APPLICATION OF OPTIMAL-TUNING PID CONTROL TO INDUSTRIAL HYDRAULIC SYSTEMS. G. P. Liu*, S. Daley + and G. R. Duan

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1 Copyright 2002 IFAC 5th Triennial Worl Congress, Barcelona, Spain APPLICATION OF OPTIMAL-TUNING PID CONTROL TO INDUSTRIAL HYDRAULIC SYSTEMS G. P. Liu*, S. Daley + an G. R. Duan * University of Nottingham U.K., guoping.liu@nottingham.ac.uk * Institute of Automation, Chinese Acaemy of Sciences, P.R.China + ALSTOM Power Technology Centre, UK, steve.aley@power.alstom.com Queen s University of Belfast, g.r.uan@qub.ac.uk Abstract: This paper is concerne with real-time optimal-tuning PID control esign for inustrial hyraulic systems. Several optimal-tuning controller esign techniques are use for a hyraulic position control system. After analysis on the system, a nonlinear ynamical moel is erive. A nonlinear PID control scheme with inverse of ea zone is introuce to overcome the ea zone in this hyraulic system. An optimal PID controller is esigne to satisfy esire time-omain performance requirements. Using the aaptive moel, an optimal-tuning PID control scheme is propose to provie optimal PID parameters even in the case where the system ynamics are time variant. To implement the optimal-tuning control in practice safely, a control performance preiction strategy is integrate. These control techniques are implemente on SPACE an are also emonstrate by a number of experiments on a hyraulic position control test rig. Copyright 2002 IFAC Keywors: Optimal-tuning control, real-time control, optimal control, hyraulic systems.. INTRODUCTION Hyraulic systems are wiely use in inustry. Recently, most of controllers for inustrial hyraulic systems are still the proportional, integral an erivative (PID) controller. The PID popularity stems from its applicability an robust performance in a wie variety of operating scenarios. Moreover, there is a wie conceptual unerstaning of the effect of the three terms involve amongst non-specialist plant operators that makes manual tuning a relatively straightforwar task. Base on frequency response methos, many PID tuning rules have evelope in the last 50 years, for example, Ziegler-Nichols rule (Ziegler an Nichols, 942), symmetric optimum rule (Kessler, 958; Voa an Lanou, 995), some-overshoot rule (Seborg et al. 989), no-overshoot rule (Seborg et al. 989), refine Ziegler-Nicholes rule (Hang et al. 99), integral of square time weighte error rule (Zhuang an Atherton, 993), an integral of absolute error rule (Pessen, 994). These methos are straightforwar to apply since they provie simple tuning formulae to etermine the PID controller parameters. However, since only a small amount of information on the ynamic behaviour of the process is use, in many situations they o not provie goo tuning or prouce satisfactory close-loop responses. In an effort to improve the performance of PID tuning for processes with changing ynamic properties, several automatic tuning an aaptive strategies have been propose (Kraus an Mayron, 984; Astrom an Hagglun, 984; Rake an Issermann, 987; McCormack an Gofrey, 998). These controllers have self-initialisation an re-calibration features to cope with little a priori knowlege an significant changes in the process ynamics. However, the PID controller parameters are still compute using the classic tuning formulae an, as note above, these o not provie a goo control performance in all situations. In some plants there can be several control loops the tuning of which can be highly time consuming an in aition, ageing an non-linear

2 effects can lea to controller parameters that are far from the optimal. This can result in inefficient plant operation, unnecessary wear an unscheule shutowns. To eal with this problem, an optimaltuning PID control scheme has been propose in recent years (Liu an Daley, 999, 2000; Daley an Liu, 999; Liu et al. 2000). The paper is to emonstrate the feasibility of using optimal-tuning PID control techniques for reucing commissioning time an achieving consistent control performance for hyraulic systems with wie operating range. The following sections etail moelling of hyraulic systems, nonlinear PID control, optimal PID controller esign, optimal-tuning PID control scheme, PID control toolbox an experimental results. 2. MODELLING OF HYDRAULIC SYSTEMS To test ynamic performance of the optimal-tuning PID controller, a hyraulic control test rig was esigne an built by ALSTOM Power Technology Centre, UK, as shown in Figure. The rig consists of two hyraulic cyliners: main cyliner an loaing cyliner. The main cyliner is couple by a shaft to the loaing cyliner. The main cyliner is controlle by a 3 way proportional valve. The loaing cyliner is controlle by a Moog servo valve to simulate variations in loa. Hyraulic pressure for operation of the rig is generate by the flui power lab hyraulic power pack an supplie by the ring main system. Computer Base Optimal-Tuning PID Controller Displacement Transucer Force Sensor represents the turbine ajusting mechanism, which may be a istributor on a Francis or pump turbine, a istributor an/or runner blaes on a Bulb or Kaplan unit, or efectors an/or injectors on Pelton turbines. The optimal-tuning PID controller is implemente in the environment that consists of SAPCE, Matlab, Simulink an Real-Time Workshop. In orer to esign an optimal controller an assess control performance for the hyraulic test rig, the theoretical an practical moelling of the rig is neee. Taking nonlinearities into account, the hyraulic system can be moelle by two components: the ea zone nonlinearilty an the linearise moel. The valve spool occlues the orifice with some overlap so that for a range of spool positions there is no flui flow. Thus, a ea zone shoul be place between the valve ynamics an actuator/loa ynamics. For the sake of simplicity, this ea zone is equivalently move to the position between the output of the controller an the input current of the valve. So, the ea zone nonlinearity may be approximately escribe as i v = i I 0 i+ I if i > I if i I if i< I () where i v the input current of the valve from the ea zone, i is the input current from the controller an I the with of the ea zone. The linearise iscrete-time moel of the hyraulic position system is given by Ayˆ ( t) = q Bi ( t) + (2) v 0 Amplifier 3 Way Proportional Valve Hyraulic Power Supply Main Cyliner Loa Cyliner Moog Servo Valve Relief Valve Loa Controller Figure The hyraulic position control system The flow of hyraulic flui to the main cyliner is controlle by a 3 way proportional valve operate through an amplifier by a controller. The position of the main cyliner along its stroke is measure using a Linear Variable Displacement Transucer (LVDT) mounte above the cyliner - this signal is use for close-loop control of the main cyliner position via a computer base optimal-tuning PID controller. In orer to gather performance information about the optimaltuning controller uner test, the sensors monitor the following: the main cyliner isplacement, the coil current of the valve an the force of the main cyliner. The rig is fitte with an emergency stop button which if activate will shut own the power pack. The hyraulic test rig is mainly use to simulate a control system of hyraulic turbines. The main cyliner represents the turbine servomotor. The loaing cyliner where A an B are the polynomials. To cope with the nonlinearity, an aaptive moel base on an ARX structure is use. The aaptive moel is calculate using a set of time varying parameters obtaine from a least squares algorithm with a forgetting factor to track parameter variations cause by unmoelle nonlinearity. The moel form can be written as yˆ ( t) T = φt θ (3) where θ is the parameter vector an φ t the inputoutput vector. The least squares algorithm with a parameter-freezing factor is $ $ $ T θ = θ + βpφ ( y( t) θ φ ) (4) t t t t t T T λ ( Pt Pt φt φt Pt ( λ+ φt Pt φt ) ) ) if β = Pt = Pt if β =0 (5) where $ θ W is the estimate parameter vector, λ [ 0, ] is a forgetting factor, an β { 0,} is the parameterfreezing factor. The freezing factor β is set manually by the esigner or automatically by a certain conition to turn aaptation on an off. It can also be use to

3 compare fixe parameter control an aaptive parameter control. In practice, it has been foun that the ynamics of the hyraulic position system is ifferent in forwar an backwar irections. Thus, a non-linear ARX moel for the system may be expresse by Af yˆ( t) = q Ab yˆ( t) = q f b B f iv ( t) + B i ( t) + b v of ob forwar backwar (6) where A f, A b, B f an B b are the polynomials. Base on the physical moel erive above, the ynamics of each irection is moelle by a fifth-orer ARX moel, respectively. The accuracy of the moel was teste. The system ran at the operating point that the amplitue, perio an c-offset of the square-wave reference input were 20mm, 0s an 75mm. Using a non-linear PID controller (K P =, K I =0, K D =0 an I =0.05), which will be introuce in Section 5, the moelling performance of the hyraulic system is shown in Figure 2.. It is clear that the moel output is very close to the output of the plant. The inverse of ea zone is escribe by i i = i + I 0 I if i > 0 if i = 0 if i < 0 (8) If the parameter I is exactly known, the ea-zone inverse cancels the ea-zone effect, that is, i v = i (9) So, the ea zone in the system is remove by the inverse of ea zone in the controller. The PID controller esign for the nonlinear system is simplifie to the PID esign for a linear system. Since