LQG predictive optimal control for adaptive applications
A k a r a c t - - T h e LQG optimal control philosophy is extended so that use may be made of future set-point or reference signal information. The tracking error is defined in terms of an ideal response model so that a model following capability may be introduced. The cost function may also be augmented by the addition of prediction error terms. The controller was derived for adaptive control applications and has many of the desirable features of the new "Predictive Control" philosophy. However, there are fewer variables to choose and guaranteed stability properties are obtained for all weightings or prediction intervals. The self-tuning controller briefly described is almost as simple as existing LQG industrial self-tuners. 1. INTRODUCTION
LQG Predictive Optimal Control for Adaptive Applications*
M. J. G R I M B L E t
The solution of the LQG optimal control problem for systems where future reference values are known leads to implications for the choice of cost functions.
are derived. The first minimizes the usual L Q G cost function (with dynamic weightings) but assumes knowledge of p future reference or set-point values. This provides a two-degrees of freedom controller with no increase in complexity over the standard "causal" reference case. This is termed the LQG Future (LQGF) control problem. Mosca and Zappa (1987) have described a related controller, for use in the MUSMAR adaptive algorithm, but this was based on a more restrictive cost and system description. The second problem is termed LQG Predictive (LQGP) control and involves a cost function which includes prediction error terms. In this case not only does the optimal control take advantage of future reference signal knowledge but future predicted errors can be penalized. This will ensure more control action is taken well before a set-point change which will degrade the steady-state error plus control signal variances but should ensure the output achieves the new reference level more accurately. This increased flexibility in design must be weighted against the small increase in complexity which occurs from the presence of least squares predictors in the equations. A secondary objective of the paper is to introduce an LQG model following capability which can be used in stochastic, discrete model reference adaptive controllers (Landau, 1979). The ideal response model can of course be set to unity in the results and the usual tracking error signal, with or without future reference signal knowledge, is then minimized. The asymptotic situation is briefly considered where control weighting and noise tend to zero. It is shown that perfect tracking can be obtained if the ideal response model includes the non-minimum phase zeros of the plant. Most of the analysis is collected in Appendices
and only the main results are quoted or discussed in the body of the work. After introducing the system model in Section 2, the LOGF controller is defined in Section 3. The properties of this algorithm, including stability, sensitivity and minimum cost, are also considered. A self-tuning algorithm based on this controller is then briefly discussed. In Section 4 the LQG cost function is augmented with prediction error terms and the Байду номын сангаасOGP controller is presented. This is not described in the same detail as the LQGF controller since similar properties hold for each. Finally concluding remarks summarize the results.
* Received 24 November 1987; revised 14 December 1988; revised 16 June 1989; received in final form 21 December 1989. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. Ku~era under the direction of Editor H. Kwakernaak. t Industrial Control Unit, University of Strathclyde, Department of Electronic and Electrical Engineering, Marland House, 50 George Street, Glasgow G l IQE, Scotland, U.K. 949
SEVERAL RECENT very successful adaptive controllers (Peterka, 1984; Clarke et al., 1987a, b, c; Clarke and Mohtadi, 1987; Lelic and Zarrop, 1987; Lelic and Wellstead, 1987) have been based on the receding horizon optimal control philosophy, where the cost function is summed from t to t + T. Although this cost function leads to relatively simple controllers which take advantage of future set-point information, the controls are not optimal in a usual LQ or LQG sense. The stability properties are not identical to LQ designs and unstable solutions can arise. Nevertheless, their predictive capabilities have been shown to be very valuable in industrial control applications (Clarke et al., 1987). In the following, an attempt is made to derive a controller combining the good stability and physical cost properties of LQC designs which also includes the advantages of the predictive control philosophy. Two forms of the controller
电气工程与自动化专业英语(李军)第13章
It is worth pointing out that there are no hard-lines limits among the classification and the resulting controllers above. Here we present a personal overview on Advanced Control Methods as follows: Adaptive Control: 1. Be defined as a feedback control system intelligent enough to adjust its characteristics in a changing environment. 1. Traditional adaptive control methods, either model reference or self-tuning, usually require some kind of
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Chapter 13: Adaptive Control and Predictive Control
Adaptive Control and Predictive Control
Before introduction to the advanced control design techniques, we present a brief overview of control techniques and paradigms： The 1950s gave rise to the state-space formulation of differential equations The method of dynamic programming was developed by Bellman (1957) The maximum principle was discussed by Pontryagin (1962).
博士生研究发现新型神经退行性疾病的治疗靶点
博士生研究发现新型神经退行性疾病的治疗靶点神经退行性疾病是一类常见且严重的神经系统疾病，如阿尔茨海默病、帕金森病和亨廷顿病等。
这些疾病的特点是神经细胞的异常死亡和功能丧失，导致相关脑区和神经递质的改变。
然而，目前对于这些疾病的治疗选项非常有限。
近期的一项研究取得突破性进展，博士生研究人员发现了新型神经退行性疾病的治疗靶点，为未来的治疗提供了新的希望。
该研究由博士生李明领导的团队进行，他们针对神经退行性疾病的发病机制进行了深入研究。
通过使用先进的基因编辑技术和转基因小鼠模型，研究人员发现了一种新型的蛋白质复合物，命名为RDF （Regulator of Degenerative Factors）。
RDF在神经退行性疾病的病理过程中发挥着重要的调控作用。
研究结果显示，RDF复合物具有调控炎症反应和神经细胞死亡的功能。
在神经退行性疾病模型中，RDF的表达水平显著下降。
通过增加RDF的表达，研究人员观察到神经细胞的存活率得到显著提高，并且炎症反应得到抑制。
此外，进一步的实验发现，RDF可以与特定的信号通路分子相互作用，从而进一步调节神经细胞的存活和功能。
基于这些发现，研究人员开始寻找RDF的调控机制。
他们发现一种名为NF-κB（核因子κB）的转录因子可以结合到RDF上，从而增加其表达。
NF-κB是炎症反应的重要调控分子，与神经退行性疾病的发病机制密切相关。
通过调控NF-κB信号通路，研究人员能够有效地改变RDF的表达水平，从而对神经细胞起到保护作用。
这项研究的发现为新型神经退行性疾病的治疗提供了新的方向。
研究人员认为，通过调节RDF和NF-κB信号通路，可以改变神经退行性疾病的发病过程，从而减轻病情并延缓疾病的进展。
此外，研究人员还证实了一种名为RDF增效剂的分子，可以增加RDF的稳定性和活性，进一步提高治疗效果。
然而，尽管这一研究取得了重要的突破，但仍有许多工作需要进行。
首先，研究人员需要进一步探索RDF的分子机制，以更好地理解其在神经退行性疾病发病过程中的作用。
基于LQG控制的EPS系统最优控制研究
技术创新控制系统您的论文得到两院院士关注基于LQG 控制的EPS 系统最优控制研究Optimal control Research for Electric Power Steering System Based on LQG Controller(山东理工大学)高士香石沛林邹广德苗立东GAO Shi-xiang SHI Pei-lin ZOU Guang-de MIAO Li-dong摘要:电动助力转向系统存在着路面随机激励、转矩传感器测量噪声、模型参数不确定所引起的干扰等复杂因素,通过建立电动助力转向系统的数学模型以及加入随机干扰信号的系统状态空间表达式,设计了线性二次型高斯状态反馈控制器和Kalman 滤波器。
并以能耗及电机的实际助力电流与目标助力电流之间的误差最小为目标函数对EPS 系统进行仿真分析,仿真结果表明:采用该最优控制方法能有效的抑制系统的外部干扰,提高系统的鲁棒性,使能耗及电机的实际助力电流与目标助力电流之间的误差达到最小。
关键词:电动助力转向系统;LQG 控制器;卡尔曼滤波;最优控制中图分类号:U463.4文献标识码:A Abstract:Because of the complex factors of the random road excitation,the measurement noises of the torque sensor and the distur -bance of model errors in the electric power steering system,the mathematical models of electric power steering system were estab -lished and the state -space formula with random disturbance signal was performed.Then the LQG state feedback controller and theKalman filter were designed.Finally,take minimum the energy consumption and the error between the actual current of assist motor and the target value as the target function and simulate.The simulation results showed that this optimal control method could attenu -ate the external disturbance effectively,improve the system robustness,and minimize the energy consumption and the error between the actual assist current of motor and the target current.Key words:electric power steering system;LQG controller;Kalman filter;optimal control文章编号:1008-0570(2010)03-1-0071-021引言电动助力转向系统采用电机直接提供转向助力来帮助驾驶员转向,从而减轻驾驶员的操纵力,提高汽车操纵的轻便性和灵活性。
《2024年基于机器学习算法的超材料快速自动设计研究》范文
《基于机器学习算法的超材料快速自动设计研究》篇一一、引言超材料作为一种具有独特物理特性的新型材料，近年来在众多领域中得到了广泛的应用。
然而，超材料的设计往往需要复杂的物理模型和精细的实验验证，导致设计过程既耗时又昂贵。
因此，寻求一种高效、自动化的超材料设计方法成为了科研领域的迫切需求。
本文提出了一种基于机器学习算法的超材料快速自动设计方法，旨在通过算法的智能优化来提高超材料设计的效率和准确性。
二、研究背景及意义随着人工智能和机器学习技术的快速发展，其在材料科学领域的应用也越来越广泛。
通过机器学习算法，我们可以从大量的材料数据中学习并发现材料性质与结构之间的内在规律，为材料的快速设计和优化提供有力支持。
将机器学习算法应用于超材料设计，不仅可以提高设计效率，还可以降低实验成本，推动超材料在各个领域的应用。
三、研究方法本研究采用了一种基于深度学习的机器学习算法，通过构建超材料设计的智能优化模型，实现超材料的快速自动设计。
具体步骤如下：1. 数据准备：收集超材料的相关数据，包括材料的组成、结构、性质等，建立超材料数据库。
2. 特征提取：从超材料数据库中提取对设计有用的特征，如材料的组成比例、结构参数等。
3. 构建模型：利用深度学习算法，构建超材料设计的智能优化模型。
该模型可以根据提取的特征，预测材料的性质和性能。
4. 模型训练与验证：使用部分数据对模型进行训练和验证，确保模型的准确性和可靠性。
5. 自动设计：将模型应用于超材料的自动设计过程中，根据设计要求，自动寻找满足条件的材料组成和结构。
四、实验结果与分析通过实验，我们验证了基于机器学习算法的超材料快速自动设计方法的有效性和准确性。
实验结果如下：1. 设计效率：与传统的超材料设计方法相比，基于机器学习算法的设计方法可以大大提高设计效率。
在同样的时间内，可以设计出更多的超材料方案。
2. 设计准确性：机器学习模型可以准确预测超材料的性质和性能，降低实验验证的次数和成本。
最优跟踪控制概念
最优跟踪控制概念全文共四篇示例，供读者参考第一篇示例：最优跟踪控制（Optimal Tracking Control）是自动控制领域中的一个重要概念，旨在设计控制系统使其能够通过自动调节来跟踪某个期望轨迹或信号，以实现系统的最佳性能。
在实际应用中，最优跟踪控制被广泛应用于各种领域，如航空航天、机器人技术、汽车控制等。
本文将探讨最优跟踪控制的基本概念、方法以及在实际应用中的重要性。
最优跟踪控制的基本概念是通过控制系统来实现对系统的期望轨迹或信号的跟踪。
在实际应用中，系统通常会受到各种干扰和噪声的影响，因此设计一个稳健的控制策略成为至关重要的任务。
最优跟踪控制方法通过最小化系统性能指标（如误差或能量）来实现系统的最佳控制性能，从而使系统能够快速、准确地跟踪期望轨迹或信号。
在最优跟踪控制方法中，一种常用的方法是使用最优控制理论。
最优控制理论是一种数学优化方法，通过对系统动力学方程进行最优控制设计，以最小化系统性能指标。
在最优跟踪控制中，通常会使用线性二次型最优控制（LQ控制）或线性二次型高斯最优控制（LQG控制）等方法来实现系统的最佳控制性能。
最优跟踪控制方法还可以结合模型预测控制（MPC）等方法来优化控制策略。
模型预测控制是一种基于模型的控制方法，通过对系统的模型进行预测来优化控制策略。
在最优跟踪控制中，可以使用模型预测控制来实现对系统的期望轨迹或信号的跟踪，从而提高系统的性能和稳定性。
在实际应用中，最优跟踪控制在各种领域都发挥着重要作用。
在航空航天领域，最优跟踪控制被广泛应用于飞行器的自动驾驶系统，以实现飞行器的姿态控制和航线跟踪。
在机器人技术领域，最优跟踪控制被用于设计机器人的运动控制系统，以实现机器人在复杂环境中的准确和稳定的运动。
在汽车控制领域，最优跟踪控制被用于设计车辆的自动驾驶系统，以实现车辆的路径规划和跟踪。
最优跟踪控制是自动控制领域中的一个重要概念，通过最优化控制策略来实现系统的最佳性能。
Introduction+to+Model+Predictive+Control
Introduction to Model Predictive ControlLars ImslandAbstractThis note gives a brief introduction to Model Predictive Control(MPC),assuming the reader has been exposed to the LQR controller,and QP optimization algorithms including KKT conditions.Basic knowledge of linear algebra is also assumed.The main focus is linear MPC,while MPC using nonlinear process models is mentioned briefly(assuming some exposure to SQP algorithms for nonlinear programming). Contents1Introduction1 2MPC principle—traditional view2 3Linear MPC—Linear Quadratic Constrained Regulation33.1Model class and infinite horizon opti-mal control problem (3)3.2The unconstrained solution:The linearquadratic regulator(LQR) (3)3.3From LQR,via constrained open-loopcontrol,to MPC (4)3.4The constrained LQR:MPC (5)3.5Stability (8)4Some practical aspects104.1MPC tuning (10)4.2MPC tracking (11)4.3State estimation and integral control..124.4Other issues (13)5Explicit solution of the linear MPC controller14 6Brief introduction to nonlinear MPC(NMPC)166.1Solving the NMPC optimization problem186.2Stability of NMPC (19)7Bibliographic notes20 A Some proofs22A.1Proof of Lemma1 (22)B Exercises24C Some Matlab functions24C.1blkdiag2.m (24)1IntroductionThe term Model Predictive Control(MPC)has come to refer to a controller whicha)uses a(multivariable)process model to predict future behavior,b)uses mathematical programming for optimizing predicted future performance(typically quadratic pro-gramming(QP)in the case of linear process models,or sequential quadratic programming(SQP)for non-linear process models,but also linear-and semi-definite programming have been used),andc)handles constraints on inputs(manipulated variables)and states/controlled variables.This type of controller has become the advanced process control technology in chemical process industry, and usage is spreading to other application areas.Most widespread is the type of MPC usually termed linear MPC–using linear process models for prediction and convex QP for optimization of a quadratic performance objective.Linear MPC will be the main focus for this note.However,over the last10-15years,MPC using nonlinear process models(usually termed nonlinear MPC) has gained ing nonlinear models will typically give a non-convex nonlinear MPC optimization prob-lem.Since nonlinear optimization is much more difficult(time-consuming,error-prone)than convex opti-mization,typically QP,for linear MPC,it is important to justify the need for nonlinear MPC before it is ap-plied.Typical reasons for using nonlinear MPC are that the process operates in several steady states(“grade change”)with significantly different dynamics,that there are large disturbances that excite nonlinearities,orthat the process is a batch process.In the last few years,there have been a significant research focus on efficient implementation of nonlinear optimization techniques tailored for nonlinear MPC.Let us return to linear MPC,which in our case will refer to a MPC using linear state-space models.The ob-jective function is a quadratic weighting of future manipulated manipulated variables and setpoint errors.Al-though thefirst MPC applications optimized performance on afinite horizon,theory developed in the nineties has shown us that(for linear process models)it is just as easy to optimize performance on the infinite horizon, and due to better theoretical properties,this should be the preferred choice.In the unconstrained case,the infinite horizon LQ controller(the LQR)delivers the optimal performance on the infinite horizon.Therefore,the main reason for the widespread use of MPC in place of the LQR is that it offers a straightforward and transparent approach to handling constraints,both on manipulated variables (inputs)and controlled variables(outputs).The linear MPC formulation presented in this note emphasizes this approach to MPC,namely that linear MPC is(should be)an integration of the LQ controller with constraint handling:•The LQ controller is used for stabilization and to obtain good control performance on the infinite horizon.•The MPC QP formulation handles the constraints in an optimal manner.Before we continue,some brief notes on notation.We consider in this note discrete-time dynamic systems, and variables that change with time have a subscript that denote time instant.For example,x k denotes vector x at time instant k.We will set up optimization problems,where we will calculate future variables(“predict”)to optimize future behavior.The notation we use will not reflect the difference between these predicted variables and the ’real’variables.In reality,predicted and real variables will invariably be different,due to noise and model error.2MPC principle—traditional viewOriginally,the MPC controller was based on step-response or impulse-response models,and in most industrial implementations,this is still the case.To illustrate the MPC principle,we assume the process model is an single-input single-output(SISO)finite impulse response(FIR)model1,y k+j=N∑i=1h i u k+j−i.This model predicts that the output at a given time depends on a linear combination of past input values;the summation weights h i are the impulse response coefficients.The sum is truncated at the point where past inputs no longer influence the output.Therefore,this representation is only possible for stable plants.The MPC optimization problem is,at time instant k,to optimize future performance(as measured by a quadratic function)over a horizon N:minN∑i=0q(y k+i+1−y d k+i+1)2+r(u k+i−u d k+i)2(1a)subject to y≤y k+i+1≤y,u≤u k+i≤u,i=0,...,N.