$\mathsf{until}_{\lt t}\ f$
$\mathsf{until}_{\< t}\ f$
$a \lt b$
$a_{< t}\ f$
This is an example using a definition environment:
Let $H$ be a subgroup of a group $G$. A left coset of $H$ in $G$ is a subset of $G$ that is of the form $x H$, where $x \in G$ and $x H = \{ x h : h \in H \}$.
Similarly a right coset of $H$ in $G$ is a subset of $G$ that is of the form $H x$, where $H x = \{ h x : h \in H\}$.
Here is a link: Bibliography?
A time axis is a totally ordered set. We indicated it as $\mathbb{T},\mathbb{U},\mathbb{V}$, etc.
The real numbers $\mathbb{R}$, the naturals $\mathbb{N}$, etc.
Any subset of the real numbers, naturals, etc.
Super-dense time.
TODO: Put reference, definition here.
Given a time axis $\mathbb{T}$ and a set $A$, a signal is a map $f:\mathbb{T}\to A$. We write $A^{\mathbb{T}}$ to mean the signal space (set of all signals from $\mathbb{T}$ to $A$).
Given a signal $f:\mathbb{T}\to A$ and a threshold $t\in\mathbb{T}$, the truncations $\mathsf{until}_{\leq t}\ f$ and $\mathsf{since}_{\geq t}\ f$ are the restrictions on $\{a\in\mathbb{T}:a\leq t\}$ and $\{a\in\mathbb{T}:t\leq a\}$. Equivalently we define $\mathsf{until}_{< t}\ f$ and $\mathsf{since}_{< t}\ f$.
TODO: understand why < does not work
A system is a relation between two signal spaces.
There exists a subcategory of $\mathbf{Rel}$ called $\mathbf{Systems}$ where:
The objects are signal spaces $A^{\mathbbT}$.
A morphism $f:A^{\mathbb{T}}\to B^{\mathbb{U}}$ is a relation between $A^{\mathbb{T}}$ and $B^{\mathbb{U}}$.
TODO: Define deterministic, total etc. in the obvious way.
A causal system $S:A^{\mathbb{T}}\to B^{\mathbb{U}}$ is one for which there exists an isomorphism $\alpha:\mathbb{T}\to\mathbb{U}$ such that, for any tree signals $a,b\in A^{\mathbb{T}}$ and $c\in A^{\mathbb{U}}$ for which $aSc$ and $bSc$, it holds that:
We call it stricty causal if
We call it anti-causal if
We call it strictly anti-causal if
Prove that causal and strictly causal systems form a subcategory of $\mathbf{Systems}$.
Characterize the systems that are both causal and anti-causal.
Given a system $S:A^{\mathbb{T}}\to B^{\mathbb{U}}$ there is an opposite system $S^{\text{op}}:A^{\mathbb{T}^{\text{op}}}\to B^{\mathbb{U}^{\text{op}}}$ defined in the obvious way.*
Can you define (strictly) anti-causal from (strictly) causal using the notion of opposite system?
Given a signal $f:\mathbb{T}\to\mathbb{U}$ and an order isomorphism $\sigma:\mathbb{T}\to\mathbb{T}$ we denote as $\sigma\mathbf{;}f$ the βtranslation of $f$ by $\sigma$β.
A time invariant system $S:A^{\mathbb{T}}\to B^{\mathbb{U}}$ is one for which given an order isomorphism $\sigma:\mathbb{T}\to\mathbb{T}$ there exists an order isomorphism $\tau:\mathbb{U}\to\mathbb{U}$ such that
Consider the signal spaces $A^{\mathbb{T}}$ as categories where the objects are the signals and a morphism $\sigma:f\to g$ is an order isomorphism such that $\sigma;f=g$. Can you see the system $S:A^{\mathbb{T}}\to B^{\mathbb{U}}$ as a profunctor?
Given a system $S:A^{\mathbb{T}}\to B^{\mathbb{U}}$ and the order isomorphisms $\sigma:\mathbb{T}\to\mathbb{T}$ and $\tau:\mathbb{U}\to\mathbb{U}$ we construct the system $\sigma;S;\tau$ as the system such that $a(\sigma;S;\tau)b\Leftrightarrow(\sigma\mathbf{;}a)S(\tau\mathbf{;}b).$
Fix two objects of $\mathbf{Systems}$ $A^{\mathbb{T}},B^{\mathbb{U}}$, and letβs try to make $\mathsf{Hom}_{\mathbf{Systems}}(A^{\mathbb{T}};B^{\mathbb{U}})$ into a category. Objects are systems $S_{1},S_{2}:A^{\mathbb{T}}\to B^{\mathbb{U}}$. A morphism $\alpha:S_{1}\to S_{2}$ is a pair of isomorphisms $\langle\sigma,\tau\rangle$ such that $\sigma;S_{1};\tau=S_{2}$. Does this satisfy the requirements for a category? Define identities and composition; show unitality and associativity.
Assuming that the previous exercise gave a positive response, do the arrows-between-arrows we defined count as natural transformations in $\mathbf{Systems}$?
Consider the signal spaces $A^{\mathbb{T}}$ as categories where the objects are the signals and a morphism $\sigma:f\to g$ is a bijection $\chi:$$A\to A$ such that $f;\chi=g$. Can you see a deterministic system $S:A^{\mathbb{T}}\to B^{\mathbb{U}}$ as a functor from $A^{\mathbb{T}}$ to $B^{\mathbb{U}}$?
What are the order isomorphisms of the integers $\mathbb{Z}?$
Let $\mathbf{DiscreteTimeSystems}$ be a subcategory of $\mathbf{Systems}$ obtained by taking the restriction of the objects to those in the form $A^{\mathbb{Z}}$ where $\mathbb{Z}$ are the integers.
Define $\mathbf{StateSpaceDiscreteTS}$ as a subcategory of $\mathbf{Rel}$ where the objects are signal spaces and the morphisms are between $U^{\mathbb{Z}}$ and $Y^{\mathbb{Z}}$are defined by a triple
and $uSy$ iff there exists an $x\in X^{\mathbb{Z}}$ such that
Check that $\mathbf{StateSpaceDiscreteTS}$ is a category. Define composition, identity, etc.
Is $\mathbf{StateSpaceDiscreteTS}$ a subcategory of $\mathbf{Systems}$?
Is $\mathbf{StateSpaceDiscreteTS}$ a subcategory of $\mathbf{DiscreteTimeSystems}$?
Can you find a functor from $\mathbf{DiscreteTimeSystems}$ to $\mathbf{StateSpaceDiscreteTS}$?
If $C$ is a subcategory of $D$ and $D$ is a monoidal category, can we infer that $C$ is a sub-monoidal category?
Is $\mathbf{StateSpaceDiscreteTS}$ a monoidal category?
Consider the set of causal, deterministic, and time-invariant morphism in $\mathbf{DiscreteTimeSystems}$. Show that any of those can be written in state space form.
Consider the set of strictly causal and deterministic morphism in $\mathbf{DiscreteTimeSystems}$. Show that any of those can be written in state space form as
limiting $g$ to not depend on the last $u$.
Define the following types of finite-dimensional linear time-invariant discrete systems. In these equations the signals belong to \left(\mathbb{R}^{n}\right)^{\mathbb{{Z}}}
for some finite $\left(\mathbb{R}^{n}\right)^{\mathbb{{n$. The symbols $A,B,C,D,E$ refer to matrices of suitable dimensions.
Strictly causal:
Causal systems:
Descriptor systems:
Show that these three types of systems are categories and from top to bottom they are ordered by subcategory inclusion.
Are these monoidal categories?
Are these traced monoidal categories?
Define the following types of finite-dimensional linear time-invariant discrete systems. In these equations the signals belong to \left(\mathbb{R}^{n}\right)^{\mathbb{{Z}}}
for some finite $\left(\mathbb{R}^{n}\right)^{\mathbb{{n$. The symbols $A,B,C,D,E$ refer to matrices of suitable dimensions.
Strictly causal:
Causal systems:
Descriptor systems:
Show that these three types of systems are categories and from top to bottom they are ordered by subcategory inclusion.
Are these monoidal categories?
Are these traced monoidal categories?
TODO: we defined βlinearβ as βthose defined by linear equationsβ. Now define the black-box linear behaviors (preserves sums, scaling, etc.).
Define the operation of discretization as, given a period $\Delta,$ as .β¦
Are discretizations functors from continuous to discrete linear systems?
Are discretizations embeddings from continuous to discrete linear systems?
Can you find a bijection from discrete linear systems to continuous linear systems?
Can you find a functor from discrete linear systems to continuous linear systems?
Can you find an embedding from discrete linear systems to continuous linear systems?
TODO: Define z-transforms and their properties.
TODO: Formalize the relationship between linear discrete systems and z-transforms.
TODO: Show what classes of Z-transforms are a traced monoidal category.
TODO: Define laplace-transforms and their properties.
TODO: Formalize the relationship between linear continuous systems and laplace -transforms.
TODO: Show how we can pass from Laplace to Z-transform by change of parameters.
TODO: define property of stability.
TODO: define BIBO stability.
TODO: when is it preserved by composition?
TODO: define properties of:
observability
controllabity
detectability
stabilizability
TODO: how do they behave under the various compositions?
(Set-based uncertainty)
(probability-based uncertainty)