New Control Design Method for First Order and …researchpub.org/journal/jac/number/vol4-no2/vol4-no2-2.pdfdimensional analysis and numerical optimization ... AUTOMATION AND SYSTEMS

INTERNATIONAL JOURNAL OF CONTROL, AUTOMATION AND SYSTEMS VOL.4 NO.2 April 2015 ISSN 2165-8277 (Print) ISSN 2165-8285 (Online) http://www.researchpub.org/journal/jac/jac.html

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Abstract— This paper extends writers’ previous work and

proposes a new simple and efficient P-, PI-, PD- and PID-

controllers design method to cope with a wide range of first

order systems (FOS) and systems that can be approximated as

first order, to achieve an important design compromise,

acceptable stability, and medium fastness of response. The

proposed method is based on relating and selecting

controllers' gains based on system's parameters. Simple set of

formulas are proposed for calculating and soft tuning of

controllers' gains. For some systems, the proposed method

presents a good starting design point but may require further

fine-tuning. The proposed controllers design method was

tested using MATLAB/Simulink for different FOS and

systems that can be approximated as first order. The result

obtained show applicability and simplicity of proposed design

method and expressions.

Index Terms— Controller, Controller design, Approximation, First order system.

I. INTRODUCTION

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 into the dynamics of the plant [1-2]. An important compromise for control system design is to result in acceptable stability, and medium fastness of response. One definition of acceptable stability is when the undershoot that follows the first overshoot of the response is small, or barely observable. Beside world wide known and applied controllers design method including Ziegler and Nichols known as the “process

reaction curve” method [3] and that of Cohen and Coon

[4], Chiein-Hrones-Reswick (CHR), Wang–Juang–Chan, many controllers design methods have been proposed and can be found in different texts including [1,5-20]. Each method has its advantages, and limitations. in [12] is presented multi-criterion optimization of PID controller by means of soft computing optimization method HC12. In

[13] is introduced an improved PID tuning approach using traditional Ziegler-Nichols tuning method with the help of simulation aspects and new built in function. in [15] a unified approach has been presented that enable the parameters of PID, PI and PD controllers (with corresponding approximations of the derivative action when needed) to be computed in finite terms given appropriate specifications expressed in terms of steady-state performance, phase/gain margins and gain crossover frequency. IN [16] is proposed an IMC tuned PID controller method for the DC motor for robust operation. in [17] is proposed a procedure for tuning PID controllers with Simulink and MATLAB. In [18] is presented using dimensional analysis and numerical optimization techniques, an optimal method for tuning PID controllers for FOPDT systems. In [19] presented some derivation of IMC controllers and tuning procedures when they are applied to SOPDT processes for achieving set-point response and disturbance rejection trade off. in [20] proposed a new and simple controllers efficient model-based design method, based on relation controller's parameters and system's parameters. This paper extend previous work [20] and proposes simple and easy P, PI, PD and PID controller design method for FOS and systems that can be approximated as first order systems. The proposed method is based on relating controller(s)' parameters and plant's parameters to result in meeting an important design compromise; acceptable stability, and medium fastness of response. To achieve smoother response in terms of minimum PO%, 5T, TS, and ESS, soft tuning parameters with recommended ranges are to be introduced. To achieve approximate desired output response or a good start design point, expression for calculating corresponding controller's gains are to be proposed.

I.1 Controllers Mathematical Modelling

P-, PD-, PI- and PID controllers, will be considered in this paper; where P- term gives control system an instant response to an error, the I- term eliminates the error in the longer term, and D-term have the effect of reducing the maximum overshoot and making the system more stable by giving control system additional control action when the error changes consistently, it also makes the loop more stable (up to a point) which allows using a higher controller

New Control Design Method for First Order and

Approximated First Order Systems Farhan A. Salem1,2, Ayman A. Aly1,3, Mosleh Al-Harthi4 and Nadjim Merabtine4

1 Mechatronics Eng. Section, College of Engineering, Taif University, 888, Taif, Saudi Arabia. 2 Alpha center for Eng. Studies and Technology Researches, Amman, Jordan. 3Mechatronics Eng. Section, Dept. of Mechanical Engineering, Assuit University, 71516, Assiut, Egypt. 4 Electrical Eng. Dept., College of Engineering, Taif University, 888, Taif, Saudi Arabia.

http://www.researchpub.org/journal/jac/jac.html


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gain and a faster integral-terms (shorter integral time or higher integral gain) [21]. The transfer function of P-, PD-Controllers transfer functions are given by Eqs.(1) and (2) respectively.

