非线性模型预测控制_front-matter
非线性模型预测控制的若干问题研究
非线性模型预测控制的若干问题研究一、概述随着现代工业技术的快速发展，非线性模型预测控制（Nonlinear Model Predictive Control，NMPC）已成为控制领域的研究热点。
非线性系统广泛存在于实际工业过程中，其特性复杂、行为多样，且具有不确定性，这使得传统的线性控制策略在面对非线性系统时往往难以取得理想的效果。
研究非线性模型预测控制策略，对于提高控制系统的性能、稳定性和鲁棒性具有重要意义。
非线性模型预测控制是一种基于非线性模型的闭环优化控制策略，其核心思想是在每个采样周期，以系统当前状态为起点，在线求解有限时域开环最优问题，得到一个最优控制序列，并将该序列的第一个控制量作用于被控系统。
这种滚动优化的策略使得非线性模型预测控制能够实时地根据系统的状态变化调整控制策略，从而实现对非线性系统的有效控制。
非线性模型预测控制的研究也面临着诸多挑战。
由于非线性系统的复杂性，其预测模型的建立往往较为困难，且模型的准确性对控制效果的影响较大。
非线性模型预测控制需要在线求解优化问题，这对计算资源的需求较高，限制了其在实时性要求较高的系统中的应用。
非线性模型预测控制的稳定性和鲁棒性也是研究的重点问题。
本文旨在深入研究非线性模型预测控制的若干关键问题，包括非线性模型的建立、优化算法的设计、稳定性和鲁棒性的分析等。
通过对这些问题的研究，旨在提出一种高效、稳定、鲁棒的非线性模型预测控制策略，为实际工业过程的控制提供理论支持和实践指导。
1. 非线性模型预测控制（NMPC）概述非线性模型预测控制（Nonlinear Model Predictive Control，简称NMPC）是一种先进的控制策略，广泛应用于各种动态系统的优化控制问题中。
NMPC的核心思想是在每个控制周期内，利用系统的非线性模型预测未来的动态行为，并通过求解一个优化问题来得到最优控制序列。
这种方法能够显式地处理系统的不确定性和约束，因此非常适合于处理那些对控制性能要求较高、环境复杂多变的实际系统。
非线性控制系统中的模型预测控制技术
非线性控制系统中的模型预测控制技术一、引言现代控制理论的发展中，非线性控制系统成为了研究的关键领域。
非线性控制系统的特点是复杂性强、系统参数难以准确测量、不确定性大等。
这些因素使得非线性控制系统很难得到精准的控制。
本文将重点剖析模型预测控制技术在非线性控制系统中的应用。
二、非线性控制系统的特点一般来说，非线性控制系统具有以下几个特点：1. 系统的非线性和复杂性2. 系统存在参数不确定性，难以精确测量3. 控制输入和输出之间存在强耦合性4. 系统存在振荡、不稳定性等问题上述特点使得非线性控制系统的控制变得非常复杂，需要使用更加先进的控制算法来解决这些问题。
三、模型预测控制模型预测控制，简称MPC，是基于一个预测模型进行控制的一种方法。
在MPC中，控制器使用当前的状态以及对未来状态的预测来作出控制决策。
控制器会计算出一个控制变量序列，然后将其施加到非线性系统中。
这种方法可以提高控制系统的性能，从而降低控制误差。
MPC 的基本流程可以概括为以下几个步骤：1. 选择一个模型2. 预测下一步的状态和输出3. 计算控制变量序列，优化控制性能4. 应用当前的控制变量MPC 具有以下优点：1. 能够将未来的控制变量和权重考虑进去，使得控制系统能够更好地适应未来的变化。
2. 能够对强耦合的非线性系统进行控制。
3. 能够更好地应对系统不确定性和时变性。
因此，MPC 已经成为了非线性控制系统中的一种重要控制方法。
四、MPC 在非线性控制系统中的应用由于非线性控制系统具有非确定性和复杂性等特点，为了更好地处理这些问题，MPC 被广泛地应用在非线性控制系统中。
特别是在化工、能源等重要领域，MPC 已经成为了非线性控制系统中最常用的控制方法之一。
例如，在控制化工过程中， MFC（Model Predictive Control）技术已经广泛应用，该技术可以对复杂的化工过程中的需求进行实时调节，并对可能出现的负面效应进行修正。
模型预测控制方法在航空发动机控制中的应用
模型预测控制方法在航空发动机控制中的应用一、引言航空发动机控制是航空工业中的关键技术之一，对于航空发动机的性能和寿命具有至关重要的影响。
随着科技的不断发展和进步，模型预测控制方法越来越得到了广泛的应用，尤其是在航空发动机的控制中。
本文将介绍模型预测控制方法在航空发动机控制中的应用。
二、航空发动机控制概述航空发动机控制是一种复杂的系统工程，其主要任务是控制发动机在不同工况下的性能和行为。
航空发动机由许多复杂的机械和电子控制系统组成，需要调节和控制各种参数，如燃料流量、空气流量、涡轮转速等，以保证发动机的最佳性能和寿命。
三、模型预测控制方法概述模型预测控制方法（Model Predictive Control, MPC）是一种先进的控制技术，常用于多变量、非线性、约束控制系统中。
该方法是基于模型的控制策略，通过预测系统输出的变化规律和约束条件，来实现对系统动态响应的优化控制。
四、模型预测控制方法在航空发动机控制中的应用航空发动机控制涉及到多个参数的调节和协调工作，因此，使用模型预测控制方法可以更加准确地预测发动机行为和性能，并针对不同的工况进行相应的调节和控制。
1. 发动机空气流量控制发动机空气流量是直接影响发动机性能的重要参数之一。
使用模型预测控制方法，可以实时预测发动机空气流量变化趋势，通过调节发动机可调导叶的角度，调整进气系统的工作状态，从而优化发动机性能。
2. 发动机燃料流量控制发动机燃料流量是影响发动机工作状态的重要参数之一。
使用模型预测控制方法，可以通过预测发动机燃料流量的变化趋势并结合发动机的工况，实现在不同发动机状态下的燃油经济性和排放控制目标。
3. 发动机转速控制发动机转速是影响发动机性能的重要参数之一，尤其是在起降以及飞行过程中。
使用模型预测控制方法，可以通过预测发动机转速的变化，并实时调节发动机的涡轮调节系统的工作状态，从而控制发动机的转速，维持发动机的最佳运行状态。
5. 发动机寿命预测与维修调度对于航空发动机来说，寿命预测和维修调度是关键问题之一。
一种滑翔增程弹非线性模型预测控制方法
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修 观 ， 良明 ， 志强 王 郭
（． １ 南京理工大学 能源与动力工程学院 ， 江苏 南京 ２０ ９ ；． １０４ ２ 上海宇航 系统工程研究所 ， 上海 ２ １０ ； ０ １８ ３ 北方 自动控制技术研究所 ， ． 山西 太原 ０ ００ ） ３ ０ ６
预测控制模型结构
预测控制模型结构预测模型预测模型是预测控制模型的核心部分，它用于描述系统的动态行为，基于历史观测数据来预测未来的系统状态。
常见的预测模型有以下几种：1.线性模型：基于线性系统的假设，使用线性状态空间模型或ARMA模型等进行预测。
2.非线性模型：考虑非线性系统的特性，使用非线性回归模型、神经网络模型等进行预测。
3.神经网络模型：通过训练神经网络来拟合系统的输入输出关系，进行预测。
4.ARIMA模型：自回归滑动平均模型，用于描述时间序列数据的动态变化。
5.状态空间模型：将系统的状态和观测变量表示为状态方程和观测方程，通过状态估计和观测估计来进行预测。
控制器控制器是预测控制模型的另一个重要组成部分，它用于根据预测模型的输出进行控制决策。
常见的控制器有以下几种：1.模型预测控制器（MPC）：基于预测模型的输出，通过优化控制问题得到最优控制系列，实现对系统的控制。
2.比例积分微分（PID）控制器：通过比例、积分和微分操作来实现对系统的控制，可以根据误差信号调整控制输出。
3.神经网络控制器：使用神经网络来估计系统的输出，然后根据估计值进行控制决策。
4.最优控制器：通过求解最优化问题，得到最优控制输入，实现对系统的控制。
模型结构预测控制模型的结构是指预测模型和控制器的组合方式。
一般来说，预测模型和控制器之间存在以下两种结构：1.串级结构：预测模型和控制器按照串联的方式连接，预测模型先进行预测，然后将预测结果传递给控制器进行控制决策。
输入数据>预测模型>预测结果>控制器>控制输入2.并行结构：预测模型和控制器同时运行，预测模型负责预测系统状态，控制器负责根据预测结果进行控制决策。
输入数据>预测模型>预测结果|V控制器>控制输入。
非线性模型预测控制算法研究
非线性模型预测控制算法研究随着科技的发展，模型预测控制技术已逐渐成为控制领域的热门研究方向之一。
在传统的线性模型预测控制算法基础上，非线性模型预测控制算法已经得到了广泛应用，并取得了良好的控制效果。
本文将对非线性模型预测控制算法进行探究，并对其在实际应用中的优异表现进行分析。
一、非线性模型预测控制原理非线性模型预测控制算法的核心思想是建立非线性预测模型，然后利用该模型进行预测和控制。
与传统的线性模型预测控制算法不同的是，在非线性模型预测控制算法中，预测模型是通过非线性函数进行描述的。
这种方法能够更加准确地描述被控对象的动态特性，实现更好的前瞻性控制。
在非线性模型预测控制算法中，我们首先需要建立一个非线性模型，通常是建立一个神经网络模型或非线性回归模型。
接着，利用系统的历史数据进行训练和参数优化，得到一个可靠的预测模型。
在预测时，将模型输入预测变量，得到预测结果，然后进行控制决策。
在控制时，根据实际的运行状况和预测结果，调整控制动作，以达到预期的控制目标。
二、非线性模型预测控制算法的优势1. 能够更加准确地描述被控对象的动态特性与传统的线性模型预测控制算法相比，非线性模型预测控制算法能够更加准确地描述被控对象的动态特性。
这是由于非线性模型能够更好地逼近实际的物理过程。
这种方法能够充分挖掘系统的非线性特性，更好地描述系统的动态行为，从而实现更加准确的预测和控制。
2. 具有更强的稳定性和鲁棒性非线性模型预测控制算法具有更强的稳定性和鲁棒性。
这是由于该算法不受系统变化的影响，能够自适应地学习系统模型，并自动调整控制策略。
这种算法的控制性能更加可靠和优化，能够在实际应用中得到广泛应用。
3. 能够应对多变环境和复杂系统非线性模型预测控制算法能够应对多变环境和复杂系统。
这种算法在实际应用中表现出了很好的灵活性和鲁棒性，能够适应各种实际应用场景。
而在传统的线性模型预测控制算法中，存在线性模型无法描述非线性系统的缺陷，因此不能很好地应对复杂系统。
非线性预测及应用
非线性预测及应用随着科技的不断发展，数据处理和分析方法得到了广泛的应用，其中非线性预测在很多领域有着重要的应用价值。
本文将对非线性预测的基本原理、主要方法和应用领域进行探讨。
一、非线性预测的基本原理非线性预测是指根据已有的数据，对未来的变化趋势进行预测的方法。
与线性预测不同的是，非线性预测需要考虑到数据的复杂性和非线性规律性，能够更加准确地反映未来的趋势变化。
非线性预测的基本原理是寻找数据体系中的模式和规律，将其用数学模型进行刻画，从而实现对未来变化的预测。
这种方法适用于那些具有复杂性和非线性规律性的数据体系，如股票价格、气候变化、物理过程等。
二、非线性预测的主要方法非线性预测的主要方法包括神经网络、支持向量机、深度学习等。
神经网络是一种通过模拟人类脑神经元的方式，构建复杂非线性关系的方法。
它具有自适应性、容错性、并行性等优点，在金融、经济、气象等领域具有广泛的应用。
支持向量机是一种基于统计学习理论的分类和回归方法，其优点在于能够通过核函数将非线性问题转化为线性问题，从而提高预测的准确性。
深度学习则是一种基于神经网络的机器学习方法，它能够通过多层次的非线性变换和卷积操作对复杂数据进行高维特征提取和分类，具有在大数据处理和图像识别方面有着广泛的应用前景。
三、非线性预测在实际应用中的价值非线性预测在实际应用中具有广泛的价值，其适用于多个领域，如金融、能源、交通、医学等。
在金融领域，非线性预测能够帮助分析师和投资者，制定有效的投资策略，降低风险，提高收益。
例如，利用神经网络的非线性预测方法，可以预测股票价格、汇率趋势等金融指标；使用支持向量机的非线性预测方法，则能够对市场走势进行长期预测和策略优化。
在能源领域，非线性预测能够对能源价格、供需关系等关键指标进行预测和优化，帮助能源行业制定战略和计划。
例如，可以利用神经网络模型进行石油价格预测，对于石油公司来说，这有助于降低生产成本并制定更加科学合理的销售计划。
动力学非线性系统建模与预测控制
动力学非线性系统建模与预测控制在现代科技的飞速发展下，高科技产业的生产要求越来越高，要求对各种机电系统进行合理的建模和控制。
其中，动力学非线性系统的建模和预测控制是一个十分重要的问题。
动力学非线性系统是指其运动状态、系统输出和控制输入之间存在非线性关系，通俗的讲就是不存在一个通用的数学函数可以描述系统的行为。
这种系统在日常生活中很常见：例如，弹簧振动、地震、车辆运动轨迹等等。
由于其极其复杂的性质，能够对其进行建模和预测控制对于人类解决很多实际问题具有重要的意义。
在这方面，我们先来谈谈建模的问题。
对于非线性系统的建模，主要有时间域和频域两种方法。
时间域方法是指通过差分方程或微分方程来描述系统的状态变化，而频域方法则是通过系统的传递函数或频率响应来描述系统的输入和输出关系，即不考虑系统的状态变化。
相对来说，频域方法建模简单易懂，广泛应用也是其中的原因之一。
但是，当非线性系统的系统建模前提不能满足输出具有平稳性时，频域方法就不能使用，这时需要使用更为复杂的时间域方法。
在开始进行动力学非线性系统的建模之前，需先了解系统的基础性质，如系统是否相对稳定等，而这些性质确定了之后，才可进行相应的状态方程和输出方程的推导。
举个例子，我们来看看质量悬挂在弹簧上进行简谐振动的建模过程。
对于这个系统，可以通过牛顿第二定律F=ma得到其状态方程为（m为质量，k为弹簧系数，x为质量相对平衡点的位移）：m(d2x/dt2)+kx=0此外，可以通过观察到系统的位移x与时间t的关系，得到其输出方程为：x=Asinωt其中A表示振幅，ω表示角频率。
将其代入状态方程，可以解得系统的频率为：ω=√(k/m)通过上述推导过程，我们就成功地建立了弹簧振动的动力学非线性系统模型。
除了建立系统模型，预测控制也是非常重要的一个环节。
在许多应用中，经常需要预测未来的状态，进而为控制提供依据。
例如，对于自主驾驶汽车来说，需要对未来的交通情况进行预测，以便进行合理的驾驶。
非线性模型预测控制_Chapter1
