Lyapunov Stability Analysis

Lyapunov Stability Analysis

• Lyapunov's direct method can be used to determine the stability of an equilibrium point.
• Considering the system $\dot{x} = f(x)$, where $f(0) = 0$, $x = 0$ is an equilibrium.
• The Lyapunov function candidate is $V(x)$.

Lyapunov Function Candidate Conditions

• $V(x)$ must be continuously differentiable.
• $V(0) = 0$
• $V(x) > 0$ for all $x \neq 0$

Stability Conditions

• If $\dot{V}(x) \le 0$, the origin is stable.
• If $\dot{V}(x) < 0$, the origin is asymptotically stable.

Rate of Change of Lyapunov Function

• The rate of change of the Lyapunov function is calculated as $\dot{V}(x) = \frac{\partial V}{\partial x} \dot{x} = \frac{\partial V}{\partial x} f(x)$.

Proof of Lyapunov Stability

• Shows that any solution $x(t)$ starting sufficiently close to 0 remains close to 0 for all time.
• The solution $x = 0$ of $\dot{x} = f(x)$ is stable if for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $||x(0)|| < \delta$ then $||x(t)|| < \epsilon$ for all $t \ge 0$.

Steps

• Pick an $\epsilon > 0$ and consider $B_\epsilon = {x \in \mathbb{R}^n : ||x|| \le \epsilon }$.
• Must find a $B_\delta$ such that solutions starting in $B_\delta$ stay in $B_\epsilon$.
• Since $V(x)$ is continuous and $V(0) = 0$, the set ${x : V(x) \le c}$ is closed and bounded for sufficiently small $c$.
• Let $c = \min_{x \in \partial B_\epsilon} V(x)$ where $\partial B_\epsilon = {x \in \mathbb{R}^n : ||x|| = \epsilon }$.
• Now let $B_\delta = {x : V(x) \le c/2 }$.
• A trajectory that starts in $B_\delta$ will have $V(x(0)) \le c/2$ and, since $\dot{V}(x) \le 0$, $V(x(t))$ can only decrease.
• Therefore, $x(t)$ can never leave $B_\epsilon$.

Invariance Principle

• If a Lyapunov function $V(x)$ has $\dot{V}(x) \le 0$, then every solution starting in a compact set converges to $M$ as $t \to \infty$, where $M$ is the largest invariant set in $S = {x : \dot{V}(x) = 0 }$.
• A set $M$ is invariant if $x(0) \in M \implies x(t) \in M$ for all $t$.

Linear Systems

• Given $\dot{x} = Ax$, a Lyapunov function is $V(x) = x^T P x$ where $P = P^T > 0$.
• The rate of change is: $\dot{V}(x) = x^T (A^T P + PA) x$.
• Letting $Q = -(A^T P + PA)$, then $\dot{V}(x) = -x^T Q x$.
• Asymptotic stability occurs if $Q > 0$, then $\dot{V}(x) < 0$ for $x \neq 0$.
• Such a $P$ exists for a given $Q$ if $A$ is stable.
• Lyapunov Equation: $A^T P + PA = -Q$

Linear Algebra and Matrix Analysis

Matrix Calculus

• Deals with matrices and fundamental matrix operations
• Matrix is a rectangular collection of scalars, e.g., real or complex numbers
• Matrix calculus provides compact notation to denote and solve mathematical problems in various fields

Matrix Definition

• Matrix $A$ of size $m \times n$ is a rectangular array with $m$ rows and $n$ columns
  • $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$
• Terminology
  • $a_{ij}$: element in the $i$-th row and $j$-th column
  • $A = [a_{ij}]_{m \times n}$
  • $M_{m,n}(\mathbb{R})$: set of $m \times n$ matrices with real coefficients
  • $M_{m,n}(\mathbb{C})$: set of $m \times n$ matrices with complex coefficients
  • If $m = n$: square matrix of order $n$, denoted as $M_n(\mathbb{R})$ or $M_n(\mathbb{C})$
• Example: $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ is a $2 \times 3$ matrix.

