USE OF FILTERED SMITH PREDICTOR IN DMC

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1 Proceedins of the th Mediterranean Conference on Control and Automation - MED22 Lisbon, Portual, July 9-2, 22. USE OF FILTERED SMITH PREDICTOR IN DMC C. Ramos, M. Martínez, X. Blasco, J.M. Herrero Predictive Control and Heuristic Optimization Group Department of Systems Enineerin and Control Politechnic University of Valencia Camino de Vera 4, P.O. Box 222 E-467 Valencia, Spain fax: cramos@isa.upv.es, MED22 Conference med22@isr.ist.utl.pt Keywords: Lowpass filters, Predictive control, Prediction methods, Delay compensation, Robustness. Abstract The use of DMC in industry is very extended. Traditionally, its formulation has been based on the Dynamic Matrix model, obtained from the step response of the system. This work shows an alternative formulation that is based on a transfer function model, and divides the DMC into two parts: controller and predictor. This allows the easy substitution of one for another. Thus, is obtained by substitutin the predictor for the Smith predictor (SP) - this bein equivalent to DMC. The properties have been made more robust by addin filters. Introduction The aim of this work is to modify the classical DMC controller [4, 5], in order to improve its properties. The Smith predictor (SP) will be used [3] as the DMC predictor, and the possible advantaes of the resultin controller will be studied. Firstly, the DMC usin a transfer function prediction model is obtained []. This model is easier to deal with than the Dynamic Matrix model. Then the DMC will be divided into two parts, controller and predictor [7]. This structure does not affect the DMC behaviour and makes the exchane of each part easier. Thus, the Smith predictor will be used because it presents ood properties with lon time delay processes. It also has some drawbacks, such as the fact that it cannot be used with unstable plants. However, this is also true in the case of DMC. After comparin the DMC and, its equivalence will be shown, and the SP will modified by addin filters, to improve the properties. Finally, several examples and their results will be analysed, usin for simplicity s sake, a lon SISO time delay process. 2 DMC in transfer function The DMC formulation [2, 6] based on the Dynamic Matrix is a drawback [] as it complicates the application of the SP. Therefore, a transfer function process model would be expected and this will be obtained from the step response data, which is known by the operator. This model can be used with openloop stable systems. In Fiure the step response of a stable system is shown. The system presents a pure delay d, which will be considered in an explicit form. Considerin P + = P +2 =... = P, the followin model can be obtained, which contains many parameters, such as the Dynamic Matrix model: y (k) u (k) = z ( P P ) z P ()

2 P P P+ û= u(k) u(k+) u(k+n ) U= u(k ) u(k N+2) u(k N+) F =.7 Amplitude Γ= ( 2 ) ( N N 2 ) ( N N ) ( 3 ) ( N N 2 ) ( N N ).3 ( N ) ( N N 2 ) ( N N ) Time (sec.) Fiure : Step response of the explicit delayed system. Predicted outputs can be obtained from the model. Assumin N = d +, N 2 = d + N, bein the prediction horizon equal to the number of elements selected from the step response (P = N), and dividin the model by, results in: In this case, the predicted output ŷ(k + d k) is used instead of the output y(k). The sequence,..., d is zero if an implicit delay in the step response elements is assumed (fiure 2). Assumin N = d + N Amplitude d+ d+2 d+n d+n d+n+ ŷ (k + d + k) = ŷ (k + d k) ( N N ) u (k N + ) ŷ (k + d + 2 k) = ŷ (k + d + k) ( N N ) u (k N + 2) ŷ (k + d + N k) = ŷ (k + d + N k) ( N N ) u (k) By substitutin ŷ(k + d + k) for ŷ(k + d + 2 k), and so on, the DMC matrix form is obtained: (2) Y = G û + Γ U + F ŷ (k + d k) (3) Bein G NxN, Γ Nx(N ) and F Nx the followin matrices and Y Nx, û Nx and U (N )x the followin vectors: Y = ŷ(k+d+ k) ŷ(k+d+2 k) G= 2 ŷ(k+d+n k) N N.2. 2 d Time (sec.) Fiure 2: Step response of the implicit delayed system. as the prediction horizon, N = and N 2 = d + N, results in: Y = G û + Γ U + F y (k) (4) Bein G N xn, Γ N x(n ) and F N x the followin matrices and Y N x, û N x and U (N )x the followin vectors: Y = ŷ(k+ k) ŷ(k+2 k) ŷ(k+n k) G = 2 N N

