Low-order feedback-feedforward controller for dead-time processes with measurable disturbances

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1 Preprint, 11th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems Low-order feedback-feedforward controller for dead-time processes with measurable disturbances Carlos Rodríguez Julio E. Normey-Rico José L. Guzmán Manuel Berenguel Sebastián Dormido Department of Computer Science and Automatic Control, National University of Distance Education, Madrid, Spain ( Department of Systems and Automation, Federal University of Santa Catarina, Florianópolis, SC Brazil ( Department of Informatics, University of Almería, CIESOL-CeiA3, Almería, Spain, ( {joseluis.guzman, Abstract: This paper presents simple tuning rules for low-order feedback and feedforward controllers based on an approximation of the filtered Smith predictor with closed-loop feedforward compensation. An analytical development is introduced to control stable first-order plus dead time processes affected by measurable disturbances that cannot be completely removed from the process output due to dead-time effects. Simulation results for the ph control of a tubular photobioreactor and the steam pressure control of an industrial boiler are given to show the effectiveness of the proposed method. Keywords: Process control, Linear control systems, Feedback control, Feedforward control, Delay compensation, Disturbance rejection, PID control 1. INTRODUCTION Despite the vast amount of control strategies in literature, low-order controllers are widely used in industry, where more of the 95% are of the Proportional-Integrative- Derivative (PID) type (Åström and Hägglund, 2006). This fact is explained by the advantageous cost-benefit ratio that they are able to provide (Vilanova and Visioli, 2012). However, proper tuning of these controllers may not be a trivial problem. In fact, there are hundreds of design strategies regarding PID feedback controllers (O Dwyer, 2009). It is worth mentioning the successful internal model control tuning rules for dead-time processes, e.g., (Rivera et al., 1986; Skogestad, 2003; Åström and Hägglund, 2004), based on low-order polynomial series approximation of the time delay. Their wide application is explained by three facts: They are analytically developed, simple and depend on an unique tuning parameter. Nevertheless, the design of PID controllers for delaydominant systems is more complicated and dead-time compensating control schemes, such as the filtered Smith predictor (FSP) (Normey-Rico and Camacho, 2007), are required. There are many other modifications of the Smith predictor (see e.g. (Bresch-Pietri et al., 2012) for time-variable uncertain delays) but the FSP has already successfully been used to obtain low-order practical controllers such as PID controllers (Normey-Rico and Guzmán, 2013). Moreover, these low-order controllers can be complemented with signal filters to improve servo and regulatory responses. The design of a PID controller within a two-degree-of-freedom (2DoF) controller was addressed in several works, e.g., in (Alfaro and Vilanova, 2012; Normey-Rico and Guzmán, 2013). A whole example of this issue is given in (Hägglund, 2012), where a unified control structure based on PID feedback controllers with low-order signal filtering in set-point, disturbance, and measurement noise is proposed. The idea behind these strategies is to initially tune the PID feedback controller for a tradeoff between robustness and regulatory response for unmeasurable disturbances, and then to include signal filters to improve servo and measurable disturbance responses. This topic moved some authors to develop simple tuning rules for signal filters. Most of the latest works address the design of feedforward controllers to improve the measurable disturbance compensation problem, e.g.,(vilanova et al., 2009; Guzmán and Hägglund, 2011; Rodríguez et al., 2013; Hast and Hägglund, 2014). However, to the authors knowledge, the whole control problem, including deadtime compensating strategies and signal filtering, has not been properly treated yet. In this work, we present an analytical development to tune a low-order controller, including a PID feedback controller and reference and measurable disturbance filters, for first-order processes with dead time. The remainder of the paper is organized as follows: In Section 2, the problem statement is presented. In Section 3, the design of the controller is addressed through polynomial series approximation of time delays and simple tuning rules are derived. The effectiveness of the proposed method is shown with two illustrative examples in Section 4. Finally, Section 5 conducts the conclusions of the work. Copyright 2016 IFAC 591

2 d r F C ff P d u Σ C fb Σ P u 1 Fig. 1. 2DoF plus feedforward controller block diagram. 2.1 Past results 2. PRELIMINARIES This paper is concerned with the FSP with closed-loop feedforward compensation (Rodríguez et al., 2016), which is a modification of the FSP (Normey-Rico and Camacho, 2007) for improved measurable disturbance compensation. This controller is implemented within a 2DoF control structure (see Fig. 1) with F(s) = N rt(s)d sp (s) D rt (s)n sp (s), (1) C fb (s) = G 1 N sp (s) D sp (s) N sp (s)e λus, (2) C ff (s) = G d (s)g 1 N dr(s)d sp (s)+n sp (s)d dr (s)e λ ds D dr (s)(d sp (s) N sp (s)e λus, ) (3) where polynomials N rt (s) and D rt (s) define the servo response, N sp (s) and D sp (s) deal with the regulatory problem, N dr (s) and D dr (s) are used to achieve an adequate measurable disturbance compensation, and G and G d (s) are the delay-free transfer functions of the process, such that P k (s) = G k (s)e λ ks, k {u,d}. However, due to the presence of time delays λ u and λ d in the final controller, its practical applications are somehow limited and it is usually implemented in the discrete-time domain, leading to a high-order controller (Normey-Rico and Camacho, 2007). 2.2 Problem statement Consider the 2DoF plus feedforward controller shown in Fig.1. The objective of this work is to obtain loworder approximations of the transfer functions (1)-(3). In particular, the desired transfer functions for F(s), C fb (s) and C ff (s) are, respectively, a first-order lead-lag filter, a PI controller, and a second-order lead-lag filter 1 : F(s) = β fs+1 τ f s+1 C fb (s) = κ fb τ i s+1 τ i s C ff (s) = κ ff (β ff [1]s+1)(β ff [2]s+1) (τ ff [1]s+1)(τ ff [2]s+1) Σ y (4) (5) (6) 1 This choice is considered based on previous works, e.g., (Vilanova et al., 2009; Hägglund, 2012). However, it is necessary to include the following assumption to fulfil the design objective: Assumption 1. Transfer functions P and P d (s) are given (or approximated) by first-order plus dead time models such that its delay-free model is G k (s) = κ k τ k s+1, k {u,d}. In what follows, the design of the controller is addressed. 3.1 Controller design 3. MAIN RESULTS To reduce the complexity of the controller transfer functions (2)-(3), let us introduce a first-order Taylor series approximation of the time delay, exp( λs) 1 λs. Thus, it is obtained that C fb (s) =G 1 N sp (s) D sp (s) N sp (s)( λ u s+1), (7) C ff (s) =G d (s)g 1 N dr (s)+n sp (s)( λ d s+1) D dr (s)(d sp (s) N sp (s)( λ u s+1)). (8) First, we study the robustness of the closed-loop system through the sensitivity transfer function ε(s), which is given by 1 ε(s) = 1+C fb (s)p. (9) Introducing (7) into (9) and taking into account the polynomial approximation of the time delay, it is obtained that ε(s) = D sp(s) N sp (s)( λ u s+1). (10) D sp (s) Thus, polynomials D sp (s) and N sp (s) can be used to properly compensate for unmeasurable disturbances and to establish a certain level of robustness(normey-rico and Camacho, 2007), e.g., a desired M s value. To obtain a PI feedback controller we define N sp (s) = 1 (11) D sp (s) = τ sp s+1 (12) such that the controller is the same as the obtained with the SIMC tuning rule (Skogestad, 2003): τ u s+1 C fb (s) = (13) κ u (λ u +τ sp )s Remark 2. NotethatF(s),C ff (s)donotaffecttheclosedloop stability as long as they are stable, thus the regulatory response against unmeasurable disturbances of the proposed controller is the same as the obtained with a PI controller designed with the SIMC tuning rule. Remark 3. As commented in (Skogestad, 2003), cancelling large time constants results in a long settling time for load disturbances. To overcome this drawback, modifications of the integral time can be used (Skogestad, 2003; Grimholt and Skogestad, 2012). Next, the servo response y(s)/r(s) is improved using the reference filter F(s). We can see that 592

