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Apply Lyaponove function to design the observer Advisor: Ming-Shyan Wang Presenter: Hanh Nguyen Thi 1 Contents Principle of method Apply for servo motor Simulation results 2 Principle of method Theorem : Lyaponove function x f x V x 0 V x 0x 0 (1) (2) If (2) is satisfied, then (1) is asymtotically stable system 3 Principle of method The system is affected by the unknown disturbance. x A x bu T y c x Unknown disturbance Given value z Sx u u ~ 1 ~ A SAS , b S b ~ ~ z A z b u Canonical control form 4 Principle of method State feedback controller for canonical control form w u ~ ~ z A z b u z R Block diagram of State feedback controller u w R z u u u w Rz State feedback controller is the unknown disturbance ̂ Is the known disturbance ˆ u w Rz ˆ u w R z ~ ~ z A z b u ~ ~ ˆ z A z b w R z ~ ˆ Am z b w ~ ~ ( Am A b R ) ~ z m Am z m b w 5 Principle of method Method: ~ ˆ Lyaponove function: ~ ~ ~ V (e, ) e T Pe T e z zm ~ ~ de Am e b dt 6 Principle of method ~ T ~ dV d e d e d d ~ ~ T T Pe e P dt dt dt dt dt T ~ d ~ ~ ~ T A m e b T (x)(t ) Pe e P A m e b T (x)(t ) 2T dt ~ d ~ ~ T T T T e ( A m P PA m )e 2 b Pe dt Q ~ d ~ T e Qe 2T bT Pe dt ~ H T 7 Principle of method ~ d ~ H bT Pe ˆ d d dt T ˆ H H b Pe ˆ dt dt d d H ( ˆ ) bT Pe dt dt Choose: Satisfy: ˆ d ˆ b T P K e K edt K d e H p i d dt dt T Ki b Pe ˆ (K p K d s) sH s dV ~T ~ T e Qe 2 H 0, e 0 dt 8 Apply for servo motor :angular position :speed of motor. d :disturbance torque Mechanical model: d dt d Bm 1 1 d e dt Jm Jm Jm Bm x1 , x2 , Jm e :motor torque. J m :moment of inertia. B m :the friction coefficient. :unmodelled uncertainty , :parameter variations 1 , u e , d Jm dx1 dt x 2 dx 2 x (u ) 2 dt z 1 x dz1 dt z 2 dz 2 x (u ) 2 dt Canonical control form 9 Apply for servo motor Controller: s ˆ u w Rz x R x ˆ 1 ˆ u r1 r2 ˆ bT Pe (K p 1 1 r1 ˆ u r2 s s Ki p3e1 p4 e2 K K d s) ( K p i K d s) s sh s sH 1 p3 p4 e2 s ( K K i K s) p d sh s Observer 10 Apply for servo motor Determine A matrix: use pole placement method. 0 Am a0 a0 s1s2 1 a1 a1 s1 s2 Determine controller parameter: 1 0 ~ ~ Am A b R a a 1 0 ~ 0 1 ~ 0 A , b 1, R r1 r2 0 r1 a0 , r2 a1 11 Apply for servo motor Determine Q matrix: dV ~ ~ T e Qe 2T H dt q1 q2 e1 ~2 e1 e2 2 H q2 q4 e2 ~ q1e12 q4 e22 2q2 e1e2 2 H 2 dV/dt the larger, the better. Choose q2=0 and q1,q4, H big enough 12 Apply for servo motor Determine P matrix: 1 a12 0 a Am 11 a21 a22 a0 a1 p p2 q1 q2 P 1 , Q q q p p 4 4 3 3 Am P PAm Q T a11 a 12 a21 p1 a22 p3 p3 p4 p2 p1 p4 p3 p2 a11 a12 q1 p4 a21 a22 q3 q2 q4 q1 2a0 q1 q 4 2a0 a1 2a1 13 Simulation 1 s 1 r1 r2 s u 1 * 1 s x2 1 p3 p4 s 1 ~ s a1 s a~0 z2 1 s h PID ̂ 2 Block diagram for the disturbance observer 14 Simulation Parametter for simulation: Bm=8e-5; Jm=6.45e-5; h=500; s1=-50; %the pole s2=-400; %the pole q1=0.5; %the element of Q matrix q4=2; %the element of Q matrix Kp=100; % Kp,Ki,Kd of PID (disturbance) Ki=1500; Kd=2 15 Simulation Disturbance profile 16 Simulation Speed response at 2 rpm 17 Thank you for your attention ! 18 Model of canonical control form : x A x bu T y c x du 0 0 A 0 a0 1 0 0 1 0 0 a1 a2 0 0 0 0 , b 1 0 1 an 1 19