Berkeley Notes
  • Introduction
  • EE120
    • Introduction to Signals and Systems
    • The Fourier Series
    • The Fourier Transform
    • Generalized transforms
    • Linear Time-Invariant Systems
    • Feedback Control
    • Sampling
    • Appendix
  • EE123
    • The DFT
    • Spectral Analysis
    • Sampling
    • Filtering
  • EECS126
    • Introduction to Probability
    • Random Variables and their Distributions
    • Concentration
    • Information Theory
    • Random Processes
    • Random Graphs
    • Statistical Inference
    • Estimation
  • EECS127
    • Linear Algebra
    • Fundamentals of Optimization
    • Linear Algebraic Optimization
    • Convex Optimization
    • Duality
  • EE128
    • Introduction to Control
    • Modeling Systems
    • System Performance
    • Design Tools
    • Cascade Compensation
    • State-Space Control
    • Digital Control Systems
    • Cayley-Hamilton
  • EECS225A
    • Hilbert Space Theory
    • Linear Estimation
    • Discrete Time Random Processes
    • Filtering
  • EE222
    • Real Analysis
    • Differential Geometry
    • Nonlinear System Dynamics
    • Stability of Nonlinear Systems
    • Nonlinear Feedback Control
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  1. EE222

Differential Geometry

Definition 19

M⊂RnM\subset \mathbb{R}^nM⊂Rn is a mmm-dimensional smooth sub-manifold of Rn\mathbb{R}^nRn if ∀p∈M, ∃r>0\forall \boldsymbol{p}\in M,\ \exists r > 0∀p∈M, ∃r>0 and F:Br(p)→Rn−mF: B_r(\boldsymbol{p}) \to \mathbb{R}^{n-m}F:Br​(p)→Rn−m such that

M∩Br(p)={x∈Br(p)∣F(x)=0},F is smooth,∀xˉ∈M∩Br(p),Rank(∂F∂x∣xˉ)=n−m\begin{aligned} M \cap B_r(\boldsymbol{p}) = \{\boldsymbol{x}\in B_r(\boldsymbol{p}) | F(\boldsymbol{x}) = 0\},\\ F\text{ is smooth,}\\ \forall \bar{\boldsymbol{x}} \in M \cap B_r(\boldsymbol{p}), \text{Rank}\left(\frac{\partial F}{\partial \boldsymbol{x}} \bigg\rvert_{\bar{\boldsymbol{x}}}\right) = n - m \end{aligned}M∩Br​(p)={x∈Br​(p)∣F(x)=0},F is smooth,∀xˉ∈M∩Br​(p),Rank(∂x∂F​​xˉ​)=n−m​

By Definition 19, a manifold is essentially defined as the 0-level set of some smooth function FFF and can be thought of as a surface embedded in a higher dimension.

Definition 20

The tangent space of a manifold MMM at p∈M\boldsymbol{p}\in Mp∈M is given by

TpM=Null(∂F∂x∣p)T_{\boldsymbol{p}}M = \text{Null}\left(\frac{\partial F}{\partial \boldsymbol{x}}\bigg |_{\boldsymbol{p}}\right)Tp​M=Null(∂x∂F​​p​)

The tangent space consists of all vectors tangent to the manifold at a particular point p\boldsymbol{p}p.

Definition 21

The Tangent Bundle of a manifold MMM is the collection of all tangent spaces

TM=⋃p∈MTpMT_M = \bigcup_{\boldsymbol{p}\in M} T_{\boldsymbol{p}} MTM​=⋃p∈M​Tp​M

Definition 22

A vector field f:M→TMf:M\to T_Mf:M→TM​ on a manifold MMM is an assignment of each point p∈M\boldsymbol{p}\in Mp∈M to a vector in the tangent space in that point TpMT_{\boldsymbol{p}}MTp​M.

Therefore, a vector field can be thought of as a curve through the tangent bundle of a manifold.

Definition 23

The Lie Derivative of a function VVV with respect to a vector field fff is given by

LfV=(∇xV)⊤f(x).L_fV = (\nabla_{\boldsymbol{x}}V)^\top f(\boldsymbol{x}).Lf​V=(∇x​V)⊤f(x).

A Lie Derivative is essentially a directional derivative, and it measures how a function changes along a vector field.

Definition 24

Suppose that f(x)f(\boldsymbol{x})f(x) and g(x)g(\boldsymbol{x})g(x) are vector fields. The Lie Bracket of fff and ggg is given by

[f,g]=Lfg−Lgf[f, g] = L_fg - L_gf[f,g]=Lf​g−Lg​f

The Lie Bracket is another vector field, and it essentially measures the difference between moving along vector field fff and vector field ggg across some infinitesimal distance. Another way to think about the Lie Bracket is as a measure of the extent to which fff and ggg commute with each other. The Lie Bracket is also sometimes denoted using the adjoint map

adfg=[f,g].\text{ad}_fg = [f, g].adf​g=[f,g].

It is helpful when chaining Lie Brackets since we can denote

[f,[f,[f,⋯[f,g]]]]=adfig.[f,[f,[f,\cdots[f,g]]]] = \text{ad}_f^ig.[f,[f,[f,⋯[f,g]]]]=adfi​g.

Since the Lie Bracket is a vector field, we can look at Lie Derivatives with respect to the Lie Bracket of two vector fields.

Theorem 4

For a function hhh and vector fields fff and ggg,

L[f,g]h=LfLgh−LgLfhL_{[f,g]}h = L_fL_gh - L_gL_fhL[f,g]​h=Lf​Lg​h−Lg​Lf​h

We can also use relate repeated Lie Derivatives to doing repeated Lie Brackets.

Theorem 5

LgLfih(x)=0⇔Ladfigh(x)=0L_gL_f^ih(\boldsymbol{x}) = 0 \Leftrightarrow L_{\text{ad}_f^ig}h(\boldsymbol{x}) = 0Lg​Lfi​h(x)=0⇔Ladfi​g​h(x)=0

Definition 25

Suppose f1,f2,⋯ ,fnf_1,f_2,\cdots,f_nf1​,f2​,⋯,fn​ are vector fields. A distribution Δ\DeltaΔ is the span of the vector fields at each point x\boldsymbol{x}x:

Δ(x)=span{f1(x),f2(x),⋯ ,fn(x)}.\Delta(\boldsymbol{x}) = \text{span}\{f_1(\boldsymbol{x}), f_2(\boldsymbol{x}),\cdots,f_n(\boldsymbol{x})\}.Δ(x)=span{f1​(x),f2​(x),⋯,fn​(x)}.

At each point x, Δ(x)\boldsymbol{x},\ \Delta(\boldsymbol{x})x, Δ(x) is a subspace of the tangent space at x\boldsymbol{x}x.

Definition 26

The dimension of a distribution at a point x\boldsymbol{x}x is given by

Dim Δ(x)=Rank([f1(x)∣f2(x)∣⋯∣fn(x)])\text{Dim }\Delta(\boldsymbol{x}) = \text{Rank}\left(\begin{bmatrix} f_1(\boldsymbol{x}) & \bigg\lvert & f_2(\boldsymbol{x}) & \bigg\lvert & \cdots & \bigg\lvert & f_n(\boldsymbol{x}) \end{bmatrix}\right)Dim Δ(x)=Rank([f1​(x)​​​f2​(x)​​​⋯​​​fn​(x)​])

Distributions have different properties which are important to look at.

Definition 27

A distribution Δ\DeltaΔis nonsingular, also known as regular, if its dimension is constant.

Definition 28

A distribution Δ\DeltaΔ is involutive if

∀f,g∈Δ,[f,g]∈Δ\forall f, g\in \Delta, \quad [f, g] \in \Delta∀f,g∈Δ,[f,g]∈Δ

In involutive distributions, you can never leave the distribution by traveling along vectors inside the distribution.

Definition 29

A nonsingular KKK-dimensional distribution Δ(x)=span{f1(x),⋯ ,fk(x)}\Delta(\boldsymbol{x}) = \text{span}\{f_1(\boldsymbol{x}), \cdots, f_k(\boldsymbol{x})\}Δ(x)=span{f1​(x),⋯,fk​(x)} is completely integrable if ∃ϕ1,⋯ ,ϕn−k\exists \phi_1,\cdots,\phi_{n-k}∃ϕ1​,⋯,ϕn−k​ such that ∀i,k, Lfkϕi=0\forall i,k,\ L_{f_k}\phi_i = 0∀i,k, Lfk​​ϕi​=0 and ablaxϕiabla_{\boldsymbol{x}}\phi_iablax​ϕi​are linearly independent. \label{thm:involutive}

It turns out that integrability and involutivity are equivalent to each other.

Theorem 6 (Frobenius Theorem)

A nonsingular Δ\DeltaΔ is completely integrable if and only if Δ\DeltaΔis involutive.

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