3 y(s) r(s) = F(s) C fb (s)p 1+C fb (s)p = F(s) N sp(s) D sp (s) e λus. (14) Introducing (1), (11), and (12) into (14), it can be written y(s) r(s) = N rt(s) D rt (s) e λus. (15) To obtain a first-order lead-lag filter, we consider that N rt (s) = 1, (16) D rt (s) = τ rt s+1. (17) Thus, polynomial D rt (s) can be used to shape the servo response. Moreover, note that since the reference filter acts in open-loop, it does not influence the closed-loop robustness specification. Remark 4. Since a first-order Taylor approximation of the time delay is considered, the actual servo response may differ from the theoretical one. In fact, some overshoot usually appears (Skogestad, 2003). Finally, the measurable disturbance compensation response y(s)/d(s) is improved using the feedforward transfer function C ff (s). It can be seen that y(s) d(s) = P d(s) C ff (s)p 1+C fb P u = P d(s) ( D dr (s) N dr (s)e ) λ bs e λds, (18) D dr (s) where λ b = λ u λ d. To obtain the desired second-order lead-lag filter in the feedforward transfer function C ff (s), we define N dr (s) = β dr s+1, (19) D dr (s) = τ dr s+1. (20) Then, to accelerate the disturbance compensation, particularly with slow dynamics, β dr can be chosen to remove the undesired pole of P d (s) which appears in (18), which leads to (Rodríguez et al., 2016) β dr = τ d τ d e λ b τ d +τ dr e λ b τ d. (21) τ u κ fb = κ u (λ u +τ sp ) 2 τ i = τ sp (4) Shape the servo response using parameter τ rt for a tradeoff between speed and overshoot. (5) Tune F(s) as a first-order lead-lag filter (4) which parameters are given by β f = τ sp (22) τ f = τ rt (23) (6) Define the speed of the measurable disturbance compensation using parameter τ dr (suggested tuning λ u /4 τ dr λ u /2). (7) Set β dr = τ d τ d exp( λ b /τ d )+τ dr exp( λ b /τ d ). (8) Tune the feedforward controller C ff (s) as a secondorder lead-lag filter(6) with the following parameters: κ ff = κ d(β dr +λ d +τ sp τ dr ) κ u (λ u +τ sp ) β ff [1] = τ u β ff [2] = β drτ sp +τ dr λ d β dr +λ d +τ sp τ d τ ff [1] = τ d τ ff [2] = τ dr 4. ILLUSTRATIVE EXAMPLES 4.1 ph control of a tubular photobioreactor Consider the tubular photobioreactor model for microalgae growth described in (Berenguel et al., 2004). For control purposes, the ph of the culture must be kept around some operating point through the injection of CO 2 that allows the microalgae to perform the photosynthesis in the presence of solar radiation (see Fig. 2). The plant external description in the Laplace domain is given by Thus, the proposed feedforward controller is a secondorder lead-lag filter with tunable speed depending upon the sole parameter τ dr > 0. Note that if τ dr 0 then a faster disturbance rejection and a larger control signal peak are obtained. 3.2 Tuning guideline Therefore, the steps to design the proposed controllers are the following: (1) Obtain process transfer functions P and P d (s) following any of the well-known identification methods, e.g., (Skogestad, 2003; Åström and Hägglund, 2006). (2) Define parameter τ sp to meet a desired tradeoff between regulatory response speed and robustness (suggested tuning τ sp = λ u (Skogestad, 2003)). (3) Tune the feedback controller as a PI controller (5) with parameters given by Fig. 2. Tubular photobioreactor for microalgae growth. 2 See (Skogestad, 2003; Grimholt and Skogestad, 2012) for lagdominant processes. 593

