146 Chapter 4 Open-Loop Discrete-Time Systems 4.12 SUMMARY In this chapter we have examined various aspects of open-loop discrete-time systems. In par- ticular we discussed the starred transform and showed that it possesses the properties of the z-transform defined in Chapter 2. Next, the starred transform was used to find the pulse trans- fer function of open-loop systems. The pulse transfer function was extended to the analysis of open-loop systems containing digital filters. In order to analyze systems containing ideal time delays, the modified z-transform and its properties were derived. Then techniques for finding discrete state-variable models of open-loop sampled-data systems were presented. These devel- opments form the basis for the computer calculations of discrete state models and the pulse transfer function. The foundation built in this chapter for open-loop systems will serve as a basis for presenting the analysis of closed-loop discrete-time systems in Chapter 5. References and Further Readings [1] C. L. Phillips and J. Parr, Feedback Control Systems, 5th ed. Upper Saddle River, NJ: Prentice- Hall, 2011. [2] D. M. Look, "Direct State Space Formulation of Second-Order Coupled Systems," M.S. thesis, Auburn University, Auburn, AL, 1971. [3] J. L. Melsa and S. K. Jones, Computer Programs for Computational Assistance. New York: McGraw-Hill Book Company, 1973. [4] R. C. Ward, "Numerical Computation of the Matrix Exponential with Accuracy Estimate," SIAM J. Numer. Anal., pp. 600-610, 1977. [5] C. Moler and C. Van Loam, "Nineteen Dubious Ways to Compute the Exponential of a Matrix," SIAM Rev., pp. 801-836, 1978. [6] D. Westreich, "A Practical Method for Computing the Exponential of a Matrix and Its Integral," Commun. Appl. Numer. Methods, pp. 375-380, 1990. E(s) = [7] J. A. Cadzow and H. R. Martens, Discrete-Time and Computer Control Systems. Upper Saddle River, NJ: Prentice-Hall, 1970. [8] P. M. De Russo, R. J. Roy, C. M. Close, and A. A. Desrochers, State Variables for Engineers, 2nd ed. New York: John Wiley & Sons, Inc., 1998. (s [9] G. F. Franklin, J. D. Powell, and M. Workman, Digital Control of Dynamic Systems, 3rd ed. Half Moon Bay, CA: Ellis-Kagle Press, 2006. [10] E. I. Jury, Theory and Application of the z-Transform Method. Huntington, NY: R.E. Krieger Publishing Co., Inc., 1973. B. C. Kuo, Digital Control Systems, 2nd ed. New York: Oxford University Press, 1995. K. Ogata, Discrete-Time Control Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1995. [11] [12] Problems 4.2-1. (a) Show that a pole of E(s) in the left half-plane transforms into a pole of E(z) inside the unit circle. (b) Show that a pole of E(s) on the imaginary axis transforms into a pole of E(z) on the unit circle. (c) Show that a pole of E(s) in the right half-plane transforms into a pole of E(z) outside the unit circle. 4.2-2. Let T = 0.05 s and s+2 1)(s + 1) (a) Without calculating E(z), find its poles. (b) Give the rule that you used in part (a). (c) Verify the results of part (a) by calculating E(z). (d) Compare the zero of E(z) with that of E(s). (e) The poles of E(z) are determined by those of E(s). Does an equivalent rule exist for zeros? 4.2-3. Find the z-transform of the following functions, using z-transform tables. Compare the pole-zero locations of E(z) in the z-plane with those of E(s) and E*(s) in the s-plane (see Problems 3.4-1). Let T = 0.1 s. 20 (a) E(s) (b) E(s) 5 s(s + 1) (s + 2)(s + 5) 4.2-4. -4.3-1. (c) E(s) (e) E(s) (a) E(s) - (b) E(s) - s +2 s(s+1) = S² + 5s + 6 s(s+ 4)(s + 5) (g) Verify any partial-fraction expansions required, using MATLAB. Repeat Problem 4.2-3 with T= 0.01 s. Find the z-transforms of the following functions: (εs - 1)² €²s(s+ 1)' (0.5s + 1)(1 ε-0.25s 0.5s(s+ 0.25) E(s) == T = 0.5 s 1 S (d) E(s) = T=1s FIGURE P4.3-2 System for Problem 4.3-2. 