Equation (2) can be expressed in terms of derivative time constant TD to have the manipulated shown form. The transfer function of PI-Controller is given by Eq.(3), that can be expressed in terms of integral time constant TI, to have the from given by Eq.(4). The transfer function of PID-controller is given by Eq.(5), this equation is second order system, with two zeros and one pole at origin, and can be expressed to have the form given by Eq.(6), which indicates that PI and PD controllers are special cases of the PID controller. The ability of PI and PID controllers to compensate many practical processes has led to their wide acceptance in many industrial applications. The transfer function of PID controller, also, can also be expressed to have the manipulated form given by Eq.(7). Since PID transfer function is a second order system, it can be expressed in terms of damping ratio ζ and undamped natural frequency ωn to have the form given by Eq.(8). The transfer function of PID control given by Eq.(8) can, also, be expressed in terms of derivative time TD and integral time TI to have the form given by Eq.(9). Since in Eq. (9) the numerator has a higher degree than the denominator, the transfer function is not causal and cannot be realized, therefore this PID controller is modified through the addition of a lag to the derivative term, to have the form given by Eq.(10), where: N: determines the gain KHF of the PID controller in the high frequency range, the gain KHF

must be limited because measurement noise signal often contains high frequency components and its amplification should be limited. Usually, the divisor N is chosen in the range 2 to 20. All these controllers and their form are simulated in MATLAB/Simulink shown in Figure 2. Lead and lag Compensators are soft approximations of PD, PI-controllers respectively, are used to improve systems transient and steady state response by presenting additional poles and zeros to the system, the transfer function of compensator are given by Eq.(11), where lag compensator is soft approximations of PI, with Zo > Po, and Zo small numbers near zero , Zo =KI/KP. The smaller the value of Po, the better this controller approximates the PI controller. Lead compensator is soft approximations of PD, with Z < P, where the larger the value of P, the better the lead controller approximates PD control. PD-controller is approximated to lead controller as given by Eq.(12)

)

( p p

U sG s K

E s (1)

( ) ( ) ( )

( ) (1 s) ( ) (1 ) (1 ) (1 )1 1 /

PPD P D D D PD

D

D D DPD P D P PD P D P P

P D D

KG s K K s K s K s Z

K

K T s T sG s K K s K G s K T s K K

K T s T s N

(2)

( )( )

( )

I

P

I P I P P PI

PI P

KK s

K K s K K K s ZG s K

s s s s

(3)

11( ) (1 ) (1 )

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

I I IPI P P P P

P I I

P P

I

K K T sG s K K K K

s K s T s T s

eu t K e t e t dt u t K e

T Ti

(4) 2

2

( )

P ID

D DI D P IPID P D

K KK s s

K KK K s K s KG s K K s

s s s

(5)

( ) ( )D PI PD PD

PID D PI PD PI

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

s s

(6)

2

2

( )

( ) ( )

D PI PD D PI PD D PI PD D

PID

D PI PDD D PI PD PI PD DPID PI PD D D

K s Z s Z K s Z Z K s Z Z KG

s s

K s Z ZK s K Z Z Z Z KG Z Z K K s

s s s s

(7) 2

2 2

2 where : 2

( ) 2 ,

P ID

D n nD D P IPID n n

D D

K KK s s

K s sK K K KG s

s s K K

(8) 2

where : 11

, , , 1 .

P DI D

I

I D IPID P D P

I I P

K KT Integra

T T s T sG K T s K

T s T sl time T derivative time

K K

(9)

D

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

1

DPID P

DI

T sG K

T sT s

N

(10)

( )

( )

o

c

O

s ZG s K

s P

(11)

( )

( )

P

P D oP D

Lead P D P D c

O

K Ps

K s P K Ps s ZK K PPsG s K K K K P K

s P s P s P s P

(12)

II. FIRST ORDER PLANTS' STANDARD

The general standard transfer function form of first order systems, and systems that can be approximated as first order systems, is given by Eq.(13), and can be represented by the block diagram shown in Figure 1. These systems are characterized, mainly, by time constant T. Most complex systems have dominant features that typically can be approximated based on plant's dominant poles approximation by either a first or second order system response. Control system's response is largely dictated by those poles that are the closest to the imaginary axis, i.e. the poles that have the smallest real part magnitudes.



9

To test and verify the proposed controllers' design method, four different first order systems with different time constants, given by Eq.(14) are to be used with Simulink model shown in Figure 2. Controllers' parameters, response measures and values of tuned parameters are to be listed in Table 1.

1

G s 1Ts

(13)

1 2 3 4

10 1 0.005 0.1( ) , ( ) , ( ) , ( )

10 1 2 10 1 2 100sys sys sys sysG s G s G s G s

s s s s

(14)

Fig.1 Block diagram representation of first order system

III. PROPOSED CONTROLLERS DESIGN METHOD

The proposed (P, PI, PD, and PID) controllers design method is based on relating and selecting controllers' parameters based on process's parameters, mainly, time constant T. The methods applied for derivation proposed expressions include dimensional analysis to simplify a problem by reducing the number of variables to the smallest number of essential ones [22]. Mathematical modelling and solving step response for tracking acceptable stability and medium fastness of response in terms of minimum T, TS, and ESS, analysis of results based on relating processes' parameter to controllers parameters, considering effect of each and all parameters on overall system response and finally trial and error. For some cases the proposed method present a good starting point to get a process under control, to achieve desired and/or smoother response, soft tuning parameters (α, β and ε) with recommended ranges are to be

introduced. Simulink is to be used to test the proposed design method for different plants given by Eq.(14), to achieve desired output magnitude of 10 for 10 input (or 5 for 5 input) value. The block diagram representation of the model used is shown in Figure 2.Controllers will be used and tested in different forms given by Eqs. (1) by (10), manual switches are used to switch between controllers' types and forms, as well as, controlled systems.

III.1 P-controller design for first order systems.

For first order systems and systems that can be approximated as first order, the value of proportional gain KP, can be selected as given by E.(15), where T : plant's time constant, R: desired output value ( Reference input), α:

a tuning parameter that takes the value from 1 to 200, to speed up the response in terms of reducing TR, TS and ESS, parameter α, is softly increased.. For first order systems with small DC gain and/or small time constant, to speed up response and to minimize time of selection and tuning processes, the expression given by Eq.(16) can be applied directly, where each increase of parameter α will increase by KP 10. To result in an approximate start point for tuning α, to

achieve a desired 5T response measure (the time of reaching 99.3% of steady state final value), expression given by Eq.(17) is proposed, where Tdesired = 5T value.

Fig.2 Block diagram representation of used Simulink model with different

controllers'

PK RT (15) 10PK RT (16)

esi dd reT

T (17)

Applying expression given by Eq.(15) to calculate gain values for each three systems(1),(2),(3) given by Eq.(14) and running Simulink model for each system, for different values of tuning parameter α, will result in response curves

shown in Figure 3. Calculated gains and tuned values of parameter α are shown in Table 1. Analysing these response

curves show that, acceptable stability, and medium fastness of response with minimum steady state error can be achieved applying proposed expressions, also, the desired response can be achieved by tuning the value of only one parameter α, where increasing α will speed up response and

reduce overshoot, also applying Eq.(16) to systems with small DC gain and/or small time constant, simplifies and accelerate design process while achieving optimal response.

output

1

Unity feedback

U R

Sys(4)

U R

Sys(3)

U R

Sys(2)

U R

Sys(1)

Step input

Vin=10

Pulse

Generator

PD(s)

P, PI, PD, PID Controller

1

s

Td.s

(Td/N)s+1

Filter

1

(Td/N)s+1

Filter

Error

0.02683

1756

8.053

9.973

Control signal

du/dt

Kp

1

0

Td du/dt

0

0

0

du/dt

Kp

.

Td

1

s

Kd

,

1/Ti

1

sKi

sy2.mat

sy12.mat



10

Applying Eq.(17), for achieving desired output 5T time of 0.2 seconds, for systems(1),(2),(3), will result in response curves shown in Figure 3(f), Calculated gains and tuned values of parameter α are shown in Table 2. Analysing

these response curves show that, the desired response in terms of Tdesired is almost achieved.

Table 1: P-controller design for first order system

Fig.3(a) P-controller design for system(1)

Fig.3(b) P-controller design for system(2)

Fig.3(c) PI-controller design for system 3 applying Eq.(12).

Fig.3(d) PI-controller design for system 3 applying Eq.(13)

Table 2: P-controller design for achieving 5T=0.2.

Fig.3(f) P-controller design for achieving desired 5T applying Eq.(14)

III.2 PI-controller design for first order systems

Based on only plant's time constant T, the gains of PI-controller given by Eq.(4), can be calculated using proposed expressions given by Eq.(18), where α is a tuning parameter. Increasing parameters α will speed up response, and reduce steady state error and vise versa. A more simplified expressions can be proposed and given by Eq.(19). For first order systems with small DC gain and/or small time constant, when applying expressions given by Eq.(20), it is required to assign parameter α big values, therefore for such systems, to speed up response and to minimize design time, expressions given by Eq.(18) can be multiplied by 10, also applying expressions given by Eq.(19), for such systems, a soft tuning of parameter α will be enough. To design a PI-controller to result in an approximate start point for tuning α , to meet desired output response based

on desired output time constant (5T), expression given by Eq.(20) can be proposed to approximately calculate the tuning parameter α and then calculated KP, KI, and TI, finally and if necessary, soft tuning can be done to achieve desired response.

P

PI

PI

I

K T

K TK

T T

K TT T

K

(18)

2

/

P

I

PI

I

K T

KT

K TT T

K T

(19)

desired

*R*T

= [0.3:1]Dc T

*gain

where

(20)

Testing the proposed PI-controller design for first order systems(1)(2)(3) given by Eq.(14) a long with Simulink model shown in Figure 2. The result of applying expressions given by Eq.(19), for achieving desired output of 10 with minimum or zero ESS and minimum TS, as well as, tuned values of tuning parameter α, are shown in Table

3, the resulted response curves are shown in Figure 4, these response curves show, that a fast response, with minimum

0 0.5 10

5

10

Time (seconds)

Mag

nitu

de

Sys(1) =1

0 0.50

5

10

Time (seconds)

Mag

nitu

de

Sys(1) =5

0 0.2 0.40

5

10

Time (seconds)

Mag

nitu

de

Sys(1) =5

0 0.5 10

2

4

6

8

Time (seconds)

Magnitude

Sys(2) =1

0 0.5 10

5

10

Time (seconds)

Magnitude

Sys(2) =5

0 0.50

5

10

Time (seconds)

Magnitude

Sys(2) =10

0 500

5

10

Time (seconds)

Mag

nitu

de

Sys(2) =1

0 10 200

2

4

6

8

Time (seconds)

Mag

nitu

de

Sys(2) =5

0 0.5 10

5

10

Time (seconds)

Mag

nitu

de

Sys(2) =550

0 5 10 150

5

10

(Sec)

Magnitude

Sys3 P-design, = 1

0 5 100

5

10

(Sec)

Magnitude

Tuned: = 5

0 50

5

10

(Sec)

Magnitude

Tuned: = 10

0 0.1 0.20

5

10

Time (seconds)

Mag

nitu

de

Sys(1) =10, T-desired=0.1

0 0.1 0.20

5

10

Time (seconds)

Mag

nitu

de

Sys(1) =5, T-desired=0.5

0 0.1 0.20

5

10

Time (seconds)

Mag

nitu

de

Sys(1) =100000, T-desired=0.1



11

or zero ESS are achieved, by applying first proposed expression and then by tuning (increasing) parameter α. To achieve desired output response in terms of desired 5T, for systems (1) and (2), expression given by Eq.(20) is applied, the numerical data and performance measures are listed in Table 4, the resulted response curves are shown in Figure 5(a)(b), these curves show that desired response is achieved, without any soft tuning. pplying expression given by Eq.(19) for systems (1) and (2), the response curves are shown in Figure 5(c)(d), the numerical result are shown in table 3, the numerical result are listed in Table 5. Applying expression given by Eq.(19) for systems with small DC gain and/or small time constant in particular for system(4), the response curves are shown in Figure 5(e).

Table 3: PI-controller design for first order system

Fig.4(a) PI-controller design for system (1)

Fig.4(b) PI-controller design for system(2)

Fig.4(c) PI-controller design for system (3)

Table 4: PI-controller design for systems(1) and (2) applying Eq(18)

Fig.5(a) PI-controller design for system (1) applying Eq.(18).

Fig.5(b) PI-controller design for system (2) applying Eq.(18)

Table 5: PI-controller design for systems (1) and (2) applying Eq(20)

Fig. 5(c) PI-controller design for system (1) applying Eq.(19).

Fig. 5(d) PI-controller design for system (2) applying Eq.(19)

Fig. 5(d) PI-controller design for system (4) applying Eq.(19)

0 5 100

5

10

Time (seconds)

Mag

nitu

de

Sys1 PI-design, = 1

0 1 20

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 5

0 0.5 10

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 10

0 5 10 150

5

10

Time (seconds)

Mag

nitu

de

Sys2 PI-design, = 1

0 50

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 5

0 1 20

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 10

0 500 1000 15000

5

10

Time (seconds)

Mag

nitu

de

Sys3 PI-design, = 1

0 100 200 3000

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 5

0 10 200

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 100

0 0.5 10

5

10

Time (seconds)

data

Desired 5T=1(s),=5

0 0.5 10

5

10

Time (seconds)

data

Desired 5T=3(s),=10

0 2 4 60

5

10

Time (seconds)

data

Desired 5T=5(s),=1

0 0.5 10

5

10

Time (seconds)

data

Desired 5T=1(s),=9.9

0 2 40

5

10

Time (seconds)

data

Desired 5T=3(s),=3.3

0 5 100

5

10

Time (seconds)

data

Desired 5T=8(s),=1.2375

0 5 100

5

10

Time (seconds)

Outp

ut

Sys(1) =1

0 20 40 600

5

10

Time (seconds)

Outp

ut

Sys(1) =0.1

0 0.5 10

5

10

Time (seconds)

Outp

ut

Sys(1) =10

0 5 100

5

10

15

Time (seconds)

Outp

ut

Sys(2) =1

0 20 40 600

5

10

Time (seconds)

Outp

ut

Sys(2) =0.1

0 2 4 60

5

10

15

Time (seconds)

Outp

ut

Sys(2) =5

0 50 100 1500

5

10

Time (seconds)

Outp

ut

Sys(2) =1

0 5 100

5

10

Time (seconds)

Outp

ut

Sys(2) =10

0 5 100

5

10

Time (seconds)

Outp

ut

Sys(2) =50



12

III.3 PD-controller design for first order systems

Based on only plant's time constant T, the gains of PD-controller given by Eq.(2), can be calculated by proposed expressions given by Eq.(21), where α is a tuning parameter, increasing parameters α will speed up response, and reduce steady state error. A more simplified expressions with one tuning parameter α can also be proposed and given by Eq.(22), where increasing parameters α from unity, will speed up response, and reduce steady state error, meanwhile reducing parameters α will slow response and increase hugely steady state error. For first order systems with small DC gain and/or small time constant, expressions given by Eq. (23) are proposed. It is important to consider that, in these expressions the desired output value R, can be removed, and it will be sufficient to increase parameters α correspondingly, also applying proposed PD-controller design approach, the resulted response will be with small time constant (fast response), finally there are limitations for KD values as max KD =2.5.

