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Switched Fuzzy Systems: RepresentationModelling, Stability Analysis,and Control Design

Hong Yang 1 , Georgi M. Dimirovski 2 ,3 , and Jun Zhao 1

1 Northeastern University, School of Information Science & Technology, 110004

Shenyang, Liaoning, Peoples Republic of [email protected] , [email protected] SS Cyril and Methodius University, School of Electrical Engineering & Info.

Technologies, 1000 Skopje, Karpos 2 BB, Republic of [email protected]

3 Dogus University of Istanbul, School of Engineering, Acibadem, Zeamet Sk 21,34722 Istanbul - Kadikoy, Republic of Turkey

Summary. Stability issues for switched systems whose subsystems are all fuzzysystems, either continuous-time or discrete-time, are studied and new results derived.

Innovated representation models for switched fuzzy systems are proposed. The singleLyapunov function method has been adopted to study the stability of this classof switched fuzzy systems. Sufficient conditions for quadratic asymptotic stabil-ity are presented and stabilizing switching laws of the statedependent form aredesigned. The elaborated illustrative examples and the respective simulation exper-iments demonstrate the effectiveness of the proposed method.

1 Introduction

The large class of switched systems has attracted extensive research duringthe last couple of decades both as such and also in conjunction of the evenlarger class of hybrid systems, e.g. see [2, 8, 13, 14, 21]. For, these systemshave a wide range of potential applications. For instance, such systems arewidely used in the multiple operating point control systems, the systems of power transmission and distribution, constrained robotic systems, intelligentvehicle highway systems, etc. Thus switched systems represent one of therather important types of hybrid systems. Basically all switched systems areconsisted of a family of continuous-time or discrete-time subsystems and aswitching rule law that orchestrates the switching among them.

Recently switched systems have been extended further to encompassswitched fuzzy systems too [12, 14, 20] following the advances in fuzzy slid-ing mode control [4, 9, 11] although for long time it was known that ideal

H. Yang et al.: Switched Fuzzy Systems: Representation Modelling, Stability Analysis, and Control Design , Studies in Computational Intelligence (SCI) 109 , 155168 (2008)www.springerlink.com c Springer-Verlag Berlin Heidelberg 2008


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156 H. Yang et al.

relay switching is a time optimal control law [17]. A switched fuzzy systeminvolves fuzzy systems among its sub-systems. The extension emanated out of the remarkable developments in theory, applications, and the industrial imple-mentations of fuzzy control systems, e.g. see [1,16,18,19,23], which exploited

Lyapunov stability theory.It appeared, the class of switched fuzzy systems can describe more precisely

both continuous and discrete dynamics as well as their interactions in complexreal-world systems. In comparison to either switched or fuzzy control systems,still few stability results on switched fuzzy control systems can be found inthe literature. For the continuous-time case, in [12] a combination of hybridsystems and fuzzy multiple model systems was described and an idea of thefuzzy switched hybrid control was put forward. For the discrete-time case,in [5], a fuzzy model whose subsystems are switched systems was described.

In this model switching takes place simply based on state variables or time.Subsequently the same authors gave some extensions to output [6] and toguaranteed-cost [7] control designs.

In here, an innovated representation modelling of continuous-time anddiscrete-time switched fuzzy systems is proposed. Sufficient conditions forasymptotic stability are derived by using the method of single Lyapunov func-tion and the parallel distributed compensation (PDC) fuzzy controller schemeas well as the stabilizing statedependent switching laws.

Further this study is written as follows. In Sect. 2, the representation

modelling problem has been explored and innovated models proposed for bothcontinuous-time and discrete-time cases; note, also both autonomous and non-autonomous system are observed in this study. Section 3 gives a thoroughpresentation of the new theoretical results derived. In Sect. 4, there are pre-sented the illustrative examples along with the respective simulation resultsto demonstrate the applicability and efficiency of the new theory. Thereafter,conclusion and references are given.

2 Novel Models of Switched Fuzzy Systems

In this paper, only TakagiSugeno ( T S ) fuzzy systems representing thecategory of data based models are considered. This representation differs fromexisting ones in the literature cited: each subsystem is a T S fuzzy systemhence dening an entire class of switched fuzzy systems. This class inheritssome essential features of hybrid systems [2,14] and retains all the informationand knowledge representation capacity of fuzzy systems [16].

2.1 The Continuous-Time Case

Consider the continuous T S fuzzy model that involves N ( t ) rules of thetype as

R l ( t ) : I f 1 is M ( t )1 and p is M l ( t ) p ,

Then x = A ( t ) l x(t) + B ( t ) l u ( t ) (t), l = 1 , 2, . . . , N ( t ) (1)


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Switched Fuzzy Systems 157

where : R+ M = {1, 2, , m} is a piecewise constant function and itis representing the switching signal. In rule-based model (1), Rl ( t ) denotesthe lth fuzzy inference rule, N ( t ) is the number of inference rules, u(t) is theinput variable, and the vector x(t) = [x1(t) x2(t) xn (t)]T Rn represents thestate variables. Vector = [ 1 2 p ] represents the vector of rule antecedents(premises) variables. In the linear dynamic model of the rule consequent,the matrices A ( t ) l Rn n and B ( t ) l Rn p are assumed to have theappropriate dimensions.

The ith fuzzy subsystem can be represented as follows:

R li : I f 1 is M li 1 and p is M lip ,Then x(t) = A il x(t) + B il u i (t),l = 1 , 2, . . . , N i , i = 1 , 2, . . . , m . (2)

Thus the global model of the ith fuzzy sub-system is described by means of the equation

x(t) =N i

l=1

il ( (t)) ( Ail x(t) + B il u i (t)) , (3)

together with

il (t) =

n

=1 M l ( t )

N i

l =1

n

=1 M l ( t )

, (4a)

0 il (t) 1,N i

l=1il (t) = 1 , (4b)

where M l ( t ) denotes the membership function of the fuzzy state variable xthat belongs to the fuzzy set M l .

2.2 The Discrete-Time Case

Similarly, we can dene the discrete switched T S fuzzy model includingN (k ) pieces of rules

R l (k ) : I f 1 is M l (k )1 and p is M

l (k ) p ,

T hen x (k + 1) = A (k ) l x(k) + B (k ) l u (k ) (k), l = 1 , 2, . . . , N (k ) (5)

where (k) : {0, 1, } {1, 2, , m} is a sequence representing switchingsignal {0, Z + } {1, 2, , m}.

In turn, the ith sub fuzzy system can be represented as follows:

R li : I f 1 is M li 1 and p is M lip ,T hen x (k + 1) = A il x(k) + B il u i (k), (6)

l = 1 , 2, . . . , N i , i = 1 , 2, m.


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158 H. Yang et al.

Therefore the global model of the ith sub fuzzy system is described by meansof the equation

x(k + 1) =N i

l =1

il

(k) (Ail

x(k) + Bil

ui(k)) , (7)

together with

il (k) =

n

=1M l (k )

N i

l=1

n

=1M l (k )

, 0 il (k) 1,N i

l=1

il (k) = 1 . (8)

The representation modeling section is thus concluded.

3 New Stability Results for Switched Fuzzy Systems

First the respective denition for quadratic asymptotic stability of switchednonlinear systems, e.g. see [2,14,2426], and a related lemma in conjunctionwith stability analysis re recalled.

Denition 3.1. The system (1) is said to be quadratic stable if there exist a positive denite matrix P and a state-dependent switching law = (x)such that the quadratic Lyapunov function V (x(t)) = xT (t)P x (t) satises ddt V (x(t)) < 0 for any x(t) = 0 along the system state trajectory from arbitrary initial conditions.

Remark 3.1. A apparently analogous denition can be stated for the discrete-time case of system (5), and it is therefore omitted.

Lemma 3.1. Let aij i (1 i m, 1 j i N i ) be a group of constants

satisfying m

i =1

a ij i < 0, 1 j i N i .

Then, there exists at least one i such that

a ij i < 0, 1 j i N i .

Proof. It is trivial and therefore omitted.

3.1 Stability of Continuous-Time Switched Fuzzy Systems

First, the novel stability result for systems (1) with u 0 in the fuzzy systemrepresentation is explored.


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Theorem 3.1. Suppose there exist a positive denite matrix P and constants ij l 0, i = 1 , 2, . . . , m, j i = 1 , 2, . . . , N i such that

N i

i =1

ij i

(AT ij i

P + P Aij i

) < 0, ji = 1 , 2, N

i, (9)

then the system (1) is quadratic stable under the switching law:

(k) = arg min {V i (x) = max

j i{xT (AT ij i P + P A ij i )x < 0, j i = 1 , 2, N i }} (10)

Proof. From inequality (9) it may well be inferred that

N i

i =1 iji x

T

(AT ij i P + P A ij i )x < 0, j i =1 , 2, N i (11)

for any x = 0. Notice that (11) holds true for any j i {1, 2, N i } and ij 0. On the other hand, Lemma 3.1 asserts that there exists at least onei such that

xT (AT ij i P + P A ij i )x < 0, (12)

for any j i . Thus, the switching law (10) is a well-dened one. Next, thetime derivative of the respective quadratic [8, 10, 15, 19, 25] Lyapunov func-tion V (x(t)) = xT (t)P x (t), is to be calculated:

ddt

V (x(t)) = xT N i

l =1

il Ail

T

P + P N i

l=1

il Ail x

=N i

l =1

il xT AT il P + P A il x

Notice, here i = (x) is generated by means of switching law (10). By tak-

ing (4), (12) into account, one can deduce that dV (x(t)) /dt < 0, x = 0.Hence system (1) is quadratic stable under switching law (10), which ends upthis proof.

Now the stability result for the more important case with u = 0, is pre-sented. It is pointed out that the parallel distributed compensation (PDC)method for fuzzy controller design [22,23] is used for every fuzzy sub-system.It is shown in the sequel how to design controllers to achieve quadratic sta-bility in the closed loop and under the switching law.

Namely, local fuzzy controller and system (2) have the same fuzzy inferencepremise variables:

R lic : I f 1 is M li 1 and p is M

lip ,

T hen u i (t) = K il x(t), l = 1 , 2, N i , i = 1 , 2, m. (13)


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160 H. Yang et al.

Thus, the global control is

u i (t) =N i

l =1

il K il x(t). (14)

Then the global model of the ith sub fuzzy system is described by meansof the following equation:

x =N i

l=1

il (t)N i

r =1

ir (Ail + B il K ir )x(t). (15)

Theorem 3.2. Suppose there exist a positive denite matrix P and constants ij l 0, i = 1 , 2, . . . , m, i = 1 , 2, . . . , N i such that

N i

i =1 ij i (Aij i + B ij i K i i )T P + P (Aij i + B ij i K i i ) < 0, j i ,i = 1 , 2, N i ,

(16)

Then, the system (1) along with (13)(14) is quadratic stable under the switch-ing law:

(x) = arg min {V i (x) = max

j i , i{xT [(Aij i + B ij i K i i )T P

+ P (Aij i + B ij i K i i )]x < 0, j i ,i = 1 , 2, N i }} (17)

Proof. It is seen from (12) thatN i

i =1

ij i xT (Aij i + B ij i K i i )T P + P (Aij i + B ij i K i i ) x < 0

j l ,l = 1 , 2, . . . , N i . (18)

for any x = 0. Further, it should be noted that (18) holds for any j i ,i {1, 2, N i } and ij i 0. By virtue of Lemma 3.1, there exists at least one isuch that

xT (Aij i + B ij i K i i )T P + P (Aij i + B ij i K i i ) x < 0

for any j i ,i . Thus the switching law (17) is a well-dened one. Via similarcalculations as in Theorem 3.1 using Lyapunov function V (x) = xT (t)P x(t),on can nd:

dV (x(t))dt

= xT (t)P x (t) + xT (t)P x(t)

=N i

l=1

N i

r =1

il ir xT (t) (Ail + B il K ir )T P + P (Ail + B il K ir ) x(t)

in which i = (x) is given by law (17). Thus, system (1) along with (13)(14)is quadratic stable under switching law (17), and this completes the proof.


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3.2 Stability of Discrete-Time Switched Fuzzy Systems

Again rst the case when u = 0 in system (5) is considered.

