Observability

One will consider in this article the linear systems invariants ( SLI ) defined by the following representation of state:

\ begin {boxes} \ dowry X = HAS X + B U \ \ Y = C X + D U \ end {boxes}

A system is known as observable if the observation of its entries and exits during an time interval finished t_i; t_f makes it possible to find the initial state x (t_i) . In fact, since it is possible for the SLI to have an analytical solution, observability is thus an interesting property which enables us to affirm that one can know the state x (T) at any moment included/understood in the interval t_i; t_f .

Diagonal system

That is to say a system describes by the Représentation of state \ begin {boxes} \ dowry X = HAS X + B U \ \ Y = C X \ end {boxes} with has a diagonal Matrice. This system is observable if and only if all the elements of the matrix C are nonnull.

Criterion of $kalman for the observability of the linear systems invariants

In a more general case, the system is observable if and only if:
rang (\ mathcal {O}) = row \ begin {bmatrix} C \ \ CA \ \… \ \ CA^ {n-1} \ end {bmatrix} = n

The matrix \ mathcal {O} is called the matrix of observability, and its lines are calculated in an iterative way easily: CA^ {k+1} = CA^ {K} * A.

Duality observability/Commandabilité

There exists a principle of duality between observability and the Commandabilité: are two systems:

S: \ begin {boxes} \ dowry X = AX+BU \ \ Y=CX \ end {boxes}

S^*: \ begin {boxes} \ dowry X^* = A^TX^*+C^TU^* \ \ Y^*=B^TX^* \ end {boxes}

  • S is observable if and only if S^* is commandable

  • S is commandable if and only if S^* is observable
  • a system at the same time commandable and observable is known as minimal .

Detectability

Observability is a strong property structural of the system. It is often sufficient to use the property of detectability. This last property can be defined in several ways equivalent, a system is known as detectable if:
  • Its nonobservable poles are stable.
  • There exists a matrix of profit of Observateur of state K such that the matrix (A-KC) is Hurtwitz.

Canonical form for observability

It is often interesting to separate the observable variables of state from the others. Let us note \ chi the partition observable of the vector of state, and \ xi the remainder of the vector of state, nonobservable. The system is written then:
\ begin {bmatrix} \ dowry \ chi \ \ \ dowry \ xi \ end {bmatrix} = \ begin {bmatrix} A_ {11} & 0 \ \ A_ {21} & A_ {22} \ end {bmatrix} \ begin {bmatrix} \ chi \ \ \ xi \ end {bmatrix} + \ begin {bmatrix} B_1 \ \ B_2 \ end {bmatrix} U

there = \ begin {bmatrix} C_1 & 0 \ end {bmatrix} \ begin {bmatrix} \ chi \ \ \ xi \ end {bmatrix}

The canonical form for observability is characterized by the absence of the terms A_ {12} and C_2 which is thus null. In this form, the system is then detectable if the matrix A_ {22} is Hurtwitz.

See too

  • Automatic Commandabilité
  • Representation of Observant state
  • of state

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