5 S² + 25 + 2 (f) E(s) 4.3-2. (a) Find the system response at the sampling instants to a unit step input for the system of Fig. P4.3-2. Plot c(nT) versus time. (b) Verify your results of (a) by determining the input to the plant, m(t), and then calculating c(t) by continuous-time techniques. (c) Find the steady-state gain for a constant input (de gain), from both the pulse transfer function and from the plant transfer function. (d) Is the gain in part (c) obvious from the results of parts (a) and (b)? Why? T = 0.25 s s+2 s²(s + 1) Zero-order hold 1-8-Ts S 2 S² + 25 + 5 G(z) = 3 M(s) Plant 5s s +0.1 Problems 147 4.3-3. Repeat Problem 4.3-2 for the case that T = 0.1 s and the plant transfer function is given by: 5 (a) Gp(s) ² + 3s + 2 (b) Gp(s) 4.3-4. (a) Find the conditions on a transfer function G(z) such that its de gain is zero. Prove your result. (b) For E-TS = [1 = 6,₂(8)] Gp(s) S C(s) find the conditions of G₂(s) such that the de gain of G(z) is zero. Prove your result. (c) Normally, a pole at the origin in the s-plane transforms into a pole at z = 1 in the z-plane. The function in the brackets in part (b) has a pole at the origin in the s-plane. Why does this pole not transform into a pole at z = 1? Open-Loop Discrete-Time Systems Find the conditions on G(z) such that its de gain is unbounded. Prove your result. For G(z) as given in part (b), find the conditions of G,(s) such that the dc gain of G(z) is unbounded. Prove your result. 4.3-5. Find the system response at the sampling instants to a unit-step input for the system of Fig. P4.3-5. 148 ter 4. ● E(s) = 1 S+0.5 FIG. P4.3-5 System for Problem 4.3-5. E(s) T FIG. P4.3-6 System for Problem 4.3-6. 4.3-6. (a) Find the output c(kT) for the system of Fig. P4.3-6, for e(t) equal to a unit-step function. (b) What is the effect on c(kT) of the sampler and data hold in the upper path? Why? (c) Sketch the unit-step response c(t) of the system of Fig. P4.3-6. This sketch can be made without math- ematically solving for C(s). (d) Repeat part (c) for the case that the sampler and data hold in the upper path is removed. E(s) 2 S E(S) G₁(s) T=1s T G₁(s) FIGURE P4.3-7 Systems for Problem 4.3-7. T 4.3-7. (a) Express each C(s) and C(z) as functions of the input for the systems of Fig. P4.3-7. (b) List those transfer functions in Fig. P4.3-7 that contain the transfer function of a data hold. T 1-ε-Ts S D(z) G₂(s) 1-g-Ts S (a) (b) G3(s) 1-8-Ts S G₂(S) 3s (s + 1)(s+2) G3(S) T G4(s) C(s) C(s) C(s) C(s) E(s) FIGURE P4.3-7 (continued) M G₁(s) A/D T 1-ε-Ts S G₂(s) Power amplifier K Digital filter D(z) FIG. P4.3-8 Model of robot arm joint. (c) 4.3-8. Shown in Fig. P4.3-8 is the block diagram of one joint of a robot arm. This system is described in Problem 1.5-4. The signal M(s) is the sampler input, Ea(s) is the servomotor input voltage, Om(s) is the motor shaft angle, and the output (s) is the angle of the arm. (a) Suppose that the sampling-and-data-reconstruction process is implemented with an analog-to-dig- ital converter (A/D) and a digital-to-analog converter. Redraw Fig. P4.3-8 showing the A/D and the D/A. Ea DIA (b) Suppose that the units of ea(t) are volts, and of m(t) are rpm. The servomotor is rated at 24 V (the volt- age ea(t) should be less than or equal to 24 V in magnitude). Commercially available D/As are usually rated with output voltage ranges of ±5 V, ± 10 V, 0 to 5 V, 0 to 10 V, or 0 to 20 V. If the gain of the power amplifier is 2.4, what should be the rated voltage of the D/A? Why? (c) Derive the analog transfer function (s)/Ea(s). (d) With K = 2.4 and T = 0.1 s, derive the pulse transfer function a(z)/M(z). (e) Derive the steady-state output for m(t) constant. Justify this value from the motor characteristics. (f) Verify the results of part (d) by computer. G3(S) Servomotor 200 0.5 s + 1 Problems C(s) m 149 Gears 1 100 0. 