Chapter1Introduction1.1What Is Nonlinear Model Predictive Control?Nonlinear model predictive control(henceforth abbreviated as NMPC)is an opti-mization based method for the feedback control of nonlinear systems.Its primaryapplications are stabilization and tracking problems,which we briefly introduce inorder to describe the basic idea of model predictive control.Suppose we are given a controlled process whose state x(n)is measured at dis-crete time instants t n,n=0,1,2,....“Controlled”means that at each time instantwe can select a control input u(n)which influences the future behavior of the stateof the system.In tracking control,the task is to determine the control inputs u(n)such that x(n)follows a given reference x ref(n)as good as possible.This means thatif the current state is far away from the reference then we want to control the systemtowards the reference and if the current state is already close to the reference thenwe want to keep it there.In order to keep this introduction technically simple,weconsider x(n)∈X=R d and u(n)∈U=R m,furthermore we consider a referencewhich is constant and equal to x∗=0,i.e.,x ref(n)=x∗=0for all n≥0.With such a constant reference the tracking problem reduces to a stabilization problem;in itsfull generality the tracking problem will be considered in Sect.3.3.Since we want to be able to react to the current deviation of x(n)from the ref-erence value x∗=0,we would like to have u(n)in feedback form,i.e.,in the form u(n)=μ(x(n))for some mapμmapping the state x∈X into the set U of control values.The idea of model predictive control—linear or nonlinear—is now to utilize amodel of the process in order to predict and optimize the future system behavior.Inthis book,we will use models of the formx+=f(x,u)(1.1) where f:X×U→X is a known and in general nonlinear map which assigns to a state x and a control value u the successor state x+at the next time instant.Starting from the current state x(n),for any given control sequence u(0),...,u(N−1)with L.Grüne,J.Pannek,Nonlinear Model Predictive Control,1 Communications and Control Engineering,DOI10.1007/978-0-85729-501-9_1,©Springer-Verlag London Limited201121Introduction horizon length N≥2,we can now iterate(1.1)in order to construct a predictiontrajectory x u defined byx u(0)=x(n),x u(k+1)=fx u(k),u(k),k=0,...,N−1.(1.2)Proceeding this way,we obtain predictions x u(k)for the state of the system x(n+k) at time t n+k in the future.Hence,we obtain a prediction of the behavior of the sys-tem on the discrete interval t n,...,t n+N depending on the chosen control sequence u(0),...,u(N−1).Now we use optimal control in order to determine u(0),...,u(N−1)such that x u is as close as possible to x∗=0.To this end,we measure the distance between x u(k)and x∗=0for k=0,...,N−1by a function (x u(k),u(k)).Here,we not only allow for penalizing the deviation of the state from the reference but also—if desired—the distance of the control values u(k)to a reference control u∗,which here we also choose as u∗=0.A common and popular choice for this purpose isthe quadratic functionx u(k),u(k)=x u(k)2+λu(k)2,where · denotes the usual Euclidean norm andλ≥0is a weighting parameter for the control,which could also be chosen as0if no control penalization is desired. The optimal control problem now readsminimize Jx(n),u(·):=N−1k=0x u(k),u(k)with respect to all admissible1control sequences u(0),...,u(N−1)with x u gen-erated by(1.2).Let us assume that this optimal control problem has a solution which is given by the minimizing control sequence u (0),...,u (N−1),i.e.,minu(0),...,u(N−1)Jx(n),u(·)=N−1k=0x u (k),u (k).In order to get the desired feedback valueμ(x(n)),we now setμ(x(n)):=u (0), i.e.,we apply thefirst element of the optimal control sequence.This procedure is sketched in Fig.1.1.At the following time instants t n+1,t n+2,...we repeat the procedure with the new measurements x(n+1),x(n+2),...in order to derive the feedback values μ(x(n+1)),μ(x(n+2)),....In other words,we obtain the feedback lawμby an iterative online optimization over the predictions generated by our model(1.1).2 This is thefirst key feature of model predictive control.1The meaning of“admissible”will be defined in Sect.3.2.2Attentive readers may already have noticed that this description is mathematically idealized since we neglected the computation time needed to solve the optimization problem.In practice,when the measurement x(n)is provided to the optimizer the feedback valueμ(x(n))will only be available after some delay.For simplicity of exposition,throughout our theoretical investigations we will assume that this delay is negligible.We will come back to this problem in Sect.7.6.1.2Where Did NMPC Come from?3Fig.1.1Illustration of the NMPC step at time t nFrom the prediction horizon point of view,proceeding this iterative way the trajectories x u(k),k=0,...,N provide a prediction on the discrete interval t n,...,t n+N at time t n,on the interval t n+1,...,t n+N+1at time t n+1,on the interval t n+2,...,t n+N+2at time t n+2,and so on.Hence,the prediction horizon is moving and this moving horizon is the second key feature of model predictive control.Regarding terminology,another term which is often used alternatively to model predictive control is receding horizon control.While the former expression stresses the use of model based predictions,the latter emphasizes the moving horizon idea. Despite these slightly different literal meanings,we prefer and follow the common practice to use these names synonymously.The additional term nonlinear indicates that our model(1.1)need not be a linear map.1.2Where Did NMPC Come from?Due to the vast amount of literature,the brief history of NMPC we provide in this section is inevitably incomplete and focused on those references in the literature from which we ourselves learned about the various NMPC techniques.Furthermore, we focus on the systems theoretic aspects of NMPC and on the academic develop-ment;some remarks on numerical methods specifically designed for NMPC can be found in rmation about the use of linear and nonlinear MPC in prac-tical applications can be found in many articles,books and proceedings volumes, e.g.,in[15,22,24].Nonlinear model predictive control grew out of the theory of optimal control which had been developed in the middle of the20th century with seminal contri-butions like the maximum principle of Pontryagin,Boltyanskii,Gamkrelidze and Mishchenko[20]and the dynamic programming method developed by Bellman [2].Thefirst paper we are aware of in which the central idea of model predictive41Introduction control—for discrete time linear systems—is formulated was published by Propo˘ı[21]in the early1960s.Interestingly enough,in this paper neither Pontryagin’s max-imum principle nor dynamic programming is used in order to solve the optimal con-trol problem.Rather,the paper already proposed the method which is predominant nowadays in NMPC,in which the optimal control problem is transformed into a static optimization problem,in this case a linear one.For nonlinear systems,the idea of model predictive control can be found in the book by Lee and Markus[14] from1967on page423:One technique for obtaining a feedback controller synthesis from knowl-edge of open-loop controllers is to measure the current control process state and then compute very rapidly for the open-loop control function.Thefirst portion of this function is then used during a short time interval,after whicha new measurement of the process state is made and a new open-loop con-trol function is computed for this new measurement.The procedure is then repeated.Due to the fact that neither computer hardware nor software for the