Editing State observer (section)

== Typical observer model ==
[[File:Luenberger Observer.svg|thumb|Block diagram of Luenberger Observer. Input of observer gain L is <math>y \mathbf{-} \hat y</math>.]]
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 discrete-time system is assumed to satisfy

: <math>x(k+1) = A x(k) + B u(k)</math>

: <math>y(k) = C x(k) + D u(k)</math>

where, at time <math>k</math>, <math>x(k)</math> is the plant's state; <math>u(k)</math> is its inputs; and <math>y(k)</math> 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 mathematics|discrete]] time steps, very similar equations hold for [[continuous function|continuous]] systems).  If this system is [[Observability|observable]] then the output of the plant, <math>y(k)</math>, 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 <math>L</math>; this is then added to the equations for the state of the observer to produce a so-called ''[[David Luenberger|Luenberger]] observer'', defined by the equations below.  Note that the variables of a state observer are commonly denoted by a "hat": <math>\hat{x}(k)</math> and <math>\hat{y}(k)</math> to distinguish them from the variables of the equations satisfied by the physical system.
<!-- insert plant with observer systems schematic -->

: <math>\hat{x}(k+1) = A \hat{x}(k) + L \left[y(k) - \hat{y}(k)\right] + B u(k)</math>

: <math>\hat{y}(k) = C \hat{x}(k) + D u(k)</math>

The observer is called asymptotically stable if the observer error <math>e(k) = \hat{x}(k) - x(k)</math> converges to zero when <math> k \to \infty </math>. For a Luenberger observer, the observer error satisfies <math> e(k+1) = (A - LC) e(k)</math>.  The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix <math> A - LC </math> 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 <math>K</math>.

: <math>u(k)= -K \hat{x}(k)</math>

The observer equations then become:

: <math>\hat{x}(k+1) = A \hat{x}(k) + L \left(y(k) - \hat{y}(k)\right) - B K \hat{x}(k)</math>

: <math>\hat{y}(k) = C \hat{x}(k) - D K \hat{x}(k)</math>

or, more simply,

: <math>\hat{x}(k+1) = \left(A - B K \right) \hat{x}(k) + L \left(y(k) - \hat{y}(k)\right)</math>

: <math>\hat{y}(k) = \left(C - D K\right) \hat{x}(k)</math><!--

Substituting in the equation for <math>y(k)</math> from the plant system

: <math>\hat{x}(k+1) = \left(A - B K - L C) \right) \hat{x}(k) + L \hat{y}(k)</math>

: <math>\hat{y}(k) = \left(C - D K\right) \hat{x}(k)</math>

 -->

Due to the [[separation principle]] we know that we can choose <math>K</math> and <math>L</math> independently without harm to the overall stability of the systems.  As a rule of thumb, the poles of the observer <math>A-LC</math> are usually chosen to converge 10 times faster than the poles of the system <math>A-BK</math>.

=== 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 <math>L</math> are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when <math>A-LC</math> is a [[Hurwitz-stable matrix|Hurwitz matrix]]).

For a continuous-time linear system

: <math>\dot{x} = A x + B u, </math>

: <math>y = C x + D u, </math>

where <math>x \in \mathbb{R}^n, u \in \mathbb{R}^m ,y \in \mathbb{R}^r</math>, the observer looks similar to discrete-time case described above:

: <math>\dot{\hat{x}} = A \hat{x}+ B u + L \left(y - \hat{y}\right) </math>.

: <math>\hat{y} = C \hat{x} + D u, </math>

The observer error <math>e=x-\hat{x}</math> satisfies the equation

: <math> \dot{e} = (A - LC) e</math>.

The eigenvalues of the matrix <math>A-LC</math> can be chosen arbitrarily by appropriate choice of the observer gain <math>L</math> when the pair <math>[A,C]</math> is observable, i.e. [[observability]] condition holds. In particular, it can be made Hurwitz, so the observer error <math>e(t) \to 0</math> when <math>t \to \infty</math>.

=== Peaking and other observer methods ===

When the observer gain <math>L</math> is high, the linear Luenberger observer converges to the system states very quickly. However, high observer gain leads to a peaking phenomenon in which initial estimator error can be prohibitively large (i.e., impractical or unsafe to use).<ref name="Khalil02">{{Citation
 | last = Khalil
 | first = H.K.
 | authorlink = Hassan K. Khalil
 | year = 2002
 | edition = 3rd
 | url = http://www.egr.msu.edu/~khalil/NonlinearSystems/
 | isbn = 978-0-13-067389-3
 | title = Nonlinear Systems
 | publisher = [[Prentice Hall]]
 | location = Upper Saddle River, NJ}}</ref> As a consequence, nonlinear high-gain observer methods are available that converge quickly without the peaking phenomenon. For example, [[sliding mode control]] can be used to design an observer that brings one estimated state's error to zero in finite time even in the presence of measurement error; the other states have error that behaves similarly to the error in a Luenberger observer after peaking has subsided. Sliding mode observers also have attractive noise resilience properties that are similar to a [[Kalman filter]].<ref name="UtkinGS99">{{citation|title=Sliding Mode Control in Electromechanical Systems|last1=Utkin|first1=Vadim|last2=Guldner|first2=Jürgen|last3=Shi|first3=Jingxin|year=1999|publisher=Taylor & Francis, Inc.|location=Philadelphia, PA|isbn=978-0-7484-0116-1}}</ref><ref name="Drakunov83">{{citation|title=An adaptive quasioptimal filter with discontinuous parameters|journal=Automation and Remote Control|last1=Drakunov|first1=S.V.|year=1983|volume=44|issue=9|pages=1167–1175}}</ref>
Another approach is to apply multi observer, that significantly improves transients and reduces observer overshoot. Multi-observer can be adapted to every system where high-gain observer is applicable.<ref name="MMObserver">{{citation|doi=10.1080/00207179.2014.1000380|bibcode=2015IJC....88.1209B|title=Multi modelling as new estimation schema for High Gain Observers|journal=International Journal of Control|last1=Bernat|last2=Stepien |first1=J.|first2=S.|year=2015|volume=88|issue=6|pages=1209–1222|s2cid=8599596}}</ref>