EE C128 / ME C134 Final Exam Fall 2014

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "EE C128 / ME C134 Final Exam Fall 2014"

Transcription

1 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 calculator allowed. 3. Closed book, closed notes, closed internet. 4. Allowed: 2 sheets (each double sided) Chi Chi. 5. Additional sheets are available and may be submitted (e.g. for graphs). 6. Write your name below, and your SID on the top right corner of every page (including this one). 7. If you turn in additional sheets: ˆ Write your name and/or SID on every sheet, and ˆ Write the number of additional sheets you are turning in above where indicated. 8. Do not write on the back of any page. Part Score 1

2 1. Laplace transform, controllable canonical form (a) Derive the following Laplace property for convolution integrals L {g 1 (t)} = G 1 (s) L {g 2 (t)} = G 2 (s) { t } L g 1 (τ)g 2 (t τ)dτ = G 1 (s)g 2 (s) 0 (b) Find the solution to the following ODE with the given initial conditions: d 2 x dt 2 dx dt + 2x = t ẋ(0) = 2, x(0) = 1 0 δ(τ) sin (t τ)dτ UC Berkeley, 12/19/ of 22

3 UC Berkeley, 12/19/ of 22

4 2. Bode Plotting and Nyquist Stability Consider the following transfer function: G(s) = 100(s + 4) (s s + 20)(s + 2) (a) Sketch a Bode plot of the system (magnitude and phase). Label all slopes and points on the graph. UC Berkeley, 12/19/ of 22

5 (b) Using the Bode plots you created in Part (a), calculate the phase margin and gain margin for G(s). (c) Draw a Nyquist plot of the system and use the Nyquist stability criterion to determine if the closed loop system under unity feedback is stable. UC Berkeley, 12/19/ of 22

6 (d) Assume we have a closed loop system below where G(s) is given in Part (a) and C(s) = K. For what values of K is the system stable? UC Berkeley, 12/19/ of 22

7 3. Solution in time domain, stability (a) For the following system, explicitly determine the time-domain solution x(t) ẋ = where u(t) is a unit step function. [ ] x + [ 1 0 ] [ y = 1 0 u, x(0) = ] x [ 1 1 ] UC Berkeley, 12/19/ of 22

8 (b) Determine the transfer function G(s) for the zero initial state response, given the system in Part (a). (c) BONUS: Does the degree of your state space model and transfer function match? Why or why not? UC Berkeley, 12/19/ of 22

9 4. Electrical Circuit and Root Locus (a) For the above circuit derive the transfer function C(s) = V out V in. UC Berkeley, 12/19/ of 22

10 (b) Assume the system below, G(s), is in unity negative feedback. Determine the value of K such that the steady state error to a step response is Also determine the percent overshoot and settling time of the feedback system at this K. G(s) = K (s + 5)(s + 15) (c) Now assume that the system is described by the figure below, where C(s) and G(s) are obtained from parts (a) and (b) respectively. Draw the root locus given R 1 = 125 MΩ, C 1 = 15 µf, R 2 = 625 MΩ and C 2 = 0.1 µf. Watch your signs! UC Berkeley, 12/19/ of 22

11 (d) Label on the root locus a suitable region if the goal is to achieve a settling time (T s ) 0.4 sec and percent overshoot (%OS) 20%. UC Berkeley, 12/19/ of 22

12 5. Controller design using State-feedback Consider the mechanical system shown above. Here, V denotes the voltage applied to the motor (control input) and x(t) is the position of the mass. You may assume the back emf from the motor is negligible (EMF = 0) and the torque supplied by the motor is equal to T = IK m J m θ where, K m : Constant relating T and I J m : Inertia of the motor (a) Show that G(s) is the transfer function from V to x. To do this, you MUST derive the governing equations for the mechanical/electrical system. [( G(s) = X(s) V (s) = N ( ) ) ] 2 K m r 2 1 N2 J 1 + J 2 + J m s 2 + r 2 (Ms 2 + f v s + k) N 1 R + Ls N 1 UC Berkeley, 12/19/ of 22