4.3-9 Fig. P4.3-9 illustrates a thermal stress chamber. This system is described in Problem 1.6-1. The system output c(t) is the chamber temperature in degrees Celsius, and the control-voltage input m(t) operates a valve on a steam line. The sensor is based on a thermistor, which is a temperature-sensitive resistor. The disturbance input d(t) models the opening of the door into the chamber. With the door closed, d(t) = 0; if the door is opened at t = to, d(t) = u(t - to), a unit-step function. (a) Suppose that the sampling-and-data-reconstruction process is implemented with an analog-to-digital converter (A/D) and a digital-to-analog converter (D/A). Redraw Fig. P4.3-9 showing the A/D and the D/A. (b) Derive the transfer function C(z)/E(z). (c) A constant voltage of e(t) = 10 V is applied for a long period. Find the steady-state temperature of the chamber, with the door closed. Note that this problem can be solved without knowing the sample period T. 150 Chapter 4. Open-Loop Discrete-Time Systems (d) Find the steady-state effect on the chamber temperature of leaving the door open. (e) Find the expression for C(s) as a function of the Laplace-transform variable s, the control input, and the disturbance input. The z-transform variable z cannot appear in this expression. E(s) e(t) T 1-ε-Ts S D(s) d(t) E(s) e(t) Voltage T M(s) m(t) FIGURE P4.3-9 Block diagram for a thermal test chamber. 1-ε-Ts S FIG. P4.3-10 Block diagram for a satellite. 2.5 s +0.5 4.3-10. Given in Fig. P4.3-10 is the block diagram of a rigid-body satellite. The control signal is the voltage e(t) . The zero-order hold output m(t) is converted into a torque T(t) by an amplifier and the thrusters (see Section 1.4). The system output is the attitude angle 0(t) of the satellite. (a) Find the transfer function (z)/E(z). Chamber 2.5 s+( (b) Use the results of part (a) to find the system's unit-step response, that is, the response with e(t) = u(t). (c) Sketch the zero-order-hold output m(t) in (b). (d) Use m(t) in part (c) to find c(t) = L¯¹ [KM(s)/Js²]. (e) In part (d), evaluate c(kT). This response should equal that found in part (b). Sensor 0.04 M(s) m(t) Amplifier and thrusters K T(s) T(t) Torque Temperature (°C) c(t) Sate/lite 1 Js² ✪(s) 0(t) 4.3-11. The antenna positioning system described in Section 1.5 and Problem 1.5-1 is depicted in Fig. P4.3-11. In this problem we consider the yaw angle control system, where 0(t) is the yaw angle. The angle sensor (a digital shaft encoder and the data hold) yields vo(kT) = [0.4 0(kT)], where the units of vo(t) are volts and 0(t) are degrees. The sample period is T = 0.05 s. (a) Find the transfer function Ⓒ(z)/E(z). (b) The yaw angle is initially zero. The input voltage e(t) is set equal to 10 V at t = 0, and is zero at each sample period thereafter. Find the steady-state value of the yaw angle. (c) Note that in part (b), the coefficients in the partial-fraction expansion add to zero. Why does this occur? (d) The input voltage e(t) is set to a constant value. Without solving mathematically, give a description of the system response. E(s) e(t) T Vo(s) vo(t) 1-8-Ts S E*(s) e(kT) E(s) e(t) T FIG. P4.4-1 System for Problem 4.4-1. 0.4 FIGURE P4.3-11 Block diagram for an antenna control system. D(z) 2-z-¹ M(s) m(t) Voltage in Amplifier M(z) m(kT) +10 V A/D K Sensor (e) Suppose in part (d) that you are observing the antenna. Describe what you would see. 4.4-1. Example 4.3 calculates the step response of the system in Fig. 4-2. Example 4.4 calculates the step response of the same system preceded by a digital filter with the transfer function D(z) = (2-z-¹). This system is shown in Fig. P4.4-1. D(z) 1-8-Ts S (a) Solve for the output of the digital filter m(kT). (b) Let the response in Example 4.3 be denoted as c₁(kT). Use the results in part (a) to express the output c(kT) in Fig. P4.4-1 as a function of c₁(kT'). (c) Use the response c₁(kT) calculated in Example 4.3 and the results in part (b) to find the output c(kT) in Fig. P4.4-1. This result should be the same as in Example 4.4. (d) Use the response C₁(z) calculated in Example 4.3 and the result in part (b) to find the output C(z) in Fig. P4.4-1. This result should be the same as in Example 4.4. == -Ts 