11 Linear Parameter-Varying (LPV) Buckling Control of an Imperfect Beam: : : 109 [13]. Equation (11.13) can also be written in short form Pxm DAm.Fx/xmCBmu y DCmxm: (11.14) Due to imperfections such as predeformation, eccentric loading or clamping moments that are present in a real beam-column system, the controller needs to have an additional integral term to avoid a static controller error. Therefore, the modal state vector xm is augmented by the integral of the modal displacements xint DZ 1 tLPV qdt (11.15) starting fromtLPV to get the newŒ6 1 state vector x DŒxm; xint T. With the first derivative of the new state vector Pxint Dq D I 0 xm; (11.16) the augmented state space system including the integral term can be written as Px D" Am.Fx/ 0 I 0 0# „ ƒ‚ … Œ6 6 xC Bm 0 „ƒ‚… Œ6 2 u y D Cm 0 „ ƒ‚ … Œ2 6 x; (11.17) [14]. In short form, the final state space system (11.17) of the beam-column system in Fig. 11.1 is written as Px DA.Fx/xCBu y DCx: (11.18) 11.3.2 Quadratically Stable Gain-Scheduled LPV Control The FE state space model (11.6) as well as the final controller state space model (11.18) are LPV systems in which the system matrices AFE.Fx/ and A.Fx/ depend on axial load Fx. There are different control approaches to deal with the parameterdependency of LPV systems. One approach is to use robust control in which a single controller is used for all occurring axial load variations with respect to amplitude and time-dependency. Due to the large variation of axial loads and the transition from sub- to supercritical axial loads, robust control is not favorable and has not been used so far. The approach pursued in earlier own studies [7–10] calculated static control matrices for a number of different axial loads that were manually switched, resulting in discontinuities in control input (11.3). Now, in this investigation, active buckling control of the circular beam-column is achieved by a quadratically stable gain-scheduled LPV control. In this approach, a continuous control input according to (11.3) with u DKLPV.Fx/x; (11.19) is achieved by Œ2 6 control matrixKLPV.Fx/ as a linear function of the axial load Fx with KLPV.Fx/ D Fx;2 Fx Fx;2 Fx;1 K1 C Fx Fx;1 Fx;2 Fx;1 K2: (11.20) In (11.20) Fx;1 D300N and Fx;2 D4000N are the minimum and maximum considered axial loads for the controller that define the vertices of the LPV systemA1 DA.Fx;1/ and A2 DA.Fx;2/ and K1 and K2 are the control matrices calculated for the respective systems. The resulting controller stabilizes the beam-column for the entire range of considered axial loads [15].
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