Controllability

A system is fully controlled, when a state may be transferred in a finite time by appropriate control signals at any given new state. Tax is a system if it can be transferred from the selected initial states in selected final states. Controllability thus describes the influence of external input variables (generally, the control values) to the inner system state. A distinction is made between Ausgangssteuerbarkeit and Zustandssteuerbarkeit.

Colloquially, the word is often used in the sense of controlled variable. In technical terms, however, a distinction between controllable, observable and controllable. For a system to be controlled, it must both be possible to monitor its state, as well as to control them. For practical application, normally controllability is relevant. Steurbarkeit is one aspect of it.

The two terms controllability and observability was introduced after 1960 by Rudolf Kalman.

• 2.1 Full Ausgangssteuerbarkeit
• 2.2 Full Zustandssteuerbarkeit 2.2.1 Criterion of Kalman
• 2.2.2 Criterion of Gilbert
• 2.2.3 Criterion of Hautus

Definition

Starting point for assessing the controllability of a linear system is the state space representation

Using the system matrix, control matrix, observation matrix, the passage of the matrix, the state vector, the output vector and the control vector.

For determining the controllability, it is different depending on the form of the state space representation criteria.

Complete controllability

Fully controllable state (also called distance) is a linear system, if there is a control function for each of the initial state, which results in the system within any finite period of time in any final state.

Structural controllability

One class of systems is structurally controlled, when there is at least a system that is fully controlled.

Here are matrices in which all elements were equal to 0 marked with * since all elements equal to 0 to decide on structural observability and structural controllability. That the det S must be equal to 0.

Steuerbarkeitskriterien

Full Ausgangssteuerbarkeit

The system is exactly then completely output controllable if the rank of the matrix

Coincides with the number of output variables: condition for Ausgangssteuerbarkeit is so rank. Under the assumption that rank (C) = r we have, every state controllable system is output controllable. The converse is not true here.

Full Zustandssteuerbarkeit

Criterion of Kalman

The system is completely controllable if and only according to Kalman, if the controllability

Applies

In the special case is even invertible for controllable systems, which is a prerequisite for the use of the formula for Ackermann to pole placement for SISO systems. Zustandssteuerbarkeit by the Kalman is a special case for the complete Ausgangssteuerbarkeit.

Criterion of Gilbert

The system, the state space model is present in a canonical normal form, if and Gilbert completely controlled when the matrix does not have a zero line, and when the p rows of the matrix, are part of the canonical variables for a p- times the natural value, are linearly independent.

Is the matrix of eigenvectors.

Criterion of Hautus

The system (A, B) is completely controllable if and only after Hautus if the condition

Is satisfied for all eigenvalues ​​of the matrix A.

Controllability of scanning systems

The above relationships are also valid for scanning, if is replaced by the transition matrix and by the discrete input matrix. After the verification can be simplified by first the conditions for the continuous system are checked and then the additional condition

Is satisfied.

Control normal form ( Frobenius form)

The control normal form can among other things, the transfer function can be easily determined. For the following applies:

Or for systems without derivatives of the input variable

The particular shape of and is useful for the analysis and the construction of state controllers.

Nonlinear controllability and flatness

In Nonlinear You can not make global statements about the controllability and must always connect it to a scope. Special role of the mathematical operator ad here.

Therefore, the system property of the flatness extends controllability of the non-linear case. In the linear case controllable systems are also flat.

However, exercise care when closing on the controllability of the nonlinear system from the linearization. The linearization around a point controlled, the non-linear system is locally controlled to this point. If it is not controllable linearization, the system can nevertheless still be controllable.

Reasons for not fully controllable systems

For the non- full controllability, there are two essential reasons:

Reasons for the inquiry

The Steuerbarkeitskriterium can also be used to simplify a control task. If not, the manipulated variable, but the disturbance investigated for their controllability with respect to the control variable, it shows a uncontrollability that this part of the system is not subject to perturbation and thus this part need not be regulated, if only the fault should be suppressed. On the other hand, can not be compensated a disturbance if a system part by the disturbance but is not controllable by the manipulated variable.

The property of the uncontrollability of the disturbance is used for some control methods. So is developed in the Noise neutralization of the knob so that the manipulated variable does not depend on the disturbance.