Key Mathematical Backgrounds

Transcription

1 Ke Mathematical Background Dierential Equation Ordinar Linear Partial Nonlinear: mooth, nonmooth,, piecewie linear Map Linear Nonlinear Equilibrium/Stead-State State Solution Linearization Traner Function Stabilit Tranient State Fourier Serie

2 Linear Ordinar Dierential Equation (ODE) a n d dt d n dt n c d n a n dt n LL a Alo repreented oten a a n n n a n LL a d dt a a Firt-order (calar) linear homogeneou ODE order:, degree: need BC to olve Nth-order linear homogeneou ODE order: n, degree: n need n BC to olve a n n n a n LL a a r( t ) Nth-order non-homogeneou order: n, degree: n Need n BC to olve

3 Linear Partial Dierential Equation (PDE) Firt-order linear partial dierential equation Second-order order linear partial dierential equation

4 Nonlinear Dierential Equation Smooth d dt d dt (,t ) 4 i dierentiable Nonmooth d dt g(,t ) g i not dierentiable d dt gn( ) Piecewie Linear d dt a i b i i,,.

5 Map ( k ) ( ( k )) Oten obtained b dicretizing a dierential equation Linear Nonlinear Linear Map Nonlinear Map ' Eample () t a() t b () ( ) ( at ) t e at e a b Eample ' () t () t a i b () ( ) ( t ) t e a it e a i a i b (k ) e at (k) ( ) e at a b ( k n ) e a i T i ( k n ) ( ) e a i T b ai

6 Equilibrium/Stead-State State Solution Conider an autonomou dierential equation d dt ( ). () The equilibrium olution o the above equation i obtained b olving ( ). d dt Eample The equilibrium/tead-tate olution are and Note: I the RHS o () i (, t), then the olution o () ma be periodic/time-dependent dependent

7 Nonlinear DE & h, A (,u) (,u) Linearized model ~ & A ~ Bu ~ ~ C ~ Du ~ where ~, ~, and u ~ are mall deviation o w ( ) z,z, LL,z n w ( z,z, LL,z n ) w M ( ) w m z,z, LL,z n in variable z,z, LL,z w, and u g dz g dz M gm dz, B. The coeicient A, B, C, and D u g dz g dz M gm dz, C where the Jacobian L L M L w z h n g dzn g dzn M gm dzn, D i h u or a vector o Linearization,, and u rom their tead - tate/equilibrium value unction are the Jacobian matrice evaluated on the nominal olution :

8 Eample o Linearization () & ( ) ( ) 4 3 & & & To ind the tem equilibrium point, olve , A(,) and B( ) are the two equilibrium point. Now the linearized tem model at A. ~ ~ A & J ~. ~. ~ ~ ~ A & Thi linearized tem equation i onl applicable at the equilibrium point A

9 Traner Function Traner Function () ( ) pn pn p p zn zn z z K D B A I C G ω ω ω ω ω ω ω ω L L For the tate-pace equation Du C Bu A & the traner unction o the tem i given b Deinition: K: DC gain w z,.., w zn : zero w p,.., w pn : pole

10 Traner Function: Illutrative Eample & A C A - Bu Du B û () & A C Bu Du ŷ () [ ] C D () C( I A) B D G ( I A) ( )( ) () G ( )( ) Stem i untable

11 Traner Function: Feedback Control Eample What i Where So Ĝ() or ()/u() () G() () û () () u() () u() H() () () G()[ u() H() ()] - () () G() H() Ĝ() ŷ () () G()H() () G() u() () u() G() G()H() Ĝ()

12 Traner Function: Feedback Control Eample () u() From the previou eample we have G() Ĝ() G()H() G () ( )( ) û () - () () G() H() Ĝ() ŷ () Now uing have H() we G () ( ) Stem i table now

13 Linear Autonomou Stem: Stabilit Eigenvalue (time domain) Pole (requenc/ Laplace domain) Lapunov theor Linear Nonautonomou Stem: Floquet theor Lapunov theor Smooth Nonlinear Autonomou/Nonautonomou Stem Lapunov method Biurcation anali Floquet theor Nonmooth/Dicontinuou Nonlinear Stem Linear matri inequalit (LMI) Lapunov method Poincare map and Floquet theor

14 Stabilit (Eample) Linear Autonomou Stem: & u State-pace & equation X & X U AX BU Y ' X eig( A), Note: Stem i table becaue both eigenvalue are negative H () Note: Stem i table becaue pole are negative To prove the tem i table uing Lapunov Theor, one need to ind a poitive deinite matri, P, uch that A T P PA T Q In thi cae, or Q.5 P.5 Becaue P i a poitive-deinite matri, the tem i table

15 Stabilit (Eample) Linear/nonlinear non-autonomou tem: ( t) &, To prove the tem i table uing Lapunov Theor, one ha to how that W V t ( ) V ( t, ) W ( ) V ( t, ) W ( ) 3 ( ) W ( ) W ( ) W,, 3 are poitive deinite Eample: & ( t) Let the energ unction o the tem be V ( ) W ( ) W ( ) V ( ) V V ( t, ) ( t) W ( ) or t 3 ( ) W 3 Note: Stem i table becaue the two condition are atiied.

16 Stabilit (Eample) Smooth nonlinear tem: d dt 4 Equilibrium point o the tem: ± Let the energ unction o the tem be 4 ( ) V a & a ( ) 4 a 3 a V a, a > For ( ) a 4a V & 3 > For ( ) a 4a V & 3 < Stem untable Stem table

17 Tranient State: Deinition/Illutration α: maimum overhoot α β: underhoot γ: tead tate tolerance tr: rie time.5.95 β.9 } γ t: ettling time. tr t Unit tep repone o a Second order tem

18 Fourier Serie or periodic ignal a a b ( ) a a co( n) b in( n) where n n π π π π π π π π π n ( ) d n ( ) co( n) ( ) in( n) d d n n

19 Fourier Serie or Square Wave () a a a π π π π n π π co n π π / ( ) d ( ) ( n) co π d π / π ( n) d co( n) d co( n) π π / π / d

20 Fourier Serie or Square Wave a b n nπ 4in in( nπ / ) nπ nπ n ( ) a in( n / )co( n) b in( n) π πn n n ( ) in( nπ / )co( n) πn n n 7 π ( ) 4 co co3 co5 co K

21 Fourier Serie or Square Wave () n Spectral Repreentation

22 Fourier Serie or Square Wave.5 Fundamental 3 rd Harmonic Fundamental 3 rd & 5 th Harmonic Fundamental 3 rd, 5 th and 7 th Harmonic Fundamental 3 rd, 5 th, 7 th and 9 th Harmonic

23 Fourier Serie: Pictorial illutration or dierent waveorm