Theorem 3.3. Suppose there exist a positive denite matrix P and constants ij l 0, i = 1 , 2, . . . , m, i = 1 , 2, . . . , N i such that

N i

i =1

ij i (AT ij i P A i i P ) < 0, j i ,i = 1 , 2, N i , (19)

then the system (5) is quadratic stable under the switching law:

(k) = arg min {V i (k) = max

j i , i{xT (AT ij i P A i i P )x < 0, j i ,i = 1 , 2, N i }}

(20)

Proof. Similarly as for Theorem 3.1, from (19) it is inferred that

N i

i =1

ij i xT (AT ij i P A i i P )x < 0, j i ,i =1 , 2, N i

for any x = 0. Then there exists at least an i such that

xT

(AT ij i P A i i P )x < 0,

for any j i , i {1, 2, N i }. Now the time derivative of Lyapunov functionV (x(t)) = xT (t)P x (t) is calculated to give:

V (x(k)) = V (x(k + 1)) V (x(k))

= xT (k)N i

l=1

il (k)Ail

T

P N i

r =1

i (k)Ai P x(k)

=N i

l=1

ilN i

r =1

i xT (k) AT il P A i P x(k)

Hence the system (5) is quadratic stable under switching law (20), and thiscompletes the proof.

The more important case with u = 0 in system (5) is considered next.And again the PDC method for fuzzy controller design is used for every fuzzysub-system. Namely, it is observed that local fuzzy control and system (6)

have the same fuzzy inference premise variables:

R lic : I f 1 is M li 1 and p is M

lip ,

Then u i (k) = K il x(k), l = 1 , 2, N i , i = 1 , 2, m. (21)


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162 H. Yang et al.

Thus, the global control is

u i (k) =N i

l =1

il K il x(k). (22)

Then the global model of the ith sub fuzzy system is described by:

x(k + 1) =N i

l =1

il (k)N i

r =1

i (Ail + B il K i )x(k). (23)

Theorem 3.4. Suppose there exist a positive denite matrix P and ij l 0,i = 1 , 2, . . . , m, i = 1 , 2, . . . , N i such that

N i

i =1

ij i (Aij i + B ij i K i i )T P (Aip i + B ip i K iq i ) P < 0

j i ,i , pi , q i = 1 , 2, N i (24)

Then, the system (5) along with (21)(22) is quadratic stable under the switch-ing law:

(k) = arg min {V i (k) = max

j i , i ,p i ,q i{xT [(Aij i + B ij i K i i )

T P (Aip i + B ip i K iq i )

P ]x < 0, j i ,i , pi , q i = 1 , 2, N i }}(25)

Proof. It is very similar to that of Theorem 3.3 and thus omitted.

4 Illustrative Examples and Simulation Results

Results applying the above developed theory on two examples, one for thecontinuous-time and one the discrete-time case, and the respective simulations(e.g. using MathWorks software [27]) are given below.

4.1 A Continuous-Time Case of Autonomous System

Consider a continuous-time switched fuzzy system described as follows:

R11 : If x is M 111 , Then x(t) = A11 x(t)

R21 : If x is M 211 , Then x(t) = A12 x(t)

R12 : If y is M 121 , Then x(t) = A21 x(t)

R22 : If y is M 221 , Then x(t) = A22 x(t)


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where

A11 = 17 0.05674.93 0.983 , A12 =

10 0.216 0.0132 45.29 ;

A21 = 50 0.0420.008 10 , A22 = 40 0.08670.047 120 .

Above, the fuzzy sets M 111 , M 211 , M 121 , M 221 , respectively, are represented bymeans of the following membership functions:

111 (x) = 1 1

1 + e 2x, 211 (x) =

11 + e 2x

;

121 (y) = 1 1

1 + e( 2( y 0 .3)), 221 (y) =

11 + e( 2( y 0 .3))

.

For2

i =1

ij i (AT ij i P + P A ij i ) < 0, j i = 1 , 2

by choosing ij i = 1, one can nd (e.g. ultimately by using LMI toolbox) thefollowing P matrix

P = 0.0159 0.00010.0001 0.0071 .

Then system is quadratic asymptotically stable under the following switch-ing law

(x) = arg min {V i (x) = max

j i{xT (AT ij i P + P A ij i )x < 0, j i = 1 , 2}}.

Figure 1 above depicts the obtained simulation results for the controlled evo-lution of system state variables when, at the initial time instant, the systemis perturbed by the state vector x(0) = [2 2]T .

