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4.2.1 An Example
Let the plant and disturbance be represented by the following ARMAX model: y(t+1) = - a₁ y(t) + b₁ u(t) + b₂ u(t-1)+ c₁ e(t) + e(t+1) (4.2.2)
The reference trajectory will be y^*(t+1) (it is either stored or generated by a dynamic model from the reference).
The variance of the difference y(t+1) - y^*(t+1) is computed, and it represents the performance criterion to be minimized:
E{[y(t+1) - y^*(t+1)]²} = E{[-a₁y(t) + b₁u(t) + b₂u(t-1) + c₁e(t) - y^*(t+1)]²}
+E{e²(t+1)}+2E{e(t+1)[-a₁y(t) + b₁u(t) + b₂u(t-1) + c₁e(t) - y^*(t+1)]} (4.2.3)
The third term of the right hand side of Equation 4.2.3 is zero since e(t + 1), the white noise at instant t + 1 is independent of all signals appearing at instants t, t1,... (note that y* (t + 1) depends upon the reference r(t), r(t-1),… only, see for example Section 3.3. Equation 3.3.33). Of the two terms which then remain in the criterion given in Equation 4.2.3, E {(e² (t + 1)} does not depend upon u(t). It results that the choice of u(t) will only affect the first term which can only be positive or zero. It follows that the minimization of the criterion 4.2.3 corresponds to finding u(t) such that:
E{[-a₁ y(t) + b₁ u(t) + b₂ u(t-1) + c₁ e(t) - y^*(t+1)]²} = 0 (4.2.4)
which may be obtained by making the bracketed expression zero. The (theoretical) control law that results is:
*
y (t +1) − c¹e(t) + a¹y(t)
u(t) = (4.2.5)
−1
b₁ + b₂q
Introducing this expression into the plant output equation given by Equation 4.2.2, one obtains that:

y(t+1) - y^*(t+1) = e(t+1) and respectively	(4.2.6)
y(t) - y^*(t) = e(t)	(4.2.7)

This leads to the following remarks:

1. 
   the application of the control law given by Equation 4.2.5 leads to a minimum variance for the difference y(t) - y^*(t) which becomes a white noise ;
   
2. 
   The controller cancels the zeros of the discrete time plant model (the zeros must be stable) ;
   
3. 
   e(t) can be replaced in Equation 4.2.5 by the measurable expression given in Equation 4.2.7. This results in the control law:
   

−1 *
(1+ c1q )y (t +1) − (c1 − a1)y(t)
u(t) =
b1 + b2q−1
(4.2.8)
T(q⁻¹)y^*(t +1) − R(q⁻¹)y(t)
=
S(q⁻¹)
The structure of the minimum variance control law is the same as that for tracking and regulation with independent objectives in the deterministic case (Chapter3,
Section 3.4) if P(q^-1) = C(q^-1) (desired closed loop poles).
In fact the transfer operator between y^*(t+1) and y(t+1) is:
−1 −1 * −1
HCL (q−1) = −1 T(q−1 )q −1B *(q −1) −1 (4.2.9) A(q )S(q )+ q B (q )R(q )
In the example considered above:
A(q^-1) = 1 + a₁ q^-1
B(q^-1) = q^-1 B^*(q^-1) ; B^*(q^-1) = b₁ + b₂ q^-1 ; d = 0
T(q^-1) = C(q^-1) = 1 + c₁ q^-1
S(q^-1) = B^*(q^-1) = b₁ + b₂ q^-1 ; R(q^-1) = r₀ = c₁ - a₁ and one obtains:

HCL (q−1) = −T1(q−1)−q1 −1 −1 = T(q−1−)1q−1 = q−1 (4.2.10)
A(q )+ q R(q ) C(q )
The closed loop poles are effectively defined by C(q^-1) which characterizes the disturbance.
In other words, an optimum choice exists for closed loop poles (regulation behavior), and this choice directly depends upon the zeros of the disturbance model.
Finally, an optimal performance test can be carried out for controller tuning by means of a whiteness test applied to the sequence {y(t) - y^*(t)} for cases without time delay. In cases with a time delay of d samples at the optimum, {y(t) - y^*(t)} is a MA process of order d and thus the autocorrelation functions R(i) will be zero for indexes i ≥ d + 1 (see Equation 4.2.30).
An Interpretation of the Minimum Variance Control
Let consider the optimal predictor for the ARMAX process given by 4.2.2:
yˆ(t +1) =−^a₁ y(t) + ^b₁u(t) + ^b₂u(t −1) + ^c₁e(t)
Let find the value u(t) that imposes
yˆ(t +1) = y^*(t +1)
Then one exactly obtains the expression of u(t) given by 4.2.5 and 4.2.8 respectively.
Thus, one can consider the minimum variance control law as obtained in two stages¹:

1. 
   Computation of an optimal output predictor;
   
2. 
   Computation of a control law such that the output prediction be equal to the reference (or in general in order to satisfy a specified deterministic criterion).
   

Note that the computation of the control law for the predictor is a deterministic problem as all the variables are known at the computation stage.
This computation strategy in two stages for the control in a stochastic environment is very general (separation theorem). It can also be summarized as follows: first compute the best output prediction and afterwards consider the
problem as a deterministic control problem, by replacing the real measured output with its prediction.