4. Linearizing Control and Estimation for Nonlinear Dynamics in Financial Systems

A flatness-based adaptive fuzzy control is applied to the problem of stabilization of the dynamics of a chaotic finance system, describing interaction between the interest rate, the investment demand and the price exponent. First it is proven that the system is differentially flat. This implies that all its state variables and its control inputs can be expressed as differential functions of a specific state variable, which is a so-called flat output. It also implies that the flat output and its derivatives are differentially independent which means that they are not connected to each other through an ordinary differential equation. By proving that the system is differentially flat and by applying differential flatness diffeomorphisms, its transformation to the linear canonical (Brunovsky) is performed. For the latter description of the system, the design of a stabilizing state feedback controller becomes possible. A first problem in the design of such a controller is that the dynamic model of the finance system is unknown and thus it has to be identified with the use neurofuzzy approximators. The estimated dynamics provided by the approximators is used in the computation of the control input, thus establishing an indirect adaptive control scheme. The learning rate of the approximators is chosen from the requirement the system’s Lyapunov function to have always a negative first-order derivative. Another problem that has to be dealt with is that the control loop is implemented only with the use of output feedback. To estimate the non-measurable state vector elements of the finance system, a state observer is implemented in the control loop. The computation of the feedback control signal requires the solution of two algebraic Riccati equations at each iteration of the control algorithm. Lyapunov stability analysis demonstrates first that an H-infinity tracking performance criterion is satisfied. This signifies elevated robustness against modelling errors and external perturbations. Moreover, the global asymptotic stability is proven for the control loop.