LMI-based design of an I-PD+PD type LPV state feedback controller for a gantry crane

Abstract

In this paper, an alternative gain-scheduled PID tuning procedure is proposed for gantry crane control systems. In order to avoid excessive overshoot and aggressive control action due to the proportional kick and/or derivative kick effects, an I-PD+PD type control law is considered. The scheduling parameter is considered the cable length due to the payload lifting and lowering movements. A linear parameter-varying (LPV) gantry crane model is constructed to enable gain-scheduled controller design with a linear matrix inequalities (LMIs) framework. Based on the LPV model, a convex optimization problem is formulized to minimize L2 gain under regional pole location constraints. Then, a fixed gain L2 gain state feedback I-PD+PD type controller and a conventional pole placement state feedback I-PD+PD controller are designed to investigate the efficiency of the proposed controller. A pole placement controller is tuned to minimize the very common ITAE (integral of time multiplied by absolute error) performance index. Simulation results show that the proposed controller has superior tracking performance under time-varying cable length, when compared with nominal fixed gain controllers.

LMI-Based Robust Stabilization of a Class of Input-Constrained Uncertain Nonlinear Systems with Application to a Helicopter Model

“…A set S ⊂ R n x is said to be invariant with respect to motion of system (14), if for all initial conditions x(0) ∈ S, the motion of the system x(t) remains in S for all t > 0.…”

“…Condition (22) will be used in the sequel in order to design the robust stabilization conditions of the closed-loop nonlinear system (15) with the norm-bounded parametric uncertainties (17) under the bounded external disturbance (14) and subject to the saturation (18) of the affine statefeedback control law (19) under condition (35).…”

“…e nonlinear system (15) with normbounded parametric uncertainties (17), under the bounded external disturbance (14), is robustly stabilizable by implementing the affine state-feedback control law (19) subject to the saturation constraint (18), if for some fixed positive parameters c, ε m , δ a , δ b , δ c , δ d , and δ e , there exist a symmetric matrix P > 0, a feedback matrix K, a feedback constant m, and some positive scalars…”

“…We now consider condition (22) on the Lyapunov function V(x) and also conditions (10), (14), and (17). en, using expressions of the uncertainties matrices in (24) and relying on the Young relation (Lemma 2), we obtain the following condition:…”

“…en, the nonlinear system (15) with norm-bounded parametric uncertainties (17) and bounded disturbance (14) is robustly stabilizable by means of the saturated affine statefeedback control law (19) subject to constraint (18). Moreover, the matrix gain K of the control law (19) is…”