PM diagram of the Transfer Function and its use in the Design of Controllers

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "PM diagram of the Transfer Function and its use in the Design of Controllers"

Transcription

1 PM diagram of the Transfer Function and its use in the Design of Controllers Santiago Garrido, Luis Moreno Abstract This paper presents the graphical chromatic representation of the phase and the magnitude of the transfer function G(s) of a system and its educational use in the design of the most typical controllers. The magnitude is represented in decibels and the phase is represented by colours. This representation permits to put a face to the transfer functions, deepening our intuitive understanding of transfer functions. An important characteristic of this diagram is that permits to read the phase and the gain margins directly and the imaginary axis cut represents the Bode diagram. It permits to see intuitively the connexions among the different diagrams. It is also possible to put the grid with damping ratio ζ and frequency ω n. In summary, this PM diagram can be useful, especially in Control Education because improves the intuition about transfer functions, and it can be used in a Classical Control Course to complement the design of controllers using the Root Locus diagram. Index Terms Root Locus, PID controller, Lead network, Bode plot, Phase margin, Gain margin. I. INTRODUCTION A picture is worth a thousand words. There are many possible representations of the transfer function G(s) of a system such as the Root Locus, the Bode diagram, the Nyquist diagram and the Nichols diagram. In all of them, it is possible to see some characteristics of the system and to create different kinds of controllers to improve the behaviour of the system. Nowadays, with tools such as SISOTOOL of Matlab, it is possible to see many of these diagrams at the same time and the time response of the system to tune the controller. The graphical representation of functions is one of the most important mathematical tools because it allows us to understand the behaviour of the functions. While it is easy to represent the graph of a real function in the plane, the graph of a single variable complex function is more problematic. The reason adduced about why we can not represent the Transfer Function is because G(s) : C C and as C is equivalent to represent two real variables, it is necessary to have 4 axis to represent the Transfer Function G(s). Our brain is trained to visualize objects in three spatial dimensions, while the graphs of complex functions live in a four-dimensional space. Hence most of us are unable to imagine such an object. Complex functions have the reputation of being mysterious entities; seeing these strange objects may help to overcome the fear one might feel while dealing with them. Santiago Garrido and Luis Moreno are with the Robotics Lab., Carlos III University, Madrid, Spain. The Phase Magnitude (PM) diagram makes it possible to put a face to the functions, deepening our intuitive understanding of basic and advanced concepts in complex analysis. They reveal intrinsic structures behind the formulas, literally open our eyes to the wonderful realm of complex functions, and may serve students, teachers, scientists, and engineers as simple and efficient tools in their work. A possibility is to represent the magnitude and the phase of the function in each point with level curves. Cavicchi[6],[7] tried this solution in 996 and 23, but the problem was the bad resolution and the difficulty of reading his diagrams. But the situation has changed with the modern mathematical program computers that permit us to represent the fourth spatial dimension using the colour-coded values of the phase on the domain of the function. This is the solution adopted in this paper. Another problem with the representation of transfer functions is that they are rational and they have poles. In these points the function goes to infinity and far from the poles and zeros, the function is quite plane. In these conditions, the level curves are too close near the poles and zeros, and there are not level curves in the almost plane zones. A possible solution to solve this problem is the technique adopted in the Complex Variable books [3], [4], [5]: to represent the arctan of the function G(s), but in this case the magnitude level curves are not equally spaced and the diagram can t be used numerically. The best solution for Control applications is the solution adopted by Bode in its diagram: to use the vertical scale in decibels y = 2 log (x), because in this way the zones near the poles have less level curves and the level curves are equally spaced: the space between two consecutive magnitude level curves is the same in decibels. The solution adopted in this paper is inspired in [3], but represents the complex function by its module in decibels: 2 log G(s) and its phase in rescaled hsv coded colours. In the figures of the examples of controllers each phase band represents and each magnitude band 2 db. In this case the module level curves are equally spaced, and it is possible to read in the representation almost directly the gain and the phase that has to add the controller and use the values directly in the solution of problems and exercises. Different extensions of Root Locus and related diagrams have been used by many researchers. Ogata[] treats the Constant Gain Locus diagram and the Contour Roots in his book. Also, Kuo[2] refers to the Contour Root Locus. Cavicchi[6], [7] uses the Phase-Root diagram that represent the level curves of gain and phase. More recently, Cerone[8] proposes the Constant Magnitude Loci for Control Education.

