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笔者带更新-运动学
课程主讲教师：
Prof. Wei Zhang

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This lecture introduces basic concepts and results on Lyapunov stability of nonlinear systems

1. Background

1.1 What is Stability Analysis

• system asymptotic/ˌæsimp'tɔtik,-kəl/ 渐进的 behavior (not too much about transient/'trænzɪənt/短暂的)
• ability to return to the desired asymptotic behavior (nout just convergence)

示例-小球平衡系统与反馈控制

1.2 General ODE Models for Dynamical Systems

• ODE: $\dot{x}=f\left(x,u\right)$ , with $x\left(0\right)=x_0$
  $x\in \mathcal{X}\subseteq {\mathbb{R}}^n$ : state
  $u\in \mathcal{U}\subseteq {\mathbb{R}}^m$ : control input
  $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to {\mathbb{R}}^n$ : (time-invariant) vector field
• System output $y=g\left(x,u\right)$
• (static)Control law : $\mu :\mathcal{X}\to \mathcal{U}$ , $u=\mu \left(x\right)$
  
• Closed-loop dynamics under $\mu$ : $\dot{x}=f\left(x,\mu \left(x\right)\right)$ $\Rightarrow \dot{x}=f_{\mathrm{cl}}\left(x\right)$
• Autonomous system :
  $\dot{x}=f\left(x,u\right)$ , with $x\left(0\right)=x_0$

1.3 Example

1.3.1 Pendulum

1.3.2 Adaptive Control

Closed-loop dynamics under adaptive control:
$$\{\begin{array}{l}\dot{y}=y+u\\ u=-ky,\dot{k}=y^2\end{array}$$

1.4 Equilibrium Point of Dynamical Systems

Definition 1 (Equilibrium Point) - 平衡点
A state $x^{\ast }$ is an equilibrium point of system (1) if once $x\left(t\right)=x^{\ast }$ , it remains equal to $x^{\ast }$ at all future time.

• Mathematically : $f\left(x^{\ast }\right)=0$
• e.g. undamped pendulum with no driving force: $f\left(x\right)=\dot{x}$ velocity
  $\dot{x}=\left[\begin{array}{c}{x}_2\\ \frac{g}{l}\cos x_1\end{array}\right]=0\Rightarrow \{\begin{array}{l}{x}_2=0\\ \cos x_1=0,x_1=\frac{2k\pi +\pi }{2},k\in \mathbb{Z}\end{array}$

1.5 Invariant Set of Dynamical Systems

Definition 2 (Invariant Set) - 不变集
A set $E$ is an invariant set of system (1) if every trajectory which starts from a point $E$ remains in $E$ at all future time.

• Mathematically : If $x\left(t_0\right)\in E$ , then $x\left(t\right)\in E,\forall t⩾t_0$
• e.g. Closed-loop dynamics under adaptive control :
  
  $f\left(x\right)=\dot{x}=\left[\begin{array}{c}{x}_1-x_1{x}_2\\ {x_1}^2\end{array}\right]\Rightarrow \{\begin{array}{l}{x}_1=0\\ x_2=arbitrary\end{array}\Rightarrow E=\left\{x\in {\mathbb{R}}^2,x_1=0\right\}$

2. Lyapunov Stability Definitions

Stability :

1. about equilibrium
2. ability to stay close or return to equilibrium

2.1 Lyapunov Stability Definitions

Consider a time-invariant autonomous (with no control) nonlinear system : (on closed-loop system : $\dot{x}=f\left(x,u\left(x\right)\right)=f_{\mathrm{cl}}\left(x\right)$)
$\dot{x}=f\left(x\right),x\in {\mathbb{R}}^n$ , with I.C. $x\left(0\right)=x_0$
$f\left(x\right)$ - vector field

• Assumption :
  (i) $f$ Lipshitz continuous —— existence & uniqueness of ODE
  (ii) origin is an isolated equilibrium $f\left(0\right)=0$ —— $f\left(x^{\ast }\right)=0$
  If equilibrium $x^{\ast }$ is not at the origin define $\tilde{x}=x-x^{\ast }$ , $\dot{\tilde{x}}=\dot{x}-0=f\left(\tilde{x}+x^{\ast }\right)$
• Stability Definitions :

1. The equilibrium $x=0$ is called stable(stay close to equilibrium) in the sense of Lyapunov , if
   $ϵ-\delta$ argument —— $\forall ϵ>0,\exists \delta >0,s.t.\Vert x\left(0\right)\Vert ⩽\delta \Rightarrow \Vert x\left(t\right)\Vert ⩽ϵ,\forall t⩾0$
   
   Objective: For any $ϵ>0$ , ensure $\Vert x\left(t\right)\Vert ⩽ϵ$ for all $t$
   our choice : selecting initial state $x\left(0\right)$
   stability : objective can be ensure by choosing I.C. sufficiently small

2. asymptotically stable (stay close + convergence) if it is stable and $\delta$ can be chosen so that
   $\Vert x\left(0\right)\Vert ⩽\delta \Rightarrow \Vert x\left(t\right)\Vert \to 0$ as $t\to \infty$ (convergence)

3. exponentially stable if there exist positive constants $\delta ,\lambda ,c$ such that
   $\Vert x\left(t\right)\Vert ⩽c\Vert x\left(0\right)\Vert e^{-\lambda t},\forall \Vert x\left(0\right)\Vert ⩽\delta$
   

4. globallt asymptomtotically / exponentially stable (G.A.S / G.E.S) if the above conditions holds for all $\delta >0$

5. Region of Attraction - 吸引域 : R A ≜ { x ∈ R n : w h e v e r x ( 0 ) = x , t h e n x ( t ) → 0 } R_A\triangleq \left\{ x\in \mathbb{R} ^n:whever\,\,x\left( 0 \right) =x,then\,\,x\left( t \right) \rightarrow 0 \right\} RA​≜{x∈Rn:wheverx(0)=x,thenx(t)→0}
   Globaly asymptotically stable $R_A\triangleq {\mathbb{R}}^n$

2.2 Stability Examples using 2D Phase Portrait

• Undamped pendulum with no driving force :
  $\dot{x}=\left[\begin{array}{c}{x}_2\\ \frac{g}{l}\cos x_1\end{array}\right]=0\Rightarrow \{\begin{array}{l}{x}_2=0\\ \cos x_1=0,x_1=\frac{2k\pi +\pi }{2},k\in \mathbb{Z}\end{array}$
  
• Closed-loop dynamics under adaptive control :
  $f\left(x\right)=\dot{x}=\left[\begin{array}{c}{x}_1-x_1{x}_2\\ {x_1}^2\end{array}\right]\Rightarrow \{\begin{array}{l}{x}_1=0\\ x_2=arbitrary\end{array}\Rightarrow E=\left\{x\in {\mathbb{R}}^2,x_1=0\right\}$
  
• Does attractiveness implies stable in Lyapunov sense?
  Answer is No —— e.g. $\{\begin{array}{l}{\dot{x}}_1={x_1}^2-{x_2}^2\\ {\dot{x}}_2=2x_1{x}_2\end{array}$ —— asymptotically stable : 1. stable 2. convergence
  By inspection of its vector field, we see that $x\left(t\right)\to 0$ for all $x\left(0\right)\in {\mathbb{R}}^2$
  However, there is no $\delta$-ball satisfying the Lyapunov stability condition
  
  convergence but not stable

3. Lyapunov Stability Theorem

3.1 How to verify stability of a system?

• Find explicit solution of the ODE $x\left(t\right)$ and check stability definitions (typically not possible for nonlinear systems) —— e.g. $x\left(t\right)=e^{-t}{x}_0$
• Numerical simulations of ODE do not provide stability guarantees and offer limited insights
• Need to determine stability without explicitly solving the ODE
• Perferably, analysis only depends on the vector field
• The most powerful tool is : Lyapunov function
• State trajectory $x\left(t\right)$ governed by complex dynamics in ${\mathbb{R}}^n$ —— $\dot{x}\left(t\right)=f\left(x\left(t\right)\right)$
• Lyapunov function $V:{\mathbb{R}}^n\to \mathbb{R}$ maps $x\left(t\right)$ to a scalar function of time $V\left(x\left(t\right)\right)$
  scalar - $V\left(x\left(t\right)\right)\leftrightarrow \dot{V}\left(x\left(t\right)\right)=\frac{\mathrm{d}}{\mathrm{d}t}V\left(x\left(t\right)\right)=g\left(V\left(\right)\right)$ - scalar ODE
• If the function is designed such that : $\left[x\left(t\right)\to equilibrium\right]\iff \left[V\left(x\left(t\right)\right)\to 0\right]$. Then we can study $V\left(x\left(t\right)\right)$ as function of time $t$ to infer stability of the state trajectory in ${\mathbb{R}}^n$

