Fuzzy and 2-DOF Controllers for Processes with a Discontinuously

Variable Parameter

Alexandra-Iulia Szedlak-Stinean

, Radu-Emil Precup

and Emil M. Petriu

Department of Automation and Applied Informatics, Politehnica University of Timisoara,

Bd. V. Parvan 2, 300223, Timisoara, Romania

School of Electrical Engineering and Computer Science, University of Ottawa,

800 King Edward, K1N 6N5, Ottawa, ON, Canada

Keywords: Classical PID Controllers, Digital and Experimental Results, Flexible Drive Dynamics, Fuzzy Controllers,

Mechatronics Application, Two-Degree-of-Freedom PID Controllers, Variable Moment of Inertia.

Abstract: The application with a discontinuously variable parameter (moment of inertia) discussed in this paper is

represented by an experimental setup, namely the Model 220 Industrial Plant Emulator (M220IPE), which

allows the testing of several control solutions by performing real-time experiments. This paper suggests a

simple state feedback control structure with three position controllers developed with the aid of linearized

mathematical models and particular features of the process. The control structures contain conventional

controllers (PID) and also advanced control solutions (Takagi-Sugeno PD+I fuzzy and two-degree-of-

freedom PID controllers). The aim of these control structures is to achieve good robustness, good disturbance

control behaviours regarding the model uncertainties and also to improve set-point responses. The proposed

control structures are validated by digital and experimental results obtained for three specific values of the

moment of inertia.

1 INTRODUCTION

Over the past two decades, the development of

mechatronics systems has led to a novel stage of

engineering design. By constantly evolving, these

systems exhibit increasing performances ensuring, as

shown in (Isermann, 2005), applicative and functional

versatility, intelligence and flexibility. Since in

various fields very good performance specifications

are imposed, the design of the control systems is also

important. If several process operating conditions

(e.g., moment of inertia) are involved as variable

parameters, the imposed specifications become even

more restrictive. In this regard, the degree of

complexity of the control subsystem of a mechatronics

application differs from one application to another and

may include relatively simple and advanced control

structures as well.

The representative mechatronics system described

in this paper considers a nonlinear and complex

laboratory equipment with adjustable inertia (ECP,

2010). One of the purposes of this paper is to illustrate

how the use of laboratory equipment of medium

complexity and of different control structures can be

made accessible, easily understandable and

increasingly attractive.

In this paper design and implementation details are

given regarding a state feedback control structure

(SFCS) for M220IPE with flexible drive dynamics.

The use of an SFCS offers no guarantee for the zero

steady-state control error. Taking into account that the

transfer function (t.f.) related to the inner control loop

has two real poles and two complex conjugated poles,

including the integral term in the state feedback

control in terms of extending the state does not assure

the imposed performance requirements (reduced

settling times and phase margins of 60

). That is the

reason why the SFCS is inserted in a control loop that

contains three types of position controllers: (1) a PID

controller, (2) a Takagi-Sugeno PD+I fuzzy controller

(TS-PD+I-FC) and (3) a two-degree-of-freedom PID

controller (2-DOF PID).

This paper offers fivefold contributions: 1. the

mathematical modeling of M220IPE with flexible

drive dynamics and the interpretation of these models

as benchmark type MMs, 2. the design and

implementation of the SFCS, 3. the development and

verification through simulations and experiments of

Szedlak-Stinean, A-I., Precup, R-E. and Petriu, E.

Fuzzy and 2-DOF Controllers for Processes with a Discontinuously Variable Parameter.

DOI: 10.5220/0006468504310438

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 2, pages 431-438

ISBN: Not Available

431

the proposed SFCS with PID controllers for nine

cases, 4. the digital simulation and experimental

testing of the SFCS with PID controllers, TS-PD+I-

FCs and 2-DOF PID controllers in three most

favorable case studies (i.e., the case studies I.a, II.b

and III.c) dedicated to the position control of

M220IPE with flexible drive dynamics, and 5.

comparative analyses to prove the validity of all

control solutions. They are relevant in the field as they

offer cost-effective control structures, characterized

by a simple structure, design and implementation.

The paper is divided into the following sections:

the dynamic equations described in the process MM in

case of flexible drive dynamics and the system

parameters values are given in Section 2. The

structural properties of the process and the design and

implementation of the SFCS are offered in Section 3.

Three positioning control structures developed for

M220IPE with flexible drive dynamics are presented

in Section 4. The digital simulation and experimental

results obtained are shown in Section 5. The

conclusions are highlighted in Section 6.

2 ELECTROMECHANICAL

PLANT – M220IPE

The controlled system is a mechatronics application

that is composed of three individual subsystems,

namely the electromechanical component, the real-

time controller unit and the dedicated executive

software. The main subsystem is the

electromechanical plant, which comprises a drive

motor that is coupled to a drive disk by means of a

timing belt, a disturbance motor that is coupled via a

4:1 gear ratio to a load disk and a speed reduction

assembly that connects the drive and load discs.

The moment of inertia of the two discs can be

adjusted by adding or removing specific weights

(ECP, 2010; Stinean et al., 2013a; Stinean et al.,

2013b). The electromechanical plant can emulate a

variety of dynamic configurations tested by

employing a nonlinear MM that closely describes the

actual behaviour of the system. The schematic

structure of the controlled process is illustrated in

Figure 1.

2.1 Mathematical Models

The MM of the electromechanical plant with flexible

drive dynamics can be determined using the relations

(1) - (3), where the terms are described in (ECP, 2010;

Stinean et al., 2013a; Stinean et al., 2013b; Stinean et

al., 2015; Acho et al., 2013).

The dynamic equations that describe the process

in case of flexible drive dynamics are:

.0)(

,)(

2122

121

=+−++−

=−+−++

−−

−−−−

θθθθθ

kkgrccgrcJ

TkgrkgrgrcgrccJ

Ddr

(1)

with J

, J

, gr and gr

’

expressed as

.12/' ,/6

, ,

,)()(

___

____

22'*

dplpdp

ldwlddlddrwdrddr

backlashldpdrpp

ldpdrdr

ngrnngr

JJJJJJ

JJJJ

grJgrJJJ

+=+=

++=

−−

(2)

The state-space MM (SS-MM) of M220IPE with

flexible dynamics with θ

as the process output is

.][ ,][

,][ ,] [

4...114...11

4...1,2211







jjii

jiij

TBxAx

θθθθ

(3)

with the matrix elements shown in (4)

.0 0, 0, ,1

,0 ,0 ,/1 ,0

,/)( ,/

,/,/

1, 0, 0,,0

,/ ,/

,/)( ,/

0, 0, 1, ,0

14131211

4131

2111

1224443

1242

34333231

1224

12122

14131211

====

+−=−=

====

+−=−=

====

−−

cccc

bbJbb

JccaJka

JgrcaJkgra

aaaa

JgrcaJkgra

JgrccaJkgra

aaaa

ldld

drdr

(4)

The application of the Laplace transform to (1)

considering that the initial conditions are zero, leads

to the following t.f.:

.)/(

,)/()/(

)],/([

)(d ,

,)(

),(/])([)(/)(

211

21212121

121

122

grkckcd

grccccccgrkJ

kJdgrccJ

ccJJJd

sdsdsdsdsd

sdksccsJsTs

drld

drlddr

ldD

++++

+=++

++==

+++=

(5)

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

432

Figure 1: Electromechanical plant of M220IPE.

2.2 System Parameter Values

As given in the electromechanical plant manual

(ECP, 2010) the parameter values used for the design

of the SFCS are presented in Table 1.

Table 1: System parameter values.

