采集数据模型预测控制解析

Sampled-Data Model Predictive Control for Nonlinear Time-Varying Systems: Stability and Robustness

Fernando A. C. C. Fontes, Lalo Magni, Eva Gyurkovics PhD, Imperial College London

IEEE Transactions on Power Delivery, V ol. 8, No. 1, Jariuary 1993.

Keyword barbalat Principle Nonlinear time-varying systems Stability and robustness Summary. We describe here 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 computation of the control laws, are carried out at discrete instants of time. This framework can

address a very large class of systems, nonlinear, time-varying, and nonholonomic.

As in many others sampled-data Model Predictive Control schemes, Barbalat’s lemma has

an important role in the proof of nominal stability results. It is argued that the generalization

of Barbalat’s lemma, described here, can have also a similar role in the proof of robust stability results, allowing also to address a very general class of nonlinear, time-varying, nonholonomic systems, subject to disturbances. The possibility of the framework to accommodate 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 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 continuous-time model for the plant. Nevertheless,

any implementable MPC scheme can only measure the state and solve an optimization problem

at discrete instants of time.

In all the references cited above, Barbalat’s lemma, or a modification of it, is used as an important ste p to prove stability of the MPC schemes. (Barbalat’s 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). 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 monotone

decreasing. Then, applying Barbalat’s lemma, attractiveness of the trajectory 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 allowing discontinuous feedbacks, etc. If, in addition, the value functionpossesses some continuity properties, then Lyapunov stability (i.e. the trajectory stays arbitrarily close to the origin provided it starts close enough to the origin) can also be guaranteed (see e.g. [11]). However, this last property might not be possible to achieve for certain classes of systems, for example a car-like 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 v alid for a very general class of nonlinear time-varying 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 on-line 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 bang-bang discontinuous feedbacks (as in [9]).

2 A Sampled-Data 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 comprise 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 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 horizon and predictive horizon, Tc

and Tp, with Tp ≥ Tc > δ, and an auxiliary control law kaux : IR×IR n → IR m. The feedback control is obtained by repeatedly solving online open-loop 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 computational time.

The notation adopted here is as follows. The variable t represents real time while we reserve s

to denote 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 trajectory 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 our optimal solution to an open-loop optimal control problem. The process (x?, u?) is the closed-loop trajectory and control resulting from the MPC strategy. We call design parameters the variables present in the open-loop optimal control problem that are not from the system model (i.e. variables we are able to choose); these comprise 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 ? IR n.

The resultant control law u?is a “sampling-feedback” 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 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 accepted 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, possibly nonlinear, time-varying 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 time-invariant 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 discontinuous feedbac ks for a generalclass of controllablesystems is the concept of “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 the concept introduced in [5]. Those trajectories are, under mild conditions,

well-defined 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 sampled-data feature — appropriate use of a positive inter-sampling time, combined with an appropriate interpretation of a solution to a discontinuous differential equation.

4 Barbalat’s Lemma and Variants

Barbalat’s 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).

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 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 parameters

(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 (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 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 continuous-time MPC approaches, such as the one described here for time-varying 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 well-known form of the Barbalat’s lemma (see e.g. [17]).

5 Nominal Stability

A stability 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 shown to be monotone decreasing. More precisely, for some δ > 0small

enough and for any 。

where M is a continuous, radially unbounded, positive definite function. TheMPC value function V is defined as

where is the value function for the optimal control problem

(the optimal control problem defined where the horizon isshrank

in its initial part by ).

From (7) we can then write that for any t ≥ t0

Since is finite, we conclude that the function 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 trajectory

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; 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 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 may deduce that 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.

7 Finite Parameterizations of the Control Functions

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 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 problems (e.g. [7,Proposition 2]). However, for implementation, using any optimization algorithm, the control functions need to be described by a finite number of parameters (the so called finite parameterizations of the control functions). The control can be parameterized as piecewise constant controls (e.g. [13]), polynomials or splines described by a finite number of coeficients, bang-bang controls (e.g. [9, 10]), etc.Note that we are not considering discretization

of the model or the dynamic equation. The problems of discrete approximations are discussed in detail e.g. in [16] and [12].

But, in the proof of stability, we just have to show at some point that the optimal cost (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 previous stability results follows

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 control values is constant for all time, then both the nominal stability and robust stability results here described remain valid.

An example, is the use of discontinuous feedback control strategies of 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 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 subject to bounded disturbances.

Sampled-Data Model Predictive Control for Nonlinea

Time-Varying Systems:Stability and Robustness.

IEEE Transactions on Power Delivery, V ol. 8, No. 1, Jariuary 1993. 采集数据模型预测控制非线性时变系统的稳定性和鲁棒性

Fernando A.C.C.Fontes

英国伦敦皇家学院

摘要我们这里所叙述的是一采样数据模型预测控制的框架，使用连续时间模型，采样的实际状况以及计算控制的状态，在离散时刻的时间进行。这个框架可以解决一个非常大的非线性、时变、非完整系统。

像许多其他采样数据模型预测控制计划，barbalat引理一个重要的角色证明名义稳定的结果。这是认为泛为barbalat的引理，这里所描述的在证明的鲁棒稳定性的结果也有类似的的作用，也可以解决一个非常一般的一类非线性,时变,非完整受到扰动的系统。在那个可能性的框架内，以容纳间断的意见是必要的实现名义的稳定性和鲁棒稳定性，例如一般类别的系统。

关键词barbalat引理、非线性时变系统、稳定性和鲁棒性

1、引言

多模型预测控制(MPC)计划使用连续时间描述的文献模型和样本工厂在离散时刻的状态的时间。例如[3，7，9，13] 和[6] 。在考虑连续时间模型时有许多好处。尽管如此，任何可执行的模型预测控制计划只能衡量和解决再离散时刻时

间的优化问题。

引用上述所提到的情况， barbalat的引理，或者修改它，使其用来作为一个重要步骤，以证明稳定的MPC的计划。（barbalat引理是众所周知的并且强有力以推断的渐近稳定性的非线性系统的工具，尤其是在推断时间变量系统上面，利用Lyapunov办法; 见例如[17]的讨论和应用）。显示模型预测控制的一项战略是稳定（在名义上是如此），这表明，如果某些设计参数（例如目标函数，码头设置等）方便选定，那么价值函数就是单调递减的。然后运用barbalat引理，该吸引力轨迹的模型名义上可以建立为(i.e. x(t) →0 as t →∞).从这种稳定的状态可以推断，一个很笼统的非线性系统：包括时变系统，非完整系统，系统允许间断意见等除外，如果值函数具有一定的连续性属性，然后Lyapunov稳定性（即轨迹停留任意接近的起源提供了足

够向原产地的密切开始）也可以得到保障（见例如[11]）。不过，这为某些类别系统的最后状态可能不会实现，，例如轮式车的例子（见例[8]所讨论的问题）。

类似的做法，建立单调减少的价值功能之后,我们想要设置一些包含原点以保证状态轨迹的渐近方法。但是，困难所在是预测的轨迹只有刚好与由此产生的轨迹在特定的抽样instants时鲁棒稳定性能才可以得到，因为我们用一种广义版本的barbalat的引理。这些鲁棒稳定性结果也有效期为一个很一般类非线性时变系统的允许间断的意见。

最优控制有待解决的问题与模型预测控制的战略是在这里制定了非常笼统的受理套管制（例如，可衡量的控制职能），使结果更容易得到保证。在理论上讲，存在解决办法。不过，某种形式的有限参数的控制功能需要/可取的解决上线的优化问题。它可以证明即稳定或鲁棒性的结果在这里所描述的仍然有效，当优化进行了有限的参数化的管制，如分段常数控制（如在[13]），或帮邦间断反馈（如在[9]）。

2、数据采样MPC框架

我们会考虑一种非线性的静态具有输入与状态的限制，凡变化状态后，时间t0 ，预计由以下模型。

这个模型的数据包括了一套所有可能的初始状态，在最初的时间

矢量这是状态的测量时间，某一函数f ：

一套的尽可能控制值。

我们假设这个制度，以渐近的可控性对，并为所有我们进一步假设函数f是连续的和局部Lipschitz方面的第二个论点。

注意到，在区间控制值的选定是由一个单值决定，因此优化

的设定都是进行在区间内的。

这里采用的符号如下所示。实时变量t代表我们的储备，我们保留S来表示时间变量，用于在预测模型。向量xt是指的实际状态。核电厂的测量时间t过程

的是一对从系统模型取得的弹道/控制。那个轨迹，有时是标注为

的，当我们想作明确地依赖于初始时间、初始状态和控制功

能。两变量是指我们的最优解，这是一个开放的闭环优化控制问题。过

程中是闭环系统的轨迹和控制造成的从MPC的策略。我们要求设计参数

的变数，目前，在开环最优控制问题是没有从系统模型（即变量，我们可以选择）;这些包括控制TC ，该预测地平线总磷，运行成本和终端成本的职能升和W，辅助控制律kaux，和终端约束集是由此产生的轨迹是由

决定的

这里

和功能于是

类似的采样数据框架使用的连续时间模型和采样国家的核电厂在离散instants的时间通过了在[ 2 ， 6 ， 7 ， 8 ， 13 ] 并正成为公认的框架，连续时间的MPC。它的结果可以表明，在此框架内有可能的地址和保证稳定性和鲁棒性，由此产生的闭环控制系统是一个非常大的类系统，可能是非线性，时变的和非完整性的。

3、非完整系统的和间断的反馈意见

在实践中有许多关于物理系统的兴趣，只能适当的作为非完整性系统。有一些例子，如轮式车辆，机器人，以及其他许多机械系统。

在控制这种系统时遇到的困难是任何线性化的原起源是无法控制的，因此任何的线性控制方法是无用的。不过，可能是主要的富有挑战性的特点是对非完整系统的，，刚才时间不变连续反馈获准[ 1 ]是不可能稳定的。但是，如果我们容许间断意见，它可能并不清楚什么是解决动态微分方程。（见[ 4， 8]为进一步讨论这个问题）。

解决该问题的概念已被证明是成功的，在处理与稳定由间断的意见是一种通用类别的可控系统概念是“采样-反馈”提出的解决办法[ 5 ] 。可以看出，采样数据所描述的MPC的框架内，可结合自然与“抽样反馈法”，从而确定一个轨迹的方式，这是非常类似的概念，介绍了在[ 5 ] 。这些轨迹在温和条件下，即使反馈法是间断的也是定义良好的。

在文献中的几个工程，允许间断的反馈意见的法律的背景下货币政策委员会（见[ 8 ]的调查工作）。这些框架内，允许间断只不过是采样数据的特点，适当使用一种积极的跨采样时间，再加上一个适当的解释解决一个间断微分方程。

4、barbalat的引理和变种

barbalat的引理是众所周知的以推断渐近稳定性非线性系统有力的工具，尤其是时变系统，利用Lyapunov样办法（见例如[ 17 ]为讨论和应用）。

简单的变种，这引理已成功地用来证明稳定的结果为模型预测控制（货币政策委员会）的非线性和时变系统的[ 7 ，15 ] 。事实上，在所有采样数据MPC 框架内列举出的上述情况， barbalat的引理，或修改它，是用来作为一个重要步骤，以证明稳定MPC的计划。这表明，如果某些设计参数（目标功能，码头设置等），方便的选定，则值函数是单调递减。然后，运用barbalat的引理，吸引力的轨迹的名义模型可以建立。这种稳定的财产可以推断，一个很笼统的类非线性系统：包括时变系统的，非完整系统，系统允许间断意见等等。

最近的工作，稳健的MPC的非线性系统[ 9 ]用了一个泛化对barbalat引理的一个重要步骤，以证明稳定的算法。

不过，这是我们认为，这种泛化的引理可能提供一个有用的工具来分析稳定在其他稳健的连续时间的货币政策委员会的做法，如一个形容这里时变系统的。

再微积分范畴内的一个标准的结果是如果一个函数有界较低和减少，那么会收敛到一个极限。不过，我们不能断定其衍生物是否会减少或消失，除非我们对

f施加了一些平滑的数据（t）。于是我们有了这样一个众所周知的形式，即barbalat引理（见例如： [ 17 ] ）。

5、名义的稳定

如果设计参数方便选定（即选定，以满足某一个足够稳定的条件，例如[ 7 ] ）稳定性分析可以进行显示，然后在MPC的价值函数V是表明要单调递减的。更确切地说，

对于和

这里M是连续的，径向无界，正定功能。函数V的MPC值被定义为

这里是为最优控制问题的函数

值。

从（7）我们可以知道对任意

因为是有限的。我们得出和

因此，

，因为f是连续的，我们得出所有的条件，申请barbalat

的引理2可见该轨迹渐近收敛到原点。注意：这个概念的稳定，并不一定包括Lyapunov稳定性财产以往其他稳定的的概念，请参阅[8]。

6、鲁棒稳定性

在过去的几年中合成的MPC被认为是在不同的工程[ 14 ]。下文所述的框架是基于一个在[ 9 ] ，扩展至timevarying 的系统。我们的目标是推动一个给定的状态非线性系统受扰动

反馈MPC的策略，是在每个采样即时状态由多次获得解决上线。

在这个优化问题，我们使用公约，如果一些约束不满意，那么价值的游戏+ ∞。这可确保价值的游戏是有限的，最优控制策略保证满意的程度限制，为一切可能的干扰情况。

7、有限参数的控制功能

结果的稳定性和鲁棒稳定性，证明了用最优控制解决问题所在是控制职能，选定一个非常一般的设置（一套可衡量的职能）。这个足够证明理论的稳定结果，它甚至允许使用的结果存在一个最小的解决方案，以实现最优控制问题（如[ 7 ，命题2 ] ）。然而，对于实现任何优化算法的描述控制功能需要加以形容一个有限的参数数目（即所谓的有限参数的控制功能）。控制参数为分段常数控制（如[ 13 ] ），多项式或由有限数量的样条函数描述得到，如枪声控件（例如， [ 9 ，10 ] ）等。注意，我们是不会考虑离散模型或动态方程。问题的离散逼近，详细讨论了在[ 16 ]及[ 12 ] 。

但是，在证明稳定时，我们只要表明，在一些点就是最优成本（值函数）是低于成本的使用另一受理控制。因此，只要设定可接受的控制值U的常数所有的时间，轻而易举的事，无论如何必然前稳定结果如下如果我们考虑到一套接纳控制功能（包括辅警控制法）是一个有限parameterizable，设置这样的一套受理的控制值是不断为所有的时间，那么双方的名义稳定性和鲁棒稳定的结果，这里所描述的仍然有效。

举例来说，是不连续的使用反馈控制继电器式控制策略类型,可以通过少量的描述参数,使得问题计算易处理。控制策略的值只允许再一个极端的范围内。许多控制问题如枪声稳定控制问题丰特斯和马尼[9]描述这种参数化的应用独轮脚踏车移动机器人有界干扰。