Subordinate Regulation Control System

The system is a multi-circuit system of subordinate regulation, which consists of three circuits, each of which includes an input quantity, a regulator, and a controlled quantity. Each circuit has its own regulator. They depend on each other and they are all unknown, therefore, the main task of this system is the calculation of regulators.

x = 20v

2y' + y = x - 2z

z'' + z' + 3z = 15y

Value of the transition time tpp = 2,4

The system should consist of three control loops (values ​​x, y, z). If necessary, additional action can be introduced to compensate for the internal feedback of the control object. All regulators and additional corrective devices must be physically feasible.

The system should have the first order of astatism. The transition function of the system should have a weak oscillation and overshoot of 4-8% (which corresponds to the distribution of the poles across Butterworth).

From the first equation of the system we get the first-order link, the transfer function W01 (p), at the input of which is the variable v, and at the output is the variable x. The second equation gives us the transfer function W02 (p), the input of which is the error of the controller e5, and the output is the variable y. Also, we get negative feedback from the amplifying variable z 2 times. At the input of the adder, the variable x. From the third equation, a first-order link is obtained - W03 (p). The output of this function is z. We immediately include the variable u in the circuit in accordance with the condition v = 47u. The resulting circuit is shown in Figure 1.1.

Figure 1.1

The next step is the internal compensation feedback from the variable z. It is compensated by the introduction of positive feedback (Figure 1.2)

Figure 1.2

After the internal feedback is compensated, it is necessary to make compensation from the control object, since, by condition, we cannot change the control object itself. For transfer, standard rules for transferring a node through a link and an adder through a link are used. The resulting circuit is shown in Figure 1.3.

Figure 1.3

Now, when calculating, you can ignore feedback values, which greatly simplifies the task. The equivalent circuit is presented in Figure 1.4.

Figure 1.4

Take a section of the circuit from variable u to variable x and add to them the adder and controller R1 (p), as shown in Figure 1.5.

Figure 1.5

The entire plot obtained from x1 to x represents the Desired Function F1 (p). The current section is of second order, therefore, the desired transfer function will be of second order. We take it from the table of transfer functions for Butterworth distribution.

Now we write the formula for the first controller:

We proceed to the next circuit. Now the regulation section looks like in Figure 1.6

Figure 1.6

We follow the same algorithm as above, but the desired transfer function F2 (p):

And finally, repeat the procedure for the last contour, section x3 - z (Figure 1.7).

Figure 1.7

Now you need to calculate T3 using the formula T = tpp/τpp. 

Where tpp = 2,4 s and τpp = 6 (from the table of indicators of universal transition functions for a fourth-order system). Then:

Now we substitute this value in the formulas of all regulators and get their final expressions:

The finished scheme looks like this: