IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 2, APRIL 2001
369
Robust Fuzzy Control of Nonlinear Systems with Parametric Uncertainties
Ho Jae Lee, Jin Bae Park, and Guanrong Chen
Abstract—This paper addresses the robust fuzzy control
problem for nonlinear systems in the presence of parametric
uncertainties. The Takagi–Sugeno (T–S) fuzzy model is adopted
for fuzzy modeling of the nonlinear system. Two cases of the T–S
fuzzy system with parametric uncertainties, both continuous-time
and discrete-time cases are considered. In both continuous-time
and discrete-time cases, sufficient conditions are derived for
robust stabilization in the sense of Lyapunov asymptotic stability,
for the T–S fuzzy system with parametric uncertainties. The
sufficient conditions are formulated in the format of linear matrix
inequalities. The T–S fuzzy model of the chaotic Lorenz system,
which has complex nonlinearity, is developed as a test bed. The
effectiveness of the proposed controller design methodology
is finally demonstrated through numerical simulations on the
chaotic Lorenz system.
Index Terms—Chaotic Lorenz system, fuzzy control, linear matrix inequality, parametric uncertainties, robust stability.
I. INTRODUCTION
M
OST plants in the industry have severe nonlinearity and
uncertainties. Thus, they post additional difficulties to
the control theory of general nonlinear systems and the design
of their controllers. In order to overcome these kinds of difficulties in the design of a controller for an uncertain nonlinear
system, various schemes have been developed in the last two
decades, among which a successful approach is fuzzy control.
Recently, fuzzy control has attracted increasing attention, essentially because it can provide an effective solution to the control
of plants that are complex, uncertain, ill-defined, and have available qualitative knowledge from domain experts for their controllers design.
In spite of the usefulness of fuzzy control, its main drawback
comes from the lack of a systematic control design methodology.
Particularly, stability analysis of a fuzzy system is not easy, and
parameter tuning is generally a time-consuming procedure, due
to the nonlinear and multiparametric nature of the fuzzy control
systems. To resolve these problems, the idea that a linear system
is adopted as the consequent part of a fuzzy rule has evolved
into the innovative T–S model [1], which becomes quite popular today. For a few years, the trend of fuzzy control has been
to develop some systematic design algorithms so as to guarantee the control performance and system stability for the T–S
fuzzy-model-based controllers [2]–[14]. As a common belief, the
T–S fuzzy-model-based control technique is simple and effective
Manuscript received March 3, 2000; revised August 29, 2000. This work was
supported by Brain Korea 21 Project.
H. J. Lee and J. B. Park are with the Department of Electrical and Computer
Engineering, Yonsei University 120-749, Seoul, Korea.
G. Chen is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204 USA, and also with the Department of
Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong.
Publisher Item Identifier S 1063-6706(01)01357-1.
in the control of complexsystems with nonlinearity,such as theinverted pendulum [2]–[5] and chaotic systems [14]–[16]. Besides
stability, another important requirement for a control system is
its robustness and this remains to be a central issue in the study of
uncertain nonlinear control systems and their controllers design.
Robustness in fuzzy model-based control has been extensively
studied in the past. For example, Kiriakidis [7] studied the issue
of stability robustness against modeling errors in T–S fuzzy
model-based control.
Motivated by the aforementioned concerns, this paper deals
with the parametric uncertainties issue in a nonlinear system
with the T–S fuzzy-model-based. The parametric uncertainties
are principal factors responsible for the degraded stability and
performance of an uncertain nonlinear control system. In fact,
in many cases it is very difficult, if not impossible, to obtain the
accurate values of some system parameters. This is due to the
inaccurate measurement, unaccessibility to the system parameters or on-line variation of the parameters. Therefore, this has
promoted some active research in the last few years [17], [18].
Clearly, it is also very important to consider the robust stability
against parametric uncertainties in the T–S fuzzy-model-based
control systems.
