线性系统理论笔记

三、系统的数学描述

3.2 输入输出描述

二、初始松弛概念

瞬时系统（无记忆系统）：t1时刻的输出仅仅取决于该时刻所加的输入（仅由电阻构成的网络）

系统在t1时刻松弛：在t1时刻系统不存储能量

三、线性性质

1、定义

系统是线性的判别：$\mathbf{H}\left({\alpha }_1{u}_1+{\alpha }_2{u}_2\right)={\alpha }_{1}H{u}_1+{\alpha }_{2}H{u}_2$（可加性和齐次性）

2、线性松弛系统的脉冲响应

脉冲函数：$d_{}\left(t-t_1\right)=\{\begin{array}{ll}0,& t<t_1\\ \frac{1}{D},& t_1\le t<t_1+D\\ 0,& t>t_1+\end{array}$

用脉冲函数近似表示信号：$u\cong \sum u\left(t_i\right)d_{}\left(t-t_i\right)D$

3. The impulse response function of the linear system y=Hu

级数变成积分的形式：$y=Hu=H\sum u\left(t_i\right){\delta }_{\Delta }\left(t-t_i\right)\Delta ={\int }_{-\infty }^{+\infty }[H\delta (t-\tau )]u(\tau )d\tau$

系统脉冲响应函数：$H\delta (t-\tau )=g(t,\tau )$

4、脉冲响应矩阵

由系统脉冲响应函数组成的矩阵，其中$g_{ij}(t,\tau )$ is the impulse response of the i-th output to the j-th input

Fourth, causality

Causal (unexpected): If the output of the system at time t depends only on time t and the input before t $y(t)=H{u}_{(-\infty ,t)}$

Five, relaxation

The system is at $t_0$Time is relaxed: if and only if the system outputs $\mathrm{y}\left({\mathrm{t}}_0,\infty \right)$ uniquely by$\mathrm{u}\left(t_0,\infty \right)$所激励$y_{\left(t_0,+\infty \right)}=H{u}_{\left(t_0,\infty \right)}$

Six, time invariance

1. Definition of time-invariant system

The system is time invariant: the input signal delay $\alpha$ seconds, its response is also delayed by exactly$\alpha$秒，且波形不变$\bar{u}(t)=Q_{}u(t)=u(t-a)$

2. Impulse response functions of time-invariant, linear, and relaxed systems

$g(t,t)=H\delta (t-t)=g(t-t,0)=g(t-t)$

7. Transfer function matrix

4. Regularity and strict regularity

Regular: $g(\infty )$ is a finite constant

Strictly regular: $g(\infty )=0$

summary

3.3 State variable description

If the concept of energy is used, the movement process of the system can be regarded as the transformation process of energy, so the number of state variables is equal to and only equal to the number of independent energy storage elements in the system;

3.4 Relationship between input and output description and state variable description

2. State space description derived from input and output

传递函数：$g(s)=\frac{and(s}{u(s)}=\frac{b_m{s}^m+b_{m-1}{s}^{m-1}+\dots \dots +b_1{s}^1+b_0}{s^n+{\mathrm{a}}_{\mathrm{n}-1}{s}^{n-1}+\dots \dots +a_1{s}^1+a_{0}s}$

状态变量：$\begin{array}{l}\dot{x}=\left[\begin{array}{lrrr}0& 1& & \\ ⋮& & \ddots & \\ 0& & & 1\\ -a_0& -a_1& & -a_{n-1}\end{array}\right]x+\left[\begin{array}{l}0\\ ⋮\\ 0\\ 1\end{array}\right]u\\ y=\left[b_0,\cdots ,b_m,0,\cdots ,0\right]x\end{array}$

三、由方框图描述导出状态空间描述

1、化简方框图成规范方框图（各组成环节只为一阶惯性环节和比例放大环节）

2、指定状态变量，列些关系方程

四、由状态空间描述导出传递函数矩阵

$\mathbf{G}(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{B}+E$

3.5 组合系统的数学描述

一、时变情形

1、组合系统的输入输出描述

两个多变量系统$S_i$由下式描述：$y_i(t)={\int }_{-\infty }^t{G}_i(t,\tau )u_i(\tau )d\tau i=1,2$

