EL3210 Multivariable Feedback Control
EL3210 Multivariable Feedback ControlElling W. JacobsenDep. of Automatic Control, KTHjacobsen@kth.sehttps://people.kth.se/ jacobsen/multi 17.shtmlLecture 1: Introduction, classical SISO feedback controlLecture 1: classical SISO controlEL3210 MIMO Control1 / 44
The Practical8 lecturesslides on homepagereading assignments on homepageCourse literatureSkogestad and Postlethwaite, Multivariable Feedback Control, 2nded.Supporting text: Zhou, Doyle and Glover, Robust and OptimalControl8 homeworks, compulsorydownload from homepage after each lecture, hand in within oneweekrequire Matlab with Robust Control toolbox1-day take home open book exam, within 6 weeks after last lectureLecture 1: classical SISO controlEL3210 MIMO Control2 / 44
Course ContentFeedback control of MIMO LTI systems under model uncertaintyfrequency domain analysis and design;extension of classical SISO methods to MIMO systemsoptimal control problems formulated in input-output spaceinput-output controllability; what can be achieved with feedback ina given system?robustness: stability and performance under model uncertaintyLecture 1: classical SISO controlEL3210 MIMO Control3 / 44
Course goalsAfter completed course you should be able toquantify the performance that can be achieved with feedback for agiven systemanalyze feedback systems with respect to stability andperformance in the presence of structured and unstructuredmodel uncertaintydesign/synthesize controllers for robust performanceLecture 1: classical SISO controlEL3210 MIMO Control4 / 44
Lecture PlanL1: Introduction, classical SISO feedback control (Ch.1-2)L2: Performance limitations in SISO feedback (Ch. 5)L3: Introduction to MIMO systems, excerpts from Linear SystemsTheory (Ch. 3-4)L4: Performance limitations in MIMO feedback (Ch. 6)L5: Uncertainty and robust stability (Ch. 7-8)L6: Robust performance (Ch. 7-8)L7: Controller synthesis and design (Ch. 9-10)L8: Alternative formulations (LMIs, IQCs, .), summary (Ch. 10, 12)Lecture 1: classical SISO controlEL3210 MIMO Control5 / 44
Todays LectureThe Control ProblemA Historical PerspectiveBrief introduction to normsBrief recap of classical controlLecture 1: classical SISO controlEL3210 MIMO Control6 / 44
The Control ProblemControl problems usually formulated in terms of signal trackingy Gu Gd dy - output / controlled variableu - input / manipulated variabled - disturbancer - reference, setpoint– Regulator problem: attenuate effect of d on y– Servo problem: make y follow rControl objective: make e r y “small” using feedback u C(y , r )Lecture 1: classical SISO controlEL3210 MIMO Control7 / 44
Why Feedback?Why not u G 1 r G 1 Gd d ?1model uncertainty - uncertain knowledge of system behavior2unmeasured disturbances3instabilityCost of feedback:– potentially induce instability– feed measurement noise into processLecture 1: classical SISO controlEL3210 MIMO Control8 / 44
Fact 1: Feedback has its limitationsfeedback is a simple and potentially very powerful tool for tailoringthe dynamic behavior of a system, but with hard limitations to whatcan be achievedcontrol performance depends on controller and systemZiegler and Nichols (1943): In the application of automatic controllers, it isimportant to realize that controller and process form a unit; credit or discreditfor results obtained are attributable to one as much as the other. . . . Thefinest controller made, when applied to a miserably designed process, maynot deliver the desired performance. True, on badly designed processes,advanced controllers are able to eke out better results than older models, buton these processes, there is a definite end point which can be approached byinstrumentation and it falls short of perfection.Lecture 1: classical SISO controlEL3210 MIMO Control9 / 44
Approaches to Control Design"Traditional":1. specify desired performance2. design controller that meets specifications3. if 2 fails, try more advanced controller and repeat from 2This course:1.2.3.4.specify desired performancedetermine achievable performanceif conflict between 1 and 2, change specifications or modify systemdesign controller using your favorite methodLecture 1: classical SISO controlEL3210 MIMO Control10 / 44
