FREQUENCY DOMAIN FEEDFORWARD COMPENSATION F.J. Pérez Castelo and R. Ferreiro Garia Dept. Ingeniería Industrial. Universidad de La Coruña javierp@ud.es, Phone: 98 7.Fax: -98-7 ferreiro@ud.es, Phone: 98 67.Fax: 98 67 Abstrat: This work deals with some pratial aspets of PID ontrollers regarding multivariable non-linear feedforward ompensation. This ontribution onerns the task of ompensation for multivariable disturbanes on the ontrolled variable. The strategy onsists of establishing an adaptive funtion apable of ompensatinge by means of a feedforward strategy all disturbane variables under any dynami ondition. Keywords: QFT theory, frequeny domain design, frequeny domain identifiation, multivariable disturbanes, multivariable feedforward ompensation.. FEEDFORWARD CONTROL BACKGROUND. Conventional feedforward ontrol deals with the task of orreting the manipulated variable for disturbanes on the ontrolled proess. Most ommon industrial proesses are disturbed by more than a variable. For instane, in heat exhangers, ontrolled temperature is disturbed from flow and temperature variations of heated fluid. Furthermore, they are disturbed also by variations of the operating point beause dissipation heat may depend on the ambient and operating temperatures and by proess parameters. In onventional feedforward ontrol (Shinskey 98 an error must be deteted in a ontrolled variable before the feedbak ontroller an at to hange the manipulated variable. Therefore, disturbanes must upset the system before the feedbak ontroller an do anything. It seems very reasonable that if a disturbane entering a proess ould be deteted, a ontroller should begin to orret it before it upsets the proess. This is the basi idea of feedforward ontrol. If disturbane an be measured, this result will be used to send a signal through a feedforward ontrol algorithm that makes appropriate hanges in the manipulated variable so as to keep the ontrolled variable near its desired value. Classial industrial ontrollers offer the possibility of ompensation for only a disturbane variable entering the proess, if suh a disturbane an be measured. The real problem onerning industrial ontrol, in whih a good performane is needed, requires the ompensation task for more than a single disturbane variable inluded disturbane model parameters. In suh a ase, onventional ontrollers are not effiient and proposed adaptive ontroller takes advantage. Furthermore, disturbane variables are assoiated by non-linear funtions. Non-linear feedforward ompensator an be designed for non-linear systems. An alternative to implement feedforward ontrol systems in order to ompensate multivariable disturbanes, may be implemented by means of a frequeny analysis proedure based in disturbane model identifiation by FFT algorithm.
. SPECTRAL IDENTIFICATION In order to get information related to system frequeny response, it is neessary to implement a proedure by means of FFT algorithms. This information will be used later to design the orresponding PID ontroller. The appliation of FFT algorithms to the system dynamis by exiting it with sinusoidal signals of different frequenies permits the ahievement of the magnitude and phase angle at onrete frequenies, but also subarmoni omponents originated by external disturbanes are to be found. The identifiation task starts searhing for the ultimate frequeny (w relay with system phase angle response -π rad performing the Relay Feedbak Analysis (Åström.K.J et al., 989. This frequeny will be the key for the ontrol frequenies seletion used in the regulator design (Ferreiro et al., 995. The feedbak PID ontroller and the perturbation lead/lag feedforward ompensator (Figure, and will be designed in funtion of the performane speifiation indiated by the designer. The ontrol frequeny (w p for the feedbak ontroller and the time onstants (τa, τr for the lead/lag feedforward ompensator will be obtained as fration of the w relay and the ontrol system performane speifiations. With the FFT algorithm the following information will be found about the plant, working in open-loop onfiguration: The magnitude M = G( jw p and phase P = G( jw p at the ontrol frequeny (wp seleted for the design of the feedbak PID