1-E S 4.5(z - 0.90) z - 0.85 Digital computer D(z) Motor, gears, and pedestal 4.4-2. Consider the hardware depicted in Fig. P4.4-2. The transfer function of the digital controller implemented in the computer is given by 20 s² + 6s FIGURE P4.4-2 Hardware configuration for Problem 4.4-2. T The input voltage rating of the analog-to-digital converter is ±10 V and the output voltage rating of the digital-to-analog converter is also ±10 V. £10 V DIA ✪(s) 0(t) G₁(s) 1 +1 (a) Calculate the de gain of the controller. (b) State exactly how you would verify, using the hardware, the value calculated in part (a). Give the equip- ment required, the required settings on the equipment, and the expected measurements. Problems 151 C(s) c(t) Voltage out 152 Chapter 4. Open-Loop Discrete-Time Systems 4.4-3. For the system of Fig. P4.4-3, the filter solves the difference equation m(k)= 0.9m(k-1) + 0.2e(k) The sampling rate is 1 Hz and the plant transfer function is given by 1 s +0.2 E(s) 4.4-5. (a) Find the system transfer function C(z)/E(z). (b) Find the system dc gain from the results of part (a). (c) Verify the results of part (b) by finding the de gain of the filter using D(z) and that of the plant using Gp(s). (d) Use the results of part (b) to find the steady-state value of the unit-step response. (e) Verify the results of part (d) by calculating c(kT) for a unit-step input. (f) Note that in part (e), the coefficients in the partial-fraction expansion add to zero. Why does this occur? A/D e(KT) FIGURE P4.4-3 System for Problems 4.4-3 and 4.4-4. E(s) G₁(s) Digital filter D(z) G₁(s) A/D 4.4-4. Repeat Problem 4.4-3 for the case that the filter solves the difference equation m(k + 1) 0.5e(k+ 1) - (0.5)(0.98)e(k) + 0.995m(k), the sampling rate is 10 Hz, and the plant transfer function is given by 5 (s + 1)(s + 2) FIGURE P4.4-5 System for Problem 4.4-5. = M(z) m(kT) Gp(s): = DIA Consider the system of Fig. P4.4-5. The filter transfer function is D(z). (a) Express C(z) as a function of E. (b) A discrete state model of this system does not exist. Why? (c) What assumptions concerning e(t) must be made in order to derive an approximate discrete state model? Digital filter Plant G₁(s) DIA 0.5 Zz - 1 C(s) G₂(s) C(s) 4.4-6. Consider again the system of Fig. P4.4-5. Add a sampler for E(s) at the input. Given G₁(s) 1 s + 10 D(z) G₂(s) S 5² + 9s + 23 find c(KT) for a unit-step input with T = 0.5 s. Is this a good choice for the sampling interval? If not, what sampling rate would you recommend. Find c(kT) at your recommended sampling rate and plot both step responses using MATLAB. 4.5-1. Find the modified z-transform of the following functions: (a) E(s) (c) E(s) (e) E(s) = (c) E(s) = (a) E(s) = = (e) E(s) = (c) E(s) (e) E(s) - (a) E(s) = (e) E(s) 20 (s + 2)(s + 5) s +2 s(s+ 1) 4.5-2. Find the z-transform of the following functions. The results of Problem 4.5-1 may be useful. 20€ 0.37's (s + 2)(s + 5) - s² + 5s + 6 s(s+ 4)(s + 5) (c) E(s) = S +28 1.17s s(s + 1) (s² + 5s + 6)ε-0.3Ts s(s+ 4)(s + 5) s² + 25 + 2 s(s+ 2)² (b) E(s) S² + 2s +3 s(s+ 2)²(s + 4) (d) E(s) s²-0.3Ts + 2s +3 s(s+ 2)²(s + 4) (f) E(s) = 4.5-3. Find the modified z-transform of the following functions: 6 (s + 1)(s + 2)(s + 3) (b) E(s) = == (d) E(s) = (f) E(s) = (b) E(s) = (d) E(s) (f) E(s) (b) E(s) = - Gp(s) = 5 s(s+1) = (d) E(s) = s+2 s²(s + 1) (f) E(s) = 2 S² + 25 + 5 4.5-4. Find the z-transform of the following functions. The results of Problem 4.5-3 may be useful. (a) E(s) 6ε-0.3Ts 4ε-0.67's s(s+ 2)² (s + 1)(s + 2)(s + 3) s² + 25 + 2€-1.17s s(s+ 2)² 5ε-0.67's s(s + 1) S + 28-0.27's s²(s + 1) 28-0.75TS S² + 2s + 5 4 s(s+ 2)² (s + 2)² s² (s + 1)² 2s + 7 (s² + 2s + 5)(s + 3) (s + 2)²-0.2Ts s²(s + 1)² Problems (2s + 7)e-0.75Ts (s² + 2s + 5)(s + 3) 4.6-1. Generally, a temperature control system is modeled more accurately if an ideal time delay is added to the plant. Suppose that in the thermal test chamber of Problem 4.3-9, the plant transfer function is given by 2ε-2s s + 0.5 C(s) E(s) 153 Hence the plant has a 2-s time delay before its response to an input. For this problem, let the sample period T = 0.6s. (a) Find the unit step response for the system of Fig. P4.3-9; that is, find c(kT) with e(t) = u(t) and d(t)= 0, and with no delay. (b) Repeat part (a), with the 2-s time delay included in G₁(s), as given above. 154 Chapter 4 Open-Loop Discrete-Time Systems (c) Solve for c(t) with no delay, using the Laplace transform and the plant input m(t). (d) Find c(t) for the delay included, using the results of part (c). (e) Show that in part (c), c(t) evaluated at t = kT yields c(kT) in part (a). (f) Show that in part (d), c(t) evaluated at t = kT yields c(kT) in part (b). For the thermal test chamber of Problems 4.3-9 and 4.6-1, let T = 0.6 s. Suppose that a proportional-integral (PI) digital controller with the transfer function 4.6-2. 4.6-3. is inserted between the sampler and the zero-order hold in Fig. P4.3-9. (a) With d(t) = 0 and e(t) = u(t), solve for c(kT), with the time delay of 2 s omitted. (d) Considering the physical characteristics of a valve, what happens to the temperature in the physical test chamber? For Problem 4.4-6, assume that the total computational delay in D(z) plus the plant delay totals 3.47 sec- onds. Repeat the problem under these new conditions. 4.8-1. Find a discrete-state-variable representation for the system shown in Fig. P4.8-1. Digital filter 4.9-1. (b) Explain what happens to the temperature in the chamber in part (a). Is this result physically possible? (c) In Fig. P4.3-9, calculate m(kT), the signal that controls the valve in the steam line, for the inputs of part (a). E(s) T FIGURE P4.8-1 System for Problem 4.8-1. D(z) = 1.2 + 0.1z z-1 x(k + 1) = 0.9 0 A discrete-state-variable description of the continuous-time system is given by 0 A −1 0.1z D(z) = 1 0.9 0 1.5 Derive the state equations for this system. (b) Repeat part (a) for the system described by M(z) Use the matrix technique of Section 4.9. Continuous plant and data hold 2 0 x(k) + 0.5 2.3]x(k) y(k) = [1 4.8-2. Repeat Problem 4.8-1 after replacing the digital filter with this transfer function: z - 0.5 (z - 0.8)(z 1) 1 m(k) (a) The model of a continuous-time system with algebraic loops is given as x(t) = -2x(t) + 0.5x(t) + 3u(t) y(t) = x(t) + 4u(t) x₁ (t) = −xi(t) + 2*2(t) + u1(t) - x₂(t) = x₂(t) x₂(t) = x₁(t) + x₁(t) + u₂(t) y(t) x₂(t) Y(s) 4.9-2. (c) Verify the results of part (b) by solving the given equations for x₁(t) and x₂(t) in the standard state-variable format. The model of a continuous-time system with algebraic loops is given as x₁ (1) x₂ (1) *3(D) y(t) = −X1(t) + 2kz(t) + u1(t) -x2₂ (1) -x3(1) X3(t) x2(1) X3(t) Derive the state equations for this system. Use the matrix technique of Section 4.9. Verify the results by solving the given equations for x₁(t), x₂(t), and x3(t) in the standard state-variable format. 4.10-1. Consider the system of Fig. P4.10-1. The plant is described by the first-order differential equation E(s) dy(t) dt x₁(t) + x₁(t) + u₂(t) x₂(t) + x₁(t) + u₂(t) +0.05y(t) = 0.1m(t) Let T = 2 s. (a) Find the system transfer function Y(z)/E(z). (b) Draw a discrete simulation diagram, using the results of part (a), and give the state equations for this diagram. T FIGURE P4.10-1 System for Problem 4.10-1. (c) Draw a continuous-time simulation diagram for Go(s), and given the state equations for this diagram. (d) Use the state-variable model of part (c) to find a discrete state model for the system. The state vectors of the discrete system and the continuous-time system are to be the same. (e) Draw a simulation diagram for the discrete state model in part (d). (f) Use Mason's gain formula to find the transfer function in part (e), which must be the same as found in part (a). 