Lecture: Z-transform. Automatic Control 1. Z-transform. Prof. Alberto Bemporad. University of Trento. Academic year

Size: px

Start display at page:

Download "Lecture: Z-transform. Automatic Control 1. Z-transform. Prof. Alberto Bemporad. University of Trento. Academic year"

Victoria Dalton
7 years ago
Views:

1 Automatic Control 1 Z-transform Prof. Alberto Bemporad University of Trento Academic year Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

Z-transform Z-transform Consider a function f(k), f :, f(k) = for all k < Definition The unilateral Z-transform of f(k) is the function of the complex variable z defined by F(z) =.8.6 f(k)z k.4.

2 Z-transform Z-transform Consider a function f(k), f :, f(k) = for all k < Definition The unilateral Z-transform of f(k) is the function of the complex variable z defined by F(z) =.8.6 f(k)z k k= f(k) k Once F(z) is computed using the series, it s extended to all z for which F(z) makes sense Witold Hurewicz ( ) Z-transforms convert difference equations into algebraic equations. It can be considered as a discrete equivalent of the Laplace transform. Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

3 Z-transform Examples of Z-transforms Discrete impulse if k f(k) = δ(k) 1 if k = [δ] = F(z) = 1 Discrete step if k < f(k) = 1I(k) 1 if k [1I] = F(z) = z z 1 Geometric sequence f(k) = a k 1I(k) [f] = F(z) = z z a Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

4 Z-transform Properties of Z-transforms Linearity [k 1 f 1 (k) + k 2 f 2 (k)] = k 1 [f 1 (k)] + k 2 [f 2 (k)] Example: f(k) = 3δ(k) 5 2 k 1I(t) [f] = 3 5z z 1 2 Forward shift 1 [f(k + 1) 1I(k)] = z [f] zf() Example: f(k) = a k+1 1I(k) [f] = z z z a z = az z a 1 z is also called forward shift operator Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

5 Z-transform Properties of Z-transforms Backward shift or unit delay 2 [f(k 1) 1I(k)] = z 1 [f] Example: f(k) = 1I(k 1) [f] = z z(z 1) Multiplication by k [kf(k)] = z d [f] dz Example: f(k) = k 1I(k) [f] = z (z 1) 2 2 z 1 is also called backward shift operator Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

6 Z-transform Initial and final value theorems Initial value theorem f() = lim z F(z) Example: f(k) = 1I(k) k 1I(k) F(z) = f() = lim z F(z) = 1 z z 1 z (z 1) 2 Final value theorem lim k + f(k) = lim(z 1)F(z) z 1 Example: f(k) = 1I(k) + (.7) k 1I(t) F(z) = f(+ ) = lim z 1 (z 1)F(z) = 1 z z 1 + z z+.7 Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

7 Transfer functions Discrete-time transfer function Let s apply the Z-transform to discrete-time linear systems x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) x() = x Define X(z) = [x(k)], U(z) = [u(k)], Y(z) = [y(k)] Apply linearity and forward shift rules zx(z) zx = AX(z) + BU(z) Y(z) = CX(z) + DU(z) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

8 Transfer functions Discrete-time transfer function X(z) = z(zi A) 1 x + (zi A) 1 BU(z) Y(z) = zc(zi A) 1 x }{{} + (C(zI A) 1 B + D)U(z) }{{} Z-transform of natural response Z-transform of forced response Definition: The transfer function of a discrete-time linear system (A, B, C, D) is the ratio G(z) = C(zI A) 1 B + D between the Z-transform Y(z) of the output and the Z-transform U(z) of the input signals for the initial state x = MATLAB»sys=ss(A,B,C,D,Ts);»G=tf(sys) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

9 Transfer functions Discrete-time transfer function u(k) y(k) U(z) Y (z) A; B; C; D G(z) x = Example: The linear system x(k + 1) =.5 1 x(k) + u(k).5 1 y(k) = 1 1 x(k) with sampling time T s =.1 s has the transfer function z G(z) = z 2.25 Note: Even for discrete-time systems, the transfer function does not depend on the input u(k). It s only a property of the linear system MATLAB»sys=ss([.5 1; -.5],[;1],[1-1],,.1);»G=tf(sys) Transfer function: -z sˆ Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

10 Difference equation Difference equations Consider the n th -order difference equation forced by u a n y(k n) + a n 1 y(k n + 1) + + a 1 y(k 1) + y(k) = b n u(k n) + + b 1 u(k 1) For zero initial conditions we get the transfer function G(z) = = b n z n + b n 1 z n b 1 z 1 a n z n + a n 1 z n a 1 z b 1 z n b n 1 z + b n z n + a 1 z n a n 1 z + a n Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

11 Difference equation Difference equations Example: 3y(k 2) + 2y(k 1) + y(k) = 2u(k 1) G(z) = 2z 1 3z 2 + 2z = 2z z 2 + 2z + 3 Note: The same transfer function G(z) is obtained from the equivalent matrix form 1 x(k + 1) = x(k) + u(k) y(k) = 2 x(k) G(z) = 2 z Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

