1、无 锡 职 业 技 术 学 院 外文翻译 2 System Design an Compensation Techniques Control systems are designed to perform specific tasks.The requirements imposed on the control system are usually referred as performance imposed on the control system are usually referred as performance specifications.They generally re
2、late to accuracy,relative stability and speed of response. Generally,the performance specifications should not be more stringent than necessary to perform the given task.If the accurary at steady-state operation is of prime importance in a given control system,then we should not require unnecessaril
3、y rigid performance specifications on the transient response since such specifications will require expensive components.We should remember that the most important part of control system design is to state the performance specifications precisely so that they will yield an optimal control system for
4、 a given purpose. In this lesson,we are going to briefly introduce the design and compensation procedure of single-input-single-output-(SISO),linear time-invariant (LTI) control systems by the frequency response and root-locus approaches,Compensation is the modification of the modification of the sy
5、stem dynamics to satisfy the given specifications. Setting the gain is the first step in adjusting the system for satisfactory performance.In many cases,increasing the gain value will improve the steady-state behavior but will result in poor stability or even instability,Then it is necessary to rede
6、sign the system (by modifying the structure or by incorporating additional devices or components) to alter the overall behavior so that the system will behave as desired. Fig.8.1 shows the configuration where the compensator G(s) is placed in series with the plant.This scheme is called series compen
7、sation. Another kind or compensation is feedback compensation.Generally,series compensation may be simpler than feedback compensation, In discussing compensators,we frequently use terminology as lead network,and lag-lead network.If a sinusoidal input ei is applied to the input of a network and the s
8、teady-state output e0 (which is also sinusoidal) has a phase lead,then the network is called a lead network.Similarly, if the steady-state output e0 has a phase lag,then the network is called a lag network.In a lag-lead network,phase lag and phase lead both occur in the output but in different frequ
9、ency regions；phase lag occurs in the low-frequency region and phase lead occurs in the high-frequency region. The root-locus method is a graphical method for determining the locations of all closed-loop poles from knowledge of the locations of the locations of the open-loop poles and zeros as some p
10、arameter(usually the gain) is varied from zero to infinity.The method yields a clear indication of effects of parameter adjustment.In practice,the root-locus plot of a system may indicate that the desired performance cannot be achieved just by the adjustment of gain.Then it is necessary to reshape t
11、he root loci to meet the performance specifications. In designing a control system,we may modify the original root loci by inserting a suitable compensator Gc(s) (as shown in Fig.8.1).Once the effects on the root locus of the addition of the poles and/or zeros are fully understood,we can readily det
12、ermine the locations of the pole(s) and zero(s) of the compensator that will reshape the root locus as desired.In the design by the root-locus method,the root-loci of the system are reshaped through the use of a compensator so that a pair of dominant closed-loop poles can be placed at the desired lo
13、cations.(Usually,the damping ratio and undamped natural frequency may be specified by the location of a pair of dominant closed-loop poles.) The addition of a pole to the open-loop transfer function has the effect of pulling the root locus to the right,tending to lower the systems relative stability
14、 and to slow down the settling of the response.The addition of a zero has the effect of pulling the root locus to the left,tending to make the system more stable and to speed up the settling of the response. The root-locus approach to design is very powerful when the specifications are given in term
15、 无 锡 职 业 技 术 学 院 外文翻译 3 of time domain quantities,such as the damping ratio and undamped natural frequency,maximum overshoot,rise time and setting time. Let us consider a design problem.The original system either is unstable for all values of gain or is stable but has undesirable transient response
16、characteristics.In this case,the reshaping of the root locus is necessary in order that the dominant closed-loop poles be at desired locations in the complex plane.Inserting an appropriate lead compensator in cascade with the feed-forward transfer function may solve this problem. It is important to
17、note that in a control system design,transient-reponse performance is usually most important.In the frequency-response approach,we specify the transient-response in term of the phase and gain margin,resonant peak magnitude,the gain crossover frequency,resonant frequency response is indirect,the freq
18、uency domain specification can be met conveniently by means of Bode diagram. Design in the frequency domain is simple and straightforward.After the open loop has been designed by frequency response method,the closed loop poles and zeros can be determined.The transient response characters must be che
19、cked to see whether the designed system meets the requirements in the time domain.If it does not,the compensator has to be modified and the analysis must be repeated until a satisfactory result is obtained. Basically,there are two approaches in the frequency-domain design.One is the polar plot appro
20、ach and the other is the Bode diagram approach.It is more convenient to work with Bode diagram.A Bode diagram of the compensator can be simply added to the original Bode diagram,and thus plotting the complete Bode diagram is a simple matter.Also,if the open loop gain is varied,the magnitude curve is
21、 shifted up or down without changing the slope of the curve,and the phase curve remains the same. A common approach to the Bode diagram is that we first adjust the open loop gain so that the requirement on the steady state accuracy is met.Then we plot the magnitude and phase curves of the uncompensa
22、ted open loop.If the specification on the phase margin and gain margin are not satisfied,then a suitable compensator that will reshape the open loop transfer function is determined. In many practical cases,compensation is essentially a compromise between steady-state accuracy and relative stability.
