電氣工程及其自動(dòng)化專業(yè)畢設(shè)外文翻譯(存儲(chǔ)版)
【正文】 propriate use of a positive intersampling time, bined with an appropriate interpretation of a solution to a discontinuous differential equation. 4 Barbalat’s Lemma and Variants Barbalat’s lemma is a wellknown and powerful tool to deduce asymptotic stability of nonlinear systems, especially timevarying systems, using Lyapunovlike approaches (see . [17] for a discussion and applications). Simple variants of this lemma have been used successfully to prove stability results for Model Predictive Control (MPC) of nonlinear and timevarying systems [7, 15]. In fact, in all the sampled data MPC frameworks cited above, Barbalat’slemma, or a modification of it, is used as an important step to prove stabilityof the MPC schemes. It is shown that if certain design param eters (objectivefunction, terminal set, etc.) are conveniently selected, then the value function is monotone decreasing. Then, applying Barbalat’s lemma, attractiveness of the trajectory of the nominal model can be established (. x(t) → 0 as t → ∞ ). This stability property can be deduced for a very general class of nonlinear systems: including timevarying systems, nonholonomic systems, systems allowing discontinuous feedbacks, etc. A recent work on robust MPC of nonlinear systems [9] used a generalization of Barbalat’s lemma as an important step to prove stability of the algorithm. However, it is our believe that such generalization of the lemma might provide a useful tool to analyse stability in other robust continuoustime MPC approaches, such as the one described here for timevarying systems. A standard result in Calculus states that if a function is lower bounded and decreasing, then it converges to a limit. However, we cannot conclude whether its derivative will decrease or not unless we impose some smoothness property on f˙(t). We have in this way a wellknown form of the Barbalat’s lemma (see . [17]). 5 Nominal Stability A stability analysis can be carried out to show that if the design parameters are conveniently selected (. selected to satisfy a certain sufficient stability condition, see . [7]), then a certain MPC value function V is shown to be monotone decreasing. More precisely, for some δ 0small enough and for any 。see . [17] for a discussion and applications). To show that an MPC strategy is stabilizing (in the nominal case), it is shown that if certain design parameters (objective function, terminal set, etc.) are conveniently selected, then the value function is mono tone decreasing. Then, applying Barbalat’s lemma, attractiveness of the trajectory of the nominal model can be established (. x(t) → 0 as t → ∞ ). This stability property can be deduced for a very general class of nonlinear systems: including timevarying systems, nonholonomic systems, systems allowing discontinuous feedbacks, etc. If, in addition, the value functionpossesses some continuity properties, then Lyapunov stability (. the trajectory stays arbitrarily close to the origin provided it starts close enough to the origin) can also be guaranteed (see . [11]). However, this last property might not be possible to achieve for certain classes of systems, for example a carlike vehicle (see [8] for a discussion of this problem and this example). A similar approach can be used to deduce robust stability of MPC for systems allowing uncertainty. After establishing monotone decrease of the value function, we would want to guarantee that the state trajectory asymptotically approaches some set containing the origin. But, a difficulty encountered is thatthe predicted trajectory only coincides with the resulting trajectory at specificsampling instants. The robust stability properties can be obtained, as we show,using a generalized version of Barbalat’s lemma. These robust stability resultsare also valid for a very general class of nonlinear timevarying systems allowing discontinuous feedbacks. The optimal control problems to be solved within the MPC strategy are here formulated with very general admissible sets of controls (say, measurable control functions) making it easier to
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