In practice, finite bandwidth is an idealization associated with infinite-duration signals, whereas finite duration implies infinite bandwidth. To show this, assume that a given signal is to be band limited. Band limiting is equivalent to multiplica- tion by a pulse in the frequency domain. By the convolution theorem, multiplica- tion in the frequency domain is equivalent to convolution of the inverse Fourier transforms. Hence, the inverse transform of the band-limited function is the con- volution of the original time function with the sinc function, a function of infinite duration. We conclude that a band-limited function is of infinite duration.
A time-limited function is the product of a function of infinite duration and a pulse. The frequency convolution theorem states that multiplication in the time domain is equivalent to convolution of the Fourier transforms in the frequency domain. Thus, the spectrum of a time-limited function is the convolution of the spectrum of the function of infinite duration with a sinc function, a function of infinite bandwidth. Hence, the Fourier transform of a time-limited function has infinite bandwidth. Because all measurements are made over a finite time period, infinite bandwidths are unavoidable. Nevertheless, a given signal often has a finite “effective bandwidth” beyond which its spectral components are negligi- ble. This allows us to treat physical signals as band limited and choose a suitable sampling rate for them based on the sampling theorem.
In practice, the sampling rate chosen is often larger than the lower bound specified in the sampling theorem. A rule of thumb is to choose ws as
ωs=kωm, 5≤ ≤k 10 (2.66)
The choice of the constant k depends on the application. In many applications, the upper bound on the sampling frequency is well below the capabilities of state- of-the-art hardware. A closed-loop control system cannot have a sampling period below the minimum time required for the output measurement; that is, the sam- pling frequency is upper-bounded by the sensor delay.4 For example, oxygen sensors used in automotive air/fuel ratio control have a sensor delay of about 20 ms, which corresponds to a sampling frequency upper bound of 50 Hz. Another limitation is the computational time needed to update the control. This is becom- ing less restrictive with the availability of faster microprocessors but must be considered in sampling rate selection.
In digital control, the sampling frequency must be chosen so that samples provide a good representation of the analog physical variables. A more detailed discussion of the practical issues that must be considered when choosing the sampling frequency is given in Chapter 12. Here, we only discuss choosing the sampling period based on the sampling theorem.
4It is possible to have the sensor delay as an integer multiple of the sampling period if a state esti- mator is used, as discussed in Franklin et al. (1998).
For a linear system, the output of the system has a spectrum given by the product of the frequency response and input spectrum. Because the input is not known a priori, we must base our choice of sampling frequency on the frequency response.
The frequency response of a first-order system is
H j K
j b
ω ω ω
( )=
+1 (2.67)
where K is the DC gain and wb is the system bandwidth. The frequency response amplitude drops below the DC level by a factor of about 10 at the frequency 7wb. If we consider wm = 7wb, the sampling frequency is chosen as
ωs=kωb, 35≤ ≤k 70 (2.68)
For a second-order system with frequency response
H j K
j n n
ω ζω ω ω ω
( )=
+ −( )
2 1 2 (2.69)
and the bandwidth of the system is approximated by the damped natural frequency
ωd=ωn 1−ζ2 (2.70)
Using a frequency of 7wd as the maximum significant frequency, we choose the sampling frequency as
ωs=kωd, 35≤ ≤k 70 (2.71)
In addition, the impulse response of a second-order system is of the form y t( )= Ae−ζωntsin(ωdt+φ) (2.72) where A is a constant amplitude, and f is a phase angle. Thus, the choice of sam- pling frequency of (2.71) is sufficiently fast for oscillations of frequency wd and time to first peak p/wd.
example 2.25
Given a first-order system of bandwidth 10 rad/s, select a suitable sampling frequency and find the corresponding sampling period.
Solution
A suitable choice of sampling frequency is ws = 60, wb = 600 rad/s. The corresponding sampling period is approximately T = 2p/ws ≅ 0.01 s.
example 2.26
A closed-loop control system must be designed for a steady-state error not to exceed 5 percent, a damping ratio of about 0.7, and an undamped natural frequency of 10 rad/s.
Select a suitable sampling period for the system if the system has a sensor delay of 1. 0.02 s
2. 0.03 s Solution
Let the sampling frequency be
ω ω
ω ζ
s d
n
≥
= −
= −
= 35
35 1
350 1 0 49 249 95
2
. . rad s The corresponding sampling period is T = 2p/ws ≤ 0.025 s.
1. A suitable choice is T = 20 ms because this is equal to the sensor delay.
2. We are forced to choose T = 30 ms, which is equal to the sensor delay.
reSOurceS
Chen, C.-T., System and Signal Analysis, Saunders, 1989.
Feuer, A., and G. C. Goodwin, Sampling in Digital Signal Processing and Control, Birkhauser, 1996.
Franklin, G. F., J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, Addison-Wesley, 1998.
Goldberg, S., Introduction to Difference Equations, Dover, 1986.
Jacquot, R. G., Modern Digital Control Systems, Marcel Dekker, 1981.
Kuo, B. C., Digital Control Systems, Saunders, 1992.
Mickens, R. E., Difference Equations, Van Nostrand Reinhold, 1987.
Oppenheim, A. V., A. S. Willsky, and S. H. Nawab, Signals and Systems, Prentice Hall, 1997.
PrOBlemS
2.1 Derive the discrete-time model of Example 2.1 from the solution of the system differential equation with initial time kT and final time (k + 1)T.
2.2 For each of the following equations, determine the order of the equation and then test it for (i) linearity, (ii) time invariance, (iii) homogeneity.
(a) y(k + 2) = y(k + 1) y(k) + u(k) (b) y(k + 3) + 2 y(k) = 0
(c) y(k + 4) + y(k − 1) = u(k)
(d) y(k + 5) = y(k + 4) + u(k + 1) − u(k) (e) y(k + 2) = y(k) u(k)
2.3 Find the transforms of the following sequences using Definition 2.1.
(a) {0, 1, 2, 4, 0, 0, . . .} (b) {0, 0, 0, 1, 1, 1, 0, 0, 0, . . .} (c) {0, 2−0.5, 1, 2−0.5, 0, 0, 0, . . .}
2.4 Obtain closed forms of the transforms of Problem 2.3 using the table of z- transforms and the time delay property.
2.5 Prove the linearity and time delay properties of the z-transform from basic principles.
2.6 Use the linearity of the z-transform and the transform of the exponential function to obtain the transforms of the discrete-time functions.
(a) sin(kwT) (b) cos(kwT)
2.7 Use the multiplication by exponential property to obtain the transforms of the discrete-time functions.
(a) e−akTsin(kwT) (b) e−akTcos(kwT)
2.8 Find the inverse transforms of the following functions using Definition 2.1 and, if necessary, long division.
(a) F (z) = 1 + 3z−1 + 4z−2 (b) F (z) = 5z−1 + 4z−5
(c) F z z
z z
( )=
+ +
2 0 3. 0 02.
(d) F z z
z z
( )= −
+ +
0 1 0 04 0 25
2
.
. .
2.9 For Problems 2.8(c) and (d), find the inverse transforms of the functions using partial fraction expansion and table lookup.
2.10 Solve the following difference equations.
(a) y(k + 1) − 0.8 y(k) = 0, y(0) = 1 (b) y(k + 1) − 0.8 y(k) = 1(k), y(0) = 0
(c) y(k + 1) − 0.8 y(k) = 1(k), y(0) = 1
(d) y(k + 2) + 0.7 y(k + 1) + 0.06 y(k) =d(k), y(0) = 0, y(1) = 2
2.11 Find the transfer functions corresponding to the difference equations of Problem 2.2 with input u(k) and output y(k). If no transfer function is defined, explain why.
2.12 Test the linearity with respect to the input of the systems for which you found transfer functions in Problem 2.11.
2.13 If the rational functions of Problems 2.8.(c) and (d) are transfer functions of LTI systems, find the difference equation governing each system.
2.14 We can use z-transforms to find the sum of integers raised to various powers.
This is accomplished by first recognizing that the sum is the solution of the difference equation
f k( )= f k( −1)+a k( )
where a(k) is the kth term in the summation. Evaluate the following summations using z-transforms.
(a) k
k n
∑= 1
(b) k
k n
2
=1
∑
2.15 Find the impulse response functions for the systems governed by the following difference equations.
(a) y(k + 1) − 0.5 y(k) = u(k)
(b) y(k + 2) − 0.1 y(k + 1) + 0.8 y(k) = u(k) 2.16 Find the final value for the functions if it exists.
(a) F z z
z z
( )=
− +
2 1 2. 0 2.
(b) F z z
z z
( )=
+ +
2 0 3. 2
2.17 Find the steady-state response of the systems resulting from the sinusoidal input u(k) = 0.5 sin(0.4 k).
(a) H z z
( )= z
−0 4.
(b) H z z
z z
( )=
+ +
2 0 4. 0 03.
2.18 Find the frequency response of a noncausal system whose impulse response sequence is given by
u k u k( ) ( )=u k K k( + ) = −∞ ∞
{ , , , . . . , }
Hint: The impulse response sequence is periodic with period K and can be expressed as
u t u l mK t l mK
m l K
*( )= ( + ) ( − − )
=−∞
∞
=
− ∑
∑ d
0 1
2.19 The well-known Shannon reconstruction theorem states that any band-limited signal u(t) with bandwidth ws/2 can be exactly reconstructed from its samples at a rate ws= 2p/T. The reconstruction is given by
u t u k
t kT t kT
s
k s
( )= ( ) ( − )
( − )
=−∞
∑∞ sin
ω ω 2
2
Use the convolution theorem to justify the preceding expression.
2.20 Obtain the convolution of the two sequences {1, 1, 1} and {1, 2, 3}.
(a) Directly
(b) Using z-transformation
2.21 Obtain the modified z-transforms for the functions of Problems (2.6) and (2.7).
2.22 Using the modified z-transform, examine the intersample behavior of the functions h(k) of Problem 2.15. Use delays of (1) 0.3T, (2) 0.5T, and (3) 0.8T.
Attempt to obtain the modified z-transform for Problem 2.16 and explain why it is not defined.
2.23 The following open-loop systems are to be digitally feedback-controlled. Select a suitable sampling period for each if the closed-loop system is to be designed for the given specifications.
(a) G s
ol( )= s + 1
3 Time constant = 0.1 s (b) G s
s s
ol( )=
+ +
1
4 3
2 Undamped natural frequency = 5 rad/s, damping ratio = 0.7
2.24 Repeat problem 2.23 if the systems have the following sensor delays.
(a) 0.025 s (b) 0.03 s
cOmPuter exerciSeS
2.25 Consider the closed-loop system of Problem 2.23(a).
(a) Find the impulse response of the closed-loop transfer function, and obtain the impulse response sequence for a sampled system output.
(b) Obtain the z-transfer function by z-transforming the impulse response sequence.
(c) Using MATLAB, obtain the frequency response plots for sampling frequencies ws = kwb, k = 5, 35, 70.
(d) Comment on the choices of sampling periods of part (b).
2.26 Repeat Problem 2.25 for the second-order closed-loop system of Problem 2.23(b) with plots for sampling frequencies ws= kwd, k = 5, 35, 70.
2.27 Use MATLAB with a sampling period of 1 s and a delay of 0.5 s to verify the results of Problem 2.17 for w = 5 rad/s and a = 2 s−1.
2.28 The following difference equation describes the evolution of the expected price of a commodity5
p ke( +1)= −(1 γ) ( )p ke +γp k( )
where pe(k) is the expected price after k quarters, p(k) is the actual price after k quarters, and g is a constant.
(a) Simulate the system with g = 0.5 and a fixed actual price of one unit, and plot the actual and expected prices. Discuss the accuracy of the model prediction.
(b) Repeat part (a) for an exponentially decaying price p(k) = (0.4)k. (c) Discuss the predictions of the model referring to your simulation results.
5D. N. Gujarate, Basic Econometrics. McGraw-Hill, p. 547, 1988.
3
Modeling of Digital Control Systems
Objectives
After completing this chapter, the reader will be able to do the following:
1. Obtain the transfer function of an analog system with analog-to-digital and digital-to-analog converters including systems with a time delay.
2. Find the closed-loop transfer function for a digital control system.
3. Find the steady-state tracking error for a closed-loop control system.
4. Find the steady-state error caused by a disturbance input for a closed-loop control system.
As in the case of analog control, mathematical models are needed for the analysis and design of digital control systems. A common configuration for digital control systems is shown in Figure 3.1. The configuration includes a digital-to-analog con- verter (DAC), an analog subsystem, and an analog-to-digital converter (ADC). The DAC converts numbers calculated by a microprocessor or computer into analog electrical signals that can be amplified and used to control an analog plant. The analog subsystem includes the plant as well as the amplifiers and actuators neces- sary to drive it. The output of the plant is periodically measured and converted to a number that can be fed back to the computer using an ADC. In this chapter, we develop models for the various components of this digital control configura- tion. Many other configurations that include the same components can be similarly analyzed. We begin by developing models for the ADC and DAC, then for the combination of DAC, analog subsystem, and ADC.