Fuzzy rule based combination of linear a

Leung Centre of Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 20 Februar

Fuzzy rule-based combination of linear and switching

state-feedback controllers

H.K Lam∗, F.H.F Leung

Centre of Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic

University, Hung Hom, Kowloon, Hong Kong

Received 20 February 2004; received in revised form 2 May 2005; accepted 24 May 2005

Available online 20 June 2005

Abstract

This paper presents a fuzzy rule-base combined controller, which is a fuzzy rule-based combination of linear and switching state-feedback controllers, for nonlinear systems subject to parameter uncertainties The switching state-feedback controller is employed to drive the system states toward the origin When the system state approaches the origin, the linear state-feedback controller will gradually replace the switching state-feedback controller The smooth transition between the linear and switching state-feedback controllers is governed by the fuzzy rules By using the fuzzy rule-based combination technique, the proposed fuzzy rule-base combined controller integrates the advantages of both the linear and switching state-feedback controllers but eliminates their disadvantages As

a result, the proposed fuzzy controller provides good performance during the transient period and the chattering effect is removed when the system state approaches the origin Stability conditions will be derived to guarantee the system stability Furthermore, a saturation function is employed to replace the switching component to alleviate the chattering during the transient period By using the proposed fuzzy rule-based combination technique, the steady state error introduced by the saturation function can be eliminated Application examples will be given to show the merits of the proposed approach.

© 2005 Elsevier B.V All rights reserved.

Keywords: Fuzzy model based control; Fuzzy controller; Stability

∗Corresponding author.

E-mail address:hak_keung_lam@yahoo.com.hk (H.K Lam).

0165-0114/$ - see front matter © 2005 Elsevier B.V All rights reserved.

doi:10.1016/j.fss.2005.05.021

1 Introduction

Control of nonlinear systems is a challenging task because no systematic mathematical tools exist tohelp find necessary and sufficient conditions to guarantee the stability and performance The problembecomes more complex if some of the system’s parameters are uncertain Fuzzy control is good at handlingill-defined and complex systems However, effective stability analysis and systematic controller designmethodologies are still lacking

Recently, stability analysis of fuzzy-model-based control systems, which is made of a Takagi–Sugeno–Kang (TSK) fuzzy plant model[15,17] and a fuzzy controller, has become a hot research topic Differentstability conditions have been published In [21], it has been proven that the fuzzy control system isguaranteed to be stable if there exists a common solution to a set of Lyapunov equations Relaxed stabilityconditions have been derived in [5,6,18,20,24,27] under the assumption that the fuzzy controllers sharesthe same premises as those of the fuzzy plant models This assumption implies that the nonlinear plant

is completely known Other stability analysis methods of fuzzy-model-based control systems were alsoproposed In [28], a linear controller was designed based on the fuzzy plant model In this approach, thefuzzy plant model is separated into two parts, nominal and varying parts The nominal part is handled

by the linear controller The varying part is treated as an uncertainty Hence, when the magnitude of thevarying part is too large, the linear controller will fail to control the plant In [10,11], stability conditionswere derived based on the matrix measures [23] of the sub-system matrices In [9], a nonlinear state-feedback controller designed based on the fuzzy plant model was also proposed In [3], the stabilityconditions of a number of Riccati equations were derived Constructive algorithms were proposed to helpfind the solution of the derived stability conditions However, in most works, the parameter uncertainties

of the system which commonly appear are practically not considered

In some published work, fuzzy logic has been combined with the traditional sliding-mode controller

to combine their advantages together In [1], a fuzzy sliding-mode controller using the sliding-surfacefunction as the input of the fuzzy system has been reported Since only one variable is taken as the input

of the fuzzy system, the number of fuzzy rules can be greatly reduced In [4,22], a fuzzy system wasemployed to estimate the values of the control gains of the sliding-mode controller Adaptive laws havebeen derived to update the rules of the fuzzy systems As the switching function of the sliding-modecontroller is approximated by a continuous function, the chattering effect can be alleviated In [16], anadaptive fuzzy controller has been proposed to generate the control signals by estimating the values ofthe unknown system parameters of the systems Based on these estimated parameter values, trackingcontrol can be achieved by the sliding-mode control However, in these approaches, the way to determinethe fuzzy rules is still an open question Furthermore, the approximation error of the fuzzy systems willintroduce steady-state error to the system states or even cause the system unstable In [25], to compensatethe approximation error of the fuzzy system, switching elements were still needed in the controller.Other techniques have also been proposed to alleviate the chattering effect A saturation function wasproposed to replace the switching element or include the system states in the switching element [13] In[26,29], a two-phase variable structure controller was proposed The distance of the system states fromthe sliding surface was considered during the controller design For these approaches, the chattering effectmay disappear only when the system state reaches the equilibrium, otherwise, the steady-state error mayappear in the system states

Switching control [2,7,8] is good at handling nonlinear systems subject to parameter uncertainties.With this method a good system performance and global system stability can be guaranteed, however, an

undesired chattering effect will occur Linear state-feedback controller designed based on a linear model ofthe nonlinear system offers a simple and systematic design methodology However, this control approachguarantees the system stability locally, i.e the system state is within a small operating domain Through-out this paper, the fuzzy controller which is a fuzzy rule-based combination of the linear state-feedbackand the switching controllers, is proposed The fuzzy rule-based combination technique combines theiradvantages of both approaches and eliminates their disadvantages Consequently, a fuzzy rule-base com-bined controller, which provides good system performance, guarantees global system stability, and has nochattering effect when the system state approaches the origin, can be obtained Furthermore, a saturationfunction is employed to replace the switching component to alleviate the chattering effect during thetransient period By using the proposed fuzzy rule-based combination technique, the steady state errorintroduced by the saturation function can be eliminated.

This paper is organized as follows In Section 2, the fuzzy plant model and the proposed fuzzy rule-basecombined controller will be presented In Section 3, the design of the fuzzy rule-base combined controllerand the stability analysis will be presented In Section 4, application examples will be given A conclusionwill be drawn in Section 5

2 Fuzzy plant model and fuzzy rule-base combined controller

We consider a multivariable fuzzy-model-based control system comprising a TSK fuzzy plant modeland a fuzzy rule-base combined controller connected in closed loop

2.1 TSK fuzzy plant model

Let p be the number of fuzzy rules describing the nonlinear plant subject to bounded parameter

uncertainties Theith rule is of the following format:

Rule i: IF f1(x(t)) is M1iand and f
(x(t)) is M
i

where M i

 is a fuzzy term of rule i corresponding to the function f  (x(t)),  = 1, 2, ,
, i =

1, 2, , p,
is a positive integer; Ai ∈ n×n and Bi ∈ n×m are known constant system and

in-put matrices, respectively; x(t) ∈  n×1 is the system state vector and u(t) ∈  m×1is the input vector.

The system dynamics are described by

and M i

 (f  (x(t))),  = 1, 2, , n, which denotes the grade of membership corresponding to the fuzzy

termM  i, is a nonlinear function of the system states and the parameter uncertainties

2.2 Fuzzy rule-base combined controller

The proposed fuzzy rule-base combined controller is a fuzzy rule-based combination of linear andswitching state-feedback controllers It has two rules in the following format:

Rule 2 : IF q(x(t)) is NZ THEN u(t) =

p

j=1

where rule 1 is for the linear state-feedback controller and rule 2 is for the switching state-feedback

controller; ZE (zero) and NZ (non-zero) are the fuzzy terms; G and Gj ∈ m×n,j = 1, 2, , p, are

the feedback gains; The value of n j (x(t)) governed by the switching scheme will be discussed later; q(x(t))
0 is a function of system state to be designed The membership function corresponding to ZEwill cover the region withq(x(t)) = 0 while that of NZ will cover the region with q(x(t)) > 0 The

inferred output of the fuzzy rule-base combined controller is given by

3 Stability analysis and design of fuzzy rule-base combined controller

In this section, the system stability analysis and the design of the feedback gains will be presented.From (2) and (7), writingw i (x(t)), n j (x(t)) and m k (x(t)) as w i,n j andm k respectively, and with the

There are two cases to investigate the stability of (9) and will be detailed as follows.

