# 草稿:线性变参数控制

## 增益规划

#### 经典增益规划方法的缺点

• 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。

## 線性變參數系統

### 參數相依系統

${\displaystyle {\dot {x}}=f(x,u,t)}$

${\displaystyle {\dot {x}}=f(x(t),u(t),t),x(t_{0})}$
${\displaystyle x(t_{0})=x_{0},u(t_{0})=u_{0}}$

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 ${\displaystyle w(t),w}$ denotes the exogenous variable , and ${\displaystyle x(t)}$ denotes the modeled state, then the state equations are written as

${\displaystyle {\dot {x}}=f(x(t),w(t),{\dot {w}}(t),u(t))}$

The parameter ${\displaystyle w}$ 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.

${\displaystyle {\dot {x}}=A(w(t))x(t)+B(w(t))u(t)}$
${\displaystyle y=C(w(t))x(t)+D(w(t))u(t)}$

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. which relies on the 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

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

• 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.