Design Methods for Control Systems

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Design Methods for Control Systems"

Transcription

1 Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term

2 Schedule November 25 MSt December 2 MSt Homework # 1 December 9 MSt December 16 MSt Homework # 2 January 6 GM January 13 GM Homework # 3 January 20 GM January 27 GM Homework # 4 Some time in February: Review session

3 Overview Ch Introduction to to feedback control theory Ch Classical control system design Ch LQ, LQG and H 2 control system design Ch Design of of multivariable control systems Ch Uncertainty models and robustness Ch H optimization and µ -synthesis

4 Scope and features Mature review of classical and modern control system design techniques Linear time-invariant systems 70% SISO- 30% MIMO Continuous-time MATLAB exercises Control toolbox Mu-Tools and Robust Control toolboxes

5 Overview Ch Introduction to to feedback control theory Introduction Introduction Types Types of of control control systems systems Design Design issues issues Configurations Configurations High-gain High-gain feedback feedback Stability Stability Closed-loop Closed-loop characteristic characteristic polynomial polynomial Nyquist Nyquist criterion criterion Stability Stability margins margins Performance Performance System System functions functions Low Low and and high high frequencies frequencies Robustness Robustness Robustness Robustness functions functions Loop Loop shaping shaping Limits Limits of of performance performance Two-degree-of-freedom control control systems systems

6 Types of control systems Regulator systems Servo or positioning systems Tracking systems

7 Design issues Targets Closed-loop stability Disturbance attenuation Good command response Robustness Limitations Plant capacity Measurement noise

8 Configurations Two degrees of freedom One degree of freedom

9 High-gain feedback-1 Loop gain γ = ψ oφ Signal balance e = r γ () e Feedback equation e+ γ () e = r

10 High-gain feedback-2 Feedback equation e+ γ () e = r High gain: γ ( e ) >> e Implies: γ () e r Hence: e << r r ψ ( y) So that y ψ 1 () r In case of unit feedback y r Good tracking

11 High-gain feedback-3 Loop gain δ = ( φ) o ( ψ) Signal balance: z = d δ ( z) High gain: δ ( z ) >> z Hence: z << d Good disturbance reduction

12 High-gain feedback-4 Need closed-loop stability Good tracking and disturbance attentuation are retained as long as the closed-loop system remains stable the gain remains high Under these conditions high-gain feedback implies robustness with respect to loop uncertainty

13 Pitfalls of high-gain feedback High-gain feedback has pitfalls: Naively making the gain large easily results in an unstable feedback system Even if the feedback system is stable overly large plant inputs may occur that exceed the plant capacity Measurement noise causes loss of performance

14 Stability-1 State space representation: xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt ()

15 Stability-2 xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt () The closed-loop system is stable if its state space representation is asymptotically stable Equivalent statements: xt () 0 as t for every solution of xt &() = Axt () All eigenvalues of A have strictly negative real parts All roots of det(si A) have strictly negative real parts

16 Stability-3 The control system is BIBO stable if every bounded input signal r results in bounded output signals e, u and z. BIBO = bounded input bounded output Asymptotic stability BIBO stability The converse is not true

17 Stability-4 Internal stability Inject internal signals into each exposed interconnection of the system, and define additional internal output signals after each injection point Then the system is internally stable if it is BIBO stable with respect to all inputs (external and internal) and all (external and internal) outputs

18 Stability-5 Example

19 Stability-6 If each component system is stabilizable and detectable ( has no hidden unstable modes ) then Stability Internal stability When input-output descriptions are used (such as transfer functions) internal stability is often easier to check than asymptotic stability

20 Closed-loop characteristic polynomial-1 State space representation of the open-loop system: Characteristic polynomial: State space representation of the closed-loop system: Characteristic polynomial: xt &() = Axt () + Bet () yt () = Cxt () + Det () χ () s = det( si A) xt &() = Axt (), cl 1 Acl = A B( I + D) C χ () s = det( si A ) cl cl

21 Closed-loop characteristic polynomial-2 P(s) C(s) L(s)=P(s)C(s) plant transfer matrix compensator transfer matrix loop gain transfer matrix Then χ cl det( I + L( s)) () s = χ() s det( I + L ( ))

22 Closed-loop characteristic polynomial-3 χ cl det( I + L( s)) () s = χ() s det( I + L ( )) SISO case: N() s Y() s Ls () = PsCs () () = Ds () X() s Then (within a nonzero constant factor) χ () s = D() s X() s χ cl () s = D() s X() s + N() s Y() s

23 Nyquist criterion Consider the SISO case The locus of L( jω), ω R in the complex plane is called the Nyquist plot of the loop gain The number of unstable closedloop poles = The number of times the Nyquist plot encircles the point 1 + The number of unstable open-loop poles

24 Generalized Nyquist criterion Consider the MIMO case The number of unstable closed-loop poles = The number of times the locus of det( I + L( jω)), ω R encircles the origin + The number of unstable open-loop poles (Principle of the argument)

25 Stability margins-1 In the SISO case, the point 1 is a critical point for the Nyquist plot of the closed-loop system. If the Nyquist plot is changed so that it crosses the point 1 then the system becomes unstable If the closed-loop system is stable but the Nyquist plot passes closely by 1 then the system is near-unstable, that is, has an oscillatory response the system may become unstable by small perturbations of the plant, that is, the system is not robust

26 Stability margins-2 There exist various stability margins. They measure how close the Nyquist plot gets to 1 Gain margin k m Phase margin φ m Modulus margin τ m (Landau)

27 System functions: L and S Loop gain L Sensitivity function S L = PC z 1 = v 1{ + L S

28 System functions: R Input disturbance ( proces ) sensitivity function R 1 z = Pv = SPr { 1+ L R

29 System functions: H and T Closed-loop transfer function H Complementary sensitivity function T z L L = Fr H = L {1 + L H T F

30 System functions: U Input ( control ) sensitivity function U C u = ( Fr m v) CP U

31 Measurement noise 1 PC PC z = v+ Fr m PC PC PC S T T

32 S, R, U and T 1 P z1 1+ PC 1+ PC v1 S R v1 z = = C PC v v U T 1+ PC 1+ PC 2 2 2

33 Design interrelations S T 1 = 1 + L L = 1 + L U= T / P R= SP H = T F S and T are suitable objects for manipulation

34 Low and high frequencies-1 Typical shapes for S and T

35 Low and high frequencies-2 Loop gain L large at low frequencies: small at high frequencies: L( jω ) >> 1, S 1/ L, T 1 L( jω ) << 1, S 1, T L Crossover region: L( jω) 1

36 Low and high frequencies-3 input sensitivity C 1/ P for low frequencies U = T / P = 1+ PC C for high frequencies input disturbance sensitivity R P 1/ C for low frequencies = SP = 1+ PC P for high frequencies closed-loop transfer function H = TF F corrects T

37 Robustness functions-1 Sufficient condition for stability under perturbation: L( jω) L ( jω) < 1 + L ( jω), o ω R o

38 Robustness functions-2 Equivalently, L( jω) Lo( jω) 1 + Lo( jω) <, ω L ( jω) L ( jω) o o R or L( jω) Lo ( jω) 1 <, ω L ( jω) T ( jω) o o R

39 Robustness functions-3 Bound on the relative size of perturbations: L( jω) L ( jω) L o o ( jω ) W 1 ( jω), ω R Sufficient and necessary condition for stability under all perturbations that satisfy the bound: W 1 1 ( jω) <, ω R To ( jω )

40 Robustness functions-4 Size of the smallest perturbation that may destabilize the system: W 1 1 ( jω) =, ω R T ( jω ) o

41 Robustness functions-5 The preceding discussion focuses on preventing the Nyquist plot of the loop gain L from crossing the point 1. Preventing the inverse Nyquist plot that is, the Nyquist plot of 1/L from crossing the point 1 also guarantees stability. Sufficient condition: < 1 +, ω R L( jω) L ( jω) L ( jω) o o

42 Robustness functions-6 Equivalently, 1 1 L( jω) Lo ( jω) 1 <, ω R 1 So ( jω ) L ( jω ) o

43 Robustness functions-7 Consider perturbations such that 1 1 L( jω) Lo ( jω) 1 L ( jω ) o W 2 ( jω), ω R Sufficient and necessary condition for robust stability: W 2 1 ( jω) <, So ( j ) ω R ω

44 Robustness functions-8 Size of the smallest perturbation that may destabilize the system: W 2 1 ( jω) =, ω R So ( jω )

45 Combined robustness test-1 Define δ L ( jω) = L( jω) L ( jω) L o o ( jω ) δ L 1 ( jω) = 1 1 L( jω) Lo ( jω) 1 L ( jω ) o

46 Combined robustness test-2 Then the perturbed closed-loop system is stable if ω R δ L ( j ) 1 ω < 1 S( jω ) or δ L ( jω) < 1 T( jω ) typically satisfied at low frequencies typically satisfied at high frequencies

47 Combined robustness test-3 Critical frequency region: crossover area

48 Loop shaping Low frequencies: large loop gain High frequencies: small loop gain In the crossover region the phase is constrained because of stability

49 Bode s gain-phase relationship-1 Between break frequencies the loop gain behaves as Hence L( jω) c( jω) n L( jω) cω n π arg L( jω ) n 2 Phase and magnitude do not behave independently Bode s gain-phase relationship describes the relation more accurately

50 Bode s gain-phase relationship-2 The limitations imposed by stability on the phase in the crossover region by Bode s gain-phase relationship limit the rate at which the loop gain decreases: If, say, π arg L( jω) in the crossover region 2 then 1 L( jω) cω in the crossover region

51 Limits of performance Bode s integral The Freudenberg-Looze equalities Limitations are imposed by causality the pole-zero configuration

52 Bode s integral-1 If L has at least two more poles than zeros then 0 log S( jω) dω = π Re pi 0 The p i are the right-half plane poles of the loop gain. i Proof: Use the Poisson integral from complex function theory

53 Bode s integral-1

54 Bode s integral-1

55 Bode s integral-2 Dual result: Suppose that the loop has integrating action of at least order 2. Then 0 1 log T(1/ jω) dω = π Re 0 z The z i are the right-half plane zeros of the loop gain. i i

56 Freudenberg-Looze equality-1 Let z be any right-half plane zero of the loop gain. Poisson s formula of complex function theory leads to the equality 0 log( S( jω) ) dw ( ω) = log B ( z) 0 z 1 poles Strengthens Bode s integral W z 1 ω Imz 1 ω + Im ( ω) = arctan + arctan π Re z π Re z z Increasing function. Rises most steeply at z. B () poles s = i p p i i + s s Blaschke product

57 Freudenberg-Looze equality-2 W z for different values of arg z (a) arg z = 0 (b) arg z is almost π/2 Frequencies where W z rises most steeply contribute most to the integral

58 Freudenberg-Looze equality-3 The bounds for S hold provided µ 1 ε Wz ( ω1 ) 1 W ( ω ) z 1 B ( 1 ) poles ( z) Wz ω The dependence of the right-hand side on the various parameters may be analyzed

59 Freudenberg-Looze equality-4 Effects of right-half plane zeros on S S may be made small up to the frequency min i z i. Attempting to make S small beyond this frequency makes S peak Right-half plane poles further impair the achievable reduction of S (in particular, nearlycancelling right-half plane pole-zero pairs)

60 Freudenberg-Looze equality-5 Rederivation of the Freudenberg-Looze equality while 1 1 L replacing L with 1/L, so that S = = = 1+ L L L interchanging the roles of the poles and the zeros T leads to 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros

61 Freudenberg-Looze equality-6 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros p is any right-half plane pole of the loop gain, and B () zeros s = i z z i i + s s In the application of the equality, interchange the roles of low and high frequencies

62 Freudenberg-Looze equality-7 Effects of right-half plane poles on T T may be made small above the frequency max i p i. Attempting to make T small below this frequency makes T peak Right-half plane zeros further impair the achievable reduction of T (in particular, nearly-cancelling right-half plane pole-zero pairs)

63 Freudenberg-Looze equality-8 Consequences for S and T

64 Two-degree-of-freedom systems-1 N Y Let P =, C = D X Then the closed-loop transfer function is PC NY H = F = F 1+ PC D cl Dcl = DX + NY with D cl the closed-loop characteristic polynomial

65 Two-degree-of-freedom systems-2 Other two-degree-of-freedom configuration: H = NX D cl F has zeros at the roots of N and X H NY = F has zeros at the Dcl roots of N and Y Can the zeros of H be made independent of the feedback compensator?

66 Two-degree-of-freedom systems-3 Further two-degree-of-freedom configuration N Y Y P =, C = C = D X X Need XX 1 2 = X, YY 1 2 = Y to achieve the same loop gain as in the two previous cases Have H PC = = 1+ PC NX Y D cl

67 Two-degree-of-freedom systems-4 H is independent of the compensator if we let so that H = NX Y D 2 1 cl Y = X = 1 Y = Y, X = X N C1 =, C2 = Y, H = X D cl

68 Two-degree-of-freedom systems-5 Resulting feedback system Equivalent configuration

69 Two-degree-of-freedom systems-6 Extension May choose F polynomial so that we obtain a 1½-degree-of-freedom system

70 Two-degree-of-freedom systems-7 Further extension: F polynomial, F o rational: 2½-degree-of-freedom system F =1, F o rational: H = NF D cl F o 2-degree-of-freedom system

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral

More information

Chapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control

Chapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization non-stochastic version of LQG Application to feedback system design

More information

Let the plant and controller be described as:-

Let the plant and controller be described as:- Summary of Fundamental Limitations in Feedback Design (LTI SISO Systems) From Chapter 6 of A FIRST GRADUATE COURSE IN FEEDBACK CONTROL By J. S. Freudenberg (Winter 2008) Prepared by: Hammad Munawar (Institute

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

Analysis of SISO Control Loops

Analysis of SISO Control Loops Chapter 5 Analysis of SISO Control Loops Topics to be covered For a given controller and plant connected in feedback we ask and answer the following questions: Is the loop stable? What are the sensitivities

More information

(Continued on next page)

(Continued on next page) (Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic

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

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

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

CDS 101/110a: Lecture 8-1 Frequency Domain Design

CDS 101/110a: Lecture 8-1 Frequency Domain Design CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve

More information

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017

More information

Lecture 1: Feedback Control Loop

Lecture 1: Feedback Control Loop Lecture : Feedback Control Loop Loop Transfer function The standard feedback control system structure is depicted in Figure. This represend(t) n(t) r(t) e(t) u(t) v(t) η(t) y(t) F (s) C(s) P (s) Figure

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

Closed-loop system 2/1/2016. Generally MIMO case. Two-degrees-of-freedom (2 DOF) control structure. (2 DOF structure) The closed loop equations become

Closed-loop system 2/1/2016. Generally MIMO case. Two-degrees-of-freedom (2 DOF) control structure. (2 DOF structure) The closed loop equations become Closed-loop system enerally MIMO case Two-degrees-of-freedom (2 DOF) control structure (2 DOF structure) 2 The closed loop equations become solving for z gives where is the closed loop transfer function

More information

Robust Control 3 The Closed Loop

Robust Control 3 The Closed Loop Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002 Outline Closed Loop Transfer Functions Traditional Performance Measures Time

More information

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31 Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

