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1、<p> System Design an Compensation Techniques</p><p> Control systems are designed to perform specific tasks.The requirements imposed on the control system are usually referred as performance imposed
2、on the control system are usually referred as performance specifications.They generally relate to accuracy,relative stability and speed of response.</p><p> Generally,the performance specifications should n
3、ot 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 unnecessarily rigid performance specific
4、ations 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 s
5、o that they</p><p> 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 r
6、esponse and root-locus approaches,Compensation is the modification of the modification of the system dynamics to satisfy the given specifications.</p><p> Setting the gain is the first step in adjusting the
7、 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 redesign the system (by mo
8、difying the structure or by incorporating additional devices or components) to alter the overall behavior so that the system will behave as desired.</p><p> Fig.8.1 shows the configuration where the compens
9、ator G(s) is placed in series with the plant.This scheme is called series compensation. Another kind or compensation is feedback compensation.Generally,series compensation may be simpler than feedback compensation,</p
10、><p> 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 steady-state output e0 (which is also s
11、inusoidal) 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
12、occur in the output but in different frequency regions;phase lag occurs in th</p><p> The root-locus method is a graphical method for determining the locations of all closed-loop poles from knowledge of the
13、 locations of the locations of the open-loop poles and zeros as some parameter(usually the gain) is varied from zero to infinity.The method yields a clear indication of effects of parameter adjustment.In practice,the roo
14、t-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 the root lo</p><p> In designing a control system,we may
15、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 determine the
16、 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 tha
17、t a pair of dominant cl</p><p> 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 system's relative stability and to sl
18、ow 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.</p><p>
19、 The root-locus approach to design is very powerful when the specifications are given in term of time domain quantities,such as the damping ratio and undamped natural frequency,maximum overshoot,rise time and setting tim
20、e.</p><p> 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 characteristics.In this case,the reshaping of th
21、e 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 t
22、his problem.</p><p> It is important to 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 t
23、erm of the phase and gain margin,resonant peak magnitude,the gain crossover frequency,resonant frequency response is indirect,the frequency domain specification can be met conveniently by means of Bode diagram.</p>
24、<p> 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 charac
25、ters must be checked 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 obtaine
26、d.</p><p> Basically,there are two approaches in the frequency-domain design.One is the polar plot approach and the other is the Bode diagram approach.It is more convenient to work with Bode diagram.A Bode
27、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 shifted up or down wi
28、thout changing the slope of the curve,and the phase curve remains the same.</p><p> A common approach to the Bode diagram is that we first adjust the open loop gain so that the requirement on the steady sta
29、te accuracy is met.Then we plot the magnitude and phase curves of the uncompensated open loop.If the specification on the phase margin and gain margin are not satisfied,then a suitable compensator that will reshape the o
30、pen loop transfer function is determined.</p><p> In many practical cases,compensation is essentially a compromise between steady-state accuracy and relative stability.In order to have a high value of the v
31、elocity 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-stat
32、e accuracy requirements.For the medium-frequency region (near the gain crossover frequency wc from both directions),the slop</p><p> The basic characteristics of lead,lag,and lag-lead compensation are as fo
33、llowing.lead compensation essentially yields an appreciable improvement in transient response and a small change in steady-state accuracy.It may accentuate high-frequency noise effects.On the other hand ,lag compensation
34、 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 compen</p>&
35、lt;p> Discrete-time Systems and the z-Transform Method </p><p> Discrete-time systems,or sampled-data system,are dynamic systems in which one or more variables can change only at discrete instants of ti
36、me.These intstants,which we shall denote by kt or tk(k=0,1,2,........),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 i
37、nterval between these discrete instants can be approximated by simple interpolation.</p><p> Discrete-time systems differ from continuous-time ones in that the signals for a discrete-time system are in samp
38、le-data form.</p><p> 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
39、by several plants so that 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 ti
40、me.Sometimes,however,sampling operation,or discretization may be entirely fictitious and introdu</p><p> In this lesson,we shall be concerned with discrete-time systems which the signal representing the con
41、trol 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 list them as follows:</p><p> (1
42、)Periodic(conventional) sampling:In this case,the sampling instants are equally spaced,or tk=kt(k=1,2,3....) </p><p> Multiple-order sampling:The pattern of the tk is repeated periodically,or tk+r - tk=cons
43、tant for all k.</p><p> Multiple-order-rate sampling:In this case,two concurrent sampling operations occur at tk=pT1 and qT2,where T1,T2 are contants and p ,q are integers.</p><p> Random samp
44、ling:In this case,the sampling instants are random,or tk is a random variable.</p><p> Here we shall treat only the case which the samplng is periodic.</p><p> Quantization.The inclusion of di
45、gital computer in an otherwise 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 co
46、mputer in a control system requires the use of digital-to-digital converters.The conversion of an analog signal to the corresponding digital signal(binary number)is an approximation because the analog signal can take an
47、infinite number of values,wh</p><p> 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 divide
48、d into a finite number of disjoint intervals hi which are not necessarily equal.All magnitudes fallinjg within each interval are equated to a single value within the interval.This single value is the digital approximatio
49、n to the magnitudes of the analog input signal.Thus,if x si the analog input,the digital output is given by y=Q</p><p> Where Q is the quantizing function.</p><p> The function x(t) is a discr
50、ete-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.</p><p> Transducer.A transducer is a d
51、evice which converts an input signal into an output signal of another form.(The output signal.in general,depends on the past history of the input).</p><p> Analog transducer.An analog transducer is a device
52、 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.</p&g
53、t;<p> 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 tr
54、ansducer,are unquantized.</p><p> Digital transducer.A digital transducer is one in which the input signal is a continuous function of time and the output signal is a quantized signal which can assume only
55、certain discrete levels.</p><p> Analog-to-digital transducer.A digital-to-analog transducer is one in which the input signal is a quantized signal and the output signal is a smoothed continuous function of
56、 time.</p><p> Analog controllers and digital controllers.In considering the types of controllers which are used in industrial control system,we may divide them into the following three categories:</p>
57、;<p> Analog controllers or computers:Analog controllers or computer represent the variables in the equations by continuous physical quantities.Analog controllers can be designed which will satisfactorily serve a
