Abstract
This paper is concerned with the robust stabilization of a class of continuous-time nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, via an affine state-feedback control law using the linear matrix inequality (LMI) approach. The nonlinear dynamics is subject to norm-bounded parametric uncertainties and disturbances. In addition, the problem of actuator nonlinearity is addressed by considering the saturation effect of the control law. We demonstrate first that the synthesis problem of the saturated controller is expressed in terms of bilinear matrix inequalities (BMIs). Thanks to the Schur complement lemma and the matrix inversion lemma, we convert these BMIs into LMIs allowing the simultaneous computation of the two gains of the affine controller. Furthermore, we address in this work the estimation problem of the domain of attraction using the invariant set concept. This is solved by computing the largest attractive invariant ellipsoid. Compared with previous works, the research procedure of such ellipsoidal set is achieved in a single step with a reduced number of LMI constraints and then with less conservative conditions. A portfolio of numerical results is presented. The effectiveness and robustness of the proposed saturated controller in the stabilization of the adopted helicopter pitch model toward parametric uncertainties and disturbances are illustrated through simulation results.
Highlights
IntroductionThere is an ever-increasing demand of advanced control strategies for mechatronic systems with enhanced performances
Remark 6. e choice of the parameters of the maximum value of the control input u, umax, is based on two facts: (1) in the literature, the common choice of the saturation limit is umax 1 according to the invariance sets’ concept and (2) in the present work, we have considered a general case of the actuator saturation limit in the development of linear matrix inequality (LMI) constraints and in the design of the state-feedback controller in order to show the effect of the saturation limit umax on the largeness of the invariant attractive ellipsoid
An LMI-based approach for designing a robust affine state-feedback control law to stabilize the pitch dynamics of a helicopter model was proposed. e nonlinear dynamics of the helicopter was subject to an external disturbance and norm-bounded parametric uncertainties
Summary
There is an ever-increasing demand of advanced control strategies for mechatronic systems with enhanced performances. It is known, on the one hand, that almost all existing physical and mechatronic systems unavoidably include uncertainties and disturbances due to inaccurate modeling, measurement errors, exterior conditions, or parameter variations. The linear matrix inequality (LMI) technique [9] has been widely used to solve the robust control for uncertain linear and nonlinear systems with polytopic uncertain parameters and norm-bounded uncertain parameters. Most control synthesis problems cannot be written in a LMI form They are written in terms of a more general form known as a bilinear matrix inequality (BMI), which is usually not exploitable numerically to solve.
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