4 8 Problems 149

bull The characteristic behavior associated with poles in the z-plane is shown in Figs 421 through 423 and summarized in Fig 4 25 Responses are typically

)I1S in the determined via MATLABS impulsem or stepm lifferential bull A system represented by H (z) has a discrete frequency response to sinusoids

at wo

given by an amplitude A and phase 1jf as ence equashy

(4110)and

which can be evaluated by MATLABS bodem (4113) bull The discrete Final Value Theorem for an F(z ) that converges and has a final

c1 aBounded value is given by

lim f(k) = lim(z - l)F(z ) (4115) k~oo z---I

(435)

Zl or its stateshy 48 Problems

41 Check the followin g for stability preceded by a (a) u(k) = OSu(k - I ) - 03u(k - 2)

(b) u(k) = l6u(k - I ) - u (k - 2) (44 )

(c) u(k) = 08u(k - 1) + O4u(k - 2)

42 (a) Derive the difference equation corresponding to the approximation of integration found by fitting a parabola to the points e _ 2 e _ l e and taking the area under this k k k parabola between I = kT - T and I = kT as the approximation to the integral of e(l ) over this range

(445) (b) Find the transfer function of the resulting discrete system and plot the poles and

(446) zeros in the z-pl ane

43 Veri fy that the transfer function of the system of Fi g 48(c) is given by the same H(z)fference equations as the system of Fig 4 9(c)

44 (a) Compute and plot the unit-pul se response of the system derived in Problem 42

(b) Is this system BIBO stable (459)

45 Consider the difference equation

u (k + 2) = 02Su(k)

(a) Assume a solution u(k) = Ai l and find the characteristic equation in zmiddot

(b) Find the charac teri stic roots Zl and Z2 and decide if the equation solutions are stable(458) or unstable

(c) Assume a general solution of the form

u(k ) = Al ZI k + A 2zce matrices is and find A I and A2 to match the initial conditions u (0) = 0 U (I) = l

(4 (d) Repeat parts (a) (b) and (c) for the equation

u(k + 2) = -02Su(k) )

1shyj (

150 Chapter 4 Discrete Systems Analysis

(e) Repeat parts (a) (b) and ec) for the equation

u(k + 2) = u(k + I) - 05u(k)

46 Show that the characteristic equation

Z2 - 2r cos(e)z + r2

has t he roots

47 (a) Use the method of block-d iagram reduction applying Figs 45 46 and 47 to compute the transfer function of Fig 48(c)

(b) Repeat part (a) for the diagram of Fig 49(c)

48 Use MATLAB to determine how many roots of the following are outside the unit circle

(a) Z2 + 025 = 0

(b) Z3 - IIz2 + OOlz + 0405 = 0

(e) Z3 - 36z2 + 4z - 16 = 0

49 Compute by hand and table look-up the discrete transfer function if the G (s) in Fig 412 is

(a) K

(b) 1

(s+ 3)

(e) 3 laquo+ 1)( 3)

( +1)(d) -T (~ T I2

(e) ---r (1- )

(I) -T 3c- L$T~

(g) ( 1)(r+ 3

(h) Repeat the calcu lation of these discrete transfer func tions using MATLAB Compute for the sampling period T = 005 and T = 05 and plot the location of the poles and zeros in the z-plane

410 Use MATLAB to compute the discrete transfer function if the G(s) in Fig 4 I 2 is

(a) the two-mass system with the non-col ocated actuator and sensor ofEq (A2 1) with sampling periods T = 002 and T = 01 Plot th e ze ros and poles of the results in the z-plane Let wp = 5 ~p = OO

(b) the two-mass system wi th the colocated actuator and sensor given by Eq (A23) Use T = 002 and T = 01 Plot the zeros and poles of the results in the z-plane Let wp = 5 W = 3 ~p = ~ = O z

(e) the two-input-two-output paper machine described in Eq (A24) Let T = 01 and T = 05

411 Consider the system described by the transfer fun ction

_Y(s) = G(s) = 3 U(s) (s + 1)(s + 3)

(a) Draw the block diagram corresponding to this system in control canonical fonn define the state vector and give the corresponding description matrices F G H 1

to

Ie

12 is

npute loles

) with Its in

23) lane

I and

Jrm H J

48 Problems lSI

(b) Write G(s) in partial fractions and draw the corresponding parallel block diagram with each component part in control canonical form Define the state $ and give the corresponding state description matrices A B cD

