Feedback and Control Systems Lab Manual-

8/10/2019 Feedback and Control Systems Lab Manual-

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Experiment No. 01SYSTEM MODELING AND SIMULATION

1. Objective(s):This activity aims to1. introduce the modeling and simulation tools of MATLAB and LabVIEW to the students;2. equip the students with the skills and knowledge in using MATLAB and LabVIEW to model and

simulate systems; and3. equip the students with the skill to measure the major performance indicators of a control system,

namely: time response parameters, error performance and stability.2. Intended Learning Outcomes (ILOs): At the end of this activity, the student shall be able to

1. create MATLAB and LabVIEW programs that will simulate electrical, mechanical and position

control systems; and2. determine the effects of component values to the system’s time response parameters, error and

stability of dynamic systems.3. DiscussionOne of the steps involved in the design of a control system is to model the system itself from its schematic.The system’s model is very important since it will provide information on the system’s various parameters,such as time response, error and stability information. These parameters will then help the designer tocome up with a control system that would make the system perform at its desired state. Thus, modelingand simulation is an important step in the design of control systems.

Systems can be modeled as transfer functions using the Laplace transform of the differential equationrepresenting the system, or as state-space models which expresses the system in terms of state andoutput vectors. Solutions of both models can be highly simplified by the use of computer aided tools, suchas MATLAB and LabVIEW. In this activity, MATLAB and LabVIEW are to be used to model and simulatedynamic systems after obtaining their transfer functions.

MATLAB has the control system toolbox which can be used to create transfer functions-domain models ofdynamic systems and plot and obtain information on the systems step response. In the same manner,LabVIEW has the control design and simulation module which can be used to simulate dynamic systems.

This activity will demonstrate how these tools can be used to model and simulate dynamic systems.4. Resources:To perform this activity, a computer workstation with MATLAB R2012a or higher and LabVIEW 8.6 orhigher installed is required. For MATLAB, the control systems toolbox is required and for LabVIEW, thecontrol design and simulation module.5. Procedure: Activity 1.1 – Modeling and simulation of a series RLC electrical network.



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4. LabVIEW.Build the front panel (FP) and the block diagram (BD) as shown below, calling this VIact01-01.vi. In the BD, place aSimulation Loop. Right-click on one of the boundaries of the loop and choose

Configure Simulation Parameters . Change the Simulation Time’s Final Time to 20. Place a Step Signal , a Transfer Function , a Build Array and the SimTime Waveform functions

inside the simulation loop. Configure the transfer function block to contain the transfer functionobtained fromQ1.1(a).

In the FP, a Waveform Chart will automatically be placed. Configure theLegend on the top rightpart of the chart and name them as Input and Output as shown. Right click on the chart andchoose X Scale >> Properties . In the Display Format tab, choose Type as Floating-point , thenclickOK. Change the scale of the x-axis of the waveform chart to0-20.

Q1.4(a) Use the VI to plot the step response of the circuit above. Roughly sketch the plot below andlabel the necessary time response and error information in the plot. The plots obtained in the

previous steps must be the same.Q1.4(b) Based on the plots obtained, is the system stable? Why or why not?Q1.4(c) Change some of the parameters of the RLC circuit and obtain a new transfer function.Simulate this new transfer function. Use both MATLAB and LabVIEW. Conduct several trials (atleast ten trials) and plot the step response of the system in each trial. Comment on the effect of thevalues of the resistor, inductor and capacitor on the output capacitor voltage.Q1.4(d) Discuss the different timing options in the Configure Simulation Parameters of the



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simulation loop.Q1.4(e) Create a virtual instrument using the control design and simulation module andMathScriptnode of LabVIEW to simulate the electrical network below. Provide a screenshot of theblock diagram and the front panel of the VI on a separate sheet of paper. Plot the step response onthe space provided below.

Activity 1.2 – Modeling and simulation of mechanical systems. 1. In this part of the activity, the response of the mechanical system such as the one shown below to a

step input will be simulated.

Q2.1(a) Find the transfer functions ( ) = ( )/ ( ) and ( ) = ( )/ ( ). Fill up thespaces provided below.

Q2.1(b) Compute for the output displacement of the system () and () to a step force inputand plot them on the space provided.2. Repeat steps 2, 3 and 4 of Activity 1.1 to simulate the mechanical system given.Q2.2(a) Roughly sketch the plot of (), () and the step input as seen in the waveform chart onthe space provided.Q2.2(b) Interpret the waveforms. How does the position of the masses vary as a step force isapplied to the system at ()? (Hint: what happens when you apply a step force to the systemabove?) Q2.2(c) Determine what happens when the surface at which the masses moves on has frictionwhich is

= / for both masses. Plot the new response on a separate sheet of paper and

interpret the results.Q2.2(d) Simulate the rotational mechanical system below, plotting the responses () and () with respect to an input step torque.



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Q2.2(e) Interpret the waveforms. How does the angular position of the inertia vary as a step torqueis applied to the system at ()?



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Course: Experiment No.:Group No.: Section:Group Members: Date Performed:

Date Submitted:

Instructor:

6. Data and Results:Q1.1(a) For this circuit, find the transfer function ( ) = ( )/ ( ).

Q1.1(b) For a step input, find an expression for the output capacitor voltage.

Q1.1(c) Using this expression, plot the output capacitor voltage and roughly sketch the plot below.

Q1.3(a) Roughly sketch the plot of the transfer function of the above circuit. Use this graph todetermine the time response and error of the system.



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Q1.4(a) Use the VI to plot the step response of the circuit above. Roughly sketch the plot below andlabel the necessary time response and error information in the plot. The plots obtained in the

previous steps must be the same.

Q1.4(b) Based on the plots obtained, is the system stable? Why or why not? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.4(c) Change some of the parameters of the RLC circuit and obtain a new transfer function.Simulate this new transfer function. Use both MATLAB and LabVIEW. Conduct several trials (atleast ten trials) and plot the step response of the system in each trial. Comment on the effect of thevalues of the resistor, inductor and capacitor on the output capacitor voltage. _____________________________________________________________________________________



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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.4(d) Discuss the different timing options in the Configure Simulation Parameters of thesimulation loop. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.4(e) Create a virtual instrument using the control design and simulation module and MathScriptnode of LabVIEW to simulate the electrical network below. Provide a screenshot of the blockdiagram and the front panel of the VI on a separate sheet of paper. Plot the step response on thespace provided below.



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Q2.1(a) Find the transfer functions ( ) = ( )/ ( ) and ( ) = ( )/ ( ). Fill up thespaces provided below.

( ) = ( )/ ( )

( ) = ( )/ ( )

Q2.1(b) Compute for the output displacement of the system () and () to a step force inputand plot them on the space provided.

()

()

Plot of () and ()



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Q2.2(a) Roughly sketch the plot of (), () and the step input as seen in the waveform chart onthe space provided.

Plot of () and () in MATLAB

()

()

Plot of

() and

() in LabVIEW

Q2.2(b) Interpret the waveforms. How does the position of the masses vary as a step force isapplied to the system at ()? (Hint: what happens when you apply a step force to the systemabove?) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q2.2(c) Determine what happens when the surface at which the masses moves on has frictionwhich is = / for both masses. Plot the new response on a separate sheet of paper andinterpret the results. _____________________________________________________________________________________



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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q2.2(d) Simulate the rotational mechanical system below, plotting the responses () and () with respect to an input step torque.

Plot of () and ()

Q2.2(e) Interpret the waveforms. How does the angular position of the inertia vary as a step torqueis applied to the system at ()? _____________________________________________________________________________________

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7. Conclusion:



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8. Assessment:1. Create a MATLAB program or a LabVIEW virtual instrument that will interactively simulate electrical

and mechanical system. System configuration is your choice, just provide the schematic. The interface

must contain controls to adjust the parameters of the components of the system. Build one program orVI for each of the system (electrical, translational mechanical and rotational mechanical). Graphics canbe integrated into the program. Using these programs or VIs, investigate the effect of the componentvalues to the time response, error performance and stability of the system.

