State-space feedback 6 challenges of pole placement

Transcription

1 State-space feedbac 6 challeges of pole placemet J Rossiter

2 Itroductio The earlier videos itroduced the cocept of state feedbac ad demostrated that it moves the poles. x u x Kx Bu It was show that whe a system is fully cotrollable, the poles ca be placed arbitrarily, that is wherever the user desires. This video cosiders the repercussios of havig to place all the poles so called POLE PLCEMENT. Discrete time case uses same cocepts/algebra. x x

3 Pole placemet with caoical forms Oe ca form the closed-loop state space model by ispectio. 3 x x Kx u Bu x x Oe ca choose the parameters of the closed-loop pole polyomial directly by choosig the parameters i. a a I a a a a

4 Behaviours 4 This video will loo at the cosequeces of pole placemet.. How easily ca oe determie a good locatio for each ad every pole.. What if the target locatios are poorly chose. 3. Ca oe come up with a systematic desig methodology. It will be show that beig able to place the poles is ot the same as beig able to place the poles well.

5 NUMERICL EXMPLES

6 Example Compare the closed-loop behaviour with differet choices of outputs ad target poles. 6 K C B q p I I pq q p

7 poles = poles = - - poles = Iput sigal - poles = -3-3 poles = poles = 4 - -

8 poles = poles = - -. poles = poles = poles = poles = -. Much less differece with oe pole fixed at -..

9 Example r q p I I pqr pr qr pq r q p K B

10 poles = -.,-4,-. poles = -,-4,- poles = -,-4,- - Iput sigal poles = -3,-4, poles = -4,-4, poles =,-4, - Behaviour hugely affected by targeted poles.

11 Pole selectio With a small umber of poles as i low order systems, oe could use isights from root-loci or similar to suggest sesible closed-loop pole locatios. However, with high order systems this becomes less obvious/systematic. geeral piece of guidace is that poles should ot be moved too far from the ope-loop positios as this will probably ecessitate aggressive iputs ad also is liely to result i a sesitive feedbac loop.

12 Summary u Kx. Whe a system is i cotrollable form, every coefficiet of the closed-loop pole polyomial ca be defied as desired usig state feedbac.. This meas every closed-loop pole ca be placed exactly as desired. 3. HOWEVER this does ot imply owledge of good places to put the poles. I geeral selectig fast poles may ot imply good overall behaviour. more systematic desig approach is eeded! 4. MOREOVER, we have ot yet tacled tracig problems ad esurig the output reaches a specified target. gai, the required chages are ot obvious.

13 thoy Rossiter Departmet of utomatic Cotrol ad Systems Egieerig Uiversity of Sheffield 6 Uiversity of Sheffield This wor is licesed uder the Creative Commos ttributio. UK: Eglad & Wales Licece. To view a copy of this licece, visit or sed a letter to: Creative Commos, 7 Secod Street, Suite 3, Sa Fracisco, Califoria 94, US. It should be oted that some of the materials cotaied withi this resource are subject to third party rights ad ay copyright otices must remai with these materials i the evet of reuse or repurposig. If there are third party images withi the resource please do ot remove or alter ay of the copyright otices or website details show below the image. Please list details of the third party rights cotaied withi this wor. If you iclude your istitutios logo o the cover please iclude referece to the fact that it is a trade mar ad all copyright i that image is reserved.