# Design of the deadbeat controller with limited output/Aperiodinio reguliatoriaus su apribotu isejimu projektavimas.

Introduction

Deadbeat control system is digital control system. The deadbeat control could be used in systems where the known finite settling time is required. In this article, the presented deadbeat controller design is based on the object z-transfer function. The overall structure of the control system is shown in Fig. 1.

[FIGURE 1 OMITTED]

It is known that the deadbeat controller transfer function [W.sub.DC](z) may be written as the following [1]

[W.sub.DC](z) = Q(z) / 1-P(z), (1)

where Q(z), P(z) are the polynomials of the transfer function of the deadbeat controller. The coefficients of the polynomials Q(z), P(z) are found by using these equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [a.sub.i], [b.sub.i] are coefficients of the polynomials of the continuous object's z-transfer function [W.sub.CO](z) = B(c) / A(z); m - order of the object transfer function. The coefficients of polynomial Q(z) can be expressed as [1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Thus the initial value of manipulated variable u(0) depends only on the sum (2) of the object's z-transfer function coefficients [b.sub.i].

The main drawback of the deadbeat controller is that the sampling time [T.sub.0] and manipulated variable u(0) are inversely proportional. [T.sub.0] is conditioned by [u.sub.max], which depends from various factors.

By multiplying the numerator and the denominator polynomials of the controller's transfer function (1) by an additional polynomial C(z) [1] we get

[W.sub.DC](z) = Q(z)C(z) / 1-P(z)C(z), (4)

where C(z)=1+[c.sub.1] [z.sup.-1] . The coefficients of the deadbeat controller are found using these equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The deadbeat controller's properties depend from coefficient [c.sub.l]. If we assume that u(0) [less than or equal to] [u.sub.max], then using equations (3) and (5), the following can be written

[c.sub.10(6)] = [c.sub.1] = 1 / [U.sub.max]*([b.sub.1] + [b.sub.2] + ... +[b.sub.m]) - 1 (6)

where [c.sub.10] - a coefficient limiting the value of the manipulated variable from the top at time moment zero. So, if [c.sub.1] [greater than or equal to] [c.sub.10] then u(0) [less than or equal to] [u.sub.max].

Let us assume that -u(0) [less than or equal to] u(1) [less than or equal to] u(0), then express u(0) and u(1) in terms of qt by using (3) and (5)

[-q.sub.0] [less than or equal to][q.sub.0] + [q.sub.0][a.sub.1] + [q.sub.0][c.sub.1] [less than or equal to] [q.sub.0]. (7)

Dividing inequality (7) from [q.sub.0] ([q.sub.0] [greater than or equal to]0) we get

-2[less than or equal to] [a.sub.1] + [c.sub.1] [less than or equal to] 0. (8)

Significant (8) boundary is expression

[c.sub.11] = [c.sub.1] = [-a.sub.1] , (9)

where [c.sub.11] - a coefficient limiting the value of the manipulated variable at time moment one. So, if [c.sub.1] [less than or equal to] [c.sub.11] then [absolute value of u(1)][less than or equal to] [u.sub.max].

Let us assume that -u(0) [less than or equal to] u(2) [less than or equal to] u(0) , then express u(0) and u(2) in terms of [q.sub.i] by using (3) and (5)

[-q.sub.0] [less than or equal to] + [q.sub.0][a.sub.1] + [q.sub.0][c.sub.1] + [q.sub.0][a.sub.2] + [q.sub.0][a.sub.1] [c.sub.1] [less than or equal to] [q.sub.0]. (10)

Dividing the inequality (10) by [q.sub.0] ([q.sub.0] [greater than] 0) and rearranging we get

-2-[a.sub.1] - [a.sub.2] [less than or equal to] [c.sub.1] (1 + [a.sub.1]) [less than or equal to] -[a.sub.1] - [a.sub.2.] (11)

The expression (1+[a.sub.1]) in inequality (11) may be either positive or negative. If it is positive, then

-2 - [a.sub.1] - [a.sub.2] / 1 + [a.sub.1] [greater than or equal to][c.sub.1] [greater than or equal to] [-a.sub.1] - [a.sub.2] / 1 + [a.sub.1] [right arrow] [c.sub.121]. (12)

If (1+[a.sub.1]) is negative, then

-2 - [a.sub.1] - [a.sub.2] / 1 + [a.sub.1] [greater than or equal to][c.sub.1] [greater than or equal to] [-a.sub.1] - [a.sub.2] / 1 + [a.sub.1] [right arrow] [c.sub.122]. (13)

If [absolute value [a.sub.1]]> [absolute value of [a.sub.2]] in equation (11), then:

[c.sub.121] = -2 - [a.sub.1] - [a.sub.2] / 1 + [a.sub.1] , (14)

[c.sub.122] = [-a.sub.1] - [a.sub.2] / 1 + [a.sub.1], (15)

where [c.sub.121], [c.sub.122] - coefficients limiting the value of the manipulated variable at time moment two.

Such procedures are included in the design of deadbeat controller with limited manipulated variable.

Using equations (6), (9), (14) and (15) it is necessary to find the values of coefficients cj0, cjj, cj2j , cj22 at different sampling time T0 values and to draw dependencies [c.sub.10]=f([T.sub.0]), [c.sub.11]= f([T.sub.0]), [c.sub.121]=f([T.sub.0]), [c.sub.122]=f([T.sub.0]) (see Fig.2.).