13 [ T (b) Choosing x = x ẋ] as the state vector and x as the output, derive a state space model (matrices A, B, C and D) for the above system. Use the following parameters: R = 1, K m = 0.1, L = 0, N 2 /N 1 = 10, r = 1, J 1 = J 2 = 1, J m = 0, M = 1, k = 1, f v = 1. Note your input to the system should be V. (c) Explicitly write the observability and controllability matrices. Is the system controllable? Is it observable? (Use parameters from Part (b)) UC Berkeley, 12/19/ of 22

14 (d) Determine the eigenvalues of matrix A you found in Part (b). (e) We will now control the system[ using ] a state feedback controller as shown in the diagram below ] r 1 where K = [k 1, k 2 and r =. r 2 Write the dynamics of the closed-loop system as ẋ = Ãx + Br. That is, find both à and B in terms of the system parameters given in Part (b) and the elements of the controller gain matrix K. UC Berkeley, 12/19/ of 22

15 6. State-feedback and observer design Consider the following system: ẋ = Ax + Bu, y = Cx with A = [ (a) Compute the eigenvalues and eigenvectors for A. ], B = [ 0 1 ] [, C = 0 1 ] (b) Use state-feedback of the form of u = Kx. Determine the gain K = [k 1 k 2 ] such that the poles of the closed loop system are located at s 1,2 = 2 ± 5j. UC Berkeley, 12/19/ of 22

16 (c) Unfortunately for this system we are unable to measure all the states. In order to do state feedback we must use a Luenberger observer of the form: ˆx = Aˆx + Bu + L(y ŷ) ŷ = C ˆx and the system is controlled using state feedback, given by u = K ˆx Determine the error dynamics of the system, ė, where e = ˆx x. The result must be in terms of e only. (d) Determine the observer matrix L = [l 1 l 2 ] T such that the error dynamics have poles at s 1,2 = 2 ± 5j. UC Berkeley, 12/19/ of 22

17 (e) Comment on the performance of the state observer given the previously placed poles. What are we interested in when designing an observer and how could we improve the observer? (f) Complete the following block diagram of the system described in part(c). Plant Controller u y Observer ˆx UC Berkeley, 12/19/ of 22

18 7. Linear Quadratic Regulator Consider the LTI system ẋ = Ax + Bu x = [x 1, x 2 ] T where A = [ ] [ ] 1 1 1, B = We would like to solve an LQR problem for the system. control u (t) that minimizes the cost functional That is, we want to find the optimal J = t=0 (x 2 1(t) + u 2 (t)) dt (a) Solve the Algebraic Riccati Equation for the infinite horizon LQR. Hint: the solution of the Algebraic Riccati equation is a positive semi-definite matrix. A (2 2) matrix P is positive semi-definite everywhere when: [ ] p 11 0 p 22 0 p 2 p 11 p p 11 p 22 for P = p 12 p 22 UC Berkeley, 12/19/ of 22

19 (b) Determine the optimal feedback matrix K 1 such that the optimal control is u (t) = K 1 x(t). UC Berkeley, 12/19/ of 22

20 8. Linear Quadratic Regulator Consider the system where the dynamics are scalar: ẋ = ax + bu x is a scalar We want to create a finite horizon optimal controller given the cost function: J = tf t=0 (qx 2 (t) + ru 2 (t))dt (a) Write the Ricatti Equation for this system as well as the terminal condition. (b) Find P for the static case where t f =. Your answer should be in terms of q, r, a, and b. Note the P should be positive semi-definite everywhere. UC Berkeley, 12/19/ of 22

21 (c) For this scalar system, given any t f, the Riccati equation can be analytically solved: P (τ) = (ap (t f ) + q) sinh(βτ) + βp (t f ) cosh(βτ) ) a sinh(βτ) + β cosh(βτ) ( b 2 P (t f ) r where τ = t f t, β = a 2 + b2 q and sinh(.) and cosh(.) are the hyperbolic trigonometric r functions. Taking the limit as t f, the solution becomes: P (τ) = q a + β Show that this is equivalent to your solution from Part (b). UC Berkeley, 12/19/ of 22

22 f(t) F (s) δ(t) 1 1 u(t) s 1 tu(t) s 2 t n n! u(t) s n+1 ω sin(ωt)u(t) cos(ωt)u(t) e αt sin(ωt)u(t) e αt cos(ωt)u(t) s 2 +ω 2 s s 2 +ω 2 ω (s+α) 2 +ω 2 s+α (s+α) 2 +ω 2 Table 1: Laplace transforms of common functions sinh(θ) cosh(θ) tanh(θ) e θ e θ 2 e θ + e θ 2 sinh(θ) cosh θ = 1 e 2θ 1 + e 2θ Table 2: Trigonometric functions UC Berkeley, 12/19/ of 22

EE C128 / ME C134 Midterm Fall 2014

EE C128 / ME C134 Midterm Fall 2014 EE C128 / ME C134 Midterm Fall 2014 October 16, 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 calculator

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

EE221A Linear System Theory Final Exam

EE221A Linear System Theory Final Exam EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2016 12/16/16, 8-11am Your answers must be supported by analysis,

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

Problem Weight Score Total 100

Problem Weight Score Total 100 EE 350 EXAM IV 15 December 2010 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total

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

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

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

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

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels) GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-14 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page

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

Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10) Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane

More information

6 OUTPUT FEEDBACK DESIGN

6 OUTPUT FEEDBACK DESIGN 6 OUTPUT FEEDBACK DESIGN When the whole sate vector is not available for feedback, i.e, we can measure only y = Cx. 6.1 Review of observer design Recall from the first class in linear systems that a simple

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

Suppose that we have a specific single stage dynamic system governed by the following equation:

Suppose that we have a specific single stage dynamic system governed by the following equation: Dynamic Optimisation Discrete Dynamic Systems A single stage example Suppose that we have a specific single stage dynamic system governed by the following equation: x 1 = ax 0 + bu 0, x 0 = x i (1) where

More information

Linear Systems Theory

Linear Systems Theory ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace

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

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

EE 380 EXAM II 3 November 2011 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO

EE 380 EXAM II 3 November 2011 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO EE 380 EXAM II 3 November 2011 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total

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

Chapter 3. State Feedback - Pole Placement. Motivation

Chapter 3. State Feedback - Pole Placement. Motivation Chapter 3 State Feedback - Pole Placement Motivation Whereas classical control theory is based on output feedback, this course mainly deals with control system design by state feedback. This model-based

More information

Module 02 CPS Background: Linear Systems Preliminaries

Module 02 CPS Background: Linear Systems Preliminaries Module 02 CPS Background: Linear Systems Preliminaries Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html August

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

Subject: Optimal Control Assignment-1 (Related to Lecture notes 1-10)

Subject: Optimal Control Assignment-1 (Related to Lecture notes 1-10) Subject: Optimal Control Assignment- (Related to Lecture notes -). Design a oil mug, shown in fig., to hold as much oil possible. The height and radius of the mug should not be more than 6cm. The mug must

More information

OPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28

OPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28 OPTIMAL CONTROL Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 28 (Example from Optimal Control Theory, Kirk) Objective: To get from

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

Topic # Feedback Control Systems

Topic # Feedback Control Systems Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the

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

Homework Solution # 3

Homework Solution # 3 ECSE 644 Optimal Control Feb, 4 Due: Feb 17, 4 (Tuesday) Homework Solution # 3 1 (5%) Consider the discrete nonlinear control system in Homework # For the optimal control and trajectory that you have found

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

CONTROL DESIGN FOR SET POINT TRACKING

CONTROL DESIGN FOR SET POINT TRACKING Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded

More information

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature...

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature... Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: 29..23 Given and family names......................solutions...................... Student ID number..........................

More information

CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION

CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION Objectives Students should be able to: Draw the bode plots for first order and second order system. Determine the stability through the bode plots.

More information

EE C128 / ME C134 Fall 2014 HW 9 Solutions. HW 9 Solutions. 10(s + 3) s(s + 2)(s + 5) G(s) =

EE C128 / ME C134 Fall 2014 HW 9 Solutions. HW 9 Solutions. 10(s + 3) s(s + 2)(s + 5) G(s) = 1. Pole Placement Given the following open-loop plant, HW 9 Solutions G(s) = 1(s + 3) s(s + 2)(s + 5) design the state-variable feedback controller u = Kx + r, where K = [k 1 k 2 k 3 ] is the feedback

More information

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered

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

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

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

School of Engineering Faculty of Built Environment, Engineering, Technology & Design

