Stochastic Controledited Part 6 docx

Stochastic Control192 For a class of state estimation problems where observations on system state vectors are constrained, i.e., when it is not feasible to make observations at every moment, the question of how many observations to take must be answered. This paper models such a class of problems by assigning a fixed cost to each observation taken. The total number of observations is determined as a function of the observation cost. Extension to the case where the observation cost is an explicit function of the number of observations taken is straightforward. A different way to model the observation constraints should be investigated. More work is needed, however, to obtain improved decision rules for the problems of unconstrained and constrained optimization under parameter uncertainty when: (i) the observations are from general continuous exponential families of distributions, (ii) the observations are from discrete exponential families of distributions, (iii) some of the observations are from continuous exponential families of distributions and some from discrete exponential families of distributions, (iv) the observations are from multiparameter or multidimensional distributions, (v) the observations are from truncated distributions, (vi) the observations are censored, (vii) the censored observations are from truncated distributions. 9. Acknowledgments This research was supported in part by Grant No. 06.1936, Grant No. 07.2036, Grant No. 09.1014, and Grant No. 09.1544 from the Latvian Council of Science. 10. References Alamo, T.; Bravo, J. & Camacho, E. (2005). Guaranteed state estimation by zonotopes. Automatica, Vol. 41, pp. 1035 1043 Barlow, R. E. & Proshan, F. (1966). Tolerance and confidence limits for classes of distributions based on failure rate. Ann. Math. Stat., Vol. 37, pp. 1593 1601 Ben-Daya, M.; Duffuaa, S.O. & Raouf, A. (2000). Maintenance, Modeling and Optimization, Kluwer Academic Publishers, Norwell, Massachusetts Coppola, A. (1997). Some observations on demonstrating availability. RAC Journal, Vol. 6, pp. 1718 Epstein, B. & Sobel, M. (1954). Some theorems relevant to life testing from an exponential population. Ann. Math. Statist., Vol. 25, pp. 373  381 Gillijns, S. & De Moor, B. (2007). Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, Vol. 43, pp. 111116 Grubbs, F. E. (1971). Approximate fiducial bounds on reliability for the two parameter negative exponential distribution. Technometrics, Vol. 13, pp. 873  876 Hahn, G.J. & Nelson, W. (1973). A survey of prediction intervals and their applications. J. Qual. Tech. , Vol. 5, pp. 178 188 Ireson, W. G. & Coombs, C. F. (1988). Handbook of Reliability Engineering and Management, McGraw-Hill, New York Kalman, R. (1960). A new approach to linear filtering and prediction problems. Trans. ASME, J. Basic Eng ., Vol. 82, pp. 34  45 Kendall, M.G. & Stuart, A. (1969). The Advanced Theory of Statistics, Vol. 1 (3 rd edition), Griffin, London Ko, S. & Bitmead, R. R. (2007). State estimation for linear systems with state equality constraints. Automatica, Vol. 43, pp. 1363 1368 McGarty, T. P. (1974). Stochastic Systems and State Estimation, John Wiley and Sons, Inc., New York Nechval, N.A. (1982). Modern Statistical Methods of Operations Research, RCAEI, Riga Nechval, N.A. (1984). Theory and Methods of Adaptive Control of Stochastic Processes, RCAEI, Riga Nechval, N.A.; Nechval, K.N. & Vasermanis, E.K. (2001). Optimization of interval estimators via invariant embedding technique. IJCAS (An International Journal of Computing Anticipatory Systems ), Vol. 9, pp. 241255 Nechval, N. A.; Nechval, K. N. & Vasermanis, E. K. (2003). Effective state estimation of stochastic systems. Kybernetes, Vol. 32, pp. 666  678 Nechval, N.A. & Vasermanis, E.K. (2004). Improved Decisions in Statistics, SIA “Izglitibas soli”, Riga Norgaard, M.; Poulsen, N. K. & Ravn, O. (2000). New developments in state estimation for nonlinear systems. Automatica, Vol. 36, pp. 16271638 Savkin, A. & Petersen, L. (1998). Robust state estimation and model validation for discrete- time uncertain system with a deterministic description of noise and uncertainty. Automatica, Vol. 34, 1998, pp. 271274 Yan, J. & Bitmead, R. R. (2005). Incorporating state estimation into model predictive control and its application to network traffic control. Automatica, Vol. 41, pp. 595  604 Improved State Estimation of Stochastic Systems via a New Technique of Invariant Embedding 193 For a class of state estimation problems where observations on system state vectors are constrained, i.e., when it is not feasible to make observations at every moment, the question of how many observations to take must be answered. This paper models