Compensation of Control System1．Multiple Constrains in DesignThe performance of a feedback control system is of primary importance.We have found that a suitable control system should have some of the following properties.1)It should be stable and present acceptable response to inputcommand,i.e,the controlled variable should follow the changes in the input at a suitable speed without unduly large oscillations or overshoots.2)It should operate with as little error as possible.3)It should be able to mitigate the effect of undesirabledisturbances.A feedback control system that provided an optimum perfprmance without any necessary adjustments is rare ually it is necessary to compromise among the many conflicting and demanding specifications and to adjust the system parameters to provide a suitable and acceptable performance when it is not possible to obtain all the desired optimum specifications.The preceding chapters have showen that it is often possible to adjust the system parameters in order to provide the desired system response.When the achievement of a simple performance requirement may be met by selecting a particular value of K, the process is called gain compensation.However,we often find that it is not sufficient to adjust a system parameter and thus obtain the desired performance.Rather we are required to reconsider the structure of the system and redesign the system in order to obtain a suitable one.That is ,we must examine scheme of the system and obtain a new design that results in a suitable system.Thus the design of a control system is concerned with the arrangement of the systemstructure and the selection of suitable components and parameters.When we are not able to relax several perfprmance requirements,we must alter the system in some way.The alteration or adjustment of a control system in oreder to provider a suitable performance is called compensation.In redesigning a control system to alter the system response,an additional component or device is inserted within the structure of the feedback system to compensate for the performance deficiency.The compensating device may be electric,mechanical,hydraulic,pneumatic,or some other type of devices or networks and is often called a monly an electric network serves as a compensator in many control systems.Quite often,in practice,the best and simplest way to improve the performance of a control system is to alter,if possible,the plant itself.That is,if the system designer is able to specify and alter the design of the plant,then the performance of the system may be readily improved.For example,to improve the transient behavior of a servomechanism position controller,we often can choose a better motor for the system.Thus a control system designer should recognize that an alteration of the plant maybe result in an improved system.However,often the plant is unalterable or has been altered as much as possible and still result in unsatisfactory performance.Then the addition of compensator becomes useful for improving the performance of the system.2．Types of CompensationThe compensator is placed in a suitable location within the system,and can be done in several ways .An additional component may be interested in the forward path.This is called the cascade or serial compensation.The transfer function of the compensator is designated as Gc(s),whereas that of the original plant(or process)is denoted by Gp(s).Alternatively,the compensator may be placed in the feedbackpath .This is called the feedback compensation. A combination of these two schemes.The selection of the compensation scheme depends upon a consideration of the specifications, the power levels at various signal nodes in the system,and the compensators available for us.3.Cascade CompensationAlthough many different types of compensators can be used,the simplest among them are cascade phase-lead,phase-lag,and phase-lag-lead networks.Each of these can be realized by using an operational amplifier network.The Bode diagram is used to determine a suitable cascade compensator in preference to other frequence plots.The frequence response of the cascade compensator is added to the frequency response of the uncompensated system.It is assumed, in below discussion, that the compensator Gc(s),is used with an uncompensated system so that the overall open-loop gain can be set to satisfy the steady-state error requirement,then Gc(s) is used to adjust the system dynamics favorably without affecting the steady-stste error.For convenience,the open-loop transfer function of the uncompensated system,Gp(s)H(s),is denoted by Go(s).At first,consider a system described by the open-loop transfer functionGo(s)=k/s(0.2s+1)Suppose we wish the closed-loop system to meet the following performance requirements:a)The steady-state error for a unit ramp is to be no more than 0.00316.b)The phase margin is to be no less than 45.For the first requirement,the static velocity error constant can be cocalculated from equationεss=1/Kv=1/K≦0.00316and thus the required open-loop gain is K=Kv=3.16It may also be seen that the phase margin will be about 45.at w=5 rad/s;therefore,to meet the second requirement,the magnitude must be zero at this frequency.Obviously,it is not possible to satisfy both system performance requirements with a singular value of gain.The system needs to be modified in some way,i.e,the shape of the Bode diagram has to be altered in some way to allow it to achieve both perfprmance requirements. The system perfprmance requirements stated in the example are typical of those found in many design cases;a steady-state error determines one value of gain while a desired transient response determines anthor.Note how each requirement relates to a different region of frequency acxis in the Bode diagram.(1)The steady-state error relates to the slope and magnitude at lowfrequency.(2)The phase margin relates to the gain crossover frequency,which usuallyoccurs at higher frequency.4．