Observability is a measure of how well internal states of a
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
can be inferred from knowledge of its external outputs.
In
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, the observability and
controllability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observabil ...
of a linear system are mathematical
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...
.
The concept of observability was introduced by the Hungarian-American engineer
Rudolf E. Kálmán
Rudolf Emil Kálmán (May 19, 1930 – July 2, 2016) was a Hungarian Americans, Hungarian-American electrical engineer, mathematician, and inventor. He is most noted for his co-invention and development of the Kalman filter, a mathematical algo ...
for linear dynamic systems. A dynamical system designed to estimate the state of a system from measurements of the outputs is called a
state observer or simply an observer for that system.
Definition
Consider a physical system modeled in
state-space representation. A system is said to be observable if, for every possible evolution of
state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by
sensor
A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon.
In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
s). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.
Linear time-invariant systems
For
time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable. Consider a
SISO system with
state variables (see
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
for details about
MIMO
In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wir ...
systems) given by
:
:
Observability matrix
If and only if the column
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the ''observability matrix'', defined as
:
is equal to
, then the system is observable. The rationale for this test is that if
columns are linearly independent, then each of the
state variables is viewable through linear combinations of the output variables
.
Related concepts
Observability index
The ''observability index''
of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied:
, where
:
Unobservable subspace
The ''unobservable subspace''
of the linear system is the kernel of the linear map
given by
[Sontag, E.D., "Mathematical Control Theory", Texts in Applied Mathematics, 1998]where
is the set of continuous functions from
to
.
can also be written as
:
Since the system is observable if and only if
, the system is observable if and only if
is the zero subspace.
The following properties for the unobservable subspace are valid:
*
*
*
Detectability
A slightly weaker notion than observability is ''detectability''. A system is detectable if all the unobservable states are stable.
Detectability conditions are important in the context of
sensor networks
Wireless sensor networks (WSNs) refer to networks of spatially dispersed and dedicated sensors that monitor and record the physical conditions of the environment and forward the collected data to a central location. WSNs can measure environmental c ...
.
Linear time-varying systems
Consider the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
time-variant system
A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application. In other words, a time delay or time advance of input not only shifts the output signal in time but also changes ...
:
:
Suppose that the matrices
,
and
are given as well as inputs and outputs
and
for all
then it is possible to determine
to within an additive constant vector which lies in the
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...
of
defined by
:
where
is the
state-transition matrix
In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...
.
It is possible to determine a unique
if
is
nonsingular
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplica ...
. In fact, it is not possible to distinguish the initial state for
from that of
if
is in the null space of
.
Note that the matrix
defined as above has the following properties:
*
is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
*
is
positive semidefinite for
*
satisfies the linear
matrix differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one funct ...
::
*
satisfies the equation
::
Observability matrix generalization
The system is observable in
,
t_1if there exists
\bar \in _0,t_1/math> and a positive integer ''k'' such that[Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.]
: \operatorname \begin
& N_0(\bar) & \\
& N_1(\bar) & \\
& \vdots & \\
& N_(\bar) &
\end = n,
where N_0(t):=C(t) and N_i(t) is defined recursively as
: N_(t) := N_i(t)A(t) + \fracN_i(t),\ i = 0, \ldots, k-1
Example
Consider a system varying analytically in (-\infty,\infty) and matricesA(t) = \begin
t & 1 & 0\\
0 & t^ & 0\\
0 & 0 & t^
\end,\, C(t) = \begin
1 & 0 & 1
\end.
Then \begin
N_0(0) \\
N_1(0) \\
N_2(0)
\end
= \begin
1 & 0 & 1 \\
0 & 1 & 0 \\
1& 0 & 0
\end , and since this matrix has rank = 3, the system is observable on every nontrivial interval of \mathbb.
Nonlinear systems
Given the system \dot = f(x) + \sum_^mg_j(x)u_j , y_i = h_i(x), i \in p. Where x \in \mathbb^n the state vector, u \in \mathbb^m the input vector and y \in \mathbb^p the output vector. f,g,h are to be smooth vector fields.
Define the observation space \mathcal_s to be the space containing all repeated Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
s, then the system is observable in x_0 if and only if \dim(d\mathcal_s(x_0)) = n, where
:d\mathcal_s(x_0) = \operatorname(dh_1(x_0), \ldots , dh_p(x_0), dL_L_, \ldots , L_h_j(x_0)),\ j\in p, k=1,2,\ldots.
Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar, Kou, Elliot and Tarn, and Singh.
There also exist an observability criteria for nonlinear time-varying systems.
Static systems and general topological spaces
Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in \mathbb^n. Just as observability criteria are used to predict the behavior of Kalman filter
For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimat ...
s or other observers in the dynamic system case, observability criteria for sets in \mathbb^n are used to predict the behavior of data reconciliation and other static estimators. In the nonlinear case, observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.
See also
* Controllability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observabil ...
* Identifiability
In statistics, identifiability is a property which a model must satisfy for precise inference to be possible. A model is identifiable if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining ...
* State observer
* State space (controls)
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables w ...
References
External links
*{{planetmath reference, urlname=Observability, title=Observability
MATLAB function for checking observability of a system
Classical control theory
fr:Représentation d'état#Observabilité et détectabilité