the signal i v is not measurable, it is very ifficult to know the with I of the ea zone exactly. In practice, the with is often estimate by utilising the measurable signals i an y. There are some practical estimation metho available (see, for example, Tao an Kokotovic, 996) Reference Moel output Plant output Figure 2 Responses of the plant an moel 3. NON-LINEAR PID CONTROL It is assume that the ieal transfer function of a PID controller is given by K I Ks ()= KP + + KDs (7) s where KP, K an I KD are the PID parameters. Since there exists ea-zone in the hyraulic position system, it is ifficult to achieve high-precision tracking using only linear controllers. The approach is to a a nonlinear compensator, which inclues the inverse of ea zone, to the linear controller. The inverse of ea zone is employe to cancel the ea zone. This structure is shown in Figure 3. r + - PID Controller i Nonlinear Conpensator I i iv q B I A y 4. OPTIMAL PID CONTROLLER DESIGN Since PID controller parameters are usually esigne using either one or two measurement points of the system frequency response, their control performance may not satisfy the esire time-response requirements. To overcome this isavantage, an optimal PID controller esign is propose in the time omain. The time-omain specifications for a control system esign involve certain requirements associate with the time response of the system. The requirements are often expresse in terms of the stanar quantities on the rise time, settling time, overshoot, peak time, an steaystate error of a step response. The time response of a stanar secon-orer system is wiely use to represent these requirements. Its transfer function is G ( s) = s 2 2 ωn + 2ςω s + ω n 2 n (0) where the parameter ω n is natural frequency, an ς the amping ratio. In orer to have a goo close-loop time response, the following performance function nees to be consiere uring the esign of a PID controller: 2 J( KP, KI, KD ) = ( ystep ( t) ystep ( t)) t () 0 where y step() t is the esire step response which may be prouce by the transfer function G (s), an y step( t ) the step response of the system with the PID controller. Thus, the optimal PID controller esign may be state as min JK (, K, K ) (2) K, K, K P I D P I D Figure 3 Close-loop control system There are a number of optimisation methos to solve the above problem (Liu et al., 200). Many

4 optimisation algorithms now exist in stanar libraries of optimisation software, for example, the optimisation toolbox for use with MATLAB (Grace, 994). 5. OPTIMAL-TUNING PID CONTROL When a system has ifferent operating points with wiely iffering ynamic properties, it is not always possible to control with a fixe parameter controller, even if this is a highly robust controller. For this case, an optimal-tuning PID control scheme is propose as shown in Figure 4. It mainly consists of four parts: moel parameter estimation, esire system specifications, optimal-tuning mechanism an nonlinear PID controller. The moel parameters are estimate using the least squares ientification metho. The esire system specifications are represente by the time response of the esire stanar secon-orer system. The optimal-tuning mechanism fins optimal parameters for the PID controller so that the esire system specifications are satisfie. The operating proceure of the optimal-tuning PID control is as follows. When the system operating point or ynamics change, the new moel parameters are reestimate by switching on the estimation algorithm. Then, using this upate moel parameters, the tuning mechanism searches for the optimal parameters for the PID controller to satisfy the esire system specifications. Finally, the PID controller is set to have the obtaine optimal parameters. In this way, the PID controller may cope with all operating points of the system an the close-loop system will have similar optimal control performance. But, compare with a fixe parameter control, the isavantage of this strategy is that it nees slightly more computation to search for the optimal parameters. The optimal-tuning PID control technique is applie to the hyraulic test rig, which is representative of many inustrial systems which utilise flui power. This is a particularly apposite application of the metho since hyraulic systems are often very conservatively tune ue to the fact that the cost of getting the tuning wrong can be highly estructive an costly. comprises an external control loop an an internal control loop. The external control loop inclues the hyraulic system an a nonlinear PID controller. The internal control loop inclues the aaptive moel upate by an on-line system ientification algorithm an a nonlinear PID controller. The parameters of the two PID controllers are ajuste by the optimal-tuning