214 spring 2014.pdf

2.14/2.140 Problem Set 1

Assigned: Wed. Feb. 5, 2014 Due: Wed. Feb. 12, 2014, in class Reading: FPE Chapters 1 and 2, review as needed

This problem set is a review of material on modeling and analysis of electrical, mechanical, electromechanical, uid, and thermal dynamic systems. The problems from Franklin, Powell, and Emami-Naeini, Feedback Control of Dynamic Systems, 6th Ed. are referenced by FPE Chapter.Problem#. The Problem Archive can be downloaded as a complete pdf le from the course web page; see the problem numbers therein (pdf bookmarks will help you nd them). Note that the problem archive also contains solutions to many of the problems. Please do not consult these solutions initially. Refer to the solutions if you are stuck, or to check your analysis.

The following problems are assigned to both 2.14 and 2.140 students.

Problem 1 FPE 2.5

Problem 2 FPE 2.10

Problem 3 FPE 2.15, a only

Problem 4 FPE 2.25

Problem 5 Archive Problem 4.24




The following problems are assigned to only 2.140 students. Students in 2.14 are welcome to work these, but no extra credit will be given.

Problem G1 FPE 2.9 Write the equations of motion in state-space form in terms of state variables x, x, y, and y, and with input u, and output y. Then eliminate variables in the state equations to nd the transfer function from input u to output y. What are the locations of the poles and zeros of this transfer function?

Problem G2 FPE 2.12

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MIT OpenCourseWarehttp://ocw.mit.edu

2.14 / 2.140 Analysis and Design of Feedback Control SystemsSpring 2014

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Assigned: Thurs. Feb. 13, 2014Due: Wed. Feb. 19, 2014, in classReading: FPE sections 3.23.5




Problem 3 FPE 3.20

Problem 4 FPE 3.28

Problem 5 Consider a motor with the following parameters: torque constant K = 1 Nm/A, rotorinertia J = 102 kg m2, and coil resistance R = 5 . A torque disturbance Td acts on therotor of the motor in opposition to the motor torque Tm = Ki, where i is the motor coilcurrent. The voltage at the motor terminals is Vm = iR+K, where is the motor angularvelocity.

a) We consider the motor as a plant with input Vm and output . Draw a block diagram forthe motor system which shows these signals as well as i, Tm, Td, and the back emf voltagevb = K.

b) Now we ask you to design a proportional speed controller of the form

Vm = Kp(r ).

Here Kp is the proportional gain of the controller, and r is the speed command. Choose Kpsuch that the closed-loop system has a time constant of = 1 msec. For this value of Kpmake plots of Vm and when the reference is a unit step r(t) = 1us(t) rad/sec.

c) Now we ask you to design a proportional plus integral speed controller of the form

Vm(s) = Kp(1 +1

Tis)(r(s) (s)).

Here Kp is the proportional gain of the controller, and Ti is referred to as the integraltime. Choose Kp and Ti such that the closed-loop system has a natural frequency of n =1000 rad/sec and a damping ratio of = 0.4. For these values make plots of Vm and whenthe reference is a unit step r(t) = 1us(t) rad/sec.

Problem 6 An ideal linear voice coil actuator is configured as shown below with the actuation axisoriented vertically. This ideal actuator has no internal mass or electrical resistance. Gravityacts downward as shown in the figure. For the purposes of this problem, assume that motionin the x direction is unlimited. The actuator has F = Kf i and e = Kf x.

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The actuator mechanical port is connected to a mass m. The actuator electrical port isconnected to a resistance R in series with a switch. Consider the two cases: a) switch openand b) switch closed. At t = 0, the mass is released with x(0) = 0, x(0) = 0.

Write the equations of motion for the two cases. Take care with minus signs in your analysis:note that with the switch closed e = iR. To facilitate this analysis, draw a block diagramfor this system which includes at least the signals e, i, F , x, x, and which has the accelerationof gravity g as an input. Use this block diagram to determine the transfer function from inputg to output x.

Solve for the time trajectory x(t) for the two cases, and plot these as a function of time onthe same axes. How are the initial (t = 0+) accelerations (slope) related? Why?

Problem 7 An ideal linear voice coil actuator is configured as shown below with the actuation axisoriented vertically. This ideal actuator has no internal mass or electrical resistance. Gravityacts downward as shown in the figure. For the purposes of this problem, assume that motionin the x direction is unlimited. The actuator has F = Kf i and e = Kf x. The actuatormechanical port is connected to a mass m. We consider two cases in which the electrical portis connected to a capacitor or an inductor.

a) The actuator electrical port is connected to a capacitance C as shown below.

Write the equations of motion. Take care with minus signs in your analysis: note that

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i = Cde/dt. To facilitate this analysis, draw a block diagram for this system which includesat least the signals e, i, F , x, x, and which has the acceleration of gravity g as an input. Usethis block diagram to determine the transfer function from input g to output x.

At t = 0, the mass is released with x(0) = 0, x(0) = 0. Solve for the time trajectory x(t) andplot as a function of time. Write an expression for the acceleration; how does changing thevalue of C affect the acceleration? From the viewpoint of the mass, what mechanical elementis it connected to?

b) The actuator electrical port is connected to an inductance L as shown below.

Write the equations of motion. Take care with minus signs in your analysis: note thate = Ldi/dt. To facilitate this analysis, draw a block diagram for this system which includesat least the signals e, i, F , x, x, and which has the acceleration of gravity g as an input. Usethis block diagram to determine the transfer function from input g to output x.

At t = 0, the mass is released with x(0) = 0, x(0) = 0. Solve for the time trajectory x(t) andplot as a function of time, with the axes clearly labeled and dimensioned. How does changingthe value of L affect the trajectory? From the viewpoint of the mass, what mechanical elementis it connected to?

Problem 8 This problem considers an electrical/thermal model of a transistor circuit using anNPN bipolar junction transistor. This circuit is used in Labs 2 and 3 to give you experiencewith a temperature control system. Here we look at the details of the electrical and thermalmodels for the circuit. As shown below, an NPN bipolar transistor is a three-terminal device(base, emitter, collector), with correspondingly defined voltages and currents.

What makes a transistor useful is that it amplifies. For this NPN device, a basic model of

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this amplification is the relationship iC = iB, where in typical devices 100. Thus asmall input current at the base terminal can be amplified into a much larger current at thecollector terminal. Since the sum of currents entering the device must equal zero, we alsohave iE = iC + iB. Further, when iB > 0, we can assume vBE 0.6 V. (This problem onlymodels the transistor operation in the forward-active region, and omits internal dynamics.There is significantly more to learn about these devices for general use.)

The particular circuit of interest

drives the transistor with an input Vin through a base resistor Rb. Note also in this circuitthat vCE = Vs > 0.

An equivalent circuit for the transistor is

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We will assume that when iB is some positive value, vBE = 0.6 V, and thus require Vin 0.6.Look at the circuit equations to follow the derivation which shows that the power dissipatedin the device is Pdiss Vs(Vin vBE)/Rb. (The approximation comes in because we ignorethe very small dissipation iBvBE . Why is this reasonable?) Thus we see that the input Vincan directly control the device dissipation, via the base current, at relatively low input power.That is the key idea of amplification: large power or signals somewhere can be controlledby small power input signals. Note that we inherently also assume Vin > vBE , since powerdissipation must always be positive. (If you want negative and positive heat flows from onedevice, get a thermoelectric cooler, and go learn more about thermodynamics!)

Now for the thermal side of the model. Assume that the transistor is a lump of materialat uniform temperature. Further assume a linear power flow to ambient proportional to thedifference between transistor Tj and ambient Ta temperatures. The thermal capacitance ofthe lump Ct has units of [J/

K], and the thermal resistance Rt has units of [K/W ]. Thepower dissipation Pdiss injects heat into the transistor lump, which connects to ambient viathe thermal resistance.

The thermal capacitance is described by

CtdTjdt

=

P = Pdiss (Tj Ta)Rt

.

This relationship can be shown in the equivalent thermal circuit, where we equate electricalcurrent with power flow, and electrical voltage with temperature.

a) Draw a block diagram including at least input Vin, input Ta, Pdiss, Tj .

b) Now (finally!) we consider closed-loop control issues. For this purpose, Tj is the output to becontrolled and Vin is the input for control. In this context, Ta is a disturbance signal.

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This thermal plant will be controlled by an on/off controller with hysteresis, as shown

Temperature is measured with a sensor which converts K to volts, via Vs = 0.1(Tj273) [V].That is, the sensor measures in degrees Celsius, with a scale factor of 1 Volt per 10 degreesCelsius.

The hysteresis block has the form

where the switching points are at . For the remainder of this problem, assume the fol-lowing values of the parameters: Vref = 5 V, = 0.1 V, Rb = 20 k, = 100, Vs = 5 V,Ct = 1 J/

K, and Rt = 400 K/W . Use these numbers in your dynamic model developedabove.

