Stability and Robustness 1

Size: px
Start display at page:

Download "Stability and Robustness 1"

Transcription

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 gain theorem is introduced as a means to verify stability in presence of model uncertainty. 2. Stability of feedback systems Feedback and stability are closely connected issues. On one hand, the introduction of feedback may potentially create instability. On the other hand, properly applied feedbackisoftenthebestwaytogetridofinstabilities. There are several well known examples in the history. One is the construction ofairplanes.thefirstairplanesintheearly9sweredependingonthepilotto stabilize the dynamics manually using the control-stick. The modern fighter JAS- Gripen was built unstable for the sake of maneuverability and relies on computer control for stabilization. Figure 2. Lawrence Sperry demonstrates a stabilizing gyroscopic controller. He waves his handintheair,whilehismechaniciswalkingonthewing. Another striking example was the construction of the first electronic feedback amplifiers, that were necessary to build long distance telephone connections in the93s.inthiscase,highgainfeedbackwasneededtoreducethenonlinear signal distorsion. Stability problems became a major issue, and the development of frequency domain stability criteria was critical for successful implementation. A modern example is the maneuver test for Mercesdes A-class, the so called elk-test, that created unstable oscillations severe enough to turn the car over. The problem was solved by introducing electronic feedback control. Recall from the previous lecture that a system is called input-output stable (orl 2 -stable)ifitsl 2 -gainisbounded.atransferfunctioniscalledstableifit corresponds to an input-output stable system. The following stability criterion is available for linear time-invariant systems. THEOREM 2. ArationaltransferfunctionG(s)isstableifandonlyifallpolesofGhavenegative realpart.inparticular,if G(s) = C(sI A) B+D,itissufficientthatall eigenvalues of A have negative real part. WrittenbyA.RantzerwithcontributionsbyK.J.Åström

2 Lecture2. StabilityandRobustness Figure 2.2 The stability problems of Mercedes A-class were solved by electronic feedback Moregenerally,asystemwithimpulseresponse (t)isstableprovidedthatg(iω)= e iωt (t)dtiswelldefinedandthereisacwith G(iω) cforall ω. Example Let us determine input-output stability of the following systems (I) [ ] [ ] 2 ẋ= + u 3 2 y=[ ]x+u (II) y(t)= t ) e (u(τ) y(τ) τ t dτ (I) The first system is input-output stable due to stable eigenvalues of the systemmatrix.atwo-by-twomatrixlikethisisstableifandonlyifthetraceis negative(here 3) and the determinant is positive(here 8). This is because thetraceisthesumoftheeigenvaluesandthedeterminantistheproduct. In general, eigenvalues can be computed by the matlab command eig(a): >> eig([- 2; -3-2]) ans = i i Note that the coefficients of the characteristic polynomial should not be computed, at least for high order systems, since this generally leads to numerical difficulties. (I I) The input-output relationship can be written y= (u y) where (t)=e t,t.afterlaplacetransformation,thisgives Y(s)= s+ [U(s) Y(s)] Y(s)= s+2 U(s) Thetransferfunction/(s+2)showsthatthesystemisstable,sincethe only pole 2 is negative. Our main objective is to study stability of feedback loops. From the basic course, we recall the Nyquist criterion, which supports understanding by graphical illustrations. 2

3 2. Stability of feedback systems Σ G(s) Im.5 ω Am m.5 ω c.5.5 Re Figure 2.3 TheclosedloopsystemremainsstableaslongastheNyquistdiagramdoes notencircle.theamplitudemargina m andphasemargin φ m measurethedistancefrom instability. THEOREM 2.2 THE NYQUIST CRITERION SupposethatG(s)isstableandthattheNyquistplotG(iω), ω Rdoesnot encircle.then(+g(s)) isalsostable. The following example illustrates the use of the theorem. Example 2 As motivating example, consider a position control in a mechanical system with damping coefficient c. The controller contains a time delay: ẍ+cẋ+x=u u(t)= k[x(t T)+ẋ(t T)] The feedback loop is illustrated in the figure below. Nominal values of the parametersarek=,c=andt=.letusinvestigatehowmuchmarginthereisin each of the parameters before the system becomes unstable? k(s+) s 2 +cs+ e st For this purpose, we plot the Nyquist and Bode diagrams of the nominal transfer function(s+)/(s 2 +s+).thematlabcommand margingivesnumericalvalues for the amplitude- and phase-margins. Im.5 Nyquist diagram Re Magnitude Phase Gm = Inf, Pm = 9.47 deg (at.442 rad/sec) Frequency Figure2.4 Nyquistandbodeplotsforthenominaltransferfunction(s+)/(s 2 +s+) 3

4 Lecture2. StabilityandRobustness Fork=c=theopenlooptransferfunctionis s+ s 2 +s+ e st ForsmallvaluesofTtheNyquistplotwillnotencircle,sothesystemsremains stable.tofindoutexactlyhowlargevaluesoftareneededforinstability,notethat the phase margin 9 degrees(or 9π/8 radians) is obtained at the frequency 9π.4rad/sec.Henceatimedelayof 8.4 =.35secondscanbetoleratedfor k=c=. Toinvestigatetherobustnesstovariationsintheparameterscandk,wenote thattheclosedloopcharacteristicpolynomialfort=is (s 2 +cs+)+k(s+)=s 2 +(c+k)s++k The stability condition for a second degree polynomial is positive coefficients, so stabilityismaintainedaslongasc+k>and+k>. Having analyzed the example with respect to parametric uncertainty above, it is natural to ask about robustness to unmodelled dynamics. For example, this would be relevant to capture the difference between a rigid body and an elastic one.thisisalsothesubjectfortheremainingpartofthelecture. 2.2 Sensitivity Two transfer functions are of particular interest in the study of the feedback loop below. d n r e u Σ C(s) Σ P(s) x Σ y Figure 2.5 A simple control loop These are S(s)= +P(s)C(s) T(s)= P(s)C(s) +P(s)C(s) (the sensitivity function) (the complementary sensitivity function) ThetermcomplementaryreferstothefactthatS(s)+T(s).NotethatT(s) isthetransferfunctionfromreferencesignalrtothetheplantoutput x.another name for T(s) is therefore the closed loop transfer function of the system, while P(s)C(s) is called the open loop transfer function or just the loop transfer function. The name sensitivity function refers to the fact that S measures how small relativeerrorsin ParemappedintorelativeerrorsinT.Thisisverifiedbya simple calculation: dt dp = d dp ( ) = +PC C (+PC) 2=TS P dt/t dp/p =S 4

5 2.3 Norms of signals Im R =sup ω +C(iω)P(iω) R Re C(iω)P(iω) Figure 2.6 TheL 2 -gainofthesensitivityfunctionistheinverseofthedistancefromthe Nyquist plot to. Notice that T is a nonlinear function of P. Hence the sensitivity calculation only says something about the response to small pertubations in P. For larger perturbations, the system could have a drastically different behaviour and even become unstable. Given the Nyquist criterion, it is natural to conjecture that the robustness to unmodelled dynamics should somehow be related to the distance from the Nyquist plottothepoint.itisthereforestrikingtonotethatthel 2 -gainofthesensitivity function turns out to be exactly the inverse of this distance. Consequences are investigated in the next section. 2.3 Norms of signals InthepreviousChapterwelookedatsizesofsignals(norms)andgainsofsystems. RecallthattheL 2 -gainofasiso-systemwasthelargestmagnitudeinthebodediagram. Later in the course we will work with multivariable systems where the relation between vectors of inputs and outputs could be described as a matrix with transfer function elements, as for example [ y y 2 ] [ = 2 s+ s s 2 +s+ ] 4 [u 2s+ 3 u s+4 2 ] (2.) Forthatpurposewewillfirstintroducehowtomeasurethesizeofavector (vectornorm)andbasedonthistheinducednorm(gain)ofamatrix. Forx R n,weusethe L 2 -norm x = x T x= x 2+ +x2 n YoumaythinkofitasPythagoras theoreminr n. FormatricesM R n n,weusethe L 2 -inducednorm,thelargestratiobetween the output and the corresponding input M :=sup x Mx x =sup x xt M T Mx x T x = λ(m T M) Here λ(m T M)denotesthelargesteigenvalueof M T M.Thelargestgainis thus λ(m T M),alsocalledthelargestsingularvalueofM,denoted σ(m).the 5

