Control theory
The control theory (including control theory ) is a branch of applied mathematics. It considers dynamic systems whose behavior can be influenced by so-called input variables from the outside. Such systems are, for example, the subject of control engineering, from the control theory has emerged.
Examples of systems can be found in numerous and diverse areas of application in science, technology and medicine, economics, biology, ecology, and from the social sciences. The planet Earth, cars, people, utility rooms, cells, ecosystems and societies are examples of systems. Typical questions in control theory relate to the analysis of a given system and its specific influence by specifying appropriate input variables. Typical practical questions are for example:
- Is the system stable?
- How sensitive the system responds to disturbances and model uncertainties?
- If all system variables in safe areas?
- Is it possible to achieve a given desired target state?
- What must be chosen, the input variable to reach a target state in the shortest time and with the least effort?
Prerequisite for a precise answer to such questions is the introduction of mathematical models for the system description. Further mathematical concepts and terms for stability, controllability and observability have been developed in the control theory based on these models.
Overview of mathematical model forms
The mathematical representation of the given model is based on meaningful statements about given dynamic systems. A selection of common model forms for systems with wertekontinuierlichem behavior are:
- Ordinary Differential Equations
- Partial Differential Equations
- Stochastic Differential Equations
- Differential inclusions
Continuous ordinary differential equations by
- Block diagrams and
- Bond Count
Are shown.
The differential equations can be (eg, Hammerstein model, Wiener model) linear ( eg, state space model, transfer function ) or non-linear. Problems on the basis of non-linear models are generally more difficult to solve than those based on linear models. Examples of systems with event behavior are:
- Machines
- Petri nets
The combination of continuous and discrete event systems are called hybrid systems, such as
- Discontinuous differential equations,
- Systems with switching dynamics,
- Hybrid automata.
Cross- cutting problems of control theory
On the basis of mathematical models are searched for answers to questions in control theory, some of which are listed by way of example:
- Simulation / prediction ( solution of the initial value problem )
- Stability analysis
- Reachability analysis, Steuerbarkeitsanalyse, observability
- Safety analysis
- Robustness analysis
- Chaos / bifurcation analysis
- Imparting a desired behavior
Of current interest is the consideration of complex dynamic systems, which lead to complex problems. With complex problems, such problems are meant, whose representation and solution a "large" amount of disk space and / or computing time. Some problems of control theory lead to non- decidable mathematical problems. The reduction of complexity practically relevant issues, so that their ( approximate ) practical solvability is guaranteed, is the subject of ongoing research.
Mathematical Tools
To apply come of it, in contrast to the standard control technology, various analytical and numerical mathematical methods that are used for modeling of such usually non-linear systems:
- Solutions of differential equations
- Stability Theory by Lyapunov
- Concepts of convergence
- Signal standards, system standards, operator norms
- Riccati equations
- Calculus of variations
- Convex optimization
- Global and local optimization calculation
- Invariant quantities
Applications
Since the control theory emerged from the theoretical control technology, it is applied course in control engineering or in the entire automation technology.
Another typical application relates to fault tolerant systems. Since often the targeted modulation of complex systems is expensive and risky, according to a high overhead for monitoring and control is operated. The statements of the control theory often support decisions under uncertainty and therefore need to be accompanied by adequate risk management and an analysis of errors and influence (FMEA ). See also Fault tolerant control system.