Control theory
This article is about mathematical control theory. For criminological theories of the same name, see Criminological Control Theory. In addition, control theory stands for the theory of optimal control.
Control theory is a branch of applied mathematics. It considers dynamic systems whose behavior can be influenced by external input variables. Such systems are, for example, the subject of control engineering, from which control theory has emerged.
Examples of systems can be found in numerous and diverse fields of application in the natural sciences, technology and medicine, economics, biology, ecology and the social sciences. Planet Earth, cars, people, economies, cells, ecosystems, and societies are examples of systems. Typical questions in control theory concern the analysis of a given system as well as its targeted manipulation by specifying suitable input variables. Typical practical questions are, for example:
- Is the system stable?
- How sensitive is the system to disturbances and model uncertainties?
- Do all system variables stay within certain ranges?
- Is it possible to achieve a given desired target state?
- How must the input variable be chosen in order to achieve a target state in the shortest possible time and with the least possible effort?
A prerequisite for precisely answering such questions is the introduction of mathematical models for describing systems. Based on these models, further mathematical concepts and terms for stability, controllability and observability have been developed in control theory.
Mathematical model forms
Mathematical modeling is the basis of statements about given dynamical systems.
A selection of common model forms for systems with continuous value behavior is:
- Ordinary differential equations
- Partial differential equations
- Stochastic differential equations
- Differential enclosures
Continuous ordinary differential equations can be represented by
- Block diagrams and
- Bond graphs.
The differential equations can be linear (e.g. state space model, transfer function) or nonlinear (e.g. Hammerstein model, Wiener model). Problems based on nonlinear models are generally more difficult.
Examples of systems with discrete-event behavior are:
- Automats
- Petri Nets
The combination of continuous and discrete-event systems is called hybrid systems, for example
- discontinuous differential equations,
- Systems with switching dynamics,
- hybrid automata.
Cross Sectional Problems
Based on mathematical models, control theory seeks answers to questions such as:
- Simulation / prediction (solution of the initial value problem)
- Stability analysis
- Reachability analysis, controllability analysis, observability analysis
- Security analysis
- Robustness analysis
- Chaos / Bifurcation Analysis
- Imposing a desired behavior.
Of current interest is the consideration of complex dynamical systems, which lead to complex problems. By complex problems are meant those problems whose representation and solution require a "large" amount of memory and/or computation time. Some problems in control theory lead to non-decidable mathematical problems. Reducing the complexity of practically relevant problems so that their (approximate) practical solvability is guaranteed is the subject of ongoing research.