6 Lecture2. StabilityandRobustness.5 Input u= [.39.95] T, u =.5 u u y=gu = [ ] T, y = y (red):eigenvectors ; (blue): V ; (green): U A=U*S*V T Figure 2.7 Matlab-example showing how the output vector y = Mu changes size and directionwhentheinputsignalwith u =arechangedindifferentdirectionsaroundtheunit circle.(note scaling of axes in lower plot). fraction Mx / x ismaximizedwhenxisacorrespondingeigenvectortom T M, butnotethatweingeneralhavedifferentdirectionsoftheinputvectorxandthe outputvectormx(asxingeneralisnotaeigenvectorofm). Example: [ y y 2 ] = [ ][ u u 2 ] Matlab-code for singular value decomposition of the matrix [ ] 2 4 M= 3 Singula Value Decomposition(SVD): M=U S V whereboththematricesu andv areunitary(i.e.haveorthonormalcolumnss.t. V V = I)and Sisthediagonal matrixwith(sorteddecreasing)singularvalues σ i. MultiplyingMwithainputvectoralongthefirstcolumnin Vgives M V (:,) =USV V (:,) = [ ] =US =U (:,) σ Thatis,wegetmaximalgain σ intheoutputdirectionu (:,) ifweuseaninputindirectionv (:,) (andminimalgain σ n = σ 2 ifweusethelastcolumnv (:,n) =V (:,2) ). >> M=[2 4 ; 3] M = >> [U,S,V]=svd(M) U = S = V = >> M*V(:,) ans = >> U(:,)*S(,) ans = Nowwhenweknowhowtocalculatetheinducednormforamatrix,wecan apply this to a transfer function matrix, where the elements depend on the frequency. The magnitude plot of the Bode diagram for single-input-single-output systems has the multivariable correspondence of plotting all singular values for 6

7 2.3 Norms of signals Singular Values 2 System: G Frequency (rad/sec):.6 Singular Value (abs): 5.26 Singular Values (abs) System: G Frequency (rad/sec):..6 Singular Value (abs): Frequency (rad/sec) Figure 2.8 The singular values of the tranfer function matrix in Eq. 2.. Note that G()=[2,4 ;3]whichcorrespondstoMintheSVD-exampleabove. G = thematrixtransferfunctionasafunctionofthefrequency.inmatlabthisisdone with the command sigma, see example below. ThegainofthesystemGisdefinedas G =max ω G(iω) and is the largest magnitude(largest singular value) when sweeping over all frequenciesfor G(iω),seeFig2.8. Example: Consider the transfer matrix G(iω) from Eq. 2. G(s)= 2 s+ s s 2 +.s+ 4 2s+ 3 s+ >> s=tf( s ) >> G=[ 2/(s+) 4/(2*s+); s/(s^2+.*s+) 3/(s+)]; >> sigma(g) % plot sigma values of G wrt fq >> grid on >> norm(g,inf) % infinity norm = system gain ans =