(1b) Superscript d signifies“desired”values,or“reference”.This objective function can easily be shown to be a QP. The scalars q and r are weights that can be used as tuning parameters(in addition to the horizon-length N). The optimization problem is illustrated in Figure1.The MPC principle is to repeat this optimization at every time instant,using only thefirst of the calculated inputs as input to the process:1.At time instant k,solve the QP to obtain an optimal,feasible input sequence.2.Apply thefirst input in the input sequence(u k,u k+1,...,u k+N)to process.3.Set k=k+1(next time instant),go to step1.1Actually,this is more or less the models used in thefirst reported industrial MPC applications,[24,25,7].k +NPast Present Future kFigure 1:MPC principleA small remark is in place here:The reader will note that measurements are not used in this procedure,while the whole idea of MPC is to re-optimize when new measurements are available at each sample instant.In a real implementation of the approach used here,feedback from measurements are obtained by updating of biases on the measured outputs (see Section 4.3),which are then used in the prediction.The MPC controller we will study closer in this text,starting from next section,is based on the same principle as in the procedure above,but differs in two important aspects:It uses state-space models,and it optimizes performance on an infinite horizon (while using a finite number of degrees of freedom).3Linear MPC —Linear Quadratic Constrained Regulation 3.1Model class and infinite horizon optimal control problemThe model (2)is typically obtained ei-ther by linearization of a nonlinear processmodel,or by system identification methodsbased on measured data.The state x andinput u are deviation variables,expressingthe difference between the ’real’state (in-put)and the value of the state (input)in thedesired process equilibrium.For instance,if a nonlinear process model is found frommass and energy balances to be˙ξ=f (ξ,ν)where ξis the state and νthe input,thenwe could define x =ξ−ξd and u =ν−νdwhere ξd and νd are the desired equilib-rium,satisfying f (ξd ,νd )=0.The matri-ces A and B could be found,using forwardEuler discretization with discretization in-terval h ,from A =I +h ∂f ∂ξ (ξd ,νd ),B =h ∂f ∂ν (ξd ,νd ).We will assume that the process to be controlled is described by a discrete state space model x k +1=Ax k +Bu k ,(2)where dim x =n x and dim u =n u .The process could be stable or unstable (there are no restrictions on the eigenvalues of A ),but we assume that the pair (A ,B )is stabilizable.We will most of the time assume that the state x is measured.The objective we want to obtain by control,is to optimize perfor-mance by minimizing the infinite horizon cost J (x 0,{u k }k =0,...,∞)=∞∑k =0x T k Qx k +u T k Ru k ,(3)where future x k are given by the model (2)based on x 0and {u k }k =0,...,∞.The matrices Q and R are tuning parameters,but must fulfill that Q ≥0and R >0,and the state must be detectable through Q (that is,the pair (A ,√Q )must be detectable,where √Q is a matrix such that Q =(√Q )2).It is implicit in the cost function (3)that we treat the regulation problem,that is,we want to control x k →0.The problem of tracking time-varying references are treated in Section 4.2.3.2The unconstrained solution:The linear quadratic regulator (LQR)When there are no constraints,the optimal future controls that minimizes the infinite horizon objective func-tion (3),is given as a fixed (time-invariant)state feedbacku k =−Kx k .(4)This controller is often denoted the(discrete-time)infinite horizon LQ controller,or the Linear Quadratic Reg-ulator(LQR).The state feedback matrix K=(R+B T PB)−1B T PA is given by the positive semi-definite solution of the discrete-time algebraic Riccati equation(DARE),A T PA−P−A T PB(R+B T PB)−1B T PA+Q=0.(5)A fact that we will use in the sequel,is that when u k=−Kx k,∞∑k=0x T k Qx k+u T k Ru k=∞∑k=0x T kQ+K T RKx k=x T0Px0.(6)We do not prove this here,but remark that it is rather easy to show by noting that(5)can be written(A−BK)T P(A−BK)−P+Q+K T RK=0,and using the same trick as in the beginning of Section A.1.3.3From LQR,via constrained open-loop control,to MPCThe LQR has some important advantages;+it is inherently multivariable(it takes care of couplings in the process),+it is optimal on the infinite horizon,and+it can be shown that it has good robustness properties(due to feedback2).However,it has one important disadvantage:-it does not handle constraints on states and inputs.The optimization problem of minimizing(3)with constraints on states and inputs,is a QP with an infinite number of optimization variables.This is in general an intractable optimization problem.In the special case when there are no constraints,the solution of this QP is given by the LQR state feedback,as we learned in the previous section.Including constraints,the QP can,however,be approximated by a QP with afinite number of optimization variables.As afirst approach,consider truncating the sum in the objective function at a given’horizon length’L:L−1∑k=0x T k+1Qx k+1+u T k Ru k,(7)and using u0,u1,...,u L−1(and x1,x2,...,x L)as optimization variables.This corresponds to what thefirst MPC controllers did,and is still common.The main problem with this approach is that thisfinite horizon objective function is not what we really want to minimize,since in regulation we generally want to optimize all(infinite) future behavior.Therefore,in practice,one often chooses large values for L,for which the approximation can be close.A rule of thumb is’larger than the longest time-constant in the process’,however,this can lead to excessively large optimization problems,and one is still not guaranteed stability.However,in the nineties,it was discovered[20,6,28]that it is possible to optimize performance on the infinite horizon,with afinite number of optimization variables.(It turned out this was also the key to proving closed-loop stability for MPC approaches.More on this in Section3.5.)The simple trick is to divide the objective function into two parts;∞∑k=0x T k Qx k+u T k Ru k=L−1∑k=0x T k Qx k+u T k Ru k+∞∑k=Lx T k Qx k+u T k Ru k,and let the control moves in thefirst part be free optimization variables,while the control moves in the second part is given by the LQR controller u k=−Kx k,k≥L.The rationale is that after some time,we have’resolved’the constraints such that the LQR controller is optimal on the rest of the horizon.If we do this,we can use(6) to obtain∞∑k=Lx T k Qx k+u T k Ru k=x T L Px L,2However,implemented in combination with state estimation,e.g.the Kalmanfilter(that is,the LQG controller)it can be non-robust[11].where P is the Riccati matrix.This means that we can write∞∑k=0x T k Qx k+u T k Ru k=L−1∑k=0x T k Qx k+u T k Ru k+x T L Px L.(8)Note that this is exact,and not an approximation.However,since we now have put structure on the allowed control moves,minimizing this objective function is not necessarily the same as minimizing(3),unless the horizon is’long enough’(more on this later).In the next section we will use this objective function,but we do a change of optimization variables by letting the optimization variables on thefirst part of the horizon be perturbations to u k=−Kx k.It will then turn out that(8)takes a particularly simple form(Lemma1).Once we have such a QP with afinite number of optimization variables,we can calculate the optimizing future control at a given point,and apply them for all future(or at least for an horizon L).However,this approach is not robust,since it does not incorporate feedback.The simple principle of MPC is that feedback can be achieved by performing the open-loop optimization over again at each sample instant,when new measurements are available,and applying only thefirst part of the optimal inputs.3.4The constrained LQR:MPCIn many practical control problems,constraints on inputs and states(and/or outputs)are important.We assume here that these constraints are written compactly asD x x≤d x,D u u≤d u,(9)and that these polytopes contains the origin in the interior.Note that this constraint formulation includes the most usual’box’constraints on inputs and outputs(or states),u≤u≤¯u,y≤y=Cx≤¯y.Constraints that are’mixed’(depending on both x and u)can also easily be added,but we avoid them here as they require a slight redefinition of the MPC problem to come.The control law that will be used in optimization of future performance subject to these constraints,is:u k=−Kx k+c k,k=0,...,L−1,−Kx k,k≥L.(10)It is important to note that this is not the’real’control law,but the one that is used when predicting future performance.The real control law will be defined later in this section(Algorithm1).The c k are degrees of freedom available for constraint handling,entering as’perturbations’to a linear state feedback law.This formulation allows active constraint handling during transients,on the control horizon k=0,...,L−1.Furthermore,thefixed state feedback K affects the asymptotic behavior.Algorithms us-ing this separation of future predicted inputs is often denoted as’dual mode’.We will assume that K is the unconstrained optimal on the infinite horizon,that is,the LQR controller.In this case(3)takes the form of a quadratic function:Lemma1The objective function(3)with future x k and u k given by(2)and(10),respectively,is given byJ(x0,c)=c T W c c+x T0Σx0,(11) where c=[c T0,...,c T L−1]T,and the symmetric matrix3W c=diag(W,...,W),where W=B TΣB+R,andΣis the solution of the Lyapunov equation4Σ−(A−BK)TΣ(A−BK)=Q+K T RK.The proof of this lemma can be found in the Appendix(Section A.1).If we accept that this function must be quadratic,it is easy to see that there must be no cross-terms(terms involving both x k and c):Close to x k=0, there are no constraints active,and hence c=0must be the minimizer of J.If there were cross-terms,the minimizer would not be c=0.3The operator diag gives a block-diagonal matrix with the arguments as the blocks along the diagonal,cf.the Matlab-function blkdiag.4This Lyapunov equation can be solved in Matlab with the function dlyap.However,you might already have calculated it sinceΣ=P, where P is the solution of(5).Note that the last term of J(x0,c)is not dependent on the degrees of freedom c and can therefore be omitted when optimizing J.If there are no constraints in the problem formulation,the minimum to J(x0,c)is clearly given by c=0, giving the LQR controller,as expected.Thus,the interesting case is when there are constraints on inputs u and states x(and/or outputs).For these constraints to be added to our MPC optimization problem,we must calculate future u k and x k as functions of x0and c:x1=Ax0+B(−Kx0+c0)=(A−BK)x0+Bc0x2=Ax1+B(−Kx1+c1)=(A−BK)x1+Bc1=(A−BK)2x0+(A−BK)Bc0+Bc1...u0=−Kx0+c0u1=−Kx1+c1=−K(A−BK)x0−KBc0+c1....As we see,these functions are linear,and straightforward but somewhat tedious to define,hence we omit the detailed definitions here(an example is given in the code listing below).To ensure that constraints are fulfilled for all future time,we must calculate all future u k and x k.As optimization problems with an infinite number of constraints are undesirable,we usually stop at afinite number,and here we choose the constraint horizon to be equal to the control horizon.This is discussed closer below.Writing the linear mappings asx T1x T2···x T L T=P x1x0+P c1c,(12a)u T0u T1···u T L−1 T=P x2x0+P c2c,(12b)we can combine this with(9)to put together constraints asMx0+Nc≤b.(13) We are now ready to state the MPC algorithm:Algorithm1(LTI dual mode MPC)At each sampling instant k,perform the optimization(the QP)mincc T W c c subject to Mx k+Nc≤b.(14) Use thefirst block element of the optimal c,c0,to calculate u k=−Kx k+c0.An important observation is that if,for a given x k,c=0is feasible for Mx k+Nc≤b,then c=0is the solution to the QP problem in the algorithm above(the region of x k where this happen is defined by the polytope{x:Mx≤b}).An interpretation of this is that if u k=−Kx k is feasible,then c=0,and c=0only if u k=−Kx k is not feasible(in the sense that implementing it will lead to a constraint violation on the control horizon).We thus see that we have separated stabilization and optimal control on the infinite horizon(−Kx k) from optimized constraint handling(c).An implementation example forfinding the matrices W c and M,N and b is given in the form of a Matlab function below.The constraints are only enforced on the control horizon.It makes use of the function blkdiag2, which can be found in Appendix C.Listing1:Function for generating matrices for solving MPC problem function[Wc,M,N,b,K,Sigma]=genMPCprob(A,B,Q,R,Dx,dx,Du,du,L,A_s,b_s)%Generate matrices for MPC QP problem%Inputs:%A,B System matrices in discrete time system:x+=A x+B u%Q,R Weights in infinite horizon cost:J=sum(x’Q x+u’R u)%L Control horizon length(degrees of freedom)%Dx,dx State constraints on the horizon:Dx x<=dx%Du,du Input constraints on the horizon:Du x<=du%A_S,b_S Stability constraints on end of control horizon:%A_s x(L)<=b_S(may be omitted)%%Outputs:%Wc Resulting cost function:J=c’Wc c+const%M,N,b Constraints on horizon:M x0+N c<=b%K Unconstrained optimal state feedback%Sigma Riccati equation solution%3/3/2007Lars Imsland[K,SS]=dlqr(A,B,Q,R);Phi=A-B*K;nx=size(A,1);nu=size(B,2);%Define costfunction,c’*W_c*cSigma=dlyap(Phi’,Q+K’*R*K);%Note that Sigma=SS,calculated earlierW=B’*Sigma*B+R;Wc=blkdiag2(W,L);%Define predictions:%[x_1x_2...x_L]=Px1*x_0+Pc1*[c_0c_1...c_L-1]%[u_0u_1...u_L-1]=Px2*x_0+Pc2*[c_0c_1...c_L-1]Px1=[eye(nx);Phi];Pc1=zeros(L*nx,L*nu);Pc1(1:nx,1:nu)=B;for i=2:L,Px1=[Px1;Px1((i-1)*nx+1:i*nx,:)*Phi];for j=1:i,Pc1((i-1)*nx+1:i*nx,(j-1)*nu+1:j*nu)=...Px1((i-j)*nx+1:(i-j+1)*nx,1:nx)*B;endendPx2=blkdiag2(-K,L)*Px1(1:L*nx,:);Pc2=blkdiag2(-K,L)*([zeros(nx,L*nu);Pc1(1:(L-1)*nx,...1:(L-1)*nu),zeros((L-1)*nx,nu)])+blkdiag2(eye(nu),L);Px1=Px1(nx+1:end,:);%remove first I%Define constraints,N*c<=b-M*x0N=[blkdiag2(Dx,L)*Pc1;blkdiag2(Du,L)*Pc2];M=[blkdiag2(Dx,L)*Px1;blkdiag2(Du,L)*Px2];b=[repmat(dx,L,1);repmat(du,L,1)];if(exist(’A_S’,’var’)),%If you have"stability constraints"A_S x<=b_SPxlast=Px1(nx*(L-1)+1:nx*L,:);Pclast=Pc1(nx*(L-1)+1:nx*L,:);N=[N;A_S*Pclast];M=[M;A_S*Pxlast];b=[b;b_S];endThe following example uses this function to control a process consisting of a double integrator with state-and input constraints.Example1(Double integrator)The double integrator,two integrators in series,discretized with sample in-terval T s,can be written in state-space form(2)withA=1T s01,B=T2sT s.We will control this system using MPC,that is,minimize the infinite horizon cost(3)withQ=1000,R=1,using L=5degrees of freedom(horizon)for constraint handling.The constraints are−0.5≤x2≤0.5,and −1≤u≤1.Matlab code using the genMPCprob-function in Listing1for specifying and simulating the MPC controller is given below:Listing2:Code for simulating QP MPC example%Control problem:Double integratorTs=0.05;%Sampling timeA=[1Ts;01];B=[Tsˆ2;Ts];Q=diag([10]);R=1;L=5;nu=1;%Constraints:%-0.5<=x_2<=0.5%-1<=u<=1Dx=[01;0-1];dx=[.5;.5];Du=[1;-1];du=[1;1];[Wc,M,N,b,K]=genMPCprob(A,B,Q,R,Dx,dx,Du,du,L);i=1;x(:,i)=[-2;0];time=1:Ts:10;opt=optimset;opt.Display=’notify’;opt.Diagnostics=’off’;rgeScale=’off’;for i=1:length(time),%Find control for state x(:,i)c(:,i)=quadprog(2*Wc,zeros(1,L*nu),N,b-M*x(:,i),[],[],[],[],[],opt);u(:,i)=-K*x(:,i)+c(1:nu,i);%Simulatex(:,i+1)=A*x(:,i)+B*u(:,i);end%Phaseplotfigure(1);plot(x(1,:),x(2,:));xlabel(’x_1(position)’);ylabel(’x_2(velocity)’);title(’Phase plot double integrator’);%Input and perturbationfigure(2);subplot(211);plot(time,c(1:nu,:));xlabel(’Time[s]’);ylabel(’c’);subplot(212);plot(time,u);xlabel(’Time[s]’);ylabel(’u’);The calculated control u k and’perturbation’c k are plotted in Figure2,for a case where we start in x0= (−2,0)T.As we can see,in the beginning c k=0for handling constraints.First,the input constraint is active, but after a while,the state constraint is active(see Figure3).Eventually,we get closer to the origin and the LQ controller is optimal(c k=0).The states are plotted in a phase plot(the state-space)in Figure3.The colored region is the largest polytope (even though it might not look like a polytope)where the LQ-controller does not violate constraints for this example.The red ring is plotted at the time instant when c k=0,and as we can see,in this case5this happens when the state enters the polytope.3.5StabilityWefirst note that with an infinite number of degrees of freedom(that is,L=∞),the MPC controller in Algorithm1is nominally6stabilizing in the sense that x k→0as k→∞,if•the full state is detectable through the objective function as mentioned in Section3.1,5This will typically be the case,but not always.6Not taking into account model errors and noise.Figure 2:Upper plot:The perturbation c .Lower plot:The implemented input u .Figure 3:Phase plot.The polytope is the maximal region were u =−Kx can be applied without violating constraints.The circle marks when the state enters this polytope (that is,when c k becomes 0).