Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. IV (2009), No. 4, pp. 415-426

Fuzzy Logic Control System Stability Analysis Based on Lyapunov’s
Direct Method

R.-E. Precup, M.-L. Tomescu, Şt. Preitl

Radu-Emil Precup, Ştefan Preitl
"Politehnica" University of Timişoara
Department of Automation and Applied Informatics
Bd. V. Parvan 2, RO-300223 Timişoara, Romania
E-mail: {radu.precup, stefan.preitl}@aut.upt.ro

Marius-Lucian Tomescu
"Aurel Vlaicu" University of Arad
Computer Science Faculty
Complex Universitar M, Str. Elena Dragoi 2, RO-310330 Arad, Romania
E-mail: tom uav@yahoo.com

Abstract: A stability analysis method for nonlinear processes controlled by Takagi-
Sugeno (T-S) fuzzy logic controllers (FLCs) is proposed. The stability analysis of
these fuzzy logic control systems is done in terms of Lyapunov’s direct method. The
stability theorem presented here ensures sufficient conditions for the stability of the
fuzzy logic control systems. The theorem enables the formulation of a new stability
analysis algorithm that offers sufficient stability conditions for nonlinear processes
controlled by a class of T-S FLCs. In addition, the paper includes an illustrative
example that describes one application of this algorithm in the design of a stable
fuzzy logic control system.
Keywords: fuzzy logic controller, LaSalle’s invariance principle, Lyapunov function
candidate.

1 Introduction

Fuzzy logic controllers have been proposed for a long time and applied successfully in many applica-
tions [1, 2, 3, 13, 14, 18]. A comprehensive work on the proof of stability of fuzzy logic control systems
represents one of the challenges in fuzzy control [6, 12, 16, 17]. This paper presents a new stability
analysis method for fuzzy logic control systems comprising nonlinear processes and T-S FLCs. The
advantages of this method with respect to the state-of-the-art result from its specific features. First, it is
different to Lyapunov’s theorem in several important aspects and allows more applications. In particular,
it is well-suited to controlling processes where the derivative of the Lyapunov function candidate is not
negative definite. Therefore Lyapunov’s direct method can cope with fuzzy control of a wide area of
nonlinear dynamic systems. Second, the stability of the closed-loop system is guaranteed by the stability
in each active region of the fuzzy rules. So making use of the proposed stability analysis approach deter-
mines the inserting of new fuzzy rules become very easy because just the fulfillment of one condition in
the stability analysis theorem is needed.

The paper discusses the following topics. Section 2 deals with the description of the accepted class
of fuzzy logic control systems. The proposed stability analysis method focused on a stability theorem
based on Lyapunov’s direct method and the new stability analysis algorithm that guarantees the stability
of fuzzy logic control systems are presented in Section 3. Next, Section 4 offers a simple example to
validate the theoretical part suggesting ways of applying the proposed algorithm. The conclusions are
drawn in Section 5.

Copyright c© 2006-2009 by CCC Publications



416 R.-E. Precup, M.-L. Tomescu, Şt. Preitl

output 

y(t) 

x(0) initial 

conditions 

 

u(t) 

Reference input 

r 
Process 

 !  !  !tuxbxfx "#  

state variables 

(x1,x2,…,xn) 

Figure 1: Fuzzy logic control system structure.

2 Fuzzy Logic Control Systems

The structure of a fuzzy logic control system consisting of a process controlled by an FLC is pre-
sented in Figure 1. Let X be the universe of discourse and consider a single-input n-th order nonlinear
system of the following form representing the state-space equations of the controlled process:

ẋ = f (x) + b (x) u, x (t) = x, (1)

where:

− x ∈ X , x = [x, x, ..., xn]T is the state vector;

− f (x) = [ f (x) , f (x) , ..., fn (x)]
T , b (x) = [b (x) , b (x) , ..., bn (x)]

T are functions describing the dy-
namics of the process, f , b : D → Rn are locally Lipschitz maps from a domain D ⊂ Rn into Rn;

− u is the control signal applied to the process input;

− the time variable, t, has been omitted to simplify the further formulation;

− x (t) is the initial state at time t.

The i-th fuzzy (control) rule in the rule base of the T-S FLC base is of the form (2):

Rule i : IF x IS Xi, AND x IS Xi, AND ... AND xn IS Xi,n
THEN u = ui (x) , i = , r, r ∈ N∗, (2)

where r is the total number of rules, Xi,, Xi,, .., Xi,n are fuzzy sets that describe the linguistics terms (LTs)
of the input variables xk, k = , n, u = ui (x) is the control signal of rule i, similar to the case of parallel
distributed compensation, and the function AND is a t-norm. ui can be a single value or a function of the
state vector, x.

The structure presented in Figure 1 cam be viewed as a nonlinear state-feedback control system.
However other input variables (to the FLC) can be considered as well instead of the state variables xk, k =
, n. One simple design of the fuzzy logic control system can be done in terms of parallel distributed
compensation.

Each fuzzy rule generates the firing strength defined in (3):

αi (x)= AND (µi, (x) , µi, (x) . . . µi,n (xn)) ∈ [, ] ,∀x ∈ X , i = , r. (3)



Fuzzy Logic Control System Stability Analysis Based on Lyapunov’s Direct Method 417

It is assumed that for any x ∈ X there exists among all rules at least one αi ∈ (, ] , i = , r. The control
signal u is a function of αi and ui. Applying the weighted sum defuzzification method the output of the
FLC is given by

u =

r∑
i=

αiui
r∑

i=
αi

. (4)

Definition 1. For any input x ∈ X if the firing strength αi (x) corresponding to the fuzzy rule i is zero,
that fuzzy rule i, i = , r, is called an inactive fuzzy rule for the input x; otherwise, it is called an active
fuzzy rule.

It should be noted that with x = x an inactive fuzzy rule will not affect the controller output u (x).
Hence (4) can be rewritten as follows aiming the consideration of all active fuzzy rules only:

u (x) =

r∑
i=,αi 6=

αi (x) ui (x)

r∑
i=,αi 6=

αi (x)
. (5)

Definition 2. An active region of the fuzzy rule i is defined as a set

X Ai = {x ∈ X |αi (x) 6= } , i = , r. (6)

3 Stability Analysis

The stability analysis presented in this paper is based on LaSalle’s invariance principle cited and
analyzed in [11]. This Section is concentrated on the formulation and proof of Theorem 1 that ensures
sufficient conditions for the stability of nonlinear processes controlled by T-S FLCs.

The Lyapunov function candidate V : Rn → R, V (x) = xT Px is considered. It is positive and un-
bounded, where P ∈ Rn×n is a positive definite matrix. Considering the state trajectories fulfilling (1) in
order to obtain the closed-loop system dynamics, it results that V has continuous partial derivatives and
the derivatives of V with respect to time expressed in terms of (7):

V̇ (x) = ẋT Px + xT Pẋ = ( f (x) + b (x) u (x))T Px+
+xT P ( f (x)+ b (x)u (x)) = F (x) + B (x) u (x) ,

(7)

where:
F (x) = f (x)T Px + xT P f (x) , B (x) = b (x)T Px + xT Pb (x) . (8)

The following sets are defined to be used in the stability analysis:

B = {x ∈ X |B (x) =  } , B+ = {x ∈ X |B (x) >  } , B− = {x ∈ X |B (x) <  } . (9)

The main result is given by the following Theorem.

Theorem 3. Let the process be described by (1) with x =  ∈ Rn an equilibrium point. If there exists a
function V : Rn→ R, V (x)= xT Px, P∈ Rn×n, positive definite, unbounded and fulfilling 1, 2 and 3:

1. F (x) ≤ ,∀x ∈ B,

2. ui (x) ≤ − F(x)B(x) for x ∈ X Ai ∩B+ and ui (x) ≥ −
F(x)
B(x) for x ∈ X Ai ∩B−, i = , r,



418 R.-E. Precup, M.-L. Tomescu, Şt. Preitl

3. the set
{

x ∈ X
∣∣V̇ (x) = 

}
contains no state trajectories except the trivial one, x (t) =  for t ≥ ,

then the closed-loop system composed by the T-S FLC and the process (1) will be globally asymptotically
stable in the sense of Lyapunov at the origin.

Proof. By the definition of V it results that V () = , V (x) > ,∀x 6=  and V (x) = xT Px → ∞ as
‖x‖ → ∞. Further on, it will be proved that V̇ is negative semi-definite with respect to time employing
(7). An arbitrary initial state vector x ∈ X is accepted. Then the following three cases are possible.

