Sensitivity Analysis as a Tool for Optimization in Fluid Power Simulation

Transcription

1 The 9 th International Conference on Hydraulics and Pneumatics, Prague, September 29 October, 28 Sensitivity Analysis as a Tool for Optimization in Fluid Power Simulation Riedel, Christian ; Stammen, Christian 2 ; Liermann, Matthias 3 & Murrenhoff, Hubertus 4 Dip.-Ing., IFAS at RWTH Aachen University, Steinbachstr. 53, 5274 Aachen, Germany, Christian.Riedel@ifas.rwth-aachen.de, 2 Dr.-Ing., Montanhydraulik GmbH, Bahnhofstr. 39, Holzwickede, Germany, Stammen@montanhydraulik.com, 3 Dr.-Ing., IFAS at RWTH Aachen University, Steinbachstr. 53, 5274 Aachen, Germany, Matthias.Liermann@ifas.rwth-aachen.de, 4 Prof. Dr.-Ing., IFAS at RWTH Aachen University, Steinbachstr. 53, 5274 Aachen, Germany, post@ifas.rwth-aachen.de, Abstract: Digital simulation is frequently used in the design of fluid power systems. However, due to variations of certain parameters it is often difficult to find a good parameter set for a simulation model to achieve a robust system behaviour of a drive. As an approach to solve this problem, this paper presents a procedure which consists of two steps. In a first step a sensitivity analysis of certain design parameters on the system behaviour is done using nonlinear system simulation and methods of design of experiment (DOE). This first step serves to increase the understanding of a system. The example system used for demonstration of this procedure in this paper is the self-energizing electro-hydraulic brake. In a second step an optimization of selected design parameters towards robustness against changes of certain parameters is carried out. The tool for the optimization of design parameters is presented in this paper. It uses the maximum possible distance of a parameter value to the boundaries where quality criteria are violated as the measure for robustness. Keywords: hydraulics, sensitivity analysis, optimization, robustness, simulation, selfenergization, DSHplus, design of experiment Introduction. Motivation Digital simulation is frequently used in the design of fluid power systems. Many factors influence the static and dynamic behaviour of fluid power applications in simulation as well as in reality. Usually geometry parameters of the mechanical design are well known. Small changes due to production tolerances are of importance in some cases e.g. at the metering edges of valves. For some of those parameters the catalogue data is only insufficient. Especially information about friction and its progression over speed and pressure is not available. Extensive rig testing is necessary to obtain accurate values for such parameters. The precise parameterization of a simulation model to achieve a robust design of a drive behaviour can be complicated due to changes of parameters caused for example by wear. This concerns in particular the friction at seals and guides of cylinder drives, the leak tightness and flow behaviour of valves and decreasing efficiency over life time of displacement units. A fundamental factor with high uncertainty is the pressure-dependent bulk modulus of the fluid.

2 It affects the dynamic behaviour of systems at low pressure levels. Another value that changes during the operation is the temperature, which affects in particular viscosity, but also the fluid bulk modulus /Murr5/. For a set of parameters the quantitative influence of changes on the system s performance is difficult to predict from experience only. Since no tool exists so far, which allows to examine the range of tolerance in which the system parameters may vary without violating quality criteria, it is unknown at which point changes lead to intolerable system behaviour. The automated digital simulation of fluid power systems is suitable for the investigation of these relations. Although uncertainties in the parameterization and simplifications of the model result in differences between digital simulation and behaviour of the real system, it is accurate enough to indicate in which range the parameters lead to acceptable system behaviour..2 Approach A preliminary sensitivity analysis of changes of design parameters on the system behaviour is performed using nonlinear system simulation in DSHplus and methods of design of experiment (DOE) to increase the understanding of an example system which we describe in section 2. This serves to determine the influences and interactions of system parameters on relevant quality criteria and to illustrate them in an appropriate way for the user. At first, we specify quality criteria and choose the considered parameters. In case of a force control system the quality criteria may be e.g. offset, oscillation amplitude, overshoot and time delay until 9% of desired force is reached. In accordance with the design task relevant parameters must be selected afterwards. In order to get a good understanding of the system s behaviour it is advisable to conduct a preliminary parameter study which includes a simulation of an entire parameter range with the methods of DOE. We describe this procedure in section 3. The defined goal of the first step is to locate a suitable parameter starting point for a further optimization process (second step) which we describe in section 4 of this paper. The second step is the optimization of selected design parameters towards robustness against changes of certain parameters. A software tool for the optimization of desing parameters is presented. It uses the maximum possible distance of a parameter value to the boundaries where quality criteria are violated as the measure for robustness. A suitable algorithm automatically locates violations of certain quality criteria in a systematic way around a starting point. Then it finds new design parameters at which the range of tolerance is greater. This approach is considered best practice for preliminary design of a hydraulic system. 2 Example system The example system for demonstration of the proposed procedure is the simulation model of the Self-energising Electro-Hydraulic brake (SEHB). This new brake concept combines high force-to-weight ratio of hydraulic actuation with flexibility of digital control and efficiency by using the principle of self-energisation. It has been developed at the Institute for Fluid Power Drives and Controls (IFAS at RWTH Aachen University) within a research project funded by the DFG (German Research Foundation). The SEHB concept has been introduced in /Lier6, Lier7/. No external hydraulic power but only low electric power for the operation of hydraulic valve(s), pressure sensors and controller electronics is required to operate the brake. One of the major advantages of this concept is the possibility to control the actual braking torque, while conventional friction brakes can only control the perpendicular force. The idea of SEHB is that the pressure needed for actuation of a hydraulic disc brake is gained from hydraulic support of the friction force. Unlike conventional brakes, where the brake caliper is fixed, in the SEHB concept it is movable tangential to the friction contact. A hydraulic supporting cylinder connects the caliper to the bogie structure, thus fixing it