1-ε-Ts S M(s) Problems 155 Plant Gp(s) Y(s) 4.10-2. Repeat Problem 4.10-1 for the plant described by the second-order differential equation d²y(t) dt² dy(t) +0.15- + 0.005y(t) 0.1m(t) dt 4.10-3. Consider the robot arm system of Fig. P4.3-8. Let T = 0.1 s. (a) Find system transfer function a(z)/M(z). (b) Draw a discrete simulation diagram, using the results of part (a), and give the state equations for this diagram. (c) Draw a continuous-time simulation diagram for amplifier-servomotor-gears system and give the state equations for this diagram. (d) Use the state-variable model of part (c) to find a discrete state model for the system. The states of the discrete systems are the same as those of the continuous-time system. (e) Draw a simulation diagram for the discrete state model in part (d). (f) Use Mason's gain formula to find the transfer function in part (e), which must be the same as found in part (a)./n 112 Problems 3.2-1. (a) Give the definition of the starred transform. (b) Give the definition of the z-transform. (c) For a function e(t), derive a relationship between its starred transform E*(s) and its z-transform E(z). A signal e(t) is sampled by the ideal sampler as specified by (3-3). (a) List the conditions under which e(t) can be completely recovered from e*(t), that is, the conditions under which no loss of information by the sampling process occurs. (b) State which of the conditions listed in part (a) can occur in a physical system. Recall that the ideal sampler itself is not physically realizable. (c) Considering the answers in part (b), state why we can successfully employ systems that use sampling. Use the residue method of (3-10) to find the starred transform of the follow functions. 20 (b) E(s) 5 s(s+ 1) (s + 2)(s + 5) s+2 s(s+ 1) 3.3-1. -3.4-1. Chapter 3. Sampling and Reconstruction (a) E(s) = (c) E(s) -3.4-6. (e) E(s) (a) e(t) (c) e(t) 3.4-3. For e(t) = S² + 5s +6 s(s+ 4)(s + 5) (d) E(s) = (f) E(s) 3.4-2. Find E* (s) for each of the following functions. Express E*(s) in closed form. 8-2Ts ɛat Find E*(s), with T = 0.5 s, for (b) E(s) = (d) e(t) = ga(t-2T)u(t - 2T) ε-3t (a) Express E*(s) as a series. (b) Express E*(s) in closed form. (c) Express E*(s) as a series which is different from that in part (a). 3.4-4. Express the starred transform of e(t - kT)u(t-kT), k an integer, in terms of E*(s), the starred transform of e(t). Base your derivation on (3-3). 3.4-5. Find E*(s) for E(s) E(s) s+2 s²(s + 1) 2 s² + 25 + 5 - = sa Ea(t-T/2)u(t - T/2) 1- &¹ -Ts s(s+1) (1- ε-0.5s)2 0.5s² (s + 1) 3.4-7. Suppose that E*(s) = [e(t)]* = 1. (a) Find e(kT) for all k. (b) Can e(t) be found? Justify your answer. (c) Sketch two different continuous-time functions that satisfy part (a). (d) Write the equations for the two functions in part (c). 3.6-1. (a) Find E*(s), for T = 0.1 s, for the two functions below. Explain why the two transforms are equal, first from a time-function approach, and then from a pole-zero approach. (i) e(t)= cos(4πt) (ii) e₂(t)= cos(16mt) (b) Give a third time function that has the same E*(s). Problems 113 3.6-2. Compare the pole-zero locations of E*(s) in the s-plane with those of E(s), for the functions given in Problem 3.4-1. 3.7-1. 3.7-2. Suppose that the signal e(t) = cos[(ws/2)t +0] is applied to an ideal sampler and zero-order hold. (a) Show that the amplitude of the time function out of the zero-order hold is a function of the phase angle by sketching this time function. (b) Show that the component of the signal out of the data hold, at the frequency w = w,/2, is a function of the phase angle 0 by finding the Fourier series for the signal. 3.7-3. (a) A sinusoid with a frequency of 2 Hz is applied to a sampler/zero-order hold combination. The sampling rate is 10 Hz. List all the frequencies present in the output that are less than 50 Hz. (b) Repeat part (a) if the input sinusoid has a frequency of 8 Hz. (c) The results of parts (a) and (b) are identical. Give three other frequencies, which are greater than 50 Hz, that yield the same results as parts (a) and (b). 3.7-4. Given the signal e(t) = 3 sin 4t + 2 sin 7t. 3.7-5. 