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 approximamay 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-