- Email: [email protected]
Dual-rate adaptive control
Aummica,
PII:
Vol. 32, No. 7, pp. 1027-10343, 19% Copyright 0 1996 &evier Science Ltd Printed in Great Britain. All rkhts reserved cmo5-1098/% 0.00
sooo5-1098(%)ooo34-9
ho
+
Brief Paper
Dual-rate Adaptive Control* PEDRO
ALBERTOS,?
JULIAN
SALT? and JOSEP TORNEROt
Key Words-Sampled-data systems; dual-rate systems; multirate modelling; input-output self-tuning regulator; adaptive control.
2. Modelling step Let us first introduce the new dual-rate operator and its relationship with the classical single-rate discrete models. 2.1. Dual-rate operator description. Two sampling rates are involved. The control and measurement periods are T and NT respectively. The discrete operator associated with the interval T is the well-known forward operator 4. U(q) represents the sequence {u(k)}, where the time interval between two consecutive elements is T. Thus 4”’ is able to operate with the discrete sequence of output measurements. In the same framework, Y(qN) represents the sequence {y(Nk)}, where the time interval between two consecutive elements is NT, and similarly for U(qN). Ui(qN) represents a time-interval-NT sequence, delayed by i periods of time T. Given a single-input single-output continuous-time process, modelled by the transfer function G(s), the following discrete-time (DT) models may be attached to it.
called the dual-rate transfer Abstract-An operator, function, describing a dual-rate discretized continuous-time system is presented. It can be easily related to single-rate transfer functions of the same system, and its parameters can be experimentally estimated. Based on these models, two adaptive control strategies are outlined, and some results are illustrated by a simple example. Copyright 0 1996 Elsevier Science Ltd. 1. Introduction Digital control techniques offer many possibilities that are unavailable with linear continuous-time invariant controllers. Finite-time control is one of these, (Isermann, 1981). On the other hand, sampling of continuous-time signals implies loss of some information, and some unexpected problems can appear in the intersampling period. These two features are reinforced if extra control actions are taken in that period. It is a classical result (see e.g. Astrom and Wittenmark, 1984), that the closed-loop discrete-time transfer function zeros may be arbitrarily placed with either a periodic feedback gain or a generalized type of hold circuit. Stronger discrete-time results are obtained if generalized sample-data hold functions or periodic controls are implemented, as explained in Chen and Francis (1991), and Kabamba (1987) respectively. Recent work of Goodwin and Feuer (1992) points out that the intersampling behaviour of the actual process output is sometimes unacceptable. In some industrial applications, and mainly if the controlled output is measured by means of an analyzer-like device, samples of the output are obtained at a rate that is slower than the possible updating control rate. Thus, under these circumstances, dual-rate control is suitable. Lu and Fisher (1992) and Guillandoust et al. (1987), for example, have proposed models for this kind of processes. In the contributions of Salt et al. (1993,1994), a new approach to deal with this problem has been presented. The purpose of this paper is to analyse the features of this new model and to show its suitability to implement an adaptive control scheme, where the output is available at a rate lower than that of the control updating. Obviously, the above-mentioned intersampling-period problems also could appear. Although the available sequences of input and output data are not equally spaced, they have the maximum information about the process behaviour. The proposed approach tries to model the process using this information. In a second stage, a single-rate model is derived, allowing the use of any standard controller design technique. If the designed controller is a slow-rate one, a periodic structure will allow the achievement of higher performances, such as, for instance, allocation of transfer function zeros.
Definition 1. Fast-sampling
DT model. The input-period-T discrete transfer function of the process is given by
where n is the process model order. The DT transfer function poles are denoted by oi.p That is, D,(q) = ny=, (q - oi.r). Definition 2. Slow-sampling DT model. The output-periodNT discrete transfer function of the process may be
expressed as
(2) In this case, the poles Q&IN) = n:=, (qN - %.NT)-
are
denoted
by %VT and
Definition 3. Dual-rate DT (DRDT) model. It is possible to introduce a dual-rate operator able to express the relationship between two different spaced-time sequences: {y(Nk)} and {u(k)}. The mathematical form of this model is (Salt et al., 1993) Y(qN) -=G(q,N)=s U(q)
%=“,& Tq-’ =l + C:=, a,,,,q --iN’
*Received 14 September 1994; revised 15 March 1995; received in final form 18 December 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor C. C. Hang. Corresponding author Professor P. Albertos. Tel. +34 6 387 95 70; Fax +34 6 387 95 79: E-mail [email protected]. t Departamento de Ingenierfa de Sistemas, Computadores y Automatica. Universidad Politecnica de Valencia, P.O. Box 22012, 46071 Valencia, Spain.
where it is easy to prove that P1.r = b1.r Definition
4. Block
DRDT
(BDRDT)
(4) model.
The
NT-
period output sequence of the process may be expressed as
(5) 1027
lllf0%:I-F
description:
1028
Brief Papers
where the BDRDT model, &(qN) is expressed as &IN)
= @,(s?
‘.
MN)
obtained from the dual-rate operator e(q, is as follows.
Gv(@Yl
Gj(qN)represents the contribution to the output of Uj(q”‘), the period-NT sequence of input values taken at time instants (kN + j)T. From the DRDT model, by grouping all the equally delayed coefficients, it is easy to derive &qN)
I
= z%i pi?++.fl+
D,tq$
, i= 1,. . ,n.
(6)
Given a period-T sequence {u(k)}, let us define the period-NT sequence {iI( of blocks of inputs, where the sequence elements are
ii(k)
=
Denominator. As has been described before, the coefficients, and therefore the NT discrete poles (~~.~r, are known. The poles for period T are obtained as their respective lowest l/N power, (li,r. So, W(q) can be built by use of (12). Numerator.
This requires a deconvolution between the dual-rate operator numerator N(q, N) and the polynomial W(q), (13). It should be noted that the first numerator coefficient b,., is always known from the operator numerator.
[I WQN) 1 UKN+I
UK:+*
(7)
N). The algorithm
2.3. Reduced-order model formulas. If a second-order process (n = 2) and N = 2 is considered, the procedure explained before can be substantially simplified. Let us write the dual-rate operator G(q, N) as
UKN+N
By means of the shift operator, expressed as
U(qN) =
this sequence
(8)
Lemma 1. The coefficients of the numerator polynomial of the slow sampling DT model, GN(qN), are given by
j=l,...,n
&r,
(9)
Proof: This is by direct model application.
Let us assume that, under dual-rate operation, the input is assumed to be constant between any two consecutive measurements, as in the case of slow-sampling DT. The same output is obtained 0 by means of both models: G(q, N) and GN(qN). Corollary 1. The slow-sampling
DT model, GN(qN), can be easily obtained from the dual-rate operator. The denominator polynomials are the same, and Lemma 1 provides the relation between the numerator polynomials. Lemma 2. The numerator and denominator of the DRDT model can be obtained by convolution of those of the fast-sampling DT model with the polynomial W(q), (Guilandoust et al., 1987), described by W(q) = fi (qN-’ + cyi,fl+*
+.
.+
a?f').
(10)
i=l
Proof. It should be noted that the polynomial W(q) may be expressed as
(II) Taking into account that the pole oi,NT for a period NT is the Nth power of a pole ai,T for period T, it follows that W(q) = ,~+x~~=~.
(12)
1. Thus Y(9) -= u(q)
Pi = h,,r.
By Lemma 1,
UN(4N)
2.2. Single-rate equivalent models. At this point, let us describe the way in which the dual-rate operator is related to the single-rate discrete transfer functions.
bj,Nr = ,,_$+,
’
where
U2kN) :
[
may be
N&M’(q) _ N&M’(q) D,(9)w(9)DN(qN)
Y(q)DN(qN) gives the output at NT intervals, Y(qN), based on the input sequence U(q). Then
b I.ZT = PI +
P2,
b2.2r
h,
Corollary
2 The
fast-sampling
DT model G,(q)
+
T,
(-1)3h=
(-a,,T)(-‘%T)($=)>
1.T
(14)
and b,,, is always known; thus b,,, is easily obtained. 3. Dual-rate controller design Let us consider the control system design problem when the control action may be updated at a frequency (period T) faster than the output sampling rate (period NT). Given a CT process model, the following possibilities may be considered. 3.1. High-rate contoller design. In this case, the fast DT transfer function should be obtained. Period-T specifications are assumed for the controller design. In the implementation stage, the actual output measurement can only be used at intervals NT. A possible solution, as suggested in Albertos (1993) is to implement a weighted-model-based control. The calculated G,(q) plays the role of an observer, its output being used at the sampling times where the plant output is not available. Figure 1 shows the described procedure, where the variable A is equal to one at r = NT, and zero otherwise. 3.2. Periodic controller design. Another approach is to design a low-rate controller based on the NT discrete transfer function. In this case a high rate in the control updating may be foreseen to adjust the closed-loop zero locations (Astrom and Wittenmark, 1984). If the control period is split into N units, keeping the low-frequency gain constant, this objective could be achieved. Therefore this strategy leads to a multirate scheme similar to the one previously described (Salt et al., 1994). If the N control action components of a block are taken in such a way that KI ii(k) =
(13) 0 can be
bs
leading to the discrete transfer function GN(qN). Application of Lemma 2 will give G,(q). An alternative way can be considered for this reducedorder case. First, as has been described, from the 2T discrete poles (o,,rr, a2,2T), the T poles ((Y,,~, CX*,~)are calculated. In this case the comulex solutions needed for building W(a) will be the complex conjugates (of o;r). As the operator numerator order is greater than 2, it is possible to derive
But
fi(q, N) = N,(q)W(q)
=
?
u(kN),
(15)
I:]KN qkN)
=
QJkN)
(1’5)
1029
Brief Papers
G,(s)
REFERENCE
Fig. 1. Weighted-model-based
then the following single-rate function will be obtained:
slow-sampling
DT transfer
$ =j$Kje,(9? =Z-(9”). It should be pointed out that if G(sY
K, =
(17)
1 Vi then
= G,(9?.
(18)
Therefore, in order to retain low-rate discrete-time process behaviour, it will be assumed that c,“=, K; = N in all cases. The previous approach may be extended to any other technique of periodic-controller design, given the initial CT process model. 4. Adaptive control strategy We now consider an indirect adaptive control scheme involving the above controller design options. By an appropriate arrangement of process measurements, the identification step will lead to the DRDT model G(9, N). Based on the results of Lemmas 1 and 2, some model conversions are possible, as shown in Fig. 2. The certainty equivalence principle is assumed. 4.1. High-rate control application. The dual-rate operator G(9, N) is identified. Then the W(9) polynomial is calculated, and, following Corollary 2, the fast sampling DT model is obtained. It should be pointed out that if a second-order process is being considered, the zero is very easily obtained from (14). It is also interesting to note that in this case the zero is usually real and certainly very different from the poles. Afterwards, the control design step can be performed as usual-for instance by the self-tuning regulator approach (Isermann, 1981). 4.2. Periodic control application. In this case, once the dual-rate operator G(9, N) has been identified, and following Lemma 1, the slow-sampling-rate DT transfer function at the evolving period NT, G,49N), is obtained. The slow-rate CONTINUOUS PLANT
LS-ESTIMATION I LEMMAi
&xN)
I
4 ($2
1
1
. W(q)
1 LEMMA2
G(q)
Fig. 2. Model parameters computation.
control.
control design step is a conventional one. The slow-rate controller can be implemented in block form, allowing transfer-function-zero location without changing the poles. If we select a fast sampling period T, the gain elements K~, (15), can be calculated on-line in order to get the required zero location. 5. Example Let us consider the case of a continuous-time transfer function
system with
1 G(s) = (10s + 1)(2Os + 1). If an output sampling rate of 1 s is assumed and N = 2 (T = 0.5). the following results are obtained: 0.00069 + 0.00059 G,(9) = 92 - 1.9269 + 0.927 ’ G2(9*) =
0.00249* + 0.0023 94 - 1.85619* + 0.8607 ’
which may be rewritten in pole-zero
form as
0.0024(9 + 0.95) Gk?*)= (9 - 0.9048)(9 - 0.9512) The dual-rate operator is G(~ 2) = i.319~+ &9* + P39 + b 94 - 1.85619* + 0.8607 ’ where /I, = 0.0006, p3 = 0.0017,
p2 = 0.0018, p4 = 0.0006.
In this case, the partial transfer functions, Gi(sN), would be: G,(s2) =
P29* + P4 94 - I.85619* + 0.8607
G,(9? =
P19*+Ps 94 - 1.85619* + 0.8607’
5.1. Adaptive high-rate control. In order to illustrate the whole procedure, an adaptive control system involving the estimation block, as in Fig. 2, has been implemented using a pole-assignment self-tuning regulator, as described in Teng and Sirisena (1988). The selected closed loop-behaviour has been defined by a settling time of 5s and a maximum overshoot of 6 5 lo%, leading to a pole location such as 0.79 f 0.21j. The output unavailability has been solved with a model-based control, as in Fig. 1. Figure 3 shows the process output response to a step change in the reference. Model mismatching is due to an additive measurement noise. Figure 4 plots the estimated parameter evolution for numerator and denominator. 5.2. Adaptive period control. In this case a slow-rate controller has been designed, with the same CT closed-loop
Brief Papers
0
10 Time
0
20
40
60 Time
80
1
Fig. 3. Step response of adaptive high-rate controlled system. Fig. 5. Low-rate adaptive control with constant gain (solid line, a) and dual rate (dotted line). process specifications. The zero c of the high-rate discrete transfer function could be modified by the selection of appropriated K, and K~ gains, by means of the expression 2(& -&c) K’ = - 0% - P4) + c(Pz - Pi) Obviously, K* = 2 - K,. In this case, if the purpose is to move the zero to the new position c = -0.1, the gains are K, = 2.69 and ~z = -0.69. Figure 5 shows the process output response to a step change in the reference. Curve a is the result of applying the equivalent constant gain to the original process.
10 5
x 16'
I
I
il~____________~__~__~_~~______~
0
.
x 10“ 10r
10
30
40
I
-5’1 0
20
I
I 10
20
30
40
6. Conclusions A new approach for handling dual-rate control situations has been presented. A control updating period shorter than the output sampling one has been considered. This is the usual case in practical applications if the measurement device is not fast enough, while good actuators are available. The introduced dual-rate operator gives an initial DT model of the process that is suitable for use in the classical parameter estimation algorithms. Single-rate models at both rates can be derived from it. This feature allows one to implement an adaptive control strategy, with minor model transformation computations, following any classical singlerate controller design techniques, to get the basic controller. Design of both model reference and periodic controllers allows one to take advantage of the dual rate operating conditions. Some results obtained in an academic example illustrate the validity of the approach. References
Albertos, P. (1993). Weighted model-based control system. In Proc. 2nd European Control Conf, Groningen, Vol. 4, pp. 19351939. Astrom, K. and B. Wittenmark (1984). Computer Controlled Systems. Prentice-Hall, Englewood Cliffs, NJ. Chen, T. and B. A. Francis, (1991). H2 optimal sampled data control. IEEE Trans. Autom. Control, AC&i, 387-398. Goodwin G. C. and A. Feuer, (1992). Linear periodic control: a frequency domain viewpoint. Syst. Control. Lett., 19,379-390.
q_1 -2
- --_-----_-j_____________:_____________:
0
10
20
Guilandoust, M. T., A. J. Morris and M. T. Tham (1987). Adaptive inferential control. IEEE Proc., Pt D, 134, 171-179. Isermann. R. (1981). Digital Control Systems. SpringerVerlag, Berlin. Kabamba, P. T. (1987). Control of linear systems using generalized sample-data hold functions. IEEE Trans.
____ ________
30
40
30
40
I 10
20 NumberofSamtAes
Fig. 4. Parameter convergence of process model G,(q).
Autom. Control, AC-32,772-783.
Lu, W. and G. Fisher (1992). Multirate adaptive inferential estimation. IEE Proc., Pt D, 139, 182-189. Salt, J., P. Albertos and J. Tornero, (1993) Modelling of non-conventional sampled data systems. In Proc. 2nd IEEE Conf on Control Applications, Vancouver, pp. 631-635. Salt, J., P. Albertos and J. Tornero, (1994) Digital controller improvement by multirate control. In Proc. 3rd IEEE Conf. on Control Applications, Glasgow, pp. 1459-1464. Teng Fong-Chwee and H. R. Sirisena (1988). Self-tuning PID controllers for dead time processes. IEEE Trans. Indust. Electron., IE-35, 119-125.
PII:
Vol. 32, No. 7, pp. 1027-10343, 19% Copyright 0 1996 &evier Science Ltd Printed in Great Britain. All rkhts reserved cmo5-1098/% 0.00
sooo5-1098(%)ooo34-9
ho
+
Brief Paper
Dual-rate Adaptive Control* PEDRO
ALBERTOS,?
JULIAN
SALT? and JOSEP TORNEROt
Key Words-Sampled-data systems; dual-rate systems; multirate modelling; input-output self-tuning regulator; adaptive control.
2. Modelling step Let us first introduce the new dual-rate operator and its relationship with the classical single-rate discrete models. 2.1. Dual-rate operator description. Two sampling rates are involved. The control and measurement periods are T and NT respectively. The discrete operator associated with the interval T is the well-known forward operator 4. U(q) represents the sequence {u(k)}, where the time interval between two consecutive elements is T. Thus 4”’ is able to operate with the discrete sequence of output measurements. In the same framework, Y(qN) represents the sequence {y(Nk)}, where the time interval between two consecutive elements is NT, and similarly for U(qN). Ui(qN) represents a time-interval-NT sequence, delayed by i periods of time T. Given a single-input single-output continuous-time process, modelled by the transfer function G(s), the following discrete-time (DT) models may be attached to it.
called the dual-rate transfer Abstract-An operator, function, describing a dual-rate discretized continuous-time system is presented. It can be easily related to single-rate transfer functions of the same system, and its parameters can be experimentally estimated. Based on these models, two adaptive control strategies are outlined, and some results are illustrated by a simple example. Copyright 0 1996 Elsevier Science Ltd. 1. Introduction Digital control techniques offer many possibilities that are unavailable with linear continuous-time invariant controllers. Finite-time control is one of these, (Isermann, 1981). On the other hand, sampling of continuous-time signals implies loss of some information, and some unexpected problems can appear in the intersampling period. These two features are reinforced if extra control actions are taken in that period. It is a classical result (see e.g. Astrom and Wittenmark, 1984), that the closed-loop discrete-time transfer function zeros may be arbitrarily placed with either a periodic feedback gain or a generalized type of hold circuit. Stronger discrete-time results are obtained if generalized sample-data hold functions or periodic controls are implemented, as explained in Chen and Francis (1991), and Kabamba (1987) respectively. Recent work of Goodwin and Feuer (1992) points out that the intersampling behaviour of the actual process output is sometimes unacceptable. In some industrial applications, and mainly if the controlled output is measured by means of an analyzer-like device, samples of the output are obtained at a rate that is slower than the possible updating control rate. Thus, under these circumstances, dual-rate control is suitable. Lu and Fisher (1992) and Guillandoust et al. (1987), for example, have proposed models for this kind of processes. In the contributions of Salt et al. (1993,1994), a new approach to deal with this problem has been presented. The purpose of this paper is to analyse the features of this new model and to show its suitability to implement an adaptive control scheme, where the output is available at a rate lower than that of the control updating. Obviously, the above-mentioned intersampling-period problems also could appear. Although the available sequences of input and output data are not equally spaced, they have the maximum information about the process behaviour. The proposed approach tries to model the process using this information. In a second stage, a single-rate model is derived, allowing the use of any standard controller design technique. If the designed controller is a slow-rate one, a periodic structure will allow the achievement of higher performances, such as, for instance, allocation of transfer function zeros.
Definition 1. Fast-sampling
DT model. The input-period-T discrete transfer function of the process is given by
where n is the process model order. The DT transfer function poles are denoted by oi.p That is, D,(q) = ny=, (q - oi.r). Definition 2. Slow-sampling DT model. The output-periodNT discrete transfer function of the process may be
expressed as
(2) In this case, the poles Q&IN) = n:=, (qN - %.NT)-
are
denoted
by %VT and
Definition 3. Dual-rate DT (DRDT) model. It is possible to introduce a dual-rate operator able to express the relationship between two different spaced-time sequences: {y(Nk)} and {u(k)}. The mathematical form of this model is (Salt et al., 1993) Y(qN) -=G(q,N)=s U(q)
%=“,& Tq-’ =l + C:=, a,,,,q --iN’
*Received 14 September 1994; revised 15 March 1995; received in final form 18 December 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor C. C. Hang. Corresponding author Professor P. Albertos. Tel. +34 6 387 95 70; Fax +34 6 387 95 79: E-mail [email protected]. t Departamento de Ingenierfa de Sistemas, Computadores y Automatica. Universidad Politecnica de Valencia, P.O. Box 22012, 46071 Valencia, Spain.
where it is easy to prove that P1.r = b1.r Definition
4. Block
DRDT
(BDRDT)
(4) model.
The
NT-
period output sequence of the process may be expressed as
(5) 1027
lllf0%:I-F
description:
1028
Brief Papers
where the BDRDT model, &(qN) is expressed as &IN)
= @,(s?
‘.
MN)
obtained from the dual-rate operator e(q, is as follows.
Gv(@Yl
Gj(qN)represents the contribution to the output of Uj(q”‘), the period-NT sequence of input values taken at time instants (kN + j)T. From the DRDT model, by grouping all the equally delayed coefficients, it is easy to derive &qN)
I
= z%i pi?++.fl+
D,tq$
, i= 1,. . ,n.
(6)
Given a period-T sequence {u(k)}, let us define the period-NT sequence {iI( of blocks of inputs, where the sequence elements are
ii(k)
=
Denominator. As has been described before, the coefficients, and therefore the NT discrete poles (~~.~r, are known. The poles for period T are obtained as their respective lowest l/N power, (li,r. So, W(q) can be built by use of (12). Numerator.
This requires a deconvolution between the dual-rate operator numerator N(q, N) and the polynomial W(q), (13). It should be noted that the first numerator coefficient b,., is always known from the operator numerator.
[I WQN) 1 UKN+I
UK:+*
(7)
N). The algorithm
2.3. Reduced-order model formulas. If a second-order process (n = 2) and N = 2 is considered, the procedure explained before can be substantially simplified. Let us write the dual-rate operator G(q, N) as
UKN+N
By means of the shift operator, expressed as
U(qN) =
this sequence
(8)
Lemma 1. The coefficients of the numerator polynomial of the slow sampling DT model, GN(qN), are given by
j=l,...,n
&r,
(9)
Proof: This is by direct model application.
Let us assume that, under dual-rate operation, the input is assumed to be constant between any two consecutive measurements, as in the case of slow-sampling DT. The same output is obtained 0 by means of both models: G(q, N) and GN(qN). Corollary 1. The slow-sampling
DT model, GN(qN), can be easily obtained from the dual-rate operator. The denominator polynomials are the same, and Lemma 1 provides the relation between the numerator polynomials. Lemma 2. The numerator and denominator of the DRDT model can be obtained by convolution of those of the fast-sampling DT model with the polynomial W(q), (Guilandoust et al., 1987), described by W(q) = fi (qN-’ + cyi,fl+*
+.
.+
a?f').
(10)
i=l
Proof. It should be noted that the polynomial W(q) may be expressed as
(II) Taking into account that the pole oi,NT for a period NT is the Nth power of a pole ai,T for period T, it follows that W(q) = ,~+x~~=~.
(12)
1. Thus Y(9) -= u(q)
Pi = h,,r.
By Lemma 1,
UN(4N)
2.2. Single-rate equivalent models. At this point, let us describe the way in which the dual-rate operator is related to the single-rate discrete transfer functions.