Parameter values

Parameters Values Remarks

d_dr

0.00040 [kgm

]

d_ld

0.0065 [kgm

]

backlash

0.000031 [kgm

]

w_dr

0.0021 [kgm

]

4⋅0.2 kg at r

w_dr

=0.05 m

w_dr

0.00561 [kgm

]

4⋅0.5 kg at r

w_dr

=0.05 m

w_ld

0.00824 [kgm

]

4⋅0.2 kg at r

w_ld

=0.1 m

w_ld

0.0206 [kgm

]

4⋅0.5 kg at r

w_ld

=0.1 m

p_dr

or J

p_ld

0.000008 [kgm

] n

p_d

=24 or n

p_l

=24

p_dr

or J

p_ld

0.000039 [kgm

] n

p_d

=36 or n

p_l

=36

0.004 [Nm/rad/s]

0.05 [Nm/rad/s]

0.017 [Nm/rad/s]

k 8.45 [Nm/rad]

Because the laboratory application does not allow

a continuous moment of inertia variation, the proposed

control solutions that will be implemented on

M220IPE, analyzed and tested through digital

simulations and experiments will be developed for

three specific values of inertia of the load disk, J

(Stinean et al., 2013a; Stinean et al., 2013b; Stinean

et al., 2015): the minimum value J

ld,min

=0.0065 kgm

(load disk with no weights on it), the average value

ld,avg

=0.01474 kgm

(four 0.2 kg weights positioned

at 0.1m from the load disk center) and the maximum

value J

ld,max

=0.0271 kgm

(four 0.5 kg weights

positioned at 0.1m from the load disk center).

There are totally nine possible combinations

between the process and controller parameters. The

t.f. in (5) and the matrix coefficients for the three

specific moment of inertia values are presented in

Table 2.

3 STATE FEEDBACK CONTROL

STRUCTURE

The use of the SFCS is based upon three main reasons:

1. the simultaneous control of all the essential process

variables (state variables) and the design of the

structure in relation to the evolution of these variables,

2. the stabilization of unstable processes, and 3. the

possibility to achieve the zero static control error by

extending the base structure.

The process structural properties are examined

considering the linearized SS-MMs and accounting

for the specific features of the nonlinearities. In this

regard, the controllability test of the linearized SS-

MMs (3) is conducted, using particularized parameter

values corresponding to defined operating points. The

state feedback controller contains a supplementary

amplifier with the k

gain. Since the state feedback

gain matrix k

is of proportional type, the use of SFCS

does not prove to be effective regarding the zero

steady-state control error. The pole placement method

is applied to develop our SFCS, namely to actually

determine the parameter values of k

, using three sets

of poles for each value of the moment of inertia of the

load disk Using e

-y

, y

x and u=k

, the SS-

MM of the SFCS is







BxAx

(6)

where A

=A-Bk

is the system matrix of the inner

loop, and k

=1. The expression of k

],[

4321 cccc

kkkk=

(7)

where T represents the transpose matrix and the gain

values are given in Table 3, columns 5, 6, 7 and 8.

The poles for the closed-loop system are also given in

Table 3 columns 1, 2, 3 and 4. The SS-MM of the inner

control loop leads to the t.f.

Fuzzy and 2-DOF Controllers for Processes with a Discontinuously Variable Parameter

433

)21)(1)(1(

)21(

)()(

sTsTsTsT

sTsTk

ssH

aaa

bbbSFCS

xSFCS

++++

=−=

−

BAIC

(8)

where T

, T

are the large time constants, T

is the

equivalent time constant, and ζ

is the damping

coefficient.

4 POSITIONING CONTROL

STRUCTURES FOR M220IPE

WITH FLEXIBLE DRIVE

DYNAMICS

This section presents the three proposed position

control structures developed to achieve the zero

steady-state control error specification, fulfilled by

the integral component of the controllers.