•and the optimization problem is feasible for the initial state.This is straightforward to prove,but a detailed treatment of stability is outside the scope of this rmally,due to the infinite horizon and since we assumed no noise/model error,the implemented control calculated by Algorithm 1for all future k will be given by the optimal solution at the first step,and since the optimization problem is assumed to be feasible,the objective function must be finite,implying that x k →0as k →∞.Since the objective function in (11)is the same as (3),Algorithm 1will also be stabilizing under the same assumptions using the same arguments,unless constraints will be violated after the end of the control horizon (or the constraint horizon,if this is different from the control horizon).Thus,guaranteeing stability amounts to ensuring that the constraints will hold on the infinite horizon,and not only on the horizon where we enforce them.With the formulation above,we cannot ensure this,since we only check constraints on the control horizon,that is,for 1,...,L .Since one often has to choose rather short horizon L to limit computational complexity,stability can in many cases be a practical problem (and not only theoretically).It is straightforward to increase the constraint horizon beyond L by changing the specification of (12),and one can actually guarantee that constraints will not be broken by choosing this constraint horizon long enough.This was recognized in [28,6],where an upper bound on the constraint horizon was also found.The problem with this approach is that this upper bound is rather conservative,such that a large number of extra constraints might have to be added.Another way to guarantee that constraints are not broken,without increasing the computational burden excessively,is via what is often called ’quasi-infinite horizon’.This idea was originally proposed for nonlinear MPC [5],but applies equally well (if not better)for linear MPC.For linear MPC this approach amounts to:1.Calculate the (maximum)region where the LQ control law u k =−Kx k is unconstrained and stabiliz-ing.This set is often called the maximal admissible set,or maximal feasible set.This set is rather straightforward to calculate,and one can show that it will always be a polytope,that is,on the form {x :A S x ≤b S }[13].2.Add A S x L ≤b S as an additional (linear)constraint to the constraints in Algorithm 1(that is,include them in (13)).This is often called a terminal state constraint .The last state on the control horizon,x L ,is found as a (linear)function of x 0and c as in (12).The function shown in Listing 1has A S and b S as optional inputs.This method can guarantee stability for even small L ,but a possible drawback is that the feasibility region (see below)might become small.We summarize in a theorem:Theorem 1If we apply the MPC algorithm (Algorithm 1)to the system (2),we are guaranteed that x k →0as k →∞7if•the MPC optimization problem is feasible at k =0,•(A ,√Q )is detectable,•and we can guarantee that the constraints holds on the infinite horizon,either by–choosing L long enough,or–adding a terminal state constraint to ensure constraint satisfaction on the infinite horizon.We do not prove this here,but refer to e.g.[19].The region of the state space where the MPC optimization problem is feasible,is called the feasibility region ,or the feasible set,and is defined by S ={x :∃c s.t.Mx +Nc ≤b }.4Some practical aspects 4.1MPC tuningIt is seldom that we know exactly which Q and R that correspond to the desired closed-loop behavior.Tuning of the MPC controller is the procedure where we change the MPC parameters to obtain acceptable closed-loop performance,but also,importantly,the region where the MPC controller is feasible (the feasibility region).7Wecan say more:The closed loop is asymptotically stable in the sense of Lyapunov,which implies x k →0as k →∞.Since the MPC control law is nonlinear (we will see that it is piecewise affine)we cannot use a definition of stability in terms of eigenvalues.The main parameters for tuning are the weighting matrices Q and R,and the horizon length L8.The tuning of Q and R is similar to the LQ case,that is,it is customary to choose them as diagonal matrices,and increase elements in Q(relative to elements in R)to get more aggressive closed loop behavior.Similarly,detuning is obtained by increasing the elements in R.Beyond these simple and obvious remarks,the best way to gain insight in tuning,is by trial-and-error(closed-loop simulation).More can be said about the choice of horizon length L.In addition to affecting closed-loop performance,this parameter also affects computational complexity.Generally,shorter L gives lower complexity optimization problem and hence lower computational load for the online optimization problem,while longer L gives better performance,at the cost of higher computational load.In the nominal case,one could argue that one should choose L as large as computational limitations permit, as this will make the closed-loop behave closer to the infinite horizon controller9,which is our measure of good control.Importantly,the use of an infinite prediction horizon allows to weigh also states beyond the control horizon length,thus one could choose control horizon lengths shorter than the longest time constants.In the real world,the issue of model-plant mismatch(due to nonlinearities,modeling errors and simplifi-cations,and process noise)is always important,and will quite often imply that one should not choose too long horizon lengths.One reason for this is that model uncertainties may tend to be amplified as one predicts far into the future.Lastly,forfixed Q and R,increasing L will generally(nominally)increase the feasibility region.This is especially the case if a terminal state constraint is added to the MPC optimization problem.4.2MPC trackingThe theory presented in Section3concerns regulation,that is controlling the system to a constant setpoint. In this section we look at how we can expand the controller to tracking,including,for example,changing set points.Assume that the variables y k that we want to track some given time-varying reference are given as a linear mapping of the states,y k=Cx k.Note that generally,these variables does not need to be the same as the measurements,but to keep notation simple we assume this here.Let the time-varying reference be denoted with y d k.For this reference,generate matching reference states x d k and inputs u d k such that10x d k+1=Ax d k+Bu d k,y d k=Cx d k,(15) where A and B are the same as the process model.Example2The simplest example of tracking is tracking a constant reference,that is,y dk =r where r is aconstant.Then also x d k and u d k must be constant.Then,y d k=Cx d k=C(I−A)−1Bu d k,meaning we canfindu d k=C(I−A)−1B−1r,x d k=(I−A)−1BC(I−A)−1B−1r.Note,however,that when changing between(constant)references,it is often a good idea to let the reference change smoothly to the new value.The objective function we want to minimize in tracking is∞∑k=0(y k−y d k)T˜Q(y k−y d k)+(u k−u d k)T R(u k−u d k)=∞∑k=0(x k−x d k)T Q(x k−x d k)+(u k−u d k)T R(u k−u d k)(16)8One could use different Q and R in calculation of W c and K(for example,allowing shorter L(for lower computational complexity)or larger feasibility region by having a more detuned K),but we will not consider this further.9Surprisingly,this is not always the case[9].10This can be interpreted as a setpointfilter,which is generally always a good idea.。
压电执行器动态迟滞建模与LQG最优控制器设计
压电执行器动态迟滞建模与LQG最优控制器设计佚名【摘要】为提高空间天文望远镜稳像系统中压电快摆镜(Fast Steering Mirror,FSM)的动态性能,对压电执行器(Piezoelectric Actuator,PZT)动态迟滞补偿和控制进行研究.鉴于基于广义Play算子Prandtl-Ishlinskii(PI)模型的求逆复杂性和迟滞曲线的非对称性,构造一种基于广义Stop算子PI逆模型来补偿压电执行器迟滞非线性.采用Hammerstein模型对压电执行器动态迟滞特性进行建模,以广义PI模型和自回归遍历模型(Auto-regressive Exogenous Model,ARX)分别表征Hammerstein迟滞模型中的静态非线性和率相关性,并针对迟滞率相关模型不确定性问题,提出一种前馈补偿和线性二次型Gauss最优控制算法(Linear Quadratic Gaussian,LQG)相结合的复合控制策略.利用自适应差分进化算法(Adaptive Differential Evolution algorithm,ADE)辨识和整定模型及控制器参数.实验结果表明:该动态迟滞模型能够有效描述1～100 Hz频率范围内压电执行器迟滞曲线,拟合均方根误差为0.077 1μm(@1 Hz)～0.512 3 μm(@100 Hz),相对误差为0.31％(@1 Hz)～2.09％(@100 Hz);实时跟踪幅值为24.5 μm的变频目标位移,LQG控制算法的跟踪精度相比于直接前馈控制和PID控制分别提高48.6％和27.02％.【期刊名称】《光学精密工程》【年(卷),期】2018(026)011【总页数】10页(P2744-2753)【关键词】压电执行器;动态迟滞特性;Hammerstein模型;广义Stop算子;LQG最优控制【正文语种】中文【中图分类】TL3611 引言空间天文望远镜主要通过快摆镜(Fast Steering Mirror，FSM)机构补偿卫星平台不稳定引起的像移，以实现稳像从而提高探测精度[1]。
自适应偏心窗口平滑滤波图像修复
自适应偏心窗口平滑滤波图像修复
秦川;黄素娟;王朔中
【期刊名称】《计算机工程》
【年(卷),期】2008(034)005
【摘要】提出一种自适应平滑滤波数字图像修复新算法,采用变动中心的非线性平滑滤波算子,克服了固定中心点算子产生的边缘模糊问题.为了兼顾处理效果和计算效率,根据待修复的像素是位于图像平滑区还是边缘突变区,自适应地切换处理方法.实验表明,该算法对2种类型的像素均能取得较好的修复效果.提出用"分块校验和"方法在需要保护的图像中嵌入脆弱数字水印,一旦图像受损,即可精确得到损坏像素的定位模板,在修复中不再需要人工干预.
