Optimal Estimation and Control: A Fresh Look

DURATION: FIVE DAYS
COURSE NO.: 1080

LQG theory for regulators and Kalman theory for observers both require you to believe that all process and measurement noise is white and Gaussian, when no physical noise has either property. Further, LQG theory forces you to accept that minimizing the trace of a weighted covariance of the state variables is whats good for you, while Kalman theory relies on minimizing the trace of the measurement error of weighted residuals, whose relation to engineering desires is tenuous at best. This course is designed to show you how to design optimal regulators and observers for stationary systems, based on knowledge of the all the noise power spectra, and using performance measures that are much more sensitive to your actual engineering needs. You will also see that, by your own measure, performance is likely to be greatly improved, relative to LQG or Kalman-derived feedback gains. While several papers and seminars have been given on this subject, this is the first course offered on how to do it, along with the current status of the theory. Throughout, obscure matrix relations are introduced as needed. Caveat: This theory is all analog. Its the oversampling limit of a system with infinite computational resources. A discussion will be given on directions for future research.

COURSE MATERIALS:
Include extensive notes and reference materials, plus a set of the graphics displayed during the course.

Estimation and controls practitioners who wish to get much better performance from their systems by merely using better feedback gain matrices, and those interested in continuing the research. While the material is heavily mathematical, the essential ideas in statistics, power spectra, and matrix algebra and calculus will all be covered.

That LQG and Kalman theory are history, what the new theory is all about, where its likely to go and how to use it to design your own regulators and observers.

1. On the Properties of Noise.
   

   Random processes, ergodicity, expectation, stationarity, autocorrelation, one-sided Fourier transforms, theoretical power spectra, the noise effect integrals.

2. The Perfect Regulator.
   

   Terminal state covariance, settling time, the optimal gains, LQG theory, last rites for Certainty-Equivalence, the general first order problem, examples.

3. The Optimal Observer.
   

   Dynamic estimation, the Separation Theorem, terminal error covariance, settling time, Kalman theory, the optimal gains, examples.

4. The Combined Observer and Regulator.
   

   The one-way Separation Theorem, settling time, terminal state covariance, the optimal gains, examples.

5. A Few Words on Software.
   

   Scaling, optimizing functions with discontinuous derivatives, Lyapunov equations.

6. My Crystal Ball.
   

   Non-stationary systems, discrete systems.