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CALTECH CDS 101 - Loop Design

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Chapter 8Loop Design8.1 IntroductionThis is the first Chapter that deals with design and we will therefore start bysome general aspects on design of engineering systems. Design is complicatedbecause there are many issues that have to be considered. Much researchand development has been devoted to development of design procedures andthere have been numerous attempts to formalize the design problem. It isuseful to think about a design problem in terms of specifications, trade-offs,limitations, design parameters. In additions it is useful to be aware of thefact that because of the richness of design problem there are some propertiesthat are captured explicitly by the design method and other properties thatmust be investigated when the design is completed.Specifications are an attempt to express the requirements formally. Forcontrol systems they typically include: attenuation of load disturbances,measurement noise, process uncertainty and command signal following. Spec-ifications are typically expressed by quantities that capture these features.For control systems the fundamental quantities are typically transfer func-tions or time responses and specifications are some representative featuresof these functions. A design fundamentally involves trade-offs. It is alsoextremely important to be aware of fundamental limitations to avoid unre-alistic specifications and impossible trade-offs. Sometimes it is attempted tocapture the trade-offs in a single optimization criterion, but such a criterionalways includes parameters that have to be chosen. It is often useful tomake the trade-offs explicit and to have design parameters that gives thedesigner control of them. A design method typically focuses on some aspectsof a design problem but it is often difficult to capture all aspects of a design129130 CHAPTER 8. LOOP DESIGNproblem formally. There will frequently be several properties that must beinvestigated separately when the design is completed. In control system de-sign it is common practice to verify a design by extensive simulation thatmay include hardware in the loop. When discussing design in this and thefollowing chapters we have made an attempt to emphasize the fundamentaltradeoffs and the design parameters.In this Chapter we will present a design method which is focused onthe loop transfer function of the system. The method is focused on designa feedback controller that satisfies can deal with disturbances and processuncertainty. The response to reference signals can later be dealt with usingfeedforward which is discussed in the next chapter. The idea of loop shapingis outlined in Section 8.2 which present classical results that were developedwhen control emerged in the early 1940s. This section also contains somesimple compenstators like lead-lag compensation, which is closely related toPID control. Section ?? presents another classical result due to Bode, whodeveloped the concept of an ideal loop transfer function for electronic am-plifiers. In the following sections it is shown how the loop transfer functionrelates to important properties of the closed loop system. Section ?? treatsrobustness properties. It is shown that the estimates developed earlier canbe refined to give more precise information. Two graphical tools, the Halldiagram and the Nichols chart are also introduced. Section ?? deals withload disturbances. An estimate the norm of the transfer function from loaddisturbances to process output based on the crossover frequency is derived.The result shows that it is typically advantageous to have a high crossoverfrequency. Section ?? deals with measurement noise. An estimate of thenorm of the transfer function from measurement noise to the control signalis developed. It is shown that a high crossover frequency is typically associ-ated with high controller gains at frequencies above the crossover frequency.In Section ?? it is shown that non-minimum phase dynamics imposes severeconstraints on the admissible crossover frequencies. Summarizing the find-ings of this section with the results of Sections ??, ??, and ?? we are in agood position to discuss the major trade-offs in the choice of gain crossoverfrequency. First the choice is severely restricted if process dynamics is notminimum phase. The choice is then governed by a compromise betweenrobustness, attenuation of load disturbances and injection of measurementnoise. It turns out that the compromise can be captured in a single diagram.This diagram also indicates the complexity of the controller that is requiredfor different choices. This is shown in Section ??.8.2. COMPENSATION 1318.2 CompensationThe idea of loop shaping is to find a controller so that the loop transferfunction has desired properties. Since the loop transfer function L = P Cis the product of the transfer functions of the process and the controllerit is easy to see how the loop transfer function is influenced by the con-troller. Compensation is typically done by successive modifications of theloop transfer function starting with proportional control. If the desired prop-erties cannot be obtained by proportional control the controller is modifiedby multiplying the controller transfer function by compensating networkshaving simple transfer functions. The design is conveniently visualized us-ing bode diagrams. Since the Bode diagram is logarithmic multiplicationby a compensating network corresponds to additions in the Bode diagram.We start by discussing a desired properties of a loop transfer function, thenwe present some simple compensators and we end by a few examples. Inthe following sections we will present a more systematic treatment wherewe discuss how th different properties of the loop are influenced by the looptransfer function.Properties of the Loop Transfer FunctionThe Bode diagram of a typical loop transfer function is shown in Figure 8.1.Recall that the lowest frequency where the loop transfer function has gain1 is called gain crossover frequency, see Section ??. This parameter is animportant design parameter in loop shaping. When we talk about high andlow frequencies in the following we are using the crossover frequency as areference. The gain curve of a typical loop transfer function is decreasing asshown in Figure 8.1. The behavior at low frequencies determine attenuationof load disturbances and behavior of tracking of low frequency referencesignals. The behavior around the crossover frequency determines robustnessand sensitivity to modeling


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