3.7-6. A system is defined as linear if the principle of superposition applies. Is a sampler/zero-order-hold device linear? Prove your answer. (a) List all frequencies less than = 50 rad/s that are present in e(t). (b) The signal e(t) is sampled at the frequency w, = 22 rad/s. List all frequencies present in e*(t) that are less than 50 rad/s. (c) The signal e*(t) is applied to a zero-order hold. List all frequencies present in the hold output that are less than = 50 rad/s. (d) Repeat part (c) for e* (t) applied to a first-order hold. A signal e(t) = 4 sin 7t is applied to a sampler/zero-order-hold device, with = 4 rad/s. (a) What is the frequency component in the output that has the largest amplitude? (b) Find the amplitude and phase of that component. (c) Sketch the input signal and the component of part (b) versus time. (d) Find the ratio of the amplitude in part (b) to that of the frequency component in the output at w 7 rad/s (the input frequency). It is well known that the addition of phase lag to a closed-loop system is destabilizing. A sampler/data-hold device adds phase lag to a system, as described in Section 3.7. A certain analog control system has a bandwidth of 10 Hz. By this statement we mean that the system (approximately) will respond to frequen- cies less than 10 Hz, and (approximately) will not respond to frequencies greater than 10 Hz. A sampler/ zero-order-hold device is to be added to this control system. (a) It has been determined that the system can tolerate the addition of a maximum of 10° phase lag within the system bandwidth. Determine the approximate minimum sampling rate allowed, along with the sample period T. (b) Repeat part (a) for a maximum of 5° phase lag. (c) Repeat part (a) for a maximum of 20° phase lag. 3.7-7. A sinusoid is applied to a sampler/zero-order-hold device, with a distorted sine wave appearing at the out- put, as shown in Fig. 3-15. (a) With the sinusoid of unity amplitude and frequency 2 Hz, and with f 12 Hz, find the amplitude and phase of the component in the output at fi = 2 Hz. = (b) Repeat part (a) for the component in the output at (fs - fi) = 10 Hz. (c) Repeat parts (a) and (b) for a sampler-first-order-hold device. (d) Comment on the distortion in the data-hold output for the cases considered in parts (a), (b), and (c). 114 Chapter 3. Sampling and Reconstruction 3.7-8. A polygonal data hold is a device that reconstructs the sampled signal by the straight-line approximation shown in Fig. P3.7-8. Show that the transfer function of this data hold is Is this data hold physically realizable? Input to sampler ε(1 - ε-Ts)² G(s) = Ts² L 2T 0 T FIGURE P3.7-8 Response of a polygonal data hold. 1 37 T Data-hold output I 4T Input to sampler Z Data-hold output 0 27 3T FIGURE P3.7-9 Response of a polygonal data hold with time delay. 3.7-9. A data hold is to be constructed that reconstructs the sampled signal by the straight-line approximation shown in Fig. P3.7-9. Note that this device is a polygonal data hold (see Problem 3.7-8) with a delay of T seconds. Derive the transfer function for this data hold. Is this data hold physically realizable? 5T 4T t 3.7-10. Plot the ratio of the frequency responses (in decibels) and phase versus for the data holds of Problems 3.7-8 and 3.7-9. Note the effect on phase of making the data hold realizable. 3.7-11. Derive the transfer function of the fractional-order hold [see (3-39)]. 3.7-12. Shown in Fig. P3.7-12 is the output of a data hold that clamps the output to the input for the first half of the sampling period, and returns the output to a value of zero for the last half of the sampling period. (a) Find the transfer function of this data hold. (b) Plot the frequency response of this data hold. (c) Comparing this frequency response to that of the zero-order hold, comment on which would be better for data reconstruction. Sampler input hin: T 3T 2T 5T 3T 7T 2 2 2 FIGURE P3.7-12 Data hold output for Problem 3.7-12. Data-hold Problems 115 output t