Zbekiston Respublikasi Oliy ta’lim, fan va innovatsiyalar vazirligi

4.2.2 General Case
Taking into account the similarity with the deterministic tracking and regulation with independent objectives, the controller computation will be exactly the same by choosing:
P(q^-1) = C(q^-1) (4.2.11)
and this is summarized in Figure 4.15.
The process plus the disturbance is described in the general case by:
A(q⁻¹)y(t) = q^−dB(q⁻¹)u(t) +C(q⁻¹)e(t) (4.2.12)
in which
A(q⁻¹) =1+ a₁q⁻¹ +...+ a_nAq^−nA (4.2.13)
B(q−1) = b1q−1 +...+ bnBq−nB = q−1B*(q−1) (4.2.14)
C(q⁻¹) = 1+ c₁q⁻¹ + ... + c_nC q^−nC (4.2.15)

T(q^−^1)y^*(t + d +1) − R(q^−^1)y(t) u(t) = −1 S(q ) in which the reference trajectory is defined by:	(4.2.16)
y*(t + d +1) = Bm (q−1) r(t) Am (q−1)	(4.2.17)
Bm (q^−^1) = bm0 + bm1q^−^1+... Am (q^−^1) =1+ am q^−^1+ am q^−^2+...	(4.2.18) (4.2.19)

Note that B(q^-1) must have stable zeros as well as C(q^-1) that specify the closed loop poles. Moreover C(q^-1) is always stable if the disturbance is stationary. The controller is given by:
Figure 4.15. Minimum variance tracking and regulation
The closed loop transfer function without T(q^-1)) is (see Figure 4.15):
_{−1 q}−(d+1)_B*(_q−1) _q−(d+1)
H BF₁ (q ) = ₋1 −1 −(d+1) * −1 −₁ = ₋1

q−(d+1)B*(q−1) = B*(q−1)C(q−1) As for tracking and regulation with independent objectives:	(4.2.20)
S(q^-1) = B^*(q^-1) S^'(q^-1) with	(4.2.21)
S^'(q^-1) = 1 + s^'1 q^-1 +...+ s^'d q^-d and:	(4.2.22)
R(q−1) = r0 + r1q−1 +...+ rn −1q−nA+1	(4.2.23)

A(q )S(q ) + q B (q )R(q ) C(q )
In order to compute R(q^-1) and S(q^-1) the following equation must be solved:
A(q^-1) S(q^-1) + q^-(d+1) B^*(q^-1) R(q^-1)= B^*(q^-1) C(q^-1) (4.2.24)
Taking into account the structure of S(q^-1), Equation 4.2.24 becomes:
A(q^-1) S^'(q^-1) + q^-(d+1) R(q^-1) = C(q^-1) (4.2.25)
and the solution is the same as for the deterministic case, taking P(q^-1) = C(q^-1).
For solving 4.2.25 the functions predisol.sci (Scilab) and predisol.m (MATLAB^®) available on the book website can be used.
The precompensator T(q^-1) will have to compensate the closed loop poles and thus:
T(q^-1) = C(q^-1) (4.2.26)
Let study the effect of the minimum variance control law, given by Equation 4.2.16 on the error [y(t) - y^*(t)].
Equation 4.2.12 can be rewritten as:
A(q^-1) y(t+d+1) = B^*(q^-1) u(t) + C(q^-1) e(t+d+1) (4.2.27)
Introducing the control law u(t) given by Equation 4.2.16 and multiplying by S(q^-1) both sides, one obtains:
A(q^-1) S(q^-1) y(t+d+1) = B^*(q^-1) C(q^-1) y^*(t+d+1) +
- q^-(d+1) B^*(q^-1) R(q^-1)y(t+d+1)+ S(q^-1) C(q^-1) e(t+d+1) (4.2.28)
Regrouping the terms in y(t+d+1) and considering Equations 4.2.24 and 4.2.21, Equation 4.2.28 becomes:
B^*(q^-1) C(q^-1) y(t+d+1) = B^*(q^-1) C(q^-1) y^*(t+d+1) +
+ S^'(q^-1) B^*(q^-1) C(q^-1) e(t+d+1) (4.2.29)
and dividing by B^*(q^-1) C(q^-1) one obtains:
y(t+d+1) - y^*(t+d+1) = S^'(q^-1) e(t+d+1) (4.2.30)
in other words, the tracking (or regulation) error is a MA process of order d (for d=0, [y(t+d+1) - y^*(t+d+1)] is a white noise).
This corresponds to the minimization of the variance of the error [y(t) - y^*(t)]. In fact, using Equation 4.2.25, it is possible to write:
C(q^-1) y(t+d+1) = [A(q^-1) S^'(q^-1) + q^-(d+1) R(q^-1)] y(t+d+1) (4.2.31)
Taking into account also Equation 4.2.27 and dividing by C(q^-1) both sides, one obtains²:
y(t + d +1) = R(q⁻¹) y(t)+ S′(q⁻¹)B^*(q⁻¹)u(t)+ S′(q−1)e(t + d +1) (4.2.32)
C(q⁻¹) C(q⁻¹)
from which it results:
E^⎧_⎨[y(t + d +1) − y^*(t + d +1)] ^{2 ⎫}_⎬ =
⎩ ⎭
⎧⎪⎡R(q−1) S(q−1) * ⎤2 ⎫⎪
E⎨⎢ ₋₁ y(t) + ₋₁ u(t) − y (t + d +1)⎥ ⎬
⎪⎩⎢⎣C(q ) C(q ) ⎥⎦ ⎪⎭ (4.2.33)
+ E^⎧_⎨[S′(q⁻¹)e(t + d +1)] ^{2 ⎫}_⎬
⎩ ⎭
⎧⎪⎡R(q−1) S(q−1) * ⎤ [ ′(q−1)e(t + d +1)]⎬⎪⎫
+ 2E⎨⎢ ₋₁ y(t) + ₋₁ u(t) − y (t + d +1)⎥ ⋅ S
⎪⎩⎢⎣C(q ) C(q ) ⎥⎦ ⎪⎭
The third term of the right hand side member will be zero since S^'(q^-1) e(t+d+1) contains e(t+1), e(t+2).....e(t+d+1), which are all independent of y(t), y(t-1),..., u(t), u(t-1) ... y^*(t+d+1), y^*(t+d).... The second term does not depend on u(t) and, finally, by using the control law given by Equation 4.2.16 with T(q^-1) = C(q^-1), the first term of the second member is zero, which corresponds to the minimization of the variance of [y(t) - y^*(t)].
Note that Equation 4.2.30 allows a practical test for the optimal tuning of a digital controller to be defined, if the time delay d is known, since in this case the error must be a moving average of order d. Defining:
N
R(i) = 1 _∑[y(t) − y^*(t)] [⋅ y(t − i) − y^*(t −i)] i = 0,1,2,... (4.2.34)
N
t=1
and RN(i) = R(i)/R(0) respectively, one theoretically must obtain for large N:
RN(i) ≈ 0 i ≥ d + 1 (4.2.35)
In practice, based on finite length data, one considers as an acceptable value:
| RN (i) | ≤ 2.17 / N ; i ≥ d + 1 (4.2.36)
where N is the number of samples used for computing RN(i) (for N = 256, RN(i) ≤0. 136, i ≥ d + 1).
For more details on independence tests with finite length data see also Chapter 6, Section 6.2.
Finally, we remind that this design method only applies to plants having discrete-time models with stable zeros since the controller cancels the plant zeros. In the case of a discrete time plant model with unstable zeros one uses:
− either an approximation of the minimum variance tracking and regulation control law using the pole placement with a particular choice of the desired closed loop poles;
− or a control law based on a criterion that introduces a weight on the control signal energy.
Auxiliary Poles
In some applications the poles corresponding to C(q^-1) can be too fast with respect to the open loop system dynamics. This can lead to an unacceptable stress on the actuator or to unacceptable robustness margins. In this case, one can either use generalized minimum variance tracking and regulation design (see below Section 4.3), or add auxiliary poles.
If one chooses to add additional poles, the polynomial defining the desired closed loop poles becomes:
P(q^-1) = C(q^-1) P_F(q^-1)
where P_F (q^-1) represents the polynomial corresponding to the additional poles.
This corresponds to the polynomial P(q^-1) to be used in Equation 4.2.25 instead of C(q^-1) and in this case, consequently, T(q^-1) = C(q^-1) P_F(q^-1).
This modification corresponds to minimize the variance of the regulation error filtered by P_F (q^-1), i.e.:
min E {[P_F (q^-1) [y(t+d+1) - y^*(t+d+1)]]²}
4.2.3 Minimum Variance Tracking and Regulation: Example
The considered plant model is the same as the one used for the tracking and regulation with independent objectives in Section 3.4.4 (the desired tracking performance is also the same). The results of the minimum variance tracking and regulation design are summarized in Table 4.1.


