Robust Fuzzy Filter Design for a Class of ... - Semantic Scholar
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 2, APRIL 2007
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Robust Fuzzy Filter Design for a Class of Nonlinear Stochastic Systems Chung-Shi Tseng, Senior Member, IEEE
Abstract—This paper describes the robust fuzzy filtering design for a class of nonlinear stochastic systems. The system dynamic is modelled by Itô-type stochastic differential equations. filter can be For general nonlinear stochastic systems, the obtained by solving a second-order nonlinear Hamilton–Jacobi inequality. In general, it is difficult to solve the second-order nonlinear Hamilton–Jacobi inequality. In this paper, using fuzzy approach [Takagi–Sugeno (T–S) fuzzy model], the fuzzy filtering design for the nonlinear stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a second-order Hamilton–Jacobi inequality. When the worst-case fuzzy disturbance is considered, a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error. The near minimum variance fuzzy filtering problem under the worst-case fuzzy disturbance is also characterized in terms of linear matrix inequality problem (LMIP), which can be efficiently solved by the convex optimization techniques. Simulation examples are provided to illustrate the design procedure and expected performance. Index Terms— fuzzy filtering, Itô-type stochastic differential equations, linear matrix inequalities (LMIs), nonlinear stochastic systems, second-order nonlinear Hamilton–Jacobi inequality (HJI), Takagi–Sugeno (T–S) fuzzy model.
I. INTRODUCTION
T
HE filtering problem is to design an estimator to estimate the unknown state combination via output measurement, which guarantees the gain (from the external disturbance to the estimation error) less than a prescribed level [1]–[6]. In contrast with the well-known Kalman filter, one of filtering is that, it is not necesthe main advantages of sary to know exactly the statistical properties of the external disturbance, but only assumes the external disturbance to have filtering bounded energy. Some practical applications of in signal processing can be found from [7]–[9]. filtering and control On the other hand, the stochastic problems with system models expressed by Itô-type stochastic differential equations have become a popular research topic, and have gained extensive attention [10]–[15]. Most of the aforementioned works are limited to the linear stationary stochastic systems, while [14] investigated the same problem for a class of special nonlinear stochastic systems and [12] and [15] discontrol probcussed the linear and nonlinear stochastic lems. Unlike the deterministic case, the Hamilton–Jacobi inManuscript received March 2, 2005; revised July 28, 2005 and October 20, 2005. This work was supported by National Science Council under Contract NSC 92-2213-E-159-003. The author is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, 304 Hsin Feng, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.881446
filter is equality (HJI) associated with nonlinear stochastic a second-order (not first-order) nonlinear partial differential inequality due to the effect of the diffusion term, which makes the filtering problem more complex [16], [17]. In stochastic general, it is very difficult to solve the second-order nonlinear HJI. Recently, there have been many applications of fuzzy systems theory in various fields. In most of these applications, the fuzzy systems were thought of as universal approximators for any nonlinear systems. The Takagi and Sugeno (T-S) fuzzy model [18] which has been proved to be a very good representation for a certain class of nonlinear dynamic systems was extensively studied in control systems and signal processing [19]–[23]. In this study, using Takagi–Sugeno (T–S) fuzzy model] fuzzy apfuzzy filtering design for a class of nonlinear proach, the stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a second-order Hamilton-Jacobi inequality. First, a T-S fuzzy model is proposed to approximate a class of nonlinear stochastic systems. Next, based on the T-S fuzzy filtering design for the nonlinear fuzzy model, the stochastic systems is characterized in terms of minimizing the attenuation level subject to some LMIs, which is also called eigenvalue problem (EVP) [24]. When the worst-case fuzzy disturbance is considered, a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error. The near minimum variance fuzzy filtering problem under the worst-case fuzzy disturbance is also characterized in terms of linear matrix inequality problem (LMIP), which can be efficiently solved by the LMI toolbox in Matlab [25]. Simulation examples are provided to illustrate the design procedure and expected performance. settings for nonlinear The paper is organized as follows: stochastic systems are given in Section II. In Section III, fuzzy filtering design for a class of nonlinear stochastic systems is introduced, while near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance is presented in Section IV. Some simulation examples are provided in Section V. Finally, concluding remarks are made in Section VI. For convenience, we adopt the following notations throughout : the trace (transpose) of matrix . this paper: : positive–semidefinite (positive–definite) ma: the Euclidean trix . : identity matrix. : the 2-norm of the -dimensional real vector . space of nonanticipative stochastic processes with respect satisfying . to filtration : class of functions twice continuously differexcept possibly at the origin. ential with respect to
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II. SETTINGS FOR NONLINEAR STOCHASTIC SYSTEMS Consider a class of nonlinear stochastic systems governed by Itô-type stochastic differential equations
where denotes the infinitesimal generator of (5), then we obtain
