| A. Decomposition/Partitioning/Discretization of the State-Space
(= Classification of States)
|
| B. Decomposition of States into Vectors
(= Analysis of the State-Space into Dimensions)
.---+--__
| | S2 |----.
| S1 |____| S4 |
`--_/ S3 | _-'
`-____|-' |
For modeling, S's state-space can be partitioned
into discrete sub-spaces SS = {S1, ..., Sn}.
These sub-spaces, taken as sets of states,
induce a classification on the system's states,
called the system's modes.
Each particular state s = S(t) of S at a time t
belongs to one of these modes / sub-spaces.
Formally, eg., S(t)  S1.
A state-space can be partitioned (states can be classified) in several ways.
The question is whether the boundary between two sub-spaces of the state space continuum
was drawn arbitrarily (ad-hoc division),
or in a way that matches a discontinuity (phase-shift, qualitative jump, ...)
in the actual state space of the real system S
(ontologically founded, or natural division).
SS
/ _S2 \
| S1 (_) _ |
| (*) _ (_) |
| (_) S4 |
\ S3 / |
If the description of this model does not distinguish the different states in the same sub-space,
then the described abstract model M can only be said in an Si-state or not
- the sub-spaces have become abstract states,
they are the states of abstract model M.
Formally, M(t) = S1.
The abstract states can be shown discretely, eg., as circles,
with the current state indicated by some mark.
The level of abstraction can be varied by partitioning the state-space recursively,
i.e., by a hierachical classification of states:
_____________
.' |
| `. S11 | S2
| `······|
| S12 : |
| : S13 |----
`-___: / S3
`___/
`·__
` |
spatial diagram:
SS
/ S1 S2_ \
| / S11 \ (_) |
| | _ (*) | _ |
| | (_) S13 | (_) |
| | S12 (_) | _ S3|
| \_________/ (_) |
\ S4 /
associative diagram:
SS
____^______
| | | |
S1 S2 S3 S4
___^___
| | |
S11 S12 S13 |
The entire state-space SS can be regarded
as the top in this classification hierarchy.
The subset relationship between S's state sub-spaces becomes
a specialization relationship between M's abstract states:
If M is in state S11 at time t,
this implies that it is also in state S1.
(Since we cannot have M(t) = S11
and M(t) = S1 at the same time,
"the state" of M must be agreed to mean
the most detailed abstract state available in the model description.)
|
^ .······..
| : `····.
| : ·s :
| `··. .·'
| / `·.....·'
|/
+-----------------> |
For modeling, S's state-space can analyzed into different dimensions
f1, ..., fn.
Then a particular state s = M(t) of M at time t
is a state vector (v1, ..., vn).
SS
| speed |
| | --> | temp. |
| | 15C | |
| color |
| | brown | |
| | |
Each dimension can be treated individually as a state variable of the system,
such that fi(t) is the value in the state-dimension fi at time t.
The state can then be shown as a collection of values of the state variables.
A state-space can be analyzed in several ways into dimensions.
The question is whether a state-variable is arbitrary,
or whether it corresponds to
- an actual (ontic) property of the real system S
(e.g., speed, weight, height, temperature, color, ...).
- the state of a subsystem (real, ontic part) of S
-- That is, the state-space analysis into dimensions
can be related to the decomposition of the system into subsystems
-> see system architecture
A dimension does not necessarily have to be scalar;
it may be recursively analyzable into sub-dimensions. E.g.
- composed property speed = × speed wrt. each spatial dimension,
- additive property weight =  weight of each subsystem,
- average property temperature = average of temp. of each subsystem,
- mixture property color =  red-, green- and blue-intensities
(or color = average of color of each subsystem).
The entire state-space SS can be regarded as the top
in this composition hierarchy.
spatial diagram: associative diagram:
SS
SS <>
| color _ | .----^------.
| | | [_] | speed ... color
| | [R%] [G% [B%] | _ | <> <>
| | | [_] | / | \ / | \
| | x' y' z' R% G% B%
|
|
| with dimensions ...
| with partitions ...
|
One abstract state can be analyzed into several dimensions,
whose value range is divided individually into sub-spaces.
(This can also be applied to the entire state-space SS,
as the most general of the abstract states.)
spatial diagram associative diag.
SS
___^___
SS | | ...
/ S1 \ S1
| / S1a: (a) (b) (c) \ | <>
| \ S2b: (d) (e) (f) / | ___|___
| _ _ _ | | |
| (_)S2 (_)S3 (_)S4 | S1a S1b
\ / ^ ^ ^ ^ ^ ^
| | | | | |
a b c d e f |
Note that each abstract state can be analyzed differently
into dimensions (state variables).
|
The value range (state-space) of each state-variable can separately be partitioned
into discrete values (sub-states):
SS
| ··· |
| _color_ _ _ _ |
| | (_) (_) (_) (_) (_) ... | |
| | black white red blue green | |
| | |
Also sub-dimensions can be partitioned:
SS
| color |
| | R% G% B% | |
| | | (0) (50) | | (0) (50) | | (0) (50) | | |
| | | (100) | | (100) | | (100) | | |
| | | |
| | |
|
| |
|
|
| | Transitions - temporal order [diachronous]
|
| Functional Dependence - at each time [synchronous]
|
The question which a state-space partition model begs to answer is:
If in one (abstract) state, to which other (abstract) states can the system get (next)?
| |
The question which a state vector model begs to answer is:
How is the variance in each dimension
/ of each state variable [aka `determinable']
constrained by the other dimensions/variables?
«The values some determinables take for certain objects
may be partly or wholly determined by the values of other determinables.
In such cases we speak of functional dependence
of one determinable on another.»
[Parts 344]
NB functional dependency between determinables of different objects
is an important criterion for considering these objects to form a system,
or integrated whole (as opposed to a mere aggregation)
-> systems vs. aggregates
| | | | | |
Modeling, Models, Notations:
