Skip to content Skip to navigation

Connexions

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

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

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

      What is in a lens?

      Lens makers point to Connexions 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 Connexions 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
  • E-mail the authors
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

The Concept of State

Module by: Thanos Antoulas, JP Slavinsky

Summary: Memory and state in systems.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 utut for t>t0tt0
.

Example 1

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

ddtyt+yt=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 ytyt:

yt=yt0ⅇt0−t y t y t0 t0 t (2)

As we need the information contained in yt0yt0 for this solution, ytyt defines the state.

Example 2

The differential equation describing an unforced system is:

d2dt2yt+3ddtyt+2yt=0 t2 y t 3 t1 y t 2 y t 0 (3)

Finding the qsqs function, we have

qs=s2+3s+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:

yt=c1ⅇ-t+c2ⅇ-2t 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 y0y0 and ddty0 t1 y 0 we could find two equations, and we could then solve for ytyt. Therefore the system's state, xtxt, is

xt=ytddtyt 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 ytyt and ddtyt t1 y t .

Important!:

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

Comments, questions, feedback, criticisms?

Send feedback