58、s nondecision making controllers.</p><p> Digital controllers or computers:These operate only on numbers.Decision-making is an important function in digital controllers,and they are currently being used for
59、 the solution of problems</p><p> Involving the optimal overall operation of industrial plants.</p><p> Analog-digital controllers or computers:These are often called hybrid controllers.They a
60、re combinations of amalog controllers and digital controllers.Some of high performance controllers are of this type.</p><p> Advantages of digital controllers over analog controller.Some of the advantages o
61、f digital controllers over amalog controllers may be summarized as follows:</p><p> (1)Digital controllers are capable of performing complex computations with constant accuracy at high speed.Digital compute
62、rs can have almost any desired accuracy in computations at relatively little increase in cost.On the other hand,the cost of analog computers increases rapidly as the complexity of the computations increase if constant ac
63、curacy is to be maintained.</p><p> (2)Digital controllers are extremely versatile. By merely issuing a new program,one can completely change the operations being performed.This feature is particularly imp
64、ortant if the control system is to receive operating information or instructions from some computing center,where economic analysis and optimization studies are being made.</p><p> Because of the inability
65、of conventional techniques to adequately handle complex control problems,it has been customary to subdivide a process into smaller units and handle each of these as a separate control problem. Human operators are normall
66、y used to coordinate the operation of units.Recent advances in computer control systems have caused changes in this use of industrial process controls.Recent developments in large-scale computers and mathematical methods
67、 provide a basis for use of all ava</p><p> Computer control of complex systems.Current trends in the control of large-scale systems are to consolidate the multiplicity of independently controlled units int
68、o single optimally controlled processes.In industrial process control system,it is,in general,not practical to operate for a very long time at steady state because certain changes in production requirements,raw materials
69、,economic factors,and processing equipment and techniques,may occur.Thus,the transient behavior of industrial processe</p><p> Note that a system capable of controlling a process as completely as possible w
70、ill have to solve complex equations.The more complete the control of industrial processes.</p><p> Note that a system capable capable of controlling a process as completely as possible will have to solve co
71、mplex equations.The more complete the control,the more important it is that the correct relations between operating variables be known and be used.The system must be capable of accepting instructions from such varied sou
72、rces as computer and human operators and must also be capable of changing its control subsystem completely in a short time.</p><p> Introductions to PID controllers</p><p> PID controllers can
73、 be stand-alone controllers (also called songle loop controllers), controllers in PLCs, embedded controllers, or software in Visual Basic or C# computer programs.</p><p> PID controllers are process control
74、lers with the following characteristics:</p><p> ·Continuous process control </p><p> ·Analog output(also known as “measurement” or “Process Variable” or “PV”) </p><p>
75、 ·Analop output (referred to simply as “output”)</p><p> ·Setpoint(SP)</p><p> ·Proportional(P), Integral(I), and/or Derivative(D) constants</p><p> Examples of “c
76、ontinuous process control” are temperature, pressure, flow, and level control. For example, controlling the heating of a tank. For simple control, you have temperature limit sensors(one low and one high) and then switch
77、the heater on when the low temperature limit sensor turns on and then turn the heater off when the temperature rises to the high temperature limit sensor. This is similar to most home air conditioning & heating therm
78、ostats.</p><p> In contrast, the PID controller would receive input as actual temperature and control a valve that regulates the flow of gas to the heater, The PID controller automatically finds the correct
