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Finite Dimensional Controllability

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Mathematics of Complexity and Dynamical Systems

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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

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Abbreviations

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 :

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 :

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:

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 :

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.

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Author information

Authors and Affiliations

  1. Institut Elie Cartan, Vandoeuvre-lès-Nancy, France

    Lionel Rosier

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  1. Lionel Rosier
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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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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