外文翻譯-非線性時變系統(tǒng)的穩(wěn)定性和魯棒性

1、<p><b>　　外文資料翻譯</b></p><p>　　學(xué)院(系)：　　電子信息工程系 　　　　</p><p>　　?！　I(yè)：　　 自動化　　</p><p>　　學(xué)生姓名：　　 盧 艷 　　　 </p><p>　　班 級：　　 092202H 　　<

2、;/p><p>　　學(xué) 號: 200922060212</p><p>　　采樣數(shù)據(jù)模型預(yù)測控制 非線性時變系統(tǒng)的：穩(wěn)定性和魯棒性</p><p>　　概要:我們這里所敘述的是采樣數(shù)據(jù)模型預(yù)測控制的框架，使用連續(xù)時間模型，但采樣的實際狀況以及為計算控制的狀態(tài)，進行了在離散instants的時間。在此框架內(nèi)可以解決一 個非常大的

3、一類系統(tǒng)，非線性，時變的，非完整。 </p><p>　　如同在許多其他采樣數(shù)據(jù)模型預(yù)測控制計劃，barbalat的引理一個重要的角色，在證明的名義穩(wěn)定的結(jié)果。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，也允許以解決一個很一般類非線性，時變的，非完整系統(tǒng)，受到的干擾。那個的可能性的框架內(nèi)，以容納間斷的意見是必要的實現(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)。&

4、lt;/p><p><b>　　1 引言</b></p><p>　　許多模型預(yù)測控制（MPC）計劃描述，在文獻上使用連續(xù)時間的模型和樣本狀態(tài)的在離散的instants 時間。見例如[3，7，9，13] ，也是[6] 。有許多好處，在考慮連續(xù)時間模型。不過，任何可執(zhí)行的模型預(yù)測控制計劃只能措施，狀態(tài)和解決的優(yōu)化問題在離散instants的時間。</p>&

5、lt;p>　　在所有的提述，引用上述情況， barbalat的引理，或修改它，是用來作為一個重要步驟，以證明穩(wěn)定的MPC的計劃。（ barbalat的引理是眾所周知的和有力的工具，以推斷的漸近穩(wěn)定性的非線性系統(tǒng)，尤其是時間變系統(tǒng)，利用Lyapunov樣的辦法; 見例如[17]為討論和應(yīng)用） 。顯示模型預(yù)測控制的一項戰(zhàn)略是穩(wěn)定（在名義如此） ，這表明，如果某些設(shè)計參數(shù)（目標(biāo)函數(shù)，碼頭設(shè)置等） ，方便的選定，然后價值函數(shù)是單調(diào)遞減。然

6、后，運用barbalat的引理，吸引力該軌跡的名義模型可以建立(i.e. x(t) → 0 as t → ∞).這種穩(wěn)定的狀態(tài)可以推斷，一個很籠統(tǒng)的類非線性系統(tǒng)：包括時變系統(tǒng)的，非完整系統(tǒng)，系統(tǒng)允許間斷意見，等此外，如果值函數(shù)具有一定的連續(xù)性屬性，然后Lyapunov穩(wěn)定性（即軌跡停留任意接近的起源提供了足夠的密切開始向原產(chǎn)地）也可以得到保障（見例如[11]） 。不過，這最后的財狀態(tài)可能否則就不可能實現(xiàn)，為某些類別的系統(tǒng)，例如汽車一樣

7、， 車輛（見[8]為討論這個問題，這個例子） 。</p><p>　　類似的做法，可以用來推斷魯棒穩(wěn)定的貨幣政策委員會系統(tǒng)允許的不確定性。后建立的單調(diào)減少的價值功能，我們會要保證狀態(tài)的軌跡漸近辦法訂定一些載有原產(chǎn)地。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 。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。這些魯棒穩(wěn)定性結(jié)果也有效期為一個