Matrix Operations

Matrix Addition

• For matrices $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{m \times n}$ of the same size, their sum is $C = A + B = [c{ij}]{m \times n}$, defined by: $c{ij} = a_{ij} + b_{ij}, \forall i,j$
• $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$

Scalar Multiplication

• For matrix $A = [a_{ij}]{m \times n}$ and scalar $\alpha$, the product is $B = \alpha A = [b{ij}]{m \times n}$, defined by: $b{ij} = \alpha a_{ij}, \forall i,j$

Matrix Multiplication

• Definition: For matrices $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{n \times p}$, the product is $C = AB = [c{ij}]{m \times p}$, defined by: $c{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}, \forall i,j$
• Requirements: The number of columns of $A$ must equal the number of rows of $B$
• Example: $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$ . Product is $ \begin{bmatrix} 1\cdot5 + 2\cdot7 & 1\cdot6 + 2\cdot8 \ 3\cdot5 + 4\cdot7 & 3\cdot6 + 4\cdot8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$

Matrix Transposition

• For matrix $A = [a_{ij}]{m \times n}$, the transpose $A^T$ is $B = [b{ij}]{n \times m}$, where $b{ij} = a_{ji}, \forall i,j$
• If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 3 & 5 \ 2 & 4 & 6 \end{bmatrix}$

Properties of Matrix Operations

• Associativity of addition: $(A + B) + C = A + (B + C)$
• Commutativity of addition: $A + B = B + A$
• Distributivity over Scalar Addition: $\alpha(A + B) = \alpha A + \alpha B$
• Distributivity over Scalar Multiplication: $(\alpha + \beta)A = \alpha A + \beta A$
• Associativity of Multiplication: $(AB)C = A(BC)$
• Distributivity of Multiplication over Addition: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
• Transposition of a Product: $(AB)^T = B^T A^T$

Special Matrices

Identity Matrix

• $n \times n$ matrix, denoted $I_n$, with ones on the main diagonal and zeros elsewhere
• Formula: $I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix}$
• Property: For any matrix $A \in M_{m,n}(\mathbb{R})$, $AI_n = A$ and $I_mA = A$.

Diagonal Matrix

• Square matrix where all elements outside the main diagonal are zeros

Triangular Matrix

• Upper Triangular: all elements below the main diagonal are zeros
• Lower Triangular: all elements above the main diagonal are zeros

Confusion Matrix Elements

• Verdad: Describe si una instancia fue clasificada correctamente o no.
• Positiva Verdadera (TP): Casos en los que predijimos SI y la realidad era SI.
• Negativa Verdadera (TN): Casos en los que predijimos NO y la realidad era NO.
• Falsos Positivos (FP): Casos en los que predijimos SI y la realidad era NO. (Error Tipo I)
• Falsos Negativos (FN): Casos en los que predijimos NO y la realidad era SI. (Error Tipo II)

Métricas

• Calcular métricas de rendimiento del modelo de clasificación.
• Exactitud (Accuracy): $\frac{TP + TN}{TP + TN + FP + FN}$
• Precisión (Precision): $\frac{TP}{TP + FP}$
• Recuperación (Recall): $\frac{TP}{TP + FN}$
• Puntuación F1 (F1 Score): $2 * \frac{Precision * Recall}{Precision + Recall}$

Material properties of AA 6061

Composition of AA6061 (% by weight)

• Silicon (Si): 0.4 to 0.8
• Iron (Fe): 0.7 max
• Copper (Cu): 0.15 to 0.4
• Manganese (Mn): 0.15 max
• Magnesium (Mg): 0.8 to 1.2
• Zinc (Zn): 0.25 max
• Chromium (Cr): 0.04 to 0.35
• Titanium (Ti): 0.15 max
• Aluminum (Al): Balance

Typical applications of AA6061

• Aircraft components.
• Camera lenses.
• Couplings.
• Marine fittings and hardware.
• Heavy-duty structures.
• Truck and marine transportation.