[FIGURE 2 OMITTED]

[c.sub.10] curve is the lower boundary of u(0) and the area

below the [c.sub.10] curve is highlighted; [c.sub.11] curve is the upper boundary of u(1) and the area above the cjj curve is highlighted. The same steps are applied to [c.sub.121] and [c.sub.122] curves that limit u(2).

The non-highlighted area is the area from which it is possible to choose a value for [c.sub.1] and then, by using it, define a suitable sampling time [T.sub.0]. [c.sub.1] value may be chosen from the middle of the non-highlighted area, but in this case, the manipulated variable u will be limited by this inequality: [u.sub.max] > [absolute value of u(0)] > [absolute value of u(J)] > [absolute value of u(2)]. If [c.sub.1] value is chosen from the leftmost point of the non-highlighted area (see Fig.2.), the manipulated variable will be limited by this inequality: [u.sub.max] = [absolute value of u(0)] = [absolute value of u(J)] > [absolute value of u(2)]. After the value of [c.sub.1] is chosen it is possible to define a suitable sampling time [T.sub.0]. Once this is done, the coefficients of the deadbeat controller are found using (5) and the design procedure is over.

Simulation and experiments of the deadbeat control system

Performance of the a forementioned controler design procedure can be found through the simulations and experiments.

A third order continuous object [2] was chosen and its structure is shown in Fig. 3.

[FIGURE 3 OMITTED]

If we consider the parameters of a chosen object indicated in Fig. 3, we can write continuous transfer function

W(s) = 1 / (5s + 1) * (s + 1) * (5s + 1). (16)

The object response y, simulated by Matlab software, to the unit step input u, is depicted in Fig. 4.

The continuous object is connected to the PLC Modicon 140 CPU 113 03, which has a realized deadbeat controller and measures the output y signal of the object. A single pole low pass input filter processes the object's output signal y; the resolution of the A/D converter of the analogue input module is 15 Bit; the signal voltage is measured with an absolute accuracy error at 25 [degrees]C that equals [+ or -] 0.03%. 12 Bit D/A converter of the analogue output module processes the output signal u of the controller; the voltage output range of the analogue output module is 0-10 VDC, output accuracy error at 25 [degrees]C equals [+ or -] 0.15% of the full scale.

[FIGURE 4 OMITTED]

Continuous object response y to the unit step voltage input u, is shown in Fig. 5.

[FIGURE 5 OMITTED]

By comparing the real object response with the one simulated by Matlab, it can be seen that they are fairly similar. The real object response is with a disturbance. The transfer function given in the expression (16) will henceforth be used to design the deadbeat controller.

The design of the deadbeat controller is executed in Matlab [3]. The continuous object transfer function is converted to discrete time assuming a zero order hold on the inputs. By using equations (1) and (2), while holding that u(0)=3.0, we get the transfer function of the deadbeat controller

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

By using a method, which includes Matlab simulation and experimenting with the PLC, we get the deadbeat control system (see Fig. 1.) responses y to the unit step input reference signal g.

Fig. 6. shows that the system response has a finite setling time. The response ends in 8.91 s (the response remain within a 2% of its final value) after three steps of the control signal, because this is a third order object. The system response does not have any indications of overshooting. Fig. 7. shows that the disturbances affecting the real object, also influence the system response and the controller's output, which depends on the system response. The system response has a overshoot of 7%, while the settling time takes 5 sampling time units.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The next step will be design of the deadbeat controller with limited output using equations (1) and (5). By using the additional procedure shown in Fig. 2, for choosing coefficient cj, we find the transfer function of the deadbeat controller, holding that u(0)= 3.0=u(1), because [c.sub.10] = [c.sub.11] = 1.2826.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

By using a method, which includes Matlab simulation, we get the deadbeat control system response y to the unit step input reference signal g.

Fig. 8. shows shows that the system response has a finite setling time. The response ends in 8 s after four steps of the control signal. The system response does not have any indications of overshooting. The value of the control signal u(2) is negative (then g=1) and thus cannot be realized with PLC analogue output module.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

So experiment with the PLC includes investigation of the deadbeat system response to step reference signal, when system output is y=1.

Fig. 9. shows that the system response has a overshoot of 7%, while the transition takes more than 6 sampling time units.

It can be concluded that the responses of the experiments follows the system's response to the Matlab simulation with an error.

Conclusions

The presented procedure for choosing the deadbeat controller parameter [c.sub.1] allows for a decrease in the sampling time [T.sub.0], without increasing the maximum allowed value of the control signal.

The following observations were made based on such results. Simulation results show that even though the control increases by one-step, the length of the settling time of the system response can be lower than that of the deadbeat controller without any modifications.

It was found that the deadbeat controller is not of high quality when taking into account the changes in the parameters of an object. The quality of the deadbeat control system is influenced by the accuracy of the transfer function of the object. That is why it is crucial to have a reliable method for recognizing the transfer function of the object.

References

[1.] Isermann R. Digital control systems. - Springer-Verlag New York, 1996.--350 p.

[2.] Balasevicius L., Rimkus K System parameters identification in the state space // Proceedings of International Conference Electrical and Control Technologies'2009. Kaunas: Technologija, 2009.--P. 103-106.

[3.] Hetthessy J., Barta A., Bars R. Dead beat controller design.--Ebook, 2004.--5 p.