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笔者带更新-运动学
课程主讲教师:
Prof. Wei Zhang
This lecture introduces basic concepts and results on Lyapunov stability of nonlinear systems
asymptotic/ˌæsimp'tɔtik,-kəl/ 渐进的
behavior (not too much about transient/'trænzɪənt/短暂的
)Closed-loop dynamics under adaptive control:
{ y ˙ = y + u u = − k y , k ˙ = y 2 \begin{cases} \dot{y}=y+u\\ u=-ky,\dot{k}=y^2\\ \end{cases} {
y˙=y+uu=−ky,k˙=y2
Definition 1 (Equilibrium Point) - 平衡点
A state x ∗ x^* x∗ is an equilibrium point of system (1) if once x ( t ) = x ∗ x\left( t \right) =x^* x(t)=x∗ , it remains equal to x ∗ x^* x∗ at all future time.
Definition 2 (Invariant Set) - 不变集
A set E E E is an invariant set of system (1) if every trajectory which starts from a point E E E remains in E E E at all future time.
Stability :
Consider a time-invariant autonomous (with no control) nonlinear system : (on closed-loop system : x ˙ = f ( x , u ( x ) ) = f c l ( x ) \dot{x}=f\left( x,u\left( x \right) \right) =f_{\mathrm{cl}}\left( x \right) x˙=f(x,u(x))=fcl(x))
x ˙ = f ( x ) , x ∈ R n \dot{x}=f\left( x \right) ,x\in \mathbb{R} ^n x˙=f(x),x∈Rn , with I.C. x ( 0 ) = x 0 x\left( 0 \right) =x_0 x(0)=x0
f ( x ) f\left( x \right) f(x) - vector field
The equilibrium x = 0 x=0 x=0 is called stable(stay close to equilibrium) in the sense of Lyapunov , if
ϵ − δ \epsilon -\delta ϵ−δ argument —— ∀ ϵ > 0 , ∃ δ > 0 , s . t . ∥ x ( 0 ) ∥ ⩽ δ ⇒ ∥ x ( t ) ∥ ⩽ ϵ , ∀ t ⩾ 0 \forall \epsilon >0,\exists \delta >0,s.t.\left\| x\left( 0 \right) \right\| \leqslant \delta \Rightarrow \left\| x\left( t \right) \right\| \leqslant \epsilon ,\forall t\geqslant 0 ∀ϵ>0,∃δ>0,s.t.∥x(0)∥⩽δ⇒∥x(t)∥⩽ϵ,∀t⩾0
Objective: For any ϵ > 0 \epsilon >0 ϵ>0 , ensure ∥ x ( t ) ∥ ⩽ ϵ \left\| x\left( t \right) \right\| \leqslant \epsilon ∥x(t)∥⩽ϵ for all t t t
our choice : selecting initial state x ( 0 ) x\left( 0 \right) x(0)
stability : objective can be ensure by choosing I.C. sufficiently small
asymptotically stable (stay close + convergence) if it is stable and δ \delta δ can be chosen so that
∥ x ( 0 ) ∥ ⩽ δ ⇒ ∥ x ( t ) ∥ → 0 \left\| x\left( 0 \right) \right\| \leqslant \delta \Rightarrow \left\| x\left( t \right) \right\| \rightarrow 0 ∥x(0)∥⩽δ⇒∥x(t)∥→0 as t → ∞ t\rightarrow \infty t→∞ (convergence)
exponentially stable if there exist positive constants δ , λ , c \delta ,\lambda ,c δ,λ,c such that
∥ x ( t ) ∥ ⩽ c ∥ x ( 0 ) ∥ e − λ t , ∀ ∥ x ( 0 ) ∥ ⩽ δ \left\| x\left( t \right) \right\| \leqslant c\left\| x\left( 0 \right) \right\| e^{-\lambda t},\forall \left\| x\left( 0 \right) \right\| \leqslant \delta ∥x(t)∥⩽c∥x(0)∥e−λt,∀∥x(0)∥⩽δ
globallt asymptomtotically / exponentially stable (G.A.S / G.E.S) if the above conditions holds for all δ > 0 \delta >0 δ>0