4.1 PID Controllers

The first state feedback control solution uses a

classical PID controller with the continuous-time t.f.

and tuning equations

,,),2/(1

,/)1)(1()(

22112

TTTTTkk

ssTsTksH

rrSFCSr

rrrr

===

++=

(9)

with k

– the controller gain, and T

and T

– the

controller time constants. After discretizing the

continuous-time PID controller using the backwards

difference method and setting the value of the

sampling period to h=0.004 s, the discretized t.f. is

,,,

),2(

),/()()(

102

hphpKq

KhKq

KhKhKq

zppzqzqqzH

dip

−===

+−=

++=

+++=

−−−−

(10)

where K

), K

and K

. The

PID controller parameter values for three significant

operating points obtained by Kessler’s Modulus

Optimum method (Åström and Hägglund, 1995) are

shown in Table 4, column 3.

4.2 Takagi-Sugeno PD+I Fuzzy

Controllers

Fuzzy controllers can be considered as easily

understandable initial controllers (Guerra et al., 2015;

Precup et al., 2015). This paper considers their design

and tuning such as to merge separately designed linear

controllers using the linearized process MM at several

operating points, justified as our controllers behave

like bumpless interpolators between linear controllers,

which is important due to the nonlinear input-output

map of fuzzy controllers and their adaptation to

process parameter changes.

The Takagi-Sugeno PD+I fuzzy block contains

the PD fuzzy controller in the parallel structure with

an integral (I) controller, i.e., the integral component

is implemented separately such that u

PDk

knowing that u

represents the output of this I

component. The PD quasi-continuous digital

controller and its parameters are derived from the PD

Table 2: State-space MM matrices and transfer functions expressions for M220IPE with flexible drive dynamics.

Moment of

inertia

Matrices A, B and C Process transfer function θ

(s)/T

(s)

ld,min

]0001[,

13850

307.10

13.10

1300

5036

654.0

068.12

325

1259

























−−

= CBA

)220782.267737.22(

)1300307.10(13850

+++

ssss

ld,avg

]0001[,

13850

59.4

13.10

579

5036

3.0

068.12

145

1259

























−−

= CBA

)4.98277.189365.16(

)57959.4(13850

+++

ssss

ld,max

]0001[,

13850

47.2

13.10

312

5036

157.0

068.12

9.77

1259

























−−

= CBA

)4.52904.159953.14(

)31247.2(13850

+++

ssss

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

434

Table 3: Numerical values of the selected poles and state feedback gain matrix.

Moment

of inertia

Flexible drive dynamics

Selected poles State feedback gain matrix

1 2 3 4 5 6 7 8

ld,min

-12.26 -48.49 -28.32+59.3317i -28.32-59.3317i 0.3234 0.0069 -0.7223 0.0247

ld,avg

-8.33 -26.32 -17.52+38.4817i -17.52-38.4817i 0.0749 0.0038 -0.1030 0.0124

ld,max

-4.95 -16.46 -17.38+31.3767i -17.38-31.3767i 0.0280 0.0030 -0.0155 0.0104

component by using

,//, ,/

),(

1221

121

dppd

kkkkPDk

KhKkkKkhKk

eekekeku

==α==

α+Δ=+Δ=

(11)

where Δe

-e

k-1

represents the increment of the

control error, e

is the control error and u

PDk

represents the output of the PD fuzzy controller. The

developed TS-PD+I-FC substitutes the linear PD

component in the linear PID controller presented in

Sub-section 4.1. The fuzzification in TS-PD-FC is

realized using for each input, three input linguistic

terms LT

and LT

∆ek

∈

{N, ZE, P} with trapezoidal

and triangular membership functions μ

, μ

Δe

. The

involved operators from the inference engine are

SUM and PROD and for defuzzification the weighted

average method is employed. In total for the complete

rule base, there are nine rules:

, then )N is and P is (If

, then )N is and E is (If

, then )N is and N is (If

, then ) ZEis and P is (If

, then ) ZEis and ZEis (If

, then ) ZEis and N is (If

, then )P is and

P is (If

, then )P is and ZEis (If

, then )P is and N is (If

PDkPDkkk

uuee

uueZe

uuee

=Δ

(12)

where

},,{ cba∈

indicates the indices of the linear

PD controllers, namely a for C-J

ld,min

, b for C-J

ld,avg

and c for C-J

ld,max

./)]([ heeKu

kkdPDk

χχ

+Δ=

(13)

The parameter B

∆e

is obtained by the modal

equivalence principle using the tuning equation

Δe

=min(α

, in which the B

parameter is specified

by the designer. The numerical values related to TS-

PD+I-FCs for three specific values of the load disk

moment of inertia are summarized in Table 4, column

Figure 2: TS-PD-FC structure (a) and input membership

functions (b).

