【正文】 see [8] for a discussion. 6 Robust Stability In the last years the synthesis of robust MPC laws is considered in different works [14]. The framework described below is based on the one in [9], extended to timevarying systems. Our objective is to drive to a given target set the state of the nonlinear system subject to bounded disturbances Since is finite, we conclude that the function is bounded and then that is also bounded. Therefore is bounded and, since f is continuous and takes values on bounded sets of is also bounded. Using the fact that x? is absolutely continuous and coincides with ?x at all sampling instants, we may deduce that and are also bounded. We are in the conditions to apply the previously established Generalization of Barbalat’s Lemma 3, yielding the assertion of the theorem. 7 Finite Parameterizations of the Control Functions The results on stability and robust stability were proved using an optimal control problem where the controls are functions selected from a very general set (the set of measurable functions taking values on a set U, subset of Rm). This is adequate to prove theoretical stability results and it even permits to use the results on existence of a minimizing solution to optimal control problems (. [7,Proposition 2]). However, for implementation, using any optimization algorithm, the control functions need to be described by a finite number of parameters (the so called finite parameterizations of the control functions). The control can be parameterized as piecewise constant controls (. [13]), polynomials or splines described by a finite number of coeficients, bangbang controls (. [9, 10]), that we are not considering discretization of the model or the dynamic equation. The problems of discrete approximations are discussed in detail . in [16] and [12]. But, in the proof of stability, we just have to show at some point that the optimal cost (the value function) is lower than the cost of using another admissible control. So, as long as the set of admissible control values U is constant for all time, an easy, but nevertheless important, corollary of the previous stability results follows If we consider the set of admissible control functions (including the auxiliary control law) to be a finitely parameterizable set such that the set of admissible control values is constant for all time, then both the nominal stability and robust stability results here described remain valid. An example, is the use of discontinuous feedback control strategies of bangbang type, which can be described by a small number of parameters and so make the problem putationally tractable. In bangbang feedback strategies, the controls values of the strategy are only allowed to be at one of the extremes of its range. Many control problems of interest admit a bangbang stabilizing control. Fontes and Magni [9] describe the application of this parameterization to a unicycle mobile robot subject to bounded disturbances. 。 these prise the control horizon Tc, the prediction horizon Tp, the running cost and terminal costs functions L and W, the auxiliary control law kaux, and the terminal constraint set S ? IRn. The resultant control law u? is a “samplingfeedback” control since during each sampling interval, the control u? is dependent on the state x?(ti). More precisely the resulting trajectory is given by and the function t _→ _ t_π gives the last sampling instant before t, that is Similar sampleddata frameworks using continuoustime models and sampling the state of the plant at discrete instants of time were adopted in [2, 6, 7, 8, 13] and are being the accepted framework for continuoustime MPC. It can be shown that with this framework it is possible to address —and guarantee stability, and robustness, of the resultant closedloop system — for a very large class of systems, possibly nonlinear, timevarying and nonholonomic. 3 Nonholonomic Systems and Discontinuous Feedback There are many physical systems with interest in practice which