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 ≜ R n R_A\triangleq \mathbb{R} ^n RA≜Rn
Assume that 0 ∈ D ⊆ R n 0\in D\subseteq \mathbb{R} ^n 0∈D⊆Rn
[Lyapunov Theorem] : Let D ⊆ R n D\subseteq \mathbb{R} ^n D⊆Rn be a set containing an open neighborhood of the origin. If there exists a C 1 \mathcal{C} ^1 C1 (continuous differentiable) function V : D → R V:D\rightarrow \mathbb{R} V:D→R (observable condition - e.g. V ( x ) = x 1 2 , x = [ x 1 x 2 ] = [ 0 100 ] , V ( x ) = 0 i s P S D 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) =0 is\,\,PSD V(x)=x12,x=[x1x2]=[0100],V(x)=0isPSD) such that
{ V i s P D V ˙ ( x ) ≜ ∇ V ( x ) T f ( x ) i s N S D \begin{cases} V\,\,is\,\,PD\\ \dot{V}\left( x \right) \triangleq \nabla V\left( x \right) ^{\mathrm{T}}f\left( x \right) \,\,is\,\,NSD\\ \end{cases} {
VisPDV˙(x)≜∇V(x)Tf(x)isNSD
the value of V V V along sys state trajectory nonincreasing V ˙ ( x ( t ) ) = ( ∂ V ∂ x ) T ∂ x ∂ t = ∇ V ( x ) T f ( x ) , ∇ V ( x ) [ ∂ V ∂ x 1 ∂ V ∂ x 2 ⋮ ∂ V ∂ x n ] \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}\\ \vdots\\ \frac{\partial V}{\partial x_{\mathrm{n}}}\\ \end{array} \right] V˙(x(t))=(∂x∂V)T∂t∂x=∇V(x)Tf(x),∇V(x)
∂x1∂V∂x2∂V⋮∂xn∂V
, ∇ V ( x ) T f ( x ) ≜ L f [ V ] \nabla V\left( x \right) ^{\mathrm{T}}f\left( x \right) \triangleq Lf\left[ V \right] ∇V(x)Tf(x)≜Lf[V] Lie derivative of V ( ⋅ ) V\left( \cdot \right) V(⋅) with vetor field f f f
then the origin is stable. If in addition ,
V ˙ ( x ) ≜ ∇ V ( x ) T f ( x ) i s N D \dot{V}\left( x \right) \triangleq \nabla V\left( x \right) ^{\mathrm{T}}f\left( x \right) \,\,is\,\,ND V˙(x)≜∇V(x)Tf(x)isND
then the origin is asymptotically stable —— Value of V V V along sys state trajectory is decreasing
Remarks:
A PD C 1 \mathcal{C} ^1 C1 function satisfying above equation will be called a Lyapunov function (1+2 or 1+3)
Under condition 3 , if V V V is also radially unbounded —— globally asympotically stable (G.A.S)
Main idea : 1+2 —— stability
Fact : suppose V V 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 ( 0 ) ∈ Ω b x\left( 0 \right) \in \varOmega _{\mathrm{b}} x(0)∈Ωb fro some b ⩾ 0 b\geqslant 0 b⩾0 , we have V ( x ( t ) ) ⩽ V ( x ( 0 ) ) ⩽ b V\left( x\left( t \right) \right) \leqslant V\left( x\left( 0 \right) \right) \leqslant b V(x(t))⩽V(x(0))⩽b ⇒ x ( t ) ∈ Ω b \Rightarrow x\left( t \right) \in \varOmega _{\mathrm{b}} ⇒x(t)∈Ωb
Proof of stability : Given ε > 0 \varepsilon >0 ε>0 , goal is to find δ > 0 \delta >0 δ>0, such that ∥ x ( 0 ) ∥ ⩽ δ ⇒ ∥ x ( t ) ∥ ⩽ ε \left\| x\left( 0 \right) \right\| \leqslant \delta \Rightarrow \left\| x\left( t \right) \right\| \leqslant \varepsilon ∥x(0)∥⩽δ⇒∥x(t)∥⩽ε
Sketch of proof of Lyapunov Stability theorem:
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 ( x ( t ) ) V\left( x\left( t \right) \right) V(x(t)) nonincreasing along system trajectory ⇒ \Rightarrow ⇒ if x 0 ∈ Ω b x_0\in \varOmega _{\mathrm{b}} x0∈Ωb , then x ( t ) ∈ Ω b x\left( t \right) \in \varOmega _{\mathrm{b}} x(t)∈Ωb, ∀ t \forall t ∀t
Given arbitrary ε > 0 \varepsilon >0 ε>0 , if we can find δ , b \delta ,b δ,b such that B ( 0 , δ ) ⊆ Ω b ⊆ B ( 0 , ε ) B\left( 0,\delta \right) \subseteq \varOmega _{\mathrm{b}}\subseteq B\left( 0,\varepsilon \right) B(0,δ)⊆Ωb⊆B(0,ε). Then the Lyapunov stability conditions are satisfied. Below is to show how we can find such b b b and δ \delta δ
V V V is continuous ⇒ \Rightarrow ⇒ m = min ∥ x ∥ = ε V ( x ) m=\min _{\left\| x \right\| =\varepsilon}V\left( x \right) m=min∥x∥=εV(x) exists (due to Weierstrass theorem). In addition, V V V is PD ⇒ \Rightarrow ⇒ m > 0 m>0 m>0. Therefore, if we choose b ∈ ( 0 , m ) b\in \left( 0,m \right) b∈(0,m) , then Ω b ⊆ B ( 0 , ε ) \varOmega _{\mathrm{b}}\subseteq B\left( 0,\varepsilon \right) Ωb⊆B(0,ε)
V ( x ) V\left( x \right) V(x) is continuous at origin ⇒ \Rightarrow ⇒ for any b > 0 b>0 b>0 , there exists δ > 0 \delta >0 δ>0 such that ∣ V ( x ) − V ( 0 ) ∣ = V ( x ) < b , ∀ x ∈ B ( 0 , δ ) \left| V\left( x \right) -V\left( 0 \right) \right|=V\left( x \right) <b,\forall x\in B\left( 0,\delta \right) ∣V(x)−V(0)∣=V(x)<b,∀x∈B(0,δ) . This implies that B ( 0 , δ ) ⊆ Ω b B\left( 0,\delta \right) \subseteq \varOmega _{\mathrm{b}} B(0,δ)⊆Ωb
We know V ( x ( t ) ) V\left( x\left( t \right) \right) V(x(t)) decreases monotonically as t → ∞ t\rightarrow \infty t→∞ and V ( x ( t ) ) ⩾ 0 V\left( x\left( t \right) \right) \geqslant 0 V(x(t))⩾0, ∀ t \forall t ∀t. Therefore, c = lim t → ∞ V ( x ( t ) ) c=\lim _{t\rightarrow \infty}V\left( x\left( t \right) \right) c=limt→∞V(x(t)) exists . So it suffices to show c = 0 c=0 c=0. Let us use a contradiction argument.
Suppose c ≠ 0 c\ne 0 c=0. Then c > 0 c>0 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} , ∀ t \forall t ∀t . We can choose β > 0 \beta >0 β>0 such that B ( 0 , β ) ⊆ Ω c B\left( 0,\beta \right) \subseteq \varOmega _{\mathrm{c}} B(0,β)⊆Ωc (due to continuity of V V V at 0 0 0)
Now let a = − max β ⩽ ∥ x ∥ ⩽ ε V ˙ ( x ) a=-\max _{\beta \leqslant \left\| x \right\| \leqslant \varepsilon}\dot{V}\left( x \right) a=−maxβ⩽∥x∥⩽εV˙(x). Since V V V is ND, then a > 0 a>0 a>0
V ( x ( t ) ) = V ( x ( 0 ) ) + ∫ 0 t V ˙ ( x ( s ) ) d s ⩽ V ( x ( 0 ) ) − a ⋅ t < 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\leqslant V\left( x\left( 0 \right) \right) -a\cdot t<0 V(x(t))=V(x(0))+∫0tV˙(x(s))ds⩽V(x(0))−a⋅t<0 for sufficiently large t t t. ⇒ \Rightarrow ⇒ contradiction !