School of Engineering Faculty of Built Environment, Engineering, Technology & Design Module Name and Code : ENG60803 Real Time Instrumentation Semester and Year : Semester 5/6, Year 3 Lecture Number/ Week : Lecture 3, Week 3 Learning Outcome (s) : LO5 Module Co-ordinator/Tutor : Dr. Phang

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

EE102 Homework 2, 3, and 4 Solutions

EE102 Homework 2, 3, and 4 Solutions EE12 Prof. S. Boyd EE12 Homework 2, 3, and 4 Solutions 7. Some convolution systems. Consider a convolution system, y(t) = + u(t τ)h(τ) dτ, where h is a function called the kernel or impulse response of

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

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

EE 16B Final, December 13, Name: SID #:

EE 16B Final, December 13, Name: SID #: EE 16B Final, December 13, 2016 Name: SID #: Important Instructions: Show your work. An answer without explanation is not acceptable and does not guarantee any credit. Only the front pages will be scanned

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

Introduction to Controls

Introduction to Controls EE 474 Review Exam 1 Name Answer each of the questions. Show your work. Note were essay-type answers are requested. Answer with complete sentences. Incomplete sentences will count heavily against the grade.

More information

5. Observer-based Controller Design

5. Observer-based Controller Design EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1

More information

9 Controller Discretization

9 Controller Discretization 9 Controller Discretization In most applications, a control system is implemented in a digital fashion on a computer. This implies that the measurements that are supplied to the control system must be

More information

University of Toronto Faculty of Applied Science and Engineering. ECE212H1F - Circuit Analysis. Final Examination December 16, :30am - noon

University of Toronto Faculty of Applied Science and Engineering. ECE212H1F - Circuit Analysis. Final Examination December 16, :30am - noon , LAST name: First name: Student ID: University of Toronto Faculty of Applied Science and Engineering ECE212H1F - Circuit Analysis Final Examination December 16, 2017 9:30am - noon Guidelines: Exam type:

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 9. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

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

ME 132, Fall 2015, Quiz # 2

ME 132, Fall 2015, Quiz # 2 ME 132, Fall 2015, Quiz # 2 # 1 # 2 # 3 # 4 # 5 # 6 Total NAME 14 10 8 6 14 8 60 Rules: 1. 2 sheets of notes allowed, 8.5 11 inches. Both sides can be used. 2. Calculator is allowed. Keep it in plain view

More information

Lecture 4 Continuous time linear quadratic regulator

Lecture 4 Continuous time linear quadratic regulator EE363 Winter 2008-09 Lecture 4 Continuous time linear quadratic regulator continuous-time LQR problem dynamic programming solution Hamiltonian system and two point boundary value problem infinite horizon

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

EE 3CL4: Introduction to Control Systems Lab 4: Lead Compensation

EE 3CL4: Introduction to Control Systems Lab 4: Lead Compensation EE 3CL4: Introduction to Control Systems Lab 4: Lead Compensation Tim Davidson Ext. 27352 davidson@mcmaster.ca Objective To use the root locus technique to design a lead compensator for a marginally-stable

More information

Lecture 2: Discrete-time Linear Quadratic Optimal Control

Lecture 2: Discrete-time Linear Quadratic Optimal Control ME 33, U Berkeley, Spring 04 Xu hen Lecture : Discrete-time Linear Quadratic Optimal ontrol Big picture Example onvergence of finite-time LQ solutions Big picture previously: dynamic programming and finite-horizon

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

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

Test 2 SOLUTIONS. ENGI 5821: Control Systems I. March 15, 2010

Test 2 SOLUTIONS. ENGI 5821: Control Systems I. March 15, 2010 Test 2 SOLUTIONS ENGI 5821: Control Systems I March 15, 2010 Total marks: 20 Name: Student #: Answer each question in the space provided or on the back of a page with an indication of where to find the

More information

LECTURE NOTES ON CONTROL

LECTURE NOTES ON CONTROL Department of Control for Transportation and Vehicle Systems Faculty of Transportation Engineering and Vehicle Engineering Budapest University of Technology and Economics Tamás Tettamanti PhD., Qiong Lu