并联系统：$G(t,\tau )=G_1(t,\tau )+G_2(t,\tau )$

串联系统：$G(t,\tau )={\int }_{\tau }^t{G}_2(t,\nu )G_1(\nu ,\tau )d\nu$

反馈系统：$G(t,\tau )=G_1(t,\tau )-{\int }_{\tau }^t{G}_1(t,\tau ){\int }_{\tau }^{\nu }{G}_2(\nu ,s)G(s,t)dsdn$

2. State variable description of combined system

Second, the time-invariant situation

1. Transfer function of combined system

Parallel system: $G(S)=G_1(S)+G_2\left(S\right)$

Series system: $G\left(S\right)=G_1\left(S\right)\ast G_2\left(S\right)$

反馈系统：$G(s)=G_1(s){\left(I_p+G_2(s)G_1(s)\right)}^{-1}={\left(I_q+G_1(s)G_2(s)\right)}^{-1}{G}_1(s)$（其中$det\left(I_q+G_1(s)G_2(s)\right)\ne 0$）

注意：$\det \left(I+E_1{E}_2\right)\ne 0\rightharpoonup det\left(I_q+G_1(s)G_2(s)\right)\ne 0$

2、适定性问题

判别系统是正则的：

传递函数矩阵中的$\left[{\mathrm{I}}_{\mathrm{q}}+{\mathrm{G}}_1(\infty ){\mathrm{G}}_2(\infty )\right]$是非奇异的，则$G(s)=G_1(s){\left(I_p+G_2(s)G_1(s)\right)}^{-1}$是正则的

注意：

这里$\left[{\mathrm{I}}_{\mathrm{q}}+{\mathrm{G}}_1(\infty ){\mathrm{G}}_2(\infty )\right]$和$\left[{\mathrm{I}}_{\mathrm{q}}+{\mathrm{G}}_2(\infty ){\mathrm{G}}_1(\infty )\right]$非奇异与状态变量描述时的条件$\left(I+E_1{E}_2\right)$非奇异是相同的

组合系统是适定的：

每个子系统是正则的，从任意作为输入端的点至沿着邮箱路径的每一个其他的闭环传递函数存在且正则

举例：

判别条件：当且仅当系统不具有在$s=\infty$时其纯回路增益为"1"的组合回路（不会有重合顶点），系统才是适定的。

四、线性动态方程和脉冲响应矩阵

4.2 线性动态方程的解

The solution of the input-output description equation: $y(t)={\int }_{t_0}^{t}G(t,\tau )u\left(\tau \right)d\tau$

1. The solution of the homogeneous equation

Existence and uniqueness of solutions: From the equation of the initial value condition, there can only be one solution that satisfies

$\dot{Ps}(t)=\mathbf{A}(t)\mathbf{\Psi }(t)$

方程$\frac{dx}{dt}=The\; set\; of\; all\; solutions\; of\; A\left(t\right)x$ forms an n-dimensional vector space over the field of real numbers

2. Fundamental Matrix and State Transition Matrix

1. Basic Matrix

Definition (not unique):

$\left[\begin{array}{llll}{Ps}^1(t)& {Ps}^2(t)& \cdots & {Ps}^n(t)\end{array}\right]=\Psi \left(t\right),t\in (-\infty ,+\infty )$ whereΨ${Ps}^i(t)$是$\dot{x}(t)=$n linear independent solutions of$A$$($$t$$)$$x$$($$t$$)$

nature:

方程$\dot{x}\left(t\right)=The\; fundamental\; matrix\; of\; A\left(t\right)x\left(t\right)$ for$(-\infty ,+\infty )$ where all t are non-singular matrices

2. State transition matrix

definition:

$\Psi (t)$是$\dot{x}\left(t\right)=\mathbf{A}(t)x(t)$的基本矩阵，则$\mathbf{\Phi }\left(t,t_0\right)=\mathbf{\Psi }(t){\mathbf{\Psi }}^{-1}\left(t_0\right)$是对应的状态转移矩阵

主要性质：

①$\mathbf{\Phi }(t,t)=\mathbf{I}$

②$\frac{d\mathbf{\Phi }\left(t,t_0\right)}{dt}=\mathbf{A}(t)\mathbf{\Phi }\left(t,t_0\right)$（利用了$\dot{\mathbf{\Psi }}(t)=\mathbf{A}(t)\mathbf{\Psi }(t)$）