necessary“very rapid”computation were available at that time,for a while this observation had little practical impact.In the late1970s,due to the progress in algorithms for solving constrained linear and quadratic optimization problems,MPC for linear systems became popular in control engineering.Richalet,Rault,Testud and Papon[25]and Cutler and Ramaker [6]were among thefirst to propose this method in the area of process control,in which the processes to be controlled are often slow enough in order to allow for an online optimization,even with the computer technology available at that time. It is interesting to note that in[25]the method was described as a“new method of digital process control”and earlier references were not mentioned;it appears that the basic MPC principle was re-invented several times.Systematic stability investigations appeared a little bit later;an account of early results in that direction for linear MPC can,e.g.,be found in the survey paper of García,Prett and Morari [10]or in the monograph by Bitmead,Gevers and Wertz[3].Many of the techniques which later turned out to be useful for NMPC,like Lyapunov function based stability proofs or stabilizing terminal constraints were in factfirst developed for linear MPC and later carried over to the nonlinear setting.The earliest paper we were able tofind which analyzes an NMPC algorithm sim-ilar to the ones used today is an article by Chen and Shaw[4]from1982.In this paper,stability of an NMPC scheme with equilibrium terminal constraint in contin-uous time is proved using Lyapunov function techniques,however,the whole opti-mal control function on the optimization horizon is applied to the plant,as opposed to only thefirst part as in our NMPC paradigm.For NMPC algorithms meeting this paradigm,first comprehensive stability studies for schemes with equilibrium termi-nal constraint were given in1988by Keerthi and Gilbert[13]in discrete time and in1990by Mayne and Michalska[17]in continuous time.The fact that for non-linear systems equilibrium terminal constraints may cause severe numerical diffi-culties subsequently motivated the investigation of alternative techniques.Regional1.3How Is This Book Organized?5 terminal constraints in combination with appropriate terminal costs turned out to be a suitable tool for this purpose and in the second half of the1990s there was a rapid development of such techniques with contributions by De Nicolao,Magni and Scattolini[7,8],Magni and Sepulchre[16]or Chen and Allgöwer[5],both in discrete and continuous time.This development eventually led to the formulation of a widely accepted“axiomatic”stability framework for NMPC schemes with sta-bilizing terminal constraints as formulated in discrete time in the survey article by Mayne,Rawlings,Rao and Scokaert[18]in2000,which is also an excellent source for more detailed information on the history of various NMPC variants not men-tioned here.This framework also forms the core of our stability analysis of such schemes in Chap.5of this book.A continuous time version of such a framework was given by Fontes[9]in2001.All stability results discussed so far add terminal constraints as additional state constraints to thefinite horizon optimization in order to ensure stability.Among the first who provided a rigorous stability result of an NMPC scheme without such con-straints were Parisini and Zoppoli[19]and Alamir and Bornard[1],both in1995and for discrete time systems.Parisini and Zoppoli[19],however,still needed a terminal cost with specific properties similar to the one used in[5].Alamir and Bonnard[1] were able to prove stability without such a terminal cost by imposing a rank con-dition on the linearization on the system.Under less restrictive conditions,stability results were provided in2005by Grimm,Messina,Tuna and Teel[11]for discrete time systems and by Jadbabaie and Hauser[12]for continuous time systems.The results presented in Chap.6of this book are qualitatively similar to these refer-ences but use slightly different assumptions and a different proof technique which allows for quantitatively tighter results;for more details we refer to the discussions in Sects.6.1and6.9.After the basic systems theoretic principles of NMPC had been clarified,more advanced topics like robustness of stability and feasibility under perturbations,per-formance estimates and efficiency of numerical algorithms were addressed.For a discussion of these more recent issues including a number of references we refer to thefinal sections of the respective chapters of this book.1.3How Is This Book Organized?The book consists of two main parts,which cover systems theoretic aspects of NMPC in Chaps.2–8on the one hand and numerical and algorithmic aspects in Chaps.9–10on the other hand.These parts are,however,not strictly separated;in particular,many of the theoretical and structural properties of NMPC developed in thefirst part are used when looking at the performance of numerical algorithms.The basic theme of thefirst part of the book is the systems theoretic analysis of stability,performance,feasibility and robustness of NMPC schemes.This part starts with the introduction of the class of systems and the presentation of background material from Lyapunov stability theory in Chap.2and proceeds with a detailed61Introduction description of different NMPC algorithms as well as related background information on dynamic programming in Chap.3.A distinctive feature of this book is that both schemes with stabilizing terminal constraints as well as schemes without such constraints are considered and treated in a uniform way.This“uniform way”consists of interpreting both classes of schemes as relaxed versions of infinite horizon optimal control.To this end,Chap.4first de-velops the theory of infinite horizon optimal control and shows by means of dynamic programming and Lyapunov function arguments that infinite horizon optimal feed-back laws are actually asymptotically stabilizing feedback laws.The main building block of our subsequent analysis is the development of a relaxed dynamic program-ming framework in Sect.4.3.Roughly speaking,Theorems4.11and4.14in this section extract the main structural properties of the infinite horizon optimal control problem,which ensure•asymptotic or practical asymptotic stability of the closed loop,•admissibility,i.e.,maintaining the imposed state constraints,•a guaranteed bound on the infinite horizon performance of the closed loop,•applicability to NMPC schemes with and without stabilizing terminal constraints. The application of these theorems does not necessarily require that the feedback law to be analyzed is close to an infinite horizon optimal feedback law in some quantitative sense.Rather,it requires that the two feedback laws share certain prop-erties which are sufficient in order to conclude asymptotic or practical asymptotic stability and admissibility for the closed loop.While our approach allows for inves-tigating the infinite horizon performance of the closed loop for most schemes under consideration—which we regard as an important feature of the approach in this book—we would like to emphasize that near optimal infinite horizon performance is not needed for ensuring stability and admissibility.The results from Sect.4.3are then used in the subsequent Chaps.5and6in order to analyze stability,admissibility and infinite horizon performance properties for NMPC schemes with and without stabilizing terminal constraints,respectively. Here,the results for NMPC schemes with stabilizing terminal constraints in Chap.5 can by now be considered as classical and thus mainly summarize what can be found in the literature,although some results—like,e.g.,Theorems5.21and5.22—generalize known results.In contrast to this,the results for NMPC schemes without stabilizing terminal constraints in Chap.6were mainly developed by ourselves and coauthors and have not been presented before in this way.While most of the results in this book are formulated and proved in a mathemat-ically rigorous way,Chap.7deviates from this practice and presents a couple of variants and extensions of the basic NMPC schemes considered before in a more survey like manner.Here,proofs are occasionally only sketched with appropriate references to the literature.In Chap.8we return to the more rigorous style and discuss feasibility and robust-ness issues.In particular,in Sects.8.1–8.3we present feasibility results for NMPC schemes without stabilizing terminal constraints and without imposing viability as-sumptions on the state constraints which are,to the best of our knowledge,either1.3How Is This Book Organized?7 entirely new or were so far only known for linear MPC.These resultsfinish our study of the properties of the nominal NMPC closed-loop system,which is why it is followed by a comparative discussion of the advantages and disadvantages of the various NMPC schemes presented in this book in Sect.8.4.The remaining sec-tions in Chap.8address the robustness of the stability of the NMPC closed loop with respect to additive perturbations and measurement errors.Here we decided to present a selection of results we consider representative,partially from the literature and partially based on our own research.These considerationsfinish the systems theoretic part of the book.The numerical part of the book covers two central questions in NMPC:how can we numerically compute the predicted trajectories needed in NMPC forfinite-dimensional sampled data systems and how is the optimization in each NMPC step performed numerically?Thefirst issue is treated in Chap.9,in which we start by giving an overview on numerical one step methods,a classical numerical technique for solving ordinary differential equations.After having looked at the convergence analysis and adaptive step size control techniques,we discuss some implementa-tional issues for the use of this methods within NMPC schemes.Finally,we investi-gate how the numerical approximation errors affect the closed-loop behavior,using the robustness results from Chap.8.The last Chap.10is devoted to numerical algorithms for solving nonlinearfi-nite horizon optimal control problems.We concentrate on so-called direct methods which form the currently by far preferred class of algorithms in NMPC applications. In these methods,the optimal control problem is transformed into a static optimiza-tion problem which can then be solved by nonlinear programming algorithms.We describe different ways of how to do this transformation and then give a detailed introduction into some popular nonlinear programming algorithms for constrained optimization.The focus of this introduction is on explaining how these algorithms work rather than on a rigorous convergence theory and its purpose is twofold:on the one hand,even though we do not expect our readers to implement such algorithms, we still think that some background knowledge is helpful in order to understand the opportunities and limitations of these numerical methods.On the other hand,we want to highlight the key features of these algorithms in order to be able to explain how they can be efficiently used within an NMPC scheme.This is the topic of the final Sects.10.4–10.6,in which several issues regarding efficient implementation, warm start and feasibility are investigated.Like Chap.7and in contrast to the other chapters in the book,Chap.10has in large parts a more survey like character,since a comprehensive and rigorous treatment of these topics would easilyfill an entire book.Still,we hope that this chapter contains valuable information for those readers who are interested not only in systems theoretic foundations but also in the practical numerical implementation of NMPC schemes.Last but not least,for all examples presented in this book we offer either MAT-LAB or C++code in order to reproduce our numerical results.This code is available from the web page81Introduction Both our MATLAB NMPC routine—which is suitable for smaller problems—as well as our C++NMPC package—which can also handle larger problems withreasonable computing time—can also be modified in order to perform simulationsfor problems not treated in this book.In order to facilitate both the usage and themodification,the Appendix contains brief descriptions of our routines.Beyond numerical experiments,almost every chapter contains a small selectionof problems related to the more theoretical results.Solutions for these problemsare available from the authors upon request by email.Attentive readers will notethat several of these problems—as well as some of our examples—are actually lin-ear problems.Even though all theoretical and numerical results apply to generalnonlinear systems,we have decided to include such problems and examples,be-cause nonlinear problems hardly ever admit analytical solutions,which are neededin order to solve problems or to work out examples without the help of numericalalgorithms.Let usfinally say a few words on the class of systems and NMPC problemsconsidered in this book.Most results are formulated for discrete time systems onarbitrary metric spaces,which in particular coversfinite-and infinite-dimensionalsampled data systems.The discrete time setting has been chosen because of its no-tational and conceptual simplicity compared to a continuous time formulation.Still,since sampled data continuous time systems form a particularly important class ofsystems,we have made considerable effort in order to highlight the peculiaritiesof this system class whenever appropriate.This concerns,among other topics,therelation between sampled data systems and discrete time systems in Sect.2.2,thederivation of continuous time stability properties from their discrete time counter-parts in Sect.2.4and Remark4.13,the transformation of continuous time NMPCschemes into the discrete time formulation in Sect.3.5and the numerical solutionof ordinary differential equations in Chap.9.Readers or lecturers who are inter-ested in NMPC in a pure discrete time framework may well skip these parts of thebook.The most general NMPC problem considered in this book3is the asymptotictracking problem in which the goal is to asymptotically stabilize a time varyingreference x ref(n).This leads to a time varying NMPC formulation;in particular,the optimal control problem to be solved in each step of the NMPC algorithm ex-plicitly depends on the current time.All of the fundamental results in Chaps.2–4explicitly take this time dependence into account.However,in order to be able toconcentrate on concepts rather than on technical details,in the subsequent chapterswe often decided to simplify the setting.To this end,many results in Chaps.5–8arefirst formulated for time invariant problems x ref≡x∗—i.e.,for stabilizing an x∗—and the necessary modifications for the time varying case are discussed after-wards.3Except for some further variants discussed in Sects.3.5and7.10.1.4What Is Not Covered in This Book?9 1.4What Is Not Covered in This Book?The area of NMPC has grown so rapidly over the last two decades that it is virtually impossible to cover all developments in detail.In order not to overload this book,we have decided to omit several topics,despite the fact that they are certainly important and useful in a variety of applications.We end this introduction by giving a brief overview over some of these topics.For this book,we decided to concentrate on NMPC schemes with online opti-mization only,thus leaving out all approaches in which part of the optimization is carried out offline.Some of these methods,which can be based on both infinite hori-zon andfinite horizon optimal control and are often termed explicit MPC,are briefly discussed in Sects.3.5and4.4.Furthermore,we will not discuss special classes of nonlinear systems like,e.g.,piecewise linear systems often considered in the explicit MPC literature.Regarding robustness of NMPC controllers under perturbations,we have re-stricted our attention to schemes in which the optimization is carried out for a nom-inal model,i.e.,in which the perturbation is not explicitly taken into account in the optimization objective,cf.Sects.8.5–8.9.Some variants of model predictive con-trol in which the perturbation is explicitly taken into account,like min–max MPC schemes building on game theoretic ideas or tube based MPC schemes relying on set oriented methods are briefly discussed in Sect.8.10.An emerging and currently strongly growingfield are distributed NMPC schemes in which the optimization in each NMPC step is carried out locally in a number of subsystems instead of using a centralized optimization.Again,this is a topic which is not covered in this book and we refer to,e.g.,Rawlings and Mayne[23,Chap.6] and the references therein for more information.At the very heart of each NMPC algorithm is a mathematical model of the sys-tems dynamics,which leads to the discrete time dynamics f in(1.1).While we will explain in detail in Sect.2.2and Chap.9how to obtain such a discrete time model from a differential equation,we will not address the question of how to obtain a suitable differential equation or how to identify the parameters in this model.Both modeling and parameter identification are serious problems in their own right which cannot be covered in this book.It should,however,be noted that optimization meth-ods similar to those used in NMPC can also be used for parameter identification; see,e.g.,Schittkowski[26].A somewhat related problem stems from the fact that NMPC inevitably leads to a feedback law in which the full state x(n)needs to be measured in order to evaluate the feedback law,i.e.,a state feedback law.In most applications,this information is not available;instead,only output information y(n)=h(x(n))for some output map h is at hand.This implies that the state x(n)must be reconstructed from the output y(n)by means of a suitable observer.While there is a variety of different techniques for this purpose,it is interesting to note that an idea which is very similar to NMPC can be used for this purpose:in the so-called moving horizon state estimation ap-proach the state is estimated by iteratively solving optimization problems over a101Introduction moving time horizon,analogous to the repeated minimization of J(x(n),u(·))de-scribed above.However,instead of minimizing the future deviations of the pre-dictions from the reference value,here the past deviations of the trajectory from the measured output values are minimized.More information on this topic can be found,e.g.,in Rawlings and Mayne[23,Chap.4]and the references therein.References1.Alamir,M.,Bornard,G.:Stability of a truncated infinite constrained receding horizon scheme:the general discrete nonlinear case.Automatica31(9),1353–1356(1995)2.Bellman,R.:Dynamic Programming.Princeton University Press,Princeton(1957).Reprintedin20103.Bitmead,R.R.,Gevers,M.,Wertz,V.:Adaptive Optimal Control.The Thinking Man’s GPC.International Series in Systems and Control Engineering.Prentice Hall,New York(1990) 4.Chen,C.C.,Shaw,L.:On receding horizon feedback control.Automatica18(3),349–352(1982)5.Chen,H.,Allgöwer,F.:Nonlinear model predictive control schemes with guaranteed stabil-ity.In:Berber,R.,Kravaris,C.(eds.)Nonlinear Model Based Process Control,pp.465–494.Kluwer Academic,Dordrecht(1999)6.Cutler,C.R.,Ramaker,B.L.:Dynamic matrix control—a computer control algorithm.In:Pro-ceedings of the Joint Automatic Control Conference,pp.13–15(1980)7.De Nicolao,G.,Magni,L.,Scattolini,R.:Stabilizing nonlinear receding horizon control viaa nonquadratic terminal state penalty.In:CESA’96IMACS Multiconference:ComputationalEngineering in Systems Applications,Lille,France,pp.185–187(1996)8.De Nicolao,G.,Magni,L.,Scattolini,R.:Stabilizing receding-horizon control of nonlineartime-varying systems.IEEE Trans.Automat.Control43(7),1030–1036(1998)9.Fontes,F.A.C.C.:A general framework to design stabilizing nonlinear model predictive con-trollers.Systems Control Lett.42(2),127–143(2001)10.García,C.E.,Prett,D.M.,Morari,M.:Model predictive control:Theory and practice—a sur-vey.Automatica25(3),335–348(1989)11.Grimm,G.,Messina,M.J.,Tuna,S.E.,Teel,A.R.:Model predictive control:for want of alocal control Lyapunov function,all is not lost.IEEE Trans.Automat.Control50(5),546–558 (2005)12.Jadbabaie,A.,Hauser,J.:On the stability of receding horizon control with a general terminalcost.IEEE Trans.Automat.Control50(5),674–678(2005)13.Keerthi,S.S.,Gilbert,E.G.:Optimal infinite-horizon feedback laws for a general class ofconstrained discrete-time systems:stability and moving-horizon approximations.J.Optim.Theory Appl.57(2),265–293(1988)14.Lee,E.B.,Markus,L.:Foundations of Optimal Control Theory.Wiley,New York(1967)15.Maciejowski,J.M.:Predictive Control with Constraints.Prentice Hall,New York(2002)16.Magni,L.,Sepulchre,R.:Stability margins of nonlinear receding-horizon control via inverseoptimality.Systems Control Lett.32(4),241–245(1997)17.Mayne,D.Q.,Michalska,H.:Receding horizon control of nonlinear systems.IEEE Trans.Automat.Control35(7),814–824(1990)18.Mayne,D.Q.,Rawlings,J.B.,Rao,C.V.,Scokaert,P.O.M.:Constrained model predictive con-trol:Stability and optimality.Automatica36(6),789–814(2000)19.Parisini,T.,Zoppoli,R.:A receding-horizon regulator for nonlinear systems and a neuralapproximation.Automatica31(10),1443–1451(1995)20.Pontryagin,L.S.,Boltyanskii,V.G.,Gamkrelidze,R.V.,Mishchenko,E.F.:The MathematicalTheory of Optimal Processes.Translated by D.E.Brown.Pergamon/Macmillan Co.,New York (1964)。
基于非线性模型的预测控制技术研究
基于非线性模型的预测控制技术研究
在控制系统中，预测控制技术一直受到研究者的广泛关注。
随着工业自动化程度的提高和复杂度的增加，控制的准确性和实时性也变得越来越重要。
传统的线性控制方法已不能满足实际控制需求。
因此，基于非线性模型的预测控制技术应运而生。
基于非线性模型的预测控制技术包括神经网络预测控制、模糊神经网络预测控制、小波神经网络预测控制等。
其中，神经网络预测控制技术是一种非线性控制策略，具有广阔的应用前景。
神经网络预测控制技术是一种类似于大脑神经系统的人工智能算法。
它是通过大量数据学习、训练出神经网络，将其用于模型建立和预测控制。
神经网络预测控制技术具有智能性、自适应性、非线性映射能力等优点，可以实现对非线性系统的精确控制。
模糊神经网络预测控制技术则将模糊逻辑运用于神经网络控制中，使得神经网络能够处理不确定和不完整的信息，并进行合理的推理和决策。
它比单独的神经网络预测控制技术更具表达力和智能性。
小波神经网络预测控制技术将小波分析应用于神经网络控制中，可以采用小波基函数对非线性系统进行逼近。
这种预测控制技术具有高效性和精确性，能够应对复杂的非线性系统控制问题。
与传统的线性控制方法相比，基于非线性模型的预测控制技术具有更高的自适应性和精度，可以有效应对非线性系统的控制问题。
同时，随着神经网络硬件和计算技术的进步，基于非线性模型的预测控制技术将会得到更广泛的应用。
总之，基于非线性模型的预测控制技术是一种全新的控制策略，具有广泛的应用前景。
未来，基于非线性模型的预测控制技术将会在工业自动化、智能化控制等领域发挥重要作用。
非线性控制系统的模型预测方法研究
非线性控制系统的模型预测方法研究随着科技的不断进步和应用领域的不断扩展，控制系统已经成为现代社会中不可或缺的一部分。
其中，非线性控制系统因为可以解决许多线性系统难以应对的问题，在各个领域中被广泛应用。
而在非线性控制系统中，模型预测方法成为一种常见的控制策略。
一、非线性控制系统概述非线性系统是指不符合线性叠加原理的系统，也就是说，其输出与输入之间的关系不是线性的。
相比于线性系统，非线性系统模型更加复杂，因此在控制系统中，非线性控制系统需要采取更加复杂的控制策略才能实现对系统的有效控制。
以机器人控制为例，机器人在执行任务时面临的环境和任务是复杂多变的，如何通过控制增强机器人的灵活性、稳定性和精度就成为了难点。
这时候，非线性控制系统就能够发挥重要作用，因为模型的非线性特性能够更好地反映机器人在不同环境下的复杂状态，并且能够针对不同的任务场景动态调整控制参数，实现更高效的控制。
二、模型预测方法原理在非线性控制系统中，模型预测方法（Model Predictive Control，MPC）是一种比较常见的控制策略。
模型预测方法的基本思想是利用系统的动态模型来预测未来的系统状态，然后通过控制方法将系统状态引导到期望状态。
具体来说，模型预测方法的实现流程如下：1. 设置控制参数在模型预测方法中，需要预先设置控制参数，这些参数包括期望状态、目标输出等。
通过调整这些参数可以实现更加精确的控制。
2. 预测未来系统状态根据系统的动态模型，预测未来系统状态，同时考虑系统的环境变化和噪声干扰等因素，得出未来一段时间内的状态序列。
3. 优化控制策略利用优化算法，求解出一组最优的控制信号，使得未来一段时间内的系统状态能够达到期望状态，并且满足各种约束条件。
这一步是整个模型预测方法的核心。
4. 实施控制策略根据优化得出的控制信号，实施相应的控制策略，控制系统状态在未来一段时间内发生变化，使得系统能够达到期望状态。
三、模型预测方法的特点模型预测方法因其具有的许多特点而在非线性控制系统中被广泛使用，其主要特点包括：1. 预测能力强模型预测方法可以利用系统的动态模型对未来的系统状态进行预测，可以实现更加精确的控制。
强化学习中的模型预测控制方法
模型预测控制（MPC）是一种优化方法，它结合了模型预测和动态控制，以实现更优的控制性能。
在强化学习中，模型预测控制方法可以用于处理具有不确定性和复杂性的问题，如连续时间的动态系统、连续和离散的动作空间等。
模型预测控制的主要步骤包括：
1. 预测模型：使用系统的动态模型来预测系统的未来状态。
2. 定义约束：定义一系列约束条件，包括系统限制、资源限制和目标限制等。
3. 优化目标：优化一个或多个目标函数，通常包括最大化期望回报和最小化某些损失函数。
4. 动态控制：根据当前的预测和优化结果，生成未来的控制输入，以最大化预测性能并满足所有约束。
在强化学习中应用模型预测控制的方法可以归纳为以下几种：
1. 策略优化：通过寻找一种策略，使得未来的预测性能（如回报）最大化。
强化学习中的Q-learning、Actor-Critic等方法就使用了模型预测控制的思想。
2. 时序规划：对于具有复杂时序结构的问题，可以使用MPC方法来规划连续的动作序列。
3. 动态调整：强化学习中的许多问题都涉及到动态系统的状态转移和奖励函数，这时可以使用MPC来根据系统的状态和过去的经验动态地调整控制策略。
总的来说，模型预测控制方法在强化学习中主要用于解决具有不确定性和复杂性的问题，通过结合模型预测和动态控制，可以实现更优的控制性能。
非线性系统模型预测控制算法研究
非线性系统模型预测控制算法研究随着现代科技的飞速发展，越来越多的自动化、智能化设备出现在人们的生产、生活中。
这些设备需要跑出高效、精准的控制算法来实现它们的目标。
与此同时，非线性系统的广泛存在也使得传统的线性控制算法难以满足实际需求。
这时非线性系统模型预测控制算法便应运而生。
一、什么是非线性系统模型预测控制算法非线性系统模型预测控制算法是一种通过建立非线性系统的数学模型，预测系统响应并实现控制的方法。
它利用历史数据和对未来的预测来优化控制输出，以达到最优化的效果。
该算法本质上是一种优化算法，以最小化预测误差为目标，以提高系统性能为核心。
二、非线性系统模型预测控制算法的基本思想非线性系统模型预测控制算法的基本思想可以归纳为以下几点：1. 建立非线性系统的预测模型非线性系统的预测模型一般采用动态状态空间模型或非线性回归模型。
这个预测模型将历史数据建模，并通过对未来的预测获得最优化控制输出。
2. 进行优化控制基于预测模型，通过对未来的预测和历史数据的分析，来计算出最优控制输出。
为了使算法实现简单稳定，通常只考虑最小化预测误差，忽略约束条件等其他因素。
3. 控制器实施通过实施优化控制结果，将其转化为机器控制信号。
这种控制方法具有较高的实时性和适应性，并且可以适用于复杂的非线性系统。
三、非线性系统模型预测控制算法的研究内容非线性系统模型预测控制算法的研究内容通常包含以下几个方面：1. 建模方法的研究非线性系统的建模是非线性系统模型预测控制算法的关键，选取合适的建模方法可以提高算法的精度和实用性。
目前建模方法主要有基于ARMAX模型的方法、基于神经网络的方法和基于时滞的方法等。
2. 优化方法的研究优化方法是非线性系统模型预测控制算法的另一个关键，不同的优化方法可以影响算法的收敛速度和稳定性。
目前主流的优化方法有非线性规划方法、模型预测控制方法和演化算法等。
3. 实时性和执行效率的研究非线性系统模型预测控制算法需要具有较高的实时性和执行效率，才能适应复杂的实际场景。
模型预测控制 技术路线
模型预测控制技术路线模型预测控制（Model Predictive Control，MPC）是一种先进的控制策略，它结合了动态模型、优化方法和实时测量反馈，用于实现对动态系统的高性能控制。
MPC技术路线涉及以下几个方面：1. 动态系统建模，首先需要对待控制的动态系统进行准确的建模。
这可能涉及从物理原理出发建立微分方程模型，或者通过数据驱动的方法构建黑盒模型，如神经网络或系统辨识技术。
2. 优化问题建立，MPC使用优化方法来求解控制问题，因此需要将控制目标、系统约束和性能指标转化为数学优化问题。
这通常涉及到定义系统的状态变量、控制输入、约束条件和性能指标，并将其转化为适合优化求解的形式。
3. 预测模型设计，MPC需要使用系统模型进行预测，以便在优化问题中考虑未来时刻的系统响应。
这可能涉及将系统模型离散化，选择合适的预测时域和优化时域，并考虑系统不确定性的影响。
4. 优化求解，MPC通常使用迭代优化方法来求解控制问题，因此需要选择合适的优化算法和求解器。
常用的算法包括线性二次规划（LQP）和非线性优化方法，如序列二次规划（SQP）或者内点法等。
5. 实时实现，MPC需要在实时系统中实现，这涉及到将优化问题在线实时求解，并将优化结果转化为实际的控制指令。
这通常需要考虑计算效率、数值稳定性和实时性等方面的问题。
总的来说，MPC技术路线涉及动态系统建模、优化问题建立、预测模型设计、优化求解和实时实现等多个方面，需要综合考虑控制理论、数学优化、计算机实现等多个领域的知识和技术。
在实际应用中，还需要根据具体的控制目标和系统特性进行具体的技术路线设计和实现。
课件--模型预测控制
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模型预测控制的参数优化
模型预测控制的参数优化模型预测控制（Model Predictive Control，MPC）是一种基于数学模型的高级控制方法，通过预测模型对系统进行模拟预测和优化求解，实现对系统的精确控制。
不同模型预测控制的应用领域广泛，例如工业过程控制、机器人、交通系统等。
为了获得最佳控制效果，参数优化是MPC中非常重要的一环。
MPC的基本原理是通过建立系统模型来预测系统未来的行为，并根据预测结果选择最佳控制信号。
为了实现最佳控制，需要优化一些关键参数，这些参数包括：1.预测模型参数：优化预测模型参数是实施MPC的首要任务。
预测模型可以是线性或非线性的，参数优化的目标是使得预测模型能够最准确地描述系统的行为。
对于线性模型，常用的优化方法是最小二乘法，通过最小化预测误差来优化模型参数。
对于非线性模型，可以使用最优化算法，例如梯度下降法或遗传算法等。
2.控制器权重：MPC中的控制器权重是用来平衡各个控制目标的重要参数。
例如，在工业过程控制中，可能需要同时优化温度、压力和流量等多个目标。
优化权重可以根据不同目标的重要性来分配，以实现最佳控制效果。
权重的优化可以通过试错法或者通过经验法则来获得。
3.控制时域：控制时域是指每次控制操作的时间长度。
控制时域的选择需要考虑到系统的动态响应和计算复杂性。
较短的时域可以提高控制的灵敏度和准确性，但同时也会增加计算负担。
较长的时域可以降低计算负担，但可能导致控制器的响应时间较慢。
因此，控制时域的选择需要进行权衡和优化。
4.约束参数：约束参数是限制系统操作的条件。
在MPC中，常常会对系统状态、输入信号和输出信号等进行约束。
约束参数的优化是为了确保系统操作在安全和合理的范围内，例如保持输入信号在一定范围内、确保状态变量不会超过设定范围等。
约束参数的优化可以通过调整约束边界或者动态更新来实现。
总之，模型预测控制的参数优化是提高MPC控制效果的重要任务。
参数优化的目标是实现系统的最优控制，同时考虑到系统的动态响应、计算复杂性和约束条件等方面的综合因素。
一种快速非线性模型预测控制律研究
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非线性模型预测控制
非线性模型预测控制
非线性模型预测控制，是一种基于非线性模型的控制方法，它可以有
效地控制复杂的系统，并且可以满足多个约束条件。
NMPC的基本思想是，通过预测未来的状态，并在预测的状态下求解最优控制量，从而
实现最优控制。
NMPC的优势在于，它可以有效地控制复杂的系统，并且可以满足多个
约束条件。
NMPC可以有效地控制复杂的系统，因为它可以根据系统的
实际状态来预测未来的状态，从而更好地控制系统。
此外，NMPC可以
满足多个约束条件，因为它可以根据系统的实际状态来求解最优控制量，从而满足多个约束条件。
NMPC的应用非常广泛，它可以用于控制各种复杂的系统，如机器人、
航空航天、汽车、电力系统等。
例如，NMPC可以用于控制机器人的运动，从而实现机器人的自动化操作。
此外，NMPC还可以用于控制航空
航天系统，从而实现航空航天系统的自动化操作。
NMPC的缺点在于，它的计算复杂度较高，因为它需要预测未来的状态，并在预测的状态下求解最优控制量，从而实现最优控制。
此外，NMPC
还受到系统模型的精度限制，因为它需要根据系统的实际状态来预测
未来的状态，如果系统模型的精度不够，则可能会导致NMPC的控制效
果不佳。
总之，NMPC是一种有效的控制方法，它可以有效地控制复杂的系统，
并且可以满足多个约束条件。
但是，NMPC的计算复杂度较高，并且受
到系统模型的精度限制，因此，在使用NMPC时，需要考虑这些因素。
敏捷卫星姿态机动的非线性模型预测控制
敏捷卫星姿态机动的非线性模型预测控制范国伟;常琳;戴路;徐开;杨秀彬【摘要】针对以金字塔构型控制力矩陀螺(CMG)为执行机构的敏捷小卫星开展了先进机动控制算法的研究.在考虑控制力矩陀螺力矩约束及增量约束情况下,设计了基于非线性模型预测控制(NMPC)方法的卫星姿态快速机动控制律及操纵律.通过多种仿真分析了控制器设计参数变化对卫星姿态机动的影响,并与终端滑模控制方法进行了比较.实验结果表明,增大跟踪性能加权矩阵或延长预测时域均可以提高卫星姿态机动速度,缩短卫星姿态机动时间.设计的控制方法能够使卫星姿态在18 s内实现40°的大角度快速机动,姿态指向精度和稳定度分别为0.01°和0.04(°)/s,与终端滑模控制方法相比,机动速度及稳态性均得到提高.本文方法为敏捷小卫星的在轨应用方式提供了理论支撑.【期刊名称】《光学精密工程》【年(卷),期】2015(023)008【总页数】10页(P2318-2327)【关键词】敏捷卫星;姿态机动;控制力矩陀螺(CMG);非线性模型预测控制(NMPC)【作者】范国伟;常琳;戴路;徐开;杨秀彬【作者单位】中国科学院长春光学精密机械与物理研究所小卫星技术国家地方联合工程研究中心,吉林长春130033;中国科学院长春光学精密机械与物理研究所小卫星技术国家地方联合工程研究中心,吉林长春130033;中国科学院长春光学精密机械与物理研究所小卫星技术国家地方联合工程研究中心,吉林长春130033;中国科学院长春光学精密机械与物理研究所小卫星技术国家地方联合工程研究中心,吉林长春130033;中国科学院长春光学精密机械与物理研究所小卫星技术国家地方联合工程研究中心,吉林长春130033【正文语种】中文【中图分类】V448.221 引言随着空间航天任务的快速发展，对卫星姿态的敏捷性需求日益增加。
