草稿:线性变参数控制

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线性变参数控制(LPV control)是一种用于处理线性变参数系统控制方法。

增益规划[编辑]

在为动态系统设计反馈控制器时,人们应用了众多种类的现代,多变量控制的控制器。通常来讲,这些控制器经常设计在系统动态的不同的操作点上,并且使用了线性化的模型,并且被设计为一个或者多个参数在中间状态上的函数。这是一种可以在非线性系统上使用一类线性控制器的方法,并且这些线性控制器中每个都可以在不同的操作点上获得令人满意的控制效果。调度变量是一个或者多个可以观测的变量。人们使用调度变量来确定系统当前的操作点和合适当前使用的控制器。比如在飞行器的控制中,一系列的控制器被设计在由攻角马赫数動壓重心张成网格的不同位置上。简单来说,增益规划是通过整合一系列线性控制器来为非线性系统设计非线性控制器英语Nonlinear control的方法。这些线性系统实时的通过切换或者插值的方法混合。

对多变量的控制器进行规划有时候是一个乏味且费时的工作。而新的方法是使用线性变参数技术来对自动规划的多变量控制器进行整合。

经典增益规划方法的缺点[编辑]

  • 增益规划有個重大的缺点:在设计操作点外的操作条件,不一定可以获得足够的性能,甚至有些连稳定性都无法保证。
  • 多变量控制器的规划常常是乏味且费时的工作,尤其是在航空航天的控制中更是如此。在航空航天中,为了在更高的气动包线英语Flight envelope上获得更好的性能,参数的改变十分巨大。
  • 另一点很重要的是,選定的规划變數反映了在飛行條件變化時,飛行動力學上的變化,有可能利用增益规划將線性鲁棒控制的方法論整合到非線性控制設計中,不過設計過程不會明確的提到全域穩定性、鲁棒性以及其性能特性。

增益规划的方式很簡單,而且線性化增益规划的計算量遠小於其他的非線性設計方式,不過其先天的缺點超過其優點,因此需要一個動態系統控制的新作法。像以人工神经网络(ANN)及模糊逻辑為基礎的適應控制試圖要解決這些問題,不過因為缺乏針對整個操作參數範圍內,租定性以及性能的證明,因此需要一種可以確保其特性的,和參數相關的控制器。

線性變參數系統[编辑]

線性變參數系統(LPV)是很特殊的非線性系統,其系統特性很適合用參數的變化來表示。一般而言,LPV技巧可以提供增益規劃多變數系統的系統化作法。此方式可以允許在單一架構內整合性能、強健性及带宽限制[1][2],以下是線性變參數系統的簡介,以及其術語的說明。

參數相依系統[编辑]

控制工程中,狀態空間表示法是物理系統的数学模型,由有許多輸入、輸出狀態變數的一階微分方程所組成。一個非線性、非自主系統的動態演進可以由下式表示

若系統是時變系統,則

The state variables describe the mathematical "state" of a dynamical system and in modeling large complex nonlinear systems if such state variables are chosen to be compact for the sake of practicality and simplicity, then parts of dynamic evolution of system are missing. The state space description will involve other variables called exogenous variables whose evolution is not understood or is too complicated to be modeled but affect the state variables evolution in a known manner and are measurable in real-time using sensors. When a large number of sensors are used, some of these sensors measure outputs in the system theoretic sense as known, explicit nonlinear functions of the modeled states and time, while other sensors are accurate estimates of the exogenous variables. Hence, the model will be a time varying, nonlinear system, with the future time variation unknown, but measured by the sensors in real-time. In this case, if denotes the exogenous variable vector英语Vector (mathematics and physics), and denotes the modeled state, then the state equations are written as

The parameter is not known but its evolution is measured in real time and used for control. If the above equation of parameter dependent system is linear in time then it is called Linear Parameter Dependent systems. They are written similar to Linear Time Invariant form albeit the inclusion in time variant parameter.

Parameter-dependent systems are linear systems, whose state-space descriptions are known functions of time-varying parameters. The time variation of each of the parameters is not known in advance, but is assumed to be measurable in real time. The controller is restricted to be a linear system, whose state-space entries depend causally on the parameter’s history. There exist three different methodologies to design a LPV controller namely,

  1. Linear fractional transformations英语Linear fractional transformations which relies on the small gain theorem英语Small-gain theorem for bounds on performance and robustness.
  2. Single Quadratic Lyapunov Function (SQLF)
  3. Parameter Dependent Quadratic Lyapunov Function (PDQLF) to bound the achievable level of performance.

These problems are solved by reformulating the control design into finite-dimensional, convex feasibility problems which can be solved exactly, and infinite-dimensional convex feasibility problems which can be solved approximately . This formulation constitutes a type of gain scheduling problem and contrast to classical gain scheduling, this approach address the effect of parameter variations with assured stability and performance.

參考資料[编辑]

  1. ^ J. Balas, Gary. Linear Parameter-Varying Control And Its Application to Aerospace Systems (PDF). ICAS. [2002]. 
  2. ^ Wu, Fen. Control of Linear Parameter Varying systems. Univ. of California, Berkeley. [1995]. 

參考資料[编辑]

  • Briat, Corentin. Linear Parameter-Varying and Time-Delay Systems - Analysis, Observation, Filtering & Control. Springer Verlag Heidelberg. 2015. ISBN 978-3-662-44049-0. 
  • Roland, Toth. Modeling and Identification of Linear Parameter-Varying Systems. Springer Verlag Heidelberg. 2010. ISBN 978-3-642-13812-6. 
  • Javad, Mohammadpour; Carsten, W. Scherer (编). Control of Linear Parameter Varying Systems with Applications. Springer Verlag New York. 2012. ISBN 978-1-4614-1833-7.