bj,Nr = ,,_$+,
’
where
U2kN) :
[
may be
N&M’(q) _ N&M’(q) D,(9)w(9)DN(qN)
Y(q)DN(qN) gives the output at NT intervals, Y(qN), based on the input sequence U(q). Then
b I.ZT = PI +
P2,
b2.2r
h,
Corollary
2 The
fast-sampling
DT model G,(q)
+
T,
(-1)3h=
(-a,,T)(-‘%T)($=)>
1.T
(14)
and b,,, is always known; thus b,,, is easily obtained. 3. Dual-rate controller design Let us consider the control system design problem when the control action may be updated at a frequency (period T) faster than the output sampling rate (period NT). Given a CT process model, the following possibilities may be considered. 3.1. High-rate contoller design. In this case, the fast DT transfer function should be obtained. Period-T specifications are assumed for the controller design. In the implementation stage, the actual output measurement can only be used at intervals NT. A possible solution, as suggested in Albertos (1993) is to implement a weighted-model-based control. The calculated G,(q) plays the role of an observer, its output being used at the sampling times where the plant output is not available. Figure 1 shows the described procedure, where the variable A is equal to one at r = NT, and zero otherwise. 3.2. Periodic controller design. Another approach is to design a low-rate controller based on the NT discrete transfer function. In this case a high rate in the control updating may be foreseen to adjust the closed-loop zero locations (Astrom and Wittenmark, 1984). If the control period is split into N units, keeping the low-frequency gain constant, this objective could be achieved. Therefore this strategy leads to a multirate scheme similar to the one previously described (Salt et al., 1994). If the N control action components of a block are taken in such a way that KI ii(k) =
(13) 0 can be
bs
leading to the discrete transfer function GN(qN). Application of Lemma 2 will give G,(q). An alternative way can be considered for this reducedorder case. First, as has been described, from the 2T discrete poles (o,,rr, a2,2T), the T poles ((Y,,~, CX*,~)are calculated. In this case the comulex solutions needed for building W(a) will be the complex conjugates (of o;r). As the operator numerator order is greater than 2, it is possible to derive
But
fi(q, N) = N,(q)W(q)
=
?
u(kN),
(15)
I:]KN qkN)
=
QJkN)
(1’5)
1029
Brief Papers
G,(s)
REFERENCE
Fig. 1. Weighted-model-based
then the following single-rate function will be obtained:
slow-sampling
DT transfer
$ =j$Kje,(9? =Z-(9”). It should be pointed out that if G(sY
K, =
(17)
1 Vi then
= G,(9?.
(18)
Therefore, in order to retain low-rate discrete-time process behaviour, it will be assumed that c,“=, K; = N in all cases. The previous approach may be extended to any other technique of periodic-controller design, given the initial CT process model. 4. Adaptive control strategy We now consider an indirect adaptive control scheme involving the above controller design options. By an appropriate arrangement of process measurements, the identification step will lead to the DRDT model G(9, N). Based on the results of Lemmas 1 and 2, some model conversions are possible, as shown in Fig. 2. The certainty equivalence principle is assumed. 4.1. High-rate control application. The dual-rate operator G(9, N) is identified. Then the W(9) polynomial is calculated, and, following Corollary 2, the fast sampling DT model is obtained. It should be pointed out that if a second-order process is being considered, the zero is very easily obtained from (14). It is also interesting to note that in this case the zero is usually real and certainly very different from the poles. Afterwards, the control design step can be performed as usual-for instance by the self-tuning regulator approach (Isermann, 1981). 4.2. Periodic control application. In this case, once the dual-rate operator G(9, N) has been identified, and following Lemma 1, the slow-sampling-rate DT transfer function at the evolving period NT, G,49N), is obtained. The slow-rate CONTINUOUS PLANT
LS-ESTIMATION I LEMMAi
&xN)
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1
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. W(q)
1 LEMMA2
G(q)
Fig. 2. Model parameters computation.
control.
control design step is a conventional one. The slow-rate controller can be implemented in block form, allowing transfer-function-zero location without changing the poles. If we select a fast sampling period T, the gain elements K~, (15), can be calculated on-line in order to get the required zero location. 5. Example Let us consider the case of a continuous-time transfer function
system with
1 G(s) = (10s + 1)(2Os + 1). If an output sampling rate of 1 s is assumed and N = 2 (T = 0.5). the following results are obtained: 0.00069 + 0.00059 G,(9) = 92 - 1.9269 + 0.927 ’ G2(9*) =
0.00249* + 0.0023 94 - 1.85619* + 0.8607 ’
which may be rewritten in pole-zero
form as
0.0024(9 + 0.95) Gk?*)= (9 - 0.9048)(9 - 0.9512) The dual-rate operator is G(~ 2) = i.319~+ &9* + P39 + b 94 - 1.85619* + 0.8607 ’ where /I, = 0.0006, p3 = 0.0017,
p2 = 0.0018, p4 = 0.0006.
In this case, the partial transfer functions, Gi(sN), would be: G,(s2) =
P29* + P4 94 - I.85619* + 0.8607
G,(9? =
P19*+Ps 94 - 1.85619* + 0.8607’
5.1. Adaptive high-rate control. In order to illustrate the whole procedure, an adaptive control system involving the estimation block, as in Fig. 2, has been implemented using a pole-assignment self-tuning regulator, as described in Teng and Sirisena (1988). The selected closed loop-behaviour has been defined by a settling time of 5s and a maximum overshoot of 6 5 lo%, leading to a pole location such as 0.79 f 0.21j. The output unavailability has been solved with a model-based control, as in Fig. 1. Figure 3 shows the process output response to a step change in the reference. Model mismatching is due to an additive measurement noise. Figure 4 plots the estimated parameter evolution for numerator and denominator. 5.2. Adaptive period control. In this case a slow-rate controller has been designed, with the same CT closed-loop
Brief Papers
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Fig. 3. Step response of adaptive high-rate controlled system. Fig. 5. Low-rate adaptive control with constant gain (solid line, a) and dual rate (dotted line). process specifications. The zero c of the high-rate discrete transfer function could be modified by the selection of appropriated K, and K~ gains, by means of the expression 2(& -&c) K’ = - 0% - P4) + c(Pz - Pi) Obviously, K* = 2 - K,. In this case, if the purpose is to move the zero to the new position c = -0.1, the gains are K, = 2.69 and ~z = -0.69. Figure 5 shows the process output response to a step change in the reference. Curve a is the result of applying the equivalent constant gain to the original process.
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6. Conclusions A new approach for handling dual-rate control situations has been presented. A control updating period shorter than the output sampling one has been considered. This is the usual case in practical applications if the measurement device is not fast enough, while good actuators are available. The introduced dual-rate operator gives an initial DT model of the process that is suitable for use in the classical parameter estimation algorithms. Single-rate models at both rates can be derived from it. This feature allows one to implement an adaptive control strategy, with minor model transformation computations, following any classical singlerate controller design techniques, to get the basic controller. Design of both model reference and periodic controllers allows one to take advantage of the dual rate operating conditions. Some results obtained in an academic example illustrate the validity of the approach. References
Albertos, P. (1993). Weighted model-based control system. In Proc. 2nd European Control Conf, Groningen, Vol. 4, pp. 19351939. Astrom, K. and B. Wittenmark (1984). Computer Controlled Systems. Prentice-Hall, Englewood Cliffs, NJ. Chen, T. and B. A. Francis, (1991). H2 optimal sampled data control. IEEE Trans. Autom. Control, AC&i, 387-398. Goodwin G. C. and A. Feuer, (1992). Linear periodic control: a frequency domain viewpoint. Syst. Control. Lett., 19,379-390.
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Guilandoust, M. T., A. J. Morris and M. T. Tham (1987). Adaptive inferential control. IEEE Proc., Pt D, 134, 171-179. Isermann. R. (1981). Digital Control Systems. SpringerVerlag, Berlin. Kabamba, P. T. (1987). Control of linear systems using generalized sample-data hold functions. IEEE Trans.
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Fig. 4. Parameter convergence of process model G,(q).
Autom. Control, AC-32,772-783.
Lu, W. and G. Fisher (1992). Multirate adaptive inferential estimation. IEE Proc., Pt D, 139, 182-189. Salt, J., P. Albertos and J. Tornero, (1993) Modelling of non-conventional sampled data systems. In Proc. 2nd IEEE Conf on Control Applications, Vancouver, pp. 631-635. Salt, J., P. Albertos and J. Tornero, (1994) Digital controller improvement by multirate control. In Proc. 3rd IEEE Conf. on Control Applications, Glasgow, pp. 1459-1464. Teng Fong-Chwee and H. R. Sirisena (1988). Self-tuning PID controllers for dead time processes. IEEE Trans. Indust. Electron., IE-35, 119-125.