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CALTECH CDS 101 - Simple Control Systems

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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 Systems5. 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 FeedbackIndex4Simple Control Systems4.1 IntroductionIn this chapter we will give simple examples of analysis and design ofcontrol systems. We will start in Sections 4.2 and 4.3 with two systemsthat can be handled using only knowledge of differential equations. Sec-tion 4.2 deals with design of a cruise controller for a car. In Section 4.3we discuss the dynamics of a bicycle, many of its nice properties are dueto a purely mechanical feedback which has emerged as a result of trialand error over a long period of time. Section 3.3 is a suitable prepara-tion for Sections 4.2 and 4.3. Differential equations are cumbersome formore complicated problems and better tools are needed. Efficient meth-ods for working with linear systems can be developed based on a basicknowledge of Laplace transforms and transfer functions. Coupled withblock diagrams this gives a very efficient way to deal with linear systems.The block diagram gives the overview and the behavior of the individualblocks are described by transfer functions. The Laplace transforms makeit easy to manipulate the system formally and to derive relations betweendifferent signals. This is one of the standard methods for working withcontrol systems. It is exploited in Section 4.4, which gives a systematicway of designing PI controllers for first order systems. This section alsocontains material required to develop an intuitive picture of the proper-ties of second order systems. Section 4.5 deals with design of PI and PIDcontrollers for second order systems. A proper background for Sections 4.4and 4.5 is Section 3.4. Section 4.6 deals with the design problem for sys-tems of arbitrary order. This section which requires more mathematicalmaturity can be omitted in a first reading. For the interested reader itgives, however, important insight into the design problem and the struc-ture of stabilizing controllers. Section 4.6 summarizes the chapter and144From _Control System Design_by Karl Johan Åström, 2002Copyright 2002, Karl Johan Åström. All rights reserved.Do not duplicate or redistribute.4.2 Cruise ControlFigure 4.1 Schematic diagram of a car on a sloping road.outlines some important issues that should be considered.4.2 Cruise ControlThe purpose of cruise control is to keep the velocity of a car constant.The driver drives the car at the desired speed, the cruise control systemis activated by pushing a button and the system then keeps the speedconstant. The major disturbance comes from changes of the slope of theroad which generates forces on the car due to gravity. There are alsodisturbances due to air and rolling resistance. The cruise control systemmeasures the difference between the desired and the actual velocity andgenerates a feedback signal which attempts to keep the error small inspite of changes in the slope of the road. The feedback signal is sent toan actuator which influences the throttle and thus the force generated bythe engine.We will start by developing a mathematical model of the system. Themathematical model should tell how the velocity of the car is influencedby the throttle and the slope of the road. A schematic picture is shown inFigure 4.1ModelingWe will model the system by a momentum balance. The major part of themomentum is the product of the velocity v and the mass m of the car. Thereare also momenta stored in the engine, in terms of the rotation of the crankshaft and the velocities of the cylinders, but these are much smaller thanmv.Letθdenote the slope of the road, the momentum balance can be145Chapter 4. Simple Control SystemsSlope of roadVelocityDesired velocityThrottleFEngineControllerBody−ΣΣFigure 4.2 Block diagram of a car with cruise control.written asmdvdt+ cv = F − mgθ(4.1)where the term cv describes the momentum loss due to air resistance androlling and F is the force generated by the engine. The retarding forcedue to the slope of the road should similarly be proportional to the sineof the angle but we have approximated sinθθ. The consequence of theapproximations will be discussed later. It is also assumed that the force Fdeveloped by the engine is proportional to the signal u sent to the throttle.Introducing parameters for a particular car, an Audi in fourth gear, themodel becomesdvdt+0.02v = u − 10θ(4.2)where the control signal is normalized to be in the interval 0 ≤ u ≤ 1,where u= 1 corresponds to full throttle. The model implies that with fullthrottle in fourth gear the car cannot climb a road that is steeper than10%, and that the maximum speed in 4th gear on a horizontal road isv= 1/0.02 = 50 m/s (180 km/hour).Since it is desirable that the controller should be able to maintainconstant speed during stationary conditions it is natural to choose a con-troller with integral action. A PI controller is a reasonable choice. Such acontroller can be described byu= k(vr− v)+kiZt0(vr−v(τ))dτ(4.3)A block diagram of the system is shown in Figure 4.2. To understand howthe cruise control system works we will derive the equations for the closedloop systems described by Equations(4.2) and (4.3) Since the effect of theslope on the velocity is of primary interest we will derive an equation thattells how the velocity error e= vr− v depends on the slope of the road.1464.2 Cruise ControlAssuming that vris constant we find thatdvdt=−dedt,d2vdt2=−d2edt2It is convenient to differentiate (4.3) to avoid dealing both


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