regulator. The entral frequeny of high frequeny disturbanes. The performane speifiation inludes datas like: phase margin (φ M, settling time, overshoot and bandwidth. With suh data proportional gain, integral and derivative parameters an be ahieved deterministially (Phillips et al., 995 and Åström.K.J et al., 98. The design expressions for three types of regulators are presented in Table. The ontribution angle θ orresponds with the regulator phase angle at the ontrol frequeny (w p. Proportional gain, integral and derivative parameters are ahieved deterministially (Table for PI and PD regulators. Design riteria an be ahieved by seleting frequenies at whih the ontribution angle ahieves aeptable regulators in terms of relative stability. Then the ontrol frequeny (w p an vary in order to verify the design restritions. From frequeny analysis by means of a digital signal proessor algorithm whih implements the FFT (Deimation in Frequeny (Oppenheim et al., 989, it is possible to introdue further omputer-based alulations to identify salient harateristis, disturbane harateristis and system frequeny response with some a priori knowledge. Table :Design Equations for a onrete Phase Margin θ = G ( jw = 8 φ G ( jw PD PI PID M p Design Equations os( θ Kp = G ( jw p i T d w i = tan( θ Kp = Tw i i Kp = os( θ G ( jw i p = tan( θ os( θ G ( jw i p tan( θ tan( θ K t T d = w Ti = KT t d K 8 t Restritions π < θ < > θ >. DESIGN ALGORITHM. π π π > θ > As we an see in Figure, and the ontroller will be omposed by a PID ontroller and a Lead/Lag Feedforward ompensator. PID Feedbak Controller Identifiation, Design and Adjust Blok for the PID Feedbak Controller Figure. General onfiguration Lead/Lag Feedforward Compensator Identifiation, Design and Adjust Blok for the Feedforward PLANT
The feedbak one will be working all the time and will be adapted to give response to hanges in operating onditions. Its input will be the error obtained as the differene between the set point and the ontrolled variable. PID Parameters Dynami Adjust Operating Conditions Figure. Identifiaión, Design and Adjust Blok for the PID Controller The lead/lag feedforward ompensator will give response to any disturbane. In the identifiation proess the manipulated variable measured under different operating onditions will be stored in order to estimate the manipulated variable during the ontrol system performane. Regulator Design Frequeny Identifiation at (w p FFT Control System Speifiations (Åström.K.J et al., 98 as important part of the fuzzy adaptive proedure. Input data is divided into two types: Data onerning the definition of the performane speifiation. Data onerning the dynami system behaviour. The design proedure has to follow the next sequene of ations:. Take the system to a steady state under a onrete operating onditions. Store the value of the manipulated variable (MV.. Apply the Relay Feedbak Analysis.. Selet the ontrol frequenie 5. Identify frequeny system response (magnitude and phase applying FFT at the ontrol frequenie 6. Verifiation of the design restritions. If not ome bak to the step. 7. Appliation of the design expressions, obtaining the ontrollers parameters. 8. Choie of lead/lag time onstants for the lead/lag feedforward ompensator. Feedforward Compensator Dynami Ajust and Manipulated Magnitud Estimation Operating Conditions Operating Conditions Change Detetor Compensator Feedforwardd Design Control System Speifiations Kp Ti Td Regulator Design G(jwi G(jwi wi Exitation sin(w i t Frequeny Identifiation (w i FFT Response sin(w i t Figure. Identifiaión, Design and Adjust Blok for the Feedforward Compensator Its input (output Figure will be the error between the manipulated variable estimation when a variation in the disturbane variable is deteted and the real system manipulated variable before the perturbation. Then the design algorithm will be related to the development of both elements. The design objetive is to obtain two risp sets look-up tables where we will map a omplete set of PID parameters and the lead/lag time onstants for any ombination of operating onditions. This set of parameters will be obtained trying to give optimum responses depending on the design riteria speified for every onrete ontroller. As explained in setion the fuzzy adaptive design proedure is based in the plant identifiation by frequeny tehniques obtaining w relay and subsequentally w p,τa and τr. It is important to mention that it is neessary to obtain these frequenies for any ombination of operating onditions if we want to map the system nonliniarities. The Figure shows a PID design blok diagram Control System Speifiation: Bandwidth, settling time, phase margin, regulator type et.. Figure. PID design blok diagram During the design proess we an find that for some ontrol frequenies to obtain aeptable regulators it is not possible. In these ases it is neessary to restart the design, searhing ontrol frequenies that will generate stable ontrollers. Applying the above method for different system operating onditions we will built a risp set look-up table with a omplete set of ontroller parameters and time onstants. The defuzzyfiation method will be performed by least square regression proedure (Johansson et al., 99, Brown et al., 99 obtaining polynomials expressions for every PID parameter, lead/lag time onstants and manipulated variable estimation as funtion of the operating onditions. These expressions permit to adapt in real time during the system performane the regulator parameters as
soon as the plant mathematial models hange due to its impliit nonliniarities.. CASE STUDY. A tank system is used to hek the performane of the algorithm due to its nonlinear harateristis. The Tank system model is given by the expression (.5.5.5.5 Respuesta Freuenia wp=.95rad/sg,set point 75%, Load 5% A(h h qi, qo a g d [ A ( h h ] = qi qo = qi a gh ( dt tank setion (m tank level (m input, output liquid flow (m/se. outlet pipe setion (m gravity (9.8 m/se. A tank with the following harateristis was used to verify experimentally the ontroller. Height = m Base ross setion diameter = m Top ross setion diameter = m qimax=.5 m //se. amax=.5 m The tank setion is a funtion of the h variable. Taking it into aount the equation ( is onverted in equation ( whih represents the mathematial model of our tank system dh d h d h π = qi a gh ( dt dt dt 5 The design proedure starts identifying the system by frequeny tehniques. First of all, it will be applied the relay Feedbak Analysis (Figure 5 and seondly working with the open loop onfiguration we introdue sinusoidal stimulus to our plant, proessing its responses by the FFT algorithm (Figure 6 Salida Regulador.5.5 5 5 5 Aná lisis Relay Feedbak Set point 5% Load 5%.5.5 Respuesta de Relay Feedbak para Set point 5% Load 5% 5 5 5.5 5 5 5 5 Figure 6. Frequeny Response Analysis The objetive is to find, for every ombination of operating onditions (set point and load the frequeny (w relay with system response phase 8 deg. It has been speified a set of performane speifiations (time response, bandwidth and phase margin initially. A slow time response trying to avoid great overshoots and a phase margin of 65 deg are some of these speifiations. The ontrol frequeny (w p has been seleted as fration of w relay (. w relay in this ase and the lead/lag unit pole and zero just the same (w f =.5w relay =/τa w pf =w relay =/τr. The results of the appliation of the algorithm is presented in Table. The Feedbak ontroller is a PI and Feedforward ompensator a lead/lag unit. Frequeny Identifiation SET OINT Parameters PID SET POINT Table. Design Algorithm Results 5% w p=..7-9.7º LOAD 5% 5% 75% w relay=.5 M.V.=.95 w relay=. M.V.=.878 w relay=.8 M.V.=.8 w pf= w pf= w pf= w f=.5 w f=.5 w f=.5 w p=.8. -9º w p=.6.6-6º w relay=.9 M.V.=.66 w relay=.6 M.V.=.7 w relay=.65 M.V.=.86 w pf=.9 w p=.6 w pf=.6 w p=. w pf=.65 w f=.7. w f=..6 w f=. -9º -78.º w relay=.97 M.V.=.759 w relay=.97 M.V.=.55 w relay=.97 M.V.=.8 5% w p=.88. -8º 75% w p=.96.55 -º w pf=.97 w p=.96 w f=.8.55-9.7º w pf=.97 w p=.96 w f=.8.57-9.7º LOAD 5% 5% 75% w pf=.97 w f=.8 5% K PP=.5 5T IP=.5 K PP=.8 T IP=. K PP=.7 T IP=9. 5% K PP=.7 T IP=. K PP=.8 T IP=7. K PP=.9 T IP=. 75% K PP=6. T IP=58.7 K PP=6.5 T IP=.9 K PP=5.97 T IP=.9 Figure 5. Relay Feedbak Analysis