1

P

D

DD

P

K RT

TK

KT

K R

(21)

1

P

D

DD

P

K T

K T

KT

K

(22)

2

2

1

P

D

DD

P

K R T

TK

KT

K R

(23)

Testing the proposed PD-controller design for first order systems(1)(2)(3) given by Eq.(14) a long with Simulink model shown in Figure 2. The result of applying proposed expressions for achieving desired output of 10 with minimum TS, minimum ESS, as well as, tuned values of tuning parameter α, are shown in Table 6, the response curves are shown in Figure 6, these response curves show, that a very fast response, with small steady state error can be achieved by increasing parameter α. Since system(3), is with small both DC gain and time constant, expressions given by Eq.(23) are applied, resulting in acceptable response compromise shown in Figure 6(c). Designing PD-controller for system(3) applying expressions given by Eq.(22), will result in response curve shown in Figure 6(d), response curves show that it is recommended to assign α big values, for such systems

Table 6: Applying PD-controller design for first order system.

Fig. 6(a) PD-controller design for system 1, increasing tuning parameter

alpha reduces error and speeds response.

Fig. 6(b) PD-controller design for system 2,

Fig. 6(c) PD-controller design for system(3), applying Eq.(23)

Fig. 6(d) PD-controller design for system 3, applying Eq.(22)

III.4 PID-controller design for first order systems Two design expressions to calculate PID controller’s gains

are to be proposed.

III.4.1 First PID-Controller design method To design PID-controller for first order systems, only based on plant's time constants, expressions given by Eq.(24) are proposed. Three tuning parameters (α, β and ε) are introduced that can be tuned to speed up response,

0 0.50

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 5

0 1 20

5

10

Time (seconds)

Mag

nitu

de

Sys1 PD-design, = 1

0 0.1 0.20

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 10

0 0.50

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 5

0 1 20

5

10

Time (seconds)

Mag

nitu

deSys2 PD-design, = 1

0 0.1 0.20

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 25

0 5 100

5

10

Time (seconds)

Mag

nitu

de

Tuned: = 5

0 50

5

10

Time (seconds)

Mag

nitu

de

Sys3 PD-design, = 1

0 1 2 30

5

10

Time (seconds) M

agni

tude

Tuned: = 10

0 50 1000

0.2

0.4

0.6

0.8

Time (seconds)

Outp

ut

Sys(3) =1

0 5 100

5

10

Time (seconds)

Outp

ut

Sys(3) =300

0 1 20

5

10

Time (seconds)

Outp

ut

Sys(3) =2000



13

eliminate/reduce any resulted overshoot, and eliminate/reduce steady state error, a limits for these parameters are proposed, these parameters are tuned softly, the proposed PID-controller design procedure for first order systems is as follows:

1) Calculate plant's time constant, DC gain and desired output value R.

2) Calculate PID gains KP, KI, KD, applying proposed expressions, first assigning, tuning parameters β

=α= ε=1. 3) Check the resulted response, applying calculated

gains. 4) Achieve desired response by tuning α, β and ε

considering the following: a) Speeding up response and Reducing overshoot and

steady state error: Tuning parameter α is responsible for Speeding up response, reducing/increasing both error and resulted overshoot, proposed range [α =0 : inf], where

analysing tests' results show the following: Increasing α (e.g. greater than one) will decrease

damping and correspondingly, increase overshoot, and softly speed response and reduce error.

Assigning α big value will speeding up response while increase overshoot.

Reducing α (e.g. less than one) will reduce overshoot, increase error, while slowing response.

Assigning α very small decimal value will result

in overdamped slow response without big error.

b) Speeding up response: Tuning parameter β is

responsible for speeding up response, Where I. Increasing β will speed up response, reduce

overshoot, while increasing error and increasing response times e.g. rise time and time of reaching steady state 5T.

c) Reducing overshoot and Speeding up response: Tuning parameter ε is responsible for speeding up response, reducing error and softly reducing overshoot, proposed values [ε =0:2.2], where Analysing tests' results show that, depending on plant's parameters, assigning ε small decimal values will speed up response, reduce both error and overshoot.

5) For first order systems with small DC gain and/or small time constant, it is required to assign parameter (α) big values and parameter (ε) small values, without considering the proposed limits.

I

*

* , : [0,3]

* , : [0,2]

*

1T

*

P

I

D

DD

P

P

I

K T

K T where

K T where

K TT

K T

K T

K T

(24)

Testing the proposed PID-controller design, first order systems given by Eq.(14) are used along with Simulink model shown in Figure 2. Testing the proposed PID design for system(2) to achieve output response with T=0.4 s, with minimum ESS and TS is shown in Figure 7, where in Figure 7(a) open loop response is shown, in Figure 6(b) response curve resulted from applying proposed design with β =α=

ε=1. Calculated gains and response measures are shown in Table 7.

(a) (b) . (c)

Fig.7(a) Testing the proposed PID design for system(2), where (a) The open loop system response, (b) PID design with β=α=ε=1, (d)

Tuned PID design .

Table 7: Testing PID design for system (2) Calculated gains and response measures

To test the proposed PID design for system (1) and illustrate the effects of parameters ε, α and β ,to achieve output

response with desired output of 10 with minimum ESS and TS, the design is implemented first by tuning only parameter α to be [1,0.6,6], while β =ε = unity, the resulted response

curves are shown in Figure 8(a), calculated gains and results are listed in Table 8. Effects of tuning parameters β

to have the values [6, 0.1 ,100], while other parameters kept constant and α =ε = unity, is shown in Figure 8(b). The effects of tuning parameter ε with values [6, 0.1, 100],

while other parameters kept constant and α = β = unity, is shown in Figure 8(c). To test the proposed PID design for systems with small DC gain and/or small time constant, system(4) is considered

0 2 4 60

2

4

6

Time (seconds)

Outp

ut

Sys(2) Open loop response

0 10 20 300

5

10

Time (seconds)

Outp

ut

Sys(2) =1, =1,=1

0 1 2 30

5

10

Time (seconds)

Outp

ut

Sys(2) =22, =13,=0.01



14

with T= 0.02 second, and Dc gain= 1e-03. The response curves are shown in Figure 9. Calculated gains and results are listed in Table 9.

Fig. 8(a) PID design tuning only α to be [1,0.6,6], while β =ε = unity

Fig. 8(b) PID design, tuning only β to be [6,0.1,100], while α =ε = unity

Fig. 8(c) PID design, tuning only ε to be [2e-6, 0.1, 2.1], while α = β =

unity. Table 8 PID-controller design for system (1), responses are shown in Fig.8

Fig.9 PID design, for systems with small DC gain and/or small time

constant

Table 9 PID-controller design for system (4), responses are shown in Fig.9

III.4.2 Second PID-Controller design method Expressions given by Eq.(25) can be proposed to calculate PID gains, based on only plant's time constant T with only one tuning parameter α, where increasing this parameter

will speed up response and increase overshoot, meanwhile decreasing α will relatively, slow response and reduce overshoot. Testing the proposed expressions for system (2) for α=[1,0.1,30] will result in response curves shown in Figure 10(a). Testing the proposed expressions for systems with small DC gain and/or small time constant , in particular, system(4) for α=[1,10.1,50] will result in response curves

shown in Figure 10(b).

2

I

*

/

/ 1

*

*T

P

I

D

DD

P

P

I

K T

K T

K T

K TT

K T T

K T

K T

(25)

Fig.10(a) PID design for system(2) applying expressions given by Eq.(25)

Fig. 10(b) PID design for system(4) applying expressions given by Eq.(25)

0 5 100

5

10

15

Time (seconds)

Outp

ut

Sys(1) =1, =1,=1

0 10 20 300

5

10

Time (seconds)

Outp

ut

Sys(1) =0.5, =1,=1

0 10 20 300

5

10

15

Time (seconds)

Outp

ut

Sys(1) =6, =1,=1

0 20 400

5

10

Time (seconds)

Outp

ut

Sys(1) =1, =6,=1

0 20 400

5

10

15

Time (seconds)

Outp

ut

Sys(1) =1, =0.1,=1

0 0.50

5

10

Time (seconds)

Outp

ut

Sys(1) =1, =100,=1

0 5 100

5

10

Time (seconds)

Outp

ut

Sys(1) =1, =1,=2e-6

0 5 100

5

10

15

Time (seconds)

Outp

ut

Sys(1) =1, =1,=0.1

0 10 20 300

5

10

15

Time (seconds)

Outp

ut

Sys(1) =1, =1,=2.1

0 0.05 0.10

0.005

0.01

Time (seconds)

Out

put

Open loop response

0 200 400 6000

5

10

Time (seconds)

Out

put

Sys(1) =1, =1,=1

0 5 10 150

5

10

Time (seconds)

Out

put

Sys(1) =19, =9,=0.1

0 5 100

5

10

15

Time (seconds)

Outp

ut

Sys(2) =30

0 50 1000

5

10

Time (seconds)

Outp

ut

Sys(2) =0.1

0 10 200

5

10

15

Time (seconds)

Outp

ut

Sys(2) =1

0 50 1000

5

10

Time (seconds)

Outp

ut

Sys(4) =1

0 10 200

5

10

Time (seconds)

Outp

ut

Sys(4) =10

0 2 4 60

5

10

Time (seconds)

Outp

ut

Sys(4) =50



15

IV. TESTING AND COMPARING PROPOSED CONTROLLERS DESIGN METHOD WITH EXISTING

DESIGN METHODS.

IV.1 Case (1): Both PD and PID controllers will be tested for controlling the DC motor output angular displacement θ and angular speed ω.

DC motor transfer functions are given by Eq.(26). Depending on controlled variable (θ or ω), the resulted DC system is third (θ) or second (ω) order system, and further can be approximated as first or second system, where if we assume that the armature inductance, La, is small compared to the armature resistance, Ra, transfer function can be simplified to first and second order system to have the form given by Eq.(27). Based on this the DC motor system can be considered as first order system for controlling output angle and second order system for controlling output speed. Applying proposed PID controller design for controlling output angular speed ω for achieving output of ω=R=6.6, with tachometer constant, Ktac = 0.0667, the used approximated first order DC motor system transfer function is given by Eq.(28), with time constant T= 0.6667 seconds, the calculated gain are then used to control the original third order system, the resulted response curves of original third order system given by Eq.(26) are shown in Figure 11(a), the calculations and response measures are listed in Table 10. Applying proposed PD controller design for controlling output angular speed ω, will result in response curves are shown in Figure 11(b), the calculations and response measures are listed in table 10(b). All these response curves show achieving an important design compromise; acceptable stability, and medium fastness of response

( ) ( )

( ) ( )

t t

in in a a m m t Ba a m m t B

K Ks s

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

(26)

/ /( ) ( )

( ) ( ) 1

1

t a a t a a

in in t bt bmm

m am a

K R J K R Js s

V s V s K KK Ks bs s b

J RJ R

(27) ( ) 1.15

( ) 0.02 s + 0.03in

s

V s

(28)

Table 10(a) Applying PID design for DC motor output angular speed

Fig.11(a) applying PID controller design for first order DC motor system

(ω)

Table 10(b) Applying PD design for DC motor output angular speed

Fig.11(b) applying PD controller design for first order DC motor system

(ω)

IV.2 Case (2): Testing proposed PID design method considering fourth order plant with transfer function given by Eq. (29). This system can be approximated based on plant's dominant poles approximation as first order system with one pole at -1, and DC gain of 1, to have the transfer function given by Eq(30), therefore the proposed controller design method can be applied to this system as first order system, then the calculated gains are applied to original fourth order system.

4 3 2 s

10000( )

s + 126s + 2725s + 12600 + 10000G s

(29)

1( )

s+1G s (30)

Ziegler-Nichols step response method is used to compare the proposed P, PI, PD and PID controller design, the calculated controllers' gains applying Ziegler-Nichols are shown in Table 12(a), and applying proposed method are shown in Table 11(b), the resulted response curves are shown in Figure 9.

0 500

5

10

Time (seconds)

Outp

ut

Motor: =1,=1 ,=1

0 500

2

4

6

8

Time (seconds)

Outp

ut

Motor: =50,=33 ,=50

0 5 100

2

4

6

8

Time (seconds)

Outp

ut

Motor: =0.5,=17 ,=11.3

0 10 200

2

4

6

8

Time (seconds)

Out

put

Motor: =1

0 5 100

2

4

6

8

Time (seconds)

Out

put

Motor: =1.5

0 5 100

2

4

6

8

Time (seconds)

Out

put

Motor: =3



16

Table 11(a) Calculated controllers' gains applying Ziegler-Nichols method

Table 11(b) Calculated P, PI, PD controllers' gains applying proposed method

Fig.12(a) P-Controller design applying proposed and Ziegler-

Nichols methods

Fig.12(b) PD-Controller design applying proposed and Ziegler-

Nichols methods

Fig.12(c) PI-Controller design

applying both proposed methods Fig.12(d) PI-Controller design

applying Ziegler-Nichols

PID-controller design: Applying three different PID controller design methods, in particular, Ziegler Nichols frequency response, Ziegler-Nichols step response, and Chein-Hrones-Reswick design methods, will result in PID gains shown in Table 12 (Robert A. Paz, 2001), as shown in this table, applying different methods will result in different PID gains' values and correspondingly different system's responses (see figure 13), when subjected to step input of 10. Comparison and analysis of resulted response curves, show that the Chein-Hrones-Reswick design result in less overshoot and oscillation than Ziegler-Nichols, all three methods almost, result in the same settling time. Applying the proposed first and second PID design methods, based on plant's dominant poles approximation, result in speeding and smooth response curve, with 5T time, without overshoot, and zero steady state error, shown in figure 13(a)(b).

Table 12 Calculated PID controllers' gains applying proposed method

Fig.10(a) System step responses obtained applying different design

methodologies

Fig.10(b) System step responses obtained applying proposed I and II PID

methods

V.SUMMARY: CONTROLLER DESIGN FOR FIRST ORDER SYSTEMS

The proposed design method and expressions for controllers terms selection and design for first order systems are summarized in Table 13

Table 13 P, PI, PD, PID controllers terms for first order systems.

0 2 4 6-5

0

5

10

15

Magnitude

P-Controller

Ziegler Nichols

Proposed method

0 2 4 6-5

0

5

10

15

Magnitude

PD-Controller

Proposed method

Ziegler-Nichols

0 2 4 6-5

0

5

10

15

Magnitude

PI-Controller

Second PI method

First PI Method 0 5 100

5

10

Time(s)

Magnitude

PI- Controller

PI Ziegler-Nichols

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16

18

Time Series Plot:

Time (seconds) (seconds)

data

Original III order system

Approx. as SEOND order

Approx. as FIRST order

.