More information

Robust Control 5 Nominal Controller Design Continued

Robust Control 5 Nominal Controller Design Continued Robust Control 5 Nominal Controller Design Continued Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 4/14/2003 Outline he LQR Problem A Generalization to LQR Min-Max

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

1. Find the solution of the following uncontrolled linear system. 2 α 1 1

1. Find the solution of the following uncontrolled linear system. 2 α 1 1 Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +

More information

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition

More information

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

10ES-43 CONTROL SYSTEMS ( ECE A B&C Section) % of Portions covered Reference Cumulative Chapter. Topic to be covered. Part A

10ES-43 CONTROL SYSTEMS ( ECE A B&C Section) % of Portions covered Reference Cumulative Chapter. Topic to be covered. Part A 10ES-43 CONTROL SYSTEMS ( ECE A B&C Section) Faculty : Shreyus G & Prashanth V Chapter Title/ Class # Reference Literature Topic to be covered Part A No of Hours:52 % of Portions covered Reference Cumulative

More information

AMME3500: System Dynamics & Control

AMME3500: System Dynamics & Control Stefan B. Williams May, 211 AMME35: System Dynamics & Control Assignment 4 Note: This assignment contributes 15% towards your final mark. This assignment is due at 4pm on Monday, May 3 th during Week 13

More information

ECE Circuit Theory. Final Examination. December 5, 2008

ECE Circuit Theory. Final Examination. December 5, 2008 ECE 212 H1F Pg 1 of 12 ECE 212 - Circuit Theory Final Examination December 5, 2008 1. Policy: closed book, calculators allowed. Show all work. 2. Work in the provided space. 3. The exam has 3 problems

More information

UNIVERSITY OF BOLTON SCHOOL OF ENGINEERING. MSc SYSTEMS ENGINEERING AND ENGINEERING MANAGEMENT SEMESTER 2 EXAMINATION 2015/2016

UNIVERSITY OF BOLTON SCHOOL OF ENGINEERING. MSc SYSTEMS ENGINEERING AND ENGINEERING MANAGEMENT SEMESTER 2 EXAMINATION 2015/2016 TW2 UNIVERSITY OF BOLTON SCHOOL OF ENGINEERING MSc SYSTEMS ENGINEERING AND ENGINEERING MANAGEMENT SEMESTER 2 EXAMINATION 2015/2016 ADVANCED CONTROL TECHNOLOGY MODULE NO: EEM7015 Date: Monday 16 May 2016

More information

Conventional Paper-I-2011 PART-A

Conventional Paper-I-2011 PART-A Conventional Paper-I-0 PART-A.a Give five properties of static magnetic field intensity. What are the different methods by which it can be calculated? Write a Maxwell s equation relating this in integral

More information

ECE557 Systems Control

ECE557 Systems Control ECE557 Systems Control Bruce Francis Course notes, Version.0, September 008 Preface This is the second Engineering Science course on control. It assumes ECE56 as a prerequisite. If you didn t take ECE56,

More information

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels) GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 09-Dec-13 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page

More information

Control Systems. Design of State Feedback Control.

Control Systems. Design of State Feedback Control. Control Systems Design of State Feedback Control chibum@seoultech.ac.kr Outline Design of State feedback control Dominant pole design Symmetric root locus (linear quadratic regulation) 2 Selection of closed-loop

More information

LQR, Kalman Filter, and LQG. Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin

LQR, Kalman Filter, and LQG. Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin LQR, Kalman Filter, and LQG Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin May 2015 Linear Quadratic Regulator (LQR) Consider a linear system

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

ME 375 EXAM #1 Friday, March 13, 2015 SOLUTION

ME 375 EXAM #1 Friday, March 13, 2015 SOLUTION ME 375 EXAM #1 Friday, March 13, 2015 SOLUTION PROBLEM 1 A system is made up of a homogeneous disk (of mass m and outer radius R), particle A (of mass m) and particle B (of mass m). The disk is pinned

More information

ECE : Linear Circuit Analysis II

ECE : Linear Circuit Analysis II Purdue University School of Electrical and Computer Engineering ECE 20200 : Linear Circuit Analysis II Summer 2014 Instructor: Aung Kyi San Instructions: Midterm Examination I July 2, 2014 1. Wait for