Approaches to System DesignThe performance of a control system can be specified by requirement of certain maximum overshoot and setting time for a step input. Furthermore it is usually necessary to specify the maximum allowable steady-state error for several test signal inputs and disturbances .These performance specifications are related to the location of the poles and zeros of the closed-loop transfer function.Thus the location of the closed-loop poles and zeros can be specified.As we found in chapter4,the locus of the roots of the closed-loop system can be readily obtained for the variation of one system parameter.However,when the locus of roots does not result in a suitable root configuration,we must add a compensator to alter the locus of the roots as parameter is varied.Therefore we can use the root locus method and determine a suitsble compensator transfer function so that the resultant root ;ocus yields the desired closed-looproot configuration.Alternatively,the performance of a control system can be specified in terms of the relative resonant peak,resonant frequency,and bandwidth of the closed-loop frequency response,or in terms of the phase margin, gain margin and gain crossover frequency of the open-loop frequency response.We can add a suitable compensator,if necessary,in order to satisfy the system perfprmancr.The design of the compensator is developed in terms of the frequency response as portrayed on the polar plot,the Bode diagram,or the Nichols chart.Because a cascade transfer function is readily accounted for on a Bode diagram by adding the frequency response of the compensator,we usually prefer to approach the frequency response method by utilizing the Bode diagram.5．Phase-Lead Compensation1. Phase-Lead CompensationThe phase-lead compensator is a form of high-pass filter,through which the signals at high frequencies are amplified relatively than that at low frequencies.It introduces a gain at high frequencies,which in general is destabilizing.However,its positive phase angle is stablilzing .Hence,we must carefully choose two break frequencies so that the stabilizing effect of the positive phase angle is dominant.ments on the Applicability and ResultsPhase-lead compensation has some distinct advantages over other forms of compensation,wherease it may also be difficult to use .Some observations from the example just analyzed allow a few generalizations to be made regarding phase lead compensation.1)The phase-lead compensation method provides an additional phaselead to limit the system’s overshoot to a required value.2)The open-loop(and usually the closed-loop)bandwidths isincreased.This is usually beneficial since the inclusion ofhigher frequencies in the response results in a faster response.Itmay cause problem,however,if noise exists at the higherfrequencies.3)Problem may occur when the uncompensated phase plot has a steeoslope in the vicinity of φm.This occurs because,as the new gaincrossover point moves to the right ,larger and larger phase leadis required from the compensator,demanding very large value ofα.This is difficult to achieve when the compensator is realizedwith physical components.For this reason ,value of α>15 shouldbe avoided,and methods to compensate the system using othertechniques,such as phaselag,should be investigated.6．Phase-Lag Compensation1.Phase-Lag Compensation ProcessIn phase-lag compensation,the magnitude part of the uncompensated Bode diagram is attenuated in order to reduce the gain crossover frequency,thereby allowing the uncompensated phase plot to produce the necessary phase margin.The phase-lag compensator is used to provide an attenuation and therefore to lower the crossover frequency of the system.Furthermore,at lower crossover frequency,we usually find that the phase margin of the system is increased,and the specifications can be satisfied.Of cause,the influence of the phase lag caused by the compensator should be taken into ually,the lag phase is about 5~12 if the break frequency corresponding to the zero of the compensator is wz=(0.1~0.2)wc.ments on the Applicability and Results1)The phase-lag method provides the necessary damping ratio in orderto limit the overshoot to the required value.2)The compensation process is somewhat simpler than the phase-leadcompensation in that the selection of the break frequencies is not too critical.3)As can be seen from the compensated system,the phase-lag techniquereduces the open and hence the closed-loop bandwith,which results ina slower response.4)Unlike phase-lead compensation,theoretically,phase lag compensationmay change the phase margin by more than 90.7．Phase Lag-Lead CompensationIn the compensator design it is usual to assume that the two break frequencies of the lag portion are lower than the two break frequencies of the lead portion.Further features of the Bode diagram include the following.1)The magnitude at lower frequencies is 0 db while the magnitude athigher frequencies is 20lg(αβ)ually,the compensator provides attenuation only and no gain.2)The phase angle first lags and then leads ,but the high-andlow-frequency phases are both zeros.3)The maximum phase-lag and the maximum phase-lead occur between theirrespective break frequencies.The phase lag-lead compensator utilizes the best feature of the individual lag and lead portions,usually without their disadvantages.For example,the lag-lead compensation allows the introduction of phase lead to stabilize a system,while providing attenuation at higher frequencies to filter out noise.8．Feedback CompensationIn order to improve the system performances,besides the cascade compensation,the feedback compensation is often used as another scheme.By using local feedback compensation,almost same effect,as that of cascade compensation,can be obtained.Morever ,additional specific functions forimproving system perfprmance are obtained.。