(1) where is the system state, is the controlled output, and stands for the exoge, and nous disturbance signal. are smooth functions with . is a standard one-dimensional Wiener process defined on the probarelative to an increasing family bility space of -algebras . In (1), the state equation, in engineering terminology, can be written as [26]
(7)
(2) where is a stationary white noise. For convenience, the following lemma on the globally asympof totic stability of the equilibrium point
Since (5) is globally asymptotically stable in probability, which almost surely [28] (Doob’s implies convergence theorem), we obtain
(3) is given first, where time variable is suppressed for simplicity. Lemma 1 [27]: Assume there exists a positive Lyapunov function and satisfying (8) (4) Then, by (6), we have for all nonzero , then the equilibrium point is globally asymptotically stable in probability. In the case of in (1), i.e.,
of (3) (9) (5)
where , then we have the following proposition. Proposition 1: For system (5), if there exists a positive function and solving the following HJI
This completes the proof. The following Lemma is a special case of [16, Prop. 1], which plays an important role in this paper. Lemma 2: For system (1), if there exists a positive function and solving the following HJI
(6) of (5) is globally asympthen the equilibrium point totically stable in probability and the variance of . Proof: First, by Lemma 1 and (6), it is obvious that the of (5) is globally asymptotically stable equilibrium point in probability. Second, let us consider
(10) then: i) The equilibrium point of the system (1) is globally asymptotically stable in probability in the case of , and ii) the following inequality: (11) holds for some
if initial state
and (12) holds for some if initial state . Proof: The proof is immediately followed from [16, Prop. 1 and Lemma 1].
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
Remark 1: If we let
and thereby we obtain the stochastic integral representation
(13) one can see that, for any
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and
Rule i If Then
is
and
is
with , where
(17)
(14) That is, , the worst-case disturbance, achieves the maximal possible energy gain from the disturbance input to the controlled output . Remark 2: In general, it is difficult to solve the second-order nonlinear Hamilton-Jacobi inequality in (10). In the next secfuzzy filtering design for tion, using fuzzy approach, the the nonlinear stochastic systems can be given via solving LMIs instead of a second-order HJI. III.
FUZZY FILTERING DESIGN FOR NONLINEAR STOCHASTIC SYSTEMS
(18)
A fuzzy dynamic model has been proposed by Takagi and Sugeno [18] to represent locally linear input–output relations for nonlinear systems. This fuzzy dynamic model is described by fuzzy IF–THEN rules and will be employed here to deal with the filtering design problem for a class of nonlinear stochastic systems governed by Itô-type stochastic differential equations. The th rule of the fuzzy model for the nonlinear stochastic systems is proposed as the following form: Rule i If Then
Remark 3: Note that if is known for , then is known for and conversely. So no information as our observations instead is lost or gained by considering . However, this allows us to obtain a well-defined mathof ematical model of this situation [29]. The final output of the fuzzy system is inferred as follows:
and
(19) where
is
and
is
(15) where for denotes the state vector; is the measurement stands for the exogenous disturbance output; signal; denotes a linear combination of the state is constant matrix; is variables to be estimated; , and are known conthe fuzzy set; stant matrices with appropriate dimension; is the number of IF–THEN rules; are the premise variables; is a standard one-dimensional Wiener process defined on relative to an increasing family the probability space of -algebras ; and is a stationary white noise. in To obtain a tractable mathematical interpretation of (15), we introduce
(20) and is the grade of membership of It is assumed that
in
.
for and for all Therefore, we get for
(21)
and (16)
(22)
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Based on the fuzzy model (17), the following fuzzy estimator is proposed to deal with the state estimation for the nonlinear stochastic systems: Estimation Rule If Then
is
and
then the augmented system can be expressed as follows:
(28)
is (23)
Let
where is the fuzzy estimator gain for the th estimation rule . The overall fuzzy estiand mator is written as
(29) denote the estimator error where , then the nonfiltering can be stated as follows: Find the linear stochastic (for ) such that the following estimator gain hold. of the augmented system i) The equilibrium point (28) is globally asymptotically stable in probability in the . case of ii) For a given disturbance attenuation level , the following relation holds:
(24) Remark 4: In practice, the fuzzy estimator is implemented as follows: (25)
Remark 5: In general, the premise variables depend on the state variables, which are not available and should be estimated, and the premise depend on the variables estimated state variables. The premise variables are used to represent the fuzzy model in (18) and not needed for the fuzzy and estimator in (24). In this study, only premise variables output signal are needed to construct the fuzzy estimator in (24). Then the augmented system is of the following form
(30) where is a positive function. The following well-known lemma is useful for our design. and Lemma 3 [30]: For any matrices (or vectors) with appropriate dimensions, we have (31) where is any positive-definite symmetric matrix. In this paper, we let be an identity matrix. Then, we obtain the following result. Theorem 1: For the augmented system in (28), if there exists a symmetric positive–definite matrix solving the following inequalities:
(32)
(26)
for all , then i) of the augmented system (28) is globally asymptotically stable in proba. and ii) the following inequality: bility in the case of
Defining
(33) and
(27)
holds for some
.
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
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Proof: By Lemma 2, if there exists a positive function and such that
(36)
(34) then i) and ii) hold. Let
if
(35) (37)
By (35), (34) can be written as follows: for all . By the Schur complements [24], (37) is equivalent to
(38) Therefore, the proof is complete. Remark 6: According to Remark 1, if
(39) one obtains
(40)
by Lemma 3
In this case, can be viewed as the worst-case fuzzy disturbance. For the convenience of design, let
(41)
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then (32) can be rewritten as the following LMIs:
By substituting the worst-case fuzzy disturbance (40) into (44), we obtain
(42)
where and . Remark 7: By the assertion of (41), the matrix inequalities in (32) can be transformed into an LMIs formulation in (42). Based on the previous analysis, an fuzzy filtering design as the folis proposed by minimizing the attenuation level lowing eigenvalue problem (EVP) [24]:
subject to
and
from
(45) Assuming that (46) the augmented system in (45) is written as follows:
(43)
The design procedure for the fuzzy filtering design is summarized as follows. Design Procedure 1: Step 1) Construct the fuzzy model rules (17). and Step 2) Solve the EVP in (43) to obtain (thus . Step 3) Construct the fuzzy estimator in (25).
and
IV. NEAR MINIMUM VARIANCE FUZZY FILTERING DESIGN UNDER THE WORST-CASE FUZZY DISTURBANCE The augmented system in (28) is recalled as follows:
Following Proposition 1, we obtain the following result. Theorem 2: For the augmented system in (44), if there exists a symmetric positive–definite matrix solving the following inequalities:
(44) In the followings, the problem of near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance (for ) in is considered: Find the estimator gain (23) such that the following hold. of the augmented system i) The equilibrium point (44) is globally asymptotically stable in probability in the . case of ii) The upper bound on the variance of the estimation error is minimized under the worst-case fuzzy disturbance .
(47) for all and a given disturbance attenuation level then i) of the augmented system (44) is globally asymptotically stable in probability in the case , and ii) of
(48)
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
Fig. 1. Trajectories of
under
the
x (t) (solid line) and x^ (t) (dashdot line) for the proposed H
worst-case
fuzzy
disturbance
and, therefore, . Proof: By Proposition 1, if there exists a positive function and such that
(49) then i) and ii) hold. By substituting (46) into (49), we obtain
fuzzy filter.
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Fig. 2. Trajectories of
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 2, APRIL 2007
x (t) (solid line) and x^ (t) (dashdot line) for the proposed H
fuzzy filter.
by Lemma 3
(50) if
(51) for all . By the Schur complements [24], (51) is equivalent to
(52) Therefore, the proof is complete. Based on the previous analysis, a near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance is
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
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Fig. 3. Trajectories of integration of squared error ( ke(t)k dt; T 2 [0; 15]) for the proposed H fuzzy filter (dashdot line), for the proposed near minimum variance fuzzy filtering (solid line), and for the extended Kalman filter (dotted line) and squared disturbance ( kv(t)k dt) (black point).
proposed by minimizing the upper bound on the variance of esin advance as the timation error for a given attenuation level following minimization problem:
and
where
. Note that
where subject to
and (47)
(53)
Similarly, let
is assumed to be known. If
is always assumed, then
.
then (47) can be rewritten as the following LMIs:
Based on the previous analysis, a near minimum variance fuzzy filtering design is proposed by minimizing the upper bound on the variance of estimation error for a given attenuation in advance as the following eigenvalue problem (EVP) level [24]
subject to
and (54)
(55)
By this suboptimal approach, the variance of the estimation error is bounded by (54)
(56) .
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Fig. 4. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the proposed near minimum variance fuzzy filtering.
Remark 8: The EVP in (43) and (55) can be efficiently solved by the LMI toolbox in Matlab [25]. The design procedure for the near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance is summarized as follows. Design Procedure 2: Step 1) Construct the fuzzy model rules (17). in advance. Step 2) Given an attenuation level and Step 3) Solve the EVP in (55) to obtain (thus . Step 4) Construct the fuzzy estimator in (25). V. SIMULATION EXAMPLES
the previous section, the fuzzy filtering design for nonlinear stochastic system in (57) is given by the following steps. Step 1: Construct the fuzzy model as follows. is about -1.5, THEN Rule 1) IF
Rule 2) IF
is about 0.0, THEN
Rule 3) IF
is about 1.5, THEN
Consider the following nonlinear stochastic system:
(57) where stands for the exogenous disturbance signal and is assumed to be normal distribution noise with zero mean and variance 1.0. The design purpose is to estimate the state variables of the nonlinear stochastic system. Following the design procedure in
where and (for ) are shown in Appendix. For the convenience of simulation, triangular membership functions are used for Rules 1–3 in this example. Step 2: Fuzzy Filtering Design): Solve the EVP in Example 1 ( (43) using the LMI optimization toolbox in Matlab. In this case, we obtain
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
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Fig. 5. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the proposed near minimum variance fuzzy filtering.
