Derive the z-transforms for the time functions listed

Derive the z-transforms for the time functions listed below. Do not use any z-transform tables. Use the plan \(f(t) \rightarrow f^*(t) \rightarrow F^*(s) \rightarrow F(z)\), followed by converting F(z) into closed form making use of the fact that \(1 /\left(1-z^{-1}\right)=1+z^{-1}+z^{-2}+z^{-3}+\cdots\). Assume ideal sampling. [Section: 13.3] (a) \(e^{-a t} u(t)\) (b) u(t) (c) \(t^2 e^{-a t} u(t)\) (d) \(\cos \omega t u(t)\)

Repeat all parts of Problem 1 using MATLAB and MATLAB’s Symbolic Math Toolbox.

For each F(z), find f(kT) using partial-fraction expansion. [Section: 13.3] (a) \(F(z)=\frac{z(z+3)(z+5)}{(z-0.4)(z-0.6)(z-0.8)}\) (b) \(F(z)=\frac{(z+0.2)(z+0.4)}{(z-0.1)(z-0.5)(z-0.9)}\) (c) \(F(z)=\frac{(z+1)(z+0.3)(z+0.4)}{z(z-0.2)(z-0.5)(z-0.7)}\)

Repeat all parts of Problem 3 using MATLAB and MATLAB’s Symbolic Math Toolbox.

For each F(z) in Problem 3, do the following: [Section: 13.3] (a) Find f(kT) using the power series expansion. (b) Check your results against your answers from Problem 3.

Using partial-fraction expansion and Table 13.1, find the z-transform for each G(s) shown below if T=0.5 second. [Section: 13.3] (a) \(G(s)=\frac{(s+4)}{(s+2)(s+5)}\) (b) \(G(s)=\frac{(s+1)(s+2)}{s(s+3)(s+4)}\) (c) \(G(s)=\frac{20}{(s+3)\left(s^2+6 s+25\right)}\) (d) \(G(s)=\frac{15}{s(s+1)\left(s^2+10 s+81\right)}\)

Repeat all parts of Problem 6 using MATLAB and MATLAB’s Symbolic Math Toolbox.

Find G(z)=C(z)/R(z) for each of the block diagrams shown in Figure P13.1 if T = 0:3 second. [Section: 13.4]

Find T(z)=C(z)/R(z) for each of the systems shown in Figure P13.2. [Section: 13.5]

Find C(z) in general terms for the digital system shown in Figure P13.3. [Section: 13.5]

Find the closed-loop transfer function, T(z)=C(z)/R(z), for the system shown in Figure P13.4. [Section: 13.5]

Write a MATLAB program that can be used to find the range of sampling time, T, for stability. The program will be used for systems of the type represented in Figure P13.5 and should meet the following requirements: (a) MATLAB will convert \(G_1 (s)\) cascaded with a sample-and-hold to G(z). (b) The program will calculate the z-plane roots of the closed-loop system for a range of T and determine the value of T, if any, below which the system will be stable. MATLAB will display this value of T along with the z-plane poles of the closed-loop transfer function. Test the program on \(G_1(s)=\frac{20(s+6)}{(s+1)(s+3)(s+4)(s+8)}\)

Given the system in Figure P13.6, find the range of sampling interval, T, that will keep the system stable. [Section: 13.6]

Find the range of gain, K, to make the system shown in Figure P13.7 stable. [Section: 13.6]

Find the static error constants and the steady-state error for each of the digital systems shown in Figure P13.8 if the inputs are [Section: 13.7] (a) u(t) (b) tu(t) (c) \(\frac{1}{2} t^2 u(t)\)

Write a MATLAB program that can be used to find \(K_p\), \(K_V\), and \(K_a\) for digital systems. The program will be used for systems of the type represented in Figure P13.5. Test your program for \(G(z)=\frac{0.04406 z^3-0.03624 z^2-0.03284 z+0.02857}{z^4-3.394 z^3+4.29 z^2-2.393 z+0.4966}\) where G(z) is the pulse transfer function for \(G_1(s)\) in cascade with the z.o.h. and T=0.1 second.

