Control Systems. Time response

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1 Control Systems Time response L. Lanari

2 outline zero-state solution matrix exponential total response (sum of zero-state and zero-input responses) Dirac impulse impulse response change of coordinates (state) Lanari: CS - Time response 2

3 system x() = x u(t) input ẋ(t) =Ax(t)+Bu(t) y(t) =Cx(t)+Du(t) x(t) state y(t) output Linear Time Invariant (LTI) dynamical system (Continuous Time) x IR n u IR m y IR p Lanari: CS - Time response 3

4 system representation ẋ(t) =Ax(t)+Bu(t) x() = x implicit representation v = F m v() = v example x(t) =... v(t) =v + 1 m F ( )d explicit representation (solution) we want to study the solution of the set of differential equations in order to have a qualitative knowledge of the system motion Lanari: CS - Time response 4

5 solution: zero-input response zero-input response (i.e. with u(t) = ) ẋ(t) =Ax(t)+Bu(t) scalar case ẋ = ax x() = x x(t) =e at x matrix case ẋ = Ax x() = x x(t) =e At x dimensions n 1 n 1 n n definition e At = I + At + A 2 t2 2! + = 1 X k= A k tk k! check ẋ = d dt e At x = = Ax Lanari: CS - Time response 5

6 matrix exponential definition e At = I + At + A 2 t2 2! + = 1 X k= A k tk k! properties e At = I t= e At1 e At 2 = e A(t 1+t 2 ) e A 1t e A2t = e (A 1+A 2 )t e At 1 = e At consistency composition = () A1A2 = A2A1 equality holds iff d dt eat = Ae At Lanari: CS - Time response 6

7 solution: zero-input response the exponential matrix propagates the initial condition into state at time t x e At propagation x(t) the exponential matrix propagates the state t seconds forward in time x( ) x( + t) =e At x( ) Euler approximation exact solution x( + t) x( )+tẋ( )=(I + At)x( ) x( + t) =e At x( )=(I + At + A 2 t 2 /2! +...)x( ) Lanari: CS - Time response 7

8 solution: zero-input response phase plane x = x1 x 2 it is possible to represent the vector field ẋ = Ax ẋ(t) =Ax(t) x(t) vector field ẋ(t) = apple x(t) and build a phase portrait (geometric representation of the trajectories) Lanari: CS - Time response 8

9 solution on phase plane phase portrait Pendulum with no damping (nonlinear differential equation) #(t) solution #(t)+ g` sin #(t) = (#(), #()) = (, ) unstable equilibrium state stable equilibrium state vector field # (#(), #()) = (, ) #(t) #() (#(), #()) = (.45,.8) Lanari: CS - Time response 9

10 solution: total response scalar case ẋ = ax + bu x(t) =e at x + e a(t ) bu( )d (check) Leibniz integral rule d dt f(t, )d = d dt f(t, )d + f(t, ) =t ẋ = ae at x + ae a(t ) bu( )d + bu(t) =ax + bu Lanari: CS - Time response 1

11 solution: total response matrix case x(t) =e At x + e A(t ) Bu( )d zero-input response (ZIR) zero-state response (ZSR) y(t) =Ce At x + Ce A(t ) Bu( )d + Du(t) e At Ce At state-transition matrix output-transition matrix N.B. product is not commutative for matrices Lanari: CS - Time response 11

12 solution: total response total response = zero-input response + zero-state response x() 6= x() = u(t) = u(t) 6= two distinct contributions to the motion of a linear system a non-zero initial condition causes motion as well as a non-zero input alternative names zero-input response = free response/evolution zero-state response = forced response/evolution Lanari: CS - Time response 12

13 superposition principle x(t) =e At x + y(t) =Ce At x + e A(t ) Bu( )d Ce A(t ) Bu( )d + Du(t) if (x a,u a (t)) generates x a (t) and y a (t) if (x b,u b (t)) generates x b (t) and y b (t) then ( x a + x b, u a (t)+ u b (t)) generates x a (t)+ x b (t) only for same linear combination and y a (t)+ y b (t) Lanari: CS - Time response 13

14 Dirac s delta (t ) generalized function (t) = t 6= t Z +1 1 properties ( )d =1 " 1 " t "! approximation f(t) (t ) =f( ) (t ) (as a function of t) u( )Z Z +1 1 f( ) (t )d = f(t) sifting (sampling) property t (as a function of ) Lanari: CS - Time response 14

15 Dirac s delta: application zero-state output response Ce A(t ) Bu( )d + Du(t) = sifting property h i Ce A(t ) B + D (t ) u( )d rewritten as W (t )u( )d with W (t) =Ce At B + D (t) if W (t )u( )d u(t) = (t) W (t ) ( )d = W (t) W(t) defined as the (output) impulse response Lanari: CS - Time response 15