such a class of problems by assigning a fixed cost to each observation taken. The total number of observations is determined as a function of the observation cost. Extension to the case where the observation cost is an explicit function of the number of observations taken is straightforward. A different way to model the observation constraints should be investigated. More work is needed, however, to obtain improved decision rules for the problems of unconstrained and constrained optimization under parameter uncertainty when: (i) the observations are from general continuous exponential families of distributions, (ii) the observations are from discrete exponential families of distributions, (iii) some of the observations are from continuous exponential families of distributions and some from discrete exponential families of distributions, (iv) the observations are from multiparameter or multidimensional distributions, (v) the observations are from truncated distributions, (vi) the observations are censored, (vii) the censored observations are from truncated distributions. 9. Acknowledgments This research was supported in part by Grant No. 06.1936, Grant No. 07.2036, Grant No. 09.1014, and Grant No. 09.1544 from the Latvian Council of Science. 10. References Alamo, T.; Bravo, J. & Camacho, E. (2005). Guaranteed state estimation by zonotopes. Automatica, Vol. 41, pp. 1035 1043 Barlow, R. E. & Proshan, F. (1966). Tolerance and confidence limits for classes of distributions based on failure rate. Ann. Math. Stat., Vol. 37, pp. 1593 1601 Ben-Daya, M.; Duffuaa, S.O. & Raouf, A. (2000). Maintenance, Modeling and Optimization, Kluwer Academic Publishers, Norwell, Massachusetts Coppola, A. (1997). Some observations on demonstrating availability. RAC Journal, Vol. 6, pp. 1718 Epstein, B. & Sobel, M. (1954). Some theorems relevant to life testing from an exponential population. Ann. Math. Statist., Vol. 25, pp. 373  381 Gillijns, S. & De Moor, B. (2007). Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, Vol. 43, pp. 111116 Grubbs, F. E. (1971). Approximate fiducial bounds on reliability for the two parameter negative exponential distribution. Technometrics, Vol. 13, pp. 873  876 Hahn, G.J. & Nelson, W. (1973). A survey of prediction intervals and their applications. J. Qual. Tech. , Vol. 5, pp. 178 188 Ireson, W. G. & Coombs, C. F. (1988). Handbook of Reliability Engineering and Management, McGraw-Hill, New York Kalman, R. (1960). A new approach to linear filtering and prediction problems. Trans. ASME, J. Basic Eng ., Vol. 82, pp. 34  45 Kendall, M.G. & Stuart, A. (1969). The Advanced Theory of Statistics, Vol. 1 (3 rd edition), Griffin, London Ko, S. & Bitmead, R. R. (2007). State estimation for linear systems with state equality constraints. Automatica, Vol. 43, pp. 1363 1368 McGarty, T. P. (1974). Stochastic Systems and State Estimation, John Wiley and Sons, Inc., New York Nechval, N.A. (1982). Modern Statistical Methods of Operations Research, RCAEI, Riga Nechval, N.A. (1984). Theory and Methods of Adaptive Control of Stochastic Processes, RCAEI, Riga Nechval, N.A.; Nechval, K.N. & Vasermanis, E.K. (2001). Optimization of interval estimators via invariant embedding technique. IJCAS (An International Journal of Computing Anticipatory Systems ), Vol. 9, pp. 241255 Nechval, N. A.; Nechval, K. N. & Vasermanis, E. K. (2003). Effective state estimation of stochastic systems. Kybernetes, Vol. 32, pp. 666  678 Nechval, N.A. & Vasermanis, E.K. (2004). Improved Decisions in Statistics, SIA “Izglitibas soli”, Riga Norgaard, M.; Poulsen, N. K. & Ravn, O. (2000). New developments in state estimation for nonlinear systems. Automatica, Vol. 36, pp. 16271638 Savkin, A. & Petersen, L. (1998). Robust state estimation and model validation for discrete- time uncertain system with a deterministic description of noise and uncertainty. Automatica, Vol. 34, 1998, pp. 271274 Yan, J. & Bitmead, R. R. (2005). Incorporating state estimation into model predictive control and its application to network traffic control. Automatica, Vol. 41, pp. 595  604 Stochastic Control194 Fuzzy identication of discrete time nonlinear stochastic systems 195 Fuzzy identication of discrete time nonlinear stochastic systems Ginalber L. O. Serra 15 Fuzzy identification of Discrete Time Nonlinear Stochastic Systems Ginalber L. O. Serra Federal Institute of Education, Science and Technology (IFMA) Brasil 1. Introduction System identification is the task of developing or improving a mathematical description of dynamic systems from experimental data (Ljung (1999); Söderström & Stoica (1989)). De- pending on the level of a priori insight about the system, this task can be approached in three different ways: white box