2 (a) A single zero (b) A double zero (c) A triple zero Fig. : PM diagram of a zero, a double zero and a triple zero In Fig. are shown the PM diagram of a single zero, G(s) = s, a double zero, G(s) = s 2, and a triple zero G(s) = s 3. As it can be seen the order of the colours is the same as that the original complex plane C. In the case of a double zero, the different colours appear twice and in the case of a triple zero, the different colours appear three times. In Fig. 2 are shown the PM diagram of a single pole, G(s) = /s, a double pole, G(s) = /s 2, and a triple pole G(s) = /s 3. The different colours appear once, twice and three times but in the opposite orientation. In all the PM diagrams presented in this work, the magnitude or modulus of the transfer function G(s) is represented in decibels, i.e., it is represented 2 log G(s). In this way, the difference between two consecutive lines is the same in decibels. The other set of level curves, the coloured ones, represent the phase arg(g(s)). In this case, the scale is linear with red representing and cyan representing 8. That means that the cyan line is the Root Locus and the red line is the Inverse Root Locus. An important characteristic of this diagram is that it permits to read the phase margin directly because it is the phase distance from the actual closed poles position following the same magnitude line until the intersection with the imaginary axis and the Gain Margin that it is the magnitude distance following the cyan line that represents 8 until its intersection with the imaginary axis. The imaginary axis cut represent the Bode diagram. It is also possible to put the grid with damping ratio ζ and the frequency ω n. In the drawings, we have used linear scales for the real and imaginary axis, but if there are poles and zeros placed in different decades, it would be better to put log scales. In order to use these diagrams to calculate controllers C(s) for a system G(s), it is possible to read the phase, calculate the controller C (s) with gain that adds that phase, represent the PM diagram of C (s)g(s), and then multiply by the gain needed to have C(s) = K c C (s). The use of the SISOTOOL has a strange characteristic. You can add zeros and poles and change its positions, and change the gain until by trial and error you have the closed loop poles in the desired position. However, in the educational books about Control the process is different, you know the desired position of the closed loop poles and you have to calculate the controller in order to have the closed loop poles in the required positions. II. EXPERIMENTAL RESULTS The PM diagram permits to read directly the magnitude and the phase of the transfer function in each point. Its difference between the present closed loop poles and the desired ones gives the phase and the gain that has to be added by the controller. The phase and magnitudes can be read directly (approximately) or by clicking in the position to be given by Matlab (more precise). A. Ideal PD design. Suppose you have the system G(s) = (s + )(s + 2) This system has the closed loop poles placed in s,2 =.5 ±.86j as shown in Fig. 4 (point on the right). Its magnitude is G( j) =.5 and its phase is arg(g( j)) = 8 The desired specifications give us that the desired closed loop poles have to be placed in s,2 = 3.4 ± 3.4j, as shown in Fig. 4 (point on the left). The magnitude and phase of G(s) in these points are G( 3.4±3.4j) =.788 and arg(g( 3.4 ± 3.4j)) = If the desired controller is an ideal P D controller C(s) = K c (s + b), then the phase can be calculated using Fig. 5 as [ arctan 3.4 ] b 3.4 [ 8 arctan 3.4 ] arctan =

3 (a) A single pole (b) A double pole (c) A triple pole Fig. 2: PM diagram of a pole, a double pole and a triple pole (a) G(s) = (s+3)(s+4) (s+)(s+2) (b) G(s) = (s+)(s+2)(s+3) (c) G(s) = Fig. 3: PM diagram of different transfer functions (s+)(s+2)(s+3)(s+4) i.e. arctan 3.4 b 3.4 = 54.2 and that means that b = In order to calculate the gain, the magnitude condition can be applied: K = d d 2 d = The desired controller is ( 3.4) 2 = 3.28 C(s) = K(s + b) = 3.28(s + ) = s + b = K c = 7.8 s + = 7.8( +.9s) b The operations can be simplified using the PM diagram of the Fig. 4: The controller will be designed in two steps: The first step is the design of a controller of gain that aport the desired phase (Fig. 4) and the second step is to calculate the necessary gain. Between the cyan square and the green square in (Fig. 4) there are five and a half lines of phase (each one of ) and that means that the required phase of the controller has to be approximately φ = 54. Obviously, the best way of measuring the angle is that it appears in the data tip, as shown in Fig. 4. Applying arctan 3.4 b 3.4 = 54 the zero has to be in b =. The controller of gain that aport the necessary phase is C (s) = s + Now, we represent the function C (s)g(s) = s + = ( +.9s) (s + )(s + 2) as shown in Fig. 6. Now, we are going to design the second step: to calculate the necessary gain to pass from the point

4 Fig. 4: PM diagram of G(s) = (s+)(s+2). The original closed loop poles are s,2 =.5 ±.86j (on the right), and the desired poles are s,2 3.4 ± 3.4j (on the left) Im Fig. 6: PM diagram of the Transfer Function G(s) = of the ideal PD design s+ (s+)(s+2) 3.4j -b Fig. 5: Application of the phase condition for the ideal PD design on right of Fig. 6 that are the actual closed loop poles to the point on the left that represent the desired poles. The actual closed loop poles of this system are in s =.59 ±.68j. In this figure, these closed loop poles are in the Root Locus (cyan line) and have magnitude db =. In order to find the gain, it is necessary to read the magnitude at the desired closed loop poles, C (s)g(s) s= j = 25 db =.56, then K c = /.56 = 7.8. The desired controller is s + b C(s) = K c = 7.8 s + = 7.8( +.9s) b B. Design of a lead network. Consider the system G(s) =. It is desired to find a controller with a static coefficient of velocity error K v = 2s, and phase margin of 5. 4K Re The first step is to find the value of K in order to have the required K v K v = lim s sg(s) = lim s s 4K = 2K = 2 s(s + 2) therefore, K =. Now, we represent the PM diagram of G(s) = 4 as shown in Fig.7. In this diagram, we find the point of the imaginary axis with magnitude line of db, the phase margin is the number of phase bands between the actual position and that cyan band that represents 8. In this case there are a bit less than two bands, approximately 8. The lead network C(s) = s + z α s + p = s + T α s + αt has to add a phase φ = γ desired γ actual + 5 = = 37 and the parameter α of the network can be calculated as α = sin(φ) + sin(φ) =.24 As the lead network adds a magnitude of log(α)db in its middle point ω c = αt, we search the point with magnitude K m = log(α) = 6.2 db

5 Fig. 7: PM diagram of G(s) = 4 This point correspond to frequency ω c = 9 rad/sec. Choosing this frequency as the new frequency of transition ω c = αt, it is possible to find the corner frequencies of the lead network and T = αω c = 4.4 αt = ω c α = 8.4 In summary, the lead network is C(s) = 8.4 s s = 4.723s s Where the multiplicative constant α is to have gain equal to. The closed loop poles of the system with the lead network are placed in 6.9 ± 8j as shown in Fig. 8 and its Bode, that it are the values in the imaginary axis of this figure, is shown in Fig.9. III. PHASE MARGIN, GAIN MARGIN AND BODE DIAGRAM It is possible to use the Matlab command sgrid to put over the PM diagram in order to read the values of the damping ratio ζ and the natural frequency ω n as shown in Fig. that represent the system G(s) = (s + )(s + 2)(s + 3) Fig. 8: PM diagram of C(s)G(s) = s+4.4 Magnitude (db) Phase (deg) Bode Diagram 4 s Frequency (rad/s) Fig. 9: Bode plot of the uncompensated system G(s) = 4 (blue), the compensated system C(s)G(s) = s (green), and the lead network C(s) = s s+4.4 s+8.4 (red)

6 In this case, the closed loop poles are placed in s,2 =.8 ±.8j, because of this G(s,2 ) = = db. The cut of the imaginary axis is the Bode diagram shown in Fig.. The gain margin is the value of the intersection of the Root Locus (cyan line) with the imaginary axis, in this case GM = 4.8 db, and the Phase Margin is the number of colour bands following the same zero gain line (white Line) until the intersection with the imaginary axis, multiplied by the value of each band,. In this case there are 8 bands, approximately P M = 8. Finally, in this figure the bandwidth (BW) is the distance over the imaginary axis from the origin to the point in which G(s) = 3 db. In this example, BW =.2 rad/sec. Imag Axis Root Locus Editor for Open Loop (OL) Real Axis Magnitude (db) Phase (deg) G.M.: 4.6 db Freq: 3.32 rad/s Stable loop Open Loop Bode Editor for Open Loop (OL) P.M.: 8.3 deg Freq:.3 rad/s Frequency (rad/s) Fig. : Root Locus and Bode diagram of G(s) = (s+)(s+2)(s+3) Fig. 2: PM diagram Graphic User Interface of the PM Diagram Matlab Toolbox the Bode diagram. It also can be read the bandwidth. Finally the PM diagram permits unify the time and frequency analysis. Fig. : PM diagram of G(s) = (s+)(s+2)(s+3) The authors have written a PM Diagram Toolbox for Matlab that include an Graphic User Interface shown in Fig. 2. IV. CONCLUSIONS This paper presents a graphical chromatic representation of transfer functions that it is very visual and intuitive. In Control Education, it can be used to design controllers in an easier way and more intuitive than with the Root Locus diagram. It also give us an deeper knowledge of the transfer function and unifies the analysis in time and frequency because the cut of the PM diagram with the imaginary axis is the Bode diagram. The Phase and Gain margins can be read directly in the PM diagram in a very visual and natural way, without using REFERENCES [] Ogata, K., Modern Control Engineering, 5th. ed. Prentice Hall, 29. [2] Kuo, B. C., Automatic Control Systems, 6th ed.. Prentice Hall, 99. [3] Wegert, E., Visual Complex Functions: An Introduction With Phase Portraits. BirkhŁuser, 22. [4] Needham, T. Visual Complex Analysis. Clarendon Press, 998. [5] Mathews, J., Howell, R., Complex Analysis for Mathematics and Engineering. Jones and Bartlett Learning, 2. [6] Cavicchi, T.J., Phase-Root Locus and Relative Stability. IEEE Contr. Syst. Mag., vol 6, pp , 996. [7] Cavicchi, T.J., Phase Margin Revisited: Phase-Root Locus, Bode Plots, and Phase Shifters. IEEE Trans on Education. vol 46(). pp , 23. [8] Cerone, v., Canale, M. and Regruto, D., Loop-shaping Design with Constant Magnitude Loci in Control Education. Int. J. Engng. Ed. vol 24(), pp , 28

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 24: Compensation in the Frequency Domain Overview In this Lecture, you will learn: Lead Compensators Performance Specs Altering

More information

Homework 7 - Solutions

Homework 7 - Solutions Homework 7 - Solutions Note: This homework is worth a total of 48 points. 1. Compensators (9 points) For a unity feedback system given below, with G(s) = K s(s + 5)(s + 11) do the following: (c) Find the

More information

ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques

ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques [] For the following system, Design a compensator such

More information

r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic

r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of hand-written notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers

More information

Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10

Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:

More information

ECE382/ME482 Spring 2005 Homework 6 Solution April 17, (s/2 + 1) s(2s + 1)[(s/8) 2 + (s/20) + 1]

ECE382/ME482 Spring 2005 Homework 6 Solution April 17, (s/2 + 1) s(2s + 1)[(s/8) 2 + (s/20) + 1] ECE382/ME482 Spring 25 Homework 6 Solution April 17, 25 1 Solution to HW6 P8.17 We are given a system with open loop transfer function G(s) = 4(s/2 + 1) s(2s + 1)[(s/8) 2 + (s/2) + 1] (1) and unity negative

More information

Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.

More information

1 (20 pts) Nyquist Exercise

1 (20 pts) Nyquist Exercise EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. For each G(s), hand sketch the Nyquist diagram, determine Z = P N, algebraically

More information

Frequency (rad/s)

Frequency (rad/s) . The frequency response of the plant in a unity feedback control systems is shown in Figure. a) What is the static velocity error coefficient K v for the system? b) A lead compensator with a transfer

More information

MAK 391 System Dynamics & Control. Presentation Topic. The Root Locus Method. Student Number: Group: I-B. Name & Surname: Göksel CANSEVEN

MAK 391 System Dynamics & Control. Presentation Topic. The Root Locus Method. Student Number: Group: I-B. Name & Surname: Göksel CANSEVEN MAK 391 System Dynamics & Control Presentation Topic The Root Locus Method Student Number: 9901.06047 Group: I-B Name & Surname: Göksel CANSEVEN Date: December 2001 The Root-Locus Method Göksel CANSEVEN

More information

EE3CL4: Introduction to Linear Control Systems

EE3CL4: Introduction to Linear Control Systems 1 / 30 EE3CL4: Introduction to Linear Control Systems Section 9: of and using Techniques McMaster University Winter 2017 2 / 30 Outline 1 2 3 4 / 30 domain analysis Analyze closed loop using open loop

More information

The Relation Between the 3-D Bode Diagram and the Root Locus. Insights into the connection between these classical methods. By Panagiotis Tsiotras

The Relation Between the 3-D Bode Diagram and the Root Locus. Insights into the connection between these classical methods. By Panagiotis Tsiotras F E A T U R E The Relation Between the -D Bode Diagram and the Root Locus Insights into the connection between these classical methods Bode diagrams and root locus plots have been the cornerstone of control

More information

(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:

(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications: 1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4.

More information

ECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) =

ECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) = ECE382/ME482 Spring 25 Homework 7 Solution April 17, 25 1 Solution to HW7 AP9.5 We are given a system with open loop transfer function G(s) = K(s +.2) s 2 (s + 2)(s + 5) (1) and unity negative feedback.

More information

Root Locus Methods. The root locus procedure

Root Locus Methods. The root locus procedure Root Locus Methods Design of a position control system using the root locus method Design of a phase lag compensator using the root locus method The root locus procedure To determine the value of the gain

More information

Robust Performance Example #1

Robust Performance Example #1 Robust Performance Example # The transfer function for a nominal system (plant) is given, along with the transfer function for one extreme system. These two transfer functions define a family of plants

More information

Dynamic circuits: Frequency domain analysis

Dynamic circuits: Frequency domain analysis Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution

More information

Phase Margin Revisited: Phase-Root Locus, Bode Plots, and Phase Shifters

Phase Margin Revisited: Phase-Root Locus, Bode Plots, and Phase Shifters 168 IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 1, FEBRUARY 2003 Phase Margin Revisited: Phase-Root Locus, Bode Plots, and Phase Shifters Thomas J. Cavicchi Abstract In learning undergraduate controls,

More information

PD, PI, PID Compensation. M. Sami Fadali Professor of Electrical Engineering University of Nevada

PD, PI, PID Compensation. M. Sami Fadali Professor of Electrical Engineering University of Nevada PD, PI, PID Compensation M. Sami Fadali Professor of Electrical Engineering University of Nevada 1 Outline PD compensation. PI compensation. PID compensation. 2 PD Control L= loop gain s cl = desired closed-loop

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall K(s +1)(s +2) G(s) =.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall K(s +1)(s +2) G(s) =. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering. Dynamics and Control II Fall 7 Problem Set #7 Solution Posted: Friday, Nov., 7. Nise problem 5 from chapter 8, page 76. Answer:

More information

Richiami di Controlli Automatici

Richiami di Controlli Automatici Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

More information

Lecture 14 - Using the MATLAB Control System Toolbox and Simulink Friday, February 8, 2013

Lecture 14 - Using the MATLAB Control System Toolbox and Simulink Friday, February 8, 2013 Today s Objectives ENGR 105: Feedback Control Design Winter 2013 Lecture 14 - Using the MATLAB Control System Toolbox and Simulink Friday, February 8, 2013 1. introduce the MATLAB Control System Toolbox

More information

FREQUENCY-RESPONSE DESIGN

FREQUENCY-RESPONSE DESIGN ECE45/55: Feedback Control Systems. 9 FREQUENCY-RESPONSE DESIGN 9.: PD and lead compensation networks The frequency-response methods we have seen so far largely tell us about stability and stability margins

More information

Example on Root Locus Sketching and Control Design

Example on Root Locus Sketching and Control Design Example on Root Locus Sketching and Control Design MCE44 - Spring 5 Dr. Richter April 25, 25 The following figure represents the system used for controlling the robotic manipulator of a Mars Rover. We

More information

Desired Bode plot shape

Desired Bode plot shape Desired Bode plot shape 0dB Want high gain Use PI or lag control Low freq ess, type High low freq gain for steady state tracking Low high freq gain for noise attenuation Sufficient PM near ω gc for stability

More information

16.30/31, Fall 2010 Recitation # 2

16.30/31, Fall 2010 Recitation # 2 16.30/31, Fall 2010 Recitation # 2 September 22, 2010 In this recitation, we will consider two problems from Chapter 8 of the Van de Vegte book. R + - E G c (s) G(s) C Figure 1: The standard block diagram

More information

Compensator Design for Helicopter Stabilization

Compensator Design for Helicopter Stabilization Available online at www.sciencedirect.com Procedia Technology 4 (212 ) 74 81 C3IT-212 Compensator Design for Helicopter Stabilization Raghupati Goswami a,sourish Sanyal b, Amar Nath Sanyal c a Chairman,

More information

Analysis and Design of Analog Integrated Circuits Lecture 12. Feedback

Analysis and Design of Analog Integrated Circuits Lecture 12. Feedback Analysis and Design of Analog Integrated Circuits Lecture 12 Feedback Michael H. Perrott March 11, 2012 Copyright 2012 by Michael H. Perrott All rights reserved. Open Loop Versus Closed Loop Amplifier

More information

Solutions to Skill-Assessment Exercises

Solutions to Skill-Assessment Exercises Solutions to Skill-Assessment Exercises To Accompany Control Systems Engineering 4 th Edition By Norman S. Nise John Wiley & Sons Copyright 2004 by John Wiley & Sons, Inc. All rights reserved. No part

More information

Topic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback

Topic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall

More information

ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s)

ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s) C1 ROOT LOCUS Consider the system R(s) E(s) C(s) + K G(s) - H(s) C(s) R(s) = K G(s) 1 + K G(s) H(s) Root locus presents the poles of the closed-loop system when the gain K changes from 0 to 1+ K G ( s)

More information

Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14)

Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14) Appendix J Additional Closed-Loop Frequency Response Material (Second edition, Chapter 4) APPENDIX CONTENTS J. Closed-Loop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain

More information

Overview of Bode Plots Transfer function review Piece-wise linear approximations First-order terms Second-order terms (complex poles & zeros)

Overview of Bode Plots Transfer function review Piece-wise linear approximations First-order terms Second-order terms (complex poles & zeros) Overview of Bode Plots Transfer function review Piece-wise linear approximations First-order terms Second-order terms (complex poles & zeros) J. McNames Portland State University ECE 222 Bode Plots Ver.

More information

7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM

7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM ROOT LOCUS TECHNIQUE. Values of on the root loci The value of at any point s on the root loci is determined from the following equation G( s) H( s) Product of lengths of vectors from poles of G( s)h( s)

More information

Professor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley

Professor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the

More information

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a

More information

Root Locus Design Example #4

Root Locus Design Example #4 Root Locus Design Example #4 A. Introduction The plant model represents a linearization of the heading dynamics of a 25, ton tanker ship under empty load conditions. The reference input signal R(s) is

More information

Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year

Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21-211 1 / 39 Feedback

More information

EECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators.

EECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators. Name: SID: EECS C28/ ME C34 Final Wed. Dec. 5, 2 8- am Closed book. Two pages of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 2 2 6 3 4 4 5 6 6 7 8 2 Total

More information

School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:

School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by: Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus - 1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See

More information

DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING

DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING QUESTION BANK SUBJECT CODE & NAME: CONTROL SYSTEMS YEAR / SEM: II / IV UNIT I SYSTEMS AND THEIR REPRESENTATION PARTA [2

More information

Exam. 135 minutes + 15 minutes reading time

Exam. 135 minutes + 15 minutes reading time Exam January 23, 27 Control Systems I (5-59-L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages

More information

Learn2Control Laboratory

Learn2Control Laboratory Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should

More information

100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =

100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) = 1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot

More information

ECE137B Final Exam. There are 5 problems on this exam and you have 3 hours There are pages 1-19 in the exam: please make sure all are there.

ECE137B Final Exam. There are 5 problems on this exam and you have 3 hours There are pages 1-19 in the exam: please make sure all are there. ECE37B Final Exam There are 5 problems on this exam and you have 3 hours There are pages -9 in the exam: please make sure all are there. Do not open this exam until told to do so Show all work: Credit

More information

Performance of Feedback Control Systems

Performance of Feedback Control Systems Performance of Feedback Control Systems Design of a PID Controller Transient Response of a Closed Loop System Damping Coefficient, Natural frequency, Settling time and Steady-state Error and Type 0, Type

More information

Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control. DC Motor Control Trainer (DCMCT) Student Manual

Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control. DC Motor Control Trainer (DCMCT) Student Manual Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control DC Motor Control Trainer (DCMCT) Student Manual Table of Contents 1 Laboratory Objectives1 2 References1 3 DCMCT Plant

More information

Frequency domain analysis

Frequency domain analysis Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011

More information

Design via Root Locus

Design via Root Locus Design via Root Locus I 9 Chapter Learning Outcomes J After completing this chapter the student will be able to: Use the root locus to design cascade compensators to improve the steady-state error (Sections

More information

INTERACTIVE APPLICATIONS IN A MANDATORY CONTROL COURSE

INTERACTIVE APPLICATIONS IN A MANDATORY CONTROL COURSE INTERACTIVE APPLICATIONS IN A MANDATORY CONTROL COURSE Yves Piguet Roland Longchamp Calerga Sàrl, av. de la Chablière 35, 4 Lausanne, Switzerland. E-mail: yves.piguet@calerga.com Laboratoire d automatique,

More information

NADAR SARASWATHI COLLEGE OF ENGINEERING AND TECHNOLOGY Vadapudupatti, Theni

NADAR SARASWATHI COLLEGE OF ENGINEERING AND TECHNOLOGY Vadapudupatti, Theni NADAR SARASWATHI COLLEGE OF ENGINEERING AND TECHNOLOGY Vadapudupatti, Theni-625531 Question Bank for the Units I to V SE05 BR05 SU02 5 th Semester B.E. / B.Tech. Electrical & Electronics engineering IC6501

More information

University of Science and Technology, Sudan Department of Chemical Engineering.

University of Science and Technology, Sudan Department of Chemical Engineering. ISO 91:28 Certified Volume 3, Issue 6, November 214 Design and Decoupling of Control System for a Continuous Stirred Tank Reactor (CSTR) Georgeous, N.B *1 and Gasmalseed, G.A, Abdalla, B.K (1-2) University

More information

CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. VIII - Design Techniques in the Frequency Domain - Edmunds, J.M. and Munro, N.

CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. VIII - Design Techniques in the Frequency Domain - Edmunds, J.M. and Munro, N. DESIGN TECHNIQUES IN THE FREQUENCY DOMAIN Edmunds, Control Systems Center, UMIST, UK Keywords: Multivariable control, frequency response design, Nyquist, scaling, diagonal dominance Contents 1. Frequency

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes

More information

The Frequency-Response

The Frequency-Response 6 The Frequency-Response Design Method A Perspective on the Frequency-Response Design Method The design of feedback control systems in industry is probably accomplished using frequency-response methods

More information

Root Locus. Motivation Sketching Root Locus Examples. School of Mechanical Engineering Purdue University. ME375 Root Locus - 1

Root Locus. Motivation Sketching Root Locus Examples. School of Mechanical Engineering Purdue University. ME375 Root Locus - 1 Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus - 1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: D 0.09 Position

More information

Chemical Process Dynamics and Control. Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University

Chemical Process Dynamics and Control. Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University Chemical Process Dynamics and Control Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University 1 Chapter 4 System Stability 2 Chapter Objectives End of this

More information

Reglerteknik Allmän Kurs. Del 2. Lösningar till Exempelsamling. Läsår 2015/16

Reglerteknik Allmän Kurs. Del 2. Lösningar till Exempelsamling. Läsår 2015/16 Reglerteknik Allmän Kurs Del Lösningar till Exempelsamling Läsår 5/6 Avdelningen för Reglerteknik, KTH, SE 44 Stockholm, SWEDEN AUTOMATIC CONTROL COMMUNICATION SYSTEMS LINKÖPINGS UNIVERSITET Reglerteknik

More information

CHAPTER 7 STEADY-STATE RESPONSE ANALYSES

CHAPTER 7 STEADY-STATE RESPONSE ANALYSES CHAPTER 7 STEADY-STATE RESPONSE ANALYSES 1. Introduction The steady state error is a measure of system accuracy. These errors arise from the nature of the inputs, system type and from nonlinearities of

More information

Control Systems 2. Lecture 4: Sensitivity function limits. Roy Smith

Control Systems 2. Lecture 4: Sensitivity function limits. Roy Smith Control Systems 2 Lecture 4: Sensitivity function limits Roy Smith 2017-3-14 4.1 Input-output controllability Control design questions: 1. How well can the plant be controlled? 2. What control structure

More information

Step Response for the Transfer Function of a Sensor

Step Response for the Transfer Function of a Sensor Step Response f the Transfer Function of a Sens G(s)=Y(s)/X(s) of a sens with X(s) input and Y(s) output A) First Order Instruments a) First der transfer function G(s)=k/(1+Ts), k=gain, T = time constant

More information

a. Closed-loop system; b. equivalent transfer function Then the CLTF () T is s the poles of () T are s from a contribution of a

a. Closed-loop system; b. equivalent transfer function Then the CLTF () T is s the poles of () T are s from a contribution of a Root Locus Simple definition Locus of points on the s- plane that represents the poles of a system as one or more parameter vary. RL and its relation to poles of a closed loop system RL and its relation

More information

Notes for ECE-320. Winter by R. Throne

Notes for ECE-320. Winter by R. Throne Notes for ECE-3 Winter 4-5 by R. Throne Contents Table of Laplace Transforms 5 Laplace Transform Review 6. Poles and Zeros.................................... 6. Proper and Strictly Proper Transfer Functions...................