For the digital system shown in Figure P13.5, where \(G_1(s)=K /[(s+1)(s+5)]\), find the value of K to yield a 15% overshoot. Also find the range of K for stability. Let T=0.1 second. [Section: 13.9]

Use Simulink to simulate the step response for the system of Problem 17. Set the value of gain, K, to that designed in Problem 17 for 15% overshoot.

Use MATLAB’s LTI Viewer to determine the peak time and settling time of the closed-loop step response for System 4 in Figure P13.8.

Write a MATLAB program that can be used to design the gain of a digital control system to meet a percent overshoot requirement. The program will be used for systems of the type represented in Figure P13.5 and meet the following requirements: (a) The user will input the desired percent overshoot. (b) MATLAB will convert \(G_1 (s)\) cascaded with the sample-and-hold to G(z). (c) MATLAB will display the root locus on the z-plane along with an overlay of the percent overshoot curve. (d) The user will click with the mouse at the intersection of the root locus and percent overshoot overlay and MATLAB will respond with the value of gain followed by a display of the step response of the closed-loop system. Apply your program to Problem 17 and compare results.

For the digital system shown in Figure P13.5, where \(G_1(s)=K /[s(s+1)]\), find the value of K to yield a peak time of 2 seconds if the sampling interval, T, is 0.2 second. Also, find the range of K for stability. [Section: 13.9]

For the digital system shown in Figure P13.5, where \(G_1(s)=K /[s(s+1)(s+3)]\), find the value of K to yield a 20% overshoot if the sampling interval, T, is 0.1 second. Also, find the range of K for stability. [Section: 13.9]

For the digital system shown in Figure P13.5, where \(G_1(s)=K(s+2) \div[s(s+1)(s+3)]\), find the value of K to yield a settling time of 15 seconds if the sampling interval, T, is 1 second. Also, find the range of K for stability. [Section: 13.9]

A PID controller was designed in Example 9.5 for a continuous system with unity feedback. The system's plant was \(G(s)=\frac{(s+8)}{(s+3)(s+6)(s+10)}\) The designed PID controller was \(G_c(s)=4.6 \frac{(s+55.92)(s+0.5)}{s}\) Find the digital transfer function, \(G_c(z)\), of the PID controller in order for the system to be computer controlled if the sampling interval, T, is 0.005 second. [Section: 13.10]

A continuous unity feedback system has a forward transfer function of \(G(s)=\frac{1}{s(s+5)(s+8)}\) The system is to be computer controlled with the following specifications: Percent overshoot: 10% Settling time: 2 seconds Sampling interval: 0.01 second Design a lead compensator for the digital system to meet the specifications. [Section: 13.10]

Repeat Problem 25 using MATLAB.

An inverted pendulum mounted on a motor-driven cart (Prasad, 2012) was the subject of Problem 33, Chapter 9. In that problem you were asked to develop Simulink models for two feedback systems,one ofwhichwastocontrolthecart position, x(t). At the recommended settings, the step response of that system was expected to satisfy the following requirements: a steady-state error, \(e(\infty) <2 \%\), a peak time, \(T_P<1.2\) seconds, and a percent overshoot, %OS <20.5%. Having concluded that the steady-state error was unacceptable, you designed a PID controller and found its recommended settings. Digitize the Simulink model developed for PID control of the cart position in Part d of Problem 33, Chapter 9, by adding a zero-order-hold set to 0.01 second. Then, run a simulation to evaluate performance.

(a) Convert the heading control for the UFSS vehicle shown on the back endpapers (Johnson, 1980) into a digitally controlled system. (b) Find the closed-loop pulse transfer function, T(z), if T = 0:1 second. (c) Find the range of heading gain to keep the digital system stable.

The analog system of a disk drive is shown in Figure P13.9. Do the following: (a) Convert the disk drive to a digital system. Use a sampling time of 0.01 second. (b) Find the range of digital controller gain to keep the system stable. (c) Find the value of digital controller gain to yield 15% overshoot for a digital step response. (d) Repeat all previous parts and obtain the step response for Part c using MATLAB.

