Laboratory Essay with Online Back-calculation Anti-windup Scheme for a MTG System

Transcription

1 PID' Brescia (Italy), March 8-, WeB. Laboratory Essay with Online Back-calculation Anti-windu Scheme for a MTG System Antônio M. S. Neto, Thaise P. Damo, Antonio A. R. Coelho Federal University of Santa Catarina, Deartment of Automation and Systems, 889 Florianoolis, SC, Brazil ( antoniosalla@gmail.com, thaisedamo@gmail.com, aarc@das.ufsc.br) Abstract: This aer describes a lecture based on a laboratory exeriment of the under-graduate course of Control and Automation Engineering at the Federal University of Santa Catarina. The goal is to show to the students the differences, codes and erformance of tuning methods for the PID controller imlemented with online back-calculation anti-windu scheme alied to a MTG (motor+tacogenerator) lant. The assessed system is used in many activities of a laboratory disciline of Feedback Systems. The whole study is conducted with a real lant instead of relying on comutational numerical simulation. Analyzed tuning methods are: Ziegler-Nichols, Chien-Hrones-Reswick, Åström-Hägglund, AMIGO. Finally, the PID, tuned by the Internal Model Control (IMC), is also evaluated for different filter design arameters. The online exeriment gives a better understanding, for the students, of how different tuning methods modify the system stability and the magnitude of the control signal. Keywords: PID controllers, control system analysis, anti-windu, back-calculation, educational.. INTRODUCTION Industrial rocesses are subjected to restrictions. For examle, a controller oerates in a limited range of to 5 V or to ma, a valve cannot be oened more than % and less than % or a motor working as an actuator has a seed limit. These constraints reresent oeration limitations on the rocess. As a result, the real inut signal is temorarily different from the outut signal of the controller. When this haens, if the controller is initially designed to oerate in a linear region, the behavior of the closed-loo system dynamic deteriorates as comared to the exected linear erformance. This inadequate behavior (saturation of the direct-loo) is called windu. Figure illustrates a closed-loo control system with an ideal PID controller and a limitation of magnitude on the control effort. Fig. Closed-loo control system: constraint case. The classical form of the saturation on the magnitude of the control signal, shown in Figure, is described by the following equation: umax, ub umax u f ub, umin ub u umin, ub umin where the signals u b and u f are, resectively, the controller outut and the real inut of the rocess. To understand the windu henomenon in a control-loo, let us consider the second-order continuous system reresented through the following model: max s e G(s) (s )(5.7s ) (5.7s ) controlled by a PI+D controller with and without the resence of saturation in the control signal (u min = V, u max = 5 V). This simulation can be conducted at Simulink or ScicosLab comutational latforms. Modeling the system by a first-order differential equation, K =, τ = 5.7 s, θ = s, and alying the first rule of Ziegler-Nichols tuning, it is ossible to set the following arameters: K c = 6., T i =, T d =.5. Figures and show the ste resonse for a setoint tracking of magnitude.5. The closed-loo control system with saturation resents an excessive overshoot and a long settling time when comared with the case without saturation (observe that saturation effectively oens the loo). Therefore, this simle examle shows that the nonlinear dynamic of the actuator deteriorates the erformance of the closed-loo control system and, a control structure modification on the design of the PI+D controller is necessary to avoid, for examle, the actuator wear and an inaroriate loo behavior. () ()

