Control Systems. Dynamic response in the time domain. L. Lanari

Transcription

1 Control Systems Dynamic response in the time domain L. Lanari

2 outline A diagonalizable - real eigenvalues (aperiodic natural modes) - complex conjugate eigenvalues (pseudoperiodic natural modes) - phase plots A not diagonalizable - Jordan blocks and corresponding natural modes - special case: Re( i) = Lanari: CS - Dynamic response - Time

3 what we know S ( ẋ(t) = Ax(t)+Bu(t) y(t) = Cx(t)+Du(t) x() = x x R n u R p y R q A : n n B : n p C : q n D : q p u input ẋ = Ax + Bu y = Cx + Du state y output solution x(t) = (t)x + y(t) = (t)x + Z t Z t H(t )u( )d W (t )u( )d with (t) =e At (t) =Ce At H(t) =e At B W (t) =Ce At B + D (t) Lanari: CS - Dynamic response - Time

4 what we need we want to analyze the general solution so to qualitatively describe the motion of our system and understand some of its basic properties to go further we need to be able to easily compute the exponential e At = X k= t k k! Ak which appears in all 4 terms and thus contributions (t) =e At (t) =Ce At H(t) =e At B W (t) =Ce At B + D (t) Lanari: CS - Dynamic response - Time 4

5 how can linear algebra help? ẋ = Ax + Bu y = Cx + Du if 9 T : z = Tx det(t ) 6= same system - different state ż = e Az + e Bu y = e Cz + e Du with e e At easier to compute e At = T e e At T then e e At = e TAT t = Te At T e e At easier to compute if is also easier to compute what special structure needs to have matrix ea in order to make easier to compute? e e At Lanari: CS - Dynamic response - Time 5

6 easiest case = diag{ i } = n 7 5 k = diag{ k i } = 6 4 k k k n e t = X k= k tk k! = = 6 4 P k= k t k /k! P k= k t k /k!... P k= k nt k /k! 7 5 e t = 6 4 e t e t = diag{e it } e nt matrix exponential of a diagonal matrix is immediate Lanari: CS - Dynamic response - Time 6

7 how can linear algebra help? diagonalizable case if the matrix is diagonal then its matrix exponential is immediate we found that a square matrix A could be diagonalized iff mg( i) = ma( i) for all i with the diagonalizing change of coordinates T defined as T = U e At = T e t T = U e t U Lanari: CS - Dynamic response - Time easy to compute 7

8 how can linear algebra help? non-diagonalizable case if ea =diag{j i } is block diagonal is the matrix exponential also simplified? e diag{j i}t = X k= diag{j i } k tk k! = = 6 4 e J t e J t yes = diag{ej it } e Jrt moreover diag{e J it } has a special structure that we are going to explore (being Ji a Jordan block) Lanari: CS - Dynamic response - Time 8

9 summary A diagonalizable e At = U e t U e At A non-diagonalizable e At = T diag e J it T what s next? we are going to explore these two cases and understand the different time functions that are present so that we know how, for example, the ZIR qualitatively behaves Lanari: CS - Dynamic response - Time 9

10 matrix exponential: A diagonalizable e At = U e t U T = U = u u u n e At = u u... u n 6 4 e t e t e nt v T v T v T n spectral form of the matrix exponential e At = nx i= e it u i v T i if A diagonalizable (also true for complex pairs of eigenvalues) Lanari: CS - Dynamic response - Time

11 ( matrix exponential: ZIR (if A diagonalizable) x ZIR (t) = nx e it u i vi T x natural modes i= only function of time projection of the initial condition on the subspace generated by ui constant How the zero-input response (unforced response) varies in time depends upon the eigenvalues qualitative behavior of the system motion i real complex Lanari: CS - Dynamic response - Time

12 matrix exponential: A diagonalizable we distinguish between real and complex eigenvalues if A diagonalizable if real eigenvalue natural modes e i t if complex eigenvalues ( i, i = i ) we need to expand this i + j i apple e it e i t e or 4 i i 5t i i preferred Lanari: CS - Dynamic response - Time

13 A diagonalizable - real eigenvalues if real eigenvalue e i t i > aperiodic mode i = /e i i < t e it = e t/ i with i = i time constant (meaningful only for negative i) Lanari: CS - Dynamic response - Time

14 A diagonalizable - real eigenvalues example: real eigenvalue (n = ) x ZIR (t) = > < nx i= e it u i v T i x for n = x ZIR (t) =e t u v T x + e t u v T x projection matrices = e t u c + e t u c scalars eigenspace < x > eigenspace x u u aperiodic modes x x x x x Lanari: CS - Dynamic response - Time 4