12 Difference equation Some common transfer functions Integrator x(k + 1) = x(k) + u(k) y(k) = x(k) U(z) 1 z 1 Y (z) Double integrator x 1 (k + 1) = x 1 (k) + x 2 (k) x 2 (k + 1) = x 2 (k) + u(k) y(k) = x 1 (k) U(z) 1 z 2 2z + 1 Y (z) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

13 Difference equation Some common transfer functions Oscillator x 1 (k + 1) = x 1 (k) x 2 (k) + u(k) x 2 (k + 1) = x 1 (k) y(k) = 1 2 x 1(k) x 2(k) U(z) 1 2 z z 2 z + 1 Y (z) 2 output response 1 y(k) k Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

14 Difference equation Impulse response Consider the impulsive input u(k) = δ(k), U(z) = 1. The corresponding output y(k) is called impulse response The Z-transform of y(k) is Y(z) = G(z) 1 = G(z) Therefore the impulse response coincides with the inverse Z-transform g(k) of the transfer function G(z) Example (integrator:) u(k) = δ(k) y(k) = 1 1 z 1 = 1I(k 1) u(k) k y(k) k Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

15 Difference equation Poles, eigenvalues, modes Linear discrete-time system x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) x() = G(z) = C(zI A) 1 B + D N G(z) D G (z) Use the adjugate matrix to represent the inverse of zi A The denominator D G (z) = det(zi A)! C(zI A) 1 C Adj(zI A)B B + D = + D det(zi A) The poles of G(z) coincide with the eigenvalues of A Well, as in continuous-time, not always... Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

16 Difference equation Steady-state solution and DC gain Let A asymptotically stable ( λ i < 1). Natural response vanishes asymptotically Assume constant u(k) u r, k. What is the asymptotic value x r = lim k x(k)? Impose x r (k + 1) = x r (k) = Ax r + Bu r and get x r = (I A) 1 Bu r The corresponding steady-state output y r = Cx r + Du r is Cf. final value theorem: y r = (C(I A) 1 B + D) }{{} DC gain y r = lim y(k) = lim(z 1)Y(z) k + z 1 = lim(z 1)G(z)U(z) = lim(z 1)G(z) u rz z 1 z 1 z 1 = G(1)u r = (C(I A) 1 B + D)u r G(1) is called the DC gain of the system Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21 u r

17 Difference equation Example - Student dynamics Recall student dynamics in 3-years undergraduate course.2 1 x(k + 1) =.6.15 x(k) + u(k).8.8 y(k) =.9 x(k) DC gain: [.9 ] Transfer function: G(z) = y(k) step k z 3.43z z.24, G(1).69 MATLAB»A=[b1 ; a1 b2 ; a2 b3];»b=[1;;];»c=[ a3];»d=[];»sys=ss(a,b,c,d,1);»dcgain(sys) ans =.695 For u(k) 5 students enrolled steadily, y(k) graduate Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

18 Algebraically equivalent systems Linear algebra recalls: Change of coordinates Let {v 1,..., v n } be a basis of n (= n linearly independent vectors) 1 The canonical basis of n 1 is e 1 =..., e 2 =...,..., e n =... A vector w n can be expressed as a linear combination of the basis vectors, whose coefficients are the coordinates in the corresponding basis w = n x i e i = i=1 n z i v i The relation between the coordinates x = [x 1... x n ] in the canonical basis and the coordinates z = [z 1... z n ] in the new basis is x = Tz where T = v 1... v n (= coordinate transformation matrix) i=1 1 Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

19 Algebraically equivalent systems Algebraically equivalent systems Consider the linear system x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) x() = x Let T be invertible and define the change of coordinates x = Tz, z = T 1 x z(k + 1) = T 1 x(k + 1) = T 1 (Ax(k) + Bu(k)) = T 1 ATz(k) + T 1 Bu(k) y(k) = CTz(k) + Du(k) z = T 1 x and hence z(k + 1) = Ãz(k) + Bu(k) y(k) = Cz(k) + Du(k) z() = T 1 x The dynamical systems (A, B, C, D) and (Ã, B, C, D) are called algebraically equivalent Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

20 Algebraically equivalent systems Transfer function of algebraically equivalent systems Consider two algebraically equivalent systems (A, B, C, D) and (Ã, B, C, D) Ã = T 1 AT B = T 1 B C = CT D = D (A, B, C, D) and (Ã, B, C, D) have the same transfer functions: G(z) = C(zI Ã) 1 B + D = CT(zT 1 IT T 1 AT) 1 T 1 B + D = CTT 1 (zi A)TT 1 B + D = C(zI A) 1 B + D = G(z) The same result holds for continuous-time linear systems Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

Algebraically equivalent systems English-Italian Vocabulary Z-transform forward shift operator unit delay trasformata zeta operatore di anticipo

21 Algebraically equivalent systems English-Italian Vocabulary Z-transform forward shift operator unit delay trasformata zeta operatore di anticipo ritardo unitario Translation is obvious otherwise. Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year / 21

Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor

Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter