SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015

Size: px
Start display at page:

Download "SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015"

Transcription

1 FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman

2 Problem 1: Frequency-domain analysis and control design (15 pt) Given is a plant consisting of 3 first-order time delay elements connected in series. The time constants are T 1 = 0.01, T 2 =0.5, T 3 =0.1. The overall plant gain is K=40. a) Determine the transfer function of the plant. How many stable poles has the system? Explain your answer. Show the Bode diagram of the system. Are there sufficient amplitude and phase margins in the system? Explain your answer. (4 pt) Transfer function of the plant The system has 3 stable poles: 100, 2, 10 since all roots of characteristic polynomial (denominator of G) are in the left-hand side of s-plane. Bode diagram with the amplitude and phase margins: Both, the available amplitude margin (4.52 db) and phase margin (9.63 deg) are not sufficient: a low amplitude increase (about 1.7 times) and low phase shift (9.63 deg) would lead to a closed-loop instability. b) Design the PID controller which compensates for two corner frequencies of the plant. Write down the transfer function of the open-loop and closed-loop systems. Use MATLAB to simulate the step response of the closed-loop system. Copy the simulated response to paper. (4 pt) Both PID-control time constants compensate for two largest time constants of the plant (T 2 =0.5, T 3 =0.1). The PID control gain accounts for the plant gain and smallest time constant:... with K r = 1 / (2 K T 1 ). The open- and closed-loop transfer functions are:,... Step response of the closed-loop system:

3 c) Using the Lead-Lag compensators design the control system with the following specification: phase margin 30 deg and step response steady-state error 5 %. (4 pt) The DC gain of the closed-loop system: is : {matlab -> dcgain(g/(1+g))}. Thus the step response steady-state error is already 5 %. Design a Lead compensator only for improving the phase margin which is 9.63 deg. The required phase enhancement is m deg. Assume e.g. m =25 deg, then 1/ 2.5 this from the characteristic Lead diagram, then = 0.4. At the cross-over frequency m = c = , rad/s. The Lead-compensator transfer function:,. Then, reduce the Lead gain K p so that the open loop comes back to the original cross-over frequency c = 26.9 rad/s, -> K p = The total compensator transfer function is: d) Use Simulink to simulate and to compare the step response of the closed-loop control systems from problems 1b) and 1c). Copy the step response and block diagram from Simulink to paper. Label all blocks, signals, and axes. (3 pt) Closed-loop step responses and block diagram (Simulink) of both controllers PID and Lead: Problem 2: State-space analysis and control design (15 pt) Given is a linear time-invariant system in the state-space form , a) Determine poles of the system and draw the pole diagram in complex plane. What can be said about system stability? Which type has the system? Explain your answers. (3 pt) The poles are the eigenvalues of system matrix {matlab -> A = [ ; 4 0 0; 0 1 0]; eig(a)}. -> 1 = 0, 2 = -2, 3 = -10. The system is stable as there are no poles in the righthand side of s-plane. There is one pole in origin (0,0) of the complex s-plane -> the system has an integral behavior -> the system type (number of free integrators) is 1.

4 b) Verify whether the system is controllable and observable by using Hautus criteria. (4 pt) Hautus criterion for controllability: rank i I A, B = n, and Hautus criterion for ii A observability: rank = n, where n is the system order and i is the system pole (each C one to be evaluated separately), {matlab -> A = [ ; 4 0 0; 0 1 0]; B = [8 0 0] ; C = [0 0 1]; I = eye(3,3);} -> for all three poles 1 = 0, 2 = -2, 3 = -10 evaluate the rank of matrices above -> for all poles the rank is 3 -> the system is fully controllable and observable. c) Design the state-feedback control so that the system matrix of the closed-loop control system has the following form Design the pre-filter to ensure the steady-state accuracy. Use MATLAB to simulate and to show the step response of closed-loop control systems with state feedback and pre-filter. What is the drawback of a static pre-filter? Explain your answer. (4 pt) According to Kalman canonical decomposition, the system matrix of the closed-loop control system A c is in the modal canonical form so that the real poles (eigenvalues) can be immediately read out from the main diagonal: c, 1 = -30, c, 2 = -25, c, 3 = -20. Use pole placement command place in Matlab to determine the state feedback gains; determine the new system matrix with K feedback, determine the state-space model with the new system matrix; determine the prefilter by the explicit formula:{matlab -> K = place(a,b,[-20,-25,- 30]; Ac = A-B*K; Gc = ss(ac,b,c,d); V = (C*(B*K-A)^(-1)*B)^(-1);}. The step response of the closed-loop control systems with state feedback and pre-filter: {matlab -> step(v*gc)}: The drawback of a static pre-filter V is that its computation relies on the plant parameters, i.e. matrices A, B, and C, and there is no comparison between the reference and real (output) value in the control loop. Thus, the model uncertainties will lead to that no steady-state accuracy (zero control error) can be guaranteed. d) Assuming x 3 is the single measurable state in the system, design the Luenberger state observer. The observer poles should be approximately 5 times faster than poles of the closed-loop control system from problem 2c). Use Simulink to simulate and to show the response of observer-based state-feedback control with pre-filter from problem 2c). Use sinusoidal with angular frequency 2 rad/s as a reference value. Copy the system response and block diagram from Simulink to paper. (4 pt)

5 Determine the observer feedback gain by pole placement using place command in Matlab: {matlab -> L = place(a',c', 5*[-20,-25,-30]) ;}. Determine the state-space model of Luenberger observer {matlab -> Alc = A-L*C; Blc = [B L]; Clc = C; Dlc = D;}. The observer-based state-feedback control with pre-filter (in Simulink) and the controlled response of x 3 output value to the sinusoidal reference with 2 rad/s angular frequency: Problem 3: Digital control systems (15 pt) Given are the continuous-time transfer functions of the plant G and PD controller 1, For the digital control system with sampling and without quantization of the signals, assume the sampling time T s =0.01 sec. a) Transform the plant transfer function from the continuous-time (s) domain into the discrete-time (z) domain. Determine the difference equation (discrete time-series equation) with the input u(k) and output y(k), where k is a discrete time instant. (4 pt) According to the Tustin s method with the correspondence between s -1 and z: 1/s -> Ts/2 (z+1)/(z-1), the plant transfer function in the discrete-time domain is: This results in the input-output equation: U(z)(2.5e-5 + 5e-5z e-5z -2 )=Y(z)(1-2z -1 +z -2 ). After applying the time shift transformation X(z)z -n -> x(k-n) the resulted output equation is: y(k) = 2y(k-1) - y(k-2) + 2.5e-5 u(k) + 5e-5 u(k-1) + 2.5e-5 u(k-2). b) How the difference equation from problem 3a) can be represented by using the forward and backward shift operators? Explain you answer. Draw the corresponding block diagram of the signal flow from u(k) to y(k). (4 pt) The forward shift operator (in z-domain) means multiplication with z and backward shift operator means division by z. In the discrete time domain the backward shift operator is a time-delay (or memory operator) for which the output value is the input value delayed by one discrete time instant. The block diagram using the backward shift operator in the time domain, e.g. using Simulink modeling is:

6 c) Determine the difference equation (discrete time-series equation) of the PD controller. Draw the corresponding block diagram with the signal flow from e(k) to u(k). (4 pt) Similar as in solution 3b), the input to output equation of the PD control is: 2 z z Uz ( ) Kp Kd Ez ( ) Ez ( ) Ts z1 z1 1 1 U( z)(1 z ) Ez ( )( z ) uk ( ) 8400 ek ( ) 7600 ek ( 1) uk ( 1). The block diagram (in Simulink notation) is d) Use Simulink to simulate and to show the step response of the digital control system consisting of the discrete-time plant G and PD controller from problems 3b) and 3c). Copy the step response and block diagram from Simulink to paper. (3 pt) The block diagram in Simulink and the controlled step response are:

7 Problem 4: Miscellaneous system modeling and analysis (15 pt) In Figure below is shown an electromagnet with movable armature. The controllable input value is the terminal voltage V. The measurable output value is the armature displacement Y. Figure: Electromagnet with movable armature The linearized equations of magnetic circuits and mechanical mass-spring-damper part are:,,. The system parameters are the coil inductance L, connection resistance R, electro-mechanical coupling constant C, armature mass M, viscous damping D, and return spring stiffness K. The current in electromagnetic circuits is I. The induced magnetic force is F. a) Define the dynamic states of the system and write down the overall state-space model. (4 pt) The dynamic states of the system are the current, velocity, and position: x = [I, dy/dt, Y]. The resulted state-space model is I R/ L C/ L 0 I 1/ L dx d Ax + Bu Y C/ M D/ M K / M Y 0 V dt dt, Y Y y Cx Y I Y Y T b) Draw the block diagram of the system. Label all signals and parameters in the block diagram, including summations nodes and signal flow direction. (4 pt) The block diagram of the system, e.g. using Simulink modeling: c) Write down the transfer function of the system. Determine the parameter conditions for which the system is stable. (4 pt) To construct the system transfer function from V(s) to Y(s) consider (with help of solution 4b) two sub-transfer functions: from I to Y and from V to I. The latter is first without feedback by

8 C! G 1 (s)=y(s)/i(s) = C/(Ms 2 +Ds+K) and G 2 (s)=1/(ls+r). Then the overall transfer function is the feedforward of G 1 (s)g 2 (s) with the feedback through Cs: G(s) = G 1 G 2 /(1+ CsG 1 G 2 ) -> C Gs () LMs ( DL MR) s ( DR KL C ) s KR To derive the conditions for system to be stable apply the Routh criterion by evaluating the coefficients of characteristic polynomial a 3 =LM, a 2 =DL+MR, a 1 =DR+KL+C 2, a 0 =KR. The first column of the Routh array should have no sign changes: a 3 > 0, a 2 > 0, b 1 > 0, c 1 > 0. The b 1, c 1 coefficients should be computed according to the Routh array. d) For the given set of parameters, L=0.001, R=0.1, C=1, M=0.2, D=0.01, K=1000, show the Bode diagram and step response of the plant. Is the closed-loop system with negative feedback of the armature displacement stable? Explain your answer. (3 pt) The Bode diagram and step response of the system with assumed numerical values are: In order to see whether the closed-loop system with a negative feedback of the armature displacement is stable, use the root locus {matlab -> rlocus(g)}. One can see that the closed-loop is stable. However when increasing the feedback gain, the system becomes unstable up from a certain gain value. Evaluation scale (max 60 points): points < 5 F 5 <= points <= 11 E 12 <= points <= 23 D 24 <= points <= 35 C 36 <= points <= 47 B 48 <= points A

Contents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42

Contents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42 Contents Preface.............................................. xiii 1. Introduction......................................... 1 1.1 Continuous and Discrete Control Systems................. 4 1.2 Open-Loop

More information

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203. DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING SUBJECT QUESTION BANK : EC6405 CONTROL SYSTEM ENGINEERING SEM / YEAR: IV / II year

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

Controls Problems for Qualifying Exam - Spring 2014

Controls Problems for Qualifying Exam - Spring 2014 Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function

More information

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : CONTROL SYSTEMS BRANCH : ECE YEAR : II SEMESTER: IV 1. What is control system? 2. Define open

More information

Index. Index. More information. in this web service Cambridge University Press

Index. Index. More information.  in this web service Cambridge University Press A-type elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 A-type variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,

More information

EC6405 - CONTROL SYSTEM ENGINEERING Questions and Answers Unit - I Control System Modeling Two marks 1. What is control system? A system consists of a number of components connected together to perform

More information

Digital Control Systems

Digital Control Systems Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist

More information

R a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies.

R a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies. SET - 1 II B. Tech II Semester Supplementary Examinations Dec 01 1. a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies..

More information

Department of Electronics and Instrumentation Engineering M. E- CONTROL AND INSTRUMENTATION ENGINEERING CL7101 CONTROL SYSTEM DESIGN Unit I- BASICS AND ROOT-LOCUS DESIGN PART-A (2 marks) 1. What are the

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

R10. IV B.Tech II Semester Regular Examinations, April/May DIGITAL CONTROL SYSTEMS JNTUK

R10. IV B.Tech II Semester Regular Examinations, April/May DIGITAL CONTROL SYSTEMS JNTUK Set No. 1 1 a) Explain about the shifting and scaling operator. b) Discuss briefly about the linear time invariant and causal systems. 2 a) Write the mapping points between S-Plane and Z-plane. b) Find

More information

CBE507 LECTURE III Controller Design Using State-space Methods. Professor Dae Ryook Yang

CBE507 LECTURE III Controller Design Using State-space Methods. Professor Dae Ryook Yang CBE507 LECTURE III Controller Design Using State-space Methods Professor Dae Ryook Yang Fall 2013 Dept. of Chemical and Biological Engineering Korea University Korea University III -1 Overview States What

More information

Control Systems Design

Control Systems Design ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:

More information

VALLIAMMAI ENGINEERING COLLEGE

VALLIAMMAI ENGINEERING COLLEGE VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203 DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING QUESTION BANK V SEMESTER IC650 CONTROL SYSTEMS Regulation 203 Academic Year 207 8 Prepared

More information

EECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators.

EECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators. Name: SID: EECS C128/ ME C134 Final Wed. Dec. 14, 211 81-11 am Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth 1 points total. Problem Points Score 1 16 2 12

More information

1 An Overview and Brief History of Feedback Control 1. 2 Dynamic Models 23. Contents. Preface. xiii

1 An Overview and Brief History of Feedback Control 1. 2 Dynamic Models 23. Contents. Preface. xiii Contents 1 An Overview and Brief History of Feedback Control 1 A Perspective on Feedback Control 1 Chapter Overview 2 1.1 A Simple Feedback System 3 1.2 A First Analysis of Feedback 6 1.3 Feedback System

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

D(s) G(s) A control system design definition

D(s) G(s) A control system design definition R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure

More information

FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY

FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY Senkottai Village, Madurai Sivagangai Main Road, Madurai - 625 020. An ISO 9001:2008 Certified Institution DEPARTMENT OF ELECTRONICS AND COMMUNICATION

More information

EC 8391-CONTROL SYSTEMS ENGINEERING. Questions and Answers PART-A. Unit - I Systems Components And Their Representation

EC 8391-CONTROL SYSTEMS ENGINEERING. Questions and Answers PART-A. Unit - I Systems Components And Their Representation EC 8391-CONTROL SYSTEMS ENGINEERING Questions and Answers PART-A Unit - I Systems Components And Their Representation 1. What is control system? A system consists of a number of components connected together

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

Control System Design

Control System Design ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science

More information

EC CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING

EC CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING EC 2255 - CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING 1. What is meant by a system? It is an arrangement of physical components related in such a manner as to form an entire unit. 2. List the two types

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

Control Systems. University Questions

Control Systems. University Questions University Questions UNIT-1 1. Distinguish between open loop and closed loop control system. Describe two examples for each. (10 Marks), Jan 2009, June 12, Dec 11,July 08, July 2009, Dec 2010 2. Write

More information

Analysis of Discrete-Time Systems

Analysis of Discrete-Time Systems TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time

More information

Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2)

Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) For all calculations in this book, you can use the MathCad software or any other mathematical software that you are familiar

More information

R10 JNTUWORLD B 1 M 1 K 2 M 2. f(t) Figure 1

R10 JNTUWORLD B 1 M 1 K 2 M 2. f(t) Figure 1 Code No: R06 R0 SET - II B. Tech II Semester Regular Examinations April/May 03 CONTROL SYSTEMS (Com. to EEE, ECE, EIE, ECC, AE) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions carry

More information

FEEDBACK CONTROL SYSTEMS

FEEDBACK CONTROL SYSTEMS FEEDBAC CONTROL SYSTEMS. Control System Design. Open and Closed-Loop Control Systems 3. Why Closed-Loop Control? 4. Case Study --- Speed Control of a DC Motor 5. Steady-State Errors in Unity Feedback Control

More information

EEE 184: Introduction to feedback systems

EEE 184: Introduction to feedback systems EEE 84: Introduction to feedback systems Summary 6 8 8 x 7 7 6 Level() 6 5 4 4 5 5 time(s) 4 6 8 Time (seconds) Fig.. Illustration of BIBO stability: stable system (the input is a unit step) Fig.. step)

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

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

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67 1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure

More information

INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad ELECTRICAL AND ELECTRONICS ENGINEERING TUTORIAL QUESTION BANK

INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad ELECTRICAL AND ELECTRONICS ENGINEERING TUTORIAL QUESTION BANK Course Name Course Code Class Branch INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad -500 043 ELECTRICAL AND ELECTRONICS ENGINEERING TUTORIAL QUESTION BAN : CONTROL SYSTEMS : A50 : III B. Tech

More information

Analysis of Discrete-Time Systems

Analysis of Discrete-Time Systems TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time

More information

Feedback Control of Linear SISO systems. Process Dynamics and Control

Feedback Control of Linear SISO systems. Process Dynamics and Control Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals

More information

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 Electrical and Electronics Engineering TUTORIAL QUESTION BAN Course Name : CONTROL SYSTEMS Course Code : A502 Class : III

More information

LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM II LAB EE 693

LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM II LAB EE 693 LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM II LAB EE 693 ELECTRICAL ENGINEERING DEPARTMENT JIS COLLEGE OF ENGINEERING (AN AUTONOMOUS INSTITUTE) KALYANI, NADIA EXPERIMENT NO : CS II/ TITLE : FAMILIARIZATION

More information

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory Bangladesh University of Engineering and Technology Electrical and Electronic Engineering Department EEE 402: Control System I Laboratory Experiment No. 4 a) Effect of input waveform, loop gain, and system

More information

EE C128 / ME C134 Final Exam Fall 2014

EE C128 / ME C134 Final Exam Fall 2014 EE C128 / ME C134 Final Exam Fall 2014 December 19, 2014 Your PRINTED FULL NAME Your STUDENT ID NUMBER Number of additional sheets 1. No computers, no tablets, no connected device (phone etc.) 2. Pocket

More information

Table of Laplacetransform

Table of Laplacetransform Appendix Table of Laplacetransform pairs 1(t) f(s) oct), unit impulse at t = 0 a, a constant or step of magnitude a at t = 0 a s t, a ramp function e- at, an exponential function s + a sin wt, a sine fun

More information

CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version

CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version Norman S. Nise California State Polytechnic University, Pomona John Wiley fir Sons, Inc. Contents PREFACE, vii 1. INTRODUCTION, 1

More information

EECS C128/ ME C134 Final Thu. May 14, pm. Closed book. One page, 2 sides of formula sheets. No calculators.

EECS C128/ ME C134 Final Thu. May 14, pm. Closed book. One page, 2 sides of formula sheets. No calculators. Name: SID: EECS C28/ ME C34 Final Thu. May 4, 25 5-8 pm Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 4 2 4 3 6 4 8 5 3

More information

Exercises for lectures 13 Design using frequency methods

Exercises for lectures 13 Design using frequency methods Exercises for lectures 13 Design using frequency methods Michael Šebek Automatic control 2016 31-3-17 Setting of the closed loop bandwidth At the transition frequency in the open loop is (from definition)

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

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

Lec 6: State Feedback, Controllability, Integral Action

Lec 6: State Feedback, Controllability, Integral Action Lec 6: State Feedback, Controllability, Integral Action November 22, 2017 Lund University, Department of Automatic Control Controllability and Observability Example of Kalman decomposition 1 s 1 x 10 x

More information

MAS107 Control Theory Exam Solutions 2008

MAS107 Control Theory Exam Solutions 2008 MAS07 CONTROL THEORY. HOVLAND: EXAM SOLUTION 2008 MAS07 Control Theory Exam Solutions 2008 Geir Hovland, Mechatronics Group, Grimstad, Norway June 30, 2008 C. Repeat question B, but plot the phase curve

More information

MODERN CONTROL DESIGN

MODERN CONTROL DESIGN CHAPTER 8 MODERN CONTROL DESIGN The classical design techniques of Chapters 6 and 7 are based on the root-locus and frequency response that utilize only the plant output for feedback with a dynamic controller

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

Transient response via gain adjustment. Consider a unity feedback system, where G(s) = 2. The closed loop transfer function is. s 2 + 2ζωs + ω 2 n

Transient response via gain adjustment. Consider a unity feedback system, where G(s) = 2. The closed loop transfer function is. s 2 + 2ζωs + ω 2 n Design via frequency response Transient response via gain adjustment Consider a unity feedback system, where G(s) = ωn 2. The closed loop transfer function is s(s+2ζω n ) T(s) = ω 2 n s 2 + 2ζωs + ω 2

More information

Control Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.

Control Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. Control Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Analog vs. digital systems Continuous- and Discretetime Dynamic Models Frequency Response Transfer Functions

More information

Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint

Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint Laboratory 11 State Feedback Controller for Position Control of a Flexible Joint 11.1 Objective The objective of this laboratory is to design a full state feedback controller for endpoint position control

More information

Prüfung Regelungstechnik I (Control Systems I) Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

Prüfung Regelungstechnik I (Control Systems I) Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 29. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid

More information

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

More information

Lecture 11. Frequency Response in Discrete Time Control Systems

Lecture 11. Frequency Response in Discrete Time Control Systems EE42 - Discrete Time Systems Spring 28 Lecturer: Asst. Prof. M. Mert Ankarali Lecture.. Frequency Response in Discrete Time Control Systems Let s assume u[k], y[k], and G(z) represents the input, output,

More information

1 (30 pts) Dominant Pole

1 (30 pts) Dominant Pole EECS C8/ME C34 Fall Problem Set 9 Solutions (3 pts) Dominant Pole For the following transfer function: Y (s) U(s) = (s + )(s + ) a) Give state space description of the system in parallel form (ẋ = Ax +

More information

6.1 Sketch the z-domain root locus and find the critical gain for the following systems K., the closed-loop characteristic equation is K + z 0.

6.1 Sketch the z-domain root locus and find the critical gain for the following systems K., the closed-loop characteristic equation is K + z 0. 6. Sketch the z-domain root locus and find the critical gain for the following systems K (i) Gz () z 4. (ii) Gz K () ( z+ 9. )( z 9. ) (iii) Gz () Kz ( z. )( z ) (iv) Gz () Kz ( + 9. ) ( z. )( z 8. ) (i)

More information

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD 206 Spring Semester ELEC733 Digital Control System LECTURE 7: DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD For a unit ramp input Tz Ez ( ) 2 ( z ) D( z) G( z) Tz e( ) lim( z) z 2 ( z ) D( z)

More information

Observer Incorporated Neoclassical Controller Design: A Discrete Perspective

Observer Incorporated Neoclassical Controller Design: A Discrete Perspective Marquette University e-publications@marquette Master's Theses (2009 -) Dissertations, Theses, and Professional Projects Observer Incorporated Neoclassical Controller Design: A Discrete Perspective Winston

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect

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

The output voltage is given by,

The output voltage is given by, 71 The output voltage is given by, = (3.1) The inductor and capacitor values of the Boost converter are derived by having the same assumption as that of the Buck converter. Now the critical value of the

More information

Advanced Digital Controls

Advanced Digital Controls Advanced Digital Controls University of California, Los Angeles Department of Mechanical and Aerospace Engineering Report Author David Luong Winter 8 Table of Contents Abstract 4 Introduction..5 Physical

More information

Outline. Classical Control. Lecture 5

Outline. Classical Control. Lecture 5 Outline Outline Outline 1 What is 2 Outline What is Why use? Sketching a 1 What is Why use? Sketching a 2 Gain Controller Lead Compensation Lag Compensation What is Properties of a General System Why use?

More information

MAE 143B - Homework 7

MAE 143B - Homework 7 MAE 143B - Homework 7 6.7 Multiplying the first ODE by m u and subtracting the product of the second ODE with m s, we get m s m u (ẍ s ẍ i ) + m u b s (ẋ s ẋ u ) + m u k s (x s x u ) + m s b s (ẋ s ẋ u

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 Problem Set #4 Posted: Thursday, Mar. 7, 13 Due: Thursday, Mar. 14, 13 1. Sketch the Root

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

INTRODUCTION TO DIGITAL CONTROL

INTRODUCTION TO DIGITAL CONTROL ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a linear-time-invariant

More information

PID controllers. Laith Batarseh. PID controllers

PID controllers. Laith Batarseh. PID controllers Next Previous 24-Jan-15 Chapter six Laith Batarseh Home End The controller choice is an important step in the control process because this element is responsible of reducing the error (e ss ), rise time

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

1 x(k +1)=(Φ LH) x(k) = T 1 x 2 (k) x1 (0) 1 T x 2(0) T x 1 (0) x 2 (0) x(1) = x(2) = x(3) =

1 x(k +1)=(Φ LH) x(k) = T 1 x 2 (k) x1 (0) 1 T x 2(0) T x 1 (0) x 2 (0) x(1) = x(2) = x(3) = 567 This is often referred to as Þnite settling time or deadbeat design because the dynamics will settle in a Þnite number of sample periods. This estimator always drives the error to zero in time 2T or

More information

Control Design. Lecture 9: State Feedback and Observers. Two Classes of Control Problems. State Feedback: Problem Formulation

Control Design. Lecture 9: State Feedback and Observers. Two Classes of Control Problems. State Feedback: Problem Formulation Lecture 9: State Feedback and s [IFAC PB Ch 9] State Feedback s Disturbance Estimation & Integral Action Control Design Many factors to consider, for example: Attenuation of load disturbances Reduction

More information

Control Systems Lab - SC4070 Control techniques

Control Systems Lab - SC4070 Control techniques Control Systems Lab - SC4070 Control techniques Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.:015-2788131 TU Delft, February 16, 2015 (slides modified from

More information

ECE 486 Control Systems

ECE 486 Control Systems ECE 486 Control Systems Spring 208 Midterm #2 Information Issued: April 5, 208 Updated: April 8, 208 ˆ This document is an info sheet about the second exam of ECE 486, Spring 208. ˆ Please read the following

More information

Course Summary. The course cannot be summarized in one lecture.

Course Summary. The course cannot be summarized in one lecture. Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques

More information

Laplace Transform Analysis of Signals and Systems

Laplace Transform Analysis of Signals and Systems Laplace Transform Analysis of Signals and Systems Transfer Functions Transfer functions of CT systems can be found from analysis of Differential Equations Block Diagrams Circuit Diagrams 5/10/04 M. J.

More information

STABILITY OF CLOSED-LOOP CONTOL SYSTEMS

STABILITY OF CLOSED-LOOP CONTOL SYSTEMS CHBE320 LECTURE X STABILITY OF CLOSED-LOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 10-1 Road Map of the Lecture X Stability of closed-loop control

More information

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions Cambridge University Engineering Dept. Third Year Module 3F: Systems and Control EXAMPLES PAPER ROOT-LOCUS Solutions. (a) For the system L(s) = (s + a)(s + b) (a, b both real) show that the root-locus

More information

Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30

Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30 289 Upcoming labs: Lecture 12 Lab 20: Internal model control (finish up) Lab 22: Force or Torque control experiments [Integrative] (2-3 sessions) Final Exam on 12/21/2015 (Monday)10:30-12:30 Today: Recap

More information

Example: DC Motor Speed Modeling

Example: DC Motor Speed Modeling Page 1 of 5 Example: DC Motor Speed Modeling Physical setup and system equations Design requirements MATLAB representation and open-loop response Physical setup and system equations A common actuator in

More information

EEE 184 Project: Option 1

EEE 184 Project: Option 1 EEE 184 Project: Option 1 Date: November 16th 2012 Due: December 3rd 2012 Work Alone, show your work, and comment your results. Comments, clarity, and organization are important. Same wrong result or same

More information

Dynamic Compensation using root locus method

Dynamic Compensation using root locus method CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 9 Dynamic Compensation using root locus method [] (Final00)For the system shown in the

More information

School of Mechanical Engineering Purdue University. ME375 ElectroMechanical - 1

School of Mechanical Engineering Purdue University. ME375 ElectroMechanical - 1 Electro-Mechanical Systems DC Motors Principles of Operation Modeling (Derivation of fg Governing Equations (EOM)) Block Diagram Representations Using Block Diagrams to Represent Equations in s - Domain

More information

CONTROL * ~ SYSTEMS ENGINEERING

CONTROL * ~ SYSTEMS ENGINEERING CONTROL * ~ SYSTEMS ENGINEERING H Fourth Edition NormanS. Nise California State Polytechnic University, Pomona JOHN WILEY& SONS, INC. Contents 1. Introduction 1 1.1 Introduction, 2 1.2 A History of Control

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

Predictive Cascade Control of DC Motor

Predictive Cascade Control of DC Motor Volume 49, Number, 008 89 Predictive Cascade Control of DC Motor Alexandru MORAR Abstract: The paper deals with the predictive cascade control of an electrical drive intended for positioning applications.

More information

CONTROL OF DIGITAL SYSTEMS

CONTROL OF DIGITAL SYSTEMS AUTOMATIC CONTROL AND SYSTEM THEORY CONTROL OF DIGITAL SYSTEMS Gianluca Palli Dipartimento di Ingegneria dell Energia Elettrica e dell Informazione (DEI) Università di Bologna Email: gianluca.palli@unibo.it

More information

Power System Control

Power System Control Power System Control Basic Control Engineering Prof. Wonhee Kim School of Energy Systems Engineering, Chung-Ang University 2 Contents Why feedback? System Modeling in Frequency Domain System Modeling in

More information

ME 475/591 Control Systems Final Exam Fall '99

ME 475/591 Control Systems Final Exam Fall '99 ME 475/591 Control Systems Final Exam Fall '99 Closed book closed notes portion of exam. Answer 5 of the 6 questions below (20 points total) 1) What is a phase margin? Under ideal circumstances, what does

More information

Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods

More information

Automatic Control (TSRT15): Lecture 7

Automatic Control (TSRT15): Lecture 7 Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Outline 2 Feedforward

More information

Linear State Feedback Controller Design

Linear State Feedback Controller Design Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University

More information

Acceleration Feedback

Acceleration Feedback Acceleration Feedback Mechanical Engineer Modeling & Simulation Electro- Mechanics Electrical- Electronics Engineer Sensors Actuators Computer Systems Engineer Embedded Control Controls Engineer Mechatronic

More information

EEE582 Homework Problems

EEE582 Homework Problems EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Steady-state error Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling Analysis Design Laplace

More information

Variable-gain output feedback control

Variable-gain output feedback control 7. Variable-gain output feedback control 7.1. Introduction PUC-Rio - Certificação Digital Nº 611865/CA In designing control laws, the usual first step is to describe the plant at a given operating point

More information

Outline. Classical Control. Lecture 1

Outline. Classical Control. Lecture 1 Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction

More information