CDS 110b: State Estimation

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CDS 110b

Schedule

Project

FAQ

Reading

This lecture presents an introduction to state estimation and observers. Beginning with a definition of observability, we provide conditions under which a linear system is observable and show how to construct an observer in the case where there is no noise. We then prove the separation principle, which shows how to combine state regulation with state estimation.

Lecture Outline

1. Observability
   • Definition of observability (full nonlinear system)
   • Observability conditions for linear processes: intuition + proof
2. State Estimation
   • Luenberger observer
   • Example: ducted fan
3. Separation Principle
   • Proof of the separation principle
   • Transfer function representation
   • Example: ducted fan

Lecture Materials

• Lecture Presentation (MP3)
• Reading: AM05 Sec 6.1-6.3
• obs_dfan.m - sample computations for Caltech ducted fan

References and Further Reading

• Friedland, Chapters 7 and 8

Frequently Asked Questions

Q: Is there any relationship between observability and differential flatness?

There is indeed a relationship, of sorts. Strictly speaking, when we talk about differential flatness, we do so independent of the specific outputs for the system. A system is flat if there exist outputs such that we can characterize the trajectories of the system (states and inputs) as a function of the flat outputs and their derivatives. As one might imagine, with this choice of outputs, the system is always observable.

On the other hand, a system might be observable with a certain set of outputs, but those outputs are not necessarily differentially flat outputs. For example, consider the system

${\begin{matrix}{\dot {x}}&={\begin{bmatrix}0&1\\0&0\end{bmatrix}}x&+{\begin{bmatrix}0\\1\end{bmatrix}}u\\y&={\begin{bmatrix}1&1\end{bmatrix}}x&\end{matrix}}$

This matrix is observable (easy to check by computing the observability matrix), but with the given output, it is not possible to compute the input and state. (Note that you can compute the state given the input; just not the state and input given the output.)

The basic issue in this example is that the process has a zero and hence there is an input such that the corresponding output is identically zero.