Nonlinear Network Structures for Optimal Control

Transcription

1 tomaton & Robotcs Research Insttte RRI Nonlnear Network Strctres for Optmal Control Frank. ews and Mrad Mrad b-khalaf dvanced Controls, Sensors, and MEMS CSM grop

2 System Cost f + g 0 [ ] V Q + dt he Usal Sspects Q Q R

3 Defnton.: dmssble Controls et Ψ Ω denote the set of admssble n m controls. control : R R s defned to be admssble wth respect to the state penalty fncton Q on Ω, denoted ΨΩ, f: s contnos on Ω, 0 0, stablzes on Ω, 0 Q dt <, Ω [ + ] stablzng control may not be admssble! 3

4 NONINER QUDRIC REGUOR Generalzed HJB Eqaton GHJB V, Optmal Control SVFB Δ V f + g + Q+ R 0, V 0 0. * * V R g Hamlton-Jacob-Bellman HJB Eqaton HJB V Δ * * * * V V V f + Q gr g 0, 4 * V

5 PROBEM- HJB sally has no analytc solton SOUION- Sccessve ppromaton 0 a stablzng control V f + g + Q + 0, V 0 0 contracton map Sards + R g V Sards and Beard sed Galerkn ppro to allow for GHJB solton Converges to optmal solton Gves n SVFB form 5

6 For Constraned Controls NONINER NONQUDRIC REGUOR [ ] V Q + dt 0 wth 0 φ μ Rdμ, Nonqadratc form- yshevsky PD f φ > 0 when 0 φ 6

7 , V R d Q g f V φ V g R φ New GHJB s t constraned f φ. s a satraton fncton! tanhp p - Natral, eact, no appromaton

8 8 Problem- cannot solve HJB Solton- Use Sccessve ppromaton on GHJB 0 0 0, V R d Q g f V μ μ φ + V g R φ 0 a stablzng control Iterate:

9 emma 3.: Improved Satrated Control aw If Ψ Ω, and V satsfes the eqaton GHJB V, 0 wth the bondary condton V 0 0, then the new control derved as + V φ R g s an admssble control for the system on Ω. Moreover, f the control bond φ s monotoncally non-decreasng and V + s the nqe postve defnte fncton satsfyng the + + eqaton GHJB V, 0, wth the bondary condton V , then V V. Ω heorem 3.: Convergence of Sccessve ppromatons 0 If Ψ Ω, then. V * V nformly on Ω. Ψ Ω, 0 3. * 9

10 emma 3.4: Optmal Satrated Control has the argest Stablty Regon * he satrated control has a stablty regon that s the largest of any other satrated control that s admssble wth respect Q and the system f, g. to Note that there may be stablzng satrated controls that * have larger stablty regons than, bt are not admssble wth Q and the system f, g. respect to 0

11 Problem- Cannot solve GHJB! Solton- Neral Network to appromate V j σ j σ j V w, V σ. Select bass set σ σ. σ. σ. y y 3 σ. n σ. y m npts σ. otpts hdden layer wo-ayer Neral Network wth adjstable otpt weghts

12 V σ σ tanh p p φ tanh R g σ ln tanh R R Q g f ε σ et Cost gradent appromaton hen GHJB s Nonzero resdal!

13 Neral-network-based nearly optmal satrated control law. 3

14 4 o mnmze the resdal error n a S sense Evalate the GHJB at a nmber of ponts Ω N,...,, on, g f + σ + + ln tanh, R R Q b tanh R g σ Note, f hen, GHJB s [ ],, b ε +

15 5 Evalatng ths at N ponts gves [ ] [ ] N N b b b,,,,,, N coeffcent matr Solve by S NN ranng Set!

16 Select the N sample ponts k Unform Mesh Grd n n R Random selecton- Montecarlo ppromaton error s Barron N / n ppromaton error s N Montecarlo overcomes NP-complety problems! 6

17 SIDE- Usefl for redcng complety of fzzy logc systems? Unform grd of Separable Gassan actvaton fnctons for RBF NN 7

18 emma 3.: Eqaton 8 wll have a nqe solton when N k k, k, ρi where ρ s a postve constant, and I s the dentty matr. hs s a persstency of ectaton PE condton on, k. hs gdes the choce of the N sample vectors k NN ranng Set mst be PE 8

19 lgorthm and Proofs work for any Q n Constraned npt gven by [ ] V Q + dt 0 CONSRINED SE CONRO φ μ Rdμ, 0 Qk, k nc l Q+ l Bl αl k large and even MINIMUM-IME CONRO V tanh Q + φ μ Rdμ dt 0 0 For small R and Q >> 0 ths s appro. V t s dt, 0 9

20 Eample: near system , V, w w w w w + w + w + w + w + w w + w + w + w + w , , 0

21 Regon of asymptotc stablty for the ntal controller, 0 tanh QR

22 Regon of asymptotc stablty for the nearly optmal controller, 5 V 5 tanh,

23 Eample: Nonlnear oscllator + +, V, w + w + w + w + w w 6 + w 7 + w 8 + w 9 + w w w w3 w4 w w6 + w7 + w8 + w9 + w w + w + w3 + w tanh 5 3, 0 3

24 tanh State rajectory for the Nearly Optmal Control aw Nearly Optmal Control Sgnal wth Inpt Constrants Systems States Control Inpt mes mes 4