③$\mathbf{x}(t)=\mathbf{\Phi }\left(t,t_0\right){\mathbf{x}}_0$（$\mathbf{x}(t)=\mathbf{\Psi }(t)\mathbf{\alpha }$表示）

④$\mathbf{\Phi }\left(t,t_0\right)$由$A(t)$唯一确定，与$\Psi (t)$无关

三、非齐次方程的解

1、时变线性系统的解

根据$\begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}(t)\mathbf{x}+\mathbf{B}(t)\mathbf{u}\\ \mathbf{x}\left(t_0\right)={\mathbf{x}}_0\end{array}$The solution to the state equation can be found as $\mathbf{x}(t)=Phi\left(t,t_0\right){\mathbf{x}}_0+{\int }_{t_0}^t\Phi (t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau$

其中$Phi\left(t,t_0\right)x_0$是零内容电影，${\int }_{t_0}^t\Phi (t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau$ is the zero state response

2. Input-output relationship

动态方程的输出为：$y(t)=C(t)x(t)+E(t)u(t)=C\left(t\right)(F\left(t,t_0\right){\mathbf{x}}_0+{\int }_{t_0}^t\Phi (t,\tau \left)B\right(\tau \left)u\right(\tau \left)d\tau \right)+E(t)u(t)$

When $x(t_0)=0$ , the impulse response matrix can be obtained:

$t\ge t:\mathbf{G}(t,t)=\mathbf{C}(t)\Phi (t,t)B\left(t\right)+E\left(t\right)d(t-t)$

$t<t:\mathbf{G}(t,t)=0$

3. Solutions of linear time-invariant dynamic equations

The fundamental matrix becomes: $e^{At}$ , the state transition matrix becomes: $\mathbf{\Phi}\left(t- t_{0}\right) $

At this time, the solution of the dynamic equation is:

$\mathbf{x}(t)=e^{\mathbf{A}\left(t-t_0\right)}{\mathbf{x}}_0+{\int }_{t_0}^t{e}^{\mathbf{A}(t-\tau )}\mathbf{Bu}(\tau )d\tau$

$\mathbf{y}(t)=\mathbf{C}{e}^{\mathbf{A}\left(t-t_0\right)}{\mathbf{x}}_0+{\int }_{t_0}^t\mathbf{C}{e}^{\mathbf{A}(t-\tau )}\mathbf{Bu}(\tau )d\tau +E\mathbf{u}(t)$

对应的脉冲响应矩阵变为：

$t\ge \tau \mathbf{G}(t-t)=\mathbf{C}{e}^{A(t-\tau )}B+E\delta (t-t)$

$t<t\mathbf{G}(t,t)=0$

4. Properties and methods of linear time-invariant fundamental matrix

性质：${\lim }_{t\to 0}{e}^{\mathbf{A}t}=I$、$\frac{d}{dt}{e}^{At}=A{e}^{At}=e^{\mathbf{A}t}A$

Finding method: use $andI-The\; inverse\; of\; A$ to solve

4.3 等价动态方程

一、时不变系统的等价动态方程

①由$\overline{\mathbf{x}}=\mathbf{Px}$产生的$\overline{\mathbf{A}}=\mathbf{PA}{\mathbf{P}}^{-1}$、$\overline{\mathbf{B}}=\mathbf{PB}$、$\overline{\mathbf{C}}={\mathbf{CP}}^{-1}$、$\bar{E}=E$。其中P为非奇异矩阵

零状态等价：相同的脉冲响应矩阵和传递函数矩阵

零输入等价：相同的零输入响应

②两个动态方程维数和传递函数阵相同$\leftharpoonup $动态方程等价$\rightharpoonup $零输入和零状态等价

③两个维数不一定相同的线性时不变动态方程具有相同的传递函数的充要条件是：$\bar{E}=E$、${\mathbf{CA}}^{\mathbf{i}}\mathbf{B}=\overline{\mathbf{C}}\overline{{\mathbf{A}}^{\mathbf{i}}}\overline{\mathbf{B}}$

