State-space feedback 5 Tutorial examples and use of MATLAB

Size: px

Start display at page:

Download "State-space feedback 5 Tutorial examples and use of MATLAB"

Camron Sanders
5 years ago
Views:

1 State-pace feedback 5 Tutorial example and ue of MTLB J Roiter

2 Introduction The previou video howed how tate feedback can place pole preciely a long a the ytem u fully controllable. x u x Kx Bu Thi video illutrate and compare the three different method preented.. Uing canonical form.. Uing a tranformation to obtain a canonical form. 3. ckermann method. x ( BK ) x

3 Remark. In general, pole placement for tate pace model i not a paper and pen exercie.. With the exception of by ytem, the required algebra i tediou and tudent hould ue oftware once they are comfortable with the key principle. MTLB ue i demontrated. 3. ll the method aume controllability and that the uer can pecify, in advance, the deired cloed-loop pole polynomial. 3 p c n n n o

4 Pole placement algorithm. Find tranfer function repreentation.. Find control canonical form. 3. Find pole placement tate feedback for control canonical form. 4. Find tranformation matrix uing controllability matrice. 5. Find tate feedback for original tate pace ytem. C( I z M u cz z ˆ Kz ˆ [ M K cx ) ] Bu ˆ KT ˆ B T 4

5 ckermann approach Define M c to be the controllability matrix, then: M p ( ) K c c 5

6 6 EXMPLES

7 Example : Chooe K to give cloed-loop pole at -±j.5. ; B ; Firt find the required pole polynomial 7 p c.5 The ytem i not in control canonical form o that approach i not viable. Try ckermann and tate tranformation.

8 Example - ckermann M p ( ) K c c 8

9 State tranformation. Find control canonical form. Open loop pole polynomial i p o =..5 p c 9. Find pole placement tate feedback for control canonical form. u Kz ˆ ˆ Bˆ Kˆ.5

10 State tranformation Find tranformation matrix uing controllability matrice. M cz [ M cx ] T Find tate feedback for original tate pace ytem. K KT ˆ

11 MTLB CODE ckermann Tranformation Mx=ctrb(,B) pc=^+*+.5*eye() K=[ ]*inv(mx)*pc eig(-b*k) [hat,bhat,chat,dhat]=tf (,[ ]) Khat=[.5]+hat(,:) eig(hat-bhat*khat) Mz=[Bhat,hat*Bhat] T=Mz*inv(Mx) K=Khat*T

12 Example : Chooe K to et the cloed-loop pole all at -. Firt find the required cloed-loop pole polynomial ; 6 6 B 8 6 ) ( 3 3 p c

13 Canonical approach - example 3 ; 8 6 B BK c p c Deired cloed-loop matrix

14 Tranformation approach example 4 lready in canonical form o not needed!

15 Example - ckermann 5 K p M c c ) ( ) ( 3 I p c Not paper/pen exercie in general! ],, [ B B B M c K

16 MTLB CODE 6 ckermann dd=[ 6 8]; Mx=ctrb(,B) Pc=^3+dd()*^+dd(3)*+dd(4)*eye(3) K=[ ]*inv(mx)*pc eig(-b*k)

17 Example 3: Chooe K to et the cloed-loop pole at,-,-, ;.5.87 Firt find the required cloed-loop pole polynomial but alo note that one cannot reaonably tackle thi quetion on pen and paper B p c

18 Canonical approach - example 3 8 Sytem not in canonical form o cannot ue thi approach.

19 State tranformation. Find control canonical form. Open loop pole polynomial i: p o ˆ ; ˆ B ˆ ˆ ˆ 4 3 k k k k BK.5.5 ˆ K p c

20 State tranformation Find tranformation matrix uing controllability matrice. M cz [ M cx ] T Find tate feedback for original tate pace ytem. K KT ˆ K.54.3

21 Example 3 - ckermann K p M c c ) ( I p c ) ( 3 4 Not paper/pen exercie in general! ],,, [ 3 B B B B M c.3.54 K

22 MTLB CODE ckermann Mx=ctrb(,B) dd=poly([-,-,-,-]); Pc=^4+dd()*^3+dd(3)* ^+dd(4)*+dd(5)*eye(4) K=[ ]*inv(mx)*pc eig(-b*k) Tranformation dd=poly([-,-,-,-]); d=poly([-.5,-,-,-]); [hat,bhat,chat]=tf(,d); Khat=[dd(:end)]+hat(,:) Mz=ctrb(hat,Bhat) Mx=ctrb(,B) T=Mz*inv(Mx); K=Khat*T eig(-b*k)

23 MTLB hortcut MTLB ha built in tool to find K in a ingle line rather than the detailed code illutrate earlier. 3 place.m K = place(,b,pole) acker.m K = acker(,b,pole) Both are demontrated now on the 3 example given earlier. acker.m i enitive place.m doe not do for large number of multiple pole with a tate/poor ingle input. controllability

24 Summary 4. Given ome worked example of pole placement deign.. Shown that obtain the ame anwer with 3 different method. 3. lo made it clear that in general, thee deign are not paper and pen exercie and tudent hould ue a computer for the algebra. 4. Demontrated ue of MTLB tool.

nthony Roiter Department of utomatic Control and Sytem Engineering Univerity of Sheffield www.hef.ac.uk/ace 6 Univerity of Sheffield Thi work i licened under the Creative Common ttribution.

/uk/ or end a letter to: Creative Common, 7 Second Street, Suite 3, San Francico, California 945, US.

25 nthony Roiter Department of utomatic Control and Sytem Engineering Univerity of Sheffield 6 Univerity of Sheffield Thi work i licened under the Creative Common ttribution. UK: England & Wale Licence. To view a copy of thi licence, viit or end a letter to: Creative Common, 7 Second Street, Suite 3, San Francico, California 945, US. It hould be noted that ome of the material contained within thi reource are ubject to third party right and any copyright notice mut remain with thee material in the event of reue or repurpoing. If there are third party image within the reource pleae do not remove or alter any of the copyright notice or webite detail hown below the image. (Pleae lit detail of the third party right contained within thi work. If you include your intitution logo on the cover pleae include reference to the fact that it i a trade mark and all copyright in that image i reerved.)

7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281

7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281 72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition