[studies in computational intelligence] intelligent fractional order systems and control volume 438...

I. Pan & S. Das: Intelligent Fractional Order Systems and Control, SCI 438, pp. 257–273. springerlink.com © Springer-Verlag Berlin Heidelberg 2013

Chapter 10 Global Optimization Based Frequency Domain Design of Fractional Order Controllers with Iso-damping Characteristics

Abstract. Frequency domain design of process controllers are popular since the robustness measures like gain and phase margins can easily be assigned with such a technique. Fractional order (FO) control in frequency domain involves four set of design situations i.e. the combinations of integer order (IO) and FO controllers and plants respectively. Recently iso-damping in control system design has emerged due to the fact that the performance degradation needs to be considered with variation in system’s gain and mere stability measures like gain margin is not sufficient. In this chapter, a global optimization based technique has been illu-strated with simulation examples so as to meet user specified gain cross-over fre-quency or speed of response, phase margin or percentage overshoot while also showing constant overshoot for a range of system’s gain. The complex frequency response of integer and fractional controller as well as few model templates are derived to illustrate the design procedure, in a lucid manner.

10.1 Introduction

Iso-damped control system refers to the design of control loops which are robust against system’s gain variation. The idea is to design the controller such that open loop transfer function represents Bode’s ideal transfer function i.e. no variation in the phase margin with variation in the gain-cross over frequency. Classical PID controller design with this concept was first studied by (Barbosa et al. 2004; Chen and Moore 2005). The method additionally allows specifying the gain crossover

frequency ( gcω ) and phase margin ( mφ ) with additional control over the speed

and overshoot of the closed loop system. The idea is similar to the gain-phase-margin (GPM) based tuning of PID controllers (Ho et al. 1995). (Monje et al. 2004) first extended the concept for automatic tuning of PI controller where it was proposed that the gain variation in a control loop can be efficiently handled while solving the equation for assigning phase margin and gain cross-over frequency along with the iso-damping criteria as the derivative of phase at gain cross-over being zero. For time constant variation in a control loop the phase cross-over fre-quency and gain margin assignment equations need to be considered in addition to

258 10 Global Optimization Based Frequency Domain Design

the above mentioned equations. Later (Monje et al. 2008) extended the idea for the

tuning of fractional order PI Dλ μ controllers with a constrained optimization framework.

In most process control related reduced order modeling examples, the mathe-matical representation is done in the First Order Plus Time Delay (FOPTD) or Second Order Plus Time Delay (SOPTD) template when the original plant may have higher order dynamical characteristics. (Das et al. 2011b) proposed two new fractional order templates known as Non-Integer Order Plus Time Delay (NIOPTD) template similar to the classical FOPTD and SOPTD templates while keeping the orders flexible to represent higher order dynamics. Also further stu-dies (Das et al. 2011a) show that fractional order modeling of higher order or non-linear systems are advantageous since inaccuracies associated with the FOPTD/SOPTD modeling of such systems are higher which reduces the assigned phase margin and leads to oscillatory closed loop dynamics. In fact direct frac-tional order modeling of the plant is possible if the governing physical laws can be modeled using fractional order differential equations. Frequency domain design of fractional order or even integer order controllers can be improved with accurate fractional (non-integer) order description of the higher order or nonlinear systems; since reduction in modeling accuracy is equivalent to increase in robustness of a control loop (Das et al. 2011a). For the design of fractional order controllers, con-temporary researchers have used gradient based optimization algorithms which finds any of the local minima satisfying the set of equations for the assignment of gain/phase margin and cross-over frequencies. Since the classical GPM method of PID tuning is not an optimization problem, but a simultaneous nonlinear equation solving problem (Das et al. 2011b) (Monje et al. 2008; Monje et al. 2004) used the equation for gain cross-over frequency specification as the objective function with other equations like phase margin, iso-damping, sensitivity and complementary sensitivity functions as the equality and inequality constraints. It is quite obvious that since it is an optimization problem the classical notion of the necessity of five equations for obtaining the five parameters of a FOPID controller does not hold. With an optimization framework indeed lesser or even higher (than five) number of equations may yield stabilizing FOPID parameters by satisfying different con-trol objectives. Here, three equations have been used for assigning the gain cross-over frequency (speed) and phase margin (overshoot) in a global optimization framework since it is important that all the three equations have to be solved to produce the controller parameters.

10.2 Frequency Domain Design of PID/FOPID Controllers Using Global Optimization

(Chen 2006) discussed four class of design problems for fractional order control i.e. integer order (IO) plant controlled by IO controller, IO plant controlled by FO controller, FO plant controlled by IO controller and FO plant controlled by FO controller. In order address the above four class of problems the complex frequen-cy response of the model and the controller needs to be known. Here the design

10.2 Frequency Domain Design of PID/FOPID Controllers Using Global Optimization 259

specifications to be met by the IO/FO controller are introduced first and then the frequency response, magnitude, phase and derivative of phase of the plant model and the controller are derived.

Here, for the frequency domain design we considered the fractional or-

der PI Dλ μ controller with parallel structure:

iFOPID p d

KC K K s

sμ

λ= + + (10.1)

Here, among the five parameters of the FOPID controller{ }, ,p i dK K K are the

proportional-integral-derivative gains and { },λ μ are the integro-differential or-

ders which are used as the decision variables in the global optimization algorithm. Clearly integer order PID is a special case of the generalized FOPID controller Equation (10.1), with the order of the integral and derivative terms set to unity. Thus for PID controller the decision variables reduces to three gains for the pro-portional, integral and derivative actions respectively.

iPID p d

KC K K s

s= + + (10.2)

The plant to be controlled can be expressed in one of the following templates:

(a) First Order Plus Time Delay (FOPTD):

1

FOPTD LsKP e

Ts−=

+ (10.3)

(b) Second Order Plus Time Delay (SOPTD):

2 22

SOPTD Ls

n n

KP e

s sζω ω−=

+ + (10.4)

Here, model parameters{ }, , , ,nK T Lζ ω denote gain, time constant, damping ra-

tio, natural frequency and time delay respectively. For higher order linear models these structures give large modeling error and this proves the inadequacy of model reduction with FOPTD and SOPTD template for robust controller design. To ob-tain better accuracy of the reduced order models, two new structures, involving FO elements, have been introduced. The non-integer reduced parameter models are defined as:

(c) One Non-integer Order Plus Time Delay (NIOPTD-I):

1

NIOPTD I LsKP e

Tsα− −=

+ (10.5)


(d) Two Non-integer Orders Plus Time Delay (NIOPTD-II):

22

NIOPTD II Ls

n n

KP e

s sα βζω ω− −=

+ + (10.6)

Here, the system parameters have their classical meanings and the additional two

parameters i.e. the system orders { },α β are allowed to take any real value and

hence can be termed as flexible orders of the compressed models. As discussed above, the fractional (non-integer) order models could also have been obtained from the governing physical laws also using fractional order differential equations apart from optimization based model reduction techniques.

The frequency domain design frameworks for the above discussed four exam-ples are basically same. If ( )P s be the transfer function of the plant model, then

the objective is to find out a controller ( )C s , so that the open loop system

( ) ( ) ( )G s C s P s= meets the following design specifications:

(a) Phase margin specification:

[ ( )] [ ( ) ( )]gc gc gc

m

Arg G j Arg C j P jω ω ωπ φ

=

= − + (10.7)

where, gcω and mφ are the gain cross-over frequency and phase margin

respectively. (b) Gain crossover frequency specification:

( ) ( ) ( ) 1gc gc gcG j C j P jω ω ω= = (10.8)

(c) Robustness to gain variation (Iso-damping Property):

( )[ ( )] 0gc

dArg G j

d ω ω

ωω =

=

(10.9)

The first equation (10.7) enforces the user specified phase margin ( mφ ) at the gain

cross-over frequency ( gcω ) for the open loop system comprising of the plant with

the controller so as to control the percentage of overshoot for the closed loop sys-tem. The second equation (10.8) enforces the user specified gain cross-over fre-

quency ( gcω ) which control the rise time or the speed of response. The third

equation (10.9) says the derivative of phase of the open loop system with respect

to frequency around the gain cross-over frequency ( gcω ) is zero i.e. flat or hori-

zontal. In order to meet all the above design specifications, equations (10.7)-(10.9)needs to be solved. Since depending on the complexity of the controller and model structure the frequency response of the open loop system will be complex

10.3 Frequency Response of the Reduced Order Process Models and Controllers 261

thus analytical solution is not a feasible option to obtain controller parameters. This typical problem was solved by (Monje et al. 2008) using a constrained opti-mization for logarithm of (10.8) while taking other conditions as constraints and using a simultaneous nonlinear equation solving approach by (Das et al. 2011b). Here, we report another approach by minimizing all the equations so as to solve them numerically to find a set of controller parameters instead of solving a single equation and setting others as constraints as in (Monje et al. 2008). Here, a custom objective function ( J ) is defined as the sum of above three design criteria i.e. the phase margin and gain cross-over frequency specification and iso-damping.

( )( )

( ) ( )

( ) ( )

( ) ( ) 1

[ ( )]

( ) ( )

( ) ( ) 1

[ ( )] [ ( )]

gc

gc gc

gc m gc

gc gc m

gc gc

J Arg G j G j

dArg G j

d

Arg P j Arg C j

P j C j

d dArg P j Arg C j

d d

ω ω

ω ω ω ω

ω π φ ω

ωω

ω ω π φ

ω ω

ω ωω ω

=

= =

= + − + − +

= + + − + ⋅ − + +

(10.10)

Though with the above formulation all the control objectives may not be minimized in the same scale, it easily reduces the complexity of the problem to a simple global optimization framework with constraints only posed over the bound of the decision variables i.e. the controller parameters. Clearly, in order to design IO/FO controllers with minimization of the objective function (10.10), it is first required to know the frequency response of the plant ( )P jω and the controller ( )C jω . In the next sec-

tion the frequency responses of model templates of structure (10.3)-(10.6) along with the controller structures (10.1)-(10.2) are derived.

10.3 Frequency Response of the Reduced Order Process Models and Controllers

10.3.1 First Order Plus Time Delay (FOPTD) Model

The frequency response of the FOPTD model (10.3) is given by

( )

[ ]1

cos( ) sin( )1

FOPTD jLKP j e

j T

KL j L

j T

ωωω

ω ωω

−=+

= −+

(10.11)


From equation (10.11), the magnitude and phase of the FOPTD model can be de-rived as

2 2

2

2

( ) cos ( ) sin ( )1 ( )

1 ( )

FOPTD KP j L L

T

K

T

ω ω ωω

ω

= + +

=+

(10.12)

1 1

1

sin( )[ ( )] tan tan ( )

cos( )

tan ( )

FOPTD LArg P j T

L

L T

ωω ωω

ω ω

− −

−

−= −

= − −

(10.13)

The derivative of phase of the model (10.3) with respect to frequency (ω ) is

( ) ( )

( )

1

2

[ ( )] tan

1

FOPTDd dArg P j T L

d d

TL

T

ω ω ωω ω

ω

− = − −

= − −+

(10.14)

10.3.2 Second Order Plus Time Delay (SOPTD) Model

The frequency response of the SOPTD model (10.4) is given by

( )( )

[ ]( )

2 2

2 2

2

cos( ) sin( )

2

SOPTD j L

n n

n n

KP j e

j j

K L j L

j

ωωω ζω ω ωω ω

ω ω ζω ω

−=+ ⋅ +

−=

− +

(10.15)

From equation (10.15), the magnitude and phase of the SOPTD model can be de-rived as

( ) ( )2 22 2

( )2

SOPTD

n n

KP jω

ω ω ζω ω=

− + (10.16)

1 12 2

12 2

2sin( )[ ( )] tan tan

cos( )

2tan

SOPTD n

n

n

n

LArg P j

L

L

ζω ωωωω ω ω

ζω ωωω ω

− −

−

−= − −

= − − −

(10.17)



( )

( )( ) ( )

12 2

2 2

2 22 2

2[ ( )] tan

2

2

SOPTD n

n

n n

n n

d dArg P j L

d d

L

ζω ωω ωω ω ω ω

ζω ω ω

ω ω ζω ω

− = − − −

+= − −

− +

(10.18)

10.3.3 One Non-Integer Order Plus Time Delay (NIOPTD-I) Model

The frequency response of the NIOPTD-I model (10.5) is given by

( )( )

[ ]

[ ]

cos( ) sin( )

1 cos sin 12 2

cos( ) sin( )

1 cos sin2 2

NIOPTD I jL K L j LKP j e

T j T j

K L j L

T jT

ωα

α

α α

ω ωω

απ απω ω

ω ωαπ απω ω

− − −= =

+ + +

−= + +

(10.19)

From equation (10.19), the magnitude and phase of the NIOPTD-I model can be derived as

2 2

( )

1 cos sin2 2

NIOPTD I KP j

T Tα α

ωαπ απω ω

− = + +

(10.20)

1 1

1

sinsin( ) 2[ ( )] tan tancos( ) 1 cos

2

sin2tan

1 cos2

NIOPTD ITL

Arg P jL T

TL

T

α

α

α

α

απωωω απω ω

απωω απω

− − −

−

−= −

+

= − − +

(10.21)



( ) 1

1

2 2

sin2[ ( )] tan

1 cos2

sin2

1 cos sin2 2

NIOPTD ITd d

Arg P j Ld d T

TL

T T

α

α

α

α α

απωω ωαπω ω ω

απαω

απ απω ω

− −

−

= − −

+

= − − + +

(10.22)

10.3.4 Two Non-Integer Orders Plus Time Delay (NIOPTD-II) Model

The frequency response of the NIOPTD-II model (10.6) is given by

( )( ) ( )

[ ]( ) ( )

[ ]

[ ]

2

2

2

2

2

cos( ) sin( )

2

cos( ) sin( )

cos sin 2 cos sin2 2 2 2

cos( ) sin( )

cos 2 cos2 2

sin2

NIOPTD II j L

n n

n n

n n

n n

KP j e

j j

K L j L

j j

K L j L

j j

K L j L

j

ωα β

α βα β

α β

α β

α

ωω ζω ω ω

ω ωω ζω ω ω

ω ωαπ απ βπ βπω ζω ω ω

ω ωαπ βπω ζω ω ω

απω

− −=+ +

−=

+ +

−=

+ + + +

−=

+ +

+ + 2 sin2n

β βπζω ω

(10.23)

From equation (10.23), the magnitude and phase of the NIOPTD-II model can be derived as

22

2

( )

cos 2 cos2 2

sin 2 sin2 2

NIOPTD II

n n

n

KP j

α β

α β

ωαπ βπω ζω ω ω

απ βπω ζω ω

− = + + + +

(10.24)


1

1

2

1

2

sin( )[ ( )] tan

cos( )

sin 2 sin2 2tan

cos 2 cos2 2

sin 2 sin2 2tan

cos 2 cos2 2

NIOPTD II

n

n n

n

n n

LArg P j

L

L

α β

α β

α β

α β

ωωω

απ βπω ζω ω

απ βπω ζω ω ω

απ βπω ζω ωω απ βπω ζω ω ω

− −

−

−

−= +

− + + +

= − − + +

(10.25)


( )

( ) ( )

1

2

1

2 1 3 1

[ ( )]

sin 2 sin2 2tan

cos 2 cos2 2

2 sin2

sin 2 sin2 2

co

NIOPTD II

n

n n

n

n n

dArg P j

d

dL

d

α β

α β

α β

α β

α

ωω

απ βπω ζω ωωαπ βπω ω ζω ω ω

α β πζω α β ω

απ βπαω ω βζω ω

ω

−

−

+ −

− −

+ = − −

+ +

− −

+ + = −

22

2

s 2 cos2 2

sin 2 sin2 2

n n

n

Lβ

α β

απ βπζω ω ω

απ βπω ζω ω

− + + + +

(10.26)

10.3.5 Integer Order Proportional Integral Derivative (IOPID) Controller

The frequency response of the IOPID controller (10.2) is given by

( )2

PID i d ip d p

K K KC j K K j K j

j

ωω ωω ω

−= + + = +

(10.27)


From equation (10.2), magnitude and phase of the IOPID controller can be calcu-lated as

( )22

2PID d ip

K KC j K

ωωω

−= +

(10.28)

( )2

1tanPID d i

p

K KArg C j

K

ωωω

− − =

(10.29)

The derivative of phase of the controller (10.2) with respect to frequency (ω ) is

( )

( ) ( )

21

2

22 2

[ ( )] tan

( )

PID d i

p

p d i

p d i

K Kd dArg C j

d d K

K K K

K K K

ωωω ω ω

ω

ω ω

− − =

+=

+ −

(10.30)

10.3.6 Fractional Order Proportional Integral Derivative (FOPID) Controller

The frequency response of the FOPID controller (10.1) is given by

( )( )

( )

cos sin cos sin2 2 2 2

cos cos2 2

sin cos2 2

FOPID ip d p d d

p i d

p i d

d i

KC j K K j K K j K j

j

K K j K j

K K K

j K K

μ λ λ μ μλ

λ μ

λ μ

μ λ

ω ω ω ωω

λπ λπ μπ μπω ω

λπ μπω ω

μπ λπω ω

− −

−

−

−

= + + = + +

= + − + +

+ + = + −

(10.31)

From equation (10.31), magnitude and phase of the FOPID controller can be cal-culated as

( )

2

2

cos cos2 2

sin cos2 2

p i dFOPID

d i

K K K

C j

K K

λ μ

μ λ

λπ μπω ωω

μπ λπω ω

−

−

+ + = + −

(10.32)

10.4 Illustrative Examples 267

( ) 1sin cos

2 2tancos cos

2 2

d iFOPID

p i d

K KArg C j

K K K

μ λ

λ μ

μπ λπω ωω λπ μπω ω

−

−

−

− =

+ +

(10.33)

The derivative of the phase of the controller (10.1) with respect to frequency (ω ) is

( )

( ) ( )