与传统卫星相比，这种具有快速机动能力的卫星可以极大地增加对地观测范围，延长与地面站的通信时间，并为多种空间灵巧成像模式提供保障［1－3］。
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Communications and Control Engineering For other titles published in this series,go to/series/61Series EditorsA.Isidori J.H.van Schuppen E.D.Sontag M.Thoma M.Krstic Published titles include:Stability and Stabilization of Infinite Dimensional Systems with ApplicationsZheng-Hua Luo,Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics(Second edition)Bernard BrogliatoNonlinear Control Systems IIAlberto IsidoriL2-Gain and Passivity Techniques in Nonlinear Control Arjan van der SchaftControl of Linear Systems with Regulation and Input ConstraintsAli Saberi,Anton A.Stoorvogel and Peddapullaiah SannutiRobust and H∞ControlBen M.ChenComputer Controlled SystemsEfim N.Rosenwasser and Bernhard mpeControl of Complex and Uncertain SystemsStanislav V.Emelyanov and Sergey K.Korovin Robust Control Design Using H∞MethodsIan R.Petersen,Valery A.Ugrinovski andAndrey V.SavkinModel Reduction for Control System DesignGoro Obinata and Brian D.O.AndersonControl Theory for Linear SystemsHarry L.Trentelman,Anton Stoorvogel and Malo Hautus Functional Adaptive ControlSimon G.Fabri and Visakan KadirkamanathanPositive1D and2D SystemsTadeusz KaczorekIdentification and Control Using Volterra Models Francis J.Doyle III,Ronald K.Pearson and Babatunde A.OgunnaikeNon-linear Control for Underactuated Mechanical SystemsIsabelle Fantoni and Rogelio LozanoRobust Control(Second edition)Jürgen AckermannFlow Control by FeedbackOle Morten Aamo and Miroslav KrsticLearning and Generalization(Second edition) Mathukumalli VidyasagarConstrained Control and EstimationGraham C.Goodwin,Maria M.Seron andJoséA.De DonáRandomized Algorithms for Analysis and Controlof Uncertain SystemsRoberto Tempo,Giuseppe Calafiore and Fabrizio Dabbene Switched Linear SystemsZhendong Sun and Shuzhi S.GeSubspace Methods for System IdentificationTohru KatayamaDigital Control SystemsIoan ndau and Gianluca ZitoMultivariable Computer-controlled SystemsEfim N.Rosenwasser and Bernhard mpe Dissipative Systems Analysis and Control(Second edition)Bernard Brogliato,Rogelio Lozano,Bernhard Maschke and Olav EgelandAlgebraic Methods for Nonlinear Control Systems Giuseppe Conte,Claude H.Moog and Anna M.Perdon Polynomial and Rational MatricesTadeusz KaczorekSimulation-based Algorithms for Markov Decision ProcessesHyeong Soo Chang,Michael C.Fu,Jiaqiao Hu and Steven I.MarcusIterative Learning ControlHyo-Sung Ahn,Kevin L.Moore and YangQuan Chen Distributed Consensus in Multi-vehicle Cooperative ControlWei Ren and Randal W.BeardControl of Singular Systems with Random Abrupt ChangesEl-Kébir BoukasNonlinear and Adaptive Control with Applications Alessandro Astolfi,Dimitrios Karagiannis and Romeo OrtegaStabilization,Optimal and Robust ControlAziz BelmiloudiControl of Nonlinear Dynamical SystemsFelix L.Chernous’ko,Igor M.Ananievski and Sergey A.ReshminPeriodic SystemsSergio Bittanti and Patrizio ColaneriDiscontinuous SystemsYury V.OrlovConstructions of Strict Lyapunov FunctionsMichael Malisoff and Frédéric MazencControlling ChaosHuaguang Zhang,Derong Liu and Zhiliang Wang Stabilization of Navier–Stokes FlowsViorel BarbuDistributed Control of Multi-agent NetworksWei Ren and Yongcan CaoLars Grüne Jürgen Pannek Nonlinear Model Predictive Control Theory and AlgorithmsLars Grüne Mathematisches Institut Universität Bayreuth Bayreuth95440Germanylars.gruene@uni-bayreuth.de Jürgen Pannek Mathematisches Institut Universität BayreuthBayreuth95440Germanyjuergen.pannek@uni-bayreuth.deISSN0178-5354ISBN978-0-85729-500-2e-ISBN978-0-85729-501-9DOI10.1007/978-0-85729-501-9Springer London Dordrecht Heidelberg New YorkBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Control Number:2011926502Mathematics Subject Classification(2010):93-02,92C10,93D15,49M37©Springer-Verlag London Limited2011Apart from any fair dealing for the purposes of research or private study,or criticism or review,as per-mitted under the Copyright,Designs and Patents Act1988,this publication may only be reproduced, stored or transmitted,in any form or by any means,with the prior permission in writing of the publish-ers,or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency.Enquiries concerning reproduction outside those terms should be sent to the publishers.The use of registered names,trademarks,etc.,in this publication does not imply,even in the absence of a specific statement,that such names are exempt from the relevant laws and regulations and therefore free for general use.The publisher makes no representation,express or implied,with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.Cover design:VTeX UAB,LithuaniaPrinted on acid-free paperSpringer is part of Springer Science+Business Media()For Brigitte,Florian and CarlaLGFor Sabina and AlinaJPPrefaceThe idea for this book grew out of a course given at a winter school of the In-ternational Doctoral Program“Identification,Optimization and Control with Ap-plications in Modern Technologies”in Schloss Thurnau in March2009.Initially, the main purpose of this course was to present results on stability and performance analysis of nonlinear model predictive control algorithms,which had at that time recently been obtained by ourselves and coauthors.However,we soon realized that both the course and even more the book would be inevitably incomplete without a comprehensive coverage of classical results in the area of nonlinear model pre-dictive control and without the discussion of important topics beyond stability and performance,like feasibility,robustness,and numerical methods.As a result,this book has become a mixture between a research monograph and an advanced textbook.On the one hand,the book presents original research results obtained by ourselves and coauthors during the lastfive years in a comprehensive and self contained way.On the other hand,the book also presents a number of results—both classical and more recent—of other authors.Furthermore,we have included a lot of background information from mathematical systems theory,op-timal control,numerical analysis and optimization to make the book accessible to graduate students—on PhD and Master level—from applied mathematics and con-trol engineering alike.Finally,via our web page we provide MATLAB and C++software for all examples in this book,which enables the reader to perform his or her own numerical experiments.For reading this book,we assume a basic familiarity with control systems,their state space representation as well as with concepts like feedback and stability as provided,e.g.,in undergraduate courses on control engineering or in courses on mathematical systems and control theory in an applied mathematics curriculum.However,no particular knowledge of nonlin-ear systems theory is assumed.Substantial parts of the