Skip to content Skip to navigation

Connexions

You are here: Home » Content » The Concept of State

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "State Space Systems"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Recently Viewed

This feature requires Javascript to be enabled.
 

The Concept of State

Module by: Thanos Antoulas, JP Slavinsky. E-mail the authors

Summary: Memory and state in systems.

In order to characterize the memory of a dynamical system, we use a concept known as state.

Important!:

A system's state is defined as the minimal set of variables evaluated at t=t0tt0 needed to determine the future evolution of the system for t>t0tt0, given the excitation u⁢tut for t>t0tt0
.

Example 1

We are given the following differential equation describing a system. Note that u⁢t=0 ut 0 .

d1y⁢tdt1+y⁢t=0 t1 y t y t 0
(1)

Using the Laplace transform techniques described in the module on Linear Systems with Constant Coefficients, we can find a solution for y⁢tyt:

y⁢t=y⁢t0⁢et0−t y t y t0 t0 t
(2)

As we need the information contained in y⁢t0yt0 for this solution, y⁢tyt defines the state.

Example 2

The differential equation describing an unforced system is:

d2y⁢tdt2+3⁢d1y⁢tdt1+2⁢y⁢t=0 t2 y t 3 t1 y t 2 y t 0
(3)

Finding the q⁢sqs function, we have

q⁢s=s2+3⁢s+2 q s s 2 3 s 2
(4)

The roots of this function are λ1=-1 λ1 -1 and λ2=-2 λ2 -2 . These values are used in the solution to the differential equation as the exponents of the exponential functions:

y⁢t=c1⁢e−t+c2⁢e-2⁢t y t c1 t c2 -2 t
(5)

where c1c1 and c2c2 are constants. To determine the values of these constants we would need two equations (with two equations and two unknowns, we can find the unknowns). If we knew y⁢0y0 and ddty⁢0 t1 y 0 we could find two equations, and we could then solve for y⁢tyt. Therefore the system's state, x⁢txt, is

x⁢t=( y⁢t d1y⁢tdt1 ) x t y t t1 y t
(6)

In fact, the state can also be defined as any two non-trivial (i.e. independent) linear combinations of y⁢tyt and ddty⁢t t1 y t .

Important!:

Basically, a system's state summarizes its entire past. It describes the memory-side of dynamical systems.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks