Article Outline
Glossary
Definition of the Subject
Introduction
Control Systems
Linear Systems
Linearization Principle
High Order Tests
Controllability and Observability
Controllability and Stabilizability
Flatness
Future Directions
Bibliography
Keywords
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- Control system :
-
A control system is a dynamical system incorporating a control input designed to achieve a control objective. It is finite dimensional if the phase space (e. g. a vector space or a manifold) is of finite dimension. A continuous‐time control system takes the form \({{\text{d}} x/{\text{d}} t=f(x,u)}\), \({x\in X}\), \({u\in U}\) and \({t\in {\mathbb{R}}}\) denoting respectively the state, the input, and the continuous time. A discrete‐time system assumes the form \({x_{k+1}=f(x_k,u_k)}\), where \({k\in {\mathbb{Z}}}\) is the discrete time.
- Open/closed loop:
-
A control system is said to be in open loop form when the input u is any function of time, and in closed loop form when the input u is a function of the state only, i. e., it takes the more restrictive form \({u=h(x(t))}\), where \({h{\colon}X\to U}\) is a given function called a feedback law .
- Controllability :
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A control system is controllable if any pair of states may be connected by a trajectory of the system corresponding to an appropriate choice of the control input.
- Stabilizability :
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A control system is asymptotically stabilizable around an equilibrium point if there exists a feedback law such that the corresponding closed loop system is asymptotically stable at the equilibrium point.
- Output function:
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An output function is any function of the state.
- Observability:
-
A control system given together with an output function is said to be observable if two different states give rise to two different outputs for a convenient choice of the input function.
- Flatness :
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An output function is said to be flat if the state and the input can be expressed as functions of the output and of a finite number of its derivatives.
Bibliography
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Rosier, L. (2012). Finite Dimensional Controllability. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_26
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