systems theoretic chapters of the book have been used by us for a lecture on nonlinear model predictive con-trol for master students in applied mathematics and we believe that the book is well suited for this purpose.More advanced concepts like time varying formulations or peculiarities of sampled data systems can be easily skipped if only time invariant problems or discrete time systems shall be treated.viiviii PrefaceThe book centers around two main topics:systems theoretic properties of nonlin-ear model predictive control schemes on the one hand and numerical algorithms on the other hand;for a comprehensive description of the contents we refer to Sect.1.3.As such,the book is somewhat more theoretical than engineering or application ori-ented monographs on nonlinear model predictive control,which are furthermore often focused on linear methods.Within the nonlinear model predictive control literature,distinctive features of this book are the comprehensive treatment of schemes without stabilizing terminal constraints and the in depth discussion of performance issues via infinite horizon suboptimality estimates,both with and without stabilizing terminal constraints.The key for the analysis in the systems theoretic part of this book is a uniform way of interpreting both classes of schemes as relaxed versions of infinite horizon op-timal control problems.The relaxed dynamic programming framework developed in Chap.4is thus a cornerstone of this book,even though we do not use dynamic programming for actually solving nonlinear model predictive control problems;for this task we prefer direct optimization methods as described in the last chapter of this book,since they also allow for the numerical treatment of high dimensional systems.There are many people whom we have to thank for their help in one or the other way.For pleasant and fruitful collaboration within joint research projects and on joint papers—of which many have been used as the basis for this book—we are grateful to Frank Allgöwer,Nils Altmüller,Rolf Findeisen,Marcus von Lossow,Dragan Neši´c ,Anders Rantzer,Martin Seehafer,Paolo Varutti and Karl Worthmann.For enlightening talks,inspiring discussions,for organizing workshops and mini-symposia (and inviting us)and,last but not least,for pointing out valuable references to the literature we would like to thank David Angeli,Moritz Diehl,Knut Graichen,Peter Hokayem,Achim Ilchmann,Andreas Kugi,Daniel Limón,Jan Lunze,Lalo Magni,Manfred Morari,Davide Raimondo,Saša Rakovi´c ,Jörg Rambau,Jim Rawl-ings,Markus Reble,Oana Serea and Andy Teel,and we apologize to everyone who is missing in this list although he or she should have been mentioned.Without the proof reading of Nils Altmüller,Robert Baier,Thomas Jahn,Marcus von Lossow,Florian Müller and Karl Worthmann the book would contain even more typos and inaccuracies than it probably does—of course,the responsibility for all remaining errors lies entirely with us and we appreciate all comments on errors,typos,miss-ing references and the like.Beyond proof reading,we are grateful to Thomas Jahn for his help with writing the software supporting this book and to Karl Worthmann for his contributions to many results in Chaps.6and 7,most importantly the proof of Proposition 6.17.Finally,we would like to thank Oliver Jackson and Charlotte Cross from Springer-Verlag for their excellent rs Grüne Jürgen PannekBayreuth,Germany April 2011Contents1Introduction (1)1.1What Is Nonlinear Model Predictive Control? (1)1.2Where Did NMPC Come from? (3)1.3How Is This Book Organized? (5)1.4What Is Not Covered in This Book? (9)References (10)2Discrete Time and Sampled Data Systems (13)2.1Discrete Time Systems (13)2.2Sampled Data Systems (16)2.3Stability of Discrete Time Systems (28)2.4Stability of Sampled Data Systems (35)2.5Notes and Extensions (39)2.6Problems (39)References (41)3Nonlinear Model Predictive Control (43)3.1The Basic NMPC Algorithm (43)3.2Constraints (45)3.3Variants of the Basic NMPC Algorithms (50)3.4The Dynamic Programming Principle (56)3.5Notes and Extensions (62)3.6Problems (64)References (65)4Infinite Horizon Optimal Control (67)4.1Definition and Well Posedness of the Problem (67)4.2The Dynamic Programming Principle (70)4.3Relaxed Dynamic Programming (75)4.4Notes and Extensions (81)4.5Problems (83)References (84)ix5Stability and Suboptimality Using Stabilizing Constraints (87)5.1The Relaxed Dynamic Programming Approach (87)5.2Equilibrium Endpoint Constraint (88)5.3Lyapunov Function Terminal Cost (95)5.4Suboptimality and Inverse Optimality (101)5.5Notes and Extensions (109)5.6Problems (110)References (112)6Stability and Suboptimality Without Stabilizing Constraints (113)6.1Setting and Preliminaries (113)6.2Asymptotic Controllability with Respect to (116)6.3Implications of the Controllability Assumption (119)6.4Computation ofα (121)6.5Main Stability and Performance Results (125)6.6Design of Good Running Costs (133)6.7Semiglobal and Practical Asymptotic Stability (142)6.8Proof of Proposition6.17 (150)6.9Notes and Extensions (159)6.10Problems (161)References (162)7Variants and Extensions (165)7.1Mixed Constrained–Unconstrained Schemes (165)7.2Unconstrained NMPC with Terminal Weights (168)7.3Nonpositive Definite Running Cost (170)7.4Multistep NMPC-Feedback Laws (174)7.5Fast Sampling (176)7.6Compensation of Computation Times (180)7.7Online Measurement ofα (183)7.8Adaptive Optimization Horizon (191)7.9Nonoptimal NMPC (198)7.10Beyond Stabilization and Tracking (207)References (209)8Feasibility and Robustness (211)8.1The Feasibility Problem (211)8.2Feasibility of Unconstrained NMPC Using Exit Sets (214)8.3Feasibility of Unconstrained NMPC Using Stability (217)8.4Comparing Terminal Constrained vs.Unconstrained NMPC (222)8.5Robustness:Basic Definition and Concepts (225)8.6Robustness Without State Constraints (227)8.7Examples for Nonrobustness Under State Constraints (232)8.8Robustness with State Constraints via Robust-optimal Feasibility.2378.9Robustness with State Constraints via Continuity of V N (241)8.10Notes and Extensions (246)8.11Problems (249)References (249)9Numerical Discretization (251)9.1Basic Solution Methods (251)9.2Convergence Theory (256)9.3Adaptive Step Size Control (260)9.4Using the Methods Within the NMPC Algorithms (264)9.5Numerical Approximation Errors and Stability (266)9.6Notes and Extensions (269)9.7Problems (271)References (272)10Numerical Optimal Control of Nonlinear Systems (275)10.1Discretization of the NMPC Problem (275)10.2Unconstrained Optimization (288)10.3Constrained Optimization (292)10.4Implementation Issues in NMPC (315)10.5Warm Start of the NMPC Optimization (324)10.6Nonoptimal NMPC (331)10.7Notes and Extensions (335)10.8Problems (337)References (337)Appendix NMPC Software Supporting This Book (341)A.1The MATLAB NMPC Routine (341)A.2Additional MATLAB and MAPLE Routines (343)A.3The C++NMPC Software (345)Glossary (347)Index (353)。