3 û = Γ = u(k) u(k+) u(k+n ) U = u(k ) u(k N +2) u(k N +) F = ( 2 ) ( N N 2 ) ( N N ) ( 3 ) ( N N 2 ) ( N N ) ( N ) ( N N 2 ) ( N N ) The difference is that in this case the prediction depends on the present output y(k). 3 DMC for time delay processes. Separation of structures After obtainin the prediction model, the DMC in the transfer function is divided into two parts and expressed as shown in Fiure 3 [7]. The explicit time delay model is used in order to obtain the controller. The predicted output ŷ(k + d k) needs to be known by the controller. This value is provided by the predictor usin the implicit time delay model. Fiure 3: DMC divided into controller and predictor. 3. Controller formulation Expressin the followin index in a matrix form, bein N = Nu results in: J = N2 α i [ŷ (k + i k) w (k + i)] 2 + i=n + Nu (5) λ j [ u (k + j )] 2 j= J = (Y W ) T A (Y W ) + û T λ û (6) with A NxN and λ NxN diaonal matrices with α i and λ j factors, and: Y = [ŷ (k + N k),..., ŷ (k + N 2 k)] T û = [ u (k),..., u (k + N )] T W = [w (k + N ),..., w (k + N 2 )] T The DMC controller is obtained by substitutin the prediction and minimizin the index (6): û = ( G T AG + λ ) G T A [W Γ U F ŷ (k + d k)] (7) The Recedin Horizon stratey is assumed, so obtainin a linear controller. Denotin the first row of matrix "(G T AG + λ) G T A" by h xn results in: u (k) = hw h Γ U h F ŷ (k + d k) (8) where, with a transformed Z interpretation: ( N hw = h i z ) w(k+n i )=H(z) z (N ) w(z) i= ( N h Γ U= h i γ i z ) u(k)=r(z i ) z u(z) i= h F ŷ(k+d k)= ( N h i ) ŷ(k+d k) i= Bein γ i the i-th column of matrix Γ and the value hγ i the product of h by γ i. The future references in the prediction horizon w(k + N ),..., w(k + N 2 ) must be known in advance by the controller. If this is not possible, the reference will be assumed to be constant in the prediction horizon and therefore hw = Kw(z), bein w(z) the reference at k = N. u (z) = Kw (z) R (z ) z u (z) Kŷ (k + d k) u (z) = K w (z) K ŷ (k + d k) ( + R (z ) z ) 3.2 Predictor formulation (9) () Once the optimum controller has been shown, the predictor will be obtained. The predicted output ŷ(k + d k) usin the implicit time delay model is needed. Bein,..., d worth zero results in: ŷ (k + d k) = [ ( d+ ) ( d+2 2 ) ( (N+d) (N+d) )] [ u (k ) u (k 2) u (k (N + d) + )] T + y (k) () ŷ (k + d k) = ( (N+d) i= l i z i ) u (k) + y (k) (2) ŷ (k + d k) = V ( z ) z u (z) + y (k) (3)