Case 1: B (x) is strictly positive. From the condition 2 of Theorem 1 it results that:

ui (x) ≤ −
F (x)
B (x)

⇒ u (x) =

r∑
i=,αi 6=

αi (x) ui (x)

r∑
i=,αi 6=

αi (x)
≤

−
F(x)
B(x)

r∑
i=,αi 6=

αi (x)

r∑
i=,αi 6=

αi (x)
= −

F (x)
B (x)

⇒

⇒ V̇ (x) = F (x) + B (x) u (x) ≤ F (x) + B (x)
(

−
F (x)
B (x)

)
= . (10)

Therefore,

ui (x) ≤ −
F (x)
B (x)

⇒ V̇ (x) ≤ . (11)

Case 2: B (x) is strictly negative. Once more, from the condition 2 of Theorem 1 it results that

ui (x) ≥ −
F (x)
B (x)

⇒ u (x) =

r∑
i=,αi 6=

αi (x) ui (x)

r∑
i=,αi 6=

αi (x)
≥

−
F(x)
B(x)

·
r∑

i=,αi 6=
αi (x)

r∑
i=,αi 6=

αi (x)
= −

F (x)
B (x)

⇒

⇒ V̇ (x) = F (x) + B (x) u (x)≤ F (x) + B (x)
(

−
F (x)
B (x)

)
= . (12)

Therefore,

ui (x) ≥ −
F (x)
B (x)

⇒ V̇ (x) ≤ . (13)

Case 3: x ∈ B. In this case using the condition 1 in Theorem 1 the result will be F (x) ≤ . Hence,

V̇ (x) = F (x) + B (x) u (x) = F (x) ≤ . (14)

From the above three cases it is obtained that

V̇ (x) ≤ ,∀x ∈ X . (15)

In conclusion, the derivative with respect to time of the Lyapunov function candidate, V̇ , is negative
semi-definite.

The condition 3 ensures the fulfilment of LaSalle’s invariance principle. This justifies the fact that
the equilibrium point at the origin is globally asymptotically stable.

The proof is now complete.

The stability theorem presented here ensures sufficient conditions for the stability of the fuzzy logic
control system described in Section 2. So it has been proved that if the Lyapunov function candidate is
negative semi-definite in the active region of each fuzzy rule then, the closed-loop system will be globally
asymptotically stable in the sense of Lyapunov.



Fuzzy Logic Control System Stability Analysis Based on Lyapunov’s Direct Method 419

The conditions 1 and 2 in Theorem 1 guarantee that the function V̇ is negative semi-definite in the
active region of each fuzzy rule. The condition 3 proves that the set {} is the largest invariance set in{

x ∈ X
∣∣V̇ (x) = 

}
. By LaSalle’s invariance principle it has been guaranteed that the fuzzy logic control

system, comprising the nonlinear process described by (1) and the T-S FLC, is globally asymptotically
stable in the sense of Lyapunov at the origin.

The stability analysis algorithm ensuring the stability of the class of fuzzy logic control systems
considered in Section 2 is based on Theorem 1. It consists of the following steps:

1. Set the Lyapunov function candidate V (i.e. set P).

2. Check that the set
{

x ∈ X
∣∣V̇ (x) = 

}
contains no state trajectories except the trivial one, x (t) = 

for t ≥ .
3. Determine F (x) , B (x) , B, B−, B+.

4. If F (x) ≤ ,∀x ∈ B then go to step 5. Else go to step 1.
5. For each fuzzy control rule i determine ui such that ui (x) ≤ − F(x)B(x) for x ∈ X Ai ∩B+ and ui ≥ −

F(x)
B(x)

for x ∈ X Ai ∩B−, i = , r.
The application of this algorithm will be illustrated in the next Section.

