PID Control SystemPID Control SystemAnalysis and DesignAnalysis and Design

Presented by Presented by

Binu Mammen Binu Mammen

Noah BerhanuNoah Berhanu

Souvik BhattacharyaSouvik Bhattacharya

Vishal Raj KarunalaVishal Raj Karunala

IntroductionIntroduction

Proportional-integral-derivative (PID) control provides a Proportional-integral-derivative (PID) control provides a generic and efficient solution to real world control generic and efficient solution to real world control problemsproblems

Presents remedies for problems involving the integral Presents remedies for problems involving the integral and derivative terms. PID design objectives,methods, and derivative terms. PID design objectives,methods, and future directions are discussed.and future directions are discussed.

What is PIDWhat is PID

PID stands for Proportional, Integral, Derivative. PID stands for Proportional, Integral, Derivative. Controllers are designed to eliminate the need for Controllers are designed to eliminate the need for continuous operator attention.continuous operator attention.

Cruise control in a car and a house thermostat are Cruise control in a car and a house thermostat are common examplescommon examples

Error is defined as the difference between set-point and Error is defined as the difference between set-point and measurement. (error) = (set-point) - (measurement) measurement. (error) = (set-point) - (measurement)

The output of PID controller will change in response to The output of PID controller will change in response to the errorthe error

What IS PID What IS PID ProportionalProportional

With proportional band, the controller output is With proportional band, the controller output is proportional to the error or a change in measurement.proportional to the error or a change in measurement.

(controller output) = (error)*100/(proportional band) (controller output) = (error)*100/(proportional band)

Drawbacks -With a proportional controller offset is Drawbacks -With a proportional controller offset is present. present.

Increasing the controller gain will make the loop go Increasing the controller gain will make the loop go unstable. unstable.

Integral action was included in controllers to eliminate Integral action was included in controllers to eliminate this offset this offset

What IS PID What IS PID IntegralIntegral

With integral action, the controller output is proportional With integral action, the controller output is proportional to the amount of time the error is present. Integral action to the amount of time the error is present. Integral action eliminates offset.eliminates offset.

Controller Output = (1/INTEGRAL) (Integral of)e(t)d(t) Controller Output = (1/INTEGRAL) (Integral of)e(t)d(t)

Integral action gives the controller a large gain at low Integral action gives the controller a large gain at low frequencies frequencies

What IS PID What IS PID DerivativeDerivative

The controller output is proportional to the rate of The controller output is proportional to the rate of change of the measurement or errorchange of the measurement or error

CONTROLLER OUTPUT = DERIVATIVE - dm/dtCONTROLLER OUTPUT = DERIVATIVE - dm/dt

STANDARD STRUCTURES OF PID STANDARD STRUCTURES OF PID CONTROLLERS CONTROLLERS

Parallel Structure and Three-Term FunctionalityParallel Structure and Three-Term Functionality

T.F-: T.F-: GGPIDPID(s) (s) ==U(s)/E(s) U(s)/E(s) = = KKPP(1 +1/(1 +1/TT11s s + + TTDDs)…..(1)s)…..(1)U(s) U(s) is the control signalis the control signal

E(s)E(s)=Error signal. constant,=Error signal. constant,

TTD D =Derivative time constant =Derivative time constant

TTI I =Integral time constant=Integral time constant

SS =Argument of the Laplace transform.=Argument of the Laplace transform.

Control Signal can be expressed in three terms asControl Signal can be expressed in three terms as U(s) U(s) = = KKPPE(s) E(s) + + KKII1/sE(s) 1/sE(s) + + KKDDsE(s)sE(s)

U(s) U(s) = = UUPP(s) (s) + + UUII(s) (s) + + UUDD(s),(s),

STANDARD STRUCTURES OF STANDARD STRUCTURES OF PID CONTROLLERS PID CONTROLLERS

Parallel Structure and Three-Term FunctionalityParallel Structure and Three-Term Functionality Where Where KKII = = KKPP/T/TII is the integral gain and is the integral gain and KKDD = = KKPPTTDD is the is the

derivative gain. derivative gain. The three-term functionalities include:The three-term functionalities include:

The proportional term provides an overall control action The proportional term provides an overall control action proportional to the error signal through the all pass gain proportional to the error signal through the all pass gain factor.factor.

