New system theory and its impact on control theory

Abstract

Real systems can include two types of state variables – dynamic and static. While dynamic state variables are a common part of each system, static variables are not and their presence in a system may cause some problems if standard system theories are used. In this paper, it is shown that, due to a new system theory (NST), it is possible to work correctly with systems and subsystems which include not only dynamic state variables, but also static state variables. If standard system theories are used, static variables in the real system cause not only problems in describing systems but also some challenges in control theory. These challenges involve, for example, some questions of controllability, reachability, or observability of a plant that includes static variables or the optimal control design of a plant that includes statical state variables. Some of the challenges mentioned are addressed in this paper after a brief introduction of the NST.

Keywords:

• system theory
• controllability
• reachability
• optimal control
• static dependence

Notes

^1. Two previous sentences define the Principle of Causality.

^2. It holds, that if for any time instant , , then for all time instants , .

^3. The notion which is used in this subsection is identical to the notion that is used in Definition 2.4.

^4. The symbol represents the Gaussian distribution, where the mean value of this distribution is the vector a and covariance is matrix b.

^5. It is supposed that subsystem contains at least one static attribute and that static attributes of this subsystem are ‘suitably indexed’. If subsystem does not contain any static attribute, then controllability and reachability of such a subsystem is judged by using standard criteria mentioned e.g. in (Åström and Wittenmark 1997).

^6. It is possible to impose various restrictions on set .

^7. It means that the causal dependence exists for the pertinent l and , .

^8. It means that subsystem variable at time instant exists and causal dependencies , exist for the pertinent and , , , . Or, some subsystem variables at time instant exist and causal dependencies , , exist for the pertinent and , , , , , , etc.

^9. The batch control strategy means that a control at time instant is generated at once as the vector (it is a common control strategy). The set of all admissible controllers which can generate only the batch control strategy is marked and it holds that .