8、很一般類非線性時變系統(tǒng)的允許間斷的意見。</p><p>　　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。不過，某種形式的有限參數(shù)的控制功能需要/可取的解決上線的優(yōu)化問題。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、[9]）。</p><p>　　2 采樣數(shù)據(jù)MPC的框架內(nèi)</p><p>　　我們會考慮一種非線性的靜態(tài)具有輸入與狀態(tài)的限制，凡變化的狀態(tài)后，時間t0 ，預(yù)計由以下模型。</p><p>　　數(shù)據(jù)模型，這包括了一套包含所有可能的初始狀態(tài)在最初的時間 ，矢量這是狀態(tài)的測量時間，某一函數(shù)f ：一套的盡可能控制值。</p><p>　　我們假設(shè)

10、這個制度，以漸近的可控性對 ，并為所有我們進一步假設(shè)函數(shù)f是連續(xù)的和局部Lipschitz方面的第二個論點。</p><p>　　注意到，在區(qū)間控制值的選定是由單身人士因此，優(yōu)化的決定，都是進行在區(qū)間與預(yù)期的效益，在計算時間。</p><p>　　樂譜在這里通過的是如下?？勺儑嵈淼膶崟r同時，我們保留S來表示的時間變量，用于在預(yù)測模型。那個矢量xt是指的實際狀況核電廠的測量時間t過程的是一

11、對彈道/控制取得了從系統(tǒng)模型。那個軌跡，有時是標(biāo)注為的，當(dāng)我們想作明確地依賴于初始時間，初始狀態(tài)，和控制功能。 兩人的是指我們的最優(yōu)解，以一個開放的閉環(huán)優(yōu)化控制問題。過程中是閉環(huán)系統(tǒng)的軌跡和控制造成的從貨幣政策委員會的策略。我們要求設(shè)計參數(shù)的變數(shù)，目前，在開環(huán)最優(yōu)控制問題是沒有從系統(tǒng)模型（即變量，我們可以選擇） ;這些包括控制豪華的TC ，該預(yù)測地平線總磷，運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</

12、p><p>　　正是由此產(chǎn)生的軌跡是由</p><p><b>　　這里</b></p><p><b>　　和功能于是</b></p><p>　　類似的采樣數(shù)據(jù)框架使用的連續(xù)時間模型和采樣國家的核電廠在離散instants的時間通過了在[ 2 ， 6 ， 7 ， 8 ， 13 ] 并正成為公認(rèn)的框架

13、，連續(xù)時間的貨幣政策委員會。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。</p><p>　　3 非完整系統(tǒng)的和間斷的反饋意見</p><p>　　有許多物理系統(tǒng)的興趣，在實踐中，只能為藍(lán)本適當(dāng)作為非完整系統(tǒng)。一些例子是輪式車輛，機器人，以及其他許多機械系統(tǒng)。</p><p&

14、gt;　　一遇到的困難，在控制這種系統(tǒng)是任何線性周圍的原產(chǎn)地是無法控制的，因此任何的線性控制方法是無用的，以解決這些問題。不過，可能是主要的富有挑戰(zhàn)性的特點對非完整系統(tǒng)的是，這是不可能穩(wěn)定的話，剛才時間不變連續(xù)反饋獲準(zhǔn)[ 1 ] 。但是，如果我們?nèi)菰S間斷意見，它可能并不清楚什么是解決動態(tài)微分方程。 （見[ 4日， 8日]為進一步討論這個問題）。</p><p>　　解決的概念，已被證明是成功的在處理與穩(wěn)定由間斷的

15、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 ] ?？梢钥闯?， 即采樣數(shù)據(jù)所描述的貨幣政策委員會的框架內(nèi)，可結(jié)合自然與“抽樣反饋法” ，從而確定一個軌跡的方式，這是非常類似的概念，介紹了在[ 5 ] 。這些軌跡，溫和條件下， 清楚界定，甚至當(dāng)反饋法是間斷。</p><p>　　有在文獻中的幾個工程，允許間斷的反饋意見的法律的背景下貨幣政策委員會。 （見[ 8 ]為一項調(diào)查，這些工程）的本

16、質(zhì)特征。這些框架，允許間斷只不過是采樣數(shù)據(jù)的特點- 適當(dāng)使用一種積極的跨采樣時間，再加上一個適當(dāng)?shù)慕忉尳鉀Q一個間斷微分方程。</p><p>　　4 barbalat的引理和變種</p><p>　　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)，尤其是時間變系統(tǒng)，利用Lyapunov樣辦法（見例如[ 17 ]為討論和應(yīng)用） 。</p><

17、;p>　　簡單的變種，這引理已成功地用來證明穩(wěn)定的結(jié)果為模型預(yù)測控制（貨幣政策委員會）的非線性和時變系統(tǒng)的[ 7 ， 15 ] 。事實上，在所有采樣數(shù)據(jù)貨幣政策委員會框架舉出上述情況， barbalat的引理，或修改它，是用來作為一個重要步驟，以證明穩(wěn)定貨幣政策委員會的計劃。這表明，如果某些設(shè)計參數(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。然后，運用barbalat的引理，吸引力的軌跡的名義模型可以建立。 這種穩(wěn)

18、定的財產(chǎn)可以推斷，一個很籠統(tǒng)的類非線性系統(tǒng)： 包括時變系統(tǒng)的，非完整系統(tǒng)，系統(tǒng)允許間斷意見，等等。</p><p>　　最近的工作，穩(wěn)健的貨幣政策委員會的非線性系統(tǒng)[ 9 ]用了一個泛化對barbalat引理的一個重要步驟，以證明穩(wěn)定的算法。不過，這是我們認(rèn)為，這種泛化的引理可能提供一個有用的工具來分析穩(wěn)定在其他穩(wěn)健的連續(xù)時間的貨幣政策委員會的做法， 如一個形容這里時變系統(tǒng)的。</p><p&

19、gt;　　一個標(biāo)準(zhǔn)的結(jié)果，在微積分的國家，如果一個功能是較低的范圍和減少，那么收斂到一個極限。不過，我們不能斷定是否及其衍生物會減少或沒有，除非我們施加了一些平滑的財產(chǎn)關(guān)于F 。我們在這樣一個眾所周知的形式的barbalat的引理（見例如： [ 17 ] ） 。</p><p><b>　　5 名義的穩(wěn)定</b></p><p>　　穩(wěn)定性分析可以進行顯示，如果設(shè)

20、計參數(shù)方便的選定（即選定，以滿足某一個足夠穩(wěn)定條件下，例如見[ 7 ] ） ，然后在某貨幣政策委員會的價值函數(shù)V是表明要單調(diào)遞減。更確切地說，</p><p><b>　　對于和</b></p><p>　　這里M是連續(xù)的，徑向無界，正定功能。函數(shù)V的MPC值被定義為</p><p>　　這里是為最優(yōu)控制問題 的函數(shù)值。</p>

21、<p>　　從（7）我們可以知道對任意</p><p>　　因為是有限的。我們得出和因此，</p><p>　　，因為是連續(xù)的，我們得出所有的條件，申請barbalat的引理2會見，高產(chǎn)，該軌跡漸近收斂到原點。注意： 這個概念的穩(wěn)定，并不一定包括Lyapunov穩(wěn)定性財產(chǎn)是慣常在其他的概念，穩(wěn)定;見[ 8 ]為了討論。</p><p><b>　

22、　6 魯棒穩(wěn)定性</b></p><p>　　在過去的幾年中合成的強勁貨幣政策委員會的法律被認(rèn)為是在不同的工程[ 14 ] 。 框架下文所述是基于一個在[ 9 ] ，延長至timevarying 系統(tǒng)。 我們的目標(biāo)是開車到某一所定的目標(biāo)θ （ ? irn ）國家的非線性系統(tǒng)受界擾動</p><p>　　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，

