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Control Systems I
Emam Fathyemail: [email protected]
http://www.aast.edu/cv.php?disp_unit=346&ser=68525
Lecture 11
PID
1
Introductionβ’ The usefulness of PID controls lies in their general
applicability to most control systems.
β’ In particular, when the mathematical model of the plantis not known and therefore analytical design methodscannot be used, PID controls prove to be most useful.
β’ In the field of process control systems, it is well knownthat the basic and modified PID control schemes haveproved their usefulness in providing satisfactory control,although in many given situations they may not provideoptimal control.
3
Introductionβ’ It is interesting to note that more than half of theindustrial controllers in use today are PID controllers ormodified PID controllers.
β’ Because most PID controllers are adjusted on-site, manydifferent types of tuning rules have been proposed inthe literature.
β’ Using these tuning rules, delicate and fine tuning of PIDcontrollers can be made on-site.
4
5
Four Modes of Controllers
β’ Each mode of control has specific advantages andlimitations.
β’ On-Off (Bang Bang) Control
β’ Proportional (P)
β’ Proportional plus Integral (PI)
β’ Proportional plus Derivative (PD)
β’ Proportional plus Integral plus Derivative (PID)
8
Proportional Control (P)β’ In proportional mode, there is a continuous linear relation
between value of the controlled variable and position of thefinal control element.
β’ Output of proportional controller is
β’ The transfer function can be written as
-
π(π‘)
π(π‘)
π(π‘)πΎπ
ππ(π‘) = πΎππ(π‘)
πππππππ‘ππππππΆπππ‘πππ
πππππ‘
πΉπππππππ
π(π‘)
ππ(π‘) = πΎππ(π‘)
πΆπ(π )
πΈ(π )= πΎπ
9
Proportional Controllers (P)β’ As the gain is increased the system responds faster to
changes in set-point but becomes progressivelyunderdamped and eventually unstable.
10
Proportional Plus Integral Controllers (PI)
β’ Integral control describes a controller in which the outputrate of change is dependent on the magnitude of theinput.
β’ Specifically, a smaller amplitude input causes a slowerrate of change of the output.
11
Proportional Plus Integral Controllers (PI)
β’ The major advantage of integral controllers is that they havethe unique ability to return the controlled variable back to theexact set point following a disturbance.
β’ Disadvantages of the integral control mode are that itresponds relatively slowly to an error signal and that it caninitially allow a large deviation at the instant the error isproduced.
β’ This can lead to system instability and cyclic operation. Forthis reason, the integral control mode is not normally usedalone, but is combined with another control mode.
12
Proportional Plus Integral Control (PI)
πππ π‘ = πΎππ π‘ + πΎπ π π‘ ππ‘
-
π(π‘)
π(π‘)
π(π‘)πΎπ
πΎπ π(π‘) ππ‘
πππππ‘
πΉπππππππ
π(π‘)
πΎπβ«
++
πΎππ(π‘) πππ π‘
13
Proportional Plus Integral Control (PI)
β’ The transfer function can be written as
πΆππ(π )
πΈ(π )= πΎπ + πΎπ
1
π
πππ π‘ = πΎππ π‘ + πΎπ π π‘ ππ‘
14
Proportional Plus derivative Control (PD)
πππ π‘ = πΎππ π‘ + πΎπππ(π‘)
ππ‘
-
π(π‘)
π(π‘)
π(π‘)πΎπ
πΎπππ(π‘)
ππ‘
πππππ‘
πΉπππππππ
π(π‘)
πΎππ
ππ‘
++
πΎππ(π‘) πππ π‘
15
Proportional Plus derivative Control (PD)
β’ The transfer function can be written as
πΆππ(π )
πΈ(π )= πΎπ + πΎππ
πππ π‘ = πΎππ π‘ + πΎπππ(π‘)
ππ‘
16
Proportional Plus derivative Control (PD)β’ The stability and overshoot problems that arise when a
proportional controller is used at high gain can be mitigated byadding a term proportional to the time-derivative of the error signal.The value of the damping can be adjusted to achieve a criticallydamped response.
17
Proportional Plus derivative Control (PD)
β’ The higher the error signal rate of change, the sooner the finalcontrol element is positioned to the desired value.
β’ The added derivative action reduces initial overshoot of themeasured variable, and therefore aids in stabilizing the processsooner.
β’ This control mode is called proportional plus derivative (PD) controlbecause the derivative section responds to the rate of change of theerror signal
18
Proportional Plus Integral Plus Derivative Control (PID)
ππππ π‘ = πΎππ π‘ + πΎπ π(π‘) ππ‘ + πΎπππ(π‘)
ππ‘
-
π(π‘)
π(π‘)
π(π‘)πΎπ
πΎπππ(π‘)
ππ‘
πππππ‘
πΉπππππππ
π(π‘)
πΎππ
ππ‘
++
πΎππ(π‘) ππππ π‘
πΎπβ«πΎπ π(π‘) ππ‘
+
19
Proportional Plus Integral Plus Derivative Control (PID)
ππππ π‘ = πΎππ π‘ + πΎπ π(π‘) ππ‘ + πΎπππ(π‘)
ππ‘
πΆπππ(π )
πΈ(π )= πΎπ + πΎπ
1
π +πΎπ π
Proportional Plus Integral Plus Derivative Control (PID)
β’ Although PD control deals neatly with the overshoot and ringingproblems associated with proportional control it does not cure theproblem with the steady-state error. Fortunately it is possible toeliminate this while using relatively low gain by adding an integralterm to the control function which becomes
20
CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-S ERROR
Kp Decrease Increase Small Change Decrease
Ki Decrease Increase Increase Eliminate
KdSmall
ChangeDecrease Decrease
Small Change
The Characteristics of P, I, and D controllers
21
Tips for Designing a PID Controller
1. Obtain an open-loop response and determine what needs to be improved
2. Add a proportional control to improve the rise time
3. Add a derivative control to improve the overshoot
4. Add an integral control to eliminate the steady-state error
5. Adjust each of Kp, Ki, and Kd until you obtain a desired overall response.
β’ Lastly, please keep in mind that you do not need to implement all three
controllers (proportional, derivative, and integral) into a single system, if
not necessary. For example, if a PI controller gives a good enough response
(like the above example), then you don't need to implement derivative
controller to the system. Keep the controller as simple as possible.
22
PID Tuningβ’ The transfer function of PID controller is given as
β’ It can be simplified as
β’ Where
πΆπππ(π )
πΈ(π )= πΎπ + πΎπ
1
π +πΎπ π
πΆπππ π πΈ π
= πΎπ(1 +1
πππ +ππ π )
ππ =πΎπ
πΎπππ =
πΎππΎπ
24
integral time constant
integral gainderivative gain
derivative time constant
PID Tuning
β’ The process of selecting the controller parameters(πΎπ, ππ and ππ) to meet given performance specificationsis known as controller tuning.
β’ Ziegler and Nichols suggested rules for tuning PIDcontrollers experimentally.
β’ Which are useful when mathematical models of plantsare not known.
β’ These rules can, of course, be applied to the design ofsystems with known mathematical models.
25
PID Tuning
β’ Such rules suggest a set of values of πΎπ, ππ and ππ that willgive a stable operation of the system.
β’ However, the resulting system may exhibit a large maximumovershoot in the step response, which is unacceptable.
β’ In such a case we need series of fine tunings until anacceptable result is obtained.
β’ In fact, the ZieglerβNichols tuning rules give an educatedguess for the parameter values and provide a starting pointfor fine tuning, rather than giving the final settings forπΎπ, ππ and ππ in a single shot.
26
Zeigler-Nicholβs PID Tuning Methods
β’ Ziegler and Nichols proposed rules for determining valuesof the πΎπ, ππ and ππ based on the transient responsecharacteristics of a given plant.
β’ Such determination of the parameters of PID controllers or tuning of PID controllers can be made by engineers on-site by experiments on the plant.
β’ There are two methods called ZieglerβNichols tuningrules:
β’ First method (open loop Method)
β’ Second method (Closed Loop Method)27
Zeigler-Nicholβs First Method
β’ In the first method, weobtain experimentallythe response of theplant to a unit-stepinput.
β’ If the plant involvesneither integrator(s) nordominant complex-conjugate poles, thensuch a unit-stepresponse curve may lookS-shaped
28
Zeigler-Nicholβs First Methodβ’ This method applies if the response to a step input exhibits an
S-shaped curve.
β’ Such step-response curves may be generated experimentallyor from a dynamic simulation of the plant.Table-1
29
Zeigler-Nicholβs Second Method
β’ In the second method, we first set ππ = β and ππ = 0.
β’ Using the proportional control action only (as shown in figure),increase Kp from 0 to a critical value Kcr at which the outputfirst exhibits sustained oscillations.
β’ If the output does not exhibit sustained oscillations forwhatever value Kp may take, then this method does not apply.
30
Zeigler-Nicholβs Second Method
β’ Thus, the critical gain Kcr
and the correspondingperiod Pcr are determined.
Table-2
31
Example
β’ Consider the control system shown in following figure.
β’ Apply a ZieglerβNichols tuning rule for the determinationof the values of parameters πΎπ, ππ and ππ.
35
Example
β’ Transfer function of the plant is
β’ Since plant has an integrator therefore Ziegler-Nicholβsfirst method is not applicable.
β’ According to second method proportional gain is variedtill sustained oscillations are produced.
β’ That value of Kc is referred as Kcr.
πΊ π =1
π (π + 1)(π + 5)
36
Exampleβ’ Here, since the transfer function of the plant is known we can
find πΎππ using
β Root Locus
β Routh-Herwitz Stability Criterion
β’ By setting ππ = β and ππ = 0 closed loop transfer function isobtained as follows.
πΆ(π )
π (π )=
πΎπ
π π + 1 π + 5 + πΎπ
πΎπ
37
Exampleβ’ The value of πΎπ that makes the system marginally unstable so
that sustained oscillation occurs can be obtained as
β’ The Routh array is obtained as
π 3 + 6π 2 + 5π + πΎπ = 0
β’ Examining the coefficients of firstcolumn of the Routh array we findthat sustained oscillations willoccur if πΎπ = 30.
β’ Thus the critical gain πΎππ is
πΎππ = 30 38
Exampleβ’ With gain πΎπ set equal to 30, the characteristic equation
becomes
β’ To find the frequency of sustained oscillations, we substituteπ = ππ into the characteristic equation.
β’ Further simplification leads to
π 3 + 6π 2 + 5π + 30 = 0
(ππ)3+6(ππ)2+5ππ + 30 = 0
6(5 β π2) + ππ(5 β π2) = 0
6(5 β π2) = 0
π = 5 πππ/π ππ39
Example
β’ Hence the period of sustained oscillations πππ is
β’ Referring to Table-2
π = 5 πππ/π ππ
πππ =2π
π
πππ =2π
5= 2.8099 π ππ
πΎπ = 0.6πΎππ = 18
ππ = 0.5πππ = 1.405
ππ = 0.125πππ = 0.35124 40
Example
β’ Transfer function of PID controller is thus obtained as
πΎπ = 18 ππ = 1.405 ππ = 0.35124
πΊπ(π ) = πΎπ(1 +1
πππ +ππ π )
πΊπ(π ) = 18(1 +1
1.405π + 0.35124π )
41