3.1 Uncertain input matrix

The fuzzy plant model of (2) is assumed to have the following property:

where Bm ∈ n×m is a constant matrix; (x(t)) is an unknown non-zero scalar (because w i (x(t)) is

unknown) but with single sign and known bounds, i.e | (x(t))| ∈ [ min max] It should be noted thatbecause (x(t)) = 0 is required, B(x(t)) = 0 is assumed From (9) and (10) and writing  (x(t)) as , wehave

ofw i To investigate the stability of (11), the following Lyapunov function candidate is considered:

iP + PHi ) are constant symmetric positive definite matrices;

min(·) denotes the minimum eigenvalue of the input argument Let min(Q − (
HTP + P
H))
 > 0

whereis a constant scalar and ·  denotes the l2vector or induced matrix norm[23] Let the switchinglaw ofnj be defined as,

Furthermore, ZE(q(x(t))) is designed such that min(Q − (
HTP + P
H))
 > 0 for the value of

ZE(q(x(t))) > 0 (its normalized grade of membership m1 = 0) Qi,i = 1, 2, , p is deigned to be

symmetric positive definite Hence, it can be concluded that ˙V0 (equality holds when x(t) = 0) This

implies that x(t) → 0 as t → ∞ The analysis result is summarized by the following theorem.

Theorem 1 The closed-loop system of (9), with B (x(t)) =p i=1 w i (x(t))B i = (x(t))B m and B m is a

constant matrix, is guaranteed to be asymptotically stable if there exists a constant symmetric matrix P

such that the following linear matrix inequalities hold:

with a constant K > 1 and |  (x(t))| ∈ [ min max] The membership function ZE(q(x(t))) is designed

such that min(Q−(
HTP +P
H))
 > 0 for the value of ZE(q(x(t))) > 0 where
H=p i=1 w i (A i−

Ao ) +p i=1 w i (B i− Bo )G.

3.2 Constant input matrix

The case that B(x(t)) =p i=1 w iBi = B which is a constant input matrix is considered From (9)

From (17) and (18), we have,

As K > 1 and w j ∈ [0 1], we have −K + w j < 0 for all j Hence, ˙V0 (equality holds when

x(t) = 0) This implies that x(t) → 0 as t → ∞ The analysis result is summarized by the following

where H= Ao+ BG; Hi = Ai+ BGi, Ao =p i=1 w i (0)A i are constant matrices The switching law

of the switching controller is designed as

n i = −Ksgn(x(t)TPBGix(t)), i = 1, 2, , p

with a constant K > 1 The membership function ZE(q(x(t))) is designed such that min(Q − (
ATP+

P
A))
 > 0 for the value of ZE(q(x(t))) > 0 where
A=p i=1 w i (A i− Ao ).

It should be noted that the membership functions of the fuzzy rule-base combined controller aredesigned such that m1 = 0 and m2 = 0 in the operating domain around the origin and m1 = 0 and

m2 = 0 in the operating domain far away from the origin Hence, in the operating domain that m1 = 0andm2 = 0, the switching state-feedback controller will be employed to drive the system state to theorigin When the system state inside the operating domain that both m1 andm2 = 0, both the linearand switching state-feedback controllers will be employed The ratio of contribution of each controller isdetermined by the values ofm1andm2 The linear state-feedback controller will become dominant andthe contribution of the switching state-feedback controller will vanish when the system state is inside theoperating domain around the origin (i.e.m1 = 0 and m2 = 0) Inside this domain, the chattering effectintroduced by the switching state-feedback controller totally disappears as only the linear state-feedbackcontroller takes place to handle the nonlinear plant To alleviate the chattering effect during the transientperiod, a saturation function will be employed to replace the sign function The saturation function is

T otherwise,

(20)

where T is a non-zero positive scalar to be designed The saturation function may introduce

steady-state error to the system steady-state The magnitude of the steady-steady-state error is related to the value ofT The

value of T should be designed such that the system state is able to be driven into the region with the

value ofZE(q(x(t))) > 0 ( min(Q − (
HTP + P
H))
 > 0 for the uncertain input matrix case or

min(Q − (
ATP + P
A))
 > 0 for the constant input matrix case inside this region) Once the system

states are inside this region, the linear state-feedback controller will gradually replace the switchingstate-feedback controller As a result, the chattering effect and the steady-state error will be eliminatedeventually when the linear state-feedback controller completely dominates the control process

