CN105892288A

CN105892288A - State feedback design method

Info

Publication number: CN105892288A
Application number: CN201410746701.XA
Authority: CN
Inventors: 王书舟
Original assignee: Tianjin Polytechnic University
Current assignee: Tianjin Polytechnic University
Priority date: 2014-12-05
Filing date: 2014-12-05
Publication date: 2016-08-24

Abstract

针对一类关于快系统是线性的、慢系统可部分输入输出线性化的奇异摄动系统，设计了使整个闭环系统渐近稳定的状态反馈控制器。针对奇异摄动系统的特点，分别设计使闭环慢系统和边界层系统渐近稳定的状态反馈控制律，通过构造的系统Lapunov函数得到了对于充分小的摄动参数ε，闭环系统反馈镇定需要满足的条件。边界层系统状态能够快速收敛到指定流形，并最终实现整个系统的渐近稳定。For a class of singularly perturbed systems in which the fast system is linear and the slow system can be partially linearized, a state feedback controller is designed to make the whole closed-loop system asymptotically stable. According to the characteristics of the singular perturbation system, the state feedback control laws to make the closed-loop slow system and the boundary layer system asymptotically stable are respectively designed. Through the constructed system Lapunov function, it is obtained that for a sufficiently small perturbation parameter ε, the feedback stabilization of the closed-loop system needs to satisfy conditions of. The state of the boundary layer system can quickly converge to the specified manifold, and finally realize the asymptotic stability of the whole system.

Description

A Design Method of State Feedback

一、技术领域1. Technical field

本发明属于非线性系统控制领域，本发明涉及系统稳定性、控制系统设计等方法。The invention belongs to the field of nonlinear system control, and the invention relates to methods such as system stability and control system design.

二、背景技术2. Background technology

控制系统设计的基本目标是保持系统具有良好的性能和稳定性，其中闭环系统的H_∞范数通常被认为是一项重要的性能指标。近年来线性奇异摄动系统的H_∞控制研究领域取得了一些重大成就和成功的应用。其中频域方法的主要焦点是模型匹配问题，许多反馈控制问题，如跟踪、鲁棒稳定和干扰抑制等问题均可以转化为模型匹配问题，即使闭环系统的频率响应匹配已给定的模型频率响应，并极小化频率峰值误差的H_∞范数。目前为止，非线性奇异摄动系统研究还主要是基于Lyapunov函数方法的稳定性分析，有关H_∞控制问题的研究极为有限，由于非线性系统本身不能定义L₂诱导范数，也就不存在传递函数的H_∞范数，很多针对线性系统的方法不能直接推广到非线性情形。The basic goal of control system design is to keep the system with good performance and stability, and the H _∞ norm of the closed-loop system is usually considered as an important performance index. In recent years, some great achievements and successful applications have been made in the research field of H _∞ control for linear singular perturbation systems. Among them, the main focus of the frequency domain method is the model matching problem. Many feedback control problems, such as tracking, robust stability and disturbance suppression, can be transformed into model matching problems, even if the frequency response of the closed-loop system matches the given model frequency response. , and minimize the H _∞ norm of the frequency peak error. So far, the research on nonlinear singularly perturbed systems is mainly based on the stability analysis of the Lyapunov function method, and the research on the H _∞ control problem is extremely limited. Since the nonlinear system itself cannot define the L ₂ induced norm, there is no transfer The H _∞ norm of the function, many methods for linear systems cannot be directly extended to nonlinear situations.

三、专利内容：3. Patent content:

1、专利目的1. Purpose of the patent

主要目的是研究并解决非线性奇异摄动系统的鲁棒H_∞控制问题。The main purpose is to study and solve the problem of robust H _∞ control for nonlinear singularly perturbed systems.

2、技术解决方案2. Technical solutions

首先求出第一个子系统的严格耗散不等式，然后递推得到整个系统的严格耗散不等式，控制器的设计过程避免了Hamilton-Jacobi方程的求解，该方法为解决非线性奇异摄动系统的鲁棒H_∞控制问题提供了一种途径。Firstly, the strict dissipation inequality of the first subsystem is obtained, and then the strict dissipation inequality of the whole system is obtained recursively. The design process of the controller avoids the solution of the Hamilton-Jacobi equation. This method is to solve the nonlinear singular perturbation system The robust H _∞ control problem provides a way.

3、技术效果及优点3. Technical effects and advantages

本发明构造了系统满足γ耗散不等式的能量存储函数，并可证明了在更一般意义下系统具有小于γ_r+||o(ε)||的L₂增益。The present invention constructs an energy storage function that the system satisfies the γ dissipation inequality, and can prove that the system has an L ₂ gain smaller than γ _r +||o(ε)|| in a more general sense.

四、具体实施方式4. Specific implementation

一类具有下述形式的奇异摄动系统：A class of singularly perturbed systems of the form:

$\overset{\cdot &Center Dot;}{x x} = = {F f}_{11} ((x x,, z z,, u u)) = = {f f}_{11} ((x x)) + + {Q Q}_{11} ((x x)) z z + + {g g}_{11} ((x x)) u u$

$ϵ ϵ \overset{\cdot \cdot}{z z} = = {F f}_{22} ((x x,, z z,, u u)) = = {f f}_{22} ((x x)) + + {Q Q}_{22} ((x x)) z z + + {g g}_{22} ((x x)) u u - - - - - - ((11))$

y＝h(x)y=h(x)

考虑具有如下形式的控制律：Consider a control law of the form:

$u u = = \overset{~ ~}{u u} + + K K ((x x)) z z - - - - - - ((22))$

K(x)的选取要保证Q₂(x)+g₂(x)K(x)是一致稳定阵。The selection of K(x) should ensure that Q ₂ (x)+g ₂ (x)K(x) is a consistent stable matrix.

在形如(2)的控制律作用下，系统(1)具有如下形式：Under the control law of the form (2), the system (1) has the following form:

$\frac{dx dx}{dt dt} = = {f f}_{11} ((x x)) + + [[{Q Q}_{11} ((x x)) + + {g g}_{11} ((x x)) K K ((x x))]] z z + + {g g}_{11} ((x x)) \overset{~ ~}{u u} - - - - - - ((33))$

$ϵ ϵ \frac{dz dz}{dt dt} = = {f f}_{22} ((x x)) + + [[{Q Q}_{22} ((x x)) + + {g g}_{22} ((x x)) K K ((x x))]] z z + + {g g}_{22} ((x x)) \overset{~ ~}{u u} - - - - - - ((44))$

对其进行标准的双时间刻度分解，得到快子系统：Decompose it on a standard two-time scale to get the tachyon system:

$\frac{dz dz}{dτ dτ} = = {f f}_{22} ((x x)) + + [[{Q Q}_{22} ((x x)) + + {g g}_{22} ((x x)) K K ((x x))]] z z + + {g g}_{22} ((x x)) \overset{~ ~}{u u} - - - - - - ((55))$

引进一个新的向量y＝z-z^s，其中Introduce a new vector y=zz ^s where

${z z}^{s the s} = = {[[{Q Q}_{22} ((x x)) + + {g g}_{22} ((x x)) K K ((x x))]]}^{- - 11} [[{f f}_{22} ((x x)) + + {g g}_{22} ((x x)) \overset{~ ~}{u u}]]$

y是闭环系统快动态的准稳定状态，即y is the fast dynamic quasi-stable state of the closed-loop system, namely

$\frac{dy dy}{dτ dτ} = = [[{Q Q}_{22} ((x x)) + + {g g}_{22} ((x x)) K K ((x x))]] y the y - - - - - - ((66))$

因为Q₂(x)+g₂(x)K(x)是稳定阵，所以闭环系统的快动态就有一个指数稳定的平衡点流形z^s，当快系统的状态迅速降至指定流形z^s，慢系统具有如下形式：Because Q ₂ (x)+g ₂ (x)K(x) is a stable matrix, the fast dynamics of the closed-loop system has an exponentially stable equilibrium point manifold z ^s , when the state of the fast system rapidly drops to the specified manifold z ^s , the slow system has the following form:

$\begin{matrix} \frac{dx dx}{dt dt} = = \overset{~ ~}{F f} ((x x)) + + \overset{~ ~}{G G} ((x x)) \overset{~ ~}{u u} \\ {y the y}^{s the s} = = h h ((x x)) \end{matrix} - - - - - - ((77))$

y^s表示闭环慢系统的输出。式(7)中：y ^s represents the output of the closed-loop slow system. In formula (7):

$\overset{~ ~}{F f} ((x x)) = = {f f}_{11} ((x x)) - - [[{Q Q}_{11} ((x x)) + + {g g}_{11} ((x x)) K K ((x x))]] {[[{Q Q}_{22} ((x x)) + + {g g}_{22} ((x x)) K K ((x x))]]}^{- - 11} {f f}_{22} ((x x))$

$\overset{~ ~}{G G} ((x x)) = = {g g}_{11} ((x x)) - - [[{Q Q}_{11} ((x x)) + + {g g}_{11} ((x x)) K K ((x x))]] {[[{Q Q}_{22} ((x x)) + + {g g}_{22} ((x x)) K K ((x x))]]}^{- - 11} {g g}_{22} ((x x))$

系统(1)具有相对阶{r₁，…，r_m}，也就是说，对所有的1≤i，j≤m，k≤r_i-1 x∈B_x，有且分布总是对合的，m×m矩阵 $A (x) = (a_{ij} (x)) = (L_{\tilde{G}} L_{\tilde{F}}^{r_{i} - 1} h_{i} (x))$ 对所有x∈B_x是非奇异的。System (1) has relative order {r ₁ ,...,r _m }, That is, for all 1≤i, j≤m, k≤r _i -1 x∈B _x , we have and distributed Always involute, m×m matrix $A (x) = (a_{ij} (x)) = (L_{\tilde{G}} L_{\tilde{f}}^{r_{i} - 1} h_{i} (x))$ is nonsingular for all _x ∈ B x.

在此假设下，选取一个如下的坐标变换：使得系统(1)在新坐标下具有如下的Brunowsky规范结构：Under this assumption, choose a coordinate transformation as follows: make the system (1) have the following Brunowsky canonical structure in the new coordinates:

$\overset{\cdot \cdot}{η η} = = Q Q ((η η,, ξ ξ))$

$\overset{\cdot \cdot}{ξ ξ} = = Aξ Aξ + + B B [[F f ((η η,, ξ ξ)) + + G G ((η η,, ξ ξ)) \overset{~ ~}{u u}]] - - - - - - ((88))$

y^s＝Cxy ^s =Cx

其中：in:

$A A = = diag diag {{{A A}_{11},, \cdot \cdot \cdot \cdot \cdot \cdot,, {A A}_{m m}}},, {A A}_{i i} = = {[\begin{matrix} 00 & 11 & . . . . . . & 00 \\ . . . . . . & . . . . . . & . . . . . . & . . . . . . \\ 00 & 00 & . . . . . . & 11 \\ 00 & 00 & . . . . . . & 00 \end{matrix}]}_{{r r}_{i i} \times \times {r r}_{i i}} - - - - - - ((99))$

$B B = = diag diag {{{B B}_{11},, \cdot \cdot \cdot &Center Dot; \cdot \cdot,, {B B}_{m m}}},, {B B}_{i i} = = {[\begin{matrix} 00 & . . . . . . & 00 & 11 \end{matrix}]}^{T T}_{{r r}_{i i} \times \times 11} - - - - - - ((1010))$

$C C = = diag diag {{{C C}_{11},, . . . . . .,, {C C}_{m m}}},, {C C}_{i i} = = {[\begin{matrix} 11 & 00 & . . . . . . & 00 \end{matrix}]}_{11 \times \times {r r}_{i i}} - - - - - - ((1111))$