More information

Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year

Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year Linear Control Systems Lecture #3 - Frequency Domain Analysis Guillaume Drion Academic year 2018-2019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closed-loop system

More information

3 Stabilization of MIMO Feedback Systems

3 Stabilization of MIMO Feedback Systems 3 Stabilization of MIMO Feedback Systems 3.1 Notation The sets R and S are as before. We will use the notation M (R) to denote the set of matrices with elements in R. The dimensions are not explicitly

More information

CDS 101/110a: Lecture 10-1 Robust Performance

CDS 101/110a: Lecture 10-1 Robust Performance CDS 11/11a: Lecture 1-1 Robust Performance Richard M. Murray 1 December 28 Goals: Describe how to represent uncertainty in process dynamics Describe how to analyze a system in the presence of uncertainty

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

MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan

MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan Outline Motivation & Background: H2 Tracking Performance Limits: new paradigm Explicit analytical solutions with examples H2 Regulation

More information

EE Control Systems LECTURE 9

EE Control Systems LECTURE 9 Updated: Sunday, February, 999 EE - Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of state-space

More information

Control Systems I Lecture 10: System Specifications

Control Systems I Lecture 10: System Specifications Control Systems I Lecture 10: System Specifications Readings: Guzzella, Chapter 10 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture

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

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability

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

Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems

Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations

More information

Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes

Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes Emmanuel Edet Technology and Innovation Centre University of Strathclyde 99 George Street Glasgow, United Kingdom emmanuel.edet@strath.ac.uk

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

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

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

Exercise 1 (A Non-minimum Phase System)

Exercise 1 (A Non-minimum Phase System) Prof. Dr. E. Frazzoli 5-59- Control Systems I (Autumn 27) Solution Exercise Set 2 Loop Shaping clruch@ethz.ch, 8th December 27 Exercise (A Non-minimum Phase System) To decrease the rise time of the system,

More information

Exam. 135 minutes, 15 minutes reading time

Exam. 135 minutes, 15 minutes reading time Exam August 15, 2017 Control Systems I (151-0591-00L) Prof Emilio Frazzoli Exam Exam Duration: 135 minutes, 15 minutes reading time Number of Problems: 44 Number of Points: 52 Permitted aids: Important:

More information

Chapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of

Chapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter Stability

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

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2) Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω)

More information

Problem Set 4 Solution 1

Problem Set 4 Solution 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback

More information

Control Systems. Root Locus & Pole Assignment. L. Lanari

Control Systems. Root Locus & Pole Assignment. L. Lanari Control Systems Root Locus & Pole Assignment L. Lanari Outline root-locus definition main rules for hand plotting root locus as a design tool other use of the root locus pole assignment Lanari: CS - Root

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

Exam. 135 minutes, 15 minutes reading time

Exam. 135 minutes, 15 minutes reading time Exam August 6, 208 Control Systems II (5-0590-00) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.

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

Exercise 1 (A Non-minimum Phase System)

Exercise 1 (A Non-minimum Phase System) Prof. Dr. E. Frazzoli 5-59- Control Systems I (HS 25) Solution Exercise Set Loop Shaping Noele Norris, 9th December 26 Exercise (A Non-minimum Phase System) To increase the rise time of the system, we

More information

ECE317 : Feedback and Control

ECE317 : Feedback and Control ECE317 : Feedback and Control Lecture : Routh-Hurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling

More information

Classify a transfer function to see which order or ramp it can follow and with which expected error.

Classify a transfer function to see which order or ramp it can follow and with which expected error. Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

More information

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08 Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.

More information

Lecture 7 (Weeks 13-14)

Lecture 7 (Weeks 13-14) Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.

More information

Analysis and Synthesis of Single-Input Single-Output Control Systems

Analysis and Synthesis of Single-Input Single-Output Control Systems Lino Guzzella Analysis and Synthesis of Single-Input Single-Output Control Systems l+kja» \Uja>)W2(ja»\ um Contents 1 Definitions and Problem Formulations 1 1.1 Introduction 1 1.2 Definitions 1 1.2.1 Systems

More information

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected

More information

Theory of Machines and Automatic Control Winter 2018/2019

Theory of Machines and Automatic Control Winter 2018/2019 Theory of Machines and Automatic Control Winter 2018/2019 Lecturer: Sebastian Korczak, PhD, Eng. Institute of Machine Design Fundamentals - Department of Mechanics http://www.ipbm.simr.pw.edu.pl/ Lecture

More information

Topic # Feedback Control Systems

Topic # Feedback Control Systems Topic #20 16.31 Feedback Control Systems Closed-loop system analysis Bounded Gain Theorem Robust Stability Fall 2007 16.31 20 1 SISO Performance Objectives Basic setup: d i d o r u y G c (s) G(s) n control

More information

Control Systems I. Lecture 9: The Nyquist condition

Control Systems I. Lecture 9: The Nyquist condition Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control

More information

Unit 11 - Week 7: Quantitative feedback theory (Part 1/2)

Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) X reviewer3@nptel.iitm.ac.in Courses» Control System Design Announcements Course Ask a Question Progress Mentor FAQ Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) Course outline How to access

More information

Topic # Feedback Control Systems

Topic # Feedback Control Systems Topic #19 16.31 Feedback Control Systems Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19

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

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

The stability of linear time-invariant feedback systems

The stability of linear time-invariant feedback systems The stability of linear time-invariant feedbac systems A. Theory The system is atrictly stable if The degree of the numerator of H(s) (H(z)) the degree of the denominator of H(s) (H(z)) and/or The poles

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. 8. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid

More information

Chapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a

Chapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 9 Robust

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

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

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

CDS 101/110a: Lecture 10-2 Control Systems Implementation

CDS 101/110a: Lecture 10-2 Control Systems Implementation CDS 101/110a: Lecture 10-2 Control Systems Implementation Richard M. Murray 5 December 2012 Goals Provide an overview of the key principles, concepts and tools from control theory - Classical control -

More information

Aircraft Stability & Control

Aircraft Stability & Control Aircraft Stability & Control Textbook Automatic control of Aircraft and missiles 2 nd Edition by John H Blakelock References Aircraft Dynamics and Automatic Control - McRuler & Ashkenas Aerodynamics, Aeronautics

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

FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO

More information

Chapter 7 - Solved Problems

Chapter 7 - Solved Problems Chapter 7 - Solved Problems Solved Problem 7.1. A continuous time system has transfer function G o (s) given by G o (s) = B o(s) A o (s) = 2 (s 1)(s + 2) = 2 s 2 + s 2 (1) Find a controller of minimal

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

MAE 143B - Homework 9

MAE 143B - Homework 9 MAE 43B - Homework 9 7.2 2 2 3.8.6.4.2.2 9 8 2 2 3 a) G(s) = (s+)(s+).4.6.8.2.2.4.6.8. Polar plot; red for negative ; no encirclements of, a.s. under unit feedback... 2 2 3. 4 9 2 2 3 h) G(s) = s+ s(s+)..2.4.6.8.2.4

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

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

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

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

Stability and Robustness 1

Stability and Robustness 1 Lecture 2 Stability and Robustness This lecture discusses the role of stability in feedback design. The emphasis is notonyes/notestsforstability,butratheronhowtomeasurethedistanceto instability. The small

More information

A brief introduction to robust H control

A brief introduction to robust H control A brief introduction to robust H control Jean-Marc Biannic System Control and Flight Dynamics Department ONERA, Toulouse. http://www.onera.fr/staff/jean-marc-biannic/ http://jm.biannic.free.fr/ European

More information

Full State Feedback for State Space Approach

Full State Feedback for State Space Approach Full State Feedback for State Space Approach State Space Equations Using Cramer s rule it can be shown that the characteristic equation of the system is : det[ si A] 0 Roots (for s) of the resulting polynomial

More information

MIMO analysis: loop-at-a-time

MIMO analysis: loop-at-a-time MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Plant: P (s) = 1 s 2 + α 2 s α 2 α(s + 1) α(s + 1) s α 2. (take α = 10 in the following numerical analysis) Controller:

More information

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.1-3, Emilio Frazzoli

Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.1-3, Emilio Frazzoli Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Guzzella 9.1-3, 13.3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 3, 2017 E. Frazzoli (ETH)

More information

Uncertainty and Robustness for SISO Systems

Uncertainty and Robustness for SISO Systems Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical

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

Lecture 4 Stabilization

Lecture 4 Stabilization Lecture 4 Stabilization This lecture follows Chapter 5 of Doyle-Francis-Tannenbaum, with proofs and Section 5.3 omitted 17013 IOC-UPC, Lecture 4, November 2nd 2005 p. 1/23 Stable plants (I) We assume that

More information

Singular Value Decomposition Analysis

Singular Value Decomposition Analysis Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control

More information

EL2520 Control Theory and Practice

EL2520 Control Theory and Practice EL2520 Control Theory and Practice Lecture 8: Linear quadratic control Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden Linear quadratic control Allows to compute the controller

More information

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard Control Systems II ETH, MAVT, IDSC, Lecture 4 17/03/2017 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded Control

More information

Lecture 6. Chapter 8: Robust Stability and Performance Analysis for MIMO Systems. Eugenio Schuster.

Lecture 6. Chapter 8: Robust Stability and Performance Analysis for MIMO Systems. Eugenio Schuster. Lecture 6 Chapter 8: Robust Stability and Performance Analysis for MIMO Systems Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 6 p. 1/73 6.1 General

More information

MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/8/25 Outline Closed Loop Transfer Functions

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

Control for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e

Control for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Control for Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Motion Systems m F Introduction Timedomain tuning Frequency domain & stability Filters Feedforward Servo-oriented

More information

Control of Electromechanical Systems

Control of Electromechanical Systems Control of Electromechanical Systems November 3, 27 Exercise Consider the feedback control scheme of the motor speed ω in Fig., where the torque actuation includes a time constant τ A =. s and a disturbance

More information

Discussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough

Discussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough Discussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough H.L. Trentelman 1 The geometric approach In the last

More information

Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli

Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 22, 2017 E. Frazzoli

More information

Chapter 6 - Solved Problems

Chapter 6 - Solved Problems Chapter 6 - Solved Problems Solved Problem 6.. Contributed by - James Welsh, University of Newcastle, Australia. Find suitable values for the PID parameters using the Z-N tuning strategy for the nominal

More information

Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli

Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli

More information

ECE 388 Automatic Control

ECE 388 Automatic Control Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage:

More information

CDS Solutions to Final Exam

CDS Solutions to Final Exam CDS 22 - Solutions to Final Exam Instructor: Danielle C Tarraf Fall 27 Problem (a) We will compute the H 2 norm of G using state-space methods (see Section 26 in DFT) We begin by finding a minimal state-space

More information

Limiting Performance of Optimal Linear Filters

Limiting Performance of Optimal Linear Filters Limiting Performance of Optimal Linear Filters J.H. Braslavsky M.M. Seron D.Q. Mayne P.V. Kokotović Technical Report CCEC 97-414 Center for Control Engineering and Computation University of California

More information

Some solutions of the written exam of January 27th, 2014

Some solutions of the written exam of January 27th, 2014 TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following

More information

Lecture plan: Control Systems II, IDSC, 2017

Lecture plan: Control Systems II, IDSC, 2017 Control Systems II MAVT, IDSC, Lecture 8 28/04/2017 G. Ducard Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded

More information

Robust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization

Robust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization A.2 Sensitivity and Feedback Performance A.3

More information

Control Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani

Control Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 2-3 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 28, 2018 J. Tani, E.

More information

Problem Set 5 Solutions 1

Problem Set 5 Solutions 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 5 Solutions The problem set deals with Hankel

More information