二、时变系统的等价动态方程

由$\overline{\mathbf{x}}(t)=\mathbf{P}(t)\mathbf{x}(t)$产生的$\overline{\mathbf{A}}(t)=[\mathbf{P}(t)\mathbf{A}(t)+\dot{\mathbf{P}}(t)]{\mathbf{P}}^{-1}(t)$、$\overline{\mathbf{B}}(t)=\mathbf{P}(t)\mathbf{B}(t)$、$\overline{\mathbf{C}}(t)=\mathbf{C}(t){\mathbf{P}}^{-1}(t)$、$\overline{E(t)}=E(t)$

三、经等价变换之后的基本矩阵和状态转移矩阵

基本矩阵：$\mathbf{P}(t)\Psi (t)$ 状态转移矩阵：$\mathrm{P}(t)\Phi \left(t,t_0\right){\mathrm{P}}^{-1}\left(t_0\right)$

四、具有周期的线性时变动态方程

4.4 脉冲响应矩阵与动态方程

一、由动态方程到输入/输出描述

在$x(t_0)=0$时，dynamic equation的解的$y(t)=C(t){\int }_{t_0}^t\Phi (t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau +E(t)u(t)={\int }_{t_0}^t(\mathbf{C}(t)\Phi (t,t\left)B\right(t)+E(t\left)d\right(t-\tau ))u\left(\tau \right)d\tau$

so:

$t\ge t:\mathbf{G}(t,t)=\mathbf{C}(t)\Phi (t,t)B\left(t\right)+E\left(t\right)d(t-t)$

$t<t:G(t,t)=0$

2. From input/output description to dynamic equation (time-varying system)

If and only if $\mathbf{G}(t,\tau )$ decompose into$G(t,t)=M\left(t\right)N\left(\tau \right)+E\left(t\right)d(t-\tau )$ shows that the state transition matrix can be realized by finite-dimensional dynamic equations.

3. From input/output description to dynamic equation (time-invariant system)

When $When\; G\left(s\right)$ is a regular rational function matrix, it means that$G\left(s\right)$ can be realized by finite-dimensional linear dynamic equations

5. Controllability and Observability of Linear Dynamic Equations

5.1 Introduction

Admissibility control: defined in $[t_0,+\infty )$ control vectors composed of continuous or piecewise continuous functions

5.2 Linear independence of time functions

2. Gram matrix

1. Definition

$\mathbf{W}{\left(t_1,t_2\right)}_{n\times n}={\int }_{t_1}^{t_2}\mathbf{F}(t){\mathbf{F}}^{\ast }\left(t\right)dt$where F matrix is ​​composed of$f_1,f_2...f_n$(1×p dimension) matrix

f 1 , f 2 . . $f_1,f_2...f_n$（1×p维）线性无关的充要条件是$\mathbf{W}{\left(t_1,t_2\right)}_{n\times n}={\int }_{t_1}^{t_2}\mathbf{F}(t){\mathbf{F}}^{\ast }(t)dt$非奇异

二、一些有用的判别准则

①

先求矩阵各阶导，然后找存在点来判断是否线性无关（n-1阶）

缺点：很难找到某个点来满足条件，所以是充分条件

②解决的问题：难找点

先找任意点，然后求导来判断线性无关（后面由无穷多个，直到找到n阶）

5.3 线性动态方程的可控性

一、可控性的定义及判别定理

1、定义

可控定义：在$t_1$时刻把$x(t_0)$转移到0

可达定义：在$[t_0,t_1]$ make the state$x(t_1)=0$ transferred to$x\left(t_0\right)$

2. Controllable general criteria

Equation of state at $t_0$Necessary and sufficient conditions for controllability: $\Phi (t_0,\tau )\; The\; n\; rows\; ofB(\tau )$ are linearly independent in time

3. From $x\left(t_0\right)$ to$x\left(t_1\right)$ input$u(t)$：

$u(t)=-{\mathbf{B}}^{\ast }(t){\mathbf{\Phi }}^{\ast }\left(t_0,t\right){\mathbf{W}}^{-1}\left(t_0,t_1\right)\left[x_0-\mathbf{\Phi }\left(t_0,t_1\right)x_1\right]$

4、可控性的实用判据

解决的问题：求状态转移矩阵太难了

状态方程在$t_0$可控的条件：$rank[M_0(t),M_1(t),M_2(t),...,M_{n-1}(t)]=n$

$M_0(t)=B(t)$、$M_k\left(t\right)=-A(t)M_{k-1}\left(t\right)+\frac{\mathrm{d}{M}_{k-1}(t)}{\mathrm{d}t}$

n-dimensional equation in $(-\infty ,+\infty )$ is differentially controllable for every t:$rank[M_0(t_0),M_1(t_0),M_2(t_0),...,M_{n-1}(t_0),...]=n$

