Robust Control of Mechanical Systems

Control of mechanical systems has been an important problem since several years ago. For free-motion systems, the dynamics is often modeled by ordinary differential equations arising from classical mechanics. Controllers based on feedback linearization, adaptive, and robust techniques have been proposed to control this class of systems (Brogliato et al., 1997; Slotine & Li, 1988; Spong & Vidyasagar, 1989). Many control algorithms proposed for these systems are based on models where practical situations like parameter uncertainty, external disturbances, or friction force terms are not taken into account. In addition, a complete availability of the state variables is commonly assumed (Paden & Panja, 1988; Takegaki & Arimoto, 1981; Wen & Bayard, 1988). In practice, however, the position is usually the only available measurement. In consequence, the velocity, which may play an important role in the control strategy, must be calculated indirectly, often yielding an inaccurate estimation. In (Makkar et al., 2007), a tracking controller that includes a new differentiable friction model with uncertain nonlinear terms is developed for Euler-Lagrange systems. The technique is based on a model and the availability of the full state. In (Patre et al., 2008), a similar idea is presented for systems perturbed by external disturbances. Moreover, some robust controllers have been proposed to cope with parameter uncertainty and external disturbances. H∞ control has been a particularly important approach. In this technique, the control objective is expressed as a mathematical optimization problem where a ratio between some norms of output and perturbation signals is minimized (Isidori & Astolfi, 1992). It is used to synthesize controllers achieving robust performance of linear and nonlinear systems. In general, the control techniques mentioned before yield good control performance. However, the mathematical operations needed to calculate the control signal are rather complex, possibly due to the compensation of gravitational, centrifugal, or Coriolis terms, or the need to solve a Hamilton-Jacobi-Isaacs equation. In addition, if an observer is included in the control system, the overall controller may become rather complex. Another method exhibiting good robustness properties is the sliding mode technique (Perruquetti & Barbot, 2002; Utkin, 1992). In this method, a surface in the state space is made attractive and invariant using discontinuous terms in the control signal, forcing the system to converge to the desired equilibrium point placed on this surface, and making the controlled dynamics independent from the system parameters. These controllers display good performance for regulation and tracking objectives (Utkin et al., 1999; Weibing & Hung, 1993; 8