modeling, black box modeling and gray box modeling. These models can be used for simulation, prediction, fault detection, design of controllers (model based control), and so forth. Nonlinear system identification (Aguirre et al. (2005); Serra & Bottura (2005); Sjöberg et al. (1995); ?) is becomming an important tool which can be used to improve control performance and achieve robust behavior (Narendra & Parthasarathy (1990); Serra & Bottura (2006a)). Most processes in industry are characterized by nonlinear and time-varying behavior and are not amenable to conventional modeling approaches due to the lack of precise, formal knowledge about it, its strongly nonlinear behavior and high degree of uncertainty. Methods based on fuzzy models are gradually becoming established not only in academic view point but also because they have been recognized as powerful tools in industrial applications, facil- iting the effective development of models by combining information from different sources, such as empirical models, heuristics and data (Hellendoorn & Driankov (1997)). In fuzzy models, the relation between variables are based on if-then rules such as IF < antecedent > THEN < consequent >, where antecedent evaluate the model inputs and consequent pro- vide the value of the model output. Takagi and Sugeno, in 1985, developed a new approach in which the key idea was partitioning the input space into fuzzy areas and approximating each area by a linear or a nonlinear model (Takagi & Sugeno (1985)). This structure, so called Takagi-Sugeno (TS) fuzzy model, can be used to approximate a highly nonlinear function of simple structure using a small number of rules. Identification of TS fuzzy model using exper- imental data is divided into two steps: structure identification and parameter estimation. The former consists of antecedent structure identification and consequent structure identification. The latter consists of antecedent and consequent parameter estimation where the consequent parameters are the coefficients of the linear expressions in the consequent of a fuzzy rule. To be applicable to real world problems, the parameter estimation must be highly efficient. Input and output measurements may be contaminated by noise. For low levels of noise the least squares (LS) method, for example, may produce excellent estimates of the consequent param- eters. However, with larger levels of noise, some modifications in this method are required to overcome this inconsistency. Generalized least squares (GLS) method, extended least squares (ELS) method, prediction error (PE) method, are examples of such modifications. A problem 11 Stochastic Control196 with the use of these methods, in a fuzzy modeling context, is that the inclusion of the pre- diction error past values in the regression vector, which defines the input linguistic variables, increases the complexity of the fuzzy model structure and are inevitably dependent upon the accuracy of the noise model. To obtain consistent parameter estimates in a noisy environ- ment without modeling the noise, the instrumental variable (IV) method can be used. It is known that by choosing proper instrumental variables, it provides a way to obtain consis- tent estimates with certain optimal properties (Serra & Bottura (2004; 2006b); Söderström & Stoica (1983)). This paper proposes an approach to nonlinear discrete time systems identifica- tion based on instrumental variable method and TS fuzzy model. In the proposed approach, which is an extension of the standard linear IV method (Söderström & Stoica (1983)), the cho- sen instrumental variables, statistically uncorrelated with the noise, are mapped to fuzzy sets, partitioning the input space in subregions to define valid and unbiased estimates of the con- sequent parameters for the TS fuzzy model in a noisy environment. From this theoretical background, the fuzzy instrumental variable (FIV) concept is proposed, and the main statistical characteristics of the FIV algorithm such as consistency and unbias are derived. Simulation results show that the proposed algorithm is relatively insensitive to the noise on the measured input-output data. This paper is organized as follow: In Section 2, a brief review of the TS fuzzy model formu- lation is given. In Section 3, the fuzzy NARX structure is introduced. It is used to formulate the proposed approach. In Section 4, the TS fuzzy model consequent parameters estimate problem in a noisy environment is studied. From this analysis, three Lemmas and one Theo- rem are proposed to show the consistency and unbias of the parameters estimates in a noisy environment with the proposed approach. The fuzzy instrumental variable concept is also proposed and considerations about how the FIV should be chosen are given. In Section 5, off- line and on-line schemes of the fuzzy instrumental variable algorithm are derived. Simulation results showing the efficiency of the FIV approach in a noisy environment are given in Section 6. Finally, the closing remarks are given in Section 7. 2. Takagi-Sugeno Fuzzy Model The TS fuzzy inference system is composed by a set of IF-THEN rules which partitions the in- put space, so-called universe of discourse, into fuzzy regions described by the rule antecedents in which consequent functions are valid. The consequent of each rule i is a functional expres- sion y i = f i (x) (King (1999); Papadakis & Theocaris (2002)). The i-th TS fuzzy rule has the following form: R i|i=1,2, ,l : IF x 1 is F i 1 AND ··· AND x q is F i q THEN y i = f i (x) (1) where l is the number of rules. The vector x ∈  q contains the antecedent linguistic variables, which has its own universe of discourse partitioned into fuzzy regions by the fuzzy sets repre- senting the linguistic terms. The variable x j belongs to a fuzzy set F i j with a truth value given by a membership function µ i F j :  → [0, 1]. The truth value h i for the complete rule i is com- puted using the aggregation operator, or t-norm, AND, denoted by ⊗ : [0, 1] × [0, 1] → [0, 1], h i (x) = µ i 1 (x 1 ) ⊗ µ i 2 (x 2 ) ⊗ . . . µ i q (x q ) (2) Among the different t-norms available, in this work the algebraic product will be used, and h i (x) = q ∏ j=1 µ i j (x j ) (3) The degree of activation for rule i is then normalized as γ i (x) = h i (x) ∑ l r =1 h r (x) (4) This normalization implies that l ∑ i=1 γ i (x) = 1 (5) The response of the TS fuzzy model is a weighted sum of the consequent functions, i.e., a convex combination of the local functions (models) f i , y = l ∑ i=1 γ i (x) f i (x) (6) which can be seen as a linear parameter varying (LPV) system. In this sense, a TS fuzzy model can be considered as a mapping from the antecedent (input) space to a convex region (poli- tope) in the space of the local submodels defined by the consequent parameters, as shown in Fig. 1 (Bergsten (2001)). This property simplifies the analysis of TS fuzzy models in a robust polytope model 4 model 3 model 2 model n Antecedent space (IF) submodels space (THEN) Rules model 1 Fig. 1. Mapping to local submodels space. linear system framework for identification, controllers design with desired closed loop char- acteristics and stability analysis (Johansen et al. (2000); Kadmiry & Driankov (2004); Tanaka et al. (1998); Tong & Li (2002)). 3. Fuzzy Structure Model The nonlinear input-output representation is often used for building TS fuzzy models from data, where the regression vector is represented by a finite number of past inputs and outputs of the system. In this work, the nonlinear autoregressive with exogenous input (NARX) struc- ture model is used. This model is applied in most nonlinear identification methods such as neural networks, radial basis functions, cerebellar model articulation controller (CMAC), and Fuzzy identication of discrete time nonlinear stochastic systems 197 with the use of these methods, in a fuzzy modeling context, is that the inclusion of the pre- diction error past values in the regression vector, which defines the input linguistic variables, increases the complexity of the fuzzy model structure and are inevitably dependent upon the accuracy of the noise model. To obtain consistent parameter estimates in a noisy environ- ment without modeling the noise, the instrumental variable (IV) method can be used. It is known that by choosing proper instrumental variables, it provides a way to obtain consis- tent estimates with certain optimal properties (Serra & Bottura (2004; 2006b); Söderström & Stoica (1983)). This paper proposes an approach to nonlinear discrete time systems identifica- tion based on instrumental variable method and TS fuzzy model. In the proposed approach, which is an extension of the standard linear IV method (Söderström & Stoica (1983)), the cho- sen instrumental variables, statistically uncorrelated with the noise, are mapped to fuzzy sets, partitioning the input space in subregions to define valid and unbiased estimates of the con- sequent parameters for the TS fuzzy model in a noisy environment. From this theoretical background, the fuzzy instrumental variable (FIV) concept is proposed, and the main statistical characteristics of the FIV algorithm such as consistency and unbias are derived. Simulation results show that the proposed algorithm is relatively insensitive to the noise on the measured input-output data. This paper is organized as follow: In Section 2, a brief review of the TS fuzzy model formu- lation is given. In Section 3, the fuzzy NARX structure is introduced. It is used to formulate the proposed approach. In Section 4, the TS fuzzy model consequent parameters estimate problem in a noisy environment is studied. From this analysis, three Lemmas and one Theo- rem are proposed to show the consistency and unbias of the parameters estimates in a noisy environment with the proposed approach. The fuzzy instrumental variable concept is also proposed and considerations about how the FIV should be chosen are given. In Section 5, off- line and on-line schemes of the fuzzy instrumental variable algorithm are derived. Simulation results showing the efficiency of the FIV approach in a noisy environment are given in Section 6. Finally, the closing remarks are given in Section 7. 2. Takagi-Sugeno Fuzzy Model The TS fuzzy inference system is composed by a set of IF-THEN rules which partitions the in- put space, so-called universe of discourse, into fuzzy regions described by the rule antecedents in which consequent functions are valid. The consequent of each rule i is a functional expres- sion y i = f i (x) (King (1999); Papadakis & Theocaris (2002)). The i-th TS fuzzy rule has the following form: R i|i=1,2, ,l : IF x 1 is F i 1 AND ··· AND x q is F i q THEN y i = f i (x) (1) where l is the number of rules. The vector x ∈  q contains the antecedent linguistic variables, which has its own universe of discourse partitioned into fuzzy regions by the fuzzy sets repre- senting the linguistic terms. The variable x j belongs to a fuzzy set F i j with a truth value given by a membership function µ i F j :  → [0, 1]. The truth value h i for the complete rule i is com- puted using the aggregation operator, or t-norm, AND, denoted by ⊗ : [0, 1] × [0, 1] → [0, 1], h i (x) = µ i 1 (x 1 ) ⊗ µ i 2 (x 2 ) ⊗ . . . µ i q (x q ) (2) Among the different t-norms available, in this work the algebraic product will be used, and h i (x) = q ∏ j=1 µ i j (x j ) (3) The degree of activation for rule i is then normalized as γ i (x) = h i (x) ∑ l r =1 h r (x) (4) This normalization implies that l ∑ i=1 γ i (x) = 1 (5) The response of the TS fuzzy model is a weighted sum of the consequent functions, i.e., a convex combination of the local functions (models) f i , y = l ∑ i=1 γ i (x) f i (x) (6) which can be seen as a linear parameter varying (LPV) system. In this sense, a TS fuzzy model can be considered as a mapping from the antecedent (input) space to a convex region (poli- tope) in the space of the local submodels defined by the consequent parameters, as shown in Fig. 1 (Bergsten (2001)). This property simplifies the analysis of TS fuzzy models in a robust polytope model 4 model 3 model 2 model n Antecedent space (IF) submodels space (THEN) Rules model 1 Fig. 1. Mapping to local submodels space. linear system framework for identification, controllers design with desired closed loop char- acteristics and stability analysis (Johansen et al. (2000); Kadmiry & Driankov (2004); Tanaka et al. (1998); Tong & Li (2002)). 3. Fuzzy Structure Model The nonlinear input-output representation is often used for building TS fuzzy models from data, where the regression vector is represented by a finite number of past inputs and outputs of the system. In this work, the nonlinear autoregressive with exogenous input (NARX) struc- ture model is used. This model is applied in most nonlinear identification methods such as neural networks, radial basis functions, cerebellar model articulation controller (CMAC), and Stochastic Control198 also fuzzy logic (Brown & Harris (1994)). The NARX model establishes a relation between the collection of past scalar input-output data and the predicted output y (k + 1) = F[y(k) , . . . ,y(k − n y + 1), u(k), . . ., u(k − n u + 1)] (7) where k denotes discrete time samples, n y and n u are integers related to the system’s order. In terms of rules, the model is given by R i : IF y(k) is F i 1 AND ··· AND y(k − n y + 1) is F i n y AND u(k) is G i 1 AND ··· AND u(k − n u + 1) is G i n u THEN ˆ y i (k + 1) = n y ∑ j=1 a i,j y(k − j + 1) + n u ∑ j=1 b i,j u(k − j + 1) + c i (8) where a i,j , b i,j and c i are the consequent parameters to be determined. The inference formula of the TS fuzzy model is a straightforward extension of (6) and is given by y (k + 1) = ∑ l i =1 h i (x) ˆ y i (k + 1) ∑ l i =1 h i (x) (9) or y (k + 1) = l ∑ i=1 γ i (x) ˆ y i (k + 1) (10) with x = [y(k), . . ., y(k − n y + 1), u(k), . . ., u(k − n u + 1)] (11) and h i (x) is given as (3). This NARX model represents multiple input and single output (MISO) systems directly and multiple input and multiple output (MIMO) systems in a de- composed form as a set of coupled MISO models. 