This work deals with some practical aspects of fuzzy PID
controllers regarding multivariable non-linear cascade
compensation. This contribution concerns to the task of
compensation for multivariable disturbances on the
manipulated variable. The strategy consists in establishing a
fuzzy rule base for all disturbance variables which afect the
manipulated variable and compensate the control variable by
cascade action by means of the proposed fuzzy rule base
properly adjuste

Ferreiro García, Ramón

Pérez Castelo, Francisco Javier

Repositorio da Universidade da Coruña

MULTIVARIABLE FUZZY CASCADE COMPENSATIONR. Ferreiro García, F. J. Pérez Castelo, J. Vidal PazDep. Industrial Engineering. University of A CorunaE.S Marina Civil, Paseo de Ronda 51. 15011- A Coruna. Spain fax: 34 81 167101e-mail: ferreiro@udc.es†ECC2001 Conference e-mail: ecc2001@fe.up.pthttp://www.fe.up.pt/ecc2001/Keywords: Non-linear cascade, multivariable disturbances,multivariable cascade compensation, fuzzy cascadecompensation.AbstractThis work deals with some practical aspects of fuzzy PIDcontrollers regarding multivariable non-linear cascadecompensation. This contribution concerns to the task ofcompensation for multivariable disturbances on themanipulated variable. The strategy consists in establishing afuzzy rule base for all disturbance variables which afect themanipulated variable and compensate the control variable bycascade action by means of the proposed fuzzy rule baseproperly adjusted.1 IntroductionConventional cascade control deals with the task of correctthe manipulated variable disturbances affecting the servo-actuataron or some properties inherent to manipulatedvariable. Most common industrial processes are disturbed bymore than a variable. For instance, in heat exchangers,manipulated variable can be disturbed by input pressure andtemperature of the heating fluid.If the manipulated variable affects one variable and then thisvariable affects a second controlled variable, the structureleads to series cascade control [4,6]. The effect of a cascadecontroller is the one of a inner feedback loop. Feedback innerloop data, can be measured directly or calculated by means ofa proper math-model. Modelling erros can introduce a sourceof disturbances. When modelling errors are relevant, thenexperimental data vía parameter identification is stronglyrecomended. In any case, uncertainties inherent to thevariables and parameters related to the manipulated variablegenerate powerfull reasons to develop fuzzy cascadecontrollers.Figures 1 and 2 show the basic structure for cascade controlapplied on two industrial typical processes. Then the mastercontroller output supplies the set point, that is the desiredtorque. In Figure 1 cascade is applied to achieve the torque asfunction of rudder angle and ship speed. In Figure 2 cascadeis applied to achieve the heat flow of heating fluid as funcitonof mass flow, input and output temperatures of heating fluid.The master controller output supplies the set point, that istheheat flow of heating fluid.Figure 1: Cascade structure for a ship course control.Figure 2: Cascade structure for a heat exchanger.2 Application of fuzzy cascade compensation toheading control2.1 Ship modelUsing Newton’s equations to the describe ship motion, it willbe illustrated how the speed and influence the ship headingmotion. In order to simplify yet more the ship motionequations for analysis purposes only, the transfer functionmay be expressed by a Nomoto model [1] as:)( assb+=Ψδ (1)where coefficients a and b were achieved from experimentalsea trials [1] and are defined as:PID Steering-Gear servoTorqueShipCascadecompen.rudder angleShip speedSPPID ServoValveHeat flowHeatingprocessCascadecompen.in. tempout. tempmass flowTemp.SPtempDlAlubluaa.5.0;20 == (2)where A is the rudder area or rudder control surface (in squaremeters), D is the ship displacement (in cubic meters), u is theship velocity and l is the ship length.In order to define the ship transfer function for a steeringsystem whose output is the steering servo torque or turningmoment exerted on the ship hull, a math-model relating therudder angle and rudder torque is defined from information of[2,5] as function of ship speed by:δ2...12 ulAN = (3)According Newton’s equations of motion applied on sameships, a similar transfer function model with same coefficientsbut different meaning and values may be achieved as:)( assbN +′=Ψ(4)where b’ is achieved by introducing (3) into (1) to give (4).The result of mentioned manipulation yields the coefficient b’as:Dlb2.241=′ (5)which does not depend on longitudinal ship velocity.Nevertheless, the problem solving strategy is very muchdifferent at variable low speed that is when more manoeuvringcapacity is needed. It is due to the interaction between ruddereffect, and hull with propeller action. The interaction model isa non-linear function of ship speed, propeller action andrudder angle. So that, the results of mentioned interactionaffect the modelling errors inherent to the identification task.As consequence, the use of ship longitudinal speed into anadaptive algorithm at low variable speeds, is not effectivebecause of the uncertainties and an approach for comparativeanalysis will be based in the Newtonian non-linear dynamicmodel associated to propulsion and disturbances model.2.2 Cascade compensation structureSimplified achieved