2. Time-invariant system controllability criterion

1. Seven criteria: AB criterion, PBH criterion

3. The vibration shape (mode) and mode of the time-invariant system

1. Definition

Mode shapes (modes): eigenvalues ​​of the matrix A

Mode: $\dot{x}=Ax+Bu$

Controllable vibration type: PBH criterion

4. Simplified controllability conditions

If $rankB=r$ , 则$(A,B)$ The necessary and sufficient condition for controllability is$rank{U}_{n-r}=rank[BAB...A^{n-r}B]=n$

The controllability index is: j+1

5.4 Observability of linear dynamic equations

1. Observability definition

Definition: Input and output can uniquely determine the initial state $x(t_0)$

2. General criterion of observability

① The state equation is at $[t_0,t_1]$ Controllable necessary and sufficient conditions:$C(t)\Phi (t,t_0)$的n个列在时间内线性无关

②

其中：$N_k(t)=N_{k-1}(t)A(t)+\frac{\mathrm{d}{N}_{k-1}(t)}{\mathrm{d}t}$、$N_0(t)=C(t)$

③n维方程在$(-\infty ,+\infty )$对每一个t均微分可观的充要条件是：

三、可重构性

可重构性是通过过去信息判断现在状态，而可观测性是用未来的信息判别现在的状态

判别方法：

$C\left(\tau \right)\Phi (\tau ,t_0)$ is linearly independent, that is,${\int }_{t_1}^{t_2}{Phi}^{\ast }(\tau ,t_0)C^{\ast }\left(\tau \right)C\left(\tau \right)\Phi (\tau ,t_0)dt$ non-singular

4. Duality of linear systems

$\left(I\right)$ at$t_0$Controllable (reachable) $⟺$ $\left(II\right)$ At$t_0$k observable (reconfigurable)

$\left(I\right)$ at$t_0$Observable (reconfigurable) $⟺$ $\left(II\right)$ At$t_0$k controllable (reachable)

5. Observability Criterion for Linear Time-Invariant Systems

Commonly used: PBH criterion, rank criterion, Gram non-singularity, column linear independence

5.5 Gauge decomposition of linear time-invariant systems

1. The nature of equivalent transformation

Under any equivalent transformation, the controllability and observability invariance of linear time-invariant systems

2. Decomposition of dynamic equations according to controllability

1. Definition

The canonical form after controllable decomposition transformation:

The dynamic equation of the controllable part has the same transfer function matrix as the original system (does not reflect the uncontrollable part)

2. Steps

① Column controllability matrix: $U=rank[BAB...A^{n-1}B]=n_1$

② Take $n_1$A linearly independent column vector, and then add $n-n_1$A linearly independent column vector, forming $P^{-1}$

③Do transformation: $\overline{\mathbf{A}}=\mathbf{PA}{\mathbf{P}}^{-1}$、$\overline{\mathbf{B}}=PB$、$\overline{\mathbf{C}}={\mathbf{CP}}^{-1}$、$\bar{E}=E$

3. Observability Decomposition of Dynamic Equations

1. Definition

The canonical form after observable decomposition transformation:

The dynamic equation of the observable part has the same transfer function matrix as the original system (does not reflect the unobservable part)

2. Steps

① Column controllability matrix: $U=rank[CCA...C{A}^{n-1}{]}^T=n_2$

② Take $n_2$A linearly independent row vector, and then add $n-n_2$linearly independent column vectors, forming $P$

③Do transformation: $\overline{\mathbf{A}}=\mathbf{PA}{\mathbf{P}}^{-1}$、$\overline{\mathbf{B}}=PB$、$\overline{\mathbf{C}}={\mathbf{CP}}^{-1}$、$\bar{E}=E$

4. Gauge Decomposition Theorem

Judging the controllable and observable states in the state variables by observing the B and C matrices

Notice:

The state behavior of these parts that do not appear in the transfer function matrix will inevitably affect the stability and quality of the system, which we should pay special attention to in system design.

