Zhihong Xiu - A Novel Nonlinear PID Controller

7/27/2019 Zhihong Xiu - A Novel Nonlinear PID Controller

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A Novel Nonlinear PID ControllerDesigned By Takagi-Sugeno Fuzzy Model*

Zhihong Xiu1,2, Wei Wang1

1)Research Center of Information and Control 2)Department of Command ControlDalian University of Technology Dalian Naval Academy

Dalian, Liaoning Province, China Dalian, Liaoning Province, China

[email protected] [email protected]

*This work is partially supported by China Postdoctoral Science Foundation Grant 2005038255.

Abstract - TS-PID fuzzy controller is proposed as a novelframework for nonlinear PID controller design in this paper. Theso-called TS-PID fuzzy controllers are a class of Tagagi-Sugeno(TS) fuzzy controllers whose rule consequences employ PID

expressions. Based on Lyapunov theory, a design and stabilityanalysis method of TS-PID fuzzy controllers is presented. Thisapproach utilizes some analytical techniques of linear controltheory to analyze and design a TS-PID fuzzy controller. Thesimulation results of a nonlinear mass-spring-damper systemshow that the performance of a TS-PID fuzzy controller is betterthan that of the conventional TS fuzzy controller.

Index Terms - Fuzzy control, nonlinear PID controller, TS-PID fuzzy controller, stability, analytical design.

I. INTRODUCTION

Conventional proportional-integral-derivative (PID)controllers are widely used in industry process control, due toits simplicity in structure and ease of implementation [1].However, since the controller parameters are fixed duringcontrol after they have been tuned or chosen in a certainoptimal way, a conventional PID controller cannot alwayseffectively control systems with changing parameters, andmay need frequent on-line retuning. On the other hand, in

recent years, fuzzy logical controllers (FLC), especially fuzzyPID controllers have been widely used for industrial processesowing to their heuristic nature associated with simplicity andeffectiveness for both linear and nonlinear systems [2-9].Because of the nonlinear property of control gains, fuzzy PIDcontrollers possess the potential to improve and achieve bettersystem performance over the conventional PID controller ifthe nonlinearity can be suitable utilized. Whereas, due to theexistence of nonlinearity, it is usually difficult to conducttheoretical analyses to explain why fuzzy PID controllers canachieve better performance. Consequently, it is important,from both the theoretical and practical points of view, toexplore the essential nonlinear control properties of fuzzy PID

controllers and find out appropriate design methods which willassist the control engineers to confidently utilize thenonlinearity of the fuzzy PID controllers to improve theclosed-loop performance.

Many researchers recently attempted to combineconventional PID controllers with fuzzy control, some of theminvestigated to employ a Mamdani type fuzzy controllerequivalent to a PID controller [2,3,4], others attempted to

employ a fuzzy expert system at supervisory controller level ina form of fuzzy rule-based readjusting the conventional PIDgains [5,6]. Kazemian [7] investigated the application of aself-organizing fuzzy PID controller which is the extension ofthe rule-based fuzzy controller with an additional learningmechanism. Especially, Zhao [8] exploit the Mamdani type offuzzy rules and reasoning to generate the parameters of PIDcontroller. The main difficulty in using most of these existedtypes of fuzzy PID controllers is that the design approachrequires ad hoc expertise, and there is a lack of analyticalstability analysis and design method. In order to solve these

problems, we propose a new type of fuzzy PID controllers,that is the so-called TS-PID fuzzy controllers, by combiningthe conventional PID controller with the Tagagi-Sugeno (TS)fuzzy model in this paper. A TS-PID fuzzy controller iscomposed of the conventional PID controllers in conjunctionwith a set of TS fuzzy rules (knowledge base) and a fuzzyreasoning mechanism. The PID gains are changing on-line interms of the knowledge base and fuzzy inference. By virtue ofthe gain scheduling property, TS-PID fuzzy controllers canadapt themselves to varying environments. In this paper we

present the design and stability analysis methods of TS-PIDfuzzy controllers based on Lyapunov theory.

This paper is organized as follows. Section II describesthe model of TS-PID fuzzy controllers, and outlines itsapplications. In Section III, we present an analytical designand stability method of TS-PID fuzzy controllers based onLyapunov theory. In Section IV, an example illustrates the

particular design procedures of a TS-PID fuzzy controllerapplied to a nonlinear mass-spring-damper system. Theconclusions are given in the last part of the paper.

II. TS-PID FUZZY CONTROLLERS AND APPLICATIONS

A. TS-PID Fuzzy ControllerTS-PID fuzzy controllers we proposed in this paper are a

class of TS fuzzy controllers whose rule consequences employ

PID expressions. The rules of a TS-PID fuzzy controller takethe form as:

IF z1 isiM1 and and zp is

i

pM

THEN eKedtKeKu iD

i

I

i

P

i++= , i=1,2,,l (1)

where ldenotes the number of inference rules,

z(t)=[z1,z2,,zp]T Rp denotes the premise variable vector

1-4244-0332-4/06/$20.00 2006 IEEE3724

Proceedings of the 6th World Congress on Intelligent Control

and Automation, June 21 - 23, 2006, Dalian, China

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of TS-PID fuzzy controllers,i

jM (i=1,2,, l) denote fuzzy sets of the premise variablezj,

ui denotes the output of the ith local PID controller,i

PK ,i

IK ,i

DK represent the parameters of the ith local PID

expression.By using a singleton fuzzifier, the product inference and a

center-average defuzzifier, the global model of a TS-PID

fuzzy controller can be obtained as follows:

==

=l

i

i

l

i

i

i wuwu11

)()( zz } =

++=l

i

i

D

i

I

i

Pi eKedtKeKh1

)( z

eKhedtKheKhl

i

i

Di

l

i

i

Ii

l

i

i

Pi

+

+

= === 111

)()()( zzz

eKedtKeK DIP *** ++= (2)

where =

=p

k

k

i

ki tzMw1

))(()(z is the firing strength of the ith

rule, =

=l

i

iii wwh1

)()()( zzz , and

==l

i

i

PiP KhK1

*

)(z , (3)

=

=l

i

i

IiIKhK

1

* )(z , (4)

=

=l

i

i

DiD KhK1

* )(z . (5)

From (2) to (5), we can see that TS-PID fuzzy controllersare a kind of nonlinear PID controllers in essence, and the PID

parameters are varying with the premise variable z. TS-PIDfuzzy controllers may take the advantages of both fuzzycontrollers and PID controllers, through converting the PID

parameters tuning experiences into TS fuzzy reasoning rules.

B. Applications of TS-PID Fuzzy ControllersEq. (1) is the general form of TS-PID fuzzy controllers.

According to the practical instance, the premise variables of aTS-PID fuzzy controller may take various forms, hence thedesign methods of which are different. There are two cases ofapplications of TS-PID fuzzy controllers in common used:

Case A: If the control plant is a nonlinear or model-

varying dynamic system, the system dynamics can be captured

into a set of TS fuzzy implications which characterize local

relations in the premise state space. The premise variables of

the rule base can employ the nonlinear variables or the varying

parameters of the system. The TS fuzzy modeling method of a

plant can be found in [10]. Then, a TS-PID fuzzy controller

can be designed to control this plant. If a TS-PID fuzzycontroller employ same premise variables and share the same

fuzzy sets with the TS fuzzy model in the premise parts, the

corresponding local PID controller for each linear subsystem

of TS fuzzy plant model can be designed by using the linear

control theory, and the global model of a TS-PID fuzzy

controller can be obtained by composing these local PID

controllers together according to (2). In this case, a TS-PID

fuzzy controller can be regarded as an adaptive PID controller.

Case B: If the control plant is a linear or unknown

dynamic system, as well as other systems, the premise

variables of a TS-PID fuzzy controller can employ one or

more of the error e, the derivative of e and the integral of e. In

this case, a TS-PID fuzzy controller can be regarded as a self-

adjusting PID controller, so that can achieve better

performance than a conventional PID controller. If there are

perfect manual control experiences, we can convert the

manual control experience into the rules of TS-PID fuzzycontroller in designing. If lack of perfect manual control

experiences, we can design a TS-PID fuzzy controller by

computer optimization, such as by Genetic Algorithms (GAs)

[11].

In Section III, we mainly discuss the case A.

III. STABILITY ANALYSIS AND DESIGN OF TS-PID FUZZYCONTROLLERS

A. Stability Analysis of TS Fuzzy Control SystemsIf the consequences of a TS fuzzy model adopt state

equations, its rules take the form as follows:

IFz1

is i

M1and and z

pis i

pMTHEN )()()( tutt ii BxAx += , i=1,2,,l (6)

where ldenotes the number of inference rules,i

jM ( i=1,2,,l;j=1,2,,p) denote input fuzzy sets,

T

21 )](),...,(),([)( txtxtxt n=x Rn denotes the state vector of

the system,

u(t) denotes the control input,

z(t)=[z1, z2,, zp]T Rp denotes premise variable vector of

the fuzzy system,

Ai, Bi represent the ith local model parameters of the fuzzy

system, and are of appropriate dimension.

