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