Some new notions in the stability investigations and design of controllers for nonlinear systems

Design of an adaptive controller for complex dynamic systems is a big challenge faced by the researchers. In this paper, we introduce a novel concept of dynamic pole motion (DPM) for the design of an error-based adaptive controller (E-BAC). The purpose of this novel design approach is to make the system response reasonably fast with no overshoot, where the system may be timevarying and nonlinear with only partially known dynamics. The E-BAC is implanted in a system as a nonlinear controller with two dominant dynamic parameters, the dynamic position feedback and the dynamic velocity feedback. For illustrating the strength of this new approach, in this paper we give an example of a flexible robot with nonlinear dynamics. In the design of this feedback adaptive controller, parameters of the controller are designed as a function of the system error. The position feedback K_p(e, t) and the velocity feedback K_v(e, t) are continuously varying as a function of the system error e(t). In these feedback parameters, the position feedback K_p(e, t) controls the system bandwidth, thereby the rise time in step response, whereas the velocity feedback K_v(e, t) controls the damping ratio of the system thereby the overshoot in the step response. For large errors, K_p(e, t) is large which increases the bandwidth of the system thereby a smaller rise time. Whereas, for decreasing errors, K_p(e, t) is continuously decreased to a small value with decreasing bandwidth of the system. Similarly, but contrarily, K_v(e, t) is kept very small for large errors, and it is continuously increased to a large value for decreasing error. Hence, in the design of the proposed adaptive controller, the position feedback K_p(e, t) and the velocity feedback K_v(e, t) are formulated as functions of the system error, and this approach for formulating the adaptive controller yields a very fast response with no overshoot.