With the information of Table it is possible, applying numerial analysis perform a deffuzyfiation proedure by means of polynomial expressions. These expressions have as independent variables the set point and the load and as dependent variables the ontroller parameters. Then the defuzzyfiation proess will be performed by least square regression proedure, obtaining the polynomial expressions. The expressions ( represents the funtions that relates the operating onditions to the manipulated variable estimation (MV, the PID parameters and the limit yle frequeny. Kpp = -.85 9.x -8.68x 6.7y -66.xy 5.7x y.8y -.5xy.x y / Tip = -.67.79x -.85x.9757y -7.87xy 7.78x y -.598y 9.5xy.x y w = -.8 9.69 x -9.696x 9.99y -7.8xy 5.6 x y 5. y relay.8 xy.9 x y MV = -.656.76x -.6x.778 y.xy.588x y -.y.6976xy -.8x y ( The variables x and y represent the operating onditions set_point and load. The universe of disourse for both variables is [,]. Now the identifiation and ontroller design proedures are finished. The ontrol system behaviour under different operating onditions and disturbanes will be tested. In this first experiment we just try to evaluate the feedbak ontroller element without using the perturbation feedforward ompensator. Trying to observe how we an obtain quik time responses a ontrol frequeny w p =.w relay is seleted and designed the orresponding set of ontrollers. The results are represented in Figure 7. The seond experiment was designed to analyze at the same time the ontroller performane under different operating onditions and disturbanes. These onditions are set point (5%, load (5% and a load perturbation..6...8.6.. Respuesta Temporal Perturbaion de Carga Feedforward Load Perturbation Regulator Output 5 6 Figure 8. Feedbak Controller Time Response under Disturbanes.6. Time Response with Feedforward Compensator. RESULTS AND CONCLUSIONS. Initially the system time response is tested modifying by steps the set point and the load at the same time during the experiment. Three ombinations of set point and load have been tested (.5,.5, (.5,.5 and (.75,.75. Respuesta Temporal sin Feedforward 5 6 Manipulated Variable 5 6 Figure7. System Time Response Level..8.6.. Lead/Lag Feedforward Ation 5 6 Time Figure 9. Feedforward Compensator Time Response In Figure 8 and 9 the effet of the load perturbation with and without lead/lag feedforward ompensator is represented. Its lear the orretion effet of the lead/lag unit avoiding system long transient periods out of the operating point under load disturbane onditions. The results show how valve modulation ativity is orret in the three onditions where energy demanded for rapid following is ahieved under good valve modulation, depending on the required traking
speed. It is possible to ahieve performane while keeping robustness of a ontrolled system under load hanges in aeptable limits as per time response of results, where feedforward ompensation of loads and set points with inherent modelling errors do not distort too muh the response and avoids the limitation of the integral ation. The adaptive frequeny method has been revealed as effetive in feedforward dynami ompensation where unertainties from environmental onditions are met, and some points are to be raised as follows: Low man mahine interation is needed for the adjustment task Aeptable time response to disturbanes Robustness in both ases, that is under parameter variations and relative stability REFERENCES Åström.K.J, Wittenmark B, Adaptive Control, Addison-Wesley Publishing. Chap, pp. - 7,7-95, 989 Åström.K.J and T. Hagglund, Automati Tunning of Simple Regulators with Speifiations on Phase and Amplitude Margins, IFAC 98 Automatia Vol. No. 5 pp 65-65, 98 Browm Martin and HarrisChris 99 Neurofuzzy Adaptive Modelling and Control Prentie Hall Ltd pp6-6 pp - Ferreiro Garía R. And Pérez Castelo F.J., Adaptive PID Controller Applied on Marine DP Control Using Frequeny Tehniques, Proeeedings of the rd IFAC-CAMS 95, 995 Johansson Rolf, System Modeling Identifiation Prentie-Hall. pp 78-9, 99 Oppenheim Alan V. and Ronald W. Sahafer Disrete-Time Signal Proessing, Prentie-Hall Chap 58-65 Chap, pp 696-7, 989 Phillips Charles L. and H. Troy Nagle, Jr.,. Digital Control Systems. Analysis and Design, Prentie- Hall, In. Englewood Cliffs,N.J., U.S.A. pp. 5-9 pp.-, 995 Shinskey F.G. Proess Control Systems Mgraw-Hill 98