test

M  .task/backlog.data

M  .task/completed.data

M  .task/pending.data

M  .task/undo.data

M  "Zettelkasten/Permanent Notes/20250829114522-hybrid-systems.md"

A  "Zettelkasten/Permanent Notes/20250911165736-switched-systems.md"

A  "Zettelkasten/Permanent Notes/20250911170650-lipschitz-continuous.md"

M  "Zettelkasten/Permanent Notes/Literature Notes/LIT-20250911143337-multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems.md"

.task/backlog.data

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.task/completed.data

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.task/pending.data

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.task/undo.data

@@ -109,15 +109,10 @@ time 1757621896
 old [description:"Read Branicky Lyapunov paper" entry:"1757615669" modified:"1757620907" project:"Thesis" start:"1757620907" status:"pending" tags:"reading" tags_reading:"x" uuid:"f4d32147-ec74-4f68-b8d5-146c0b6f35df"]
 new [description:"Read Branicky Lyapunov paper" entry:"1757615669" modified:"1757621896" project:"Thesis" status:"pending" tags:"reading" tags_reading:"x" uuid:"f4d32147-ec74-4f68-b8d5-146c0b6f35df"]
 ---
-time 1757864178
+time 1757625029
+new [description:"Write zettel about lipschitz continuity" entry:"1757625029" modified:"1757625029" status:"pending" tags:"zk" tags_zk:"x" uuid:"b7f68988-8c06-4d18-bf77-91d7e39fd55f"]
+---
+time 1757627393
 old [description:"Read Branicky Lyapunov paper" entry:"1757615669" modified:"1757621896" project:"Thesis" status:"pending" tags:"reading" tags_reading:"x" uuid:"f4d32147-ec74-4f68-b8d5-146c0b6f35df"]
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----
-time 1757864205
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----
-time 1757885608
-old [dep_8e7a8e19-9197-4008-b7d9-521ffcf7ba91:"x" dep_96c76e6b-5c33-4f54-a156-5c59e718f01a:"x" dep_f4d32147-ec74-4f68-b8d5-146c0b6f35df:"x" depends:"8e7a8e19-9197-4008-b7d9-521ffcf7ba91,96c76e6b-5c33-4f54-a156-5c59e718f01a,f4d32147-ec74-4f68-b8d5-146c0b6f35df" description:"Write first draft of state of the art" due:"1757649600" entry:"1757449978" modified:"1757864205" project:"ERLM" start:"1757864205" status:"pending" tags:"writing" tags_writing:"x" uuid:"4bec1530-18bc-43cb-9f0b-61e35dbf1730"]
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 ---

Zettelkasten/Permanent Notes/20250829114522-hybrid-systems.md

@@ -3,22 +3,55 @@ id: 20250829114522
 title: Hybrid Systems
 type: permanent
 created: 2025-08-29T15:45:22Z
-modified: 2025-08-29T15:48:22Z
+modified: 2025-09-11T21:46:53Z
 tags: []
 ---
 
 # Hybrid Systems
 
-Hybrid systems are those that mix continuous dynamics with
-discrete dynamics. This can mean physical phenomena
-controlled by something discrete like a logic based
-controller, or more interestingly controllers that switch
-control laws based on different circumstances.
+I'm borrowing a lot from
+[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]].
+
+Hybrid systems are those that combine continuous and
+discrete dynamics together. This is usually some sort of
+finite automata + differential equations.
+
+Hybrid systems can be written like:
+
+$$\dot{x}(t) = \xi(t), \quad t\geq 0$$
+
+where $x(t)$ is the continuous component of the state.
+$\xi(t)$ is a vector field that depends on $x(t)$ and the
+hybrid dynamics.
+
+Switching between modes (aka discontinuities in $\xi(\cdot)$)
+can happen in one of two ways:
+
+1. **Autonomous Switching** - Autonomous switches happen
+   depending on state values of $x(t)$.
+
+2. **Controlled Switching** - $\xi(\cdot)$ changes abruptly
+   in response to a control command.
+
+One may write a continuous time autonomous hybrid system
+like this:
+
+$$\dot{x}(t) = f(x(t), q(t))$$
+$$q(t) = \nu(x(t), q(t^-))$$
+
+where:
+- $x(t) \in R^n$
+- $q(t) \in Q \simeq {1,...,N}$
+- $f(\cdot,q): R^n \rightarrow R^n,q \in Q$, with each
+[[lipschitz-continuous]]. These are the *continuous
+dynamics*.
+- $\nu: R^n \times Q \rightarrow Q$ is the *finite dynamics*
+
+A controlled system might be written as:
+$$\dot{x}(t) = f(x(t), q(t), u(t))$$
+$$q(t) = \nu(x(t), q(t^-), u(t))$$
+
+where:
+- $u(t) \in R^m$
+
 
-Gemini AI says a good example of a hybrid controller is a
-robot that uses a movement controller until it reaches a
-target, and then switching to a force controller thereafter.
-The continuous dynamics and physics have not changed, but
-the derived model of the physics has entirely in a discrete
-way. This kind of 'mode' switching is called being
-event-driven.

Zettelkasten/Permanent Notes/20250911165736-switched-systems.md

@@ -0,0 +1,35 @@
+---
+id: 20250911165736
+title: Switched Systems
+type: permanent
+created: 2025-09-11T20:57:36Z
+modified: 2025-09-11T21:09:55Z
+tags: []
+---
+
+# Switched Systems
+
+Switched systems are those that mix continuous and discrete
+dynamics. They are systems that are 'multimodal'. This means
+that they can have different continuous dynamic modes.
+
+I'm borrowing form
+[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]],
+but here's a short description of how they work.
+
+A prototypical switched system is as follows:
+$$\dot{x}(t)=f_i ( x(t)), \quad i \in Q \simeq {1,...,N}$$
+
+with two conditions:
+1. Each $f_i$ is globally [[Lipschitz Continuous]]
+2. The i's are picked in a way that there are finite
+   switches in finite time.
+
+
+There's also this thing called a *continuous switched
+system*. A continuous switched system is one that does not
+change continuous states when a switch occurs. That is to
+say when switching from $i$ to $i'$:
+
+$$f_i(x(t_i),t_i) = f_{i'}(x(t_{i'}),t_{i'})$$
+

Zettelkasten/Permanent Notes/20250911170650-lipschitz-continuous.md

@@ -0,0 +1,10 @@
+---
+id: 20250911170650
+title: Lipschitz Continuous
+type: permanent
+created: 2025-09-11T21:06:50Z
+modified: 2025-09-11T21:06:50Z
+tags: []
+---
+
+# Lipschitz Continuous

Zettelkasten/Permanent Notes/Literature Notes/LIT-20250911143337-multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems.md

@@ -3,7 +3,7 @@ id: LIT-20250911143337
 title: Multiple Lyapunov Functions and Other Analysis Tools for Swtiched and Hybrid Systems
 type: literature
 created: 2025-09-11T18:33:37Z
-modified: 2025-09-11T20:55:15Z
+modified: 2025-09-11T21:53:20Z
 citekey: 
 ---
 
@@ -46,4 +46,7 @@ Complex ideas written simply. Easy to read.
 **Weak Points? Strong Points?**
 
 ## Zettels Created
+[[Switched Systems]]
+[[hybrid-systems]]
+[[lipschitz-continuous]]