23、 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 。</p><p>　　在這個優(yōu)化問題，我們使用公約，如果一些約束是不是滿意，那么價值的游戲+ ∞ 。這可確保當(dāng)價值的游戲是有限的，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。</p><p>　　7 有限參數(shù)的控制功能</p><p>　　結(jié)果的穩(wěn)定性和魯

24、棒穩(wěn)定性，證明了用最優(yōu)控制問題所在是控制職能，選定由一個非常一般設(shè)置（一套可衡量的職能） 。這個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，就存在一個最小的解決方案，以最優(yōu)控制問題（如[ 7 ， 命題2 ] ） 。不過，對于執(zhí)行，使用任何優(yōu)化算法， 控制功能需要加以形容一個有限的參數(shù)數(shù)目（ 所謂有限參數(shù)的控制功能） 。控制可參數(shù)為分段常數(shù)控制（如[ 13 ] ） ，多項式或樣條所描述的一個有限的數(shù)目coeficients ，砰-砰

25、管制（例如， [ 9 ， 10 ] ）等。 注意，我們是不會考慮的離散模型或動態(tài)方程。問題的離散逼近，詳細(xì)討論了如在[ 16 ]及[ 12 ] 。</p><p>　　但是，在證明穩(wěn)定，我們只是要表明，在一些點就是最優(yōu)成本（值函數(shù)）是低于成本的使用另一受理控制。因此，只要設(shè)定可接受的控制值U的常數(shù)所有的時間，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</p><p>　　如果我

26、們考慮到一套接納控制功能（包括輔警控制法）是一個有限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。</p><p>　　舉例來說，是利用間斷的反饋控制策略榜榜類型，可以說是由少數(shù)參數(shù)等，使問題的計算tractable 。在邦邦反饋策略， 管制的價值觀的策略是只允許在其中一個極端它的范圍。許多控制問題感興趣的承認(rèn)，一幫邦穩(wěn)定控

27、制。 fontes和magni [ 9 ]描述的應(yīng)用，這參數(shù)是一個unicycle移動機器人須有界擾動。</p><p>　　Sampled-Data Model Predictive Control for Nonlinear Time-Varying Systems: Stability and Robustness</p><p>　　Summary. We describe her

28、e a sampled-data Model Predictive Control framework that uses continuous-time models but the sampling of the actual state of the plant as well as the comp- utation of the control laws, are carried out at discrete instant

29、s of time. This framework can address a very large class of systems, nonlinear, time-varying, and nonholonomic.</p><p>　　As in many others sampled-data Model Predictive Control schemes, Barbalat’slemma has

30、</p><p>　　an important role in the proof of nominal stability results. It is arguedthat the generalization </p><p>　　of Barbalat’s lemma, described here, can have also a similar role in the proo

31、f of robust stab-</p><p>　　ility results, allowing also to address a very general class of nonlinear, time-varying, nonho- lonomic systems, subject to disturbances. Thepossibility of the framework to accommo

32、date discontinuous feedbacks is essential to achieve both nominal stability and robust stability for such general classes of systems.</p><p>　　1 Introduction</p><p>　　Many Model Predictive Contr

33、ol (MPC) schemes described in the literature use continuous-time models and sample the state of the plant at discrete instants of time. See e.g. [3, 7, 9, 13] and also [6]. There are many advantages in considering a cont

34、inuous-time 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.</p><p>　　In all the references cited

35、 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 well-known and Power- ful tool to deduce asymptotic stability of nonlinear s

36、ystems, especially time-varying systems, using Lyapunov-like approaches;see e.g. [17] for a discussion and applications). To show that an MPC strategy is stabilizing (in the nominal case), it is shown that if certain des

37、ign parameters</p><p>　　(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

38、 of the nominal model can be established (i.e. x(t) → 0 as t → ∞). This stability property can be deduced for a very general class of nonlinear systems: including time-varying systems, nonholonomic systems, systems allo