Fact 2: Models are always uncertainModels (G, Gd ) always inaccurate, e.g., true systemGp G Ewith E “uncertainty”, or “perturbation” (unknown)Definitions for closed loop:Nominal stability (NS): stable with no model uncertaintyNominal performance (NP): satisifies performance requirementswith no model uncertaintyRobust stability (RS): stable for “all” possible perturbations ERobust performance (RP): satisifies performance requirementsfor “all” possible perturbations ELecture 1: classical SISO controlEL3210 MIMO Control11 / 44
System RepresentationsState-space representationẋ Ax(t) Bu(t),y (t) Cx(t) Du(t),x Rn , u Rpy RlTransfer-functionY (s) G(s)U(s) ;G(s) C(sI A) 1 B DFrequency responseY (jω) G(jω)U(jω)Sometimes we write ẋA B x ;yC D uLecture 1: classical SISO control G EL3210 MIMO ControlA BC D 12 / 44
A Brief History of ControlClassical, 30’s-50’s: frequency domain methodsBode, Nyquist, Nichols, . . . yields insight (loop shaping) address model uncertainty (gain and phase margins) only applicable to SISO systems”Modern”, 60’s-70’s: state-space optimal controlBellman, Pontryagin, Kalman, . . . control cast as time-domain optimization problem applicable to MIMO systems (LQG) can not accomodate for unmodeled dynamics LQG has no guaranteed stability margins no clear link to classical methodsLecture 1: classical SISO controlEL3210 MIMO Control13 / 44
Famous paper (and abstract.)Lecture 1: classical SISO control(IEEE Trans AC, 1978)EL3210 MIMO Control14 / 44
A Brief History of Control”Postmodern”, 80’s-90’s: robust controlZames, Francis, Doyle, . . . frequency domain methods for MIMO systems explicitly adress model uncertainty control cast as optimization problem (H2 , H ) links classical and modern approaches; “formulate and analyze ininput-output domain, compute in state-space” high order controllers, computational issues, .It was the introduction of norms in control, in particular the H -norm,that paved the way for analyzing fundamental limitations androbustness in MIMO systemsLecture 1: classical SISO controlEL3210 MIMO Control15 / 44
A Brief History of ControlPost 90’s-analysis/synthesis using convex optimization, e.g., LMIscombining H2 for performance with H for robustnessbeyond LTI systems, e.g., Integral Quadratic Constraints (IQCs).Lecture 1: classical SISO controlEL3210 MIMO Control16 / 44
Control Structures1-Degree of freedom2-Degrees of freedomLecture 1: classical SISO controlEL3210 MIMO Control17 / 44
Control StructuresGeneral control structureP - generalized system, K - controllerw - exogeneous inputs (d, r , n)u - manipulated inputsz - exogeneous outputs (e, u)v - measurements, setpointsObjective: minimize gain from w to z. With appropriateweights/scaling, make gain smaller than 1Lecture 1: classical SISO controlEL3210 MIMO Control18 / 44
Brief on Norms (more in Lec 3)A real valued function k · k on a linear space H, over the field of real orcomplex numbers, is called a norm on H if it satisfies(i) kxk 0(ii) kxk 0 if and only if x 0(iii) kaxk a kxk for any scalar a(iv) kx y k kxk ky kfor any x, y HLecture 1: classical SISO controlEL3210 MIMO Control19 / 44
Vector and Matrix NormsFor x Cn the p-norm is1/pkxkp (Σni 1 xi p )We will mainly consider p 2, the Euclidian norm kxk2 x x H xFor A Cm n the (induced) p-norm iskAkp kAxkpx Cn ,x6 0 kxkpsupWe will mainly consider p 2kAk2 σ̄(A) maxiLecture 1: classical SISO controlEL3210 MIMO Controlqλi (AH A)20 / 44
Operator NormsFor a vector valued signal x(t) the Lp -norm is Z p 1/pΣi xi (τ ) dτkx(t)kp We will mainly consider p 2, i.e., the L2 -normsZsZ x(τ )T x(τ )dτ kx(t)k2 x 2 dτ In frequency domain (by Parsevals thm)sZ 1kxk2 x(iω)H x(iω)dω2π Lecture 1: classical SISO controlEL3210 MIMO Control21 / 44
Operator NormsFor a transfer-function G(s)the H2 -norm isskG(s)k2 12πZ tr(G(jω)H G(jω))dω and the H -norm iskG(s)k sup σ̄(G(jω))ωH denotes Hardy space, H (H2 ) is the set of stable and (strictly) propertransfer-functionsLecture 1: classical SISO controlEL3210 MIMO Control22 / 44
