【正文】 rbalat的引理一個重要的角色，在證明的名義穩(wěn)定的結(jié)果。魯棒穩(wěn)定性能可以得到，因為我們顯示， 用一種廣義的版本 barbalat的引理。它可以結(jié)果表明，與在此框架內(nèi)是有可能的地址和保證穩(wěn)定，魯棒性，由此產(chǎn)生的閉環(huán)控制系統(tǒng) 為一個非常大的類系統(tǒng)，可能是非線性，時變的和非完整。然后，運用 barbalat的引理，吸引力的軌跡的名義模型可以建立。 我們的目標是開車到某一所定的目標 θ （ ? irn ）國家的非線性系統(tǒng)受界擾動 強勁的反饋貨幣政策委員 會的策略，是由多次獲得解決上線， 在每個采樣即時鈦， Min Max 的優(yōu)化問題，磷，以選取反饋 kti ，每一次使用當(dāng)前措施，該國的核電廠 xti 。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 guarantee, in theoretical terms, the existence of solution. However, some form of finite parameterization of the control functionsis required/desirable to solve online the optimization problems. It can be shown that the stability or robustness results here described remain valid when the optimization is carried out over a finite parameterization of the controls, such as piecewise constant controls (as in [13]) or as bangbang discontinuous feedbacks (as in [9]). 2 A SampledData MPC Framework We shall consider a nonlinear plant with input and state constraints, where the evolution of the state after time t0 is predicted by the following model. The data of this model prise a set containing all possible initial states at the initial time t0, a vector xt0 that is the state of the plant measured at time t0, a given function of possible control values. We assume this system to be asymptotically controllable on X0 and that for all t ≥ 0 f(t, 0, 0) = 0. We further assume that the function f is continuous and locally Lipschitz with respect to the second argument. The construction of the feedback law is acplished by using a sampleddata MPC strategy. Consider a sequence of sampling instants π := {ti}i≥0 with a constant intersampling time δ 0 such that ti+1 = ti+δ for all i ≥ 0. Consider also the control horizon and predictive horizon, Tc and Tp, with Tp ≥ Tc δ, and an auxiliary control law kaux : IRIRn → IRm. The feedback control is obtained by repeatedly solving online openloop optimal control problems P(ti, xti, Tc, Tp) at each sampling instant ti ∈ π, every time using the current measure of the state of the plant xti . Note that in the interval [t + Tc, t + Tp] the control value is selected from a singleton and therefore the optimization decisions are all carried out in the interval [t, t + Tc] with the expected benefits in the putational time.