Lesson 16: Observability

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Systems and Control Theory Lecture Notes

Laura Giarr´ e

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Lesson 16: Observability

Observability (DT)

Observability (CT)

Kalman standard form

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Observability

a state x = q is said unobservable over [0 T ] , if for every input u, the output y

_q

obtained with initial condition

x(0) = q is the same as the output y

₀

obtained with initial condition x(0) = 0.

A dynamic system is said unobservable if it contains at least an unobservable state, observable otherwise.

Note that observability can be established assuming zero input.

The initial state x(0) can be uniquely determined from

input/output measurements iﬀ the system is observable

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Discrete-time Observability

Given a system described by the n state-space model

x(k + 1) =Ax(k) + Bu(k) y(k) =Cx(k) + Du(k)

Suppose we are given u(k) and y(k) for 0 ≥ t < T .

⎡

⎢ ⎢

⎢ ⎣

y(0) y(1) .. . y(T − 1)

⎤

⎥ ⎥

⎥ ⎦ ==

⎡

⎢ ⎢

⎢ ⎣

C CA .. . CA

^{T −1}

⎤

⎥ ⎥

⎥ ⎦ x(0)+

⎡

⎢ ⎢

⎢ ⎣

D 0 0 . . . 0

CB D 0 . . . 0

.. . .. . .. . .. . CA

^{T −2}

B CA

^{T −3}

B . . . 0

⎤

⎥ ⎥

⎥ ⎦

⎡

⎢ ⎢

⎢ ⎣

u

₀

u

₁

.. .

u(T − 1)

⎤

⎥ ⎥

⎥ ⎦

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Discrete-time Observability

Now the second term on the right ( the forced response ) is known, so we can subtract it from the vector of measured outputs to get

¯y =

⎡

⎢ ⎢

⎢ ⎣

¯y(0)

¯y(1) .. .

¯y(T − 1)

⎤

⎥ ⎥

⎥ ⎦ =

⎡

⎢ ⎢

⎢ ⎣

C CA .. . CA

^{T −1}

⎤

⎥ ⎥

⎥ ⎦ x(0) = O

_T

x(0)

O

_T

is the T -step observability matrix

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Observability (DT): Properties

Theorem

The set of states that is unobservable over T steps is precisely N (O

_T

), and is therefore a subspace.

Notice also that

N (O

_k

) ⊇ N (O

_k+1

)

N (O

_n

) = N (O

_n+l

), l ≥ n

Theorem

If x(0) = ξ is unobservable over n steps, then it is

unobservable over any number of steps. Equivalently, the

system is observable if and only if rank(O

_n

) = n.

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Observability Gramian (DT)

We begin by deﬁning the k-step observability Gramian as

Q

_k

= O

_k^T

O

_K

=

k−1

i=0

(A

ⁱ

)

^T

C

^T

CA

ⁱ

The unobservable space over k steps is evidently the nullspace of Q

_k

.

The system is observable if and only if rank(Q

_n

) = n

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Continuous-time observability

For continuous-time systems, the following are equivalent:

q is unobservable in time T is unobservable in any time.

Observability Gramian:

Q

_t

=

_t

0

(e

^Aτ

)

^T

C

^T

Ce

^Aτ

dτ

The system is then observable if and only if rank(Q

_t

) = n

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Other results

Essentially, observability results are similar to their reachability counterparts, when considering (A

^T

, C

^T

) as opposed to

(A, B). ( Duality Theorem)

(A, C) is unobservable if Cv

i

= 0 for some (right) eigenvector v

_i

of A.

(A, C) is unobservable if sI?A drop ranks for some s = λ, this λ is an unobservabe eigenvalue for the system.

The dual of controllability to the origin is constructability:

same considerations as in the reachability/controllability case

hold.

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Standard form (Observability)

We can construct a matrix T

_¯o^n×o

, whose columns span the nullspace of the observability matrix O

n

.

We may then construct T

_¯o

by selecting (n − o) linearly independent vectors, such that

rank(T

^n×n

) = rank[T

¯o

T

_o

] = n.

Since T is invertible, we can perform a similarity transform to generate an equivalent system:

AT = A[T

_¯o

T

_o

] = T ¯A = [T

_¯o

T

_o

]

A

₁

A

₂

0 A

₃

, CT = C [T

_¯o

T

_o

] = ¯C = [0 C

1

]

The presence of the zero blocks in the transformed system and output matrices follows from an argument similar to that used for the reachable canonical form coupled with the fact the the unobservable space is also A-invariant.

O = rank(O )

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Thanks