5. Irreducible dynamic equations

1. Definition

If there is no system with the same transfer function matrix but lower dimension, it is irreducible

2、

The necessary and sufficient condition for irreducibility is that the dynamic equation is controllable and observable

If there are two different realizations of dynamic equations for the same transfer function matrix, then there is a transformation relationship $P$ matrix

5.6 Controllability and Observability of Jordanian Dynamic Equations

Glossary:

**Controllability criterion: **Linear independence

**Observability criterion: **Column is linearly independent

Note: When there is only one block, it only needs to be not all 0

5.7 Output controllability

1. Definition

There is an output $y(t_0)=0$ transferred to$y\left(t_1\right)=y_1$Input $u[t_0,t_1]$ , then the system is said to be at$t_0$is output controllable

2. Criterion

Time-varying: continuous impulse response matrix $G(t,\tau )$ all rows are linearly independent

Time invariant: $[CBCAB...C{A}^{n-1}B]$full rank

5.8.4 Maintain controllable and observable conditions after time discretization of continuous systems

$G=e^{AT}$、$H={\int }_0^T{e}^{AT}dtB$

A sufficient condition for a time-discretized system to remain controllable and observable is:

Sampling period value, for everything satisfying $Re[l_i-l_j]=For\; 0$ functions, the function$\mathrm{T}\ne \frac{2l}{\mathrm{Im}\left(l_i-l_j\right)}$

Six, can not be reduced to achieve

6.1 Introduction

System implementation: with a specified rational transfer function matrix G ( s ) G(s)The linear time-invariant dynamical equations (infinitely many) of G ( s ) are called$G$$($$s$$)$ G(s)Realization of$G$$($$s$$)$

Realization in standard form: a set of formulas for realization in standard form

Minimal Realization: Irreducible Dynamic Equation Realization

6.2 Characteristic polynomials and degrees of regular rational matrices

The dynamic equation realization is an irreducible condition: $det(pI-A)=k$（g(s)的分母）,$dim(A)=deg\left(g\right(s))$

$Degree\; of\; G\left(s\right)$ : regular rational matrixG ( s ) G(s)The degree of the least common denominator (characteristic polynomial) of all subforms of$G$$($$s$$)$

6.3 Irreducible Realization of Regular Rational Functions

1. Univariate system

1. Realization of controllable standard form

Transfer Function:

$g(s)=\frac{y(s)}{u(s)}=\frac{{\beta }_1{s}^{n-1}+{\beta }_2{s}^{n-2}+\dots \dots +{\beta }_{n-1}{s}^1+{\beta }_n}{s^n+{\mathrm{a}}_1{s}^{n-1}+\dots \dots +a_{n-1}{s}^1+a_{n}s}$

$\begin{array}{rl}\dot{x}& =\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& & 1& & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& 1\\ -a_n& -a_{n-1}& -a_{n-2}& \cdots & -a_1\end{array}\right]x+\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 1\end{array}\right]u\\ y& =\left[\begin{array}{lllll}{b}_n& b_{n-1}& \cdots & b_2& b_1\end{array}\right]x+eu\end{array}$

2. Steps to find the controllable standard form

① Calculate the controllability matrix $U=rank[BAB...A^{n-1}B]$

(2) Calculation $U^{-1}$ , and take the last row h

Three-dimensional $\mathbf{P}={\left[\begin{array}{c}\mathbf{h}\\ \mathbf{hA}\\ \mathbf{h}{\mathbf{A}}^2\\ ⋮\\ h{A}^{n-1}\end{array}\right]}_{n\times n}$

④ $\overline{\mathbf{A}}=\mathbf{PA}{\mathbf{P}}^{-1}$、$\overline{\mathbf{B}}=PB$、$\overline{\mathbf{C}}={\mathbf{CP}}^{-1}$

3. Realization of considerable standard form

4. Steps to find the considerable standard form

①Find the observable matrix to judge that it is observable. If it is observable, it can be transformed into an observable standard form

② Write the dual system of the original system $(A\to A^T)、(b\to c^T)、(c\to b^T)$

③Do controllable standard form conversion ${\overline{A}}_1=\mathbf{P}{\mathbf{A}}^{\mathbf{T}}{\mathbf{P}}^{-1}$、${\overline{\mathbf{b}}}_1=\mathbf{P}{\mathbf{c}}^T$、${\overline{\mathbf{c}}}_1={b^{T}P}^{-1}$

