On Nonlinear Adaptive Control of a Handling Crane EUGEN BOBAŞU, DAN SELIŞTEANU, DAN POPESCU, DORIN ŞENDRESCU Department of Automatic Control University of Craiova A.I. Cuza Str. No. 13, RO-200585 Craiova ROMANIA http://www.automation.ucv.ro Abstract: - This paper presents some nonlinear adaptive control techniques for the control of a handling crane. The nonlinear model of the handling crane is widely analyzed and an exactly linearizing feedback control law is designed. This nonlinear control law, aggregated with our nonlinear system achieves input-output linearization. When some parameters of the system are imprecisely known or unknown, an adaptive control strategy is designed. Several computer simulations are included to demonstrate some theoretical aspects and the performances of the controlled system. Key-Words: - Nonlinear systems, Adaptive control, Linearizing control, Handling crane 1 Introduction In the last years, significant advances have been made in the development of ideas such as feedback linearizing techniques. The problem of exact linearization via feedback and diffeomorphism consists in transforming a nonlinear system into a linear one using a state feedback and a coordinate transformation of the systems state [5], [7]. Several application of the exactly linearization technique are reported for the control of the electric motors, chemical and biochemical reactors, robotic manipulators and so on [2], [3], [4], [5]. In this paper, by using the feedback linearizing technique, a nonlinear control law for a handling crane is obtained. The control goal is the regulation of the position of the load of a handling crane. This nonlinear control method provides an alternative solution to existing classical linear methods for the control of handling cranes. For the implementation of the nonlinear control law we suppose that all states are measurable (otherwise a state observer can be used in order to estimate the state variables). In many practical situations, some handling crane parameters (such as load, height) are unknown; therefore an adaptive control strategy is required in order to maintain the performances of the controlled system (for a general point of view regarding the adaptive control theory see [1], [8]). In this paper, an adaptive control law based on reference model for the exactly linearized model is designed. The paper is organized as follows: in Section 2, the mathematical theoretical fundaments of the exact linearization technique are briefly presented, while the Section 3 deals with the nonlinear model of a handling crane. Section 4 presents the input – output linearization technique for the handling crane, used in order to obtain the nonlinear control law; also the adaptive control law is designed. Some computer simulation results for different parameters of the controlled system are presented. Finally, in Section 5 some concluding remarks are collected. 2 Theoretical fundaments of the exact linearization technique The nonlinear system that we consider is described in state space by equations of the following kind: m j x h y u x g x f x j j i m i i ... 1 ) ( ) ( ) ( 1 = = + = ∑ = & (1) in which (x) g (x),...., (x), g f(x), g m 2 1 are smooth vector fields. The exact linearization via feedback and diffeomorphism consists in transforming the nonlinear system (1) into a linear one using a state feedback and a coordinate transformation of the systems state. We do not develop the details of input-output linearization techniques (for details see [7]) but directly show the application on the handling crane. This can be done introducing the Lie derivative of a function R R x h n → : ) ( along a vector field )] ( ),... ( [ ) ( 1 x f x f x f n = ∑ = ∂ ∂ = n i i i f x f x x h x h L 1 ) ( ) ( ) ( (2) Proceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 17-19, 2005 (pp123-128)