4. Consequent Parameters Estimate The inference formula of the TS fuzzy model in (10) can be expressed as y (k + 1) = γ 1 (x k )[a 1,1 y(k) + . . . + a 1,ny y(k −n y + 1) + b 1,1 u(k) + . . . + b 1,nu u(k −n u + 1) + c 1 ] + γ 2 (x k )[a 2,1 y(k) + . . . + a 2,ny y(k −n y + 1) + b 2,1 u(k) + . . . + b 2,nu u(k −n u + 1) + c 2 ] + . . . + γ l (x k )[a l,1 y(k) + . . . + a l,ny y(k −n y + 1) + b l,1 u(k) + . . . + b l,nu u(k −n u + 1) + c l ] (12) which is linear in the consequent parameters: a, b and c. For a set of N input-output data pairs {(x k , y k )|i = 1, 2, . . . , N} available, the following vetorial form is obtained Y = [ψ 1 X, ψ 2 X, . . . , ψ l X]θ + Ξ (13) where ψ i = diag(γ i (x k )) ∈  N×N , X = [y k , . . ., y k−ny+1 , u k , . . ., u k−nu+1 , 1] ∈  N×(n y +n u +1) , Y ∈  N×1 , Ξ ∈  N×1 and θ ∈  l(n y +n u +1)×1 are the normalized membership degree matrix of (4), the data matrix, the output vector, the approximation error vector and the estimated parameters vector, respectively. If the unknown parameters associated variables are exactly known quantities, then the least squares method can be used efficiently. However, in practice, and in the present context, the elements of X are no exactly known quantities so that its value can be expressed as y k = χ T k θ + η k (14) where, at the k-th sampling instant, χ T k = [γ 1 k (x k + ξ k ), . . ., γ l k (x k + ξ k )] is the vector of the data with error in variables, x k = [y k−1 , . . ., y k−n y , u k−1 , . . ., u k−n u , 1] T is the vector of the data with exactly known quantities, e.g., free noise input-output data, ξ k is a vector of noise associated with the observation of x k , and η k is a disturbance noise. The normal equations are formulated as [ k ∑ j=1 χ j χ T j ] ˆ θ k = k ∑ j=1 χ j y j (15) and multiplying by 1 k gives { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } ˆ θ k = 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]y j Noting that y j = χ T j θ + η j , { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } ˆ θ k = 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] [ γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T θ + 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]η j (16) and ˜ θ k = { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } −1 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]η j (17) Fuzzy identication of discrete time nonlinear stochastic systems 199 also fuzzy logic (Brown & Harris (1994)). The NARX model establishes a relation between the collection of past scalar input-output data and the predicted output y (k + 1) = F[y(k) , . . . ,y(k − n y + 1), u(k), . . ., u(k − n u + 1)] (7) where k denotes discrete time samples, n y and n u are integers related to the system’s order. In terms of rules, the model is given by R i : IF y(k) is F i 1 AND ··· AND y(k − n y + 1) is F i n y AND u(k) is G i 1 AND ··· AND u(k − n u + 1) is G i n u THEN ˆ y i (k + 1) = n y ∑ j=1 a i,j y(k − j + 1) + n u ∑ j=1 b i,j u(k − j + 1) + c i (8) where a i,j , b i,j and c i are the consequent parameters to be determined. The inference formula of the TS fuzzy model is a straightforward extension of (6) and is given by y (k + 1) = ∑ l i =1 h i (x) ˆ y i (k + 1) ∑ l i =1 h i (x) (9) or y (k + 1) = l ∑ i=1 γ i (x) ˆ y i (k + 1) (10) with x = [y(k), . . ., y(k − n y + 1), u(k), . . ., u(k − n u + 1)] (11) and h i (x) is given as (3). This NARX model represents multiple input and single output (MISO) systems directly and multiple input and multiple output (MIMO) systems in a de- composed form as a set of coupled MISO models. 4. Consequent Parameters Estimate The inference formula of the TS fuzzy model in (10) can be expressed as y (k + 1) = γ 1 (x k )[a 1,1 y(k) + . . . + a 1,ny y(k −n y + 1) + b 1,1 u(k) + . . . + b 1,nu u(k −n u + 1) + c 1 ] + γ 2 (x k )[a 2,1 y(k) + . . . + a 2,ny y(k −n y + 1) + b 2,1 u(k) + . . . + b 2,nu u(k −n u + 1) + c 2 ] + . . . + γ l (x k )[a l,1 y(k) + . . . + a l,ny y(k −n y + 1) + b l,1 u(k) + . . . + b l,nu u(k −n u + 1) + c l ] (12) which is linear in the consequent parameters: a, b and c. For a set of N input-output data pairs {(x k , y k )|i = 1, 2, . . . , N} available, the following vetorial form is obtained Y = [ψ 1 X, ψ 2 X, . . . , ψ l X]θ + Ξ (13) where ψ i = diag(γ i (x k )) ∈  N×N , X = [y k , . . ., y k−ny+1 , u k , . . ., u k−nu+1 , 1] ∈  N×(n y +n u +1) , Y ∈  N×1 , Ξ ∈  N×1 and θ ∈  l(n y +n u +1)×1 are the normalized membership degree matrix of (4), the data matrix, the output vector, the approximation error vector and the estimated parameters vector, respectively. If the unknown parameters associated variables are exactly known quantities, then the least squares method can be used efficiently. However, in practice, and in the present context, the elements of X are no exactly known quantities