=
u(k) = - .
=
We can defineU(z)= by comparison with Eq. (4.135) and note that u(k) =
but, by the theorem, the response to isH Therefore we can write
24 unchanged in shapeby through the linear constant that an
eigenfunction of
4.6 Propenies of the z-Transform 147
Thus is thetransfer function, which is the ratio of the transforms ofe(k) andu(k)as well as the amplitude response to inputs of the form
This derivation begins with linearity and stationarity and derives the transform as the natural tool of analysis from the fact that input signals in the form
produce an output that has the same It is somewhat more satisfying to derive the necessary transform than to start with the transform and see what systems it is good for. Better to start with the problem and find a tool than start with a tool and look for a problem. Unfortunately. the direct approach requires extensive use of the inversion integral and more sophisticated analysis to develop the main result, which is Eg. (4.138).Chanin son
where
Can we represent a general signal as alinear sum(integral) of such We can. by the inverse integral derived above, as follows
for signals with r < R for which Eq. (4.136) converges. We call the ofelk). and the (closed) path of integration is in the annular re-gion of convergence of Eq. (4.136). Ifelk) = 0, k <O. thenR DO,and this region is the whole outsidea circle of finite radius.
The consequences of linearity are that the response to a sum of signals is the sum of the responses as given in Eg. (4.131). Although Eg. (4.135) is the limit of a sum, still holds. and we can write
(4.131)
(4.132)
(4.133)
(4.134) for allj.
for all j
for allj V{e(k»)= u(k).
=
=
=
V{e(k
+
= lI(k+
j)= u(k
+
j)= j
+
if
From a comparison of Eqs. (4.133) and (4.134). it follows that Theorem
From Eq. (4.132). we must have IfVis linear. then
lI(k)= V{e(k)).
IfV is linear and time invariant and is given an input for a value for which the output is finite at timek.then the output will be of the formH
In general. ife(k)= then an arbitrary finite response can be written u(k) =
Consider (k)= = for some fixed j. From Eq. (4.131), if we let
= it follow that
LetV be a discrete system which maps an input sequence.(e(k)}.into an output sequence Then. expressing this an operator onelk).we have
Ifthe system is time invariant. a shift inelk)toe(k
+
j) must result in no other effects but a shift in the response.II.We writederivation L.A. in 1952 at Columbia University.
that is, H does not depend on the second argument and can be written Thus for the elemental signalelk) = we have a solution u(k) of the same (exponential) shape but modulated by a ratio u(k)=
4.6.3 * Another Derivation of the Transfer Function
Chapter 4 Discrete Systems Analysis
148 Chapler 4 Discrete Systems
4.7 Summary
(4.115)
lim f(k) =
-4.1 the following for stability:
(a) = 0.5u(k - I) - 0.3u(k - 2) (b) u(k)=1.6u(k-I)-u(k-2}
(e) =0.8u(k - I)+O.-lulk 2)
4.2 (al Derive the equation corresponding to the approximation of integration found by fitting a parabola to the points _, . and taking the area under this parabola betweent
=
T - Tandt=
Tas to the integral of e(t) this range.(b) Find the transfer function of the resulting discrete system and plot the and zeros in the
4.3 Verify that the transfer function of the system of Fig. -l.8(c) is given by the same as the system of Fig. -l.9Ic).
4.4 (a) Compute and plot lhe unit-pulse response ofthe system derived in Problem -l.2.
(b) Is this system HIBO stable?
4.5 Consider difference equation
• The characteristic behavior associated with poles in the is shown in Figs. 4.21 through 4.23 and summarized in Fig. 4.25. Responses are typically detennined via MATLAB'Simpulse.morstep,m.
• Asystem represented byH(z)has a discrete frequency response to sinusoids atw",given by an amplitude. A. and phase. as
A
=
T)I and=
(4.110)which can be evaluated by MATLAB'S bode.m.
• The discrete Final Value Theorem. for anF(z)that converges and has a final value. is given by
l/(k+2)=
48 Problems 149
tal Assume a solution = and find the characteristic equation in
(b) Find the characteristic and decide if equation solutions are stable or unstable.
(e) Assume a general solution of the form
= +
and findAl and to match the initial conditions = II(I)= I.
(d) Repeat parts (a). (b). and te) for the equation
+2)=
• The can be used to solve discrete difference equations in the same way that the Laplace transfonn is used to solve continuous differential equations.
• The key property of the that allows solution of difference equa-tions is
preceded by a zero-order-hold, the discrete state-space difference equations are
• Adiscrete system can be defined by its transfer function (in or its state-space difference equation.
• The of the samples of a continuous systemG(s)preceded by a zero-order-hold (ZOH) is
• Asystem will be stable in the sense that a Bounded Input will yield a Bounded Output(BIBOstability) if
= (I - { }
which is typically evaluated using MATLAB's c2d.m.
• For the continuous state-space model
= eFT
r
= (4.58)which can be evaluated by MATLAB'S c2d. m.
• The discrete transfer function in tenns of the state-space matrices is
Y(z)
=
_r.
(4.64)U(z)
which can be evaluated in MATLAB by thetffunction.
+ - +0.81)
= _
+
0 -+
O. I)8 'Draw a cascade realization. using observer canonical forms for second-order blocks and in such a way that the coefficients as shown inH above are the parameters of the block diagram.
F(z}=
4.8 Problems 151
(b) Write G(s) in partial fractions and draw the corresponding parallel block diagram with each component part in control canonical form. Denne the state and give the corresponding state description matricesA. B.C.D.
(c) By finding the transfer functions U and U of part (a) in partial fraction form. express and in terms and . Write these relations the two-by-two transformationTsuch thatx=
(d) Verify that the matrices you have found are related by the formulas A= T-1FT.
B= T-1G.
C=HT.
D=J
(b) Use the one-sided transform to solve for the transforms of the Fibonacci numbers by writing Eg. (4.4) as = + Let11 0= ", = I.[You will needto compute the transform ofi(k+ 2).]
(c) Compute the location of the poles of the transform of the Fibonacci numbers.
(d) Compute the inverse transform of the numbers.
(e) Show that if the kth Fibonacci number, then the ratio will goto (I+ the golden ratio of the Greeks.
(f) Show that if we add a forcing term.elk).to Eq. (4.4) we can generate the Fibonacci numbers by a system that canbeanalyzed by the two-sided transform: i.e.. let
Uk= + +ekand let =Do(k) (Do(k)= I atk= 0 and zero elsewhere).
Take the two-sided transform and show the same results asInpart (b).
4.14 SubstituteII= ande = into Egs. (4.2) and (4.7) and show that the transfer functions. Egs. (4.15) and (4.14), canbefound in this way.
4.15 Consider the transfer function
Z[f(k+III= (a) Show that the one-sided transform ofi(k+1)is 4.12 The first-order system 1- has a zero =
(a) Plot the step response for this system for = 0.8. 0.9. 1.1. 1.2.2.
(b) Plot the overshoot of this system on the same coordinates as those appearing in Fig. 4.30 for 1< < I.
(e) ln what way is the step response ofthis system unusual for >
4.13 The one-sided is defined (e) Repeat parts tb). and (c) for the equation
+2) = +I) - 0.5u(k)
4.6 Show that equation
2r +
4.7 (a) the method of block-diagram reduction. applying Figs. 4.5. 4.6. and 4.7 to compute the transfer function of Fig. 4.8(c).
(b) Repeat pan (a) for the diagram of Fig. 4.9(c).
4.8 to how roots of the following are outside the unit circle.
