Auto sync: 2025-09-14 20:27:41 (8 files changed)
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"
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[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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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"]
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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"]
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---
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time 1757625029
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new [description:"Write zettel about lipschitz continuity" entry:"1757625029" modified:"1757625029" status:"pending" tags:"zk" tags_zk:"x" uuid:"b7f68988-8c06-4d18-bf77-91d7e39fd55f"]
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---
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time 1757627393
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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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new [description:"Read Branicky Lyapunov paper" end:"1757627393" entry:"1757615669" modified:"1757627393" project:"Thesis" status:"completed" tags:"reading" tags_reading:"x" uuid:"f4d32147-ec74-4f68-b8d5-146c0b6f35df"]
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---
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@ -3,22 +3,55 @@ id: 20250829114522
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title: Hybrid Systems
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type: permanent
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created: 2025-08-29T15:45:22Z
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modified: 2025-08-29T15:48:22Z
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modified: 2025-09-11T21:46:53Z
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tags: []
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---
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# Hybrid Systems
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Hybrid systems are those that mix continuous dynamics with
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discrete dynamics. This can mean physical phenomena
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controlled by something discrete like a logic based
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controller, or more interestingly controllers that switch
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control laws based on different circumstances.
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I'm borrowing a lot from
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[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]].
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Hybrid systems are those that combine continuous and
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discrete dynamics together. This is usually some sort of
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finite automata + differential equations.
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Hybrid systems can be written like:
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$$\dot{x}(t) = \xi(t), \quad t\geq 0$$
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where $x(t)$ is the continuous component of the state.
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$\xi(t)$ is a vector field that depends on $x(t)$ and the
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hybrid dynamics.
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Switching between modes (aka discontinuities in $\xi(\cdot)$)
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can happen in one of two ways:
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1. **Autonomous Switching** - Autonomous switches happen
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depending on state values of $x(t)$.
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2. **Controlled Switching** - $\xi(\cdot)$ changes abruptly
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in response to a control command.
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One may write a continuous time autonomous hybrid system
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like this:
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$$\dot{x}(t) = f(x(t), q(t))$$
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$$q(t) = \nu(x(t), q(t^-))$$
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where:
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- $x(t) \in R^n$
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- $q(t) \in Q \simeq {1,...,N}$
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- $f(\cdot,q): R^n \rightarrow R^n,q \in Q$, with each
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[[lipschitz-continuous]]. These are the *continuous
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dynamics*.
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- $\nu: R^n \times Q \rightarrow Q$ is the *finite dynamics*
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A controlled system might be written as:
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$$\dot{x}(t) = f(x(t), q(t), u(t))$$
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$$q(t) = \nu(x(t), q(t^-), u(t))$$
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where:
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- $u(t) \in R^m$
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Gemini AI says a good example of a hybrid controller is a
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robot that uses a movement controller until it reaches a
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target, and then switching to a force controller thereafter.
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The continuous dynamics and physics have not changed, but
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the derived model of the physics has entirely in a discrete
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way. This kind of 'mode' switching is called being
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event-driven.
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---
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id: 20250911165736
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title: Switched Systems
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type: permanent
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created: 2025-09-11T20:57:36Z
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modified: 2025-09-11T21:09:55Z
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tags: []
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---
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# Switched Systems
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Switched systems are those that mix continuous and discrete
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dynamics. They are systems that are 'multimodal'. This means
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that they can have different continuous dynamic modes.
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I'm borrowing form
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[[multiple-lyapunov-functions-and-other-analysis-tools-for-swtiched-and-hybrid-systems]],
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but here's a short description of how they work.
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A prototypical switched system is as follows:
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$$\dot{x}(t)=f_i ( x(t)), \quad i \in Q \simeq {1,...,N}$$
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with two conditions:
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1. Each $f_i$ is globally [[Lipschitz Continuous]]
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2. The i's are picked in a way that there are finite
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switches in finite time.
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There's also this thing called a *continuous switched
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system*. A continuous switched system is one that does not
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change continuous states when a switch occurs. That is to
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say when switching from $i$ to $i'$:
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$$f_i(x(t_i),t_i) = f_{i'}(x(t_{i'}),t_{i'})$$
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@ -0,0 +1,10 @@
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---
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id: 20250911170650
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title: Lipschitz Continuous
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type: permanent
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created: 2025-09-11T21:06:50Z
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modified: 2025-09-11T21:06:50Z
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tags: []
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---
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# Lipschitz Continuous
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@ -3,7 +3,7 @@ id: LIT-20250911143337
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title: Multiple Lyapunov Functions and Other Analysis Tools for Swtiched and Hybrid Systems
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type: literature
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created: 2025-09-11T18:33:37Z
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modified: 2025-09-11T20:55:15Z
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modified: 2025-09-11T21:53:20Z
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citekey:
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---
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@ -46,4 +46,7 @@ Complex ideas written simply. Easy to read.
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**Weak Points? Strong Points?**
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## Zettels Created
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[[Switched Systems]]
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[[hybrid-systems]]
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[[lipschitz-continuous]]
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