3.2 Sign Definite Functions

Assume that $0\in D\subseteq {\mathbb{R}}^n$

• $g:D\to \mathbb{R}$ is called positive semidefinite (PSD) on $D$ if $g\left(0\right)=0$ and $g\left(0\right)⩾0,\forall x\in D$
  For quadratic function : $g\left(x\right)=x^{\mathrm{T}}Px:\left[gisPSD\right]\iff \left[PisaPSDmatrix\right]$
  e.g. : $g\left(x\right)={\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\mathrm{T}}P\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]={x_1}^2+x_1{x}_2+3{x_2}^2$
• $g:D\to \mathbb{R}$ is called positive definite (PD) on $D$ if $g\left(0\right)=0$ and g ( x ) > 0 , ∀ x ∈ D \ { 0 } g\left( x \right) >0,\forall x\in D\backslash\{0\} g(x)>0,∀x∈D\{0}
  Similarly, if $g\left(x\right)=x^{\mathrm{T}}Px$ is quadratic, then $\left[gisPD\right]\iff \left[PisaPDmatrix\right]$
• $g$ is negative semidefinite (NSD) if $-g$ is PSD
• $g:{\mathbb{R}}^n\to \mathbb{R}$ is radically unbounded if $g\left(x\right)\to \infty$ as $\Vert x\Vert \to \infty$
  

3.3 Lyapunov Stability Theorem

[Lyapunov Theorem] : Let $D\subseteq {\mathbb{R}}^n$ be a set containing an open neighborhood of the origin. If there exists a ${\mathcal{C}}^1$ (continuous differentiable) function $V:D\to \mathbb{R}$ (observable condition - e.g. $V\left(x\right)={x_1}^2,x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}0\\ 100\end{array}\right],V\left(x\right)=0isPSD$) such that
$$\{\begin{array}{l}VisPD\\ \dot{V}\left(x\right)\triangleq \nabla V{\left(x\right)}^{\mathrm{T}}f\left(x\right)isNSD\end{array}$$
the value of $V$ along sys state trajectory nonincreasing $\dot{V}\left(x\left(t\right)\right)={\left(\frac{\partial V}{\partial x}\right)}^{\mathrm{T}}\frac{\partial x}{\partial t}=\nabla V{\left(x\right)}^{\mathrm{T}}f\left(x\right),\nabla V\left(x\right)\left[\begin{array}{c}\frac{\partial V}{\partial x_1}\\ \frac{\partial V}{\partial x_2}\\ ⋮\\ \frac{\partial V}{\partial x_{\mathrm{n}}}\end{array}\right]$ , $\nabla V{\left(x\right)}^{\mathrm{T}}f\left(x\right)\triangleq Lf\left[V\right]$ Lie derivative of $V\left(\cdot \right)$ with vetor field $f$

then the origin is stable. If in addition ,
$$\dot{V}\left(x\right)\triangleq \nabla V{\left(x\right)}^{\mathrm{T}}f\left(x\right)isND$$
then the origin is asymptotically stable —— Value of $V$ along sys state trajectory is decreasing

Remarks:
A PD ${\mathcal{C}}^1$ function satisfying above equation will be called a Lyapunov function (1+2 or 1+3)
Under condition 3 , if $V$ is also radially unbounded —— globally asympotically stable (G.A.S)

3.4 Proof of Lyapunov Stability Theorem

Main idea : 1+2 —— stability

• Fact : suppose $V$ function satisfies 1+2 , then the sub level set Ω b ( V ) ≜ { x ∈ R n : V ( x ) ⩽ b } \varOmega _{\mathrm{b}}\left( V \right) \triangleq \left\{ x\in \mathbb{R} ^n:V\left( x \right) \leqslant b \right\} Ωb​(V)≜{x∈Rn:V(x)⩽b} is forward invariant
  Proof Fact : if $x\left(0\right)\in {\Omega }_{\mathrm{b}}$ fro some $b⩾0$ , we have $V\left(x\left(t\right)\right)⩽V\left(x\left(0\right)\right)⩽b$ $\Rightarrow x\left(t\right)\in {\Omega }_{\mathrm{b}}$

• Proof of stability : Given $\epsilon >0$ , goal is to find $\delta >0$, such that $\Vert x\left(0\right)\Vert ⩽\delta \Rightarrow \Vert x\left(t\right)\Vert ⩽\epsilon$
  

1. Ω b { 0 } \varOmega _{\mathrm{b}}\left\{ 0 \right\} Ωb​{0} if $b=0$ (due to P.D. of $V$)
2. As $b$ increases , level set ${\Omega }_{\mathrm{b}}$ grows in size until bitting $Ball-\epsilon$ then fix $b=\hat{b}$
3. Find $Ball-\delta$ inside ${\Omega }_{\mathrm{b}}$ (because $V$ is continuous at $0$) then $x_0\in Ball-\delta \Rightarrow x_0\in {\Omega }_{\mathrm{b}}\Rightarrow x\left(t\right)\in {\Omega }_{\mathrm{b}}$ (${\Omega }_{\mathrm{b}}$ is invariant)

Sketch of proof of Lyapunov Stability theorem:

• First show stability under condition 2

Define sublevel set Ω b = { x ∈ R n : V ( x ) ⩽ b } \varOmega _{\mathrm{b}}=\left\{ x\in \mathbb{R} ^n:V\left( x \right) \leqslant b \right\} Ωb​={x∈Rn:V(x)⩽b}. Condition 2 implies $V\left(x\left(t\right)\right)$ nonincreasing along system trajectory $\Rightarrow$ if $x_0\in {\Omega }_{\mathrm{b}}$ , then $x\left(t\right)\in {\Omega }_{\mathrm{b}}$, $\forall t$
Given arbitrary $\epsilon >0$ , if we can find $\delta ,b$ such that $B\left(0,\delta \right)\subseteq {\Omega }_{\mathrm{b}}\subseteq B\left(0,\epsilon \right)$. Then the Lyapunov stability conditions are satisfied. Below is to show how we can find such $b$ and $\delta$
$V$ is continuous $\Rightarrow$ $m={\min }_{\Vert x\Vert =\epsilon }V\left(x\right)$ exists (due to Weierstrass theorem). In addition, $V$ is PD $\Rightarrow$ $m>0$. Therefore, if we choose $b\in \left(0,m\right)$ , then ${\Omega }_{\mathrm{b}}\subseteq B\left(0,\epsilon \right)$
$V\left(x\right)$ is continuous at origin $\Rightarrow$ for any $b>0$ , there exists $\delta >0$ such that $\left|V\left(x\right)-V\left(0\right)\right|=V\left(x\right)<b,\forall x\in B\left(0,\delta \right)$ . This implies that $B\left(0,\delta \right)\subseteq {\Omega }_{\mathrm{b}}$

• Second, show asymptotic stability under condition 3:

We know $V\left(x\left(t\right)\right)$ decreases monotonically as $t\to \infty$ and $V\left(x\left(t\right)\right)⩾0$, $\forall t$. Therefore, $c={\lim }_{t\to \infty }V\left(x\left(t\right)\right)$ exists . So it suffices to show $c=0$. Let us use a contradiction argument.
Suppose $c\ne 0$. Then $c>0$. Therefore, x ( t ) ∉ Ω c = { x ∈ R n : V ( x ) ⩽ c } x\left( t \right) \notin \varOmega _{\mathrm{c}}=\left\{ x\in \mathbb{R} ^n:V\left( x \right) \leqslant c \right\} x(t)∈/Ωc​={x∈Rn:V(x)⩽c} , $\forall t$ . We can choose $\beta >0$ such that $B\left(0,\beta \right)\subseteq {\Omega }_{\mathrm{c}}$ (due to continuity of $V$ at $0$)
Now let $a=-{\max }_{\beta ⩽\Vert x\Vert ⩽\epsilon }\dot{V}\left(x\right)$. Since $V$ is ND, then $a>0$
$V\left(x\left(t\right)\right)=V\left(x\left(0\right)\right)+{\int }_0^t\dot{V}\left(x\left(s\right)\right)\mathrm{d}s⩽V\left(x\left(0\right)\right)-a\cdot t<0$ for sufficiently large $t$. $\Rightarrow$ contradiction !

3.5 Exponential Lyapunov Function

Definition 3 (Exponential Lyapunov Function) —— Important for application
$V:D\to \mathbb{R}$ is called an Exponential Lyapunov Function (ELF) on $D\subset {\mathbb{R}}^n$ if $\exists k_1,k_2,k_3,\alpha >0$ such that
$\{\begin{array}{l}{k}_1{\Vert x\Vert }^{\alpha }⩽V\left(x\right)⩽k_2{\Vert x\Vert }^{\alpha }\\ {\mathcal{L}}_{\mathrm{f}}V\left(x\right)⩽-k_{3}V\left(x\right)\end{array}$

Lyapunov stability $\exists {\mathcal{C}}^1$ func $V$
$V$ is PD - deserable ; $\dot{V}$ is ND/NSD
$k_1{\Vert x\Vert }^{\alpha }⩽V\left(x\right)⩽k_2{\Vert x\Vert }^{\alpha }$ $\Rightarrow V$ is PD (radially unbounded)
${\mathcal{L}}_{\mathrm{f}}V\left(x\right)⩽-k_{3}V\left(x\right)$ $\Rightarrow \dot{V}$ is ND, $\dot{V}⩽-k_{3}V$

Droof sketch :
recall : $z\in {\mathbb{R}}^1,\dot{z}=-k_{3}z\Rightarrow z\left(t\right)=e^{-k_{3}t}z\left(0\right)$
By comparison theorem : $\dot{V}⩽-k_{3}V\Rightarrow V\left(t\right)⩽e^{-k_{3}t}V\left(0\right)$
$\Rightarrow {\Vert x\left(t\right)\Vert }^{\alpha }⩽\frac{1}{k_1}V\left(x\left(t\right)\right)⩽\frac{1}{k_1}{e}^{-k_{3}t}V\left(x\left(0\right)\right)⩽\frac{k_2}{k_1}{e}^{-k_{3}t}{\Vert x\left(0\right)\Vert }^{\alpha }\Rightarrow {\Vert x\left(t\right)\Vert }^{\alpha }⩽c{e}^{-\beta t}{\Vert x\left(0\right)\Vert }^{\alpha }$

Theorem 1 (ELF Theorem)
If system 2 has an ELF, then it is exponentially stable

3.6 Stability Analysis Examples

• Example 1 :
  $\{\begin{array}{l}{\dot{x}}_1=-x_1+x_2+x_1{x}_2\\ {\dot{x}}_2=x_1-x_2-{x_1}^2-{x_2}^3\end{array}$ Try $V\left(x\right)={\Vert x\Vert }^2={x_1}^2+{x_2}^2$
  equilibrium : $\dot{x}=0\Rightarrow \left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$
  check Lyapunov conditions

1. $V\left(x\right)={x_1}^2+{x_2}^2=x^{\mathrm{T}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]x$ is PD and ${\mathcal{C}}^1$
2. ${\mathcal{L}}_{\mathrm{f}}V\left(x\right)={\left(\frac{\partial V}{\partial x}\right)}^{\mathrm{T}}f\left(x\right)={\left[\begin{array}{c}2x_1\\ 2x_2\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{f}_1\left(x\right)\\ f_2\left(x\right)\end{array}\right]=2x_1\left(-x_1+x_2+x_1{x}_2\right)+2x_2\left(x_1-x_2-{x_1}^2-{x_2}^3\right)=-2{\left(x_1-x_2\right)}^2-2{x_2}^4\Rightarrow ND$

$\Rightarrow$ system is asymptotically stable

• Example 2 :
  $\{\begin{array}{l}{\dot{x}}_1=-x_1+x_1{x}_2\\ {\dot{x}}_2=-x_2\end{array}$
  Can we find a simple quadratic Lyapunov function ? First try $V\left(x\right)={x_1}^2+{x_2}^2$

1. $V$ is PD
2. ${\mathcal{L}}_{\mathrm{f}}V\left(x\right)=-2\left({\left(x_2-4\right)}^2-8\right)$ Not ND

In fact the system does not have any (global polynomial Lyapunov function.) But it is GAS with a Lyapunov function $V\left(x\right)=\ln \left(1+{x_1}^2\right)+{x_2}^2$

4. Lyapunov Stability of Linear Systems

4.1 Stability of Linear Systems

Consider autonomous linear system : $\dot{x}=f\left(x\right)=Ax$

• Recall solution to the linear system : $x\left(t\right)=e^{At}x\left(0\right)$
• If isolated equilibrium only possible equilibrium is origin $x=0$
  $f\left(x\right)=0\Rightarrow Ax=0$ : 1. if $A$ is nonsingular $\Rightarrow x=0$ . 2. If $A$ is singular, ?? $A$ is the set of equilibrium
• Fact : Origin asympt. stable $\iff$ $\mathrm{Re}\left({\lambda }_{\mathrm{i}}\right)<0$ for all eigenvalues ${\lambda }_{\mathrm{i}}$ of $A$
  suppose we have isolated equilibrium $x^{\ast }=0$
  For simplicity, consider a simple case when $A$ is diagonalizable : $A=TD{T}^{-1}$ where $D=\left[\begin{array}{cccc}{\lambda }_1& & & \\ & {\lambda }_2& & \\ & & \ddots & \\ & & & {\lambda }_{\mathrm{n}}\end{array}\right]$ $\Rightarrow e^{At}=T{e}^{Dt}{T}^{-1}=T\left[\begin{array}{cccc}{e}^{{\lambda }_{1}t}& & & \\ & e^{{\lambda }_{2}t}& & \\ & & \ddots & \\ & & & e^{{\lambda }_{\mathrm{n}}t}\end{array}\right]T^{-1}$ . If $\mathrm{Re}\left({\lambda }_{\mathrm{i}}\right)<0$ , for all $i$ , every entry of $e^{At}\to 0$ $\Rightarrow e^{At}x\left(0\right)$ expenentially
• Discrete time system : $x\left(k+1\right)=Ax\left(k\right)$ os asymptotically stable if $eig\left(A\right)$ inside unit circle

4.1 Lyapunov Function of Linear Systems

• Consider a quadratic Lyapunov function candidate : $V\left(x\right)=x^{\mathrm{T}}Px$ , with $P\in {\mathbb{R}}^{n\times n}$
  V is PD $\Rightarrow P\succ 0$ ( $P$ is a PD matrix)
  ${\mathcal{L}}_{\mathrm{f}}V$ is ND $\Rightarrow$ ${\mathcal{L}}_{\mathrm{f}}V\triangleq {\left(\frac{\partial V}{\partial x}\right)}^{\mathrm{T}}Ax={\left(2Px\right)}^{\mathrm{T}}Ax=2x^{\mathrm{T}}{P}^{\mathrm{T}}Ax$ or equirelatly, $\dot{V}\left(x\left(t\right)\right)={\dot{x}}^{\mathrm{T}}Px+x^{\mathrm{T}}P\dot{x}=x^{\mathrm{T}}{A}^{\mathrm{T}}Px+x^{\mathrm{T}}PAx$ —— Is $2P^{\mathrm{T}}A=A^{\mathrm{T}}P+PA$ ? —— $x^{\mathrm{T}}{P}^{\mathrm{T}}Ax=x^{\mathrm{T}}{A}^{\mathrm{T}}Px$

$\Rightarrow V$ is LF if $P$ is PD and $A^{\mathrm{T}}P+PA$ is ND

Fact ： for Linear System , quadratic form of LF , ai all we need to consider. —— $A$ is asym stable if and only if ??

In proof of the above function , we assumed $P$ is symmetric so $P^{\mathrm{T}}A=PA$
e.g. $P^{\mathrm{T}}A=PA$ , $Q=\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right],g\left(x\right)=x^{\mathrm{T}}Qx={\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]={x_1}^2+{x_2}^2\Rightarrow {\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$ ， $\hat{Q}=\frac{1}{2}Q+\frac{1}{2}{Q}^{\mathrm{T}}$

Fact ： $A$ is asym stable if and only if

1. $\exists P\succ 0$ , such that $A^{\mathrm{T}}P+PA\prec 0$
2. equivalently , for any $Q\succ 0,\exists P$ such that $A^{\mathrm{T}}P+PA=-Q$ (Lyapunov equation)

4.2 Stability Conditions for Linear Systems

Theorem 1 (Stability Conditions for Linear System)
For an autonomous Linear system $\dot{x}=Ax$. The following statements are equivalent.

