A Hydraulic Steering Gear Simulator for Analysis and Control

Transcription

1 A Hyraulic teering Gear imulator for Analysis an Control Nikolaos I. Xiros (), Vasilios P. Tsourapas (), Kyriakos K. Mourtzouchos () () chool of Naval Architecture an Marine Engineering National Technical University of Athens Iroon Polytechniou 9, 570 Zografos, Athens GREECE () Department of Naval Architecture an Marine Engineering University of Michigan NA&ME Builing, 600 Draper Roa, Ann Arbor, Michigan UA Abstract: - Hyraulic systems are wiely use for marine applications. This paper unertakes the mathematical an visual moeling of such systems, aiming to improve precision an spee of response. Mathematical moeling inclues moeling of control valve an controller, leaing, thus, to a full hyraulic servomechanism moel. Then it is examine how the replacement of commonly use ON/OFF controller by an appropriately tune PI controller can improve the behavior an response of a typical hyraulic servomechanism. The general methoology is applie to the steering gear of a ship. Finally, the mathematical moel is integrate in a visual steering gear simulator. Key-Wors: - PI, ON/OFF, control, hyraulic, valve, steering gear, reliability Introuction tuy an improvement of shipboar mechanisms an systems is of essential importance ue to the continuously increasing automation an accuracy requirements. ystems executing a specific function by multiplying the actuating force or power are generally calle servomechanisms. This paper concerns hyraulic servomechanisms with position feeback; a typical case is the steering gear use onboar ships. The basic structure of a typical hyraulic servomechanism, is shown in Fig.. It inclues the hyraulic motor, which can be rectilinear or rotational, the control valve controls the hyraulic oil flow through the motor, an, the controller, which through motor position feeback, manipulates the control valve. In general, hyraulic servomechanisms with position feeback are classifie in one of the following categories []:. Constant volumetric flow rate systems, which employ constant volumetric flow rate pumps connecte in series to the rest of the system.. Constant pressure systems, for which, although the pump volumetric flow rate is variable, the pressure is constant. The major avantage of constant volumetric flow rate systems is their insensitivity against the ernally applie loa force or torque. A typical example is the system riving a ship s ruer; in this case, spee an positioning accuracy o not epen on ernal force or torque fluctuations. Automatic feeback control of hyraulic systems is achieve by aition of a control valve an a controller. In general, a position feeback controller, as the one examine in the present paper, manipulates the error signal for generating the riving signal of the control valve. The error signal is proportional to eviation of the payloa s position from the esire one. This eviation is transforme to an electrical signal (i.e. a voltage or a current) by use of a potentiometer as transucer. The commonly use type of feeback control for marine hyraulic systems is the ON/OFF controller. The ease of manufacturing an the minimal tuning effort requirements are the major reasons for its wie use []. However, accompanying isavantages are not negligible when highprecision an spee of response actuating systems are sought after for marine applications like e.g. the exhaust valves of the camshaftless low-spee, irectly-couple, marine engine. This type of marine engine has been introuce for increasing efficiency an reucing pollutant emissions. Its operation, however, relies on high-performance, computercontrolle hyraulics, like the ones examine in this

2 paper. Failure of the control hyraulics leas to unacceptable egraation of the engine as a whole. Commonly use ON/OFF controllers are capable of setting the control valve of the hyraulic system to one out of three preetermine positions:. Position : valve fully open right. Position-: valve at neutral position 3. Position-3: valve fully open left Hyraulic systems with ON/OFF control usually emonstrate steay state error an/or severe oscillation of either the control valve or the hyraulic motor or both. A erive rawback from the above is that any esire increase in spee of response may not be implemente otherwise but with an accompanying counter-effect in increase of steay state error. A irect consequence is reuce system reliability, ue to persisting control valve oscillation, which ecreases MTBF. It is argue that such problems can be, at least, partially avoie by replacing the commonly use ON/OFF controller with a PI one. As emonstrate later, an appropriately tune PI controller may cause steay state error to completely vanish, without oscillatory transient response. Notation ~ L P Ρ i a C i A ν ι b Hyraulic oil volumetric flow rate through the pump pecific per ra hyraulic volumetric flow through the pump, where _ ϑ V s = π n Vp = π t ϑ an V p : Volume/ra (specific volume) [3] Hyraulic oil volumetric flow through loa Total pressure on the circuit from control valve inlet to outlet Pressure rop across i-th valve port Cylinric area of valve opening Discharge coefficient, equal to π/(π+) for turbulent flow [4] Volumetric flow rate through the i-th control port of valve Piston s area Oil s velocity through the i-th port Circular cross-section of control valve Hyraulic oil ensity x* Position of control valve, x * [ U,U ] x Rate position of valve x [,] Σ F External resistance force Σ F H Hyraulic system moving force Μ Payloa mass Ι Moment of inertia. y Piston velocity For constant volumetric flow rate systems, pump volumetric flow rate is constant, whilst for constant pressure systems, pump pressure P s is constant. The equivalent circuit of the typical hyraulic system, introuce earlier, is shown in Fig. ; the through-flow an across-pressure-rop variables for each branch of the circuit are also inicate. 3 Mathematical moeling of the typical hyraulic system The mathematical moeling of the typical hyraulic system is performe by formulating an solving the governing equations. Base on the principle of mass conservation, which is equivalent to Kirchoff s current law for electric circuits, the following algebraic equations are obtaine + 3 = () + 4 = () 4-3 = L (3) - = L (4) Base on the principle of energy conservation, which is equivalent to Kirchoff s voltage law for electric circuits, the following algebraic equations are obtaine P + P = P (5) P3 + P4 = P (6) As inicate in literature [], the escriptive (constitutive) equations for the four ports of the control valve, when moele as lumpe circuit elements, are * = αν = π r U ± x C Pi ( ) i i P i = i, i =,,3,4 (πr U± x C ) ( ) *

3 P i = i, i =,,3,4 (πru x C ) ( ± ) (7) As can be seen the above equations, connecting flow an pressure variables of each branch of the equivalent brige circuit, are nonlinear. AP y(s) = x(s) F (s) ( ) ( ) ( Σ ) Ms + A R P s Ms + A R P s 0 0 with B B = an R 0 = πruc A ( ) (4) Fig. Typical Hyraulic ervomechanism From the efinition of volumetric flow rate it is obtaine that =. L A y (8) Finally, the secon law of Newton for the motor loa is expresse as follows.. M y = ΣF Σ FH (9) Using Taylor s expansion for the nonlinear escriptive relationships (7) for each port s variables the following linearise relationships are obtaine δpi = R 0 0 x+ R 0 0 δ i for i =,,3,4(0) Inex 0 implies equilibrium of the system, whilst prefix δ implies small eviation of the corresponing variable from its equilibrium value, i.e. i = 0 + δi () Pi = P0 + δ Pi () Thus, the linearise ynamics of the typical hyraulic servo system are epicte by the following equations in the Laplace transform omain. a) Constant volumetric flow rate system for rectilinear motion B y(s) = x(s) F (s) s ( ) ( ) ( Σ ) M s + B s M s + B s with B = (3) ( πruc ) b) Constant pressure system for rectilinear motion Fig. Diagram of a hyraulic system an control valve For rotational motion systems, the analogous equations are formulate as follows a) Constant volumetric flow rate system for rotational motion B y(s) = x(s) M (s) s ( ) ( ) ( Σ ) I s + B s I s + B s with B = (5) ( πruc ) b) Constant pressure system for rotational motion AP y(s) = x(s) ΣM (s) Is + A R P s Is + A R P s ( ) ( ) ( ) 0 0 with B B = an R 0 = A ( πruc ) (6) As can be seen from the ynamical equations above, the payloa position is a function of valve position x an ernal loa force or torque. Consequently apart from the parameters of each system, which appear in the linearise moel equation above, a ynamical expression for ernal loa force or torque is require, in orer to be able to preict payloa position.

4 4 Control valve s mathematical moeling The control valve ynamics can be moele, as appearing in Fig. 3, similar to a spring-amper system. Fig. 3 Control Valve Moel The governing ifferential equation for a springamper system is the following... mx+ bx+ kx= F cv (7) where m is the mass of the valve, b, k is the amper coefficient an spring stiffness, respectively, an F cv is the acting force on the control valve. Assuming that mass of the valve m is negligible, the Laplace transform of the above gives the following transfer function, which epicts the non-ieal valve ynamics, as a first orer system T(s) = τ s + (8) where τ=b/k is the time constant of the valve[5]. In effect, the control valve position x(s), appearing in the ynamical equations (3), (4), (5) an (6) is replace by the following expression x(s) = X(s) (9) τ s+ The above transfer function is use in the sequel to precisely simulate the elay introuce between the control signal generate by the controller an the control action of the non-ieal valve. For example, the moel of the rotational-motion, constant-volumetric-flow-rate system with non-ieal control valve obtains the form B y(s) = X(s) M (s) s ( ) ( ) ( ) Σ I s B A s τ s + + I s + B s B = (0) ( πruc ) 5 Hyraulic system controller moelling It is now attempte to introuce an appropriate ynamical moel of the hyraulic system controller. The controller types to be examine are a. ON/OFF controller, with control law + A,e + Z u= 0,e < Z A,e Z where the interval [-A, +A], efines the controller s ea zone amplitue. In ON/OFF controllers the ea zone amplitue is actually the maximum allowable amount of steay state error. Introuction of ea zone is necessary in orer to avoi unesire oscillation in transient response of the close-loop system. The amount of ea zone usually encountere in marine applications is between -5% of the rate input signal value. Any attempt to restore steay state error to lower values increases the amount of oscillation appearing in transient response. Fig.4 ON/OFF controller block iagram () The ON/OFF controller block iagram, evelope for simulation within the Matlab/imulink package, is shown in Fig. 4. The ea zone amplitue is har-wire in the switches shown in Fig. 4; in effect, if the input signal value (In) is larger than A, then the output (Out) is either or +. b. PI(D) controller, with control law in the general case where a D-term is also inclue t u( t ) = K p e( t ) + Ki e( ξ)ξ + K e( t ) () t 0 In marine applications, however, the feeback measurements available from the system contain a significant amount of process an measurement noise. It is well known that ifferentiation amplifies any noise component appearing in a measurement

5 signal. In effect, control options have to be limite to PI control instea of the full PID control law given in the above. On the other han, a system with P-controller only, emonstrates steay state error, which although smaller than the one emonstrate by a system with ON/OFF control, is in most cases not negligible. Finally, the introuction of the I-term in the control law eliminates steay state error, on one han, but on the other one, it increases the orer of the close-loop transfer function by one [6]. The full PID controller transfer function is obtaine by using the Laplace transform, as follows Ki G( s ) = K p + + K s (3) s In Fig. 5 the PI (without the D-term set zero) controller block iagram, as implemente in Matlab/imulink, is shown. Fig. 5 PI Controller Block-iagram As can be seen, a saturator has been intermitte between the control valve an the controller. This element moels the saturating behavior of the control valve, which is exhibite when the valve position reaches the evice s physical limits. Taking this saturating behavior into account an important esign issue for the PI controller is emerge. In inustrial an marine practice, most actuators exhibit saturation, just like the control valve in the typical hyraulic servo system. As saturation is actually a nonlinear constraint on the available power to the actuating evice, spee of large transient response is practically etermine by the saturation threshol. Reuce spee of response ue to saturation, in turn, leas inescapably to accumulation of error in the integrator of the I-term of any PI controller. In effect, if no countermeasure is taken, the accumulate integral error will generate large unesire overshoot an oscillation, when the control action is restore back into the acceptable range. Nonlinear control theory provies with an effective metho to eal with such nonlinearly saturate actuating evices [7]. It relies on the introuction of the inverse nonlinearity in the controller. In the specific case of PI contoller for the hyraulic servo-system, the inverse nonlinearity has taken the form of the saturation of the integral error fe to the I-term of the controller. Inee, the integrator s limite output is introuce in orer to restrict overshoot in large transient response, which may be emonstrate ue to the continuously increasing value of integral error that is cause when hitting the valve physical limits. 6 Moeling application to steering gear In general, marine steering gears use onboar moern ships, are hyraulic constant volumetric flow rate systems. The enormous hyroynamic resistance force, which is evelope on the ruer, when the ship is turning, make high-pressure hyraulics the only available technology nowaays for ruer actuation. The major categories of moern steering gears are [8] a) Rotational arrangements (Fig. 6), where the hyraulic motor is rotational an irectly applies a torque to the ruer axis. This type is use mainly onboar small ships, e.g. fishing vessels, ue to its relatively limite maximum value of torque that it can evelop. b) Piston Rapson - lie arrangements (Fig. 7), where pairs of opposing pistons ( or pairs) unertake to move the ruer axis, by applying a force to a lever, which is, thus, translate to turning torque. The large values of torque that this steering gear type can evelop is the major reason for which it is use onboar even the largest moern vessels (e.g. ULCCs an containerships) [9]. On the other han, the increasing requirements for precision in ruer motion, promote the automatic control system of the steering gear as an essential part of esign. Although ON/OFF control has conventionally been employe for steering gear control applications, it is foreseen that the above requirements will rener this type of feeback control rather ineffective in the near future. In orer for the mathematical moel of a marine steering gear to be configure, the value of a number of constants an parameters is require, e.g. circulation pump volumetric flow rate, hyraulic oil viscosity etc.. The ernally applie resistance force

6 an torque is also neee, but can be calculate rather simply from either sea trial ata an empirical formulae, as can be seen in stanar naval architecture literature [0]. As a typical example, the hyroynamic resistance torque as function of eflection angle for a small vessel (Table ) appears in Fig. 8. Fig. 7 Rapson-lie teering Gear Fig. 6 Rotational teering Gear The epenence of resistance torque on eflection angle was erive by using experimental ata for the particular ship avancing forwar at service spee V s. The expression for ernal torque results, then by using linear interpolation M e ( t ) = 59.9ϕ + D( t ) [N.m] (4) where φ is ruer eflection angle (eg) an D(t) is the ranom torque fluctuation, which is superimpose to the nominal value ictate in Eq. (4). This fluctuation conveniently packs all ranom, an in effect unpreictable, fluctuations ue not only to sea/weather conitions, but also ue to moeling errors introuce to the linear interpolation scheme use to yiel the analytic expression of hyroynamic torque. The steering gear moel, after replacing the values of all constants for the particular ship in han, is y(s) = x(s) ( 5. s s ) 0.4 s + D( s ) 5. s s ( ) (5) It is finally mentione that the steering gear of the particular ship is rotational. 7 teering gear simulator The above-erive mathematical moel of the steering gear was inclue to a visual moel/simulator, allowing enhance an easier capability of testing various control options. A typical screenshot of the visual simulator is shown in Fig. 9. The visual moel was evelope on an interplatform basis, using Visual Basic for the graphical part an Matlab/imulink for the mathematical moel. The graphical elements constituting the application s screenshot at Fig. 9 are the following:. Control valve: This is the hyraulic system s actuating evice, which is irectly controlle by either the ON/OFF or PI controller (ON/OFF or PI), an allows or inhibits oil flow to the hyraulic motor.. Ruer position selector: By placing the selector inicator, using the mouse, the esire ruer position is issue to the steering gear Torque (N.m) Angle Fig. 8 Resistance moment of a fishing boat steering gear

7 L OA 30.5 M L BP 5.34 m Β 7.0 m D 3.6 m Τ.9 m 85.8 ton C B V. kn Table Particulars of a typical fishing boat controller; the esire position is limite, as usually in marine practice, in the range [-35 0, ]. Three buttons above the selector allow the user to instantaneously set the esire position to one of the three values 35 0, 0 0 an teering gear/ruer an control valve visual snapshot: The rotational hyraulic motor an the ruer have been rawn using Rhinoceros 3D. In the rawing of the screenshot, the ruer, the rotational motor, the control valve an the two pumps of the gear are shown. The user can easily inspect the ruer an motor motions when the esire ruer position is change an the gear attempts to follow. 4. Inication of control valve position: This is where the rate position (%) of the control valve is shown; +/ signs inicate right or left, respectively position of the control valve spool with respect to neutral position. 5. Inication of esire ruer position: This is where the esire ruer position, as fixe by the user, is shown. 6. Inication of actual ruer position: This is where the actual ruer position, as etermine by the controller-system interaction ynamics is reporte. 7. Application termination button: Marke with X allows the user to terminate the application. 8. Control valve position plot: This is where control valve spool position (%) as function of running time (sec) is shown. 9. Ruer eflection angle position plot: This is where ruer eflection angle (egrees) as function of running time (sec) is shown. In orer to emonstrate steering gear operation with both ON/OFF an PI controllers, there are two separate executables. Ruer_pi.exe : for PI controller. Ruer_onoff.exe : for ON/OFF controller Thus the user, by executing initially one of the files, selects the esirable ruer eflection angle, through the selector or the three emergency action buttons. At the same time, motion of the ruer, motor an control valve spool can be observe. Also, the plots of the control valve position an ruer angle are continuously upate. Finally, by reaing the appropriate inicators, the user can observe control valve an ruer positions; these inications have been inclue in orer to facilitate accurate etermination of steay state error. Finally, all time series generate by the evelope applications are store in files for possible later use or processing. The user may easily moify the constants of the mathematical system moel, execute in the back en of the simulator, in orer to test or verify performance of the steering gear harware or controller setups. The complete mathematical moel of the close-loop fishing vessel steering gear with PI control, as evelope in Matlab/imulink, appears in Fig. 0. As can be seen in Fig., the PI controller performs mach better that the conventional ON/OFF one. Inee, the steering gear when couple to the PI controller emonstrates no steay state error an no oscillation. Note that integral error saturation has been employe in orer to avoi the potential problems that they may arise ue to the saturation of the control valve. On the other han, the steering gear with the ON/OFF controller emonstrates both jiggling in valve spool position, as well as, significant steay state error. 8 Conclusions In this work moeling an simulation of marine hyraulic systems with emphasis on ship steering gear has been tackle, by use of linearise mathematical analysis of hyraulic systems response. The mathematical moel evelope has been successfully integrate in a visual steering gear simulator. Finally, by performing simulations with the software evelope it was proven that employment of an appropriately tune PI controller, instea of the conventional ON/OFF one, may be require in orer to meet tough transient performance requirement, without compromising system reliability. References: [] J. F. Blackburn, Flui Power Control, The MIT Press, 97

8 [] D.A. Taylor, Marine Engineering Practice, Butterworths & Co,987 [3] G. Reethof, Flui Power Control, The MIT Press, 97 [4] T.J. Viersma, ynthesis an Design of Hyraulic ervomechanisms, TU Delft, 990 [5] J. Nagrath, M. Gopal, Control ystems Engineering, John Wiley & ons, 986 [6] B.J. Kuo, Automatic Control ystems, Prentice Hall Inc., 987 [7] J. Tao, P. V. Kokotovic, Aaptive Control of ystems with Actuator an ensor Nonlinearities [8] J.R. Fawcett, Hyraulic ervomechanisms an their Applications, Trae an Technical Press Lt., 975 [9] A.D. Blank, A. Block, Introuction to Naval Engineering, Naval Institute Press, 985 [0] W.. Paulin, D.J. Fowler, Marine Engineering, Vol., teering Gear, The Institute Of Marine Engineers, 98 Fig. 0 Mathematical moel of a fishing boat steering gear Fig. 9 Visual Moel of Rotational teering Gear

9 Fig. Comparison between ON/OFF an PI controlle steering gear by using the simulator