4.3 2-DOF PID Controllers

The advantages of using 2-DOF control solutions are

well known and concern reference tracking,

disturbance rejection and simultaneous good feedback

properties. Since multi-objective problems must be

solved to design control systems, a 2-DOF controller

has several advantages over a 1-DOF controller,

shown in (Leva and Bascetta, 2006). The 2-DOF

control structure illustrated in Figure 3 in a

formulation based on PID controllers is referred to as

2-DOF PID controller structure. Over the past two

decades many 2-DOF PID structures have been

developed (Leva and Bascetta, 2006; Alfaro et al.,

2009; Kevickzy and Banyasz, 2015), but in this paper

this structure was chosen for two main reasons: 1. the

simplicity of the structure, and 2. the ease of

transforming the PID controller into a 2-DOF

controller and vice versa. Using (Precup and Preitl,

2007; Precup et al., 2014), the main PID controller

component C

(s) and the additional block C

(s),

highlighted in Figure 3, are described by the t.f.s

)(

])([k(s)

,)]()1(

)-[(1(s)C

CFB

sDC

−++=

μτ

(14)

where k

, T

are the tuning parameters for the

2-DOF PID controller, k

(1-τ)=K

, T

and

(1-μ)=K

Fuzzy and 2-DOF Controllers for Processes with a Discontinuously Variable Parameter

435

Figure 3: 2-DOF PID structure.

The parameter values employed in the design of

this control solution are presented in Table 4, column

5 DIGITAL AND

EXPERIMENTAL RESULTS

The SFCS described in Section 3 was developed and

tested on the M220IPE laboratory equipment in the

framework of three proposed position control

solutions. The parameter referred to as hardware gain

=5.81 Nm/rad has been inserted in order to achieve

a higher controller internal numerical resolution. This

parameter also improves the encoder pulse period

measurement (ECP, 2010).

In order to avoid the oscillations produced by the

pair of complex conjugated poles and by the process

nonlinearities, four real poles were considered for the

proposed state feedback control solution. Because the

results are approximately similar with the ones

presented in this paper, this situation will not be

discussed any further, but the idea how the poles of the

inner loop can be modified may be the subject of future

research.

Analyzing the comparative simulation and

experimental results presented in Figure 4, the

following aspects can be concluded: in terms of the

best quality indices, both simulation and experimental

results show that the best performances have been

achieved in the cases I.a, II.b and III.c. The case I.a is

more effective – concerning the settling time value

≈0.8661s) and also overshoot value (σ

≈0%) – than

the cases I.b and I.c; regarding the overshoot values,

the cases I.b and I.c exceed 22%; the case I.a offers

the highest performance in both digital and

experimental results and the worse result is obtained

by the case I.c; the case II.b exhibits the highest

performance regarding the settling time value

≈1.1031) compared to the cases II.a (t

≈1.6604s)

and II.c (t

≈2.3932s); the overshoot value in this case

is around 8.65%; the case III.c has reached the highest

performance indices concerning the settling time

value (t

≈1.8006s) and first settling time value

≈0.2836s) compared to the cases III.a (t

≈2.9723,

≈2.9521s) and III.b (t

≈2.0981s, t

≈2.0853); the case

III.a was the slowest one by exhibiting the largest

settling time value.

The three proposed position control solutions

described in Section 4 were tested and validated for the

electromechanical plant with flexible dynamics only in

the three most favourable cases – case I.a, case II.b and

case III.c – and the results are presented in Figure 5.

