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O‘zbekiston Respublikasi Oliy ta’lim, fan va innovatsiyalar vazirligi
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| bet | 1/4 | | Sana | 08.06.2023 | | Hajmi | 4.5 Mb. | | #70973 |
Bog'liq Qalandarov Javohir, Qo\'yliyev Umid javoh 3-lab, wepik-gear-up-unleashing-the-power-of-multimedia-hardware-20240404124242P7c5
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
“Selection (Estimation) of the Model Complexity” mavzuda
MUSTAQIL ISHi
Bajardi: III kurs
152-20 TJIChAB (o‘zb) guruhi talabalari
Qo’yliyev Umid va Qalandarov Javohir
Qabul qildi: prof. Avazov Yu.Sh.
Toshkent - 2023
General Forms of Linear Discrete-time Models
nA
nB
A linear discrete-time model is generally described as
nA
i=1
i=1
y(t)= - ∑ ai y(t-1) + ∑ bi u(t-d-i) (2.3.20)
in which d corresponds to a pure time delay which is an integer multiple of the sampling period.
Let us introduce the following notations:
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A
i=1
1+ ∑ aiq-1 = A(q-1)=1+q-1 A*(q-1) (2.3.21)
nA
A
A*(q-1)=a1+a2q-1 + . . . +a n q-n +1 (2.3.22)
i=1
∑ bi q-1 = B(q-1) = q-1 B*(q-1) (2.3.23)
B
B
B* (q-1) = b1+b2q-1 + .... + bn q-n +1 (2.3.24)
By using the delay operator q-1 in Equation 2.3.20 and taking into account the notations of Equations 2.3.21 to 2.3.24, the Equation 2.3.20 describing the discretetime system is written as
A(q-1) y(t) = q-d B(q-1) u(t) (2.3.25)
or in the predictive form (by multiplying both sides by qd)
A(q-1) y(t+d) = B(q-1) u(t) (2.3.26)
Equation 2.3.25 can also be written in a compact form using the pulse transfer operator
y(t) = H(q-1) u(t) (2.3.27)
where the pulse transfer operator is given by
A(q-1)
q-d+B(q-1)
H (q-1) = (2.3.28)
z-d B(z-1)
The pulse transfer function characterizing the system described by Equation 2.3.20 is obtained from the pulse transfer operator given in Equation 2.3.28 by replacing q-1 with z-1 2
A(z-1)
H(z-1)= (2.3.29)
n= max (nA, nB + d)
n represents the discrete-time system order (the higher power of a term in z in the pulse transfer function denominator).
Z-3(b1z-1+b2z-2)
Example 1:
1+a1z-1
H(z-1) =
n= max (1, 5) = 5
b1 z+b2
z5 + a1z4
H(z) =
Example 2:
b1 z-1 + b2 z-2
1+ a1z-1+a2z-2
H(z-1) =
b1z+ b2
n=max (2,2) = 2
z2 + a1z + a2
H(z) =
One notes that the order n of an irreducible pulse transfer function also corresponds to the number of states for a minimal state space system representation associated to the transfer function (See Appendix C).
Stability of Discrete-time Systems
The stability of discrete-time systems can be studied either from the recursive (differences) equation describing the discrete-time system in the time domain, or from the interpretation of difference equations solutions as sums of discretized exponentials. We shall use examples to illustrate both these approaches.
Let us assume that the recursive equation is
y(t) = -a1 y(t-1) ; y(0) = y0 (2.3.30)
which is obtained from Equation 2.3.3 when the input u(t) is identically zero. The free response of the system is written as
y(1) = -a1 y0 ; y(2) = (-a1) y0 ; y(t) = (-a1)t y0 (2.3.31)
The asymptotic stability of the system implies
∞
t
lim y(t) = 0 (2.3.32)
The condition of asymptotic stability thus results from Equation 2.3.31. It is necessary and sufficient that
\a1\ < 1 (2.3.33)
On the other hand, it is known that the solution of the recursive (difference) equations is of the form (for a first-order system):
y(t) = KesTS = Kzt (2.3.34)
By introducing this solution into Equation 2.3.30, and taking into account Equation 2.3.15, one obtains
(2.3.35)
(1 + a1e)KesTJ = (1 + a1z-1 )Kzt = 0
jѡTs
σTs
(σ +jw)Ts
sTs
from which it follows that
z=e = e = e e = -a1 (2.3.36)
For this solution to be asymptotically stable, it is necessary that a = Re s < 0 which implies that e°Ts < 1 and respectively \z\ < 1 (or \a1\ < 1)
However, the term (1 + a1 z-1) is nothing more than the denominator of the pulse transfer function related to the system described by Equation 2.3.3 (see Equation 2.3.19).
The result obtained can be generalized. For a discrete-time system to be asymptotically stable, all the roots of the transfer function denominator must be inside the unit circle (see Figure 2.14):
1 + a1 z-1 +……. + an z-n = 0 |z| < 1 (2.3.37)
In contrast, if one or several roots of the transfer function denominator are in the region defined by \z\ > 1 (outside the unit circle), this implies that Re s > 0 and thus the discrete-time system will be unstable.
