may utilize nonlinear function approximation. One such application is system identification, which is the process of constructing a mathematic model of a dynamic system using experimental data from that system. Let g denote the physical system that we wish to identify. The training set G is defined by the experimental input-output data In linear system identification, a model is often used where J(k)=20a,x(k-1+20,u(k-D) and u(k)and y(k) are the system input and output at time k20. Notice that you will need to specify appropriate initial conditions. In this case f(x0), which is not a fuzzy system, is defined by f(x0)=0'x where x(k)=[y(k-1),,y(k-q),u(k)2…,u(k-q) (3.9) [an,O2…, Let N=q+p+1 so that x(k) and 0 are NxI vectors. Linear system identification amounts to adjusting 0 using information from G so that g(x)=f(xo)+ e(x)where e(x) is small for all xEX Similar to conventional linear system identification for fuzzy identification we will utilize an appropriately defined"regression vector x as specified in Equation(3.9), and we will tune a fuzzy system f(xo) so that e(x)is small. Our hope is that since the fuzzy system f(e) has more functional capabilities(as characterized by the universal approxima
may utilize nonlinear function approximation. One such application is system identification, which is the process of constructing a mathematical model of a dynamic system using experimental data from that system. Let g denote the physical system that we wish to identify. The training set G is defined by the experimental input-output data. In linear system identification, a model is often used where 1 1 () ( ) ( q q i bi i i y k yk i uk ) θ θ α = = = −+ ∑ ∑ − i (3.8) and u(k) and y(k) are the system input and output at time . Notice that you will need to specify appropriate initial conditions. In this case k 0 ≥ f (x θ), which is not a fuzzy system, is defined by ( ) T f x x θ =θ where ( ) [ ( 1),..., ( ), ( ),..., ( )]T x k yk yk q uk uk q =− − − (3.9) 1 1 ,..., , ,..., T a b aq bq θ θ θθ θ = ⎡ ⎣ ⎤ ⎦ (3.10) Let N=q+p+1 so that x (k) and θ are N 1× vectors. Linear system identification amounts to adjusting θ using information from G so that g(x) = f (x θ) + e(x) where e(x) is small for all x∈X. Similar to conventional linear system identification, for fuzzy identification we will utilize an appropriately defined "regression vector" x as specified in Equation (3.9), and we will tune a fuzzy system f ( ) x θ so that e(x) is small. Our hope is that since the fuzzy system f (x θ) has more functional capabilities (as characterized by the universal approxima-
tion property described in Section 2.3.8 on page 72) than the linear map defined in Equation (3.8), we will be able to achieve more accurate identification for nonlinear systems by appropriate adjustment of its parameters 0 of the fuzzy system Next, consider how to view the construction of a parameter(or state)estimator as a function approximation problem. To do this, suppose for the sake of illustration that we seek to construct an estimator for a single parameter in a system g. Suppose further that we conduct a set of experiments with the system g in which we vary a parameter in the system-say, a. For instance, suppose we know that the parameter a lies in the range [amin, ama] but we do not know where it lies and hence we would like to estimate it Generate a data set G with data pairs (x, aEG where the a' are a range of values over the interval [amin, am] and the corresponding to each a'is a set of input-output data from the system g in the form of Equation(3.9)that results from using a' as the parameter value in g. Let a denote the fuzzy system estimate of a. Now, if we construct a function a=f(xo) from the data in G, it will serve as an estimator for the parameter a. Each time a new x vector is encountered the estimator f will interpolate between the known associations (x', a)eG to produce the estimate a. Clearly, if the data set G is"rich" enough, it will have enough (x', a) pairs so that when the estimator is presented with an x+x', it will have a good idea of what a to specify
tion property described in Section 2.3.8 on page 72) than the linear map defined in Equation (3.8), we will be able to achieve more accurate identification for nonlinear systems by appropriate adjustment of its parameters θ of the fuzzy system. Next, consider how to view the construction of a parameter (or state) estimator as a function approximation problem. To do this, suppose for the sake of illustration that we seek to construct an estimator for a single parameter in a system g. Suppose further that we conduct a set of experiments with the system g in which we vary a parameter in the system— say, α . For instance, suppose we know that the parameterα lies in the range but we do not know where it lies and hence we would like to estimate it. Generate a data set G with data pairs min max [a , a ] (, ) i i x α ∈G where the i α are a range of values over the interval min max [ , α α ] and the i x corresponding to each i α is a set of input-output data from the system g in the form of Equation (3.9) that results from using i α as the parameter value in g. Let α denote the fuzzy system estimate of α . Now, if we construct a function α = f (x θ) from the data in G, it will serve as an estimator for the parameter α . Each time a new x vector is encountered, the estimator f will interpolate between the known associations (, ) i i x α ∈G to produce the estimate α . Clearly, if the data set G is "rich" enough, it will have enough ( , ) i i x α pairs so that when the estimator is presented with an i x ≠ x , it will have a good idea of what a to specify
because it will have many x' that are close to x that it does know how to specify a for. We will study several applications of parameter estimation in this chapter and in the problems at the end of the chapter To apply function approximation to the problem of how to construct a predictor for a parameter(or state variable) in a system, we can proceed in a similar manner to how we did for the parameter estimation case above. The only significant difference lies in how to specify the data set G. In the case of prediction, suppose that we wish to estimate a parameter a(k+ D), D time steps into the future. In this case we will need to have available training data pairs (x, a(k+D))EG that associate known future values of a with available data x'. a fuzz system constructed from such data will provide a predicted value a(k+D)=f(re) forgiven values of x Overall. notice that in each case-identification. estimation and prediction--we rely on the existence of the data set G from which to construct the fuzzy system. Next, we discuss issues in how to choose the data set g 3.2.3 Choosing the data Set While the method for adjusting the parameters 0 of f(x0) is critical to the overall success of the approximation method, there is virtually no way that you can succeed at having f approximate g if there is not appropriate information present in the training data set G. Basically, we
because it will have many i x that are close to x that it does know how to specify α for. We will study several applications of parameter estimation in this chapter and in the problems at the end of the chapter. To apply function approximation to the problem of how to construct a predictor for a parameter (or state variable) in a system, we can proceed in a similar manner to how we did for the parameter estimation case above. The only significant difference lies in how to specify the data set G. In the case of prediction, suppose that we wish to estimate a parameter α(k D+ ) , D time steps into the future. In this case we will need to have available training data pairs ( , ( )) i x α kD G + ∈ that associate known future values of a with available data i x . A fuzzy system constructed from such data will provide a predicted value α( ) () k D fx + = θ forgiven values of x . Overall, notice that in each case-identification, estimation, and prediction—we rely on the existence of the data set G from which to construct the fuzzy system. Next, we discuss issues in how to choose the data set G. 3.2.3 Choosing the Data Set While the method for adjusting the parameters θ of f (x θ) is critical to the overall success of the approximation method, there is virtually no way that you can succeed at having f approximate g if there is not appropriate information present in the training data set G. Basically, we