15 initial conditions x ZIR (t) = nx i= e it u i v T i x how to see the contribution of an initial condition to each natural mode x ( u i vi T use the projection matrices = P i u i vi T x = P i x = c i u i or x = nx i= c i u i express the initial condition in the base given by the eigenvectors Lanari: CS - Dynamic response - Time 5

16 examples v = m F A = apple Mass-Spring-Damper (MSD) m s + µṡ + ks = u A = apple k m µ m real eigenvalues when high damping µ p km compute eigenvalues and check, for the real case, the sign discuss how the eigenvalues and therefore the ZIR varies with µ in the real eigenvalues case Lanari: CS - Dynamic response - Time 6

17 example: chemical reaction first order chemical reactions reversible reaction C A C B concentration component A concentration component B A k d! B B k i! A dc A dt dc B dt = k d C A + k i C B = k d C A k i C B (find eigenvalues, interpret) Ċ A + ĊB = mass conservation C A (t)+c B (t) =C A () + C B () Lanari: CS - Dynamic response - Time 7

18 example: chemical reaction A k! B k! C dc A dt dc B dt dc C dt = k C A = k C A k C B = k C B (find eigenvalues, interpret) C A (t)+c B (t)+c C (t) = initial value Lanari: CS - Dynamic response - Time 8

19 example: chemical reaction parallel reaction A k! B A k! C A k! D dc A dt dc B dt dc C dt dc D dt = (k + k + k ) C A = k C A = k C A = k C A Lanari: CS - Dynamic response - Time 9

20 A diagonalizable - complex eigenvalues if A diagonalizable and i = i + j!i the free state response is (ZIR) we need to: 9 T R such that T R AT R = apple x ZIR (t) =e At x o = T R i i i i 4 i! i e! i i (real form) 5t T R x compute e 4 i i 5t i i find the change of coordinates TR that puts a generic ( x ) matrix A with complex eigenvalues into the real form apple T R AT R = i i i i Lanari: CS - Dynamic response - Time

21 A diagonalizable - complex eigenvalues ( i, i ) pair with i = i + j i e 4 i i 5t i i = e = 4 i 5t+ i 4 i 5t i = e it I 4 i i 4 i e i X apple i i k= 5tA 5t k t k k!! since matrices commute definition of exponential =... = e i t apple cos i t sin i t sin i t cos i t recognize known series e 4 i i 5t i i = e i t apple cos i t sin i t sin i t cos i t Lanari: CS - Dynamic response - Time

22 A diagonalizable - complex eigenvalues change of coordinates for the real system representation if complex eigenvalues i = i + j!i eigenvector complex vector u i = u re + ju im real vectors T R = u re u im non-singular also linearly independent write the initial condition as x = c a u re + c b u im = u re apple c u a im = T c R b apple ca c b T R x = apple ca c b define the quantities m R and ' R as m R = q c a + c b sin R = cos R = c a p c a + c b c b p c a + c b c a = m R sin c b = m R cos R R Lanari: CS - Dynamic response - Time

23 A diagonalizable - complex eigenvalues e At x = T e 4 i i 5t i i Tx = u re u im e it apple cos i t sin i t sin i t cos i t apple mr sin R m R cos R depends upon the initial condition ZIR e At x = m R e it [sin( i t + R )u re + cos( i t + R )u im ] vectors exponential x periodic Lanari: CS - Dynamic response - Time

24 natural modes (complex - A diagonalizable) if complex eigenvalues ( i, i ) pseudoperiodic mode e At x = m R e it [sin( i t + R )u re + cos( i t + R )u im ] e i t t e t t e i t t i > i = i < time functions of the form e t sin(!t) e t cos(!t) from and! we have a qualitative behaviour of the ZIR Lanari: CS - Dynamic response - Time 4

25 natural modes (complex - A diagonalizable) pseudoperiodic mode x u im x x u im u re u re i > i = i < (prove how it turns) Lanari: CS - Dynamic response - Time 5

26 natural modes (complex - A diagonalizable) pseudoperiodic mode i = + j alternative characterization w.r.t. real and imaginary part! n = p +! = p +! natural frequency damping =sin# Im contribution in a characteristic polynomial ( i)( i )= +! n +! n! n #! inverse relations p =! n! =! n p so / =! n ± j! n =! n ± j p Re time functions e! nt sin(! n p t) e! nt cos(! n p t) Lanari: CS - Dynamic response - Time 6

