Additive state decomposition occurs when a system is decomposed into two or more subsystems with the same dimension as that of the original system.
[1][2] A commonly used decomposition in the control field is to decompose a system into two or more lower-order subsystems, called lower-order subsystem decomposition here.
In contrast, additive state decomposition is to decompose a system into two or more subsystems with the same dimension as that of the original system.
[3] Taking a system P for example, it is decomposed into two subsystems: Pp and Ps, where dim(Pp) = np and dim(Ps) = ns, respectively.
The lower-order subsystem decomposition satisfies By contrast, the additive state decomposition satisfies Consider an 'original' system as follows: where
are defined as follows: Then the secondary system can be further written as follows: From the definition (3), it follows The process is shown in this picture: In fact, the idea of the additive state decomposition has been implicitly mentioned in existing literature.
An existing example is the tracking controller design, which often requires a reference system to derive error dynamics.
By additive state decomposition Since the tracking error e(t) can be analyzed by ep(t) and es(t) separately.
Fortunately, note that (6) is a linear time-invariant system and is independent of the secondary system (7), for the analysis of which many tools such as the transfer function are available.
By contrast, the transfer function tool cannot be directly applied to the original system (5) as it is time-varying.
Here, the task yp → 0 is assigned to the linear time-invariant system (9) (a linear time-invariant system being simpler than a nonlinear one).
On the other hand, the task xs → 0 is assigned to the nonlinear system (10) (a stabilizing control problem is simpler than a tracking problem).
The basic idea is to decompose an original system into two subsystems in charge of simpler subtasks.
The process is shown in this picture: A well-known example implicitly using additive state decomposition is the superposition principle, widely used in physics and engineering.The superposition principle states: For all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.
For a simple linear system: the statement of the superposition principle means x = xp + xs, where Obviously, this result can also be derived from the additive state decomposition.
Moreover, the superposition principle and additive state decomposition have the following relationship.