2 PID' Brescia (Italy), March 8-, WeB. to be integrated and the integral term becomes too big, this means, continues to be greatly increased (windu). For the controller to come back to oerate in the linear region it is necessary to decrease the integral term. So, it is exected that the error measurement changes its signal and, for a long eriod of time, aly in the controller inut a signal of oosite error. There are several ways to avoid the integrator windu. Next, is reviewed the oular technique called backcalculation (with the basic idea of stoing the integrator being continuously increased when the saturation haens). Figure illustrates the block diagram of the anti-windu PID controller with back-calculation scheme. Fig. Closed-loo ste resonse with PID controller without saturation. Fig. Closed-loo ste resonse with PID controller with saturation. Assuming that control education must be concetual and exerimental and as an attemt to give the ractical loo characteristics of the PID tuning in the back-calculation antiwindu scheme, this aer shows a laboratory lecture of a disciline, called Feedback Systems, at the Federal University of Santa Catarina of an under-graduate course in Control and Automation Engineering. The goal is to give to the students not only ideas of the system stability asects using different PID tuning sets but also online numerical code, energy factor of the control signal and loo saturation using setoint changes. In addition, the PID controller is the most adoted controller in the industry because of its referable cost/benefit ratio when comared to other control techniques. The study of forms of configuration and arameter tuning are imortant asects for teaching in engineering courses (Ang et al., 5; Åström and Hägglund, 6).. ANTI-WINDUP TECHNIQUE Valves and motors as actuators resent oerating limits. When the control signal reaches the maximum or minimum limit of the actuator, control saturation haens. This henomenon makes the feedback-loo inaccurate, because the actuator remains on its maximum or minimum limit indeently of the rocess outut. The consequence is that the transitory resonse becomes oscillatory, which is extremely unsatisfactory in industrial rocesses. If a controller with integral action is utilized, the error continues Fig. Anti-windu PID controller toology. The basic idea of the back-calculation technique is: when the outut of the actuator saturates, the integral term is again calculated in a way that its value stays inside the linear limit of the actuator. It is advantageous making this correction not instantly, but dynamically with a time constant T aw. In Figure, the system resents an additional feedback-loo. The difference between the inut and the outut of the actuator constitutes an error e aw that is added to the inut of the integrator with a gain of /T aw. When the saturation does not exist, the error e aw is null and the controller is oerating in the linear region. In other words, the signal u f is not saturated. If there is saturation, e aw is different from zero. The time taken by the integrator inut to t to zero is determined by the gain /T aw, where T aw can be interreted as the time constant that determines how fast the inut of the integrator becomes zero. The selection of small values for T aw can be advantageous. However, a small value choice for T aw should be carefully made, esecially for systems with derivative action. What may haen is that the measurement noise can take the outut of the controller into saturation state, resulting in a fast actuation of the anti-windu loo and making the inut of the controller undesirably zero. In ractice, T aw should be bigger than T d and smaller than T i. An emiric selection rule suggested is T aw = TT i d or T aw = T i (Åström and Hägglund, 995; Visioli, 6). Figures 5 and 6 show the ste resonse for a setoint with magnitude.5, using the system model as in (), where the lant is controlled by a PI+D controller (K c = 6., T i =, T d =.5, T aw = ) assessed with and without the anti-windu

3 PID' Brescia (Italy), March 8-, WeB. comensation. The control system with anti-windu resents small oscillation and better settling time. Fig. 7 MTG exerimental rototye. The inut signal for the lant is a voltage for the DC motor and the outut is also a voltage corresonding to the angular seed. The voltage is ranging from to 5 V and the samling eriod is. s. Plant model arameters are calculated from an exerimental essay (ste resonse) with the control magnitude of V, as shown in Figure 8. Fig. 5 Closed-loo dynamic with PID controller without anti-windu. Fig. 8 MTG resonse for a ste of V. The estimated model that reresents the MTG dynamic is given by Fig. 6 Closed-loo dynamic with PID controller with antiwindu. s K e.s Vout(s).e G(s) V (s) ( s ) (.s ) in () The anti-windu technique rovides a shorter saturation time in the control signal. This is advantageous from the oerating viewoint, it means, imroves the closed-loo stability, reduces the wear of instrumentation and exts the actuator life-time. There are many references exlaining the theory of anti-windu techniques, which can rovide a more formal descrition and insight about the imlementation (Bohn and Atherton, 995; Peng et al. 996).. PID CONTROLLER EVALUATION ON A PRACTICAL ESSAY The exerimental lant, called MTG, to be utilized in the analysis of the PID controller with the objective of reducing the windu effect in the control loo, consists of a DC motor couled by a small belt to another DC motor which is resonsible for generating the tachometer voltage (seed measurement), as shown in Figure 7. Table illustrates the imlementation, in a machine cycle, the real-time PID controller with saturation and without antiwindu technique. The control law, before the saturation, u b is calculated by the following arts: roortional, integral and derivative bands are given by u, u i and u d. Since the aim is to teach exerimental PID control education and to have the PID structure in Figures and realizable, the integral art is adjusted with the forward rectangular aroach while the derivative with the first-order difference (Bobál et al., 5). Table Basic cycle of the PID signal with saturation e(k) = yr(k) y(k); u(k) = kc*e(k); ui(k) = ui(k ) + ((kc*ts)/ti)*e(k ); ud(k) = ((kc*td)/ts)*(e(k) e(k-)); ub(k) = u(k) + ui(k) + ud(k); if ub(k) <= umin; uf(k) = umin; elseif ub(k) >= umax; uf(k) = umax; else uf(k) = ub(k); Inut Voltage Disturbance Motor Generator First-Order Dynamic Constant Filter + Outut Voltage Table describes the classical tuning methods for imlementation of the PID controller and with T aw = TT i d (O Dwyer, ).