16 impulse response for any input u(t), the zero-state output response is a convolution integral of the impulse response W(t) with the input u(t) output ZSR = W (t )u( )d example: mass + friction, measuring velocity C = 1 v = µ m v + 1 m F W (t) =Ce At B = 1 m e µ m t 1.8 zero-state response to sin(5t) y(t) u(t).6 for u(t) =sin!t output y(t) = = 1 m W (t )u( )d = 1 2 +! 2 h µ m sin!t! cos!t +!e µ m ti time µ m Lanari: CS - Time response

17 Dirac s delta: application zero-state response (state) H(t )u( )d if u(t) = (t) H(t ) ( )d H( ) (t )d = H(t) sifting property t = =! = t = t! = H(t) state impulse response e At B one interpretation H(t) =e At B e At x x = B the impulse is transferring formally looks like with the state instantaneously from to B (ZIR) as if it were starting from x = B Lanari: CS - Time response 17

18 impulse response impact hammer Impulse Response Amplitude Time (sec) impulse response (here not experimental) Hubble Space Telescope Lanari: CS - Time response 18

19 solution: total response more compact notation x(t) = (t)x + y(t) = (t)x + H(t )u( )d W (t )u( )d state (t) =e At H(t) =e At B output (t) =Ce At W (t) =Ce At B + D (t) transition matrix impulse response Dirac impulse Lanari: CS - Time response 19

20 change of coordinates (A, B, C, D) ( A, e B, e C, e D) e z = Tx det(t ) 6= ẋ = Ax + Bu ż = Az e + Bu e y = Cx + Du same system y = Cz e + Du e (proof) input u & output y do not change, only state is chosen differently ea = TAT 1 e B = TB e C = CT 1 e D = D equivalent system representation Lanari: CS - Time response 2

21 impulse response 1 experiment 1 response ẋ = Ax + Bu y = Cx + Du same system (different representation) ż = e Az + e Bu y = e Cz + e Du W (t) =Ce At B + D (t) f W (t) = e Ce e At e B + e D (t) W (t) = f W (t) (proof) i.e. independent from the chosen set of coordinates (state) Lanari: CS - Time response 21

22 general solution (recap) x(t) = (t)x + y(t) = (t)x + H(t )u( )d W (t )u( )d (t) =e At H(t) =e At B (t) =Ce At W (t) =Ce At B + D (t) the matrix exponential appears everywhere do we need to compute the exponential using its definition? e At = I + At + A 2 t2 2! + = 1 X k= A k tk k! Lanari: CS - Time response 22

23 matrix exponential SIAM REVIEW c 23 Society for Industrial and Applied Mathematics Vol. 45, No. 1, pp. 3 Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later Cleve Moler Charles Van Loan there are many different ways to compute the matrix exponential... Abstract. In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others, but that none are completely satisfactory. Most of this paper was originally published in An update, with a separate bibliography, describes a few recent developments. Key words. matrix, exponential, roundoff error, truncation error, condition AMS subject classifications. 15A15, 65F15, 65F3, 65L99 PII. S Introduction. Mathematical models of many physical, biological, and economic processes involve systems of linear, constant coefficient ordinary differential equations ẋ(t) =Ax(t). Here A is a given, fixed, real or complex n-by-n matrix. A solution vector x(t) is sought which satisfies an initial condition x() = x. In control theory, A is known as the state companion matrix and x(t) isthesystem response. In principle, the solution is given by x(t) =e ta x where e ta can be formally defined by the convergent power series e ta = I + ta + t2 A ! Published electronically February 3, 23. A portion of this paper originally appeared in SIAM Review, Volume2,Number4,1978,pages The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA (moler@mathworks.com). Department of Computer Science, Cornell University, 413 Upson Hall, Ithaca, NY (cv@cs.cornell.edu). 1 Lanari: CS - Time response 23

24 matrix exponential we will use changes of coordinates (state) z = T x ẋ = Ax + Bu y = Cx + Du e At = T 1 e At e T easier to compute if 9 T : z = Tx det(t ) 6= then ż = Az e + Bu e y = Cz e + Du e with e e At easier to compute e At e = e TAT 1t = Te At T 1 (proof) Lanari: CS - Time response 24

25 since it will be used frequently, we prove that if A = T 1 AT e then e At = T 1 e At e T from ea = TAT 1 we obtain A = T 1 AT e e At = e T 1 e AT t = Lanari: CS - Time response 25 = 1X k= = T 1 " 1 X 1X k= t k k! T 1 ( e A) k T k= = T 1 e e At T t k k! ( e A) k t k k! (T 1 e AT ) k # T

26 vocabulary English phase plane ZIR/ZSR vector field (state) impulse response convolution integral transition matrix sampling property superposition principle Italiano piano delle fasi evoluzione libera/forzata campo vettoriale risposta impulsiva (nello stato) integrale di convoluzione matrice di transizione proprietà di campionamento principio di sovrapposizione Lanari: CS - Time response 26

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