A compensated motor position control system is shown in Fig. 4.29. Assume that the

Give three advantages of feedback in control

Give two disadvantages of feedback in control.

A temperature control system is found to have zero error to a constant tracking input and an error of 0.5 C to a tracking input that is linear in time, rising at the rate of 40C/sec. What is the system type of this control system and what is the relevant error constant (Kp or Kv or etc.)?

What are the units of Kp , Kv , and Ka?

What is the definition of system type with respect to reference inputs?

What is the definition of system type with respect to disturbance inputs?

Why does system type depend on where the external signal enters the system?

What is the main objective of introducing integral control?

What is the major objective of adding derivative control?

Why might a designer wish to put the derivative term in the feedback rather than in the error path?

What is the advantage of having a tuning rule for PID controllers?

Give two reasons to use a digital controller rather than an analog controller

Give two disadvantages to using a digital controller.

Give the substitution in the discrete operator z for the Laplace operator s if the approximation to the integral in Eq. (4.98) is taken to be the rectangle of height e(kTs) and base Ts .

If S is the sensitivity of the unity feedback system to changes in the plant transfer function and T is the transfer function from reference to output, show that S + T = 1.

We define the sensitivity of a transfer function G to one of its parameters k as the ratio of percent change in G to percent change in k. The purpose of this problem is to examine the effect of feedback on sensitivity. In particular, we would like to compare the topologies shown in Fig. 4.23 for connecting three amplifier stages with a gain of K into a single amplifier with a gain of 10. (a) For each topology in Fig. 4.23, compute i so that if K= 10, Y= 10R. (b) For each topology, compute when G= Y/R . [Use the respective i values found in part (a).] Which case is the least sensitive? (c) Compute the sensitivities of the systems in Fig. 4.23(b,c) to 2 and 3 . Using your results, comment on the relative need for precision in sensors and actuators. Figure 4.23 Three-amplifier topologies for Problem 4.2

Compare the two structures shown in Fig. 4.24 with respect to sensitivity to changes in the overall gain due to changes in the amplifier gain. Use the relation as the measure. Select H1 and H2 so that the nominal system outputs satisfy F1 = F2 , and assume KH1 > 0. Figure 4.24 Block diagrams for Problem 4.3

A unity feedback control system has the open-loop transfer function (a) Compute the sensitivity of the closed-loop transfer function to changes in the parameter A. (b) Compute the sensitivity of the closed-loop transfer function to changes in the parameter a. (c) If the unity gain in the feedback changes to a value of 1, compute the sensitivity of the closed-loop transfer function with respect to .

Compute the equation for the system error for the filtered feedback system shown in Fig. 4.4.

If S is the sensitivity of the filtered feedback system to changes in the plant transfer function and T is the transfer function from reference to output, compute the sum of S + T. Show that S + T = 1 if F = H. (a) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the plant transfer function, G. (b) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the controller transfer function, Dcl . (c) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the filter transfer function, F. (d) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the sensor transfer function, H.

Consider the DC-motor control system with rate (tachometer) feedback shown in Fig. 4.25(a). (a) Find values for K and so that the system of Fig. 4.25(b) has the same transfer function as the system of Fig. 4.25(a). (b) Determine the system type with respect to tracking r and compute the system Kv in terms of parameters K and . (c) Does the addition of tachometer feedback with positive kt increase or decrease Kv?

Consider the system shown in Fig. 4.26, where Figure 4.25 Control system for Problem 4.7 Figure 4.26 Control system for Problem 4.8 (a) Prove that if the system is stable, it is capable of tracking a sinusoidal reference input r = sin o t with zero steady-state error. (Look at the transfer function from R to E and consider the gain at o .) (b) Use Rouths criterion to find the range of K such that the closed-loop system remains stable if o = 1 and = 0.25.

Consider the system shown in Fig. 4.27, which represents control of the angle of a pendulum that has no damping. (a) What condition must D (s) satisfy so that the system can track a ramp reference input with constant steady-state error? (b) For a transfer function D(s) that stabilizes the system and satisfies the condition in part (a), find the class of disturbances w(t) that the system can reject with zero steady-state error. Figure 4.27 Control system for Problem 4.9

Consider the second-order system We would like to add a transfer function of the form in series with G(s) in a unity feedback structure. (a) Ignoring stability for the moment, what are the constraints on K, a, and b so that the system is Type 1? (b) What are the constraints placed on K, a, and b so that the system is both stable and Type 1? (c) What are the constraints on a and b so that the system is both Type 1 and remains stable for every positive value for K?