math-models of the ship and itsrespective steering machines, described by expressions (1-5)associated to a closed-loop controller will be applied ondynamic simulation.A general scheme to describe the simulation model is shownin Figures 3 and 4.In Figure 3 it is shown the conventional rudder angle feedbackservo commanded by a PID controller to act on the model ofthe ship previously described by expression (1). Cascadecompensation is based on the rudder angle only.Figure 3: Simulation model for rudder conventional cascade.In Figure 4 it is shown the rudder angle feedback servocommanded by a PID controller to act on the model of theship previously described by expression (4). Cascadecompensation is based on rudder angle and ship velocity.Figure 4: Simulation model for fuzzy cascade steering gear.Above models shown in Figures 3 and 4 belong to the cargocontainer ship described in [1].In order to compare dynamic results by simulation, a generalblock diagram implemented in simulink is shown in Figure 5.PID Steeringgear )()./(..5,0 2assDlAu+ΨS  ΨPID Steeringgear )()..24/(1 2assDl+TorqueΨΨSPFuzzycascade ship speed vrudder angle δb=b0(v/L)^2; b0=-1.63a=a0(v/L); a0=0.19  v=5; L=161.In1In2Out1steering grearIn1In2Out1ship1In1In2Out1shipheding1const0.99Vel-K-TorqueSum2SumStepIn1Out1ServoIn1 Out1PID0In1 Out1PIDMuxMuxanglevelocitytorqueFuzzyFigure 5: Simulink diagram for comparison purposes by simulationCascade compensation is implemented by taking into accountthe rudder angle and ship speed to give the rudder torque. Therule base in matrix form is shown in Table 1.-24NH-18NM-12NL-6NVL0Z6VL12L18M24H1 VL -2.3 -1.7 -1.16 -0.6 0 0.6 1.16 1.7 2.32 L -21 -16 -10 -5 0 5 10 16 215 M -58 -43.5 -29 -14.5 0 14.5 29 43.5 587 H -114 -85 -57 -28.5 0 28.5 57 85 114Table 1: Fuzzy rule base of cascade compensation.3 Simulation resultsSimulations on both ship models are performed to show thevalidity of the preliminary study for the proposed steeringgear servo, operating at low or zero cruise ship’s velocities.An automatic pilot with a PID control algorithm is applied.Controller parameters are adjusted so that a proper response isachieved at normal cruise speed. The layout of the simulationinterface permits the adjustment of ship dynamic parameterssuch as ship speed in m/s, heading set point in degrees andcontroller parameters.Figures 6, 7 and 8 shows the heading response at variablespeed, varying from too low speed to a normal speed.Excitation variable is a zigzag applied to the heading setpoint. The results show how a simple servo implementationcan improve manoeuvring capacity at low critic manoeuvringspeeds. It has been shown that an acceptable heading motionresponse is achieved under speed restricted ship condition,particularly at near zero manoeuvring ahead /speeds.Nevertheless, astern speed is not applicable because thepropeller interaction with rudder is only effective at aheadpropulsion forces [5].The main advantage in using this method is to keep thesteering capacity at low ship velocities, which make thecontribution particularly useful in manoeuvring tasks. The lowcost implementation brings the possibility of using with someadvantages an alternative steering equipment, which permitsthe scheduling under a simple strategy of both steering geartypes, that are classical and new proposed methods accordingactual measured situation (manoeuvring conditions, track-keeping, sea-keeping or dynamic positioning).It has been shown that at very low speed classic anglefeedback servo is not effective while torque feedback servoachieves proper performance. At higher speeds, theperformance of proposed steering gear servo is closely similarto classic solutions.Figure 6: Heading response (degrees) at 2 kn. speed forconventional and fuzzy cascade.Figure 7: Heading response (degrees) at 3 kn. speed forconventional and fuzzy cascade.Figure 8: Heading response (degrees) at 6 kn. speed forconventional and fuzzy cascade.References[1] Astrom K.J., Wittenmark B. “Adaptive Control”,Chap.9, pp.355-358, Addison-Wesley PublishingCompany (1989).201000                             500                          1000                    time s.fuzzy cascadeclassic cascade201000                             500                         1000                     time s.fuzzy cascadeclassic cascade201000                              500                        1000                    time s.fuzzy cascadeclassic cascade[2] F.H. “Todd Resistance and Propulsion”, Principles ofNaval Architecture, pp.373-397, 31st edition. Society ofNaval Architects and Marine Engineers. U.S.A.(1967).[3] King R.E. “Computational Intellingence in ControlEngineering”, Chap.8, pp.105-118, Marcel Decker, Inc.New York, U.S.A. (1999).[4] Luyben W.L. “Process modeling, simulation, and controlfor chemical engineers”, Chap.11, pp.383-391,McGraw-Hill, Inc U.S.A. (1989).[5] Mandel P. “Ship Manoeuvring and Control”, Principlesof Naval Architecture, pp.489-510, Society of NavalArchitects and Engineers. U.S.A. Society of NavalArchitecture and Marine Engineers. U.S.A. (1967).[6] Shinskey F.G.. “Process control systems. Applications,design and tuning”, Chap.7, pp.267-309, McGraw-Hill,Inc U.S.A. (1996).

Multivariable fuzzy cascade compensation

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Multivariable fuzzy cascade compensation

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