4.2 A Cases of Discrete-Time Non-Autonomous System

Now, let consider a discrete-time switched fuzzy system that is represented asfollows:

R11 : If x(k) is M 111 , Then x (k + 1) = A11 x(k) + B11 u(k),

R21 : If x(k) is M 211 , Then x (k + 1) = A12 x(k) + B12 u(k),

R12 : If y(k) is M 121 , Then x (k + 1) = A21 x(k) + B21 u(k),

R22 : If y(k) is M 221 , Then x (k + 1) = A22 x(k) + B22 u(k),


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164 H. Yang et al.

Fig. 1. The evolution time-histories for system state variables x 1 (t ), x 2 (t ) after theperturbation by x(0) = [2 2] T at the initial time instant

where

A11 = 0 1

0.0493 1.0493 , A12 = 0 1

0.0132 0.4529 ,

A21 = 0 1

0.2 0.1 , A22 = 0.2 1 0.8 0.9 ;

B11 = 0

0.4926 , B21 =01 , B12 =

00.1316 , B22 =

01 .

The above fuzzy sets M 111 , M 211 , M 121 , M 221 , respectively, are represented bymeans of the following membership functions:

111 (x(k)) = 1 11 + e 2x (k ), 211 (x(k)) = 11 + e 2x (k )

,

121 (y(k)) = 1 1

1 + e( 2( y (k ) 0 .3)), 221 (y(k)) =

11 + e( 2( y (k ) 0 .3))

.

The state feedback gains of subsystems are obtained as

K 11 = [ 0.131 0.1148], K 12 = [ 0.0623 2.302],K 21 = [1.8 1.9] , K 22 = [ 0.7 1.3].

For2

i =1

ij i (Aij i + B ij i K i i )T P (Aip i + B ij i K iq i ) P < 0, j i ,i , pi , q i = 1 , 2,


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Fig. 2. The evolution time-histories for system state variables x 1 (k ), x 2 (k ) afterthe perturbation by x(0) = [5 1] T at the initial time instant

by choosing ij i = 1, for the P matrix one can obtain the following result:

P = 0.3563 0.0087

0.0087 0.1780.

Then, the system is asymptotically stable under the following switch-ing law

(k ) = arg min {V i (k ) = max

j i , i ,p i ,q i{x T [(A ij i + B ij i K i i )

T P (A ip i + B ip i K iq i ) P ]x < 0,

j i , i , p i , q i = 1 , 2}} .

Figure 2 below depicts the obtained simulation for the transients of controlledsystem state variables when, at the initial time instant, the system is per-turbed by the state vector x(0) = [5 1]T ; the sampling period was setup toT s = 0 .05s.

As known from the literature, the combined switching control along withthe state feedback gains does produce a reasonably varying control effort (seeFig. 3) that can be sustained by actuators.

5 Conclusion

Innovated representation models for the class of switched fuzzy systems, bothcontinuous-time and discrete-time, based on TakagiSugeno fuzzy models wereproposed. For both cases, new sufficient conditions for quadratic asymptoticstability of the control system with the given switching laws (Theorems 3.1


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166 H. Yang et al.

0.140 5 10 15

0.12

0.1

0.08

0.06

0.04

0.02

0

0.02

Fig. 3. The evolution time-history of the switching based state-feedback gain controlof the plant system perturbed by x(0) = [5 1] T at the initial time instant

thru 3.4) have been derived via the common Lyapunov function approach. Fol-lowing these new stability results, only the stability of a certain combinationof subsystem matrices has to be checked, which is easier to carry out.

On the grounds of introducing the appropriate switching laws the stabi-lizing control in the state-variable dependent form has been synthesized forboth these cases of fuzzy switched systems. Simulation results demonstratethat a control performance of considerable quality has been achieved in theclosed loop thus promising the applicability in real-world problems.

The twofold future research is envisaged towards, rstly, reducing the con-servatism of the obtained theorems and, secondly, towards deriving more so-phisticated thus delicate switching based control laws.

Acknowledgments

The authors are grateful to their respective university institutions for thecontinuing support of their academic co-operation. In particular, Georgi M.Dimirovski would like to express his special thanks to his distinguished friendsLot A. Zadeh, the legend of fuzzy systems, and Janusz Kacprzyk, renown forhis fuzzy based decisions support systems, for their rather useful suggestions.Also, he is thankful to the young colleague Daniel Liberson whose discussion

regarding the switching control helped to clarify the presentation in thispaper.


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