In Problem 49, Chapter 9, and Problem 39, Chapter 10, we considered the radial pickup position control of a DVD player. A controller was designed and placed in cascade with the plant in a unit feedback configuration to stabilize the system. The controller was given by \(M(s)=\frac{0.5(s+1.63)}{s(s+0.27)}\) and the plant by (Bittanti, 2002) \(P(s)=\frac{0.63}{\left(1+\frac{0.36}{305.4} s+\frac{s^2}{305.4^2}\right)\left(1+\frac{0.04}{248.2} s+\frac{s^2}{248.2^2}\right)}\) It is desired to replace the continuous system by an equivalent discrete system without appreciably affecting the system performance. (a) Find an appropriate sampling frequency for the discretization. (b) Using the chosen sampling frequency, translate the continuous compensator into a discrete compensator. (c) Use Simulink to simulate the continuous and discrete systems on the same graph. Assume a unit step input. Are there significant differences in the system’s performance?

In Problem 24, Chapter 11, we discussed an EVAD, a device that works in parallel with the human heart to help pump blood in patients with cardiac conditions. The device has a transfer function \(G(s)=\frac{P_{a o}(s)}{E_m(s)}=\frac{1361}{s^2+69 s+70.85}\) where \(E_m(s)\) is the motor's armature voltage, and \(P_{a o}(s)\) is the aortic blood pressure (Tasch, 1990). Using continuous techniques, a cascaded compensator is designed in a unity feedback configuration with a transfer function \(G_c(s)=\frac{0.5(s+1)}{s+0.05}\) Selecting to control the device using a microcontroller, a discrete equivalent has to be found for \(G_c(s)\). Do the following: (a) Find an appropriate sampling frequency for the discretization. (b) Translate the continuous compensator into a discrete compensator using the sampling frequency found in Part a. (c) Use Simulink to simulate the continuous and discrete systems on the same graph for a unit step input. There should be little difference between the compensated continuous and discrete systems.

In Problem 47, Chapter 9, a steam-driven turbine governor system was implemented by a unity feedback system with a forward-path transfer function (Khodabakhshian, 2005) \(G(s)=\frac{K}{(s+0.08)(s+2)(s+5)}\) (a) Use a sampling period of T = 0:5 s and find a discrete equivalent for this system. (b) Use MATLAB to draw the root locus. (c) Find the value of K that will result in a stable system with a damping factor of \(\zeta=0.7\). (d) Use the root locus found in Part a to predict the step-response settling time, \(T_s\), and peak time, \(T_p\). (e) Calculate the final value of the closed-loop system unit step response. (f) Obtain the step response of the system using Simulink. Verify the predictions you made in Parts c and d.

Discrete time controlled systems can exhibit unique characteristics not available in continuous controllers. For example, assuming a specific input and some conditions, it is possible to design a system to achieve steady state within one single time sample without overshoot. This scheme is well known and referred to as deadbeat control. We illustrate deadbeat control design with a simple example. For a more comprehensive treatment see (Ogata, 1987). Assume in Figure 13.25(a) that \(G_p(s)=\frac{1}{s+1}\). The purpose of the design will be to find a compensator, \(G_c(z)\), such that for a step input the system achieves steady state within one sample. We start by translating the system into the discrete domain to obtain the equivalent of Figure 13.25(c). The pulse transfer function, \(G_p(z)=\frac{\left(1-e^{-T}\right) z^{-1}}{1-e^{-T} z^{-1}}\), is found using Eq. (13.40), since it is assumed that the compensator will be followed by a zero-order hold. In Figure 13.25(c), the closed-loop transfer function is given by \(\frac{C(z)}{R(z)}=T(z)=\frac{G_c(z) G_p(z)}{1+G_c(z) G_p(z)}\), or, solving for the compensator, we get \(G_c(z)=\frac{1}{G_p(z)} \frac{T(z)}{1-T(z)}\). The desired system output is a unit step delayed by one unit sample. Thus, \(C(z)=\frac{z}{z-1} z^{-1}=\frac{1}{z-1}\). Since the input is a unit step, \(R(z)=\frac{z}{z-1}\); the desired closed-loop transfer function is \(\frac{C(z)}{R(z)}=T(z)=z^{-1}\), and the resulting compensator, found by direct substitution, is given by \(G_c(z)=\frac{1}{1-e^{-T}} \frac{\left(z-e^{-T}\right)}{z-1}\). Assume now that the plant is given by \(G_p(s)=\frac{1}{s}\), and a sampling period of T=0.05 second is used. (a) Design a deadbeat compensator to reach steady state within one time sample for a step input. (b) Calculate the resulting steady-state error for a unit-slope ramp input. (c) Simulate your system using Simulink. (Hint: Following Figure 13.25, the forward path will consist of the cascading of \(G_C(z)\), a zero-order hold, and \(\left.G_p(s).\right)\) Show that the system reaches steady state after one sample. Also verify your steady-state error ramp result.