(c) By findin g the transfer funct ions X I I U and X21 U of part (a) in partial fraction form express x and x2 in terms of $ and $2 Write these relations as the two-by-two transfonnat ion T such that x = T$

(d) Verify that the matrices you have found are related by the formulas

A = T - JFT

B = T-IG

C=HT

D=J

412 The first-order system (z - a) (l - a)z has a zero at Z = a

(a) Plo t the step response for this sys tem for a = 08 0 9 11 J22

(b) Plot the overshoot of thi s sys tem on the same coordinates as those appearing in Fig 430 for - I lt a lt l

(c) In what way is the step response of this system unusual for a gt I

413 The one-sided z-transform is defined as

F(z) = L00

I(k )z - k a

(a) Show that the one-sided transform of I (k + 1) is

Zf(k + J)l = zF(z) - zl(O)

(b) Use the one-s ided transfonn to solve for the transforms of the Fibonacci numbers by writing Eq (44) as Uk = + Uk Let Uo = il = l [You will need to +2 Uk+ 1 compu te the transform of I(k + 2))

(c) Compute the location of the poles of the transfonn of the Fibonacci numbers

(d) Compute the inverse transform of the numbers

(e) Show that if Uk is the kth Fibonacci number then the ratio u k+ I I U k will go to

(l + -S) 2 the golden ratio of the Greeks

(f) Show that if we add a forcing term e(k) to Eq (44) we can generate the Fibonacci numbers by a sys tem that can be anal yzed by the two-sided transform ie let U k = Uk_I + u _ + e and let e = Do(k) (Do(k) = I at k = 0 and zero elsewhere) k 2 k k Take the two-sided transform and show the same U(z) results as in part (b)

414 Substitute U = Al and e= B into Eqs (42) and (47) and show that the transfer functions Eqs (4 15) and (4 14) can be found in this way

415 Consider the transfer fun ction

(z + 1)(z2 - l3z + 0 81) H~) =

(Z2 - 12z + 05)(Z2 - l4z + 081)

Draw a cascade realization using observer canonical forms for second-order blocks and in such a way that the coefficients as shown in H (z) above are the parameters of the block diagram

152 Chapter 4 Discrete Systems Analysis

416 (a) Write the H(z ) of Problem 415 in partial fractions in two tenl1S of second order each and draw a parallel realization using the observer canonical form for each block and showing the coefficients of the partial-frac tion expansion as the parameters of the realization

(b) Suppose the two factors in the denominator of H (z) were identical (say we change the 14 to 12 and the 081 to 05) What would the parallel realization be in this case

417 Show that the observer canonical form of the system equations shown in Fig 49 can be written in the state-space form as given by Eq (427)

418 Draw out each block of Fig 410 in (a) control and (b) observer canonical fo rm Write out the state-description matri ces in each case

419 For a second-order system with damping ratio 05 and poles at an ang le in the z-plane of e= 30deg what percent overshoot to a step would you ex pect if the system had a zero at Z2 = 06)

420 Consider a signal with the transform (which converges for Iz l gt 2)

z V( z) = (z - 1)(z - 2)

(a) What value is given by the formula (Final Value Theorem) of (2 100) applied to this V(z)

(b) Find the final value of u(k) by taking the inverse transform of V (z ) using partial-fraction expansion and the tables

(e) Explain why the two results of (a) and (b) differ

421 (a) Find the z-transform and be sure to g ive the region of convergence for the g~l r lt l

[Hint Write u as the sum of two func tions one for k 0 and one for k lt O find the individual transforms and determine values of z fo r which both terms converge]

(b) If a rational function V( z) is known to converge on the unit circle Iz l = I show how partial-fraction expansion can be used to compute the inverse transform Apply your result to the transform you found in part (a)

422 Compute the inverse trans form j(k) for each of the following transforms

(a) F(z) = 1 + ~- ~ Iz l gt I

(b) F (z) = 2 (- I ) 121 gt I l -125+025

(e) F(z) = 2 _ ~+ 1 Iz l gt 1

(d) F(z) = (_(- 2) 12 lt Iz l lt2

423 Use MATLAB to plot the time sequence associated with each of the transforms in Probmiddot lem 422

424 Use the z-transform to solve the difference equation

y(k) - 3y (k - I) + 2y(k - 2) = 2u(k - 1) - 2u(k - 2)

u(k) = ~ kk lt ~ y(k) = 0 k lt o