The H2 - and H -normsThe H -norm is an induced norm from L2 to L2 , i.e., the L2 -gainy Gu ;ky k2u(t)6 0 kuk2kG(s)k supThe H2 -norm is not an induced norm. But, e.g., equalsamplification from a white noise input to the 2-norm of the outputWe will consider both norms for design later, but the fact that theH -norm is an induced norm makes it useful for analyzingperformance limitations (Lec 2) as well as robustness (Lec 5) .Lecture 1: classical SISO controlEL3210 MIMO Control23 / 44
Scaling - simplifies analysis and designUnscaled model:ê r̂ ŷŷ Ĝû Ĝd d̂ ;Scale all variables so that expected/allowed magnitude is less than 1:u ûûmax;d Introduce Dd d̂max ;d̂d̂maxy ;ŷêmaxDu ûmax ;e ;êêmax;r r̂êmaxDe êmaxy De 1 ĜDu u De 1 Ĝd Dd d {z } {z }GGdIn the scaled model, all signals should have magnitude less than1, i.e., expected d 1 and acceptable e 1Lecture 1: classical SISO controlEL3210 MIMO Control24 / 44
Example: scaled frequency responseBode plot for Gd (jω) :1100 Gd 10 110 210 310 310 2 11010ω [rad/s]010110Need disturbance attenuation for frequencies where Gd (jω) 1,i.e., for ω 0.33 rad/sOr, equivalently, we requirekSGd k 1Lecture 1: classical SISO controlEL3210 MIMO Control25 / 44
Next: classical control revisitedLecture 1: classical SISO controlEL3210 MIMO Control26 / 44
Closed-Loop Transfer Functions - 1-DOF structureClosed-loop transfer-functionsy (1 GK ) 1 GK r (1 GK ) 1 Gd d (1 GK ) 1 GK n {z} {z} {z}Tcontrol errorSTe r y Sr SGd d Tninputu KSr KSGd d KSnLecture 1: classical SISO controlEL3210 MIMO Control27 / 44
The Sensitivity FunctionsIntroduce the loop gain L GKS (1 L) 1 ;T (1 L) 1 L S T 1S - the sensitivity functionT - the complimentary sensitivity functionLecture 1: classical SISO controlEL3210 MIMO Control28 / 44
The name ”Sensitivity”Bode: relative sensitivity of T to model perturbations (uncertainty)S (dT /T )/(dG/G)But, also effect of feedback on sensitivity to disturbancesy SGd d S(jω) 1: feedback reduces disturbance sensitivity S(jω) 1: feedback increases disturbance sensitivityLecture 1: classical SISO controlEL3210 MIMO Control29 / 44
Frequency PlotsDefinitions:crossover frequency ωc : L(jωc ) 1 bandwidth ωB : S(jωB ) 1/ 2 bandwidth for T , ωBT : T (jωBT ) 1/ 2Lecture 1: classical SISO controlEL3210 MIMO Control30 / 44
Bandwidth and Crossover FrequencyEffective feedback for frequencies where S(jω) 1, i.e., up tobandwidth ωB S(jω) 1, ω [0, ωB ]Bandwidth and crossover frequencies:ωB ωc ωBTProof: see (2.53) in bookTypically assumeωc ωBLecture 1: classical SISO controlEL3210 MIMO Control31 / 44
Stability marginsGain margin GM and phase margin PM - robustness measuresBode diagramLecture 1: classical SISO controlNyquist diagramEL3210 MIMO Control32 / 44
Sensitivity peaksStability margins and performance are related S 1 1 L(jω) is distance from L(jω) to critical point 1 inNyquist diagramdefine MS maxω S(jω) ;MT maxω T (jω) , thenMS 1PMMS GMGM 1MT 1PMMT 1GM 1– obtained by considering the loop gain at L at ωc and ω180 ,respectivelyLecture 1: classical SISO controlEL3210 MIMO Control33 / 44
Controller DesignThree main approaches:1. Shaping transfer-functionsa. Loop shaping (classic): use controller K to shape loop gain L(jω)b. Shaping the closed loop: shape S, T etc, using optimizationbased methods2. Signal based approaches: minimize signals, i.e., control error eand input u, given characteristics of inputs d, r , n.3. Numerical optimization: optimize “real” control objectives, e.g.,rise time and overshoot for step responses.Lecture 1: classical SISO controlEL3210 MIMO Control34 / 44
Classic Loop Shaping - shaping L recalle 11Lnr d L} 1 {z I {z L} I {z L}SSTFundamental trade-offs:– setpoint following: L large– disturbance attenuation: L large– noise propagation: L smallAlso:u KSr KSGd d KSn– input usage: K smallLecture 1: classical SISO controlEL3210 MIMO Control35 / 44
Resolving the trade-offTypically: make L large in frequency range where disturbances andsetpoints important, and L small for higher frequencies, L 1, ω [0, ωB ] L 1, ω ωBi.e., want L to drop off steeply around ωB ωcBut, slope of L and phase arg L coupledLecture 1: classical SISO controlEL3210 MIMO Control36 / 44
Bode Relation1arg L(jω0 ) πZ ω ω0 dωd ln L lnd ln ωω ω0 ωequality for minimum phase systemswith slope N d ln L /d ln ω,arg L(jω0 ) πN(jω0 )2Thus, slope around crossover ωc should be at most 2, less to yieldsome phase margin.Lecture 1: classical SISO controlEL3210 MIMO Control37 / 44
A Procedure for Loop Shaping1. First tryL(s) ωcs yieldsy K ωc 1G (s)sωcrs ωcBut, bad disturbance rejection if Gd slow2. For disturbancese SGd dRequire SGd 1 ωcorresponds to 1 L Gd Lecture 1: classical SISO controlEL3210 MIMO Control ω38 / 44
A Procedure for Loop Shapingcont. If Gd 1 we get approximately L Gd Simple choiceL Gd K G 1 Gdand with integral actionK s ωI 1G Gds3. High frequency correctionK s ωI 1τs 1G Gd τsγs 1to improve stability, i.e., modify slope of L around ωc4. To improve setpoint tracking, add prefilter Kr (s) on setpoint 2-DOF control structureLecture 1: classical SISO controlEL3210 MIMO Control39 / 44
Shaping the Closed LoopShaping L GK is just a means of achieving a desiredclosed-loopAlternative: find controller that minimizes a weighted sensitivity,e.g., min max wP S min kwP Sk ωK– kwpSk 1Lecture 1: classical SISO control K S 1/ wp ωEL3210 MIMO Control40 / 44
Weighted SensitivityLecture 1: classical SISO controlEL3210 MIMO Control41 / 44
Performance weightTypical choice for weigthwp s/M ωBs ωB AMagnitude of 1/ wp :Control objective satisfied if kwp Sk 1Lecture 1: classical SISO controlEL3210 MIMO Control42 / 44
Next TimeFundamental performance limitations and tradeoffs in SISOfeedbackControllability analysis: what is achievable performance, e.g., ωB ,M, for a given system?Homework:Exercise 1 (download from the course homepage). Hand in nextFriday.Read Chapter 5 (and 1-2) in Skogestad and PostlethwaiteLecture 1: classical SISO controlEL3210 MIMO Control43 / 44
Skogestad and Postlethwaite, Multivariable Feedback Control, 2nd ed. Supporting text: Zhou, Doyle and Glover, Robust and Optimal Control 8 homeworks, compulsory download from homepage after each lecture, hand in within one week require Matlab with Robust Control toolbox 1-day take home open book exam, within 6 weeks after last lecture
EL3210 Multivariable Feedback Control Lecture 8: Youla parametrization, LMIs, Model Reduction and Summary [Ch. 11-12] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 8: Youla, LMIs, . EL3210 MIMO Control 1 / 33
Lectures on Multivariable Feedback Control Ali Karimpour Department of Electrical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad (September 2009) Chapter 2: Introduction to Multivariable Control 2-1 Multivariable Connections 2-2 Multivariable Poles 2-2-1 Poles from State Space Realizations 2-2-2 Poles from Transfer Functions
Relay feedback and multivariable control Abstract This doctoral thesis treats three issues in control engineering related to relay feedback and multivariable control systems. Linearsystems with relay feedback is the first topic. Such systemsare shown to exhibitseveral interesting behaviors.
376_069 Multivariable feedback control V1 1 of 428 MULTIVARIABLE CONTROL SYSTEMS DESIGN* by Ian K. Craig * These viewgraphs are based on notes prepared by Prof. Michael Athans of MIT for the course "Multivariable Control Systems 1 & 2" These viewgraphs should be read in conjunction with the textbook: S Skogestad, I Postlehwaite .
The typical structures of the multivariable feedback control systems considered in this paper are shown in Fig.1(a) and 1(b). For convenient, this kind of structures are called "layer-wrapped structures". The precompensating matrix for an N N multivariable feedback control system is implemented by cascading the determined columns P i N ( 1,2
3.4 General form of multivariable control system 3.5 Closed Loop Transfer Function Matrix with Feedback 3.6 Inverse Nyquist Diagram of ̂ () 3.7 Full State Feedback Control 3.8 Full State Feedback Observer Block Diagram 3.9 Full Order Observer and Controller Block Diagram 3.10 Compensator Arrangement Block Diagram
and graduate levels: classical loop-shaping control, an introduction to multivariable control, advanced multivariable control, robust control, controller design, control structure design and controllability analysis. The book is partly based on a graduate multivariable control course given by the
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