Zbekiston Respublikasi Oliy ta’lim, fan va innovatsiyalar vazirligi

O‘zbekiston Respublikasi Oliy ta’lim, FAN VA INNOVATSIYALAR vazirligi

Islom Karimov nomidagi toshkent davlat texnika universiteti

“Elektronika va avtomatika» fakulteti

“Ishlab chiqarish jarayonlarini avtomatlashtirish» kafedrasi

“Raqamli boshqarish tizimlari” fanidan

“Digital Control Systems Design, Identification and Implementation” mavzuda

MUSTAQIL ISH


Bajardi: III kurs sirtqi
14aS-20 TJIChAB (o‘zb) guruhi talabasi
Yusupov Mavlonjon

Qabul qildi: prof. Avazov Yu.Sh.

Toshkent - 2023

4.2 Minimum Variance Tracking and Regulation

This strategy concerns optimal controller design, ensuring a minimum variance of the controlled variable around the reference, in the case of systems subject to random disturbances. It can be applied only to discrete time plant models with stable zeros.
The objective of the minimum variance tracking and regulation is to reduce the variance (standard deviation, mean square error) of the controlled output around
the reference value either for a constant value (minimum variance control) or for a variable value (minimum variance tracking). The effect of a minimum variance control is illustrated in Figure 4.13.

Figure 4.13. Effect of a minimum variance control in the presence of random disturbances.
The interest of the variance minimization of the controlled output clearly results from the output measures histogram.
If the variance of the controlled output is large, one obtains a distribution of the measures having the form shown in Figure 4.14. In this case, an important percentage of the measured values of the controlled output is far from the reference value. Since in several applications a minimum value should be assured for the controlled output (i.e.: coat thickness, water content of paper, etc ...) one is obliged to set the reference to a value significantly greater than the necessary minimum.

Figure 4.14. Histograms of the controlled output
On the other hand, if the controller reduces significantly the variance of the controlled output, one obtains a distribution of the measures narrowed around the reference. In this case, one can not only improve the quality of the product (better uniformity), but also reduce the reference value to approach the desired minimum value (see Figure 4.14). This implies, in general, a very important reduction of costs (see Chapter 8).
Taking into account the definition of the variance of a random process (see Equation 4.1.9), it results that the objective is to compute u(t) which minimizes the following criterion:
J(u(t)) = E^⎧_⎨[y(t) − y^*(t)] ^2⎫_⎬≈ ¹ _∑[y(t) − y^*(t)] ² = min (4.2.1)
⎩ ⎭ N t=1
where y(t) - y^*(t) represents the difference between the output and the desired value y^* at the instant t.
For solving this problem, disturbance models must be considered in addition to plant models. The structure considered is the ARMAX model which incorporates both the plant and disturbance models (see Section 4.1.3). Consequently, when identifying a system, both plant model and disturbance model should be identified, in order to apply this control strategy.