1

1 1

1

[ ( )]

sin cos2 2tan

cos cos2 2

sin sin2 2

sin2

co

FOPID

d i

p i d

p d p i

i d

p i

dArg C j

d

K Kd

d K K K

K K K K

K K

K K

μ λ

λ μ

μ λ

λ μ

λ

ωω

μπ λπω ω

λπ μπω ω ω

μπ λπμω λω

λ μ πλ μ ω

ω

−

−

−

− − −

− + −

−

− =

+ +

+ +

+ + =+

2

2

s cos2 2

sin cos2 2

d

d i

K

K K

μ

μ λ

λπ μπω

μπ λπω ω−

+ + −

(10.34)

10.4 Illustrative Examples

Now, having known the frequency response of each model templates (10.3)-(10.6) and controllers (10.1)-(10.2), minimizing the objective function (10.10), the con-troller parameters can be calculated. In fact the same objective function (10.10) can be used in order to design PID and FOPID controllers having three and five independent parameters respectively. Here the controller parameters are con-

strained within an interval of { } [ ], , 1,100 ;p i dK K K ∈ { } [ ], 0,2λ μ ∈ since

the controller gains may take very large values within the unbounded optimization process and hence may create difficulty in hardware realization. Genetic algorithm has been used here as a global optimization algorithm with a population size of 50, to ensure a rigorous search such that the true global minimum is found. Also, the best results of 25 independent runs are reported in the representative examples of a FOPTD and a NIOPTD-II plant.


10.4.1 Control of FOPTD Plant

(Monje et al. 2004) studied PI λ control of a sugar cane raw juice neutralization process which can be expressed as a FOPTD model with the following parame-ters 0.55, 10, 62K L T= = = . The frequency domain specification are taken

from (Monje et al. 2004) as 0.02rad/s, 65 deggc mω φ= = . The resulting

FOPID and PID controllers are given in (10.35) and (10.36) along with minimum of the objective functions. It is evident from the Bode diagrams (Fig. 10.1 and Fig. 10.3) that both the the FOPID and PID controllers perfectly meet the design specifications along with the objective of keeping the phase flat around the gain cross-over frequency. But the robustness against gain variation is higher for FOPID controllers compared to that obtained by a PID controller, as can be seen from the range of gain variation in the time response curves (Fig. 10.2 and Fig. 10.4).

0.0477min1.1319

0.0320.4387 1.6464 , 0.064FOPTD

FOPIDC s Js

= + + = (10.35)

min

0.07171.2987 48.1238 , 0.011FOPTD

PIDC s Js

= + + = (10.36)

Fig. 10.1 Bode diagram of FOPTD plant controlled by FOPID with iso-damping


Fig. 10.2 Time response of FOPTD plant controlled by FOPID with iso-damping

Fig. 10.3 Bode diagram of FOPTD plant controlled by PID with iso-damping


Fig. 10.4 Time response of FOPTD plant controlled by PID with iso-damping

10.4.2 Control of NIOPTD-II Plant

Next a NIOPTD-II plant has been considered from (Das et al. 2011b) which is a compact representation of a higher order plant. The NIOPTD-II plant has the fol-lowing parameters

4.4659, 1.2337, 2.119, 2.4673, 1.0201, 0.1217nK Lζ ω α β= = = = = = .

The design specification are considered as 1rad/s, 70 deggc mω φ= = which

yields the iso-damped FOPID and PID controllers as (10.37) and (10.38) respectively.

1.0363min1.1507

1.19010.9577 0.0741 , 0.0582NIOPTD II

FOPIDC s Js

− = + + = (10.37)

min

1.5150.6858 0.4166 , 0.0461NIOPTD II

PIDC s Js

− = + + = (10.38)

It is observed that the both the control loops have almost similar robustness against process gain variation though increase in overshoot occurs in a much faster rate for a PID control loop compared to a FOPID control loop (Fig. 10.6 and Fig. 10.8), when both the loops are tuned with the same frequency domain specifi-cation. It is also obvious that range of flatness in phase curve or robustness against gain variation decreases with increase in the gain cross-over frequency or speed (Fig. 10.5 and Fig. 10.7).


Fig. 10.5 Bode diagram of NIOPTD-II plant controlled by FOPID with iso-damping

Fig. 10.6 Time response of NIOPTD-II plant controlled by FOPID with iso-damping


Fig. 10.7 Bode diagram of NIOPTD-II plant controlled by PID with iso-damping

Fig. 10.8 Time response of NIOPTD-II plant controlled by PID with iso-damping

10.5 Summary

Global optimization based frequency domain controller design has been enun-ciated in this chapter. Gain cross-over frequency and phase margin assignment in the controller tuning process, controls the speed of closed loop response and the

References 273

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References

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