Consider the system shown in Fig. 4.28(a). (a) What is the system type? Compute the steady-state tracking error due to a ramp input r(t) = ro t1 (t). (b) For the modified system with a feed-forward path shown in Fig. 4.28(b), give the value of Hf so the system is Type 2 for reference inputs and compute the Ka in this case. (c) Is the resulting Type 2 property of this system robust with respect to changes in Hf? i.e., will the system remain Type 2 if Hf changes slightly? Figure 4.28 Control system for Problem 4.12

A controller for a satellite attitude control with transfer function G = 1/s 2 has been designed with a unity feedback structure and has the transfer function (a) Find the system type for reference tracking and the corresponding error constant for this system. (b) If a disturbance torque adds to the control so that the input to the process is u + w, what is the system type and corresponding error constant with respect to disturbance rejection?

A compensated motor position control system is shown in Fig. 4.29. Assume that the sensor dynamics are H(s)= 1. Figure 4.29 Control system for Problem 4.14 (a) Can the system track a step reference input r with zero steady-state error? If yes, give the value of the velocity constant. (b) Can the system reject a step disturbance w with zero steady-state error? If yes, give the value of the velocity constant. (c) Compute the sensitivity of the closed-loop transfer function to changes in the plant pole at 2. (d) In some instances there are dynamics in the sensor. Repeat parts (a) to (c) for and compare the corresponding velocity constants.

The general unity feedback system shown in Fig. 4.30 has disturbance inputs w1 , w2 , and w3 and is asymptotically stable. Also, (a) Show that the system is of Type 0, Type l1 , and Type (l1 + l2 ) with respect to disturbance inputs w1 , w2 , and w3 respectively. Figure 4.30 Single inputsingle output unity feedback system with disturbance inputs

One possible representation of an automobile speed-control system with integral control is shown in Fig. 4.31. Figure 4.31 System using integral control (a) With a zero reference velocity input (vc = 0), find the transfer function relating the output speed v to the wind disturbance w. (b) What is the steady-state response of v if w is a unit ramp function? (c) What type is this system in relation to reference inputs? What is the value of the corresponding error constant? (d) What is the type and corresponding error constant of this system in relation to tracking the disturbance w?

For the feedback system shown in Fig. 4.32, find the value of that will make the system Type 1 for K = 5. Give the corresponding velocity constant. Show that the system is not robust by using this value of and computing the tracking error e = r y to a step reference for K = 4 and K = 6. Figure 4.32 Control system for Problem 4.17

Suppose you are given the system depicted in Fig. 4.33(a), where the plant parameter a is subject to variations. Figure 4.33 Control system for Problem 4.18 (a) Find G(s) so that the system shown in Fig. 4.33(b) has the same transfer function from r to y as the system in Fig. 4.33(a). (b) Assume that a = 1 is the nominal value of the plant parameter. What is the system type and the error constant in this case? (c) Now assume that a = 1 + a, where a is some perturbation to the plant parameter. What is the system type and the error constant for the perturbed system?

Two feedback systems are shown in Fig. 4.34. (a) Determine values for K1 , K2 , and K3 so that (i) both systems exhibit zero steady-state error to step inputs (that is, both are Type 1), and (ii) their static velocity error constant Kv = 1 when K0 = 1. Figure 4.34 Two feedback systems for Problem 4.19 (b) Suppose K0 undergoes a small perturbation: K0 K0 + K0 . What effect does this have on the system type in each case? Which system has a type which is robust? Which system do you think would be preferred?

