【正文】 t, xt, u) when we want to make explicit the dependence on the initial time, initial state, and control function. The pair (ˉx, ˉu) denotes our optimal solution to an openloop optimal control problem. The process (x?, u?) is the closedloop trajectory and control resulting from the MPC strategy. We call design parameters the variables present in the openloop optimal control problem that are not from the system model (. variables we are able to choose)。 但是，在證明穩(wěn)定，我們只是要表明，在一些點(diǎn)就是最優(yōu)成本（值函數(shù)） 是低于成本的使用另一受理控制。 在這個(gè)優(yōu)化問題，我們使用公約，如果一些約束是不是滿意，那么價(jià)值的游戲 + ∞ 。 函數(shù) V 的 MPC 值被定義為 這里 是 為最優(yōu)控制問題 的函數(shù)值。 這種穩(wěn)定的財(cái)產(chǎn)可以推斷，一個(gè)很籠統(tǒng)的類非線性系統(tǒng)： 包括時(shí)變系統(tǒng)的，非完整系統(tǒng)，系統(tǒng)允許間斷意見，等等。這些軌跡，溫和條件下， 清楚界定，甚至當(dāng)反饋法是間斷。 3 非完整系統(tǒng)的和間斷的反饋意見 有許多物理系統(tǒng)的興趣，在實(shí)踐中，只能為藍(lán)本適當(dāng)作為非完整系統(tǒng)?？勺儑嵈淼膶?shí)時(shí)同時(shí)，我們保留 S來表示的時(shí)間變量，用于在預(yù)測模型。這些魯棒穩(wěn)定性結(jié)果也有效期為一個(gè)很一般類非線性時(shí)變系統(tǒng)的允許間斷的意見。 見例如 [17]為討論和應(yīng)用） 。這是爭辯這泛 barbalat的引理，形容這里，可以有也類似的的作用，在證明的魯棒穩(wěn)定性的結(jié)果，也允許以解決一個(gè)很一般類非線性，時(shí)變的，非完整系統(tǒng)，受到的干擾。那個(gè)的可能性的框架內(nèi)，以容納間斷的意見是必要的實(shí)現(xiàn)名義的穩(wěn)定性和魯棒穩(wěn)定性，例如一般類別的系統(tǒng)。顯示模型預(yù)測控制的一 項(xiàng)戰(zhàn)略是穩(wěn)定（在名義如此） ，這表明，如果某些設(shè)計(jì)參數(shù)（目標(biāo)函數(shù)，碼頭設(shè)置等） ，方便的選定，然后價(jià)值函數(shù)是單調(diào)遞減。 最優(yōu)控制有待解決的 問題與模型預(yù)測控制的戰(zhàn)略是在這里制定了非?；\統(tǒng)的受理套管制（例如，可衡量的控制職能） ，使更容易保證，在理論上講，存在的解決辦法。那個(gè)矢量 xt是指的實(shí)際狀況核電廠的測量時(shí)間 t過程 的是一對 彈道 /控制取得了從系統(tǒng)模型。一些例子是輪式車輛，機(jī)器人，以及其他許多機(jī)械系統(tǒng)。 有在文獻(xiàn)中的幾個(gè)工程，允許間斷的反饋意見的法律的背景下貨幣政策委員會。 最近的工作，穩(wěn)健的貨幣政策委員會的非線性系統(tǒng) [ 9 ]用了一個(gè)泛化對 barbalat引理的一個(gè)重要步驟，以證明穩(wěn)定的算法。 從（ 7）我們可以知道對任意 因?yàn)?是有限的。這可確保 當(dāng)價(jià)值的游戲是有限的，最優(yōu)控制策略保證滿意的程度的限制，為一切可能的干擾情況。因此，只要設(shè)定可接受的控制值 U 的常數(shù)所有的時(shí)間，輕而易舉的事，但無論如何，重要的是，必然前穩(wěn)定結(jié)果如下 如果我們考慮到一套接納控制功能（包括輔警控制法）是一個(gè)有限 parameterizable設(shè)置這樣的一套受理的控制值是不斷為所有的時(shí)間，那么雙方的名義穩(wěn)定性和魯棒穩(wěn)定的結(jié)果，這里所描述的仍然有效。 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 can only be modelled appropriately as nonholonomic systems. Some examples are the wheeled vehicles, robot manipulators, and many other mechanical systems. A difficulty encountered in controlling this kind of systems is that any linearization around the origin is uncontrollable and therefore any linear control methods are useless to tackle them. But, perhaps the main challenging characteristic of the nonholonomic systems is that it is not possible to stabilize it if just timeinvariant continuous feedbacks are allowed [1]. However, if we allow discontinuous feedbacks, it might not be clear what is the solution of the dynamic differential equation. (See [4, 8] for a further discussion of this issue). A solution concept that has been proved successful in dealing with stabilization by disconti nuous feedbacks for a generalclass of controllablesystems is the concept of “samplingfeedback” solution proposed in [5]. It can be seenthat sampleddata MPC framework described can be bined naturally with a “samplingfeedback” law and thus define a trajectory in a way which is verysimilar to the concept introduced in [5]. Those trajectories are, under mild conditions, welldefined even when the feedback law is discontinuous. There are in the literature a few works allowing discontinuous feedback laws in the context of MPC. (See [8] for a survey of such works.) The essential feature of those frameworks to allow discontinuities is simply the sampleddata feature — appropriate use