【总页数】3页(P213-215)
【作者】秦川;黄素娟;王朔中
【作者单位】上海大学通信与信息工程学院,上海,200072;上海大学通信与信息工程学院,上海,200072;上海大学通信与信息工程学院,上海,200072
【正文语种】中文
【中图分类】TP391.41
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1.基于GDI+的K领域自适应平滑滤波算法优化与实现 [J], 吴俊杰;吴昌钱
2.基于GDI＋的K领域自适应平滑滤波算法优化与实现 [J], 吴俊杰;吴昌钱;
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电气工程与自动化专业英语 第13章
Chapter 13: Adaptive Control and Predictive Control
Adaptive Control and Predictive Control
Before introduction to the advanced control design techniques, we present a brief overview of control techniques and paradigms： The 1950s gave rise to the state-space formulation of differential equations The method of dynamic programming was developed by Bellman (1957) The maximum principle was discussed by Pontryagin (1962).
2
Adaptive Control and Predictive Control
Kalman demonstrated that when the system dynamic equations are linear and the performance criterion is quadratic (LQ control) Produced linear-quadratic-Gaussian (LQG) control. the concept of the H- norm and -synthesis theory. Artificial Neural Network for control and Fuzzy Control are the typical AI control design techniques.
预见控制理论及应用研究进展
第42卷第2期自动化学报Vol.42,No.2 2016年2月ACTA AUTOMATICA SINICA February,2016预见控制理论及应用研究进展甄子洋1摘要预见控制因其能够利用已知的未来目标信息或干扰信息来改善控制系统的性能,在众多实际工程领域得到越来越多的应用,正日益受到研究者的广泛关注.本文较全面概述了预见控制理论自20世纪60年代提出以后的发展状况,重点概述线性最优预见控制、线性鲁棒预见控制、非线性预见控制等相关理论的研究进展,并且总结预见控制在车辆主动悬挂系统、机电伺服系统、机器人、飞行器、电网发电系统等领域的应用现状.最后,总结了预见控制的研究成果,探讨了目前存在的主要问题并展望了未来发展方向.本文的综述旨在通过抛砖引玉,促进预见控制理论及应用的全面发展.关键词研究进展,预见控制,最优控制,鲁棒控制,车辆主动悬挂系统,机器人,飞行器引用格式甄子洋.预见控制理论及应用研究进展.自动化学报,2016,42(2):172−188DOI10.16383/j.aas.2016.c150408Research Development in Preview Control Theory and ApplicationsZHEN Zi-Yang1Abstract Preview control can improve control system performance by utilizing future desired or external disturbance information.Therefore,more and more applications are seen in various real engineeringfields,attracting a wide spread attention of researchers.This article comprehensively overviews the preview control research achievement since1960s, and especially introduces the theoretical research progress of linear optimal preview control,linear robust preview control and nonlinear preview control.Furthermore,the application developments of preview control in vehicle active suspension systems,electromechanical servo systems,robots,aircraft and power generation systems are summarized.Finally,the research achievements of the preview control are summarized,and the main problems in researches are discussed to forecast the future development direction.The author hopes to promote the integrated development of the theory and applications of the preview control.Key words Research development,preview control,optimal control,robust control,vehicle active suspension system, robot,aircraftCitation Zhen Zi-Yang.Research development in preview control theory and applications.Acta Automatica Sinica, 2016,42(2):172−188现实中有很多问题具有不确定性,比如天气、生物寿命、比赛结果、股票和汇率等,我们往往通过预测未来情况来决定当前行为.然而,现实中同样有很多问题,未来信息通常完全或部分已知,比如车辆行驶路径和视野内路况、飞行器预定飞行航线、机床加工轨迹乃至社会、经济和人生的未来发展预定目标等,我们常常根据既定目标、已知干扰等未来信息来决定当前行为.最直接的例子就是,驾驶员开车时通过观察前面道路信息(包括目标轨迹、已知路况等),提前操纵方向盘、油门、刹车,才能控制汽车平稳、快速行驶,避免急刹、紧急打方向盘等问题.这就体收稿日期2015-06-29录用日期2015-10-19Manuscript received June29,2015;accepted October19,2015国家自然科学基金(61304223,61473144),教育部高等学校博士学科点专项科研基金(20123218120015)资助Supported by National Natural Science Foundation of China (61304223,61473144)and Specialized Research Fund for the Doctoral Program of Higher Education(20123218120015)本文责任编委谢永芳Recommended by Associate Editor XIE Yong-Fang1.南京航空航天大学自动化学院南京2111061.College of Automation Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing211106现了预见控制(Preview control)的思想,即结合了根据未来信息实施的超前控制和当前信息实施的反馈控制,而传统控制通常仅利用当前信息进行反馈控制,如同驾驶员看着后视镜驾驶,势必影响动态响应性能.预见控制的思想最初起源于20世纪60年代.美国麻省理工学院著名教授、国际上机器人与远程控制领域的先驱Sheridan于1966年率先提出预见控制的三种模型[1].美国学者Bender 于1968年提出了用于车辆悬挂系统的线性最优预见控制[2].日本学者Ichikawa等于1969年提出了利用目标值改善系统跟踪性能的预见控制方法[3].此后经过五十年的发展,预见控制的设计方法经历了卷积和微分方程法(1960∼1970)、动态规划方法(1970∼1980)、最优与鲁棒控制理论(1980∼2000)、非线性与智能控制理论(2000∼今)等发展过程,并在汽车、机器人、飞行器、机电伺服系统、发电系统等领域得到了广泛的应用研究[4].控制是一种决策,决策则离不开信息,预见控制2期甄子洋:预见控制理论及应用研究进展173是一种未来信息前馈补偿加上当前信息反馈控制的方法.更准确地说,预见控制是一种利用已知期望目标和干扰信息来改善系统动态响应,抑制外界扰动,提高系统跟踪精度的控制方法,特别适用于未来控制目标和干扰等信息已知的系统,尤其对解决非最小相位系统控制问题有其独特的优势[4].正因为具备这一独特优点,预见控制理论研究及应用正日益得到国内外研究者们的广泛关注.基于线性二次型(Linear quadratic,LQ)理论的预见控制最早是1970s提出的,目前仍是预见控制理论中最重要且应用最广泛的方法,因此最优预见控制也属于最优控制理论范畴.