Consider the system shown in Fig. 4.36. (a) Find the transfer function from the reference input to the tracking error. (b) For this system to respond to inputs of the form r(t) = t n1(t) (where n < q) with zero steady-state error, what constraint is placed on the open-loop poles p1 , p2 , . . . , pq? Figure 4.36 Control system for Problem 4.21

A linear ODE model of the DC motor with negligible armature inductance (La = 0) and with a disturbance torque w was given earlier in the chapter; it is restated here, in slightly different form, as where m is measured in radians. Dividing through by the coefficient of m, we obtain where With rotating potentiometers, it is possible to measure the positioning error between and the reference angle r or e = ref m. With a tachometer we can measure the motor speed m. Consider using feedback of the error e and the motor speed m. in the form where K and TD are controller gains to be determined. (a) Draw a block diagram of the resulting feedback system showing both m and m as variables in the diagram representing the motor. (b) Suppose the numbers work out so that a1 = 65, b0 = 200, and c0 = 10. If there is no load torque (w = 0), what speed (in rpm) results from va = 100 V? (c) Using the parameter values given in part (b), let the control be D = kp + kDs and find kp and kD so that, using the results of Chapter 3, a step change in ref with zero load torque results in a transient that has an approximately 17% overshoot and that settles to within 5% of steady-state in less than 0.05 sec. (d) Derive an expression for the steady-state error to a reference angle input, and compute its value for your design in part (c) assuming ref = 1 rad. (e) Derive an expression for the steady-state error to a constant disturbance torque when ref = 0 and compute its value for your design in part (c) assuming w = 1.0.

We wish to design an automatic speed control for an automobile. Assume that (1) the car has a mass m of 1000 kg, (2) the accelerator is the control U and supplies a force on the automobile of 10 N per degree of accelerator motion, and (3) air drag provides a friction force proportional to velocity of 10 N sec/m. (a) Obtain the transfer function from control input U to the velocity of the automobile. (b) Assume the velocity changes are given by where V is given in meters per second, U is in degrees, and W is the percent grade of the road. Design a proportional control law U = kpV that will maintain a velocity error of less than 1 m/sec in the presence of a constant 2% grade. (c) Discuss what advantage (if any) integral control would have for this problem. (d) Assuming that pure integral control (that is, no proportional term) is advantageous, select the feedback gain so that the roots have critical damping ( = 1).

Consider the automobile speed control system depicted in Fig. 4.37. Figure 4.37 Automobile speed-control system (a) Find the transfer functions from W(s) and from R(s) to Y(s). (b) Assume that the desired speed is a constant reference r, so that R(s) = ro/s . Assume that the road is level, so w(t) = 0. Compute values of the gains K, Hr , and Hf to guarantee that Include both the open-loop (assuming Hy = 0) and feedback cases (Hy 0) in your discussion. (c) Repeat part (b) assuming that a constant grade disturbance W(s) = wo/s is present in addition to the reference input. In particular, find the variation in speed due to the grade change for both the feed-forward and feedback cases. Use your results to explain (1) why feedback control is necessary and (2) how the gain kp should be chosen to reduce steady-state error. (d) Assume that w(t) = 0 and that the gain A undergoes the perturbation A + A. Determine the error in speed due to the gain change for both the feed-forward and feedback cases. How should the gains be chosen in this case to reduce the effects of A?

Consider the multivariable system shown in Fig. 4.38. Assume that the system is stable. Find the transfer functions from each disturbance input to each output and determine the steady-state values of y1 and y2 for constant disturbances. We define a multivariable system to be type k with respect to polynomial inputs at wi if the steady-state value of every output is zero for any combination of inputs of degree less than k and at least one input is a non zero constant for an input of degree k. What is the system type with respect to disturbance rejection at w1? At w2? Figure 4.38 Multivariable system

The transfer functions of speed control for a magnetic tape-drive system are shown in Fig. 4.39. The speed sensor is fast enough that its dynamics can be neglected and the diagram shows the equivalent unity feedback system. (a) Assuming the reference is zero, what is the steady-state error due to a step disturbance torque of 1 N m? What must the amplifier gain K be in order to make the steady-state error ess 0.01 rad/sec? (b) Plot the roots of the closed-loop system in the complex plane, and accurately sketch the time response of the output for a step reference input using the gain K computed in part (a). (c) Plot the region in the complex plane of acceptable closed-loop poles corresponding to the specifications of a 1% settling time of ts 0.1 sec and an overshoot Mp 5%. (d) Give values for kp and kD for a PD controller, which will meet the specifications. (e) How would the disturbance-induced steady-state error change with the new control scheme in part (d)? How could the steady-state error to a disturbance torque be eliminated entirely? Figure 4.39 Speed-control system for a magnetic tape-drive

