Control Systems I
Control Systems ILecture 11: PID ControlReadings: A&M, Ch. 10, Guzzella, Chapter 11.2,Emilio FrazzoliInstitute for Dynamic Systems and ControlD-MAVTETH ZürichDecember 1st, 2017E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20171 / 31
Tentative schedule#1234DateSept. 22Sept. 29Oct. 6Oct. 13567Oct. 20Oct. 27Nov. 389Nov. 10Nov. 171011121314Nov. 24Dec. 1Dec. 8Dec. 15Dec. 22E. Frazzoli (ETH)TopicIntroduction, Signals and SystemsModeling, LinearizationAnalysis 1: Time response, StabilityAnalysis 2: Diagonalization, Modal coordinates.Transfer functions 1: Definition and propertiesTransfer functions 2: Poles and ZerosAnalysis of feedback systems: internal stability,root locusFrequency responseAnalysis of feedback systems 2: the NyquistconditionSpecifications for feedback systemsPID ControlLoop ShapingImplementation issuesRobustnessLecture 11: Control Systems I1/12/20172 / 31
Today’s learning objectivesLearn what a PID control is and how to design one:Proportional control: what it is, pro’s and con’sDerivative control: what it is, pro’s and con’sIntegral control: what it is, pro’s and con’tTuning strategies for PID controllers.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20173 / 31
A nice intro to PID controlhttps://www.youtube.com/watch?v 4Y7zG48uHRoE. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20174 / 31
Recall: Control SpecificationsType of the system (order of ramp to track with zero steady-state error):Number of integrators in L(s)Time domain specifications (max overshoot, settling time, rise time, .):Location of dominant closed-loop poles (damping ratio and real part)Frequency domain specifications (command tracking, disturbance/noiserejection, closed-loop bandwidth):Bode obstacle course (low/high frequency)Crossover frequencyControl synthesis: how do we choose a feedback control system thatachieves these objectives?E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20175 / 31
Controller design methodsWhat other methods do exist to design controllers C (s) that meet designspecifications? Many approaches, among them:PID, Loop Shaping, LQR, LQG-LTR, H , Discrete-time optimal control,continuous-time optimal control, model predictive control,.Today we look at the most widely used approach for SISO systems: PIDcontrol.PID - control (proportional-integral-derivative control) is the most widelyapplied controller design because it is able to cope well with the majority ofcases encountered in practice.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20176 / 31
Proportional ain1s 1outputScopeTransfer FcnmeasurementnoiseBand-LimitedWhite NoiseThe control input tries to move the system in a direction that is opposite tothe error, and is proportional to the error in magnitude.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20177 / 31
Proportional gain selection1.2k 2k 5k 10k 501y0.80.60.40.20012345678910tAs the proportional gain increases,The closed-loop system remains stable;The steady-state error decreases;The response becomes faster;The sensitivity to noise increases.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20178 / 31
Proportional gain selectionRoot Locus30.999Imaginary Axis (seconds-1 Real Axis (seconds-1 )Closed-loop transfer functionT (s) 1L(s) ,1 kL(s)s 1 ki.e., the closed-loop pole is at s 1 k (see root locus above).11Steady-state error to a unit step: ess lims 0 1 kL(s) 1 kE. Frazzoli (ETH)Lecture 11: Control Systems I1/12/20179 / 31
Proportional gain selectionBode DiagramMagnitude (dB)40200-20Phase (deg)-400k 2k 5k 10k 50-45-9010 -210 -110 010 110 2Frequency (rad/s)As the proportional gain increases,Phase margin remains 90 ;The crossover frequency increases;The low-frequency gain increases;The high-frequency gain increases;E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201710 / 31
Introducing an urbanceI Gaincontrol21s 1controlP GainoutputmeasurementsimoutTo WorkspaceTransfer FcnnoiseBand-LimitedWhite NoiseIntegrating the error allows one to detect potential ”biases” in the systembehavior.An integral control action tries to move the response in order to reduce thedetected biases.PI control:Z tu(t) kP e(t) kIe(τ )dτ,0C (s) kP E. Frazzoli (ETH)kIkP s kIkP /kI · s 1 kI.sssLecture 11: Control Systems I1/12/201711 / 31
Integral gain selection1.6k P 2, k I 2k P 2, k I 21.4k P 2, k I 5k P 10, k I 501.2y10.80.60.40.20012345678910tAs the integral gain increases,The steady-state error is zero (as long as kI is not zero)The response becomes more oscillatory (warning!)The sensitivity to noise does not change!E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201712 / 31
Integral gain selectionRoot Locus30Imaginary Axis (seconds-1 )20100-10-20-30-90-80-70-60-50-40-30-20-10010Real Axis (seconds-1 )Steady-state error to a unit step: ess lims 011 C (s)L(s) 0The root locus shows us that as the integral gain increases, the closed-looppoles go from being “slow” and overdamped to being “fast” but with lowdamping!E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201713 / 31
Integral gain selectionBode Diagram80Magnitude (dB)6040200-20-40Phase (deg)-60-90-120-150-18010 -210 -110 010 110 210 3Frequency (rad/s)As the integral gain increases,Phase margin decreases;The crossover frequency increases;The low-frequency gain increases — but goes to infinity near 0 in all cases ;The high-frequency gain does not change.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201714 / 31
Proportional Control — Higher order controlP Gain2s2 2s 2simoutoutputTo WorkspaceTransfer FcnmeasurementnoiseBand-LimitedWhite NoiseHow do the previous consideration extend to higher-order systems, e.g., 2ndorder?E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201715 / 31
Proportional gain s the proportional gain increases,The closed-loop system become more oscillatory (warning!);The steady-state error decreases;The response becomes faster;The sensitivity to noise increases.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201716 / 31
Proportional gain selectionRoot Locus150.170.1150.0850.0560.0360.0161412Imaginary Axis (seconds-1 .115-20.085-1.50.056-10.0360.016-0.51400.5Real Axis (seconds-1 )The root locus shows that as the proportional gain increases, the closed-looppoles have decreasing damping ratio.Steady-state error to a unit step: ess lims 0E. Frazzoli (ETH)Lecture 11: Control Systems I11 kL(s) 11 k1/12/201717 / 31
Proportional gain selectionBode Diagram40Magnitude (dB)200-20-40-60Phase (deg)-800-45-90-135-18010 -210 -110 010 110 2Frequency (rad/s)As the proportional gain increases,Phase margin gets smaller and smaller!The crossover frequency increases;The low-frequency gain increases;The high-frequency gain increases;E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201718 / 31
Introducing a differentiator50DerivativeD trolP Gain2s2 2s 2outputmeasurementsimoutTo WorkspaceTransfer FcnnoiseBand-LimitedWhite NoiseDifferentiating the error allows one to “predict” what the error will do in thenear future.An derivative control action tries to avoid overshooting, hence damping thesystem.PD control:u(t) kP e(t) kD ė(t)C (s) kP kD s.Note that this is not a causal transfer function (not physically realizable insgeneral). This is typically fixed by approximating the derivative as s cs 1for some large c.)E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201719 / 31
Derivative gain selection1.4k P 50, k D 2k P 50, k D 51.2k P 50, k D 10k P 50, k D 501y0.80.60.40.20012345678910tAs the derivative gain increases,The steady-state error not affected;The response becomes less oscillatory, but potentially slowerThe sensitivity to noise increases!E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201720 / 31
Derivative gain selectionRoot Locus30Imaginary Axis (seconds-1 )20100-10-20-30-120-100-80-60-40-20020Real Axis (seconds-1 )Steady-state error to a unit step: ess lims 011 C (s)L(s) 11 kP L(0)The root locus shows us that as the derivative gain increases, the closed-looppoles are “pulled” into the left half plane!E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201721 / 31
Derivative gain selectionBode Diagram40k P 50, k D 2k P 50, k D 5k P 50, k D 10k P 50, k D 50Magnitude (dB)200-20-40Phase (deg)-600-45-90-135-18010 -210 -110 010 110 210 3Frequency (rad/s)As the derivative gain increases,Phase margin increases;The crossover frequency increases;The low-frequency gain does not change;The high-frequency gain increases.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201722 / 31
Proportional-Integral-Derivative Control5DerivativeD ceD Gain110control2s2 2s 2controlP GainsimoutoutputTo WorkspaceTransfer FcnmeasurementnoiseBand-LimitedWhite NoiseOne can also combine the effects of an integrator and of a differentiator withthe basic proportional controller.PID control:Zu(t) kP e(t) kIte(τ )dτ kD ė(t),0C (s) kP E. Frazzoli (ETH)kIkD s 2 kP s kI kD s .ssLecture 11: Control Systems I1/12/201723 / 31
PID TuningPID tuning corresponds to choosing the parameters kp , ki and kd to reach thefeedback control design specifications.PID tuning can be done with tuning rules by hand or numerically usingMATLAB or other tools (the latter requires a system model).There exist heuristic methods to tune a PID controller without a model ofthe plant P(s), e.g. the tuning rules proposed by Ziegler and Nichols.My recommendation: think of a PID asC (s) kRL(s z1 )(s z2 )si.e., as two zeros and one pole at the origin. Decide where you want thesezeros (in the complex plane, or in terms of natural frequency and dampingratio on the Bode plot), and what you want the (root-locus) gain to be.Finally, compute the corresponding kP , kI , kD .E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201724 / 31
SummaryProportional controlDecrease the steady-state error;Increase the closed-loop bandwidth;Increase sensitivity to noise;Can reduce stability margins for higher-order systems (2nd order or more).Integral controlEliminates the steady-state error to a step (if the closed-loop is stable);Reduces stability margins, can make a higher-order system unstable.Derivative controlReduce overshooting, increase damping;Improves stability margins;Increase sensitivity to noise.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201725 / 31
Today’s learning objectivesLearn what a PID control is and how to design one:Proportional control: what it is, what it does, pro’s and con’sDerivative control: what it is, what it does, pro’s and con’sIntegral control: what it is, what it does, pro’s and con’sTuning strategies for PID controllers.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201726 / 31
Ziegler Nichols Tuning RulesAssumption: Plant can be approximated by the transfer functionP(s) ke Tsτs 1with T /(T τ ) small.Apply the controller C (s) kp to the system starting at kp 0 and increasekp until the system is in a steady-state oscillation, then note the ”critical kp ”called kp and the corresponding critical oscillation period T .Use kp and T to calculate the control gains:typePPIPDPIDE. Frazzoli (ETH)kp0.5 · kp 0.45 · kp 0.55 · kp 0.6 · kp Ti · T 0.85 · T · T 0.5 · T Lecture 11: Control Systems ITd0 · T 0 · T 0.15 · T 0.125 · T 1/12/201727 / 31
Ziegler Nichols Tuning RulesGraphically:11 (LinoGuzzella ”Analysis and Synthesis of Single-Input Single-Output Control Systems)E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201728 / 31
Ziegler Nichols Tuning Example1(s 1) · (s 2 2s 2)0.5· e 0.01sApproximation: Papprox 0.5 · s 1Plant: P(s) E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201729 / 31
Ziegler Nichols Tuning ExampleSet Ti , Td 0, τ 0 and increase gain kp .Critical gain kp 10 with critical oscillation period T E. Frazzoli (ETH)Lecture 11: Control Systems I2πω 2π2 π1/12/201730 / 31
Ziegler Nichols Tuning ExampleP, PI, PD, PID controller according to Ziegler and Nichols tuning rules.PID controller derived with MATLAB sisotool.Ziegler Nichols tuning rules can be useful when no model of the plant isavailable but generally other tuning rules provide better results.E. Frazzoli (ETH)Lecture 11: Control Systems I1/12/201731 / 31
PID Tuning PID tuning corresponds to choosing the parameters k p;k i and k d to reach the feedback control design speci cations. PID tuning can be done with tuning rules by hand or numerically using MATLAB or other tools (the latter requires a system model). There exist heuristic methods to
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1. CONTROL SYSTEMS: BASICS 1 1.1 What is Control Systems 1 1.2 Classification of Systems 1 1.3 Classification Based on the Parameters 2 1.4 Analysis of Control Systems 3 1.5 General Classification: Open and Closed-Loop Systems 3 1.6 Elements of Automatic or Feedback Control Systems 5 1.7 Requirements of Automatic Control Systems 6 2.
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