Important mechanical properties of AA6061

• Density: 2.70 g/cm3.
• Ultimate Tensile Strength: 310 MPa.
• Tensile Yield Strerngth: 276MPa.
• Elastic Modulus: 68.9 GPa.
• Poisson's Ratio: 0.33.
• Shear Modulus: 26 GPa.
• Hardness, Brinell: 95 HB.
• Fatigue Strength: 96.5 MPa.
• Machinability: good.

Important thermal properties of AA6061

• Thermal Conductivity: 167 W/m-K.
• Coefficient of Thermal Expansion: 23.6 µm/m-°C.
• Melting Point: 582 - 652°C.
• Solidus: 582°C.
• Liquidus: 652C.

Important electrical properties of AA6061

• Electrical Conductivity: 43% IACS.
• Electrical Resistivity: 0.040 µΩ-cm

Processing/Fabrication properties of AA6061

• Can be work hardened but has limited formability.
• Possesses good weldability using common methods.
• Is heat treatable.
  • Includes Solution Heat Treatment, Quenching, and Precipitation Hardening (Aging).

Corrosion resistance features of AA6061

• Exhibits good corrosion resistance, particularly marine.
• Can be enhanced with anodizing.

What is static electricity?

• Build-up of electrical charges on an object's surface.
• It stays put until it's discharged.

How does static electricity form?

• Most objects are electrically neutral with an equal amount of positive and negative charge.
• Static electricity forms when rubbing two objects produces the shift of electrons from one surface to another.
• The material that looses electrons get a positive charge while the other becomes negative.

Different methods of creating electricity

• Rubbing a balloon on your hair creates negative electrical charges on the balloon and positive in the hair.
• Walking across a carpet shifts electrons from the carpet to your feet, which creates negative charge. When you touch a metal, the electrical discharge releases and creates a static shock.

Atomic structure basics

–Atom: Basic matter component.

• Proton: Positive charge, located in the nucleus.
• Neutron: No charge, located in the nucleus.
• Electron: Negative charge, orbits the nucleus.

Triboelectric sequences

• Rank substances in their propensity to gain or loose electrons.
• Material high in the list loose electrons and result in + charge while those lower gain electrons and turn -.

Material and Polarity summary

–Material: Air/Human Skin/Leather/ rabbit Fur/ Glass/Quartz/Mica/Nylon/Wool/Cat Fur/Silk/Paper/cotton/Wood etc , Polarity: positive

• Material: Amber/hard Rubber/Nickel, Copper/brass Silver/Gold Platinium/polyester/styrene/saran wrap/polyurethande/polyethylene/PVC/Silicon/Teflon,
• Polarity: Negative

Charge Interaction

• Similar repel and opposites attract.

Conductors vs Insulators

• Conductors let electrons to shift easily (e.g., metals). -Insulators resist electrons to shift (e.g., plastic, glass).

Charging Techniques

–Shifting of electrons when rubbing (friction).

• Switching electrons by direct touch (conduction).
• Shifting of electrical charges because of an existing charge(Induction).

Importance of grounding

• Removes and controls build up of excess electrical charge by connecting the object to the Earth for balance.

Static electrical discharge

• Electrical shift when a potential electrical difference occurs.
• Usually, a spark occurs with cracking.

Static electricity practical uses

• Electrostatic dust removers that clear particles.
• Electrostatic paints where electrical charges help paintings. Photocopiers and laser prints with static electricity on drums

Static electrical danger

–Electrical discharge can damage electrical components.

• Flammable materials are prone to ignition with this charge.

Precautions: Mitigating Static Electricity

• Humidity control that eliminates static build up.
• Grounding techniques prevent build up.
• Anti static electrical sprays that decrease static build up.

Fundamental Coulomb's Law

–Relationship of charge and force, $F = k \frac{|q_1q_2|}{r^2}$

• F = Electrostatic force
• k = Coulomb's constant ($8.99 \times 10^9 Nm^2/C^2$)
• q_1 and q_2 = Magnitude of the charges
• r = Distance between the charges

Algorithmic Trading

• Automated order execution based on pre-programmed instructions.
• Algorithms consider factors like price, timing, and volume.

