Controllers and Control Algorithms: Selection and Time Domain Design Techniques Applied in...

8/10/2019 Controllers and Control Algorithms: Selection and Time Domain Design Techniques Applied in Mechatronics System

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International Journal of Engineering Sciences, 2(5) May 2013, Pages: 160-190

TI Journals

International Journal of Engineering Scienceswww.waprogramming.com

ISSN2306-6474

* Corresponding author.

Email address: [email protected]

Controllers and Control Algorithms:Selection and Time Domain Design Techniques Applied in

Mechatronics Systems Design

(Review and Research) Part I

Farhan A. Salem1,2

1Mechatronics Sec. Dept. of Mechanical Engineering, Faculty of Engineering, Taif University, 888, Taif, Saudi Arabia.

2Alpha Center for Engineering Studies and Technology Researches, Amman, Jordan.

A R T I C L E I N F O A B S T R A C T

Keywords:

Mechatronics

Controller

Control Algorithm

Controller Design

The most critical decision in Mechatronics design process is the selection, design and integration in

overall system, of two directly related to each other sub-systems; control unit and controlalgorithm. This paper provides simple and user friendly controllers and design guide that illustrates

the basics of controllers and control algorithms, their elements, effects, selection and design

procedures , it is intended for research purposes, application in educat ional processes, as well as, anintroductory material for author's proposed control systems design procedures introduced in parts II

and III.

2013 Int. j. eng. sci. All rights reserved for TI Journals.

1. Introduction

The modern advances in information technology and decision making, as well as the synergetic integration of different fundamentalengineering domains caused the engineering problems to get harder, broader, and deeper. Problems are multidisciplinary and require a

multidisciplinary engineering systems approach to solve them, such approach is called mechatronics approach, and such modernmultidisciplinary systems are called mechatronics systems. Mechatronics is defined as multidisciplinary concept, it is synergisticintegration of precision engineering mechanical engineering, electric engineering, electronic systems, information technology, intelligentcontrol system, and computer hardware and software to manage complexity, uncertainty, and communication through the design andmanufacture of products and processes from the very start of the design process, thus enabling complex decision making , exceptionallevels of accuracy and speed of high-tech equipment including ability to perform complicated and precise movements of high quality.Mechatronics systems are supposed to operate with high accuracy and speed despite adverse effects of system nonlinearities anduncertainties, since achieving and verifying accuracy in Mechatronics systems' performance is of concern, the most critical decision in the

Mechatronics design process is the selection and design of two directly related to each other sub-systems; control unit (physical controller)and control algorithm.

There are many control strategies options that may be more or less appropriate to a specific type of application each has its advantages anddisadvantages. The designer must select the best one for specific application, most are to introduced and discussed, tested in many texts

including [1-15]. Controllers' options including but not limited to: Microcontroller/microprocessor (e.g. PIC-microcontroller),Programmable logic controller (PLC), computer control, desktop/laptop, Digital Signal Processing (DSP) integrated circuits. Also,algorithms options including but not limited to: ON-OFFcontrol, P, PI, PD and PIDcontrol, lead, lag intelligent control, Fuzzy control,

adaptive control, Neural network control. In this paper we ill introduce main of them, their structures, indicate their main properties andtheir design procedures. There is several control system design and analysis techniques, including; numerical, analytical and graphical, the

three primarily simple and direct graphical methodsare Root-Locus, Bode plots and Nyquist diagrams.

The term control system design refers to the process of selecting feedback gains, poles and zeros that meet design specifications in a

closed-loop control system. Most design methods are iterative, combining parameter selection with analysis, simulation, and insight intothe dynamics of the plant [15-16]. The goal of control design is to obtain the configuration, specifications, and identification of the key

parameters of a proposed system to meet and satisfy all the design specifications. Control system design involves the following three steps;

(1) Determine the design specifications.(2) Determine control algorithm, and controller/compensator configuration. (3) Determine theparameter values of the controller to achieve the design goals, shortly, after formulating the problem and establishing the control goals, the

controller configuration is chosen, were the designer must select a controller and strategy that will satisfy all the design specifications, andfinally, the next task is to select (design) controller parameter values so that all design specifications are achieved.

The accuracy control system design (accuracy of selected gains, poles and zeros) to meet all desired specifications, depends on manyfactors including; the accuracy of derived mathematical model, the accuracy and limitations of applied design methodology and tools, and


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designer's skills and experience, in the following discussion assuming the mathematical model in terms of transfer function is accurateenough to processed to control design process. The following three primarily graphical methodsare available tothe control system analysisand design: (1) The root-locus method, (2) Bode- plot representations, (3) Nyquist diagrams.

The term control system analysis concerns itself with the impact that a given controller has on a given system when they interact in anapplied configurations.The term synthesisrefers tothe process by which new physical configurations are created, to combine separateelements or devices and construct controllers with certain properties.

The term design specifications referto statements that explicitly state and describe what the system should do? How to do it? Howwell and

accurately it is Done?. The design specifications are unique to each individual application and often include specifications about relativestability, steady-state accuracy (error), transient-response characteristics, andfrequency-response characteristics. In some applicationsthere may be additional specifications on sensitivity to parameter variations, that is, robustness, or disturbance rejection. Standard

measures of performance used include; Time constant T, Rise time TR, Settling time Ts, Peak time, TP, Maximum overshootMP, maximumundershoot Mu, Percent overshoot OS%, Delay time Td, The decay ratioDR, Damping period TOand frequency of any oscillations in the

response, the swiftness of the response and the steady state error ess[17].

1.1 Controllers Configurations:Most of the conventional design methods in control systems rely on the so-calledfixed-configuration designin that the designer at theoutset decides the basic configuration of the overall designed system and decides where the controller is to be positioned relative to thecontrolled process. Most control efforts involve the modification or compensation of the system-performance characteristics, the generaldesign using fixed configuration is also called compensation [18]. The five commonly used system configurations with controller

compensation, are ( see Figure 1)(1)Series (cascade) compensation; it is the most common control system topology, were the controllerplaced in series with the controlled process(plant), with cascade compensation the error signal is found, and the control signal is developedentirely from the error signal. (2)Feedback compensation, the controller is placed in the minor inner feedback path in parallel with thecontrolled process. (3) State-feedback compensationthe system generates the control signal by feeding back the state variables throughconstant real gains. (4)Series-feedback compensationa series controller and a feedback controller are used(5) Feedforward compensation:the controller is placed in series with the closed-loop system, which has a controller in the forward path the Feedforward controller is

placed in parallel with the forward path.

1.2 Control systemstrategies,selection, & design methodologiesThe control system strategies available for control-system design are bounded only by one's imagination, there are many control strategies

that may be more or less appropriate to a specific type of application, each has its advantages and disadvantages; the designer must selectthe best one for specific application [14-15]. Engineering practice usually dictates that one chooses the simplest controller that meets all thedesign specifications. In most cases, the more complex a controller is, the more it costs, the less reliable it is, and the more difficult it is to

design. Choosing a specific controller for a specific application is often based on the designer's past experience and sometimes intuition,and it entails as much art as it does science[18] the choice of the controller type is an integral part of the overall controller design, taking

into account that the final aim is to obtain the best cost/benefit ratioand therefore the simplest controller capable to obtain a satisfactoryperformance should be preferred.The main factors that might influence the decision on selecting certain control unit and algorithm include;simplicity, space and integration, processing power, environment (e.g. industrial, soft.. ), precision, robustness, unit cost, cost of final

product, programming language, safety criticality of the application, required time to market, reliability, number of products to be producedand designer's past experience and sometimes intuition. Based on all mentioned, the following simplified guide for control algorithm

selection, can be suggested; (1) for processes that can operate with continuous cycling, the relatively inexpensive two position controller isadequate.(2) For processes that cannot tolerate continuous cycling, a P-controller is often employed. (3) For processes that can tolerateneither continuous cycling nor offset error, a PI controller can be used. (4) For processes that need improved stability and can tolerate anoffset error, a PD-controller is employed. (5) However, there are some processes that cannot tolerate offset error, yet need good stability,the logical solution is to use a control mode that combines the advantages of the three controllers' action [19].

system to be

controlledG(s)

Reference

Input command

R(s)

Controlled

variable

C(s)+-

Primary feedback B(s)

controller

ERROR

E(s)The actuating signal

U(s)

Feedback element

Sensor /Transducer

H(s)

Disturbances

D(s)

, R(s) Output , C(s)Compensator

H(s)

G(s)

Fixed PlantDesignable controller

Figure 1 (a) Figure 1 (b)

Figure 1(a)Series or cascade compensation and Components


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Input , R(s) Output , C(s)G(s)

Fixed Plant

Compensator

Designable controller

GP(s)

y(t)u(t)

+ -

x(t)

K

Feedback

Cr(t)

Figure 1(c)Feedback compensation Figure 1(d)State feedback

E(t)u(t)

G(s)r(t)

++-

-

y(t)Gc(s)

GH(s)

y(t)r(t) e(t)

+-

u(t)PlantGf (s) Gc (s)

Figure 1(e)Series-Feedback compensation (2DOF) Figure 1(f)Forward with series compensations

r(t) y(t)+-

Plant

Gf (s)

Gc (s) +

u(t)

Figure 1(f)Forward compensations

2. Proportional Control, P-Controller

P-Controller is one of the simplest and widely used methods of control for many kinds of systems, it is always recommended to be selected

and applied first in control system selection and design process . The control action of P-controlleris proportional to the error, where P-controllerpushes the system in the direction oppositethe error, with a magnitude that is proportionalto the magnitude of the error, P-controller action, provides an instantaneous response to the control error; this action is used to improve the response of the s table system.The relation between the output of controller, (control Effort), u(t)and the actuating error signal e(t) is given by Eq.(1), taking Laplace-transform and manipulating Eq.(1), for transfer function gives:

p Kpu t K e t U s E s (1)Gp(s) = U(s )/E(s) = Kp (2)

The output of P-controller is equal to the error, e(t) , multiplied by the constant proportional gain KP, this describes a pure proportionalrelationship between inputR(s) and output KP*E(s), in effect P-Controller is an amplifier and Kpis simple ratio (non-zero term).

2.1 Properties and limitations of P-controllersIn transient mode:P-controllers are useful for improving the response of a stable system, but cannot control an unstable system by itself.The proportional controller has no sense of time, and its action is determined by the presentinstantaneousvalue of the error. Proportional

control has a tendency to make a system faster, ( is used to speed up system response), itwill have the effect of reducingthe rise time TR,small changes to settling timeTs,and increasesthe overshootMP.

In steady state mode:P-controller will reduce but nevereliminate the steady-state error ess, In a proportional controller, steady state errortends to depend inversely upon the proportional gain, so if the gain is made larger the error goes down, therefore, with only P-controller,

there will always be a small offset-error between the reference input and the measured variable, this is the main disadvantage of P-controller, to remove this offset-error, integral controlhas to be used with proportional controller, resulting in (PI- Controller).There are

practical limits as to how large the gain can be made, where very high gains lead to instabilities, but if the process has a low-order

dynamics the proportional gain can be set to a high value in order to provide a fast response and a low steady-state error.


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2.2 The effect of P-controller on error e(t)Assuming that the steady state output,Nssis proportional to the control effortu(t), the output is then given by Eq. (3) , substituting Eq.(1) inEq.(3) , gives Eq.(4) :

Nss= DC Gain * u(t) (3)Nss= DC Gain * Kpe(t ) (4)e(t )=(Nss-DC Gain)/ Kp (5)

The error e(t ), is the difference between the actual measured output and the desired input , the and the actual measured output is just the

output of the sensor Ks, therefore the errore(t ),is given by:e(t )= Input actual measured output

e(t )= R(s) - Ks*Nss (6)

Where: Ks, is sensor constant gain. Substituting in Eq.(4) and solving for the steady state output Nss, gives:

Nss= DC Gain * Kp(R(s) - Ks* Nss)

DC Gain * * ( )Steady state output,

1 DC Gain * *

p

ss

s P

K R sN

K K

(7)

Now, considering the case when KP, is set to be so large, we have:

Steady state output , ( ) /ss s

N R s K

Now, considering the case when Ks=1, (unity feedback), we have the output equal to the input:

Steady state output, ( )ss

N R s

The steady state error is calculated as follows:

e(t )= R(s) -Nss

DC Gain * * ( ) ( ) 1 DC Gain * * DC Gain * * ( )( ) ( )

1 DC Gain * * 1 DC Gain * *

p s P p

s P s P

K R s R s K K K R se s R s

K K K K

( )( )

1 DC Gain * *s P

R se s

K K

(8)

This equation shows that increasing KP, will reduce steady state error, but never eliminate it.

2.3 The effect of P-controller on transient specifications TR, Ts and TPThe proportional controller is a pure gain controller; the design is accomplished by choosing gain value, where a single gain is varied from

zero to infinity to results in a satisfactory transient response. Based on desired response specifications the proportional gain can be

designed .The general form of second order system in terms of damping ratio , and undamped natural frequency n,is given by Eq.(9):

2 2

2 2 2 2

*( ) ( )

2 2

n P n

plant open

n n n n

KG s G s

s s s s

2 2

2 2 2 2 2

( ) * *( )

1 ( ) ( ) 2 2 ( 1)

P P n P n

P n P n n n n P

K G s K K T s

K G s H s s s K s s K

Also, for second order plant transfer function, given by Eq.(9), the closed loop TF, damping ratio and undamped natural nfrequency are

given as follows:2

2 2 2

2

( )1( ) ( )

( 1) 1 ( ) ( ) 2 2

, 2 2 1/ 1 /

P P n

plant

P P n n

n P n P n n P

K G s K G s T s

s s K G s H s s s K s s

K K K

(9)

In the closed loop transfer function T(s),the term 2nis given in terms of proportional gain Kpand thus we can potentially make n,verylarge by choosing Kpto be very large, thereby speeding up the system, e.g. settling time is given by Ts= 4T = 4/ n, rise time is given byTR=(2.16+0.6)/n, the peak time TP= / (n1-

2), where: T: time constant, all these expressions show that the larger the values ofn


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the faster the system will response, the proportional controller will have the effects ofReducingthe rise time Ts, Increases the overshootOS%, andReduce ,but never eliminate, the steady-state error, ess The following MATLAB code, can be used to demonstrate the effect of

increasing proportional gain KP=[ 1 10 100]; for an open loop system given in next code, the resulted responses are shown in Figure 2

>> K =[ 1 10 100] ; t =0: 0. 001: 5; f or i =1: 3 num= K( i ) ; den=[ 1 3 K(i ) ] ; Gopen=t f ( num, den) , Gcl osed = f eedback(Gopen, 1) ; sys= f eedback(Gcl osed, 1) ; y(: , i ) =st ep( sys, t ) ;Gopen =t f ( num, den) , Gcl osed = f eedback( Gopen, 1) , pause ( 1) , end, pl ot ( t , y(: , 1) , t , y( : , 2) , ' - -' , t , y( : , 3) , ' : ' ) ; l egend( ' K=1' , ' K=10' , ' K=100' , -1 )

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K=1

K=10

K=100

Figure 2(b)The effect of increasing (changing) proportional gain KP

2.4Physical analog realization of ControllerAny controller can be built with passive components only (resistors and capacitors), and thus is easily implemented in analog controlsystems, also the operational amplifier circuit shown in Figure 3, can be used as a building block to implement physical realization ofcontrollers, by judicious choice and configurations of two impedances Z1(s) and Z2(s), (resistor and/or capacitor) any controller can be

built.

2.4.1 Implementing P-ControllerWe can physically realizeP-Controller, by judicious choice of two impedances Z1(s) and Z2(s), based on this proportional controller can beimplemented as shown in Figure 4. When the currents are summed at the inverting input, an equation including the input and outputvoltages is obtained. The final equation shows the system is a simple multiplier, or amplifier. The gain of the amplifier is determined by

the ratio of the input and feedback resistors. Inverting configuration, formula for obtaining the output value and proportional gain, (KP), isgiven by as follows: The voltage at the non-inverting input will be 0V; by design the voltage at the inverting input will be the same.

0 0V V and V V V

The currents at the inverting input can be summed.

2 2

1 2 1 2 1 1

000out out in in inV out in

V V VV V V R V RI V V

R R R R R R

This is the mathematical model of operational amplifier in the form of zeroth order. The proportional gain is given by:

2

1

out

P

in

V RK

V R

2.4.2 Digital Proportional ControllersImplementing control algorithm in a digital system is done using programming codes, the most used programming language is C language,an example code is written next.

Read KP, set poi nt / / r ead pr opor t i onal gai n and desi r ed out put , setpoi ntdoubl e err or , ef f or twhi l e ( )y = ADC_r ead ( ) ; / / r ead val ue of cont r ol l ed var i abl e f r om sensorer r or = set poi nt y; / / comput e new erroref f ort =Kp*err or / / cal cul ate contr ol ef f ortend


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Figure 3.Op-amp configuration Figure 4. Physical implementation of proportional controller

2.5 Control systems design methodologies

The purpose of a control system is to reshape the response of the closed loop system to meet the desired response; the response depends onclosed loop poles' location on complex plane, for first and second order systems it is easy to determine the poles of closed loop system.

response of higher order systems is largely dictated by those poles that are the closest to the imaginary axis, i.e. the poles that have thesmallest real part magnitudes, such poles are called thedominant poles, many times, it is possible to identify a single pole, or a pair of

poles, as the dominant poles, based on this most complex higher order systemsthat have dominant features can be approximated by either afirst or secondorder system response. In such cases, a fair idea of the control system's performance can be obtained from only timeconstant of the dominant pole for first order system and from the damping ratio and undamped natural frequency of the two dominant

poles. The approximation conditions; for dominant one first order pole: the pole closest to the imaginary axis is the one that tend todominate the response. For higher-order than second system, if the realpole isfivetime-constants, 5T,fartherto the left than the dominant

poles, we assume that the system is represented by its dominant second-order pairof poles, for example Considering a third order systemwith onerealroot, and apairof complex conjugate rootsgiven by :

2 2( )

2n n

KG s

s s s

This system can be considered as consisting of two systems; first and second order systems; that it has three poles one real pole ,at pole= , and two complex poles, the condition for dominant one first order pole , or two second order poles, is given below:

2

2 2

/ 10

/ 10 sec

2

n

n

n

n n

KApproximated as first order system

s

KApproximated as ond order system

s s

There are many control methods (techniques) for control system design, including trial and error, gain adjustments, direct pole placement,

comparison technique of standard and obtained transfer function, graphical tools, as well as using computer softwares e.g. MATLAB.

2.5.1 Direct pole placement

The objective of pole placement method is to place the closed loop poles at desired locations, to meet desired design specification. For firstand second order systems, it is possible to place allclosed loop poles with any controller type. When the process is of higher order, this is

not possible anymore, and it is necessary to make approximations to obtain a fist or second order model, based on pole's location differentdesigns exist, including for desired response type and performance.

2.5.2 Dominant poles design

2.5.2.1 P-Controller design for desired response type and desired performance

For a given second order system, by factoring the denominator using the quadratic equation to find the polesof the closed loop transferfunction, in terms of proportional gain KP, as KPvaries, the closed loop poles given by Eq.(11), move through the four ranges of operation

of a second-order system response types: undamped, overdamped, critically damped, and underdamped. Using the relation between thediscriminantand the response types,we can find the most suitable value of gain KPthat will result in response type, for example: a systemopen loop and the closed loop transfer functions are by Eq.(10): finding the closed loop poles by applying quadratic equation, the poles will

be given in terms of proportional gain KP, and given by Eq.(11), depending on numerical value of KP, the following different responsetypes results, that depends on the value of the discriminant of quadratic equation: (1) Overdamped response for 4 > 4K, ( positivediscriminant), (2) Underdamped response for 4 < 4K, , ( negative discriminant), (3) Critically damped response for 4 = 4K. , (discriminant

= zero). Depending on required response type and performance specifications, we can choose the condition and determine the value of gainKP that will result in the required response. Assuming that it is required to design this system to have an underdamped response withdamping ration of 0.7071, this means we choose the condition, that return complex conjugate poles, ( negativediscriminant), this can beaccomplished as shown next:


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2

1( ) ( )

( 1) 2

Pplant closed

P

KG s T s

s s s s K

(10)

2 2 2 22 2P n ns s K as bs c s s

2 22 222

1,2

1,2

2 4 ( ) 12 (2 ) 44 * *1

2* 2 2

2 4 41 1

2

n nn n n

n n

P

P

b b a cP j

a

KP K

(11)

4 4 4 4 0 4 4 1P P P PK K K K

Figure 5

The response will be underdamped for all values of KPgreater than one, taking in consideration that there are practical limits as to howlarge the gain can be made.

P-Controller design For desired performance:Based on pole's value, designer can calculate the exact value of proportional gain Kp, that

will result in desired damping ratio, and correspondingly response type, by setting the magnitudes of the real and imaginary parts of thepoles equal to each other and solving , as shown next: based on Eq. (11) and for system given by Eq.(10), for achieving damping of0.7071, equating real and imaginary parts of the poles, and solve for Kp, gives:

2 2 2 2 21,2 1 1 1 2 1 0.5 0.7071n n n nP j

By equating real and imaginary parts of system's closed loop pole and knowing that the real and imaginary parts must have equalmagnitude, solving, gives the value of KP

1,2

2 4 41 1 1 1 2

2

P

P P P

KP K K K

to select the exactproportional gain Kpvalue, to achieve desired response, based on desired performance specifications in terms of dampingratio and undamped natural frequency, the following expression for proportional gain, for second order systems can be proposed

2 22

2

2

4 441

2 2 2 4

P P

n n

b b aK b aKb b ac bj

a a a a

2 2 2

2 2 2 2 2 2 2 2

2 2 2

4 4(1 )

2 4 4 4

P P

n n n n n P n

b aK b aK b bj K a

a a a a

For example, for system given by Eq.(10), for achieving damping of 0.7071 and undamped natural frequency of 1.4142 , the proportionalgain is found to be KP=2, resulting in achieving desired response

2.5.2.2 P-Controller design by coefficients comparison technique2.5.2.1.1 P-Controller design for desired performance by coefficients comparison techniqueDesign bycoefficients comparisongives an easy design approach for first and second order systems, and become more difficult as the orderof the system increases. Based on desired performance specifications; desired damping ratio or percent overshoot %OS, or settling time, oress, the proportional gainKp, can be obtained by comparison the closed loop transfer function with standard form of closed loop transfer


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function. for second order systems, solving the closed loop transfer function given by Eq.(9) for n, we have proportional gainKpin termsof damping ratio and undamped natural frequency:

n PK , 1 / PK ,2( / 1 )

2 2

ln(% / 100)% ,

ln (% /100)

OSOS e

OS

(12)

Also, settling time is given byTs= 4T = 4/ n, solving for n, gives:

4 4 4 2 =

s P P

n sP s

T K KTK T

, (13)

In terms of desired steady state error, we can calculate the corresponding proportional gain Kpby solving:

( ) ( )

DC Gain * * ( )P

s

R s e sK

K e s

(14)

For first order system, the closed loop transfer function, and correspondingly, the pole and time constant T, are given in terms of KP, for

desired time constant we can selectKP

1 G s T s a+

s a s a+

T 1 / P 1 / a 1 / a

P

P

P

P P

KP K

K

K K

In terms of desired steady state error; the ess, can be written in terms of proportional gain, then we can calculate the corresponding

proportional gain Kpas shown in both Eqs (14) ,(15)

0( ) 1 1

( ) lim ( ) ( )1 ( ) 1 (0) 1 /s open P P

sR se e e

G s K G K a

(15)

For higher order systems, root locus or comparison methods can be applied, to select the exact proportional gain Kpvalue, to achievedesired response. To calculate Kpthat will result in ess 3%, we substitute in Eq.(14) to have Kp= 21.286. Another example, for systemgiven by transfer function given by Eq.(16) to calculate proportional gain, Kp, that will give a steady state error e ss of 5%, can be

accomplished as follows: the DC gain of controlled system is 1/0.5 =2, with unity feedback, Ks =1, and unity step input, substituting inEq.(14), gives Kp=95, to calculate Kpthat will result in overshoot OS% 10%, we first calculate closed loop transfer function, since this issecond order system, using Eq.(12) from desired overshoot we find damping ratio , we rewrite the denominator in terms of damping ratioand undamped natural frequency n, and by comparison we find the values of gain Kp

2 2

1( ) ( )

0.1 0.6 0.5 0.1 0.6 (0.5 )

P

P

KG s T s

s s s s K

(16)2 2 2

0.1 0.6 (0.5 ) 2P n ns s K s s 20.6 2 , (0.5 )n P nK

2.5.2.1.2 P-Controller design for desired performance by selection of both or either of gain KPand/or parameterRewriting Eq.(10) to have the below form, with undefined system parameter P, bothsystem parameter, and proportional gain KP, can be

selected to meet design specificat ions e.g. Ts, OS% , as follows: by finding closed loop transfer function and comparing it with standardsecond order transfer function , gives

2

2 2 2

1( ) ( )

( ) 2

P n

plant closed

P n n

KG s T s

s s P s Ps K s s

By comparison bothsystem parameter P, and proportional gain KP, can be designed by:

2 2P n n

K and P


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2.5.3 P-Controller design by the Root LocusConsidering that transient response specifications are obtained under the assumption that a given system has a pair of dominant complex

conjugate closed-loop poles, the root locus allows us to choose suitable gain proportional KP, to meet the required transient responsespecifications. Using root locus, we are limited to those responses that exist along the root locus , where as the gain is varied we move

through different regions of response, and bysetting the gain at a particular value yields the t ransient response dictated by the poles at thatpoint on the root locus. Proportional gain KPat that point is found by the magnitude criterion; dividing the product of the pole lengths bythe product of the zero lengths, as given by:

=K1

G(s)H(s) =

The product of pole lengths

The product of zerolengths=

polelengths

zero lengths

Based on Figure 5, the damping ratio line can be found by the following equation: = cos, and =cos-1For desired damping ratio, overshoot or settling time, rise time, the damping ratio can be found and correspondingly damping ratio line and

finally proportionalgain KPat intersection point between root locus and damping line, based on Fig 5 , the equations are given next

n , S n2 2

ln(% /100)T 1 / T 4 / ,

ln (% / 100)

OS

OS

= cos-12

n n n n tan j (1 )cl n d clP j P

3. Integral controller , I-controllers

An integral control integrates the error signal to generate the controller output signal. The I-Controller output signal, (control Effort, u(t)),ischanged at a rate proportional to the integral of error signal e(t), and equal to the integral of the error multiplied by the integral gain KI, thiscan be described as follows:

.

( )I

du t du t K e t dt K e t dt

dt dt

u t K e t dt

(17)

Integrals give information concerning the past, that is why integrals provide stability and are always have a tendency to get stuck in thepast, and being late. Taking Laplace transform of Eq.(17) and rearranging for transfer function gives, the transfer function of integralcontroller:

1

( ) , ( )

I

I

E s K

U s K E s s U s s (18)

3.1 Properties of I-controllersAdding an I-controller will effect stability, speed of response and other performance measures, where the I-controller integrates the errorand eliminate it, this is why I-controller has the unique ability to return the process back to the exact setpoint.

Root locus: Adding an I-controllerisequivalent to adding open-looppole at the originas shown by Eq.(10) (to the right of the right mostpole in the system) in the forward path, in result increasing the system type by one and steady-state error isreducedto zero (ess=0), also

resulting inshifting root-locus to the lefttending to lower the system's relativestability and slowsthe response times.

In transient mode:the major disadvantage of I-controller is in that it allows a large deviation at the instantthe error is produced , WHEREbased on fact that integration is a continual summing, integration of error over time means summing up the complete controller error

historyup to the present time, this means I-controller can initially allow a large deviationat the instantthe error is produced allowing the

oscillatory and slow transient behavior that can lead to system instability and cyclic operation, this all means the following; since the errormust accumulate, before a significant response is output from the controller, the integral control is not normally used alone, but is combined

with another control mode, I-control has a tendency to slow the response times by increasing TR, andOS% makes the transient responseworse.

In steady state mode:themajor advantage of integral controllers is in that it return the controlled variable back to the exact setpoint, wherethe I- control integrates the error and eliminates it (ess=0).

It is important to note the following: (1) largevalues of the integral gain (KI) unsterilized the response.(2) The Differential controller infeedback path is equivalent to an Integral controller in the forward path. (3) The Integral controller in feedback path is equivalent to adifferential controller in the forward path (4) The integral control is not normally used alone, but is combined with another control mode.To more clearly understand integral control, we can recall integral in math ;an integral is really the area under a curve, negative area cansubtract from positive area, lowering the value of an integral, Integration is a continual summing as time goes on, the area accumulates, E.G.

for (Adt= At) , the integrator acts to transform the step change into a gradually changing signal every sample time. Integration of error


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/D Du t K de t dt U s K s E s

G(s) = KDs (19)

4.1 Properties of D-controllerAs shown in Eq.(19), adding Differential Controller is equivalent to addition of zeroto the open looptransfer function.

In the transient mode:The D-control action mainly works in transient mode, D-controller will have the effect of improving the stabilityofthe system, and improving the transient response by providing a fast response; where adding D-control result in reducingthe overshoot Mp,

settling time TS, small changes on both rise time TR and steady state error, D-controller predicts, the large overshoot and makes theadjustment needed.

In steady state mode:If the steady-state error of a system is unchanged, (constant), in the time domain, the derivative control has no effect,since the time derivative of a constant is zero .

4.2 Remedies for Derivative action; D-controller cascaded with a first-order low-pass filterThe D-controller based on past and present states, extrapolates the current slope of the error (see Figure 11), therefore has very high gainthis means a sudden rapid change in setpoint (and hence error) will cause the derivative controller to become very large, also for high

frequency signals would differentiate high frequency noise (noise is small, random, rapid changes), and thus provide a derivative kicktothe final control, this is undesirable which can cause problems including instability. To implement D-controller, in processes with noise,Pure differentiator approximation (Pure differentiator cascaded with a first-order low-pass filter, of the next form: (1/s+1), with small time

constant e.g. shorter than 1/5of derivative time TD, is recommended this has the ef fect of attenuating (filtering) the high frequency noiseentering the D-controller.

1 1 _ , . . 0.20 0.1

s 1 T s 1D

D

ilter D filter very smal nuumber e g or T

And the derivative controller will have the following form:

( )T s 1

D

D

D

T sG s

Since D-controller works on the derivative of the er ror, derivative action is completely unable to control a process on its own , where if theerror is constant and doesn't change, de/dt =0, derivative will not do anything ,as a result derivative action is always used in conjunctionwith one or more of the other control modes, PD, PID. It is important to note the following: (1) The D-controllerin feedbackpath is

equivalent to aI- controllerin theforwardpath, (2) TheI-controllerinfeedbackpath is equivalent to aD-controllerin theforwardpath,(3) A Tachometer is an example of differential Control. In order to use the Derivative control the transfer function must be proper, that is

the degree of denominator is greater or equal to the degree of numerator, this is often requires a pole to be added to the controller.

4.3Physical analog realization of D-ControllersThe operational amplifier can be used as a building block to implement physical realization of I-Controllers shown in Figure 8. By

judicious choice of resistor and capacitor the D- controller can be implemented. The Differentiator basically works as follows: whatevercurrentIyou get flowing in C, gets differentiated across the resistorR, The output voltage Vout, is simply the voltage across resistor R. Ifthere is constant DC voltage applied as input then output voltage is 0 . If input voltage is changing from 0 to negative going voltage outputvoltage is positive DC.If input voltage applied is changing from zero to positive going voltage then output is negative DC. Formula forobtaining the output value and differentiator gain, (derivative gain KD), is given by:

( )( ) ( ) ( )

1( ) ,

in

out out in

I I

dV tV t RC V s RCV s s

dtG s RCs K K RC

s

Figure 8.Operational Amplifier differentiator circuit and output formula

4.4Digital realization of D-ControllerThe D-control system computes the error, derivates the error using some standard integration algorithm, and then generates an output

control signal from that integration, an example code is written next.


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Read KD, setpoi nt / / r ead propor t i onal gai n and desi r ed output , set poi ntdoubl e err or , ef f ortwhi l e ( )

y = ADC_r ead ( ) ; / / r ead of cont rol l ed vari abl e f r om sensorerror = setpoi nt y; / / comput e new errorErr or I nt = Err or I nt + ( err or) / dtef f ort =KD* Err or I nt / / cont rol ef f ort

end

4.5 Pseudo-Derivative Feedback Control or Rate feedback control; D-Controller as rate feedback (feedback compensation)As noted, a simple form of control systems is to place the controller in the forward loop ( cascade) in the front of the system to be

controlled. Another configuration is the design procedures for feedback compensation can be more complicated than for cascadecompensation. On the other hand, feedback compensation can yield faster responses. Thus, the engineer has the luxury of designing fasterresponses into portions of a control loop in order to provide isolation [8]. a simple controller that is always used in the feedbackloop isknown as the rate feedback controller(also called Pseudo-Derivative Feedback), where in 1977 Phelan [20-21] published a book, whichemphasizes a simple yet effective control structure, a structure that provides all the control aspects of PID control, but without system

zeros, and correspondingly removing negative zeros effect upon system response. Phelan named this structure "Pseudo-derivative feedback(PDF) control from the fact that the rate of the measured parameter is fed back without having to calculate a derivative. The rate feedback

controller is obtained by feeding back the derivative (rate) of the output of a second-order system (or a system which can be approximatedby a second-order system, i.e. a system with dominant complex conjugate poles) according to the block diagram given in Figure 9. The ratefeedback control helps to increase the system damping, decreases both the response settling time and overshoot [22]. The closed looptransfer function, for system without any controllerin the forward loop is given by:

2

2 2

1( ) ( )

( 1) 2

n

plant closed

n n

G s T ss s s s

But, the closed loop transfer function, for rate feedback controller is given by:

2

2 2( )

2 0.5

n

closed

Rate n n n

T ss K s

Comparing these two closed loop transfer functions, to find the relation between damping ratios, shows that the damping ratio is increasedapplying the rate feedback , and the undamped natural frequency is unchanged, resulting in improving transient response in terms ofreducing in settling time and overshoot

0.5Rate Rate n

K

The derivative gain can be calculated as follows:

2( )D Rate

n

K

(20)

This means, based on damping ratio of original system closed loop transfer function without controller; we can design a rate feedbackcontroller to achieve a desired damping.

G(s)

Krates

C(s)R(s) +

+ --

Figure 9.Block diagram for D-Controller as rate feedback

5. Proportional-Derivative, PD-controller

The output control signal of PD-Controller controller u(t),is equal to the sum of two signals (see Figure 10); The signal obtained bymultiplying the error signal by a constant gain KP and the signal obtained by differentiating and multiplying the error signal by gain KD, andgiven by Eq.(21) , taking Laplace transform and solving for transfer function, gives Eq.(22) :


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( )( ) ( ) ( ) ( ) ( )

P D P D

de tu t K e t K U s K E s K sE s

dt (21)

( ) ( ) ( )PPD P D D D PD

D

KG s K K s K s K s Z

K (22)

Where: ZPD = KP/KD, is the PD-controller zero,

5.1 Properties of PD-controllerThe transfer given by Eq. (22), shows that PD controller is equivalent to the addition of a simple zeroat ZPD = KP/KD, to the open-looptransfer function. The addition of zeros to the open-loop transfer function has the effect of pulling the root locus to the left, or farther fromthe imaginary axis, resulting in more stablesystem and improving the transient response.

In the transient mode:PD-controller improves (speed up) the transient response, it will decay faster resulting in less settling time TS, lesstime constant T, less peak time TP, and reduced maximum overshoot MP.

In steady state mode:has minimum effect, f rom a different point of view, the PD controller may also be used to improve the steady-state

error onlywhen error changes with respect to t ime, because it anticipates the direction of large errors and attempts corrective action beforethey with large overshoot occur (see Figure 11).

(noise) Filtering the D-controller: The main disadvantages are in that the PD controller, given by: C(s) = KP+ KDs, is not physicallyimplementable, since it is not proper, alsoD-term in D-controller, has very high gain, where for high frequency signals would differentiate

high frequency noise, thereby producing large kicks in output, this means for particular systems, the addition of PD zero may causeovershoot in the transient response for the closed loop system. In order to avoid this and use PD-controller, three main solutions; (1) Toreplaced the PD controller, with leadcompensator, whichis a softapproximation of PD controller, (2) the D-control the transfer functionmust be proper, that is the degree of denominator is greater or equal to the degree of numerator, this is often requires a pole to be added tothe controller correspondingly, Eq.(22) can be manipulated to have the following form:

( )( ) (1 ) (1 ) ( ) ( ( ) )DPD P D P P d P d

P

K de tG s K K s K s K T s u t K e t T

K dt

This is not proper transfer function, since the numerator has a higher degree than the denominator, the transfer function is not causal andcan not be realized, and therefore the PD controller is modified through the addition of a lag to the derivative term, to have a proper formgiven by:

( ) (1 )

1

d

PD P

d

T sG s K

T s

(23)

Where:dT /D PK K , is the time constants of the derivative actions,(Derivative time) extrapolating the error dT time units into the

future using the tangent to the error curve, this is shown in Figure 11. The approximation acts as a derivativefor low-frequency signals andas a constantgain for the high frequency signals. The transfer function of a PD controller with a filtered derivative term is given by:

( ) (1 )1 /

d

PD P

d

T sG s K

T s N

(24)

N: With the range of 2 to 20, it determines the gain KHFof the PID controller in the high frequency range, the gain KHFmust be limitedbecause measurement noise signal often contains high frequency components and its amplification should be limited.

Figure 10.PD controller arrangements. Figure 11.PD action[23]


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5.2Physical analog realization of PD-ControllersPD controller circuit is shown in Figure 12, we can implement this circuit for designed values of KPand KD, corresponding to R1, R2and C.

Figure 12.two different PD controller circuit

5.3PD-Controller with D-Controller as rate feedback(feedback compensation,rate sensor)This is accomplished according to the block diagram given in Figure 8.22, resulting in improving transient response in terms of reducing insettling time and overshoot, the closed loop transfer function is given by:

( )

1 ( ) ( ) ( )( )

1 ( ) ( ) 1 ( ) ( ) 1 ( )

1 ( )

P

D P P

D P D P D P

D

K G s

G s K s K G s K G sT s

G s K s G s K G s K s G s K G s K s K

G s K s

For a type 1 system, ratefeedback decreases the ramp-error constant Kvbut does not affect the step-error constant KP.An example of applying PD-Controller with D-Controller as rate feedback, for electric motor output angular position control, where themeasured output is angle and the rate of measured output angle is angular speed, which is to be feedback, this is shown in Figure 13 now, Ifwe place a tachometer, it will output a voltage proportional to angular speed, this can be feed back to the regulator. Tachometer-feedbackcontrol has exactly the same effect as the PD control shown Figure 10, the response of the system with tachometer feedback is uniquelydefined by the characteristic equation, whereas the response of the system with the PD control also depends on the zero at Z = -KP/KD,which could have a significant effect on the overshoot of the step response [7].

G(s)

KDs

KPC(s)R(s) E(s)

++-

-

G(s)

KRate

KP(s)R(s) E(s)

++-

-(s) 1

s

Figure 13 Block diagram for PD-Controller with D-Controller as rate feedback

Figure 13 Electric motor controls.

5.4 Design of PD-controller with deadbeat responseDeadbeat response means the response that proceeds rapidly to the desired level and holds at that level with minimal overshoot,. Adeadbeat response has the following characteristics, (1) Steady-state error = 0, (2) Fast response; minimum both rise timeTRand settlingtimeTs, (3) 0.1% < percent overshoot


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The system overall closed loop transfer function, T(s) , from input signal to sensor , potentiometer, output is given by:

3 2

( )( )( )

( ) ( ) ( ) ( )

p D t

in a m a m m a a m t b D t pot p t pot

K K s K sT s

V s L J s R J b L s R b K K K K K s K K K

Referring to [13], The controller gains KPand K

Ddepend on the physical parameters of the actuator drives, to determine K

Pand K

Dthat

yield optimal deadbeat response, the overall closed loop transfer function T(s) is compared with standard third order transfer function givenby below equation, knowing that =1.9, =2.2 and nTs=4.04 are known coefficients of system with deadbeat response given by table 1,

and choosing TSto be less than 2 seconds, gives the following:

3

n n3 2 2 3( ) , *0.5 4.82, 4.82 / 2 2.41n

n n n

G ss s s

3 2

( ) /( )( )

( )( ) ( )

p D t a m

a m t b D t pot p t pot in a m m a

a m a m a m

K K s K L J sT s

R b K K K K K K K KV s R J b Ls s s

L J L J L J

2

3

( ) ( )n a m a m t b

D

t pot

n a m

pt pot

L J R b K KK

K K

L JK

K K

The desired Standard secondorder closed-loop transfer function for achieving desired deadbeat response specifications is given by:

2

2 2T s n

n ns s

Table 1.The coefficients of the normalized standard transfer function

Optimal coefficients Percent

Overshoot

Percent

Undershoot

Rise,90% Rise,100% SettlingSystem

order

OS% PU% TR TR TS

2nd 1.82 0.10% 0.00% 3.47 6.58 4.82

3nd 1.90 2.20 1.65% 1.36% 3.48 4.32 4.04

4nd 2.20 3.50 2.80 0.89% 0.95% 4.16 5.29 4.81

5nd 2.70 4. 90 5.40 3.40 1.29% 0.37% 4.84 5.73 5.43

6nd 3.15 6.50 7.55 7.55 4.05 1.63% 0.94% 5.49 6.31 6.04

5.6 Analytical PD-controller design approach, based on comparison techniqueFor first and second order systems and systems that can be approximated as first or second orders, based on desired performance

specifications, the PD controller gains can be calculated as follows:

a) Based on desired performance specifications , find desired damping ratio and undamped natural frequencyb) Find overall closed loop transfer function in terms of KP, KDc) Compare closed loop TF, with standard second order transfer function written in terms of damping ratio and undamped natural

frequency, ( or first order written in terms of time constant), particularly compare both characteristic equations to separate PDcontroller gains KP, KDin terms of damping ratio and undamped natural frequency

d)

Substitute values , and find KP, KDe) For example , for transfer function given by below transfer function, to have =1 and n=4, the PD controller gains are calculated

as follows:

2

( )( ) ( )

( ) ( ) 2

P D

D P

a K KaG s T s

s s a s a aK s K

Comparing the characteristic equations and solving, gives:2 2 2

2

( ) 2 2

2,

2

D P n n

n n

D P

s a aK s K s s

aK K

a


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5.6 PD-controller design by root locus5.6.1 straight-forward PD-controller design by root locus

PD controller is used to design a response that has a desirable percent overshoot and a shorter settling time. The procedure for thegraphical root-locus PD compensatordesign can be accomplished by following the next steps:

a) Construct an accurateroot-locus plot, (or, simplyplot pole-zero diagramof the open-loopplant transfer function)b) Obtain the desired location of the closed loop dominant poles P

1,2, from desired transient performance specifications e.g.

damping ratio or OS% ,time constant T or settling time Ts by the following equations:

n , S n2 2

ln(% /100)T 1 / T 4 / ,

ln (% /100)

OS

OS

= cos-1 and Pcl= njd2

n n n n tan j (1 )clP

c) Mark the location of the dominant pole P1,in the pole zero diagram.d) Find the location of the PD controller zero Zo, such that the angle criterion as given by next equation is satisfied (The angles is

measure counterclockwise):

1 1

0 1 2 1 2

180 , 1, 3......

...... ...... 180

i i

m n

z P

i i

Z Z Z P P

r where r

Two main ways to findPD controller zero Zo

d-1) Find the PD zero location using angle criterion by drawing line from the desired location of the dominantclosed-loop poles s1,to thereal axis ,with the PD controller angle of zero zo.

d-2) Applying trigonometry, referring to Figure 14, the PD zero locat ion can be obtained using any of the following equation:

0

2n

0

tan( )

tan( ) (1 )tan( )

d

n

c

c

c

Z

Z

Figure 14

a)

Find derivative gain KDapplying magnitude criterion; estimate the vector lengths from P1 to all poles and zeros and apply themagnitude criterion to find KD.

b) Find proportional gain KPby:

0 0

P

P D

D

KZ K K Z

K

c) Find PD transfer function : by substituting the value of theZoor KP and KD in the PD controller transfer function :

0( ) ( ) ( )P

PD P D D D

D

KG s K K s K s K s Z

K

d) Analyzing the closed loop response with PD controller added, and if necessary, modifies the design to meet the desiredspecifications.

e) Speeding up the time response can be done by either or both, 1) Move zero closer to the imaginary-axis, 2) Increase theproportional gain.

5.6.2 PD-controller pole cancelling design procedure by root locus

a) PD controller is can be written in the form K*(s), that can be written in the next form:

( ) ( )K s k s Z Where :K = Kc* (the multiplication factor of plant numerator)

b) PD zero design ; Find the plant's pole closestto the origin that pushes the root locus to the right and cancelit effect by designingPI zero equal to plant's closest pole.

c) Find gain K , in K*(s), by applying magnitude criterion.

d) If the system still slow , cancel the next slowest plant's pole , and the compensator transfer function will have the form

1 2( ) ( )( )K s k s Z s Z


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6. Proportional-Integral, PI-controller

The integral of the error as well as the error itself are used for control. The output control action signal u(t),of PI-Controller controller isproportional to the error and the integral of error. The control action u(t) is equal to the sum of two signals ( see Figure 15(a)); The integralof the error, e(t),multiplied by the integral gain KI, and the error e(t),multiplied by the proportional gain KP, and given by Eq.(25), takingLaplace transform, and solving for transfer function gives Eq.(26):

1( ) ( ) ( ) ( ) ( ) ( ) ( ) IP I P I PKu t K e t K de t dt U s K E s K E s E s K

s s

(25)

( )( )

( )

I

P

I P I P P PI

PI P

KK s

K K s K K K s Z G s K

s s s s

(26)

Where: /PI I P

Z K K , is the PI-controller zero. Equation (26) can be rewritten, in terms of integral time constant T I, to have thefollowing given by (333), and implemented as shown in Figure 15(b) :

1 1( ) (1 ) (1 ) ( ) ( ( ) ( ) )I I

PI P P P P

P I

K KG s K K K u t K e t e t dt

s K s T s Ti

( ) ( )Pe

u t K e

Ti

(27)

This means if constant error exists, the controller action will keep increasing, until the error is zero, where: T I=KP/KI, is the time constantsof the integral actions, or integral time ,Both KPand TIare adjustable,a change in the value of KPaffect both the proportional and integral

parts of control action.

Figure 15(a) Figure 15(b)

Figure 15,16.UIU PI-Controller arrangement

6.1Physical analog realization of PI-ControllersPI controller circuit is shown in next Figure 16, we can implement this circuit for designed values of KPand KD, corresponding toR1, R2and C, formula for obtaining the output value and gains is given by:

12 2

2 1 1 1 1

1( ) 1 1 1 1

( )

out

in

RV s R RsC

V s R R s R C R s R C

(28)

Figure 16.two physical analog realizations of PI-Controllers


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6.2 Properties and limitations of PI-controllerIf the zero steady-state error is an essential control requirement, then PI controller is the simplest choice to use, PI controller is capable to

provide an acceptable performance for the vast majority of the process control tasks (especially if the dominant process dynamics is of firstorder) and it is indeed the most adopted controller in the industrial context

Transfer function given by Eq. (26), shows that PI-controller is equivalent to addition of apoleat the origin, P=0, and a stablezeroatZ= -KI/ Kp, to the open loop path, (the zero placed near the pole), the addition of zero pulls the root-locus to the left, meanwhile the addition of

pole to the open-loop transfer has the effect of pulling the root locus to the right, resulting in; (1) Lower the system's relative stability, (2)worse transient response, slow dawn the setting of response, (3) Improve steady-state error.

In transient mode : The presence of zeros in either the system will significantly inversely effect the response and may cause worsetransient response, slows dawn the setting and overshoot of response of the closed loop system, (this means PI control does not perform

well in sluggish systems), the size depending upon the relative position of the zeros and closed loop poles within the complex plane, [21]some times and depending on controlled system, the transient response parameters with the PI controller are almost the same as those for

the original system.

In steady state mode:PI controller is used to drastically improveor eliminatethe steady-state errors, where PI controllerzeroincreases thesystem type by one.

It is important to note the following: (1) Integral term in the feedback path is equivalent to a differentiator in the forward path, (2) PIcontroller by it self is unstable, pure integrators not easy to physically implement; this is why PI controller is approximated to result in lagcompensator.

6.3 Filtering PI-controllerTo avoid the inverse effect of the added zero, the zero should be cancelled, this can be accomplished, mainly, by one of the following two

main ways; applying prefilter or PI Control with Proportional in the Feedback Loop.

6.3.1 Systems design with prefilter

The negative effect PI-controller zero can be cancelled by adding a low-pass prefilter with pole equal to the zero of PI controller. Prefilteris defined as low-pass f ilter with a transfer function Gp(s) that filters the input signalR(s) priorto calculating the error signal. The required

prefilter transfer function to cancel the zero is given by Eq.(29). In general, the prefilter is added for systems with leadnetworks, PIor PIDcompensators that introduce zeros and which may lead to worse transient response and overshoot. Pre-filters in general will not prevent

overshoot due to disturbance

Pr _Pr ( ) ( )

O PI

efilter PI efilter

O PI

Z ZG s G s

s Z s Z

(29)

The design with prefilter procedure is simple, and consist of the following steps:(1) Determine the zeros of the closed loop system, (2)

Design a prefilter in which the poles match the zeros of the closed loop system, (3) Apply the prefilter to the input command outside theclosed loop system, as shown in Figure 17

6.4 PI Controller design with Proportional in the Feedback LoopThe proportional term in PI-controller may not be desirable in cascade with controlled system, and is preferred to be in the feedback pathwith controlled system (see Figure 18(a)) , this structure removesPIzero Z= - KI/ Kp, ( as proposed by Phelan [20]) and feed's back a

proportional component of the output rather than the error, resulting in a smoother with improved transient response without or withminimum overshoot in output response, The transfer function for this modified PI=controller is now given by:

( ) ( ( )) ( )IP

KU s K C s E s

s

For plant of first order given by Eq.(30), the over all closed loop transfer function, is given by Eq.(31), this equation is identical to closedloop equation of PI-control in cascade, the difference being that the PI compensated system contains a zero introduced by the compensator

atZ= - KI/ Kp

( )a

G ss a

(30)

2( )

( )( ) 1

I

P I

KC sT s

R s s K as K

(31)

Here notice that, there is note zero added, and correspondingly there is no PI-zero effect. However If overshoot response to commandedinput cannot be tolerated in the system, apre-filtercan be used on the system input.


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Figure 17.Systems design with prefilter Figure 18.(a) PI design with Proportional in the Feedback Loop

6.5 Analytical PI-controller design approach, based on comparison techniqueThe same procedure described for PD controller in 5.6, can be applied for PI controller design

6.6 PI-Controller design by root locusPI-Controller is used to improve (eliminate) steady state responsewithout affecting the transient response. Different procedures for thegraphical root-locus PI controllerdesign; each has its effects, designer can apply any of these procedures depending at desi red response,considering that most of steps are allmostly similar.

6.6.1 First PI-controller design procedure

a) Construct an accurateroot-locus plot, (or, simplyplot pole-zero diagramof the open-loopplant transfer function)

b) Find PI controller zero , Zosuch that the angle criterion as given by equation below is satisfied:

1 1

0 1 2 1 2

180 , 1, 3......

...... ...... 180

i i

m n

z P

i i

Z Z Z P P

r where r

Two main ways to findPI controller zero Zob-1)The exact location of PI controller zero ,Zo , can be found using the dominant pole location and trigonometry , tanfunction.

b-2) PI controller zero ,Zo can be select to add the controllerzero, Zo, close to origin, to be at 4 to 10 times to the right of the

dominant closed-loop poles Pcl, and given by:.

o Z

4 10

n

to

c) Find proportional gain Kp, applying the magnitude criterion; Estimate the vector lengthsfrom dominant P1 to all poles andzeros.

d) Obtain the integral gain KI, by:

0 I o K ZI

P

P

KZ K

K

e) Find PI controller transfer function : by substituting the value of the Zoor KP and KI in the PI controller transfer function :

0

( )( )

( )

I

P

PI P I P

PI P

KK s

K s ZK K s K K G s K

s s s s

f) Analyzing the closed loop response with PI controller added, and if necessary, modify the design to meet the desiredspecifications.

6.6.2 Second shorthand PI-controller design procedurea)

Apply any of covered proportional controller KPdesign techniques (including comparison or root locus), to design proportional

controller KPto meet the desired transient response specifications ( Ts, TR, OS%), that is place the dominant closed-loop systempoles at a desired location to satisfy specifications.

2

n nj (1 )

clP

b) Add a PI controller with a zero at n/10.

c)

Tune the gain of the system to move the closed-loop polecloser to Pcl.d)

For some system, designer can apply pole cancelling, to cancel plant closest to origin pole by adding corresponding similar PIzero.

e)

To speed up response you can increase KP gain

6.6.3 Third simple shorthand PI-controller design procedure

This simple technique has two features; Transient performance remains essentiallyunchanged and it generally takes a long time to reachzero value of steady state (ess=0)

a) Set the value of PI controller zero equals ZO = 0.1b) Find proportional gain Kp, applying the magnitude criterion; Estimate the vector lengthsfrom dominant P1 to all poles and

zeros.

c) From equationZ0 =KI/ KP, we find integral gain as follows:


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I P I P0.1 K / K K 0.1 * K

d) Find the proportional gain KP, applying angle criterion .

6.6.3 Fourth extremely simple PI-controller design procedureVery often PI-controller is implemented to have Kp=1, this implementation is sufficient to justify its main purpose of reducing the steadystate error, and PI-controller transfer function is given by:

0( )( )PI

s ZG ss

a) Apply Kp=1,b) Set the PI controllers pole at the origin and locate its zero arbitrarily close to the pole, e.g. 0.1 or 0.01

c) If necessary, adjust for Kpto compensate for the case when Kpis different from one.

6.6.4 Fifth PI-controller pole cancelling design procedure

a) PIcontroller is can be written in the form K*(s), that can be written in the next form:

( )s Z

K s ks

Where : K = Kc* (the multiplication factor of plant numerator)

b) PI zero design ; Find the plant's pole closestto the origin that pushes the root locus to the right and cancelit effect by designingPI zero equal to plant's closest pole.

c) Find gain K , in K*(s), by applying magnitude criterion.

6.6.5 PI controller tuning design procedure

a) Initially apply no integralgain, KI=0.b) Increase proportional gain Kp, until get satisfactory response.

c) Start to add in integral KI, until the steady state error is removedin satisfactory time.d) If the combination becomes oscillatory, May need to reduce Kp.

7. Proportional-Integral- Derivative PID-controller

Combining all three controllers, results in the PID controller, the output of PID controller is equal to the sum of three signals: The signal

obtained by multiplying the error signal by a constant gain KP, and The signal obtained by differentiating and multiplying the error signalby KDand The signal obtained by integrating and multiplying the error signal by KI, and given by Eq.(32), taking Laplace transform, and

solving for transfer function , gives Eq.(33)

( ) 1( ) ( ) ( ) ( ) ( ) ( ) ( )P D I P D I

de tu t K e t K K e t dt U s K E s K E s s K E sdt s

( ) ( ) ( )I IP D PID P DK K

U s E s K K s G s K K ss s

(32)

This equation can be manipulated to result in the following form

2

2

( )

P ID

D DI D P IPID P D

K KK s s

K KK K s K s K G s K K s

s s s

(33)

Equation (33) is second order system, with two zeros and one pole at origin, and can be expressed to have the following form:

( ) ( )

D PI PD PD

PID D PI PD PI

K s Z s Z s Z G K s Z G s G s

s s

(34)

Which indicates that PID transfer function is the product of transfer functions PI and PD , Implementing these two controllers jointly andindependently will take care of both controller design requirements.

The transfer function of PID controller, GPID(s) ,can also be expressed as:

2 ( )D PI PD D PI PD D PI PD DPID

K s Z s Z K s Z Z K s Z Z K G

s s

Rearranging, we have:

2 ( ) ( )PI PD DD PI PD D PI PD DPID PI PD D D

Z Z K sK s Z Z K Z Z K G Z Z K K s

s s s s


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180

Substituting the following, 1 2 3, ( ),PI PD D PI PD D DK Z Z K K Z Z K K K , gives:

21 3PID

KG K K s

s (35)

Since PID transfer function is a second order system, it can be expressed in terms of damping ratio and undamped natural frequency to have

the following form:

22 22

( )

P ID

D n nD D

PID

K KK s s

K s sK KG s

s s

(36)

Where: 2 In

D

K

K and 2 Pn

D

K

K

The transfer function of PID control given by Eq.(32) can, also, be expressed in terms of derivative time and integral time to have thefollowing form:

2 111 I D I

PID P D P

I I

T T s T sG K T s K

T s T s

(37)

Where: IThe integral time, T /P IK K , The derivative time, /D D PT K K

/I P I

K K T ,D P D

K K T

Since in Eq. (37) the numerator has a higher degree than the denominator, the transfer function is not causal and can not be realized,

therefore this PID controller is modified through the addition of a lag to the derivative term, to have the following form:

D

11 , T /N - time constant of the added lag

1

D

PID P

DI

T sG K

T sT s

N

N: determines the gain KHFof the PID controller in the high frequency range, the gain KHFmust be limited because measurement noisesignal often contains high frequency components and its amplification should be limited. Usually, the divisor N is chosen in the range 2 to20. If no D-controller, then we have PI controller, given by Eq. (38), it is clear that, PI and PD controllers are special cases of the PID

controller.

111 I

PI P P

I I

T sG K K

T s T s

(38)

The addition of the proportional and derivative components effectively predicts the error value at TDseconds (or samples) in the future,assuming that the loop control remains unchanged. The integral component adjusts the error value to compensate for the sum of all past

errors, with the intention of completely eliminating them in TIseconds (or samples). The resulting compensated single error value is scaledby the single gain KP.

7.1 Properties of PID-controllerWhen three controllers combined we get a system that responds quicklyto change (D-controller), generally track required positions (P-controller), and will eventually reduce errors (I-controller). More that 50% of industrial controllers in use utilize PID controller, PID

controller can be analog PID or digital PID, analog PID are mostly hydraulic, pneumatic, electric and electronic types or their combination,

many PID controllers are transformed into digital forms thought the use of microprocessor.

Filtering PID controller : two way including ; (1) PID introduce azerointo the closed loop transfer function, the presence of zero maycause overshoot in the transient response for the closed loop system, to filter PID controller and eliminate the overshoot, aprefilteris used,

the same procedure is used; seesystems design with prefilter. (2) Since it not be desirable to implement the controller as given above; inpractice, all signals will contain high frequency noise, and differentiating noise (by D-controller) will once again create signals with largemagnitudes. To avoid this, the derivative term KDsis usually implemented in conjunction with a low-pass filter of the form: (1/s+1), withsmall time constant e.g. shorter than 1/5of derivative time TD, for some small, this has the effect of attenuating the high frequency noiseentering the D-controller, and produces the following controller proper transfer function:


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181

( ) 11

D

PID P

I D

TIG s K

T s s

The transfer function of a PID controller with a filtered derivative is given by:

( ) 11 /

DPID

I D

TIG s KT s s N

7.2 PID Control with Derivative in the Feedback Loop, PI-D controller Derivative kick is very similar in origin to proportional kick, where any change in setpoint causes an instantaneous change in error, thisnumber is fed into the PID equation which results in an undesirable kick in the output. Since there is always a jump (Kick) in the error

signal, when system is subjected to step input, the derivative term in PID-controller may not be desirable in cascade with controlled system,and is preferred (to remove the D-term negative effect) to be in the feedback path with controlled system, where PID controller is

restructure, by placing the derivative term, D-Controller, into the feedback path, as shown in figure 19, PI terms is applied on error, whileD terms is applied on controlled variable, this is therefore a standard feature of most commercial controllers, this controller is called PI-Dcontroller, the simplified transfer function is given by:

( ) ( ) ( )1

I D

P

K K sU s K E s C s

s Ts

The PI transfer function in terms of integral time is given by Eq.(40) , The D-controller transfer function in terms of derivative time is

given is given by Eq.(39):

( )1 /

d

D

d

T sG s

T s N

(39)

1( ) (1 ) (1 )I I

PI P P P

P I

K KG s K K K

s K s T s (40)

The controller and feedback transfer functions can be equivalently written as next; moving inner a summing junction in Figure 19(a), to theleft, gives two feedback loops the equivalent to inner loop is given by:

1

*1 1 / 1(1 ) 1 1

d d

d dP P

I I

T s T s

T s N T sK KT s T s N

Further simplification gives, the following in the feedback [24], (see Figure 19(b)):

21 )

, ( )

1 1

D

I D P I P

D

P I

TKpT T s K T s K

N NFeedback H s

T sK T s

N

(41)

(a) (b)

Figure 19.PID Control with Derivative in the Feedback Loop


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182

7.3Physical analog realization of PID-ControllersPID controller circuit is shown in next Figure 20(a), Proportional term transfer function is given by Eq.(42), Integral term transfer function

is given by Eq.(43), Derivative term transfer function is given by Eq.(44), , f inally, the transfer function of the PID op-amp circuit transferfunction and corresponding transfer function are given by Eq.(45):

2

1

( )( )

( )

P

P

E s RG s

E s R (42)

( ) 1( )

( )

I

I

I I

E sG s

E s R C s (43)

( )( )

( )

D

D D D

E sG s R C S

E s (44)

2

1

( ) 1( )

( )

o I

PID D D P D

I I

E s R KG s R C S K K s

E s R R C s s (45)

Based on circuit shown in Figure 20(b), the following transfer function can be derived:

2 12 1

1 2 2

1( )PID

I

R CG s R C S

R C R C s

(46)

(a) (b)

Figure 20. Two Physical analog realizations of PID controller circuit

7.4Digital realization of PID-ControllerAn example code of the PID-control system is written next.

Read KP, KI, KPprevious_error = 0;

integral = 0;Read target_value

while ( )Read current_ value ; .

error = target_ value current_ value ; // calculate errorproportional = KP* error; // error times proportional gainintegral = integral + error*dt; //integral stores the accumulated error

integral = integral* KI;derivative = (error - previous_error)/dt; //stores change in error to derivate, dt is sampling period

derivative = KD*derivative;PID_action = proportional + integral + derivative;

previous_error =error; //Update error

end


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8. Demonstrating controllers' effects

Based on P, PD, PI, and PID derived transfer functions in terms of derivative time, and integral time given by Eqs.( 40)( 23)( 37), a

MATLAB code, is written and can be used to demonstrate the effect of changing (decreasing and increasing) one gain in terms ofproportional gain KP, derivative timeTDintegral time TI, and fixing both others equal to unity , for system given by Eq.(47), the results areshown in Figure 21, these results show the effect of each term on transient and steady-sate responses.

3 2

1( )

3 3 1G s

s s s

(47)

0 5 10 150

0.5

1

Icreasing Kp, Applying only P-controller

(sec)

Amplitude

0 50 1000

1

2

Icreasing Ti, Applying only PI-controller

(sec)

Amplitude

0 5 100

0.5

1Icreasing Td, Applying PD-controller

(sec)

Amplitude

0 10 20 30 400

1

2Icreasing Td, Applying PID-controller

(sec)

Amplitude

0 20 40 60 800

1

2Icreasing Ti, Applying PID-controller

Time (sec)

Am

plitude

0 10 20 300

1

2Icreasing Kp, Applying PID-controller

Time (sec)

Am

plitude

Figure 21.Demonstrating controllers' effects

9. Compensators: Lead, lag and lag-lead compensators.

The desired transient and steady state performance specifications for given a control plant, can be achieved using controllers; P, I, D, PI,PD, and PID, as well as by compensators. a compensator is an additional component or circuit that is inserted into a control system to

compensate for a deficient performance, to improve systems transient and steady state response by presenting additional poles and zeros tothe system, General form of the compensator is given by Eq. (48). As noted, in order to avoid controller disadvantages, controllers areapproximated, where; Lag compensator is soft approximation of PI Controller,LeadCompensator is soft approximation of PD Controller,Lag-Lead Compensator, is soft approximation of PID Controller. A first-order compensator having equal numbers of poles and zeroes thatis a single zero and polein its transfer function given by Eq. (49):

1

1

( )

n

i

i

C n

i

i

s Z

G s K

s P

(48)

( ) O

O

s ZG s

s P

(49)

Eq. (49) shows that compensators introduce, a located in the lefthalf s-plane, polezero pair into the open loop transfer function.


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184

9.1 Analog realization of compensators, TF , pole and zero distributionThe circuit shown in Figure 22(a) is active network architecture of electric compensator, by writing its mathematical model and simplifying

to find the transfer function of compensator, we find when it can be used to be lag or leadcompensator:

1 12 4 4 1 1

1 3 2 2 3 2

2 2

1 11

* * *1 11

o O

i O

s sR C sE s ZR R R C R C TK KE R R R C s R C s P

s sR C T

Where : Zo = R2C2 , and Po =R1C1

If R1C1> R2C2 , that is Zo > Po , then the compensator is known as the lag compensator,The transfer function angle given by c=Zo-Po, is negative .The pole-zero configuration for lagcompensator is shown in Figure 22(b)

If R1C1< R2C2 , that is Zo < Po , the compensator is known as the leadcompensator, the transfer function angle given by c=Zo-Po, ispositive, The pole zero configurations for leadcompensator is shown in Figure 22(c)

(a) Compensator two different electric circuits

(b) lagcompensator R1C1< R2C2 pole and zero distribut ion (c) leadcompensator If R1C1> R2C2pole and zero distribution

Figure 22.Compensators and pole zero distribution

10. Lead compensator

Lead compensator is a soft approximation of PD-controller, The PD controller given by transfer function, GPD(s) = KP+ KDs , is notphysically implementable, since it is not proper, and it would differentiate high frequency noise, thereby producing large swings in output,to avoid this, PD-controller is approximated to lead controller of the following form[25]:

( ) ( )PD Lead P D

PsG s G s K K

s P

The larger the value of P, the better the lead controller approximates PD control, rearranging gives:

( ) P D

Lead P D

K s P K PsPsG s K K

s P s P

( )

P

P D P P D

Lead P D

K Ps

K K P s K P K K PG s K K P

s P s P


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Now, letC P D

K K K P andP

P D

K PZ

K K P

, we obtain the following approximated controller transfer function of PD controller,

and called lead compensator:

( )Lead Cs Z

G s Ks P

(50)

Where : Zo< Po , If Z < P : this controller is called a lead compensator, and If Z > P : this controller is called a lag compensator.Lead

compensator transfer function can be written to be:

( )Lead

P s ZG s

Z s P

Where: P/Z, is called the lead ratio.

10.1 Properties of lead compensatorLead compensation is a softapproximat

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