The synthesis of discrete control the continuous systems units is executed one of the followings methods:
• on the basis of continuous object of adjusting synthesize continuous control units, and then will transform them in a discrete form (method of continuous models);
• build the discrete models of adjusting object and on their basis synthesize discrete control units. Consequently, in both cases it is needed to be able to find discrete approximations of analog transmission functions. Set the problem it can execute such methods:
- by z-transformation;
- by replacement of operator of analog integration of 1/s one of operators of digital integration;
- by replacement of zeros and poles on a s-plane by the proper zeros and poles on a z-plane.
We will assume that we have the continuous dynamic system with the entrance signal of u(t) and output signal at(t), which is described in area of variable of Laplace a transmission function (TF)
(1.1)where
Z=[z1 z2 ....zm] - vector of a zeros;
P=[p1 p2 ....pn] - vector of a poles;
K=Bm.
A task consists of determination at the set period of quantum of equivalent discrete transmission function (DTF)
(1.2)that in the construction of discrete model of continuous object. Under an equivalence in this case understand the coincidence of reactions of the continuous system and its discrete model on any entrance influence. More frequent than all under the coincidence of reactions understand that y[k] = y(tk) at u[k] = u(tk) , where tk = kT , k – is a number of quantum step.
In general case the put task does not have an exact decision. It is related to that during discrete of entrance signal information is lost about his value between the knots of quantum. Consequently, exit of discrete model from these values of does not can, while the reaction of the continuous system depends on all values of entrance signal.
But situations in which a discrete model can be exact in understanding are all the same possible, izlozhennom higher. Is it for this purpose necessary, that values of process of u(t) at t k - 1 ≤ t < t k to simply determined the sequence [u(t0), u(t1), ...u(tk-1),] .Eto characteristically for the impulsive systems with peak-impulsive modulation of the first family and for digital control system, if entrance process, formed by computer.
In last case a discrete entrance process will be transformed in continuous by oder hold [8, 9].
More frequent than of Zero Order Hold (ZOH), ) which will transform the latticed function of time in a step, that
u(t) = u(tk) при t k - 1 ≤ t < t k (1.3)
and has a transmission function
(1.4)
In the classic theory of management a decision over of task of diskretizacii of continuous object is brought with ekstrapolyatorom of a zero order on an entrance by z-transformation which is named step-invariant [1-3]:
(1.5)
For determination of discrete transmission functions by z-transformation for the comparatively simple continuous systems it is possible to use the tables of z-transformations [1, 5-7], and for more difficult – it is needed at first to decompose PF to the amount of vulgar fractions. If
(1.6)
unknown coefficients determine
(1.7)
(1.8)
At the use in place of ZOH (FOH – First Order Hold), which forms piece-linear approximation of discrete signal
after z-transformation
get the discrete system in which the values of signals in transients are equal to the semisum of values of the proper analog signals on the ends of period of quantum:
(1.9)
For confirmation of substantive theoretical provisions will consider an example in which a continuous object has a transmission function
(1.10)
with three poles
p = [-0.1075+0.4417i, -0.1075-0.4417i, -0.2016], |p| = [0.4546, 0.4546, 0.2016]
and two zeros
z = [0, -0.5].
Discrete transmission functions, got from PF (1.18) the method of the considered z-transformations at T = 2, look like:
(1.11)
(1.12)
Transitional functions of the initial continuous system with TF (1.10) ha(t) and discrete systems with DTF (1.11) hz(t) and (1.12) hF(t) ) is represented on a fig. 1.1. On this picture and further frequency of discrete (1/T=0.5) for evidentness is chosen near to frequency of poles of the continuous system, that in principle is not recommended.
From the analysis of the resulted charts evidently, that the variant of z-transformation needs to be chosen depending on the type of entrance signal which is put to the continuous object, to the subject to discrete.
Close transformations approach the considered z-transformations in a greater or less degree, however they settle accounts considerably simpler. By another advantage of the use of close methods of diskrete of continuous dynamic objects there is that, that, in a difference from exact z-transformations, general DTF of the consistently united continuous objects is equal to work of discrete transmission functions, found for each of continuous TF separately.
At close determination of discrete transmission functions (DTF) by this method, it is possible to use the followings substitutions:
s = (z-1) / T (2.1)
s = (z-1) / Tz (2.2)
s = 2(z-1) / T(z+1) (2.3)
Formula (2.1) -(2.3) related to the different algorithms of numeral integration, namely: Forward Euler, Backward Euler that Trapezoidal. Forward Euler, Backward Euler та Trapezoidal. Formula (2.1) -(2.3) related to the different algorithms of numeral integration, namely: Forward Euler, Backward Euler that Trapezoidal.
For demonstration подстановочных digitization methods on fig. 2.1 The transitive are shown Functions of continuous integrator W (s) = 1/s (the top number), continuous aperiodic Link with ПФ W (s) = 1 / (4s+1) (the bottom number) and their discrete approximations found Methods of substitutions.
At digitization aperiodic link receive following DTF:
Materials on the theme of master's work: Main