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CALTECH CDS 101 - PID Control

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Title PageContents1. Introduction1.1 Introduction1.2 A Brief HistoryThe Centrifugal GovernorThe Emergence of ControlThe Second Wave1.3 Process Control1.4 Manufacturing1.5 Robotics1.6 Power1.7 Aeronautics1.8 Electronics and Communications1.9 Automotive1.10 Computing1.11 Mathematics1.12 Physics1.13 Biology1.14 Summary2. Feedback2.1 Introduction2.2 Simple Forms of Feedback2.3 Representations of Feedback Systems2.4 Properties of Feedback2.5 Stability2.6 Open and Closed Loop Systems2.7 Feedforward2.8 Summary3. Dynamics3.1 Introduction3.2 Two views on dynamics3.3 Ordinary Differential Equations3.4 Laplace Transforms3.5 Frequency Response3.6 State Models3.7 Linear Time-Invariant Systems3.8 Summary4. Simple Control Systems4.1 Introduction4.2 Cruise Control4.3 Bicycle Dynamics4.4 Control of First Order Systems4.5 Control of Second Order Systems4.6 Control of Systems of High Order5. Feedback Fundamentals5.1 Feedback Fundamentals5.2 The Basic Feedback Loop5.3 The Gang of Six5.4 Disturbance Attenuation5.5 Process Variations5.6 When are Two Processes Similar5.7 The Sensitivity Functions5.8 Reference Signals5.9 Fundamental Limitations5.10 Electronic Amplifiers5.11 Summary6. PID Control7. Specifications8. Feedforward Design9. State FeedbackIndex6PID Control6.1 IntroductionThe PID controller is the most common form of feedback. It was an es-sential element of early governors and it became the standard tool whenprocess control emerged in the 1940s. In process control today, more than95% of the control loops are of PID type, most loops are actually PI con-trol. PID controllers are today found in all areas where control is used.The controllers come in many different forms. There are stand-alone sys-tems in boxes for one or a few loops, which are manufactured by thehundred thousands yearly. PID control is an important ingredient of adistributed control system. The controllers are also embedded in manyspecial-purpose control systems. PID control is often combined with logic,sequential functions, selectors, and simple function blocks to build thecomplicated automation systems used for energy production, transporta-tion, and manufacturing. Many sophisticated control strategies, such asmodel predictive control, are also organized hierarchically. PID control isused at the lowest level; the multivariable controller gives the setpointsto the controllers at the lower level. The PID controller can thus be saidto be the “bread and butter’t’t of control engineering. It is an importantcomponent in every control engineer’s tool box.PID controllers have survived many changes in technology, from me-chanics and pneumatics to microprocessors via electronic tubes, transis-tors, integrated circuits. The microprocessor has had a dramatic influenceon the PID controller. Practically all PID controllers made today are basedon microprocessors. This has given opportunities to provide additional fea-tures like automatic tuning, gain scheduling, and continuous adaptation.216From _Control System Design_by Karl Johan Åström, 2002Copyright 2002, Karl Johan Åström. All rights reserved.Do not duplicate or redistribute.6.2 The Algorithm6.2 The AlgorithmWe will start by summarizing the key features of the PID controller. The“textbook” version of the PID algorithm is described by:u(t)=Ke(t)+1TitZ0e(τ)dτ+Tdde(t)dt(6.1)where y is the measured process variable, r the reference variable, u isthe control signal and e is the control error(e = ysp− y). The referencevariable is often called the set point. The control signal is thus a sum ofthree terms: the P-term(which is proportional to the error), the I-term(which is proportional to the integral of the error), and the D-term (whichis proportional to the derivative of the error). The controller parametersare proportional gain K , integral time Ti, and derivative time Td.Theintegral, proportional and derivative part can be interpreted as controlactions based on the past, the present and the future as is illustratedin Figure 2.2. The derivative part can also be interpreted as predictionby linear extrapolation as is illustrated in Figure 2.2. The action of thedifferent terms can be illustrated by the following figures which show theresponse to step changes in the reference value in a typical case.Effects of Proportional, Integral and Derivative ActionProportional control is illustrated in Figure 6.1. The controller is givenby(6.1) with Ti=∞and Td= 0. The figure shows that there is alwaysa steady state error in proportional control. The error will decrease withincreasing gain, but the tendency towards oscillation will also increase.Figure 6.2 illustrates the effects of adding integral. It follows from(6.1)that the strength of integral action increases with decreasing integral timeTi. The figure shows that the steady state error disappears when integralaction is used. Compare with the discussion of the “magic of integralaction” in Section Section 2.2. The tendency for oscillation also increaseswith decreasing Ti. The properties of derivative action are illustrated inFigure 6.3.Figure 6.3 illustrates the effects of adding derivative action. The pa-rameters K and Tiare chosen so that the closed-loop system is oscillatory.Damping increases with increasing derivative time, but decreases againwhen derivative time becomes too large. Recall that derivative action canbe interpreted as providing prediction by linear extrapolation over thetime Td. Using this interpretation it is easy to understand that derivativeaction does not help if the prediction time Tdis too large. In Figure 6.3the period of oscillation is about 6 s for the system without derivative217Chapter 6. PID Control0 5 10 15 20010 5 10 15 20−22K = 5K= 2K= 1K= 5K= 2K= 1Figure 6.1 Simulation of a closed-loop system with proportional control. The pro-cess transfer function is P(s)=1/(s + 1)3.0 5 10 15 20010 5 10 15 20012Ti= 1Ti= 2Ti= 5Ti=∞Ti=1Ti=2Ti=5Ti=∞Figure 6.2 Simulation of a closed-loop system with proportional and integral con-trol. The process transfer function is P(s)=1/(s + 1)3, and the controller gain isK= 1.action. Derivative actions ceases to be effective when Tdis larger thana1s(one sixth of the period). Also notice that the period of oscillationincreases when derivative time is increased.A PerspectiveThere is much more to PID than is revealed by (6.1). A faithful imple-mentation of the equation will actually not result in a good controller. Toobtain a good PID controller it is also


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