Barrier Lyapunov Function Based Constrained Robust Nonlinear Output Regulation

On one hand, in engineering practice, physical systems often subject to somephysical or geometrical constraints. One example of this is the torsional MEMSmicromirror that in micro-scale. Due to the so-called pull-in phenomena, the movablemirror plate might crash abruptly into the fixed bottom electrode when it is rotatedbeyond some range, which brings damage to the surface of the mirror plate and thuslowers its lifetime. The durability of the mirror plate to a large extent determines thereliability and longevity of the micromirror trauma surgery device. On the other hand,the control coefficients of practical systems are usually not available, which makes thecontrol design more difficult.Considering all these, this paper aims to deal with the global robust nonlinearoutput regulation problem with an unknown control direction, a nonlinear exosystemand a tracking error constraint. Firstly, by designing a proper internal model andemploying an input and coordinate transformation, the output regulation problem ofthe original system subject to a tracking error constraint can be converted into thestabilization problem of the augment system subject to an output constraint. Secondly,since the traditional quadratic Lyapunov function cannot handle the problem oftracking error constraint, a new kind of Lyapunov function called Barrier LyapunovFunction will be introduced to handle this problem. Also, the Nussbaum gaintechnique will be introduced to tackle the problem of unknown control direction. Andthe backstepping technique is utilized to design the feedback controller. This designsolves the stabilization problem of the augment system, and thus directly leads to thesolution of the output regulation problem for the original system, that is, all the closedloop signals of the original system are bounded and the tracking error asymptoticallyconverges to zero. Moreover, the tracking error will keep within its given limit barrierduring its transient period if the initial value of the tracking error is selected within thelimit barrier. Finally, a numerical example is provided to illustrate the effectiveness ofthe feedback control design, and the control design is applied to achieve theconstrained tracking control of a2D torsional MEMS micromirror with sidewallelectrodes.The highlight of this paper is as follows. Firstly, we note the facts that, inpractical application, control coefficients are difficult to obtain and oftentimes systems subject to some physical and geometrical constraints. And then it is ourconsideration of the global robust nonlinear output regulation problem with anunknown control direction and a tracking error constraint, to solve which thetechniques of Barrier Lyapunov Function and Nussbaum gain are combined, such thatall the closed loop signals are bounded and the tracking error not only converges tozero asymptotically, but also not violates its limit barrier when its initial value ischosen in the given limit barrier. Finally, the proposed control design is successfullyapplied to avoid the contact between the mirror plate and the bottom electrodes of a2D torsional MEMS micromirror with sidewall electrodes, and also expand theoperational range of the micromirror’s torsional angles.