The main contribution of this paper is some sufficient conditions in the linear-matrix inequality (LMI) format and a systematic design procedure for the controller design for a general
nonlinear system with parametric uncertainties, for both continuous-time and discrete-time T–S fuzzy systems. Specifically,
this paper proposes some new solutions to the robust stabilization problem for a class of nonlinear systems with time-varying,
but norm-bounded parametric uncertainties. The chaotic Lorenz
system is used as a platform for illustration of the proposed
ideas, techniques, and procedures. Based on the exact fuzzy
modeling technique [19], [20], the T–S fuzzy model is developed. Here, the word “exact” means that the defuzzified output
of the constructed T–S fuzzy model is mathematically identical
to that of the original nonlinear system. Here, again, the Lorenz
system is used for study, chaos is not the main concern; the developed continuous-time and discrete-time T–S fuzzy models
of the Lorenz system are used for controller design. The Lorenz
system is a good platform to use because of its long-time unpredictable dynamical behavior [21]. To that end, some sufficient conditions for stabilization of both the developed continuous-time and discrete-time T–S fuzzy models are derived, subject to system parametric uncertainties. The stability conditions
for nominal nonlinear systems given in [20] are extended here to
general nonlinear systems with parametric uncertainties, which
are likewise formulated in the LMI format. The overall proposed
design methodology presents a systematic and effective framework for continuous-time and discrete-time control of complex
dynamic systems such as chaotic systems.
1063–6706/01$10.00 © 2001 IEEE
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The paper is organized as follows. Section II reviews both
the continuous-time and the discrete-time T–S fuzzy models.
The controller design method for robust stabilization of continuous/discrete-time T–S fuzzy systems in the presence of parametric uncertainties is proposed in Section III. Section IV shows
some controller design examples and simulation results. Finally,
conclusions are given in Section V with some discussion.
II. PRELIMINARIES
In this section, we first discuss two kinds of T–S fuzzy
models—continuous-time and discrete-time models.
In order to consider parametric uncertainties in the T–S fuzzy
systems, we propose the continuous-time T–S fuzzy system in
which the th rule is formulated in the following form:
Continuous-time T-S fuzzy model:
Plant Rule :
is
and
and
Next, a fuzzy model of a state-feedback controller for the continuous-time T–S fuzzy model is formulated as follows:
Controller Rule :
is
and
and
is
(4)
are constant control gains to be determined.
where
Similarly to the continuous-time case, the discrete-time T–S
fuzzy model and the corresponding discrete-time T–S fuzzy
model-based state-feedback controller is constructed as follows:
Discrete-time fuzzy model:
Plant Rule :
is
and
and
is
(5)
Controller Rule :
is
and
is
and
is
(1)
(6)
is a fuzzy set,
is the state vector,
where
is the control input vector,
and
are system matrix and input matrix, respectively,
and
are time-varying matrices with appropriate dimensions, which
represent parametric uncertainties in the plant model, and is
the number of rules of this T–S fuzzy model.
The defuzzified output of the T–S fuzzy system (1) is represented as follows:
denote the indexes of the time steps.
where and
Since the plant rules (1) and (5) have time-varying uncertain
matrices, it is not easy to design the controller gain matrices.
In order to find these gain matrices , the uncertain matrices
should be removed under some reasonable assumptions. Hereand
forth we assume, as usual, that the uncertain matrices
are admissibly norm-bounded and structured.
Assumption 1: The parameter uncertainties considered here
are norm-bounded, in the form
(2)
where
,
, and
are known real constant matrices
is an unknown matrix
of appropriate dimensions, and
function with Lebesgue-measurable elements and satisfies
, in which is the identity matrix of appropriate dimension.
where
III. ROBUST STABILIZATION OF THE T–S FUZZY MODEL
This section presents some sufficient conditions that guarantee the global asymptotic stability of the controlled T–S fuzzy
system in the presence of parametric uncertainties.
in which
is the grade of membership of
are
Some basic properties of
in
.
A. The Continuous-Time Case
(3)
Before proceeding, we recall the following matrix inequality
which will be needed throughout the proof.
Lemma 1 [18]: Given constant matrices and and a symmetric constant matrix of appropriate dimensions, the following inequality holds:
It is clear that
where
satisfies
, if and only if for some
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Consider a continuous-time T–S fuzzy model, with parametric uncertainties, described by the following state-space
equation:
Proof: Consider the Lyapunov function candidate
(12)
(7)
is a time-invariant, symmetric and positive definite
where
is positive definite and radially unmatrix. Clearly,
is
bounded. The time derivative of
The objective is to design a T–S fuzzy-model-based state-feedback controller for robust stabilization of the system (7) in the
form
(13)
(8)
The closed-loop system of (7) and (8) is found in (9), shown
at the bottom of the page.
The main result on the global asymptotic stability of the continuous-time T–S fuzzy model with parametric uncertainties is
summarized in the following theorem
Theorem 1: If there exist a symmetric and positive definite
, and some scalars
, (
matrix , some matrices
such that the following LMIs are satisfied, then the
continuous-time T–S fuzzy system (7) is asymptotically stabilizable via the T–S fuzzy model-based state-feedback controller
(8):
By substituting (9) into (13), we get (14), shown at the bottom of
the next page. If the time derivative of (13) is negative definite
and for all
except at
then
uniformly for all
the controlled fuzzy system (9) is asymptotically stable about its
zero equilibrium. Therefore, if it is possible to assume each sum
of the second equation in (14) to be negative definite, respectively, then the controlled continuous-time T–S fuzzy system is
asymptotically stable.
First, assume that the first sum of the last equation in (14) is
negative definite
(15)
Then, applying Assumption 1 to (15) yields
(a)
(16)
(10)
where
(b)
(11)
According to Lemma 1, the matrix inequality (16) holds for all
satisfying
, if and only if there exists a
such that
constant
where
and
and
, where denotes the transposed elements in the symmetric positions.
(17)
(9)
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where
Applying Schur complement to (17) results in
(18)
The matrix inequality (18) is not an LMI but a quadratic-matrix inequality (QMI). In order to use the convex optimization
technique, the QMI must be converted to an LMI via some variable changes or transformations. For this purpose, define the following transformation matrix as:
Using Lemma 1 repeatedly, the matrix inequality (21) holds for
satisfying
all
if and only if there exists a constant
such that
and take a congruence transformation. This yields
(22)
Applying Schur complement to (22) and taking the congruence
easily verify
transformation with
(19)
and
yields the first LMI (10)
Denoting
in Theorem 1.
The second LMI (11) can be established through a similar
procedure. Assume (20), shown at the bottom of the next page.
Then, using Assumption 1, (20) can be represented as
(23)
and
yields the second LMI
Letting
(11). This completes the proof of the theorem.
Q.E.D.
Remark 1: The second inequalities in Theorem 1 for the pair
, such that
, for all
, do not
have to be solved in determining the system stability.
B. The Discrete-Time Case
(21)
This section deals with the controller design problem for the
discrete-time T–S fuzzy model with parametric uncertainties.
(14)
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The state–space representation of the fuzzy system and its corresponding T–S fuzzy model-based state-feedback controller can
be described as follows:
(24)
(25)
The closed-loop system is given by (26), shown at the bottom of
the page. The following theorem provides a sufficient condition
for robust stabilization of the controlled discrete-time T–S fuzzy
system (26) in the presence of parametric uncertainties.
Theorem 2: If there exists a symmetric and positive definite matrix , some matrices , and some scalars ,
, such that the following LMIs are satisfied, then the
discrete-time T–S fuzzy system (24) is asymptotically stabilizable by the T–S fuzzy-model-based state feedback controller
(25):
where
,
, and denotes the transposed
elements in the symmetric positions.
Proof: Given in the Appendix.
Remark 2: The second inequalities in Theorem 2 for the pair
, such that
, for all , do not have
to be solved in determining the system stability.
and
in Theorems 1 and 2 can
Remark 3: The matrices
be arbitrarily chosen, to express parametric uncertainties. Note,
and
generally influence on
however, that the choices of
the performance of the controller.