PID algorithm. Roughly speaking, implementation of this PID control strategy consists of two stages. The first stage is to run a nonlinear PID controller on the on-line ientifie moel to preict the performance of the PID controller before it is use on the rig. The secon stage is to apply the PID controller to the hyraulic system using the PID parameters which are verifie to be safe on the aaptive moel. In this way, unnecessary amage which coul result from the wrong PID parameters can be prevente though a slight increase in computational loa. 6. PID CONTROL DESIGN TOOLBOX The PID control esign toolbox has specially been evelope in MATLAB to enable the esigner to have flexibility in the off-line esign of PID controllers for the hyraulic test rig. The toolbox has two sub-menus, as shown in Figure 5. One is the function menu an the other is the set-up menu. The former mainly covers real-time ata acquisition, non-linear system ientification, moel valiity test, optimal PID controller esign, off-line control performance preiction, esire an real response comparison an PID parameter isplay. The latter mainly inclues the setting of the reference input, esire time response specifications, PID initial proportional, integral an erivative gains. + - Nonlinear PID u(t) Hyraulic System y(t) r(t) Optimal-Tuning Mechanism System Ientification System Specifications + - Nonlinear PID Aaptive Moel Figure 4 Optimal-tuning PID control scheme To implement the optimal-tuning PID control in practice safely, a control performance preiction scheme is integrate, as shown in Figure 4. It mainly Figure 5 Optimal-tuning PID toolbox menu. The function for the real-time ata acquisition collects real-time input-output ata from the real plant through AD/DA converters.

5 The function for the non-linear system ientification uses the input-output ata of the hyraulic system to estimate the parameters of a non-linear moel, which represents the hyraulic system quite well. The function for the moel valiity test provies test results of the non-linear moel, which is estimate by the non-linear system ientification, using a set of input-output ata which are not use for the parameter estimation. The function for the optimal controller esign searches the optimal PID parameters in forwar an backwar irections of the hyraulic system to satisfy the esire time response specifications, base on the ientifie non-linear moel. The function for the off-line control performance preiction preicts the close-loop performance of the hyraulic system when a PID controller is applie. It consiers various nonlinearities which there exists in the systems, such as control input saturation, non-linear moel, an sensor nonlinearities. The function for the esire an real responses compares the real response of the system using the optimal PID parameters with the esire response to show how close both two responses. The function for PID parameter isplay shows the latest PID parameters in the PID parameter isplay box. The function for the reference input setting gives the shape of the square reference input by typing in the frequency, amplitue an mean level of the signal. The function for the esire time response specifications etermines the esire time response of the close-loop system by typing in the natural frequency an/or amping ratio of the first- or seconorer system. The function for proportional, integral an erivative gains makes it possible to change the PID parameters manually. isplacement an velocity are given in the two top subfigures in Figure 6, where the grey lines enote the real responses of the system an the ark lines the responses of the moel. They are also use for isplaying valiity test results. The off-line close-loop simulation performance is shown in the left lower subfigure in Figure 6. The progress of the optimisation esign for PID parameters for the forwar irection is given in the right lower subfigure in Figure EXPERIMENTAL RESULTS To effectively assess the performance of the propose tuning metho, three experimental cases have been consiere. For all three cases, the amplitue, perio an c offset of the square-wave reference input are 7.5mm, 5 secons an 75mm, respectively. Here, only optimal-tuning PID controller for the forwar irection is illustrate. Case A: The parameters of the PID controller were K P =0.8, K I =0.0 an K D =0.05. This controller was applie to both the plant an the moel. Their responses are given in Figure 7. It is clear that the responses are away from the esire response Figure 7 Responses of the plant, moel an tracking moel (Case A) Case B: The optimal-tuning mechanism was switche on. The optimal PID parameters were foun to be K P =.923, K I =0.0 an K D =0.02. This optimal PID controller was applie to the moel. The close-loop response of the moel with the optimal PID controller is very close to the esire response, as shown in Figure Figure 6 Display winow of the toolbox. The toolbox also provies a interface winow to isplay the esign performance, as shown in Figure 6. The non-linear ientification results on both Figure 8 Responses of the plant, moel an tracking moel (Case B) Case C: The optimal PID controller (K P =.923, K I =0.0 an K D =0.02) This optimal PID was applie to

