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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 utut for t>t0tt0
.

Example 1

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

d1ytdt1+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=yt0et0t 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:

d2ytdt2+3d1ytdt1+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=c1et+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=( yt d1ytdt1 ) 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.

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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?

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

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