ME/EE 561 Design Of Digital Control Systems

ME/EE 561 Design of Digital Control SystemsWinter 2012Professor Huei PengG036 Auto Labhpeng@umich.edu734-936-0352Office hours: Monday 3:35-4:30, Wednesday4:30-5:35or by appt.Copyright G.Chiu and H.PengME561 Lecture1- 1

Lecture 1-- Introduction and Motivation Why Digital Control (Computer Controlled)Systems About this course––––ObjectivesCourse contentGrading policyHW, Exams, Final project Basics of Digital Control SystemsCopyright G.Chiu and H.PengME561 Lecture1- 2

ME461EE460ME561EE561Copyright G.Chiu and H.PengME561 Lecture1- 3

Motivation for “Controls” Feedback control has a long history which began with theearly desire of humans to harness the forces of nature,improve productivity, and to avoidhazardous/laborious/repetitive work. Watt’s Fly Ball Governor had a major impact on theindustrial revolution (mechanical control). Most modern systems (aircrafts, automobiles, productionlines, CD players, etc.) could not operate without the aid ofcontrol systems.Photos from G. Goodwin’s lecture slidesCopyright G.Chiu and H.PengME561 Lecture1- 4

Evolution of Control Systems Mechanical control systems– e.g., Watt’s Fly Ball Governor Analog (electrical/electronics) control systems– e.g., Op-amp plus RC circuits based controllers. Digital (computer) control systems– Apparently a lot more flexible than the two types above– computer, DSP, microprocessorCopyright G.Chiu and H.PengME561 Lecture1- 5

Computer Controlled Systems Direct Digital Control (1960s)—implementationof PID control algorithms on large systems(chemical, power plants, space, military). Now, complex and intelligent algorithms.From: W. Powers, AVEC 2000Copyright G.Chiu and H.PengME561 Lecture1- 6

Microprocessor Based Control Systems--example on motor controlSource: Copyright G.Chiu and H.PengME561 Lecture1- 7

AD and DA“Digital” control system deals with digitized(discretized) signals.AD/DA devices translate continuous signals to/fromdigital signals.OutputyuPlantD/AComputerA/De –rReferenceClockCopyright G.Chiu and H.PengME561 Lecture1- 8

Two Forms of DiscretizationsSampling (discretize in time)Quantization (discretize in Average u(t)1234567Copyright G.Chiu and H.Peng8910kTME561 Lecture1- 9

Inherently Discrete-Time (Sampled)Systems Economic Systems Radar, GPS Accurate emission measurement of internalcombustion engines, or some medical sensors Internal combustion engines control in general Signals transmitted in nervous systems Systems with embedded computers,microprocessors or clocksCopyright G.Chiu and H.PengME561 Lecture1- 10

Objective of This Course To introduce the concepts of sampling, discretetime signals/systems, and the analysis andsynthesis of digital control systems– Upon completion of this course, you should be able toconstruct discrete-time models, design digital controlalgorithms and analyze the openloop and closed-loopbehavior.Copyright G.Chiu and H.PengME561 Lecture1- 11

This Course is NOT EECS 461 Embedded Control Systems ME 552 Mechatronics System Design MIT 6.111 Introductory Digital right G.Chiu and H.PengME561 Lecture1- 12

Course Material There is no textbook—course pack provided Reference books– Computer Controlled Systems-Theory and Design, Prentice-Hall, 1997(3rd edition), Astrom, K. J. and Wittenmark, 33148998/103-5327290-9742228?v glance– Discrete-Time Control Systems, Prentice-Hall, 1995 (2nd edition), l/-/0130342815/103-5327290-9742228?v glanceCopyright G.Chiu and H.PengME561 Lecture1- 13

Major course contentChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Introduction to Computer Controlled SystemsSampled Data AnalysisThe Z-Transform and the Difference EquationsDiscrete-Time System RepresentationAnalysis of Discrete-Time SystemsDesign of Discrete Time Controller—Input/Output ApproachesDesign of Discrete Time Controller—Polynomial ApproachesDesign of Discrete Time Controller—State Space ApproachesLinear Quadratic Optimal ControlOptimal Linear Feedback of Stochastic SystemsOptimal Design Methods: Input/Output ApproachCopyright G.Chiu and H.PengME561 Lecture1- 14

CTool Course Web Site https://ctools.umich.edu/portalCopyright G.Chiu and H.PengME561 Lecture1- 15

Grading Policy Grading: Homeworks45%Midterm20%Final35%Copyright G.Chiu and H.PengME561 Lecture1- 16

Grading Policy (cont.) Homework: 7-8 HW assignments. Due at the endof class on the due date. Homework will be acceptedup to 48 hours late with a 25% penalty for each 24hours. Standard Michigan Honor Code applies.tentative Exams: Midterm:Mar. 5 (Tue.) in classFinal:April. 25 (Wed.) 1:30-3:30pmNo make-up exam.Regrade request within 3 days in writing(no exceptions!!)Copyright G.Chiu and H.PengME561 Lecture1- 17

Competing Resources Better SensorsProvide better Vision Better ActuatorsProvide more Muscle Better ControlProvides more finesse by combining sensors andactuators in more intelligent waysCopyright G.Chiu and H.PengME561 Lecture1- 18

m/Copyright G.Chiu and 1 Lecture1- 19

Ex1 0 A Useful MATLAB CommandForce - VoltKfd4feed- ftVf1KfsKaVcfVolt am pZero-OrderHol d1-K-tau fs.s 1SaturationFeed per tooth1Feed sl ide2f(u)Sam pl ing1Feed ForceFeed forceF stiffnessSfVz2KzsKaVczVolt am p11stau zs.s 1Saturation1Zero-OrderHold1Z sl ide12f(u)12Z ForceZ forceIntegrator Sam p3d nomnom ianl depthdelta ZZ sti ffnessSz3Disturbance depthExample from 2001 machining project[A c, B c, C c, D c] linmod(‘Ex1 0’)[A d, B d, C d, D d] dlinmod(‘Ex1 0’, Ts)4x3 (why?)3x3 (why?)Note the possibility to linearize at non-zero x and u([A,B,C,D] LINMOD('SYS',X,U), see ‘help linmod’)Copyright G.Chiu and H.PengME561 Lecture1- 20

Discrete-Time Design by Emulation A simple way to design a discrete-time controlsystem is to start with classical techniques todesign a continuous-time compensator for aplant. The continuous-time compensator can then beapproximated by a discrete-time, sampled-datasystem. This process is known as emulation.Copyright G.Chiu and H.PengME561 Lecture1- 21

Emulation (cont.)yuPlantG(s)eControllerC(s) r–Emulation (indirect design)Sample and hold )A/D re(t)T–ClockCopyright G.Chiu and H.PengME561 Lecture1- 22

Sample and HoldDiscrete-Time Control Systems, OgataCopyright G.Chiu and H.PengME561 Lecture1- 23

A/D Several circuit design types––––Successive-approximation—n clock cycles for n-bit accuracy(Single-slope) IntegrationCounter (voltage to frequency)Parallel (Flash) encodingFigure 1-9 from Discrete-Time Control Systems, OgataThe Art of Electronics, Horowitz and Hill, pp.415Copyright G.Chiu and H.PengME561 Lecture1- 24

D/A Weighted resistors R-2R ladder circuitThe Art of Electronics, Horowitz and Hill, pp.410-411Copyright G.Chiu and H.PengME561 Lecture1- 25

Emulation Methods Approximate continuous-time operations(e.g., differentiation) with discrete-timeoperations (e.g., difference).––––(Forward) EulerTrapezoidal with Pre-warpMatched pole/zeroZero-Order Hold (ZOH)Copyright G.Chiu and H.PengME561 Lecture1- 26

Euler Approximation x t 0 tx limx ( k ) x ( k 1) x ( k )TwhereT tk 1 tk (the sampling period in seconds),tk kT ( for a constant sampling period ),k is an integer,x ( k ) is the value of x at time tk , andx ( k 1) is the value of x at time tk 1.Copyright G.Chiu and H.PengME561 Lecture1- 27

Ex1 1 Emulation Using Euler ApproximationyPlantG(s)uControllerC(s)e rC( s ) –s aU ( s) Ks bE( s)u b u K ( e a e )( s b )U ( s ) K ( s a ) E ( s )Euler approximation:x ( k 1) x ( k )x ( k ) Tu( k 1) u( k )e( k 1) e( k ) b u( k ) K a e( k )TTLMNOPQu( k 1) (1 bT ) u( k ) K ( aT 1) e( k ) K e( k 1)oru ( k ) (1 b T ) u ( k 1) K ( a T 1) e ( k 1) K e ( k )Copyright G.Chiu and H.PengME561 Lecture1- 28

Ex1 2 Simulation of Euler-Emulated ControllerLead compensatoryPlantG(s)ueControllerC(s) r–G( s) x1s( s 1)oC( s ) 50xs 2s 10xImplement Euler-emulated digital controller at 5Hz, 10Hz and 30HzCopyright G.Chiu and H.PengME561 Lecture1- 29

