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 ...
, a state observer or state estimator is a system that provides an estimate of the
internal state of a given real system, from measurements of the
input and output of the real system. It is typically computer-implemented, and provides the basis of many practical applications.
Knowing the system state is necessary to solve many
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 ...
problems; for example, stabilizing a system using
state feedback. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
, it is possible to fully reconstruct the system state from its output measurements using the state observer.
Typical observer model
Linear, delayed, sliding mode, high gain, Tau, homogeneity-based, extended and cubic observers are among several observer structures used for state estimation of linear and nonlinear systems. A linear observer structure is described in the following sections.
Discrete-time case
The state of a linear, time-invariant physical discrete-time system is assumed to satisfy
:
:
where, at time
,
is the plant's state;
is its inputs; and
is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current states and the current inputs. (Although these equations are expressed in terms of
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
time steps, very similar equations hold for
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 ...
systems). If this system is
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
then the output of the plant,
, can be used to steer the state of the state observer.
The observer model of the physical system is then typically derived from the above equations. Additional terms may be included in order to ensure that, on receiving successive measured values of the plant's inputs and outputs, the model's state converges to that of the plant. In particular, the output of the observer may be subtracted from the output of the plant and then multiplied by a matrix
; this is then added to the equations for the state of the observer to produce a so-called ''
Luenberger observer'', defined by the equations below. Note that the variables of a state observer are commonly denoted by a "hat":
and
to distinguish them from the variables of the equations satisfied by the physical system.
:
:
The observer is called asymptotically stable if the observer error
converges to zero when
. For a Luenberger observer, the observer error satisfies
. The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix
has all the eigenvalues inside the unit circle.
For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix
.
:
The observer equations then become:
:
:
or, more simply,
:
:
Due to the
separation principle In control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved b ...
we know that we can choose
and
independently without harm to the overall stability of the systems. As a rule of thumb, the poles of the observer
are usually chosen to converge 10 times faster than the poles of the system
.
Continuous-time case
The previous example was for an observer implemented in a discrete-time LTI system. However, the process is similar for the continuous-time case; the observer gains
are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when
is a
Hurwitz matrix In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Hurwitz matrix and the Hurwitz stability criterion
Namely, given a ...
).
For a continuous-time linear system
:
:
where
, the observer looks similar to discrete-time case described above:
:
.
:
The observer error
satisfies the equation
:
.
The eigenvalues of the matrix
can be chosen arbitrarily by appropriate choice of the observer gain
when the pair