The integral term reduces steady-state errors throughThe integral term reduces steady-state errors through low-frequency compensation.low-frequency compensation.

The derivative term improves transient response through The derivative term improves transient response through High-frequency compensation.High-frequency compensation.

STANDARD STRUCTURES STANDARD STRUCTURES OF PID CONTROLLERSOF PID CONTROLLERS

A PID controller is a phase lead-lag compensator A PID controller is a phase lead-lag compensator with one pole at the origin and the other at infinity.with one pole at the origin and the other at infinity.

PI-Phase lag.PI-Phase lag.

PD-Phase-lead compensators.PD-Phase-lead compensators.

Optimum perfomance can be achieved KOptimum perfomance can be achieved Kp p KKI I KKd d

are tuned togetherare tuned together

Effect of Independent P,I and D Effect of Independent P,I and D closed response closed response

STANDARD STRUCTURES OF STANDARD STRUCTURES OF PID CONTROLLERSPID CONTROLLERS

Series StructureSeries Structure

GGPIDPID(s) (s) = = (α (α + + TTDDs) Ks) KPP (1 +1/ (1 +1/αTαT11s)…………..(3)s)…………..(3)

GGPIDPID(s)=G(s)=GPDPD(s)G(s)GPIPI(s)(s)

GGPDPD(s)G(s)GPIPI(s) (s) are the factored PD and PI partsare the factored PD and PI parts

Integral TermIntegral Term Destabilizing Effect of the Integral TermDestabilizing Effect of the Integral Term

Adding an integral term to a Adding an integral term to a pure proportional term pure proportional term increases the gain by a factor.increases the gain by a factor.

Increases the phase-lagIncreases the phase-lag

Gain margin (GM) and phase Gain margin (GM) and phase margin (PM) aremargin (PM) are

reduced, and the closed-loop reduced, and the closed-loop system becomes more system becomes more oscillatory and potentially oscillatory and potentially unstableunstable

Integrator WindupIntegrator Windup

If the actuator realizes the control action has saturated If the actuator realizes the control action has saturated range limits, and the saturations are neglected in a linear range limits, and the saturations are neglected in a linear control design, the integrator may suffer from windup; control design, the integrator may suffer from windup; this causes low-frequency oscillations and leads to this causes low-frequency oscillations and leads to instability instability

Windup is due to the controller states becoming Windup is due to the controller states becoming inconsistent with the saturated control signal, and future inconsistent with the saturated control signal, and future

correction is ignored until the actuator desaturatescorrection is ignored until the actuator desaturates

Integrator Windup RemediesIntegrator Windup Remedies

Antiwindup can be achieved implicitly through automatic Antiwindup can be achieved implicitly through automatic reset.reset.

Explicit Antiwindup implemented explicitly through Explicit Antiwindup implemented explicitly through internal negative feedback. internal negative feedback.

Another Solution to antiwindup is to reduce the Another Solution to antiwindup is to reduce the possibilities for saturation by reducing the control signal, possibilities for saturation by reducing the control signal, as in linear quadratic optimal control schemes that as in linear quadratic optimal control schemes that minimize the tracking error and control signal through a minimize the tracking error and control signal through a weighted objective function.weighted objective function.

Integral TermIntegral TermDestabilizing Effect of the Integral TermDestabilizing Effect of the Integral Term

UUII(s) (s) =1/=1/TTIIs (Ks (KPPE(s) E(s) −−U(s) U(s) −−UUcapcap(s)/(s)/ r(gamma), r(gamma),

Derivative TermDerivative Term

ContentsContents

General form and usesGeneral form and uses

Drawbacks Drawbacks

RemediesRemedies

General formGeneral form PD = ( 1 + TPD = ( 1 + Tdds )s )

Frequency response = ( 1 + jwTFrequency response = ( 1 + jwTdd))

Gain = | 1 + jwTGain = | 1 + jwTdd||

UsesUses

Improved damping ratioImproved damping ratio

Fast recovery from disturbanceFast recovery from disturbance

Strong signal for error signalStrong signal for error signal

Drawbacks an exampleDrawbacks an example

G = (KeG = (Ke-Ls-Ls) / ( 1 + T s )) / ( 1 + T s )

| G (jw)G| G (jw)Gpdpd(jw) | > 1 (jw) | > 1 for all w if for all w if K Kpp > 1/K and T > 1/K and Tdd > T / K K > T / K Kpp

tan -1 ( wTtan -1 ( wTdd) tan-1 (Tw) – Lw ) tan-1 (Tw) – Lw

phase anglephase angle < -180 < -180

Unstable systemUnstable system

RemediesRemedies

Involves use of filtersInvolves use of filters

Linear low pass filterLinear low pass filter

Velocity FeedbackVelocity Feedback

SetPoint FilterSetPoint Filter

Nonlinear median filterNonlinear median filter

Linear low pass filterLinear low pass filter

Second order Butterworth filterSecond order Butterworth filter

GGd d (s) = K(s) = KppTTd d s / ( 1 + Ts / ( 1 + Tdd/ bs )/ bs )

Value of b [ 8,16 ]Value of b [ 8,16 ]

Cascaded to the PD only or to the whole PID controller ( Cascaded to the PD only or to the whole PID controller ( slow slow transient responsetransient response))

Velocity feedbackVelocity feedback

Known as PI-D or Type B controllerKnown as PI-D or Type B controller PD placed in the feedbackPD placed in the feedback

+YG(s)

Kd d( . ) dt

Kp (.) + Ki ∫ (. )+

--

e+

Set point filterSet point filter Known also as P-ID or Type C controllerKnown also as P-ID or Type C controller Similar to Type BSimilar to Type B Gives good overshot performance for a good choice of bGives good overshot performance for a good choice of b

G(s)

Kd d ( . ) dt

Kp

Ki ∫ ( . )

b

+

-

rye

-

+

+

+ -

Median filterMedian filter

Often used in DIPOften used in DIP

Setting the current value to the median Setting the current value to the median values of nearby data pointsvalues of nearby data points

Removes spikesRemoves spikes

drawbackdrawback Excessive smoothness for under damped Excessive smoothness for under damped

systemsystem

DESIGN OBJECTIVES DESIGN OBJECTIVES AND METHODSAND METHODS

DESIGN OBJECTIVES AND DESIGN OBJECTIVES AND EXISTING METHODSEXISTING METHODS

Matters concerning commission and Matters concerning commission and maintenance(such as pre- and post-processing as maintenance(such as pre- and post-processing as well as fault tolerance) also need to be well as fault tolerance) also need to be considered in a complete PID design.considered in a complete PID design.

Controller parameters are tuned so that so that Controller parameters are tuned so that so that the closed loop system meets the following the closed loop system meets the following five five objectivesobjectives::

1.1. stability and stability robustness, usually stability and stability robustness, usually measured in the frequency domain.measured in the frequency domain.

2.2. transient response, including rise time, transient response, including rise time, overshoot, and settling time.overshoot, and settling time.

3.3. steady-state accuracy.steady-state accuracy.

…………..CONTD..CONTD

4. 4. disturbance attenuation and robustness disturbance attenuation and robustness against environmental uncertainty, often against environmental uncertainty, often at steady stateat steady state

5. robustness against plant modeling 5. robustness against plant modeling uncertainty, usually measured in the uncertainty, usually measured in the frequency domain.frequency domain.

Most methods target one objective or a Most methods target one objective or a weighted composite of the objectives weighted composite of the objectives listed above.listed above.

Heuristic MethodsHeuristic Methods Heuristic methods evolve from Heuristic methods evolve from

empirical tuning (such as the ZN empirical tuning (such as the ZN tuning rule), often with a tradeoff tuning rule), often with a tradeoff among design objectives. among design objectives. Heuristic search involves expert Heuristic search involves expert systems, fuzzy logic, neural systems, fuzzy logic, neural networks, and evolutionary networks, and evolutionary computation computation

Frequency Response MethodsFrequency Response Methods

Frequency-domain constraints, such as Frequency-domain constraints, such as GM, PM, and sensitivities, are used to GM, PM, and sensitivities, are used to synthesize PID controllers offline. synthesize PID controllers offline.

For real-time applications, frequency-For real-time applications, frequency-domain measurements require time-domain measurements require time-frequency, localization-based methods frequency, localization-based methods such as waveletssuch as wavelets..

Analytical MethodsAnalytical Methods Because of the simplicity of PID Because of the simplicity of PID

control, parameters can be derived control, parameters can be derived analytically using algebraic relations analytically using algebraic relations between a plant model and a between a plant model and a targeted closed-loop transfer targeted closed-loop transfer function with an indirect function with an indirect performance objective, such as pole performance objective, such as pole placement, IMC, or lambda tuningplacement, IMC, or lambda tuning

Numerical Optimization MethodsNumerical Optimization Methods Optimization-based methods can be regarded Optimization-based methods can be regarded

as a special type of optimal control.as a special type of optimal control. PID parameters are obtained by numerical PID parameters are obtained by numerical

optimization for a weighted objective in the optimization for a weighted objective in the time domain.time domain.

a self-learning evolutionary algorithm (EA) a self-learning evolutionary algorithm (EA) can also be used to search for both the can also be used to search for both the parameters and their associated structure or parameters and their associated structure or to meet multiple design objectives in both the to meet multiple design objectives in both the time and frequency domainstime and frequency domains

These designs are suitable for adaptive These designs are suitable for adaptive tuning as some of the designs can be tuning as some of the designs can be computerized ,so that are automatically computerized ,so that are automatically performed online once the plant is identified.performed online once the plant is identified.

most widely adopted initial tuning methods most widely adopted initial tuning methods are based on the Z-N empirical formulas and are based on the Z-N empirical formulas and their extensions.their extensions.

Computerized Simulation Computerized Simulation ApproachApproach

By using a computerized approach, By using a computerized approach, multiple design methods can be multiple design methods can be combined within a single software or combined within a single software or firmware package to support various firmware package to support various plant types and PID structures.plant types and PID structures.

PIDeasy is a software package that PIDeasy is a software package that uses automatic simulations to search uses automatic simulations to search globally for controllers that meet all globally for controllers that meet all five design objectives in both the time five design objectives in both the time and frequency domains.and frequency domains.

First-Order Delayed PlantsFirst-Order Delayed Plants

While requirements of fast transient While requirements of fast transient response, no overshoot,and zero steady-response, no overshoot,and zero steady-state error are accommodated by state error are accommodated by timedomain criteria, PIDeasy’s timedomain criteria, PIDeasy’s multiobjective goals provide frequency-multiobjective goals provide frequency-domain margins in the range of 9–11 dB domain margins in the range of 9–11 dB and 65–66 degrees.and 65–66 degrees.

To assess the robustness of design using To assess the robustness of design using PIDeasy, GMs and PMs resulting from PIDeasy, GMs and PMs resulting from designs for plants with various designs for plants with various L/T L/T ratios ratios are shown in the figure.are shown in the figure.

SETPOINT-SCHEDULED PID SETPOINT-SCHEDULED PID NETWORKNETWORK

Consider the constant-temperature reaction processConsider the constant-temperature reaction process

wherey(t) = concentration in the outlet stream (mol/l),u(t) = flow rate of the feed stream (l/h),K = rate of reaction (l/mol-h),V = reactor volume (l),d = concentration in the inlet stream (mol/l).

The set point, equilibrium, or steady-state The set point, equilibrium, or steady-state operating trajectory of the plant is governed operating trajectory of the plant is governed byby

Placement at y=0.49 using the maximum Placement at y=0.49 using the maximum distance from the nonlinear trajectory to the distance from the nonlinear trajectory to the linear projection linking the starting and linear projection linking the starting and ending points of the operating envelope.ending points of the operating envelope.

Obtaining the individual PID controllers by using PIDeasy Obtaining the individual PID controllers by using PIDeasy or other PID software or jointly by an evolutionary or other PID software or jointly by an evolutionary algorithm, without linearization.algorithm, without linearization.