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i) Assume Ta = 25C, and that the system has been off for a long time, such that Tj(0+) = Ta. At

t = 0 the control is engaged and begins operating. Solve for and plot Tj(t) and Vin(t). Whatis the steady-state switching frequency and duty cycle (ratio of on time to cycle period)?

i) Now assume Ta = 0C, and that the system has been off for a long time, such that Tj(0+) = Ta.

At t = 0 the control is engaged and begins operating. Solve for and plot Tj(t) and Vin(t).What is the steady-state switching frequency and duty cycle? Compare with the earlier case.

Does this lower ambient temperature cause any error in the average value of Tj? You cananswer this question qualitatively on the basis of your time plots. You do not need to formallycompute this average.

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Assigned: Wed. Feb. 19, 2014Due: Wed. Feb. 26, 2014, in classReading: FPE Chapter 5; Lecture notes on PID control and root locus.

For the root locus problems from FPE, attempt to solve them by hand with the root locus sketchingrules. Only use Matlab where requested, or if you cant see an analytical solution. Its veryimportant that you learn how to sketch moderate order root loci with pencil and paper.


Problem 1 FPE 5.2

Problem 2 FPE 5.3

Problem 3 FPE 5.13

Problem 4 FPE 5.14

Problem 5 FPE 5.22. Use Matlab computational tools to help with all steps in this problem.

Problem 6 The four plots below show the pole and zero locations of the return ratio of a feedbacksystem, informally called the open-loop poles and zeros. Each of these loops also have avariable gain K > 0, which is used to move the poles along the root locus branches. For thefour systems shown below, sketch the approximate shape of the root locus plot for K > 0.Note that you will need to pay particular attention to the angle criteria in the vicinity of thecomplex poles and zeros. If the complex pairs are lightly damped, which of these systemspresents a danger of instability as the loop gain is varied? This analysis has practical relevancefor the situation where a notch filter is used to help stabilize a system with a lightly-dampedpair of poles.

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Problem 7 The block diagram for a feedback loop has a forward path transfer function G(s) =Ka(s)/b(s), and a feedback path transfer function H(s) = c(s)/d(s) as shown below. Provethat the closed-loop zeros are located at: 1) the zeros of the forward path and 2) the poles ofthe feedback path, independent of the loop gain K.

Problem 8 FPE 4.23

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The following problems are assigned to only 2.140 students. Students in 2.14 arewelcome to work these, but no extra credit will be given.

Problem G1 FPE 5.35

Problem G2

This problem was used as the ME quals systems written exam in 2005. It considers modelingand control issues associated with positioning systems driven with a piezoelectric actuator. Apiezoelectric positioner is driven with an electrical input in order to produce a mechanical outputand vice versa.

The figure below shows a model of a system incorporating a piezoelectric device.

Here the electrical terminals of the device are defined as having an input voltage e(t) and an inputcurrent i(t). The piezo crystal is sandwiched between end plates, with the bottom plate connectedto mechanical ground. The motion of the upper plate in the vertical direction is defined as x(t).A mechanical load consisting of a damper with value b is connected between the upper plate andmechanical ground. We model the piezoelectric device and plates as massless. Motion is consideredto be constrained to the x direction.

The electrical/mechanical coupling of the device is modeled as shown below

Here, the piezoelectric actuator is modeled internally as a dependent force source F (t) in parallelwith a spring k. The force depends linearly upon the input voltage as F (t) = Ke(t), where Kis a scale factor with units of N/V. The spring k models the internal stiffness of the piezoelectricactuator. The force F is applied to the massless upper plate which connects the damper and spring.

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The electrical portion of the model is shown on the left in the figure. Here, a dependent currentsource has a value i(t) = Kx(t). The system is driven with a voltage source Vin(t).

a) Calculate the transfer function X(s)/Vin(s). Clearly show the steps in your development.

b) Assume initial rest conditions. Let the input voltage be a unit step: Vin(t) = u(t). Calculate aclosed-form solution for the resulting displacement x(t) and make a graph of x(t) versus time.

c) Develop a closed-form expression for the input electrical power P (t) = e(t)i(t) associated withthe transient you solved for in part b) above, and make a graph of P (t) versus time.

We now learn that the top plate has finite mass, and so the model developed earlier needs to beaugmented. Measurements indicate that the input/output transfer function is now given by

Gp(s) X(s)Vin(s)

=1

102s2 + 10s + 106.

This experimentally-adjusted model is to be used to design the feedback loop shown below.

The controller for this loop is an integral controller Gc(s) G/s, where G is an adjustable gainassociated with the integrator.

d) Sketch a root locus plot for this control loop as the gain G > 0 is varied. For what range ofgains G is the loop stable?

e) The loop transfer function (sometimes called the return ratio) for this loop is given by L(s) =Gc(s)Gp(s). Make a careful hand sketch of the Bode plot for L(s), showing the effect of G asa parameter, and using the numerical representation of Gp(s) given above.

f) What value of G will result in a loop crossover frequency c = 100 rad/sec? (Recall that thecrossover frequency is the frequency for which the loop transfer function magnitude crossesthrough unity, that is, we require |L(j100)| = 1.)What are the phase margin and gain margin for the loop with this crossover frequency?Indicate these parameters on a Bode plot for the loop.

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Assigned: Wed. Feb. 26, 2014Due: Wed. March 5, 2014, in classReading: Lecture notes on loop shaping, frequency response, and Bode plots.


Problem 1 Problem Archive 11.5 Pole Zero Plots. Sketch the indicated Bode plots by hand usingthe sketching rules presented in the lecture and notes. Do not use Matlab to generate theseplots.

Problem 2 The Bode plot given below is for a notch filter, which can be used as part of compen-sation for attenuating resonances. This is called a notch filter because it cuts out a range offrequencies. On the given Bode plot, mark the straight asymptotes, and the phase staircasefunction. Use this analysis to determine the filter transfer function and draw a pole-zero mapfor the filter.

Problem 3 Archive Problem 13.1 Rooftop Antenna

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Problem 4 This problem considers the paint stirrer shown below. A brushed DC motor is usedto stir a bucket of paint as shown in the figure. The motor has a torque constant K [Nm/A]and the motor plus stirrer inertia is J [kg-m2]. The motor is considered ideal in that it has noinductance and no resistance; the only resistance in the electrical circuit is R as shown. Therotor is rigidly linked to the stirrer, which has no flexibility; the combined assembly has arotational velocity rad/sec. The effect of the paint on the stirrer is modeled as a rotationaldamper B [Nms/rad].

a) Solve for the transfer function Gp(s) = (s)/Vi(s) in terms of the system parameters. Now setthe parameter values as K = 0.5 Nm/A, R = 5 , B = 0.15 Nms/rad, and J = 0.2 kgm2.For these parameter values, make a hand sketch of the Bode plot for Gp.

b) Now, let the system input Vi be driven by a PI controller of the form

Gc(s) = Kp

(1 +

Kis

).

This system has a speed reference input r and output as shown in the block diagram below

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Choose the controller gain values Kp and Ki to set the loop crossover frequency c =10 rad/sec, with a phase margin m = 45

. Show your calculations. You should be ableto accomplish this design using hand-sketched Bode plots, and then using Matlab for confir-mation.

c) Create a Simulink simulation of the loop with the controller implemented as shown in the blockdiagram above. For now, let the integrator have an unbounded output (leave the integratorlimit output box unchecked). Set your simulation to use the variable step solver with a maxstep size limit of 0.01 sec, and use the ode45 solver. Note that the standard simulation windowhas a simulation stop time of 10 sec. You may want to adjust this value.

Let the input reference r take a step from zero to 10 rad/sec at t = 0, from initial restconditions. Run this as a Simulink simulation and record and plot the responses Vi(t) and(t). What is the maxiumum value of the Vi during this transient? What is the steady-statevalue of Vi after the transient has settled?

d) Now set the integrator limits to 1.5 times the steady-state value of Vi from part c), and checkthe integrator limit box. (This insures that the integrator term is able to supply the requiredsteady-state control effort for a speed of 10 rad/sec, but not much more.) Again let the inputreference r take a step from zero to 10 rad/sec at t = 0, from initial rest conditions. Runthis as a Simulink simulation and record and plot the responses Vi(t) and (t). How do thesediffer from the unlimited case in part c)?

e) Be sure to keep a copy of your calculations and this model, as we may continue studying it inthe next problem set.

Problem 5 Archive Problem 17.21 Pole Zero Bode Matching

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Problem G1 This problem was used as the ME quals systems oral exam in 2003. In the invertedpendulum system shown below, note that gravity acts in the downward direction as shown.A rigid massless link constrains the mass M to rotate about the pivot shown. The distancefrom the pivot point to the center of gravity of the mass is L m. The mass has negligiblemoment of inertia. The horizontal position of the mass in meters is x(t). The system is drivenby a position input w(t) m. The input is connected to the mass via the spring k with unitsof N/m, and via the damper b shown with units of Ns/m. In the problem, we only considermodels which describe motions in which x is very small; do not attempt to write models forlarge motions.

Mg

Mass position x(t)Position input w(t)

b

k

L

pivot

a) For this system, derive a mathematical model which describes the system for small motions.The resulting linear model should have an input w(t) and output x(t). Be sure to take into ac-count the effect of gravity. Explain your reasoning. What is the transfer function X(s)/W (s)?

b) For what range of k and b is the system stable?

c) Now consider the controlled pendulum system shown below.

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Mg

Mass position x(t)

Force output F(t)

L

pivot

Reference input w(t)

Controller

The controller has inputs of the mass position x(t) and reference command w(t). The con-troller outputs a force F (t) which is applied to the mass as shown. Develop a controller whichprovides an indentical small-motion transfer function X(s)/W (s) to that derived in part a)above. You can feel free to use any linear operation within the controller block. Explain yourreasoning for the controller you develop.

d) Now assume that a time delay of T sec is introduced into the position measurement before it isreceived by the controller. How does this affect the stability of the controlled system? Why?Show your analytical reasoning. Hint: An approach looking at phase margin is the best wayto understand the effect of the time delay.

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Assigned: Wed. Mar. 5, 2014Due: Wed. March 12, 2014, in classReading: FPE sections 6.16.8; lecture notes on loop-shaping and the Nyquist test

The following problems are assigned to both 2.14 and 2.140 students. Except where noted,you are free to use Matlab or any other computational tools to aid your solutions. There areno additional problems for 2.140 students. Note that Quiz 1 will be held in class on Wed.3/12, and is closed-book, but you may bring one page of notes, which can be two-sided. Onlyhand calculators may be used. Recognize that in Quiz 1, all requested plots will need to behand-sketches, so you should work these problems accordingly.

Problem 1 A unity-gain feedback loop is shown below; note that the plant has a pair of poleslocated on the imaginary axis.

a) Sketch a root locus plot for K > 0.

b) Use a Bode plot of the return ratio L(s) to choose the controller gain K such that the loopcrossover frequency is 100 rad/sec. What gain K is required? What is the resulting phasemargin m?

c) Make a Nyquist plot for this loop, which is accurately drawn in the region where m is defined.Be sure to label the corresponding points on the diagram. Note that your Nyquist analysiswill require detours around the poles at j10. Please clearly show the portion of the Nyquistcontour corresponding to these detours.

d) Make a sketch of the closed-loop step response for your design.

Problem 2 A unity-gain feedback loop is shown below. The controller in this loop is a propor-tional gain K, but it is implemented on a computer which requires a delay time of T secondsto produce the control output.

In all of this problem, assume that the time delay has a value of T = 10 msec.

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a) Use a Bode plot of the return ratio L(s) to determine the maximum controller gain Kmax suchthat for 0 < K < Kmax the loop is stable. For this maximal value of K = Kmax sketch aNyquist plot for the loop.

b) What value of K < Kmax will result in a stable loop with a phase margin of 20? What is the

resulting crossover frequency? What is the gain margin for this loop? For this value of Ksketch a Nyquist plot for the loop which shows this information.

Problem 3 A feedback loop for a linearized magnetic suspension is shown below with a controllerGc(s). The position of the suspended object is Y , the control reference value is R, and thereis a disturbance D acting on the system.

In this problem, assume the controller takes the form

Gc(s) = K10s+ 1

s+ 1.

a) Choose the values of K and such that the loop has a crossover frequency of 1000 rad/sec anda phase margin which is maximized for this controller. Sketch a Nyquist plot which confirmsthat the loop is stable.

b) For the value of K determined above, use Matlab to plot the response Y (t) to a unit step inreference R(t). Also plot the response Y (t) to a unit step in disturbance D(t).

c) This magnetic suspension is open-loop unstable, and thus there is a minimum controller gainK > Kmin required to stabilize the loop. Using the value of determined in part a), whatis the numerical value of Kmin at this marginally-stable condition? By what ratio can Kbe reduced with respect to the nominal design of part a) to bring the loop to the brink ofinstability? Note that this ratio can be defined as the gain-reduction margin, as distinguishedover the standard gain margin which is defined as an allowable gain-increase margin.

Problem 4 Archive Problem 11.1 Circuit Bode Plots. It is important to review how to solvecircuits for a particular transfer function. To solve for the requested transfer functions, useseries/parallel and voltage/current divider relationships with impedances representing eachcircuit element. Sketch the indicated Bode plots by hand using the sketching rules presentedin the lecture and notes. Do not use Matlab to generate these plots.


Problem 6 FPE Problem 5.5 c,d

Problem 7 FPE Problem 5.7 c,d

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Assigned: Wed. March 12, 2014Due: Wed. March 19, 2014, in classReading: Lecture notes on Operational Amplifiers

The following problems are assigned to both 2.14 and 2.140 students. The first two prob-lems review some basic circuit calculations. The remainder concern operational amplifiercircuits, and studying the stability of these connections.

Problem 1 For the circuits shown below, calculate the equivalent resistance seen at the indicatedterminals, using the series and parallel resistance formulae.

Problem 2 The circuit shown below is driven by a current source It at the terminals B-B. Theresulting voltage across these terminals is defined as Vt.

a) What is the equivalent resistance Req seen at the terminals B-B? Viewed another way, what isthe ratio Vt(t)/It(t) = Req? The easiest way to solve this problem is to use series and parallelequivalents.

b) Now we modify the circuit by removing the resistor R at terminals A-A, and replacing thiswith the capacitor C shown in the figure below.

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What is the equivalent impedance Zeq seen at the terminals B-B? Viewed another way, whatis the ratio Vt(s)/It(s) = Zeq(s)? Note that here weve switched to the Laplace domainand work in the more general concept of impedances, since the capacitor adds dynamics tothe system. The easiest way to solve this problem is to use series and parallel equivalentimpedances.

Problem 3 Archive Problem 9.2For this circuit, make a sketch of the Bode plot for the negative of the loop transmission L(s).What are the loop crossover frequency and phase margin? Also, create a Nyquist plot for theloop, and indicate the loop phase margin and gain margin on your plot. For what value of Cwill the loop be marginally stable? (Note that there is a small typo: the variables C and C

refer to the same capacitor.)

Problem 4 Archive Problem 9.5This circuit can be used as an analog PI controller, for example in a closed-loop current-controlled power amplifier (see Problem G1 below). Calculate the transfer function Vo(s)/Vi(s),under the assumption that the op amp has infinite gain. Use this transfer function to answerthe questions posed in this problem.

Problem 5 This problem considers the op amp circuit shown below

Here the resistors take values of R, R, and R, respectively. The resistors R set the idealgain of the circuit, which is 1. The resistor R can be used to detune the bandwidth ofthe opamp loop when the amplifier dynamics are too hot for the circuit configuration. Inparticular, like all real systems, op amps have additional high frequency dynamics which can

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give bad stability in some circuits. To understand this, assume

a(s) =5 107

s(107s+ 1)2.

The additional pair of poles represent high-frequency dynamics in the internal opamp circuit.

a) First, we remove the resistor R by letting (infinite resistance is an open circuit).For this configuration, calculate the loop crossover frequency and phase margin. Use Matlabto plot the unit step response of the circuit. You should find that this system is unstable.Compare the ring frequency of the unstable step response signal with the natural frequencyof the closed loop dominant pole pair and with the loop crossover frequency. You should seethat n c. (When making ring frequency measurements on the step response, be sure toconvert from Hz to rad/sec.)

b) Conduct a Nyquist analysis for the loop in part a), which should indicate the loop instability.Be sure to clearly show the number of encirclements, and resulting number of right half planeclosed-loop poles. Indicate the loop phase margin (negative) and gain margin (less than one)on the Nyquist plot.

c) Draw a root locus plot for the loop of part a). Indicate the closed-loop root locations for thenominal design with .

d) Now pick a value of which results in a loop phase margin m = 45. For this value of , what

is the loop crossover frequency c? Use Matlab to plot the unit step response of this detunedcircuit. Compare the natural frequency of the closed loop dominant pole pair with the loopcrossover frequency.

e) Conduct a Nyquist analysis for the loop in part d), which should indicate the loop stability.Indicate the loop phase margin and gain margin on the Nyquist plot.

f) Draw a root locus plot indicating where the closed-loop poles are located for the value of frompart d).

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Problem G1 In the lab you will see power amplifiers used to control the voltage on a load such asa voice coil or motor. In practice, it is also common to use power amplifiers in a feedback loopto control the current through a load. This problem investigates the design of such current-controlled amplifier configurations. In the circuit shown below, the load is represented bya coil with inductance Lc and resistance Rc. The load current ic is supplied by the poweramplifier A2. The current is measured by passing it through the sense resistor Rs; thevoltage Vf is then proportional to the load current. Thus if Vf is kept at a desired level, theload current can be regulated. The buffer amplifier A3 is used to avoid loading the main coilcircuit; that is, current i2 is supplied by amplifier A3. The control amplifier A1 is used tocompare the reference voltage Vr with the feedback voltage Vb, and then to add dynamics tothe loop via R3 and C. That is, amplifier A1 is the compensation for the current feedbackloop. Note that all the op amps are modeled as having infinite gain. All voltages labeled onthe circuit are with respect to circuit ground.

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a) The current feedback loop may be represented in the form of the block diagram shown below.

Develop expressions for each of the transfer functions in the blocks of this block diagram, andfill them in on your own version of the block diagram. Carefully note the system variablesalready shown on the block diagram, and fill in the transfer functions appropriately.

b) Suppose the input Vr has the constant value Vr = V0. What is the steady-state load current iCfor this constant reference input? Explain your results.

c) Now let the parameters take the values R1 = R2 = R4 = 10 k, R5 = 50 k, Rs = 1 ,Lc = 100 mH, and Rc = 10 . Chose the values of R3 and C such that the closed-loopsystem poles have a natural frequency of n = 1000 rad/sec and a damping ratio of = 0.4.

d) Make a sketch of the Bode plot for the negative of the loop transmission L(s). What are theloop crossover frequency and phase margin? How do these relate to the closed-loop naturalfrequency and damping ratio from part c)?

e) For these parameter values, let the input be a negative unit step Vr = us(t). Make a graph ofthe response in current iC(t).

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Assigned: Wed. Mar. 19, 2014Due: Wed. April 2, 2014, in classReading: Lecture notes on Operational Amplifiers; section from Chapter 2 of Operational Amplifiers

which describes static nonlinearities in feedback loops.

The full text Operational Amplifiers can be found online at:http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/.Video lectures by Prof. Roberge can be found at this same location. In particular, for this problemset, video lecture 2 may be helpful to understand the effect of feedback on nonlinearities. Thereis also a video course manual with blackboards for all the lectures along with solutions to selectedproblems from the text.


Problem 1 The circuit shown below implements a first-order all-pass filter. Note that we assumean infinite gain model for the op-amp.

The term all-pass is used because the circuit response magnitude is unity for all frequencies.

a) Show that the circuit transfer function is

VoVi

(s) =RCs 1RCs+ 1

.

Note the non-minimum phase (RHP) zero.

b) Make a hand sketch of the Bode plot for this filter. You should see that the phase varies from+180 at DC to 0 at high frequencies.

c) This circuit can be used as an adjustable phase shifter where R is a variable resistor used to setthe phase. Let C = 0.1F. What value of R will set the phase equal to +90 at a frequencyof 1000 rad/sec?

1

Problem 2 This problem considers a feedback loop with a static nonlinearity as shown below

The nonlinear element has an input/output characteristic given by

The example from Chapter 2 of the Roberge textbook will be helpful in understanding thisproblem.

a) Assume that the input Vi takes the form of a unit ramp Vi(t) = t. Calculate and sketch theresulting waveforms vA(t) and vB(t). These will be piecewise linear waveforms. Be sure toclearly label the relevant breakpoints. Also label the slope of each line segment.

2

Problem 3 In the loop shown below, the differential amplifier drives an RC low-pass filter. Theoutput of the filter is fed back through a unity-gain buffer. Note that we assume an infinitegain model for both of the op amps in this loop.

a) Draw a block diagram which includes Vi, va, vo, and Vf , and where the blocks have expressions interms of the system parameters. Solve for the transfer functions vo(s)/Vi(s) and va(s)/Vi(s).

b) Let R1 = 1k, R2 = 100k, R = 10k, and C = 1 F. Also let the input be a unit stepVi = us(t). Calculate analytically and plot the closed-loop step responses va(t) and vo(t).

3

Problem 4 The system shown below is a micropositioner, in which a stage guided by flexural legsmoves in the x direction as driven by the force from a voice coil actuator.

The combined moving mass of the stage and voice coil is m. The flexural legs have a springstiffness k. The actuator is assumed ideal, with a force constant K, and thus the actuatorforce is F = Ki, and e = Kx. The actuator is driven by a power amplifier with a gain of 3,through a series resistor R.

a) Draw a block diagram which includes the signals Vi, va, e, i, x, and x.

b) Use this block diagram to solve for the transfer function X(s)/Vi(s) in terms of the systemvariables.

c) Assume the system parameters take the following values: K = 10 N/A, R = 10 , k = 107 N/m,and m = 0.1 kg. Calculate the numerical values of the transfer function X(s)/Vi(s).

d) Let the input Vi take a 1 Volt step, i.e. Vi(t) = 1us(t) V. Use Matlab to calculate and plot theoutput step response. What are the system natural frequency and damping ratio?

e) Calculate and plot the system Bode plot for X(s)/Vi(s).

f) We now design a closed-loop controller for this system of the form

4

Here, position is measured with a capacitive position sensor, with a gain of 3 V/m. Choosethe value of Go which allows a gain margin of 2. What is the corresponding crossover fre-quency and phase margin? Construct a Nyquist plot which shows these values. For thiscontroller, also use Matlab to make a plot of the sensitivity S = (1 L.T.)1.

g) Because the resonance is so lightly damped, the system closed-loop bandwidth determined aboveis quite limited. However, it is possible to phase-stabilize this loop by adding an additionalhigh-frequency pole. That is, we do not require that the loop transmission remain below unitymagnitude in the vicinity of the resonance if the loop phase is such that the correspondingloop in the Nyquist plot does not encircle the 1 point. As a way to understand this, assumewe add a compensator pole to the loop. That is, let the compensator become Go/s(s+ 1).For the present analysis, let = 5/n, where n is the plant natural frequency. What is thelargest value of G0 for which the sensitivity curve does not exceed a peak value of 10 dB?(You can use Matlab to help find this value.) Make a Bode plot of the negative of the looptransmission L(j), the sensitivity S(j), and also make a Nyquist plot showing how the1 point is avoided. How many crossings of unity gain are there in the loop transmission?Compare the accuracy of command following for this phase-stabilized loop with the pureintegral compensated loop from part f). (Hint: the sensitivity curve tells this story.)


Problem G1 The OP77 data sheet is available separately from the course web page. In particular,the device transfer function a(s) is shown in Bode plot form in Figure 10 on page 8 of thedata sheet. For this problem, assume we connect the OP77 as a unity gain buffer.

a) Consider a model of the form

a(s) =G0

(s+ 1)esT

where the esT term is a time delay used to model the excess high-frequency phase. Whatvalues of G0, , and T allow the best fit to the data sheet Bode plot?

b) For the unity-gain buffer configuration, using this model, what are the loop crossover and phasemargin?

c) Use Matlab to simulate the closed-loop step response for this configuration. How is this timeresponse consistent with the loop crossover and phase margin?

Problem G2 This problem considers the AC-coupled feedback loop shown below.

The loop is referred to as AC-coupled since it has zero gain at DC, due to the zeros at s = 0.

5

a) Sketch a root locus for K > 0. Be sure to indicate key values on the locus, and accurately showdeparture and arrival angles.

b) For the remainder of the problem assume that K = 1. Make a carefully-dimensioned sketch ofthe Bode plot of the return ratio.

c) Conduct a Nyquist analysis which indicates whether the loop is stable for the value of K = 1.How does this analysis correlate with your root locus from part a)?

6


Assigned: Fri. Apr. 4, 2014Due: Wed. April 9, 2014, in classReading: Section 3.5 of Roberges Operational Amplifiers, attached to this problem set.

This excerpt catalogs relationships for time- and frequency-domain parameters.


Problem 1 The circuit shown below is adapted from the Lab 5 circuit to include a resistor Rfwhich allows adding a zero to the loop transfer function L(s). This allows independent ad-justment of the loop crossover frequency and phase margin. Note that we assume an infinitegain model for the op-amp. Also, as before we assume that Rs

p does this occur? What are the closed-loop damping ratio and natural frequency nof the dominant pair in Vf/Vi? How do these values compare with those predicted via theapproximations m/100 and n c?

g) Let Vi(t) = us(t), a unit step. Use Matlab to compute Vo(t) and Vf (t) assuming zero initialconditions. Explain the character of the two responses in terms of the poles and zeros ofthe respective transfer functions. With respect to the response Vf (t) what is the 1090%rise time tr? (See Roberge Figure 3.17). What is the ratiometric peak overshoot P0? Whatrise time would be predicted from tr = 2.2/h (Roberge equation 3.57) on the basis of hdetermined in part e) above? How does this compare with the actual step response rise time?The relationship tr = 2.2/h is extremely useful.

Problem 2 This problem considers a one degree of freedom magnetic levitation system using aLorentz actuator as shown below

In this figure, the motion of the levitated mass is given by xs relative to inertial space. Thesupporting base which carries the magnet structure moves relative to inertial space as xb. Thisbase motion can be used to model floor vibration, which is important to consider in precisionapplications. We assume no gravity in this problem. The objective of the levitation systemis to control xS to follow a desired trajectory without interference from base vibrations. Inthis problem, we consider the effect of using voltage- or current-control for the actuator driverelative to the effect of base vibrations.

The Lorentz actuator is considered to be ideal, and thus to have no inductance or resistance,with coil current i and back emf voltage e. The actuator is driven by a voltage source Vithrough a series resistance R. The coupled electromechanical system can be represented asshown below

a) Voltage Drive Draw a block diagram which includes Vi, i, e, F , xs, xs, xs and xb.

2

Use this block diagram to calculate the transfer function Xs(s)/Xb(s) in terms of the systemparameters. You may assume Vi = 0 to aid in block diagram reduction, but note that this isnot strictly necessary, given that the system model is linear.

On the basis of this transfer function, what equivalent mechanical element is connected be-tween the base and the levitated mass? How does this result in transmission of base vibrationinto the levitated mass?

Hand sketch a Bode plot for Xs(s)/Xb(s) with values labeled in terms of the system param-eters. Indicate how the base vibration transmission is shown in the Bode plot.

b) Current Drive We now close a current control loop on the coil by driving Vi on the basis ofmeasurement of coil current i. In this context, we assume that i is directly measured, andthat Vi is a dependent voltage source whose value is set by the current controller, i.e., a poweramplifier. In this simple model, we will assume that the coil has no inductance, and thus thatthe current can be controlled by a pure integral controller Gc(s) = g0/s. Draw a modifiedblock diagram which includes the same signals as in part a), but adds the current referenceir, current error ie, and controller transfer function Gc(s) = g0/s.

c) We will assume that g0 is large enough that the current-loop dynamics are very fast relative tothe mechanical subsystem dynamics. Under this assumption, we assert that the current loopcan be designed with the mass motion set to zero, xs = 0. Calculate the open loop planttransfer function I(s)/Vi(s), and determine its high-frequency asymptotic behavior. Sketcha Bode plot and show the high-frequency region.

Use this high-frequency model to write an approximate expression for the current-loop crossoverfrequency and phase margin as they depend upon g0, and thus argue that the current loopwill be stable for this controller and for a wide range of gains g0.

d) Use your modified block diagram to calculate the transfer function Xs(s)/Xb(s) in terms of thesystem parameters with the current control loop active. You may assume ir = 0 to aid inblock diagram reduction, but note that this is not strictly necessary, given that the systemmodel is linear.

On the basis of this transfer function, what equivalent mechanical element is connected be-tween the base and the levitated mass? How does this this element vary with current loopgain g0?

Hand sketch a Bode plot for Xs(s)/Xb(s) with values labeled in terms of the system param-eters, and with several overlaid plots for varying g0. How does the current loop bandwidthaffect the ability of base vibrations to affect the levitated platform motion xs?

3

92 Linear System Response

quency scale in Fig. 3.14a. These factors are combined to yield the Bode plot for the complete transfer function in Fig. 3.14b. The same information is presented in gain-phase form in Fig. 3.15.

3.5 RELATIONSHIPS BETWEEN TRANSIENT RESPONSE AND FREQUENCY RESPONSE

It is clear that either the impulse response (or the response to any other transient input) of a linear system or its frequency response completely characterize the system. In many cases experimental measurements on a closed-loop system are most easily made by applying a transient input. We may, however, be interested in certain aspects of the frequency response of the system such as its bandwidth defined as the frequency where its gain drops to 0.707 of the midfrequency value.

Since either the transient response or the frequency response completely characterize the system, it should be possible to determine performance in one domain from measurements made in the other. Unfortunately, since the measured transient response does not provide an equation for this response, Laplace techniques cannot be used directly unless the time re-sponse is first approximated analytically as a function of time. This section lists several approximate relationships between transient response and fre-quency response that can be used to estimate one performance measure from the other. The approximations are based on the properties of first-and second-order systems.

It is assumed that the feedback path for the system under study is fre-quency independent and has a magnitude of unity. A system with a fre-quency-independent feedback path fo can be manipulated as shown in Fig. 3.16 to yield a scaled, unity-feedback system. The approximations given are valid for the transfer function Va!/Vi, and V, can be determined by scaling values for V0 by 1/fo.

It is also assumed that the magnitude of the d-c loop transmission is very large so that the closed-loop gain is nearly one at d-c. It is further assumed that the singularity closest to the origin in the s plane is either a pole or a complex pair of poles, and that the number of poles of the function exceeds the number of zeros. If these assumptions are satisfied, many practical systems have time domain-frequency domain relationships similar to those of first- or second-order systems.

The parameters we shall use to describe the transient response and the frequency response of a system include the following.

(a) Rise time t,. The time required for the step response to go from 10 to 90 % of final value.

108

107

106

105

tM

104

103

102 -

10 -

1

0.1 -270* -18 0o -90*

Figure 3.15 Gain phase plot of s(O.1s + 107(10- 4 s + 1)

1)(s 2 /101 2 + 2(0.2)s/10 + 1)'

93

94 Linear System Response

a(s> vo

10 0

(a)

sff - aV.a

(b)

Figure 3.16 System topology for approximate relationships. (a) System with frequency-independent feedback path. (b) System represented in scaled, unity-feedback form.

(b) The maximum value of the step response P0 . (c) The time at which Po occurs t,. (d) Settling time t,. The time after which the system step response re-

mains within 2 % of final value. (e) The error coefficient ei. (See Section 3.6.) This coefficient is equal

to the time delay between the output and the input when the system has reached steady-state conditions with a ramp as its input.

(f) The bandwidth in radians per second wh or hertz fh (fh = Wch/ 2 r). The frequency at which the response of the system is 0.707 of its low-frequency value.

(g) The maximum magnitude of the frequency response M,. (h) The frequency at which M, occurs w,.

These definitions are illustrated in Fig. 3.17.

95 Relationships Between Transient Response and Frequency Response

For a first-order system with V(s)/Vi(s) = l/(rs + 1), the relationships are

2.2 = 0.35 (3.51) t,.=2.2T (351

Wh fh

PO = M,= 1 (3.52) t, = oo (3.53) ta = 4r (3.54) ei = r (3.55)

W, = 0 (3.56)

For a second-order system with V,(s)/Vi(s) = 1/(s 2/,, 2 + 2 s/w, + 1) and 6 A cos-'r (see Fig. 3.7) the relationships are

2.2 0.35 (357 W h fh

P0 = 1 + exp = 1 + e-sane (3.58)V1 - ( t7 - (3.59)

Wn #1-2 ~on sin6 4 4 t, ~ =(3.60)o cos

i= =2 cos (3.61) Wn (on

1 _ 1 M, - sI < 0.707, 6 > 45' (3.62)2 1_2 sin 20

, = co V1 - 2 2 = w, V-cos 26 < 0.707, 6 > 450 (3.63) Wh = fn(l-22 N2 -4 2+ 4 ) 12(3.64)

If a system step response or frequency response is similar to that of an approximating system (see Figs. 3.6, 3.8, 3.11, and 3.12) measurements of tr, Po, and t, permit estimation of wh, w,, and M, or vice versa. The steady-state error in response to a unit ramp can be estimated from either set of measurements.

- -- -- -- - --

t vo(t)

P0

1 0.9

0.1 tp ts t: tr

(a)

Vit Vi (jco)

1 0.707 ---

01 C (b)

vt K V0 (tW

e1 t: (c)

Figure 3.17 Parameters used to describe transient and frequency responses. (a) Unit-step response. (b) Frequency response. (c) Ramp response.

96

97 Error Coefficients

One final comment concerning the quality of the relationship between 0.707 bandwidth and 10 to 90% step risetime (Eqns. 3.51 and 3.57) is in order. For virtually any system that satisfies the original assumptions, in-dependent of the order or relative stability of the system, the product trfh is within a few percent of 0.35. This relationship is so accurate that it really isn't worth measuringfh if the step response can be more easily determined.

3.6 ERROR COEFFICIENTS The response of a linear system to certain types of transient inputs may

be difficult or impossible to determine by Laplace techniques, either be-cause the transform of the transient is cumbersome to evaluate or because the transient violates the conditions necessary for its transform to exist. For example, consider the angle that a radar antenna makes with a fixed reference while tracking an aircraft, as shown in Fig. 3.18. The pointing angle determined from the geometry is

= tan-' t (3.65)

Line of flight

Aircraft velocity = v

Radar antenna ,0

Length = I

Figure 3.18 Radar antenna tracking an airplane.


Assigned: Wed. Apr. 9, 2014Due: Wed. April 16, 2014, in classReading: FPE Sections 7.17.4


Problem 1 This problem revisits the current control circuit from Problem 1 of Problem Set 8. Asbefore we assume that Rs

Problem 2 This problem revisits the one degree of freedom magnetic levitation system from Prob-lem 2 of Problem Set 8.

For all of this problem, we assume that we close a current control loop on the coil by driv-ing Vi on the basis of measurement of coil current i. In this context, we assume that i isdirectly measured, and that Vi is a dependent voltage source whose value is set by the cur-rent controller, i.e., a power amplifier. In this simple model, we will assume that the coilhas no inductance, and thus that the current can be controlled by a pure integral controllerGc(s) = g0/s.

a) Write a state-space model for this system in terms of state variable x = [xs xs Vi]. Note thatwe have selected Vi as a state since it is the output of the integral controller. This state-spacemodel will have inputs ir and xb. For the floor vibration, you may assume that you have thesignals xb, xb, and xb available as inputs. Which of these do you need to use in your statemodel? Also, set up your state model to have outputs i and xs.

b) Now assume that the levitation system parameters take the numerical values m = 2 kg, K =5 N/A, and R = 8 . For these values, and again assuming that the electrical current loop

2

bandwidth is much larger than the mechanical dynamics, choose a value of g0 such that thecurrent loop crosses over at c = 10

4 rad/sec. What is the resulting current loop phasemargin m?

c) For your value of g0 chosen above, enter the state-space model into Matlab, and use Matlabcommands to generate the step responses from the two inputs to the two outputs. That is,you will generate four step responses, two for the input ir taking a step while xb = 0, andtwo for the input xb taking a step while ir = 0.

Problem 3 FPE 7.2

Problem 4 FPE 7.6

Problem 5 FPE 7.18


Problem G1 FPE 7.19

Problem G2 FPE 7.20

3


Assigned: Wed. Apr. 16, 2014 Due: Wed. April 23, 2014, in class Reading: FPE Sections 8.18.4


Problem 1 FPE 8.1

Problem 2 FPE 8.2

Problem 3 FPE 8.5

Problem 4 FPE 8.6

The following problems are assigned to only 2.140 students. Students in 2.14 are welcome to work these, but no extra credit will be given.

Problem G1 FPE 8.8

1


Assigned: Wed. Apr. 23, 2014 Due: Wed. April 30, 2014, in class Reading: FPE Sections 8.18.4

Problem 1: FPE 8.7 For this problem, in part a), calculate by hand the requested magnitude expression |H(z1)| where z1 = ej1T , and 1 = 3 rad/sec. (The problem statement omits the subscript 1 in the rst sentence.) In part b), you may use Matlab to generate the requested Bode plots. How do these compare to what you would expect from the continuous-time system?

Problem 2: This problem considers designing a continuous-time controller for a single-axis magnetic levitation device. This controller is then mapped into an approximating discrete-time controller.

The plant to be controlled is a mass driven by a voice-coil actuator. Assume that this plant has a transfer function

Gp(s) = 1000/s2

a) For this plant, design a continuous-time PID controller using the series-lag-lead form 1 s + 1

Gc(s) = Kp 1 + TI s s + 1

to achieve a crossover frequency of c = 1000 rad/sec, with a phase margin of m = 45 .

Your continuous-time design is to meet specications for the specied continuous time plant, with no consideration of the later sampling operation when implemented as a discrete-time approximation. You should be able to accomplish this design using hand-calculations and Bode plot sketches. Clearly show your design eort. Indicate the loop crossover frequency, phase margin and gain margin.

b) Now map this controller to discrete-time using the Tustin transformation, and assuming a sample rate of 6 kHz. We recommend using the operator notation introduced in class to aid in this step. Show your hand calculations that give the discrete-time controller. We recommend that you discretize the three terms of the PID controller separately: 1) gain, 2) lag, 3) lead. This will be easier than discretizing the full second-order controller. Give a block diagram for the discrete-time controller showing the transfer functions in this block diagram. What are the resulting dierence equations for each element? Carefully show your design approach and calculations. (For the purposes of this problem, do the approximations by hand, rather than directly using Matlab to generate the transformations.)

c) Use Matlab/Simulink to simulate the step response of your closed-loop system with i) the continuous-time PID controller, and ii) the approximating discrete-time controller which you designed. Please include plots of the control signal (plant input) and the plant output for each case. How do these responses compare?

Note that you will need to choose the integration solver routines and sample times in Simulink to properly simulate the continuous-time loop. That is, dont use a xed-step solver with 6 kHz

1

in Simulink when you want to simulate the continuous-time loop. If these considerations are unfamiliar, please consult any of the online Simulink tutorials to understand the solvers and time-steps used to simulate continous-time and mixed discrete-time/continuous-time systems.

In your Simulink simulations, reduce the discrete-time sample rate from 6 kHz, and comment on the eect of longer sample times in the resulting signals. Include relevant plots to show these eects. At about what reduced sample rate does the system go unstable?

Recognize that you can implement the Simlink model block diagram in the series lag lead form, replacing each block with its discrete equivalent.

When we ask you to simulate the continuous-time loop, this should be done with the continuous-time controller. When we ask you to simulate the discrete-time loop, use discrete-time transfer functions in Simulink, and input the dierence equation parameters that you calculate when discretizing.

2

2.14/2.140 Final Design Problem

Assigned: April 28, 2014Due: week of May 12, 2014, at checkoffs

1 Overview

This problem considers the control of a fast tool servo (FTS), and is based upon the work done byXiaodong Lu in his Ph.D. thesis. A copy of this thesis is available on the course web page. Thedesign problem focuses on:

High-bandwidth current controller for driving the actuator. Identification of the electromechanical plant dynamics from a measured Bode plot. Design of a controller to minimize following error for a sinusoidal reference trajectory, in the

presence of sensor noise.

(2.140 students only) Design of an add-on Adaptive Feedforward Cancellation (AFC) con-trol block at the reference trajectory frequency in order to eliminate following error at thatfrequency.

We do not give you exact specifications for the servo performance. Rather, you are to apply yourbest effort to get good performance. The measure of perfomance in this problem is reduction ofthe following error due to the reference trajectory and to sensor noise.

The problem starts with designing the controller for the current loop, and then moves on to designof the FTS position controller.

2 Current control loop

The current control loop circuit is shown in Figure 1. The circuit includes the back emf voltagefrom the actuator as a dependent source. The force supplied by the actuator is F = Kf ic, and thecorresponding back emf voltage is Vbemf = Kf x. A simple model of the electromechanical systemof the FTS is shown in Figure 2, with the actuator force F applied to the mass m1.

In this problem, assume the following component values: Rc = 4 , Lc = 2104 H, Kf = 20 N/A.

In designing the current controllers for power amplifiers, it is typical to neglect the actuator backemf in the current loop dynamics. (The back emf is quite important in modeling nonlinear issues

1

such as amplifier saturation, but we do not consider that issue in this problem.) We ask you in thissection to show that the back emf can be ignored, and then to design a current loop controller onthat basis.

Calculate the plant transfer function for the current loop with the back emf voltage included.Note that in this model, the FTS mechanical dynamics will be reflected into the electricaltransfer function.

Calculate the plant transfer function for the current loop with the back emf voltage assumedequal to zero. Under this assumption, the FTS mechanical dynamics will have no effect onthe electrical transfer function.

Show that for high frequencies, both transfer functions are equal. Explain physically why thisis the case.

On this basis, recognize that you can design the current control loop assuming the back emfvoltage is zero.

Design the current control loop such that the crossover frequency is c = 6105 rad/sec, witha phase margin m 60. We also require that an input voltage of Vset = 10 V will resultin a steady-state coil current of ic = 5 A. That is, the current drive has an input/output DCgain of Ga ic/Vset|DC = 0.5 A/V. For these design specifications, what are the resultingvalues of R2, R3, C1, and C2? Note that R1 is given as 10 k. Further note, as shown onthe schematic, that we assume that no current enters the input resistors of the differentialamplifier, since these resistors are large compared with the 0.2 sense resistor.

Provide a Bode plot for the loop return ratio showing the crossover frequency and phasemargin.

3 Actuator model

The actuator conceptual model is shown in Figure 2. Here, the actuator force F = Kf ic acts on aspring/mass/damper system representing the moving components of the fast tool servo.

A block diagram for the plant dynamics is given in Figure 3. Position x is measured with acapacitance probe with a gain of G1 = 5 105 V/m. The sensor has an associated noise voltageVn.

From this point on in the problem, we assume that the current drive dynamics are so fast thatthere is a static gain coefficient Ga A/V which relates Vset and ic. That is, we now assume thatthere are no dynamics associated with the current drive. This is shown on the block diagram whereic depends statically upon Vset via the gain Ga.

2

4 Plant measured Bode plot

The measured plant Bode plot is shown in Figure 4. As we see from the Bode plot, the plantdynamics are more complicated than would be expected from the simple model of Figure 2. Thelowest mode in the Bode plot is due to the spring/mass/damper system, but there are higherfrequency modes and a time delay.

This Bode plot and associated data files for magnitude and phase are available on the course webpage in the .zip file GpPlantFrequencyData.zip. Use these data files as part of your model buldingprocess. In the data file, we have included a .mat file for entry into Matlab. This file includes thecolumn vectors Gpmag, which is the magnitude of Gp(j) in straight magnitude units, Gpphase,which is the phase of Gp(j) in degrees, and ww, which is the associated frequency vector in rad/sec.

On the basis of the low-frequency dynamics and first mode, what are the values of m1, b1,and k1?

Fit a dynamical model to the measured Bode plot. Provide a pole-zero plot for your model,and show a Bode plot of your model frequency response overlaid on the measured frequencyresponse. (In extracting data from the Matlab bode command, you will notice extra dimen-sions, which can be eliminated with the squeeze command.)

You will use this model to design your controller in the following section.

5 Controller Design

The control loop configuration is shown in Figure 5. The controller configuration is shown inFigure 6. Here, the integrator is explicitly shown, and transfer function Cls(s) is a loop shapingcontroller of your design.

Throughout this problem, we assume

xref = 105 sin 3000t [m].

We also assume throughout that

Vn = 5 102 sin 105t [V].

Design a controller with as good performance as you can achieve. The controller must mini-mize the following error xe due to the reference trajectory and the sensor noise.

You must show us Bode, Nyquist, and Sensitivity plots for your FTS controller design. Ofcourse, the loop must be stable! Also, the Sensitivity magnitude plot must be under +10 dBfor all frequencies. These plots are for the loop broken at the plant input Vset.

3

Also provide time-domain plots of the position error xe(t). What is the magnitude of theerror due to the reference trajectory xref (t)? What is the magnitude of the error due to thesensor noise Vn(t)? How have you addressed minimizing these errors in your loop design?

Also, you must run a live simulation on Matlab during checkoffs to answer questions aboutthe loop. Thus, you will need to have a Matlab-based model of the FTS position control looprunning at your checkoff, so that we can ask you questions about its performance. Pleasecome to lab in advance of your checkoff in order to have time to get your model runningbefore the checkoff starts.

A template in the form of an m-file (Template.m) has been posted on Stellar. You are requiredto use this template to present your design during the check-offs on Monday, May 12, andTuesday, May 13. A checkoff signup sheet will be posted in advance of checkoffs. To use thetemplate, download a copy of it from Stellar. Please read the instructions provided below aswell as the comments within the m-file on how to use the template.

Within the template, we have defined the parameters values given in the problem set, suchas LC and Rc. The controller and plant transfer functions, which need to be designed andidentified by you, have been declared as blank ( [] in MATLAB). You must initialize thesetransfer functions by adding your own code to the m-file. Once you have initialized thetransfer functions, all of the plots required for the check-off can be generated by running theTemplate.m file. You can add your design code to this file. It is important that you use theexact same variable names declared within the template. You are allowed to declare yourown variables but do not overwrite any variables already in use by the template. Please testyour code on either your own laptop or the lab computers ahead of time and confirm that itruns correctly before your check-off.

For 2.140 students, you will need to document your design as it works without the AFC, andthen document with AFC. For this purpose, please prepare two models from the templatefile, or provide some means to switch between models so that we can see the effect of usingor not using AFC.

Your design project grade will depend upon your checkoff results, and upon our evaluationof the quality of your design efforts. Please prepare a lab report detailing your design work,and showing how you went about optimizing the design for the given specifications. Includerelevant calculations, data, and plots.

6 AFC Controller Design (2.140 students only)

This section is only to be completed by 2.140 students.

In order to eliminate steady-state following error, we can add an Adaptive Feedforward Cancellationcontrol block. A reference for the design of this type of controller is given in the Ph.D. thesis ofXiaodong Lu, which is available on the course web page. In particular, look at section 7.3 of Lusthesis. The controller you will implement will use only one AFC block, in the topology shown inFigure 712b of the thesis. The controller configuration with a single AFC control block is shownin Figure 7.

4

Figure 1: Current loop circuit.

Here, the transfer function A1(s) is a resonator with a natural frequency chosen to match the inputposition reference frequency of 3000 rad/sec. That is,

A1(s) =K1s

s2 + 21,

where K1 is an AFC gain to be chosen for adequate stability, and 1 = 3000 sets the resonator tolie at the reference trajectory frequency.

Design the AFC-based controller, and show that the following error at 3000 rad/sec is elim-inated. Show plots of the loop Bode, Nyquist and sensitivity curves. The sensitivity mustremain below +10 dB for all frequencies. All these plots should be for the loop broken at theplant input Vset.

Make a Bode plot of your controller including AFC; you should see that the controller gainis infinite at the resonant frequency.

The approach given in section 7.3.3 of Lus thesis, and in particular the design rule on page244 will help you with this design task. Explore the rate of error convergence as K1 is varied;how does K1 affect convergence? What is the upper limit of K1 for loop stability?

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Figure 2: Model of actuator driving fast tool servo.

Figure 3: Block diagram of fast tool servo with current setpoint input Vset, sensor noise inputVn, position output x, and sensor output voltage Vsense. Plant dynamics including flexibility arerepresented in Gp(s).

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180

160

140

120

100

80

60

Mag

nitud

e (d

B)

102 103 104 105 106900

810

720

630

540

450

360

270

180

90

0

Phas

e (d

eg)

Bode plot for FTS plant transfer function Gp(jw)

Frequency (rad/sec)

Figure 4: Experimentally-measured plant transfer function for Gp(j). This plot and associateddata files for magnitude and phase are available on the course web page.

Figure 5: Control loop configuration.

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Figure 6: Controller configuration with isolated integrator.

Figure 7: Controller configuration with isolated integrator and a single AFC controller block.

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1

Instruction for Dynamic Signal Analyzer (DSA)

1. Make your closed loop system stable by implementing controller, such as P or PI, and wait until

it settles down to steady state (Fig.1).

Figure 1 Closed loop system stabilized with proportional control

2. Go to the Dynamic Signal Analyzer tab (Fig. 2) and fill out specifications.

Figure 2 Dynamic Signal Analyzer tab

2

Dynamic Signal Analyzer (blue circle in Fig. 3) can measure frequency response of a system that

you are interested in experimentally. It generates one output signal (SRC) to excite the loop, and

takes two input signals (CH1 and CH2) to measure frequency response of a system between the

two channels. DSA compares CH1 and CH2 signals to calculate magnitude ratio and phase

difference between them, which tells you the frequency response of a system between two

channels.

Figure 3 - DSA implemented in the Block Diagram

SRC can be injected into the loop through any summing junctions; in this particular lab, we are

injecting it through a summing junction for reference and feedback signals (the left most red circle

in Fig. 3). And we are tapping two signals from CH1 and CH2 to measure frequency response

between the two channels. For example, if you want to measure frequency response of thermal

plant (Tout/Vin), you can wire control effort signal to CH1 and temperature signal to CH2 (other red

circles in Fig. 3). If you want to measure loop return ratio (Tout/E), then you can wire error signal

to CH1 and temperature signal to CH2.

Frequency Range

We recommend use 0.1 Hz for initial frequency, 1 Hz

for final frequency, and 10 for number of bins. If you

want, you can put more bin numbers and wider

frequency range, which will give you finer result but

take more time.

Amplitude

For linear system, its frequency response does not

change with respect to sinusoid amplitude. However, Figure 4 - Specification of DSA

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you cannot put too large value because it could saturate the actuator, which makes the whole

system nonlinear. Also, you might not want to put too small value because that leads to poor

signal to noise ratio.

Cycles

To measure frequency response of a system, it needs to be in steady state, so DSA spends several

settling cycles to get to steady state. After that, DSA takes several cycles to calculate averaged

value.

CH1 and CH2 Offset

We recommend you subtract DC offset manually from CH1 and CH2 signals to get better result.

(Typical DSA do that automatically, but current version of LabVIEW DSA does not have that

functionality)

3. Now, hit the RUN button!! Then you can see a pop-up window as below.

Figure 5 Identification Window of DSA

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4. Once it finishes the analysis, you can right click each plot to export data to excel.

Figure 6 - Export data to Excel

5. Import the data to MATLAB to design the controller by loop shaping. Please refer to

LoopShaping.m example file for the next steps.

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Appendix A: Troubleshooting if your dont see a pop-up window

1) If you have trouble seeing the pop-up window, its possible that your computer is blocking the

server that opens up this VI. You can resolve this by going to Tools >> Options (Figure A-1)

Figure A-1: Options menu

6

2) Under Categories on the left-hand side, select VI server. Make sure TCP/IP box is checked

(Figure A-2)

Figure A-2: Configuring VI Server

7

3) Scroll down the menu to Machine Access. Type an asterisk (*) in the Machine name/address

box (see blue circle in Figure A-3), make sure Allow access is chosen, then press the Add

button (see yellow circle in Figure A-3). Do the same thing for Exported VIs. You should see a

check mark next to an asterisk under Machine access list (see red circle). Its okay if you have

other items listed.