Benefits of Algorithmic Trading

• Reduced transaction costs.
• Improved order execution.
• Enhanced liquidity.
• Enables complex trading strategies.

Common Algorithmic Trading Strategies

Trend Following

• Capitalizing on assets moving in a consistent direction.
• Moving Averages: Using moving averages to trade in line with the prevailing trend.
• Breakout Strategies: Buying when prices exceed resistance or selling when they fall below support.

Mean Reversion

• Based on the concept that prices will revert to their average over time.
• Pairs Trading: Exploiting price differences between correlated assets with the expectation of convergence.
• Bollinger Bands: Identifying overbought or oversold conditions and trading contrarian.

Arbitrage Strategies

• Exploiting price differences in different markets or forms of the same asset.
• Statistical Arbitrage: Using statistical models to identify mispricing across a basket of assets.
• Triangular Arbitrage: Profiting from price discrepancies between three currencies in the foreign exchange market.

Sentiment Analysis

• Gauging market sentiment from news and social media to inform trades.
• News Sentiment Analysis: Trading based on positive or negative signals extracted from news headlines.
• Social Media Sentiment: Aligning trades with market sentiment indicated by social media trends.

Machine Learning

• Using machine learning algorithms to predict price movements.
• Supervised Learning: Training models on historical data to predict future price movements.
• Reinforcement Learning: Learning optimal trading strategies through interaction with the market.

Example of Algorithmic Trading Programmed Rules

• An algorithm buys shares of stock if the 50-day moving average exceeds the 200-day moving average.

Note

• Algorithmic trading strategies have advantages but carry risks.

Atomic Habits: Introduction

• Book about building good habits and breaking bad ones.
• By James Clear.

Dijkstra's algorithm

• Given by Edsger W. Dijkstra.
• Calculates the shortest path from a source node to a target node in a graph.
• Allows to find paths to every other node.
• Used in route planners, for example.

Description of the Algorithmus

• Directed on graph $G = (V, E)$ with the edge weights $w: E \to \R_{\ge 0}$.
• Weights indicate distance.
• Finds the shortest path from a source $s \in V$ to every other node $v \in V$.

Key Algorithim Initialization steps.

• Initalize distances to all nodea as "infinite" except source which is zero.
• Create a set with all nodes of the graph.

Process steps

• while the set of unvisited nodes exists, do the following:
  • Select the one that is unvisited with the least distance. The Source nodes does this.
  • Then, iterate through each neighboring node to compute total length and update accordingly.
  • Update the lengths on neighbor if smaller from existing.

Key steps in thermodynamics; definitions:

• A phase refers to a substance that is uniform and has similar physcial properties.
• Compressing a object refers to a scenario where magnetic, gravitation and tension are negligible.
• Pure substance: A substance that is uniform and unchangeable when it comes to composition.

Phase Diagrams

• Include the phase diagrams to track different phases.

Essential Properties of Diagrams to Consider

-T-v diagram: Temperature vs specific volume

• P-v diagram: Pressure vs specific volume
• P-T diagram: Pressure vs Temperature

Understanding of Property Tables

• Saturated liquid
• Saturated vapor
• Compressed liquid
• Superheated vapor
• 2-Phase mixture

Applying properties to Quality

$$ x = \frac{m_{vapor}}{m_{total}} $$

Specific volume considerations

$$ v = v_f + x v_{fg} $$

Estimating of internal energy

$$ u = u_f + x u_{fg} $$

Estimating Enthalpy

$$ h = h_f + x h_{fg} $$

Applying it to entropy

$$ s = s_f + x s_{fg} $$

Understanding Ideal Gas Equation of State

$$ Pv = RT $$

• P is the absolute pressure
• v is the specific volume
• R is the gas constant
• T is the absolute temperature

Compressibility Factor defined as

$$ Z = \frac{Pv}{RT} $$

Van der Waals state and equation example

$$ P = \frac{RT}{v-b} - \frac{a}{v^2} $$

Redlich-Kwong Equation of State details

$$ P = \frac{RT}{v-b} - \frac{a}{v(v+b)T^{1/2}} $$