The feedback gains of the linear and switching state-feedback controllers can be embedded in the LMIstability conditions of Theorems 1 and 2[6,20] Hence, the feedback gains can be obtained by solvingthe LMIs The details can be found in [6,20] and will not be reiterated in this paper

In this paper, a framework of the fuzzy rule-base combined controller, which combines the linear andswitching state-feedback controllers, is proposed for nonlinear systems subject to parameter uncertain-ties In general, any controller can be combined by using the fuzzy rule-based combination techniques.However, the main concern is to consider the characteristics of the controllers to be combined to ben-efit the control process Taking the fuzzy rule-based combination of linear and switching controllers as

an example, the linear state-feedback controller provides good local stabilization ability without ter effect and weaker robustness property while the switching state-feedback controller provides goodglobal stabilization ability with chattering effect and stronger robustness property It can be seen thatthe characteristics of these two controllers can complement the deficiencies of each other Hence, thefuzzy rule-based combination technique can be employed to combine their advantages which are good

chat-to the control process By employing the proposed fuzzy rule-based combination technique chat-to combinethe conventional controllers, the good characteristics of the conventional controllers can be combined toform a better solution for controlling the nonlinear systems However, it can be seen that the LMI stabilityconditions are derived under the proposed framework of the fuzzy rule-base combined controller Hence,the LMI stability conditions are needed to be derived for combining different kinds of controllers.The proposed fuzzy rule-base combined controller is subject to the following limitations: One, thenonlinear system should be represented by a fuzzy plant model in form of (2) Two, the input matrix

should exhibit the property that B(x(t)) =p i=1 w i (x(t))B i= (x(t))B mwhere Bmis a constant matrix

(Theorem 1) or B(x(t)) = p i=1 wi (x(t))Bi = B where B is a constant matrix (Theorem 2) If these

conditions are satisfied, Theorems 1 or 2 can be applied to design a fuzzy rule-base combined controllerfor the nonlinear systems

4 Application examples

Two application examples will be given in this section Their details are given as follows

M mg

x=

l

u

x .=  .

Fig 1 Cart-pole typed inverted pendulum.

4.1 Uncertain input matrix: inverted pendulum

An application example on stabilizing a cart-pole typed inverted pendulum[8] is given in this section.The proposed fuzzy rule-base combined controller will be employed to control the plant Fig 1 showsthe diagram of the inverted pendulum Its dynamic equation is given by

¨

 (t) = g sin(  (t)) − aml˙  (t)2sin(2  (t))/2 − a cos(  (t))u(t)

where (t) is the angular displacement of the pendulum, g = 9.8 m/s2is the acceleration due to gravity,

m ∈ [2 5] kg is the mass of the pendulum, M ∈ [8 20] kg is the mass of the cart, a = 1/(m + M),

2l = 1 m is the length of the pendulum, and u(t) is the force applied to the cart The objective of this

application example is to design a fuzzy rule-base combined controller for the nonlinear plant of (21)such that (t) = 0 at steady state It is reported in[8] that the nonlinear plant of (21) can be represented

by the following fuzzy rules:

Rule i: IF f1(x(t)) is M i

1ANDf2(x(t)) is M i

2

THEN˙x(t) = A ix(t) + B i u(t) for i = 1, 2, 3, 4 (22)

so that the system dynamical behavior is described by



, B2 = B4 =

0

f2max

and Bm=

01



;

f1min = 9 and f1max = 18, f2min = −0.1765 and f2max = −0.0468 of which f1minf1(x(t))f1max and

f2minf2(x(t))f2 maxthat can be obtained analytically;

2(f2(x(t))) = 1 − M1(f2(x(t))) for  = 2, 4 The fuzzy plant model

and its parameters are obtained by the algorithm which is detailed in[19]

The linear state-feedback controller of the proposed fuzzy rule-base combined controller is designedbased on the following system and input matrices:

linear state-feedback gain is designed as G = [258.0952 39.2857] such that H = A o + BoG The feedback gains of the switching state-feedback controller is designed as G1 = G2 = [−10 − 3] and

G3 = G4 = [−19 − 3] such that H = Hi = Ai + BmGi, i = 1, 2, 3, 4 The switching law is

defined as in Theorem 1 withK = 3.5 which is obtained by trial-and-error for good performance and

min = |f2max| = 0.0468 With the help of the MATLAB LMI toolbox, the solution to the stability

conditions of Theorem 1 can be solved numerically We have P=1.8333

such that the stability conditions of Theorem 1 are satisfied With the help of genetic algorithm [12],

it can be found that min(Q − (
HTP+
HP))
 = 0.3035 > 0 for |x1 (t)|0.25 and |x2(t)|0.5.