27 natural modes (complex - A diagonalizable) pseudoperiodic mode i = + j =sin#! n! n! n # Im!! n # Im! Re Re =! n same same! n = study how does the time behavior change when we compare and Lanari: CS - Dynamic response - Time 7

28 phase plane examples 5 Phase portrait (lambda = -, lambda = ) 4 Phase portrait (lambda = -.5, lambda = -) 4 Phase portrait (lambda = -.5, lambda = ) 4 x x x x x x = = =.5 = =.5 = 4 Phase portrait (complex conjugate case) 4 Phase portrait (complex conjugate case) 5 Phase portrait (lambda =.5, lambda =.) 4 x x x x / =.5 ± j x / =. ± j x =.5 =. Lanari: CS - Dynamic response - Time 8

29 natural modes (Mass - Spring - Damper) pseudoperiodic mode s k u m s + µṡ + ks = u µ A = apple k m µ m eigenvalues real eigenvalues when high damping µ p km if > over damping if = critical damping complex eigenvalues when low damping µ< p km if < under damping n = natural frequency r k m = damping coefficient µ p km mechanical friction coefficient Lanari: CS - Dynamic response - Time 9

30 natural modes (Mass - Spring - Damper) pseudoperiodic mode m s + µṡ + ks = u Normalized Zero Input Response - Oscillator harmonic oscillator s(t)/s() µ = µ crit µ = µ crit µ =.5µ crit µ = µ crit µ =.7µ crit µ =.µ crit µ = µ crit = p km t normalized (s(t)/s()) plot of the ZIR under-damped critically damped over-damped Lanari: CS - Dynamic response - Time

31 natural modes (diagonalizable case) example: aperiodic + pseudoperiodic mode block-diagonal 4 5 same eigenvalues / = ± j = Lanari: CS - Dynamic response - Time

32 natural modes (non diagonalizable case) example: one Jordan block of dimension J = 4 i i i 5 = J J 4 i {z } {z } i i diagonal nilpotent commute in the product e (J +J )t = e J t e J t (proof) e Jt = 4 e i t te i t t e i t e i t te i t e i t 5 e it,te it, t e i t new time functions maximum exponent depends on the dimension of the Jordan block Lanari: CS - Dynamic response - Time

33 natural modes (non diagonalizable case) we have e At = T diag e J it T define nk (dimension of the largest Jordan block associated to i) is defined as index of i new time functions (natural modes) appear e it,te it,..., t n k (n k )! e i t depend on the dimension of the Jordan blocks (nk: index of i) Lanari: CS - Dynamic response - Time

34 natural modes (non diagonalizable case) what kind of contribution these new terms give? t k k! e i t Functions t k /k! e t k = 4 5 if i is real negative, exponential wins! t example 4 Phase portrait, ma =, mg = (lambda = -.5) A =.5.5 x x Lanari: CS - Dynamic response - Time 4

35 natural modes (non diagonalizable case) if we have complex eigenvalues with geometric multiplicity greater than then the following time functions will also appear e it sin i t, te it sin i t,..., t n k (n k )! e it sin i t Functions t^k/k! e^{ t} \sin t k = t Lanari: CS - Dynamic response - Time 5

36 natural modes (non diagonalizable case) special cases Re( i) = examples i = 4 5 t diverging t i = j!i j i 6 j i 4 j i j i 7 5 or 6 4! i! i 7! 5 i! i sin!it t sin!it diverging Lanari: CS - Dynamic response - Time 6

37 natural modes (non diagonalizable case) example x x -.5 (x, x) plane.8 - Re( i) = i = projection on the (x,x ) plane x.5 x A A = e At x A t t t A x only constant mode is selected x(t) A x x A constant and t mode are selected x(t) = t A x Lanari: CS - Dynamic response - Time 7

38 summary A diagonalizable mg( i) = ma( i) for all i real i complex i = i + j!i aperiodic mode e i t pseudoperiodic mode e it [sin( i t + R )u re + cos( i t + R )u im ] e At = nx e it u i v T i spectral form i= A not diagonalizable mg( i) < ma( i) real i complex i = i + j!i...,..., t n k (n k )! e i t t n k (n k )! e it sin i t Lanari: CS - Dynamic response - Time 8

39 vocabulary English natural mode aperiodic/pseudoperiodic natural mode natural frequency damping coefficient spectral form Italiano modo naturale modo naturale aperiodico/ pseudoperiodico pulsazione naturale coefficiente di smorzamento forma spettrale Lanari: CS - Time response 9