Ex1 2 MATLAB Implementation of theContinuous-Time System% Construct the plant transfer function:G num 1;G den [1 1 0];G tf(G num, G den);G( s) % Construct the compensator transfer function:C num 50*[1 2];C den [1 10];C tf(C num, C den);1s( s 1)C( s ) 50s 2s 10% The open-loop transfer function can be simply calculated by:OP G*C;% The closed-loop transfer function is:CL feedback(OP,1,-1);% Or you can use the following arithmetic:CL1 OP / (1 OP);% OrCL2 OP * inv(1 OP);yPlantG(s)uControllerC(s)e r–% The unit step response of the closed-loop system:[y,t] step(CL);Copyright G.Chiu and H.PengME561 Lecture1- 30

Ex1 2 SIMULINK Implementation of theContinuous-Time SystemtScopeyResponse (Output)ClockTime1s 2 s50(s 2)PlantLead CompensatorCopyright G.Chiu and H.Peng(s 10)StepME561 Lecture1- 31

Ex1 2 Discrete-Time ControllerMATLAB: see notesTime step TCoefficients dependon Tt digitalClockScopey digitalResponse(Output)1s 2 sPlantZero-OrderHoldCopyright G.Chiu and H.PengTime50z 50*(-0.9333)1z-0.66671Discrete Implementationof the Lead CompensatorSamplingStepME561 Lecture1- 32

1.21.211Unit Step Response10HzUnit Step ResponseEx1 2 Results0.80.60.40.830Hz0.60.4Analog ControlDigital Control (30 Hz)Analog ControlDigital Control (10 Hz)0.200.200.51Time (sec)01.500.51Time (sec)1.51.61.21.41.2Unit StepResponse5HzUnit StepResponse10.80.610.80.60.4Analog ControlDigital Control (5 Hz)0.200.4Analog ControlSampled Digital Output (5 Hz)Actual Output0.200.51Time (sec)( )1.5Copyright G.Chiu and H.Peng000.51Time (sec)(b)1.5ME561 Lecture1- 33

Issues for Digital Control Systems Sampling time for emulated designs need to beat least about 20-30 times closed-loopbandwidth. Else inter-sampling behaviorbecomes questionable Clock Dependency. Aliasing (Though briefly discussed in Chapter 1,explained in more details in Chapter 2) Time delay due to Sample/Hold. Need for discrete-time models.Copyright G.Chiu and H.PengME561 Lecture1- 34

Ex1 3 Clock Dependency A/DC ( s) Tas aClock1Step ResponseStep Response10.80.60.4Reference StepContinuous-Time ResponseDigital Response0.200123450.40.201234560123Time4561Step ResponseStep Response0.60610.80.60.40.200.80123Time45Copyright G.Chiu and H.Peng60.80.60.40.20ME561 Lecture1- 35

Common MythSince sampling will lose information in betweensamples, digital control will always be inferior to itscontinuous-time counter part, i.e. we cannotexpect better performance from digital control The above argument is true for emulationimplementationAlthough sampling inevitably loses information,digital control does pose some unique designflexibilities that are not achievable throughcontinuous-time LTI controlDead-beat control is one example.Copyright G.Chiu and H.PengME561 Lecture1- 36

Ex1 4 Dead-Beat ControlPlant:G (s) k G A (s) Control law:U (s) kJ s2bKs bR(s) KY (s)as a1Y ( s)2 s 0 1GCL ( s ) R( s ) s 0 3 2 s 0 2 2 s 0 1uControllerC(s)r1.5PositionAmplifierkOpen-Loop Frequency Response10.500123456789100.8Analog Control ResponseDigital Control ResponseSampled 12345Time6789101Control Inputy0.50-0.5-1Copyright G.Chiu and H.Peng0ME561 Lecture1- 37

Ex1 5 Aliasing/Beating4.9 Hz sine 10246Time8 46810e(t)A/DT10 Hz samplingClock10.5Sampled right G.Chiu and H.Peng6810ME561 Lecture1- 38

4.9 Hz sine wave and 10Hz SamplingPeriod 1/4.9 sec 204 .50.60.70.80.91t 0:0.001:1;y sin(9.8*pi*t);t2 0:0.1:1;y2 sin(9.8*pi*t2);plot(t, y, t2, y2, 'ro')Copyright G.Chiu and H.PengME561 Lecture1- 39

Time Delay Due to Sample/HolduH(t)u(kT)uAverage u(t)123456789kT10Delay T/2 which deteriorates phase margin anddamping (see Example 1 6)Copyright G.Chiu and H.PengME561 Lecture1- 40

Ex1 6uZOH signal uH(t)u(kT) T u (t ) u t 2 L u(t)U ( s) eAveraged signal u(t)123456789 10kTe Tj 2T s2 U ( s ) 1 and e Tj 2T 2 ZOH introduces additional phase lag, which reduces thephase margin of the continuous-time design– The amount of the phase lag is proportional to the frequency Device in digital feedback loop that reduces phase margin– Hold circuits (D/A)– Low-pass (anti-aliasing) filterCopyright G.Chiu and H.PengME561 Lecture1- 41