Linear Systems Analysis in the Time Domain

Transcription

1 Liar Sysms Aalysis i h Tim Domai

2 Firs Ordr Sysms di vl = L, vr = Ri, d di L + Ri = () d R x= i, x& = x+ ( ) L L X() s I() s = = = U() s E() s Ls+ R R L s + R u () = () =, i() = L i () = R R

3 Firs Ordr Sysms Y() s =, U( s) =, u( ) = U() s Ts+ s T : im cosa Y() s = = T Ts + s s Ts + y () =, y & () = T T T

4 Firs Ordr Sysms R() = r Rs () = r s T T Ys () = r = r + Ts + s s s s + (/ T) T y () = r ( T+ T ) T () = R () y () = rt( ) ( ) = rt

5 Scod Ordr Sysms Rs () Cs () G(s) Gs () ω = s + ζω s+ ω ω Rs () = ( spipu), Cs () = s s + ζω s + ω s s + ζω s+ ω =, s= ζω ± ζ ω

6 Udrdampd cas < ζ < ζ = Scod Ordr Sysms s = ζω ± ζ ω ω = ω ζ i, ( d ) ω s + ζω Cs () = = ( s+ ζϖ ) + ω ( ζ ) s s ( s+ ζω ) + ω d s + ζω ζω = s ( s+ ζω ) + ω ( s+ ζω ) + ω d d ζω ζω ζ ζω C ( ) = cosωd siω si( ) d= ω d+ η ζ ζ η = a Criically dampd cas ζ ζ ω Rs () =, Cs () = s ( s+ ζω) s ω c () = ( + ω )

7 Ovrdampd cas Cs () = ζ > Scod Ordr Sysms ω ( + ζω + ω ζ )( + ζω ω ζ ) s s s c() ζ ( ζ + ζ ) ζ ( ζ ζ ) ( + ) ω ( ) ω () = + ζ ζ ζ ζ ( ) ( ) ζ + ζ ω ζ ζ ω ω = + ( ) ( ) ζ ζ + ζ ω ζ ζ ω Approximaio (Afr h fasr rm disappard) Cs () ζω ω ζ = Rs () s + ζω ω ζ c = ζ ζ ω ( ) ()

8 Exprimal Drmiaio of Dampig Raio mx && + bx& + kx =, x& () = && x+ ζω x& + ω x= b ζ = = ω m ( s+ ζω ) b b k mk + ζωs+ ω [ ] sxs ( ) sx() x& () + ζω sxs ( ) x() + ω Xs ( ) = X () s = s x() ζω () ζ x ζω ζ x ( ) = x()si ω ()cos cos a + x ω = ω ζ d d ζ d ζ m ζω x = = ζω ( + ( ) T) x ( ) ζω T

9 Exprimal Drmiaio of Dampig Raio Logarihmic dcrm x π πζ x l = ζωt = ζω = = l x ω d ζ x x x l = ( ) ζ = ζω T x l x x 4π + l x

10 Esima of Rspos Tim x() ζ w ζ x ( ) = cos ω d a ζ ζ ω = ω ζ d ζ ω = = = d ω ζ,, ωd η ζω ζ ζω ζ π a = η ζ

11 Scod Ordr Trasis Sp ipu rspos ζω ζω ζ ζω C ( ) = cosωd siω si( ) d= ω d+ η ζ ζ η = a ζ ζ

12 Scod Ordr Trasis ) Pak ovrshoo M p dx ζ ζω ζω = ω si( ) cos( ), ( ) ωd+ φ ω ωd+ φ = ωd = ω ζ d ζ ζ a( ω d + φ ) = ζ =, ωd = π, π, L π π p = = ω d ω ζ M = y( ) p p M s.63 Ts ζ π ζω ω ζ π ζπ ω ζ φ = si + = + xp ζ ω ζ ζ M p y s ζπ pr ui ovrshoo M o = = xp y s ζ

13 Scod Ordr Trasis ) Slig im : Th im rquird for h oscillaios o dcras o a spcifid absolu prcag rror. T s ζω = y r = si( ω ) d+ φ ζ x) = % or 5% % cas, Ts 4 T = 4 ζω 5% cas, T 3 T y() s 3) Ris im r..9. r T s

14 Scod Ordr Trasis 4) Frqucy of oscillaio of h rasi ω ω ζ ω m k m d =, =, k ωd = ζ m + ζω m+ ω = m = ζω ± jω ζ = σ ± jω d