Given \(G(s)=\frac{8}{s+4}\) Use the LabVIEW Control Design and Simulation Module to (1) convert G(s) to a digital transfer function using a sampling rate of 0.25 second; and (2) plot the step responses of the discrete and the continuous transfer functions.

Given \(G(z)=\frac{K(z+0.5)}{(z-0.25)(z-0.75)}\) Use the LabVIEW Control Design and Simulation Module and the MathScript RT Module to (1) obtain the value of K that will yield a damping ratio of 0.5 for the closed-loop system in Figure 13.20, where H(z)=1; and (2) display the step response of the closed-loop system in Figure 13.20 where H(z)=1. Compare your results with those of Skill-Assessment Exercise 13.8.

Obtaining an exact shape in metal forming can be tricky because of material springback. A feedback system has been devised in which critical deviations from specifications are measured as soon as a part is formed and automatic incremental corrections to the forming tools are made before the next part is formed. Eventually the system compensates for material springback and results in parts compliant with specifications. A unity-feedback digital system with a forward path \(G(z)=\frac{k}{z-1}\) can be used as a simplified representation of the system (Fu, 2013). a. Make a sketch of the system’s root locus. b. Find the range of k for which the system is closed loop stable. c. Find the system’s steady-state error for a step input. d. Find the value of k that will result in the fastest possible response.

The purpose of an artificial pacemaker is to regulate heart rate in those patients in which the natural feedback system malfunctions. Assume a unity-feedback system with a forward path, \(G(s)=\frac{1352 k}{s(s+8)(s+20)}\) as a simplified model of a pacemaker (Neogi, 2010). a. Convert the pacemaker model to a discrete system with a sampling rate of 0.01 second. b. Draw the root locus of the system using a computer program. c. Use the root locus in Part b to find the range of k for which the system is closed-loop stable. d. Use the root locus from Part b to find the value of k that will yield a 5% overshoot for a step input. e. Simulate The unit-step input of your discretized system to verify your design.

A linear model of the \(\alpha\)-subsystem of a grid-connected voltage-source converter with a Y-Y transformer (Mahmood, 2012) was presented in Problem 69, Chapter 8, and Problem 51, Chapter 10. The system was represented with unity-feedback and a forward path consisting of the cascading of a compensator and a plant. The plant is given by \(G_P(s)=\frac{V_a(s)}{M_a(s)}=\frac{(s+2200)}{(s+220)\left(s^2+120 s+16 \times 10^6\right)}\) This system is now to be digitally controlled with the following specifications: percent overshoot, %OS=10%; settling time, \(T_S=0.1\) second; and sampling interval, T=0.001 second. Design a lead compensator for that system to meet these specifications. [Section: 13.10]