and the fuzzy estimation gains are found to be
Step 3: Construct the fuzzy estimator as follows:
The initial condition in the simulation is assumed to be , where and are random initial values with zero mean and variance 1.0, and and are always assumed, i.e., . Figs. 1 and 2 show the trajectories (and ) and (and ), respectively, using of fuzzy filter. The trajectories of integration of the proposed squared error and squared disturbance are shown in Fig. 3. The simulations are performed for 200 times to approximately evaluate . Example 2 (Near Minimum Variance Fuzzy Filtering Design Under the Worst-Case Fuzzy Disturbance): Solve the EVP in (55) using the LMI optimization toolbox in Matlab. In this case, is given beforehand, which is also a minimum if attenuation level for a feasible solution in this case, we obtain
and the fuzzy estimation gains are found to be
Figs. 4 and 5 show the trajectories of (and ) (and ), respectively, using the proposed near and minimum variance fuzzy filtering. The trajectories of inand tegration of squared error squared disturbance are shown in Fig. 3. Similarly, the simulations are performed for 200 times to approximately evaluate . For comparison, all states in (57) are estimated by the wellknown extended Kalman filter. However, the dynamic equation in (57) is not a standard form for the extended Kalman filter. Hence, an alternative form instead of (57) is expressed as follows:
(58) where and with the assumptions of zero mean of
and
and ; and and are all independent one another. In practice, it is very difficult to know exactly the statistical properties of the noises.
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Fig. 6. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the extended Kalman filter.
Fig. 7. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the extended Kalman filter.
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
Based on the dynamic equation in (58), the extended Kalman filter is developed, however it is still tested by the dynamic equation in (57) in this simulation. For the extended Kalman are shown in Fig. 6, while filter, the trajectories of , and are shown in Fig. 7. The trajectothe trajectories of , and ries of integration of squared error and squared disturbance are shown in Fig. 3. Similarly, the simulations are performed for 200 times to approximately evaluate . Obviously, the performance of the proposed fuzzy filter is better than that of the extended Kalman filter. VI. CONCLUSION fuzzy filIn this paper, based on a T–S fuzzy model, the tering problems for a class of nonlinear stochastic systems governed by Itô-type stochastic differential equations are studied. fuzzy filtering design for the Using fuzzy approach, the nonlinear stochastic systems can be given via solving LMIs instead of a second-order HJI. When the worst-case fuzzy disturbance is considered, a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error. The near minimum variance fuzzy filtering problem under the worst-case fuzzy disturbance is also characterized in terms of LMIP. filtering design from linear stoThis study extends the chastic systems to a class of nonlinear stochastic systems using fuzzy techniques. LMI-based design procedure for the fuzzy filtering problems for the nonlinear stochastic systems is developed systematically. The proposed design procedure is very simple and can be performed efficiently using the LMI optimization toolbox in Matlab. Simulation examples are provided to illustrate the design procedure and expected performance. Therefore, the proposed method is very suitable for practical applications in the nonlinear stochastic systems. APPENDIX
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[26] W. M. Wonham, “Random differential equations in control theory,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha Reid, Ed. New York: Academic, 1970, vol. 2, pp. 131–212. [27] R. Z. Has’minskii, Stochastic Stability of Differential Equations. Alphen, The Netherlands: Sijtjoff and Noordhoff, 1980. [28] J. L. Doob, Stochastic Processes. New York: Wiley, 1953. [29] B. Øksendal, Stochastic Differential Equations: An Introduction With Applications, 5th ed. New York: Springer-Verlag, 1998. [30] K. Zhou and P. P. Khargonekar, “Robust stabilization of linear systems with norm bounded time varying uncertainty,” Syst. Control Lett., vol. 10, pp. 17–20, 1988.
Chung-Shi Tseng (M’01–SM’05) received the B.S. degree from the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, the M.S. degree from the Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, and the Ph.D. degree in the electrical engineering from National Tsing-Hua University, Hsin-Chu, Taiwan, in 1984, 1987, and 2001, respectively. He is currently a full Professor at Ming Hsin University of Science and Technology, Hsin-Feng, Taiwan. His research interests are in nonlinear robust control, adaptive control, fuzzy control, fuzzy signal processing, and robotics.