23、In order to have a high value of the velocity error constant and yet satisfactory relative stability,we find it necessary to reshape the open loop frequency response curve.The gain in the low-frequency region should be large enough to satisfy the steady-state accuracy requirements.For the medium-fre
24、quency region (near the gain crossover frequency wc from both directions),the slope of the log-magnitude curve in the Bode diagram should be -20dB per decade. This slope should extend over a sufficient wide frequency band to assure a proper phase margin.For the high-frequency region,the gain should
25、be attenuated as rapidly as possible to minimize the effects of noise. The basic characteristics of lead,lag,and lag-lead compensation are as following.lead compensation essentially yields an appreciable improvement in transient response and a small change in steady-state accuracy.It may accentuate
26、high-frequency noise effects.On the other hand ,lag compensation yields an appreciable improvement in steady-state accuracy at the expense of increasing the transient-response time.Lag compensation will suppress the effects of high-frequency noise signals.Lag-lead compensation combines the character
27、istics of both lead compensation and lag compensation. Discrete-time Systems and the z-Transform Method Discrete-time systems,or sampled-data system,are dynamic systems in which one or more variables can change only at discrete instants of time.These intstants,which we shall denote by kt or tk(k=0,1
28、,2,.),may specify the time at which some physical measurement is performed or the time at which the memory of a digital computer is read out,etc.The time interval between these discrete instants can be approximated by simple interpolation. 无 锡 职 业 技 术 学 院 外文翻译 4 Discrete-time systems differ from con
29、tinuous-time ones in that the signals for a discrete-time system are in sample-data form. Discrete-time systems arise in practice whenever the measurements neccessary for control are obtained in an intermittent fashion,or a large scale controller or computer is time-shared by several plants so that
30、a control signal is sent out to each plant only periodically or whenever a digital computer is used to perform computations necessary for control.Many modern industrial control systems are in time.Sometimes,however,sampling operation,or discretization may be entirely fictitious and introduced only t
31、o simplify the analysis of control system which actually contains only continuous elements.whose inputs and/or outputs are in time.Sometimes,however,sampling operation,or discretization,may be entirely fictitious and introduced only to simplify the analysis of control system which actually contain o
32、nly continuous elements. In this lesson,we shall be concerned with discrete-time systems which the signal representing the control efforts is piecewise constant and changes only st discrete points in time.Since there are several different types of sampling operation of practical importance,we shall
33、list them as follows: (1)Periodic(conventional) sampling:In this case,the sampling instants are equally spaced,or tk=kt(k=1,2,3.) Multiple-order sampling:The pattern of the tk is repeated periodically,or tk+r - tk=constant for all k. Multiple-order-rate sampling:In this case,two concurrent sampling
34、operations occur at tk=pT1 and qT2,where T1,T2 are contants and p ,q are integers. Random sampling:In this case,the sampling instants are random,or tk is a random variable. Here we shall treat only the case which the samplng is periodic. Quantization.The inclusion of digital computer in an otherwise
35、 analog system produces in digital form(usually as binary numbers) in part of the system.The system then takes the form of a mixed digital-analog combination.The introduction of a digital computer in a control system requires the use of digital-to-digital converters.The conversion of an analog signa
36、l to the corresponding digital signal(binary number)is an approximation because the analog signal can take an infinite number of values,whereas the variety of different numbers which can be formed by a finite number of values,whereas the variety of different numbers which can be formed by a finite s
37、et of digits is limited.This approximation process is called quantization. The process of quantizing (converting a signal in analog form to digital form)may be fulfilled by means of some specific circuits.The range of input magnitudes is divided into a finite number of disjoint intervals hi which ar
38、e not necessarily equal.All magnitudes fallinjg within each interval are equated to a single value within the interval.This single value is the digital approximation to the magnitudes of the analog input signal.Thus,if x si the analog input,the digital output is given by y=Q(x ) Where Q is the quant
39、izing function. The function x(t) is a discrete-time function.The operation of digital control systems involves quantization both in amplitude and in time.We s hall next present the definitions of several terms. Transducer.A transducer is a device which converts an input signal into an output signal
40、 of another form.(The output signal.in general,depends on the past history of the input). Analog transducer.An analog transducer is a device which converts an input signal into an ouput signals occur only at discrete instants of time (usually periodic),but the magnitudes of these signals may be any value within the physical limitations of the system. Sampled-data transducer.This is a transducer in which the input and output signals occur only at discrete instants of time(usually periodic),but the magnitudes of the signal,as in the case of the analog transducer,are unquantized.
