Numerical algorithm for feedback linearizable systems

Abstract

Abstract.A numerical algorithm that achieves asymptotic stability for feedback linearizable systems is presented. The nonlinear systems can be represented in various forms that include differential equations, simulated physical models or lookup tables. The proposed algorithm is based on a quotient method and proceeds iteratively. At each step, the dynamic system is desensitized with respect to the current input vector field. Control is obtained by tracking a desired value along the input vector field at each step. The numerical algorithm uses the direction on the tangent manifold at a given point and its variation around that point. This enables the algorithm to produce control values simply using a simulator of the nonlinear system. Keywords: Numerical algorithms, feedback-linearization, quotient, control law design, nonlinear dynamical systems PACS: 05.10.-a Feedback linearization is an effective method to handle nonlinear systems. However, it requires exact knowledge of the system and, furthermore, there exist conditions for a nonlinear system to be feedback linearizable (FL). Moreover, for feedback linearization, an output function of relative degree n must be known [1,2], where n is the dimension of the system. Computing such an output requires a systematic procedure, such as the one proposed in [3,4], which proceeds by successively generating quotients that are desensitized with respect to the input vector field. This method has been extended to produce a control design technique applicable to FL input-affine single-input systems of the formFor non-FL systems, the method requires approximations [6,7]. However, for some non-FL systems, the system of equations turns out to be so complicated that, even after approximations, the method is not applicable due to the lack of closed-form solutions to some of the equations. This difficulty has motivated the use of numerical algorithms to compute the control input. This paper presents a numerical algorithm that computes the input for a given state of the system. The algorithm requires the time derivatives of the states at particular state and input values. The time derivatives can be obtained from the system's differential equations (through traditional modeling) or through advanced computer-generated physical modeling. Computer-generated physical modeling is an interactive technique that connects electro-mechanical components to simulate dynamical systems [8]. Simscape™ is such a simulation tool. With it, a designer can simulate complicated electro-mechanical systems without having full knowledge of the underlying differential equations. The inability of these tools to provide the governing differential equations make them unsuited for developing nonlinear control laws. Using the proposed numerical algorithm, it is possible to harness the simulation capabilities of such tools. Moreover, if measured data are available for a system, it is also possible to compute the derivatives via numerical differentiation and use that for the proposed numerical algorithm NUMERICAL...