4 Illustrative example

This Section is dedicated to the validation of the theoretical results derived in Section 3 by the de-
sign of a stable fuzzy logic control system with T-S FLC controlling a nonlinear process, the inverted
pendulum on a cart system. This simple mechanical system is representative to model a class of attitude
control problems whose goal is to maintain permanently the desired vertically oriented position. Since
the inverted pendulum is a nonlinear system, the basic balance equations for the system are derived firstly
and put into the standard state-space form. Given an inverted pendulum mounted on a cart as shown in
Figure 2, the first principle nonlinear equations are applied in the sequel. Assuming that the rod is mass-
less and that the cart mass and the point mass at the upper end of the inverted pendulum are denoted as M
and m, respectively, there is an externally x-directed force on the cart, F(t), and the gravity force acts on
the point mass at all times. The coordinate system is defined according to Figure 2, where x(t) represents
the cart position and θ (t) is the tilt angle referenced to the vertical upward direction.

Figure 2: Variables related to the inverted pendulum on a cart system.

The differential equation that describes the behavior of the simplified system, playing the role of
controlled process, is usually written as

(m + M)·l · θ̈ − (m + M)·l ·g·sin (θ ) = −u, (16)



420 R.-E. Precup, M.-L. Tomescu, Şt. Preitl

where:

M - the mass of the cart,

m - the mass of the pendulum,

l - the length of pendulum (distance to the center of mass),

x - the cart position coordinate,

θ - the pendulum angle with respect to the vertical position,

u - the control signal, equal to the externally x-directed force, u = F .

The state vector consists of the angle, θ , and the angular velocity of the pendulum, θ̇ . Therefore, the
two state variables are defined as z and z, where z ∈ [−, ], z ∈ [−, ], z (t) = θ (t) and
z (t) = θ̇ (t). In order to write equation (19) in terms of state variables, they are substituted resulting in

ż = f (z) + b (z) u, (17)

where: z =
[

z
z

]
- state vector, f (z) =

[
z

g
l sin (z)

]
, b (z) =

[


− 
(m+M)l

]
.

The goal of fuzzy logic control system design, to be presented as follows, is to ensure the upright
stabilization of the pendulum aiming the setpoint value of z, z = . The design starts with setting the
fuzzification module of the T-S FLC. Figures 3 and 4 illustrates the membership functions corresponding
to the LTs of the linguistic variables z and z. The three LTs representing Positive, Zero and Negative
values are noted by P, Z and N, respectively.

Figure 3: Membership functions of z.

Figure 4: Membership functions of z.



Fuzzy Logic Control System Stability Analysis Based on Lyapunov’s Direct Method 421

The inference engine of the FLC employs the MIN and MAX operators and it is assisted by the
complete rule base illustrated in Table 1. The weighted sum defuzzification method is used in the T-S
FLC structure. Summarizing, the only parameters to be calculated are the consequents ui in the 9 fuzzy
control rules.

Table 1
Fuzzy Control Rule Base

Rule Antecedent Consequent
z z u

1 P P u
2 N N u
3 P N u
4 N P u
5 P Z u
6 N Z u
7 Z P u
8 Z N u
9 Z Z u

The algorithm presented in Section 3 will be applied as follows in order to find the values of ui for which
the system (19) can be stabilized by the above described T-S FLC.

Step 1: The Lyapunov function candidate V (z) = zT Pz = z + z

 is considered, where P =

(
 
 

)
.

Therefore V is positive. If ‖z‖ → ∞ then V (z) → ∞. V (z) > ,∀z 6= .
Step 2: The derivative is V̇ (z)= z

(
z +

g
l sin (z) −


(m+M)l u

)
and V̇ () = . Assume that there is a

trajectory with z (t) =  and z (t) 6= . Then ddt z (t) =
g
l sin (z (t)) −


(m+M)l u (t) 6= , which means that

z (t) can not stay constant. Hence, z (t) =  is the only possible state trajectory for which V̇ (z) = . So
the set

{
z ∈ X

∣∣V̇ (z) = 
}

contains no trajectory of the system except the trivial trajectory z (t) =  for
t ≥ .
Step 3: The expressions of F and B are:

F (z) = z
(

z +
g
l

sin (z)
)

, B (z) = −
z

(m + M) l
. (18)

The following elements necessary in Theorem 1 obtain the particular values expressed in (23) to (26):

B = {(z, )∈ X |z∈ [−, ] } , B+= {(z, z)∈ X |z <  } , B−= {(z, z)∈ X |z >  } , (19)

−
F (z)
B (z)

= l (m + M) (zl + g sin (z)) . (20)

Step 4: If z ∈ B then z =  and F (z) = .
Step 5: Each rule will be analyzed further on. This is not a complex task since only 9 rules are involved.