8 Lecture2. StabilityandRobustness 2.4 Robustness via the small gain theorem r e S e 2 r 2 S 2 In this section, we will investigate stability robustness using the following theorem,basedonthenotionofinput-outputl 2 -gain.forsimplicity,calculations are done assuming zero initial conditions. THEOREM 2.3 THE SMALL GAIN THEOREM AssumethatS ands 2 areinput-outputstablesystemswith L 2 -gain S and S 2.If S S 2 <,thenthel 2 -gainfrom(r,r 2 )to(e,e 2 )intheclosedloop system is finite. Proof. Define y T = T e =S 2 (e 2 )+r e 2 =S (e )+r 2 y(t) 2 dt.then S(y) T S y T. e =r +S 2 (r 2 +S (e )) ) e T r T + S 2 ( r 2 T + S e T e T r T + S 2 r 2 T S S 2 Boundedgainfrom(r,r 2 )toe followsast.thegaintoe 2 isboundedin the same way. To demonstrate how the small gain theorem can be used for robustness analysis,considerthefeedbackloopinfigure2.4,wheretheplant P(s)hasbeen replaced by[ + (s)]p(s). v (s) w C(s) P(s) Figure 2.9 Loop diagram with perturbed plant[ + (s)]p(s) Thetransferfunctionfromwtovisequalto T(s),thecomplementarysensitivity function. Hence, by the small gain theorem, the feedback system remains stableaslongas T < 8

9 2.4 Robustness via the small gain theorem Note that the small gain theorem does not assume linearity or time-invariance. HencetheclosedloopsystemwillremainstableforallplantsoftheformP(s)[+ (s)] where has L 2 -gain smaller than[sup ω T(iω) ], even for that are nonlinear or time-varying. Asasecondexample,letusderiveastabilitycriterionforthecasethatthe perturbation appears additively, i.e. P(s) is replaced by P(s) + (s). v (s) w C(s) P(s) Figure 2. Loop diagram with perturbed plant P(s) + (s) ThenthetransferfunctionfromwtovisequaltoC(s)S(s),sothesmallgain theorem shows stability for all perturbations satisfying CS < For linear time-invariant perturbations, this criterion can be nicely illustrated in the Nyquist diagram. Clearly, the condition C < +PC guaranteesthatthenyquistplotof(p+ )Cdoesnotencircle. +PC C 9

Lecture 2. FRTN10 Multivariable Control. Automatic Control LTH, 2018

Lecture 2. FRTN10 Multivariable Control. Automatic Control LTH, 2018 Lecture 2 FRTN10 Multivariable Control Course Outline L1 L5 Specifications, models and loop-shaping by hand 1 Introduction 2 Stability and robustness 3 Specifications and disturbance models 4 Control synthesis

More information

Systems Analysis and Control

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

More information

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

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

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

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

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

More information

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

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

Theory of Robust Control

Theory of Robust Control Theory of Robust Control Carsten Scherer Mathematical Systems Theory Department of Mathematics University of Stuttgart Germany Contents 1 Introduction to Basic Concepts 6 1.1 Systems and Signals..............................

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

Design Methods for Control Systems

Design Methods for Control Systems Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9

More information

Module 5: Design of Sampled Data Control Systems Lecture Note 8

Module 5: Design of Sampled Data Control Systems Lecture Note 8 Module 5: Design of Sampled Data Control Systems Lecture Note 8 Lag-lead Compensator When a single lead or lag compensator cannot guarantee the specified design criteria, a laglead compensator is used.

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

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

Step Response Analysis. Frequency Response, Relation Between Model Descriptions

Step Response Analysis. Frequency Response, Relation Between Model Descriptions Step Response Analysis. Frequency Response, Relation Between Model Descriptions Automatic Control, Basic Course, Lecture 3 November 9, 27 Lund University, Department of Automatic Control Content. Step

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

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

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. 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

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

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

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

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

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

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the

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

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

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using

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

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

Power System Control

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

More information

Digital Control Systems

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

More information

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

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

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

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

K(s +2) s +20 K (s + 10)(s +1) 2. (c) KG(s) = K(s + 10)(s +1) (s + 100)(s +5) 3. Solution : (a) KG(s) = s +20 = K s s

K(s +2) s +20 K (s + 10)(s +1) 2. (c) KG(s) = K(s + 10)(s +1) (s + 100)(s +5) 3. Solution : (a) KG(s) = s +20 = K s s 321 16. Determine the range of K for which each of the following systems is stable by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify

More information

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

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

More information

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 Design

Control Systems Design ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output

More information

Time Response Analysis (Part II)

Time Response Analysis (Part II) Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary

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

MAE 143B - Homework 9

MAE 143B - Homework 9 MAE 143B - Homework 9 7.1 a) We have stable first-order poles at p 1 = 1 and p 2 = 1. For small values of ω, we recover the DC gain K = lim ω G(jω) = 1 1 = 2dB. Having this finite limit, our straight-line

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

Data Based Design of 3Term Controllers. Data Based Design of 3 Term Controllers p. 1/10

Data Based Design of 3Term Controllers. Data Based Design of 3 Term Controllers p. 1/10 Data Based Design of 3Term Controllers Data Based Design of 3 Term Controllers p. 1/10 Data Based Design of 3 Term Controllers p. 2/10 History Classical Control - single controller (PID, lead/lag) is designed

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

Stability Margin Based Design of Multivariable Controllers

Stability Margin Based Design of Multivariable Controllers Stability Margin Based Design of Multivariable Controllers Iván D. Díaz-Rodríguez Sangjin Han Shankar P. Bhattacharyya Dept. of Electrical and Computer Engineering Texas A&M University College Station,

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

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 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

Outline. Control systems. Lecture-4 Stability. V. Sankaranarayanan. V. Sankaranarayanan Control system

Outline. Control systems. Lecture-4 Stability. V. Sankaranarayanan. V. Sankaranarayanan Control system Outline Control systems Lecture-4 Stability V. Sankaranarayanan Outline Outline 1 Outline Outline 1 2 Concept of Stability Zero State Response: The zero-state response is due to the input only; all the

More information

(a) Find the transfer function of the amplifier. Ans.: G(s) =

(a) Find the transfer function of the amplifier. Ans.: G(s) = 126 INTRDUCTIN T CNTR ENGINEERING 10( s 1) (a) Find the transfer function of the amplifier. Ans.: (. 02s 1)(. 001s 1) (b) Find the expected percent overshoot for a step input for the closed-loop system

More information

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

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

More information

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

FREQUENCY-RESPONSE DESIGN

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

More information

Recitation 11: Time delays

Recitation 11: Time delays Recitation : Time delays Emilio Frazzoli Laboratory for Information and Decision Systems Massachusetts Institute of Technology November, 00. Introduction and motivation. Delays are incurred when the controller

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

AA/EE/ME 548: Problem Session Notes #5

AA/EE/ME 548: Problem Session Notes #5 AA/EE/ME 548: Problem Session Notes #5 Review of Nyquist and Bode Plots. Nyquist Stability Criterion. LQG/LTR Method Tuesday, March 2, 203 Outline:. A review of Bode plots. 2. A review of Nyquist plots

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

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

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

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

EL2520 Control Theory and Practice

EL2520 Control Theory and Practice So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal

More information

Control Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli

Control Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)

More information

Exercises Automatic Control III 2015

Exercises Automatic Control III 2015 Exercises Automatic Control III 205 Foreword This exercise manual is designed for the course "Automatic Control III", given by the Division of Systems and Control. The numbering of the chapters follows

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

EEE 184: Introduction to feedback systems

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

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 22: The Nyquist Criterion Overview In this Lecture, you will learn: Complex Analysis The Argument Principle The Contour

More information

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

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

More information

Robust Performance Example #1

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

More information

Nonlinear Control Systems

Nonlinear Control Systems Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes

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 7

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

More information

LMI Methods in Optimal and Robust Control

LMI Methods in Optimal and Robust Control LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 14: LMIs for Robust Control in the LF Framework ypes of Uncertainty In this Lecture, we will cover Unstructured,

More information

CDS 101: Lecture 4.1 Linear Systems

CDS 101: Lecture 4.1 Linear Systems CDS : Lecture 4. Linear Systems Richard M. Murray 8 October 4 Goals: Describe linear system models: properties, eamples, and tools Characterize stability and performance of linear systems in terms of eigenvalues