最优预见控制与LQ最优跟踪控制的相同之处在于都需要利用未来信息,不同之处在于前者是滚动时域内的局部最优化,属于在线规划,而后者是整体时域内的全局最优化,属于离线规划.预见控制与工业过程控制领域广泛应用的预测控制既相似又有区别.相似之处在于它们都是利用了未来信息作为前馈,都以性能指标最小化为控制目标.区别之处在于:1)前者可以直接利用可预见的系统未来信息,而后者则是通过预测模型预测系统未来信息;2)前者一般假设对象模型已知,而后者对模型精度要求不高,有在线校正模型参数功能;3)前者一般用全状态反馈控制结构,而后者更多是输出反馈控制结构;4)前者适用于模型已知、动态响应速度要求较快、未来信息已知的系统,而后者更适用于模型不精确、具有受约束和非线性等特性的系统.因此,两者各具优点,两者结合也具有很好的互补性.本文目的是将众多研究者在IEEE Transac-tions on Automatic Control、Automatica、IEEE Conference on Decision and Control、《自动化学报》等国内外重要期刊和会议上发表的分散成果,从理论和应用角度进行系统地总结和提炼,以形成一套较完整的预见控制理论并促进应用的发展.本文内容如下:首先从理论角度概述预见控制的发展现状;其次从应用角度概述预见控制的发展现状;最后对预见控制的研究进行总结,并对未来发展方向进行展望.1预见控制的理论研究进展下面首先概述对预见控制起到奠定性作用的早期文献成果,然后重点概述典型预见控制理论研究的发展状况.1.1早期研究概述Sheridan在1966年首次探讨了人类智能或人工智能控制器响应的描述和预测方法,该智能系统特点在于实际输入为有限预见,并且连续目标值视为非均匀重要性,提出了三种预见控制模型:1)扩展线性卷积模型;2)控制元件被控对象动态过程快速试验模型,实质上是一种快速迭代预测与实时控制模型;3)基于动态规划思想的预见区间最优轨迹迭代计算模型[1].随后,Bender于1968年应用连续时间维纳滤波原理解决了一类线性预见控制问题,并应用于车辆悬挂系统[2].日本国内自1960s开始,对预见控制理论和应用展开了许多研究,Hayase等将预见信息引入到控制系统中,提出了最优性能指标下的预见控制方法[3].然而,它要求车辆加速度与跟踪轨迹之间的传递函数不稳定,这在实际中并不可行.Tomizuka于1973年在博士论文中提出了LQ最优预见跟踪控制问题,然而需要建立的增广模型状态维数随着预见步数增加而增加,使得高维Riccati方程求解很困难.随后他于1975年利用动态规划方法解决了连续线性系统的无限预见最优跟踪控制问题,同年也解决了离散线性系统的无限预见最优跟踪控制问题[5−6].Miller于1976年提出了针对无限预见最优控制问题的有限预见近似最优控制方法[7].Balzer于1981年解决了干扰有限预见下的离散线性系统最优控制问题,并应用于车辆悬挂系统[8].Katayama等于1985∼1987年应用线性二次型积分技术解决了离散线性系统和连续线性系统的最优预见控制问题,该控制器结构包括带积分状态反馈项和预见前馈项,并应用于伺服系统[9−10].Sternad等于1988年基于多项式线性二次型高斯(Linear quadratic Gaussian,LQG)方法设计干扰可预见的最优前馈控制器以抑制系统扰动,并首次应用于非最小相位系统[11].而Trudnowski 等于1989年验证了预见信息在有限整定时间系统中的作用[12].归纳起来,早期出现的预见控制主要基于滤波理论[2]、状态空间方法[5−6]、多项式LQ方法[11,13]、矩阵分式LQ方法[14−16]等,通常以LQ函数作为性能指标函数,而文献[12]则基于其他的关于输出的函数,文献[17]则以最小l1、l2、l∞范数跟踪误差作为性能指标函数.虽然多数成果是针对单输入单输出(Single-input single-output,SISO)系统,而且往往针对目标值或者干扰值为阶跃函数的情形,但是早期的研究成果奠定了预见控制在最优控制范畴内的重要地位,也为20世纪90年代开始的预见控制理论研究高潮奠定了基础.1.2线性最优预见控制研究离散域和连续域的线性最优预见控制理论已比较成熟,下面概述一些重要的研究成果.174自动化学报42卷1.2.1离散域最优预见控制线性最优预见控制问题的离散域方面,土谷武士、江上正、廖福成、谭跃钢等在20世纪80年代开始深入研究[18−24],提出了离散型线性最优预见控制的三种设计方法[18]:偏微分最优化法、扩大误差系统法和逐次最优化法.其中,偏微分最优化法先假设控制律形式再求得性能指标函数的极小点;扩大误差系统法把最优预见控制问题转化为最优调节问题来求解;而逐次最优化法则利用最优性原理.三种方法设计的控制系统结构基本相同,且理论推导与设计过程都显得比较复杂.线性最优预见控制系统基于误差系统方程设计,其基本结构如图1所示,主要由如下四部分组成:状态反馈器、目标前馈补偿器、干扰前馈补偿器以及积分环节.图1传统最优预见控制系统Fig.1Traditional optimal preview control system定理1[18].离散时间线性定常系统的状态空间模型为x (k +1)=Ax x (k )+Bu u (k )+Ed d (k )(1)y (k )=Cxx (k )(2)构造如下误差系统x (k +1)=Axx (k )+B ∆u (k )+E ∆d (k )+F ∆y ∗(k +1)(3)e (k )=Cxx (k )(4)假设未来k +1∼k +k f 时刻的目标值和k ∼k +k f −1时刻的干扰值均已知,定义预见控制LQ 性能指标函数J =k f j =1[ e (k +j ) 2Q e + ∆u (k +j −1) 2R ](5)式中,Q e ,R 为对称正定权重矩阵,k f 为预见步数,基于三种求解方法,求得最优控制增量序列为∆u (k )=F x x (k )+k f j =1F y ∗(j )∆y ∗(k +j )+k f j =1F d (j −1)∆d (k +j −1)(6)式中,F x 为状态反馈矩阵,F y ∗为目标预见前馈矩阵,F d 为干扰预见前馈矩阵.该预见控制系统的特点在于[18]:1)具有内模原理;2)闭环系统极点由反馈项决定,前馈项用于改善系统的相位特性;3)当控制能量权重趋于零时,性能指标评价值渐近趋于零.随后,廖福成课题组进一步将最优预见控制推广应用于多重采样离散线性系统[25−29]、状态或输入时滞离散线性系统[28−33]、离散广义线性系统[33−34].以上研究采用了扩大误差系统法,把多重采样系统转化为单一采样系统,把时滞系统转化为一般系统,把广义系统的预见控制问题转化为广义系统的最优控制问题,得以解决.近年来,甄子洋、王志胜等提出了一种新的离散时间最优预见控制设计方法:信息融合估计法[35−38].信息融合估计主要利用多源信息来估计未知量,它是信息融合领域的重要理论基础.控制、估计等都属于决策问题,利用对象动态方程约束信息、输出测量信息、未来预见信息、二次型性能指标约束信息等有用信息,基于融合估计理论,求取控制序列的最优估计,即为信息融合预见控制.其基本原理结构如图2所示,主要由两部分组成:协状态融合滤波器和控制量融合估计器.图2信息融合预见控制系统IFig.2Information fusion preview control system I定理2[36].针对离散时间线性系统的LQ 最优预见控制问题,基于信息融合估计理论,求得控制量融合估计为ˆu(k )=[R +B T P −1(k +1)B ]−1B T P −1(k +1)·[ˆx(k +1)−Ax x (k )−Ed d (k )](7)且协状态融合滤波方程为2期甄子洋:预见控制理论及应用研究进展175ˆx(k +j )=P (k +j )A T [P (k +j +1)+BR −1B T ]−1·[ˆx (k +j +1)−Ed d (k +j )]+P (k +j )C T Qyy ∗(k +j )(8)P −1(k +j )=C T QC +A T [P (k +j +1)+BR −1B T ]−1A(9)信息融合预见控制经过进一步转化,可得到传统预见控制类似的结构形式,如定理3和图3所示.图3信息融合预见控制系统IIFig.3Information fusion preview control system II定理3[36].针对上述离散时间线性系统的最优预见控制问题,得到信息融合预见控制律的另一种形式为∆ˆu(k )=K x x (k )+k f j =1K y ∗(k +j )y ∗(k +j )+k fj =1K d (k +j −1)d (k +j −1)(10)式中,K x 为状态反馈矩阵,K y ∗为目标前馈矩阵,K d 为干扰前馈矩阵.若把当前参考输入或期望输出当作是未来所有时刻的预见信息,则信息融合预见控制还可以推广应用于伺服跟踪控制问题[39]、解耦控制问题[40]、最优调节与跟踪系统问题[41−43],这也证实了预见控制与最优控制、跟踪控制的同一性.概括来讲,信息融合预见控制与传统预见控制的异同之处在于:1)前者既可以基于误差系统设计,也可以基于原系统设计,后者通常采用误差系统设计.基于原系统设计的信息融合预见控制因为矩阵维数降低,使得结构更加简单,计算量更小.2)前者基于信息融合估计理论,推导过程简便且物理意义明确,因为它定义了信息量来表征未来预见信息的权重和作用大小,而后者基于经典数学方法设计,缺乏清晰的物理概念.3)两者的渐近特性具有相同的结论,即在控制无约束情况下,LQ 最优预见性能指标都趋近于零,即完全消除干扰的影响,达到无静差跟踪;同时,两者的最优性能指标、状态反馈系数、预见前馈系数以及脉冲转递函数阵的渐近值均相同[36].实际应用中由于控制量受限的原因,无静差跟踪往往难以达到.信息融合预见控制系统的稳态误差是控制矩阵K x ,K y ∗,K d 的函数,而这些参数又由预见步数K f 和权重矩阵Q 、R 完全确定,即影响信息融合预见控制的重要参数实质上是K f ,Q ,R .