so that its value can be expressed as y k = χ T k θ + η k (14) where, at the k-th sampling instant, χ T k = [γ 1 k (x k + ξ k ), . . ., γ l k (x k + ξ k )] is the vector of the data with error in variables, x k = [y k−1 , . . ., y k−n y , u k−1 , . . ., u k−n u , 1] T is the vector of the data with exactly known quantities, e.g., free noise input-output data, ξ k is a vector of noise associated with the observation of x k , and η k is a disturbance noise. The normal equations are formulated as [ k ∑ j=1 χ j χ T j ] ˆ θ k = k ∑ j=1 χ j y j (15) and multiplying by 1 k gives { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } ˆ θ k = 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]y j Noting that y j = χ T j θ + η j , { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } ˆ θ k = 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] [ γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T θ + 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]η j (16) and ˜ θ k = { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } −1 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]η j (17) Stochastic Control200 where ˜ θ k = ˆ θ k −θ is the parameter error. Taking the probability in the limit as k → ∞, p.lim ˜ θ k = p.lim { 1 k C −1 k 1 k b k } (18) with C k = k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T b k = k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]η j Applying Slutsky’s theorem and assuming that the elements of 1 k C k and 1 k b k converge in probability, we have p.lim ˜ θ k = p.lim 1 k C −1 k p.lim 1 k b k (19) Thus, p.lim 1 k C k = p.lim 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] [ γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T p.lim 1 k C k = p.lim 1 k k ∑ j=1 (γ 1 j ) 2 (x j + ξ j )(x j + ξ j ) T + . . . + p.lim 1 k k ∑ j=1 (γ l j ) 2 (x j + ξ j )(x j + ξ j ) T Assuming x j and ξ j statistically independent, p.lim 1 k C k = p.lim 1 k k ∑ j=1 (γ 1 j ) 2 [x j x T j + ξ j ξ T j ] + . . . +p.lim 1 k k ∑ j=1 (γ l j ) 2 [x j x T j + ξ j ξ T j ] p.lim 1 k C k = p.lim 1 k k ∑ j=1 x j x T j [(γ 1 j ) 2 + . . . + (γ l j ) 2 ] + p.lim 1 k k ∑ j=1 ξ j ξ T j [(γ 1 j ) 2 + . . . + (γ l j ) 2 ] (20) with ∑ l i =1 γ i j = 1. Hence, the asymptotic analysis of the TS fuzzy model consequent pa- rameters estimation is based in a weighted sum of the fuzzy covariance matrices of x and ξ. Similarly, p.lim 1 k b k = p.lim 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]η j p.lim 1 k b k = p.lim 1 k k ∑ j=1 [γ 1 j ξ j η j , . . ., γ l j ξ j η j ] (21) Substituting from (20) and (21) in (19), results p.lim ˜ θ k = {p.lim 1 k k ∑ j=1 x j x T j [(γ 1 j ) 2 + . . . + (γ l j ) 2 ]+ p.lim 1 k k ∑ j=1 ξ j ξ T j [(γ 1 j ) 2 + . . . + (γ l j ) 2 ]} −1 p.lim 1 k k ∑ j=1 [γ 1 j ξ j η j , . . . ,γ l j ξ j η j ] (22) with ∑ l i =1 γ i j = 1. For the case of only one rule (l = 1), the analysis is simplified to the linear one, with γ i j | i=1 j =1, ,k = 1. Thus, this analysis, which is a contribution of this article, is an extension of the standard linear one, from which can result several studies for fuzzy filtering and modeling in a noisy environment, fuzzy signal enhancement in communication channel, and so forth. Provided that the input u k continues to excite the process and, at the same time, the coefficients in the submodels from the consequent are not all zero, then the output y k will exist for all k observation intervals. As a result, the fuzzy covariance matrix ∑ k j =1 x j x T j [(γ 1 j ) 2 + . . . + (γ l j ) 2 ] will also be non-singular and its inverse will exist. Thus, the only way in which the asymptotic error can be zero is for ξ j η j identically zero. But, in general, ξ j and η j are correlated, the asymptotic error will not be zero and the least squares estimates will be asymptotically biased to an extent determined by the relative ratio of noise to signal variances. In other words, least squares method is not appropriate to estimate the TS fuzzy model consequent parameters in a noisy environment because the estimates will be inconsistent and the bias error will remain no matter how much data can be used in the estimation. 4.1 Fuzzy instrumental variable (FIV) To overcome this bias error and inconsistence problem, generating a vector of variables which are independent of the noise inputs and correlated with data vetor x j from the system is required. If this is possible, then the choice of this vector becomes effective to remove the asymptotic bias from the consequent parameters estimates. The fuzzy least squares estimates is given by: { 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )][γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T } ˆ θ k = 1 k k ∑ j=1 [γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )]{[γ 1 j (x j + ξ j ), . . ., γ l j (x j + ξ j )] T θ + η j } [...]