Vocabulary S(t) is the state vector U(t) is the input vector Ym(t) is the measurement vector (output) Model 1 is the « state model » Model 2 is the «observation model»	Hypothesis S(t) can’t be measured properly (theoretically, in practice, measured but with too much noise, …) Model 1 and  Model 2 are continous and differenciable
Question Knowing Model 1, model 2, U(t-k), Y(t), Ym(t), IS IT POSSIBLE TO RECONSTRUCT S(t)  ?	Definition : virtual sensor If the answer of the above question is yes, then one can say that this system is a VIRTUAL SENSOR of S(t)

FAQ 1 : if you know Model 1, why don't you simply use it to find S(t) ?
If you wish to simply use Model 1 in order to reconstruct S(t), then you need to know first U(t-k) AND S(t-1).
U(t-k) is known (inputs of the system), but concerning S(t-1), it's not that simple ! ...
(remind that you don't have a sensor for S => you cannot know S(t-1) ... ).
In some applications, it is possible to put the system in an "obvious" state (example : starting point for a car : before for unlock the  parck brake : speed = 0, longitudinal acceleration = 0, ...). It means that in such a case, S(0) is known without needing sensors.
Then, application of Model 1 will lead succesively to S(1), S(2), ..., S(t-1), S(t), ...
NB : knowing S(t-1) and U(t-k), application of Model 1 is a feedforward process : the modeling error cumulates among iterations.
It means that the more the iterations, the less the precision of estimation.

FAQ 2 : if you know Model 2, why don't you simply inverse this model in order to build S(t) from Ym(t) ?
Ifever this is possible, then YES, you should inverse Model 2.
Generally, Model 2 is injective : one only measure a few consequences of S(t).
In such a case, there exists several inputs of Model 2 that lead to the same output value, and "more" equations are needed in order to choose the right input value.
Because S(t) is produced by Model 1 ... THIS IS THE ROLE OF Model 1 ! to give more equations that S(t) MUST fit into
Example : self coherence among time.
Condition is then to verify that supplementary equations given by Model 1 are independant on equations given by Model 2. When this condition is verified, one say that S(t) is "observable" through the virtual sensor (see below "observability").

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A very simple "handmade" example



S1(t) and S2(t) are estimated, although one never measured them : this is called a virtual sensor of vector S.

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The iterative locked loop estimation idea

The idea is to compare Y(t) with Ym(t) and to "automatically adjust" Model 1 in order to minimize the error. Under conditions of "observability" (see below), one consider that X(t) is well estimated when error is very near zero.

In this diagram, the bloc "?" is the minimization of the output error. The locked loop allows to verify that estimated values of S(t) lead to the proper output Ym(t). If not, then it means that estimated S(t) should be modified.
Because S(t) is produced by Model 1, one can consider that a modification of S(t) is in fact a modification of Model 1.

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The idea of "not copying noise"

The idea is to compare Y(t) with Ym(t) and to "automatically adjust" Model 1 in order to minimize the error. Ifever there is noise added with Ym(t), then minimizing the error will lead to "copy" the noise from Ym(t) to Y(t) ...
Then taking into account existence of noise (knowledge based approach) may lead to tolerate a residual error on Y(t).

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General case of linear systems

Observability condition
State equation : X(t) = A.X(t-1) + B.U(t-p)
Observation equation : Y(t) = C.X(t)
A is a n x n matrix.
One can demonstrate that the system is observable if (and only if) the matrix :
(CT | AT.CT | (AT)2.CT | … | (AT)(n-1).CT)
is a rank n matrix.

NB : in special case when A is written in the Eigen vectors base, then A is a diagonal matrix, and condition of observability becomes more simple : C must not have a null column.

Interesting way of writing equations
State equation : S(t) = A.S(t-1) + B.U(t-p)
Observation equation : Y(t) = C.S(t)
S is the state vector
U is the input vector
Y is the output vector
Ŝ_i/j is the predicted value for vector S at time i from the knowledge of every signals at time j
Ŷ_i/j is the predicted value for vector Y at time i from the knowledge of every signals at time j

Diagram :


First phase, at time k, consists in applying the models for predicting S at time (k+1), knowing every signal at time k :
Ŝ_k+1/k = A.Ŝ_k/k +  B.U_k-i      and       Ŷk+1/k =  C.Ŝk+1/k
Diagram :

