{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

topic21

# topic21 - Topic#21 16.31 Feedback Control Systems...

This preview shows pages 1–6. Sign up to view the full content.

Topic #21 16.31 Feedback Control Systems Robustness Analysis Model Uncertainty Robust Stability (RS) tests RS visualizations Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Fall 2007 16.31 21–1 Model Uncertainty Want to consider what happens if our model of the system is incorrect actual system dynamics G A ( s ) are in one of the sets Multiplicative model G p ( s ) = G N ( s )(1 + Δ( s )) Additive model G p ( s ) = G N ( s ) + Δ( s ) where 1. G N ( s ) is the nominal dynamics ( known ) 2. Δ( s ) is the modeling error not known directly , but bound E 0 ( s ) known (assumed stable) where | Δ( j ω ) | ≤ | E 0 ( j ω ) | ∀ ω If E 0 ( j ω ) small, our confidence in the model is high nominal model is a good representation of the actual dynamics If E 0 ( j ω ) large, our confidence in the model is low nominal model is not a good representation of the actual dynamics 10 -1 10 0 10 1 10 2 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Freq (rad/sec) |G| G N multiplicative uncertainty Figure 1: Typical system TF with multiplicative uncertainty November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007 16.31 21–2 Simple example: Assume we know that the actual dynamics are G A ( s ) = ω 2 n s 2 ( s 2 + 2 ζω n s + ω 2 n ) but we take the nominal model to be G N = 1 /s 2 . Can explicitly calculate the error E ( s ) , and it is shown in the plot. 10 -1 10 0 10 1 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 Freq (rad/sec) |G| G N E=G A /G N -1 E 0 G A G A G N E E 0 Figure 2: Various TF’s for the example system Can also calculate an LTI overbound E 0 ( s ) of the error. Since E ( s ) is not normally known, it is the bound E 0 ( s ) that is used in our analysis tests. November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Fall 2007 16.31 21–3 10 -1 10 0 10 1 10 -6 10 -4 10 -2 10 0 10 2 10 4 Freq (rad/sec) |G| G N Possible G’s given E 0 G A G N Figure 3: G N with one partial bound. Can add many others to develop the overall bound that would completely include G A . Usually E 0 ( j ω ) not known, so we would have to develop it from our approximate knowledge of the system dynamics. Want to demonstrate that the system is stable for any possible perturbed dynamics in the set G p ( s ) Robust Stability November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007 16.31 21–4 Unstructured Uncertainty Model Standard error model lumps all errors in the system into the actu- ator dynamics.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 15

topic21 - Topic#21 16.31 Feedback Control Systems...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online