4 Therefore, the final block diaram is shown in Fiure 4, bein equivalent to the classical DMC formulation. The process is expressed by G p (z)z dp. Fiure 5: block diaram. 4 Fiure 4: DMC block diaram. In this section, the DMC predictor will be substituted by the Smith predictor [8, 9, 7, 3], whose prediction model is: y (k) u (k) = ( h z h N z N) z d = G (z) z d (4) bein h i the factors of the system impulse response. Therefore, the open-loop prediction for instant k + d and k is: ŷ (k + d k) = G (z) u (k) (5) ŷ (k) = G (z) z d u (k) (6) The offset is the difference between the real output and the predicted output, both at the present instant. The final prediction will be adjusted by addin this offset, resultin in: offset = y (k) ŷ (k) (7) ŷ (k + d k) = ŷ (k + d k) + offset (8) same in both controllers in order to demonstrate that DMC and are equivalent. That is: ŷ (k + d k) = V ( z ) z u (z) + y (k) ŷ (k + d k) = G (z) ( z d) u (k) + y (k) (2) Expandin both polynomials results in equivalence, so that both explicit and implicit models are equivalent to SP model if the interators are left out. Besides, the prediction in both predictors is adjusted by addin the offset. In the implicit delay model, the,..., d sequence is worth zero, and the d+,..., d+n sequence is equivalent to the,..., N sequence from the explicit delay model. So V (z ) can be expressed as: V ( z ) = { d (2) ( d+ ) ( N N d ) ( N N d ) ( N N )} And both polynomials V (z ) z and G (z) ( z d) are equivalent, resultin in: z + ( 2 ) z 2 + ( 3 2 ) z ( d d ) z d + [( d+ ) d ] z (d+) + + [( d+2 2 ) ( d+ )] z (d+2) + + [( N N d ) ( N 2 N d 2 )] z (N ) + ŷ (k [ + d k) = G (z) u (k) + ] + y (k) G (z) z d u (k) (9) + [( N N d ) ( N N d )] z N + + [( N N d+ ) ( N N d )] z (N+) + The block diaram is shown in Fiure 5. As the controller is the same in DMC and, it is only necessary to check that ŷ(k + d k) is the + [( N N ) ( N N 2 )] z (N+d ) ( N N ) z (N+d) (22)

5 5 Improved. Use of filters As the DMC and are equivalent, the aim of this section is to improve their properties. Robustness and ood behaviour in presence of disturbances is souht. Two alternatives are presented to chane the. The first consists of untunin the optimum controller. This is complicated and affects the nominal response in presence of setpoint chanes. The second alternative consists of modifyin the optimum predictor. Because the controller has not been untuned, this improves robustness without affectin the nominal response. Reardin the second alternative, the addition of filter R(z) is proposed as shown in Fiure 6. Fiure 7: Consideration of uncertainties on the block diaram. The riht-hand part of the manitude is expected to be as lare as possible. Low-pass filter R improves robustness, despite slowin the disturbance rejection response. 6 Examples 6. DMC desin Fiure 6: block diaram. The next transfer function is obtained from the block diaram: y (z) w (z) = CG p z dp + CG + CR ( G p z dp Gz d) (23) The R(z) filter enables filterin the discrepancies between the model and the process. So, if G(z)z d = G p (z)z dp the filter does not take effect. A robustness index will be constructed to analyse results, based on the application of the Small Gain Theorem. So an unstructured additive model G p = G + G is considered for takin into account the process uncertainties []. This has been selected but it is not the only possible model. From this point of view, a robustness index for each frequency is calculated. This considers the maximum error manitude from the additive model in order to assure closed-loop stability. The followin index is obtained from Fiure 7: + CG + CGz d (R ) w a w > (24) CR A lon SISO time delay process is selected for comparin the DMC and filtered, as a pilot plant is located in the Department of Systems Enineerin and Control. The process presents a temperature loop, whose transfer function model is: G (s) = s + e s (25) The output variable is the water temperature after the heat exchane ( o C) and the input variable is the fan voltae (%). The sample period (T s ) equals.3sec. and the followin desin parameters are assumed: N = 3, N 2 = 75, Nu = 6, α =, λ =. The process operatin point is u = 3.8, y =, and the input constraints are u max =, u min =. To avoid non-linearities, constraints have not been considered. A unitary reference step at time t = sec. is assumed in all simulations. 6.2 Example. Uncertainty absence Fiure 8 shows the reference step response to uncertainties and disturbances absence. Both controllers present nominal behaviour, as filter R does not untune the controller, and the filter does not take effect because the feedback sinal y(k) ŷ(k) is zero.