By using a singleton fuzzifier, the product inference and a

center-average defuzzifier, the following global dynamic

model can be obtained:

{ }=

+=l

i

iii tutht1

)()()()( BxAzx (7)

where =

=p

k

k

i

ki tzMw1

))(()(z is the firing strength of the ith

rule, and =

=l

i

iiiwwh

1

)()()( zzz .

When u(t)=0, the fuzzy system (7) becomes an input-free

system:

==l

iii

tht1

)()()( xAzx (8)

Based on the Lyapunov direct method, Tanaka et al.[12,13] studied the stability problem of an input-free TS fuzzysystems and a theorem was obtained as follows:

Theorem 1: The equilibrium of (8) is asymptotically

stable in the large if there exists a common positive-definite

matrix Psuch that

0T


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In practical applications, some system dynamics arecaptured by a set of TS fuzzy implications described by (6)which characterize local relations in the state space. Thecorresponding TS fuzzy controller can be designed via paralleldistributed compensation (PDC) [13,14]. In the PDC concept,each control rule is distributively designed for thecorresponding rule of a TS fuzzy model. The TS fuzzycontroller shares the same fuzzy sets with the fuzzy model in

the premises.For (6), let Ki denotes the state feedback gain of the ith

local model, the local control laws of a TS fuzzy state

feedback controller are as follows:

IFz1 isiM1 and and zp is

i

pM

THEN u= Kix, i=1,2,,l. (10)The global model of a TS fuzzy controller can be inferred

as follows:

=

=l

i

iihu1

xK . (11)

If a TS fuzzy system (6) is with the common input matrix

B, that is, B1=B2==Bl=B, the global closed-loop fuzzy

control system described by (7) and (11) is as follows:

=

=l

i

iiih

1

)( xBKAx . (12)

Let Gi=AiBKi, from (12) we have

=

=l

i

iih

1

xGx . (13)

(13) has the same form as the input-free system model

described by (8), so we can utilize Theorem 1 to check the

stability of this class of closed-loop TS fuzzy control systems

in the design process.

If a TS fuzzy system (6) whose input matrices B1,B2,,Blare not all the same, the global closed-loop fuzzy control

system described by (7) and (11) is as follows:

= =

=l

i

jiij

l

j

ihh1 1

)( xKBAx . (14)

We cannot directly utilize Theorem 1 to check thestability of this class of closed-loop TS fuzzy control systems.Therefore, based on Theorem 2 and Theorem 3 in [13], astability analysis method of the closed-loop TS fuzzy systemswas proposed in [14] as follows.

Theorem 2: The equilibrium of a fuzzy control system

(14) is asymptotically stable in the large if there exists a

common positive-definite symmetric matrix P such that the

following condition (C1) or (C2) is satisfied:

(C1): 0T


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IF z1 is11M and z2 is

1

2M

THEN uxx = 02.0IF z1 is

1

1M and z2 is2

2M

THEN uxxx = 02.0225.0 IF z1 is

21M and z2 is

1

2M

THEN uxx = 5275.1IF z1 is

2

1M and z2 is2

2M

THEN uxxx = 5275.1225.0 Where xz =1 , xz =2 , the membership functions of their

fuzzy sets are obtained as follows [16]:

25.21)(

21

1

1

1

zzM = ,

25.2)(

21

1

2

1

zzM = ,

25.21)(

22

2

1

2

zzM = ,

25.2)(

22

2

2

2

zzM = .

Fig. 1 and Fig. 2 show the fuzzy sets in the premise parts.

From (17), we can see that the four linear subsystems areall second-order plants. According to the linear system theory[17], we know that it can achieve a satisfying performance toapply the PD type controller to a second-order plant. By someexperiences of tuning PID gains, we can select different PIDgains in the deferent subsystems to achieve a better

performance. For example, we can increase the proportionalgain KP if the error is large (such as in the third and the fourthsubsystems) to improve the rise time, and increase thederivative KD to reduce the overshoot if the error is little butthe derivative of error is large (such as in second subsystem).Finally, we design the rules of a TS-PID fuzzy controller forthe mass-spring-damper system as follows:


1

2M

THEN eeu 0.498.31 =IF z1 is

1

1M and z2 is2

2M

THEN eeu 775.698.32 =IF z1 is

2

1M and z2 is1

2M

THEN eeu 0.34725.53 =

IF z1 is2

1M and z2 is

2

2MTHEN eeu 775.24725.54 =

where e(t)=r(t)x(t), r(t) is the set position of the mass.

We can verify the stability of the system by using the

Lyapunov approach presented in Section III.

The TS fuzzy model (17) can be rewritten by introducing

matrix representation as follows:


1

2M THEN uBxAx += 1IF z1 is

11M and z2 is

2

2M THEN uBxAx += 2

IF z1 is2

1M and z2 is12M THEN uBxAx += 3


2

2M THEN uBxAx += 4

wherex

=[x1,x2]

T

,x1= z1=x,x2= z2=x

,

=002.0

101A

,

=002.0

1225.02A

,

=05275.1

103A

,

=05275.1

1225.01A

, B=[0, 1]T.

Ifr(t)=0, then e(t)=x(t). The TS-PID fuzzy controller (18)

can be rewritten in the form of a fuzzy state feedback

controller as follows:

IF z1 is1

1M and z2 is1

2M THEN xK11 =u

IF z1 is1

1M and z2 is22M THEN xK2

2 =u

IF z1 is2

1M and z2 is12M THEN xK3

3 =u

IF z1 is 21M and z2 is 22M THEN xK44 =uwhere the state feedback gain Ki of the linear subsystems

can be derived from (18) as follows: K1 =[3.98, 4.0], K2=[3.98, 6.775], K3=[5.4725, 3.0], K4 =[5.4725, 2.775].

Since all the linear subsystems have the same input matrix

B, the closed-loop state matrices of these subsystems can be

derived from Gi=AiBKi as follows:

=44

101G

,

=

775.64

1225.02G

,

=37

103G

,

=

775.27

1225.04G

.

We can conclude that the close-loop fuzzy control systemof the mass-spring-damper system is stable by Theorem 1, forwe have found a common positive-definite symmetric matrixsatisfying the condition of Theorem 1 via the LMI approach asfollows:

=

575.1087.0

087.02303.1P .

We simulated the mass-spring-damper system (16)

controlled by TS-PID fuzzy controller (18) using various

Fig.2. The membership functions ofz2

(17)

z1

z2

11M

2

1M

12M

22M

(18)

(20)

Fig.1. The membership functions ofz1

(19)

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initial conditions. The simulation results show that this system

is stable under all initial conditions. The output response of

this system may be compared with that obtained from a TS

fuzzy state feedback controller designed by Tanaka et al. [16].

The output responses of the system under the same initial

conditions 1)0( =x , 1)0( =x are plotted in Fig. 3 and Fig. 4.It is obvious that the responses of mass-spring-damper

system controlled by TS-PID fuzzy controller are much betterthan that controlled by the TS state feedback controllerdesigned in [16].

V. CONCLUSION

The TS-PID fuzzy controller is a novel framework fornonlinear PID controller design. Comparing with aconventional PID controller, TS-PID fuzzy controllers possessthe potential ability to achieve better system performance ifthe nonlinearity can be suitable utilized. This paper presentsthe stability analysis and design approaches to TS-PID fuzzycontrollers. Based on the Lyapunov theory, we present adesign and stability analysis method of TS-PID fuzzycontrollers. This approach utilizes some analytical techniques

of linear control theory to analyze and design a TS-PID fuzzycontroller. An example illustrates the particular procedures ofdesign of TS-PID fuzzy controllers applied to a nonlinear

plant. The simulation results show that the performance of aTS-PID fuzzy controller is better than that of the conventionalTS fuzzy controller. Owing to their simple structures and goodcontrol performances, TS-PID fuzzy controllers proposed inthis paper can be widely used in other control processes.

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Fig. 3. Output response x of the mass-spring-damper system

controlled by the TS-PID fuzzy controller (solid line) and the TS statefeedback controller [16] (broken line)

Fig. 5. Output response x of the mass-spring-damper system

controlled by the TS-PID fuzzy controller (solid line) and the TSstate feedback controller [16] (broken line)

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Zhihong Xiu - A Novel Nonlinear PID Controller