ACT4E
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Definition

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 $H x = \{ h x : h \in H\}$.

Definition (Time axis)

A time axis is a totally ordered set. We indicated it as $\mathbb{T},\mathbb{U},\mathbb{V}$, etc.

Example

The real numbers $\mathbb{R}$, the naturals $\mathbb{N}$, etc.

Example

Any subset of the real numbers, naturals, etc.

Example

Super-dense time.

Definition (Signals).

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 $A$).

Definition (Truncation of a signal).

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 $\mathsf{since}_{< t}\ f$.

Definition (System).

A system is a relation between two signal spaces.

Definition (The category of systems).

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 $B^{\mathbb{U}}$.

Definition (Causal system).

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

Exercise

Prove that causal and strictly causal systems form a subcategory of $\mathbf{Systems}$.

Exercise

Characterize the systems that are both causal and anti-causal.

Lemma

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.*

Exercise

Can you define (strictly) anti-causal from (strictly) causal using the notion of opposite system?

Definition

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 “translation of $f$ by $\sigma$”.

Definition

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

Exercise

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?

Definition

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 $a(\sigma;S;\tau)b\Leftrightarrow(\sigma\mathbf{;}a)S(\tau\mathbf{;}b).$

Exercise

Fix two objects of $\mathbf{Systems}$ $A^{\mathbb{T}},B^{\mathbb{U}}$, and let’s 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.

Exercise

Assuming that the previous exercise gave a positive response, do the arrows-between-arrows we defined count as natural transformations in $\mathbf{Systems}$?

Exercise

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 $B^{\mathbb{U}}$?

Exercise

What are the order isomorphisms of the integers $\mathbb{Z}?$

Definition

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.

Exercise

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 $\mathbf{StateSpaceDiscreteTS}$?

Exercise

If $C$ is a subcategory of $D$ and $D$ is a monoidal category, can we infer that $C$ is a sub-monoidal category?

Exercise

Is $\mathbf{StateSpaceDiscreteTS}$ a monoidal category?

Exercise

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.

Exercise

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 $u$.

Definition

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.

1. Strictly causal:

   $$\begin{aligned} x_{k+1} & =Ax_{k}+Bu_{k}\\ y_{k} & =Cx_{k}\end{aligned}$$

2. Causal systems:

   $$\begin{aligned} x_{k+1} & =Ax_{k}+Bu_{k}\\ y_{k} & =Cx_{k}+Du_{k}\end{aligned}$$

3. Descriptor systems:

   $$\begin{aligned} Ex_{k+1} & =Ax_{k}+Bu_{k}\\ y_{k} & =Cx_{k}+Du_{k}\end{aligned}$$

Exercise

Show that these three types of systems are categories and from top to bottom they are ordered by subcategory inclusion.

Exercise

Are these monoidal categories?

Exercise

Are these traced monoidal categories?

Definition

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.

1. Strictly causal:

   $$\begin{aligned} \dot{x} & =Ax_{t}+Bu_{t}\\ y_{t} & =Cx_{t}\end{aligned}$$

2. Causal systems:

   $$\begin{aligned} \dot{x} & =Ax_{t}+Bu_{t}\\ y_{t} & =Cx_{t}+Du_{t}\end{aligned}$$

3. Descriptor systems:

   $$\begin{aligned} E\dot{x} & =Ax_{t}+Bu_{t}\\ y_{t} & =Cx_{t}+Du_{t}\end{aligned}$$

Exercise

Show that these three types of systems are categories and from top to bottom they are ordered by subcategory inclusion.

Exercise

Are these monoidal categories?

Exercise

Are these traced monoidal categories?

Definition

Define the operation of discretization as, given a period $\Delta,$ as .…

Exercise

1. Are discretizations functors from continuous to discrete linear systems?

2. Are discretizations embeddings from continuous to discrete linear systems?

Exercise

1. Can you find a bijection from discrete linear systems to continuous linear systems?

2. Can you find a functor from discrete linear systems to continuous linear systems?

3. Can you find an embedding from discrete linear systems to continuous linear systems?