As for the continuous-time case, some stability criteria are available (Jury criterion, Routh-Hurwitz criterion applied after the change of variable w = (z + 1)/(z-1)) for establishing the existence of unstable roots for a polynomial in the variable z with no explicit calculation of the roots (Astrom and Wittenmark 1997).
A helpful tool to test z-polynomial stability is derived from a necessary condition for the stability of a z-1 -polynomial. This condition states: the evaluations of the polynomial A(z-1) given by Equation 2.3.37 in z = 1, (A(1)) and in z = -1 (A(-1)) must be positive (the coefficient of A(q-1) corresponding to z0 is supposed to be positive).
Example:
A(z-1) = 1 - 0.5 z-1 (stable system)
A(1) = 1 - 0.5 = 0.5 > 0 ; A(-1) = 1 + 0.5 = 1.5 > 0
A(z-1) = 1 - 1.5 z-1; (unstable system)
A(1) = - 0. 5 < 0 ; A(-1) = 2.5 > 0
Steady-state Gain
In the case of continuous-time systems, the steady-state gain is obtained by making s = 0 (zero frequency) in the transfer function. In the discrete case, s = 0 corresponds to
s = 0 ^ z = esT = 1 (2.3.38)
and thus the steady-state gain G(0) is obtained by making z = 1 in the pulse transfer function. Therefore for the first-order system one obtains:
nB
1+a1
b1
=
z=1
1+ a1 z-1
b1z-1
G(o) =
∑ bi
Generally speaking, the steady-state gain is given by the formula
i=1
z-d B(z-1)
nA
∑ai
z=1
A(z-1)
z=1
G(o) = H(1) = H(z-1) = = (2.3.39)
1 +
i=1
In other words, the steady-state gain is obtained as the ratio between the sum of the numerator coefficients and the sum of the denominator coefficients. This formula is quite different from the continuous-time systems, where the steady-state gain appears as a common factor of the numerator (if the denominator begins with 1).
The steady-state gain may also be obtained from the recursive equation describing the discrete-time models, the steady-state being characterized by u(t) = const. andy(t) = y(t-1) = y(t-2)....
From Equation 2.3.3, it follows that
(1 + aj) y(t) = b1 u(t)
b1
and respectively
1+ a1
y(t) = u(t) = G (o) u(t)
Models for Sampled-data Systems with Hold
Up to this point we have been concerned with sampled-data systems models corresponding to the discretization of inputs and outputs of a continuous-time system. However, in a computer controlled system, the control applied to the plant is not continuous. It is constant between the sampling instants (effect of the zero- order hold) and varies discontinuously at the sampling instants, as is illustrated in Figure 2.16.
It is important to be able to relate the model of the discretized system, which gives the relation between the control sequence (produced by the digital controller) and the output sequence (obtained after the analog-to-digital converter), to the transfer function H(s) of the continuous-time system. The zero-order hold, whose operation is reviewed in Figure 2.17 introduces a transfer function in cascade with H(s).
(2.3.40)
T he hold converts a Dirac pulse given by the digital-to-analog converter at the sampling instant into a rectangular pulse of duration Ts, which can be interpreted as the difference between a step and the same step shifted by Ts. As the step is the integral of the Dirac pulse, it follows that the zero-order hold transfer function is
2.3.7 Analysis of First-order Systems with Time Delay
The continuous-time model is characterized by the transfer function
H(s) 1st (2.3.42)
where G is the gain, T is the time constant and is the pure time delay. If Ts is the sampling period, then is expressed as
= d Ts + L ; 0 < L < Ts (2.3.43)
where L is the fractional time delay and d is the integer number of sampling periods included in the delay and corresponding to a sampled delay of d-periods. From Table 2.4, one derives the transfer function of the corresponding sampled model (when a zero-order hold is used)
-1 zd ( b z1 b2z2) zd1( b b2z1) 1 a1z1 1 a1z1
(2.3.44)
with
The effect of the fractionaltime delay can be seen in the appearance of the coefficient b2 in the transfer function. For L = 0, one gets b2 = 0. On the other hand, if L = Ts, it follows that b1 = 0, which correspond to an additional delay of one sampling period. For L<0.5Ts one has b2 < b1, and for L>0.5Ts one has b2 > b1. For L=0.5Ts b2 ≈ b1. Therefore, a fractional delay introduces a zero in the pulse transfer function. For L > 0.5 Ts the relation |b2| > |b1| holds and the zero is outside the unit circle (unstable zero) 5.
The pole-zero configuration in the z plane for the first-order system with ZOH is represented in Figure 2.18. The term z-d-1 introduces d+1 poles at the origin [ H( z) = ( b1z + b2) / zd+1 ( z + a1)].
5 The presence of unstable zeros has no influence on the system stability, but it imposes constraints on the use of controller design techniques based on the cancellation of model zeros by controller poles.
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