2. The standard form of a multivariate system

1. Luenberger controllable canonical form

2. Transformation steps

① Assuming that column B is full rank, list the controllability matrix $U=rank[BAB...A^{n-1}B]$

②Select n linearly independent vectors from left to right

3-dimensional $P^{-1}=\left[{\mathbf{b}}_1\mathbf{A}{\mathbf{b}}_1\cdots {\mathbf{A}}^{m_1-1}{\mathbf{b}}_1{\mathbf{b}}_{2}A{b}_2\cdots A^{m_2-1}{b}_2\cdots b_{p}A{b}_p\cdots A^{m_p-1}{b}_p\right]$

④ Calculate $P_1$, with $h_1$means $P_1$Row $\mu_1$ of

⑤构造变换阵$P_2=\left[\begin{array}{c}{h}_1\\ h_{1}A\\ ⋮\\ h_1{A}^{m_1-1}\\ h_2\\ ⋮\\ h_2{A}^{m_2-1}\\ ⋮\\ h_p\\ ⋮\\ h_p{A}^{m_P-1}\end{array}\right]$

⑥Do a series of transformations $\overline{A}=P_2\mathbf{A}{\mathbf{P}}_2^{-1}$、$\overline{\mathbf{B}}=P_{2}B$、$\overline{\mathbf{C}}={\mathbf{CP}}_2^{-1}$

3. Minimal Realization of Regularized Rational Transfer Function

1. Minimum order realization of controllable canonical form

2. Minimum order realization of observable canonical form

3. Implementation steps by Hankel matrix

①观察传递函数$g(s)=\frac{y(s)}{u(s)}=\frac{{\beta }_0{s}^n+{\beta }_1{s}^{n-1}+b_2{s}^{n-2}+\dots \dots +b_{n-1}{s}^1+b_n}{s^n+{\mathrm{a}}_1{s}^{n-1}+\dots \dots +a_{n-1}{s}^1+a_{n}s}$

② Set of formulas: use iterative method to divide polynomials to find coefficients, see example

③ Set matrix:

④Use $aH(n+1,n)=0$ to determine the last layer coefficient$a$

4. If the standard form is realized

see examples

6.4 Realization of multivariable systems

1. The relationship between the controllability and observability of the dynamic equation and the transfer function matrix

If G ( s ) G(s)There is no non-constant common factor between the numerator and denominator of$G$$($$s$$)\; ,\; then\; the\; system\; is\; controllable\; and\; observable$

The necessary and sufficient condition for the system to be observable and controllable is that the pole polynomial of G(s) is equal to the characteristic polynomial of A

Second, the realization of the vector transfer function

1. The row denominator expands considerably in standard form

2. The controllable canonical form of column denominator expansion

3. Realization of transfer function matrix G(s)

1. Column expansion steps: (controllable standard implementation)

2. Row expansion steps: (observable standard implementation)

Notice:

1. The N of the B matrix is ​​written from the constant item

2. If it is uncontrollable later, a controllable decomposition transformation is required, and then a controllable and considerable part must be left

3. Jordan-shaped expansion

7. State Feedback and State Observer

7.2 Status Feedback

1. State feedback has a considerable impact on controllability

State feedback does not change the controllability of the system, but may change the observability of the system

2. The impact of output feedback on controllable and considerable

For continuous linear time-invariant systems, output feedback can keep the system controllable and observable

3. Pole Configuration of Single Input System

Condition: The mode that the system needs to change is controllable

calculation steps:

①计算$det(pI-(A+bK))$

② Calculate the expected polynomial of the given eigenvalues

③ equal to solve the equation

Block diagram:

Fourth, the influence of state feedback on the zero point of transfer function

The state feedback can move the pole, which generally does not affect the zero point, but there may be zero-pole cancellation (affecting the zero-pole)

Seven, calm problem

Judgment condition: the eigenvalues ​​of the system all have negative real parts (controllable systems or uncontrollable modes of uncontrollable systems have negative real parts)