39、wing discontinuous feedbacks, etc. If, in addition, the value functionpossesses some continuity p</p><p>　　However, this last property might not be possible to achieve for certain classes of systems, for ex

40、ample a car-like vehicle (see [8] for a discussion of this problem and this example).</p><p>　　A similar approach can be used to deduce robust stability of MPC for systems allowing uncertainty. After establ

41、ishing monotone decrease of the value function, we would want to</p><p>　　guarantee that the state trajectory asymptotically approaches some set containing the origin. But, a difficulty encountered is thatth

42、e 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

43、stability resultsare also valid for a very general class of nonlinear time-varying systems allowing discontinuous feedbacks.</p><p>　　The optimal control problems to be solved within the MPC strategy are her

44、e 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

45、 of the control functionsis required/desirable to solve on-line the optimization problems. It can be shown that the stability or robustness results here described remain valid when the optimization is carrie</p>&

46、lt;p>　　piecewise constant controls (as in [13]) or as bang-bang discontinuous feedbacks (as in [9]).</p><p>　　2 A Sampled-Data MPC Framework</p><p>　　We shall consider a nonlinear plant with

47、input and state constraints, where the evolution of the state after time t0 is predicted by the following model.</p><p>　　The data of this model comprise a set containing all possible initial states at the i

48、nitial time t0, a vector xt0 that is the state of the plant measured at time t0, a given function </p><p>　　of possible control values.</p><p>　　We assume this system to be asymptotically contro

49、llable 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.</p><p>　　The construction of the feedback la

50、w is accomplished by using a sampleddata MPC strategy. Consider a sequence of sampling instants π := {ti}i≥0 with a constant inter-sampling time δ > 0 such that ti+1 = ti+δ for all i ≥ 0. Consider also the control hor

51、izon and predictive horizon, Tc and Tp, with Tp ≥ Tc > δ, and an auxiliary control law kaux : IR×IRn → IRm. The feedback control is obtained by repeatedly solving online open-loop optimal control problems P(ti, x

52、ti, Tc, Tp) at each sampling instant</p><p>　　Note that in the interval [t + Tc, t + Tp] the control value is selected from a singleton and</p><p>　　therefore the optimization decisions are all

53、carried out in the interval [t, t + Tc] with the expected benefits in the computational time.</p><p>　　The notation adopted here is as follows. The variable t represents real time while we reserve s to denot

54、e the time variable used in the prediction model. The vector xt denotes the actual state of the plant measured at time t. The process (x, u) is a pair trajectory/control obtained from the model of the system. The traject

55、ory is sometimes denoted as s _→ x(s; 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</p><p>　　prediction horizon Tp

56、, the running cost and terminal costs functions L and W, the auxiliary control law kaux, and the terminal constraint set S ? IRn.</p><p>　　The resultant control law u? is a “sampling-feedback” control since

57、during each sampling interval, the control u? is dependent on the state x?(ti). More precisely the resulting trajectory is given by</p><p>　　and the function t _→ _t_π gives the last sampling instant before

58、t, that is</p><p>　　Similar sampled-data frameworks using continuous-time models and sampling the state of the plant at discrete instants of time were adopted in [2, 6, 7, 8, 13] and are becoming the accepte

59、d framework for continuous-time MPC. It can be shown that with this framework it is possible to address —and guarantee stability, and robustness, of the resultant closed-loop system — for a very large class of systems, p

60、ossibly nonlinear, time-varying and nonholonomic.</p><p>　　3 Nonholonomic Systems and Discontinuous Feedback</p><p>　　There are many physical systems with interest in practice which can only be

61、modelled</p><p>　　appropriately as nonholonomic systems. Some examples are the wheeled vehicles, robot manipulators, and many other mechanical systems.</p><p>　　A difficulty encountered in contr

62、olling 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 nonho

63、lonomic systems is that it is not possible to stabilize it if just time-invariant continuous feedbacks are allowed [1]. However, if we allow discontinuous feedbacks, it might not be clear what is the solution of the dyna