Secons proposed PID method

First proposed PID method

Ziegler-Nichols

Chein-Hrones-Resw ick

0 5 10-5

0

5

10

15

Time (seconds)

Out

put

Motor: =3

0 5 10-5

0

5

10

15

Time (seconds)

Out

put

PID II aproach: =10

0 5 100

5

10

Time (seconds)

Out

put

I aproach: =3, =10,=2



17

VI.CONCLUSIONS

A new simple and efficient time domain P, PI, PD and PID controllers design method is proposed, to cope with a wide range of first order systems (FOS) and systems that can be approximated as first order, to achieve an important design compromise; acceptable stability, and medium fastness of response, the proposed method is based on selecting controllers' parameters based on plant's parameters. To achieve smoother response in terms of minimum PO%, 5T,

TS, and ESS, soft tuning parameters with recommended ranges are introduced. To achieve approximate desired output response or a good start design point. Expressions for calculating corresponding controller's gains are proposed. The proposed controllers design method was test for different first order system using MATLAB/Simulink software. The results obtained show applicability and simplicity of proposed design expressions.

REFERENCES

[1] Katsuhiko Ogata, modern control engineering, third edition, Prentice hall, 1997.

[2] Ahmad A. Mahfouz, Mohammed M. K., Farhan A. Salem, Modeling, Simulation and Dynamics Analysis Issues of Electric Motor, for Mechatronics Applications, Using Different Approaches and Verification by MATLAB/Simulink", IJISA, vol.5, no.5, pp.39-57, 2013.

[3] . G. Ziegler, N. B. Nichols. “Process Lags in Automatic Control Circuits”, Trans. ASME, 65, pp. 433-444, 1943.

[4] G. H. Cohen, G. A. Coon. “Theoretical Consideration of Related

Control”, Trans. ASME, 75, pp. 827-834, 1953. [5] Astrom K,J, T. Hagllund, PID controllers Theory, Design and Tuning

, 2nd edition, Instrument Society of America,1994. [6] Ashish Tewari, Modern Control Design with MATLAB and

Simulink, John Wiley and sons, LTD, 2002 England. [7] Norman S. Nise, Control system engineering, Sixth Edition John

Wiley & Sons, Inc,2011. [8] Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini,

Feedback Control of Dynamic Systems, 4th Ed., Prentice Hall, 2002. [9] Dale E. Seborg, Thomas F. Edgar, Duncan A. Mellichamp ,Process

dynamics and control, Second edition, Wiley 2004. [10] Dingyu Xue, YangQuan Chen, and Derek P. Atherton, Linear

Feedback Control". 2007 by the Society for Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics the Society for Industrial and Applied Mathematics, 2007.

[11] Chen C.L, A Simple Method for Online Identification and Controller Tuning , AIChe J,35,2037 ,1989.

[12] R. Matousek, HC12: Efficient PID Controller Design, Engineering Letters, pp 41-48, 20:1, 2012

[13] K. J. Astrom and T. Hagglund, The Future of PID Control, IFAC J. Control Engineering Practice, Vol. 9, 2001.

[14] Susmita Das, Ayan Chakraborty,, Jayanta Kumar Ray, Soumyendu Bhattacharjee, Biswarup Neogi, Study on Different Tuning Approach with Incorporation of Simulation Aspect for Z-N (Ziegler-Nichols) Rules, International Journal of Scientific and Research Publications, Volume 2, Issue 8, August 2012.

[15] L. Ntogramatzidis, A. Ferrante, Exact tuning of PID controllers in control feedback design, IET Control Theory and Applications, 2010.

[16] M.Saranya , D.Pamela , A Real Time IMC Tuned PID Controller for DC Motor design is introduced and implemented, International

Journal of Recent Technology and Engineering (IJRTE), Volume-1, Issue-1, April 2012.

[17] Fernando G. Martons, Tuning PID controllers using the ITAE criterion, Int. J. Engng Ed. Vol 21, No 3 pp.000-000-2005.

[18] Saeed Tavakoli, Mahdi Tavakoli, optimal tuning of PID controllers for first order plus time delay models using dimensional analysis, The Fourth International Conference on Control and Automation (ICCA’03), 10-12 June 2003, Montreal, Canada.

[19] Juan Shi and Wee Sit Lee, Set Point Response and Disturbance Rejection Tradeoff for Second-Order Plus Dead Time Processes ,5th Asian Control Conference, pp 881-887, 2004 .

[20] Farhan A. Salem, ,New controllers efficient model-based design method, European Scientific Journal May 2013 edition vol.9, No.15 , 2013.

[21] Jacques F. Smuts , Process Control for Practitioners , Opti Controls Inc ,October 14, 2011.

[22] M. Zlokarnik. Dimensional Analysis and Scale-up in Chemical Engineering, Springer-Verlag, Berlin, 1991.


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