multivariable multirate system description and design

MULTIVARIABLE MULTIRATE SYSTEM DESCRIPTION AND DESIGN Ryszard Gessing Silesian Technical University, Institute of Automatic Control, ul. Akademicka 16, 44–101 Gliwice, Poland fax: +4832371165, email: rgessing @ ia.gliwice.edu.pl

Abstract— It is shown that for a multirate (MR) system the discrete-time (DT) transfer function (TF) description exists only in a rather seldom case when the sampling period of input is integer multiple of that of the output. It is also noticed that in other cases, especially in the case of multivariable (MV) systems the whole system cannot be described by using TF method. For the MV MR system a continuous-time (CT) model is proposed which makes it possible to obtain the controllers by using some appropriate CT method of design and the DT Tustin approximation. Finally , the example of twovariable, tworate system illustrating the proposed methods is considered. R´esum´e: On peut constater que pour le syst`eme avec des p´eriodes diverses d’´echantillonage de l’entr´ee et de la sortie du syst`eme, une description de transmittance discret n’existe que dans le cas, quand la p´eriode d’´echantillonage de l’entr´ee est une propre multiple de la p´eriode d’´echantillonage de la sortie. Dans le cas du syst´eme multivariable avec des p´eriodes diverses d’´echantillonage, le syst`eme integral ne peut pas eˆ tre d´ecrit par une transmittance discr`ete. Dans ce cas, on propose un mod`ele continu de l’objet qui facilite la synth`ese des regulateurs a` la base du m´ethode de temps continu et d’approximation de Tustin. A la fin, on a appliqu´e la m´ethode propos´ee pour un exemple de ce syst`eme avec deux e´ tats variables et deux p´eriodes diverses d’´echantillonage.

Keywords. Multivariable systems; multirate systems; systems design.

1. INTRODUCTION Discrete-time (DT), multirate (MR) systems having samplers with various sampling periods have rather a long history. In second half of fifties Kranc (1957) proposed his operator for analysing these systems. The systems have been described by Kuzin (1962) in his excellent book. A recapitulation in which the state of art related to these systems is given in the survey paper of Araki (1993). It should be stressed that the MR systems for the case when the sample periods are not integer multiples of each other usually can’t be described by the DT transfer functions (TF)’s since these systems are nonstationary. In connection with this some difficulties arise in describing the closed-loop (CL) systems. In the present paper •

the cases when the open-loop (OL) TF exists and not exists are discussed;

•

in the case of multivariable (MV) MR systems when the TF description does not exist, the proposition is given based on applying the continuous-time (CT) models and adapting the method of design described by Gessing (1995a). 2. MULTIVARIABLE MULTIRATE SYSTEM

Consider the MV MR system shown in Fig.1, composed of CT multi-input and multi-output plant described by the matrix TF   G11 (s), ... G1m (s)  G(s) =  ........ ... ........ (1) Gm1 (s), ... Gmm (s) and m independent feedback channels with samplers having different sampling periods hi , zero order holds ZOHi and ¯ i (z), i = 1, 2, ..., m. DT controllers described by DT TF’s R The different sampling periods applied in different channels can be justified by different velocities of particular channels, as well as by limitations resulting from applied measurement devices. The considered system is linear, but the whole system can not be described by using TF method. Really, the TF description of the whole system from Fig.1 would be possible if would exist the TF for the system shown in Fig.2 in the case of different sampling periods hi , hj . Thus there arises the question: whether and when there exists the TF for the system from Fig.2 ? It should be stressed that in the available literature there is no exact answer to this question, though the problem of possible nonexistence of a TF is noticed e.g. in (Program CC, 1991).

3. PROBLEM OF TRANSFER FUNCTION EXISTENCE Consider the three cases: Case 1 : hi = h, hj = N h, N −an integer; Case 2 : hi = M h, hj = h, M −an integer; Case 3 : hi = M h, hj = N h, M, N −some mutual first integers. Here h denotes some basic time interval (greatest common devisor of hi , hj ).

Y h (z) = Zh [y(nh)], ∞ X U N h (z N ) = u(lN h)(z N )−l

(7)

l=0

Ghij = (1 − z −N )Zh [ˆ gij (nh)]

Fig. 1.

Thus, for the Case 1 there exists the TF (8) which relates the DT output Y h (z) to any DT input U N h (z N ). From the other hand, in the Case 2 it can be noticed that then,does not exist the TF which relates the input and output signals. The cause is that the system is then nonstationary. Really, if for any input uj (nh) the system output is yi (nM h), n = 0, 1, 2, ..., then for the shifted by h in time input uj (nh − h) the output yi is not shifted by h since the minimal shift in time for the output is M h. Also in the Case 3 the TF of the system shown in Fig.2 does not exist since the system is nonstationary, too. From above considerations it results that the whole MV MR system shown in Fig.1 can not be described by using the TF method. This creates an essential difficulty in designing this kind of systems.