4 8.05 Reference No feedforward Basic tuning Proposed (τ dr = λ u /2) ph 8 CO2(t) [%] Is at(t) [W/m 2 ] time [h] No feedforward Basic tuning Proposed (τ dr = λ u /2) time [h] S atu rated rad iation time [h] Fig. 3. Simulation results for ph control of a tubular photobioreactor. ph(s) = κ τs+1 + κ r τ r s+1 I sat(s) ω 2 n s 2 +2ζω n s+ωn 2 e λs CO 2 (s) (24) where ph(s) is the ph of the culture, CO 2 (s) is the percentage of CO 2 valve aperture and I sat (s) is the solar radiation. This linear model has been properly validated by the authors around the desired operating point, obtaining the following values of the parameters (Pawlowski et al., 2014): κ = 0.08 ph % 1, τ = 28 min, ω n = rad s 1, ζ = 0.042, λ = 7 min, κ r = ph m 2 W 1 and τ r = 182 min. For control purposes, we disregard the non-dominant oscillatory behaviour caused by the recirculation, that is P = s+1 e 7s P d (s) = s+1 The capability of system (24) to compensate for solar radiationduringawholesummerdayisdepictedinfig.3.in this example, three controllers were simulated: A simple PI feedback controller designed accordingly the SIMC tuning rule for τ sp = λ u (blue line); the same PI controller plus the basic feedforward compensator C ff (s) = G d (s)g 1 (green line); and the aforementioned P I controller with the proposed feedforward compensator for τ dr = λ u /2 (red line). For the proposed controller, the reduction in the error 1-norm and 2-norm is 24% compared to the controller with basic feedforward tuning and around 64% with respect to the controller without feedforward action (see Table 1). In general, the proposed controller allows for smaller integral errors at the cost of larger control signal changes despite the modeling errors. Table 1. Numerical results for photobioreactor example. Design r y 1 r y 2 u 1 No feedforward Basic Proposed (τ dr = λ u/2) Steam pressure control in an industrial boiler Consider the industrial boiler for steam generation introduced in (Pellegrinetti and Bentsman, 1996). For control purposes, the steam pressure y 1 must be kept at a desired operatingpointdespitethesteamdemandd 1 manipulating the input fuel rate u 1 (see Fig. 4). The external description of the model at some operating point is given by (Fernández et al., 2011) y 1 (s) = s+1 e 6.75s u 1 (s) s+1 d 1(s) (25) Fig. 5 shows the servo response and the measurable disturbance response of system (25) against step changes. In 594

5 u 1 Fig. 4. Industrial steam generation plant. this case, three controllers were simulated: A PI feedback controller designed with the SIMC tuning rule for τ sp = λ u plus a feedforward controller tuned accordingly C ff (s) = G d (s)g 1 (blue line); and two more controllers designed using the proposed methodology including the same PI controller plus a reference filter with τ r = 0.75λ u (to accelerate the servo response) and a disturbance feedforward filter with τ dr = λ u /2 and τ dr = λ u /4 (green and red lines), respectively. For both proposed controllers, significant reductions in the error 1-norm and 2-norm are obtained (see Table 2) compared to the controller with basic feedforward tuning. Table 2. Numerical results for industrial boiler example. Design r y 1 r y 2 u 1 Basic Proposed (τ dr = λ u/2) Proposed (τ dr = λ u/4) y 1 5. CONCLUSION In this paper, a novel procedure for PI feedback and leadlag filters design has been introduced. The main advantage of the proposed controller is its practical utility since it makes use of a robust and well-known PI design strategy (the SIMC tuning rule) that is complemented with both filters for improved reference and measurable disturbance responses. It has been shown that the proposed control provides smaller integral error when perfect measurable disturbance compensation is not possible due to dead times. Simple heuristics tuning rules are proposed to meet a desirable tradeoff between robustness, regulatory response and control effort. In contrast, finding an optimal value for the free parameters of the proposed controller depends on the control specifications of each problem and requires further study. Future work will also focus on the extension of the proposed controller to other type of processes, such as those described by integrating and unstable first-order models. d 1 ACKNOWLEDGEMENTS This work has been partially funded by the following projects: PHB financed by the Spanish Ministry of Education; CNPq-BRASIL; CAPES-DGU 220/2010; and Spanish Ministry of Economy and Competitiveness and EU-ERDF funds under contracts DPI C2-1-R and DPI C2-1-R. REFERENCES Alfaro, V.M. and Vilanova, R. (2012). Model-reference robust tuning of 2Dof PI controllers for first-and secondorder plus dead-time controlled processes. Journal of Process Control, 22(2), Åström, K.J. and Hägglund, T. (2004). Revisiting the Ziegler Nichols step response method for PID control. Journal of Process Control, 14(6), Åström, K.J. and Hägglund, T. (2006). Advanced PID Control. ISA-The Instrumentation, Systems, and Automation Society, Research Triangle Park, NC. Berenguel, M., Rodríguez, F., Acién, F.G., and García, J.L. (2004). Model predictive control of ph in tubular photobioreactors. Journal of Process Control, 14(4), Bresch-Pietri, D., Chauvin, J., and Petit, N.(2012). Adaptive control scheme for uncertain time-delay systems. Automatica, 48(8), Fernández, I., Rodríguez, C., Guzman, J.L., and Berenguel, M. (2011). Control predictivo por desacoplo con compensación de perturbaciones para el benchmark de control Revista Iberoamericana de Automática e Informática Industrial RIAI, 8(2), Grimholt, C. and Skogestad, S.(2012). Optimal PI-control and verification of the SIMC tuning rule. In IFAC Conference on Advances in PID Control. Brescia, Italy. Guzmán, J.L. and Hägglund, T. (2011). Simple tuning rules for feedforward compensators. Journal of Process Control, 21(1), Hägglund, T. (2012). Signal filtering in PID control. In IFAC Conference on Advances in PID Control. Brescia, Italy. Hast, M. and Hägglund, T. (2014). Low-order feedforward controllers: Optimal performance and practical considerations. Journal of Process Control, 24(9), Normey-Rico, J.E. and Camacho, E.F. (2007). Control of Dead-time Processes. Springer, London. Normey-Rico, J.E. and Guzmán, J.L. (2013). Unified PID tuning approach for stable, integrative, and unstable dead-time processes. Industrial & Engineering Chemistry Research, 52(47), O Dwyer, A. (2009). Handbook of PI and PID Controller Tuning Rules. Imperial College Press, London, 3rd edition. Pawlowski, A., Fernández, I., Guzmán, J.L., Berenguel, M., Acién, F.G., and Normey-Rico, J.E. (2014). Eventbased predictive control of ph in tubular photobioreactors. Computers & Chemical Engineering, 65, Pellegrinetti, G. and Bentsman, J. (1996). Nonlinear control oriented boiler modeling-a benchmark problem for controller design. IEEE Transactions on Control Systems Technology, 4(1), Rivera, D.E., Morari, M., and Skogestad, S. (1986). Internal model control: PID controller design. Industrial 595

6 63 Reference Basic tuning Proposed (τ dr = λ u /2) Proposed (τ dr = λ u /4) y1(t)[%] time [s] 100 Basic tuning Proposed (τ dr = λ u /2) Proposed (τ dr = λ u /4) u1(t)[%] time [s] 100 Steam demand d 1(t)[%] time [s] Fig. 5. Simulation results for steam pressure control in an industrial boiler. & Engineering Chemistry Process Design and Development, 25(1), Rodríguez, C., Guzmán, J.L., Berenguel, M., and Hägglund, T. (2013). Generalized feedforward tuning rules for non-realizable delay inversion. Journal of Process Control, 23(9), Rodríguez, C., Normey-Rico, J.E., Guzmán, J.L., and Berenguel, M. (2016). On the filtered Smith predictor with feedforward compensation. Journal of Process Control. In Press. Skogestad, S. (2003). Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control, 13(4), Vilanova, R., Arrieta, O., and Ponsa, P. (2009). IMC based feedforward controller framework for disturbance attenuation on uncertain systems. ISA Transactions, 48(4), Vilanova, R. and Visioli, A. (2012). PID Control in the Third Millennium: Lessons Learned and New Approaches. Springer, London. 596