3 between two columns of oil, see figure. In the case of braking, the friction force acts on the supporting cylinder causing a pressure difference. Independent of which chamber is pressurized and which is released, the configuration of four check valves assures that the lower line has the higher pressure and the upper line has low pressure. A control valve connects high- and low-pressure side with the chambers of the brake actuator. If the right flow-scheme of the control valve is active, high-pressure is applied on the piston face, while the ring side of the brake actuator is connected to low pressure, thus opening the positive feedback of supporting pressure on braking pressure. With a certain ratio between piston areas self-energization takes place and the brake force is increasing until the brake disc is blocked unless the valve is closed before. If the left flow-scheme of the control valve is active, the piston face chamber is relieved, while the piston ring side is charged, reducing the actuator force yielding a negative feedback of the supporting pressure on the braking pressure. Figure Principle of Self-energizing Electro-Hydraulic Brake (SEHB) 3 Step : Sensitivity analysis In order to get an overview of the system s behaviour it is advisable to conduct a preliminary study of the most important parameters and target values. Thus, a suggestive starting point can be found for a further optimization process. At first, target values and parameters must be chosen according to the particular task as described in section.2. For the SEHB exemplary the parameters friction coefficient µ, overlapping of metering edges of the control valve y, proportional factor of controller K p and starting stroke of braking cylinder s are varied. The friction coefficient µ changes in reality e.g. due to dirt or lubrication. The metering edges are selected to investigate the change of the system s control behaviour with an increasing overlap y. As proportional valves are not entirely free of leakage this promises better results, but complicates controlling. Even though the overlap y is a design parameter it is inexact due production tolerances or wear. The proportional factor K p is a crucial parameter for the control of the system. The starting position of the cylinder proceeds due to the wear of the brake pad. For the sake of simplicity a full factorial experimental plan is chosen as the basis for the analysis. Thus, all possible combinations of variations of the selected parameters are simulated, disregarding the increased computing time compared to a partly factorial experimental plan /Band94/. Often, characteristic quality criteria such as the time between actuation and ninety percent force t npf of the brake, overshoot s, oscillation amplitude and control deviation are used to analyse the performance of a system /Murr8, Abel5/. In the special case of SEHB we focus on t npf and overshoot F os as main quality factors.

4 Overshoot [N] t npf [s] Fig. 2 and 3 show examples of the conducted study. In figure 2 the time between actuation and ninety percent force t npf is illustrated over the friction µ and the control s proportional gain K p. While the latter value can be adjusted, the friction coefficient µ is not precisely known in practice and is subject to change. It becomes clear that decreasing values of µ lead to an exponential rise of the time needed to reach ninety percent of the actuation force, which reduces the performance of the braking system µ [-] Kp [V/kN] xe-3 Figure 2 Time at ninety percent force (t npf ) as a function of µ and K p Figure 3 shows the initial overshoot of the brake force dependent on the friction coefficient µ and the control s proportional gain K p. Evidently there is no overshoot for small parameter values in both cases. For larger values the simulation shows overshoot of up to N that would lead to a significant increase in deceleration of a vehicle and thus impair comfort. As we compare the simulations for the two quality criteria t npf and overshoot we come to the conclusion that an optimal proportional gain K p for the best performance of the brake can be found at about.6 to.8 V/kN. Furthermore this graph shows that K p should be adapted according to the actual friction coefficient as it shows strong interactions between the parameters µ [-] Kp [V/kN] xe-3 Figure 3 Overshoot as a function of µ and Kp This example also shows that for a benchmark of quality criteria in course of an optimization there is a need for a scaling factor that enables comparison by assigning higher relevance to certain criteria.

5 4 Step 2: Optimization towards robustness If all ranges of tolerance for uncertain parameters in the design process were of interest in which quality criteria correspond with the system s requirements, all possible combinations had to be simulated for the entire parameter range (full factorial experimental plan). As correlations can be highly non-linear the step size for simulation must be very small. Therefore the detailed simulation of a complete parameter area requires extensive computing time. In order to optimize chosen parameter sets of the model towards robustness, it is more efficient to simulate and evaluate only certain parameters sets around this starting point. This applies to both the well-known parameters, which are set specifically by the mechanical or control design, as well as to uncertain parameters. If for example the robustness of the design parameters due to changes of friction coefficient values at a cylinder needs to be optimized, this analysis could be conducted. Starting from an initial set of parameters the study would be expanded to neighbouring points in the parameter area until the quality criteria would be violated. This approach substantially reduces the number of simulations and computing time, as well as memory allocation. 4. Systematic scanning of parameter range For the efficiency of the described procedure the properties of the necessary algorithms are crucial. In order to find the range of tolerance as measure for robustness it is reasonable to simulate and evaluate a set of parameters around a starting point developing one on another. Firstly, specified step sizes and extreme values are defined for the parameters. Then the neighbouring points are examined (step V = ). Afterwards the points of the next shell are computed (step V = 2). Since the individual points of an order must have the same distance to the starting point, the ideal scanning form is a circle in the R² or a n-dimensional sphere in the R n (see figure 4). If a variable step size is allowed for the parameter variations, the n- dimensional circle can be chosen. If a discrete dot matrix is used, an accurate illustration of these forms does not exist however. We pursue the latter discrete dot matrix as for reasons of simplicity and therefore accept less accuracy in our results. As shown in figure 4, further forms exist, in which the parameter points can be scanned, the interior square and the exterior square. Figure 4 Systematic scanning of parameter range In the two-dimensional case the following relation results for the computation of the perimeter of the geometrical objects: 2 2 a a 4 a Interior square < Circle < Exterior square ()

6 This relation clarifies that the distance to the optimum of the circle is smaller for the interior square, as that of the exterior square to the circle. In higher dimensions the relationship continues to improve in favour of the interior square. Furthermore, the interior square always forms an estimation to the safe side, so that each point of an order has a smaller distance to the starting point, as a point of the next order. Therefore, an algorithm is used which searches the points around the starting point (i j k ) in expanding bodies. In addition the algorithm must be able to accomplish this for arbitrarily high dimensions. The following equation describes the algorithm for the formation of the body in the threedimensional case, which results in an octahedron form: Point surface i i i j k j V i i V... V j with j ( V i )...( V i ) k s k ( V i j )...( V i s j V s 2 ( V i j ) j ) (2) In the regarded three-dimensional case nested loops are run through for the coordinates i, j, k outgoing from the starting point. The beginning and final points of the respective loops result according to equation 2 from the step number V of the body minus the number of the preceding coordinates. To ensure that only the outer points of the body are searched, the step size s of the k-coordinate depends on the step number V and the coordinates i and j. For the outer points of the body the sum of the absolute values of the coordinates is equal to the order. In this case the step size s is equal to one. For higher dimensions of the regarded parameter areas the number of coordinates and loops increase. If the surface is processed without violation of a quality factor, the order is increased by one and the surface of the next larger body is scanned. If the range of tolerance around a design point is sought, the algorithm serves its purpose by finding a point outside the tolerance. Therefore, the tolerance width is known by the order of the body. 4.2 Approach for optimization of robustness In a further step an optimization searches for a new design point, at which the robustness is maximal. If the body described above is expanded to one or more points which do not comply with criteria of the initial values or are located at the edge of the defined parameter area, the algorithm uses the location of the faults to search for a new starting point of the iteration described in section 3.. Between the starting point C and the located violations of quality criteria F n a resulting vector F Res is generated, as shown in figure 5. The vector F opt corresponds to the inverted vector F Res and points to the new starting point. The nearest point in the dot matrix is selected as a new starting point, resulting in the formation of a new body. The algorithm skips values that have already been computed. The search for new starting points repeats as long, as the order of the body increases or the new starting point has already been computed. Thus the penultimate starting value represents an optimum. Since the search does not necessarily stand for the global optimum around a selected starting point, the algorithm starts at several different points within the permissible parameter range.

7 Figure 5 Proceeding of the algorithm 4.3 Structure of algorithm Firstly the parameter set is generated for the calculation of a starting point. Then the simulation is started. The results are exported to Matlab, which calculates the selected quality criteria and examines them on possible violation of the chosen tolerances. The necessary data about parameters and quality criteria which are to be varied, are obtained from correspondent files. The diagram in figure 6 shows the structure of the optimization model with Matlab SimuLink and an integrated DSHplus model. The entire sequence is summarized in a Matlab program. Figure 6 Procedure of the Algorithm

8 After the optimization the result is visualized. The specific points of the body are demonstrated in the parameter area, which is shown for an exemplary investigation in figure 7. The black coloured starting points of the specific bodies are connected by arrows for purpose of illustration. The points of violation of the quality criteria F to F n are marked red. The outer points of the last octahedron are connected by lines in order to illustrate the permissible range of tolerance. In this example an optimization of the proportional gain K p and the starting stroke of the braking actuator S was performed. Furthermore, the uncertain parameters friction coefficient µ and overlap y were analyzed, but not optimized. The latter parameter is not shown in figure 7 as the fourth dimension cannot be displayed. 2 S [mm] µ [-] x -3 Kp [-] Figure 7 Results of Optimization Process 5 Conclusion A procedure for a sensitivity analysis on the basis of simulation models is presented at the example of a Self-energizing Electro-Hydraulic Brake to examine its basic system behaviour. A preliminary parameter study is conducted to analyze the interactions between certain quality criteria and design parameters and to find a suitable design point which can be optimized in a further step. The presented algorithm for an optimization of the robustness of a system towards parameter changes allows a search for violation of certain quality criteria around a design point in the R n. Hence, an optimization can be derived directly, with which a new robust design point for the system can be found. This approach offers a support to the user for a robust design of system parameters. The illustrations of the influences and interactions are also suitable for the knowledge transfer, a better understanding of the fluid power system and the model used for the simulation. Observed differences between simulation and test stand can inspire a further development of the simulation model. 8 References /Abel5/ Abel, D. Mess- und Regelungstechnik [Measurment and Control], 3st edition, Verlag Mainz, 27 /Band94/ Bandemer, H. Bellmann, A. Statistische Versuchsplanung [Design of Experiment], 4. Aufl., Teubner-Verlag, 994 /Lier6/ Liermann, M.; Selbstverstärkende hydraulische Bremse für

9 /Lier7/ Stammen, C. Liermann, M.; Stammen, C.; Murrenhoff, H. Schienenfahrzeuge - Intelligentes, Integriertes Einzelrad- Antriebs-Brems-Modul [Self-energising Electro-Hydraulik Brake for Rail Vehicles], Ölhydraulik und Pneumatik, O+P /26, 5-57 Development of A Self-energising Electro-Hydraulic Brake (SEHB) for Rail Vehicles, The Tenth Scandinavian International Conference on Fluid Power, SICFP 7, May 2-23, 27, Tampere, Finland /Murr5/ Murrenhoff, H. Grundlagen der Fluidtechnik Teil : Hydraulik [Fundamentals of Fluid Power], 4th edition, Shaker-Verlag, 25 /Murr8/ Murrenhoff, H. Servohydraulik geregelte hydraulische Antriebe [Servo Hydraulics], 3rd edition, Shaker-Verlag, 28 Information for editors (not published): File format: Microsoft Word Authors statement for List of authors: Order Surname First Name Institution and Country Riedel Christian IFAS at RWTH Aachen, Germany 2 Stammen Christian Montanhydraulik GmbH, Germany 3 Liermann Matthias IFAS at RWTH Aachen, Germany 4 Murrenhoff Hubertus IFAS at RWTH Aachen, Germany Christian.Riedel@ifas.rwt h-aachen.de Stammen@montanhydrau lik.com Matthias.Liermann@ifas. rwth-aachen.de post@ifas.rwth-aachen.de