DT multivariable multirate system

Fig. 2.

(8)

DT system with hi 6= hj

4. MULTIVARIABLE MULTIRATE SYSTEM DESIGN Let us introduce the following notation h

Y (z) = Zh [y(nh)] =

∞ X

y(nh)z

−n

(2)

n=0

i.e. the symbol Zh denotes the Z-transform for the sampling period h. In the Case 1 for any DT series [u(nN h)] and [y(nh)], n = 0, 1, 2, ... it is yi (nh) =

∞ X

[u(lN h) − u((l − 1)N h)] ·

1.1. Design algorithm

l=0

·ˆ yij (nh − lN h)

(3)

where yˆij (t) = L−1



 1 Gij (s) s

(4)

and L−1 denotes the symbol of inverse Laplace transform. From applying Z-transform to both sides of (3) it results Y h (z) =

∞ X

[u(lN h) − u((l − 1)N h)] ·

l=0

·Zh [ˆ gij (nh)]z −lN = = (1 − z −N )U N h (z)Zh [ˆ gij (nh)]

(5)

Y h (z) = Ghij (z)U N h (z N )

(6)

or

where

Consider the system shown in Fig.1. Assume that in the control structure shown there, the input ui of the plant is appropriately suited to the output yi , i = 1, 2, ..., m , so that both appear in the i-th control channel. Since the whole system can not be described by means of a DT TF it is reasonable to use an appropriate CT model of the plant with ZOH, described for the SISO case in (Gessing, 1995a). Using this model the following design algorithm of DT MV systems can be proposed.

1) For given TF’s Gij (s) and assumed sampling periods hj create the modified TF’s - with the derivative Gdij (s) = (1 − shj //2)Gij (s) or with delay Geij (s) = exp(−shj //2)Gij (s), i, j = 1, 2, ..., m; 2) Using one from the known methods of design of CT MV systems choose the CT controllers Rj (s), j = 1, 2, ..., m , for the CT plant described by modified with the derivative (or with the delay) TF’s; 3) Check if the sampling periods hj are sufficiently small, i.e. if the sampling frequency ωj = 2π//hj is approximately ten times greater than the appropriately calculated bandwidth frequency ωj∗ of the j-th loop; if not - decrease the sampling period hj and repeat the steps 1) and 2); ¯ j (z) result from using the Tustin 4) The DT controllers R approximation of the CT controllers Rj (s), j = 1, 2, ..., m, i.e. ¯ j (z) = Rj R



2 z−1 nz+1



(9)

The bandwidth frequency ωj∗ for each j- th loop is calculated with accounting the remaining m − 1 loops of the system closed. Note. In order to choose the proper sampling periods hj assuring the fulfilment of the condition ωj ≥ ωj∗ , ′ first the bandwidth frequencies ωj∗ can be determined for the CT system with the controllers Rj (s) designed for the multivariable plant G(s). Next, we can assume that ′ ωj∗ ≈ ωj∗ and determine hj . Since the models with derivative (Gessing, 1995a) are simpler they will be preferred for using in the case of CT plants without delay. In the case of CT plants with delay the use of the model with delay is justified since then no increase of design complexity is obtained.

5. A METHOD OF DESIGN OF CT MV SYSTEMS

Fig.1 in which the plant matrix TF is determined by   10 7 , (s+2)(s+3) s(s+1)  G(s) =  (10) 5 7 , s(s+1) (s+2)(s+3) ¯ 1 (z) and R ¯ 2 (z) are to be designed which The DT TF’s R determine the algorithms for the DT controllers of both loops. After some preliminary trials it is assumed h1 = 0.18 and h2 = 0.24 . The model with derivative takes the form  −0.9s+10 −0.84s+7  , s(s+1) (s+2)(s+3)  (11) Gd (s) =  −0.45s+5 −0.84s+7 , s(s+1) (s+2)(s+3) First, the CT controllers Rj (s) j = 1, 2 are designed using the method described in the previous section and illustrated ′ in (Gessing, 1995b). In the first step the controller R1 (s) d is designed for the plant G11 (s). In the second step the ′ controller R2 (s) is designed for the replacement SISO plant of the second loop described by the TF ′

Now, consider the CT MV system shown in Fig.1 in which the samplers and ZOH’s disappear and the DT controllers ¯ j (z) are replaced with the CT ones Rj (s). Here, the R approach will be proposed which makes it possible to apply a one variable design method, repeatedly. The approach is based on a repetitive procedure in which the CT controller Rj (s) in the j-th loop is designed for the replacement SISO plant resulting from accounting the remaining loops closed (those which already have designed controllers). Thus, first the controller R1 (s) is designed for the plant G11 (s); the remaining loops are open since they have no designed controllers, yet. Second controller R2 (s) is designed for the 2–nd replacement SISO plant resulting from accounting the first loop closed and the remaining open; tird – for the 3–rd replacement SISO plant with accounting two first loops closed and the remaining open; finally, the mth controller Rm (s)–for the m-th replacement SISO plant resulting from accounting all the remaining loops closed. Then, we repeat our design procedure starting from the first loop and designing the controller Rj (s) in the j-th loop, j = 1, 2, ..., m for the j–th replacement SISO plant resulting from accounting all the remaining loops closed with the controllers designed in the latter steps. If needed, we repeat the design procedure until the further steps give no improvement of the control. It should be stressed that the proposed design procedure is usually fastly convergent. This is the result of the fact that the change of the controller Rj (s) in the j-th loop influences mainly the characteristics of the j-th open loop. The characteristics of the remaining open loops are significantly less influenced, since for them the loop with Rj (s) is closed and – less sensitive to parameters change. 6. EXAMPLE Consider the DT twovariable tworate system shown in

G∗2 (s) = Gd22 − Gd12

R1 Gd ′ 1 + R1 Gd11 21

(12)

′′

In the third step the controller R1 (s) is designed for the replacement SISO plant of the first loop described by ′

G∗1 (s)

=

Gd11

−

Gd21

R2 Gd ′ 1 + R2 Gd22 12

(13)

′′

In the forth step the controller R2 (s) for the replacement ′ ′ ′′ SISO plant G∗2 (determined by (11) with R1 replaced by R1 ) is designed which however gives no essential improvement of the control. Finally, the CT controllers are described by the TF’s:

Fig. 3.

Step responses of the first output

0.8s + 1 2.6 , R2 (s) = (14) 0.1s + 1 s The bandwidth frequencies of both the loops with appropriate replacement SISO plants are ω1∗ = 3.57 and ω2∗ = R1 (s) = 0.4

Fig. 4.

Step responses of the second output

Fig. 5.

Time responses of y2 for w1 = 1(t)

Fig. 6.

Time responses of y1 for w2 = 1(t)

1.27. Since ω1 = 2π//0.18 = 34.9 and ω2 = 2π//0.24 = 26.1 then ω1 //ω1∗ = 9.78 and ω2 //ω2∗ == 20.55. The DT controllers are determined by the TF’s resulting from the Tustin approximation of (13) 1.873684(z − 0.7977528) , z − 0.05263158 0.312(z + 1) R2 (s) = z−1 R1 (z) =

(15)

In Fig. 3, 4 the step responses of both the loops and in Fig. 5, 6 the interaction responses of the CT system described by (12) and (14) are compared with those of DT system described by (10) and (15). Since, for the DT system the TF description does not exist the appropriate transients were determined by means of SIMULINK program, ver.1.3c. It is seen that both transients of the CT model with derivative and of the DT system are mutually very close.

R EFERENCES 7. FINAL CONCLUSIONS For MR SISO systems the DT TF description exists only in some rather seldom cases when the sampling period of input is integer multiple of that of the output. In other cases the MR SISO systems can not be described by means of TF. The MV MR systems can not be described by the TF method, which creates some essential difficulties in their description and design. The proposed method of design of MV MR systems, based on the approximate plant TF’s Gdij (s) or Geij (s) and the Tustin approximation of the designed CT controllers makes it possible to design the MV MR systems, practically without knowledge concerning the description of DT systems. ACKNOWLEDGEMENT The paper was partially supported by Science Research Committee grant No 8 T11A 031 10.

[1] Araki M. (1993). Recent Developments in Digital Control Theory. Preprints of the 12-th IFAC World Congress, Sydney, July 1993, pp. 251-260. [2] Kranc, G.M. (1957). Compensation of an Error Sampled System by a Multi-rate Controller. Trans. AIEE, 76, Pt.III, pp.149-159. [3] Kuzin, L.T. (1962). Calculation and Design of Discrete- Time Systems. Moskow (in Russian). [4] Program CC (1991). Tutorial and User’s Guide. System Technology Inc. California. [5] Gessing, R. (1995a). Comments on ”A Modification and the Tustin Approximation” with a Concluding Proposition. IEEE Trans. Autom. Contr., 40, no. 5, pp. 942- 944. [6] Gessing, R. (1995b). Discrete-Time Multivariable System Design by Means of Continuous-Time Methods. Preprints of the IFAC/IFIP/ IMACS Symposium: Large Scale Systems Theory and Applications, London, pp.135-139. [7] SIMULINK (1993). Dynamic System Simulation Software User’s Guide. The Math. Works Inc.