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The Concept of State

Module by: Thanos Antoulas, JP Slavinsky

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 .

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.

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