Second phase, at time (k+1), consists in computing the error between Ŷk+1/k and the measure Yk+1
Ỹk+1/k  =  Yk+1 -  Ŷk+1/k     
This error is a "knowledge" : one predicted the value of Yk+1Yk , and The idea is to make a correction of Ŝ in order to take into account the error Ỹk+1 that was produced by Ŝ_k+1/k
The correction is in proportion to the error, and leads to an estimation of S at time (k+1), knowing every other signals until time (t+1) :
 Ŝ_k+1/k+1 =  Ŝ_k+1/k  + L_k+1 . Ỹk+1                              (3)
L is called the correction gain.
Replacing  Ỹk+1  by its expression into 3 leads to :
Ŝ_k+1/k+1 =  Ŝ_k+1/k  + L_k+1 . (Yk+1 -  Ŷk+1/k)=  A.Ŝ_k/k  + B.BUk-i  +  L_k+1 . (Yk+1 -  Ŷk+1/k) 
Ŝ_k+1/k+1 =  A.Ŝ_k/k  + B.Uk-i  +  L_k+1 . (Yk+1 - C.Ŝ_k+1/k )
Ŝ_k+1/k+1 =  A.Ŝ_k/k  + B.Uk-i  +  L_k+1 . (Yk+1 - C.(A.Ŝ_k/k +  B.Uk-i ) )
Ŝ_k+1/k+1 =  (I - L_k+1.C).A.Ŝ_k/k  + (I - L_k+1.C).B.Uk-i  +  L_k+1.Yk+1
Diagram :
                                            
One can see that the correction leads to a modification of the state model.
There exists several methods to compute the correction gain L_k+1
(see below)
Third phase, still at time (t+1), consists in applying the corrected model
The complete diagram is then :

One uses to record the residual error : Ỹk+1/k+1  =  Yk+1 -  Ŷk+1/k+1 
If one decide to keep the new state model for next step, then one can call the system an "adaptive" virtual sensor.
If not, one computes next "first phase" using initial model.

NB : this way of decomposing the linear estimation of S(t) into 3 phases :
 - prediction leading to an error between measurement (that is considered as a desired output) and computed output
 - model 1 modification (in order to minimize error between desired output and computed output)
 - estimation (application of the model)
allows to make a link between linear virtual sensors theory and other applied maths techniques such as neural networks, for instance, that involve exactly the same 3 phases (cf. NEURAL NETS).

Estimation of a good correction gain L

Asymptotic methods :
Let us consider the error made on S estimation at time (k+1) :
Šk+1/k  =           Sk+1              -     Ŝk+1/k+1
Šk+1/k  =  (A.Sk + B.Uk-i )    -     (  (I - L_k+1.C).A.Ŝ_k/k  + (I - L_k+1.C).B.Uk-i  +  L_k+1.Yk+1  )
Šk+1/k  =  (A.Sk + B.Uk-i )    -     (  (I - L_k+1.C).A.Ŝ_k/k  + (I - L_k+1.C).B.Uk-i  +  L_k+1.Yk+1  )
after some obvious manipulation, one get :
Šk+1/k  =  (I - L_k+1.C).A. Šk/k 
This recurrence is asymptotic if the Eigen values of matrix (I - L_k+1.C).A have their module smaller than 1 ( such a matrix is called a Schur matrix ).
It means that there exists "many" methods to choose the correction gain.
Some famous methods are :
 - Luenberger method : change of variable in order to get a canonical form of the system that leads to a simple way of choosing a gain L_k+1 S
 - Minimization of a quadratic linear form
 -  ...

Stochastic methods :
This approach considers that models are not perfect. Their imperfection is considered as a random noise :
 - a state model noise
 - an observation model noise.
Every iteration, noise is amplified by the C matrix, and then Ŷk+1/k  is corrupted with noise.
In such a case, it is obvious that the error (Yk+1 - Ŷk+1/k )  has not the same meaning : a null error means that one tries to copy the noise : if you try to estimate Y including noise of the models, then (because models are pure deterministic systems) the only solution is to add noise to S too ...
The idea of stochastic methods is to take into account imperfection of models and trying to go right in the middle of the noise distribution.
Intuitively, while output error is smaller than noise, then it is not necessary to apply a correction gain. This idea leads to other methods based on sets theory (not explained in this page) ... Stochastic approach should propose a model of noise and use it in order to minimize the influence of noise.
Estimation theory allows to write correction as the average value of prediction error :
Ŝ_k+1/k+1 =  Ŝ_k+1/k  + E{ Ŝ_k+1/k /  Ŷk+1/k }    (4) : mathematic belief of S, knowing the prediction of Y
(4) simply means that correction should be the average value of error made when Yk+1 is predicted with the value Ŷk+1/k
Comparison between (3) and (4) using conditional belief and probability density distribution leads to an iterative estimation of L_k+1  (but densities of noise probability has to be known).