More information

Software Engineering/Mechatronics 3DX4. Slides 6: Stability

Software Engineering/Mechatronics 3DX4. Slides 6: Stability Software Engineering/Mechatronics 3DX4 Slides 6: Stability Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on lecture notes by P. Taylor and M. Lawford, and Control

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

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall 2007

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall 2007 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering.4 Dynamics and Control II Fall 7 Problem Set #9 Solution Posted: Sunday, Dec., 7. The.4 Tower system. The system parameters are

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

Andrea Zanchettin Automatic Control AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Linear systems (frequency domain)

Andrea Zanchettin Automatic Control AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Linear systems (frequency domain) 1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Linear systems (frequency domain) 2 Motivations Consider an LTI system Thanks to the Lagrange s formula we can compute the motion of

More information

Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types

Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Venkata Sonti Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 562 This

More information

Chapter 5. Standard LTI Feedback Optimization Setup. 5.1 The Canonical Setup

Chapter 5. Standard LTI Feedback Optimization Setup. 5.1 The Canonical Setup Chapter 5 Standard LTI Feedback Optimization Setup Efficient LTI feedback optimization algorithms comprise a major component of modern feedback design approach: application problems involving complex models

More information

ROBUST STABILITY AND PERFORMANCE ANALYSIS OF UNSTABLE PROCESS WITH DEAD TIME USING Mu SYNTHESIS

ROBUST STABILITY AND PERFORMANCE ANALYSIS OF UNSTABLE PROCESS WITH DEAD TIME USING Mu SYNTHESIS ROBUST STABILITY AND PERFORMANCE ANALYSIS OF UNSTABLE PROCESS WITH DEAD TIME USING Mu SYNTHESIS I. Thirunavukkarasu 1, V. I. George 1, G. Saravana Kumar 1 and A. Ramakalyan 2 1 Department o Instrumentation

More information

Lecture 5 Classical Control Overview III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 5 Classical Control Overview III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lecture 5 Classical Control Overview III Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore A Fundamental Problem in Control Systems Poles of open

More information

Exam in Systems Engineering/Process Control

Exam in Systems Engineering/Process Control Department of AUTOMATIC CONTROL Exam in Systems Engineering/Process Control 27-6-2 Points and grading All answers must include a clear motivation. Answers may be given in English or Swedish. The total

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

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

ECE317 : Feedback and Control

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

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

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

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

Stability of CL System

Stability of CL System Stability of CL System Consider an open loop stable system that becomes unstable with large gain: At the point of instability, K( j) G( j) = 1 0dB K( j) G( j) K( j) G( j) K( j) G( j) =± 180 o 180 o Closed

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

ECSE 4962 Control Systems Design. A Brief Tutorial on Control Design

ECSE 4962 Control Systems Design. A Brief Tutorial on Control Design ECSE 4962 Control Systems Design A Brief Tutorial on Control Design Instructor: Professor John T. Wen TA: Ben Potsaid http://www.cat.rpi.edu/~wen/ecse4962s04/ Don t Wait Until The Last Minute! You got

More information

L2 gains and system approximation quality 1

L2 gains and system approximation quality 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility

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

NPTEL Online Course: Control Engineering

NPTEL Online Course: Control Engineering NPTEL Online Course: Control Engineering Ramkrishna Pasumarthy Assignment-11 : s 1. Consider a system described by state space model [ ] [ 0 1 1 x + u 5 1 2] y = [ 1 2 ] x What is the transfer function

More information

Dr Ian R. Manchester

Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

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

CDS 101/110: Lecture 3.1 Linear Systems

CDS 101/110: Lecture 3.1 Linear Systems CDS /: Lecture 3. Linear Systems Goals for Today: Describe and motivate linear system models: Summarize properties, examples, and tools Joel Burdick (substituting for Richard Murray) jwb@robotics.caltech.edu,

More information