More information

Essence of the Root Locus Technique

Essence of the Root Locus Technique Essence of the Root Locus Technique In this chapter we study a method for finding locations of system poles. The method is presented for a very general set-up, namely for the case when the closed-loop

More information

2.010 Fall 2000 Solution of Homework Assignment 8

2.010 Fall 2000 Solution of Homework Assignment 8 2.1 Fall 2 Solution of Homework Assignment 8 1. Root Locus Analysis of Hydraulic Servomechanism. The block diagram of the controlled hydraulic servomechanism is shown in Fig. 1 e r e error + i Σ C(s) P(s)

More information

UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences. EE105 Lab Experiments

UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences. EE105 Lab Experiments UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences EE15 Lab Experiments Bode Plot Tutorial Contents 1 Introduction 1 2 Bode Plots Basics

More information

EEL2216 Control Theory CT1: PID Controller Design

EEL2216 Control Theory CT1: PID Controller Design EEL6 Control Theory CT: PID Controller Design. Objectives (i) To design proportional-integral-derivative (PID) controller for closed loop control. (ii) To evaluate the performance of different controllers

More information

Appendix 3B MATLAB Functions for Modeling and Time-domain analysis

Appendix 3B MATLAB Functions for Modeling and Time-domain analysis Appendix 3B MATLAB Functions for Modeling and Time-domain analysis MATLAB control system Toolbox contain the following functions for the time-domain response step impulse initial lsim gensig damp ltiview

More information

Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3.. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

More information

An Internal Stability Example

An Internal Stability Example An Internal Stability Example Roy Smith 26 April 2015 To illustrate the concept of internal stability we will look at an example where there are several pole-zero cancellations between the controller and

More information

Alireza Mousavi Brunel University

Alireza Mousavi Brunel University Alireza Mousavi Brunel University 1 » Control Process» Control Systems Design & Analysis 2 Open-Loop Control: Is normally a simple switch on and switch off process, for example a light in a room is switched

More information

Experiment 13 Poles and zeros in the z plane: IIR systems

Experiment 13 Poles and zeros in the z plane: IIR systems Experiment 13 Poles and zeros in the z plane: IIR systems Achievements in this experiment You will be able to interpret the poles and zeros of the transfer function of discrete-time filters to visualize

More information

ME 375 Final Examination Thursday, May 7, 2015 SOLUTION

ME 375 Final Examination Thursday, May 7, 2015 SOLUTION ME 375 Final Examination Thursday, May 7, 2015 SOLUTION POBLEM 1 (25%) negligible mass wheels negligible mass wheels v motor no slip ω r r F D O no slip e in Motor% Cart%with%motor%a,ached% The coupled

More information

Root Locus Design Example #3

Root Locus Design Example #3 Root Locus Design Example #3 A. Introduction The system represents a linear model for vertical motion of an underwater vehicle at zero forward speed. The vehicle is assumed to have zero pitch and roll

More information

Lecture 1 Root Locus

Lecture 1 Root Locus Root Locus ELEC304-Alper Erdogan 1 1 Lecture 1 Root Locus What is Root-Locus? : A graphical representation of closed loop poles as a system parameter varied. Based on Root-Locus graph we can choose the

More information

Lecture 7:Time Response Pole-Zero Maps Influence of Poles and Zeros Higher Order Systems and Pole Dominance Criterion

Lecture 7:Time Response Pole-Zero Maps Influence of Poles and Zeros Higher Order Systems and Pole Dominance Criterion Cleveland State University MCE441: Intr. Linear Control Lecture 7:Time Influence of Poles and Zeros Higher Order and Pole Criterion Prof. Richter 1 / 26 First-Order Specs: Step : Pole Real inputs contain

More information

9/9/2011 Classical Control 1

9/9/2011 Classical Control 1 MM11 Root Locus Design Method Reading material: FC pp.270-328 9/9/2011 Classical Control 1 What have we talked in lecture (MM10)? Lead and lag compensators D(s)=(s+z)/(s+p) with z < p or z > p D(s)=K(Ts+1)/(Ts+1),

More information

FREQUENCY-RESPONSE ANALYSIS

FREQUENCY-RESPONSE ANALYSIS ECE450/550: Feedback Control Systems. 8 FREQUENCY-RESPONSE ANALYSIS 8.: Motivation to study frequency-response methods Advantages and disadvantages to root-locus design approach: ADVANTAGES: Good indicator

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

More information

Tradeoffs and Limits of Performance

Tradeoffs and Limits of Performance Chapter 9 Tradeoffs and Limits of Performance 9. Introduction Fundamental limits of feedback systems will be investigated in this chapter. We begin in Section 9.2 by discussing the basic feedback loop

More information

Distributed Real-Time Control Systems

Distributed Real-Time Control Systems Distributed Real-Time Control Systems Chapter 9 Discrete PID Control 1 Computer Control 2 Approximation of Continuous Time Controllers Design Strategy: Design a continuous time controller C c (s) and then

More information

Chapter 6 Steady-State Analysis of Continuous-Time Systems

Chapter 6 Steady-State Analysis of Continuous-Time Systems Chapter 6 Steady-State Analysis of Continuous-Time Systems 6.1 INTRODUCTION One of the objectives of a control systems engineer is to minimize the steady-state error of the closed-loop system response

More information

6.302 Feedback Systems Recitation 16: Compensation Prof. Joel L. Dawson

6.302 Feedback Systems Recitation 16: Compensation Prof. Joel L. Dawson Bode Obstacle Course is one technique for doing compensation, or designing a feedback system to make the closed-loop behavior what we want it to be. To review: - G c (s) G(s) H(s) you are here! plant For

More information

Study on Control of First Order Plus Delay Using Smith Predictor

Study on Control of First Order Plus Delay Using Smith Predictor Journal of Al-Nahrain University Vol.18 (2), June, 2015, pp.10-17 Science Study on Control of First Order Plus Delay Using Smith Predictor Saad A. Ahmed Department of Chemistry, College of Education-Ibn

More information

Autonomous Mobile Robot Design

Autonomous Mobile Robot Design Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:

More information

Quantitative Feedback Theory based Controller Design of an Unstable System

Quantitative Feedback Theory based Controller Design of an Unstable System Quantitative Feedback Theory based Controller Design of an Unstable System Chandrima Roy Department of E.C.E, Assistant Professor Heritage Institute of Technology, Kolkata, WB Kalyankumar Datta Department

More information

Feedback design for the Buck Converter

Feedback design for the Buck Converter Feedback design for the Buck Converter Portland State University Department of Electrical and Computer Engineering Portland, Oregon, USA December 30, 2009 Abstract In this paper we explore two compensation

More information

EXAMPLE PROBLEMS AND SOLUTIONS

EXAMPLE PROBLEMS AND SOLUTIONS Similarly, the program for the fourth-order transfer function approximation with T = 0.1 sec is [num,denl = pade(0.1, 4); printsys(num, den, 'st) numlden = sa4-2o0sa3 + 1 80O0sA2-840000~ + 16800000 sa4

More information

x(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =!

x(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =! 1 Time-Delay Systems 1.1 Introduction Recitation Notes: Time Delays and Nyquist Plots Review In control systems a challenging area is operating in the presence of delays. Delays can be attributed to acquiring

More information

Lab # 4 Time Response Analysis

Lab # 4 Time Response Analysis Islamic University of Gaza Faculty of Engineering Computer Engineering Dep. Feedback Control Systems Lab Eng. Tareq Abu Aisha Lab # 4 Lab # 4 Time Response Analysis What is the Time Response? It is an

More information

Homework 11 Solution - AME 30315, Spring 2015

Homework 11 Solution - AME 30315, Spring 2015 1 Homework 11 Solution - AME 30315, Spring 2015 Problem 1 [10/10 pts] R + - K G(s) Y Gpsq Θpsq{Ipsq and we are interested in the closed-loop pole locations as the parameter k is varied. Θpsq Ipsq k ωn

More information

Introduction to Feedback Control

Introduction to Feedback Control Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

More information

Homework Assignment 3

Homework Assignment 3 ECE382/ME482 Fall 2008 Homework 3 Solution October 20, 2008 1 Homework Assignment 3 Assigned September 30, 2008. Due in lecture October 7, 2008. Note that you must include all of your work to obtain full

More information

PID Tuning of Plants With Time Delay Using Root Locus

PID Tuning of Plants With Time Delay Using Root Locus San Jose State University SJSU ScholarWorks Master's Theses Master's Theses and Graduate Research Summer 2011 PID Tuning of Plants With Time Delay Using Root Locus Greg Baker San Jose State University

More information

Sinusoidal Forcing of a First-Order Process. / τ

Sinusoidal Forcing of a First-Order Process. / τ Frequency Response Analysis Chapter 3 Sinusoidal Forcing of a First-Order Process For a first-order transfer function with gain K and time constant τ, the response to a general sinusoidal input, xt = A

More information

1 Chapter 9: Design via Root Locus

1 Chapter 9: Design via Root Locus 1 Figure 9.1 a. Sample root locus, showing possible design point via gain adjustment (A) and desired design point that cannot be met via simple gain adjustment (B); b. responses from poles at A and B 2

More information

A MATLAB Toolbox for Teaching Model Order Reduction Techniques

A MATLAB Toolbox for Teaching Model Order Reduction Techniques A MATLAB Toolbox for Teaching Model Order Reduction Techniques Authors: Ali Eydgahi, Department of Engineering and Aviation Sciences, University of Maryland Eastern Shore, Princess Anne, MD 853, aeydgahi@mail.umes.edu

More information

16.31 Homework 2 Solution

16.31 Homework 2 Solution 16.31 Homework Solution Prof. S. R. Hall Issued: September, 6 Due: September 9, 6 Problem 1. (Dominant Pole Locations) [FPE 3.36 (a),(c),(d), page 161]. Consider the second order system ωn H(s) = (s/p

More information

Comparative Study of Zero Effects in Pole-Placement Control System Design Via the Shift and Delta Transforms

Comparative Study of Zero Effects in Pole-Placement Control System Design Via the Shift and Delta Transforms FACTA UNIVRSITATIS (NIŠ) SR.: LC. NRG. vol. 8, no. 3, December 25, 439-45 Comparative Study of Zero ffects in Pole-Placement Control System Design Via the Shift and Delta Transforms Dedicated to Professor

More information

Transient Response of a Second-Order System

Transient Response of a Second-Order System Transient Response of a Second-Order System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a well-behaved closed-loop

More information