(a) + 025 = 0 (b)
(e) - + 1.6= 0
4.9 Compute by hand and table look-up the di.screte transfer function if the G(s) in Fig. 4.12 (a)
(h) Repeat the calculation of these discrete transfer functions usingMATLAB.Compute for the sampling periodT = 0.05 and T = 0.5 and plot the location of the poles and zeros in the
4.10 to compute the discrete function if the G(s) in Fig. 4.12 the two-mass system with the non-colocated actuator and sensor of Eg. (A.2l) with sampling periodsT = 0.02 and T = 0.1. Plot the zeros and poles of the results in
the Let = 5. = 0.01.
(b) the two,mass with the colocated actuator and sensor given by Eg. (A.23).
T = 0.02 and T = 0.1. Plot the zeros and poles of the resulLs in the LetwI'= 5.w.= 3. = =o.
(cl the two-input-two-output paper machine described in Eg. (A.24). LetT = 0.1 and T=0.5.
4.11 Consider the system described by the transfer function
Yes) 3
(s+IHs+3)'
(a) Draw the block diagram corresponding to this in control canonical form, denne the state vector. and give the corresponding description matrices F. G. H.J.
150 Chapter 4 Dlscrete
152 Chapter4 Discrete Systems Analysis
4.16 (a) Write H of Problem 4.15 in partial fractions in two tenns of second order each. and draw a parallel realization. using the observer canonical form for each block and showing the coefficients of the partial-fraction expansion as the parameters oftherealization.
(b) Suppose the two faclOrs in the denominawr ofH were identical we change to 1.2 and the 0.81 to 0.5). What would the parallel realizationbein this case?
4.17 Show that observer canonical fonn of the system equations shown in Fig. can he written in the state-space fonn as given by E'I. (4.27).
4.18 Draw out each block of Fig. 4.10 in(a)control and (b) observer canonical fonn. Write out the state-description matrices in each case.
4.19 For a second-order system with damping ratio 0.5 and poles at an angle in the :-plane of
= 30".what percent overshoot to a step would you expect if the system had a zero at
=0.6?
4.20 Consider a signal with the transfonn converges for >2)
=
- 2)
(a) What valueisgiven by the fonnula (Final Value Theorem) of (2.100) applied to this
(b) Find final value of by taking the transfonn of using partial-fraction expansion and tables.
(e) Explain the two results of (a) and differ.
4.21 (a) Find the and be sure to the region of convergence for the signal r < I.
[Hint:Write as the of two functions. one fork 0 and one fork<O.
find the individual transfonns. and detennine values of for whichbothterms converge.)
(b) Ifa rational function is known10converge on the unit circle = I. show how partial-fraction expansion can be used to compute the inverse transform. Apply your result to the transfonn you found in part (a).
4.22 Compute the transform.Irk).for each of the following transfonns:
(a) 1+'-:. >
(b) = [:
(e)
=
. > I:(d) = 1/2< < 2.
4.23 Use MATLAB to plot the rime sequence associated with each of the transfonns in Prob-lem 4.22.
4.24 Use the to solve the difference equation
- 3y(k - I)+ - 2)= 2u(k - I) - 2u(k - 2).
{
k. k> 0
= 0 0
y(k)= O. k< O.
4.8 Problems 153
4.25 For the difference equation in Problem 4.24. using MATLAB.
4.26 Compute by hand and table look-up the discrete transfer function if the G(s) in Fig. is
!Of"+I)
GiS)= +
and the sample period isT= 10msec. Verify the calculation using 4.27 Find the discrete state-space model for the system in Problem 4.26.
4.28 Compute by hand and table look-up the discrete transfer function if theGI in Fig. 4.12 is
+1) s'+s+1O
and the sample isT=10 Verify the calculation using lind DC gain of both the Gis) and the
4.29 Find the discrete state-space model for the system in Problem 4.28. Then compute the eigenvalues of and thc transmission zeros of the state-space model.
4.30 Find the state-space model for Fig. I 2 with Gis) I
where there is a one cycle delay after the AID converter.
- '