Taking this into account, by comparing the achieved

results, following conclusions can be drawn: (1) the

PID control solution was used for comparison and also

for the design of the other solutions; it was the least

favourable one achieving poor results in both settling

time and first settling time; (2) the 2-DOF PID control

solution is more effective – in all three cases

concerning the settling time and the first settling time

– in comparison with the PID control solution, but

worst in comparison to the TS-PD+I-FC control

solution; (3) all proposed solutions showed good

reference tracking behaviour and (4) the suggested

position controllers contribute in average to both

robustness and good dynamic performance regarding

at least one process parameter. Other effects will be

pointed out for other relevant applications including

those presented in (Filip, 2008; Mazdin et al., 2016).

6 CONCLUSIONS

This paper provided details regarding the development

of SFCSs aimed to control the position for a

mechatronics system built round the M220IPE

laboratory equipment. Since an SFCS doesn’t

guarantee the zero steady-state control error,

therefore it is included in a control loop that contains

three possible control structures with PID, TS-PD+I-

FC and 2-DOF PID controllers. Digital and

experimental results are given for three specific load

disk moment of inertia values.

The step responses in relation to the reference input

are used in the control structures comparison. The

comparison shows that the suggested solutions are

transparent and relatively easy to understand and to

employ, and the best reference tracking and control

system performance has been obtained in the cases

I.a, II.b and III.c. The controller C-J

ld,min

is suitable

for J

ld,min

and less suitable for J

ld,avg

and J

ld,max

. The

controller C-J

ld,avg

is suitable for J

ld,avg

and less

suitable for J

ld,min

and J

ld,max

. The controller C-J

ld,max

suitable for J

ld,max

and less suitable for J

ld,min

and J

ld,avg

The performance indices improvement for the

proposed control solutions using both model-based and

model-free tuning techniques will be the target of

future research. In addition, the pole placement will be

replaced with optimal parameter tuning.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

436

ACKNOWLEDGEMENTS

This work was supported by grants from the

Romanian Executive Agency for Higher Education,

Research, Development and Innovation Funding

(UEFISCDI), project number PN-II-RU-TE-2014-4-

0207, the Partnerships in priority areas – PN II

program of UEFISCDI, project numbers PN-II-PT-

PCCA-2013-4-0544 and PN-II-PT-PCCA-2013-4-

0070, and the NSERC of Canada.

Table 4: SFCS t.f.s. and numerical values of PID, TS-PD+I-FC and 2-DOF PID controllers.

Moment

inertia

Flexible drive dynamics

SFC structure t.f. H

SFCS

(s)

PID TS-PD+I-FC 2-DOF PID

3 4 5

1 2 K

ld,min

)000231.00131.01)(0815.01)(0206.01(

)00077.00079.01(9503.6

ssss

++++

0.1123 1.1 0.0018 0.4617 0.1123 0.2433 0.1127 0.1024 0.0074

ld,avg

)000559.00196.01)(12.01)(0379.01(

)00172.00079.01(4588.20

ssss

++++

0.1105 0.7 0.0032 0.7981 0.1105 0.1384 0.1108 0.1584 0.0142

ld,max

)000777.00270.01)(202.01)(0607.01(

)0032.00079.01(2229.41

ssss

++++

0.0919 0.35 0.0043 1.0729 0.0919 0.0857 0.0925 0.2643 0.0287

Figure 4: Digital (a) and experimental (b) results concerning the behaviour of SFCs with PID controllers designed for

M220IPE with flexible drive dynamics: cases I.a-I.c, cases II.a-II.c and cases III.a-III.c.

Fuzzy and 2-DOF Controllers for Processes with a Discontinuously Variable Parameter

437

Figure 5: Digital (a) and experimental (b) results concerning the behaviour of SFCS with three proposed control solutions

developed for M220IPE with flexible drive dynamics: case I.a, case II.b and case III.c.

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