Figure A-3: Configuring machine access

Instructions for installing LabVIEW software for 2.14 labs

1. Go to http://www.ni.com/academic/download.htm. You should see the screen in Figure 1.

Figure 1: Website for software download

2. Expand the categories and download the software highlighted in red in Figure 2

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Figure 2: Software to download

3. Once youve downloaded the myRIO DVD 1 and DVD2, run the setup executable of DVD1. Install the components as seen in Figure 3. During the install, you will be asked for a serial #. If

you havent been provided with one in class, choose evaluation mode, otherwise, use the

serial # youve been given.

Figure 3: Items to install during myRIO Software Suite setup

4. Run the ELVISmx installer and install ELVISmx. You are done!

2

2.14/2.140 Lab 1

Assigned: Week of Feb. 10, 2014 Due: Week of Feb. 17, in your lab session

For Lab 1, we ask that you install and run Labview on your own laptop computer. You will also need to install the CD&Sim and Mathscript RT modules. Please see instructions posted on the course web page.

Please load and run the le URACTLR.vi. This le lets you implement on/o and linear control of 4 dierent plants. You can also test automatic (machine) on/o and linear control. The system runs as a 10 second simulation. During this run, your goal is to use the control input to keep the plant output at a level of 0.5. Deviations from 0.5 are errors; the square of these errors is integrated over the 10 second run, and displayed on a gauge and as a numerical output. The lower the integrated square error, the better.

By the end of the lab session, conduct your own experiments with this control system. Look at how each system is controlled by the machine automatic control. How well do you (human) control the 4 dierent systems? How well does the machine control work? For the second-order system, explore changes in n and , and see how these aect the control. Before leaving lab, meet with one of the lab sta one-on-one to explain what youve observed. Be prepared to explain the loop behavior in light of the plant dynamics for each plant.

A report documenting your experiments is due the following week (week of Feb 17) at the start of your assigned lab session. Reports will not be accepted more than 10 minutes late, i.e., after 2:15 pm. We will not accept reports turned in to other lab sections. This report should show time traces from some of your best runs and for your best tunings of the automatic controller in on/o and linear modes for each of the four plants. Explain the observed signals (control eort and plant output) in terms of the plant dynamics and controller charcteristics.

Your Lab grade will be based 50% upon your understanding as shown to in the face-to-face lab meetings and 50% upon the data and understanding shown in your written report. Grades for each of these components will range from 0 (worst/not there) to 5 (excellent) for a total lab grade of 0 to 10.

We will not make up lab sessions. If you miss a lab session, except for medical or emergency reasons, your lab grade will be 0.

1

2.14/2.140 Lab 2

Assigned: Week of Feb. 24, 2013Due: Week of Mar. 3, in your lab session.

Lab 2 studies a thermal control loop. The mechanical configuration of the thermal plant is shownbelow.

Here a thermistor is attached to a 2N3904 NPN transistor. The thermistor is packaged in a roundepoxy bead; we have sanded off one side of the bead in order to create a flat spot to improve heattransfer. The joint thermal conductivity is improved by using thermal grease in the joint. The pairis held together with heat-shrink tubing, and then a styrofoam bead is used as thermal insulation.

The electrical configuration is shown below

1

This system will be the focus of the next two labs. The goal of the control system is to regulate themeasured temperature to 40 C. This is done by maintaining the transistor heating at a correctvalue using the control input at the base resistor. When operating at 40 C, for small changesthe system can be viewed as linear, since temperature can be increased and/or decreased using thecontrol input. (Strictly speaking, the changes of the control input are linearly related to the changesof the temperature, where these changes are measured relative to the steady-state operating pointvalues.)

For this week, we ask you to do the following:

1. In open loop, manually adjust the control effort so as to set the steady operating temperatureat 40 C. Experimentally save and plot the step response for small control changes aboutthis equilibrium. How well does the measured step response match a fitted first order stepresponse?

2. Experiment with human ON/OFF control. Try this with and without the styrofoam insula-tion in place. Save and plot an interesting set(s) of data (both control effort and resultingtemperature).

3. Experiment with machine ON/OFF control. Try this with and without the styrofoam insu-lation in place. Save and plot an interesting set(s) of data (both control effort and resultingtemperature).

4. Experiment with human linear control. That is, you manually move the control effort sliderto control the measured temperature to a setpoint of 40 C. Notice how the control effortmust be actuated to make a relatively fast change in the controlled temperature. Try thiswith and without the styrofoam insulation in place. Also notice the effect of disturbancessuch as blowing air on or touching the transistor with the insulation removed. Save and plotan interesting set(s) of data (both control effort and resulting temperature).

5. Experimentally tune P and PI linear controller. First tune the loop with only a P term. Thenadd in an I term. Can you see the effect of the integral control? Save and plot the closed-loopstep response to reference temperature changes of the loop operating with these controllers(both control effort and resulting temperature), and record the control gains used. Also,impose some disturbance on the system, for instance, by removing the styrofoam insulation,or putting your fingers on the transistor, or blowing on the device, etc. Save and plot someinteresting set(s) of data (both control effort and resulting temperature).

6. For graduate students only: Create a thermal model competent to explain the observeddynamics. Estimate the thermal mass on the basis of the system dimensions and somereasonable assumptions of specific heat. Use experiments to determine the thermal lossmodeled as a resistance to ambient, both with and without the styrofoam peanut insulation.Fit parameters to this model and compare with your experimental step measurements.

Checkoff in lab session: In this weeks lab session demonstrate to one of the teaching staffyour working controller and some experimental results. Progress shown in this checkoff will counttowards half of your lab grade.

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Answer sheets: At the start of next weeks lab session you must submit your lab writeup. Thesewrite-ups are due at the start of the lab session, and will not be accepted late. Please attach tothis lab assignment clearly labeled answers and plots for all the questions above. This lab reportwill count towards half of your lab grade.

It is key that you submit your lab report on time at the start of the lab session next week, as wewill begin work on Lab 3 during that session.

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1.1. FIRST-ORDER SYSTEMS 21

Figure 1.15: Sketch of bulb and relevant thermal elements.

1.1.4 Thermal first-order system

For an example thermal system we study the desk lamp shown in the picture(to be added). This lamp bulb is electrically heated via the bulb filament.The resulting bulb temperature is measured with the infrared sensor shownin the figure (to be added). A sketch of the light bulb in the lamp is shownin the line drawing of Figure 1.15.

We left the lamp on for a long enough time to reach steady state, andthen turned o the lamp and measured the decay of temperature back toambient. Data taken from this system is shown in tabular and graphicalform in Figure 1.16. By inspection of this data, the bulb system is well-fitby a first-order model of the form of (1.1). An estimate of the associatedtime constant is about 3 minutes. But we need to have in seconds, so thesystem time constant is formally given as = 180 sec.

An abstraction to a lumped model of this system is shown in Figure 1.17.Here the thermal capacitance of the bulb is summarized by the block ofmaterial labeled with the capacitance Cb with units of [J/K]. The blockis assumed to have a uniform temperature Tb [K]. This block has a totalstored thermal energy Wb = CbTb [J]. The change of thermal stored energyhappens via heat flow

qb =dWbdt

= CbdTbdt

. (1.23)

Here qb in units of watts represents heat flow into the bulb. As shown inthe figure, we assume that the block is insulated on three sides, and so the

22 CHAPTER 1. NATURAL RESPONSE

Figure 1.16: Data from light bulb cooling experiment.

"ULBTHERMALCAPACITANCE

4HERMALRESISTANCETOOUTSIDEWORLD

#B

4B 2B 44A

Figure 1.17: Lumped model for bulb cooling experiment.

1.1. FIRST-ORDER SYSTEMS 23

heat flow through those sides is zero. The block is connected to the outsideambient temperature via the thermal resistance Rb, such that

qb =Ta TbRb

. (1.24)

This resistance represents the flow of heat into the bulb as a linear function ofthe temperature dierence4 between the ambient and the bulb temperatures.

Setting equality between the last two equations gives

CbdTbdt

=Ta TbRb

. (1.25)

Now, its convenient to define a variable to represent the temperature dif-ference between the bulb and ambient: T Tb Ta. Since the ambienttemperature is constant, dT/dt = dTb/dt. Making these substitutions andmultiplying (1.25) through by Rb yields

RbCbdT

dt+ T = 0. (1.26)

If we define = RbCb, this is in the form of (1.1). The natural response isthus as calculated in section 1.1, with its associated figures. Specifically, ifthe initial temperature dierence of the bulb is defined as T (0) = T0, thenthe temperature dierence as a function of time varies as

T (t) = T0et/RbCb [K]. (1.27)

If you want

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