15 Soluio of Liar (Tim Ivaria) Sa Equaio u u M u m sysm x, x, L, x y y M y l x & () = Ax () + Bu () y () = Cx () + Du () [ ] x () = x x L x T

16 Scalar Fucio u () = i) x& = ax Basic Marix Liar Algbra -Homogous Soluio x () = C a =, x() = C x () = x() a x x x a ( ( ) = ) ( ), =, ( ) k ( ) ( ) ii) a = xp( a) = + a + a + L+ a + L!! k! Homogous Soluio i) x=ax & A:, x: d x( ) = xp A( ) x( ), ( ) = A d A ii) How o valua [ ] A A

17 Sa Trasiio Marix A( x( ) = ) x( ) =Φ( )x( ) [ ] A ( ) Φ ( ) = = xp A( ) : Sa rasiio marix (STM) : fudamal marix of h sysm Propris of STM. Φ( ) Φ( ) =Φ( ) for ay,,. Φ () = I ( ( ) ) A A 3. Φ() Φ () = Φ () = Φ () Q = =Φ () 4. Φ () = Φ ( ) 5. Φ( ) g Φ () =Φ( g) is osigular for all fii valus of (ivrs xiss)

18 Compl Soluio of h Sa Equaio x & = A x+ B u, x & A x = B u, igraig facor A A A A d A A x& A x = B u, x = B u( ) d ( ) A Aτ x( ) x() = B u( τ) dτ A A( τ ) x( ) x() B ( τ) = + u dτ =Φ ( ) x() + Φ( τ) B u( τ) dτ

19 Compl Soluio of h Sa Equaio A A [ ] = Aτ for,, x( ) x( ) B u( τ ) dτ x( ) =Φ( ) x( ) + Φ( τ) B u( τ) dτ x( ) =Φ ( ) x() + Φ( τ) B u( τ) dτ - Zro-ipu - Zro-sa rspos rspos - fr rspos - forcd rspos chagof variabl τ, l β = τ, τ = β = τ = β = d β = d τ x( ) =Φ ( ) x() + Φ( β) B u( β) dβ

20 Marix Expoial x( () = A x() L( α ), x& = Ax L( β) Marix xpoial A = L! 3! A 3 3 I A A A L (*) Q a 3 ( a) ( a) = + a+ + + L! 3! d A ( ) A A A L A I A A d 3 = = = A A L d A A x( & ) = ( ) x() = A x() = A x( ) d ( α) is h soluio of h marix diffrial quaio ( β )

21 Marix Expoial k A A = xp(a ) = = k! A ( ) d = A d A ( A + B ) A B k = ( A+ B) A B = if AB = BA k if AB BA 3 ( A+ B) (A+ B) (A+ B) 3 = I + (A+ B) + + +L! 3! A A B B A B = I+ A+ + + L I+ B+ + + L! 3!! 3! A = I + (A+ B) + + AB! B B A B AB B L! 3!!! 3!

22 A How o Evalua Hc, ( A BA AB BA ABA B A BAB A B AB + B) A B = + + L! 3! ( A+ B) A B Th diffrc bw ad vaishs, if A ad B commu.

23 How o Evalua A (*) a) Diagoalizd Form λ A = λ λ 3 3 λ λ A 3 = I + A+ λ λ + + L 3! λ 3 λ λ + λ + λ + L λ 3! λ = O = λ3

24 How o Evalua A b) Jorda Form λ λ λ = = A A λ λ, λ λ λ A, λ λ λ λ λ A λ λ = λ =

25 How o Evalua A c) Gral A : diagoaliz!! [ λ ] x& = Ax + B u i) Q ( λ ) = d I A = : ii) λ : i A: igvalu of A disic igvalus. sol = λ i=, L, iii) ( λ I A) p = p : ig vcor i i i Α p = λ p i i i i Characrisic quaio. λ A [ p L p] = [ p L p ] O, AP= PΛ λ λ - P AP = O =Λ Diagoalizabl if λi : disi igvalus λ

26 How o Evalua A Α p =λ p i i i T if p p =, p is a ormalizd igvcor. i i i Orhogoaliy of igvcor T if A= A T T A p i = λ i p i pj A pi = λi pj pi = T T A pj = λ jp j pi A pj = λ jpi pj = λ p L p = p L p O λ [ ] [ ] A T PAP=Λ

27 How o Evalua A l Pxˆ = x, x & = Pxˆ& x& = Ax+ Bu Px& ˆ = APxˆ + B u ˆ& = ˆ+ u =Λ ˆ+ x P APx P B x P B Λ Λ( τ ) x( ˆ ) x() ˆ P B ( ) u τ dτ = + Λ Λ( τ ) Px( ˆ ) P P x() P P B ( ) u τ dτ = + Λ Λ( τ ) x( ) P P x() P P B ( τ ) = + u dτ u

28 d) Gral A (-> Jorda form) λ λ i rpad : mulipl igvalu [ λ ] ( λ λ )( λ λ ) A How o Evalua d I A = A : 3 3 i ( λ I A) p = ( λ I A) p = if rak ( λ I A) = h p? Fid p p 3 p suchha λ AP = P λ A p = λp, λ A p = λp A p = p + λ p, (A λ Ι ) p = p 3 3 3

29 Mulipl Eigvalu - Diagoal Form x) λ A= Q( λ) = d λi A = λ λ igvalu : λ =, λ =, λ = 3 ( λ ) ( λ ) = ( λ ) igvcor : Ap = λ p I A p = i i i i i λ, =, p, = choos p =, p = λ 3, p p = = = 3 3

30 Mulipl Eigvalu - Diagoal Form P= [ p p p3] =, P = x& - P AP= =Λ = Ax xˆ& = P APxˆ =Λxˆ ˆ ˆ ˆ Λ Λ x( ) = x(), P x = x P x( ) = P x() = = Λ A x( ) P P x() x() A = P P =

31 Mulipl Eigvalu - Jorda Form λ A= 3 Q( λ) = d λi A = λ 3 λ igvalu : λ =, λ =, 3 ( λ ) ( λ ) = ( ) igvcor : A p = λ p λ I A p = i i i i i λ, =, 3 p, = p = 5 λ3 =, 3 p3 = p3 = 3 W ca fid oly igvcor for λ =

32 Mulipl Eigvalu - Jorda Form If o abl o fid 3 idpd vcors, fid which rasforms A as Jorda form. λ AP = PJ P = λ λ 3 λ A[ p p p3] = [ p p p 3] λ λ 3 [ A p A p A p ] [ pλ p λ p λ p ] = + So ha, [ λ ] p A p = λ p I A p = p [ λ ] A p = p + λ p A I p = p p [ λ ] A p = λ p A I p = p

33 Mulipl Eigvalu - Jorda Form a A λi p = 3 p, l, p b = = c [ ] b c c c b choos p + =, 3 =, = =, = 5 5 P = 3, P = P AP= = Jorda form

34 Mulipl Eigvalu - Jorda Form A if x& = 3 x = Ax, x( ) = x()? l Pxˆ = x, x & = Pxˆ& & x( ˆ() = P APx ˆ = J x ˆ = x ˆ J J =, x( ˆ ) = x() ˆ J - A x( ) P = P x() = x()

35 Mulipl Eigvalu - Jorda Form 5 5 A J = P P = = 3 3 ( ) = ( )

36 Summary. Diagoal Form. Jorda Form λ λ A λ A = λ = λ3 λ 3 λ λ λ A λ A= λ =, λ λ λ A λ λ λ λ λ A λ λ = λ =

37 Summary 3. Gral A & - x Ax Bu A pi λipi, P AP = + = =Λ l ˆ x=px ˆ& ˆ ˆ x= P APx + P B u= Λ x+ P B ˆ Λ Λ( τ ) x( ) x() P B ( ) u ˆ u τ dτ ˆ = + Λ Λ( τ ) Px( ) x( ) P P x() P P B ( ) = = + u τ dτ P Λ P = A

38 Laplac Trasformaio Mhod A A( τ ) x& = Ax+ B u, x( ) = x + B u( τ) dτ Laplac Trasformaio sx( s) x = A X( s) + B u( s), X( s) = ( si A) x + ( si A) B u( s) = L ( si A) A - ( s I A) = I A s s = I+ A+ A + A s s s s 3 3 L 3 = I+ A+ A + A L 3 4 s s s s

39 Laplac Trasformaio Mhod Laplac rasformaio abl d L [] =, L, f ( ) ( ) F( s) s = = ( )! L s ds L ( s I A) = L I+ A+ A + A L 3 4 s s s s 3 3 = I+ A+ A + A L! 3! I A A A! 3! A 3 3 = L

40 Soluio of Liar (Tim Ivaria) Sa Equaio Mhod. Diagoalizaio Mhod. Laplac Trasformaio Mhod 3. Sylvsr's Irpolaio Formula