... Fuzzy Systems, Vol 6, No 2, May-1998, 250– 265 , ISSN 1 063 -67 06 Tong, S & Li, H (2002) Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties Fuzzy Sets and Systems, Vol 131, No 2, Oct-2002, 165 –184, ISSN 0 165 -0114 Fuzzy frequency response for stochastic linear parameter varying dynamic systems 217 12 15 Fuzzy frequency response for stochastic linear parameter varying dynamic... 1.9302uk + 0. 961 4u2 k ˆ R2 : IF yk is F2 THEN yk = 1.0142 − 1.9308uk + 0. 961 8u2 k ˆ R3 : IF yk is F3 THEN yk = 1.01 26 − 1.9177uk + 0.9555u2 k ˆ R4 : IF yk is F4 THEN yk = 1.0123 − 1.9156uk + 0.9539u2 k Fuzzy identification of discrete time nonlinear stochastic systems 211 Global approach: ˆ R1 : IF yk is F1 THEN yk = 1.0147 − 1.9310uk + 0. 961 3u2 k ˆ R2 : IF yk is F2 THEN yk = 1.0129 − 1.9196uk + 0.9570u2... 0.8717u2 k ˆ R2 : IF yk is F2 THEN yk = 0.7 466 − 1.2077uk + 0.5872u2 k ˆ R3 : IF yk is F3 THEN yk = 0.8938 − 1.1831uk + 0.5935u2 k Global approach: ˆ R4 : IF yk is F4 THEN yk = 1.0853 − 1.4776uk + 0.7397u2 k ˆ R1 : IF yk is F1 THEN yk = 0. 062 1 − 0. 463 0uk + 0.2272u2 k ˆ R2 : IF yk is F2 THEN yk = 0.3729 − 0.3 068 uk + 0.1534u2 k ˆ R3 : IF yk is F3 THEN yk = 0.7 769 − 0.3790uk + 0.1891u2 k ˆ R4 : IF yk is... algorithm, Proceedings of 14th IEEE Conference on Fuzzy Systems, pp.1 062 –1 067 , ISBN 0-7803-9159-4, NV, May 2005, Reno Serra, G L O & Bottura, C.P (2006a) Multiobjective evolution based fuzzy PI controller design for nonlinear systems, Engineering Applications of Artificial Intelligence, Vol 19, No 2, Mar-20 06, 157– 167 Serra, G L O & Bottura, C P (2006b) An IV-QR algorithm for neuro-fuzzy multivariable on-line... Takagi-Sugeno fuzzy models, IEEE Transactions on Fuzzy Systems, Vol 8, No 3, Jun 2000, 297–313, ISSN 1 063 -67 06 Kadmiry, B & Driankov, D (2004) A fuzzy gain-scheduler for the attitude control of an unmanned helicopter, IEEE Transactions on Fuzzy Systems, Vol 12, No 3, Aug 2004, 502– 515, ISSN 1 063 -67 06 Kasabov, N.K & Song, Q (2002) DENFIS: dynamic evolving neural-fuzzy inference system and its application... Fig 3 Approximation of the polynomial function 0 0.5 1 uk 1.5 2 210 Stochastic Control Membership functions for LS estimation Membership degree 1.5 F2 F1 1 F4 0.5 0 −0.4 −0.2 0 0.2 0.4 0 .6 yk 0.8 1 1.2 1.4 1 .6 1.4 1 .6 Membership functions derived for FIV estimation 1.5 Membership degree F3 F1 1 F2 F3 F4 0.5 0 −0.4 −0.2 0 0.2 0.4 0 .6 0.8 yfiltered 1 1.2 Fig 4 Antecedent membership functions Local approach:... 500 points is created from (50) The linguistic variables partitions obtained by the ECM method are shown in Fig 5 The TS fuzzy model consequent parameters recursive estimate result is shown in 0.8 0 .6 0.4 0.2 0 membership degree F2 F1 1 −2 −1 0 1 2 yk 3 4 G2 G1 1 5 0.8 0 .6 0.4 0.2 0 −2 −1 0 1 yk−1 2 3 4 5 Fig 5 Antecedent membership functions Fig 6 The coefficient of determination, widely used in analysis... on Fuzzy Systems, Vol 15; No 2, Apr-2007, 200–210, ISSN 1 063 -67 06 Sjöberg, J.; Zhang, Q.; Ljung, L.; Benveniste, A.; Delyon, B.; Glorennec, P.; Hjalmarsson, H & Juditsky, A (1995) Nonlinear black-box modeling in system identification : an unified overview, Automatica: Special issue on trends in system identification, Vol 31, No 12, Dec-1995, 169 1–1724, ISSN 0005-1098 Söderström, T and Stoica, P (1989)... Springer 2 16 Stochastic Control Takagi, T & Sugeno, M (1985) Fuzzy identification of systems and its aplication to modeling and control, IEEE Transactions on Systems, Man and Cybernetics, Vol 15, No 1, 1 16 132, ISSN: 1083-4419 Tanaka, K.; Ikeda, T & Wang, H (1998) Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs, IEEE Transactions on Fuzzy Systems, Vol 6, No 2,... application for time-series prediction, IEEE Transactions on Fuzzy Systems, Vol 10, No 2, Apr 2002, 144–154, ISSN 1 063 -67 06 King, R.E (1999) Computational intelligence in control engineering, Marcel Dekker Ljung, L (1999) System Identification: Theory for the user, Prentice Hall Narendra, K.S & Parthasarathy, K (1990) Identification and control of dynamical systems using neural networks, IEEE Transactions . K. N. & Vasermanis, E. K. (2003). Effective state estimation of stochastic systems. Kybernetes, Vol. 32, pp. 66 6  67 8 Nechval, N.A. & Vasermanis, E.K. (2004). Improved Decisions. K. N. & Vasermanis, E. K. (2003). Effective state estimation of stochastic systems. Kybernetes, Vol. 32, pp. 66 6  67 8 Nechval, N.A. & Vasermanis, E.K. (2004). Improved Decisions. function. Stochastic Control210 −0.4 −0.2 0 0.2 0.4 0 .6 0.8 1 1.2 1.4 1 .6 0 0.5 1 1.5 Membership functions for LS estimation y k Membership degree −0.4 −0.2 0 0.2 0.4 0 .6 0.8 1 1.2 1.4 1 .6 0 0.5 1 1.5 Membership