Model Predictive Control of Under-actuated Mechanical Systems

Model Predictive Control (MPC) is an optimization-based control technique that has received an increasing research interest and has been widely applied in industry. Similarly to optimal control, MPC has an inherent ability to deal naturally with constraints on the inputs and on the state. Moreover, since the controls generated are closed-loop strategies obtained by optimizing some criterion, the method possesses some desirable performance properties [8] and also intrinsic robustness properties [7]. Thousands of applications have been reported [9], making MPC being classified as the only advanced control technique with a substancial impact on industrial control [6]. Until recently, a technical difficulty prevented the continuous-time MPC approaches to be used to stabilize important classes of nonlinear systems, such as under-actuated mechanical systems which frequently appear in robotics. Such class of nonlinear systems cannot be stabilized by a continuous time-invariant feedback. As a consequence, the trajectories cannot be described by classical (Caratheodory) solutions to differential equations.This difficulty is overcame in the framework proposed in [1] that describes how a continuous-time MPC framework using a positive inter-sampling time, combined with the use of an appropriate concept of solution to a differential equation, can address nonholonomic systems. An implementation of an MPC strategy to control a wheeled mobile robot using the results above is reported in [4, 5]. An MPC framework with guaranteed robustness properties addressing a general class of nonlinear systems is discussed in [2, 3]. Preliminary results on the implementation of a robust MPC strategy, devised in the above references, to a wheeled robot are reported. Conditions under which steering to a set is guaranteed are established.