64、mic differential equation. </p><p>　　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

65、“sampling-feedback” solution proposed in [5]. It can be seenthat sampled-data MPC framework described can be combined naturally with a “sampling-feedback” law and thus define a trajectory in a way which is verysimilar to

66、 the concept introduced in [5]. Those trajectories are, under mild conditions,</p><p>　　well-defined even when the feedback law is discontinuous.</p><p>　　There are in the literature a few works

67、 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 sampled-data feature — appropriate use o

68、f a positive inter-sampling time, combined with an appropriate interpretation of a solution to a discontinuous differential equation.</p><p>　　4 Barbalat’s Lemma and Variants</p><p>　　Barbalat’s

69、 lemma is a well-known and powerful tool to deduce asymptotic stability of nonlinear systems, especially time-varying systems, using Lyapunov-like approaches (see e.g. [17] for a discussion and applications).</p>

70、<p>　　Simple variants of this lemma have been used successfully to prove stability results for Model Predictive Control (MPC) of nonlinear and time-varying systems [7, 15]. In fact, in all the sampled -data MPC fra

71、meworks 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.) ar

72、e conveniently selected, then the value function is monotone decreasing</p><p>　　A recent work on robust MPC of nonlinear systems [9] used a generalization of Barbalat’s lemma as an important step to prove s

73、tability 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 continuous-time MPC approaches, such as the one described here

74、for time-varying systems.</p><p>　　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 wi

75、ll decrease or not unless we impose some smoothness property on f˙(t). We have in this way a well-known form of the Barbalat’s lemma (see e.g. [17]).</p><p>　　5 Nominal Stability</p><p>　　A stab

76、ility analysis can be carried out to show that if the design parameters are conveniently selected (i.e. selected to satisfy a certain sufficient stability condition, see e.g. [7]), then a certain MPC value function V is

77、shown to be monotone decreasing. More precisely, for some δ > 0small enough and for any。</p><p>　　where M is a continuous, radially unbounded, positive definite function. TheMPC value function V is define

78、d as</p><p>　　where is the value function for the optimal control problem</p><p>　　(the optimal control problem defined where the horizon isshrank in its initial part by).</p><p>　

79、　From (7) we can then write that for any t ≥ t0</p><p>　　Since is finite, we conclude that the function is bounded and then thatds is also bounded. Therefore is bounded and, since f is continuous and take

80、s 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 ne

81、cessarily include the Lyapunov stability property as is usual in other notions of stability; see [8] for a discussion.</p><p>　　6 Robust Stability</p><p>　　In the last years the synthesis of rob

82、ust MPC laws is considered in different works [14].</p><p>　　The framework described below is based on the one in [9], extended to timevarying systems.</p><p>　　Our objective is to drive to a gi

83、ven target set the state of the nonlinear system subject to bounded disturbances</p><p>　　Since is finite, we conclude that the function is</p><p>　　bounded and then that is also bounded. There

84、fore is</p><p>　　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 m

85、ay 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.</p><p>　　7 Finite Parameterizat

86、ions of the Control Functions</p><p>　　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

87、 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 prob

88、lems (e.g. [7,Proposition 2]). However, for implementation, using any optimization algorithm, the control funct</p><p>　　But, in the proof of stability, we just have to show at some point that the optimal c

89、ost (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 p

90、revious stability results follows</p><p>　　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 co

91、ntrol values is constant for all time, then both the nominal stability and robust stability results here described remain valid.</p><p>　　An example, is the use of discontinuous feedback control strategies o

92、f bang-bang type, which can be described by a small number of parameters and so make the problem computationally tractable. In bang-bang feedback strategies, the controls values of the strategy are only allowed to be at

93、 one of the extremes of its range. Many control problems of interest admit a bang-bang stabilizing control. Fontes and Magni [9] describe the application of this parameterization to a unicycle mobile robot subje</p>