ZE(q(x(t))) and NZ(q(x(t))) are designed as shown in Fig 2 with

the system state is outside the domain, i.e.|x1(t)| > 0.25 and/or |x2(t)| > 0.5.

The linear and the switching state-feedback controllers will be combined to form the proposed fuzzyrule-base combined controller which will be applied to the nonlinear plant of (21) The system state

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2 0.4 0.6 0.8 1

To test the robustness property of the proposed fuzzy rule-base combined controller, the values ofm and

M are changed to 5 and 20 kg, respectively Fig 5 shows system state responses under the same initial

conditions withm = 5 kg and M = 20 kg Fig 6 shows the control signal under x(0) =22

45 0T

FromFigs 3 to 6, it can be seen that the inverted pendulum under different values ofm and M can be stabilized

0 1 2 3 4 5 6 7 8 9 10 -6000

-4000 -2000 0 2000 4000 6000 8000

x2

(b) (a)

Fig 5 System responses of the inverted pendulum with the proposed fuzzy rule-base combined controller underm = 5 kg and

M = 20 kg: (a) x1(t); (b) x2(t).

successfully Furthermore, when the system state approaches the origin, the chattering effect introduced

by the switching state-feedback controller disappears

For comparison purpose, a sliding-mode controller [2] given in Appendix A and a fuzzy controller[24] (which is a fuzzy rule-base combined linear state-feedback controllers) given in Appendix B will

be employed to handle the inverted pendulum Figs 7 and 8 show the system responses and the control

signals under x(0) = 22

45 0T

,m = 5 kg and M = 20 kg Referring to Fig 7, it can be seen that

the performance of sliding-mode controller is slightly better in terms of transient response time Thecontrol signal of the sliding-mode controller switches between±6000 N while that of the proposed fuzzy

rule-base combined controller lies in the range of±5000 N It should be noted that although both the

0 1 2 3 4 5 6 7 8 9 10 -6000

-4000 -2000 0 2000 4000 6000 8000

andM = 20 kg.

sliding-mode and the fuzzy rule-base combined controllers exhibit chattering effect during the transientperiod, however, the chattering effect will be totally eliminated in the fuzzy rule-base combined controllerwhen the system state nears the origin (It can be seen in Fig 6 that the chattering effect disappearsafter 2 s) Referring to Fig 8, it can be seen that the published fuzzy controller fails to stabilize theinverted pendulum A saturation function with T = 1 is employed to replace the sign function in the

proposed fuzzy rule-base combined controller to alleviate the chattering effect during the transient period.Figs 9 and 10 show the system responses and control signals of the proposed fuzzy rule-base combinedcontroller with saturation function It can be seen that the chattering effect in the system state and controlsignals are significantly alleviated The steady-state error introduced by the saturation function is alsoeliminated

4.2 Constant input matrix: single-link flexible joint

A single-link flexible joint shown in Fig 11 will be given as an application example The dynamicalequations [14] governing its behavior are as follows:

where m (t) and  l (t) denote the angular displacements of the rotor and the thin link respectively; J m=

0.4 kg m2denotes the rotor inertia;B m = 0.15 Nm s/rad denotes the rotor friction; m ∈ [10 11.5] kg and

l = 3 m denote the mass and length of the link, respectively; g = 9.8 m/s2 is the acceleration due togravity;k = 0.1 Nm/rad denotes the joint stiffness; g r = 10 denotes the gear ratio The objective of thisapplication example is to design the proposed fuzzy rule-base combined controller to stabilize the plant

Time (sec)

(c) Fig 7 System responses of the sliding-mode (solid lines) and the proposed fuzzy rule-base combined (dotted lines) controllers

and control signal of the sliding-mode controller for the inverted pendulum under x(0) =22

THEN˙x(t) = A ix(t) + Bu(t) for i = 1, 2, 3, 4 (27)

so that the system dynamical behavior is described by