Tracking control of variable stiffness hysteretic-systems using linear-parameter-varying gain-scheduled controller

Tracking control of systems with variable stiffness hysteresis using a gain-scheduled (GS) controller is developed in this paper. Variable stiffness hysteretic system is represented as quasi linear parameter dependent system with known bounds on parameters. Assuming that the parameters can be measured or estimated in real-time, a GS controller that ensures the performance and the stability of the closed-loop system over the entire range of parameter variation is designed. The proposed method is implemented on a spring-mass system which consists of a semi-active independently variable stiffness (SAIVS) device that exhibits hysteresis and precisely controllable stiffness change in real-time. The SAIVS system with variable stiffness hysteresis is represented as quasi linear parameter varying (LPV) system with two parameters: linear time-varying stiffness (parameter with slow variation rate) and stiffness of the friction-hysteresis (parameter with high variation rate). The proposed LPV-GS controller can accommodate both slow and fast varying parameter, which was not possible with the controllers proposed in the prior studies. Effectiveness of the proposed controller is demonstrated by comparing the results with a fixed robust <TEX>$\mathcal{H}_{\infty}$</TEX> controller that assumes the parameter variation as an uncertainty. Superior performance of the LPV-GS over the robust <TEX>$\mathcal{H}_{\infty}$</TEX> controller is demonstrated for varying stiffness hysteresis of SAIVS device and for different ranges of tracking displacements. The LPV-GS controller is capable of adapting to any parameter changes whereas the <TEX>$\mathcal{H}_{\infty}$</TEX> controller is effective only when the system parameters are in the vicinity of the nominal plant parameters for which the controller is designed. The robust <TEX>$\mathcal{H}_{\infty}$</TEX> controller becomes unstable under large parameter variations but the LPV-GS will ensure stability and guarantee the desired closed-loop performance.