Consider the system shown in Fig. 4.40 with PI control. (a) Determine the transfer function from R to Y. (b) Determine the transfer function from W to Y. (c) What is the system type and error constant with respect to reference tracking? (d) What is the system type and error constant with respect to disturbance rejection? Figure 4.40 Control system for Problem 4.27

Consider the second-order plant with transfer function and in a unity feedback structure. (a) Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P [D = kp], PD [D = kp + kDs], and PID [D = kp + kI/s + kDs] controllers. Let kp = 19, kI = 0.5, and kD =4/ 19. (b) Determine the system type and error constant of the system with respect to disturbance inputs for each of the three regulators in part (a) with respect to rejecting polynomial disturbances w (t) at the input to the plant. (c) Is this system better at tracking references or rejecting disturbances? Explain your response briefly. (d) Verify your results for parts (a) and (b) using MATLAB by plotting unit step and ramp responses for both tracking and disturbance rejection.

The DC-motor speed control shown in Fig. 4.41 is described by the differential equation + 60y = 600va 1500w, where y is the motor speed, va is the armature voltage, and w is the load torque. Assume the armature voltage is computed using the PI control law where e = r y. (a) Compute the transfer function from W to Y as a function of kp and kI . (b) Compute values for kp and kI so that the characteristic equation of the closed-loop system will have roots at 60 60j. Figure 4.41 DC Motor speed-control block diagram for Problems 4.29 and 4.30

Consider the satellite-attitude control problem shown in Fig. 4.42 where the normalized parameters are J = 10 spacecraft inertia, Nmsec 2/rad r = reference satellite attitude, rad. Figure 4.42 Satellite attitude control = actual satellite attitude, rad. Hy = 1 sensor scale, factor V/rad. Hr = 1 reference sensor scale factor, V/rad. w = disturbance torque. Nm (a) Use proportional control, P, with D(s) = kp , and give the range of values for kp for which the system will be stable. (b) Use PD control and let D(s) = (kp +kDs) and determine the system type and error constant with respect to reference inputs. (c) Use PD control, let D(s) = (kp + kDs) and determine the system type and error constant with respect to disturbance inputs. (d) Use PI control, let D(s) = (kp + kI/s), and determine the system type and error constant with respect to reference inputs. (e) Use PI control, let D(s) = (kp + kI/s), and determine the system type and error constant with respect to disturbance inputs. (f) Use PID control, let D(s) = D(s) = (kp + kI/s + kDs) and determine the system type and error constant with respect to reference inputs. (g) Use PID control, let D(s) = D(s) = D(s) = (kp + kI/s + kDs) and determine the system type and error constant with respect to disturbance inputs.

The unit-step response of a paper machine is shown in Fig. 4.43(a) where the input into the system is stock flow onto the wire and the output is basis weight (thickness). The time delay and slope of the transient response may be determined from the figure. (a) Find the proportional, PI, and PID-controller parameters using the ZeiglerNichols transient-response method. (b) Using proportional feedback control, control designers have obtained a closed-loop system with the unit impulse response shown in Fig. 4.43(b). When the gain Ku = 8.556, the system is on the verge of instability. Determine the proportional-, PI-, and PID-controller parameters according to the ZeiglerNichols ultimate sensitivity method

A paper machine has the transfer function where the input is stock flow onto the wire and the output is basis weight or thickness. (a) Find the PID-controller parameters using the ZeiglerNichols tuning rules. (b) The system becomes marginally stable for a proportional gain of Ku = 3.044 as shown by the unit impulse response in Fig. 4.44. Find the optimal PID-controller parameters according to the ZeiglerNichols tuning rules. Figure 4.43 Paper-machine response data for Problem 4.32 Figure 4.44 Unit impulse response for the paper machine in Problem 4.33

Compute the discrete equivalents for the following possible controllers using the trapezoid rule of Eq. (4.104). Let Ts = 0.05 in each case. (a) D1 (s) = (s + 2)/2

Give the difference equations corresponding to the discrete controllers found in Problem 4.34 respectively. (a) part 1 (b) part 2 (c) part 3 (d) part 4