6 the plant. It can be seen from Figure 9 that the closeloop responses of both the plant an the moel with the optimal PID controller are very close to the esire response Figure 9 Responses of the plant, moel an tracking moel (Case C) 8. DISCUSSIONS AND CONCLUSIONS The optimal-tuning PID controller esign has been consiere for the hyraulic position control system. Several esign techniques have been aresse. Since the ynamics of the system for each movement irection is ifferent, a non-linear irection epenent ARX moel was use. As there exists ea-zone in the hyraulic position system, it is ifficult to achieve highprecision tracking using only linear controllers. A nonlinear PID controller, which inclues the inverse of ea zone, was introuce to cancel the ea zone. The optimal-tuning PID esign strategy mainly consists of four parts: non-linear moel estimation, efinition of esire specifications, optimal-tuning mechanism an PID controller. The non-linear moel for the process is estimate using the recursive least squares metho. The esire system specifications are represente by the time response of a esire secon-orer system. The optimal-tuning mechanism fins optimal parameters for the PID controller so that the esire specifications are satisfie. To implement the optimal-tuning PID control safely, a control performance preiction scheme has been integrate. It mainly comprises an external control loop an an internal control loop. A PID controller runs firstly in the internal loop to preict its performance before it is use in the real plant. In this way, unnecessary amage which coul result from the wrong PID parameters can be prevente. The PID control esign toolbox written in MATLAB enables the esigner to have flexibility in the off-line esign of PID controllers for hyraulic systems. These techniques above have been successfully applie to the hyraulic position control system. The experimental results have shown that they can significantly improve system performance. 9. REFERENCES Astrom, K.J. an T. Hagglun (984). Automatic tuning simple regulators with specifications on phase an amplitue margins. Automatica, vol. 20, no. 5, pp Daley, S. an G.P. Liu, Optimal PID tuning using irect search algorithm, IEE Proc. Computing an Control Engineering Journal, vol., no. 4, pp.5-56, 999. Hang, C.C., K.J. Astrom an W.K. Ho (99). Refinements of the Ziegler-Nichols tuning formula. IEE Proceeings-D, vol. 38, no. 2, pp. -8. Grace, A. (994). Optimisation Toolbox for Use with MATLAB. The MathWorks Inc. Kessler, C. (958). Das symmetrische optimum. Regelungstetechnik, vol. 6, no., pp Kraus, T.W. an T.J. Mayron (984). Self-tuning PID controllers base on a pattern recognition approach. Control Engineering, pp Liu, G.P. an S. Daley (999). Optimal-tuning PID controller esign in the frequency omain with application to a rotary hyraulic system, Control Engineering Practice, vol. 7, pp Liu, G.P. an S.Daley (2000). Optimal-tuning nonlinear PID control for hyraulic systems, Control Engineering Practice, vol. 8, no. 9, pp Liu, G.P., R. Dixon an S.Daley (2000). Multiobjective optimal-tuning proportional-integral controller esign for ALSTOM gasifier problem, Proceeings of ImechE Part I, vol. 24, pp Liu, G.P., J.B. Yang an J.F. Whiborne (200). Multiobjective Optimisation an Control, Research Stuies Press Lt. ISBN: McCormack, A.S. an K. Gofrey (998). Rule-base autotuning base on frequency omain ientification. IEEE Transactions on Control Systems Technology, vol. 6, no., pp Pessen, D.W. (994). A new look at PID-controller tuning. Trans. Amer. Soc. Mech. Eng., J. Dynamic Syst., Meas., Contr., vol. 6, pp Rake, F. an R. Isermann (987). A parameteraaptive PID controller with stepwise parameter optimisation. Automatica, vol. 23, pp Seborg, D.E., T.F. Egar an D.A. Mellichamp (989). Process Dynamics an Control. New York: Wiley. Tao, G. an P.V. Kokotovic, (996). Aaptive Control of Systems with Actuator an Sensor Nonlinearities, New York: Wiley. Voa, A. an I.D. Lanau (995). A metho for the auto-calibration of PID controllers. Automatica, vol. 3, no., pp. 4-53, 995. Zhuang, M. an D.P. Atherton (993). Automatic tuning of optimum PID controllers. IEE Proceeings-D, vol. 40, no. 3, pp Ziegler, J.G. an N.B. Nichols (942). Optimum settings for automatic controllers. Trans. ASME, vol. 64, pp