More information

'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ. EGR 224 Spring Test II. Michael R. Gustafson II

'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ. EGR 224 Spring Test II. Michael R. Gustafson II 'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ EGR 224 Spring 2018 Test II Michael R. Gustafson II Name (please print) In keeping with the Community Standard, I have neither provided nor received any

More information

Chapter 7. Digital Control Systems

Chapter 7. Digital Control Systems Chapter 7 Digital Control Systems 1 1 Introduction In this chapter, we introduce analysis and design of stability, steady-state error, and transient response for computer-controlled systems. Transfer functions,

More information

ECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University

ECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University ECEn 483 / ME 431 Case Studies Randal W. Beard Brigham Young University Updated: December 2, 2014 ii Contents 1 Single Link Robot Arm 1 2 Pendulum on a Cart 9 3 Satellite Attitude Control 17 4 UUV Roll

More information

1 Steady State Error (30 pts)

1 Steady State Error (30 pts) Professor Fearing EECS C28/ME C34 Problem Set Fall 2 Steady State Error (3 pts) Given the following continuous time (CT) system ] ẋ = A x + B u = x + 2 7 ] u(t), y = ] x () a) Given error e(t) = r(t) y(t)

More information

Lecture 9. Introduction to Kalman Filtering. Linear Quadratic Gaussian Control (LQG) G. Hovland 2004

Lecture 9. Introduction to Kalman Filtering. Linear Quadratic Gaussian Control (LQG) G. Hovland 2004 MER42 Advanced Control Lecture 9 Introduction to Kalman Filtering Linear Quadratic Gaussian Control (LQG) G. Hovland 24 Announcement No tutorials on hursday mornings 8-9am I will be present in all practical

More information

CHAPTER 1 Basic Concepts of Control System. CHAPTER 6 Hydraulic Control System

CHAPTER 1 Basic Concepts of Control System. CHAPTER 6 Hydraulic Control System CHAPTER 1 Basic Concepts of Control System 1. What is open loop control systems and closed loop control systems? Compare open loop control system with closed loop control system. Write down major advantages

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 5. 2. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

More information

CYBER EXPLORATION LABORATORY EXPERIMENTS

CYBER EXPLORATION LABORATORY EXPERIMENTS CYBER EXPLORATION LABORATORY EXPERIMENTS 1 2 Cyber Exploration oratory Experiments Chapter 2 Experiment 1 Objectives To learn to use MATLAB to: (1) generate polynomial, (2) manipulate polynomials, (3)

More information

'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ. EGR 224 Spring Test II. Michael R. Gustafson II

'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ. EGR 224 Spring Test II. Michael R. Gustafson II 'XNH8QLYHUVLW\ (GPXQG73UDWW-U6FKRRORI(QJLQHHULQJ EGR 224 Spring 2017 Test II Michael R. Gustafson II Name (please print) In keeping with the Community Standard, I have neither provided nor received any

More information

Lecture 9: Discrete-Time Linear Quadratic Regulator Finite-Horizon Case

Lecture 9: Discrete-Time Linear Quadratic Regulator Finite-Horizon Case Lecture 9: Discrete-Time Linear Quadratic Regulator Finite-Horizon Case Dr. Burak Demirel Faculty of Electrical Engineering and Information Technology, University of Paderborn December 15, 2015 2 Previous

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

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

Introduction to Modern Control MT 2016

Introduction to Modern Control MT 2016 CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear

More information

Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis

Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of

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

ẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)

ẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7) EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and

More information

Quadratic Stability of Dynamical Systems. Raktim Bhattacharya Aerospace Engineering, Texas A&M University

Quadratic Stability of Dynamical Systems. Raktim Bhattacharya Aerospace Engineering, Texas A&M University .. Quadratic Stability of Dynamical Systems Raktim Bhattacharya Aerospace Engineering, Texas A&M University Quadratic Lyapunov Functions Quadratic Stability Dynamical system is quadratically stable if

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

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

Lecture 10 Linear Quadratic Stochastic Control with Partial State Observation

Lecture 10 Linear Quadratic Stochastic Control with Partial State Observation EE363 Winter 2008-09 Lecture 10 Linear Quadratic Stochastic Control with Partial State Observation partially observed linear-quadratic stochastic control problem estimation-control separation principle

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