79、 (constant) flow of gas to the heater that kepps the temperature steady at the setpoint. Instead of the temperature bouncing back and forth between two points, the temperature is held steady. If the setpoint is lowered,
80、then the PID controller automatically reduces the amount of gas flowing to the heater. If </p><p> The analog input (measurement) is called the “process variable” or “PV”. You want the PV to be a highly acc
81、urate indication of the process parameter you are trying to control. For example, if you want to maintain a temperature of + or – one degree then we typically strive for at least ten times that or one-tenth of a degree.
82、If the analog input is a 12 bit analog input and the temperature range for the sensor is 0 to 400 degrees then our “theoretical” accuracy is calculated to be 400 degrees div</p><p> The analog output is oft
83、en simply referred to as “output”. Often this is given as 0~ 100 percent. In this heating example, it would mean the valve is totally closed (0%) or totally open (100%).</p><p> The setpoint (SP) is simply-
84、what process value do you want. In this example-what temperature do you want the process at?</p><p> The PID controller’s jod is to maintain the output at a level so that there is no difference (error) betw
85、een the process variable (PV) and the setpoint (SP)</p><p> In Fif.1, the valve could be controlling the gas going to a heater, the chilling of a cooler, the pressure in a pipe, the flow through a pipe, the
86、 level in a tank, or any other process control system.</p><p> What the PID controller is looking at is the difference (or “error”) between the PV and the SP. It looks at the absolute error and the rate of
87、change of error. Absolute error means- is there a big difference in the PV and SP or a little difference? Rate of change of error means- si the difference between the PV or SP getting smaller or larger as time goes on.&l
88、t;/p><p> When there is a “process upset”, meaning, when the process variable or the setpoint quickly changes- the PID controller has to quickly change the output to get the process variable back equal to the
89、setpoint. If you have a walk-in cooler with a PID controller and someone opens the door and walk in, the temperature (process variable) could rise very quickly. Therefore the PID controller has to increase the cooling (o
90、utput) to compensate for this rise in temperature.</p><p> Once the PID controller has the process variable equal to the setpoint, a good PID controller wlii not vary the output. You want the output to be v
91、ery steady (not changing). If the valve (motor, or other control element) is constantly changing, instead of maintaing a constant value, this could cause more wear on the control element.</p><p> So there a
92、re these two contradictory goals. Fast response (fast change in output) when there is a “process” upset, but slow response (steady output) when the PV is close to the setpoint. </p><p><b> 系統(tǒng)設計和補償技術&l
93、t;/b></p><p> 控制系統(tǒng)被設計用來執(zhí)行特定任務。對控制系統(tǒng)的要求通常被稱為系統(tǒng)的性能指標。它們通常和系統(tǒng)精確度、相對穩(wěn)定性及響應速度有關。</p><p> 一般地,系統(tǒng)的性能指標不應該比系統(tǒng)執(zhí)行給定任務事所必須達到的指標更加苛刻。對于某一個給定的系統(tǒng)而言,如果穩(wěn)態(tài)運行精度是最為重要的,那么,我們就不應該提出不必要的過高的暫態(tài)性能指標要求。滿足這些過高的暫態(tài)性能指標
94、往往需要昂貴的部件。我們應該牢記,控制系統(tǒng)設計過程中最重要的一個環(huán)節(jié)就是把性能要求精確的表達出來,這樣才會設計出對于給定的任務而言最優(yōu)的控制系統(tǒng)。</p><p> 在本課中,我們將要簡單的介紹使用頻率響應法和跟軌跡法對但輸入但輸出線性定常系統(tǒng)進行設計和補償?shù)姆椒?。補償是指改變系統(tǒng)的動態(tài)特性以滿足給定的指標。</p><p> 調節(jié)一個系統(tǒng)以得到滿意性能的第一步是設定它的增益。在很多情
95、況下,增加增益值將改善系統(tǒng)的穩(wěn)態(tài)性能,但是也將使系統(tǒng)穩(wěn)定性變差,甚至變得不穩(wěn)定。于是必須重新設計系統(tǒng)(修改結構或者增加裝置或是部件),改變總體特性,使系統(tǒng)按照我們所希望的那樣運行。</p><p> 補償器G(s)和被控對象串聯(lián)連接。這種方法稱為串聯(lián)補償。另外一種補償是反饋補償。串聯(lián)補償通常比反饋補償簡單。</p><p> 在討論補償器時,我們經(jīng)常使用的術語是超前網(wǎng)絡、滯后網(wǎng)絡以及滯
96、后超前網(wǎng)絡。如果一個正弦信號ei加到一個網(wǎng)絡上,它的穩(wěn)態(tài)輸出eo(也是正弦信號)相位超前,則該網(wǎng)絡稱為超前網(wǎng)絡。在滯后超前網(wǎng)絡中,相位滯后和相位超前兩種情況都會出現(xiàn),但是出現(xiàn)在不同的頻率范圍內;相位滯后出現(xiàn)在低頻段,相位超前出現(xiàn)在高頻段。</p><p> 跟軌跡法是一種圖解的方法。已知開環(huán)零點和極點的位置,當某個參數(shù)(通常是增益)的值從零變化到無窮大時,可以確定所有可能的閉環(huán)極點的位置。本方法清楚地顯示出參數(shù)
97、調節(jié)的效果。實際上,系統(tǒng)的跟軌跡圖表明,僅僅通過調節(jié)增益并不能獲得理想的性能。于是,必須改變跟軌跡的形狀來滿足性能指標。</p><p> 在設計控制系統(tǒng)時,我們可以通過插入一個合適的補償器G(s)來改變原來的跟軌跡。一旦完全了解了增加極點和/或零點對于跟軌跡的影響,我們就可以很方便的確定補償器零點極點的位置,以使跟軌跡變成我們所希望的形狀。在用跟軌跡法設計的過程中,通過使用補償器改變系統(tǒng)跟軌跡的形狀,以使閉環(huán)
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