Control of HIV/AIDS. In Chapter 11, a continuous cascaded compensator for a unity feedback system was designed for the treatment of the HIV-infected patient treated with RTIs (Craig, 2004). The transfer function of the designed compensator was \(G_c(s)=\frac{-2 \times 10^{-4}\left(s^2+0.04 s+0.0048\right)}{s(s+0.02)}\) The linearized plant was given by \(P(s)=\frac{Y(s)}{U_1(s)}=\frac{-520 s-10.3844}{s^3+2.6817 s^2+0.11 s+0.0126}\) The compensated system is overdamped with an approximate settling time of 100 seconds. This system must be discretized for practical reasons: (1) HIV patient cannot be monitored continuously and (2) medicine dosage cannot be adjusted continuously. a. Show that a reasonable sampling period for this system is T=8 days (medicine dosage will be updated on a weekly basis). b. Use Tustin's method and T=8 days to find a discrete equivalent to \(G_c(s)\). c. Use Simulink to simulate the continuous and discrete compensated systems for a unit step input. Plot both responses on the same graph.

Hybrid vehicle. In Problem 67, Chapter 7 (Figure P7.31), the block diagram of a cascade scheme for the speed control of an HEV (Preitl, 2007) was represented as a unity feedback system. In that diagram the output of the system is the speed transducer's output voltage, \(C(s)=K_{s s} V(s)\). In Part b of Problem 33, Chapter 11, where a compensator was designed for this problem, we discussed the feasibility of achieving full pole-zero cancellation when we place a PI speed controller's zero, \(Z_I\), on top of the uncompensated system's real pole, closest to the origin (located at -0.0163 ). Noting that perfect pole-zero cancellation may not be maintained, we studied a case where the PI-controller's zero changed by +20%, moving to -0.01304. In that case, the transfer function of the plant with a PI speed controller, which has a proportional gain =K, was given by \(G(s)=\frac{K(s+0.6)(s+0.01304)}{s(s+0.0163)(s+0.5858)}\) Assuming that \(G_1(s)\) in Figure P13.5 equals the transfer function, G(s), given above for the vehicle with the speed controller: (a) Develop a MATLAB M-file that would allow you to do the following: (Hint: Refer to the M-files you developed for Problems 12 and 20 of this chapter) (1) Convert \(G_1(s)\) cascaded with a sample-and-hold to G(z); (2) Search over the range 0<T<5 second for the largest sampling period \(T_{\max }\) below which the system is stable. Calculate the z-plane roots of the closed-loop system for the whole range of the sampling time, T. Subsequently set \(T=0.75 T_{\max }\); (3) Design the gain of a digital control system to meet a percent overshoot requirement, %OS, allowing the user to input the value of the desired %OS and the value of the PI speed controller's proportional gain, K; (4) Plot the step response of that digital system (in per unit, p.u., vs. time in seconds). (b) Run the M-file you developed in Part a and enter the values of the desired percent overshoot, %OS=0, and the PI speed controller's proportional gain, K=61; (c) Select a point in the graphics window displaying the root locus, such that all poles of the closed-loop transfer function, \(T_z\), are inside the unit circle. (d) Write the sampled-data transfer functions obtained, \(G_z\) and \(T_z\), indicating the corresponding value of the sampling time, T, and all poles, r, of the closed-loop transfer function, \(T_z\); (e) Plot the step response of that digital system (in per unit, p.u., vs. time in seconds) noting the following characteristics: final value, rise time, and settling time.

Parabolic trough collector. In Problem 34, Chapter 11, a zero steady-state error for a unit-step input was achieved through the design of a lag compensator with integral control. In that problem, the open-loop transmission can be written as \(L(s)=G_c(s) G(s)\), where the parabolic trough plant is given by (Camacho, 2012) \(G(s)=\frac{137.2 \times 10^{-6}}{s^2+0.0224 s+196 \times 10^{-6}} e^{-39 s}\) and the lag compensator is given by \(G_c(s)=1.12 \frac{(s+0.01)}{s}\) We want to substitute for the continuous compensator with a digital one. (a) Find a suitable sampling period for the system. (b) Find the equivalent compensator’s transfer function in z-domain. (c) Use Simulink to simulate the digital compensator with the continuous plant. Compare the resulting response with that of the original system using the continuous compensator on the same graph.