6 35 35 & Output (ºC).5 & Output (ºC) Fiure 8: Nominal response. 6.3 Example 2. Gain uncertainty In this case, the process ain is supposed to be.5 times reater than the model ain, so the feedback sinal y(k) ŷ(k) will be different from zero. On applyin the Discrete Fourier transform, the sinal main frequency becomes f p =.28Hz. By selectin the cut-off frequency f c = f p /3 a first-order lowpass filter is obtained: R =.87 (z + ) z.9826 (26) Fiure 9 shows the system response usin the filter. The system output and input are smoother and the response slower. If a reater uncertainty is assumed, the ood behaviour of the filter is verified. Assumin that the process ain is 2.5 times reater than the model ain, the unfiltered becomes unstable, but the filter stabilizes the system as shown in Fiure. 6.4 Example 3. Time delay uncertainty Assumin the process time delay is 2sec., the model time delay is sec. and in the absence of ain uncertainty, the response of Fiure is obtained. The use of the filter stabilizes the system, but instability appears if it is left out. To observe the effect of the filter on robustness, Fiure 2 shows the riht-hand part of the manitude of equation (24) and the additive uncertainty. The use of the filter increases the manitude for hih frequencies, except Fiure 9: reater). Gain uncertainty response (.5 times for a small band. Hih manitude for all frequencies is souht. This can be taken into account for the filter desin. 6.5 Example 4. Process noise In this case, the n(k) noise will be assumed, which is caused by white noise ξ(k) of amplitude ±.5 passin throuh an interator and the model denominator as shown in the CARIMA model: n (k) = ξ (k) (27) A In the presence of this disturbance, and in the absence of uncertainties, the filter worsens the behaviour of the system, as shown in Fiure 3. Similar conclusions are obtained if the noise of amplitude ±.5 expressed by the followin ARIMA model is assumed, as shown in Fiure 4. 7 Conclusions n (k) = ξ (k) (28) The controller is presented as the merer of SP and DMC. A DMC based on a transfer function prediction model for time delay processes has been

7 & Output (ºC) Manitude (db) 5 5 Bode Diaram Additive uncertainty Frequency (rad/sec) Fiure : reater). Gain uncertainty response (2.5 times Fiure 2: Checkin robust stability with additive uncertainty. & Output (ºC) Fiure : Time delay uncertainty response. & Output (ºC) Fiure 3: System response with CARIMA model noise.

8 & Output (ºC) Fiure 4: System response with ARIMA model noise. obtained. This provides an alternative to the use of the Dynamic Matrix model. Another noteworthy contribution is the division of the DMC into two parts, controller and predictor. This is the equivalent to the classical DMC and eases the exchane of each part, as shown in the with the Smith predictor. The equivalence between DMC and, as they use equivalent prediction models is presented as a fundamental conclusion. Finally, the filtered improves the system robustness, despite slowin the disturbance rejection response. [3] F.X. Blasco, M.A. Martínez, J.S. Senent, and J. Sanchis. Automatic Systems (In Spanish). Universidad Politécnica de Valencia, 2. [4] E.F. Camacho and C. Bordóns. Model Predictive Control in the Process Industry. Spriner, 995. [5] C.R. Cutler and D.L. Ramaker. Dynamic matrix control - a computer control alorithm. In Proceedins of the joint automatic control conference (JACC), San Francisco, CA, 98. [6] B.L. Harkins. The dmc controller. ISA, (9):853, 99. [7] J.E. Normey. Predicción para Control. PhD thesis, Universidad de Sevilla, 999. [8] J.E. Normey and E.F. Camacho. A smith predictor based eneralized predictive controller. Technical Report GAR 996/2, University of SevilleNASA, 996. [9] J.E. Normey and E.F. Camacho. Robustness effects of a prefilter in smith predictor based eneralised predictive controller. In Control Theory and Applications. IEEE Proceedins (in press), 999. [] S. Skoestad and I. Postlethwaite. Multivariable feedback control. Analysis and Desin. John Wiley and Sons, 996. Acknowledements This work has been partially financed by European FEDER funds, project FD C2-2. s [] F.J. Alonso, J. Acedo, and R. González. Basic, Advanced and Multivariable Control. Sección Española de ISA, Repsol, Petronor, 2 (In Spanish). [2] A.O. Babatunde. Dynamic matrix control: A nonstochastic, industrial process control technique with parallels in applied statistics. Ind. En. Chem. Fundam., 25(4):72 78, 986.

Ch 14: Feedback Control systems

Ch 14: Feedback Control systems Ch 4: Feedback Control systems Part IV A is concerned with sinle loop control The followin topics are covered in chapter 4: The concept of feedback control Block diaram development Classical feedback controllers