261
Robust Fuzzy Filter Design for a Class of Nonlinear Stochastic Systems Chung-Shi Tseng, Senior Member, IEEE
Abstract—This paper describes the robust fuzzy filtering design for a class of nonlinear stochastic systems. The system dynamic is modelled by Itô-type stochastic differential equations. filter can be For general nonlinear stochastic systems, the obtained by solving a second-order nonlinear Hamilton–Jacobi inequality. In general, it is difficult to solve the second-order nonlinear Hamilton–Jacobi inequality. In this paper, using fuzzy approach [Takagi–Sugeno (T–S) fuzzy model], the fuzzy filtering design for the nonlinear stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a second-order Hamilton–Jacobi inequality. When the worst-case fuzzy disturbance is considered, a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error. The near minimum variance fuzzy filtering problem under the worst-case fuzzy disturbance is also characterized in terms of linear matrix inequality problem (LMIP), which can be efficiently solved by the convex optimization techniques. Simulation examples are provided to illustrate the design procedure and expected performance. Index Terms— fuzzy filtering, Itô-type stochastic differential equations, linear matrix inequalities (LMIs), nonlinear stochastic systems, second-order nonlinear Hamilton–Jacobi inequality (HJI), Takagi–Sugeno (T–S) fuzzy model.
I. INTRODUCTION
T
HE filtering problem is to design an estimator to estimate the unknown state combination via output measurement, which guarantees the gain (from the external disturbance to the estimation error) less than a prescribed level [1]–[6]. In contrast with the well-known Kalman filter, one of filtering is that, it is not necesthe main advantages of sary to know exactly the statistical properties of the external disturbance, but only assumes the external disturbance to have filtering bounded energy. Some practical applications of in signal processing can be found from [7]–[9]. filtering and control On the other hand, the stochastic problems with system models expressed by Itô-type stochastic differential equations have become a popular research topic, and have gained extensive attention [10]–[15]. Most of the aforementioned works are limited to the linear stationary stochastic systems, while [14] investigated the same problem for a class of special nonlinear stochastic systems and [12] and [15] discontrol probcussed the linear and nonlinear stochastic lems. Unlike the deterministic case, the Hamilton–Jacobi inManuscript received March 2, 2005; revised July 28, 2005 and October 20, 2005. This work was supported by National Science Council under Contract NSC 92-2213-E-159-003. The author is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, 304 Hsin Feng, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.881446
filter is equality (HJI) associated with nonlinear stochastic a second-order (not first-order) nonlinear partial differential inequality due to the effect of the diffusion term, which makes the filtering problem more complex [16], [17]. In stochastic general, it is very difficult to solve the second-order nonlinear HJI. Recently, there have been many applications of fuzzy systems theory in various fields. In most of these applications, the fuzzy systems were thought of as universal approximators for any nonlinear systems. The Takagi and Sugeno (T-S) fuzzy model [18] which has been proved to be a very good representation for a certain class of nonlinear dynamic systems was extensively studied in control systems and signal processing [19]–[23]. In this study, using Takagi–Sugeno (T–S) fuzzy model] fuzzy apfuzzy filtering design for a class of nonlinear proach, the stochastic systems can be given via solving linear matrix inequalities (LMIs) instead of a second-order Hamilton-Jacobi inequality. First, a T-S fuzzy model is proposed to approximate a class of nonlinear stochastic systems. Next, based on the T-S fuzzy filtering design for the nonlinear fuzzy model, the stochastic systems is characterized in terms of minimizing the attenuation level subject to some LMIs, which is also called eigenvalue problem (EVP) [24]. When the worst-case fuzzy disturbance is considered, a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error. The near minimum variance fuzzy filtering problem under the worst-case fuzzy disturbance is also characterized in terms of linear matrix inequality problem (LMIP), which can be efficiently solved by the LMI toolbox in Matlab [25]. Simulation examples are provided to illustrate the design procedure and expected performance. settings for nonlinear The paper is organized as follows: stochastic systems are given in Section II. In Section III, fuzzy filtering design for a class of nonlinear stochastic systems is introduced, while near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance is presented in Section IV. Some simulation examples are provided in Section V. Finally, concluding remarks are made in Section VI. For convenience, we adopt the following notations throughout : the trace (transpose) of matrix . this paper: : positive–semidefinite (positive–definite) ma: the Euclidean trix . : identity matrix. : the 2-norm of the -dimensional real vector . space of nonanticipative stochastic processes with respect satisfying . to filtration : class of functions twice continuously differexcept possibly at the origin. ential with respect to
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II. SETTINGS FOR NONLINEAR STOCHASTIC SYSTEMS Consider a class of nonlinear stochastic systems governed by Itô-type stochastic differential equations
where denotes the infinitesimal generator of (5), then we obtain
(1) where is the system state, is the controlled output, and stands for the exoge, and nous disturbance signal. are smooth functions with . is a standard one-dimensional Wiener process defined on the probarelative to an increasing family bility space of -algebras . In (1), the state equation, in engineering terminology, can be written as [26]
(7)
(2) where is a stationary white noise. For convenience, the following lemma on the globally asympof totic stability of the equilibrium point
Since (5) is globally asymptotically stable in probability, which almost surely [28] (Doob’s implies convergence theorem), we obtain
(3) is given first, where time variable is suppressed for simplicity. Lemma 1 [27]: Assume there exists a positive Lyapunov function and satisfying (8) (4) Then, by (6), we have for all nonzero , then the equilibrium point is globally asymptotically stable in probability. In the case of in (1), i.e.,
of (3) (9) (5)
where , then we have the following proposition. Proposition 1: For system (5), if there exists a positive function and solving the following HJI
This completes the proof. The following Lemma is a special case of [16, Prop. 1], which plays an important role in this paper. Lemma 2: For system (1), if there exists a positive function and solving the following HJI
(6) of (5) is globally asympthen the equilibrium point totically stable in probability and the variance of . Proof: First, by Lemma 1 and (6), it is obvious that the of (5) is globally asymptotically stable equilibrium point in probability. Second, let us consider
(10) then: i) The equilibrium point of the system (1) is globally asymptotically stable in probability in the case of , and ii) the following inequality: (11) holds for some
if initial state
and (12) holds for some if initial state . Proof: The proof is immediately followed from [16, Prop. 1 and Lemma 1].
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
Remark 1: If we let
and thereby we obtain the stochastic integral representation
(13) one can see that, for any
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and
Rule i If Then
is
and
is
with , where
(17)
(14) That is, , the worst-case disturbance, achieves the maximal possible energy gain from the disturbance input to the controlled output . Remark 2: In general, it is difficult to solve the second-order nonlinear Hamilton-Jacobi inequality in (10). In the next secfuzzy filtering design for tion, using fuzzy approach, the the nonlinear stochastic systems can be given via solving LMIs instead of a second-order HJI. III.
FUZZY FILTERING DESIGN FOR NONLINEAR STOCHASTIC SYSTEMS
(18)
A fuzzy dynamic model has been proposed by Takagi and Sugeno [18] to represent locally linear input–output relations for nonlinear systems. This fuzzy dynamic model is described by fuzzy IF–THEN rules and will be employed here to deal with the filtering design problem for a class of nonlinear stochastic systems governed by Itô-type stochastic differential equations. The th rule of the fuzzy model for the nonlinear stochastic systems is proposed as the following form: Rule i If Then
Remark 3: Note that if is known for , then is known for and conversely. So no information as our observations instead is lost or gained by considering . However, this allows us to obtain a well-defined mathof ematical model of this situation [29]. The final output of the fuzzy system is inferred as follows:
and
(19) where
is
and
is
(15) where for denotes the state vector; is the measurement stands for the exogenous disturbance output; signal; denotes a linear combination of the state is constant matrix; is variables to be estimated; , and are known conthe fuzzy set; stant matrices with appropriate dimension; is the number of IF–THEN rules; are the premise variables; is a standard one-dimensional Wiener process defined on relative to an increasing family the probability space of -algebras ; and is a stationary white noise. in To obtain a tractable mathematical interpretation of (15), we introduce
(20) and is the grade of membership of It is assumed that
in
.
for and for all Therefore, we get for
(21)
and (16)
(22)
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Based on the fuzzy model (17), the following fuzzy estimator is proposed to deal with the state estimation for the nonlinear stochastic systems: Estimation Rule If Then
is
and
then the augmented system can be expressed as follows:
(28)
is (23)
Let
where is the fuzzy estimator gain for the th estimation rule . The overall fuzzy estiand mator is written as
(29) denote the estimator error where , then the nonfiltering can be stated as follows: Find the linear stochastic (for ) such that the following estimator gain hold. of the augmented system i) The equilibrium point (28) is globally asymptotically stable in probability in the . case of ii) For a given disturbance attenuation level , the following relation holds:
(24) Remark 4: In practice, the fuzzy estimator is implemented as follows: (25)
Remark 5: In general, the premise variables depend on the state variables, which are not available and should be estimated, and the premise depend on the variables estimated state variables. The premise variables are used to represent the fuzzy model in (18) and not needed for the fuzzy and estimator in (24). In this study, only premise variables output signal are needed to construct the fuzzy estimator in (24). Then the augmented system is of the following form
(30) where is a positive function. The following well-known lemma is useful for our design. and Lemma 3 [30]: For any matrices (or vectors) with appropriate dimensions, we have (31) where is any positive-definite symmetric matrix. In this paper, we let be an identity matrix. Then, we obtain the following result. Theorem 1: For the augmented system in (28), if there exists a symmetric positive–definite matrix solving the following inequalities:
(32)
(26)
for all , then i) of the augmented system (28) is globally asymptotically stable in proba. and ii) the following inequality: bility in the case of
Defining
(33) and
(27)
holds for some
.
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Proof: By Lemma 2, if there exists a positive function and such that
(36)
(34) then i) and ii) hold. Let
if
(35) (37)
By (35), (34) can be written as follows: for all . By the Schur complements [24], (37) is equivalent to
(38) Therefore, the proof is complete. Remark 6: According to Remark 1, if
(39) one obtains
(40)
by Lemma 3
In this case, can be viewed as the worst-case fuzzy disturbance. For the convenience of design, let
(41)
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then (32) can be rewritten as the following LMIs:
By substituting the worst-case fuzzy disturbance (40) into (44), we obtain
(42)
where and . Remark 7: By the assertion of (41), the matrix inequalities in (32) can be transformed into an LMIs formulation in (42). Based on the previous analysis, an fuzzy filtering design as the folis proposed by minimizing the attenuation level lowing eigenvalue problem (EVP) [24]:
subject to
and
from
(45) Assuming that (46) the augmented system in (45) is written as follows:
(43)
The design procedure for the fuzzy filtering design is summarized as follows. Design Procedure 1: Step 1) Construct the fuzzy model rules (17). and Step 2) Solve the EVP in (43) to obtain (thus . Step 3) Construct the fuzzy estimator in (25).
and
IV. NEAR MINIMUM VARIANCE FUZZY FILTERING DESIGN UNDER THE WORST-CASE FUZZY DISTURBANCE The augmented system in (28) is recalled as follows:
Following Proposition 1, we obtain the following result. Theorem 2: For the augmented system in (44), if there exists a symmetric positive–definite matrix solving the following inequalities:
(44) In the followings, the problem of near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance (for ) in is considered: Find the estimator gain (23) such that the following hold. of the augmented system i) The equilibrium point (44) is globally asymptotically stable in probability in the . case of ii) The upper bound on the variance of the estimation error is minimized under the worst-case fuzzy disturbance .
(47) for all and a given disturbance attenuation level then i) of the augmented system (44) is globally asymptotically stable in probability in the case , and ii) of
(48)
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
Fig. 1. Trajectories of
under
the
x (t) (solid line) and x^ (t) (dashdot line) for the proposed H
worst-case
fuzzy
disturbance
and, therefore, . Proof: By Proposition 1, if there exists a positive function and such that
(49) then i) and ii) hold. By substituting (46) into (49), we obtain
fuzzy filter.
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Fig. 2. Trajectories of
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x (t) (solid line) and x^ (t) (dashdot line) for the proposed H
fuzzy filter.
by Lemma 3
(50) if
(51) for all . By the Schur complements [24], (51) is equivalent to
(52) Therefore, the proof is complete. Based on the previous analysis, a near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance is
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
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Fig. 3. Trajectories of integration of squared error ( ke(t)k dt; T 2 [0; 15]) for the proposed H fuzzy filter (dashdot line), for the proposed near minimum variance fuzzy filtering (solid line), and for the extended Kalman filter (dotted line) and squared disturbance ( kv(t)k dt) (black point).
proposed by minimizing the upper bound on the variance of esin advance as the timation error for a given attenuation level following minimization problem:
and
where
. Note that
where subject to
and (47)
(53)
Similarly, let
is assumed to be known. If
is always assumed, then
.
then (47) can be rewritten as the following LMIs:
Based on the previous analysis, a near minimum variance fuzzy filtering design is proposed by minimizing the upper bound on the variance of estimation error for a given attenuation in advance as the following eigenvalue problem (EVP) level [24]
subject to
and (54)
(55)
By this suboptimal approach, the variance of the estimation error is bounded by (54)
(56) .
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Fig. 4. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the proposed near minimum variance fuzzy filtering.
Remark 8: The EVP in (43) and (55) can be efficiently solved by the LMI toolbox in Matlab [25]. The design procedure for the near minimum variance fuzzy filtering design under the worst-case fuzzy disturbance is summarized as follows. Design Procedure 2: Step 1) Construct the fuzzy model rules (17). in advance. Step 2) Given an attenuation level and Step 3) Solve the EVP in (55) to obtain (thus . Step 4) Construct the fuzzy estimator in (25). V. SIMULATION EXAMPLES
the previous section, the fuzzy filtering design for nonlinear stochastic system in (57) is given by the following steps. Step 1: Construct the fuzzy model as follows. is about -1.5, THEN Rule 1) IF
Rule 2) IF
is about 0.0, THEN
Rule 3) IF
is about 1.5, THEN
Consider the following nonlinear stochastic system:
(57) where stands for the exogenous disturbance signal and is assumed to be normal distribution noise with zero mean and variance 1.0. The design purpose is to estimate the state variables of the nonlinear stochastic system. Following the design procedure in
where and (for ) are shown in Appendix. For the convenience of simulation, triangular membership functions are used for Rules 1–3 in this example. Step 2: Fuzzy Filtering Design): Solve the EVP in Example 1 ( (43) using the LMI optimization toolbox in Matlab. In this case, we obtain
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
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Fig. 5. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the proposed near minimum variance fuzzy filtering.
and the fuzzy estimation gains are found to be
Step 3: Construct the fuzzy estimator as follows:
The initial condition in the simulation is assumed to be , where and are random initial values with zero mean and variance 1.0, and and are always assumed, i.e., . Figs. 1 and 2 show the trajectories (and ) and (and ), respectively, using of fuzzy filter. The trajectories of integration of the proposed squared error and squared disturbance are shown in Fig. 3. The simulations are performed for 200 times to approximately evaluate . Example 2 (Near Minimum Variance Fuzzy Filtering Design Under the Worst-Case Fuzzy Disturbance): Solve the EVP in (55) using the LMI optimization toolbox in Matlab. In this case, is given beforehand, which is also a minimum if attenuation level for a feasible solution in this case, we obtain
and the fuzzy estimation gains are found to be
Figs. 4 and 5 show the trajectories of (and ) (and ), respectively, using the proposed near and minimum variance fuzzy filtering. The trajectories of inand tegration of squared error squared disturbance are shown in Fig. 3. Similarly, the simulations are performed for 200 times to approximately evaluate . For comparison, all states in (57) are estimated by the wellknown extended Kalman filter. However, the dynamic equation in (57) is not a standard form for the extended Kalman filter. Hence, an alternative form instead of (57) is expressed as follows:
(58) where and with the assumptions of zero mean of
and
and ; and and are all independent one another. In practice, it is very difficult to know exactly the statistical properties of the noises.
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Fig. 6. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the extended Kalman filter.
Fig. 7. Trajectories of x (t) (solid line) and x ^ (t) (dashdot line) for the extended Kalman filter.
TSENG: ROBUST FUZZY FILTER DESIGN FOR A CLASS OF NONLINEAR STOCHASTIC SYSTEMS
Based on the dynamic equation in (58), the extended Kalman filter is developed, however it is still tested by the dynamic equation in (57) in this simulation. For the extended Kalman are shown in Fig. 6, while filter, the trajectories of , and are shown in Fig. 7. The trajectothe trajectories of , and ries of integration of squared error and squared disturbance are shown in Fig. 3. Similarly, the simulations are performed for 200 times to approximately evaluate . Obviously, the performance of the proposed fuzzy filter is better than that of the extended Kalman filter. VI. CONCLUSION fuzzy filIn this paper, based on a T–S fuzzy model, the tering problems for a class of nonlinear stochastic systems governed by Itô-type stochastic differential equations are studied. fuzzy filtering design for the Using fuzzy approach, the nonlinear stochastic systems can be given via solving LMIs instead of a second-order HJI. When the worst-case fuzzy disturbance is considered, a near minimum variance fuzzy filtering problem is also solved by minimizing the upper bound on the variance of the estimation error. The near minimum variance fuzzy filtering problem under the worst-case fuzzy disturbance is also characterized in terms of LMIP. filtering design from linear stoThis study extends the chastic systems to a class of nonlinear stochastic systems using fuzzy techniques. LMI-based design procedure for the fuzzy filtering problems for the nonlinear stochastic systems is developed systematically. The proposed design procedure is very simple and can be performed efficiently using the LMI optimization toolbox in Matlab. Simulation examples are provided to illustrate the design procedure and expected performance. Therefore, the proposed method is very suitable for practical applications in the nonlinear stochastic systems. APPENDIX
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[26] W. M. Wonham, “Random differential equations in control theory,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha Reid, Ed. New York: Academic, 1970, vol. 2, pp. 131–212. [27] R. Z. Has’minskii, Stochastic Stability of Differential Equations. Alphen, The Netherlands: Sijtjoff and Noordhoff, 1980. [28] J. L. Doob, Stochastic Processes. New York: Wiley, 1953. [29] B. Øksendal, Stochastic Differential Equations: An Introduction With Applications, 5th ed. New York: Springer-Verlag, 1998. [30] K. Zhou and P. P. Khargonekar, “Robust stabilization of linear systems with norm bounded time varying uncertainty,” Syst. Control Lett., vol. 10, pp. 17–20, 1988.
Chung-Shi Tseng (M’01–SM’05) received the B.S. degree from the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, the M.S. degree from the Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, and the Ph.D. degree in the electrical engineering from National Tsing-Hua University, Hsin-Chu, Taiwan, in 1984, 1987, and 2001, respectively. He is currently a full Professor at Ming Hsin University of Science and Technology, Hsin-Feng, Taiwan. His research interests are in nonlinear robust control, adaptive control, fuzzy control, fuzzy signal processing, and robotics.