7.3 State Observer

2. The existence of the state observer, n is the state observer

Condition: the system is observable

3. State Observer for Single-Input-Single-Output System

① calculusdet $det(pI-A)$ and the desired characteristic polynomial

2-dimensional P function $\left[\begin{array}{ccccc}{a}_{n-1}& a_{n-2}& \cdots & a_1& 1\\ a_{n-2}& & & 1& \\ ⋮& & ⋮& & \\ a_1& 1& & 0& \\ 1& & & & \end{array}\right]\left[\begin{array}{c}\mathbf{c}\\ \mathbf{cA}\\ ⋮\\ {cA}^{n-1}\end{array}\right]$

③列$\bar{l}=\left[\begin{array}{c}{\bar{a}}_n-a_n\\ {\bar{a}}_{n-1}-a_{n-1}\\ ⋮\\ ⋮\\ {\bar{a}}_1-a_1\end{array}\right]$Then bring it into $l=P^{-1}\bar{l}$

④求的$\dot{\hat{x}}=(A-lC)\hat{x}+bu+ly$

The block diagram is as follows:

5. Minimum dimensional state observer

7.4 Connection of state feedback and state observer

1. The composition of the state feedback system including the observer

Conditions: the system is controllable and considerable

① Estimator: $\dot{\hat{x}}=(A-lC)\hat{x}+bu+ly$

②State feedback: $u=r+K\hat{x}$

组合：$\dot{\hat{x}}=(A-lC+bK)\hat{x}+br+ly$

2. Characteristics of State Feedback Systems Including Observers

1. The dimension of the combined system: 2n

2. The dynamic equation after equivalent transformation:

It can be seen that the estimator does not affect the transfer function of the system

3. Separation principle

Separation principle: If the system (A, B, C) is controllable and observable, for a state feedback system including an estimator, the design of the state feedback law and the design of the estimator can be carried out independently

8. Stability of linear systems

8.1 Lee Stability

Stable condition: $det(pI-A)=The\; characteristic\; roots\; of\; 0$ are all in the negative real part

special:

where when having $s=$When the repeated root of$0$$n-rank(l_{i}I-A)$ ) when stable

Requirements:

Lee Stability: $det(pI-A)$ The elementary factor corresponding to the root whose real part is zero is once (or the geometric multiplicity is equal to the algebraic multiplicity), and the remaining roots have negative real parts

Asymptotically stable: $det(pI-$All roots of$A$$)\; have\; negative\; real\; parts$

8.2 Stability Analysis of Linear Time-Invariant Systems

Controlled Decomposition Standard Type:

$\begin{array}{c}\dot{\bar{x}}=\left[\begin{array}{cc}{\overline{\mathbf{A}}}_c& {\overline{A}}_{12}\\ 0& {\overline{A}}_{\bar{c}}\end{array}\right]\bar{x}+\left[\begin{array}{c}{\overline{\mathbf{B}}}_c\\ 0\end{array}\right]u\\ y=\left[{\overline{\mathbf{C}}}_c{\overline{C}}_{\bar{c}}\right]\bar{x}+\mathbf{E}u\end{array}$

Observable decomposed standard form:

$\begin{array}{c}\dot{\overline{\mathbf{x}}}=\left[\begin{array}{cc}{\overline{\mathbf{A}}}_o& 0\\ {\overline{A}}_{21}& {\overline{A}}_{\bar{o}}\end{array}\right]\overline{x}+\left[\begin{array}{l}{\overline{\mathbf{B}}}_o\\ {\overline{B}}_{\bar{o}}\end{array}\right]u\\ y=\left[\begin{array}{ll}{\overline{\mathbf{C}}}_0& 0\end{array}\right]\bar{x}+\mathbf{E}u\end{array}$

State, output formula:

$x(t)=\Phi (t,t_0)x_0+{\int }_{t_0}^t\Phi (t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau$

$y(t)=C(t)\Phi (t,t_0)x_0+{\int }_{t_0}^{t}C\left(t\right)\Phi (t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau$

Judging conditions:

BIBS is stable ( $x(t_0)=0$）$⟺Controllable$ partial convergence

BIBS full stability ( $x\left(t_0\right)$ optional)$⟺The$ controllable part converges, and the uncontrollable part does not diverge

BIBO is stable ( $x(t_0)=0$）$⟺Controllable$ and considerable partial convergence

BIBO is fully stable ( $x\left(t_0\right)$ optional)$⟺The$ controllable and appreciable part converges, and the appreciable and uncontrollable part does not diverge

3. Overall stability (T stable)

T stable = BIBS fully stable

relation chart