考虑状态反馈Consider status feedback

$\overset{~ ~}{u u} = = - - {[[A A ((x x))]]}^{- - 11} [[b b ((x x)) + + c c ((x x))]] - - - - - - ((1212))$

$b b ((x x)) = = {[[{L L}_{\overset{~ ~}{F f}}^{{r r}_{11} - - 11} {h h}_{11} ((x x)),, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; {L L}_{\overset{~ ~}{F f}}^{{r r}_{m m} - - 11} {h h}_{m m} ((x x))]]}^{T T} - - - - - - ((1313))$

$c c ((x x)) = = {[[{c c}_{11} ((x x)),, \cdot \cdot \cdot &Center Dot; \cdot &Center Dot;,, {c c}_{m m} ((x x))]]}^{T T} = = {[[{Σ Σ}_{k k = = 11}^{{r r}_{11}} {β β}_{k k}^{11} {L L}_{\overset{~ ~}{F f}}^{k k - - 11} {h h}_{11} ((x x)),, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, {Σ Σ}_{k k = = 11}^{{r r}_{m m}} {β β}_{k k}^{11} {L L}_{\overset{~ ~}{F f}}^{k k - - 11} {h h}_{m m} ((x x))]]}^{T T} - - - - - - ((1414))$

其中是一些选取参数，满足多项式的根都位于左半平面。in are selected parameters that satisfy the polynomial The roots of are located in the left half-plane.

闭环系统在经过这个坐标变换之后，将由如下的方程组所描述：After this coordinate transformation, the closed-loop system will be described by the following equations:

$\begin{matrix} \overset{\cdot \cdot}{η η} = = Q Q ((η η,, ξ ξ)) \\ \overset{\cdot \cdot}{ξ ξ} = = \overset{&OverBar; &OverBar;}{A A} ξ ξ \end{matrix} - - - - - - ((1515))$

$\overset{&OverBar; &OverBar;}{A A} = = diag diag {{{\overset{&OverBar; &OverBar;}{A A}}_{11},, . . . . . .,, {\overset{&OverBar; &OverBar;}{A A}}_{m m}}},, {\overset{&OverBar; &OverBar;}{A A}}_{i i} = = {[\begin{matrix} 00 & 11 & . . . . . . & 00 \\ . . . . . . & . . . . . . & . . . . . . & . . . . . . \\ 00 & 00 & . . . . . . & 11 \\ {- - β β}_{11}^{i i} & {- - β β}_{22}^{i i} & . . . . . . & {- - β β}_{{r r}_{i i}}^{i i} \end{matrix}]}_{{r r}_{i i} \times \times {r r}_{i i}} - - - - - - ((1616))$

这种方法针对奇异摄动系统的特点，分别设计使闭环慢系统和边界层系统渐近稳定的状态反馈控制律，通过构造的系统Lapunov函数得到了对于充分小的摄动参数ε，闭环系统反馈镇定需要满足的条件。边界层系统状态能够快速收敛到指定流形，并最终实现整个系统的渐近稳定。According to the characteristics of the singular perturbation system, this method designs the state feedback control law to make the closed-loop slow system and the boundary layer system asymptotically stable, and obtains the closed-loop system feedback for sufficiently small perturbation parameter ε through the constructed system Lapunov function Conditions that need to be met for sedation. The state of the boundary layer system can quickly converge to the specified manifold, and finally realize the asymptotic stability of the whole system.

Claims

1. A state feedback design method for a singular perturbation system, characterized in that: for sufficiently small perturbation parameter ε, the condition that the feedback stabilization of the closed-loop system needs to be satisfied is obtained by constructing the system Lapunov function.

2. a kind of singular perturbation system state feedback design method described in claim 1 is characterized in that: aiming at the characteristics of singular perturbation system, respectively designed the state feedback control that makes closed-loop slow system and boundary layer system asymptotically stable law.