【正文】 is bounded and then that ds is also bounded. Therefore is bounded and, since f is continuous and takes values on bounded sets of is also bounded. All the conditions to apply Barbalat’s lemma 2 are met, yielding that the trajec tory asymptotically converges to the origin. Note that this notion of stability does not necessarily include the Lyapunov stability property as is usual in other notions of stability。 SampledData Model Predictive Control for Nonlinear TimeVarying Systems: Stability and Robustness Summary. We describe here a sampleddata Model Predictive Control framework that uses continuoustime models but the sampling of the actual state of the plant as well as the p utation of the control laws, are carried out at discrete instants of time. This framework can address a very large class of systems, nonlinear, timevarying, and nonholonomic. As in many others sampleddata Model Predictive Control schemes, Barbalat’slemma has an important role in the proof of nominal stability results. It is arguedthat the generalization of Barbalat’s lemma, described here, can have also a similar role in the proof of robust stab ility results, allowing also to address a very general class of nonlinear, timevarying, nonho lonomic systems, subject to disturbances. Thepossibility of the framework to acmodate discontinuous feedbacks is essential to achieve both nominal stability and robust stability for such general classes of systems. 1 Introduction Many Model Predictive Control (MPC) schemes described in the literature use continuoustime models and sample the state of the plant at discrete instants of time. See . [3, 7, 9, 13] and also [6]. There are many advantages in considering a continuoustime model for the plant. Neverthe less, any implementable MPC scheme can only measure the state and solve an optimization pro blem at discrete instants of time. In all the references cited above, Barbalat’s lemma, or a modification of it, is used as an impo rtant step to prove stability of the MPC schemes. (Barbalat’s lemma is a wellknown and Power ful tool to deduce asymptotic stability of nonlinear systems, especially timevarying systems, using Lyapunovlike approaches。 注意，我們是不會(huì)考慮的離散模型或動(dòng)態(tài)方程。 框架下文所述是基于一個(gè)在 [ 9 ] ，延長至 timevarying 系統(tǒng)。 5 名義的穩(wěn)定 穩(wěn)定性分析可以進(jìn)行顯示，如果設(shè)計(jì)參數(shù)方便的選定（即選定，以滿足某一個(gè)足夠穩(wěn)定條件下，例如見 [ 7 ] ） ，然后在某貨幣政策委員會(huì)的價(jià)值函數(shù) V 是表明要單調(diào)遞減。這表明，如果某些設(shè)計(jì)參數(shù)（目標(biāo)功能，碼頭設(shè)置等） ，方便的選定，則值函數(shù)是單調(diào)遞減。 解決的概念，已被證明是成功的在處理與穩(wěn)定由間斷的意見為是一種通用類別的可控系統(tǒng)概念是 “ 采樣 反饋 ” 提出的解決辦法 [ 5 ] 。這些包括控制豪華的 TC ，該預(yù)測地平線總磷，運(yùn)行成本和終端成本的職能升和 W ， 輔助控制律 kaux ，和終端約束集 正是由此產(chǎn)生的軌跡是由 這里 和功能 于是 類似的采樣數(shù)據(jù)框架使用的連續(xù)時(shí)間模型和采樣國家的核電廠在離散 instants的時(shí)間通過了在 [ 2 ， 6 ， 7 ， 8 ， 13 ] 并正成為公認(rèn)的框架，連續(xù)時(shí)間的貨幣政策委員會(huì)。 我們假設(shè)這個(gè)制度，以漸近的可控性對(duì) ，并為所有 我們進(jìn)一步假設(shè)函數(shù) f是連續(xù)的和局部 Lipschitz方面的第二個(gè)論點(diǎn)。但是，遇到的困難是， 預(yù)測的軌跡，只有剛好與由此產(chǎn)生的軌跡在特定的抽樣 instants 。 在所有的提述，引用上述情況， barbalat的引理，或修改它，是用來作為一個(gè)重要步驟，以證明穩(wěn)定的 MPC的計(jì)劃。在 此框架內(nèi)可以解決一個(gè)非常大的一類系統(tǒng)，非線性，時(shí)變的，非完整。見例如 [3， 7， 9， 13] ，也是 [6] 。不過，這最后的財(cái)狀態(tài)可能 否則就不可能實(shí)現(xiàn)，為某些類別的系統(tǒng)，例如汽車一樣， 車輛（見 [8]為討論這個(gè)問題，這個(gè)例子） 。它可以證明即穩(wěn)定或魯棒性的結(jié)果在這里所描述的仍然有效，當(dāng)優(yōu)化進(jìn)行了有限的參數(shù)化的管制，如分段常數(shù)控制（如在 [13]） ，或幫邦間斷反饋（如在 [9]）。 兩人 的是指我們的最優(yōu)解，以一個(gè)開放的閉環(huán)優(yōu)化控制問題。不過，可能是主要的富有挑戰(zhàn)性的特點(diǎn)對(duì)非完整系統(tǒng)的是，這是不可能穩(wěn)定的話，剛才時(shí)間不變連續(xù)反饋獲準(zhǔn) [ 1 ] 。 4 barbalat的引理和變種 barbalat的引理是眾所周知的和有力的工具，以推斷的漸近穩(wěn)定性非線性系統(tǒng)， 尤其是時(shí)間變系統(tǒng)，利用 Lyapunov樣辦法（見例如 [ 17 ]為討論和應(yīng)用） 。 一個(gè)標(biāo)準(zhǔn)的結(jié)果，在微積分的國家，如果一個(gè)功能是較低的范圍和減少，那么收斂到一個(gè)極限。注意： 這個(gè)概念的穩(wěn)定，并不一定包括 Lyapunov 穩(wěn)定性財(cái)產(chǎn)是慣常在其他的概念，穩(wěn)定 。這個(gè)是足夠的證明理論的穩(wěn)定結(jié)果，它甚至允許使用的結(jié)果，就 存在一個(gè)最小的解決方案，以最優(yōu)控制問題（如 [ 7 ， 命題 2 ] ） 。在邦邦反饋策略， 管制的價(jià)值觀的策略是只允許在其中一個(gè)極端它的范圍。 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