Table 4.1. Minimum Variance Tracking and regulation

Plant: d = 0 B(q^-1) = 0.2 q^-1 + 0.1 q^-2 A(q^-1) = 1 - 1.3 q^-1 + 0.42 q^-2 Tracking dynamics  Ts = 1s, ω0 = 0.5 rad/s, ζ = 0.9 Bm = +0.0927 +0.0687 q^-1 Am = 1 - 1.2451 q^-1 + 0.4066 q^-2 Disturbance polynomial  C(q^-1) = 1 -1.34 q^-1 + 0.49 q^-2 Pre-specifications: Integrator *** CONTROL LAW *** S(q^-1) u(t) + R(q^-1) y(t) = T(q^-1) y^*(t+d+1) y^*(t+d+1) = [(Bmq^-1)/Am(q^-1)] . ref(t) Controller: R(q^-1) = 0.96 - 1.23 q^-1 + 0.42 q^-2 S(q^-1) = 0.2 - 0.1 q^-1 - 0.1 q^-2 T(q^-1) = C(q^-1) Gain margin: 2.084 Phase margin: 61.8 deg Modulus margin: 0.520 (- 5.68 dB) Delay margin: 1.3 s

The controller design results for R(q^-1) and S(q^-1) are given in the lower part of Table 4.1. Before starting the simulation, the variance and mean value of the white noise generating the disturbance (through the filter C(q^-1)/A(q^-1)) must be specified. The simulations results shown in Figure 4.16 illustrate the operation of the minimum controller in regulation and tracking. One can see that introduction of the controller effectively reduce the variance of the output.

Foydalanilgan adabiyotlar :

For the description of stochastic disturbances by means of ARMA(X) models and the minimum variance control see: Box G.E.P., Jenkins G.M. (1970) Time Series Analysis, Forecasting and Control, Holden Day, S. Francisco. Åström K.J. (1970) Introduction to Stochastic Control Theory, Academic Press, N.Y. and also Åström K.J., Wittenmark B. (1997) Computer Controlled Systems - Theory and Design, 3rd edition, Prentice-Hall, Englewood Cliffs, N.J. The idea of the generalized minimum variance control has been introduced by Clarke D.W., Gawthrop P.J., (1975) A self-tuning controller, Proc. IEEE, vol. 122, pp. 929-934. and further developed in: Clarke D.W., Gawthrop P.J. (1979) Self-tuning Control, Proc. IEEE, vol. 126, pp. 633-40. For a unified approach to the control design in a stochastic and deterministic environment see: Landau I.D. (1981) Model Reference Adaptive Controllers and Stochastic Selftuning Regulators, A Unified Approach, Trans. A.S.M.E, J. of Dyn. Syst. Meas. and Control, vol. 103, n°4, pp. 404-416. Control design techniques presented in Chapter 3 (with the time domain interpretation given in the Appendix B) and in Chapter 4 belong to the category of methods called « one step ahead predictive control » (more exactly d+1 steps). There also exist multi-step predictive control techniques (generalized predictive control). See Appendix B as well as: Landau I.D., Lozano R., M’Saad M. (1997) Adaptive Control, (Chapter 7), Springer, London, UK. Camacho, E.F., Bordons, C. (2004) Model Predictive Control, 2nd edition, Springer, London.

1

2