2. A position control system can be represented by a block diagram shown below, whereK is a constantgain andG(s) is the transfer function of the plant, which takes the form

G(s) = Ks(s+a)

where

K and

a are constants. (Note that

G(s) has the same form as the transfer function of a

servomotor.)

Create a MATLAB program or a LabVIEW virtual instrument to implement and simulate the positioncontrol system as shown. Using this simulator, investigate the effect of the constantsK, K and a tothe time response, error and stability of the system.



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Assessment rubric for the activity’s intended learning outcomes

INTENDEDLEARNING

OUTCOMES1 2 3 4 Points

Create aprogram that

will simulate theperformance of

electricalsystem(MP 1)

A program forelectrical system wascreated but does notwork or results vary

significantly fromreality

The program worksbut the results are

doubtful or areinconsistent when

trials are repeatedlydone

The program works,the results arecredible but the

interface is difficult touse

The program returnsresults that are

expected and withlittle or no variationfrom reality and with

user friendlyinterface.

Determine theeffect of

componentvalues of

electricalsystem to the

performance ofthe system

(MP 1)

An investigation wasmade but containsinsufficient data or

limited trials.

A sufficient amount ofdata were gathered

but was notsynthesized properly.

The data gatheredwas analyzed but

does not support theconclusions made.

The data gatheredwas analyzed very

well and validconclusions were

drawn.



translationalmechanical

system(MP 1)

A program fortranslational

mechanical systemwas created but does

not work or resultsvary significantly from

reality



trials are repeatedly

done





user friendly interface



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componentvalues of

translational

mechanicalsystem to theperformance of

the system(MP 1)

An investigation wasmade but contains

insufficient data orlimited trials.







drawn.



rotationalmechanical

system(MP 1)

A program forrotational mechanicalsystem was createdbut does not work or

results varysignificantly from

reality



trials are repeatedly

done







componentvalues ofrotational

mechanicalsystem to the


(MP 1)


limited trials.







drawn.



a positioncontrol system

(MP 2)

A program for aposition control

system was createdbut does not work or

results varysignificantly from

reality



trials are repeatedlydone







componentvalues of

position controlsystem to the


(MP 2)


limited trials.







drawn.

Total Score



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Mean Score = (Total Score /8)

Percentage Score = (Total Score / 32) x 100%

Assessment rubric for the conduct of laboratory experiments

Performance Indicators 1 2 3 Points

Conduct experiments inaccordance with good andsafe laboratory practice.

Members do not followgood and safe laboratory

practice in the conductof experiments.

Members follow goodand safe laboratory

practice most of the timein the conduct of

experiments.


practice at all times inthe conduct ofexperiments.

Operate equipment andinstruments with ease

Members are unable tooperate the equipment

and instruments.

Members are able tooperate equipment and

instrument withsupervision.

Members are able tooperate the equipmentand instruments with

ease and with minimumsupervision.

Analyze data, validateexperimental values against

theoretical values todetermine possible

experimental errors, andprovide valid conclusions.

The group hasincomplete data.

The group has completedata but has no analysis

and valid conclusion.

The group has completedata, validates

experimental valuesagainst theoretical

values, and providesvalid conclusion.

Total Score





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Experiment No. 02TIME RESPONSE OF DYNAMIC SYSTEMS

1. Objective(s):This activity aims to

1. demonstrate the use of computer aided tools to determine the poles and zeros, and the response

of the system to various inputs of dynamic systems;2. equip the students with the knowledge and skills in obtaining the pole-zero plot, time response plot

and information, and determining the relationship of the time response parameters of the system inrelation to its pole location; and

3. provide the students with the knowledge of designing component values to meet time responseobjectives and simulating the design to verify its correctness.

2. Intended Learning Outcomes (ILOs): At the end of this activity, the students shall be able to:

1. determine and, on the complexs-plane, plot the poles and zeros of a dynamic system;2. plot the time response of, and interpret the time response characteristics of dynamic systems

represented as transfer functions; and3. design components of dynamic systems to achieve time response parameter objectives.

3. Discussion After obtaining a model of the system, the system is analyzed for its transient and steady-state responses.It was learned in the discussion that the response of the system is highly dependent on the location of thesystem poles. Thus, the location of the poles gives a vivid picture of the form of the response, as well ashow fast the response is.

For first-order system, or system with only one pole and no zero, the response has only one form and is

given as c(t) = A+Be

whereA and B are the residues of the partial fraction expansion of the rational Laplace transform of theresponse of the system. In the discussion, the parameters time constant, rise time and settling time aredefined, which are all dependent on the pole location–a.



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For second-order system, the response depends on the location of the poles also. In the discussion, asecond-order system with no zeros can have an overdamped, underdamped, undamped, or criticallydamped response depending on the location of the poles. Two related specifications: the damping ratio andthe natural frequency are defined in order to relate these to the type of the response of the system.

Higher-ordered systems containing dominant complex poles as well as with zeros can be approximated assecond-order system when certain conditions are met. These approximations can be validated usingcomputer aided tools. This activity shows the use of MATLAB and LabVIEW in obtaining the time responseparameters of dynamic systems.4. Resources:To perform this activity, a computer workstation with MATLAB R2012a or higher and LabVIEW 8.6 orhigher installed is required. For MATLAB, the control systems toolbox is required and for LabVIEW, thecontrol design and simulation module.5. Procedure: Activity 2.1 – Poles and Zeros 1. MATLAB. Use the command pzmap() to determine the poles and zeros, as well as plot the pole-

zero map of a system whose transfer function is defined in the objectsys . Use the following format:

>> [p z] = pzmap(sys)

and MATLAB will return the location of the poles in vectorp and the zeros in vectorz , as well as afigure will show the pole-zero map.

2. LabVIEW.Create a VI calledact02-01.vi. Build the FP and BD as shown below.



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For the Pole-Zero Map, set the upper and lower limits of the imaginary axs to +10 and -10 respectively, and the real axis from+10 to -20. Then disable the autoscaling of the plot.

3.

Manually compute for the poles and zeros of the transfer function given in the table below, then plotthem on the complexs-plane. Complete the table below.Transfer Function Poles and Zeros Pole-Zero Plot

G(s) = s +2s+ 2s + 6s + 4s +7s +2

4. Use MATLAB and LabVIEW to determine the poles and zeros and to plot them on the complexs-plane. Record the results below.

Transfer Function Poles and Zeros Pole-Zero Plot

In MATLAB:

G(s) = s +2s+ 2

s + 6s + 4s +7s +2

In LabVIEW:

G(s) = s +2s+ 2s + 6s + 4s +7s +2



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Q1.4(a)Complete the table below, using MATLAB and LabVIEW. Verify the results using handcalculations on separate sheets of paper. Q1.4(b) Using the poles and zeros obtained in the previous questions, write the general form of the

step responses of the system whose transfer functions are given below.Q1.4(c) How does the location of the poles and zeros relate to the general form of the step responseof the system? Activity 2.2 – Time Response of Dynamic Systems1. MATLAB. To obtain the step response parameters of systems represented by transfer functionsys

use the command step() and stepinfo() which plots the step response of the system, andprovides the step response parameters of the system such as the settling time, rise time, peak timeand percent overshoot. Enter the commands in the following formats

>> step(sys)>>stepinfo(sys) The command damp() gives information on the poles of the transfer function, as well as theassociated damping ratio and natural frequencies. Enter the command in the following format

>> damp(sys)

The command ltiview() can also be used to plot the time response of linear systems. On thecommand window, type in

>> help ltiview

for more information on this command. In this case, use the following format

>>ltiview({‘step’,’pzmap’},sys)

to display the step response, as well as the pole-zero map of the transfer functionsys .2. LabVIEW.Replicate act02-01.vi and name the other copy as act02-02a.vi. Add components as

shown below.



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Q2.1(a) Use the MATLAB commands and the LabVIEW VI to complete the table below. On aseparate sheet, roughly sketch the time response of each of the system.Q2.1(b) Comment on the results for the system (a) and (b). Check the values obtained using the VIwith the results of the formula. Are the values for the damping ratio and the natural frequenciesvalid? Comment also on the validity of the values for the peak time and percent overshoot.Q2.1(c) Comment on the results for the systems (c) through (f). Determine the form of the responsewith respect to the value of the damping ratio.Q2.1(d) Systems (g) through (i) have additional real poles, aside from two complex poles. Commenton the results. Which exhibits a near second-order response? What is the relationship between thedominant complex poles and the real third pole for a third-order system to exhibit an approximatesecond-order response?Q2.1(e) Systems (j) and (k) have zeros. Which of the system exhibit non-minimum phase behavior?



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Discuss the non-minimum phase behavior based on the time response plot.Q2.1(f) Systems (l) through (o) are systems with additional poles and with zeros. Which of these canbe approximated into a second-order response using pole-zero cancellation? What general rule canbe established which will allow such approximation?Q.2.1(g) Plot the responses of systems (a) through (o) on separate sheets of paper. Screenshots forsuch plots may be provided.3. Complex systems can also be modeled and simulated in MATLAB and LabVIEW. As an example the

system whose block diagram is shown below has the transfer functionsG(s) = ( )and H(s) =

.

4. MATLAB. To obtain the closed-loop equivalent the above transfer function, use the commandfeedback() , as in the following format

>> T = feedback(G,H)

whereT is the object representing the closed-loop transfer function,G is the forward transfer functionand H the feedback transfer function. Read more on the commandsfeedback() , as well as on thecommands parallel() and series() by using thehelp command of MATLAB.

Q4.1(a) Using MATLAB, determine the poles and zeros, plot and analyze the time responsecharacteristics of the system given above. Complete the table below.

5. LabVIEW. Modifyact02-02a.vi and rename this as act02-02b.vi. Add a CD Construct TransferFunction.vi block to defineH(s) and connect G(s) withH(s) using the CD Feedback.vi. Use theHelp to obtain more information about theModel Interconnection palette.

Q5.1(a) Using LabVIEW, determine the poles and zeros, plot and analyze the time responsecharacteristics of the system given above. Complete the table below. Course: Experiment No.:Group No.: Section:Group Members: Date Performed:

Date Submitted:Instructor:

6. Data and Results:Q1.4(a)Complete the table below, using MATLAB and LabVIEW. Verify the results using handcalculations on separate sheets of paper.

Transfer Function Poles and Zeros Pole-Zero Plot



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G(s) = 2s+2

G(s) = 5(s+3)(s+6)



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G(s) = s+5(s+10)

G(s) = s + 7s + 24s +24

s +10s +35s + 50s +24

G(s) = s + 2s+ 10s + 38s +515s + 2950s +6000

Q1.4(b) Using the poles and zeros obtained in the previous questions, write the general form of thestep responses of the system whose transfer functions are given below.

Transfer Function General Form of the Step Response

G(s) = s +2s +2s + 6s + 4s + 7s+ 2



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G(s) = 2s+2

G(s) = 5(s+3)(s+6)

G(s) = s+5(s+10)

G(s) = s + 7s + 24s +24s +10s +35s + 50s +24

G(s) = s + 2s+ 10s + 38s + 515s + 2950s+ 6000

Q1.4(c) How does the location of the poles and zeros relate to the general form of the step responseof the system? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q2.1(a) Use the MATLAB commands and the LabVIEW VI to complete the table below. On aseparate sheet, roughly sketch the time response of each of the system.

Transfer FunctionDamping Ratio

/ NaturalFrequency

Time Response Parameters

%



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a)

G(s) = 5s+5

b)

G(s) = 20s +20

c)

G(s) = 20

s +6s+144

d)

G(s) = 9s +9s+ 9

e)

G(s) = 100s +100

f)

G(s) = 225(s+15)

g)

G(s) = 24.542s + 4s+24.542



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h)

G(s) = 245.42(s+ 10)(s + 4s+ 24.542)

i)

G(s) = 73.626(s+ 3)(s + 4s+ 24.542)

j)

G(s) = s+2s +3s+36

k)

G(s) = s−2s +3s+36

l)

G(s) = s +3(s+2)(s +3s+10)

m)

G(s) = s +2.5(s+2)(s +4s+20)

n)

G(s) = s+2.1(s+2)(s +s+5)



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o)

G(s) = s+2.01(s+2)(s +5s+20)

p)

G(s) = s +2s +10s + 38s +515s +2950s +6000

Q2.1(b) Comment on the results for the system (a) and (b). Check the values obtained using the VIwith the results of the formula. Are the values for the damping ratio and the natural frequenciesvalid? Comment also on the validity of the values for the peak time and percent overshoot. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Q2.1(c) Comment on the results for the systems (c) through (f). Determine the form of the responsewith respect to the value of the damping ratio. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ ____________________________________________________________________________________ _

Q2.1(d) Systems (g) through (i) have additional real poles, aside from two complex poles. Commenton the results. Which exhibits a near second-order response? What is the relationship between thedominant complex poles and the real third pole for a third-order system to exhibit an approximatesecond-order response? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________



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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ ____________________________________________________________________________________ _

Q2.1(e) Systems (j) and (k) have zeros. Which of the system exhibit non-minimum phase behavior?Discuss the non-minimum phase behavior based on the time response plot. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ ____________________________________________________________________________________ _

Q2.1(f) Systems (l) through (o) are systems with additional poles and with zeros. Which of these canbe approximated into a second-order response using pole-zero cancellation? What general rule canbe established which will allow such approximation? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ ____________________________________________________________________________________ _

Q.2.1(g) Plot the responses of systems (a) through (o) on separate sheets of paper. Screenshots forsuch plots may be provided.Q4.1(a) Using MATLAB, determine the poles and zeros, plot and analyze the time responsecharacteristics of the system given above. Complete the table below.

Closed-loop Transfer Function Poles and Zeros Pole-Zero Plot



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Time Response Characteristics

%

Time Response Plot

Q5.1(a) Using LabVIEW, determine the poles and zeros, plot and analyze the time responsecharacteristics of the system given above. Complete the table below.

Closed-loop Transfer Function Poles and Zeros Pole-Zero Plot

Time Response Characteristics

%

Time Response Plot



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7. Conclusion:

8. Assessment:1. In typical conventional aircraft, longitudinal flight model linearization results in transfer functions with two

pairs of complex conjugate poles. Consequently, the natural response for these airplanes has twomodes in their natural response. The “short period” mode is relatively well-damped and has a high-frequency oscillation. The “plugoid mode” is lightly damped and its oscillation frequency is relatively low.For example, in a specific aircraft the transfer function from wing elevator deflection to nose angle (angleof attack) is (McRuer, 1973)

θ(s)δ(s) = − 26.12(s+0.0098)(s+1.371)

(s +8.99×10s+3.97×10)(s +4.21s+18.23)

a. Determine the poles and zeros of this system and plot them on the complexs-plane.b. Sketch the step response of the wing elevator deflection to a step nose angle input. Determine the

time response parameters.c. On the plot of the response, label which is the short period mode and which is the phugoid mode.

Which of the poles cause the short period and the phugoid responses, respectively?

2. Assume that the motor whose transfer function isG(s) = ( ) is used in a position control system.

a. Obtain the system’s pole-zero plot, the type of the response of the system to the step input, the plot

of the step response, and obtain the system’s damping ratio and natural frequency, and ifapplicable, the settling time, percent overshoot, rise time and peak time.

b. It is wanted that the time response of the position control system be improved. In order to do justthat, an amplifier and a tachometer are inserted into the loop, as shown in the figure below.Investigate the effects of the addition of the amplifier and the tachometer on the response of thesystem.



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c. Find the values ofK and K to yield a16% overshoot and a settling time of0.2 seconds.Determine the rest of the time response parameters.


INTENDEDLEARNINGOUTCOMES

1 2 3 Points



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Determine and,on the complex

s-plane, plotthe poles and

zeros of a

dynamicsystem.(MP1a, MP2a)

The student was notable to determine and

plot the poles and zeros

of the system.

The student was able todetermine and plot thepoles and zeros of thesystem but there are

some errors.

The student was able todetermine and plot thepoles and zeros of the

system.

Plot the timeresponse of,and interpret

the timeresponse

characteristicsof dynamic

systems

represented astransferfunctions.

(MP1b, MP1c,MP2a)

The student was notable to plot and

interpret the timeresponse

characteristics of thesystems.

The student was able toplot the time response

characteristics ofsystems but was notable to interpret the

characteristics.

The student was able toplot and interpret the

time responsecharacteristics of

systems.

Designcomponents of

dynamicsystems to

achieve timeresponse

parameterobjectives.(MP2b, MP2c)

The student was notable to design the

components of systemsto achieve objectives.

The student was able todesign some of the

component values butdoes not achieve the

design objectives.

The student was able todesign component

values of systems andachieves design

objectives.

Total Score







33


Members do not followgood and safe laboratorypractice in the conduct

of experiments.



experiments.





and instruments.














Total Score





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Experiment No. 03STABILITY AND STEADY-STATE ERROR ANALYSIS AND DESIGN OF SYSTEMS


1. demonstrate the use of computer aided tools in analyzing the stability and steady-state error oflinear systems; and

2. equip the students with the skills and knowledge in designing systems with the aid of tools toachieve transient response and steady-state error requirements of systems while ensuring stability.

2. Intended Learning Outcomes (ILOs): At the end of this activity, the students shall be able to

1. analyze the stability and steady-state error of dynamic systems described by transfer functions;2. design component values of systems to meet steady-state error requirements while ensuring

stability.

3. DiscussionStability is the most important requirement of any control system. If the system is unstable, it cannot bedesigned for transient response and steady-state error. It also poses threat to life and property, asinstability can mean a motor that has uncontrollable speed, or too much heat produced by a heater. In thediscussion, two definitions of stability are offered:

A system is stable if the natural response approaches zero as time approaches infinity. A system is stable if every bounded input yields a bounded output (the bounded-input bounded-

output or BIBO requirement).It was also discussed that stability is also related to the location of the closed-loop poles. In the discussion,the following were concluded:

A system is stable if all of its closed-loop poles are in the left-half of the complexs-plane. A system is marginally stable if it has poles of multiplicity one at thejω-axis. A system is unstable if it has at least one pole on the right-half of the complexs-plane or has

multiple poles on a single location at thejω-axis.To find how the poles are distributed on the complexs-plane, the Routh-Hurwitz criterion is being used,although tools such as MATLAB and LabVIEW can compute the exact location of closed-loop poles of ahigher-ordered system.

Steady-state error is the difference of the actual output to the desired output of the system. It can beevaluated using the closed-loop transfer function or an equivalent unity feedback system. In the discussion,the latter approach was preferred, since it also provides perspective on the static error constants whichrelates to the error of the system.

In this activity, the analysis and design of systems related to stability and steady-state error using MATLABand LabVIEW will be explored.4. Resources:To perform this activity, a computer workstation with MATLAB R2012a or higher and LabVIEW 8.6 or



35

higher installed is required. For MATLAB, the control systems toolbox is required and for LabVIEW, thecontrol design and simulation module.5. Procedure: Activity 3.1 – Stability via pole location 1. Use the Routh-Hurwitz criterion to determine the pole location distribution of the system whose

configuration is shown below. Complete the table.

SystemClosed-loop pole distribution(via Routh-Hurwitz criterion)

Left-half plane Right-half plane -axis3 2

Q1.1(a) What can be said about the stability of the system?2. MATLAB. The pole-zero map of the closed-loop transfer function can be plotted and from there, the

number of poles on the left-half, right-half and thejω-axis of the complexs-plane. The commandroots() computes the roots of a polynomial whose coefficients are written as a row matrix. If thepolynomial has the form

P(s) = as + a s +⋯+as+a then the command is entered in the following manner

>> roots([a n a n-1 ... a 1 a 0 ]) Q1.2(a) Using MATLAB, complete the table below. Sketch the pole-zero plot of the closed-looptransfer function. Indicate the number of poles, as well as the exact location of the poles undereach region of the -plane.Q1.2(b) Does the results returned by MATLAB agree with the results generated by the Routh table.Is the conclusion about the stability of system the same when the results generated by MATLABwere interpreted?3. LabVIEW. Build theact03-01.vi VI as shown below. The VI analyzes the stability of the system whose

configuration is shown in the front panel.



36

Q1.3(a) Using the VI, complete the table below. Sketch the pole-zero plot of the closed-loop transferfunction. Indicate the number of closed-loop poles, as well as the exact location of these polesunder each region of the -plane.Q1.3(b) Will you reach the same conclusions about the stability of the system when the LabVIEWvirtual instrument is used?Q1.3(c) Use MATLAB and LabVIEW to complete the table below. Indicate the number of closed-loop

poles, as well as their exact location under each region of the complex -plane. Under “Remarks”,tell whether the system is stable, unstable or marginally stable. On separate sheets of paper, sketchthe closed-loop pole-zero map of each of the systems. Verify the results using Routh table. Activity 3.2 – Analysis of steady-state error. 1. The steady-state error will be evaluated using the configuration below. Refer to the lecture on the

formulas to be used in evaluating the static error constants and the error for step, ramp and parabolictest inputs.

Remember that the system must be tested first for stability before analyzing it for transient response orsteady-state error. Thus, the techniques learned in Activity 3.1 can be applied first before proceeding.

2. MATLAB. To use MATLAB, the object representingG(s) must be converted first to a symbolic object.If G contains the transfer function object, use the following commands to convertG into a symbolicobjectGsym .



37

>> [num den] = tfdata(G);>>syms s>>Gsym = poly2sym(cell2mat(num),s)/poly2sym(cell2mat(den),s)

Gsym is now a symbolic math object. The functionlimit() can now be used to evaluate the static

error constants, which will be then used to evaluate the error for various test inputs. As an example, ifGsym is the symbolic object representing the open-loop transfer function of the unity feedback systemas shown in step one of this sub-activity, then the static error constantK and the error due to the stepinpute (∞) are evaluated as>>Kp = limit(Gsym,0)>>estep = 1/(1+Kp)

Q2.1(a) What does the following functions in MATLAB do: tfdata() , syms , poly2sym() ,cell2mat() . Discuss the syntax and the required arguments of each function.

Q2.2(b) Use MATLAB to evaluate the static error constants and steady-state errors of the systemsshown below. Complete the table. Verify the values obtained using manual calculations. For the lastsystem, assume that the input and output are the same quantity.3. LabVIEW.Build theact03-02.vi as shown below. TheArray of Polynomial Coefficients to Formula

String.vi can be obtained from your instructor or from this link:https://decibel.ni.com/content/docs/DOC-22590 if you have an available internet connection.

https://decibel.ni.com/content/docs/DOC-22590



38

Q2.3(a) Use the VI above to evaluate the static error constants and steady-state errors of thesystems shown below. Complete the table. For the last system, assume that the input and outputare the same quantity.Course: Experiment No.:Group No.: Section:

Group Members: Date Performed:Date Submitted:Instructor:

6. Data and Results:Q1.1(a) What can be said about the stability of the system? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Q1.2(a) Using MATLAB, complete the table below. Sketch the pole-zero plot of the closed-looptransfer function. Indicate the number of poles, as well as the exact location of the poles undereach region of the -plane.

SystemClosed-loop pole distribution

(via MATLAB)Left-half plane Right-half plane -axis



39

Pole-zero Map

Q1.2(b) Does the results returned by MATLAB agree with the results generated by the Routh table.Is the conclusion about the stability of system the same when the results generated by MATLABwere interpreted? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Q1.3(a) Using the VI, complete the table below. Sketch the pole-zero plot of the closed-loop transferfunction. Indicate the number of closed-loop poles, as well as the exact location of these polesunder each region of the -plane.

SystemClosed-loop pole distribution

(via LabVIEW)Left-half plane Right-half plane -axis

Pole-zero Map



40

Q1.3(b) Will you reach the same conclusions about the stability of the system when the LabVIEWvirtual instrument is used? _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

_____________________________________________________________________________________Q1.3(c) Use MATLAB and LabVIEW to complete the table below. Indicate the number of closed-looppoles, as well as their exact location under each region of the complex -plane. Under “Remarks”,tell whether the system is stable, unstable or marginally stable. On separate sheets of paper, sketchthe closed-loop pole-zero map of each of the systems. Verify the results using Routh table.

SystemClosed-loop pole distribution and location

LHP RHP -axis

T(s) = 34s +10s +35s + 50s +34

with



41

Q2.1(a) What does the following functions in MATLAB do: tfdata() , syms , poly2sym() ,cell2mat() . Discuss the syntax and the required arguments of each function. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q2.2(b) Use MATLAB to evaluate the static error constants and steady-state errors of the systemsshown below. Complete the table. Verify the values obtained using manual calculations. For the lastsystem, assume that the input and output are the same quantity.

System (check if stable)Static error constants Steady-state errors

(∞) (∞) (∞)

Q2.3(a) Use the VI above to evaluate the static error constants and steady-state errors of thesystems shown below. Complete the table. For the last system, assume that the input and outputare the same quantity.



42

System (check if stable)Static error constants Steady-state errors

(∞) (∞) (∞)

7. Conclusion:

8. Assessment:

1.

For the system shown below, do the following:(a) AtK = 10, is the system stable? Find the system type, the static error constant and the error ofthe system at this gain, then plot and determine the time response parameters if possible.

(b) Repeat part (a) atK = 10.(c) Plot the value of the static error constant and the steady-state error as a function of the gainK for

the range at which the system is stable. What conclusions can be drawn from the plot?



43

2. The open-loop transfer function of a swivel controller and plant for an industrial robot is given as

G(s) = ω(s)V(s) =

K(s+10)(s +4s +10)

whereω(s) is the Laplace transform of the robot’s angular swivel velocity andV(s) is the inputvoltage to the controller. AssumeG(s) is the forward transfer function of a velocity control loop withan input transducer and a sensor, each represented by a constant gain of3 (Schneider, 1992), do thefollowing:(a) Plot the value of the error of the system as a function of the gainK at the range ofK for which the

system is stable.

(b) Design the value of the gainK to minimize the steady-state error between the input commandedangular swivel velocity and the output actual angular swivel velocity. Show that the system is stillstable at the design point.

(c) For the chosen value of the gain at part (b), determine the system type, steady-state error and thetransient response of the system.


INTENDEDLEARNINGOUTCOMES

1 2 3 Points

Analyze the stabilityand steady-state errorof dynamic systems

described by transferfunctions.

(MP 1)

The student was not

able to analyze thestability and steady-state error of the

system.

The student was ableto analyze the stabilityof the system but notthe steady-state error,or was able to obtainthe steady-state errorbut did not check for

stability.

The student was able

to analyze the stabilityand steady-state errorof the system

properly.



44

Design componentvalues of systems tomeet steady-stateerror requirements

while ensuringstability.(MP 2)

The student was notable to design the

component values ofthe systems.

The student was ableto design componentvalues but does not

fully meet the steady-state error

requirements or thatthe design was not

verified.

The student was ableto correctly design

component values ofthe system that meetsstability and steady-

state errorrequirements and thedesign is correctly

verified.

Total Score






Members do not followgood and safe laboratory

practice in the conductof experiments.



experiments.





and instruments.







45










Total Score





46

Experiment No. 04ROOT LOCUS ANALYSIS OF SYSTEMS

1. Objective(s):This activity aims to equip the students with the skills and knowledge in analyzing control systems using the

root locus approach.2. Intended Learning Outcomes (ILOs): At the end of this activity, the students shall be able to obtain transient response, steady-state error andstability information on feedback control systems using the root locus approach.3. DiscussionRoot locus is the graphical representation of the paths of the closed-loop poles as a parameter of thesystem is varied. Commonly, this parameter is the forward gain of the system. A generalized root locusapproach was also presented in the discussion with which an open-loop pole was being varied and thelocus of the closed-loop pole was tracked.

The root locus gives the control engineer to analyze and design higher-ordered system in graphicalapproach. The use of computer-aided tools such as MATLAB and LabVIEW will also simplify the analysisand design process, since these tools will take away the laborious mathematics and have the designerfocus more on interpreting the results.4. Resources:To perform this activity, a computer workstation with MATLAB R2012a or higher and LabVIEW 8.6 orhigher installed is required. For MATLAB, the control systems toolbox is required and for LabVIEW, thecontrol design and simulation module.5. Procedure:Note: The following steps can be accomplished either inMATLAB or LabVIEW. In LabVIEW, the

MathScript tool can be accessed via the Welcome window on theTools menu, then choose MathScriptWindow. The Command Window of this tool works in the same manner as that of theMATLABcommandwindow.

1. The unity feedback system shown below will be analyzed using root locus techniques. Note that theroot locus is the plot of the open-loop transfer functionKG(s)H(s) as K or the gain is varied.

In this case, G(s) = ( )( )( )( )and H(s) = 1. Define these transfer function in MATLAB or

LabVIEW using the commands



47

>> Gnum = poly([3 5]); Gden = poly([-1 -2]);Hnum = [1];Hden = [1]; >> G = tf(Gnum,Gden); H = tf(Hnum,Hden);

Note that the variableK is not included in the transfer functionG(s). This is because the root locuscommand of MATLAB and LabVIEW automatically assigns

K as being multiplied to

G(s).

2. The rlocus() command plotsKG(s)H(s) as the function ofK. To get more information about thecommand, type inhelp rlocus on the command window. In this case, use the command>>rlocus(G*H,0:0.01:1000);

which plots the root locus of the open-loop transfer functionKG(s)H(s) from0 < K < 1000 withincrements of0.01 per point.

Q1.2(a) Sketch the root locus as shown in the plot generated by MATLAB or LabVIEW. Indicate thelocation of the open-loop poles and zeros as applicable. Q1.2(b) Based on the root locus, will the system break into oscillation at some gain ? Will it also

be unstable? Label on the sketch above these regions.3. The root locus sketch contains important points and information. Use the commandrlocfind() to

locate important points in the root locus. (Tip: you might want to zoom into the point of interest firstbefore using therlocfind() command for better accuracy.) The format of the command is >> [K p] = rlocfind(G*H)

whereG and H are the objects that represent the forward and the feedback transfer functionsrespectively and K will contain the value of the gain at the selected point andp the location of theclosed-loop poles at that gainK.Note: The commandrlocfind() produces an interactive root locus graph for both MATLAB andLabVIEW. In MATLAB, you will be allowed to choose a point on the root locus, and the value of thegain and the closed-loop poles are returned by the command. In LabVIEW, you are allowed to select aparticular gain or drag the closed-loop poles (represented by red x’s) into your desired location. Still thecommand returns the gain at the chosen point and the closed-loop poles at that particular gain.

Q1.3(a) Using the rlocfind() command, find the following points as indicated in the tablebelow. Verify the values using manual calculations.Q1.3(b) What is the range of gain so that the system is stable? Discuss how this can be obtainedin the root locus. Q1.3(c) What are the rules for sketching the root locus that can be observed from the sketch?

Discuss each and how these rules are manifested in the sketch.4. Sketch the root locus of the system shown below. This system is to be analyzed for transient responseand steady-state error. The system is to operate with a1.52% overshoot.



48

Q1.4(a) What is the damping ratio that corresponds to the operating overshoot of the systemabove? This will be your

MATLAB.Superimpose the d line that corresponds to the operating overshoot of the system usingthe command sgrid(dr,wn) where dr is the one computed above, andwn equals zero, so thatthe

ω circle will be suppressed.

LabVIEW.With therlocfind() called, the complexs-plane grid is automatically displayed. Youmight have to zoom and drag repeatedly to find thed line.Once the d line is drawn, locate the intersection of the line to the root locus.

Q1.4(b) The root locus and the line will intersect at three points. Find each of those points, thendetermine the gain and the closed-loop and open-loop poles for each case. Fill up the table below.Q1.4(c) In each cases, explain which case has a valid second-order approximation. 5. With the gain in each case known, the closed-loop transfer function can now be computed. From here,

the step response parameters and the steady-state error can now be obtained.

Q1.5(a) Use MATLAB or LabVIEW to complete the table below. The programs and techniquesdeployed in the previous activities can be used to complete the table.Q1.5(b) Use the control design and simulation module of LabVIEW to simulate each of the casesabove and plot their step responses on separate sheets of paper.Course: Experiment

No.:Group No.: Section:Group Members: Date

Performed:Date

Submitted:Instructor:

6. Data and Results:Q1.2(a) Sketch the root locus as shown in the plot generated by MATLAB or LabVIEW. Indicate thelocation of the open-loop poles and zeros as applicable.



49

Q1.2(b) Based on the root locus, will the system break into oscillation at some gain ? Will it alsobe unstable? Label on the sketch above these regions. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.3(a) Using the rlocfind() command, find the following points as indicated in the table

below. Verify the values using manual calculations.

Point of interestGain at the point of

interest

Closed-loop polesat the point of

interestBreakaway point from

the real axisBreak-in point into the

real axisThe point at which thesystem is oscillating

Q1.3(b) What is the range of gain so that the system is stable? Discuss how this can be obtainedin the root locus. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________



50

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Q1.3(c) What are the rules for sketching the root locus that can be observed from the sketch?Discuss each and how these rules are manifested in the sketch. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q1.4(a) What is the damping ratio that corresponds to the operating overshoot of the systemabove? This will be your

Q1.4(b) The root locus and the line will intersect at three points. Find each of those points, thendetermine the gain and the closed-loop and open-loop poles for each case. Fill up the table below.

GainDominant

complex closed-

loop poles

Third-orderclosed-loop pole

Closed-loop zero

Case 1Case 2Case 3

Q1.4(c) In each cases, explain which case has a valid second-order approximation. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q1.5(a) Use MATLAB or LabVIEW to complete the table below. The programs and techniquesdeployed in the previous activities can be used to complete the table.

Case 1 Case 2 Case 3Dominant complex closed-loop



51

polesThird-order closed-loop pole

Closed-loop zeroSecond-order approximation

ok?Gain

Percent overshootSettling time

Peak timeRise time

Static error constantError

Q1.5(b) Use the control design and simulation module of LabVIEW to simulate each of the casesabove and plot their step responses on separate sheets of paper.

7. Conclusion:

8. Assessment:1. The block diagram of a six-degree of freedom industrial robot’s swing motion system is shown below. If

K = 64,510, use the root locus to predict the time response and steady-state error parameters of thissystem (Hardy, 1967).



52

2. It is important to precisely control the amount of organic fertilizer applied to a specific crop area in orderto provide specific nutrient quantities and to avoid unnecessary environmental pollution. A precisedelivery liquid manure machine has been developed for this purpose (Saeys, 2008). The systemconsists of a pressurized tank, a valve and a rheo-logical flow sensor. After simplification, the systemcan be modeled as a closed-loop negative feedback system with a forward-path transfer function

G(s) = 2057.38K(s −120s +4800)s(s+13.17)(s +120s +4800)

consisting of an electrohydraulic system in cascade with the gain of the manue flow valve and avariable gainK. The feedback path is comprised of

H(s) = 10(s −4s +5.333)(s+10)(s +4s+5.333)

(a) Sketch the root locus of the system.(b) Find the range of the gainK for closed-loop stability.(c) Find the value ofK that will result in the smallest settling time for this system.(d) For the value of the gainK found in part (c), determine the time response and steady-state error

specifications of the system.



53


INTENDEDLEARNING


Obtain transientresponse,

steady-stateerror andstability

information onfeedback controlsystems usingthe root locus

approach. (MP 1)

The student wasnot able to sketchthe root locus of

the system

The student wasable to sketch the

root locus butcannot obtain anyparameters from

it.

The student wasable to obtain

some parametersof the system fromits root locus butsome of them areincorrect and thestudent was notable to interpret

them.

The student wasable to use theroot locus of thesystem to obtain

its transientresponse, steady-

state error andstability

information.

Obtain transientresponse,steady-state

error andstability

information onfeedback controlsystems usingthe root locus

approach. (MP 2)

The student wasnot able to sketchthe root locus of

the system

The student wasable to sketch the

root locus butcannot obtain anyparameters from

it.

The student wasable to obtainsome parametersof the system fromits root locus butsome of them areincorrect and thestudent was notable to interpret

them.

The student wasable to use theroot locus of thesystem to obtain

its transientresponse, steady-

state error andstability

information.

Total Score





54




Members do not follow

good and safe laboratorypractice in the conduct

of experiments.



experiments.





and instruments.






theoretical values to

determine possibleexperimental errors, andprovide valid conclusions.

The group has

incomplete data.




experimental values

against theoreticalvalues, and providesvalid conclusion.

Total Score





55

Experiment No. 05DESIGN OF CASCADE COMPENSATORS USING ROOT LOCUS TECHNIQUES: PID CONTROL


1. demonstrate the operation of proportional-integral (PI), proportional-derivative (PD) andproportional-integral-derivative (PID) control.

2. equip the students with the skills and knowledge in using root locus techniques to design acascade compensator to improve the transient and steady-state response of a system.

2. Intended Learning Outcomes (ILOs): At the end of this activity, the students shall be able to design PI, PD and PID cascade compensators, asapplicable, to improve the transient response and steady-state error performance of feedback controlsystems.3. DiscussionThe transient response and steady-state error performance of a given control system can be improved bycompensating the system, either in cascade or in feedback. This activity focuses on the cascadecompensation of the system.

In the discussion, a constant gain cascaded to the plant provides faster transient response and improvedsteady-state error performance. This is called proportional control or proportional compensator. This isbecause the constant gain amplifies the error or actuating signal which drives the plant faster, making itsoutput follow the input faster. However, further improvements in the transient response and steady-stateerror can be achieved when differentiators and integrators are used.

To drive the error to zero, a pure integral term can be multiplied to the plant, which increases its system

type. Such can be accomplished via a cascade compensator which is a parallel combination of a pure gainand an integrator. This compensator is called a proportional-integral (PI) controller. It drives the steady-state error to zero but does not generally improve the transient response of the system.

Improvement in the transient response of the system can be achieved by cascading a parallel connectedpure gain and a differentiator. The differentiator causes the error between the input and the output, whosedifference is largest at the start, to be differentiated. The process of differentiation produces the slope of thefunction, which is very large when the error is also large. Thus the output of the differentiator is large whenthe error is large, which, in addition to the proportional gain, drives the plant faster than when there is only apure gain in the loop. This method of control used to improve the transient response is called a

proportional-derivative (PD) control. This method of control does not generally, although in most cases itdoes, improve the steady-state error performance of the system.

To improve the steady-state error and the transient response performance of the feedback system, bothmethods are used simultaneously. This leads to a method of control called the proportional-integral-derivative (PID) control.

The use of PI, PD and PID controller changes the root locus of the system. This allows more flexibility in



56

choosing the operating points of the system compared to the use of a pure proportional gain as controller,which limits the operating point within the root locus of the system.

This activity first demonstrates the effect of proportional, PI, PD and PID controllers in a feedback systemthen proceed with the computer aided design of cascade compensators.

4. Resources:To perform this activity, a computer workstation with MATLAB R2012a or higher and LabVIEW 8.6 orhigher installed is required. For MATLAB, the control systems toolbox is required and for LabVIEW, thecontrol design and simulation module.5. Procedure: Activity 5.1 – Proportional, PI, PD and PID controllers. 1. LabVIEW. Build the front panel and the block diagram as follows. Name this VI asact05-01.vi.

Set the Configure Simulation Parameters as shown.



57

Set the Pulse Signal Configuration parameters to start time to 0, amplitude to 2, offset to -1,duty cycle to 50 and period to 200.

Set the default values ofKp, Ki and Kd as 1, 0, and 0 respectively. Configure theWaveform Chart so that you can clearly see the input, output response and the

error plots.MATLAB. Setup the system as shown in the VI above. In MATLAB, you will check the stability andtransient response information of the system first before implementing it in LabVIEW.

2. The proportional control will be explored in this step. Run theact05-01.vi. Set the Switch button toTrue. Determine the error and the step response parameters of the system whenKp = 1 and Ki and

Kd are zero. Record the results below.Kp = 1, Ki = 0, Kd = 0

Dominant complex closed-looppoles

Higher-ordered closed-looppole/s

Closed-loop zero/sSecond-order approximation

ok?

Percent overshootSettling timePeak timeRise time




58

Q1.2(a) Adjust the proportional gain while maintaining Ki and Kd to zero. What happens to thetransient response and steady-state error of the system? Make sure that the system is still stablewhen you record your observations. Q1.2(b) Using the root locus of the system, determine the value of the gain when the systemoperates with a damping ratio of

= . for its dominant complex conjugate poles. Also

determine the rest of the time response and steady-state error performance. Sketch the stepresponse below.Q1.2(c) Discuss the effects of proportional gain to the transient response and steady-state errorperformance of the system.3. The proportional-integral (PI) control can be accomplished by adding a pole at the origin and a zero

nearby. The transfer function of the PI controller is

G(s) = Kp+Kis = Kps+

s

Q1.3(a) If the transfer function of the PI controller to be used in the system is ( ) = .,

determine the value of and so that the system has a damping ratio of = . .Complete the table below and sketch the step response of the system.Q1.3(b) What was the effect of PI control to the steady-state error of the system? How was itpossible? Q1.3(c) What was the effect of PI control to the transient response of the system? Are theseobservations generally true?Q1.3(d) Place the Switch button to False to control the input manually. Turn the Input Position knoband observe the Output Position knob, as well as the plot of the response. Take note of the time

response parameters such as overshoot and settling time, as well as the error. Comment on yourobservations. Place the Switch button back to True after drawing your observations.4. Set the Ki to zero. The proportional-derivative (PD) controller has the transfer function

G(s) = Kds+Kp = Kd s+KpKd

The PD control will be applied to the unity feedback system whose forward transfer function isG(s) =( )( )( ). Change the parameters in the VI accordingly.

Q1.4(a) Determine the required so that the system will have a = .. At this , determinethe transient response and steady-state error performance of the system. Also sketch the step

response of the system.5. Compute the requiredKd’s and Kp’s when a compensator zero is to be placed at−2, −3 and −4.

Then adjust the value ofKd accordingly.Q1.5(a) Complete the table below, which corresponds to the performance of the PD compensatedsystem. Q1.5(b) Discuss the effects of PD control to the system to the transient response and steady-stateerror performance of the system based on the data gathered above.



59

6. The PID controller has the transfer function

G (s) = Kp+Kis +Kds = Kd s + s+

s

which puts two zeros and a pole at the origin. In this step, the effect of PID control on the unity

feedback system whose forward transfer function isG(s) = ( )( )( ) is to be explored.Q1.6(a) Determine the value of the proportional gain so that the system above has an overshoot of

%. From here, determine the step response and steady-state error performance of the systemwith just the proportional gain.7. A PID compensator is to be cascaded to the plant having the transfer functionG (s) =

( .)( .). Determine the values ofKp, Ki and Kd to implement such controller.

Q1.7(a) Adjust the values of , and accordingly. Evaluate the PID compensated systemperformance and complete the table below.Q1.7(b) Observe the effects of the PID compensator to the feedback system. Is there an

improvement in the transient response and steady-state error performance of the system with onlythe proportional gain? Discuss. Activity 5.2 – Design of a PID compensator 1. A PID controller is to be designed for the unity feedback system shown below. The desired operating

point of the compensated system should be two-thirds of the peak time of the uncompensated systemwith20% overshoot and zero steady-state error for a step input.

MATLAB.Use the variableG to define the forward transfer function.

Then open the SISO Design GUI of MATLAB by typing in>>sisotool

in the command window. In theControl and Estimation Tool Manager , set the Control Architectureto the default as shown,



60

In the System Data menu, select the system G to be imported from the workspace (you should havedefined G as a transfer function prior to this step), by clicking theBrowse button and choosing theavailable transfer function model forG. Set the rest of the parameters to unity.In theGraphical Tuning tab, set the options to the ones shown below.

then click theShow Design Plot button. This will now show the root locus of the system. To determinethe required gain so that the system will exhibit an overshoot of20%, right-click on the root locus thenchoose Design Requirements >> New. Choose the Design requirement type as Percent overshoot and type in 20 under the Design requirement parameters . Click OK. The 20% overshoot lineappears on the root locus. Drag the closed-loop poles towards the intersection of the line and the rootlocus (you may need to adjust the limits of the axis to see the intersection – right-click on the graphthen choose Properties , then adjust the limits on theLimits tab; you may also want to zoom into theintersection so that you can pinpoint it more accurately – just use theZoom In and Zoom Out tools onthe tool bar).Store the design by clicking onStore Design button. Export the design to the workspace by clicking onFile >> Export. Choose all the variables, choose the correct design to be exported in theDesignpulldown then clickExport to Workspace . Just clickOK when a prompt to overwrite a modelappears. Notice that there are now objectsC, F , and H in the workspace.

Q2.1(a) Determine the required proportional gain so that the system operates with % overshoot.Determine the rest of the parameters and record the results below.2. A PD controller is to be designed to meet transient response specifications. The design must include

the zero location and the loop gain. The compensator zero is to be added so that the PD compensated

root locus intersects the required operating point. Q2.2(a) What will be the new complex dominant poles of the PD compensated system to meet thetransient response requirements?Q2.2(b) Locate the compensator zero so that the root locus of the compensated system passesthrough this new operating point.3. Add this compensator zero to the root locus by right-clicking on the root locus, then choose Edit

Compensator. On the Compensator Editor window, right-click on the Dynamics pane and choose Add



61

Pole/Zero >> Real Zero. Choose the location of the zero as specified above. Take note of the changein the root locus.

Q2.3(a) What is the transfer function of the PD compensator?Q2.3(b) Simulate the compensated system. Determine the time response parameters of thecompensated system. Were the objectives of the design met?4. After designing the PD compensator and meeting the transient response requirements, the PI

compensator is to be designed. Add a pole at the origin and a zero near the origin so that the steady-state error will become zero because of an increase in system type.

Q2.4(a) Write down the transfer function of the PI controller you will deploy for the given system.Q2.4(b) Simulate the PID compensated system and complete the table below. Were therequirements met?Q2.4(c) What is the transfer function of the PID compensator used to achieve the requirements set?From there, determine the value of the proportional, integral and derivative gains of the controller.Q2.5(c) Plot the step responses of the uncompensated, PD compensated and PID compensated

systems on a separate sheet of paper. Put the plots on a single graph and then compare.Course: Experiment No.:Group No.: Section:Group Members: Date Performed:

Date Submitted:Instructor:

6. Data and Results:Q1.2(a) Adjust the proportional gain while maintaining Ki and Kd to zero. What happens to thetransient response and steady-state error of the system? Make sure that the system is still stablewhen you record your observations. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.2(b) Using the root locus of the system, determine the value of the gain when the system

operates with a damping ratio of = . for its dominant complex conjugate poles. Alsodetermine the rest of the time response and steady-state error performance. Sketch the stepresponse below.

= . , Ki = 0, Kd = 0Dominant complex closed-loop

polesHigher-ordered closed-loop



62

pole/sClosed-loop zero/s

Second-order approximationok?

Proportional Gain Percent overshoot

Settling timePeak timeRise time


Step response plot

Q1.2(c) Discuss the effects of proportional gain to the transient response and steady-state errorperformance of the system. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Q1.3(a) If the transfer function of the PI controller to be used in the system is ( ) = .,

determine the value of and so that the system has a damping ratio of = . .Complete the table below and sketch the step response of the system.

= . , Kd = 0Dominant complex closed-loop




63



Proportional Gain Integral Gain


Peak timeRise time


Step response plot

Q1.3(b) What was the effect of PI control to the steady-state error of the system? How was itpossible?

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.3(c) What was the effect of PI control to the transient response of the system? Are theseobservations generally true?

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________



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_____________________________________________________________________________________Q1.3(d) Place the Switch button to False to control the input manually. Turn the Input Position knoband observe the Output Position knob, as well as the plot of the response. Take note of the timeresponse parameters such as overshoot and settling time, as well as the error. Comment on yourobservations. Place the Switch button back to True after drawing your observations. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Q1.4(a) Determine the required so that the system will have a = .. At this , determine

the transient response and steady-state error performance of the system. Also sketch the stepresponse of the system.

= ., Ki = 0, Kd = 0Dominant complex closed-loop







Step response plot



65

Q1.5(a) Complete the table below, which corresponds to the performance of the PD compensatedsystem.

= ., Ki = 0,

= − = ., Ki = 0,

= − = ., Ki = 0,

= − Dominant complex closed-

loop polesHigher-ordered closed-

loop pole/sClosed-loop zero/s

Second-orderapproximation ok?

Proportional Gain Derivative gain Percent overshoot



Q1.5(b) Discuss the effects of PD control to the system to the transient response and steady-stateerror performance of the system based on the data gathered above. _____________________________________________________________________________________ _____________________________________________________________________________________

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q1.6(a) Determine the value of the proportional gain so that the system above has an overshoot of

%. From here, determine the step response and steady-state error performance of the systemwith just the proportional gain.

% = , Ki = 0, Kd = 0




ok?



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Step response plot

Q1.7(a) Adjust the values of , and accordingly. Evaluate the PID compensated systemperformance and complete the table below.

% = Dominant complex closed-loop




Proportional gain Integral gain

Derivative gain Percent overshoot

Settling time

Peak timeRise time




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Step response plot

Q1.7(b) Observe the effects of the PID compensator to the feedback system. Is there animprovement in the transient response and steady-state error performance of the system with onlythe proportional gain? Discuss. _____________________________________________________________________________________

_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________Q2.1(a) Determine the required proportional gain so that the system operates with % overshoot.Determine the rest of the parameters and record the results below.

Uncompensated System




ok?Gain

Percent overshoot



Q2.2(a) What will be the new complex dominant poles of the PD compensated system to meet thetransient response requirements?



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Q2.2(b) Locate the compensator zero so that the root locus of the compensated system passesthrough this new operating point.

Q2.3(a) What is the transfer function of the PD compensator?

Q2.3(b) Simulate the compensated system. Determine the time response parameters of thecompensated system. Were the objectives of the design met? ________________________________

Uncompensated System PD Compensated SystemDominant complex closed-loop



Second-order approximationok?Gain


Peak time

Rise timeStatic error constant

ErrorQ2.4(a) Write down the transfer function of the PI controller you will deploy for the given system.

Q2.4(b) Simulate the PID compensated system and complete the table below. Were therequirements met? ____________________________________________________________________

UncompensatedSystem

PD CompensatedSystem

PID CompensatedSystem

Dominant complexclosed-loop poles

Higher-orderedclosed-loop pole/sClosed-loop zero/s

Second-order



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approximation ok?Gain


Peak timeRise time

Static errorconstant

ErrorQ2.4(c) What is the transfer function of the PID compensator used to achieve the requirements set?From there, determine the value of the proportional, integral and derivative gains of the controller.

PID Controller TransferFunctionProportional Gain Kp

Integral Gain KiDerivative Gain Kd

Q2.5(c) Plot the step responses of the uncompensated, PD compensated and PID compensatedsystems on a separate sheet of paper. Put the plots on a single graph and then compare. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________7. Conclusion:

8. Assessment:1. The block diagram below shows the droop control of an ac/dc conversion and power distribution

system to stabilize the dc-bus voltage. Here,G(s) is the transfer function of the controller,G(s) is



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the transfer function of the controlled plant, which is a conversion and power distribution unit andH(s) is the feedback low-pass filter. Evaluate the performance of the uncompensated system when thecontroller is a simple gain block, i.e.G(s) = K, at 4.4% overshoot. Then design a cascadecompensator so that the compensated system will operate with a percent overshoot of at most4.4%, apeak time

20% smaller than that of the uncompensated system and zero steady-state error.

Summarize the transient response and steady-state error performance of the uncompensated and thecompensated systems, as well as produce a plot of the time response of both systems. (Nise, 2008)

2. The transfer function for an AFTI/F-16 aircraft relating angle of attack,α(t), to elevator deflection,

δ(t), is given by (Monahemi, 1992)

G(s) = α(s)δ(s) = (s+23)(s +0.05s +0.04)

(s−0.7)(s+1.7)(s +0.08s+0.04)

Assume the block diagram shown above for controlling the angle of attackα, design a cascadecompensator to yield zero steady-state error, a settling time of about0.05 sec and a percentovershoot not greater than20%.



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INTENDEDLEARNING


Design PI, PDand PID cascadecompensators,

as applicable, toimprove the

transientresponse andsteady-state

errorperformance of

feedback control

systems. (MP 1)

The student wasnot able to design

anything.

The student wasable to obtainsome of the

parameters of thePD or PI but

cannot design thePID compensator

as a whole.

The student wasable to design a

PID compensatorbut was not able

to meet the designobjectives


PID compensatorwhich meets the

requiredimprovements in

the transientresponse and

steady-state errorperformance of

the system.

Design PI, PDand PID cascadecompensators,

as applicable, toimprove the

transientresponse andsteady-state

errorperformance of

feedback controlsystems. (MP 2)

The student wasnot able to design

anything.

The student wasable to obtainsome of the

parameters of thePD or PI but

cannot design thePID compensator

as a whole.


PID compensatorbut was not able

to meet the designobjectives


PID compensatorwhich meets the

requiredimprovements in

the transientresponse and

steady-state error

performance ofthe system.

Total Score








Members do not follow

good and safe laboratorypractice in the conduct

of experiments.



experiments.





and instruments.






theoretical values to

determine possibleexperimental errors, andprovide valid conclusions.

The group has

incomplete data.




experimental values

against theoreticalvalues, and providesvalid conclusion.

Total Score



Feedback and Control Systems Lab Manual-

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