IV. COMPUTER SIMULATIONS
In this section, to show the effectiveness of the proposed
system controller design techniques, we simulate the control of
the chaotic Lorenz system with parametric uncertainties. The
control objective is to drive its chaotic trajectory to the origin.
A. Fuzzy Modeling of the Chaotic Lorenz Systems
The Lorenz equations are as follows:
(29)
(a)
(27)
(b)
To construct a T–S fuzzy model for the Lorenz system, the
and
in (29)
quadratic nonlinear terms—
must be expressed as weighted linear sums of some linear functions.
Fact 1: The nonlinear term
can be represented by a weighted sum of linear functions of the
form
(28)
(20)
(26)
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Fig. 1. Trajectory of the chaotic Lorenz system.
The discretized T–S fuzzy model is then obtained as follows:
where
Plant Rules:
Rule 1:
and
Rule 2:
is about
is about
where
(30)
From Fact 1 and since all nonlinear terms in (29) are functions
, we can construct an exact T–S fuzzy model of the
of
system (29) as follows:
Plant Rules:
Rule 1:
Rule 2:
is about
is about
and
is the sampling time and, for simplicity, the third and
higher order terms are truncated.
where
B. The Continuous-Time Case
We can use (30) as membership functions by choosing
as
, because
is seemingly
in Fig. 1 and satisfies (3).
bounded within
Next, we can obtain the discretized T–S fuzzy model of the
chaotic Lorenz system using following simple relationship between continuous-time and discrete-time LTI system,
This section presents a controller design example of the continuous-time T–S fuzzy model, to show the effectiveness of the
proposed robust stabilization technique. To the end, the chaotic
Lorenz system with parametric uncertainties are used as a test
bed.
The T–S fuzzy model of the chaotic Lorenz system (29) as
follows:
Plant Rules:
Rule 1:
Rule 2:
is about
is about
where
(31)
(32)
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Fig. 2. The membership functions for the T–S fuzzy model of the Lorenz system.
and the membership functions are
By applying Theorem 1 and solving the corresponding LMIs,
we obtain the following controller gain matrices:
The membership functions for the plant rules are shown in
Fig. 2.
and
are arbitrarily chosen as
The input matrices
In checking the global stability of the T–S fuzzy model-based
control system, we found the common positive definite matrix
to be
which preserves the system controllability and
just for simplicity.
are (10, 28, 8/3) for chaos
The nominal values of
to emerge. In this paper, we assume that all system parameters
are uncertain but bounded within 30% of their nominal values.
Based on Assumption 1, we define
The system parameters
, and are randomly varied within
30% of their nominal values. The initial values of states are
. The simulation time is 10 s.
The control result is shown in Fig. 3. The control input is acs for comparison. Before the control input
tivated at
was activated, the phase trajectory of the Lorenz system was
chaotic. However, after the control input is activated, the phase
trajectory is quickly directed to the origin. Fig. 4 shows the simulation result, for the Lorenz system without parametric uncertainties but with the same feedback controller. The control input
s. Similarly to the case of the parais also activated at
metric uncertain system, after control input was activated, the
system state is rapidly guided to the origin. Two simulation results show that the T–S fuzzy model-based controller, designed
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Fig. 3. The controlled phase trajectory of the Lorenz system with parametric uncertainties (all system parameters are randomly varied within 30% of the nominal
values).
Fig. 4.
The phase trajectory of the controlled Lorenz system without parametric uncertainties.
by using Theorem 1, is robust against norm-bounded parametric
uncertainties.
where
C. The Discrete-Time Case
In this section, we design the discrete-time T–S fuzzy-modelbased controller, which is based on the scheme described in
Section III-B. The discretized T–S fuzzy model of the chaotic
Lorenz system is as follows:
and
s is the sampling time. Without loss of controllability, the input matrices are chosen as
Plant Rules:
Rule 1:
Rule 2:
is about
is about
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Fig. 5. The phase trajectory of the controlled discretized system of the chaotic Lorenz system in the presence of parametric uncertainties (all system parameters
are varied within the 30% of the nominal values).