4 IFAC Conference on Advances in PID Control PID' Brescia (Italy), March 8-, WeB. behavior gives a large saturation area on the control signal with a short settling time. Table Classical tuning for the PID controller Tuning Method K c T i T d Ziegler-Nichols Chien-Hrones- Reswick Åström- Hägglund AMIGO. K.6 K.9 K..5 K Using the estimated model described in () and the tuning rules from Table, it is ossible to obtain the gains of Table, for the PID controller (Åström and Hägglund, 6). Table PID controller gains Tuning Method K c T i T d T aw Ziegler-Nichols..5. Chien-Hrones- Reswick Åström-Hägglund AMIGO Fig. System resonse to PID tuned by Chien-Hrones- Reswick method. The Chien-Hrones-Reswick tuning method does not resent overshoot. It shows the longest settling time, the second less aggressive control and the lowest saturation area System resonses for each tuning method are illustrated in Figures 9 to Fig. System resonse to PID tuned by Åström-Hägglund method Fig. 9 System resonse to PID tuned by Ziegler-Nichols method. The Ziegler-Nichols tuning method rovides the most aggressive control between the four methods. This kind of Fig. System resonse to PID tuned by AMIGO method.

5 IFAC Conference on Advances in PID Control PID' Brescia (Italy), March 8-, WeB. The Åström-Hägglund tuning method resents a very similar resonse comared to the Ziegler-Nichols method but with a much less aggressive control. It also has less overshoot. The settling time and saturation area is very close to the Ziegler- Nichols method. The AMIGO tuning method is the one that resents the second lowest saturation area between the four methods. The combination of arameters results in a good settling time according to the saturation area of the other methods. The settling time is still larger than the Ziegler-Nichols and Åström-Hägglund methods, but this is comensated by the smaller overshoot and the less aggressive control.. IMC BASED PID TUNING The IMC tuning method is becoming largely alied in the industry. So, essays based on the IMC-PID tuning method are resented as an alternative solution for the anti-windu PID strategy reviewed in this aer. This calibration method can rovide good results not only decreasing the saturation level and overshoot of the system resonse but also by keeing the system seed at reasonable levels more theoretical background of the IMC PID tuning can be found in Rivera et al. (986) and Zulkeflee et al. (). Next, the PID controller for the MTG lant is tuned by the IMC-PID settings. The goal of the control system is to achieve a fast and accurate setoint tracking. IMC-PID tuning arameters for first-order lus dead-time rocesses are described in Table. Table PID controller settings with IMC strategy PID Parameters IMC Tuning. 5 K c K (.5 ) T i T d. 5 Due to K c being inversely related to the filter constant λ (associated with the lant resonse seed and the energy factor of the control), different essays with different values of λ, to verify the system resonse, are imlemented. The arameter λ must be greater than.8θ because of the model uncertainty due to the Padé aroximation. The integral time and derivative time of the PID controller are constants and with values T i =.5, T d =.78, resectively, and T aw =.5. Parameter K c changes according to λ as dislayed in Table 5. Table 5 Parameter K c of the PID according to λ λ K c.8θ θ.995 Closed-loo system results of the exeriment, according to the variation of λ, are dislayed in Figures and Fig. System resonse to PID tuned by IMC-PID method for λ =.8θ Fig. System resonse to PID tuned by IMC-PID method for λ = 9.7θ. Comaring the IMC-PID tuning with the other four methods dislayed in Section, it can be observed that the saturation area is very small for greater values of K c and almost absent for smaller values. The system also does not resent overshoot for any value of λ. Comaring the results with the other method that resents no overshoot, Chien-Hrones- Reswick tuning, it is ossible to note that the settling time of the IMC-PID tuning is smaller. Utilizing λ = 9.7θ the system achieved a control without saturation, but has slower behavior, and therefore, being the only method of the ones analyzed that can remove the saturation level on the control signal. These exeriments can be imlemented with the Matlab numerical code resented in Aix A. 5. CONCLUSIONS Through different tuning methods the rocess control engineering students can understand and observe how the

6 PID' Brescia (Italy), March 8-, WeB. PID arameters affect the system dynamic under an on-line numerical code and anti-windu scheme. This ractical case study, combined with the fact that the exeriment is conducted over a real lant, not over a comuter simulated model, oens the mind of the students on how a real system works, considering that an exerimental system has measurement noise and does not have the exact behavior of the model obtained from the ste resonse identification. Costing less than some engineering textbooks, each university can build a MTG lant for control lab lessons and reinforcing classroom concets. Another imortant oint is that the numerical code and PID tuning methods resented in this aer do not aly only to the MTG lant, but for any stable system with first-order aroximated behavior. ACKNOWLEDGMENT This research was suorted by the DAS of the Federal University of Santa Catarina and CNPq. REFERENCES Ang, K.H.; Chong, G.; Li, Y. (5). PID Control System Analysis, Design, and Technology, IEEE Trans. on Control Systems Technology, Åström, K.; Hägglund, T. (995). PID Controllers: Theory, Design, and Tuning, ISA. Åström, K.J.; Hägglund, T. (6). Advanced PID Control, Instrument Society of America. Bohn C.; Atherton, D.P. (995). An Analysis Package Comaring PID Anti-Windu Strategies, IEEE Control Systems,. -. Bobál, V.; Böhm, J.; Fessl, J.; Machácek, J. (5). Digital Self-Tuning Controllers, Sringer. O Dwyer, A. (). A Summary of PI and PID Controller Tuning Rules for Processes with Time Delay. Part : PID Controller Tuning Rules, IFAC Digital Control: Past, Present and Future of PID Control,. -6. Peng, Y.; Vrancic, D.; Hanus, R. (996). Anti-Windu, Bumless, and Conditioned Transfer Techniques for PID Controllers, IEEE Control Systems, Rivera, D.E.; Morari, M.; Skogestad, S. (986). Internal Model Control: PID Controller Design, Industrial & Engineering Chemistry Process Design and Develoment, Visioli, A. (6). Practical PID Control, Sringer. Zulkeflee, S.A.; Shaari, N.; Aziz, N. (). Online Imlementation of IMC Based PID Controller in Batch Esterification Reactor, 5 th International Symosium on Design, Oeration and Control of Chemical Processes,.6-6. Aix A. MATLAB SIMULATION CODE Matlab code of the PID controller clear all; close all; clc; % Initial conditions nit = 5; ts =.;umin = ; umax =.9; ub(:) = ; uf(:) = ; erro(:) = ; y(:) = ; u(:) = ; ui(:) = ; ud(:) = ; eaw(:) = ; % Reference yr(:8) =.5; yr(8:6) =.5; yr(6:5) =.5; % Plant model arameters k =.; tau =.; teta =.; lambda = 9.7*teta; % the IMC-PID tuning % PID controller arameters method = 5; % Tuning method for the PID switch method case % Ziegler-Nichols kc = (.*tau)/(k*teta); ti = *teta; td =.5*teta; case % Chien-Hrones-Reswick kc = (.6*tau)/(k*teta); ti = tau; td =.5*teta; case % Åström-Hägglund kc = (.9*tau)/(k*teta); ti = *teta; td =.5*teta; case % AMIGO kc = (/k)*(. +.5*(tau/teta)); ti = ((.8*tau +.*teta)/(.*tau + teta))*teta; td = (.5*tau*teta)/(tau +.*teta); case 5 % IMC-PID kc = (tau+.5*teta)/(k*(lambda+.5*teta)); ti = tau+.5*teta; td = tau*teta/(*tau+teta); taw = sqrt(ti*td); % Start the data acquisition board inicializa_laca(5); % Closed-loo exeriment for k = :nit % Outut and error y(k) = recebe_dado(); erro(k) = yr(k) - y(k); % Control law with anti-windu u(k) = kc*erro(k); ui(k) = ui(k-) + ((kc*ts)/ti)*erro(k-) + (ts/taw)*eaw(k-); ud(k) = ((kc*td)/ts)*(erro(k) - erro(k-)); ub(k) = u(k) + ui(k) + ud(k); % Saturation if ub(k) <= umin uf(k) = umin; elseif ub(k) >= umax uf(k) = umax; else uf(k) = ub(k); eaw(k) = uf(k) - ub(k); % S control signal to the board envia_dado(,uf(k)); atraso_ms(*ts); % End the data acquisition board finaliza_laca; % Simulation results t = :nit; sublot(,,), lot(t,y(t),t,yr(t),'linewidth', ), ylabel(''), xlabel(''); sublot(,,), lot(t,ub(t),t,uf(t),'linewidth', ), ylabel(''), xlabel('');