The control theory (also control theory ) is a branch of applied mathematics . They considered dynamic systems whose behavior through input variables can be influenced from the outside. Such systems are e.g. B. The subject of control engineering from which control theory emerged.
Examples of systems can be found in numerous and varied application areas from natural sciences , technology and medicine , economics , biology , ecology and from the social sciences . The planet earth , cars , people, economic spaces, 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 influencing by specifying suitable input variables. Typical practical questions are, for example:
- Is the system stable ?
- How sensitive does the system react to disturbances and model uncertainties?
- Do all system variables stay in certain areas?
- Is it possible to achieve a given desired goal state?
- How must the input variable be selected in order to achieve a target state in the shortest possible time and with the least amount of effort?
A prerequisite for precisely answering such questions is the introduction of mathematical models to describe the system. Based on these models, further mathematical concepts and terms for stability, controllability and observability were developed in control theory.
Mathematical model forms
The mathematical modeling is the basis of statements about given dynamic systems.
A selection of common model forms for systems with value-continuous behavior are:
- Ordinary differential equations
- Partial differential equations
- Stochastic differential equations
- Differential Inclusions
Continuous ordinary differential equations can be represented by
- Block diagrams and
- Bond graph.
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:
The combination of continuous and discrete-event systems is called hybrid systems , for example
- discontinuous differential equations,
- Systems with switching dynamics,
- hybrid automata.
On the basis of the mathematical models in control theory, answers z. B. searched for the following questions:
- Simulation / prediction (solution of the initial value problem )
- Stability analysis
- Accessibility analysis, controllability analysis , observability analysis
- Security analysis
- Robustness analysis
- Chaos / Bifurcation Analysis
- Imprinting a desired behavior.
Of current interest is the consideration of complex dynamic systems , which lead to complex problems. Complex problems are those problems whose representation and solution require a “large” amount of storage space and / or computing time. Some problems in control theory lead to undecidable mathematical problems. The reduction of the complexity of practically relevant problems so that their (approximate) practical solvability is guaranteed is the subject of ongoing research.
In contrast to standard control engineering, various analytical and numerical methods are used to model such mostly non-linear systems:
- Solution concepts of differential equations
- Stability theory according to Lyapunov
- Convergence terms
- Signal standards , system standards , operator standards
- Riccati equations
- Calculus of variations
- Convex optimization
- Global and local optimization calculation
- Invariant sets .
Since control theory emerged from theoretical control engineering, it is applied in control engineering and in all automation engineering.
Another typical application relates to system fault tolerance. Since the targeted influencing of complex systems is often expensive and risky, a correspondingly high effort is made for observation and control. The statements of control theory often support decisions under uncertainty and must therefore be accompanied by appropriate risk management and an analysis of the possibility of errors and influences ( FMEA ). See also fault tolerant rule system .
- ^ Eduardo D. Sontag: Mathematical Control Theory. Deterministic Finite Dimensional Systems (= Texts in Applied Mathematics. 6). 2nd Edition. Springer, New York NY et al. 1998, ISBN 0-387-98489-5 .
- ↑ Vincent D. Blondel, John N. Tsitsiklis: A survey of computational complexity results in systems and control. In: Automatica. Vol. 36, No. 9, 2000, pp. 1249-1274, doi : 10.1016 / S0005-1098 (00) 00050-9 .