuous-time models to determine measured-state feedback gains (using pole placement or LQR), and ignored the fact that the coil voltage on the maglev's electromagnet (the u in our model of the "plant") was only updated every 200 microseconds.

Perhaps we grew annoyed at re-calibrating the coil current measurements again and again, or were frustrated with trying to denoise the velocity estimates, but now we are ready for an alternative to measuring all the states of our maglev system. And there is an alternative, we can use observer-state feedback, and then only need to measure the umbrella position. More specificially,

  • Evolve the observer-estimated state using the state-space model,
  • Use the difference between the measured and observer-predicted umbrella position to correct the observer's estimated state,
  • Use the observer-estimated state instead of the measured state when adjusting the coil current command.

But exactly how do we "evolve" the observer-estimated state? If we are still in continuous time, then "evolving" means solving differential equations to update the observer states, and that is expensive to do on a microcontroller. But if we are in discrete-time, then "evolving" means multiplying the current state by a matrix, as we will see.

So, it is time to switch to discrete time. But we can not just "switch" to discrete-time, our physical system is in continuous time, only our controller is in discrete-time. The two do connect, through sampling and a zero-order hold. The controller "samples" the state and output of the physical system every \Delta T seconds. That is,

y[n] = y(n \Delta T),
where n is being used as the sample index. The u(t) the controller sends to the plant is only updated every \Delta T seconds, but it is "held" between samples as,
u(t) = u[\lfloor \frac{t}{\Delta T} \rfloor],
where the notation \lfloor \tau \rfloor means "greatest integer less than \tau. A typical zero-order hold continuous waveform is shown in the figure below:

Given the physical system is in continuous time, our continuous time plant model is still valid,

\frac{d}{dt}\textbf{x}(t) = \textbf{A}\textbf{x}(t) + \textbf{B}u(t),
y(t) = \textbf{C}\textbf{x}(t),
though we now understand that if u(t) is generated by a microcontroller, it is piecewise-constant in time. And in addition, we are only interested in samples of y(t) for t = n \Delta T. In particular,
\textbf{x}[n] = \textbf{A}_d\textbf{x}[n-1] + \textbf{B}_du[n-1],
and
y[n] = \textbf{C}\textbf{x}[n],
where
\textbf{A}_d = e^{\textbf{A} \Delta T} \approx \textbf{I} + \Delta T \textbf{A}
and
\textbf{B}_d = \left(e^{\textbf{A} \Delta T} - \textbf{I}\right) \textbf{A}^{-1} \textbf{B} \approx \Delta T \textbf{B},
where the approximations on the right side of the above two equations become exact in the limit as \Delta T \rightarrow 0.

The Bigger Picture

We have a path to practical observer-based control, a path we have been marking off all term long (as we hope to make clear in the examples below and in the observer lab). That is:

  1. Combine physical insight, block diagrams, and composition strategies to construct a parameterized-transfer-function system model.
  2. Use ad-hoc control strategies to enable the collection of calibration data (step- and frequency-responses).
  3. Convert the calibrated transfer function model to a CT State-Space model.
  4. Compute the assocated DT State-Space Model (assuming ZOH on u).
  5. Determine state feedback gains (using LQR).
  6. Determine observer gains (using pole placement or LQR).

We will start with an example of discrete-time state-space control, followed by a few basic problems and a description of the discrete observer approach (please read).

Below are two extra-credit prelab problems, one on a more detailed analysis of the CT to DT process, and one on examining LQR in the scalar discrete-time setting. There is also an extra credit check-off at the end of the observer lab, and in that checkoff you will examine the connection between the ideas of phase margin, state-feedback, and state-feedback with observers. We think these extra credit problems are helpful, but not essential, see the home page for how they will affect the grading.