For rule 1: z IS P, z IS P. So X A = (, ] × (, ], X A ∩ B+ = /0 and X A ∩ B− = (, ] × (, ].
Thus, u (z) ≥ − F(z)B(z) = l (m + M) (zl + g sin (z)). It is taken u (z) = l (m + M) (zl + g), and this func-
tion fulfills the condition 2 in Theorem 1.

For rule 2: z IS N, z IS N. So X A = [−, )× [−, ), X A ∩ B+ = [−, )× [−, ) and X A ∩
B− = /0. Thus u ≤ − F(z)B(z) = l (m + M) (zl + g sin (z)). It is taken u (z) = l (m + M) (zl − g), and this
function fulfills the condition 2 in Theorem 1.



422 R.-E. Precup, M.-L. Tomescu, Şt. Preitl

For rule 3: z IS P, z IS N. So X A = (, ]×(−, ], X A ∩B− = /0 and X A ∩B+ = (, ]×(−, ].
Thus u (z) ≤ − F(z)B(z) = l (m + M) (zl + g sin (z)). It is taken u (z) = −z, and this function fulfills again
the given condition.

For rule 4: z IS N, z IS P. So X A = [−, )×(, ], X A ∩B+ = /0 and X A ∩B− = [−, )×(, ].
Thus u (z) ≥ − F(z)B(z) = l (m + M) (zl + g sin (z)). It is taken u (z) = −z, and this function fulfills the
given condition.

For rule 5: z IS P, z IS Z. So X A = (, ]×(−, ). Thus, two possible cases will occur:

a) for z ∈ X A ∩B− = (, ]×[, ) ⇒ u (z) ≥ − F(z)B(z) = l (m + M) (zl + g sin (z)) and

b) for z ∈ X A ∩B+ = (, ]×(−, ] ⇒ u (z) ≤ − F(z)B(z) = l (m + M) (zl + g sin (z)).

In order to satisfy both conditions it is chosen u (z) = l (m + M) (zl + g sin (z)).
For rule 6: z IS N, z IS Z. So X A = [−, )×(−, ). The result will be:

a) for z ∈ X A ∩B− = [−, )×(, ) ⇒ u (z) ≥ − F(z)B(z) = l (m + M) (zl + g sin (z)) and

b) for z ∈ X A ∩B+ = [−, )×(−, ) ⇒ u (z) ≤ − F(z)B(z) = l (m + M) (zl + g sin (z)).

In order to satisfy both conditions it is chosen u (z) = l (m + M) (zl + g sin (z)).
For rule 7: z IS Z, z IS P. So X A = (−, ) × (, ], X A ∩ B+ = /0 and X A ∩ B− = (−, ) ×

(, ] . Thus u (z) ≥ − F(z)B(z) = l (m + M) (zl + g sin (z)). It is set u (z) = l (m + M) (zl + g) to fulfill
the condition 2 in Theorem 1.

For rule 8: z IS Z, z IS N. So X A = (−, ) × [−, ), X A ∩ B+ = (−, ) × [−, ) and
X A ∩ B− = /0. Thus u (z) ≤ − F(z)B(z) = l (m + M) (zl + g sin (z)). It is set u (z) = l (m + M) (zl − g) to
fulfill the condition 2 in Theorem 1.

For rule 9: z IS Z, z IS Z. So X A = (−, )×(−, ). This will yield:

a) for z ∈ X A ∩B+ = (−, )×(−, ) ⇒ u (z) ≤ − F(z)B(z) = l (m + M) (zl + g sin (z)) and

b) for z ∈ X A ∩B− = (−, )×(, ) ⇒ u (z) ≥ − F(z)B(z) = l (m + M) (zl + g sin (z)).

In order to satisfy both conditions it is set u (z) = l (m + M) (zl + g sin (z)).

Concluding, from Theorem 1 it results that the closed-loop system composed by the nonlinear pro-
cess modeled in (19) and the T-S FLC designed here is globally asymptotically stable in the sense of
Lyapunov at the origin. Considering the values of process parameters m = ., M = ., l = , g = .,
the responses of z and z versus time in the closed-loop system are presented in Figures 5 to 8 for
different initial conditions.

5 Summary and Conclusions

A new approach to the global asymptotic stability analysis of fuzzy logic control systems employing
T-S FLCs dedicated to a class of nonlinear processes has been introduced. The example proves how the
stability analysis algorithm suggested here can be applied to the design of a stable fuzzy logic control
system for a nonlinear process. The new stability approach can be applied also in situations when the
system has an equilibrium point different to the origin and / or the setpoint is nonzero by an appropriately
defined state transformation [15].



Fuzzy Logic Control System Stability Analysis Based on Lyapunov’s Direct Method 423

[s]

Figure 5: State variables versus time for fuzzy logic control system with T-S FLC in the condition
z () =  and z () = −.

[s] 

Figure 6: State variables versus time for fuzzy logic control system with T-S FLC in the condition
z () =  and z () = .

[s] 

Figure 7: State variables versus time for fuzzy logic control system with T-S FLC in the condition
z () =  and z () = −.

[s] 

Figure 8: State variables versus time for fuzzy logic control system with T-S FLC in the condition
z () = − and z () = .

The stability analysis algorithm suggested in this paper canbe applied also when the rule base (2)
of the T-S FLC is not complete. However interpolation techniques [10, 19] are needed in the imple-



424 R.-E. Precup, M.-L. Tomescu, Şt. Preitl

mentation of the T-S FLC. They require the re-assessment of the stability conditoins derived prior to the
implementation.

Further research will be concentrated on new applications of the proposed algorithm to several classes
of processes [4, 5, 7, 8, 9, 17, 20]. The complex applications require the computer-aided design of the
Takagi-Sugeno fuzzy logic controllers employing the stability analysis algorithm proposed in this paper
to strive for increased generality.

Acknowledgement
The support from the CNCSIS and CNMP of Romania is acknowledged.

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Radu-Emil Precup was born in Lugoj, Romania, in 1963. He received the Dipl.Ing. degree
(Hons.) in Automation and Computers from the "Traian Vuia" Polytechnic Institute of Timişoara,
Romania, in 1987, the diploma in Mathematics from the West University of Timişoara in 1993,
and the PhD degree in Automatic Systems from the "Politehnica" University of Timisoara (PUT),
Romania, in 1996. From 1987 to 1991, he was with Infoservice S.A., Timişoara. He is currently
with the PUT, where he became a Professor in the Department of Automation and Applied
Informatics, in 2000, and he is currently a Doctoral Supervisor of Automation and Systems
Engineering. He is also an Honorary Professor with the Budapest Tech Polytechnical Institution,
Budapest, Hungary. He is the author or coauthor of more than 120 papers published in various
scientific journals, refereed conference proceedings, and contributions to books.

Marius L. Tomescu was born in Arad, Romania, in 1967. He received his MSc degree in
Computer Science from the West University of Timişoara, Romania, in 1993, and the PhD degree
in Automation from the "Politehnica" University of Timişoara, Romania, in 2008. From 1993 to
2008, he was with Grup Service S.R.L., Arad. He is with the "Aurel Vlaicu" University of Arad,
where he has been a Lecturer in the Department of Computer Science since 2002. His research
interests include fuzzy control applications and stability analysis of fuzzy control systems. He
is the author or coauthor of more than 30 papers published in various scientific journals and
refereed conference proceedings. Dr. Tomescu is a member of the Romanian Society of Control
Engineering and Technical Informatics.



426 R.-E. Precup, M.-L. Tomescu, Şt. Preitl

Ştefan Preitl was born in Timişoara, Romania, in 1943. He received the Dipl.Ing. degree in
Electrical Engineering, and the PhD degree in Measurement Techniques from the "Traian Vuia"
Polytechnic Institute of Timişoara in 1966 and 1983, respectively. From 1967 to 1972, he was
with Electromotor S.A., Timişoara. He is currently with the "Politehnica" University of Tim-
işoara, Timişoara, where he became a Professor in the Department of Automation and Applied
Informatics, in 1992, and he is currently a Doctoral Supervisor of Automation and Systems En-
gineering. He is also an Honorary Professor with the Budapest Tech Polytechnical Institution,
Budapest, Hungary. He is the author or coauthor of more than 170 papers published in various
scientific journals, refereed conference proceedings, and books in the field of automatic control.