• (Linear) System is (globally) asmptotically stable
• (Linear) System is (globally) exponentially stable
• $\mathrm{Re}\left({\lambda }_{\mathrm{i}}\right)<0$ for all eigenvalues ${\lambda }_{\mathrm{i}}$ of $A$ —— lie on open left half complex plane (OLHP)
• System has a quadratic Lyapunov function $V\left(x\right)=x^{\mathrm{T}}Px$
• For ant symmetric $Q\succ 0$ , there exists a symmetric $P\succ 0$ that solves the following Lyapunov equation :
  $$A^{\mathrm{T}}P+PA=-Q$$
  $Q\succ 0$ is given , $P$ is the variale to be solved , and $V\left(x\right)=x^{\mathrm{T}}Px$ is Lyapunov function of the system

5. Converse Lyapunov Function

When there is a Lyapunov Function?

• Converse Lyapunov Theorem for Asymptotic Stability
  origin asymptotically stable ; $f$ is locallt Lipschitz on D with region of attration $R_A$ $\Rightarrow Vs.t.$ $V$ is continuuos and PD on $R_A$ ; $L_{\mathrm{f}}V$ is ND on $R_A$ ; $V\left(x\right)\to \infty$ as $x\to \partial R_{\mathrm{A}}$
  convex result that is not constructive

• Converse Lyapunov Theorem for Exponential Stability
  origin exponentially stable on $D$ ; $f$ is ${\mathcal{C}}^1$ $\Rightarrow \exists$ an ELF $V$ on $D$

• For nonlinear sys , $\exists V\Rightarrow$ stability (sufficient condition)

• Proofs are involved especially for the converse theorem for asymptotic stability

• Important : proofs of converse theorems often assume the knowledge of system solution and hence are not constructive

6. Extension of Discrete-Time System

6.1 What about Discrete Time Systems?

• So far, all our definitions, results, examples are given using continuous dynamical system models.
• All of them have discrete-time counterparts. The ideas and conclusions are the “same” (in sprit)
• For example, given autonomous discrete-time system : $x\left(k+1\right)=f\left(x\left(k\right)\right)$ with $f\left(0\right)=0$ (origin is an isolated equilibrium)
  Rate of change of a function $V\left(x\right)$ along system trajectory can be defined as : ${\Delta }_{\mathrm{f}}V\left(x\right)=V\left(f\left(x\right)\right)-V\left(x\right)\Leftarrow V\left(x\left(k+1\right)\right)-V\left(x\left(k\right)\right)$ , where $L_{\mathrm{f}}V\left(x\right)={\left(\frac{\partial V}{\partial x}\right)}^{\mathrm{T}}f\left(x\right)$
  Asymptotically stable requires : $V$ is PD(observable all the bad behavior of ‘x’ shows up in $V$) and ${\Delta }_{\mathrm{f}}V$ is ND —— ${\Delta }_{\mathrm{f}}V\left(x\right)\prec 0$ for all x ∈ R n / { 0 } x\in \mathbb{R} ^n/\left\{ 0 \right\} x∈Rn/{0}
  Exponentially stable requires : $k_1{\Vert x\Vert }^{\alpha }⩽V\left(x\right)⩽k_2{\Vert x\Vert }^{\alpha }and{\Delta }_{\mathrm{f}}V\left(x\right)⩽-k_{3}V\left(x\right)$

6.2 Concluding Remarks

• We have learned different notions of internal stability, e.g. stability in Lyapunov sense, asymptotic stability, globally asymptotic stability (G.A.S), exponential stability, globally exponential stability(G.E.S)
• Sufficient condition to ensure stability is often the existence of a properly defined Lyapunov function
• Key requirements for a Lyapunov function :
  Positive definite and is zero at the system equilibrium
  Descease along system trajectory
• For linear system : G.A.S $\iff$ G.E.S $\iff$ Existence of a quadratic Lyapunov function
• The definitions and results in this lecture have sometimes been stated in simplified form to facilitate presentation. More general version can be found in standard textbooks on nonlinear systems
• Next Lecture : Semidefinite Programming and computational stability analysis