Fig. 6. The phase trajectory of the controlled discretized system of the chaotic Lorenz system without parametric uncertainties.
and, for simplicity, the associated uncertain matrices are assumed to be
From Theorem 2, the controller gain matrices are obtained as
The system parameters , , are the same as those in the
continuous-time case. Also, the parameter uncertainties are assumed to be bounded within 30% of their nominal values. Based
,
,
,
,
, and
on Assumption 1, the matrices
are defined as
The common positive definite matrix
is found to be
which guarantees the global stability of the controlled discrete-time T–S fuzzy system (26).
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The simulation time is 10 s, and the trajectory of each simula. During the simtion starts with
ulation time, all system parameters are randomly varied within
the bounds of 30% of their nominal values.
The controlled trajectory is shown in Fig. 5. Similar to
the continuous-time case, the control input is activated at
s for the purpose of comparison. Before
s, the trajectory does not go to the origin, but exhibits a
s, the trajectory
chaos-like irregular behavior. After
is controlled very quickly to the origin. The simulation
result, without parametric uncertainties, is depicted in Fig. 6.
s,
As soon as the control input is activated, at
the system phase trajectory is well guided to the origin
by the controller designed according to Theorem 2. From
two simulation results, we see that the designed T–S
fuzzy-model-based state-feedback controller not only can
stabilize the nonlinear systems, but has strong robustness
against admissible parametric uncertainties.
V. CONCLUSION
In this paper, we have developed and analyzed a new robust
fuzzy controller design methodology for the control of both
continuous-time and discrete-time T–S fuzzy models with parametric uncertainties. The basic approach is based on the rigorous
Lyapunov stability theory and the basic tool is the linear matrix
inequality.Thedesigned controllercanindeedtolerateadmissible
(norm bounded and structured) parametric uncertainties, namely,
it can globally asymptotically stabilize the closed-loop T–S fuzzy
system subject to all admissible parametric uncertainties.
Some convenient sufficient conditions for robust stabilization
of the T–S fuzzy model, containing time-varying parametric
uncertainties, were derived for both continuous-time and
discrete-time cases. The conditions are formulated in the LMI
format. To demonstrate the effectiveness of the proposed
controller design method, the T–S fuzzy model of the chaotic
Lorenz system was developed as a platform and test bed for illustration and its discrete version was also obtained. The design
scheme was applied to the stabilizing control of the Lorenz
system, guiding its chaotic trajectories to the origin. Simulation
results have verified and confirmed the effectiveness of the
new approach in controlling nonlinear systems with parametric
uncertainties.
The approach presented in this paper has virtually generalized and extended some existing fuzzy control methods based
on the T–S model and the LMI criterion. It is believed that this
approach is useful for the control of ill-modeled, uncertain, nonlinear, and complex systems.
(34)
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(39)
APPENDIX
PROOF FOR THEOREM 2, SECTION III-B
Proof: The proof is analogous to the proof of Theorem 1.
Consider the Lyapunov function candidate
(33)
is
which is positive definite. The rate of increases of
in (34), shown at the bottom of the previous page. Therefore, if
the two sums in (34) are both uniformly negative definite for all
and for all
, then
is negative definite so the
controlled system is asymptotically stable.
Assume that the first sum of the last equality in (34) is negative definite
(35)
By applying Schur complement and Assumption 1, (35) is
equivalent to
Schur complement and Lemma 1, and then taking the congruence transformation, we obtain (28), which completes the proof
of the theorem.
Q.E.D.
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H
(36)
satisfying
From Lemma 1, (36) holds for all
if and only if there exists a constant
such that
(37)
where
Applying Schur complement to (37) and taking the congruence
result in
transformation with diag
H
(38)
and
yields (27).
Denoting
The second LMI (28) can also be established through a similar procedure. Let the following inequality be assumed in (39),
shown at the top of the page. Using Assumption 1, applying
H
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