Addition of two more controllers at nodes or setpoints 1 and 3.Addition of two more controllers at nodes or setpoints 1 and 3. Formation of a pseudo-linear controller network comprised of three Formation of a pseudo-linear controller network comprised of three

PIDs to be interweighted by scheduling functions PIDs to be interweighted by scheduling functions SS11(y)(y), , SS22(y)(y), and , and SS33(y).(y).

Due to nonlinearity, these functions are often asymmetric. Due to nonlinearity, these functions are often asymmetric. Similar to gain scheduling, linear interpolation meets the Similar to gain scheduling, linear interpolation meets the

requirements for setpoint scheduling.requirements for setpoint scheduling.

The resulting PID network is given byThe resulting PID network is given by

where where p p denotes the derivative operator.denotes the derivative operator.

Performance of the pseudo-linear PID network applied to the Performance of the pseudo-linear PID network applied to the nonlinear process example.nonlinear process example.

To validate tracking performance, another To validate tracking performance, another setpoint setpoint r r = 0.53 mol/= 0.53 mol/l l is used to test the is used to test the control system. control system.

The controller network tracks this setpoint The controller network tracks this setpoint change accurately without oscillation.change accurately without oscillation.

A 10% load disturbance occurring during A 10% load disturbance occurring during [3, 3.5] h,is rejected confirming load [3, 3.5] h,is rejected confirming load disturbance rejection at steady state. disturbance rejection at steady state.

SUMMARYSUMMARY What is PID?What is PID? PID Controller stands for Proportional-Integral-Derivative PID Controller stands for Proportional-Integral-Derivative

Controller. It is a type of feedback controller. It can also be Controller. It is a type of feedback controller. It can also be referred to as the “Tuner”.referred to as the “Tuner”.

Why should we use the PID controller?Why should we use the PID controller? 1. The Controller provides the excitation for the plant.1. The Controller provides the excitation for the plant.

2. Designed to control the overall system behavior.2. Designed to control the overall system behavior.

What is Tuning?What is Tuning? Tuning is nothing but the individual adjustment of the Tuning is nothing but the individual adjustment of the

proportional, integral and derivative terms.proportional, integral and derivative terms.

What exactly does it do?What exactly does it do? It helps to achieve the output (velocity, temperature, position) It helps to achieve the output (velocity, temperature, position)

desired, in a short time, with minimal overshoot, and with little desired, in a short time, with minimal overshoot, and with little error.error.

PID equationPID equation

Kp – Proportional GainKp – Proportional Gain Ki – Integral GainKi – Integral Gain Kd- Derivative GainKd- Derivative Gain

EXAMPLES:EXAMPLES:

A motor driving a gear train A motor driving a gear train A thermal system - A Heater A thermal system - A Heater Mechanical Devices.Mechanical Devices.

DRAWBACKS:DRAWBACKS:

PID control can be costly to implement and support.PID control can be costly to implement and support.

It requires frequent valve- and damper-position It requires frequent valve- and damper-position readjustment and this nearly continuous repositioning readjustment and this nearly continuous repositioning shortens actuator life, adds to maintenance costs, and shortens actuator life, adds to maintenance costs, and makes control stability a question.makes control stability a question.

ENHANCEMENTENHANCEMENT::

NEWPORT MICRO-INFINITY®: Most sophisticated form NEWPORT MICRO-INFINITY®: Most sophisticated form of discrete control available today.of discrete control available today.

The NEWPORT MICRO-INFINITY® is a full function The NEWPORT MICRO-INFINITY® is a full function “Auto tune” (or self-tuning) PID controller which “Auto tune” (or self-tuning) PID controller which combines proportional control with two additional combines proportional control with two additional adjustments, which help the unit automatically adjustments, which help the unit automatically compensate to changes in the system. compensate to changes in the system.

These adjustments, integral and derivative, are These adjustments, integral and derivative, are expressed in time-based units.expressed in time-based units.

CONCLUSIONSCONCLUSIONS::

Cost effectiveness: Division of self-contained stand-Cost effectiveness: Division of self-contained stand-alone instructional units around standard PID structures.alone instructional units around standard PID structures.

Automation by including system identification techniques Automation by including system identification techniques and modular code blocks can be made available for and modular code blocks can be made available for timely application and adaptation in real-time.timely application and adaptation in real-time.