Definition 3 (Exponential Lyapunov Function) —— Important for application
V : D → R V:D\rightarrow \mathbb{R} V:D→R is called anExponential Lyapunov Function (ELF)
on D ⊂ R n D\subset \mathbb{R} ^n D⊂Rn if ∃ k 1 , k 2 , k 3 , α > 0 \exists k_1,k_2,k_3,\alpha >0 ∃k1,k2,k3,α>0 such that
{ k 1 ∥ x ∥ α ⩽ V ( x ) ⩽ k 2 ∥ x ∥ α L f V ( x ) ⩽ − k 3 V ( x ) \begin{cases} k_1\left\| x \right\| ^{\alpha}\leqslant V\left( x \right) \leqslant k_2\left\| x \right\| ^{\alpha}\\ \mathcal{L} _{\mathrm{f}}V\left( x \right) \leqslant -k_3V\left( x \right)\\ \end{cases} { k1∥x∥α⩽V(x)⩽k2∥x∥αLfV(x)⩽−k3V(x)
Lyapunov stability ∃ C 1 \exists \mathcal{C} ^1 ∃C1 func V V V
V V V is PD - deserable ; V ˙ \dot{V} V˙ is ND/NSD
k 1 ∥ x ∥ α ⩽ V ( x ) ⩽ k 2 ∥ x ∥ α k_1\left\| x \right\| ^{\alpha}\leqslant V\left( x \right) \leqslant k_2\left\| x \right\| ^{\alpha} k1∥x∥α⩽V(x)⩽k2∥x∥α ⇒ V \Rightarrow V ⇒V is PD (radially unbounded)
L f V ( x ) ⩽ − k 3 V ( x ) \mathcal{L} _{\mathrm{f}}V\left( x \right) \leqslant -k_3V\left( x \right) LfV(x)⩽−k3V(x) ⇒ V ˙ \Rightarrow \dot{V} ⇒V˙ is ND, V ˙ ⩽ − k 3 V \dot{V}\leqslant -k_3V V˙⩽−k3V
Droof sketch :
recall : z ∈ R 1 , z ˙ = − k 3 z ⇒ z ( t ) = e − k 3 t z ( 0 ) z\in \mathbb{R} ^1,\dot{z}=-k_3z\Rightarrow z\left( t \right) =e^{-k_3t}z\left( 0 \right) z∈R1,z˙=−k3z⇒z(t)=e−k3tz(0)
By comparison theorem : V ˙ ⩽ − k 3 V ⇒ V ( t ) ⩽ e − k 3 t V ( 0 ) \dot{V}\leqslant -k_3V\Rightarrow V\left( t \right) \leqslant e^{-k_3t}V\left( 0 \right) V˙⩽−k3V⇒V(t)⩽e−k3tV(0)
⇒ ∥ x ( t ) ∥ α ⩽ 1 k 1 V ( x ( t ) ) ⩽ 1 k 1 e − k 3 t V ( x ( 0 ) ) ⩽ k 2 k 1 e − k 3 t ∥ x ( 0 ) ∥ α ⇒ ∥ x ( t ) ∥ α ⩽ c e − β t ∥ x ( 0 ) ∥ α \Rightarrow \left\| x\left( t \right) \right\| ^{\alpha}\leqslant \frac{1}{k_1}V\left( x\left( t \right) \right) \leqslant \frac{1}{k_1}e^{-k_3t}V\left( x\left( 0 \right) \right) \leqslant \frac{k_2}{k_1}e^{-k_3t}\left\| x\left( 0 \right) \right\| ^{\alpha}\Rightarrow \left\| x\left( t \right) \right\| ^{\alpha}\leqslant ce^{-\beta t}\left\| x\left( 0 \right) \right\| ^{\alpha} ⇒∥x(t)∥α⩽k11V(x(t))⩽k11e−k3tV(x(0))⩽k1k2e−k3t∥x(0)∥α⇒∥x(t)∥α⩽ce−βt∥x(0)∥α
Theorem 1 (ELF Theorem)
If system 2 has an ELF, then it is exponentially stable
⇒ \Rightarrow ⇒ system is asymptotically stable
In fact the system does not have any (global polynomial Lyapunov function.) But it is GAS with a Lyapunov function V ( x ) = ln ( 1 + x 1 2 ) + x 2 2 V\left( x \right) =\ln \left( 1+{x_1}^2 \right) +{x_2}^2 V(x)=ln(1+x12)+x22
Consider autonomous linear system : x ˙ = f ( x ) = A x \dot{x}=f\left( x \right) =Ax x˙=f(x)=Ax
⇒ V \Rightarrow V ⇒V is LF if P P P is PD and A T P + P A A^{\mathrm{T}}P+PA ATP+PA is ND
Fact : for Linear System , quadratic form of LF , ai all we need to consider. —— A A A is asym stable if and only if ??
In proof of the above function , we assumed P P P is symmetric so P T A = P A P^{\mathrm{T}}A=PA PTA=PA
e.g. P T A = P A P^{\mathrm{T}}A=PA PTA=PA , Q = [ 1 1 − 1 1 ] , g ( x ) = x T Q x = [ x 1 x 2 ] T [ 1 1 − 1 1 ] [ x 1 x 2 ] = x 1 2 + x 2 2 ⇒ [ x 1 x 2 ] T [ 1 0 0 1 ] [ x 1 x 2 ] Q=\left[ \begin{matrix} 1& 1\\ -1& 1\\ \end{matrix} \right] , g\left( x \right) =x^{\mathrm{T}}Qx=\left[ \begin{array}{c} x_1\\ x_2\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} 1& 1\\ -1& 1\\ \end{matrix} \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{matrix} 1& 0\\ 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} x_1\\ x_2\\ \end{array} \right] Q=[1−111],g(x)=xTQx=[x1x2]T[1−111][x1x2]=x12+x22⇒[x1x2]T[1001][x1x2] , Q ^ = 1 2 Q + 1 2 Q T \hat{Q}=\frac{1}{2}Q+\frac{1}{2}Q^{\mathrm{T}} Q^=21Q+21QT
Fact : A A A is asym stable if and only if
- ∃ P ≻ 0 \exists P\succ 0 ∃P≻0 , such that A T P + P A ≺ 0 A^{\mathrm{T}}P+PA\prec 0 ATP+PA≺0
- equivalently , for any Q ≻ 0 , ∃ P Q\succ 0,\exists P Q≻0,∃P such that A T P + P A = − Q A^{\mathrm{T}}P+PA=-Q ATP+PA=−Q (Lyapunov equation)
Theorem 1 (Stability Conditions for Linear System)
For an autonomous Linear system x ˙ = A x \dot{x}=Ax x˙=Ax. The following statements are equivalent.
- (Linear) System is (globally) asmptotically stable
- (Linear) System is (globally) exponentially stable
- R e ( λ i ) < 0 \mathrm{Re}\left( \lambda _{\mathrm{i}} \right) <0 Re(λi)<0 for all eigenvalues λ i \lambda _{\mathrm{i}} λi of A A A —— lie on open left half complex plane (OLHP)
- System has a quadratic Lyapunov function V ( x ) = x T P x V\left( x \right) =x^{\mathrm{T}}Px V(x)=xTPx
- For ant symmetric Q ≻ 0 Q\succ 0 Q≻0 , there exists a symmetric P ≻ 0 P\succ 0 P≻0 that solves the following Lyapunov equation :
A T P + P A = − Q A^{\mathrm{T}}P+PA=-Q ATP+PA=−Q
Q ≻ 0 Q\succ 0 Q≻0 is given , P P P is the variale to be solved , and V ( x ) = x T P x V\left( x \right) =x^{\mathrm{T}}Px V(x)=xTPx is Lyapunov function of the system
Converse Lyapunov Theorem for Asymptotic Stability
origin asymptotically stable ; f f f is locallt Lipschitz on D with region of attration R A R_A RA ⇒ V s . t . \Rightarrow V\,\,s.t. ⇒Vs.t. V V V is continuuos and PD on R A R_A RA ; L f V L_{\mathrm{f}}V LfV is ND on R A R_A RA ; V ( x ) → ∞ V\left( x \right) \rightarrow \infty V(x)→∞ as x → ∂ R A x\rightarrow \partial R_{\mathrm{A}} x→∂RA
convex result that is not constructive
Converse Lyapunov Theorem for Exponential Stability
origin exponentially stable on D D D ; f f f is C 1 \mathcal{C} ^1 C1 ⇒ ∃ \Rightarrow \exists ⇒∃ an ELF V V V on D D D
For nonlinear sys , ∃ V ⇒ \exists V\Rightarrow ∃V⇒ 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
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