




版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
1、nlinear controller2.2 Digital PID algorithm and its modified forms 2.2.1 Principle of PID controlIn continuous time system, PID is one of the most popular algorithms_+c(t)r(t)PIDplantIt features in :nthe output of control is a linear combination of Proportional, Integral and Derivative of nregulate
2、according to the error )()()(tctrte)(teni.e. .2.1)()(1)()(0tdipdttdeTdtteTtektu11)()()(sTsTksEsUsDdipcor: in the transfer function formwhere: : proportional coefficient. : integral time constant. : derivative time constant.pkiTdT(1) proportional: generating a control proportional to the error . (3)
3、Derivative indication of the rate of error changes. introducing an earlier compensation signal to speed up system response, reducing regulation time.(2) Integral: producing a control that eliminates steady state error.nControl functions:eeiTeffect of integral part As long as the error exits, the con
4、trol takes effect.pupu)(te2.2.2 Digital PID algorithmIn general, there are two types of digital PID -Position -IncrementalnPosition Look at the Eqt.2.1, to implement the digital PID with computers, problems occur: The data is only valid at sampling instant. Therefore, we need to modify the equations
5、 for integral, derivative calculations.Assuming: Sampling interval is sufficiently short. Ideal sampling switch and Hold are applied.)()(kTeteThen: 2.2tkjkjjTeTTjTedtte000)()()( 2.3)()(,) 1(jTeteTjtjTTTkTekTedttde)()()( 2.4te(t) 0T0Error of approximation)1()()()()(0kjdipkekeTTjeTTkekkukjdipkekekjekk
6、ekku0)1()()()()( (2.5)or:.(2.6)where:ipiTTkk . integral coefficient. TTkkdpdderivative coefficient. Further: let Ze(k) = E(z) then according to:The properties of z - transform, we have:)()1(1zEzkeZ101)( )(zzEjeZkjTherefore:)7 . 2.(.).()(1)()()(11zEzzEKzzEKzEKzUdipRewrite Eqt (2.7):)1 (11)()()(11zKzK
7、KzEzUzDdipcz transfer function of digital PID.Figuratively:Output of the position PID controller is normally a manipulation variable, corresponding to position of actuators e. g. position of spool )7 . 2).()(1)()()(11zEzzEKzzEKzEKzUdipProblems with the position PID controller: 1) may have saturation
8、 problem; 2) in case of computer failure large change of u(k) (magnitude) large change of position of actuator may cause mechanical (as well as electrical) damage.Incremental. The distinctive change is that the output of the controller is increments of control signal(or manipulation variables). i.e.
9、 u(k), rather than u(k). e.g. stepper motor. )9 . 2.(.).1()()()()2() 1(2)()()1()()(kekeKkeKkeKkekekeKkeKkekeKkudipdip) 1()()(kekekenThe incremental PID algorithm can be derived from position PID algorithm. At t=(k-1)T, we have: Subtracting Eqt.(2.8) from Eqt.(2.6): Where:)8 . 2.().2() 1()() 1() 1(10
10、kekeKjeKkeKkudkjipAlternatively, Eqt.(2.9) can be rewritten as:Where: )2() 1()()(kCekBekAekuTTKCTTKBTTTTKAdpdpdip)21()1(Only the latest three measurements are needed for calculation providing the are fixed.dipKKK, No fundamental difference between the position and the incremental algorithms. Improve
11、ment due to incremental algorithm. Output increments of manipulation variable: the manipulation variable will be hold in case of computer malfunction. no accumulation, avoid saturation problem easy to calculate. n2.2.3 Modified digital PID algorithms. Implemented with software - flexible -Integral a
12、ction. Purpose of using “integral”: eliminating steady state error. If experienced a large error that lasts for a considerable period of time, integral action may cause big overshoot.- Integral-separated PID algorithms. Basic ideas:- set a threshold 0- when e(k) , PD control applied (reducing oversh
13、oot and speeding up system response )- when e(k) , PID control applied (integral returns to improve steady state accuracy ) )1()()()()(0kekeTTjeTTkekkudkjipTherefore the output of PID controller can be mathematically expressed as:Where: )(1)(0kekeFiguratively, the effect of the modified PID algorith
14、m: - Anti windup Another way to improve the integral effectAssignment: the principle of the anti windup algorithm - Derivative action The purpose of using “derivative”: Speeding up the system response Previously, pure derivative was used.i.e, )1()()1()()()(kekekkekeTTkkuddpdDiscussions:1) Sensitive
15、to disturbances; 2) Suppose a sudden change of error appears, e.g., let e(k) be a step change, e(k)= 1, k=0, 1, 2, Then ud(k) is, )1()()1()()()(kekekkekeTTkkuddpdud(0)=kdud(1)=ud(2)=0i.e. the derivative is only active at the first period of control, while ud(k) equals zero for the following interval
16、s.Consequently, Not sufficient for systems with large inertia Amplitude (and/or rate of change) of ud (=kd) may cause data overflow in computation, or mechanical damage to actuatorsModification:making the derivative effect last longerThe gain of the derivative must be limited(we aim at)Introducing a
17、 first order lag ( low-pass filter) into PID controllerThere are generally two ways to do that i.e., two configurations:How the derivative performance is improved.(For Configuration A)nIn transfer function form:( )( )( )( )( )1ppdppIdifkk T sU skE sUsUsUsTsT snIn which:( )( )1pddfk T sUsE sT sRewrit
18、ten in differential eq. form:( )( )( )ddfpddutde tutTk TdtdtUsing difference eq. to replace the derivatives,( )(1)( )(1)( )dddfpdukuke ke kukTk TTTFurther, we have( )(1) ( )(1)2.11fp dddffTk Tu ku ke ke kTTTTLet then ffTTT1fTTT 111( )(1) ( )(1)(1)2.12dddu kke ke ku k ddpTkkTNotice: and Therefore:whe
19、reif is unit step function, i.e. ( )e k( )1,0,1,e kk(0)(1) (0)( 1)( 1)(1)ddddukeeuk(1)(1) (1)(0)(0)(0)ddddukeeuu2(2)(1)(0)( )(1)(0)dddkddduuuukukuThe consequences are: the magnitude of ( the output in the first sampling interval ) is reduced to ;dk(1)dkthe output of effects in the following interval
20、s( )duk( )(0),1kdukunSchematicallyOriginalModifiedThere are other types of modified PID controllers :n with dead-zone (band), n derivative lead, n variable gain , n self-tuning , 2.3 Parameters tuning of digital PID controllerController = algorithm + para. settingPIDHow to get the PID para. settings
21、?There are generally two ways to get the settingsMathematical models of the controlled plant unknown- experimental methodMathematical models of the controlled plant known- analytical methodThe following methods are applicable to continuous time systems, but can be extended to discrete time systems p
22、roviding that the sampling intervals is very short Bode diagram, root-locus, - Ziegler- Nichols methoda) Step-response methodFeature:Experiment on open-loop systemProcedures: assume the plant can be described (approximated) by a first-order lag with time delay i.e. 1)(TskesGs inject a step input int
23、o the open-loop system determine the parameters from the measurements of the response curveLet, PID controller,)11()(sTsTksGdipcThen, Ziegler-Nichols Eq. gives,5.022.1dipTTkTk: apparent dead timeT: apparent time constantk: gain b) Ultimate-sensitivity methodFeature:experiment on closed-loop systemProcedures: pure proportional control to the closed-loop system increase the gain of controller until the stability limit of the closed-loo
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 2025年湖南省长沙市中考历史真题(原卷版)
- 量子行业发展基础分析
- 流动司法便民服务站工作制度
- 老年护理课件中职
- 老年护理课件
- 老年护理服务培训课件
- 老年心理健康概述
- 2025年安检排爆市场调研报告
- 财务报表分析报告应用合同
- 不动产抵押合同变更协议
- 2023年上海历史高考试题(含答案)
- 2020年北京实习律师面试题库(通用部分)
- 2023年莒县小升初英语考试模拟试题及答案解析
- 金相的基础知识
- 个人餐饮技术服务合同(4篇)
- HF-01型电除尘器高频电源使用说明书
- 询价单(表格模板)
- 病原学检测阴性肺结核诊断流程T∕CHATA 008-2020
- 广东省大学生校外实践教学基地
- 生物质气化耦合高效发电与多联产技术20161127 (1) (1)
- 正畸头影测量分析
评论
0/150
提交评论