预见步数的增加能提高控制系统的响应速度,减小稳态跟踪误差并趋于零,且在系统动态调节过程中的控制能量要小些[36].通过对线性系统控制的结果显示,性能指标函数值随着预见步数的增加单调递减,直至收敛到定值,表明随着预见步数的增加,性能指标迅速减小并趋于定值,反映了较近未来信息对提高控制性能的作用较大,而较远的未来信息的作用较小,最终信息量趋于饱和[36,44].因此,在满足性能要求前提下,预见步数取较小值即可,以避免计算量随预见步数的增加而增大.权重矩阵具有相对关系,Q 越大,控制精度越高,但是控制能量消耗越大,所以它们的选取需要充分考虑控制受限问题,通常固定一个权重矩阵,调节另一个权重矩阵,且都选取为对角矩阵.因此,如何选取控制参数K f ,Q ,R 使得预见闭环系统的稳态误差为极小的问题,可定义为预见控制系统设计的逆问题,可考虑用梯度法或智能优化算法来迭代求解.1.2.2连续域最优预见控制连续时间预见控制系统的结构如图4所示,与全信息控制系统相似,即都是将外部输入(包括参考输入、外界干扰)和系统状态作为控制器的输入,区别在于前者的外部输入在未来h 时间段内是可预见的,而且控制器利用了未来所有预见信息.图4中采用了对外部输入延时的等效方式,避免了时延算子给控制器设计带来的麻烦.图4连续型最优预见控制系统Fig.4Continuous optimal preview control system设连续时间线性系统状态空间模型为176自动化学报42卷˙x(t)=Ax x(t)+B1w(t)+B2u(t)(11)y(t)=Cx x(t)+Du u(t)(12)定义无限时域LQ性能指标函数J=∞x T(t)Qx x(t)+u T(t)Ru u(t)d t(13)则通过引入一个预见信息积分补偿器,与原系统构成一个增广系统,基于算子Riccati方程最小化指标函数,得到最优控制律为[45]u(t)=−R−1B T2[Mx x(t)+he(A−B2B T2M)TβMB1w(t+β)dβ](14)式中,M为代数Riccati方程的稳态解,h为干扰预见时间.还可以进行初始状态不确定下的最差情况计算分析.定义表征外部输入w与输出y之间的H2性能函数为γ=∞tr(F T(t)F(t))d t(15)式中,F(t)为系统脉冲响应矩阵.最小化该函数,求得如下H2预见全信息控制律[46]u(t)=−B T2[Mx x(t)+he(A−B2B T2M)TβMB1w(t+β)dβ](16)而H2函数的极小值与微分Riccati方程的稳态解有关.该控制律包含状态反馈项和预见前馈项,对比式(14)和式(16)发现,文献[45]中的控制律比文献[46]增加了控制能量加权矩阵R作为调节参数.然而,文献[45]和文献[46]都是带预见信息的最优调节器问题.对于连续线性系统(A,B,C)的最优跟踪控制问题,定义如下有限时域LQ性能指标函数J=12e T(tα)e(tα)+12tαt0[e T(t)Q e e(t)+x T(t)Q x x(t)+u T(t)Q u u(t)+˙u T(t)R˙u(t)](17)构建一个引入误差信号、预见信号的增广系统,将最优跟踪问题转化为最优调节问题,利用极小值原理,求得最优预见跟踪控制律为[47]u(t)=−f−1(t)·Tt0f(σ)R−1[p T13(σ)e(σ)+p T23(σ)x(σ)+[00I]·tαΛ(σ+h)σΦ(σ,τ)P(τ)D˙y∗(τ)dτ]dσ(18)式中,相关变量定义参考文献[47].文献[48]进一步研究了连续线性系统的H2前馈干扰抑制问题.针对含多输入输出时延的连续线性系统的H2预见控制问题,则采用类似方法将其转化为LQ问题,再利用最优性原理来求解[49].文献[46]、文献[48]和文献[49]的共同点在于:将优化时间区间划分为标准无限时域LQ问题和非标准有限时域LQ问题,其中,无限时域部分等同于最优控制状态反馈项,有限时域部分采用了极小值原理,其结果等同于最优控制器的有限脉冲响应部分.这种分割优化区间的思想来源于基于间隙度量的鲁棒控制理论[50].然而,上述方法需要求解一个微分Riccati 方程,未能清晰表述预见时间对H2性能的影响.为此,文献[51]提出了针对连续时间系统在单输入、多输入、多预见时间情形下的多种H2预见控制算法,其中多输入多预见时间下的预见控制系统结构分别如图5所示[51].图中,Λ=diag{e−sh1,···,e−sh nw}.图5多输入多预见时间H2预见控制系统Fig.5Multiple input multiple preview time H2previewcontrol system文献[51]方法相比于文献[46]、文献[48]和文献[49]方法,具有如下优点:1)它只需求解Lya-punov方程,比求解Riccati方程更加简单,从而减少了计算量;2)它给出了预见信息、预见时间对H2性能影响的解析表达式,并且得到了无限时间预见信息下的最佳性能值;3)它还能处理多预见时间情形,这在其他文献中均未涉及.上述方法都是解决外界干扰到系统输出之间的H2函数最小化问题,H2预见调节器实质上是干扰补偿问题,而未能对H2预2期甄子洋:预见控制理论及应用研究进展177见跟踪控制问题进行深入研究.并且,可预见干扰形式多为δ函数形式.文献[46]、文献[48−49]研究的问题也被称之为单边模型匹配问题(One-side model matching problem).而双边模型匹配问题,即给定真有理传递函数矩阵G1,G2,G3,常数h,寻找K∈H∞,最小化H2范数 T 2,且T=e−sh G1−G3KG2∈H2∩H∞(19)当G2=G3=I时即为单边模型匹配.双边模型匹配问题在实际应用中经常遇到,主动悬挂系统就是一个典型实例.然而,近年极少有关于含预见信息的双边模型匹配问题得到研究,文献[52]进行了这方面的首创工作,并获得了最优解和最佳性能值.但是该最优解需要求解两个带约束矩阵Sylvester 方程和两个代数Riccati方程,计算比较困难,而且Riccati方程无法保证有稳定解.同时,该方法基于传递函数矩阵设计,而非基于状态空间模型设计.非最小相位特性在实际系统中比较常见,如机器人、飞行器等.非最小相位零点在系统带宽增大的时候会引起闭环灵敏度函数出现大的峰值,导致系统带宽受限制、跟踪精度不能任意小.而仅仅采用反馈控制难以实现完全跟踪,非因果稳定逆理论给解决这一问题提供了可行方法,该理论依赖于未来期望轨迹,由于在线计算量较大导致难以应用.预见控制能够改善非最小相位零点系统的控制器带宽,为此,文献[53]和文献[54]提出了预见稳定逆(Preview-based stable-inversion)方法,给出了逆输入与未来期望轨迹之间的定量关系,使稳定逆方法在非最小相位系统中应用变得可行,也证明了只要有足够的预见时间,系统跟踪精度能够达到任意小.然而,该方法对系统的不确定性和动态变化比较敏感.而文献[55]专门研究了反馈控制下系统跟踪性能的受限问题,利用最优插值方法(Optimal interpolation approach)定量分析了预见控制对改善系统跟踪性能的作用,从而验证了预见控制在非最小相位系统中的独特优势.综上所述,线性最优预见控制的研究成果主要集中在精确建模、状态反馈控制和无约束条件基础上,如何将其推广应用于模型难以精确建立的情形以及输出反馈控制、状态/输入受约束控制等复杂问题,是未来值得研究的重要方向.1.3线性鲁棒预见控制研究近年来,线性鲁棒预见控制问题得到深入研究,尤其是鲁棒H∞预见控制问题.离散域的鲁棒H∞预见控制问题,主要采用博弈论方法解决[56−61],而离散域的鲁棒LQ预见控制则主要采用线性矩阵不等式(Linear matrix inequality,LMI)方法解决[62−63].连续域的鲁棒H∞预见控制有如下几种解法:1)博弈论和鞍点方法,通常需要求解Ric-cati方程[64−68];2)函数解析(Functional-analytic)法,它将问题转化为无限维形式加以解决[69−70];3) Hamilton矩阵方法,通常也需要求解Riccati方程[71−75];4)基于双侧拉普拉斯变换的传递函数方法,因为预见实质上是一种无限维系统,将H∞优化预见问题可以转化为因果和有限维H∞优化问题得以解决[76].且前三种方法都是基于状态空间模型,第四种方法基于传递函数.1.3.1离散域H∞预见控制参考输入或期望输出这一目标信号的跟踪问题一直是控制理论的基础问题,也是应用最为广泛的实际问题.若要实现最优跟踪,需要利用目标信号的未来值.起初H∞跟踪控制只是把目标信号当作扰动,因此很少关心该信号的作用,更不会用它的预见信息,直到提出了针对连续时间线性系统的利用未来参考输入的H∞跟踪控制方法.受此启发,文献[56−57]将其推广于离散时间线性系统,并针对已知控制时域内所有目标值、仅仅已知当前目标值、已知预见窗口目标值三种情形,基于博弈论分别设计了H∞鲁棒跟踪控制器.上述研究都是针对确定性线性系统,随后该课题组利用同样方法,解决了系统参数存在范数有界不确定性的线性时变系统的H∞跟踪控制问题[58].然而该方法无法应用于参数在不确定区间非一致性分布的情形,而且它假定参数输入信号是L2信号,也将H∞-Riccati方程存在半正定解作为前提条件,显得较为苛刻.上述前提条件对于追求鲁棒LQ性能的离散时间多面体不确定系统则往往是失效的,因为这类系统采用LMI方法分析鲁棒性能,而LMI条件无法转化为Riccati方程.此时,可以构造增广系统将带积分鲁棒预见跟踪控制问题转化为鲁棒LQ调节器问题,得到的控制器结构也包含预见前馈补偿器、状态反馈控制器和积分环节[62].随后,文献[63]将该方法应用于带执行器故障的离散时间多面体不确定系统中.由于系统参数不确定性通常建模为白噪声随机过程,这在现实中也普遍存在.增益调度方法在调度参数受到测量噪声污染时也是如此.文献[59]利用文献[57]中的博弈论方法,针对带乘性噪声多变量离散时间线性系统,就同样的三种不同预见信息模式,分别设计了状态反馈H∞预见跟踪控制算法.预见跟踪控制问题本质上可以理解为固定时滞平滑问题(Fixed-lag smoothing problem)的对偶问题.在预见跟踪控制问题中,预见信息被描述为外部输入的一种延时,此时全状态包含了系统状态和。